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alg-geom/9303006

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https://arxiv.org/abs/alg-geom/9303006

alg-geom/9303006

Roberto Paoletti

Roberto Paoletti

Seshadri constants, gonality of space curves and restriction of stable bundles

36 pages, amslatex

We define the Seshadri constant of a space curve and consider ways to estimate it. We then show that it governs the gonality of the curve. We use an argument based on Bogomolov's instability theorem on a threefold. The same methods are then applied to the study of the behaviour of a stable vector bundle on P^3 under restriction to curves and surfaces.

alg-geom

\section{\bf {Introduction}} There exist many situations in algebraic geometry where the extrinsic geometry of a variety is reflected in clear restrictions in the way that it can map to projective spaces. For example, it is well-known that the gonality of a smooth plane curve $C$ of degree $d$ is $d-1$, and that every minimal pencil has the form $\cal O_C(H-P)$, where $H$ denotes the hyperplane class and $P\in C$. In fact, there exist to date various statements of this kind concerning the existence of morphisms from a divisor to $\bold P^1$. The first general results in this direction are due to Sommese (\cite{so:amp}) and Serrano (\cite{se:ext}). Reider (\cite{re:app}) then showed that at least part of Serrano's results for surfaces can be obtained by use of vector bundle methods based on the Bogomolov-Gieseker inequality for semistable vector bundles on a surface. In \cite{pa}, a generalization of these methods to higher dimensional varieties is used to obtain the following statement: \begin{thm} Let $X$ be a smooth projective $n$-fold, and let $Y\subset X$ be a reduced irreducible divisor. If $n\ge 3$ assume that $Y$ is ample, and if $n=2$ assume that $Y^2>0$ (so that in particular it is at least nef). Let $\phi :Y@>>>\bold P^1$ be a morphism, and let $F$ denote the numerical class of a fiber. \noindent (i) If $$F\cdot Y^{n-2}<\sqrt {Y^n}-1,$$ then there exists a morphism $\psi :X@>>>\bold P^1$ extending $\phi$. Furthermore, the restriction $$H^0(X,\psi ^{*}\cal O_{\bold P^1}(1))@>>> H^0(Y,\phi ^{*}\cal O_{\bold P^1}(1))$$ is injective. In particular, $\psi$ is linearly normal if $\phi$ is. \noindent (ii) If $$F\cdot Y^{n-2}=\sqrt {Y^n}-1$$ and $Y^n\neq 4$, then either there exists an extension $\psi :X@>>>\bold P^1$ of $\phi$, or else we can find an effective divisor $D$ on $X$ such that $(D\cdot Y^{n-1})^2=(D^2\cdot Y^{n-2})Y^n$ and $D\cdot Y^{n-1}=\sqrt {Y^n}$, and an inclusion $$\phi ^{*}\cal O_{\bold P^1}(1)\subset \cal O_Y(D).$$ \end{thm} \bigskip However, a much less understood range of situations is the one where $codim(Y)\ge 2$. In some particular cases there are rather precise statements. In curve theory, in particular, one has a clear picture of the gonality of Castelnuovo extremal curves (\cite{acgh}). In even degree, for example, if $C\subset \bold P^3$ is a smooth complete intersection of a smooth quadric and a hypersurface of degree $a\ge 2$, the gonality is attained by restricting to $C$ the rulings on the quadric. More generally, unpublished work of Lazarsfeld shows that if $C\subset \bold P^3$ is a smooth complete intersection of type $(a,b)$, with $a\ge b$, then $gon(C)\ge a(b-1)$. Lazarsfeld's argument is also based on Bogomolov's instability theorem. In a somewhat more general direction, Ciliberto and Lazarsfeld have studied linear series of low degree on various classes of space curves (\cite{cl}). Their method is based on the number of conditions imposed by a linear series on another. Naturally enough, one is led to investigate more general situations. We shall focus on the gonality of space curves, and then show how the methods developped apply to other circumstances as well. In the codimension $1$ case we have seen that the self intersection of the divisor governs the numerical constraint on a free pencil on $Y$. Loosely speaking, in the higher codimension case a similar role is played by the Seshadri costant of the curve. This is defined as follows. Consider a smooth curve $C\subset \bold P^3$ and denote by $$f:X_C@>>>\bold P^3$$ the blow up of $\bold P^3$ along $C$, and by $$E=f^{-1}C$$ the exceptional divisor. The {\it Seshadri constant} of $C$ is \begin{center} $\epsilon (C)=sup\{\eta \in \bold Q| f^{*}H-\eta E$ is ample$\}$. \end{center} This is a very delicate invariant, and it gathers classical information such as what secants the curve has and the minimal degree in which powers of $\cal J_C$ are globally generated. For example, if $C\subset \bold P^3$ is a complete intersection of type $(a,b)$, with $a\ge b$, then $\epsilon (C)=\frac 1a$. More generally, if $C\subset \bold P^3$ is defined as the zero locus of a regular section of a rank two vector bundle $\cal E$, then we have an estimate $\epsilon (C)\ge \gamma (\cal E)$, where $\gamma (\cal E)$ is the Seshadri constant of $\cal E$, defined as \bigskip \begin{center} $\gamma (\cal E)=sup\{\frac nm|S^n\cal E^{*}(m)$ is globally generated$\}$. \end{center} \bigskip \noindent It is always true that $\epsilon (C)\ge \frac 1d$. However, the problem of finding general optimal estimates $\epsilon (C)$ for an arbitrary curve seems to be a hard one. Something can be said, for example, as soon as $C$ can be expressed as an irreducible component of a complete intersection of smooth surfaces. Interest in Seshadri constants, of course, is not new. In fact, if $Y$ is a subvariety of any projective variety $X$, one can define in an obvious way the Seshadri constant of $Y$ with respect to any polarization $H$ on $X$. Seshadri constants of points, in particular, have received increasing attention recently, partly in relation to the quest for Fujita-type results. A differential geometric interpretation has been given by DeMailly (\cite{de}). Seshadri constants of points on a surface have been investigated by Ein and Lazarsfeld (\cite{el}), who have proved the surprising fact that they can be bounded away from zero at all but countably many points of $S$. However, Seshadri constants of higher dimensional subvarieties have apparently never been put at use. What a bound on the gonality of a space curve might look like is suggested by Lazarsfeld's result. In fact, we may write $a(b-1)=deg(C)-\frac 1{\epsilon (C)}$, so that for a complete intersection we have the optimal bound $$gon(C)\ge d-\frac 1{\epsilon (C)}.$$ Keeping the notation above, let us define $$H_{\eta}=f^{*}H-\eta E$$ and $$\delta _{\eta}(C)=\eta \cdot deg(N)-d,$$ where $N$ is the normal bundle of $C$. For example, for a complete intersection of type $(a,b)$ with $a\ge b$ we have $\delta _{1/a}(C)=b^2$. $\delta _{\eta}(C)$ has a simple geometric meaning, that we explain at the end of Chapter 3. Our result is \begin{thm} Let $C\subset \bold P^3$ be a smooth curve of degree $d$ and Seshadri constant $\epsilon (C)$. Set $\alpha =min \big \{1,\sqrt d\big (1-\epsilon (C)\sqrt d\big )\big \}$. Then $$gon(C)\ge min \Big \{\frac {\delta _{\epsilon (C)} (C)}{4\epsilon (C)}, \alpha \Big (d-\frac {\alpha}{\epsilon (C)}\Big )\Big \}.$$ \end{thm} This reproduces Lazarsfeld's result if $a\ge b+3$. As another example, it says that if $a\gg b$ and $C$ is residual to a line in a complete intersection of type $(a,b)$, then $gon(C)=ab-(a+b-2)$ (consider the pencil of planes through the line). In view of the above, one would expect the above bound to hold with $\alpha =1$ always, but I have been unable to prove it. The idea of the proof is as follows. If $A$ is a minimal pencil on $C$, and if $\pi :E@>>>C$ is the induced projection, one can define a rank two vector bundle on $X_C$ by the exactness of the sequence $$0@>>>\cal F@>>>H^0(C,A)\otimes \cal O_{X_C}@>>>\pi ^{*}A@>>>0.$$ The numerical assuptions then force $\cal F$ to be Bogomolov unstable w.r.t. $H_{\epsilon (C)}$ (see $\S0$) and therefore a maximal destabilizing line bundle $$\cal O_{X_C}(-D)\subset \cal F$$ comes into the picture. $D$ and $A$ are related by the inequalities coming from the instability of $\cal F$, and from this one can show that $deg(A)$ is forced to satisfy the above bound. By its general nature, this argument can be applied to the study of linear series on arbitrary smooth subvarieties of $\bold P^r$. We will not detail this generalization here. \bigskip In another direction, similar methods have been used by Bogomolov (\cite{bo:78} and \cite{bo:svb}) to study the behaviour of a stable bundle on a surface under restriction to a curve $C$ that is linearly equivalent to a multiple of the polarization at hand. For example, it follows from Bogomolov's theorem that if $S$ is a smooth surface with $Pic(S)\simeq \bold Z$ and $\cal E$ is a stable rank two vector bundle on $S$, then $\cal E|_C$ is also stable, for every irreducible curve $C\subset S$ such that $C^2>4c_2(\cal E)^2$. A more complicated statement holds for arbitrary surfaces. One can see, in fact, that this result implies a similar one for surfaces in $\bold P^3$. In the spirit of the above discussion, one is then led to consider the problem of the behaviour under restriction to subvarieties of higher codimension. The inspiring idea, suggested by the divisor case, should be that when some suitable invariants, describing some form of "positivity" of the subvariety, become large with respect to the invariants of the vector bundle, then stability is preserved under restriction. Furthermore, if in the divisor case one needs the hypothesis that $\cal E$ be $\cal O_S(C)$-stable, in the higher codimension case one should still expect some measure of the relation between the geometry of the subvariety and the stability of the vector bundle to play a role in the solution to the problem. In fact, in the case of space curves the same kind of argument that proves the theorem about gonality can be applied to this question. Before explaining the result, we need the following definition. Recall that if $X$ is a smooth projective threefold, $\cal F$ is a vector bundle on $X$ and $L$ and $H$ are two nef line bundles on $X$, $\cal F$ is said to be $(H,L)$-stable if for every nontrivial subsheaf $\cal G\subset \cal F$ we have $(fc_1(\cal G)-gc_1(\cal F))\cdot H\cdot L< 0$, where $f=rank(\cal F)$ and $g=rank(\cal G)$. Let then $\cal E$ be a rank two vector bundle on $\bold P^3$, and consider a curve $C\subset \bold P^3$. Let us define the {\it stability constant of $\cal E$ with respect to $C$} as \bigskip \begin{center} $\gamma (C,\cal E)=sup\{\eta \in [0,\epsilon (C)]| f^{*}\cal E$ is $(H,H_{\eta})$-stable$\}.$ \end{center} \bigskip \noindent For example, if $C$ is a complete intersection of type $(a,b)$ and the restriction of $\cal E$ to one of the two surfaces defining $C$ is stable (with respect to the hyperplane bundle) then $\gamma (C,\cal E)=\epsilon (C)$. Then we have \begin{thm} Let $\cal E$ be a stable rank two vector bundle on $\bold P^3$ with $c_1(\cal E)=0$. Let $C\subset \bold P^3$ be a smooth curve of degree $d$ and Seshadri constant $\epsilon (C)$, and let $\gamma =\gamma (C,\cal E)$ be the stability constant of $\cal E$ w.r.t. $C$. Suppose that $\cal E|_C$ is not stable. Then $$c_2(\cal E)\ge min\Big \{\frac {\delta _{\gamma}(C)} 4, \alpha \gamma \Big (d-\frac {\alpha}{\gamma}\Big ) \Big \},$$ where $\alpha =:min\Big \{1,\sqrt d \Big (\sqrt {\frac 34}-\gamma \sqrt d \Big )\Big \}$. \end{thm} The problem of the behaviour of stable bundles on $\bold P^r$ under restriction to curves has been studied by many researchers. In particular, a well-known fundamental theorem of Mehta and Ramanathan (\cite{mr:res}) shows that $\cal E|_C$ is stable if $C$ is a {\it general} complete intersection curve of type $(a_1,a_2,\cdots)$, and all the $a_i\gg 0$. Flenner (\cite{fl}) has then given an explicit bound on the $a_i$s in term of the invariants of $\cal E$ which makes the conclusion of Mehta and Ramanathan's Theorem true. On the other hand, here we give numerical conditions that imply stability for $\cal E|_C$, with no generality assumption and without restricting $C$ to be a complete intersection. We have the following applications: \begin{cor} Let $\cal E$ be a stable rank two vector bundle on $\bold P^3$ with $c_1(\cal E)=0$ and $c_2(\cal E)=c_2$. Suppose that $b\ge c_2+2$. If $V\subset \bold P^3$ is a smooth surface of degree $b$, then $\cal E|_V$ is $\cal O_V(H)$-stable. \end{cor} \begin{cor} Let $\cal E$ be a stable bundle on $\bold P^3$ with $c_1(\cal E)=0$ and $c_2(\cal E)\neq 1$. Suppose that $C= V_a\cap V_b\subset \bold P^3$ is an irreducible smooth complete intersection curve and that $V_a$ is smooth. Assume furthermore that $a\ge \frac 43b+\frac {10}3$ that and that $b\ge c_2(\cal E)+2$. Then $\cal E|_C$ is stable. \end{cor} \begin{cor} Let $c_2\ge 0$ be an integer and let $\cal M(0,c_2)$ denote the moduli space of stable rank two vector bundles on $\bold P^3$. If $a\gg b\gg c_2$ and $C\subset \bold P^3$ is an irreducible smooth complete intersection of type $(a,b)$, then $\cal M(0,c_2)$ embeds in the moduli space of stable vector bundles of degree $0$ on $C$. \end{cor} \bigskip \bigskip This paper covers part of the content of my Phd thesis at UCLA. I want to thank Robert Lazarsfeld, my advisor, for introducing me to Algebraic Geometry and taking continuous interest in my progress. I am also endebted to a number of people for valuable comments and discussions; among them, O. Garcia-Prada, D. Gieseker, M. Green, J. Li and A. Moriwaki. \section{\bf {Preliminaries}}\label{section:preliminaries} In this section we state some results that will be used in the sequel. The following fact is well-known: \begin{lem} Let $X$ be a smooth projective variety and let $Y\subset X$ be a divisor. Suppose that we have an exact sequence: $$0@>>>\cal F@>>>\cal E@>>>A@>>>0,$$ where $A$ is a line bundle on $Y$ and $\cal E$ is a rank two vector bundle on $X$. Let $[Y]\in A^1(X)$ be the divisor class of $Y$ and let $[A]\in A^2(X)$ be the image of the divisor class of $A$ under the push forward $A^1(Y)@>>>A^2(X)$. Then $\cal F$ is a rank two vector bundle on $X$, having Chern classes $c_1(\cal F)=c_1(\cal E)-[Y]$ and $c_2(\cal F)=c_2(\cal E)+[A] -Y\cdot c_1(\cal E)$. \label{lem:eltr} \end{lem} {\it Proof.} The first statement follows by considering local trivializations. As to the Chern classes of $\cal F$, we could prove the statement by directly computing $$c_t(\cal F)=c_t(\cal E)\cdot c_t(A)^{-1}.$$ However, the following shorter argument proves that the above equalities hold numerically, after multiplying both sides by $n-2$ nef divisor classes (which is what we need). First of all, the morphism $\cal F@>>>\cal E$ drops rank along $Y$, and therefore $c_1(\cal F)=c_1(\cal E)-Y$. Let us consider the second equality. If $X$ is a surface, the proof is reduced to a Riemann-Roch computation. If $dim(X)=3$, let $H$ be any very ample divisor on $X$, and let $S\in |H|$ be a general smooth surface. By generality, we may assume that $C=S\cap Y$ is a smooth irreducible curve. Then by restriction we obtain an exact sequence on $S$: $0@>>>\cal F|_S@>>>\cal E|_S@>>>A|_C@>>>0$. By applying the statement for the surface case, we then obtain $(c_2(\cal F)-c_2(\cal E)-[A] +Y\cdot c_1(\cal E))\cdot H=0$. But then the expression between brackets has to be killed by all ample divisors, and so it is numerically trivial. The general case is similar. $\sharp$ \bigskip \begin{lem} Let $X$ be a smooth projective threefold, and let $C\subset X$ be a smooth curve in $X$. Denote by $f:X_C@>>>X$ the blow up of $X$ along $C$, and let $E$ be the exceptional divisor. Then $E^3=-deg(N)$, where $N$ is the normal bundle of $C$ in $X$. Furthermore, let $A$ be any line bundle on $X$, and by abuse of language let $A$ also denote its pull-back to $X_C$. Then $E^2\cdot A=-C\cdot A$. \label{lem:segre} \end{lem} {\it Proof.} Both statements follow from a simple Segre class computation (see for example \cite{fu}). $\sharp$ \bigskip We now recall some known results about instability of rank two vector bundles on projective manifolds, which are one of the main tools in the following analysis. Recall the following notation. \begin{defn} If $S$ is a smooth projective surface, $N(S)$ is the vector space of the numerical equivalence classes of divisors in $S$; $K^{+}(S) \subset N(S)$ is the (positive) cone spanned by those divisors $D$ such that $D^2>0$ and $D\cdot H>0$ for some polarization on $S$. In general, if $X$ is a smooth projective $n$-fold and $H$ is a polarization on it, we shall denote by $K^{+}(X,H)$ the cone of all numerical classes $D$ in $N(X)$ such that $D^2\cdot H^{n-2}>0$ and $D\cdot H^{n-1}>0$ (or, equivalently, $D\cdot R\cdot H^{n-2}>0$ for any other polarization $R$ on $X$). \label{defn:poscone} \end{defn} \begin{defn} Let $X$ be a smooth projective $n$-fold, and let $\cal E$ be a rank two vector bundle on $X$, with Chern classes $c_1(\cal E)$ and $c_2(\cal E)$. The {\it discriminant} $\Delta (\cal E)\in A^2(X)$ is $$\Delta (\cal E)=c_1(\cal E)^2-4c_2(\cal E).$$ \label{defn:discr} \end{defn} \begin{lem} Let $X$ be a smooth projective $n$-fold, and let $\cal E$ be a rank two vector bundle on $X$. Fix a polarization $H$ on $X$. Suppose that $\cal L_1, \cal L_2\subset \cal E$ are two line bundles in $\cal E$. Let us denote by $l_1$ and $l_2$ their $H$-degrees, respectively (i.e., $l_i=\cal L_i\cdot H^{n-1}$) and let $e=deg_H(\cal E)=\wedge ^2\cal E\cdot H^{n-1}$ be the $H$-degree of $\cal E$. Suppose that $2l_i>e$ for $i=1$ and $i=2$ (in other words, $\cal L_1$ and $\cal L_2$ make $\cal E$ $H$-unstable). If $\cal L_2$ is saturated in $\cal E$, then $\cal L_1\subseteq \cal L_2$. \label{lem:contains} \end{lem} {\it Proof.} Set $l=min \{l_1,l_2\}$. By assumption, we have $$2l-e>0.$$ \begin{claim} If the statement is false, the morphism of vector bundles $$\phi :\cal L_1\oplus \cal L_2@>>>\cal E$$ is generically surjective. \end{claim} {\it Proof} Set $\cal Q=\cal E/\cal L_2$. Then $\cal Q$ is a rank one torsion free sheaf. The morphism $\cal L_1@>>>\cal Q$ is therefore either identically zero or generically nonzero. If $\cal L_1 \not \subset \cal L_2$ the morphism $\cal L_1@>>>\cal Q$ is then generically nonzero. But this implies that $\phi$ is generically surjective. $\sharp$ \bigskip Therefore, $\wedge ^2\cal E\otimes \cal L_1^{-1}\otimes \cal L_2 ^{-1}$ is an effective line bundle; it follows that $$0\le e-(l_1+l_2)\le e-2l,$$ a contradiction. $\sharp$ \bigskip \begin{cor} Let $X$ and $\cal E$ be as above, and let $\cal A\subset \cal E$ be a saturated $H$-destabilizing line bundle. Then $\cal A$ is the maximal $H$-destabilizing line bundle. \label{cor:max} \end{cor} \begin{cor} Let $X$ be a smooth projective $n$-fold, and fix a very ample linear series $|V|$ on $X$, with $V\subset H^0(X,H)$. Suppose that $\cal E$ is a rank two vector bundle on $X$ which is $H$-unstable. Let $C\subset X$ be a general complete intersection of $n-1$ divisors in $|V|$. Then the maximal destabilizing line bundle of $\cal E|_C$ is the restriction to $C$ of the maximal destabilzing line bundle of $\cal E$. \label{cor:res} \end{cor} {\it Proof.} Let $\cal A$ be the maximal destabilizing line bundle of $\cal E$. Then the inclusion $ \psi :\cal A@>>>\cal E$ drops rank in codimension two, because $\cal A$ is saturated in $\cal E$. Let $Z$ be the locus where $\psi$ drops rank. For a general complete intersection curve, we have $C\cap Z=\emptyset$. Hence $\cal A|_C$ is the maximal destabilizing line bundle of $\cal E|_C$. $\sharp$ \bigskip The basic result is the following \begin{thm} (Bogomolov) Let $S$ be a smooth projective surface, and let $\cal E$ be a rank two vector bundle on $S$. Let $c_1(\cal E)$ and $c_2(\cal E)$ be its Chern classes, and suppose that $$c_1(\cal E)^2-4c_2(\cal E)>0.$$ Then there exists an exact sequence $$0@>>>A@>>>\cal E@>>>\cal J_Z\otimes B@>>>0,$$ where $A$ and $B$ are line bundles on $S$ and $Z$ is a codimension two (possibly empty) local complete intersection subscheme, with the property that $A-B\in K^{+}(S)$. \label{thm:bog} \end{thm} For a proof, see \cite{bo:st}, \cite{mi:cc}, \cite{re:vbls}, \cite{gi} or \cite{la:svbt}. \begin{cor} Let $S$ and $\cal E$ be a smooth projective surface and a rank two vector bundle on it such that the hypothesis of the theorem are satisfied. Let $\cal A$ and $\cal B$ be the line bundles in the above exact sequence. Then the following inequalities hold: $$(\cal A-\cal B)\cdot H>0$$ for all polarizations $H$ on $S$, and $$(\cal A-\cal B)^2\ge c_1(\cal E)^2-4c_2(\cal E).$$ \label{cor:devissage} \end{cor} {\it Proof.} The first inequality follows from the condition $A-B\in K^{+}(S)$. To obtain the second, just use the above exact sequence to compute $c_1(\cal E)$ and $c_2(\cal E)$: we obtain $$c_1(\cal E)^2-4c_2(\cal E)=(A+B)^2-4A\cdot B-4deg[Z] \le (A-B)^2.$$ $\sharp$ \bigskip \begin{cor} Let $S$ and $\cal E$ satisfy the hypothesis of Bogomolov's theorem, and let $H$ be any polarization on $S$. Then $\cal E$ is $H$-unstable, and $\cal A$ is the maximal $H$-destabilizing subsheaf of $\cal E$. \end{cor} Recall the fundamental theorem of Mumford-Mehta-Ramanatan (cfr \cite{mi:cc}): \begin{thm} Let $X$ be a smooth projective $n$-fold,and let $H$ be a polarization on $X$. Consider a vector bundle $\cal E$ on $X$. If $m\gg 0$ and $V\in |mH|$ is general, then the maximal $H|_V$-destabilizing subsheaf of $\cal E|_V$ is the restriction of the maximal $H$-destabilizing subsheaf of $\cal E$. \label{thm:mumera} \end{thm} This theorem is very powerful, because it detects global instability from instability on the general complete intersection curve. \begin{thm} Let $X$ be a smooth projective $n$-fold, and let $H$ be a polarization on $X$. Consider a rank two vector bundle $\cal E$ on $X$, and suppose that $$(c_1(\cal E)^2-4c_2(\cal E))\cdot H^{n-2}>0.$$ Then there exists an exact sequence $$0@>>>\cal A@>>>\cal E@>>>\cal B\otimes \cal J_Z@>>>0,$$ where $\cal A$ and $\cal B$ are invertible sheaves and $Z$ is a locally complete intersection of codimension two (possibly empty) such that $$\cal A-\cal B\in K^{+}(X,H) $$ and $$(\cal A-\cal B)^2\cdot H^{n-2}\ge (c_1(\cal E)^2-4c_2(\cal E)) \cdot H^{n-2}.$$ Furthermore, $\cal A$ is the maximal $(H,\cdots ,H,L)$-destabilizing subsheaf of $\cal E$, for every ample line bundle $L$ on $X$. \label{thm:main} \end{thm} {\it Proof.} The case $n=2$ is just the content of Theorem \ref{thm:bog}; for $n\ge 3$, the statement follows by induction using theorem \ref{thm:mumera}. $\sharp$ \bigskip \begin{defn} If $\cal E$ satisfies the hypothesis of the theorem, we shall say that $\cal E$ is {\it Bogomolov-unstable with respect to $H$}. \label{defn:bogunst} \end{defn} \bigskip \begin{lem} Let $f:X@>>>Y$ be a morphism of projective varieties. Let $\cal F$ and $A$ be, respectively, a vector bundle and an ample line bundle on $X$. For $y\in Y$, let $X_y=f^{-1}y$ and denote by $\cal J_{X_y}$ the ideal sheaf of $X_y$. Then there exists $k>0$ such that $$H^i(X,\cal F\otimes A^n\otimes \cal J_{X_y})=0$$ for all $i>0$, $n\ge k$ and for all $y\in Y$. \label{lem:fs} \end{lem} {\it Proof.} For all $y\in Y$, there is an exact sequence $$0@>>>\cal F\otimes A^n\otimes \cal J_{X_y}@>>> \cal F\otimes A^n@>>>\cal F\otimes A^n|_{X_y}@>>>0.$$ Furthermore, there exists a {\it flattening stratification} of $Y$ w.r.t. $f$, $Y= \coprod _{l=1}^r Y_l$, with the following property (\cite{mu:cs}). The $Y_l$ are locally closed subschemes of $Y$, and if $X_l=:f^{-1}Y_l$, $l=1,\cdots,r$, and $f_l:X_l@>>>Y_l$ is the restriction of $f$, then $f_l$ is a flat morphism. Let us then start by finding $k_1$ such that for all $n\ge k_1$ we have $$H^i(X,\cal F\otimes A^n )=0$$ and $$H^i(X,\cal F\otimes A^n\otimes \cal J_{X_l})=0$$ for all $i>0$ and for all $l=1,\cdots,r$. Then it is easy to see that the statement is equivalent to saying that there is $k\ge k_1$ such that for all $n\ge k$ the restriction maps $$H^0(X,\cal F\otimes A^n)@>\phi _y>>H^0(X_y,\cal F\otimes A^n|_{X_y})$$ are all surjective, and that $$H^i(X_y,\cal F\otimes A^n|_{X_y})=0,$$ for all $y\in Y$ and for all $i>0$. If $y\in Y_l$ and $\cal J^{X_l}_{X_y}$ denotes the ideal sheaf of $X_y$ in $X_l$, then we have an exact sequence $$0@>>>\cal J_{X_l}@>>>\cal J_{X_y}@>>>\cal J^{X_l}_{X_y}@>>>0.$$ \begin{claim} The lemma is true if there exists $k$ such that for all $n\ge k$, for $l=1,\cdots,r$ and for all $y\in Y_l$ we have that $H^i(X_l,\cal F\otimes A^n|_{X_l}\otimes \cal J^{X_l}_{X_y})=0$ for $i>0$. \end{claim} {\it Proof.} It follows from the exact sequences $$ \CD H^i(X,\cal F\otimes A^n\otimes \cal J_{X_l})@>>>H^i(X,\cal F\otimes A^n \otimes \cal J_{X_y}) @>>>H^i(X,\cal F\otimes A^n\otimes \cal J^{X_l}_{X_y})\\ @| @. @| \\ 0 @. @. 0 \endCD $$ for $i>0$. $\sharp$ \bigskip This means that we can reduce to the case where $f$ is flat. For $y_0\in Y$, we can find $k_0$ such that for $n\ge k_0$ and for $i>0$ we have $$H^i(X,\cal F\otimes A^n\otimes \cal J_{X_{y_0}})=0.$$ Therefore, the morphism $$\lim_{y_0\in U}H^0(f^{-1}U,\cal F\otimes A^n)@>\beta _{y_0}>> H^0(X_y,\cal F\otimes A^n) $$ is onto, and then so is $$\psi _{y_0}=:\beta_{y_0} \otimes k(y_0): f_{*}(\cal F\otimes A^n)(y)@>>>H^0(X_{y_0},\cal F\otimes A^n |_{X_{y_0}}).$$ By Grauert's theorem (\cite{ha:ag}) we then have that $\psi _{y_0}$ is an isomorphism, and that the same holds for $\psi _y$, for $y$ in a suitable open neighbourhood $U_0$ of $y_0$. Therefore the restriction morphism $$H^0(X,\cal F\otimes A^n)@>>>H^0(X_y,\cal F\otimes A^n|_{X_y})$$ come from a morphism of sheaves, and hence they are onto for all $y\in V_0$, for a suitable open set $V_0\subset U_0$. We can then invoke the quasi-compactness of $Y$ to conclude that there exists $k$ such that $H^1(X,\cal F\otimes A^n\otimes \cal J_{X_y})=0$ for all $y\in Y$. As to $i\ge 2$, we have isomorphisms $$H^i(X_y,\cal F\otimes A^n|_{X_y}) \simeq H^{i+1}(X,\cal F\otimes A^n\otimes \cal J_{X_y})=0$$ for all $i>0$, and so we need to show that $H^i(X_y,\cal F\otimes A^n)=0$ for $n\gg 0$, $i>0$ and for all $y\in Y$. But for $n\gg 0$ we have $$R^if_{*}(\cal F\otimes A^n)=0$$ if $i>0$ and then this implies $h^i(X_y,\cal F\otimes A^n|_{X_y})=0$ for all $y$ (\cite{mu:cs}). $\sharp$ \bigskip We record here a trivial numerical lemma that will be handy in the sequel: \begin{lem} If $s\ge \alpha$, $a \ge 2s$ and $b \ge a s-s^2$, then $b \ge a \alpha-\alpha ^2$. \label{lem:trivial} \end{lem} {\it Proof.} $a s-s^2$ is increasing in $s$ if $a \ge 2s$. The statement follows. $\sharp$ \bigskip \section{\bf {Seshadri Constants of Curves}}\label{section:sc} Let $C\subset \bold P^3$ be a smooth curve and let $H$ denote the hyperplane bundle on $\bold P^3$. We shall let $f:X_C@>>>\bold P^3$ be the blow up of $\bold P^3$ along $C$, and $E=f^{-1}C$ be the exceptional divisor. \begin{defn} The Seshadri constant of $C$ is \begin{center} $\epsilon (C)=: sup \{\eta \in \bold Q|f^{*}H-\eta E$ is ample$\}$. \end{center} \label{defn:sc} \end{defn} \noindent In other terms, $\epsilon (C)$ is the supremum of the ratios $\frac nm$, where $n$ and $m$ are such that $mH-nE$ is ample (or, equivalently, very ample). In the sequel we shall use the short hand $$H_{\eta}=:H-\eta E$$ for $\eta \in \bold Q$; furthermore, we shall generally write $H$ for $f^{*}H$ (as we just did). \begin{lem} $H_{\eta}$ is ample if and only if $0<\eta <\epsilon (C)$. It is nef if and only if $\eta \in [0,\epsilon (C)]$. \label{lem:nef} \end{lem} {\it Proof.} Since the ample cone of a projective variety is convex, the line $H-tE\subset N^1(X)$ intersects $K^{+}(X)$ in a segment $(H-t_1E,H-t_2E)$. Let $F$ denote the numerical class of a fiber of $\pi :E@>>>C$. Then $H_{\eta}\cdot F=\eta$, and therefore if $H_{\eta}$ is ample we must have $\eta >0$. Hence $t_1\ge 0$. On the other hand, it is well known that $H-tE$ is ample for $t>0$ sufficiently small, and therefore $t_1=0$. By definition, $t_2=\epsilon _2(C)$. The remaining part of the statement is clear. $\sharp$ \bigskip \begin{cor} We have \begin{center} $\epsilon (C)=sup\{\eta |H_{\eta}\cdot D\ge 0$ for all curves $D\subset X_C\}.$ \label{cor:nef} \end{center} \end{cor} \begin{lem} Let $C\subset \bold P^3$ be a smooth curve, and let $\cal J_C$ be its ideal sheaf. Let $m$ and $n$ be nonnegative integers. Then $\cal O_{X_C}(mH-nE)$ is globally generated if $\cal J_C^n(m)$ is. \label{lem:gg} \end{lem} {\it Proof.} Let us suppose that $\cal J_C^n(m)$ is globally generated, and let $$F_1,\cdots,F_k\in H^0(\bold P^3,\cal J_C^n(m))$$ be a basis. Let $P\in C$ and let $U$ be some open neighbourhood of $P$. By assumption, $F_1,\cdots,F_k$ generate $\cal J_C$ in $U$. By abuse of language, let us write $F_i$ for the pull-backs $f^{*}F_i$. Then if $e$ is a local equation for $E$ in a Zariski open set $V\subset f^{-1}U$, then the ideal generated by the $F_i$s is $<\{F_i\}>=(e^n)$. Hence we can write $$\sum _{i=1}^kP_iF_i=e^n$$ for some $P_i$s regular on $V$. However, by construction we can write $F_i=\tilde F_ie^n$, and therefore we have $$\sum _{i=1}^k\tilde F_iP_i=1$$ in $V$. Hence the $\tilde F_i$ are base point free, and they can be extended to global sections of $\cal O_{X_C}(mH-nE)$, which is therefore globally spanned. $\sharp$ \bigskip \begin{cor} Let $C\subset \bold P^3$ be a smooth curve. Then \begin{center} $\epsilon (C)\ge sup \{\frac nm|\cal J_Y^n(m)$ is globally generated$\}$. \end{center} \label{cor:gg} \end{cor} Let us look at some examples. \begin{exmp} If $L\subset \bold P^3$ is a line, then $\cal J_L(1)$ is globally generated. Therefore $\epsilon (L)\ge 1$. On the other hand, let $\Lambda \subset \bold P^3$ be a hyperplane containing $L$ and let $D\subset \Lambda$ be any irreducible curve distinct from $L$. Then $H_1\cdot \tilde D=deg(D)-L\cdot _{\Lambda}D=0$, where $\tilde D\subset Bl_L(\bold P^3)$ is the proper transform of $L$. Hence $\epsilon (L)=1$. As we shall see shortly, this generalizes to the statement that if $C\subset \bold P^3$ is a smooth complete intersection of type $(a,b)$ and $a\ge b$, then $\epsilon (C)=\frac 1a$. \end{exmp} \begin{exmp} If $C\subset \bold P^3$ has an $l$-secant line, then $\epsilon (C)\le \frac 1l$. To see this, let $L$ be the $l$-secant; denoting by $\tilde L\subset X_C$ the proper transform of $L$ in $Bl_C(\bold P^3)$ we have $H\cdot \tilde L=1$ and $E\cdot \tilde L=l$. Hence $0\le H_{\epsilon}\cdot \tilde L$ implies $\epsilon \le \frac 1l$. \end{exmp} \begin{lem} Let $C\subset \bold P^3$ be a smooth curve of degree $d$. Then $$\frac 1{\sqrt d}\ge \epsilon (C)\ge \frac 1d$$ \label{lem:deg} \end{lem} {\it Proof.} It is well-known that a smooth subvariety of degree $d$ of projective space is cut out by hypersurfaces of degree $d$. Hence $\cal J_C(d)$ is globally generated, and this proves the second inequality. As to the first, we must have $0\le H\cdot H_{\epsilon}^2=1-\epsilon ^2d$, by a simple Segre class computation. $\sharp$ \bigskip The right inequality is sharp if the curve is degenerate; the left one is sharp for a complete intersection curve of type $(a,a)$. If the curve is nondegenerate, however, one can say something more. \begin{defn} Let $C\subset \bold P^3$ be a smooth curve, and let $\cal J_C$ be its ideal sheaf. $C$ is said to be {\it $l$-regular} if $$H^i(\bold P^3,\cal J_C(l-i))=0$$ for all $i>0$. The {\it regularity} of $C$, denoted by $m(C)$, is the smallest $l$ such that $C$ is $l$-regular (\cite{ca}, \cite{mu}, \cite{gr}). \label{defn:reg} \end{defn} \begin{rem} By a celebrated theorem of Castelnuovo, we have $m(C)\le d-1$ (\cite{ca}, \cite{glp}). \label{rem:cast} \end{rem} \begin{prop} Let $C \subset \bold P^3$ be a smooth space curve, and let $m=m(C)$ be its regularity. Then $$\dfrac 2{m-1}\ge \epsilon (C)\ge \dfrac 1m.$$ \label{prop:reg} \end{prop} {\it Proof.} By a classical theorem of Castelnuovo-Mumford, the homogeneuos ideal of $C$ is saturated in degree $m(C)$ and therefore $\epsilon (C)\ge \frac 1{m(C)}$. By definition, to prove the first inequality it is enough to show that $H^i(\bold P^3,\cal J_C(k))=0$ for $k\ge \lceil \frac 2{\epsilon (C)} \rceil -3$ because this implies $m(C)\le \frac 2{\epsilon (C)}+1$ and then the statement. To prove the above vanishing, observe that $$\Big \{2/ (\lceil 2/\epsilon (C)\rceil +1) \Big \} <\epsilon (C)$$ and therefore $$\Big (\Big \lceil \frac 2{\epsilon (C)}\Big \rceil +1\Big )H-2E$$ is an ample integral divisor in $X_C$. Since $\omega _{X_C}=\cal O_{X_C}(-4H+E)$, the Kodaira vanishing theorem gives: $$H^i(X_C,\cal O_{X_C}((\lceil 2/\epsilon (C) \rceil -3)H-E))=0$$ for $i>0$, as desired. $\sharp$ \bigskip \begin{rem} Using vanishing theorems on the blow up to obtain bounds on the regularity is a well-known technique: see \cite{bel} for various results in this direction. \end{rem} \begin{rem} It is not possible, in the above vanishing, to replace the condition on $k$ by $k\ge \lceil \frac 1{\epsilon}\rceil$. To see this, suppose that $C$ is a complete intersection of type $(a,b)$ so that we have a Koszul resolution $$0@>>>\cal O_{\bold P^3}(-b)@>>> \cal O_{\bold P^3}\oplus \cal O_{\bold P^3}(a-b)@>>>\cal J_C(a)@>>>0.$$ It follows that $H^2(\bold P^3,\cal J_C(a))\simeq H^3(\bold P^3, \cal O_{\bold P^3}(-b))\neq 0$, for $b\ge 4$. \end{rem} \begin{cor} Let $C\subset \bold P^3$ be a nondegenerate smooth curve. Then $\epsilon (C)\ge \frac 1{d-1}$. \end{cor} \noindent Equality is attained in the previous corollary in the case of a twisted cubic. \bigskip It is convenient to introduce the following definition. \begin{defn} Let $C\subset \bold P^3$ be a smooth curve. For an irreducible curve $D\subset \bold P^3$ different from $C$ let $\tilde D$ be its proper transform in the blow up of $\bold P^3$ along $C$. Define \begin{center} $\epsilon _1(C)=:sup \{\eta \in \bold Q| (H-\eta E)|_E$ is ample$\}$ \end{center} and \begin{center} $\epsilon _2(C)=:sup\{\eta \in \bold Q| H_{\eta}\cdot \tilde D\ge 0 \forall$ irreducible curves $D\neq C$\}. \end{center} \label{defn:12} \end{defn} \begin{rem} $\epsilon (C)=min\{\epsilon _1(C), \epsilon _2(C)\}$. \label{rem:12} \end{rem} We are interested in estimating the Seshadri constant of a space curve $C$. It is convenient to examine $\epsilon _1(C)$ and $\epsilon _2(C)$ separately. We shall see that $\epsilon _1(C)$ is determined by the structure of the normal bundle, while $\epsilon _2(C)$ depends on the "linkage" of $C$, and is generally much harder to estimate. We start with an analysis of $\epsilon _1(C)$. \begin{defn} Let $C$ be a smooth projective curve and let $\cal E$ be a rank two vector bundle on it. For all finite morphisms $f:\tilde C@>>>C$ and all exact sequences of locally free shaves on $\tilde C$ of the form $0@>>>L@>>>f^{*}\cal E@>>>M@>>>0$, consider the ratios $\frac {deg(L)}{deg(f)}$. Let $\Sigma _{\cal E}$ denote the set of all the numbers obtained in this way. Define $$s(\cal E)=:sup\Sigma _{\cal E}.$$ \label{defn:wahl} \end{defn} \begin{rem} As in \cite{w}, $s(\cal E)$ can be interpreted as a measure of the instability of $\cal E$. More precisely, we have $$s(\cal E)=\frac 12deg(\cal E)$$ if $\cal E$ is semistable and $$s(\cal E)=deg(L)$$ if $\cal E$ is unstable, and $L\subset \cal E$ is the maximal destabilzing line subundle of $\cal E$. In other words, $s(\cal E)-\frac 12deg(\cal E)\ge 0$ always, and equality holds if and only if $\cal E$ is semistable. \label{rem:wahl} \end{rem} We then have \begin{prop} Let $C\subset \bold P^3$ be a smooth curve. Denote by $N$ the normal bundle of $C$ in $\bold P^3$, and let $\epsilon _1(C)$ be as above. Then $$\epsilon _1(C)=\frac {deg(C)}{s(N)}.$$ \label{prop:e1} \end{prop} {\it Proof.} Let $X_C=:Bl_C(X)@>f>>X$ be the blow up of $X$ along $C$ and let $E$ be the exceptional divisor; recall that $E$ can be identified with the relative projective space of lines in the vector bundle $N$. Set $\pi =f|_E$ and denote by $F$ a fiber of $\pi$. Let $D\subset E$ be any reduced irreducible curve. If $D$ is a fiber of $\pi$, then $\eta >0$ ensures that $H_{\eta}\cdot D>0$. Hence we may assume that $D@>>>C$ is a finite map, whose degree is given by $a=D\cdot F$. Let $ \psi : \tilde D@>>>D\subset X_C$ be the normalization of $D$, and let $p:\tilde D@>>>C$ be the induced morphism. Then $\psi$ is equivalent to the assignment of a sub-line bundle $L\subset p^{*}N$, given by $L=\psi ^{*}\cal O_{\bold PN}(-1)$. Since $\cal O_{\bold PN}(-1)\simeq \cal O_E(E)$, we have $deg(L)=D\cdot E$. Hence $H_{\eta}\cdot D=aH\cdot C-\eta \cdot deg(L)$; the condition $\eta \le \epsilon _1(C)$ translates therefore in the condition $\eta \le inf\{\dfrac {H\cdot C}{deg(L)/a}\}$. In other words, then, it is equivalent to $\eta \le \dfrac {H\cdot C}{s(N)}$. $\sharp$ \bigskip \begin{exmp} Let $C\subset \bold P^3$ be a smooth complete intersection curve of type $(a,b)$, with $a\ge b$. Then we have a Koszul resolution of the ideal sheaf of $C$, from which it is easy to conclude that $\epsilon (C)\ge \frac 1a$. On the other hand, $s(N)=a^2b$ and therefore by Proposition \ref{prop:e1} $\epsilon _1(C)=\frac 1a$. Hence we have $\epsilon (C)=\frac 1a$. \end{exmp} \begin{exmp} Let $C\subset \bold P^3$ be given as the zero locus of a regular section of a rank two vector bundle $\cal E$ on $\bold P^3$. It is well known that this is always the case provided that the determinant of the normal bundle $N$ extends. The Koszul resolution then is $$0@>>>det(\cal E)^{-1}@>>> \cal E^{*}@>>>\cal J_C@>>>0.$$ By Corollary \ref{cor:gg} and Proposition \ref{prop:e1}, we then conclude that $$\frac {H\cdot C}{s(\cal E|_C)}\ge \epsilon (C)\ge \epsilon (\cal E)$$ where \begin{center} $\epsilon (\cal E)=sup \{\frac nm|S^n\cal E^{*}(m)$ is spanned$\}$. \end{center} \label{exmp:zl} \end{exmp} We shall call $\epsilon (\cal E)$ the Seshadri constant of the vector bundle $\cal E$. It has the following geometric interpretation. Let $\bold P\cal E$ be the relative projective space of lines in $\cal E$. $Pic (\bold P\cal E)$ is generated by two line bundles $H$ and $\cal O(1)$, where $H$ is the pull-back of the hyperplane bundle on $\bold P^3$. Let $R$ be some divisor associated to the line bundle $\cal O(1)$. It is well known that the rational divisor $H+\eta R$ is ample, for sufficiently small $\eta \in \bold Q^{+}$ (\cite{ha:ag}). \begin{prop} $\epsilon (\cal E)=sup\{\eta \in \bold Q| H+\eta R\in Div _{\bold Q}(\bold P\cal E)$ is ample$\}$. \label{prop:scvb} \end{prop} {\it Proof.} Provisionally denote by $\gamma (\cal E)$ the right hand side of the statement. Also, for brevity let us set $X=\bold P\cal E$ and let $X_z$ stand for the fiber over a point $z\in \bold P^3$. Let us first prove that $\epsilon (\cal E)\le \gamma (\cal E)$. Suppose then that $\eta =\frac nm<\epsilon (C)$, where $n$ and $m$ are such that $S^n\cal E^{*}(m)$ is globally generated. Since $$S^n\cal E^{*}(m)=f_{*}\cal O_X(mH+nR),$$ we have the identifications $$H^0(X,\cal O_X(mH+nR))\simeq H^0(\bold P^3, S^n\cal E^{*}(m))$$ and $$H^0(X_z,\cal O_{X_z}(mH+nR))\simeq S^n\cal E^{*}(m)(z).$$ With this in mind, we then have a surjection $$H^0(X,\cal O_X(mH+nR))@>>>H^0(X_z,\cal O_{X_z}(mH+nR))$$ for all $z\in \bold P^3$, and since $\cal O_X(mH+nR)$ is generated along the fibers, it is also globally generated. Let us now prove that $\gamma (\cal E)\le \epsilon (\cal E)$. Let $\eta =\frac nm <\gamma (\cal E)$, where $n$ and $m$ have been chosen so that $mH+nR$ is ample. After perhaps multiplying $m$ and $n$ by some large positive integer we may suppose that $mH+nR$ is very ample and that $$H^i(X,\cal J_{X_z}(mH+nR))=0$$ for all $i>0$ and all $z\in \bold P^3$ (see Lemma \ref{lem:fs}). But then we have surjective restriction maps $$H^0(X,\cal O_X(mH+nR))@>>>H^0(X_z,\cal O_{X_z}(mH+nR))$$ for all $z\in \bold P^3$, and the lemma then follows from the above identifications. $\sharp$ \begin{rem} The inequality $\epsilon (C)\ge \epsilon (\cal E)$ from Example \ref{exmp:zl} can then be explained as follows. For each $n\ge 0$ we have surjective morphisms $S^n\cal E^{*}@>>>\cal J_C^n$, and therefore we have a surjection of sheaves of graded algebras $$\bigoplus _{n\ge 0} S^n\cal E^{*}@>>> \bigoplus _{n\ge 0}\cal J_C^n,$$ which yields a closed embedding $$i:X_C\hookrightarrow \bold P\cal E.$$ On the other hand, $i^{*}\cal O_{\bold P\cal E}(R)=\cal O_{X_C}(-E)$ and the above ineqality is just saying that if $H+\eta R$ is ample, it restricts to an ample divisor on $X_C$. \end{rem} \bigskip We now consider ways to estimate $\epsilon _2(C)$. $\epsilon _2(C)$ gathers more global information than $\epsilon _1(C)$, because it relates to how $C$ is "linked" to the curves in $\bold P^3$. Recall that our definition was: \begin{center} $\epsilon _2(C)=:sup\{\eta \in \bold Q| H_{\eta}\cdot \tilde D\ge 0$ $\forall$ irreducible curves $D\subset \bold P^3$, $D\neq C \}$. \end{center} As usual, $\tilde D$ denotes the proper transform of $D$ in the blow up of $C$. There does not seem to be much that one can say about $\epsilon _2(C)$ in general; with some extra assumptions, however, we can obtain an estimate. Let us make the following definiton: \begin{defn} Let $D\subset \bold P^3$ be a reduced irreducible curve, and let $t:D_n@>>>D\subset \bold P^3$ be its normalization. If the derivative $dt:T_{D_n}@>>>t^{*}T_{\bold P^3}$ never drops rank, we shall say that $D$ has only ordinary singularities. \label{defn:os} \end{defn} \begin{prop} Let $C\subset \bold P^3$ be a smooth curve. Suppose that $C$ is contained in the intersection of two distinct reduced and irreducible hypersurfaces $V_a$ and $V_b$ of degree $a$ and $b$, respectively. Suppose that all the residual curves to $C$ in the complete intersection $V_a\cap V_b$ are reduced and that at least one of the two hypersurfaces is smooth. Then $$ \epsilon _2(C)\ge \frac 1{a+b-2}.$$ If all the residual curves have ordinary singularities, then equality holds if and only if the residual curve is a union of disjoint lines. \label{prop:e2} \end{prop} \begin{exmp} It is well-known that a curve which is linked to a line $L$ in a complete intersection of type $(a,b)$ is cut out by the hypersurfaces $V_a$ and $V_b$ and by a third equation of degree $a+b-2$. Therefore its ideal sheaf is generated in degree $a+b-2$, so that $\epsilon (C)\ge \frac 1{a+b-2}$. On the other hand, it is easy to check that $\tilde L\cdot E=a+b-2$. Therefore in this case we find directly that $\epsilon (C)=\frac 1{a+b-2}$. More generally, the same argument works whenever $C$ is linked to a union of (reduced) disjoint lines. \label{exmp:sharp} \end{exmp} \begin{exmp} The assumption that the residual curves be all reduced is necessary. To see this, let $L\subset \bold P^3$ be a line, and let $V$ be a smooth surface of degree $v$ through $L$. We have $L\cdot _VL=2-v$. Let $H$ be the hyperplane bundle restricted to $V$. Then for $s\gg 0$ the linear series $|sH-2L|$ is very ample. Choose a smooth curve $C\in |sH-2L|$. Then $C$ is linked to a double line supported on $L$ in the complete intersection $V\cap W$, where $W$ is a suitable hypersurface of degree $s$ in $\bold P^3$. We have $$\tilde L\cdot E_C=(sH-2L)\cdot _VL=s-2L^2=s+2v-4,$$ and so $\epsilon _2(C)\le \dfrac 1{s+2v-4}$. \end{exmp} {\it Proof.} We need to show that for $\eta \le\frac 1{a+b-2}$ we have $\tilde D\cdot H_{\eta}\ge 0$, whenever $D\subset \bold P^3$ is some irreducible curve distinct from $C$. Clearly we may assume that $D$ is reduced. Let us start with the following simple observation. Let $C$ and $D$ be reduced curves in $\bold P^3$, and let $D_n@>t>>D\subset \bold P^3$ be the normalization of $D$. If $X_C@>f>>\bold P^3$ is the blow up of $C$ and $E_C$ is the exceptional divisor, clearly $t$ factors through $f$, i.e. there exists $u:D_n@>>>X_C$ such that $t=f\circ u$. On the other hand, $t^{-1}C=u^{-1}f^{-1} C=u^{-1}E_C$ and therefore \begin{equation} \tilde D\cdot E_C=D_n\cdot _uE_C=deg\{t^{-1}C\}. \label{eq:norm} \end{equation} Given the geometric situation, we start testing the desired positivity condition on the curves that are not contained in $V_a\cap V_b$. \begin{lem} Let $C\subset \bold P^3$, $V_a$ and $V_b$ be as in the statement of the Proposition. Suppose that $a\ge b$, and let $\eta \le \frac 1a$. Then for every irreducible curve $D\not\subset V_a\cap V_b$ we have $\tilde D\cdot H_{\eta}\ge 0$. \label{lem:nonres} \end{lem} {\it Proof of the Lemma.} Let $D$ be reduced and have degree $s$, and set $G=:V_a\cap V_b$. $G$ is a complete intersection curve, and then we know from the Koszul resolution of its ideal sheaf that its Seshadri constant satisfies $\epsilon (G)\ge \frac 1a$. Let $X_G@>>>\bold P^3$ be the blow up of $\bold P^3$ along $G$, and let $E_G$ be the exceptional divisor. For $\eta \in \bold Q$, let $H^{\prime}_{\eta} = g^{*}H-\eta E_G$. By what we have just said, $H^{\prime}_{\frac 1a}$ is a nef divisor on $X_G$. Therefore, if we let $D^{\prime}\subset X_G$ denote the proper transform of $D$ in $X_G$, we have $D^{\prime}\cdot H^{\prime}_{\eta}\ge 0$, and this can be rewritten as $D^{\prime}\cdot E_G\le as$. Now let as above $t:D_n@>>>D\subset \bold P^3$ be the normalization of $D$, and let $\tilde D\subset X_C$ denote the proper transform of $D$ in the blow up of $C$. Then by equation (\ref{eq:norm}) $$\tilde D\cdot E_C=deg\{t^{-1}C\}\le deg\{t^{-1}G\} =D^{\prime}\cdot E_G,$$ since $G\supset C$ as schemes. Therefore, \begin{equation} H_{\frac 1a}\cdot \tilde D\ge H^{\prime}_{\frac 1a}\cdot D^{\prime}\ge 0, \end{equation} and the statement follows. $\sharp$ \bigskip We now need to consider the condition $H_{\eta}\cdot \tilde D_i\ge 0$, where the $D_i$s are the irreducible components of the residual curve to $C$ in the complete intersection $V_a\cap V_b$. Let us drop the index $i$, and let $D$ be one of the the $D_i$s. We have to show that $H_{\eta}\cdot \tilde D\ge 0$ for $\eta \le \frac 1{a+b-2}$. We shall be using case (b) of the following lemma, but it may be worthwhile to state it in more generality: \begin{lem} Let $C$ and $D$ be reduced irreducible space curves, and suppose that either one of the following conditions holds: (a) $C$ and $D$ are both smooth, or (b) $C$ and $D$ lie in a smooth hypersurface $S\subset \bold P^3$. Then $\tilde D\cdot E_C=\tilde C\cdot E_D$, where $\tilde D$ (resp., $\tilde C$) is the proper transform of $D$ in the blow up of $C$ (resp., the proper transform of $C$ in the blow up of $D$). \label{lem:blowups} \end{lem} {\it Proof.} Let us first suppose that $(b)$ holds. Let $t: C_n@>>>C\subset S$ be the normalization of $C$. By (\ref{eq:norm}), we know that $\tilde C\cdot E_D=deg\{t^{-1}D\}=deg\{t^{*}\cal O_S(D)\} =C\cdot _SD$ and similarly for $\tilde C\cdot E_D$. If (a) holds, the situation is almost the same, because at each intersection point $P$ of $C$ and $D$ we can still locally view $C$ and $D$ as lying in some smooth open surface in an neighbourhood of $P$, and the problem is local in $P$. Explicitly, the argument is the following. Suppose that $C\cap D$ is supported on $P_1,\cdots,P_k$. We "measure" the intersection of $C$ and $D$ in the following way (cfr \cite{sev}): let $\pi:\bold P^3--\to \bold P^2$ be a general projection, and set \begin{equation} C{*}D=:\sum _{i=1}^ki(\pi (P_i),\pi (C),\pi (D)), \label{eq:int} \end{equation} where $i$ denotes the ordinary intersection multiplicity. Using the projection formula, one can easily check the following: \begin{claim} Let $P\in \bold P^3$ be chosen generally, and let $C_P$ be the cone on $C$ with vertex $P$. Then $$C{*}D=\sum _{i=1}^k i(P_i,D,C_P).$$ \label{claim:cones} \end{claim} Observe that these intersection multiplicities are generally constant by the principle of continuity. Given that $C*D$ is symmetric, Lemma \ref{lem:blowups} will follow once we establish that $C*D=\tilde D\cdot E_C$. Since $C$ is smooth, it is defined scheme-theoretically by the cones through it (\cite{mu}). Hence for the proof of Lemma 3.5 we are reduced to the following: \begin{lem} Let $C$ and $D$ be distinct reduced irreducible curves in $\bold P^3$. Suppose that $C\cap D$ is supported at points $P_1,\cdots,P_k$. Let $\cal C\subset H^0(\bold P^3, \cal J_C(m))$ be an irreducible family of hypersurfaces. Suppose that the linear series $V=|\cal C|$ spanned by $\cal C$ globally generates $\cal J_C(m)$ (in other words, $C$ is cut out scheme-theoretically by the elements of $\cal C$). Then for a general $F\in \cal C$ we have $$\tilde D\cdot E_C=\sum _{i=1}^k i(P_i,D,F).$$ \label{lem:blowups1} \end{lem} {\it Proof.} The assumption implies in particular that $\cal C\not\subset H^0(\bold P^3,\cal J_C^2(m))$, i.e. that the general $F\in \cal C$ is generically smooth along $C$. For such a general $F$, then, if $\tilde F$ denotes the proper transform in $X_C$ we have $$\tilde F\in |f^{*}F-E|.$$ Furthermore, the family of all such $\tilde F$ has to be base point free, so there is $F\in \cal C$ which is generically smooth along $C$ and such that $\tilde F$ does not meet any of the intersection points of $\tilde D$ and $E_C$. Let us denote by a subscript $(\cdot ,\cdot )_{P}$ the contribution to a given intersection product on $X_C$ coming from the points lying over $P\in \bold P^3$. Then by construction and the projection formula we have $$(\tilde D\cdot E_C)_{P_i}= (\tilde D\cdot f^{*}F)_{P_i}=i(P_i,D,F)$$ and this proves the lemma. $\sharp$ \bigskip Let then $X_D@>>>\bold P^3$ be the blow up of $\bold P^3$ along $D$, and let $G$ be the complete intersection $V_a\cap V_b$. Then $C$ is a component of the effective cycle $G-D$, and furthermore $G-D$ does not have any component supported on $D$. Hence we may consider the proper transform $\tilde {G-D}\subset X_D$, which is an effective cycle in $X_D$ containing $\tilde C$ as a component. Suppose, say, that $V_a$ is smooth. Then we are in case (b) of lemma 3.5, and therefore we have \begin{equation} \tilde D\cdot E_C=\tilde C\cdot E_D\le (\tilde {G-D})\cdot E_D. \label{eq:fulton} \end{equation} In the hypothesis of the proposition, at a generic point of $D$ $V_a$ and $V_b$ are both smooth and meet transversally (for otherwise $D$ would not be reduced). Therefore $\tilde V_a\equiv f^{*}V_a-E$ and $\tilde V_b\equiv f^{*}V_b -E$, and no component of $\tilde V_a\cap \tilde V_b$ maps dominantly to $D$. Furthermore if, say, $V_a$ is smooth then $\tilde V_a\simeq V_a$ does not contain any fiber of $\pi$. Therefore $\tilde {(G-D)}=\tilde V_a\cap \tilde V_b$, and so $$\tilde {(G-D)}\cdot E_C= (f^{*}V_a-E)\cdot (f^{*}V_b-E) \cdot E_C.$$ Let $N$ denote the normal bundle to the complete intersection $G$. {}From intersection theory, the latter term is known to be \begin{equation} \{c(N)\cap s(D,\bold P^3)\}_0=s(D,\bold P^3)_0+(a+b)H\cap s(D,\bold P^3)_1 \label{eq:fulton1} \end{equation} where $c(N)$ denotes the total Chern class of $N$, and $s(D,\bold P^3)$ is the Segre class of $D$ in $\bold P^3$ (\cite{fu}, $\S$9 ). Summing up, we have \begin{equation} \tilde D\cdot E_C\le s(D,\bold P^3)_0+(a+b)H\cap s(D,\bold P^3)_1 \label{eq:fulton2} \end{equation} and equality holds if and only if $D$ does not meet any component of $G-C$ different from $C$. \begin{lem} We have $s(D,\bold P^3)_1=[D]$ and $s(D,\bold P^3)_0\le -2deg(D)$; if $C$ only has ordinary singularities then equality holds if and only if $D$ is a line. \end{lem} {\it Proof.} If either (a) or (b) in the statement of Lemma X holds, then $D$ is a local complete intersection, and therefore it has a normal bundle $N$ in $\bold P^3$. Hence $s(D,\bold P^3)=c(N)^{-1}\cap [D]$, and the statement is then reduced to the inequality $deg(N)\ge 2deg(D)$. Let $t:D_n@>>>D$ be the normalization of $D$. We then have a generically surjective morphism $t^{*}T_{\bold P^3}@>>>t^{*}N$. On the other hand, $T_{\bold P^3}(-1)$ is globally generated, and therefore we must have $deg(N(-1))\ge 0$, i.e. $deg(N) \ge 2d$. If furthermore $D$ only has ordinary singularities, we have an exact sequence $0@>>>T_D@>>>t^{*}T_{\bold P^3} @>>>N$ and this shows that equality holds if and only if $g=0$ and $d=1$. $\sharp$ \bigskip We then have $\tilde D\cdot E_C\le (a+b-2)deg(D)$, and if $D$ has only ordinary singularities then equality holds if and only if $D$ is a line not meeting any component of $G-D$ different from $C$. The Proposition follows. $\sharp$ \bigskip We know define two auxiliary invariants related to the Seshadri constant that will be useful shortly. \begin{defn} Let $C\subset \bold P^3$ be a smooth curve of degree $d$ and let $\epsilon (C)$ be its Seshadri constant. Let $N$ be the normal bundle of $C$ in $\bold P^3$. For $0\le \eta \le \epsilon (C)$ a rational number, define $$\delta _{\eta}(C)=:\eta \cdot deg(N)-d$$ and $$\lambda _{\eta}(C)=:\eta ^2d^2-\delta _{\eta}(C).$$ \label{defn:dandl} \end{defn} It is easy to check that \begin{equation} \delta _{\eta}(C)=:E^2\cdot H_{\eta}. \label{eq:dint} \end{equation} More explicitly, suppose that $0<\eta <\epsilon (C)$ and let $m$ and $n$ be large positive integers such that $\eta =\frac nm$ and $mH-nE$ is very ample. Then for a general $S\in |mH-nE|$ the intersection $C^{\prime}=E\cap S$ is an irreducible smooth curve, and the induced morphism $C^{\prime}@>>>C$ has degree $n$. Then \begin{equation} \delta _{\eta}(C)=\frac {C^{\prime}\cdot _SC^{\prime}} {H\cdot _SH}. \label{eq:dint1} \end{equation} Similarly, \begin{equation} \lambda _{\eta}(C)=\frac {(H\cdot C^{\prime})^2}{(H\cdot _SH)^2}- \frac {C^{\prime}\cdot _SC^{\prime}} {H\cdot _SH}. \label{eq:lint} \end{equation} \begin{rem} If we let $x=\eta d$, we have $\lambda _{\eta}(C)=f(x)$, where $$f(x)=x^2-\Big (4+\frac {2g-2}d\Big )x+d.$$ For $C$ subcanonical, $f$ is the polynomial introduced by Halphen in his celebrated {\it speciality theorem} (\cite{gp}), given by $$g(X)=x^2-(4+e)x+d$$ where $e=max\{k|H^1(C,\cal O_C(k))\neq 0\}.$ Observe that $e\le (2g-2)/d$ always. \end{rem} \begin{cor} Suppose that there exists an irreducible surface of degree $m$ through $C$, having multiplicity $n$ along $C$. If $\eta =\frac nm$, then $\lambda _{\eta}(C)\ge 0$. In particular, $\lambda _{\eta}(C)\ge 0$ for all $0\le \eta \le \epsilon (C)$. Equality holds if and only if $\cal O_S(C^{\prime})$ is numerically equivalent to a multiple of $\cal O_S(H)$. In particular, $\lambda _{\epsilon (C)}(C)\ge 0$ and equality holds if $C$ is a complete intersection. If $C$ is subcanonical and $\eta d$ is an integer, then $\lambda _{\eta}(C)=0$ forces $C$ to be a complete intersection. \label{cor:lpos} \end{cor} {\it Proof.} A straightforward application of the Hodge index theorem. The last part follows from the corresponding statement of the speciality theorem (see \cite{gp}). $\sharp$ \bigskip \begin{cor} We have: $$g\le \frac 12 d^2\epsilon (C)+d\Big (\frac 1{2\epsilon (C)}-2 \Big )+1.$$ \end{cor} The right-hand side of the above inequality is a decreasing function of $\epsilon$ in the interval $(1/d, 1/{\sqrt d})$. In other words, higher Seshadri constants impose tighter conditiond on the genus. For a Castelnuovo extremal curve of even degree we have $\epsilon =\frac 2d$ and the right hand side, as a function of $d$, is asimptotic to $\frac {d^2}4$. \begin{cor} Let $D$ be a divisor on $X_C$, and set $s=D\cdot H_{\eta}\cdot H$. Then for $0\le \eta \le \epsilon (C)$ we have $$D^2\cdot H_{\eta}-D\cdot H_{\eta}\cdot E\le s^2-s\eta d.$$ \label{cor:sl} \end{cor} {\it Proof.} Write $$D=xH+yE.$$ Then $$D^2\cdot H_{\eta}=x^2+y^2\delta _{\eta}(C)+2xyd$$ and $$D\cdot H_{\eta}\cdot E=x\eta d+y\delta _{\eta}(C).$$ {}From this we obtain $$D^2\cdot H_{\eta}-D\cdot H_{\eta}\cdot E =s^2-s\eta d-\lambda _{\eta}(C)(y^2-y).$$ Since $y$ is an integer, the statement then follows from Corollary \ref{cor:lpos}. $\sharp$ \bigskip \begin{rem} From the inequality (see Remark \ref{rem:wahl}) $$s(N)\ge \frac 12 deg(N)$$ and the definition of $\delta _{\eta}(C)$, it is easy to see that $$d\ge \delta _{\eta}(C).$$ \label{rem:dd} \end{rem} \section{\bf {Gonality of space curves and free pencils on projective varieties}}\label{section:gon} We have seen that if $C\subset S$ is a smooth curve with $C^2>0$, then one can give lower bounds on the gonality of $C$. We deal here with the next natural question: if $C\subset \bold P^3$, what can be said about $gon(C)$ in terms of the invariants of this embedding, and exactly which invariants should one expect to play a direct role? A hint to this is given by Lazarsfeld's result, to the effect that if $C$ is nondegenerate complete intersection of type $(a,b)$ with $a\ge b$ then $gon(C)\ge a(b-1)$. For $C\subset \bold P^r$ a smooth curve, we let $$\delta _{\eta}(C)=E^2\cdot H_{\eta}^2.$$ We then have $\delta _{\eta}(C)=\eta ^{r-3}(\eta deg(N)-deg(C))$. \begin{thm} Let $C\subset \bold P^r$ be a smooth curve of degree $d$, $r\ge 3$. Let $\epsilon (C)$ be the Seshadri constant of $C$, and set $\alpha =min \Big \{1,\sqrt {\epsilon (C) ^{r-3}d}\Big (1-\epsilon (C)\sqrt {\epsilon (C)^{r-3} d}\Big )\Big \}$. Then $$gon(C)\ge min\Big \{ \frac {\delta _{\epsilon (C)}(C)}{4\epsilon (C) ^{r-2}}, \alpha \Big (deg(C)-\frac {\alpha}{\epsilon (C)^{r-2}}\Big )\Big \}.$$ \end{thm} Although we state the result for curves in $\bold P^r$ for the sake of simplicity, it is easy to see that the same considerations apply when $\bold P^r$ is replaced by a general smooth projective manifold $X$ with $Pic(X)\simeq \bold Z$. Later in this section we shall indicate how these results generalize to higher dimensional varieties in $\bold P^r$. {\it Proof.} To avoid cumbersome notation, we shall assume that $r=3$. The proof applies to higher value of $r$, with no significant change. We want then to show that \begin{equation} gon(C)\ge min \Big \{\frac {\delta _{\epsilon (C)}(C)} {4\epsilon (C)}, \alpha \Big (d-\frac {\alpha}{\epsilon (C)}\Big ) \Big \}, \label{eq:spcv} \end{equation} where $\alpha =min \{1,\sqrt d(1-\epsilon (C)\sqrt d)\}$. Suppose, to the contrary, that the statement is false: if $k=gon(C) $, then $k$ is strictly smaller than both terms within the braces in the last inequality. For $\eta <\epsilon (C)$ sufficiently close to $\epsilon (C)$ the same inequality holds. More precisely, if let $\alpha _{\eta}=min \{1,\sqrt d(1-\eta \sqrt d)\}$, we have: \begin{equation} k<\dfrac {\delta _{\eta}(C)}{4\eta} \label{eq:in1} \end{equation} and \begin{equation} k< \alpha _{\eta}\Big (d-\dfrac {\alpha _{\eta}}{\eta}\Big ). \label{eq:in2} \end{equation} Pick a minimal pencil $A\in Pic^k(C)$, and set $V=:H^0(C, A)$. Then $V$ is a two-dimensional vector space. On $C$ we have an exact sequence of locally free sheaves $0@>>> -A@>>>V\otimes \cal O_C@>>>A@>>>0$. Consider the blow up diagram: \begin{equation} \CD E @>>> X_C=Bl_C(X) \\ @V\pi VV @VVfV \\ C@>>> X \endCD \label{eq:cd} \end{equation} (here $E$ clearly denotes the exceptional divisor). Define \begin{equation} \cal F=: Ker(\psi :V\otimes \cal O_{X_C}@>>>\pi ^{*}A). \label{eq:F} \end{equation} $\pi ^{*}A$ is a line bundle on $E$, and $\psi$ is surjective. Since $E$ is a Cartier divisor in $X_C$, $\cal F$ is a rank two vector bundle on $X_C$. As usual we set $H_{\eta}=H-\eta E$, where $\eta$ is a rational number. \begin{claim} Let $\eta$ be a rational number in the interval $(0,\epsilon (C))$. If $k<\dfrac {\delta _{\eta}(C)}{4\eta}$, then $\cal F$ is Bogomolov-unstable with respect to $H_{\eta}$. \end{claim} {\it Proof.} By Lemma \ref{lem:eltr}, the Chern classes of $\cal F$ are $c_1(\cal F)=-E$ and $c_2(\cal F)=\pi ^{*}[A]$, where $[A]$ denotes the divisor class in $A^1(C)$ of any element in $|V|$, and we implicitly map $A^1(E)$ to $A^2(X_C)$. Then the discriminant of $\cal F$ (definition \ref{defn:discr}) is given by $$ \Delta (\cal F)=E^2-4[A].$$ Therefore by the assumption we have \begin{equation} \Delta (\cal F)\cdot H_{\eta}= \delta _{\eta}(C)-4\eta k>0, \label{eq:basin2} \end{equation} which implies that $\cal F$ is Bogomolov-unstable with respect to $H_{\eta}$. $\sharp$ \bigskip Therefore, by Theorem \ref{thm:main}, there exists a unique saturated invertible subsheaf $\cal L\subset \cal F$ satisfying the following properties: (i) $\cal L$ is the maximal destabilizing subsheaf of $\cal F$ with respect to any pair $(H_{\eta},R)$, with $R$ an arbitrary ample divisor on $X_C$. In particular, for any such pair we have: $(2c_1(\cal L)-c_1(\cal F))\cdot H_{\eta}\cdot R>0$. Incidentally, this implies that $\cal L$ is the same for all the values of $0< \eta <\epsilon (C)$ which make the hypothesis of the claim true. (ii) $(2c_1(\cal L)-c_1(\cal F))^2\cdot H_{\eta}\ge \Delta (\cal F)\cdot H_{\eta}$. \bigskip Given the inclusions $\cal L\subset \cal F\subset \cal O_{X_C}^2$, we have \begin{equation} \cal L=\cal O_{X_C}(-D) \label{eq:eff} \end{equation} for some effective divisor $D$ on $X_C$. We can write $$D=xH+yE,$$ with $x$ and $y$ integers and $x\ge 0$. Set $$s=:D\cdot H_{\eta}\cdot H=x+y\eta d.$$ Since $\cal F$ has no sections, $D\neq 0$. The same applies for the restriction to any ample surface. Hence $s\ge 0$ for $0<\eta <\epsilon (C)$. \begin{lem} Assume that $s\ge \alpha$. Then $k\ge \alpha(d-\dfrac {\alpha} {\eta})$. \end{lem} {\it Proof.} Given (\ref{eq:eff}), from (ii) and (\ref{eq:basin2}) we get \begin{equation} (E-2D)^2\cdot H_{\eta}\ge \delta _{\eta}(C)-4\eta k. \label{eq:basin3} \end{equation} Since $E^2\cdot H_{\eta}=\delta _{\eta}(C)$, this can be rewritten $$D^2\cdot H_{\eta}-D\cdot H_{\eta}\cdot E\ge -\eta k.$$ By Corollary \ref{cor:sl}, we then have \begin{equation} s^2-s\eta d\ge -\eta k. \label{eq:basin4} \end{equation} On the other hand, we have the destabilizing condition (i) \begin{equation} (E-2D)\cdot H_{\eta}\cdot H\ge 0. \label{eq:dest} \end{equation} Now $$E\cdot H_{\eta}\cdot H=\eta d$$ and therefore (\ref{eq:dest}) can be written \begin{equation} \eta d\ge 2s. \label{eq:dest1} \end{equation} Therefore we can apply Lemma \ref{lem:trivial} with $a=\eta d$ and $b=\eta k$ to obtain $$\eta k\ge \eta d \alpha-\alpha ^2.$$ This proves the Lemma. $\sharp$ \bigskip The proof of the theorem is then reduced to the following Lemma. \begin{lem} $s\ge \alpha$. \end{lem} {\it Proof.} We shall argue that $s\ge \alpha _{\eta}$ for all rational $\eta <\epsilon (C)$ such that the inequalities (\ref{eq:in1}) and (\ref{eq:in2}) hold. For all such $\eta$ we are then in the situation of Claim 4.1. \begin{claim} $\cal L$ is saturated in $V\otimes \cal O_X$. \label{claim:sat} \end{claim} {\it Proof.} By construction, $\cal L =\cal O_X(-D)$ is saturated in $\cal F$. Therefore, if the Claim is false then the inclusion $\cal L\subset V\otimes \cal O_X$ drops rank along $E$. Hence there exists an inclusion $\cal O_X(E-D) \subset \cal O_X^2$. This implies that $D-E$ is effective, and in particular $(D-E)\cdot H_{\eta}^2\ge 0$. Together with the instability condition $(E-2D)\cdot H_{\eta}^2>0$, this would imply $D\cdot H_{\eta}^2<0$, against the fact that $D$ is effective. $\sharp$ \bigskip By Claim \ref{claim:sat}, there is an exact sequence $$0@>>>\cal O_X(-D)@>>>V\otimes \cal O_X@>>>\cal O_X(D)\otimes \cal J_Y@>>>0,$$ where $Y$ is a closed subscheme of $X$ of codimension two or empty. Computing $c_2(\cal O_X^2)=0$ from this sequence, we obtain $D^2=[Y],$ and therefore $D^2\cdot H\ge 0$. On the other hand, $D^2\cdot H=x^2-y^2d$, and so $$x\ge |y|\sqrt d.$$ Now, $$s=x+y\eta d\ge x-|y|\eta d\ge |y|\sqrt d(1-\eta \sqrt d).$$ By construction, $H^0(X,\cal F)=0$, and therefore $D\neq 0$. Hence, if $y=0$ then $s=x\ge 1$. If $y\neq 0$, then the above inequality shows that $s\ge \sqrt d(1-\eta \sqrt d)$. $\sharp$ \bigskip This completes the proof of the Theorem. $\sharp$ \bigskip \begin{exmp} Let $C\subset \bold P^3$ be a smooth complete intersection curve of type $(a,b)$, with $a\ge b+3$, $b\ge 2$. Then $gon(C)\ge a(b-1)$. \label{exmp:ci} \end{exmp} \begin{rem} This shows that the result is generally optimal. However, the theorem is void for a complete intersection of type $(a,a)$. But for complete intersections one knows more than just the Seshadri constant: not only $\epsilon (C)=\frac 1a$, but in fact the linear series $|aH-E|$ is base point free, and the general element is smooth. An ad hoc argument proves that $gon(C)\ge a(b-1)$ (\cite{la:unp}). \label{rem:ci} \end{rem} \begin{exmp} Let $C$ be a nondegenerate smooth complete curve in $\bold P^3$ that is linked to a line in a complete intersection of type $(a,b)$. Then for $a\gg b\gg 0$ we obtain $gon (C)\ge deg(C)-(a+b-2)$. This is clearly optimal, because a base point free linear series of that degree is obtained by considering the pencil of planes through the line. The same considerations as in Remark \ref{rem:ci} apply. \end{exmp} \begin{rem} An analysis of "small" linear series on special classes of space curves is carried out by Ciliberto and Lazarsfeld in \cite{cl}. It would be interesting to know whether the present method can be adapted to give a generalization of their results. \end{rem} {}From the Theorem, we immediately get \begin{cor} Let $X\subset \bold P^r$ be a smooth projective variety. Let $d$ be the degree of $X$ and $\epsilon (X)$ be its Seshadri constant. Suppose that $A$ is a line bundle on $X$ with a pencil of sections $V\subset H^0(X,A)$ whose base locus has codimension at least two. Let $F$ be any divisor in the linear series $|A|$. Then $$deg(F)\ge min \Big \{ \frac 1{4\epsilon (C)^{r-2}} \Big [\epsilon (X) \big (c_1(N)\cdot _XH^{n-1}+ (n-1)d \big )-d\Big ], \alpha \Big (d-\frac {\alpha}{\epsilon (X)}\Big )\Big \},$$ where $\alpha =\Big \{1,\sqrt {\eta ^{r-3}d} \Big (1-\epsilon (X)\sqrt {d\epsilon (C)^{r-3}} \Big )\Big \}$. \end{cor} {\it Proof.} Let $C\subset X$ be a curve of the form $X\cap \Lambda$, where $\Lambda\subset \bold P^3$ is a linear subspace of dimension $c+1$, with $c$ the codimension of $X$. Then $V$ restricts to a base point free pencil on $C$, and the result follows by applying the theorem. $\sharp$ \bigskip Given the general nature of the above arguments, one clearly expects that they should be applicable to a wider range of situations. In fact, we give now the generalization of theorem 3.1 to arbitrary smooth projective varieties in $\bold P^r$. The proof is exactly the same as the one for theorem 3.1, the only change consisting in a more involved notation. \begin{thm} Let $Y\subset \bold P^r$ be a projective manifold of degree $d$ and codimension $c$. Let $\epsilon (Y)$ be its Seshadri constant, and suppose $0\le \eta \le \epsilon (Y)$. If $A$ is base point free pencil on $Y$, then $$\pi ^{*}[A]\cdot H_{\eta}^{r-2}\ge min \Big \{\frac {\delta _{\eta}(Y)}4, \frac 1{H^2H_{\eta}^{r-2}} \alpha \Big (H\cdot E\cdot H_{\eta}^{r-2}- \alpha \Big ) \Big \},$$ where $\alpha =min \Big \{H^2\cdot H_{\eta} ^{r-2}, \sqrt {\eta ^{c-2} deg(Y)} \Big (H^2\cdot H_{\eta}^{r-2}-\dfrac {H\cdot E\cdot H_{\eta}^{r-2}} {\sqrt {\eta ^{c-2} deg(Y)}}\Big ) \Big \}$. \end{thm} \section{\bf {Stability of restricted bundles}}\label{section:stability} We deal here with the following problem: \begin{prob} Let $\cal E$ be a rank two vector bundle on $\bold P^3$, and let $C\subset \bold P^3$ be a smooth curve. If $\cal E$ is stable, what conditions on $C$ ensure that $\cal E|_C$ is also stable? \end{prob} \begin{rem} This question has been considered by Bogomolov (\cite{bo:78} and \cite{bo:svb}) in the case of vector bundles on surfaces. In particular, he shows that if $S$ is a smooth projective surface with $Pic(S)\simeq \bold Z$, $\cal E$ is a stable rank two vector bundle on $S$ with $c_1(\cal E)=0$ and $C\subset S$ is a smooth curve with $C^2>4c_2(\cal E)^2$, then $\cal E|_C$ is stable. \label{rem:surfacecase} \end{rem} After a suitable twisting, we may also assume that $\cal E$ is {\it normalized}, i.e. $c_1(\cal E)=0$ or $-1$. We shall suppose here that $c_1(\cal E)=0$, the other case being similar. As usual we adopt the following notation: $$f:X_C@>>>\bold P^3$$ is the blow up of $\bold P^3$ along $C$, $$E=f^{-1}C$$ is the exceptional divisor, and $$\pi :E@>>>C$$ is the induced projection. Recall that for $\eta \in \bold Q$ we set $$H_{\eta}=:H-\eta E,$$ where we write $H$ for $f^{*}H$. If $0<\eta <\epsilon (C)$, $H_{\eta}$ is a polarization on $X_C$. \begin{defn} We define the {\it stability constant} of $\cal E$ w.r.t. $C$ as \begin{center} $\gamma (C,\cal E)= sup \{\eta \in [0,\epsilon (C)]| f^{*}\cal E$ is $(H_{\eta}, H)$-stable$\}$. \end{center} \label{defn:gamma} \end{defn} \begin{rem} Recall that $f^{*}\cal E$ is $(H,H_{\eta})$-stable if for all line bundles $\cal L\subset f^{*}\cal E$ we have $\cal L\cdot H\cdot H_{\eta}<0$. \end{rem} \begin{lem} Suppose $0\le \eta <\epsilon (C)$. Then $f^{*}\cal E$ is $(H,H_{\eta})$-semistable if and only if $\eta \le \gamma (C,\cal E)$. If $\eta <\gamma (C,\cal E)$, $f^{*}\cal E$ is $(H,H_{\eta})$-stable. \end{lem} {\it Proof.} The collection of the numerical classes of nef divisors $D$ with respect to which $f^{*}\cal E$ is $(H,D)$-semistable (or stable) is convex, hence it contains the segment $[H,H_{\gamma (C,\cal E)}]$. Since $f^{*}\cal E$ is $(H,H)$-stable the second statement follows. $\sharp$ \bigskip \begin{lem} Suppose that $V\subset \bold P^3$ is a smooth surface of degree $a$ containing $C$, and that $\cal E|_V$ is $\cal O_V(H)$-stable. Then $$\gamma (C,\cal E)\ge min\big \{\frac 1a,\epsilon (C)\big \}.$$ \label{lem:bound} \end{lem} {\it Proof.} Let $\tilde V$ be the proper transform of $V$ in $X_C$. We have $\tilde V\simeq V$ and $$\tilde V\in |aH_{\frac 1a}|.$$ The hypothesis implies that for every line-bundle $$\cal L\subset f^{*}\cal E$$ we have $$\cal L\cdot H_{\frac 1a}\cdot H<0.$$ Hence the same holds for every $\eta$ with $0\le \eta \le \frac 1a$. $\sharp$ \bigskip \begin{rem} Note that the same argument actually proves the following stronger statement: let $V\supset C$ be a reduced irreducible surface through $C$ having degree $m$ and multiplicity $n$ along $C$, and such that $f^{*}\cal E|_{\tilde V}$ is $\cal O_{\tilde V}(H)$-stable. Then $\gamma (C,\cal E)\ge min\{\frac nm,\epsilon (C)\}$. \end{rem} \begin{lem} Fix $c_2\ge 0$ an integer. Then then there exists an integer $k$ with the following property. If $\cal E$ is a stable rank two vector bundle on $\bold P^3$ with $c_1(\cal E)=0$ and $c_2(\cal E)=c_2$, and if $V\subset \bold P^3$ is a smooth surface of degree $a\ge k$, then $ \cal E|_V$ is $\cal O_V(H)$-stable. \label{lem:surfres} \end{lem} {\it Proof.} We start by finding $s$ such that for a general surface $S$ of degree $s$ we have $Pic(S)\simeq \bold ZH$ ($s\ge 4$ will do) and furthermore the restriction $\cal E|_S$ is $\cal O_S (H)$-stable. Bogomolov's theorem (remark \ref{rem:surfacecase}) then says that for any curve $C\subset S$ such that $C^2>4c_2(\cal E)^2s^2$ the restriction $\cal E|_C$ is also stable. Let now $a>0$ be such that $a^2>4c_2(\cal E)^2s$. Suppose that $V$ is a smooth surface of degree $a$ and that $\cal E|_V$ is not stable. Then the same is true for $C=V\cap S$. For a general choice of $S$, $C$ is a smooth curve, and since $C\cdot _SC=a^2s>4c_2(\cal E)^2s^2$, we have a contradiction. $\sharp$ \bigskip We can in fact restate the previous lemma as follows: {\it Let $s$ be the smallest positive integer such that for a general surface of degree $s$ we have $Pic(S)\simeq \bold Z$ and $\cal E|_S$ stable. If $a>2c_2(\cal E)\sqrt s$ and $V\subset \bold P^3$ is any smooth surface of degree $a$, then $\cal E|_V$ is $\cal O_V(H)$-stable. } \begin{cor} Let $\cal E$ be a rank two stable bundle on $\bold P^3$ with $c_1(\cal E) =0$ but $c_2(\cal E)\neq 1$ (i.e., $\cal E$ is not a null correlation bundle (\cite{oss}). If $a>2c_2(\cal E)$ and $V\subset \bold P^3$ is a smooth hypersurface of degree $a$, then $\cal E|_V$ is $\cal O_V(H)$-stable. \label{cor:surfres} \end{cor} {\it Proof.} In fact, a theorem of Barth says that in this case we can take $s=1$ (\cite{ba}). $\sharp$ \bigskip \begin{rem} In light of Barth's restriction theorem, by induction these statements generalize to $\bold P^r$ for any $r\ge 2$ (for $r=2$ this is just Bogomolov's theorem, and the hypothesis $c_2\neq 1$ is not needed). \end{rem} \begin{rem} In the proof of Corollary 5.1, we use stability on the hyperplane section to deduce stability on the whole surface. What makes this work is Bogomolov's theorem (cfr remark \ref{rem:surfacecase}), which gives us a control of the behaviour of stability under restriction to plane curves. On the other hand, if we are given an arbitrary smooth surface $V$, it may well be that $\cal E|_V$ is $H$-stable while $\cal E|_C$ is not, where $C$ is an hyperplane section of $V$. In that case, however, $\cal E|_{V\cap W}$ will be stable, if $W$ is a smooth surface of very large degree such that $V\cap W$ is a smooth curve. To improve the above result, therefore, one would need to control the behaviour of stability under restriction to curves not necessarily lying in a plane. After proving the restriction theorem \ref{thm:curveres} we shall strengthen the above corollary (cfr Corollary \ref{cor:surres1}). \end{rem} \bigskip \begin{defn} If $X$ is a smooth variety and $c_i\in A^i(X)$ for $i=1$ and $2$, let $\cal M_X(c_1,c_2)$ denote the moduli space of stable rank two vector bundles with Chern classes $c_1$ and $c_2$. \end{defn} \begin{cor} Fix an integer $c_2\ge 0$. Then for any sufficiently large positive integer $a$ the following holds: if $V\subset \bold P^r$ is a smooth hypersurface of degree $a$, then $\cal M_{\bold P^r}(0,c_2)$ embeds as an open subset of $\cal M_V(0,c_2a)$. \label{cor:surfmoduli} \end{cor} {\it Proof.} $\cal M_{\bold P^r}(0,c_2)$ forms a bounded family of vector bundles, and therefore so does the collection of the vector bundles $End (\cal E,\cal F)$, with $\cal E$, $\cal F\in \cal M_{\bold P^r}(0,c_2)$. Therefore, if $k\gg 0$, we have $$H^i(\bold P^r,End (\cal E,\cal F)(-a))=0$$ for all $i\le 2$, $a\ge k$ and for all $\cal E$, $\cal F\in \cal M_{\bold P^r}(0,c_2)$. Furthermore, by the above lemma we can assume that $\cal E|_V$ is $\cal O_V(H)$-stable for all $\cal E\in \cal M_{\bold P^r}(0,c_2)$. {}From the long exact sequence in cohomology associated to the exact sequence of sheaves $$0@>>>End(\cal E,\cal F)(-a)@>>>End(\cal E,\cal F)@>>> End(\cal E|_V,\cal F|_V)@>>>0$$ we then obtain isomorphisms $$H^0(\bold P^r,End(\cal E,\cal F))\simeq H^0(V, End (\cal E|_V,\cal F|_V))$$ and $$H^1(\bold P^r,End (\cal E,\cal F))\simeq H^1(V,End (\cal E|_V,\cal F|_V)).$$ Since there can't be any nontrivial homomorphism between two nonisomorphic stable bundles of the same slope, the first one says that $$\cal E@>>>\cal E|_V$$ is a one-to-one morphism of $\cal M_{\bold P^r}(0,c_2)$ into $\cal M_V(0,c_2a)$ and the second says that the derivative of this morphism is an isomorphism (\cite{ma}). $\sharp$ \bigskip \begin{cor} $\gamma (C,\cal E)>0$. \label{cor:gammapos} \end{cor} {\it Proof.} By Lemma 5.3, for $r\gg 0$ the restiction of $\cal E$ to any smooth surface of degree $r$ is stable with respect to the hyperplane bundle. So we just need to consider a smooth surface through $C$ of very large degree and apply Lemma 5.2. $\sharp$ \bigskip \begin{exmp} Let $$C=V_a\cap V_b\subset \bold P^3$$ be a smooth complete intersection of type $(a,b)$, with $a\ge b$. Suppose that $V_a$ is smooth, and that $\cal E|_{V_a}$ is $\cal O_{V_a}(H)$-stable. Then $$\gamma (C,\cal E)=\frac 1a=\epsilon (C).$$ In general, $0<\eta <\gamma (C,\cal E)$ if and only if for $m$ and $n$ sufficiently large integers such that $\eta =\frac nm$, and $S\in |mH-nE|$ a smooth surface, we have that $f^{*}\cal E|_S$ is $\cal O_S(H)$-stable. In other words, we have a degree $m$ hypersurface with an ordinary singularity of multiplicity $n$ along $C$, such that the pull-back of $\cal E$ to the desingularization of $S$ is $H$-stable. \end{exmp} \bigskip Our result is then the following: \begin{thm} Let $\cal E$ be a stable rank two vector bundle on $\bold P^3$ with $c_1(\cal E)=0$. Let $C\subset \bold P^3$ be a smooth curve of degree $d$ and Seshadri constant $\epsilon (C)$, and let $\gamma =\gamma (C,\cal E)$ be the stability constant of $\cal E$ w.r.t $C$. Let $\alpha =min\Big \{1,\sqrt d\Big (\sqrt {\frac 34}-\gamma \sqrt d \Big )\Big \}.$ Suppose that $\cal E|_C$ is not stable. Then $$c_2(\cal E)\ge min\Big \{\frac {\delta _{\gamma}}4, \alpha \gamma \Big (d-\frac {\alpha}{\gamma}\Big )\Big \}.$$ \label{thm:curveres} \end{thm} {\it Proof.} Suppose to the contrary that $c_2(\cal E)$ is strictly smaller that both quantities on the right hand side. We can find a rational number $\eta$ with $0<\eta <\gamma $ such that \begin{equation} c_2(\cal E)<\dfrac {\delta _{\eta}(C)}4 \label{eq:c2ineq} \end{equation} and \begin{equation} c_2(\cal E)<\alpha \eta \Big (d-\dfrac {\alpha}{\eta}\Big). \label{eq:c2ineq1} \end{equation} By assumption there exists a line bundle $L$ on $C$ of degree $l\ge 0$ sitting in an exact sequence $$0@>>>L@>>>\cal E|_C@>>>L^{-1}@>>>0.$$ Define a sheaf $\cal F$ on $X_C$ by the exactness of the sequence \begin{equation} 0@>>>\cal F@>>>f^{*}\cal E@>>>\pi ^{*}L^{-1}@>>>0. \label{eq:Fi} \end{equation} Then $\cal F$ is a rank two vector bundle on $X_C$ having Chern classes $c_1(\cal F)=-[E]$ and $c_2(\cal F)=f^{*}c_2(\cal E)-\pi ^{*}[L]$ (cfr Lemma \ref{lem:eltr}). A straightforward computation then gives \begin{equation} \Delta (\cal F)\cdot H_{\eta}= \delta _{\eta}(C)-4c_2(\cal E)+4\eta l\ge \delta _{\eta}(C)-4c_2(\cal E) \label{eq:DF} \end{equation} and this is positive by (\ref{eq:c2ineq}). Therefore $\cal F$ is Bogomolov-unstable with respect to $H_{\eta}$ (Theorem \ref{thm:main}). Let $$\cal L\subset \cal F$$ be the maximal destabilizing line bundle w.r.t. $H_{\eta}$. We shall write $$\cal L=\cal O_{X_C}(-D),$$ with $$D=xH-yE.$$ \begin{claim} $x>0$ \label{claim:xpos} \end{claim} {\it Proof.} Pushing forward the inclusion $\cal L\subset \cal F$ we obtain an inclusion $\cal O_{\bold P^3}(-x)\subset \cal E$. Therefore the statement follows from the assumption of stability on $\cal E$ and the hypothesis $c_1(\cal E)=0$. $\sharp$ \bigskip The destabilizing condition says $(2c_1(\cal L)-c_1(\cal F))\cdot H_{\eta}\cdot R\ge 0$ for all nef divisors on $X_C$, with strict inequality holding when $R$ is ample. In particular, with $R=H$ we have \begin{equation} (E-2D)\cdot H_{\eta}\cdot H\ge 0. \label{eq:resinst} \end{equation} Let us set $s=D\cdot H_{\eta}\cdot H$. Then (\ref{eq:resinst}) reads \begin{equation} \eta d\ge 2s. \label{eq:resinst1} \end{equation} On the other hand, since $\cal L$ is saturated in $\cal F$, we also have $(E-2D)^2\cdot H_{\eta}\ge \Delta (\cal F)\cdot H_{\eta}$, and with some algebra this becomes \begin{equation} c_2(\cal E)\ge D\cdot E\cdot H_{\eta}-D^2\cdot H_{\eta}+\eta l\ge D\cdot E\cdot H_{\eta}-D^2\cdot H_{\eta}. \label{eq:resinst2} \end{equation} Invoking Corollary \ref{cor:sl}, we then have \begin{equation} c_2(\cal E)\ge s\eta d-s^2. \label{eq:resinst3} \end{equation} \begin{claim} $\cal L$ is saturated in $f^{*}\cal E$. \label{claim:ressat} \end{claim} {\it Proof.} Suppose not. Then there would be an inclusion $$\cal L(E)=\cal O_{X_C}(E-D)\subset f^{*}\cal E$$ and therefore the $(H,H_{\eta})$-stability of $f^{*}\cal E$ would force $$(E-D)\cdot H_{\eta}\cdot H<0.$$ On the other hand by instability we have $E\cdot H_{\eta}\cdot H\ge 2D\cdot H_{\eta}\cdot H$ and from this we would get $$E\cdot H_{\eta}\cdot H=\eta d<0,$$ absurd. $\sharp$ \bigskip Therefore there is an exact sequence $$0@>>>\cal O_{X_C}(-D)@>>>f^{*}\cal E@>>>\cal O_{X_C}(D)\otimes \cal J_W @>>>0$$ where $W$ is a local complete intersection subscheme of $X_C$ of codimension two or empty. Computing $c_2(f^{*}\cal E)$ from the above sequence we then get $f^{*}c_2(\cal E)=W-D^2$, i.e. $$D^2\cdot H\ge -c_2(\cal E).$$ This can be rewritten $$x^2\ge y^2d-c_2(\cal E).$$ Recall that we have (remark \ref{rem:dd}) $$d\ge \delta _{\eta}(C),$$ and therefore the assumption $c_2(\cal E)< \delta _{\eta}(C)/4$ implies \begin{equation} c_2(\cal E)< \frac d4. \label{eq:c2d4} \end{equation} \begin{lem} $$s\ge min\Big \{1,\sqrt d\Big (\sqrt {\frac 34}-\eta \sqrt d\Big ) \Big \}.$$ \end{lem} {\it Proof.} If $y\le 0$ then $s=x+|y|\eta d\ge 1$. If $y> 0$, we have $s=x-y\eta d\ge y\sqrt d \Big (\sqrt {1-\frac {c_2(\cal E)}d} -\eta\sqrt d\Big )$ and therefore using (\ref{eq:c2d4}) we obtain $$s\ge \sqrt d\Big (\sqrt {\frac 34}-\eta \sqrt d \Big ).$$ $\sharp$ \bigskip Hence we can apply lemma \ref{lem:trivial} with $a=\eta d$ and $b=c_2(\cal E)$ to deduce $$c_2(\cal E)\ge \alpha \eta d-\alpha ^2,$$ which contradicts (\ref{eq:c2ineq1}). This completes the proof of the Theorem. $\sharp$ \bigskip \begin{cor} Let $\cal E$ be a stable rank two vector bundle on $\bold P^3$ with $c_1(\cal E)=0$ and $c_2(\cal E)=c_2$. If $b\ge c_2+2$ and $V\subset \bold P^3$ is a smooth hypersurface of degree $b$, then $\cal E|_V$ is $\cal O_V(H)$-stable. \label{cor:surres1} \end{cor} {\it Proof.} Let $a\gg b$; then we may assume that if $W\subset \bold P^3$ is a surface of degree $a$ then $\cal E|_W$ is $H$-stable. If $W$ is chosen generally, we may also assume that $C=W\cap V$ is a smooth curve. Then by Lemma \ref{lem:bound} we have $\gamma (C,\cal E)=\epsilon (C)= \frac 1a$. For $a$ large enough, furthermore, we also have $\alpha =1$. Hence the theorem says that if $\cal E|_C$ is not stable, then $c_2\ge b-1$. The hypothesis implies therefore that $\cal E|_C$ is stable, and this forces $\cal E|_V$ to be stable also. $\sharp$ \bigskip \begin{cor} Let $\cal E$ be as above, and let $C=V_a\cap V_b$ be a smooth complete intersection curve of type $(a,b)$, and suppose that $V_a$ is smooth. Assume that $a\ge \frac 43b+\frac {10}3$ and that $b\ge c_2+2$. Then $\cal E|_C$ is stable. \end{cor} {\it Proof.} By Corollary 5.4, $\cal E|_{V_b}$ is $H$-stable. Hence by Lemma \ref{lem:bound} $\gamma (C,\cal E)=\frac 1a$. The first hypothesis implies that $\alpha =1$. Hence if $\cal E|_C$ is not stable the theorem yields $c_2\ge b-1$, a contradiction. $\sharp$ \bigskip \begin{cor} Fix a nonnegative integer $c_2$. Then we can find positive integers $a$ and $b$ such that if $C\subset \bold P^3$ is any smooth complete intersection curve of type $(a,b)$ then $\cal M_{\bold P^3}(0,c_2)$ embeds as a subvariety of $\cal M_C(0)$. \end{cor} {\it Proof.} The argument is similar to the one in the proof of Corollary 5.2. Here one uses the Koszul resolution $$0@>>>\cal O_{\bold P^3}(-a-b)@>>>\cal O_{\bold P^3}(-a)\oplus \cal O_{\bold P^3}(-b)@>>>\cal J_C@>>>0$$ to show that $H^i(\bold P^3,End(\cal E,\cal F)\otimes \cal J_C)=0$ for $i\le 1$. $\sharp$ \bigskip \begin{rem} Using the above corollary, we obtain a compactification of $\cal M_{\bold P^3}(c_1,c_2)$, by simply taking its closure in the moduli space of semistable bundles on the curve. It would be interesting to know whether these compactifications are intrinsic, i.e. they are independent of the choice of the curve or, if not, how they depend on the geometry of the embedding $C\subset \bold P^3$. \end{rem}

9405

alg-geom/9405014

en

https://arxiv.org/abs/alg-geom/9405014

alg-geom/9405014

Eckhard Meinrenken

Eckhard Meinrenken

On Riemann-Roch Formulas for Multiplicities

21 pages, AMS-LaTeX

A Theorem due to Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of the symplectically reduced space. Combined with the shifting-trick, this gives explicit formulas for the multiplicities of the various irreducible components. One of the assumptions of the Theorem is that the reduction is regular, so that the reduced space is a smooth symplectic manifold. In this paper, we prove a generalization of this result to the case where the reduced space may have orbifold singularities. Our proof uses localization techniques from equivariant cohomology, and relies in particular on recent work of Jeffrey-Kirwan and Guillemin. Since there are no complex geometry arguments involved, the result also extends to non Kaehlerian settings.

alg-geom

\section{Introduction} Let $(M,\omega)$ be a compact K\"{a}hler manifold, and let $\tau:L\rightarrow M$ be a holomorphic line bundle over $M$ with Hermitian fiber metric $h$. $(L,h)$ is said to satisfy the quantizing condition if $-2\pi i \omega$ is the curvature of the canonical Hermitian connection on $L$. Let $H^i(M,{\cal O}(L))$ be the $i$th cohomology group for the sheaf of germs of holomorphic sections. By the Riemann-Roch Formula of Hirzebruch and Atiyah-Singer\cite{AS68a}, the Euler number \begin{equation} \mbox{Eul}(L):=\sum_i (-1)^i \dim H^i(M,{\cal O}(L))\end{equation} is equal to the characteristic number \begin{equation} \mbox{Eul}(L)=\int_M Td\,(M)\,Ch\,(L),\label{HRR}\end{equation} where ${Td}\,(M)$ is the Todd class and ${Ch}\,(L)=e^{[\omega]}$ the Chern character. Recall that if $L$ is ``sufficiently positive'', in particular if one replaces $L$ by some sufficiently high tensor power, all the cohomology groups with $i>0$ are zero by Kodaira's Theorem \cite{GH78}, so in this case (\ref{HRR}) gives a formula for the dimension of the space $H^0(M,{\cal O}(L))=\Gamma_{hol}(M,L)$ of holomorphic sections. Let $G$ be a compact, connected Lie group that acts on $M$ by K\"ahler diffeomorphisms $\Phi:G\times M\rightarrow M$, with an equivariant moment map $J:M\rightarrow {\frak g}^*$. Suppose also that $\Phi$ lifts to Hermitian bundle automorphisms of $L\rightarrow M$, according to the rules of geometric quantization \cite{GS82a}. The corresponding virtual representation of $G$ on $\sum (-1)^i\,H^i(M,{\cal O}(L))$ may then be regarded as the ``quantization'' of the classical action $\Phi$. Its character $\chi$ is the element of the representation ring $R(G)$ defined by \begin{equation} \chi(g):=\sum_i\, (-1)^i\, \mbox{tr}\big(g|H^i(M,{\cal O}(L))\big). \label{char} \end{equation} {}From the Equivariant Riemann-Roch Formula of Atiyah-Segal-Singer \cite{AS68b,AS68a}, one has an expression for $\chi(g)$ as the evaluation of certain characteristic classes on the fixed point set $M^g=\{x\in M|\,g.x=x\}$ (which is a K\"ahler submanifold of $M$): Let ${Ch}^g(L|M^g)=c_L(g)\,{Ch}(L|M^g)$, where $c_L(g)\in S^1$ is the (locally constant) action of $g$ on $L|M^g$. Denote by $N^g$ the normal bundle of $M^g$ in $M$, by $F(N^g)$ its curvature, and let \begin{equation} D^g(N^g)=\det(I-(g^\sharp)^{-1}e^{-\frac{i}{2\pi}F(N^g)})\end{equation} where $g^\sharp$ is the automorphism of $N^g$ defined by $g$. Then \begin{equation} \chi(g)=\int_{M^g} \frac{{Td}\,(M^g){Ch}^g(L|{M^g})}{D^g(N^g)}. \label{character}\end{equation} By a Theorem of Guillemin and Sternberg \cite{GS82a}, there are also Riemann-Roch Formulas for the multiplicities of the irreducible components of the above representation, at least if certain regularity assumptions are satisfied. Let $T\subset G$ be a maximal torus, and ${\frak g}={\frak t}\oplus [{\frak t},{\frak g}]$ the corresponding decomposition of the Lie algebra. Choose a fundamental Weyl chamber ${\frak t}^*_+\subset{\frak t}^*\subset {\frak g}^*$, let $\Lambda\subset {\frak t}^*$ the integral lattice, and $\Lambda_+=\Lambda\cap {\frak t}^*_+$ the dominant weights. For a given lattice point $\mu\in\Lambda_+$, let $V_\mu$ denote the corresponding irreducible representation with highest weight $\mu$, and define the multiplicity $N(\mu)$ by the alternating sum \begin{equation} N(\mu):=\sum_i (-1)^i \dim\big(V_\mu^*\otimes H^i(M,{\cal O}(L))\big)^G. \label{defmult}\end{equation} Suppose that $\mu\in\Lambda_+$ is a regular value of $J$, or equivalently that the action of the isotropy group $G_\mu$ on $J^{-1}(\mu)$ is locally free. If the action is in fact free, the reduced space $M_\mu=J^{-1}(\mu)/G_\mu$ is a smooth symplectic manifold, and it is well-known that it acquires a natural K\"ahler structure, together with a quantizing line bundle $L_\mu$. The main result of \cite{GS82a} is that the multiplicity of $\mu$ in $\Gamma_{hol}(M,L)$ is equal to the dimension of the space $\Gamma_{hol}(M_\mu,L_\mu)$, so in particular $N(\mu)$ is given by the Euler number of $L_\mu$ if $L$ is sufficiently positive: \begin{theo}[V. Guillemin, S. Sternberg \cite{GS82a}] If the action of $G_\mu$ on $J^{-1}(\mu)$ is free, and if $L$ is sufficiently positive, \begin{equation} N(\mu)=\int_{M_\mu}{Td}\,(M_\mu){Ch}\,(L_\mu). \label{GS} \end{equation} \end{theo} The ``physical'' interpretation of this Theorem is that reduction and quantization commute. In practice, one is often dealing with situations where the action is only locally free. In this case, the reduced space is in general just an orbifold (or V-manifold) in the sense of Satake \cite{S57}, which means (roughly) that it is locally the quotient of a manifold by a finite group. Moreover, the reduced line bundle is in general just an orbifold-bundle, that is, at some points the fiber of $L_\mu$ is not ${\Bbb C}$, but its quotient by a finite group. Guillemin and Sternberg conjectured that in this case, the right hand side of (\ref{GS}) has to be replaced by the expression from T. Kawasaki's Riemann-Roch Formula for orbifolds \cite{K79}. It was proved by R. Sjamaar \cite{S93} that this assertion is true if $L$ is sufficiently positive, and if $L_\mu$ is an honest line bundle. In fact, his approach also covers the truly singular case where $\mu$ is not even a regular value, by using Kirwan's partial desingularization procedure to reduce it to the orbifold case. On the other hand, the condition that $L_\mu$ be a genuine line bundle is rather restrictive. It is the aim of the present paper to give a different proof of the Guillemin-Sternberg conjecture (for $\mu$ a regular value), without having to make this assumption. The method we use is motivated by recent work of V. Guillemin \cite{G94}, who used localization techniques from equivariant cohomology to establish the connection between the Multiplicity Formula (\ref{GS}) and a certain formula for counting lattice points in polytopes. This formula is known to be true in various interesting cases, and for these gives a new proof of (\ref{GS}) without using any complex geometry arguments. (In particular, it also works for {\em almost} K\"ahler polarizations.) We will adopt this utilization of equivariant cohomology, but in a slightly different guise. The main idea is to consider the rescaled problem, where we replace \begin{equation} \omega\leadsto m\omega,\,L\leadsto\,L^m,\,J\leadsto\, mJ, \,\mu\leadsto m\mu\end{equation} for $m\in{\Bbb N}$. Our starting point will be the Equivariant Riemann-Roch Formula, but instead of (\ref{character}) we will use it in a form due to Berline and Vergne \cite {BV85}, involving equivariant characteristic classes. By a stationary phase version of the Localization Formula of Jeffrey and Kirwan \cite{JK93}, we pass from equivariant characteristic classes to (ordinary) characteristic classes on the reduced spaces. This leads to the desired Multiplicity Formula for $N^{(m)}(m\mu)$, up to an error term $O(m^{-\infty})$. Since the multiplicities are integers, one easily finds that the error term is zero for large $m$. To investigate the general dependence of $N^{(m)}(m\mu)$ on $m$, we use a different expression for $N^{(m)}(m\mu)$ via the number of lattice points in certain polytopes. If $J(M)$ is contained in the set of regular elements of ${\frak g}^*$, in particular in the abelian case, this analysis turns out to be sufficiently good to show that the above error term is zero for all $m$. \section{Statement of the result} In order to state the result, we have to give a closer description of the reduced space and its singular strata. Suppose that $\mu\in\Lambda_+$ is a regular value of $J$. Recall first the shifting trick to express $M_\mu$ as a reduced space at the zero level set: Let ${O}=G.\mu$ be the coadjoint orbit through $\mu$, equipped with its usual Kirillov K\"ahler structure, and let ${O}^-$ denote ${O}$ with the opposite K\"ahler structure. The action of $G$ on ${O}$ is Hamiltonian, with moment map $\Psi$ the embedding into ${\frak g}^*$. Then $\tilde{M}=M\times {O}^-$ is a K\"ahler manifold, and the diagonal action of $G$ is Hamiltonian, with moment map $\tilde{J}=J-\Psi$. There are canonical identifications \begin{equation} M_\mu=J^{-1}(\mu)/G_\mu\cong J^{-1}({O})/G\cong \tilde{J}^{-1}(0)/G.\end{equation} By Kostant's version of the Borel-Weil-Bott Theorem, one also has a natural quantizing bundle $\Xi\rightarrow {O}$, and the irreducible representation $V_\mu$ corresponding to $\mu$ gets realized as the space of holomophic sections of $\Xi$. (The higher order cohomology groups $H^i(O,\O(\Xi))$ vanish.) The tensor product $\tilde{L}:=L\otimes \Xi^*$ quantizes $\tilde{M}$, and there is an isomorphism \begin{equation} H^i(\tilde{M},{\cal O}(\tilde{L}))\cong V_\mu^*\otimes H^i(M,{\cal O}(L)).\label{qst}\end{equation} Hence $N(\mu)={\rm Eul}(\tilde{L})$, which is the quantum counterpart of the shifting-trick. Using the shifting-trick, it is enough to consider the case $\mu=0$. The reduced space $M_0$ inherits a natural K\"ahler structure from $M$ (see \cite{GS82a}), and the reduced bundle $L_0=(L|J^{-1}(0))/G$ renders a quantizing orbifold-line bundle. Note however that $L_0$ need not be an honest line bundle, not even over the smooth part of $M_0$. Sections of an orbifold bundle are defined as coming from invariant sections for the local orbifold charts, so all sections of $L_0$ have to vanish at points were the fiber is not ${\Bbb C}$. Let us regard $P=J^{-1}(0)$ as an orbifold-principal bundle over $M_0=J^{-1}(0)/G$. Following \cite{K79,F92}, we introduce \begin{equation} \tilde{P}=\{(x,g)|\,x\in P,\,g.x=x\}\subset P\times G,\end{equation} and let $\Sigma=\tilde{P}/G$ be its quotient under the locally free action $h.(x,g)=(h.x,\,h\,g\,h^{-1})$. The projection of $\tilde{P}$ to the second factor descends to a locally constant mapping \begin{equation} \tau:\Sigma\rightarrow {\rm Conj}(G)\end{equation} to the set of conjugacy classes. For $g\in G$, let $(g)={\rm Ad}(G).g$ denote the corresponding conjugacy class, and $\Sigma_g$ its preimage under $\tau$. There is a natural identification $\Sigma_g=P^g/Z_g$, where $P^g\subset P$ is the fixed point manifold and $Z_g$ the centralizer of $g$ in $G$. Since the fixed point set $M^g\subset M$ is a K\"ahler submanifold, and the action of $Z_g$ on $M^g$ is Hamiltonian with the restriction of $J$ serving as a moment map, this makes clear that $\Sigma$ is a K\"ahler orbifold (with several components of different dimensions). Note that this K\"ahler structure does not depend on the choice of the representative for $(g)$. Observe also that $\Sigma_e\cong M_0$. The collection of bundles $(L|P^g)/Z_g$ defines a quantizing orbifold line bundle $L_\Sigma\rightarrow \Sigma$. As above, let $c_L(g)\in S^1$ be the locally constant action of $g$ on $L|P^g$, denote by $c_\Sigma:\Sigma\rightarrow S^1$ the function defined by the $c_L(g)'s$, and let ${Ch}^\Sigma(L_\Sigma)$ be the cohomology class defined by \begin{equation} Ch^\Sigma(L_\Sigma)=c_\Sigma\,e^{\omega_\Sigma}\end{equation} where $\omega_\Sigma$ is the K\"ahler form on $\Sigma$. Consider now the natural mapping $f:\Sigma\rightarrow M_0$, sending $G.(x,g)\rightarrow G.x$. In a local orbifold chart, the tangent space to $\Sigma$ at $G.(x,g)$ is isomorphic to $T_x(M^g\cap J^{-1}(0))/T_x(Z_g.x)$, while the tangent space to $M_0$ at $G.x$ is $T_x(J^{-1}(0))/T_x(G.x)$. From this, it is easy to see that $f$ is a Khler immersion. Let $N_\Sigma\rightarrow \Sigma$ be the normal bundle of this immersion, and denote by $g^\sharp$ the automorphism of $N_\Sigma|\Sigma_g$ induced by the action of $g$. Then the collection of differential forms \begin{equation} \det(I-(g^\sharp)^{-1}e^{-\frac{i}{2\pi}F(N_\Sigma)}),\end{equation} where $F(N_\Sigma)$ is the curvature of $N_\Sigma$, defines a characteristic class $D^\Sigma(N_\Sigma)$. Finally, for each connected component $\Sigma_i$ of $\Sigma$, let $d_i$ be the number of elements in a generic stabilizer for the $G$-action on the corresponding component $\tilde{P}_i$, and $d_\Sigma:\,\Sigma\rightarrow{\Bbb N} $ the function defined by the $d_i$'s. For general values $\mu\in\Lambda_+$, let $\Sigma_\mu$, $L_\mu$ etc. be defined by means of the shifting-trick. The main result of this paper is the following \begin{theo}[Multiplicity Formula] \label{multf} If $\mu\in\Lambda_+$ is a regular value of $J$, the multiplicities $N^{(m)}(m\mu)$ are for $m>>0$ given by the formula \begin{equation} N^{(m)}(m\mu)= \int_{\Sigma_\mu} \frac{1}{d_{\Sigma_\mu}}\, \frac{Td\,({\Sigma_\mu})\, {Ch}^{\Sigma_\mu}(L^m_{\Sigma_\mu})} {D^{\Sigma_\mu}(N_{\Sigma_\mu})}.\label{orbifold1}\end{equation} If the image of the moment map, $J(M)$, is contained in ${\frak g}^*_{reg}$ (the set of regular elements of ${\frak g}^*$), one may set $m=1$ in this formula: \begin{equation} N(\mu)= \int_{\Sigma_\mu} \frac{1}{d_{\Sigma_\mu}}\,\frac{{Td}\, ({\Sigma_\mu})\, {Ch}^{\Sigma_\mu}(L_{\Sigma_\mu})} {D^{\Sigma_\mu}(N_{\Sigma_\mu})},\label{orbifold}\end{equation} In particular, this is the case if $G$ is abelian. \end{theo} The following sections are aimed at proving this Theorem. We do not know whether the second part of the Theorem remains true without the condition $J(M)\subset{\frak g}^*_{reg}$.\\ \noindent{\bf Remarks.} \begin{enumerate} \item Comparing the right hand side of (\ref{orbifold}) to Kawasaki's Riemann-Roch Formula for orbifolds \cite{K79}, the Theorem says that $N(\mu)$ is equal to the Euler number of the orbifold-bundle $L_\mu\rightarrow M_\mu$. In particular, $N(\mu)$ is zero if the fiber of $L_\mu$ over the smooth stratum of $M_\mu$ is a nontrivial quotient of ${\Bbb C}$. \item Let $\Delta=J(M)\cap{\frak t}_+^*$, which is a convex polytope by a result of Guillemin-Sternberg and Kirwan, and $\Delta^*\subset \Delta$ the set of regular values. By the Duistermaat-Heckman Theorem \cite{DH82}, the diffeotype of the reduced space $M_\mu$ (and of course also of $\Sigma_\mu$) does not change as $\mu$ varies in a connected component of ${\rm int}(\Delta^*)$, and the cohomology class of the symplectic form $\omega_\mu$ varies linearly. In particular, the symplectic volume $\mbox{\rm Vol}(M_\mu)$ is a polynomial on these connected components. If the action of $G_\mu$ on $J^{-1}(\mu)$ is free, so that ${\Sigma_\mu}=M_\mu$, the right hand side in (\ref{orbifold}) is equal to a polynomial as well, since all that varies is the Chern character ${Ch}(L_\mu)=e^{\omega_\mu}$. In the orbifold case, the behaviour is slightly more complicated: For $\mu\in\Lambda_+$ in any given connected component of ${\rm int}(\Delta^*)$, and any connected component $\Sigma_{\mu,j}$ of $\Sigma_\mu$, \begin{equation} {Ch}^{\Sigma_\mu}(L_{\Sigma_\mu})|\,\Sigma_{\mu,j} =\rho_\mu(g_j^{-1})\,c_L(g_j)\,e^{\omega_{\Sigma_\mu}},\end{equation} where $g_j\in G_\mu$ represents $\tau(\Sigma_{\mu,j})$, and $\rho_\mu:G_\mu\rightarrow S^1$ is defined by $\rho_\mu(\exp(\xi))=e^{2\pi i\l \mu,\xi\rangle}$. Hence, the right hand side of (\ref{orbifold}) is of the form \begin{equation} N(\mu)=\sum_j \rho_\mu(g_j^{-1})\, c_L(g_j)\,p_j(\mu), \end{equation} where the $p_j$ are polynomials of degree $\frac{1}{2}\dim(\Sigma_{\mu,j})$. \item Since ${Ch}\,((L^m)_{m\mu})=e^{m\omega_\mu}$, the right hand side of (\ref{orbifold1}) is a polynomial in $m$ if the $G_\mu$-action on $J^{-1}(\mu)$ is free. In the orbifold-case, this is not true in general since \begin{equation} {Ch}^{\Sigma_\mu}(L^m_{\Sigma_\mu})|\,\Sigma_{\mu,j}= \rho_\mu(g_j^{-1})^m\,c_L(g_j)^m\,e^{m\omega_{\Sigma_\mu}}.\end{equation} \begin{defi}[Ehrhart \cite{E77}] A function $f:{\Bbb N}\rightarrow {\Bbb C}$ is called an arithmetic polynomial, if for some $k\in {\Bbb N}$, all the functions \begin{equation} q_j(m)=f(km-j),\,\,j=0,\ldots, k-1\end{equation} are polynomials. $k$ is called the period of $f$. \end{defi} Equivalently, $f$ is an arithmetic polynomial if and only if it can be written in the form \begin{equation} f(m)=\sum_{l=0}^{k-1}\exp(2\pi i\,\frac{l\,m}{k})\,p_l(m),\end{equation} where the $p_l$ are polynomials. Taking $k$ such that $g^k=e$ for all $(g)\in \tau({\Sigma_\mu})$, the right hand side of (\ref{orbifold1}) clearly has this property. \item Our proof of Theorem \ref{multf} does not really use the assumption that $M$ is K\"ahler. Everything will be derived from the Equivariant Riemann-Roch Formula (\ref{char}), which is of course valid in much more general situations. Suppose for instance that $M$ is an arbitrary compact symplectic manifold, equipped with a Hamiltonian $G$-action, and that these data are quantizable. Then one can always choose a compatible, invariant almost K\"ahler structure, and replace the virtual space $\sum (-1)^i \,H^i(M,{\cal O}(L))$ with the index space of some G-invariant Dirac operator for the Clifford module $L\otimes \wedge(T^{(0,1)}M)^*$ (see \cite{BGV92,G94}). As an immediate consequence of the Berline-Vergne Formula for the character, Theorem \ref{RiemannRoch} below, the multiplicities $N(\mu)$ defined in this way do not depend on the choice of the almost K\"ahler structure or of the quantizing line bundle $L$. \end{enumerate} \bigskip \noindent{\bf Example:} Let $M={\Bbb C} P(2)$, equipped with the Fubini-Study K\"ahler form $\omega_{FS}$. Let $G=S^1$ act by \[e^{i\phi}.[z_0:z_1:z_2]=[e^{i\phi}z_0:e^{-i\phi}z_1:z_2].\] This action is Hamiltonian, and has a moment map \[J([z_0:z_1:z_2])= \frac{|z_1|^2-|z_0|^2}{|z_0|^2+|z_1|^2+|z_2|^2}.\] The dual of the tautological line bundle serves as a quantizing line bundle $L$. We also consider the tensor powers $L^m$, which are quantizing line bundles for $(M,m\,\omega_{FS})$. By Kodaira's Theorem, $H^i(M,\O(L^m))=0$ for all $i>0,\,m\in{\Bbb N}$. If we identify the spaces $\Gamma_{hol}(M,L^m)$ with the homogeneous polynomials of degree $m$ on ${\Bbb C}^3$, the representation of $S^1$ is induced by the action $e^{i\phi}.(z_0,z_1,z_2)= (e^{i\phi}z_0,e^{-i\phi}z_1,z_2)$ on ${{\Bbb C}^3}$. The isotypical subspace of $\Gamma_{hol}(M,L^m)$ corresponding to the weight $l\in{\Bbb Z}$ is, for $l\ge 0$, spanned by \[ z_0^l\,z_2^{m-l},\,z_0^{l+1}\,z_1\,z_2^{m-l-2},\ldots, z_0^{l+r}\,z_1^r\,z_2^{m-l-2r}\] with $r=\left[\frac{m-l}{2}\right]$. For $l \le 0$, the roles of $z_0$ and $z_1$ are reversed. Thus \[ N^{(m)}(l)=\left\{\begin{array}{ll} {1+\left[\frac{m-|l|}{2}\right]}&{\mbox{ if $|l|\le m$}}\\ {0}&{\mbox{ otherwise}} \end{array} \right.,\] for all $l\in{\Bbb Z},\,m\in{\Bbb N}$. On the other hand, the image of the moment map $J^{(m)}=mJ$ is the interval $-m\,\le\, \mu\,\le\, m$, with critical values at $-m,\,0,\,m$. If $0< \, |l|\,< m $, the level set $(mJ)^{-1}(l)$ consists of two orbit type strata: On the set where $z_2\not=0$, the action is free, and on the set where $z_2=0$, the stabilizer is ${\Bbb Z}_2$. Writing $S=\{e,g\}$, the reduced space $M^e_\mu=M_\mu$ is an orbifold with a ${\Bbb Z}_2$ singularity (the ``teardrop-orbifold''), and $M^g_\mu$ is the singular point. Since $c_{L^m}(g)=(-1)^m$ and $\rho_l(g^{-1})=(-1)^l$, we expect the multiplicities to grow like $N^{(m)}(l)=p_e(m,l)+(-1)^{(m-l)}p_g(m,l)$ where $p_e$ is a first order polynomial and $p_g$ a constant. Comparison with the above formula shows that this is indeed the case, with $p_e(m,l)=3/4+(m-|l|)/2$ and $p_g(m,l)=1/4$. Note that for $m-|l|$ even, the fiber of $L^m_l$ at the singular point is ${\Bbb C}$, whereas form $m-|l|$ odd it is ${\Bbb C}/{\Bbb Z}_2$. This means that for $m-|l|$ odd, all holomorphic sections of $L^m_l$ have to vanish at the singular point. Again, this fits with the above explicit formulas. \section{Some equivariant cohomology}\label{section3} We start by reviewing Cartan's model for equivariant cohomology, following Berline and Vergne \cite{BV85}. Let $M$ be a compact manifold, $G$ a compact Lie group, and $\Phi:G\times M\rightarrow M$ a smooth action. Denote by ${\cal A}_G(M)$ the space of $G$-invariant polynomial mappings $\sigma:{\frak g}\rightarrow {\cal A}(M)$, that is, $\sigma(\xi)$ depends polynomially on $\xi$ and satisfies the equivariance property \begin{equation} \sigma(\mbox{Ad}_g(\xi))=\Phi_{g^{-1}}^*(\sigma(\xi)). \end{equation} The elements of ${\cal A}_G(M)$ are called {\em equivariant differential forms}, and the space ${\cal A}_G(M)$ is preserved under the {\em equivariant differential} \begin{equation} {\d_\G}:{\cal A}_G(M)\rightarrow{\cal A}_G(M),\,\,({\d_\G}\sigma)(\xi)= \d(\sigma(\xi))+2\pi i \big(\iota(\xi_M)\sigma (\xi)\big).\end{equation} Here, $\xi_M$ denotes the fundamental vector field, i.e. the generating vector field of the flow $(t,p)\mapsto \exp(-t\xi).p$. Equivariance together with Cartan's identity for the Lie derivative, $\L_Y=\iota_Y\circ \d +\d\circ \iota_Y$, implies $\d_{\frak g}^2=0$. The cohomology $H_G(M)$ of the complex $({\cal A}_G(M),{\d_\G})$ is called the {\em equivariant cohomology}. One can show that if the action of $G$ on $M$ is locally free, the pullback mapping ${\cal A}(M/G)\rightarrow {\cal A}(M)^G_{hor} \hookrightarrow {\cal A}_G(M)$ gives rise to an isomorphism $H(M/G)\rightarrow H_G(M)$. After choosing a principal connection on $M\rightarrow M/G$, the inverse is induced on the level of forms by the mapping \begin{equation} {\cal A}_G(M)\rightarrow {\cal A}(M)^G\rightarrow {\cal A}(M)^G_{hor}\cong {\cal A} (M/G)\end{equation} given by substituting $\frac{i}{2\pi}$ times the curvature in the ${\frak g}$-slot, followed by projection onto the horizontal part (for the proof, see \cite{DV93}). For what follows, it will be necessary to relax the polynomial dependence on $\xi$ to analytic dependence, possibly defined only on some neighborhood of $0\in{\frak g}$. We will denote the corresponding space of equivariant forms by ${\cal A}^\omega_G(M)$, and the cohomology by $H_G^\omega(M)$. Suppose now that ${\cal V}\rightarrow M$ is a $G$-equivariant Hermitian vector bundle over $M$, with fiber dimension $N$. Let ${\cal A}(M,{\cal V})$ be the bundle-valued differential forms, and ${\cal A}_G(M,{\cal V})$ their equivariant counterpart. For each $G$-invariant Hermitian connection $\nabla:{\cal A}(M,{\cal V})\rightarrow {\cal A}(M,{\cal V})$, the moment map $\mu\in{\cal A}_G(M,\mbox{End}({\cal V}))$ of Berline and Vergne is defined by \begin{equation} \mu(\xi).\sigma := \xi.\sigma - \nabla_{\xi_M}\sigma,\end{equation} where $\sigma\rightarrow \xi.\sigma$ denotes the representation of ${\frak g}$ on the space of sections. Geometrically, $\mu(\xi)$ is the vertical part (with respect to the connection) of the fundamental vector field $\xi_{\cal V}$ on ${\cal V}$. Let $F({\cal V})\in{\cal A}^2 (M,\mbox{End}({\cal V}))$ denote the curvature of $\nabla$. The {\em equivariant} curvature is then defined by \begin{equation} F_{\frak g}({\cal V},\xi)=F({\cal V})+2\pi i \mu(\xi),\end{equation} and it satisfies the Bianchi identity with respect to the equivariant covariant derivative $\nabla_{\frak g}=\nabla+2\pi i \iota(\xi_M)$. Suppose now that $A\rightarrow f(A)$ is the germ of a $U(N)$-invariant analytic function on ${\frak{u}}(N)$. Then $f(F_{\frak g})\in{\cal A}_G(M)$ is $\d_{\frak g}$-closed, and one can show that choosing a different connection changes $f(F_{\frak g})$ by a $\d_{\frak g}$-exact form. The corresponding cohomology classes are called the {\em equivariant characteristic classes} of ${\cal V}\rightarrow M$. If the action on $M$ is locally free, one can choose $\nabla$ in such a way that $\mu=0$, which shows that the mapping $H^\omega_G(M)\rightarrow H(M/G)$ sends the equivariant characteristic classes of ${\cal V}$ to the usual characteristic classes of the orbifold-bundle ${\cal V}/G$. The following characteristic classes will play a role in the sequel: \begin{itemize} \item[(a)] The equivariant Chern character, defined by \begin{equation} {Ch}_{\frak g}({\cal V},\xi)=\mbox{tr} (e^{\frac{i}{2\pi}F_{\frak g}({\cal V},\xi)}).\end{equation} In the above geometric quantization setting, ${\cal V}=L$ is a line bundle, and for the equivariant curvature one has $\frac{i}{2\pi}F_{\frak g}({\cal V},\xi)=\omega+2\pi i\l J,\xi\rangle $, thus \begin{equation} Ch_{\frak g}(L,\xi)=e^{\omega+2\pi i\l J,\xi\rangle}.\end{equation} More generally, if $g\in G$ acts trivially on the base $M$, one defines \begin{equation} {Ch}_{\frak g}^g({\cal V},\xi)=\mbox{tr}(g^{\cal V}\,\,e^{ \frac{i}{2\pi}F_{\frak g}({\cal V},\xi)})\end{equation} where $g^{\cal V}\in\Gamma(M,\mbox{End}({\cal V}))$ is the induced action of $g$. In the line bundle case, this is simply $c_L(g){Ch}_{\frak g}(L,\xi)$ where where $c_L(g)\in S^1$ is the action of $g$ on the fibers. \item[(b)] The equivariant Todd class, \begin{equation} {Td}_{\frak g}({\cal V},\xi)= \det\Big(\frac{\frac{i}{2\pi}F_{\frak g}({\cal V},\xi)}{1-e^{-\frac{i}{2\pi} F_{\frak g}({\cal V},\xi)}}\Big).\end{equation} The Todd class of a complex manifold is defined as the Todd class of its tangent bundle. \item[(c)] The equivariant Euler class \begin{equation}\chi_{\frak g}({\cal V},\xi)=\det(\frac{i}{2\pi}F_{\frak g}({\cal V},\xi)).\end{equation} \item[(d)] The class \begin{equation} D_{\frak g}^g({\cal V},\xi)=\det (I-(g^{-1})^{\cal V} e^{-\frac{i}{2\pi} F_{\frak g}({\cal V},\xi)}),\end{equation} for $g\in G$ acting trivially on $M$. \end{itemize} All of this also makes sense for symplectic vector bundles, since the choice of a compatible complex structure reduces the structure group to $U(n)$, and any two such choices are homotopic. The equivariant Euler class occurs in the Localization Formula of Atiyah-Bott and Berline-Vergne. \begin{theo}[Atiyah-Bott \cite{AB84}, Berline-Vergne \cite{BV83}] \label{AB84} Suppose $M$ is an orientable $G$-ma\-ni\-fold, and $\sigma\in {\cal A}_G^\omega(M)$ is ${\d_\G}$-closed. Assume that $\xi\in{\frak g}$ is in the domain of definition of $\sigma(\xi)$, and let $F$ be the set of zeroes of $\xi_M$. The connected components of $F$ are then smooth submanifolds of even codimension, and the normal bundle $N_F$ admits a Hermitian structure which is invariant under the flow of $\xi_M$. Choose orientations on $F$ and $M$ which are compatible with the corresponding orientation of $N_F$. Then \begin{equation} \int_M \sigma(\xi)=\int_{F} \frac{\iota_F^*\sigma(\xi)}{\chi_{\frak g}(N_F,\xi)},\label{localization} \end{equation} where $\iota_F:F\rightarrow M$ denotes the embedding. \end{theo} We will need this result only in the symplectic or complex case, where the above orientations are given in a natural way. Jeffrey and Kirwan \cite{JK93} have proved a different sort of Localization Formula for the case of {\em Hamiltonian} G-spaces. We will need a stationary phase version of their result. Let $(M,\omega)$ be a Hamiltonian $G$-space, with moment map $J:M\rightarrow{\frak g}^*$, and suppose $0$ is a regular value of $J$. Let $\sigma\in{\cal A}^\omega_G(M)$ be ${\d_\G}$-closed, and let $\Delta\in C^\infty_0({\frak g})$ be a cutoff-function, with $\sigma(\xi)$ defined for $\xi\in\mbox{supp}(\Delta)$, and $\Delta=1$ in a neighborhood of $0$. Consider the integral \begin{equation}\int_{\frak g}\int_M \Delta(\xi)\sigma(\xi)e^{m(\omega+2 \pi i\,\l J,\xi\rangle)} \,\d\xi,\end{equation} where $\d\xi$ is the measure on ${\frak g}$ corresponding to the normalized measure on $G$. Notice that the ``equivariant symplectic form'' $\omega+2 \pi i\,\l J,\xi\rangle$ is ${\d_\G}$-closed. Since $e^{m\omega}$ is simply a polynomial in $m$, the leading behaviour of this integral for $m\rightarrow\infty$ is determined by the stationary points of the phase function $e^{2\pi i m\l J,\xi\rangle}$. Stationarity in ${\frak g}$-direction gives the condition $J=0$, and stationarity in $M$-direction the condition $\d \l J,\xi\rangle=0$, or $\xi=0$ since the action on $J^{-1}(0)$ is locally free. Let $M_0= J^{-1}(0)/G$ be the reduced space, $\pi:J^{-1}(0)\rightarrow M_0$ the projection and $\iota:J^{-1}(0)\rightarrow M$ the embedding. Consider the mapping \begin{equation} \kappa:H^\omega_G(M)\rightarrow H(M_0),\label{kappa}\end{equation} given by composing pullback to $J^{-1}(0)$ with the mapping $H^\omega_G(J^{-1}(0))\rightarrow H(M_0)$. On the level of forms $\sigma\in {\cal A}^\omega_G(M)$, the form $\pi^*\kappa(\sigma)$ is by definition equal to the horizontal part of $\iota^*\sigma(\frac{i}{2\pi}F^\theta)$, where $F^\theta\in{\cal A}^2(J^{-1}(0),{\frak g})^G_{hor}$ is the curvature of some connection $\theta\in{\cal A}^1(J^{-1}(0),{\frak g})^G$. \begin{theo} \label{jf} For $m\rightarrow\infty$, \begin{equation} \int_{\frak g}\int_M \Delta(\xi)\sigma(\xi)e^{m(\omega+2\pi i\l J,\xi\rangle)}\,\d\xi =\frac{1}{d}\int_{M_0}\kappa(\sigma) \, e^{m\omega_0}\,\,\,+O(m^{-\infty}),\label{jklocf} \end{equation} where $d$ is the number of elements in the generic stabilizer for the $G$-action on $J^{-1}(0)$. \end{theo} {\bf Proof.}\hspace{0.5cm} This is a variation of Theorem 4.1 in Jeffrey-Kirwan \cite{JK93}, which deals with polynomial equivariant cohomology classes, and where one has a Gaussian convergence factor instead of the cutoff. Following \cite{JK93}, we will perform the integral in a local model for $M$ near $J^{-1}(0)$. By the Coisotropic Embedding Theorem of Gotay, a neighborhood of $J^{-1}(0)$ in $M$ is equivariantly symplectomorphic to a neighborhood of the zero section of the trivial bundle $J^{-1}(0)\times {\frak g}^*$, with symplectic form $\pi^*\omega_0+\d\l \alpha,\theta\rangle$, where $\alpha$ is the coordinate function on ${\frak g}^*$. In this model, $G$ acts by $g.(x,\alpha)=(g.x,\mbox{Ad}_{g^{-1}}^*(\alpha))$ , and the moment map is simply $J(x,\alpha)=\alpha$. Using the model and another cutoff-function $\Delta'(\alpha)$ on ${\frak g}^*$, equal to $1$ near the origin and with sufficiently small support, the same computation as in \cite{JK93} shows that the integral is equal to \[ m^{\dim G}\int_{J^{-1}(0)} \int_{{\frak g}^*}\int_{{\frak g}} \Delta(\xi)\Delta'(\alpha) (\iota^*\sigma)(\xi)e^{m(\pi^*\omega_0+\l \alpha,F^\theta+2\pi i \xi\rangle)}\, \d\xi\, \d\alpha\,\d g +O(m^{-\infty}).\] Here, $\d g$ denotes the (vertical) volume form on the fibers of $J^{-1}(0)\rightarrow M_0$, corresponding to the canonical identification $T_x(\mbox{fiber})\cong{\frak g}$ by means of the $G$-action. Now apply the Stationary Phase Theorem (see e.g. \cite{H90}, Theorem 7.7.5) to the $\alpha,\xi$-integral, the relevant phase function being $e^{2\pi i m \l \alpha,\xi\rangle}$. Since $e^{m\l \alpha,F^\theta\rangle}$ is simply a polynomial in $\alpha$, the stationary phase expansion terminates after finitely many terms, and the result is \[ \int_{J^{-1}(0)} e^{m \pi^*\omega_0} \sum_{r=0}^\infty \frac{1}{r!}\big(\frac{i}{2 \pi m}\big)^r\left. \Big(\sum_j\frac{\partial}{\partial \xi_j} \frac{\partial}{\partial \alpha^j}\Big)^r\right|_{\atop\stackrel{\xi=0}{\alpha=0}} \iota^*\sigma(\xi)\, e^{m\l \alpha,F^\theta\rangle } \,\,\d g +O(m^{-\infty})\] \[ =\int_{J^{-1}(0)} \pi^*\,(e^{m \omega_0})\,\iota^*\sigma(\frac{i}{2\pi} F^\theta)\,\d g +O(m^{-\infty}).\] Since $\iota^*\sigma(\frac{i}{2\pi}F^\theta)$ gets wedged with $\d g$, only its horizontal part, which by definition of $\kappa$ is $\pi^*\kappa(\sigma)$, contributes to the integral. The result (\ref{jklocf}) now follows by integration over the fiber; the factor ${1}/{d}$ appears since this is the volume of a generic fiber. \hspace{0.5cm}\bigskip\mbox{$\Box$} We now turn to the Equivariant Hirzebruch-Riemann-Roch Theorem, in the form due to Berline and Vergne \cite{BV85}. Let $M$ be a compact complex manifold, equipped with a holomorphic action of $G$, and let $L\rightarrow M$ be a G-equivariant holomorphic line bundle. Define the character $\chi\in R(G)$ as in (\ref{char}). \begin{theo}\label{RiemannRoch} For $\xi$ sufficiently close to zero, \begin{equation} \chi(e^{\xi})= \int_M {Td}_{\frak g}(M,\xi)\,{Ch}_{\frak g}(L,\xi).\label{e}\end{equation} More generally, if $g\in G$, one has for all sufficiently small $\xi\in {\frak k}$, the Lie algebra of the centralizer $Z_g$ of $g$: \begin{equation} \chi(g\, e^{\xi})= \int_{M^g} \frac{{Td}_{\frak k}(M^g,\xi)\,{Ch}_{\frak k}^g (L|\,M^g,\xi)}{ D^g_{\frak k}(N^g,\xi)}, \label{g}\end{equation} where $M^g$ is the fixed point set and $N^g\rightarrow M^g$ its normal bundle. \end{theo} To be precise, Berline and Vergne have shown how to rewrite the Equivariant Atiyah-Segal-Singer Index Theorem for Dirac operators in this style, with the equivariant $\hat{A}$-genus appearing on the right hand side. This formula, however, implies the above Theorem in the same way as the usual Atiyah-Singer Index Theorem leads to the Hirzebruch-Riemann-Roch Formula; see \cite{BGV92}, p. 152 for the calculations. \section{The stationary phase approximation} In this section, we will prove the first part of Theorem \ref{multf}. By the shifting trick, it is sufficient to consider the case $\mu=0$. The idea is to substitute the expressions from the Equivariant Hirzebruch-Riemann-Roch Theorem \ref{RiemannRoch} for $\chi^{(m)}$ in \begin{equation} N^{(m)}(0)=\int_G \chi^{(m)}(h)\d h,\label{4.5}\end{equation} and apply the Localization Formula, Theorem \ref{jf}. For this, we need to know what happens to the equivariant Todd class of $M$ under the mapping (\ref{kappa}): \begin{lemma}\label{todd} Let \begin{equation} j_{\frak g}(\xi)=\det\Big({\textstyle\frac{1-e^{-{ad}(\xi)} } {{ad}(\xi)}}\Big)\end{equation} be the Jacobian of the exponential mapping $\exp:{\frak g}\rightarrow G$. Then \begin{equation} \kappa({Td}_{\frak g}(M) j_{\frak g})={Td}\,(M_0).\end{equation} \end{lemma} {\bf Proof.}\hspace{0.5cm} Identify the vertical subbundle of $TJ^{-1}(0)$ with the trivial bundle ${\frak g}$, and the symplectic bundle ${\frak g}\oplus I{\frak g}$ (where $I$ is the complex structure of $M$) with ${\frak g}_{\Bbb C}$. Then \[ \iota^*(TM)=\pi^*(TM_0)\oplus {\frak g}_{\Bbb C}. \] Since the equivariant Todd class of ${\frak g}_{\Bbb C}$ is just $j_{\frak g}^{-1}$, this shows $\kappa({Td}_{\frak g}(M))=Td\,(M_0)\,\kappa(j_{\frak g}^{-1})$, q.e.d. \hspace{0.5cm}\bigskip\mbox{$\Box$} We will now consider the contribution to (\ref{4.5}) coming from a small ${\rm Ad}$-invariant neighborhood of a given orbit $(g)={\rm Ad}(G).g$. Let $\sigma\in C^\infty(G)$ be an $\mbox{Ad}$-invariant cutoff-function, supported in a sufficiently small neighborhood of $(g)$ and equal to 1 near $(g)$. Consider the integral \begin{equation} I_g(m)=\int_G \sigma(h)\chi^{(m)}(h)\d h.\end{equation} Since (\ref{g}) only holds for $\xi\in{\frak k}$, we want to replace this integral by an integral over $Z_g$. (Of course, this step is void in the abelian case.) Let ${\frak r}\subset{\frak g}$ be the orthogonal of ${\frak k}$ with respect to some invariant inner product (or, more intrinsically, the annihilator of $({\frak g}^*)^g\cong{\frak k}^*$). For $k\in Z_g$, let $k^{\frak{r}}$ denote the action of $k$ on ${\frak r}$. \begin{lemma} \label{lemmaGK} Let $f\in C^\infty(G)$ be $\mbox{Ad}(G)$-invariant, with support in a small neighborhood of $\mbox{Ad}(G).g$. Then, for a suitable $\mbox{Ad}(Z_g)$-invariant cutoff-function $\tilde{\sigma}\in C^\infty_0(Z_g)$, supported near $e\in Z_g$ and identically $1$ near $e$, \begin{equation} \int_G f(h)\d h=\int_{Z_g} f(gk) \det(I-(g^{-1}k^{-1})^{\frak{r}})\tilde{\sigma}(k)\,\d k. \label{GK}\end{equation} \end{lemma} {\bf Proof.}\hspace{0.5cm} From the Slice Theorem for actions of compact Lie groups, it follows that an invariant neighborhood of the orbit $\mbox{Ad}(G).g$ is equivariantly diffeomorphic to a neighborhood of the zero section of $G\times_{Z_g}{\frak k}$, where $Z_g$ acts on ${\frak k}$ by the adjoint action. Using the exponential map for $Z_g$, it follows that the mapping \begin{equation} \phi:\,G\times_{Z_g}Z_g\rightarrow G,\,\,(h,k)\mapsto \mbox{Ad}_h(gk), \end{equation} with $Z_g$ acting on itself by $\mbox{Ad}$, is an equivariant diffeomorphism from a neighborhood of the unit section to a neighborhood of the orbit. Let $\d\nu$ be the canonical measure on the group bundle $G\times_{Z_g}Z_g=:W$, constructed from the normalized invariant measures on $Z_g$ and $G/Z_g$. We have to compute the tangent mapping to $\phi$, but by equivariance it is sufficient to do this at points $(e,k)$ in the fiber over $e\,Z_g$, which is canonically isomorphic to $Z_g$. If we identify $T_{(e,k)}W\cong \frak{r}\oplus\frak{k}$, and $TG\cong G\times{\frak g}$ using left trivialization, the tangent mapping is given by \[ T_{(e,k)}\phi(\xi,\eta)=((1-\mbox{Ad}_{(gk)^{-1}})(\xi), \eta).\] This shows that the measure transforms according to \begin{equation} \phi^*\,\d g = \det(I-(g^{-1}k^{-1})^{\frak r})\,\d \nu.\end{equation} We can now perform the integral by first pulling $f$ back to $G\times_{Z_g}Z_g$, multiplying with a suitable cuttoff function $\tilde{\sigma}$, integrating over the fibers of $W\rightarrow G/Z_g$, and then integrate over the base $\mbox{Ad}(G).g\cong G/Z_g$. \hspace{0.5cm}\bigskip\mbox{$\Box$} Using the Lemma, we find that \begin{equation} I_g(m)= \int_{Z_g} \tilde{\sigma}(k)\chi^{(m)}(gk) \det(I-(g^{-1}\,k^{-1})^{\frak r})\, \d k.\end{equation} Replacing this with an integral over the Lie algebra ${\frak k}$, and using (\ref{g}) gives \begin{equation} I_g(m)=\int_{\frak k}\int_{M^g} \Delta(\xi) \frac{{Td}_{\frak k}(M^g,\xi){Ch}_{\frak k}^g(L^m|M^g,\xi)}{ D^g_{\frak k}(N^g,\xi)} \det(I-(g^{-1}e^{-\xi})^{\frak r}) j_{\frak k}(\xi)\d \xi,\label{LK}\end{equation} with $\Delta(\xi)=\tilde{\sigma}(e^{\xi})$. Let $\kappa_g:\,H_{Z_g}(M^g)\rightarrow H(\Sigma_g)$ be the mapping defined by (\ref{kappa}), with $M$ replaced by $M^g$ and $G$ by $Z_g$. By Lemma \ref{todd}, \begin{equation} \kappa_g(Ch^g_{\frak k}(L^m|M^g))\,\kappa_g(j_{\frak k})= Ch^{\Sigma}(L^m_\Sigma)|\Sigma_g.\end{equation} For $x\in P^g=M^g\cap J^{-1}(0)$, let ${\frak r}_M(x):=\{\xi_M(x)|\xi\in {\frak r}\}\cong {\frak r}$. Then \begin{equation} N^g(x)=N_\Sigma(y)\oplus{\frak r}_M(x)\oplus I{\frak r}_M(x) =N_\Sigma(y)\oplus {\frak r}_{\Bbb C},\end{equation} where $y=G.(x,g)$. But $D^g_{\frak k}({\frak r}_{\Bbb C},\xi)=\det(I-(g^{-1}e^{-\xi}))$, hence \begin{equation} \kappa_g(D_{\frak k}^g(N^g,\xi))= \kappa_g(\det(I-(g^{-1}e^{-\xi})^{\frak r}))\,\, D^\Sigma(N_\Sigma)|\Sigma_g.\end{equation} With these preperations, we apply Theorem \ref{jf} to the integral \ref{LK}, and obtain \begin{equation} I_g(m)={\sum}'_{j} \frac{1}{d_j} \int_{\Sigma_j} \frac{Td\,(\Sigma_j) {Ch}^\Sigma(L^m_\Sigma)} {D^\Sigma(N_\Sigma)}+O(m^{-\infty}),\end{equation} the sum being over the connected components of $\Sigma_g$. Summing over all contributions, we get (\ref{orbifold1}) up to an error term $O(m^{-\infty})$. As we remarked above, the right hand side of (\ref{orbifold1}) is an arithmetic polynomial as a function of $m$. But if $f:{\Bbb N}\rightarrow {\Bbb Z}$ is an integer-valued function with $\lim_{m\rightarrow\infty}(f(m)-p(m))=0$ for some polynomial $p$, then $f(m)=p(m)$ for large $m$. This shows that the error term is zero for large $m$, and finishes the proof of the first part of Theorem \ref{multf}. \section{Counting lattice points} To prove the second part of Theorem \ref{multf}, i.e. that we can set $m=1$, all we have to show is that $m\rightarrow N^{(m)}(m\mu)$ is an arithmetic polynomial. \begin{theo} \label{stepw} Suppose that $J(M)\subset {\frak g}^*_{reg}$. Then the function $m\mapsto N^{(m)}(m\mu)$ is an arithmetic polynomial for all $\mu\in\Lambda_+$. \end{theo} Before we prove this, we convert the computation of the multiplicities into a problem of counting lattice points. The next steps are based on work of Guillemin-Lerman-Sternberg \cite{GLS88} and Guillemin-Prato \cite{GP90}, except that we replace their use of the Atiyah-Bott Lefschetz Formula with the Localization Formula \ref{AB84} applied to (\ref{e}), since we do not want to assume isolated fixed points. Consider the action of the maximal torus $T\subset G$, with its moment map $J^T$ equal to $J$ followed by projection to ${\frak t}^*$. \begin{prop} \label{prop5.1} For all generic $\xi\in{\frak t}$, \begin{equation} \chi(e^{\xi})=\sum_{{\cal F}} \int_{\cal F} \frac{{Td} ({\cal F})e^{\omega + 2\pi i \l J_{\cal F},\xi\rangle}} {\det(I-e^{-\frac{i}{2\pi}F_{\frak t}(N_{\cal F},\xi)})},\label{fixedpoints}\end{equation} the sum being over the fixed points manifolds of the $T\subset G$-action, $N_{\cal F}$ the corresponding normal bundles, and $J_{\cal F}$ the constant value of $J$ on ${\cal F}$. \end{prop} {\bf Proof.}\hspace{0.5cm} By ``generic'' we mean that the zero set of $\xi_M$ is equal to the fixed point set of the $T$-action. To get ({\ref{fixedpoints}), apply the Localization Formula, Theorem \ref{AB84}, to (\ref{e}). The bundle $TM|_{\cal F}$ splits into the direct sum $T{\cal F}\oplus N_{\cal F}$. Since $T$ does not act on $T{\cal F}$, \[{Td}_{\frak t}(M,\xi)={Td}\,({\cal F})\,{Td}_{\frak t}(N_{\cal F},\xi).\] The equivariant Euler class of $N_{\cal F}$ in Proposition \ref{AB84} cancels the denominator of the equivariant Todd class, which immediately gives (\ref{fixedpoints}). \hspace{0.5cm}\bigskip\mbox{$\Box$} Although the left hand side of (\ref{fixedpoints}) is an analytic function of $\xi$, the individual summands on the right hand side have poles. Since they are not in $L^1_{loc}$, they do not a priori define distributions on ${\frak t}$. This problem can be fixed as follows \cite{D93}. By using the splitting principle (or simply a partition of unity on ${\cal F}$) if necessary, we can assume that $N_{\cal F}$ splits into a direct sum of invariant line bundles $N_{\cal F}^1,\ldots,N_{\cal F}^r$. Let $\alpha^j_{\cal F}$ be the weight for the $T$-action on $N_{\cal F}^j$, that is, $e^\xi\in T$ acts by the character $\exp(2\pi i \l \alpha_{\cal F}^j,\xi\rangle)$. Each $\alpha^j_{\cal F}$ determines an orthogonal hyperplane in ${\frak t}$, let $C$ be any fixed connected component in the complement of the union of all these hyperplanes. If we replace $\xi$ by $\xi-i\eta$ in ({\ref{fixedpoints}), with $\eta\in C$, the terms on the right hand side are analytic for all $\xi$. One can therefore regard (\ref{fixedpoints}) as an equality of distributions, with the summands on the right hand side defined defined as a distributional limit for $\eta\rightarrow 0$ in $C$. Let us now first discuss the abelian case, i.e. assume that $G=T$ is a torus. Denote by $F^j(N_{\cal F})$ the components of the curvature. By expanding $\det(I-e^{-\frac{i}{2\pi}{F}_{\frak t}(N_{\cal F},\xi-i\eta)})^{-1}$ into its Taylor series w.r.t. ${F}^j(N_{\cal F})$, we can write it as a finite sum \begin{equation} \det(I-e^{-\frac{i}{2\pi}{F}_{\frak t}(N,\xi-i\eta)})^{-1}= \sum_{s\in{\Bbb N}^r} \frac{p_s({F}^1(N_{\cal F}),\ldots, {F}^r(N_{\cal F}))} {\prod_j (1-e^{-2\pi i\l {\alpha}^j_{\cal F},\xi-i\eta\rangle})^{s_j}}, \label{5} \end{equation} where for all $s=(s_1,\ldots,s_r)$, $p_s$ is a polynomial. We now invoke the ``polarization trick'' used in ref. \cite{GLS88,GP90}. For each $\alpha^j_{\cal F}$, write \begin{equation} \check{\alpha}^j_{\cal F}=\left\{\begin{array}{ll} {\alpha}^j_{\cal F}&\mbox{ if } \l {\alpha}^j_{\cal F},\eta\rangle>0\\ -{\alpha}^j_{\cal F}&\mbox{ if } \l {\alpha}^j_{\cal F},\eta\rangle<0\end{array}\right. \label{signs}\end{equation} for any, hence all, $\eta\in C$. Let $l_j^0=0$ if $\check{\alpha}^j_{\cal F}=\alpha^j_{\cal F}$, $1$ otherwise. Then \begin{equation} \chi(e^{\xi-i\eta})=\sum_{\cal F}\sum_{s\in {\Bbb N}^r}c_{{\cal F},s} \frac{e^{2\pi i\l J_{\cal F}-\sum l_j^0 s_j\check{\alpha}^j_{\cal F},\xi-i\eta\rangle}} {\prod_j (1-e^{-2\pi i\l \check{\alpha}^j_{\cal F},\xi-i\eta\rangle})^{s_j}} \label{lhs}\end{equation} with \begin{equation} c_{{\cal F},s}= (-1)^{k_{{\cal F},s}} \int_{\cal F} {Td}\,({\cal F})\,e^\omega p_s({F}^1 (N_{\cal F}),\ldots, {F}^r(N_{\cal F})),\end{equation} where $k_{{\cal F},s}=\sum l_j^0 s_j$ is the number of sign changes. For given ${\cal F},s$, write $(a^1,\ldots,a^N)$ for the list of $\check{\alpha}^j_{\cal F}$'s, appearing with respective multiplicities $s_j$. Since \begin{equation} (1-e^{-2\pi i\l a^j,\xi-i\eta\rangle})^{-1}=\sum_{l_j=0}^\infty e^{- 2\pi i \l l_j\,a^j,\xi-i\eta \rangle},\end{equation} we get \begin{equation} \chi(e^\xi)=\sum_{{\cal F},s} c_{{\cal F},s}\sum_{l\in{\Bbb Z}^N_+} e^{2\pi i\l J_{\cal F}-\sum (l_j+l_j^0)a^j,\xi\rangle}\end{equation} (the sum over ${\Bbb Z}^N_+:=\{l\in {\Bbb Z}^N:\,l_j\ge 0\}$ is a well-defined periodic distribution). Comparing this to \begin{equation} \chi(e^\xi)=\sum_{\mu\in\Lambda} N(\mu)e^{2\pi i\l\mu,\xi\rangle}\label{abelian}\end{equation} yields \begin{equation} N(\mu) = \sum_{{\cal F},s}c_{{\cal F},s}{\frak P}_{{\cal F},s}(J_{\cal F}-\mu- \sum l_j^0 a^j)\label{latticepoints}\end{equation} where the partition function ${\frak P}_{{\cal F},s}(\nu)$ is the number of solutions $k\in {\Bbb Z}^N$ of $\sum k_j a^j=\nu$, $k_j\ge 0$. Starting from this expression, we will now show that $N^{(m)}(m\mu)$ is an arithmetic polynomial. We have to replace $\omega$ by $m\omega$, $\mu$ by $m\mu$ and $J$ by $mJ$. Since $c_{{\cal F},s}^{(m)}$ is a polynomial in $m$, it is sufficient to show that the number of integer solutions of \begin{equation} m(J_{\cal F}-\mu)=\sum (l_j+l_j^0) a^j,\,\,l_j\ge 0 \label{10}\end{equation} is an arithmetic polynomial as a function of $m$. Let us write $\nu=J_{\cal F}-\mu$, and consider $A=(a^1,\ldots,a^N)$ as a ${\Bbb Z}$-linear mapping ${\Bbb Z}^N\rightarrow {\Bbb Z}^p$, where $p=\dim(T)$. We are thus looking for integer solutions of \begin{equation} m\nu= A\,l,\,\,l_j\ge l_j^0.\label{11}\end{equation} We will need the following \begin{theo}[Ehrhart \cite{E77}] Let $L$ be a lattice, with underlying vector space $L_{\Bbb R}=L\otimes_{\Bbb Z} {\Bbb R}$, and $\Delta\subset L_{\Bbb R}$ a lattice polytope, i.e. a polytope whose vertices are all lattice points. Then, for all $r\in {\Bbb N}$, the counting function \begin{equation} f(m)=\#\big(\frac{m}{r}\Delta\cap L\big) \end{equation} is an arithmetic polynomial, with period $r$. \end{theo} Let now $x_0\in{\Bbb R}^N$ be any solution of $Ax=\nu$. The general solution of $A\,x=m\nu$ is thus given by the affine plane $E_m=mx_0+\mbox{ker}(A)$. Let $r\in {\Bbb N}$ be the smallest number such that the vertices of the polytope $\Delta:=E_r\cap {\Bbb R}^N_+$ are lattice points. If $l^0=0$, the set of solutions of (\ref{11}) is the intersection $\frac{m}{r}\Delta\cap {\Bbb Z}^N$, so the number of solutions is an arithmetic polynomial by Ehrhart's Theorem. If $l^0\not=0$, let $\Delta_j$ be the face of $\Delta$ defined by $x_j=0$, and let $\Delta'$ be the union of all $\Delta_j$ for which $l_j^0=1$. Then the solution set of (\ref{11}) is \[ \frac{m}{r}\Delta\cap {\Bbb Z}^N\,-\,\frac{m}{r}\Delta'\cap {\Bbb Z}^N,\] and this is again an arithmetic polynomial by Ehrhart's Theorem. This proves Theorem \ref{stepw} in the abelian case. Suppose now that $G$ is nonabelian, but that $J(M)$ is contained in the set of regular elements, ${\frak g}^*_{reg}=G.\mbox{int}({\frak t}^*_+)$. We will show how this reduces to the abelian case. By the Symplectic Slice Theorem \cite{GS84}, $Y_+=J^{-1}(\mbox{int} ({\frak t}^*_+))$ is a symplectic (but not necessarily K\"ahler) submanifold of $M$, and is in fact a Hamiltonian $T$-space, with the restriction of $J$ serving as a moment map. The above assumption implies that $Y_+$ is a {\em closed} submanifold, and $M=G\times_T Y_+$. The restriction $L_+=L|Y_+$ renders a quantizing bundle for $Y_+$. Consider the expression \begin{equation} \chi'(e^\xi):= \int_{Y^+} {Td}_{\frak t}(Y_+,\xi)\, {Ch}_{\frak t}(L_+,\xi).\end{equation} We claim that this is of the form \begin{equation} \chi'(e^\xi)=\sum_{\mu\in\Lambda} N'(\mu) \,e^{2\pi i \l \mu,\xi\rangle},\end{equation} where $N'(\mu)\not=0$ for only finitely many lattice points, and $N'(\mu)=0$ unless $\mu\in J(Y_+)\cap\Lambda\subset\Lambda_+$. Indeed, one can check directly that $\chi(e^\xi)$ comes from a function on $T$, given near any point $g\in T$ by the formula (\ref{g}), and then repeat the above analysis. (One can also pick a $T$-invariant almost K\"ahler structure on $M$, and then realize $\chi(e^\xi)$ as the equivariant index for some Dirac operator associated to the Clifford module $L\otimes \Lambda(T^{(0,1)}Y_+)^*$.) \begin{lemma} For all $\mu\in\Lambda_+$, $N(\mu)=N'(\mu)$. \end{lemma} Since we know that ${N'}^{(m)}(m\mu)$ is an arithmetic polynomial, this will finish the proof of Theorem \ref{multf}.\\ {\bf Proof.}\hspace{0.5cm} Let us go back to the formula (\ref{fixedpoints}) for the character. Notice that the Weyl group $W=N_G(T)/T$ acts on $M^T$ by permuting the connected components, and that $M^T$ consists of its portion in $Y_+$ and the $W$-transforms thereof. Let ${\cal F}\subset Y_+$ be a connected component of $M^T$. The normal bundle $N_{\cal F}$ of ${\cal F}$ in $M$ splits into into its part in $Y_+$, $N_{\cal F}':=N_{\cal F}\cap TY_+$, and the symplectic orthogonal of $TY_+|{\cal F}$, which is canonically isomorphic to the trivial bundle ${\frak g}/{\frak t}$. The weights for the $T$-action on ${\frak g}/{\frak t}$ are of course simply the positive roots $\beta\in {\frak t}^*$ of $G$. Therefore, by taking the trivial connection on ${\frak g}/{\frak t}$, \[ \det(I- e^{-\frac{i}{2\pi}F_{\frak t}(N_{\cal F},\xi)})=\prod_{\beta>0} (1-e^{-2\pi i \l \beta,\xi\rangle})\det(I-e^{-\frac{i}{2\pi} F_{\frak t}(N'_{\cal F},\xi)}), \] hence \[ \chi(e^\xi)=\sum_{w\in W}\frac{1}{\prod_{\beta>0} (1-e^{-2\pi i \l \beta,w^{-1}(\xi)\rangle})}\sum_{{\cal F}\subset Y_+} \int_{\cal F} \frac{{Td}\,({\cal F})e^{\omega + 2\pi i \l J_{\cal F},w^{-1}(\xi)\rangle}} {\det(I-e^{-\frac{i}{2\pi}F_{\frak t}(N'_{\cal F},w^{-1}(\xi))})}.\] To the sum \[ \sum_{{\cal F}\subset Y_+}\int_{\cal F} \frac{{Td}\,({\cal F})e^{\omega + 2\pi i \l J_{\cal F},\xi\rangle}} {\det(I-e^{-\frac{i}{2\pi}F_{\frak t}(N'_{\cal F},\xi)})},\] we can apply the Localization Formula, this time in the opposite direction, and get that it is equal to the above expression $\chi'(e^\xi)$. This gives \[\chi(e^\xi)=\sum_{w\in W} \frac{\chi'(e^{w^{-1}(\xi)})}{ \prod_{\beta>0} (1-e^{-2\pi i \l \beta,w^{-1}(\xi)\rangle})}.\] But \[ \prod_{\beta>0} (1-e^{-2\pi i \l \beta,w^{-1}(\xi)\rangle})=\det(w)\, e^{-2\pi i\l w(\delta)-\delta,\xi\rangle}\,\prod_{\beta>0} (1-e^{-2\pi i \l \beta,\xi\rangle}),\] where $\delta=\frac{1}{2}\sum_{\beta>0}\beta$ is the magic weight. Weyl's Character Formula hence shows that \[ \chi(e^\xi)=\sum_{\mu\in\Lambda_+} N'(\mu)\sum_{w\in W} \det(w)\,\frac{e^{2\pi i\l w(\delta+\mu)-\delta,\xi\rangle}} {\prod_{\beta>0} (1-e^{-2\pi i \l \beta,\xi\rangle})}=\sum_{\mu\in\Lambda_+} N'(\mu) \,\chi_\mu(e^\xi),\] where $\chi_\mu$ is the character of the irreducible representation corresponding to $\mu$. This proves $N(\mu)=N'(\mu)$. \bigskip\hspace{0.5cm}\bigskip\mbox{$\Box$} \noindent{\bf Remarks.} \begin{enumerate} \item If $J(M)\not\subset {\frak g}^*_{reg}$, it is still possible to derive a formula for $N(\mu)$ similar to (\ref{latticepoints}), following part II of Guillemin-Prato \cite{GP90}. However, this formula involves an additional ``shift'', so that (\ref{11}) gets replaced by an equation of the form \[ Al=m\nu+\sigma,\,l_j\ge l_j^0.\] In general, the number of integer solutions of such an equation is not an arithmetic polynomial for all $m\in{\Bbb N}$, even though this is true for large $m$. \item On the other hand, Theorem \ref{stepw} does not require that $\mu$ is a regular value of $J$. Even in the singular case, it is therefore sufficient to prove Multiplicity Formulas under the assumption $m>>0$. \end{enumerate} \noindent{\bf Acknowledgements.} I would like to thank V. Guillemin, J. Kalkman, E. Lerman, R. Sjamaar and C. Woodward for useful comments and discussions. I am very much indebted to Victor Guillemin, whose recent results \cite{G94} on Multiplicity Formulas of Riemann-Roch type have been the basic motivation for this work. The stationary phase version (\ref{jklocf}) of the Jeffrey-Kirwan Localization Theorem was worked out jointly with Jaap Kalkman, who has also been a great help in explaining equivariant cohomology to me. This work was carried out when I was visiting scholar at the M.I.T., and I wish to thank the Mathematics Department for its hospitality. \bigskip \noindent{\bf Postscript:} After completing this article, we learned about independent work of M. Vergne \cite{V94}, who has made a different application of equivariant cohomology to the multiplicity problem.

9405

alg-geom/9405004

en

https://arxiv.org/abs/alg-geom/9405004

alg-geom/9405004

Michael Thaddeus

Michael Thaddeus

Geometric invariant theory and flips

33 pages, LaTeX with AMS fonts

We study the dependence of geometric invariant theory quotients on the choice of a linearization. We show that, in good cases, two such quotients are related by a flip in the sense of Mori, and explain the relationship with the minimal model programme. Moreover, we express the flip as the blow-up and blow-down of specific ideal sheaves, leading, under certain hypotheses, to a quite explicit description of the flip. We apply these ideas to various familiar moduli problems, recovering results of Kirwan, Boden-Hu, Bertram-Daskalopoulos- Wentworth, and the author. Along the way we display a chamber structure, following Duistermaat-Heckman, on the space of all linearizations. We also give a new, easy proof of the Bialynicki-Birula decomposition theorem.

alg-geom

\section{#1} \setcounter{equation}{0}} \renewcommand{\theequation}{\thesection .\arabic{equation}} \newcommand{\re}[1]{\mbox{\rm \bf (\ref{#1})}} \catcode`\@=\active \catcode`\@=11 \def\@eqnnum{\hbox to .01pt{}\rlap{\bf \hskip -\displaywidth(\theequation)}} \catcode`\@=12 \newenvironment{s}[1] { \addvspace{12pt} \def\smallskipamount{6pt} \refstepcounter{equation} \noindent {\bf (\theequation) #1.} \begin{em}} {\end{em} \par \addvspace{12pt} \def\smallskipamount{6pt} } \newenvironment{r}[1] { \addvspace{12pt} \def\smallskipamount{6pt} \refstepcounter{equation} \noindent {\bf (\theequation) #1.} } {\par \addvspace{12pt} \def\smallskipamount{6pt} } \begin{document} \catcode`\@=\active \catcode`\@=11 \newcommand{\newcommand}{\newcommand} \newcommand{\vars}[2] {{\mathchoice{\mbox{#1}}{\mbox{#1}}{\mbox{#2}}{\mbox{#2}}}} \newcommand{\Aff}{\vars{\bf} \def\blb{\bf} \def\frak{} \def\mb{ A}{\blb A}} \newcommand{\C}{\vars{\bf} \def\blb{\bf} \def\frak{} \def\mb{ C}{\blb C}} \newcommand{\Hyp}{\vars{\bf} \def\blb{\bf} \def\frak{} \def\mb{ H}{\blb H}} \newcommand{\N}{\vars{\bf} \def\blb{\bf} \def\frak{} \def\mb{ N}{\blb N}} \newcommand{\Pj}{\vars{\bf} \def\blb{\bf} \def\frak{} \def\mb{ P}{\blb P}} \newcommand{\Q}{\vars{\bf} \def\blb{\bf} \def\frak{} \def\mb{ Q}{\blb Q}} \newcommand{\R}{\vars{\bf} \def\blb{\bf} \def\frak{} \def\mb{ R}{\blb R}} \newcommand{\Z}{\vars{\bf} \def\blb{\bf} \def\frak{} \def\mb{ Z}{\blb Z}} \newcommand{\oper}[1]{\mathop{\mathchoice{\mbox{\rm #1}}{\mbox{\rm #1}} {\mbox{\scriptsize \rm #1}}{\mbox{\tiny \rm #1}}}\nolimits} \newcommand{\Aut}{\oper{Aut}} \newcommand{\chr}{\oper{char}} \newcommand{\diag}{\oper{diag}} \newcommand{\End}{\oper{End}} \newcommand{\Fl}{\oper{Fl}} \newcommand{\Gr}{\oper{Gr}} \newcommand{\Hom}{\oper{Hom}} \newcommand{\NS}{\oper{NS}} \newcommand{\Par}{\oper{Par}} \newcommand{\Pic}{\oper{Pic}} \newcommand{\Proj}{\oper{Proj}} \newcommand{\Quot}{\oper{Quot}} \newcommand{\Spec}{\oper{Spec}} \newcommand{\tr}{\oper{tr}} \newcommand{\GL}[1]{{\rm GL(#1)}} \newcommand{\PSL}[1]{{\rm PSL(#1)}} \newcommand{\SL}[1]{{\rm SL(#1)}} \newcommand{\ep}{\epsilon} \newcommand{\ga}{\gamma} \newcommand{\la}{\lambda} \newcommand{\La}{\Lambda} \newcommand{\si}{\sigma} \newcommand{\A}{A} \newcommand{\aff}{{k}} \newcommand{\blowup}{\tilde{\Aff}^{\raisebox{-.5ex}{$\scriptstyle 2$}}} \newcommand{\down}{\Big\downarrow} \newcommand{\be}{{\bf E}} \newcommand{\beqas}{\begin{eqnarray*}} \newcommand{\beqa}{\begin{eqnarray}} \newcommand{\beq}{\begin{equation}} \newcommand{\bl}{\vskip 1.2ex } \newcommand{\ci}{{\Im}} \newcommand{\co}{{\cal O}} \newcommand{\E}{E} \newcommand{\eeqas}{\end{eqnarray*}} \newcommand{\eeqa}{\end{eqnarray}} \newcommand{\eeq}{\end{equation}} \newcommand{\emb}{\hookrightarrow} \newcommand{\fp}{\mbox{ $\Box$}} \newcommand{\half}{\frac{\scriptstyle 1}{\scriptstyle 2}} \newcommand{\k}{{k}} \newcommand{\kst}{{k^{\times}}} \newcommand{\lrow}{\longrightarrow} \newcommand{\m}{{\frak m}} \newcommand{\mod}{/ \! \! /} \newcommand{\pf}{{\em Proof}} \newcommand{\sans}{\backslash} \newcommand{\st}{\, | \,} \catcode`\@=12 \noindent {\LARGE \bf Geometric invariant theory and flips} \medskip \\ {\bf Michael Thaddeus }\\ Mathematical Institute, 24--29 St Giles, Oxford OX1 3LB, England \medskip \smallskip \noindent Ever since the invention of geometric invariant theory, it has been understood that the quotient it constructs is not entirely canonical, but depends on a choice: the choice of a {\em linearization} of the group action. However, the founders of the subject never made a systematic study of this dependence. In light of its fundamental and elementary nature, this is a rather surprising gap, and this paper will attempt to fill it. In a way, this neglect is understandable, because the different quotients must be related by birational transformations, whose structure in higher dimensions is poorly understood. However, it has been considerably clarified in the last dozen years with the advent of Mori theory. In particular, the birational transformations that Mori called {\em flips} are ubiquitous in geometric invariant theory; indeed, one of our main results \re{1f} describes the mild conditions under which the transformation between two quotients is given by a flip. This paper will not use any of the deep results of Mori theory, but the notion of a flip is central to it. The definition of a flip does not describe the birational transformation explicitly, but in the general case there is not much more to say. So to obtain more concrete results, hypotheses must be imposed which, though fairly strong, still include many interesting examples. The heart of the paper, \S\S4 and 5, is devoted to describing the birational transformations between quotients as explicitly as possible under these hypotheses. It turns out that there are fairly explicit smooth loci in two different quotients whose blow-ups are isomorphic. Thus the two quotients are related by a blow-up followed by a blow-down. This is somewhat at odds with the point of view of Mori theory, which views a flip as two contractions, not two blow-ups; but it facilitates explicit calculations of such things as topological cohomology or Hilbert polynomials. The last three sections of the paper put this theory into practice, using it to study moduli spaces of points on the line, parabolic bundles on curves, and Bradlow pairs. An important theme is that the structure of each individual quotient is illuminated by understanding the structure of the whole family. So even if there is one especially natural linearization, the problem is still interesting. Indeed, even if the linearization is unique, useful results can be produced by enlarging the variety on which the group acts, so as to create more linearizations. I believe that this problem is essentially elementary in nature, and I have striven to solve it using a minimum of technical machinery. For example, stability and semistability are distinguished as little as possible. Moreover, transcendental methods, choosing a maximal torus, and invoking the numerical criterion are completely avoided. The only technical tool relied on heavily is the marvelous Luna slice theorem \cite{luna}. Luckily, this is not too difficult itself, and there is a good exposition in GIT, appendix 1D. This theorem is used, for example, to give a new, easy proof of the Bialynicki-Birula decomposition theorem \re{2e}. Section 1 treats the simplest case: that of an affine variety $X$ acted on by the multiplicative group $\kst$, and linearized on the trivial bundle. This case has already been treated by Brion and Procesi \cite{bp}, but the approach here is somewhat different, utilizing $\Z$-graded rings. The four main results are models for what comes later. The first result, \re{2i}, asserts that the two quotients $X \mod \pm$ coming from nontrivial linearizations are typically related by a flip over the affine quotient $X \mod 0$. The second, \re{2d}, describes how to blow up ideal sheaves on $X \mod \pm$ to obtain varieties which are both isomorphic to the same component of the fibred product ${X \mod -} \times_{X \mod 0} {X \mod +}$. In other words, it shows how to get from $X \mod +$ to $X \mod -$ by performing a blow-up followed by a blow-down. The third result, \re{2j}, asserts that when $X$ is smooth, the blow-up loci are supported on subvarieties isomorphic to weighted projective fibrations over the fixed-point set. Finally, the fourth, \re{2k}, identifies these fibrations, in what gauge theorists would call the quasi-free case, as the projectivizations of weight subbundles of the normal bundle to the fixed-point set. Moreover, the blow-ups are just the familiar blow-ups of smooth varieties along smooth subvarieties. Sections 3, 4, and 5 are concerned with generalizing these results in three ways. First, the variety $X$ may be any quasi-projective variety, projective over an affine. Second, the group acting may be any reductive algebraic group. Third, the linearization may be arbitrary. But \S2 assumes $X$ is normal and projective, and is something of a digression. It starts off by introducing a notion of $G$-algebraic equivalence, and shows, following Mumford, that linearizations equivalent in this way give the same quotients. Hence quotients are really parametrized by the space of equivalence classes, the $G$-N\'eron-Severi group $\NS^G$. Just as in the Duistermaat-Heckman theory in symplectic geometry, it turns out \re{1b} that $\NS^G \otimes \Q$ is divided into chambers, on which the quotient is constant. The analogues of the four main results of \S1 then apply to quotients in adjacent chambers, though they are stated in a somewhat more general setting. The first two results are readily generalized to \re{1f} and \re{1g}. The second two, however, require the hypotheses mentioned above; indeed, there are two analogues of each. The first, \re{2n} and \re{1x}, make fairly strong hypotheses, and show that the weighted projective fibrations are locally trivial. The second, \re{1l} and \re{1n}, relax the hypotheses somewhat, but conclude only that the fibrations are locally trivial in the \'etale topology. Counterexamples \re{1r} and \re{2p} show that the hypotheses are necessary. The strategy for proving all four of these results is not to imitate the proofs in the simple case, but rather to reduce to this case by means of a trick. In fact, given a variety $X$ acted on by a group $G$, and a family of linearizations parametrized by $t$, we construct \re{1h} a new variety $Z$, dubbed the ``master space'' by Bertram, acted on by a torus $T$, and a family of linearizations on $\co(1)$ parametrized by $t$, such that $X \mod G(t) = Z \mod T(t)$ naturally. This reduces everything to the simple case. The final sections, \S\S6, 7, and 8, are in a more discursive style; they explain how to apply the theory of the preceding sections to some examples. In all cases, the strongest hypotheses are satisfied, so the best result \re{1x} holds. Perhaps the simplest interesting moduli problems are those of (ordered or unordered) sets of $n$ points in $\Pj^1$; these are studied in \S6. The ideas here should have many applications, but only a very simple one is given: the formula of Kirwan \cite{k} for the Betti numbers of the moduli spaces when $n$ is odd. In \S7 the theory is applied to parabolic bundles on a curve, and the results of Boden and Hu \cite{bhu} are recovered and extended. Finally, in \S8, the theory is applied to Bradlow pairs on a curve, recovering the results of the author \cite{t1} and Bertram et al.\ \cite{bdw}. While carrying out this research, I was aware of the parallel work of Dolgachev and Hu, and I received their preprint \cite{dh} while this paper was being written. Their main result is contained in the third of the four main results I describe, \re{1l}; and of course, some of the preliminary material, corresponding to my \S2, overlaps. I am indebted to them for the observation quoted just after \re{1y}, and for the result \re{1s}, though my proof of the latter is original. Dolgachev and Hu do not, however, include the results on flips or blow-ups, study the local triviality of the exceptional loci, or identify the projective bundles in the quasi-free case. For them, this is not necessary, since they appear \cite{hu} to be interested chiefly in computing Betti numbers and intersection Betti numbers of quotients, and for this, their main result suffices, together with the deep results of Beilinson, Bernstein, and Deligne \cite{bbd}. I am more interested in computing algebraic cohomology, as in \cite{t1}; for this, a precise characterization of the birational map between quotients is necessary, which led me to the present paper. In any case, even where our results coincide, our methods of proof are quite different. A few words on notation and terminology. Many of the statements involve the symbol $\pm$. This should always be read as two distinct statements: that is, $X^\pm$ means $X^+$ (resp.\ $X^-$), never $X^+ \cup X^-$ or $X^+ \cap X^-$. Similarly, $X^\mp$ means $X^-$ (resp.\ $X^+$). The quotient of $X$ by $G$ is denoted $X \mod G$, or $X \mod G(L)$ to emphasize the choice of a linearization $L$. When there is no possibility of confusion, we indulge in such abuses of notation as $X \mod \pm$, which are explained in the text. For stable and semistable sets, the more modern definitions of \cite{n} are followed, not those of \cite{mf}, which incidentally is often referred to as GIT. Points are assumed to be closed unless otherwise stated. The stabilizer in $G$ of a point $x \in X$ is denoted $G_x$. When $E$ and $F$ are varieties with morphisms to $G$, then $E \times_G F$ denotes the fibred product; but if $G$ is a group acting on $E$ and $F$, then $E \times_G F$ denotes the twisted quotient $(E \times F) / G$. Unfortunately, both notations are completely standard. All varieties are over a fixed algebraically closed field $\k$. This may have any characteristic: although we use the Luna slice theorem, which is usually said \cite{luna,mf} to apply only to characteristic zero, in fact this hypothesis is used only to show that the stabilizer must be linearly reductive. Since all the stabilizers we encounter will be reduced subgroups of the multiplicative group, this will be true in any characteristic. By the way, most of the results in \S\S1 and 3 apply not only to varieties, but to schemes of finite type over $\k$. But for simplicity everything is stated for varieties. Finally, since the experts do not entirely agree on the definition of a flip, here is what we shall use. Let $X_- \to X_0$ be a {\em small contraction} of varieties over $\k$. This means a small birational proper morphism; {\em small} means that the exceptional set has codimension greater than $1$. (This appears to be the prevailing terminology in Mori theory \cite[(2.1.6)]{ko}, but it is called {\em semismall\/} in intersection homology, where {\em small} has a stronger meaning.) Let $D$ be a $\Q$-Cartier divisor class on $X_-$ which is relatively negative over $X_0$; that is, $\co(-D)$ is relatively ample. Then the {\em $D$-flip} is a variety $X_+$, with a small contraction $X_+ \to X_0$, such that, if $g: X_- \dasharrow X_+$ is the induced birational map, then the divisor class $g_*D$ is $\Q$-Cartier, and $\co(D)$ is relatively ample over $X_0$. We emphasize the shift between ampleness of $\co(-D)$ and that of $\co(D)$. If a flip exists it is easily seen to be unique. Note that several authors, including Mori \cite{m}, require that each contraction reduce the Picard number by exactly $1$. We will not require this; indeed, it is not generally true of our flips \cite[4.7]{toric}. For convenience, $D$-flips will be referred to simply as {\em flips}. However, in the literature, the unmodified word {\em flip} has traditionally denoted a $K$-flip where $K$ is the canonical divisor of $X_-$; this is not what we will mean. \bit{The simplest case} We begin by examining the simplest case, that of an affine variety acted on by the multiplicative group of $\k$. This has been studied before in several papers, that of Brion and Procesi \cite{bp} being closest to our treatment; but we will clarify, extend, and slightly correct the existing results. Let $R$ be a finitely-generated integral algebra over the algebraically closed field $\k$, so that $X = \Spec R$ is an affine variety over $\k$. In this section only, $G$ will denote the multiplicative group of $\k$. An action of $G$ on $\Spec R$ is equivalent to a $\Z$-grading of $R$ over $\k$, say $R = \bigoplus_{i \in \Z} R_i$. We will study geometric invariant theory quotients $X \mod G$, linearized on the trivial bundle. So choose any $n \in \Z$, and define a $\Z$-grading on $R[z]$ by $R_i \subset R[z]_i$, $z \in R[z]_{-n}$. Of course $R[z]$ is also $\N$-graded by the degree in $z$, but this $\Z$-grading is different. Since $X = \Spec R = \Proj R[z]$, the $\Z$-grading is equivalent to a linearization on $\co$ of the $G$-action on $X$. The quotient is $\Proj R[z]^{G} = \Proj R[z]_0 = \Proj \bigoplus_{i \in \N} R_{ni} z^i$. For $n = 0$, this is just $\Proj R_0[z] = \Spec R_0$, the usual affine quotient \cite[3.5; GIT Thm.\ 1.1]{n}. For $n > 0$, $\Proj \bigoplus_{i \in \N} R_{ni} = \Proj \bigoplus_{i \in \N} R_{i}$ by \cite[II Ex.\ 5.13]{h} (the hypothesis there is not needed for the first statement); similarly for $n < 0$, $\Proj \bigoplus_{i \in \N} R_{ni} = \Proj \bigoplus_{i \in \N} R_{-i}$. Hence we need concern ourselves only with the quotients when $n = 0$, $1$, and $-1$; we shall refer to them in this section as $X \mod 0$, $X \mod +$ and $X \mod -$ respectively. Note that $X \mod \pm$ are projective over $X \mod 0$. \begin{s}{Proposition} \label{2h} If $X \mod + \neq \emptyset} \def\dasharrow{\to \neq X \mod -$, then the natural morphisms $X \mod \pm \to X \mod 0$ are birational. \end{s} Of course, if say $X \mod - = \emptyset} \def\dasharrow{\to$, then $X \mod +$ can be any $\k$-variety projective over $X \mod 0$. \pf. For some $d > 0$ $R_{-d}$ contains a nonzero element $t$. The function field of $X \mod 0$ is $\{ r/s \st r,s \in R_0 \}$, while that of $X \mod +$ is $\{ r/s \st r,s \in R_{di} \mbox{ \rm for some } i \geq 0 \}$. But the map $r/s \mapsto (rt^i)/(st^i)$ from the latter to the former is an isomorphism. \fp \bl Let $X^\pm \subset X$ be the subvarieties corresponding to the ideals $\langle R_i \st \mp i > 0 \rangle$ (note the change of sign), and let $X^0 = X^+ \cap X^-$; then $X^0$ corresponds to the ideal $\langle R_i \st i \neq 0 \rangle$. Also say $\lim g \cdot x = y$ if the morphism $G \to X$ given by $g \mapsto g \cdot x$ extends to a morphism $\Aff^1 \to X$ such that $0 \mapsto y$. \begin{s}{Lemma} \label{2l} As sets, $X^\pm = \{ x \in X \st \exists \lim g^{\pm 1} \cdot x \}$, and $X^0$ is the fixed-point set for the $G$-action. \end{s} \pf. A point $x$ is in $X^+$ if and only if for all $n < 0$, $R_n$ is killed by the homomorphism $R \to \k[x, x^{-1}]$ of graded rings induced by $g \mapsto g \cdot x$. This in turn holds if the image of $R$ is contained in $\k[x]$, that is, if $G \to X$ extends to $\Aff^1 \to X$. The proof for $X^-$ is similar. Hence $x \in X^0$ if and only if $\lim g^{\pm 1} \cdot x$ both exist, that is, the closure of $G \cdot x$ is a projective variety in $X$. Since $X$ is affine, this means $x$ is a fixed point of $G$. \fp \begin{s}{Proposition} \label{2a} \mbox{\rm (a)} $X^{ss}(0) = X$; \mbox{\rm (b)} $X^s(0) = X \sans (X^+ \cup X^-)$ ; \mbox{\rm (c)} $X^{ss}(\pm) = X^s(\pm) = X \sans X^\mp$. \end{s} \pf. Recall that $x \in X^{ss}(L)$ if for some $n > 0$ there exists $s \in H^0(L^n)^G$ such that $s(x) \neq 0$, and $x \in X^{s}(L)$ if the morphism $G \to X^{ss}(L)$ given by $g \mapsto g \cdot x$ is proper. For $L = 0$, $H^0(L^n)^{G} = R_0$ for all $n$, but this contains 1, which is nowhere vanishing. That is all there is to (a). The valuative criterion implies that the morphism $G \to X$ is proper if and only if the limits do not exist, which together with \re{2l} implies (b). For $L = \pm$, $H^0(L^n)^{G} = R_{\pm n}$, so $X^{ss}(\pm) = X \sans X^\mp$ follows immediately from the definition of $X^\pm$. The additional condition of properness needed for $x \in X^s(\pm)$ is equivalent, by the valuative criterion, to $\lim g \cdot x$ and $\lim g^{-1} \cdot x \notin X^{ss}(\pm)$. But one does not exist, and the other, if it exists, is fixed by $G$, so is certainly not in $X^{ss}(\pm)$. \fp \begin{s}{Corollary} \label{2m} The morphisms $X \mod \pm \to X \mod 0$ are isomorphisms on the complements of $X^{\pm} \mod \pm \to X^\pm \mod 0$. \end{s} In good cases, $X^{\pm} \mod \pm$ will be exactly the exceptional loci of the morphisms, but they can be smaller, even empty---for instance $X \mod - \to X \mod 0$ in \re{2g} below. \pf. By \re{2a}, the sets $(X \sans X^\pm) \mod \pm$ and $(X \sans X^\pm) \mod 0$ contain no quotients of strictly semistable points. They are therefore isomorphic. \fp \begin{s}{Proposition} There are canonical dominant morphisms $\pi_\pm: X^\pm \to X^0$ such that for all $x \in X^\pm$, $\pi_\pm (x) = \lim g^{\pm 1} \cdot x$. \end{s} \pf. Note first that $R_0 \cap \langle R_i \st \pm i > 0 \rangle = R_0 \cap \langle R_i \st i \neq 0 \rangle$. So $R / \langle R_i \st i \neq 0 \rangle = R_0 / \langle R_i \st i \neq 0 \rangle$ are naturally included in $R / \langle R_i \st \pm i > 0 \rangle$ as the $G$-invariant parts. Hence there are natural dominant morphisms $\pi_\pm: X^\pm \to X^0$. Because $\pi_\pm$ is induced by the inclusion of the degree 0 part in $R / \langle R_i \st \mp i > 0 \rangle$, $\pi_\pm (x)$ is the unique fixed point such that $f(\pi_\pm (x)) = f(x)$ for all $f \in R_0$. But $f \in R_0$ means it is $G$-invariant, hence constant on orbits, so the same property is satisfied by $\lim g^{\pm 1} \cdot x$, which is a fixed point in the closure of $G \cdot x$. \fp \bl The next two results digress to show what the results so far have to do with flips. \begin{s}{Proposition} \label{2i} If $X^{\pm} \subset X$ have codimension $\geq 2$, then the birational map $f: X \mod - \dasharrow X \mod +$ is a flip with respect to $\co(1)$. \end{s} \pf. The hypothesis implies that the open sets $(X \sans X^\pm) \mod \pm$ in $X \mod \pm$ have complements of codimension $\geq 2$. But by \re{2m} these open sets are identified by $f$. Hence there is a well-defined push-forward $f_*$ of divisors. For some $n > 0$ the twisting sheaves $\co(\pm n) \to X \mod \pm$ are line bundles. Indeed, they are the descents \cite{dn} from $X$ to $X \mod \pm$ of the trivial bundle $\co$, with linearization given by $n$ as at the beginning of this section. Consequently, they agree on the open sets $(X \sans X^\pm) \mod \pm$, so $f_* \co(-n) = \co(n)$. But $\co(\pm n) \to X \mod \pm$ are relatively ample over $X \mod 0$, so $f$ is a flip. \fp \begin{s}{Proposition} \label{2o} Let $Y_0$ be normal and affine over $\k$, and let $f: Y_- \dasharrow Y_+$ be a flip of normal varieties over $Y_0$. Then there exists $X$ affine over $\k$ and a $G$-action on $X$ so that $Y_0 = X \mod 0$, $Y_\pm = X \mod \pm$. \end{s} \pf. Let $L = \co(D)$, where $D$ is as in the definition of a flip. Since $Y_\pm$ are normal, and the exceptional loci of $f$ have codimension $\geq 2$, $f$ induces an isomorphism $H^0(Y_-, L^n) \cong H^0(Y_+, f_*L^n)$ for all $n$. The $\N$-graded algebras $\bigoplus_{n < 0} H^0(Y_-, L^n)$ and $\bigoplus_{n > 0} H^0(Y_+, f_*L^n)$ are the homogeneous coordinate rings of the quasi-projective varieties $Y_-$ and $Y_+$ respectively, so are finitely-generated over $\k$. Hence the same is true of the $\Z$-graded algebra $R = \bigoplus_{n \in \Z} H^0(Y_-, L^n)$. Let $X = \Spec R$ with the $G$-action coming from the grading. Then $X \mod 0 = Y_0$ and $X \mod \pm = Y_\pm$. I thank Miles Reid for pointing out this simple proof. \fp \bl In order to describe the birational map $X \mod - \dasharrow X \mod +$ more explicitly, we will next construct a variety birational to $X \mod \pm$ which dominates them both. It is admittedly true in general that any birational map can be factored into a blow-up and blow-down of some sheaves of ideals. The virtue of the present situation, however, is that these sheaves can be identified fairly explicitly, and that the common blow-up is precisely the fibred product. Choose $d > 0$ such that $\bigoplus_{i \in \Z} R_{di}$ is generated by $R_0$ and $R_{\pm d}$. Then let $\ci^\pm$ be the sheaves of ideals on $X$ corresponding to $\langle R_{\mp d} \rangle$. Let $\ci^\pm \mod \pm$ on $X \mod \pm$ and $\ci^\pm \mod 0$ on $X \mod 0$ be the ideal sheaves of invariants of $\ci^\pm$, that is, the sheaves of ideals locally generated by the invariant elements of $\ci^\pm$. Note that $\ci^\pm$ are supported on $X^\pm$, so that $\ci^\pm \mod \pm$ are supported on $X^\pm \mod \pm$. For $i, j \geq 0$, let $R_{i,j} = R_i \cdot R_{-j} \subset R_{i-j}$. \begin{s}{Lemma} \label{2b} The ideal sheaf $(\ci^+ + \ci^-) \mod 0$ is exactly $\langle R_{d,d} \rangle$, and its pullbacks by the morphisms $X \mod \pm \to X \mod 0$ are $\ci^\pm \mod \pm$. \end{s} \pf. The ideals in $R$ corresponding to $\ci^\pm$ are by definition $\langle R_{\mp d} \rangle$, and $\langle R_{\mp d} \rangle \cap R_0 = R_{d,d}$, so $(\ci^+ + \ci^-) \mod 0 = \langle R_{d,d} \rangle$. Regard $X \mod \pm$ as quotients with respect to the linearizations $\pm d$. Then $X \mod \pm = \bigoplus_{i \geq 0} R_{\pm di}$, so for any $\si \in R_{\pm d}$, $\Spec (\si^{-1} \bigoplus_{i \geq 0} R_{\pm di})_0$ is an affine in $X \mod \pm$. But $\ci^\pm \cap (\si^{-1} \bigoplus_{i \geq 0} R_{\pm di})_0 = \si^{-1} \langle R_{d,d} \rangle$, so locally $\ci^\pm \mod \pm$ is the pullback of $\langle R_{d,d} \rangle$. As $\si$ ranges over $R_{\pm d}$, these affines cover $X \mod \pm$, so the result holds globally. \fp \begin{s}{Theorem} \label{2d} Suppose $X \mod + \neq \emptyset} \def\dasharrow{\to \neq X \mod -$. Then the blow-ups of $X \mod \pm$ at $\ci_\pm \mod \pm$, and the blow-up of $X \mod 0$ at $(\ci^+ + \ci^-) \mod 0$, are all naturally isomorphic to the irreducible component of the fibred product ${X \mod -} \times_{X \mod 0} {X \mod +}$ dominating $X \mod 0$. \end{s} \pf. The blow-up of $X \mod +$ at $\ci_+ \mod +$ is $\Proj \bigoplus_n H^0((\ci_+ \mod +)^n(dn))$ for $d$ sufficiently divisible. But by \re{2b} $H^0((\ci_+ \mod +)^n(dn)) = R_{(d+1)n, n}$, so the blow-up is $\Proj \bigoplus_n R_{(d+1)n, n}$. There is a surjection of $R_0$-modules $R_{(d+1)n} \otimes_{R_0} R_{-n} \to R_{(d+1)n, n}$, so the blow-up embeds in $\Proj \bigoplus_n R_{(d+1)n} \otimes_{R_0} R_{-n}$. This is precisely the fibred product ${X \mod -} \times_{X \mod 0} {X \mod +}$, with polarization $\co(d+1,1)$. By \re{2m} this naturally contains $(X \sans X^\pm) \mod \pm$ as a nonempty open set, but so does the blow-up. The blow-up is certainly irreducible, so it is the component containing $(X \sans X^\pm) \mod \pm$. The proof for $X \mod -$ is similar. Likewise, the blow-up of $X \mod 0$ at $(\ci^+ + \ci^-) \mod 0$ is $\Proj \bigoplus_n (\ci^+ + \ci^- \mod 0)^n = \Proj \bigoplus_n R_{dn,dn} = \Proj \bigoplus_n R_{n,n}$. This embeds in the fibred product with polarization $\co(1,1)$, but by \re{2m} contains $(X \sans X^\pm) \mod \pm$ as a nonempty open set. \fp \bl The ideal sheaves $\ci^\pm$ are supported on $X^\pm$, so $\ci^\pm \mod \pm$ are supported on $X^\pm \mod \pm$. But they are not just $\ci_{X^\pm \mod \pm}$, as the following counterexample shows. \begin{s}{Counterexample} \label{2g} To show that the ideal sheaves of $X^\pm \mod \pm$ cannot generally replace $\ci_\pm \mod \pm$ in \re{2d}. \end{s} In other words, the blow-up may be weighted, not just the usual blow-up of a smooth subvariety. Let $G$ act on $X = \Aff^3$ by $\la(w,x,y) = (\la^{-1} w, \la x, \la^2 y)$; in other words, $w \in R_{-1}$, $x \in R_1$, and $y \in R_2$. Then $X \mod 0 = \Spec \k[wx,w^2y] = \Aff^2$, and $X \mod - = \Proj \k[wx,w^2y, zw] = \Aff^2$, where the $\N$-grading of every variable is 0 except $z$, which is graded by 1. However, \beqas X \mod + & = & \Proj \k[wx,w^2y,zx,z^2y] \\ & = & \Proj \k[wx,w^2y,z^2(w^2x^2),z^2(w^2y)] \\ & = & \Proj \k[u,v,zu^2,zv], \eeqas which is $\Aff^2$ blown up at the ideal $\langle u^2,v \rangle$. This has a rational double point, so is not the usual blow-up at a point. \fp The paper \cite{bp} of Brion and Procesi asserts (in section 2.3) a result very similar to \re{2d}. They state that the two quotients are related by blow-ups---``\'eclatements''---of certain subvarieties. The counterexample above shows that the blow-ups must sometimes be weighted, that is, must have non-reduced centres. Brion and Procesi do not state this explicitly, but they are no doubt aware of it. Another minor contradiction to their result is furnished by the following counterexample. \begin{s}{Counterexample} To show that the fibred product of \re{2d} can be reducible, and the blow-up one of its irreducible components. \end{s} Let $G$ act on the singular variety $X = \Spec \k[a^2,ab,b^2,c,d]/ \langle ad-bc \rangle$ where $a,b,c,d$ are of degree 1, acted on with weights $1,-1,1,-1$ respectively. Then \beqas X \mod 0 & = & \Spec \k[ab,cd,a^2d^2,b^2c^2]/\langle ad-bc \rangle \\ & = & \Spec \k[ab,cd] \\ & = & \Aff^2. \eeqas But, using the same $\N$-grading convention as in the previous example, \beqas X \mod - & = & \Proj \k[ab,cd,z^2b^2,zd] \\ & = & \Proj \k[u,v,zu,zv] \\ & = & \blowup, \eeqas that is, the blow-up of $\Aff^2$ at the origin, and by symmetry $X \mod + \cong \blowup$ as well. Taking $d =2$ gives ideal sheaves $\ci_+ = \langle a^2,c^2 \rangle$ and $\ci_- = \langle b^2,d^2 \rangle$; the sheaf of invariants of both is$\langle u^2,v^2 \rangle$, the ideal sheaf of twice the exceptional divisor. Hence blowing up $\ci_\pm \mod \pm$ does nothing. The fibred product, however, is $\blowup \times_{\Aff^2} \blowup$, which is not just $\blowup$: it has two components, isomorphic to $\blowup$ and $\Pj^1 \times \Pj^1$ respectively and meeting in a $\Pj^1$. \fp \bl Following Bialynicki-Birula \cite{bb}, define the {\em trivial $w_i$-fibration} over an affine variety $Y$ to be $\Aff^r \times Y$, with a $G$-action induced by the action on $\Aff^r$ with weights $w_i$. A {\em $w_i$-fibration} over $Y$ is a variety over $Y$, with a $G$-action over the trivial action on $Y$, which is locally the trivial $w_i$-fibration. As Bialynicki-Birula points out, a $w_i$-fibration need not be a vector bundle, because the transition functions need not be linear. But if all the $w_i$ are equal, then it is a vector bundle. Suppose now that $G$ acts on an affine variety $X$ which is {\em smooth}. Then it will be proved in (a) below that $X^0$ is also smooth. Purely for simplicity, suppose that it is also connected. (If not, the following theorem is still valid, but the fibrations involved need have only locally constant rank and weights.) The group $G$ acts on the normal bundle $N_{X^0 / X}$. Let $N^\pm_X$, or simply $N^\pm$, be the subbundles of positive and negative weight spaces for this action, with weights $w^\pm_i \in \Z$. \begin{s}{Theorem} \label{2e} Suppose $X$ is smooth. Then \mbox{\rm (a)} $X^0$ is smooth; \mbox{\rm (b)} as varieties with $G$-action, $\pi_\pm: X^\pm \to X^0$ are naturally $w^\pm_i$-fibrations; \mbox{\rm (c)} $N_{X^0 / X}$ has no zero weights, so equals $N^+ \oplus N^-$; \mbox{\rm (d)} the normal bundles $N_{X^0 / X^\pm} = N^\pm$; \mbox{\rm (e)} if all $w^\pm_i = \pm w$ for some $w$, then $\ci^\pm$ and $\ci^+ +\ci^-$ cut out the $d/w$th infinitesimal neighbourhoods of $X^\pm$ and $X^0$ respectively. \end{s} Parts (b), (c), (d) are the {\em Bialynicki-Birula decomposition theorem} \cite[Thm.\ 4.1]{bb}. Another version of this result, possibly more familiar, gives a Morse-style decomposition of a projective variety into a disjoint union of $w_i$-fibrations. It follows easily from this \cite{bb}. \pf. First consider the case of a finite-dimensional vector space $V$, acted on linearly by $G$. Then $V = \Spec S$ for $S$ a $\Z$-graded polynomial algebra. This decomposes naturally into three polynomial algebras, $S = S^- \otimes S^0 \otimes S^+$, corresponding to the subspaces of negative, zero, and positive weight. Then $\ci^\pm = \langle S^\mp_{\mp d} \rangle$, $V^\pm = \Spec S^\pm \otimes S^0$, and $V^0 = \Spec S^0$. Parts (a)--(e) all follow easily. Indeed, the fibrations are naturally trivial. To return to the general case, note first that if $U \subset X$ is a $G$-invariant open affine, then $U = \Spec F^{-1}R$ for some $F \subset R_0$. Hence $(F^{-1}R)_i = F^{-1}(R_i)$ for each $i$, so $\ci^\pm_U = \ci^\pm_X |_U$, $U^\pm = X^\pm \cap U$, $U^0 = X^0 \cap U$, and $\pi_\pm$ is compatible with restriction. Consequently, the whole theorem is local in the sense that it suffices to prove it for a collection of $G$-invariant open affines in $X$ containing $X^+ \cup X^-$. Now for any closed point $x \in X^0$, apply the Luna slice theorem \cite{luna,mf} to $X$. Since $G_x = G$, the Luna slice is a $G$-invariant open affine $U \subset X$ containing $x$, and $G \times_{G_x} N_x = T_x X$. Hence there is a strongly \' etale $G$-morphism (see \cite{luna,mf}) $\phi: U \to V = T_x X$ such that $\phi(x) = 0$. In particular, $U = U \mod 0 \times_{V \mod 0} V$. Any $y \in X^+ \cup X^-$ is contained in some such $U$: indeed, just take $x = \pi_\pm (y)$. It therefore suffices to prove the theorem for $U$, so by abuse of notation, say $U = \Spec R$ from now on. The $G$-morphism $\phi$ then corresponds to a graded homomorphism $S \to R$, where $S$ is a $\Z$-graded polynomial ring, such that $R = R_0 \otimes_{S_0} S$. In particular, $R_i = R_0 \otimes_{S_0} S_i$ for each $i$. Hence $R_{\pm d}$ and $S_{\pm d}$ generate the same ideals in $R$, so $\ci^\pm_U = \phi^{-1}\ci^\pm_V$. Also, $U^\pm = \phi^{-1} V^\pm = V^\pm \times_{V \mod 0} U \mod 0$, and $U^0 = \phi^{-1}V^0 = V^0 \times_{V \mod 0} U \mod 0$. This immediately implies (a). Since $V^\pm \to V^0$ are trivial fibrations, it also gives the local trivialization of $X^\pm \to X^0$ near $x$ needed to prove (b). Parts (c) and (d) also follow, since $\phi$, as an \' etale $G$-morphism, satisfies $\phi^* N_{V^0/V} = N_{U^0/U}$ as bundles with $G$-action, so in particular $\phi^* N^\pm_V = N^\pm_U$. The hypotheses of part (e) imply that $d$ is a multiple of $w$; the conclusion holds if and only if the map $R_{\mp w}^{d/w} \to R_{\mp d}$ is surjective. This is true for $S$, and follows in general from $R_i = R_0 \otimes_{S_0} S_i$. \fp The above methods can be used to describe the local structure of the non-reduced schemes cut out by $\ci^\pm$ even when not all $w^\pm_i = \pm w$, but we will not pursue this. \begin{s}{Corollary} \label{2j} Suppose $X$ is smooth. Then $X^\pm \mod \pm$ are locally trivial fibrations over $X^0$ with fibre the weighted projective space $\Pj(|w_i^\pm|)$. \fp \end{s} If $X \mod + \neq \emptyset} \def\dasharrow{\to \neq X \mod -$, these are the supports of the blow-up loci of \re{2d}. On the other hand, if $X \mod - = \emptyset} \def\dasharrow{\to$, then $X^+ = X$ and $X^0 = X \mod 0$, so this says the natural morphism $X \mod + \to X \mod 0$ is a weighted projective fibration. \begin{r}{Remark} \label{2f} It follows from the above corollary that, in this smooth case, ${X^- \mod -} \times_{X \mod 0} {X^+ \mod +}$ is irreducible of codimension 1 in ${X \mod -} \times_{X \mod 0} {X \mod +}$. It must therefore be exactly the exceptional divisor of each of the two blow-ups of \re{2d}. In other words, the latter fibred product is irreducible, and is isomorphic to each of the blow-ups. This implies that, when $X$ is smooth, the surjection of $R_0$-modules $R_{i} \otimes_{R_0} R_{-j} \to R_{i,j}$ is an isomorphism for $i,j > 0$ sufficiently divisible. However, I do not know of a direct algebraic proof of this fact. \end{r} \begin{s}{Theorem} \label{2k} Suppose $X$ is smooth, and that all $w_i^\pm = \pm w$ for some $w$. Then $X^\pm \mod \pm$ are naturally isomorphic to the projective bundles $\Pj(N^\pm)$ over the fixed-point set $X^0$, their normal bundles are naturally isomorphic to $\pi_\pm^* N^\mp(-1)$, and the blow-ups of $X \mod \pm$ at $X^\pm \mod \pm$, and of $X \mod 0$ at $X^0 \mod 0$, are all naturally isomorphic to the fibred product ${X \mod -} \times_{X \mod 0} {X \mod +}$. \end{s} Note that if each 0-dimensional stabilizer on $X$ is trivial, then all $w^\pm_i = \pm 1$. \pf. All the blow-ups and the fibred product are empty if either $X \mod +$ or $X \mod -$ is empty, so suppose they are not. By the observation of Bialynicki-Birula quoted above, if all $w^\pm_i = \pm w$, then the fibrations $X^\pm \to X^0$ are actually vector bundles. But any vector bundle is naturally isomorphic to the normal bundle of its zero section, so by \re{2e}(d) $X^\pm \mod \pm = \Pj(X^\pm) = \Pj(N^\pm)$, and the natural $\co(1)$ bundles correspond. By \re{2e}(e), $\ci^\pm$ cut out the $d/w$th infinitesimal neighbourhoods of $X^\pm$. This means that $R_{\mp w}^{d/w} \to R_{\mp d}$ are surjective and hence that $\ci^\pm \mod \pm$ and $(\ci^+ + \ci^-) \mod 0$ cut out the $d/w$th infinitesimal neighbourhoods of $X^\pm \mod \pm$ and $X^0 \mod 0$ respectively. Since blowing up a subvariety has the same result as blowing up any of its infinitesimal neighbourhoods, the result follows from \re{2d} and \re{2f}, except for the statement about normal bundles. To prove this, recall first that if $E$ is the exceptional divisor of the blow-up $\tilde{Y}$ of any affine variety $Y$ at $Z$, then $N_{E/\tilde{Y}} = \co_E(-1)$, and $N^*_{Z/Y} = (R^0\pi) N^*_{E/\tilde{Y}}$. Applying this to the case $Y = X \mod 0$ shows that the normal bundle to ${X^- \mod -} \times_{X^0} {X^+ \mod +}$ is the restriction of $\co(-1,-1) \to {X \mod -} \times_{X \mod 0} {X \mod +}$, which is exactly the obvious $\co(-1,-1) \to \Pj N^+ \times_{X^0} \Pj N^-$. The normal bundle to $\Pj N^\pm$ is then just $(R^0\pi_\mp \co(1,1))^* = \pi_\pm^* N^\mp(-1)$. \fp \begin{r}{Example} The simplest non-trivial example of these phenomena is also the best-known; indeed it goes back to an early paper of Atiyah \cite{a}. Let $X = \Aff^4$, and let $G$ act by $\lambda \cdot(v,w,x,y)= (\lambda v, \lambda w, \lambda^{-1} x, \lambda^{-1} y)$. Then $X \mod 0 = \Spec \k[vx,vy,wx,wy] = \Spec[a,b,c,d]/\langle ad-bc \rangle$, the affine cone on a smooth quadric surface in $\Pj^3$. But, using the $\N$-grading conventions of the previous examples, \beqas X \mod + & = & \Proj \k[vx,vy,wx,wy,zv,zw] \\ & = & \Proj \k[a,b,c,d,za,zc] / \langle ad-bc \rangle. \eeqas This is the blow-up of $X \mod 0$ at the Weil divisor cut out by $a$ and $c$. But this Weil divisor is generically Cartier, so the blow-down $X \mod + \to X \mod 0$ is generically an isomorphism even over the divisor. The exceptional set of the morphism therefore has codimension 2; indeed, it is the $\Pj^1$ lying over the cone point. Likewise, $X \mod - = \Proj \k[a,b,c,d,zb,zd]/ \langle ad-bc \rangle$, and similar remarks apply by symmetry. By \re{2k} the fibred product ${X \mod -} \times_{X \mod 0} {X \mod +}$ is the common blow-up of $X \mod \pm$ at these $\Pj^1$, and also the blow-up of $X \mod 0$ at the cone point. This is exactly the proper transform of the quadric cone in $\Aff^4$ blown up at the origin, so it has fibre $\Pj^1 \times \Pj^1$ over the cone point, as expected. \end{r} \bit{The space of linearizations} In \S\S3, 4 and 5 we will generalize the results of \S1 in three directions. First, the group $G$ may now be any reductive group over $\k$. Second, the variety $X$ may now be any quasi-projective variety over $\k$, projective over an affine variety. Finally, the linearization may be arbitrary. Before doing this, though, we will prove some general facts, in the case where $X$ is normal and projective, about the structure of the group of all linearizations. This will show how to apply our general results in this case. So in this section, suppose $X$ is normal and projective over $\k$, and that $G$ is a reductive group over $\k$ acting on $X$. We first recall a few familiar facts about the Picard group. In the Picard group $\Pic$ of isomorphism classes of line bundles, the property of ampleness depends only on the algebraic equivalence class of the bundle. Hence there is a well-defined ample subset $\A$ of the N\'eron-Severi group $\NS$ of algebraic equivalence classes of line bundles. This determines the {\em ample cone} $\A_\Q = \A \otimes_{\N} \Q_{\geq 0} \subset \NS_\Q = \NS \otimes \Q$. The N\'eron-Severi group is finitely-generated, so $\NS_\Q$ is a finite-dimensional rational vector space. We will refer to an element of $\A$ as a {\em polarization}, and an element of $\A_\Q$ as a {\em fractional polarization}. Now let $\Pic^G$ be the group of isomorphism classes of linearizations of the $G$-action (cf.\ 1, \S3 of GIT). There is a forgetful homomorphism $f: \Pic^G \to \Pic$, whose kernel is the group of linearizations on $\co$, which is exactly the group $\chi(G)$ of characters of $G$. There is an equivalence relation on $\Pic^G$ analogous to algebraic equivalence on $\Pic$; it is defined as follows. Two linearizations $L_1$ and $L_2$ are said to be {\em $G$-algebraically equivalent} if there is a connected variety $T$ containing points $t_1, t_2$, and a linearization $L$ of the $G$-action on $T \times X$ induced from the second factor, such that $L|_{t_1 \times X} \cong L_1$ and $L|_{t_2 \times X} \cong L_2$. \begin{s}{Proposition} \label{1d} If $L$ is an ample linearization, then $X^{ss}(L)$, and the quotient $X \mod G(L)$ regarded as a polarized variety, depend only on the $G$-algebraic equivalence class of $L$. \end{s} \pf. The statement about $X^{ss}(L)$ is proved exactly like Cor.\ 1.20 of GIT, except that the Picard group $P$ is replaced by $T$. The statement about $X \mod G(L)$ as a variety then follows from this, since $X \mod G(L)$ is a good quotient of $X^{ss}(L)$, hence a categorical quotient of $X^{ss}(L)$, so is uniquely determined by $X^{ss}(L)$. As for the polarization, note that, if $L_1$ and $L_2$ are $G$-linearly equivalent ample linearizations, then the linearization $L$ on $T \times X$ inducing the equivalence can be assumed ample: just tensor $L$ by the pullback of a sufficiently ample bundle on $T$. Then $\co(1) \to (T \times X) \mod G (L)$ is a family of line bundles on $X \mod G$ including $\co(1) \to X \mod G (L_1)$ and $\co(1) \to X \mod G (L_2)$, so these are algebraically equivalent. \fp So define $\NS^G$ to be the group of $G$-algebraic equivalence classes of linearizations. In light of \re{1d}, by abuse of terminology an element of $\NS^G$ will frequently be called just a linearization. The forgetful map $f$ descends to $f: \NS^G \to \NS$. \begin{s}{Proposition} \label{2q} This new $f$ has kernel $\chi(G)$ modulo a torsion subgroup. \end{s} \pf. Let $M \to \Pic_0 X \times X$ be the Poincar\'e line bundle, and let $G$ act on $\Pic_0 X \times X$, trivially on the first factor. By Cor.\ 1.6 of GIT, some power $M^n$ of $M$ admits a linearization. Since the $n$th power morphism $\Pic_0 X \to \Pic_0 X$ is surjective, this shows that any element of $\Pic_0 X$ has a linearization $G$-algebraically equivalent to a linearization on $\co$. Hence $\ker f$ is $\chi(G)$ modulo the subgroup of linearizations on $\co$ which are $G$-algebraically equivalent to the trivial linearization. We will show that this subgroup is torsion. Suppose there is a linearization $L_1$ on $\co$ which is $G$-algebraically equivalent to the trivial linearization. Then there exist $T$ containing $t_1$, $t_2$ and $L$ as in the definition of $G$-algebraic equivalence. There is an induced morphism $g: T \to \Pic_0 X$ such that $t_1, t_2 \mapsto \co$. As before, let $M^n$ be the power of the Poincar\'e bundle admitting a linearization. Then $N = (1 \times g)^* M^n \otimes L^{-n}$ is a family of linearizations on $\co \to X$. Since the isomorphism classes of such linearizations form the discrete group $\chi(G)$, $L_1^n = N_{t_1}^{-1} \otimes N_{t_2}$ is trivial as a linearization. \fp \bl Hence $\NS^G$ is finitely-generated and $\NS^G_\Q = \NS^G \otimes \Q$ is again a finite-dimensional rational vector space. We refer to an element of $\NS^G_\Q$ as a {\em fractional linearization}. The map $f: \NS^G \to \NS$ is not necessarily surjective (see 1, \S3 of GIT). But by Cor.\ 1.6 of GIT, $f_\Q: \NS^G_\Q \to \NS_\Q$ is surjective. By \re{2q}, the kernel is $\chi(G) \otimes \Q$, the group of {\em fractional characters}. (Not to be confused with $f_\Q$ is the natural surjective linear map $\NS^G_\Q(X) \to \NS_\Q(X \mod G)$: this is induced by descent, since divisor classes always descend over $\Q$.) An ample linearization $L$ is said to be {\em $G$-effective} if $L^n$ has a $G$-invariant section for some $n > 0$. This is equivalent to having a semistable point, so \re{1d} shows that $G$-effectiveness depends only on the $G$-algebraic equivalence class of the linearization. Hence there is a well-defined $G$-effective subset $\E^G \subset f^{-1}\A \subset \NS^G$, and a {\em $G$-effective cone} $\E^G_\Q = \E^G \otimes_{\N} \Q_{\geq 0} \subset \NS^G_\Q$. Now a linearization $L$ determines a quotient $X \mod G$ if $L$ is ample; then $X \mod G \neq \emptyset} \def\dasharrow{\to$ if and only if $L$ is also $G$-effective. Of course, we can also tensor by $\Q$, allowing fractional linearizations; the quotient $X \mod G$ will then be fractionally polarized. Hence any fractional linearization $L \in f_\Q^{-1}(\A_\Q) \subset \NS^G_\Q$ defines a fractionally polarized quotient, which will be nonempty if and only if $L \in \E^G_\Q$. Replacing a fractional linearization $L$ by $L^n$ for some positive $n \in \Q$ has no effect on the quotient, except to replace the fractional polarization $\co(1)$ by $\co(n)$. \bl The first result describing the dependence of the quotient $X \mod G(L)$ on the choice of $L \in \NS^G_\Q$ is the following, which is analogous to the Duistermaat-Heckman theory in symplectic geometry. \begin{s}{Theorem} \label{1b} The $G$-effective cone $\E_\Q^G$ is locally polyhedral in the ample cone $f_\Q^{-1} \A_\Q$. It is divided by homogeneous {\rm walls}, locally polyhedral of codimension 1 in $f_\Q^{-1} \A_\Q$, into convex {\rm chambers} such that, as $t$ varies within a fixed chamber, the semistable set $X^{ss}(t)$, and the quotient $X \mod G(t)$, remain fixed, but $\co(1)$ depends affinely on $t$. If $t_0$ is on a wall or walls, or on the boundary of $\E_\Q^G$, and $t_+$ is in an adjacent chamber, then there is an inclusion $X^{ss}(+) \subset X^{ss}(0)$ inducing a canonical projective morphism $X \mod G(+) \to X \mod G(0)$. \end{s} The proposition above could be proved directly, using Kempf's descent lemma \cite{dn} for the statement about $\co(1)$, and Mumford's numerical criterion \cite[4.9; GIT Thm.\ 2.1]{n} for the rest. But it will follow easily from the construction \re{1h} to be introduced in the next section, so we put off the proof until then. \bl Theorem \re{1b} asserts that the walls are locally polyhedral, and in particular, locally finite; but with a little more effort we can prove a global result. \begin{s}{Theorem} \label{1s} There are only finitely many walls. \end{s} \pf. Suppose not. Then there exists an infinite sequence $\{ C_i \}$ of chambers such that, for any $m,n \geq 0$, the convex hull of $C_n \cup C_{n+m}$ intersects the interior of $C_{n+1}$ nontrivially. Indeed, let $C_0$ be any chamber; then there exists a wall $W_0$ bounding it such that there are infinitely many chambers on the other side of $W_0$ (or more properly, the affine hyperplane containing $W_0$). Let $C_1$ be the other chamber bounded by $W_0$. Inductively, given $C_0, \dots, C_n$ such that $C_n$ is on the other side of $W_i$ from $C_i$ for all $i < n$, there is a wall $W_n$ bounding $C_n$ such that there are infinitely many chambers which for all $i \leq n$ are on the other side of $W_i$ from $C_i$. Let $C_{n+1}$ be the other chamber bounded by $W_n$. For a sequence chosen in this manner, $C_{n+m}$ is on the other side of $W_n$ from $C_n$, so the convex hull of $C_n \cup C_{n+m}$ meets the interior of $C_{n+1}$. Choose an $L_i$ in the interior of each $C_i$. For any fixed $x \in X$, the set $\{ L \in \NS_\Q \st x \in X^{ss}(L) \}$ is convex, since $s_\pm \in H^0(L_\pm)^G$, $s_\pm(x) \neq 0$ imply $s_+ \cdot s_- \in H^0(L_+ \otimes L_-)^G$, $(s_+ \cdot s_-)(x) \neq 0$. But by \re{1b} it is also a union of chambers, so by induction it includes $C_n \cup C_{n+m}$ if and only if it includes $C_{n+i}$ for all $i \leq m$. Its intersection with $\{ L_i \}$ is therefore the image of an interval in $\N$. Hence $X^{ss}(L_{i+1}) \sans X^{ss}(L_i)$ are all disjoint; but each one is open in the complement of $X^{ss}(L_0)$ and the preceding ones. Since varieties are noetherian, this implies there exists $i_0$ such that for all $i \geq i_0$, $X^{ss}(L_{i+1}) \sans X^{ss}(L_i) = \emptyset} \def\dasharrow{\to$, and hence $X^{ss}(L_{i+1}) \subset X^{ss}(L_i)$. There is therefore an infinite sequence of dominant projective morphisms $$ \cdots \to X \mod G(L_{i_0 + 2}) \to X \mod G(L_{i_0 + 1}) \to X\mod G(L_{i_0}).$$ Hence the N\'eron-Severi group of $X \mod G(L_i)$ has arbitrarily large rank for some $i$. But as mentioned before, there is a natural surjective linear map $\NS^G_\Q(X) \to \NS_\Q(X \mod G(L_i))$ for all $i$. Since $\NS^G_\Q(X)$ is finite-dimensional, this is a contradiction. \fp \bit{The general case} We now embark on our generalization of the results of \S1. So let $G$ be a reductive group over $\k$, acting on a quasi-projective variety $X$ over $\k$, projective over an affine variety. This is the largest category in which geometric invariant theory guarantees that the semistable set has a good quotient. All of the arguments in this section use the following trick. \begin{r}{Construction} \label{1h} Let $L_1 , \dots , L_{n+1}$ be ample linearizations. Let $\Delta$ be the $n$-simplex $\{ (t_i) \in \Q^{n+1} \st \sum t_i = 1 \} $. Then for any $t = (t_i) \in \Delta$, $L(t) = \bigotimes_i L_i^{t_i}$ is a fractional linearization on $X$. We refer to the set $\{ L(t) \st t \in \Delta \}$ as an {\em $n$-simplicial family} of fractional linearizations. Put $$Y = \Pj(\bigoplus_i L_i) = \Proj \sum_{j_i \in \N} H^0(\bigotimes_i L_i^{j_i}),$$ and let $q: Y \to X$ be the projection. Then $G$ acts naturally on $\bigoplus_i L_i$, hence on $Y$ with a canonical linearization on the ample bundle $\co(1)$. Likewise, the $n$-parameter torus $T = \{\la \in \k^{n+1} \st \prod_i \la_i = 1 \}$ acts on $\bigoplus_i L_i$ by $\la(u_i) = (\la_i u_i)$, and hence on $Y$. This $T$-action commutes with the $G$-action. Moreover, since it comes from $\bigoplus_i L_i$, it too is linearized on $\co(1)$. But this obvious linearization is not the only one. Indeed, any $t \in \Delta$ determines a fractional character of $T$ by $t(\la) = \prod_i \la_i^{t_i}$; then $\la(u_i) = (t^{-1}(\la) \la_i u_i)$ determines a fractional linearization depending on $t$. This gives an $n$-simplicial family $M(t)$ of fractional linearizations on $\co(1)$ of the $T$-action on $Y$, each compatible with the canonical linearization of the $G$-action. In other words, $M(t)$ is a family of fractional linearizations of the $G \times T$-action on $Y$. Let $Y^{ss}(t)$ be the semistable set for this action and linearization, and let $Y^{ss}(G)$ be the semistable set for the $G$-action alone. For any $t$, $Y^{ss}(t) \subset Y^{ss}(G)$. With respect to $M(t)$, $T$ acts trivially on $H^0(\bigotimes_i L_i^{j_i})$ if and only if $j_i = m t_i$ for some fixed $m$. Hence the subalgebra of $T$-invariants is $\sum_m H^0 ((\bigotimes_i L_i^{t_i})^m)$. The quotient $Y \mod T(t)$ is therefore $X$, but with the residual $G$-action linearized by $L(t)$. Consequently $(Y \mod T(t)) \mod G = X \mod G(t)$, the original quotients of interest. Moreover, $X^{ss}(t) = q(Y^{ss}(t))$. Let $Z$ be the quotient $Y \mod G$ with respect to the canonical linearization defined above, and let $p: Y^{ss}(G) \to Z$ be the quotient. Then the $M(t)$ descend to an $n$-simplicial family $N(t)$ of fractional linearizations on $\co(1)$ of the residual $T$-action on $Z$, and $Y^{ss}(t) = q^{-1}(Z^{ss}(t))$. When two group actions commute, the order of taking the quotient does not matter, so $(Y \mod T(t)) \mod G = (Y \mod G) \mod T(t) = Z \mod T(t)$. So we have constructed a variety $Z$, acted on by a torus $T$, and a simplicial family $N(t)$ of fractional linearizations on $\co(1)$, such that $Z \mod T(t) = X \mod G(t)$. Moreover, $X^{ss}(t) = q(p^{-1}(Z^{ss}(t)))$. \end{r} As a first application of this construction, let us prove the result asserted in the last section. \pf\ of \re{1b}. The result is relatively easy in the case where $X = \Pj^n$ and $G$ is a torus $T$. Indeed, the $T$-effective cone is globally polyhedral, as is each chamber; for details see \cite{bp, toric}. In the general case, choose a locally finite collection of simplices in $f_\Q^{-1}\A_\Q \subset \NS^G_\Q$ such that every vertex is in $\NS^G$, and for every $L \in f_\Q^{-1}\A_\Q$, some fractional power $L^m$ is in one of the simplices. By the homogeneity property mentioned just before the statement of \re{1b}, it suffices to prove the statement analogous to \re{1b} where $f_\Q^{-1}\A_\Q$ is replaced by the simplex parametrizing each of these families. The construction \re{1h} applies, so there exists $Z \subset \Pj^n$ and a simplicial family $N(t)$ in $\NS^T(\Pj^n)$ such that $X \mod G(t) = Z \mod T(t)$ for all $t \in \Delta$. The conclusions of the theorem are preserved by restriction to a $T$-invariant subvariety, so they hold for $Z$ and $Z \mod T(t)$, and hence for $X$ and $X \mod G(t)$. \fp \bl The rest of this section and all of \S\S4 and 5 will refer to the following set-up. Let $X$ and $G$ be as before. Let $L_+$ and $L_-$ be ample linearizations such that, if $L(t) = L_+^t L_-^{1-t}$ for $t \in [-1,1]$, there exists $t_0 \in (-1,1)$ such that $X^{ss}(t) = X^{ss}(+)$ for $t > t_0$ and $X^{ss}(t) = X^{ss}(-)$ for $t < t_0$. For example, \re{1b} implies that this is the case if $X$ is normal and projective, $L_\pm$ are in adjacent chambers, and the line segment between them crosses a wall only at $L(t_0)$. Even in the normal projective case, however, there are other possibilities; for example, $L_\pm$ could both lie in the same wall, or $L(t_0)$ could lie on the boundary of $\E^G_\Q$. In future, $L(t_0)$ will be denoted $L_0$. \bl The following lemma shows how to globalize the results of \S1 within this set-up. Suppose $T \cong \kst$ acts on $X$, and let $\si \in H^0(X, L_0^n)^T$ for some $n$, so that $X_\si \subset X^{ss}(0)$ is a $T$-invariant affine. \begin{s}{Lemma} \label{1e} Suppose $f(L_-) = f(L_+)$. Then \mbox{\rm (a)} $X^{ss}(\pm) \subset X^{ss}(0)$; \mbox{\rm (b)} $X_\si^{ss}(0) = X_\si \cap X^{ss}(0)$; and \mbox{\rm (c)} $X_\si^{ss}(\pm) = X_\si \cap X^{ss}(\pm)$; so there is a natural commutative diagram $$\begin{array}{ccc} X_\si \mod \pm & \emb & X \mod \pm \vspace{.7ex} \\ \down{} & & \down{} \vspace{.7ex} \\ X_\si \mod 0 & \emb & X \mod 0 \end{array} $$ whose rows are embeddings. \end{s} \pf. Put $R_m = H^0(X, L_0^m)$, so that $X = \Proj \bigoplus_{m \in \N}R_m$, and let $R_m = \bigoplus_{n \in \Z} R_{m,n}$ be the weight decomposition for the $\kst$-action. Suppose $x \in X^{ss}(+) \sans X^{ss}(0)$. Then for $m>0$, every element of $R_{m,0}$ vanishes at $x$. Since $\bigoplus_m R_m$ is finitely-generated, this implies that, for $m/n$ large enough, every element of $R_{m,n}$ vanishes at $x$. But then there exists $t>t_0$ such that $x \notin X^{ss}(t)$, contradicting the set-up. The proof for $X^{ss}(-)$ is similar. This proves (a). Without loss of generality suppose $\si \in H^0(X, L_0)$. Then $\si \in R_{1,0}$ and $X_\si = \Spec (\si^{-1}R)_0$. Since $\bigoplus_m R_{m,0}$ is finitely-generated, for $m$ large the map $R_{m,0} \to (\si^{-1} R)_{0,0}$ given by dividing by $\si^m$ is surjective. But $R_{m,0} = H^0(X, L_0^m)^T$ and $(\si^{-1} R)_{0,0} = H^0(X_\si, L_0^m)^T$ (the latter since $L_0^m$ is trivial on $X_\si$), so this implies that $X_\si^{ss}(0) = X_\si \cap X^{ss}(0)$, hence that $X_\si \mod 0$ embeds in $X \mod 0$. This proves (b). Without loss of generality take $L_+$ to be $L_0$ twisted by the fractional character $\la \mapsto \la^{1/p}$ for $p$ large. Since $R$ is finitely-generated, $R_{m,n} \to (\si^{-1} R)_{0,n}$ is surjective for $m/n$ large. But for $m = np$, $R_{m,n} = H^0(X, L_+^m)^T$ and $(\si^{-1} R)_{0,n} = H^0(X_\si, L_+^m)^T$, so this implies that $X_\si^{ss}(+) = X_\si \cap X^{ss}(+)$, hence that $X_\si \mod +$ embeds in $X \mod +$. The case of $L_-$ is similar. This proves (c). \fp Hence, in studying $\kst$-quotients where $L_+ \cong L_-$ as bundles, we may work locally, using the methods of \S1. \begin{s}{Theorem} \label{1f} If $X \mod G(+)$ and $X \mod G(-)$ are both nonempty, then the morphisms $X \mod G(\pm) \to X \mod G(0)$ are proper and birational. If they are both small, then the rational map $X \mod G(-) \dasharrow X \mod G(+)$ is a flip with respect to $\co(1) \to X \mod G(+)$. \end{s} Again, this could be proved directly, by first examining the stable sets to show birationality, then applying Kempf's descent lemma to the linearization $L_+$. But again, we will use the trick. \pf\ of \re{1f}. Perform the construction of \re{1h} on $L_+$ and $L_-$. This gives a variety $Z$ with an action of $T \cong \kst$ and a family $N(t)$ of fractional linearizations with $f_\Q(N(t))$ constant such that $Z \mod T(t) = X \mod G (t)$. The whole statement is local over $X \mod G(0)$, so by \re{1e} it suffices to prove it for affines of the form $Z_\si$, with the $T$-action and fractional linearizations $N(t)$. But this is the case considered in \S1, so \re{2h} and \re{2i} complete the proof. \fp There is a converse to \re{1f} analogous to \re{2o}, which we leave to the reader. \begin{r}{Application} For an application, suppose that $X$ is normal and projective. Choose any nonzero $M \in \NS_\Q X$, let $L(t) = L \otimes M^t$ and consider the ray $\{ L(t) \st t \geq 0 \} \subset \NS^G_\Q(X)$. By \re{1b}, the quotient $X \mod G(t)$ is empty except for $t$ in some bounded interval $[0,\omega]$, and this interval is partitioned into finitely many subintervals in whose interior $X \mod G(t)$ is fixed. But when a critical value $t_0$ separating two intervals is crossed, there are morphisms $X \mod G(t_\pm) \to X \mod G(t_0)$, which by \re{1f} are birational except possibly at the last critical value $\omega$. Since the fractional polarization on $X \mod G(t)$ is the image of $L \otimes M^{-t}$ in the natural descent map $\NS^G_\Q(X) \to \NS_\Q(X \mod G(t))$, the descents of $M$ to $\Q$-Cartier divisor classes on $X \mod G(\pm)$ are relatively ample for each morphism $X \mod G(t_+) \to X \mod G(t_0)$, and relatively negative for each morphism $X \mod G(t_-) \to X \mod G(t_0)$. So suppose that each $X \mod G(t_+) \to X \mod G(t_0)$ is small when $X \mod G(t_-) \to X \mod G(t_0)$ is small, and that each $X \mod G(t_+) \to X \mod G(t_0)$ is an isomorphism when $X \mod G(t_-) \to X \mod G(t_0)$ is divisorial. It then follows that the finite sequence of quotients $X \mod G(t)$ runs the $M$-minimal model programme \cite[(2.26)]{ko} on $X \mod G(L)$, where by abuse of notation $M$ denotes its image in the descent map. \end{r} For some $d > 0$, the ideal sheaves $\langle H^0(X, L_\pm^{nd})^G \rangle$ and $\langle H^0(X, L_\pm^{d})^G\rangle^n$ on $X$ are equal for all $n \in \N$. For such a $d$, let $\ci^\pm = \langle H^0(X, L_\mp^{d})^G \rangle$ (note the reversal of sign), and let $\ci^\pm \mod G(\pm)$ be the corresponding sheaves of invariants on $X \mod G (\pm)$. Also let $(\ci^+ + \ci^-) \mod G(0)$ be the sheaf of invariants of the ideal sheaf $\ci^+ + \ci^-$ on $X \mod G(0)$. \begin{s}{Theorem} \label{1g} Suppose $X \mod G(+)$ and $X \mod G(-)$ are both nonempty. Then the pullbacks of $(\ci^+ + \ci^-) \mod G(0)$ by the morphisms $X \mod G(\pm) \to X \mod G(0)$ are exactly $\ci^\pm \mod G(\pm)$, and the blow-ups of $X \mod G(\pm)$ at $\ci^\pm \mod G(\pm)$, and of $X \mod G(0)$ at $(\ci^+ + \ci^-) \mod G(0)$, are all naturally isomorphic to the irreducible component of the fibred product $X \mod G(-) \times_{X \mod G(0)} X \mod G(+)$ dominating $X \mod G(0)$. \end{s} \pf. Construct a variety $Z$ as in the proof of \re{1f}. Notice that for $d$ large, since $\ci^\pm_X = \langle H^0(X, L_\pm^d)^G \rangle$ on $X$ and $\ci^\pm_Z = \langle H^0(Z, N_\pm^d)^T \rangle$ on $Z$, the pullbacks of both $\ci_X^\pm$ and $\ci_Z^\pm$ to $Y$ are $\ci_Y^\pm = \langle H^0(Y, M_\pm^d)^{G \times T} \rangle$. Hence $\ci_X^\pm$ and $\ci_Z^\pm$ have the same sheaves of invariants on the quotients $Z \mod T(t) = X \mod G (t)$. It therefore suffices to prove the statement for $Z$ and $N_\pm$. All statements are local over $Z \mod T(0)$, so by \re{1e} it suffices to prove them for affines of the form $Z_\si$. But this is the case considered in \S1, so \re{2b} and \re{2d} complete the proof. \fp \bl \bit{The smooth case: strong results} In the next two sections we seek to generalize the other two main results of \S1, \re{2j} and \re{2k}. Indeed, we will give two different generalizations of each. The generalizations in \S4 make fairly strong hypotheses, and prove that, as in \S1, $X^\pm \mod G(\pm)$ are locally trivial over $X^0 \mod G(0)$. Moreover, the proofs are quite easy using the tools already at hand. Those in \S5 relax the hypotheses somewhat, but conclude only that $X^\pm \mod G(\pm)$ are locally trivial in the \'etale topology. The proofs therefore require \'etale covers and are more difficult; in fact we confine ourselves to a sketch of the \'etale generalization of \re{2k}. Let $X$, $G$, and $L_\pm$ be as in \S3. As in \re{1h}, let $Y = \Pj(L_+ \oplus L_-)$, let $T$ be the torus acting on $Y$, let $Z = Y \mod G$, and let $p: Y^{ss}(G) \to Z$ be the quotient morphism. Fix the isomorphism $T \cong \kst$ given by projection on the first factor. Define $Y^\pm$, $Y^0$, $Z^\pm$, and $Z^0$ similarly to $X^\pm$ and $X^0$. Also let $i_\pm : X \to Y$ be the embeddings given by the sections at 0 and $\infty$. Write $q : Y \to X$ for the projection as before, but let $\pi$ denote the restriction of $q$ to $Y \sans (i_+(X) \cup i_-(X))$. So in particular $\pi^{-1}(X)$ denotes $Y \sans (i_+(X) \cup i_-(X))$ itself. \begin{s}{Lemma} \label{2r} $X^{ss}(\pm) \subset X^{ss}(0)$. \end{s} \pf. This is true for $Z$ by \re{1e}(a), but $X^{ss}(\pm) = q(p^{-1}(Z^{ss}(\pm)))$ and $X^{ss}(0) = q(p^{-1}(Z^{ss}(0)))$. \fp \begin{s}{Lemma} \label{1j} \mbox{\rm (a)} $i_\pm(X) \cap Y^{ss}(0) = \emptyset} \def\dasharrow{\to$; \mbox{\rm (b)} $i_\pm(X) \cap Y^{ss}(G) = i_\pm(X^{ss}(\pm))$; \mbox{\rm (c)} $\pi^{-1}(X) \cap Y^{ss}(0) = \pi^{-1}(X) \cap Y^{ss}(G) = \pi^{-1}(X^{ss}(0))$. \end{s} \pf. For $i_\pm(x)$ to be in $Y^{ss}(0)$, it must certainly be semistable for the $T$-action on the fibre $q^{-1}(x) = \Pj^1$. But in the fractional linearization $M_0$, $T$ acts with nontrivial weight on both homogeneous coordinates of $\Pj^1$, so any invariant section of $\co(n)$ for $n > 0$ must vanish both at $0$ and $\infty$. Hence $i_\pm(x)$ are unstable, which proves (a). However, for $i_\pm(x)$ to be in $Y^{ss}(G)$ requires only that the section of $\co(n)$ which is nonzero at $x$ be $G$-invariant. Pushing down by $q$ shows that $$H^0(Y, \co(n)) = H^0(X, \bigoplus_{j=0}^n L_+^j \otimes L_-^{n-j}) = \bigoplus_{j=0}^n H^0(X, L_+^j \otimes L_-^{n-j}),$$ and a section of $\co(n)$ is nonzero at $i_\pm(x)$ if and only if its projection on $H^0(X, L_\pm^n)$ is nonzero at $x$. Hence $i_\pm(x) \in X^{ss}(G)$ if and only if $x \in X^{ss}(\pm)$, which proves (b). With respect to the fractional linearization $M(t)$, the $T$-invariant subspace in the above decomposition consists of that $H^0(X, L_+^j \otimes L_-^{n-j})$ such that $L_+^j \otimes L_-^{n-j}$ is a power of $L(t)$, and an invariant section is nonzero on $\pi^{-1}(x)$ if and only if the corresponding element of $H^0(X, L_+^j \otimes L_-^{n-j})$ is nonzero at $x$. Hence there is a $G$-invariant section of some $\co(n)$ non-vanishing on $\pi^{-1}(x)$ if and only if there is a $G$-invariant section of some $L(t)^n$ non-vanishing at $x$; this implies $\pi^{-1}(x) \subset Y^{ss}(G)$ if and only if $x \in \cup_t X^{ss}(t)$, which equals $X^{ss}(0)$ by \re{2r}. On the other hand, $x \in X^{ss}(0)$ if and only if there is a $G$-invariant section of some $L_0^n$ non-vanishing at $x$, and hence a $G \times T$-invariant section of some $M_0^n$ non-vanishing on $\pi^{-1}(x)$, that is, $\pi^{-1}(x) \subset Y^{ss}(0)$. This proves (c). \fp \bl Let $X^\pm$ and $X^0$ be the intersections with $X^{ss}(0)$ of the supports of the sheaves $\ci^\pm$ and $\ci^+ + \ci^-$, defined as in \S3. Note that this generalizes the definitions of \S1. Indeed, \beqas X^\pm & = & X^{ss}(0) \sans X^{ss}(\mp); \\ X^0 & = & X^{ss}(0) \sans (X^{ss}(+) \cup X^{ss}(-)). \eeqas \begin{s}{Lemma} \label{1u} \mbox{\rm (a)} $\pi^{-1}X^\pm = Y^\pm = p^{-1}Z^\pm$; \mbox{\rm (b)} $\pi^{-1}X^0 = Y^0 = p^{-1}Z^0$. \end{s} \pf. These follow immediately from $\pi^* \ci^\pm_X = \ci^\pm_Y = p^* \ci^\pm_Z$. \fp \bl Choose $x \in X^0$; throughout this section, we will assume the following. \begin{r}{Hypothesis} \label{1y} Suppose that $X$ is smooth at $x \in X^0$, that $G \cdot x$ is closed in $X^{ss}(0)$, and that $G_x \cong \kst$. \end{r} Note that if $G_x \cong \kst$ for {\em all} $x \in X^0$, then an orbit in $X^0$ cannot specialize in $X^0$, so it is closed in $X^0$ and hence in $X^{ss}(0)$. So the second part of the hypothesis is redundant in this case. The third part is necessary, as the counterexample \re{1r} will show. But it is always true when $G$ is a torus or when $G$ acts diagonally on the product of its flag variety with another variety \cite{dh}. \bl Since $x \in X^{ss}(0)$, $G_x$ acts trivially on $(L_0)_x$. If it acts nontrivially on $(L_+)_x$, requiring it to act with some negative weight $v_+ < 0$ fixes an isomorphism $G_x \cong \kst$. It then acts on $(L_-)_x$ with some positive weight $v_- > 0$. To obtain the first, stronger generalizations, assume that these two weights are coprime: $(v_+, v_-) = 1$. When $X$ is normal and projective and $L_\pm$ are in adjacent chambers, this additional hypothesis can be interpreted as follows. The weight of the $G_x$-action defines a homomorphism $\rho: \NS^G \to \Z$, and $L_\pm$ can be chosen within their chambers to satisfy this hypothesis, and the conditions of the set-up, if and only if $\rho$ is surjective. Again, this is always true when $G$ is a torus or when $G$ acts diagonally on the product of its flag variety with another variety. If $(v_+, v_-) = 1$, then $\pi^{-1}(x)$ is contained in a $G$-orbit, so $p(\pi^{-1}(x))$ is a single point in $Z$. \begin{s}{Lemma} \label{1v} If \re{1y} holds and $(v_+, v_-) = 1$, then $L_\pm$ can be chosen so that $G$ acts freely on $Y$ at $\pi^{-1}(x)$ and $Z$ is smooth at $p(\pi^{-1}(x))$. \end{s} \pf. Since $G_x$ acts on $(L_\pm)_x$ with weights $v_\pm$, there exist positive powers of $L_+$ and $L_-$ whose weights add to 1. Replace $L_\pm$ by these powers. Then $G_x$ acts freely on $(L_+ \otimes L_-^{-1})_x \sans 0$. But this is exactly $\pi^{-1}(x)$, so $G$ acts freely on $Y$ at $\pi^{-1}(x)$. To show that $Z$ is smooth at $p(\pi^{-1}(x))$, it therefore suffices to show that $\pi^{-1}(x) \subset Y^s(G)$, that is, that the $G$-orbit of $\pi^{-1}(x)$ is closed in $Y^{ss}(G)$. But if $y \in Y^{ss}(G)$ is in the closure of $G \cdot \pi^{-1}(x)$, then $y \notin i_\pm(X)$ by \re{1j}(b) and \re{1u}(b), so $y \in \pi^{-1}(x')$ for some $x' \in X^0$ by \re{1j}(c). Then $x'$ is in the closure of $G \cdot x \subset X^0$, so by \re{1y}, $x' \in G \cdot x$ and hence $y \in G \cdot \pi^{-1}(x)$. \fp \begin{s}{Proposition} \label{1w} If \re{1y} holds and $(v_+, v_-) = 1$, then \mbox{\rm (a)} $X^0$ is smooth at $x$; \mbox{\rm (b)} on a neighbourhood of $x$ in $X^0$, there exists a vector bundle $N$ with $\kst$-action, whose fibre at $x$ is naturally isomorphic to $N_{G \cdot x / X}$; \mbox{\rm (c)} the bundle $N^0$ of zero weight spaces of $N$ is exactly the image of $TX^0$ in $N$; \mbox{\rm (d)} the bundles $N^\pm$ of positive and negative weight spaces of $N$ are naturally isomorphic to $N_{X^0 / X^\pm}$. \end{s} \pf. By \re{2e}(a), \re{1u}(b) and \re{1v}, $Z^0$ is smooth at $p(\pi^{-1}(x))$, and $Y^0$ is locally a principal $G$-bundle over $Z^0$. Hence $Y^0$ is smooth at $\pi^{-1}(x)$, so $X^0$ is smooth at $x$. The bundle $N_Z$ is just $TZ|_{Z^0}$, so define $N_Y = p^* N_Z$. This is acted upon by $\kst$, so by Kempf's descent lemma \cite{dn} descends to a bundle $N_X$ which has the desired property. This proves (b); the proofs of (c) and (d) are similar, using \re{2e}(c) and (d). \fp \bl As in \S1, let $w_i^\pm \in \Z$ be the weights of the $\kst$-actions on $N^\pm$. \begin{s}{Theorem} \label{2n} If \re{1y} holds and $(v_+, v_-) = 1$, then over a neighbourhood of $x$ in $X^0 \mod G(0)$, $X^\pm \mod G(\pm)$ are locally trivial fibrations with fibre the weighted projective space $\Pj(|w_i^\pm|)$. \end{s} \pf. This now follows immediately from \re{2j} and \re{1v}. \fp \bl If $X \mod G(\pm)$ are both nonempty, then $X^\pm \mod G(\pm)$ are the supports of the blow-up loci of \re{1g}. But if $X \mod G(-) = \emptyset} \def\dasharrow{\to$, then $X^+ \mod G(+) = X \mod G(+)$ and $X^0 \mod G(0) = X \mod G(0)$, so \re{2n} says the natural morphism $X \mod G(+) \to X \mod G(0)$ is a locally trivial weighted projective fibration. \bl If moreover all $w_i^\pm = \pm w$ for some $w$, then for any linearization $L$ such that $L_x$ is acted on by $G_x$ with weight $-1$, the bundles $N^\pm \otimes L^{\pm w}$ are acted upon trivially by all stabilizers. So by Kempf's descent lemma \cite{dn} they descend to vector bundles $W^\pm$ over a neighbourhood of $x$ in $X^0 \mod G(0)$. \begin{s}{Theorem} \label{1x} Suppose that \re{1y} holds, that $(v_+, v_-) = 1$, and that all $w_i^\pm = \pm w$ for some $w$. Then over a neighbourhood of $x$ in $X^0 \mod G(0)$, $X^\pm \mod G(\pm)$ are naturally isomorphic to the projective bundles $\Pj W^\pm$, their normal bundles are naturally isomorphic to $\pi_\pm^* W^\mp(-1)$, and the blow-ups of $X \mod G(\pm)$ at $X^\pm \mod G(\pm)$, and of $X \mod G(0)$ at $X^0 \mod G(0)$, are all naturally isomorphic to the fibred product $X \mod G(-) \times_{X \mod G(0)} X \mod G(+)$. \end{s} \pf. First notice that, although $W^\pm$ depend on the choice of $L$, the projectivizations $\Pj W^\pm$, and even the line bundle $\co(1,1) \to \Pj W^+ \times_{X^0 \mod G(0)} \Pj W^-$, are independent of $L$. Now on $Z$, taking $L \cong \co$ yields $W^\pm = N^\pm$. But $N^\pm_Y = p^* N^\pm_Z$, so taking $L = p^* \co$ on $Y$, with the induced linearization, yields $W^\pm_Y = W^\pm_Z$. On the other hand, $N^\pm_Y = \pi^* N^\pm_X$ also, so for another choice of $L$ on $Y$, $W^\pm_Y = W^\pm_X$. Hence $\Pj W^\pm_X \cong \Pj W^\pm_Z$, and the line bundles $\co(1,1) \to \Pj W^+_X \times_{X^0 \mod G(0)} \Pj W^-_X$ and $\co(1,1) \to \Pj W^+_Z \times_{Z^0} \Pj W^-_Z$ correspond under this isomorphism; pushing down and taking duals, the bundles $\pi_\pm^* W^\mp_X (-1) \to \Pj W^\pm_X$ and $\pi_\pm^* W^\mp_Z (-1) \to \Pj W^\pm_Z$ also correspond. On the other hand, by \re{1u}(a) $X^\pm \mod G(\pm) = Z^\pm \mod T(\pm)$. The theorem then follows from \re{2k} together with \re{1e}. \fp \bl The hypothesis on $w_i^\pm$ is most easily verified as follows. \begin{s}{Proposition} \label{1z} If every 0-dimensional stabilizer is trivial near $x$, then all $w_i^\pm = \pm 1$. \end{s} \pf. If not all $w_i^\pm = \pm 1$, then by \re{2e}(b) there is a point $z \in Z$ with proper nontrivial stabilizer $T_z$ such that $p(\pi^{-1}(x))$ is in the closure of $T \cdot z$. Then any $y \in p^{-1}(z)$ has nontrivial 0-dimensional stabilizer $(G \times T)_y$, and the closure of $(G \times T) \cdot y$ contains $\pi^{-1}(x)$. But then $G_{\pi(y)} \cong (G \times T)_y$, and the closure of $G \cdot \pi(y)$ contains $x$. \fp \bl For most applications, the hypothesis \re{1y} will hold for all $x \in X^0$. Then the conclusions of \re{2n} and \re{1x} hold globally, because they are natural. Notice, however, that if $X^0$ is not connected, then the $w_i^\pm$ need be only locally constant. \bit{The smooth case: \'etale results} If $(v_+, v_-) \neq 1$, however, then the proof of \re{1v} fails, and $X^\pm \mod G(\pm)$ need not be locally trivial over $X^0 \mod G(0)$, even if \re{1y} holds: see \re{2p} for a counterexample. But they will be locally trivial in the \'etale topology. The proof uses the Luna slice theorem. The first step, however, is to check that $v_\pm$ are always nonzero. As always, let $L_\pm$ and $L_0$ be as in the set-up of \S3. \begin{s}{Lemma} \label{1t} If \re{1y} holds, then $G_x$ acts nontrivially on $(L_\pm)_x$. \end{s} \pf. If $G_x$ acts trivially on $L_+$ (and hence on $L_-$), then the embedding $\kst = \pi^{-1}(x) \subset Y$ descends to an embedding $\kst \subset Z$. This is completed by two points, which must come from two equivalence classes of $G$-orbits in $Y^{ss}(G)$. These semistable orbits cannot be $G \cdot (i_\pm(x))$, since $i_\pm(x) \notin Y^{ss}(G)$ by \re{1j}(a) and the definition of $X^0$. Hence our two classes of semistable $G$-orbits must be contained in $\pi^{-1}(\overline{G \cdot x} \sans G \cdot x)$. By \re{1j}(b) and (c) their images in $\pi$ are in $X^{ss}(0)$. But they are also in the closure of $G \cdot x$, which contradicts \re{1y}. \fp Again, requiring $G_x$ to act on $(L_+)_x$ with negative weight $v_+ < 0$ fixes an isomorphism $G_x \cong \kst$. It then acts on $(L_-)_x$ with positive weight $v_- > 0$. We no longer require $(v_+, v_-) = 1$, but assume instead the following. \begin{r}{Hypothesis} \label{2s} Suppose that either $\chr \k = 0$ or $(v_+, v_-)$ is coprime to $\chr \k$. \end{r} Now choose $y \in \pi^{-1}(x)$, and let $S = (G \times T)_y$. \begin{s}{Lemma} \label{1o} If \re{1y} holds, then there is a fixed isomorphism $S \cong \kst$, and $G_y = S \cap G$ is a proper subgroup such that $S / G_y \cong T$ naturally. Moreover, if \re{2s} holds, then $L_\pm$ may be chosen so that $G_y$ is reduced. \end{s} \pf. Since $\pi$ is the quotient morphism for the $T$-action, $S \subset G_x \times T = \kst \times T$; indeed, it is the subgroup acting trivially on $\pi^{-1}(x)$. Since by \re{1t} $\kst$ acts nontrivially, and $T$ obviously acts with weight 1, this has a fixed isomorphism to $\kst$, and its intersection with $\kst \times 1$ is a proper subgroup having the desired property. If $\chr \k = 0$, then $G_y$ is certainly reduced. Otherwise, let $v = (v_+, v_-)$, and replace $L_+$ and $L_-$ with positive powers so that $G_x \cong \kst$ acts on $\pi^{-1}(x) = (L_+ \otimes L_-^{-1})_x$ with weight $v$. Then $G_y = \Spec \k[z]/ \langle z^v - 1 \rangle$, which is reduced if $v$ is coprime to $\chr \k$. \fp \bl Notice that for any $y \in \pi^{-1}(x)$, $(G \times T) \cdot y = G \cdot y$. Together with lemma \re{1o}, this implies that any $S$-invariant complement to $T_y (G \times T) \cdot y$ in $T_y Y$ is also a $G_y$-invariant complement to $T_y (G \cdot y)$ in $T_y Y$. It follows from the definition of the Luna slice \cite{luna,mf} that a slice for the $(G \times T)$-action at $y$ is also a slice for the $G$-action at $y$. Luna's theorem then implies that there exists a smooth affine $U \subset Y$ containing $y$ and preserved by $S$, and a natural diagram $$\begin{array}{ccccc} G \times U & \lrow & G \times_{G_y} U & \lrow & Y^{ss}(G) \vspace{.7ex} \\ \down{} & & \down{} & & \down{} \vspace{.7ex} \\ U & \lrow & U / G_y & \lrow & Z, \end{array} $$ such that the two horizontal arrows on the right are strongly \'etale with respect to the actions of $$\begin{array}{ccccc} G \times S &\lrow & G \times T & \lrow & G \times T \vspace{.7ex} \\ \down{} & & \down{} & & \down{} \vspace{.7ex} \\ S &\lrow & T & \lrow & T. \end{array} $$ These actions are obvious in every case except perhaps on $G \times U$; there the $G \times S$-action is given by $(g, s)\cdot(h,u) = (gh\hat{s}^{-1}, s u)$, where $\hat{s}$ is the image of $s$ in the projection $S \to G$. Each of these actions has a 1-parameter family of fractional linearizations, pulled back from the right-hand column. For any object $V$ in the diagram, define $V^0$ and $V^\pm$ with respect to these linearizations. \begin{s}{Lemma} \label{1k} For every arrow $f: V \to W$ in the diagram, $f^{-1}(W^\pm) = V^\pm$ and $f^{-1}(W^0) = V^0$. \end{s} \pf. This is straightforward for the vertical arrows, and for the morphism $U \to U / G_y$, because they are all quotients by subgroups of the groups which act. The result for $G \times U \lrow G \times_{G_y} U$ follows from the commutativity of the diagram. As for the strongly \'etale morphisms, these are treated as in the proof of \re{2e}(b). \fp \bl With this construction, \re{1w} can now be strengthened. \begin{s}{Proposition} If \re{1y} and \re{2s} hold, then the conclusions of \re{1w} follow even if $(v_+, v_-) \neq 1$. \end{s} \pf. By \re{1o} and \re{1k}, in a neighbourhood of $\pi^{-1}(x)$ in $Y$, a point is in $Y^0$ if and only if it has stabilizer conjugate to $S$. Hence in a neighbourhood of $x$ in $X$, a point is in $X^0$ if and only if it has stabilizer conjugate to $G_x$. So if $W \subset X^0$ is the fixed-point set for the $G_x$-action on a neighbourhood of $x \in X$, then a neighbourhood of $x$ in $X^0$ is precisely the affine quotient of $W \times G$ by the diagonal action of the normalizer of $G_x$ in $G$. In particular, this is smooth as claimed in (a), because $W$ is smooth by \re{2e}(a) and the normalizer acts freely. Also, its tangent bundle has a natural subbundle consisting of the tangent spaces to the $G$-orbits. Let $N_X$ be the quotient of $TX|_{X^0}$ by this subbundle. Then $N_X$ certainly satisfies (b), and $N_Y = \pi^* N_X$, so by it suffices to prove (c) and (d) for $Y$ with its $G \times T$-action. But (c) and (d) hold for $U$ with its $S$-action by \re{2e}(c) and (d), hence for $G \times U$ with its $G \times S$-action since $N_U$ pulls back to $N_{G \times U}$. But the morphism $G \times_{G_y} U \to Y$ is \'etale, and so is the morphism $G \times U \to G \times_{G_y} U$, since $G_y$ is reduced. The result for $Y$ then follows from \re{1k}, since \'etale morphisms are isomorphisms on tangent spaces. \fp \bl Let $w_i^\pm \in \Z$ be the weights of the $\kst$-action on $N^\pm$. \begin{s}{Theorem} \label{1l} If \re{1y} and \re{2s} hold, then over a neighbourhood of $x$ in $X^0 \mod G(0)$, $X^\pm \mod G(\pm)$ are fibrations, locally trivial in the \'etale topology, with fibre the weighted projective space $\Pj(|w_i^\pm|)$. \end{s} As before, if $X \mod G(\pm)$ are both nonempty, then $X^\pm \mod G(\pm)$ are the supports of the blow-up loci of \re{1g}; but if $X \mod G(-) = \emptyset} \def\dasharrow{\to$, then \re{1l} says the natural morphism $X \mod G(+) \to X \mod G(0)$ is a weighted projective fibration, locally trivial in the \'etale topology. The proof requires the following lemma. \begin{s}{Lemma} \label{1m} If $\phi: V \to W$ is a strongly \'etale morphism of affine varieties with $\kst$-action, then $\phi \mod \pm : V \mod \pm \to W \mod \pm$ are \'etale, and $V \mod \pm = {W \mod \pm} \times_{W \mod 0} {V \mod 0}$. \end{s} \pf. Say $V = \Spec R$, $W = \Spec S$. The $\kst$-actions induce $\Z$-gradings on $R$ and $S$, and $V = W \times_{W \mod 0} V \mod 0$ implies $R = S \otimes_{S_0} R_0$. Hence $\bigoplus_{i \in \N} R_{\pm i} = \bigoplus_{i \in \N} S_{\pm i} \otimes_{S_0} R_0$, which implies the second statement. Then $\phi \mod \pm$ are certainly \'etale, since being \'etale is preserved by base change. \fp \pf\ of \re{1l}. The Luna slice $U$ associated to any $y \in \pi^{-1}(x)$ is smooth, and for $S = (G \times T)_y$, $U \mod S(t) = (U / G_y) \mod T(t)$ since $T = S / G_y$ by \re{1o}. But as stated when $U$ was constructed, $U / G_y$ is strongly \'etale over $Z$, so by \re{1m} and \re{1e} $U \mod S(t)$ is \'etale over $Z \mod T(t) = X \mod G(t)$, and $U \mod S(\pm) = X \mod G(\pm) \times_{X \mod G(0)} U \mod S(0)$. In particular, $U^0 \mod S(\pm) = X^0 \mod G(\pm) \times_{X^0 \mod G(0)} U^0 \mod S(0)$, which is exactly the pullback of $X^0 \mod G(\pm)$ by the \'etale morphism $U^0 \mod S(0) \to X^0 \mod G(0)$. The theorem therefore follows from the analogous result \re{2j} for quotients of smooth affines by $\kst$. \fp \begin{s}{Counterexample} \label{1r} To show that the hypothesis $G_x \cong \kst$ is necessary in \re{1l}. \end{s} Let $G$ be any semisimple reductive group, and let $V_+$ and $V_-$ be representations of $G$. Let $X = \Pj (V_+ \oplus V_- \oplus \k)$, and let $G \times \kst$ act on $X$, $G$ in the obvious way, and $\kst$ with weights $1,-1,0$. Then $\NS_\Q^{G \times \kst} \cong \Q$, with two chambers separated by a wall at 0. Moreover $X^\pm = \Pj(V_\pm \oplus \k)$, so $X^0 = \{ (0,0,1) \}$. But $G_{(0,0,1)} = G \times \kst$, so the hypothesis is violated. Now $X^\pm \mod \kst (\pm) = \Pj (V_\pm)$, so $X^\pm \mod (G \times \kst)(\pm) = \Pj (V_\pm) \mod G$. This certainly need not be a projective space, as the theorem would predict; see for example the discussion of the case $G = {\rm PSL(2)}$ in \S6. \fp \bl Since $G$ acts on $N^\pm$, there are quotients $N^\pm \mod G(\pm)$, which are fibrations with fibre $\Pj(|w_i^\pm|)$ over a neighbourhood of $x$ in $X^0 \mod G(0)$, locally trivial in the \'etale topology. Notice that by \re{1k}, since $N^\pm_V = N_{V^0 / V^\pm}$ for $V = X$, $Y$, $G \times_{G_y} U$, and $U$, \beqas N^\pm_U \mod S(\pm) & = & N^\pm_{G \times U} \mod (G \times S)(\pm) \\ & = & N^\pm_Y \mod (G \times T)(\pm) \times_{Y^0 \mod (G \times T)(0)} (G \times U)^0 \mod (G \times S)(0) \\ & = & N^\pm_X \mod G(\pm) \times_{X^0 \mod G(0)} U^0 \mod S(0), \eeqas which is the pullback of $N^\pm_X \mod G(\pm)$ by the \'etale morphism $U^0 \mod S(0) \to X^0 \mod G(0)$. The following result then ought to be true, but proving it conclusively is rather cumbersome, so we content ourselves with a sketch. \begin{s}{Theorem} \label{1n} Suppose that \re{1y} and \re{2s} hold, and that all $w_i^\pm = \pm w$ for some $w$. Then $X^\pm \mod G(\pm)$ are naturally isomorphic to $N^\pm \mod G(\pm)$, and the blow-ups of $X \mod G(\pm)$ at $X^\pm \mod G(\pm)$, and of $X \mod G(0)$ at $X^0 \mod G(0)$, are all naturally isomorphic to the fibred product $X \mod G(-) \times_{X \mod G(0)} X \mod G(+)$. \end{s} {\em Sketch of proof}. All the blow-ups and the fibred product are empty if either $X \mod G(+)$ or $X \mod G(-)$ is empty, so suppose they are not. Now $X^\pm \mod G(\pm) \to X^0 \mod G(0)$ are covered in the \'etale topology by $U^\pm \mod S(\pm) \to U^0 \mod S(0)$ by \re{1m}, and $N^\pm_X \mod G(\pm) \to X^0 \mod G(0)$ are covered in the \'etale topology by $N^\pm_U \mod S(\pm) \to U^0 \mod S(0)$ by the remarks above. But the analogous result for $U$ holds by \re{2k}. The theorem would therefore follow if we could display a morphism $N^\pm_X \mod G(\pm) \to X^\pm \mod G(\pm)$ compatible with the \'etale morphisms and the isomorphisms of \re{2k}. Unfortunately, this is somewhat awkward to construct. One way to do it is to imitate the argument of \re{1x}, using a bundle of tangent cones with $\kst$-action over $Z^0$, which is typically in the singular locus of $Z$. This requires generalizing the Bialynicki-Birula decomposition theorem to the mildly singular space $Z$, which can still be accomplished using the Luna slice theorem. \fp \bl The hypothesis on $w_i^\pm$ is again most easily verified as follows. \begin{s}{Proposition} If every 0-dimensional stabilizer is trivial near $x$, then all $w_i^\pm = \pm 1$. \end{s} \pf. If not all $w_i^\pm = \pm 1$, then for each Luna slice $U$ there is a point $u \in U$ with nontrivial proper stabilizer $S_u$. Then any $(g,u) \in G \times U$ satisfies $(G \times S)_{(g,u)} \cong S_u$. Since the morphism $G \times U \to Y$ is \'etale, this implies that there exists $y \in Y$ with a nontrivial 0-dimensional stabilizer $(G \times T)_y$. But then $G_{\pi(y)} = (G \times T)_y$. \fp \bl Again, for most applications, the hypothesis \re{1y}, and hence the conclusions of \re{1l} and \re{1n}, will hold globally. \bit{The first example} In this section we turn to a simple application of our main results, the much-studied diagonal action of $\PSL{2}$ on the $n$-fold product $(\Pj^1)^n$. This has $n$ independent line bundles, so it is tempting to study the quotient with respect to an arbitrary $\co(t_1, \dots, t_n)$. We will take a different approach, however: to add an $n+1$th copy of $\Pj^1$, and consider only fractional linearizations on $(\Pj^1)^{n+1}$ of the form $\co(t,1,1,\dots,1)$. This has the advantage that it does not break the symmetry among the $n$ factors. In other words, the symmetric group $S_n$ acts compatibly on everything, so in addition to $(\Pj^1)^n$, we learn about quotients by $\PSL{2}$ of the symmetric product $(\Pj^1)^n / S_n = \Pj^n$. So for any $n>2$, let $(\Pj^1)^n$ be acted on diagonally by $G = \PSL{2}$, fractionally linearized on $\co(1,1,\dots,1)$. We wish to study the quotient $(\Pj^1)^n \mod G$. The stability condition for this action is worked out in \cite[4.16; GIT Ch.\ 3]{n}, using the numerical criterion. This is readily generalized to an arbitrary linearization on $X = \Pj^1 \times (\Pj^1)^n = (\Pj^1)^{n+1}$; indeed for the fractional linearization $\co(t_0, \dots, t_n)$, it turns out that $(x_j) \in X^{ss}(t_j)$ if and only if, for all $x \in \Pj^1$, $$\sum_{j=0}^n t_j \, \delta(x,x_j) \leq \sum_{j=0}^n t_j / 2.$$ Moreover, $(x_j) \in X^s(t_j)$ if and only if the inequality is always strict. We will study the case where $t_0$ is arbitrary, but $t_j = 1$ for $j>0$. For $t_0 < 1$, it is easy to see that $\Pj^1 \times ((\Pj^1)^n)^s \subset X^{ss} \subset \Pj^1 \times ((\Pj^1)^n)^{ss}$. So the projection $X \to (\Pj^1)^n$ induces a morphism $X \mod G(t_0) \to (\Pj^1)^n \mod G$ whose fibre over each stable point is $\Pj^1$. Indeed, each diagonal in $X = \Pj^1 \times (\Pj^1)^n$ is fixed by $G$, so descends to $X \mod G(t_0)$. Hence, over the stable set in $(\Pj^1)^n \mod G$, $X \mod G(t_0)$ is exactly the total space of the universal family. Now because $G = \PSL{2}$, not $\SL{2}$, the bundle $\co(1,0,0,\dots,0)$ has no bona fide linearization, only a fractional one. However, $\co(1,1,0,\dots,0)$ does admit a bona fide linearization, as does $\co(1,1,\dots,1)$ if $n$ is odd. So these bundles descend to $X \mod G(t_0)$ for $t_0 < 1$, yielding line bundles whose restriction to each $\Pj^1$ fibre is $\co(1)$. This implies that, over the stable set in $(\Pj^1)^n \mod G$, $X \mod G(t_0)$ is a locally trivial fibration. In particular, if $n$ is odd, it is a fibration everywhere. However, if $n$ is even, there is no $S_n$-invariant line bundle having the desired property. Hence the quotient $(\Pj^1 \times \Pj^n) \mod G(t_0) = (X \mod G(t_0)) / S_n$, though it has generic fibre $\Pj^1$ over $\Pj^n \mod G$, and is generically trivial in the \'etale topology, is not even locally trivial anywhere. It is (generically) what is sometimes called a {\em conic bundle}. To apply our results to this situation, note first that for numerical reasons the stability condition only changes when equality can occur in the inequality above, that is, when $t_0 = n - 2m$ for some integer $m \leq n/2$. These will be our walls. A point $(x_j) \in X$ is in $X^0$ for one of these walls if it is semistable for $t = t_0$, but unstable otherwise. This means there exist points $x, x' \in \Pj^1$ such that $$t \, \delta(x, x_0) + \sum_{j=1}^n \delta(x,x_j) \geq \frac{t+n}{2} $$ for $t \leq t_0$, and $$t \, \delta(x', x_0) + \sum_{j=1}^n \delta(x',x_j) \geq \frac{t+n}{2} $$ for $t \geq t_0$, with equality in both if and only if $t = t_0$. This requires that all $x_j$ be either $x$ or $x'$; indeed, $x_0$ and exactly $m$ other $x_j$ must be $x$, and the $n-m$ remaining $x_j$ must be $x'$. In particular, this implies $m \geq 0$, so there are only finitely many walls, as expected. On the other hand, any $(x_j) \in X$ of this form will belong to $X^{ss}(t_0)$, provided that $x \neq x'$. So $X^0(t_0)$ consists of ${n \choose m}$ copies of $(\Pj^1 \times \Pj^1) \sans \Delta$. Hence every point $(x_j) \in X^0(t_0)$ is stabilized by the subgroup of $G$ fixing $x$ and $x'$, which is isomorphic to $\kst$. So the hypothesis \re{1y} is satisfied. Moreover, the bundle $\co(2,0,0,\dots,0)$ is acted on by this $\kst$ with weight 1, so the strong results of \S4 will apply. Finally, we claim that, even though $X^0$ is disconnected, the weights $w_i^\pm$ are globally constant, and are all $\pm 1$. Indeed, it is easy to see that the $w_i^\pm$ are independent of the component, because the action of the symmetric group $S_n$ on $X$ commutes with the $G$-action, and acts transitively on the components of $X^0$. To evaluate $w_i^\pm$, note that each component is a single orbit, and that setting $x=0$, $x' = \infty$ determines an unique point in this orbit with stabilizer $\{ \diag (\lambda^{-1}, \lambda) \st \lambda \in \kst \} / \pm 1$. This acts on $T\Pj^1$ with weight $-1$ at 0, $1$ at $\infty$; so it acts on $T_{(x_j)}X$ with $m+1$ weights equal to $-1$ and $n-m$ weights equal to $1$. But it acts on the $G$-orbit $G / \kst$ with one weight equal to 1 and one equal to $-1$, so $N$ is acted on with $m$ weights $-1$ and $n-m-1$ weights $1$. So the very strongest result \re{1x} applies. Hence $X^\pm \mod G(\pm)$ are bundles with fibre $\Pj^{n-m-2}$ and $\Pj^{m-1}$, respectively, over $X^0 \mod G(0)$. Since this is just ${n \choose m}$ points, $X^\pm \mod G(\pm)$ are disjoint unions of ${n \choose m}$ projective spaces. Moreover, the blow-ups of $X \mod G(\pm)$ at $X^\pm \mod G(\pm)$ are both isomorphic to $X \mod G(-) \times_{X \mod G(0)} X \mod G(+)$. This does not seem to say much about $(\Pj^1)^n \mod G$ itself, only about $X \mod G(t)$, which for $t$ small is (at least generically) a $\Pj^1$-bundle over it. But this is enough to compute quite a lot (cf.\ \cite{t1}). We content ourselves with just one calculation, of the Betti numbers of $(\Pj^1)^n \mod G$ and $\Pj^n \mod G$ for $n$ odd, in the case where the ground field is the complex numbers $\C$. These formulas are originally due to Kirwan \cite{k}. \begin{s}{Proposition} For $n$ odd, $$P_t ((\Pj^1)^n \mod G) = \sum_{m=0}^{(n-1)/2} {n \choose m} \frac{t^{2m} - t^{2(n-m-1)}}{1-t^4} $$ and $$P_t (\Pj^n \mod G) = \sum_{m=0}^{(n-1)/2} \frac{t^{2m} - t^{2(n-m-1)}}{1-t^4}. $$ \end{s} \pf. Let $t_0 = n - 2m$, and $t_\pm = t_0 \mp 1$. Then the blow-ups of $X \mod G(\pm)$ at $X^\pm \mod G(\pm)$ are equal. So by the standard formula for Poincar\'e polynomials of blow-ups, $$ P_t(X \mod G(-)) - P_t(X^- \mod G(-)) + P_t(E) = P_t(X \mod G(+)) - P_t(X^+ \mod G(+)) + P_t(E), $$ where $E$ is the exceptional divisor. Cancelling and rearranging yields $$ P_t(X \mod G(+)) - P_t(X \mod G(-)) = P_t(X^+ \mod G(+)) - P_t(X^- \mod G(-)). $$ But $X^\pm \mod G(\pm)$ are ${n \choose m}$ copies of $\Pj^{n-m-2}$ and $\Pj^{m-1}$ respectively, so \beqas P_t(X \mod G(+)) - P_t(X \mod G(-)) & = & {n \choose m} \left( \frac{1-t^{2(n-m-1)}}{1-t^2}-\frac{1-t^{2m}}{1-t^2} \right) \\ & = & {n \choose m} \frac{t^{2m}-t^{2(n-m-1)}}{1-t^2}. \eeqas Summed over $m$, the left-hand side telescopes, so for $t<1$ $$P_t(X \mod G(t)) = \sum_{m=0}^{(n-1)/2} {n \choose m} \frac{t^{2m}-t^{2(n-m-1)}}{1-t^2}. $$ But for $n$ odd, this is a $\Pj^1$-bundle over $(\Pj^1)^n \mod G$, and the Poincar\'e polynomial of any projective bundle splits, so $$P_t((\Pj^1)^n \mod G) = \sum_{m=0}^{(n-1)/2} {n \choose m} \frac{t^{2m}-t^{2(n-m-1)}}{1-t^4}, $$ as desired. As for $\Pj^n \mod G$, it is the quotient of $(\Pj^1)^n \mod G$ by the action of the symmetric group $S_n$. A result from Grothendieck's T\^ohoku paper then implies \cite{gr,mac} that $H^*(\Pj^n \mod G, \C)$ is the $S_n$-invariant part of $H^*((\Pj^1)^n \mod G, \C)$. But since $S_n$ acts on $X \mod G(t)$ for all $t$, the calculation above actually decomposes $H^*((\Pj^1)^n \mod G, \C)$ as a representation of $S_n$: the term with coefficient ${n \choose m}$ gives the multiplicity of the permutation representation induced by the natural action of $S_n$ on subsets of $\{ 1, \dots, n\}$ of size $m$. The trivial summand of this representation is exactly one-dimensional, so the cohomology of $\Pj^n \mod G$ is as stated. \fp \bl We round off this section by using some of the ideas discussed above to give the counterexample promised in \S5. \begin{s}{Counterexample} \label{2p} To show that the hypothesis $(v_+, v_-) = 1$ is necessary in \re{2n}. \end{s} Let $V$ be the standard representation of $\GL{2}$, and let $W = S^n V \otimes (\Lambda^2 V)^{-n/2}$, where $S^n V$ is the $n$th symmetric power for some even $n>2$. Let $X = \Pj (V \oplus V^* \oplus W)$, and let $\GL{2}$ act on $X$. Then $\NS_\Q^{\GL{2}} \cong \Q$, with two chambers separated by a wall at 0. The central $\kst \subset \GL{2}$ acts on $V$, $V^*$, and $W$ with weight $1$, $-1$, and $0$ respectively, so $X^+$ is open in $\Pj(V \oplus W)$, $X^-$ is open in $\Pj(V^* \oplus W)$, and $X^0$ is open in $\Pj W$. By construction, the scalars $\kst \subset \GL{2}$ act trivially on $\Pj W$ with the linearization $L_0$, so the action reduces to the action of $\PSL{2}$ on $\Pj^n$ considered above. A generic $x \in \Pj^n$ is stable and is acted on freely by $\PSL{2}$, so $\GL{2}_x = \kst$. Moreover, it is stable, so $\GL{2} \cdot x = \PSL{2} \cdot x$ is closed in $(\Pj^n)^{ss}$ and hence in $X^{ss}(0)$. Therefore \re{1y} holds for the $\GL{2}$-action at $x$. On the other hand, the hypothesis $(v_+, v_-) = 1$ cannot be satisfied: the tautological linearization $L_0$ on $\co(1)$ is acted on with weight 0, and the linearization $L_+$ obtained by tensoring $L_0$ with the character $\det: \GL{2} \to \kst$ is acted on with weight $2$, but together these generate $\Pic^\GL{2} X$. Now $X^+ \mod \kst (+) = \Pj V \times \Pj W$. As a variety with $\PSL{2}$-action, this is exactly $\Pj^1 \times \Pj^n$ as considered above. The $+$ linearization on $\Pj V \times \Pj W$ corresponds to the linearization given by $t<1$ on $\Pj^1 \times \Pj^n$, so $X^\pm \mod \GL{2}(+) = (\Pj V \times \Pj W) \mod \PSL{2}(+) = (\Pj^1 \times \Pj^n) \mod \PSL{2}(t)$. As mentioned above, this is a conic bundle, so it is not even locally trivial over $X^0 \mod \GL{2}(0) = \Pj W \mod \PSL{2}(0) = \Pj^n \mod \PSL{2}$ at $x$. \fp \bit{Parabolic bundles} In the last two sections we apply our main results to moduli problems of vector bundles with additional structure over a curve. Throughout these sections, $C$ will denote a smooth projective curve over $k$, of genus $g>0$. Fix a point $p \in C$. In this section we will study parabolic bundles of rank $r$ and degree $d$ over $C$, with parabolic structure at $p$. We refer to \cite{ms,s} for basic definitions and results on parabolic bundles. However, we insist for simplicity on full flags at $p$, so the weights $\ell_j \in [0,1)$ are strictly increasing. The space of all possible weights is therefore parametrized by $$W = \{ (\ell_j) \in \Q^r \st 0 \leq \ell_1 < \ell_2 < \cdots < \ell_r < 1 \}.$$ There are several constructions of the moduli space $M(\ell_j)$ of parabolic bundles semistable with respect to $(\ell_j)$. The one best suited for our purposes is due to Bhosle \cite{bho}, following Gieseker \cite{g}; so we first review his construction, then hers. Suppose without loss of generality that $d > > 0$, and let $\chi = d + r(1-g)$. Let $\Quot$ be the Grothendieck Quot scheme \cite{quot} parametrizing quotients $\phi: \co_C^\chi \to E$, where $E$ has Hilbert polynomial $\chi +ri$ in $i$, and let $\co^\chi \to \be$ be the universal quotient over $\Quot \times C$. Let $R \subset \Quot$ be the smooth open subvariety consisting of locally free sheaves $E$ such that $H^0(\co^\chi) \to H^0(E)$ is an isomorphism, and let $R^{ss}$ be the subset corresponding to semistable bundles. For $d$ large, every semistable bundle of rank $r$ and degree $d$ is represented by a point in $R$. Let $Z$ be the bundle over $\Pic^d C$, constructed as a direct image, with fibre $\Pj \Hom (H^0(\La^r \co^\chi), H^0(M))$ at $M$. The group $G = \SL{\chi}$ acts on $R$ and $Z$, and there is a natural $G$-morphism $T: R \to Z$, and a linearization $L$ on $Z$, such that $T^{-1}Z^{ss}(L) = R^{ss}$. Moreover, the restriction $T: R^{ss} \to Z^{ss}(L)$ is finite. The existence of a good quotient of $R^{ss}$ by $G$ then follows from a lemma \cite[Lemma 4.6]{g} which states that if a set has a good $G$-quotient, then so does its preimage by a finite $G$-morphism. This quotient is the moduli space of semistable bundles on $C$. To construct the moduli space of semistable parabolic bundles in an analogous way, let $\tilde{R}$ be the bundle $\Fl \be|_{R \times \{ p \} }$ of full flags in $\be_p$. This parametrizes a family of quasi-parabolic bundles; for $d$ large, any bundle which is semistable for some weights $(\ell_j)$ is represented by a point in $\tilde{R}$. Let $\tilde{R}^{ss}(\ell_j)$ be the subset corresponding to parabolic bundles semistable with respect to $(\ell_j)$. Also let $G_r$ be the product of Grassmannians $\prod_{j=1}^r \Gr(\chi - j,\chi)$. Then $G$ acts on $\tilde{R}$, $Z$, and $G_r$, and there is a $G$-morphism $\tilde{T}: \tilde{R} \to Z \times G_r$, and a family $L(\ell_j)$ of fractional linearizations on $Z \times G_r$ depending affinely on $(\ell_j)$, such that $\tilde{T}^{-1}(Z \times G_r)^{ss}(\ell_j) = \tilde{R}^{ss}(\ell_j)$. Moreover, the restriction $\tilde{T}: \tilde{R}^{ss}(\ell_j) \to (Z \times G_r)^{ss}(\ell_j)$ is finite. The existence of a good quotient $M(\ell_j)$ again follows from the lemma. In fact, we can say more. \begin{s}{Proposition} \label{2u} $T$ is an embedding. \end{s} \pf. Since $T$ is injective \cite[4.3]{g}, it suffices to show its derivative is everywhere injective. At a quotient $\phi: \co^\chi \to E$, the tangent space to $\Quot$ is given by the hypercohomology group $\Hyp^1(\End E \stackrel{\phi}{\to} E \otimes \co^\chi)$ (cf.\ \cite{bdw,t1}). Since $H^1(E \otimes \co^\chi) = \k^\chi \otimes H^1(E) = 0$, this surjects onto $H^1(\End E)$, and hence onto $H^1(\co)$, which is the tangent space to $\Pic^d C$. So it suffices to show the kernel of this surjection injects into the tangent space to $\Pj \Hom(H^0(\La^r \co^\chi), H^0(\La^r E))$. The kernel is isomorphic to the quotient of $\Hyp^1(\End_0 E \stackrel{\phi}{\to} E \otimes \co^\chi)$ by the 1-dimensional subspace generated by $\phi$ (cf.\ \cite[2.1]{t1}), where $\End_0$ denotes trace-free endomorphisms. So the kernel injects as desired if and only if the natural map $\Hyp^1(\End E \stackrel{\phi}{\to} E \otimes \co^\chi) \to \Hom(H^0(\La^r \co^\chi), H^0(\La^r E))$ is injective as well. What is this natural map? It is obtained from the derivative of $T$; since $T$ is essentially $\phi \mapsto \La^r \phi$, the derivative of $T$ at $\phi$ is essentially $\psi \mapsto (\La^{r-1} \phi) \wedge \psi$. More precisely, an element of the hypercohomology group above is determined by \v Cech cochains $g \in C^1(\End_0 E)$ and $\psi \in C^0(E \otimes \co^\chi)$ such that $g \phi = d\psi$. Since $\phi$ is surjective, the hypercohomology class of the pair is uniquely determined by $\psi$; on the other hand, a cochain $\psi$ determines the trivial hypercohomology class if and only if $\psi = f \phi$ for some $f \in C^0(\End_0 E)$. The natural map to $\Hom(H^0(\La^r \co^\chi), H^0(\La^r E))$ is then indeed given by $\psi \mapsto (\La^{r-1} \phi) \wedge \psi$; its injectivity follows from the lemma below. \fp \begin{s}{Lemma} If $\phi: \k^\chi \to \k^r$ is a linear surjection of vector spaces, and $\psi: \k^\chi \to \k^r$ is a linear map, then $(\La^{r-1} \phi) \wedge \psi = 0$ if and only if $\psi = f \phi$ for some $f \in \End_0 \k^r$. \end{s} \pf. Suppose first that $\chi = r$. Then $\phi$ is invertible, and $(\La^{r-1} \phi) \wedge \psi$ is a homomorphism of 1-dimensional vector spaces. Indeed, if $e_1, \dots, e_r$ is the standard basis for $\k^r$, then $\La^r \k^\chi$ is spanned by $\phi^{-1}e_1 \wedge \cdots \wedge \phi^{-1}e_r$. But \beqas ((\La^{r-1} \phi) \wedge \psi)(\phi^{-1}e_1 \wedge \cdots \wedge \phi^{-1}e_r) & \!\!\!\!\!\!\!= & \!\!\!\!\!\!\!1/r \sum_i e_1 \wedge \cdots \wedge e_{i-1} \wedge \psi\phi^{-1}(e_i) \wedge e_{i+1} \wedge \cdots \wedge e_r \\ & \!\!\!\!\!\!\! = & \!\!\!\!\!\!\! 1/r \, (\tr \psi\phi^{-1})(e_1 \wedge \cdots \wedge e_r), \eeqas so the lemma is true when $\chi = r$. In the general case, $(\La^{r-1} \phi) \wedge \psi = 0$ implies $\ker \psi \supset \ker \phi$, for if not, let $\phi^{-1}$ be a right inverse for $\phi$, and let $u \in \ker \phi \sans \ker \psi$. For some $i$, the coefficient of $e_i$ in $\psi(u)$ is nonzero. Then \beqas ((\La^{r-1} \phi) \wedge \psi) (\phi^{-1}e_1 \wedge \cdots \wedge \phi^{-1}e_{i-1} \wedge u \wedge \phi^{-1}e_{i+1} \wedge \cdots \wedge \phi^{-1}e_r) & & \\ & \hspace{-60ex} = & \hspace{-31ex} e_i \wedge \cdots \wedge e_{i-1} \wedge \psi(u) \wedge e_{i+1} \wedge \cdots \wedge e_r \neq 0. \eeqas So $\psi$ descends to $\k^\chi / \ker \phi$; this has dimension $r$, so the case above applies. \fp \begin{s}{Corollary} \label{2t} $\tilde{T}$ is an embedding. \end{s} \pf. $\tilde{R}$ is a bundle of flag varieties over $R$, and each fibre clearly embeds in $G_r$. \fp Let $X$ be the Zariski closure in $Z \times G_r$ of $\tilde{T} (\tilde{R})$. \begin{s}{Corollary} The moduli space $M(\ell_j)$ of semistable parabolic bundles is $X \mod G(\ell_j)$. \end{s} \pf. Since $\tilde{T}^{-1}(Z \times G_r)^{ss}(\ell_j) = \tilde{R}^{ss}(\ell_j)$, this is automatic provided there are no semistable points in $X \sans \tilde{T}(\tilde{R})$. Since $M(\ell_j)$ is already projective, any such points would be in the orbit closures of semistable points in $\tilde{T}(\tilde{R})$. Hence there would be $x \in \tilde{R}^{ss}(\ell_j)$, and a 1-parameter subgroup $\la(t) \subset G$, such that $\lim \la(t) \cdot x \notin \tilde{T}(\tilde{R})$, but $\mu^{\ell_j} (\tilde{T}(x), \la) = 0$, where $\mu^{\ell_j}$ is the valuation used in the numerical criterion \cite[4.8; GIT Defn.\ 2.2]{n}. But all the destabilizing subgroups of points in $\tilde{T}(\tilde{R})$ correspond to destabilizing subbundles, and their limits are points corresponding to the associated graded subbundles; in particular, they are in $\tilde{T}(\tilde{R})$. \fp We are therefore in a position to apply our main results. Let us first look for walls and chambers. Notice that the stability condition only changes at values where there can exist subbundles whose parabolic slope equals that of $E$. If such a subbundle has rank $r^+$, degree $d^+$, and weights $\ell_{j^+_i}$ for some $ \{ j^+_i \} \subset \{ 1, \dots, r \}$, then the slope condition is $$ \frac{d^+ + \sum_{i = 1}^{r^+} \ell_{j^+_i}}{r^+} = \frac{d + \sum_{j = 1}^r \ell_j}{r}. $$ This determines a codimension 1 affine subset of $W$, which is one of our walls. The complementary numbers $r^- = r - r^+$, $d^- = d - d^+$, and $\{ \ell_{j^-_i} \} = \{ \ell_j \} \sans \{ \ell_{j^+_i} \}$ of course determine the same wall, but no other numbers do. Also, there are only finitely many walls, since for a given $r^+$ and $\{ \ell_{j^+_i} \}$, the affine hyperplane defined by the above equation only intersects $W$ for finitely many $d^+$. The connected components of the complement of the walls are the chambers; for purely numerical reasons, the semistability condition is constant in the interior of a chamber. Now, as in the set-up of \S3, suppose $(\ell_j)$ lies on a single wall in $W$, and choose $(\ell_j^+)$ and $(\ell_j^-)$ in the adjacent chambers such that the line segment connecting them crosses a wall only at $(\ell_j)$. Then $x \in X$ belongs to $X^0$ if and only if it represents a parabolic bundle which splits as $E_+ \oplus E_-$, where $E_\pm$ are $(\ell_{j^\pm_i})$-stable parabolic bundles. This is because, to be in $X^{ss}(0) \sans X^{ss}(+)$, a parabolic bundle must have a semistable parabolic subbundle $E_+$ of rank $r^+$, degree $d^+$, and weights $\ell_{j^+_i}$. Indeed, $E_+$ must be stable, for since $(\ell_j)$ lies on only one wall, $E_+$ can have no parabolic subbundle of the same slope. For the same reason $E / E_+$ must be stable. On the other hand, to be in $X^{ss}(0) \sans X^{ss}(-)$, $E$ must have an stable parabolic subbundle $E_-$ of rank $r^-$, degree $d^-$, and weights $\ell_{j^-_i}$. Since all the weights are distinct, $E_-$ cannot be isomorphic to $E_+$; so there is a nonzero map $E_- \to E / E_+$. By \cite[III Prop.\ 9(c)]{s}, this map must be an isomorphism, so $E$ splits as $E_+ \oplus E_-$. On the other hand, if $E_+$ and $E_-$ are any stable parabolic bundles with rank, degree, and weights as above, then $E_+ \oplus E_-$ is certainly represented in $X^0$. Hence $X^0 \mod G(0) = M(\ell_{j_i^+}) \times M(\ell_{j_i^-})$, the product of two smaller moduli spaces. It is now easy to verify the hypotheses of our strongest result \re{1x}. First, $X$ is smooth on $X^{ss}(\ell_j)$, hence at $X^0$. Second, for any $x \in X^0$, the stabilizer $G_x$ is the subgroup isomorphic to $\kst$ acting on $H^0(E_+)$ with weight $\chi^- / c$ and on $H^0(E_-)$ with weight $-\chi^+ / c$, where $c$ is the greatest common divisor $(\chi^+, \chi_-)$. This is because any $g \in \GL{\chi}$ stabilizing a point in $X^0$ induces an automorphism of $\be_x$, and vice versa, so for $x \in X^0$ there is an isomorphism $\GL{\chi}_x \cong \Par \Aut (E_+ \oplus E_-) = \kst \times \kst$; but only the automorphisms acting with the weights above correspond to special linear $g$. Third, if $L_j$ is the ample generator of $\Pic \Gr(\chi - j, \chi)$, then for $x \in X^0$, $G_x \cong \kst$ acts on $(L_j)_x$ with weight $(n_j^+ \chi^- - n_j^- \chi^+) / c$, where $n_j^\pm$ are the number of $j_i^\pm$ less than or equal to $j$. But since $\chi^+ / c$ and $\chi^- / c$ are coprime, so are these weights for some two values of $j$. Bhosle gives a formula for the linearization on $Z \times \Gr$ determined by $(\ell_j)$ in terms of the $L_i$; an easy argument using this formula shows that $(\ell_j^\pm)$ can be chosen within their chambers so that $G_x$ acts with coprime weights on the corresponding linearizations. Because all semistable parabolic bundles are represented by points in $X$, and because semistability is an open condition, the universal family of parabolic bundles is a versal family near any point $x \in X^0$. Moreover, two points in $X$ represent the same parabolic bundle if and only if they are in the same orbit. The normal bundle $N_{G \cdot x / X}$ to an orbit is therefore exactly the deformation space of the parabolic bundle. For $\be_x = E_+ \oplus E_-$ as above this is $H^1(X, \Par \End (E_+ \oplus E_-))$. The stabilizer $G_x$ acts with weight $r^\pm / (r^+, r^-)$ on $E_\pm$, so $N^0 = H^1(X, \Par \End E_+) \oplus H^1(X, \Par \End E_-)$, and $N^\pm = H^1(X, \Par \Hom (E_\mp, E_\pm))$. Moreover, every element in $N^\pm$ is acted on with weight exactly $\pm(\chi^+ + \chi_-)/ c$, so $N^\pm$ descend to vector bundles $W^\pm$ over $M(\ell_{j_i^+}) \times M(\ell_{j_i^-})$. Indeed, if $\be_\pm \to M(\ell_{j_i^\pm}) \times C$ are universal bundles, then $W^\pm = (R^1 \pi) \Par \Hom(\be_\mp, \be_\pm)$. Theorem \re{1x} then states that $\Pj W^\pm$ are the exceptional loci of the morphisms $M(\ell_j^\pm) \to M(\ell_j)$. This is the result of Boden and Hu \cite{bhu}. Moreover, \re{1x} asserts that the blow-ups of $M(\ell_j^\pm)$ at $\Pj W^\pm$, and of $M(\ell_j)$ at $M(\ell_{j_i^+}) \times M(\ell_{j_i^-})$, are all naturally isomorphic to the fibred product $M(\ell_j^-) \times_{M(\ell_j)} M(\ell_j^+)$. With the obvious modifications, the same techniques and results go through for bundles with parabolic structure at several marked points, or with degenerate flags. \bit{Bradlow pairs} The moduli spaces of Bradlow pairs on our curve $C$ can be studied in the same way. The role of the weights will be played by a positive parameter $\si \in \Q$. A {\em Bradlow pair} is a pair $(E, \phi)$ consisting of a vector bundle $E$ over $C$ and a nonzero section $\phi \in H^0(X,E)$. We refer to \cite{brad,bd,t1} for basic definitions and results on Bradlow pairs. As in \cite{t1}, we confine ourselves to the study of rank 2 pairs. In this case a Bradlow pair of degree $d$ is $\si$-{\em semistable} if for all line bundles $L \subset E$, $$\begin{array}{cl} \deg L \leq d/2 - \si & \mbox{if $\phi \in H^{0}(L)$ and} \\ \deg L \leq d/2 + \si & \mbox{if $\phi \not\in H^{0}(L)$.} \end{array}$$ It is $\si$-{\em stable} if both inequalities are strict. The moduli spaces $B_d(\si)$ of $\si$-semistable rank 2 pairs were constructed in \cite{t1}. In that paper, the determinant was fixed, but to parallel the discussion of parabolic bundles above we shall now allow arbitrary determinant. With that modification, the construction goes as follows. It suffices to construct $B_d(\si)$ for $d$ sufficiently large. This is because, for any effective divisor $D$, $B_d(\si)$ will be embedded in $B_{d+ 2|D|}(\si)$ as the locus where $\phi$ vanishes on $D$. So assume that $d/2 - \si > 2g-2$, and let $\chi = d + 2(1-g)$. Let $\Quot$, $R$ and $Z$ be as in \S7 above, and let $G = \SL{\chi}$ act diagonally on $R \times \Pj^{\chi-1}$. The hypothesis $d/2 - \si > 2g-2$ implies that every $\si$-semistable pair is represented by a point in $R \times \Pj^{\chi-1}$. Let $(Z \times \Pj^{\chi-1})^{ss}(\si)$ denote the semistable set with respect to the fractional linearization $\co(\chi + 2 \si, 4 \si)$, and let $(R \times \Pj^{\chi-1})^{ss}(\si)$ denote the $\si$-semistable set in the sense of the definition above. Then the natural $G$-morphism $T \times 1: R \times \Pj^{\chi-1} \to Z \times \Pj^{\chi-1}$ satisfies $T^{-1} (Z \times \Pj^{\chi-1})^{ss}(\si) = (R \times \Pj^{\chi-1})^{ss}(\si)$. Moreover, by \re{2u}, it is an embedding. So if $X$ denotes the Zariski closure of its image, then for reasons like those given in \S7, the moduli space $B_d(\si)$ is the geometric invariant theory quotient $X \mod G(\si)$, where $\si$ denotes the fractional linearization $T^* \co(\chi + 2 \si, 4 \si)$. So again our main results apply. The stability condition only changes for $\si \in d/2 + \Z$, so these points are the walls. Fix one such $\si$. Then $x \in X$ belongs to $X^0$ if and only if it represents a pair which splits as $L \oplus M$, where $\deg L = d/2 - \si$ and $\phi \in H^0(L)$. Indeed, a subbundle $L$ of degree $d/2 - \si$ is needed to violate the first semistability condition to the right of the wall, and a subbundle $M$ of degree $d/2 + \si$ is needed to violate the second semistability condition to the left. But since $\deg M = \deg E/L$, the map $M \to E/L$ is an isomorphism, so $E$ is split. On the other hand, for any pair $(L, \phi)$ with $\deg L = d/2 - \si$ and $\phi \in H^0(L) \sans 0$, and for any line bundle $M$ with $\deg M = d/2 + \si$, certainly $(L \oplus M, \phi \oplus 0)$ is represented in $X^0$. Hence $X^0 \mod G(0) = S^iC \times \Pic^{d-i} C$, where $i = d/2 - \si$ and $S^iC$ is the $i$th symmetric product. It is now easy to verify the hypotheses of our strongest result \re{1x}. First, $X$ is smooth at $X^{ss}(0)$, hence at $X^0$. Second, for any $x \in X^0$, the stabilizer $G_x$ is the subgroup isomorphic to $\kst$ acting on $H^0(L)$ with weight $\chi(M)/c$ and on $H^0(M)$ with weight $-\chi(L)/c$, where $c$ is the greatest common divisor $(\chi(L), \chi(M))$. This is because any $g \in \GL{\chi}$ stabilizing a point in $X^0$ induces an automorphism of the corresponding pair, and vice versa, but only the automorphisms acting with the weights above correspond to special linear $g$. Third, this stabilizer $\kst$ acts on $\co(1,0)_x$ with weight $(\chi(M) - \chi(L))/c$, and on $\co(0,1)_x$ with weight $\chi(M)/c$. These are coprime, so linearizations with coprime weights can be chosen within the chambers adjacent to $\si$. As in \S7, the normal bundle to an orbit is exactly the deformation space of the Bradlow pair. For any pair $(E, \phi)$, this is the hypercohomology group $\Hyp^1(\End E \stackrel{\phi}{\to} E)$. (See \cite{bd} or \cite[(2.1)]{t1}; the slightly different formula in \cite{t1} arises because the determinant is fixed.) More naturally, the term $E$ in the complex is actually $E \otimes \co$, where $\co$ is the dual of the subsheaf of $E$ generated by $\phi$; it is acted on accordingly by $G$. For a pair $(L \oplus M, \phi \oplus 0)$ represented in $X^0$, this splits as $$\Hyp^1(\co \oplus \co \stackrel{\phi \oplus 0}{\lrow} L) \oplus \Hyp^1(LM^{-1} \to 0) \oplus \Hyp^1(ML^{-1} \stackrel{\phi}{\to} M).$$ These are acted on by the stabilizer $\kst$ with weights $0$ and $\pm(\chi(M) + \chi(L))/c$ respectively, so they are exactly $N^0$, $N^+$, and $N^-$. The expressions for $N^\pm$ can be simplified: $\Hyp^1(LM^{-1} \to 0)$ is just $H^1(LM^{-1})$, and if $D$ is the divisor of zeroes of $\phi$, the long exact sequence of $$ 0 \lrow ML^{-1} \lrow M \lrow \co_D \otimes M \lrow 0 $$ implies that $\Hyp^1(ML^{-1} \stackrel{\phi}{\to} M)$ is just $H^0(\co_D \otimes M)$. Since every element in $N^\pm$ is acted on with weight exactly $\pm (\chi(M) + \chi(L))/c$, $N^\pm$ descend to vector bundles $W^\pm$ over $S^iC \times \Pic^{d-i}C$. Indeed, if ${\bf M} \to \Pic^{d-i}C \times C$ is a Poincar\'e bundle and $\Delta \subset S^iC \times C$ is the universal divisor, then $W^+ = (R^1 \pi) {\bf M}^{-1}(\Delta)$ and $W^- = (R^0 \pi) \co_\Delta \otimes {\bf M}$. Theorem \re{1x} then states that $\Pj W^\pm$ are the exceptional loci of the morphisms $B_d(\si \pm \half) \to B_d(\si)$, and that the blow-ups of $B_d(\si \pm \half)$ at $\Pj W^\pm$, and of $B_d(\si)$ at $S^iC \times \Pic^{d-i} C$, are all isomorphic to the fibred product $B_d(\si - \half) \times_{B_d(\si)} B_d(\si + \half)$. This includes the main result (3.18) of \cite{t1}; to recover the $W^\pm$ obtained there for a fixed determinant line bundle $\La$, just substitute ${\bf M} = \La(-\Delta)$. Notice that since the construction only works for $d$ large, the result has so far only been proved in that case. For general $d$, choose disjoint divisors $D$, $D'$ of degree $|D| = |D'|$ such that $d + 2|D|$ is large enough. Then $D$ and $D'$ determine two different embeddings $B_{d+2|D|}(\si) \to B_{d+4|D|}(\si)$ whose images intersect in $B_d(\si)$. The result for $B_d(\si)$ then follows readily from the result for $B_{d+2|D|}(\si)$ and $B_{d+4|D|}(\si)$. \bl A similar argument proves the analogous result of Bertram, Daskalopoulos and Wentworth \cite{bdw} on Bradlow $n$-pairs. These are pairs $(E, \phi)$, where $E$ is as before, but $\phi$ is now a nonzero element of $H^0(E \otimes \co^n)$. The stability condition is just like that for ordinary Bradlow pairs, except that the two cases are $\phi \in H^0(L \otimes \co^n)$ and $\phi \notin H^0(L \otimes \co^n)$. There is no geometric invariant theory construction in the literature of moduli spaces of $n$-pairs, but the construction of \cite{t1} for 1-pairs generalizes in the obvious way; for example, $\Pj^{\chi-1}$ gets replaced by $\Pj^{n\chi-1}$. Again the stability condition only changes for $\si \in d/2 + \Z$, so fix one such $\si$. Assume that $d/2 - \si > 2g-2$, so that the moduli space can be constructed directly as a geometric invariant theory quotient. Then $x \in X^0$ if and only if it splits as $L \oplus M$, where $\deg L = i = d/2 - \si$ and $\phi \in H^0(L \otimes \co^n)$. Hence $X^0 \mod G(0) = \times^n_{\Pic^i C} S^i C \times \Pic^{d-i}C$, where $\times^n_{\Pic^i C}$ denotes the $n$-fold fibred product over $\Pic^i C$. The hypotheses of \re{1x} are verified exactly as before. The normal bundle to an orbit is again the deformation space, but this is now $\Hyp^1(\End E \stackrel{\phi}{\to} E \otimes \co^n)$; for a pair $(L \oplus M, \phi \oplus 0)$ represented in $X^0$, this splits as $$\Hyp^1(\co\oplus \co \stackrel{\phi\oplus 0}{\lrow} L \otimes \co^n) \oplus \Hyp^1(LM^{-1} \to 0) \oplus \Hyp^1(ML^{-1} \stackrel{\phi}{\to} M \otimes \co^n).$$ Again these are exactly $N^0$, $N^+$, and $N^-$. The expression for $N^+$ is simply $H^1(LM^{-1})$, but the expression for $N^-$ cannot be simplified very much. If $F$ is defined to be the cokernel of the sheaf injection $ML^{-1} \to M \otimes \co^n$ induced by $\phi$, then $N^- = H^0(F)$, but this is not very helpful as $F$ may not be locally free. In any case, $N^\pm$ descend as before. If $\Delta_j \subset \times^n_{\Pic^i C} S^i C \times C$ is the pullback from the $j$th factor of the universal divisor $\Delta \subset S^iC \times C$, then there is a universal map $\co \to \oplus_j \co(\Delta_j)$ of bundles on $\times^n_{\Pic^i C} S^i C \times C$, and hence a sheaf injection ${\bf ML}^{-1} \to {\bf ML}^{-1} \otimes \oplus_j \co(\Delta_j)$ of bundles on $\times^n_{\Pic^i C} S^i C \times \Pic^{d-i} C \times C$, where ${\bf M}$ and ${\bf L}$ are Poincar\'e bundles on $\Pic^i C \times C$ and $\Pic^{d-i}C \times C$ respectively. Let ${\bf F}$ be the cokernel of this injection; then $W^- = (R^0\pi) {\bf F}$. As before, $W^+ = (R^1 \pi){\bf LM}^{-1}$. Theorem \re{1x} then gives a result precisely analogous to the one stated above for 1-pairs. However, the argument passing from large $d$ to general $d$ no longer works, so this result is only valid for $i = d/2 - \si > 2g-2$. Indeed, for $i$ smaller than this, the fibred product $\times^n_{\Pic^i C} S^i C$, and hence $X^0 \mod G(0)$, have bad singularities. Nevertheless, Bertram et al.\ succeed in using this result to compute certain Gromov invariants of Grassmannians. {\em Acknowledgements.} I am grateful to Aaron Bertram, Igor Dolgachev, David Eisenbud, Jack Evans, Antonella Grassi, Stuart Jarvis, J\'anos Koll\'ar, Kenji Matsuki, Rahul Pandharipande, David Reed, Eve Simms, and especially Frances Kirwan, Peter Kronheimer, and Miles Reid for very helpful conversations and advice.

9405

alg-geom/9405003

en

https://arxiv.org/abs/alg-geom/9405003

alg-geom/9405003

Alan Hyungju Park

H. Park and C. Woodburn

An Algorithmic Proof of Suslin's Stability Theorem over Polynomial Rings

23 pages, LaTex

Let $k$ be a field. Then Gaussian elimination over $k$ and the Euclidean division algorithm for the univariate polynomial ring $k[x]$ allow us to write any matrix in $SL_n(k)$ or $SL_n(k[x])$, $n\geq 2$, as a product of elementary matrices. Suslin's stability theorem states that the same is true for the multivariate polynomial ring $SL_n(k[x_1,\ldots ,x_m])$ with $n\geq 3$. As Gaussian elimination gives us an algorithmic way of finding an explicit factorization of the given matrix into elementary matrices over a field, we develop a similar algorithm over polynomial rings.

alg-geom

\section{Introduction} Immediately after proving the famous {\em Serre's Conjecture} (the {\em Quillen-Suslin theorem}, nowadays) in 1976 \cite{suslin:serre}, A. Suslin went on \cite{suslin} to prove the following $K_1$-analogue of {\em Serre's Conjecture} which is now known as {\em Suslin's stability theorem}: \begin{quote} Let $R$ be a commutative Noetherian ring and $n\geq \max(3,\dim(R)+2)$. Then, any $n\times n$ matrix $A=(f_{ij})$ of determinant $1$, with $f_{ij}$ being elements of the polynomial ring $R[x_1,\ldots ,x_m]$, can be written as a product of elementary matrices over $R[x_1,\ldots ,x_m]$. \end{quote} \begin{definition} For any ring $R$, an $n\times n$ elementary matrix $E_{ij}(a)$ over $R$ is a matrix of the form $I+a\cdot e_{ij}$ where $i\neq j,a\in R$ and $e_{ij}$ is the $n\times n$ matrix whose $(i,j)$ component is $1$ and all other components are zero. \end{definition} For a ring $R$, let $SL_n(R)$ be be the group of all the $n\times n$ matrices of determinant $1$ whose entries are elements of $R$, and let $E_n(R)$ be the subgroup of $SL_n(R)$ generated by the elementary matrices. Then {\em Suslin's stability theorem} can be expressed as \begin{eqnarray} SL_n(R[x_1,\ldots ,x_m]) = E_n(R[x_1,\ldots ,x_m]) \quad {\rm for\ all}\ n\geq \max(3,\dim(R)+2). \end{eqnarray} In this paper, we develop an algorithmic proof of the above assertion over a field $k$. By implementing this algorithm, for a given $A\in SL_n(k[x_1,\ldots ,x_m])$ with $n\geq 3$, we are able to find those elementary matrices $E_1,\ldots ,E_t\in E_n(k[x_1,\ldots ,x_m])$ such that $A=E_1\cdots E_t$. \begin{remark} If a matrix $A$ can be written as a product of elementary matrices, we will say $A$ is {\em realizable}. \end{remark} \bigskip \begin{itemize} \item In section 2, an algorithmic proof of the normality of $E_n(k[x_1,\ldots ,x_m])$ in $SL_n(k[x_1,\ldots ,x_m])$ for $n\geq 3$ is given, which will be used in the rest of paper. \item In section 3, we develop an algorithm for the {\em Quillen Induction Process}, a standard way of reducing a given problem over a ring to an easier problem over a local ring. Using this {\em Quillen Induction Algorithm}, we reduce our realization problem over the polynomial ring $R[X]$ to one over $R_M[X]$'s, where $R=k[x_1,\ldots, x_{m-1}]$ and $M$ is a maximal ideal of $R$. \item In section 4, an algorithmic proof of the {\em Elementary Column Property}, a stronger version of the {\em Unimodular Column Property}, is given, and we note that this algorithm gives another constructive proof of the {\em Quillen-Suslin theorem}. Using the {\em Elementary Column Property}, we show that a realization algorithm for $SL_n(k[x_1,\ldots ,x_m])$ is obtained from a realization algorithm for the matrices of the following special form: $$\left(\! \begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1 \end{array}\!\right) \in SL_3(k[x_1,\ldots ,x_m]),$$ where $p$ is monic in the last variable $x_m$. \item In section 5, in view of the results in the preceding two sections, we note that a realization algorithm over $k[x_1,\ldots ,x_m]$ can be obtained from a realization algorithm for the matrices of the special form $\left(\!\begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1 \end{array}\!\right)$ over $R[X]$, where $R$ is now a local ring and $p$ is monic in $X$. A realization algorithm for this case was already found by M.P. Murthy in \cite{murthy}. We reproduce {\em Murthy's Algorithm} in this section. \item In section 6, we suggest using the {\em Steinberg relations} from algebraic $K$-theory to lower the number of elementary matrix factors in a factorization produced by our algorithm. We also mention an ongoing effort of using our algorithm in {\em Signal Processing}. \end{itemize} \section{Normality of $E_n(k[x_1,\ldots ,x_m])$ in $SL_n(k[x_1,\ldots ,x_m])$} \begin{lemma} The Cohn matrix $A=\left(\! \begin{array}{cc} 1+xy & x^2 \\ -y^2 & 1-xy \end{array}\!\right)$ is not realizable, but $\left(\! \begin{array}{cc} A & 0 \\ 0 & 1 \end{array}\!\right) \in SL_3(k[x,y])$ is. \end{lemma} \noindent {\bf Proof:\ } The nonrealizability of $A$ is proved in \cite{cohn}, and a complete algorithmic criterion for the realizability of matrices in $SL_2(k[x_1,\ldots ,x_m])$ is developed in \cite{thk}. Now consider \begin{eqnarray} \left(\! \begin{array}{cc} A & 0 \\ 0 & 1 \end{array} \!\right) =\left(\! \begin{array}{ccc} 1+xy & x^2 & 0 \\ -y^2 & 1-xy & 0 \\ 0 & 0 & 1 \end{array}\!\right). \end{eqnarray} Noting that $\left(\!\begin{array}{ccc} 1+xy & x^2 & 0 \\ -y^2 & 1-xy & 0 \\ 0 & 0 & 1 \end{array}\!\right)=I+\left(\! \begin{array}{c} x\\ -y \\ 0 \end{array}\!\right)\cdot (y,x,0)$, we see that the realizability of this matrix is a special case of the following {\bf Lemma 3}. \hspace*{\fill}{\bf $\Box$} \bigskip \\ \begin{definition} Let $n\geq 2$. A {\bf Cohn-type matrix} is a matrix of the form $$ I+a{\bf v} \cdot (v_j{\bf e_i}-v_i{\bf e_j}) $$ where ${\bf v}=\left(\!\begin{array}{c}v_1\\ \vdots\\ v_n\end{array}\! \right)\in (k[x_1,\ldots ,x_m])^n$, $i<j\in\{ 1,\ldots ,n\}$, $a\in k[x_1,\ldots ,x_m]$, and ${\bf e_i}=(0,\ldots ,0,1,0,\ldots,0)$ with $1$ occurring only at the $i$-th position. \end{definition} \begin{lemma} \label{lem;Cohn} Any Cohn-type matrix for $n\geq 3$ is realizable. \end{lemma} \noindent {\bf Proof:\ } First, let's consider the case $i=1,j=2$. In this case, \begin{eqnarray} B & = & I+a\left(\!\begin{array}{c} v_1\\ \vdots\\ v_n \end{array}\!\right) \cdot (v_2, -v_1,0,\ldots , 0) \nonumber\\ & = & \left(\!\begin{array}{ccccc} 1+av_1v_2 & -av_1^2 & 0 & \cdots & 0 \\ av_2^2 & 1-av_1v_2 & 0 & \cdots & 0 \\ av_3v_2 & -av_3v_1 & & & \\ \vdots & \vdots & & I_{n-2} & \\ av_nv_2 & -av_nv_1 & & & \end{array}\!\right) \nonumber\\ & = & \left(\!\begin{array}{ccccc} 1+av_1v_2 & -av_1^2 & 0 & \cdots & 0 \\ av_2^2 & 1-av_1v_2 & 0 & \cdots & 0 \\ 0 & 0 & & & \\ \vdots & \vdots & & I_{n-2} & \\ 0 & 0 & & & \end{array}\!\right) \prod_{l=3}^nE_{l1}(av_lv_2)E_{l2}(-av_lv_1), \end{eqnarray} So, it's enough to show that \begin{eqnarray} A=\left( \!\begin{array}{ccc} 1+av_1v_2 & -av_1^2 & 0 \\ av_2^2 & 1-av_1v_2 & 0 \\ 0 & 0 & 1 \end{array}\!\right) \end{eqnarray} is realizable for any $a,v_1,v_2\in k[x_1,\ldots ,x_m]$. Let ``$\rightarrow$'' indicate that we are applying elementary operations, and consider the following: \begin{eqnarray} A & = & \left(\! \begin{array}{ccc} 1+av_1v_2 & -av_1^2 & 0 \\ av_2^2 & 1-av_1v_2 & 0 \\ 0 & 0 & 1\end{array}\!\right) \rightarrow \left(\!\begin{array}{ccc} 1+av_1v_2 & -av_1^2 & v_1 \\ av_2^2 & 1-av_1v_2 & v_2\\ 0 & 0 & 1\end{array}\!\right)\nonumber\\ & \rightarrow & \left( \!\begin{array}{ccc} 1 & -av_1^2 & v_1\\ 0 & 1-av_1v_2 & v_2 \\ -av_2 & 0 & 1\end{array}\!\right) \rightarrow \left(\! \begin{array}{ccc} 1 & 0 & v_1 \\ 0 & 1 & v_2 \\ -av_2 & av_1 & 1 \end{array} \!\right) \rightarrow \left( \!\begin{array}{ccc} 1 & 0 & v_1 \\ 0 & 1 & v_2 \\ 0 & av_1 & 1+av_1v_2 \end{array}\!\right)\nonumber\\ & \rightarrow & \left(\! \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & v_2 \\ 0 & av_1 & 1+av_1v_2 \end{array}\!\right) \rightarrow \left(\! \begin{array}{ccc} 1 & 0 &0 \\ 0 & 1 & v_2 \\ 0 & 0 & 1 \end{array}\! \right) \rightarrow \left(\!\begin{array}{ccc} 1 & 0 &0 \\ 0 & 1 &0 \\ 0 & 0 & 1\end{array}\! \right). \end{eqnarray} Keeping track of all the elementary operations involved, we get \begin{eqnarray} A=E_{13}(-v_1)E_{23}(-v_2)E_{31}(-av_2)E_{32}(av_1)E_{13}(v_1) E_{23}(v_2)E_{31}(av_2)E_{32}(-av_1). \end{eqnarray} In general (i.e., for arbitrary $i<j$), \begin{eqnarray} B & = & I+a\left(\!\begin{array}{c} v_1 \\ \vdots \\v_n \end{array}\!\right) \cdot (0,\ldots ,0,v_j,0,\ldots ,0,-v_i,0,\ldots ,0) \nonumber\\ & & \mbox{(Here,\ $v_j$\ occurs\ at\ the\ $i$-th\ position\ and\ $-v_i$\ occurs\ at\ the}\nonumber\\ & & j\mbox{-th\ position.)}\nonumber\\ & = & \left(\!\begin{array}{ccccccc} 1 & \cdots & av_1v_j & \cdots & -av_1v_i & \cdots & 0 \\ & \ddots & \vdots & & \vdots & & 0 \\ & & 1+av_iv_j & & -av_i^2 & & \\ & & \vdots & & \vdots & & \\ & & av_j^2 & & 1-av_iv_j & & \\ & & \vdots & & \vdots & & \\ & & v_nv_j & & -v_nv_i & & 1 \end{array}\!\right)\nonumber\\ & = & \left(\!\begin{array}{ccccccc} 1 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\ & \ddots & \vdots & & \vdots & & 0 \\ & & 1+av_iv_j & & -av_i^2 & & \\ & & \vdots & & \vdots & & \\ & & av_j^2 & & 1-av_iv_j & & \\ & & \vdots & & \vdots & & \\ & & 0 & & 0 & & 1 \end{array}\!\right)\nonumber\\ & & \cdot\prod_{1\leq l\leq n, l\neq i,j} E_{li}(av_lv_j)E_{lj}(-av_lv_i) \nonumber\\ & = & E_{it}(-v_i)E_{jt}(-v_j)E_{ti}(-av_j)E_{tj}(av_i)E_{it}(v_i) E_{jt}(v_j)E_{ti}(av_j)E_{tj}(-av_i) \nonumber\\ & & \cdot\prod_{1\leq l\leq n, l\neq i,j} E_{li}(av_lv_j)E_{lj}(-av_lv_i). \end{eqnarray} In the above, $t\in \{1,\ldots, n\}$ can be chosen to be any number other than $i$ and $j$. \hspace*{\fill}{\bf $\Box$} \bigskip \\ Since a {\em Cohn-type matrix} is realizable, any product of {\em Cohn-type matrices} is also realizable. This observation motivates the following generalization of the above lemma. \begin{definition} Let $R$ be a ring and ${\bf v}=(v_1,\ldots ,v_n)^t \in R^n$ for some $n\in I\!\!N$. Then ${\bf v}$ is called a {\em unimodular column vector} if its components generate $R$, i.e. if there exist $g_1,\ldots ,g_n\in R$ such that $v_1g_1+\cdots +v_ng_n=1$. \end{definition} \begin{cor} Suppose that $A \in SL_n(k[x_1, \ldots ,x_m])$ with $n\geq 3$ can be written in the form $A=I+{\bf v}\cdot {\bf w}$ for a unimodular column vector $\ {\bf v}$ and a row vector $\ {\bf w}$ over $\ k[x_1,\ldots ,x_m]\ $ such that ${\bf w}\cdot {\bf v}=0$. Then $A$ is realizable. \end{cor} \noindent {\bf Proof:\ } Since ${\bf v}=(v_1,\ldots ,v_n)^t$ is unimodular, we can find $g_1,\ldots ,g_n \in k[x_1,\ldots ,x_m]$ such that $v_1g_1+\cdots +v_ng_n=1$. We can use the {\em effective Nullstellensatz} to explicitly find these $g_i$'s (See \cite{fitchas:galligo}). This combined with ${\bf w}\cdot {\bf v}=w_1v_1+\cdots +w_nv_n=0$ yields a new expression for ${\bf w}$: \begin{eqnarray} {\bf w}=\sum_{i<j}a_{ij}(v_j{\bf e_i}-v_i{\bf e_j}) \end{eqnarray} where $a_{ij}=w_ig_j-w_jg_i$. Now, \begin{eqnarray} A=\prod_{i<j}\left(I+{\bf v}\cdot a_{ij}(v_j{\bf e_i}-v_i{\bf e_j}) \right). \end{eqnarray} Each component on the right hand side of this equation is a {\em Cohn-type matrix} and thus realizable, so $A$ is also realizable. \hspace*{\fill}{\bf $\Box$} \bigskip \\ \begin{cor} \quad $BE_{ij}(a)B^{-1}$ is realizable for any $\ B\in GL_n(k[x_1, \ldots ,x_m])$ with $n\geq 3$ and $\ a\in k[x_1, \ldots ,x_m]$. \end{cor} \noindent {\bf Proof:\ } Note that $i\neq j$, and $$BE_{ij}(a)B^{-1}=I+(i\mbox{-th\ column\ vector\ of}\ B)\cdot a \cdot (j\mbox{-th\ row\ vector\ of}\ B^{-1}).$$ Let ${\bf v}$ be the $i$-th column vector of $B$ and ${\bf w}$ be $a$ times the $j$-th row vector of $B^{-1}$. Then $(i$-th row vector of $B^{-1})\cdot {\bf v}=1$ implies ${\bf v}$ is unimodular, and ${\bf w}\cdot {\bf v}$ is clearly zero since $i\neq j$. Therefore, $BE_{ij}(a)B^{-1}=I+{\bf v}\cdot {\bf w}$ satisfies the condition of the above corollary, and is thus realizable. \hspace*{\fill}{\bf $\Box$} \bigskip \\ \begin{remark} One important consequence of this corollary is that for $n\geq 3$, $E_n(k[x_1,\ldots ,x_m])$ is a normal subgroup of $SL_n(k[x_1,\ldots ,x_m])$, i.e. if $A\in SL_n(k[x_1,\ldots ,x_m])$ and $E\in E_n(k[x_1,\ldots ,x_m])$, then the above corollary gives us an algorithm for finding elementary matrices $E_1,\ldots ,E_t$ such that $A^{-1}EA=E_1\cdots E_t$. \end{remark} \section{Glueing of Local Realizability} Let $R=k[x_1,\ldots , x_{m-1}],\ X=x_m$ and $M\in$ Max($R$) =\{ maximal ideals of $R$\}. For $A\in SL_n(R[X])$, we let $A_M\in SL_n(R_M[X])$ be its image under the canonical mapping $SL_n(R[X])\rightarrow SL_n(R_M[X])$. Also, by induction, we may assume $SL_n(R)=E_n(R)$ for $n\geq 3$. Now consider the following analogue of Quillen's theorem for elementary matrices; \begin{quote} Suppose $n\geq 3$ and $A\in SL_n(R[X])$. Then $A$ is realizable over $R[X]$ if and only if $A_M\in SL_n(R_M[X])$ is realizable over $R_M[X]$ for every $M\in$ Max($R$). \end{quote} While a non-constructive proof of this assertion is given in \cite{suslin} and a more general functorial treatment of this {\em Quillen Induction Process} can be found in \cite{knus}, we will attempt to give a constructive proof for it here. Since the necessity of the condition is clear, we have to prove the following; \begin{thm}\label{thm;glueing} (Quillen Induction Algorithm) For any given $A\in SL_n(R[X])$, if $A_M\in E_n(R_M[X])$ for every $M\in {\rm Max}(R)$, then $A\in E_n(R[X])$. \end{thm} \begin{remark} In view of this theorem, for any given $A\in SL_n(R[X])$, now it's enough to have a realization algorithm for each $A_M$ over $R_M[X]$. \end{remark} \noindent {\bf Proof:\ } Let ${\bf a_1}=(0,\ldots ,0)\in k^{m-1}$, and $M_1=\{ g\in k[x_1,\ldots ,x_{m-1}]\mid g({\bf a_1})=0\}$ be the corresponding maximal ideal. Then by the condition of the theorem, $A_{M_1}$ is realizable over $R_{M_1}[X]$. Hence, we can write \begin{eqnarray} A_{M_1}=\prod_jE_{s_jt_j}\left(\frac{c_j}{d_j}\right) \end{eqnarray} where $c_j,d_j\in R, d_j\not\in M_1$. Letting $r_1=\prod_jd_j\notin M_1$, we can rewrite this as \begin{eqnarray} A_{M_1}=\prod_jE_{s_jt_j}\left(\frac{c_j\prod_{k\neq j}d_k}{r_1}\right) \in E_n(R_{r_1})\subset E_n(R_{M_1}). \end{eqnarray} Denote an algebraic closure of $k$ by $\bar{k}$. Inductively, let ${\bf a_j} \in {\bar{k}}^{m-1}$ be a common zero of $r_1,\ldots ,r_{j-1}$ and $M_j=\{ g\in k[x_1,\ldots ,x_{m-1}]\mid g({\bf a_j})=0\}$ be the corresponding maximal ideal of $R$ for each $j\geq 2$. Define $r_j\notin M_j$ in the same way as in the above so that \begin{eqnarray} A_{M_j}\in E_n(R_{r_j}[X]). \end{eqnarray} Since ${\bf a_j}$ is a common zero of $r_1,\ldots ,r_{j-1}$ in this construction, we immediately see $r_1,\ldots ,r_{j-1}\in M_j=\{g\in R\mid g({\bf a_j})=0\}$. But noting $r_j\notin M_j$, we conclude that $r_j\notin r_1R+\cdots +r_{j-1}R$. Now, since the Noetherian condition on $R$ guarantees that we will get to some $L$ after a finite number of steps such that $r_1R+\cdots +r_LR=R$, we can use the usual {\em Ideal Membership Algorithm} to determine when $1_R$ is in the ideal $r_1R+\cdots +r_LR$. Let $l$ be a {\em large} natural number (It will soon be clear what {\em large} means). Then since $r_1^lR+\cdots +r_L^lR=R$, we can use the {\em effective Nullstellensatz} to find $g_1,\ldots ,g_L\in R$ such that $r_1^lg_1+\cdots +r_L^lg_L=1$. Now, we express $A(X)\in SL_n(R[X])$ in the following way: \begin{eqnarray} A(X) & = & A(X-Xr_1^lg_1) \cdot [ A^{-1}(X-Xr_1^lg_1) A(X) ]\nonumber\\ & = & A(X-Xr_1^lg_1-Xr_2^lg_2) \cdot [ A^{-1}(X-Xr_1^lg_1-Xr_2^lg_2) A(X-Xr_1^lg_1) ] \nonumber\\ & & \cdot [ A^{-1}(X-Xr_1^lg_1) A(X) ]\nonumber\\ & = & \cdots \nonumber\\ & = & A(X-\sum_{i=1}^LXr_i^lg_i) \cdot [ A^{-1}(X-\sum_{i=1}^LXr_i^lg_i) A(X-\sum_{i=1}^{L-1}Xr_i^lg_i) ] \cdots \nonumber \\ & & \cdots [ A^{-1}(X-Xr_1^lg_1) A(X) ]. \end{eqnarray} Note here that the first matrix $A(X-\sum_{i=1}^LXr_i^lg_i)=A(0)$ on the right hand side is in $SL_n(R)=E_n(R)$ by the induction hypothesis. What will be shown now is that for a sufficiently large $l$, each expression in the brackets in the above equation for $A$ is actually in $E_n(R[X])$, so that $A$ itself is in $E_n(R[X])$. To this end, by letting $A_{M_i}=A_i$ and identifying $A\in SL_n(R[X])$ with $A_i\in SL_n(R_{M_i}[X])$, note that each expression in the brackets is in the following form: \begin{eqnarray} A_i^{-1}(cX)A_i((c+r_i^lg)X). \end{eqnarray} \newline {\bf *Claim:} For any $c,g\in R$, we can find a sufficiently large $l$ such that $A_i^{-1}(cX)A_i((c+r_i^lg)X)\in E_n(R[X])$ for all $i=1,\ldots ,L$. \newline Let \begin{eqnarray} D_i(X,Y,Z)=A_i^{-1}(Y\cdot X)A_i((Y+Z)\cdot X)\in E_n(R_{r_i}[X,Y,Z]) \end{eqnarray} and write $D_i$ in the form \begin{eqnarray} D_i=\prod_{j=1}^hE_{s_jt_j}(b_j+Zf_j) \end{eqnarray} where $b_j\in R_{r_i}[X,Y]$ and $f_j\in R_{r_i}[X,Y,Z]$. {}From now on, the elementary matrix $E_{s_jt_j}(a)$ will be simply denoted as $E^j(a)$ for notational convenience. Now define $C_p$ by \begin{eqnarray} C_p=\prod_{j=1}^pE^j(b_j)\in E_n(R_{r_i}[X,Y]). \end{eqnarray} Then the $C_p$'s satisfy the following recursive relations; \begin{eqnarray} E^1(b_1) & = & C_1 \nonumber \\ E^p(b_p) & = & C_{p-1}^{-1}C_p\quad (2\leq p\leq h)\nonumber \\ C_h & = & I. \end{eqnarray} Hence, using $E_{ij}(a+b)=E_{ij}(a)E_{ij}(b)$, \begin{eqnarray} D_i & = & \prod_{j=1}^hE^j(b_j+Zf_j) \nonumber \\ & = & \prod_{j=1}^hE^j(b_j)E^j(Zf_j) \nonumber \\ & = & [E^1(b_1)E^1(Zf_1)][E^2(b_2)E^2(Zf_2)] \ \cdots\ [E^h(b_h)E^h(Zf_h)] \nonumber \\ & = & [C_1E^1(Zf_1)][C_1^{-1}C_2E^2(Zf_2)]\ \cdots\ [C_{h-1}^{-1}C_hE^h(Zf_h)] \nonumber \\ & = & \prod_{j=1}^hC_jE^j(Zf_j)C_j^{-1}. \end{eqnarray} Now in the same way as in the proof of {\bf Corollary 1} and {\bf Corollary 2} of section 2, we can write $C_jE^j(Zf_j)C_j^{-1}$ as a product of Cohn-type matrices, i.e. for any given $j\in \{1,\ldots ,h\}$, let ${\bf v}=\left(\!\begin{array}{c} v_1 \\ \vdots \\ v_n \end{array}\!\right)$ be the $s_j$-th column vector of $C_j$. Then \begin{eqnarray} C_jE_{s_jt_j}(Zf_j)C_j^{-1}=\prod_{1\leq \gamma < \delta \leq n} [I+{\bf v}\cdot Zf_j\cdot a_{\gamma \delta}(v_{\gamma}{\bf e_{\delta}}-v_{\delta}{\bf e_{\gamma}})] \end{eqnarray} for some $a_{\gamma \delta}\in R_{r_i}[X,Y]$. Also we can find a natural number $l$ such that \begin{eqnarray} v_{\gamma}=\frac{v_{\gamma}'}{r_i^l},\quad a_{\gamma \delta}=\frac{a_{\gamma \delta}'}{r_i^l},\quad f_j=\frac{f_j'}{r_i^l} \end{eqnarray} for some $v_{\gamma}',a_{\gamma \delta}'\in R[X,Y],\ f_j'\in R[X,Y,Z]$. Now, replacing $Z$ by $r_i^{4l}g$, we see that all the Cohn-type matrices in the above expression for $C_jE^j(Zf_j)C_j^{-1}$ have denominator-free entries. Therefore, \begin{eqnarray} C_jE^j(r_i^{4l}gf_j)C_j^{-1}\in E_n(R[X,Y]). \end{eqnarray} Since this is true for each $j$, we conclude that for a sufficiently large $l$, \begin{eqnarray} D_i(X,Y,r_i^{l}g)=\prod_{j=1}^hC_jE^j(r_i^{l}gf_j) C_j^{-1}\in E_n(R[X,Y]). \end{eqnarray} Now, letting $Y=c$ proves the claim. \hspace*{\fill}{\bf $\Box$} \bigskip \\ \section{Reduction to $SL_3(k[x_1,\ldots ,x_m])$} Let $A\in SL_n(k[x_1,\ldots ,x_m])$ with $n\geq 3$, and ${\bf v}$ be its last column vector. Then ${\bf v}$ is unimodular. (Recall that the cofactor expansion along the last column gives a required relation.) Now, if we can reduce ${\bf v}$ to ${\bf e_n}=(0,0,\ldots ,0,1)^t$ by applying elementary operations, i.e. if we can find $B\in E_n(k[x_1,\ldots ,x_m])$ such that $B{\bf v}={\bf e_n}$, then \begin{eqnarray} BA=\left(\! \begin{array}{cccc} & & & 0 \\ & \tilde{A} & & \vdots \\ & & & 0 \\ p_1 &\ldots & p_{n-1} & 1 \end{array}\! \right) \end{eqnarray} for some $\tilde{A} \in SL_{n-1}(k[x_1,\ldots ,x_m])$ and $p_i \in k[x_1,\ldots ,x_m]$ for $i=1,\ldots ,n-1$. Hence, \begin{eqnarray} BAE_{n1}(-p_1)\cdots E_{n(n-1)}(-p_{n-1})= \left(\!\begin{array}{cc} \tilde{A} & 0 \\ 0 & 1 \end{array}\!\right). \end{eqnarray} Therefore our problem of expressing $A\in SL_n(k[x_1,\ldots ,x_m])$ as a product of elementary matrices is now reduced to the same problem for $\tilde{A}\in SL_{n-1}(k[x_1,\ldots ,x_m])$. By repeating this process, we get to the problem of expressing $A= \left(\! \begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1 \end{array}\! \right) \in SL_3(k[x_1,\ldots ,x_m])$ as a product of elementary matrices, which is the subject of the next section. In this section, we will develop an algorithm for finding elementary operations that reduce a given unimodular column vector ${\bf v} \in (k[x_1,\ldots ,x_m])^n$ to ${\bf e_n}$. Also, as a corollary to this {\em Elementary Column Property}, we give an algorithmic proof of the {\em Unimodular Column Property} which states that for any given unimodular column vector ${\bf v} \in (k[x_1,\ldots ,x_m])^n$, there exists a unimodular matrix $B$, i.e. a matrix of constant determinant, over $k[x_1,\ldots ,x_m]$ such that $B{\bf v}={\bf e_n}$. Lately, {\sl A. Logar, B. Sturmfels} in \cite{logar:sturmfels} and {\sl N. Fitchas, A. Galligo} in \cite{fitchas:galligo}, \cite{fitchas} have given different algorithmic proofs of this {\em Unimodular Column Property}, thereby giving algorithmic proofs of the {\em Quillen-Suslin theorem}. Therefore, our algorithm gives another constructive proof of the {\em Quillen-Suslin theorem}. The second author has given a different algorithmic proof of the {\em Elementary Column Property} based on a localization and patching process in \cite{cynthia}. \begin{definition} For a ring $R$, ${\rm Um}_n(R)=\{ n$-dimensional unimodular column vectors over $R\}$. \end{definition} \begin{remark} Note that the groups $GL_n(k[x_1,\ldots ,x_m])$ and $E_n(k[x_1,\ldots ,x_m])$ act on the set ${\rm Um}_n(k[x_1,\ldots ,x_m])$ by matrix multiplication. \end{remark} \begin{thm}\label{thm;reduction} (Elementary Column Property) For $n\geq 3$, the group $E_n(k[x_1,\ldots ,x_m])$ acts transitively on the set Um$_n(k[x_1,\ldots ,x_m])$. \end{thm} \begin{remark} According to this theorem, if ${\bf v,v'}$ are $n$-dimensional unimodular column vectors over $k[x_1,\ldots ,x_m]$, then we can find $B\in E_n(k[x_1,\ldots ,x_m])$ such that $B{\bf v}={\bf v'}$. Letting ${\bf v'}={\bf e_n}$ gives a desired algorithm. \end{remark} \begin{cor} (Unimodular Column Property) For $n\geq 2$, the group $GL_n(k[x_1,\ldots ,x_m])$ acts transitively on the set Um$_n(k[x_1,\ldots ,x_m])$. \end{cor} \noindent {\bf Proof:\ } For $n\geq 3$, the {\em Elementary Column Property} cleary implies the {\em Unimodular Column Property} since a product of elementary matrices is always unimodular, i.e. has a constant determinant. If $n=2$, for any ${\bf v}=(v_1,v_2)^t\in \mbox{Um}_2(k[x_1,\ldots ,x_m])$, find $g_1,g_2\in k[x_1,\ldots ,x_m]$ such that $v_1g_1+v_2g_2=1$. Then the unimodular matrix $U_{\bf v}= \left(\!\begin{array}{cc} v_2 & -v_1 \\ g_1 & g_2 \end{array}\!\right)$ satisfies $U_{\bf v}\cdot {\bf v}={\bf e_2}$. Therefore we see that, for any ${\bf v}, {\bf w}\in \mbox{Um}_2(k[x_1,\ldots ,x_m])$, $U_{\bf w}^{-1}U_{\bf v}\cdot {\bf v}={\bf w}$ where $U_{\bf w}^{-1}U_{\bf v}\in GL_2(k[x_1,\ldots ,x_m])$. \hspace*{\fill}{\bf $\Box$} \bigskip \\ Let $R=k[x_1,\ldots ,x_{m-1}]$ and $X=x_m$. Then $k[x_1,\ldots ,x_m]=R[X]$. By identifying $A\in SL_2(R[X])$ with $\left(\! \begin{array}{cc} A & 0 \\ 0 & I_{n-2} \end{array}\!\right)\in SL_n(R[X])$, we can regard $SL_2(R[X])$ as a subgroup of $SL_n(R[X])$. Now consider the following theorem. \begin{thm}\label{thm;link} Suppose ${\bf v}(X)= \left(\!\begin{array}{c} v_1(X) \\ \vdots \\ v_n(X) \end{array}\!\right) \in {\rm Um}_n(R[X])$, and $v_1(X)$ is monic in $X$. Then there exists $B_1\in SL_2(R[X])$ and $B_2\in E_n(R[X])$ such that $B_1B_2\cdot {\bf v}(X)={\bf v}(0)$. \end{thm} \noindent {\bf Proof:\ } Later \hspace*{\fill}{\bf $\Box$} \bigskip \\ We will use this theorem to prove the {\bf Theorem~\ref{thm;reduction}}, now. \medskip \noindent {\bf Proof of Theorem~\ref{thm;reduction}:} Since the {\em Euclidean division algorithm} for $k[x_1]$ proves the theorem for $m=1$ case, by induction, we may assume the statement of the theorem for $R= k[x_1,\ldots ,x_{m-1}]$. Let $X=x_m$ and ${\bf v}=\left(\!\begin{array}{c} v_1\\ \vdots\\ v_n\end{array}\!\right) \in {\rm Um}_n(R[X])$. We may also assume that $v_1$ is monic by applying a change of variables (as in the well-known proof of the {\em Noether Normalization Lemma}). Now by the above {\bf Theorem~\ref{thm;link}}, we can find $B_1\in SL_2(R[X])$ and $B_2\in E_n(R[X])$ such that \begin{eqnarray} B_1B_2\cdot {\bf v}(X)={\bf v}(0)\in R. \end{eqnarray} And then by the inductive hypothesis, we can find $B'\in E_n(R)$ such that \begin{eqnarray} B'\cdot {\bf v}(0)={\bf e_n}. \end{eqnarray} Therefore, we get \begin{eqnarray} {\bf v}=B_2^{-1}B_1^{-1}B'^{-1}{\bf e_n}. \end{eqnarray} By the normality of $E_n(R[X])$ in $SL_n(R[X])$ ({\bf Corollary 2}), we can write $B_1^{-1}B'^{-1}=B''B_1^{-1}$ for some $B''\in E_n(R[X])$. Since \begin{eqnarray} B_1^{-1}=\left(\! \begin{array}{ccccc} p & q& 0 &\ldots & 0 \\ r & s & 0 & \ldots &0\\ 0 & 0 &&& \\\vdots & \vdots & & I_{n-2} & \\ 0 & 0 &&& \end{array}\!\right) \end{eqnarray} for some $p,q,r,s \in R[X]$, we have \begin{eqnarray} {\bf v} & = & B_2^{-1}B_1^{-1}B'^{-1}{\bf e_n}\nonumber\\ & = & (B_2^{-1}B'')B_1^{-1}{\bf e_n}\nonumber\\ & = & (B_2^{-1}B'')\left(\! \begin{array}{ccccc} p & q& 0 &\ldots & 0 \\ r & s & 0 & \ldots &0\\ 0 & 0 &&& \\\vdots & \vdots & & I_{n-2} & \\ 0 & 0 &&& \end{array}\!\right)\left(\! \begin{array}{c} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{array} \!\right)\nonumber\\ & = & (B_2^{-1}B''){\bf e_n} \end{eqnarray} where $B_2^{-1}B''\in E_n(R[X])$. Since we have this relationship for any ${\bf v}\in {\rm Um}_n(R[X])$, we get the desired transitivity. \hspace*{\fill}{\bf $\Box$} \bigskip \\ Now, we need one lemma to construct an algorithm for the {\bf Theorem~\ref{thm;link}}. \begin{lemma}\label{lem;link} Let $f_1,f_2,b,d\in R[X]$ and $r$ be the resultant of $f_1$ and $f_2$. Then there exists $B\in SL_2(R[X])$ such that \begin{eqnarray} B\left(\! \begin{array}{c} f_1(b) \\f_2(b) \end{array}\!\right) =\left( \!\begin{array}{c} f_1(b+rd) \\f_2(b+rd) \end{array}\!\right). \end{eqnarray} \end{lemma} \noindent {\bf Proof:\ } By the property of the resultant of two polynomials, we can find $g_1,g_2\in R[X]$ such that $f_1g_1+f_2g_2=r$. Also let $s_1,s_2,t_1,t_2\in R[X,Y,Z]$ be the polynomials defined by \begin{eqnarray} f_1(X+YZ) & = & f_1(X)+Ys_1(X,Y,Z)\nonumber\\ f_2(X+YZ) & = & f_2(X)+ Ys_2(X,Y,Z)\nonumber\\ g_1(X+YZ) & = & g_1(X)+Yt_1(X,Y,Z) \nonumber\\ g_2(X+YZ) & = & g_2(X)+ Yt_2(X,Y,Z). \end{eqnarray} Now, let \begin{eqnarray} B_{11} & = & 1+s_1(b,r,d)\cdot g_1(b)+t_2(b,r,d)\cdot f_2(b) \nonumber\\ B_{12} & = & s_1(b,r,d)\cdot g_2(b)-t_2(b,r,d)\cdot f_1(b) \nonumber \\ B_{21} & = & s_2(b,r,d)\cdot g_1(b)-t_1(b,r,d)\cdot f_2(b) \nonumber \\ B_{22} & = & 1+s_2(b,r,d)\cdot g_2(b)+t_1(b,r,d)\cdot f_1(b). \end{eqnarray} Then one checks easily that $B=\left(\! \begin{array}{cc} B_{11} & B_{12} \\ B_{21} & B_{22} \end{array}\!\right)$ satisfies the desired property and that $B\in SL_2(R[X])$. \hspace*{\fill}{\bf $\Box$} \bigskip \\ {\bf Proof of Theorem~\ref{thm;link}:} Let ${\bf a_1}=(0,\ldots ,0)\in k^{m-1}$. Define $M_1=\{ g\in k[x_1,\ldots ,x_{m-1}]\mid g({\bf a_1})=0\} $ and $k_1=R/M_1$ as the corresponding maximal ideal and residue field, respectively. Since ${\bf v}\in (R[X])^n$ is a unimodular column vector, its image ${\bf \bar{v}}$ in $(k_1[X])^n=((R/M_1)[X])^n$ is also unimodular. Since $k_1[X]$ is a principal ideal ring, the minimal Gr\"{o}bner basis of its ideal $<\bar{v}_2,\ldots ,\bar{v}_n>$ consists of a single element, $G_1$. Then $\bar{v}_1$ and $G_1$ generate the unit ideal in $k_1[X]$ since $\bar{v}_1,\bar{v}_2,\ldots ,\bar{v}_n$ generate the unit ideal. Using the Euclidean division algorithm for $k_1[X]$, we can find $E_1\in E_{n-1}(k_1[X])$ such that \begin{eqnarray} E_1\left( \!\begin{array}{c} \bar{v}_2 \\ \vdots \\ \bar{v}_n \end{array} \right)= \left( \begin{array}{c} G_1 \\ 0 \\ \vdots \\ 0 \end{array}\! \right). \end{eqnarray} By identifying $k_1$ with a subring of $R$, we may regard $E_1$ to be an element of $E_n(R[X])$ and $G_1$ to be an element of $R[X]$. Then, \begin{eqnarray} \left( \!\begin{array}{cc} 1 & 0 \\ 0 & E_1 \end{array} \right) {\bf v}=\left( \begin{array}{c} v_1 \\ G_1+q_{12} \\ q_{13} \\ \vdots \\ q_{1n} \end{array} \!\right) \end{eqnarray} for some $q_{12},\ldots, q_{1n}\in M_1[X]$. Now, define $r_1\in R$ by \begin{eqnarray} r_1 & = & {\rm Res}(v_1, G_1+q_{12}) \nonumber\\ & = & {\rm the\ resultant\ of}\ v_1\ {\rm and}\ G_1+q_{12} \end{eqnarray} and find $f_1,h_1\in R[X]$ such that \begin{eqnarray} f_1\cdot v_1+h_1\cdot (G_1+q_{12})=r_1. \end{eqnarray} Since $v_1$ is monic, and $\bar{v}_1$ and $G_1\in k_1[X]$ generate the unit ideal, we have \begin{eqnarray} \bar{r}_1 & = & \overline{{\rm Res}(v_1, G_1+q_{12})}\nonumber\\ & = & {\rm Res}(\bar{v}_1, G_1)\nonumber\\ & \neq & 0. \end{eqnarray} Therefore, $r_1\notin M_1$. Denote an algebraic closure of $k$ by $\bar{k}$. Inductively, let ${\bf a_j}\in {\bar{k}}^{m-1}$ be a common zero of $r_1,\ldots ,r_{j-1}$ and $M_j$ be the corresponding maximal ideal of $R$ for each $j\geq 2$. Define $r_j\notin M_j$ in the same way as in the above. Define also, $E_j\in E_{n-1}(k_j[X]), G_j\in k_j[X], f_j,h_j\in R[X]$, and $q_{j2}, \ldots , q_{jn}\in M_j[X]$ in an analogous way. Since we let ${\bf a_j}$ be a common zero of $r_1,\ldots ,r_{j-1}$ in this construction, we see $r_1,\ldots ,r_{j-1}\in M_j=\{g\in R\mid g({\bf a_j})=0\}$. But noting $r_j\notin M_j$, we conclude that $r_j\notin r_1R+\cdots +r_{j-1}R$. Now, since $R$ is Noetherian, after a finite number of steps, we will get to some $L$ such that $r_1R+\cdots +r_LR=R$. We can use the {\em effective Nullstellensatz} to explicitly find those $g_i$'s in $R$ such that $r_1g_1+\cdots +r_Lg_L= 1$. Define, now, $b_0,b_1,\ldots ,b_L\in R[X]$ in the following way: \begin{eqnarray} b_0 & = & 0 \nonumber\\ b_1 & = & r_1g_1X \nonumber\\ b_2 & = & r_1g_1X+r_2g_2X \nonumber\\ & \vdots & \nonumber\\ b_L & = & r_1g_1X+r_2g_2X+\cdots +r_Lg_LX=X. \end{eqnarray} Then these $b_i$'s satisfy the recursive relations: \begin{eqnarray} b_0 & = & 0 \nonumber\\ b_i & = & b_{i-1} + r_ig_iX \quad {\rm for}\ i=1,\ldots ,L. \end{eqnarray} {\bf *Claim:} For each $i\in \{ 1,\ldots ,L\}$, there exists $B_i\in SL_2(R[X])$ and $B_i'\in E_n(R[X])$ such that ${\bf v}(b_i)=B_iB_i'{\bf v}(b_{i-1})$. If this claim is true, then using $E_n(R[X])\cdot SL_2(R[X])\subseteq SL_2(R[X])\cdot E_n(R[X])$ (Normality of $E_n(R[X])$; {\bf Corollary 2}), we inductively get \begin{eqnarray} {\bf v}(X) & = & {\bf v}(b_L)\nonumber\\ & = & B_LB_L'{\bf v}(b_{L-1})\nonumber\\ & \vdots & \nonumber\\ & = & BB'{\bf v}(b_0)\nonumber\\ & = & BB'{\bf v}(0) \end{eqnarray} for some $B\in SL_2(R[X])$ and $B'\in E_n(R[X])$. Therefore it's enough to prove the above claim. For this purpose, let $\tilde{G_i} =G_i+q_{i2}$. Then \begin{eqnarray} \left(\! \begin{array}{cc} 1 & 0 \\ 0 & E(X) \end{array} \!\right) {\bf v}(X)=\left(\! \begin{array}{c} v_1(X) \\ \tilde{G_i}(X) \\ q_{i3}(X) \\ \vdots \\ q_{in}(X) \end{array} \!\right). \end{eqnarray} For $3\leq l \leq n$, we have \begin{eqnarray} q_{il}(b_i)-q_{il}(b_{i-1}) & \in & (b_i - b_{i-1})\cdot R[X] \nonumber\\ & = & r_ig_iX \cdot R[X]. \end{eqnarray} Since $r_i\in R$ doesn't depend on $X$, we have \begin{eqnarray} r_i & = & f_i(X)v_1(X)+h_i(X)\tilde{G_i}(X)\nonumber\\ & = & f_i(b_{i-1})v_1(b_{i-1})+h_i(b_{i-1})\tilde{G_i} (b_{i-1})\nonumber\\ & = & {\rm a\ linear\ combination\ of}\ v_1(b_{i-1})\ {\rm and}\ \tilde{G_i}(b_{i-1})\ {\rm over}\ R[X]. \end{eqnarray} Therefore, we see that for $3\leq l \leq n$, \begin{eqnarray} q_{il}(b_i)=q_{il}(b_{i-1})+{\rm a\ linear\ combination\ of}\ v_1(b_{i-1})\ {\rm and}\ \tilde{G_i}(b_{i-1})\ {\rm over} R[X].\nonumber \end{eqnarray} Hence we can find $C\in E_n(R[X])$ such that \begin{eqnarray} C \left(\! \begin{array}{cc} 1 & 0 \\ 0 & E(b_{i-1}) \end{array} \!\right) {\bf v}(b_{i-1}) & = & C \left(\! \begin{array}{c} v_1(b_{i-1}) \\ \tilde{G}_i(b_{i-1}) \\ q_{i3}(b_{i-1}) \\ \vdots \\ q_{in}(b_{i-1}) \end{array} \!\right) \nonumber\\ & = & \left(\! \begin{array}{c} v_1(b_{i-1})\\ \tilde{G}_i(b_{i-1}) \\ q_{i3}(b_{i}) \\ \vdots \\ q_{in}(b_{i}) \end{array} \!\right). \end{eqnarray} Now, by the {\bf Lemma~\ref{lem;link}}, we can find $\tilde{B} \in SL_2(R[X])$ such that \begin{eqnarray} \tilde{B} \left(\! \begin{array}{c} v_1(b_{i-1}) \\ \tilde{G_i}(b_{i-1}) \end{array} \!\right)=\left(\! \begin{array}{c} v_1(b_i) \\ \tilde{G_i}(b_i) \end{array} \!\right). \end{eqnarray} Finally, define $B\in SL_n(R[X])$ as follows: \begin{eqnarray} B=\left(\! \begin{array}{cc} 1 & 0 \\ 0 & E(b_i)^{-1} \end{array} \!\right) \left(\! \begin{array}{cc} \tilde{B} & 0 \\ 0 & I_{n-2} \end{array} \!\right) \cdot C\cdot \left(\!\begin{array}{cc} 1 &0 \\ 0& E(b_i)\end{array}\!\right). \end{eqnarray} Then this $B$ satisfies \begin{eqnarray} B{\bf v}(b_{i-1})={\bf v}(b_i), \end{eqnarray} and by using the normality of $E_n(R[X])$ again, we see that \begin{eqnarray} B\in SL_2(R[X])E_n(R[X]) \end{eqnarray} and this proves the claim. \hspace*{\fill}{\bf $\Box$} \bigskip \\ \section{Realization Algorithm for $SL_3(R[X])$} Now, we want to find a realization algorithm for the matrices of the special type in $SL_3(k[x_1,\ldots ,x_m])$, i.e. matrices of the form $\left(\!\begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1 \end{array}\!\right) \in SL_3(k[x_1,\ldots ,x_m])$. Again, by applying a change of variables, we may assume that $p\in k[x_1,\ldots ,x_m]$ is a monic polynomial in the last variable $x_m$. In view of the {\em Quillen Induction Algorithm} developed in the section 3, we see that it's enough to develop a realization algorithm for the matrices of the form $\left(\!\begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1 \end{array}\! \right) \in SL_3(R[X])$, where $R$ is now a commutative local ring and $p\in R[X]$ is a monic polynomial. A realization algorithm for this case was obtained by M.P. Murthy, and we present in the below a slightly modified version of the {\bf Lemma 3.6} in \cite{murthy} {\sl Suslin's Work on Linear Groups over Polynomial Rings and Serre Problem} by S.K. Gupta and M.P. Murthy. \begin{lemma}\label{lemma:split} Let $L$ be a commutative ring, and $a,a',b\in L$. Then, the followings are true. \begin{enumerate} \item $(a,b)$ and $(a',b)$ are unimodular over $L$ if and only if $(aa',b)$ is unimodular over $L$. \item For any $c,d\in L$ such that $aa'd-bc=1$, there exist $c_1,c_2,d_1,d_2\in L$ such that $ad_1-bc_1=1,\ a'd_2-bc_2=1$, and $$\left(\!\begin{array}{ccc} aa' & b & 0 \\ c & d & 0 \\ 0 & 0 & 1 \end{array}\!\right)\equiv \left(\!\begin{array}{ccc}a&b&0\\ c_1&d_1&0\\ 0&0&1 \end{array}\!\right)\cdot \left(\!\begin{array}{ccc}a'&b&0\\ c_2&d_2&0\\ 0&0&1 \end{array}\!\right) \pmod {E_3(L)}.$$ \end{enumerate} \end{lemma} \noindent {\bf Proof:\ } (1) If $(aa',b)$ is unimodular over $L$, there exist $h_1,h_2\in L$ such that $h_1\cdot (aa')+h_2\cdot b=1$. Now $(h_1a')\cdot a+h_2\cdot b=1$ implies $(a,b)$ is unimodular, and $(h_1a)\cdot a'+h_2\cdot b=1$ implies $(a',b)$ is unimodular. Suppose, now, that $(a,b)$ and $(a',b)$ are unimodular over $L$. Then, we can find $h_1,h_2,h_1',h_2'\in L$ such that $h_1a+h_2b=1,\ h_1'a'+h_2'b=1$. Now, let $g_1=h_1h_1',\ g_2=h_2'+a'h_2h_1'$, and consider \begin{eqnarray} g_1aa'+g_2b & = & h_1h_1'aa'+(h_2'+a'h_2h_1')b \nonumber\\ & = & h_1'a'(h_1a+h_2b)+h_2'b\nonumber\\ & = & h_1'a'+h_2'b\nonumber\\ & = & 1. \end{eqnarray} So we have a desired unimodular relation. \medskip \noindent (2) If $c,d\in L$ satisfy $aa'd-bc=1$, then $(aa',b)$ is unimodular, which in turn implies that $(a,b)$ and $(a',b)$ are unimodular. Therefore, we can find $c_1,d_1,d_1,d_2\in L$ such that $ad_1-bc_1=1$ and $a'd_2-bc_2=1$. For example, we can let \begin{eqnarray} c_1=c_2=c,\quad d_1=a'd, \quad d_2=ad. \end{eqnarray} Now, consider \begin{eqnarray} \left(\!\begin{array}{ccc} aa' & b & 0 \\ c & d & 0 \\ 0 & 0 & 1 \end{array}\!\right) & = & E_{21}(cd_1d_2-d(c_2+a'c_1d_2)) \left(\!\begin{array}{ccc}aa'&b& 0\\ c_2+a'c_1d_2&d_1d_2 &0 \\ 0&0&1 \end{array}\!\right)\nonumber\\ & = & E_{21}(cd_1d_2-d(c_2+a'c_1d_2))E_{23}(d_2-1)E_{32}(1)E_{23}(-1) \nonumber\\ & & \left(\!\begin{array}{ccc} a & b & 0\\ c_1 & d_1 & 0\\ 0 & 0 & 1 \end{array}\!\right)E_{23}(1)E_{32}(-1)E_{23}(1) \left(\!\begin{array}{ccc}a'&b&0\\ c_2&d_2&0 \\ 0&0&1\end{array} \!\right)\nonumber\\ & & E_{23}(-1)E_{32}(1)E_{23}(a-1)E_{31}(-a'c_1)E_{32}(-d_1). \end{eqnarray} This explicit expression tells us that \begin{eqnarray} \left(\!\begin{array}{ccc} aa' & b & 0 \\ c & d & 0 \\ 0 & 0 & 1 \end{array}\!\right) & \equiv & \left(\!\begin{array}{ccc} a&b&0\\ c_1&d_1&0 \\ 0&0&1\end{array}\!\right) \cdot \left(\!\begin{array}{ccc}a'&b&0 \\ c_2&d_2&0\\ 0&0&1\end{array}\!\right)\pmod {E_3(L)}. \end{eqnarray} \hspace*{\fill}{\bf $\Box$} \bigskip \\ \begin{thm} Suppose $(R,M)$ is a commutative local ring, and $A=\left(\!\begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1 \end{array}\!\right) \in SL_3(R[X])$ where $p$ is monic. Then $A$ is realizable over $R[X]$. \end{thm} \noindent {\bf Proof:\ } By induction on $\deg (p)$. If $\deg (p)=0$, then $p=0\ \mbox{or}\ 1$, and $A$ is clearly realizable. Now, suppose $\deg(p)=d>0$ and $\deg(q)=l$. Since $p\in R[X]$ is monic, we can find $f,g\in R[X]$ such that \begin{eqnarray} q & = & fp+g,\quad \deg(g)<d. \end{eqnarray} Then, \begin{eqnarray} AE_{12}(-f) & = & \left(\!\begin{array}{ccc}p&q-fp&0\\ r&s-fr&0\\ 0&0 & 1 \end{array}\!\right)=\left(\!\begin{array}{ccc}p&g&0\\ r&s-fr&0\\ 0&0 & 1 \end{array}\!\right). \end{eqnarray} Hence we may assume $\deg(q)<d$. Now, we note that either $p(0)$ or $q(0)$ is a unit in $R$, otherwise, we would have $p(0)s(0)-q(0)r(0)\in M$ that contradicts to $ps-qr=p(0)s(0)-q(0)r(0)=1$. Let's consider these two cases, separately. \medskip \noindent Case 1: When $q(0)$ is a unit. \newline Using the invertibility of $q(0)$, we have \begin{eqnarray} AE_{21}(-q(0)^{-1}p(0)) & = & \left(\!\begin{array}{ccc} p-q(0)^{-1}p(0)q & q&0\\ r-q(0)^{-1}p(0)s& s &0\\ 0&0&1 \end{array}\!\right). \end{eqnarray} So, we may assume $p(0)=0$. Now, write $p=Xp'$. Then, by the above {\bf Lemma}~\ref{lemma:split}, we can find $c_1,d_1,c_2,d_2\in R[X]$ such that $Xd_1-qc_1=1,\ p'd_2-qc_2=1$ and \begin{eqnarray} \left(\!\begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1 \end{array}\!\right) & \equiv & \left(\!\begin{array}{ccc}X&q&0\\ c_1&d_1&0 \\ 0&0&1\end{array}\!\right) \cdot \left(\!\begin{array}{ccc}p'&q&0 \\ c_2&d_2&0\\ 0&0&1\end{array}\!\right)\pmod {E_3(R[X])} \end{eqnarray} Since $\deg(p')<d$, the second matrix on the right hand side is realizable by the induction hypothesis. As for the first one, we may assume that $q$ is a unit of $R$ since we can assume $\deg(q)<\deg(X)=1$ and $q(0)$ is a unit. And then invertibility of $q$ leads easily to an explicit factorization of $\left(\!\begin{array}{ccc}X&q&0\\ c_1&d_1&0 \\ 0&0&1\end{array}\!\right)$ into elementary matrices. \medskip \noindent Case 2: When $q(0)$ is not a unit. \newline First we claim the following; there exist $p',q'\in R[X]$ such that $\deg(p')<l,\deg(q')<d$ and $p'p-q'q=1$. To prove this claim, we let $r\in R$ be the resultant of $p$ and $q$. Then, there exist $f,g\in R[X]$ with $\deg(f)<l,\deg(g)<d$ such that $fp+gq=r$. Since $p$ is monic and $p,q\in R[X]$ generate the unit ideal, we see that $r\notin M$, i.e. $r\in A^{*}$. Now, letting $p'=f/r,\ q'=-g/r$ shows the claim. Also note that the two relations, $p'(0)p(0)-q'(0)q(0)=1$ and $q(0)\in M$, imply $p'(0)\notin M$. This means $q(0)+p'(0)$ is a unit. Now, consider the following. \begin{eqnarray} \left(\!\begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1 \end{array}\!\right) & = & E_{21}(rp'-sq') \left(\!\begin{array}{ccc} p & q & 0 \\ q' & p' & 0 \\ 0 & 0 & 1 \end{array}\!\right)\nonumber\\ & = & E_{21}(rp'-sq')E_{12}(-1) \left(\!\begin{array}{ccc} p+q'& q+p'&0\\ q' & p' & 0 \\ 0 & 0 & 1 \end{array}\!\right). \end{eqnarray} Noting that the last matrix on the right hand side is realizable by the Case~1 since $q(0)+p'(0)$ is a unit and $\deg(p+q')=d$, we see that $\left(\!\begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1 \end{array}\!\right)$ is also realizable. \hspace*{\fill}{\bf $\Box$} \bigskip \\ \section{Eliminating Redundancies} When applied to a specific matrix, the algorithm presented in this paper will produce a factorization into elementary matrices, but this factorization may contain more factors than is necessary. The {\em Steinberg relations} \cite{milnor} from algebraic $K$--theory provide a method for improving a given factorization by eliminating some of the unnecessary factors. The {\em Steinberg relations} that elementary matrices satisfy are \begin{enumerate} \item $E_{ij}(0) = I$ \item $E_{ij}(a)E_{ij}(b) = E_{ij}(a+b)$ \item For $i\neq l$, $[E_{ij}(a),E_{jl}(b)]=E_{ij}(a)E_{jl}(b)E_{ij}(-a)E_{jl}(-b) = E_{il}(ab)$ \item For $j\neq l$, $[E_{ij}(a),E_{li}(b)]= E_{ij}(a)E_{li}(b)E_{ij}(-a)E_{li}(-b) = E_{lj}(-ab)$ \item For $i\neq p$, $j\neq l$, $[E_{ij}(a),E_{lp}(b)]= E_{ij}(a)E_{lp}(b)E_{ij}(-a) E_{lp}(-b)=I.$ \end{enumerate} The first author is in the process of implementing the realization algorithm of this paper, together with a {\em Redundancy Elimination Algorithm} based on the above set of relations, using existing computer algebra systems. As suggested in \cite{thk}, an algorithm of this kind has application in {\em Signal Processing} since it gives a way of expressing a given multidimensional filter bank as a cascade of simpler filter banks. \section{Acknowledgement} The authors wish to thank A. Kalker, T.Y. Lam, R. Laubenbacher, B. Sturmfels and M. Vetterli for all the valuable support, insightful discussions and encouragement.

9405

alg-geom/9405013

en

https://arxiv.org/abs/alg-geom/9405013

alg-geom/9405013

Vadim Schechtman

Vladimir Hinich and Vadim Schechtman

Deformation theory and Lie algebra homology

amslatex (Replacement of the previous version. Minor corrections are made)

A description of a ring of functions on the base of a universal formal deformation for several moduli problems is given. The answer is given in terms of a homology group of a certain dg Lie algebra canonically (up to an essentially unique quasi-isomorphism) associated with a problem.

alg-geom

\section{Introduction} \subsection{} \label{pose} Let $X$ be a smooth proper scheme $X$ over a field $k$ of characteristic $0$, $G$ an algebraic group over $k$ and $p: P\lra X$ a $G$-torsor over $X$. Consider the following deformation problems. {\bf Problem 1.} Flat deformations of $X$. {\bf Problem 2.} Flat deformations of the pair $(X,P)$. {\bf Problem 3.} Deformations of $P$ ($X$ being fixed). According to Grothendieck, one can assign to a deformation problem of the above type a sheaf of Lie algebras over $X$ which can be defined as "a sheaf of infinitesimal automorphisms" of the corresponding deformation functor (cf. Section ~\ref{univers}). Let us describe the sheaves corresponding to our problems. For Problem 1 it is a tangent sheaf $\CA_1=\CT_X$. For Problem 2 it is a sheaf $\CA_2=\CA_P$ defined as follows. For a Zariski open $U\subset X$, $\Gamma(U,\CA_P)$ is the space of $G$-invariant vector fields on $p^{-1}(U)$. The map $p$ induces (epimorhic) map $\epsilon:\CA_P\lra\CT_X$. For Problem 3, $\CA_3=\fg_P:= \ker(\epsilon)$. The last sheaf may be also defined as a sheaf of sections of a vector bundle associated with $P$ and the adjoint representation of $G$ on its Lie algebra. Note that the sheaves $\CA_i$ are locally free $\CO_X$-modules of finite type (but the bracket is not $\CO_X$-linear for $i=1,2$). Suppose that $H^0(X,\CA_i)=0$. Then it is known that there exists a universal formal deformation space $\fS_i=\Spf(R_i)$ for Problem $i$ (see Section ~\ref{univers}). Here $R_i$ is a complete local $k$-algebra with residue field $k$. Let $\bm_{R_i}$ denote its maximal ideal. We have the Kodaira-Spencer isomorphism \begin{equation} \label{ksg} \kappa^1: T_{\fS_i,s}=(\bm_{R_i}/\bm_{R_i}^2)^*\iso H^1(X,\CA_i) \end{equation} describing the tangent space of $\fS_i$ at the closed point $s$. The main goal of the present paper is to describe, in case when $\fS_i$ is smooth, the whole ring $R_i$ in terms of the sheaf $\CA_i$. \subsection{} \label{sullivan} To formulate the answer, we need a certain cohomological construction. Let $\fg$ be a sheaf of $k$-Lie algebras over $X$ which is also a quasicoherent $\CO_X$-module. Consider an affine open covering $\CU$ of $X$ and the corresponding complex of \v{C}ech cochains $\CHC(\CU,\fg)$. Using a generalization of Thom-Sullivan construction \footnote{introduced in ~\cite{hlha}} used in Rational homotopy theory one can construct (see Section ~\ref{direct}) a certain differential graded Lie algebra $R\Gamma^{Lie}(X,\fg)$ canonically quasi-isomorphic to $\CHC(\CU,\fg)$. This dg Lie algebra does not depend, up to (essentially) unique quasi-isomorphism, on a covering, hence we omit it from the notation. Now one can apply to $R\Gamma^{Lie}(X,\fg)$ the Quillen functor $C$ (which is a generalization to dg Lie algebras of the Chevalley complex computing the homology of a Lie algebra with trivial coefficients) and get the complex $C(R\Gamma^{Lie}(X,\fg))$. This complex carries the canonical increasing filtration $\{ F_nC(R\Gamma^{Lie}(X,\fg))\}$ with graded quotients isomorphic to symmetric powers $S^n(R\Gamma^{Lie}(X,\fg)[1])$ (for details, see \ref{quillen}). Let us define homology spaces $$ H^{Lie}_i(R\Gamma^{Lie}(X,\fg)):=H^{-i}(C(R\Gamma^{Lie}(X,\fg)));\ F_nH^{Lie}_i(R\Gamma^{Lie}(X,\fg)):=H^{-i}(F_nC(R\Gamma^{Lie}(X,\fg))). $$ These spaces depend only on $X$ and on the sheaf $\fg$. We have evident maps $$ \ldots\lra F_{n-1}H^{Lie}_i(R\Gamma^{Lie}(X,\fg))\lra F_nH^{Lie}_i(R\Gamma^{Lie}(X,\fg))\lra\ldots\lra H^{Lie}_i(R\Gamma^{Lie}(X,\fg)) $$ and $$ F_nH^{Lie}_i(R\Gamma^{Lie}(X,\fg))\lra H^{n-i}(\Lambda^iR\Gamma^{Lie}(X,\fg)) $$ ($\Lambda^i$ denotes the exterior power). \subsection{} Return to the assumptions ~\ref{pose}. Pick $i=1,2$ or $3$. Recall that we suppose that $H^0(X,\CA_i)=0$. Suppose also that $\fS_i$ is {\em smooth} i.e. isomorphic to a formal power series ring over $k$. Let $R^*_i$ denote the space of continuous $k$-linear maps $R_i\lra k$ ($k$ is equipped with the discrete topology). The main result of this paper is (see Thm.~\ref{complet}). \subsubsection{} \label{thm-intro} \begin{thm}{} One has compatible canonical isomorphisms \begin{equation} \label{isomor} \kappa: R^*_i\iso H^{Lie}_0(R\Gamma^{Lie}(X,\CA_i));\ \end{equation} \begin{equation} \kappa^{\leq n}: (R_i/\bm^{n+1}_{R_i})^*\iso F_nH^{Lie}_0(R\Gamma^{Lie}(X,\CA_i)) \end{equation} After passing to the graded quotients, isomorphisms $\kappa^{\leq n}$ induce isomorphisms $$ S^n(T_{\fS_i,s})\cong (\bm^{n}/\bm^{n+1})^*\iso H^{n}(\Lambda^nR\Gamma^{Lie}(X,\CA_i))\cong S^nH^1(X,\CA_i) $$ which coincide with $(-1)^n\kappa^1$. \end{thm} \subsection{} Let us describe the contents of the paper in a more detail. Our construction of the isomorphism $\kappa$ is based on the construction of {\bf higher Kodaira-Spencer morphisms} --- they are relative versions of ~(\ref{isomor}). Similarly to a usual KS map they are not necessarily isomorphisms and are defined for arbitrary --- not necessarily universal --- deformation\footnote{this idea is used in ~\cite{bs}}. In the same way as the usual KS map comes from a certain boundary homomorphism of a short exact sequence of complexes, our higher KS map comes from a map, which we call {\bf connecting morhism}. It arises from an extension of {\em dg Lie algebras} $$ 0\lra\fg\lra\fa\lra\ft\lra 0 $$ and maps the enveloping algebra of the cone of the map $\fg\ra\fa$ (which is quasi-isomorphic to $U(\ft)$) to the standard complex $C(\fg)$. The construction is given in Section ~\ref{envstand}, cf. ~\ref{constr-thm}. For applications we need a generalization of this construction to {\em dg Lie algebroids}. The simplest example of a sheaf of Lie algebroids is the tangent sheaf $\CT_S$ over a variety $S$. We borrow their definition from ~\cite{bb} and ~\cite{bfm}. Following {\em loc.cit.}, one defines an analogue of enveloping algebra for Lie algebroids --- we call them {\em twisted enveloping algebras}. For example, the twisted enveloping algebra of $\CT_S$ will be the sheaf $\Diff_S$ of differential operators provided $S$ is smooth. These sheaves are sources of higher KS maps. Section ~\ref{twisted} generalizes the principal construction of Section ~\ref{envstand} to (sheaves of) dg Lie algebroids. The construction of connecting morphisms is our first point. (They may be of an independent interest.) Section ~\ref{formal} is technical; we discuss there some basic properties of differential operators on formal schemes. \subsection{} In Sections~\ref{direct} and~\ref{thoms} we develop certain formalism of {\bf Homotopy Lie algebras}. We define there for (locally noetherian) schemes $X$ categories $\Holie^{qc}(X)$ which contain sheaves of dg Lie algebras with bounded cohomology and flat quasicoherent components, and have good functorial properties. For example, the functor $R\Gamma^{Lie}$ mentioned in ~\ref{sullivan} is a particular case of the {\bf direct image} functor --- its construction is the main result of these Sections. Although it is a technical tool, we regard the contents of Sect.~\ref{direct} and ~\ref{thoms} as a {\bf second main outcome} of the present paper. We believe that unlike usual types of algebras, {\em Homotopy Lie algebras} are correctly defined only as objects of appropriate {\em Homotopy categories}. The main property one needs from them is good functorial behaviour. This is what Thom-Sullivan functor does. The success of the construction is due to the fact that Thom-Sullivan functor has excellent exactness, base change, etc. properties. They are proved in ~\ref{thoms}. Another approach to the definition and functoriality of Homotopy Lie algebras has been developed in ~\cite{hla}. In {\em loc.cit.} a HLA was a "dg Lie algebra up to higher homotopies". There are strong indications that both definitions give rise to equivalent Homotopy categories. It seems that one can use also cosimplicial Lie algebras for the construction of the same Homotopy category. From this point of view, different versions of HLA are nothing but different "models" of these Homotopy categories. We believe that every "model" might be useful in applications. Of course these remarks apply also to other types of algebras. \subsection{} In Section ~\ref{kodaira} we take all the results together and start the machine of Section ~\ref{twisted} which cooks up higher KS maps for us. In Section ~\ref{univers} we discuss universal formal deformations. \subsection{} One can generalize the above approach and get a description of sheaves of differential operators on moduli spaces acting on natural vector bundles (for example, determinant bundles) in terms of Lie algebra homology (with non-trivial coefficients). For example, for operators of order $\leq 1$ on determinant bundles one gets a result equivalent to ~\cite{bs}. Also there are strong indications that combining our description with Serre duality it is possible to get a "global counterpart" of the results of ~\cite{bd}. We will return to these subjects in future publications. \subsection{} The general main idea that {\bf the completion of a local ring of a moduli space at a given point $X$ is isomorphic to the dual of the $0$-th homology group of the "Lie algebra of infinitesimal automorphisms of $X$"} was spelled out very clearly a few years ago by Drinfeld, Deligne, Feigin (cf. ~\cite{d}, ~\cite{del}, \cite{f}). We knew this idea from Drinfeld. For deformations of complex structures a result analogous to Theorem ~\ref{thm-intro} was proven in ~\cite{gm} (by different argument). The present paper develops further the results of ~\cite{hdt}. Note also some recent related work: ~\cite{ev}, \cite{r}. This paper owes much to various ideas of A.Beilinson and V.Drinfeld. We express to them our deep gratitude. We thank H.~Esnault, V.~Ginzburg and E.~Viehweg for useful discussions. We are especially grateful to Professor Han Sah who made possible several visits of V.H. to Stony Brook. \section{Enveloping algebras and standard complexes} \label{envstand} \subsection{Preliminaries} \subsubsection{} Let us fix some notations and sign conventions (cf. ~\cite{de}, 1.1). Throughout this Section we fix a commutative ground ring $k$ of containing $\Bbb Q$. $\Mod (k),\ \Gr (k)$ will denote a category of $k$-modules and that of $\Bbb Z$-graded $k$-modules respectively. $\CC (k)$ denotes the category of complexes of $k$-modules (all differentials have degree $+1$). We have an obvious forgetful functor $\CC (k)\lra \Gr (k)$. If $X\in \CC (k)$ (or $\Gr (k)$), $x\in X^p$, we refer to $p$ as to degree of $x$, and denote it $|x|$. We identify $k$-modules with complexes concentrated in degree $0$. If $n\in \Bbb Z$, $X\in \Gr (k)$, $X[n]$ will denote the shifted module $X[n]^p=X^{p+n}$. If $X\in \CC(k)$, we define $X[n]\in \CC(k)$ which is as above as a graded module, the differential being $d_{X[n]}=(-1)^nd_X$. {\em A map of degree $n$}, $f: X\lra Y$ in $\CC(k)$ is by definition a morphism $f: X\lra Y[n]$ in $\Gr (k)$. For such an $f$ we set $df=d_Y\circ f+ (-1)^nf\circ d_X: X\lra Y[n+1]$. The category $\CC (k)$ has a tensor structure --- for $X,Y\in \CC (k)$, $X\otimes Y$ is the usual tensor product of complexes over $k$. We have natural associativity and commutativity isomorphisms \begin{equation} \label{assoc} a_{X,Y,Z}:(X\otimes Y)\otimes Z\iso X\otimes (Y\otimes Z),\ (x\otimes y)\otimes z\mapsto x\otimes (y\otimes z) \end{equation} and \begin{equation} \label{r} R_{X,Y}: X\otimes Y\iso Y\otimes X \end{equation} defined by the formula $R_{X,Y}(x\otimes y)=(-1)^{|x||y|}y\otimes x$. This formula, as well as other formulas in the "dg world" are obtained by implementing {\em the Quillen sign rule}: "whenever something of degree $p$ is moved past something of degree $q$ the sign $(-1)^{pq}$ accrues", {}~\cite{q}, p. 209. Isomorphisms ~(\ref{assoc}) and ~(\ref{r}) endow $\CC(k)$ with a structure of a {\em symmetric monoidal category} in the sense of MacLane, ~\cite{mac}. We have canonical {\em shifting} isomorphisms \begin{equation} \label{shift} X[n]\otimes Y[m]\iso (X\otimes Y)[n+m] \end{equation} sending $x\otimes y$, $x\in X^i,\ y\in Y^j$ to $(-1)^{im}x\otimes y$. (They may be obtained by identifying (following ~\cite{de}, 1.1) $X[n]$ with $k[n]\otimes X$ , and applying $R_{X,k[m]}$). {\em Algebras.} The structure of a symmetric monoidal category allows one to define usual types of algebras in $\CC (k)$. We shall refer to them by adding "dg" to their original name. More specifically, we shall use the following algebras (cf. ~\cite{q}). --- A {\em dg Lie algebra} is a complex $X$ together with a bracket $[,]: X\otimes X\lra X$ which is skew symmetric, i.e. $[,]\circ R_{X,X}=-[,]$, and satisfies the Jacobi identity $$ [x,[y,z]]+(-1)^{|x|(|y|+|z|)}[y,[z,x]]+(-1)^{|z|(|x|+|y|)}[z,[x,y]]=0 $$ We shall denote $\Dglie=\Dglie_k$ the category of dg Lie algebras over $k$. --- A {\em dg coalgebra} is a complex $X$ together with an coassociative comultiplication $\Delta: X\lra X\otimes X$ and a counit $\epsilon: X\lra k$. $X$ is called {\em cocommutative} if $\Delta= R_{X,X}\circ \Delta$. We shall denote $\Dgcoalg=\Dgcoalg_k$ the category of cocommutative dg coalgebras over $k$. --- A {\em dg algebra} is a complex $X$ together with an associative multiplication $\mu: X\otimes X\lra X$ and the unit $1\in X^0$ (which may be considered as a map $k\lra X$). $X$ is called {\em commutative} is $\mu=\mu\circ R_{X,X}$. --- A {\em dg Hopf algebra} is a complex $X$ together with a multiplication $\mu$, comultiplication $\Delta$, a unit and a counit, making it a dg algebra and a dg coalgebra, and such that these two structures are compatible in the standard way. In particular, $\Delta$ is a map of dg algebras, where the multiplication in $X\otimes X$ is defined by the rule $(x\otimes y)(x'\otimes y')=(-1)^{|y||x'|}xx'\otimes yy'$. $X$ is called (co)commutative if the underlying (co)algebra is. An element $x\in X$ is called {\em primitive} if $\Delta (x)=x\otimes 1+1\otimes x$. \subsubsection{} \label{connected} An element $u$ of a dg coalgebra $C$ is called {\em group-like} if the following conditions are fulfilled: (i) $du=0$ (ii) $\Delta(u)=u\otimes u$ (iii) $\epsilon(u)=1\in k$ A group-like element $u\in C$ defines a decomposition $C=k\cdot u\oplus C^+$ with $C^+=\ker(\epsilon)$. Let $\pi_u:C\ra C^+$ be the projection onto the second summand. For a positive integer $n$ define the map $$ \Delta^{(n)}: C\lra C^{\otimes n} $$ by induction on $n$: set $\Delta^{(1)}=\id_C, \ \Delta^{(n)}=(\Delta\otimes \id_{C^{\otimes(n-2)}})\circ \Delta^{(n-1)}$. The choice of a group-like element $u$ defines an increasing filtration $F^u=\{F^u_n\}$ of $C$ by the formula $$ F^u_n=\ker\left(C\overset{\Delta^{(n+1)}}{\lra}C^{\otimes n+1} \overset{\pi_u^{\otimes n+1}}{\lra}(C^+)^{\otimes n+1}\right).$$ \begin{defn}{unital} 1. A group-like element $u\in C$ is called {\em a unit} if the corresponding filtration $F^u$ is exhaustive: $C=\cup_{i=0}^{\infty} F_i$. 2. {\em A unital coalgebra} is a pair $(C,1_C)$ where $C$ is a coalgebra and $1_C\in C$ is a unit it in. \end{defn} The category $\Dgcu(k)$ has as objects unital cocommutative dg $k$-coalgebras; a morphism in $\Dgcu(k)$ is a coalgebra morphism preserving the units. \subsubsection{} Let $X\in \CC (k)$. For an integer $n\geq 1$ denote $T^n(X)$ its $n$-th tensor power $X\otimes\ldots \otimes X$. Set $T^0X=k$. The direct sum $T(X)=\oplus_{n\geq 0} T^n(X)$ has a natural structure of a dg algebra --- the {\em tensor algebra} of $X$. The commutativity isomorphisms ~(\ref{r}) define the action of the symmetric group on $n$ letters $\Sigma_n$ on $T^n(X)$. We denote $S^n(X)$, $\Lambda^n(X)$ the complexes of coinvariants (resp., coantiinvariants) of this action and $\pi_{S,n}: T^n(X)\lra S^n(X)$, $\pi_{\wedge,n}: T^n(X)\lra \Lambda^n(X)$ the canonical projections. For $x_1,\ldots, x_n\in X$ we set $$ x_1\cdot\ldots\cdot x_n=\pi_{S,n}(x_1\otimes\ldots \otimes x_n)\in S^n(X); $$ $$ x_1\wedge\ldots\wedge x_n=\pi_{\wedge,n}(x_1\otimes\ldots \otimes x_n)\in \Lambda^n(X) $$ The projections $\pi_{S,n},\ \pi_{\wedge,n}$ have canonical sections \begin{equation} \label{sym} i_{S,n}: S^n(X)\lra T^n(X),\ i_{S,n}(x_1\cdot\ldots\cdot x_n)= \frac{1}{n!}\sum_{\sigma\in \Sigma_n}\sigma(x_1\otimes\ldots\otimes x_n) \end{equation} \begin{equation} \label{antisym} i_{\wedge,n}: \Lambda^n(X)\lra T^n(X),\ i_{\wedge,n}(x_1\cdot\ldots\cdot x_n)= \frac{1}{n!}\sum_{\sigma\in \Sigma_n}(-1)^{|\sigma|} \sigma(x_1\otimes\ldots\otimes x_n) \end{equation} The isomorphisms ~(\ref{shift}) induce canonical isomorphisms $$ a_n: T^n(X[1])\cong T^n(X)[n] $$ such that for $\sigma\in \Sigma_n$, $a_n\circ \sigma=(-1)^{|\sigma|}\sigma \circ a_n$. After passing to coinvariants, we get canonical shifting ({\em "d\'{e}calage"}) isomorphisms \begin{equation} \label{dec} \dec_n:S^n(X[1])\cong \Lambda^n(X)[n] \end{equation} Explicit formula: for $x_i\in X^{p_i},\ i=1,\ldots , n$, \begin{equation} \label{decalage} \dec_n(x_1\cdot\ldots\cdot x_n)=(-1)^{\sum_{i=1}^n (n-i)p_i} (x_1\wedge\ldots\wedge x_n) \end{equation} \subsubsection{} The direct sum $S(X)=\sum_{n\geq 0} S^n(X)$ is naturally a commutative dg algebra. Let us define the counit $S(X)\lra k=S^0(X)$ as a canonical projection, and a map $\Delta: S(X)\lra S(X)\otimes S(X)$ by two conditions which characterize it uniquely: (i) $\Delta(x)=x\otimes 1 + 1\otimes x$ for $x\in X=S^1(X)$ (ii) $\Delta$ is a map of dg algebras. This makes $S(X)$ a commutative and cocommutative dg Hopf algebra, cf. {}~\cite{q}, App B, 3.3. The argument of {\em loc.cit.} shows that $S(X)$ is a unital dg coalgebra with $1_{S(X)}=1\in k=S^0(X)$. The corresponding filtration is $F_iS(X)=\oplus_{p=0}^iS^p(X)$. \subsubsection{Universal property of $S(X)$} \label{coalg-s} Let $C\in\Dgcu$ and suppose we are given a map of unital dg coalgebras $f: C\lra S(X)$. Set $f_n=p_n\circ f: C\lra S^n(X)$ where $p_n: S(X)\lra S^n(X)$ is the projection. We have $f_0=\epsilon_C$. \begin{lem}{} For $n\geq 1$ we have $$ f_n=\frac{1}{n!}\pi_{S,n}\circ f_1^{\otimes n}\circ \Delta^{(n)} $$ \end{lem} \begin{pf} This follows from the compatibility of $f$ with the comultiplication. \end{pf} As a consequence, we get \begin{prop}{} (\cite{q}, App. B, 4.4) The assignment $f\mapsto f_1$ establishes a bijection between the set of all unital dg coalgebra maps $f: C\lra S(X)$ and the set of maps $f_1: C\lra X$ in $\CC (k)$ such that $f_1(1_C)=0$. \end{prop} This follows from the previous lemma and the remark that for $x\in F_iC$ we have $f_n(x)=0$ for $n>i$. $\Box$ \subsubsection{Universal enveloping algebra} \label{envel} If $\fg$ is a dg Lie algebra, its {\em universal enveloping algebra $U(\fg)$} is a dg algebra which is a quotient of the tensor algebra $T(\fg)$ by the two-sided dg ideal generated by all elements $$ xy-(-1)^{|x||y|}yx -[x,y], $$ $x,y\in \fg=T^1(\fg)$ homogeneous. The composition $\fg=T^1(\fg)\hra T(\fg)\lra U(\fg)$ is injective; one identifies $\fg$ with the subcomplex of $U(\fg)$. Evidently, $U(\fg)$ is generated by $\fg$ as a dg algebra. We shall denote $F_iU(\fg)\subset U(\fg)$ the subspace spanned by all products of $\leq i$ elements of $\fg$. We have $F_0U(\fg)=k\cdot 1$. $U(\fg)$ has a canonical structure of a unital cocommutative dg Hopf algebra, the comultiplication being defined uniquely by the requirement that $\fg$ consists of primitive elements. (See ~\cite{q} for details.) $U(\fg)^+\subset U(\fg)$ will denote the kernel of the counit ({\em augmentation ideal}); one has a canonical decomposition $U(\fg)\cong k\cdot 1\oplus U(\fg)^+$. \subsection{Quillen standard complex} \label{quillen} \subsubsection{} Let $X\in \CC (k)$. Consider the dg coalgebra $C(X):=S(X[1])$. Using the d\'{e}calage isomorphisms we identify $C(X)$ with $\oplus_{n\geq 0} \Lambda^n(X)[n]$. Let us write down the precise formula for the comultiplication in $C(X)$. Let $x_i\in X^{\alpha_i},\ i=1,\ldots , n$. For a finite subset $I=\{ p_1,\ldots , p_i\}\subseteq\{ 1,\ldots , n\};\ p_1<\ldots <p_i$, set $$ x_{I}=x_{p_1}\wedge x_{p_2}\wedge\ldots \wedge x_{p_i}\in \Lambda^i(X), \ x_{\emptyset}=1, $$ $\bar I:=\{1,\ldots,n\}-I$. Then \begin{equation} \label{comult-in-c} \Delta(x_1\wedge\ldots\wedge x_n)=\sum_I(-1)^{s(I)}x_I\otimes x_{\bar I} \end{equation} where the summation is over all subsets $I\subseteq \{1,\ldots,n\}$, and signs $s(I)$ are defined by the rule $$ x_1\wedge\ldots\wedge x_n=(-1)^{s(I)}x_I\wedge x_{\bar I} $$ For future, we shall denote this sign $s(I;\alpha_1,\ldots \alpha_n)$. Set $$ C(X)^{pq}=(\Lambda^{-p}(X))^q $$ (we agree that $\Lambda^p=0$ for $p<0$). It is clear that $C(X)^n=\oplus_{p+q=n} C(X)^{pq}$. The differential in $C(X)$ has bidegree $(0,1)$. For future, let us denote it $d_{II}$. Note that $d_{II}^{-p,*}:\Lambda^p(X)^*\lra \Lambda^p(X)^{*+1}$ is equal to $(-1)^p$ times the differential on $\Lambda^p(X)$ induced by that on $X$. \subsubsection{} Now suppose that $\fg$ is a dg Lie algebra. Let us define maps $$ d_n: \Lambda^n(\fg)\lra \Lambda^{n-1}(\fg) $$ by the formula $$ d_n(x_1\wedge\ldots \wedge x_n)=\sum_{1\leq i<j\leq n}(-1)^{s(\{ i,j\}; \alpha_1,\ldots, \alpha_n)} [x_i,x_j]\wedge x_{\ol{\{ i,j\}}} $$ where $x_i\in X^{\alpha_i}$. In particular, (i) {\em the composition $d_2\circ \pi_{\wedge, 2}$ coincides with the bracket on $\fg$.} Set $$ d_I^{pq}=(d_{-p})^q: C(\fg)^{pq}\lra C(\fg)^{p+1,q};\ d_I=\sum_{p,q} d_I^{pq} $$ Set $d=d_I+d_{II}$; it is an endomorphism of degree $1$ of the graded $k$-module $C(\fg)$. \begin{prop}{} (a) The map $d$ has the following properties. (ii) $d^2=0$. In particular, $d_I^2=0$ and $d_Id_{II}+d_{II}d_I=0$. (iii) The comultiplication $\Delta: C(\fg)\lra C(\fg)\otimes C(\fg)$ is compatible with $d$. (b) Given $d_{II}$, the differential $d_I$ is uniquely determined by the properties (i), (ii), and (iii). (c) Conversely, given $d=d_I+d_{II}$ satisfying (ii) and (iii), define the bracket $\fg\otimes \fg\lra \fg$ as the composition $(d_I)^{-2,*}\circ \pi_{\wedge, 2}$. This endows $\fg$ with the structure of a dg Lie algebra. \end{prop} $\Box$ So, for a dg Lie algebra $\fg$ we get a unital cocommutative dg coalgebra $C(\fg)$ which is called {\em the standard complex} of $\fg$. It was introduced by Quillen, \cite{q}, App. B, no. 6. \subsubsection{Maurer-Cartan condition} Let $A$ be a unital cocommutative dg coalgebra, and $f: A\lra C(\fg)$ be a map in $\Dgcu$. Let us denote $f_i: A\lra S^i(\fg[1])$ the composition $p_i\circ f$ where $ p_i$ is the projection $p_i: C(\fg)\lra S^i(\fg[1])$. Define the map $[f_1,f_1]: A\lra \fg[2]$ in $\Gr (k)$ as follows: if $x\in A,\ \Delta (x)=\sum y_i\otimes z_i$, set $[f_1,f_1](x)=\sum (-1)^{|y_i|} [f_1(y_i),f_1(z_i)]$. On the other hand, consider $df_1=d_{\fg}\circ f_1+f_1\circ d_A: A\lra \fg[2]$. For $x\in A$ we have $f(dx)=f_0(dx)+f_1(dx)+\ldots$. Since $f(dx)=d\circ f(x)=(d_I+d_{II})\circ f(x)$, we have $$ f_1(dx)=d_I\circ f_2(x)+d_{II}\circ f_1(x)= d_I\circ f_2(x)-d_{\fg}\circ f_1(x) $$ On the other hand, if $\Delta (x)=\sum y_i\otimes z_i$, we have by {}~\ref{coalg-s} $$ f_2(x)=\frac{1}{2}\sum f_1(y_i)\cdot f_2(z_i)=\frac{1}{2}\sum (-1)^{|y_i|+1} f_1(y_i)\wedge f_1(z_i) $$ whence $$ d_I\circ f_2 =-\frac{1}{2}[f_1,f_1] $$ Hence we get \begin{equation} \label{mc} df_1+\frac{1}{2}[f_1,f_1]=0 \end{equation} --- the {\em Maurer-Cartan} equation. Let us denote $MC(A,\fg)$ the set of all maps $f_1: A\lra \fg[1]$ in $\Gr (k)$ satisfying ~(\ref{mc}) and such that $f_1(1_A)=0$. \subsubsection{} \begin{prop}{} (~\cite{q}, App. B, 5.3) The assignment $f\mapsto f_1$ yields a bijection $$ \Hom_{\Dgcu}(A,C(\fg))\cong MC(A,\fg) $$ \end{prop} $\Box$ \subsubsection{} \label{formula} Suppose that $A$ is a dg Hopf algebra, $a_1,\ldots a_n\in A^0$ primitive elements. For a subset $I\subset \{1,\ldots,n\}$ set $a_I=a_{i_1}\cdot a_{i_2}\cdot\ldots\cdot a_{i_s}$ where $I=\{ i_1,\ldots i_s\}$; $i_1<i_2<\ldots < i_s$; $a_{\emptyset}=1$. Let us call a {\em $p$-partition} of $\{1,\ldots,n\}$ a sequence $P=(I_1,\ldots ,I_p)$ of subsets $I_j\subset \{1,\ldots,n\}$ such that $\{1,\ldots,n\}$ is the disjoint union $I_1\cup\ldots\cup I_p$. Denote $\CP_p(n)$ the set of all $p$-partitions. One computes without difficulty that $$ \Delta^{(p)}(a_1\cdot\ldots\cdot a_n)=\sum_{P=(I_1,\ldots ,I_p)\in \CP_p(n)} a_{I_1}\otimes\ldots\otimes a_{I_p} $$ Suppose we are given a dg coalgebra map $f: A\lra C(\fg)$. It follows form {}~\ref{coalg-s} that $$ f_p(a_1\cdot\ldots\cdot a_n)=\frac{1}{p!} \sum_{P=(I_1,\ldots ,I_p)\in \CP_p(n)} f_1(a_{I_1})\cdot\ldots\cdot f_1(a_{I_p}) \in S^p(\fg^1) $$ In particular, \begin{equation} f_n(a_1\cdot\ldots\cdot a_n)=f_1(a_1)\cdot\ldots\cdot f_1(a_n) \end{equation} \subsection{Connecting morphism} \subsubsection{Conic dg Lie algebras} \label{conic} Let $\fg$ be a dg Lie algebra; $\fh \subset \fg$ a dg Lie ideal. Denote by $i: \fh \lra \fg$ the embedding. Define a dg Lie algebra $\fX$ as follows. Set $\fX^n=\fh^{n+1}\oplus \fg^n$. The differential $d:\fX^n\lra \fX^{n+1}$ sends $(h,g)$ to $(-dh,i(h)+dg)$. So, as a complex, $\fX$ is the usual cone of $i$. The bracket in $\fX$ is defined as follows. We have $\fX=\fh [1]\oplus \fg$ (as graded modules). The bracket $\fX\otimes \fX\lra \fX$ has components: $\fg\otimes \fg\lra \fg$ is the bracket in $\fg$; $\fh [1]\otimes\fg\cong (\fh\otimes\fg)[1]\lra \fh[1]$ and $\fg\otimes\fh [1]\cong (\fg\otimes\fh)[1]\lra \fh[1]$ are compositions of the shifting isomorphisms ~(\ref{shift}) and the adjoint action of $\fg$ on $\fh$. Explicitely: \begin{equation} \label{bracket} [(h,g),(h',g')]=((-1)^a[g,h']+[h,g'],[g,g']) \end{equation} for $g\in \fg^a$. Define maps in $\Gr (k):\ \phi: \fX\lra \fh[1],\ \phi (h,g)=h;\ \theta: \fX\lra \fg,\ \theta (h,g)=g.$ Note that $\phi$ is a map of complexes. On the other hand, $\theta$ is a morphism of graded (not dg) Lie algebras. We have \begin{equation} \label{dtheta} d\theta=-i\circ \phi \end{equation} Consequently, $\theta$ induces the map $\Theta: U(\fX)\lra U(\fg)$ of enveloping algebras which is a morphism of graded (not dg) Hopf algebras. \subsubsection{Construction of the connecting morphism} \label{constr} Define the map $\tilde c_1: T(\fX) \lra \fh[1]$ in $\Gr (k)$ as follows. Set $$ \tilde c_1|_{T^0(\fX)}=0,\ \tilde c_1|_{T^1(\fX)}=\phi $$ Suppose we have defined $\tilde c_1$ on $T^n(\fX)$ for $n\geq 1$. For $u\in T^n(\fX),\ x\in \fX$ set \begin{equation} \label{tc} \tilde c_1(xu)= (-1)^{|x|}\ad_{\fg}(\theta (x))(\tilde c_1(u)) \end{equation} This defines $\tilde c_1$ on $T^{n+1}(\fX)$. Here $\ad_{\fg}$ denotes the adjoint action of $\fg$ on $\fh$: $\ad_{\fg}(g)(h)=[g,h]$. Let us denote $\ad_{U(\fg)}$ the induced action of $U(\fg)$ on $\fh$. {}~(\ref{tc}) implies the important equality: \begin{equation} \label{imp} \tilde c_1(uv)=(-1)^{|u|}\ad_{U(\fg)}(\Theta(u))(\tilde c_1(v)) \end{equation} for all $u\in T(\fX), v\in T^+(\fX):=\sum_{n>0} T^n(\fX)$. \subsubsection{} \begin{thm}{constr-thm} (i) The map $\tilde c_1$ vanishes on the kernel of the projection $T(\fX)\lra U(\fX)$, and hence it induces the map \begin{equation} \label{c1} c_1: U(\fX)\lra \fh[1] \end{equation} (ii) $c_1$ satisfies the Maurer-Cartan equation ~(\ref{mc}). Consequently, $c_1$ defines the map of unital dg coalgebras \begin{equation} \label{conn} c: U(\fX)\lra C(\fh) \end{equation} \end{thm} We will call $c$ {\bf connecting morphism}. \begin{pf} (i) We have to prove that \begin{equation} \label{toprove} \tilde c_1(uxyv-(-1)^{ab}uyxv-u[x,u]v)=0 \end{equation} for all $u,v\in T(\fX),\ x\in \fX^a,\ y\in \fX^b$. If $v\in T^+(\fX)$ this follows from ~(\ref{imp}); so we can suppose $v=1$. Again, from ~(\ref{imp}) follows that it suffices to prove ~(\ref{toprove}) for $u=1$. This reduces to proving that $$ (-1)^a[\theta(x),\phi(y)]-(-1)^{(a+1)b}[\theta(y),\phi(x)]=\phi([x,y]) $$ which is equivalent to ~(\ref{bracket}). (ii) Recall the canonical filtration $F_nU(\fX)$ from ~\ref{envel}. Let us prove by induction on $n$ that \begin{equation} \label{mcprove} (dc_1+\frac{1}{2}[c_1,c_1])(u)=0 \end{equation} for all $u\in F_nU(\fX)$. If $n\leq 1$ then both summands in ~(\ref{mcprove}) are $0$. Suppose we have $x\in \fX^a,\ u\in U(\fX)^b$ such that $\epsilon(u)=0$ where $\epsilon: U(\fX)\lra k$ is the counit. We have \begin{multline} (dc_1)(xu)=d(c_1(xu))+c_1(d(xu))=(-1)^ad(\theta(x),c_1(u)])+ c_1(dx\cdot u)+ (-1)^ac_1(x\cdot du)= \\ =(-1)^ad([\theta(x),c_1(u)])+(-1)^{a+1}[\theta(dx),c_1(u)]+[\theta(x),c_1(du)] \\ \label{eq1} \end{multline} Note that \begin{eqnarray} [\theta(x),c_1(du)]=[\theta(x),(dc_1)(u)]-[\theta(x),d(c_1(u))]= \nonumber \\ =[\theta(x),(dc_1)(u)]+(-1)^{a+1}d([\theta(x),c_1(u)])+ (-1)^a[d(\theta(x)),c_1(u)] \nonumber \end{eqnarray} (we have used that $d([g,h])=[dg,h]+(-1)^a[g,dh]$ for $g\in \fg^a$). Substituting this in ~(\ref{eq1}) we get \begin{equation} \label{dcone} (dc_1)(xu)=(-1)^a[d(\theta(x))-\theta(dx),c_1(u)]=(-1)^{a+1}[\phi(x),c_1(u)] \end{equation} (we have used ~(\ref{dtheta})). Suppose that $\Delta(u)=u\otimes 1+1\otimes u+\sum_iu'_i\otimes u''_i$. Let $b,b'_i,b''_i$ be the degrees of $u,u'_i,u''_i$ respectively. Then $$ \Delta(xu)=xu\otimes 1+1\otimes xu+x\otimes u+(-1)^{ab}u\otimes x+ \sum_ixu'_i\otimes u''_i+\sum_i(-1)^{ab'_i}u'_i\otimes xu''_i $$ so that \begin{multline} \label{conecone} [c_1,c_1](xu)=2(-1)^a[\phi(x),c_1(u)]+\sum_i(-1)^{a+b'_i}[c_1(xu'_i),c_1(u''_i)] + \\ +\sum_i(-1)^{(a+1)b'_i}[c_1(u'_i),c_1(xu''_i)]=2(-1)^a[\phi(x),c_1(u)]+ \ad_{\fg}(\theta(x))([c_1,c_1](u)) \end{multline} Adding up ~(\ref{dcone}) and ~(\ref{conecone}) we get $$ (dc_1+\frac{1}{2}[c_1,c_1])(xu)=\ad_{\fg}(\theta(x)) ((dc_1+\frac{1}{2}[c_1,c_1])(u)) $$ Now ~(\ref{mcprove}) follows by induction on $n$. This proves the theorem. \end{pf} \subsubsection{} Let $\fa$ be a graded Lie algebra, $M$ a graded $\fa$-module, $\phi: \fa\lra M$ a $1$-cocycle of $\fa$ with values in $M$, i.e. $$ \phi([a,b])=a\phi(b)-(-1)^{|a||b|}b\phi(a) $$ for $a,b\in \fa$. Then there exists a unique map of graded modules $$ c_1^+: U(\fa)^+\lra M $$ such that (i) $c_1^+|_{\fa}=\phi$; (ii) $c_1^+$ commutes with the action of $\fa$, where the $\fa$ acts on $U(\fa)^+$ by the left multiplication. Taking the composition with the projection $U(\fa)\lra U(\fa)^+$ we get a map $$ c_1: U(\fg)\lra M $$ The map ~(\ref{c1}) is obtained by applying this remark to $\fa=\fX,\ M=\fh[1]$, the action of $\fX$ on $\fh[1]$ being induced from the adjoint action of $\fg$ on $\fh[1]$ through the graded Lie algebra morphism $\theta:\fX\lra \fg$. \section{Twisted enveloping algebras and connecting morphism} \label{twisted} \subsection{} \label{dif-smo} From now on until the end of the paper we fix a ground field $k$ of characteristic $0$. Let $(X,\CO_X)$ be a topological space equipped with a sheaf of commutative $k$-algebras $\CO_X$. Define the {\em tangent sheaf} $\CT_X$ as the sheaf of $\CO_X$-modules associated with the presheaf $$ U\mapsto \operatorname{Der}_k(\Gamma(U,\CO_X),\Gamma(U,\CO_X)) $$ (the space of $k$-derivations). $\CT_X$ is a sheaf of $k$-Lie algebras. We will say that $X$ is {\em differentially smooth} if there exists an open covering $X=\bigcup U_i$ such that for each $U_i$ the restriction $\CT_X|_{U_i}$ is a free $\CO_X|_{U_i}$-module admitting a finite basis of {\em commuting} sections $\dpar_1,\ldots,\dpar_n\in\Gamma(U_i,\CT_X)$. \subsection{} If $\CF$ a sheaf on $X$, the notation $t\in \CF$ will mean that $t$ is a local section of $\CF$. We will use below the straightforward "sheaf" versions of the definitions and results from Sect.~\ref{envstand}. In particular, we will use the notion of a {\em dg $\CO_X$-Lie algebra} (the bracket is supposed to be $\CO_X$-linear). If $\CF^{\cdot}$ is a complex of sheaves, $\CZ^i(\CF^{\cdot})$ will denote sheaves of $i$-cocycles, $\CH^i(\CF^{\cdot})$ cohomology sheaves (not to be confused with cohomology {\em spaces} $H^i(X,\CF)$). If $\fg$ is a dg $\CO_X$-Lie algebra, we will consider its standard complex $C(\fg)=C_{\CO_X}(\fg)$ (this is a complex of sheaves of dg $\CO_X$-coalgebras) over $X$) and its canonical filtration $F_iC(\fg)$; we set $\gr_iC(\fg)=F_iC(\fg)/F_{i-1}C(\fg)$. We will use notations \begin{equation} \label{homology} \CH_i^{Lie}(\fg):=\CH^{-i}(C(\fg));\ \CF_j\CH_i^{Lie}(\fg):=\CH^{-i}(F_jC(\fg)) \end{equation} \subsection{Dg Lie algebroids} Below we will use some definitions and constructions from ~\cite{bb}, 1.2 (see also ~\cite{bfm}, 3.2) and their generalization to a dg-situation. \begin{defn}{} A {\em dg Lie algebroid} over $X$ is a sheaf of dg $k$-Lie algebras $\CA$ on $X$ together with a structure of a left $\CO_X$-module on it and a map $$ \pi: \CA\lra \CT_X $$ of dg $k$-Lie algebras and $\CO_X$-modules such that $$ [a,fb]=f[a,b]+\pi(a)(f)b $$ for all $a,b\in \CA,\ f\in \CO_X$. We denote $\CA_{(0)}:=\ker \pi$. It is an $\CO_X$-Lie algebra and a dg Lie ideal in $\CA$. $\CA$ is called {\em transitive} if $\pi$ is epimorphic (i.e. $\pi^0: \CA^0\lra \CT_X$ is epimorphic). A Lie algebroid is a dg Lie algebroid concentrated in degree $0$. \end{defn} Dg Lie algebroids over $X$ form a category in an obvious way, with the final object $(\CT_X, \id_{\CT_X})$. A dg Lie algebroid with $\pi=0$ is the same as a dg $\CO_X$-Lie algebra. \subsection{Modules} Let $\CA$ be a dg Lie algebroid over $X$. A {\em dg $\CA$-module} is a complex of left $\CO_X$ modules $M$ which is (as a complex of $k$-modules) equipped with the action of the dg $k$-Lie algebra $\CA$ such that \begin{equation} \label{eq1-mod} f(am)=(fa)m \end{equation} \begin{equation} \label{eq2-mod} a(fm)=f(am)+\pi(a)(f)m \end{equation} for all $f\in \CO_X,\ a\in \CA,\ m\in M$. We will say in this situation that we have an {\em action} of a dg Lie algebroid $\CA$ on $M$. \subsection{Twisted enveloping algebras} \label{twist-env} Let $(\CA, \pi)$ be a dg Lie algebroid over $X$. Let $U_k(\CA)$ denote the enveloping algebra of $\CA$ considered as a dg $k$-Lie algebra, $U_k(\CA)^+\subseteq U_k(\CA)$ the augmentation ideal. Let us consider $U_k(\CA)^+$ as a sheaf of dg algebras without unit. Denote by $U_{\CO_X}(\CA)^+$ its quotient by the two-sided dg ideal generated by all elements $$ a_1\cdot fa_2-fa_1\cdot a_2-\pi(a_1)(f)a_2, $$ $a_1,a_2\in \CA,\ f\in \CO_X$. Set $U_{\CO_X}(\CA)=U_{\CO_X}(\CA)^+\oplus \CO_X$ and define the structure of a dg algebra with unit by the rule $$ f\cdot a=fa,\ a\cdot f=fa+\pi(a)(f),\ f\in \CO_X,a\in \CA $$ We have a canonical algebra map $$ \CO_X\lra U_{\CO_X}(\CA) $$ providing $U_{\CO_X}(\CA)$ with a structure of an $\CO_X$-bimodule, and a map of left $\CO_X$-modules and dg $k$-Lie algebras (with the structure of a dg Lie algebra on $U_{\CO_X}(\CA)$ given by the commutator) $$ i: \CA\lra U_{\CO_X}(\CA) $$ which is a composition of evident maps $\CA\lra U_k(\CA)^+\lra U_{\CO_X}(\CA)$. The map $i$ induces an equivalence of the category of dg modules over $U_{\CO_X}(\CA)$ (as an associative dg algebra) and that of $\CA$-modules. We define the canonical filtration $F_iU_{\CO_X}(\CA)\subset U_{\CO_X}(\CA)$ as the image of $F_iU_k(\CA)^+\oplus \CO_X$. \subsection{Coalgebra structure} Let $\CA$ be a dg Lie algebroid over $X$. Consider the comultiplication $\Delta: U_k(\CA)\lra U_k(\CA)\otimes_k U_k(\CA)$. For $x\in U_k(\CA)^+$, $\Delta(x)=x\otimes 1+1\otimes x+\Delta^+(x)$ where $\Delta^+(x)\in U_k(\CA)^+\otimes_k U_k(\CA)^+$. Consider the composition \begin{equation} \label{comp} U_k(\CA)^+\overset{\Delta^+}{\lra}U_k(\CA)^+\otimes_k U_k(\CA)^+ \lra U_{\CO_X}(\CA)^+\otimes_{\CO_X} U_{\CO_X}(\CA)^+ \end{equation} (we consider the tensor square of $U_{\CO_X}(\CA)^+$ as a {\em left} $\CO_X$-module). One checks directly that $\ker (U_k(\CA)^+\lra U_{\CO_X}(\CA))$ is contained in the kernel of ~(\ref{comp}). Hence, ~(\ref{comp}) induces the map $$ \Delta^+:U_{\CO_X}(\CA)^+\lra U_{\CO_X}(\CA)^+\otimes_{\CO_X} U_{\CO_X}(\CA)^+ $$ Define $$ \Delta:U_{\CO_X}(\CA)\lra U_{\CO_X}(\CA)\otimes_{\CO_X} U_{\CO_X}(\CA) $$ as follows: for $f\in \CO_X,\ \Delta(f)=f\otimes 1$; for $x\in U_{\CO_X}(\CA)^+,\ \Delta(x)=x\otimes 1+1\otimes x+\Delta^+(x)$. Define the counit $U_{\CO_X}(\CA)\lra \CO_X$ to be the canonical projection. This defines a structure of a unital cocommutative dg $\CO_X$-coalgebra on $U_{\CO_X}(\CA)$. \subsection{Differential operators} Let $(X,\CO_X)$ be a differentially smooth $k$-ringed space. \begin{defn}{} The sheaf of differential operators (resp., that of operators of order $\leq n$) on $X$ is $$ \Diff_X=U_{\CO_X}(\CT_X);\ \Diff^{\leq n}_X=F_nU_{\CO_X}(\CT_X)\ . $$ \end{defn} \subsection{Poincare-Birkhoff-Witt condition} The associated graded algebra $$ \gr U_{\CO_X}(\CA):=\oplus_{i\geq 0} F_iU_{\CO_X}(\CA)/F_{i-1}U_{\CO_X}(\CA) $$ is a commutative dg $\CO_X$-algebra. Hence, we have a canonical surjective morphism \begin{equation} \label{pbw-map} j: S_{\CO_X}{\CA}\lra \gr U_{\CO_X}(\CA) \end{equation} We will say that {\em $\CA$ satisfies the Poincare-Birkhoff-Witt (PBW) condition} if ~(\ref{pbw-map}) is isomorphism. \subsection{} \begin{thm}{pbw-algebr} Let $(X,\CO_X)$ be differentially smooth and let $\CA$ be a transitive dg Lie algebroid over $X$. Then $\CA$ satisfies PBW. \end{thm} {\em Proof} will be done in several steps. \subsubsection{} \label{pbw-lie} \begin{lem}{} Let $\fg$ be a dg $\CO_X$-Lie algebra. Then one has a canonical isomorphism of unital dg $\CO_X$-coalgebras $$ e: S_{\CO_X}(\fg)\lra U_{\CO_X}(\fg) $$ defined by the formula $$ e(x_1\cdot\ldots\cdot x_n)=\frac{1}{n!}\sum_{\sigma\in \Sigma_n}\pm i(x_1)\cdot \ldots i(x_n) $$ where $i:\fg\lra U(\fg)$ is the canonical map, the sign $\pm$ is inserted according to the Quillen rule. \end{lem} \begin{pf} The argument of the proof of ~\cite{q}, App. B, Thm. 3.2 works in our situation. \end{pf} Let us continue the proof of ~\ref{pbw-algebr}. We can forget about differentials. The question is local on $X$, so we can suppose that $\CT_X$ is freely generated over $\CO_X$ by $n$ commuting global vector fields $\dpar_1,\ldots, \dpar_n\in \CT_X(X)$ which can be lifted to global sections $\gamma_1,\ldots,\gamma_n\in \CA^0(X)$. Consider the map $$ \mu: U_{\CO_X}(\CA_{(0)})(X)\otimes_k k[\gamma_1,\ldots,\gamma_n]\lra U_{\CO_X}(\CA)(X) $$ given by the multiplication. Here $k[\gamma_1,\ldots,\gamma_n]$ denotes the free left $k$-module on the basis $\gamma_1^{d_1},\ldots,\gamma_n^{d_n}$, $d_i$ being nonnegative integers. \subsubsection{} \begin{lem}{} $\mu$ is an isomorphism of filtered graded $k$-modules. \end{lem} Here the filtration on $U_{\CO_X}(\CA_{(0)})(X)$ is the canonical one; on $k[\gamma_1,\ldots,\gamma_n]$ the filtration by the total degree, and the filtration on their tensor product is the tensor product of filtrations: $F_i(A\otimes B)=\sum_{p+q=i}\Ima(F_pA\otimes F_qB\lra A\otimes B)$. \begin{pf} We apply the idea of Serre's proof of the PBW theorem, see {}~\cite{se}, proof of Thm. 3, p. I, ch. III.4. First, it is clear that $\mu$ is surjective. To prove the injectivity, consider the free left $U_{\CO_X}(\CA_{(0)})(X)$-module $F$ with the basis $\{ x_M\}$ indexed by finite non-decreasing sequences $M=(i_1,\ldots,i_d)$ with $i_j\in \{1,\ldots,n\}$. One can introduce on $M$ the structure of an $\CA(X)$-module as follows. For $M$ as above, call $d$ the length of $M$, and denote it $l(M)$. For $i\in \{1,\ldots,n\}$ say that $i\leq M$ if $i\leq i_1$, and denote $iM$ the concatenation $(i,i_1,\ldots,i_d)$. Note that $\CA(X)$ is generated as an abelian group by elements $a\gamma_i,\ a\in \CA_{(0)}(X)$, so it suffices to define $\gamma_i\cdot ux_M,\ u\in U_{\CO_X}(\CA_{(0)})(X)$. Let us do this by induction. We set $$ \gamma_i\cdot ux_M=u\gamma_i\cdot x_M-\ad_{\gamma_i}(u)\cdot x_M $$ so it suffices to define $\gamma_i\cdot x_M$. Suppose we have defined $\gamma_i\cdot ux_N$ for all $N:\ l(N)<l(M)$ and $\gamma_j\cdot ux_N$ for all $j<i,\ l(N)=l(M)$; we suppose that these elements are linear combinations over $U_{\CO_X}(\CA_{(0)})(X)$ of $x_{N'}$ with $l(N')\leq l(N)+1$. Set $$ \gamma_i\cdot x_M=\left\{ \begin{array}{ll} x_{iM} & \mbox{if $i\leq M$} \\ \gamma_j\cdot (\gamma_i\cdot x_N)+ [\gamma_i,\gamma_j]\cdot x_N & \mbox{if $M=jN,\ i>j$} \end{array} \right. $$ the right hand side being defined by induction (note that $[\gamma_i,\gamma_j]\in \CA_{(0)}(X)$). One checks that this definition is correct. Using this, one proves, as in {\em loc. cit.} that all $x_M$ form the $U_{\CO_X}(\CA_{(0)})(X)$-basis of $U_{\CO_X}(\CA)(X)$, which implies the injectivity of $\mu$. \end{pf} To finish the proof of ~\ref{pbw-algebr}, it remains to note that $$ \gr (\mu): S_{\CO_X}(\CA_{(0)})(X)\otimes_k k[\gamma_1\ldots,\gamma_n]\lra \gr U_{\CO_X}(\CA)(X) $$ coincides with $j(X)$. Theorem ~\ref{pbw-algebr} is proven. $\Box$ \subsection{} \label{quasi-iso} \begin{cor}{} Let $(X,\CO_X)$ be differentially smooth and $f: \CA\lra \CB$ be a map of transitive dg Lie algebroids satisfying one of the two assumprions: (i) $f$ locally $\CO_X$-homotopy equivalence; (ii) $f$ is quasiisomorphism, all components $\CA^i,\ \CB^i$ are flat over $\CO_X$, $\CH^i(\CA)=\CH^i(\CB)=0$ for big $i$. Then the induced map $U_{\CO_X}(f): U_{\CO_X}(\CA)\lra U_{\CO_X}(\CB)$ is a filtered quasi-isomorphism. \end{cor} \begin{pf} Each of our hypotheses implies that $T_{\CO_X}(f): T_{\CO_X}(\CA)\lra T_{\CO_X}(\CB)$ is a quasi-isomorphism; hence this is true for $S_{\CO_X}(f)$. Now by ~\ref{pbw-algebr} the same is true for $U_{\CO_X}(f)$. \end{pf} \subsection{Pushout} Let $\CA$ be a dg Lie algebroid over $X$ and $\fg$ be a dg $\CO_X$-Lie algebra which is also an $\CA$-module. An {\em $\CA$-morphism} $\psi: \CA_{(0)}\lra \fg$ is a morphism of $\CO_X$-Lie algebras which commutes with the $\CA$-action, where the action of $\CA$ on $\CA_{(0)}$ is the adjoint one, and such that $$ a\cdot x=[\psi(a),x] $$ for all $a\in \CA_{(0)},\ x\in \fg$. Given such a morphism, the dg Lie algebroid $\CA_{\psi}$ is defined as follows. Consider the dg Lie algebra semi-direct product $\CA \semid \fg$ (so, the bracket is $[(a,x),(b,y)]=([a,b],-(-1)^{|b||x|}b\cdot x+a\cdot y+[x,y]),\ a,b\in \CA,\ x,y\in fg$). By definition, $\CA_{\psi}$ is the quotient of $\CA \semid \fg$ by the dg Lie ideal $\CA_{(0)}\hra \CA \semid \fg,\ a\mapsto (a, -\psi(a))$. The map $\pi_{\CA_{\psi}}:\CA_{\psi}\lra \CT_X$ maps $(a,x)$ to $\pi_{\CA}(a)$. If $\CA$ is transitive then so is $\CA_{\psi}$, and $\CA_{\psi (0)}=\fg$. \subsection{Boundary morphism} \subsubsection{Conic Lie algebroids} Let $(\CA,\pi)$ be a dg Lie algebroid over $X$, $\fh$ a dg Lie algebra and a left $\CO_X$-module, $i: \fh\hra \CA$ an embedding of dg Lie algebras and of $\CO_X$-modules such that $i(\fh)$ is a dg Lie ideal in $\CA$, and $\pi\circ i=0$. This implies that $\fh$ is a dg $\CO_X$-Lie algebra. Let us consider the complex $\fA=\Cone (i)$. According to ~\ref{conic}, $\fA$ has a canonical structure of a dg $k$-Lie algebra. Together with the evident structure of an $\CO_X$-module and $\pi_{\fA}:\fA\overset{\theta}{\lra} \CA\overset{\pi}{\lra} \CT_X$, $\fA$ becomes a dg Lie algebroid. Let us consider $\fA$ as a dg $k$-Lie algebra, and apply to it (the sheaf version of) the construction ~\ref{constr}. We get a map of sheaves of graded $k$-modules \begin{equation} \label{conn-prep} c_{1/k}^+: U_k(\fA)^+\lra \fh[1] \end{equation} satisfying the Maurer-Cartan equation. \subsubsection{} \begin{thm}{} \label{conn-thm} The map $c_{1/k}^+$ factors through the canonical map $U_k(\fA)^+\lra U_{\CO_X}(\fA)^+$ and hence induces the map $$ c_1^+:U_{\CO_X}(\fA)^+\lra \fh[1]. $$ Taking its composition with the projection $U_{\CO_X}(\fA)\lra U_{\CO_X}(\fA)^+$, we get $$ c_1:U_{\CO_X}(\fA)\lra \fh[1]. $$ This map is a morphism of left graded $\CO_X$-modules and it satisfies the Maurer-Cartan equation. Consequently, it induces the map of filtered dg $\CO_X$-coalgebras \begin{equation} c:U_{\CO_X}(\fA)\lra C_{\CO_X}(\fh) \end{equation} \end{thm} The map $c$ will be called {\bf boundary morphism} associated to $\fA$. \begin{pf} We have only to check that \begin{equation} \label{check} ux(fy)v-u(fx)yv-u(\pi_{\fA}(x)(f)y)v\in \ker(c_{1/k}^+) \end{equation} for all $x,y\in \fA,\ u,v\in U_k(\fA),\ f\in \CO_X$. Note that $\fh$ is an $\CA$-module with respect to the adjoint action (NB! this is not true for $\CA$: the Axiom ~(\ref{eq1-mod}) does not hold). It follows that the action of $U_k(\CA)$ on $\fh[1]$ factors through $U_{\CO_X}(\CA)$, hence ~(\ref{check}) holds true for $v\in U_k(\fA)^+$; so we can suppose $v=1$; obviously, it is enough to check ~(\ref{check}) for $u=1$.In that case it is a direct check (note that the maps $\phi$ and $\theta$ are $\CO_X$-linear): $$ c_{1/k}^+(x(fy)-(fx)y-\pi_{\fA}(x)(f)y)=[\theta(x),f\phi(y)]- [f\theta(x),\phi(y)]-\pi_{\CA}(\theta(x))(f)\phi(y)=0 $$ \end{pf} \subsection{Connecting morphisms} \label{abstr-ksmaps} Here $X$ is supposed to be differentially smooth. \subsubsection{} Let $\CA$ be a transitive dg Lie algebroid over $X$. Set $\fh:=\CA_{(0)}$, so we have an exact sequence \begin{equation} \label{fund-abstr} 0\lra \fh\overset{i}{\lra} \CA\overset{\pi}{\lra} \CT_X\lra 0\ . \end{equation} It induces the map \begin{equation} \label{ks1-abstr} \kappa^1: \CT_X\lra \CH^1(\fh)\ . \end{equation} Set $\fA=\Cone (i)$. By Thm.~\ref{quasi-iso} $\pi$ induces a filtered quasi-isomorphism $U_{\CO_X}(\fA)\lra U_{\CO_X}(\CT_X)=\Diff_X$, whence isomorphisms $$ \CH^0(U_{\CO_X}(\fA))\cong \Diff_X;\ \CH^0(F_nU_{\CO_X}(\fA))\cong \Diff_X^{\leq n}\ . $$ On the other hand, Thm.~\ref{conn-thm} gives the filtered map $$ c: U_{\CO_X}(\fA)\lra C_{\CO_X}(\fh)\ . $$ By taking $\CH^0(c)$, we get maps \begin{equation} \label{abstr-ks} \kappa:\Diff_X\lra \CH_0^{Lie}(\fh) \end{equation} as well as \begin{equation} \label{abstr-ks-i} \kappa^{\leq n}: \Diff^{\leq n}_X\lra \CF_n\CH_0^{Lie}(\fh) \end{equation} which are called {\bf connecting morphisms}. \subsubsection{} We have by definition $$ \gr_nC_{\CO_X}(\fh)=S^n_{\CO_X}(\fh[1]) $$ hence the maps \begin{equation} \label{proj-lie} \CF_n\CH_0^{Lie}(\fh)\lra \CH_0(S^n_{\CO_X}(\fh[1]))\ . \end{equation} On the other hand, the embeddings $S^n(\fh^1)\subset S^n(\fh[1])$ induce embedding of cocycles $S^n\CZ^1(\fh)\hra \CZ^0(S^n(\fh[1]))$ which pass to cohomology and give the maps \begin{equation} \label{emb} S^n_{\CO_X}(\CH^1(\fh))\lra \CH^0(S^n(\fh[1]))\ . \end{equation} \subsubsection{} \label{main-thm} \begin{thm}{} The connecting morphisms ~(\ref{abstr-ks}) and {}~(\ref{abstr-ks-i}) have the following properties. (i) The squares $$\begin{array}{ccc} \Diff^{\leq n-1}_X & \overset{\kappa^{\leq n-1}}{\lra} & \CF_{n-1}\CH_0^{Lie}(\fh) \\ \downarrow & \; & \downarrow \\ \Diff^{\leq n}_X & \overset{\kappa^{\leq n}}{\lra} & \CF_{n}\CH_0^{Lie}(\fh) \\ \end{array}$$ commute. We have $$ \kappa=\lim_{\ra}\kappa^{\leq n}\ . $$ (ii) The squares $$\begin{array}{ccccc} \Diff^{\leq n}_X &\; & \overset{\kappa^{\leq n}}{\lra} &\; & \CF_{n}\CH_0^{Lie}(\fh) \\ \downarrow & \; & \; &\; & \downarrow \\ S^n_{\CO_X}(\CT_X) & \overset{(-1)^nS^n(\kappa^1)}{\lra} & S^n_{\CO_X} (\CH^1(\fh)) & \overset{(\ref{emb})}{\lra} & \CH^0(S^n(\fh[1])) \\ \end{array}$$ commute. Here the left vertical arrow is the symbol map, and the right one is ~(\ref{proj-lie}). \end{thm} The property (i) is obvious. (ii) will be proven in the next Subsection. \subsection{Explicit formulas} \subsubsection{} \label{expl1} In the previous assumptions, suppose we have local sections $\dpar_1,\ldots ,\dpar_n\in \CT_X$. Let us pick $0$-cocycles in $\fA$ lifting them: $a_p=(-\alpha_p,\gamma_p)\in \fA^0,\ \gamma_p\in \CA^0,\alpha_p\in\fh^1,\ \pi(\gamma_p)=\dpar_p;\ d_{\CA}(\gamma_p)=i(\alpha_p)$. Consider the map $c_{1/k}^+$, ~(\ref{conn-prep}). For $I$ as in ~\ref{formula} define $a_I$ as there, and set $\dpar_I=\dpar_{i_1}\cdot\ldots\cdot\dpar_{i_s}$, $$ \alpha(\dpar_I)=c_{1/k}^+(a_I)=\ad (\gamma_{i_1})\circ\ldots\circ \ad(\gamma_{i_{s-1}})(-\alpha_{i_s}) $$ Define elements (we use the notations of ~\ref{formula}) $$ \kappa^{(p)}_n=\frac{1}{p!} \sum_{P=(I_1,\ldots,I_p)\in\CP_p(n)}\alpha(\dpar_{I_1})\cdots \alpha(\dpar_{I_p})\in S^p_{\CO_X}(\fh^1), $$ $p=1,\ldots,n$. In particular, \begin{equation} \label{kappa-nn} \kappa^{(n)}_n=(-1)^n\alpha_1\cdots \alpha_n. \end{equation} \subsubsection{} \label{expl2} \begin{prop}{} The class $\kappa^{\leq n}(\dpar_1\cdots\dpar_n)$ is represented by the cocycle $$ (0,\kappa^{(1)}_n,\ldots,\kappa^{(n)}_n)\in\oplus_{p=0}^nS^p_{\CO_X}(\fh^1) \subset F_nC_{\CO_X}(\fh)^0. $$ \end{prop} \begin{pf} This follows from ~\ref{formula} applied to $c_{1/k}$ (note that $\fA$ is a Hopf algebra). \end{pf} As a corollary, we get the claim (ii) of ~\ref{main-thm} which follows from ~(\ref{kappa-nn}). \subsubsection{Schur polynomials} Let us define polynomials $P_n(\alpha_1,\alpha_2,\ldots)$ by means of the generating function $$ \exp (\sum_{p=1}^{\infty}\alpha_p\frac{t^p}{p!})= \sum_{n=0}^{\infty}P_n(\alpha_1,\ldots)\frac{t^n}{n!} $$ It is easy to see that $P_n(\alpha_1,\ldots)=P_n(\alpha_1,\ldots,\alpha_n)$ and \begin{equation} \label{schur} P_n(\alpha_1,\ldots,\alpha_n)=\sum_{(n_1,\ldots,n_s):\sum_j jn_j=n} \frac{n!}{(1!)^{n_1}\cdot\ldots\cdot (s!)^{n_s} n_1!\cdot\ldots\cdot n_s!} \alpha_1^{n_1}\cdot\ldots\cdot\alpha_s^{n_s} \end{equation} The first polynomials are: $P_0=1,\ P_1=\alpha_1,\ P_2=\frac{\alpha_1^2}{2}+ \alpha_2$. \subsubsection{} Now suppose we are given a local section $\dpar\in \CT_X$ together with a lifting $a=(-\alpha,\gamma)\in \fA^0$ as in ~\ref{expl1}. For $i\geq 1$ set $$ \alpha_i=(\ad (\gamma))^{i-1}(-\alpha)\in \fh^1 $$ \subsubsection{} \label{explicit} \begin{prop}{} The class $\kappa^{\leq n}(\dpar^n)$ is represented by the cocycle $$ P_n(\alpha_1,\ldots,\alpha_n)\in F_nS_{\CO_X}(\fh^1)\subseteq F_nC_{\CO_X}(\fh)^0. $$ \end{prop} \begin{pf} From ~\ref{expl2} follows that $\kappa^{\leq n}(\dpar^n)$ can be represented by the polynomial $Q(\alpha_1,\ldots,\alpha_n)$ where the coefficient of $Q$ at $\alpha_1^{n_1}\cdots\alpha_s^{n_s}$ is equal to the number of partitions of the set $\{1,\ldots,n\}$ containing $n_1$ of $1$-element subsets, $n_2$ of $2$-element subsets, ..., $n_s$ of $s$-element subsets. From ~(\ref{schur}) follows that $Q=P_n$. \end{pf} \subsubsection{} Summing up the expressions ~\ref{explicit} over $n$, we can rewrite our formulas as $$ \kappa(\exp(t\dpar))=\exp (\frac{\exp(\ad(\gamma))-\Id}{\ad(\gamma)}(\alpha)). $$ This was pointed out to us by I.T.Leong. Cf. Deligne's formula {}~\cite{gmd}, p. 51, (1-1). \section{Differential calculus on formal schemes} \label{formal} The aim of this Section is to extend (a part of) the classical Grothendieck's language of differential calculus, \cite{ega} IV, \S\S 16, 17, to formal schemes. \subsection{} Recall that we have fixed a ground field $k$ of characteristic $0$. We will work in the category $\Fsch$ of separated locally noetherian formal schemes over $k$. All necessary definitions and facts about them are contained in \cite{ega} I, \S10. Objects of $\Fsch$ will be called simply {\em formal schemes}. Inside $\Fsch$, we will consider a full subcategory $\Sch$ of separated locally noetherian schemes of over $k$ whose objects will be called simply (usual) {\em schemes}. By definition, a formal scheme is a topological space $\fX$ equipped with a sheaf of topological rings $\CO_{\fX}$. $\fX$ is a union of affine formal schemes $\Spf(A)$ where $A$ is a noetherian $k$-algebra complete in the $I$-adic topology for some ideal $I\subset A$. As a topological space, $\Spf(A)=\Spec(A/I)$, and $\Gamma(\Spf(A),\CO_{\Spf(A)})=A$. There exists a sheaf of ideals $\fI\subset\CO_{\fX}$ such that for sufficiently small affine $U\subset \fX$ the ideals $\Gamma(U,\fI)^n,\ n\geq 1$, form a base for the topology of $\Gamma(U,\CO_{\fX})$. Such a sheaf is called the {\em ideal of definition} of $\fX$. All ringed spaces $$ X_n:=(\fX,\CO_{\fX}/\fI^{n+1}) $$ are (usual) schemes, and we have \begin{equation} \label{indlim} \fX=\dirlim\ X_n \end{equation} cf. {\em loc.cit.}, 10.11. Among ideals of definition there exists a unique maximal one. Let $f:\fX\lra \fS$ be a morphism of formal schemes, $\fK\subseteq\CO_{\fS}$ an ideal of definition of $\fS$ and $\fI$ the maximal ideal of definition of $\fX$. Then $f^*(\fK)\subseteq\fI$, so $f$ induces maps of schemes $f_n:X_n\lra S_n$ such that \begin{equation} \label{adic} f=\dirlim\ f_n \end{equation} cf. {\em loc.cit.}, 10.6.10. \subsubsection{} The category $\Fsch$ has fibered products, {\em loc.cit.}, 10.7. \subsection{} \label{aff} Let $\fX$ be affine, $\fX=\Spf(A)$. We have a canonical functor $$ \Delta: \Mod(A)\lra \Mod(\CO_{\fX}),\ M\mapsto M^{\Delta} $$ from the category of $A$-modules to the category of sheaves $\CO_{\fX}$-modules. If $A$ is noetherian then $\Delta$ establishes an equivalence between the category of $A$-modules of finite type and that of coherent $\CO_{\fX}$- modules, {\em loc.cit.}, 10.10.2. \subsection{} Let $\fX$ be a formal scheme, $\CM$ a coherent $\CO_{\fX}$-module. If $\fX$ is represented as in ~(\ref{indlim}) then \begin{equation} \label{indlim-m} \CM\cong \invlim\ M_n \end{equation} for a suitable inverse system of coherent $X_n$-modules $M_n$, {\em loc.cit.}, 10.11.3. We will consider $\CM$ as a sheaf of topological $\CO_{\fX}$ modules equipped with the topology defined in {\em loc.cit.}, 10.11.6. Note that for every affine open $U\subseteq\fX$ the module $\Gamma(U,\CM)$ is complete. If $\CN$ is another coherent $\CO_{\fX}$-module, we have isomorphisms $$ \CM\otimes_{\CO_{\fX}}\CN\cong\invlim\ (M_n\otimes_{\CO_{X_n}}N_n) $$ and $$ \CHom_{\CO_{\fX}}(\CM,\CN)\cong \invlim\ \CHom_{\CO_{X_n}}(M_n,N_n), $$ cf. {\em loc.cit.}, 10.11.7. \subsection{Jets} Let $f: \fX\lra \fS$ be a morphism of formal schemes, $f=\dirlim\ f_i$ a representation as in ~(\ref{adic}). Let us consider the diagonal $$ \Delta_f: \fX\hra \fX\times_{\fS}\fX $$ We have $$ \fX\times_{\fS}\fX=\dirlim\ X_i\times_{S_i}X_i $$ and $\Delta_f=\dirlim\ \Delta_{f,i}$ where $$ \Delta_{f,i}: X_i\hra X_i\times_{S_i}X_i $$ are diagonal mappings. If $\CI_i\subseteq\CO_{X_i\times_{S_i}X_i}$ is the ideal of $\Delta_{f,i}$ then $$ \fI=\invlim\CI_i\subseteq\invlim\ \CO_{X_i\times_{S_i}X_i}= \CO_{\fX\times_{\fS}\fX} $$ is the ideal of $\Delta_f$. Let $p_j:\fX\times_{\fS}\fX\lra \fX,\ j=1,2$ be projections. For any integer $n\geq 0$ set $$ \fX^{(n)}_f=(\Delta_f(\fX),\CO_{\fX\times_{\fS}\fX}/\CI^{n+1}) $$ --- it is a closed formal subscheme of $\fX\times_{\fS}\fX$. Consider canonical projections $p_i^{(n)}:\fX^{(n)}\lra \fX,\ i=1,2$. \subsubsection{} \label{defn-jets} \begin{defn}{} (Cf. \cite{ega} IV 16.3.1, 16.7.1.) We define a sheaf of rings over $\fX$ which is called the {\em sheaf of $n$-jets} as $$ \CP^n_f=\CP^n_{\fX/\fS}:=p_{1*}(\CO_{\fX\times_{\fS}\fX}/\fI^{n+1})= p_{1*}^{(n)}p^{(n)*}_2(\CO_{\fX}) $$ We have two canonical morphisms of sheaves of topological rings \begin{equation} \label{first} \CO_{\fX}\lra \CP_f^n,\ x\mapsto x\otimes 1 \end{equation} and \begin{equation} \label{second} d^n_{\fX/\fS}:\CO_{\fX}\lra \CP_f^n,\ x\mapsto 1\otimes x \end{equation} by means of which one introduces a structure of left (resp., right) $\CO_{\fX}$-module on $\CP^n_f$. More generally, for a coherent sheaf $\CM$ over $\fX$ set $$ \CP^n_{\fX/\fS}(\CM):=p_{1*}^{(n)}p^{(n)*}_2(\CM) $$ \end{defn} We have $$ \CP^n_{\fX/\fS}(\CM)=\CP^n_{\fX/\fS}\otimes_{\CO_{\fX}}\CM $$ where the right $\CO_{\fX}$-module structure on $\CP^n_{\fX/\fS}$ is used on the right-hand side, cf. {\em loc.cit.}, 16.7.2.1. The $\CO_{\fX}$-bimodule structure on $\CP^n_f$ induces an $\CO_{\fX}$-bimodule structure on $\CP^n_{\fX/\fS}(\CM)$. Following {\em loc.cit.}, if we do not specify the structure of an $\CO_{\fX}$-module on $\CP^n_{\fX/\fS}(\CM)$, we mean that of a {\em left} module. \subsubsection{} If $\CM=\invlim\ M_i$ as in ~(\ref{indlim-m}) then \begin{equation} \label{invlim-p} \CP^n_{\fX/\fS}(\CM)=\invlim\ \CP^n_{X_i/S_i}(M_i)\ . \end{equation} \subsubsection{} Consider affine open formal subschemes $U=\Spf(B)\subseteq\fX,\ V=\Spf(A)\subseteq \fS$ such that $f(U)\subseteq V$, so that $A,B$ are adic rings, and $B$ is a topological $A$-algebra. Let $I=\ker(B\otimes_AB\lra B)$ be the kernel of the multiplication, and $$ P_{B/A}=(B\otimes_AB)/I^{n+1} $$ considered as a $B$-module by means of a map $x\mapsto x\otimes 1$. It is naturally a topological $B$-module, and we can consider its completion, $\hP_{B/A}$. We have \begin{equation} \CP^n_U\cong (\hP_{B/A})^{\Delta} \end{equation} \subsubsection{} \begin{claim}{} $d^n_{\fX/\fS}(\CO_{\fX})$ topologically generates $\CO_{\fX}$-module $\CP^n_f$. \end{claim} (cf. \cite{ega} IV, 16.3.8) $\Box$ \subsubsection{} We have evident projections \begin{equation} \label{cofilt} \CP^n_f\lra\CP^{n-1}_f\ . \end{equation} Consider a projection \begin{equation} \label{proj} \CP^n_f\lra \CP^0_f=\CO_{\fX} \end{equation} and let $\bCP^n_f$ denote its kernel. A map $$ d_1: \CO_{\fX}\lra \CP^n_f,\ x\mapsto x\otimes 1, $$ is left inverse to ~(\ref{proj}), so we have a splitting $$ \CP^n_f\cong\bCP^n_f\oplus\CO_{\fX}. $$ \subsection{Differentials} We define $$ \Omega^1_f=\Omega^1_{\fX/\fS}:=\bCP^1_f\ . $$ If $\fX=\Spf(B),\ \fS=\Spf(B)$ then we will use the notation $\Omega^1_{B/A}$ for $\Gamma(\fX,\Omega^1_{\fX/\fS})$. We have natural isomorphisms $$ \Omega^1_f\cong p_{1*}(\CI/\CI^2)\cong p_{2*}(\CI/\CI^2)\ . $$ We have a canonical continuous $\CO_{\fS}$-linear map \begin{equation} \label{difl} d:\CO_{\fX}\lra \Omega^1_{\fX/\fS} \end{equation} induced by the map $x\mapsto x\otimes 1-1\otimes x$. \subsubsection{} From ~(\ref{invlim-p}) follows that we have a natural isomorphism \begin{equation} \Omega^1_{\fX/\fS}\cong\invlim\ \Omega^1_{X_i/S_i} \end{equation} \subsection{Example} \label{power} Suppose $\fS=\Spf(A)$, and $\fX=\Spf(A_n)$ where $A_n=A\{ T_1,\ldots,T_n\}$ is the completion of the polynomial ring $A[T_1,\ldots,T_n]$ in the $J$-adic topology $J$, being an ideal generated by an ideal of definition of $A$ and $T_1,\ldots, T_n$. Then $\Omega^1_{A_n/A}$ is a free $A_n$-module with the basis $dT_1,\ldots,dT_n$. \subsection{Derivations} Let $\CM$ be a coherent $\CO_{\fX}$-module. We define {\em the sheaf of derivations} $$ \Der_{\fS}(\CO_{\fX},\CM):=\CHom_{\CO_{\fX}}(\Omega^1_{\fX/\fS},\CM). $$ The map ~(\ref{difl}) induces a canonical embedding $$ \Der_{\fS}(\CO_{\fX},\CM)\hra\CHom_{\CO_{\fS}}(\CO_{\fX},\CM) $$ which identifies $\Der_{\fS}(\CO_{\fX},\CM)$ with the sheaf of local $\CO_{\fS}$-homomorphisms $\alpha:\CO_{\fX}\lra\CM$ such that $$ \alpha(xy)=x\alpha(y)+y\alpha(x). $$ We set $$ \CT_{\fX/\fS}:=\Der_{\fS}(\CO_{\fX},\CO_{\fX}) $$ and call this sheaf the {\em tangent sheaf of $\fX$ with respect to $\fS$}, cf. {\em loc.cit.}, 16.5.7. It is a sheaf of $\CO_{\fS}$-Lie algebras. \subsection{Differential operators} Let $\CM,\ \CN$ be coherent $\CO_{\fX}$-modules. We define {\em the sheaf of differential operators of order $\leq n$ from $\CM$ to $\CN$}: $$ \Diff^{\leq n}_{\fX/\fS}(\CM,\CN)=\CHom_{\CO_{\fX}}(\CP^n_{\fX/\fS}(\CM),\CN). $$ The structure of $\CO_{\fX}$-bimodule on $\CP^n({\CM})$ induces the structure of $\CO_{\fX}$-bimodule on $\Diff^{\leq n}_{\fX/\fS}(\CM,\CN)$. The projections ~(\ref{cofilt}) induce embeddings $$ \Diff^{\leq n-1}_{\fX/\fS}(\CM,\CN)\hra\Diff^{\leq n}_{\fX/\fS}(\CM,\CN)\ . $$ We set $$ \Diff_{\fX/\fS}(\CM,\CN)=\dirlim\ \Diff^{\leq n}_{\fX/\fS}(\CM,\CN)\ . $$ We denote $$ \Diff^{\leq n}_{\fX/\fS}:=\Diff^{\leq n}_{\fS}(\CO_{\fX},\CO_{\fX});\ \Diff_{\fX/\fS}:=\Diff_{\CO_{\fS}}(\CO_X,\CM)\ . $$ As in the case of schemes, we have compositions $$ \Diff^{\leq n}_{\fX/\fS}\otimes_{\CO_{\fS}}\Diff^{\leq m}_{\fX/\fS}\lra \Diff^{\leq n+m}_{\fX/\fS} $$ (where in the tensor product the 1st (resp., 2nd) factor has a structure of a right (resp., left) $\CO_{\fX}$-module). \subsection{} Note that we have canonically $$ \Diff^{\leq 1}_{\fX/\fS}\cong\CO_{\fX}\oplus\CT_{\fX/\fS}\ . $$ Thus, the multiplication induces a canonical map \begin{equation} \label{twist-dif} U_{\CO_{\fX}}(\CT_{\fX/\fS})\lra \Diff_{\fX/\fS} \end{equation} --- cf. ~\ref{twist-env}. \subsection{} \label{functor} Suppose we have a commutative square of formal schemes $$\begin{array}{ccc} \fX & \overset{u}{\lra} & \fX' \\ \downarrow & \; & \downarrow \\ \fS & \lra & \fS' \end{array}$$ It induces natural maps $$ \nu_n: u^*\CP^n_{\fX'/\fS'}\lra \CP^n_{\fX/\fS} $$ If the square is cartesian, these maps are isomorphisms, cf. {\em loc.cit.}, 16.4.5. \subsection{} \label{point} Let $\fX$ be a formal scheme, $x:\Spec(k)\lra \fX$ a point. Then one has a canonical isomorphism of $k(x)$-algebras $$ (\CP^n_{\fX/k})_x\otimes_{\CO_x}k(x)\cong \CO_x/\fm_x^{n+1} $$ --- cf. {\em loc.cit.}, 16.4.12. \subsection{Formally smooth morphisms} Let us call a morphism $f:\fX\lra \fS$ of formal schemes {\em formally smooth} if for every commutative square $$\begin{array}{ccc} Y & \lra & \fX \\ \downarrow & \; & \downarrow \\ T & \lra & \fS \end{array}$$ where $Y=\Spec(B)\lra T=\Spec(A)$ is a closed embedding of affine (usual) schemes corresponding to an epimorphism $A\lra B$ whose kernel $I$ satisfies $I^2=0$, there exists a lifting $T\lra\fX$. \subsubsection{} Suppose that $f=\dirlim\ f_n$, where all $f_n:X_n\lra S_n$ are smooth morphisms of (usual) schemes. Then $f$ is formally smooth. This follows from ~\cite{ega} 0$_{\mbox{IV}}$, 19.4.1. \subsection{} Let $\fX\overset{f}{\lra}\fY\overset{g}{\lra}\fS$ be morphisms of formal schemes. By ~\ref{functor} they induce maps $f^*\Omega^1_{\fY/\fS}\lra\Omega^1_{\fX/\fS}$ and $\Omega^1_{\fX/\fS}\lra\Omega^1_{\fX/\fY}$. They in turn induce the maps in the sequence \begin{equation} \label{seq} 0\lra\CT_{\fX/\fY}\overset{g'}{\lra}\CT_{\fX/\fS} \overset{f'}{\lra} f^*\CT_{\fY/\fS}\ . \end{equation} \subsubsection{} \label{seq-thm} \begin{thm}{} The sequence ~(\ref{seq}) is exact. If $f$ is formally smooth then $f'$ is surjective. \end{thm} \begin{pf} Follows from ~\cite{ega} 0$_{\mbox{IV}}$, 20.7.18. \end{pf} \subsection{Smooth morphisms} \label{smooth} \begin{defn}{} A morphism $f:\fX\lra\fS$ is called {\em smooth} if it is formally smooth and locally of finite type (\cite{ega} I, 10.13). \end{defn} \subsubsection{} \begin{lem}{} If $f$ is smooth then $\Omega^1_f$ is locally free of finite type. \end{lem} \begin{pf} Everything is reduced to the case of affine formal schemes, $\fX=\Spf(B),\ \fS=\Spf(A)$, and $B$ is a noetherian topological $A$-algebra topologically of finite type over $A$. First, $\Omega^1_{B/A}$ is of finite type by ~\ref{power} and ~\ref{seq-thm} since $B$ is a quotient of some $A\{ T_1,\ldots, T_n\}$. Let us prove that $\Omega^1_{B/A}$ is projective $B$-module. Suppose we have a diagram $$\begin{array}{ccc} \; & \; & \Omega^1_{B/A}\\ \; & \; & \downarrow\\ M & \overset{\psi}{\lra}& N\\ \end{array}$$ with epimorhic $\psi$ and finitely generated $M$ . Let $I$ be an ideal of definition in $B$. By \\ \cite{ega}~0$_{\mbox{IV}}$, 20.4.9, we can find liftings $\phi_n:\Omega^1_{B/A}\lra M/I^nM$ for each $n$. We claim that we can choose $\phi_n$'s in such a way that for all $n$ the composition $\Omega^1_{B/A}\overset{\phi_{n+1}}{\lra} M/I^{n+1}M\lra M/I^nM$ is equal to $\phi_n$. Indeed, we do it step by step, using again the lifting property and the fact that canonical maps $$ M/I^{n+1}M\lra M/I^nM\times_{N/I^nN}N/I^{n+1}N $$ are surjective. Since $B$ is complete, $M=\invlim\ M/I^nM$, hence there exists a lifting $\phi:\Omega^1_{B/A}\lra M$. Now, since $\Omega^1_{B/A}$ is of finite type, we can find an epimorphic map $F\lra \Omega^1_{B/A}$ from a finitely generated free $B$-module $F$; using the lifting property proved above, we conclude that $\Omega^1_{B/A}$ is a direct summand of $F$. \end{pf} \subsection{Differentially smooth morphisms} \begin{defn}{} A morphism $f:\fX\lra\fS$ of formal schemes is called {\em differentially smooth} if $\Omega^1_f$ is locally free of finite rank. \end{defn} For an arbitrary $f$, define the graded ring $$ \CGr.(\CP_f)=\oplus_{n=0}^{\infty}\CGr_n(\CP_f);\ \CGr_n(\CP_f)=\fI^n/\fI^{n+1}, $$ $\fI$ being as in ~\ref{defn-jets}. Evidently $\CGr_1(\CP_f)=\Omega^1_f$ and we have the canonical surjective morphism from the symmetric algebra \begin{equation} \label{symm} S^._{\CO_{\fX}}(\Omega^1_f)\lra\CGr_.(\CP_f)\ . \end{equation} \subsubsection{} \begin{lem}{} If $f$ is differentially smooth then ~(\ref{symm}) is isomorphism. \end{lem} \begin{pf} The same as in ~\cite{ega} IV 16.12.2 (cf. {\em loc.cit.}, 16.10). \end{pf} \subsubsection{} \label{basis} \begin{thm}{} Let $f$ be differentially smooth, $U\subseteq\fX$ an open, $\{ t_i\}_{i\in I}\in\Gamma(U,\CO_{\fX})$ a set of sections such that $\{ dt_i\}_{i\in I}$ is the basis of $\Omega^1_{U/\fS}$. Let $\{\dpar_i\}\in\Gamma(U,\CT_{\fX/\fS})$ be the dual basis. Then (i) all $\dpar_i$ commute with each other; (ii) $\Diff^{\leq n}_{\fX/\fS}$ is freely generated by all monomials on $\dpar_i$ of degrees $\leq n$. \end{thm} \begin{pf} The same as in {\em loc.cit.}, 16.11.2. \end{pf} Thus, if $\fS=\Spec(k)$, if $f$ is differentially smooth then $\fX$ is differentially smooth in the sense of ~\ref{dif-smo}. \subsection{} \begin{cor}{} If $f$ is differentially smooth then the map ~(\ref{twist-dif}) is isomorphism. \end{cor} \begin{pf} Follows immediately from \Thm{basis} and \Thm{pbw-algebr}: the map~(\ref{twist-dif}) preserves filtrations and induces an isomorphism on the associated graded rings. \end{pf} Hence, if $\fS=\Spec(k)$, our sheaf $\Diff_{\fX/\fS}$ coincides with $\Diff_{\fX}$ defined in Section ~\ref{twisted}. \subsection{Examples of differentially smooth morphisms} \subsubsection{} Smooth morphisms. This follows from {\em loc.cit.}, 16.10.2. \subsubsection{} The structure morphism $\Spf(A\{ T_1,\ldots,T_n\})\lra\Spf(A)$ (see ~\ref{power}). \section{Homotopy Lie algebras and direct image functor} \label{direct} In this Section we develop a formalism of homotopy Lie algebras which is sufficiently good for our needs. In~\ref{holie} we define the category $\Holie$ of homotopy Lie algebras and a functor from it to the filtered derived category given by the Quillen standard complex (see~\ref{quillen}). In order to define higher direct images for $\Holie$ in~\ref{dihla}, we provide in~\ref{s-derham} a construction of Thom-Sullivan functor from cosimplicial modules to complexes, and some homotopical properties of it in~\ref{cob}. The proof of the properties of the Thom-Sullivan functor is given in Section~\ref{thoms}. The main result of this Section is~\Thm{main-5}. \subsection{Homotopy Lie algebras} \label{holie} \subsubsection{} \label{cat} Let $X$ be a formal scheme. We define $\Dglie(X)$ (resp., $\Dglie^{qc}(X),\ \Dglie^{c}(X)$) as a category of dg $\CO_X$-Lie algebras $\fg$ such that --- (i) all components $\fg^i$ are $\CO_X$-flat (and quasicoherent or coherent over $\CO_X$ respectively); (ii) one has $\CH^i(\fg)=0$ for sufficiently big $i$. A morphism $f: \fg\lra \fh$ in this category is a map of complexes of $\CO_X$-modules compatible with brackets. Let us call $f$ {\em a quasi-isomorphism} if it induces an isomorphism of all cohomology sheaves $\CH^i(f):\CH^i(\fg)\lra \CH^i(\fh)$. By definition, the category $\Holie^{(*)}(X)$, is the localization of $\Dglie^{(*)}(X)$ with respect to the class of all quasi-isomorphisms. Objects of $\Holie(X)$ are called {\em homotopy Lie algebras}. So, each dg $\CO_X$-Lie algebra defines a homotopy Lie algebra. \subsubsection{} If $f: X\lra Y$ is a flat morphism, the inverse image functor for $\CO_X$-modules induces the functor \begin{equation} \label{inv-im-dglie} f^{*Lie}:\Dglie^{(*)}(Y)\lra \Dglie^{(*)}(X) \end{equation} It takes quasi-isomorphisms to quasi-isomorphisms, and hence induces the functor (to be denoted by the same letter) \begin{equation} \label{inv-im-holie} f^{*Lie}:\Holie^{(*)}(Y)\lra \Holie^{(*)}(X) \end{equation} \subsubsection{Filtered derived categories} (For more details, see ~\cite{i}, ch. V, no.1 where the case of finite filtrations is considered.) Let $\CC(X)$ denote the category of complexes of $\CO_X$-modules, and $\CCF(X)$ the category whose objects are complexes of $\CO_X$-modules $A$ together with a filtration by $\CO_X$-subcomplexes $\ldots\subseteq F_iA\subseteq F_{i+1}A\subseteq\ldots,\ i\in \Bbb Z$ such that $F_iA=0$ for sufficiently small $i$ and $A=\cup_i F_iA$; the morphisms being morphisms of complexes compatible with filtrations. We set $\gr_i(A)=F_iA/F_{i-1}A$. For $a,b\in \Bbb Z$, let $\CCF_{[a,b]}(X)$ be the full subcategory consisting of complexes with $F_{a-1}A=0,F_{b}A=A$. A morphism $f:A\lra B$ in $\CCF(X)$ is called {\em a filtered quasi-isomorphism} if the induced maps $\gr_i(f): \gr_i(A)\lra \gr_i(B)$ are quasi-isomorphisms. From our assumptions on filtrations follows that a filtered quasi-isomorphism is a quasi-isomorphism. We denote by $\CD(X)$ the localization of $\CC(X)$ with respect to quasi-isomorphisms, and $\CDF(X)$, $\CDF_{[a,b]}(X)$ the localization of $\CCF(X)$ (resp., $\CCF_{[a,b]}(X)$) with respect to filtered quasi-isomorphisms. \subsubsection{} \begin{lem}{} Suppose that $f:\fg\lra \fh$ is a quasi-isomorphism in $\Dglie(X)$. Then the induced morphism $C(f): C(\fg)\lra C(\fh)$ is a filtered quasi-isomorphism. \end{lem} \begin{pf} It suffices to prove that all $\gr_i(f): S^i(\fg[1])\lra S^i(\fh[1])$ are quasi-isomorphisms. Repeated application of the K\"{u}nneth spectral sequence ~\cite{g}, Ch.~I, 5.5.1, shows that $f^{\otimes i}:\fg^{\otimes i}\lra \fh^{\otimes i}$ are quasi-isomorphisms (we use the assumptions ~\ref{cat} (i)-(ii)). After passing to $\Sigma_n$-invariants, we get the desired claim. \end{pf} \subsubsection{} It follows that the functors $\fg\mapsto C(\fg)$, $\fg\mapsto F_nC(\fg)$ induce functors between homotopy categories $$ C: \Holie(X)\lra \CDF(X) $$ $$ F_nC:\Holie(X)\lra \CDF_{[0,n]}(X) $$ For $\fg\in \Holie(X)$ we define homology sheaves $$ \CH^{Lie}_n(\fg)=\CH^{-n}(C(\fg));\ \CF_m\CH_n^{Lie}(\fg)=H^{-n}(F_mC(\fg)) $$ --- these are sheaves of $\CO_X$-modules. If $X=\Spec (k)$, we denote them $H^{Lie}_n(\fg),\ F_mH_n^{Lie}(\fg)$ respectively --- these are $k$-vector spaces. \subsubsection{} \label{sec-spectr-seq} Let $\fg\in \Holie(X)$. We have a spectral sequence \begin{equation} \label{spectr-seq} E_1^{pq}=\CH^q(\Lambda^{-p}_{\CO_X}(\fg)) \Lra \CH^{Lie}_{-p-q}(\fg) \end{equation} If all sheaves $\CH^i(\fg)$ are $\CO_X$-flat then by K\"{u}nneth formula, we have \begin{equation} \label{kunneth} \CH^q(\Lambda^{n}_{\CO_X}(\fg))\cong \left(\sum_{q_1+\ldots+q_n=q}\CH^{q_1}(\fg)\otimes_{\CO_X}\ldots\otimes_{\CO_X} \CH^{q_n}(\fg)\right)^{\Sigma_n,-} \end{equation} where $(\cdot)^{\Sigma_n,-}$ denotes the subspace of anti-invariants of the symmetric group $\Sigma_n$. \subsection{Thom-Sullivan complex} \label{s-derham} \subsubsection{} In this subsection $\Delta$ will denote the category of totally ordered finite sets $[n]=\{ 0,\ldots, n\}$, $n\geq 0,$ and non-decreasing maps. For any category $C$, we denote $\Delta^0C,\ \Delta C$ the categories of simplicial and cosimplicial objects in $C$ respectively. For $A\in \Delta^0 C$ (resp., $B\in \Delta C$) we denote $\alpha^*: A_m\lra A_n$ (resp., $\alpha_*: A^n\lra A^m$) the map in $C$ corresponding to a morphism $\alpha: [n]\lra[m]$ in $\Delta$. $\Ens$ will denote the category of sets, $\Delta[n]\in \Delta^0\Ens$ the standard $n$-simplex. \subsubsection{} (For details, see \cite{bug}). Denote by $R_n$ the commutative $k$-algebra {}~{$k[t_0,\ldots,t_n]/(\sum_{i=0}^nt_i-1)$} ($t_i$ being independent variables). Set $\Bbb A[n]=\Spec R_n$. Together with the standard face and degeneracy maps, algebras $R_n,\ n\geq 0,$ form a simplicial algebra $R$. Let $\Omega_n$ denote the algebraic de Rham complex of $R_n$ over $k$, $$ \Omega_n=\Gamma(\Bbb A[n],\Omega_{\Bbb A[n]/k}) $$ It is a commutative dg $k$-algebra which may be identified with $$ R_n[dt_0,\ldots,dt_n]/(\sum dt_i), $$ with $\deg(dt_i)=1$ and the differential given by the formula $d(t_i)=dt_i$. The algebras $\Omega_n,\ n\geq 0$ together with coface and codegeneracy maps induced from $R$, form a simplicial commutative dg algebra $\Omega=\{\Omega_n\}$. So, for a fixed $p,\ \Omega^p$ is a simplicial vector space, and $\Omega_p$ is a complex. \subsubsection{} \label{invtensor} (Cf. ~\cite{hlha}, 3.1.3) Let $C$ be a small category. Let us denote $\Mor(C)$ the category whose objects are morphisms $f: x\lra y$ in $C$, a map $f\lra g$ is given by a commutative diagram $$\begin{array}{ccc} \cdot & \overset{f}{\lra} & \cdot \\ \uparrow & \; & \downarrow \\ \cdot & \overset{g}{\lra} & \cdot \\ \end{array}$$ The composition is defined in the evident way. For $f$ as above set $s(f)=x,\ t(f)=y$. Let $A$ be a commutative $k$-algebra, $X: C^0\lra \Mod(k),\ Y:C\lra \Mod(A)$ two functors ($C^0$ denotes the opposite category). Denote by $$ X\otimes Y: \Mor(C)\lra \Mod(A) $$ the functor given by $X\otimes Y(f)=X(s(f))\otimes_kY(t(f))$ and defined on morphisms in the evident way. Set $$ X\invtimes Y=\invlim (X\otimes Y)\in \Mod(A) $$ In other words, $X\invtimes Y$ is an $A$-submodule of $\prod_{c\in Ob(C)}X(c)\otimes Y(c)$ consisting of all $\{a(c)\}_{c\in Ob(C)},$ $ a(c)\in X(c)\otimes Y(c)$ such that for every $f:b\lra c\in \Mor(C)$ $$ (f^*\otimes 1)(a(c))=(1\otimes f_*)(a(b)). $$ \subsubsection{Thom-Sullivan complex} (Cf. ~\cite{hlha}, \S 4) Applying the previous construction to $C=\Delta,\ X=\Omega^p\in \Delta^0\Mod(k), \ Y\in\Delta\Mod(A)$, we get $A$-modules $$ \Omega^p(Y)=\Omega^p\invtimes Y $$ When $p$ varies, they form a complex $\Omega(Y)\in \CC(A)$ which is called {\em Thom-Sullivan complex} of $Y$. If $Y$ is a constant cosimplicial object, then \begin{equation} \label{const} \Omega(Y)\cong Y \end{equation} canonically. For example, if $Y$ is the cosimplicial space of $k$-valued cochains of a simplicial set $X$, $\Omega(Y)$ is the Thom-Sullivan complex of $X$ described in ~\cite{bug}. If $Y$ is a {\em complex} in $\Delta\Mod(A)$ or, what is the same, a cosimplicial complex of $A$-modules, then, applying the previous construction componentwise we get a {\em bicomplex} of $A$-modules; we will denote the corresponding simple complex again $\Omega(Y)$. This way we get a functor \begin{equation} \label{thom} \Omega: \Delta\Mod(A)\lra \CC(A). \end{equation} \subsubsection{} \begin{lem}{} The functor $\Omega$ is exact. \end{lem} For a proof, see \ref{exact-6}. \subsubsection{Normalization} For $Y\in \Delta\Mod(A)$ denote by $N(Y)\in \CC(A)$ its normalization, i.e. set $N(Y)^i\subset Y^i$ to be the intersection of kernels of all codegeneracies $Y^i\lra Y^{i-1}$, the differential $N(Y)^i\lra N(Y)^{i+1}$ being the alternating sum of the cofaces. ($N(Y)^i=0$ for $i<0$). We say that $Y$ is {\em finite dimensional} if $N(Y)^i=0$ for $i>>0$. This way we get a functor $$ N:\Delta\Mod(A)\lra \CC(A) $$ For each $n\geq 0$ denote by $\CZ^{\cdot}_n=C^{\cdot}(\Delta[n],k)$ the complex of normalized $k$-valued cochains of the standard simplex. When $n$ varies, the complexes $\CZ^{\cdot}_n$ form a simplicial object $\CZ=\CZ^{\cdot}_{\cdot}\in \Delta^0\CC(k)$. Given $Y\in \Delta\Mod(A)$, we can apply the construction {}~\ref{invtensor} to each $\CZ^n_{\cdot}$ and $Y$, and obtain $\CZ\invtimes Y\in \CC(A)$. It follows from the definitions that we have a natural isomorphism $$ \CZ\invtimes Y\cong N(Y) $$ (cf. ~\cite{hlha}, 2.4). \subsubsection{} For each $n$ we have the {\em integration} map $$ \int: \Omega_n\lra C^*(\Delta[n],k) $$ in $\CC(k)$, ~\cite{bug},\S 2. Taken together, they give rise to the morphism $$ \int: \Omega\lra \CZ $$ in $\Delta^0C(k)$. It induces natural maps \begin{equation} \label{int-y} \int_Y:\Omega(Y)\lra N(Y) \end{equation} for every $Y\in \Delta\Mod(A)$. \subsubsection{"De Rham theorem"} \label{derham-thm} \begin{lem}{} For every $Y\in \Mod(A)$ the map $\int_Y$ is a quasi-isomorphism. \end{lem} \begin{pf} See ~\cite{hlha}, 4.4.1. \end{pf} \subsubsection{Base change} Let $A'$ be a commutative $A$-algebra, $Y\in \Delta\Mod(A)$, whence $Y\otimes_AA'\in \Delta\Mod(A')$. We have an evident base change morphism \begin{equation} \label{bchange-map} \Omega(Y)\otimes_AA'\lra \Omega(Y\otimes_AA'). \end{equation} \subsubsection{} \label{base} \begin{lem}{} If $Y$ is finite dimensional then ~(\ref{bchange-map}) is isomorphism. \end{lem} \subsubsection{} \label{flat} \begin{lem}{} Let $A$ be noetherian and $Y\in \Delta\Mod(A)$; suppose that all $Y^n$ are flat over $A$. Then for every $p,\ \Omega^p(Y)$ is flat over $A$. \end{lem} The proof of ~\ref{base} and ~\ref{flat} will be given in the next Section, see ~\ref{base-6},~\ref{flat-6}. \subsubsection{} \label{cosimpl-lie} Suppose that $B$ is a commutative dg algebra and $\fg$ is a dg Lie algebra. Then $B\otimes \fg$ is naturally a dg Lie algebra, the bracket being defined as $$ [a\otimes x,b\otimes y]=(-1)^{|x||b|} ab\otimes [x,y]. $$ Let $\fg$ be a cosimplicial dg $A$-Lie algebra. Then all $\Omega_n\otimes \fg^n$ are dg $A$-Lie algebras; hence their inverse limit $\Omega(\fg)$ is. This way we get a functor $\Omega$ from the category of cosimplicial dg $A$-Lie algebras to dg $A$-Lie algebras. \subsubsection{} Let $X$ be a (formal) scheme. We can sheafify Thom-Sullivan construction. Denote by $\Mod(\CO_X)$ the category of sheaves of $\CO_X$-modules. For $\CF\in \Delta\Mod(\CO_X)$ we get $N(\CF), \Omega(\CF)\in \CC(X)$. We call $\CF$ {\em finite dimensional} if $N(\CF)^i=0$ for $i>>0$. We denote $\Delta(\Mod(\CO_X))^f\subseteq \Delta(\Mod(\CO_X))$ the full subcategory consisting of finite dimensional cosimplicial sheaves. By~\ref{qc} if $\CF$ is finite dimensional, and all $\CF^i$ are quasicoherent then all $\Omega^i(\CF)$ are quasicoherent. By~\ref{flat} if all $\CF^i$ are $\CO_X$-flat then all $\Omega^i(\CF)$ are $\CO_X$-flat. If $\fg$ is a cosimplicial dg $\CO_X$-Lie algebra, then, applying {}~\ref{cosimpl-lie} we get a dg $\CO_X$-Lie algebra $\Omega(\fg)$. \subsubsection{} \label{derham-dir-im} Let $f:X\lra Y$ be a map of schemes, $\CF\in\Delta\Mod(\CO_X)$. Then we have an evident equality $$ f_*\Omega(\CF)=\Omega(f_*\CF) $$ \subsection{Cosimplicial homotopies} \label{cob} Let $\CA$ be a category with finite products. \subsubsection{} \begin{defn}{path} Define the {\em path functor} $X\mapsto X^I$ from $\Delta{\CA}$ to itself as follows: $$ (X^I)^n=\prod_{s:[n]\ra[1]}X^n,$$ the map $f_*:(X^I)^m\ra(X^I)^n$ for any $f:[m]\ra[n]$ being defined by the formula $$ f_*(\{x_s\})_t=f_*(x_{tf}),\ t:[n]\ra[1].$$ \end{defn} The path functor is endowed with natural transformations $$i_X:X\ra X^I,\ \pr_{i,X}:X^I\ra X (i=0,1)$$ given by the formulas $$ i_X(x)_s=x;\ \pr_{i,X}(\{x_s\})=x_{s_i}$$ where $s_i:[n]\ra[1]$ denotes the constant map with value $i$. \Defn{path} is a special case of constructions given in~\cite{q2}, ch.~2, see Prop. 2 for the dual statement. In particular, one can define in the same way a functor $X\mapsto X^S$ for any finite simplicial set $S$, this construction is functorial on $S\in\Delta^0\Ens$ and the natural transformations $i_X,\pr_{i,X}$ are induced by the corresponding maps $\pi:I\ra *\text{ and }\iota_i:*\ra I (i=0,1)$ in $\Delta^0\Ens$. \subsubsection{} \begin{defn}{strict-homo} Let $X,Y\in\Delta{\CA}$. Maps $f_i:X\ra Y \ (i=0,1)$ are said to be {\em strictly homotopic} if there exists a map $F:X\ra Y^I$ such that $f_i=\pr_i\circ F$. \end{defn} \subsubsection{} \begin{exa}{=} Let $X\in\Delta{\CA}$. The maps $\id,\ i\circ\pr_0:X^I\ra X^I$ are strictly homotopic. In fact, the maps $\id_I$ and $\iota_0\circ\pi$ are strictly homotopic in $\Delta^0\Ens$. \end{exa} \subsubsection{} Let now $\CA$ be additive. Recall that for $X\in\Delta{\CA}$ the total complex $\Tot(X)\in \CC({\CA})$ is defined by the properties $$ \Tot(X)^n=X^n,$$ $$d=\sum (-1)^i\delta^i:\Tot(X)^n\ra\Tot(X)^{n+1}.$$ This defines a functor $\Tot:\Delta{\CA}\ra \CC({\CA})$. \begin{lem}{homo-homo} A strict homotopy $H:X\ra Y^I$ between $f$ and $g$ induces a (chain) homotopy $h:\Tot(X)\ra\Tot(Y)[-1]$ between $\Tot(f)$ and $\Tot(g)$. \end{lem} \begin{pf} For $x\in X^n$ define $$ h(x)=\sum_{i=0}^{n-1}(-1)^i\sigma^i(y_{\alpha_i})$$ where $y_{\alpha}$ are defined by $H(x)=\{y_{\alpha}\}$ and \begin{equation} \alpha_i(t)=\begin{cases} 0& \text{ if } t\leq i\\ 1& \text{ if } t>i. \end{cases} \end{equation} A direct calculation shows that $h$ is the chain homotopy we need. \end{pf} \subsubsection{} We apply the above constructions to homotopy Lie algebras. Fix a commutative ring $A\supseteq{\Bbb Q}$ and put ${\CA}=\Dglie(A)$. \begin{cor}{glueh} Let $X,Y\in\Delta\Dglie(A)$. Let $f,g:X\ra Y$ be strictly homotopic. Then the maps $\Omega(f), \Omega(g):\Omega(X)\ra\Omega(Y)$ induce equal maps in the homotopy category $\Holie(A)$. \end{cor} \begin{pf} It suffices to check that $\Omega(\pr_{0,Y})=\Omega(\pr_{1,Y})$ in the homotopy category. Since the both maps split $\Omega(i_Y)$, it sufficies to check that the latter one is a quasi-isomorphism. For this we can substitute the functor $\Omega$ with $\Tot$ (since they are naturally quasi-isomorphic). We can also substitute the category $\Dglie(A)$ with the category ${\CC}(A)$ since the forgetful functor $\#:\Dglie(A)\ra{\CC}(A)$ commutes with direct products. Then~\Exa{=} and~\Lem{homo-homo} prove even more that we actually need. \end{pf} \subsection{Direct image of homotopy Lie algebras} \label{dihla} \subsubsection{\v{C}ech resolutions} Let $X$ be a topological space, $\CF$ an abelian sheaf over $X$, $\CU=\{ U_i\}_{i\in I}$ an open covering of $X$. For each $n\geq 0$ set $$ \CHC^n(\CU,\CF)=\prod_{(i_0,\ldots,i_n)\in I^{n+1}}j_{i_0\ldots i_n*} j_{i_0\ldots i_n}^*\CF $$ where $j_{i_0\ldots i_n}:U_{i_0}\cap\ldots\cap U_{i_n}\hra X$. Together with the standard cofaces and codegeneracies, the sheaves $\CHC^n(\CU,\CF),\ n\geq 0,$ form a cosimplicial sheaf $\CHC(\CU,\CF)$. It is finite dimensional if the covering $\CU$ is finite. We have an embedding $\CF\lra \CHC^0(\CU,\CF)$; it induces the augmentation map \begin{equation} \label{augm} \CF\lra\CHC(\CU,\CF) \end{equation} where $\CF$ is considered as a constant cosimplicial sheaf. The induced map \begin{equation} \label{augm-n} \CF\lra N(\CHC(\CU,\CF)) \end{equation} is a quasi-isomorphism, \cite{g}, ch.~II, 5.2.1. \subsubsection{} Suppose that --- either $X$ is a scheme and $\CF$ is a quasicoherent sheaf of $\CO_X$-modules, --- or $X$ is a formal scheme and $\CF$ is a coherent sheaf. Choose an affine covering $\CU$. Then the complex $\Gamma(X,N(\CHC(\CU,\CF)))$ represents $R\Gamma(X,\CF)$, ~\cite{h}, III, 4.5; ~\ref{aff}. If $f:X\lra Y$ is a morphism of (formal) schemes then $f_*N(\CHC(\CU,\CF))$ represents $Rf_*(\CF)$. \subsubsection{} Applying to ~(\ref{augm}) the functor $\Omega$, and using {}~(\ref{const}), we get a canonical map of complexes \begin{equation} \label{augm-omega} \CF\lra \Omega(\CHC(\CU,\CF)) \end{equation} which is a quasi-isomorphism by the above and ~\ref{derham-thm}. \subsubsection{} Suppose we are in one of the following situations: {\bf Case 1.} $f: X\lra Y$ is a flat morphism of schemes, $\fg\in \Dglie^{qc}(X)$. {\bf Case 2.} $f: X\lra Y$ is a flat morphism of formal schemes, $\fg\in \Dglie^{c}(X)$. Choose an open affine covering $\CU$ of $X$, and consider the cosimplicial complex of $\CO_Y$-modules $f_*\CHC(\CU,\fg)$; it has an evident structure of a cosimplicial dg $\CO_Y$-Lie algebra. Applying the Thom-Sullivan functor, we get a dg $\CO_Y$-Lie algebra $\Omega(f_*\CHC(\CU,\fg))$. Note that by ~\ref{derham-dir-im} this is the same as $f_*\Omega(\CHC(\CU,\fg))$. Let us denote it $f_{*,\CU}^{Lie}(\fg)$. \subsubsection{} Let now $\CU=\{U_i\}_{i\in I}$ and $\CV=\{V_j\}_{j\in J}$ be two open coverings of $X$ so that cosimplicial dg Lie algebras $\CHC(\CU,\fg),\ \CHC(\CV,\fg)$ are defined. Let maps $f,g:I\ra J$ satisfy the conditions $U_i\subseteq V_{f(i)},\ U_i\subseteq V_{g(i)}.$ \begin{lem}{to5.} The maps from $\CHC(\CV,\fg)$ to $\CHC(\CU,\fg)$ induced by $f$ and $g$ are strictly homotopic. \end{lem} \begin{pf} Immediate. \end{pf} \begin{cor}{} In the notations above the maps from $f_{*,\CV}^{Lie}(\fg)$ to $f_{*,\CU}^{Lie}(\fg)$ induced by $f$ and by $g$, coincide in $\Holie(Y)$. \end{cor} \begin{pf} Compare~\ref{to5.} with~\ref{glueh}. \end{pf} \subsubsection{} If $\fg=f^{*\Lie}\fh$ for some $\fh\in\Dglie(Y)$ then the augmentation ~(\ref{augm-omega}) $$ f^{*\Lie}\fh\lra \Omega(\CHC(\CU,f^{*\Lie}\fh)) $$ and the adjunction map $\fh\lra f_*f^*\fh$ induce the map of dg $\CO_X$-Lie algebras \begin{equation} \label{adj} \fh\lra f_{*,\CU}^{Lie}(f^{*\Lie}\fh). \end{equation} On the other hand, for any $\fg\in\Holie(X)$ a map \begin{equation} f^{*\Lie}f_*^{\Lie}(\fg)\lra\fg \end{equation} in $\Holie(X)$ is defined as the composition \begin{equation} f^{*\Lie}f_*^{\Lie}(\fg)=f^{*\Lie}f_*\Omega(\CHC(\CU,\fg))\ra \Omega(\CHC(\CU,\fg))\ra\fg \end{equation} the last map being inverse to the composition of the quasi-isomorphisms~(\ref{augm-n}) and~(\ref{int-y}). Putting together the above considerations, we get the following \subsubsection{} \begin{thm}{main-5} (i) In Case 1 (resp., Case 2) $f_{*,\CU}^{Lie}(\fg)$ belongs to $\Dglie^{qc}(Y)$ (resp., $\Dglie(Y)$). (ii) The class of $f_{*,\CU}^{Lie}(\fg)$ in $\Holie^{qc}(Y)$ (resp., $\Holie(Y)$) does not depend, up to a unique isomorphism, on $\CU$. (iii) The functor $f_{*,\CU}^{Lie}$ takes quasi-isomorphisms to quasi-isomorphisms; thus it induces the functor $$ f_*^{Lie}:\Holie^{qc}(X)\lra \Holie^{qc}(Y) $$ in Case 1, and $$ f_*^{Lie}:\Holie^{c}(X)\lra \Holie(Y) $$ in Case 2 respectively, such that the square $$\begin{array}{ccc} \Holie^{(*)}(X) & \overset{f_*^{Lie}}{\lra} & \Holie^{(*')}(Y) \\ \downarrow & \; & \downarrow \\ \CD(X) & \overset{f_*}{\lra} & \CD(Y) \\ \end{array}$$ the vertical arrows being forgetful functors, 2-commutes in both cases. This means that there is a natural isomorphism between the two functors from $\Holie^{(*)}(X)$ to $\CD(Y)$. (iv) In Case 1, maps ~(\ref{adj}) induce the natural transformation $\Id_{\Holie(Y)}\lra f_*^{Lie}f^{*Lie}$ which makes the functor $f_*^{Lie}$ right adjoint to $f^{*Lie}$. $\Box$ \end{thm} \subsubsection{} In case $Y=\Spec(k),\ f:X\lra Y$ the structure morphism, we will denote $f_*^{Lie}(\fg)$ also by $\Gamma^{Lie}(X,\fg)$. \section{Thom-Sullivan functor} \label{thoms} In this Section we will compute more explicitely the Thom-Sullivan functor and prove some fundamental properties of it. \subsection{} We keep the assumptions and notations from ~\ref{s-derham}. In particular, $k$ is a base field of characteristic zero and $A$ denotes a commutative $k$-algebra. Expressing $t_0$ as $t_0=1-\sum_{i=1}^nt_i$ we identify $k$-algebras $R_n$ with $k[t_1,\ldots,t_n]$ and commutative dg $k$-algebras $\Omega_n$ with $R_n[dt_1,\ldots,dt_n]$. The standard simplicial morphisms of dg $k$-algebras $$ d_i:\Omega_n\lra\Omega_{n-1},\ s_i:\Omega_n\lra \Omega_{n+1}, $$ $i=0,\ldots, n$ are defined by the formulas \begin{equation} d_i(t_j)= \begin{cases} t_j & \text{ if } j<i,\\ 0 & \text{ if } j=i,\\ t_{j-1}& \text{ if } j>i \end{cases} \label{d_i} \end{equation} for $i>0$ and \begin{equation} d_0(t_j)= \begin{cases} t_{j-1} & \text{ if } j>1,\\ 1-\sum t_i& \text{ if } j=1. \end{cases} \label{d_0} \end{equation} \begin{equation} s_i(t_j)= \begin{cases} t_j & \text{ if } j<i,\\ t_i+t_{i+1} & \text{ if } j=i,\\ t_{j+1}& \text{ if } j>i \end{cases} \label{s_i} \end{equation} for $i>0$ and \begin{equation} s_0(t_j)=t_{j+1}. \label{s_0} \end{equation} These maps satisfy standard simplicial identities. \subsection{} Let $X\in\Delta\Mod(A)$. So $X$ is a set of $A$-modules $X^p,\ p\geq 0$, together with maps $$ \delta^i:X^{p-1}\lra X^{p},\ \sigma^i:X^{p+1}\lra X^{p}, $$ $i=0,\ldots,p$, satisfying the standard cosimplicial identities. By definition, $\Omega(X)$ is a complex of $A$-modules $$ 0\lra\Omega^0(X)\overset{d}{\lra}\ldots\overset{d}{\lra}\Omega^n(X) \overset{d}{\lra}\ldots $$ where $\Omega^n(X)$ is the space of all collections $\{x_p\in\Omega_p^n\otimes X^p\}_{p\geq 0}$ satisfying the following conditions \begin{equation} (1\otimes\delta^i)(x_p)=(d_i\otimes 1)(x_{p+1}) \label{d-eq} \end{equation} \begin{equation} (s_i\otimes 1)(x_p)=(1\otimes\sigma^i)(x_{p+1}) \label{s-eq} \end{equation} for all $p,i$ --- see the diagram below. \begin{center} \begin{picture}(14,6) \put(5,0){\makebox(4,2){$\Omega^n_{p+1}\otimes X^{p+1}$}} \put(5,4){\makebox(4,2){$\Omega^n_p\otimes X^{p}$}} \put(0,2){\makebox(4,2){$\Omega^n_{p+1}\otimes X^{p}$}} \put(10,2){\makebox(4,2){$\Omega^n_p\otimes X^{p+1}$}} \put(5.5,4.5){\vector(-2,-1){2}} \put(5.5,1.5){\vector(-2,1){2}} \put(8.5,1.5){\vector(2,1){2}} \put(8.5,4.5){\vector(2,-1){2}} \put(3.5,4){\makebox(1,0.5){$\scriptsize s_i\otimes 1$}} \put(3.5,1.5){\makebox(1,0.5){$\scriptsize 1\otimes\sigma^i$}} \put(9.5,1.5){\makebox(1,0.5){$\scriptsize d_i\otimes 1$}} \put(9.5,4){\makebox(1,0.5){$\scriptsize 1\otimes\delta^i$}} \end{picture} \end{center} Until the end of this Section, let us fix $n\geq 0$. Our aim will be an explicit computation of $\Omega^n(X)$. \subsection{} In what follows ${\Bbb N}=\{0,1,\ldots\}$. Let $p\in\Bbb N$. For $a\in{\Bbb N}^p,\alpha\in\{0,1\}^p$ denote $$ \omega_{a,\alpha}=t_1^{a_1}\cdots t_p^{a_p}dt_1^{\alpha_1}\wedge \cdots\wedge dt_p^{\alpha_p}\in\Omega_p. $$ We have $\deg\omega_{a,\alpha}=\sum\alpha_i$. Let $I^p\subset {\Bbb N}^p\times\{0,1\}^p$ consist of all pairs $(a,\alpha)$ such that $\sum\alpha_i=n$. Evidently, forms $\omega_{a,\alpha},\ (a,\alpha)\in I^p$, make up a basis of $\Omega_p^n$. Set $I=\coprod_p I^p$. An arbitrary element $x_p\in\Omega_p^n\otimes X^p$ takes form $$ x_p=\sum_{(a,\alpha)\in I^p}\omega_{a,\alpha}\otimes x_{a,\alpha} $$ with $x_{a,\alpha}\in X^p$. This way we get a mapping \begin{equation} \label{mapping} x\mapsto\{ x_{a,\alpha}\}_{(a,\alpha)\in I} \end{equation} from $\Omega^n(X)$ to the set of collections \begin{equation} \{ x_{a,\alpha}\}_{(a,\alpha)\in I},\ x_{a,\alpha}\in X^p\ \mbox{for}\ (a,\alpha)\in I^p \label{collect} \end{equation} \subsubsection{} Let us introduce two operations on $I$. For $(a,\alpha)\in I^p$ denote by $\eta_i(a,\alpha)\in I^{p+1},\ i=1,\ldots,p+1$ the element obtained from $(a,\alpha)$ by inserting $(0,0)$ on the place $i$. Further, denote by $\zeta_i(a,\alpha)\in I^{p-1},\ i=1,\ldots,p-1$, the pair $(a',\alpha')$ with $a'=(\ldots,a_i+a_{i+1},\ldots)$ and $\alpha'=(\ldots,\alpha_i+\alpha_{i+1},\ldots)$--- this operation is defined only if $\alpha_i+\alpha_{i+1}\leq 1$. \subsubsection{} Let $ x_p=\sum_{(a,\alpha)\in I^p}\omega_{a,\alpha}\otimes x_{a,\alpha}\in \Omega^{n}_p\otimes X^p$ and $ x_{p+1}=\sum_{(b,\beta)\in I^{p+1}}\omega_{b,\beta}\otimes x_{b,\beta}\in \Omega^{n}_{p+1}\otimes X^{p+1}$. The condition~(\ref{d-eq}) is equivalent to the following formulas~(\ref{d_i-eq}) and~(\ref{d_0-eq}): \begin{equation} x_{\eta_i(a,\alpha)}=\delta^ix_{a,\alpha},\ \mbox{for }i\geq 1 \label{d_i-eq} \end{equation} \begin{equation} \delta^0(x_{a,\alpha})= \sum\begin{Sb} e_0;\ e_i\leq a_i\\ \epsilon_i\leq\alpha_i\\ \beta_1:=\sum\epsilon_i\leq 1 \end{Sb} (-1)^{e+\beta_1+\sum_{i\geq j\geq 2}\epsilon_i\beta_j}{b_1!\over {e_0!\cdots e_p!}} x_{b,\beta} \label{d_0-eq} \end{equation} where the (big) sum is taken over non-negative $e_i,\epsilon_j$ satisfying the conditions written and $b_1=\sum_{i=0}^pe_i, \ e=\sum_{i=1}^pe_i,\ b_{i+1}=a_i-e_i,\ \beta_1=\sum\epsilon_i, \beta_{i+1}=\alpha_i-\epsilon_i.$ The condition~(\ref{s-eq}) is equivalent to the following formulas~(\ref{s_i-eq}) and~(\ref{s_0-eq}): \begin{equation} \sigma^i(x_{b,\beta})=\begin{cases} 0 &\text{ if } \beta(i)=\beta(i+1)=1\\ \binom{b_i+b_{i+1}}{b_i}x_{\zeta_i(b,\beta)}& \text{ otherwise } \end{cases} \label{s_i-eq} \end{equation} for $i\geq 1$; \begin{equation} \sigma^0(x_{b,\beta})=\begin{cases} x_{a,\alpha}& \text{ if } (b,\beta)=\eta_1(a,\alpha)\\ 0 & \text{ otherwise.} \end{cases} \label{s_0-eq} \end{equation} Let us denote $\bT(X)$ the set of all collections ~(\ref{collect}) satisfying {}~(\ref{d_i-eq})---~(\ref{s_0-eq}). Let us call a collection ~(\ref{collect}) {\em locally finite} if for every $p$ the set $\{ (a,\alpha)\in I^p|x_{a,\alpha}\neq 0\}$ is finite. Let us denote by $\bT^{lf}(X)\subset\bT(X)$ the subset consisting of all locally finite collections. The above argument proves \subsubsection{} \begin{lem}{} The mapping ~(\ref{mapping}) defines an isomorphism $$ \rho:\Omega^n(X)\iso\bT^{lf}(X) $$ \end{lem} $\Box$ Elements $x_{a,\alpha},\ a,\alpha\in I$ are coordinates of $x\in\Omega^n(X)$. Now our strategy will be: using relations ~(\ref{d_i-eq})---~(\ref{s_0-eq}) to express these coordinates in terms of smaller subsets of coordinates. \subsection{} An element $(a,\alpha)\in I^p$ is called {\em reduced} if none of $(a_i,\alpha_i),\ i=1,\ldots, p,$ is equal to $(0,0)$. An element $(a,\alpha)\in I^p$ is called {\em special} if it is reduced and $(a_1,\alpha_1)=(1,0)$. An element $(a,\alpha)\in I^p$ is called {\em d-free} if it is reduced and not special. The set of all d-free elements in $I^p$ will be denoted by ${\cal F}^p$. We set ${\cal F}=\cup{\cal F}^p\subset I$. Let us denote by $\bT_{\CF}(X)$ the set of all collections \begin{equation} \label{coll-f} \{ x_{b,\beta}\}_{(b,\beta)\in\CF}, \mbox{ where } x_{b,\beta}\in X^p \mbox{ for } (b,\beta)\in\CF^p \end{equation} satisfying following conditions: \begin{align} \sigma_0(x_{b,\beta})&=0; \nonumber\\ \sigma_i(x_{b,\beta})&=0 \text{ if } \beta(i)=\beta(i+1)=1\\ \sigma_i(x_{b,\beta})&=\binom{b_i+b_{i+1}}{b_i}x_{\zeta_i(b,\beta)} \text{ otherwise } \nonumber \label{props} \end{align} \subsubsection{} \begin{lem}{} The natural projection defines an isomorphism $$ \pi_1: \bT(X)\iso\bT_{\CF}(X) $$ \end{lem} {\bf Proof.} The formula~(\ref{d_i-eq}) allows one to express $x_{b,\beta}$ for non-reduced $(b,\beta)\in I^{p+1}$ through $x_{a,\alpha},(a,\alpha)\in I^p$. Furthermore, the formula~(\ref{d_0-eq}) allows one to calculate $x_{b,\beta}$ for special $(b,\beta)\in I^{p+1}$ by induction as follows. Endow $I^p$ with the partial order in which $(a,\alpha)\geq (a',\alpha')$ iff $a_i\geq a_i'$ and $\alpha_i=\alpha'_i$ for all $i$. We determine the value of $x_{b,\beta}$ for special $(b,\beta)\in I^{p+1}$ by induction on $(a,\alpha)$ such that $b=a\cup 1_1,\ \beta=\alpha \cup 0_1$ in the obvious notation. For this one should consider the equation~(\ref{d_0-eq}) and see that all special summands in the right hand side except of $x_{b,\beta}$ correspond to smaller values of $(a,\alpha)$. This immediately implies that the map which takes $x=\{x_{a,\alpha}\}_{(a,\alpha)\in I^{\cdot}}\in\Th(X)^n$ to the collection $\{x_{b,\beta}\}_{(b,\beta)\in {\cal F}^{\cdot}}$, is injective. To prove bijectivity, we proceed by induction on $p$. Suppose that, apart from $x_{b,\beta}$ with d-free $(b,\beta)$, all elements $x_{a,\alpha}$ with $({a,\alpha})\in I^i,\ i\leq p$, are constructed and satisfy the equations~(\ref{d_i-eq})--(\ref{s_0-eq}). In order to make the next induction step, we have to check the following claims: 1) If $(b,\beta)=\eta_i(a,\alpha)=\eta_j(a',\alpha')$ then $\delta^ix_{a,\alpha}=\delta^jx_{a',\alpha'}.$ 2) The condition~(\ref{d_0-eq}) is satisfied for any $(a,\alpha)\in I^p$ (and not only for reduced $(a,\alpha)$). 3) The conditions~(\ref{s_i-eq}) and~(\ref{s_0-eq}) are satisfied for any $(b,\beta)\in I^{p+1}$ (and not only for d-free $(b,\beta)$). These claims can be checked by a direct calculation using standard identities for the morphisms in the category $\Delta$. $\Box$ \subsection{} An element $(a,\alpha)\in I$ is called {\em basic} if --- $(a_1,\alpha_1)=$ either $(2,0)$ or $(0,1)$; --- $(a_i,\alpha_i)=$ either $(1,0)$ or $(0,1)$ for $i>1$. If $(a_1,\alpha_1)=(2,0)$ then $(a,\alpha)$ is called {\em basic element of the first kind}, otherwise --- {\em basic element of the second kind}. We denote $\CB$ the set of all basic elements; we define $\CB^p=\CB\cap I^p$. Evidently $\CB^p\subset\CF^p\subset I^p$. \subsubsection{} \begin{lem}{} Suppose we are given $x=(x_{a,\alpha})_{(a,\alpha)\in I}\in \bT(X)$. The elements $x_{a,\alpha},\ (a,\alpha)\in\CB$ satisfy the following relations: \begin{equation} \label{rel-1} \sigma^0(x_{a,\alpha})=0 \end{equation} \begin{equation} \sigma^i(x_{a,\alpha})=0 \text{ if } \alpha_i=\alpha_{i+1}=1 \end{equation} \begin{equation} \sigma^i(x_{a,\alpha})=\sigma^i(x_{a',\alpha'}) \end{equation} for $i>1$ where $$ a=(\ldots,0,1,\ldots), a'=(\ldots,1,0,\ldots)$$ $$ \alpha=(\ldots,1,0,\ldots), \alpha'=(\ldots,0,1,\ldots).$$ (switching the places $i,i+1$); \begin{equation} \sigma^1(x_{a,\alpha})=(\sigma^1)^2(x_{a',\alpha'}) \label{rel-3} \end{equation} where $$ a=(2,0,\ldots),\ a'=(0,1,1,\ldots)$$ $$ \alpha=(0,1,\ldots),\ \alpha'=(1,0,0,\ldots).$$ \end{lem} {\bf Proof.} Direct. $\Box$ \subsubsection{} We denote by $\bT_{\CB}(X)$ the set of all collections $\{ x_{a,\alpha}\}_{(a,\alpha)\in \CB}$ satisfying the relations ~(\ref{rel-1}) -- ~(\ref{rel-3}). Thus we get a map \begin{equation} \label{map-pi2} \pi_2:\bT_{\CF}(X)\lra \bT_{\CB}(X) \end{equation} \subsubsection{} \begin{lem}{} The map ~(\ref{map-pi2}) is an isomorphism. \end{lem} {\bf Proof.} One can easily see that any $d$-free element may be obtained from an element from $\CB$ by applying operations $\zeta_i$. This proves injectivity of ~(\ref{map-pi2}). The proof of surjectivity is standard. $\Box$ Summing up, we get a sequence of natural maps \begin{equation} \label{map-pi} \Omega^n(X)\iso \bT^{lf}(X)\hra\bT(X)\iso\bT_{\CF}(X)\iso\bT_{\CB}(X) \end{equation} \subsection{} Let us call a collection $\{x_{a,\alpha}\}_{(a,\alpha)\in{\cal B}^{\cdot}}$ {\em locally finite} if $$ \forall p\in{\Bbb N}\ \exists m\in{\Bbb N}\ \forall(a,\alpha)\in {\cal B}^q_{m'} \text{ with }m'\geq m\ \forall f:[q]\ra[p]\in\Delta\text{ one has } f(x_{a,\alpha})=0. $$ Let us denote by $\bT^{lf}_{\CB}(X)\subset\bT_{\CB}(X)$ the subspace of all locally finite collections. \subsubsection{} \begin{lem}{} The map ~(\ref{map-pi}) induces an isomorphism $$ \pi:\Omega^n(X)\iso \bT^{lf}_{\CB}(X) $$ \end{lem} {\bf Proof.} Direct check. $\Box$ \subsection{} Set $\CB^p_m=\{ (a,\alpha)\in\CB^p|\sum a_i=m\}$; $\CB_m=\cup_p\ \CB_m^p$. By definition, $$ \CB_m=\CB_m^{m+n-1}\cup\CB_m^{m+n}, $$ the first (resp., second) subset consisting of all elements of the first (resp., second) kind. Let us introduce the following total order on $\CB_m$: $(a,\alpha)>(a',\alpha')$ iff $a>a'$ in the lexicographic order. Let us denote by $\bT_{\CB,m}(X)$ the set of all collections $\{ x_{a,\alpha}\}_{(a,\alpha)\in \CB_m}$ satisfying the relations ~(\ref{rel-1}) -- ~(\ref{rel-3}). Given $(b,\beta)\in\CB_m$, denote by $\bT_{\leq (b,\beta)}(X)$ the space of all collections $\{ x_{a,\alpha}\}_{(a,\alpha)\in \CB_m,\ (a,\alpha) \leq (b,\beta)}$. We have obvious maps $$ \bT_{\leq (b,\beta)}(X)\lra\bT_{\leq (a,\alpha)}(X) $$ for $(a,\alpha)\leq (b,\beta)$. \subsubsection{} \label{inverse} \begin{lem}{} We have $$ \bT_{\CB,m}(X)=\invlim\ \bT_{\leq (b,\beta)}(X) $$ the inverse limit over $\CB_m$. \end{lem} {\bf Proof.} Obvious. $\Box$ \subsection{} \label{step} \begin{lem}{} Let $(a,\alpha)\in{\cal B}_m$. Let $(b,\beta)\in{\cal B}^l_m$ be the first element such that $(b,\beta)>(a,\alpha)$. Then the map $$\bT_{\leq(b,\beta)}(X)\ra\bT_{\leq(a,\alpha)}(X)$$ is surjective and its kernel is isomorphic to a direct summand of $X^l$. \end{lem} {\bf Proof} In order to lift an element of $\bT_{a,\alpha}(X)$, we have to find an element $x_{b,\beta}$ having prescribed values for some of $\sigma^i(x_{b,\beta})$. 1. Existence of the lifting: One has the following conditions on $\sigma^i(x_{b,\beta})$: (0) $\sigma^0(x_{b,\beta})=0$ --- always. (a) if $\beta_i=\beta_{i+1}=1$. Then one has $\sigma^i(x_{b,\beta})=0$. (b) if $(\beta_i,\beta_{i+1})=(0,1)$ and $i>1$. Then the condition is $\sigma^i(x_{b,\beta})=\sigma^i(x_{b',\beta'})$ where the pair $(b',\beta')$ is obtained from $(b,\beta)$ by switching the places $(i,i+1)$. (c) if $(\beta_1,\beta_2)=(0,1)$. Then the condition is $\sigma^1(x_{b,\beta})=(\sigma^1)^2(x_{a,\alpha})$ where $(\alpha_1,\alpha_2,\alpha_3)=(1,0,0),\alpha_i=\beta_{i-1}\text{ for } i>3$ and $a_i$ are defined uniquely by $\alpha_i$. Let $I\subseteq[0,l-1]$ be the set of indices $i$ such that $\sigma^i(x_{b,\beta})$ is defined by the conditions (0)---(c) above. Let $y^i\in A^{l-1},\ i\in I,$ be the right hand sides of the conditions (0)---(c) so that they take form $$ \sigma^i(x_{b,\beta})=y^i,\ i\in I.$$ The first observation is that for any couple $i<j$ in $I$ one has \begin{equation} \sigma^{j-1}y^i=\sigma^iy^j. \label{compatibility} \end{equation} This can be checked easily by an explicit calculation. Now consider two different cases: (1st case) $I$ does not coincide with $[0,l-1]$. In this case \Lem{kan} below asserts the existence of a solution for the system of equations for $x_{b,\beta}$. (2nd case) $I=[0,l-1]$. This is possible only in two cases: --- $\beta_i=1$ for all $i$. Then $y^i=0$ for all $i$. --- $\beta_1=0;\beta_i=1$ for $i>1$. Then $y^i=0$ for $i\not=1$ and the condition~(\ref{compatibility}) gives that $y^1\in N^{l-1}X$. In both cases \Lem{more} assures the existence of a solution. 2. The kernel of the map. The kernel of the map in question has always form $$ N^p_I(X):=\{a\in X^p| \sigma^i(a)=0\text{ for all }i\in I\}$$ for some subset $I\subseteq[0,p]$. According to~\Lem{kan2} below, this is a direct summand of $X^p$. $\Box$ \subsection{} Let $\cal M$ be the set of all non-decreasing functions $f:{\Bbb N}\ra{\Bbb N}$ equipped with a partial order: $$ f\geq g\text{ iff }f(n)\geq g(n)\text{ for each }n. $$ \subsubsection{} \begin{defn}{growth} Given $f\in{\cal M}$, an element $x\in\bT(X)$ {\em has growth $\leq f$} if $\pi_1(x)=\{x_{b,\beta}\}_{(b,\beta)\in{\cal F}}$ satisfies the property $$ x_{b,\beta}=0 \text{ if } (b,\beta)\in{\cal F}^p\text{ and } \sum b_i>f(p). $$ Denote by $\bT^f(X)\subset \bT(X)$ the subspace of all elements of growth $\leq f$. \end{defn} We have $\bT^f(X)\subseteq \bT(X)$. Moreover, \subsubsection{} \label{directlim} \begin{lem}{} We have $$ \bT^{lf}(X)=\dirlim\ \bT^f(X) $$ the limit taken over $f\in\CM$. \end{lem} {\bf Proof.} Obvious. \subsection{} For $X\in\Delta\Mod(A)$ and $d\in{\Bbb N}\cup\{ -1,\infty\}$ define $X_{>d}\in\Delta\Mod(A)$ as follows. Set $X_{>-1}=X;\ X_{>\infty}=0$. For $d\in \Bbb N$ set \begin{equation} X_{>d}^i=\{x\in X^i|f(x)=0\ \forall f:[i]\ra[d]\in\Delta\} \end{equation} We have obviously $X_{>d}^i=0\text{ for }i\leq d$, and $X_{>d}^{d+1}=N^{d+1}X$ where $N^iX$ denotes the normalization (see ~\ref{normal} below). \subsubsection{} Given $f\in\CM$, define a function $f^{\circ}: \Bbb N\lra \Bbb N\cup\{ -1,\infty\}$ by a formula $$ f^{\circ}(m)=p\text{ iff } f(p)<m\leq f(p+1). $$ For $f\in\CM$ let us denote $$ \bT_{\CB}^f(X)=\pi_2\pi_1(\bT^f(X)) $$ \subsubsection{} \begin{lem}{} We have $$ \bT_{\CB}(X)=\prod_{m=0}^{\infty}\bT_{\CB,m}(X) $$ \end{lem} {\bf Proof.} Evident: the equations~(\ref{rel-1}--\ref{rel-3}) are homogeneous on $m=\sum a_i$. $\Box$ \subsubsection{} \begin{lem}{product} For any $f\in{\cal M}$ one has \begin{equation} \label{eq-prod} \bT^f_{\CB}(X)=\prod_{m=0}^{\infty}\bT_{\CB,m}(X_{>f^{\circ}(m)}) \end{equation} \end{lem} \begin{pf} An element $x\in\bT(X)$ has growth $\leq f$ iff $\pi_1(x)=\{x_{b,\beta}\}_{(b,\beta)\in{\cal F}}$ satisfies the property $$ x_{b,\beta}=0 \text{ if } (b,\beta)\in{\cal F}^p\text{ and } p\leq f^{\circ}(\sum b_i). $$ Then the formulas~(\ref{props}) give immediately the result. \end{pf} Recall that $X$ is called {\em finite dimensional} if there exists $d\in\Bbb N$ such that $N^i(X)=0$ for $i>d$ (we say that $\dim(X)\leq d$. If this is the case then $X_{>d}=0$, hence the product in ~(\ref{eq-prod}) is finite. Combining all the previous results together, we get \subsection{} \begin{thm}{} For every $n\in\Bbb N$ the $A$-module $\Omega^n(X)$ may be obtained from $X$ applying the following operations (naturally in $X$): (1) taking modules $\bT_{\CB,m}(X)$ which have a natural finite filtration with graded factors isomorphic to direct summands of modules $X^l,\ l\in\Bbb N$; (2) taking direct products over $\Bbb N$; if $\dim(X)<\infty$ then products are finite; (3) passing to a filtered direct limit. \end{thm} {\bf Proof.} Follows immediately from ~\ref{directlim}, ~\ref{product}, {}~\ref{step} and ~\ref{inverse}. $\Box$ We have the following easy corollaries which are the main properties of the functor $\Omega$. \subsection{} \label{exact-6} \begin{cor}{} The functor $X\mapsto\Omega(X)$ is exact. $\Box$ \end{cor} \subsection{} \label{base-6} \begin{cor}{} Suppose $X\in\Delta\Mod(A)$ is finite dimensional. Let $A'$ be a (commutative) $A$-algebra. Then the natural base change map $$ \Omega(X)\otimes_AA'\lra\Omega(X\otimes_AA') $$ is an isomorphism. $\Box$ \end{cor} \subsection{} \label{flat-6} \begin{cor}{} Let $X\in\Delta\Mod(A)$ and suppose that either $A$ is noetherian or $X$ is finite dimensional. If all $X^i$ are flat $A$-modules so are all $\Omega^n(X)$. $\Box$ \end{cor} \subsection{} \label{qc} \begin{cor}{} Let $S$ be topological space endowed with a sheaf $\CO_S$ of commutative $\Bbb Q$-algebras. Let $X\in\Delta\Mod({\CO}_S)$ be finite. If all $X^i$ are quasi-coherent ${\CO}_S$-modules so are all $\Omega^n(X)$. $\Box$ \end{cor} In the remaining part of this Section we will prove some facts about cosimplicial modules needed above. Most of them are more or less standard. In fact, mostly we will discuss an explicit form of Dold-Puppe correspondence. \subsection{} \label{normal} Let $X$ be a cosimplicial $A$-module. For any $i\geq 0$ define $$ N^i(X)=\{x\in X^i|\sigma^j(x)=0\text{ for all }j\}.$$ Define a subcategory $\Lambda$ in $\Delta$ as follows. $\Lambda$ has the same objects as $\Delta$; The set of morphisms of $\Lambda$ is generated by the faces $\delta^i:[n]\ra[n+1]$ with $i=0,\ldots, n$ (that is: (a) only faces appear; (b) the last face $\delta^{n+1}:[n]\ra[n+1]$ disappear). Define {\em a shift functor} $ s:\Delta\ra\Delta$ by the formulas $$s[n]=[n+1],\ s(\delta^i)=\delta^{i+1},\ s(\sigma^i)=\sigma^{i+1}.$$ \subsubsection{} \begin{prop}{decomposition} For any cosimplicial $A$-module $X$ one has $$ X^n=\bigoplus_{m\geq 0}\bigoplus_{f:[m]\ra[n]\in\Lambda}f(N^m(X)). $$ \end{prop} \begin{pf} We will prove the claim by induction. For $n=0$ the claim is trivial. Suppose it is true for degrees $< n$ and for all cosimplicial modules $X$. Apply this to the shift $Y=Xs$ of $X$. We have by the inductive hypothesis $$ X^n=Y^{n-1}=\bigoplus_{m\geq 0} \bigoplus_{f:[m]\ra[n-1]\in{\Lambda}}f(M^m)$$ where $M^i=N^i(Y)$ is the normalization of $Y=Xs$. An element $x\in X^{m+1}$ belongs to $M^m$ iff $\sigma^i(x)=0$ for $i>0$. Write $x=y+\delta^0\sigma^0(x)$ where $y=x-\delta^0\sigma^0(x)$. One checks that $y\in N^{m+1}(X)$ and (of course) $\sigma^0(x)\in X^m$. Thus by induction (note that $m\leq n-1$) $$\sigma^0(x)=\sum_k\sum_{g:[k]\ra[m]\in\Lambda}g(z_g)$$ with $z_g\in N^k(X)$. If $x'\in M^m$ is the element corresponding to $x\in X^{m+1}$ then $f(x')=(sf)(x)$ and therefore $$ f(x')=(sf)(x)=(sf)(y)+(sf)\delta^0\sum g(z_g)=(sf)(y)+\sum\delta^0 fg(z_g)$$ which proves that $$ X^n=\sum_m\sum_{f:[m]\ra[n]\in\Lambda}f(N^m(X)).$$ Let us finally prove the uniqueness of the presentation of an element $x\in X^n$ into sum \begin{equation} x=\sum_m\sum_{f:[m]\ra[n]\in\Lambda}f(x_f) \label{presentation} \end{equation} with $x_f\in N^m(X)$. For any $f:[m]\ra[n]\in\Lambda$ define a left-inverse $f^l:[n]\ra[m]$ as follows: if $f=\delta^{i_1}\cdots\delta^{i_{n-m}}$ with $i_1>\ldots>i_{n-m}$ then $$ f^l=\sigma^{i_{n-m}}\cdots\sigma^{i_1}.$$ We will check the uniqueness of the elements $x_f$ in the presentation~(\ref{presentation}) by induction on $m$. Suppose that $x_f$ are defined uniquely for all $f:[m']\ra[n]$ with $m'<m$ (this is true, say, for $m=0$). Define an order on the set $\Hom_{\Lambda}([m],[n])$ saying that $\delta^{i_1}\cdots\delta^{i_{n-m}}\geq\delta^{j_1}\cdots\delta^{j_{n-m}}$ iff $(i_1,\ldots,i_{n-m})\geq(j_1,\ldots,j_{n-m})$ in the lexicographic order. One immediately checks the following \begin{lem}{} If $f>g$ then $f^lg\not=\id_{[m]}$. \end{lem} By the inductive hypothesis we can suppose that $x_f=0$ for $f:[m']\ra[n]$ with $m'<m$. Then \begin{equation} f^l(g(x_g))=\begin{cases} x_g&\text{ if }f=g\\ 0& \text{ if } f>g \end{cases} \end{equation} Thus, the transition matrix expressing the values of $f^l(x)$ for different $f$ through $x_g$ is upper-triangular and hence invertible. The proposition is proven. \end{pf} It is very convenient to rewrite the statement of~\Prop{decomposition} as follows. Let ${\Bbb Q}\Lambda_{mn}$ be the rational vector space spanned by the set $\Hom_{\Lambda}([m],[n])$. Then one has \begin{cor}{} One has a canonical on $X\in\Delta\Mod(A)$ isomorphism $$ X^n=\bigoplus_{m\geq 0}{\Bbb Q}\Lambda_{mn}\otimes_{\Bbb Q}N^m(X).$$ \end{cor} \subsection{} \begin{lem}{kan} Let $I\subset[0,n]$ be a proper subset and let $y_i\in X^{n-1},\ i\in I$ be given. Then the system of equations $$ \sigma^i(x)=y^i,\ i\in I,$$ has a solution if (and only if) $y^i$ satisfy the compatibilities $$ \sigma^{j-1}y^i=\sigma^iy^j\text{ for }i<j\in I.$$ \end{lem} \begin{pf} Consider the functor $$ i:\Delta^0\ra\Delta $$ defined by the formulas $$ i([n])=[n+1],\ i(\delta^i)=\sigma^i,\ i(\sigma^i)=\delta^{i+1}$$ --- see Gabriel-Zisman's functor II, ~\cite{gz}, 3.1.1). If $X$ is a cosimplicial module then $Xi$ is a simplicial module. The property of $X$ we have to prove just means that $Xi$ is a Kan simplicial set. This is well-known to be always true (see, e.g., ~\cite{q2}, Prop. II.3.1). \end{pf} \subsection{} \begin{lem}{more}For any system $y^i\in N^{n-1}X,\ i\in[0,n-1]$ there exists an element $x\in X^n$ such that $\sigma^ix=y^i$ for all $i\in[0,n-1]$. \end{lem} \begin{pf} We will look for $x=\sum\delta^i(z^i)$ with $z^i\in N^{n-1}X$. Then the conditions on $x$ are expressed by the equations $y^i=\sigma^i(x)=z^i+z^{i+1}$ which are clearly solvable. \end{pf} \subsection{} {\em Notation.} For $I\subseteq[0,n]$ denote $$ N^i_I(X)=\{x\in X^i|\sigma^j(x)=0\text{ for all }j\in I\}.$$ \subsubsection{} \begin{lem}{kan2} Let $I\subseteq[0,n]$. There exist a collection of vector subspaces ${\Bbb Q}\Lambda^I_{m,n}\subseteq{\Bbb Q}\Lambda_{m,n}$ such that $$ N^n_I(X)=\bigoplus_m{\Bbb Q}\Lambda^I_{m,n}\otimes_{\Bbb Q}N^m(X) $$ \end{lem} Here is a more general statement. \subsubsection{} \begin{lem}{kan3} Fix $d\leq n\in{\Bbb N}$ and a subset $\Phi\subseteq\Hom_{\Delta}([n],[d])$. Define $$X^n_{\Phi}=\{x\in X^n|f(a)=0\text{ for all }f\in\Phi\}.$$ There exists a collection of vector subspaces ${\Bbb Q}\Lambda^{\Phi}_{m,n}\subseteq{\Bbb Q}\Lambda_{m,n}$ such that $$ X^n_{\Phi}=\bigoplus_m{\Bbb Q}\Lambda^{\Phi}_{m,n}\otimes_{\Bbb Q}N^m(X).$$ \end{lem} \begin{pf} of~\ref{kan3}. Let $t:[n]\ra[k]\in\Delta$. The condition $tx=0$ for $x=\sum_ff(x_f)$ takes form $$0=tx=\sum_ftf(x_f)=\sum_gg(\sum_{f: tf=g}x_f)$$ which is equivalent to the system of equations $$ \sum_{f: tf=g}x_f=0$$ numbered by $g\in\Mor\Delta$. The lemma immediately follows from this observation. \end{pf} \section{Higher Kodaira-Spencer morphisms} \label{kodaira} \subsection{} \label{laawm} (Cf. \cite{bs}, 1.2.) Let $S$ be a differentially smooth formal scheme (for example, a smooth scheme). Let $\pi: X\lra S$ be a smooth separated map of formal schemes, ~\ref{smooth}, for example a smooth morphism of usual schemes. We have the exact sequence ~(\ref{seq}): $$ 0\lra \CT_{X/S}\lra \CT_X\overset{\epsilon_{\CT}}{\lra} \pi^*\CT_S\lra 0 $$ The first embedding makes $\CT_{X/S}$ a Lie algebroid over $X$. Note that the sheaf $\pi^*\CT_S$ is not a sheaf of Lie algebras. Let $\pi^{-1}$ denotes the functor of set-theoretical inverse image, so that $\pi^*=\CO_X\otimes_{\pi^{-1}\CO_S}\pi^{-1}$. The subsheaf $\pi^{-1}\CT_S\subset \pi^*\CT_S$ is a $\pi^{-1}\CO_S$-Lie algebra. Set $\CT_{\pi}:=\epsilon_{\CT}^{-1}(\pi^{-1}\CT_S)$ - this is a sheaf of $k$-Lie algebras and $\pi^{-1}\CO_S$-modules (consisting of vector fields with the constant projection to $S$ along fibers of $\pi$). We have an exact sequence of $k$-Lie algebras and $\pi^{-1}\CO_S$-modules $$ 0\lra\CT_{X/S}\lra \CT_{\pi}\lra \pi^{-1}\CT_S\lra 0 $$ Let $\epsilon: \CA\lra \CT_X$ be a transitive Lie algebroid over $X$. We set $\CA_{/S}:=\epsilon^{-1}(\CT_{X/S})\subset\CA_{\pi}:= \epsilon^{-1}(\CT_{\pi})\lra \CA$. These are subsheaves of Lie algebras. $\CA_{/S}$ is a subsheaf of $\CO_X$-modules, and a Lie algebroid over $X$ included into an exact sequence $$ 0\lra \CA_{(0)}\lra \CA_{/S}\lra \CT_{X/S}\lra 0 $$ $\CA_{\pi}$ is a subsheaf of $\pi^{-1}\CO_S$-modules. We have an exact sequence of $k$-Lie algebras and $\pi^{-1}\CO_S$-modules \begin{equation} \label{a} 0\lra\CA_{/S}\lra\CA_{\pi}\lra \pi^{-1}\CT_S\lra 0 \end{equation} \subsection{} \label{dglie-fund} Pick a finite affine open covering $\CU$ of $X$. We will suppose that $\CA$ is a locally free $\CO_X$-module of finite type, hence, so is $\CA_{/S}$. Let us apply the functor $\pi_*\CHC(\CU,\cdot)$ to the exact sequence {}~(\ref{a}). We have $R^1\pi_*\CHC^i(\CU,\CA_{/S})=0$; so we get an exact sequence of cosimplicial $\CO_S$-modules $$ 0\lra \pi_*\CHC(\CU,\CA_{/S})\lra \pi_*\CHC(\CU,\CA_{\pi}) \lra \pi_*\CHC(\CU,\pi^{-1}\CT_S)\lra 0 $$ Next, applying the Thom-Sullivan functor $\Omega$ we get an exact sequence of complexes \begin{equation} \label{ex-derham} 0\lra \pi^{Lie}_{*,\CU}(\CA_{/S})\lra \Omega(\pi_*\CHC(\CU,\CA_{\pi})) \lra \Omega(\pi_*\CHC(\CU,\pi^{-1}\CT_S))\lra 0 \end{equation} Note that $\Omega(\pi_*\CHC(\CU,\CA))$ is naturally a dg $k$-Lie algebra. We have a canonical adjunction map $$ \CT_S\lra \Omega(\pi_*\CHC(\CU,\pi^{-1}\CT_S)) $$ so, taking the pullback of ~(\ref{ex-derham}) we get an exact sequence \begin{equation} \label{fund} 0\lra \pi^{Lie}_{*,\CU}(\CA_{/S})\lra \CA^{\pi}_{\CU}\lra \CT_S\lra 0 \end{equation} By definition, $$ \CA^{\pi}_{\CU}=\Omega(\pi_*\CHC(\CU,\CA_{\pi})) \times_{\Omega(\pi_*\CHC(\CU,\pi^{-1}\CT_S))}\CT_S, $$ and this sheaf inherits the structure of a dg $k$-Lie algebra and $\CO_S$-module from $\Omega(\pi_*\CHC(\CU,\CA_{\pi}))$ and $\CT_S$. One sees that this way we get a structure of a transitive dg Lie algebroid on $\CA^{\pi}_{\CU}$. \subsection{} \label{higher} Now we can apply to ~(\ref{fund}) the construction {}~\ref{abstr-ksmaps}. We get the maps: \begin{equation} \label{ks-class} \kappa^1:\CT_S\lra R^1\pi_*(\CA_{/S}) \end{equation} --- the "classical" KS map; \begin{equation} \label{ks} \kappa:\Diff_S\lra \CH_0^{Lie}(\pi_*^{Lie}(\CA_{/S})) \end{equation} and \begin{equation} \label{ks-leq-n} \kappa^{\leq n}:\Diff^{\leq n}_S\lra \CF_n\CH^{Lie}_0(\pi_*^{Lie}(\CA_{/S})) \end{equation} satisfying the compatibilities ~\ref{main-thm}. These maps are called {\bf higher Kodaira-Spencer morphisms.} \subsection{Split case} \label{dglie-fund-const} Suppose that $X=Y\times S$, and $\pi$ is a projection to the second factor. In this case we have canonical embeddings $$ \pi^{-1}\CT_S\hra\pi^*\CT_S\hra \CT_X $$ By taking the pull-back of the exact sequence $$ 0\lra\CA_{(0)}\lra \CA\lra\CT_X\lra 0, $$ we get an exact sequence \begin{equation} \label{a-const} 0\lra \CA_{(0)}\lra \bar{\CA}\lra\pi^{-1}\CT_S\lra 0 \end{equation} Now we can repeat the construction of ~\ref{dglie-fund}, replacing the sequence ~(\ref{a}) by ~(\ref{a-const}). This will provide a dg Lie algebroid \begin{equation} \label{fund-const} 0\lra\CA_{(0)}\lra\bar{\CA}^{\pi}_{\CU}\lra\CT_S\lra 0 \end{equation} Again we can apply the construction ~\ref{abstr-ksmaps}, and get the KS maps analogous to ~(\ref{ks-class}) -~(\ref{ks-leq-n}): \begin{equation} \label{ks-class0} \kappa^1_{(0)}:\CT_S\lra R^1\pi_*(\CA_{(0)}) \end{equation} --- the "classical" KS map; \begin{equation} \label{ks0} \kappa_{(0)}:\Diff_S\lra \CH_0^{Lie}(\pi_*^{Lie}(\CA_{(0)})) \end{equation} and \begin{equation} \label{ks-leq-n0} \kappa^{\leq n}_{(0)}:\Diff^{\leq n}_S\lra \CF_n\CH^{Lie}_0(\pi_*^{Lie} (\CA_{(0)})) \end{equation} satisfying the same compatibilities. \subsection{} \label{isom} \begin{thm}{} Suppose that --- either we are in the situation ~\ref{higher}, $\pi_*\CA_{/S}=0$, and $\kappa^1$ is an isomorphism, --- or we are in the situation ~\ref{dglie-fund-const}, $\pi_*\CA_{(0)}=0$ and $\kappa^1_{(0)}$ is an isomorphism. Then all $\kappa^{\leq n}$ (resp., $\kappa^{\leq n}_{(0)}$) are isomorphisms. \end{thm} {\bf Proof.} Suppose for definiteness that we are in the first situation. The claim is proved by induction on $n$, using commutative diagrams \begin{center} \begin{picture}(20,4) \put(0,0){\makebox(1,1){$0$}} \put(2.5,0){\makebox(4,1){$\CF_{n-1}\CH^{Lie}_0(\pi^{Lie}_*(\CA_{/S}))$}} \put(8,0){\makebox(4,1){$\CF_{n}\CH^{Lie}_0(\pi^{Lie}_*(\CA_{/S}))$}} \put(13.5,0){\makebox(4,1){$S^n(R^1\pi_*(\CA_{/S}))$}} \put(19,0){\makebox(1,1){$0$}} \put(0,3){\makebox(1,1){$0$}} \put(2.5,3){\makebox(4,1){$\Diff^{\leq n-1}_S$}} \put(8,3){\makebox(4,1){$\Diff_S^{\leq n}$}} \put(13.5,3){\makebox(4,1){$S^n(\CT_S)$}} \put(19,3){\makebox(1,1){$0$}} \put(4.5,3){\vector(0,-1){2}} \put(10,3){\vector(0,-1){2}} \put(15.5,3){\vector(0,-1){2}} \put(1,0.5){\vector(1,0){1}} \put(7,0.5){\vector(1,0){1}} \put(12.5,0.5){\vector(1,0){1}} \put(17.8,0.5){\vector(1,0){1}} \put(1,3.5){\vector(1,0){2}} \put(5.5,3.5){\vector(1,0){3}} \put(11,3.5){\vector(1,0){3}} \put(17,3.5){\vector(1,0){2}} \put(2.5,1){\makebox(2,2){$\kappa^{\leq n-1}$}} \put(8,1){\makebox(2,2){$\kappa^{\leq n}$}} \put(16,1){\makebox(2,2){$(-1)^nS^n(\kappa^1)$}} \end{picture} \end{center} Note that our assumptions imply that $$H^{-1}(F_nC(\pi^{Lie}_*(\CA_{/S}))/F_{n-1}C(\pi^{Lie}_*(\CA_{/S})))=0$$ and $$H^{0}(F_nC(\pi^{Lie}_*(\CA_{/S}))/F_{n-1}C(\pi^{Lie}_*(\CA_{/S})))= S^n(R^1\pi_*(\CA_{/S})).$$ $\Box$ \subsection{Deformations of schemes.} \label{def-schemes} Set $\CA=\CT_X$. Applying the previous construction, we get the maps \begin{equation} \label{ks-class-1} \kappa^1: \CT_S\lra R^1\pi_*(\CT_{X/S}) \end{equation} --- the classical Kodaira-Spencer map; \begin{equation} \label{ks-1} \kappa: \Diff_S\lra \CH_0^{Lie}(\pi_*^{Lie}(\CT_{X/S})) \end{equation} and \begin{equation} \label{ks-leq-n-1} \kappa^{\leq n}: \Diff_S^{\leq n}\lra \CF_n\CH_0^{Lie}(\pi_*^{Lie}(\CT_{X/S})) \end{equation} satisfying the compatibilities ~\ref{main-thm}. \subsection{Deformations of $G$-torsors} \label{def-tors} (a) Let $G$ be an algebraic group over $k$, $\fg=\Lie(G)$, and $p:P\lra X$ a $G$-torsor over $X$. We define sheaves of Lie algebras $\CA_P$ and $\fg_P$ as in ~\ref{pose}. $\CA_P$ is naturally a transitive Lie algebroid over $X$, with $\CA_{P(0)}=\fg_P$. Applying the previous construction, we get the maps analogous to {}~(\ref{ks-class-1}) - ~(\ref{ks-leq-n-1}), with $\CT_{X/S}$ replaced by $\CA_{P/S}$, subject to the same compatibilities. (b) Suppose in addition that $X=Y\times S$ as in ~\ref{dglie-fund-const}. Then we can apply the construction of {\em loc. cit.} to $\CA=\CA_P$, and get higher KS maps taking value in $\CF_n\CH_0^{Lie}(\fg_P)$. \section{Universal deformations} \label{univers} \subsection{} Let us fix a smooth scheme $X$, an algebraic group $G$ and a $G$-torsor $P$ over $X$. Consider the following deformation problems. To each problem we assign a sheaf of $k$-Lie algebras $\CA_i,\ i=1,2,3$ over $X$. {\bf Problem 1.} Flat deformations of $X$; $\CA_1=\CT_X$. {\bf Problem 2.} Flat deformations of the pair $(X,P)$; $\CA_2=\CA_P$, cf. \ref{def-tors}. {\bf Problem 3.} Deformations of $P$ ($X$ being fixed); $\CA_3=\fg_P$. Accordig to Grothendieck, to each problem corresponds a (2-)functor of infinitesimal deformations $$ F_i:\Artin_k\lra\Groupoids $$ from the category of local artinian $k$-algebras with the residue field $k$ to the (2-)category of groupoids. In each case, $\CA_i$ is "a sheaf of infinitesimal automorphisms" corresponding to $F_i$ (in the sense of ~\cite{sga1}, Exp.III, 5, especially Cor. 5.2 for Problem 1; for the other problems the meaning is analogous). In particular, we have the Kodaira-Spencer-Grothendieck isomorphisms $$ \ft_{\bF_i}\cong H^1(X,\CA_i). $$ where $$ \bF_i:\Artin_k\lra \Ens $$ is the composition of $F_i$ and the connected components functor $\pi_0:\Groupoids\lra \Ens$. Here for a functor $$ F:\Artin_k\lra\Ens $$ $\ft_F$ denotes the tangent space to $F$: $$ \ft_F=F(\Spec(k[\epsilon]/(\epsilon^2)), $$ cf. ~\cite{gr}, Exp. 195. \subsection{} One can verify that in each case the conditions of Schlessinger's Theorem 2.11, ~\cite{sch}, are fullfilled, and there exists a versal formal deformation $\fS_i$. \subsubsection{} \begin{lem}{} Suppose that $H^0(X,\CA_i)=0$. Then $\fS_i$ is a universal deformation, i.e. $\bF_i$ is prorepresentable. \end{lem} \begin{pf} This fact is presumably classic. We give a sketch of a proof for completeness. We have $\fS_i=\Spf(R)$, and we have a canonical morphism $x: h_R\lra F$ (we use notations of ~\cite{sch}). Consider the functor $$ G:\Artin_R\lra\Ens $$ from the category of local artinian $R$-algebras with the residue field $k$, defined as $G(\alpha:R\lra A))=\Aut_{F_i(A)}(\alpha_*(x))$. {\bf Claim 1.} $G$ is prorepresentable. Indeed, one can check that the hypotheses of {\em loc.cit.}, 2.11 hold true for $G$. Let $\Spf(T)$ prorepresents $G$, where $T$ is a complete local $R$-algebra. {\bf Claim 2.} The structure morphism $\phi: R\lra T$ is injective. In fact, this morphism has a section since groups of automorphisms have identity. Now, we have $$ \bm_T/\bm^2_T+\bm_R=H^0(X,\CA_i)=0 $$ hence $R=T$, whence $\bF_i$ is prorepresentable by {\em loc.cit.} 3.12. \end{pf} \subsection{} \label{complet} \begin{thm}{} Suppose that $H^0(X,\CA_i)=0$; let $\fS=\Spf(R)$ be the base of the universal formal deformation for Problem $i$. Suppose that $\fS$ is formally smooth. Then we have a canonical isomorphism $$ R^*=H^{Lie}_0(R\Gamma^{Lie}(X,\CA_i)) $$ where $R^*$ denotes the space of continuous $k$-linear maps $R\lra k$ ($k$ considered in the discrete topology). \end{thm} \begin{pf} Let us treat Problem 2 for definiteness (for other problems the proof is the same). Since $\fS$ is formally smooth, $R$ is isomorphic to a power series algebra $k[[T_1,\ldots,T_n]]$, hence $\fS$ is differentially smooth. Let $\pi: \fX\lra\fS$ be the universal deformation and $\fP$ the universal $G$-torseur over $\fX$. Applying ~\ref{def-tors}, we get the map $$ \kappa:\Diff_{\fS}\lra \CH^{Lie}_0(\pi_*^{Lie}(\CA_{\fP/\fS})) $$ Since $\fS$ is universal, the usual KS map $\kappa^1:\CT_{\fS}\lra R^1\pi_*(\CA_{\fX/\fS})$ is isomorpism. Note that $\CA_{\fP/\fS}|_X\cong \CA_i$. Since $H^0(X,\CA_i)$, we have $\pi_*(\CA_{\fX/\fS})=0$; hence by {}~\ref{isom} $\kappa$ is isomorphism. Now let us take the geometric fiber of $\kappa$ at the closed point $s:\Spec(k) \hra \fS$. We have $\Diff(\fS)_{k(s)}\cong R^*$ by ~\ref{point} and $$ \CH^{Lie}_0(\pi_*^{Lie}(\CA_{\fP/\fS}))_{k(s)}\cong H^{Lie}_0(R\Gamma^{Lie}(X,\CA_i)) $$ by ~\ref{base}. The theorem follows. \end{pf}

9405

alg-geom/9405005

en

https://arxiv.org/abs/alg-geom/9405005

alg-geom/9405005

Dr. Yakov Karpishpan

Yakov Karpishpan

Higher-order differentials of the period map and higher Kodaira-Spencer classes

18 pages, LaTeX

In \cite{K} we introduced two variants of higher-order differentials of the period map and showed how to compute them for a variation of Hodge structure that comes from a deformation of a compact K\"ahler manifold. More recently there appeared several works (\cite{BG}, \cite{EV}, \cite{R}) defining higher tangent spaces to the moduli and the corresponding higher Kodaira-Spencer classes of a deformation. The $n^{th}$ such class $\kappa_n$ captures all essential information about the deformation up to $n^{th}$ order. A well-known result of Griffiths states that the (first) differential of the period map depends only on the (first) Kodaira-Spencer class of the deformation. In this paper we show that the second differential of the Archimedean period map associated to a deformation is determined by $\kappa_2$ taken modulo the image of $\kappa_1$, whereas the second differential of the usual period map, as well as the second fundamental form of the VHS, depend only on $\kappa_1$ (Theorems 2, 5, and 6 in Section~3). Presumably, similar statements are valid in higher-order cases (see Section~4).

alg-geom

\section{Constructing linear maps out of connections} We start by reviewing the definitions of higher-order differentials of the period map from \cite{K}, using a slightly different approach. Let $S$ be a polydisc in ${\bf C}^s$ centered at 0. Consider a free ${\cal O}_S$-module $\cal V$ with a decreasing filtration by ${\cal O}_S$-submodules $\ldots\subseteq{\cal F}^p\subseteq{\cal F}^{p-1}\subseteq\ldots$ and an integrable connection $$\nabla:{\cal V}\rightarrow{\cal V}\otimes\Omega_S^1$$ satisfying the Griffiths transversality condition $\nabla({\cal F}^p)\subseteq{\cal F}^{p-1}\otimes\Omega_S^1$. \begin{lemma} (a) $\nabla$ induces an \ ${\cal O}_S$-linear map \begin{eqnarray*} \Theta_S & \longrightarrow & {\cal H}om_{{\cal O}_S} ({\cal F}^p,{\cal F}^{p-1}/{\cal F}^p)\\ \xi & \longmapsto & \nabla_{\xi} \bmod {\cal F}^p \end{eqnarray*} (b) Analogously, we also have an \ ${\cal O}_S$-linear symmetric map \begin{eqnarray*} \Theta_S^{\otimes 2} & \longrightarrow & {\cal H}om_{{\cal O}_S} ({\cal F}^p,{\cal F}^{p-2}/{\cal F}^p+span\,\{\nabla_{\eta} ({\cal F}^p)\,|\ all\ \eta\in\Gamma(S,\Theta_S)\}) \\ \zeta\otimes\xi & \mapsto & \nabla_{\zeta}\nabla_{\xi} \bmod {\cal F}^p + span\,\{\nabla_{\eta}({\cal F}^p)\}\ . \end{eqnarray*} \end{lemma} \ \\ \noindent {\bf Proof.\ \ } Both (a) and (b) are proved by straightforward computations; the fact that the map in (b) is symmetric follows from the integrability of $\nabla$: $$\nabla_{\zeta}\nabla_{\xi}- \nabla_{\xi}\nabla_{\zeta}=\nabla_{[\zeta,\xi]}\ ,$$ and so $$\nabla_{\zeta}\nabla_{\xi}\equiv\nabla_{\xi}\nabla_{\zeta} \bmod span\,\{\nabla_{\eta}({\cal F}^p)\,|\ \eta\in\Theta_S\}\ .$$ \ $\Box$\\ \ \par We will apply the Lemma to two connections arising from a deformation of a compact K\"{a}hler manifold $X$, \begin{eqnarray} \nonumber {\cal X} & \supset & X\\ \pi\ \downarrow & & \downarrow\\ \nonumber S & \ni & 0 \end{eqnarray} \noindent 1. The usual Gauss-Manin connection $\nabla$ on ${\cal H}=R^m\pi_{\ast}{\bf C}_{\cal X}$\ . In this case we denote the map in (a) $$d\Phi: \Theta_S\rightarrow {\cal H}om_{{\cal O}_S} ({\cal F}^p,{\cal F}^{p-1}/{\cal F}^p)\ . $$ This is {\em the (first) differential of the (usual) period map}. The same notation and terminology will be applied to the induced map $$\Theta_S\rightarrow\bigoplus_p{\cal H}om_{{\cal O}_S} ({\cal F}^p/{\cal F}^{p+1},{\cal F}^{p-1}/{\cal F}^p)\ .$$ The map given by part (b) of the Lemma is called {\em the second differential of the (usual) period map} and denoted $$d^2\Phi: \Theta_S^2\longrightarrow{\cal H}om_{{\cal O}_S} ({\cal F}^p,{\cal F}^{p-2}/{\cal F}^p+span\,\{\nabla_{\eta} ({\cal F}^p)\,|\ \eta\in\Theta_S\})\ .$$ \noindent 2. The Archimedean Gauss-Manin connection $\nabla=\nabla_{ar}$ on $${\cal H}\otimes B_{ar}= R^m\pi_{\ast}{\bf C}_{\cal X}[T,T^{-1}]$$ (see Appendix). The corresponding map from (a) is called {\em the (first) differential of the {\em Archimedean} period map}, denoted $$d\Psi:\Theta_S\longrightarrow {\cal H}om_{{\cal O}_S}({\cal H}_{ar}, {\cal F}_{ar}^{-1}/{\cal H}_{ar})\ .$$ Again, we will abuse notation and write $d\Psi$ to denote the induced map \begin{equation} \Theta_S\rightarrow {\cal H}om_{{\cal O}_S} (Gr_{{\cal F}_{ar}}^0, Gr_{{\cal F}_{ar}}^{-1})\ . \end{equation} Finally, the map in part (b) of the Lemma, for $\nabla=\nabla_{ar}$, is {\em the second differential of the Archimedean period map} and will be denoted $$d^2\Psi:\Theta_S^2\longrightarrow {\cal H}om_{{\cal O}_S} ({\cal H}_{ar},{\cal F}_{ar}^{-1}/{\cal H}_{ar}+ span\,\{\nabla_{\eta}({\cal H}_{ar})\,|\ \eta\in\Theta_S\})\ .$$ We have an identification of ${\cal O}_S$-modules $$Gr_{{\cal F}_{ar}}^0({\cal H}\otimes B_{ar})\cong {\bf R}^m\pi_{\ast}(Gr_F^0(\Omega^{\textstyle\cdot}_{{\cal X}/S}\otimes B_{ar}))\cong{\bf R}^m\pi_{\ast}\Omega^{\textstyle\cdot}_{{\cal X}/S}= {\cal H}\ ,$$ obtained from the obvious isomorphism of sheaf complexes $$\begin{array}{ccccc} \longrightarrow & \Omega^p_{{\cal X}/S}.T^p & \stackrel{\bf d}{\longrightarrow} & \Omega^{p+1}_{{\cal X}/S}.T^{p+1} & \longrightarrow\\ & \simeq\downarrow & & \downarrow\simeq & \\ \longrightarrow & \Omega^p_{{\cal X}/S} & \stackrel{d}{\longrightarrow} & \Omega^{p+1}_{{\cal X}/S} & \longrightarrow \end{array}$$ \noindent (``dropping the T's"). Similarly, $Gr_{{\cal F}_{ar}}^{-1}\cong {\cal H}$. We use these identifications to obtain a version of (2) ``without T's," $$\overline{d\Psi}:\Theta_S\longrightarrow{\cal E}nd_ {{\cal O}_S}({\cal H})\ .$$ Analogously, we also define $$\overline{d^2\Psi}:\Theta_S^{\otimes 2}\longrightarrow{\cal E}nd_ {{\cal O}_S}({\cal H})\ ,$$ with $\overline{d^2\Psi}(\zeta,\xi)$ being the composition $${\cal H}\cong Gr_{{\cal F}_{ar}}^0 \stackrel{d^2\Psi(\zeta,\xi)}{\longrightarrow}{\cal F}_{ar}^{-2}/{\cal H}_{ar}+span\,\{\nabla_{\eta}({\cal F}_{ar}^{-1})\,|\ \eta\in\Theta_S\}\,\longrightarrow\hspace{-12pt Gr_{{\cal F}_{ar}}^{-2}\cong{\cal H}\ .$$ \ \\ \noindent {\bf Remark.\ \ } Let ${\bf t}=(t_1,\ldots,t_s)$ be a coordinate system on $S$ centered at 0. Then, in the notation of \cite{K}, $d\Psi(\partial/\partial t_i)|_{t=0}$ is $\widetilde{L}^{(i)}$, $\overline{d\Psi}(\partial/\partial t_i)|_{t=0}$ is $\overline{L}^{(i)}$, $d^2\Psi(\partial/\partial t_i\otimes \partial/\partial t_j)|_{t=0}$ is $\widetilde{L}^{(ij)}$, and $\overline{d^2\Psi}(\partial/\partial t_i\otimes \partial/\partial t_j)|_{t=0}$ is $\overline{L}^{(ij)}$. \ ${\cal O}_S$-linearity of $d\Psi$, $d^2\Psi$, etc. is essential for the ability to restrict to 0 in $S$. To formulate the next Lemma, we bring out the natural $C_S^{\infty}$-linear identification $$h:{\cal H}=\bigoplus_{p+q=m}{\cal H}^{p,q} \stackrel{\simeq}{\longrightarrow} \bigoplus_p Gr_{\cal F}^p{\cal H}=: Gr_{\cal F}^{\textstyle\cdot}{\cal H}\ .$$ \begin{lemma} (a) For any \ $\xi\in\Theta_S$ and all \ $p$ we have $$\overline{d\Psi}(\xi)({\cal F}^p{\cal H})\subset {\cal F}^{p-1}{\cal H}\ ,$$ the induced endomorphism of degree \ $-1$ of \ $Gr_{\cal F}^{\textstyle\cdot}{\cal H}$ coincides with \ $d\Phi(\xi)$ and, in fact, $$\overline{d\Psi}(\xi)=h^{-1}\circ d\Phi(\xi)\circ h\ .$$ (b) For any \ $\zeta,\xi\in\Theta_S$ and all \ $p$ we have $$\overline{d^2\Psi}(\zeta,\xi)({\cal F}^p{\cal H})\subset {\cal F}^{p-2}{\cal H}\ ,$$ and $$d^2\Phi(\zeta,\xi):{\cal F}^p\longrightarrow{\cal F}^{p-2}/ {\cal F}^p+span\,\{\nabla_{\eta}({\cal F}^p)\,|\ \eta\in\Theta_S\}$$ factors through \ $\overline{d^2\Psi}(\zeta,\xi): {\cal F}^p\longrightarrow{\cal F}^{p-2}$. \end{lemma} \ \\ \noindent {\bf Proof.\ \ } (a) For any $\xi\in\Theta_S$ $$\nabla_{\xi}{\cal H}^{p,q}\subset {\cal H}^{p,q}\oplus{\cal H}^{p-1,q+1}$$ and, correspondingly, $$\nabla_{\xi}^{ar}{\cal H}^{p,q}.T^p\subset {\cal H}^{p,q}.T^p\oplus{\cal H}^{p-1,q+1}.T^p\ .$$ Therefore, $d\Psi(\xi)$ maps ${\cal H}^{p,q}.T^p$ to its image under $\nabla_{\xi}^{ar}$ modulo ${\cal H}_{ar}={\cal F}_{ar}^0$, i.e. into ${\cal H}^{p-1,q+1}.T^p$. Hence $$\overline{d\Psi}(\xi)({\cal H}^{p,q})\subset {\cal H}^{p-1,q+1}\ ,$$ which implies every statement in part (a) of the Lemma. Part (b) is established by similar reasoning.\ $\Box$\\ \ \par The connection $\nabla$ on $\cal H$ naturally induces a connection $\nabla$ on ${\cal E}nd_{{\cal O}_S}({\cal H})$ subject to the rule $$\nabla(Ax)=(\nabla A)x+A\nabla x$$ for any $A\in{\cal E}nd_{{\cal O}_S}({\cal H})$ and $x\in{\cal H}$. In accordance with Lemma 1, for any $\zeta\in\Theta_S$ the covariant derivative $\nabla_{\zeta}$ on ${\cal E}nd({\cal H})$ determines an ${\cal O}_S$-linear map $${\cal E}_{\zeta}: im\,(\overline{d\Psi})\longrightarrow {\cal E}nd({\cal H})/im\,(\overline{d\Psi})\ .$$ \begin{lemma} For any \ $\zeta,\xi\in\Theta_S$ (a) \ $\nabla_{\zeta}(\overline{d\Psi}(\xi))({\cal F}^p)\subset {\cal F}^{p-1}$ for all \ $p$. (b) \ ${\cal E}_{\zeta}(\overline{d\Psi}(\xi))$ determines an element of \ ${\cal E}nd_{{\cal O}_S}^{(-1)}(Gr^{\textstyle\cdot} _{\cal F}{\cal H})/im\,(d\Phi)$. \end{lemma} \ \\ \noindent {\bf Proof.\ \ } (a) $\nabla_{\zeta}(\overline{d\Psi}(\xi))\omega= \nabla_{\zeta}(\overline{d\Psi}(\xi)\omega)- \overline{d\Psi}(\xi)\nabla_{\zeta}\omega$ for any $\omega\in{\cal H}$. Assume $\omega\in{\cal H}^{p,q}$. We want to show that the $(p-2,q+2)$-component of the right-hand side is 0. But this component is $$(\nabla_{\zeta}\nabla_{\xi}\omega)_{(p-2,q+2)}- (\nabla_{\xi}\nabla_{\zeta}\omega)_{(p-2,q+2)}= (\nabla_{[\xi,\zeta]}\omega)_{(p-2,q+2)}=0\ !$$ (b) follows from (a) and the relation between $\overline{d\Psi}(\xi)$ and $d\Phi(\xi)$ established in part (a) of the previous Lemma.\ $\Box$\\ \ \par \begin{dfntn} The {\em second fundamental form of the VHS} $${\rm II}:\Theta_S^{\otimes 2}\longrightarrow {\cal E}nd_{{\cal O}_S}(Gr^{\textstyle\cdot}_{\cal F})/im\,(d\Phi)$$ is defined by $${\rm II}(\zeta,\xi):=h\circ\nabla_{\zeta}(h^{-1}d\Phi(\xi)h)\circ h^{-1}\bmod im\,(d\Phi)\ .$$ \end{dfntn} \ \\ \noindent {\bf Remark.\ \ } ${\rm II}|_{t=0}$ was denoted $d^2\Phi$ in \cite{K}. \ We will omit the identification $h$ in what follows. \begin{prop}[(2.4) in \cite{K}] $${\rm II}(\zeta,\xi)\equiv\overline{d^2\Psi}(\zeta,\xi)- \overline{d\Psi}(\zeta) \circ \overline{d\Psi}(\xi) \bmod im\,(d\Phi)\ .$$ \end{prop} \ \\ \noindent {\bf Proof.\ \ } $\nabla_{\zeta}(\overline{d\Psi}(\xi))\omega= \nabla_{\zeta}(\overline{d\Psi}(\xi)\omega)- \overline{d\Psi}(\xi)\nabla_{\zeta}\omega$ for any $\omega\in{\cal H}$. Now, let $\mbox{\boldmath $\omega$}$ be the element of $Gr_{{\cal F}_{ar}}^0({\cal H}\otimes B_{ar})$ corresponding to $\omega$ under the isomorphism ${\cal H}\cong Gr_{{\cal F}_{ar}}^0({\cal H}\otimes B_{ar})$. Using a similar identification of ${\cal H}$ with $Gr_{{\cal F}_{ar}}^{-2}({\cal H}\otimes B_{ar})$, we have the following correspondences: $$\nabla_{\zeta}(\overline{d\Psi}(\xi)\omega)\longleftrightarrow (\nabla_{\zeta}(d\Psi(\xi)\mbox{\boldmath$\omega$}) \bmod {\cal F}_{ar}^{-1})=(d^2\Psi(\zeta,\xi) \mbox{\boldmath$\omega$}\bmod {\cal F}_{ar}^{-1}) \in Gr_{{\cal F}_{ar}}^{-2}\ ,$$ and $$\overline{d\Psi}(\xi)\nabla_{\zeta}\omega\longleftrightarrow (d\Psi(\zeta)\circ d\Psi(\xi)\mbox{\boldmath$\omega$} \bmod {\cal F}_{ar}^{-1})\in Gr_{{\cal F}_{ar}}^{-2}\ .$$ It remains to pass to ${\cal H}$ on the right-hand side, i.e. put bars on $d\Psi$ and $d^2\Psi$.\ $\Box$\\ \ \par \section{The second Kodaira-Spencer class} First, let us recall the construction of the (first) Kodaira-Spencer map $\kappa_1=\kappa$ of the deformation (1): it is the connecting morphism in the higher-direct-image sequence \begin{equation} \longrightarrow\pi_{*}\Theta_{\cal X}\longrightarrow\Theta_S \stackrel{\kappa}{\longrightarrow}R^1\pi_{*}\Theta_{{\cal X}/S} \longrightarrow \end{equation} associated with the short exact sequence \begin{equation} 0\longrightarrow\Theta_{{\cal X}/S}\longrightarrow \Theta_{\cal X}\longrightarrow\pi^{*}\Theta_S\longrightarrow 0\ . \end{equation} Given $\xi\in\Theta_S$, the corresponding covariant derivative of the Gauss-Manin connection $$\nabla_{\xi}:{\bf R}^m\pi_{*}\Omega^{\textstyle\cdot}_{{\cal X}/S} \longrightarrow{\bf R}^m\pi_{*}\Omega^{\textstyle\cdot}_{{\cal X}/S}$$ is computed as follows (see \cite{Del}, \cite{KO}, or \cite{K}). Choose a Stein covering ${\cal U}=\{U_i\}$ of $X$. Then $\{W_i=U_i\times S\}$ constitute a Stein covering $\cal W$ of $\cal X$. Consider a class in $\Gamma(S,{\bf R}^m\pi_{*}\Omega^{\textstyle\cdot}_{{\cal X}/S})$ represented by the \v{C}ech cocycle $\omega=\{\omega_Q\in \Gamma({\cal W}_Q,\Omega^p_{{\cal X}/S})\}$, where $Q=(i_1<\ldots<i_q)$ and $p+q+1=m$. Let the same symbol $\omega_Q$ denote a pull-back of $\omega_Q\in \Gamma({\cal W}_Q,\Omega^p_{{\cal X}/S})$ to an element of $\Gamma({\cal W}_Q,\Omega^p_{\cal X})$. Let $v=\{v_i\}$ denote liftings of $\xi\in\Theta_S$, or rather, in $\pi^{*}\Theta_S$, to $\Gamma(W_i,\Theta_{\cal X})$. Then $$\nabla_{\xi}[\omega]=[\check{\pounds}_v\omega]$$ where the brackets denote cohomology classes in ${\bf R}^m\pi_{*}\Omega^{\textstyle\cdot}_{{\cal X}/S}$ (more precisely, in $\Gamma(S,{\bf R}^m\pi_{*}\Omega^{\textstyle\cdot}_{{\cal X}/S})$), and $\check{\pounds}_v$ is the Lie derivative on $\check{C}^m({\cal W},\Omega^{\textstyle\cdot}_{\cal X}) $ with respect to $v=\{v_i\}\in \check{C}^0({\cal W},\Theta_{\cal X})$: \begin{equation} \check{\pounds}_v\omega= \{\check{D}v_{i_1}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega_{i_1,\ldots,i_q}+ v_{i_1}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\check{D}\omega_{i_1,\ldots,i_q}\} \end{equation} with $\check{D}=d\pm\delta$, $\delta=\check{\delta}$ being the \v{C}ech differential, as usual. Now, when $\xi\in\Theta_S$ lifts to all of $\cal X$, i.e. $\xi$ lies in the image of $\pi_*\Theta_{\cal X}\rightarrow\Theta_S$ ($= ker\,(\kappa)$ !), the cochain $v\in\check{C}^0({\cal W},\Theta_{\cal X})$ lifting $\xi$ is a cocycle, i,e, $\delta v=0$. But then formula (5) reduces to $$\check{\pounds}_v\omega= \{dv_{i_1}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega_{i_1,\ldots,i_q}+ v_{i_1}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\, d\omega_{i_1,\ldots,i_q}\}= \{\pounds_{v_{i_1}}\omega_{i_1,\ldots,i_q}\} ,$$ where $\pounds_{v_{i_1}}$ now denotes the usual Lie derivative with respect to the vector field $v_{i_1}$. Evidently, in this case $\nabla_{\xi}{\cal F}^p\subset{\cal F}^p$. Consequently, the (first) differential of the period map $$d\Phi:\Theta_S\longrightarrow {\cal H}om({\cal F}^p,{\cal H}/{\cal F}^p)$$ factors through $\Theta_S/\,im\,\{\pi_*\Theta_{\cal X}\rightarrow\Theta_S\}= \Theta_S/\,ker\,(\kappa)$, and thus we arrive at \begin{thm}[Griffiths] There is a commutative diagram $$\begin{array}{ccccc} \Theta_S & \stackrel{d\Phi}{\longrightarrow} & \bigoplus_p{\cal H}om({\cal F}^p,{\cal H}/{\cal F}^p) & \,\longrightarrow\hspace{-12pt & {\cal E}nd^{(-1)}(Gr^{\textstyle\cdot}_{\cal F}{\cal H}) \\ & & & & \\ & \searrow\kappa & \uparrow & \smile\nearrow & \\ & & & & \\ & & {\bf T}^1_{{\cal X}/S} & & \end{array}$$ where \ ${\bf T}^1_{{\cal X}/S}:=R^1\pi_*\Theta_{{\cal X}/S}$ and the northeast arrow sends \ $x$ to the map \ $x\smile$ \ \ (the cup product with \ $x$). \end{thm} Analogous results hold for $d\Psi$ and $\overline{d\Psi}$ (see \cite{K}). We seek a similar statement for $d^2\Phi$ and $d^2\Psi$. First we need to review the construction of the ``second-order tangent space to the moduli." Here we are following (a relativized version of) the presentation in \cite{R}. We will make the simplifying assumption that $X$ has no global holomorphic vector fields, i.e. $\pi_*\Theta_{{\cal X}/S}=0$. Consider the diagram \begin{eqnarray} \nonumber{\cal X}\times_S{\cal X} & \stackrel{g}{\longrightarrow} & {\cal X}_2/S \\ f\searrow & & \swarrow p \\ \nonumber & S & \end{eqnarray} where ${\cal X}_2/S$ denotes the symmetric product of $\cal X$ with itself over $S$ (fiberwise). Let ${\cal K}^{\textstyle\cdot}$ denote the complex of sheaves on ${\cal X}_2/S$ $$\begin{array}{ccc} {\scriptstyle -1} & & {\scriptstyle 0} \\ (g_*(\Theta_{{\cal X}/S}^{\makebox[0in][l]{$\scriptstyle\times$ 2}))^{-} & \stackrel{[\, ,\,]}{\longrightarrow} & \Theta_{{\cal X}/S} \end{array}$$ where $\makebox[0in][l]{$\times$}\raisebox{-1pt}{$\Box$}$ stands for the exterior tensor product on ${\cal X}\times_S{\cal X}$, \ \ $(\ \ )^{-}$ denotes the anti-invariants of the ${\bf Z}/2{\bf Z}$-action, and the differential is the restriction to the diagonal $\Delta\subset{\cal X}\times_{S}{\cal X}$, followed by the Lie bracket of vector fields. \begin{dfntn} ${\bf T}^{(2)}_{{\cal X}/S}:={\bf R}^1p_*{\cal K}^{\textstyle\cdot}$ is the sheaf on $S$ whose fiber over each $t\in S$ is the {\em second-order (Zariski) tangent space to the base $V_t$ of the miniversal deformation of} $X_t$, i.e. ${\bf T}^{(2)}_{X_t}\cong({\bf m}_{V_t,0}/{\bf m}_{V_t,0}^3)^*$. \end{dfntn} This should not be confused with the sheaf ${\bf T}^{2}_{{\cal X}/S}=R^2\pi_*\Theta_{{\cal X}/S}$ whose fiber over each $t\in S$ is the {\em obstruction space}\ \ ${\bf T}^{2}_{X_t}$ for deformations of $X_t$. In fact, we have this exact sequence: \begin{equation} 0\longrightarrow{\bf T}^1_{{\cal X}/S}\longrightarrow {\bf T}^{(2)}_{{\cal X}/S}\longrightarrow Sym^2{\bf T}^1_{{\cal X}/S}\stackrel{o}{\longrightarrow} {\bf T}^{2}_{{\cal X}/S}\ , \end{equation} where $o$ is the first obstruction map, given by the ${\cal O}_S$-linear graded Lie bracket: $$Sym^2R^1\pi_*\Theta_{{\cal X}/S}\stackrel{[\, ,\,]}{\longrightarrow} R^2\pi_*\Theta_{{\cal X}/S}\ .$$ We will find it easier to deal with an ``unsymmetrized version" of ${\bf T}^{(2)}_{{\cal X}/S}$. \begin{dfntn} $\widetilde{\bf T}^{(2)}_{{\cal X}/S}:={\bf R}^1f_* \widetilde{\cal K}^{\textstyle\cdot}$, where $\widetilde{\cal K}^{\textstyle\cdot}$ is the complex on ${\cal X}\times_S{\cal X}$, $$\begin{array}{ccc} {\scriptstyle -1} & & {\scriptstyle 0} \\ \Theta_{{\cal X}/S}^{\makebox[0in][l]{$\scriptstyle\times$ 2} & \stackrel{[\, ,\,]}{\longrightarrow} & \Theta_{{\cal X}/S}\ . \end{array}$$ \end{dfntn} $\widetilde{\bf T}^{(2)}_{{\cal X}/S}$ fits in the commutative diagram with exact rows: $$\begin{array}{cccccccc} 0\rightarrow & {\bf T}^1_{{\cal X}/S} & \longrightarrow & \widetilde{\bf T}^{(2)}_{{\cal X}/S} & \longrightarrow & ({\bf T}^1_{{\cal X}/S})^{\otimes 2} & \stackrel{\widetilde{o}}{\longrightarrow} & {\bf T}^2_{{\cal X}/S} \\ & \| & & \downsurj & & \downsurj & & \| \\ 0\rightarrow & {\bf T}^1_{{\cal X}/S} & \longrightarrow & {\bf T}^{(2)}_{{\cal X}/S} & \longrightarrow & Sym^2{\bf T}^1_{{\cal X}/S} & \stackrel{o}{\longrightarrow} & {\bf T}^2_{{\cal X}/S} \end{array}$$ \begin{dfntn} $T_S^{(2)}:={\cal D}_S^{(2)}/{\cal O}_S$ will denote the sheaf of {\em second-order tangent vectors}\ \ on $S$. \end{dfntn} As part of a more general construction in \cite{EV}, there is {\em the second Kodaira-Spencer map} associated to every deformation as in (1): $$\kappa_2:T_S^{(2)}\longrightarrow{\bf T}^{(2)}_{{\cal X}/S}\ .$$ We will work with a natural lifting $\widetilde{\kappa}_2$ of $\kappa_2$: \begin{equation} \begin{array}{ccc} \Theta_S\oplus\Theta_S^{\otimes 2} & \stackrel{\widetilde{\kappa}_2}{\longrightarrow} & \widetilde{\bf T}^{(2)}_{{\cal X}/S} \\ \downsurj & & \downsurj \\ T_S^{(2)} & \stackrel{\kappa_2}{\longrightarrow} & {\bf T}^{(2)}_{{\cal X}/S} \end{array} \end{equation} or, rather, with the restriction of $\widetilde{\kappa}_2$ to $\Theta_S^{\otimes 2}$. It is easy to describe $\widetilde{\kappa}_2$ explicitly. Let $$\kappa:\Theta_S\longrightarrow {\bf T}^1_{{\cal X}/S}= R^1\pi_*\Theta_{{\cal X}/S}$$ be the ({\em relative, first}\ ) Kodaira-Spencer map of the family $\pi$ as in (1). It is equivalent to the datum of a section of $\Gamma(S,\Omega_S^1\otimes R^1\pi_*\Theta_{{\cal X}/S})$. This section can be represented by a $\check{C}^1({\cal U},\Theta_X)$-valued one-form on $S$, \begin{equation} \theta({\bf t})d{\bf t}:= \sum_{\ell=1}^s\theta({\bf t})_{\ell}dt_{\ell}= \sum_{\ell=1}^s\sum_{I\in{\bf Z}_+^s}^s \theta_{\ell}^{(I)}{\bf t}^Idt_{\ell}\ \ \ ({\bf Z}_+:=\{0\}\cup{\bf N})\ . \end{equation} Here each $\theta_{\ell}^{(I)}=\{\theta_{ij,\ell}^{(I)}\}_{ij}$ is a cochain in $\check{C}^1({\cal U},\Theta_X)$ and each $\theta({\bf t})_{\ell}$ is a cocycle on $X_t$ for every value of $\bf t$, but only the leading coefficients $\theta_{\ell}^{(0)}\ \ (\ell=1,\ldots,s)$ are \v{C}ech {\em cocycles}\ on $X$\ \ $(t=0)$. The rest satisfy the ``deformation equation" \begin{equation} \delta\left(\frac{\partial\theta({\bf t})_{\ell}} {\partial_{t_k}}\right)= [\theta({\bf t})_{\ell},\theta({\bf t})_k] \end{equation} When $s=1$, this reduces to $\delta\dot{\theta}(t)=[\theta(t),\theta(t)]$. \noindent Thus, it is natural to make the following \begin{dfntn} $\widetilde{\kappa}_2:\Theta_S^{\otimes 2}\longrightarrow \widetilde{\bf T}^{(2)}_{{\cal X}/S}$ sends $\frac{\partial}{\partial t_k}\otimes \frac{\partial}{\partial t_{\ell}}$ to the cohomology class of the cocycle \begin{equation} (\theta({\bf t})_k\times \theta({\bf t})_{\ell}, \frac{\partial\theta({\bf t})_{\ell}} {\partial_{t_k}}) \in \check{C}^1({\cal W}\times_S{\cal W}, \widetilde{\cal K}^{\textstyle\cdot})\ . \end{equation} \end{dfntn} For example, if \begin{equation} \theta({\bf t})d{\bf t}=\sum_{\ell=1}^s(\theta_{\ell}^{(0)}+ \sum_{k=1}^s\theta_{\ell}^{(k)}t_k)dt_{\ell}+O({\bf t}^2) \end{equation} is the expansion of $\theta({\bf t})d{\bf t}$ at 0 up to order two, then $\widetilde{\kappa}_2|_{0}:\Theta_S^{\otimes 2}|_{0}\rightarrow \widetilde{\bf T}_X^{(2)}$ sends $\frac{\partial}{\partial t_k}\otimes \frac{\partial}{\partial t_{\ell}}$ to the cohomology class of the cocycle \begin{equation} (\theta_k^{(0)}\times\theta_{\ell}^{(0)}, \theta_{\ell}^{(k)})\in \check{C}^1({\cal U}\times{\cal U}, \widetilde{\cal K}^{\textstyle\cdot}|_{0})\ . \end{equation} Indeed, for the definition of $\widetilde{\kappa}_2$ to make any sense, we must have the following commutative diagram with exact rows: \begin{equation} \begin{array}{cccccccc} 0\rightarrow & \Theta_S & \rightarrow & \Theta_S\oplus\Theta_S^{\otimes 2} & \rightarrow & \Theta_S^{\otimes 2} & \rightarrow & 0 \\ & \kappa_1\ \downarrow & & \downarrow \widetilde{\kappa}_2 & & \downarrow \kappa_1^2 & & \\ 0\rightarrow & {\bf T}_{{\cal X}/S}^1 & \rightarrow & \widetilde{\bf T}_{{\cal X}/S}^{(2)} & \rightarrow & ({\bf T}_{{\cal X}/S}^1)^{\otimes 2} & \stackrel{o}{\rightarrow} & {\bf T}_{{\cal X}/S}^2\ . \end{array} \end{equation} The square on the right induces a commutative triangle \begin{equation} \begin{array}{ccc} \Theta_S^{\otimes 2} & \stackrel{\widetilde{\kappa}_2}{\longrightarrow} & \widetilde{\bf T}_{{\cal X}/S}^{(2)} \\ & \kappa_1^2\ \searrow & \downarrow \\ & & ({\bf T}_{{\cal X}/S}^1)^{\otimes 2}\ . \end{array} \end{equation} Therefore, $\widetilde{\kappa}_2(\frac{\partial}{\partial t_k}\otimes \frac{\partial}{\partial t_\ell})$ must project onto $$\kappa_1(\frac{\partial}{\partial t_k})\otimes \kappa_1(\frac{\partial}{\partial t_\ell})= [\theta({\bf t})_k]\otimes[\theta({\bf t})_\ell]\ .$$ Since $$\check{C}^1({\cal W}\times_S{\cal W},\widetilde{\cal K}^{\textstyle\cdot})= \check{C}^2({\cal W}\times_S{\cal W},\widetilde{\cal K}^{-1}) \oplus\check{C}^1(\widetilde{\cal K}^0)\ ,$$ and $$\check{C}^2({\cal W}\times_S{\cal W},\widetilde{\cal K}^{-1}) \simeq\check{C}^1({\cal U},\Theta_{{\cal X}/S})^{\otimes 2}\ ,$$ this means that the $\check{C}^2(\widetilde{\cal K}^{-1})$-component of a representative of $\widetilde{\kappa}_2(\frac{\partial}{\partial t_k}\otimes \frac{\partial}{\partial t_\ell})$ in $\check{C}^1(\widetilde{\cal K}^{\textstyle\cdot})$ may be taken to be $$\theta({\bf t})_k\times\theta({\bf t})_{\ell}\ .$$ And, in view of (10), the cochain (11) is indeed a {\em cocycle} \ in $\check{C}^1(\widetilde{\cal K}^{\textstyle\cdot})$. We still need to check that $\widetilde{\kappa}_2$ is well-defined. \begin{prop}\ \ \ $\widetilde{\kappa}_2:\Theta_S^{\otimes 2}\longrightarrow \widetilde{\bf T}^{(2)}_{{\cal X}/S}$ can be presented as a connecting morphism in the higher-direct-image sequence of a short exact sequence. \end{prop} \ \\ \noindent {\bf Proof.\ \ } The starting point is the sequence of ${\cal O}_{\cal X}$-modules (4), whose direct-image sequence (3) gives the first Kodaira-Spencer mapping $\kappa_1$ as a connecting morphism. Now, (4) contains an exact subsequence \begin{equation} 0\longrightarrow\Theta_{{\cal X}/S} \stackrel{\alpha}{\longrightarrow}\widetilde{\Theta}_{\cal X} \stackrel{\beta}{\longrightarrow}\pi^{-1}\Theta_S \longrightarrow 0 \end{equation} of $\pi^{-1}{\cal O}_S$-modules, whose direct-image sequence also has $\kappa_1$ as a connecting morphism. From now on we will work with (16) in place of (4). We can splice two sequences produced from (16) by exterior tensor products with $\Theta_{{\cal X}/S}$ and with $\pi^{-1}\Theta_S$, respectively: $$\begin{array}{cccc} & & 0 & \\ & & \uparrow & \\ & & \pi^{-1}\Theta_S\displayboxtimes\pi^{-1}\Theta_S & \\ & & \uparrow & \\ & & \pi^{-1}\Theta_S\displayboxtimes \widetilde{\Theta}_{\cal X} & \\ & \nearrow & \uparrow & \\ 0\longrightarrow\Theta_{{\cal X}/S}\displayboxtimes \Theta_{{\cal X}/S} \longrightarrow \widetilde{\Theta}_{\cal X}\displayboxtimes \Theta_{{\cal X}/S} & \rightarrow & \pi^{-1}\Theta_S\displayboxtimes\Theta_{{\cal X}/S} & \longrightarrow 0 \\ & & \uparrow & \\ & & 0 & \end{array}$$ The resulting four-term exact sequence \begin{equation} 0\rightarrow\Theta_{{\cal X}/S}^{\makebox[0in][l]{$\scriptstyle\times$ 2} \rightarrow\widetilde{\Theta}_{\cal X} \displayboxtimes\Theta_{{\cal X}/S} \stackrel{\beta\makebox[0in][l]{$\scriptstyle\times$\alpha}{\longrightarrow} \pi^{-1}\Theta_S\displayboxtimes\widetilde{\Theta}_{\cal X} \rightarrow(\pi^{-1}\Theta_S)^{\makebox[0in][l]{$\scriptstyle\times$ 2} \rightarrow 0 \end{equation} can be extended to a commutative diagram \begin{equation} \begin{array}{ccccc} 0\rightarrow & \Theta_{{\cal X}/S}^{\makebox[0in][l]{$\scriptstyle\times$ 2} & \rightarrow & \widetilde{\Theta}_{\cal X} \displayboxtimes\Theta_{{\cal X}/S} & \rightarrow \pi^{-1}\Theta_S\displayboxtimes\widetilde{\Theta}_{\cal X} \rightarrow(\pi^{-1}\Theta_S)^{\makebox[0in][l]{$\scriptstyle\times$ 2} \rightarrow 0 \\ & \downarrow & & \downarrow & \\ & \Theta_{{\cal X}/S} & = & \Theta_{{\cal X}/S} & \end{array} \end{equation} where the vertical maps are composed of the restriction to the diagonal $\Delta\subset{\cal X}\times_S{\cal X}$ followed by Lie brackets. \ \\ \noindent {\bf Remark.\ \ } Here we use the fact that the restriction of the Lie bracket $$[\ ,\ ]:\Theta_{\cal X}^{\makebox[0in][l]{$\scriptstyle\times$ 2} \longrightarrow\Theta_{\cal X}$$ to $\widetilde{\Theta}_{\cal X}\makebox[0in][l]{$\times$}\raisebox{-1pt}{$\Box$}\Theta_{{\cal X}/S}$ takes values in $\Theta_{{\cal X}/S}$ (see \cite{BS}, and also \cite{EV}). We note that the first column of the diagram (18) constitutes the complex $\widetilde{\cal K}^{\textstyle\cdot}$ computing $\widetilde{\bf T}^{(2)}_{{\cal X}/S}$. Let ${\cal L}^{\textstyle\cdot}$ denote the complex $$\begin{array}{ccc} {\scriptstyle -1} & & {\scriptstyle 0} \\ \widetilde{\Theta}_{\cal X}\displayboxtimes \Theta_{{\cal X}/S} & \stackrel{\ell}{\longrightarrow} & (\pi^{-1}\Theta_S\displayboxtimes \widetilde{\Theta}_{\cal X})\oplus\Theta_{{\cal X}/S} \end{array}$$ with $\ell=(\beta\makebox[0in][l]{$\times$}\raisebox{-1pt}{$\Box$}\alpha,[\ ,\ ])$\ . Then we can rewrite (18) as a short exact sequence of {\em complexes}\ \ on ${\cal X}\times_S{\cal X}$: \begin{equation} 0\longrightarrow\widetilde{\cal K}^{\textstyle\cdot}\longrightarrow {\cal L}^{\textstyle\cdot}\longrightarrow (\pi^{-1}\Theta_S)^{\makebox[0in][l]{$\scriptstyle\times$ 2}\longrightarrow 0 \end{equation} The associated direct-image sequence yields \begin{equation} \longrightarrow{\bf R}^0f_*{\cal L}^{\textstyle\cdot} \longrightarrow\Theta_S^{\otimes 2} \stackrel{\widetilde{\kappa}_2}{\longrightarrow} \widetilde{\bf T}^{(2)}_{{\cal X}/S}\longrightarrow\ \ . \end{equation} Tracing out the definition of a connecting morphism (bearing in mind that if $\zeta\in\Theta_{\cal X}$ is a local lifting of $\partial/\partial t_k$, and $\theta({\bf t})$ is any element of $\Theta_{{\cal X}/S}$, then $[\zeta,\theta({\bf t})]= \frac{\partial\theta({\bf t})}{\partial t_k}$) shows that it is indeed the same as $\widetilde{\kappa}_2$ given by the explicit Definition in coordinates given above. \ $\Box$\\ \ \par \ \\ \noindent {\bf Remark.\ \ } The explicit construction above shows how the data, up to second order, of the (first) Kodaira-Spencer mapping $$\kappa:\Theta_{S,0}/{\bf m}_{S,0}^2\Theta_{S,0}\longrightarrow {\bf T}^1_{{\cal X}/S,0}/{\bf m}_{S,0}^2{\bf T}^1_{{\cal X}/S,0}$$ determines the second Kodaira-Spencer class $$\widetilde{\kappa}_2:\Theta_S^{\otimes 2}|_0= \Theta_{S,0}^{\otimes 2}/{\bf m}_{S,0}\Theta_{S,0}^{\otimes 2} \longrightarrow \widetilde{\bf T}^{(2)}_X$$ (see (12)). Conversely, if $\zeta,\xi\in\Theta_S|_0$, and $\widetilde{\kappa}_2(\zeta\otimes\xi)$ is represented by a cocycle $(\widehat{\zeta}\times\widehat{\xi},\theta)\in \check{C}^1(\widetilde{\cal K}^{\textstyle\cdot}|_0)$, then we can choose coordinates $\bf t$ on $S$ so that $\zeta=\partial/\partial t_k$, $\xi=\partial/\partial t_{\ell}$, and the Kodaira-Spencer mapping of the deformation in question is represented in ${\bf T}^1_{{\cal X}/S,0}\otimes\Omega^1_{S,0}/ {\bf m}_{S,0}^2{\bf T}^1_{{\cal X}/S,0}\otimes\Omega^1_{S,0}$ by the form $$\sum_{\mu=1}^s(\theta_{\mu}^{(0)}+\sum_{\nu=1}^s \theta_{\mu}^{(\nu)}t_k)dt_{\ell} $$ with $\theta_k^{(0)}=\widehat{\zeta}$, $\theta_{\ell}^{(0)}=\widehat{\xi}$, and $\theta_{\ell}^{(k)}=\theta$. \section{Main results} There is a natural composition map \begin{equation} \Theta_S^{\otimes 2}\hookrightarrow \Theta_S\oplus\Theta_S^{\otimes 2}\,\longrightarrow\hspace{-12pt T_S^{(2)}\ . \end{equation} However, this map is not ${\cal O}_S$-linear. For example, $x\otimes y-y\otimes x$ is mapped to $[x,y]$, whereas for any $f\in{\cal O}_S$ $$f.(x\otimes y-y\otimes x)=(f.x)\otimes y- y\otimes(f.x) $$ is sent to $f.[x,y]-y(f).x$ \ . Nevertheless, (21) induces an ${\cal O}_S$-linear map $$\Theta_S^{\otimes 2}\,\longrightarrow\hspace{-12pt T_S^{(2)}/\Theta_S\ \ (\simeq Sym^2\Theta_S)\ .$$ The latter fits in a commutative square of ${\cal O}_S$-linear maps obtained from (8), \begin{equation} \begin{array}{ccc} \Theta_S^{\otimes 2} & \stackrel{\overline{\widetilde{\kappa}}_2}{\longrightarrow} & \widetilde{\bf T}^{(2)}_{{\cal X}/S}/ \, im\, (\kappa_1) \\ \downsurj & & \downsurj \\ T_S^{(2)}/\Theta_S & \stackrel{\overline{\kappa}_2}{\longrightarrow} & {\bf T}^{(2)}_{{\cal X}/S}/\,im\,(\kappa_1) \end{array} \end{equation} \begin{thm} The second differential of the Archimedean period map $d^2\Psi$ factors through the diagonal of (22), $$\overline{\widetilde{\kappa}}_2: \Theta_S^{\otimes 2}\longrightarrow \widetilde{\bf T}_{{\cal X}/S}^{(2)}/\,im\,(\kappa_1)\ . $$ \end{thm} \ \\ \noindent {\bf Proof.\ \ } Since the statement deals with ${\cal O}_S$-linear maps, it is enough to prove it pointwise, for each $t\in S$. It suffices to restrict to $0\in S$. We need to show that $d^2\Psi(y)=0$ for any $y\in \Theta_S^{\otimes 2}|_0$ with $\widetilde{\kappa}_2(y)\in im\,(\kappa_1)$. The condition on $y$ implies that $\widetilde{\kappa}_2(y)\in \widetilde{\bf T}^{(2)}_X$ can be represented by a cocycle of the form \begin{equation} (0,\theta)\in \check{C}^2(\Theta_X^{\makebox[0in][l]{$\scriptstyle\times$ 2}[1])\oplus \check{C}^1(\Theta_X)=\check{C}^1(\widetilde{\cal K}^{\textstyle\cdot}|_0)\ , \end{equation} where $\theta$ is a {\em cocycle} \ in $\check{C}^1(\Theta_X)$ representing $\kappa_1(\eta)$ for some $\eta\in \Theta_S|_0$. At this point we ``recall" two theorems from \cite{K}. \begin{thm}[(5.4) in \cite{K}] If \ $\kappa\in {\bf T}^1_{{\cal X}/S,0}\otimes\Omega_{S,0}^1/ {\bf m}_{S,0}^2{\bf T}^1_{{\cal X}/S,0}\otimes\Omega_{S,0}^1$ is represented by the form $$\sum_{\mu=1}^s(\theta_{\mu}^{(0)}+\sum_{\nu=1}^s \theta_{\mu}^{(\nu)}t_k)dt_{\ell} $$ with $\theta_k^{(0)}=\widehat{\zeta}$, $\theta_{\ell}^{(0)}=\widehat{\xi}$, and $\theta_{\ell}^{(k)}=\theta$, then the second differential of the Archimedean period map $$d^2\Psi(\frac{\partial}{\partial t_k}\otimes \frac{\partial}{\partial t_{\ell}}): H_{ar}\longrightarrow F_{ar}^{-2}/ H_{ar}+span\,\{\nabla_{\eta}|_0({\cal H}_{ar})\ |\ \eta\in\Theta_S\}$$ is induced by the map \begin{eqnarray*} \lefteqn{H_{ar}=H_{ar}^m\longrightarrow} \\ & & {\bf H}^m(\Omega^{\textstyle\cdot}_X\otimes B_{ar}/ F_{ar}^0(\Omega^{\textstyle\cdot}_X\otimes B_{ar})+ span\,\{ \check{\boldpounds}_{\eta}|_0{\cal F}_{ar}^0\ |\ \eta\in\Theta_S\}) \end{eqnarray*} given on the cochain level by \begin{eqnarray} \nonumber \lefteqn{\omega_{i_1,\ldots,i_q}.T^p=\omega_Q.T^p\mapsto}\\ & & \widehat{\zeta}_{i_{-1}i_0}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\widehat{\xi}_{i_0i_1}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\, \omega_Q.T^p-\widehat{\xi}_{i_0i_1}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\, \check{\varpounds}_{\widehat{\zeta}_{i_0i_1}} \omega_Q.T^{p+1}+ \theta_{i_0i_1}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega_Q.T^p\ . \end{eqnarray} \end{thm} \ \begin{thm}[(5.7) in \cite{K}] $d^2\Psi$ on $\Theta_S^{\otimes 2}|_0$ is determined by $$\kappa\in {\bf T}^1_{{\cal X}/S,0}\otimes\Omega_{S,0}^1/ {\bf m}_{S,0}^2{\bf T}^1_{{\cal X}/S,0}\otimes\Omega_{S,0}^1\ .$$ \end{thm} Reading the two theorems in light of the Remark at the end of Section~2, Theorem 4 shows that $d^2\Psi(y)$ is determined by $\widetilde{\kappa}_2(y)=[(0,\theta)]\in \widetilde{\bf T}^{(2)}_X$, and Theorem 3 says that $d^2\Psi(y)$ is induced on the cochain level by the contraction with the {\em cocycle} \ $\theta$ representing $\kappa_1(\eta)$. This contraction is equivalent to $\nabla_{\eta}|_0$ modulo $H_{ar}$ (in fact, it is none other than $d\Psi(\eta)$), and so we have proved that $d^2\Psi(y)=0$. \ $\Box$\\ \ \par \begin{thm} The graded version of the second differential of the Archimedean period map \ $\overline{d^2\Psi}$, as well as the second differential of the usual period map \ $d^2\Phi$ and the second fundamental form of the VHS, \ {\rm II}, \ all factor through $$\kappa_1^2:\Theta_S^{\otimes 2}\longrightarrow ({\bf T}^1_{{\cal X}/S})^{\otimes 2}\ ,$$ and thus depend on \ $\kappa_1$ only. \end{thm} \ \\ \noindent {\bf Proof.\ \ } Again it suffices to restrict to $0\in S$. Suppose $$\widetilde{\kappa}_2(y)=[(\widehat{\zeta}\times \widehat{\xi},\theta)]\in \widetilde{\bf T}^{(2)}_X$$ for some $y=\zeta\otimes\xi\in\Theta_S^{\otimes 2}|_0$. Examining formula (24), we observe that the term involving $\theta$ lies in $F_{ar}^{-1}$. Therefore, $\overline{d^2\Psi}(y)$ depends only on $\kappa_1^2(y):=\kappa_1(\zeta)\otimes\kappa_1(\xi)$. This proves the Theorem for $\overline{d^2\Psi}$. The statements for $d^2\Phi$ and II follow from this by Lemma 2 (b) and Proposition 1, respectively. \ $\Box$\\ \ \par Finally, all the maps in question are symmetric, and so we may pass from $\kappa_1^2$ to $Sym^2\kappa_1$ and from $\overline{\widetilde{\kappa}}_2$ to $\overline{\kappa}_2$ (see (8)). Referring to the following symmetrized version of (14), \begin{equation} \begin{array}{cccccccc} 0\rightarrow & \Theta_S & \rightarrow & {\bf T}^{(2)}_S & \rightarrow & Sym^2\,\Theta_S & \rightarrow & 0 \\ & \kappa_1 \downarrow & & \downarrow \kappa_2 & & \downarrow Sym^2\kappa_1 & & \\ 0\rightarrow & {\bf T}_{{\cal X}/S}^1 & \rightarrow & {\bf T}_{{\cal X}/S}^{(2)} & \rightarrow & Sym^2{\bf T}_{{\cal X}/S}^1 & \stackrel{o}{\rightarrow} & {\bf T}_{{\cal X}/S}^2\ , \end{array} \end{equation} we conclude with \begin{thm} $d^2\Psi$ factors through $$\overline{\kappa}_2:Sym^2\Theta_S\longrightarrow {\bf T}_{{\cal X}/S}^{(2)}/\,im\,(\kappa_1),$$ whereas \ $\overline{d^2\Psi}$, $d^2\Phi$ and \ ${\rm II}$ \ factor through $$Sym^2\kappa_1:Sym^2\Theta_S\longrightarrow Sym^2{\bf T}_{{\cal X}/S}^1\ .$$ \end{thm} \ \\ \noindent {\bf Remark.\ \ } When the deformation is versal, i.e. $im\,(\kappa_1)$ is all of ${\bf T}_{{\cal X}/S}^1$, there is no difference between $\overline{\kappa}_2$ and $Sym^2\kappa_1$. \section{The higher-order cases} The definition of the second differential of the period map in Section 1 easily generalizes to higher-order cases (cf. \cite{K}). All three papers mentioned in the introduction define ``tangent spaces to the moduli" ${\bf T}_{{\cal X}/S}^{(n)}$ of all orders $n$. However, these definitions seem more complicated than in the case $n=2$. Still, we should have a diagram analogous to (25), \begin{equation} \begin{array}{cccccccc} 0\rightarrow & {\bf T}^{(n-1)}_S & \rightarrow & {\bf T}^{(n)}_S & \rightarrow & Sym^n\,\Theta_S & \rightarrow & 0 \\ & \kappa_{n-1} \downarrow & & \downarrow \kappa_n & & \downarrow Sym^n\kappa_1 & & \\ 0\rightarrow & {\bf T}_{{\cal X}/S}^{(n-1)} & \rightarrow & {\bf T}_{{\cal X}/S}^{(n)} & \rightarrow & Sym^n{\bf T}_{{\cal X}/S}^1 & \stackrel{o_n}{\rightarrow} & {\bf T}_{{\cal X}/S}^2\ , \end{array} \end{equation} where $o_n$ is the $n^{th}$ obstruction map, and we expect that the $n^{th}$ differential of the Archimedean period map $d^n\Psi$ factors through the $n^{th}$ Kodaira-Spencer map $\kappa_n$ modulo the image of $\kappa_{n-1}$, whereas the $n^{th}$ differential of the usual period map $d^n\Phi$ and the $n^{th}$ fundamental form of the VHS I$n$I factor through $Sym^n\kappa_1$. \section{Appendix: Archimedean cohomology} In this section we summarize what we need about Archimedean cohomology. For more information on this subject we refer to \cite{Den}. \begin{dfntn} $B_{ar}={\bf C}[T,T^{-1}], \ \ {\bf L}={\bf C}[T^{-1}]$. \ $B_{ar}$ is filtered by the ${\bf L}$-submodules $F^p=T^{-p}.{\bf L}$. \end{dfntn} Thus, if $X$ is a compact K\"{a}hler manifold, $H^m(X)\otimes_{\bf C}B_{ar}$ receives the filtration $F^{\textstyle\cdot}_{ar}$ obtained as the tensor product of the Hodge filtration on $H^m(X,{\bf C})$ and the filtration $F^{\textstyle\cdot}$ on $B_{ar}$. $F^{\textstyle\cdot}_{ar}$ is a decreasing filtration with infinitely many levels, and $$Gr_{F_{ar}}^k\cong\bigoplus_{p+q=m}H^{p,q}.T^{p-k}\ .$$ \begin{dfntn} The {\em Archimedean cohomology}\ of $X$ is $$H^m_{ar}(X):=F_{ar}^0(H^m(X,{\bf C})\otimes B_{ar})\ .$$ \end{dfntn} Consider the complex of sheaves $\Omega_X^{\textstyle\cdot}\otimes_{\bf C}B_{ar}$ with the differential $${\bf d}(\omega.T^k):=d\omega.T^{k+1}\ .$$ This complex is also filtered by the tensor product of the stupid filtration on $\Omega^{\textstyle\cdot}_X$ and $F^{\textstyle\cdot}$ on $B_{ar}$, and we have \begin{eqnarray*} \lefteqn{H^m_{ar}(X)=F^0{\bf H}^m(X,\Omega^{\textstyle\cdot}_X\otimes B_{ar}) \cong} \\ & & {\bf H}^m(X,F^0(\Omega^{\textstyle\cdot}_X\otimes B_{ar})) \cong {\bf H}^m(X,\Omega^{\textstyle\cdot}_X)\otimes{\bf L} \cong H^m(X,{\bf C})\otimes{\bf L}\ . \end{eqnarray*} Note that ${\bf H}^m(X,\Omega^{\textstyle\cdot}_X\otimes B_{ar})$ is a complex infinite-dimensional Hodge structure (of weight $m$), and $(\Omega^{\textstyle\cdot}_X\otimes B_{ar},{\bf d})$ is a Hodge complex. Hence $$Gr_{F_{ar}}^k{\bf H}^m(X,\Omega^{\textstyle\cdot}_X\otimes B_{ar})\cong {\bf H}^m(X,Gr_{F_{ar}}^k(\Omega^{\textstyle\cdot}_X\otimes B_{ar}))\ .$$ We will write boldface $\check{\bf D}$ for the differential in the \v{C}ech cochain complex computing ${\bf H}^m(X,\Omega^{\textstyle\cdot}_X\otimes B_{ar})$, and $$\check{\boldpounds}_v:=\check{\bf D}v\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\, + v\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\check{\bf D}$$ for the corresponding Lie derivative with respect to a vector field $v$ on $X$. These constructions extend without any difficulty to the relative situation. In particular, given a flat family $\pi:{\cal X}\longrightarrow S$ of compact K\"{a}hler manifolds, the bundle $${\cal H}\otimes B_{ar}={\bf R}^m\pi_*(\Omega^{\textstyle\cdot}_{{\cal X}/S}\otimes B_{ar})$$ is filtered by ${\cal F}^{\textstyle\cdot}_{ar}$, and the Gauss-Manin connection $\nabla$ extends to $$\nabla_{ar}:{\cal H}\otimes B_{ar}\longrightarrow {\cal H}\otimes B_{ar}\otimes\Omega_S^1\ ,$$ with the usual Griffiths' transversality property $$\nabla_{ar}({\cal F}^p_{ar})\subset{\cal F}^{p-1}_{ar}\otimes \Omega_S^1\ .$$ Specifically, if $x$ is a section of $\cal H$, then $$\nabla_{ar}(x.T^p)=\nabla x.T^p\ .$$ The real difference arises when one examines the definition of $\nabla_{ar}$ on the cochain level, due to the fact that $\bf d$ increases the exponent at $T$.

9405

alg-geom/9405008

en

https://arxiv.org/abs/alg-geom/9405008

alg-geom/9405008

Klaus Altmann

Klaus Altmann

Infinitesimal Deformations and Obstructions for Toric Singularities

26 pages, LaTeX (uses diagram.sty)

The obstruction space T^2 and the cup product T^1 x T^1 -> T^2 are computed for toric singularities.

alg-geom

\section{#1} \protect\setcounter{secnum}{\value{section}} \protect\setcounter{equation}{0} \protect\renewcommand{\theequation}{\mbox{\arabic{secnum}.\arabic{equation}}}} \setcounter{tocdepth}{1} \begin{document} \title{Infinitesimal Deformations and Obstructions for Toric Singularities} \author{Klaus Altmann\footnotemark[1]\\ \small Dept.~of Mathematics, M.I.T., Cambridge, MA 02139, U.S.A. \vspace{-0.7ex}\\ \small E-mail: [email protected]} \footnotetext[1]{Die Arbeit wurde mit einem Stipendium des DAAD unterst\"utzt.} \date{} \maketitle \begin{abstract} The obstruction space $T^2$ and the cup product $T^1\times T^1\to T^2$ are computed for toric singularities. \end{abstract} \tableofcontents \sect{Introduction}\label{s1} \neu{11} For an affine scheme $\,Y= \mbox{Spec}\; A$, there are two important $A$-modules, $T^1_Y$ and $T^2_Y$, carrying information about its deformation theory: $T^1_Y$ describes the infinitesimal deformations, and $T^2_Y$ contains the obstructions for extending deformations of $Y$ to larger base spaces.\\ \par In case $Y$ admits a versal deformation, $T^1_Y$ is the tangent space of the versal base space $S$. Moreover, if $J$ denotes the ideal defining $S$ as a closed subscheme of the affine space $T^1_Y$, the module $\left( ^{\displaystyle J}\! / \! _{\displaystyle m_{T^1} \,J} \right) ^\ast$ can be canonically embedded into $T^2_Y$, i.e. $(T_Y^2)^\ast$-elements induce the equations defining $S$ in $T^1_Y$.\\ \par The vector spaces $T^i_Y$ come with a cup product $T_Y^1 \times T^1_Y \rightarrow T^2_Y$. The associated quadratic form $T^1_Y \rightarrow T^2_Y$ describes the quadratic part of the elements of $J$, i.e. it can be used to get a better approximation of the versal base space $S$ as regarding its tangent space only.\\ \par \neu{12} In \cite{T1} we have determined the vector space $T^1_Y$ for affine toric varieties. The present paper can be regarded as its continuation - we will compute $T^2_Y$ and the cup product. \\ These modules $T^i_Y$ are canonically graded (induced from the character group of the torus). We will describe their homogeneous pieces as cohomology groups of certain complexes, that are directly induced from the combinatorial structure of the rational, polyhedral cone defining our variety $Y$. The results can be found in \S \ref{s3}.\\ \par Switching to another, quasiisomorphic complex provides a second formula for the vector spaces $T^i_Y$ (cf. \S \ref{s6}). We will use this particular version for describing these spaces and the cup product in the special case of three-dimensional toric Gorenstein singularities (cf. \S \ref{s7}).\\ \par \sect{$T^1$, $T^2$, and the cup product (in general)}\label{s2} In this section we will give a brief reminder to the well known definitions of these objects. Moreover, we will use this opportunity to fix some notations.\\ \par \neu{21} Let $Y \subseteq \,I\!\!\!\!C^{w+1}$ be given by equations $f_1,\dots,f_m$, i.e. its ring of regular functions equals \[ A=\;^{\displaystyle P}\!\! / \! _{\displaystyle I} \quad \mbox{ with } \begin{array}[t]{l} P = \,I\!\!\!\!C[z_0,\dots, z_w]\\ I = (f_1,\dots,f_m)\, . \end{array} \] Then, using $d:^{\displaystyle I}\! / \! _{\displaystyle I^2} \rightarrow A^{w+1}\;$ ($d(f_i):= (\frac{\partial f_i}{\partial z_0},\dots \frac{\partial f_i}{\partial z_w})$), the vector space $T^1_Y$ equals \[ T^1_Y = \;^{\displaystyle \mbox{Hom}_A(^{\displaystyle I}\!\! / \! _{\displaystyle I^2}, A)} \! \left/ \! _{\displaystyle \mbox{Hom}_A(A^{w+1},A)} \right.\; . \vspace{1ex} \] \par \neu{22} Let ${\cal R}\subseteq P^m$ denote the $P$-module of relations between the equations $f_1,\dots,f_m$. It contains the so-called Koszul relations ${\cal R}_0:= \langle f_i\,e^j - f_j \,e^i \rangle$ as a submodule.\\ Now, $T^2_Y$ can be obtained as \[ T^2_Y = \;^{\displaystyle \mbox{Hom}_P(^{\displaystyle {\cal R}}\! / \! _{\displaystyle {\cal R}_0}, A)} \! \left/ \! _{\displaystyle \mbox{Hom}_P(P^m,A)} \right.\; . \vspace{1ex} \] \par \neu{23} Finally, the cup product $T^1\times T^1 \rightarrow T^2$ can be defined in the following way: \begin{itemize} \item[(i)] Starting with an $\varphi\in \mbox{Hom}_A(^{\displaystyle I}\! / \! _{\displaystyle I^2}, A)$, we lift the images of the $f_i$ obtaining elements $\tilde{\varphi}(f_i)\in P$. \item[(ii)] Given a relation $r\in {\cal R}$, the linear combination $\sum_ir_i\,\tilde{\varphi}(f_i)$ vanishes in $A$, i.e. it is contained in the ideal $I\subseteq P$. Denote by $\lambda(\varphi)\in P^m$ any set of coefficients such that \[ \sum_i r_i \, \tilde{\varphi}(f_i) + \sum_i \lambda_i(\varphi)\, f_i =0\quad \mbox{ in } P. \] (Of course, $\lambda$ depends on $r$ also.) \item[(iii)] If $\varphi, \psi \in \mbox{Hom}_A(^{\displaystyle I}\! / \! _{\displaystyle I^2}, A)$ represent two elements of $T^1_Y$, then we define for each relation $r\in {\cal R}$ \[ (\varphi \cup \psi)(r) := \sum_i \lambda_i (\varphi)\, \psi(f_i) + \sum_i \varphi(f_i)\, \lambda_i(\psi)\; \in A\, . \vspace{1ex} \] \end{itemize} {\bf Remark:} The definition of the cup product does not depend on the choices we made: \begin{itemize} \item[(a)] Choosing a $\lambda'(\varphi)$ instead of $\lambda(\varphi)$ yields $\lambda'(\varphi) - \lambda(\varphi) \in {\cal R}$, i.e. in $A$ we obtain the same result. \item[(b)] Let $\tilde{\varphi}'(f_i)$ be different liftings to $P$. Then, the difference $\tilde{\varphi}'(f_i) - \tilde{\varphi}(f_i)$ is contained in $I$, i.e. it can be written as some linear combination \[ \tilde{\varphi}'(f_i) - \tilde{\varphi}(f_i) = \sum_j t_{ij}\, f_j\, . \] Hence, \[ \sum_i r_i \,\tilde{\varphi}'(f_i) = \sum_i r_i \,\tilde{\varphi}(f_i) + \sum_{i,j} t_{ij}\, r_i\, f_j\,, \] and we can define $\lambda'_j(\varphi):= \lambda_j(\varphi) - \sum_it_{ij}\,r_i$ (corresponding to $\tilde{\varphi}'$ instead of $\tilde{\varphi}$). Then, we obtain for the cup product \[ (\varphi\cup\psi)'(r) - (\varphi\cup\psi)(r) = -\sum_ir_i\cdot \left( \sum_j t_{ij}\, \psi(f_j)\right)\, , \] but this expression comes from some map $P^m\rightarrow A$. \vspace{3ex} \end{itemize} \sect{$T^1$, $T^2$, and the cup product (for toric varieties)}\label{s3} \neu{31} We start with fixing the usual notations when dealing with affine toric varieties (cf. \cite{Ke} or \cite{Oda}): \begin{itemize} \item Let $M,N$ be mutually dual free Abelian groups, we denote by $M_{I\!\!R}, N_{I\!\!R}$ the associated real vector spaces obtained by base change with $I\!\!R$. \item Let $\sigma=\langle a^1,\dots,a^N\rangle \subseteq N_{I\!\!R}$ be a rational, polyhedral cone with apex - given by its fundamental generators. \\ $\sigma^{\scriptscriptstyle\vee}:= \{ r\in M_{I\!\!R}\,|\; \langle \sigma,\,r\rangle \geq 0\} \subseteq M_{I\!\!R}$ is called the dual cone. It induces a partial order on the lattice $M$ via $[\,a\geq b$ iff $a-b \in \sigma^{\scriptscriptstyle\vee}\,]$. \item $A:= \,I\!\!\!\!C[\sigma^{\scriptscriptstyle\vee}\cap M]$ denotes the semigroup algebra. It is the ring of regular functions of the toric variety $Y= \mbox{Spec}\; A$ associated to $\sigma$. \item Denote by $E\subset \sigma^{\scriptscriptstyle\vee}\cap M$ the minimal set of generators of this semigroup ("the Hilbert basis"). $E$ equals the set of all primitive (i.e. non-splittable) elements of $\sigma^{\scriptscriptstyle\vee}\cap M$. In particular, there is a surjection of semigroups $\pi:I\!\!N^E \longrightarrow\hspace{-1.5em}\longrightarrow \sigma^{\scriptscriptstyle\vee}\cap M$, and this fact translates into a closed embedding $Y\hookrightarrow \,I\!\!\!\!C^E$.\\ To make the notations coherent with \S \ref{s2}, assume that $E=\{r^0,\dots,r^w\}$ consists of $w+1$ elements. \vspace{2ex} \end{itemize} \neu{32} To a fixed degree $R\in M$ we associate ``thick facets'' $K_i^R$ of the dual cone \[ K_i^R := \{r\in \sigma^{\scriptscriptstyle\vee}\cap M \, | \; \langle a^i, r \rangle < \langle a^i, R \rangle \}\quad (i=1,\dots,N) . \vspace{2ex} \] \par {\bf Lemma:}{\em \begin{itemize} \item[(1)] $\cup_i K_i^R = (\sigma^{\scriptscriptstyle\vee}\cap M) \setminus (R+ \sigma^{\scriptscriptstyle\vee})$. \item[(2)] For each $r,s\in K_i^R$ there exists an $\ell\in K_i^R$ such that $\ell\geq r,s$. Moreover, if $Y$ is smooth in codimension 2, the intersections $K^R_i\cap K^R_j$ (for 2-faces $\langle a^i,a^j\rangle <\sigma$) have the same property. \vspace{1ex} \end{itemize} } \par {\bf Proof:} Part (i) is trivial; for (ii) cf. (3.7) of \cite{T1}. \hfill$\Box$\\ \par Intersecting these sets with $E\subseteq \sigma^{\scriptscriptstyle\vee}\cap M$, we obtain the basic objects for describing the modules $T^i_Y$: \begin{eqnarray*} E_i^R &:=& K_i^R \cap E = \{r\in E\, | \; \langle a^i,r \rangle < \langle a^i, R \rangle \}\, ,\\ E_0^R &:=& \bigcup_{i=1}^N E_i^R\; ,\mbox{ and}\\ E^R_{\tau} &:=& \bigcap_{a^i\in \tau} E^R_i \; \mbox{ for faces } \tau < \sigma\,. \end{eqnarray*} We obtain a complex $L(E^R)_{\bullet}$ of free Abelian groups via \[ L(E^R)_{-k} := \bigoplus_{\begin{array}{c} \tau<\sigma\\ \mbox{dim}\, \tau=k \end{array}} \!\!L(E^R_{\tau}) \] with the usual differentials. ($L(\dots)$ denotes the free Abelian group of integral, linear dependencies.) \\ The most interesting part ($k\leq 2$) can be written explicitely as \[ L(E^R)_{\bullet}:\quad \cdots \rightarrow \oplus_{\langle a^i,a^j\rangle<\sigma} L(E^R_i\cap E^R_j) \longrightarrow \oplus_i L(E_i^R) \longrightarrow L(E_0^R) \rightarrow 0\,. \vspace{1ex} \] \par \neu{33} {\bf Theorem:} {\em \begin{itemize} \item[(1)] $T^1_Y(-R) = H^0 \left( L(E^R)_{\bullet}^\ast \otimes_{Z\!\!\!Z}\,I\!\!\!\!C\right)$ \item[(2)] $T^2_Y(-R) \supseteq H^1 \left( L(E^R)_{\bullet}^\ast \otimes_{Z\!\!\!Z}\,I\!\!\!\!C\right)$ \item[(3)] Moreover, if $Y$ is smooth in codimension 2 (i.e.\ if the 2-faces $\langle a^i, a^j \rangle < \sigma$ are spanned by a part of a $Z\!\!\!Z$-basis of the lattice $N$), then \[ T^2_Y(-R) = H^1 \left( L(E^R)_{\bullet}^\ast \otimes_{Z\!\!\!Z}\,I\!\!\!\!C\right)\, . \] \item[(4)] Module structure: If $x^s\in A$ (i.e. $s\in \sigma^{\scriptscriptstyle\vee}\cap M$), then $E_i^{R-s}\subseteq R_i^R$, hence $L(E^R)_{\bullet}^\ast \subseteq L(E^{R-s})^\ast_{\bullet}$. The induced map in cohomology corresponds to the multiplication with $x^s$ in the $A$-modules $T^1_Y$ and $T^2_Y$. \vspace{2ex} \end{itemize} } \par The first part was shown in \cite{T1}; the formula for $T^2$ will be proved in \S \ref{s4}. Then, the claim concerning the module structure will become clear automatically.\\ \par {\bf Remark:} The assumption made in (3) can not be dropped: \\ Taking for $Y$ a 2-dimensional cyclic quotient singularity given by some 2-dimensional cone $\sigma$, there are only two different sets $E_1^R$ and $E_2^R$ (for each $R\in M$). In particular, $H^1 \left( L(E^R)_{\bullet}^\ast \otimes_{Z\!\!\!Z}\,I\!\!\!\!C\right)=0$.\\ \par \neu{34} We want to describe the isomorphisms connecting the general $T^i$-formulas of \zitat{2}{1} and \zitat{2}{2} with the toric ones given in \zitat{3}{3}.\\ \par $Y\hookrightarrow\,I\!\!\!\!C^{w+1}$ is given by the equations \[ f_{ab}:= \underline{z}^a-\underline{z}^b\quad (a,b\in I\!\!N^{w+1} \mbox{ with } \pi(a)=\pi(b) \mbox{ in } \sigma^{\scriptscriptstyle\vee} \cap M), \] and it is easier to deal with this infinite set of equations (which generates the ideal $I$ as a $\,I\!\!\!\!C$-vector space) instead of selecting a finite number of them in some non-canonical way. In particular, for $m$ of \zitat{2}{1} and \zitat{2}{2} we take \[ m:= \{ (a,b)\in I\!\!N^{w+1}\timesI\!\!N^{w+1}\,|\;\pi(a)=\pi(b)\}\,. \] The general $T^i$-formulas mentioned in \zitat{2}{1} and \zitat{2}{2} remain true.\\ \par {\bf Theorem:} {\em For a fixed element $R\in M$ let $\varphi: L(E)_{\,I\!\!\!\!C}\rightarrow \,I\!\!\!\!C$ induce some element of \[ \left(\left. ^{\displaystyle L(E_0^R)}\!\!\right/ \!_{\displaystyle \sum_i L(E_i^R)} \right)^\ast \otimes_{Z\!\!\!Z} \,I\!\!\!\!C \cong T^1_Y(-R)\quad \mbox{(cf. Theorem \zitat{3}{3}(1)).} \] Then, the $A$-linear map \begin{eqnarray*} ^{\displaystyle I}\!\!/\!_{\displaystyle I^2} &\longrightarrow& A\\ \underline{z}^a-\underline{z}^b & \mapsto & \varphi(a-b)\cdot x^{\pi(a)-R} \end{eqnarray*} provides the same element via the formula \zitat{2}{1}. }\\ \par Again, this Theorem follows from the paper \cite{T1} - accompanied by the commutative diagram of \zitat{4}{3} in the present paper. (Cf. Remark \zitat{4}{4}.)\\ \par {\bf Remark:} A simple, but nevertheless important check shows that the map $(\underline{z}^a-\underline{z}^b) \mapsto \varphi(a-b)\cdot x^{\pi(a)-R}$ goes into $A$, indeed:\\ Assume $\pi(a)-R \notin \sigma^{\scriptscriptstyle\vee}$. Then, there exists an index $i$ such that $\langle a^i, \pi(a)-R \rangle <0$. Denoting by "supp $q$" (for a $q\in I\!\!R^E$) the set of those $r\in E$ providing a non-vanishing entry $q_r$, we obtain \[ \mbox{supp}\,(a-b) \subseteq \mbox{supp}\,a \cup \mbox{supp}\, b \subseteq E^R_i\, , \] i.e. $\varphi(a-b)=0$.\\ \par \neu{35} The $P$-module ${\cal R}\subseteq P^m$ is generated by relations of two different types: \begin{eqnarray*} r(a,b;c) &:=& e^{a+c,\,b+c}- \underline{z}^c\, e^{a,b}\quad (a,b,c\in I\!\!N^{w+1};\, \pi(a)=\pi(b))\quad \mbox{ and}\\ s(a,b,c) &:=& e^{b,c} - e^{a,c} + e^{a,b}\quad (a,b,c\in I\!\!N^{w+1};\, \pi(a)=\pi(b)=\pi(c))\,.\\ &&\qquad\qquad(e^{\bullet,\bullet} \mbox{ denote the standard basis vectors of } P^m.) \vspace{1ex} \end{eqnarray*} \par {\bf Theorem:} {\em For a fixed element $R\in M$ let $\psi_i: L(E_i^R)_{\,I\!\!\!\!C}\rightarrow \,I\!\!\!\!C$ induce some element of \[ \left( \frac{\displaystyle \mbox{Ker}\,\left( \oplus_i L(E_i^R) \longrightarrow L(E'^R)\right)}{\displaystyle \mbox{Im}\, \left( \oplus_{\langle a^i,a^j\rangle<\sigma} L(E_i^R\cap E_j^R) \rightarrow \oplus_i L(E_i^R)\right)} \right)^\ast \otimes_{Z\!\!\!Z}\,I\!\!\!\!C \subseteq T^2_Y(-R) \quad \mbox{(cf. \zitat{3}{3}(2)).} \] Then, the $P$-linear map \begin{eqnarray*} ^{\displaystyle {\cal R}}\!\!/\!_{\displaystyle {\cal R}_0} &\longrightarrow & A\\ r(a,b;c) & \mapsto & \left\{ \begin{array}{ll} \psi_i(a-b)\, x^{\pi(a+c)-R} & \mbox{for } \pi(a)\in K_i^R;\; \pi(a+c)\geq R\\ 0 & \mbox{for }\pi(a)\geq R \mbox{ or } \pi(a+c)\in\bigcup_i K_i^R \end{array}\right.\\ s(a,b,c) &\mapsto & 0 \end{eqnarray*} is correct defined, and via the formula \zitat{2}{2} it induces the same element of $T^2_Y$. } \vspace{2ex} \\ \par For the prove we refer to \S \ref{s4}. Nevertheless, we check the {\em correctness of the definition} of the $P$-linear map $^{\displaystyle {\cal R}}\!/\!_{\displaystyle {\cal R}_0} \rightarrow A$ instantly: \begin{itemize} \item[(i)] If $\pi(a)$ is contained in two different sets $K_i^R$ and $K_j^R$, then the two fundamental generators $a^i$ and $a^j$ can be connected by a sequence $a^{i_0},\dots,a^{i_p}$, such that \begin{itemize} \item[$\bullet$] $a^{i_0}=a^i,\, a^{i_p}=a^j,$ \item[$\bullet$] $a^{i_{v-1}}$ and $a^{i_v}$ are the edges of some 2-face of $\sigma$ ($v=1,\dots,p$), and \item[$\bullet$] $\pi(a)\in K^R_{i_v}$ for $v=0,\dots,p$. \end{itemize} Hence, $\mbox{supp}\,(a-b)\subseteq E^R_{i_{v-1}}\cap E^R_{i_v}$ ($v=1,\dots,p$), and we obtain \[ \psi_i(a-b)=\psi_{i_1}(a-b)=\dots=\psi_{i_{p-1}}(a-b)=\psi_j(a-b)\,. \] \item[(ii)] There are three types of $P$-linear relations between the generators $r(\dots)$ and $s(\dots)$ of ${\cal R}$: \begin{eqnarray*} 0 &=& \underline{z}^d\,r(a,b;c) -r(a,b;c+d) + r(a+c,b+c;d)\,,\\ 0 &=& r(b,c;d) - r(a,c;d) + r(a,b;d) - s(a+d,b+d,c+d) + \underline{z}^d\, s(a,b,c)\,,\\ 0 &=& s(b,c,d) - s(a,c,d) + s(a,b,d) - s(a,b,c)\,. \end{eqnarray*} Our map respects them all. \item[(iii)] Finally, the typical element $(\underline{z}^a-\underline{z}^b)e^{cd} - (\underline{z}^c-\underline{z}^d)e^{ab} \in {\cal R}_0$ equals \[ -r(c,d;a)+r(c,d;b)+r(a,b;c)-r(a,b;d) - s(a+c,b+c,a+d) - s(a+d,b+c,b+d)\,. \] It will be sent to 0, too. \vspace{2ex} \end{itemize} \par \neu{36} The cup product $T^1_Y\times T^1_Y\rightarrow T^2_Y$ respects the grading, i.e. it splits into pieces \[ T^1_Y(-R)\times T^1_Y(-S) \longrightarrow T^2_Y(-R-S)\quad (R,S\in M)\,. \] To describe these maps in our combinatorial language, we choose some set-theoretical section $\Phi:M\rightarrowZ\!\!\!Z^{w+1}$ of the $Z\!\!\!Z$-linear map \begin{eqnarray*} \pi: Z\!\!\!Z^{w+1} &\longrightarrow& M\\ a&\mapsto&\sum_v a_v\,r^v \end{eqnarray*} with the additional property $\Phi(\sigma^{\scriptscriptstyle\vee}\cap M)\subseteq I\!\!N^{w+1}$.\\ \par Let $q\in L(E)\subseteqZ\!\!\!Z^{w+1}$ be an integral relation between the generators of the semigroup $\sigma^{\scriptscriptstyle\vee}\cap M$. We introduce the following notations: \begin{itemize} \item $q^+,q^-\inI\!\!N^{w+1}$ denote the positive and the negative part of $q$, respectively. (With other words: $q=q^+-q^-$ and $\sum_v q^-_v\,q^+_v=0$.) \item $\bar{q}:=\pi(q^+)=\sum_v q_v^+\,r^v = \pi(q^-)=\sum_v q_v^-\,r^v \in M$. \item If $\varphi,\psi: L(E)\rightarrowZ\!\!\!Z$ are linear maps and $R,S\in M$, then we define \[ t_{\varphi,\psi,R,S}(q):= \varphi(q)\cdot \psi \left( \Phi(\bar{q}-R)+\Phi(R)-q^-\right) + \psi(q)\cdot \varphi\left( \Phi(\bar{q}-S)+\Phi(S)-q^+\right)\,. \vspace{2ex} \] \end{itemize} \par {\bf Theorem:} {\em Assume that $Y$ is smooth in codimension 2.\\ Let $R,S\in M$, and let $\varphi,\psi: L(E)_{\,I\!\!\!\!C}\rightarrow\,I\!\!\!\!C$ be linear maps vanishing on $\sum_i L(E_i^R)_{\,I\!\!\!\!C}$ and $\sum_i L(E_i^S)_{\,I\!\!\!\!C}$, respectively. In particular, they define elements $\varphi\in T^1_Y(-R),\,\psi\in T^1_Y(-S)$ (which involves a slight abuse of notations).\\ Then, the cup product $\varphi\cup\psi\in T^2_Y(-R-S)$ is given (via \zitat{3}{3}(3)) by the linear maps $(\varphi\cup\psi)_i: L(E_i^{R+S})_{\,I\!\!\!\!C}\rightarrow\,I\!\!\!\!C$ defined as follows: \begin{itemize} \item[(i)] If $q\in L(E_i^{R+S})$ (i.e. $\langle a^i,\mbox{supp}\,q\rangle < \langle a^i,R+S\rangle$) is an integral relation, then there exists a decomposition $q=\sum_k q^k$ such that \begin{itemize} \item $q^k\in L(E_i^{R+S})$, and moreover \item $\langle a^i, \bar{q}^k\rangle < \langle a^i,R+S\rangle$. \end{itemize} \item[(ii)] $(\varphi\cup\psi)_i\left( q\in L(E_i^{R+S})\right):= \sum_k t_{\varphi,\psi,R,S}(q^k)$. \vspace{2ex} \end{itemize} } \par It is even not obvious that the map $q\mapsto \sum_k t(q^k)$ \begin{itemize} \item does not depend on the representation of $q$ as a particular sum of $q_k$'s (which would instantly imply linearity on $L(E_i^{R+S})$), and \item yields the same result on $L(E_i^{R+S}\cap E_j^{R+S})$ for $i,j$ corresponding to edges $a^i, a^j$ of some 2-face of $\sigma$. \end{itemize} The proof of these facts (cf.\ \zitat{5}{4})and of the entire theorem is contained in \S \ref{s5}.\\ \par {\bf Remark 1:} Replacing all the terms $\Phi(\bullet)$ in the $t$'s of the previous formula for $(\varphi\cup\psi)_i(q)$ by arbitrary liftings from $M$ to $Z\!\!\!Z^{w+1}$, the result in $T^2_Y(-R-S)$ will be unchanged as long as we obey the following two rules: \begin{itemize} \item[(i)] Use always (for all $q$, $q^k$, and $i$) the {\em same liftings} of $R$ and $S$ to $Z\!\!\!Z^{w+1}$ (at the places of $\Phi(R)$ and $\Phi(S)$, respectively). \item[(ii)] Elements of $\sigma^{\scriptscriptstyle\vee}\cap M$ always have to be lifted to $I\!\!N^{w+1}$. \vspace{2ex} \end{itemize} {\bf Proof:} Replacing $\Phi(R)$ by $\Phi(R)+d$ ($d\in L(E)$) at each occurence changes all maps $(\varphi\cup\psi)_i$ by the summand $\psi(d)\cdot\varphi(\bullet)$. However, this additional linear map comes from $L(E)^\ast$, hence it is trivial on $\mbox{Ker}\left(\oplus_iL(E_i^{R+S})\rightarrow L(E_0^{R+S})\subseteq L(E)\right)$.\\ \par Let us look at the terms $\Phi(\bar{q}-R)$ in $t(q)$ now: Unless $\bar{q}\geq R$, the factor $\varphi(q)$ vanishes (cf. Remark \zitat{3}{4}). On the other hand, the expression $t(q)$ is never used for those relations $q$ satisfying $\bar{q}\geq R+S$ (cf. conditions for the $q^k$'s). Hence, we may assume that \[ (\bar{q}-R)\geq 0\; \mbox{ and, moreover, } (\bar{q}-R)\in \bigcup_i K_i^S\,. \] Now, each two liftings of $(\bar{q}-R)$ to $I\!\!N^{w+1}$ differ by an element of $\mbox{Ker}\,\psi$ only (apply the method of Remark \zitat{3}{4} again), in particular, they cause the same result for $t(q)$. \hfill$\Box$\\ \par {\bf Remark 2:} In the special case of $R\geq S\kgeq0$ we can choose liftings $\Phi(R)\geq \Phi(S) \geq 0$ in $I\!\!N^{w+1}$. Then, there exists an easier description for $t(q)$: \begin{itemize} \item[(i)] Unless $\bar{q}\geq R$, we have $t(q)=0$. \item[(ii)] In case of $\bar{q}\geq R$ we may assume that $q^+\geq\Phi(R)$ is true in $I\!\!N^{w+1}$. (The general $q$'s are differences of those ones.) Then, $t$ can be computed as the product $t(q)=\varphi(q)\,\psi(q)$. \vspace{2ex} \end{itemize} \par {\bf Proof:} (i) As used many times, the property $\bar{q}\in\bigcup_iE_i^R$ implies $\varphi(q)=0$. Now, we can distinguish between two cases:\\ {\em Case 1: $\bar{q}\in\bigcup_iE_i^S$.} We obtain $\psi(q)=0$, in particular, both summands of $t(q)$ vanish.\\ {\em Case 2: $\bar{q}\geq S$.} Then, $\bar{q}-S,\,S\in \sigma^{\scriptscriptstyle\vee}\cap M$, and $\Phi$ lifts these elements to $I\!\!N^{w+1}$. Now, the condition $\bar{q}\in\bigcup_iE_i^R$ implies that $\varphi\left( \Phi(\bar{q}-S)+\Phi(S)-q^+\right)=0$.\\ \par (ii) We can choose $\Phi(\bar{q}-R):=q^+-\Phi(R)$ and $\Phi(\bar{q}-S):=q^+-\Phi(S)$. Then, the claim follows straight forward. \hfill$\Box$\\ \par \sect{Proof of the $T^2$-formula}\label{s4} \neu{41} We will use the sheaf $\Omega^1_Y=\Omega^1_{A|\,I\!\!\!\!C}$ of K\"ahler differentials for computing the modules $T^i_Y$. The maps \[ \alpha_i: \mbox{Ext}^i_A\left( \;^{\displaystyle\Omega_Y^1}\!\!\left/\!_{\displaystyle \mbox{tors}\,(\Omega_Y^1)}\right. , \, A \right) \hookrightarrow \mbox{Ext}^i_A\left( \Omega^1_Y,\,A\right) \cong T^i_Y\quad (i=1,2) \] are injective. Moreover, they are isomorphisms for \begin{itemize} \item $i=1$, since $Y$ is normal, and for \item $i=2$, if $Y$ is smooth in codimension 2. \vspace{2ex} \end{itemize} \par \neu{42} As in \cite{T1}, we build a special $A$-free resolution (one step further now) \[ {\cal E}\stackrel{d_E}{\longrightarrow}{\cal D}\stackrel{d_D}{\longrightarrow} {\cal C}\stackrel{d_C}{\longrightarrow}{\cal B} \stackrel{d_B}{\longrightarrow} \;^{\displaystyle\Omega_Y^1}\!\!\left/\!_{\displaystyle \mbox{tors}\,(\Omega_Y^1)}\right. \rightarrow 0\,. \vspace{2ex} \] With $L^2(E):=L(L(E))$, $L^3(E):=L(L^2(E))$, and \[ \mbox{supp}^2\xi:= \bigcup_{q\in supp\,\xi} \mbox{supp}\,q\quad (\xi\in L^2(E)),\quad \mbox{supp}^3\omega:= \bigcup_{\xi\in supp\,\omega}\mbox{supp}^2\xi\quad (\omega\in L^3(E)), \] the $A$-modules involved in this resolution are defined as follows: \[ \begin{array}{rcl} {\cal B}&:=&\oplus_{r\in E} \,A\cdot B(r),\qquad {\cal C}\,:=\,\oplus_{\!\!\!\!\!\begin{array}[b]{c}\scriptstyle q\in L(E) \vspace{-1ex}\\ \scriptstyle\ell\geq supp\, q\end{array}} \!\!\!A\cdot C(q;\ell),\\ {\cal D}&:=&\left( \oplus_{\!\!\!\!\!\!\!\begin{array}{c}\scriptstyle q\in L(E)\vspace{-1ex}\\ \scriptstyle\eta\geq\ell\geq supp\, q\end{array}} \!\!\!\!A\cdot D(q;\ell,\eta) \right) \oplus \left( \oplus_{\!\!\!\!\begin{array}{c}\scriptstyle\xi\in L^2(E)\vspace{-1ex}\\ \scriptstyle\eta\geq supp^2 \xi\end{array}} \!\!\!A\cdot D(\xi;\eta) \right),\;\mbox{ and}\\ {\cal E}&:=& \begin{array}[t]{r} \left( \oplus_{\!\!\!\!\!\!\!\!\begin{array}{c}\scriptstyle q\in L(E)\vspace{-1ex}\\ \scriptstyle\mu\geq\eta\geq\ell\geq supp\, q\end{array}} \!\!\!\!\!A\cdot E(q;\ell,\eta,\mu) \right) \oplus \left( \oplus_{\!\!\!\!\!\!\!\begin{array}{c}\scriptstyle\xi\in L^2(E)\vspace{-1ex}\\ \scriptstyle\mu\geq\eta\geq supp^2 \xi\end{array}} \!\!\!A\cdot E(\xi;\eta,\mu) \right) \oplus \qquad \\ \oplus \left( \oplus_{\!\!\!\!\begin{array}{c}\scriptstyle\omega\in L^3(E)\vspace{-1ex}\\ \scriptstyle\omega\geq supp^3 \omega\end{array}} \!\!\!\! A\cdot E(\omega;\mu)\right) \end{array} \end{array} \] ($B,C,D,$ and $E$ are just symbols). The differentials equal \[ \begin{array}{cccl} d_B: &B(r)&\mapsto &d\,x^r\vspace{1ex}\\ d_C: &C(q;\ell)&\mapsto &\sum_{r\in E} q_r\,x^{\ell-r}\cdot B(r)\vspace{1ex}\\ d_D: &D(q;\ell,\eta)&\mapsto &C(q;\eta) - x^{\eta-\ell}\cdot C(q,\ell)\\ d_D: &D(\xi;\eta)&\mapsto& \sum_{q\in L(E)}\xi_q\cdot C(q,\eta)\vspace{1ex}\\ d_E: &E(q;\ell,\eta,\mu)&\mapsto& D(q;\eta,\mu)-D(q;\ell,\mu)+ x^{\mu-\eta}\cdot D(q;\ell,\eta) \\ d_E: &E(\xi;\eta,\mu)&\mapsto &D(\xi;\mu) - x^{\mu-\eta}\cdot D(\xi;\eta) - \sum_{q\in L(E)} \xi_q\cdot D(q;\eta,\mu)\\ d_E: &E(\omega;\mu)&\mapsto &\sum_{\xi\in L^2(E)} \omega_{\xi}\cdot D(\xi;\mu)\, . \vspace{2ex} \end{array} \] \par Looking at these maps, we see that the complex is $M$-graded: The degree of each of the elements $B$, $C$, $D$, or $E$ can be obtained by taking the last of its parameters ($r$, $\ell$, $\eta$, or $\mu$, respectively).\\ \par {\bf Remark:} If one prefered a resolution with free $A$-modules of finite rank (as it was used in \cite{T1}), the following replacements would be necessary: \begin{itemize} \item[(i)] Define succesively $F\subseteq L(E)$, $G\subseteq L(F) \subseteq L^2(E)$, and $H\subseteq L(G)\subseteq L^2(F) \subseteq L^3(E)$ as the finite sets of normalized, minimal relations between elements of $E$, $F$, or $G$, respectively. Then, use them instead of $L^i(E)$ ($i=1,2,3$). \item[(ii)] Let $\ell$, $\eta$, and $\mu$ run through finite generating (under $(\sigma^{\scriptscriptstyle\vee}\cap M)$-action) systems of all possible elements meeting the desired inequalities. \end{itemize} The disadvantages of those treatment are a more comlplicated description of the resolution, on the one hand, and difficulties to obtain the commutative diagram \zitat{4}{3}, on the other hand.\\ \par \neu{43} Combining the two exact sequences \[ ^{\displaystyle {\cal R}}\!/\!_{\displaystyle I\,{\cal R}} \longrightarrow A^m \longrightarrow ^{\displaystyle I}\!\!/\!_{\displaystyle I^2}\rightarrow 0\quad \mbox{and}\quad ^{\displaystyle I}\!\!/\!_{\displaystyle I^2}\longrightarrow \Omega^1_{\,I\!\!\!\!C^{w+1}}\otimes A \longrightarrow \Omega_Y^1 \rightarrow 0\,, \] we get a complex (not exact at the place of $A^m$) involving $\Omega_Y^1$. We will compare in the following commutative diagram this complex with the previous resolution of $^{\displaystyle\Omega_Y^1}\!\!\left/\!_{\displaystyle \mbox{tors}\,(\Omega_Y^1)}\right.$: \vspace{-5ex}\\ \[ \dgARROWLENGTH=0.8em \begin{diagram} \node[5]{^{\displaystyle I}\!\!/\!_{\displaystyle I^2}} \arrow{se,t}{d}\\ \node[2]{^{\displaystyle {\cal R}}\!/\!_{\displaystyle I\,{\cal R}}} \arrow[2]{e} \arrow{se,t}{p_D} \node[2]{A^m} \arrow{ne} \arrow[2]{e} \arrow[2]{s,l}{p_C} \node[2]{\Omega_{\,I\!\!\!\!C^{w+1}}\!\otimes \!A} \arrow{e} \arrow[2]{s,lr}{p_B}{\sim} \node{\Omega_Y} \arrow[2]{s} \arrow{e} \node{0}\\ \node[3]{\mbox{Im}\,d_D} \arrow{se}\\ \node{{\cal E}} \arrow{e,t}{d_E} \node{{\cal D}} \arrow[2]{e,t}{d_D} \arrow{ne} \node[2]{{\cal C}} \arrow[2]{e,t}{d_C} \node[2]{{\cal B}} \arrow{e} \node{^{\displaystyle\Omega_Y^1}\!\!\!\left/\!\!_{\displaystyle \mbox{tors}\,(\Omega_Y^1)}\right.} \arrow{e} \node{0} \end{diagram} \] \par Let us explain the three labeled vertical maps: \begin{itemize} \item[(B)] $p_B: dz_r \mapsto B(r)$ is an isomorphism between two free $A$-modules of rank $w+1$. \item[(C)] $p_C: e^{ab} \mapsto C(a-b;\pi(a))$. In particular, the image of this map is spanned by those $C(q,\ell)$ meeting $\ell\geq \bar{q}$ (which is stronger than just $\ell\geq\mbox{supp}\,q$). \item[(D)] Finally, $p_D$ arises as pull back of $p_C$ to $^{\displaystyle {\cal R}}\!/\!_{\displaystyle I\,{\cal R}}$. It can be described by $r(a,b;c)\mapsto D(a-b; \pi(a),\pi(a+c))$ and $s(a,b,c)\mapsto D(\xi;\pi(a))$ ($\xi$ denotes the relation $\xi=[(b-c)-(a-c)+(a-b)=0]$). \vspace{2ex} \end{itemize} \par {\bf Remark:} Starting with the typical ${\cal R}_0$-element mentioned in \zitat{3}{5}(iii), the previous description of the map $p_D$ yields 0 (even in ${\cal D}$).\\ \par \neu{44} By \zitat{4}{1} we get the $A$-modules $T^i_Y$ by computing the cohomology of the complex dual to those of \zitat{4}{2}.\\ \par As in \cite{T1}, denote by $G$ one of the capital letters $B$, $C$, $D$, or $E$. Then, an element $\psi$ of the dual free module $(\bigoplus\limits_G \,I\!\!\!\!C[\check{\sigma}\cap M]\cdot G)^\ast$ can be described by giving elements $g(x)\in\,I\!\!\!\!C[\check{\sigma}\cap M]$ to be the images of the generators $G$ ($g$ stands for $b$, $c$, $d$, or $e$, respectively).\\ \par For $\psi$ to be homogeneous of degree $-R\in M$, $g(x)$ has to be a monomial of degree \[ \deg g(x)=-R+\deg G. \] In particular, the corresponding complex coefficient $g\in \,I\!\!\!\!C$ (i.e. $g(x)=g\cdot x^{-R+\deg G}$) admits the property that \[ g\neq 0\quad\mbox{implies}\quad -R+\deg G\ge 0\quad (\mbox{i.e.}\; -R+\deg G\in\check{\sigma}). \vspace{2ex} \] \par {\bf Remark:} Using these notations, Theorem \zitat{3}{3}(1) was proved in \cite{T1} by showing that \begin{eqnarray*} \left(\left. ^{\displaystyle L(E_0^R)}\!\!\right/ \!_{\displaystyle \sum_i L(E_i^R)} \right)^\ast \otimes_{Z\!\!\!Z} \,I\!\!\!\!C &\longrightarrow& ^{\displaystyle \mbox{Ker}({\cal C}^\ast_{-R}\rightarrow {\cal D}^\ast_{-R})}\!\!\left/ \!_{\displaystyle \mbox{Im}({\cal B}^\ast_{-R}\rightarrow{\cal C}^\ast_{-R})}\right.\\ \varphi &\mapsto& [\dots,\, c(q;\ell):=\varphi(q),\dots] \end{eqnarray*} is an isomorphism of vector spaces.\\ Moreover, looking at the diagram of \zitat{4}{3}, $e^{ab}\in A^m$ maps to both $\underline{z}^a-\underline{z}^b\in ^{\displaystyle I}\!\!/\! _{\displaystyle I^2}$ and $C(a-b;\pi(a))\in {\cal C}$. In particular, we can verify Theorem \zitat{3}{4}: Each $\varphi:L(E)_{\,I\!\!\!\!C} \rightarrow\,I\!\!\!\!C$, on the one hand, and its associated $A$-linear map \begin{eqnarray*} ^{\displaystyle I}\!\!/\!_{\displaystyle I^2} &\longrightarrow& A\\ \underline{z}^a-\underline{z}^b & \mapsto & \varphi(a-b)\cdot x^{\pi(a)-R}, \end{eqnarray*} on the other hand, induce the same element of $T^1_Y(-R)$. \\ \par \neu{45} For computing $T_Y^2(-R)$, the interesting part of the dualized complex $\zitat{4}{2}^\ast$ in degree $-R$ equals the complex of $\,I\!\!\!\!C$-vector spaces \[ {\cal C}^{\ast}_{-R} \stackrel{d_D^{\ast}}{\longrightarrow} {\cal D}^{\ast}_{-R} \stackrel{d_E^{\ast}}{\longrightarrow} {\cal E}^{\ast}_{-R} \] with coordinates $\underline{c}$, $\underline{d}$, and $\underline{e}$, respectively: \begin{eqnarray*} {\cal C}^{\ast}_{-R} &=& \{\underline{c(q;\ell)}\, |\; c(q;\ell)=0 \;\mbox{for}\;\ell-R\notin\check{\sigma}\}\\ {\cal D}^{\ast}_{-R} &=& \{[\underline{d(q;\ell,\eta)},\underline{d(\xi;\eta)}]\;|\; \begin{array}[t]{ccccl} d(q;\ell,\eta)&=&0& \mbox{for} &\eta-R\notin\check{\sigma}\mbox{, and}\\ d(\xi;\eta)&=&0& \mbox{for} &\eta-R\notin\check{\sigma} \} \end{array}\\ {\cal E}^{\ast}_{-R} &=& \{ [\underline{e(q;\ell,\eta,\mu)}, \underline{e(\xi;\eta,\mu)}, \underline{e(\omega ;\mu)}]\,|\; \mbox{each coordinate vanishes for } \mu - R \notin \check{\sigma} \}. \vspace{1ex} \end{eqnarray*} \par The differentials $d_D^{\ast}$ and $d_E^{\ast}$ can be described by \[ \begin{array}{lcll} d(q;\ell,\eta)&=&c(q;\eta)-c(q;\ell),&\\ d(\xi;\eta)&=&\sum\limits_{q\in F}\xi_q\cdot c(q;\eta),& \mbox{and}\\ e(q;\ell,\eta,\mu) &=& d(q;\eta,\mu) - d(q;\ell,\mu) + d(q;\ell,\eta),\\ e(\xi;\eta,\mu) &=& d(\xi;\mu) - d(\xi;\eta) - \sum_{q\in F} \xi_q\cdot d(q;\eta,\mu),\\ e(\omega ;\mu) &=& \sum\limits_{\xi\in G} \omega_{\xi}\cdot d(\xi;\mu). \end{array} \vspace{1ex} \] \par Denote $V:= \mbox{Ker}\,d^{\ast}_E \subseteq {\cal D}_{-R}^{\ast}\,$ and $\,W:= \mbox{Im}\,d_D^{\ast}\subseteq V$, i.e. \begin{eqnarray*} V&=& \{ [\underline{d(q;\ell,\eta)};\,\underline{d(\xi;\eta)}]\,|\; \begin{array}[t]{l} q\in L(E), \;\eta\geq\ell\geq\mbox{supp}\,q\mbox{ in }M;\\ \xi\in L^2(E), \;\eta\geq\mbox{supp}^2\xi; \vspace{0.5ex}\\ d(q;\ell,\eta) = d(\xi;\eta) = 0 \mbox{ for } \eta -R \notin \check{\sigma},\\ d(q;\ell,\mu) = d(q;\ell,\eta) + d(q;\eta,\mu) \; (\mu\geq\eta\geq\ell\geq \mbox{supp}\, q),\\ d(\xi;\mu)= d(\xi;\eta) + \sum_q \xi_q \cdot d(q;\eta,\mu)\; (\mu\geq\eta\geq \mbox{supp}^2 \xi),\\ \sum_{\xi\in G}\omega_{\xi} \,d(\xi;\mu) =0 \mbox{ for }\omega \in L^3(E) \mbox{ with } \mu \geq \mbox{supp}^3\omega\,\}, \end{array}\\ W&=& \{ [\underline{d(q;\ell,\eta)};\,\underline{d(\xi;\eta)}]\,|\; \mbox{there are $c(q;\ell)$'s with} \begin{array}[t]{l} c(q,\ell)=0 \mbox{ for } \ell-R\notin\check{\sigma},\\ d(q;\ell,\eta) = c(q;\eta)-c(q;\ell),\\ d(\xi;\eta)= \sum_q\xi_q\cdot c(q;\eta)\,\}. \end{array} \end{eqnarray*} By construction, we obtain \[ V\!\left/\!_{\displaystyle W}\right. = \mbox{Ext}^i_A\left( \;^{\displaystyle\Omega_Y^1}\!\!\left/\!_{\displaystyle \mbox{tors}\,(\Omega_Y^1)}\right. , \, A \right)(-R) \subseteq T^2_Y(-R) \] (which is an isomorphism, if $Y$ is smooth in codimension 2).\\ \par \neu{46} Let us define the much easier vector spaces \begin{eqnarray*} V_1&:=& \{[\underline{x_i(q)}_{(q\in L(E_i^R))}]\,|\; \begin{array}[t]{l} x_i(q)=x_j(q) \mbox{ for } \begin{array}[t]{l} \bullet\, \langle a^i, a^j \rangle < \sigma \mbox{ is a 2-face and}\\ \bullet\, q\in L(E_i^R\cap E_j^R)\,, \end{array}\\ \xi\in L^2(E_i^R) \mbox{ implies } \sum_q\xi_q\cdot x_i(q)=0 \,\}\;\mbox{ and} \end{array} \vspace{1ex} \\ W_1&:=& \{[\underline{x(q)}_{(q\in \cup_i L(E_i^R))}] \,|\; \begin{array}[t]{l} \xi\in L(\bigcup_i L(E_i^R)) \mbox{ implies } \sum_q\xi_q\cdot x(q)=0 \,\}. \vspace{2ex} \end{array} \end{eqnarray*} \par {\bf Lemma:} {\em The linear map $V_1\rightarrow V$ defined by \begin{eqnarray*} d(q;\ell,\eta) &:=& \left\{ \begin{array}{lll} x_i(q) &\mbox{ for }& \ell\in K_i^R,\;\; \eta \geq R\\ 0 &\mbox{ for }& \ell \geq R \;\mbox{ or } \;\eta \in \bigcup_i K_i^R\,; \end{array} \right. \\ d(\xi;\eta) &:=& 0\, \end{eqnarray*} induces an injective map \[ V_1\!\left/\!_{\displaystyle W_1}\right. \hookrightarrow V\!\left/\!_{\displaystyle W}\right.\,. \] If $Y$ is smooth in codimension 2, it will be an isomorphism. } \\ \par {\bf Proof:} 1) The map $V_1 \rightarrow V$ is {\em correct defined}: On the one hand, an argument as used in \zitat{3}{5}(i) shows that $\ell\in K_i^R\cap K_j^R$ would imply $x_i(q)=x_j(q)$. On the other hand, the image of $[x_i(q)]_{q\in L(E_i^R)}$ meets all conditions in the definition of $V$. \vspace{1ex} \\ 2) $W_1$ maps to $W$ (take $c(q,\ell):=x(q)$ for $\ell\geq R$ and $c(q,\ell):=0$ otherwise). \vspace{1ex} \\ 3) The map between the two factor spaces is {\em injective}: Assume for $[x_i(q)]_{q\in L(E_i^R)}$ that there exist elements $c(q,\ell)$, such that \begin{eqnarray*} c(q;\ell) &=& 0 \; \mbox{ for } \ell \in \bigcup_i K^R_i\, ,\\ x_i(q) &=& c(q;\eta) - c(q;\ell) \; \mbox{ for } \eta \geq \ell,\, \ell\in K_i^R,\, \eta\geq R\,,\\ 0 &=& c(q;\eta) - c(q;\ell)\; \mbox{ for } \eta\geq \ell \mbox{ and } [\ell\geq R\mbox{ or } \eta\in \bigcup_iK_i^R]\, , \mbox{ and}\\ 0 &=& \sum_q \xi_q \cdot c(q;\eta) \; \mbox{ for } \eta \geq \mbox{supp}^2\xi\, . \end{eqnarray*} In particular, $x_i(q)$ do not depend on $i$, and these elements meet the property \[ \sum_q \xi_q \cdot x_{\bullet}(q) = 0 \; \mbox{ for } \xi\in L(\bigcup_i L(E_i^R)). \] 4) If $Y$ is smooth in codimension 2, the map is {\em surjective} :\\ Given an element $[d(q;\ell,\eta),\,d(\xi;\eta)]\in V$, there exist complex numbers $c(q;\eta)$ such that: \begin{itemize} \item[(i)] $d(\xi;\eta) = \sum_q\xi_q\cdot c(q;\eta)\,$ , \item[(ii)] $c(q;\eta)=0 \mbox{ for } \eta\notin R+\sigma^{\scriptscriptstyle\vee}\, (\mbox{i.e. }\eta\in \bigcup_iK_i^R)\,$. \end{itemize} (Do this separately for each $\eta$ and distinguish between the cases $\eta\in R +\sigma^{\scriptscriptstyle\vee}$ and $\eta\notin R+\sigma^{\scriptscriptstyle\vee}$.)\\ In particular, $[c(q;\eta) - c(q;\ell),\, d(\xi;\eta)]\in W$. Hence, we have seen that we may assume $d(\xi;\eta)=0$.\\ \par Let us choose some sufficiently high degree $\ell^\ast\geq E$. Then, \[ x_i(q):= d(q;\ell,\eta) - d(q;\ell^\ast\!,\eta) \] (with $\ell\in K_i^R$, $\ell\geq \mbox{supp}\,q$ (cf.\ Lemma \zitat{3}{2}(2)), and $\eta\geq\ell,\ell^\ast\!,R$) defines some preimage: \begin{itemize} \item[(i)] It is independent from the choice of $\eta$: Using a different $\eta'$ generates the difference $d(q;\eta,\eta')-d(q;\eta,\eta')$. \item[(ii)] It is independent from $\ell\in K_i^R$: Choosing another $\ell'\in K_i^R$ with $\ell'\geq\ell$ would add the summand $d(q;\ell,\ell')$, which is 0; for the general case use Lemma \zitat{3}{2}(2). \item[(iii)] If $\langle a^i,a^j\rangle < \sigma$ is a 2-face with $\mbox{supp}\,q \subseteq L(E^R_i)\cap L(E_j^R)$, then by Lemma \zitat{3}{2}(2) we can choose an $\ell\in K_i^R\cap K_j^R$ achieving $x_i(q)=x_j(q)$. \item[(iv)] For $\xi\in L^2(E_i^R)$ we have \[ \sum_q \xi_q\cdot d(q;\ell,\eta) = \sum_q \xi_q\cdot d(q;\ell^\ast\!,\eta) = 0\,, \] and this gives the corresponding relation for the $x_i(q)'$s. \item[(v)] Finally, if we apply to $[\underline{x_i(q)}]\in V_1$ the linear map $V_1\rightarrow V$, the result differs from $[d(q;\ell,\eta),0]\in V$ by the $W$-element built from \[ c(q;\ell) := \left\{ \begin{array}{ll} d(q;\ell,\eta) - d(q;\ell^\ast\!,\eta) & \mbox{ if } \ell\geq R \\ 0 & \mbox{ otherwise }. \end{array} \right. \vspace{-2ex} \] \end{itemize} \hfill$\Box$\\ \par \neu{47} Now, it is easy to complete the proofs for Theorem \zitat{3}{3} (part 2 and 3) and Theorem \zitat{3}{5}:\\ \par First, for a tuple $[\underline{x_i(q)}]_{q\in L(E_i^R)}$, the condition \[ \xi\in L^2(E_i^R) \mbox{ implies } \sum_q\xi_q\cdot x_i(q)=0 \] is equivalent to the fact the components $x_i(q)$ are induced by elements $x_i\in L(E_i^R)_{\,I\!\!\!\!C}^\ast$.\\ The other condition for elements of $V_1$ just says that for 2-faces $\langle a^i,a^j\rangle<\sigma$ there is $x_i=x_j$ on $L(E_i^R\cap E_j^R)_{\,I\!\!\!\!C}=L(E_i^R)_{\,I\!\!\!\!C}\cap L(E_j^R)_{\,I\!\!\!\!C}$. In particular, we obtain \[ V_1= \mbox{Ker}\left( \oplus_i L(E_i^R)_{\,I\!\!\!\!C}^\ast \rightarrow \oplus_{\langle a^i,a^j\rangle <\sigma} L(E_i^R\cap E_j^R)_{\,I\!\!\!\!C}^\ast \right)\,. \] In the same way we get \[ W_1 = \left( \sum_i L(E^R_i)_{\,I\!\!\!\!C}\right)^\ast\,, \] and our $T^2$-formula is proven.\\ \par Finally, if $\psi_i:L(E_i^R)_{\,I\!\!\!\!C}\rightarrow \,I\!\!\!\!C$ are linear maps defining an element of $V_1$, they induce the following $A$-linear map on ${\cal D}$ (even on $\mbox{Im}\,d_D$): \begin{eqnarray*} D(q;\ell,\eta) &\mapsto& \left\{ \begin{array}{lll} \psi_i(q)\cdot x^{\eta-R} &\mbox{ for }& \ell\in K_i^R,\;\; \eta \geq R\\ 0 &\mbox{ for }& \ell \geq R \;\mbox{ or } \;\eta \in \bigcup_i K_i^R \end{array} \right.\\ D(\xi;\eta &\mapsto& 0\,. \end{eqnarray*} Now, looking at the diagram of \zitat{4}{3}, this translates exactly into the claim of Theorem \zitat{3}{5}.\\ \par \sect{Proof of the cup product formula}\label{s5} \neu{51} Fix an $R\in M$, and let $\varphi\in L(E)^\ast_{\,I\!\!\!\!C}$ induce some element (also denoted by $\varphi$) of $T^1_Y(-R)$. Using the notations of \zitat{2}{3}, \zitat{3}{4}, and \zitat{3}{6} we can take \[ \tilde{\varphi}(f_{\alpha\beta}):= \varphi(\alpha-\beta)\cdot \underline{z}^{\Phi(\pi(\alpha)-R)} \] for the auxiliary $P$-elements needed to compute the $\lambda(\varphi)$'s (cf. Theorem \zitat{3}{4}).\\ \par Now, we have to distinguish between the two several types of relations generating the $P$-module ${\cal R}\subseteq P^m$: \begin{itemize} \item[(r)] Regarding the relation $r(a,b;c)$ we obtain \begin{eqnarray*} \sum_{(\alpha,\beta)\in m} r(a,b;c)_{\alpha\beta}\cdot \tilde{\varphi}(f_{\alpha\beta}) &=& \tilde{\varphi}(f_{a+c,b+c}) - \underline{z}^c\,\tilde{\varphi}(f_{ab}) \\ &=& \varphi(a-b)\cdot \left( \underline{z}^{\Phi(\pi(a+c)-R)} - \underline{z}^{c+\Phi(\pi(a)-R)} \right) \\ &=& \varphi(a-b)\cdot f_{\Phi(\pi(a+c)-R),\,c+\Phi(\pi(a)-R)}\,. \end{eqnarray*} In particular, \[ \lambda_{\alpha\beta}^{r(a,b;c)}(\varphi) = \left\{\begin{array}{ll} \varphi(a-b) & \mbox{ for } [\alpha,\beta] = [c+\Phi(\pi(a)-R),\, \Phi(\pi(a+c)-R)]\\ 0 & \mbox{ otherwise}\,. \end{array} \right. \] \item[(s)] The corresponding result for the relation $s(a,b,c)$ is much nicer: \begin{eqnarray*} \sum_{(\alpha,\beta)\in m} s(a,b,c)_{\alpha\beta}\cdot \tilde{\varphi}(f_{\alpha\beta}) &=& \tilde{\varphi}(f_{bc})- \tilde{\varphi}(f_{ac})+ \tilde{\varphi}(f_{ab})\\ &=& [\varphi(b-c)-\varphi(a-c)+\varphi(a-b)]\cdot \underline{z}^{\Phi(\pi(a)-R)}\\ &=& 0\,. \end{eqnarray*} In particular, $\lambda^{s(a,b,c)}(\varphi)=0$. \vspace{2ex} \end{itemize} \par \neu{52} Now, let $R,S,\varphi$, and $\psi$ as in the assumption of Theorem \zitat{3}{6}. Using formula \zitat{2}{3}(iii), our previous computations yield $(\varphi\cup\psi)(s(a,b,c))=0$ and \[ \begin{array}{l} (\varphi\cup\psi)(r(a,b;c))= \sum_{\alpha,\beta}\lambda^{r(a,b;c)}_{\alpha\beta}(\varphi) \cdot \psi(f_{\alpha\beta}) + \sum_{\alpha,\beta} \varphi(f_{\alpha\beta})\cdot \lambda^{r(a,b;c)}_{\alpha\beta}(\psi) \vspace{2ex}\\ \qquad= \begin{array}[t]{r} \varphi(a-b)\cdot \psi\left( c^{}+\Phi(\pi(a)-R)-\Phi(\pi(a+c)-R)\right)\cdot x^{\pi(c+\Phi(\pi(a)-R))-S} +\qquad\\ +\psi(a-b)\cdot \varphi\left( c+\Phi(\pi(a)-S)-\Phi(\pi(a+c)-S)\right)\cdot x^{\pi(c+\Phi(\pi(a)-S))-R} \end{array} \vspace{2ex}\\ \qquad= \begin{array}[t]{r} \left[ \varphi(a-b)\cdot \psi\left( c+\Phi(\pi(a)-R)-\Phi(\pi(a+c)-R)\right) +\right. \qquad\qquad\qquad\qquad\qquad\\ \left. + \psi(a-b)\cdot \varphi\left( c+\Phi(\pi(a)-S)-\Phi(\pi(a+c)-S)\right) \right] \cdot x^{\pi(a+c)-R-S}\,. \vspace{1ex} \end{array} \end{array} \] \par {\bf Remark:} Unless $\pi(a+c)\geq R+S$, both summand in the brackets will vanish. For instance, on the one hand, $\pi(a)\in\bigcup_iK_i^R$ would cause $\varphi(a-b)=0$, and, on the other hand, $\pi(a)-R\geq 0$ and $\pi(c+\Phi(\pi(a)-R))\in \bigcup_iK_i^S$ imply $\psi(c+\Phi(\pi(a)-R)-\Phi(\pi(a+c)-R))=0$.\\ \par To apply Theorem \zitat{3}{5} we would like to remove the argument $c$ from the big coefficient. This will be done by adding a suitable coboundary $T$.\\ \par \neu{53} Let us start with defining for $(\alpha,\beta)\in m$ \[ t(\alpha,\beta):= \begin{array}[t]{r} \varphi(\alpha-\beta)\cdot \psi \left( \Phi(\pi(\alpha)-R)+\Phi(R)-\beta\right)+\qquad\qquad\qquad\\ + \psi(\alpha-\beta) \cdot \varphi \left( \Phi(\pi(\alpha)-S)+\Phi(S)-\alpha\right)\,. \end{array} \] (This expression is related to $t_{\varphi,\psi,R,S}$ from \zitat{3}{6} by $t(q)=t(q^+,q^-)$.) \\ \par {\bf Lemma:}{\em Let $\alpha,\beta,\gamma\inI\!\!N^E$ with $\pi(\alpha)=\pi(\beta)=\pi(\gamma)$. \begin{itemize} \item[(1)] $t(\alpha,\beta)=t(\alpha-\beta)$ as long as $\pi(\alpha)\in\bigcup_i K_i^{R+S}$. \item[(2)] $t(\beta,\gamma)-t(\alpha,\gamma)+t(\alpha,\beta)=0$. \vspace{2ex} \end{itemize} } \par {\bf Proof:} (1) It is enough to show that $t(\alpha+r,\beta+r)=t(\alpha,\beta)$ for $r\in I\!\!N^E$, $\pi(\alpha+r)\in\bigcup_iK_i^{R+S}$. But the difference of these two terms has exactly the shape of the coefficient of $x^{\pi(a+c)-R-S}$ in \zitat{5}{2}. In particular, the argument given in the previous remark applies again.\\ \par (2) By extending $\varphi$ and $\psi$ to linear maps $\,I\!\!\!\!C^E\rightarrow\,I\!\!\!\!C$, we obtain \[ t(\alpha,\beta) = \begin{array}[t]{r} [\varphi(\alpha-\beta)\,\psi\left( \Phi(\pi(\alpha)-R) + \Phi(R)\right) + \psi(\alpha-\beta) \, \varphi\left( \Phi(\pi(\alpha)-S)+\Phi(S)\right)]+\,\\ +[\varphi(\beta)\,\psi(\beta)-\varphi(\alpha)\,\psi(\alpha)]. \end{array} \] Now, since $\pi(\alpha)=\pi(\beta)=\pi(\gamma)$, both types of summands add up to 0 separately in $t(\beta,\gamma)-t(\alpha,\gamma)+t(\alpha,\beta)$. \hfill$\Box$\\ \par {\bf Remark:} The previous lemma does not imply that $t(q)$ is $Z\!\!\!Z$-linear in $q$. The assumption for $\pi(\alpha)$ made in (1) is really essential.\\ \par Now, we obtain a $P$-linear map $T\in \mbox{Hom}(P^m,A)$ by \[ T: e^{\alpha\beta}\mapsto \left\{ \begin{array}{ll} t(\alpha,\beta)\,x^{\pi(\alpha)-R-S} & \mbox{ for } \pi(\alpha)\geq R+S\\ 0 & \mbox{ otherwise}\,. \end{array} \right. \] Pulling back $T$ to ${\cal R}\subseteq P^m$ yields (in case of $\pi(a+c)\geq R+S$) \begin{eqnarray*} T(r(a,b;c)) &=& \left\{ \begin{array}{ll} [t(a+c,b+c)-t(a,b)]\cdot x^{\pi(a+c)-R-S} & \mbox{ for } \pi(a)\geq R+S\\ t(a+c,b+c)\cdot x^{\pi(a+c)-R-S} & \mbox{ otherwise} \end{array} \right.\\ &=& \left\{ \begin{array}{ll} -(\varphi\cup\psi)(r(a,b;c)) & \mbox{ for } \pi(a)\geq R+S\\ t(a,b)\,x^{\pi(a+c)-R-S} -(\varphi\cup\psi)(r(a,b;c)) & \mbox{ otherwise}\, \end{array} \right. \end{eqnarray*} and $T(s(a,b,c))=0$ (by (2) of the previous lemma).\\ \par On the other hand, $T$ yields a trivial element of $T^2_Y(-R-S)$, i.e. inside this group we may replace $\varphi\cup\psi$ by $(\varphi\cup\psi)+T$ to obtain \begin{eqnarray*} (\varphi\cup\psi)(r(a,b;c)) &=& \left\{ \begin{array}{ll} t(a,b)\cdot x^{\pi(a+c)-R-S}&\mbox{ for } \pi(a)\in \bigcup_i K_i^{R+S};\; \pi(a+c)\geq R+S\\ 0 & \mbox{ otherwise}\,, \end{array} \right. \vspace{1ex}\\ (\varphi\cup\psi)(s(a,b,c)) &=& 0\,. \vspace{1ex} \end{eqnarray*} \par Having Theorem \zitat{3}{5} in mind, this formula for $\varphi\cup\psi$ is exactly what we were looking for:\\ Given an $r(a,b;c)$ with $\pi(a)\in K_i^{R+S}$, let us compute $(\varphi\cup\psi)_i(q:=a-b)$ following the recipe of (i), (ii) of Theorem \zitat{3}{6}. We do not need to split $q=a-b$ into a sum $q=\sum_k q^k$ - the element $q$ itself already satisfies the condition \[ \langle a^i,\bar{q}\rangle \leq \langle a^i, \pi(a) \rangle < \langle a^i,R+S\rangle. \] In particular, with $(\varphi\cup\psi)_i(a-b)=t(a-b)=t(a,b)$ we will obtain the right result - if the recipe is assumed to be correct. \\ \par \neu{54} We will fill those remaining gap now, i.e. we will show that \begin{itemize} \item[(a)] each $q\in L(E_i^{R+S})$ admits a decomposition $q=\sum_k q^k$ with the desired properties, \item[(b)] $\sum_k q^k=0$ (with $\bar{q}^k\in K_i^{R+S}$) implies $\sum_k t(q^k)=0$, and \item[(c)] for adjacent $a^i,a^j$ the relations $q\in L(E_i^{R+S}\cap E_j^{R+S})$ admit a decomposition $q=\sum_kq^k$ that works for both $i$ and $j$. \end{itemize} (In particular, this answers the questions arised right after stating the theorem in \zitat{3}{6}.)\\ \par Let us fix an element $i\in \{1,\dots,N\}$. Since $\sigma^{\scriptscriptstyle\vee}\cap M$ contains elements $r$ with $\langle a^i,r\rangle =1$, some of them must be contained in the generating set $E$, too. We choose one of these elements and call it $r(i)$.\\ Now, to each $r\in E$ we associate some relation $p(r)\in L(E)$ by \[ p(r):= e^r - \langle a^i, r \rangle\cdot e^{r(i)} + [\mbox{suitable element of } Z\!\!\!Z^{E\cap (a^i)^\bot}]\,. \] The two essential properties of these special relations are \begin{itemize} \item[(i)] $\langle a^i, \bar{p}(r)\rangle = \langle a^i, r\rangle$, and \item[(ii)] if $q\in L(E)$ is any relation, then $q$ and $\sum_{r\in E}q_r\cdot p(r)$ differ by some element of $L(E\cap (a^i)^\bot)$ only. \vspace{1ex} \end{itemize} \par In particular, this proves (a). For (b) we start with the following\\ \par {\em Claim:} Let $q^k\in L(E)$ be relations such that $\sum_k \langle a^i,\bar{q}^k\rangle < \langle a^i, R+S\rangle$. Then, $\sum_k t(q^k)=t(\sum_k q^k)$.\\ \par {\em Proof:} We can restrict ourselves to the case of two summands, $q^1$ and $q^2$. Then, by Lemma \zitat{5}{3}, \begin{eqnarray*} t(q^1)+t(q^2) &=& t\left((q^1)^+,(q^1)^-\right) + t\left((q^2)^+,(q^2)^-\right)\\ &=& t\left((q^1)^++(q^2)^+,(q^1)^-+(q^2)^+\right) + t\left((q^2)^++(q^1)^-,(q^2)^-+(q^1)^-\right)\\ &=& t\left((q^1)^++(q^2)^+,(q^2)^-+(q^1)^-\right)\\ &=& t(q^1+q^2)\,. \hspace{9cm} \Box \end{eqnarray*} \par In particular, if $\sum_kq^k=0$ (with $\bar{q}^k\in K_i^{R+S}$), then this applies for the special decompositions \[ q^k=\sum_r q^k_r\cdot p(r) + q^{0,k} \quad (q^{0,k}\in L(E\cap(a^i)^\bot)) \] of the summands $q^k$ themselves. We obtain \[ \sum_{q^k_r>0}q^k_r\cdot t\left(p(r)\right) + t(q^{0,k}) = t\left( \sum_{q^k_r>0}q^k_r\,p(r)+q^{0,k}\right) =: t(q^{1,k}) \] and \[ \sum_{q^k_r<0}q^k_r\cdot t\left(p(r)\right)= t\left( \sum_{q^k_r<0}q^k_r\,p(r)\right)=:t(q^{2,k})\,. \] Up to elements of $E\cap (a^i)^\bot$, the relations $q^{1,k}$ and $q^{2,k}$ are connected by the common \[ (q^{1,k})^-=-q^{1,k}_{r(i)}\cdot e^{r(i)}=\langle a^i,\bar{q}^k\rangle \cdot e^{r(i)}=q^{2,k}_{r(i)}\cdot e^{r(i)}=(q^{2,k})^+\,. \] Hence, Lemma \zitat{5}{3} yields \[ \sum_r q^k_r\cdot t\left(p(r)\right) + t(q^{0,k}) = t(q^{1,k}) + t(q^{2,k}) = t\left( q^{1,k}+q^{2,k}\right) = t(q^k)\,, \] and we conclude \begin{eqnarray*} \sum_k t(q^k) &=& \sum_k \left(\sum_r q^k_r\cdot t\left(p(r)\right) + t(q^{0,k})\right)\\ &=& \sum_r \left( \sum_k q^k_r \right) t\left(p(r)\right) + t\left(\sum_k q^{0,k}\right) \quad (\mbox{cf. previous claim})\\ &=& 0+ t\left( \sum_k q^k - \sum_{k,r} q^k_r\,p(r) \right)\\ &=& 0\,. \vspace{2ex} \end{eqnarray*} \par Finally, only (c) is left. Let $a^i$, $a^j$ be two adjacent edges of $\sigma$. We adapt the construction of the elementary relations $p(r)$. Instead of the $r(i)$'s, we will use elements $r(i,j)\in E$ characterized by the property \[ \langle a^i, r(i,j)\rangle = 1\,,\; \langle a^j, r(i,j)\rangle = 0\,. \] (Those elements exist, since $Y$ is assumed to be smooth in codimension 2.)\\ Now, we define \[ p(r):= e^r - \langle a^i,r\rangle \cdot e^{r(i,j)} - \langle a^j,r \rangle \cdot e^{r(j,i)} + [\mbox{suitable element of }Z\!\!\!Z^{E\cap(a^i)^\bot\cap(a^j)^\bot}]\,. \] These special $p(r)$'s meet the usual properties (i) and (ii) - but for the two different indices $i$ and $j$ at the same time. In particular, if $q\in L(E)$ is any relation, then $q$ and $\sum_{r\in E}q_r\cdot p(r)$ differ by some element of $L(E\cap(a^i)^\bot\cap(a^j)^\bot)$ only.\\ \par \sect{An alternative to the complex $L(E^R)_{\bullet}$}\label{s6} \neu{61} Let $R\in M$ be fixed for the whole \S \ref{s6}. The complex $L(E^R)_{\bullet}$ introduced in \zitat{3}{2} fits naturally into the exact sequence \[ 0\rightarrow L(E^R)_{\bullet} \longrightarrow (Z\!\!\!Z^{E^R})_{\bullet} \longrightarrow \mbox{span}(E^R)_{\bullet}\rightarrow 0 \] of complexes built in the same way as $L(E^R)_{\bullet}$, i.e. \[ (Z\!\!\!Z^{E^R})_{-k} := \oplus\!\!\!\!\!\!_{\begin{array}{c} \scriptstyle\tau<\sigma\vspace{-1ex} \\ \scriptstyle dim\, \tau=k \end{array}} \!\!\!\!Z\!\!\!Z^{E^R_{\tau}} \qquad \mbox{and}\qquad \mbox{span}(E^R)_{-k} := \oplus\!\!\!\!\!\!_{\begin{array}{c} \scriptstyle\tau<\sigma\vspace{-1ex} \\ \scriptstyle dim\, \tau=k \end{array}} \!\!\!\!\mbox{span}(E^R_{\tau})\,. \] \par {\bf Lemma:}{\em The complex $(Z\!\!\!Z^{E^R})_{\bullet}$ is exact.\\ } \par {\bf Proof:} The complex $(Z\!\!\!Z^{E^R})_{\bullet}$ can be decomposed into a direct sum \[ (Z\!\!\!Z^{E^R})_{\bullet} = \bigoplus_{r\in M} (Z\!\!\!Z^{E^R})(r)_{\bullet} \] showing the contribution of each $r\in M$. The complexes occuring as summands are defined as \begin{eqnarray*} (Z\!\!\!Z^{E^R})(r)_{-k} &:=& \oplus\!\!\!\!\!\!_{\begin{array}{c} \scriptstyle\tau<\sigma\vspace{-1ex}\\ \scriptstyle dim\, \tau=k \end{array}} \!\!\!\! \left\{ \begin{array}{ll} Z\!\!\!Z=Z\!\!\!Z^{\{r\}} & \mbox{ for } r\in E^R_{\tau}\\ 0 & \mbox{ otherwise} \end{array} \right\}\\ &=& Z\!\!\!Z^{\#\{\tau\,|\; dim\,\tau=k; \, r\in E^R_{\tau}\}}\,. \end{eqnarray*} Denote by $H^+$ the halfspace \[ H^+ := \{ a\in N_{I\!\!R}\,|\; \langle a,r\rangle < \langle a, R\rangle\} \subseteq N_{I\!\!R}. \] Then, for $\tau \neq 0$, the fact that $r\in E^R_{\tau}$ is equivalent to $\tau \setminus \{0\} \subseteq H^+$. On the other hand, $r\in E^R_0$ corresponds to the condition $\sigma \cap H^+ \neq \emptyset$.\\ In particular, $(Z\!\!\!Z^{E^R})(r)_{\bullet}$, shifted by one place, equals the complex for computing the reduced homolgy of the topological space $\cup \{\tau\,|\;\tau \setminus \{0\} \subseteq H^+\} \subseteq \sigma$ cut by some affine hyperplane. Since this space is contractable, the complex is exact. \hfill$\Box$\\ \par {\bf Corollary:}{\em The complexes $L(E^R)_{\bullet}^\ast$ and $\mbox{span}(E^R)_{\bullet}^\ast[1]$ are quasiisomorphic. In particular, under the usual assumptions (cf. Theorem \zitat{3}{3}), we obtain \[ T^i_Y(-R) = H^i\left( \mbox{span}(E^R)_{\bullet}^\ast\otimes _{Z\!\!\!Z}\,I\!\!\!\!C\right)\,. \vspace{2ex} \] } \par \neu{62} We define the $I\!\!R$-vector spaces \begin{eqnarray*} V^R_i &:= &\mbox{span}_{I\!\!R}(E_i^R)=\left\{ \begin{array}{l@{\quad\mbox{for}\;\:}l} 0 &\langle a^i,R\rangle\le 0\\ \left[ a^i=0\right] \subseteq M_{I\!\!R} & \langle a^i,R\rangle =1\\ M_{I\!\!R}=I\!\!R^n & \langle a^i,R\rangle\ge2 \end{array} \right.\\ && \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (i=1,\ldots,N),\; \mbox{ and}\\ V^R_{\tau} &:=& \cap_{a^i\in \tau} V^R_i \supseteq \mbox{span}_{I\!\!R}(E^R_{\tau}) \quad (\mbox{for faces } \tau<\sigma)\,. \end{eqnarray*} $\,$ \vspace{-2ex}\\ \par {\bf Proposition:}{\em With ${\cal V}^R_{-k}:= \oplus\!\!\!\!\!\!_{\begin{array}{c} \scriptstyle\tau<\sigma\vspace{-1.5ex} \\ \scriptstyle dim\, \tau=k \end{array}} \!\!\!\!V_{\tau}^R$ we obtain a complex ${\cal V}^R_{\bullet} \supseteq \mbox{span}_{I\!\!R}(E^R)_{\bullet}$. Moreover, if $Y$ is smooth in codimension $k$, then both complexes are equal at $\geq\!(-k)$. } \\ \par {\bf Proof:} $V^R_{\tau} = \mbox{span}_{I\!\!R}(E^R_{\tau})$ is true for smooth cones $\tau<\sigma$ (cf.(3.7) of \cite{T1}). \hfill$\Box$\\ \par {\bf Corollary:}{\em \begin{itemize} \item[(1)] If $Y$ is smooth in codimension 2, then $T^1_Y(-R)= H^1\left(({\cal V}^R_{\bullet})^\ast \otimes_{I\!\!R} \,I\!\!\!\!C \right)$. \item[(2)] If $Y$ is smooth in codimension 3, then $T^2_Y(-R)= H^2\left(({\cal V}^R_{\bullet})^\ast \otimes_{I\!\!R} \,I\!\!\!\!C \right)$. \vspace{1ex} \end{itemize} } \par The formula (1) for $T^1_Y$ (with a more boring proof) was already obtained in (4.4) of \cite{T1}.\\ \par \sect{3-dimensional Gorenstein singularities}\label{s7} \neu{71} We want to apply the previous results for the special case of an isolated, 3-dimensional, toric Gorenstein singularity. We start with fixing the notations.\\ \par Let $Q=\mbox{conv}(a^1,\dots,a^N)\subseteq I\!\!R^2$ be a lattice polygon with primitive edges \[ d^i:= a^{i+1}-a^i\in Z\!\!\!Z^2\,. \] Embedding $I\!\!R^2$ as the affine hyperplane $[a_3=1]$ into $N_{I\!\!R}:=I\!\!R^3$, we can define the cone \[ \sigma:= \mbox{Cone}(Q) \subseteq N_{I\!\!R}\,. \] The fundamental generators of $\sigma$ equal the vectors $(a^1,1),\dots,(a^N,1)$, which we will also denote by $a^1,\dots,a^N$, respectively. \\ \par The vector space $M_{I\!\!R}$ contains a special element $R^\ast:=[0,0;1]$: \begin{itemize} \item $\langle \bullet,R^\ast\rangle = 1$ defines the affine hyperplane containing $Q$, \item $\langle \bullet,R^\ast\rangle = 0$ describes the vectorspace containing the edges $d^i$ of $Q$. \end{itemize} The structure of the dual cone $\sigma^{\scriptscriptstyle\vee}$ can be described as follows: \begin{itemize} \item $[c;\eta]\in M_{I\!\!R}$ is contained in $\sigma^{\scriptscriptstyle\vee}$, iff $\langle Q,-c\rangle \leq \eta$. \item $[c;\eta]\in \partial \sigma^{\scriptscriptstyle\vee}$ iff there exists some $i$ with $\langle a^i,-c\rangle = \eta$. \item The set $E$ contains $R^\ast$. However, $E\setminus \{R^\ast\}\subseteq \partial \sigma^{\scriptscriptstyle\vee}$. \vspace{1ex} \end{itemize} \par {\bf Remark:} The toric variety $Y$ built by the cone $\sigma$ is 3-dimensional, Gorenstein, and regular outside its 0-dimensional orbit. Moreover, all those singularities can be obtained in this way.\\ \par \neu{72} Let $V$ denote the $(N-2)$-dimensional $I\!\!R$-vector space \[ V:=\{(t_1,\dots,t_N)\,|\; \sum_i t_i\,d^i=0\}\subseteq I\!\!R^N\,. \] The non-negative tuples among the $\underline{t}\in V$ describe the set of Minkowski summands $Q_{\underline{t}}$ of positive multiples of the polygon $Q$. ($t_i$ is the scalar by which $d^i$ has to be multiplied to get the $i$-th edge of $Q_{\underline{t}}$.)\\ \par We consider the bilinear map \[ \begin{array}{cclcl} V&\times& I\!\!R^E & \stackrel{\Psi}{\longrightarrow}& I\!\!R\\ \underline{t}&,&[c;\eta]\in E &\mapsto& \left\{ \begin{array}{ll} 0& \mbox{ if } c=0\quad(\mbox{i.e. } [c;\eta]=R^\ast)\\ \sum_{v=1}^{i-1} t_v\cdot \langle d^v,-c\rangle & \mbox{ if }\langle a^i,-c\rangle =\eta\,. \end{array} \right. \end{array} \] Assuming both $a^1$ and the associated vertices of all Minkowski sumands $Q_{\underline{t}}$ to coincide with $0\inI\!\!R^2$, the map $\Psi$ detects the maximal values of the linear functions $c$ on these summands \[ \Psi(\underline{t},[c;\eta]) = \mbox{Max}\,(\langle a,-c\rangle\,|\; a\in Q_{\underline{t}})\,. \] In particular, $\Psi(\underline{1},[c;\eta])=\eta$, i.e. $\Psi$ induces a map \[ \Psi: \quad^{\displaystyle V}\!\!/\!_{\displaystyle I\!\!R\cdot\underline{1}} \times L_{I\!\!R}(E) \longrightarrow I\!\!R\,. \] The results of \cite{Gor} and \cite{Sm} imply that $\Psi$ provides an isomorphism \[ ^{\displaystyle V_{\,I\!\!\!\!C}}\!\!/\!_{\displaystyle \,I\!\!\!\!C\cdot\underline{1}}\stackrel{\sim}{\longrightarrow} \left(\left. ^{\displaystyle L(E_0^{R^\ast})}\!\!\right/ \!_{\displaystyle \sum_i L(E_i^{R^\ast})} \right)^\ast \otimes_{Z\!\!\!Z} \,I\!\!\!\!C \cong T^1_Y(-R^\ast)= T^1_Y\,. \] In particular, $\mbox{dim}\,T^1_Y = N-3$.\\ \par \neu{73} Let $R\in M$. Combining the general results of \S \ref{s6} with the fact \[ \bigcap_i V_i^R = \mbox{Ker}\left[ \oplus_i(V_i^R\cap V^R_{i+1})\longrightarrow \oplus_i V^R_i\right] \] coming from the special situation we are in, we obtain the handsome formula \[ T^2_Y(-R)= \left[ \left.^{\displaystyle \bigcap_i (\mbox{span}_{\,I\!\!\!\!C} E_i^R)}\!\!\! \right/ \!\!\! _{\displaystyle \mbox{span}_{\,I\!\!\!\!C} (\bigcap_i E_i^R)} \right] ^\ast\,. \] $T^1_Y$ is concentrated in the degree $-R^\ast$. Hence, for computing $T^2_Y$, the degrees $-kR^\ast$ ($k\geq 2$) are the most interesting (but not only) ones. In this special case, the vector spaces $V^{kR^\ast}_i$ equal $M_{I\!\!R}$, i.e. \[ T^2_Y(-kR^\ast)= \left[ \left. ^{\displaystyle M_{\,I\!\!\!\!C}}\!\!\! \right/ \!\!\! _{\displaystyle \mbox{span}_{\,I\!\!\!\!C} (\bigcap_i E_i^{kR^\ast})} \right] ^\ast \subseteq \left[ \left. ^{\displaystyle M_{\,I\!\!\!\!C}}\!\!\! \right/ \!\!\! _{\displaystyle \,I\!\!\!\!C\cdot R^\ast}\right] ^\ast = \mbox{span}_{\,I\!\!\!\!C}(d^1,\dots,d^N) \subseteq N_{\,I\!\!\!\!C}\,. \vspace{2ex} \] \par {\bf Proposition:} {\em For $c\in I\!\!R^2$ denote by \[ d(c):= \mbox{Max}\,(\langle a^i,c\rangle\,|\; i=1,\dots,N) - \mbox{Min}\,(\langle a^i,c\rangle\,|\; i=1,\dots,N) \] the diameter of $Q$ in $c$-direction. If \[ k_1:= \!\!\begin{array}[t]{c} \mbox{Min} \vspace{-1ex}\\ \scriptstyle c\inZ\!\!\!Z^2\setminus 0 \end{array} \!\!d(c) \quad \mbox{ and } \quad k_2:= \!\!\!\begin{array}[t]{c} \mbox{Min} \vspace{-1ex}\\ \scriptstyle c,c'\inZ\!\!\!Z^2 \vspace{-1ex}\\ \scriptstyle lin.\, indept. \end{array} \!\!\!\mbox{Max}\,[ d(c), d(c')]\,, \] then $\quad\begin{array}[t]{lll} \dim T^2_Y(-kR^\ast) = 2 & \mbox{ for } & 2\leq k \leq k_1\,,\\ \dim T^2_Y(-kR^\ast) = 1 & \mbox{ for } & k_1+1\leq k \leq k_2\,,\mbox{ and}\\ \dim T^2_Y(-kR^\ast) = 0 & \mbox{ for } & k_2+1\leq k \,. \end{array} \vspace{2ex} $ } \par {\bf Proof:} We have to determine the dimension of $\;\mbox{span}_{\,I\!\!\!\!C}\left( \bigcap_i E_i^{kR^\ast}\right)\!\!\left/\!\!_{\displaystyle \,I\!\!\!\!C\cdot R^\ast}\right.$. Computing modulo $R^\ast$ simply means to forget the $\eta$ in $[c;\eta]\in M$. Hence, we are done by the following observation for each $c\in Z\!\!\!Z^2\setminus0$: \[ \begin{array}{rcl} \exists \eta\inZ\!\!\!Z: \;[c,\eta]\in \bigcap_iK_i^{kR^\ast} & \Longleftrightarrow & \exists \eta\inZ\!\!\!Z: \;(k-1)R^{\ast}\geq [c;\eta] \geq 0\\ & \Longleftrightarrow & d(c) \leq k-1\,. \end{array} \vspace{-3ex} \] \hfill$\Box$\\ \par {\bf Corollary:} {\em Unless $Y=\,I\!\!\!\!C^3$ or $Y=\mbox{cone over }I\!\!P^1\timesI\!\!P^1$, we have \[ T^2_Y(-2R^\ast)= \mbox{span}_{\,I\!\!\!\!C}(d^1,\dots,d^N), \] i.e. $\dim T^2_Y(-2R^\ast)=2$.} \\ \par \neu{74} {\bf Proposition:} {\em Using both the isomorphism $\;V_{\,I\!\!\!\!C}\!\!\left/\!\!_{\displaystyle \,I\!\!\!\!C\cdot\underline{1}}\right. \stackrel{\sim}{\rightarrow} T^1_Y$ and the injection $T^2_Y(-2R^\ast)\hookrightarrow \mbox{span}_{\,I\!\!\!\!C}(d^1,\dots,d^N)$, the cup product $T^1_Y\times T^1_Y \rightarrow T^2_Y$ equals the bilinear map \[ \begin{array}{ccccc} V_{\,I\!\!\!\!C}\!\!\left/\!\!_{\displaystyle \,I\!\!\!\!C\cdot\underline{1}}\right. & \times & V_{\,I\!\!\!\!C}\!\!\left/\!\!_{\displaystyle \,I\!\!\!\!C\cdot\underline{1}}\right. & \longrightarrow & \mbox{span}_{\,I\!\!\!\!C}(d^1,\dots,d^N)\\ \underline{s} & , & \underline{t} & \mapsto & \sum_i s_i\,t_i\,d^i\,. \end{array} \vspace{1ex} \] } \par {\bf Proof:} {\em Step 1:} To apply Theorem \zitat{3}{6} we will combine the isomorphisms for $T^2_Y$ presented in \S \ref{s6} and \zitat{7}{3}. Actually, we will describe the dual map by associating to each $r\in M$ an element $[q^1(r),\dots,q^N(r)]\in\oplus_iL(E_i^{2R^\ast})$.\\ First, for every $i=1,\dots,N$, we have to write $r\in M = (\mbox{span}\, E_i^{2R^\ast})\cap (\mbox{span}\, E_{i+1}^{2R^\ast})$ as a linear combination of elements from $E_i^{2R^\ast}\cap E_{i+1}^{2R^\ast}$. This set contains a $Z\!\!\!Z$-basis for $M$ consisting of \begin{itemize} \item $r^i:=$ primitive element of $\sigma^{\scriptscriptstyle\vee}\cap (a^i)^\bot \cap (a^{i+1})^\bot$, \item $R^\ast$, and \item $r(i):= r(i,i+1)$ (cf. notation at the end of \zitat{5}{4}), i.e. $\begin{array}[t]{l} \langle a^i, r(i)\rangle = 1 \mbox{ and}\\ \langle a^{i+1}, r(i) \rangle = 0\,. \end{array} $ \end{itemize} In particular, we can write \[ r= g^i(r)\cdot r^i + \langle a^{i+1},r\rangle\cdot R^\ast + \left( \langle a^i,r \rangle - \langle a^{i+1},r\rangle \right) \cdot r(i) \] with some integer $g^i(r)\in Z\!\!\!Z$.\\ \par Now, we have to apply the differential in the complex $(Z\!\!\!Z^{E^{2R^\ast}})_{\bullet}$, i.e. we map the previous expression via the map \[ \oplus_i Z\!\!\!Z^{E_i^{2R^\ast}\cap E_{i+1}^{2R^\ast}} \longrightarrow \oplus_i Z\!\!\!Z^{E_i^{2R^\ast}}\, . \] The result is (for every $i$) the element of $L(E_i^{2R^\ast})$ \[ \begin{array}{l} g^i(r)\, e^{r^i} - g^{i-1}(r)\, e^{r^{i-1}} + \langle a^i-a^{i+1},r\rangle\cdot e^{r(i)} - \langle a^{i-1} - a^i,r\rangle \cdot e^{r(i-1)} + \langle a^{i+1}-a^i, r \rangle\cdot e^{R^\ast} \vspace{1ex}\\ \qquad = \langle d^i,r\rangle \cdot \left( e^{R^\ast} - e^{r(i)}\right) + [(a^i)^\bot \mbox{-summands}] =: q^i(r)\,. \end{array} \vspace{2ex} \] \par {\em Step 2:} Defining \[ q^i:= e^{R^\ast}-e^{r(i)} + [(a^i)^\bot \mbox{-summands}] \in L(E_i^{2R^\ast})\quad (i=1,\dots,N)\,, \] we use Theorem \zitat{3}{6} and the second remark of \zitat{3}{6} to obtain \[ (\underline{s}\cup\underline{t})_i \left( q^i(r) \right) = \langle d^i,r \rangle \cdot t_{\Psi(\underline{s},\bullet), \Psi(\underline{t},\bullet), R^\ast,R^\ast} (q^i) = \Psi(\underline{s},q^i)\cdot \Psi(\underline{t},q^i)\,. \] To compute those two factors, we take a closer look at the $q^i$'s. Let \[ q^i= e^{R^\ast}-e^{r(i)} + \sum_v \lambda^i_v \,e^{[c^v; \eta^v]}\,, \vspace{-1ex} \] and the sum is taken over those $v$'s meeting the property $\langle a^i,-c^v\rangle = \eta^v$. Then, by definition of $\Psi$ in \zitat{7}{2}, \[ \Psi(\underline{s},q^i)= \sum_{j=1}^{(i+1)-1}s_j\,\langle d^j, r(i)\rangle - \sum_v \lambda^i_v \cdot \left( \sum_{j=1}^{i-1}s_j\, \langle d^j,c^v\rangle \right)\,. \] On the other hand, we know that $q^i$ is a relation, i.e. the equation \[ R^\ast - r(i) + \sum_v \lambda^i_v [c^v; \eta^v] =0 \vspace{-1ex} \] is true in $M$. Hence, \[ \begin{array}{rcl} \Psi(\underline{s},q^i)&=& \sum_{j=1}^i s_j\,\langle d^j, r(i)\rangle - \sum_{j=1}^{i-1} s_j \langle d^j, r(i)\rangle \vspace{0.5ex}\\ &=& s_i\cdot \langle d^i, r(i)\rangle \vspace{0.5ex}\\ &=& -s_i\,. \end{array} \vspace{-3ex} \] \hfill$\Box$ \vspace{2ex}\\ \par $T^1_Y\subseteq \,I\!\!\!\!C^N$ is the tangent space of the versal base space $S$ of our singularity $Y$. It is given by the linear equation $\sum_i t_i\cdot d^i=0$.\\ On the other hand, the cup product $T^1_Y\times T^1_Y\rightarrow T^2_Y$ shows the quadratic part of the equations defining $S\subseteq\,I\!\!\!\!C^N$. By the previous proposition, it equals $\sum_i t_i^2 \cdot d^i$.\\ \par These facts suggest an idea how the equations of $S\subseteq\,I\!\!\!\!C^N$ could look like. In \cite{Vers} we have proved this conjecture; $S$ is indeed given by the equations \[ \sum_{i=1}^N t_i^k \cdot d^i =0\quad (k\geq 1)\,. \vspace{3ex} \] \par

9405

alg-geom/9405002

en

https://arxiv.org/abs/alg-geom/9405002

alg-geom/9405002

Ron Stern

Ronald Fintushel and Ronald Stern

The blowup formula for Donaldson invariants

16 pages, AMS-LaTeX

In this paper we present a formula which relates the Donaldson invariants of a 4-manifold X with the Donaldson invariants of its blowup X#-CP(2). This blow-up formula is independent of X and involves sigma-functions associated to a naturally arising elliptic function. This, the final version, corrects some earlier misconceptions regarding signs.

alg-geom

\section{Introduction} Since their introduction in 1984 \cite{Donpoly}, the Donaldson invariants of smooth $4$-manifolds have remained as mysterious as they have been useful. However, in the past year there has been a surge of activity pointed at comprehension of the structure of these invariants \cite{KM,FS}. One key to these advances and to future insights lies in understanding the relation of the Donaldson invariants of a $4$-manifold $X$ and those of its blowup $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$. It is the purpose of this paper to present such a blowup formula. This formula is independent of $X$ and is given in terms of an infinite series \[ B(x,t) = \sum\limits_{k=0}^{\infty}B_k(x){t^k\over k!} \] which is calculated in \S4 below. This formula has been the target of much recent work. The abstract fact that there exists such a formula which is independent of $X$ was first proved by C. Taubes using techniques of \cite{Reds}. J. Bryan \cite{B} and P. Ozsvath \cite{Ozs} have independently calculated the coefficients through $B_{10}(x)$. Quite recently, J. Morgan and Ozsvath have announced a scheme which can recursively compute all of the $B_k(x)$. The special case of the blowup formula for manifolds of ``simple type'' (see \S5 below) was first given by P. Kronheimer and T. Mrowka. However, none of the techniques in these cases approach the simplicity of that offered here. Before presenting the formula, we shall first establish notation for the Donaldson invariants of a simply connected $4$-manifold $X$ with $b^+>1$ and odd. (The hypothesis of simple connectivity is not necessary, but makes the exposition easier.) An orientation of $X$, together with an orientation of $H^2_+(X;\bold{R})$ is called a {\em homology orientation} of $X$. Such a homology orientation determines the ($SU(2)$) Donaldson invariant, a linear function \[ D=D_X:\bold{A}(X)=\text{Sym}_*(H_0(X)\oplus H_2(X))\to \bold{R} \] which is a homology orientation-preserving diffeomorphism invariant. Here \[\bold{A}(X)=\text{Sym}_*(H_0(X)\oplus H_2(X))\] is viewed as a graded algebra where $H_i(X)$ has degree $\frac12(4-i)$. We let $x\in H_0(X)$ be the generator $[1]$ corresponding to the orientation. Then as usual, if $a+2b=d>\frac34(1+b^+_X)$ and $\alpha\in H_2(X)$, \[ D(\alpha^ax^b)=\langle\mu(\alpha)^a\nu^b,[\cal M_X^{2d}]\rangle \] where $[\cal M_X^{2d}]$ is the fundamental class of the (compactified) $2d$-dimensional moduli space of anti-self-dual connections on an $SU(2)$ bundle over $X$, $\mu:H_i(X)\to H^{4-i}(\cal B^*_X)$ is the canonical map to the cohomology of the space of irreducible connections on that bundle \cite{Donpoly}, and $\nu=\mu(x)$. The extension of the definition to smaller $d$ is given in \cite{MMblowup} (and is accomplished, in fact, from the knowledge of the lowest coefficient in the $SO(3)$ blowup formula). Since an $SU(2)$ bundle $P$ over $X$ has a moduli space of dimension \[\dim\cal M_X(P) = 8c_2(P)-3(1+b^+_X) \] it follows that such moduli spaces $\cal M_X^{2d}$ can exist only for $d\equiv\frac12(1+b^+_X)\pmod4$. Thus the Donaldson invariant $D$ is defined only on elements of $\bold{A}(X)$ whose total degree is congruent to $\frac12(1+b^+_X)\pmod4$. By definition, $D$ is $0$ on all elements of other degrees. We can now state the blowup formula. Let $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$ and let $e\in H_2(X)$ denote the homology class of the exceptional divisor. Since $b^+_X=b^+_{\hat{X}}$, the corresponding Donaldson invariants $D=D_X$ and $\hat{D}=D_{\hat{X}}$ have their (possible) nonzero values in the same degrees $\pmod4$. We first show that there are polynomials $B_k(x)$ satisfying \[ \hat{D}(e^k\,z) = D(B_k(x)\,z) \] for all $\,z\in \bold{A}(X)$ and then define the formal power series $B(x,t)$ as above. Our result is that \[ B(x,t)=e^{-{t^2x\over6}}\sigma_3(t) \] where $\sigma_3$ is a particular quasi-periodic Weierstrass sigma-function \cite{Ak} associated to the $\wp$-function which satisfies the differential equation \[(y')^2=4y^3-g_2y-g_3 \] where \[ g_2=4\,(\frac{x^2}{3}-1)\, , \hspace{.25in} g_3={8x^3-36x\over 27} \,. \] There are also Donaldson invariants associated to $SO(3)$ bundles $V$ over $X$. To define these invariants one needs, along with a homology orientation of $X$, an integral lift of $w_2(V)$. If $c\in H_2(X;\bold{Z})$ is the Poincar\'e dual of the lift, the invariant is denoted $D_c$ or $D_{X,c}$ if the manifold $X$ is in doubt. $D_c$ is nonzero only in degrees congruent to $-c\cdot c+\frac12(1+b^+)\pmod4$. If $c'\equiv c\pmod2$ then \[ D_{c'}=(-1)^{({c'-c\over2})^2}D_c\,. \] The $SO(3)$ blowup formula states that there are polynomials $S_k(x)$ such that \[ \hat{D}_e(e^k) = D(S_k(x)) \] and if \[ S(x,t) = \sum\limits_{k=0}^{\infty}S_k(x){t^k\over k!} \] then \[ S(x,t)=e^{-{t^2x\over6}}\sigma(t) \] where $\sigma(t)$ is the standard Weierstrass sigma-function \cite{Ak} associated to $\wp$. The coefficients $S_k(x)$ for $k\le 7$ were earlier computed by T. Leness \cite{Leness}. The discriminant of the cubic equation $4y^3-g_2y-g_3=0$ turns out to be ${x^2-4\over 4}$. Thus, when (viewed as a function on $\bold{A}(X)$) $D(x^2-4)=0$, the Weierstrass sigma-functions degenerate to elementary functions, and the blowup formula can be restated in terms of these functions. This is done in the final section. It is interesting to note that the condition $D(x^2z)=4\,D(z)$ is the {\em simple type} condition introduced by Kronheimer and Mrowka \cite{KM}. Our formulas are proved by means of a simple relation satisfied by $D(\tau^4z)$ where $\tau\in H_2(X;\bold{Z})$ is represented by an embedded $2$-sphere of self-intersection $-2$. When this relation is applied to $\tau=e_1-e_2$, the difference of the two exceptional classes of the double blowup $X\#2\overline{\bold{CP}}^{\,2}$, one obtains a differential equation for $B(x,t)$. Solving this equation gives our formulas. \section{Some Relations among Donaldson Invariants} The key to the blowup formula lies in a few simple relations which are useful for evaluating Donaldson invariants on classes represented by embedded spheres of self-intersection $-2$ and $-3$. We begin by studying the behavior of the Donaldson invariant of a $4$-manifold with a homology class $\tau$ represented by an embedded $2$-sphere $S$ of self-intersection $\tau\cdot \tau=-2$. Let $\langle \tau\rangle^\perp$ denote $\{\alpha\in H_2(X)|\tau\cdot\alpha=0\}$ and let \[\bold{A}(\tau^\perp)=\bold{A}_X(\tau^\perp)=\text{Sym}_*(H_0(X)\oplus \langle \tau\rangle^\perp)\,.\] \begin{thm}{\em (Ruberman\; \cite{R})} Suppose that $\tau\in H_2(X;\bold{Z})$ with $\tau\cdot \tau=-2$ is represented by an embedded sphere $S$. Then for $\,z \in \bold{A}(\tau^\perp)$, we have $D(\tau^2\,z)=2\,D_{\tau}(\,z)$. \label{Ruber}\end{thm} \begin{pf} Write $X=X_0 \cup N$ where $N$ is a tubular neighborhood of $S$, and note that $\partial N$ is the lens space $L(2,-1)$. Since $b^+_{X_0}>0$, generically there are no reducible anti-self-dual connections on $X_0$. However, since $b^+_{N}=0$, there are nontrivial reducible anti-self-dual connections arising from complex line bundles $\lambda^m$, $m \in\bold{Z}$, where $\langle c_1(\lambda),\tau\rangle=-1$. The corresponding moduli spaces $\cal M_N(\lambda^m\oplus\bar{\lambda}^m)$ have dimensions $4m^2-3$ and have boundary values $\zeta^m$ where $\zeta$ generates the character variety of $SU(2)$ representations of $\pi_1(\partial N)=\bold{Z}_2$ mod conjugacy. (Of course, $\zeta^{2m}$ is trivial, and $\zeta^{2m+1}=\zeta$.) Since $\langle \tau\rangle^\perp=H_2(X_0)$, we need to evaluate the Donaldson invariant on two copies of $\tau$ and classes in $H_2(X_0)$. After cutting down moduli spaces by intersecting with transverse divisors representing the images under $\mu$ of these classes in $H_2(X_0)$ and using the given homology orientation, we may assume without loss of generality that there are no such classes and that we are working with a $4$-dimensional moduli space $\cal M_X$. Let $V_1$ and $V_2$ be divisors representing $\mu(\tau)$, coming from general positioned surfaces in $N$. The Donaldson invariant is the signed intersection number \[ D(\tau^2)=\#(\cal M_X\cap V_1\cap V_2). \] A standard dimension counting argument (cf. \cite{Donpoly}) shows that if we choose a metric on $X$ with long enough neck length, $\partial N\times [0,T]$, then all the intersections take place in a neighborhood $\cal{U}$ of the grafted moduli space $\cal M_{X_0}[\zeta]\# \{A_{\lambda}\}$ where $A_{\lambda}$ is the reducible anti-self-dual connection on $\lambda\oplus \bar{\lambda}$, and $\cal M_{X_0}[\zeta]$ is the $0$-dimensional cylindrical end moduli space on $X_0$ consisting of anti-self-dual connections which decay exponentially to the boundary value $\zeta$. Let $m_{X_0}$ be the signed count of points in $\cal M_{X_0}[\zeta]$. A neighborhood of $A_{\lambda}$ in the moduli space $\cal M_N(\lambda\oplus\bar{\lambda})$ is diffeomorphic to $(\bold C \times_{S^1} SO(3))/SO(3) \cong\bold C/S^1\cong[0,\infty)$. Here $S^1$ acts on $SO(3)$ so that $SO(3)/S^1=S^2$ and on $\bold C$ with weight $-2$. Thus the neighborhood $\cal{U}$ is \[(\tilde{\cal M}_{X_0}[\zeta]\times (\bold C \times_{S^1} SO(3)))/SO(3)\] where ``$\tilde{\cal M}_{X_0}[\zeta]$'' denotes the based moduli space. Now $\tilde{V}_1\cap (\bold C \times_{S^1} SO(3))=\{0\} \times_{S^1} SO(3)$, and the intersection of $V_1$ with all of $\cal M_X$ is \[(\cal M_{X_0}[\zeta] \times(\{0\} \times_{S^1} SO(3)))/SO(3)=\Delta .\] Fix a point $p\in\cal M_{X_0}[\zeta]$, let $SO(3)\cdot p$ denote its orbit in $\tilde{\cal M}_{X_0}[\zeta]$, and let \[\Delta_p=SO(3)\cdot p\times (\{0\} \times_{S^1} SO(3)))/SO(3)\cong S^2.\] Identify $\Delta_p$ with a transversal in $\tilde{\Delta}_p$ and compute the intersection number $\tilde{V}_2\cdot \Delta_p=\iota_p$. Since $\iota_p$ is independent of $p\in \cal M_{X_0}[\zeta]$, we have $D(\tau^2)=\iota_p\cdot m_{X_0}$. The constant $\iota_p$ is computed in \cite{FMbook} as follows. Note that $\Delta_p=\{0\}\times_{S^1} SO(3) \subset \bold C\times_{S^1} SO(3)$ is a \,zero-section of the $c_1=-2$ complex line bundle over $S^2$ and $\tilde{V}_2$ is another section. Thus $\tilde{V}_2\cdot \Delta_p=-2$; and so $D(\tau^2)=-2\,m_{X_0}$. To identify the relative invariant $m_{X_0}$, view $\cal M_{X_0}[\zeta]$ as $\cal M_{X_0,0}[\text{ad}(\zeta)]$, an $SO(3)$ moduli space. Since $\text{ad}(\zeta)$ is the trivial $SO(3)$-representation, we may graft connections in $\cal M_{X_0,0}[\text{ad}(\zeta)]$ to the trivial $SO(3)$ connection over $N$, and since $b_N^+=0$, there is no obstruction to doing this. We obtain an $SO(3)$ moduli space over $X$ corresponding to an $SO(3)$ bundle over $X$ with $w_2$ Poincar\'e dual to $\tau$. (This is the unique nonzero class in $H^2(X;\bold{Z}_2)$ which restricts trivially to both $N$ and $X_0$.) Thus for $\,z \in \bold{A}(\tau^\perp)$, we have $D(\tau^2\,z)=\pm2\,D_{\tau}(\,z)$. (Note that since $\tau\cdot\tau=-2$, we have $D_{-\tau}=D_\tau$.) To determine the sign in this equation, we need to compare orientations on the moduli spaces which are involved. Let $A_0\in \cal M_{X_0}[\zeta]$. The way that a sign is attached to this point is described in \cite{Donor,K}. By addition and subtraction of instantons, $A_0$ is related to a connection $B_0$ in a reducible bundle $E$ over $X_0$, and $B_0$ can be connected by a path to a reducible connection $R$ which comes from a splitting $E\cong L_0\oplus\bar{L}_0$. There is a standard orientation for the determinant line of the operator $d_R^+\oplus d_R^*$, and this can be followed back to give an orientation for the determinant line at $A_0$. This determinant line is canonically oriented because the cohomology $H^*_{A_0}$ vanishes. Comparing the two orientations gives a sign, $\varepsilon$. To determine the sign at the grafted connection $A_0\# A_{\lambda}$, note that the same sequence of instanton additions and subtractions as above relates $A_0\# A_{\lambda}$ to $B_0\# A_{\lambda}$ which can be connected to $R\# A_{\lambda}$, a reducible connection on the bundle $L\oplus\bold{R}$ over $X$, where the Mayer-Vietoris map $H^2(X)\to H^2(X_0)\oplus H^2(N)$ carries $c=c_1(L)$ to $c_1(L_0)+c_1(\lambda)$. Since $R\# A_{\lambda}$ is reducible, there is an orientation of the determinant line, and it relates to the orientation which can be pulled back from the trivial connection by $(-1)^{c\cdot c}$. Thus pulling the orientation back over $A_0\# A_{\lambda}$ gives the sign $\varepsilon\cdot (-1)^{c\cdot c}$. To get the sign for $A_0\#\Theta$ we first pass to $SO(3)$, and then $\text{ad}(A_0)$ is related as above to the reducible connection $\text{ad}(R)$ which lives in the line bundle $L_0^2$. Grafting to the trivial connection $\Theta_N$, we get $\text{ad}(A_0)\#\Theta_N$ which is connected to the reducible connection $\text{ad}(R)\#\Theta_N$. This lives in the grafted line bundle $L_0^2\#\bold{R}$ which has $c_1=2c_1(L_0)$. (Note that although $c_1(L_0)$ is not a global class, $2c_1(L_0)$ is.) The class $2c_1(L_0)$ restricts trivially to $X_0$ and to $N$ (mod $2$); so its mod $2$ reduction is the same as that of $\tau$. (We are here identifying $\tau$ and its Poincar\'e dual.) Since $\tau=2c_1(\lambda)$, the difference in these reductions is $2c_1(L_0)-\tau=2(c-\tau)$. The corresponding orientations compare via the parity of $(c-\tau)\cdot(c-\tau)\equiv c\cdot c\pmod2$. Thus the sign which is attached to $A_0\#\Theta$ is $\varepsilon\cdot (-1)^{c\cdot c}$, the same as for $A_0\# A_{\lambda}$, and the sign in the formula above is `${}\,+\,{}$'. \end{pf} For the case of the $SO(3)$ invariants the proof of Theorem~\ref{Ruber} can be easily adapted to show: \begin{thm} Suppose that $\tau\in H_2(X;\bold{Z})$ with $\tau\cdot\tau=-2$ is represented by an embedded sphere $S$. Let $c\in H_2(X;\bold{Z})$ satisfy $c\cdot\tau\equiv0\pmod2$. Then for $\,z \in \bold{A}(\tau^\perp)$ we have $D_c(\tau^2\,z)=2\,D_{c+\tau}(z)$. \ \ \ \qed \label{RuberSO3}\end{thm} We next need to review some elementary facts concerning the Donaldson invariants of blowups. These can be found, for example in \cite{FMbook,Ko,Leness}. Let $X$ have the Donaldson invariant $D$, and let $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$ have the invariant $\hat{D}$. \begin{lem} Let $e\in H_2(\overline{\bold{CP}}^{\,2};\bold{Z})\subset H_2(\hat{X};\bold{Z})$ be the exceptional class, and let $c\in H_2(X;\bold{Z})$. Then for all $\,z\in \bold{A}(X)$: \begin{enumerate} \item $\hat{D}_c(e^{2k+1}\,z)=0$ for all $k\ge 0$. \item $\hat{D}_c(\,z)=D_c(\,z)$. \item $\hat{D}_c(e^2\,z)=0$. \item $\hat{D}_c(e^4\,z)=-2\,D_c(\,z)$. \item $\hat{D}_{c+e}(e^{2k}\,z)=0$ for all $k\ge 0$. \item $\hat{D}_{c+e}(e\,z)=D_c(\,z)$. \item $\hat{D}_{c+e}(e^3\,z)=-D_c(x\,z)$. \end{enumerate} \label{blowuplow} \end{lem} \begin{pf} Items (1)--(5) are standard and are explained in \cite{FMbook}. Both (1) and (5) follow because the automorphism of $H_2(X\#\overline{\bold{CP}}^{\,2};\bold{Z})$ given by reflection in $e$ is realized by a diffeomorphism. Items (2) and (3) follow from counting arguments, and (4) follows from simple arguments as in the proof of Theorem~\ref{Ruber} above \cite{FMbook}. Item (6) is due to D. Kotschick \cite{Ko}. A proof of (7) is given in \cite{Leness}. (However, the sign there differs from ours since item (6) is stated in \cite{Leness} with an incorrect sign.) We sketch a proof here. Consider a neighborhood $N$ of the exceptional curve, and let $X_0=\hat{X}\setminus N \cong X\setminus B^4$. As in the proof of Ruberman's theorem we lose no generality by assuming that we are evaluating $\hat{D}$ only on $e^3$. A dimension counting argument shows that if we stretch the neck between $X_0$ and $N$ to have infinite length by taking a sequence of generic metrics $\{ g_n\}$, and if $V_i$ are transverse divisors representing $\mu(e)$, then any sequence of connections \[ A_n\in \cal M_{\hat{X},c+e}(g_n)\cap V_1\cap V_2\cap V_3\] must converge to the sum of a connection in the $4$-manifold $\cal M_{X_0,c}$ and the unique reducible connection on $N$ corresponding to the line bundle $\lambda$ over $N$ whose Euler class is Poincar\'e dual to $e$. The based moduli space $\tilde{\cal M}_N(\lambda)$ is the orbit of this reducible connection, a $2$-sphere, $S^2_e$. Let $v$ denote the (positive) generator of the equivariant cohomology $H^*_{SO(3)}(S^2_e)\cong H^*(\bold{CP}^{\infty})$ in dimension $2$. The class $\mu(e)$ lifts to the equivariant class $-\frac12\langle c_1(\lambda),e\rangle\,v=\frac12 v\in H^2_{SO(3)}(S^2_e)$. The connections in $S^2_e$ are asymptotically trivial and this induces an $SO(3)$ equivariant push-forward map \[ \partial_*(N): H^*_{SO(3)}(S^2_e)\to H^*_{SO(3)}({1})=H^*(BSO(3)).\] If $u\in H^*_{SO(3)}({1})$ is the generator in dimension $4$ then $\partial_*(N)(v^{2k+1})=2\,u^k$. So $\partial_*(N)(e^3)=\frac14 u$. Since each connection in $\cal M_{X_0,c}$ is also asymptotically trivial, there is an induced map $\partial^*(X_0):H^*_{SO(3)}({1})\to H^*_{SO(3)}(\tilde{\cal M}_{X_0,c})$. It follows from \cite{Reds,AB} that $\hat{D}_{c+e}(e^3)$ is obtained by evaluating \[\langle \partial^*(X_0)\,\partial_*(N)(v^3),[\tilde{\cal M}_{X_0,c}]\rangle = \frac14\,\langle \partial^*(X_0)(u),[\tilde{\cal M}_{X_0,c}]\rangle = \frac14\,\langle \pi_*\partial^*(X_0)(u), [\cal M_{X_0,c}] \rangle \] where basepoint fibration $\beta$ over $X_0$ is \[\pi:\tilde{\cal M}_{X_0,c}\to\cal M_{X_0,c}\, ,\] the last equality because the $SO(3)$ action on $\tilde{\cal M}_{X_0,c}$ is free. But Austin and Braam \cite{AB}, for example, show that $\pi_*\partial^*(X_0)(u)=p_1(\beta)$. Since $\nu=-\frac14 p_1(\beta)$, we get $\langle\nu,[\cal M_{X_0,c}]\rangle = -D_c(x)$. \end{pf} We next consider embedded $2$-spheres of self-intersection $-3$. \begin{thm} Suppose that $\tau\in H_2(X;\bold{Z})$ is represented by an embedded 2-sphere $S$ with self-intersection $-3$. Let $\omega\in H_2(X;\bold{Z})$ satisfy $\omega\cdot\tau\equiv0\pmod2$. Then for all $\,z\in\bold{A}(\tau^\perp)$ we have \[D_\omega(\tau\,z) = -D_{\omega+\tau}(\,z).\] \label{3curve}\end{thm} \begin{pf} The proof is similar in structure to that of Theorem~\ref{Ruber}. Write $X=X_0\cup N$ where $N$ is a tubular neighborhood of $S$. Then $\partial N=L(3,-1)$. Let $\eta$ generate the character variety of $SO(3)$ representations of $\pi_1(\partial N)$. Reducible anti-self-dual $SO(3)$ connections on $N$ arise from complex line bundles $\lambda^m$, $m\in\bold{Z}$, where $\langle c_1(\lambda),\tau\rangle=-1$. The corresponding moduli spaces $\cal M_N(\lambda^m\oplus\bold{R})$ have boundary values $\eta^m$ and dimensions $\frac{2}{3}m^2-3$ if $m\equiv 0\pmod3$ and ${2m^2+1\over3}-2$ if $m\not\equiv0\pmod3$. Since it is easiest to work with an $\omega$ which satisfies $\partial\omega_{X_0}=0\in H_1(\partial X;\bold{Z})=\bold{Z}_3$, we simply work with $\rho=3\,\omega$ rather than $\omega$. This is no problem, since $D_{3\omega}=(-1)^{\omega\cdot\omega}D_{\omega}$. Thus we may write $\rho=\rho_0+\rho_N\in H_2(X_0;\bold{Z})\oplus H_2(N;\bold{Z})$. As in our previous arguments, we assume that we are evaluating $D_{\rho}$ only on $\tau$. A dimension counting argument shows that $D_{\rho}(\tau)$ is the product of relative invariants $D_{X_0}[\eta^m]$ coming from a $0$-dimensional cylindrical end moduli space over $X_0$ with terms coming from nontrivial reducible connections on $N$. These reducible connections must live in moduli spaces of dimension $\le 1$, and the corresponding line bundles must have $c_1\equiv \rho_N\pmod2$. Our hypothesis, $\omega\cdot\tau\equiv0\pmod2$ implies that $\rho_N\cdot\tau\equiv0\pmod2$; so the line bundle in question must be an even power of $\lambda$. Recalling the constraint that the dimension of the corresponding moduli space be $\le1$, the only possibility is $\cal M_N(\lambda^2\oplus\bold{R})$. Consider an anti-self-dual connection $A_0$ lying in the finite $0$-dimensional moduli space $\cal M_{X_0}[\eta^2]$, and let $A_{\lambda^2}$ be the reducible anti-self-dual connection on $N$. A neighborhood of the $SO(3)$ orbit of $A_{\lambda^2}$ in the based moduli space $\tilde{\cal M}_N(\lambda^2\oplus\bold{R})$ is modelled by $SO(3)\times_{S^1}\bold{C}$ and the (based) divisor for $\tau$ is $-\frac12 \langle c_1(\lambda^2),\tau\rangle (SO(3)\times_{S^1}\{0\})=SO(3)\times_{S^1}\{0\}$. The based connections obtained from grafting the orbit $SO(3)_{A_0}$ of $A_0$ to the orbit of $A_{\lambda^2}$ are given by the fibered product of these orbits over the 2-sphere in $SO(3)$ consisting of representations of $\pi_1(\partial N)$ which are in the conjugacy class $\eta^2$. By cutting this down by the divisor for $\tau$ we obtain (up to sign) the fibered product of $SO(3)_{A_0}$ with $S^2$ over $S^2$; i.e. simply $SO(3)_{A_0}$. Taking the quotient by $SO(3)$, \[ D_{\rho}(\tau)=\pm D_{X_0}[\eta^2].\] Since $\eta^2=\eta$ in the character variety (a copy of $\bold{Z}_3$), we can graft anti-self-dual connections $A_0$ to the unique (reducible) connection $A_{\lambda}$ lying in the moduli space $\cal M_N(\lambda\oplus\bold{R})$ of formal dimension $-1$. As the glued-together bundle has $w_2$ which is Poincar\'e dual to $\rho+\tau\pmod2$, we have \[ D_{\rho_0}[\eta^2]=\pm D_{\rho+\tau},\] so our result is proved up to a sign. To get this sign, we need to compare signs induced at $A_0\# A_{\lambda^2}$ and $A_0\# A_{\lambda}$ using a fixed homology orientation of $X$ and the integral lifts $\rho$ and $\rho+\tau$ of the corresponding Stiefel-Whitney classes. By an excision argument \cite{Donor}, the difference in signs depends only on the part of the connections over the neighborhood $N$. Thus the sign is universal, and may be determined by an example. For this, let $X$ be the $K3$ surface and $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$. Let $s$ be any class in $H_2(X)$ of square $-2$ represented by an embedded $2$-sphere (e.g. a section), and let $\tau=s+e$. Note that $s-2e\in\bold{A}(\tau^{\perp})$. Then using Theorems \ref{Ruber} and \ref{blowuplow}, \begin{eqnarray*} \hat{D}((s-2e)\tau)&=&D(s^2)=2\,D_s\\ \hat{D}_{\tau}(s-2e)&=&-2\,D_s \end{eqnarray*} so the overall sign is `${}\,-\,{}$'. \end{pf} Next we combine our two relations to obtain a relation which is crucial in obtaining the general blowup formula. This relation was first proved by Wojciech Wieczorek using different methods. His proof will appear in his thesis \cite {Wiecz}. \begin{cor} Suppose that $\tau\in H_2(X;\bold{Z})$ is represented by an embedded $2$-sphere with self-intersection $-2$, and let $c\in H_2(X;\bold{Z})$ with $c\cdot\tau\equiv0\pmod2$. Then for all $\,z\in\bold{A}(\tau^\perp)$ \[ D_c(\tau^4\,z)=-4\,D_c(\tau^2 \, x\,z)-4\,D_c(\,z). \] \label{4ofem}\end{cor} \begin{pf} In $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$ the class $\tau+e$ is represented by a $2$-sphere of self-intersection $-3$, and $(\tau-2e)\cdot(\tau+e)=0$. From Lemma~\ref{blowuplow} we get \[ \hat{D}_c((\tau-2e)^3 \,(\tau+e)\,z) = D_c(\tau^4\,z)-8\,\hat{D}_c(e^4\,z) = D_c(\tau^4\,z)+16\,D_c(\,z). \] On the other hand, by Theorems~\ref{RuberSO3} and \ref{3curve} and by Lemma~\ref{blowuplow}, \begin{eqnarray*} \hat{D}_c((\tau-2e)^3 \,(\tau+e)\,z)&=&-\hat{D}_{c+\tau+e}((\tau-2e)^3\,z) =6\,\hat{D}_{c+\tau+e}(\tau^2\, e\,z)+8\,\hat{D}_{c+\tau+e}(e^3\,z)\\ &=&6\,D_{c+\tau}(\tau^2\,z)-8\,D_{c+\tau}(x\,z) = 12\,D_c(\,z)-4\,D_c(\tau^2\, x\,z) \end{eqnarray*} and the result follows. \end{pf} \bigskip \section{The blowup equation} Let $X$ be a simply connected oriented $4$-manifold and let $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$. Let $c\in H_2(X;\bold{Z})$. Of course $H_2(\hat{X};\bold{Z})=H_2(X;\bold{Z})\oplus\bold{Z} e$ with $e$ the exceptional class. It follows from Lemma~\ref{blowuplow}(1),(5) that we can write \[\hat{D}_c=\sum\beta_{c,k}\,E^{2k}\] where $E$ denotes the $1$-form given by $E(y)=e\cdot y$ and $\beta_{c,k}({\alpha^d})=\hat{D}_c(\alpha^d \, e^{2k})$ for any $\alpha\in H_2(X)$. Similarly \[\hat{D}_{c+e}=\sum\gamma_{c,k}\,E^{2k+1}.\] Consider $\bar{X}=X\#2\overline{\bold{CP}}^{\,2}$ with exceptional classes $e_1,\, e_2\in H_2(\bar{X};\bold{Z})$, and let $\bar{D}$ denote its Donaldson invariant. Then $e_1+e_2$ has self-intersection $-2$ and is represented by an embedded $2$-sphere. Furthermore, the intersection $(e_1-e_2)\cdot (e_1+e_2)=0$; so we can apply Corollary~\ref{4ofem} to get \begin{equation} \bar{D}_c((e_1-e_2)^r \, (e_1+e_2)^4\,z)= -4\,\bar{D}_c((e_1-e_2)^r \, (e_1+e_2)^2\,x\,z)-4\,\bar{D}_c((e_1-e_2)^r\,z) \label{blowuprecursion} \end{equation} for all $\,z\in \bold{A}(X)$. \begin{lem} There are polynomials, $B_{k}(x)$, independent of $X$, so that for any $c\in H_2(X;\bold{Z})$ and $\,z\in \bold{A}(X)$ we have $\hat{D}_c(e^k\,z)= D_c(B_k(x)\,z)$.\end{lem} \begin{pf} Lemma~\ref{blowuplow}(1) implies that $\beta_{c,0}=D_c$. Thus we have $B_0=1$. Assume inductively that for $j\le k$, $\hat{D}_c(e^{j}z)= D_c(B_{j}(x)z)$. Expanding \eqref{blowuprecursion} via the induction hypothesis we have \begin{multline*} \bar{D}_c((e_1-e_2)^{k-3}\,(e_1+e_2)^4\,z) = -4\,D_c(\,z\,\sum_{i=0}^{k-3}\binom{k-3}{i}\{ xB_{i+2}(x)B_{k-3-i}(x) \\ -2x B_{i+1}(x)B_{k-2-i}(x) + x B_i(x)B_{k-1}(x) + B_i(x)B_{k-3-i}(x)\}\,) = D_c(P(x)\,z) \end{multline*} for some polynomial $P$. On the other hand, expanding the argument of \[ \bar{D}_c((e_1-e_2)^{k-3}(e_1+e_2)^4\,z) \] and using the induction hypothesis in a similar fashion, we get \[\bar{D}_c((e_1-e_2)^{k-3}(e_1+e_2)^4\,z) = 2\,\hat{D}_c(e^{k+1}\,z) + D_c(R(x)\,z) \] for another polynomial $R$. The lemma follows.\end{pf} \begin{lem} There are polynomials, $S_{k}(x)$, independent of $X$, so that for any $c\in H_2(X;\bold{Z})$ and $\,z\in \bold{A}(X)$ we have $\hat{D}_{c+e}(e^k\,z)= D_c(S_k(x)\,z)$. \end{lem} \begin{pf} By Theorem~\ref{Ruber} we have for any even $k>0$, \[ \bar{D}_c((e_1+e_2)^k(e_1-e_2)^2)=2\,D_{c+e_1-e_2}((e_1+e_2)^k)=-2\,D_{c+e_1+e_2}((e_1+e_2)^k) \] This formula can then be used as above to inductively calculate $S_{k-1}(x)$ in terms of $S_1(x),\dots,S_{k-3}(x)$ and $B_0(x),\dots,B_{k+2}(x)$. \end{pf} We now explicitly determine the polynomials $B_k(x)$ and $S_k(x)$. Set \[ B(x,t)=\sum_{t=0}^\infty B_k(x)\frac{t^k}{k!}\hspace{.25in}\text{and}\hspace{.25in} S(x,t)=\sum_{t=0}^\infty S_k(x)\frac{t^k}{k!}\] Note that \begin{equation} \begin{split}\frac{d^n}{dt^n}\,\hat{D}(\exp(te)\,z) & =\hat{D}(\,z\,\sum e^{k+n}\frac{t^k}{k!})= D(B_{k+n}(x)\frac{t^k}{k!}\,z) \\ &=\frac{d^n}{dt^n}\,D(B(x,t)\,z) = D(B^{(n)}(x,t)\,z) \end{split} \label{diff} \end{equation} where the last differentiation is with respect to $t$. On $\bar{X} = X\#2\overline{\bold{CP}}^{\,2}$, we get $\bar{D}(\exp(t_1e_1+t_2e_2)z)=D(B(x,t_1)\,B(x,t_2)z)$. Now apply Corollary \ref{4ofem} to $e_1-e_2\in H_2(\bar{X};\bold{Z})$. Since for any $t\in\bold{R}$ the class $te_1+te_2\in\langle e_1-e_2\rangle^\perp$, we have the equation \begin{equation} \begin{split}\bar{D}(\exp(te_1+te_2)\,(e_1-e_2)^4\,z) & + 4\, \bar{D}(x\,\exp(te_1+te_2)\,(e_1-e_2)^2\,z) \\ &+4\,\bar{D}(\exp(te_1+te_2)\,z) = 0 \end{split} \label{pre} \end{equation} But, for example, \[ e_1^4\,\exp(te_1+te_2) = \left(\sum e_1^{k+4}\frac{t^k}{k!}\right) \left(\sum e_2^k\frac{t^k}{k!}\right) = \frac{d^4}{dt^4}(\exp(te_1))\,\exp(te_2) \] Arguing similarly and using \eqref{diff} \begin{equation*} \begin{split} & \bar{D}(\exp(te_1+te_2)\,(e_1-e_2)^4\,z) \\ &=D((2\,B^{(4)}(x,t)\,B(x,t)-8\,B'''(x,t)\,B'(x,t) +6\,(B''(x,t))^2)\,z)\\ &= 2\,D((B^{(4)}\,B-4\,B'''\,B'+3\,(B'')^2 )\,z) \end{split} \end{equation*} where $B=B(x,t)$. Completing the expansion of \eqref{pre} we get \[2\,D((B^{(4)}\,B-4\,B'''\,B'+3\,(B'')^2 +4x\,(B''B-(B')^2)+2\,B^2)\,z)=0 \] for all $\,z\in \bold{A}(X)$. This means that the expression \[ B^{(4)}\,B-4\,B'''\,B'+3\,(B'')^2 +4x\,(B''B-(B')^2)+2\,B^2 \] lies in the kernel of $D:\bold{A}(X)\to\bold{R}$. Thus the ``blowup function'' $B(x,t)$ satisfies the differential equation \[ B^{(4)}\,B-4\,B'''\,B'+3\,(B'')^2 +4x\,(B''B-(B')^2)+2\,B^2 =0 \] modulo the kernel of $D$. Of course, the fact that this equation holds only modulo the kernel of $D$ is really no constraint, since our interest in $B(x,t)$ comes from the equation $\hat{D}(\exp(te)z)=D(B(x,t)z)$. Now let $B=\exp(f(t))$. \begin{prop} Modulo the kernel of $D$, the logarithm $f(t)$ of $B(x,t)$ satisfies the differential equation \[ f^{(4)}+6\,(f'')^2+4xf''+2=0\] with the initial conditions $f=f'=f''=f'''=0$. \ \ \qed \label{DE}\end{prop} The initial conditions follow from Lemma~\ref{blowuplow}. \bigskip \section{The blowup formula} In order to solve the differential equation of Proposition~\ref{DE}, we set $u=f''$. Then the differential equation becomes \[ u''+6u^2+4xu+2=0 \] with initial conditions $u(0)=u'(0)=0$. This is equivalent to the equation \begin{equation} (u')^2=-4u^3-4xu^2-4u \label{uDE}\end{equation} as can be seen by differentiating both sides of the last equation with respect to $t$. Replacing $u$ by $-v$ and completing the cube yields \[ (v')^2=4(v-\frac{x}{3})^3-\frac43 vx^2 +\frac{4x^3}{27}+4v \] Finally, letting $y=v-\frac{x}{3}$ we get \begin{equation} (y')^2=4y^3-g_2y-g_3 \hspace{.2in} \text{where} \hspace{.2in} g_2=4\,(\frac{x^2}{3}-1) \hspace{.1in} \text{and} \hspace{.1in} g_3={8x^3-36x\over 27} . \label{yDE}\end{equation} This is the equation which defines the Weierstrass $\wp$-function. In fact, if we rewrite \eqref{yDE} as \[ \frac{dt}{dy}={1\over\sqrt{4y^3-g_2y-g_3}}\] then \[t=\int_y^\infty{ds\over\sqrt{4s^3-g_2s-g_3}}=\wp^{-1}(y)\] and we see that for arbitrary constants $c$, $y=\wp(t+c)$ gives all solutions to \eqref{yDE}, and so $f''=u=-(\wp(t+c)+{x\over3})$ is the general solution of \eqref{uDE}. The roots of the cubic equation \[ 4s^3-g_2s-g_3=0 \] are \begin{equation} e_1=\frac{x}{6}+{\sqrt{x^2-4}\over2}, \hspace{.15in} e_2=\frac{x}{6}-{\sqrt{x^2-4}\over2}, \hspace{.15in} e_3=-\frac{x}{3} \label{roots} \end{equation} where we have followed standard notation (cf. \cite{Ak}). These correspond to the half-periods $\omega_i=\wp^{-1}(e_i)$ of the $\wp$-function. The initial condition $f''(0)=0$ implies that $\wp(c)=-\frac{x}{3}=e_3$; so $c=\omega_3+2\varpi$, where $2\varpi=2m_1\omega_1+2m_3\omega_3$, with $m_1$, $m_3\in\bold{Z}$, is an arbitrary period. (Note that the initial condition $f'''(0)=0$ follows because the half-periods are \,zeros of $\wp'$.) The Weierstrass \,zeta-function satisfies $\zeta'=-\wp$; thus $f'(t)=\zeta(t+\omega_3+2\varpi)-{tx\over3}+ a$. The constant $a$ is determined by the initial condition $f'(0)=0$;\; $a=-\zeta(\omega_3+2\varpi)$. Since the logarithmic derivative of the Weierstrass sigma-function is $\zeta$, integrating one more time gives $f(t)=\log\sigma(t+\omega_3+2\varpi)-t\zeta(\omega_3+2\varpi)-{t^2x\over6}+b$, and the initial condition $f(0)=0$ shows that $b=-\log\sigma(\omega_3+2\varpi)$. Thus \[ B(x,t)=e^{f(t)}=e^{-{t^2x\over6}}e^{-t\zeta(\omega_3+2\varpi)}\,{\sigma(t+\omega_3+2\varpi)\over\sigma(\omega_3+2\varpi)}. \] For $\omega=\omega_1$ or $\omega_3$ and $\eta=\zeta(\omega)$ we have the formulas \[\zeta(u+2m\omega)=2m\eta+\zeta(u)\, ,\hspace{.2in} \sigma(u+2m\omega)=(-1)^me^{2\eta(mu+m^2\omega)}\sigma(u)\] (which follow easily from \cite[p.199]{Ak}). Using them, our formula for $B(x,t)$ becomes \[ B(x,t)=e^{-{t^2x\over6}}e^{-\eta_3 t}\,{\sigma(t+\omega_3)\over\sigma(\omega_3)}.\] The above addition formula for the sigma-function implies that \[\sigma(t+\omega_3)=\sigma((t-\omega_3)+2\omega_3)=-e^{2\eta_3t}\sigma(t-\omega_3).\] Thus \[ B(x,t)=-e^{-{t^2x\over6}}e^{\eta_3 t}\,{\sigma(t-\omega_3)\over\sigma(\omega_3)}=e^{-{t^2x\over6}}\sigma_3(t), \] the last equality by the definition of the quasi-periodic function $\sigma_3$. In conclusion, \begin{thm} Modulo the kernel of $D$, the blowup function $B(x,t)$ is given by the formula \[ B(x,t)=e^{-{t^2x\over6}}\sigma_3(t).\ \ \ \qed\] \label{blowup}\end{thm} \noindent The indexing of the Weierstrass functions $\sigma_i$ depends on the ordering of the roots $e_i$ of the equation $4s^3-g_2s-g_3=0$. This can be confusing. The important point is that the sigma-function we are using corresponds to the root $-{x\over3}$. One can now obtain the individual blowup polynomials from the formula for $B(x,t)$. For example, $B_{12}=-512\,x^4-960\,x^2-408$ and (for fun), \begin{multline*} \!B_{30}(x)\!=\!134,217,728\,x^{13}+4,630,511,616\,x^{11}+ 68,167,925,760\,x^9-34,608,135,536,640\,x^7\\ -39,641,047,695,360\,x^5-9,886,101,110,784\,x^3+543,185,367,552\,x \end{multline*} (We thank Alex Selby for help with some computer calculations.) We also have \begin{thm} Modulo the kernel of $D$, the blowup function $S(x,t)$ is given by the formula \[ S(x,t)=e^{-{t^2x\over6}}\sigma(t).\] \label{SO3blowup}\end{thm} \begin{pf} As usual we let $\bar{X}=X\#2\overline{\bold{CP}}^{\,2}$ with exceptional classes $\varepsilon_1$ and $\varepsilon_2$. (We have temporarily changed notation to avoid confusion with the roots $e_i$ of $4s^3-g_2s-g_3=0$.) Consider the class $\varepsilon_1-\varepsilon_2$ which is represented by a sphere of self-intersection $-2$. By Theorem~\ref{Ruber} we have \[\bar{D}(\exp(t\varepsilon_1+t\varepsilon_2)\,(\varepsilon_1-\varepsilon_2)^2\,z)=2\,D_{\varepsilon_1-\varepsilon_2}(\exp(t\varepsilon_1+t\varepsilon_2)\,z)= -2\,D_{\varepsilon_1+\varepsilon_2}(\exp(t\varepsilon_1+t\varepsilon_2)\,z)\] for all $\,z\in \bold{A}(X)$. Equivalently we get $D((2\,B''B-2\,(B')^2)\,z)=-2\,D(S^2\,z)$. In other words, \[ S^2=e^{-{t^2x\over3}}({x\over3}\sigma_3^2+(\sigma_3')^2-\sigma_3\sigma_3'')\] Write $\sigma_3(t)=\exp(h(t))$. Then \[ S^2= e^{-{t^2x\over3}}e^{2h}({x\over3}-h'') \] i.e. \[ S=\pm e^{-{t^2x\over6}}\sigma_3(t)({x\over3}-h'')^{\frac12}. \] Since $\exp(h)=\sigma_3(t)=\sigma(t)(\wp(t)-e_3)^{\frac12}$, it follows that $h=\log\sigma(t)+\frac12\log(\wp(t)-e_3)$. Then \[h'=\zeta(t)+\frac12{\wp'(t)\over\wp(t)-e_3}=\zeta(t)+\frac12(\zeta(t+\omega_3)+\zeta(t-\omega_3)-2\zeta(t))\] by \cite[p.41]{Ak}. Thus $h'= \frac12(\zeta(t+\omega_3)+\zeta(t-\omega_3))$, and \[ h''=\frac12(-\wp(t+\omega_3)-\wp(t-\omega_3))=-\wp(t+\omega_3).\] Thus \[ S=\pm e^{-{t^2x\over6}}\sigma_3(t)(\wp(t+\omega_3)-e_3)^{\frac12}= \pm e^{-{t^2x\over6}}\sigma_3(t)\left({(e_3-e_1)(e_3-e_2)\over \wp(t)-e_3}\right)^{\frac12} \] (\cite[p.200]{Ak}). However, $(e_3-e_1)(e_3-e_2)=1$ (see\eqref{roots}); so \[ S =\pm e^{-{t^2x\over6}}{\sigma_3(t)\over\sqrt{\wp(t)-e_3}}=\pm e^{-{t^2x\over6}}\sigma(t)\, . \] To determine the sign, note that (fixing $x$) $S'(0)=S_1=1$ by Theorem~\ref{blowuplow}(6). But from our formula $S'(0)=\pm\sigma'(0)$, and $\sigma'(0)=1$. \end{pf} \bigskip \section{The blowup formula for manifolds of simple type} A $4$-manifold is said to be of {\em simple type} \cite{KM} if for all $z\in \bold{A}(X)$ the relation $D(x^2z)=4\,D(z)$ is satisfied by its Donaldson invariant. It is clear that if $X$ has simple type, then $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$ does as well. In this case, following \cite{KM}, one considers the invariant $\bold{D}$ defined by \[\bold{D}(\alpha)=D((1+{x\over2})\exp(\alpha))\] for all $\alpha\in H_2(X)$. $\bold{D}$ is called the {\em Donaldson series} of $X$. Note that the simple type condition implies that for any $z\in\bold{A}(X)$, \[D((1+{x\over2})\,z\,x)=2\,D((1+{x\over2})\,z)\, ,\] i.e. $x$ acts as multiplication by $2$ on $D(1+{x\over2})$. The blowup formula in this case has been determined previously by Kronheimer and Mrowka. In this section, we derive that formula by setting $x=2$ in Theorems~\ref{blowup} and \ref{SO3blowup}. This gives a degenerate case of the associated Weierstrass functions. All the formulas below involving elliptic functions can be found in \cite{Ak}. The squares $k^2$, ${k'}^2$ of the modulus and complementary modulus of our Weierstrass functions are given by \[ k^2={x-\sqrt{x^2-4}\over x+\sqrt{x^2-4}} \hspace{.35in} {k'}^2={2\sqrt{x^2-4}\over x+\sqrt{x^2-4}}. \] Thus $k^2=1$ and ${k'}^2= 0$ when $x=2$. The corresponding complete elliptic integrals of the first kind are \begin{eqnarray*} K&=&\int_0^1{ds\over\sqrt{(1-s^2)(1-k^2s^2)}}= \int_0^1{ds\over1-s^2}\\ K'&=& \int_0^1{ds\over\sqrt{(1-s^2)(1-{k'}^2s^2)}}= \int_0^1{ds\over\sqrt{1-s^2}} \end{eqnarray*} Thus $K=\infty$ and $K'={\pi\over2}$ when $x= 2$. Also, when $x=2$ we have $g_2=\frac43$ and $g_3=-\frac{8}{27}$; so the roots of $4s^3-g_2s-g_3=0$ are $e_1=e_2=\frac13$ and $e_3=-\frac23$. This means that when $x=2$ the basic periods are \[ \omega_1={K\over\sqrt{e_1-e_3}}= K=\infty \hspace{.35in} \omega_3={iK'\over\sqrt{e_1-e_3}}= iK' ={i\pi\over2}\,.\] In this situation, \[\sigma(t)={2\omega_3\over\pi}e^{{1\over6}({\pi t\over2\omega_3})^2}\sin{\pi t\over2\omega_3}= e^{-{t^2\over6}}\sinh t \] and \[\wp(t)=-{1\over3}({\pi\over2\omega_3})^2+({\pi\over2\omega_3})^2{1\over\sin^2({\pi t\over2\omega_3})} ={1\over3}+{1\over\sinh^2t} \, .\] So \[ \sigma_3(t)=\sigma(t)\sqrt{\wp(t)-e_3}=e^{-{t^2\over6}}\sinh t\sqrt{1+{1\over\sinh^2t}} = e^{-{t^2\over6}}\cosh t. \] \begin{thm} If $X$ has simple type, the Donaldson series of $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$ is \[ \hat{\bold{D}}=\bold{D}\cdot e^{-{E^2\over2}}\cosh E \] where $E$ is the form dual to the exceptional class $e$, i.e. $E(\,z)=e\cdot \,z$ for all $\,z\in H_2(\hat{X})$. Also \[ \hat{\bold{D}}_e=-\bold{D}\cdot e^{-{E^2\over2}}\sinh E\,. \] \end{thm} \begin{pf} For $\alpha\in H_2(X)$ we calculate \begin{eqnarray*} \hat{\bold{D}}(\alpha+te) &=& \hat{D}((1+{x\over2})\exp(\alpha)\exp(te))=D((1+{x\over2})\exp(\alpha)B(x,t))\\ &=& D((1+{x\over2})\exp(\alpha)e^{-{t^2x\over6}} e^{-{t^2\over6}}\cosh t) \end{eqnarray*} The simple type condition implies that $D((1+{x\over2})e^{-{t^2x\over6}})= D((1+{x\over2})e^{-{t^2\over3}})$. Hence \[ \hat{\bold{D}}(\alpha+te)=\bold{D}(\alpha)e^{-{t^2\over2}}\cosh t=\bold{D}(\alpha)(e^{-{E^2\over2}}\cosh E)(te)\] as desired. The formula for $\hat{\bold{D}}_e$ follows similarly since $\sinh(E)(te)=-\sinh(t)$. \end{pf} A $4$-manifold $X$ is said to have $c$-{\em simple type} if, for $c\in H_2(X;\bold{Z})$, $D_c(x^2\,z)=4\,D_c(z)$ for all $z\in \bold{A}(X)$. It is shown in \cite{FS}, and also by Kronheimer and Mrowka, that if $X$ has simple type, then it has $c$-simple type for all $c\in H_2(X;\bold{Z})$. As above we have, \begin{thm}If $X$ has $c$-simple type, \begin{eqnarray*} \hat{\bold{D}}_c&=&\bold{D}_c\cdot e^{-{E^2\over2}}\cosh E\,. \\ \hat{\bold{D}}_{c+e}&=&-\bold{D}_c\cdot e^{-{E^2\over2}}\sinh E\,.\ \ \ \qed \end{eqnarray*} \end{thm} \newpage

9510

alg-geom/9510006

en

https://arxiv.org/abs/alg-geom/9510006

alg-geom/9510006

Yekutieli Amnon

Amnon Yekutieli

Some Remarks on Beilinson Adeles

AMSLaTeX 1.1, 6 pages, to appear in: Proc. AMS (replaced only to fix a latex problem)

This paper contains two remarks on Beilinson's adeles with values in the De Rham complex of a scheme. The first is an interpretation, in terms of adeles, of the decomposition of the De Rham complex on a scheme defined modulo $p^{2}$ (the result of Deligne-Illusie). The second remark is about the possible relation between adeles and Hodge decomposition. We work out a counter example.

alg-geom

\subsection{Introduction} In this note we consider two aspects of Beilinson adeles on schemes. Let $X$ be a scheme of finite type over a field $k$. Given a quasi-coherent sheaf $\cal{M}$ let $\underline{\Bbb{A}}^{q}_{\mathrm{red}}(\cal{M})$ be the sheaf of reduced Beilinson adeles of degree $q$ (see \cite{Be}, \cite{Hr}, \cite{HY1}). It is known that $\underline{\Bbb{A}}^{q}_{\mathrm{red}}(\cal{M}) \cong \underline{\Bbb{A}}^{q}_{\mathrm{red}}(\cal{O}_{X}) \otimes_{\cal{O}_{X}} \cal{M}$. For any open set $U \subset X$ \begin{equation} \label{eqn0.1} \Gamma(U, \underline{\Bbb{A}}^{q}_{\mathrm{red}}(\cal{M})) \subset \prod_{\xi \in S(U)^{\mathrm{red}}_{q}} \cal{M}_{\xi} \end{equation} where $S(U)^{\mathrm{red}}_{q}$ is the set of reduced chains of points in $U$ of length $q$, and $\cal{M}_{\xi}$ is the Beilinson completion of $\cal{M}$ along the chain $\xi$ (cf.\ \cite{Ye1}). For $q=0$ and $\cal{M}$ coherent one has $\cal{M}_{(x)} = \widehat{\cal{M}}_{x}$, the $\frak{m}_{x}$-adic completion, and (\ref{eqn0.1}) is an equality. Let $\Omega^{{\textstyle \cdot}}_{X/k}$ be the De Rham complex on $X$, relative to $k$. As shown in \cite{HY1}, setting $\cal{A}_{X}^{p, q} := \underline{\Bbb{A}}^{q}_{\mathrm{red}}(\Omega^{p}_{X/k})$ and $\cal{A}_{X}^{i} := \bigoplus_{p+q=i} \cal{A}_{X}^{p, q}$ we get a differential graded algebra (DGA), which is quasi-isomorphic to $\Omega^{{\textstyle \cdot}}_{X/k}$ and is flasque. Thus $\mathrm{H}^{{\textstyle \cdot}}(X, \Omega^{{\textstyle \cdot}}_{X/k}) = \mathrm{H}^{{\textstyle \cdot}} \Gamma(X, \cal{A}_{X}^{{\textstyle \cdot}})$. In particular if $X$ is smooth, we get the De Rham cohomology $\mathrm{H}^{{\textstyle \cdot}}_{\mathrm{DR}}(X/k)$. More generally, let $\frak{X}$ be a formal scheme, of formally finite type (f.f.t.) over $k$ (see \cite{Ye2}). Then applying the adelic construction to the complete De Rham complex $\widehat{\Omega}^{{\textstyle \cdot}}_{\frak{X}/k}$ we get a DGA $\cal{A}_{\frak{X}}^{{\textstyle \cdot}}$. If $X \subset \frak{X}$ is a smooth formal embedding (op.\ cit.) and $\operatorname{char} k = 0$ then $\mathrm{H}^{{\textstyle \cdot}} \Gamma(X, \cal{A}_{\frak{X}}^{{\textstyle \cdot}})= \mathrm{H}^{{\textstyle \cdot}}_{\mathrm{DR}}(X/k)$. There is an analogy between the sheaf $\cal{A}_{X}^{p,q}$ on a smooth $n$-dimensional variety $X$ and the sheaf of smooth $(p,q)$-forms on a complex manifold. The coboundary operator $\mathrm{D}$ of $\cal{A}_{X}^{{\textstyle \cdot}}$ is defined as a sum $\mathrm{D} := \mathrm{D}' + \mathrm{D}''$, and $\mathrm{D}'' : \cal{A}_{X}^{p, q} \rightarrow \cal{A}_{X}^{p, q+1}$ plays the role of the anti-holomorphic derivative. The map $\int_{X} = \sum_{\xi} \operatorname{Res}_{\xi} : \Gamma(X, \cal{A}_{X}^{2n}) \rightarrow k$ is the counterpart of the integral ($\operatorname{Res}_{\xi}$ is the Parshin-Lomadze residue along the maximal chain $\xi$ in $X$, see \cite{Ye1}). This analogy to the complex manifold picture is quite solid; for example, in \cite{HY2} there is an algebraic proof of the Bott residue formula, which in some parts is just a translation of the original proof of Bott to the setting of adeles. The main purpose of this note is to examine the potential applicability of adeles for the study of algebraic De Rham cohomology. In \S 1 the construction of Deligne-Illusie \cite{DI} is rewritten in terms of adeles. In \S 2 we consider a possibility to relate adeles to Hodge theory, and show by example its failure. \subsection{Lifting Modulo $p^{2}$} We interpret, in terms of adeles, the result of Deligne-Illusie on the decomposition of the De Rham complex in characteristic $p$. In this part we shall follow closely the ideas and notation of \cite{DI}. Let $k$ be a perfect field of characteristic $p$. Write $\tilde{k} := W_{2}(k)$. Let $F_{k} : \operatorname{Spec} k \rightarrow \operatorname{Spec} k$ be the Frobenius morphism, i.e.\ $F^{*}_{k}(a) = a^{p}$ for $a \in k$. By pullback along $F_{k}$ we get a scheme $X' := X \times_{k} k$ and a finite, bijective $k$-morphism $F = F_{X/k} : X \rightarrow X'$. Assume we are given some lifting $\tilde{X}$ of $X$ to $\tilde{k}$. By this we mean a smooth scheme $\tilde{X}$ over $\tilde{k}$ s.t.\ $X \cong \tilde{X} \times_{\tilde{k}} k$. Using the Frobenius $F_{\tilde{k}}$ we also define a scheme $\tilde{X}'$, and a $\tilde{k}$-morphism $F_{\tilde{X}} : \tilde{X} \rightarrow \tilde{X}'$. For any point $x \in X$ the relative Frobenius homomorphism $F^{*}_{x} : \cal{O}_{X', F(x)} \rightarrow \cal{O}_{X, x}$ can be lifted to a $\tilde{k}$-algebra homomorphism $\tilde{F}^{*}_{x} : \cal{O}_{\tilde{X}', F(x)} \rightarrow \cal{O}_{\tilde{X}, x}$ (cf.\ \cite{DI}). In view of (\ref{eqn0.1}), the collection $\{ \tilde{F}^{*}_{x} \}_{x \in X}$ induces a homomorphism of sheaves of DG $\tilde{k}$-algebras \[ \tilde{F}^{*} : \underline{\Bbb{A}}^{0}_{\mathrm{red}}(\Omega^{{\textstyle \cdot}}_{\tilde{X}' / \tilde{k}}) \rightarrow F_{*} \underline{\Bbb{A}}^{0}_{\mathrm{red}}(\Omega^{{\textstyle \cdot}}_{\tilde{X} / \tilde{k}}) . \] \begin{lem} \label{lem1} The liftings $\{ \tilde{F}^{*}_{x} \}_{x \in X}$ determine $\cal{O}_{X'}$-linear homomorphisms \[ f : \Omega^{1}_{X' / k} \rightarrow F_{*} \cal{A}^{1, 0}_{X} \] \[ h : \Omega^{1}_{X' / k} \rightarrow F_{*} \cal{A}^{0, 1}_{X} \] such that \[ \mathrm{D} (f + h) = 0 . \] \end{lem} \begin{pf} Let $\nmbf{p} : \Omega^{{\textstyle \cdot}}_{X/k} \stackrel{\simeq}{\rightarrow} p \Omega^{{\textstyle \cdot}}_{\tilde{X}/\tilde{k}}$ be multiplication by $p$. This extends to an $\underline{\Bbb{A}}^{0}_{\mathrm{red}}(\cal{O}_{X})$-linear isomorphism \[ \nmbf{p} : \cal{A}^{{\textstyle \cdot}, 0}_{X} = \underline{\Bbb{A}}^{0}_{\mathrm{red}}(\Omega^{{\textstyle \cdot}}_{X/k}) \stackrel{\simeq}{\rightarrow} p \underline{\Bbb{A}}^{0}_{\mathrm{red}}(\Omega^{{\textstyle \cdot}}_{\tilde{X}/\tilde{k}}) . \] Just as in \cite{DI} we get a homomorphism $f$ making the diagram \[ \setlength{\unitlength}{0.20mm} \begin{array}{ccc} \Omega^{1}_{\tilde{X}'/\tilde{k}} & \lrar{\tilde{F}^{*}} & p F_{*} \underline{\Bbb{A}}^{0}_{\mathrm{red}}(\Omega^{1}_{\tilde{X}/\tilde{k}}) \\ \ldar{} & & \luar{\nmbf{p}} \\ \Omega^{1}_{X'/k} & \lrar{f} & F_{*} \underline{\Bbb{A}}^{0}_{\mathrm{red}}(\Omega^{1}_{X/k}) \end{array} \] commutative. Next, for any chain of points $(x_{0}, x_{1})$ in $X$ and a local section $a \in \cal{O}_{\tilde{X}'}$ we have \[ \mathrm{D}'' \tilde{F}^{*}(a) = \tilde{F}^{*}_{x_{0}}(a) - \tilde{F}^{*}_{x_{1}}(a) \in p \cal{O}_{\tilde{X}, (x_{0}, x_{1})} . \] Therefore \[ \mathrm{D}'' \tilde{F}^{*} : \cal{O}_{\tilde{X}'} \rightarrow p F_{*} \underline{\Bbb{A}}^{1}_{\mathrm{red}}(\cal{O}_{\tilde{X}}) \] is a derivation which kills $p \cal{O}_{\tilde{X}'}$, and we get an $\cal{O}_{X'}$-linear homomorphism $h$ s.t.\ the diagram \[ \setlength{\unitlength}{0.20mm} \begin{array}{ccc} \cal{O}_{\tilde{X}'} & \lrar{\mathrm{D}'' \tilde{F}^{*}} & F_{*} p \underline{\Bbb{A}}^{1}_{\mathrm{red}}(\cal{O}_{\tilde{X}}) \\ \ldar{\mathrm{d}} & & \luar{\nmbf{p}} \\ \Omega^{1}_{X'/k} & \lrar{h} & F_{*} \underline{\Bbb{A}}^{1}_{\mathrm{red}}(\cal{O}_{X}) \end{array} \] commutes. Reinterpreting the calculations of \cite{DI} in terms of adeles we see that the following hold: for each point $x_{0} \in X$, $\mathrm{D}' f = 0$ in $\Omega^{1}_{X/k, (x_{0})}$; for each chain $(x_{0}, x_{1}, x_{2})$ in $X$, $\mathrm{D}'' h = 0$ in $\cal{O}_{X, (x_{0}, x_{1}, x_{2})}$; lastly, for each chain $(x_{0}, x_{1})$, $\mathrm{D}'' f = - \mathrm{D}' h$ in $\Omega^{1}_{X/k, (x_{0}, x_{1})}$. This implies that on the level of sheaves $\mathrm{D}(f + h) = 0$. \end{pf} \begin{prop} \label{prop1} The liftings $\{ \tilde{F}^{*}_{x} \}_{x \in X}$ determine an $\cal{O}_{X'}$-linear homomorphism of complexes \[ \psi_{\tilde{X}} : \bigoplus_{i = 0}^{n} \underline{\Bbb{A}}^{{\textstyle \cdot}}_{\mathrm{red}}(\Omega^{i}_{X' / k})[-i] \rightarrow F_{*} \cal{A}^{{\textstyle \cdot}}_{X} \] making the diagram \begin{equation} \label{eqn6} \setlength{\unitlength}{0.25mm} \begin{array}{ccc} \bigoplus_{i} \Omega^{i}_{X' / k} & \lrar{C^{-1}} & F_{*} \mathrm{H}^{{\textstyle \cdot}} \Omega^{{\textstyle \cdot}}_{X / k} \\ \ldar{} & & \ldar{} \\ \bigoplus_{i} \mathrm{H}^{{\textstyle \cdot}} \underline{\Bbb{A}}^{{\textstyle \cdot}}_{\mathrm{red}}(\Omega^{i}_{X' / k})[-i] & \lrar{\mathrm{H}^{{\textstyle \cdot}}(\psi_{\tilde{X}})} & F_{*} \mathrm{H}^{{\textstyle \cdot}} \cal{A}^{{\textstyle \cdot}}_{X} \end{array} \end{equation} commute. Here $C^{-1}$ is the Cartier operation, and the vertical arrows are the canonical isomorphisms. Therefore $\psi_{\tilde{X}}$ is a quasi-isomorphism. \end{prop} \begin{pf} Since \[ \underline{\Bbb{A}}^{j}_{\mathrm{red}}(\Omega^{i}_{X' / k}) \cong \underline{\Bbb{A}}^{j}_{\mathrm{red}}(\cal{O}_{X'}) \otimes_{\cal{O}_{X'}} \Omega^{i}_{X' / k} , \] and since $F^{*} : \underline{\Bbb{A}}^{{\textstyle \cdot}}_{\mathrm{red}}(\cal{O}_{X'}) \rightarrow F_{*} \underline{\Bbb{A}}^{{\textstyle \cdot}}_{\mathrm{red}}(\cal{O}_{X})$ commutes with $\mathrm{D}''$ and is killed by $\mathrm{D}'$ (i.e.\ $\mathrm{D}' F^{*} = 0$) it suffices to define $\cal{O}_{X'}$-linear homomorphisms $\psi_{\tilde{X}}^{i} : \Omega^{i}_{X' / k} \rightarrow F_{*} \cal{A}^{i}_{X}$ s.t.\ $\mathrm{D} \psi_{\tilde{X}}^{i} = 0$. Define $\psi_{\tilde{X}}^{0} := F^{*}$, and $\psi_{\tilde{X}}^{1} := f+h$ as in Lemma \ref{lem1}. For $1 \leq i \leq n$ let $\nmbf{a} : \Omega^{i}_{X' / k} \rightarrow (\Omega^{1}_{X' / k})^{\otimes i}$ be the anti-symmetrizing operator (this makes sense since $n < p$; cf.\ \cite{DI}), and define $\psi_{\tilde{X}}^{i}$ by \[ \setlength{\unitlength}{0.30mm} \begin{array}{ccc} (\Omega^{1}_{X' / k})^{\otimes i} & \lrar{(\psi_{\tilde{X}}^{1})^{\otimes i}} & (F_{*} \cal{A}^{1}_{X})^{\otimes i} \\ \luar{\nmbf{a}} & & \ldar{\mathrm{product}} \\ \Omega^{i}_{X' / k} & \lrar{\psi_{\tilde{X}}^{i}} & F_{*} \cal{A}^{i}_{X} \end{array} \] Let $a \in \cal{O}_{\tilde{X}}$ be a local section, with corresponding pullback $a \otimes 1 \in \cal{O}_{\tilde{X}'}$, and with image $a_{0} \in \cal{O}_{X}$. Then according to the calculations in \cite{DI}, we have $\tilde{F}^{*}(a \otimes 1) = a^{p} + \nmbf{p} u$ for some local section $u \in \underline{\Bbb{A}}^{0}_{\mathrm{red}}(\cal{O}_{X})$. Therefore $f (\mathrm{d} a_{0} \otimes 1) = a_{0}^{p-1} \mathrm{d} a_{0} + \mathrm{D}' u$ and $h (\mathrm{d} a_{0} \otimes 1) = \mathrm{D}'' u$, so \[ \psi_{\tilde{X}} (\mathrm{d} a_{0} \otimes 1) = a_{0}^{p-1} \mathrm{d} a_{0} + \mathrm{D} u . \] This means that \[ \mathrm{H}^{1} (\psi_{\tilde{X}}) = C^{-1} : \Omega^{1}_{X' / k} \stackrel{\simeq}{\rightarrow} F_{*} \mathrm{H}^{1} \Omega^{{\textstyle \cdot}}_{X / k} \cong F_{*} \mathrm{H}^{1} \cal{A}^{{\textstyle \cdot}}_{X} . \] Clearly in degree $0$, $\mathrm{H}^{0} (\psi_{\tilde{X}}) = F^{*} = C^{-1}$. Since the vertical arrows in diagram (\ref{eqn6}) are isomorphisms of (sheaves of) graded algebras, it follows that $\mathrm{H}^{{\textstyle \cdot}} \cal{A}^{{\textstyle \cdot}}_{X}$ is a graded-commutative algebra, and therefore \[ \mathrm{H}^{{\textstyle \cdot}}(\psi_{\tilde{X}}) : \bigoplus_{i} \Omega^{i}_{X' / k} \rightarrow F_{*} \mathrm{H}^{{\textstyle \cdot}} \cal{A}^{{\textstyle \cdot}}_{X} \] is a homomorphism of graded algebras. But then $\mathrm{H}^{{\textstyle \cdot}}(\psi_{\tilde{X}}) = C^{-1}$ in all degrees, and it's an isomorphism. \end{pf} Of course in the derived category the map $\psi_{\tilde{X}}$ is independent of the choices of Frobenius liftings. \subsection{A Hodge-type Decomposition?} The second aspect is a naive attempt to use adeles for a Hodge-type decomposition of De Rham cohomology. Suppose $\operatorname{char} k = 0$ and $X$ is smooth over $k$, of dimension $n$. For any $0 \leq p, q \leq n$ define a canonical subspace \begin{equation} \mathrm{H}^{p,q} := \frac{ \Gamma(X, \cal{A}^{p,q}_{X}) \cap \operatorname{Ker} D } { \Gamma(X, \cal{A}^{p,q}_{X}) \cap \operatorname{Im} D } \subset \mathrm{H}^{p+q}_{\mathrm{DR}}(X/k) \end{equation} (cf.\ \cite{GH} p.\ 116). Since the sheaves $\cal{A}^{p, q}_{X}$ imitate the Dolbeault sheaves on a complex manifold so nicely, one can imagine that \[ \mathrm{H}^{i}_{\mathrm{DR}}(X/k) = \bigoplus_{p+q=i} \mathrm{H}^{p, q} \] if $X$ is proper. Yet this is {\em false}, as can be seen from the example below. What we get is a serious breakdown in the analogy to smooth forms on a complex manifold. I should mention that even in \cite{HY2} there was a breakdown in this analogy; there it was not possible to define a connection on the adelic sections of a vector bundle, and hence an auxiliary algebraic device, the sheaf $\tilde{\cal{A}}_{X}^{{\textstyle \cdot}}$ of Thom-Sullivan adeles, had to be introduced. \begin{prob} Is it true that for $X$ smooth, the filtration on $\cal{A}_{X}^{{\textstyle \cdot}}$ by the subcomplexes $\cal{A}_{X}^{{\textstyle \cdot}, \geq q}$ induces the coniveau filtration on $\mathrm{H}^{{\textstyle \cdot}}_{\mathrm{DR}}(X/k)$? \end{prob} \begin{exa} \label{exa1} Suppose $k$ is algebraically closed and $X$ is an elliptic curve. Then $\operatorname{dim} \mathrm{H}^{1}_{\mathrm{DR}}(X/k) = 2$. Consider the nondegenerate pairing on $\mathrm{H}^{1}_{\mathrm{DR}}(X/k)$ given by \[ \langle [\alpha] , [\beta] \rangle = \int_{X} [\alpha] \smile [\beta] = \sum_{\xi} \operatorname{Res}_{\xi} (\alpha \cdot \beta) \] for adeles $\alpha, \beta \in \cal{A}^{1}_{X}$. We see that $\langle \mathrm{H}^{1, 0} , \mathrm{H}^{1, 0} \rangle = \langle \mathrm{H}^{0, 1} , \mathrm{H}^{0, 1} \rangle = 0$. Therefore if $\mathrm{H}^{1}_{\mathrm{DR}}(X) = \mathrm{H}^{1, 0} + \mathrm{H}^{0, 1}$, then $\operatorname{dim} \mathrm{H}^{1, 0} = \operatorname{dim} \mathrm{H}^{0, 1} = 1$. It is easy to find $0 \neq [\alpha] \in \mathrm{H}^{1, 0}$; take any $0 \neq [\alpha] \in \Gamma(X, \Omega^{1}_{X/k})$. On the other hand an adele \[ \beta = (b_{(\mathrm{gen}, x)}) \in \Gamma(X, \cal{A}^{0, 1}_{X}) = \Bbb{A}^{1}_{\mathrm{red}}(X, \cal{O}_{X}) \] (where $x$ runs over the set $X_{0}$ of closed points, and $\mathrm{gen}$ is the generic point) satisfies $\mathrm{D} \beta = 0$ iff $\mathrm{d} b_{(\mathrm{gen}, x)} = 0$ for every $x$. This forces $b_{(\mathrm{gen}, x)} \in k$. But taking $b = (b_{(\mathrm{gen})}, b_{(x)}) \in \Bbb{A}^{0}_{\mathrm{red}}(X, \cal{O}_{X})$, with $b_{(\mathrm{gen})} = 0$, $b_{(x)} = b_{(\mathrm{gen}, x)}$ we get $\beta = \mathrm{D} b$. Hence $\mathrm{H}^{0, 1} = 0$. \end{exa} \begin{prob} \label{prob2} For $\alpha$ as above find explicitly a cocycle $\beta \in \Gamma(X, \cal{A}^{1}_{X})$ s.t.\ $\langle \alpha , \beta \rangle = 1$. \end{prob} The best I can do is: \begin{prop} \label{prop2} Suppose $X$ is a smooth proper curve and $k$ is algebraic\-ally closed. Let $\alpha_{(\mathrm{gen})} \in \Omega^{1}_{k(X) / k}$ be a differential of the $2$-nd kind, namely $\operatorname{Res}_{(\mathrm{gen}, x)} \alpha_{(\mathrm{gen})} = 0$ for every $x \in X_{0}$. Then it defines a cocycle $\alpha \in \Gamma(X, \cal{A}^{1}_{X})$ whose component at $(\mathrm{gen})$ is $\alpha_{(\mathrm{gen})}$. Every cohomology class in $\mathrm{H}^{1}_{\mathrm{DR}}(X / k)$ is gotten in this way. The Hodge filtration is induced by the differentials of the $1$-st kind. \end{prop} \begin{pf} The adele $\alpha$ will be given by its bihomogeneous components, $\alpha = \alpha^{1,0} + \alpha^{0,1}$. We set $\alpha^{1,0} := (\alpha_{(\mathrm{gen})}, \alpha_{(x)})$ where for $x \in X_{0}$, $\alpha_{(x)} = 0$. Since $\operatorname{Res}_{(\mathrm{gen}, x)} \alpha_{(\mathrm{gen})} = 0$ there is some $a_{(\mathrm{gen}, x)} \in k(X)_{(\mathrm{gen}, x)}$ (unique up to adding a constant) s.t.\ $\mathrm{d} a_{(\mathrm{gen}, x)} = \alpha_{(\mathrm{gen})}$. Set $\alpha^{0,1} := ( a_{(\mathrm{gen}, x)} )$. Then $\alpha$ is evidently a cocycle. If $\alpha_{(\mathrm{gen})}$ is of the $1$-st kind then actually we get $a_{(\mathrm{gen}, x)} \in \cal{O}_{X, (x)}$; call this element also $a_{(x)}$. So we can define an adele $\tilde{\alpha} = \tilde{\alpha}^{1,0} + \tilde{\alpha}^{0,1}$ with $\tilde{\alpha}^{1,0} := (\alpha_{(\mathrm{gen})}, \mathrm{d} a_{(x)} )$ and $\tilde{\alpha}^{0,1} := 0$. We get a cocycle (cohomologous to $\alpha$), and conversely any cocycle in $\Gamma(X, \cal{A}^{1,0}_{X})$ looks like this. Consider the niveau spectral sequence of De Rham homology (cf.\ \cite{Ye3}). A comparison of dimensions shows that this degenerates at the $E_{2}$ term. Also the niveau filtration on $\mathrm{H}_{1}^{\mathrm{DR}}(X / k)$ is trivial. Hence we get \[ \begin{array}{rcl} \mathrm{H}_{1}^{\mathrm{DR}}(X / k) & = & \operatorname{Ker} \left( \mathrm{H}^{1} \Omega^{{\textstyle \cdot}}_{k(X) / k} \rightarrow \bigoplus_{x \in X_{0}} k \right) \\ & \cong & (\text{forms of the $2$-nd kind}) / (\text{exact forms}) . \end{array} \] Now the map $\mathrm{H}^{1}_{\mathrm{DR}}(X / k) \rightarrow \mathrm{H}_{1}^{\mathrm{DR}}(X / k)$, $[\alpha] \mapsto [\alpha] \frown [X] = \pm [\mathrm{C}_{X} \cdot \alpha]$ is bijective. A direct inspection reveals that the adele $\alpha = \alpha^{1,0} + \alpha^{0,1}$ is sent to the differential of the second kind $\alpha_{(\mathrm{gen})} \in \Omega^{1}_{k(X)/k}$. \end{pf}

9510

alg-geom/9510003

en

https://arxiv.org/abs/alg-geom/9510003

alg-geom/9510003

Nakajima Hiraku

Hiraku Nakajima

Instantons and affine Lie algebras

12 pages, AMSLaTeX v 1.1

Nucl.Phys.Proc.Suppl.46:154-161,1996

10.1016/0920-5632(96)00017-5

Various constructions of the affine Lie algebra action on the homology group of moduli spaces of instantons on 4-manifolds are discussed. The analogy between the local-global principle and the role of mass is also explained. The detailed proofs are given in separated papers \cite{Na-algebra,Na-Hilbert}.

alg-geom

\section{Introduction} Vafa and Witten \cite{VW} introduced topological invariants\footnote{ The author does not know how to define their invariants in a mathematically rigorous way. The difficulty lies in the lack of the compactness of relevant moduli spaces.} for $4$-manifolds using $N = 4$ topological supersymmetric Yang-Mills theory. Then the $S$-duality conjecture implies that the generating function of those invariants is a modular form of certain weight, where the summation runs over all $\operatorname{SU}(2)$ or $\operatorname{SO}(3)$-principal bundles of any topological types. (In general, it has the modular invariance only for $\Gamma_0(4)$, a subgroup of $\operatorname{SL}(2,\Bbb Z)$.) For some $4$-manifolds, they identify those invariants with the Euler numbers of the instanton moduli spaces. Then they can check the modular invariance for various $4$-manifolds, using mathematical results, Yoshioka's formulae \cite{Yo} for ${\Bbb P}^2$ and the blow up, G\"ottsche and Huybrechts's result \cite{GotHu} for the K3 surface, and the author's result for ALE spaces. On the other hand, the author's motivation of the study \cite{Na-quiver,Na-gauge,Na-algebra} of the homology groups of the instanton moduli spaces on ALE spaces is totally different. The author's motivation was trying to understand Ringel \cite{Ri} and Lusztig's \cite{Lu1,Lu2} constructions of the lower triangular part ${\bold U}_q^-$ of the quantized enveloping algebra. They used the moduli spaces of representations of quivers, and their cotangent bundle\footnote{They are not cotangent bundles rigorously. The situation is very much similar to the relation between the moduli space of vector bundles over a curve and Hitchin's moduli space of Higgs bundles.} can be identified with the instanton moduli spaces on ALE spaces, via the ADHM description \cite{KN}. The author showed that the generating function of the Euler numbers of the instanton moduli spaces on ALE spaces becomes the character of the affine Lie algebra, which has been known to have modular transformation property by Kac-Peterson (see \cite{Kac}). The definition of the affine Lie algebra representation on the homology group of the instanton moduli spaces is very geometric, and seems to be generalized, at least, to projective surfaces. The results of similar direction are announced recently by Ginzburg-Kapranov-Vasserot \cite{GKV} and Grojnowski \cite{Gj}. Unfortunately, our construction depends heavily on the complex structure of the base manifold. It is a challenging problem to generalize the construction to more general $4$-manifolds. One may need to reformulate the homology group of the moduli spaces, etc..... \subsection*{Acknowledgements} The author's understanding of the analogy between the local-global principle and the theory of mass came from lectures given by the seminar on Seiberg-Witten theory organized by K.~Ueno. He would like to speakers, especially S.-K.Yang. He also thank to G.~Moore and K.~Yoshioka for valuable discussions. \section{The Hilbert scheme of points and the Heisenberg algebra: Twist around points} In this section, we study the relationship between the Hilbert scheme of points and the Heisenberg algebra. The reasons why we study the Hilbert scheme are (a) it is a toy model for moduli spaces of instantons, (b) it appears in the boundary of the compactification of the instanton moduli spaces over projective surfaces, and (c) its homology group is isomorphic to that of a moduli space for some special cases \cite{GotHu}. We explain the reason~(b) a little bit more. Since the instanton moduli spaces are usually noncompact, one must compactify them to consider their Euler numbers. When the base manifold is a projective surface, the results of Donaldson and Uhlenbeck-Yau enable us to identify the instanton moduli space with the moduli space of $\mu$-stable holomorphic vector bundles (Hitchin-Kobayashi correspondence). Then the one of the most natural compactifications seems to be Gieseker-Maruyama's compactifications $\overline{\frak M}$\nobreak {}~\footnote{The Gieseker-Maruyama's compactifications are not smooth in general. In fact, Vafa-Witten's formula for the K3 surface gives the fractional Euler number. This may be the contribution of the singularities.}, namely moduli spaces of semi-stable torsion free sheaves. If $\cal E$ is a torsion free sheaf which is not locally free, its double dual ${\cal E}^{\vee\vee}$ is a locally free sheaf and we have an exact sequence \begin{equation*} 0 @>>> {\cal E} @>>> {\cal E}^{\vee\vee} @>>> {\cal E}^{\vee\vee}/{\cal E} @>>> 0. \end{equation*} Thus ${\cal E}$ can be determined by (a) ${\cal E}^{\vee\vee}$ and (b) ${\cal E}^{\vee\vee}\to{\cal E}^{\vee\vee}/{\cal E}$. The double dual ${\cal E}^{\vee\vee}$ is contained in the interior of $\overline{\frak M}$, but in the different component with lower second Chern number. Thus it is natural to expect that those studies can be decomposed into two parts, the interior~(a) and the quotient map~(b). And the variety of the quotient map~(b), which depends only on the rank of $\cal E$ and the length of ${\cal E}^{\vee\vee}/{\cal E}$, looks very much like the Hilbert scheme of points. In fact, the Hilbert scheme is the special case ${\cal E}^{\vee\vee} = {\cal O}$. The Betti numbers of the variety was computed by Yoshioka~\cite[0.4]{Yo}. Let $X$ be a projective surface defined over $\Bbb C$. Let $\Hilbn{X}$ be the component of the Hilbert scheme of $X$ parameterizing the ideals of ${\cal O}_X$ of colength $n$. It is smooth and irreducible. Let $S^n X$ denotes the $n$-th symmetric product of $X$. It parameterizes formal linear combinations $\sum n_i [x_i]$ of points $x_i$ in $X$ with coefficients $n_i\in{\Bbb Z}_{> 0}$ with $\sum n_i = n$. There is a canonical morphism $\pi\colon \Hilbn{X}\to S^n X$ defined by \begin{equation*} \pi({\cal J}) = \sum_{x\in X} \operatorname{length}({\cal O}_X/{\cal J})_x [x]. \end{equation*} It is known that $\pi$ is a resolution of singularities. G\"ottsche \cite{Got} computed the generating function of the Poincar\'e polynomials \begin{equation*} \begin{split} & \sum_{n=0}^\infty q^n P_t(\Hilbn{X}) \\ = &\prod_{m=1}^\infty \prod_{i=0}^4 \, (1 - (-t)^{2m - 2 + i}q^m)^{(-1)^{i+1}b_i(X)}\, , \end{split} \end{equation*} where $b_i(X)$ is the Betti number of $X$. It was shown that the Euler number of $\Hilbn{X}$ is equal to the orbifold Euler number of $S^n X$ by Hirzebruch-Hofer \cite{HH}. It was also pointed out by Vafa and Witten that this is equal to the character of the Fock space. We shall construct the representation of the Heisenberg and Clifford algebras in a geometric way. The key point is to introduce appropriate ``Hecke correspondence'' which give the generators of the Heisenberg/Clifford algebra. Take a basis of $H_*(X)$ and assume that each element is represented by a (real) compact submanifold $C^a$. ($a$ runs over $1, 2, \dots, \dim H_*(X)$.) Take a dual basis for $H_*(X)$, and assume that each element is also represented by a submanifold $D^a$. (Those assumptions are only for the brevity. The modification to the case of cycles is clear.) For each $a = 1,2,\dots,\dim H_*(X)$, $n = 1,2,\dots$ and $i=1,2,\dots$, we introduce cycles of products of the Hilbert schemes by \begin{equation*} \begin{split} &E_i^a(n) = \{\, ({\cal J}_1,{\cal J}_2) \in\HilbX{n-i}\times\HilbX{n} \mid {\cal J}_1\supset {\cal J}_2 \\ & \quad\text{and $\Supp({\cal J}_1/{\cal J}_2) = \{ p\}$ for some $p\in D^a$}\; \}, \\ &F_i^a(n) = \{\, ({\cal J}_1,{\cal J}_2)\in\HilbX{n+i}\times\HilbX{n} \mid {\cal J}_1\subset {\cal J}_2 \\ & \quad\text{and $\Supp ({\cal J}_2/{\cal J}_1) = \{ p \}$ for some $p\in C^a$}\;\}. \end{split} \end{equation*} Then we define an endomorphism $H_*(\HilbX{n}) \to H_*(\HilbX{n-i})$ by \begin{equation*} c\mapsto (p_1)_* ([E_i^a(n)]\cap p_2^* c), \end{equation*} where $p_1$, $p_2$ are projections of the first and second factor of $\HilbX{n-i}\times\HilbX{n}$ and $p_2^* c = [\HilbX{n-i}]\times c$ and $(p_1)_*$ is a push-forward. Similarly, we have an endomorphism $H_*(\HilbX{n}) \to H_*(\HilbX{n+i})$ using $F_i^a(n)$. Collecting the operators with respect to $n$, we have operators $[E_i^a]$, $[F_i^a]$ acting on the direct sum $\bigoplus_n H_*(\HilbX{n})$. Then \begin{Theorem}. The following relations hold as operators on $\bigoplus_{n} H_*(\HilbX{n})$. \begin{gather*} [E_i^a] [E_j^b] = (-1)^{\dim D^a\dim D^b}[E_j^b] [E_i^a]\\ [F_i^a] [F_j^b] = (-1)^{\dim C^a\dim C^b}[F_j^b] [F_i^a]\\ [E_i^a] [F_j^b] = (-1)^{\dim D^a\dim C^b}[F_j^b] [E_i^a]\\ \qquad\qquad+ \delta_{ab}\delta_{ij}c_i \operatorname{Id}, \end{gather*} where $c_i$ is a nonzero integer depending only on $i$ \rom(independent of $X$\rom). \label{th:main}\end{Theorem} In particular, for each fixed $a$, $[E_i^a]$, $[F_i^a]$ ($i = 1,\dots$) define the action of the Heisenberg or Clifford algebra according to the parity of $\dim C^a$. Moreover, comparing with G\"ottsche's formula, we can conclude our representation is irreducible. The definition of the correspondence $E_i^a$, $F_i^a$ can be naturally generalized to the case of moduli spaces of torsion free sheaves (see \cite{Gj}). However, the author has no idea to generalize to more general $4$-manifolds. \section{Elementary transformation: Twist along embedded submanifolds} The opearator of the previous section twists sheaves around points. There is another kind of operator which twists sheaves along an embedded $2$-dimensional submanifold. This operation is called the elementary transformation in the literature. Suppose $C$ is a holomorphic curve embedded in a projective surface $X$. Let $i$ denote the inclusion map. Let $\frak M$ be the moduli space of $\operatorname{U}(r)$-instantons, namely Einstein-Hermitian connections. We identify it with the moduli space of holomorhic vector bundles over $X$ by the Hitchin-Kobayashi correspondence. It decomposes by the first and second Chern classes. For each integer $d$, we also consider the following moduli space ${\frak P}$ of parabolic bundles $({\cal E_1},{\cal E_2},\varphi)$ where $\cal E_i$ is a holomorphic vector bundle over $X$, and $\varphi\colon{\cal E_1} \to {\cal E_2}$ is an injection which is an isomorphism outside $C$. In order to define the moduli space, we need to introduce the notion of the stability to parabolic bundles (see \cite{MY}), for this we need to choose an ample line bundle $L$ or the K\"ahler metric which is a curvature of $L$. Moreover, it is necessary to consider the Gieseker-Maruyama compactification of moduli spaces, as explained in the previous section. But we do not go in detail. There is a morphism $f\colon {\frak P}\to {\frak M}\times {\frak M}$. Then we can define two operators on the homology group exactly as in the previous section: \begin{equation*} \begin{split} c &\mapsto (p_1)_* (f_*[{\frak P}]\cap p_2^* c),\\ c &\mapsto (p_2)_* (p_1^* c\cap f_*[{\frak P}]) \end{split} \end{equation*} Note that the first and second Chern classes are not preserved this operator. Strictly speaking, we do not have a globally defined morphism since the stability conditions for the parabolic bundles and their underlying vector bundles are not equivalent in general. But it is enough for our purpose to have $f_*[{\frak P}]$ as an element of homology group of ${\frak M}\times {\frak M}$. For example, $f$ could be a meromorphic map. Since we do not know what is the right setting for general projective surfaces\footnote{If one could define the Hecke operators using Kronheimer-Mrowka's singular anti-self-dual connections, they might be the right setting.}, we focus on particular examples, namely ALE spaces. The ALE spaces are the minimal resolution of simple singularities ${\Bbb C}^2/\Gamma$, where $\Gamma$ is a finite subgroup of $\operatorname{SU}(2)$. The second homology group $H_2$ of the ALE space is spanned by the irreducible components $\Sigma_1,\dots, \Sigma_n$ of the exceptional set, which are the projective line. The intersection matrix is the negative of the Cartan matrix of type ADE. The classification of simple singularities are given by the Dynkin graphs in this way. In particular, there is a bijection between simple singularities and simple Lie algebra of type ADE. The rank is equal to the number of the number of the irreducible components, namely $n$. By the work of Kronheimer, it is known that they have hyper-K\"ahler metrics. There are a variant of the ADHM description, which identifies the framed moduli spaces of instantons, or more precisely, torsion-free sheaves with the cotangent bundles of the moduli space of representations of quivers of affine Dynkin graphs. Since we shall work on non-compact spaces, we have extra discrete parameters which parameterizes the boundary condition. We consider instantons which converge to a flat connection at the end of the ALE space $X$. The flat connection on the end can be classified by its monodromy, namely a representation $\rho$ of the finite group $\Gamma$. Let $\rho_0$, $\rho_1$, \dots, $\rho_n$ be the irreducible representations of $\Gamma$ with $\rho_0$ the trivial representation. By the McKay correspondence, there is a bijection between the vertices of the affine Dynkin graph and the irreducible representations (see \cite{Na-gauge} for more detail). The monodromy representation $\rho$ is decomposed as $\rho = \bigoplus_{k=0}^n \rho_k^{\oplus w_k}$, where $w_k$ is the multiplicity. This datum will be preserved under the Hecke operator. Corresponding to each irreducible component $\Sigma_k$, we take a component of the moduli space of parabolic bundles where ${\cal E_2}/\varphi({\cal E_1})$ is rank $1$ and degree $-1$. We then define operators $e_k$ and $f_k$ on the homology group of the moduli space as above. We also have an operator $\alpha_k^\vee$ which is the multiplication by $-\langle c_1, [\Sigma_k]\rangle$ on the homology class belonging to the component with the first Chern class $c_1$. For $k = 0$, we can define similar operators $e_0$, $f_0$ by replacing ${\cal O}_{\Sigma_k}(-1)$ by a sheaf ${\cal O}_{\bigcup \Sigma_k}$. The operator $\alpha_0^\vee$ is defined so that \begin{equation*} \sum_{k=0}^n \dim\rho_k \alpha_k^\vee = \operatorname{rank} E\, \operatorname{Id}. \end{equation*} Finally define the operator $d$ to detect the instanton number. Namely the mulitiplication by \begin{equation*} -\int_X \operatorname{ch}({\cal E}). \end{equation*} on the homology group of the each component of the moduli spaces. \begin{Theorem}. Operators $\alpha_k^\vee$, $e_k$, $f_k$ \rom($k = 0, \dots, n$\rom), $d$ satisfy the relation of the affine Lie algebra corresponding to the extended Dynkin graph. Moreover, the representation on $H_*({\frak M})$ is integrable. \end{Theorem} The irreducible decomposition of the representation is complicated, but we have one irreducible factor whose geometric meaning is clear. \begin{Theorem}. If we take the middle degree part of the homology group $H_*({\frak M})$ \rom(since the dimension of $\frak M$ are changing on components, the middle degree also changes\rom), it is preserved by the affine Lie algebra action. Moreover, it is the integrable highest weight representation with the highest weight vector $\,{}^{t}(w_0,\dots,w_n)$. The level is equal to the rank of the vector bundle. \end{Theorem} The highest weight vector lies in the paticularly chosen moduli space which consists of a single point. \section{Local-Global Principle and the Mass} Historically the Hecke operators were originally introduced in the theory of modular forms. There are also analogoues operators in the theory of the moduli spaces vector bundles over curves \cite{NR}, which are used in the geometric Langlands program. Our operators can be considered as natural complex $2$-dimensional analogue of these operators. The importance of the Hecke operators comes from the fact that they lie in the heart of the ``local-global principle''. We shall explain it only very briefly. The interested reader should consult to good literatures about the Hecke operators and the modular forms (see e.g., \cite{Langlands}). The local-global principle roughly says that a global problem could be studied as a collection of local problems. The basic example is the Hasse principle: A quadratic from with integer coefficients has a nontrivial integer solution if and only if it has real solution and a $p$-adic solution. In this case the global problem is to find an integer solution and the local problems are find solutions in $\Bbb R$ and $\Bbb Q_p$. Thus the base manifold, which is the parameter space of the local places, is the set of prime numbers plus infinity $\Bbb R$. The theory of modular forms are also examples of the global-local principle. Consider the space of modular forms of weight $k$, which can be considered as functions of lattices in $\Bbb C$ with homogeneous degree $-k$, i.e., $F(\lambda L) = \lambda^{-k}F(L)$. Then for each prime number $p$, we define the Hecke operator $T(p)$ by \begin{equation*} (T(p) F)(L) = p^{k-1} \sum_{[ L' : L] = p} F(L'), \end{equation*} where the summation runs over the set of sublattices of $L$ with index $p$. These operators commute each other. If a modular form is a simultaneous eigenfunction, its $L$-function has an Euler product expansion. The analogy between the modular forms and $4$-dimensional gauge theory our theory are given in the table. \begin{table}[hbt] \begin{tabular}{l|l} \hline prime numbers & points in a $4$-manifold \\ & \ and submanifolds $C$\\ \hline the space of & the homology gruop \\ \ modular forms & \ of moduli spaces \\ \hline Hecke operators & our Hecke operators\\ \hline \end{tabular} \end{table} In physics, there is a good explanation why the local-global principle holds in some topological field theories\footnote{Topological field theories of cohomological type according to the terminology in \cite{CMR}}. In these theories, topological invariants, like Donaldson's invariants, are expressed as correlation functions. It is not by no means obvious that the results are topological invariants, since one needs to introduce a Riemannian metric for the definition of the Lagrangian. However, by a clever choice of the Lagrangian, the resulted correlation functions are independent of the choice of the metric. The mechanism is just like the fact that the euler class, which is defined as the pfaffian of the curvature, is a topological invariant. Thus one can take a family of Riemannian metrics $g_t = tg$ with $t > 0$, and consider the limiting behaviour for $t\to \infty$. In the limit, the distance of two different points goes to infinity. Hence if the ``mass'' of all particles is not zero, there are no interaction between two points. Then one can compute the correlation functions by integrating local contributions over the base manifold. In this sense, the local-global principle holds in this theory. In $N = 1$ topological supersymmetric Yang-Mills theory, it is believed that all particles have non-trivial mass. However in $N = 2$ topological supersymmetric Yang-Mills theory, which is relevant to Donaldson's invariants, it is no longer true. Hence there may be massless particles which make interaction even when the distance between two points are very large. Anyway, if all particles would have non-trivial mass, Donaldson's invariants would depend only on homology classes of the underlying manifolds. On a K\"ahler manifold $X$ with a non-trivial holomorphic $2$-form $\omega$, Witten \cite{Wi-SUSY} used a perturabation from the $N = 2$ theory to the $N = 1$ theory adding a term depending on $\omega$. The remarkable observation was that there remain particles which have zero mass where $\omega$ vanishes. Unless the manifold $X$ is a K3 surface or a torus, $\omega$ vanishes along a divisor $C$. Thus the local-global principle holds in the $N = 2$ theory with only one modification; there are non-trivial contribution from $C$. In other words, the $2$-dimensional submanifold $C$ cannot be divided any more, and should be considered as a point. Similarly, in \cite{VW}, again for the same class of K\"ahler manifolds, the $N = 4$ theory was perturbed to the $N = 1$ theory, and the correlation function was calculated in a similar way. And finally, Witten conjectured \cite{Wi-monopole} that the Kronheimer-Mrowka's basic classes \cite{KrMr} coincide with homology classes whose Seiberg-Witten invariants are nonzero. It means that the local-global principle fails exactly along basic classes. Now it becomes clear why we must introduce two kinds of Hecke correspondences, twist along a point and twist along a $2$-dimensional submanifold. For the Hilbert scheme on a projective surface, it is apparent that the local-global principle holds without the introduction of $C$. This should be the basic reason why the homology group of the Hilbert scheme is generated only by the first kind of the Hecke correspondence. For higher rank case, we mighty need the second type of the Hecke correspondence in order to get all homology classes in the moduli spaces. However, the relation between two kinds of Hecke operators is not clear, at this moment. Moreover, there might be other types of correspondences which is useful to describe the homology group of the moduli space. For example, G\"ottsche and Huybrechts \cite{GotHu} used an interesting correspondence in order to relate moduli spaces of rank $2$ bundles and Hilbert schemes. Finally, we would like to point out the difference between our situation and the classical one (i.e., the Hecke operators on modular forms). The first kind of the Hecke operator is independent of the choice of the representative $C^a$ of the homology class. This is because we are studing the homology group of the moduli space. It is not clear that the second kind of the Hecke operator depends only on the homology class of $C$. But it depends only on the rational equivalence class.

9510

alg-geom/9510010

en

https://arxiv.org/abs/alg-geom/9510010

alg-geom/9510010

Zhenbo Qin

Zhenbo Qin and Yongbin Ruan

Quantum cohomology of projective bundles over $\Pee^n$

AMS-TEX Version 2.1

Results in the preliminary version have been strengthed. In addition, Batyrev's conjectural formula for quantum cohomology of projective bundles associated to direct sum of line bundles over $\Pee^n$ is partially verified.

alg-geom

\section{1. Introduction} Quantum cohomology, proposed by Witten's study \cite{16} of two dimensional nonlinear sigma models, plays a fundamental role in understanding the phenomenon of mirror symmetry for Calabi-Yau manifolds. This phenomenon was first observed by physicists motivated by topological field theory. A topological field theory starts with correlation functions. The correlation functions of sigma model are linked with the intersection numbers of cycles in the moduli space of holomorphic maps from Riemann surfaces to manifolds. For some years, the mathematical construction of these correlation functions remained to be a difficult problem because the moduli spaces of holomorphic maps usually are not compact and may have wrong dimension. The quantum cohomology theory was first put on a firm mathematical footing by \cite{12,13} for semi-positive symplectic manifolds (including Fano and Calabi-Yau manifolds), using the method of symplectic topology. Recently, an algebro-geometric approach has been taken by \cite{8,9}. The results of \cite{12,13} have been redone in the algebraic geometric setting for the case of homogeneous spaces. The advantage of homogeneous spaces is that the moduli spaces of holomorphic maps always have expected dimension and their compactifications are nice. Beyond the homogeneous spaces, one can not expect such nice properties for the moduli spaces. The projective bundles are perhaps the simplest examples. However, by developing sophisticated excessive intersection theory, it is possible that the algebro-geometric method can work for any projective manifolds. In turn, it may shed new light to removing the semi-positive condition in the symplectic setting. Although we have a solid foundation for quantum cohomology theory at least for semi-positive symplectic manifolds, the calculation remains to be a difficult task. So far, there are only a few examples which have been computed, e.g., Grassmannian \cite{14}, some rational surfaces \cite{6}, flag varieties \cite{4}, some complete intersections \cite{3}, and the moduli space of stable bundles over Riemann surfaces \cite{15}. One of the common feature for these examples is that the relevant moduli spaces of rational curves have expected dimension. Then, one can use the intersection theory. We should mention that there are many predications based on mathematically unjustified mirror symmetry (for Calabi-Yau 3-folds) and linear sigma model (for toric varieties). In this paper, we attempt to determine the quantum cohomology of projective bundles over the projective space $\Pee^n$. In contrast to the previous examples, the relevant moduli spaces in our case frequently do not have expected dimensions. It makes the calculation more difficult. We overcome this difficulty by using excessive intersection theory. There are two main ingredients in our arguments. The first one is a result of Siebert and Tian (the Theorem 2.2 in \cite{14}), which says that if the ordinary cohomology $H^*(X; \Zee)$ of a symplectic manifold $X$ with the symplectic form $\omega$ is the ring generated by $\alpha_1, \ldots, \alpha_s$ with the relations $f^1, \ldots, f^t$, then the quantum cohomology $H^*_\omega(X; \Zee)$ of $X$ is the ring generated by $\alpha_1, \ldots, \alpha_s$ with $t$ new relations $f_\omega^1, \ldots, f_\omega^t$ where each new relation $f_\omega^i$ is just the relation $f^i$ evaluated in the quantum cohomology ring structure. It was known that the quantum product $\alpha \cdot \beta$ is the deformation of ordinary cup product by the lower order terms called quantum corrections. The second ingredient is that under certain numerical conditions, most of the quantum corrections vanishes. Moreover, the nontrivial quantum corrections seem to come from Mori's extremal rays. Let $V$ be a rank-$r$ bundle over $\Pee^n$, and $\Pee(V)$ be the corresponding projective bundle. Let $h$ and $\xi$ be the cohomology classes of a hyperplane in $\Pee^n$ and the tautological line bundle in $\Pee(V)$ respectively. For simplicity, we make no distinction between $h$ and $\pi^*h$ where $\pi: \Pee(V) \to \Pee^n$ is the natural projection. Denote the product of $i$ copies of $h$ and $j$ copies of $\xi$ in the ordinary cohomology ring by $h_i \xi_j$, and the product of $i$ copies of $h$ and $j$ copies of $\xi$ in the quantum cohomology ring by $h^i \cdot \xi^j$. For $i = 0, \ldots, r$, put $c_i(V) = c_i \cdot h_i$ for some integer $c_i$. It is well known that $-K_{\Pee(V)} = (n + 1 - c_1)h + r\xi$ and the ordinary cohomology ring $H^*(\Pee(V); \Zee)$ is the ring generated by $h$ and $\xi$ with the two relations: $$h_{n+1} = 0 \qquad \text{and} \qquad \sum_{i=0}^r (-1)^i c_i \cdot h_i \xi_{r-i} = 0. \eqno (1.1)$$ In particular, $H^{2(n+r-2)}(\Pee(V); \Zee)$ is generated by $h_{n-1}\xi_{r-1}$ and $h_{n}\xi_{r-2}$, and its Poincar\'e dual $H_2(\Pee(V); \Zee)$ is generated by $(h_{n-1}\xi_{r-1})_*$ and $(h_{n}\xi_{r-2})_*$ where for $\alpha \in H^*(\Pee(V); \Zee)$, $\alpha_*$ stands for its Poincar\'e dual. We have $$-K_{\Pee(V)}(A) = a(n + 1 - c_1) + r \cdot \xi(A) = a(n + 1 - c_1) + r(ac_1 + b) \eqno (1.2)$$ for $A = (a h_{n-1}\xi_{r-1}+ b h_{n}\xi_{r-2})_* \in H_2(\Pee(V); \Zee)$. By definition, $V$ is an ample (respectively, nef) bundle if and only if the tautological class $\xi$ is an ample (respectively, nef) divisor on $\Pee(V)$. Assume that $V$ is ample such that either $c_1 \le (n+1)$ or $c_1 \le (n + r)$ and $V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef. Then both $\xi$ and $-K_{\Pee(V)}$ are ample divisors. Thus, $\Pee(V)$ is a Fano variety, and its quantum cohomology ring is well-defined \cite{13}. Here we choose the symplectic form $\omega$ on $\Pee(V)$ to be the Kahler form $\omega$ such that $[\omega] = -K_{\Pee(V)}$. Let $f_\omega^1$ and $f_\omega^2$ be the two relations in (1.1) evaluated in the quantum cohomology ring $H^*_{\omega}(\Pee(V); \Zee)$. Then by the Theorem 2.2 in \cite{14}, the quantum cohomology $H^*_{\omega}(\Pee(V); \Zee)$ is the ring generated by $h$ and $\xi$ with the two relations $f_\omega^1$ and $f_\omega^2$: $$H_\omega^*(\Pee(V); \Zee) = \Zee [h, \xi]/(f_\omega^1, f_\omega^2) \eqno (1.3)$$ By Mori's Cone Theorem \cite{5}, $\Pee(V)$ has exactly two extremal rays $R_1$ and $R_2$. Up to an order of $R_1$ and $R_2$, the integral generator $A_1$ of $R_1$ is represented by lines in the fibers of the projection $\pi$. We shall show that under certain numerical conditions, the nontrivial homology classes $A \in H_2(\Pee(V); \Zee)$ which give nontrivial quantum corrections are $A_1$ and $A_2$, where $A_2$ is represented by some smooth rational curves in $\Pee(V)$ which are isomorphic to lines in $\Pee^n$ via $\pi$. In general, it is unclear whether $A_2$ generates the second extremal ray $R_2$. However, we shall prove that under further restrictions on $V$, $A_2$ generates the extremal ray $R_2$. These analyses enable us to determine the quantum cohomology ring $H^*_{\omega}(\Pee(V); \Zee)$. The simplest ample bundle over $\Pee^n$ is perhaps the direct sum of line bundles $V=\oplus^r_{i=1} {\Cal O}(m_i)$ where $m_i>0$ for every $i$. Since we can twist $V$ by ${\Cal O}(-1)$ without changing $\Pee(V)$, we can assume that $\hbox{min}\{ m_1, \ldots, m_r \}=1$. In this case, $\Pee(V)$ is a special case of toric variety. Batyrev \cite{2} conjectured a general formula for quantum cohomology of toric varieties. Furthermore, he computed the contributions from certain moduli spaces of holomorphic maps which have expected dimensions. In our case, the contributions Batyrev computed are only part of the data to compute the quantum cohomology. As we explained earlier, the difficulty in our case lies precisely in computing the contributions from the moduli spaces with wrong dimensions. Nevertheless, in our case, Batyrev's formula (see also \cite{1}) reads as follows. \vskip 0.1in \noindent {\bf Batyrev's Conjecture:} {\it Let $V=\oplus^r_{i=1} {\Cal O}(m_i)$ where $m_i > 0$ for every $i$. Then the quantum cohomology ring $H^*_{\omega}(\Pee(V); \Zee)$ is generated by $h$ and $\xi$ with two relations} $$h^{n+1}=\prod^r_{i=1}(\xi-m_ih)^{m_i-1} \cdot e^{-t(n+1+r-\sum_{i=1}^r m_i)} \qquad {and} \qquad \prod^r_{i=1}(\xi-m_ih) = e^{-tr}.$$ \vskip 0.1in Our first result partially verifies Batyrev's conjecture. \theorem{A} Batyrev's conjecture holds if $$\sum_{i=1}^r m_i < \text{min}(2r, (n+1+2r)/2, (2n+2+r)/2).$$ \endproclaim Note that under the numerical condition of Theorem A, only extremal rational curves with fundamental classes $A_1$ and $A_2$ give the contributions to the two relations in the quantum cohomology. The moduli space of rational curves $\frak M(A_2, 0)$ with fundamental class $A_2$ does not have expected dimension in general. But it is compact. This fact simplifies a great deal of the excessive intersection theory involved. To remove the numerical condition, we have to consider other moduli spaces (for example $\frak M(kA_2, 0)$ with $k > 1$ and its excessive intersection theory). These moduli spaces are not compact in general. Then, we have an extra difficulty of the compactification and the appropriate excessive intersection theory with it. It seems to be a difficult problem and we shall not pursue here. In general, ample bundles over $\Pee^n$ are not direct sums of line bundles. We can say much less about its quantum cohomology. However, we obtain some result about its general form and compute the leading coefficient. \theorem{B} {\rm (i)} Let $V$ be a rank-$r$ ample bundle over $\Pee^n$. Assume either $c_1 \le n$ or $c_1 \le (n + r)$ and $V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef so that $\Pee(V)$ is Fano. Then the quantum cohomology $H^*_{\omega}(\Pee(V); \Zee)$ is the ring generated by $h$ and $\xi$ with two relations $$h^{n+1} = \sum_{i+j \le (c_1 - r)} a_{i,j} \cdot h^i \cdot \xi^j \cdot e^{-t(n+1-i-j)}$$ $$\sum_{i=0}^r (-1)^i c_i \cdot h^i \cdot \xi^{r-i} = e^{-tr} + \sum_{i+j \le (c_1-n-1)} b_{i,j} \cdot h^i \cdot \xi^{j} \cdot e^{-t(r-i-j)}$$ where the coefficients $a_{i, j}$ and $b_{i, j}$ are integers depending on $V$; {\rm (ii)} If we further assume that $c_1<2r$, then the leading coefficient $a_{0,c_1-r}=1$. \endproclaim It is understood that when $c_1 \le n$, then the summation $\sum_{i+j \le (c_1-n-1)}$ in the second relation in Theorem B (i) does not exist. In general, it is not easy to determine all the integers $a_{i, j}$ and $b_{i, j}$ in Theorem B (i). However, it is possible to compute these numbers when $(c_1 - r)$ is relatively small. For instance, when $(c_1 - r) = 0$, then necessarily $V = \Cal O_{\Pee^n}(1)^{\oplus r}$ and it is well-known that the quantum cohomology $H^*_{\omega}(\Pee(V); \Zee)$ is the ring generated by $h$ and $\xi$ with the two relations $h^{n+1} = e^{-t(n+1)}$ and $\sum_{i=0}^r (-1)^i c_i \cdot h^i \cdot \xi^{r-i} = e^{-tr}$. When $(c_1 - r) = 1$ and $r < n$, then necessarily $V = \Cal O_{\Pee^n}(1)^{\oplus (r - 1)} \oplus \Cal O_{\Pee^n}(2)$. When $(c_1 - r) = 1$ and $r = n$, then $V = \Cal O_{\Pee^n}(1)^{\oplus (r - 1)} \oplus \Cal O_{\Pee^n}(2)$ or $V = T_{\Pee^n}$ the tangent bundle of $\Pee^n$. In these cases, $V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef. In particular, the direct sum cases have been computed by Theorem A. We shall prove the following. \proposition{C} The quantum cohomology ring $H^*_{\omega}(\Pee(T_{\Pee^n}); \Zee)$ with $n \ge 2$ is the ring generated by $h$ and $\xi$ with the two relations: $$h^{n+1} = \xi \cdot e^{-tn} \qquad \text{and} \qquad \sum_{i=0}^n (-1)^i c_i \cdot h^i \cdot \xi^{n-i} = (1 + (-1)^n) \cdot e^{-tn}.$$ \endproclaim Recall that for an arbitrary projective bundle over a general manifold, its cohomology ring is a module over the cohomology ring of the base with the generator $\xi$ and the second relation of (1.1). Naively. one may think that the quantum cohomology of projective bundle is a module over the quantum cohomology of base with the generator $\xi$ and the quantanized second relation. Our calculation shows that one can not expect such simplicity for its quantum cohomology ring. We hope that our results could shed some light on the quantum cohomology for general projective bundles, which we shall leave for future research. Our paper is organized as follows. In section 2, we discuss the extremal rays and extremal rational curves. In section 3, we review the definition of quantum product and compute some Gromov-Witten invariants. In the remaining three sections, we prove Theorem B, Theorem A, and Proposition C respectively. \medskip\noindent {\bf Acknowledgements:} We would like to thank Sheldon Katz, Yungang Ye, and Qi Zhang for valuable helps and stimulating discussions. In particular, we are grateful to Sheldon Katz for bringing us the attention of Batyrev's conjecture. \section{2. Extremal rational curves} Assume that $V$ is ample such that either $c_1 \le (n+1)$ or $c_1 \le (n + r)$ and $V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef. In this section, we study the extremal rays and extremal rational curves in the Fano variety $\Pee(V)$. By Mori's Cone Theorem (p.25 in \cite{5}), $\Pee(V)$ has precisely two extremal rays $R_1 = \Bbb R_{\ge 0} \cdot A_1$ and $R_2 = \Bbb R_{\ge 0} \cdot A_2$ such that the cone $\hbox{NE}(\Pee(V))$ of curves in $\Pee(V)$ is equal to $R_1 + R_2$ and that $A_1$ and $A_2$ are the homology classes of two rational curves $E_1$ and $E_2$ in $\Pee(V)$ with $0 < -K_{\Pee(V)}(A_i) \le \hbox{dim}(\Pee(V)) + 1$. Up to orders of $A_1$ and $A_2$, we have $A_1 = (h_{n}\xi_{r-2})_*$, that is, $A_1$ is represented by lines in the fibers of $\pi$. It is also well-known that if $V = \oplus_{i=1}^r \Cal O_{\Pee^n}(m_i)$ with $m_1 \le \ldots \le m_r$, then $A_2 = [h_{n-1}\xi_{r-1}+ (m_1 - c_1) h_{n}\xi_{r-2}]_*$ which is represented by a smooth rational curve in $\Pee(V)$ isomorphic to a line in $\Pee^n$ via $\pi$. However, in general, it is not easy to determine the homology class $A_2$ and the extremal rational curves representing $A_2$. Assume that $$V|_\ell = \oplus_{i = 1}^r \Cal O_\ell(m_i) \eqno (2.1)$$ for generic lines $\ell \subset \Pee^n$ where we let $m_1 \le \ldots \le m_r$. Since $V$ is ample, $m_1 \ge 1$. \lemma{2.2} Let $A = [h_{n-1}\xi_{r-1} + (m_1 - c_1) h_{n}\xi_{r-2}]_*$. Then, \roster \item"{(i)}" $A$ is represented by a smooth rational curve isomorphic to a line in $\Pee^n$; \item"{(ii)}" $A_2 = A$ if and only if $(\xi- m_1 h)$ is nef; \item"{(iii)}" $A_2 = A$ if $2c_1 \le (n + 1)$; \item"{(iv)}" $A$ can not be represented by reducible or nonreduced curves if $m_1 = 1$. \endroster \endproclaim \proof (i) Let $\ell \subset \Pee^n$ be a generic line. Then we have a natural projection $V|_\ell = \oplus_{i = 1}^r \Cal O_\ell(m_i) \to \Cal O_\ell(m_1)$. By the Proposition 7.12 in Chapter II of \cite{7}, this surjective map $V|_\ell \to \Cal O_\ell(m_1) \to 0$ induces a morphism $g: \ell \to \Pee(V)$. Then $g(\ell)$ is isomorphic to $\ell$ via the projection $\pi$. Since $h([g(\ell)]) = 1$ and $\xi([g(\ell)]) = m_1$, we have $$[g(\ell)] = [h_{n-1}\xi_{r-1}+ (m_1 - c_1) h_{n}\xi_{r-2}]_* = A.$$ (ii) First of all, if $A_2 = [h_{n-1}\xi_{r-1}+ (m_1 - c_1) h_{n}\xi_{r-2}]_*$, then for any curve $E$, $[E] = a(h_{n}\xi_{r-2})_* + b[h_{n-1}\xi_{r-1}+ (m_1 - c_1) h_{n}\xi_{r-2}]_*$ for some nonnegative numbers $a$ and $b$; so $(\xi- m_1h)([E]) = a \ge 0$; therefore $(\xi- m_1h)$ is nef. Conversely, if $(\xi- m_1h)$ is nef, then $0 \le (\xi- m_1h)([E]) = ac_1 + b - am_1$ where $[E] = (ah_{n-1}\xi_{r-1}+ b h_{n}\xi_{r-2})_*$ for some curve $E$; thus $[E] = (ac_1 + b - am_1)(h_{n}\xi_{r-2})_* + a[h_{n-1}\xi_{r-1}+ (m_1 - c_1) h_{n}\xi_{r-2}]_*$; it follows that $A_2 = [h_{n-1}\xi_{r-1}+ (m_1 - c_1) h_{n}\xi_{r-2}]_* = A$. (iii) Let $A_2 = (ah_{n-1}\xi_{r-1} + bh_{n}\xi_{r-2})_*$. Since $A_1 = (h_{n}\xi_{r-2})_*$ and $a = h(A_2) \ge 0$, $a \ge 1$. If $a > 1$, then since $2c_1 \le (n + 1)$, we see that $$\align -K_{\Pee(V)}(A_2) &= (n+1-c_1)a +r \cdot \xi(A_2) \ge 2(n+1-c_1) + r \\ &> n + r = \text{dim}(\Pee(V)) + 1; \\ \endalign$$ but this contradicts with $-K_{\Pee(V)}(A_2) \le \hbox{dim}(\Pee(V)) + 1$. Thus $a = 1$ and $A_2 = (h_{n-1}\xi_{r-1} + bh_{n}\xi_{r-2})_*$. Now $[\pi(E_2)] = \pi_*(A_2) = (h_{n-1})_*$. So $\pi(E_2)$ is a line in $\Pee^n$. Since $V|_\ell = \oplus_{i = 1}^r \Cal O_\ell(m_i)$ for a generic line $\ell \subset \Pee^n$, $V|_{\pi(E_2)} = \oplus_{i = 1}^r \Cal O_{\pi(E_2)}(m_i')$ where $m_i' \ge m_1$ for every $i$. Thus, $\xi(A_2) \ge m_1$, and so $c_1 + b \ge m_1$. It follows that $$A_2 = [h_{n-1}\xi_{r-1} + (m_1- c_1) h_{n}\xi_{r-2}]_* + (c_1 + b - m_1) \cdot (h_{n}\xi_{r-2})_*.$$ Therefore, $A_2 = [h_{n-1}\xi_{r-1}+ (m_1 - c_1) h_{n}\xi_{r-2}]_* = A$. (iv) Since $\xi(A) = m_1 = 1$ and $\xi$ is ample, the conclusion follows. \endproof Next, let $\frak M(A,0)$ be the moduli space of morphisms $f: \Pee^1 \to \Pee(V)$ with $[\hbox{Im}(f)] = A$. In the lemma below, we study the morphisms in $\frak M(A,0)$ when $A = [h_{n-1}\xi_{r-1}+ (m - c_1) h_{n}\xi_{r-2}]_*$. Note that $\xi(A) = m$. \lemma{2.3} Let $A = [h_{n-1}\xi_{r-1}+ (m - c_1) h_{n}\xi_{r-2}]_*$. \roster \item"{(i)}" If $\frak M(A, 0) \ne \emptyset$, then $m \ge m_1$ and $\frak M(A, 0)$ consists of embeddings $f: \ell \to \Pee(V)$ induced by surjective maps $V|_\ell \to \Cal O_\ell(m) \to 0$ where $\ell$ are lines in $\Pee^n$; \item"{(ii)}" If $m = m_1$ and $m_1 = \ldots = m_k < m_{k+1} \le \ldots \le m_r$, then the moduli space $\frak M(A, 0)$ has (complex) dimension $(2n +k)$; \item"{(iii)}" If $m \ge m_r$, then $\frak M(A, 0)$ has dimension $(2n+r+rm-c_1)$. \endroster \endproclaim \proof (i) Let $f: \Pee^1 \to \Pee(V)$ be a morphism in $\frak M(A, 0)$. Then $[\hbox{Im}(f)] = A = [h_{n-1}\xi_{r-1}+ (m - c_1) h_{n}\xi_{r-2}]_*$. Since $h(A) = 1$, $\pi^*H \cap f(\Pee^1)$ consists of a single point for any hyperplane $H$ in $\Pee^n$. Thus, $\pi|_{f(\Pee^1)}: f(\Pee^1) \to (\pi \circ f)(\Pee^1)$ is an isomorphism and $\ell = (\pi \circ f)(\Pee^1)$ is a line in $\Pee^n$. Since $h([\ell]) = 1$, $(\pi \circ f): \Pee^1 \to \ell = (\pi \circ f)(\Pee^1)$ is also an isomorphism, and so is $f: \Pee^1 \to f(\Pee^1)$. Replacing $f: \Pee^1 \to \Pee(V)$ by $f \circ (\pi \circ f)^{-1}: \ell \to \Pee(V)$, we conclude that $\frak M(A, 0)$ consists of embeddings $f: \ell \to \Pee(V)$ such that $[\hbox{Im}(f)] = A$, $\ell$ are lines in $\Pee^n$, and $\pi|_{f(\ell)}: f(\ell) \to \ell$ are isomorphisms. In particular, these embeddings $f: \ell \to \Pee(V)$ are sections to the natural projection $\pi|_{\Pee(V|_\ell)}: \Pee(V|_\ell) \to \ell$. Thus, by the Proposition 7.12 in Chapter II of \cite{7}, these embeddings are induced by surjective maps $V|_\ell \to \Cal O_\ell(m) \to 0$. By (2.1), the splitting type of the restrictions of $V$ to generic lines in $\Pee^n$ is $(m_1, \ldots, m_r)$ with $m_1 \le \ldots \le m_r$; thus we must have $V|_\ell = \oplus_{i = 1}^r \Cal O_\ell(m_i')$ where $m_i' \ge m_1$ for every $i$. It follows that $m \ge \hbox{min} \{ m_1', \ldots, m_r' \} \ge m_1$. (ii) Note that all the lines in $\Pee^n$ are parameterized by the Grassmannian $G(2, n+1)$ which has dimension $2(n -1)$. For a fixed generic line $\ell \subset \Pee^n$, the surjective maps $V|_\ell \to \Cal O_\ell(m_1) \to 0$ are parameterized by $$\Pee(\hbox{Hom}(V|_\ell, \Cal O_\ell(m_1))) \cong \Pee(\oplus_{i=1}^r H^0(\ell, \Cal O_\ell(m_1-m_i))) \cong \Pee^{k - 1};$$ It follows from (i) that as the generic line $\ell$ varies, the morphisms $f: \ell \to \Pee(V)$ induced by these surjective maps $V|_\ell \to \Cal O_\ell(m_1) \to 0$ form an open dense subset of $\frak M(A, 0)$. Thus, $\hbox{dim}(\frak M(A, 0)) = 3+ 2(n -1) + (k - 1) = 2n +k$. (iii) As in the proof of (ii), for a fixed generic line $\ell \subset \Pee^n$, the surjective maps $V|_\ell \to \Cal O_\ell(m) \to 0$ are parameterized by a nonempty open subset of $$\Pee(\hbox{Hom}(V|_\ell, \Cal O_\ell(m))) \cong \Pee(\oplus_{i=1}^r H^0(\ell, \Cal O_\ell(m-m_i))) \cong \Pee^{(rm - c_1 +r) - 1}.$$ As the generic line $\ell$ varies, the morphisms $f: \ell \to \Pee(V)$ induced by these surjective maps $V|_\ell \to \Cal O_\ell(m) \to 0$ form an open dense subset of $\frak M(A, 0)$. It follows that $\frak M(A, 0)$ has dimension $(2n+r+rm-c_1)$. \endproof \section{3. Calculation of Gromov-Witten invariants} In this section, we shall compute some Gromov-Witten invariants of $\Pee(V)$. First of all, we recall that for two homogeneous elements $\alpha$ and $\beta$ in $H^*(\Pee(V); \Zee)$, the quantum product $\alpha \cdot \beta \in H^*(\Pee(V); \Zee)$ can be written as $$\alpha \cdot \beta = \sum_{A \in H_2(\Pee(V); \Zee)} (\alpha \cdot \beta)_A \cdot e^{t \cdot K_{\Pee(V)}(A)} \eqno (3.1)$$ where $(\alpha \cdot \beta)_A$ has degree $\hbox{deg}(\alpha) + \hbox{deg}(\beta) + 2K_{\Pee(V)}(A)$ and is defined by $$(\alpha \cdot \beta)_A(\gamma_*) = \Phi_{(A, 0)}(\alpha, \beta, \gamma)$$ for a homogeneous cohomology class $\gamma \in H^*(\Pee(V); \Zee)$ with $$\hbox{deg}(\gamma) = -2K_{\Pee(V)}(A) + 2(n+r-1) - \hbox{deg}(\alpha) - \hbox{deg}(\beta). \eqno (3.2)$$ Furthermore, for higher quantum products, we have $$\alpha_1 \cdot \alpha_2 \cdot \ldots \cdot \alpha_k = \sum_{A \in H_2(\Pee(V); \Zee)} (\alpha_1 \cdot \alpha_2 \cdot \ldots \cdot \alpha_k)_A \cdot e^{t \cdot K_{\Pee(V)}(A)} \eqno (3.3)$$ where $(\alpha_1 \cdot \alpha_2 \cdot \ldots \cdot \alpha_k)_A$ is defined as $(\alpha_1 \cdot \alpha_2 \cdot \ldots \cdot \alpha_k)_A(\gamma_*) = \Phi_{(A,0)}(\alpha_1, \alpha_2, \ldots, \alpha_k, \gamma)$. Thus, $\alpha_1 \cdot \alpha_2 \cdot \cdots \cdot \alpha_k = \alpha_1\alpha_2\dots\alpha_k + \text{(lower\ order\ terms)}$, where $\alpha_1\alpha_2\dots\alpha_k$ stands for the ordinary cohomology product of $\alpha_1, \alpha_2, \ldots, \alpha_k$, and the degree of a lower order term is dropped by $2K_{\Pee(V)}(A)$ for some $A \in H_2(\Pee(V); \Zee)$ which is represented by a nonconstant effective rational curve. There are two explanations for the Gromov-Witten invariant $\Phi_{(A, 0)}(\alpha, \beta, \gamma)$ defined by the second author \cite{12}. Recall that the Gromov-Witten invariant is only defined for a generic almost complex structure and that $\frak M(A,0)$ is the moduli space of morphisms $f: \Pee^1 \to \Pee(V)$ with $[\hbox{Im}(f)] = A$. Assume the genericity conditions: \roster \item"{(i)}" $\frak M(A, 0)/PSL(2; \Cee)$ is smooth in the sense that $h^1(N_f) = 0$ for every $f \in \frak M(A, 0)$ where $N_f$ is the normal bundle, and \item"{(ii)}" the homology class $A$ is only represented by irreducible and reduced curves. \endroster \noindent Then the complex structure is already generic and one can use algebraic geometry to calculate the Gromov-Witten invariants. Moreover, $\frak M(A, 0)/PSL(2; \Cee)$ is compact with the expected complex dimension $$-K_{\Pee(V)}(A) + (n+r-1) - 3. \eqno (3.4)$$ The first explanation for $\Phi_{(A, 0)}(\alpha, \beta, \gamma)$ is that when $\alpha, \beta, \gamma$ are classes of subvarieties $B, C, D$ of $\Pee(V)$ in general position, $\Phi_{(A, 0)}(\alpha, \beta, \gamma)$ is the number of rational curves $E$ in $\Pee(V)$ such that $[E] = A$ and that $E$ intersects with $B, C, D$ (counted with suitable multiplicity). The second explanation for $\Phi_{(A, 0)}(\alpha, \beta, \gamma)$ is that $$\Phi_{(A, 0)}(\alpha, \beta, \gamma) = \int_{\frak M(A, 0)} e_0^*(\alpha) \cdot e_1^*(\beta) \cdot e_2^*(\gamma) $$ where the evaluation map $e_i: \frak M(A, 0) \to \Pee(V)$ is defined by $e_i(f) = f(i)$. Assume that the genericity condition (i) is not satisfied but $h^1(N_f)$ is independent of $f \in \frak M(A, 0)$ and $\frak M(A, 0)/PSL(2; \Cee)$ is smooth with dimension $$-K_{\Pee(V)}(A) + (n+r-1) - 3+h^1(N_f).$$ Then one can form an obstruction bundle $COB$ of rank $h^1(N_f)$ over the moduli space $\frak M(A, 0)$. Moreover, if the genericity condition (ii) is satisfied, then by the Proposition 5.7 in \cite{11}, we have $$\Phi_{(A, 0)}(\alpha, \beta, \gamma) = \int_{\frak M(A, 0)} e_0^*(\alpha) \cdot e_1^*(\beta) \cdot e_2^*(\gamma) \cdot e(COB) \eqno (3.5)$$ where $e(COB)$ stands for the Euler class of the bundle $COB$. We remark that in general, the cohomology class $h_i \xi_j$ may not be able to be represented by a subvariety of $\Pee(V)$. However, since $\xi$ is ample, $s \xi$ is very ample for $s \gg 0$. Thus, the multiple $t h_i \xi_j$ with $t \gg 0$ can be represented by a subvariety of $\Pee(V)$ whose image in $\Pee^n$ is a linear subspace of codimension $i$. Since $\Phi_{(A, 0)}(\alpha, \beta, h_i \xi_j) = {1/t} \cdot \Phi_{(A, 0)}(\alpha, \beta, t \cdot h_i \xi_j)$ for $\alpha$ and $\beta$ in $H^*(\Pee(V); \Zee)$, it follows that to compute $\Phi_{(A, 0)}(\alpha, \beta, h_i \xi_j)$, it suffices to compute $\Phi_{(A, 0)}(\alpha, \beta, t \cdot h_i \xi_j)$. In the proofs below, we shall assume implicitly that $t = 1$ for simplicity. Now we compute the Gromov-Witten invariant $\Phi_{((h_{n}\xi_{r-2})_*, 0)}(\xi, \xi_{r-1}, h_{n}\xi_{r-1})$. \lemma{3.6} $\Phi_{((h_{n}\xi_{r-2})_*, 0)}(\xi, \xi_{r-1}, h_{n}\xi_{r-1}) = 1$. \endproclaim \proof First of all, we notice that $A = (h_{n} \xi_{r-2})_*$ can only be represented by lines $\ell$ in the fibers of $\pi$. In particular, there is no reducible or nonreduced effective curves representing $A$. Thus, $\frak M(A, 0)/PSL(2; \Cee)$ is compact and has dimension: $$\text{dim}(\Pee^n) + \text{dim} G(2, r)= n + 2(r-2) = n + 2r - 4$$ which is the expected dimension by (3.4) (here we use $G(2, r)$ to stand for the Grassmannian of lines in $\Pee^{r-1}$). Next, we want to show that $\frak M(A, 0)/PSL(2; \Cee)$ is smooth. Let $p = \pi(\ell)$. Then from the two inclusions $\ell \subset \pi^{-1}(p) \subset \Pee(V)$, we obtain an exact sequence relating normal bundles: $$0 \to N_{\ell|\pi^{-1}(p)} \to N_{\ell|\Pee(V)} \to (N_{\pi^{-1}(p)|\Pee(V)})|_\ell \to 0.$$ Since $N_{\ell|\pi^{-1}(p)} = N_{\ell|\Pee^{r-1}} = \Cal O_{\ell}(1)^{\oplus (r-2)}$ and $N_{\pi^{-1}(p)|\Pee(V)} = (\pi|_{\pi^{-1}(p)})^*T_{p, \Pee^n}$, the previous exact sequence is simplified into the exact sequence $$0 \to \Cal O_{\ell}(1)^{\oplus (r-2)} \to N_{\ell|\Pee(V)} \to (\pi|_\ell)^*T_{p, \Pee^n} \to 0.$$ It follows that $H^1(\ell, N_{\ell|\Pee(V)}) = 0$. Thus, $\frak M(A, 0)/PSL(2; \Cee)$ is smooth. Finally, the Poincar\'e dual of $h_n \xi_{r-1}$ is represented by a point $q_0 \in \Pee(V)$. If a line $\ell \in \frak M(A,0)$ intersects $q_0$, then $\ell \subset \pi^{-1}(\pi(q_0))$. Since the restriction of $\xi$ to the fiber $\pi^{-1}(\pi(q_0)) \cong \Pee^{r-1}$ is the cohomology class of a hyperplane in $\Pee^{r-1}$, we conclude that $\Phi_{((h_{n}\xi_{r-2})_*, 0)}(\xi, \xi_{r-1}, h_{n}\xi_{r-1}) = 1$. \endproof Next we show the vanishing of some Gromov-Witten invariant. \lemma{3.7} Let $A = b(h_n \xi_{r-2})_*$ with $b \ge 1$ and $\alpha \in H^*(\Pee(V); \Zee)$. Then, $$\Phi_{(A, 0)}(h_{p_1}\xi_{q_1}, h_{p_2}\xi_{q_2}, \alpha) = 0$$ if $p_1, q_1, p_2, q_2$ are nonnegative integers with $(q_1+q_2) < r$. \endproclaim \proof We may assume that $\alpha$ is a homogeneous class in $H^*(\Pee(V); \Zee)$. By (3.2), $$\align {1 \over 2} \cdot \hbox{deg}(\alpha) &= (n+r-1) - K_{\Pee(V)}(A) - (p_1+p_2+q_1+q_2) \\ &= (n+r+br-1)-(p_1+p_2+q_1+q_2). \\ \endalign$$ Let $\alpha = h_{(n+r+br-1)-(p_1+p_2+q_1+q_2+q_3)} \xi_{q_3}$ with $0 \le q_3 \le (r - 1)$. Let $B, C, D$ be the subvarieties of $\Pee(V)$ in general position, whose homology classes are Poincar\'e dual to $h_{p_1}\xi_{q_1}, h_{p_2}\xi_{q_2}, \alpha$ respectively. Then the homology classes of $\pi(B), \pi(C), \pi(D)$ in $\Pee^n$ are Poincar\'e dual to $h_{p_1}, h_{p_2}, h_{(n+r+br-1)-(p_1+p_2+q_1+q_2+q_3)}$ respectively. Since $(q_1+q_2+q_3) < (2r -1)$, we have $p_1 + p_2 + [(n+r+br-1)-(p_1+p_2+q_1+q_2+q_3)] = (n+r+br-1)-(q_1+q_2+q_3) > n$. Thus, $\pi(B) \cap \pi(C) \cap \pi(D) = \emptyset$. Notice that the genericity conditions (i) and (ii) mentioned earlier in this section are not satisfied for $b\ge 2$. However, we observe that these conditions can be relaxed by assuming: \roster \item"{(i$'$)}" $h^1(N_f)=0$ for every $f\in \frak M(A, 0)$ such that $\hbox{Im}(f)$ intersects $B, C, D$, and \item"{(ii$'$)}" there is no reducible or nonreduced effective (connected) curve $E$ such that $[E]=A$ and $E$ intersects $B, C, D$. \endroster \noindent In fact, we will show that there is no effective connected curve $E$ at all representing $A$ and intersecting $B, C, D$. It obviously implies (i$'$), (ii$'$) and $$\Phi_{(A, 0)}(h_{p_1}\xi_{q_1}, h_{p_2}\xi_{q_2}, \alpha) = 0.$$ Suppose that $E=\sum a_i E_i$ is such an effective connected curve where $a_i > 0$ and $E_i$ is irreducible and reduced. Then, $\sum a_i [E_i] = [E] = A$. Since $(h_n \xi_{r-2})_*$ generates an extremal ray for $\Pee(V)$, $[E_i]=b_i (h_n \xi_{r-2})_*$ for $0<b_i\le b$. Thus the curves $E_i$ are contained in the fibers of $\pi$. Since $E$ is connected, all the curves $E_i$ must be contained in the same fiber of $\pi$. So $\pi(E)$ is a single point. Since $E$ intersects $B, C, D$, $\pi(E)$ intersects with $\pi(B), \pi(C), \pi(D)$. It follows that $\pi(B) \cap \pi(C) \cap \pi(D)$ contains $\pi(E)$ and is nonempty. Therefore we obtain a contradiction. \endproof Finally, we show that if $c_1 < 2r$ and $A = [h_{n-1}\xi_{r-1} + (1 - c_1) h_{n}\xi_{r-2}]_*$, then $\Phi_{(A, 0)}(h, h_n, h_n \xi_{2r-c_1-1}) = 1$. Since $c_1 < 2r$, we see that for a generic line $\ell \subset \Pee^n$, $$V|_{\ell} = \Cal O_{\ell}(1)^{\oplus k} \oplus \Cal O_{\ell}(m_{k+1}) \oplus \ldots \oplus \Cal O_{\ell}(m_r)$$ where $k \ge 1$ and $2 \le m_{k+1} \le \ldots \le m_r$. We remark that even though the moduli space $\frak M(A, 0)/PSL(2; \Cee)$ is compact by Lemma 2.2 (iv), it may not have the correct dimension by Lemma 2.3 (ii). The proof is lengthy, but the basic idea is that we shall determine the obstruction bundle and use the formula (3.5). \lemma{3.8} Let $V$ be a rank-$r$ ample vector bundle over $\Pee^n$ satisfying $c_1<2r$ and the assumption of Theorem B (i). If $A = [h_{n-1}\xi_{r-1} + (1 - c_1) h_{n}\xi_{r-2}]_*$, then $$\Phi_{(A, 0)}(h, h_n, h_n \xi_{2r-c_1-1}) = 1.$$ \endproclaim \noindent {\it Proof.} Note that by Lemma 2.2 (iv), the moduli space $\frak M(A, 0)/PSL(2; \Cee)$ is compact. Let $B, C, D$ be the subvarieties of $\Pee(V)$ in general position, whose homology classes are Poincar\'e dual to $h, h_n, h_n \xi_{2r-c_1-1}$ respectively. Then the homology classes of $\pi(B), \pi(C)$, $\pi(D)$ in $\Pee^n$ are Poincar\'e dual to $h, h_n, h_n$ respectively. Thus $\pi(C)$ and $\pi(D)$ are two different points in $\Pee^n$. Let $\ell_0$ be the unique line passing $\pi(C)$ and $\pi(D)$. Let $V|_{\ell_0} = \Cal O_{\ell_0}(1)^{\oplus k} \oplus \Cal O_{\ell_0}(m_{k+1}) \oplus \ldots \oplus \Cal O_{\ell_0}(m_r)$ where $2 \le m_{k+1} \le \ldots \le m_r$. Since $c_1 < 2r$, $k \ge 1$. Let $f: \ell \to \Pee(V)$ be a morphism in $\frak M(A, 0)$ for some line $\ell \in \Pee^n$. If $\hbox{Im}(f)$ intersects with $B, C$, and $D$, then $\ell = \ell_0$. As in the proof of Lemma 2.3 (ii), the morphisms $f: \ell_0 \to \Pee(V)$ in $\frak M(A, 0)$ are parameterized by $\Pee(\hbox{Hom}(V|_{\ell_0}, \Cal O_{\ell_0}(1))) \cong \Pee^{k - 1}$; moreover, $\hbox{Im}(f)$ are of the form: $$\ell_0 \times \{ q \} \subset \ell_0 \times \Pee^{k - 1} = \Pee(\Cal O_{\ell_0}(1)^{\oplus k}) \subset \Pee(V|_{\ell_0}) \subset \Pee(V) \eqno (3.9)$$ where $q$ stands for points in $\Pee^{k - 1} \subset \Pee^{r - 1} \cong \pi^{-1}(\pi(D))$. Note that $\ell_0 \times \{ q \}$ always intersects with $B$ and $C$, and that $D$ is a dimension-$(c_1 - r)$ linear subspace in $\Pee^{r - 1} \cong \pi^{-1}(\pi(D))$. Thus, $\ell_0 \times \{ q \}$ intersects with $B, C, D$ simultaneously if and only if $\ell_0 \times \{ q \}$ intersects with $D$, and if only only if $$q \in \Pee^{c_1+k-2r} \overset \hbox{def} \to = \Pee^{k - 1} \cap D \subset \Pee^{r - 1} \cong \pi^{-1}(\pi(D)). \eqno (3.10)$$ It follows that $\frak M/PSL(2; \Cee) \cong \Pee^{c_1+k-2r}$ where $\frak M$ consists of morphisms $f \in \frak M(A, 0)$ such that $\hbox{Im}(f)$ intersects with $B, C, D$ simultaneously. If $c_1+k-2r = 0$, then $a_0 = \Phi_{(A, 0)}(h, h_n, h_n \xi_{2r-c_1-1}) = 1$. But in general, we have $c_1+k-2r \ge 0$. We shall use (3.5) to compute $a_0 = \Phi_{(A, 0)}(h, h_n, h_n \xi_{2r-c_1-1})$. Let $N_f = N_{\ell_0 \times \{ q \}|\Pee(V)}$ be the normal bundle of $\hbox{Im}(f) = \ell_0 \times \{ q \}$ in $\Pee(V)$. If $h^1(N_f)$ is constant for every $f \in \frak M$, then by (3.5), $\Phi_{(A, 0)}(h, h_n, h_n \xi_{2r-c_1-1})$ is the Euler number $e(COB)$ of the rank-$(c_1+k-2r)$ obstruction bundle $COB$ over $$\frak M/PSL(2; \Cee) \cong \Pee^{c_1+k-2r}.$$ Thus we need to show that $h^1(N_f)$ is constant for every $f \in \frak M$. First, we study the normal bundle $N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)}$. The three inclusions $$\ell_0 \times \Pee^{c_1+k-2r} \subset \ell_0 \times \Pee^{k - 1} = \Pee(\Cal O_{\ell_0}(1)^{\oplus k}) \subset \Pee(V|_{\ell_0}) \subset \Pee(V) \eqno (3.11)$$ give rise to two exact sequences relating normal bundles: $$0 \to N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V|_{\ell_0})} \to N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)} \to N_{\Pee(V|_{\ell_0})|\Pee(V)} \to 0$$ $$0 \to N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(\Cal O_{\ell_0}(1)^{\oplus k})} \to N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V|_{\ell_0})} \to N_{\Pee(\Cal O_{\ell_0}(1)^{\oplus k})|\Pee(V|_{\ell_0})} \to 0$$ Notice that $N_{\Pee(V|_{\ell_0}) | \Pee(V)} = (\pi|_{\Pee(V|_{\ell_0})})^*(N_{\ell_0 | \Pee^n}) = \Cal O_{\ell_0}(1)^{\oplus (n-1)}$ and that $$N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(\Cal O_{\ell_0}(1)^{\oplus k})} = N_{\ell_0 \times \Pee^{c_1+k-2r}|\ell_0 \times \Pee^{k-1}} = \Cal O_{\Pee^{c_1+k-2r}}(1)^{\oplus (2r-c_1-1)}.$$ Since $V|_{\ell_0} = \Cal O_{\ell_0}(1)^{\oplus k} \oplus \oplus_{i=k+1}^r \Cal O_{\ell_0}(m_i)$, $\xi|_{\ell_0 \times \Pee^{k-1}} = \Cal O_{\ell_0}(1) \otimes \Cal O_{\Pee^{k-1}}(1)$ and $$\align N_{\Pee(\Cal O_{\ell_0}(1)^{\oplus k}) | \Pee(V|_{\ell_0})} &= \oplus_{i=k+1}^r \Cal O_{\ell_0}(-m_i) \otimes \xi|_{\ell_0 \times \Pee^{k-1}} \\ &= \oplus_{i=k+1}^r \Cal O_{\ell_0}(1-m_i) \otimes \Cal O_{\Pee^{k-1}}(1).\\ \endalign$$ Thus the previous two exact sequences are simplified to: $$0 \to N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V|_{\ell_0})} \to N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)} \to \Cal O_{\ell_0}(1)^{\oplus (n-1)} \to 0 \eqno (3.12)$$ $$0 \to \Cal O_{\Pee^{c_1+k-2r}}(1)^{\oplus (2r-c_1-1)} \to N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V|_{\ell_0})} \to$$ $$\oplus_{i=k+1}^r \Cal O_{\ell_0}(1-m_i) \otimes \Cal O_{\Pee^{c_1+k-2r}}(1) \to 0 \eqno (3.13)$$ Now (3.13) splits since for $k+1 \le i \le r$, we have $m_i \ge 2$ and $$\align &\quad \text{Ext}^1(\Cal O_{\ell_0}(1-m_i) \otimes \Cal O_{\Pee^{c_1+k-2r}}(1), \Cal O_{\Pee^{c_1+k-2r}}(1)) \\ &= H^1(\ell_0 \times \Pee^{c_1+k-2r}, \Cal O_{\ell_0}(m_i - 1)) = 0.\\ \endalign$$ Thus, the normal bundle $N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V|_{\ell_0})}$ is isomorphic to $$\oplus_{i=k+1}^r \Cal O_{\ell_0}(1-m_i) \otimes \Cal O_{\Pee^{c_1+k-2r}}(1) \oplus \Cal O_{\Pee^{c_1+k-2r}}(1)^{\oplus (2r-c_1-1)},$$ and the exact sequence (3.12) becomes to the exact sequence: $$0 \to \oplus_{i=k+1}^r \Cal O_{\ell_0}(1-m_i) \otimes \Cal O_{\Pee^{c_1+k-2r}}(1) \oplus \Cal O_{\Pee^{c_1+k-2r}}(1)^{\oplus (2r-c_1-1)} \to$$ $$N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)} \to \Cal O_{\ell_0}(1)^{\oplus (n-1)} \to 0 \eqno (3.14)$$ Restricting (3.14) to $\ell_0 \times \{ q \}$ and taking long exact cohomology sequence result $$\oplus_{i=k+1}^r H^1(\Cal O_{\ell_0}(1-m_i)) \otimes \Cal O_{\Pee^{c_1+k-2r}}(1)|_{q} \to$$ $$H^1((N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)})|_{\ell_0 \times \{ q \}}) \to 0. \eqno (3.15)$$ Next, we determine $N_f$ and show that $h^1(N_f) \le c_1+k-2r$. The two inclusions $\ell_0 \times \{ q \} \subset \ell_0 \times \Pee^{c_1+k-2r} \subset \Pee(V)$ give an exact sequence $$0 \to N_{\ell_0 \times \{ q \} | \ell_0 \times \Pee^{c_1+k-2r}} \to N_{\ell_0 \times \{ q \}|\Pee(V)} \to (N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)})|_{\ell_0 \times \{ q \}} \to 0.$$ Since $N_{\ell_0 \times \{ q \} | \ell_0 \times \Pee^{c_1+k-2r}} = T_{q, \Pee^{c_1+k-2r}}$, the above exact sequence becomes $$0 \to T_{q, \Pee^{c_1+k-2r}} \to N_f \to (N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)})|_{\ell_0 \times \{ q \}} \to 0. \eqno (3.16)$$ Thus, $h^1(N_f) = h^1((N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)})|_{\ell_0 \times \{ q \}})$. By (3.15), we obtain $$\align h^1(N_f) &= h^1((N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)})|_{\ell_0 \times \{ q \}}) \le \sum_{i=k+1}^r h^1(\Cal O_{\ell_0}(1-m_i)) \\ &= \sum_{i=k+1}^r (m_i - 2) = c_1 + k - 2r.\\ \endalign$$ Finally, we show that $h^1(N_f) = c_1 + k - 2r$. It suffices to prove that $h^1(N_f) \ge c_1 + k - 2r$. Since $\ell_0$ is a generic line in $\Pee^n$ and $V|_{\ell_0} = \Cal O_{\ell_0}(1)^{\oplus k} \oplus \oplus_{i = k+1}^r \Cal O_{\ell_0}(m_i)$, $\hbox{dim} \frak M(A, 0) = (2n + k)$ by Lemma 2.3 (ii). Since $h^0(N_f)$ is the dimension of the Zariski tangent space of $\frak M(A, 0)/PSL(2; \Cee)$ at $f$, $h^0(N_f) \ge (2n + k-3)$. Thus, $$h^1(N_f) = h^0(N_f) - \chi(N_f) \ge (2n + k-3) - (2n + 2r-c_1-3) = k + c_1 -2r.$$ Therefore, $h^1(N_f) = c_1 + k - 2r$. In particular, $h^1(N_f)$ is independent of $f \in \frak M$. To obtain the obstruction bundle $COB$ over $\Pee^{c_1+k-2r}$, we notice that (3.15) gives $$\oplus_{i=k+1}^r H^1(\Cal O_{\ell_0}(1-m_i)) \otimes \Cal O_{\Pee^{c_1+k-2r}}(1)|_{q} \cong H^1((N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)})|_{\ell_0 \times \{ q \}}).$$ Thus by the exact sequence (3.16), we conclude that $$\align H^1(N_f) &\cong H^1((N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)})|_{\ell_0 \times \{ q \}}) \\ &\cong \oplus_{i=k+1}^r H^1(\Cal O_{\ell_0}(1-m_i)) \otimes \Cal O_{\Pee^{c_1+k-2r}}(1)|_{q}. \tag 3.17 \endalign $$ It follows that $COB = \Cal O_{\Pee^{c_1+k-2r}}(1)^{\oplus (c_1+k-2r)}$. By (3.5), we obtain $$a_0 = \Phi_{(A, 0)}(h, h_n, h_n \xi_{2r-c_1-1}) = e(COB) = 1. \qed$$ \section{4. Proof of Theorem B} In this section, we prove Theorem B which we restate below. \theorem{4.1} {\rm (i)} Let $V$ be a rank-$r$ ample bundle over $\Pee^n$. Assume either $c_1 \le n$ or $c_1 \le (n + r)$ and $V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef so that $\Pee(V)$ is Fano. Then the quantum cohomology $H^*_{\omega}(\Pee(V); \Zee)$ is the ring generated by $h$ and $\xi$ with two relations $$h^{n+1} = \sum_{i+j \le (c_1 - r)} a_{i,j} \cdot h^i \cdot \xi^j \cdot e^{-t(n+1-i-j)} \eqno (4.2)$$ $$\sum_{i=0}^r (-1)^i c_i \cdot h^i \cdot \xi^{r-i} = e^{-tr} + \sum_{i+j \le (c_1-n-1)} b_{i,j} \cdot h^i \cdot \xi^{j} \cdot e^{-t(r-i-j)} \eqno (4.3)$$ where the coefficients $a_{i, j}$ and $b_{i, j}$ are integers depending on $V$; {\rm (ii)} If we further assume that $c_1<2r$, then the leading coefficient $a_{0,c_1-r}=1$. \endproclaim \noindent {\it Proof.} (i) First, we determine the first relation $f_\omega^1$ in (1.3). By Lemma 3.7, $$h \cdot h_p = h_{p +1} + \sum_{A \in H_2'} (h \cdot h_p)_A \cdot e^{t K_{\Pee(V)}(A)} \eqno (4.4)$$ where $p \ge 1$ and $H_2'$ stands for $H_2(\Pee(V); \Zee) - \Zee \cdot (h_n\xi_{r-2})_*$. Thus, $$h^{n - p} \cdot h_{p +1} = h^{n - p + 1} \cdot h_{p} - \sum_{A \in H_2'} h^{n - p} \cdot (h \cdot h_p)_A \cdot e^{t K_{\Pee(V)}(A)}.$$ If $(h \cdot h_p)_A \ne 0$, then $A = [E]$ for some effective curve $E$. So $a = h(A) \ge 0$. Since $A \in H_2'$, $a \ge 1$. We claim that $-K_{\Pee(V)}(A) \ge (n+1-c_1 + r)$ with equality if and only if $A = [h_{n-1}\xi_{r-1} + (1 - c_1) h_{n}\xi_{r-2}]_* \overset \hbox{def} \to = A_2$. Indeed, if $c_1 \le n$, then $-K_{\Pee(V)}(A) = (n+1-c_1)a + r \cdot \xi(A) \ge (n+1-c_1 + r)$ with equality if and only if $a = \xi(A) = 1$, that is, if and only if $A = A_2$; if $c_1 \le (n + r)$ and $(\xi -h)$ is nef, then again $-K_{\Pee(V)}(A) = (n+1+r-c_1)a + r \cdot (\xi - h)(A) \ge (n+1-c_1 + r)$ with equality if and only if $a = 1$ and $(\xi - h)(A) = 0$, that is, if and only if $A = A_2$. Thus, $\hbox{deg}((h \cdot h_p)_A) = 1 + p + K_{\Pee(V)}(A) \le (p - n +c_1 -r)$, and $\hbox{deg}(h^{n - p} \cdot (h \cdot h_p)_A) \le (c_1 - r)$. Using induction on $p$ and keeping track of the exponential $e^{t K_{\Pee(V)}(A)}$, we obtain $$0 = h_{n+1} = h^{n+1} - \sum_{i+j \le (c_1 - r)} a_{i,j} \cdot h^i \cdot \xi^j \cdot e^{-t(n+1-i-j)}.$$ Therefore, the first relation $f_\omega^1$ for the quantum cohomology ring is: $$h^{n+1} = \sum_{i+j \le (c_1 - r)} a_{i,j} \cdot h^i \cdot \xi^j \cdot e^{-t(n+1-i-j)}.$$ Next, we determine the second relation $f_\omega^2$ in (1.3). We need to compute the quantum product $h^i \cdot \xi^{r-i}$ for $0 \le i \le r$. First, we calculate the quantum product $\xi^r$. Note that if $A = (bh_{n}\xi_{r-2})_*$ with $b \ge 1$, then $-K_{\Pee(V)}(A) = br \ge r$ with $-K_{\Pee(V)}(A) = r$ if and only if $A = (h_{n}\xi_{r-2})_* \overset \hbox{def} \to = A_1$. Thus for $p \ge 1$, $$\xi \cdot \xi_p = \cases \xi_{p+1} + \sum_{A \in H_2'} (\xi \cdot \xi_p)_A \cdot e^{t K_{\Pee(V)}(A)}, &\text{if $p < r -1$}\\ \xi_{r} + (\xi \cdot \xi_{r-1})_{A_1} \cdot e^{-tr} + \sum_{A \in H_2'} (\xi \cdot \xi_{r-1})_A \cdot e^{t K_{\Pee(V)}(A)}, &\text{if $p = r -1$.}\\ \endcases $$ Note that $(\xi \cdot \xi_{r-1})_{A_1}$ is of degree zero; by Lemma 3.6, we obtain $(\xi \cdot \xi_{r-1})_{A_1} = \Phi_{(A_1, 0)}(\xi, \xi_{r-1}, h_{n}\xi_{r-1}) = 1$. Therefore for $p \ge 1$, $$\xi \cdot \xi_p = \cases \xi_{p+1} + \sum_{A \in H_2'} (\xi \cdot \xi_p)_A \cdot e^{t K_{\Pee(V)}(A)}, &\text{if $p < r -1$}\\ \xi_{r} + e^{-tr} + \sum_{A \in H_2'} (\xi \cdot \xi_{r-1})_A \cdot e^{t K_{\Pee(V)}(A)}, &\text{if $p = r -1$.}\\ \endcases \eqno (4.5)$$ Now, for $i \ge 1$ and $j \ge 1$ with $i + j \le r$, we have $$h_i \cdot \xi_j = \cases h_i \xi_{j} + \sum_{A \in H_2'} (h_i \cdot \xi_j)_A \cdot e^{t K_{\Pee(V)}(A)}, &\text{if $i + j < r$}\\ h_i \xi_{j} + (h_i \xi_{j})_{A_1} \cdot e^{-tr} + \sum_{A \in H_2'} (h_i \cdot \xi_j)_A \cdot e^{t K_{\Pee(V)}(A)}, &\text{if $i + j = r$;}\\ \endcases $$ when $i + j = r$, $(h_i \xi_{j})_{A_1}$ is of degree zero; by Lemma 3.7, we have $(h_i \cdot \xi_j)_{A_1} = \Phi_{(A_1, 0)}(h_i, \xi_{r-i}, h_{n}\xi_{r-1}) = 0$. Therefore for $i \ge 1$ and $j \ge 1$ with $i + j \le r$, $$h_i \cdot \xi_j = h_i \xi_{j} + \sum_{A \in H_2'} (h_i \cdot \xi_j)_A \cdot e^{t K_{\Pee(V)}(A)}. \eqno (4.6)$$ From the proof of the first relation $f_\omega^1$, we see that if $\alpha$ and $\beta$ are homogeneous elements in $H^*(\Pee(V); \Zee)$ with $\hbox{deg}(\alpha) + \hbox{deg}(\beta) = m \le r$, then $\hbox{deg}((\alpha \cdot \beta)_A) \le m - (n+1-c_1 + r)$ for $A \in H_2'$. Thus if $\gamma$ is a homogeneous element in $H^*(\Pee(V); \Zee)$ with $\hbox{deg}(\gamma) = r-m$, then $\hbox{deg}(\gamma \cdot (\xi \cdot \xi_p)_A) \le (c_1 - n - 1)$. Since $\sum_{i=0}^r (-1)^i c_i \cdot h_i \xi_{r-i} = 0$, it follows from (4.4), (4.5), and (4.6) that the second relation $f_\omega^2$ is $$\sum_{i=0}^r (-1)^i c_i \cdot h^i \cdot \xi^{r-i} = e^{-tr} + \sum_{i+j \le (c_1-n-1)} b_{i,j} \cdot h^i \cdot \xi^{j} \cdot e^{-t(r-i-j)}.$$ (ii) From the proof of the first relation in (i), we see that $-K_{\Pee(V)}(A) \ge (n+1-c_1 + r)$ with equality if and only if $A = A_2$; moreover, the term $\xi^{c_1-r}$ can only come from the quantum correction $(h \cdot h_n)_{A_2}$. Now $$(h \cdot h_n)_{A_2} = (\sum_{i = 0}^{c_1 - r} a_i' h_i \xi_{c_1 - r - i}) \cdot e^{-t(n+1-c_1 + r)}$$ where $a_0' = \Phi_{(A_2, 0)}(h, h_n, h_n \xi_{2r-c_1-1})$. Since $c_1 < 2r$, $(c_1 - r)< r$. By (4.4), (4.5), and (4.6), we conclude that $h_i \xi_{c_1 - r - i} = h^i \cdot \xi^{c_1 - r - i} + \text{(lower degree terms)}$. Thus $a_{0,c_1-r} = a_0' = \Phi_{(A_2, 0)}(h, h_n, h_n \xi_{2r-c_1-1})$. By Lemma 3.8, $a_{0,c_1-r} = 1.\qed$ It is understood that when $c_1 \le n$, then the summations on the right-hand-sides of the second relations (4.3) and (4.9) below do not exist. Next, we shall sharpen the results in Theorem 4.1 by imposing additional conditions on $V$. Let $V$ be a rank-$r$ ample vector bundle over $\Pee^n$. Then $c_1 \ge r$. Thus if $c_1<2r$ and if either $2c_1 \le (n + r)$ or $2c_1 \le (n + 2r)$ and $V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef, then the conditions in Theorem 4.1 are satisfied. \corollary{4.7} {\rm (i)} Let $V$ be a rank-$r$ ample vector bundle over $\Pee^n$ with $c_1<2r$. Assume that either $2c_1 \le (n + r)$ or $2c_1 \le (n + 2r)$ and $V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef so that $\Pee(V)$ is a Fano variety. Then the first relation (4.2) is $$h^{n+1}= \left ( \sum_{i=0}^{c_1-r} a_i \cdot h^i \cdot \xi^{c_1-r-i} \right ) \cdot e^{-t(n+1+r-c_1)} \eqno (4.8)$$ where the integers $a_i$ depend on $V$. Moreover, $a_0 = 1$. {\rm (ii)} Let $V$ be a rank-$r$ ample vector bundle over $\Pee^n$. Assume that $2c_1 \le (2n + r+1)$ and $V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef so that $\Pee(V)$ is Fano. Then the second relation (4.3) is $$\sum_{i=0}^r (-1)^i c_i \cdot h^i \cdot \xi^{r-i} = e^{-tr} + \sum_{i=0}^{c_1-n-1} b_{i} \cdot h^i \cdot \xi^{c_1-n-1-i} \cdot e^{-t(n+1+r-c_1)} \eqno (4.9)$$ where the integers $b_i$ depend on $V$. \endproclaim \proof (i) From the proof of Theorem 4.1 (i), we notice that it suffices to show that the only homology class $A \in H_2' = H_2(\Pee(V); \Zee) - \Zee \cdot (h_n\xi_{r-2})_*$ which has nonzero contributions to the quantum corrections in (4.4) is $A = [h_{n-1}\xi_{r-1} + (1 - c_1) h_{n}\xi_{r-2}]_* \overset \hbox{def} \to = A_2$. In other words, if $A = (ah_{n-1}\xi_{r-1}+ b h_{n}\xi_{r-2})_*$ with $a \ne 0$ and if $\Phi_{(A, 0)}(h, h_p, \alpha) \ne 0$ for $1 \le p \le n$ and $\alpha \in H^*(\Pee(V); \Zee)$, then $A = A_2$. First of all, we show that $a = 1$. Suppose $a \ne 1$. Then $a \ge 2$. By (3.2), $$\align {1 \over 2} \cdot \hbox{deg}(\alpha) &= (n+r-1) - K_{\Pee(V)}(A) - 1 - p\cr &= (n+r-1) + [(n + 1-c_1)a + r \cdot \xi(A)] - 1-p\cr &\ge \hbox{dim}(\Pee(V)) + [(n + 1-c_1)a + r \cdot \xi(A)] - 1-n.\cr \endalign$$ If $2c_1 \le (n + r)$, then $c_1 \le n$, and $[(n + 1-c_1)a + r \cdot \xi(A)] - 1-n \ge 2(n + 1-c_1) + r - 1-n > 0$. If $2c_1 \le (n + 2r)$ and $(\xi - h)$ is nef, then $c_1 \le n + r$, and $[(n + 1-c_1)a + r \cdot \xi(A)] - 1-n = [(n + 1 + r-c_1)a + r \cdot (\xi-h)(A)] - 1-n \ge 2(n + 1 + r-c_1)-1 -n > 0$. Thus, $[(n + 1-c_1)a + r \cdot \xi(A)] - 1-n > 0$, and so $\hbox{deg}(\alpha)/2 > \hbox{dim}(\Pee(V))$. But this is absurd. Next, we prove that $b = (1 - c_1)$, or equivalently, $\xi(A) = 1$. Suppose $\xi(A) \ne 1$. Then $\xi(A) \ge 2$. By (3.2), $$\align {1 \over 2} \cdot \hbox{deg}(\alpha) &= (n+r-1) + [(n + 1-c_1) + r \cdot \xi(A)] - 1-p\cr &\ge \hbox{dim}(\Pee(V)) + [(n + 1-c_1) + 2r] - 1-n\cr &> \hbox{dim}(\Pee(V))\cr \endalign$$ since $c_1 < 2r$. But once again this is absurd. (ii) We follow the previous arguments for (i). Again it suffices to show that if $A = (ah_{n-1}\xi_{r-1}+ b h_{n}\xi_{r-2})_*$ with $a \ne 0$ and if $\Phi_{(A, 0)}(\alpha_1, \alpha_2, \alpha) \ne 0$ for some $\alpha_1, \alpha_2, \alpha \in H^*(\Pee(V); \Zee)$ with $\hbox{deg}(\alpha_1) + \hbox{deg}(\alpha_2) \le r$, then $A = A_2$. Indeed, if $a \ne 1$ or if $a = 1$ but $\xi(A) \ne 1$, then we must have $\hbox{deg}(\alpha)/2 > \hbox{dim}(\Pee(V))$. But this is impossible. Therefore, $a = 1$ and $\xi(A) = 1$. So $A = A_2$. \endproof Now we discuss the relation between the quantum corrections and the extremal rays of the Fano variety $\Pee(V)$. Let $V$ be a rank-r ample vector bundle over $\Pee^n$ with $c_1<2r$ and $2c_1 \le (n + r)$. By (4.8) and (4.3), the quantum cohomology ring $H^*_{\omega}(\Pee(V); \Zee)$ is the ring generated by $h$ and $\xi$ with two relations $$h^{n+1}= \left ( \sum_{i=0}^{c_1-r} a_i \cdot h^i \cdot \xi^{c_1-r-i} \right ) \cdot e^{-t(n+1+r-c_1)} \eqno (4.10)$$ $$\sum_{i=0}^r (-1)^i c_i \cdot h^i \cdot \xi^{r-i} = e^{-tr}. \eqno (4.11)$$ From the proof of Theorem 4.1 (i), we notice that the quantum correction to the second relation (4.11) comes from the homology class $A_1 = (h_n\xi_{r-2})_*$ which is represented by the lines in the fibers of $\pi: \Pee(V) \to \Pee^n$. Also, we notice from the proof of Corollary 4.7 (i) that the quantum correction to the first relation (4.10) comes from the homology class $A_2 = [h_{n-1}\xi_{r-1} + (1 - c_1) h_{n}\xi_{r-2}]_*$; from the proof of Lemma 3.8, $A_2$ can be represented by a smooth rational curve isomorphic to lines in $\Pee^n$ via $\pi$. Now $A_1$ generates one of the two extremal rays of $\Pee(V)$. It is unclear whether $A_2$ generates the other extremal ray. By Lemma 2.2 (iii), if we further assume that $2c_1 \le (n + 1)$, then indeed $A_2$ generates the other extremal ray of $\Pee(V)$. By Lemma 2.2 (ii), $A_2$ generates the other extremal ray of $\Pee(V)$ if and only if $(\xi - h)$ is nef, that is, $V \otimes \Cal O_{\Pee^n}(-1)$ is a nef vector bundle over $\Pee^n$. \section{5. Direct sum of line bundles over $\Pee^n$} In this section, we partially verify Batyrev's conjecture on the quantum cohomology of projective bundles associated to direct sum of line bundles over $\Pee^n$. We shall use (3.5) to compute the necessary Gromov-Witten invariants. Our first step is to recall some standard materials for the Grassmannian $G(2, n+1)$ from \cite{3}. Then we determine certain obstruction bundle and its Euler class. Finally we proceed to determine the first and second relations for the quantum cohomology. On the Grassmannian $G(2, n+1)$, there exists a tautological exact sequence $$0 \to S \to (\Cal O_{G(2, n+1)})^{\oplus (n + 1)} \to Q \to 0 \eqno (5.1)$$ where the sub- and quotient bundles $S$ and $Q$ are of rank $2$ and $(n -1)$ respectively. Let $\alpha$ and $\beta$ be the virtual classes such that $\alpha + \beta = -c_1(S)$ and $\alpha \beta = c_2(S)$. Then $$\hbox{cl}(\{ \ell \in G(2, n+1)| \ell \cap h_p \ne \emptyset \}) = {\alpha^p - \beta^p \over \alpha - \beta} \eqno (5.2)$$ where $\hbox{cl}(\cdot)$ denotes the fundamental class and $h_p$ stands for a fixed linear subspace of $\Pee^n$ of codimention $p$. If $P(\alpha, \beta)$ is a symmetric homogeneous polynomial of degree $(2n - 2)$ (so that $P(\alpha, \beta)$ can be written as a polynomial of maximal degree in the Chern classes of the bundle $S$), then we have $$\int_{G(2, n+1)} P(\alpha, \beta) = \left ( \text{the coefficient of } \alpha^n \beta^n \text{ in} -{1 \over 2}(\alpha - \beta)^2 P(\alpha, \beta) \right ). \eqno (5.3)$$ Let $F_n = \{ (x, \ell) \in \Pee^n \times G(2, n+1)| x \in \ell \}$, and $\pi_1$ and $\pi_2$ are the two natural projections from $F_n$ to $\Pee^n$ and $G(2, n+1)$ respectively. Then $F_n = \Pee(S^*)$ where $S^*$ is the dual bundle of $S$, and $(\pi_1^*\Cal O_{\Pee^n}(1))|_{F_n}$ is the tautological line bundle over $F_n$. Let $\hbox{Sym}^m(S^*)$ be the $m$-th symmetric product of $S^*$. Then for $m \ge 0$, $$\pi_{2*} (\pi_1^*\Cal O_{\Pee^n}(m)|_{F_n}) \cong \hbox{Sym}^m(S^*). \eqno (5.4)$$ By the duality theorem for higher direct image sheaves (see p.253 in \cite{7}), $$\align R^1 \pi_{2*} (\pi_1^*\Cal O_{\Pee^n}(-m)|_{F_n}) &\cong (\pi_{2*} (\pi_1^*\Cal O_{\Pee^n}(m - 2)|_{F_n}))^* \otimes (\hbox{det}S^*)^* \\ &\cong \hbox{Sym}^{m-2}(S) \otimes (\hbox{det}S) \tag 5.5 \\ \endalign$$ Now, let $V = \oplus_{i = 1}^r \Cal O_{\Pee^n}(m_i)$ where $1 = m_1 = \ldots = m_k < m_{k+1} \le \ldots \le m_r$. Assume that $k \ge 1$ and $\Pee(V)$ is Fano. Then the two extremal rays of $\Pee(V)$ are generated by the two classes $A_1 = (h_{n}\xi_{r-2})_*$ and $A_2 = [h_{n-1}\xi_{r-1}+ (1 - c_1) h_{n}\xi_{r-2}]_*$. From the proof of Lemma 2.3 (ii), we see that $$\frak M(A_2, 0)/PSL(2; \Cee) = G(2, n+1) \times \Pee^{k -1}. \eqno (5.6)$$ Let a morphism $f \in \frak M(A_2, 0)$ be induced by some surjective map $V|_\ell \to \Cal O_\ell(1) \to 0$ such that the image $\hbox{Im}(f)$ of $f$ is of the form $$\hbox{Im}(f) = \ell \times \{q\} \subset \ell \times \Pee^{k-1} \subset \Pee^n \times \Pee^{k-1}. $$ Then by arguments similar to the proof of (3.17), we have $$H^1(N_f) \cong \oplus_{u=k+1}^r H^1(\Cal O_{\ell}(1-m_u)) \otimes \Cal O_{\Pee^{k-1}}(1)|_{q}. \eqno (5.7)$$ It follows that the obstruction bundle $COB$ over $\frak M(A_2, 0)/PSL(2; \Cee)$ is $$COB \cong \oplus_{u=k+1}^r R^1 \pi_{2*} (\pi_1^*\Cal O_{\Pee^n}(1-m_u)|_{F_n}) \otimes \Cal O_{\Pee^{k-1}}(1). \eqno (5.8)$$ Since $c_1(S) = -(\alpha + \beta)$ and $c_2(S) = \alpha \beta$, we obtain from (5.5) the following. \lemma{5.9} The Euler class of the obstruction bundle $COB$ is $$e(COB) = \prod_{u = k +1}^r \prod_{v=0}^{m_u - 3} [(1+v)(-\alpha) + (m_u - 2 -v)(-\beta) + \tilde h] \eqno (5.10)$$ where $\tilde h$ stands for the hyperplane class in $\Pee^{k-1}$. \qed \endproclaim Next assuming $c_1 < 2r$, we shall compute the Gromov-Witten invariant $$W_i \overset \hbox{def} \to = \Phi_{(A_2, 0)}(h_{\tilde n}, h_{n+1 - \tilde n}, h_{n-i}\xi_{2r-c_1 - 1+i}) \eqno (5.11)$$ where $0 \le i \le (c_1 - r)$ and $\tilde n = \left [{n+1 \over 2} \right ]$ is the largest integer $\le (n+1)/2$. \lemma{5.12} Assume $c_1 < \text{min}(2r, (n+1+2r)/2)$ and $0 \le i \le (c_1 - r)$. Then $W_i$ is the coefficient of $t^i$ in the power series expansion of $$\prod_{u = 1}^r (1 - m_u t)^{m_u-2}.$$ \endproclaim \noindent {\it Proof.} Note that the restriction of $\xi$ to $\Pee^n \times \Pee^{k-1} = \Pee(\Cal O_{\Pee^n}(1)^{\oplus k})$ is $(h + \tilde h)$. Thus, $$\align h_{n-i}\xi_{2r-c_1 - 1+i}|_{\Pee^n \times \Pee^{k-1}} &= \sum_{j = 0}^{2r-c_1-1+i} {2r-c_1 - 1+i \choose j} h_{n-i+j} \tilde h_{2r-c_1-1+i-j}\\ &= \sum_{j = 0}^{i} {2r-c_1 - 1+i \choose j} h_{n-i+j} \tilde h_{2r-c_1-1+i-j}. \\ \endalign$$ So by (3.5) (replacing $\frak M(A_2, 0)$ by $\frak M(A_2, 0)/PSL(2; \Cee)$), (5.2), and Lemma 5.9, $$W_i = \int_{G(2, n+1) \times \Pee^{k-1}} \tilde P(\alpha, \beta) \eqno (5.13)$$ where $\tilde P(\alpha, \beta)$ is the symmetric homogeneous polynomial of degree $(2n - 2) + (k-1)$: $$\align \tilde P(\alpha, \beta) &= {\alpha^{\tilde n} - \beta^{\tilde n} \over \alpha - \beta} \cdot {\alpha^{n + 1- \tilde n} - \beta^{n + 1- \tilde n} \over \alpha - \beta} \\ &\qquad \cdot \sum_{j = 0}^{i} {2r-c_1 - 1+i \choose j} {\alpha^{n-i+j} - \beta^{n-i+j} \over \alpha - \beta} \cdot \tilde h_{2r-c_1-1+i-j} \\ &\qquad \cdot \prod_{u = k +1}^r \prod_{v=0}^{m_u - 3} [(1+v)(-\alpha) + (m_u - 2 -v)(-\beta) + \tilde h] \\ &= \sum_{j = 0}^{i} {2r-c_1 - 1+i \choose j} {\alpha^{n+1} - \alpha^{n + 1- \tilde n}\beta^{\tilde n} - \alpha^{\tilde n} \beta^{n + 1- \tilde n} + \beta^{n+1} \over (\alpha - \beta)^2}\\ &\qquad \cdot \sum_{t= 0}^{n-i+j-1} \alpha^{t} \beta^{n-i+j-1-t} \cdot \tilde h_{2r-c_1-1+i-j}\\ &\qquad \cdot \prod_{u = k +1}^r \prod_{v=0}^{m_u - 3} [(1+v)(-\alpha) + (m_u - 2 -v)(-\beta) + \tilde h]. \\ \endalign$$ By (5.3) and (5.13), we conclude from straightforward manipulations that: $$\align W_i &= \sum_{j = 0}^{i} {2r-c_1 - 1+i \choose j} \cdot (-1)^{i-j} \\ &\qquad \cdot \sum_{j_{k+1} + \ldots + j_r = i -j} \quad \prod_{u = k+1}^r {m_u-2 \choose j_u} (m_u -1)^{j_u} \\ &= \sum_{j = 0}^{i} {2r-c_1 - 1+i \choose i-j} \cdot (-1)^{j} \\ &\qquad \cdot \sum_{j_{k+1} + \ldots + j_r = j} \quad \prod_{u = k+1}^r {m_u-2 \choose j_u} (m_u -1)^{j_u}. \\ \endalign$$ Thus $W_i$ is the coefficient of $t^i$ in the polynomial $$\align &\qquad (1+t)^{2r-c_1 - 1+i} \cdot \prod_{u = k+1}^r [1 - (m_u -1) t]^{m_u-2}\\ &= (1+t)^{2r-c_1 - 1+i} \cdot \prod_{u = k+1}^r [(1+t) - m_u t]^{m_u-2}\\ &= (1+t)^{2r-c_1 - 1+i} \cdot \sum_{j=0}^{c_1-2r+k} \sum_{j_{k+1} + \ldots + j_r = j} \\ &\qquad \cdot \prod_{u = k+1}^r {m_u-2 \choose j_u} (-m_ut)^{j_u} \cdot (1+t)^{m_u-2-j_u}\\ &= \sum_{j=0}^{c_1-2r+k} \sum_{j_{k+1} + \ldots + j_r = j} \quad \prod_{u = k+1}^r {m_u-2 \choose j_u} (-m_ut)^{j_u} \cdot (1+t)^{i+k-1-j}\\ \endalign$$ since $\sum_{u = k+1}^r (m_u - 2 - j_u) = c_1 -2r+k - j$. So $W_i$ is the coefficient of $t^i$ in $$\align \prod_{u = k+1}^r (1-m_u t)^{m_u-2} \cdot \sum_{j=0}^{+ \infty} {j +k-1 \choose k-1} t^j &= \prod_{u = k+1}^r (1-m_u t)^{m_u-2} \cdot {1 \over (1 - t)^k}\\ &= \prod_{u = 1}^r (1 - m_u t)^{m_u-2}. \qed\\ \endalign$$ \proposition{5.14} Let $V = \oplus_{i = 1}^r \Cal O_{\Pee^n}(m_i)$ where $m_i \ge 1$ for each $i$ and $$\sum_{i=1}^r m_i < \text{min}(2r, (n+1+2r)/2).$$ Then the first relation $f^1_\omega$ for the quantum cohomology ring $H^*_{\omega}(\Pee(V); \Zee)$ is $$h^{n+1} = \prod_{u = 1}^r (\xi - m_u h)^{m_u-1} \cdot e^{-t(n+1+r-\sum_{i=1}^r m_i)}. \eqno (5.15)$$ \endproclaim \noindent {\it Proof.} We may assume that $1 = m_1 = \ldots = m_k < m_{k+1} \le \ldots \le m_r$. Since the conclusion clearly holds when $k = r$, we also assume that $k < r$. Let $c_1 = \sum_{i=1}^r m_i$. Notice that the conditions in Corollary 4.7 (i) are satisfied. Thus, $$h^{n+1}= \left ( \sum_{i=0}^{c_1-r} a_i \cdot h^i \cdot \xi^{c_1-r-i} \right ) \cdot e^{-t(n+1+r-c_1)}.$$ More directly, putting $\tilde n = \left [{n+1 \over 2} \right ]$, then $\tilde n < -K_{\Pee(V)}(A_2) = (n+1+r -c_1)$, and $(n+1-\tilde n) < -K_{\Pee(V)}(A_2)$ unless $n$ is even and $c_1 = (n + 2r)/2$. From the proofs in Theorem 4.1 and Corollary 4.7 (i) for the first relation $f^1_\omega$, we have $h^{\tilde n} = h_{\tilde n}$, and $h^{n+1-\tilde n} = h_{n+1-\tilde n}$ unless $n$ is even and $c_1 = (n + 2r)/2$. Moreover, if $n$ is even and $c_1 = (n + 2r)/2$, then $h^{n+1-\tilde n} = h \cdot h^{n-\tilde n} = h \cdot h_{n-\tilde n} = h_{n+1-\tilde n} + (h \cdot h_{n-\tilde n})_{A_2} \cdot e^{-t(n+1+r-c_1)}$. Since $(h \cdot h_{n-\tilde n})_{A_2}$ is of degree zero, $(h \cdot h_{n-\tilde n})_{A_2} = \Phi_{(A_2, 0)}(h, h_{n-\tilde n}, h_n\xi_{r-1})$. Since $1 \le k <r$, we can choose a point $q_0$ in $\Pee(V)$ representing the homology class $(h_n\xi_{r-1})_*$ such that the point $q_0$ is not contained in the $(k-1)$-dimensional linear subspace $$\Pee^{k -1} = \Pee((\Cal O_{\Pee^n}(1)^{\oplus k})|_{\pi(q_0)}) \subset \Pee(V|_{\pi(q_0)}) \cong \Pee^{r-1}.$$ Note that for every $f \in \frak M(A_2, 0)$, $\hbox{Im}(f) = \ell \times \{ q \}$ for some line $\ell \subset \Pee^n$ and some point $q \in \Pee^{k -1}$. Thus $\hbox{Im}(f)$ can not pass $q_0$. As in the proof of Lemma 3.7, we conclude that $\Phi_{(A_2, 0)}(h, h_{n-\tilde n}, h_n\xi_{r-1}) = 0$. Therefore, $h^{n+1-\tilde n} = h_{n+1-\tilde n}$. So $$h^{n+1} = h^{\tilde n} \cdot h^{n+1-\tilde n} = h_{\tilde n} \cdot h_{n+1-\tilde n}.$$ By similar arguments in the proofs of Theorem 4.1 and Corollary 4.7 (i) for the first relation $f^1_\omega$, we see that if $(h_{\tilde n} \cdot h_{n+1-\tilde n})_A \ne 0$, then $A = 0, A_2$. Thus $$h^{n+1} = h_{n+1} + (h_{\tilde n} \cdot h_{n+1-\tilde n})_{A_2} \cdot e^{-t(n+1+r-c_1)} = (h_{\tilde n} \cdot h_{n+1-\tilde n})_{A_2} \cdot e^{-t(n+1+r-c_1)}.$$ So it suffices to show that $(h_{\tilde n} \cdot h_{n+1-\tilde n})_{A_2} = \prod_{u = 1}^r (\xi - m_u h)^{m_u-1}$. Note that $$\prod_{u = 1}^r (\xi - m_u h)^{m_u-1} = \prod_{u = 1}^r (\xi - m_u h)_{m_u-1}$$ where the right-hand-side stands for the product in the ordinary cohomology. Thus we need to show that $(h_{\tilde n} \cdot h_{n+1-\tilde n})_{A_2} = \prod_{u = 1}^r (\xi - m_u h)_{m_u-1}$, or equivalently, $$\Phi_{(A_2, 0)}(h_{\tilde n}, h_{n+1 - \tilde n}, h_{n-i}\xi_{2r-c_1 - 1+i}) = \prod_{u = 1}^r (\xi - m_u h)_{m_u-1} h_{n-i}\xi_{2r-c_1 - 1+i} \eqno (5.16)$$ for $0 \le i \le (c_1 - i)$. The left-hand-side of (5.16) is computed in Lemma 5.12. Denote the right-hand-side of (5.16) by $\tilde W_i$. Let $s_i$ be the $i$-th Segre class of $V$. Then we have $s_i = (-1)^i \cdot \sum_{j_1 + \ldots + j_r = i} \prod_{u=1}^r m_u^{j_u}$ and $$\sum_{i=0}^{+ \infty} (-1)^{i} s_{i} t^{i} = \prod_{u=1}^r {1 \over 1-m_u t}. \eqno (5.17)$$ Moreover from the second relation in (1.1), we obtain for $i \ge r$, $$\xi_i = (-1)^{i - (r-1)} s_{i-(r-1)} \xi_{r-1} + \text{ (terms with exponentials of } \xi \text{ less than } (r-1)).$$ It follows from the right-hand-side of (5.16) that $\tilde W_i$ is equal to $$\align &\quad \sum_{j=0}^{c_1-r} \quad \sum_{j_1 + \ldots + j_r = j} \quad \prod_{u=1}^r {m_u-1 \choose j_u} \xi_{m_u-1-j_u} (-m_uh)_{j_u} h_{n-i}\xi_{2r-c_1 - 1+i} \\ &= \sum_{j=0}^{i} \quad \sum_{j_1 + \ldots + j_r = j} \quad \prod_{u=1}^r {m_u-1 \choose j_u} (-m_u)^{j_u} h_{n-i+j}\xi_{r - 1+i-j} \\ &= \sum_{j=0}^{i} (-1)^{i-j} s_{i-j} \quad \sum_{j_1 + \ldots + j_r = j} \quad \prod_{u=1}^r {m_u-1 \choose j_u} (-m_u)^{j_u}. \\ \endalign$$ Therefore, the formal power series $\sum_{i=0}^{+ \infty} \tilde W_i t^i$ is equal to $$\align &\quad \sum_{i=0}^{+ \infty} \sum_{j=0}^{i} (-1)^{i-j} s_{i-j} t^{i-j} \quad \sum_{j_1 + \ldots + j_r = j} \quad \prod_{u=1}^r {m_u-1 \choose j_u} (-m_u t)^{j_u} \\ &= \sum_{j=0}^{+ \infty} \sum_{i=j}^{+ \infty} (-1)^{i-j} s_{i-j} t^{i-j} \quad \sum_{j_1 + \ldots + j_r = j} \quad \prod_{u=1}^r {m_u-1 \choose j_u} (-m_u t)^{j_u} \\ &= \sum_{j=0}^{+ \infty} \sum_{i=0}^{+ \infty} (-1)^{i} s_{i} t^{i} \quad \sum_{j_1 + \ldots + j_r = j} \quad \prod_{u=1}^r {m_u-1 \choose j_u} (-m_u t)^{j_u} \\ &= \sum_{j=0}^{+ \infty} \prod_{u=1}^r {1 \over 1-m_u t} \quad \sum_{j_1 + \ldots + j_r = j} \quad \prod_{u=1}^r {m_u-1 \choose j_u} (-m_u t)^{j_u} \\ &= \prod_{u=1}^r {1 \over 1-m_u t} \sum_{j=0}^{+ \infty} \quad \sum_{j_1 + \ldots + j_r = j} \quad \prod_{u=1}^r {m_u-1 \choose j_u} (-m_u t)^{j_u} \\ &= \prod_{u=1}^r {1 \over 1-m_u t} \quad \prod_{u=1}^r (1-m_u t)^{m_u-1}\\ &= \prod_{u=1}^r (1-m_u t)^{m_u-2}\\ \endalign$$ where we have applied (5.17) in the third equality. By Lemma 5.12, $\tilde W_i = W_i$ for $0 \le i \le (c_1 - r)$. Hence the formule (5.16) and (5.15) hold. \qed It turns out that under certain conditions on the integers $m_i$, the second relation $f^2_\omega$ for the quantum cohomology ring $H^*_{\omega}(\Pee(V); \Zee)$ is much easier to be determined. Note that the second relation $f^2$ in (1.1) can be rewritten as $$\prod_{i=1}^r (\xi-m_i h) = 0 \eqno (5.18)$$ where the left-hand-side stands for the product in the ordinary cohomology ring. \proposition{5.19} Let $V = \oplus_{i = 1}^r \Cal O_{\Pee^n}(m_i)$ where $m_i \ge 1$ for each $i$, $m_i = 1$ for some $i$, and $\sum_{i=1}^r m_i < (2n+2+r)/2$. Then the second relation $f^2_\omega$ for the quantum cohomology ring $H^*_{\omega}(\Pee(V); \Zee)$ is $$\prod_{i=1}^r (\xi-m_i h) = e^{-tr} \eqno (5.20)$$ where the left-hand-side stands for the product in the quantum cohomology ring. \endproclaim \noindent {\it Proof.} We may assume that $1 = m_1 = \ldots = m_k < m_{k+1} \le \ldots \le m_r$. So $k \ge 1$. We notice that the conditions in Corollary 4.7 (ii) are satisfied. From the proofs of Theorem 4.1 (i) and Corollary 4.7 (ii), we see that the quantum corrections to the second relation (5.18) can only come from the classes $A_1, A_2$; moreover, the quantum correction from $A_1$ is $e^{-tr}$. Thus it suffices to show that the quantum correction from $A_2$ is zero. In view of (3.3), it suffices to show that $$\Phi_{(A_2, 0)}(\xi-m_1 h, \ldots, \xi-m_r h, \alpha) = 0$$ for every $\alpha \in H^*(\Pee(V); \Zee)$. For $1 \le i \le r$, let $V_i$ be the subbundle of $V$: $$V_i = \Cal O_{\Pee^n}(m_1) \oplus \ldots \oplus \Cal O_{\Pee^n}(m_{i-1}) \oplus \Cal O_{\Pee^n}(m_{i+1}) \oplus \ldots \oplus \Cal O_{\Pee^n}(m_r),$$ and let $B_i = \Pee(V_i)$ be the codimension-$1$ subvariety of $\Pee(V)$ induced by the projection $V \to V_i \to 0$. Then the fundamental class of $B_i$ is $(\xi-m_i h)$. As in the proof of Lemma 3.7, we need only to show that if $f \in \frak M(A_2, 0)$, then the image $\hbox{Im}(f)$ can not intersect with $B_1, \ldots, B_r$ simultaneously. In fact, we will show that $\hbox{Im}(f)$ can not intersect with $B_1, \ldots, B_k$ simultaneously. Indeed, $\hbox{Im}(f)$ is of the form $$\hbox{Im}(f) = \ell \times \{q\} \subset \ell \times \Pee^{k-1} \subset \Pee^n \times \Pee^{k-1} = \Pee(\Cal O_{\Pee^n}(1)^{\oplus k})$$ for some line $\ell \subset \Pee^n$, and $B_i|_{\pi^{-1}(\ell)} = \Pee(V_i|_\ell)$. Put $p = \pi(q) \in \Pee^n$, and $$V|_p = \oplus_{i=1}^k \Cee \cdot e_i \oplus (\oplus_{i=k+1}^r \Cal O_{\Pee^n}(m_i)|_p)$$ where $e_i$ is a global section of $\Cal O_{\Pee^n}(m_i) = \Cal O_{\Pee^n}(1)$ for $i \le k$. Now the point $q$ is identified with $\Cee \cdot v$ for some nonzero vector $v \in \oplus_{i=1}^k \Cee \cdot e_i$. Let $v = \sum_{i=1}^k a_i e_i$. Since $\ell \times \{q\}$ and $B_i$ ($1 \le i \le k$) intersect, the one-dimensional vector space $\Cee \cdot v$ is also contained in $(V_i)|_p$. It follows that $a_i = 0$ for every $i$ with $1 \le i \le k$. But this is impossible since $v$ is a nonzero vector. \qed In summary, we partially verify Batyrev's conjecture. \theorem{5.21} Let $V = \oplus_{i = 1}^r \Cal O_{\Pee^n}(m_i)$ where $m_i \ge 1$ for each $i$ and $$\sum_{i=1}^r m_i < \text{min}(2r, (n+1+2r)/2, (2n+2+r)/2).$$ Then the quantum cohomology $H^*_{\omega}(\Pee(V); \Zee)$ is generated by $h$ and $\xi$ with relations $$h^{n+1}=\prod^r_{i=1}(\xi-m_ih)^{m_i-1} \cdot e^{-t(n+1+r-\sum_{i=1}^r m_i)} \qquad {and} \qquad \prod^r_{i=1}(\xi-m_ih) = e^{-tr}.$$ \endproclaim \proof Follows immediately from Propositions 5.14 and 5.19. \endproof \section{6. Examples} In this section, we shall determine the quantum cohomology of $\Pee(V)$ for ample bundles $V$ over $\Pee^n$ with $2 \le r \le n$ and $c_1 = r + 1$. In these cases, $V|_\ell = \Cal O_\ell(1)^{\oplus (r - 1)} \oplus \Cal O_\ell(2)$ for every line $\ell \subset \Pee^n$. In particular, $V$ is a uniform bundle. If $r < n$, then by the Theorem 3.2.3 in \cite{10}, $V = \Cal O_{\Pee^n}(1)^{\oplus (r - 1)} \oplus \Cal O_{\Pee^n}(2)$; if $r = n$, then by the results on pp.71-72 in \cite{10}, $V = \Cal O_{\Pee^n}(1)^{\oplus (n - 1)} \oplus \Cal O_{\Pee^n}(2)$ or $V = T_{\Pee^n}$ the tangent bundle of $\Pee^n$. When $V = \Cal O_{\Pee^n}(1)^{\oplus (r - 1)} \oplus \Cal O_{\Pee^n}(2)$ with $r \le n$, the conditions in Theorem 5.21 are satisfied, so the quantum cohomology ring $H^*_{\omega}(\Pee(V); \Zee)$ is the ring generated by $h$ and $\xi$ with two relations $$h^{n+1}= (\xi-2h) \cdot e^{-t(n+1+r-c_1)} \qquad \hbox{and} \qquad (\xi-h)^{r-1} (\xi-2h) = e^{-tr}.$$ In the rest of this section, we compute the quantum cohomology of $\Pee(T_{\Pee^n})$. It is well-known that $(\xi-h)$ is a nef divisor on $\Pee(T_{\Pee^n})$, and the two extremal rays of $\Pee(T_{\Pee^n})$ are generated by $A_1 = (h_{n}\xi_{n-2})_*$ and $A_2 = (h_{n-1}\xi_{n-1} - n h_{n}\xi_{n-2})_*$. Moreover, $A_2$ is represented by smooth rational curves in $\Pee(T_{\Pee^n})$ induced by the surjective maps $T_{\Pee^n}|_\ell \to \Cal O_\ell(1) \to 0$ for lines $\ell \subset \Pee^n$. Since $c_1 = n+1$ and $n \ge 2$, the assumptions in Corollary 4.7 are satisfied, so the quantum cohomology ring $H^*_{\omega}(\Pee(T_{\Pee^n}); \Zee)$ is the ring generated by $h$ and $\xi$ with two relations $$h^{n+1}= (a_1 h + \xi) \cdot e^{-tn} \quad \text{and} \quad \sum_{i=0}^n (-1)^i c_i \cdot h^i \cdot \xi^{n-i} = (1 + b_0) \cdot e^{-tn}. \eqno (6.1)$$ More precisely, putting $H_2' = H_2(\Pee(V); \Zee) - \Zee \cdot (h_n\xi_{n-2})_*$, then we see from the proof of Corollary 4.7 (i) that the only homology class $A \in H_2'$ which has nonzero contributions to the quantum corrections in (4.4) is $A = A_2$. Thus by (4.4), $$h \cdot h_p = \cases h_{p+1}, &\text{if $p \le n -2$}\\ h_n + a_1' \cdot e^{-tn}, &\text{if $p = n -1$}\\ h_{n+1} + (a_2'h+a_3'\xi) \cdot e^{-tn}, &\text{if $p = n$}.\\ \endcases \eqno (6.2)$$ where $a_1' = \Phi_{(A_2, 0)}(h, h_{n-1}, h_n \xi_{n-1})$, $a_3' = \Phi_{(A_2, 0)}(h, h_{n}, h_n \xi_{n-2})$, and $$a_2' = \Phi_{(A_2, 0)}(h, h_{n}, h_{n-1} \xi_{n-1})-c_1 a_3'.$$ By Lemma 3.8, $a_3' = 1$. Thus $a_1 = (a_1'+a_2')$ and the first relation $f_\omega^1$ in (6.1) is $$h^{n+1}= ((a_1'+a_2')h + \xi) \cdot e^{-tn} \eqno (6.3)$$ Similarly, from the proof of Corollary 4.7 (ii), we see that the only homology class $A \in H_2'$ which has nonzero contributions to the quantum corrections in (4.5) and (4.6) is also $A = A_2$. By (4.5), $\xi \cdot \xi_p = \xi_{p+1}$ if $p < n -1$, and $\xi \cdot \xi_{n-1} = \xi_{n} + e^{-tn} + b_2^{(n)} \cdot e^{-tn}$ where $b_2^{(n)} = \Phi_{(A_2, 0)}(\xi, \xi_{n-1}, h_n \xi_{n-1})$. Thus, $$\xi^p = \cases \xi_{p}, &\text{if $p < n$}\\ \xi_{n} + (1 + b_2^{(n)}) \cdot e^{-tn}, &\text{if $p = n$}\\ \endcases \eqno (6.4)$$ By (6.2), we have $h \cdot h_p = h_{p+1}$ if $p < n -1$, and $h \cdot h_{n-1} = h_{n} + b_2^{(0)} \cdot e^{-tn}$ where $b_2^{(0)} = a_1' = \Phi_{(A_2, 0)}(h_{n-1}, h, h_n \xi_{n-1})$. Thus, we obtain $$h^p = \cases h_{p}, &\text{if $p < n$}\\ h_{n} + b_2^{(0)} \cdot e^{-tn}, &\text{if $p = n$}\\ \endcases \eqno (6.5)$$ By (4.6), for $1 \le i \le (n-1)$, $h_{n-i} \cdot \xi_i = h_{n-i}\xi_i + b_2^{(i)} \cdot e^{-tn}$ where $b_2^{(i)} = \Phi_{(A_2, 0)}(h_{n-i}, \xi_i, h_n \xi_{n-1})$. Thus by (6.4) and (6.5), we have $$h^{n-i} \cdot \xi^i = h_{n-i} \cdot \xi_i = h_{n-i}\xi_i + b_2^{(i)} \cdot e^{-tn}. \eqno (6.6)$$ Since $\sum_{i=0}^n (-1)^i c_i \cdot h_i \xi_{n-i} = 0$, it follows from (6.4), (6.5), (6.6) that $$\sum_{i=0}^n (-1)^i c_i \cdot h^i \cdot \xi^{n-i} = (1 + \sum_{i=0}^n (-1)^i c_i b_2^{(n-i)}) \cdot e^{-tn}. \eqno (6.7)$$ Next, we compute the above integers $a_1', a_2'$, and $b_2^{(i)}$ where $0 \le i \le n$. \lemma{6.8} Let $V = T_{\Pee^n}$ with $n \ge 2$ and $A_2 = (h_{n-1}\xi_{n-1} - nh_{n}\xi_{n-2})_*$. \roster \item"{(i)}" $\Phi_{(A_2, 0)}(h, h_n, h_{n-1} \xi_{n-1}) = n$; \item"{(ii)}" Let $\alpha = h_j\xi_k$ and $\beta = h_s\xi_t$ where $j, k, s, t$ are nonnegative integers such that {\rm max}$(j, k) > 0$, {\rm max}$(s, t) > 0$, and $(j+k+s+t) = n$. Then, $$\Phi_{(A_2, 0)}(\alpha, \beta, h_n \xi_{n-1}) = 1.$$ \endroster \endproclaim \noindent {\it Proof.} (i) By Lemma 2.2 (iv), $\frak M({A_2}, 0)/PSL(2; \Cee)$ is compact. By (3.17), we have $h^1(N_f) = 0$ for every $f \in \frak M({A_2}, 0)$. Thus, $\frak M({A_2}, 0)/PSL(2; \Cee)$ is also smooth. Fix a line $\ell_0$ in $\Pee^n$. Let $g: \ell_0 \to \Pee(T_{\Pee^n}|_{\ell_0}) \subset \Pee(T_{\Pee^n})$ be the embedding induced by the natural projection $T_{\Pee^n}|_{\ell_0} = \Cal O_{\ell_0}(1)^{\oplus (n - 1)} \oplus \Cal O_{\ell_0}(2) \to \Cal O_{\ell_0}(2) \to 0.$ Since $h([g(\ell_0)]) = 1$ and $\xi([g(\ell_0)]) = 2$, we have $[g(\ell_0)] = [h_{n-1}\xi_{n-1} - (n - 1) h_{n}\xi_{n-2}]_*$. So $h_{n-1}\xi_{n-1} = [g(\ell_0)]_* + (n - 1) h_{n}\xi_{n-2}$, and $$\Phi_{({A_2}, 0)}(h, h_{n}, h_{n-1} \xi_{n-1}) = \Phi_{({A_2}, 0)}(h, h_{n}, [g(\ell_0)]_*) + (n - 1) \Phi_{({A_2}, 0)}(h, h_{n}, h_{n}\xi_{n-2}).$$ By Lemma 3.8, it suffices to show that $\Phi_{(A_2, 0)}(h, h_{n}, [g(\ell_0)]_*) = 1$. Let $B$ and $C$ be the subvarieties of $\Pee(T_{\Pee^n})$ in general position, whose homology classes are Poincar\'e dual to $h$ and $h_{n}$ respectively. Then the homology classes of $\pi(B)$ and $\pi(C)$ in $\Pee^n$ are Poincar\'e dual to $h$ and $h_n$ respectively. Let $f: \ell \to \Pee(T_{\Pee^n})$ be a morphism in $\frak M(A_2, 0)$ induced by a surjective map $T_{\Pee^n}|_\ell \to \Cal O_{\ell}(1) \to 0$ for some line $\ell \subset \Pee^n$. If the image $\hbox{Im}(f)$ intersects with $B, C$, and $g(\ell_0)$, then $\ell$ intersects with $\pi(B)$, $\pi(C)$, and $\pi(g(\ell_0)) = \ell_0$. In other words, $\ell$ passes through the point $\pi(C)$ and intersects with $\ell_0$. Moreover, putting $p = \ell \cap \ell_0$ and noticing that every surjective map $T_{\Pee^n}|_\ell \to \Cal O_{\ell}(1) \to 0$ factors through the natural projection $T_{\Pee^n}|_\ell = \Cal O_{\ell}(1)^{(n-1)} \oplus \Cal O_{\ell}(2) \to \Cal O_{\ell}(1)^{(n-1)}$, we conclude that the $(n-1)$-dimensional subspace $(\Cal O_{\ell}(1)^{(n-1)})|_p$ in $(T_{\Pee^n}|_\ell)|_p = T_{p, \Pee^n}$ must contain the $1$-dimensional subspace $(\Cal O_{\ell_0}(2))|_p$ in $(T_{\Pee^n}|_{\ell_0})|_p = T_{p, \Pee^n}$. Conversely, let $p \in \ell_0$ and let $\ell_p$ be the unique line connecting the two points $\pi(C)$ and $p$. If the $(n-1)$-dimensional subspace $(\Cal O_{\ell_p}(1)^{(n-1)})|_p$ in $(T_{\Pee^n}|_{\ell_p})|_p = T_{p, \Pee^n}$ contains the $1$-dimensional subspace $(\Cal O_{\ell_0}(2))|_p$ in $(T_{\Pee^n}|_{\ell_0})|_p = T_{p, \Pee^n}$, then there exists a unique surjective map $T_{\Pee^n}|_{\ell_p} \to \Cal O_{\ell_p}(1) \to 0$ such that the image of the induced morphism $f: \ell_p \to \Pee(T_{\Pee^n})$ intersects $g(\ell_0)$ at the point $g(p)$. Since there exists a unique point $p \in \ell_0$ such that the $(n-1)$-dimensional subspace $(\Cal O_{\ell_p}(1)^{(n-1)})|_p$ in $(T_{\Pee^n}|_{\ell_p})|_p = T_{p, \Pee^n}$ contains the $1$-dimensional subspace $(\Cal O_{\ell_0}(2))|_p$ in $(T_{\Pee^n}|_{\ell_0})|_p = T_{p, \Pee^n}$, it follows that $$\Phi_{(A_2, 0)}(h, h_{n}, [g(\ell_0)]_*) = 1.$$ (ii) It is well-known (see p.176 of \cite{7}) that there is an exact sequence $$0 \to \Cal O_{\Pee^n} \to \Cal O_{\Pee^n}(1)^{\oplus (n+1)} \to T_{\Pee^n} \to 0. \eqno (6.9)$$ The surjective map $\Cal O_{\Pee^n}(1)^{\oplus (n+1)} \to T_{\Pee^n} \to 0$ induces the inclusion $\phi: \Pee(T_{\Pee^n}) \subset \Pee^n \times \Pee^n$ such that $\xi$ is the restriction of the $(1, 1)$ class in $\Pee^n \times \Pee^n$. Let $B, C, q_0$ be the subvarieties of $\Pee(T_{\Pee^n})$ in general position, whose homology classes are Poincar\'e dual to $\alpha, \beta, h_n \xi_{n-1}$ respectively. Then $q_0$ is a point. Put $p_0 = \pi(q_0) \in \Pee^n$. Now the morphisms in $\frak M(A_2, 0)$ are of the forms $f: \ell \to \Pee(T_{\Pee^n})$ induced by surjective maps $T_{\Pee^n}|_\ell \to \Cal O_{\ell}(1) \to 0$ for lines $\ell \subset \Pee^n$. If the image $\hbox{Im}(f)$ passes $q_0$, then the line $\ell$ passes $p_0$ and $q_0$ is contained in the hyperplane $$\Pee^{n-2} = \Pee((\Cal O_\ell(1)^{\oplus (n-1)})|_{p_0}) \subset \Pee((T_{\Pee^n}|_\ell)|_{p_0}) = \pi^{-1}(p_0) = \Pee^{n-1}.$$ Conversely, if $\ell$ passes $p_0$ and $q_0$ is contained in the hyperplane $$\Pee^{n-2} = \Pee((\Cal O_\ell(1)^{\oplus (n-1)})|_{p_0}) \subset \Pee((T_{\Pee^n}|_\ell)|_{p_0}) = \pi^{-1}(p_0) = \Pee^{n-1}, \eqno (6.10)$$ then there exists a unique $f \in \frak M(A_2, 0)$ of the form $f: \ell \to \Pee(T_{\Pee^n})$ such that $\hbox{Im}(f)$ passes $q_0$; moreover, putting $q_0 = (p_0, p_0') \in \Pee^n \times \Pee^n$ such that $\pi$ is the first projection of $\Pee^n \times \Pee^n$, then $\hbox{Im}(f) = \ell \times \{ p_0' \} \subset \Pee^n \times \Pee^n$. The set of all lines $\ell$ passing $p_0$ such that $q_0$ is contained in the hyperplane (6.10) is parameterized by an $(n - 2)$-dimensional linear subspace $\Pee^{n - 2}$ in $\Pee^n$ (the first factor in $\Pee^n \times \Pee^n$). It follows that the images $\hbox{Im}(f) \subset \Pee(T_{\Pee^n})$ sweep a hyperplane $$H \overset \hbox{def} \to = \Pee^{n - 1} \times \{ p_0' \} \subset \Pee^n \times \{ p_0' \}. \eqno (6.11)$$ Since $\xi$ is the restriction of the $(1, 1)$ class in $\Pee^n \times \Pee^n$, $\xi|_H$ is the hyperplane class $\tilde h$ in $H = \Pee^{n - 1} \times \{ p_0' \} \cong \Pee^{n - 1}$. Thus $\alpha|_H = \tilde h_{j + k}$ and $\beta|_H = \tilde h_{s+t}$. Since $(j+k+s+t) = n$ and $B$ and $C$ are in general position, there is a unique line in $H$ passing $q_0 = (p_0, p_0')$ and intersecting with $B$ and $C$. Therefore, $$\Phi_{(A_2, 0)}(\alpha, \beta, h_n \xi_{n-1}) = 1. \qed$$ Finally, we summarize the above computations and prove the following. \proposition{6.12} The quantum cohomology ring $H^*_{\omega}(\Pee(T_{\Pee^n}); \Zee)$ with $n \ge 2$ is the ring generated by $h$ and $\xi$ with the two relations: $$h^{n+1} = \xi \cdot e^{-tn} \qquad \text{and} \qquad \sum_{i=0}^n (-1)^i c_i \cdot h^i \cdot \xi^{n-i} = (1 + (-1)^n) \cdot e^{-tn}.$$ \endproclaim \proof By Lemma 6.8 (ii), $a_1' = 1$. By Lemma 3.8, $a_3' = 1$. By Lemma 6.8 (i), $$a_2' = \Phi_{(A_2, 0)}(h, h_{n}, h_{n-1} \xi_{n-1}) -c_1 a_3'= -1.$$ Thus by (6.3), the first relation $f_\omega^1$ is $h^{n+1}= \xi \cdot e^{-tn}$. By Lemma 6.8 (ii), $b_2^{(i)} = 1$ for $0 \le i \le n$. By (6.7), the second relation $f_\omega^2$ is $\sum_{i=0}^n (-1)^i c_i \cdot h^i \cdot \xi^{n-i} = (1 + \sum_{i=0}^n (-1)^i c_i) \cdot e^{-tn}$. From the exact sequence (6.9), $c_i = {n+1 \choose i}$ for $0 \le i \le n$. Therefore, the relation $f_\omega^2$ is $\sum_{i=0}^n (-1)^i c_i \cdot h^i \cdot \xi^{n-i} = (1 + (-1)^n) \cdot e^{-tn}$. \endproof \Refs \ref \no {1} \by A. Astashkevich, V. Sadov \paper Quantum cohomology of partial flag manifolds $F_{n_1 \ldots n_k}$ \jour Preprint \endref \ref \no {2} \by V.V. Batyrev \paper Quantum cohomology rings of toric manifolds \jour Preprint \endref \ref \no {3} \by A. Beauville \paper Quantum cohomology of complete intersections \jour Preprint \endref \ref \no {4} \by I. Ciocan-Fontanine \paper Quantum cohomology of flag varieties \jour Preprint \endref \ref \no {5} \by H. Clemens, J. Kolla\'r, S. Mori \book Higher dimensional complex geometry. {\rm (Asterisque, Vol. 166) Paris: Soc. Math. Fr.} \yr 1988 \endref \ref \no {6} \by B. Crauder, R. Miranda \paper Quantum cohomology of rational surfaces \inbook The moduli space of curves \eds R. Dijkgraaf, C. Faber, G. van der Geer \bookinfo Progress in Mathematics {\bf 129} \publ Birkh\" auser \publaddr Boster Basel Berlin \yr 1995 \endref \ref \no {7} \by R. Hartshorne \book Algebraic Geometry \publ Springer \publaddr Berlin-Heidelberg-New York\yr 1978 \endref \ref \no {8} \by M. Kontsevich, Y. Manin \paper Gromov-Witten classes, quantum cohomology, and enumerative geometry \jour Preprint \endref \ref \no {9} \by J. Li, G. Tian \paper Quantum cohomology of homogeneous varieties \jour Preprint \endref \ref \no {10} \by C. Okonek, M. Schneider, H. Spindler \book Vector bundles on complex projective spaces, {\rm Progress in Math.} \publ Birkh{\" a}user \yr 1980 \endref \ref \no {11} \by Y. Ruan \paper Symplectic topology and extremal rays \jour Geom. Func. Anal. \vol 3 \pages 395-430 \yr 1993 \endref \ref \no {12} \bysame \paper Topological sigma model and Donaldson type invariants in Gromov theory \jour To appear in Duke Math. J. \endref \ref \no {13} \by Y. Ruan, G. Tian \paper A mathematical theory of quantum cohomology \jour To appear in J. Diffeo. Geom. \yr 1995 \endref \ref \no {14} \by B. Siebert, G. Tian \paper On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator \jour Preprint \endref \ref \no {15} \bysame \paper Quantum cohomology of moduli space of stable bundles \jour In preparation \endref \ref \no{16} \by E. Witten \paper Topological sigma models \jour Commun. Math. Phys. \vol 118 \pages 411-449 \yr 1988 \endref \endRefs \enddocument

9510

alg-geom/9510014

en

https://arxiv.org/abs/alg-geom/9510014

alg-geom/9510014

V. Batyrev

Victor V. Batyrev and Yuri Tschinkel

Manin's conjecture for toric varieties

45 pages, LaTeX

We prove an asymptotic formula conjectured by Manin for the number of $K$-rational points of bounded height with respect to the anticanonical line bundle for arbitrary smooth projective toric varieties over a number field $K$.

alg-geom

\section{Algebraic tori and toric varieties} Let $X_K$ be an algebraic variety defined over a number field $K$ and $E/K$ a finite extension of number fields. We will denote the set of $E$-rational points of $X_K$ by $X(E)$ and by $X_E$ the $E$-variety obtained from $X_K$ by base change. We sometimes omit the subscript in $X_E$ if the respective field of definition is clear from the context. Let ${\bf G}_{m,E}= {\rm Spec}( E[x,x^{-1}])$ be the multiplicative group scheme over $E$. \begin{dfn} {\rm A linear algebraic group $T_K$ is called a {\em $d$-dimen\-sio\-nal algebraic torus} if there exists a finite extension $E/K$ such that $T_{E}$ is isomorphic to $({\bf G}_{m,E})^d$. The field $E$ is called the {\em splitting field } of $T$. For any field $E$ we denote by $\hat{T}_E = {\rm Hom}\,( T, E^*)$ the group of regular $E$-rational characters of $T$. } \label{opr.tori} \end{dfn} \begin{theo} {\rm \cite{grothendieck,ono1,vosk}} There is a contravariant equivalence between the category of algebraic tori defined over a number field $K$ and the category of torsion free ${\rm Gal}(E/K)$-modules of finite rank over ${\bf Z}$. The functors are given by $$ M \rightarrow T = {\rm Spec}(K\lbrack M \rbrack); \;\; T \rightarrow \hat{T}_E. $$ The above contravariant equivalence is functorial under field extensions of $K$. \label{represent} \end{theo} Let ${\rm Val}(K)$ be the set of all valuations of a global field $K$. Denote by $S_{\infty}$ the set of archimedian valuations of $K$. For any $v \in {\rm Val}(K)$, we denote by $K_v$ the completion of $K$ with respect to $v$. Let $E$ be a finite Galois extension of $K$. Let ${\cal V}$ be an extension of $v$ to $E$, $E_{\cal V}$ the completion of $E$ with respect to ${\cal V}$. Then \[ {\rm Gal}(E_{\cal V}/ K_v ) \cong G_v \subset G, \] where $G_{v}$ is the decomposition subgroup of $G$ and $ K_v \otimes_K E \cong \prod_{{\cal V} \mid v} E_{\cal V}. $ Let $T$ be an algebraic torus over $K$ with the splitting field $E$. Denote by $T(K_v)=$ the $v$-adic completion of $T(K)$ and by $T({\cal O}_v)\subset T(K_v)$ its maximal compact subgroup. \begin{dfn} {\rm Denote by $T({\bf A}_K)$ the adele group of $T$. Define \[T^1({\bf A}_K) = \{ {\bf t} \in T({\bf A}_K) \, : \, \prod_{v \in {\rm Val}(K)} \mid m(t_v) \mid_v = 1, \; {\rm for \; all}\; m \in \hat{T}_K \subset M \}. \] Let \[ {\bf K}_T = \prod_{v \in {\rm Val}(K)} T({\cal O}_v), \] be the maximal compact subgroup of $T({\bf A}_K)$. } \end{dfn} \begin{prop} {\rm \cite{ono1}} The groups $T({\bf A}_K)$, $T^1({\bf A}_K)$, $T(K)$, ${\bf K}_T$ have the following properties: {\rm (i)} $T({\bf A}_K)/T^1({\bf A}_K) \cong {\bf R}^t$, where $t$ is the rank of $\hat{T}_K$; {\rm (ii)} $T^1({\bf A}_K)/T(K)$ is compact; {\rm (iii)} $T^1({\bf A}_K)/ T(K)\cdot {\bf K}_T $ is isomorphic to the direct product of a finite group ${\bf cl}(T_K)$ and a connected compact abelian topological group which dimension equals the rank $r'$ of the group of ${\cal O}_K$-units in $T(K)$; {\rm (iv)} $W(T) = {\bf K}_T \cap T(K)$ is a finite group of all torsion elements in $T(K)$. \label{subgroups} \end{prop} \begin{dfn} {\rm We define the following cohomological invariants of the algebraic torus $T$: $$ h(T)={\rm Card}[H^1(G,M)], $$ $$ {\rm III}(T)={\rm Ker}\, \lbrack H^1(G, T(K)) \rightarrow \prod_{v\in {\rm Val}(K)} H^1(G_v, T(K_v)) \rbrack, $$ $$ i(T)= {\rm Card}[{\rm III}(T)]. $$ \label{coh.inv} } \end{dfn} \begin{dfn} {\rm Let $\overline{T(K)}$ be the closure of $T(K)$ in $T({\bf A}_K)$ in the {\em direct product topology}. Define the {\em obstruction group to weak approximation} as $$ A(T)= T({\bf A}_K)/\overline{T(K)}. $$ \label{weak0} } \end{dfn} \begin{rem} {\rm It is known that over the splitting field $E$ one has $A(T_E)=0$. } \end{rem} Let us recall standard facts about toric varieties over arbitrary fields \cite{danilov,demasur,fulton,oda,BaTschi}. \begin{dfn} {\rm A finite set $\Sigma$ consisting of convex rational polyhedral cones in $N_{\bf R} = N \otimes {\bf R}$ is called a {\em $d$-dimensional fan} if the following conditions are satisfied: (i) every cone $\sigma \in \Sigma$ contains $0 \in N_{\bf R}$; (ii) every face $\sigma'$ of a cone $\sigma \in \Sigma$ belongs to $\Sigma$; (iii) the intersection of any two cones in $\Sigma$ is a face of both cones. } \end{dfn} \begin{dfn} {\rm A $d$-dimensional fan $ \Sigma $ is called {\em complete and regular} if the following additional conditions are satisfied: (i) $N_{\bf R}$ is the union of cones from $\Sigma$; (ii) every cone $\sigma \in \Sigma$ is generated by a part of a ${\bf Z}$-basis of $N$.\\ We denote by $\Sigma(j)$ the set of all $j$-dimensional cones in $\Sigma$. For each cone $\sigma \in \Sigma$ we denote by $N_{{\sigma}, \bf R}$ the minimal linear subspace containing $\sigma$. } \label{def.fan} \end{dfn} \noindent Let $T_K$ be a $d$-dimensional algebraic torus over $K$ with splitting field $E$ and $G = {\rm Gal}\, (E/K)$. Denote by $M$ the lattice $\hat{T}_E$ and by $N ={\rm Hom}\, (M, {\bf Z})$ the dual abelian group. \begin{theo} A complete regular $d$-dimensional fan $ \Sigma $ defines a smooth equivariant compactification ${\bf P}_{ \Sigma ,E}$ of the $E$-split algebraic torus $T_E$. The {\em toric variety} ${\bf P}_{ \Sigma ,E}$ has the following properties: (i) There is a $T_E$-invariant open covering by affine subsets $U_{\sigma,E}$: \[ {\bf P}_{\Sigma,E} = \bigcup_{ \sigma \in \Sigma} U_{\sigma,E}. \] The affine subsets are defined as $U_{\sigma,E} = {\rm Spec}(E \lbrack M \cap \check{\sigma} \rbrack$), where $\check{\sigma}$ is the cone in $M_{\bf R}$ which is dual to $\sigma$. (ii) There is a representation of ${\bf P}_{\Sigma,E}$ as a disjoint union of split algebraic tori $T_{\sigma,E}$ of dimension ${\rm dim}\, T_{\sigma,E} = d - {\rm dim}\, \sigma $: \[ {\bf P}_{\Sigma,E} = \bigcup_{ \sigma \in \Sigma } T_{\sigma,E}. \] For each $j$-dimensional cone $\sigma \in \Sigma{(j)}$ we denote by $T_{\sigma,E}$ the kernel of a homomorphism $T_E \rightarrow ({\bf G}_{m,E})^j$ defined by a ${\bf Z}$-basis of the sublattice $N \cap N_{{\sigma},{\bf R}} \subset N$. \end{theo} \noindent To construct compactifications of non-split tori $T_K$ over $K$, we need a complete fan $\Sigma$ of cones having an additional combinatorial structure: an {\em action of the Galois group } $G={\rm Gal}(E/K)$ \cite{vosk1}. The lattice $M=\hat{T}_E$ is a $G$-module and we have a representation $\rho: G \rightarrow {\rm Aut}(M)$. Denote by $\rho^*$ the induced dual representation of $G$ in ${\rm Aut}(N) \cong {\rm GL}(d,{\bf Z})$. \begin{dfn} {\rm A complete fan $\Sigma \subset N_{\bf R}$ is called {\em $G$-invariant} if for any $g \in G$ and for any $\sigma \in \Sigma$, one has $\rho^*(g) (\sigma) \in \Sigma$. Let $N^G$ (resp. $M^G$, $N_{\bf R}^G$, $M_{\bf R}^G$ and $ \Sigma ^G$) be the subset of $G$-invariant elements in $N$ (resp. in $M$, $N\otimes {\bf R}$, $M\otimes {\bf R}$ and $ \Sigma $). Denote by $\Sigma_G \subset N_{\bf R}^G$ the fan consisting of all possible intersections $\sigma \cap N_{\bf R}^G$ where $\sigma$ runs over all cones in $\Sigma$. } \label{opr.invar} \end{dfn} \noindent The following theorem is due to Voskresenski\^i \cite{vosk1}: \begin{theo} Let $\Sigma$ be a complete regular $G$-invariant fan in $N_{\bf R}$. Assume that the complete toric variety ${\bf P}_{\Sigma,E}$ defined over the splitting field $E$ by the $G$-invariant fan $\Sigma$ is projective. Then there exists a unique complete algebraic variety ${\bf P}_{\Sigma,K}$ over $K$ such that its base extension ${\bf P}_{\Sigma,K} \otimes_{{\rm Spec} (K)} {\rm Spec}(E)$ is isomorphic to the toric variety ${\bf P}_{\Sigma,E}$. The above isomorphism respects the natural $G$-actions on ${\bf P}_{\Sigma,K} \otimes_{{\rm Spec}(K)} {\rm Spec}(E)$ and ${\bf P}_{\Sigma,E}$. \end{theo} \begin{rem} {\rm Our definition of heights and the proof of the analytic properties of height zeta functions do not use the projectivity of respective toric varieties. We note that there exist non-projective compactifications of split algebraic tori. We omit the technical question of existence of non-projective compactifications of non-split tori. } \end{rem} We proceed to describe the algebraic geometric structure of the variety ${\bf P}_{ \Sigma ,K}$ in terms of the fan with Galois-action. Let ${\rm Pic}({\bf P}_{ \Sigma ,K}) $ be the Picard group and $\Lambda_{\rm eff}$ the cone in ${\rm Pic}({\bf P}_{ \Sigma ,K})$ generated by classes of effective divisors. Let ${\cal K}$ be the canonical line bundle of ${\bf P}_{ \Sigma ,K}$. \begin{dfn} {\rm A continuous function $\varphi\; : \; N_{\bf R} \rightarrow {\bf R}$ is called {\em $\Sigma$-piecewise linear} if the restriction of $\varphi$ to every cone $\sigma \in \Sigma$ is a linear function. It is called {\em integral} if $\varphi(N) \subset {\bf Z}$. Denote the group of $ \Sigma $-piecewise linear integral functions by $PL( \Sigma )$. } \end{dfn} We see that the $G$-action on $M$ (and $N$) induces a $G$-action on the free abelian group $PL( \Sigma )$. Denote by $e_1, \ldots, e_n$ the primitive integral generators of all $1$-dimensional cones in $\Sigma$. A function $\varphi\in PL( \Sigma )$ is determined by its values on $e_i,\, (i=1,...,n)$. Let $T_{i}$ be the $(d-1)$-dimensional torus orbit corresponding to the cone ${\bf R}_{\geq 0}e_i \in \Sigma(1)$ and $\overline{T}_i$ the Zariski closure of $T_i$ in ${\bf P}_{\Sigma,E}$. \begin{prop} Let ${\bf P}_{ \Sigma ,K}$ be a smooth toric variety over $K$ which is an equivariant compactification of an algebraic torus $T_K$ with splitting field $E$ and $ \Sigma $ the corresponding complete regular fan with $G={\rm Gal}(E/K)$-action. Then: (i) There is an exact sequence $$ 0 \rightarrow M^G \rightarrow PL( \Sigma )^G \rightarrow {\rm Pic}({\bf P}_{ \Sigma ,K}) \rightarrow H^1(G,M) \rightarrow 0. $$ (ii) Let $$ \Sigma (1)= \Sigma _1(1)\cup ...\cup \Sigma _r(1) $$ be the decomposition of $ \Sigma (1)$ into a union of $G$-orbits. The cone of effective divisors $ \Lambda _{\rm eff}$ is generated by classes of $G$-invariant divisors $$ D_j = \sum_{{\bf R}_{\geq 0}e_i \in \Sigma _j(1)} \overline{T}_i \,\,\, (j=1,...,r). $$ (iii) The class of the anticanonical line bundle ${\cal K}^{-1}\in {\rm Pic}({\bf P}_{ \Sigma ,K})$ is the class of the $G$-invariant piecewise linear function $\varphi_{ \Sigma }\in PL( \Sigma )^G$ given by $\varphi_{ \Sigma }(e_j)=1$ for all $j=1,...,n$. \label{nonsplit.geom} \end{prop} \begin{theo} {\rm \cite{vosk,ct}} Let $T$ be an algebraic torus over $K$ with splitting field $E$. Let ${\bf P}_{ \Sigma ,K}$ be a complete smooth equivariant compactification of $T$. There is an exact sequence: \[ 0 \rightarrow A(T) \rightarrow Hom (H^1(G,{\rm Pic}({\bf P}_{ \Sigma ,E})),{\bf Q}/{\bf Z}) \rightarrow {\rm III}(T) \rightarrow 0. \] \label{weak} \end{theo} \begin{rem} {\rm The group $H^1(G,{\rm Pic}({\bf P}_{ \Sigma ,E}))$ is canonically isomorphic to the non-trivial part of the Brauer group ${\rm Br}({\bf P}_{\Sigma,K})/{\rm Br}(K)$, where ${\rm Br}({\bf P}_{\Sigma,K}) = H^2_{\rm et}({\bf P}_{\Sigma,K}, {\bf G}_m)$. This group appears as the obstruction group to the Hasse principle and weak approximation in \cite{manin,ct}. } \end{rem} \begin{coro} {\rm Let $\beta({\bf P}_{\Sigma})$ be the cardinality of $H^1(G,{\rm Pic}({\bf P}_{ \Sigma ,E}))$. Then \[ {\rm Card} \lbrack A(T) \rbrack = \frac{\beta({\bf P}_{\Sigma})}{i(T)}. \] \label{weak1} } \end{coro} \section{Tamagawa numbers} In this section we recall the definitions of Tamagawa numbers of tori following A. Weil \cite{weil} and of algebraic varieties with a metrized canonical line bundle following E. Peyre \cite{peyre}. The constructions of Tamagawa numbers depend on a choice of a finite set of valuations $S\subset {\rm Val}(K)$ containing archimedian valuations and places of bad reduction, but the Tamagawa numbers themselves do not depend on $S$. Let $X$ be a smooth algebraic variety over $K$, $X(K_v)$ the set of $K_v$-rational points of $X$. Then a choice of local analytic coordinates $x_1, \ldots, x_d$ on $X(K_v)$ defines a homeomorphism $\phi\,: \, U \rightarrow K_v^d$ in $v$-adic topology between an open subset $U \subset X(K_v)$ and $\phi(U) \subset K_v^d$. Let $dx_1 \cdots dx_d$ be the Haar measure on $K_v^d$ normalized by the condition \[ \int_{{\cal O}_v^d} dx_1 \cdots dx_d = \frac{1}{(\sqrt{\delta_v})^d} \] where $\delta_v$ is the absolute different of $K_v$. Denote by $dx_1 \wedge \cdots \wedge dx_d$ the standard differential form on $K_v^{d}$. Then $f = \phi^*(dx_1 \wedge \cdots \wedge dx_d)$ is a local analytic section of the canonical sheaf ${\cal K}$. If $\| \cdot \|$ is a $v$-adic metric on ${\cal K}$, then we obtain the $v$-adic measure on $U$ by the formula \[ \int_{U'} \omega_{{\cal K},v} = \int_{\phi(U')} \| f(\phi^{-1}(x)) \|_v dx_1 \cdots dx_d, \] where $U'$ is arbitrary open subset in $U$. The measure $\omega_{{\cal K},v}$ does not depend on the choice of local coordinates and extends to a global measure on $X(K_v)$ \cite{peyre}. \begin{dfn} {\rm \cite{ono1} Let $T$ be an algebraic torus defined over a number field $K$ with splitting field $E$. Denote by \[ L_S(s, T;E/K) = \prod_{v \in {\rm Val}(K)} L_v(s, T ;E/K) \] the Artin $L$-function corresponding to the representation \[ \rho \; :\; G= {\rm Gal}(E/K) \rightarrow {\rm Aut}(\hat{T}_E) \] and a finite set $S \subset {\rm Val}(K)$ containing all archimedian valuations and all non-archimedian valuations of $K$ which are ramified in $E$. By definition, $L_v(s,T;E/K) \equiv 1$ if $v \in S$, $L_v(s,T;E/K)= {\rm det}(Id - q^{-s}_v F_v)^{-1}$ if $v \not\in S$, where $F_v \in {\rm Aut}(\hat{T}_E)$ is a representative of the Frobenius automorphism. } \end{dfn} Let $T$ be an algebraic torus of dimension $d$ and $\Omega$ a $T$-invariant algebraic $K$-rational differential $d$-form. The form $\Omega$ defines an isomorphism of the canonical sheaf on $T$ with the structure sheaf on $T$. Since the structure sheaf has a canonical metrization, using the above construction, we obtain a $v$-adic measure $\omega_{\Omega,v}$ on $T(K_v)$. Moreover, according to A. Weil \cite{weil}, we have \[ \int_{T({\cal O}_v)} \omega_{\Omega,v} = \frac{{\rm Card} \lbrack T(k_v) \rbrack}{q^d_v} = L_v(1, T; E/K)^{-1} \] for all $v\not\in S$. We put $d\mu_v = L_v(1, T; E/K) \omega_{\Omega,v}$ for all $v\in {\rm Val}(K)$. Then the local measures $d\mu_v$ satisfy $$ \int_{T({\cal O}_v)} d\mu_v = 1 $$ for all $v\not\in S$. \begin{dfn} {\rm We define the {\em canonical measure} on the adele group $T({\bf A}_K)$ $$ \omega_{\Omega,S} = \prod_{v \in {\rm Val}(K)} L_v(1, T; E/K) \omega_{\Omega,v} = \prod_{v \in {\rm Val}(K)} d\mu_v. $$ \label{can.meas} } \end{dfn} \noindent By the product formula, $\omega_{\Omega,S} $ does not depend on the choice of $\Omega$. Let ${\bf dx}$ be the standard Lebesgue measure on $T({\bf A}_K)/T^1({\bf A}_K)$. There exists a unique Haar measure $\omega^1_{\Omega,S}$ on $T^1({\bf A}_K)$ such that $\omega^1_{\Omega,S} {\bf dx} = \omega_{\Omega,S}$. \bigskip We proceed to define {\em Tamagawa measures} on algebraic varieties following E. Peyre \cite{peyre}. Let $X$ be a smooth projective algebraic variety over $K$ with a metrized canonical sheaf ${\cal K}$. We assume that $X$ satisfies the conditions $h^1(X, {\cal O}_X) = h^2(X, {\cal O}_X) = 0$. Under these assumptions, the N{\'e}ron-Severi group $NS(X)$ (or, equivalently, the Picard group ${\rm Pic}(X)$ modulo torsion) over the algebraic closure $\overline{K}$ is a discrete continuous ${\rm Gal}(\overline{K}/K)$-module of finite rank over ${\bf Z}$. Denote by $T_{NS}$ the corresponding torus under the duality from \ref{represent} and by $E_{NS}$ a splitting field. \begin{dfn} {\rm \cite{peyre} The {\em adelic Tamagawa measure} $\omega_{{\cal K},S}$ on $X({\bf A}_K)$ is defined by $$ \omega_{{\cal K},S} = \prod_{v \in {\rm Val}(K)} L_v(1, T_{NS}; E_{NS}/K)^{-1}\omega_{{\cal K},v}. $$ } \end{dfn} \begin{dfn} {\rm Let $t$ be the rank of the group of $K$-rational characters $\hat{T}_K$ of $T$. Then the {\em Tamagawa number of } $T$ is defined as \[ \tau(T) = \frac{b_S(T)}{l_S(T)} \] where \[ b_S(T) = \int_{T^1({\bf A}_K)/T(K)} \omega^1_{\Omega,S} , \] \[ l_S(T) = \lim_{s \rightarrow 1} (s-1)^t L_S(s, T; E/K). \]} \label{tamagawa1} \end{dfn} \begin{dfn} {\rm \cite{peyre} Let $k$ be the rank of the N{\'e}ron-Severi group of $X$ over $K$, and $\overline{X(K)}$ the closure of $X(K) \subset X({\bf A}_K)$ in the direct product topology. Then the {\em Tamagawa number} of $X$ is defined by \[ \tau_{\cal K}(X) = \frac{b_S(X)}{l_S(X)} \] where \[ b_S(X) = \int_{\overline{X(K)}} \omega_{{\cal K},S} \] whenever the adelic integral converges, and \[ l_S^{-1}(X) = \lim_{s \rightarrow 1} (s-1)^k L_S(s, T_{NS}; E_{NS}/K). \] } \label{tamagawa2} \end{dfn} \begin{rem} {\rm Notice that there is a difference in the choice of convergence factors for the Tamagawa measure on an algebraic variety $X$ and for the Tamagawa measure on an algebraic torus $T$. In the first case, we choose $L_v^{-1}(1, T_{NS};E_{NS}/K)$ whereas in the second case one uses $L_v(1, T; E/K)$. This explains the difference in the definitions of $l_S(X)$ and $l_S(T)$. } \end{rem} \begin{rem} {\rm For a toric variety ${\bf P}_{ \Sigma }\supset T$ one can take $E_{NS}=E$, where $E$ is a splitting field of $T$. \label{ENS} } \end{rem} \begin{rem} {\rm It is clear that in both definitions the Tamagawa numbers do not depend on the choice of the finite subset $S\subset {\rm Val}(K)$. E. Peyre ($\cite{peyre}$) proves the existence of the Tamagawa number for Fano varieties by using the Weil conjectures. The same method shows the existence of the Tamagawa number for smooth complete varieties $X$ satisfying the conditions $h^1(X, {\cal O}_X) = h^2(X, {\cal O}_X) = 0$. } \end{rem} \begin{theo} {\rm \cite{ono2}} Let $T$ be an algebraic torus defined over $K$. The Tamagawa number $\tau (T)$ doesn't depend on the choice of a splitting field $E/K$. We have $$ \tau (T)=h(T)/i(T). $$ The constants $h(T),i(T)$ were defined in \ref{coh.inv}. \label{tamagawa} \end{theo} We see that the Tamagawa number of an algebraic torus is a rational number. We have $\tau({\bf G}_m(K)) =1$. The Tamagawa number of a Fano variety with a metrized canonical line bundle is certainly not rational in general. For ${\bf P}^1_{{\bf Q}}$ with our metrization we have $\tau_{\cal K}({\bf P}^1_{{\bf Q}}) =1/\zeta_{{\bf Q}}(2)$. \begin{prop} {\rm \cite{BaTschi}} One has $$ \int_{\overline{T(K)}} \omega_{{\cal K},S} = \int_{\overline{{\bf P}_{ \Sigma }(K)}} \omega_{{\cal K},S}. $$ \label{two-integrals} \end{prop} \section{Heights and their Fourier transforms} Let $\varphi \in PL(\Sigma)^G_{\bf C}$. Using the decomposition of $ \Sigma (1)$ into a union of $G$-orbits $$ \Sigma (1)= \Sigma _1(1)\cup ...\cup \Sigma _r(1), $$ we can identify $\varphi$ with a $T$-invariant divisor with complex coefficients $$ D_{\varphi} = s_1 D_1 + \cdots + s_r D_r $$ where $s_j = \varphi(e_j) \in {{\bf C}}$ and $e_j$ is a primitive lattice generator of some cone $ \sigma \in \Sigma _j(1)$ $(j =1, \ldots, r)$. It will be convenient to identify an element $\varphi =\varphi_{\bf s}\in PL(\Sigma)^G_{\bf C}$ with the vector ${\bf s} = (s_1, \ldots, s_r)$ of its complex coordinates. Let us recall the definition of heights on toric varieties from \cite{BaTschi}. For our purposes it will be sufficient to describe the restrictions of heights to the Zariski open subset $T\subset {\bf P}_{ \Sigma ,K}$. \begin{prop} Let $v\in {\rm Val} (K) $ be a valuation and $G_v\subset G$ the decomposition group of $v$. There is an injective homomorphism $$ \pi_v: T(K_v)/T({\cal O}_v)\hookrightarrow N_v, $$ which is an isomorphism for all but finitely many $v\in {\rm Val}(K)$. Here $N_v=N^{G_v}\subset N$ for non-archimedian $v$ and $N_v=N_{{\bf R}}^{G_v}$ for archimedian valuations $v$. For every non-archimedian valuation we can identify the image of $\pi_v$ with a sublattice of finite index in $N_v$. \label{pi-image} \end{prop} \begin{dfn} {\rm Let ${\bf s } \in {{\bf C}}^r$ be a complex vector defining a complex piecewise linear $G$-invariant function $\varphi \in PL(\Sigma)^G_{\bf C}$. For any point $x_v \in T(K_v) \subset {\bf P}_{\Sigma}(K_v)$, denote by $\overline{x}_v$ the image of $x_v$ in $N_v$, where $N_v$ is considered as a canonical lattice in the real space $N_{\bf R}^{G_v}$ for non-archimedian valuations (resp. as the real Lie-algebra $N_{{\bf R},v}$ of $T(K_v)$ for archimedian valuations). Define the {\em complexified local Weil function} $H_{ \Sigma ,v}(x_v, {\bf s})$ by the formula \[H_{ \Sigma ,v}(x_v, {\bf s}) = e^{\varphi(\overline{x}_v)\log q_v }\] where $q_v$ is the cardinality of the residue field $k_v$ of $K_v$ if $v$ is non-archimedian and $\log q_v = 1$ if $v$ is archimedian. } \end{dfn} \begin{theo} {\rm \cite{BaTschi}} The complexified local Weil function $H_{ \Sigma ,v}(x_v, {\bf s} )$ satisfies the following properties: {\rm (i)} $H_{ \Sigma ,v}(x_v,{\bf s})$ is $T({\cal O}_v)$-invariant. {\rm (ii)} If ${\bf s} = 0$, then $H_{ \Sigma ,v}(x_v,{\bf s}) = 1$ for all $x_v \in T(K_v)$. {\rm (iii)} $H_{ \Sigma ,v}(x_v, {\bf s} + {\bf s}') = H_{ \Sigma ,v}(x_v,{\bf s}) H_{ \Sigma ,v}(x_v,{\bf s}')$. {\rm (iv)} If ${\bf s}=(s_1,...,s_r)\in {\bf Z}^r$, then $H_{ \Sigma ,v}(x_v, {\bf s})$ is a classical local Weil function corresponding to a Cartier divisor $D_{\bf s} = s_1 D_1 + \cdots + s_r D_r$ on ${\bf P}_{ \Sigma ,K}$. \label{local.f} \end{theo} \begin{dfn} {\rm For a piecewise linear function $\varphi_{\bf s} \in PL(\Sigma)^G_{\bf C}$ we define the {\em complexified height function on $T(K)\subset {\bf P}_{ \Sigma ,K}(K)$} by \[ H_{\Sigma}(x, {\bf s}) = \prod_{v \in {\rm Val}(K)} H_{ \Sigma ,v}(x_v, {\bf s}). \]} \end{dfn} \begin{rem} {\rm Although the local heights are defined only as functions on $PL( \Sigma )_{{\bf C}}^G \cong {{\bf C}}^r$, the product formula implies that for $x\in T(K)$ the global complexified height function descends to the Picard group ${\rm Pic}({\bf P}_{ \Sigma ,K})_{{\bf C}}$. Moreover, since $H_{\Sigma}(x, {\bf s})$ is the product of local complex Weil functions $H_{ \Sigma ,v}(x, {\bf s})$ and since for all $x_v \in T({\cal O}_v)$ we have $H_{ \Sigma ,v}(x_v, {\bf s}) = 1$ for all $v$, we can immediately extend $H_{\Sigma}(x,{\bf s})$ to a function on $T({\bf A}_K)\times PL( \Sigma )^G_{{\bf C}}$. } \end{rem} \begin{dfn} {\rm Let $ \Sigma (1) = \Sigma _1(1) \cup \cdots \cup \Sigma _l(1) $ be the decomposition of $ \Sigma (1)$ into a disjoint union of $G_v$-orbits. Denote by $d_j$ the length of the $G_v$-orbit $ \Sigma _j(1)$ $(d_1 + \cdots + d_l = n)$. We establish a 1-to-1 correspondence $ \Sigma _j(1) \leftrightarrow u_j$ between the $G_v$-orbits $ \Sigma _1(1), \ldots, \Sigma _l(1)$ and independent variables $u_1, \ldots, u_l$. Let $\sigma \in \Sigma ^{G_v}$ be any $G_v$-invariant cone and $ \Sigma _{j_1}(1) \cup \cdots \cup \Sigma _{j_k}(1)$ the set of all $1$-dimensional faces of $\sigma$. We define the rational function $R_{\sigma}(u_1, \ldots, u_l)$ corresponding to $\sigma$ as follows: \[ R_{\sigma}(u_1, \ldots, u_l) : = \frac{u_{j_1}^{d_{j_1}} \cdots u_{j_k}^{d_{j_k}} }{(1 - u_{j_1}^{d_{j_1}}) \cdots (1 - u_{j_k}^{d_{j_k}}) }. \] Define the polynomial $Q_{ \Sigma }(u_1, \ldots, u_l)$ by the formula \[\sum_{\sigma \in \Sigma ^{G_v}} R_{\sigma}(u_1, \ldots, u_l) = \frac{Q_{ \Sigma }(u_1, \ldots, u_l)} {(1 - u_1^{d_1}) \cdots (1- u_l^{d_l}) }. \] } \end{dfn} \begin{prop} {\rm \cite{BaTschi}} Let $\Sigma$ be a complete regular $G_v$-invariant fan. Then the polynomial \[ Q_{ \Sigma } (u_1, \ldots, u_l) - 1 \] contains only monomials of degree $\geq 2$. \label{p-function} \end{prop} Let $\chi$ be a topological character of $T({\bf A}_K)$ such that its $v$-component $\chi_v\, : \, T(K_v) \rightarrow S^1 \subset {\bf C}^*$ is trivial on $T({\cal O}_v)$. For each $ j \in \{ 1, \ldots, l\}$, we denote by $n_j$ one of $d_j$ generators of all $1$-dimensional cones of the $G_v$-orbit $ \Sigma _j(1)$; i.e., $G_vn_j$ is the set of generators of $1$-dimensional cones in $ \Sigma _j(1)$. Recall ($\ref{pi-image}$) that for non-archimedian valuations, $n_j$ represents an element of $T(K_v)$ modulo $T({\cal O}_v)$. Therefore, $\chi_v(n_j)$ is well-defined. By ($\ref{pi-image}$) we know that the homomorphism $$ \pi_v : T(K_v)/T({\cal O}_v) \rightarrow N_v $$ is an isomorphism for almost all $v$. We call these valuations {\em good}. \begin{dfn} {\rm Denote by $\hat{H}_{\Sigma,v} (\chi_v, -{\bf s})$ the value at $\chi_v$ of the {\it multiplicative} Fourier transform of the local Weil function $H_{ \Sigma ,v}(x_v,-{\bf s})$ with respect to the $v$-adic Haar measure $d\mu_v$ on $T(K_v)$ normalized by $\int_{T({\cal O}_v)} d\mu_v = 1$.} \end{dfn} \begin{prop} {\rm \cite{BaTschi}} Let $v$ be a good non-archimedian valuation of $K$ . For any topological character $\chi_v$ of $T(K_v)$ which is trivial on the subgroup $T({\cal O}_v)$ and a piecewise linear function $\varphi = \varphi_{\bf s} \in PL( \Sigma )^G_{{\bf C}}$ one has \[ \hat{H}_{\Sigma,v} (\chi_v, -{\bf s}) = \int_{T(K_v)} H_{ \Sigma ,v}(x_v,-{\bf s}) \chi_v(x_v) d\mu_v = \] \[ = \frac{Q_{ \Sigma }\left( \frac{\chi_{v}(n_1)}{{q_v}^{\varphi(n_1)}}, \ldots, \frac{\chi_{v}(n_l)}{{q_v}^{\varphi(n_l)}} \right)} {(1- \frac{\chi_{v}(n_1)}{{q_v}^{\varphi(n_1)}} ) \cdots (1 - \frac{\chi_{v}(n_l)}{{q_v}^{\varphi(n_l)}} ) }. \] \label{integral.1} \end{prop} \begin{coro} {\rm \cite{BaTschi}} {\rm Let $v$ be a good non-archimedian valuation of $K$. The restriction of \[ \int_{T(K_v)} H_{ \Sigma ,v}(x_v,-{\bf s}) d \mu_v \] to the line $s_1 = \cdots = s_r = s$ is equal to \[ L_v( s,T; E/K)\cdot L_v(s, T_{NS}, E/K) \cdot Q_{ \Sigma } (q_v^{-s}, \ldots, q_v^{-s}). \] } \label{loc-int} \end{coro} \begin{rem} {\rm It is difficult to calculate the Fourier transforms of local heights for the finitely many "bad" non-archimedian valuations $v$, because there is only an embedding of finite index $$ T(K_v)/T({\cal O}_v)\hookrightarrow N_v. $$ However, for our purposes it will be sufficient to use upper estimates for these local Fourier transforms. One immediately sees that for all non-archimedian valuations $v$ the local Fourier transforms of $H_{ \Sigma ,v}(x_v,-{\bf s})$ can be bounded absolutely and uniformly in all characters by a finite combination of multidimensional geometric series in $q_v^{-1/2}$ in the domain ${\rm Re}({\bf s})\in {\bf R}_{>1/2}$. \label{badreduction} } \end{rem} Now we assume that $v$ is an archimedian valuation. By ($\ref{pi-image}$), we have $T(K_v)/T({\cal O}_v) = N_{\bf R}^{G_v} \subset N_{\bf R}$ where $G_v$ is the trivial group for the case $K_v = {\bf C}$, and $G_v = {\rm Gal}({\bf C}/ {\bf R}) \cong {\bf Z}/2{\bf Z}$ for the case $K_v = {\bf R}$. Let $\langle \cdot,\cdot\rangle $ be the pairing between $N_{{\bf R}}$ and $M_{{\bf R}}$ induced from the duality between $N$ and $M$. Let $y$ be an arbitrary element of the dual ${\bf R}$-space $M_{\bf R}^{G_v} = Hom(T(K_v)/T({\cal O}_v), {\bf R})$. Then $\chi_y(x_v) = e^{- i \langle \overline{x}_v,y \rangle}$ is a topological character of $T(K_v)$ which is trivial on $T({\cal O}_v)$. We choose the Haar measure $d\mu_v$ on $T(K_v)$ as the product of the Haar measure $d\mu_v^0$ on $T({\cal O}_v)$ and the Haar measure $d\overline{x}_v$ on $T(K_v)/T({\cal O}_v)$. We normalize the measures such that the $d\mu_v^0$-volume of $T({\cal O}_v)$ equals $1$ and $d\overline{x}_v$ is the standard Lebesgue measure on $N_{\bf R}^{G_v}$ normalized by the full sublattice $N^{G_v}$. \begin{prop} {\rm \cite{BaTschi}} Let $v$ be an archimedian valuation of $K$. The Fourier transform $\hat{H}_{ \Sigma ,v}(\chi_y,-{\bf s})$ of a local archimedian Weil function $$ H_{ \Sigma ,v} (x_v,-{\bf s}) = e^{-\varphi_{\bf s}(\overline{x}_v)} $$ is a rational function in variables $s_j = \varphi_{\bf s}(e_j)$ for ${\rm Re}({\bf s}) \in {\bf R}_{>0}$. \label{archim.tr} \end{prop} {\it Proof.} Let us consider the case $K_v = {\bf C}$. One has a decomposition of the space $N_{\bf R}$ into a union of $d$-dimensional cones $N_{\bf R} = \bigcup_{\sigma \in \Sigma (d)} \sigma$. We calculate the Fourier transform as follows: \[ \hat{H}_{ \Sigma ,v}(\chi_y,-{\bf s}) = \int_{N_{\bf R}} e^{-\varphi_{\bf s}(\overline{x}_v) - i \langle \overline{x}_v,y \rangle} d\overline{x}_v= \] \[ = \sum_{\sigma \in \Sigma(d)} \int_{\sigma} e^{-\varphi_{\bf s}(\overline{x}_v) - i \langle \overline{x}_v,y \rangle} d\overline{x}_v= \sum_{\sigma \in \Sigma(d)} \frac{1}{\prod_{e_j \in \sigma} (s_j + i \langle e_j,y \rangle)}. \] The case $K_v = {\bf R}$ can be reduced to the above situation. \hfill $\Box$ \medskip \section{Poisson formula} Let ${\bf P}_{ \Sigma } $ be a toric variety and $H_{ \Sigma }(x,{\bf s})$ the height function constructed above. \begin{dfn} {\rm We define the zeta-function of the complex height-function $H_{\Sigma}(x, {\bf s})$ as \[ Z_{\Sigma}({\bf s}) = \sum_{x \in T(K)} H_{\Sigma}(x,{\bf -s}). \]} \end{dfn} \begin{theo} The series $Z_{\Sigma}({\bf s})$ converges absolutely and uniformly for ${\bf s}$ contained in any compact in the domain ${\rm Re}({\bf s})\in {\bf R}^r_{>1}$. \label{convergence} \end{theo} \noindent {\em Proof.} It was proved in \cite{ono1} that we can always choose a finite set $S$ such that the natural map $$ \pi_{S}\; : \; T(K) \rightarrow \bigoplus_{v \not\in S} T(K_v)/T({\cal O}_v) = \bigoplus_{v \not\in S} N_v $$ is surjective. Denote by $T({\cal O}_S)$ the kernel of $\pi_S$ consisting of all $S$-units in $T(K)$. Let $W(T) \subset T({\cal O}_S)$ the subgroup of torsion elements in $T({\cal O}_S)$. Then $T({\cal O}_S)/W(T)$ has a natural embedding into the finite-dimensional logarithmic space $$ N_{{\bf R},S} = \bigoplus_{v \in S} T(K_v)/T({\cal O}_v) \otimes {{\bf R}} $$ as a sublattice of codimension $t$. Let $\Gamma$ be a full sublattice in $N_{{\bf R},S}$ containing the image of $T({\cal O}_S)/W(T)$. Denote by $\Delta$ a bounded fundamental domain of $\Gamma$ in $N_{{\bf R},S}$. For any $x \in T(K)$ we denote by $\overline{x}_S$ the image of $x$ in $N_{{\bf R},S}$. Define $\phi(x)$ to be the element of $\Gamma$ such that $\overline{x}_S - \phi(x) \in \Delta$. Thus, we have obtained the mapping $$ \phi\; : \; T(K) \rightarrow \Gamma. $$ Define a new function $\tilde{H}_{ \Sigma }(x, {\bf s})$ on $T(K)$ by $$ \tilde{H}_{ \Sigma }(x, {\bf s}) = \prod_{v \in S} H_{ \Sigma ,v}(\phi(x)_v, {\bf s}) \prod_{v \not\in S} H_{ \Sigma ,v}(x_v, {\bf s}). $$ If ${\bf K}\subset {\bf C}^r$ is a compact in the domain ${\rm Re}({\bf s}) \in {\bf R}^r_{>1}$, then there exist two positive constants $C_1({\bf K}) < C_2({\bf K})$ such that $$ 0 < C_1({\bf K}) < \frac{\tilde{H}_{ \Sigma }(x, {\bf s})}{H_{ \Sigma }(x, {\bf s})} < C_2({\bf K}) \;\; \mbox{\rm for ${\bf s} \in {\bf K}, \,x \in T(K)$}, $$ since $\overline{x}_v - \phi(x)_v$ belongs to some bounded subset $\Delta_v$ in $N_{{\bf R},v}$ for any $x \in T(K)$ and $v \in S$. Therefore, it is sufficient to prove that the series $$ \tilde{Z}_{ \Sigma }({\bf s}) = \sum_{x \in T(K)} \tilde{H}_{ \Sigma }(x, - {\bf s}) $$ is absolutely converent for ${\bf s} \in {\bf K}$. Notice that $\tilde{Z}_{ \Sigma }({\bf s})$ can be estimated from above by the the following Euler product $$ \left( \sum_{ \gamma \in \Gamma} \prod_{v \in S}H_{ \Sigma ,v}( \gamma_v, -{\bf s}) \right) \prod_{v \not\in S} \left( \sum_{z \in N_v} H_{ \Sigma ,v}(z, - {\bf s}) \right). $$ The sum $$ \sum_{ \gamma \in \Gamma} \prod_{v \in S}H_{ \Sigma ,v}( \gamma_v, -{\bf s}) $$ is an absolutely convergent geometric series for ${\rm Re}({\bf s}) \in {\bf R}^r_{>1}$. On the other hand, the Euler product $$ \prod_{v \not\in S} \left( \sum_{z \in N_v} H_{ \Sigma ,v}(z, - {\bf s}) \right) $$ can be estimated from above by the product of zeta-functions $$ C_3({\bf K}) \prod_{j =1}^r \zeta_{K_j}(s_j),$$ where $C_3({\bf K})$ is some constant depending on ${\bf K}$. Since each $ \zeta_{K_j}(s_j)$ is absolutely convergent for ${\rm Re}(s_j) > 1$, we obtain the statement. \hfill $\Box$ We need the Poisson formula in the following form: \begin{theo} Let ${\cal G}$ be a locally compact abelian group with Haar measure $dg, {\cal H}\subset {\cal G} $ a closed subgroup with Haar measure $dh$. The factor group ${\cal G}/{\cal H}$ has a unique Haar measure $dx$ normalized by the condition $dg=dx\cdot dh$. Let $F\,:\, {\cal G} \rightarrow {\bf R} $ be an ${L}^1$-function on ${\cal G}$ and $\hat{F}$ its Fourier transform with respect to $dg$. Suppose that $\hat{F}$ is also an ${L}^1$-function on ${\cal H}^{\perp}$, where ${\cal H}^{\perp}$ is the group of topological characters $\chi \,: \, {\cal G} \rightarrow S^1$ which are trivial on ${\cal H}$. Then $$ \int_{\cal H} F(x)dh=\int_{{\cal H}^{\perp}}\hat{F}(\chi) d\chi, $$ where $d\chi$ is the orthogonal Haar measure on ${\cal H}^{\perp}$ with respect to the Haar measure $dx$ on ${\cal G}/{\cal H}$. \label{poi} \end{theo} We will apply this theorem in the case when ${\cal G}=T({\bf A}_K)$ and ${\cal H}=T(K)$, $dg = \omega_{\Omega,S}$, and $dh$ is the discrete measure on $T(K)$. \begin{theo} (Poisson formula) For all ${\bf s}$ with ${\rm Re}({\bf s})\in {\bf R}^r_{>1}$ we have the following formula: $$ Z_{\Sigma}({\bf s})=\frac{1}{(2\pi )^t b_S(T)} \int_{(T({\bf A}_K)/T(K))^*} \left( \int_{T(A_F)}H_{ \Sigma }(x,-{\bf s})\chi(x)\omega_{\Omega,S} \right) d\chi, $$ where $\chi \in (T({\bf A}_K)/T(K))^* $ is a topological character of $T({\bf A}_K)$, trivial on the closed subgroup $T(K)$ and $d\chi$ is the orthogonal Haar measure on $(T({\bf A}_K)/T(K))^*$. The integral converges absolutely and uniformly to a holomorphic function in ${\bf s}$ in any compact in the domain ${\rm Re}({\bf s})\in {\bf R}^r_{>1}$. \label{poiss} \end{theo} {\em Proof.} Because of \ref{convergence} we only need to show that the Fourier transform $\hat{H}_{ \Sigma }(\chi,-{\bf s})$ of the height function is an $ L^1$-function on $(T({\bf A}_K)/T(K))^*$. By \ref{integral.1} and uniform estimates at places of bad reduction \ref{badreduction}, we know that the Euler product $$ \prod_{v\not\in S_{\infty}}\hat{H}_{ \Sigma ,v}(\chi_v,-{\bf s}) $$ converges absolutely and is uniformly bounded by a constant $c({\bf K})$ for all characters $\chi$ and all ${\bf s}\in {\bf K}$, where ${\bf K}$ is some compact in the domain ${\rm Re}({\bf s})\in {\bf R}_{>1}^r$. Since the height function $H_{ \Sigma ,v}(x,-{\bf s})$ is invariant under $T({\cal O}_v)$ for all $v$, the Fourier transform of $\hat{H}_{ \Sigma }(\chi,-{\bf s})$ equals zero for characters $\chi$ which are non-trivial on the maximal compact subgroup ${\bf K}_T$. Denote by ${\cal P}$ the set of all such characters $\chi\in (T({\bf A}_K)/T(K))^* $. We have a non-canonical splitting of characters $\chi =\chi_l\cdot \chi_y$, where $\chi_l\in (T^1({\bf A}_K)/T(K))^*$ and $\chi_y\in (T({\bf A}_K)/T^1({\bf A}_K))^*$. Let us consider the logarithmic space $$ N_{{\bf R},\infty}=\bigoplus_{v\in S_{\infty}}T(K_v)/T({\cal O}_v)= \bigoplus_{v\in S_{\infty}}N_{{\bf R},v}. $$ It contains the lattice $T({\cal O}_K)/W(T)$ of ${\cal O}_K$-integral points of $T(K)$ modulo torsion. Denote by $ M_{{\bf R},\infty}=\bigoplus_{v\in S_{\infty}}M_{{\bf R},v} $ the dual space. It has a decomposition as a direct sum of vector spaces $M_{{\bf R},\infty}=M_L\oplus M_Y$, such that the space $M_L$ contains the dual lattice $L:=(T({\cal O}_K)/W(T))^*$ as a full sublattice and the space $M_Y$ is isomorphic to $(T({\bf A}_K)/T^1({\bf A}_K))^*= \hat{T}_K\otimes {\bf R}$. By \ref{subgroups}, we have an exact sequence $$ 0 \rightarrow {\bf cl}^*(T) \rightarrow {\cal P} \rightarrow {\cal M} \rightarrow 0, $$ where ${\cal M}$ is the image of the projection of ${\cal P}$ to $M_{{\bf R},\infty}$ and ${\bf cl}^*(T)$ is a finite group. We see that the character $\chi\in {\cal P}$ is determined by its archimedian component up to a finite choice. Denote by $y(\chi)\in {\cal M}\subset M_{{\bf R},\infty}$ the image of $\chi\in {\cal P}$. The following lemmas will provide the necessary estimates of the Fourier transform of local heights at archimedian places. This allows to apply the Poisson formula \ref{poi}. \hfill $\Box $ \begin{lem} {\rm \cite{BaTschi}} {\rm Let $\Sigma\subset N_{{\bf R}}$ be a complete fan in a real vector space of dimension $d$. Denote by $M_{{\bf R}}$ the dual space. For all $m\in M_{{\bf R}}$ we have the following estimate $$ |\sum_{ \sigma \in \Sigma (d)} \frac{1}{\prod_{e_j\in \sigma }(s_j+i<e_j,m>)}|\le \frac{1}{(1 + \|m\|)^{1+1/d}}. $$ } \end{lem} \begin{coro} {\rm Consider \[ \hat{H}_{ \Sigma ,\infty}(\chi, -{\bf s}) = \prod_{v \in S_{\infty}} \hat{H}_{ \Sigma ,v}(\chi, -{\bf s}) \] as a function on \[ {\cal M}\subset M_{{\bf R}, \infty} = \bigoplus_{v \in S_{\infty}} M_{{\bf R},v}. \] Let $d'$ be the dimension of $M_{{\bf R}, \infty}$. We have a direct sum decomposition $M_{{\bf R}, \infty}=M_L\oplus M_Y$ of real vector spaces. Let $M'_Y\subset M_Y$ be any affine subspace, $dy'$ the Lebesgue measure on $M'_Y$ and $L'\subset M_L$ any lattice. Let $g(y,-{\bf s})$ be a function on $M_{{\bf R}, \infty}$ satisfying the inequality $|g(y,-{\bf s})|\le c \|y\|^{\delta } $ for all $y\in M_{{\bf R}, \infty}$, all ${\bf s}$ in some compact domain in ${\rm Re}({\bf s})\in {\bf R}_{>1/2}$, some $0<\delta < 1/d'$ and some constant $c>0$. Then the series \[ \sum_{y(\chi) \in L} \int_{M'_Y} g(y,-{\bf s})\hat{H}_{ \Sigma ,\infty}(y(\chi), -{\bf s}) dy' \] is absolutely and uniformly convergent to a holomorphic function in ${\bf s}$ in this domain. \label{infconver} } \end{coro} {\em Proof.} We apply \ref{archim.tr} and observe that on the space $N_{{\bf R}, \infty}$ we have a fan $ \Sigma _{\infty}$ obtained as the direct product of fans $ \Sigma ^{G_v}$ for $v\in S_{\infty}$ (i.e., every cone in $ \Sigma _{\infty}$ is a direct product of cones in $ \Sigma ^{G_v}$). \hfill $\Box $ \section{${\cal X}$-functions of convex cones} Let $(A, A_{\bf R}, \Lambda ) $ be a triple consisting of a free abelian group $A$ of rank $k$, a $k$-dimensional real vector space $A_{\bf R} = A \otimes {\bf R}$ containing $A$ as a sublattice of maximal rank, and a convex $k$-dimensional cone $\Lambda \subset A_{{\bf R}}$ such that $\Lambda \cap - \Lambda = 0 \in A_{{\bf R}}$. Denote by $ \Lambda ^{\circ}$ the interior of $ \Lambda $ and by ${ \Lambda }_{\bf C}^{\circ} = { \Lambda }^{\circ} + iA_{{\bf R}}$ the complex tube domain over ${ \Lambda }^{\circ}$. Let $( A^*, A^*_{{\bf R}}, \Lambda ^*) $ be the triple consisting of the dual abelian group $A^* = {\rm Hom}(A, {\bf Z})$, the dual real vector space $A^*_{{\bf R}} = {\rm Hom}(A_{{\bf R}}, {\bf R})$, and the dual cone $ \Lambda ^* \subset A^*_{{\bf R}}$. We normalize the Haar measure $ {\bf d}{\bf y}$ on $A_{{\bf R}}^*$ by the condition: ${\rm vol}(A^*_{{\bf R}}/A^*)=1$. \begin{dfn} {\rm The {\em ${\cal X}$-function of} ${ \Lambda }$ is defined as the integral \[ {\cal X}_{ \Lambda }({\bf s}) = \int_{{ \Lambda }^*} e^{- \langle {\bf s}, {\bf y} \rangle} {\bf d}{\bf y}, \] where ${\bf s} \in { \Lambda }_{\bf C}^{\circ}$. } \end{dfn} \begin{rem} {\rm ${\cal X}$-functions of convex cones have been investigated in the theory of homogeneous cones by M. K\"ocher, O.S. Rothaus, and E.B. Vinberg \cite{koecher,rothaus,vinberg}. In these papers ${\cal X}$-functions were called {\em characteristic functions of cones}, but we find such a notion rather misleading in view of the fact that ${\cal X}_{ \Lambda }({\bf s})$ is the Fourier-Laplace transform of the standard set-theoretic characteristic function of the dual cone $ \Lambda ^*$. } \end{rem} \noindent The function ${\cal X}_{ \Lambda }({\bf s})$ has the following properties \cite{rothaus,vinberg}: \begin{prop} {\rm (i)} If ${\cal A}$ is any invertible linear operator on ${\bf C}^k$, then \[ {\cal X}_{ \Lambda } ({\cal A}{\bf s}) = \frac{{\cal X}_{ \Lambda }({\bf s})} {{\rm det}{\cal A}}; \] {\rm (ii)} If ${ \Lambda } = {\bf R}^k_{\geq 0}$, then \[ {\cal X}_{ \Lambda }({\bf s}) = (s_1 \cdots s_k)^{-1}, \;{\rm for } \;{\rm Re}(s_i) > 0 ; \] {\rm (iii)} If ${\bf s} \in { \Lambda }^{\circ}$, then \[ \lim_{{\bf s} \rightarrow \partial { \Lambda }} {\cal X}_{ \Lambda }({\bf s}) = \infty; \] {\rm (iv)} ${\cal X}_{ \Lambda }({\bf s}) \neq 0$ for all ${\bf s} \in { \Lambda }_{\bf C}^{\circ}$. \label{zeta.cone} \end{prop} \begin{prop} If ${ \Lambda }$ is a $k$-dimensional finitely generated polyhedral cone, then ${\cal X}_{ \Lambda }({\bf s})$ is a rational function $$ {\cal X}_{ \Lambda }({\bf s}) = \frac{P({\bf s})}{Q({\bf s})}, $$ where $P$ is a homogeneous polynomial, $Q$ is a product of all linear homogeneous forms defining the codimension $1$ faces of $ \Lambda $, and ${\rm deg}\, P - {\rm deg}\, Q = -k$. \label{merom} \end{prop} \noindent {\em Proof.} We subdivide the dual cone ${ \Lambda }^*$ into a finite union of simplicial subcones $ \Lambda _j^*$ $(j \in J)$. Let $ \Lambda _j \subset A_{{\bf R}}$ be the dual cone to $ \Lambda _j^*$. Then $$ {\cal X}_{ \Lambda }({\bf s}) = \sum_{j \in J} {\cal X}_{ \Lambda _j}({\bf s}). $$ By \ref{zeta.cone}(i) and (ii), $${\cal X}_{ \Lambda _j}({\bf s}) = \frac{P_j({\bf s})}{Q_j({\bf s})} \;\; ( j \in J),$$ where $P_j$ is a homogeneous polynomial of degree $0$ and $Q_j$ is the product of $k$ homogeneous linear forms defining the codimension $1$ faces of $ \Lambda _j$. Therefore, ${\cal X}_{ \Lambda }({\bf s})$ can be uniquely represented up to constants as a ratio of two homogeneous polynomials $P({\bf s})/Q({\bf s})$ with $g.c.d.(P,Q)=1$ where $Q$ is a product of linear homogeneous forms defining some faces of $ \Lambda _j$ of codimension $1$. Since $Q$ does not depend on a choice of a subdivision of $ \Lambda ^*$ into a finite union of simplicial cones $ \Lambda ^*_j$, only linear homogeneous forms which vanish on codimension $1$ faces of $ \Lambda $ can be factors of $Q$. On the other hand, by \ref{zeta.cone}(iii), every linear homogeneous form vanishing on a face of $ \Lambda $ of codimension $1$ must divide $Q$. \hfill $\Box$ \begin{theo} Let $(A, A_{{\bf R}}, \Lambda )$ and $(\tilde{A}, \tilde{A}_{{\bf R}}, \tilde{ \Lambda })$ be two triples as above, $k = {\rm rk}\, A$ and $\tilde{k} = {\rm rk}\, \tilde{A}$, and $\psi\;:\; A \rightarrow \tilde{A}$ a homomorphism of free abelian groups with a finite cokernel $A'$ (i.e., the corresponding linear mapping of real vector spaces $\psi \;:\; A_{{\bf R}} \rightarrow \tilde{A}_{{\bf R}}$ is surjective), and $\psi( \Lambda ) = \tilde{ \Lambda }$. Let $B= {\rm Ker}\, \psi \subset A$, ${\bf d}{\bf b}$ the Haar measure on $B_{{\bf R}} = B \otimes {{\bf R}}$ normalized by the condition ${\rm vol}(B_{{\bf R}}/B)=1$. Then for all ${\bf s}$ with ${\rm Re}({\bf s}) \in \Lambda^{\circ}$ the following formula holds: $$ {\cal X}_{\tilde{ \Lambda }}(\psi({\bf s})) = \frac{1}{(2\pi)^{k-\tilde{k}}|A'|} \int_{B_{{\bf R}}} {\cal X}_{{ \Lambda }} ({\bf s} + i {\bf b}) {\bf db}, $$ where $|A'|$ is the order of the finite abelian group $A'$. \label{char0} \end{theo} {\em Proof.} We have the dual injective homomorphisms of free abelian groups $\psi^*\;:\; \tilde{A}^* \rightarrow A^*$ and of the corresponding real vector spaces $\psi^*\;:\; \tilde{A}^*_{{\bf R}} \rightarrow A^*_{{\bf R}}$. Moreover, $\tilde{ \Lambda }^* = \Lambda ^* \cap \tilde{A}^*_{{\bf R}}$. Let $C_{ \Lambda ^*}({\bf y})$ be the set-theoretic characteristic function of the cone $ \Lambda ^* \subset A^*_{{\bf R}}$ and $C_{ \Lambda ^*}(\tilde{\bf {y}})$ the restriction of $C_{ \Lambda ^*}({\bf y})$ to $\tilde{A}_{{\bf R}}^*$ which is the set-theoretic characteristic function of $\tilde{ \Lambda }^* \subset \tilde{A}_{{\bf R}}^*$. Then $$ {\cal X}_{\tilde{ \Lambda }}(\psi({\bf s})) = \int_{\tilde{A}^*_{{\bf R}}} C_{ \Lambda ^*}(\tilde{\bf {y}}) e^{- \langle \psi({\bf s}), \tilde{\bf y} \rangle} {\bf d}\tilde{\bf y}. $$ Now we apply the Poisson formula to the last integral. For this purpose we notice that any additive topological character of ${A}^*_{{\bf R}}$ which vanishes on the subgroup $\tilde{A}^*_{{\bf R}} \subset {A}^*_{{\bf R}}$ has the form $$ e^{- i \langle {\bf b}, {\bf y} \rangle}, \;\;\; \mbox{ \rm where ${\bf b} \in B_{{\bf R}}$}. $$ Moreover, $$ \frac{{\bf db}}{(2\pi)^{k-\tilde{k}}|A'|} $$ is the orthogonal Haar measure on $B_{{\bf R}}$ with respect to the Haar measures ${\bf d}\tilde{\bf y}$ and ${\bf d}{\bf y}$ on $\tilde{A}^*_{{\bf R}}$ and $A^*_{{\bf R}}$ respectively. It remains to notice that $${\cal X}_{{ \Lambda }} ({\bf s} + i {\bf b}) = \int_{{A}^*_{{\bf R}}} C_{ \Lambda ^*}({\bf {y}}) e^{- \langle {\bf s} + i{\bf b}, {\bf y} \rangle} {\bf d}{\bf y} $$ is the value of the Fourier transform of $C_{ \Lambda ^*}({\bf {y}}) e^{- \langle {\bf s} , {\bf y} \rangle}$ on the topological character of $A_{{\bf R}}^*/\tilde{A}_{{\bf R}}^*$ corresponding to an element ${\bf b} \in B_{{\bf R}} \subset A_{{\bf R}}$. \hfill $\Box$ \begin{coro} {\rm Assume that in the above situation ${\rm rk}\,= k - \tilde{k} =1$ and $\tilde{A} = A/B$. Denote by $ \gamma$ a generator of $B$. Then $$ {\cal X}_{\tilde{ \Lambda }}({\psi}({\bf s})) = \frac{1}{2\pi i}\int_{{\rm Re}(z) = 0} {\cal X}_{ \Lambda }({\bf s} + z \cdot \gamma) dz, $$ where $z = x + iy \in {{\bf C}}$. \label{char1} } \end{coro} \begin{coro}{\rm Assume that a $\tilde{k}$-dimensional rational finite polyhedral cone $\tilde{\Lambda} \subset \tilde{A}_{{\bf R}}$ contains exactly $r$ one-dimensional faces with primitive lattice generators $a_1, \ldots, a_r \in \tilde{A}$. We set $k := r$, $A := {{\bf Z}}^r$ and denote by $\psi$ the natural homomorphism of lattices ${\bf Z}^r \rightarrow \tilde{A}$ which sends the standard basis of ${{\bf Z}}^r$ into $a_1, \ldots, a_r \in \tilde{A}$, so that $\tilde{ \Lambda }$ is the image of the simplicial cone ${\bf R}^r_{\ge 0}\subset {\bf R}^r$ under the surjective map of vector spaces $\psi\; : \; {\bf R}^r \rightarrow A_{{\bf R}}$. Denote by $M_{{\bf R}}$ the kernel of $\psi$ and set $M := {{\bf Z}}^r \cap M_{{\bf R}}$. Let ${\bf s}=(s_1,...,s_r)$ be the standard coordinates in ${\bf C}^r$. Then $$ {\cal X}_{ \Lambda }(\psi({\bf s}))=\frac{1}{(2\pi )^{r-k}|A'|} \int_{M_{{\bf R}}}\frac{1}{\prod_{j=1,n}(s_j+iy_j)} {\bf d}{\bf y} $$ where ${\bf dy}$ is the Haar measure on the additive group $M_{{\bf R}}$ normalized by the lattice $M$, $y_j$ are the coordinates of ${\bf y}$ in ${\bf R}^r$, and $|A'|$ is the index of the sublattice in $\tilde{A}$ generated by $a_1, \ldots, a_r$. \label{int.formula} } \end{coro} \begin{exam} {\rm Consider an example of a non-simplicial convex cone which appears as the cone of effective divisors of the split toric Del Pezzo surface $X$ of anticanonical degree 6. The cone ${\Lambda}_{{\rm eff}}$ has 6 generators corresponding to exceptional curves of the first kind on $X$. We can construct $X$ as the blow up of 3 points $p_1, p_2, p_3$ in general position in ${\bf P}^2$. Denote the exceptional curves by $C_1, C_2, C_3, C_{12}, C_{13}, C_{23}$, where $C_{ij}$ is the proper pullback of the line joining $p_i$ and $p_j$. Let ${\bf s} = s_1 [C_1] + s_2 [C_2] + s_3 [C_3] + s_{12}[C_{12}] + s_{13}[C_{13}] + s_{23} [C_{23}] \in \Lambda _{\rm eff}^{\circ}$ be an element in the interior of the cone of effective divisors. The sublattice $M \subset {\bf Z}^6$ of rank $2$ consisting of principal divisors is generated by $ \gamma_1 = C_1 + C_{13} - C_2 - C_{23}$ and $ \gamma_2 = C_1 + C_{12} - C_3 - C_{23} = 0$. In our case, the integral formula in \ref{int.formula} is a 2-dimensional integral ($r =6$) which can be computed by applying twice the residue theorem to two $1$-dimensional integrals like the one in \ref{char1}. We obtain the following formula for the characteristic function of ${\Lambda}_{\rm eff}$: } \[ {\cal X}_{\Lambda}(\psi({\bf s})) = \frac{ s_1 + s_2 + s_3 + s_{12} + s_{13} + s_{23} } {(s_1 + s_{23}) (s_2 + s_{13})(s_3 + s_{12})(s_1 + s_2 + s_3 ) (s_{12} + s_{13} + s_{23})}. \] \end{exam} \begin{dfn} {\rm Let $X$ be a smooth proper algebraic variety. Consider the triple $({\rm Pic}(X), {\rm Pic}(X) \otimes{\bf R}, \Lambda _{\rm eff})$ where $ \Lambda _{\rm eff} \subset {\rm Pic}(X)\otimes {\bf R}$ is the cone generated by classes of effective divisors on $X$. Assume that the anticanonical class $ \lbrack {\cal K}^{-1} \rbrack \in {\rm Pic}(X)_{\bf R}$ is contained in the interior of $ \Lambda _{\rm eff}$. We define the constant $\alpha(X)$ by \[ \alpha(X) = {\cal X}_{ \Lambda _{\rm eff}}( \lbrack {\cal K}^{-1} \rbrack). \] } \end{dfn} \begin{coro}{\rm If ${ \Lambda }_{\rm eff}$ is a finitely generated polyhedral cone, then $\alpha(X)$ is a rational number. } \end{coro} \section{Some technical statements} Let $E$ be a number field and $\chi$ an unramified Hecke character on ${\bf G}_m(A_E)$. Its local components $\chi_v$ for all valuations $v$ are given by: $$ \chi_v: {\bf G}_m(E_v)/{\bf G}_m({\cal O}_{v}) \rightarrow S^1 $$ $$ \chi_v(x_v)=|x_v|_v^{it_v}. $$ \begin{dfn} {\rm Let $\chi$ be an unramified Hecke character. We set $$ y(\chi) : = \{ t_v \}_{v \in S_{\infty}(E)} \in {{\bf R}}^{r_1 + r_2}, $$ where $r_1$ (resp. $ r_2$) is the number of real (resp. pairs of complex) valuations of $E$. We also set $$ \| y(\chi) \| := \max_{v \in S_{\infty}(E)} |t_v|. $$ \label{y-comp} } \end{dfn} We will need uniform estimates for Hecke $L$-functions in vertical strips. They can be deduced using the Phragmen-Lindel\"of principle \cite{rademacher}. \begin{theo} For any $\varepsilon > 0$ there exists a $\delta>0 $ such that for any $0<\delta_1<\delta $ there exists a constant $c(\varepsilon,\delta_1) > 0$ such that the inequality $$ | L_E(s, \chi) | \leq c(\varepsilon) ( 1 + |{\rm Im}(s)| + \|y(\chi)\| )^{\varepsilon} $$ holds for all $s$ with $ \delta_1<|{\rm Re}(s) -1|< \delta$ and every Hecke L-function $L_E(s, \chi)$ corresponding to an unramified Hecke character $\chi$. \label{estim} \end{theo} \begin{coro}{\rm For any $\varepsilon >0 $ there exists a $ \delta>0$ such that for any compact ${\bf K}$ in the domain $ 0<| {\rm Re}\,( s) - 1| <\delta$ there exists a constant $C({\bf K},\varepsilon)$ depending only on ${\bf K}$ and $\epsilon$ such that \[ | L_E(s,\chi) | \leq C({\bf K},\varepsilon) (1 + \|y(\chi)\|)^{\varepsilon} \] for $s \in {\bf K}$ and every unramified character $\chi$. \label{m.estim} } \end{coro} Let $ \Sigma $ be the Galois-invariant fan defining ${\bf P}_{ \Sigma }$ and $ \Sigma (1)= \Sigma _1(1)\cup ...\cup \Sigma _r(1)$ the decomposition of the set of one-dimensional generators of $ \Sigma $ into $G$-orbits. Let $e_j$ be a primitive integral generator of $\sigma_j$, $G_j \subset G$ the stabilizer of $e_j$. Denote by $K_j \subset E$ the subfield of $G_j$-fixed elements. Consider the $n$-dimensional torus $$ T'=\prod_{j=1}^r R_{K_j/K}({\bf G}_m). $$ Let us recall the exact sequence of Galois-modules from Proposition \ref{nonsplit.geom}: $$ 0 \rightarrow M^G \rightarrow PL( \Sigma )^G \rightarrow {\rm Pic}({\bf P}_{ \Sigma }) \rightarrow H^1(G,M) \rightarrow 0. $$ It induces a map of tori $T' \rightarrow T$ and a homomorphism $$ a\,:\,\,T'({\bf A}_K)/T'(K) \rightarrow T({\bf A}_K)/T(K). $$ So we get a dual homomorphism for characters $$ a^*:\, (T({\bf A}_{K})/T(K))^* \rightarrow \prod_{j =1}^r ({\bf G}_{m}({\bf A}_{K_j})/{\rm G}_m(K_j))^*. $$ \begin{prop}{\rm \cite{drax1}} The kernel of $a^*$ is dual to the obstruction group to weak approximation $A(T) $ defined in \ref{weak0}. \label{kera} \end{prop} Let $\chi\in (T({\bf A}_{K})/T(K))^*$ be a character. Then $\chi\circ a$ defines $r$ Hecke characters of the idele groups \[ \chi_j \; :\; {\bf G}_m({\bf A}_{K_j}) \rightarrow S^1 \subset {\bf C}^*, \; j =1, \ldots, r. \] If $\chi$ is trivial on ${\bf K}_T$, then all characters $\chi_j$ $(j =1, \ldots, r)$ are trivial on the maximal compact subgroups in ${\bf G}_m({\bf A}_{K_j})$. We denote by $L_{K_j}(s,\chi_j)$ the Hecke $L$-function corresponding to the unramified character $\chi_j$. \begin{prop} Let $\chi=(\chi_v)$ be a character and $\hat{H}_{ \Sigma ,v}(\chi_v,-{\bf s})$ the local Fourier transform of the complex local height function $H_{ \Sigma ,v}(x_v,-{\bf s})$. For any compact ${\bf K}$ contained in the domain ${\rm Re}({\bf s})\in {\bf R}^r_{>1/2} $ there exists a constant $c({\bf K})$ such that \[ \prod_{v\not\in S}\hat{H}_{ \Sigma ,v}(\chi_v,-{\bf s})\cdot \prod_{i=1}^r L^{-1}_{K_j}(s_j,\chi_j) \le c({\bf K})\] for all characters $\chi\in (T({\bf A}_{K})/T(K))^*$. \label{dmethod} \end{prop} The proof follows from explicit computations of local Fourier transforms \ref{integral.1} and is almost identical with the proof of Proposition 3.1.3 in \cite{BaTschi}. \begin{prop} There exists an $\varepsilon >0$ such that for any open $U\subset {\bf C}^r$ contained in the domain $ 0< |{\rm Re}(s_j)-1|<\varepsilon $ for $j=1,...,r$ the integral $$ \int_{(T({\bf A}_{K})/T(K))^*}\hat{H}_{ \Sigma }(\chi, -{\bf s})d\chi $$ converges absolutely and uniformly to a holomorphic function for ${\rm Re}({\bf s})\in U$. \label{analytic} \end{prop} {\em Proof.} Using uniform estimates of Fourier transforms for non-archimedian places of bad reduction (\ref{badreduction}) and the proposition above we need only to consider the following integral $$ \int_{(T({\bf A}_{K})/T(K))^*}\hat{H}_{ \Sigma ,\infty}(\chi,-{\bf s}) \prod_{j=1}^r L_{K_j}(s_j,\chi_j)d\chi. $$ Observe that there exist constants $c_1>0$ and $c_2>0$ such that we have the following inequalities: $$ c_1\|y(\chi)\|\le \sum_{j=1}^r\|y(\chi_j)\|\le c_2\|y(\chi)\|. $$ Here we denoted by $\|y(\chi)\|$ the norm of $y(\chi)\in M_{{\bf R},\infty}$. Recall that since we only consider $\chi$ which are trivial on the maximal compact subgroup ${\bf K}_T$, all characters $\chi_j$ are unramified. To conclude, we apply uniform estimates of Hecke L-functions from Corollary \ref{m.estim} and the Corollary \ref{infconver}. \hfill $\Box $ The rest of this section is devoted to the proof of our main technical result. Let ${{\bf R}}\lbrack {\bf s} \rbrack$ (resp. ${{\bf C}}\lbrack {\bf s} \rbrack$) be the ring of polynomials in $s_1, \ldots, s_r$ with coefficients in ${{\bf R}}$ (resp. in ${{\bf C}}$), ${{\bf C}}\lbrack \lbrack {\bf s} \rbrack \rbrack$ the ring of formal power series in $s_1, \ldots, s_r$ with complex coefficients. \begin{dfn} {\rm Two elements $f({\bf s}),\, g({\bf s})\in {{\bf C}}\lbrack \lbrack {\bf s} \rbrack \rbrack$ will be called {\em coprime}, if $g.c.d.(f({\bf s}),\, g({\bf s})) =1$. } \end{dfn} \begin{dfn} {\rm Let $f({\bf s})$ be an element of ${{\bf C}}\lbrack \lbrack {\bf s} \rbrack \rbrack$. By the {\em order} of a monomial $s_1^{ \alpha _1}...s_r^{ \alpha _r}$ we mean the sum of the exponents $ \alpha _1+...+ \alpha _r$. By {\em multiplicity $\mu(f({\bf s}))$ of $f({\bf s})$ at ${\bf 0} = (0, \ldots, 0)$} we always mean the minimal order of non-zero monomials appearing in the Taylor expansion of $f({\bf s})$ at ${\bf 0}$ . } \label{mult1} \end{dfn} \begin{dfn} {\rm Let $f({\bf s})$ be a meromorphic at ${\bf 0}$ function. Define the {\em multiplicity $\mu(f({\bf s}))$ } of $f({\bf s})$ at ${\bf 0}$ as \[ \mu(f({\bf s})) = \mu(g_1({\bf s})) - \mu(g_2({\bf s})) \] where $g_1({\bf s})$ and $g_2({\bf s})$ are two coprime elements in ${{\bf C}}\lbrack \lbrack {\bf s} \rbrack \rbrack$ such that $f = g_1/g_2$. } \label{mult2} \end{dfn} \begin{rem} {\rm It is easy to show that for any two meromorphic at ${\bf 0}$ functions $f_1({\bf s})$ and $f_2({\bf s})$, one has (i) $\mu(f_1 \cdot f_2) = \mu(f_1) \cdot \mu(f_2)$ (in particular, one can omit "coprime" in Definition \ref{mult2}); (ii) $\mu(f_1 + f_2) \geq \min \{ \mu(f_1), \mu(f_2) \}$; (iii) $\mu(f_1 + f_2) = \mu(f_1)$ if $\mu(f_2) > \mu(f_1)$. } \label{mult3} \end{rem} Using the properties \ref{mult3}(i)-(ii), one immediately obtains from Definition \ref{mult1} the following: \begin{prop} Let $f_1({\bf s})$ and $f_2({\bf s})$ be two analytic at ${\bf 0}$ functions, $l({\bf s})$ a homogeneous linear function, $ \gamma = ( \gamma_1, \ldots, \gamma_r) \in {{\bf C}}^r$ an arbitrary complex vector with $l( \gamma) \neq 0$, and $g({\bf s}) := f_1({\bf s})/f_2({\bf s})$. Then the multiplicity of $$ \tilde{g}({\bf s}): = \left(\frac{\partial}{\partial z}\right)^k g({\bf s} + z \cdot \gamma) |_{z = - l({\bf s})/l( \gamma)} $$ at ${\bf 0}$ is at least $\mu(g) - k$. \label{mult4} \end{prop} Let $\Gamma \subset {{\bf Z}}^r$ be a sublattice, $\Gamma_{{\bf R}} \subset {{\bf R}}^r$ (resp. $\Gamma_{{\bf C}} \subset {{\bf C}}^r$) the scalar extension of $\Gamma$ to a {{\bf R}}-subspace (resp. to a {{\bf C}}-subspace). We always assume that $\Gamma_{{\bf R}} \cap {{\bf R}}_{\geq 0}^r = 0$. We set $V_{{\bf R}}: = {{\bf R}}^r/\Gamma_{{\bf R}}$ and $V_{{\bf C}}: = {{\bf C}}^r/\Gamma_{{\bf C}}$. Denote by $\psi$ the canonical ${{\bf C}}$-linear projection ${\bf C}^r \rightarrow V_{{\bf C}}$. \begin{dfn} {\rm A complex analytic function $f({\bf s})= f(s_1, \ldots, s_r): U \rightarrow {\bf C}$ defined on an open subset $U \subset {\bf C}^r$ is said to {\em descend to $V_{{\bf C}}$} if for any vector $ \alpha \in \Gamma_{{\bf C}}$ and any ${\bf u}= (u_1, \ldots, u_r) \in U$ one has \[ f({\bf u}+ z \cdot \alpha ) = f({\bf u}) \;\; \mbox{\rm for all $\{ z \in {\bf C}\, :\,{\bf u}+ z \cdot \alpha \in U\}$}.\] } \end{dfn} \begin{rem} {\rm By definition, if $f({\bf s})$ descends to $V_{{\bf C}}$, then there exists an analytic function $g$ on $\psi(U) \subset V_{{\bf C}}$ such that $f = g \circ \psi$. Using Cauchy-Riemann equations, one immediatelly obtains that $f$ descends to $V_{{\bf C}}$ if and only if for any vector $ \alpha \in \Gamma_{{\bf R}}$ and any ${\bf u}= (u_1, \ldots, u_r) \in U$, one has \[ f({\bf u}+ iy \cdot \alpha ) = f({\bf u})\; \; \mbox{\rm for all $\{ y \in {\bf R}\, :\,{\bf u}+ iy \cdot \alpha \in U\}$}. \]} \label{desc} \end{rem} \begin{dfn} {\rm An analytic function $W({\bf s})$ in the domain ${\rm Re}({\bf s}) \in {{\bf R}}_{>0}^r$ is called {\em good with respect to $\Gamma$} if it satisfies the following conditions: {(i)} $W({\bf s})$ descends to $V_{{\bf C}}$; {(ii)} There exist pairwise coprime linear homogeneous polynomials $$ l_1({\bf s}), \ldots, l_p({\bf s}) \in {{\bf R}}\lbrack {\bf s} \rbrack$$ and positive integers $k_1, \ldots, k_p$ such that for every $j \in \{1, \ldots, p \}$ the linear form $l_j({\bf s})$ descends to $V_{{\bf C}}$, $l_j({\bf s})$ does not vanish for ${\bf s} \in {{\bf R}}_{>0}^n$, and $$ P({\bf s}) = W({\bf s}) \cdot \prod_{j =1}^p l_j^{k_j}({\bf s}) $$ is analytic at ${\bf 0}$. (iii) There exist a non-zero complex number $C(W)$ and a decomposition of $P({\bf s})$ into the sum $$ P({\bf s}) = P_0({\bf s}) + P_1({\bf s}) $$ so that $P_0({\bf s})$ is a homogeneous polynomial of degree $\mu(P)$, $P_1({\bf s})$ is an analytic function at ${\bf 0}$ with $\mu(P_1) > \mu(P_0)$, both functions $P_0$, $P_1$ descend to $V_{{\bf C}}$, and $$ \frac{P_0({\bf s})}{\prod_{j =1}^p l_j^{k_j}({\bf s})} = C(W) \cdot {\cal X}_{ \Lambda }(\psi({\bf s})), $$ where ${\cal X}_{ \Lambda }$ is the ${\cal X}$-function of the cone $ \Lambda = \psi({{\bf R}}^r_{\geq 0}) \subset V_{{\bf C}}$; } \end{dfn} \begin{dfn} {\rm If $W({\bf s})$ is a good with respect to $\Gamma$ as above, then the meromorphic function $$ \frac{P_0({\bf s})}{\prod_{j =1}^p l_j^{k_j}({\bf s})} $$ will be called the {\em principal part of $W({\bf s})$ at ${\bf 0}$} and the non-zero constant $C(W)$ the {\em principal coefficient of $W({\bf s})$ at ${\bf 0}$}. } \end{dfn} Suppose that $\Gamma \neq {{\bf Z}}^r$. Let $ \gamma \in {{\bf Z}}^r$ be an element which is not contained in $\Gamma$, $\tilde{\Gamma}: = \Gamma \oplus {\bf Z} < \gamma >$, $\tilde{\Gamma}_{{\bf R}} := \Gamma_{{\bf R}} \oplus {\bf R} < \gamma >$, $\tilde{V}_{{\bf R}} := {{\bf R}}^r /\tilde{\Gamma}_{{\bf R}}$ and $\tilde{V}_{{\bf C}} := {{\bf R}}^r /\tilde{\Gamma}_{{\bf C}}$. The following easy statement will be helpful in the sequel: \begin{prop} Let $f({\bf s})$ be an analytic at ${\bf 0}$ function, $l({\bf s})$ a homogeneous linear function such that $l( \gamma) \neq 0$. Assume that $f({\bf s})$ and $l({\bf s})$ descend to $V_{{\bf C}}$. Then $$ \tilde{f}({\bf s}) : = f\left({\bf s} - \frac{l({\bf s})}{l( \gamma)} \cdot \gamma\right) $$ descends to $\tilde{V}_{{\bf C}}$. \label{desc2} \end{prop} \begin{theo} Let $W({\bf s})$ be a good function with respect to $\Gamma$ as above, $$ \Phi({\bf s}) = \prod_{j\;:\; l_j( \gamma)=0} l_j^{k_j}({\bf s}) $$ the product of those linear forms $l_j({\bf s})$ $j \in \{ 1, \ldots, p\}$ which vanish on $ \gamma$. Assume that $\tilde{\Gamma}_{{\bf R}} \cap {{\bf R}}_{\geq 0}^r = 0$ and the following statements hold: {\rm (i)} The integral $$ \tilde{W}({\bf s}) : = \int_{{\rm Re}(z) = 0} W({\bf s} + z \cdot \gamma) dz , \;\; z \in {\bf C} $$ converges absolutely and uniformly on any compact in the domain ${\rm Re}({\bf s}) \in {{\bf R}}_{>0}^r$; {\rm (ii)} There exists $ \delta > 0$ such that the integral $$ \int_{{\rm Re}(z) = \delta } \Phi({\bf s}) \cdot W({\bf s} + z \cdot \gamma) dz $$ converges absolutely and uniformly in an open neighborhood of ${\bf 0}$. Moreover, the multiplicity of the meromorphic function $$ \tilde{W}_{ \delta }({\bf s}): = \int_{{\rm Re}(z) = \delta } W({\bf s} + z \cdot \gamma) dz $$ at ${\bf 0}$ is at least $1 + {\rm rk}\, \tilde{\Gamma} - r$; {\rm (iii)} For any ${\rm Re}({\bf s}) \in {{\bf R}}_{>0}^r$, the function $$ \phi(t,{\bf s}) = \sup_{0\leq {\rm Re}(z) \leq \delta ,\,{\rm Im}(z)=t }|W({\bf s}+ z\cdot \gamma)| $$ tends to $0$ as $|t| \rightarrow + \infty$. Then $\tilde{W}({\bf s})$ is a good function with respect to $\tilde{\Gamma}$, and $C(\tilde{W}) = 2 \pi i \cdot C(W)$. \label{desc3} \end{theo} \noindent {\em Proof.} Assume that $l_j( \gamma) < 0$ for $j=1, \ldots, p_1$, $l_j( \gamma) = 0$ for $j=p_1 +1, \ldots, p_2$, and $l_j( \gamma) > 0$ for $j=p_2 +1, \ldots, p$. In particular, one has \[ \Phi({\bf s}) = \prod_{j = p_1 + 1}^{p_2} l_j^{k_j}({\bf s}). \] Denote by $z_j$ the solution of the equation \[ l_j({\bf s}) + z l_j( \gamma) = 0,\;\;j =1, \ldots, p_1. \] Let $U$ be the intersection of ${{\bf R}}^r_{>0}$ with an open neighborhood of ${\bf 0}$ where $$ \Phi({\bf s}) \cdot \tilde{W}_{ \delta }({\bf s}) $$ is analytic. Then both functions $\tilde{W}_{ \delta }({\bf s})$ and $\tilde{W}({\bf s})$ are analytic in $U$. Moreover, the integral formulas for $\tilde{W}_{ \delta }({\bf s})$ and $\tilde{W}({\bf s})$ show that the equalities $\tilde{W}_{ \delta }({\bf u}+ iy \cdot \gamma) =\tilde{W}_{ \delta }({\bf u})$ and $\tilde{W}({\bf u}+ iy \cdot \gamma) =\tilde{W}({\bf u})$ hold for any $y \in {\bf R}$ and ${\bf u},{\bf u}+ iy \cdot \gamma \in U$. Therefore, both functions $\tilde{W}_{ \delta }({\bf s})$ and $\tilde{W}({\bf s})$ descend to $\tilde{V}_{\bf C}$ (see Remark \ref{desc}). Using assumptions (i)-(iii) of Theorem, we can apply the residue theorem and obtain \[ \tilde{W}({\bf s}) - \tilde{W}_{ \delta }({\bf s}) = 2 \pi i\cdot \sum_{j=1}^{p_1} {\rm Res}_{z = z_j} W({\bf s} + z \cdot \gamma)\] for ${\bf s} \in U$. We denote by $U( \gamma)$ the open subset of $U$ defined by the inequalities $$ \frac{l_j({\bf s})}{l_j( \gamma)} \neq \frac{l_{m}({\bf s})}{l_{m}( \gamma)}\;\; \mbox{\rm for all $j \neq m$, $\;\;j,m \in \{ 1, \ldots, p\}$.} $$ The open set $U( \gamma)$ is non-empty, since we assume that $g.c.d.(l_j, l_{m})=1$ for $j \neq m$. For ${\bf s} \in U( \gamma)$, we have $$ {\rm Res}_{z = z_j} W({\bf s} + z \cdot \gamma) = \frac{1}{(k_j-1)!} \left( \frac{\partial}{\partial z} \right)^{k_j-1} \frac{l_{j}({\bf s} + z \cdot \gamma)^{k_j} P({\bf s} + z \cdot \gamma)}{l_j^{k_j} ( \gamma) \cdot \prod_{m =1}^p l_{m}^{k_m}({\bf s} + z \cdot \gamma) }|_{z = z_j}, $$ where $$ z_j = - \frac{l_j({\bf s})}{l_j( \gamma)}. $$ Let $$ W({\bf s}) \cdot \prod_{j =1}^p l_j^{k_j}({\bf s}) = {P}({\bf s}) = {P}_0({\bf s}) + {P}_1({\bf s}), $$ where ${P}_0({\bf s})$ is a uniquely determined homogeneous polynomial and ${P}_0({\bf s})$ is an analytic at ${\bf 0}$ function such that $\mu({P}) = \mu({P}_0) < \mu({P}_1)$ and $$ \frac{P_0({\bf s})}{\prod_{j =1}^p l_j^{k_j}({\bf s})} = C(W) \cdot {\cal X}_{ \Lambda }({\bf s}) $$ (${\cal X}_{ \Lambda }({\bf s})$ is the ${\cal X}$-function of the cone $ \Lambda = \psi({{\bf R}}^r_{\geq 0})$). We set $$ R_0({\bf s}) : = \frac{P_0({\bf s})}{\prod_{j =1}^p {l}^{k_j}_j({\bf s})}, \;\; R_1({\bf s}) : = \frac{P_1({\bf s})}{\prod_{j =1}^p {l}_j^{k_j}({\bf s})}. $$ Then $\mu(W)= \mu (R_0) < \mu (R_1)$. Moreover, $\mu(W) = - {\rm dim} V_{{\bf R}} = r - {\rm rk}\, \Gamma$. Define $$ \tilde{R}_0({\bf s}):= 2\pi i \cdot \sum_{j=1}^{p_1} {\rm Res}_{z = z_j} R_0({\bf s}+ z\cdot \gamma) $$ and $$ \tilde{R}_1({\bf s}):= 2\pi i \cdot \sum_{j=1}^{p_1} {\rm Res}_{z = z_j} R_1({\bf s}+ z\cdot \gamma). $$ By Proposition \ref{mult4}, we have $\mu (\tilde{R}_1) \geq 1+ \mu(R_1) \geq 2 + \mu(R_0)= 1+ {\rm rk}\, \tilde{\Gamma} - r $. We claim $$ \tilde{R}_0({\bf s}) = 2 \pi i \cdot C(W) {\cal X}_{\tilde{ \Lambda }} (\tilde{\psi}({\bf s})) $$ in particular $\mu (\tilde{R}_0) = \mu(R_0) + 1 = {\rm rk}\, \tilde{\Gamma}$. Indeed, repeating for ${\cal X}_{ \Lambda }(\psi({\bf s}))$ the same arguments as for $W({\bf s})$ we obtain $$ \int_{{\rm Re}(z) = 0} {\cal X}_{ \Lambda }(\psi({\bf s} + z \cdot \gamma)) dz - \int_{{\rm Re}(z) = \delta } {\cal X}_{ \Lambda }(\psi({\bf s} + z \cdot \gamma)) dz $$ $$ = 2\pi i \cdot \sum_{j=1}^{k_1} {\rm Res}_{z = z_j} {\cal X}_{ \Lambda }(\psi({\bf s} + z_j \cdot \gamma)). $$ Moving the contour of integration ${\rm Re}(z) = \delta $, by residue theorem, we obtain $$ \int_{{\rm Re}(z) = \delta } {\cal X}_{ \Lambda }(\psi({\bf s} + z \cdot \gamma)) dz =0. $$ On the other hand, $$ {\cal X}_{\tilde{ \Lambda }}(\tilde{\psi}({\bf s})) = \frac{1}{2\pi i}\int_{{\rm Re}(z) = 0} {\cal X}_{ \Lambda }(\psi({\bf s} + z \cdot \gamma)) dz $$ (see Theorem \ref{char1}). By \ref{mult2}(iii), using the decomposition $$ \tilde{W}({\bf s}) = \tilde{W}_{ \delta }({\bf s}) + \tilde{R}_0({\bf s}) + \tilde{R}_1({\bf s}) $$ and our assumption $\mu(\tilde{W}_{ \delta }) \geq 1 + {\rm rk}\, \tilde{\Gamma} - r$, we obtain that $\mu (\tilde{W}) = \mu (\tilde{R}_0) = {\rm rk}\, \tilde{\Gamma} - r$. By \ref{desc2}, the linear forms \[ h_{m,j}({\bf s}):= l_{m}({\bf s} + z_j \cdot \gamma) = l_{m}({\bf s}) - \frac{l_j({\bf s})}{l_{j}( \gamma)} l_{m}( \gamma) \] and the analytic in the domain $U( \gamma)$ functions \[ {\rm Res}_{z = z_j} W({\bf s} + z \cdot \gamma), \;\; {\rm Res}_{z = z_j} R_0({\bf s} + z \cdot \gamma) \] descend to $\tilde{V}_{{\bf C}}$. For any $j \in \{ 1, \ldots, p_1\}$, let us denote $$ Q_j ({\bf s}) = \prod_{m \neq j, m=1}^p h_{m,j}^{k_m}({\bf s}). $$ It is clear that $$ Q_j^{k_j}({\bf s}) \cdot {\rm Res}_{z = z_j} W({\bf s} + z \cdot \gamma)\; \;\mbox{\rm and}\;\; Q_j^{k_j}({\bf s}) \cdot {\rm Res}_{z = z_j} R_0({\bf s} + z \cdot \gamma) $$ are analytic at ${\bf 0}$ and $\Phi({\bf s})$ divides each $Q_j ({\bf s})$. So we obtain that $$ \tilde{W}({\bf s}) \prod_{j=1}^{p_1} Q_j^{k_j}({\bf s}) = \left( \tilde{W}_{ \delta }({\bf s}) + 2 \pi i\cdot \sum_{j=1}^{p_1} {\rm Res}_{z = z_j} W({\bf s} + z \cdot \gamma) \right) \prod_{j=1}^{p_1} Q_j^{k_j}({\bf s}) $$ and $$ \tilde{R}_0({\bf s}) \prod_{j=1}^{p_1} Q_j^{k_j}({\bf s}) = \left( 2 \pi i\cdot \sum_{j=1}^{p_1} {\rm Res}_{z = z_j} R_0({\bf s} + z \cdot \gamma) \right) \prod_{j=1}^{p_1} Q_j^{k_j}({\bf s}) $$ are analytic at ${\bf 0}$. Define the set $\{ \tilde{l}_1({\bf s}), \ldots, \tilde{l}_q({\bf s}) \}$ as a subset of pairwise coprime elements in the set of homogeneous linear forms $\{ h_{m,j}({\bf s}) \}$ $(m \in \{1, \ldots, p\}, \; j \in \{1, \ldots, p_1\})$ such that there exist positive integers $n_1, \ldots, n_q$ and a representation of the meromorphic functions $\tilde{W}({\bf s})$ and $\tilde{R}_0({\bf s})$ as quotients \[ \tilde{W}({\bf s}) = \frac{\tilde{P}({\bf s})}{\prod_{j =1}^q \tilde{l}^{n_j}_j({\bf s})},\;\; \tilde{R}_0({\bf s}) = \frac{\tilde{P}_0({\bf s})}{\prod_{j =1}^q \tilde{l}^{n_j}_j({\bf s})},\] where $\tilde{P}({\bf s})$ is analytic at ${\bf 0}$, $\tilde{P}_0({\bf s})$ is a homogeneous polynomial and none of the forms $\tilde{l}_1({\bf s}), \ldots, \tilde{l}_q({\bf s})$ vanishes for $ {\bf s} \in {{\bf R}}_{>0}^r$ (the last property can be achieved, because both functions $\tilde{W}({\bf s})$ and $\tilde{R}_0({\bf s})$ are analytic in $U$). Define $$ \tilde{P}_1({\bf s}) = \left( \tilde{W}_{ \delta }({\bf s}) + \tilde{R}_1({\bf s}) \right) \cdot \prod_{j =1}^q \tilde{l}^{n_j}_j({\bf s}). $$ Then $$ \tilde{P}({\bf s}) = \tilde{P}_0({\bf s}) + \tilde{P}_1({\bf s}) $$ where $\tilde{P}_0({\bf s})$ is a homogeneous polynomial and $\tilde{P}_1({\bf s})$ is an analytic at ${\bf 0}$ function such that $\mu(\tilde{P}) = \mu(\tilde{P}_0) < \mu(\tilde{P}_1)$ and $$ \frac{\tilde{P}_0({\bf s})}{\prod_{j =1}^q \tilde{l}_j^{n_j}({\bf s})} = 2\pi i \cdot C(W) \cdot {\cal X}_{\tilde{ \Lambda }}(\tilde{\psi}({\bf s})). $$ \section{Main theorem} Let us set $$W_{ \Sigma }({\bf s}) := Z_{ \Sigma }(\varphi_{\bf s} + \varphi _{ \Sigma }) = Z_{ \Sigma }(s_1 +1, \ldots, s_r +1).$$ By Theorem \ref{convergence}, $W_{ \Sigma }({\bf s})$ is an analytic function in the domain ${\rm Re}({\bf s}) \in {\bf R}^r_{>0}$. \begin{theo} The analytic function $W_{ \Sigma }({\bf s})$ is good with respect to the lattice $M^G \subset PL( \Sigma )^G = {\bf Z}^r$. \label{analytic.cont} \end{theo} \noindent {\em Proof.} By Theorem \ref{poiss}, we have the following integral representation for $Z_{ \Sigma }({\bf s}) $ in the domain ${\rm Re}({\bf s}) \in {\bf R}^r_{>1}$ $$ Z_{\Sigma}({\bf s})=\frac{1}{(2\pi )^t b_S(T)} \int_{(T({\bf A}_K)/T(K))^*}\hat{H}_{ \Sigma }(\chi, -{\bf s})d\chi $$ We need only to consider characters $\chi$ which are trivial on the maximal compact subgroup ${\bf K}_T\subset T^1({\bf A}_K)$, because for all other characters the Fourier transform $\hat{H}_{ \Sigma }(\chi, -{\bf s})$ vanishes. Choosing a non-canonical splitting of characters corresponding to some splitting of the sequence $$ 0 \rightarrow T^1({\bf A}_K) \rightarrow T({\bf A}_K) \rightarrow T({\bf A}_K)/T^1({\bf A}_K) \rightarrow 0 $$ we obtain $$ Z_{\Sigma}({\bf s})=\frac{1}{(2\pi )^t b_S(T)} \int_{(T({\bf A}_K)/T^1({\bf A}_K))^*}d\chi_y \int_{(T^1({\bf A}_K)/T(K))^*}\hat{H}_{ \Sigma }(\chi, -{\bf s})d\chi_l $$ We have an isomorphism $M^G_{{\bf R}}\simeq(T({\bf A}_K)/T^1({\bf A}_K))^*$ and the measure $d\chi_y $ coincides with the usual Lebesgue measure on $M^G_{{\bf R}}$. Recall that a character $\chi\in (T({\bf A}_K)/T(K))^*$ defines $r$ Hecke characters $\chi_1,...,\chi_r$ of the idele groups ${\bf G}_m({\bf A}_{K_j})$. In particular, we get $r$ characters $\chi_{1,y},...,\chi_{r,y}$. We have an embedding $M^G\subset PL( \Sigma )^G$, which together with explicit formulas for Fourier transforms of local heights shows that the integral $$ A_{ \Sigma }({\bf s},\chi_y):=\frac{1}{b_S(T)} \int_{(T^1({\bf A}_K)/T(K))^*} \hat{H}_{ \Sigma }(\chi, -({\bf s}+{\bf 1}))d\chi_l $$ is a function on $PL( \Sigma )^G_{{\bf C}}$ and we have $$ A_{ \Sigma }({\bf s},\chi_y)=A_{ \Sigma }({\bf s}+i{\bf y})= A_{ \Sigma }(s_1+iy_1,...,s_r+iy_r). $$ Denote by $\Gamma:=M^G$ the lattice of $K$-rational characters of $T$. Let $t$ be the rank of $\Gamma$. The case $t =0$ corresponds to an anisotropic torus $T$. It has been considered already in \cite{BaTschi}. So we assume $t >0$. For any element $ \gamma \in \Gamma\subset {{\bf Z}}^r$ we denote by $l( \gamma)$ the number of its coordinates which are zero $(0 \leq l( \gamma) \leq r)$. Let $l(\Gamma)$ be the minimum of $l( \gamma)$ among $ \gamma \in \Gamma$. Notice that $l(\Gamma) \leq r - t - 1$. Indeed, if we had $l(\Gamma) \geq r -t$, then $M^G$ would be contained in the intersection of $r - t$ linear coordinate hyperplanes $s_j = 0$ (the latter contradicts the condition $M^G_{{\bf R}} \cap {\bf R}^r_{\geq 0} = 0$). We can always choose a ${{\bf Z}}$-basis $ \gamma^1, \ldots , \gamma^t$ of $\Gamma$ in such a way that $l(\Gamma) = l( \gamma^u)$ $(u =1, \ldots, t)$. Without loss of generality we assume that $\Gamma$ is contained in the intersection of coordinate hyperplanes $s_j = 0$, $j \in \{1, \ldots, l(\Gamma) \}$. We set $$\Phi({\bf s}) := \prod_{j =1}^{l(\Gamma)} s_j. $$ For any $u \leq t$ we define a subgroup $\Gamma^{(u)} \subset \Gamma$ of rank $u$ as $$\Gamma^{(u)}:= \bigoplus_{j =1}^u {{\bf Z}}< \gamma_j>.$$ We introduce some auxiliary functions $$ W^{(u)}_{ \Sigma }({\bf s}) = \int_{\Gamma^{(u)}_{{\bf R}}} A_{ \Sigma }({\bf s}+i{\bf y}^{(u)}){\bf dy}^{(u)} $$ where $ {\bf dy}^{(u)}$ is the induced measure on $\Gamma^{(u)}_{{\bf R}}\subset PL( \Sigma )^G_{{\bf R}}$. Denote $V^{(u)}_{{\bf C}} = {{\bf C}}^r/\Gamma_{{\bf C}}^{(u)}$. We prove by induction that $W^{(u)}_{ \Sigma }({\bf s})$ is good with respect to $\Gamma^{(u)} \subset {{\bf Z}}^r$. By \ref{infconver}, $W^{(u)}_{ \Sigma }({\bf s})$ is an analytic function in the domain ${\rm Re}({\bf s}) \in {{\bf R}}^r_{>0}$. There exist $ \delta _1,..., \delta _t > 0$ such that the integral $$ \int_{{\rm Re}(z) = \delta _u} \Phi({\bf s}) \cdot W_{ \Sigma }^{(u-1)}({\bf s} + z \cdot \gamma^u) dz $$ converges absolutely and uniformly in an open neighborhood of ${\bf 0}$. This can be seen as follows: For any $ \varepsilon $ with $0< \varepsilon <1/rd'$, where $d'=\dim M_{{\bf R},\infty}$, we can choose a ball $B_{e_1}\subset {\bf R}$ of radius $e_1$ around ${\bf 0}$ such that for any ball $B_{e_2}\subset B_{e_1} $ of radius $e_2$ ($0<e_2 <e_1$) around $0$ we can uniformly bound the Hecke $L$-functions $L_{K_j}(s_j+1,\chi_j)$ appearing in $\hat{H}_{ \Sigma }(\chi,{\bf s})$ by $$ c_j(e_2 )(\|y(\chi_j)\|+ |{\rm Im}(s_j)|+1)^{ \varepsilon } $$ with some constants $c_j(e_2 )$ for all ${\bf s}$ in the domain ${\rm Re}(s_j)\in B_{e_1}\backslash B_{e_2}$ for $j=1,...,r$ (see \ref{estim}). By \ref{infconver}, this assures the absolute and uniform convergence of the integral $$ \int_{\Gamma^{(u)}_{{\bf R}}} A_{ \Sigma }({\bf s}+i{\bf y}^{(u)}){\bf dy}^{(u)} $$ for all ${\bf s}$ contained in a compact in ${\bf C}^r$ such that ${\rm Re}(s_j)\in B_{ \varepsilon _1}\backslash B_{ \varepsilon _2}$ for $j=1,...r$. We know that the coordinates $ \gamma_j^u$ of the vectors $ \gamma^u=( \gamma_1^u,..., \gamma^u_r)\in {\bf Z}^r$ are not equal to zero for $l(\Gamma )<j \le r$. Therefore, we can now choose such real $ \delta _u>0$ that $ \delta _u \gamma^u_j$ are all contained in the {\em open} ball $B_{e_1}$. So there must exist some $e_2>0$ such that $ \delta _u \gamma^u_j\not\in B_{e_2}$ for all $u=1,...,t$ and all $l(\Gamma )<j\le r$. It follows that there exists an open neighborhood of ${\bf 0}$, such that for all ${\bf s}$ contained in this neighborhood we have ${\rm Re}(s_j+ \delta _u \gamma^u_j)\in B_{e_1}\backslash B_{e_2}$ for all $l(\Gamma )<j\le r$. Since we remove the remaining poles by multiplying with $\Phi({\bf s})$ we obtain the absolute and uniform convergence of $W^{(u)}_{ \Sigma }({\bf s})$ to a holomorphic function in ${\bf s}$ in this neighborhood. Moreover, the multiplicity of the meromorphic function $$ \tilde{W}_{ \delta _u}^{(u)}({\bf s}): = \int_{{\rm Re}(z) = \delta _u} W^{(u-1)}_{ \Sigma }({\bf s} + z \cdot \gamma_u) dz $$ at ${\bf 0}$ is at least $1 + {\rm rk}\, {\Gamma} - r \geq 1 + {\rm rk}\, {\Gamma}^{(u)} - r$. We apply Theorem \ref{desc3}. The property (iii) follows from estimates \ref{m.estim} and \ref{infconver}. This concludes the proof. \hfill $\Box$ \begin{theo} Denote by $\hat{H}_{ \Sigma ,S}(\chi,-{\bf s})$ the multiplicative Fourier transform of the height function with respect to the measure $\omega_{\Omega, S}$ (see \ref{can.meas}). The principal coefficient $C( \Sigma )$ of $$ A_{ \Sigma }({\bf s})=\frac{1}{b_S(T)} \int_{(T^1({\bf A}_K/T(K){\bf K}_T)^*} \hat{H}_{ \Sigma ,S}(\chi_l,-{\bf s})d\chi_l $$ at $s_1=...=s_r=1$ is equal to $\beta({\bf P}_{ \Sigma }) \tau_{\cal K}({\bf P}_{ \Sigma })$. \label{beta.tau} \end{theo} {\em Proof.} We follow closely the exposition of the proof of theorem 3.4.6 in \cite{BaTschi}. Since $M^G\hookrightarrow PL( \Sigma )^G$ we have an embedding of characters $$ (T({\bf A}_K)/T^1({\bf A}_K))^* = M^G_{{\bf R}}\hookrightarrow \prod_{j=1}^r ({\bf G}_m({\bf A}_{K_j})/{\bf G}_m^1({\bf A}_{K_j}))^*. $$ Recall that the kernel of $$ a^*\,:\, (T({\bf A}_K)/T({\bf A}_K))^* \rightarrow \prod_{j=1}^r ({\bf G}_m({\bf A}_{K_j})/{\bf G}_m({\bf A}_{K_j}))^* $$ is dual to the obstruction group to weak approximation $A(T)=T({\bf A}_K)/\overline{T(K)}$. We have a splitting $$ \overline{T(K)} = \overline{T(K)}_S \times T(A_{K,S}). $$ Here we denoted by $\overline{T(K)}_S$ the image of $\overline{T(K)}$ in $\prod_{v \in S} T(K_v)$ and $T(A_{K,S})=T({\bf A}_K)\cap \prod_{v\not\in S}T(K_v)$. The pole of the highest order $r$ of $\hat{H}_{ \Sigma ,S}(\chi_l,-{\bf s}) $ at $s_1=...=s_r=1$ appears from characters $\chi_l$ such that the corresponding $\chi_1,...,\chi_r$ are trivial characters of the groups ${\bf G}_m({\bf A}_{K_j})/{\bf G}_m({K_j})$, i.e., $\chi_l$ is a character of the finite group $A(T)=\prod_{v\in S}T(K_v)/\overline{T(K)}_S$, and is trivial on the group $T({\bf A}_{K,S})$. For ${\bf s}\in {\bf R}_{>1}^r$ we can again apply the Poisson formula to $A(T)$. By \ref{weak1}, the order of $A(T)$ equals $\beta({\bf P}_{ \Sigma })/i(T)$. We obtain $$ \frac{1}{b_S(T)} \sum_{\chi \in (A(T))^* } \hat{H}_{ \Sigma ,S}(\chi_l, - {\bf s})= \frac{\beta({\bf P}_{\Sigma})}{i(T) b_S(T)} \int_{\overline{T(K)}} H_{ \Sigma }(x,-{\bf s}) \omega_{\Omega,S} \] (see \ref{weak1}). We restrict to the line $s_1=...=s_r=s$ and we want to compute the limit $$ \lim_{s \rightarrow 1} (s-1)^r \int_{\overline{T(K)}} H_{ \Sigma }(x,-{\bf s}) \omega_{\Omega,S}. $$ We have \begin{equation} \int_{\overline{T(K)}} H_{ \Sigma }(x,-{\bf s}) \omega_{\Omega,S} = \label{const1} \end{equation} $$ = \int_{\overline{T(K)}_S} \prod_{v \in S} H_{ \Sigma ,v}(x_v,-{\bf s}) \omega_{\Omega,v} \cdot \prod_{v \not\in S} \int_{T(K_v)} H_{ \Sigma ,v}(x_v,-{\bf s}) d\mu_v $$ (recall that $\omega_{\Omega,v}=\prod_{v\in {\rm Val}(K)} d\mu_v$ and $d\mu_v = L_v(1,T;E/K) \omega_{\Omega,v}$ for all $v$ and $L_v(1,T;E/K) =1$ for $v \in S$). {}From our calculations of the Fourier transform of local height functions for $v \not\in S$ (\ref{loc-int}), we have \begin{equation} \prod_{v \not\in S} \int_{T(K_v)} H_{ \Sigma ,v}(x_v,-{\bf s}) d\mu_v = \label{const2} \end{equation} \[ = L_S(s, T;E/K) \cdot L_S(s, T_{NS}; E/K) \prod_{v \not\in S} Q_{ \Sigma }(q_v^{-s}, \ldots, q_v^{-s}). \] By \ref{p-function}, \[ \prod_{v \not\in S} Q_{ \Sigma }(q_v^{-s}, \ldots, q_v^{-s}) \] is an absolutely convergent Euler product for $s =1$. Moreover, the limits $$ \lim_{s \rightarrow 1} (s-1)^t L_S(s, T;E/K) $$ $$ \lim_{s \rightarrow 1} (s-1)^{(r-t)} L_S(s, T_{NS};E/K) $$ exist and equal the non-zero constants $l_S(T)$ and $l^{-1}_S({\bf P}_{ \Sigma })$ ($r=t+k$). By \ref{badreduction}, \[ \int_{\overline{T(K)}_S} \prod_{v \in S} H_{ \Sigma ,v}(x_v,-{\bf s}) \omega_{\Omega,v} \] is absolutely convergent for $s_1=,,,=s_r=1$. Using (\ref{const1}) and (\ref{const2}), we obtain: \begin{equation} \lim_{s \rightarrow 1} (s-1)^r \int_{\overline{T(K)}} H_{ \Sigma }(x,-{\bf s}) \omega_{\Omega,S} = \label{const3} \end{equation} \[ = \frac{l_S(T)}{l_S({\bf P}_{ \Sigma })} \int_{\overline{T(K)}_S} \prod_{v \in S} H_{ \Sigma ,v}(x_v,-{\bf s}) \omega_{\Omega,v} \cdot \prod_{v \not\in S} Q_{ \Sigma }(q_v^{-1}, \ldots, q_v^{-1}).\] Now recall (\ref{loc-int}), that for $v\not\in S$ we have $$ Q_{ \Sigma }(q_v^{-1}, \ldots, q_v^{-1})= \int_{T(K_v)}L_v^{-1}(1,T_{NS};E/K) H_{ \Sigma ,v}(x_v,-{\bf 1})\omega_{\Omega,v}. $$ It was proved in \cite{BaTschi} Proposition 3.4.4 that the restriction of the $v$-adic measure $\omega_{{\cal K},v}$ to $T(K_v) \subset {\bf P}_{\Sigma}(K_v)$ coincides with the measure \[ H_{ \Sigma ,v}(x, -{\bf 1}) \omega_{\Omega,v}. \] Here ${\cal K}$ is the canonical sheaf on the toric variety ${\bf P}_{\Sigma}$ metrized as above. We also have \begin{equation} \int_{\overline{T(K)}_S} \prod_{v \in S} H_{ \Sigma ,v}(x_v,-{\bf 1}) \omega_{\Omega,v} = \int_{\overline{T(K)}_S} \prod_{v \in S} \omega_{{\cal K},v}. \label{const6} \end{equation} Using the splitting $\overline{T(K)} = \overline{T(K)}_S \times T(A_{K,S})$ and multiplying the above equations we get $$ \int_{\overline{T(K)}} \omega_{{\cal K},S} = \int_{\overline{T(K)}_S} \prod_{v \in S} \omega_{{\cal K},v} \cdot \prod_{v \not\in S} \int_{{T(K_v)}} L_v^{-1}(1,T_{NS};E/K) \omega_{{\cal K},v}. $$ On the other hand, it was proved in \cite{BaTschi} Proposition 3.4.5 that we have \[ \int_{\overline{T(K)}} \omega_{{\cal K},S} = \int_{\overline{{\bf P}_{ \Sigma }(K)}} \omega_{{\cal K},S} = b_S({\bf P}_{ \Sigma }). \] Therefore, \[ b_S({\bf P}_{ \Sigma }) = \int_{\overline{T(K)}_S} \prod_{v \in S} H_{ \Sigma }(x,-\varphi_{\Sigma}) \omega_{\Omega,v} \cdot \prod_{v \not\in S} Q_{ \Sigma }(q_v^{-1}, \ldots, q_v^{-1}) .\] Collecting terms, we obtain $$ C( \Sigma )= \frac{\beta({\bf P}_{ \Sigma })}{i(T) b_S(T)} \cdot \frac{l_S(T)}{l_S({\bf P}_{ \Sigma })} \cdot b_S({\bf P}_{ \Sigma }). $$ By \ref{tamagawa1} and \ref{tamagawa}, we have the following equality $$ i(T) b_S(T) = h(T) l_S(T). $$ It remains to notice that we have an exact sequence of lattices $$ 0 \rightarrow M^G \rightarrow PL( \Sigma )^G \rightarrow {\rm Pic}({\bf P}_{ \Sigma }) \rightarrow H^1(G,M) \rightarrow 0 $$ and that the number $h(T)= |H^1(G,M)|$ appears in the integral formula for the ${\cal X}$-function of the cone ${ \Lambda }_{\rm eff}\subset {\rm Pic}({\bf P}_{ \Sigma })$. We apply Theorem \ref{char0} and obtain that $$ W_{ \Sigma }({\bf s})=\frac{1}{(2\pi)^t b_S(T)} \int_{M^G_{{\bf R}}}A_{ \Sigma }({\bf s}+i{\bf y}){\bf dy} $$ is good with respect to the lattice $M^G$ and that $$ C( \Sigma )= \beta( {\bf P}_{ \Sigma })\tau_{\cal K}({\bf P}_{ \Sigma }) $$ is the principal coefficient of $W_{ \Sigma }({\bf s})$ at ${\bf 0}$. \hfill $\Box $ \begin{theo} There exists a $\delta >0$ such that the height zeta-function $\zeta_{ \Sigma }(s)$ obtained by restiction of the zeta-function $Z_{ \Sigma }({\bf s})$ to the complex line $s_j = \varphi (e_j)=s$ for all $j=1,...,r$ has a representation of the form $$ \zeta_{ \Sigma }(s)= \frac{\Theta( \Sigma )}{(s-1)^k} + \frac{g(s)}{(s-1)^{k-1}} $$ with $k= r-t = {\rm rk}\, {\rm Pic}({\bf P}_{ \Sigma })$ and some holomorphic function $g(s)$ in the domain ${\rm Re}(s)>1-\delta$. Moreover, we have $$ \Theta(\Sigma) = \alpha({\bf P}_{\Sigma})\beta({\bf P}_{\Sigma}) \tau_{\cal K}({\bf P}_{\Sigma}). $$ \end{theo} {\it Proof.} Since $W_{ \Sigma }({\bf s})$ is good with respect to the lattice $M^G \subset {{\bf Z}}^r$, we have the following representation of $W_{ \Sigma }({\bf s})$ in a small open neighborhood of ${\bf 0}$: $$ W_{ \Sigma }({\bf s}) = \frac{ P({\bf s})}{\prod_{j =1}^p l_j^{k_j}({\bf s})} $$ where $P({\bf s}) = P_0({\bf s}) + P_1({\bf s})$, $\mu(P_1) > \mu(P_0)$ and $$ \frac{P_0({\bf s})}{\prod_{j =1}^p l_j^{k_j}({\bf s})} = \beta({\bf P}_{ \Sigma }) \tau_{\cal K}({\bf P}_{ \Sigma }) \cdot {\cal X}_{ \Lambda _{\rm eff}}(\psi({\bf s})), $$ where ${\cal X}_{ \Lambda _{\rm eff}}$ is the ${\cal X}$-function of the cone $ \Lambda _{\rm eff} = \psi({{\bf R}}^r_{\geq 0}) \subset {\rm Pic}({\bf P}_{ \Sigma })_{{\bf R}}$. If we restrict $$ \frac{P_0({\bf s})}{\prod_{j =1}^p l_j^{k_j}({\bf s})} $$ to the line $s_j = s - 1$ $(j =1, \ldots, r)$, then we get the meromorphic function $\Theta (s-1)^{-k}$ with $\Theta = \alpha({\bf P}_{\Sigma})\beta({\bf P}_{\Sigma}) \tau_{\cal K}({\bf P}_{\Sigma})$. Moreover, the order of the pole at $s =1$ of the restriction of $$ \frac{P_1({\bf s})}{\prod_{j =1}^p l_j^{k_j}({\bf s})} $$ to the line $s_j = s - 1$ $(j =1, \ldots, r)$ is less than $k$. Therefore, this restriction can be written as $g(s)/(s-1)^{k-1}$ for some analytic at $s =1 $ function $g(s)$. \begin{coro} {\rm Let $T$ be an algebraic torus and ${\bf P}_{\Sigma}$ its smooth projective compactification. Let $k$ be the rank of ${\rm Pic}({\bf P}_{\Sigma})$. Then the number of $K$-rational points $x \in T(K)$ having the anticanonical height $H_{{\cal K}^{-1}}(x) \leq B$ has the asymptotic \[ N(T,{\cal K}^{-1}, B) = \frac{\Theta(\Sigma)}{(k-1)!} \cdot B (\log B)^{k-1}(1+o(1)), \hskip 0,3cm B \rightarrow \infty.\] } \end{coro} \noindent {\em Proof.} We apply a Tauberian theorem to $\zeta_{ \Sigma }(s)$. \hfill $\Box$

9510

alg-geom/9510012

en

https://arxiv.org/abs/alg-geom/9510012

alg-geom/9510012

Selman Akbulut

Selman Akbulut (Michigan State University)

Lectures on Seiberg-Witten Invariants

AMSLaTeX, 26 pages with 1 figure

These are yet another lecture notes on Seiberg-Witten invariants, where no claim of originality is made, they contain a discussion of some related results from the recent literature.

alg-geom

\section{Introduction} Every compact oriented smooth $4$-manifold has a $Spin_{c}$ structure, i.e. the second Steifel-Whitney $w_{2}(X)\in H^{2}(X;\Z_{2})\;$ has an integral lifting. This is because: $w_{2}(X)$ can be represented by an imbedded surface $F\subset X$. If $F$ is orientable then clearly the homology class $[F]$ comes from an integral class; if not then it suffices to show the circle $S\subset F$ representing $w_{1}(F)$ is null homologous in $H_{1}(X;\Z)$, because the Bockstein $\delta[F]=[S]$ in the coefficient exact sequence: $$ ..\to H_{2}(X;\Z)\stackrel{\times 2}{\longrightarrow} H_{2}(X;\Z)\stackrel{\rho}{\longrightarrow} H_{2}(X;\Z_{2})\stackrel{\delta}{\to} H_{1}(X;\Z)\to..$$ where $\rho $ is the reduction map. Now if $\delta[F]\neq 0$, we can choose an imbedded oriented $3$-manifold $\Sigma \subset X$ representing the Poincare dual of $\delta[F]$, which is transverse to $F$. Then $T=F\cap \Sigma \subset F$ has a trivial normal bundle $\nu (T,F)$ since $$\nu (T,X) = \nu (T,F)\oplus \nu (T,\Sigma)$$ and the two other normal bundles in the above equality are trivial. This gives a contradiction, since in $F$ the $1$-manifold $T$ meets $S$ transversally at one point and $[S]=w_{1}(F)$ implies $\nu (T,F)$ must necessarily be nontrivial $\;\;\;\Box$. \vspace{.05in} \begin{eqnarray*}\mbox{Recal:}\;\;\;\;\;\;\; Spin(4)&=&SU (2)\times SU (2)\\ Spin_{c}(4)&=&(\;SU (2)\times SU(2)\times S^{1}\;)/\Z_{2} =(\;Spin(4)\times S^{1})/\Z_{2}\\ SO(4)&=&(\;SU (2)\times SU (2)\;)/\Z_{2}\\ U(2)&=&(\;SU (2)\times S^{1}\;)/\Z_{2}\end{eqnarray*} \noindent We have fibrations: $$S^{1}\longrightarrow Spin_{c}(4)\to SO(4)$$ $$\Z_{2}\longrightarrow Spin_{c}(4)\to SO(4)\times S^{1}$$ \vspace{.01in} \noindent We can also identify $Spin_{c}(4)=\{ (A,B)\in U_{2}\times U_{2}\; |\; det(A)=det(B)\;\}$ by $$(A,B)\leadsto (A.(det A)^{-1/2}\; I\;,\; B.(det B)^{-1/2}\; I\;,\; (det A)^{1/2} )$$ We also have $2$ fold cover $Spin_{c}(4)\to SO(4)\times S^{1}$ .The fibrations above extend to fibrations: $$S^{1}\to Spin_{c}(4)\to SO(4)\to K(\Z,2)\to BSpin_{c}(4)\to BSO(4)\to K(\Z,3)$$ The last map in the sequence is given by the Bokstein of the second Steifel-Whitney class $\delta (w_{2})$ which explains why lifting of $w_{2}$ to an integral class corresponds to a $Spin_{c}(4)$-structure. We also have the fibration: $$\Z_{2}\to Spin_{c}(4)\to SO(4)\times S^{1}\to K(\Z_{2},1)\to BSpin_{c}(4)\to BSO(4)\times BS^{1}\to K(\Z_{2},2) $$ The last map in this sequence is given by $\;w_{2}\times 1 + 1\times \rho(c_{1})\;$ which clearly vanishes exactly when $\delta (w_{2})=0$ . Finally we have the fibration: $$\Z_{2}\to Spin(4)\times S^{1}\to Spin_{c}(4) \to K(\Z_{2},1)\to BSpin(4)\times BS^{1}\to BSpin_{c}(4)\to K(\Z_{2},2) $$ \noindent The last map is given by $w_{2}$. This sequence says that locally a $\;Spin_{c}(4)$ bundle consists a pair of a $Spin(4)$ bundle and a complex line bundle. Also recall $\;\;H^{2}(X;\Z)=[X,K(\Z,2)]=[X,BS^{1}]= \{\mbox{complex line bundles on X}\}$ \vspace{.15in} \noindent {\bf Definition}: Let $ L\longrightarrow X$ be a complex line bundle over a smooth oriented $4$-manifold with $c_{1}(L)=w_{2}(TX)$ (i.e. $L$ is a characteristic line bundle). A $Spin_{c}(4)\;$ structure on $X$, corresponding $L$, is a principal $\;Spin_{c}(4)$-bundle $\;P\longrightarrow X\;$ such that the associated framed bundles of $TX$ and $L$ satisfy: $$P_{SO(4)}(TX)=P\times_{{\rho}_{0}} SO(4) $$ $$P_{S^{1}}(L)=P\times_{{\rho}_{1}} S^{1} $$ \noindent where $(\rho_{0}, \rho_{1}): Spin_{c}(4)\to SO(4)\times S^{1}$ are the obvious projections $$\begin{array}{ccccc} & & Spin(4)\times S^{1} & & \\ &&&&\\ & & \downarrow \pi & &\\ &&&&\\ SO(4) & \stackrel{\rho_{0}} {\longleftarrow } &Spin_{c}(4)& \stackrel{\rho_{1}} { \longrightarrow} & S^{1}\\ &&&&\\ &&&&\\ & \swarrow \rho_{+} && \rho_{-}\searrow & \\ &&&&\\ U(2) && \downarrow \tilde{\rho}_{+}\;\;\;\;\;\;\tilde{\rho}_{-}\downarrow & & U(2)\\ &&&&\\ & Ad \searrow&& \swarrow Ad& \\ &&&&\\ & &SO(3) & & \\ \end{array}$$ So $\;\tilde{\rho}_{\pm}= Ad\circ \rho _{\pm}\;$, also call $\;\bar{\rho}_{\pm}= \rho_{\pm}\circ \pi\;$. For $x\in \H = \R^{4}$ we have \begin{eqnarray*} \rho_{1}[\;q_{+},q_{-},\lambda \;]&=&\lambda^{2} \\ \rho_{0}[\;q_{+},q_{-},\lambda \;]&=&[\;q_{+},q_{-}\;]\;\;\;\;\;,\;\mbox{i.e.} \;\;\;x\longmapsto q_{+}xq_{-}^{-1}\\ \rho_{\pm}[\;q_{+},q_{-},\lambda \;]&=&[\;q_{\pm},\lambda \;] \;\;\;\;\;\;\;,\;\mbox{i.e.} \;\;\;x\longmapsto q_{\pm} x \lambda^{-1} \\ \tilde{\rho}_{\pm}[\;q_{+},q_{-},\lambda \;]&=& Ad\circ q_{\pm} \;\;\;\;\;\;\;,\; \mbox{i.e.}\;\;\;x\longmapsto q_{\pm}xq_{\pm}^{-1}\\ \bar{\rho}_{\pm}(\;q_{+},q_{-},\lambda \;) &=& \lambda q_{\pm} \end{eqnarray*} \noindent Apart from $TX$ and $L$, $Spin_{c}(4)$ bundle $P\to X$ induces a pair of $U(2)$ bundles: $$W^{\pm}=P\times _{\rho_{\pm}}\C^{2}\longrightarrow X$$ Let $\Lambda^{p}(X)=\Lambda^{p}T^{*}(X) $ be the bundle of exterior $p$ forms. If $X$ is a Riemanian manifold (i.e. with metric), we can construct the bundle of self(antiself)-dual 2-forms $\;\Lambda_{\pm}^{2}(X)$ which we abbreviate by $ \Lambda^{\pm}(X) \;$. We can identify $ \Lambda ^{2}(X) \;$ by the Lie algebra $so(4)$-bundle $$\Lambda ^{2}(X)=P(T^{*}X)\times _{ad}so(4)\;\;\;\;\;\;\mbox{by}\;\;\; \Sigma \;a_{ij}\;dx^{i}\wedge dx^{j}\;\longleftrightarrow \; (a_{ij})$$ where $ad:SO(4)\to so(4)\;$ is the adjoint representation. The adjoint action preserves the two summands of $so(4)=spin(4)=so(3)\times so(3)=\R^{3}\oplus \R^{3}$. By above identification it is easy to see that the $\pm 1$ eigenspaces $ \Lambda^{\pm}(X) \;$ of the star operator $*:\Lambda (X)\to \Lambda (X)$ corresponds to these two ${\R}^{3}$-bundles; this gives: $$\Lambda^{\pm}(X)= P\times _{\tilde{\rho}_{\pm}} \R^{3} $$ If the $Spin_{c}(4)$ bundle $ P\to X$ lifts to $Spin(4)$ bundle $ \bar{P}\to X$ (i.e. when $w_{2}(X)=0$), corresponding to the obvious projections $ p_{\pm}: Spin(4)\to SU(2) $, $p_{\pm}(q_{-},q_{+})=q_{\pm}$ we get a pair of $SU(2)$ bundles: $$V^{\pm}= P\times_{p_{\pm}} \C^{2} $$ Clearly since $\;x\longmapsto q_{\pm} x \lambda ^{-1}= q_{\pm} x \;(\lambda^{2})^{-1/2}\; $ in this case we have: $$\;W^{\pm}= V^{\pm}\otimes L^{-1/2}\;$$ \subsection {Action of $\;\Lambda^*(X) \;\;$on$\;\; W_\pm$ } From the definition of $Spin_{c}(4)$ structure above we see that $$T^{*}(X)=P\times \H/ (p,v)\sim (\tilde{p},q_{+}v\;q_{-}^{-1})\;\;,\;\mbox{where}\;\;\; \tilde{p}=p[\;q_{+},q_{-},\lambda\;]$$ We define left actions (Clifford multiplications), which is well defined by $$ T^{*}(X) \otimes W^{+}\longrightarrow W^{-}\;\;,\;\mbox{by}\;\;\;\; [\;p,v\;]\otimes[\;p,x\;]\longmapsto [\;p,-\bar{v}x\;]$$ $$T^{*}(X) \otimes W^{-} \longrightarrow W^{+}\;\;,\;\mbox{by}\;\;\;\; [\;p,v\;]\otimes[\;p,x\;] \longmapsto [\;p,vx\;]$$ From identifications, we can check the well definededness of these actions, e.g.: $$[p,v\;]\otimes [\;p,x\;]\sim [\;\tilde{p},q_{+}v\;q_{-}^{-1}\;] \otimes [\;\tilde{p}, q_{+}x\lambda^{-1}\;] \longmapsto [\;\tilde{p}, \; q_{-}(-\bar{v}x)\lambda^{-1}\;] \sim[\;p,-\bar{v}x\;]$$ \vspace{.05in} \noindent By dimension reason complexification of these representation give $$ \rho:T^{*}(X)_\C\stackrel{\cong }{\longrightarrow} Hom(W^{\pm}, W^{\mp})\equiv W^{\pm}\otimes W^{\mp}$$ \noindent We can put them together as a single representation (which we still call $\rho $) $$ \rho: T^{*}(X)\longrightarrow Hom(W^{+}\oplus W^{-})\;\;,\;\;\mbox{by} \;\;\;v\;\longmapsto \; \rho (v)=\left(\begin{array}{cc} 0& v \\ -\bar{v} & 0\end{array} \right) $$ We have $\;\rho (v)\circ\rho (v)=-|v|^{2}I\;$. By universal property of the Clifford algebra this representation extends to the Clifford algebra $C(X)=\Lambda^{*}(X)$ (exterior algebra) $$\begin{array}{ccc} \Lambda^{*}(X)& & \\ & & \\ \downarrow &\searrow & \\ & & \\ T^{*}(X) & \longrightarrow & Hom\;(W^{+}\oplus W^{-}) \end{array}$$ \noindent One can construct this extension without the aid of the universal property of the Clifford algebra, for example since $$\Lambda^{2}(X)=\left \{\; v_{1}\wedge v_{2}=\frac{1}{2}(v_{1}\otimes v_{2}-v_{2}\otimes v_{1})\;|\; v_1,v_{2}\in T^{*}(X)\;\right \}$$ The action of $T^{*}(X)$ on $W^{\pm}$ determines the action of $\;\Lambda^{2}(X)=\Lambda^{+}(X)\otimes\Lambda^{-}(X) $, and since $\;2Im\;(v_{2}\bar{v}_{1})=-v_{1}\bar{v}_{2} + v_{2}\bar{v}_{1} $ we have the action $\rho$ with property: $$\; \Lambda^{+}(X)\otimes W^{+} \longrightarrow W^{+}\;\;\;\;\mbox{to be}\;\;\; [\;p,v_{1}\wedge v_{2}\;]\otimes [\;p,x\;] \longrightarrow [\;p,Im\;(v_{2}\bar{v}_{1})x\;] $$ $$\rho: \Lambda^{+} \longrightarrow Hom(W^{+}, W^{+})$$ \begin{eqnarray}\rho (v_{1}\wedge v_{2})&=& \;\frac{1}{2}\;[\;\rho(v_{1}),\rho (v_{2})\;] \end{eqnarray} \vspace{.1in} Let us write the local descriptions of these representations: We first pick a local orthonormal basis $\;\{e^{1},e^{2},e^{3},e^{4}\}\;$ for $T^{*}(X)$, then we can take $$\;\{\;f_{1}=\frac{1}{2}( e^{1}\wedge e^{2} \pm e^{3}\wedge e^{4}),\; f_{2}=\frac{1}{2}( e^{1}\wedge e^{3} \pm e^{4}\wedge e^{2}),\; f_{3}=\frac{1}{2}( e^{1}\wedge e^{4} \pm e^{2}\wedge e^{3})\;\}$$ to be a basis for $\Lambda^{\pm}(X)$. After the local identification $T^{*}(X)=\H$ we can take $e^{1}=1,\;e^{2}=i,\;e^{3}=j,\; e^{4}=k$. Let us identify $W^{\pm}=\C^{2}=\{z+jw\;|\; z,w\in \C\;\}$, then the multiplication by $1,i,j,k$ (action on $\C^{2}$ as multiplication on left) induce the representations $\rho (e^{i})\;,\;i=1,2,3,4$. From this we see that $\Lambda^{+}(X)$ acts trivially on $W^{-}$; and the basis $f_{1},f_{2},f_{3}$ of $\Lambda^{+}(X)$ acts on $W^{+}$ as multiplication by $i,j,k$, respectively (these are called Pauli matrices). $$\begin{array}{cc} \rho(e^{1})=\left( \begin{array} {cccc} & &1 & 0\\ & & 0 & 1 \\-1 & 0 & & \\ 0 & -1 & & \end{array} \right) & \;\;\;\; \rho(e^{2})=\left( \begin{array}{cccc} &&i&0\\ &&0&-i \\i&0&&\\ 0&-i&& \end{array} \right) \\ &\\ &\\ \rho(e^{3})=\left( \begin{array}{cccc} &&0&-1\\ &&1&0 \\0&-1&&\\1&0&& \end{array} \right) &\;\;\; \rho(e^{4})=\left( \begin{array}{cccc} &&0&-i\\ &&-i&0 \\0&-i&&\\-i&0&& \end{array} \right) \\ \end{array}$$ \vspace{.15in} $$\begin{array}{ccc} \rho(f_1)=\left( \begin{array}{cc} i&0\\ 0&-i \end{array} \right)& \rho(f_2)=\left( \begin{array}{cc} 0&-1\\ 1&0 \end{array} \right)& \rho(f_3)=\left( \begin{array}{cc} 0&-i\\ -i&0 \end{array} \right) \\ \end{array}$$ \vspace{.15in} In particular we get an isomorphism $\Lambda^{+}(X)\longrightarrow su\;(W^{+})$ (traceless skew adjoint endemorphism of $W^{+}$); which after complexifying extends to an isomorphism $\rho:\Lambda^{+}(X)_\C\cong sl\;(W^{+})$ (traceless endemorphism of $W^{+}$) $$\begin{array}{ccc} \Lambda^{+}(X)&\stackrel{\cong}{\longrightarrow}& su\;(W^{+})\\ &&\\ \bigcap & & \bigcap\\ &&\\ \Lambda^{+}(X)_\C&\stackrel{\rho}{\longrightarrow} & sl\;(W^{+}) \end{array}$$ \vspace{.1in} Recall $\;Hom(W^{+}, W^{+})\cong W^{+}\otimes (W^{+})^{*}\;$; we identify the dual space $(W^{+})^{*}$ naturally with $\bar{W}^{+}$ (= $W^{+}$ with scalar multiplication $\;c.v=\bar {c}v$) by the pairing $$W^{+}\otimes \bar{W}{+}\longrightarrow \C $$ given by $\;z\otimes w\to z\bar{w}$. Usually $sl\;(W^{+})$ is denoted by $(W^{+}\otimes \bar{W}^{+})_0\;$ and the trace map gives the identification: $$W^{+}\otimes \bar{W}^{+}=(W^{+}\otimes \bar{W}^{+})_0\oplus \C= \Lambda^{+}(X)_\C\oplus \C$$ Let $\sigma :W^{+}\longrightarrow\Lambda^{+}(X)\; $ be the map $\;[\;p,x\;]\longmapsto [\;p,\;\frac{1}{2}(x i \;\bar{x})\;] $. By local identification as above $ W^{+}=\C^{2}$ and $\Lambda^{+}(X)=\R\oplus \C$, we see $\sigma$ corresponds to $$\;(z,w)\longmapsto i\;\left(\frac{|z|^{2}-|w|^{2}}{2}\right) -k\; Re(z\bar{w})+ j\;Im(z\bar{w}) = \left(\frac{|w|^{2}-|z|^{2}}{2}\right) + z\bar{w} $$ \noindent We identify this by the element $i\sigma(z,w)$ of $su\;(W^{+})$ (by Pauli matrices) where: \begin{eqnarray}\;(z,w)\; \longmapsto \sigma(z,w)\;=\left( \begin{array}{cc} (\;|z|^{2}-|w|^{2})/2& {z}\bar{w}\\ \bar{z}{w}& (\;|w|^{2}-|z|^{2})/2 \end{array} \right)\end{eqnarray} $\sigma$ is the projection of the diagonal elements of $W^{+}\otimes \bar{W}^{+}$ onto $(W^{+}\otimes \bar{W}^{+})_0$ \vspace{.05in} \noindent We can check: \begin{eqnarray} i\;\sigma(z,w)=\rho \;[\;\frac{|z|^{2}-|w|^{2}}{2}\; f_{1} + Im(z\bar{w}) \;f_{2} - Re(z\bar{w})\; f_{3}\;] \end{eqnarray} From these identifications we see: \begin{eqnarray} |\;\sigma(\psi) \;|^{2}&=&\frac{1}{4} |\;\psi \;|^{4} \\ <\;\sigma(\psi)\;\psi,\psi\;> &=& \frac{1}{2} |\;\psi \;|^{4}\\ <\rho(\omega)\;\psi\;,\;\psi>&=&2i\;<\rho (\omega)\;,i\; \sigma(\psi)> \end{eqnarray} Here the norm in $\;su(2)\;$ is induced by the inner product $<A,B>=\frac{1}{2}trace(AB)\;$. \\ By calling $\sigma(\psi, \psi)=\sigma (\psi)$ we extend the definition of $\sigma $ to $ W^{+}\otimes \bar{W}^{+} $ by $$<\rho(\omega)\;\psi\;,\;\varphi>=2i\;<\rho (\omega)\;,i\; \sigma(\psi, \varphi)> $$ \vspace{.02in} $$\begin{array}{cccc} \Lambda^{+}(X)\;\;\;\;\;=&su (W^{+})&\;\;\stackrel{i\;\sigma}{\longleftarrow}& W^{+}\\ &&&\\ &\bigcap & & \bigcap\\ &&&\\ (W^{+}\otimes \bar{W}^{+})_0\;\; =&sl\;(W^{+})&\stackrel{i\;\sigma}{\longleftarrow} & W^{+}\otimes \bar{W}^{+} \end{array}$$ \vspace{.15in} \noindent{\bf Remark}: A $Spin_{c}(4)$ structure can also be defined as a pair of $U(2)$ bundles: $$W^{\pm}\longrightarrow X\;\;\mbox{with}\;\;det(W^{+})=det(W^{-})\longrightarrow X \;\;\mbox{(a complex line bundle), }$$ $$\mbox{ and an action}\;\; c_{\pm}:T^{*}(X)\longrightarrow Hom(W^{\pm},W^{\mp})\;\; \mbox{with}\; \;c_{\pm}(v)c_{\mp}(v)=-|v|^{2}I $$ \vspace{.1in} \noindent The first definition clearly implies this, and conversely we can obtain the first definition by letting the principal $Spin_{c}(4)$ bundle to be: $$P=\{\;(p_{+},p_{-})\in P(W^{+})\times P(W^{-})\;|\; det(p_{+})=det(p_{-})\;\} $$ Clearly, $\;Spin_{c}(4)=\{ (A,B)\in U_{2}\times U_{2}\; |\; det(A)=det(B)\;\}$ acts on $P$ freely. \vspace{.15in} \noindent This definition generalizes and gives way to the following definition: \vspace{.1in} \noindent{\bf Definition}: A Dirac bundle $W\longrightarrow X$ is a Riemanian vector bundle with an action $ \rho:T^{*}(X)\longrightarrow Hom(W,W)\;$ satisfying $\;\rho (v)\circ \rho (v)=-|v|^{2}I $. $W$ is also equipped with a connection $D$ satisfying: $$<\rho (v)x,\rho (v)y>=<x,y>$$ $$D_{Y}(\rho (v)s)=\rho (\nabla_{X} v) s + \rho (v) D_{Y}(s)$$ \noindent where $\nabla $ is the Levi-Civita connection on $T^{*}(X)$, and $Y$ is a vector filed on $X$ \vspace{.1in} An example of a Dirac bundle is $\;W=W^{+}\oplus W^{-}\longrightarrow X\;$ and $\;D=d+d^{*}\;$ with $W^{+}=\oplus\Lambda^{2k}(X)\; $ and $W^{-}=\oplus\Lambda^{2k+1}(X)\; $ where $\rho (v)= v\wedge . + v \;\bot ..$ (exterior $+$ interior product with $v$). In this case $\rho : W^{\pm}\to W^{\mp} $. In the next section we will discuss the natural connections $D$ for $Spin_{c}$ structures $W^{\pm}$ . \section{ Dirac Operator} Let $\cal{A}(L)$ denote the space of connections on a $\;U(1)$ bundle $L\longrightarrow X$. Any $A\in \cal{A}(L)$ and the Levi-Civita connection $A_{0}$ on the tangent bundle coming from Riemanian metric of $X$ defines a product connection on $P_{SO(4)}\times P_{S^{1}}$. Since $Spin_{c}(4)$ is the two fold covering of $SO(4)\times S^{1}$, they have the same Lie algebras $spin_{c}(4)=so(4)\oplus i\;\R$. Hence we get a connection $\tilde{A}$ on the $Spin_{c}(4)$ principle bundle $P\longrightarrow X$. In particular the connection $\tilde{A}$ defines connections to all the associated bundles of P, giving back $A,\;A_{0}$ on $L, \;T(X)$ respectively, and two new connections $A^{\pm}$ on bundles $W^{\pm}$. We denote the corresponding covariant derivatives by $\nabla_{A}$. $$\nabla _{A} :\Gamma(W^{+})\to \Gamma(\;T^{*} X\otimes W^{+})$$ Composing this with the Clifford multiplication $\Gamma(\;T^{*} X\otimes W^{+} )\to \Gamma(W^{-}) $ gives the Dirac operator $$D\!\!\!\!/\, _{A} :\Gamma(W^{+})\to \Gamma(W^{-})$$ Locally, by choosing orthonormal tangent vector field $e=\{e_{i}\;\}_{i=1}^{4}$ and the dual basis of $1$-forms $\{e^{i}\;\}_{i=1}^{4}$ in a neighborhood $U$ of a point $x\in X$ we can write $$D\!\!\!\!/\, _{A}=\sum \rho (e^{i}) \nabla_{e_{i}}$$ where $\nabla_{e_{i}}: \Gamma(W^{+})\to \Gamma(W^{+})$ is the covariant derivative $\nabla _{A}$ along $e_{i}$. Also locally $W^{\pm}=V^{\pm}\otimes L^{1/2} $, hence by Leibnitz rule, the connection $A$ and the untwisted Dirac operator $$\partial\!\!\!/\, : \Gamma(V^{+})\to \Gamma(V^{-})$$ determines $D\!\!\!\!/\,_{A}$. Notice that as in $W^{\pm}$, forms $\Lambda^{*}(X)$ act on $V^{\pm}$. Now let $\omega=(\omega_{ij})$ be the Levi-Civita connection $1$-form, i.e. $so(4)$-valued ``equivariant" $1$-form on $P_{SO(4\;)}(X)$ and $\tilde{\omega } =(\tilde{\omega }_{ij})=e^{*}(\omega) $ be the pull-back $1$-form on $U$. Since $P_{SO(4\;)}(U)=P_{Spin(4\;)}(U)$ the orthonormal basis $e\in P_{SO(4)}(U)$ determines an orthonormal basis $\sigma=\{\sigma^{k}\}\in P_{SU_{2}}(V^{+})$, then (e.g. [{\bf LM}]) $$\partial\!\!\!/\,(\sigma^{k})=\frac{1}{2}\sum_{i<j} \rho( \tilde{\omega}_{ji})\;\rho( e^{i}) \rho ( e^{j})\;\sigma^{k} $$ Metrics on $T(X)$ and $L$ give metrics on $W^{\pm}$ and $T^{*}(X) \otimes W^{\pm} $, hence we can define the adjoint $\nabla _{A}^{*}:\Gamma(T^{*} X\otimes W^{-})\to \Gamma(W^{+})$. Similarly we can define $D\!\!\!\!/\, _{A} :\Gamma(W^{-})\to \Gamma(W^{+}) $ which turns out to be the adjoint of the previous $D\!\!\!\!/\,_{A}$ and makes the following commute (vertical maps are Clifford multiplications): $$\begin{array}{ccccc} \Gamma(W^{+})&\stackrel{ \nabla _{A}}{\longrightarrow}&\Gamma(\;T^{*} X \otimes W^{+})& \stackrel{ \nabla _{A}}{\longrightarrow}&\Gamma(\;T^{*} X \otimes T^{*} X\otimes W^{+})\\ \parallel & & \downarrow & &\downarrow \\ \Gamma(W^{+}) & \stackrel{ D\!\!\!\!/\, _{A}}{\longrightarrow} &\Gamma(W^{-})& \stackrel{ D\!\!\!\!/\, _{A}}{\longrightarrow}&\Gamma(W^{+})\\ \end{array}$$ \noindent Let $F_{A}\in \Lambda^{2}(X)$ be the curvature of the connection $A$ on $L$, and $F^{+}_{A}\in \Lambda^{+}(X)$ be the self dual part of this curvature, and $s$ be the scalar curvature of $X$. Weitzenbock formula says that: \begin{eqnarray} D\!\!\!\!/\,_{A}^{2}(\psi)&=& \nabla _{A}^{*}\nabla _{A}\psi +\frac{s}{4}\psi + \frac{1}{4}\rho(F_{A}^{+})\psi \end{eqnarray} To see this we we can assume $\nabla_{e_{i}}(e^{j})=0$ at the point $x$ \begin{eqnarray*} D\!\!\!\!/\,_{A}^{2}\psi&=& \sum \rho( e^{i}).\;\nabla_{e_{i}}\;[\;\sum \rho (e^{j}).\;\nabla_{e_{j}}\psi\;]\;\\ &=& \nabla^{*}_{A}\nabla_{A}\psi+\frac{1}{2}\sum_{i,j}\rho (e^{i})\;\rho (e^{j})\;(\nabla_{e_{i}}\nabla_{e_{j}} - \nabla_{e_{j}}\nabla_{e_{i}})\;\psi\\ &=& \nabla^{*}_{A}\nabla_{A}\psi+\frac{1}{2} \sum_{i,j}\rho (e^{i})\;\rho (e^{j})\; \Omega_{ij}^{A} \;\psi \end{eqnarray*} $\Omega_{ij}^{A}=R_{ij}+ \frac{1}{2} F_{ij}\;$ is curvature on $V^{+}\otimes L^{1/2}$, i.e. $R_{ij}$ is the Riemanian curvature and the imaginary valued $2$-form $F_{ij}$ is the curvature of $A$ for the line bundle $L$ (endemorphisims of $W^{+}$). So if $\psi=\sigma\otimes \alpha \in \Gamma(V^{+}\otimes L^{1/2})\;$, then \begin{eqnarray*}\frac{1}{2}\sum_{i,j}\rho (e^{i})\;\rho (e^{j})\; \Omega_{ij}^{A}\;(\sigma\otimes \alpha)&=&\frac{1}{2} (\sum \rho (e^{i})\;\rho (e^{j})\;R_{ij} \;\sigma )\otimes \alpha \\ && + \;\frac{1}{4}\sum \rho (e^{i})\;\rho (e^{j}) \sigma \otimes ( F_{ij} \alpha)\\ &&\\ &=& \frac{1}{8} \;\sum \rho (e^{i})\;\rho (e^{j})\; \rho (e^{k})\; \rho (e^{l})\;R_{ijkl} \;(\psi) \\ && + \;\frac{1}{4}\;\rho \;(\sum F_{ij} \; e^{i}\wedge e^{j}) \;(\psi ) \end{eqnarray*} The last identity follows from (1). It is a standard calculation that the first term is $s/4$ ( e.g.[{\bf LM}], pp. 161), and since $\Lambda ^{-} (X)$ act as zero on $W^{+}$, the second term can be replaced by $$\frac{1}{4}\;\rho(F_{A}^{+})\;\psi=\frac{1}{4}\;\rho \;(\sum F_{ij}^{+} \; e^{i}\wedge e^{j}) \;\psi $$ \subsection {A Special Calculation} In Section 4 we need some a special case (7). For this, suppose $$V^{+}=L^{1/2}\oplus L^{-1/2}$$ where $L^{1/2}\longrightarrow X$ is some complex line bundle with $L^{1/2}\otimes L^{1/2}=L$. Hence $W^{+}=(L^{1/2}\oplus L^{-1/2})\otimes L^{-1/2}=L^{-1}\oplus \C$. In this case there is a unique connection $\frac{1}{2}A_{0}$ in $L^{-1/2}\to X$ such that the induced Dirac operator $D_{A_{0}}$ on $W^{+}$ restricted to the trivial summand $\underline \C\to X$ is the exterior derivative $d$. This is because for $\sigma _{\pm}\in \Gamma (L^{\pm 1/2})$, the following determines $\nabla_{\frac{A_{0}}{2}}(\sigma_{-})$ : \begin{eqnarray*} \nabla_{A_{0}} (\sigma_{+}+0)\otimes \sigma_{-}&=& \partial\!\!\!/\, (\sigma_{+}+0 )\otimes \sigma_{-} + (\sigma_{+}+ 0 )\otimes \nabla_{\frac{A_{0}}{2}}(\sigma_{-}) \\ =\nabla_{A_{0}}(\sigma_{+}\otimes \sigma_{-})&=& d(\sigma_{+}\otimes \sigma_{-}) \end{eqnarray*} The following is essentially the Leibnitz formula for Laplacian applied to Weitzenbock formula (7) \vspace{.12in} \noindent{\bf Proposition}: Let $A, A_{0} \in \cal{A}(L^{-1}) $ and $i\;a=A-A_{0}$. Let $\nabla _{a}=d +i\;a $ be the covariant derivative of the trivial bundle $\underline{\C}\longrightarrow X$, and $\alpha :X\to \C$. Let $u_{0}$ be a section of $W^{+}=L^{-1}\oplus{\C}$ with a constant $\C $ component and $D\!\!\!\!/\,_{A_{0}}(u_{0})=0$ then: \begin{eqnarray} D\!\!\!\!/\,_{A}^{2}(\alpha u_{0})= (\nabla _{a}^{*}\nabla _{a} \alpha )u_{0} + \frac{1}{2} \rho (F_{a} )\;\alpha \;u_{0} -2<\nabla _{a}\alpha \;, \nabla_{A_{0}}(u_{0})> \end{eqnarray} Proof: By writing $\nabla_{A}=\nabla ^{A}$ for the sake of not cluttering notations, and abbreviating $\;\nabla_{e_{j}}=\nabla_{j}\;$ and $\;\nabla^{a}_{j}(\alpha)= \nabla_{j}(\alpha) + i\; a_{j} \alpha \;$, and leaving out summation signs for repeated indices (Einstein convention) we calculate: \begin{eqnarray} \nabla^{A}(\alpha u_{0})&=&\nabla^{A}(\alpha)u_{0}+\alpha \nabla^{A}(u_{0})\nonumber \\ &=& e^{j}\otimes\nabla_{j}(\alpha)u_{0}+ \alpha (\nabla^{A_{0}}(u_{0}) +i\; e^{j}\otimes a_{j}\;u_{0}) \nonumber \nonumber \\ &=& e^{j}\otimes(\nabla_{j}(\alpha) +i\; a_{j} \alpha\;) u_{0} + \alpha \nabla^{A_{0}}(u_{0}) \nonumber \\ D\!\!\!\!/\,_{A}(\alpha u_{0} )&=&\rho ( e^{j}) \;\nabla ^{a}_{j}(\alpha)\; u_{0} + \alpha \;D\!\!\!\!/\,_{A_{0}}(u_{0}) = \rho ( e^{j}) \;\nabla ^{a}_{j}(\alpha)\; u_{0} \end{eqnarray} \vspace{.1in} \noindent By abbreviating $\;\mu=\nabla ^{a}_{j}(\alpha)\; $ we calculate: \begin{eqnarray} \nabla^{A} ( \rho ( e^{j}) \;\mu\; u_{0} )&=& e^{k}\otimes \rho ( e^{j}) \;\nabla_{k}(\mu ) u_{0} + e^{k}\otimes\; \rho(e^{j})\;\mu \; (\nabla^{A_{0}}_{k}(u_{0}) +i\; a_{k}\;u_{0}) \nonumber \\ &=& e^{k}\otimes \rho(e^{j}) \;\nabla^{a}_{k}(\mu)\; u_{0}\; + e^{k}\otimes \rho(e^{j})\;\mu\;\nabla^{A_{0}}_{k}(u_{0}) \nonumber\\ D\!\!\!\!/\,_{A}( \rho ( e^{j}) \;\mu\; u_{0})&=& \rho (e^{k} )\rho(e^{j})\;\nabla^{a}_{k}(\mu)\; u_{0}\; + \rho (e^{k} )\rho(e^{j})\;\mu\;\nabla^{A_{0}}_{k}(u_{0}) \nonumber \\ & = & -\nabla^{a}_{j}(\mu)\; u_{0}\; + \frac{1}{2} \sum_{k,j} \rho (e^{k} )\rho(e^{j}) (\nabla^{a}_{k}(\mu)- \nabla^{a}_{j}(\mu) )u_{0} \nonumber\\ & & - \mu\;\nabla^{A_{0}}_{j}(u_{0}) - \mu\;\rho(e^{j}) \sum_{k\neq j} \rho (e^{k} ) \nabla^{A_{0}}_{k}(u_{0}) \end{eqnarray} Since $0=D\!\!\!\!/\,_{A_{0}}(u_{0})=\sum \rho (e^{k} ) \nabla^{A_{0}}_{k}(u_{0}) $ the last term of (3) is $-\mu\;\nabla^{A_{0}}_{j}(u_{0}) $. \vspace{.1in} \noindent By plugging $\mu=\nabla ^{a}_{j}(\alpha)\; $ in (10) and summing over $j$, from (2) we see \begin{eqnarray*} D\!\!\!\!/\,_{A}^{2}(\alpha u_{0})=-\nabla ^{a}_{j}\nabla ^{a}_{j}(\alpha) u_{0} +\frac{1}{2} \rho(\sum F^{a}_{k,j}\; e^{k}\wedge e^{j})\;\alpha \;u_{0}-2\sum \nabla ^{a}_{j}(\alpha) \nabla^{A_{0}}_{j}(u_{0})\;\;\;\;\Box \end{eqnarray*} \noindent{\bf Remark}: Notice that since $u_{0}$ has a constant $\C$ component and $\nabla_{A_{0}}$ restricts to the usual $d$ the $\C$ component, the term $<\nabla _{a}\alpha \;, \nabla_{A_{0}}(u_{0})>$ lies entirely in $L$ component of $W^{+}$ \section{Seiberg-Witten invariants} Let $X$ be a closed oriented Riemanian manifold, and $L\longrightarrow X$ a characteristic line bundle. Seiberg -Witten equations are defined for $(A,\psi)\in {\cal A}(L)\times \Gamma(W^{+})$, \begin{eqnarray} D\!\!\!\!/\,_{A}(\psi)&=&0\\ \rho(F_{A}^{+})&=&\sigma(\psi) \end{eqnarray} Gauge group $\;{\cal G}(L)=Map(X,S^{1})\;$ acts on $\;\tilde{\cal B}(L)={\cal A}(L)\times \Gamma(W^{+})\;$ as follows: for $\;s=e^{if}\in {\cal G}(L) $ $$s^{*}(A, \psi)= (s^{*} A, s^{-1}\psi)=(A+s^{-1}ds\;,\; s^{-1} \psi)= (A+i\; df \; ,\;s^{-1}\psi) $$ \noindent By locally writing $W^{\pm}= V^{\pm}\otimes L^{1/2} $, and $\psi= \varphi\otimes \lambda\in \Gamma ( V^{\pm} \otimes L^{1/2})$ and from: \begin{eqnarray*} D\!\!\!\!/\, _{s^{*}A}( \varphi\otimes \lambda)=\; \partial\!\!\!/\, (\varphi)\otimes \lambda + [\; \varphi\otimes D_{A}(\lambda) + i\;df\; ( \varphi \otimes \lambda)\;]\; \end{eqnarray*} we see that $D\!\!\!\!/\, _{s^{*}A}\; (s^{-1}\psi) =s^{-1} D\!\!\!\!/\,_{A} (\psi) $, and from definitions $$\rho(F_{s^{*}A}^{+})=s^{-1}\rho(F_{A}^{+})\;s = \rho(F_{A}^{+})=\sigma(\psi) =\sigma(s^{-1}\psi) $$ \noindent Hence the solution set $\;\tilde{\cal M}(L)\subset \tilde{\cal B}(L)\;$ of Seiberg-Witten equations is preserved by the action $\;(A,\psi)\longmapsto s^{*}( A,\psi )\;$ of ${\cal G }(L)$ on $\tilde{\cal M}(L)$. Define $$ {\cal M}(L)=\tilde{\cal M}(L)/{\cal G }(L) \;\;\subset\;\; {\cal B}(L)= \tilde{\cal B}(L)/{\cal G} (L)$$ We call a solution $(A,\psi)$ of (11) and (12) an irreducible solution if $\psi\neq 0 $. $\;{\cal G}(L)$ acts on the subset $\tilde{\cal M}^{*}(L)$ of the irreducible solutions freely, we denote $$ {\cal M}^{*}(L)=\tilde{\cal M}^{*}(L)/{\cal G }(L) $$ \vspace{.005in} Any solution $(A,\psi) $ of Seiberg -Witten equations satisfies the $C^{0}$ bound \begin{eqnarray} |\psi|^{2}\leq \mbox{max}(0,-2s) \end{eqnarray} where $s$ is the scalar curvature function of $X$. This follows by plugging (12) in the Weitzenbock formula (7). \begin{eqnarray} D\!\!\!\!/\,_{A}^{2}(\psi)&=& \nabla _{A}^{*}\nabla _{A}\psi +\frac{s}{4}\psi + \frac{1}{4}\sigma (\psi)\psi \end{eqnarray} Then at the points where where $|\psi|^{2}$ is maximum, we calculate \begin{eqnarray*} 0\leq \frac{1}{2}\Delta |\psi |^{2}&=& \frac{1}{2} d^{*}d<\;\psi,\psi\;> = \frac{1}{2} d^{*} (\; <\nabla_{A} \psi,\psi>+<\psi,\nabla_{A} \psi> \;) \\ &=& \frac{1}{2} d^{*} ( \;\bar{<\psi, \nabla _{A} \psi>}+ <\psi,\nabla_{A} \psi>\; )= d^{*} < \psi\;, \nabla_{A} \psi>_{\;\R}\\ &=& <\psi, \nabla_{A} ^{*}\nabla_{A} \psi>-|\nabla_{A} \psi|^{2} \;\leq \; <\psi, \nabla_{A} ^{*}\nabla _{A} \psi>\\ &\leq & - \; \frac{s}{4} |\; \psi \; |^{2} - \frac{1}{8} |\;\psi \; |^{4} \end{eqnarray*} The last step follows from (14), (11) and (5), and the last inequality gives (13) \vspace{.15in} \noindent{\bf Proposition 3.1} $\;{\cal M}(L)$ is compact \vspace{.12in} Proof: Given a sequence $[\;A_{n},\psi_{n}\;] \in {\cal M}(L) $ we claim that there is a convergent subsequence (which we will denote by the same index), i.e. there is a sequence of gauge transformations $g_{n}\in {\cal G}(L)$ such that $g_{n}^{*}(A_{n},\psi_{n})$ converges in $C^{\infty}$. Let $A_{0}$ be a base connection. By Hodge theory of the elliptic complex: $$\Omega^{0}(X)\stackrel {d^{0}}{\longrightarrow} \Omega^{1}(X) \stackrel{d^{+}}{\longrightarrow} \Omega^{2}_{+}(X) $$ $$A-A_{0}=h_{n}+a_{n}+b_{n}\in {\cal H}\oplus im (d^{+})^{*}\oplus im (d) $$ where $\cal{H}$ are the harmonic $1$-forms. After applying gauge transformation $g_{n}$ we can assume that $b_{n}=0$, i.e. if $b_{n}=i\;d f _{n}$ we can let $g_{n}=e^{if}$. Also $$h_{n}\in {\cal H}=H^{1}(X;\R)\;\;\;\mbox{and a component of}\;\;{\cal G}(L)\; \;\mbox{is}\;\; H^{1}(X;\Z)$$ Hence after a gauge transformation we can assume $ h_{n}\in H^{1}(X;\R)/ H^{1}(X;\Z)$ so $h_{n}$ has convergent subsequence. Consider the first order elliptic operator: $$D =d^{*} \oplus d^{+} : \Omega^{1}(X)_{L^{p}_{k}}\longrightarrow \Omega^{0}(X)_{L^{p}_{k-1}} \oplus \Omega^{2}_{+}(X)_{L^{p}_{k-1}}$$ \noindent The kernel of $D$ consists of harmonic $1$-forms, hence by Poincare inequality if $a$ is a $1$-form orthogonal to the harmonic forms, then for some constant $C$ $$ ||a ||_{L^{p}_{k}}\leq C ||D(a)||_{L^{p}_{k-1}} $$ \noindent Now $a_{n}=(d^{+})^{*} \alpha_{n}$ implies $d^{*}(a_{n})=0 $. Since $\alpha_{n}$ is orthogonal to harmonic forms, and by calling $A_{n}=A_{0}+a_{n}$ we see : $$ || a_{n}||_{L^{p}_{1}}\leq C\; ||D(a_{n}) ||_{L^{p}} \leq C || d^{+} a_{n} ||_{L^{p}}= C \;|| F_{A_{n}}^{+}-F_{A_{0}}^{+} ||_{L^{p}}$$ Here we use C for a generic constant. By (12), (4) and (13) there is a $C$ depending only on the scalar curvature $s$ with \begin{eqnarray}|| a_{n}||_{L^{p}_{1}}\leq C \end{eqnarray} By iterating this process we get $ || a_{n}||_{L^{p}_{k}}\leq C $ for all $k$ , hence $|| a_{n}||_{\infty}\leq C$. From the elliptic estimate and $D\!\!\!\!/\,_{A_{n}}(\psi_{n})=0$ : \begin{eqnarray} ||\psi_{n}||_{L^{p}_{1}}&\leq & C(\; ||D\!\!\!\!/\, _{A_{0}}\psi_{n}\;||_{L^{p}} + || \psi _{n} ||_{L^{p}})= C(\; ||a_{n}\psi_{n}\;||_{L^{p}}+|| \psi _{n} ||_{L^{p}}) \nonumber \\ ||\psi_{n}||_{L^{p}_{1}} &\leq & C(\; ||a_{n}||_{\infty} ||\psi _{n} ||_{L^{p}} + ||\psi _{n} ||_{L^{p}} ) \leq C \end{eqnarray} By repeating this (boothstrapping) process we get $||\psi_{n}||_{L^{p}_{k}}\leq C $, for all $k$, where C depends only on the scalar curvature $s$ and $A_{0}$. By Rallich theorem we get convergent subsequence of $\;(a_{n},\psi_{n})\;$ in ${L^{p}_{k-1}} $ norm for all $k$. So we get this convergence to be $C^{\infty}$ convergence. \hspace{2in}$\Box$ \vspace{.15in} It is not clear that the solution set of Seiberg-Witten equations is a smooth manifold. However we can perturb the Seiberg-Witten equations (11), (12) by any self dual $2$-form $\delta \in \Omega^{+}(X)$, in a gauge invariant way, to obtain a new set of equations whose solutions set is a smooth manifold: \begin{eqnarray} D\!\!\!\!/\,_{A}(\psi)&=&0\\ \rho(F_{A}^{+} + i\;\delta )&=&\sigma(\psi) \end{eqnarray} \vspace{.15in} Denote this solution space by $\tilde{\cal M}_{\delta}(L)$, and parametrized solution space by $$\tilde{\bf{\cal M}}=\bigcup_{\delta \in \Omega^{+}} \tilde{\cal M}_{\delta} (L)\times \{\;\delta\;\}\subset {\cal A}(L)\times \Gamma(W^{+}) \times \Omega ^{+}(X)$$ $${\cal M}_{\delta}(L) = \tilde{\cal M}_{\delta}(L)\;/{\cal G}(L) \;\;\subset\;\; {\bf{\cal M}} =\tilde{\bf{\cal M}}\;/{\cal G}(L) $$ \vspace{.08in} \noindent Let $\tilde{\cal M}_{\delta}(L)^{*}\subset\tilde{\bf{\cal M}}^{*}$ be the corresponding irreducible solutions, and also let $ {\cal M}_{\delta}(L)^{*} \subset {\bf{\cal M}}^{*}$ be their quotients by Gauge group. The following theorem says that for a generic choice of $\delta $ the set ${\cal M}_{\delta}(L)^{*}$ is a closed smooth manifold. \vspace{.15in} \noindent {\bf Proposition 3.2} $\;{\cal M}^{*}$ is a smooth manifold. Projection $\pi:{\cal M}^{*}\longrightarrow \Omega^{+}(X) $ is a proper surjection of Fredholm index: $$d(L)=\frac{1}{4}[\;c_{1}(L)^{2}-(2\chi +3\sigma)\;]$$ where $\chi$ and $\sigma$ are Euler characteristic and the signature of $X$. \vspace{.12in} Proof: The linearization of the map $\;(A,\psi,\delta)\longmapsto (\rho(F_{A}^{+} + i\;\delta )- \sigma(\psi) ,D\!\!\!\!/\,_{A}(\psi)\;) $ at $(A_{0},\psi_{0},\delta_{0})$ is given by: $$ P: \Omega^{1}(X)\oplus \Gamma(W^{+})\oplus \Omega^{+}(X)\longrightarrow su(W^{+})\oplus \Gamma(W^{-})$$ $$ P(a,\psi, \epsilon)=(\rho (d^{+}a + i\; \epsilon ) -2 \;\sigma(\psi,\psi_{0})\;,\; D\!\!\!\!/\,_{A_{0}}\psi +\rho(a)\psi_{0}) $$ To see that this is onto we pick $ (\kappa,\theta)\in su(W^{+})\oplus \Gamma(W^{-})$, by varying $\epsilon$ we can see that $(\kappa,0) $ is in the image of $P$. To see $(0,\theta)$ is in the image of $P$, we prove that if it is in the orthogonal complement to $\;image(P)\;$ then it is $(0,0)$; i.e. assume $$<D\!\!\!\!/\,_{A_{0}}\psi, \theta> +<\rho(a)\psi_{0}, \theta >=0$$ for all $a$ and $\psi$, then by choosing $\psi=0$ we see $<\rho(a)\psi_{0}, \theta >=0$ for all $a$ which implies $\theta=0$ \vspace{.05in} By implicit function theorem $\;\tilde {\cal M}\;$ is a smooth manifold, and by Sard's theorem $\;\tilde{\cal M}_{\delta}(L)\;$ are smooth manifolds, for generic choice of $\delta $'s. Hence their free quotients $\;{\bf{\cal M}}^{*\;}$ and $\;{\cal M}_{\delta}(L)^{*}\;$ are smooth manifolds. After taking ``gauge fixig" account, the dimension of $ {\cal M}_{\delta}(L)$ is given by the index of $\;P+ d^{*}$ (c.f. [{\bf DK}]). $\;P+ d^{*}$ is the compact perturbation of $$ S: \Omega^{1}(X)\oplus \Gamma(W^{+})\longrightarrow [\;\Omega^{0}(X)\oplus \Omega_{+}^{2}(X)\;]\oplus \Gamma(W^{-}) $$ $$S=\left( \begin{array}{cc} d^{*}\oplus d^{+} & 0 \\ 0 & D\!\!\!\!/\,_{A_{0}} \end{array} \right)$$ By Atiyah-Singer index theorem \begin{eqnarray} \mbox{ dim }{\cal M}_{\delta} (L)=\mbox{ind} (S)&=& \mbox{index} (d^{*}\oplus d^{+}) + \mbox{index} _{\R}\;D\!\!\!\!/\,_{A_{0}}\nonumber\\ &=& -\frac{1}{2} (\chi + \sigma )+\frac{1}{4}( c_{1}(L)^{2}-\sigma ) \nonumber \\ &=& \frac{1}{4}\;[\;c_{1}(L)^{2}-(2\chi + 3 \sigma )\;]\nonumber \\ &=& \frac{c_{1}(L)^{2}-\sigma }{4}\;-\;(1+b^{+}) \end{eqnarray} where $b^{+}$ is the dimension of positive define part $\;H^{2}_{+}\;$ of $\;H^{2}(X;\Z)$. Notice that when $b^{+}$ is odd this expression is even, since $L$ being a characteristic line bundle we have $c_{1}(L)^{2}=\sigma\; \mbox{mod}\; 8 $ \hspace{3in} $\Box$ \vspace{.15in} Now assume that $H^{1}(X)=0$, then ${\cal G}(L)=K(\Z,1)$. Than being a free quotient of a contractible space by ${\cal G}(L)$ we have $${\cal B}^{*}(L)=K(\Z,2)=\C\P^{\infty}$$ The orientation of $H^{2}_{+}$ gives an orientation to ${\cal M}_{\delta}(L)$. Now By (19) if $ b^{+} $ is odd ${\cal M}_{\delta}(L)\subset{\cal B}^{*}(L) $ is an even dimensional $2d$ smooth closed oriented submanifold, then we can define Seiberg-Witten invariants as: $$SW_{L}(X)=<{\cal M}_{\delta}(L)\;,\; [\;\C\P^{d}\;]> $$ As in the case of Donaldson invariants ([{\bf DK}]), even though ${\cal M}_{\delta}(L)$ depends on metric (and on the perturbation $\delta$) the invariant $SW_{L}(X)$ is independent of these choices, provided $\;b^{+}\ge 2\;$, i.e. there is a generic metric theorem. Also by (13) if $X$ has nonnegative scalar curvature then all the solutions are reducible, i.e. $\psi=0$. This implies that $A$ is anti-self-dual, i.e. $F_{A}^{+}=0$; but just as in [{\bf DK}] , If $b^{+}\geq 2$ and $L$ nontrivial, for a generic metric $L$ can not admit such connections. Hence $\tilde{\cal M}=\emptyset$ which implies $SW_{L}(X)=0$. Similar to Donaldson invariants there is a ``connected sum theorem" for Seiberg-Witten invariants: If $X_{i}\;i=1,2\;$ are oriented compact smooth manifolds with common boundary, which is a $3$-manifold with a positive scalar curvature; then gluing these manifolds together along their boundaries produces a manifold $X=X_{1}\smile X_{2}$ with vanishing Seiberg-Witten invariants (cf [{\bf F}],[{\bf FS}]). There is also conjecture that only $0$-dimensional moduli spaces ${\cal M}_{\delta}(L)$ give nonzero invariants $SW_{L}(X)$. \section{Almost Complex and Symplectic Structures} Now assume that $X$ has an almost complex structure. This means that there is a principal $GL(2,\C)$-bundle $Q\longrightarrow X$ such that $$T(X)\cong Q\times_{GL(2,\C)} \C^{2}$$ By choosing Hermitian metric on $T(X)$ we can assume $Q\longrightarrow X$ is a $U(2)$ bundle, and the tangent frame bundle $P_{SO(4)}(TX)$ comes from $Q$ by the reduction map $$U(2)=(\;S^{1}\times SU (2) \;)/\Z_{2}\hookrightarrow (\;SU (2)\times SU (2)\;)/\Z_{2} =SO(4)$$ Equivalently there is an endemorphism $I\in \Gamma(End(TX))$ with $I^{2}=-Id$ $$\begin{array}{ccc} T(X)& \stackrel{I}{\longrightarrow} & T(X)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\searrow & & \swarrow\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ & X & \end{array}$$ The $\;\pm i \;$ eigenspaces of $I$ splits the complexified tangent space $T(X)_{\C}$ $$T(X)_{\C}\cong T^{1,0}(X)\oplus T^{0,1}(X)=\Lambda^{1,0}(X)\oplus \Lambda^{0,1}(X)$$ This gives us a complex line bundle which is called the canonical line bundle: $$K=K_{X}=\Lambda^{2,0}(X)=\Lambda^{2}(T^{1,0})\longrightarrow X$$ Both $K^{\pm}$ are characteristic; corresponding to line bundle $K \longrightarrow X$ there is a canonical $Spin_{c}(4)$ structure on $X$, given by the lifting of $f[\lambda , A ]=([\lambda ,A], \lambda^{2})$ $$\begin{array}{ccc} && Spin_{c}(4)\\ & & \\ & F \nearrow & l\downarrow\\ & & \\ U(2) &\stackrel{f}{\longrightarrow} & SO(4)\times S^{1} \end{array}$$ $\;F[\lambda ,A]=[\lambda , A,\lambda]$. Transition function $\lambda^{2}$ gives $K$, and the corresponding ${\C}^{2}$-bundles are given by: \begin{eqnarray*} W^{+}&= & \Lambda ^{0,2}(X)\oplus\Lambda^{0,0}(X) = K^{-1}\oplus{\C}\\ W^{-}&=&\Lambda^{0,1}(X) \end{eqnarray*} We can check this from the transition functions, e.g. for $W^{+}$, $x=z+jw\in{\H}$ $$x\longmapsto \lambda x \lambda^{-1} =\lambda (z+jw) \bar{\lambda } = z + jw \bar{\lambda} \bar{ \lambda} = z +jw \lambda ^{-2}$$ Since we can identify $\bar{\Lambda}^{0,1}(X)\cong \Lambda^{1,0}(X)$, and $\Lambda^{0,2}(X)\otimes \Lambda^{1,0}(X) \cong \Lambda^{0,1}(X)$ we readily see the decomposition $T(X)_{\C}\cong W^{+}\otimes \bar{W}^{-}$. As real bundles we have $$ \Lambda^{+}(X)\cong K\oplus {\R}$$ \noindent We can verify this by taking $\;\{e^{1},e^{2}=I(e^{1}),e^{3},e^{4}=I(e^{3} )\}\;$ to be a local orthonormal basis for $T^{*}(X)$, then $$\Lambda^{1,0}(X)=\;<e^{1}-ie^{2},e^{3}-ie^{4}>\;\;\;,and\;\;\;\; \Lambda^{0,1}(X)=\;<e^{1}+ie^{2},e^{3}+ie^{4}>$$ $$K =\;< f =(e^{1}-ie^{2})\wedge (e^{3}-ie^{4})>$$ $$\Lambda^{+}(X)=\;<\omega=\frac{1}{2}( e^{1}\wedge e^{2} + e^{3}\wedge e^{4}),\;f_{2}= \frac{1}{2}( e^{1}\wedge e^{3} + e^{4}\wedge e^{2}),\; f_{3}= \frac{1}{2}( e^{1}\wedge e^{4} + e^{2}\wedge e^{3})> $$ $\omega $ is the global form $ \omega(X,Y)=g(X,IY) $ where $ g $ is the hermitian metric (which makes the basis $\{e^{1},e^{2},e^{3},e^{4}\} $ orthogonal). Also since $f= 2(f_{2} -i f_{3}) $, we see as ${\R}^{3}$-bundles $\Lambda^{+}(X)\cong K\oplus {\R}(\omega)$. We can check: $$W^{+}\otimes \bar{W}^{+}\cong{\C}\oplus{\C}\oplus K\oplus \bar{K}= (K\oplus{\R})_{\C}\oplus {\C}$$ As before by writing the sections of $W^{+}$ by $z+jw \in \Gamma ({\C}\oplus K^{-1})$ we see that $\omega, f_{2},f_{3}$ act as Pauli matrices; in particular \begin{eqnarray*} \omega & \longmapsto & {\left(\begin{array}{cc} i &0 \\ 0 &- i \end{array}\right)}\\ f &\longmapsto &{2\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)- 2i\left(\begin{array}{cc} 0 & -i \\- i & 0 \end{array}\right)= \left(\begin{array}{cc} 0 & -4 \\ 0 & 0 \end{array}\right)}\\ \bar{f} &\longmapsto &{2\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)+ 2i\left(\begin{array}{cc} 0 &- i \\ -i & 0 \end{array}\right)= \left(\begin{array}{cc} 0 & 0 \\ 4 & 0 \end{array}\right)}\\ \end{eqnarray*} So in particular, if we write $\psi\in \Gamma (W^{+})=\Gamma({\C}\oplus K^{-1})$ by $\psi=\alpha u_{0} + {\beta} \;$, where $\beta$ is a section of $K^{-1}$, and $\alpha: X\to {\C} $ and $u_{0}$ is a fixed section of the trivial bundle $\underline{\C}\to X$ with $||u_{0}||=1$, then $$\rho(\omega )\; u_{0}= iu_{0} \;\;\;\; \rho(\omega )\;\beta=-i\beta\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$ $$\rho(\beta)\; u_{0}=4\beta\;\;\;\; \rho(\beta )\;\beta=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(*)$$ $$\;\;\;\;\;\;\rho(\bar{\beta})\; u_{0}=0 \;\;\;\;\;\;\;\;\;\rho(\bar{\beta} ) \;\beta =-4\;|\beta |^{2}u_{0} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$ \noindent We see these by locally writing $\psi$ in terms of basis $\;\psi = \alpha u_{0} + \lambda \bar{f} $, where $\beta= \lambda \bar{f}$ with $\;||\bar{f}||=1 $. Writing Formula (3) in terms of the basis $\{ \omega, f, \bar{f}\}$ we get: \begin{eqnarray}i\; \sigma ( \alpha, \lambda)& =& \rho\;[\;\frac{|\alpha |^{2}-|\lambda |^{2}}{2}\;\omega - \frac{i}{4}\;\alpha \bar {\lambda} f +\frac{i}{4} \;\bar{\alpha}{\lambda} \bar{f}\;] \nonumber\\ \sigma (\psi)&=& \rho\;[\;\frac{|\beta |^{2}-|\alpha |^{2}}{2}\;i\; \omega - \frac{1}{4}\;\alpha \; \bar {\beta} +\frac{1}{4} \;\bar{\alpha}\; \beta \;] \end{eqnarray} If we consider the decomposition $\;F_{A}^{+}=F^{2,0}_{A} +F^{0,2}_{A} +F^{1,1}_{A}\; $ the equation $\rho(F_{A})=\sigma(\psi)$ gives Witten's formulas: \begin{eqnarray} F^{2,0}_{A}&=& -\frac{1}{4}\alpha \;\bar{\beta}\\ F^{0,2}_{A}&=& \;\frac{1}{4}\bar{\alpha} \;\beta\\ F^{1,1}_{A}&=& \frac{|\beta |^{2}-|\alpha |^{2}}{2}\;i\;\omega \end{eqnarray} \vspace{.18in} In case $\;X\;$ is a Kahler surface the Dirac operator is given by (c.f.[{\bf LM}]) $$D\!\!\!\!/\,_{A}=\bar{\partial\!\!\!/\,}^{*}_{A} + \bar{\partial\!\!\!/\,}_{A} :\Gamma (W^{+})\to \Gamma (W^{-})$$ Hence from the Dirac part of the Seiberg-Witten equation (17) we have \begin{eqnarray} \bar{\partial\!\!\!/\,}_{A}^{*}(\beta) + \bar{\partial\!\!\!/\,}_{A}(\alpha u_{0})&=&0 \nonumber \\ \bar{\partial\!\!\!/\,}_{A} \bar{\partial\!\!\!/\,}^{*}_{A}(\beta) + \bar{\partial\!\!\!/\,}_{A}\bar{\partial\!\!\!/\,}_{A}(\alpha u_{0})&=&0 \end{eqnarray} \noindent The second term is $\; \bar{\partial\!\!\!/\,}_{A}\bar{\partial\!\!\!/\,}_{A}(\alpha u_{0})= F^{0,2}_{A} \alpha u_{0}= \frac{1}{4} |\alpha|^{2}\beta $. By taking inner product both sides of (24) by $\;\beta \;$ and integrating over $X$ we get the $L^{2}$ norms satisfy \begin{eqnarray}||\alpha||^{2} ||\beta||^{2}=0\;\;& \Longrightarrow \;\;\alpha= 0\; \mbox{ or }\;\beta=0 \end{eqnarray} This argument eventually calculates $\;SW_{K}(X)=1\;$ ([{\bf W}]). We will not repeat this argument here, instead we will review a stronger result of C.Taubes for symplectic manifolds below, which implies this result. \vspace{.18in} We call an almost complex manifold with Hermitian metric $\{X, I,g\}$ syplectic if $d\omega=0$. Clearly a nondegenerate closed form $\omega$ and a hermitian metric determines the almost complex structure $I$. Given $\omega$ then $I$ is called an almost complex structure taming the symplectic form $\omega$ \vspace{.1in} By Section 2.1 there is a unique connection $A_{0}$ in $\; K\longrightarrow X \;$ such that the induced Dirac operator $D_{A_{0}}$ on $W^{+}$ restricted to the trivial summand $\underline {\C}\to X$ is the exterior derivative $d$. Let $u_{0}$ be the section of $W^{+}$ with constant ${\C}$ component and $||u_{0}||=1$. Taubs's first fundamental observation is $$D\!\!\!\!/\,_{A} (u_{0})=0 \;\;\;\; \mbox{ if and only if}\;\;\;\;\; d\omega=0 $$ This can be seen by applying the Dirac operator to both sides of $ i u_{0} = \rho(\omega). u_{0}$, and observing that by the choice of $u_{0}$ the term $\nabla_{A_{0}}(u_{0})$ lies entirely in $K^{-1}$ component: \begin{eqnarray*} i D\!\!\!\!/\, _{A_{0}}(u_{0})&=&\sum \rho(e^{i})\nabla_{i}\;(\rho (\omega )\;u_{0}) \\ &=&\sum \rho (e^{i}) \;[\ \nabla_{i}\;(\rho(\omega) )\;u_{0} + \rho(\omega)\;\nabla_{i}\;(u_{0})\;]\\ &=& \sum \rho (e^{i}) \nabla_{i}\;(\rho(\omega) ) \;u_{0} -i\;\sum\rho (e^{i}\;)\nabla _{i}\;(u_{0})\\ 2i\;D\!\!\!\!/\, _{A_{0}}(u_{0}) &=& \sum \rho (e^{i}) \nabla_{i}\;(\rho(\omega) ) \;u_{0}= \rho ( (d +d^{*}) \omega)\;u_{0}=\rho ((d-*d)\omega)\;u_{0} \end{eqnarray*} Last equality holds since $\omega \in \Lambda^{+}(X)_{\C}\oplus {\C}$, and by naturality, the Dirac operator on $ \Lambda^{*}(X)_{\C} $ is $\;d+d^{*}$, and since $d=-*d*$ on $2$ forms and $\omega$ is self dual $$2i\;D\!\!\!\!/\, _{A_{0}}(u_{0})=-\rho (*d \omega ) u_{0}$$ \vspace{.15in} \noindent{\bf Theorem (Taubes) }: Let $(X,\omega)$ be a closed symplectic manifold such that $b_{2}(X)^{+}\geq2$, then $SW_{K}(X)=\pm1 $. \vspace{.1in} Proof: $\mbox{Write}\;\;\;\psi=\alpha u_{0} + {\beta} \in \Gamma( W^{+})= \Gamma ({\C}\oplus K^{-1}) $ where $\alpha: X\to {\C} $, and $u_{0}$ is the section as above. Consider the perturbed Seiberg-Witten equations : For $(A,\psi)\in {\cal A}(L)\times \Gamma (W^{+})$ : \begin{eqnarray} D\!\!\!\!/\, _{A}(\psi)&=&0\\ \rho (F_{A}^{+})&=&\rho (F_{A_{0}}^{+})+r\;[\; \sigma(\psi)+i\;\rho(\omega)\;] \end{eqnarray} By (20) the second equation is equivalent to: \begin{eqnarray} F_{A}^{+}-F_{A_{0}}^{+} &=&r\;\ [\; ( \;\frac{|\beta |^{2}-|\alpha|^{2}}{2} +1 )\; i \omega - \frac{1}{4}\alpha \bar{\beta} +\frac{1}{4} \bar{\alpha} \beta \; ] \end{eqnarray} We will show that up to gauge equivalence there is a unique solution to these equations. Write $A=A_{0}+a$, after a gauge transformation we can assume that $a$ is coclosed, i.e. $d^{*}(a)=0$. Clearly $ (A,\psi)=(A_{0},u_{0})$, and $r=0$ satisfy these equations. It suffices to show that for $r\longmapsto \infty $ these equations admit only $(A_{0},u_{0})$ as a solution. From Weitzenbock formulas (7), (8) and abbreviating $\nabla_{A_{0}}(u_{0})=b$ we get \begin{eqnarray} D\!\!\!\!/\,_{A}^{2}(\psi)= D\!\!\!\!/\,_{A}^{2}(\beta)+(\nabla _{a}^{*}\nabla_{a}\alpha)u_{0} -2<\nabla_{a} \alpha,b> + \frac{1}{2}\alpha\; \rho(F_{A}^{+}-F_{A_{0}}^{+})\;u_{0}\\ D\!\!\!\!/\,_{A}^{2}(\beta) =(\nabla _{A}^{*}\nabla_{A}\;\beta) +\frac{s}{4}\;\beta +\frac{1}{4} \rho ( F_{A_{0}}^{+})\;\beta +\frac{1}{4} \rho (F_{A}^{+}-F_{A_{0}}^{+}) \;\beta \hspace{.3in} \end{eqnarray} \noindent From (28) and (*) we see that \begin{eqnarray}\frac{1}{2}\alpha \;\rho(F_{A}^{+}-F_{A_{0}}^{+})u_{0}&=& \frac {r}{4}\alpha \;(\;|\alpha|^{2}-|\beta|^{2}-2)\;u_{0} + \frac{r}{2} \;|\alpha |^{2}\beta \\ \frac{1}{4}\;\rho(F_{A}^{+}-F_{A_{0}}^{+})\;\beta &= & - \;\frac{r}{8}\;(\;|\alpha|^{2}-|\beta|^{2}-2) \;\beta +\frac{r}{4}\alpha |\beta|^{2} u_{0} \end{eqnarray} \noindent By substituting (31) in (29) we get \begin{eqnarray}D\!\!\!\!/\,_{A}^{2}(\psi -\beta)&=& [\;\nabla _{a}^{*}\nabla_{a}\alpha + \frac{r}{4}\alpha \;(|\alpha|^{2}-|\beta|^{2}-2)\;]\;u_{0}\nonumber\\ && -2 <\nabla_{a} \alpha,b> +\frac{r}{2}\; |\alpha\;|^{2} \beta \end{eqnarray} \noindent By substituting (32) in (30), then substituting (30) in (33) we obtain: \begin{eqnarray}0=D\!\!\!\!/\,_{A}^{2}(\psi)&=& [\;\nabla _{a}^{*}\nabla_{a}\alpha + \frac{r}{4}\alpha \;(|\alpha|^{2}-2)\;] \;u_{0}-2 <\nabla_{a} \alpha,b> \nonumber \\ & & +[\;\nabla _{A}^{*}\nabla_{A}+\frac{s}{4} + \frac{1}{4}\;\rho(F_{A_{0}}^{+})+\frac{r}{8} (\;3|\alpha|^{2}+|\beta |^{2}+2)\;]\;\beta \end{eqnarray} \noindent By recalling that $\beta$ and $u_{0}$ are orthogonal sections of $W^{+}$, we take inner product of both sides of (8) with $\beta$ and integrate over $X$ and obtain: \begin{eqnarray*} \int_{X}(\;|\nabla_{A}\beta\;|^{2} +\frac{r}{8}\;|\beta|^{4} + \frac{r}{4}|\beta|^{2}+\frac{3r}{8}|\alpha|^{2}|\beta|^{2}\;)&=& \\ 2\int_{X}(<<\nabla_{a} \alpha \;,\;b>\;,\;\beta> -\;\frac{s}{4}\;|\beta|^{2} -\;\frac{1}{4}<\rho(F_{A_{0}}^{+})\beta \;,\; \beta> & & \end{eqnarray*} $$\mbox{Hence}\;\; \int_{X} \;|\nabla_{A}\beta\;|^{2} +\frac{r}{8}\;|\beta|^{4} + \frac{r}{4}|\beta|^{2}+\frac{3r}{8}|\alpha|^{2}|\beta|^{2}\; \leq \int_{X}c_{1}|\beta|^{2}+ c_{2}|\beta | |\nabla_{a} \alpha| $$ where $c_1$ and $c_2$ are positive constants depending on the Riemanian metric and the base connection $A_{0}$. Choose $r\gg 1$, by calling $c_{2}=2c_{3}$ we get : $$ \int_{X}(\;|\nabla_{A}\beta\;|^{2} +\frac{r}{8}\;|\beta|^{4} + \frac{r}{8}|\beta|^{2}+\frac{3r}{8}|\alpha|^{2}|\beta|^{2}\;) \leq \int_{X}(c_{1}-\frac{r}{8})\;|\beta\;|^{2}+ 2c_{3}| \beta\; | |\nabla_{a} \alpha| $$ $$=- \left[ \;(r/8-c_{1})^{1/2}\;|\beta\;|-c_{3}(r/8-c_{1})^{-1/2}\; |\nabla_{a} \alpha|\;\right]^{2}+ \frac{c_{3}^{2}}{(r/8-c_{1})} |\nabla_{a} \alpha|^{2} \leq \frac{C}{r}|\nabla_{a} \alpha|^{2}$$ For some $C$ depending on the metric and $A_{0}$. In particular we have \begin{eqnarray*} \int_{X}r \;|\beta|^{2} -\frac{8C}{r}\;|\nabla_{a} \alpha|^{2}&\leq &0 \\ 8c_{2}\;|\beta \;|\;|\nabla_{a} \alpha|- \frac{8C}{r}|\nabla_{a} \alpha|^{2}&\leq& \int_{X}({r}-8c_{1})\;|\beta\;|^{2} \end{eqnarray*} \begin{eqnarray}\mbox{Hence}\;\;\;\;\;\;\;\; c_{2}\;|\beta \;|\;|\nabla_{a} \alpha|- \frac{2C}{r}|\nabla_{a} \alpha|^{2}&\leq &0 \end{eqnarray} \vspace{.1in} Now by self adjointness of the Dirac operator, and by $\alpha u_{0}=\psi-\beta $ we get: \begin{eqnarray} <D\!\!\!\!/\,_{A}^{2}(\psi)\;,\;\alpha u_{0}>&=&<D\!\!\!\!/\,_{A}^{2}(\psi -\beta)\;,\;\alpha u_{0}>+ <D\!\!\!\!/\,_{A}^{2}\;(\beta)\;,\;\alpha u_{0}>\nonumber\\ &=&<D\!\!\!\!/\,_{A}^{2}(\psi -\beta)\;,\;\alpha u_{0}>+ <\beta\;,\;D\!\!\!\!/\,_{A}^{2}(\psi-\beta)> \end{eqnarray} \noindent We can calculate (36) by using (33) and obtain the inequalities: \begin{eqnarray*} 0=<D\!\!\!\!/\,_{A}^{2}(\psi)\;,\;\alpha u_{0}>&=&|\nabla_{a}\alpha|^{2} + \frac{r}{4}|\alpha|^{4}- \frac{r}{4}|\alpha|^{2}|\beta|^{2}- \frac{r}{2}|\alpha|^{2}\\ && + \;\frac{r}{2}\;|\alpha|^{2}|\beta|^{2}-2 <<\nabla_{a} \alpha,b>, \beta> \end{eqnarray*} \begin{eqnarray*} \int_{X}|\nabla_{a}\alpha|^{2} +\frac{r}{4}|\alpha|^{4}-\frac{r}{2}|\alpha|^{2} &\leq &\int_{X}2 <<\nabla_{a} \alpha,b>, \beta> -\frac{r}{4} |\alpha|^{2}|\beta|^{2}\\ &\leq& \int_{X}2 <<\nabla_{a} \alpha,b>, \beta>\leq \int_{X}c_{2} |\nabla_{a} \alpha|\; |\beta\; | \end{eqnarray*} By choosing $c_{4}=1-{2C}/{r}$ and by (35), we see \begin{eqnarray} \int_{X}c_{4}\;|\nabla_{a}\alpha|^{2} +\frac{r}{4}|\alpha|^{4}-\frac{r}{2}|\alpha|^{2} &\leq &0 \end{eqnarray} Since for a connection $A$ in $K\longrightarrow X$ the class $({i}/{2\pi}) F_{A}$ represents the Chern class $c_{1}(K)$, and since $\omega$ is a self dual two form we can write: $$\int_{X}\omega \wedge F_{A}=-2\pi\;i\; \omega c_{1}(K) \;\;\;\;\;\; \int_{X}\omega \wedge F_{A}=\int_{X}\omega \wedge F_{A}^{+}$$ $$\int_{X} \omega \wedge (F_{A}^{+}-F_{A_{0}}^{+})=0$$ By (28) this implies: \begin{eqnarray} \frac{r}{2} \int_{X} (2-|\alpha|^{2}+|\beta|^{2})=0 \end{eqnarray} By adding (38) to (37) we get \begin{eqnarray} \int_{X}c_{4}\;|\nabla_{a}\alpha|^{2} + \frac{r}{2}|\beta|^{2} +r (1 - \frac{1}{2}|\alpha|^{2})^{2}\leq 0 \end{eqnarray} Assume $r\gg 1$, then $c_{4}\geq 0$ and hence $\nabla_{a}\alpha=0$ and $\beta=0$ and $|\alpha|=\sqrt{2}$, hence: $$\beta =0\;\;\mbox{and}\;\;\alpha =\sqrt{2}e^{i\;\theta}\;\;\mbox{ and}\;\; \nabla_{a}(\alpha)=d(e^{i\;\theta})+i\;a\;e^{i\;\theta}=0$$ Hence $a= d(-\theta)$, recall that we also have $d^{*}(a)=0$ which gives $$0 = <d^{*}d(\theta),\theta> = <d(\theta),d(\theta>)>=||d(\theta)||^{2} $$ Hence $\;a=0\;$ and $\;\alpha \;$=constant. So up to a gauge equivalence $(A,\psi)=(A_{0},u_{0})$ \hspace{4.5in} $\Box$ \section{Applications} Let $X$ be a simply connected closed smooth $4$-manifold. By J.H.C.Whitehead the intersection form $$ q_{X}:H_{2}(X;\Z)\otimes H_{2}(X;\Z) \longrightarrow \Z$$ determines the homotopy type of $X$. By C.T.C Wall in fact $q_{X}$ determines the $h$-cobordism class of $X$. Donaldson (c.f. [{\bf DK}]) showed that if $q_{X}$ is definite then it is dioganalizable, i.e. $$q_{X}=p<1>\oplus q<-1>$$ We call $q_{X}$ is even if q(a,a) is even for all $a$, otherwise we call $q_{X}$ odd. Since integral liftings $c$ of the second Steifel Whitney calass $w_{2}$ of $X$ are characterized by $\;c.a=a.a\;$ for all $a\in H_{2}(X;\Z)$, the condition of $q_{X}$ being even is equivalent to $X$ being spin. From classification of unimodular even integral quadratic forms and the Rohlin theorem it follows that the intersection form of a closed smooth spin manifold is in the form: \begin{eqnarray}q_{X}=2k E_{8}\oplus lH \end{eqnarray} where $E_{8}$ is the $8\times 8$ intersection matrix given by the Dynikin diagram \begin{figure}[htb] \vspace{.8in} \special{picture 1 scaled 700} \caption{} \end{figure} \noindent and $H$ is the form $\; H=\left(\begin{array}{cc}0&1\\1&0 \end{array} \right)\;$. The intersection form of the manifold $S^{2}\times S^{2}$ realizes the form $H$, and the K3 surface (quadric in ${\C}{\P}^{3}$) realizes $ 2E_{8}\oplus 3H$. Donaldson had shown that if $k=1$, then $l\geq 3$ ([{\bf D}]). Clearly connected sums of K3 surface realizes $\;2kE_{8}\oplus 3k H$. In general it is a conjecture that in (40) we must necessarily have $l\geq 3k$ (sometimes this is called $11/8$ conjecture). Recently by using Seiberg-Witten theory M.Furuta has shown that \vspace{.15in} \noindent{\bf Theorem (Furuta) }: Let $X$ be a simply connected closed smooth spin \\ $4$-manifold with the intersection form $q_{X}=2k E_{8}\oplus lH $, then $l\geq 2k+1$ \vspace{.15in} Proof: We will only sketch the proof of $\;l\geq 2k$. We pick $L\longrightarrow X$ to be the trivial bundle (it is characteristic since $X$ is spin). Notice that the spinor bundles $$V^{\pm}=P\times _{\rho_{\pm}}{\C}^{2}\longrightarrow X$$ $\rho_{\pm}: x\longmapsto q_{\pm} x $ , are quoternionic vector bundles. That is, there is an action $j: V^{\pm}\to V^{\pm}$ defined by $[p,x]\to [p,xj]$, which is clearly well defined. This action commutes with $$\partial\!\!\!/\,: \Gamma(V^{+})\longrightarrow \Gamma(V^{-}) $$ Let $A_{0}$ be the trivial connection, and write $\;\pm A=A_{0}\pm i\;a\in \cal{A}(L)$ \begin{eqnarray*} \partial\!\!\!/\, _{A}(\psi j)&=&\sum \rho (e^{k}) \;[\;\nabla _{k} +i\ a\;]\; (\psi j)= \sum \rho (e^{k}) \;[\;\nabla _{k}(\psi)j+ \psi j ia\;] \\ &=& \sum \rho (e^{k}) \;[\;\nabla _{k}(\psi)j- ia \psi j\;]= \partial\!\!\!/\,_{-A}(\psi)j \end{eqnarray*} $\Z_{4}\;$ action $\;(A , \psi)\longmapsto (-A , \psi j)\;$ on $\;\Omega^{1}(X)\times \Gamma(V^{+})\;$ preserves the compact set $${\cal M}_{0}=\tilde {\cal M}\cap \; ker( d^{*}) \oplus \Gamma (V^{+})$$ \vspace{.05in} \noindent where $\;\;\tilde {\cal M}=\{ (a, \psi)\in \Omega^{1}(X)\oplus \Gamma(V^{+})\;|\; \partial\!\!\!/\,_{A}(\psi)=0 \;,\;\;\;\rho(F_{A}^{+})=\sigma(\psi) \;\} $ \\ For example from the local description of $\sigma $ in (2) we can check $$\sigma (\psi j)=\sigma (z+jw)j = \sigma(-\bar{w} + j\bar{z})=-\sigma(\psi)= -\;F_{A}^{+}=F_{-A}^{+}$$ This $\Z_{4}$ in fact extends to an action of the subgroup $G$ of $SU(2)$ which is generated by $<S^{1}\;,\;j>$, where $S^{1}$ acts trivially on $\Omega^{*}$ and by complex multiplication on $\Gamma(V^{+})$, and $j$ acts by $-1$ on $\Omega^{*}$ and by quaternionic multiplication on $\Gamma(V^{+})$ In particular we get a $G$-equivariant map $\;\varphi = L + \theta : {\cal V}\to {\cal W} \;$ where: $$ \varphi : {\cal V}=ker(d^{*})\oplus \Gamma(V^{+})\longrightarrow {\cal W}= \Omega_{+}^{2}\oplus \Gamma(V^{-}) $$ $$L=\left( \begin{array}{cc} d^{+} & 0 \\ 0 & \partial\!\!\!/\, \end{array} \right)\;\;\; \mbox{and}\;\;\; \theta (a,\psi)=(\;\sigma (\psi)\;,\; a \psi\;)$$ with $\varphi^{-1}(0)= {\cal M}_{0}$ and $\varphi (v)=L(v) + \theta (v)$ with $L$ linear Fredholm and $\theta $ quadratic. We apply the ``usual" Kuranishi technique (cf [{\bf L}]) to obtain a finite dimensional local model $\;V\longmapsto W\;$ for $\varphi$. We let ${\cal V}=\oplus V_{\lambda }$ and ${\cal W}=\oplus W_{\lambda }$, where $V_{\lambda }$ and $W_{\lambda }$ be $\lambda $ eigenspaces of $L^{*}L :V\to V$ and $LL^{*}:W\to W$ repectively. Since $L L^{*}$ is a multiplication by $\lambda $ on $V_{\lambda}$, for $\lambda > 0$ we have isomorphisms $L :V_{\lambda}\stackrel{\cong }{\longrightarrow} W_{\lambda }$. Now pick $\Lambda >0$ and consider projections: $$\oplus_{\lambda\leq \Lambda}W_{\lambda}\;\;\stackrel{p_{\Lambda}}{\longleftarrow} \;\; W \;\; \stackrel{1-p_{\Lambda}}{\longrightarrow}\;\; \oplus_{\Lambda > \Lambda} W_{\lambda}$$ Consider the local diffeomorphism $f_{\Lambda}: V \stackrel{ }{\longrightarrow} V$ given by: $$u=f_{\Lambda}(v)=v+L^{-1}(1-p_{\Lambda})\theta (v) \;\;\Longleftrightarrow \;\;L(u)=L(v)+(1-p_{\Lambda})\theta (v)$$ The condition $\varphi(v)=0$ is equivalent to $\;p_{\Lambda}\;\varphi (v)=0 $ and $ (1-p_{\Lambda})\;\varphi (v)=0 $, but \begin{eqnarray*} (1-p_{\Lambda})\;\varphi (v)=0\;\Longleftrightarrow \; (1-p_{\Lambda})\;L(v) + (1-p_{\Lambda})\;\theta(v)=0 \;\Longleftrightarrow\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\; \\ (1-p_{\Lambda})\;L(v) + L(u)-L(v) =0 \; \Longleftrightarrow\;L(u)=p_{\Lambda}\;L(v) \; \Longleftrightarrow\; u\in \oplus_{\lambda\leq \Lambda}V_{\lambda} \end{eqnarray*} Hence $\;\varphi (v)=0\;\Longleftrightarrow\;p_{\Lambda}\;\varphi (v)=0\; $ and $\;u\in \oplus_{\lambda\leq \Lambda}V_{\lambda}\;$, let $$ \varphi _{\Lambda}:V=\oplus_{\lambda\leq \Lambda} V_{\lambda}\longrightarrow W= \oplus_{\lambda\leq \Lambda}W_{\lambda}\hspace{.15in} \mbox{where}\hspace{.15in} \varphi _{\Lambda}(u)=p_{\Lambda}\;\varphi\; f_{\Lambda}^{-1}(u)$$ Hence in the local diffeomorphism $f_{\Lambda}: {\cal O}\stackrel{ \approx}{\longrightarrow} {\cal O} \subset {\cal V }$ takes the piece of the compact set $f_{\lambda}({\cal O}\cap {\cal M}_{0} )$ into the finite dimensional subspace $V\subset {\cal V}$, where ${\cal O}$ is a neighborhood of $(0,0)$. As a side fact note that near $(0,0)$ we have $${\cal M}(L) \approx{\cal M}_{0}(L)/S^{1} $$ We claim that for $\lambda \gg 1 \;$, the local diffeomorphism $f_{\Lambda}: {\cal O}\stackrel{ \approx}{\longrightarrow} {\cal O} \subset V $ extends to a ball $B_{R}$ of large radius $R$ containig the compact set ${\cal M}_{0}(L) $, i.e. we can make the zero set $\varphi_{\Lambda}^{-1}(0)$ a compact set. We see this by applying the Banach contraction principle. For example for a given $u\in B_{R}$, showing that there is $v\in V$ such that $f_{\Lambda}(v)=u$ is equivalent of showing that the map $\;T_{u}(v)=u- L^{-1}(1-p_{\Lambda})\theta (v)\;$ has a fixed point. Since $L^{-1}(1-p_{\Lambda})$ has eigenvalues $1/\lambda$ on each $W_{\lambda}$ in appropriate Sobolev norm we can write $$|| T_{u}(v_{1})- T_{u}(v_{2}) ||\leq \frac{C}{\Lambda }||\theta(v_{1})- \theta (v_{2})|| \leq \frac{C}{\Lambda }||v_{1}-v_{2}||$$ Vector subspaces $V_{\lambda}$ and $W_{\lambda}$ are either quaternionic or real depending on whether they are subspaces of $\Gamma (V^{\pm})$ or $\Omega^{*}(X)$. For a generic metric we can make the cokernel of $\partial\!\!\!/\,$ zero hence the dimension of the kernel (as a complex vector space) is $\;ind(\partial\!\!\!/\,)=-\sigma /8=2k\;$, and since $H^{1}(X)=0$ the dimension of the cokernel of $d^{+}$ (as a real vector space) is $b^{+}=l$. Hence $\;\varphi _{\Lambda}\;$ gives a $G$-equivariant map $$\varphi: {\H}^{k+y}\oplus {\R}^{x}\longrightarrow {\H}^{y}\oplus {\R}^{l+x}$$ with compact zero set. From this Furuta shows that $l \geq 2l+1$. Here we give an easier argument of D.Freed which gives a slightly weaker result of $l\geq 2k$. Let $E_{0}$ and $E_{1}$ be the complexifications of the domain an the range of $\varphi $; consider $E_{0}$ and $E_{1}$ as bundles over a point $x_{0}$ with projections $\pi_{i}:E_{i}\to x_{0}$, and with $0$-sections $s_{i}:x_{0}\to E_{i}\;, i=0,1$. Recall $K_{G}(x_{0})=R(G)$, and we have Bott isomorphisms $\beta (\rho)=\pi _{i}^{*}(\rho)\;\lambda_{E_{i}}\;$, for $\;i=0,1$ where $\;\lambda_{E_{i}}\;$ are the Bott classes. By compactness we get an induced map $\varphi ^{*}$: $$\begin{array} {ccc} K_{G}(B(E_{1}),S(E_{1}))&\stackrel{\varphi ^{*}} \longrightarrow & K_{G}(B(E_{0}),S(E_{0}))\\ &&\\ \approx \;\uparrow \beta & & \approx \;\uparrow \beta \\ R(G) && R(G) \end{array}$$ Consider $s_{i}^{*}(\lambda_{E_{i}})= \sum(-1)^{k}\Lambda^{k}(E_{i})=\Lambda_{-1}(E_{i})\in R(G)$, then by some $\rho$ we have $$\Lambda_{-1}(E_{1})=s_{1}^{*}(\lambda_{E_{1}})=s_{0}^{*} \varphi^{*}(\lambda_{E_{1}}) =s_{0}^{*} (\pi _{0}^{*}(\rho)\;\lambda_{E_{0}})=\rho \;\Lambda_{-1}(E_{0})$$ So in particular $tr_{j}(\Lambda_{-1}(E_{0}))$ divides $tr_{j}(\Lambda_{-1}(E_{1}))$. By recalling $\;j:E_{i}\to E_{i}\;$ $$tr_{j}(\Lambda_{-1}(E_{i}))=det (I-j)\;\;\;\;\;\mbox{for}\;\;i=0,1$$ Since $(z,w)j=(z+jw)j=-\bar{w}+j\bar{z}=(-\bar{w},\bar{z})\;$ $j$ acts on ${\H}\otimes{\C}$ by matrix $$ A=\left(\begin{array}{cccc}0 &0 &1 &0\\ 0&0&0&-1\\-1&0&0&0\\0&1&0&0 \end{array}\right) $$ so $\;det (I-A)=4$, and $j$ acts on $\;{\R}\otimes {\C}\;$ by $j(x)=-x$ so $det (I-(-I))=2\;$. Hence $\;4^{k+y}\;2^{x}\;$ divides $\;4^{y}\;2^{l+x}\;$ which implies $\;l\geq 2k\;$ \hspace{1.4in} $\Box$ \vspace{.15in} There is another nice application of Seiberg-Witten invariants: It is an old problem whether the quotient of a simply connected smooth complex surface by an antiholomorphic involution $\sigma:\tilde{X}\to \tilde{X}\;$ (an involution which anticommutes with the almost complex homomorphism $\sigma _{*}J=-J\sigma_{*}$) is a "standard" manifold (i.e. connected sums of $S^{2}\times S^{2}$ and $\;\pm \C\P^{2}\;$). A common example of a antiholomorphic involution is the complex conjugation on a complex projective algebraic surface with real coefficients. It is known that the quotient of $\C\P^{2}$ by complex conjugation is $S^{4}$ (Arnold, Massey, Kuiper); and for every $d$ there is a curve of degree $d$ in $\C\P^{2}$ whose two fold branched cover has a standard quotient ([{\bf A}]). This problem makes sense only if the antiholomorphic involution has a fixed point, otherwise the quotient space has fundamental group $\Z_{2}$ and hence it can not be standard. By ``connected sum" theorem, Seiberg-Witten invariants of ``standard" manifolds vanish, so it is natural question to ask whether Seiberg-Witten invariants of the quotients vanish. Shugang Wang has shown that this is the case for free antiholomorphic involutions. \vspace{.15in} \noindent {\bf Theorem} ({\bf S.Wang}) Let $\tilde{X}$ be a minimal Kahler surface of general type, and $\sigma: \tilde{X}\to \tilde{X}$ be a free antiholomorphic involution, then the quotient $X=\tilde{X}/\sigma$ has all Seiberg-Witten invariants zero \vspace{.12in} Proof: Let $h$ be the Kahler metric on $\tilde{X}$, i.e. $\omega(X,Y)=h(X,JY)$ is the Kahler form. Then $\;\tilde{g}=h+\sigma^{*}h\;$ is an invariant metric on $\tilde {X}$ with the Kahler form $\;\tilde{\omega}=\omega - \sigma^{*}\omega \;$. Let $g$ be the ``push-down" metric on $X$. Now we claim that all $SW_{L}(X)=0$ for all $L\to X$, in fact we show that there are no solutions to Seiberg-Witten equations for $X$: Otherwise if $L\longrightarrow (X,g)$ is the characteristic line bundle supporting a solution $(A,\psi)$, then the pull-back pair $(\tilde{A},\tilde{\psi})$ is a solution for the pull-back line bundle $\tilde{L}\longrightarrow \tilde{X}$ with the pull-back $Spin_{c}$ structure, hence $$0\leq dim {\cal M_{\tilde{L}}}(\tilde{X})=\frac{1}{4}c_{1}^{2}(\tilde{L})- \frac{1}{4} (3\sigma(\tilde{X})+2 \chi (\tilde{X}))$$ But $\tilde{X}$ being a minimal Kahler suface of general type $3\sigma(\tilde{X})+2 \chi (\tilde{X})=K^{2}_{\tilde{X}}>0\;$, hence $\;c_{1}^{2}(\tilde{L})>0 $. This implies that $(\tilde{A},\tilde{\psi})\; $ must be an irreducible solution (i.e. $\psi \neq 0 $), otherwise $F^{+}_{\tilde{A}}=0$ would imply $\;c_{1}^{2}(\tilde{L})<0\;$. Now by (25) the nonzero solution $\;\psi= \alpha u_{0} + \beta$ must have either one of ${\;\alpha\;}$ or $\;\beta\;$ is zero (so the other one is nonzero), and since $\tilde{\omega}\wedge \tilde{\omega}$ is the volume element: $$\tilde{\omega}.c_{1}(\tilde{L})=\frac{i}{2\pi} \int \tilde{\omega}\wedge F^{+}_{\tilde{A}}= \frac{i}{2\pi}\int \tilde{\omega}\wedge(\frac{|\beta |^{2}- |\alpha |^{2}}{2})\;i\;\tilde{\omega} \neq 0$$ But since $\;\sigma^{*} (\tilde{\omega})=- \tilde{\omega}\;$, $\;\sigma^{*} c_{1}(\tilde{L})= c_{1}(\tilde{L})\;$, and $\;\sigma \;$ is an orientation preserving map we get a contradiction \\ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \tilde{\omega}.c_{1}(\tilde{L})=\sigma^{*}(\tilde{\omega}.c_{1}(\tilde{L}))= -\tilde{\omega}.c_{1}(\tilde{L}) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Box$$

9510

alg-geom/9510011

en

https://arxiv.org/abs/alg-geom/9510011

alg-geom/9510011

Pablo A. Gastesi

Indranil Biswas, Pablo Gastesi and Suresh Govindarajan

Parabolic Higgs bundles and Teichm\"uller spaces for punctured surfaces

AMSLaTeX v 2.09, 13 pages, DVI file available at http://www.imsc.ernet.in/~pablo/

Trans.Am.Math.Soc. 349 (1997) 1551-1560

TIFR/TH/95-50

In this paper we study the relation between parabolic Higgs bundles and irreducible representations of the fundamental group of punctured Riemann surfaces established by Simpson. We generalize a result of Hitchin, identifying those parabolic Higgs bundles that correspond to Fuchsian representations. We also study the Higgs bundles that give representations whose image is contained, after conjugation, in SL($k,\Bbb R$). We compute the real dimension of one of the components of this space of representations, which in the absence of punctures is the generalized Teichm\"uller space introduced by Hitchin, and which in the case of $k=2$ is the Teichm\"uller space of the Riemann surface.

alg-geom

\section{Introduction} In the well-known paper \cite{H1}, Hitchin introduced Higgs bundles, and established a one-to-one correspondence between equivalence classes of irreducible GL($2,\Bbb{C}$) representations of the fundamental group of a compact Riemann surface and isomorphism classes of rank two stable Higgs of degree zero. In \cite{S2}, Simpson defined parabolic Higgs bundles, which generalized Hitchin's correspondence to the case of open Riemann surfaces (see also \cite{S1}). Here, by an open Riemann surface we mean the complement of finitely many points in a compact surface. More precisely, Simpson identified what he calls filtered local systems with parabolic Higgs bundles. In \cite{H2}, Hitchin identified the Higgs bundles corresponding to the Fuchsian representations. Our main aim here is to generalize his results to the case of open Riemann surfaces. Before giving more details, we describe the result of Hitchin on Fuchsian representations. Let $X$ be a compact Riemann surface of genus $g \geq 2$, and let $L$ be a line bundle on $X$ such that $L^2 = K_X$, that is $L$ is a square root of the canonical bundle of $X$. Define $$ E \, := \, L^* \oplus L$$ which is a rank $2$ bundle on $X$. For $a\in H^0(X,K^2)$, let $${\theta}(a) \, :=\, \left(\begin{matrix}0 & 1\\ a & 0\\ \end{matrix}\right) \, \in \,H^0\big({\bar X}, End(E)\otimes K\big)$$ be the Higgs field. Hitchin proved that the conjugacy classes of Fuchsian representations of $\pi_1(X)$ (homomorphisms of ${\pi}_1(X)$ into PSL$(2,\Bbb R$) such that the quotient of the action on the upper half plane is a compact Riemann surface of genus $g$) correspond to the Higgs bundles of the form $(E,{\theta}(a))$ defined above. Moreover, the Higgs bundle $(E,{\theta}(0))$ corresponds to the Fuchsian representation for the Riemann surface $X$ itself. Consider now an open Riemann surface $X = {\bar X} - \{ p_1, \ldots, p_n \}$, where $\bar X$ is a compact surface of genus $g$, and $p_1, \ldots, p_n$ are $n>0$ distinct points of $\bar X$. Let $D$ denote the divisor given by these points, {\it i.e.\/}\ $D=\{p_1,\dots,p_n\}$. We will further assume that $2g-2+n>0$, which is equivalent to the condition that the universal covering space of $X$ is (conformally equivalent to) the upper half plane. Consider the bundle $$E\,\, :=\,\, (L\otimes {\cal O}_X(D))^*\, \oplus \,L $$ and give parabolic weight $1/2$ to the fiber $E_{p_i}$, $1\leq i \leq n$. For $a\in H^0({\bar X},K^2\otimes {\cal O}_{\bar X}(D))$ let $${\theta}(a) \, :=\, \left(\begin{matrix}0 & 1\\ a & 0\\ \end{matrix}\right) \, \in \,H^0\big({\bar X}, End(E)\otimes K\otimes {\cal O}_X(D)\big) $$ be a parabolic Higgs field on the parabolic bundle $E$. We prove that under the identification between filtered local systems and parabolic Higgs bundles, Fuchsian representations of $n$-punctured Riemann surfaces are in one-one correspondence with parabolic Higgs bundles of the type $(E,{\theta}(a))$ defined above. Moreover, the parabolic Higgs bundle $(E,{\theta}(0))$ corresponds to the Fuchsian representation of the punctured surface $X$ itself. Thus this is a direct generalization of the result of Hitchin on Fuchsian representations of compact Riemann surfaces to the punctured case. In section $3$, we generalize the above results to the case of representations of the fundamental group of the surface $X$ into PSL($k,\Bbb R$), for $k>2$. More precisely, we consider a parabolic bundle $W_k$, obtained by tensoring the $(k-1)$th symmetric product of the bundle $E$ defined above with an appropriate power of ${\cal O}_{\bar X}(D)$. The Higgs fields we consider are generalizations of the $2$-dimensional case, namely they are of the form $$\theta(a_2,\ldots,a_{k-1}):=\left(\begin{matrix}0 & 1 & \cdots & 0 \\ 0 & 0 & 1 & \vdots \\ \vdots & & & 0 \\ a_k & \cdots & a_2 & 0 \end{matrix}\right) ,$$ $a_j$ is a section of the line bundle $K^j \otimes ({\cal O}_{{\bar X}(D)})^{j-1}$. As in section $2$, we have that the pair $(W_k,\theta(a_2, \ldots, a_k))$ is a stable parabolic bundle of parabolic degree $0$. It is not difficult to see that the parabolic dual of $W_k$ is naturally isomorphic to the parabolic bundle $W_k$ itself. This implies that the holonomy of the flat connection corresponding to these bundles is contained in a split real form of SL($k,\Bbb C$), which is isomorphic to SL($k,\Bbb R$). We prove that one of the components of the space of representations of $\pi_1(X)$ into SL$(k,\Bbb R)$, with fixed conjugacy class of monodromy around the punctures, has real dimension equal to $2(k^2-1)(g-1)+k(k-1)n$. Observe that for $k=2$, this dimension is $2(3g-3+n)$, which is precisely the real dimension of ${\cal T}_g^n$, the Teichm\"uller space of compact surfaces of genus $g$ with $n$ punctures. It is therefore natural, following \cite{H2}, to call the above component the {\it Teichm\"uller component} of the corresponding space of representations. Further study of this set is worthwhile. \section{Higgs bundles for Fuchsian representations} Let $\bar X$ be a compact Riemann surface of genus $g$, and let $$D \,\, := \,\, \{p_1,p_2,\ldots ,p_n\}$$ be $n$ distinct points on $\bar X$. Define $X := {\bar X} - D$ to be the punctured Riemann surface given by the complement of the divisor $D$. We will assume that $2g-2+n > 0$, that is, the surface $X$ supports a metric of constant curvature $(-4)$. The degree of the holomorphic cotangent bundle $K$, of $\bar X$ is $2g-2$. Therefore, there is a line bundle $L$ on $\bar X$ such that $L^2 = K$. Fix such a line bundle $L$. Note that any two of the $4^g$ possible choices of $L$ differ by a line bundle of order $2$. Using $L$ we will construct a parabolic Higgs bundle on $\bar X$, as follows. Let $\xi = {\cal O}_{\bar X}(D)$ denote the line bundle on $\bar X$ given by the divisor $D$. Define \begin{equation} E\,\, :=\,\, (L\otimes \xi)^*\, \oplus \,L \label{eq:bundle}\end{equation} to be the rank 2 vector bundle on $\bar X$. To define a parabolic structure on $E$ (we will follow the definition of parabolic Higgs bundle in \cite{S2}), on each point $p_i\in D$, $1\leq i \leq n$, we consider the trivial flag $$ E_{p_i} \, \supset \, 0,$$ and give parabolic weight $1/2$ to $E_{p_i}$. This gives a parabolic structure on $E$. Note that \begin{equation} Hom(L , L^{*}\otimes {\xi}^*)\otimes K\otimes\xi \, =\, {\cal O} \, \subset \, End(E)\otimes K\otimes \xi \label{eq:hom}\end{equation} Let $1$ denote the section of $\cal O$ given by the constant function $1$. So from (\ref{eq:hom}) we have \begin{equation} \theta \, :=\, \left(\begin{matrix}0 & 1\\ 0 & 0\\ \end{matrix}\right) \, \in \,H^0\big({\bar X}, End(E)\otimes K\otimes\xi\big) \label{eq:theta} \end{equation} \begin{lemma} The parabolic Higgs bundle $(E, \theta)$ is a parabolic stable Higgs bundle of parabolic degree zero.\label{lemma:stable}\end{lemma} \begin{pf} From the definition of parabolic degree (see \cite[definition 1.11]{MS} or \cite{S2}) we immediately conclude that the parabolic degree of $E$ is zero. To see that $(E, \theta)$ is stable, first note that there is only one sub-bundle of $E$ which is invariant under $\theta$, namely the summand $(L\otimes \xi)^*$ in (2.1). (A sub-bundle $F \subset E$ is called {\it invariant} under $\theta$ if $\theta (F) \subset F\otimes K\otimes \xi$.) The degree of $(L\otimes \xi)^*$ is $1-g -n$. So the parabolic degree of $(L\otimes \xi)^*$, for the induced parabolic structure, is $1-g-n/2$. But, from our assumption that $2g-2+n > 0$ we have $1-g-n/2 < 0$. So $(E,\theta)$ is stable. \end{pf} {}From the proof of the \lemref{lemma:stable} it follows that $(E,\theta)$ constructed above is stable if and only if $2g-2+n > 0$. We will show later that this corresponds to the fact that $X$ admits a complete metric of constant negative curvature if and only if $2g-2+n > 0$. {}From the main theorem of \cite[pg. 755]{S2} we know that there is a tame harmonic metric on the bundle $E$. (See the Synopsis of that paper for the definition of tame harmonic metric.) It is well-known that there is an unique complete K\"ahler metric on $X$, known as the {\it Poincar\'e} {\it metric}, such that its curvature is $(-4)$. Both the bundles $L$ and $(L\otimes \xi)^*$ are equipped with metrics induced by the tame harmonic metric on $E$. So $$ Hom(L,(L\otimes \xi)^*) \, =\, L^2\otimes{\xi}^* \, =\, T\otimes \xi$$ is equipped with a metric. The restriction to $X$ of the line bundle $\xi$, and hence ${\xi}^*$, on $\bar X$ has a canonical trivialization. Therefore we have a hermitian metric on $T_X$ the tangent bundle of $X$. We will denote this hermitian metric on $T_X$ by $H$. Note that $H$ is singular at $D$, {\it i.e.\/}\ does not induce a hermitian metric on $T_{\bar X}$. \begin{lemma} The hermitian metric $H$ on the holomorphic tangent bundle on $X$ obtained above is the Poincar\'e metric. \label{lemma:metric}\end{lemma} \begin{pf} We recall the Hermitian-Yang-Mills equation which gives the harmonic metric on $E$ \cite{S2}. This equation was first introduced in \cite{H1}. Let $\nabla$ denote the holomorphic hermitian connection on the restriction of $E$ to $X$ for the harmonic metric. Then the Hermitian-Yang-Mills equation of the curvature of $\nabla$ is the following: \begin{equation} K(\nabla) \, :=\, {\nabla}^2 \, = \, -\, [\theta, {\theta}^*]\label{eq:yang}\end{equation} If the decomposition (\ref{eq:bundle}) is orthogonal with respect to the metric, then $[\theta, {\theta}^*]$ is a 2-form with values in the diagonal endomorphisms of $E$ (diagonal for the decomposition (\ref{eq:bundle})). Using this, the equation (\ref{eq:yang}) reduces to the following equation on $X$ \begin{equation} F_H \,=\, -2{\bar H}, \label{eq:curv}\end{equation} where $H$ is a hermitian metric on $T_X$ and $\bar H$ is the $(1,1)$-form on $X$ given by $H$. Observe that $\bar H$ also denotes the K\"ahler $2$-form for the metric $H$. A metric $H'$ on $T_X$ induces a metric on $L$. Since the bundle $\xi$ has a natural trivialization over $X$, the metric $H'$ also induces a metric on $(L\otimes\xi)^*$, and therefore also on $E$. If $H'$ satisfies the equation (\ref{eq:curv}) then the metric on $E$ obtained this way satisfies (\ref{eq:yang}). Now from the uniqueness of the solution of (\ref{eq:yang}) (\cite{S2}), we have that such metric is obtained from the solution of (\ref{eq:curv}) in the above fashion. {}From the computation in Example (1.5) of \cite[pg. 66]{H1}, we conclude that the K\"ahler metric $H$ on $X$ has Gaussian curvature $(-4)$. So in order to complete the proof of the lemma we must show that the K\"ahler metric on $X$ is complete. Recall the asymptotic behavior of the harmonic metric near the punctures given in Section 7 of \cite{S2}. First of all, observe that the fiber of $K\otimes \xi$ at any $p_i \in D$ is canonically isomorphic to $\Bbb C$. So the fiber $(End(E)\otimes K\otimes \xi)_{p_i}$ is $End(E_{p_i})$. The evaluation of the section ${\theta}$ at $p_i$ as an element of $End(E_{p_i})$ is defined to be the residue of $\theta$ at $p_i$. For the Higgs field $\theta$, we have that the residue at each $p_i$ is $$N \, := \, \left(\begin{matrix}0 & 1\\ 0 & 0\\ \end{matrix}\right).$$ In \cite[pg. 755]{S2}, Simpson studies parabolic Higgs bundles with residue $N$ as above. Consider the displayed equation in page 758 of \cite{S2}, which describes the asymptotic behavior of the harmonic metric. Using the fact that the parabolic weight of $E_{p_i}$ is $1/2$ we conclude that for the metric on $L$ induced by the tame harmonic metric on $E$, both $a_i$ and $n_i$ in the equation in page 758 of \cite{S2} are $1/2$. (We also use the fact that, in the notation of \cite[pg. 755]{S2}, $L \subset W_1$ and $L$ is not contained in $W_0$.) In other words, in a suitable trivialization of $L$ on an open set containing a puncture $p_i \in D$, and with holomorphic coordinate $z$ around $p_i$, the hermitian metric on $L$ obtained by restricting the harmonic metric on $E$ is $$r^{1/2}|{\mathrm{log}}(r)|^{1/2},$$ where $r = |z|$. Similarly, for $(L\otimes\xi)^*$, the $a_i$ and $n_i$ in the equation \cite[pg. 758]{S2} are $1/2$ and $-1/2$ respectively. So the metric on $Hom(L,(L\otimes\xi)^*)$ is $({\mathrm{ log}}|(r)|)^{-1}$. Recall the earlier remark that ${\xi}^*$ has a natural trivialization on $X$. The section of ${\xi}^*$ on $X$ has a pole of order $1$ at the points of $D$, when it is considered as a meromorphic section of ${\xi}^*$ on $\bar X$. This implies that the hermitian metric on $T = L^{-2}$ is \begin{equation} r^{-1}|{\mathrm{log}}(r)|^{-1}. \label{eq:metric}\end{equation} But this is the expression the Poincar\'e metric of the punctured disk in $\Bbb C$. This proves that the K\"ahler metric on $X$ induced by $H$ is indeed complete. This completes the proof of the lemma. \end{pf} {}From the decomposition (\ref{eq:bundle}) it follows that \begin{equation} Hom(L^{*}\otimes {\xi}^* ,L)\otimes K\otimes\xi \, =\, K^2\otimes {\xi}^2 \, \subset \, End(E)\otimes K\otimes \xi \label{eq:hom2}\end{equation} Note that the bundle $\xi$ has a natural section, which we will denote by $1_{\xi}$. We may imbedd $H^0({\bar X} ,K^2\otimes \xi)$ into $H^0({\bar X} ,K^2\otimes {\xi}^2)$ by the homomorphism $s \longmapsto s\otimes 1_{\xi}$. So using (\ref{eq:hom2}) we have a natural homomorphism \begin{equation} \rho \, :\, H^0({\bar X}, K^2\otimes \xi) \, \longrightarrow \, H^0({\bar X} ,End(E)\otimes K\otimes \xi) \label{eq:rho} \end{equation} Note that the image of $\rho$ is contained in the image of the inclusion $$H^0({\bar X}, End(E)\otimes K) \, \longrightarrow \,H^0({\bar X}, End(E)\otimes K\otimes \xi)$$ With a slight abuse of notation, for any $a\in H^0({\bar X},K^2\otimes \xi)$, the corresponding element in $H^0({\bar X} ,End(E)\otimes K)$ will also be denoted by $\rho (a)$. The following theorem is a generalization of theorem (11.2) of \cite{H1} to the case of open Riemann surfaces. \begin{thm} For any $a \in H^0({\bar X} ,K^2\otimes \xi)$, the Higgs structure $${\theta}_a \, :=\,{\theta} + {\rho}(a) \, =\, \left(\begin{matrix}0 & 1\\ 0 & 0\\ \end{matrix}\right) + {\rho}(a)$$ on the parabolic bundles $E$ (defined in (\ref{eq:bundle})) makes $(E,{\theta}_a)$ a parabolic stable Higgs bundle of parabolic degree zero. Let $H_a$ denote the harmonic metric (given by the main theorem of \cite{S2}) on the restriction of $E$ to $X$, and let $h$ denote the K\"ahler metric on $X$ induced by the tame harmonic metric ${H}_{a}$ as in \lemref{lemma:metric}. Then the following holds : \begin{enumerate} \item{} The section of the $2$-nd symmetric power of the complex tangent bundle $$h_a \, :=\, a+h+{\bar a}+a{\bar a}/h \, \in \, {\Omega}^0(X, S^2T^*\otimes\Bbb C)$$ is a Riemannian metric on $X$. \item{} The metric $h_a$ is a complete Riemannian metric of constant Gaussian curvature $(-4)$. The Riemann surface structure on $X$ given by metric $h_a$ is a Riemann surface with punctures, {\it i.e.\/}\ there are no holes. (A Riemann surface with a hole is a complement of a disk in a compact Riemann surface.) \item{} Associating to $a \in H^0({\bar X} ,K^2\otimes \xi)$ the complex structure on the $C^{\infty}$ surface $X$ given by the metric $h_a$, the map obtained from $H^0({\bar X} ,K^2\otimes \xi)$ to the Teichm\"uller space ${\cal T}^n_g$ of surfaces of genus $g$ and $n$ punctures is a bijection. \end{enumerate} \label{thm:main}\end{thm} \begin{pf} To prove that $(E,{\theta}_a)$ is stable we use a trick of \cite{H2}. For $\mu >0$, define an automorphism of $E$ by $$T\, := \, \left( \begin{matrix}1 & 0\\ 0 & \mu\\ \end{matrix}\right) .$$ The parabolic Higgs bundle $(E,{\theta}_a)$ is isomorphic to $(E,T^{-1}\circ {\theta}_a\circ T)$, and hence $(E,T^{-1}\circ {\theta}_a\circ T)$ is parabolic stable if and only if $(E,{\theta}_a)$ is so. Since $\mu \neq 0$, we have $(E,T^{-1}\circ {\theta}_a\circ T)$ is parabolic stable if and only if $(E,\frac{1}{\mu} T^{-1}\circ {\theta}_a\circ T)$ is parabolic stable. Now $$1/\mu T^{-1}\circ {\theta}_a\circ T\,=\,\left(\begin{matrix} 0 & 1 \\ 0 & 0\\ \end{matrix}\right) +{\rho}(a) /\mu \, = \, {\theta}_{a/\mu}.$$ But from the openness of the stability condition we have that since $(E, \theta)$ is stable [\lemref{lemma:stable}], there is a non-empty open set $U$ in $H^0({\bar X} ,K^2\otimes \xi)$ containing the origin such that for any $a\in U$, the bundle $(E,{\theta}_a)$ is parabolic stable. Taking $\mu$ to be sufficiently large so that ${\theta}_{a/\mu} \in U$, we conclude that any $(E,{\theta}_a)$ is parabolic stable. The bundle $E$ is equipped with the harmonic metric $H_a$, and $K$ has a metric induced by $h_a$. Using these metrics we construct a hermitian metric on $End(E)\otimes K$. Since $\rho(a) \in H^0({\bar X} ,End(E)\otimes K)$, we may take its pointwise norm. To prove the statement ($1$) we first want to calculate the behavior of $||\rho(a)||$ near the punctures. Since $\rho(a) \in H^0({\bar X} ,End(E)\otimes K)$, we have $$ {\mathrm{residue}}\, ({\theta}_a)\, =\, {\mathrm{residue}}\, ({\theta}) \, =\, N.$$ So the two hermitian metrics $H_0$ (corresponding to $a=0$) and $H_a$ on $E$ are mutually bounded, {\it i.e.\/}\ $C_1.H_0 \leq H_a \leq C_2.H_0$ for some constants $C_1$ and $C_2$. (Recall that the metric in \lemref{lemma:metric} was induced by $H_0$.) From this it is easy to check that around any puncture $p_i$, the norm $||\rho(a)||$ is bounded by $r|{\mathrm{log}}(r)|^{3/2}$. This implies that $||\rho(a)||$ converges to zero as we approach a puncture. Arguing as in ($11.2$) of \cite{H1}, if $h_a$ is not a metric then $$1\, - \, ||\rho(a)|| \, \leq\, 0$$ at some point $x\in X$. Since $||\rho(a)||$ converges to zero as we approach a puncture, the infimum of the function $1 - ||\rho(a)||$ on $X$ must be attained somewhere, say at $x_0\in X$. Let $\Delta$ denote the Laplacian operator acting on smooth functions on $X$. Since the operator ${\cal L} := - \Delta -4.||\rho(a)||^2$ is uniformly elliptic on $X$, we may apply \cite[Section VI.3., Proposition 3.3]{JT} for the operator ${\cal L}$ and the point $x_0$. We conclude that either $1 - ||\rho(a)|| >0$ or $1 - ||\rho(a)||$ is a constant function. This proves that $h_a$ is a Riemannian metric on $X$. {}From the computation in the proof of Theorem (11.3)(ii) of \cite[pg. 120]{H1}, we conclude that $h_a$ is a metric of curvature $(-4)$. To complete the proof of the statement ($2$) we must show that $h_a$ is complete and it has finite volume. (If the volume of the Poincar\'e metric on a Riemann surface is finite then the Riemann surface is a complement of finite number of points in a compact Riemann surface. In particular, the Riemann surface can not have any holes.) The above established fact that the metrics $H_0$ and $H_a$ on $E$ are mutually bounded, together with \lemref{lemma:metric} imply that the Riemannian metric $h_a$ and the Poincar\'e metric on $X$ are mutually bounded. Since the Poincar\'e metric is complete and of finite volume, the same must hold for $h_a$. To prove the statement ($3$) we have to show that map from $H^0({\bar X} ,K^2\otimes \xi)$ to the Teichm\"uller space ${\cal T}^n_g$ obtained in ($2$) is surjective. This will follow from Section $3$ where we will prove that the image is both open and closed, and hence it must be surjective as ${\cal T}^n_g$ is connected. However we may also use the argument in \cite[Theorem (11.2)(iii)]{H1} to prove statement (3). Let $h_0$ denote the Poincar\'e metric on $X$. Indeed, to make the argument work all we need to show is the following generalization of the Eells-Sampson theorem to punctured Riemann surfaces: given a complete Riemannian metric $h$ of constant curvature $(-4)$ and finite volume on the $C^{\infty}$ surface $X$, there is a unique diffeomorphism $f$, of $X$ homotopic to the identity map, such that $f$ is a harmonic map from $(X,h_0)$ to $(X,h)$. This follows from the generalization of the theorem of Corlette, \cite{C}, to the non-compact case as mentioned in \cite[pg. 754]{S2}. Let $(V,\nabla)$ be the flat rank two bundle given by the Fuchsian representation for the Riemann surface $(X,g)$. Let $H$ be the harmonic metric on $V$ given by the main theorem of \cite{S2} (pg. $755$) for the flat bundle $(V,\nabla)$ on the Riemann surface $(X,h_0)$. In other words, $H$ gives a section, denoted by $s$, of the associated bundle with fiber SL$(2,{\Bbb R})/$SO($2$) = $\Bbb H$, where $\Bbb H$ is the upper half plane. This section $s$ gives the harmonic map $f$ mentioned above. This completes the proof of the theorem. \end{pf} The vector space $H^0({\bar X} ,K^2\otimes \xi)$ has a natural complex structure. So does the Teichm\"uller space ${\cal T}_g^n$. The identification of $H^0({\bar X} ,K^2\otimes \xi)$ with ${\cal T}^n_g$ given by Theorem 2.11 does not preserve the complex structures. Indeed, ${\cal T}^n_g$ is known to be biholomorphic to a bounded domain in ${\Bbb C}^{3g-3+n}$. Since any bounded holomorphic function on an affine space must be constant, the identification in Theorem 2.11 is never holomorphic. \noindent {\bf Remark}\, The parabolic dual of the parabolic bundle $E$ is $E^*\otimes {\xi}^*$ with trivial parabolic flag and parabolic weight $1/2$ at the parabolic points $p_i$, $1\leq i \leq n$. So the parabolic dual of $E$ is $E$ itself. Any parabolic Higgs bundle $(E, {\theta}_a)$ (as in \thmref{thm:main}) is naturally isomorphic to the parabolic Higgs bundle $(E^*, {\theta}^*_a)$, where $E^*$ is the parabolic dual of $E$. This implies that the holonomy of the flat connection on $X$ corresponding to the Higgs bundle $(E,{\theta}_a)$ is contained (after conjugation) in SL$(2,\Bbb R$). This of course is also implied by \thmref{thm:main} since the image of a Fuchsian representation is contained in PSL$(2,\Bbb R$). \section{Higgs bundles for SL(${\load{\normalsize}{\it}k},\Bbb R$) representations} Recall the bundle $E$ of section 2, which was defined by $E = (L \otimes \xi)^* \oplus L$, where $L$ is a (fixed) square root of the canonical bundle $K$, and $\xi={\cal O}_{\bar X}(D)$. The ($k-1$)-th symmetric product of ${\Bbb C}^2$ produces an embedding of SL($2,\Bbb R$) into SL($k,\Bbb R$), via action on homogeneous polynomials of degree $k$. Let $V_k$ denote the bundle given by the ($k-1$)-th symmetric product of $E$, that is $V_k:=S^{k-1}(E)$. At each point $p_i\in D$ we have the trivial flag $(V_k)_{p_i}\supset 0$, $1\leq p_i \leq n$, with weight equal to $\frac{k-1}{2}$. In order to construct a parabolic bundle, we need to reduce the weight to a number in the interval $[0,1)$. We do this by tensoring $V_k$ with $\xi^{m(k)}$, where $m(k)$ is equal to $\frac{k}{2}-1$, if $k$ is even, or $\frac{k-1}{2}$, if $k$ is odd. We will denote the bundle $V_k\otimes\xi^{m(k)}$ by $W_k$. At each point $p_i\in D$, we take the trivial flag $(W_k)_{p_i} \supset 0$ of $W_k$, with weight equal to $\frac{1}{2}$, if $k$ is even, or $0$, if $k$ is odd. Considering $1$ as the section of $\cal O$ given by the constant function $1$, we can define \begin{equation} \theta(0,\ldots,0):=\left(\begin{matrix}0 & 1 & \cdots & 0 \\ 0 & 0 & 1 & \vdots \\ \vdots & & & 1 \\ 0 & 0 & \cdots & 0 \end{matrix}\right) ,\label{eqn:theta0}\end{equation} which represents an element of $H^0\big( {\bar X}, End(W) \otimes K \otimes \xi)$. \begin{lemma}The bundle $(W_k,\theta(0,\ldots,0))$ is a parabolic stable Higgs bundle of para-\newline bolic degree zero.\end{lemma} \begin{pf} If $k$ is even, we have that the parabolic degree of $W_k$ is equal to $\frac{k(k+1)}{2}n + \frac{k}{2}(k+1)n=0$. In the case of odd $k$, it is easy to see that the degree (as a bundle) of $W_k$ is $0$, and since the weight is equal to $0$, we get that the parabolic degree of $W_k$ is zero. The invariant proper sub-bundles of \ref{eqn:theta0} are $L^{1-k}\otimes \xi^{-k/2},\ldots, L^{1-k}\otimes \xi^{-k/2}\oplus\cdots\oplus L^{(k/2)-1}\otimes\xi^{(k-4)/2}$, if $k$ is even; or $L^{1-k}\otimes \xi^{(1-k)/2},\ldots, L^{1-k}\otimes \xi^{(1-k)/2}\oplus\cdots\oplus L^{k-3}\otimes\xi^{(k-3)/2}$, if $k$ is odd. It is not difficult to see that all these sub-bundles have negative parabolic degree. \end{pf} Using the natural section $1_\xi$ of $\xi$, we embed the spaces $H^0({\bar X},K^j \otimes \xi^{j-1})$, $j=2,\ldots,k$, into $H^0({\bar X},End(W_k)\otimes K \otimes \xi)$. By an abuse of notation, if $a_j \in H^0({\bar X},K^j \otimes \xi^{j-1})$, we understand the above embedding as producing an element \begin{equation} \theta(a_2,\ldots,a_{k-1}):=\left(\begin{matrix}0 & 1 & \cdots & 0 \\ 0 & 0 & 1 & \vdots \\ \vdots & & & 1 \\ a_k & \cdots & a_2 & 0 \end{matrix}\right)\label{eqn:thetaa}\end{equation} of $H^0({\bar X},End(W_k) \otimes K \otimes \xi)$. Now, by the arguments of Hitchin, based on the openness of the stability of bundles, we get that the pair $(W_k, \theta (a_2,\ldots,a_k))$ is a stable parabolic Higgs bundle of parabolic degree $0$. Using these special Higgs bundles, one can obtain some information about the space of representations of the fundamental group of $X$ into SL($k,\Bbb R$). More precisely, our result is as follows. \begin{prop} The space of representations of the fundamental group of $X$ in SL($k,\Bbb R$), with fixed conjugacy class of monodromy around the punctures, has a component of real dimension $2(k^2-1)(g-1)+k(k-1)n$. \end{prop} \begin{pf} By the work of Simpson \cite{S2} and Balaji Srinivasan \cite{B}, we have a one-to-one continuous correspondence between the space $M$ of stable parabolic Higgs bundles of degree zero, and the space of representations of the fundamental group of $X$ into SL($k,\Bbb C$). Consider the parabolic dual of $W_k$, which is constructed as follows. First, take the dual bundle $W_k^*$ of $W_k$. If $k$ is odd, since the weight of the flag is $0$, we have that the parabolic dual of $W_k$ is $W_k^*$, with trivial flag at the points $p_i\in D$, and weight equal to zero. If $k$ is even, we have a weight of $-\frac{1}{2}$ associated to the trivial flag of $W_k^*$. Tensor $W_k^*$ with $\xi$ to obtain that the parabolic dual of $W_k$ is $W_k^* \otimes \xi$. So we always have that the parabolic dual of the bundle $W_k$ is $W_k$ itself. This implies that the image of the fundamental group under the representation induced by $(W_k,\theta)$ lies in SL($k,\Bbb R$). Since $a_j$ is a section of $K^j \otimes \xi^{j-1}$, we have that the residue of the Higgs field is invariant, {\it i.e.\/}\ $${\mathrm{residue}}\, (\theta(a_2,\ldots,a_{k-1})) = {\mathrm{residue}}\, (\theta(0,\ldots,0)) = \left(\begin{matrix}0 & 1 & \cdots & 0 \\ 0 & 0 & 1 & \vdots \\ \vdots & & & 1 \\ 0 & 0 & \cdots & 0 \end{matrix}\right) .$$ This implies that in the above representation, the conjugacy class of the elements corresponding to small loops around the punctures of $X$ is invariant. By the embedding of SL($2,\Bbb R$) into SL($k,\Bbb R$), we have that this is the class of the element \begin{equation} \left(\begin{matrix}1 & 1 & \cdots & 0 \\ 0 & 1 & 1 & \vdots \\ \vdots & & & 1 \\ 0 & 0 & \cdots & 1 \end{matrix}\right) .\end{equation} Using a bases, $\{p_1,\ldots,p_{k-1}\}$, for the set of invariant polynomials of the Lie algebra of SL($k,\Bbb C$), we can construct a continuous mapping $p:M\rightarrow \bigoplus_{j=2}^{k}H^0(\bar{X},K^j\otimes\xi^j)$, given by assigning to the Higgs field $(W_k,\Phi)$ the elements $(p_1(\Phi),\ldots,p_{k-1}(\Phi))$. The Higgs fields of the form (\ref{eqn:thetaa}) produce a section $s$ of $p$, defined over the closed subspace $\bigoplus_{j=2}^{k}H^0(\bar{X},K^j\otimes\xi^{j-1})$. Therefore, we have that the image of $s$ is closed. One can easily compute that the dimension (over $\Bbb R$) of the space of sections $\bigoplus_{j=2}^{k}H^0(\bar{X},K^j\otimes\xi^{j-1})$ is equal to $$\sum_{j=2}^k 2(2j-1)(g-1)+2\sum_{j=2}^k(j-1)n=2(k^2-1)(g-1)+k(k-1)n.$$ On the other hand, the dimension of the space of representations of the fundamental group of $X$ into SL($k,\Bbb R$), with the condition that the monodromy around the punctures lies in the above conjugacy class, can be computed as follows. The fundamental group of $X$ can be identified with a group of M\"obius transformations (or elements of SL($2,\Bbb R$)), generated by elements $\{c_1,d_1,\ldots,c_g,d_g,e_1,\ldots,e_n\}$, and with one relation of the form $\prod_{j=1}^g[c_j,d_j]\prod_{j=1}^n e_j=id$, where $[c,d]=cdc^{-1}d^{-1}$ denotes the commutator of the elements $c$ and $d$. In classical terms, the transformations $c_j$'s and $d_j$'s are hyperbolic, that is conjugate to dilatations, while the $e_j$'s are parabolic, or conjugate to translations. In terms of loops on $X$, we have that the $c_j$'s and $d_j$'s can be identified with paths around the handles of $X$, while the $e_j$'s are simple loops around the punctures. The image of the elements $c_j$ and $d_j$ depends on dim(SL($k,\Bbb R$))=$k^2-1$ parameters. In order to compute the number of parameters of the elements $e_j$, first observe that these transformations belong to the conjugacy classes of elements of $U=\{$regular unipotent elements of PSL($2, \Bbb R$)$\}$. Any matrix of SL($k,\Bbb R$) can be written as $ldu$, where $l$ is unipotent and lower triangular, $d$ is diagonal, and $u$ is unipotent upper triangular. We therefore have $(ldu)U(ldu)^{-1}=lUl^{-1}$. So the conjugacy class of $U$ depends on $k(k-1)$ parameters. Therefore, we have that the real dimensions of $\bigoplus_{j=2}^{k}H^0(\bar{X},K^j\otimes\xi^{j-1})$ and the space of representations of $\pi_1(X)$, with fixed conjugacy class for the monodromy elements around the punctures, are equal. Standard arguments using the invariance of domain theorem complete the proof.\end{pf} \ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi

9510

alg-geom/9510005

en

https://arxiv.org/abs/alg-geom/9510005

alg-geom/9510005

Richard Wentworth

Georgios Daskalopoulos, and Richard Wentworth

On the Brill-Noether Problem for Vector Bundles

LaTeX 2e (amsart)

On an arbitrary compact Riemann surface, necessary and sufficient conditions are found for the existence of semistable vector bundles with slope between zero and one and a prescribed number of linearly independent holomorphic sections. Existence is achieved by minimizing the Yang-Mills-Higgs functional.

alg-geom

\section{Introduction} In this note we exhibit the existence of semistable vector bundles on compact Riemann surfaces with a prescribed number of linearly independent holomorphic sections. This may be regarded as a higher rank version of the classical Brill-Noether problem for line bundles. Fix a compact Riemann surface $\Sigma$ of genus $g\geq 2$ and integers $r$ and $d$ satisfying \begin{equation} 0\leq d \leq r\ , \quad r\geq 2\ . \label{d-range} \end{equation} Then the main result may be stated as follows: \begin{Main} Let $k$ be a positive integer and suppose that $r$ and $d$ satisfy \eqref{d-range}. Then the necessary and sufficient conditions for the existence of a semistable bundle on $\Sigma$ with at least $k$ linearly independent holomorphic sections are $k\leq r$ and if $d\neq 0$, $r\leq d+(r-k)g$. \end{Main} By analogy with the classical situation of special divisors (cf.\ \cite{ACGH,N}) one can define the higher rank version of the Brill-Noether number: \begin{equation} \rho^{k-1}_{r,d}= r^2(g-1)+1-k(k-d+r(g-1))\ . \end{equation} Then $\rho^{k-1}_{r,d}$ is the formal dimension of the locus $W^{k-1}_{r,d}$ in the moduli space of semistable bundles of rank $r$ and degree $d$. $W^{k-1}_{r,d}$ is defined as the closure of the set of stable bundles with at least $k$ linearly independent sections. Note that the condition in the Main Theorem implies that $\rho^{k-1}_{r,d}\geq 1$, except in the trivial case $d=0$ where $W^{k-1}_{r,d}$ is necessarily empty. The converse, in general, is not true. Thus, unlike the case of divisors, there are situations where $\rho^{k-1}_{r,d}\geq 0$ and $W^{k-1}_{r,d}=\emptyset$. It would be interesting to improve the Main Theorem to a statement concerning stable bundles; however, our method does not immediately imply such a result except in special cases. We do have the following: \begin{MainCor} {\rm (i) (see \cite[Thm III.2.4]{S})} For $d>0$ and any rank $r$, there exists a stable bundle of rank $r$ and degree $d$ with a nontrivial holomorphic section. {\rm (ii)} If $0<d<r$ and $r\leq d+g$, then there exists a stable bundle of rank $r$ and degree $d$ with precisely $r-1$ linearly independent sections. \end{MainCor} Instead of the constructive approach to theorems of this type taken in references \cite{S,T}, we use a variational method. More precisely, we study the Morse theory of the Yang-Mills-Higgs functional (cf.\ \cite{B}). The idea is simply the following: Let $(A^i,\vec\varphi^i)$ be a minimizing sequence with respect to the Yang-Mill-Higgs functional~\eqref{YMH}. Here, $\vec\varphi_i=(\varphi^i_1,\ldots,\varphi^i_k)$ is a $k$-tuple of linearly independent holomorphic sections with respect to $A^i$. The minimal critical values correspond to solutions to the $k$-$\tau$-vortex equations, which for an appropriate choice of $\tau$ imply that the limiting holomorphic structure is semistable (cf.\ \cite{BDW}). If the sequence is assumed to converge to a nonminimal critical value, then we show that under the assumptions of the Main Theorem there exist ``negative directions" which contradict the fact that the sequence is minimizing. The energy estimates used closely follow \cite{D}. However, an extra combinatorial argument is needed to ensure that the bundles constructed have the correct number of holomorphic sections, and this is where the assumption $r\leq d+(r-k)g$ is needed. We have been informed that the Main Theorem stated above has been proven using somewhat different methods in \cite{BGN}. \medskip \noindent {\it Acknowledgements.} The authors would like to thank L. Brambila Paz for introducing them to this problem and for several useful discussions during the preparation of this manuscript. They are also grateful for the warm hospitality of UAM, Mexico and the Max-Planck Institute in Bonn, where a portion of this work was completed. \section{The Yang-Mills-Higgs Functional} \label{S:functional} Let $\Sigma$, $d$, and $r$ be as in the Introduction, and let $k$ be a positive integer. Let $E$ be a fixed hermitian vector bundle on $\Sigma$ of rank $r$ and degree $d$. Let $\mathcal{A}$ denote the space of hermitian connections on $E$, $\Omega^0(E)$ the space of smooth sections of $E$, and $\mathcal{H}\subset \mathcal{A}\times \Omega^0(E)^{\oplus k}$ the subspace consisting of holomorphic $k$-pairs. Thus, $$ \mathcal{H}=\left\{ \left(A, \vec\varphi=(\varphi_1,\ldots,\varphi_k)\right) : D''_A\varphi_i=0 , \ i=1,\ldots,k\right\}\ . $$ Given a real parameter $\tau$, we define the Yang-Mills-Higgs functional: \begin{align} f_\tau &: \mathcal{A}\times\Omega^0(E)^{\oplus k}\longrightarrow \Bbb R \notag \\ f_\tau(A,\vec\varphi)&= \Vert F_A\Vert^2 + \sum_{i=1}^k\Vert D_A\varphi_i\Vert^2 +\frac{1}{4}\left\Vert \sum_{i=1}^k \varphi_i\otimes\varphi_i^\ast-\tau {\bf I}\, \right\Vert^2 -2\pi\tau d \label{YMH} \end{align} In the above, the $\Vert\cdot\Vert$ denotes $L^2$ norms. Using a Weitzenb\"ock formula we obtain (cf.\ \cite[Theorem 4.2]{B}) $$ f_\tau(A,\vec\varphi)=2\sum_{i=1}^k\Vert D''_A \varphi_i\Vert^2+\left\Vert\sqrt{-1}\Lambda F_A+\frac{1}{2}\sum_{i=1}^k\varphi_i\otimes\varphi_i^\ast-\frac{\tau}{2}{\bf I}\, \right\Vert^2\ , $$ and therefore the absolute minimum of $f_\tau$ consists of holomorphic $k$-pairs satisfying the $k$-$\tau$-vortex equations discussed in \cite{BDW}. \begin{Prop} \label{P:gradient} {\rm (i)} The $L^2$-gradient of $f_\tau$ is given by \begin{align*} \left(\nabla_{(A,\vec\varphi)} f_\tau\right)_1 &= D_A^\ast F_A+\frac{1}{2}\sum_{j=1}^k\left( D_A\varphi_j\otimes\varphi_j^\ast-\varphi_j\otimes D_A\varphi_j^\ast\right) \\ \left(\nabla_{(A,\vec\varphi)} f_\tau\right)_{2,i} &= \Delta_A\varphi_i-\frac{\tau}{2}\varphi_i+\frac{1}{2}\sum_{j=1}^k\langle \varphi_i,\varphi_j\rangle\varphi_j \end{align*} {\rm (ii)} If $(A,\vec\varphi)\in\mathcal{H}$, then under the usual identification $\Omega^0(\Sigma,{\rm ad}\, E)\simeq$ \break $\Omega^{0,1}(\Sigma,{\rm End}\, E)$, we have \begin{align*} \left(\nabla_{(A,\vec\varphi)} f_\tau\right)_1 &= -D''_A\left(\sqrt{-1}\Lambda F_A+\frac{1}{2}\sum_{j=1}^k\varphi_j\otimes\varphi_j^\ast\right) \\ \left(\nabla_{(A,\vec\varphi)} f_\tau\right)_{2,i} &= \sqrt{-1}\Lambda F_A(\varphi_i)-\frac{\tau}{2}\varphi_i+\frac{1}{2}\sum_{j=1}^k\langle \varphi_i,\varphi_j\rangle\varphi_j \end{align*} {\rm (iii)} If $(A,\vec\varphi)\in\mathcal{H}$ is a critical point of $f_\tau$, then either {\rm (I)} $\vec\varphi\equiv 0$ and $A$ is a direct sum of Hermitian-Yang-Mills connections (not necessarily of the same slope), or {\rm (II)} $A$ splits as $A=A'\oplus A_Q$ on $E=E'\oplus E_Q$, where $(A',\vec\varphi)$ solves the $k$-$\tau$-vortex equations and $A_Q$ is a direct sum of Hermitian-Yang-Mills connections (not necessarily of the same slope). \end{Prop} \begin{proof} (i) is a standard calculation, and (ii) follows from (i) via the K\"ahler identities. We are going to prove (iii). If $(A,\vec\varphi)$ is critical, then since $ \sqrt{-1}\Lambda F_A+\frac{1}{2}\sum_{j=1}^k\varphi_j\otimes\varphi_j^\ast $ is a self-adjoint holomorphic endomorphism, it must give a splitting $A=A_0\oplus\cdots\oplus A_\ell$ according to its distinct (constant) eigenvalues $\sigma_0,\ldots,\sigma_\ell$. Write $$ \sqrt{-1}\Lambda F_A= \begin{pmatrix} -\frac{1}{2}\sum_{j=1}^k\varphi_j\otimes\varphi_j^\ast +\sigma_0\ {\bf I} & 0 & \cdots & 0 \\ 0& \sigma_1\ {\bf I} & & \vdots \\ \vdots && \ddots & \\ 0 & \cdots && \sigma_\ell\ {\bf I} \end{pmatrix} \quad . $$ Thus, \begin{align*} 0 &= \sqrt{-1}\Lambda F_A(\varphi_i)-\frac{\tau}{2}\varphi_i+\frac{1}{2}\sum_{j=1}^k\langle \varphi_i,\varphi_j\rangle\varphi_j\\ &= -\frac{1}{2}\sum_{j=1}^k\langle\varphi_i, \varphi_j\rangle\varphi_j+\sigma_0\varphi_i -\frac{\tau}{2}\varphi_i+\frac{1}{2}\sum_{j=1}^k\langle \varphi_i,\varphi_j\rangle\varphi_j\\ &= \left(\sigma_0-\frac{\tau}{2}\right)\varphi_i\ , \end{align*} from which we obtain either Case I or Case II, depending upon whether $\vec\varphi\equiv 0$. \end{proof} Next, recall that $\mathcal{H}$ is an infinite dimensional complex analytic variety whose tangent space is given by the kernel of a certain differential defined in \cite[3.15]{BDW}. Moreover, $\mathcal{H}$ is preserved by the action of the complex gauge group ${\G}^{\CBbb}$. We have the following: \begin{Prop} \label{P:tangent} If $(A,\vec\varphi)\in \mathcal{H}$, then $\nabla_{(A,\vec\varphi)}f_\tau$ is tangent to the orbits of ${\G}^{\CBbb}$. In particular, $\nabla_{(A,\vec\varphi)}f_\tau$ is tangent to $\mathcal{H}$ itself. \end{Prop} \begin{proof} Set $ u=\sqrt{-1}\Lambda F_A+\frac{1}{2}\sum_{j=1}^k\varphi_j\otimes\varphi_j^\ast-\frac{\tau}{2}{\bf I} $. By Proposition \ref{P:gradient} (ii) we have that $\nabla_{(A,\vec\varphi)}f_\tau=d_1(u)$, where $d_1$ is the differential defined in \cite[3.15]{BDW}. The Proposition follows. \end{proof} Because of Proposition \ref{P:tangent}, the critical points of the functional $f_\tau$ restricted to $\mathcal{H}$ are characterized by Proposition \ref{P:gradient} (iii). A solution $(A(t),\vec\varphi(t))$, $t\in [0,t_0)$ to the initial value problem \begin{equation} \label{flow} \left(\frac{\partial A}{\partial t}, \frac{\partial\vec\varphi}{\partial t}\right) = -\nabla_{(A,\vec\varphi)}f_\tau\ , \quad \left( A(0),\vec\varphi(0)\right) = (A_0,\vec\varphi_0)\ , \end{equation} is called the $L^2$-gradient flow of $f_\tau$ starting at $(A_0,\vec\varphi_0)$. Notice that \begin{equation} \label{E:decay} \frac{d}{dt}f_\tau(A(t),\vec\varphi(t)) =-\left\Vert\nabla_{(A(t),\vec\varphi(t))} f_\tau\right\Vert^2\ , \end{equation} and so the energy decreases along the $L^2$-gradient flow. \begin{Prop} \label{P:shorttime} Given $(A_0,\vec\varphi_0)\in\mathcal{H}$, there is a $t_0>0$ such that the $L^2$-gradient flow exists for $0\leq t< t_0$. \end{Prop} \begin{proof} The proof is an application of the implicit function theorem as in \cite{R}. \end{proof} \section{Technical Lemmas} \label{S:technical} In this section we collect several results needed for the proof of the Main Theorem. Throughout, $E$ will denote a holomorphic bundle of rank $r$ and degree $d$ on the compact Riemann surface $\Sigma$. \begin{Lem} \label{L:bounds} Let $E$ be as above with $0\leq d\leq r$ and $h^0(E)=k$. If either {\rm (i)} $E$ is semistable, or {\rm (ii)} $E$ satisfies the k-$\tau$-vortex equation for some $0<\tau<1$ and $E$ does not contain the trivial bundle as a split factor; then $k\leq r$ and if $d\neq 0$, $r\leq d+(r-k)g$. \end{Lem} \begin{proof} We first show that $k\leq r$. Suppose $k\geq r$. Thus, $E$ has at least $r$ linearly independent holomorphic sections. If the sections generate $E$ at every point, then $E\simeq\mathcal{O}^{\oplus r}$; in which case $d=0$ and $k=r$. Suppose the sections fail to generate at every point. Then we can find a point $p\in\Sigma$ and a section of $E$ vanishing at $p$. Thus $E$ contains $\mathcal{O}(p)$ as a subsheaf, which is a contradiction to (ii). If (i) is assumed, then $E$ is strictly semistable with $d=r$, and the bound $k\leq r$ follows from induction on the rank. Note that the second inequality is also satisfied in this case. Assume $0< d< r$. In both cases (i) and (ii) we obtain $0\to \mathcal{O}^{\oplus k} \xrightarrow{\pi} E\to F\to 0$, where $F$ is locally free. By dualizing and taking the resulting long exact sequence in cohomology, we find $$ 0\longrightarrow H^0(F^\ast)\longrightarrow H^0(E^\ast) \xrightarrow{\delta} H^0(\mathcal{O}^{\oplus k}) \longrightarrow H^1(F^\ast) \ . $$ We are going to show that $H^0(E^\ast)=0$. The result then follows by the Riemann-Roch formula. For (i), $H^0(E^\ast)=0$ by semistability. For (ii), note first that $\delta=0$. For if not, there would be a section $s:\mathcal{O}\to E^\ast$ with $\pi^\ast\circ s=\sigma\neq 0$. But $\sigma$ could not have any zeros, and so $\mathcal{O}$ would be a split factor in $E^\ast$; hence, also in $E$. Secondly, we show that $H^0(F^\ast)=0$. Let $L\subset F^\ast$ be a subbundle. Then $\tau$-stability immediately implies $c_1(L^\ast) > \tau >0$. Thus, in particular, $F^\ast$ cannot contain $\mathcal{O}$ as a subsheaf. This completes the proof. \end{proof} \begin{Lem} \label{L:extension} Let $E_1$, $E_2$ be holomorphic bundles of rank $r_1, r_2$ and degree $d_1, d_2$, satisfying $0\leq \mu_1=d_1/r_1 < d_2/r_2=\mu_2\leq 1$. Suppose $h^0(E_1)=k_1 \leq r_1$, $h^0(E_2)=k_2\leq r_2$, and $$ d_2+(r_2-k_2-1)g < r_2\leq d_2+(r_2-k_2)g\ . $$ Furthermore, \begin{itemize} \item If $d_1\neq 0$ assume $r_1\leq d_1+(r_1-k_1)g$. \item If $d_1=0$ and $k_1=r_1$, assume $r_2 < d_2+(r_2-k_2)g$. \end{itemize} Then there exists a nontrivial extension $0\to E_1\to E\to E_2\to 0$ such that $h^0(E)= k_1+k_2$. \end{Lem} \begin{proof} If $k_2=0$, the result follows from Riemann-Roch. Suppose $k_2\geq 1$. The condition that the $k_2$ sections of $E_2$ lift for some nontrivial extension is $k_2 h^1(E_1) < h^1(E_1\otimes E_2^\ast)$. Notice that \begin{align*} h^1(E_1) &= h^0(E_1)-d_1+r_1(g-1)=k_1-d_1+r_1(g-1) \\ h^1(E_1\otimes E_2^\ast) &= h^0(E_1\otimes E_2^\ast)+r_1 r_2(\mu_2-\mu_1+g-1) \\ &\geq r_1 r_2(\mu_2-\mu_1+g-1)\ , \end{align*} hence, it suffices to show that $$ k_2(k_1-d_1+r_1(g-1)) < r_1 r_2(\mu_2-\mu_1+g-1)\ , $$ or equivalently, that \begin{equation} \label{E:one} r_1(d_2-r_2+(r_2-k_2)g)-r_2 d_1 + k_2 d_1 -k_1 k_2 + k_2 r_1 > 0\ . \end{equation} Now if $k_2=r_2=d_2$, then \eqref{E:one} is trivially satisfied by the hypotheses. Similarly for $d_1=0$. So assume $k_2\leq r_2-1$, $d_1\neq 0$. Write $d_2=r_2-(r_2-k_2)g+p$, where $0\leq p < g$ by assumption. On the other hand, $$ d_1 < r_1\frac{d_2}{r_2}\leq \left(d_1+(r_1-k_1)g\right)\frac{r_2-(r_2-k_2)g+p}{r_2}\ , $$ which is equivalent to $$ -\frac{d_1 p}{g}+k_1 p + (r_1-k_1)(r_2-k_2)(g-1) < r_1 p-r_2d_2+k_2 d_1-k_1 k_2+k_2 r_1 \ . $$ Therefore, \eqref{E:one} will follow from \begin{equation} \label{E:two} -\frac{d_1 p}{g}+k_1 p + (r_1-k_1)(r_2-k_2)(g-1)\geq 0\ . \end{equation} Now if $p=0$ then \eqref{E:two} is trivially satisfied. Assume that $1\leq p\leq g-1$. Then \begin{align*} -\frac{d_1 p}{g} &+k_1 p + (r_1-k_1)(r_2-k_2)(g-1) \\ &\geq -d_1+r_1 p-(r_1-k_1)p+(r_1-k_1)(r_2-k_2)(g-1) \\ &\geq (r_1-d_1)+(r_1-k_1)(r_2-k_2-1)(g-1) \\ &\geq 0 \ , \end{align*} which proves \eqref{E:two}, \eqref{E:one}, and hence the Lemma. \end{proof} In order to get an upper bound on the infimum of the Yang-Mills-Higgs functional in the next section, we shall need the following construction and energy estimate: \begin{Lem} \label{L:special} Assume $0< d < r$, $k\geq 1$, and $r\leq d+(r-k)g$. Let $F$ be a holomorphic bundle of degree $d$ and rank $r-1$ with $h^0(F)=k-1$. Then there exists a non-split extension $0\to\mathcal{O}\to E\to F\to 0$ with $h^0(E)=k$. \end{Lem} \begin{proof} The condition for all of the sections of $F$ to lift is \begin{align*} (k-1)h^1(\mathcal{O}) < h^1(F^\ast)\ &\iff\quad g(k-1) < d+(r-1)(g-1) \\ &\iff\quad r< d+(r-k)g+1\ , \end{align*} and hence the result. \end{proof} \begin{Prop}[{cf.\ \cite[Prop. 3.5]{D}}] \label{P:energyestimate} Let $E_1, E_2$ be hermitian bundles with slope $\mu_1, \mu_2$. Let $A_1, A_2$ be hermitian connections on $E_1, E_2$, and $\vec\varphi^1, \vec\varphi^2$ be $k_1$ and $k_2$ tuples of holomorphic sections. Set $k=k_1+k_2$. Let $\tau_1, \tau_2$ and $\tau$ be real numbers satisfying $\mu_1\leq\tau_1\leq \tau < \mu_2 \leq\tau_2$, and assume that $(A_1,\vec\varphi^1)$ and $(A_2,\vec\varphi^2)$ satisfy the $\tau_1$ and $\tau_2$ vortex equations, respectively. Set $E=E_1\oplus E_2$, $\varphi_i=(\varphi^1_i,0)$ for $i=1,\ldots, k_1$, and $\varphi_{k_1+i}=(0,\varphi^2_i)$ for $i=1,\ldots, k_2$. Then there exist constants $\varepsilon_1, \varepsilon_2 , \eta >0$ such that for all $$ \beta\in H^{0,1}\left(\Sigma, {\rm Hom}(E_2,E_1)\right)\ ,\quad \vec\psi\in\Omega^0(E)^{\oplus k}\ , $$ with $\Vert\beta\Vert =\varepsilon_1$, $\Vert\vec\psi\Vert\leq \varepsilon_2$, and $$ \left( A_\beta= \begin{pmatrix} A_1 & \beta \\ 0 & A_2 \end{pmatrix}, \vec\varphi+\vec\psi\right)\in\mathcal{H}\ , $$ it follows that $ f_\tau(A_\beta,\vec\varphi+\vec\psi) < f_\tau(A_1\oplus A_2,\vec\varphi)-\eta $. \end{Prop} \begin{proof} By assumption, $$ \sqrt{-1}\Lambda F_{A_\ell}+\frac{1}{2}\sum_{j=1}^{k_\ell} \varphi^{\ell}_j\otimes(\varphi^{\ell}_j)^\ast = \frac{\tau}{2}{\bf I}_{\ell}\ ,\quad \ell=1,2\ . $$ It follows that $$ \sqrt{-1}\Lambda F_{A_1\oplus A_2}+\frac{1}{2}\sum_{j=1}^{k_1}\varphi^1_j\otimes(\varphi^1_j)^\ast +\frac{1}{2}\sum_{j=1}^{k_2}\varphi^2_j \otimes(\varphi^2_j)^\ast-\frac{\tau}{2}{\bf I}= \begin{pmatrix} \frac{\tau_1-\tau}{2}{\bf I}_1 & 0 \\ 0 & \frac{\tau_2-\tau}{2}{\bf I}_2 \end{pmatrix} $$ The argument of \cite[pp.\ 715-716]{D} shows that there is a constant $\varepsilon_1$ such that for $\beta$ and $A_\beta$ as in the statement, $$ f_\tau\left( A_\beta,\varphi^1_1,\ldots,\varphi^1_{k_1}, \varphi^2_1,\ldots,\varphi^2_{k_2}\right) < f_\tau\left( A_1\oplus A_2,\varphi^1_1,\ldots,\varphi^1_{k_1}, \varphi^2_1,\ldots,\varphi^2_{k_2}\right) \ . $$ Now if we take $\varepsilon_2$ sufficiently small the Proposition follows (note that which norms we use is irrelevent, since $\beta$ and $\vec\varphi+\vec\psi$ satisfy elliptic equations, and hence the $L^2$ norm is equivalent to any other). \end{proof} \section{Proof of the Main Theorem} \label{S:proof} Necessity of the conditions follows from Lemma \ref{L:bounds}, and sufficiency for $d=0$ or $d=r$ is clear as well. To prove sufficiency in general, we shall proceed by induction on the rank. The case $r=2$, $d=1$ is clear from a direct construction. Assume the Main Theorem holds for bundles of rank $< r$. We show that it holds for $r$ as well. Let $\mathcal{H}^\ast\subset\mathcal{H}$ denote the subspace of $k$-pairs $\left(A,\vec\varphi=(\varphi_1,\ldots,\varphi_k)\right)$ such that the sections $\varphi_1,\ldots,\varphi_k$ are linearly independent. Fix $\tau$ as in Assumption 1 of \cite{BD}, i.e. $\mu(E)<\tau=\mu(E)+\gamma < \mu_+$, where $\mu_+$ is the smallest possible slope greater that $\mu=\mu(E)$ of a subbundle of $E$ (note that $0<\tau<1$ and that we also normalize the volume of $\Sigma$ to be $4\pi$). \begin{Lem} \label{L:inf} Let $m=\inf_{\mathcal{H}^\ast} f_\tau$. Then $0\leq m<\pi/(r-1)$. \end{Lem} \begin{proof} Let $F$ be a vector bundle of degree $d$ and rank $r-1$. Then by the inductive hypothesis, we may assume there exist hermitian connections $A_1$ and $A_2$ on $\mathcal{O}$ and $F$, respectively, and holomorphic sections $\varphi_1\neq 0$ in $H^0(\Sigma,\mathcal{O})$, and $\varphi_2,\ldots,\varphi_k$ linearly independent sections in $H^0(\Sigma,F)$, such that $(A_1,\varphi_1)$ and $(A_2,\varphi_2,\ldots,\varphi_k)$ satisfy the $\tau_1$ and $\tau_2$ vortex equations, respectively, for $\tau_1=\tau$, $\tau_2=d/(r-1)+\gamma$. It follows from Lemma \ref{L:special} and Proposition \ref{P:energyestimate} that there is a nontrivial extension $\beta: 0\to\mathcal{O}\to E\to F\to 0$, and $\vec\psi$ such that $(A_\beta,\vec\varphi+\vec\psi)\in\mathcal{H}^\ast$ and \begin{align*} f_\tau(A_\beta,\vec\varphi+\vec\psi)&<f_\tau(A_1\oplus A_2,\varphi_1,\ldots,\varphi_k)-\eta \\ &= \left\Vert\frac{1}{2}\left(\frac{d}{r-1}-\frac{d}{r}\right){\bf I_F}\, \right\Vert^2-\eta < \frac{\pi}{r-1}\ . \end{align*} \end{proof} Let $(A^i,\vec\varphi^i)$ be a minimizing sequence in $\mathcal{H}^\ast$. Thus, $f_\tau(A^i,\vec\varphi^i)\to m$. By weak compactness (more precisely, see the argument in \cite[Lemma 5]{BD}) there is a subsequence converging to $(A^\infty,\vec\varphi^\infty)$ in the $C^\infty$ topology. By the continuity of $f_\tau$ with respect to the $C^\infty$ topology, Propositions \ref{P:shorttime} and \ref{P:tangent}, and equation \eqref{E:decay}, it follows that $(A^\infty,\vec\varphi^\infty)$ is a critical point of $f_\tau$. If the holomorphic structure $E^\infty$ defined by $A^\infty$ is semistable, then by semicontinuity of cohomology we are finished. We therefore assume $E^\infty$ is unstable and derive a contradiction. According to Proposition \ref{P:gradient} (iii) we must consider the following cases: \begin{align*} \vec\varphi^\infty &= 0\ , \quad E^\infty=E_1\oplus\cdots\oplus E_\ell \tag{I} \\ \vec\varphi^\infty &\neq 0\ , \quad E^\infty=E_{\varphi}\oplus E_1\oplus\cdots\oplus E_\ell \tag{II} \end{align*} Set $\mu_j=\mu(E_j)$, and assume $\mu_1 < \cdots < \mu_\ell$. If $\mu_\ell > 1$ (or similarly, $\mu_1 < 0$), then $$ f_\tau\left(A^\infty,\vec\varphi^\infty\right)\geq \pi(\mu_\ell-\tau)^2 r_\ell \geq \pi(\mu_\ell-1)^2 r_\ell \geq\frac{\pi}{r_\ell}\geq\frac{\pi}{r-1} > m\ , $$ contradicting Lemma \ref{L:inf}. We therefore rule out this possibility. We will consider Cases I and II separately. \medskip \noindent \emph{Case I}\@. Let $k_i=h^0(E_i)$. By semicontinuity of cohomology, $\sum_{i=1}^\ell k_i\geq k$. If $\mu_\ell =1$, then we may replace $E_\ell$ by a Hermitian-Yang-Mills bundle $\widehat E_\ell$ with exactly $\hat k_\ell=r_\ell$ sections. Hence, we may assume that $$ d_\ell + (r_\ell -\hat k_\ell-1)g < r_\ell \leq d_\ell +(r_\ell-\hat k_\ell)g\ {}. $$ For $1<i<\ell$, the inductive hypothesis implies that we may replace $E_i$ by a Hermitian-Yang-Mills bundle $\widehat E_i$ with $$ h^0(\widehat E_i)=\hat k_i=\left[ \frac{d_i+r_i(g-1)}{g}\right]\ , $$ the maximal number of sections allowed for $d_i, r_i$, and $g$. Note that \begin{equation} \label{E:max} d_i + (r_i -\hat k_i-1)g < r_i \leq d_i +(r_i-\hat k_i)g\ . \end{equation} If $\mu_1\neq 0$, then we can replace $E_1$ by $\widehat E_1$ as above. If $\mu_1=0$, we may replace $E_1$ with $\mathcal{O}^{\oplus r_1}$, with $\hat k_1=r_1\geq k_1$ sections. By our choices of $\hat k_i$, $\sum_{i=1}^\ell \hat k_i\geq \sum_{i=1}^\ell k_i\geq k$. Let $0\leq\mu_1<\cdots<\mu_s\leq \mu <\mu_{s+1} <\cdots <\mu_\ell\leq 1$. Suppose first that $\mu_s\neq 0$. By Lemma \ref{L:extension} there is a nontrivial extension $0\to\widehat E_s\to G\to \widehat E_{s+1}\to 0$, with $h^0(G)=\hat k_s+\hat k_{s+1}$. Thus, $$ h^0\left(\widehat E_1\oplus\cdots\oplus\widehat E_{s-1}\oplus G\oplus \widehat E_{s+1}\oplus \cdots\oplus \widehat E_\ell\right)=\sum_{i=1}^\ell \hat k_i\geq k\ . $$ On the other hand, by Proposition \ref{P:energyestimate} there is a hermitian connection on $\widehat E_1\oplus\cdots\oplus\widehat E_{s-1}\oplus G\oplus \widehat E_{s+1}\oplus \cdots\oplus \widehat E_\ell$ and linearly independent sections $\varphi_1,\ldots, \varphi_k$ such that $f_\tau(A,\vec\varphi)< f_\tau(A_\infty, 0)=m$, contradicting the minimality of $(A_\infty, 0)$. Now suppose $\mu_s=\mu_1=0$, $\mu < \mu_i$ for $2\leq i\leq \ell$. If for any $2\leq i\leq \ell$ we have $r_i < d_i+(r_i-\hat k_i)g$, then by Lemma \ref{L:extension} there is a nontrivial extension $0\to\widehat E_1\to G\to \widehat E_i\to 0$, with $h^0(G)=\hat k_1+\hat k_i$, and Proposition \ref{P:energyestimate} yields a contradiction as before. Suppose that for all $2\leq i\leq \ell$, $r_i=d_i+(r_i-\hat k_i)g$. We claim that $\sum_{i=1}^\ell \hat k_i > k$. For if $\sum_{i=1}^\ell \hat k_i = k$, then $\sum_{i=2}^\ell (r_i-\hat k_i)= r-k$, and hence $$ r > \sum_{i=2}^\ell r_i = \sum_{i=2}^\ell d_i+ (r_i-\hat k_i)g = d+ (r-k)g\ ; $$ a contradiction. Thus, we may replace $\widehat E_1$ by a bundle $\widehat E_1^\prime$ having $\hat k_1^\prime=\hat k_1-1$ sections. According to Lemma \ref{L:extension} there is a nontrivial extension $0\to\widehat E_1^\prime \to G\to \widehat E_2\to 0$, with $h^0(G)=\hat k_1^\prime +\hat k_2$, $\hat k_1^\prime +\sum_{i=2}^\ell\hat k_i\geq k$, and Proposition \ref{P:energyestimate} yields a contradiction as before. \medskip \noindent \emph{Case II}\@. First notice that by the invariance of the Yang-Mills-Higgs equations under the natural action by U($k$), we may assume that $\varphi_1^\infty, \ldots, \varphi_k^\infty$ form an $L^2$-orthogonal set of sections. In particular, we may assume that there exists $s\leq k$ such that $\varphi_1^\infty,\ldots,\varphi_s^\infty$ are linearly independent and $\varphi_{s+1}^\infty,\ldots,\varphi_k^\infty\equiv 0$. Write $E_\varphi=E_\varphi^\prime\oplus\mathcal{O}^{\oplus t}$, where $E_\varphi^\prime $ contains no split factor of $\mathcal{O}$. Set $k_i=h^0(E_i)$, $k_\varphi=h^0(E_\varphi)$, $k_\varphi^\prime =h^0(E_\varphi^\prime)=k_\varphi-t$. By semicontinuity of cohomology, $k_\varphi+\sum_{i=1}^\ell k_i\geq k$. As in Case I, we may replace each $E_i$ by a Hermitian-Yang-Mills bundle $\widehat E_i$ such that $h^0(\widehat E_i)=\hat k_i\geq k_i$, and \eqref{E:max} is satisfied for $i=1,\ldots,\ell$. On the other hand, since $E_\varphi$ satisfies the $k$-$\tau$-vortex equation for $\tau=\mu+\gamma$ as above, it follows that $E_\varphi^\prime$ is $\tau$-stable. Therefore, $0\neq \mu(E_\varphi^\prime)\leq \mu=\mu(E)$; and since $\tau < 1$, we obtain from Lemma \ref{L:bounds} that $r_\varphi^\prime \leq d_\varphi+(r_\varphi^\prime-k_\varphi^\prime)g$. Finally, notice that since $E^\infty$ is unstable, $\mu_\ell > \mu$. We may now apply Lemma \ref{L:extension} to $E_\varphi^\prime$ and $\widehat E_\ell$ to obtain a nontrivial extension $0\to E_\varphi^\prime\to G\to \widehat E_\ell\to 0$, with $h^0(G)=k_\varphi^\prime+\hat k_\ell$. It follows that $$ h^0\left( G\oplus\mathcal{O}^{\oplus t}\oplus \widehat E_1\oplus\cdots\oplus\widehat E_{\ell-1} \right) = k_\varphi +\sum_{i=1}^\ell \hat k_i \geq k\ . $$ By Proposition \ref{P:energyestimate} there is a hermitian connection $A$ on $G\oplus\mathcal{O}^{\oplus t}\oplus \widehat E_1\oplus\cdots\oplus\widehat E_{\ell-1}$ and linearly independent sections $\varphi_1,\ldots,\varphi_k$ extending $\varphi_1^\infty,\ldots,\varphi_s^\infty$ such that $f_\tau(A,\vec\varphi)< f_\tau(A_\infty,\vec\varphi^\infty)=m$, again contradicting the minimality of $m$. This completes the proof of the Main Theorem. \section{Stable Bundles} We conclude by proving the Corollary stated in the Introduction. Consider first part (ii). The upper bound follows from Lemma \ref{L:bounds}. By the Main Theorem, it suffices to show that if $E$ is semistable with $0<\mu < 1$ and $h^0(E)=r-1$, then $E$ is stable. Suppose to the contrary. Then we can find a semistable subbundle $E'$ with $Q=E/E'$ stable and $\mu(E')=\mu(Q)=\mu$. By Lemma \ref{L:bounds}, $h^0(E')\leq r'-1$, and $h^0(Q)\leq r_Q-1$; contradiction. Now consider part (i). Tensoring by ample line bundles allows us to restrict to the case $0<d\leq r$. Let $\mathfrak{B}_\tau$ be the set of gauge equivalence classes of solutions to the (one section) $\tau$-vortex equation for bundles of rank $r$ and degree $d$. By the proof of the Main Theorem, $\mathfrak{B}_\tau\neq\emptyset$. One can therefore show as in \cite{BDW,BD} that $\mathfrak{B}_\tau$ is a smooth projective variety of dimension $(r^2-r)(g-1)+d$ with a morphism $\psi : \mathfrak{B}_\tau\to\mathfrak{M}(r,d)$, where $\mathfrak{M}(r,d)$ is the moduli space of semistable bundles of rank $r$ and degree $d$. The image of $\psi$ is precisely the set of isomorphism classes of semistable bundles $E$ with $h^0(E)\geq 1$. \begin{Lem} Suppose that there exists a semistable (resp.\ stable) bundle $E_0$ of rank $r$, degree $d$, $0< d\leq r$, and $h^0(E_0)=k\geq 1$. Then there exists a semistable (resp.\ stable) bundle $E$ of the same rank and degree with $h^0(E)=k-1$. \end{Lem} \begin{proof} By Lemma \ref{L:bounds}, $k\leq r$. The case where $d=r$ and $E_0$ is strictly semistable is trivial. In the other cases, $k<r$, and we may write $$ \beta_0 : 0\to\mathcal{O}^{\oplus k}\to E_0\to F\to 0\ , $$ where by assumption the connecting homomorphism $\delta_0: H^0(F)\to H^1(\mathcal{O}^{\oplus k})$ is injective. Consider $\{L_t : t\in D\}$ a smooth local family of line bundles parametrized by the open unit disk $D\subset \Bbb C$ and satisfying $L_0=\mathcal{O}$ and $H^0(L_t)=0$, $t\neq 0$. Set $G_t=\mathcal{O}^{k-1}\oplus L_t$. The semistability of $E_0$ implies that $ H^0(F^\ast\otimes G_t)=0$. Hence, $\{ H^1(F^\ast\otimes G_t) : t\in D\}$ defines a smooth vector bundle $V$ over $D$. Let $\beta=\{\beta(t) : t\in D\}$ be a nowhere vanishing section of $V$ with $\beta(0)=\beta_0$. Then $\beta$ defines a smooth family of nonsplit extensions $0\to G_t\to E_t\to F\to 0$ and a smooth family of connecting homomorphisms $$ \delta_t : H^0(F)\longrightarrow H^1(G_t)\subset\Omega^{0,1}(U)\ , $$ where $U$ is the trivial rank $k$, $C^\infty$ vector bundle on $\Sigma$. By assumption, $\delta_0$ is injective; hence, $\delta_t$ is injective for small $t$. It follows that $h^0(E_t)=k-1$ for small $t$. Furthermore, since $E_0$ is semistable (resp.\ stable) then $E_t$ is also semistable (resp.\ stable) for small $t$. \end{proof} \noindent Since the condition $h^0(E)=1$ is open in $\mathfrak{B}_\tau$, by the Lemma there exists some component $\mathfrak{B}_\tau^\prime$ of $\mathfrak{B}_\tau$ containing an open dense set $\mathfrak{B}_\tau^\ast$ consisting of pairs $[E,\varphi]$ with $h^0(E)=1$. Let $W^\ast=\psi(\mathfrak{B}_\tau^\ast)$. We will assume that $W^\ast$ is contained in the strictly semistable locus of $\mathfrak{M}(r,d)$ and derive a contradiction. We may assume that each irreducible component of $W^\ast$ is contained a subvariety $S$ parametrized by bundles of the form $E_1\oplus E_2$, where $E_1$ is stable with $h^0(E_1)\geq 1$, $E_2$ is semistable, and $\mu(E_1)=\mu(E_2)=\mu(E)$. It follows that \begin{equation} \label{E:dim1} \begin{aligned} \dim S&=(r_1^2-r_1)(g-1)+d_1+r_2^2(g-1)+1 \\ &=r^2(g-1)-2r_1r_2(g-1)-r_1(g-1)+d_1+1\ . \end{aligned} \end{equation} For $[E]\in W^\ast\subset S$, we have \begin{equation} \label{E:dim2} \dim_{[E,\varphi]}\mathfrak{B}_\tau\leq \dim_{[E]} S +\dim \psi^{-1}([E]) \ . \end{equation} The dimension of the fiber of $\psi$ is given by $h^1(E_2\otimes E_1^\ast)$. Assume first that a generic $E_1$ is not isomorphic to any factor of $E_2$. Then the fiber dimension is $r_1 r_2(g-1)$. Thus, we obtain from \eqref{E:dim1} and \eqref{E:dim2} that $$ (r^2-r)(g-1)+d \leq r^2(g-1)-r_1r_2(g-1)-r_1(g-1)+d_1+1\ , $$ or, $$ r_2(r_1-1)(g-1)+ d_2-1 \leq 0\ . $$ This yields a contradiction in all cases other than $r=d=2$. The latter situation is covered by the following construction which is verified by straightforward dimension counting: \begin{Prop} For generic line bundles $L$ of degree 2 and generic extensions $0\to\mathcal{O}\to E\to L\to 0$, $E$ is a stable rank 2 bundle of degree 2 with a nontrivial holomorphic section. \end{Prop} \noindent In case $E_1$ is isomorphic to some factor of $E_2$, the fiber dimension increases by 1. On the other hand, in this case $W^\ast$ is contained in a strict subvariety of $S$, so by \eqref{E:dim2} the same argument applies. Isomorphisms with more factors are handled similarly. This completes the proof of the Corollary.

9510

alg-geom/9510009

en

https://arxiv.org/abs/alg-geom/9510009

alg-geom/9510009

Ludmil Katzarkov

Nilpotent groups and universal coverings of smooth projective varieties

LaTeX 2.09, 15 pages

In this paper we prove that the universal cover of a smooth projective variety with nilpotent fundamental group is holomorphically convex.

alg-geom

\section{Introduction} Characterizing the universal coverings of smooth projective varieties is an old and hard question. Central to the subject is a conjecture of Shafarevich according to which the universal cover $\widetilde{X}$ of a smooth projective variety is holomorphically convex, meaning that for every infinite sequence of points without limit points in $\widetilde{X}$ there exists a holomorphic function unbounded on this sequence. \medskip In this paper we try to study the universal covering of a smooth projective variety $X$ whose fundamental group $\pi_{1}(X)$ admits an infinite image homomorphism \[ \rho : \pi_{1}(X) \longrightarrow L \] into a complex linear algebraic group $L$. We will say that a nonramified Galois covering $X' \rightarrow X$ corresponds to a representation $\rho : \pi_{1}(X) \rightarrow L$ if its group of deck transformations is ${\rm im}(\rho)$. \begin{defi} We call a representation $\rho : \pi_{1}(X) \rightarrow L$ linear, reductive, solvable or nilpotent if the Zariski closure of its image is a linear, reductive, solvable or nilpotent algebraic subgroup in $L$. We call the corresponding covering linear, reductive, solvable or nilpotent respectively. The natural homomorphism $\pi_{1}(X,x) \rightarrow \fgc{{\rm uni}}{X,x}$ to Malcev's pro-uni\-po\-tent completion will be called the Malcev representation and the corresponding covering the Malcev covering. \end{defi} One may ask not only if the universal covering of $X$ is holomorphically convex but also if some special intermediate coverings that correspond to representations $\rho : \pi_{1}(X) \longrightarrow L$ are holomorphically convex. \bigskip In case $X$ is an algebraic surface and $\rho : \pi_{1}(X) \longrightarrow L$ is a reductive representation this question has been answered in \cite{KR}. The author and M. Ramachandran proved there that if $X' \longrightarrow X$ is a Galois covering of a smooth projective surface corresponding to a reductive representation of $\pi_{1}(X)$ and such that ${\rm Deck}(X'/X)$ does not have two ends, then $X'$ is holomorphically convex. The proof is based on two major developments in K\"{a}hler geometry that occured in the last decade. The first is a correspondence, established through the work of Hitchin \cite{HI}, Corlette \cite{C} and Simpson \cite{SC}, between Higgs bundles, representations of the fundamental group of a smooth projective variety $\rho : \pi_{1}(X) \rightarrow G$ (here $G$ is a linear algebraic group over ${\Bbb C}$) and $\rho$ equivariant harmonic maps from the universal covering of $X$ to the corresponding symmetric space for $G$. This correspondence is called now - non-abelian Hodge theory. The second is the theory of harmonic maps to buildings developed by Gromov and Schoen \cite{GS}. This theory gives the $p$-adic version of the theory of Higgs bundles developed by Corlette, Hitchin and Simpson and can be thought as of a $p$-adic non-abelian Hodge theory. \medskip These two ideas are used simultaneously in \cite{KR} in order to get more information about $\pi_{1}(X)$. The proof in \cite{KR} uses also some very powerful ideas of Lasell, Ramachandran \cite{BR} and Napier \cite{N1}, which can be interpreted as a non-abelian strictness property. These ideas provide a bridge and make the Nonabelian Hodge theory suitable for questions related to the Shafarevich conjecture. \medskip In this paper we elaborate further on the idea that the answer to certain uniformization questions depends heavily on the fundamental group of the variety. We study the question if solvable or nilpotent coverings $X' \rightarrow X$ are holomorphically convex for $X$ smooth projective variety. First we prove the following: \begin{theo} The Malcev covering of any smooth projective $X$ is holomorphically convex. \end{theo} As an immediate consequence of this statement we get: \begin{theo} Let $X$ be a smooth projective variety with a virtually nilpotent fundamental group. Then the Shafarevich conjecture is true for $X$. \end{theo} (Recall that a finitely generated group is nilpotent if its lower central series has finitely many terms. A group is virtually nilpotent if it has a finite index subgroup which is nilpotent.) \bigskip The proof of Theorem 1.1 uses the functorial Mixed Hodge Structure (MHS) on $\pi_{1}(X)$ combined with some new ideas of J\'anos Koll\'ar from \cite{K1} and \cite{K2}. At the end of section 2 we give a different proof of Theorem 1.2, which combined with the strictness property for the nonabelian Hodge theory seems to be a very promising idea ( see \cite{LM}). Observe that what helps us prove Theorem 1.1 is the use of all nilpotent representations of $\pi_{1}(X)$ at the same time. We can ask even more basic question than the Shafarevich conjecture: \bigskip \noindent {\bf Question 1.} Are there any nonconstant holomorphic functions on the universal covering $X$ of any smooth projective variety? \bigskip Clearly it is enough to restrict ourselves to the case when $\pi_{1}(X)$ is an infinite group. To study this question in bigger generality we add some more Hodge theoretic tools - the results of Arapura \cite{A}, Beauvile \cite{BE}, Green, Lazarsfeld \cite{GL} and Simpson \cite{SA} about characterizing the absolute sets in the moduli space of rank one local systems. We also need the following variant of the result of Arapura and Nori \cite{AN} saying that linear solvable K\"{a}hler groups are nilpotent. \begin{theo} Let $\Gamma$ be a quotient of a K\"{a}hler group $\pi_{1}(X)$ so that $\Gamma$ is a ${\Bbb Q}$-linear solvable group, then there are two possibilities - either $\Gamma$ is virtually nilpotent or $\pi_{1}(X)$ surjects onto the fundamental group of a curve of genus bigger than zero. \end{theo} The above theorem gives a way of constructing new examples of non-K\"{a}hler groups. In particular any group $\Gamma$ with infinite $H^{1}([\Gamma,\Gamma], {\Bbb Q})$ possesing a solvable linear quotient defined over ${\Bbb Q}$ that is not virtually nilpotent cannot be K\"{a}hler. Unfortunately we could not prove a solvable variant of the theorem 1.1. The maximum we were able to say is how much the solvable coverings differ from the nilpotent ones. We show the following. \begin{theo} If $\Gamma$ is a quotient of a K\"{a}hler group $\pi_{1}(X)$ so that $\Gamma$ is a complex linear solvable group, then there are two possibilities - either $\Gamma$ is deformable to a virtually nilpotent representation of $\pi_{1}(X)$ or $\pi_{1}(X)$ surjects onto the fundamental group of a curve of genus bigger than zero. \end{theo} This theorem gives a way of constructing new examples of non-K\"{a}hler groups. In particular any group $\Gamma$ with infinite $H^{1}([\Gamma,\Gamma], {\Bbb Q})$ possessing a solvable linear quotient defined over ${\Bbb Q}$ that is not virtually nilpotent cannot be K\"{a}hler. In \cite{LM} it is proved that the linear covering are holomorphically convex for $X$ an algebraic surface. Of course this implies a solvable variant of the theorem 1.1 for algebraic surfaces. The above theorem implies immediately: \begin{corr} Let $\rho : \pi_{1}(X) \longrightarrow S({\Bbb C})$ be a Zariski dense representation of the fundamental group of a smooth projective variety $X$ to the complex points of an affine solvable group defined over ${\Bbb Q}$. Then the image of $\pi_{1}(X)$ in the Malcev completion of $\pi_{1}(X)$ is infinite. \end{corr} In particular this implies that the first Betti number of $X$ is nonzero so the universal covering of $X$ $\widetilde{X}$ admits nonconstant holomorphic functions. The above corollary can be proved of course in a different way too. If we restrict ourselves to the case when $X$ is an algebraic surface we get a stronger statement. \begin{theo} Let $ X $ be a smooth projective surface with an infinite complex linear representation of its fundamental group. Then there exist non-constant holomorphic functions on $\widetilde{X}$. \end{theo} In some sense the above theorem says that the universal coverings are different from arbitrary coverings. The well known example of Cousin (see e.g. \cite{N1}) gives a ${\Bbb Z}$-covering of the two dimensional torus which does not admit holomorphic functions. The Theorem 1.5 raises a natural question: \bigskip \noindent {\bf Question 2.} Are there examples of infinite $\pi_{1}(X)$ without any infinite linear representation? \bigskip There are known examples of groups with this properties, e.g. Higman's four generator group. The question is if they can be fundamental groups. Even more interesting question was asked by J. Koll\'ar and C. Simpson. \bigskip \noindent {\bf Question 3.} Are there examples of infinite residually finite $\pi_{1}(X)$ without any infinite linear representation? \bigskip As it was pointed out to me by S. Gersten the answer of this question is positive if we are looking for an arbitrary group not for $\pi_{1}(X)$. There are the groups of Grigorchuk and Gupta-Sidki which are finitely generated infinite torsion groups. These groups are known to be residually finite ( see e.g. \cite{BU}). \medskip A negative answer to this question could have a great impact on the answer to Shafarevich conjecture for residually finite groups (see \cite{LM}, \cite{LP}). From another side a recent paper of Bogomolov and the author \cite{BL} shows that things can get quite exotic even for $\pi_{1}(X)$. In some sense the examples constructed in \cite{BL} indicate that if the answer of {\bf Question 2.} is negative then the statement of Theorem 1.5 could be the best statement in such a generality. Theorem 1.5 and Corollary 1.1 suggest the following: \begin{con} Let $ X $ be a smooth projective variety with an infinite linear representation of its fundamental group. Then there exist non-constant holomorphic functions on $\widetilde{X}$. \end{con} All of this strongly suggests that Hodge theory has a lot to offer in studying uniformization questions. We stop at the border line, before we introduce the next level of Hodge theoretic considerations, the theory of Nonabelian Mixed Hodge Structures- a theory that is giving us a way of working with all linear representations at the same time to get maximal information about $\pi_{1}(X)$. The first steps in this theory are done in \cite{SL}, \cite{SIM1}, \cite{SIM2}, \cite{SIM3} and \cite{LP} and it is far from being sufficiently developed. In any case it has fast consequences even on a very primitive level. Using these very first steps we prove in \cite{LM} the Shafarevich conjecture for surfaces with linear fundamental groups. The same method implies that the coverings that correspond to any linear representation are holomorphically convex. The proof uses basically only the mixed Hodge structure on the relative completion of $\pi_{1}(X)$ with respect to some complex variation of mixed Hodge structures. Our feeling is that this is just the beginning. \bigskip \noindent {\bf Acknowledgements:} I would like to thank A. Beilinson F. Bogomolov, J. Carlson, K. Corlette, R. Donagi, M. Gromov, S. Gersten, M. Larsen, M. Nori, T. Pantev, C. Simpson, D. Toledo and S. Weinberger for the useful conversations and H. Clemens, P. Deligne, R. Hain, J. Koll\'ar and M. Ramachandran for teaching me all ingredients of the technique used in this paper. Special thanks to Professor J. Koll\'ar for inviting me to visit University of Utah, where most of this work was done. \section{The Malcev covering} In this section we prove Theorem 1.1. and give some applications. We start with some ideas of J\'anos Koll\'ar from \cite{K1} and \cite{K2}. In \cite{K1} Koll\'ar observed that the Shafarevich conjecture is equivalent to: 1) There exists a normal variety ${\bf Sh}(X)$ and a proper map with connected fibers ${\bf Sh} : X \longrightarrow {\bf Sh}(X) $, which contracts precisely the subvarieties $Z$ in $X$ with the property that $ {\rm im} [\pi_{1}(Z')\longrightarrow \pi_{1}(X)]$ is finite. Here $Z'$ denotes a desingularization of $Z$. 2) ${\bf Sh}(\widetilde{X} )$ is a Stein space. Here we denote by ${\bf Sh}(\widetilde{X} )$ the Grauert- Remmert reduction of ${\bf Sh}(\widetilde{X} )$. In our notations ${\bf Sh}(X)={\bf Sh}(\widetilde{X} ) / \pi_{1}(X)$. The action of $\pi_{1}(X)$ may have fixed points on ${\bf Sh}(\widetilde{X} )$ but we can still take a quotient. One can consider also a relative version of condition 1). Let $H \triangleleft \pi_{1}(X)$ be a normal subgroup. We will say that a subgroup $R \subset \pi_{1}(X)$ is almost contained in $H$ if the intersection $R \cap H$ has finite index in $R$ and we will write $R \lesssim H$. We have the following condition. \medskip \begin{enumerate} \item There exists a normal variety ${\bf Sh}^{H}(X)$ and a proper map with connected fibers ${\bf Sh}^{H}: X \longrightarrow {\bf Sh}^{H}(X),$ which contracts exactly the subvarieties $Z$ in $X$ having the property that ${\rm im}[\pi_{1}(Z')\longrightarrow \pi_{1}(X)] \lesssim H$. Again $Z'$ denotes a desingularization of $Z$. The relative version of 2) is the following: \item ${\bf Sh}^{H}(\widetilde{X} )$ is a Stein space. Here we denote by ${\bf Sh}^{H}(\widetilde{X} )$ the Grauert- Remmert reduction of ${\bf Sh}(\widetilde{X} )$. In our notations ${\bf Sh}^{H}(X) = {\bf Sh}^{H}(\widetilde{X} ) / (\pi_{1}(X)/H)$. \end{enumerate} This was also independently observed by F. Campana in \cite{CM}. \bigskip Our approach is that if there is a natural candidate for ${\bf Sh}(X) $ it is enough to check condition 1) only for $Z$ - an algebraic curve. This certainly is the case when $\pi_{1}(X)$ is a nilpotent group. In the simplest case when $\pi_{1}(X)$ is virtually abelian one uses for ${\bf Sh}(X) $ the Albanese variety ${\rm Alb} (X)$. It is clear (see e,g, \cite{CT}) that for a smooth projective variety $X$ with $\pi_{1}(X)$ an infinite nilpotent group the Albanese map: \[ {\rm Alb} : X \longrightarrow {\rm Alb}(X) \] has nontrivial image. In other words $\dim_{\Bbb{C}}({\rm im}({\rm Alb}))>0.$ Moreover if we denote by $S$ the Stein factorization of the Albanese map, then this is a natural candidate for ${\bf Sh}(X) $ in case $\pi_{1}(X)$ is a nilpotent group. Observe that the map \[ X \longrightarrow S \] contracts all subvarieties $Z$ with the property that $ {\rm im} [H_{1}(Z, {{\Bbb Q}})\longrightarrow H_{1}(X, {{\Bbb Q}})]$ is trivial. Now using that $\pi_{1}(X)$ is a nilpotent group and the theory of Mixed Hodge Structures on its Malcev completion we show that the fact that $ {\rm im} [H_{1}(Z, {{\Bbb Q}}) \longrightarrow H_{1}(X, {{\Bbb Q}})]$ is trivial is equivalent to the fact that $ {\rm im} [\pi_{1}(Z)\longrightarrow \pi_{1}(X)]$ is finite for $Z$ an algebraic curve. We finish the proof by reducing the argument for $Z$ of arbitrary dimension to $Z$ an algebraic curve. To prove Theorem 1.1 we need to show again that there is natural candidate for ${\bf Sh}^{H}(X) $, where $H={\rm ker}(\rho : \pi_{1}(X) \longrightarrow \fgc{{\rm uni}}{X,x}$ of the Malcev representation. Again this candidate is $S$ the Stein factorization of the Albanese map. At the end of section we give a different proof of Theorem 1.1, which is basically spelling of the proof we have given already in the language of equivariant harmonic maps. \subsection{Mixed Hodge Structure considerations} In this subsection we explain why if $\pi_{1}(X)$ is a nilpotent group the theory of Mixed Hodge Structures on it implies that ${\rm im} [H_{1}(Z, {{\Bbb Q}}) \longrightarrow H_{1}(X, {{\Bbb Q}})]$ is trivial is equivalent to the fact that $ {\rm im} [\pi_{1}(Z)\longrightarrow \pi_{1}(X)]$ is finite for $Z$ an algebraic curve. For some background one can look at \cite{D}, \cite{DG} or \cite{H}. For the proof of Theorem 1.1 we need to work with $X$ smooth but for completeness in this section we will require only the MHS on $H^{1}(X)$ is of weights $> 0$. \begin{lemma} If Z is a compact nodal curve and $f:Z \longrightarrow X$ is a map to a variety such that MHS on $H^{1}(X)$ is of weights $> 0$ then the map \[ f_{* }: L(Z,x) \longrightarrow L(X,f(x)) \] is trivial if and only if the map \[f^{*} : H^{1}(X,{\Bbb Q}) \longrightarrow H^{1}(Z,{\Bbb Q})\] is trivial. Here $L(Z,x)$ and $L(X,f(x))$ are the corresponding Lie algebras of the unipotent completions $\fgc{{\rm uni}}{Z,x}$ and $\fgc{{\rm uni}}{ X,f(x)}$ of the fundamental groups $\pi_{1}(Z,x)$ and $\pi_{1}(X,f(x))$ respectively and $x$ is a point in $Z$. \end{lemma} {\bf Proof.} Observe that the map in unipotent completions determines and is determined by a map on the corresponding Lie algebras: \[L(Z,x) \longrightarrow L(X,f(x)).\] First let us consider the case where $H_1(Z)$ is pure of weight $-1$. This is the case when the dual graph of $Z$ is a tree. By a standard strictness argument the weight filtration on $L(Z,x)$ is its lower central series, and the associated graded Lie algebra is generated by $Gr_{-1} L(Z,x) = H_{1}(Z, {\Bbb Q})$. Since \[L(Z,x) \longrightarrow L(X,f(x))\] is a morphism of MHS, it is non-zero if and only if the map \[Gr L(Z,x) \longrightarrow Gr L(X,f(x))\] on weight graded quotients is. Since \[Gr_{-1} L(X,f(x)) = H_{1}(X,{\Bbb Q})/W_{-2},\] and since $H_{1}(Z,{\Bbb Q}) \longrightarrow H_{1}(X,{\Bbb Q})$ is trivial, it follows that $L(Z,x) \longrightarrow L(X,f(x))$ is trivial. \medskip To prove the general case, we take a partial normalization \[Z' \longrightarrow Z\] with the property that $Z'$ is connected and such that $H^{1}(Z')$ is a pure MHS of weight 1. This can be done as follows. Take a maximal tree $T$ in the dual graph of $Z$ and normalize only those double points corresponding to edges not in $ T$. Then $H_{1}(Z) $ is pure MHS of weight -1. The previous argument implies that \[L(Z',x) \longrightarrow L (X,f(x))\] is trivial. To complete the proof, note that we have an exact sequence \[1 \longrightarrow N \longrightarrow \pi_{1}(Z,x) \longrightarrow \pi_{1}(\Gamma,*) \longrightarrow 1,\] where $\Gamma$ denotes the dual graph of $Z $ and $N$ is the normal subgroup of $\pi_{1}(Z)$ generated by $\pi_{1}(Z',x)$. After passing to unipotent completions, we obtain an exact sequence \[0 \longrightarrow (L(Z'))\longrightarrow L(Z,x) \longrightarrow L(\Gamma,*) \longrightarrow 0.\] This is an exact sequence in the category of Malcev Lie algebras with MHS. The ideal $(L(Z'))$ generated by $L(Z')$ is exactly $W_{-1} L(Z)$, so the MHS induced on $L(\Gamma,*)$ is pure of weight 0. It follows that the homomorphism $L(Z,x) \longrightarrow L(X,f(x))$ induces a homomorphism \[L(\Gamma,*) \longrightarrow L(X,f(x)).\] This is a morphism of MHS of (0,0) type. It is injective if and only if the map \[L(\Gamma,*) = Gr L(\Gamma,*) \longrightarrow Gr L(X,f(x))\] is also injective. Since $H_{1}(X)$ has weights $< 0$ and $L(\Gamma,*)$ has weight zero, it follows that \[L(\Gamma,*) = Gr L(\Gamma,*) \longrightarrow Gr L(X,f(x))\] is zero. This proves the statement in general. Namely, we have that for any nodal curve (singular, reducible) the map \[ f_{* }: L(Z,x) \longrightarrow L(X,f(x)) \] is trivial if and only if the map \[f^{*} : H^{1}(X,{\Bbb Q}) \longrightarrow H^{1}(Z,{\Bbb Q})\] is trivial. \hfill $\Box$ \begin{lemma} Let $X$ be a smooth projective variety with a nilpotent fundamental group $\pi_{1}(X)$. Then for any algebraic curve $Z \subset X $ the fact $ {\rm im} [H_{1}(Z, {{\Bbb Q}})\longrightarrow H_{1}(X, {{\Bbb Q}})]$ is trivial is equivalent to the fact that $ {\rm im} [\pi_{1}(Z)\longrightarrow \pi_{1}(X)]$ is finite. \end{lemma} {\bf Proof.} Since we can always find a partial normalization $\widetilde{Z} \rightarrow Z$ with $\widetilde{Z}$-nodal and $\pi_{1}(\widetilde{Z},\tilde{x}) \rightarrow \pi_{1}(Z,x)$ surjective it follows from the previous lemma that the map \[ f_{* }: L(Z,x) \longrightarrow L(X,f(x)) \] is the zero map. Furthermore, if $\pi_{1}(X)$ is a torsion free nilpotent group then by definition it embeds in $\pi_{un}(X, f(x))$. It is easy to see that torsion elements of a nilpotent group generate a finite group and hence \[\pi_{1}(X,f(x)) \longrightarrow \fgc{{\rm uni}}{X,f(x)}\] is an embedding up to torsion which proves the lemma. \hfill $\Box$ We have actually proved more: \begin{lemma} Let $X$ be a smooth projective variety and $\rho : \pi_{1}(X) \longrightarrow L(X,f(x))$ be the Malcev representation of $\pi_{1}(X)$. Then for any algebraic curve $Z \subset X $ the fact $ {\rm im} [H_{1}(Z, {{\Bbb Q}})\longrightarrow H_{1}(X, {{\Bbb Q}})]$ is trivial is equivalent to the fact that $ {\rm im} [\pi_{1}(Z) \longrightarrow \pi_{1}(X) / H ]$ is finite. Here $H$ is the kernel of the Malcev representation. \end{lemma} \subsection{A reduction to the case of an algebraic curve} In this section we show how to reduce the argument for $Z$ of arbitrary dimension to $Z$ an algebraic curve. \begin{lemma} Let $F$ be a connected subvariety in $X$ then we can find a curve $Z \subset F$ such that $\pi_{1}(Z) $ surjects on $\pi_{1}(F) $. \end{lemma} {\bf Proof.} If $F$ is smooth variety the above lemma is just the Lefschetz hyperplane section theorem. Let $F= F_{1}+ \ldots + F_{i}$ be singular and with many components of different dimension. Denote by $n$ the normalization $n:F' \longrightarrow F $ of $F$. In every component of $F'$ after additional desingularization we can find finitely many points $x_{k}, y_{k}$ such that $n(x_{k})=n(y_{k})$ and $\pi_{1}(F'/ x_{k}=y_{k})$ surjects onto $\pi_{1}(F)$. The way to do that is to take the Whitney stratification of $F$ and put the points $x_{k},y_{k}$ in every stratum in a way that all loops that come from singularities pass through these points. Now following \cite{GM}(ii, 1.1) we take hypesurfaces with big degrees that pass through the points $x_{k}, y_{k}$ and intersect every component of $F'$, $F'_{l}$ in a curve $Z_{l}$ such that $Z'= \cup Z_{l}$ and $\pi_{1}(Z')$ surjects on $\pi_{1}(F') $. We make $Z=n(Z')$. Observe that $Z$ might be singular and have many components but it will be connected. \hfill $\Box$ Now we are ready to finish the proof of Theorem 1.1. We start with the Stein factorization of the Albanese map for $X$ \[ {\rm Alb} : X \longrightarrow S \longrightarrow {\rm im}({\rm Alb}) \subset {\rm Alb}(X).\] Denote by $S'$ the fiber product of the universal covering $\widetilde{{\rm Alb}( X)}$ of ${\rm Alb}(X)$ and $S$ over ${\rm Alb}(X)$. By definition the map \[ S' \longrightarrow \widetilde{{\rm Alb}(X)} \] is a covering map and since $\widetilde{{\rm Alb}(X)}$ is a Stein manifold $S'$ is a Stein manifold as well. It follows from the definition of the Albanese morphism that the fibers of the map \[ {\rm Alb} : X \longrightarrow S \] are all subvarieties $F$ in $X$ for which the map $H_{1}(F, {{\Bbb Q}})\longrightarrow H_{1}(X, {{\Bbb Q}})$ is trivial. We willshow that if the fact that $H_{1}(F, {{\Bbb Q}}) \longrightarrow H_{1}(X, {{\Bbb Q}})$ is trivial implies that $ {\rm im} [\pi_{1}(F)\longrightarrow \pi_{1}(X) / H ]$ is finite. We have shown this in Lemma 2.3 when $F$ is an algebraic curve. Now if $dim_{\Bbb{C}}(F)>1$ we apply Lemma 2.4 to find a curve $Z$ in $F$ such that $\pi_{1}(Z) $ surjects on $\pi_{1}(F) $. The argument of Lemma 2.1.2 implies that $\pi_{1}(F) $ goes to a finite group in $\pi_{1}(X)/H $ since $\pi_{1}(Z) $ goes to afinite group in $\pi_{1}(X)/H $. Observe that the curve $Z$ is also contained in the fiber $F$ of the map \[ {\rm Alb} : X \longrightarrow S .\] Therefore the map $H_{1}(Z, {{\Bbb Q}})\longrightarrow H_{1}(X, {{\Bbb Q}})$ is also trivial. To finish the proof of Theorem 1.1 we need to observe that $S$ satisfies the conditions for being the Shafarevich variety of $X$, $S= {\bf Sh}^{H}(X)$. Namely 1) There exists a holomorphic map with connected fibers $ X \longrightarrow S $, which contracts only the subvarieties $Z$ in $X$ with the property that $ {\rm im} [\pi_{1}(Z)\longrightarrow \pi_{1}(X)/H]$ is finite. 2) $ {\bf Sh}^{H}(\widetilde{X})=S'$ is a Stein space. \hfill $\Box$ To prove Theorem 1.2 we use the same argument as above but $H$ is a finite group. Actually we have shown more: \begin{corr} Let $X$ be a smooth projective variety with a virtually residually nilpotent fundamental group. Then the Shafarevich conjecture is true for $X$. \end{corr} \subsection{Some examples} In this subsections we give some examples and geometric applications of our method. We start with the following result that was also proved by Campana in \cite{CM1}. \begin{corr} Let $X$ be a smooth projective surface and $\Gamma$ is the image of $\pi_{1}(X)$ in $L(X,f(x))$. Let as before $S$ be the Stein factorization of the map $X \longrightarrow {\rm im}({\rm Alb}(X))$. After taking an etale finite covering $X'' \longrightarrow X$ the homomorphism $\pi_{1}(X'') \longrightarrow \Gamma$ factors through the map $\pi_{1}(S) \longrightarrow \Gamma$. \end{corr} {\bf Proof.} According to \cite{K2} 4.8 after taking some etale finite covering $X'' \longrightarrow X$, $\pi_{1}(X'')$ is the same as the fundamental group of $\pi_{1}(S)$. This follows from the fact that residually nilpotent groups are residually finite. \hfill $\Box$ Nilpotent K\"{a}hler groups were constructed by Sommese and Van de Ven \cite{SV}, and Campana \cite{CM1}. The construction goes as follows: Start with a finite morphism from an abelian variety $A$ to ${\Bbb P}^{n}$. Now take the preimage $X$ in $A$ of any abelian d-fold in ${\Bbb P}^{n}$. A double cover of $X$ has as fundamental group a nonsplit central extension of an abelian group by ${\Bbb Z}$. Let us following \cite{SV} give more explicit example. We start with a four dimensional abelian variety $A$ and a finite morphism $f$ to ${\Bbb P}^{4}$. Take the Mumford-Horrocks abelian surface $Z$ in ${\Bbb P}^{4}$ and pull it back to $A$. Let us call the new surface $f^{-1}(Z)$. The following exact sequence was established in \cite{SV} \[\pi_{2}(A)\oplus \pi_{2}(Z) \longrightarrow \pi_{2}({\Bbb P}^{4}) \longrightarrow \pi_{1}(f^{-1}(Z)) \longrightarrow \pi_{1}(A)\oplus \pi_{1}(Z) \longrightarrow 0.\] In our case this sequence reads as: \[0 \longrightarrow {\Bbb Z} \longrightarrow \pi_{1}(f^{-1}(Z)) \longrightarrow {\Bbb Z}^{12} \longrightarrow 0\] and shows that $f^{-1}(Z)$ has a two steps nilpotent fundamental group. Actually we know more. By theorem of Arapura and Nori \cite{AN} all K\"{a}hler linear solvable groups are virtually nilpotent. So we have the following: \begin{corr} Let $X$ be a smooth projective variety with a linear solvable fundamental group. Then the universal covering $\widetilde{X}$ is holomorphically convex. \end{corr} Now we will use the technique from Lemmas 2.3 and 2.4 to show that the theorem of Arapura and Nori \cite{AN} is the marginal statement meaning that there exists a residually solvable linear group which does not embed in its Malcev completion. By residually solvable we mean a group that embeds in its completion with respect to all finitely generated solvable representations. The following example came out from a discussion with D. Arapura, J\'anos Koll\'ar, M. Nori, T. Pantev, M. Ramachandran and D. Toledo. Consider nontrivial smooth family of smooth abelian varieties of dimension $N$ over curve $C$. Let us denote this family by $X$. The fundamental group of $X$ is given by the following exact sequence \[ 0 \longrightarrow {\Bbb Z}^{2N} \longrightarrow \pi_{1}(X) \longrightarrow \pi_{1}(C) \longrightarrow 0.\] The group $\pi_{1}(X) $ is a semidirect product of the groups ${\Bbb Z}^{2N}$ and $\pi_{1}(C)$. For generic enough family we can make the monodromy action \[M: \pi_{1}(C) \longrightarrow Sp(2N, {\Bbb Z}) \] to be irreducible and from here one can get that the image of ${\Bbb Z}^{2N}$ in $H^{1}(X,{\Bbb Z})$ is trivial. The the group $\pi_{1}(X)$ is linear. To see that we consider the morphism \[l: \pi_{1}(X) \longrightarrow SL(2,{\Bbb C}) \times [ GL(V) \ltimes V ].\] Here $V$ is a vector space over $\Bbb{C}$ of dimension $N$ on which ${\Bbb Z}^{2N}$ acts discretely. It is easy to see that $L$ is an injection. The group $\pi_{1}(X)$ is also virtually residually solvable. This can be seen as follows: Choose a prime number $p$. Since the group $\pi_{1}(X)$ is linear, namely it embeds in $GL(T)$ for some vector space $T$ we can embed it up to a finite index in a series of finite solvable groups $GL(T_{p^{q}})$ for $q=1,2, \ldots$. {}From another point $\pi_{1}(X)$ does not embed in its Malcev completion up to a finite index. Assume that $\pi_{1}(X)$ does embed in its Malcev completion. Then lemma 2.3 and lemma 2.4 imply that if the image of ${\Bbb Z}^{2N}$ in $H^{1}(X,{\Bbb Z})$ is trivial then the image of ${\Bbb Z}^{2N}$ in the Malcev completion of $\pi_{1}(X)$ is trivial which is not the case in our situation. Therefore our technique does not answer the question if the universal coverings of $X$ or of generic hyperplane sections of it is holomorphically convex. Of course this is true and can be seen as follows: \begin{prop} The universal covering of any smooth family of Abelian varieties or of any generic hyperplane section of them is holomorphically convex. \end{prop} {\bf Proof.} It follows from \cite{K1}, Theorem 6.3 that every smooth family of abelian varieties over a curve has a linear fundamental group since according to 6.3 \cite{K1} after a finite etale covering it is birational to a family of a smooth abelian varieties. But the universal covering of a family of smooth abelian varieties or a generic hyperplane section of it is holomorphically convex since it is a Stein space since. It embeds in $(SIEG \times C^{N})$, where $SIEG $ is the Siegel upperhalf plane. \hfill $\Box$ It also follows from \cite{LM} where more powerful technique, the theory of Nonabelian Mixed Hodge Structures, is used. We formulate: \begin{corr} Let $X$ be a smooth projective variety with an infinite virtually nilpotent fundamental group and such that ${\rm rank}\, {\rm Pic}(X) = 1$ (or better ${\rm rank}\, NS(X)=1$). Then for every subvariety $Z$ in $X$ we have that $ {\rm im} [\pi_{1}(Z)\longrightarrow \pi_{1}(X)]$ is infinite. \end{corr} The proof is an easy consequence of \cite{K2} (Chapter 1). We demonstrate a quick application of the above corollary. Denote by $A$ a four dimensional abelian variety. Let us say that $X$ is a hypersurface in it with an isolated singular point $s$ of the following type $xy=zt$. Denote by $X'$ the blow of $X$ in $s$. We glue in $X$ ${\Bbb P}^{1} \times {\Bbb P}^{1}$ instead of $s$. It is easy to see that $X'$ is smooth and that ${\Bbb P}^{1}$ can be blown down. The new space $X''$, obtained after blowing down one of the above ${\Bbb P}^{1}$, is smooth, ${\rm rank} NS(X'')=1$ and $\pi_{1}(X)={\Bbb Z}^{4}$. Therefore the Shafarevich conjecture should be true for $X''$. But as we can see $X''$ contains ${\Bbb P}^{1}$. This contradicts the above corollary and we conclude that $X''$ is not projective. Of course all this can be seen in many different ways. This is another spelling of the fact that ${\Bbb P}^{1}$, that remains in $X''$, should be homologically nontrivial if $X''$ is projective. We give now an idea of an alternative proof of Theorem 1.1 which came from conversations with M. Ramachandran. It is based on the use of $\pi_{1}(X)$ equivariant harmonic maps to the universal coverings to Higher Albanese varieties defined in \cite{HZ}. Combined with the strictness property for the nonabelian Hodge theory this seems to be a very promising idea ( see \cite{LM}). Denote by $G_{s}$ the complex simply connected group $\pi_{1}(X) / \Gamma^{s+1}$, where $\Gamma^{i}$ are the groups from the lower central series for $\pi_{1}(X)$ and $\Gamma^{s}$ is the smallest nontrivial one. The corresponding Lie algebra $g_{s}$ has MHS. Denote by $F^{0}G_{s}$ the closed subgroup in $G_{s}$ group that corresponds to $F^{0}g_{s}$. Since the group $\pi_{1}(X) / \Gamma^{s+1}$ is unipotent then as it is easy to see we have a free action of the corresponding to $G_{s}$ lattice $G_{s}({\Bbb Z})$ on $G_{s}/F^{0}G_{s}$. Therefore in the same way as in \cite{KR} we obtain a $\pi_{1}(X) $ equivariant proper horizontal holomorphic map ( see \cite{HZ}) \[ \widetilde{X} \longrightarrow G_{s}/F^{0}G_{s}.\] According to \cite{H1} $G_{s}/F^{0}G_{s}$ is biholomorphic to $\Bbb{C}^{N}$. Therefore $\widetilde{X}$ is holomorphically convex. \begin{rem}{\rm The above argument is weaker then the argument we have used in the first proof. It cannot be generalized to the case of residually nilpotent groups since in this case $G_{s}/F^{0}G_{s}$ will not be a manifold.} \end{rem} \section{Solvable coverings} We would like to obtain the analog of Theorem 1.1 for solvable coverings. The analog of Theorem 1.2 for solvable groups - Corollary 2.3 was proved in the previous section as a consequence of the result of Arapura and Nori. We cannot prove solvable analog of theorem 1.1. The maximum we can do is to realize how close the solvable representations come to nilpotent ones. To be able to do so we need to generalize slightly the result of Arapura and Nori. First we prove Theorem 1.3. \noindent {\bf Proof.} (The idea of the proof was suggested to me by T. Pantev.) Denote by $\Gamma$ the image of the solvable representation $\rho : \pi_{1}(X) \rightarrow L$. We need to show that either $\Gamma$ is virtually nilpotent or there exists a holomorphic map with connected fibers $f : X \rightarrow C$ to a smooth curve $C$ of genus $\geq 1$. First we introduce some notations. For a finitely generated group $\Gamma$ denote by $\Sigma(\Gamma)$ the set of all special characters of $\Gamma$. That is \[ \Sigma(\Gamma) := \left\{ \alpha : \Gamma \rightarrow {\Bbb C}^{\times} \left| H^{1}(\Gamma, {\Bbb C}_{\alpha}) \neq 0 \right. \right\}, \] where ${\Bbb C}_{\alpha}$ is the one dimensional $\Gamma$-module associated to $\alpha$. Now we recall the following: \begin{prop} [Arapura-Nori \cite{AN}] Let $\Gamma$ be a finitely generated ${\Bbb Q}$-linear solvable group. Then the following are equivalent \begin{enumerate} \item $\Gamma$ is virtually nilpotent. \item $\Sigma(\Gamma)$ consists of finitely many torsion characters. \end{enumerate} \end{prop} Due to this proposition it is enough to show that either $\Sigma(\Gamma)$ consists of finitely many torsion characters or $X$ has a non-trivial map to a curve of genus bigger than zero. Now, since $\pi_{1}(X)$ surjects on $\Gamma$ it follows that $\Sigma(\Gamma) \subset \Sigma(\pi_{1}(X))$ and hence it suffices to show that either $\Sigma(\pi_{1}(X))$ consists of finitely many torsion characters or $X$ has an irrational pencil. \medskip For a smooth projective variety $X$ denote by $M(X)$ the moduli space of homomorphisms from $\pi_{1}(X)$ to ${\Bbb C}^{\times}$. The locus of special characters is a jump locus in $M(X)$ and hence it is a subscheme in a natural way. It turns out that $\Sigma(\pi_{1}(X))$ is actually a smoooth subvariety having very special geometric properties which we are going to exploit. Since the subvariety $\Sigma(\pi_{1}(X)) \subset M(X)$ is completely canonical one expects it to have an intrinsic description. One way to construct natural subvarieties in $M(X)$ is via pullbacks. Namely, given any surjective morphism $\varphi : X \rightarrow Y$ we can pullback the moduli space of characters of $\pi_{1}(Y)$ to get a subvariety $\varphi^{*}M(Y) \subset M(X)$. According to \cite{SA}, Lemma 2.1 and Theorem 6.1 every connected component $\Sigma$ of the subvariety $\Sigma(\pi_{1}(X)) \subset M(X)$ is of this kind. More specifically for every such $\Sigma$ there exists a torsion character $\sigma \in \Sigma$ and a connected abelian subvariety $P \subset {\rm Alb}(X)$ so that $\Sigma$ is the translation of $\varphi^{*}M({\rm Alb}(X)/P) \subset M(X)$ by $\sigma$. Here $\varphi : X \rightarrow {\rm Alb}(X) \rightarrow {\rm Alb}(X)/P$ is the composition of the Albanese map and the natural quotient morphism. In particular, $\Sigma(\pi_{1}(X))$ has a positive dimensional component if and only if its intersection with the set of all unitary characters has a positive dimensional component. Now the Hodge decomposition of the cohomology of a unitary local system implies that unless $\Sigma(\pi_{1}(X))$ consists of finitely many torsion characters the subvariety of all special line bundles in ${\rm Pic}^{\tau}(X)$ has a positive dimensional component. Indeed, for a unitary character $\alpha$ denote by ${\Bbb L}_{\alpha}$ the corresponding rank one local system and by $L_{\alpha} = {\Bbb L}_{\alpha}\otimes_{\Bbb C} {\cal O}_{X}$ the corresponding holomorphic line bundle. Now by the Hodge theorem \[h^{1}(\pi_{1}(X),{\Bbb C}_{\alpha}) = h^{1}(X, {\Bbb L}_{\alpha}) = h^{1}(X,L_{\alpha})+h^{0}(X,\Omega^{1}_{X} \otimes L_{\alpha}) = 2h^{1}(X,L_{\alpha}),\] i.e. $\alpha$ is a special character iff the line bundle $L_{\alpha}$ is special. Furthermore a theorem of Beauville (\cite{BE}, Proposition 1) asserts that the subvariety of ${\rm Pic}^{0}(X)$ consisting of special line bundles is a union of a finite set and the subvarieties of the form $f^{*}{\rm Pic}^{0}(B)$ where $f : X \rightarrow B$ is a morphism with connected fibers to a curve $B$ of genus $\geq 1$. Thus $X$ posseses irrational pencils which finishes the proof of Theorem 1.3 \hfill $\Box$ \medskip The above theorem can be seen as the solvable analog of the theorem of Simpson's that $SL(n,{\Bbb Z})$ is not a K\"{a}hler group, $n>2$. This theorem gives a way of constructing new examples of non-K\"{a}hler groups. In particular any group $\Gamma$ with infinite $H^{1}([\Gamma,\Gamma], {\Bbb Q})$ possessing a solvable linear quotient defined over ${\Bbb Q}$ that is not virtually nilpotent cannot be K\"{a}hler. \medskip Now we prove theorem 1.4. \noindent {\bf Proof.} We would like to use theorem 3.1. Therefore we need an infinite solvable representation defined over ${\Bbb Q}$. First we show \begin{lemma} Let $S$ be a connected affine solvable group defined over ${\Bbb Q}$. If $\rho : \pi_{1}(X) \rightarrow S({\Bbb C})$ is a Zariski dense represenattion into the group of complex points of $S$, then $\rho$ can be deformed to a representation $\nu : \pi_{1}(X) \rightarrow S(\overline{{\Bbb Q}})$ having an infinite image. \end{lemma} {\bf Proof.} Denote by $\Lambda$ the image of $\rho:\pi_{1}(X) \to S$. Let $\phi:\Lambda \to S(\Bbb{C})$ be a homomorphism from $\Lambda$ to the group of complex points of $S$ with an infinite image. We want to find a subgroup $\Lambda'$ of $ \Lambda$ of finite index and a homomorphism $\phi':\Lambda'\to S (\overline{{\Bbb Q}})$ with infinite image, such that $\phi '$ is arbitrarily close to $\phi$. Let $S_{0} := S$, and consider the (upper) derived series $S_{i+1} := [S_{i},S_{i}]$ for $S$. We choose the maximal $i$ such that $\Lambda\cap S_{i(\Bbb{C})}$ is of finite index of $\Lambda$. We replace $G$ by $\Lambda' = \Lambda\cap S_i(\Bbb{C})$. The image of $\Lambda'$ in $S_i(\Bbb{C})/S_{i+1}(\Bbb{C})$ is infinite. The group $A = S_{i}/S_{i+1}$ is either a torus or a vector space group. Let $X$ be the affine variety of homomorphisms $\Lambda' \to A$, $Y$ the affine variety $Hom(\Lambda',S_{i}), X' $ the image of $Y$ in $X$. Thus $X'$ is the affine subvariety of $X$ consisting of homomorphisms which factor through $S_{i}$. We want to find points on $X'(\overline{{\Bbb Q}})$ arbitarily close in $X'(\Bbb{C})$ to the point defined by $\phi$. If the original point is defined over $\overline{{\Bbb Q}}$, we are done. If not, the original point cannot be isolated, since $X'$ is defined over $\overline{{\Bbb Q}}$. Thus we have arbitrarily close points defined over $\overline{{\Bbb Q}}$. To find a $\overline{{\Bbb Q}}$ point with an infinite image we consider two cases: 1) $A$ is a vector space group. Let $g$ be an element of $\Lambda'$ such that $\phi(g) $ maps to a non-trivial element of $A$. Then for every nearby representation $\phi'$ the element $\phi'(g)$ is non-trivial in $A$, therefore it is of infinite order in $A$. 2) $A$ is a torus $T$. We fix an element $g$ in $\Lambda'$ such that $\phi(g)$ maps to a point of infinite order on $T$. Let $Z \subset T$ denote the image of $X'$ in $T$ under the map which takes each homomorphism $\Lambda'\to T$ to the image of $g$ in $T$. By definition $X' \to Z$ is a surjective map and it is defined over $\overline{{\Bbb Q}}$, so every $\overline{{\Bbb Q}}$ -point of $Z$ comes from a $\overline{{\Bbb Q}}$-point of $X'$ and therefore from a $\overline{{\Bbb Q}}$ -point of $Y$, i.e. an actual $\bar Q$ -homomorphism from $\Lambda'$ to $S_{i}$. So it is enough to find a $\overline{{\Bbb Q}}$ -point of $Z$ which is close to the image of the original homomorphism $\phi$ but which is also of infinite order. We prove the following: \begin{claim} Let $T$ be a torus, $Z$ a $\overline{{\Bbb Q}}$-affine subvariety of $T$, $p$ a point in $Z(C) $ of infinite order. Then $p$ is in the closure of the subset of $Z(\overline{{\Bbb Q}}) $ consisting of points of infinite order. \end{claim} {\bf Proof.} If $p$ is in $Z(\overline{{\Bbb Q}})$, we are done. If not, there exists a character $\chi: Z \to GL(1,\Bbb{C})$ such that $\chi(p)$ is not in $ \overline{{\Bbb Q}}$ . As $Z, T$ and $\chi$ are defined over $ \overline{{\Bbb Q}}$ , and $GL(1,\Bbb{C})$ is 1-dimensional, it follows that $\chi(Z)$ is an open subset of $GL(1,\Bbb{C})$. In particular, every neighborhood of $p$ in $Z$ maps to a neighborhood of $\chi(p)$ containing non-torsion elements. If $ q \in Z(\overline{{\Bbb Q}})$ maps to a non-torsion element, then of course $q$ is a point of infinite order in $T(\Bbb{C})$. This finishes the proof of the claim and the lemma. \hfill $\Box$ \medskip To get an infinite solvable representation defined over ${\Bbb Q}$ consider the affine solvable group $\widetilde{S}$ obtained from $S$ by restriction of scalars, i.e. $\widetilde{S} := {\rm res}_{\overline{{\Bbb Q}}/ {\Bbb Q}}S$. The representation $\nu$ induces a representation $\tilde{\nu}: \pi_{1}(X) \rightarrow \widetilde{S}({\Bbb Q})$ which has an image isomorphic to the image of $\nu$. \hfill $\Box$ \hfill $\Box$ Now to prove Corollary~1.1 we have to consider the following two alternatives 1) The group $\tilde{\nu}(\pi_{1}(X)$ is virtually nilpotent so it has a subgroup of finite index which is nilpotent. 2) There exists an holomorphic map with connected fibers $f : X \rightarrow C$ to a curve of genus one or higher. But then we know that $ \pi_{1}(C)$ embeds in its Malcev completion. In both cases there exists a finite \'{e}tale cover of $X$ which has a non-trivial Albanese variety, which is what we need. Theorem 1.5 follows easily from Theorem 1.1 and the result from \cite{KR}. {\bf Proof.} Let us start with a complex linear representation ${\rm im}[\pi_{1}(X) \rightarrow L]$. Then we have the following three possibilities. (a) The image of $\pi_{1}(X)$ in $L/R^{u}L$ does not have zero or two ends. Then we can apply \cite{KR} to conclude that $\widetilde{X}$ has a non-constant holomorphic function. (b) The image ${\rm im}[\pi_{1}(X) \rightarrow L/R^{u}L]$ has two ends. Then by the theorem of Hopf and Freudenthal it follows that ${\rm im}[\pi_{1}(X) \rightarrow L/R^{u}L]$ has a subgroup of finite index that is isomorphic to ${\Bbb Z}$. Therefore the abealianization of $\pi_{1}(X)$ is not finite. This implies that the Malcev representation is not trivial and we apply Theorem 1.1 to finish the proof. (c) The image of $\pi_{1}(X)$ in $L/R^{u}L$ has zero ends. Then $L/R^{u}L$ is a finite group. So the Malcev representation is not trivial and we are taken applying Theorem 1.1. \hfill $\Box$ \medskip Theorem 1.5 follows from \cite{LM} as well. To be able to attack conjecture 1.1 we should be able to analyze the real issue, the semisimple representations. Some initial steps in this direction are done in \cite{LM}. \medskip What should we do if the answer of {\bf Question 3} is positive? We hope using \cite{LP} to be able to handle the case when the image of $ \pi_{1}(X)$ in its proalgebraic completion is infinite. \medskip What should we do if the answer of {\bf Question 2} is positive and the $ \pi_{1}(X)$ in question has finite image in its proalgebraic completion. At the moment this case seems to be out of reach.

9510

alg-geom/9510007

en

https://arxiv.org/abs/alg-geom/9510007

alg-geom/9510007

Yekutieli Amnon

Amnon Yekutieli

Smooth Formal Embeddings and the Residue Complex

33 pages, AMSLaTeX, final version (some corrections, section on D-modules omitted), to appear in Canadian Math. J

Let \pi : X -> S be a finite type morphism of noetherian schemes. A smooth formal embedding of X (over S) is a bijective closed immersion X -> \frak{X}, where \frak{X} is a noetherian formal scheme, formally smooth over S. An example of such an embedding is the formal completion \frak{X} = Y_{/X} where X \subset Y is an algebraic embedding. Smooth formal embeddings can be used to calculate algebraic De Rham (co)homology. Our main application is an explicit construction of the Grothendieck residue complex when S is a regular scheme. By definition the residue complex is the Cousin complex of \pi^{!} \cal{O}_{S}. We start with Huang's theory of pseudofunctors on modules with 0-dimensional support, which provides a graded sheaf \cal{K}^{.}_{X/S}. We then use smooth formal embeddings to obtain the coboundary operator on \cal{K}^{.}_{X / S}. We exhibit a canonical isomorphism between the complex (\cal{K}^{.}_{X/S}, \delta) and the residue complex of Grothendieck. When \pi is equidimensional of dimension n and generically smooth we show that H^{-n} \cal{K}^{.}_{X/S} is canonically isomorphic to the sheaf of regular differentials of Kunz-Waldi. Another issue we discuss is Grothendieck Duality on a noetherian formal scheme \frak{X}. Our results on duality are used in the construction of \cal{K}^{.}_{X/S}.

alg-geom

\section{Introduction} It is sometimes the case in algebraic geometry, that in order to define an object associated to a singular variety $X$, one first embeds $X$ into a nonsingular variety $Y$. One such instance is algebraic De Rham cohomology $\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}}(X) = \mrm{H}^{{\textstyle \cdot}}(Y, \widehat{\Omega}^{{\textstyle \cdot}})$, where $\widehat{\Omega}^{{\textstyle \cdot}}$ is the completion along $X$ of the De Rham complex $\Omega_{Y / k}^{{\textstyle \cdot}}$ (relative to a base field $k$ of characteristic $0$; cf.\ \cite{Ha}). Now $\widehat{\Omega}^{{\textstyle \cdot}}$ coincides with the complete De Rham complex $\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X} / k}$, where $\mfrak{X}$ is the formal scheme $Y_{/ X}$. It is therefore reasonable to ask what sort of embedding $X \subset \mfrak{X}$ into a formal scheme would give rise to the same cohomology. The answer we provide in this paper is that any {\em smooth formal embedding} works. Let us define this notion. Suppose $S$ is a noetherian base scheme $S$ and $\pi : X \rightarrow S$ is a finite type morphism. A smooth formal embedding of $X$ consists of morphisms $X \rightarrow \mfrak{X} \rightarrow S$, where $X \rightarrow \mfrak{X}$ is a closed immersion of $X$ into a noetherian formal scheme $\mfrak{X}$, which is a homeomorphism of the underlying topological spaces; and $\mfrak{X} \rightarrow S$ is a {\em formally smooth} morphism. A smooth formal embedding $X \subset \mfrak{X} = Y_{/ X}$ like in the previous paragraph is said to be algebraizable. But in general $X \subset \mfrak{X}$ will not be algebraizable. Smooth formal embeddings enjoy a few advantages over algebraic embeddings. First consider an embedding $X \subset \mfrak{X}$ and an \'{e}tale morphism $U \rightarrow X$. Then it is pretty clear (cf.\ Proposition \ref{prop2.4}) that there is an \'{e}tale morphism of formal schemes $\mfrak{U} \rightarrow \mfrak{X}$ and a smooth formal embedding $U \subset \mfrak{U}$, s.t.\ $U \cong \mfrak{U} \times_{\mfrak{X}} X$. Next suppose $X \subset \mfrak{X}, \mfrak{Y}$ are two smooth formal embeddings, and we are given either a closed immersion $\mfrak{X} \rightarrow \mfrak{Y}$ or a formally smooth morphism $\mfrak{Y} \rightarrow \mfrak{X}$, which restrict to the identity on $X$. Then locally on $X$, \begin{equation} \label{eqn0.1} \mfrak{Y} \cong \mfrak{X} \times \operatorname{Spf} \mbb{Z} [\sqbr{t_{1}, \ldots, t_{n}}] \end{equation} (Theorem \ref{thm2.2}). As mentioned above, De Rham cohomology can be calculated by smooth formal embeddings. Indeed, when $\operatorname{char} S = 0$, $\mrm{H}^{q}_{\mrm{DR}}(X / S) = \mrm{R} \pi_{*} \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X} / S}$, where $X \subset \mfrak{X}$ is any smooth formal embedding (Corollary \ref{cor2.1}). Moreover, in \cite{Ye3} it is proved that De Rham homology $\mrm{H}_{{\textstyle \cdot}}^{\mrm{DR}}(X)$ can also be calculated by smooth formal embeddings, when $S = \operatorname{Spec} k$, $k$ a field. According to the preceding paragraph, given an \'{e}tale morphism $g : U \rightarrow X$ there is a homomorphism $g^{*} : \mrm{H}_{{\textstyle \cdot}}^{\mrm{DR}}(X) \rightarrow \mrm{H}_{{\textstyle \cdot}}^{\mrm{DR}}(U)$, and we conclude that homology is contravariant w.r.t.\ \'{e}tale morphisms. See Remark \ref{rem2.4} for an application to $\mcal{D}$-modules on singular varieties. The main application of smooth formal embeddings in the present paper is for an {\em explicit construction of the Grothendieck residue complex} $\mcal{K}^{{\textstyle \cdot}}_{X / S}$, when $S$ is any regular scheme. By definition $\mcal{K}^{{\textstyle \cdot}}_{X / S}$ is the Cousin complex $\mrm{E} \pi^{!} \mcal{O}_{S}$, in the notation of \cite{RD} Sections IV.3 and VII.3. Recall that Grothendieck Duality, as developed by Hartshorne in \cite{RD}, is an abstract theory, stated in the language of derived categories. Even though this abstraction is suitable for many important applications, often one wants more explicit information. In particular a significant amount of work was directed at finding an explicit presentation of duality in terms of differential forms and residues. Mostly the focus was on the dualizing sheaf $\omega_{X}$, in various circumstances. The structure of $\omega_{X}$ as a coherent $\mcal{O}_{X}$-module and its variance properties are thoroughly understood by now, thanks to an extended effort including \cite{KW}, \cite{Li}, \cite{HK1}, \cite{HK2}, \cite{LS1} and \cite{HS}. Regarding an explicit presentation of the full duality theory of dualizing complexes, there have been some advances in recent years, notably in the papers \cite{Ye1}, \cite{SY}, \cite{Hu}, \cite{Hg1} \cite{Sa} and \cite{Ye3}. The later papers \cite{Hg2}, \cite{Hg3} and \cite{LS2} somewhat overlap our present paper in their results, but their methods are quite distinct; specifically, they do not use formal schemes. We base our construction of $\mcal{K}^{{\textstyle \cdot}}_{X / S}$ on I-C.\ Huang's theory of pseudofunctors on modules with zero dimensional support (see \cite{Hg1}). Suppose $\phi : A \rightarrow B$ is a residually finitely generated homomorphism between complete noetherian local rings, and $M$ is a discrete $A$-module (i.e.\ $\operatorname{dim} \operatorname{supp} M = 0$). Then according to \cite{Hg1} there is a discrete $B$-module $\phi_{\#} M$, equipped with certain variance properties (cf.\ Theorem \ref{thm6.1}). In particular when $\phi$ is residually finite there is a map $\operatorname{Tr}_{\phi} : \phi_{\#} M \rightarrow M$. Huang's theory is developed using only methods of commutative algebra. Now given a point $x \in X$ with $s := \pi(x) \in S$, consider the local homomorphism $\phi : \widehat{\mcal{O}}_{S, s} \rightarrow \widehat{\mcal{O}}_{X, x}$. Define $\mcal{K}_{X / S}(x) := \phi_{\#} \mrm{H}^{d}_{\mfrak{m}_{s}} \widehat{\mcal{O}}_{S, s}$, where $d := \operatorname{dim} \widehat{\mcal{O}}_{S, s}$, $\mfrak{m}_{s}$ is the maximal ideal and $\mrm{H}^{d}_{\mfrak{m}_{s}}$ is local cohomology. Then $\mcal{K}_{X / S}(x)$ is an injective hull of $k(x)$ as $\mcal{O}_{X, x}$-module. As a graded $\mcal{O}_{X}$-module we take $\mcal{K}^{{\textstyle \cdot}}_{X / S} := \bigoplus_{x \in X} \mcal{K}_{X / S}(x)$, with the obvious grading. Then for any scheme morphism $f : X \rightarrow Y$, we deduce from Huang's theory a homomorphism of graded sheaves $\operatorname{Tr}_{f} : f_{*} \mcal{K}^{{\textstyle \cdot}}_{X / S} \rightarrow \mcal{K}^{{\textstyle \cdot}}_{Y / S}$. The problem is to exhibit a coboundary operator $\delta : \mcal{K}^{q}_{X / S} \rightarrow \mcal{K}^{q + 1}_{X / S}$, and to determine that the complex we obtain is indeed isomorphic to $\mrm{E} \pi^{!} \mcal{O}_{S}$. For this we use smooth formal embeddings, as explained below. In Section 5 we discuss Grothendieck Duality on formal schemes, extending the theory of \cite{RD}. We propose a definition of dualizing complex $\mcal{R}^{{\textstyle \cdot}}$ on a noetherian formal scheme (Definition \ref{dfn5.1}), and prove its uniqueness (Theorem \ref{thm5.1}). It is important to note that the cohomology sheaves $\mrm{H}^{q} \mcal{R}^{{\textstyle \cdot}}$ are discrete quasi-coherent $\mcal{O}_{\mfrak{X}}$-modules, and in general {\em not coherent}. We define the Cousin functor $\mrm{E}$ associated to $\mcal{R}^{{\textstyle \cdot}}$, and show that $\mrm{E} \mcal{R}^{{\textstyle \cdot}} \cong \mcal{R}^{{\textstyle \cdot}}$ in the derived category, and $\mrm{E} \mcal{R}^{{\textstyle \cdot}}$ is a residual complex. On a regular formal scheme $\mfrak{X}$ the (surprising) fact is that $\mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{X}}$ is a dualizing complex, and not $\mcal{O}_{\mfrak{X}}$ (Theorem \ref{thm5.3}). Now let $U \subset X$ be an affine open set and suppose $U \subset \mfrak{U}$ is a smooth formal embedding. Say $n := \operatorname{rank} \widehat{\Omega}^{1}_{\mfrak{U} / S}$, so $\widehat{\Omega}^{n}_{\mfrak{U} / S}$ is a locally free $\mcal{O}_{\mfrak{U}}$-module of rank $1$, and $\mrm{R} \underline{\Gamma}_{\mrm{disc}} \widehat{\Omega}^{n}_{\mfrak{U} / S}[n]$ is a dualizing complex. Since the Cousin complex is a sum of local cohomology modules, there is a natural identification of graded $\mcal{O}_{\mfrak{U}}$-modules $\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \widehat{\Omega}^{n}_{\mfrak{U} / S}[n] \cong \mcal{K}^{{\textstyle \cdot}}_{\mfrak{U} / S}$. This makes $\mcal{K}^{{\textstyle \cdot}}_{\mfrak{U} / S}$ into a complex. Since $\mcal{K}^{{\textstyle \cdot}}_{U / S} \cong \mcal{H}om_{\mfrak{U}} \left( \mcal{O}_{U}, \mcal{K}^{{\textstyle \cdot}}_{\mfrak{U} / S} \right)$ we come up with an operator $\delta$ on $\mcal{K}^{{\textstyle \cdot}}_{U / S} = \mcal{K}^{{\textstyle \cdot}}_{X / S}|_{U}$. Given another smooth formal embedding $U \subset \mfrak{V}$ we have to compare the complexes $\mcal{K}^{{\textstyle \cdot}}_{\mfrak{U} / S}$ and $\mcal{K}^{{\textstyle \cdot}}_{\mfrak{V} / S}$. This is rather easy to do using the following trick. Choosing a sequence $\underline{a}$ of generators of some defining ideal of $\mfrak{U}$, and letting $\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a})$ be the associated Koszul complex, we obtain an explicit presentation of the dualizing complex, namely \[ \mrm{R} \underline{\Gamma}_{\mrm{disc}} \widehat{\Omega}^{n}_{\mfrak{U} / S}[n] \cong \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes \widehat{\Omega}^{n}_{\mfrak{U} / S}[n] \] (cf.\ Lemma \ref{lem4.3}). By the structure of smooth formal embeddings we may assume there is a morphism $f : \mfrak{U} \rightarrow \mfrak{V}$ which is either formally smooth or a closed immersion. Then choosing relative coordinates (cf.\ formula \ref{eqn0.1}) and using Koszul complexes we produce a morphism $\mrm{R} \underline{\Gamma}_{\mrm{disc}} \widehat{\Omega}^{n}_{\mfrak{U} / S}[n] \rightarrow \mrm{R} \underline{\Gamma}_{\mrm{disc}} \widehat{\Omega}^{m}_{\mfrak{V} / S}[m]$. Applying the Cousin functor $\mrm{E}$ we recover $\operatorname{Tr}_{f} : \mcal{K}^{{\textstyle \cdot}}_{\mfrak{U} / S} \rightarrow \mcal{K}^{{\textstyle \cdot}}_{\mfrak{V} / S}$ as a map of complexes! We conclude that $\delta$ is independent of $\mfrak{U}$ and hence it glues to a global operator (Theorem \ref{thm6.2}). If $f : X \rightarrow Y$ is a finite morphism, then the trace map $\operatorname{Tr}_{f} : f_{*} \mcal{K}^{{\textstyle \cdot}}_{X / S} \rightarrow \mcal{K}^{{\textstyle \cdot}}_{Y / S}$, which is provided by Huang's theory, is actually a homomorphism of complexes (Theorem \ref{thm7.6}). We show this by employing the same trick as above of going from Koszul complexes to Cousin complexes, this time inserting a ``Tate residue map'' into the picture. We use Theorem \ref{thm7.6} to prove directly that if $\pi : X \rightarrow S$ is equidimensional of dimension $n$ and generically smooth, then $\mrm{H}^{-n} \mcal{K}^{{\textstyle \cdot}}_{X / S}$ coincides with the sheaf of regular differentials $\tilde{\omega}^{n}_{X / S}$ of Kunz-Waldi \cite{KW} (Theorem \ref{thm7.4}). Finally in Theorem \ref{thm8.10} we exhibit a canonical isomorphism $\zeta_{X}$ between the complex $\mcal{K}^{{\textstyle \cdot}}_{X / S}$ constructed here and the complex $\pi^{\triangle} \mcal{O}_{S} = \mrm{E} \pi^{!} \mcal{O}_{S}$ of \cite{RD}. Given a morphism of schemes $f : X \rightarrow Y$ the isomorphisms $\zeta_{X}$ and $\zeta_{Y}$ send Huang's trace map $\operatorname{Tr}_{f} : f_{*} \mcal{K}^{{\textstyle \cdot}}_{X / S} \rightarrow \mcal{K}^{{\textstyle \cdot}}_{Y / S}$ to the trace $\operatorname{Tr}^{\mrm{RD}}_{f} : f_{*} \mrm{E} \pi^{!}_{X} \mcal{O}_{S} \rightarrow \mrm{E} \pi^{!}_{Y} \mcal{O}_{S}$ of \cite{RD} Section VI.4. In particular it follows that for $f$ proper, $\operatorname{Tr}_{f}$ is a homomorphism of complexes (Corollary \ref{cor8.1}). Sections 1 and 3 of the paper contain the necessary supplements to \cite{EGA}. Perhaps the most noteworthy result there is Theorem \ref{thm1.10}, which states that formally finite type morphisms are stable under base change. This was also proved in \cite{AJL2}. \medskip \noindent {\em Acknowledgments.}\ The author wishes to thank L.\ Alonso, I-C.\ Huang, R.\ H\"{u}bl, A.\ Jerem\'{\i}as, J.\ Lipman and P.\ Sastry for helpful discussions, some of which took place during a meeting in Oberwolfach in May 1996. \tableofcontents \section{Formally Finite Type Morphisms} In this section we define formally finite type morphisms between noetherian formal schemes. This mild generalization of the finite type morphism of \cite{EGA} I \S 10 has the advantage that it includes the completion morphism $\mfrak{X} \rightarrow \mfrak{X}_{/ Z}$ (cf.\ Proposition \ref{prop1.12}), and still is preserved under base change (Theorem \ref{thm1.10}). We follow the conventions of \cite{EGA} $0_{\mrm{I}}$ \S 7 on {\em adic} rings. Thus an adic ring is a commutative ring $A$ which is complete and separated in the $\mfrak{a}$-adic topology, for some ideal $\mfrak{a} \subset A$. As for formal schemes, we follow the conventions of \cite{EGA} I \S 10. Throughout the paper all formal schemes are by default noetherian (adic) formal schemes. We write $A \sqbr{\underline{t}} = A \sqbr{t_{1}, \ldots, t_{n}}$ for the polynomial algebra with variables $t_{1}, \ldots,$ \linebreak $t_{n}$ over a ring $A$. The easy lemma below is taken from \cite{AJL2}. \begin{lem} \label{lem1.11} Let $A \rightarrow B$ be a continuous homomorphism between noetherian adic rings, and let $\mfrak{b} \subset B$ be a defining ideal. Then the following are equivalent: \begin{enumerate} \rmitem{i} $A \rightarrow B / \mfrak{b}$ is a finite type homomorphism. \rmitem{ii} For some homomorphism $f : A \sqbr{\underline{t}} \rightarrow B$ extending $A \rightarrow B$ one has $\mfrak{b} = B \cdot f^{-1}(\mfrak{b})$ and $A \sqbr{\underline{t}} \rightarrow B / \mfrak{b}$ is surjective. \end{enumerate} \end{lem} \begin{proof} (i) $\Rightarrow$ (ii): Say $b_{1}, \ldots, b_{m}$ generate $\mfrak{b}$ as a $B$-module, and the images of $b_{m+1}, \ldots, b_{n}$ generate $B / \mfrak{b}$ as an $A$-algebra. Then the homomorphism $A \sqbr{\underline{t}} \rightarrow B$, $t_{i} \rightarrow b_{i}$ has the required properties. \noindent (ii) $\Rightarrow$ (i): Trivial. \end{proof} \begin{dfn} \label{dfn1.10} Let $A \rightarrow B$ be a continuous homomorphism between adic noetherian rings. We say that $A \rightarrow B$ is of {\em formally finite type} (f.f.t.) if the equivalent conditions of Lemma \ref{lem1.11} hold. We shall also say that $B$ is a formally finite type $A$-algebra. \end{dfn} \begin{exa} \label{exa1.9} Let $I \subset A$ be any open ideal, and let $B := \lim_{\leftarrow i} A / I^{i}$. Then $A \rightarrow B$ is f.f.t. \end{exa} Recall that if $A'$ and $B$ are adic $A$-algebras, with defining ideals $\mfrak{a}'$ and $\mfrak{b}$, the complete tensor product $A' \widehat{\otimes}_{A} B$ is the completion of $A' \otimes_{A} B$ w.r.t.\ the topology defined by the image of $(\mfrak{a}' \otimes_{A} B) \oplus (A' \otimes_{A} \mfrak{b})$. \begin{prop} \label{prop1.11} Let $A, A'$ and $B$ be noetherian adic rings, $A \rightarrow B$ a f.f.t.\ homomorphism, and $A \rightarrow A'$ any continuous homomorphism. Then $B' := A' \widehat{\otimes}_{A} B$ is a noetherian adic ring, and $A' \rightarrow B'$ is a f.f.t.\ homomorphism. \end{prop} \begin{proof} Choose a homomorphism $f : A \sqbr{\underline{t}} \rightarrow B$ satisfying condition (ii) of Lemma \ref{lem1.11}. Let $\mfrak{b} \subset B$ and $\mfrak{a}' \subset A'$ be defining ideals. Write $C := A' \otimes_{A} B$ and $\mfrak{c} := \mfrak{a}' \cdot C + C \cdot \mfrak{b}$, so $B' = \lim_{\leftarrow i} C / \mfrak{c}^{i}$. Consider the homomorphism $f' : A' \sqbr{\underline{t}} \rightarrow C$, and let $\mfrak{c}' := {f'}^{-1}(\mfrak{c})$ and $\widehat{A' \sqbr{\underline{t}}} := \lim_{\leftarrow i} A' \sqbr{\underline{t}} / {\mfrak{c}'}^{i}$. Since $\mfrak{c} = C \cdot \mfrak{c}'$, it follows from \cite{CA} Section III.2.11 Proposition 14 that $\widehat{A' \sqbr{\underline{t}}} \rightarrow B'$ is surjective. Hence $B'$ is a noetherian adic ring with the $\mfrak{b}'$-adic topology, where $\mfrak{b}' = B' \cdot \mfrak{c}$. Furthermore $A' \sqbr{\underline{t}} \rightarrow B' / \mfrak{b}'$ is surjective, and we conclude that $A' \rightarrow B'$ is f.f.t. \end{proof} In the next three examples $A$ is an adic ring with defining ideal $\mfrak{a}$. \begin{exa} \label{exa1.2} Recall that for $a \in A$, the complete ring of fractions $A_{\{a\}}$ is the completion of the localized ring $A_{a}$ w.r.t.\ the $\mfrak{a}_{a}$-adic topology. Then $A_{\{a\}} \cong A \widehat{\otimes}_{\mbb{Z} \sqbr{t}} \mbb{Z} \sqbr{t, t^{-1}}$, which proves that $A \rightarrow A_{\{a\}}$ is f.f.t. \end{exa} \begin{exa} \label{exa1.6} Given indeterminates $t_{1}, \ldots, t_{n}$, the ring of restricted formal power series $A\{ \underline{t} \} = A\{ t_{1}, \ldots, t_{n} \}$ is the completion of the polynomial ring $A \sqbr{ \underline{t} }$ w.r.t.\ the $(A \sqbr{ \underline{t} } \cdot \mfrak{a})$-adic topology. Hence $A \{ \underline{t} \} \cong A \widehat{\otimes}_{\mbb{Z}} \mbb{Z} \sqbr{ \underline{t} }$, which demonstrates that $A \rightarrow A \{ \underline{t} \}$ is f.f.t. \end{exa} \begin{exa} \label{exa1.1} Consider the adic ring $A \widehat{\otimes}_{\mbb{Z}} \mbb{Z} [\sqbr{ \underline{t} }]$, where $\mbb{Z} [\sqbr{ \underline{t} }] = \mbb{Z} [\sqbr{ t_{1}, \ldots, t_{n} }]$ is the ring of formal power series, with the $(\underline{t})$-adic topology. Since inverse limits commute, we see that $A \widehat{\otimes}_{\mbb{Z}} \mbb{Z} [\sqbr{ \underline{t} }] \cong A [\sqbr{ \underline{t} }]$, the ring of formal power series over $A$, endowed with the $(A [\sqbr{ \underline{t} }] \cdot (\mfrak{a} + \underline{t}))$-adic topology. By Prop.\ \ref{prop1.11}, $A \rightarrow A [\sqbr{ \underline{t} }]$ is f.f.t. \end{exa} Let $A \rightarrow B$ be a f.f.t\ homomorphism between adic rings. Choose a defining ideal $\mfrak{b} \subset B$, and set $B_{i} := B / \mfrak{b}^{i+1}$. For $n \geq 0$ define \[ \widehat{\Omega}^{n}_{B / A} := \lim_{\leftarrow i} \Omega^{n}_{B_{i} / A} \cong \lim_{\leftarrow i} B_{i} \otimes_{B} \Omega^{n}_{B / A} \] (cf.\ \cite{EGA} $0_{\mrm{IV}}$ 20.7.14). Let $\widehat{\Omega}^{{\textstyle \cdot}}_{B / A} := \bigoplus_{n \geq 0} \widehat{\Omega}^{n}_{B / A}$, which is a topological DGA (differential graded algebra), with $\widehat{\Omega}^{0}_{B / A} = B$. This definition is independent of the ideal $\mfrak{b}$. Since $\Omega^{n}_{B_{i} / A}$ is finite over $B_{i}$ it follows that $\widehat{\Omega}^{n}_{B / A}$ is finite over $B$. \begin{rem} If $A \rightarrow B$ is f.f.t.\ then $\widehat{\Omega}^{{\textstyle \cdot}}_{B/A} \cong \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B/A}$, where $\Omega^{{\textstyle \cdot}, \mrm{sep}}_{B/A}$ is the separated algebra of differentials defined in \cite{Ye1} \S 1.5 for semi-topo\-logical algebras. Also $\widehat{\Omega}^{{\textstyle \cdot}}_{B/A}$ is the universally finite differential algebra in the sense of \cite{Ku}. \end{rem} \begin{prop} \label{prop1.1} Let $L \rightarrow A \rightarrow B$ be f.f.t.\ homomorphisms between adic noetherian rings. \begin{enumerate} \item $A \rightarrow B$ is formally smooth relative to $L$ iff the sequence \[ 0 \rightarrow B \otimes_{A} \widehat{\Omega}^{1}_{A / L} \xrightarrow{v} \widehat{\Omega}^{1}_{B / L} \xrightarrow{u} \widehat{\Omega}^{1}_{B / A} \rightarrow 0 \] is split exact. \item $A \rightarrow B$ is formally \'{e}tale relative to $L$ iff $B \otimes_{A} \widehat{\Omega}^{1}_{A / L} \rightarrow \widehat{\Omega}^{1}_{B / L}$ is bijective. \end{enumerate} \end{prop} \begin{proof} Use the results of \cite{EGA} $0_{\mrm{IV}}$ Section 20.7, together the fact that these are finite $B$-modules. \end{proof} \begin{prop} \label{prop1.3} Let $f: A \rightarrow B$ be a formally smooth, f.f.t.\ homomorphism between noetherian adic rings. Then $B$ is flat over $A$ and $\widehat{\Omega}^{1}_{B/A}$ is a projective finitely generated $B$-module. \end{prop} \begin{proof} For flatness it suffices to show that if $\mfrak{n}$ is a maximal ideal of $B$ and $\mfrak{m} := f^{-1}(\mfrak{n})$, then $\widehat{A}_{\mfrak{m}} \rightarrow \widehat{B}_{\mfrak{n}}$ is flat ($\widehat{B}_{\mfrak{n}}$ is the completion of $B_{\mfrak{n}}$ with the $\mfrak{n}$-adic topology). Now $\mfrak{n}$ is open, and hence so is $\mfrak{m}$. Both $A \rightarrow \widehat{A}_{\mfrak{m}}$ and $B \rightarrow \widehat{B}_{\mfrak{n}}$ are formally \'{e}tale, therefore $\widehat{A}_{\mfrak{m}} \rightarrow \widehat{B}_{\mfrak{n}}$ is formally smooth. Because $A \rightarrow B$ is f.f.t.\ it follows that $A / \mfrak{m} \rightarrow B / \mfrak{n}$ is finite type, and hence finite (and $\mfrak{m}$ is a maximal ideal). By \cite{EGA} $0_{\mrm{IV}}$ Thm.\ 19.7.1, $\widehat{B}_{\mfrak{n}}$ is flat over $\widehat{A}_{\mfrak{m}}$. The second assertion follows from \cite{EGA} $0_{\mrm{IV}}$ Thm.\ 20.4.9. \end{proof} \begin{prop} \label{prop1.4} Let $f : A \rightarrow B$ be a f.f.t., formally smooth homomorphism of noetherian adic rings, and let $\mfrak{q} \in \operatorname{Spf} B$. Suppose $\operatorname{rank} \widehat{\Omega}^{1}_{\widehat{B}_{\mfrak{q}} / A} = n$. Then: \begin{enumerate} \item For some $b \in B - \mfrak{q}$ there is a formally \'{e}tale homomorphism $\tilde{f} : A\sqbr{\underline{t}} = A\sqbr{t_{1}, \ldots, t_{n}} \rightarrow B_{ \{b\} }$ extending $f$. \item For any $\mfrak{q}' \in \operatorname{Spf} B_{ \{b\} }$ let $\mfrak{r} := \tilde{f}^{-1}(\mfrak{q}')$. Then $\widehat{A\sqbr{\underline{t}}}_{\mfrak{r}} \rightarrow \widehat{B}_{\mfrak{q}'}$ is finite \'{e}tale. \item Let $\mfrak{p} := f^{-1}(\mfrak{q})$. Assume $\widehat{A}_{\mfrak{p}}$ is regular of dimension $m$, and $\operatorname{tr.deg}_{k(\mfrak{p})} k(\mfrak{q}) = l$. Then $\widehat{B}_{\mfrak{q}}$ is regular of dimension $n + m - l$. \end{enumerate} \end{prop} \begin{proof} 1.\ By Prop.\ \ref{prop1.3} we can find $b$ s.t.\ $\widehat{\Omega}^{1}_{B_{ \{b\} } / A} \cong B_{ \{b\} } \otimes_{B} \widehat{\Omega}^{1}_{B / A}$ is free, say with basis $\mrm{d} b_{1}, \ldots, \mrm{d} b_{n}$. Then we get a homomorphism $A \sqbr{ \underline{t} } \rightarrow B_{ \{b\} }$, $t_{i} \mapsto b_{i}$. In order to stay inside the category of adic rings we may replace $A \sqbr{\underline{t}}$ with its completion $A \{ \underline{t} \}$ (cf.\ Examples \ref{exa1.2} - \ref{exa1.1} for the notation). According to Proposition \ref{prop1.1} we see that $A \sqbr{\underline{t}} \rightarrow B_{ \{b\} }$ is formally \'{e}tale relative to $A$. But since $A \rightarrow B_{ \{b\} }$ is formally smooth, this implies that $A \sqbr{\underline{t}} \rightarrow B_{ \{b\} }$ is actually (absolutely) formally \'{e}tale. \noindent 2.\ Consider the formally \'{e}tale homomorphism $k(\mfrak{r}) \rightarrow \widehat{B}_{\mfrak{q}'} / \mfrak{r} \widehat{B}_{\mfrak{q}'}$. Since $\mfrak{q}'$ is an open prime ideal it follows that $A \rightarrow B / \mfrak{q}'$ is a finite type homomorphism, so the field extension $k(\mfrak{r}) \rightarrow k(\mfrak{q}')$ is finitely generated. By \cite{Hg1} Lemma 3.9 we see that in fact $\widehat{B}_{\mfrak{q}'} / \mfrak{r} \widehat{B}_{\mfrak{q}'} = k(\mfrak{q}')$, so $k(\mfrak{r}) \rightarrow k(\mfrak{q}')$ is finite separable. Hence $\widehat{A\sqbr{\underline{t}}}_{\mfrak{r}} \rightarrow \widehat{B}_{\mfrak{q}'}$ is finite \'{e}tale. \noindent 3.\ Take $\mfrak{q}' := \mfrak{q}$. Under the assumption the ring $\widehat{A\sqbr{\underline{t}}}_{\mfrak{r}}$ is regular, and according to \cite{Ma} \S 14.c Thm.\ 23, $\operatorname{dim} \widehat{A\sqbr{\underline{t}}}_{\mfrak{r}} = m + n - l$. By part 2, $\widehat{B}_{\mfrak{q}}$ is also regular, and $\operatorname{dim} \widehat{B}_{\mfrak{q}} = \operatorname{dim} \widehat{A\sqbr{\underline{t}}}_{\mfrak{r}}$. \end{proof} Let us now pass to formal schemes. Given a noetherian formal scheme $\mfrak{X}$, choose a defining ideal $\mcal{I} \subset \mcal{O}_{\mfrak{X}}$, and set \begin{equation} \label{eqn1.1} X_{n} := (\mfrak{X}, \mcal{O}_{\mfrak{X}}/\mcal{I}^{n+1}) . \end{equation} $X_{n}$ is a noetherian (usual) scheme, and $\mfrak{X} \cong \lim_{n \rightarrow} X_{n}$ in the category of formal schemes. One possible choice for $\mcal{I}$ is the largest defining ideal, in which case one has $X_{0} = \mfrak{X}_{\mrm{red}}$, the reduced closed subscheme (cf.\ \cite{EGA} I \S 10.5). \begin{lem} \label{lem1.12} Suppose $f : \mfrak{X} \rightarrow \mfrak{Y}$ is a morphism between noetherian formal schemes. There are defining ideals $\mcal{I} \subset \mcal{O}_{\mfrak{X}}$ and $\mcal{J} \subset \mcal{O}_{\mfrak{Y}}$ s.t.\ $f^{-1} \mcal{J} \cdot \mcal{O}_{\mfrak{X}} \subset \mcal{I}$. Letting $X_{n}$ and $Y_{n}$ be the corresponding schemes \textup{(}cf.\ \textup{(\ref{eqn1.1})} above\textup{)}, we get morphisms of schemes $f_{n} : X_{n} \rightarrow Y_{n}$, with $f = \lim_{n \rightarrow} f_{n}$. \end{lem} \begin{proof} See \cite{EGA} I \S 10.6. For instance, one could take $\mcal{I}$ to be the largest defining ideal and $\mcal{J}$ arbitrary. \end{proof} \begin{dfn} \label{dfn1.2} Let $f: \mfrak{X} \rightarrow \mfrak{Y}$ be a morphism of noetherian (adic) formal schemes. We say that $f$ is of {\em formally finite type} (or that $\mfrak{X}$ is a formally finite type formal scheme over $\mfrak{Y}$) if the morphism $f_{0} : X_{0} \rightarrow Y_{0}$ in Lemma \ref{lem1.12} is finite type, for some choice of defining ideals of $\mfrak{X}$ and $\mfrak{Y}$. \end{dfn} Observe that if the morphism $f_{0}$ is finite type then so are all the $f_{n}$, and the definition doesn't depend on the defining ideals chosen. \begin{rem} \label{rem1.10} The definition of f.f.t.\ morphism we gave in an earlier version of the paper was more cumbersome, though equivalent. The present Definition \ref{dfn1.2} is taken from \cite{AJL2}, where the name is ``pseudo-finite type morphism'', and I wish to thank A.\ Jerem\'{\i}as for bringing it to my attention. \end{rem} Here are a couple of examples of f.f.t.\ morphisms: \begin{exa} A finite type morphism $\mfrak{X} \rightarrow \mfrak{Y}$ (in the sense of \cite{EGA} I \S 10.13) is f.f.t. \end{exa} \begin{exa} \label{exa1.7} Let $X$ be a scheme of finite type over a noetherian scheme $S$, and let $X_{0} \subset X$ be a locally closed subset. Then the completion $\mfrak{X} = X_{/X_{0}}$ (see \cite{EGA} I \S 10.8) is of f.f.t.\ over $S$. Such a formal scheme is called {\em algebraizable}. \end{exa} \begin{dfn} \label{dfn1.1} A f.f.t.\ morphism $f : \mfrak{X} \rightarrow \mfrak{Y}$ is called {\em formally finite} (resp.\ {\em formally proper}) if the morphism $f_{0} : X_{0} \rightarrow Y_{0}$ in Lemma \ref{lem1.12} is finite (resp.\ proper), for some choice of defining ideals. \end{dfn} \begin{exa} \label{exa1.8} If in Example \ref{exa1.7} the subset $X_{0} \subset X$ is closed, then $\mfrak{X} \rightarrow X$ is formally finite. If $X_{0} \rightarrow S$ is proper, then $\mfrak{X} \rightarrow S$ is formally proper. \end{exa} \begin{prop} \label{prop1.2} \begin{enumerate} \item An immersion $\mfrak{X} \rightarrow \mfrak{Y}$ is f.f.t. \item If $\mfrak{X} \rightarrow \mfrak{Y}$ and $\mfrak{Y} \rightarrow \mfrak{Z}$ are f.f.t., then so is $\mfrak{X} \rightarrow \mfrak{Z}$. \item Let $\mfrak{U} = \operatorname{Spf} B$ and $\mfrak{V} = \operatorname{Spf} A$. Then a morphism $\mfrak{U} \rightarrow \mfrak{V}$ is f.f.t.\ iff the ring homomorphism $A \rightarrow B$ is f.f.t. \end{enumerate} \end{prop} \begin{proof} Consider morphisms of schemes $X_{0} \rightarrow Y_{0}$ etc.\ as in Lemma \ref{lem1.12}. For part 3 use condition (i) of Lemma \ref{lem1.11}. \end{proof} \begin{prop} \label{prop1.12} Let $\mfrak{X}$ be a noetherian formal scheme and $Z \subset \mfrak{X}$ a locally closed subset. Then there is a noetherian formal scheme $\mfrak{X}_{/Z}$, with underlying topological space $Z$, and the natural morphism $\mfrak{X}_{/Z} \rightarrow \mfrak{X}$ is f.f.t. \end{prop} \begin{proof} Pick an open subset $\mfrak{U} \subset \mfrak{X}$ s.t.\ $Z \subset \mfrak{U}$ is closed, and choose a defining ideal $\mcal{I}$ of $Z$. Let $\mcal{O}_{\mfrak{Z}} := \lim_{\leftarrow i} \mcal{O}_{\mfrak{U}} / \mcal{I}^{i}$. According to \cite{EGA} I Section 10.6, $\mfrak{X}_{/Z} := (Z, \mcal{O}_{\mfrak{Z}})$ is a noetherian formal scheme. Clearly $\mfrak{X}_{/Z} \rightarrow \mfrak{X}$ is f.f.t. \end{proof} In \cite{EGA} I \S 10.3 it is shown that finite type morphisms between noetherian formal schemes are preserved by base change. This is true also for f.f.t.\ morphisms: \begin{thm} \label{thm1.10} Suppose $\mfrak{X}$, $\mfrak{Y}$ and $\mfrak{Y}'$ are noetherian formal schemes, $\mfrak{X} \rightarrow \mfrak{Y}$ is a f.f.t.\ morphism, and $\mfrak{Y}' \rightarrow \mfrak{Y}$ is an arbitrary morphism. Then $\mfrak{X}' := \mfrak{X} \times_{\mfrak{Y}} \mfrak{Y}'$ is also noetherian, and the morphism $\mfrak{X}' \rightarrow \mfrak{Y}'$ is f.f.t. \end{thm} \begin{proof} First note that the formal scheme $\mfrak{X}' = \mfrak{X} \times_{\mfrak{Y}} \mfrak{Y}'$ exists (\cite{EGA} I \S 10.7). For any affine open sets $\mfrak{U} = \operatorname{Spf} B \subset \mfrak{X}$, $\mfrak{V}' = \operatorname{Spf} A' \subset \mfrak{Y}'$ and $\mfrak{V} = \operatorname{Spf} A \subset \mfrak{Y}$ such that $\mfrak{U} \rightarrow \mfrak{V}$ and $\mfrak{V}' \rightarrow \mfrak{V}$, one has $\mfrak{U}' = \mfrak{U} \times_{\mfrak{V}} \mfrak{V}' = \operatorname{Spf} B \widehat{\otimes}_{A} A'$, and $\mfrak{U}' \subset \mfrak{X}'$ is open. By Propositions \ref{prop1.11} and \ref{prop1.2}, $\mfrak{U}'$ is a noetherian formal scheme, and $\mfrak{U}' \rightarrow \mfrak{V}'$ is f.f.t. But finitely many such $\mfrak{U}'$ cover $\mfrak{X}'$. \end{proof} \begin{cor} \label{cor1.10} If $\mfrak{X}_{1}$, $\mfrak{X}_{2}$ and $\mfrak{Y}$ are noetherian and $\mfrak{X}_{i} \rightarrow \mfrak{Y}$ are f.f.t.\ morphisms, then $\mfrak{X}_{3} := \mfrak{X}_{1} \times_{\mfrak{Y}} \mfrak{X}_{2}$ is also noetherian, and $\mfrak{X}_{3} \rightarrow \mfrak{Y}$ is f.f.t. \end{cor} \begin{rem} \label{rem1.4} I do not know an example of a f.f.t.\ formal scheme $\mfrak{X}$ over a scheme $S$ which is not locally algebraizable. (Locally algebraizable means there is an open covering $\mfrak{X} = \bigcup \mfrak{U}_{i}$, with $\mfrak{U}_{i} \rightarrow S$ algebraizable, in the sense of Example \ref{exa1.7}.) \end{rem} \begin{dfn} \label{dfn1.3} A morphism of formal schemes $\mfrak{X} \rightarrow \mfrak{Y}$ is said to be {\em formally smooth} (resp.\ {\em formally \'{e}tale}) if, given a (usual) affine scheme $Z$ and a closed subscheme $Z_{0} \subset Z$ defined by a nilpotent ideal, the map $\operatorname{Hom}_{\mfrak{Y}}(Z, \mfrak{X}) \rightarrow \operatorname{Hom}_{\mfrak{Y}}(Z_{0}, \mfrak{X})$ is surjective (resp.\ bijective). \end{dfn} This is the definition of formal smoothness used in \cite{EGA} IV Section 17.1. We shall also require the next notion. \begin{dfn} \label{dfn1.4} A morphism $g: \mfrak{X} \rightarrow \mfrak{Y}$ between noetherian formal schemes is called {\em \'{e}tale} if it is of finite type (see \cite{EGA} I \S 10.13) and formally \'{e}tale. \end{dfn} Note that if $\mfrak{Y}$ is a usual scheme, then so is $\mfrak{X}$, and $g$ is an \'{e}tale morphism of schemes. According to \cite{EGA} I Prop.\ 10.13.5 and by the obvious properties of formally \'{e}tale morphisms, if $\mfrak{U} \rightarrow \mfrak{X}$ and $\mfrak{V} \rightarrow \mfrak{X}$ are \'{e}tale, then so is $\mfrak{U} \times_{\mfrak{X}} \mfrak{V} \rightarrow \mfrak{X}$. Hence for fixed $\mfrak{X}$, the category of all \'{e}tale morphisms $\mfrak{U} \rightarrow \mfrak{X}$ forms a site (cf.\ \cite{Mi} Ch.\ II \S 1). We call this site the small \'{e}tale site on $\mfrak{X}$, and denote it by $\mfrak{X}_{\mrm{et}}$. \section{Smooth Formal Embeddings and De Rham Cohomology} Fix a noetherian base scheme $S$ and a finite type $S$-scheme $X$. \begin{dfn} \label{dfn2.1} A {\em smooth formal embedding} (s.f.e.) of $X$ (over $S$) is the following data: \begin{enumerate} \rmitem{i} A noetherian formal scheme $\mfrak{X}$. \rmitem{ii} A formally finite type, formally smooth morphism $\mfrak{X} \rightarrow S$. \rmitem{iii} An $S$-morphism $X \rightarrow \mfrak{X}$, which is a closed immersion and a homeomorphism between the underlying topological spaces. \end{enumerate} We shall refer to this by writing ``$X \subset \mfrak{X}$ is a s.f.e.'' \end{dfn} \begin{exa} \label{exa2.1} Suppose $Y$ is a smooth $S$-scheme, $X \subset Y$ a locally closed subset, and $\mfrak{X} = Y_{/X}$ the completion. Then $X \subset \mfrak{X}$ is a smooth formal embedding. Such an embedding is called an {\em algebraizable embedding} (cf.\ Remark \ref{rem1.4}). \end{exa} The smooth formal embeddings of $X$ form a category, in which a morphism of embeddings is an $S$-morphism of formal schemes $f : \mfrak{X} \rightarrow \mfrak{Y}$ inducing the identity on $X$. Note that any morphism of embeddings $f: \mfrak{X} \rightarrow \mfrak{Y}$ is affine (cf.\ \cite{EGA} I Prop.\ 10.6.12), and the functor $f_{*} : \mathsf{Mod}(\mfrak{X}) \rightarrow \mathsf{Mod}(\mfrak{Y})$ is exact. Let $\mfrak{X}$ and $\mfrak{Y}$ be two smooth formal embeddings of $X$. Consider the formal scheme $\mfrak{X} \times_{S} \mfrak{Y}$. Then the diagonal $\Delta : X \rightarrow \mfrak{X} \times_{S} \mfrak{Y}$ is an immersion (we do not assume our formal schemes are separated!). \begin{prop} \label{prop2.1} The completion $(\mfrak{X} \times_{S} \mfrak{Y})_{/ X}$ of $\mfrak{X} \times_{S} \mfrak{Y}$ along $\Delta(X)$ is a smooth formal embedding of $X$, and moreover it is a product of $\mfrak{X}$ and $\mfrak{Y}$ in the category of smooth formal embeddings. \end{prop} \begin{proof} By Theorem \ref{thm1.10} and Proposition \ref{prop1.12} it follows that $(\mfrak{X} \times_{S} \mfrak{Y})_{/ X}$ is formally finite type over $S$, so in particular it is noetherian. Clearly $(\mfrak{X} \times_{S} \mfrak{Y})_{/ X} \rightarrow S$ is formally smooth. \end{proof} The benefit of using formal rather than algebraic embeddings is in: \begin{prop} \label{prop2.4} Let $X \subset \mfrak{X}$ be a smooth formal embedding \textup{(}over $S$\textup{)} and $g : U \rightarrow X$ an \'{e}tale morphism. Then there exists a noetherian formal scheme $\mfrak{U}$ and an \'{e}tale morphism $\widehat{g} : \mfrak{U} \rightarrow \mfrak{X}$ s.t.\ $U \cong \mfrak{U} \times_{\mfrak{X}} X$. $\widehat{g} : \mfrak{U} \rightarrow \mfrak{X}$ is unique \textup{(}up to a unique isomorphism\textup{)}, and moreover $U \rightarrow \mfrak{U}$ is a smooth formal embedding. \end{prop} \begin{proof} This is essentially the ``topological invariance of \'{e}tale morphisms'', cf.\ \cite{EGA} IV \S 18.1 (or \cite{Mi} Ch.\ I Thm.\ 3.23). Let $\mcal{I} := \operatorname{Ker}(\mcal{O}_{\mfrak{X}} \rightarrow \mcal{O}_{X})$ and $X_{i} := (\mfrak{X}, \mcal{O}_{\mfrak{X}} / \mcal{I}^{i+1})$; so $X = X_{0}$. For every $i$ there is a unique \'{e}tale morphism $g_{i} : U_{i} \rightarrow X_{i}$ s.t.\ $U \cong U_{i} \times_{X_{i}} X$. Identifying the underlying topological spaces of $U_{i}$ and $U$ we get an inverse system of sheaves $\{ \mcal{O}_{U_{i}} \}$ on $U$. Setting $\mcal{O}_{\mfrak{U}} := \lim_{\leftarrow i} \mcal{O}_{U_{i}}$ we get a noetherian formal scheme $\mfrak{U}$ having the proclaimed properties (cf.\ \cite{EGA} I \S 10.6). \end{proof} Thus we can consider $\mfrak{X}_{\mrm{et}}$ as a ``smooth formal embedding'' of $X_{\mrm{et}}$. If $\mcal{M}$ is a sheaf on $X_{\mrm{et}}$ and $U \rightarrow X$ is an \'{e}tale morphism, we denote by $\mcal{M}|_{U}$ the restriction of $\mcal{M}$ to $U_{\mrm{Zar}}$. \begin{cor} \label{cor2.3} Let $X \subset \mfrak{X}$ be a smooth formal embedding over $S$. Then there is a sheaf of DGAs $\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}} / S}$ on $X_{\mrm{et}}$, with the property that for each $g: U \rightarrow X$ in $X_{\mrm{et}}$ and corresponding $\widehat{g}: \mfrak{U} \rightarrow \mfrak{X}$ in $\mfrak{X}_{\mrm{et}}$, one has $\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}} / S} |_{U} \cong \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{U} / S}$. \end{cor} \begin{proof} By Prop.\ \ref{prop1.1}, $\widehat{\Omega}^{p}_{\mfrak{U} / S} \cong \widehat{g}^{*} \widehat{\Omega}^{p}_{\mfrak{X} / S}$. Now $\widehat{\Omega}^{p}_{\mfrak{X} / S}$ is coherent, so we can use \cite{Mi} Ch.\ II Cor.\ 1.6 (which applies to our \'{e}tale site $\mfrak{X}_{\mrm{et}}$). \end{proof} For smooth formal embeddings, closed immersions and smooth morphisms are locally trivial, in the following sense. Recall that for an adic algebra $A$, the ring of formal power series $A [\sqbr{ \underline{t} }] = A [\sqbr{ t_{1}, \ldots, t_{n} }]$ is adic (cf.\ Example \ref{exa1.1}). \begin{thm} \label{thm2.2} Let $f : \mfrak{X} \rightarrow \mfrak{Y}$ be a morphism of smooth formal embeddings of $X$ over $S$. Assume $f$ is a closed immersion \textup{(}resp.\ formally smooth\textup{)}. Then, given a point $x \in X$, there are affine open sets $U \subset X$ and $W \subset S$, with $x \in U$ and $U \rightarrow W$, satisfying condition \textup{($*$)} below. \begin{enumerate} \item[\textup{($*$)}] Let $W = \operatorname{Spec} L$, and let $\operatorname{Spf} A \subset \mfrak{Y}$ and $\operatorname{Spf} B \subset \mfrak{X}$ be the affine formal schemes supported on $U$. Then there is an isomorphism of topological $L$-algebras $A \cong B [\sqbr{\underline{t}}]$ \textup{(}resp.\ $B \cong A [\sqbr{\underline{t}}]$\textup{)} such that $f^{*} : A \rightarrow B$ is projection modulo $(\underline{t})$ \textup{(}resp.\ the inclusion\textup{)}. \end{enumerate} \end{thm} \begin{proof} 1.\ Assume $f$ is a closed immersion. According to \cite{EGA} $0_{\mrm{IV}}$ Thm.\ 19.5.3 and Cor.\ 20.7.9, by choosing $U = \operatorname{Spec} C$ small enough, and setting $I := \operatorname{Ker}(f^{*} : A \rightarrow B)$, we obtain an exact sequence \[ 0 \rightarrow I / I^{2} \rightarrow B \otimes_{A} \widehat{\Omega}^{1}_{A/L} \rightarrow \widehat{\Omega}^{1}_{B/L} \rightarrow 0 \] of free $B$-modules. Choose $a_{1}, \ldots, a_{n}, b_{1}, \ldots, b_{m} \in A$ s.t.\ $\{ a_{i} \}$ is a $B$-basis of $I / I^{2}$, and $\{ \mrm{d} b_{i} \}$ is a $B$-basis of $\widehat{\Omega}^{1}_{B/L}$. By the proof of Prop.\ \ref{prop1.4} the homomorphisms $L \sqbr{\underline{s}} \rightarrow B$, $L \sqbr{\underline{s}, \underline{t}} \rightarrow A$ and $L \sqbr{\underline{s}, \underline{t}} \rightarrow B [\sqbr{\underline{t}}]$, sending $s_{i} \mapsto b_{i}$ and $t_{i} \mapsto a_{i}$, are all formally \'{e}tale. Take $\mfrak{a} := \operatorname{Ker}(A \rightarrow C)$, which is a defining ideal of $A$, containing $A \cdot (\underline{t}) = I$. Let $\mfrak{b} := \mfrak{a} \cdot B$, which is a defining ideal of $B$. Hence the ideal $\mfrak{c} = B [\sqbr{\underline{t}}] \cdot (\mfrak{b}, \underline{t})$ is a defining ideal of $B [\sqbr{\underline{t}}]$. By formal \'{e}taleness of $L \sqbr{\underline{s}, \underline{t}} \rightarrow A$ and $L \sqbr{\underline{s}, \underline{t}} \rightarrow B [\sqbr{\underline{t}}]$, the isomorphism $A / \mfrak{a} \cong B [\sqbr{\underline{t}}] / \mfrak{c} \cong C$ lifts uniquely to an isomorphism $A \cong B [\sqbr{\underline{t}}]$. \noindent 2.\ Now assume $f$ is formally smooth. Let $\mfrak{b} := \operatorname{Ker}(B \rightarrow C)$, which is a defining ideal of $B$. Since $A \rightarrow B / \mfrak{b}$ is surjective it follows that $(B / \mfrak{b}) \otimes_{B} \widehat{\Omega}^{1}_{B/A}$ is generated by $\mrm{d}(\mfrak{b})$. By Nakayama's Lemma we see that $\widehat{\Omega}^{1}_{B/A} = B \cdot \mrm{d}(\mfrak{b})$. After shrinking $U$ sufficiently we get $\widehat{\Omega}^{1}_{B/A} = \bigoplus_{i = 1}^{n} B \cdot \mrm{d} b_{i}$ with $b_{i} \in \mfrak{b}$, and the homomorphism $A [\sqbr{\underline{t}}] \rightarrow B$, $t_{i} \mapsto b_{i}$, is formally \'{e}tale. Continuing like in part 1 of the proof we conclude that this is actually an isomorphism. \end{proof} \begin{thm} \label{thm2.1} Suppose $S$ is a noetherian scheme of characteristic $0$, and $X$ is a finite type $S$-scheme. Let $f : \mfrak{X} \rightarrow \mfrak{Y}$ be a morphism of smooth formal embeddings of $X$. Then the DGA homomorphism $f^{*} : \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{Y} / S} \rightarrow \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X} / S}$ is a quasi-isomorphism. Moreover, if $g : \mfrak{X} \rightarrow \mfrak{Y}$ is any other morphism, then $\mrm{H}(f^{*}) = \mrm{H}(g^{*})$. \end{thm} \begin{proof} The assertions of the theorem are both local, and they will be proved in three steps. \noindent Step 1.\ Assume $f$ is a closed immersion. By Thm.\ \ref{thm2.2} it suffices to check the case $f : \operatorname{Spf} B = \mfrak{U} \rightarrow \operatorname{Spf} A = \mfrak{V}$ with $A \cong B [\sqbr{\underline{t}}]$ as topological $L$-algebras. We must show that $\widehat{\Omega}^{{\textstyle \cdot}}_{A/L} \rightarrow \widehat{\Omega}^{{\textstyle \cdot}}_{B/L}$ is a quasi-isomorphism. But since $\mbb{Q} \subset L$, this is the well known Poincar\'{e} Lemma for formal power series (cf.\ \cite{Ha} Ch.\ II Prop.\ 1.1, or \cite{Ye3} Lemma 7.5). \noindent Step 2.\ Suppose $f_{1}, f_{2}: \mfrak{X} \rightarrow \mfrak{Y}$ are two morphisms. We wish to show that $\mrm{H}(f_{1}^{*}) = \mrm{H}(f_{2}^{*})$. First consider \[ \mfrak{Y} \xrightarrow{\mrm{diag}} (\mfrak{Y} \times_{k} \mfrak{Y})_{/X} \xrightarrow{p_{i}} \mfrak{Y} . \] Since the diagonal immersion is closed, we can apply the result of the previous paragraph to it. We conclude that $\mrm{H}(p_{1}^{*}) = \mrm{H}(p_{2}^{*})$, and that these are isomorphisms. But looking at \[ \mfrak{X} \xrightarrow{\mrm{diag}} (\mfrak{X} \times_{k} \mfrak{X})_{/X} \xrightarrow{f_{1} \times f_{2}} (\mfrak{Y} \times_{k} \mfrak{Y})_{/X} \xrightarrow{p_{i}} \mfrak{Y} \] we see that our claim is proved. \noindent Step 3.\ Consider an arbitrary morphism $f: \mfrak{X} \rightarrow \mfrak{Y}$. Take any affine open set $U \subset X$, with corresponding affine formal schemes $\operatorname{Spf} B = \mfrak{U} \subset \mfrak{X}$ and $\operatorname{Spf} A = \mfrak{V} \subset \mfrak{Y}$. The definition of formal smoothness implies there is some morphism of embeddings $g : \mfrak{V} \rightarrow \mfrak{U}$. This morphism will not necessarily be an inverse of $f|_{\mfrak{U}}$, but nonetheless, according to Step 2, $\mrm{H}(g^{*})$ and $\mrm{H}(f|_{\mfrak{U}}^{*})$ will be isomorphisms between $\mrm{H} \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{U} / S}$ and $\mrm{H} \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{V} / S}$, inverse to each other. \end{proof} In \cite{Ha} the relative De Rham cohomology $\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}} (X / S)$ was defined. In the situation of Example \ref{exa2.1}, where $X \subset Y$ is a smooth algebraic embedding of $S$-schemes, $\mfrak{X} = Y_{/X}$ and $\pi : \mfrak{X} \rightarrow S$ is the structural morphism, the definition is $\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}} (X / S) = \mrm{H}^{{\textstyle \cdot}} \mrm{R} \pi_{*} \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X} / S}$. Even if $X$ is not globally embeddable, $\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}} (X / S)$ can still be defined, by taking a system of local embeddings $\{ U_{i} \subset V_{i} \}$, $X = \bigcup U_{i}$, and putting together a ``\v{C}ech-De Rham'' complex (cf.\ \cite{Ha} pp.\ 28-29; it seems one should also assume $X$ separated and the $U_{i}$ are affine). \begin{cor} \label{cor2.1} Suppose $S$ has characteristic $0$. Let $X \subset \mfrak{X}$ be any smooth formal embedding \textup{(}not necessarily algebraizable\textup{)}. Then $\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}} (X / S) = \mrm{H}^{{\textstyle \cdot}} \mrm{R} \pi_{*} \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X} / S}$ as graded $\mcal{O}_{S}$-algebras. \end{cor} \begin{proof} Assume for simplicity that a global smooth algebraic embedding exists. The general case, involving a system of embeddings, only requires more bookkeeping. Say $X \subset Y$ is the given algebraic embedding, and let $\mfrak{Y} := Y_{/X}$. Now the two formal embeddings $\mfrak{X}$ and $\mfrak{Y}$ are comparable: their product $(\mfrak{X} \times_{S} \mfrak{Y})_{/X}$ maps to both. By the theorem we get quasi-isomorphic DGAs on $X$. \end{proof} \begin{rem} \label{rem2.1} From Corollaries \ref{cor2.3} and \ref{cor2.1} we see that there is a sheaf of DGAs $\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}} / S}$ on $X_{\mrm{et}}$, with the property that for any $U \rightarrow X$ \'{e}tale, $\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}}(U/S) =$ \linebreak $\mrm{H}^{{\textstyle \cdot}} \Gamma(U, \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}} / S})$. As will be shown in \cite{Ye4}, the DGA $\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X} / S}$ has an adelic resolution $\mcal{A}^{{\textstyle \cdot}}_{\mfrak{X} / S}$, where $\mcal{A}^{p,q}_{\mfrak{X} / S} = \underline{\mbb{A}}^{q}_{\mrm{red}}( \widehat{\Omega}^{p}_{\mfrak{X} / S})$, Beilinson's sheaf of adeles. The adeles calculate cohomology: $\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}}(X/S) = \mrm{H}^{{\textstyle \cdot}} \Gamma(X, \mcal{A}^{{\textstyle \cdot}}_{\mfrak{X} / S})$. Furthermore the adeles extend to an \'{e}tale sheaf $\mcal{A}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}} / S}$. \end{rem} \begin{rem} \label{rem2.3} Suppose $S = \operatorname{Spec} k$, a field of characteristic $0$. In \cite{Ye3} a complex $\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}$, called the De Rham-residue complex, is defined. One has $\mrm{H}^{i}(X, \mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}) = \mrm{H}_{-i}^{\mrm{DR}}(X)$, the De Rham homology. Moreover there is a sheaf $\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}}}$ on $X_{\mrm{et}}$, which directly implies that the De Rham homology is contravariant for \'{e}tale morphisms. Furthermore $\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}$ is naturally a DG $\mcal{A}^{{\textstyle \cdot}}_{\mfrak{X}}$-module, \end{rem} \begin{rem} \label{rem2.4} Smooth formal embeddings can be also used to define the category of $\mcal{D}$-modules on a singular scheme $X$ (in characteristic $0$). Say $X \subset \mfrak{X}$ is such an embedding. Then a formal version of Kashiwara's Theorem (cf.\ \cite{Bo} Theorem VI.7.11) implies that $\msf{Mod}_{\mrm{disc}}(\mcal{D}_{\mfrak{X}})$, the category of discrete modules over the ring of differential operators $\mcal{D}_{\mfrak{X}}$ is, as an abelian category, independent of $\mfrak{X}$. \end{rem} \section{Quasi-Coherent Sheaves on Formal Schemes} Let $\mfrak{X}$ be a noetherian (adic) formal scheme. By definition, a quasi-coherent sheaf on $\mfrak{X}$ is an $\mcal{O}_{\mfrak{X}}$-module $\mcal{M}$, such that on sufficiently small open sets $\mfrak{U} \subset \mfrak{X}$ there are exact sequences $\mcal{O}_{\mfrak{U}}^{(J)} \rightarrow \mcal{O}_{\mfrak{U}}^{(I)} \rightarrow \mcal{M}|_{\mfrak{U}} \rightarrow 0$, for some indexing sets $I,J$ (cf.\ \cite{EGA} $0_{\mrm{I}}$ \S 5.1). We shall denote by $\mathsf{Mod}(\mfrak{X})$ (resp.\ $\mathsf{Coh}(\mfrak{X})$, resp.\ $\mathsf{QCo}(\mfrak{X})$) the category of $\mcal{O}_{\mfrak{X}}$-modules (resp.\ the full subcategory of coherent, resp.\ quasi-coherent, modules). It seems that the only important quasi-coherent sheaves are the coherent and the discrete ones (Def.\ \ref{dfn3.1}). Nevertheless we shall consider all quasi-coherent sheaves, at the price of a little extra effort. \begin{rem} There is some overlap between results in this section and \cite{AJL2}. \end{rem} Let $A$ be a noetherian adic ring, and let $\mfrak{U} := \operatorname{Spf} A$ be the affine formal scheme. Then there is an exact functor $M \mapsto M^{\triangle}$ from the category $\mathsf{Mod}_{\mrm{f}}(A)$ of finitely generated $A$-modules to $\mathsf{Mod}(\mfrak{U})$. It is an equivalence between $\mathsf{Mod}_{\mrm{f}}(A)$ and $\mathsf{Coh}(\mfrak{U})$ (see \cite{EGA} I \S 10.10). \begin{prop} \label{prop3.2} The functor $M \mapsto M^{\triangle}$ extends uniquely to a functor $\mathsf{Mod}(A) \rightarrow \mathsf{Mod}(\mfrak{U})$, which is exact and commutes with direct limits. The $\mcal{O}_{\mfrak{U}}$-module $M^{\triangle}$ is quasi-coherent. For any $\mcal{O}_{\mfrak{U}}$-module $\mcal{M}$ the following are equivalent: \begin{enumerate} \rmitem{i} $\mcal{M} \cong M^{\triangle}$ for some $A$-module $M$. \rmitem{ii} $\mcal{M} \cong \lim_{\alpha \rightarrow} \mcal{M}_{\alpha}$ for some directed system $\{ \mcal{M}_{\alpha} \}$ of coherent $\mcal{O}_{\mfrak{U}}$-modules. \rmitem{iii} For every affine open set $\mfrak{V} = \operatorname{Spf} B \subset \mfrak{U}$, one has $\Gamma(\mfrak{V}, \mcal{M}) \cong B \otimes_{A} \Gamma(\mfrak{U}, \mcal{M})$. \end{enumerate} \end{prop} \begin{proof} Take any $A$-module $M$ and write it as $M = \lim_{\alpha \rightarrow} M_{\alpha}$ with finitely generated modules $M_{\alpha}$. Define a presheaf $M^{\triangle}$ on $\mfrak{U}$ by $\Gamma(\mfrak{V}, M^{\triangle}) := \lim_{\alpha \rightarrow} \Gamma(\mfrak{V}, M^{\triangle}_{\alpha})$, for $\mfrak{V} \subset \mfrak{U}$ open. Since $\mfrak{U}$ is a noetherian topological space it follows that $M^{\triangle}$ is actually a sheaf. By construction $M \mapsto M^{\triangle}$ commutes with direct limits. Since the functor is exact on $\mathsf{Mod}_{\mrm{f}}(A)$, it's also exact on $\mathsf{Mod}(A)$. The implication (i) $\Rightarrow$ (ii) is because $M^{\triangle}_{\alpha}$ is coherent. (ii) $\Rightarrow$ (iii): for such $B$ one has $\Gamma(\mfrak{V}, \mcal{M}_{\alpha}) \cong B \otimes_{A} \Gamma(\mfrak{U}, \mcal{M}_{\alpha})$; now apply $\lim_{\alpha \rightarrow}$. (iii) $\Rightarrow$ (i): set $M := \Gamma(\mfrak{U}, \mcal{M})$. Then for every affine $\mfrak{V}$ we have $\Gamma(\mfrak{V}, \mcal{M}) = B \otimes_{A} M = \Gamma(\mfrak{V}, M^{\triangle})$, so $\mcal{M} = M^{\triangle}$. Finally the module $M$ has a presentation $A^{(I)} \rightarrow A^{(J)} \rightarrow M \rightarrow 0$. By exactness we get a presentation for $M^{\triangle}$. \end{proof} It will be convenient to write $\mcal{O}_{\mfrak{U}} \otimes_{A} M$ instead of $M^{\triangle}$. \begin{rem} I do not know whether Serre's Theorem holds, namely whe\-ther {\em every} quasi-coherent $\mcal{O}_{\mfrak{U}}$-module $\mcal{M}$ is of the form $\mcal{M} \cong \mcal{O}_{\mfrak{U}} \otimes_{A} M$. Thus it may be that $\mathsf{QCo}(\mfrak{U})$ is not closed under direct limits in $\mathsf{Mod}(\mfrak{U})$ (cf.\ Lemma \ref{lem4.1}). \end{rem} \begin{cor} \label{cor3.1} Let $\mcal{M}$ be a quasi-coherent $\mcal{O}_{\mfrak{X}}$-module and $x \in \mfrak{X}$ a point. Then there is an open neighborhood $\mfrak{U} = \operatorname{Spf} A$ of $x$ s.t.\ $\mcal{M}|_{\mfrak{U}} \cong \mcal{O}_{\mfrak{U}} \otimes_{A} \Gamma(\mfrak{U}, \mcal{M})$. For such $\mfrak{U}$ one has $\mrm{H}^{1}(\mfrak{U}, \mcal{M}) = 0$. \end{cor} \begin{proof} Choose $\mfrak{U}$ affine such that $\mcal{M}|_{\mfrak{U}}$ has a presentation $\mcal{O}_{\mfrak{U}}^{(J)} \xrightarrow{\phi} \mcal{O}_{\mfrak{U}}^{(I)} \xrightarrow{\psi} \mcal{M}|_{\mfrak{U}} \rightarrow 0$. Define $M := \operatorname{Coker}(\phi: A^{(I)} \rightarrow A^{(J)})$. Applying the exact functor $\mcal{O}_{\mfrak{U}} \otimes_{A}$ to $A^{(I)} \xrightarrow{\phi} A^{(J)} \rightarrow M \rightarrow 0$ we get $\mcal{M}|_{\mfrak{U}} \cong \mcal{O}_{\mfrak{U}} \otimes_{A} M$. By the Proposition $M \cong \Gamma(\mfrak{U}, \mcal{M})$. As for $\mrm{H}^{1}(\mfrak{U}, - )$, use the fact that it vanishes on coherent sheaves. \end{proof} \begin{prop} \label{prop3.3} Let $\mcal{M}$ be coherent and $\mcal{N}$ quasi-coherent \textup{(}resp.\ coherent\textup{)}. Then $\mcal{H}om_{\mcal{O}_{\mfrak{X}}}(\mcal{M}, \mcal{N})$ is quasi-coherent \textup{(}resp.\ coherent\textup{)}. \end{prop} \begin{proof} For small enough $\mfrak{U} = \operatorname{Spf} A$ we get $\mcal{M}|_{\mfrak{U}} \cong \mcal{O}_{\mfrak{U}} \otimes_{A} M$ and $\mcal{N}|_{\mfrak{U}} \cong \mcal{O}_{\mfrak{U}} \otimes_{A} N$. Now for any $\mfrak{V} = \operatorname{Spf} B \subset \mfrak{U}$, $A \rightarrow B$ is flat; so \[ \operatorname{Hom}_{B}(B \otimes_{A} M, B \otimes_{A} N) \cong B \otimes_{A} \operatorname{Hom}_{A}(M, N) . \] Hence \[ \mcal{H}om_{\mcal{O}_{\mfrak{X}}}(\mcal{M}, \mcal{N})|_{\mfrak{U}} \cong \mcal{O}_{\mfrak{U}} \otimes_{A} \operatorname{Hom}_{A}(M,N) . \] \end{proof} Recall that a subcategory $\mathsf{B}$ of an abelian category $\mathsf{A}$ is called a thick abelian subcategory if for any exact sequence $M_{1} \rightarrow M_{2} \rightarrow N \rightarrow M_{3} \rightarrow M_{4}$ in $\mathsf{A}$ with $M_{i} \in \mathsf{B}$, also $N \in \mathsf{B}$. \begin{prop} \label{prop3.1} The category $\mathsf{QCo}(\mfrak{X})$ is a thick abelian subcategory of $\mathsf{Mod}(\mfrak{X})$. \end{prop} \begin{proof} First observe that the kernel and cokernel of a homomorphism $\mcal{M} \rightarrow \mcal{N}$ between quasi-coherent sheaves is also quasi-coherent. This is immediate from Cor.\ \ref{cor3.1} and Prop.\ \ref{prop3.2}. So it suffices to prove: $0 \rightarrow \mcal{M}' \rightarrow \mcal{M} \rightarrow \mcal{M}'' \rightarrow 0$ exact, $\mcal{M}', \mcal{M}''$ quasi-coherent $\Rightarrow$ $\mcal{M}$ quasi-coherent. For a sufficiently small affine open formal subscheme $\mfrak{U} = \operatorname{Spf} A$ we will get, by Cor.\ \ref{cor3.1}, that $\mrm{H}^{1}(\mfrak{U}, \mcal{M}') = 0$. Hence the sequence \[ 0 \rightarrow \Gamma(\mfrak{U}, \mcal{M}') \rightarrow M= \Gamma(\mfrak{U}, \mcal{M}) \rightarrow \Gamma(\mfrak{U}, \mcal{M}'') \rightarrow 0 \] is exact. This implies that $\mcal{M}|_{\mfrak{U}} \cong \mcal{O}_{\mfrak{U}} \otimes_{A} M$. \end{proof} \begin{dfn} \label{dfn3.1} Let $\mcal{M}$ be an $\mcal{O}_{\mfrak{X}}$-module. Define \[ \underline{\Gamma}_{\mrm{disc}} \mcal{M} := \lim_{n \rightarrow} \mcal{H}om_{\mcal{O}_{\mfrak{X}}}(\mcal{O}_{\mfrak{X}} / \mcal{I}^{n}, \mcal{M}) \subset \mcal{M} \] where $\mcal{I} \subset \mcal{O}_{\mfrak{X}}$ is any defining ideal. $\mcal{M}$ is called {\em discrete} if $\underline{\Gamma}_{\mrm{disc}} \mcal{M} = \mcal{M}$. \end{dfn} \begin{prop} \label{prop3.4} Let $\mcal{M}$ be a quasi-coherent $\mcal{O}_{\mfrak{X}}$-module. Then $\underline{\Gamma}_{\mrm{disc}} \mcal{M}$ is quasi-coherent, and in fact is a direct limit of discrete coherent $\mcal{O}_{\mfrak{X}}$-modules. \end{prop} \begin{proof} Let $X_{n}$ be as in formula (\ref{eqn1.1}) and $\mcal{M}_{n} := \mcal{H}om_{\mcal{O}_{\mfrak{X}}}(\mcal{O}_{X_{n}}, \mcal{M})$, so $\underline{\Gamma}_{\mrm{disc}} \mcal{M} = \lim_{n \rightarrow} \mcal{M}_{n}$. If $\mcal{M}$ is quasi-coherent, then $\mcal{M}_{n}$ is a quasi-coherent $\mcal{O}_{X_{n}}$-module (by Prop.\ \ref{prop3.3}), and hence is a direct limit of coherent modules. \end{proof} \section{Some Derived Functors of $\mcal{O}_{\mfrak{X}}$-Modules} Denote by $\mathsf{Mod}_{\mrm{disc}}(\mfrak{X})$ (resp.\ $\mathsf{QCo}_{\mrm{disc}}(\mfrak{X})$) the full subcategory of $\mathsf{Mod}(\mfrak{X})$ consisting of discrete modules (resp.\ discrete quasi-coherent modules). These are thick abelian subcategories. In this section we study injective objects in the category $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$, and introduce the discrete Cousin functor $\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}}$. \begin{lem} \label{lem4.1} $\msf{Mod}_{\mrm{disc}}(\mfrak{X})$ is a locally noetherian category, with enough injectives. \end{lem} \begin{proof} A family of noetherian generators consists of the sheaves $\mcal{O}_{U}$, where $X \subset \mfrak{X}$ is a closed subscheme, $U \subset X$ is an open set, and $\mcal{O}_{U}$ is extended by $0$ to all of $X$ (cf.\ \cite{RD} Theorem II.7.8). If $\mcal{J} \in \msf{Mod}(\mfrak{X})$ is injective then $\underline{\Gamma}_{\mrm{disc}} \mcal{J}$ is injective in $\msf{Mod}_{\mrm{disc}}(\mfrak{X})$. \end{proof} Given a point $x \in \mfrak{X}$ let $J(x)$ be an injective hull of the residue field $k(x)$ over the local ring $\mcal{O}_{\mfrak{X},x}$, and let $\mcal{J}(x)$ be the corresponding $\mcal{O}_{\mfrak{X}}$-module. Then $\mcal{J}(x)$ is a discrete quasi-coherent sheaf, constant on $\overline{\{ x \}}$, and it is injective in $\msf{Mod}(\mfrak{X})$. \begin{prop} \label{prop4.1} \begin{enumerate} \item $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$ is a locally noetherian category with enough injectives. \item Let $\mcal{J} \in \msf{QCo}_{\mrm{disc}}(\mfrak{X})$ be an injective object. Then $\mcal{J}$ is injective in $\msf{Mod}_{\mrm{disc}}(\mfrak{X})$ and injective on $\msf{Coh}(\mfrak{X})$. For any $\mcal{M} \in \msf{Mod}_{\mrm{disc}}(\mfrak{X})$ or $\mcal{M} \in \msf{Coh}(\mfrak{X})$ the sheaf $\mcal{H}om_{\mfrak{X}}(\mcal{M}, \mcal{J})$ is flasque. \end{enumerate} \end{prop} \begin{proof} 1.\ Let $\mcal{N} \in \msf{QCo}_{\mrm{disc}}(\mfrak{X})$. Choose a defining ideal $\mcal{I}$ of $\mfrak{X}$ and let $X_{0}$ be the scheme $(\mfrak{X}, \mcal{O}_{\mfrak{X}} / \mcal{I})$. Define $\mcal{N}_{0} := \mcal{H}om_{\mfrak{X}}(\mcal{O}_{X_{0}}, \mcal{N})$, which is a quasi-coherent $\mcal{O}_{X_{0}}$-module. Then the injective hull of $\mcal{N}_{0}$ in $\msf{Mod}(X_{0})$ is isomorphic to $\bigoplus_{\alpha} \mcal{J}_{0}(x_{\alpha})$ for some $x_{\alpha} \in X_{0}$. According to Proposition \ref{prop3.4}, $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$ is locally noetherian, and this implies that $\bigoplus_{\alpha} \mcal{J}(x_{\alpha})$ is an injective object in it. Now $\mcal{N}_{0} \subset \mcal{N}$ and $\mcal{N}_{0} \subset \bigoplus_{\alpha} \mcal{J}(x_{\alpha})$ are essential submodules, so there is some homomorphism $\mcal{N} \rightarrow \bigoplus_{\alpha} \mcal{J}(x_{\alpha})$, which is necessarily injective and essential. \noindent 2.\ If $\mcal{N} = \mcal{J}$ is injective in $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$, it follows that $\mcal{J} \rightarrow \bigoplus_{\alpha} \mcal{J}(x_{\alpha})$ is an isomorphism. Since $\msf{Mod}_{\mrm{disc}}(\mfrak{X})$ is locally noetherian it follows that $\mcal{J}$ is injective in it. Given $\mcal{M} \in \msf{Mod}_{\mrm{disc}}(\mfrak{X})$ and open sets $\mfrak{V} \subset \mfrak{U} \subset \mfrak{X}$ consider the sheaves $\mcal{M}|_{\mfrak{V}} \subset \mcal{M}|_{\mfrak{U}} \subset \mcal{M}$ (extension by $0$). Then $\mcal{H}om_{\mfrak{X}}(\mcal{M}|_{\mfrak{U}}, \mcal{J}) \rightarrow \mcal{H}om_{\mfrak{X}}(\mcal{M}|_{\mfrak{V}}, \mcal{J})$ is surjective. The category $\msf{Coh}(\mfrak{X})$ is noetherian, and therefore the functor $\operatorname{Hom}_{\mfrak{X}}(-, \mcal{J})$ is exact on it. Given $\mcal{M} \in \msf{Coh}(\mfrak{X})$ we have $\mcal{H}om_{\mfrak{X}}(\mcal{M}, \mcal{J}) \cong \bigoplus \mcal{H}om_{\mfrak{X}}(\mcal{M}, \mcal{J}(x_{\alpha}))$ which is clearly flasque. \end{proof} \begin{cor} \label{cor4.1} Let $\mcal{J}^{{\textstyle \cdot}} \in \msf{D}^{+}(\msf{QCo}_{\mrm{disc}}(\mfrak{X}))$ be a complex of injectives. Then for any $\mcal{M}^{{\textstyle \cdot}} \in \msf{Mod}_{\mrm{disc}}(\mfrak{X})$ or $\mcal{M}^{{\textstyle \cdot}} \in \msf{Coh}(\mfrak{X})$ one has \[ \begin{aligned} \mrm{R} \mcal{H}om_{\mfrak{X}}(\mcal{M}^{{\textstyle \cdot}}, \mcal{J}^{{\textstyle \cdot}}) & \cong \mcal{H}om_{\mfrak{X}}(\mcal{M}^{{\textstyle \cdot}}, \mcal{J}^{{\textstyle \cdot}}) \\ \mrm{R} \operatorname{Hom}_{\mfrak{X}}(\mcal{M}^{{\textstyle \cdot}}, \mcal{J}^{{\textstyle \cdot}}) & \cong \operatorname{Hom}_{\mfrak{X}}(\mcal{M}^{{\textstyle \cdot}}, \mcal{J}^{{\textstyle \cdot}}) \cong \Gamma(\mfrak{X}, \mcal{H}om_{\mfrak{X}}(\mcal{M}^{{\textstyle \cdot}}, \mcal{J}^{{\textstyle \cdot}})) . \end{aligned} \] \end{cor} \begin{proof} The first equality follows from Proposition \ref{prop4.1} (cf.\ \cite{RD} Section I.6). Since each sheaf $\mcal{H}om_{\mfrak{X}}(\mcal{M}^{p}, \mcal{J}^{q})$ is flasque we obtain the second equality. \end{proof} The functor $\underline{\Gamma}_{\mrm{disc}} : \msf{Mod}(\mfrak{X}) \rightarrow \msf{Mod}_{\mrm{disc}}(\mfrak{X})$ has a derived functor \[ \mrm{R} \underline{\Gamma}_{\mrm{disc}} : \msf{D}^{+}(\msf{Mod}(\mfrak{X})) \rightarrow \msf{D}^{+}(\msf{Mod}_{\mrm{disc}}(\mfrak{X})) , \] which is calculated by injective resolutions. There is another way to compute cohomology with supports. Let $t$ be an indeterminate. Define $\mbf{K}^{{\textstyle \cdot}}(t)$ to be the Koszul complex $\mbb{Z}\sqbr{t} \xrightarrow{\cdot t} \mbb{Z}\sqbr{t}$, in dimensions $0$ and $1$, and let $\mbf{K}^{{\textstyle \cdot}}_{\infty}(t) := \lim_{i \rightarrow} \mbf{K}^{{\textstyle \cdot}}(t^{i})$. Given a sequence $\underline{t} = (t_{1}, \ldots, t_{n})$ define $\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{t}) := \mbf{K}^{{\textstyle \cdot}}_{\infty}(t_{1}) \otimes \cdots \otimes \mbf{K}^{{\textstyle \cdot}}_{\infty}(t_{n})$, a complex of flat $\mbb{Z} \sqbr{\underline{t}}$-modules (in fact it's a commutative DGA). If $A$ is a noetherian commutative ring and $\underline{a} = (a_{1}, \ldots, a_{n}) \in A^{n}$, then we write $\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a})$ instead of $\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{t}) \otimes_{\mbb{Z} \sqbr{\underline{t}}} A$. Now suppose $\mfrak{a} \subset A$ is an ideal, and $\underline{a}$ are generators of $\mfrak{a}$. Then for any $M^{{\textstyle \cdot}} \in \msf{D}^{+}(\msf{Mod}(A))$ there is a natural isomorphism \begin{equation} \label{eqn4.1} \mrm{R} \Gamma_{\mfrak{a}} M^{{\textstyle \cdot}} \cong \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes M^{{\textstyle \cdot}} \end{equation} in $\msf{D}(\msf{Mod}(A))$. We refer to \cite{LS1}, \cite{Hg1} and \cite{AJL1} for full details and proofs. For sheaves one has: \begin{lem} \label{lem4.3} Suppose $\underline{a} \in \Gamma(\mfrak{U}, \mcal{O}_{\mfrak{U}})^{n}$ generates a defining ideal of the formal sche\-me $\mfrak{U}$. Then for any $\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{+}(\msf{Mod}(\mfrak{U}))$ there is a natural isomorphism \[ \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}} \cong \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes \mcal{M}^{{\textstyle \cdot}} . \] \end{lem} \begin{proof} Let $\mcal{I} := \mcal{O}_{\mfrak{U}} \cdot \underline{a}$. Then $\underline{\Gamma}_{\mrm{disc}} = \underline{\Gamma}_{\mcal{I}}$, and we may use \cite{AJL1} Lemma 3.1.1. \end{proof} \begin{prop} \label{prop4.2} Let $X$ be a noetherian scheme, $X_{0} \subset X$ a closed subset, $\mfrak{X} = X_{/ X_{0}}$ and $g : \mfrak{X} \rightarrow X$ the completion morphism. Then for any $\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{+}_{\mrm{qc}}(\msf{Mod}(X))$ there is a natural isomorphism $g^{*} \mrm{R} \underline{\Gamma}_{X_{0}} \mcal{M}^{{\textstyle \cdot}} \cong \mrm{R} \underline{\Gamma}_{\mrm{disc}} g^{*} \mcal{M}^{{\textstyle \cdot}}$. In particular for a single quasi-coherent sheaf $\mcal{M}$ one has $g^{*} \underline{\Gamma}_{X_{0}} \mcal{M} \cong \underline{\Gamma}_{\mrm{disc}} g^{*} \mcal{M}$. \end{prop} \begin{proof} Let $\mcal{M}^{{\textstyle \cdot}} \rightarrow \mcal{J}^{{\textstyle \cdot}}$ be a resolution by quasi-coherent injectives. Since $g$ is flat we get \[ \phi : g^{*} \mrm{R} \underline{\Gamma}_{X_{0}} \mcal{M}^{{\textstyle \cdot}} = g^{*} \underline{\Gamma}_{X_{0}} \mcal{J}^{{\textstyle \cdot}} \rightarrow \underline{\Gamma}_{\mrm{disc}} g^{*} \mcal{J}^{{\textstyle \cdot}} \rightarrow \mrm{R} \underline{\Gamma}_{\mrm{disc}} g^{*} \mcal{J}^{{\textstyle \cdot}} = \mrm{R} \underline{\Gamma}_{\mrm{disc}} g^{*} \mcal{M}^{{\textstyle \cdot}} . \] Locally on any affine open $U \subset X$, with $U_{0} = U \cap X_{0}$ and $\mfrak{U} = U_{/U_{0}}$, we can find $\underline{a}$ in $\Gamma(U, \mcal{O}_{U})$ which define $U_{0}$. It's known that $\underline{\Gamma}_{U_{0}} (\mcal{J}^{{\textstyle \cdot}}|_{U}) \rightarrow \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes (\mcal{J}^{{\textstyle \cdot}}|_{U})$ is a quasi-isomorphism. Since $g$ is flat we obtain quasi-isomorphisms \begin{multline*} \hspace{10mm} \phi|_{\mfrak{U}} : g^{*} \underline{\Gamma}_{U_{0}} (\mcal{J}^{{\textstyle \cdot}}|_{U}) \rightarrow g^{*} \left( \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes (\mcal{J}^{{\textstyle \cdot}}|_{U}) \right) \\ \cong \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes g^{*} (\mcal{J}^{{\textstyle \cdot}}|_{U}) = \mrm{R} \underline{\Gamma}_{\mrm{disc}} g^{*} (\mcal{J}^{{\textstyle \cdot}}|_{U}) . \hspace{10mm} \end{multline*} It follows that $\phi$ is an isomorphism. \end{proof} Denote by $\msf{D}^{+}_{\mrm{d}}(\msf{Mod}(\mfrak{X}))$ the subcategory of complexes with discrete cohomologies. \begin{lem} \label{lem4.2} \begin{enumerate} \item If $\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{+}_{\mrm{d}}(\msf{Mod}(\mfrak{X}))$ then $\mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M} \rightarrow \mcal{M}$ is an isomorphism. \item If $\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{+}_{\mrm{qc}}(\msf{Mod}(\mfrak{X}))$ then $\mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M} \in \msf{D}^{+}_{\mrm{qc}}(\msf{Mod}_{\mrm{disc}}(\mfrak{X}))$. \end{enumerate} \end{lem} \begin{proof} From Lemma \ref{lem4.3} we see that the functor $\mrm{R} \underline{\Gamma}_{\mrm{disc}}$ has finite cohomological dimension. By way-out reasons (cf.\ \cite{RD} Section I.7) we may assume $\mcal{M}^{{\textstyle \cdot}}$ is a single discrete (resp.\ quasi-coherent) sheaf. Then the claims are obvious (use Proposition \ref{prop3.4} for 2). \end{proof} \begin{thm} \label{thm4.1} The identity functor $\msf{D}^{+}(\msf{QCo}_{\mrm{disc}}(\mfrak{X})) \rightarrow \msf{D}^{+}_{\mrm{dqc}}(\msf{Mod}(\mfrak{X}))$ is an equivalence of categories. In particular any $\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{+}_{\mrm{dqc}}(\msf{Mod}(\mfrak{X}))$ is isomorphic to a complex of injectives $\mcal{J}^{{\textstyle \cdot}} \in \msf{D}^{+}(\msf{QCo}_{\mrm{disc}}(\mfrak{X}))$. \end{thm} \begin{proof} According to Lemma \ref{lem4.2} we see that $\msf{D}^{+}_{\mrm{qc}}(\msf{Mod}_{\mrm{disc}}(\mfrak{X})) \rightarrow \msf{D}^{+}_{\mrm{dqc}}(\msf{Mod}(\mfrak{X}))$ is an equivalence with quasi-inverse $\mrm{R} \underline{\Gamma}_{\mrm{disc}}$. Next, by Proposition \ref{prop4.1} and by \cite{RD} Proposition I.4.8, the functor $\msf{D}^{+}(\msf{QCo}_{\mrm{disc}}(\mfrak{X})) \rightarrow \msf{D}^{+}_{\mrm{qc}}(\msf{Mod}_{\mrm{disc}}(\mfrak{X}))$ is an equivalence. \end{proof} \begin{rem} \label{rem6.10} In \cite{AJL2} it is proved that $\msf{D}(\msf{QCo}_{\mrm{disc}}(\mfrak{X})) \rightarrow \msf{D}_{\mrm{dqc}}(\msf{Mod}(\mfrak{X}))$ is an equivalence, using the quasi-coherator functor. \end{rem} Suppose there is a codimension function $d : \mfrak{X} \rightarrow \mbb{Z}$, i.e.\ a function satisfying $d(y) = d(x) + 1$ whenever $(x, y)$ is an immediate specialization pair. Then there is a filtration $\cdots \supset Z^{p} \supset Z^{p+1} \supset \cdots$ of $\mfrak{X}$, with $Z^{p} := \{ F \subset \mfrak{X} \mid F \text{ closed}, d(F) \geq p \}$. Here $d(F) := \operatorname{min} \{ d(x) \mid x \in F \}$. This filtration determines a Cousin functor \begin{equation} \mrm{E} : \msf{D}^{+}(\msf{Ab}(\mfrak{X})) \rightarrow \msf{C}^{+}(\msf{Ab}(\mfrak{X})) \end{equation} where $\msf{C}^{+}$ denotes the abelian category of bounded below complexes (cf.\ \cite{RD} \S IV.1). Given a point $x \in \mfrak{X}$ and a sheaf $\mcal{M} \in \msf{Ab}(\mfrak{X})$ we let $\Gamma_{x} \mcal{M} := (\underline{\Gamma}_{\, \overline{\{x\}}\, } \mcal{M})_{x}$ $\subset \mcal{M}_{x}$. The derived functor $\mrm{R} \Gamma_{x} : \msf{D}^{+}(\msf{Ab}(\mfrak{X})) \rightarrow \msf{D}(\msf{Ab})$ is calculated by flasque sheaves. Let us write $\mrm{H}_{x}^{q} \mcal{M} := \mrm{H}^{q} \mrm{R} \Gamma_{x} \mcal{M}$, the local cohomology, and let $i_{x} : \{x\} \rightarrow \mfrak{X}$ be the inclusion According to \cite{RD} \S IV.1 Motif F one has a natural isomorphism \begin{equation} \label{eqn4.3} \mrm{E}^{p} \mcal{M}^{{\textstyle \cdot}} = \mcal{H}_{Z^{p} / Z^{p+1}}^{p} \mcal{M}^{{\textstyle \cdot}} \cong \bigoplus_{d(x) = p} i_{x *} \mrm{H}_{x}^{p} \mcal{M}^{{\textstyle \cdot}} . \end{equation} Observe that if $\mcal{M} \in \msf{D}^{+}(\msf{Mod}(\mfrak{X}))$ then $\mrm{E} \mcal{M}^{{\textstyle \cdot}} \in \msf{C}^{+}(\msf{Mod}(\mfrak{X}))$ and $\mrm{R} \Gamma_{x} \mcal{M} \in$ \newline $\msf{D}^{+}(\msf{Mod}(\mcal{O}_{\mfrak{X}, x}))$. Unlike an ordinary scheme, on a formal scheme the topological support of a quasi-coherent sheaf does not coincide with its algebraic support. But for discrete sheaves these two notions of support do coincide. This suggests: \begin{dfn} Given $\mcal{M} \in \msf{D}^{+}(\msf{Mod}(\mfrak{X}))$ its {\em discrete Cousin complex} is \newline $\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}}$. \end{dfn} \begin{thm} \label{thm4.2} For any $\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{+}_{\mrm{qc}}(\msf{Mod}(\mfrak{X}))$ the complex $\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}}$ consists of discrete quasi-coherent sheaves. So we get a functor \[ \mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} : \msf{D}^{+}_{\mrm{qc}}(\msf{Mod}(\mfrak{X})) \rightarrow \msf{C}^{+}(\msf{QCo}_{\mrm{disc}}(\mfrak{X})). \] \end{thm} \begin{proof} According to Theorem \ref{thm4.1} we may assume $\mcal{N}^{{\textstyle \cdot}} = \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}}$ is in \newline $\msf{D}^{+}(\msf{QCo}_{\mrm{disc}}(\mfrak{X}))$. On any open formal subscheme $\mfrak{U} = \operatorname{Spf} A$ we get $\mcal{N}^{{\textstyle \cdot}} = \mcal{O}_{\mfrak{U}} \otimes_{A} N^{{\textstyle \cdot}}$, where $N^{q} = \Gamma(\mfrak{U}, N^{q})$ (cf.\ Propositions \ref{prop3.4} and \ref{prop3.2}) Then for $x \in \mfrak{U}$, \[ \mrm{R} \Gamma_{x} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}} = \mrm{R} \Gamma_{x} \mcal{N}^{{\textstyle \cdot}} = \mrm{R} \Gamma_{\mfrak{p}} N^{{\textstyle \cdot}}_{\mfrak{p}} \] where $\mfrak{p} \subset A$ is the prime ideal of $x$. Hence $\mrm{H}^{q}_{x} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}} = \mrm{H}^{q}_{\mfrak{p}} N^{{\textstyle \cdot}}_{\mfrak{p}}$ is $\mfrak{p}$-torsion. So the sheaf corresponding to $x$ in (\ref{eqn4.3}) is quasi-coherent and discrete. \end{proof} \section{Dualizing Complexes on Formal Schemes} In this section we propose a theory of duality on noetherian formal sche\-mes. There is a fundamental difference between this theory and the duality theory on schemes, as developed in \cite{RD}. A dualizing complex $\mcal{R}^{{\textstyle \cdot}}$ on a scheme $X$ has coherent cohomology sheaves; this will not be true on a general formal scheme $\mfrak{X}$, where $\mrm{H}^{q} \mcal{R}^{{\textstyle \cdot}}$ are discrete quasi-coherent sheaves (Def.\ \ref{dfn5.1}). We prove uniqueness of dualizing complexes (Thm.\ \ref{thm5.1}), and existence in some cases (Prop.\ \ref{prop5.8} and Thm.\ \ref{thm5.3}). Before we begin here is an instructive example due to J.\ Lipman. \begin{exa} \label{exa5.2} Consider the ring $A = k[\sqbr{t}]$ of formal power series over a field $k$. Let $\mfrak{X} := \operatorname{Spf} A$, which has a single point. The modules $A$ and $J = \mrm{H}^{1}_{(t)} A$ both have finite injective dimension and satisfy $\operatorname{Hom}_{A}(A, A) = \operatorname{Hom}_{A}(J, J) = A$. Which one is a dualizing complex on $\mfrak{X}$? We will see that $J$ is the correct answer (Def.\ \ref{dfn5.1}), and $A$ is a ``fake'' dualizing complex (Thm.\ \ref{thm5.3}). The relevant relation between them is: $J = \mrm{R} \Gamma_{\mrm{disc}} A [1]$. \end{exa} Suppose $\mcal{N}^{{\textstyle \cdot}} \in \msf{D}^{+}(\msf{Mod}_{\mrm{disc}}(\mfrak{X}))$. We say $\mcal{N}^{{\textstyle \cdot}}$ has finite injective dimension on $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$ if there is an integer $q_{0}$ s.t.\ for all $q > q_{0}$ and $\mcal{M} \in \msf{QCo}_{\mrm{disc}}(\mfrak{X})$, $\mrm{H}^{q} \mrm{R} \operatorname{Hom}_{\mfrak{X}} (\mcal{M}, \mcal{N}^{{\textstyle \cdot}}) = 0$. \begin{dfn} \label{dfn5.1} A {\em dualizing complex} on $\mfrak{X}$ is a complex $\mcal{R}^{{\textstyle \cdot}} \in \msf{D}^{\mrm{b}}_{\mrm{dqc}}(\msf{Mod}(\mfrak{X}))$ satisfying: \begin{enumerate} \rmitem{i} $\mcal{R}^{{\textstyle \cdot}}$ has finite injective dimension on $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$. \rmitem{ii} The adjunction morphism $\mcal{O}_{\mfrak{X}} \rightarrow \mrm{R} \mcal{H}om_{\mfrak{X}} (\mcal{R}^{{\textstyle \cdot}}, \mcal{R}^{{\textstyle \cdot}})$ is an isomorphism. \rmitem{iii} For some defining ideal $\mcal{I}$ of $\mfrak{X}$, $\mrm{R} \mcal{H}om_{\mfrak{X}}(\mcal{O}_{\mfrak{X}} / \mcal{I}, \mcal{R}^{{\textstyle \cdot}})$ has coherent cohomology sheaves. \end{enumerate} \end{dfn} \begin{lem} \label{lem5.2} Let $\mcal{N}^{{\textstyle \cdot}} \in \msf{D}^{+}_{\mrm{dqc}}(\msf{Mod}(\mfrak{X}))$. Then $\mcal{N}^{{\textstyle \cdot}}$ has finite injective dimension on $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$ iff it is isomorphic to a bounded complex of injectives in $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$. \end{lem} \begin{proof} Because of Theorem \ref{thm4.1}, the proof is just like \cite{RD} Prop.\ I.7.6. \end{proof} In light of this, we can, when convenient, assume the dualizing complex $\mcal{R}^{{\textstyle \cdot}}$ is a bounded complex of discrete quasi-coherent injectives. \begin{prop} \label{prop5.2} Let $\mcal{R}^{{\textstyle \cdot}}$ be a dualizing complex on $\mfrak{X}$. Then for any $\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{\mrm{b}}_{\mrm{c}}(\msf{Mod}(\mfrak{X}))$ the morphism of adjunction \[ \mcal{M}^{{\textstyle \cdot}} \rightarrow \mrm{R} \mcal{H}om_{\mfrak{X}} ( \mrm{R} \mcal{H}om_{\mfrak{X}} (\mcal{M}^{{\textstyle \cdot}}, \mcal{R}^{{\textstyle \cdot}}), \mcal{R}^{{\textstyle \cdot}}) \] is an isomorphism. \end{prop} \begin{proof} We can assume $\mfrak{X}$ is affine, and so replace $\mcal{M}^{{\textstyle \cdot}}$ with a complex of coherent sheaves. By ``way-out'' arguments (cf.\ \cite{RD} Section I.7) we reduce to the case $\mcal{M}^{{\textstyle \cdot}} = \mcal{O}_{\mfrak{X}}$, which property (ii) applies. \end{proof} \begin{lem} \label{lem5.4} Suppose $\mcal{R}^{{\textstyle \cdot}}$ is a dualizing complex on $\mfrak{X}$. Let $\mcal{I}$ be any defining ideal of $\mfrak{X},$ and let $X_{0}$ be the scheme $(\mfrak{X}, \mcal{O}_{\mfrak{X}} / \mcal{I})$. Then $\mrm{R} \mcal{H}om_{\mfrak{X}} (\mcal{O}_{X_{0}}, \mcal{R}^{{\textstyle \cdot}})$ is a dualizing complex on $X_{0}$. \end{lem} \begin{proof} We can assume $\mcal{R}^{{\textstyle \cdot}}$ is a bounded complex of injectives in $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$, so $\mcal{R}^{{\textstyle \cdot}}_{0} := \mcal{H}om_{\mfrak{X}} (\mcal{O}_{X_{0}}, \mcal{R}^{{\textstyle \cdot}})$ is a complex of injectives on $X_{0}$. Property (iii) implies that $\mcal{R}^{{\textstyle \cdot}}_{0}$ has coherent cohomology sheaves. Now \[ \mcal{H}om_{X_{0}}(\mcal{R}^{{\textstyle \cdot}}_{0}, \mcal{R}^{{\textstyle \cdot}}_{0}) \cong \mcal{H}om_{\mfrak{X}} ( \mcal{H}om_{\mfrak{X}} (\mcal{O}_{X_{0}}, \mcal{R}^{{\textstyle \cdot}}), \mcal{R}^{{\textstyle \cdot}}) \cong \mcal{O}_{X_{0}} , \] so $\mcal{R}^{{\textstyle \cdot}}_{0}$ is dualizing. \end{proof} \begin{thm} \label{thm5.1} \textup{(Uniqueness)}\ Suppose $\mcal{R}^{{\textstyle \cdot}}$ and $\tilde{\mcal{R}}^{{\textstyle \cdot}}$ are dualizing complexes and $\mfrak{X}$ is connected. Then $\tilde{\mcal{R}}^{{\textstyle \cdot}} \cong \mcal{R}^{{\textstyle \cdot}} \otimes \mcal{L}[n]$ in $\msf{D}(\msf{Mod}(\mfrak{X}))$, for some invertible sheaf $\mcal{L}$ and integer $n$. \end{thm} \begin{proof} We can assume both $\mcal{R}^{{\textstyle \cdot}}$ and $\tilde{\mcal{R}}^{{\textstyle \cdot}}$ are bounded complexes of injectives in $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$. Choose a defining ideal $\mcal{I}$ and let $X_{m}$ be the scheme $(\mfrak{X}, \mcal{O}_{\mfrak{X}} / \mcal{I}^{m+1})$. Define a complex $\mcal{R}^{{\textstyle \cdot}}_{m} := \mcal{H}om_{\mfrak{X}} (\mcal{O}_{X_{m}}, \mcal{R}^{{\textstyle \cdot}})$ and likewise $\tilde{\mcal{R}}^{{\textstyle \cdot}}_{m}$. These are dualizing complexes on $X_{m}$, so by \cite{RD} Thm.\ IV.3.1 there is an isomorphism \[ \phi_{m} : \mcal{R}^{{\textstyle \cdot}}_{m} \otimes \mcal{L}_{m}[n_{m}] \rightarrow \tilde{\mcal{R}}^{{\textstyle \cdot}}_{m} \] in $\msf{D}(\msf{Mod}(X_{m}))$, for some invertible sheaf $\mcal{L}_{m}$ and integer $n_{m}$. Writing $\mcal{M}_{m}^{{\textstyle \cdot}} := \mcal{H}om_{X_{m}}( \mcal{R}^{{\textstyle \cdot}}_{m}, \tilde{\mcal{R}}^{{\textstyle \cdot}}_{m})$ we have $\mcal{M}_{m}^{{\textstyle \cdot}} \cong \mcal{L}_{m}[n_{m}]$ in $\msf{D}(\msf{Mod}(X_{m}))$. Now \[ \mcal{M}_{m}^{{\textstyle \cdot}} \cong \mcal{H}om_{X_{m+1}}( \mcal{H}om_{X_{m+1}}(\mcal{O}_{X_{m}}, \mcal{R}^{{\textstyle \cdot}}_{m+1}), \mcal{R}^{{\textstyle \cdot}}_{m+1})) \otimes \mcal{L}_{m+1}[n_{m+1}] \] as complexes of $\mcal{O}_{X_{m+1}}$-modules, so by the dualizing property of $\mcal{R}^{{\textstyle \cdot}}_{m+1}$ we deduce an isomorphism $\mcal{M}_{m}^{{\textstyle \cdot}} \cong \mcal{O}_{X_{m}} \otimes \mcal{L}_{m+1}[n_{m+1}]$ in $\msf{D}(\msf{Mod}(X_{m+1}))$. We conclude that $n_{m} = n_{m+1}$ and $\mcal{L}_{m} \cong \mcal{O}_{X_{m}} \otimes \mcal{L}_{m+1}$. Set $n := n_{m}$ and $\mcal{L} := \lim_{\leftarrow m} \mcal{L}_{m}$. Next, since $\mcal{R}^{q}_{m} \subset \mcal{R}^{q}_{m+1}$ and $\tilde{\mcal{R}}^{q}_{m+1}$ is injective in $\msf{Mod}(X_{m+1})$, we see that $\mcal{M}_{m+1}^{q} \rightarrow \mcal{M}_{m}^{q}$ is surjective for all $q,m$. Furthermore, $\mrm{H}^{q} \mcal{M}^{{\textstyle \cdot}}_{m+1} \rightarrow \mrm{H}^{q} \mcal{M}^{{\textstyle \cdot}}_{m}$ is also surjective, since $\mrm{H}^{q} \mcal{M}^{{\textstyle \cdot}}_{m} = \mcal{L}_{m}$ or $0$. Define \[ \mcal{M}^{{\textstyle \cdot}} := \mcal{H}om_{\mfrak{X}}( \mcal{R}^{{\textstyle \cdot}}, \tilde{\mcal{R}}^{{\textstyle \cdot}}) \cong \lim_{\leftarrow m} \mcal{M}^{{\textstyle \cdot}}_{m} . \] According to \cite{Ha} Cor.\ I.4.3 and Prop.\ I.4.4 it follows that $\mrm{H}^{q} \mcal{M}^{{\textstyle \cdot}} = \lim_{\leftarrow m} \mrm{H}^{q} \mcal{M}^{{\textstyle \cdot}}_{m}$. This implies that $\mcal{H}om_{\mfrak{X}}( \mcal{R}^{{\textstyle \cdot}} \otimes \mcal{L}[n], \tilde{\mcal{R}}^{{\textstyle \cdot}})) \cong \mcal{O}_{\mfrak{X}}$ in $\msf{D}(\msf{Mod}(\mfrak{X}))$, so by Corollary \ref{cor4.1} \[ \mrm{H}^{0} \operatorname{Hom}_{\mfrak{X}} (\mcal{R}^{{\textstyle \cdot}}\otimes \mcal{L}[n], \tilde{\mcal{R}}^{{\textstyle \cdot}}) \cong \Gamma(\mfrak{X}, \mcal{O}_{\mfrak{X}}) . \] Choose a homomorphism of complexes $\phi : \mcal{R}^{{\textstyle \cdot}} \otimes \mcal{L}[n] \rightarrow \tilde{\mcal{R}}^{{\textstyle \cdot}}$ corresponding to $1 \in \Gamma(\mfrak{X}, \mcal{O}_{\mfrak{X}})$. Backtracking we see that for every $m$, $\phi$ induces a homomorphism $\mcal{R}^{{\textstyle \cdot}}_{m} \otimes \mcal{L}[n] \rightarrow \tilde{\mcal{R}}^{{\textstyle \cdot}}_{m}$ which represents $\phi_{m}$ in $\msf{D}(\msf{Mod}(X_{m}))$. So $\phi = \lim_{m \rightarrow} \phi_{m}$ is a quasi-isomorphism. \end{proof} \begin{prob} Let $\mcal{R}^{{\textstyle \cdot}}$ be a dualizing complex. Is it true that the following conditions on $\mcal{N}^{{\textstyle \cdot}} \in \msf{D}^{\mrm{b}}_{\mrm{dqc}}(\msf{Mod}(\mfrak{X}))$ are equivalent? \begin{enumerate} \rmitem{i} $\mcal{N}^{{\textstyle \cdot}} \cong \mrm{R} \mcal{H}om_{\mfrak{X}}^{{\textstyle \cdot}}( \mcal{M}^{{\textstyle \cdot}}, \mcal{R}^{{\textstyle \cdot}})$ for some $\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{\mrm{b}}_{\mrm{c}}(\msf{Mod}(\mfrak{X}))$. \rmitem{ii} For any $\mcal{M}$ discrete coherent, $\mrm{R} \mcal{H}om_{\mfrak{X}}^{{\textstyle \cdot}}( \mcal{M}, \mcal{N}^{{\textstyle \cdot}}) \in \msf{D}^{\mrm{b}}_{\mrm{c}}(\msf{Mod}(\mfrak{X}))$. \end{enumerate} \end{prob} Recall that for a point $x \in \mfrak{X}$ we denote by $J(x)$ an injective hull of $k(x)$ over $\mcal{O}_{\mfrak{X}, x}$, and $\mcal{J}(x)$ is the corresponding quasi-coherent sheaf. \begin{lem} \label{lem5.6} Suppose $\mcal{R}^{{\textstyle \cdot}}$ is a dualizing complex on $\mfrak{X}$. For any $x \in \mfrak{X}$ there is a unique integer $d(x)$ s.t.\ \[ \mrm{H}^{q}_{x} \mcal{R}^{{\textstyle \cdot}} \cong \begin{cases} J(x) & \text{ if } q = d(x)\\ 0 & \text{ otherwise}. \end{cases} \] Furthermore $d$ is a codimension function. \end{lem} \begin{proof} We can assume $\mcal{R}^{{\textstyle \cdot}}$ is a bounded complex of injectives in $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$. Then as seen before $\mrm{H}^{q}_{x} \mcal{R}^{{\textstyle \cdot}} = \mrm{H}^{q} \Gamma_{x} \mcal{R}^{{\textstyle \cdot}}$. Define schemes $X_{m}$ and complexes $\mcal{R}^{{\textstyle \cdot}}_{m}$ like in the proof of Thm.\ \ref{thm5.1}. Since $\mcal{R}^{{\textstyle \cdot}}_{m}$ is dualizing it determines a codimension function $d_{m}$ on $X_{m}$ (cf.\ \cite{RD} Ch.\ V \S 7). But the arguments used before show that $d_{m} = d_{m+1}$. Finally $\mrm{H}^{q} \Gamma_{x} \mcal{R}^{{\textstyle \cdot}} = \lim_{m \rightarrow} \mrm{H}^{q} \Gamma_{x} \mcal{R}^{{\textstyle \cdot}}_{m}$, and $\mrm{H}^{q} \Gamma_{x} \mcal{R}^{{\textstyle \cdot}}_{m} \cong J_{m}(x)$, an injective hull of $k(x)$ over $\mcal{O}_{X_{m}, x}$. \end{proof} \begin{dfn} \label{dfn5.3} A residual complex on the noetherian formal scheme $\mfrak{X}$ is a dualizing complex $\mcal{K}^{{\textstyle \cdot}}$ which is isomorphic, as $\mcal{O}_{\mfrak{X}}$-module, to $\bigoplus_{x \in \mfrak{X}} \mcal{J}(x)$. \end{dfn} \begin{prop} \label{prop5.7} Say $\mcal{R}^{{\textstyle \cdot}}$ is a dualizing complex on $\mfrak{X}$. Let $d$ be the codimension function above, and let $\mrm{E}$ be the associated Cousin functor. Then $\mcal{R}^{{\textstyle \cdot}} \cong \mrm{E} \mcal{R}^{{\textstyle \cdot}}$ in $\msf{D}(\msf{Mod}(\mfrak{X}))$, and $\mrm{E} \mcal{R}^{{\textstyle \cdot}}$ is a residual complex. \end{prop} \begin{proof} By Lemma \ref{lem5.6} $\mcal{R}^{{\textstyle \cdot}}$ is a Cohen-Macaulay complex, in the sense of \cite{RD} p.\ 247, Definition. So there exists some isomorphism $\mcal{R}^{{\textstyle \cdot}} \rightarrow \mrm{E} \mcal{R}^{{\textstyle \cdot}}$ in $\msf{D}^{\mrm{b}}(\msf{Mod}(\mfrak{X}))$. \end{proof} To conclude this section we consider some situations where a dualizing complex exists. If $f : \mfrak{X} \rightarrow \mfrak{Y}$ is a morphism then $(\mfrak{Y}, f_{*} \mcal{O}_{\mfrak{X}})$ is a ringed space, and $\bar{f} : \mfrak{X} \rightarrow (\mfrak{Y}, f_{*} \mcal{O}_{\mfrak{X}})$ is a morphism of ringed spaces. \begin{prop} \label{prop5.8} Let $f : \mfrak{X} \rightarrow \mfrak{Y}$ be a formally finite morphism, and assume $\mcal{K}^{{\textstyle \cdot}}$ is a residual complex on $\mfrak{Y}$. Then $\bar{f}^{*} \mcal{H}om_{\mfrak{Y}}(f_{*} \mcal{O}_{\mfrak{X}}, \mcal{K}^{{\textstyle \cdot}})$ is a residual complex on $\mfrak{X}$. \end{prop} \begin{proof} Let $f_{n} : X_{n} \rightarrow Y_{n}$ be morphisms as in Lemma \ref{lem1.12}, and let $\mcal{K}_{n}^{{\textstyle \cdot}}$ \linebreak $:= \mcal{H}om_{\mfrak{Y}}(\mcal{O}_{Y_{n}}, \mcal{K}^{{\textstyle \cdot}})$. Since $f_{n}$ is a finite morphism, $\bar{f}_{n}^{*} \mcal{H}om_{Y_{n}}(f_{n *} \mcal{O}_{X_{n}}, \mcal{K}_{n}^{{\textstyle \cdot}})$ is a residual complex on $X_{n}$. As in the proof of Thm.\ \ref{thm5.1}, \[ \bar{f}^{*} \mcal{H}om_{\mfrak{Y}}(f_{*} \mcal{O}_{\mfrak{X}}, \mcal{K}^{{\textstyle \cdot}}) \cong \lim_{n \rightarrow} \bar{f}_{n}^{*} \mcal{H}om_{Y_{n}}(f_{n *} \mcal{O}_{X_{n}}, \mcal{K}_{n}^{{\textstyle \cdot}}) \] is residual. \end{proof} \begin{exa} \label{exa5.1} Suppose $X_{0} \subset X$ is closed, $\mfrak{X} = X_{/ X_{0}}$ and $g : \mfrak{X} \rightarrow X$ is the completion morphism. Let $\mcal{K}^{{\textstyle \cdot}}$ be a residual complex on $X$. In this case $g = \bar{g}$, and by Proposition \ref{prop4.2} \[ g^{*} \mcal{H}om_{X}(g_{*} \mcal{O}_{\mfrak{X}}, \mcal{K}^{{\textstyle \cdot}}) \cong \lim_{n \rightarrow} g^{*} \mcal{K}_{n}^{{\textstyle \cdot}} \cong g^{*} \underline{\Gamma}_{X_{0}} \mcal{K}^{{\textstyle \cdot}} \cong \underline{\Gamma}_{\mrm{disc}} g^{*} \mcal{K}^{{\textstyle \cdot}} \] is a residual complex. We see that if $\mcal{R}^{{\textstyle \cdot}}$ is any dualizing complex on $X$ then $\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} g^{*} \mcal{R}^{{\textstyle \cdot}}$ is dualizing on $\mfrak{X}$. \end{exa} We call a formal scheme $\mfrak{X}$ {\em regular} of all its local rings $\mcal{O}_{\mfrak{X}, x}$ are regular. \begin{lem} \label{lem5.7} Suppose $\mfrak{X}$ is a regular formal scheme. Then $d(x) := \operatorname{dim} \mcal{O}_{\mfrak{X},x}$ is a bounded codimension function on $\mfrak{X}$. \end{lem} \begin{proof} Let $\mfrak{U} = \operatorname{Spf} A \subset \mfrak{X}$ be a connected affine open set. If $x \in \mfrak{U}$ is the point corresponding to an open prime ideal $\mfrak{p}$, then $\widehat{A}_{\mfrak{p}} \cong \widehat{\mcal{O}}_{\mfrak{X}, x}$. Therefore $A_{\mfrak{p}}$ is a regular local ring. Now in the adic noetherian ring $A$ any maximal ideal $\mfrak{m}$ is open. Hence, by \cite{Ma} \S 18 Lemma 5 (III), $A$ is a regular ring, of finite global dimension equal to its Krull dimension. Now let $U := \operatorname{Spec} A$, so as a topological space, $\mfrak{U} \subset U$ is the closed set defined by any defining ideal $I \subset A$. Since $U$ is a regular scheme, $\mcal{O}_{U}$ is a dualizing complex on it. The codimension function $d'$ corresponding to $\mcal{O}_{U}$ satisfies $d'(y) = \operatorname{dim} \mcal{O}_{U, y}$. Thus $0 \leq d'(y) \leq \operatorname{dim} U$. But clearly $d|_{\mfrak{U}} = d'|_{\mfrak{U}}$. By covering $\mfrak{X}$ with finitely many such $\mfrak{U}$ this implies that $d$ is a bounded codimension function. \end{proof} \begin{thm} \label{thm5.3} Suppose $\mfrak{X}$ is a regular formal scheme. Then $\mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{X}}$ is a dualizing complex on $\mfrak{X}$. \end{thm} \begin{proof} By the proof of Theorem \ref{thm4.2} and known properties of regular local rings, for any $x \in \mfrak{X}$ \[ \mrm{H}^{q}_{x} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{X}} \cong \mrm{H}^{q}_{\mfrak{m}_{x}} \widehat{\mcal{O}}_{\mfrak{X}, x} \cong \begin{cases} J(x) & \text{ if } q = d(x)\\ 0 & \text{ otherwise} \end{cases} \] where $\mfrak{m}_{x} \subset \widehat{\mcal{O}}_{\mfrak{X}, x}$ is the maximal ideal, and $J(x)$ is an injective hull of $k(x)$. Since $d$ is bounded it follows that $\mcal{K}^{{\textstyle \cdot}} := \mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{X}}$ is a bounded complex of injectives in $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$. Like in the proof of Proposition \ref{prop5.7}, $\mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{X}} \cong \mcal{K}^{{\textstyle \cdot}}$ in $\msf{D}(\msf{Mod}(\mfrak{X}))$. To complete the proof it suffices to show that for any affine open set $\mfrak{U} = \operatorname{Spf} A \subset \mfrak{X}$ the complex $\mcal{K}^{{\textstyle \cdot}}|_{\mfrak{U}}$ is residual on $\mfrak{U}$. Let $U := \operatorname{Spec} A$ and let $g : \mfrak{U} \rightarrow U$ be the canonical morphism Let $U_{0} \subset U$ be the closed set $g(\mfrak{U})$, so that $\mfrak{U} \cong U_{/ U_{0}}$. Define $\mcal{K}^{{\textstyle \cdot}}_{U} := \mrm{E} \mcal{O}_{U}$, which is a residual complex on $U$. Then according to Proposition \ref{prop4.2} \[ \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{U}} \cong g^{*} \mrm{R} \underline{\Gamma}_{U_{0}} \mcal{O}_{U} \cong g^{*} \underline{\Gamma}_{U_{0}} \mcal{K}^{{\textstyle \cdot}}_{U} . \] As in Example \ref{exa5.1} this is a dualizing complex, so $\mcal{K}^{{\textstyle \cdot}}|_{\mfrak{U}} \cong \mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{U}}$ is a residual complex. \end{proof} \begin{rem} According to \cite{RD} Thm.\ VI.3.1, if $f : X \rightarrow Y$ is a finite type morphism between finite dimensional noetherian schemes, and if $\mcal{K}^{{\textstyle \cdot}}$ is a residual complex on $Y$, then there is a residual complex $f^{\triangle} \mcal{K}^{{\textstyle \cdot}}$ on $X$. Now suppose $f : \mfrak{X} \rightarrow \mfrak{Y}$ is a f.f.t.\ morphism and $f_{n} : X_{n} \rightarrow Y_{n}$ are like in Lemma \ref{lem1.12}. In the same fashion as in Prop.\ \ref{prop5.8} we set $f^{\triangle} \mcal{K}^{{\textstyle \cdot}} := \lim_{n \rightarrow} f_{n}^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{n}$. This is a residual complex on $\mfrak{X}$. If $f$ is formally proper then $\operatorname{Tr}_{f} = \lim_{n \rightarrow} \operatorname{Tr}_{f_{n}}$ induces a duality \[ \mrm{R} f_{*} \mcal{M}^{{\textstyle \cdot}} \rightarrow \mrm{R} \mcal{H}om_{\mfrak{Y}}(\mrm{R} f_{*} \mrm{R} \mcal{H}om_{\mfrak{X}}(\mcal{M}^{{\textstyle \cdot}}, f^{\triangle} \mcal{K}^{{\textstyle \cdot}}), \mcal{K}^{{\textstyle \cdot}}) \] for every $\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{\mrm{b}}(\msf{Coh}(\mfrak{X}))$. The proofs are standard, given the results of this section. \end{rem} \section{Construction of the Complex $\mcal{K}^{{\textstyle \cdot}}_{X/S}$} In this section we work over a regular noetherian base scheme $S$. We construct the relative residue complex $\mcal{K}^{{\textstyle \cdot}}_{X/S}$ on any finite type $S$-scheme $X$. The construction is explicit and does not rely on \cite{RD}. Let $A, B$ be complete local rings, with maximal ideals $\mfrak{m}, \mfrak{n}$. Recall that a local homomorphism $\phi : A \rightarrow B$ is called residually finitely generated if the field extension $A / \mfrak{m} \rightarrow B / \mfrak{n}$ is finitely generated. Denote by $\msf{Mod}_{\mrm{disc}}(A)$ the category of $\mfrak{m}$-torsion $A$-modules (equivalently, modules with $0$-dimensional support). Suppose $A \sqbr{\underline{t}} = A \sqbr{t_{1}, \ldots, t_{n}}$ is a polynomial algebra and $\mfrak{p} \subset A \sqbr{\underline{t}}$ is some maximal ideal. Then $A \rightarrow B = \widehat{A \sqbr{\underline{t}}}_{\mfrak{p}}$ is formally smooth of relative dimension $n$ and residually finite. Let $b_{i} \in B / \mfrak{n}$ be the image of $t_{i}$ and $\bar{q}_{i} \in (A / \mfrak{m}) \sqbr{b_{1}, \ldots, b_{i-1}}\sqbr{t_{i}}$ the monic irreducible polynomial of $b_{i}$, of degree $d_{i}$. Choose a monic lifting $q_{i} \in A \sqbr{t_{1}, \ldots, t_{i}}$. Then for a discrete $A$-module $M$ one has \[ \mrm{H}^{n}_{\mfrak{p}} \left( \widehat{\Omega}^{n}_{B / A} \otimes_{A} M \right) \cong \bigoplus_{1 \leq i_{l}} \ \bigoplus_{0 \leq j_{l} < d_{l}} \gfrac{ t_{1}^{j_{1}} \cdots t_{n}^{j_{n}} \mrm{d} t_{1} \cdots \mrm{d} t_{n} } { q_{1}^{i_{1}} \cdots q_{n}^{i_{n}} } \otimes M . \] As in \cite{Hg1} Section 7 define the Tate residue \begin{equation} \label{eqn6.6} \operatorname{res}_{t_{1}, \ldots, t_{n}; A, B} : \mrm{H}^{n}_{\mfrak{p}} \left( \widehat{\Omega}^{n}_{B / A} \otimes_{A} M \right) \rightarrow M \end{equation} by the rule \[ \gfrac{ t_{1}^{j_{1}} \cdots t_{n}^{j_{n}} \mrm{d} t_{1} \cdots \mrm{d} t_{n} } { q_{1}^{i_{1}} \cdots q_{n}^{i_{n}} } \otimes m \mapsto \begin{cases} m & \text{ if } i_{l} = 1, j_{l} = d_{l} - 1 \\ 0 & \text{ otherwise} \end{cases} \] (cf.\ \cite{Ta}). Observe that any residually finite homomorphism $A \rightarrow C$ factors into some $A \rightarrow B = \widehat{A \sqbr{\underline{t}}}_{\mfrak{p}} \rightarrow C$. \begin{thm} \label{thm6.1} \textup{(Huang)}\ Consider the category $\msf{Loc}$ of complete noetherian local rings and residually finitely generated local homomorphisms. Then: \begin{enumerate} \item For any morphism $\phi : A \rightarrow B$ in $\msf{Loc}$ there is a functor \[ \phi_{\#} : \msf{Mod}_{\mrm{disc}}(A) \rightarrow \msf{Mod}_{\mrm{disc}}(B) . \] For composable morphisms $A \xrightarrow{\phi} B \xrightarrow{\psi} C$ there is an isomorphism $(\psi \phi)_{\#} \cong \psi_{\#} \phi_{\#}$, and $(1_{A})_{\#} \cong 1_{\msf{Mod}_{\mrm{disc}}(A)}$. These data form a pseudofunctor on $\msf{Loc}$ \textup{(}cf.\ \cite{Hg1} Def.\ \textup{4.1)}. \item If $\phi : A \rightarrow B$ is formally smooth of relative dimension $q$, and $n = \operatorname{rank} \widehat{\Omega}^{1}_{B / A}$, then there is an isomorphism, functorial in $M \in \msf{Mod}_{\mrm{disc}}(A)$, \[ \phi_{\#} M \cong \mrm{H}^{q}_{\mfrak{n}}( \widehat{\Omega}^{n}_{B/A} \otimes_{A} M) . \] \item If $\phi : A \rightarrow B$ is residually finite then there is an $A$-linear homomorphism, functorial in $M \in \msf{Mod}_{\mrm{disc}}(A)$, \[ \operatorname{Tr}_{\phi} : \phi_{\#} M \rightarrow M , \] which induces an isomorphism $\phi_{\#} M \cong \mrm{Hom}^{\mrm{cont}}_{A}(B, M)$. For composable homomorphisms $A \xrightarrow{\phi} B \xrightarrow{\psi} C$ one has $\operatorname{Tr}_{\psi \phi} = \operatorname{Tr}_{\phi} \operatorname{Tr}_{\psi}$ under the isomorphism of part \textup{1}. \item If $B = \widehat{A \sqbr{\underline{t}}}_{\mfrak{p}}$ then $\operatorname{Tr}_{\phi} = \operatorname{res}_{t_{1}, \ldots, t_{n}; A, B}$ under the isomorphism of part \textup{2}. \end{enumerate} \end{thm} \begin{proof} Parts 1 and 2 are \cite{Hg1} Thm.\ 6.12. Parts 3 and 4 follow from \cite{Hg1} Section 7. \end{proof} \begin{dfn} \label{dfn6.3} Suppose $L$ is a regular local ring of dimension $q$, with maximal ideal $\mfrak{r}$. Given a homomorphism $\phi : L \rightarrow A$ in $\msf{Loc}$, define \[ \mcal{K}(A / L) := \phi_{\#} \mrm{H}^{q}_{\mfrak{r}} L , \] the {\em dual module of $A$ relative to $L$}. \end{dfn} Since $\mrm{H}^{q}_{\mfrak{r}} L$ is an injective hull of the field $L / \mfrak{r}$, it follows that $\mcal{K}(A / L)$ is an injective hull of $A / \mfrak{m}$ (cf.\ \cite{Hg1} Corollary 3.10). \begin{cor} \label{cor6.1} If $\psi : A \rightarrow B$ is a residually finite homomorphism, then there is an $A$-linear homomorphism \[ \operatorname{Tr}_{\psi} = \operatorname{Tr}_{B / A} : \mcal{K}(B / L) \rightarrow \mcal{K}(A / L) . \] Given another such homomorphism $B \rightarrow C$, one has $ \operatorname{Tr}_{C / A} = \operatorname{Tr}_{B / A} \operatorname{Tr}_{C / B}$. \end{cor} \begin{rem} \label{rem6.1} One can show that when $L$ is a perfect field, there is a functorial isomorphism between $\mcal{K}(A / L) = \phi_{\#} L$ above and the dual module $\mcal{K}(A)$ of \cite{Ye2}, which was defined via Beilinson completion algebras. \end{rem} Suppose $\pi : \mfrak{X} \rightarrow S$ is a formally finite type (f.f.t.) formally smooth morphism. According to Proposition \ref{prop1.4}, $\mfrak{X}$ is a regular formal scheme. When we write $n = \operatorname{rank} \widehat{\Omega}^{1}_{\mfrak{X}/S}$ we mean that $n$ is a locally constant function $n : \mfrak{X} \rightarrow \mbb{N}$. \begin{lem} \label{lem6.1} Given a f.f.t.\ morphism $\pi : \mfrak{X} \rightarrow S$ and a point $x \in \mfrak{X}$, let $s := \pi(x)$, and define \[ d_{S}(x) := \operatorname{dim} \widehat{\mcal{O}}_{S, s} - \operatorname{tr.deg}_{k(s)} k(x) . \] Then: \begin{enumerate} \item $d_{S}$ is a codimension function. \item If $\pi$ is formally smooth then \[ d_{S}(x) = \operatorname{dim} \widehat{\mcal{O}}_{\mfrak{X}, x} - \operatorname{rank} \widehat{\Omega}^{1}_{\mfrak{X}/S} . \] \end{enumerate} \end{lem} \begin{proof} We shall prove 2 first. Let $L := \widehat{\mcal{O}}_{S, s}$ and $A := \widehat{\mcal{O}}_{\mfrak{X}, x}$. By Prop.\ \ref{prop1.4}, \[ \operatorname{rank} \widehat{\Omega}^{1}_{A/L} = \operatorname{dim} A - \operatorname{dim} L + \operatorname{tr.deg}_{L / \mfrak{r}} A / \mfrak{m} . \] We see that $d_{S}$ is the codimension function associated with the dualizing complex $\mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{X}}[n]$ (see Theorem \ref{thm5.3}). As for 1, the property of being a codimension function is local. But locally there is always a closed immersion $\mfrak{X} \subset \mfrak{Y}$ with $\mfrak{Y} \rightarrow S$ formally smooth. \end{proof} We shall use the codimension function $d_{S}$ by default. \begin{dfn} \label{dfn6.1} Let $\pi : \mfrak{X} \rightarrow S$ be a formally finite type morphism. Given a point $x \in \mfrak{X}$, consider $\phi : L = \widehat{\mcal{O}}_{S ,\pi(x)} \rightarrow A = \widehat{\mcal{O}}_{\mfrak{X}, x}$, which is a morphism in $\msf{Loc}$. Since $L$ is a regular local ring, the dual module $\mcal{K}(A / L)$ is defined. Let $\mcal{K}_{\mfrak{X} / S}(x)$ be the quasi-coherent sheaf which is constant on $\overline{\{x\}}$ with group of sections $\mcal{K}(A / L)$, and define \[ \mcal{K}_{\mfrak{X} / S}^{q} := \bigoplus_{d_{S}(x) = q} \mcal{K}_{\mfrak{X} / S}(x) . \] \end{dfn} In Theorem \ref{thm6.2} we are going to prove that on the graded sheaf $\mcal{K}_{X / S}^{{\textstyle \cdot}}$ there is a canonical coboundary operator $\delta$ which makes it into residual complex. \begin{dfn} \label{dfn6.2} Let $f : \mfrak{X} \rightarrow \mfrak{Y}$ be a morphism of formal schemes over $S$. Define a homomorphism of graded $\mcal{O}_{\mfrak{Y}}$-modules $\operatorname{Tr}_{f} : f_{*} \mcal{K}_{\mfrak{X} / S}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{\mfrak{Y} / S}^{{\textstyle \cdot}}$ as follows. If $x \in \mfrak{X}$ is closed in its fiber and $y = f(x)$, then $A = \widehat{\mcal{O}}_{\mfrak{Y}, y} \rightarrow B = \widehat{\mcal{O}}_{\mfrak{X}, x}$ is a residually finite $L$-algebra homomorphism. The homomorphism $\operatorname{Tr}_{B / A} : \mcal{K}(B / L) \rightarrow \mcal{K}(A / L)$ of Cor.\ \ref{cor6.1} gives a map of sheaves \[ \operatorname{Tr}_{f} : f_{*} \mcal{K}_{\mfrak{X} / S}(x) \rightarrow \mcal{K}_{\mfrak{Y} / S}(y) . \] If $x$ is not closed in its fiber, we let $\operatorname{Tr}_{f}$ vanish on $f_{*} \mcal{K}_{\mfrak{X} / S}(x)$. \end{dfn} \begin{prop} \label{prop6.3} \begin{enumerate} \item $\operatorname{Tr}_{f}$ is functorial: if $g : \mfrak{Y} \rightarrow \mfrak{Z}$ is another morphism, then $\operatorname{Tr}_{gf} = \operatorname{Tr}_{g} \operatorname{Tr}_{f}$. \item If $f$ is formally finite \textup{(}see Def.\ \textup{\ref{dfn1.1})}, then $\operatorname{Tr}_{f}$ induces an isomorphism of graded sheaves \[ f_{*} \mcal{K}_{\mfrak{X} / S}^{{\textstyle \cdot}} \cong \mcal{H}om_{\mfrak{Y}}(f_{*} \mcal{O}_{\mfrak{X}}, \mcal{K}_{\mfrak{Y} / S}^{{\textstyle \cdot}}) . \] \item If $g : \mfrak{U} \rightarrow \mfrak{X}$ is an open immersion, then there is a natural isomorphism $\mcal{K}_{\mfrak{U} / S}^{{\textstyle \cdot}} \cong g^{*} \mcal{K}_{\mfrak{X} / S}^{{\textstyle \cdot}}$. \end{enumerate} \end{prop} \begin{proof} Part 3 is trivial. Part 1 is a consequence of Cor.\ \ref{cor6.1}. As for part 2, $f$ is an affine morphism, and fibers of $f$ are all finite, so all points of $X$ are closed in their fibers. \end{proof} Suppose $\underline{a} = (a_{1}, \ldots, a_{n})$ is a sequence of elements in the noetherian ring $A$. Let us write $\tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{a})$ for the subcomplex $\mbf{K}^{\geq 1}_{\infty}(\underline{a})$, so we get an exact sequence \begin{equation} 0 \rightarrow \tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \rightarrow \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \rightarrow A \rightarrow 0 . \end{equation} For any $M^{{\textstyle \cdot}} \in \msf{D}^{+}(\msf{Mod}(A))$ let $\mcal{M}^{{\textstyle \cdot}}$ be the complex of sheaves $\mcal{O}_{X} \otimes M^{{\textstyle \cdot}}$ on $X := \operatorname{Spec} A$, and let $U \subset X$ be the open set $\bigcup \{ a_{i} \neq 0 \}$. Then \[ \mrm{R} \Gamma(U, \mcal{M}^{{\textstyle \cdot}}) \cong \tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{a})[1] \otimes M^{{\textstyle \cdot}} \] in $\msf{D}(\msf{Mod}(A))$. In fact $\tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes \mcal{O}_{X}$ is a shift by $1$ of the \v{C}ech complex corresponding to the open cover of $U$. \begin{lem} \label{lem6.2} Let $A$ be an adic noetherian ring and $M^{{\textstyle \cdot}} \in \msf{D}^{+}(\msf{Mod}(A))$. Define $\mfrak{U} := \operatorname{Spf} A$ and $\mcal{M}^{{\textstyle \cdot}} := \mcal{O}_{\mfrak{U}} \otimes M^{{\textstyle \cdot}}$. \begin{enumerate} \item Let $x \in \mfrak{U}$ with corresponding open prime ideal $\mfrak{p} \subset A$. Suppose the sequence $\underline{a}$ generates $\mfrak{p}$. Then \[ \mrm{R} \Gamma_{x} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}} \cong \mrm{R} \Gamma_{\mfrak{p}} M^{{\textstyle \cdot}}_{\mfrak{p}} \cong \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes M^{{\textstyle \cdot}}_{\mfrak{p}} \] in $\msf{D}^{+}(\msf{Mod}(A_{\mfrak{p}})).$ \item Suppose $y \in \mfrak{U}$ is an immediate specialization of $x$, and its ideal $\mfrak{q}$ has generators $\underline{a}, \underline{b}$. Then \[ \mrm{R} \Gamma_{x} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}} \cong \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes \tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{b})[1] \otimes M^{{\textstyle \cdot}}_{\mfrak{q}} \] in $\msf{D}^{+}(\msf{Mod}(A_{\mfrak{q}})).$ \item Assume $d$ is a codimension function on $\mfrak{U}$. Then in the Cousin complex $\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}}$ the map \[ \mrm{H}^{d(x)}_{x} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}} \rightarrow \mrm{H}^{d(y)}_{y} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}} \] is given by applying $\mrm{H}^{d(y)}$ to \[ \left( \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes \tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{b}) \rightarrow \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}, \underline{b}) \right) \otimes M^{{\textstyle \cdot}}_{\mfrak{q}} . \] \end{enumerate} \end{lem} \begin{proof} Part 1 follows immediately from formula (\ref{eqn4.1}). Parts 2 and 3 are true because $\operatorname{Spec} (A / \mfrak{p})_{\mfrak{q}} = \{ \mfrak{p}, \mfrak{q} \}$. \end{proof} As a warm up for Thm.\ \ref{thm6.2}, here is: \begin{prop} \label{prop6.4} If $\pi : \mfrak{X} \rightarrow S$ is formally smooth, with $n = \operatorname{rank} \widehat{\Omega}^{1}_{\mfrak{X} / S}$, then there is a canonical isomorphism of graded sheaves \[ \mcal{K}_{\mfrak{X} / S}^{{\textstyle \cdot}} \cong \mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \widehat{\Omega}^{n}_{\mfrak{X} / S}[n] . \] This makes $\mcal{K}_{\mfrak{X} / S}^{{\textstyle \cdot}}$ into a residual complex. \end{prop} \begin{proof} Take any point $x$, and with the notation of Def.\ \ref{dfn6.1} let $p := \operatorname{dim} L$ and $q := \operatorname{dim} A$. Then by Lemma \ref{lem6.2} part 1 and \cite{Hg1} Proposition 2.6 we have a canonical isomorphism \[ \mrm{H}^{d(x)}_{x} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \widehat{\Omega}^{n}_{\mfrak{X} / S}[n] \cong \mrm{H}^{q}_{\mfrak{m}} \widehat{\Omega}^{n}_{A / L} \cong \mrm{H}^{q-p}_{\mfrak{m}} \left( \widehat{\Omega}^{n}_{A / L} \otimes_{L} \mrm{H}^{p}_{\mfrak{r}} L \right) \cong \mcal{K}(A / L) . \] According to Theorem \ref{thm5.3} and Proposition \ref{prop5.7}, $\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \widehat{\Omega}^{n}_{\mfrak{X} / S}[n]$ is a residual complex. \end{proof} In particular taking $\mfrak{X} = S$ we get $\mcal{K}_{S / S}^{{\textstyle \cdot}} = \mrm{E} \mcal{O}_{S}$. \begin{lem} \label{lem6.3} Suppose $X \subset \mfrak{X}$ and $X \subset \mfrak{Y}$ are s.f.e.'s and $f : \mfrak{X} \rightarrow \mfrak{Y}$ is a morphism of embeddings. Then $\operatorname{Tr}_{f} : \mcal{K}^{{\textstyle \cdot}}_{\mfrak{X}} \rightarrow \mcal{K}^{{\textstyle \cdot}}_{\mfrak{Y}}$ is a homomorphism of complexes. \end{lem} \begin{proof} Factoring $f$ through $(\mfrak{X} \times_{S} \mfrak{Y})_{/ X}$ we can assume that $f$ is either a closed immersion, or that it is formally smooth. At any rate $f$ is an affine morphism, so we can take $\mfrak{X} = \operatorname{Spf} B$, $\mfrak{Y} = \operatorname{Spf} A$ and $S = \operatorname{Spec} L$. By Theorem \ref{thm2.2} we can suppose one of the following holds: (i) $B \cong A [\sqbr{\underline{t}}]$ for a sequence of indeterminates $\underline{t} = (t_{1}, \ldots, t_{l})$, and $A \rightarrow B$ is the inclusion; or (ii) $A \cong B [\sqbr{\underline{t}}]$ and $A \rightarrow B$ is the projection modulo $\underline{t}$. We shall treat each case separately. \noindent (i)\ Choose generators $\underline{a}$ for a defining ideal of $A$. Let $m := \operatorname{rank} \widehat{\Omega}^{1}_{A / L}$ and $n := \operatorname{rank} \widehat{\Omega}^{1}_{B / L}$, so $n = m + l$. Define an $A$-linear map $\rho : \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{t}) \otimes \widehat{\Omega}^{l}_{B / A}[l] \rightarrow A$ by $\rho(\underline{t}^{(-1, \ldots, -1)} \mrm{d} \underline{t}) = 1$ and $\rho(\underline{t}^{\underline{i}}\, \mrm{d} \underline{t}) = 0$ if $\underline{i} \neq (-1, \ldots, -1)$. Extend $\rho$ linearly to \[ \rho : \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}, \underline{t}) \otimes \widehat{\Omega}^{n}_{B / L}[n] \rightarrow \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes \widehat{\Omega}^{m}_{A / L}[m] . \] This $\rho$ sheafifies to give a map of complexes in $\msf{Ab}(X)$ \[ \tilde{\rho} : \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}, \underline{t}) \otimes \widehat{\Omega}^{n}_{\mfrak{X} / S}[n] \rightarrow \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes \widehat{\Omega}^{m}_{\mfrak{Y} / S}[m] . \] By Lemma \ref{lem6.2} and \cite{Hg1} \S 5, for any point $x \in X$, $\mrm{H}^{d(x)}_{x}(\tilde{\rho})$ recovers $\operatorname{Tr}_{f} : \mcal{K}_{\mfrak{X} / S}(x) \rightarrow \mcal{K}_{\mfrak{Y} / S}(x)$. Thus $\operatorname{Tr}_{f} = \mrm{E}(\tilde{\rho})$ is a homomorphism of complexes. \noindent (ii)\ Now $l = m - n$. Take $\underline{a}$ to be generators of a defining ideal of $B$. Define a $B$-linear map $\rho' : B \rightarrow \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{t}) \otimes \widehat{\Omega}^{l}_{A / L}[l]$ by $\rho'(1) = \underline{t}^{(-1, \ldots, -1)} \mrm{d} \underline{t}$. Extend $\rho$ linearly to \[ \rho' : \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes \widehat{\Omega}^{n}_{B / L}[n] \rightarrow \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}, \underline{t}) \otimes \widehat{\Omega}^{m}_{A / L}[m] . \] Again this extends to a map of complexes of sheaves $\tilde{\rho}'$ in $\msf{Ab}(X)$, and checking punctually we see that $\operatorname{Tr}_{f} = \mrm{E}(\tilde{\rho}')$. \end{proof} \begin{thm} \label{thm6.2} Suppose $X \rightarrow S$ is a finite type morphism. There is a unique operator $\delta : \mcal{K}_{X / S}^{q} \rightarrow \mcal{K}_{X / S}^{q+1}$, satisfying the following local condition:\\[2mm] \textup{\bf (LE)}\ \blnk{4mm} \begin{minipage}{11cm} Suppose $U \subset X$ is an open subset, and $U \subset \mfrak{U}$ is a smooth formal embedding. By Proposition \textup{\ref{prop6.3}} there is an inclusion of graded $\mcal{O}_{U}$-modules $\mcal{K}_{X / S}^{{\textstyle \cdot}}|_{U} \subset \mcal{K}_{\mfrak{U} / S}^{{\textstyle \cdot}}$. Then $\delta|_{U}$ is compatible with the coboundary operator on $\mcal{K}_{\mfrak{U} / S}^{{\textstyle \cdot}}$ coming from Proposition \textup{\ref{prop6.4}}. \end{minipage}\\[2mm] Moreover $(\mcal{K}_{X / S}^{{\textstyle \cdot}}, \delta)$ is a residual complex on $X$. \end{thm} \begin{proof} Define $\delta|_{U}$ using {\bf LE}. According to Lemma \ref{lem6.3}, $\delta|_{U}$ is independent of $\mfrak{U}$, so it glues. We get a bounded complex of quasi-coherent injectives on $X$. By Proposition \ref{prop6.4} it follows that it is residual. \end{proof} \begin{rem} This construction of $\mcal{K}_{X / S}^{{\textstyle \cdot}}$ actually allows a computation of the operator $\delta$, given the data of a local embedding. The formula is in part 3 of Lemma \ref{lem6.2}, with $M^{{\textstyle \cdot}} = \widehat{\Omega}^{n}_{A / L}[n]$. The formula for changing the embedding can be extracted from the proof of Lemma \ref{lem6.3}. Of course when $\operatorname{rank} \widehat{\Omega}^{1}_{\mfrak{X} / S}$ is high these computations can be nasty. \end{rem} \begin{rem} \label{rem6.6} The recent papers \cite{Hg2}, \cite{Hg3} and \cite{LS2} also use the local theory of \cite{Hg1} as a starting point for explicit constructions of Grothendieck Duality. Their constructions are more general than ours: Huang constructs $f^{!} \mcal{M}^{{\textstyle \cdot}}$ for a finite type morphism $f : X \rightarrow Y$ and a residual complex complex $\mcal{M}^{{\textstyle \cdot}}$; and Lipman-Sastry even allow $\mcal{M}^{{\textstyle \cdot}}$ to be any Cousin complex. \end{rem} \section{The Trace for Finite Morphisms} In this section we prove that $\operatorname{Tr}_{f}$ is a homomorphism of complexes when $f$ is a finite morphism. The proof is by a self contained calculation involving Koszul complexes and a comparison of global and local Tate residue maps. In Theorem \ref{thm7.4} we compare the complex $\mcal{K}_{X / S}^{{\textstyle \cdot}}$ to the sheaf of regular differentials of Kunz-Waldi. Throughout $S$ is a regular noetherian scheme. \begin{thm} \label{thm7.6} Suppose $f : X \rightarrow Y$ is finite. Then $\operatorname{Tr}_{f} : f_{*} \mcal{K}_{X / S}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y / S}^{{\textstyle \cdot}}$ is a homomorphism of complexes. \end{thm} The proof appears after some preparatory work, based on and inspired by \cite{Hg1} \S7. \begin{rem} \label{rem7.1} In Section 8 we prove a much stronger result, namely Corollay \ref{cor8.1}, but its proof is indirect and relies on the Residue Theorem of \cite{RD} Chapter VII. We have decided to include Theorem \ref{thm7.6} because of its direct algebraic proof. \end{rem} Let $A$ be an adic noetherian ring with defining ideal $\mfrak{a}$. Suppose $p \in A \sqbr{t}$ is a monic polynomial of degree $e > 0$. Define an $A$-algebra \begin{equation} \label{eqn7.3} B := \lim_{\leftarrow i} A \sqbr{t} / A \sqbr{t} \cdot p^{i} . \end{equation} Let $\mfrak{b} :=B \mfrak{a} + B p$; then $B \cong \lim_{\leftarrow i} B / \mfrak{b}^{i}$, so that $B$ is an adic ring with the $\mfrak{b}$-adic topology. The homomorphism $\phi : A \rightarrow B$ is f.f.t.\ and formally smooth, and $\widehat{\Omega}^{1}_{B / A} = B \cdot \mrm{d} t$. Furthermore $p \in B$ is a non-zero-divisor, and by long division we obtain an isomorphism \begin{equation} \label{eqn7.2} \mrm{H}^{1}_{(p)} B = \mrm{H}^{1} \left(\mbf{K}^{{\textstyle \cdot}}_{\infty}(p) \otimes B \right) \cong \bigoplus_{1 \leq i}\ \bigoplus_{0 \leq j < e} A \cdot \gfrac{t^{j}}{p^{i}} . \end{equation} Define an $A$-linear homomorphism $\operatorname{Res}_{B / A} : \mrm{H}^{1}_{(p)} \widehat{\Omega}^{1}_{B / A} \rightarrow A$ by \[ \operatorname{Res}_{B / A} \left( \gfrac{t^{j} \mrm{d}t}{p^{i}} \right) := \begin{cases} 1 & \text{ if } i=1, j=e-1 \\ 0 & \text{ otherwise} . \end{cases} \] We call $\operatorname{Res}_{B / A}$ the {\em global Tate residue}. It gives rise to a map of complexes in $\msf{Mod}(A)$: \begin{equation} \label{eqn7.9} \operatorname{Res}_{B / A} : \mbf{K}^{{\textstyle \cdot}}_{\infty}(p)[1] \otimes \widehat{\Omega}^{1}_{B / A} \rightarrow A . \end{equation} Note that both the algebra $B$ and the map $\operatorname{Res}_{B / A}$ depend on $t$ and $p$. Suppose $\mfrak{q} \subset B$ is an open prime ideal and $\mfrak{p} = \phi^{-1}(\mfrak{q}) \subset A$. Then the local homomorphism $\phi_{\mfrak{q}} : \widehat{A}_{\mfrak{p}} \rightarrow \widehat{B}_{\mfrak{q}}$ is formally smooth of relative dimension $1$ and residually finite. Let $\tilde{\mfrak{q}} := \mfrak{q} \cap \widehat{A}_{\mfrak{p}} \sqbr{t}$, and denote by $\bar{\mfrak{q}}$ the image of $\tilde{\mfrak{q}}$ in $k(\mfrak{p}) \sqbr{t}$, so $k(\mfrak{p}) \sqbr{t} / \bar{\mfrak{q}} = k(\mfrak{q})$. For a polynomial $q \in \widehat{A}_{\mfrak{p}} \sqbr{t}$ let $\bar{q}$ be its image in $k(\mfrak{p}) \sqbr{t}$. Suppose $q$ satisfies: \begin{equation} \label{eqn7.5} q \text{ is monic, and the ideal } (\bar{q}) \subset k(\mfrak{p}) \sqbr{t} \text{ is } \bar{\mfrak{q}}\text{-primary.} \end{equation} Then $\widehat{B}_{\mfrak{q}} \cdot \mfrak{q} = \sqrt{\widehat{B}_{\mfrak{q}} \cdot (\mfrak{p}, q)} \subset \widehat{B}_{\mfrak{q}}$, and \[ \widehat{B}_{\mfrak{q}} \cong \lim_{\leftarrow i} \widehat{A}_{\mfrak{p}} \sqbr{t} / \tilde{\mfrak{q}}^{i} \cong \lim_{\leftarrow i} \widehat{A}_{\mfrak{p}} \sqbr{t} / \widehat{A}_{\mfrak{p}} \sqbr{t} \cdot q^{i} . \] Hence $q$ is a non-zero-divisor in $\widehat{B}_{\mfrak{q}}$ and $\widehat{B}_{\mfrak{q}} / \widehat{B}_{\mfrak{q}} \cdot q$ is a free $\widehat{A}_{\mfrak{p}}$-module with basis $1, t, \ldots, t^{d-1}$, where $d = \operatorname{deg} q$. We see that a decomposition like (\ref{eqn7.2}) exists for $\mrm{H}^{1}_{(q)} \widehat{B}_{\mfrak{q}}$. Suppose we are given a discrete $\widehat{A}_{\mfrak{p}}$-module $M$. Then one gets \[ \mrm{H}^{1}_{\mfrak{q}} \left( \widehat{\Omega}^{1}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}} \otimes_{\widehat{A}_{\mfrak{p}}} M \right) \cong \left( \mrm{H}^{1}_{(q)} \widehat{\Omega}^{1}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}} \right) \otimes_{\widehat{A}_{\mfrak{p}}} M \cong \bigoplus_{1 \leq i}\ \bigoplus_{0 \leq j < d} \gfrac{t^{j} \mrm{d} t}{q^{i}} \otimes M \] (cf.\ \cite{Hg1} pp.\ 41-42). Define the {\em local Tate residue map} \[ \operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}} : \mrm{H}^{1}_{\mfrak{q}} \left( \widehat{\Omega}^{1}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}} \otimes_{\widehat{A}_{\mfrak{p}}} M \right) \rightarrow M \] by \[ \operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}} \left( \gfrac{t^{j} \mrm{d}t \otimes m}{q^{i}} \right) := \begin{cases} m & \text{ if } i=1, j=d-1 \\ 0 & \text{ otherwise} . \end{cases} \] Clearly $\operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}}$ is functorial in $M$, and it depends on $t$. \begin{lem} \label{lem7.7} $\operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}}$ is independent of $q$. It coincides with the residue map $\operatorname{res}_{t; \widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}}$ of \textup{(\ref{eqn6.6})}, i.e.\ of \cite{Hg1} Definition \textup{8.1}. \end{lem} \begin{proof} Suppose the polynomials $q_{1}, q_{2} \in \widehat{A}_{\mfrak{p}} \sqbr{t}$ satisfy (\ref{eqn7.5}). Then so does $q_{3} := q_{1} q_{2}$. Let $\operatorname{deg} q_{h} = d_{h}$, and let $\operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}; q_{h}}$ be the residue map determined by $q_{h}$. Pick any $1 \leq i$ and $0 \leq j < d_{1}$, and write $q_{2}^{i} = \sum_{l = 0}^{i d_{2}} a_{l} t^{l}$, so $a_{i d_{2}} = 1$. By the rules for manipulating generalized fractions (cf.\ \cite{Hg1} \S 1) we have \begin{equation} \label{eqn7.1} \operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}; q_{3}} \left( \gfrac{t^{j} \mrm{d} t \otimes m}{q_{1}^{i}} \right) = \sum_{l = 0}^{i d_{2}} \operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}; q_{3}} \left( \gfrac{t^{l + j} \mrm{d} t \otimes a_{l} m}{q_{3}^{i}} \right) . \end{equation} If $i \geq 2$ or $j \leq d_{1} - 2$ one has $l + j \leq i d_{3} -2$, and therefore each summand of the right side of (\ref{eqn7.1}) is $0$. When $i = 1$ and $j = d_{1} - 1$ the only possible nonzero residue there is for $l = d_{2}$, and this residue is $m$. We conclude that $\operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}; q_{3}} = \operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}; q_{1}}$. Clearly also $\operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}; q_{3}} = \operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}; q_{2}}$. If we take $q$ such that $(\bar{q}) = \bar{\mfrak{q}}$, this is by definition the residue map of (\ref{eqn6.6}). \end{proof} \begin{lem} \label{lem7.8} Let $F$ be the set of prime ideals in $B / (p)$ lying over $\mfrak{p}$. Then for any $M \in \msf{Mod}_{\mrm{disc}}(\widehat{A}_{\mfrak{p}})$ one has \[ \left( \mrm{H}^{1}_{(p)} \widehat{\Omega}^{1}_{B / A} \right) \otimes_{A} M \cong \bigoplus_{\mfrak{q}' \in F} \mrm{H}^{1}_{\mfrak{q}'} \left( \widehat{\Omega}^{1}_{\widehat{B}_{\mfrak{q}'} / \widehat{A}_{\mfrak{p}}} \otimes_{\widehat{A}_{\mfrak{p}}} M \right) , \] and w.r.t.\ this isomorphism, \[ \operatorname{Res}_{B / A} \otimes 1 = \sum_{\mfrak{q}' \in F} \operatorname{Res}_{\widehat{B}_{\mfrak{q}'} / \widehat{A}_{\mfrak{p}}} . \] \end{lem} \begin{proof} The isomorphism of modules is not hard to see. Let $\bar{p} = \prod_{\mfrak{q}' \in F} \bar{p}_{\mfrak{q}'}$ be the primary decomposition in $k(\mfrak{p}) \sqbr{t}$ (all the $\bar{p}_{\mfrak{q}'}$ monic). By Hensel's Lemma this decomposition lifts to $p = \prod_{\mfrak{q}' \in F} p_{\mfrak{q}'}$ in $\widehat{A}_{\mfrak{p}} \sqbr{t}$. Since each polynomial $p_{\mfrak{q}'}$ satisfies condition (\ref{eqn7.5}) for the prime ideal $\mfrak{q}'$, we can use it to calculate $\operatorname{Res}_{\widehat{B}_{\mfrak{q}'} / \widehat{A}_{\mfrak{p}}}$. \end{proof} \begin{proof} (of Thm.\ \ref{thm7.6})\ This claim is local on $Y$, so we may assume $X$, $Y$ and $S$ are affine, say $X = \operatorname{Spec} \bar{B}$, $Y = \operatorname{Spec} \bar{A}$ and $S = \operatorname{Spec} L$. By the functoriality of $\operatorname{Tr}$ we can assume $\bar{B} = \bar{A} \sqbr{b}$ for some element $b \in \bar{B}$. It will suffice to find suitable s.f.e.'s $X \subset \mfrak{X}$ and $Y \subset \mfrak{Y}$ with a morphism $\widehat{f} : \mfrak{X} \rightarrow \mfrak{Y}$ extending $f$, and to check that $\operatorname{Tr}_{\widehat{f}} : \widehat{f}_{*} \mcal{K}^{{\textstyle \cdot}}_{\mfrak{X} / S} \rightarrow \mcal{K}^{{\textstyle \cdot}}_{\mfrak{Y} / S}$ commutes with $\delta$. Pick any s.f.e.\ $Y \subset \mfrak{Y} = \operatorname{Spf} A$, so $\mfrak{a} := \operatorname{Ker}(A \rightarrow \bar{A})$ is a defining ideal. Let $A \sqbr{t} \rightarrow \bar{B}$ be the homomorphism $t \mapsto b$. Choose any monic polynomial $p(t) \in A \sqbr{t}$ s.t.\ $p(b) = 0$, and define the adic ring $B$ as in formula (\ref{eqn7.3}). So $\mfrak{X} := \operatorname{Spf} B$ is the s.f.e.\ of $X$ we want. Let $(y_{0}, y_{1})$ be an immediate specialization pair in $Y$, and let $F_{i} := f^{-1}(y_{i}) \subset X$. Let $\mfrak{p}_{0} \subset \mfrak{p}_{1} \subset A$ be the prime ideals corresponding to $(y_{0}, y_{1})$. Pick a sequence of generators $\underline{a}$ for $\mfrak{p}_{0}$, and generators $(\underline{a}, \underline{a}')$ for $\mfrak{p}_{1}$. Let $m := \operatorname{rank} \widehat{\Omega}^{1}_{A / L}$. Consider the commutative diagram of complexes \[ \begin{CD} \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}, p)[1] \otimes \tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{a}') \otimes (\widehat{\Omega}^{m + 1}_{B / L})_{\mfrak{p}_{1}} @> \operatorname{Res}_{B / A} \otimes 1 >> \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes \tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{a}') \otimes (\widehat{\Omega}^{m}_{A / L})_{\mfrak{p}_{1}} \\ @VVV @VVV \\ \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}, \underline{a}', p)[1] \otimes (\widehat{\Omega}^{m + 1}_{B / L})_{\mfrak{p}_{1}} @> \operatorname{Res}_{B / A} \otimes 1 >> \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}, \underline{a}') \otimes (\widehat{\Omega}^{m}_{A / L})_{\mfrak{p}_{1}} \end{CD} \] gotten from tensoring the map $\operatorname{Res}_{B / A}$ of (\ref{eqn7.9}) with $A_{\mfrak{p}_{1}} \otimes \widehat{\Omega}^{m}_{A / L}$ and the various $\mbf{K}^{{\textstyle \cdot}}_{\infty}$. Applying $\mrm{H}^{i}$ to this diagram, where $i := \operatorname{dim} \widehat{A}_{\mfrak{p}_{1}}$, and using Lemmas \ref{lem6.2} and \ref{lem7.8} we obtain a commutative diagram \[ \begin{CD} \bigoplus_{\mfrak{q}_{0} \in F_{0}} \mrm{H}^{1}_{\mfrak{q}_{0}} \left( \widehat{\Omega}^{1}_{ \widehat{B}_{\mfrak{q}_{0}} / \widehat{A}_{\mfrak{p}_{0}}} \otimes \mrm{H}^{i - 1}_{\mfrak{p}_{0}} \widehat{\Omega}^{m}_{ \widehat{A}_{\mfrak{p}_{0}} / L} \right) @> \sum \operatorname{Res} >> \mrm{H}^{i - 1}_{\mfrak{p}_{0}} \widehat{\Omega}^{m}_{ \widehat{A}_{\mfrak{p}_{0}} / L} \\ @VVV @VVV \\ \bigoplus_{\mfrak{q}_{1} \in F_{1}} \mrm{H}^{1}_{\mfrak{q}_{1}} \left( \widehat{\Omega}^{1}_{ \widehat{B}_{\mfrak{q}_{1}} / \widehat{A}_{\mfrak{p}_{1}}} \otimes \mrm{H}^{i}_{\mfrak{p}_{1}} \widehat{\Omega}^{m}_{ \widehat{A}_{\mfrak{p}_{1}} / L} \right) @> \sum \operatorname{Res} >> \mrm{H}^{i}_{\mfrak{p}_{1}} \widehat{\Omega}^{m}_{ \widehat{A}_{\mfrak{p}_{1}} / L} . \end{CD} \] In this diagram $\operatorname{Res} = \operatorname{Res}_{ \widehat{B}_{\mfrak{q}_{0}} / \widehat{A}_{\mfrak{p}_{0}}}$ etc. Using the definitions this is the same as \[ \begin{CD} \bigoplus_{x_{0} \in F_{0}} f_{*} \mcal{K}_{\mfrak{X} / S}(x_{0}) @> \operatorname{Tr}_{f} >> \mcal{K}_{\mfrak{Y} / S}(y_{0}) \\ @V \delta VV @V \delta VV \\ \bigoplus_{x_{1} \in F_{1}} f_{*} \mcal{K}_{\mfrak{X} / S}(x_{1}) @> \operatorname{Tr}_{f} >> \mcal{K}_{\mfrak{Y} / S}(y_{1}) . \end{CD} \] \end{proof} According to \cite{KW}, if $\pi : X \rightarrow S$ is equidimensional of dimension $n$ and generically smooth, and $X$ is integral, then the {\em sheaf of regular differentials} $\tilde{\omega}^{n}_{X/S}$ (relative to the DGA $\mcal{O}_{S}$) exists. It is a coherent subsheaf of $\Omega^{n}_{k(X)/k(S)}$. \begin{thm} \label{thm7.4} Suppose $\pi : X \rightarrow S$ is equidimensional of dimension $n$ and generically smooth, and $X$ is integral. Then $\mcal{K}^{-n}_{X/S} = \Omega^{n}_{k(X)/k(S)}$, and \[ \tilde{\omega}^{n}_{X/S} = \mrm{H}^{-n} \mcal{K}^{{\textstyle \cdot}}_{X/S} . \] \end{thm} First we need: \begin{lem} \label{lem7.1} Suppose $L_{0} \rightarrow A_{0} \rightarrow B_{0}$ are finitely generated field extensions, with $L_{0} \rightarrow A_{0}$ and $L_{0} \rightarrow B_{0}$ separable, $A_{0} \rightarrow B_{0}$ finite, and $\operatorname{tr.deg}_{L_{0}} A_{0} = n$. Then $\mcal{K}(A_{0} / L_{0}) = \Omega^{n}_{A_{0} / L_{0}}$, $\mcal{K}(B_{0} / L_{0}) = \Omega^{n}_{B_{0} / L_{0}}$, and $\operatorname{Tr}_{B_{0} / A_{0}} : \mcal{K}(B_{0} / L_{0}) \rightarrow \mcal{K}(A_{0} / L_{0})$ coincides with $\sigma^{L_{0}}_{B_{0} / A_{0}} : \Omega^{n}_{B_{0} / L_{0}} \rightarrow \Omega^{n}_{A_{0} / L_{0}}$ of \cite{Ku} \S \textup{16}. \end{lem} \begin{proof} Since $L_{0} \rightarrow A_{0}$ is formally smooth, we get $\mcal{K}(A_{0} / L_{0}) = \Omega^{n}_{A_{0} / L_{0}}$. The same for $B_{0}$. Consider the trivial DGA $L_{0}$. Then the universal $B_{0}$-extension of $\Omega^{{\textstyle \cdot}}_{A_{0} / L_{0}}$ is $\Omega^{{\textstyle \cdot}}_{B_{0} / L_{0}}$, so $\sigma^{L_{0}}_{B_{0} / A_{0}}$ makes sense. To check that $\sigma^{L_{0}}_{B_{0} / A_{0}} = \operatorname{Tr}_{B_{0} / A_{0}}$ we may reduce to the cases $A_{0} \rightarrow B_{0}$ separable, or purely inseparable of prime degree, and then use the properties of the trace. \end{proof} \begin{proof} (of the Theorem)\ Given any point $x \in X$ there is an open neighborhood $U$ of $x$ which admits a factorization $\pi|_{U} = h g f$, with $f : U \rightarrow Y$ an open immersion; $g : Y \rightarrow Z$ finite; and $h : Z \rightarrow S$ smooth of relative dimension $n$ (in fact one can take $Z$ open in $\mbf{A}^{n} \times S$). This follows from quasi-normalization (\cite{Ku} Thm.\ B20) and Zariski's Main Theorem (\cite{EGA} IV 8.12.3; cf.\ \cite{Ku} Thm.\ B16). We can also assume $Y, Z, S$ are affine, say $Y = \operatorname{Spec} B$, $Z = \operatorname{Spec} A$ and $S = \operatorname{Spec} L$. Let us write $\tilde{\omega}^{n}_{B / L} := \Gamma(Y, \tilde{\omega}^{n}_{Y/S})$ and $\mcal{K}^{{\textstyle \cdot}}_{B / L} := \Gamma(Y, \mcal{K}^{{\textstyle \cdot}}_{Y/S})$. Also let us write $B_{0} := k(Y)$, $A_{0} := k(Z)$ and $L_{0} := k(S)$. By \cite{KW} \S 4, \[ \tilde{\omega}^{n}_{B / L} = \{ \beta \in \Omega^{n}_{B_{0} / L_{0}} \mid \sigma^{L_{0}}_{B_{0} / A_{0}} (b \beta) \in \Omega^{n}_{A / L} \text{ for all } b \in B \} . \] One has \[ \mcal{K}^{-n}_{B / L} = \mcal{K}(B_{0} / L_{0}) = \Omega^{n}_{A_{0} / L_{0}} \] and the same for $A$. According to Prop.\ \ref{prop6.4} there is a quasi-isomorphism $\Omega^{n}_{A / L}[n]$ \linebreak $ \rightarrow \mcal{K}^{{\textstyle \cdot}}_{A / L}$. From the commutative diagram \[ \begin{CD} 0 @> >> \mrm{H}^{-n} \mcal{K}^{{\textstyle \cdot}}_{B / L} @> >> \mcal{K}^{-n}_{B / L} @>{\delta}>> \mcal{K}^{-n+1}_{B / L} \\ & & @VVV @V{\operatorname{Tr}_{g}}VV @V{\operatorname{Tr}_{g}}VV \\ 0 @>>> \Omega^{n}_{A / L} @>>> \mcal{K}^{-n}_{A / L} @>{\delta}>> \mcal{K}^{-n+1}_{A / L} \end{CD} \] and the isomorphism \[ \mcal{K}^{-n+1}_{B / L} \cong \operatorname{Hom}_{A} ( B, \mcal{K}^{-n+1}_{A / L}) \] induced by $\operatorname{Tr}_{g}$ we conclude that $\tilde{\omega}^{n}_{B / L} = \mrm{H}^{-n} \mcal{K}^{{\textstyle \cdot}}_{B / L}$. Since $\tilde{\omega}^{n}_{Y / S}$ and $\mrm{H}^{-n} \mcal{K}^{{\textstyle \cdot}}_{Y / S}$ are coherent sheaves and $f : U \rightarrow Y$ is an open immersion, this shows that $\tilde{\omega}_{U/S} = \mrm{H}^{-n} \mcal{K}^{{\textstyle \cdot}}_{U/S}$. \end{proof} \begin{cor} If $X$ is a Cohen-Macaulay scheme then the sequence \[ 0 \rightarrow \tilde{\omega}^{n}_{X / S} \rightarrow \mcal{K}^{-n}_{X / S} \rightarrow \cdots \rightarrow \mcal{K}^{m}_{X / S} \rightarrow 0 \] \textup{(}$m = \operatorname{dim} S$\textup{)} is exact. \end{cor} \begin{proof} $X$ is Cohen-Macaulay iff any dualizing complex has a single nonzero cohomology sheaf. \end{proof} \begin{exa} \label{exa7.1} Suppose $X$ is an $(n+1)$-dimensional integral scheme and $\pi : X \rightarrow \operatorname{Spec} \mbb{Z}$ is a finite type dominant morphism (i.e.\ $X$ has mixed characteristics). Then $\pi$ is flat, equidimensional of dimension $n$ and generically smooth. So \[ \tilde{\omega}^{n}_{X / \mbb{Z}} = \mrm{H}^{-n} \mcal{K}^{{\textstyle \cdot}}_{X / \mbb{Z}} \subset \Omega^{n}_{k(X) / \mbb{Q}} . \] \end{exa} \begin{rem} \label{rem7.7} In the situation of Thm.\ \ref{thm7.4} there is a homomorphism \[ \mrm{C}_{X} : \Omega^{n}_{X / S} \rightarrow \mcal{K}^{-n}_{X / S} \] called the {\em fundamental class of} $X / S$. According to \cite{KW}, when $\pi$ is flat one has $\mrm{C}_{X}(\Omega^{n}_{X / S}) \subset \tilde{\omega}^{n}_{X/S}$; so $\mrm{C}_{X} : \Omega^{n}_{X / S}[n] \rightarrow \mcal{K}^{{\textstyle \cdot}}_{X / S}$ is a homomorphism of complexes. \end{rem} \begin{rem} In \cite{LS2} Theorem 11.2 we find a stronger statement than our Theorem \ref{thm7.4}: $S$ is only required to be an excellent equidimensional scheme without embedded points, satisfying Serre's condition $\mrm{S}_{2}$; and $\pi$ is finite type, equidimensional and generically smooth. Moreover, for $\pi$ proper, the trace is compared to the integral of \cite{HS} (cf.\ Remark \ref{rem8.1}). The price of this generality is that the proofs in \cite{LS2} are not self-contained but rely on rather complicated results from other papers. \end{rem} \section{The Isomorphism $\mcal{K}^{{\textstyle \cdot}}_{X / S} \protect \cong \pi^{!} \mcal{O}_{S}$} In this section we describe the canonical isomorphism between the complex $\mcal{K}^{{\textstyle \cdot}}_{X / S}$ constructed in Section 6, and the twisted inverse image $\pi^{!} \mcal{O}_{S}$ of \cite{RD}. Recall that for residual complexes there is an inverse image $\pi^{\triangle}$, and $\pi^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{S / S} = \mrm{E} \pi^{!} \mcal{O}_{S}$, where $\mrm{E}$ is the Cousin functor corresponding to the dualizing complex $\pi^{!} \mcal{O}_{S}$. For an $S$-morphism $f : X \rightarrow Y$ denote by $\operatorname{Tr}^{\mrm{RD}}_{f}$ the homomorphism of graded sheaves \[ \operatorname{Tr}^{\mrm{RD}}_{f} : f_{*} \pi^{\triangle}_{X} \mcal{K}^{{\textstyle \cdot}}_{S / S} \cong f_{*} f^{\triangle} \pi^{\triangle}_{Y} \mcal{K}^{{\textstyle \cdot}}_{S / S} \rightarrow \pi^{\triangle}_{Y} \mcal{K}^{{\textstyle \cdot}}_{S / S} \] of \cite{RD} Section VI.4. \begin{thm} \label{thm8.10} Let $\pi : X \rightarrow S$ be a finite type morphism. Then there exists a unique isomorphism of complexes \[ \zeta_{X} : \mcal{K}^{{\textstyle \cdot}}_{X / S} \rightarrow \pi^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{S / S} \] such that for every morphism $f : X \rightarrow Y$ the diagram \begin{equation} \label{eqn8.4} \begin{CD} f_{*} \mcal{K}^{{\textstyle \cdot}}_{X / S} @>{\operatorname{Tr}_{f}}>> \mcal{K}^{{\textstyle \cdot}}_{Y / S} \\ @V{f_{*} (\zeta_{X})}VV @V{\zeta_{Y}}VV \\ f_{*} \pi^{\triangle}_{X} \mcal{K}^{{\textstyle \cdot}}_{S / S} @>{\operatorname{Tr}^{\mrm{RD}}_{f}}>> \pi^{\triangle}_{Y} \mcal{K}^{{\textstyle \cdot}}_{S / S} \end{CD} \end{equation} is commutative. \end{thm} The proof of Thm.\ \ref{thm8.10} is given later in this section, after some preparation. Here is one corollary: \begin{cor} \label{cor8.1} If $f : X \rightarrow Y$ is proper then $\operatorname{Tr}_{f}$ is a homomorphism of complexes, and for any $\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{-}_{\mrm{qc}}(\msf{Mod}(X))$ the induced morphism \[ f_{*} \mcal{H}om_{X}(\mcal{M}^{{\textstyle \cdot}}, \mcal{K}^{{\textstyle \cdot}}_{X / S}) \rightarrow \mcal{H}om_{X}(\mrm{R} f_{*} \mcal{M}^{{\textstyle \cdot}}, \mcal{K}^{{\textstyle \cdot}}_{Y / S}) \] is an isomorphism. \end{cor} \begin{proof} Use \cite{RD} Theorem VII.2.1 and Corollary VII.3.4. \end{proof} \begin{rem} \label{rem8.1} In \cite{Hg3} and \cite{LS2} the authors prove that in their respective constructions the trace $\operatorname{Tr}_{f} : f_{*} f^{!} \mcal{N}^{{\textstyle \cdot}} \rightarrow \mcal{N}^{{\textstyle \cdot}}$ is a homomorphism of complexes for any proper morphism $f$ and residual (resp.\ Cousin) complex $\mcal{N}^{{\textstyle \cdot}}$ (cf.\ Remark \ref{rem6.6}). \end{rem} Let $Y = \operatorname{Spec} A$ be an affine noetherian scheme, $X := \mbf{A}^{n} \times Y =$ \newline $\operatorname{Spec} A \sqbr{t_{1}, \ldots, t_{n}}$ and $f : X \rightarrow Y$ the projection. Fix a point $x \in X$, and let $y := f(x)$, $Z_{0} := \overline{\{x\}}_{\mrm{red}}$. Assume $Z_{0} \rightarrow Y$ is finite. \begin{lem} \label{lem8.10} There exists an open set $U \subset Y$ containing $y$ and a flat finite morphism $g : Y' \rightarrow U$ s.t.: \begin{enumerate} \rmitem{i} $g^{-1}(y)$ is one point, say $y'$. \rmitem{ii} Define $X' := \mbf{A}^{n} \times Y'$, and let $f' : X' \rightarrow Y'$, $h : X' \rightarrow X$. Then for every point $x' \in h^{-1}(x)$ there is some section $\sigma_{x'} : Y' \rightarrow X'$ of $f'$ with $x' \in \sigma_{x'}(Y')$. \end{enumerate} \end{lem} \begin{proof} Choose any finite normal field extension $K$ of $k(y)$ containing $k(x)$. Define recursively open sets $U_{i} = \operatorname{Spec} A_{i} \subset Y$ and finite flat morphisms $g_{i} : Y_{i} = \operatorname{Spec} A'_{i} \rightarrow U_{i}$ s.t.\ $g_{i}^{-1}(y) = \{y_{i}\}$ and $k(y_{i}) \subset K$, as follows. Start with $U_{0} = Y_{0} := Y$ and $A'_{0} = A_{0} := A$. If $k(y_{i}) \neq K$ take some $\bar{b} \in K - k(y_{i})$ and let $\bar{p} \in k(y_{i}) \sqbr{t}$ be the monic irreducible polynomial of $\bar{b}$. Choose a monic polynomial $p \in \mcal{O}_{Y_{i}, y_{i}} \sqbr{t}$ lifting $\bar{p}$. There is some open set $U_{i + 1} = \operatorname{Spec} A_{i + 1} \subset U_{i}$ s.t.\ $p \in (A'_{i} \otimes_{A_{i}} A_{i + 1}) \sqbr{t}$. Define $A'_{i + 1} := (A'_{i} \otimes_{A_{i}} A_{i + 1}) \sqbr{t} / (p)$ and $Y_{i + 1} = \operatorname{Spec} A'_{i + 1}$. For $i = r$ this stops, and $k(y_{r}) = K$. For every point $x' \in \operatorname{Spec} (K \otimes_{k(y)} k(x))$ and $1 \leq i \leq n$ let $\bar{a}_{i, x'} \in k(x') \cong k(y_{r})$ be the image of $t_{i}$, and let $a_{i, x'} \in \mcal{O}_{Y_{r}, y_{r}}$ be a lifting. Take an open set $U = \operatorname{Spec} A_{r + 1} \subset U_{r}$ s.t.\ each $a_{i, x'} \in A' = (A'_{r} \otimes_{A_{r}} A_{r + 1})$, and define $Y' := \operatorname{Spec} A'$. So for each $x'$ the homomorphism $B' = A' \sqbr{t} \rightarrow A'$, $t_{i} \mapsto a_{i, x'}$ gives the desired section $\sigma_{x'} : Y' \rightarrow X'$. \end{proof} Let $Z_{i}$ be the $i$-th infinitesimal neighborhood of $Z_{0}$ in $X$, so $f_{i} : Z_{i} \rightarrow Y$ is a finite morphism. Suppose we are given a quasi-coherent $\mcal{O}_{Y}$-module $\mcal{M}$ which is supported on $\overline{\{ y \}}$. One has \[ \mcal{H}^{n}_{Z_{0}} \left( \Omega^{n}_{X / Y} \otimes f^{*} \mcal{M} \right) \cong \lim_{i \rightarrow} \mcal{E}xt^{n}_{X} \left( \mcal{O}_{Z_{i}}, \Omega^{n}_{X / Y} \otimes f^{*} \mcal{M} \right) \] and by \cite{RD} Thm.\ VI.3.1 \[ \mcal{E}xt^{n}_{X} \left( \mcal{O}_{Z_{i}}, \Omega^{n}_{X / Y} \otimes f^{*} \mcal{M} \right) = \mcal{H}^{0} f_{i}^{!} \mcal{M} . \] Note that we can also factor $f_{i}$ through $\mbf{P}^{n} \times Y$, so $f_{i}$ is projectively embeddable, and by \cite{RD} Thm.\ III.10.5 we have a map \begin{equation} \label{eqn8.10} \operatorname{Tr}_{f}^{\mrm{RD}} : f_{*} \mcal{H}^{n}_{Z_{0}} \left( \Omega^{n}_{X / Y} \otimes f^{*} \mcal{M} \right) \rightarrow \mcal{M} . \end{equation} Now define $\widehat{A} := \widehat{\mcal{O}}_{Y , y}$ and $\widehat{B} := \widehat{\mcal{O}}_{X , x}$, with $\mfrak{n} \subset \widehat{B}$ the maximal ideal and $\phi = f^{*} : \widehat{A} \rightarrow \widehat{B}$. Set $M := \mcal{M}_{y}$, which is a discrete $\widehat{A}$-module. We then have a natural isomorphism of $\widehat{A}$-modules \begin{equation} \label{eqn8.1} \left( f_{*} \mcal{H}^{n}_{Z_{0}} \left( \Omega^{n}_{X / Y} \otimes f^{*} \mcal{M} \right) \right)_{y} \cong \mrm{H}^{n}_{\mfrak{n}} \left( \widehat{\Omega}^{n}_{\widehat{B} / \widehat{A}} \otimes_{\widehat{A}} M \right) \cong \phi_{\#} M . \end{equation} \begin{lem} \label{lem8.1} Under the isomorphism \textup{(\ref{eqn8.1})}, \[ \operatorname{Tr}_{f}^{\mrm{RD}} = \operatorname{Tr}_{\phi} : \phi_{\#} M \rightarrow M . \] \end{lem} \begin{proof} The proof is in two steps.\\ Step 1.\ Assume there is a section $\sigma : Y \rightarrow X$ to $f$ with $x \in W_{0} = \sigma(Y)$. The homomorphism $\sigma^{*} : B = A \sqbr{\underline{t}} \rightarrow A$ chooses $a_{i} = \sigma^{*}(t_{i}) \in A$, so after the linear change of variables $t_{i} \mapsto t_{i} - a_{i}$ we may assume that $\sigma$ is the $0$-section (i.e.\ $\mcal{O}_{W_{0}} = \mcal{O}_{X} / \mcal{O}_{X} \cdot \underline{t}$). Let $W_{i}$ be the $i$-th infinitesimal neighborhood of $W_{0}$. Since $f : W_{i} \rightarrow Y$ is projectively embeddable, there is a trace map \[ \operatorname{Tr}_{f}^{\mrm{RD}} : f_{*} \mcal{H}^{n}_{W_{0}} \Omega^{n}_{X / Y} \rightarrow \mcal{O}_{Y} . \] For any $a \in A$ one has \begin{equation} \label{eqn8.11} \operatorname{Tr}_{f}^{\mrm{RD}} \left( \gfrac{a \mrm{d} t_{1} \wedge \cdots \wedge \mrm{d} t_{n}} {t_{1}^{i_{1}} \cdots t_{n}^{i_{n}}} \right) = \begin{cases} a & \text{ if } \underline{i} = (1, \ldots, 1) \\ 0 & \text{ otherwise} . \end{cases} \end{equation} This follows from properties R6 (normalization) and R7 (intersection) of the residue symbol (\cite{RD} Section III.9). Alternatively this can be checked as follows. Note that $\operatorname{Tr}_{f}^{\mrm{RD}}$ factors through $\mrm{R} f_{*} \Omega^{n}_{\mbf{P}^{n}_{Y} / Y}$. For the case $\underline{i} = (1, \ldots, 1)$ use \cite{RD} Proposition III.10.1. For $\underline{i} \neq (1, \ldots, 1)$ consider a change of coordinates $t_{i} \mapsto \lambda_{i} t_{i}$, $\lambda_{i} \in A$. By \cite{RD} Corollary III.10.2, $\operatorname{Tr}_{f}^{\mrm{RD}}$ is independent of homogeneous coordinates, so it must be $0$. Now since $W_{0} \cap f^{-1}(y) = Z_{0}$ we have \[ \mcal{H}^{n}_{Z_{0}} \left( \Omega^{n}_{X / Y} \otimes f^{*} \mcal{M} \right) \cong \mcal{H}^{n}_{W_{0}} \left( \Omega^{n}_{X / Y} \otimes f^{*} \mcal{M} \right) \] and so the formula for $\operatorname{Tr}_{f}^{\mrm{RD}}$ in (\ref{eqn8.10}) is given by (\ref{eqn8.11}). But the same formula is used in \cite{Hg1} to define $\operatorname{Tr}_{\phi}$.\\ Step 2.\ The general situation: take $g : Y' \rightarrow Y$ as in Lemma \ref{lem8.10}, and set $Z_{0}' := Z_{0} \times_{Y} Y'$. The flatness of $g$ implies there is a natural isomorphism of $\mcal{O}_{Y'}$-modules \[ g^{*} f_{*} \mcal{H}^{n}_{Z_{0}} \left( \Omega^{n}_{X / Y} \otimes f^{*} \mcal{M} \right) \cong f_{*}' \mcal{H}^{n}_{Z_{0}'} \left( \Omega^{n}_{X' / Y'} \otimes {f'}^{*} \mcal{M}' \right) \] (where $\mcal{M}' := g^{*} \mcal{M}$) and by \cite{RD} Thm.\ III.10.5 property TRA4 we have \begin{equation} \label{eqn8.14} g^{*}(\operatorname{Tr}_{f}^{\mrm{RD}}) = \operatorname{Tr}_{f'}^{\mrm{RD}} . \end{equation} Let $\widehat{A}' := \widehat{\mcal{O}}_{Y', y'} \cong A' \otimes_{A} \widehat{A}$, so $\widehat{A} \rightarrow \widehat{A}'$ is finite flat. Therefore \begin{equation} \label{eqn8.2} \widehat{A}' \otimes_{\widehat{A}} \mrm{H}^{n}_{\mfrak{n}} \left( \widehat{\Omega}^{n}_{\widehat{B} / \widehat{A}} \otimes_{\widehat{A}} M \right) \cong \bigoplus_{\mfrak{n}' \in Z_{0}'} \mrm{H}^{n}_{\mfrak{n}'} \left( \widehat{\Omega}^{n}_{\widehat{B}_{\mfrak{n}'} / \widehat{A}'} \otimes_{\widehat{A}'} M' \right) . \end{equation} Here $M' := \mcal{M}'_{y'} \cong \widehat{A}' \otimes_{\widehat{A}} M$ and $\prod_{\mfrak{n}' \in Z_{0}'} \widehat{B}_{\mfrak{n}'}$ is the decomposition of $A' \otimes_{A} \widehat{B}$ to local rings. Write $\phi'_{\mfrak{n}'} : \widehat{A}' \rightarrow \widehat{B}_{\mfrak{n}'}$. Direct verification shows that under the isomorphism (\ref{eqn8.2}), \begin{equation} \label{eqn8.15} 1 \otimes \operatorname{Tr}_{\phi} = \sum_{\mfrak{n}' \in Z_{0}'} \operatorname{Tr}_{\phi'_{\mfrak{n}'}} . \end{equation} Since $\widehat{A} \rightarrow \widehat{A}'$ is faithfully flat it follows that $M \rightarrow M'$ is injective. In view of the equalities (\ref{eqn8.14}) and (\ref{eqn8.15}), we conclude that it suffices to check for each $\mfrak{n}' = x' \in Z_{0}$ that $\operatorname{Tr}_{\phi'_{\mfrak{n}'}} = \operatorname{Tr}^{\mrm{RD}}_{f'}$ on $\mrm{H}^{n}_{\mfrak{n}'} \left( \widehat{\Omega}^{n}_{\widehat{B}_{\mfrak{n}'} / \widehat{A}'} \otimes_{\widehat{A}'} M' \right)$. But there is a section $\sigma_{x'} : Y' \rightarrow X'$, so we can apply step 1. \end{proof} \begin{proof} (of Thm.\ \ref{thm8.10}.)\\ Step 1. (Uniqueness)\ Suppose $\zeta_{X}' : \mcal{K}^{{\textstyle \cdot}}_{X / S} \rightarrow \pi^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{S / S}$ is another isomorphism satisfying $\operatorname{Tr}_{\pi} = \operatorname{Tr}^{\mrm{RD}}_{\pi} \pi_{*}(\zeta_{X}')$. Then $\zeta_{X}' = a \zeta_{X}$ for some $a \in \Gamma(X, \mcal{O}_{X}^{*})$, and by assumption for any closed point $x \in X$ and $\alpha \in \mcal{K}_{X / S}(x)$ there is equality $\operatorname{Tr}_{\pi}(\alpha) = \operatorname{Tr}_{\pi} (a \alpha)$. Now writing $s := \pi(x)$, it's known that \[ \operatorname{Hom}_{\mcal{O}_{S, s}} \left( \mcal{K}_{X / S}(x), \mcal{K}_{S / S}(s) \right) \] is a free $\widehat{\mcal{O}}_{X, x}$-module with basis $\operatorname{Tr}_{\pi}$. Therefore $a = 1$ in $\widehat{\mcal{O}}_{X, x}$. Because this is true for all closed points we see that $a = 1$.\\ Step 2.\ Assume $X = \mbf{A}^{n} \times S$ and $f = \pi$. In this case there is a canonical isomorphism of complexes \[ \mcal{K}_{X / S}^{{\textstyle \cdot}} \cong \mrm{E} \Omega^{n}_{X / S}[n] \cong \mrm{E} \pi^{!} \mcal{O}_{S} \cong \pi^{\triangle} \mcal{K}_{S / S}^{{\textstyle \cdot}} \] (cf.\ \cite{RD} Thm.\ VI.3.1 and our Prop.\ \ref{prop6.4}), which we use to define $\zeta_{X} : \mcal{K}^{{\textstyle \cdot}}_{X / S} \rightarrow \pi^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{S / S}$. Consider $x \in X$, $Z := \overline{\{x\}}_{\mrm{red}}$, $s := \pi(x)$ and assume $x$ is closed in $\pi^{-1}(s)$. By replacing $S$ with a suitable open neighborhood of $s$ we can assume $Z \rightarrow S$ is finite. Then we are allowed to apply Lemma \ref{lem8.1} with $Y = S$, $\mcal{M} = \mcal{K}_{S / S}(s)$. It follows that (\ref{eqn8.4}) commutes on $\pi_{*} \mcal{K}_{X / S}(x) \subset \pi_{*} \mcal{K}^{{\textstyle \cdot}}_{X / S}$.\\ Step 3.\ Let $X$ be any finite type $S$-scheme. For every affine open subscheme $U \subset X$ we can find a closed immersion $h : U \rightarrow \mbf{A}^{n}_{S}$. Write $Y := \mbf{A}^{n}_{S}$ and let $\pi_{U}$ and $\pi_{Y}$ be the structural morphisms. Now $\operatorname{Tr}_{h}$ induces an isomorphism \[ \mcal{K}^{{\textstyle \cdot}}_{U / S} \cong \mcal{H}om_{Y}(\mcal{O}_{U}, \mcal{K}^{{\textstyle \cdot}}_{Y / S}) , \] and $\operatorname{Tr}^{\mrm{RD}}_{h}$ induces an isomorphism \[ \pi_{U}^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{S / S} \cong \mcal{H}om_{Y}(\mcal{O}_{U}, \pi_{Y}^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{S / S}) . \] So the isomorphism $\zeta_{Y}$ of Step 2 induces an isomorphism $\zeta_{U} : \mcal{K}^{{\textstyle \cdot}}_{U / S} \rightarrow \pi_{U}^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{S / S}$, which satisfies $\operatorname{Tr}_{\pi_{U}} = \operatorname{Tr}^{\mrm{RD}}_{\pi_{U}} \pi_{U *}(\zeta_{U})$. According to Step 1 the local isomorphisms $\zeta_{U}$ can be glued to a global isomorphism $\zeta_{X}$.\\ Step 4.\ Let $f : X \rightarrow Y$ be any $S$-morphism. To check (\ref{eqn8.4}) we may assume $X$ and $Y$ are affine, and in view of step 3 we may in fact assume $Y = \mbf{A}^{m} \times S$ and $X = \mbf{A}^{n} \times Y \cong \mbf{A}^{n + m} \times S$. Now apply Lemma \ref{lem8.1} with $x \in X$ closed in its fiber and $\mcal{M} := \mcal{K}_{Y / S}(y)$. \end{proof}

9510

alg-geom/9510018

en

https://arxiv.org/abs/alg-geom/9510018

alg-geom/9510018

Ezra Getzler

Ezra Getzler

Mixed Hodge structures of configuration spaces

AmSLaTeX 1.v2, 18 pages, revised version

Preprint 96-61, Max-Planck-Institut f. Mathematik, Bonn

The symmetric group S_n acts freely on the configuration space of n distinct points in a quasi-projective variety. In this paper, we study the induced action of the symmetric group S_n on the de Rham cohomology of this space, using mixed Hodge theory, combined with methods from the theory of symmetric functions. (We prove a motivic version of this as well.) As an application of our results, we calculate the S_n-equivariant Hodge polynomial of the Fulton-MacPherson compactification X[n] of the configuration space.

alg-geom

\subsection*{Acknowledgments} The author wishes to thanks the Department of Mathematics at the Universit\'e de Paris-VII for its hospitality during the inception of this paper. He is partially supported by a research grant of the NSF, a fellowship of the A.P. Sloan Foundation, and the Max-Planck-Institut f\"ur Mathematik in Bonn. \section{Symmetric functions and $\lambda$-rings} \subsection{Symmetric functions} In this section, we recall some results on symmetric functions and representations of $\SS_n$ which we need later. For the proofs of these results, we refer to Macdonald \cite{Macdonald}. The ring of symmetric functions is the inverse limit $$ \Lambda = \varprojlim \mathbb{Z}[x_1,\dots,x_k]^{\SS_k} . $$ It is is a polynomial ring in the complete symmetric functions $$ h_n = \sum_{i_1\le\dots\le i_n} x_{i_1}\dots x_{i_n} . $$ The power sums (also known as Newton polynomials) $$ p_n = \sum_i x_i^n $$ form a set of generators of the polynomial ring $\Lambda_\mathbb{Q}=\Lambda\o\mathbb{Q}$. This is shown by means of the elementary formula \begin{equation} \label{P-H} P_t = t \frac{d}{dt} \log H_t , \end{equation} where $$ H_t = \sum_{n=0}^\infty t^n h_n = \prod_i (1-tx_i)^{-1} \quad\text{and}\quad P_t = \sum_{n=0}^\infty t^n p_n = \sum_i (1-tx_i)^{-1} . $$ Written out explicitly, we obtain Newton's formula relating the two sets of generators: $$ nh_n = p_n + h_1p_{n-1} + \dots + h_{n-1}p_1 . $$ We may also invert \eqref{P-H}, obtaining the formula \begin{equation} \label{H-P} H_t = \exp \Bigl( \sum_{n=1}^\infty \frac{t^np_n}{n} \Bigr) . \end{equation} A partition $\lambda$ is a decreasing sequence $(\lambda_1\ge\dots\ge\lambda_\ell)$ of positive integers; we write $\lambda\vdash n$, where $n=\lambda_1+\dots+\lambda_\ell$, and denote the length of $\lambda$ by $\ell(\lambda)$. Identifying $\Lambda$ with the ring of characters of the Lie algebra $\gl_\infty = \varprojlim \gl_k$, we see that partitions correspond to dominant weights, and thus $\Lambda$ has a basis of consisting of the characters of the irreducible representations of $\gl_\infty$. These characters, given by the Weyl character formula $$ s_\lambda = \lim_{k\to\infty} \frac{\det(x_i^{\lambda_j+k-j})_{1\le i,j\le k}} {\det(x_i^{k-j})_{1\le i,j\le k}} , $$ are known as the Schur functions. In terms of the polynomial generators $h_n$, they are given by the Jacobi-Trudy formula $s_\lambda=\det\bigl(h_{\lambda_i-i+j}\bigr)_{1\le i,j\le\ell(\lambda)}$. There is a non-degenerate integral bilinear form on $\Lambda$, denoted $\<f,g\>$, for which the Schur functions $s_\lambda$ form an orthonormal basis. (This is sometimes called the Hall inner product.) The adjoint of multiplication by $f\in\Lambda$ with respect to this inner product is denoted $D(f)$. Written in terms of the power sums $p_n$, the operator $D(f)$ has the formula $D(p_n)=n\partial/\partial p_n$, while the inner product $\<f,g\>$ has the formula $$ \< f , g \> = D(f)g \Big|_{p_n=0,n\ge1} . $$ \subsection{Pre-$\lambda$-rings} A pre-$\lambda$-ring is a commutative ring $R$, together with a morphism of commutative rings $\sigma_t:R\to R\[t\]$ such that $\sigma_t(a)=1+ta+O(t^2)$. Expanding $\sigma_t$ in a power series $$ \sigma_t(a) = \sum_{n=0}^\infty t^n \sigma_n(a) , $$ we obtain endomorphisms $\sigma_n$ of $R$ such that $\sigma_0(a)=1$, $\sigma_1(a)=a$, and $$ \sigma_n(a+b) = \sum_{k=0}^n \sigma_{n-k}(a) \sigma_k(b) . $$ There are also operations $\lambda_k(a)=(-1)^k\sigma_k(-a)$, with generating function \begin{equation} \label{invert} \lambda_t(a) = \sum_{n=0}^\infty t^n \lambda_n(a) = \sigma_{-t}(a)^{-1} . \end{equation} The $\lambda$-operations are polynomials in the $\sigma$-operations with integral coefficients, and vice versa. In this paper, we take the $\sigma$-operations to be more fundamental; nevertheless, following custom, the structure they define is called a pre-$\lambda$-ring. Given a pre-$\lambda$-ring $R$, there is a bilinear map $\Lambda\o R\to R$, which we denote $f\circ a$, defined by the formula $$ (h_{n_1}\dots h_{n_k})\circ a = \sigma_{n_1}(a)\dots\sigma_{n_k}(a) . $$ The image of the power sum $p_n$ under this map is the operation on $R$ known as the Adams operation $\psi_n$. We denote the operation corresponding to the Schur function $s_\lambda$ by $\sigma_\lambda$. Note that \eqref{H-P} implies the relation $$ \sigma_t(a) = \exp \Bigl( \sum_{n=1}^\infty \frac{t^n\psi_n(a)}{n} \Bigr) . $$ The following formula (I.4.2 of \cite{Macdonald}) is known as Cauchy's formula: \begin{equation} \label{Cauchy} H_t(...,x_iy_j,...) = \prod_{i,j} (1-tx_iy_j)^{-1} = \sum_{\lambda\vdash n} s_\lambda(x) \o s_\lambda(y) = \exp\Bigl( \sum_{k=1}^\infty \frac{p_k(x) \o p_k(y)}{k} \Bigr) . \end{equation} From it, the following result is immediate. \begin{proposition} If $R$ and $S$ are pre-$\lambda$-rings, their tensor product $R\o S$ is a pre-$\lambda$-ring, with $\sigma$-operations $$ \sigma_n(a\o b) = \sum_{\lambda\vdash n} \sigma_\lambda(a) \o \sigma_\lambda(b) , $$ and Adams operations $\psi_n(a\o b) = \psi_n(a) \o \psi_n(b)$. \end{proposition} For example, $\sigma_2(a\o b) = \sigma_2(a)\o\sigma_2(b) + \lambda_2(a)\o\lambda_2(b)$. \subsection{$\lambda$-rings} The polynomial ring $\mathbb{Z}[x]$ is a pre-$\lambda$-ring, with $\sigma$-operations characterized by the formula $\sigma_n(x^i)=x^{ni}$. Taking tensor powers of this pre-$\lambda$-ring with itself, we see that the polynomial ring $\mathbb{Z}[x_1,\dots,x_k]$ is a pre-$\lambda$-ring. The $\lambda$-operations on this ring are equivariant with respect to the permutation action of the symmetric group $\SS_k$ on the generators, hence the ring of symmetric functions $\mathbb{Z}[x_1,\dots,x_k]^{\SS_k}$ is a pre-$\lambda$-ring. Taking the limit $k\to\infty$, we obtain a pre-$\lambda$-ring structure on $\Lambda$. \begin{definition} A $\lambda$-ring is pre-$\lambda$-ring such that if $f,g\in\Lambda$ and $x\in R$, \begin{equation} \label{lambda-ring} f\circ(g\circ x)=(f\circ g)\circ x . \end{equation} \end{definition} By definition, the pre-$\lambda$-ring $\Lambda$ is a $\lambda$-ring; in particular, the operation $f\circ g$, called plethysm, is associative. The following result (see Knutson, \cite{Knutson}) is the chief result in the theory of $\lambda$-rings. \begin{theorem} \label{universal} $\Lambda$ is the universal $\lambda$-ring on a single generator $h_1$. \end{theorem} This theorem makes it straighforward to verify identities in $\lambda$-rings: it suffices to verify them in $\Lambda$. As an application, we have the following corollary. \begin{corollary} The tensor product of two $\lambda$-rings is a $\lambda$-ring. \end{corollary} \begin{proof} We need only verify this for $R=\Lambda$. A torsion-free pre-$\lambda$-ring whose Adams operations are ring homomorphisms which satisfy $\psi_m(\psi_n(a))=\psi_{mn}(a)$ is a $\lambda$-ring. It is easy to verify these conditions for $\Lambda\o\Lambda$, since $\psi_n(a\o b) = \psi_n(a) \o \psi_n(b)$. \end{proof} In the definition of a $\lambda$-ring, it is usual to adjoin the axiom $$ \sigma_n(xy) = \sum_{\lambda\vdash n} \sigma_\lambda(a) \o \sigma_\lambda(y) . $$ However, this formula follows from our definition of a $\lambda$-ring: by universality, it suffices to check it for $R=\Lambda\o\Lambda$, $x=h_1\o1$ and $y=1\o h_1$, for which it is evident. \section{Complete $\lambda$-rings} A filtered $\lambda$-ring $R$ is a $\lambda$-ring with decreasing filtration $$ R = F^0R \supset F^1R \supset \dots , $$ such that \begin{enumerate} \item $\bigcap_k F^kR = 0$ (the filtration is discrete); \item $F^mR\*F^nR\subset F^{m+n}R$ (the filtration is compatible with the product); \item $\sigma_m(F^nR)\subset F^{mn}R$ (the filtration is compatible with the $\lambda$-ring structure). \end{enumerate} The completion of a filtered $\lambda$-ring is again a $\lambda$-ring; define a complete $\lambda$-ring to be a $\lambda$-ring equal to its completion. For example, the universal $\lambda$-ring $\Lambda$ is filtered by the subspaces $F^n\Lambda$ of polynomials vanishing to order $n-1$, and its completion is the $\lambda$-ring of symmetric power series, whose underlying ring is the power series ring $\mathbb{Z}\[h_1,h_2,h_3,\dots\]$. The tensor product of two filtered $\lambda$-rings is again a filtered $\lambda$-ring, when furnished with the filtration $$ F^n(R\o S) = \sum_{k=0}^n F^{n-k}R \o F^kS . $$ If $R$ and $S$ are filtered $\lambda$-rings, denote by $R\Hat{\otimes} S$ the completion of $R\o S$. Let $\mathcal{R}$ be a Karoubian rring\xspace over a field of characteristic zero, and consider the complete $\lambda$-ring $\Lambda\Hat{\otimes} K_0(\mathcal{R})$, where $K_0(\mathcal{R})$ has the discrete filtration. This $\lambda$-ring has a natural realization, as the Grothendieck group of the Karoubian rring\xspace $$ \hom{\mathcal{R}} = \prod_{n=0}^\infty [\SS_n,\mathcal{R}] , $$ whose objects are the $\SS$-modules in $\mathcal{R}$. In this rring\xspace, the sum and product are given by the same formulas as in the rring\xspace $[\SS,\mathcal{R}]$ of bounded $\SS$-modules. Without assuming the existence of infinite sums in $\mathcal{R}$, plethysm does not extend to a monoidal structure on $\hom{\mathcal{R}}$. However, $\mathcal{X}\circ\mathcal{Y}$ is well-defined in $\[\SS,\mathcal{R}\]$ under either of the following two hypotheses: $$ \text{i) $\mathcal{X}$ is bounded, or ii) $\mathcal{Y}(0)=0$.} $$ The first of these situations allows us to construct a $\lambda$-ring structure on the Grothendieck group of $\hom{\mathcal{R}}$, by the same method as for $[\SS,\mathcal{R}]$, while the second will be needed in the proof of our main theorem. Introducing the notation $\hom[k]{\mathcal{R}}$ for the subcategory of $\hom{\mathcal{R}}$ consisting of $\SS$-modules $\mathcal{X}$ such that $\mathcal{X}(n)=0$ for $n<k$, we see that plethysm extends to a symmetric monoidal structure on $\hom[1]{\mathcal{R}}$. Denote the Grothendieck group of the full subcategory $\hom[1]{\mathcal{R}}\subset\hom{\mathcal{R}}$ by $\Check{K}^\SS_0(\mathcal{R})$. Since $\Check{K}^\SS_0(\mathcal{R})$ is a (non-unital) $\lambda$-ring, we may define a bilinear operation $$ \circ : \Hat{K}^\SS_0({\mathsf{Proj}}) \o \Check{K}^\SS_0(\mathcal{R}) \to \check{K}_\SS(\mathcal{R}) , $$ satisfying \eqref{lambda-ring}. This operation may be extended to a bilinear operation (which we denote by the same symbol), $$ \circ : \Hat{K}^\SS_0(\mathcal{R}) \o \Check{K}^\SS_0(\mathcal{R}) \to \check{K}_\SS(\mathcal{R}) , $$ using the Peter-Weyl Theorem: to define $x\circ y$, we expand $x$ in a series $x=\sum_\lambda x_\lambda\*s_\lambda$, and define $$ x\circ y = \sum_\lambda x_\lambda \* \sigma_\lambda(y) . $$ The interest of this operation lies in the following lemma, which is a simple consequence of the definition of $x\circ y$. \begin{lemma} \label{plethysm} If $\mathcal{X}$ and $\mathcal{Y}$ are objects of $\hom{\mathcal{R}}$ and $\hom[1]{\mathcal{R}}$ respectively, $[\mathcal{X}\circ\mathcal{Y}]=[\mathcal{X}]\circ[\mathcal{Y}]$. \end{lemma} If $R$ is a complete $\lambda$-ring, the operation $$ \Exp(a) = \sum_{n=0}^\infty \sigma_n(a) : R \to 1+F_1R $$ is an analogue of exponentiation, whose logarithm is given by a formula of Cadogan \cite{Cadogan}. \begin{proposition} \label{Cadogan} On a complete filtered $\lambda$-ring $R$, the operation $\Exp:R\to1+F_1R$ has inverse $$ \Log(1+a) = \sum_{n=1}^\infty \frac{\mu(n)}{n} \log(1+\psi_n(a)) . $$ \end{proposition} \begin{proof} Expanding $\Log(1+a)$, we obtain $$ \Log(1+a) = - \sum_{n=1}^\infty \frac{1}{n} \sum_{d|n} \mu(d) \psi_d(-a)^{n/d} = \sum_{n=1}^\infty \Log_n(a) . $$ Let $\chi_n$ be the character of the cyclic group $C_n$ equalling $e^{2\pi i/n}$ on the generator of $C_n$. The characteristic of the $\SS_n$-module $\Ind^{\SS_n}_{C_n}\chi_n$ equals $$ \frac{1}{n} \sum_{k=0}^{n-1} e^{2\pi ik/n} p_{(k,n)}^{n/(k,n)} = \frac{1}{n} \sum_{d|n} \mu(d) p_d^{n/d} , $$ while the characteristic of the $\SS_n$-module $\Ind^{\SS_n}_{C_n}\chi_n\o\varepsilon_n$, where $\varepsilon_n$ is the sign representation of $\SS_n$, equals $$ \frac{1}{n} \sum_{d|n} \mu(d) \bigl( (-1)^{d-1}p_d \bigr)^{n/d} = \frac{(-1)^n}{n} \sum_{d|n} \mu(d) (-p_d)^{n/d} . $$ It follows that $(-1)^{n-1}\Log_n$ is the operation associated to the $\SS_n$-module $\Ind^{\SS_n}_{C_n}\chi_n\o\varepsilon_n$, and hence defines a map from $F_1R$ to $F_nR$. To prove that $\Log$ is the inverse of $\Exp$, it suffices to check this for $R=\Lambda$ and $x=h_1$. We must prove that $$ \Exp\left( \sum_{n=1}^\infty \frac{\mu(n)}{n} \log(1+p_n) \right) = 1+h_1 . $$ The logarithm of the expression on the left-hand side equals $$ \exp \Bigl( \sum_{k=1}^\infty \frac{p_k}{k} \Bigr) \circ \Bigl( \sum_{n=1}^\infty \frac{\mu(n)}{n} \log(1+p_n) \Bigr) = \sum_{n=1}^\infty \sum_{d|n} \mu(d) \frac{\log(1+p_n)}{n} = \log(1+p_1) , $$ and the formula follows. \end{proof} \begin{example} \label{Log(t)} If $a\in F^1R$ is a line bundle in the complete $\lambda$-ring $R$ (that is, $\sigma_n(a)=a^n$ for all $n\ge0$), we see that $$ \Exp(a) = \frac{1}{1-a} . $$ In particular, this shows that $\Exp(t^n)=(1-t^n)^{-1}$, and that $$ \Exp(t-t^2) = \frac{\Exp(t)}{\Exp(t^2)} = \frac{1-t^2}{1-t} = 1 + t . $$ It follows that $\Log(1-t)=t$ and that $\Log(1+t)=t-t^2$. \end{example} We now introduce the operations on $\lambda$-rings which will arise in the calculation of the Serre polynomials of the local systems $\mathsf{F}(X,n)\times_{\SS_n}V_\lambda$. We start by considering the case $X=\mathbb{C}$. \begin{proposition} $$ \sum_\lambda s_\lambda \o \Serre(\mathsf{F}(\mathbb{C},n),V_\lambda) = \prod_{k=1}^\infty (1+p_k)^{\frac{1}{k}\sum_{d|k}\mu(k/d)\mathsf{L}^d} \in \Lambda \Hat{\otimes} \mathbb{Z}[\mathsf{L}] $$ \end{proposition} \begin{proof} It is proved in Lehrer-Solomon \cite{LS} that \begin{equation} \label{Lehrer-Solomon} \sum_{n=0}^\infty \sum_{i=0}^\infty (-x)^i \ch_n(H^i(\mathsf{F}(\mathbb{C},n),\mathbb{C})) = \prod_{k=1}^\infty (1+x^kp_k)^{\frac{1}{k}\sum_{d|k}\mu(k/d)x^{-d}} , \end{equation} where $H^i(\mathsf{F}(\mathbb{C},n),\mathbb{C})$ is the $\SS_n$-module associated to the de Rham cohomology of degree $i$. By Poincar\'e duality, we see that $$ \sum_{n=0}^\infty \sum_{i=0}^\infty (-x)^i \ch_n(H_c^i(\mathsf{F}(\mathbb{C},n),\mathbb{C})) = \prod_{k=1}^\infty (1+p_k)^{\frac{1}{k}\sum_{d|k}\mu(k/d)x^d} . $$ But the mixed Hodge structure on the cohomology group $H^i_c(\mathsf{F}(\mathbb{C},n),\mathbb{C})$ is pure of weight $2i$, and indeed $H^i_c(\mathsf{F}(\mathbb{C},n),\mathbb{C})=H^i_c(\mathsf{F}(\mathbb{C},n),\mathbb{C})^{i,i}$, proving the result. \end{proof} Motivated by this proposition, we define operations $\Phi_\lambda$ in a $\lambda$-ring $R$, parametrized by partitions $\lambda$, by means of the generating function \begin{equation} \label{Phi} \Phi(x) \equiv \sum_\lambda s_\lambda\o\Phi_\lambda(x) = \prod_{k=1}^\infty (1+p_k)^{\frac{1}{k}\sum_{d|k}\mu(k/d)\psi_d(x)} \in \Lambda \Hat{\otimes} R . \end{equation} \begin{theorem} \label{explicit} We have the formula $\Phi(x) = \Exp(\Log(1+p_1)x)$. In particular, the operations $\Phi_\lambda$ are defined on any $\lambda$-ring. \end{theorem} \begin{proof} Applying $\Log$ to the definition of $\Phi(x)$, we obtain \begin{align*} \Log(\Phi(x)) &= \sum_{n=1}^\infty \frac{\mu(n)}{n} \psi_n \log(\Phi(x)) \\ &= \sum_{n=1}^\infty \frac{\mu(n)}{n} \psi_n \sum_{k=1}^\infty \frac{1}{k}\sum_{d|k}\mu(k/d) \log(1+p_k) \psi_d(x) \\ &= \sum_{n,d,e=1}^\infty \frac{\mu(n)\mu(e)}{nde} \psi_{nd}(x) \log(1+p_{nde}) \\ &= \sum_{e=1}^\infty \frac{\mu(e)}{e} \log(1+p_e) x , \end{align*} by M\"obius inversion. On applying $\Exp$, we obtain the desired formula. \end{proof} Using this theorem, we can prove more explicit formulas for $\Phi_n$ and $\Phi_{1^n}$. \begin{corollary} \label{braid} $$ \sum_{n=0}^\infty t^n \Phi_n(y) = \frac{\sigma_t(y)}{\sigma_{t^2}(y)} \text{ and } \Phi_{1^n}(y) = \lambda_n(y) $$ \end{corollary} \begin{proof} We obtain $\sum_{n=0}^\infty t^n \Phi_n(y)$ from $\Phi(x)$ by replacing $p_n$ by $t^n$. By Theorem \ref{explicit}, it follows that $$ \sum_{n=0}^\infty t^n \Phi_n(y) = \Exp(\Log(1+t)x) = \Exp((t-t^2)x) = \frac{\sigma_t(x)}{\sigma_{t^2}(x)} , $$ since $\Log(1+t)=t-t^2$ by Example \ref{Log(t)}. The proof of the second formula is similar, except that we replace $p_n$ by $(-t)^n$, and apply the formula $\Log(1-t)=-t$. \end{proof} \section{Representations of finite groups in Karoubian rrings\xspace} Let $(\mathcal{R},\otimes,{1\!\!1})$ be a symmetric monoidal category with coproducts, denoted $X\oplus Y$. We say that $\mathcal{R}$ is a \textbf{rring\xspace} (this is our abbreviation for the usual term \emph{ring category}) if there are natural isomorphisms $$ (X\oplus Y)\o Z \cong (X\o Z)\oplus(Y\o Z) \quad\text{and}\quad X\o 0 \cong 0 $$ which describe the distributivity of the tensor product over the sum, satisfying the coherence axioms of Laplaza \cite{Laplaza}. If $\o$ is the categorical product, we say that $\mathcal{R}$ is a Cartesian rring\xspace. The Grothendieck group $K_0(-)$ is a functor from rrings\xspace to commutative rings. Given an object $X$ of a rring\xspace $\mathcal{R}$, denote by $[X]$ its isomorphism class; then $K_0(\mathcal{R})$ is generated as an abelian group by the isomorphism classes of objects, with the relation $$ [X] + [Y] = [X\oplus Y] . $$ The product on $K_0(\mathcal{R})$ is given by the formula $[X]\*[Y]=[X\o Y]$. (Here, we suppose that the isomorphism classes of objects of $\mathcal{R}$ form a set; this hypothesis will always be fulfilled in this paper.) If $\mathcal{R}$ and $\mathcal{S}$ are two rrings\xspace, $\mathcal{R}\o\mathcal{S}$ is a rring\xspace whose objects are formal sums of tensor products $X\o Y$, where $X$ and $Y$ are objects of $\mathcal{R}$ and $\mathcal{S}$ respectively; note that $K_0(\mathcal{R}\o\mathcal{S})\cong K_0(\mathcal{R})\o K_0(\mathcal{S})$. Recall that an additive category over a commutative ring $R$ is a category $\mathcal{R}$ such that the set of morphisms $\mathcal{R}(X,Y)$ is a $R$-module for all objects $X$ and $Y$, the composition maps $\mathcal{R}(Y,Z)\o_K\mathcal{R}(X,Y)\to\mathcal{R}(X,Z)$ are $R$-linear, and every finite set of objects has a direct sum. A \textbf{Karoubian category} over a ring $R$ is an additive category over $R$ such that every idempotent has an image, denoted $\im(p)$. (Karoubian categories are also sometimes known as pseudo-abelian categories.) \begin{definition} A \textbf{Karoubian rring\xspace} $\mathcal{R}$ is a rring\xspace which is a Karoubian category, and whose sum $X\oplus Y$ is the direct sum. \end{definition} An example of a Karoubian rring\xspace is the category ${\mathsf{Proj}}$ of finitely generated projective $R$-modules. If $\mathcal{R}$ is a Karoubian rring\xspace and $G$ is a group, let $[G,\mathcal{R}]$ be the Karoubian rring\xspace of $G$-modules in $\mathcal{R}$, that is, functors from $G$ to $\mathcal{R}$. If $X$ and $Y$ are objects of $[G,\mathcal{R}]$, the $R$-module of morphisms $\mathcal{R}(X,Y)$ carries a natural $R[G]$-module structure, given by the formula $f^g=g^{-1}\*f\*g$. There is a natural bifunctor $V\boxtimes X$, the external tensor product, from $[G,{\mathsf{Proj}}]\times[G,\mathcal{R}]$ to $[G,\mathcal{R}]$, characterized by the identity of $R[G]$-modules $$ \mathcal{R}(V\boxtimes X,Y) \cong V \o \mathcal{R}(X,Y) . $$ For the finitely generated free module $R[G]^n$, we have $$ R[G]^n\boxtimes X = \bigoplus_{g\in G} X^{\oplus n} . $$ For general $V$, we realize $V$ as the image of an idempotent $p$ in a free module $R[G]^n$, and define $V\boxtimes X$ to be the image of the corresponding idempotent in $R[G]^n\boxtimes X$. Using the external tensor product, we may embed $[G,{\mathsf{Proj}}]$ into $[G,\mathcal{R}]$ by the functor $V\DOTSB\mapstochar\to V\boxtimes{1\!\!1}$. If $G$ is a group whose order is invertible in $R$, the functor $(-)^G$ of $G$-invariants from $[G,\mathcal{R}]$ to $\mathcal{R}$ is defined by taking the image of the idempotent automorphism of $\mathcal{R}$ $$ p = \frac{1}{|G|} \sum_{g\in G} g . $$ From now on, we restrict attention to groups satisfying this condition. If $H$ is a subgroup of $G$, the induction functor $\Ind^G_H:[H,\mathcal{R}]\to[G,\mathcal{R}]$ is defined by the formula $$ \Ind^G_H X = (R[G]\boxtimes X)^H . $$ Here, we use the $G\times H$-module structure of $R[G]$, where $G$ acts on the left and $H$ acts on the right. The following is a generalization of the Peter-Weyl theorem to Karoubian categories. \begin{theorem}[Peter-Weyl] \label{Peter-Weyl} If $\mathcal{R}$ is a Karoubian rring\xspace over a commutative ring $R$ and $G$ is a group whose order is invertible in $R$, the composition $$ [G,{\mathsf{Proj}}] \o \mathcal{R} \hookrightarrow [G,{\mathsf{Proj}}] \o [G,\mathcal{R}] \xrightarrow{\boxtimes} [G,\mathcal{R}] $$ is an equivalence of categories. \end{theorem} \begin{proof} Since the order of $G$ is invertible in $R$, the group algebra $R[G]$ is semi-simple, and may be written $$ R[G] \cong \bigoplus_a \End(V_a) \cong \bigoplus_a V_a \o V_a^* , $$ where we sum over the isomorphism classes of irreducible representations $\{V_a\}$ of $G$. This permits us to rewrite the induction functor as $$ \Ind_H^GX = (R[G]\o X)^H \cong \bigoplus_a V_a \o (V_a^*\boxtimes X)^H . $$ Taking $H=G$, and recalling that $\Ind_G^G$ is equivalent to the identity functor, we obtain the desired equivalence between $[G,{\mathsf{Proj}}]\o\mathcal{R}$ and $[G,\mathcal{R}]$. \end{proof} \section{$\SS$-modules in Karoubian rrings\xspace} Let $\SS$ be the category of permutations $\coprod_{n=0}^\infty\SS_n$ and let $\mathcal{R}$ be a rring\xspace. A bounded $\SS$-module in $\mathcal{R}$ is an object $\mathcal{X}$ of $$ [\SS,\mathcal{R}] = \bigoplus_{n=0}^\infty \, [\SS_n,\mathcal{R}] , $$ in other words, a sequence $\{\mathcal{X}(n)\mid n\ge0\}$ of $\SS_n$-modules in $\mathcal{R}$ such that $\mathcal{X}(n)=0$ for $n\gg0$. Let ${1\!\!1}_n$ denote the $\SS$-module such that ${1\!\!1}_n(n)$ is the trivial $\SS_n$-module, while ${1\!\!1}_n(k)=0$ for $k\ne n$. The category $[\SS,\mathcal{R}]$ is itself a rring\xspace: \begin{enumerate} \item the sum of two $\SS$-modules is $(\mathcal{X}\oplus\mathcal{Y})(n)=\mathcal{X}(n)\oplus\mathcal{Y}(n)$; \item the product of two $\SS$-modules is defined using induction: $$ (\mathcal{X}\o\mathcal{Y})(n) = \bigoplus_{j+k=n} \Ind_{\SS_j\times\SS_k}^{\SS_n} \mathcal{X}\o\mathcal{Y} ; $$ \item the unit of the product is ${1\!\!1}_0$. \end{enumerate} Denote the Grothendieck group of the rring\xspace $[\SS,\mathcal{R}]$ by $K_0^\SS(\mathcal{R})$. There is another monoidal structure $\mathcal{X}\circ\mathcal{Y}$ on $[\SS,\mathcal{R}]$, called plethysm. If $\lambda$ is a partition of $n$, let $\SS_\lambda = \SS_{\lambda_1}\times\dots\times\SS_{\lambda_{\ell(\lambda)}}\subset\SS_n$, and let $N(\SS_\lambda)$ be the normalizer of $\SS_\lambda$ in $\SS_n$. The quotient $W(\SS_\lambda)=N(\SS_\lambda)/\SS_\lambda$ may be identified with $$ \{ \sigma\in\SS_{\ell(\lambda)} \mid \text{$\lambda_{\sigma(i)}=\lambda_i$ for $1\le i\le\ell(\lambda)$} \} \subset \SS_{\ell(\lambda)} . $$ Given bounded $\SS$-modules $\mathcal{X}$ and $\mathcal{Y}$, we obtain an action of $N(\SS_\lambda)$ on the tensor product $$ \mathcal{X}(\ell(\lambda)) \o \bigotimes_{1\le i\le\ell(\lambda)} \mathcal{Y}(\lambda_i) . $$ \textbf{Plethysm} is the monoidal structure (not symmetric) defined by the formula $$ (\mathcal{X}\circ\mathcal{Y})(n) = \bigoplus_{\lambda\vdash n} \bigoplus_{k=0}^\infty \Ind^{\SS_n}_{N(\SS_\lambda)} \biggl( \mathcal{X}(\ell(\lambda)+k) \o \bigotimes_{1\le i\le\ell(\lambda)} \mathcal{Y}(\lambda_i) \o \mathcal{Y}(0)^{\o k} \biggr)^{\SS_k} , $$ and with unit ${1\!\!1}_1$. \begin{lemma} Let $\mathcal{R}$ be a Karoubian rring\xspace over a field of characteristic zero. The Grothendieck group $K_0^\SS(\mathcal{R})$ is a pre-$\lambda$-ring, with $\sigma$-operations characterized by the formula $$ \sigma_n([\mathcal{X}]) = \bigl[{1\!\!1}_n\circ\mathcal{X}\bigr] , $$ where $\mathcal{X}$ is a bounded $\SS$-module. \end{lemma} \begin{proof} We must prove that for bounded $\SS$-modules $\mathcal{X}$ and $\mathcal{Y}$, \begin{equation} \label{pre} \sigma_n([\mathcal{X}]+[\mathcal{Y}]) = \sum_{i=0}^n \sigma_i([\mathcal{X}]) \* \sigma_{n-i}([\mathcal{Y}]) . \end{equation} Observe that ${1\!\!1}_n\circ(\mathcal{X}\oplus\mathcal{Y})$ equals $$ \bigoplus_{i=0}^n \bigoplus_{\lambda\vdash i} \bigoplus_{\mu\vdash n-i} \bigoplus_{j,k=0}^\infty \Ind^{\SS_n}_{N(\SS_\lambda)\times N(\SS_\mu)} \biggl( \bigotimes_{1\le i\le\ell(\lambda)} \mathcal{X}(\lambda_i) \o \mathcal{X}(0)^{\o j} \o \bigotimes_{1\le i\le\ell(\mu)} \mathcal{Y}(\mu_i) \o \mathcal{Y}(0)^{\o k} \biggr)^{\SS_j\times\SS_k} $$ Since $$ \Ind^{\SS_n}_{N(\SS_\lambda)\times N(\SS_\mu)}V\o W = \Ind^{\SS_n}_{\SS_i\times\SS_{n-i}} \Bigl( \Ind^{\SS_i}_{N(\SS_\lambda)} V \o \Ind^{\SS_{n-i}}_{N(\SS_\mu)} W \Bigr) , $$ it follows that $$ {1\!\!1}_n\circ(\mathcal{X}\oplus\mathcal{Y}) \cong \bigoplus_{i=0}^n ({1\!\!1}_i\circ\mathcal{X})\o({1\!\!1}_{n-i}\circ\mathcal{Y}) , $$ proving \eqref{pre} for elements of $K_0^\SS(\mathcal{R})$ of the form $[\mathcal{X}]$ and $[\mathcal{Y}]$. The definition of the sigma operations on virtual elements $[\mathcal{X}_0]-[\mathcal{X}_1]$ is now forced by \eqref{invert}: $$ \sigma_n([\mathcal{X}_0]-[\mathcal{X}_1]) = \sum_{k=0}^\infty \sum_{\substack{j_\ell>0 \\ i+j_1+\dots+j_\ell=n}} (-1)^k \sigma_i([\mathcal{X}_0]) \sigma_{j_1}([\mathcal{X}_1]) \dots \sigma_{j_k}([\mathcal{X}_1]) . $$ \def{} \end{proof} \begin{lemma} \label{atiyah} There is an isomorphism of $\lambda$-rings $K_0^\SS({\mathsf{Proj}})\cong\Lambda$. \end{lemma} \begin{proof} The pre-$\lambda$-ring $K_0^\SS({\mathsf{Proj}})$ is the sum of abelian groups $ K_0^\SS(\mathcal{R}) = \bigoplus_{n=0}^\infty R(\SS_n) , $ where $R(\SS_n)=K_0([\SS_n,{\mathsf{Proj}})$ is the abelian group underlying the virtual representation ring of $\SS_n$. The identification of $K_0^\SS({\mathsf{Proj}})$ with $\Lambda$ is via the Frobenius characteristic $\ch:R(\SS)\to\Lambda$, which sends the irreducible representation $V_\lambda$ associated to the partition $\lambda$ to the Schur function $s_\lambda$. The Frobenius characteristic is given by the explicit formula $$ \ch_n(V) = \frac{1}{n!} \sum_{\sigma\in\SS_n} \Tr_V(\sigma) p_\sigma , $$ where $p_\sigma$ is the monomial in the power sums obtained by taking one factor $p_k$ for each cycle of $\sigma$ of length $k$. For the proof that $\ch(\dots)$ is a map of $\lambda$-rings, see Knutson \cite{Knutson} or Appendix~A of Macdonald \cite{Macdonald}. \end{proof} Using these lemmas and the Peter-Weyl Theorem, we will show that $K_0^\SS(\mathcal{R})$ is a $\lambda$-ring for any Karoubian rring\xspace over a field of characteristic zero. First, we prove some simple lemmas which are of interest in their own right. Plethysm is distributive on the left with respect to sum. \begin{lemma} \label{additive} $ (\mathcal{X}_1\oplus\mathcal{X}_2)\circ\mathcal{Y} \cong (\mathcal{X}_1\circ\mathcal{Y}) \oplus (\mathcal{X}_2\circ\mathcal{Y}) $ \end{lemma} \begin{proof} Clear. \end{proof} It is also distributive on the left with respect to product. \begin{lemma} \label{multiplicative} $ (\mathcal{X}_1\o\mathcal{X}_2)\circ\mathcal{Y} \cong (\mathcal{X}_1\circ\mathcal{Y}) \o (\mathcal{X}_2\circ\mathcal{Y}) $ \end{lemma} \begin{proof} By Lemma \ref{additive}, it suffices to check this formula when $\mathcal{X}_1(j)=X_1$, $\mathcal{X}_2(k)=X_2$, $\mathcal{X}_1(i)=0$, $i\ne j$ and $\mathcal{X}_1(i)=0$, $i\ne k$. We have \begin{multline*} \bigl((\mathcal{X}_1\o\mathcal{X}_2) \circ \mathcal{Y}\bigr)(n) \\ = \bigoplus_{q=0}^n \bigoplus_{\substack{\lambda\vdash n \\ \ell(\lambda)+q=j+k}} \Ind^{\SS_n}_{N(\SS_\lambda)} \biggl( \Ind^{\SS_{j+k}}_{\SS_j\times\SS_k} \bigl( X_1\o X_2 \bigr) \o \bigotimes_{1\le i\le\ell(\lambda)} \mathcal{Y}(\lambda_i) \o \mathcal{Y}(0)^{\o q} \biggr)^{\SS_q} . \end{multline*} But we have \begin{multline*} \bigoplus_{\substack{\lambda\vdash n \\ \ell(\lambda)+q=j+k}} \biggl( \Ind^{\SS_{j+k}}_{\SS_j\times\SS_k} \bigl( X_1\o X_2 \bigr) \o \bigotimes_{1\le i\le\ell(\lambda)} \mathcal{Y}(\lambda_i) \o \mathcal{Y}(0)^{\o q} \biggr)^{\SS_q} \cong \bigoplus_{p=0}^q \bigoplus_{i=0}^n \\ \bigoplus_{\substack{\mu\vdash i \\ \ell(\mu)+p=j}} \biggl( X_1 \o \bigotimes_{1\le i\le\ell(\mu)} \mathcal{Y}(\lambda_i) \o \mathcal{Y}(0)^{\o p} \biggr)^{\SS_q} \o \bigoplus_{\substack{\lambda\vdash n-i \\ \ell(\lambda)+q-p=k}} \biggl( X_2 \o \bigotimes_{1\le i\le\ell(\lambda)} \mathcal{Y}(\lambda_i) \o \mathcal{Y}(0)^{\o q} \biggr)^{\SS_q} , \end{multline*} from which the lemma follows. \end{proof} \begin{lemma} \label{associative} If $\mathcal{V}$ is a bounded $\SS$-module in ${\mathsf{Proj}}$ and $\mathcal{X}$ is a bounded $\SS$-module in $\mathcal{R}$, $$ \ch(\mathcal{V}) \circ [\mathcal{X}] = [\mathcal{V}\circ\mathcal{X}] . $$ \end{lemma} \begin{proof} By Lemma \ref{additive}, we may assume that $\mathcal{V}$ is an irreducible $\SS_n$-module $V_\lambda$. It remains to show that $\sigma_\lambda([\mathcal{X}])=[V_\lambda\circ\mathcal{X}]$ for all partitions $\lambda$. By Lemma \ref{multiplicative}, we see that for any partition $\mu$ with $\ell=\ell(\mu)$, we have $$ \bigl( {1\!\!1}_{\mu_1} \o \dots \o {1\!\!1}_{\mu_\ell} \bigr) \circ \mathcal{X} \cong \bigl( {1\!\!1}_{\mu_1} \o \mathcal{X} \bigr) \o \dots \o \bigl( {1\!\!1}_{\mu_\ell} \circ \mathcal{X} \bigr) . $$ Taking the class in $K_0^\SS(\mathcal{R})$ of both sides, we see that $$ \bigl[ \bigl( {1\!\!1}_{\mu_1} \o \dots \o {1\!\!1}_{\mu_\ell} \bigr) \circ \mathcal{X} \bigr] = \sigma_{\mu_1}([\mathcal{X}]) \dots \sigma_{\mu_\ell}([\mathcal{X}]) . $$ The irreducible representation $V_\lambda$ is a linear combination of representations ${1\!\!1}_{\mu_1}\o\dots\o{1\!\!1}_{\mu_\ell}$ with integral coefficients, and by Lemma \ref{atiyah}, the Schur function $s_\lambda$ is a linear combination of symmetric functions $h_{\mu_1}\o\dots\o h_{\mu_\ell}$ with the same coefficients; the proof is completed by application of Lemma \ref{additive}. \end{proof} \begin{theorem} The Grothendieck group $K_0^\SS(\mathcal{R})$ of a Karoubian rring\xspace $\mathcal{R}$ over a field of characteristic zero is a $\lambda$-ring. \end{theorem} \begin{proof} If $f=\ch(\mathcal{V})$ and $g=\ch(\mathcal{W})$, where $\mathcal{V}$ and $\mathcal{W}$ are bounded $\SS$-modules in ${\mathsf{Proj}}$, and $x=[\mathcal{X}]$, where $\mathcal{X}$ is a bounded $\SS$-module in $\mathcal{R}$, it follows from Lemma \ref{associative} that $$ f \circ \bigl( g \circ x \bigr) = \ch\bigl( \mathcal{V} \circ (\mathcal{W}\circ\mathcal{X}) \bigr) = \ch\bigl( (\mathcal{V}\circ\mathcal{W}) \circ \mathcal{X} \bigr) = \ch(\mathcal{V}\circ\mathcal{W}) \circ x . $$ Since $\ch$ is a morphism of $\lambda$-rings, we see that $\ch(\mathcal{V}\circ\mathcal{W})=f\circ g$, and from which we obtain the formula \eqref{lambda-ring} characterizing $\lambda$-rings in this case: $$ f\circ(g\circ x) = (f\circ g) \circ x . $$ It only remains to extend \eqref{lambda-ring} to virtual elements $g=\ch(\mathcal{W}_0)-\ch(\mathcal{W}_1)$ and $x=[\mathcal{X}_0]-[\mathcal{X}_1]$. Both sides of \eqref{lambda-ring} are polynomial functions of $g\in\Lambda$ and $x\in K_0^\SS(\mathcal{R})$ and hence must coincide, since they are equal on a cone with non-empty interior. \end{proof} It follows that the Grothendieck group $K_0(\mathcal{R})$ is a $\lambda$-ring, namely the sub-$\lambda$-ring of $K_0^\SS(\mathcal{R})$ consisting of virtual objects such that $X(n)=0$, $n>0$. The Peter-Weyl Theorem now has the following consequence. \begin{theorem} If $\mathcal{R}$ is a Karoubian rring\xspace over a field of characteristic zero, there is an isomorphism $K_0^\SS(\mathcal{R})\cong\Lambda\o K_0(\mathcal{R})$ of $\lambda$-rings. \end{theorem} \begin{proof} The Peter-Weyl Theorem gives isomorphisms of rings $$ \Lambda \o K_0(\mathcal{R}) \xleftarrow{\ch\o1} K_0^\SS({\mathsf{Proj}}) \o K_0(\mathcal{R}) \xrightarrow{\boxtimes} K_0^\SS(\mathcal{R}) . $$ The first of these arrows is an isomorphism of $\lambda$-rings by Lemma \ref{atiyah}. As rings, both $K_0^\SS({\mathsf{Proj}})\o K_0(\mathcal{R})$ and $K_0^\SS(\mathcal{R})$ are generated by $K_0(\mathcal{R})$ and $[{1\!\!1}_n]$, $n\ge1$, and $\boxtimes$ respects the $\sigma$-operations of these elements, proving that it is a map of $\lambda$-rings. \end{proof} \section{The main result} If $\mathcal{R}$ is a Karoubian rring\xspace, denote by $\mathcal{R}[\mathbb{N}]$ the Karoubian ring of bounded sequences $$ (A^0,A^1,A^2,\dots\mid\text{$A^n=0$ for $n\gg0$}) . $$ The sum on $\mathcal{R}[\mathbb{N}]$ is defined by $(A\oplus\mathcal{Y})^n=A^n\oplus\mathcal{Y}^n$, while the product is defined by $$ (A\o B)^n = \bigoplus_{i+j=n} A^i \o B^j . $$ \begin{definition} A \textbf{K\"unneth functor} with values in the Karoubian rring\xspace $\mathcal{R}$ is a rring\xspace functor $\GS$ from the Cartesian {rring\xspace} ${\mathsf{Var}}$ of quasi-projective varieties and open embeddings to $\mathcal{R}[\mathbb{N}]$. \end{definition} In other words, a functor $\GS:{\mathsf{Var}}\to\mathcal{R}[\mathbb{N}]$ is a K\"unneth functor if there are natural isomorphisms \begin{gather*} \textstyle \GS^i(X\coprod Y)\cong\GS^i(X)\oplus\GS^i(Y) , \\ \GS^n(X\times Y) \cong \bigoplus_{n=i+j} \GS^i(X)\o\GS^j(Y) . \end{gather*} If $\GS=\{\GS^n\}$ is a K\"unneth functor, denote by $\Serre(X)$ the associated Euler characteristic $$ \Serre(X) = \sum_{n=0}^\infty (-1)^n [\GS^n(X)] $$ in the Grothendieck group $K_0(\mathcal{R})$. \begin{definition} A \textbf{Serre functor} with values in $\mathcal{R}$ is a K\"unneth functor $\GS$ such that for any closed sub-variety $Z$ of $X$, $$ \Serre(X) = \Serre(X\setminus Z) + \Serre(Z) . $$ \end{definition} If $\GS$ is a Serre functor and $X=X^0\subset X^1\subset X^2\subset\dots$ is a filtered quasi-projective variety such that $X^n=\emptyset$ for $n\gg0$, we have \begin{equation} \label{gr} \Serre(\gr X) \equiv \sum_n \Serre(\gr^nX) = \Serre(X) . \end{equation} Here are two examples of Serre functors: \begin{enumerate} \item The category of mixed Hodge structures over $\mathbb{C}$ is a rring\xspace, whose Grothendieck group may be identified with the polynomial ring $\mathbb{Z}[u,v]$ by means of the Serre polynomial \eqref{Serre}. The functor $\GS^n(X)$ which takes a quasi-projective variety $X$ to the mixed Hodge structure $(H^n_c(X,\mathbb{C}),F,W)$ over $\mathbb{C}$ is a Serre functor. The associated characteristic $\Serre(X)$ may be identified with the Serre polynomial. \item Gillet and Soul\'e \cite{GS} have constructed a functor to the homotopy category of chain complexes of (pure effective rational) Chow motives; let $\GS^n(X)$ be the $n$th cohomology of this complex. \end{enumerate} If $\mathcal{R}$ is a rring\xspace, let $\mathsf{T}:\mathcal{R}\to\[\SS,\mathcal{R}\]$ be the rring\xspace functor with $\mathsf{T}(X,n)=X^n$. (More precisely, $\mathsf{T}(X,n)$ is defined by induction: $\mathsf{T}(X,0)={1\!\!1}$, and $\mathsf{T}(X,n)=\mathsf{T}(X,n-1)\o X$.) The following result is a generalization of Macdonald's formula \cite{Macdonald-symmetric} for the Poincar\'e polynomial of the symmetric power $S^nX=X^n/\SS_n$. \begin{proposition} \label{Macdonald} If $X$ is a quasi-projective variety, $$ \Serre(\mathsf{T}(X)) = \Exp\bigl(p_1\Serre(X)\bigr) \in \Hat{K}^\SS_0(\mathcal{R}) . $$ Here, $\Serre(\mathsf{T}(X))$ denotes the class $n\DOTSB\mapstochar\to\Serre(\mathsf{T}(X,n))$ in the Grothendieck group $\Hat{K}^\SS_0(\mathcal{R})$. \end{proposition} \begin{proof} Since $\GS$ is a rring\xspace-functor, $\GS\*\mathsf{T}=\mathsf{T}\*\GS$. By the Peter-Weyl Theorem, $$ \GS(\mathsf{T}(X,n)) = \mathsf{T}[\GS(X)](n) = \bigoplus_{\lambda\vdash n} V_\lambda \boxtimes \bigl( V_\lambda^*\o \GS(X)^{\o n} \bigr)^{\SS_n} . $$ Descending to the Grothendieck group, we see that $$ \Serre(\mathsf{T}(X,n)) = \bigoplus_{\lambda\vdash n} s_\lambda \o \sigma_\lambda(\Serre(X)) \in \Lambda_n \o K_0(\mathcal{R}) \subset K_0^\SS(\mathcal{R}) . $$ Summing over $n\ge0$, and applying Cauchy's formula \eqref{Cauchy}, we see that $$ \Serre(\mathsf{T}(X)) = \exp \Bigl( \sum_{k=1}^\infty \frac{p_k\o\psi_k\Serre(X)}{k} \Bigr) \in \Hat{K}^\SS_0(\mathcal{R}) . $$ The proposition now follows by the definition of $\Exp(\dots)$. \end{proof} Consider the following decreasing filtration on the $\SS$-module $\mathsf{T}(X)$, where $X$ is a quasi-projective variety: $$ \mathsf{T}^i(X)(n) = \{ (z_1,\dots,z_n) \in X^n \mid \text{$\{z_1,\dots,z_n\}$ has cardinality at most $n-i$} \} . $$ Let $\gr^i\mathsf{T}(X)=\mathsf{T}^i(X)\setminus\mathsf{T}^{i+1}(X)$ be the associated graded $\SS$-module. \begin{lemma} \label{gr-T} Let ${\mathsf{Z}}$ be the object of $\hom[1]{{\mathsf{Var}}}$ $$ {\mathsf{Z}}(n) = \begin{cases} \mathbb{A}^0 , & n>0 , \\ \emptyset , & n=0 . \end{cases}$$ Then $\gr\mathsf{T}(X)=\mathsf{F}(X)\circ{\mathsf{Z}}$; in particular, $\gr^0\mathsf{T}(X)=\mathsf{F}(X)$. \end{lemma} \begin{proof} This lemma reflects the fact that an element of $\gr^i\mathsf{T}(X,n)$ determines, and is determined by, a partition of the set $\{1,\dots,n\}$ into $n-i$ disjoint subsets, together a point in $\mathsf{F}(X,n-i)$. \end{proof} We now arrive at the main theorem of this paper. \begin{theorem} \label{MAIN} Let $X$ be a quasi-projective variety over $\mathbb{C}$. If $\GS$ is a Serre functor and $V_\lambda$ is an irreducible representation of $\SS_n$, $$ \Serre(\mathsf{F}(X,n),V_\lambda) = \Phi_\lambda(\Serre(X)) . $$ \end{theorem} \begin{proof} If $\GS$ is Serre functor, \eqref{gr} and Lemma \ref{gr-T} show that $$ \Serre(\mathsf{T}(X)) = \Serre(\gr\mathsf{T}(X)) = \Serre(\mathsf{F}(X)) \circ \Serre({\mathsf{Z}}) . $$ To calculate $\Serre(\mathsf{F}(X))$, we invert the operation $-\circ\Serre({\mathsf{Z}})$ on $\Hat{K}^\SS_0(\mathcal{R})$. Indeed, $\Serre({\mathsf{Z}})=\Exp(p_1)-1$ and by Lemma \ref{Cadogan}, \begin{align*} \Serre(\mathsf{F}(X)) &= \Serre(\mathsf{F}(X)) \circ \bigl( \Exp(p_1) - 1 \bigr) \circ \Bigl( \sum_{n=1}^\infty \frac{\mu(n)}{n} \log(1+p_n) \Bigr) \\ &= \Serre(\mathsf{T}(X)) \circ \Bigl( \sum_{n=1}^\infty \frac{\mu(n)}{n} \log(1+p_n) \Bigr) . \end{align*} By Proposition \ref{Macdonald}, this equals \begin{multline*} \exp \Bigl( \sum_{k=1}^\infty \frac{p_k\*\psi_k\Serre(X)}{k} \Bigr) \circ \Bigl( \sum_{\ell=1}^\infty \frac{\mu(\ell)}{\ell} \log(1+p_\ell) \Bigr) = \exp \Bigl( \sum_{k=1}^\infty \sum_{\ell=1}^\infty \frac{\mu(\ell)}{k\ell} \log(1+p_{k\ell}) \psi_k\Serre(X)\Bigr) \\ = \exp \Bigl( \sum_{n=1}^\infty \sum_{d|n} \frac{\mu(n/d)}{n} \log(1+p_n) \psi_d\Serre(X) \Bigr) , \end{multline*} from which the theorem follows by extracting the coefficient of the Schur function $s_\lambda$ on both sides. \end{proof} The concise formulation $$ \Serre(\mathsf{F}(X)) = \Exp(\Log(1+p_1)\Serre(X)) $$ of this result makes the resemblance with the formula $\Serre(\mathsf{T}(X))=\Exp(p_1\Serre(X))$ clearer. In the special cases $\lambda=(n)$ or $\lambda=(1^n)$, when $\Phi_\lambda$ is given by the explicit formula of Corollary \ref{braid}, we obtain the following corollary. \begin{corollary} If $\Serre(X)=\sum_{p,q}h_{pq}u^pv^q$ is the Serre polynomial of $X$, then \begin{gather*} \sum_{n=0}^\infty t^n \Serre(\mathsf{F}(X,n)/\SS_n) = \frac{\sigma_t(X)}{\sigma_{t^2}(X)} = \prod_{p,q=0}^\infty \left( \frac{1-t^2u^pv^q}{1-tu^pv^q} \right)^{h_{pq}} , \\ \sum_{n=0}^\infty t^n \Serre(\mathsf{F}(X,n),\varepsilon) = \sigma_{-t}(X)^{-1} = \prod_{p,q=0}^\infty (1+tu^pv^q)^{h_{pq}} . \end{gather*} \end{corollary} For example, if $X=\mathbb{C}$, $\mathsf{F}(\mathbb{C},n)/SS_n$ is the classifying space $K(B_n,1)$ of the braid group $B_n$ on $n$ strands. Our formula becomes in this case $$ \sum_{n=0}^\infty t^n \Serre(\mathsf{F}(\mathbb{C},n)/\SS_n) = \frac{1-t^2uv}{1-tuv} = 1 + t\mathsf{L} + t^2(\mathsf{L}^2-\mathsf{L}) + t^3(\mathsf{L}^3-\mathsf{L}^2) + \dots , $$ reflecting the isomorphism of rational cohomology groups $H^\bullet(B_n,\mathbb{Q})\cong H^\bullet(\mathbb{G}_m,\mathbb{Q})$ as mixed Hodge structures. \section{The Fulton-MacPherson compactification} Fulton and MacPherson \cite{FM} have introduced a sequence of functors $X\DOTSB\mapstochar\to X[n]$ from ${\mathsf{Var}}$ to $[\SS_n,{\mathsf{Var}}]$, with the following properties. \begin{enumerate} \item If $X$ is projective, then so is $X[n]$. \item There is natural transformation of functors $\mathsf{F}(X,n)\hookrightarrow X[n]$, which is an embedding. \item The complement $X[n]\setminus\mathsf{F}(X)$ is a divisor with normal crossings. \end{enumerate} In this section, we calculate the equivariant Serre polynomial $\Serre(X[n])$. Denote by $\FM(X)$ the functor $X\DOTSB\mapstochar\to(n\DOTSB\mapstochar\to X[n])$ from ${\mathsf{Var}}$ to $\[\SS,{\mathsf{Var}}\]$, \subsection{Trees and $\SS$-modules} Let $\Gamma(n)$, $n\ge2$, be the set of isomorphism classes of labelled rooted trees with $n$ leaves, such that each vertex has at least two branches. It is easily seen that $\Gamma(n)$ is finite: in fact, the generating function \begin{equation} \label{enumerate} x + \sum_{n=2}^\infty \frac{x^n |\Gamma(n)|}{n!} \end{equation} is the inverse under composition of $x-x^2-x^3-x^4-\dots$. Given a tree $T\in\Gamma(n)$, denote by $\VERT(T)$ the set of vertices of $T$; given a vertex $v\in\VERT(T)$, denote by $n(v)$ the valence of $v$ (its number of branches). Given a tree $T\in\Gamma(n)$ and an $\SS$-module $\mathcal{V}$ in the rring\xspace $\mathcal{R}$, let $\mathcal{V}(T)$ be the object \begin{equation} \label{V(T)} \mathcal{V}(T) = \bigotimes_{v\in\VERT(T)} \mathcal{V}(n(v)) , \end{equation} and let ${\mathbb{T}}\mathcal{V}(n)$ be the $\SS_n$-module $$ {\mathbb{T}}\mathcal{V}(n) = \bigoplus_{T\in\Gamma(n)} \mathcal{V}(T) . $$ Thus, ${\mathbb{T}}$ is a functor from $\hom[2]{\mathcal{R}}$ to itself. (Recall that $\hom[2]{\mathcal{R}}$ is the full subcategory of $\SS$-modules such that $X(0)=X(1)=0$.) A proof of the following formula for $\mathcal{R}={\mathsf{Proj}}$ may be found in \cite{modular}; however, the same proof works in general. Observe that this theorem may be used to prove \eqref{enumerate}. \begin{theorem} \label{revert} The elements $$ f = h_1 - \sum_{n=2}^\infty [\mathcal{V}] \quad\text{and}\quad g = h_1 + \sum_{n=2}^\infty [{\mathbb{T}}\mathcal{V}] $$ of $\Check{K}^\SS_0(\mathcal{R})$ satisfy the formula $f\circ g = g\circ f=h_1$. \end{theorem} \subsection{The varieties $\protect\overset{\circ}{\mathsf{P}}_k(n)$} The algebraic groups $\mathbb{C}^k$ and ${\mathbb{G}}_m$ act on the affine space $\mathbb{C}^k$ by translation and dilatation respectively; by functoriality, these actions extend to $\mathsf{F}(\mathbb{C}^k,n)$. Denote by $G_k=\mathbb{C}^k\rtimes{\mathbb{G}}_m$ the semidirect product of these groups, and by $\overset{\circ}{\mathsf{P}}_k(n)$, $n>1$, the quotient of the configuration space $\mathsf{F}(\mathbb{C}^k,n)$ by the free $G_k$-action. This action is $\SS_n$-equivariant, and $\overset{\circ}{\mathsf{P}}_k(n)$ is a smooth $\SS_n$-variety of dimension $nk-k-1$. For example, $\overset{\circ}{\mathsf{P}}_k(2)$ is naturally isomorphic to the projective space $\mathbb{CP}^{k-1}$, with trivial $\SS_2$-action. \begin{proposition} $$ \Serre\bigl( \overset{\circ}{\mathsf{P}}_k(n),\SS_n \bigr) = \frac{\Serre(\mathsf{F}(\mathbb{C}^k,n),\SS_n)}{\Serre(\mathbb{C}^k)\Serre({\mathbb{G}}_m)} = \frac{\Serre(\mathsf{F}(\mathbb{C}^k,n),\SS_n)}{\mathsf{L}^k(\mathsf{L}-1)} $$ \end{proposition} \begin{proof} We start with a lemma. \begin{lemma} Let $G$ be an algebraic group and $P$ be a $G$-torsor with base $X=P/G$. If the projection $P\to X$ is locally trivial in the Zariski topology, $\Serre(P) = \Serre(G) \Serre(X)$. \end{lemma} \begin{proof} We stratify $X$ by locally closed subvarieties $X_i$ of codimension $i$ over which the torsor $P$ is trivial. The strata are chosen inductively: $X_{-1}$ is empty, while $X_i$ is a Zariski-open subset of $X\setminus X_{i-1}$ over which $P$ is trivial. The formula follows, since $$ \Serre(P) = \sum_i \Serre(P_i) = \sum_i \Serre(G) \Serre(X_i) . $$ \def{} \end{proof} The action of $\mathbb{C}^k$ on $\mathsf{F}(\mathbb{C}^k,n)$ is not just locally, but globally, trivial: a global section is given by $(z_1,\dots,z_n) \DOTSB\mapstochar\to (z_1-{\bar{z}},\dots,z_n-{\bar{z}})$, where ${\bar{z}} = \frac{1}{n} \sum_{i=1}^n z_i$. On the other hand, any free action of ${\mathbb{G}}_m$ on a variety is locally trivial in the Zariski topology: free actions with quotient $X$ are classified by $H^1(X_{\text{fl}},{\mathbb{G}}_m)$, locally trivial free actions with quotient $X$ are classified by $H^1(X,{\mathbb{G}}_m)$, and these two groups are isomorphic by Hilbert's Theorem 90 (see Proposition XI.5.1 of Grothendieck \cite{SGA1}). \end{proof} \subsection{Stratification of $\FM(X)$} The $\SS$-variety $\overset{\circ}{\mathsf{P}}_k$ has a natural compactification to a smooth projective $\SS$-variety $\P_k$, which has a natural stratification. The strata are labelled by trees $T\in\Gamma(n)$, and the stratum associated to $T$ is isomorphic to $\overset{\circ}{\mathsf{P}}_k(T)$, in the notation of \eqref{V(T)}. It follows from Theorem \ref{revert} that $\Serre(\P_k)$ is the inverse of \begin{align*} h_1 - \Serre\bigl( \overset{\circ}{\mathsf{P}}_k \bigr) &= p_1 - \frac{\prod_{n=1}^\infty (1+p_n)^{\frac{1}{n}\sum_{d|n}\mu(n/d)\mathsf{L}^{kd}} - 1 - \mathsf{L}^kp_1}{\mathsf{L}^k(\mathsf{L}-1)} \\ & = \frac{\mathsf{L}^{k+1}p_1 + 1 - \prod_{n=1}^\infty (1+p_n)^{\frac{1}{n}\sum_{d|n}\mu(n/d)\mathsf{L}^{kd}}}{\mathsf{L}^k(\mathsf{L}-1)} . \end{align*} under plethysm. The importance of the spaces $\P^k(n)$ comes from the following result of Fulton and MacPherson. \begin{proposition} The $\SS$-module $\FM(X)$ has a filtration such that $$ \gr\FM(X) \cong \mathsf{F}(X)\circ\P_k . $$ \end{proposition} Since $X[n]$ is a projective $Q$-variety (it has singularities which are quotients of affine space by a finite group), $\Serre(\FM(X))(n)$ equals the $\SS_n$-equivariant Hodge polynomial of $X[n]$. The above proposition shows that $\Serre(\FM(X))=\Serre(\mathsf{F}(X))\circ\Serre(\P_k)$, and leads to a practical algorithm for the calculation of the $\SS_n$-equivariant Hodge numbers of $X[n]$. On forgetting the action of the symmetric groups $\SS_n$, we recover the formula of Fulton and Macpherson for the Poincar\'e polynomials of $\FM(X,n)$, in a form stated by Manin \cite{Manin:1}. On replacing $h_n$ by $x^n/n!$, we obtain $$ 1 + \sum_{n=1}^\infty x^n \Serre(X[n]) = (1+x)^{\Serre(X)} \circ \biggl( \frac{\mathsf{L}^{k+1}x+1-(1+x)^{\mathsf{L}^k}}{\mathsf{L}^k(\mathsf{L}-1)} \biggr)^{-1} $$ In this formula, we may take the limit $\mathsf{L}\to1$ using L'H\^opital's rule, obtaining a formula for the Euler characteristic of $\FM(X,n)$: $$ 1 + \sum_{n=1}^\infty x^n \chi(X[n]) = (1+x)^{\chi(X)} \circ \bigl( (k+1)x - k(1+x)\log(1+x) \bigr)^{-1} . $$ The one dimensional case has special interest, since the spaces $\overset{\circ}{\mathsf{P}}_1(n)$ and $\P_1(n)$ are naturally isomorphic to the moduli spaces $\mathcal{M}_{0,n+1}$ and $\overline{\mathcal{M}}_{0,n+1}$; this isomorphism comes about because the translations and dilatations in one dimension generate the isotropy group of the point $\infty\in\mathbb{CP}^1$ with respect to the action of the group $\PSL(2,\mathbb{C})$. This identification means that the action of $\SS_n$ on these spaces is the restriction of an action of $\SS_{n+1}$. We have calculated the $\SS_{n+1}$-equivariant Serre polynomials of these spaces in \cite{gravity}; in a sequel to this paper, we calculate the $\SS_n$-equivariant Serre polynomial of $\overline{\mathcal{M}}_{1,n}$.

9510

alg-geom/9510017

en

https://arxiv.org/abs/alg-geom/9510017

alg-geom/9510017

Frank Sottile

Frank Sottile

Explicit Enumerative Geometry for the Real Grassmannian of Lines in Projective Space

LaTeX 2e, 18 pages with three figures

Duke Math. J., 87 (1997) 59-85

We extend the classical Schubert calculus of enumerative geometry for the Grassmann variety of lines in projective space from the complex realm to the real. Specifically, given any collection of Schubert conditions on lines in projective space which generically determine a finite number of lines, we show there exist real generic conditions determining the expected number of real lines. Our main tool is an explicit description of rational equivalences which also constitutes a novel determination of the Chow rings of these Grassmann varieties.

alg-geom

\section{Introduction} A basic problem in algebraic geometry is to describe the common zeroes of a set of polynomials. This is more difficult over non-algebraically closed fields. For systems of polynomials with few monomials on a complex torus, Khovanskii~\cite{Khovanskii_fewnomials} showed that the real zeroes are at most a small fraction of the complex zeroes. Fulton (\cite{Fulton_introduction_intersection}, \S7.2) asked how many solutions to a problem of enumerative geometry can be real; in particular, how many of the 3264 conics tangent to five general real conics can be real. He later showed that all, in fact, can be real. Recently, this was independently rediscovered by Ronga, Tognoli, and Vust~\cite{Ronga_Tognoli_Vust}. Robert Spesier suggested the classical Schubert calculus of enumerative geometry would be a good testing ground for these questions. For any problem of enumerating lines in ${\bf P}^n$ incident on real linear subspaces in general position, we show that all solutions can be real. \medskip A flag and a partition $\lambda = (\alpha,\beta)$ determine a Schubert subvariety of the Grassmannian of lines in ${\bf P}^n$ of type $\lambda$, which has codimension $|\lambda| = \alpha+\beta$. Any generically transverse intersection of Schubert varieties is rationally equivalent to a sum of Schubert varieties. The classical Schubert calculus gives algorithms for determining how many of each type. For partitions $\lambda^1,\ldots,\lambda^m$, we describe a cycle $\Omega({\cal T})$ (depending upon $\lambda^1, \ldots,\lambda^m$) which is a sum of distinct Schubert varieties. Let ${\cal G}= {\cal G}(\lambda^1,\ldots,\lambda^m)$ be the set of points of the Chow variety representing generically transverse intersections of Schubert varieties of types $\lambda^1,\ldots,\lambda^m$. In \S 4, we show that ${\cal G}$ is unirational. Cycles represented by the points of a rational curve on the Chow variety are rationally equivalent. In \S 3, we show \medskip \noindent{\bf Theorem A.} \ {\em Let $X\in {\cal G}$. Then there is a chain of rational curves between $X$ and the cycle $\Omega({\cal T})$. Furthermore, these curves may be explicitly described and each lies in the Zariski closure of ${\cal G}$. In particular, the point representing $\Omega({\cal T})$ is in the Zariski closure of $\,{\cal G}$.} \medskip The proof of Theorem A constitutes an explicitly geometric determination of the Schubert calculus of enumerative geometry for lines in ${\bf P}^n$. In fact, it shows these `Schubert-type' enumerative problems may be solved {\em without} reference to the Chow ring, a traditional tool in enumerative geometry. We use it to compute products in the Chow ring. Let $\sigma_\lambda$ be the rational equivalence class of a Schubert variety of type $\lambda$. \medskip \noindent{\bf Theorem B. } {\em Let $c^\lambda$ be the number of components of $ \,\Omega({\cal T})$ of type $\lambda$. Then $$ \prod_{i=1}^m \sigma_{\lambda^i} = \sum_\lambda c^\lambda \sigma_\lambda. $$ } Thus we derive the structure of these Chow rings in a strong sense: All products among classes from the Schubert basis are expressed as linear combinations of basis elements and these expressions are obtained by exhibiting rational equivalences between a generically transverse intersection of Schubert varieties and the cycle $\Omega({\cal T})$. We believe this is the first non-trivial explicit description of rational equivalences giving all products among a set generators of the Chow group for any variety. When $k= \mbox{\bf R}$, we show \medskip \noindent{\bf Theorem C.} \ {\em Let $\lambda^1,\ldots,\lambda^m$ be partitions with $|\lambda^1|+\cdots+|\lambda^m|$ equal to the dimension of the Grassmannian of lines in $\mbox{\bf P}^n$. Then there exists a nonempty classically open subset in the product of $m$ real flag manifolds whose corresponding Schubert varieties meet transversally, with all points of intersection real. } \medskip To the best of our knowledge, this is the first result showing that a large class of non-trivial enumerative problems can have all of their solutions real. The construction of the cycles $\Omega({\cal T})$ and rational curves of Theorem~A use a `calculus of tableaux' outlined in \S\ref{sec:calculus_of_tableaux} and extended in \S6, where we define a non-commutative associative algebra with additive basis the set of Young tableaux. This algebra has surjections to the Chow rings of Grassmann varieties and the algebra of symmetric functions. However, it differs fundamentally from the plactic algebra of Lascoux and Sch\"utzenberger~\cite{Lascoux_Schutzenberger_monoid_plactic}. In \S7, we ask which enumerative problems may be solved over which (finite) fields and give the answer for two classes of Schubert-type enumerative problems. We also show how some of our constructions may be carried out over finite fields. The rational equivalences we construct are a modification of the classical method of degeneration. This method may fail when applied to more than a few conditions; an intersection typically becomes improper before the conditions become special enough to completely determine the intersection. Considering deformations of intersection cycles, rather than of conditions, an idea of Chaivacci and Escamilla-Castillo~\cite{Chiavacci_Escamilla-Castillo}, enables us to deform generically transverse intersections into sums of distinct Schubert varieties. Theorem C is from our 1994 Ph.D. Thesis from the University of Chicago, written under the direction of William Fulton. We would like to thank William Fulton for suggesting these problems, for his thoughtful advice, and above all for introducing us to algebraic geometry. \section{Preliminaries} Let $k$ be an infinite field. Varieties will be closed, reduced, projective (not necessarily irreducible), and defined over $k$. When $k={\mbox{\bf R}}$, let $X(\mbox{\bf R})$ be the points of $X$ with residue field $\bf R$. We use the classical topology on $X(\mbox{\bf R})$. A subset $Y$ (not necessarily algebraic) of a variety $X$ is {\em unirational} if $Y$ contains the image $U$ of a dense open subset of affine space under an algebraic morphism and $Y\subset \overline{U}$. A subset $Y\subset X(\mbox{\bf R})$ is {\em real unirational} if $Y$ contains the image $U$ of a dense open subset of $\mbox{\bf R}^n$ under a real algebraic map and $Y\subset \overline{U}$. Let $X$ be a smooth variety, $U$ and $W$ subvarieties of $X$, and set $Z = U\cap W$. Then $U$ and $W$ meet {\em generically transversally} if $U$ and $V$ meet transversally at the generic point of each component of $Z$. Then $Z$ is generically reduced, the fundamental cycle $[Z]$ of $Z$ is multiplicity free, and in the Chow ring $A^*X$ of $X$: $$ [U] \cdot [W] = [U\cap W] = [Z] = \sum_{i=1}^r\, [Z_i], $$ where $Z_1,\ldots,Z_r$ are the irreducible components of $Z$. \vspace{.5in} \subsection{Chow Varieties}\label{sec:Chow} Let $X$ be a projective variety. The Chow variety $\mbox{\it Chow}\, X$ is a projective variety parameterizing positive cycles on $X$. Let $U$ be a smooth variety and $W$ a subvariety of $X\times U$ with equidimensional fibres over $U$. Then there is an dense open subset $U'$ of $U$ such that the association of a point $u$ of $U$ to the fundamental cycle of the fibre $W_u$ determines a morphism $U'\rightarrow \mbox{\it Chow}\, X$. If $U$ is a smooth curve, then $U' = U$. Moreover, if $X$, $U$, and $W$ are defined over $k$, then so are $\mbox{\it Chow}\, X$, $U'$, and the map $U' \rightarrow\mbox{\it Chow}\, X$ (\cite{Samuel}, \S I.9). Cycles represented all points on a rational curve in $\mbox{\it Chow}\, X$ are rationally equivalent. We will use the same notation for a subscheme of $X$, its fundamental cycle and the point representing that cycle in $\mbox{\it Chow}\, X$. \vspace{.5in} \subsection{Grassmannians and Schubert Subvarieties} For $S\subset {\mbox{\bf P}^n}$, let $\Span{S}$ be its linear span. For a vector space $V$, let ${\bf P} V$ be the projective space of all one dimensional subspaces of $V$. Suppose $K = {\mbox{\bf P}}U$ and $M={\mbox{\bf P}}W$. Set $\mbox{Hom}(K,M) = \mbox{Hom}(U,W)$, the space of linear maps from $U$ to $W$. If $K\subset M$, set $M/K = {\mbox{\bf P}}(W/U)$. A complete flag ${F\!_{\DOT}}$ is a collection of subspaces $F_n\subset\cdots\subset F_1\subset F_0 = {\mbox{\bf P}^n}$, where $\dim F_i = n-i$. If $p>n$, set $F_p = \emptyset$. For $0\leq s\leq n$, let $\mbox{\bf G}_s{\mbox{\bf P}^n}$ be the Grassmannian of $s$-dimensional subspaces of ${\mbox{\bf P}^n}$. We sometimes write $\mbox{\bf G}^{n-s}{\mbox{\bf P}^n}$ for this variety. Its dimension is $(n-s)(s+1)$. For a partition $\lambda=(\alpha,\beta)$, let $\mbox{\bf Fl}(\lambda)$ denote the variety of partial flags of {\em type} $\lambda$; those $K\subset M$ with $K$ a $(n-\alpha-1)$-plane and $M$ a $(n-\beta)$-plane. A partial flag $K\subset M$ determines a {\em Schubert variety} $\Omega(K,M)$; those lines contained in $M$ which also meet $K$. The {\em type} of $\Omega(K,M)$ is the type, $\lambda=(\alpha,\beta)$, of $K\subset M$ and its codimension is $|\lambda| =\alpha+\beta$. If $\alpha=\beta$, then $\Omega(K,M) = \mbox{\bf G}_1M$, the Grassmannian of lines in $M$. If $\beta=0$, so $M={\mbox{\bf P}^n}$, then we write $\Omega_K$ for this Schubert variety. The tangent space to $\ell \in {\mbox{\bf G}_1{\mbox{\bf P}^n}}$ is naturally identified with the linear space $\mbox{Hom}(\ell,{\mbox{\bf P}^n}/\ell)$. It is not hard to verify the following Lemma, whose proof we omit. \subsection{Lemma.}\label{lemma:one} {\em \begin{enumerate} \item The smooth locus of $\Omega(K,M)$ consists of those $\ell$ with $\ell \not\subset K$. For such $\ell$, $$ T_{\ell}\Omega(K,M) = \{ \phi\in \mbox{Hom}(\ell,{\mbox{\bf P}^n}/\ell)\,|\, \phi(\ell)\subset M/\ell\mbox{ and } \phi(\ell\cap K)\in (K+\ell)/\ell\}. $$ \item Let $K,M\subset {\bf P}^n$. Then $\Omega_K\bigcap\mbox{\bf G}_1M = \Omega(K\cap M,\, M)$, and this is transverse at the smooth points of $\,\Omega(K\cap M,\, M)$ if and only if $K$ and $M$ meet properly in ${\mbox{\bf P}^n}$. \item Let $K_i\subset M_i$, for $i=1,2$. Then the intersection $\Omega(K_1,M_1)\bigcap \Omega(K_2,M_2)$ is improper unless $M_i$ meets both $K_j$ and $M_j$ properly for $i\neq j$. \end{enumerate} } \medskip An intersection of two Schubert varieties may be generically transverse and reducible: \subsection{ Lemma.}\label{lemma:component_calculation} {\em Let $F,P,N,H$ be linear subspaces of ${\bf P}^n$ and suppose $H$ is a hyperplane not containing $P$ or $N$, $F\subsetneq P\cap H$, and $N$ meets $F$ properly. Set $L=N\cap H$. Then $\Omega(F,P)$ and $\Omega_L$ meet generically transversally, $$ \Omega(F,P)\bigcap\Omega_L \ =\ \Omega(N\cap F,P) +\Omega(F,P\cap H)\bigcap\Omega_N, $$ and the second component is itself a generically transverse intersection. } \medskip \noindent{\bf Proof:} The right hand side is a subset of the left, we show the other inclusion. Let $\ell \in\Omega(F,P)\bigcap\Omega_L$. If $\ell$ meets $L\cap F= N\cap F$, then $\ell \in \Omega(N\cap F,P)$. Otherwise, $\ell$ is spanned by its intersections with $F$ and $L$, hence $\ell\subset \Span{F,L}\cap P \subset P\cap H$. Since $N\cap P\cap H = L\cap P\cap H$, we see that $\ell \in \Omega(F,P\cap H)\bigcap\Omega_N$. Verifying these intersections are generically transverse is left to the reader. \QED \subsection{A Calculus of Tableaux}\label{sec:calculus_of_tableaux} The {\em Young diagram} of a partition $\lambda = (\alpha,\beta)$ is a two rowed array of boxes with $\alpha$ boxes in the first row and $\beta$ in the second. Note that $\alpha\geq \hfl{|\lambda|+1}\geq \hfl{|\lambda|} \geq \beta$. We make no distinction between a partition and its Young diagram. A {\em Young tableau} $T$ of {\em shape} $\lambda$ is a filling of the boxes of $\lambda$ with the integers $1,2,\ldots,|\lambda|$. These integers increase left to right across each row and down each column. Thus the $j$th entry in the second row of $T$ must be at least $2j$. Call $|\lambda|$ the {\em degree} of $T$, denoted $|T|$. If $\alpha = \beta$, then $T$ is said to be {\em rectangular}. Let $T$ be a tableau and $\alpha$ a positive integer. Define $T* \alpha$ to be the set of all tableaux of degree $|T|+\alpha$ whose first $|T|$ entries comprise $T$ and last $\alpha$ entries occur in distinct columns, increasing from left to right. For example: $$ \setlength{\unitlength}{1.3pt}% \begin{picture}(35,20)(0,8) \thicklines \put(0, 0){\line(1,0){10}} \put(0,10){\line(1,0){30}} \put(0,20){\line(1,0){30}} \put( 0, 0){\line(0,1){20}} \put(10, 0){\line(0,1){20}} \put(20,10){\line(0,1){10}} \put(30,10){\line(0,1){10}} \put( 2,11.8){\bf 1} \put(12,11.8){\bf 2} \put(22,11.8){\bf 3} \put(2, 1.8){\bf 4} \end{picture} \ *\ 4 \ =\ \left\{\rule{0pt}{16pt}\ \setlength{\unitlength}{1.3pt}% \begin{picture}(70,20)(0,8) \thicklines \put(0, 0){\line(1,0){10}} \put(0,10){\line(1,0){70}} \put(0,20){\line(1,0){70}} \put( 0, 0){\line(0,1){20}} \put(10, 0){\line(0,1){20}} \put(20,10){\line(0,1){10}} \put(30,10){\line(0,1){10}} \put(40,10){\line(0,1){10}} \put(50,10){\line(0,1){10}} \put(60,10){\line(0,1){10}} \put(70,10){\line(0,1){10}} \put(0,0){\usebox{\Shading}} \put(32,11.8){\bf 5} \put(42,11.3){\bf 6} \put(52,11.3){\bf 7} \put(62,11.3){\bf 8} \end{picture} \,,\ \setlength{\unitlength}{1.3pt}% \begin{picture}(60,20)(0,8) \thicklines \put(0, 0){\line(1,0){20}} \put(0,10){\line(1,0){60}} \put(0,20){\line(1,0){60}} \put( 0, 0){\line(0,1){20}} \put(10, 0){\line(0,1){20}} \put(20, 0){\line(0,1){20}} \put(30,10){\line(0,1){10}} \put(40,10){\line(0,1){10}} \put(50,10){\line(0,1){10}} \put(60,10){\line(0,1){10}} \put(0,0){\usebox{\Shading}} \put(12, 1.6){\bf 5} \put(32,11.8){\bf 6} \put(42,11.3){\bf 7} \put(52,11.3){\bf 8} \end{picture} \,,\ \setlength{\unitlength}{1.3pt}% \begin{picture}(50,20)(0,8) \thicklines \put(0, 0){\line(1,0){30}} \put(0,10){\line(1,0){50}} \put(0,20){\line(1,0){50}} \put( 0, 0){\line(0,1){20}} \put(10, 0){\line(0,1){20}} \put(20, 0){\line(0,1){20}} \put(30, 0){\line(0,1){20}} \put(40,10){\line(0,1){10}} \put(50,10){\line(0,1){10}} \put(0,0){\usebox{\Shading}} \put(12, 1.6){\bf 5} \put(22, 1.3){\bf 6} \put(32,11.8){\bf 7} \put(42,11.3){\bf 8} \end{picture}\ \right\}. $$ Let $T(\alpha)\in T*\alpha$ be the tableau whose last $\alpha$ entries lie in the first row. Let $T^{+\alpha}\in T*\alpha$ be the tableau whose last $\alpha$ entries lie in the second row, if that is possible. Write $T^+$ for $T^{+1}$ and define $T^{+\alpha}(\beta)$ to be $(T^{+\alpha})(\beta)$. Then $$ T*\alpha = \{T(\alpha),\, T^{+}(\alpha-1),\ldots,T^{+\alpha}\}. $$ Given a set ${\cal T}$ of tableaux, define ${\cal T}*\alpha$ to be $\bigcup_{T\in {\cal T}} T*\alpha$, a disjoint union. Similarly define ${\cal T}(\alpha)$, ${\cal T}^{+\alpha}$, and ${\cal T}^{+\alpha}(\beta)$. For $0\leq s\leq \alpha$, set ${\cal T}_{s,\alpha} = {\cal T}(\alpha) \bigcup {\cal T}^{+}(\alpha -1) \bigcup\cdots\bigcup {\cal T}^{+s}(\alpha - s)$. It follows that ${\cal T}_{s,\alpha} = {\cal T}_{s-1,\alpha} \bigcup \left( {\cal T}^{+s} (\alpha-s)\right)$ and ${\cal T}*\alpha = {\cal T}_{\alpha,\alpha}$. Finally, for positive integers $\alpha_1,\ldots,\alpha_m$, define $\alpha_1*\cdots*\alpha_m$ to be $\emptyset*\alpha_1*\cdots*\alpha_m$, where $\emptyset$ is the unique tableau of shape $(0,0)$. \subsection{Arrangements}\label{sec:arrangements} An {\em arrangement} ${\cal F}$ is a collection of $2n-3$ hyperplanes $H_2,\ldots,H_{2n-2}$ and a complete flag ${F\!_{\DOT}}$ in ${\mbox{\bf P}^n}$, which satisfy some additional assumtions. For a tableau $T$ of degree at most $2n-2$, let $H_T$ be $\bigcap \{H_j\,|\, j \in\mbox{ the second row of $T$}\}$. For $p=2,3,\ldots,2n-2$, the linear spaces in an arrangement are required to satisfy: \begin{enumerate} \item $H_p \cap F_{\hfl{p}} = F_{\hfl{p}+1}$, \item For tableaux $S, T$ of degrees at most $p-1$, $H_T\cap H_S\subset H_p \Rightarrow H_T\cap H_S=F_{\hfl{p}+1}$. \end{enumerate} \noindent The only obstruction to constructing arrangements over $k$ is that $k$ have enough elements to find $H_p$ satisfying condition 2. In particular, there are arrangements for any infinite field. In \S\ref{thm:arrangement_finite_field}, we give an estimate over which finite fields it is possible to construct arrangements. Let ${\cal F}$ be an arrangement and $T$ a tableau of shape $(\alpha,\beta)$ with $\alpha + \beta \leq 2n-2$. Define $\Omega(T) = \Omega(F_{\alpha+1}, H_T)$. If $\alpha\geq n$, then $F_{\alpha+1} = \emptyset$, so $\Omega(T) = \emptyset$. \subsection{Lemma}\label{lemma:arrangements} {\em Let ${\cal F}$ be an arrangement, $S,T$ tableaux of degree $l\leq 2n{-}2$ and suppose $T$ has shape $(\alpha,\beta)$ with $\alpha\leq n-1$. Then \begin{enumerate} \item $F_{\alpha+1} \subset F_{\hfl{l+1}+1} \subset H_T$ and $F_{\alpha+1}\subset H_T$ is a partial flag of type $(\alpha,\beta)$. \item If $\beta>0$, then $H_T\neq F_\beta$. \item If either $\Omega(T) =\Omega(S)$ or $H_T=H_S$, then $T=S$. \end{enumerate} } \medskip \noindent{\bf Proof:} Since $\alpha\geq \hfl{l{+}1}$ and for $p\leq l$, $F_{\hfl{l{+}1}}\subset H_p$, we see $F_{\alpha+1}\subset F_{\hfl{l{+}1}{+}1}\subset H_T$, by the definition of arrangement. We show the codimension of $H_T$ is $\beta$ by induction on $\beta$. When $\beta = 0$ or $1$, this is clear. Removing, if necessary, extra entries from the first row of $T$, we may assume $l$ is in the second row of $T$. Let $S$ be the tableau obtained from $T$ by removing $l$. By induction, $H_S$ has codimension $\beta-1$. By the definition of arrangement, $H_S \not\subset H_l$. Noting $H_T = H_S\cap H_l$ proves (1). For (2), if $H_T=F_\beta$, then $F_\beta\subset H_{p}$, where $p$ is the $\beta$th entry in the second row of $T$. Thus $F_\beta \subset F_{\hfl{p}+1}$ so $\beta\geq\hfl{p}{+}1$. But this contradicts $p \geq 2\beta$. For (3), since $\Omega(T)\neq \emptyset$, $H_T$ is the union of all lines in $\Omega(T)$. Thus either assumption implies $H_T=H_S$. Suppose $|S|+|T|$ is minimal subject to $S\neq T$ but $H_S = H_T$. Let $s = |S|$ and $t=|T|$, then $s$ and $t$ are necessarily in the second rows of $S$ and $T$, respectively. If $s\neq t$, suppose $s>t$. Then $H_T = H_S\cap H_s$, which contradicts $H_T \not\subset H_s$. Thus $s=t$. Let $S'$ and $T'$ be the tableaux obtained by removing $s$ from each of $S$ and $T$. Then $H_{S'}\neq H_{T'}$, but $H_{S'}\cap H_{T'} = H_S \subset H_s$. Thus $H_S = F_{\hfl{s}+1}$ a contradiction, as the codimension of $H_S$ is at most $\hfl{s}$. \QED \subsection{The Cycles $\Omega({\cal T})$ and $\Omega({\cal T}_{s,\alpha};L)$} Let ${\cal F}$ be an arrangement. For a set ${\cal T}$ of tableaux with common degree $l\leq 2n{-}2$, define $\Omega({\cal T})$ to be the sum over $T\in {\cal T}$ of the Schubert varieties $\Omega(T)$. By Lemma~\ref{lemma:arrangements}(3), these are distinct, so $\Omega({\cal T})$ is a multiplicity free cycle. Let $0\leq s\leq \alpha$ be integers and $L$ a subspace of codimension $\alpha-s+1$ which meets the subspaces $F_{\hfl{l+s}+1},\ldots,F_{l+1}$ properly with $L\cap F_{l+1} = F_{l+\alpha-s+2}$ and if $|T| = l$, then $L$ meets $H_{T^{+(s-1)}}$ and $H_{T^{+s}}$ properly. When this occurs, we shall say that $L$ {\em meets ${\cal F}_{l,s}$ properly}. Let $T\in {\cal T}_{s-1,\alpha}$. Then $T$ is a tableau whose first row has length $b\geq\hfl{l+s}+1+\alpha-s$ and no entry in its second row exceeds $l+s-1$. Define $$ \Omega(T;L) = \left\{\begin{array}{ll} \Omega(F_{b+1},H_T) & \mbox{ if } b\geq l+1\\ \Omega(F_{b-\alpha+s}\cap L, H_T) & \mbox{ otherwise } \end{array} \right. . $$ Then we define $\Omega({\cal T}_{s-1,\alpha};L)$ to be the sum over $T\in {\cal T}_{s-1,\alpha}$ of the varieties $\Omega(T;L)$. By Lemma~\ref{lemma:arrangements}(3), these are distinct, so $\Omega({\cal T}_{s-1,\alpha};L)$ is a multiplicity free cycle. \subsection{The sets $U_{i,s}$ and ${\cal G}_{i,s}$.}% \label{sec:sets_of_cycles} Let $\alpha_1,\ldots,\alpha_m$ be positive integers and ${\cal F}$ an arrangement. Fix $1\leq i\leq m$. Let ${\cal T}=\alpha_1*\cdots*\alpha_{i-1}$ and $l$ be the common degree of tableaux in ${\cal T}$. Let $\Sigma_{i,0}\subset \mbox{\bf G}_1{\mbox{\bf P}^n}\times\prod_{j=i}^m\mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}$ be the subvariety whose fibre over $(K_i,\ldots,K_m)$ is $$ \Omega({\cal T}) \bigcap \Omega_{K_i}\bigcap\cdots\bigcap\Omega_{K_m}. $$ Let $U_{i,0} \subset \prod_{j=i}^m\mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}$ be those points for which this intersection is generically transverse. Let ${\cal G}_{i,0}\subset \mbox{\it Chow}\, \mbox{\bf G}_1{\mbox{\bf P}^n}$ be the fundamental cycles of fibres of $\Sigma_{i,0}$ over $U_{i,0}$. For $1\leq s\leq \alpha_i$, let $\Sigma_{i,s}\subset \mbox{\bf G}_1{\mbox{\bf P}^n}\times \mbox{\bf G}^{\alpha_i-s+1}({\mbox{\bf P}^n}/F_{l+\alpha_i-s+2}) \times \prod_{j=i+1}^m\mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}$ be the subvariety where $L$ meets ${\cal F}_{l,s}$ properly and whose fibre over $(L,K_{i+1},\ldots,K_m)$ is $$ \left[\Omega({\cal T}_{s-1,\alpha_i};L)+ \Omega({\cal T}^{+s})\bigcap\Omega_L\right] \bigcap \Omega_{K_{i+1}} \bigcap\cdots\bigcap\Omega_{K_m}. $$ Define $U_{i,s}$ and ${\cal G}_{i,s}$ analogously to $U_{i,0}$ and ${\cal G}_{i,0}$. Set ${\cal G}_{m+1,0}$ to be the singleton $\{ \Omega(\alpha_1*\cdots*\alpha_m)\}$. When $k = \mbox{\bf R}$, let ${\cal G}_{i,s;\mbox{\scriptsize\bf R}}$ be the fundamental cycles of fibres of $\Sigma_{i,s}$ over $U_{i,s}(\mbox{\bf R})$. \section{Main Results} In \S4, we prove \medskip \noindent{\bf Theorem D.}\ {\em Let $\alpha_1,\ldots,\alpha_m$ be positive integers and ${\cal F}$ any arrangement. Then \begin{enumerate} \item For all $1\leq i\leq m$, and $0\leq s\leq \alpha_i$, $U_{i,s}$ is a dense open subset of the corresponding product of Grassmannians. \item For all $1\leq i\leq m$, $0\leq s\leq \alpha_i$, ${\cal G}_{i,s}$ is a unirational subset of $\mbox{\it Chow}\,{\mbox{\bf G}_1{\mbox{\bf P}^n}}$. When $k=\mbox{\bf R}$, ${\cal G}_{i,s;\mbox{\scriptsize\bf R}}$ is a real unirational subset of $\mbox{\it Chow}\,{\mbox{\bf G}_1{\mbox{\bf P}^n}}(\mbox{\bf R})$. \end{enumerate} } \medskip Let $U$ be an open subset of ${\mbox{\bf P}^1}$ and $\phi:U \rightarrow U_{i,s}$. Then $\phi^*\Sigma_{i,s}$ has equidimensional fibres over $U$. As in \S\ref{sec:Chow}, the association of a point $u$ of $U$ to the fundamental cycle of the fibre $(\phi^*\Sigma_{i,s})_u$ is an algebraic morphism, which we denote $\phi_* : U\rightarrow \mbox{\it Chow}\, {\mbox{\bf G}_1{\mbox{\bf P}^n}}$. Let $\phi_*$ be the unique extension of $\phi_*$ to ${\mbox{\bf P}^1}$, as well. In \S5, we prove: \medskip \noindent{\bf Theorem E.}\ {\em Let $\alpha_1,\ldots,\alpha_m$ be positive integers and ${\cal F}$ any arrangement. Then \begin{enumerate} \item For all $1\leq i\leq m$, if $X$ is a closed point of ${\cal G}_{i+1,0}$, then there is an open subset $U$ of ${\mbox{\bf P}^1}- \{0\}$ and a map $\phi: U \rightarrow U_{i,\alpha_i}$ such that $X= \phi_*(0)$. \item For all $1\leq i\leq m$ and $0\leq s\leq \alpha_i$, if $X$ is a closed point of ${\cal G}_{i,s+1}$, then there is an open subset $U$ of $\,{\mbox{\bf P}^1}- \{0\}$ and a map $\phi: U \rightarrow U_{i,s}$ such that $X= \phi_*(0)$. \end{enumerate} }\medskip \subsection{Proof of Theorem A} In the situation of Theorem E, let ${\cal T} =\alpha_1*\cdots*\alpha_m$. Then ${\cal G}_{1,0}$ is the set ${\cal G}$ and ${\cal G}_{m+1,0}$ is the cycle $\Omega({\cal T})$. By Theorem~D part 2, any two points of ${\cal G}_{i,s}$ are connected by a chain of rational curves, each lying within the closure of ${\cal G}_{i,s}$. Downward induction in the lexicographic order on pairs $(i,s)$ gives a chain of rational curves between $\Omega({\cal T})$ and a cycle $X\in {\cal G}$. Thus Theorem E implies Theorem A when $\beta_i = 0$ for $1\leq i \leq m$. Suppose $\lambda^i = (\alpha_i+\beta_i,\beta_i)$ for $1\leq i \leq m$. Let $M_0\subset {\mbox{\bf P}^n}$ have codimension $\beta=\beta_1+\cdots+\beta_m$ and ${\cal F}$ be any arrangement in $M_0$. Define $U_{i,s}$ and ${\cal G}_{i,s}$ as in \S\ref{sec:sets_of_cycles}, with $M_0$ replacing ${\mbox{\bf P}^n}$. Let $X\in {\cal G}$, so $X$ is a generically transverse intersection $$ \Omega(K_1,M_1)\bigcap\cdots\bigcap\Omega(K_m,M_m), $$ where $K_i\subset M_i$ has type $\lambda^i$ for $1\leq i\leq m$. Set $M = M_1\cap\cdots\cap M_m$. Iteration of Lemma~\ref{lemma:one} shows that $M$ has codimension $\beta$ and $L_i = M \cap K_i$ has codimension $\alpha_i +1$ in $M$. Thus $$ X = \Omega_{L_1}\bigcap\cdots\bigcap \Omega_{L_m} $$ is a generically transverse intersection in ${\mbox{\bf G}_1} M$. Let $\gamma$ be any automorphism of ${\mbox{\bf P}^n}$ with $\gamma M = M_0$ and $\Gamma$ a one parameter subgroup containing $\gamma$. The orbit $\Gamma \cdot X$ is a rational curve (or a point) in $\mbox{\it Chow}\, {\mbox{\bf G}_1{\mbox{\bf P}^n}}$ containing $\gamma(X)$. Since $\gamma(X)$ is in ${\cal G}_{i,0}$, previous arguments have shown there exists a chain of rational curves between $\gamma(X)$ and $\Omega({\cal T})$, each contained within the closure of ${\cal G}_{i,0}$. \QED \subsection{The Schubert Calculus.} Let $\lambda^1,\ldots,\lambda^m$ be partitions with $\lambda^i = (\alpha_i+\beta_i,\beta_i)$. Set $\beta = \beta_1+\cdots+\beta_m$ and $s_i = \alpha_i+\cdots+\alpha_{i-1}$. For a partition $\lambda$ with $|\lambda| = s_m$, define ${\cal C}^\lambda = {\cal C}^\lambda(\alpha_1,\ldots,\alpha_m)$ to be those tableaux of shape $\lambda$ such that for $1\leq i\leq m$ the integers $s_i+1,\ldots, s_i+\alpha_i$ occur in distinct columns increasing from left to right. Then $\alpha_1*\cdots*\alpha_m = \bigcup_{|\lambda| = s_m} {\cal C}^\lambda$. Let $c^\lambda_{\alpha_1,\ldots,\alpha_m} = \#{\cal C}^\lambda$. Interpreting Theorems A and E in terms of products in the \medskip Chow ring of ${\mbox{\bf G}_1{\mbox{\bf P}^n}}$, we have: \noindent{\bf Theorem B$'$.}\ {\em \begin{enumerate} \item ${\displaystyle \sigma_{\alpha_1}\cdots\sigma_{\alpha_m} = \sum_{|\lambda| = s_m} c^\lambda_{\alpha_1,\ldots,\alpha_m} \sigma_\lambda}$. \item ${\displaystyle \sigma_{\lambda^1}\cdots\sigma_{\lambda^m} = \sum_{\stackrel{\mbox{\scriptsize $\lambda = (\beta+a,\beta+b)$}}{a+b = s_m}} c^{(a,b)}_{\alpha_1,\ldots,\alpha_m} \sigma_\lambda}$. \end{enumerate} } \medskip In particular, if $\alpha_1+\cdots +\alpha_m = 2n-2$, then the only non-zero term on the right hand side of (1) is $c^{(n-1,n-1)}_{\alpha_1,\ldots,\alpha_m}\sigma_{(n-1,n-1)}$, or $c^{(n-1,n-1)}_{\alpha_1,\ldots,\alpha_m}$ times the class of a point (line). Thus \subsection{ Corollary} {\em The number of lines meeting general $(n-\alpha_i - 1)$-planes for $1\leq i\leq m$ is equal to the number of tableaux of shape $(n-1,n-1)$ such that for $1\leq i\leq m$, the integers $s_i+1,\ldots, s_i+\alpha_i$ occur in distinct columns increasing from left to right. } \medskip This number, $c^{(n-1,n-1)}_{\alpha_1,\ldots,\alpha_m}$, is also known as a Kostka number~\cite{Sagan}. \subsection{Enumerative Geometry of the Real Grassmannian.} Let ${\cal G}_{\mbox{\scriptsize\bf R}}$ consist of the fundemental cycles of generically transverse intersections of Schubert varieties of types $\lambda^1,\ldots,\lambda^m$ defined by real flags. \medskip \noindent{\bf Theorem C$'$.} \ {\em Let $\lambda^1,\ldots,\lambda^m$ be partitions, suppose $\lambda^i = (\alpha_i+\beta_i,\beta_i)$, and set $\beta=\sum_{i=1}^m \beta_i$. Let $M \subset {\mbox{\bf P}^n}$ be a real $(n-\beta)$-plane and ${\cal F}$ an arrangement in $M$. \begin{enumerate} \item $\Omega(\alpha_1*\cdots*\alpha_m)$ is in the closure of $\,{\cal G}_{\mbox{\scriptsize\bf R}}$. \item If $\,|\lambda^1|+\cdots+|\lambda^m|=2n-2$, then there is a nonempty classically open subset in the product of $m$ real flag manifolds whose corresponding Schubert varieties meet transversally, with all points of intersection real. \end{enumerate} } \medskip \noindent{\bf Proof:} Suppose that $\lambda^i = (\alpha_i+\beta_i,\beta_i)$. Define $U_{i,s}$ and ${\cal G}_{i,s}$ as in \S\ref{sec:sets_of_cycles} for the arrangement ${\cal F}$ in $M$ and the integers $\alpha_1,\ldots,\alpha_m$. Arguing as in the proof of Theorem~A shows ${\cal G}_{1,0;\mbox{\scriptsize\bf R}}\subset {\cal G}_{\mbox{\scriptsize\bf R}}$. Restricting to the real points of the varieties in Theorem E shows ${\cal G}_{i,s;\mbox{\scriptsize\bf R}}\subset \overline{{\cal G}_{1,0;\mbox{\scriptsize\bf R}}}$. The case $(i,s) = (m+1,0)$ is part 1. For 2, let $d = c^{(n-1,n-1)}_{\alpha_1,\ldots,\alpha_m}$. Then $\Omega(\alpha_1*\cdots*\alpha_m)$ is $d$ distinct real lines. Hence ${\cal G}_{i,s} \subset S^d{\mbox{\bf G}_1{\mbox{\bf P}^n}}$, the Chow variety of effective degree $d$ zero cycles on ${\mbox{\bf G}_1{\mbox{\bf P}^n}}$. The real points $S^d{\mbox{\bf G}_1{\mbox{\bf P}^n}}(\mbox{\bf R})$ of $S^d{\mbox{\bf G}_1{\mbox{\bf P}^n}}$ are effective degree $d$ zero cycles stable under complex conjugation. The dense subset of $S^d{\mbox{\bf G}_1{\mbox{\bf P}^n}}(\mbox{\bf R})$ of multiplicity free cycles has a component ${\cal M}$ parameterizing sets of $d$ distinct real lines and $\Omega(\alpha_1*\cdots*\alpha_m)\in {\cal M}$. By part 1, $\Omega(\alpha_1*\cdots*\alpha_m)\in \overline{{\cal G}_{\mbox{\scriptsize\bf R}}}$, which shows ${\cal G}_{\mbox{\scriptsize\bf R}}\bigcap{\cal M}\neq \emptyset$, a restatement of 2. \QED \section{Generically Transverse Intersections} \subsection{Lemma.}\label{lemma:gic} {\em Let $\lambda^1,\ldots,\lambda^m$ be partitions. Then the set $U$ of partial flags $K_1\subset M_1,\,\ldots,$ $\,K_m\subset M_m$ for which the intersection $$ \Omega(K_1,M_1)\bigcap\cdots\bigcap\Omega(K_m,M_m) $$ is generically transverse is a dense open subset of $\,\prod_{i=1}^m \mbox{\bf Fl}(\lambda^i)$. } \medskip \noindent{\bf Proof:} For $1\leq i\leq m$, let $K_i\subset M_i$ be a partial flag of type $\lambda^i= (\alpha_i+\beta_i,\beta_i)$ and suppose the corresponding Schubert varieties meet generically transversely. By Lemma~\ref{lemma:one}, $$ \Omega(K_1,M_1)\bigcap\cdots\bigcap\Omega(K_m,M_m) \ =\ \mbox{\bf G}_1M\bigcap \Omega_{L_1}\bigcap\cdots\bigcap\Omega_{L_m}, $$ where $M = M_1\cap\cdots\cap M_m$, $K_i = L_i\cap M_i$ where $L_i$ meets $M_i$ properly, and $M$ has codimension $\beta = \beta_1+\cdots+\beta_m$. \medskip Fix a codimension $\beta$ subspace $M$ of ${\mbox{\bf P}^n}$. As $U$ is stable under the diagonal action of $Gl_{n+1}$, it is the union of the translates of $V = U\cap X$, where $X$ consists of those $m$-tuples of flags with $M \subset M_i, 1\leq i\leq m$. Moreover, $U$ is open if and only if $V$ is open in $X$. Let $Y\subset X$ be those flags where $M = M_1\cap\cdots\cap M_m$ and $K_i$ meets $M$ properly. The product of maps defined by $(K_i,M_i) \mapsto K_i\cap M=L_i$ exhibits $Y$ as a fibre bundle over the product $\prod_{i=1}^m \mbox{\bf G}^{\alpha_i+1}M$, and $V$ is the inverse image of the set $W$ consisting of those $(L_1,\ldots,L_m)$ for which $\Omega_{L_1}\bigcap\cdots\bigcap\Omega_{L_m}$ is generically transverse. Thus we may assume $\beta_i = 0$. \medskip Let $\Sigma\subset({\mbox{\bf P}^n})^m\times\mbox{\bf G}_1{\mbox{\bf P}^n}\times\prod_{i=1}^m\mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$ consist of those $(p_1,\ldots,p_m,\ell,L_1,\ldots,L_m)$ such that $p_i\in \ell \cap L_i$ for $1\leq i\leq m$. The projection of $\Sigma$ to $({\mbox{\bf P}^n})^m\times \mbox{\bf G}_1{\mbox{\bf P}^n}$ exhibits $\Sigma$ as a fibre bundle with fibre $\prod_{i=1}^m\mbox{\bf G}^{\alpha_i+1}({\mbox{\bf P}^n}/p_i)$ and image those $(p_1,\ldots,p_m,\ell)$ with each $p_i\in \ell$. This image has dimension $m + 2n-2$. Thus $\Sigma$ is irreducible of dimension $$ m+2n-2+\sum_{i=1}^m(n-\alpha_i-1)(\alpha_i+1)\ = \ 2n-2 -\sum_{i=1}^m \alpha_i+ \sum_{i=1}^m (n-\alpha_i)(\alpha_i+1). $$ The image of $\Sigma$ in $\prod_{i=1}^m\mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$ consists of those $(L_1,\ldots,L_m)$ whose corresponding Schubert varieties have nonempty intersection. This image is a proper subvariety if $2n-2 < \sum_{i=1}^m \alpha_i$. In this case, $U$ is the complement of this image. Suppose $2n-2 \geq \sum_{i=1}^m \alpha_i$. Let $W\subset \Sigma$ consist of those points where $\Omega_{L_1},\ldots,\Omega_{L_m}$ meet transversally at $\ell$. By Lemma~\ref{lemma:one}, $W$ consists of those points such that \begin{enumerate} \item $\ell \not\subset L_i$ for $1\leq i\leq m$, thus $p_i = \ell\cap L_i$ and $\ell$ is a smooth point of $\Omega_{L_i}$. \item The tangent spaces $T_{\ell}\Omega_{L_i}$ meet transversally. \end{enumerate} Thus $W$ is an open subset of $\Sigma$. We show $W\neq \emptyset$. \smallskip Fix $\ell \in \mbox{\bf G}_1{\mbox{\bf P}^n}$ and distinct points $p_1,\ldots,p_m$ of $\ell$. Define $f : \mbox{Hom}(\ell,{\mbox{\bf P}^n}/\ell)-\{0\} \rightarrow ({\mbox{\bf P}^n}/\ell)^m$ by $\phi \mapsto (\phi(p_1),\ldots,\phi(p_m))$. Let $G\subset Gl_{n+1}$ fix $\ell$ pointwise. Then $G^m$ acts transitively on $({\mbox{\bf P}^n}/\ell)^m$. Choose $(L_1,\ldots,L_m)\in \prod_{i=1}^m\mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$ with $p_i = \ell \cap L_i$. By Theorem 2 (i) of~\cite{Kleiman}, there is a dense open subset $V$ of $G^m$ consisting of those {\boldmath $g$} such that either $f^{-1}(\mbox{\boldmath $g$} ((L_1+\ell)/\ell \times\cdots\times (L_m+\ell)/\ell))$ is empty or its codimension equals that of $(L_1+\ell)/\ell \times\cdots\times (L_m+\ell)/\ell$ in ${\mbox{\bf P}^n}$, which is $\sum_{i=1}^m \alpha_i$. Let {\boldmath $g$} $ = (g_1,\ldots,g_m)\in V$ and set $L_i' = g_i L_i$. Then $f^{-1}((L_1'+\ell)/\ell \times\cdots\times (L_m'+\ell)/\ell)\cup \{0\}$ is the intersection of the tangent spaces $T_\ell\Omega_{L_i'}$ for $1\leq i\leq m$. Since $\alpha_i$ is the codimension of $T_\ell\Omega_{K_i'}$ in $T_{\ell}{\mbox{\bf G}_1{\mbox{\bf P}^n}}$ and $\sum_{i=1}^m \alpha_i \leq 2n-2$, we see that $\Omega_{L_1'},\ldots,\Omega_{L_m'}$ meet transversally at $\ell$. Thus $W\neq \emptyset$. \medskip Let $Z = \Sigma - W$ and $\pi$ be the projection $\Sigma\rightarrow \prod_{i=1}^m \mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$. The desired set $U$ consists of those $(L_1,\ldots,L_m)$ with $\dim(\pi^{-1}(L_1,\ldots,L_m) \bigcap Z)< 2n-2-\sum_{i=1}^m \alpha_i$. $U$ is open and non-empty, for otherwise $\dim Z = \dim \Sigma$, which implies $Z = \Sigma$ and contradicts $W\neq\emptyset$. \QED \subsection{Lemma.}\label{lemma:good_dimension} {\em Let $d, \alpha_1,\ldots,\alpha_m$ be positive integers and $Z$ a subscheme of ${\mbox{\bf G}_1{\mbox{\bf P}^n}}$ with $\dim(Z)<d$. Then the set $W \subset \prod_{i=1}^m\mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$ consisting of those $(K_1,\ldots,K_m)$ for which $\dim(Z\bigcap \Omega_{K_1}\bigcap\cdots\bigcap\Omega_{K_m})< d-\sum_{i=1}^m\alpha_i$ is open and dense. } \medskip \noindent{\bf Proof:} Let $\Sigma\subset Z\times \prod_{i=1}^m\mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$ be the subscheme whose fibre over $(K_1,\ldots,K_m)$ is $Z\bigcap \Omega_{K_1}\bigcap\cdots\bigcap\Omega_{K_m}$. By the upper semicontinuity of fibre dimension, $W$ is open. If $W$ were empty, then all fibres of the projection to $\prod_{i=1}^m\mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$ would have dimension at least $d-\sum_{i=1}^m\alpha_i$ and so $\dim\Sigma \geq d-\sum_{i=1}^m\alpha_i +\sum_{i=1}^m (n-\alpha_i)(\alpha_i+1)$. Projecting to $Z$ exhibits $\Sigma$ as a fibre bundle with fibre over a point $\ell$ of $Z$ equal to $X_1(\ell)\times\cdots \times X_m(\ell)$, where $X_i(\ell)\subset \mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$ is the set of those $K_i\subset\mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$ which meet $\ell$, which has codimension $\alpha_i$. Thus $\Sigma$ has dimension $$ \dim Z - \sum_{i=1}^m \alpha_i + \sum_{i=1}^m (n-\alpha_i)(\alpha_i+1). $$ Since $d> \dim Z$, $W$ must be non-empty. \QED \subsection{Proof of Theorem D, part 1}\label{sec:Proof_D} We show that for each $1\leq i\leq m$ and $0\leq s\leq \alpha_i$, the sets $U_{i,s}$ are open dense subsets of the corresponding products of Grassmannians. Let ${\cal T} = \alpha_1*\cdots*\alpha_{i-1}$. Recall that $U_{i,0}$ consists of those $(K_i,\ldots,K_m)$ such that the intersection $$ \Omega({\cal T}) \bigcap \Omega_{K_i}\bigcap\cdots\bigcap\Omega_{K_m} $$ is generically transverse. Such a cycle has dimension $d = 2n-2 -\sum_{i=1}^m \alpha_i$. Let $Z$ be the singular locus of $\Omega({\cal T})$. The above intersection is generically transverse if $\dim (Z \bigcap \Omega_{K_i}\bigcap\cdots\bigcap\Omega_{K_m}) <d$ and if, for every component $\Omega(T)$ of $\Omega({\cal T})$, the intersection $\Omega(T)\bigcap \Omega_{K_i}\bigcap\cdots\bigcap\Omega_{K_m}$ is generically transverse. By Lemma~\ref{lemma:arrangements}, $\Omega(T) = \Omega(S)\neq \emptyset$ implies that $T=S$, thus $Z$ is a union of intersections of components and the singular loci of components, and hence $\dim(Z) < \dim(\Omega({\cal T}))$. By Lemma~\ref{lemma:good_dimension}, there is an open subset $W$ of $\prod_{j=i}^m \mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}$ consisting those $(K_1,\ldots,K_m)$ for which $\dim(Z\bigcap \Omega_{K_i}\bigcap\cdots\bigcap\Omega_{K_m})<d$. For $T\in {\cal T}$, let $U_T\subset \prod_{j=i}^m\mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}$ be those $(K_i,\ldots,K_m)$ where the intersection $$ \Omega(T)\bigcap \Omega_{K_i}\bigcap\cdots\bigcap\Omega_{K_m} $$ is generically transverse. It suffices to show that for each $T\in {\cal T}$, $U_T$ is a dense open subset of $\prod_{j=i}^m\mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}$, since $U_{i,0} = W\cap\bigcap_{T\in {\cal T}} U_T$. Suppose $T$ has shape $\lambda = (\alpha,\beta)$ and let $V\subset \mbox{\bf Fl}(\lambda)\times\prod_{j=i}^m\mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}$ be those flags for which the intersection $$ \Omega(F,H)\bigcap \Omega_{K_i}\bigcap\cdots\bigcap\Omega_{K_m} $$ is generically transverse. By Lemma~\ref{lemma:gic}, $V$ is dense and open. Note that $$ \{F_{\alpha+1}\subset H_T\}\times U_T = V\bigcap\left( \{F_{\alpha+1}\subset H_T\}\times \prod_{j=i}^m\mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}\right), $$ so $U_T$ is open. Since $\mbox{\bf Fl}(\lambda) = Gl_{n+1} \cdot \{F_{\alpha+1}\subset H_T\}$, and $V$ is stable under the diagonal action of $Gl_{n+1}$, $V = Gl_{n+1} \cdot (\{F_{\alpha+1}\subset H_T\}\times U_T)$. Thus $U_T$ is non-empty. \medskip The case of $U_{i,s}$ for $s>0$ follows by similar arguments. \QED \subsection{Unirationality of ${\cal G}_{i,s}$} \mbox{ }\medskip \noindent{\bf Lemma.}\label{lemma:unirationality} {\em Let $X$ be a projective variety, $U$ a dense open subset of a variety $Y$ which is covered by affine spaces, and suppose that $\Sigma \subset X\times U$ is closed and the projection to $U$ has generically reduced fibres of pure dimension. Then the set ${\cal G}\subset \mbox{\it Chow}\, X$ of fundamental cycles of the fibres of $\,\Sigma$ is unirational. When $k=\mbox{\bf R}$, let ${\cal G}_{\mbox{\scriptsize\bf R}}$ be those cycles lying over $U(\mbox{\bf R})$, then ${\cal G}_{\mbox{\scriptsize\bf R}}$ is real unirational. } \medskip Part 2 of Theorem D is a consequence of this Lemma. \noindent{\bf Proof:} Let $\pi$ be the projection $\Sigma \rightarrow U$. As in \S 2.1, let $U'\subset U$ be the open set where the map $\phi$ which associates a point of $U$ to the fundamental cycle of the fibre of $\pi$ at $x$ is an algebraic morphism. Then $\phi(U')\subset {\cal G}$. We show ${\cal G}\subset \overline{\phi(U')}$. Let $x\in U$ and choose a map $f:{\bf A}^1 \rightarrow Y$ with $f(0) = x$ and $f^{-1}(U')\neq \emptyset$. This is possible as $Y$ has a covering by affine spaces. Then $f^{-1}\Sigma \rightarrow f^{-1}(U)$ is a family over a smooth curve with generically reduced fibres of pure dimension. The association of a point $u$ of $f^{-1}(U)$ to the fundamental cycle of the fibre $\pi^{-1}(f(u))$ gives a map $\psi: f^{-1}(U) \rightarrow \mbox{\it Chow}\, X$ agreeing with $\phi\circ f$ on $f^{-1}(U')$. Thus the fundamental cycle of $\pi^{-1}(x)$ is in $\phi(U')$. If $k = \mbox{\bf R}$, these maps show $\phi(U'(\mbox{\bf R}))\subset{\cal G}_{\mbox{\scriptsize\bf R}} \subset \overline{\phi(U'(\mbox{\bf R}))}$, thus ${\cal G}_{\mbox{\scriptsize\bf R}}$ is real unirational. \QED \section{Construction of Explicit Rational Equivalences} We use the following to parameterize the explicit rational equivalences we construct. \subsection{Lemma.}\label{lemma:limits_are_good} {\em Let ${F\!_{\DOT}}$ be a complete flag in ${\mbox{\bf P}^n}$. Suppose $L_{\infty}$ is a hyperplane not containing $F_n$. Then there exists a pencil of hyperplanes $L_t$, for $t\in \mbox{\bf P}^1 = {\bf A}^1\bigcup \{\infty\}$, such that if $t\neq 0$, then $L_t$ meets the subspaces in ${F\!_{\DOT}}$ properly, and, for each $i\leq n-1$, the family of codimension $i+1$ planes induced by $L_t\bigcap F_i$, for $t\neq 0$ has fibre $F_{i+1}$ over $0$. } \medskip \noindent{\bf Proof:} Let $x_0,\ldots,x_n$ be coordinates for ${\mbox{\bf P}^n}$ such that $L_{\infty}$ is given by $x_n=0$ and $F_i$ by $x_0=\cdots=x_{i-1}=0$. Let $e_0,\ldots,e_n$ be a basis for ${\mbox{\bf P}^n}$ dual to these coordinates and define $$ L_t = \Span{te_j + e_{j+1}\,|\, 0\leq j\leq n-1}. $$ For $t\neq 0$, $L_t \bigcap F_i = \Span{te_j + e_{j+1}\,|\, i\leq j\leq n-1}$ and so has codimension $i+1$. The fibre of this family at $0$ is $\Span{e_{j+1}\,|\, i\leq j\leq n-1} = F_{i+1}$. \QED In the situation of Lemma~\ref{lemma:limits_are_good}, we write $\lim_{t\rightarrow 0} L_t\cap F_i = F_{i+1}$. \medskip For the remainder of this section, fix an arrangement ${\cal F}$. Set ${\cal T} = \alpha_1*\cdots*\alpha_{i-1}$ and let $l$ be the common degree of tableaux in ${\cal T}$. \subsection{Proof of Theorem E, part 1.}\label{sec:proof_E_1} Let $1\leq i\leq m$, and suppose $X_0$ is a cycle in ${\cal G}_{i+1,0}$: $$ X_0 = \Omega({\cal T}*\alpha_i) \bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}. $$ Let $L_\infty$ be any hyperplane which meets ${\cal F}_{l,\alpha_i}$ properly. By Lemma~\ref{lemma:limits_are_good} applied to the flag induced by ${F\!_{\DOT}}$ in ${\mbox{\bf P}^n}/F_{l+2}$ and the hyperplane $L_\infty/F_{l+2}$, there is a pencil $L_t$ of hyperplanes such that if $t\neq 0$ and $i\leq l+1$, then $L_t$ meets $F_i$ properly and $\lim_{t\rightarrow 0} L_t\cap F_i = F_{i+1}$. Let ${\cal X}\subset \mbox{\bf P}^1\times {\mbox{\bf G}_1{\mbox{\bf P}^n}}$ be the subscheme whose fibre at $t\neq 0$ is $$ X_t = \left[ \Omega({\cal T}_{\alpha_i-1,\alpha_i};L_t) + \rule{0pt}{13pt} \Omega({\cal T}^{+\alpha})\right] \bigcap \Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}. $$ Since $L_\infty$ meets ${\cal F}_{l,\alpha_i}$ properly, the set $U'\subset {\mbox{\bf P}^1}$ of $t$ where $L_t$ meets ${\cal F}_{l,\alpha_i}$ properly is open and dense. We claim that $X_0$ is the fibre of ${\cal X}$ over $0$. In that case, let $U''\subset \mbox{\bf P}^1$ be the open subset of those $t$ for which $X_t$ is generically reduced. Since $X_0$ is generically reduced, $0 \in U''$ so $U'' \neq \emptyset$. For $t\in U'\cap U''$, the fibre $X_t\in {\cal G}_{i-1,\alpha_i}$, as $\Omega_{L_t} = {\mbox{\bf G}_1{\mbox{\bf P}^n}}$. The restriction of ${\cal X}$ to $U\cup \{0\}$ gives a family over a smooth curve with generically reduced equidimensional fibres. Thus the association of a point of $U\cup \{0\}$ to its fibre gives a map $\phi: U\cup \{0\}\rightarrow \mbox{\it Chow}\,{\mbox{\bf G}_1{\mbox{\bf P}^n}}$ with $\phi(U)\subset {\cal G}_{i-1,\alpha_i}$ and $\phi(0) = X_0$, proving part 1. For $T\in {\cal T}_{\alpha_i-1,\alpha_i}$, let ${\cal X}_T\subset \mbox{\bf P}^1\times {\mbox{\bf G}_1{\mbox{\bf P}^n}}$ be the subscheme whose fibre at $t\neq 0$ is $$ (X_T)_t = \Omega(T;L_t) \bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}. $$ Since ${\cal X} = \sum_{T\in {\cal T}_{\alpha_i-1,\alpha_i}} {\cal X}_T \ +\ \mbox{\bf P}^1\times \Omega({\cal T}^{+\alpha_i}) \bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$, to show that $X_0$ is the fibre of ${\cal X}$ over $0$ it suffices to show that for each $T\in {\cal T}_{\alpha_i-1,\alpha_i}$, the fibre of ${\cal X}_T$ at 0 is $\Omega(T)\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$. Let $T\in {\cal T}_{\alpha_i-1,\alpha_i}$. If the first row of $T$ has length exceeding $l+1$, then $\Omega(T;L_t) = \Omega(T)$, so ${\cal X}_T$ is the constant family $\mbox{\bf P}^1\times\Omega(T)\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$. Now suppose the first row of $T$ has length $b\leq l+1$. Then, for $t\neq 0$, $\Omega(T;L_t) = \Omega(F_b\cap L_t,H_T)$. Since $\lim_{t\rightarrow 0}F_b\bigcap L_t = F_{b+1}$, we see that $\Omega(T)$ is the fibre over 0 of the family over ${\bf P}^1$ whose fibre over $t\neq 0$ is $\Omega(T;L_t)$. Since $\Omega(T)\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$ is generically transverse, there is an open subset $U_T\subset {\bf P}^1$ such that for $t\in U_T-\{0\}$, $\Omega(T;L_t)\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$ is generically transverse. This shows that the fibre over 0 of ${\cal X}_T$ is $\Omega(T)\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$. \medskip \subsection{Proof of Theorem E, part 2} Let $0\leq s\leq \alpha_i-1$ and suppose $X_0\in {\cal G}_{i,s+1}$ $$ X_0 = \left[\Omega({\cal T}_{s,\alpha_i};N)+ \Omega({\cal T}^{+s+1})\bigcap\Omega_N \right]\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}. $$ Then $N$ has codimension $\alpha_i-s$ in ${\mbox{\bf P}^n}$ and meets ${\cal F}_{l,s+1}$ properly, and the above intersection is generically transverse. We make a useful calculation. Let $L_0 = N\cap H_{l+s+1}$. \medskip \noindent{\bf Lemma. } $$ \Omega({\cal T}^{+s}(\alpha_i-s);N)+\Omega({\cal T}^{+s+1})\bigcap\Omega_N = \Omega({\cal T}^{+s})\bigcap \Omega_{L_0}. $$ \medskip Since ${\cal T}_{s,\alpha_i} = {\cal T}_{s-1,\alpha_i} \bigcup {\cal T}^{+s}(\alpha_i-s)$, we see that $$ X_0 = \left[\Omega({\cal T}_{s-1,\alpha_i};N)+ \Omega({\cal T}^{+s})\bigcap \Omega_{L_0} \right]\bigcap \Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}. $$ \noindent{\bf Proof:} Let $T\in {\cal T}^{+s}$ and suppose that $b$ is the length of the first row of $T$. Then $\hfl{l+s+1}\leq b\leq l$. The degree of $T$ is $l+s$, so $L_0 = N\cap H_{l+s+1}$ meets $H_T$ properly, because $H_T\cap H_{l+s+1}$ equals either $H_{T^+}$ or $F_{\hfl{l+s+1}+1}$, each of which meets $N$ properly. If $T$ is rectangular, $b = \frac{l+s}{2}$ and $\Omega(T) = \mbox{\bf G}_1 H_T$. Thus, $\Omega(T)\bigcap \Omega_{L_0} = \Omega( H_T\cap L_0, H_T)$. Since $L_0$ meets $H_T$ properly, this is generically transverse, by Lemma~\ref{lemma:one}. We calculate $ H_T\cap L_0$. First note that $ H_T\cap H_{l+s+1}=F_{\hfl{l+s+1}+1} = F_{\hfl{l+s}+1}$. So $L_0\cap H_T = N\cap H_{l+s+1}\cap H_T = N\cap F_{\hfl{l+s}+1}$. As $H_T = H_{T(\alpha_i-s)}$, $$ \Omega(F_{b+1},H_T)\bigcap \Omega_{L_0} = \Omega(N\cap F_{\hfl{l+s}+1},H_{T(\alpha_i-s)}) = \Omega(T(\alpha_i-s);N). $$ Suppose $T$ is not rectangular. Since $F_{b+1}\subset F_{\hfl{l+s+1}+1}\subset H_{l+s+1}$, Lemma~\ref{lemma:component_calculation} implies $$ \Omega(F_{b+1},H_T)\bigcap \Omega_{L_0} = \Omega(F_{b+1}\cap N,H_T) + \Omega(F_{b+1},H_T\cap H_{l+s+1}) \bigcap\Omega_N. $$ But this is $\Omega(T(\alpha_i-s);N) + \Omega(T^+)\bigcap \Omega_N$. Summing over $T\in {\cal T}^{+s}$ completes the proof. \QED Let ${N_{\DOT}}$ be any complete flag in $N/F_{l+\alpha_i-s+2}$ refining the images of $$ F_{l+\alpha_i-s+1}\subset N\cap F_l\subset \cdots\subset N\cap F_{\hfl{l+s}+1}\subset L_0. $$ Let $L_\infty$ be any hyperplane of $N$ which meets ${\cal F}_{l,s}$ properly. By Lemma~\ref{lemma:limits_are_good} applied to ${N_{\DOT}}$ in $N/F_{l+\alpha_i-s+2}$, there is a pencil $L_t$ of hyperplanes of $N$ each containing $F_{l+\alpha_i-s+2}$, and for $\hfl{l+s}+1\leq j\leq l$ and $t\neq 0$, $L_t$ meets $N\cap F_j$ properly, with $\lim_{t\rightarrow 0} L_t\cap N\cap F_j = N\cap F_{j+1}$. Since $L_t\cap N\cap F_j = L_t\cap F_j$ for $j$ in this range, $L_t$ meets $F_j$ properly. Let ${\cal X}\subset {\mbox{\bf P}^1}\times{\mbox{\bf G}_1{\mbox{\bf P}^n}}$ be the subscheme whose fibre over $t\in {\mbox{\bf P}^1}$ is $$ X_t = \left[\Omega({\cal T}_{s-1,\alpha_i};L_t) + \Omega({\cal T}^{+s})\bigcap \Omega_{L_t}\right] \bigcap \Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}. $$ Since $L_\infty$ meets ${\cal F}_{l,\alpha_i}$ properly, The set $U'\subset {\mbox{\bf P}^1}$ of $t$ where $L_t$ meets ${\cal F}_{l,\alpha_i}$ properly is open and dense. We claim that $X_0$ is the fibre of ${\cal X}$ over $0$. In that case, let $U''\subset \mbox{\bf P}^1$ be the open subset of those $t$ for which $X_t$ is generically reduced. Since $X_0$ is generically reduced, $0 \in U''$ so $U'' \neq \emptyset$. For $t\in U'\cap U''$, the fibre $X_t\in {\cal G}_{i,s}$. The restriction of ${\cal X}$ to $U\cup \{0\}$ gives a family over a smooth curve with generically reduced equidimensional fibres. Thus the association of a point of $U\cup \{0\}$ to its fibre gives a map $\phi: U\cup \{0\}\rightarrow \mbox{\it Chow}\,{\mbox{\bf G}_1{\mbox{\bf P}^n}}$ with $\phi(U)\subset {\cal G}_{i,s}$ and $\phi(0) = X_0$, proving part 1. For $T\in {\cal T}_{s-1,\alpha_i}$ let ${\cal X}_T$ be the subscheme of ${\mbox{\bf P}^1}\times {\mbox{\bf G}_1{\mbox{\bf P}^n}}$ whose fibre over $t\neq 0$ is $$ ({\cal X}_T)_t = \Omega(T;L_t)\bigcap \Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}. $$ Arguing as at the end of \S\ref{sec:proof_E_1}, we may conclude that $\Omega(T;N)\bigcap \Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$ is the fibre of ${\cal X}_T$ at 0. For $S\in {\cal T}^{+s}$, let ${\cal X}_S$ be the subscheme of ${\mbox{\bf P}^1}\times {\mbox{\bf G}_1{\mbox{\bf P}^n}}$ whose fibre over $t$ is $$ ({\cal X}_S)_t = \Omega(S)\bigcap \Omega_{L_t} \bigcap \Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}. $$ Arguing as at the end of \S\ref{sec:proof_E_1}, we may conclude that $\Omega(S)\bigcap \Omega_{L_0} \bigcap \Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$ is the fibre of ${\cal X}_S$ at 0. Since ${\cal X} = \sum_{T\in {\cal T}_{s-1,\alpha_i}} {\cal X_T} +\sum_{S\in {\cal T}^{+s}}{\cal X}_S$, we conclude that the fibre of ${\cal X}$ at 0 is $X_0$. \QED \section{An Algebra of Tableaux} The Schubert classes, $\sigma_{\lambda}$, form an integral basis for the Chow ring of any Grassmann variety. Thus there exist integral constants $c^{\lambda}_{\mu\,\nu}$ defined by the identity: $$ \sigma_{\mu}\cdot\sigma_{\nu} = \sum_\lambda c^{\lambda}_{\mu\,\nu}\sigma_{\lambda}. $$ In 1934, Littlewood and Richardson~\cite{Littlewood_Richardson} gave a conjectural formula for these constants, which was proven in 1978 by Thomas~\cite{Thomas_schensted_construction}. Lascoux and Sch\"utzenberger~\cite{Lascoux_Schutzenberger_monoid_plactic} constructed the ring of symmetric functions as a subalgebra of a non-commutative associative ring called the plactic algebra whose additive group is the free abelian group $\Lambda$ with basis the set of Young tableaux. For that, each tableau $T$ of shape $\lambda$ determines a monomial summand of the Schur function, $s_{\lambda}$. Evaluating $s_{\lambda}$ at Chern roots of the dual to the tautological bundle of the Grassmannian gives the Schubert class $\sigma_\lambda$. Non-symmetric monomials in these Chern roots are not defined, so individual Young tableaux are not expected to appear in the geometry of Grassmannians. In this context, the crucial use we made of the Schubert varieties $\Omega(T)$ is surprising. A feature of our methods is the correspondence between an iterative construction of the set $\alpha_1*\cdots*\alpha_m$ and the rational curves in the proof of Theorem~E. This suggests an alternate non-commutative associative product $\circ$ on $\Lambda$. The resulting algebra has surjections to the ring of symmetric functions and to Chow rings of Grassmannians. Additional combinatorial preliminaries for this section may be found in~\cite{Sagan}. Here, partitions $\lambda$, $\mu$, and $\nu$ may have any number of rows. Suppose $T$ and $U$ are, respectively, a tableau of shape $\mu$ and a skew tableau of shape $\lambda/\mu$. Let $T\bigcup U$ be the tableau of shape $\lambda$ whose first $|\mu|$ entries comprise $T$, and remaining entries comprise $U$, with each increased by $|\mu|$. For tableaux $S$ and $T$ where the shape of $S$ is $\lambda$, define $$ S \circ T = \sum S\bigcup U, $$ the sum over all $\nu$ and all skew tableaux $U$ of shape $\nu/\lambda$ Knuth equivalent to $T$. For example: \begin{eqnarray*} \setlength{\unitlength}{1.3pt}% \begin{picture}(30,10)(0,12) \thicklines \put(0, 0){\line(1,0){10}} \put(0,10){\line(1,0){30}} \put(0,20){\line(1,0){30}} \put( 0, 0){\line(0,1){20}} \put(10, 0){\line(0,1){20}} \put(20,10){\line(0,1){10}} \put(30,10){\line(0,1){10}} \put( 2,11.8){\bf 1} \put(12,11.8){\bf 2} \put(22,11.8){\bf 3} \put( 2, 1.8){\bf 4} \end{picture} \ \circ\ \begin{picture}(40,10)(0,2) \thicklines \put(0, 0){\line(1,0){40}} \put(0,10){\line(1,0){40}} \put( 0, 0){\line(0,1){10}} \put(10, 0){\line(0,1){10}} \put(20, 0){\line(0,1){10}} \put(30, 0){\line(0,1){10}} \put(40, 0){\line(0,1){10}} \put( 2, 1.8){\bf 1} \put(12, 1.8){\bf 2} \put(22, 1.8){\bf 3} \put(32, 1.8){\bf 4} \end{picture} &=& \setlength{\unitlength}{1.3pt}% \begin{picture}(70,10)(0,12) \thicklines \put(0, 0){\line(1,0){10}} \put(0,10){\line(1,0){70}} \put(0,20){\line(1,0){70}} \put( 0, 0){\line(0,1){20}} \put(10, 0){\line(0,1){20}} \put(20,10){\line(0,1){10}} \put(30,10){\line(0,1){10}} \put(40,10){\line(0,1){10}} \put(50,10){\line(0,1){10}} \put(60,10){\line(0,1){10}} \put(70,10){\line(0,1){10}} \put(0,0){\usebox{\Shading}} \put(32,11.8){\bf 5} \put(42,11.3){\bf 6} \put(52,11.3){\bf 7} \put(62,11.3){\bf 8} \end{picture} \ +\ \setlength{\unitlength}{1.3pt}% \begin{picture}(60,10)(0,12) \thicklines \put(0, 0){\line(1,0){20}} \put(0,10){\line(1,0){60}} \put(0,20){\line(1,0){60}} \put( 0, 0){\line(0,1){20}} \put(10, 0){\line(0,1){20}} \put(20, 0){\line(0,1){20}} \put(30,10){\line(0,1){10}} \put(40,10){\line(0,1){10}} \put(50,10){\line(0,1){10}} \put(60,10){\line(0,1){10}} \put(0,0){\usebox{\Shading}} \put(12, 1.6){\bf 5} \put(32,11.8){\bf 6} \put(42,11.3){\bf 7} \put(52,11.3){\bf 8} \end{picture} \ +\ \setlength{\unitlength}{1.3pt}% \begin{picture}(50,10)(0,12) \thicklines \put(0, 0){\line(1,0){30}} \put(0,10){\line(1,0){50}} \put(0,20){\line(1,0){50}} \put( 0, 0){\line(0,1){20}} \put(10, 0){\line(0,1){20}} \put(20, 0){\line(0,1){20}} \put(30, 0){\line(0,1){20}} \put(40,10){\line(0,1){10}} \put(50,10){\line(0,1){10}} \put(0,0){\usebox{\Shading}} \put(12, 1.6){\bf 5} \put(22, 1.3){\bf 6} \put(32,11.8){\bf 7} \put(42,11.3){\bf 8} \end{picture} \ + \raisebox{-12pt}{\rule{0pt}{5pt}} \\ & & \setlength{\unitlength}{1.3pt}% \begin{picture}(70,20)(0,12) \thicklines \put(0,-10){\line(1,0){10}} \put(0, 0){\line(1,0){10}} \put(0,10){\line(1,0){60}} \put(0,20){\line(1,0){60}} \put( 0,-10){\line(0,1){30}} \put(10,-10){\line(0,1){30}} \put(20,10){\line(0,1){10}} \put(30,10){\line(0,1){10}} \put(40,10){\line(0,1){10}} \put(50,10){\line(0,1){10}} \put(60,10){\line(0,1){10}} \put(0,0){\usebox{\Shading}} \put( 2,-8.2){\bf 5} \put(32,11.3){\bf 6} \put(42,11.3){\bf 7} \put(52,11.3){\bf 8} \end{picture} \ +\ \setlength{\unitlength}{1.3pt}% \begin{picture}(60,20)(0,12) \thicklines \put(0,-10){\line(1,0){10}} \put(0, 0){\line(1,0){20}} \put(0,10){\line(1,0){50}} \put(0,20){\line(1,0){50}} \put( 0,-10){\line(0,1){30}} \put(10,-10){\line(0,1){30}} \put(20, 0){\line(0,1){20}} \put(30,10){\line(0,1){10}} \put(40,10){\line(0,1){10}} \put(50,10){\line(0,1){10}} \put(0,0){\usebox{\Shading}} \put( 2,-8.2){\bf 5} \put(12, 1.8){\bf 6} \put(32,11.3){\bf 7} \put(42,11.3){\bf 8} \end{picture} \ +\ \setlength{\unitlength}{1.3pt}% \begin{picture}(50,20)(0,12) \thicklines \put(0,-10){\line(1,0){10}} \put(0, 0){\line(1,0){30}} \put(0,10){\line(1,0){40}} \put(0,20){\line(1,0){40}} \put( 0,-10){\line(0,1){30}} \put(10,-10){\line(0,1){30}} \put(20, 0){\line(0,1){20}} \put(30, 0){\line(0,1){20}} \put(40,10){\line(0,1){10}} \put(0,0){\usebox{\Shading}} \put( 2,-8.2){\bf 5} \put(12, 1.3){\bf 6} \put(22, 1.8){\bf 7} \put(32,11.3){\bf 8} \end{picture} \ . \raisebox{-35pt}{\rule{0pt}{5pt}} \end{eqnarray*} This product is related to the composition $*$ of \S\ref{sec:calculus_of_tableaux}: Let $Y_\alpha$ be the unique standard tableau of shape $(\alpha,0)$. Then $T*\alpha$ consists of the summands of $T\circ Y_{\alpha}$ with at most two rows. \medskip \noindent{\bf Theorem F.}\ {\em The product $\circ $ determines an associative non-commutative $\bf Z$-algebra structure on $\Lambda$ with unit the empty tableau $\emptyset$. Moreover, $\circ$ is not the plactic product. } \medskip \noindent{\bf Proof:} In the plactic algebra, the product of two tableaux is always a third, showing $\circ$ is not the plactic product. For any tableau $T$, $\emptyset \circ T = T\circ \emptyset = T$. Note $$ \setlength{\unitlength}{1.3pt}% \begin{picture}(10,10)(0,3) \thicklines \put(0, 0){\line(1,0){10}} \put(0,10){\line(1,0){10}} \put( 0, 0){\line(0,1){10}} \put(10, 0){\line(0,1){10}} \put( 2, 1.8){\bf 1} \end{picture} \ \circ\ \begin{picture}(20,8)(0,3) \thicklines \put(0, 0){\line(1,0){20}} \put(0,10){\line(1,0){20}} \put( 0, 0){\line(0,1){10}} \put(10, 0){\line(0,1){10}} \put(20, 0){\line(0,1){10}} \put( 2, 1.8){\bf 1} \put(12, 1.8){\bf 2} \end{picture} \ = \ \begin{picture}(30,10)(0,3) \thicklines \put(0, 0){\line(1,0){30}} \put(0,10){\line(1,0){30}} \put( 0, 0){\line(0,1){10}} \put(10, 0){\line(0,1){10}} \put(20, 0){\line(0,1){10}} \put(30, 0){\line(0,1){10}} \put( 2, 1.8){\bf 1} \put(12, 1.8){\bf 2} \put(22, 1.8){\bf 3} \end{picture} \ +\ \setlength{\unitlength}{1.3pt}% \begin{picture}(20,20)(0,8) \thicklines \put(0, 0){\line(1,0){10}} \put(0,10){\line(1,0){20}} \put(0,20){\line(1,0){20}} \put( 0, 0){\line(0,1){20}} \put(10, 0){\line(0,1){20}} \put(20,10){\line(0,1){10}} \put( 2,11.8){\bf 1} \put(12,11.8){\bf 3} \put( 2, 1.8){\bf 2} \end{picture} \ \ \ \neq\ \ \ \begin{picture}(30,10)(0,3) \thicklines \put(0, 0){\line(1,0){30}} \put(0,10){\line(1,0){30}} \put( 0, 0){\line(0,1){10}} \put(10, 0){\line(0,1){10}} \put(20, 0){\line(0,1){10}} \put(30, 0){\line(0,1){10}} \put( 2, 1.8){\bf 1} \put(12, 1.8){\bf 2} \put(22, 1.8){\bf 3} \end{picture} \ +\ \setlength{\unitlength}{1.3pt}% \begin{picture}(20,20)(0,8) \thicklines \put(0, 0){\line(1,0){10}} \put(0,10){\line(1,0){20}} \put(0,20){\line(1,0){20}} \put( 0, 0){\line(0,1){20}} \put(10, 0){\line(0,1){20}} \put(20,10){\line(0,1){10}} \put( 2,11.8){\bf 1} \put(12,11.8){\bf 2} \put( 2, 1.8){\bf 3} \end{picture} \ =\ \begin{picture}(20,10)(0,3) \thicklines \put(0, 0){\line(1,0){20}} \put(0,10){\line(1,0){20}} \put( 0, 0){\line(0,1){10}} \put(10, 0){\line(0,1){10}} \put(20, 0){\line(0,1){10}} \put( 2, 1.8){\bf 1} \put(12, 1.8){\bf 2} \end{picture} \circ \setlength{\unitlength}{1.3pt}% \begin{picture}(10,10)(0,3) \thicklines \put(0, 0){\line(1,0){10}} \put(0,10){\line(1,0){10}} \put( 0, 0){\line(0,1){10}} \put(10, 0){\line(0,1){10}} \put( 2, 1.8){\bf 1} \end{picture} \ ,\raisebox{-20pt}{\rule{0pt}{5pt}} $$ so $\circ$ is non-commutative. To show associativity, let $R$, $S$, and $T$ be tableaux. Then $$ R\circ (S\circ T) = \sum R \bigcup W, $$ the sum over $W$ Knuth equivalent to $S\bigcup V$, where $V$ is Knuth equivalent to $T$. Let $U'$ be the first $|S|$ entries in $W$, and $V'$ the last $|T|$ entries, each decreased by $|S|$, thus, $$ R\circ (S\circ T) = \sum R \bigcup U' \bigcup V', $$ the sum over $U'$ Knuth equivalent to $S$ and $V'$ to $T$, which is $(R\circ S)\circ T$. \QED Let $m<n$. For a tableau $T$ of shape $\lambda$, let $\phi(T)$ be the Schur function $s_\lambda$. Define $\phi_{m,\,n}(T)$ to be 0 if $\lambda_1+m \geq n$ or $\lambda_{m+1}\neq 0$ and $\sigma_\lambda$ otherwise. Then $\phi$ and $\phi_{m,n}$ are, respectively, additive surjections from $\Lambda$ to the algebra of symmetric functions and to $A^*\mbox{\bf G}_m{\bf P}^n$. \medskip \noindent{\bf Theorem G.} {\em The maps $\phi$ and $\phi_{m,\,n}$ are $\bf Z$-algebra homomorphisms. } \medskip \noindent{\bf Proof:} For any tableaux $S$ and $T$ of shape $\nu$ and partitions $\lambda$ and $\mu$, there is a natural bijection (given by dual equivalence of Haiman~\cite{Haiman_dual_equivalence}) between the set of tableaux with shape $\lambda/\mu$ Knuth equivalent to $S$ and those Knuth equivalent to $T$, and this common number is $c^\lambda_{\mu\,\nu}$. This shows that $\phi$ is an algebra homomorphism. It follows that $\phi_{m,\,n}$ is as well. \QED \section{Enumerative Geometry and Arrangements Over Finite Fields} A main result of this paper, Theorem~C, shows that any Schubert-type enumerative problem concerning lines in projective space may be solved over $\bf R$. By `solved' over a field $k$, we mean there are flags in ${\bf P}^n_k$ determining Schubert varieties which meet transversally in finitely many points, all of which are defined over $k$. Given an enumerative problem, we feel it is legitimate to inquire over which (finite) fields it may be solved. We present two families of enumerative problems for which this question may be resolved, and consider the problem of finding arrangements over finite fields. \subsection{The $n$ lines meeting four $(n-1)$-planes in ${\bf P}^{2n-1}$.} Given three non-intersecting $(n-1)$-planes $L_1,L_2,$ and $L_3$ in ${\bf P}^{2n-1}$, there are coordinates $x_1,\ldots,x_{2n}$ such that \begin{eqnarray*} L_1 &:& x_1 = x_2 = \cdots = x_n = 0\\ L_2 &:& x_{n+1} = \cdots = x_{2n} = 0\\ L_3 &:& x_1 - x_{n+1} = \cdots = x_n - x_{2n}= 0 \end{eqnarray*} One may check that $ \Omega_{L_1}\bigcap\Omega_{L_2}\bigcap\Omega_{L_3}$ is a transverse intersection, and if $\Sigma_{1,n-1}\subset {\bf P}^{2n-1}$ is the union of the lines meeting each of $L_1,L_2,$ and $L_3$, then $\Sigma_{1,n-1}$ is the image of the standard Segre embedding of ${\bf P}^1\times{\bf P}^{n-1}$ into ${\bf P}^{2n-1}$ (cf.~\cite{Harris_geometry}): $$ \psi:[a,b]\times[y_1,\ldots,y_n] \longmapsto [ay_1,\ldots,ay_n,by_1,\ldots,by_n]. $$ The lines meeting $L_1$, $L_2$, and $L_3$ are the images of ${\bf P}^1\times \{p\}$, for $p\in{\bf P}^{n-1}$. $\Sigma_{1,n-1}$ has degree $n$, so a general $(n-1)$-plane $L_4$ meets $\Sigma_{1,n-1}$ in $n$ distinct points, each determining a line meeting $L_1,\ldots,L_4$. These lines, $\ell_1,\ldots,\ell_n$, meet $L_1$ in distinct points which span $L_1$. Changing coordinates if necessary, we may assume $\ell_j$ is the span of $x_j$ and $x_{n+j}$. For $1\leq j\leq n$, let $p_j = [\alpha_j,\beta_j]\in {\bf P}^1$ be the first coordinate of $\psi^{-1}(\ell_j\cap L_4)$. Then $$ L_4 \quad :\quad \beta_1x_1 -\alpha_1 x_{n+1}=\cdots= \beta_nx_n - \alpha_n x_{2n}=0. $$ Also, $p_1,\ldots,p_n$ are distinct; otherwise $L_4\cap \Sigma_{1,n-1}$ contains a line. Thus, if $k$ has at least $n-1$ elements, this enumerative problem may be solved over $k$. \subsection{The $n$ lines meeting a fixed line and $n+1$ $(n-1)$-planes in ${\bf P}^{n+1}$.} A line $\ell$ and $(n-1)$-planes $K_1,\ldots,K_n$ in ${\bf P}^{n+1}$ are independent if for every $p\in \ell$, the hyperplanes $\Gamma_i(p) = \Span{p,K_i}$, for $1\leq i\leq n$, meet in a line. In this case, the union $$ S_{1,n-1} = \bigcup_{p\in \ell} \Gamma_1(p)\cap \cdots\cap \Gamma_n(p) $$ is a rational normal surface scroll. Moreover, the lines meeting each of $\ell$, $K_1,\ldots,K_n$ are precisely those lines $\lambda(p) = \Gamma_1(p)\cap \cdots\cap \Gamma_n(p)$ for $p\in \ell$. Since $S_{1,n-1}$ has degree $n$, a general $(n-1)$-plane $K_{n+1}$ meets $S_{1,n-1}$ in $n$ distinct points, each determining a line $\lambda(p)$ which meets $\ell,K_1,\ldots,K_{i+1}$. If $k$ is finite with $q$ elements, there are only $q+1$ lines $\lambda(p)$ defined over $k$. Thus it is necessary that $q\geq n-1$ to solve this problem over $k$. We show this condition suffices. All rational normal surface scrolls are projectively equivalent, (cf.~\cite{Harris_geometry}, \S9), thus we may assume that $S_{1,n-1}$ has the following standard form. Let $x_1,x_2,y_1,\ldots,y_n$ be coordinates for ${\bf P}^{n+1}$ where $\ell$ has equation $y_1 = \cdots = y_n = 0$. Then for $p = [a,b,0,\ldots,0] \in \ell$, $\lambda(p)$ is the linear span of $p$ and the point $[0,0,a^{n-1},a^{n-2}b,\ldots,ab^{n-2},b^{n-1}]$. Let $\alpha_1,\ldots,\alpha_n\in {\bf P}^1$ be distinct points. Let $F = \sum_{i=0}^n A_i b^i a^{n-i}$ be a form on ${\bf P}^1$ vanishing at $\alpha_1,\ldots,\alpha_n$. Define $K_{n+1}$ by the vanishing of the two linear forms $$ \Lambda_1 \ :\ x_2 - y_1\ \ \ \ \ \ \Lambda_2 \ :\ A_0 x_1 + A_1y_1 + \cdots + A_n y_n. $$ The intersection of $S_{1,n-1}$ and the hyperplane defined by $\Lambda_1$ is the rational normal curve $$ \psi : [a,b] \longmapsto [a^n,a^{n-1}b,a^{n-1}b, a^{n-2}b^2,\ldots,a b^{n-1},b^n]. $$ Since $\psi^* (\Lambda_2) = F$, the lines meeting each of $\ell,K_1,\ldots,K_{n+1}$ are $\lambda(\alpha_1),\ldots,\lambda(\alpha_n)$. Thus, if $k$ has at least $n-1$ elements, this enumerative problem may be solved over $k$. \medskip These two families are the only non-trivial examples of Schubert-type enumerative problems for which we know an explicit description of their solutions. Each of these problems can be solved over any field $k$ where $\#{\bf P}^1_k$ exceeds the number of solutions. It would be interesting to find explicit solutions to other enumerative problems to test whether this holds more generally. \subsection{Arrangements over Finite Fields} In \S\ref{sec:arrangements} we remarked it is possible to construct arrangements over some finite fields. Here we show how. Recall that an arrangement is complete flag ${F\!_{\DOT}}$ and $2n-3$ hyperplanes $H_2,\ldots,H_{2n-2}$ such that for any $p$, \begin{enumerate} \item $H_p \cap F_{\hfl{p}} = F_{\hfl{p}+1}$, \item For tableaux $S, T$ of degrees at most $p-1$, if $H_T\cap H_S \subset H_p$, then $H_T\cap H_S = F_{\hfl{p}+1}$. \end{enumerate} We give an equivalent set of conditions. For any subset $A\neq \emptyset$ of $\{2,3,\ldots,2n-2\}$, let $H_A$ be $\bigcap\{H_i\,|\, i\in A\}$. Set $H_{\emptyset} = {\mbox{\bf P}^n}$. \medskip \noindent{\bf Lemma.}\label{lemma:alt_def_arrangement} {\em A complete flag ${F\!_{\DOT}}$ and hyperplanes $\,H_2,\ldots,H_{2n}$ constitute an arrangement if and only if for each $m = 1,\ldots,n-1$, they satisfy \begin{enumerate} \item [1$'$.] $H_{2m}\cap F_m = F_{m+1}$ and $F_{m+1}\subset H_{2m+1}$. \item [2$'$.] For any $A\subset \{2,3,\ldots,2m\}$ where $H_A$ has codimension $\#A \leq m$, $H_A\not\subset H_{2m+1}$. \end{enumerate} } \medskip \noindent{\bf Proof:} Let $H_2,\ldots,H_{2n-2}$ and ${F\!_{\DOT}}$ satisfy 1$'$ and 2$'$. We show they constitute an arrangement by induction on $m$. Suppose that for $p<2m$ conditions 1 and 2 for arrangements are satisfied. We show that 1 and 2 are satisfied for $p = 2m$ and $2m+1$. For $p=2m$, 1 and 1$'$ are equivalent. Moreover, if $S,T$ are tableaux of degree less than $2m$, then $F_m \subset H_T\cap H_S$, so $H_T\cap H_S\not\subset H_{2m}$, so 2 is satisfied. Thus $H_2,\ldots,H_{2m}$, $F_1,\ldots,F_{m+1}$ constitute an arrangement in ${\bf P}^n/F_{m+2}\simeq {\bf P}^{m+1}$. Then by Lemma~\ref{lemma:arrangements}, if $T$ has shape $(\alpha,\beta)$ with $\alpha+\beta < 2m$, $H_T$ has codimension $\beta$ and is not equal to $F_\beta$. Let $S,T$ be tableaux of degree at most $2m$ and suppose $F_{m+1}\neq H_S\cap H_T$. Let $B$ be the union of the second rows of $S$ and $T$. Since $H_B = H_S\cap H_T$ has codimension $s \leq m$, there is a set $A\subset B$ of order $s$ with $H_A = H_B$. By 2$'$, $ H_S\cap H_T = H_A \not\subset H_{2m+1}$. So $H_{2m+1}$ satisfies 1 and 2. Conversely, suppose $H_2,\ldots,H_{2n-2}$ and ${F\!_{\DOT}}$ constitute an arrangement. These satisfy 1$'$. To show they satisfy 2$'$, let $A\subset \{2,\ldots,2m\}$ where $H_A$ has codimension $s = \# A$. Suppose $A = a_1<\cdots<a_s$ and let $j$ be the largest index such that $a_j<2j$. If $j=0$, then $A$ is the second row of a tableau $T$, so $H_A = H_T \not\subset H_{2m+1}$. If $j\neq 0$, then $j\geq 2$. Since $p<2j$ implies $F_j\subset H_p$ and $H_A$ has codimension $s$, we have $F_j = H_{a_1}\cap \cdots\cap H_{a_j}$. An induction using condition 1$'$ shows that $F_j = H_3\cap H_2\cap H_4\cap \cdots\cap H_{2j-2}$. Thus $H_A = H_{A'}$ where $$ A' = 3,2<\cdots<2j-2<a_{j+1}<\cdots<a_s, $$ and if $i>j$, $a_i\geq 2i$. Let $B: 2<\cdots<2j-2<a_{j+1}<\cdots<a_s$. then we see that $B$ is the second row a tableau $T$ of degree at most $2s$. Let $S$ be the tableau of degree 3 whose second row consists of $3$. Since $s\leq m$, $F_{m+1} \neq H_S\cap H_T$, so $H_A = H_S\cap H_T \not\subset H_{2m+1}$. \QED \subsection{Corollary.}\label{sec:refined_condition} {\em Condition $2'$ may be replaced by \begin{enumerate} \item[2$''$.] If $A: 3,2<\cdots<2j-2<a_{j+1}<\cdots<a_m\leq 2m$ satisfies $i>j$ implies $a_i\geq 2i$, then $H_A\not\subset H_{2m+1}$. \end{enumerate}} \medskip \noindent{\bf Proof:} Suppose $H_2,\ldots,H_{2n-2}$ and ${F\!_{\DOT}}$ satisfy 1$'$ and 2$'$ and $A : 3,2<\cdots<2j-2<a_{j+1}<\cdots<a_m\leq 2m$ satisfies $i>j$ implies $a_i \geq 2i$. Then $2<\cdots<2j-2<a_{j+1}<\cdots<a_m$ is the second row of a tableau $T$. By Lemma~\ref{lemma:arrangements}, $H_{T}$ has codimension $m-1$ and does not equal $F_{m-1}$. Thus $H_T\not\subset H_3$ so $H_A = H_T\cap H_3$ has codimension $m$. Conversely, suppose $H_2,\ldots,H_{2n-2}$ and ${F\!_{\DOT}}$ satisfy 1$'$ and 2$''$. {}From the proof of the previous Lemma, it suffices to know 2$'$ for those subsets $A$ of the form $3,2<\cdots<2j-2<a_{j+1}<\cdots<2s\leq 2m$, where $i>j$ implies $a_i\geq 2i$. If $B = A\cup\{2s+2,\ldots,2m\}$, then $B$ also has the form in 2$''$. So $H_B \not\subset H_{2m+1}$. Since $H_B\subset H_A$, $H_A\not\subset H_{2m+1}$, showing 2$'$ holds for $H_2,\ldots,H_{2n-2},{F\!_{\DOT}}$. \QED We estimate the size of a field $k$ necessary to construct an arrangement. \subsection{Theorem.}\label{thm:arrangement_finite_field} {\em There exists an arrangement in ${\bf P}^n_k$ if the order of $k$ is at least $$ \frac{(2n-4)!}{(n-2)!(n-1)!}\ +\ \sum_{i=1}^{n-4} \frac{(2i)!}{i! (i+1)!}. $$ } \medskip \noindent{\bf Proof:} Consider the problem of inductively constructing an arrangement in ${\bf P}^n_k$ satisfying 1$'$ and 2$''$ of \S\S\ref{lemma:alt_def_arrangement} and~\ref{sec:refined_condition}. Since it is always possible to find a hyperplane not containing any particular proper linear subspace of ${\mbox{\bf P}^n}$, the only possible obstruction is the selection of hyperplanes $H_{2m+1}\supset F_{m+1}$ satisfying 2$''$ for $m=0,1,\ldots,n-2$. Let $\check{\bf P}^m$ be the set of hyperplanes defined over $k$ containing $F_{m+1}$. Every codimension $m$ subspace $H_A$ containing $F_{m+1}$ determines a hyperplane $\check{H}_A$ in $\check{\bf P}^m$ consisting of those hyperplanes $H$ of ${\bf P}^n$ containing $H_A$. Thus there exists a hyperplane $H_{2m+1}$ satisfying 2$'$ if, as sets of $k$-points, $$ X = \check{\bf P}^m - \bigcup_{A\in {\cal S}} \check{H}_A\neq \emptyset, $$ Where ${\cal S}$ is the set of those sequences $3,2<\cdots<2j-2<a_{j+1}<\cdots<a_m\leq 2m$ such that if $i>j$, then $a_i\geq 2i$. We claim $\#{\cal S} = s_m = \frac{(2m)!}{m!(m+1)!} + \sum_{i=1}^{m-2} \frac{(2i)!}{i! (i+1)!}$. Suppose $k$ has $q\geq s_{n-2}$ elements. Since $\check{\bf P}^m$ has $(q^{m+1}-1)/(q-1)$ elements and each $\check{H}_A$ has $(q^m-1)/(q-1)$ elements, $X$ is non-empty if $q^{m+1}-1 > (q^m-1)s_m$. This holds as $$ \left\lfloor\frac{q^{m+1}-1}{q^m-1}\right\rfloor \geq q \geq s_{n-2}\geq s_m. $$ To enumerate ${\cal S}$, let $\{3,2<\cdots<2j-2<a_{j+1}<\cdots<a_m\} \in {\cal S}$. If $b_j = a_{j+i} - 2j$, then $b_1,\ldots, b_{m-j}$ is the second row of a tableau of shape $(m-j,\,m-j)$. Conversely, if $b_1,\ldots, b_{m-j}$ is the second row of a tableau of shape $(m-j,\,m-j)$, then $$ \{3, 2<\cdots<2j-2<b_1+2j<\cdots<b_{m-j}+2j\} \in {\cal S}. $$ Let ${\cal T}_s$ be the set of tableaux of shape $(s,s)$. These considerations show there is a bijection $$ {\cal S} \longleftrightarrow {\cal T}_m\cup{\cal T}_{m-2}\cup{\cal T}_{m-3}\cup\ldots\cup{\cal T}_0. $$ Noting that $\#{\cal T}_s = \frac{(2s)!}{s!(s+1)!}$, by the hook length formula of Frame, Robinson, and Thrall~\cite{FRT}, shows that the order of ${\cal S}$ is $\frac{(2m)!}{m!(m+1)!}+\sum_{i=1}^{m-2} \frac{(2i)!}{i! (i+1)!}$. \QED This result is not the best possible: For ${\bf P}^4$, this gives $q\geq 5$, but arrangements in ${\bf P}^4$ may be constructed over the field with three elements.

9510

alg-geom/9510002

en

https://arxiv.org/abs/alg-geom/9510002

alg-geom/9510002

Lev Borisov

Lev A. Borisov

Finiteness Theorem for Sp(4,Z)

We consider Siegel upper half space of rank two ${\cal H}^2$ and different subgroups $H\subseteq {\bf Sp(4,Z)}$ of finite index. The purpose of this paper is to prove that the field of rational functions of ${\cal H}^2/H$ has general type for all but the finite number of $H$.

alg-geom

\section{Introduction} The Siegel upper half space of rank two consists of complex symmetric two by two matrices whose imaginary part is positive definite. It will be denoted by ${\cal H}$ throughout the paper. It is the moduli space of principally polarized marked abelian surfaces. The group ${\rm Sp(4,{\bf Z})}$ acts on ${\cal H}$ by the automorphisms of the marking. This group consists of four by four integer matrices of the form $\pmatrix{A&B \cr C&D\cr}$ where $A,B,C,$ and $D$ are two by two matrices that obey $A{}^tB=B{}^tA,C{}^tD=D{}^tC,A{}^tD-B{}^tC={\bf 1}.$ Written in coordinates, this action becomes $$\pmatrix{A&B\cr C&D\cr}\cdot M=(AM+B)(CM+D)^{-1}.$$ It is a natural generalization of the usual upper half plane with the action of ${\rm Sl(2,{\bf Z})}$. It is related to various moduli spaces of abelian surfaces in the same way the usual upper half plane is related to moduli spaces of elliptic curves. We shall be concerned mostly with quotients of ${\cal H}$ by the action of subgroups $H$ of finite index in ${\rm Sp(4,{\bf Z})}.$ These quotients are known to be algebraic varieties of dimension $3$. They have been studied extensively since the end of last century. Some of these varieties have extremely rich and beautiful geometry, see for instance \cite{Geer},\cite{Lee} and \cite{GeerII}. The goal of this paper is to prove the following statement, see proposition \ref{fintheorem}. {\bf Finiteness Theorem.} There are only finitely many subgroups $H\subseteq {\rm Sp(4,{\bf Z})}$ of finite index such that ${\cal H}/H$ is not of general type. The important corollary of this result is that there are only finitely many subgroups $H$ such that the quotient ${\cal H}/H$ is rational. Varieties of general type can be viewed as the generalization to higher dimension of curves of genus two or more. It is reasonable to expect that they do not have any special geometric properties, and thus all interesting quotients ${\cal H}/H$ can be in principle listed. This theorem is analogous to the result of J.G. Thompson (see \cite{Thompson}) for the usual upper half plane. More accurate estimates for certain classes of subgroups of ${\rm Sp(4,{\bf Z})}$ have been proved in \cite{Grady,Gritsenko,Hulek}. The method of the proof is roughly the following. It is known that $H$ contains a principal congruence subgroup $\Gamma(n)$ of some level $n$. The quotient ${\cal H}/\Gamma(n)$ admits a well understood smooth compactification, constructed in the paper of Igusa \cite{Igusa}. Our aim is to construct global sections of the multicanonical line bundle on the desingularization of the compactification of $({\cal H}/\Gamma(n))/(H/\Gamma(n))$ from the sections of certain line bundles on the Igusa compactification of ${\cal H}/\Gamma(n)$. We will use standard facts about singular algebraic varieties, which are collected in Section 7. The results of Sections 2 and 4 are probably known to specialists in the field, although there are not many convenient references. Section 3 and 5 are the key sections of the paper. The former is a purely combinatorial calculation, and the latter is an algebra-geometrical one. In both sections we assume that $n$ is a power of a prime, and Section 6 allows us to drop this restriction. This paper is essentially my University of Michigan thesis. Major part of it was done when I was still in Moscow. It is influenced a lot by my advisor Vasilii Iskovskikh who taught me the basics of algebraic geometry as well as some singularity theory which comes in very handy in the paper. I would like to thank Osip Shvartsman for many stimulating discussions on the subject of this paper. My thesis advisor Igor Dolgachev has been a constant source of inspiration for my studies of algebraic geometry at the University of Michigan. I also wish to thank Gopal Prasad for several valuable conversations and Melvin Hochster for providing a useful reference. \section{Algebraic cycles on Satake and Igusa compactifications} The purpose of this section is to recall the basic facts about some special algebraic cycles on the Satake and Igusa compactifications of ${\cal H}/\Gamma(n)$ and to find a nice combinatorial description of their components. We consider the principal congruence subgroup $\Gamma(n)$ of level $n$ inside ${\rm Sp(4,{\bf Z})}$. For the rest of the section $n$ is fixed and is greater than two. The group $\Gamma(n)$ acts on the Siegel upper half space of rank two ${\cal H}$ according to the formula $$\pmatrix{A&B\cr C&D\cr}\cdot\tau=(A\tau+B)(C\tau+D)^{-1}.$$ The quotient ${\cal H}/\Gamma(n)$ is a nonsingular algebraic variety. It is a Zariski open subset of the compact singular algebraic variety called the Satake compactification of ${\cal H}/\Gamma(n)$. The exact references can be found in \cite{Igusa}. The monoidal transformation of the Satake compactification along the singular locus is nonsingular. This variety was first considered by Igusa in \cite{Igusa}, and is called the Igusa compactification. We denote it by $X_n$. Points of ${\cal H}/\Gamma(n)$ are referred to as the {\it finite}\/ part of the compactification and the rest is the part {\it at infinity.} The part at infinity of the Satake compactification consists of a finite number of curves that intersect in a finite number of {\it cusp}\/ points. The part at infinity of the Igusa compactification is a divisor $D=\sum_iD_i$, which has simple normal crossings. Its components are elliptic fibrations over the curves at infinity of the Satake compactification. The group $G=\Gamma(1)/\Gamma(n)$ acts on both compactifications, and the map between them is equivariant. The group $G$ is isomorphic to ${\rm Sp(4,{\bf Z}/n{\bf Z})}$, and $\pm{\bf 1}$ act as the identity. There are two more types of divisors on the Igusa compactification that will be important to our discussion. First of all, there are divisors $E_i$ that are conjugates of the closure of the image in ${\cal H}/\Gamma(n)$ of the set of diagonal matrices in ${\cal H}$. They are disjoint and are isomorphic to the product of two modular curves (see \cite{Yamazaki}). We denote their sum by $E$. We also consider divisors that are conjugates of the closure of the image of the set of matrices $\pmatrix{x&y\cr y&x\cr}$ in ${\cal H}$. Geometrically, these matrices correspond to Jacobians of genus two curves with an extra involution, see \cite{Bolza}. We denote them by $F_i$ and their sum by $F$. They do intersect with each other and their geometry is somewhat more complicated. We prove the necessary statements regarding these at the end of this section. We abuse notation somewhat to denote ${\rm Sp(4,{\bf Z})}$-conjugates of the sets $\pmatrix{x&0\cr 0&z\cr}$ and $\pmatrix{x&y\cr y&x\cr}$ by $E_i$ and $F_j$ as well. Let us introduce the abelian group $V$ of column vectors of height four with coefficients in ${\bf Z}/n{\bf Z}$ provided with the skew form $\langle ~,~\rangle $ defined by the formula $\langle {}^t(x^1,...,x^4),{}^t(y^1,...,y^4)\rangle =x^1y^3+x^2y^4-x^3y^1-x^4y^2.$ The group $G$ acts naturally on $V$ by left multiplication. Our goal here is to construct $G$-equivariant correspondences between components of cycles on the Satake and Igusa compactifications mentioned above and some objects defined in terms of the group $V$. \begin{prop} { The infinity divisors of the Igusa compactification (or equivalently, the curves at infinity of the Satake compactification) are in one-to-one $G$- equivariant correspondence with the primitive $\pm$vectors $\pm v$ in $V$. Here we call a vector $v$ primitive iff its order is exactly $n$. The $\pm$ means that we identify opposite vectors.} \label{indexD} \end{prop} {\em Proof.} It is known (see \cite{Igusa}) that all components of $D$ are $G$-conjugate. It can be shown that the group $G$ also acts transitively on the set of primitive $\pm$vectors. It remains to notice that the stabilizer of the $\pm$vector ${}^t(0,1,0,0)$ coincides with the stabilizer of $D_0$, where $D_0$ is the {\it standard}\/ divisor that corresponds to the basis of open subsets $\{\pmatrix{x&y\cr y&z\cr},~{\rm Im}(z)\to+\infty\}$ of ${\cal H}.$ The description of the stabilizer of $D_0$ can be derived from \cite{Igusa}. It consists of matrices of the form $$\pm\pmatrix{a&0&b&m_3\cr m_1&1&m_2&m_4\cr c&0&d&m_5\cr 0&0&0&1\cr}({\rm mod}n),$$$$~ad-bc=1({\rm mod}n), ~bm_1+m_3=am_2({\rm mod}n),~dm_1+m_5=cm_2({\rm mod}n).$$ This allows us to construct a bijective correspondence between infinity divisors on the Igusa compactification and $\pm$vectors in $V.$ We will use the notation $\pm v_\alpha$ for the $\pm$vector that corresponds to the divisor $D_\alpha$ and vice versa.\hfill$\Box$ \begin{prop} { Cusp points $Q_i$ of the Satake compactification are in one-to-one $G$-equivariant correspondence with the following pairs $(W,\pm f)$. We consider all possible $W\subset V$ and $f:W\times W\to {{\bf Z}/n{\bf Z}}$ such that (1) $W$ is a subgroup of $\,V$ isomorphic to $({{\bf Z}/n{\bf Z})}^2$, (2) $\langle ,\rangle |_W=0$, (3) $f$ is a non-degenerate skew form on $W$ with values in ${{\bf Z}/n{\bf Z}}$, where non-degeneracy means $f(W\times W)\ni 1(n)$.} \label{indexptsSatake} \end{prop} {\em Proof.} All cusp points are conjugates of the one described by the basis of open sets $$\{\pmatrix{x&y\cr y&z\cr},~{\rm Im}(\pmatrix{x&y\cr y&z\cr})\to+\infty\}$$ (see \cite{Igusa}). We call this point {\it standard}\/. The stabilizer of the standard point consists of matrices of the form $$\{\pmatrix{A&B\cr {\bf 0}&{}^tA^{-1}\cr}, A{}^tB=B{}^tA, det(A)=\pm1(n)\}$$ However, this is exactly the stabilizer of the {\it standard}\/ pair $$(W,\pm f)=({}^t(*,*,0,0),f({}^t(1,0,0,0),{}^t(0,1,0,0))=1(n)).$$ It can be shown that any pair $(W,f)$ is a $G$-conjugate of the standard pair. As a result, we can define the required $G$-equivariant correspondence. We will use the notation $(W_\alpha,\pm f_\alpha)$ for the pair that corresponds to the point $Q_\alpha$ and vice versa. \hfill$\Box$ \begin{prop} { The curve at infinity of the Satake compactification that corresponds to the divisor $D_\alpha$ contains the cusp point $Q_\beta$ iff $v_\alpha\in W_\beta$.} \label{indexcurvethroughpointonSatake} \end{prop} {\em Proof.} Consider the action of the group that stabilizes the standard curve. It acts transitively on the set of cusp points of this curve, which are exactly the $Q_i$'s. Therefore, all inclusion pairs are acted upon transitively. The standard curve passes through the standard point, and ${}^t(0,1,0,0)\in {{}^t(}*,*,0,0)$, which proves the only if part of the statement. On the other hand, the stabilizer of the standard point acts transitively on the $\pm$vectors in ${}^t(*,*,0,0)$, which proves the if part. \hfill$\Box$ \begin{prop} { Two infinity divisors $D_\alpha$ and $D_\beta$ intersect over the point $Q_\delta$ iff $v_\alpha,v_\beta \in W_\delta$ and $f_\delta(v_\alpha,v_\beta)=\pm 1(n)$. In this case the intersection is isomorphic to ${\bf P}^1$.} \label{indexDD} \end{prop} {\em Proof.} Because of transitivity of the action, the point $Q_\delta$ may be considered standard. We follow the argument of \cite{Igusa} for the case where $g_0=0$ and $g_1=2$. Curves of the intersection of the two infinity divisors are conjugate to one of the curves obtained by taking the limits of the points $\pmatrix{x&y\cr y&z\cr}$, with imaginary parts of two out of three normal coordinates $y,(-x-y),(-z-y)$ going to $-\infty$ and the remaining one being bounded. They are pairwise intersections of the divisors that correspond to the limits where exactly one of the imaginary parts goes to $-\infty$ and the other two are bounded. The divisor that corresponds to $Im(z+y)\to\infty$ is exactly the standard divisor, because ${\rm Im}(y)$ is bounded. The other two divisors are obtained from the standard one by the action of $\{\pmatrix{A&{\bf 0}\cr {\bf 0}&{}^tA^{-1}\cr}\}$ with $A=\pmatrix{1&-1\cr 0&1\cr}$ and $A=\pmatrix{0&1\cr 1&0\cr}$ respectively. Therefore, these divisors correspond to the $\pm$vectors $\pm {}^t(0,1,0,0)$, $\pm {}^t(-1,1,0,0)$, and $\pm {}^t(1,0,0,0).$ This proves the "only if" part. The "if" part follows from the transitivity of the action of $G$ on the combinatorial data on the right hand side of the statement. The fact that each irreducible component of the intersection is isomorphic to ${\bf P}^1$ is proven in \cite{Igusa}, and the uniqueness of the irreducible component can be derived easily from the description of the divisors in terms of the above limits. \hfill$\Box$ \begin{prop} { Three infinity divisors $D_\alpha, D_\beta, D_\gamma$ intersect over the point $Q_\delta$ iff $v_\alpha,v_\beta,v_\gamma \in W_\delta$, the set $\{\pm v_\alpha \pm v_\beta \pm v_\gamma\}$ contains $0$, and $f_\delta(v_\alpha,v_\beta)=\pm 1(n)$. In this case the intersection point is unique.} \label{indextDDD} \end{prop} {\em Proof.} As in the previous proposition, we prove that all points of triple intersection are conjugates of the intersection point of the divisors that correspond to $\pm {}^t(0,1,0,0)$, $\pm {}^t(-1,1,0,0)$, and $\pm {}^t(1,0,0,0).$ Then we notice that any triple of $\pm$vectors with the above properties can be transformed to the triple $(\pm {}^t(0,1,0,0)$, $\pm {}^t(-1,1,0,0)$, $\pm {}^t(1,0,0,0)).$ \hfill$\Box$ \begin{prop} { Divisors $E_i$ are in one-to-one $G$-equivariant correspondence with unordered pairs $(W_1,W_2)$ such that (1) $W_1$ and $W_2$ are subgroups of $V$ isomorphic to $({{\bf Z}/n{\bf Z}})^2$ each, (2) $W_1 \perp W_2=V$.} \label{indexE} \end{prop} {\em Proof.} All divisors $E_i$ are conjugates of the {\it standard}\/ one defined as the closure of the image of the set of diagonal matrices. The stabilizer of this standard divisor is described in \cite{Yamazaki}. It is exactly the stabilizer of the {\it standard}\/ pair $({}^t(*,0,*,0),{}^t(0,*,0,*)).$ It is easy to show that every pair $(W_1,W_2)$ with above properties is conjugate to this standard one, which completes the proof. For a given $E_\alpha$ the corresponding pair will be denoted by $(W_{\alpha1},W_{\alpha2})$ and vice versa. \hfill$\Box$ \begin{prop} { The divisor $E_\alpha$ intersects the divisor $D_\beta$ iff $v_\beta$ lies in one of the subgroups $W_{\alpha1},W_{\alpha2}$. In this case the intersection is isomorphic to the modular curve of principal level $n$.} \label{indexED} \end{prop} {\em Proof.} We assume that the divisor $E_\alpha$ is a standard one. Then the statement of the proposition follows from the description of the action of the group $\Gamma(n)$ in a neighborhood of the set of diagonal matrices (see \cite{Yamazaki}). \hfill$\Box$ There is an alternative way to describe divisors $E_i$. \begin{prop} { Divisors $E_i$ are in one-to-one $G$-equivariant correspondence with conjugates of the involution $$\pm\pmatrix{1&0&0&0\cr 0&-1&0&0\cr 0&0&1&0\cr 0&0&0&-1\cr}$$ in the group ${\rm Sp(4,{\bf Z}/n{\bf Z})/\{\pm 1\}}.$} \label{indexinvE} \end{prop} {\em Proof.} The action of $$\pm\pmatrix{1&0&0&0\cr 0&-1&0&0\cr 0&0&1&0\cr 0&0&0&-1\cr}$$ on ${\cal H}$ is defined by $\pmatrix{x&y\cr y&z\cr}\to \pmatrix{x&-y\cr -y&z\cr}$, so it fixes exactly the points of the standard divisor $E_0.$ This gives a one-to-one correspondence between $\Gamma(1)$ conjugates of this involution and $\Gamma(1)$ conjugates of the diagonal in ${\cal H}.$ This correspondence survives when we mod out by $\Gamma(n)$, and then we use surjectivity of $\Gamma(1)/\Gamma(n)\to {\rm Sp(4,{\bf Z}/n{\bf Z})}.$ \hfill$\Box$ This alternative description is related to the original one as follows. \begin{prop} { The involution $\varphi_\alpha$ that fixes all points of the divisor $E_\alpha$ is defined by (1) $\varphi_\alpha |_{W_{\alpha1}}={\rm id} |_{W_{\alpha1}}$, (2) $\varphi_\alpha |_{W_{\alpha2}}=-{\rm id} |_{W_{\alpha2}}$. This definition makes sense, because the switch of the order of two subgroups $W_{\alpha1},W_{\alpha2}$ results in the change of sign of the involution $\varphi_\alpha$.} \label{twodescrE} \end{prop} {\em Proof.} It is true for the standard divisor, and the rest follows from the transitivity of the action of the group $G$. \hfill$\Box$ We can describe divisors $F_i$ in the same fashion, because there is also an involution in ${\rm Sp(4,{\bf Z})}$, whose fixed points on ${\cal H}$ are exactly the matrices $\pmatrix{x&y\cr y&x\cr}$ that form the standard divisor $F_0.$ \begin{prop} { Divisors $F_i$ are in one-to-one $G$-equivariant correspondence with conjugates in ${\rm Sp(4,{\bf Z}/n{\bf Z})/\{\pm 1\}}$ of the involution $$\pm\pmatrix{0&1&0&0\cr 1&0&0&0\cr 0&0&0&1\cr 0&0&1&0\cr}.$$} \label{indexinvF} \end{prop} {\em Proof.} It is completely analogous to the proof of \ref{indexinvE}. \hfill$\Box$ Now we are going to discuss the geometry of the divisor $F$. \begin{prop} { Divisors $F_i$ are smooth surfaces of general type if $n$ is sufficiently big. Moreover, ${\rm dim}H^0(F_i,K_{F_i}) > 0$.} \label{gentypeF} \end{prop} {\em Proof.} Because $F_\alpha$ is an irreducible component of the set of fixed points of an involution on $X$, it is a smooth surface. The finite part of $F_\alpha$ is isomorphic to the quotient of ${\cal H}^1/\Gamma_1(2n)\times{\cal H}^1/\Gamma_1(2n)$ by the diagonal action of the group $\Gamma_1(n)/\Gamma_1(2n)$, where ${\cal H}^1$ is the usual upper half plane, and $\Gamma_1(n)$ is the principal congruence subgroup of ${\rm Sl(2,{\bf Z})}.$ This can be shown by direct calculation, using an element of ${\rm Sp(4,{\bf R})}$ that maps a matrix $\pmatrix{x&y\cr y&x\cr}$ to the matrix $\pmatrix{x-y&0\cr 0&x+y\cr}$. As a result, $F_\alpha$ admits a finite morphism to $({\cal H}^1/\Gamma_1(n))^2$, which is of general type and has global $2$-forms, if $n$ is sufficiently big. \hfill$\Box$ The divisor $F+D$ does not have simple normal crossings. \begin{prop} { There are exactly $n$ divisors $F_\gamma$ on $X_n$ that contain any given curve $l_{\alpha\beta}=D_\alpha\cap D_\beta$.} \label{nFl} \end{prop} {\em Proof.} We assume that the line $l_{\alpha\beta}$ is standard. Let us consider the involution that fixes all points of the divisor $F_i$. It fixes all points of the line $l_{\alpha\beta}$. This implies that the matrix of this involution equals $$\pm\pmatrix{0&1&0&b\cr 1&0&-b&0\cr 0&0&0&1\cr 0&0&1&0\cr}.$$ We can lift these involutions to $\Gamma(1)$ so that they map $\pmatrix{x&y\cr y&z\cr}$ to $\pmatrix{z+b&y\cr y&x-b\cr}$. The corresponding divisors $F_i$ are $\pmatrix{x&y\cr y&x-b\cr}.$ The number of $\Gamma(n)$-inequivalent divisors of this form is equal to $n$. \hfill$\Box$ \begin{prop} { If a divisor $F_\gamma$ contains a line $l_{\alpha\beta}$, then $c_1(F_\gamma)l_{\alpha\beta}=-2$.} \label{Fintl} \end{prop} {\em Proof.} The line $l_{\alpha\beta}$ may be assumed to be standard. Calculations in the local coordinates show that the normal bindle to $l_{\alpha\beta}$ inside $X_n$ is isomorphic to ${\cal O}(2)\oplus{\cal O}(2)$, and the normal bundle to $l_{\alpha\beta}$ inside $F_\gamma$ is the subbundle of the form $(x,e^{2\pi i b/n}x)$.\hfill{$\Box$} \section{Upper bounds on the indices of subgroups of ${\rm Sp(4,{\bf Z}/p^t{\bf Z})}$} This is the key section of the paper. Its purpose is to estimate the index of the subgroup $H\subseteq {\rm Sp(4,{\bf Z}/n{\bf Z})}$ if $H$ contains sufficiently many elements of a special type. We additionally assume that $n=p^t$ for some prime number $p$ and integer $t$. We fix $H$ and assume that $H\ni\pm{\bf 1}$ throughout the rest of the section. We use the notation $[x]_p$ with $x\in {\bf R}_{\geq 1}$ for the maximum number of form $p^t,t\in {\bf N}$ that does not exceed $x$. We first discuss subgroups that contain many elements that fix $D_i$ pointwise. \begin{dfn} { For any primitive vector $v$ we consider the subgroup ${\rm Ram}_G(v)$ of $G={\rm Sp(4,{\bf Z}/n{\bf Z})}$ that consists of transvections, which are operators of the form $$r_{v,\alpha}: w \to w+\alpha\langle w,v\rangle v,~\alpha\in{{\bf Z}/n{\bf Z}}.$$ Because $v$ is primitive, ${\rm Ram}_G(v)\simeq {{\bf Z}/n{\bf Z}}$. We denote $${\rm Ram}_H(v)=H\cap {\rm Ram}_G(v),~{\rm ram}_H(v)=|{\rm Ram}_H(v)|/n.$$ Clearly, ${\rm ram}_H(-v)={\rm ram}_H(v)$.} \label{dfnramD} \end{dfn} \begin{rem} { We shall see later in Proposition \ref{ramdiv} that ${\rm Ram}_G(v_\alpha)$ is exactly the group that fixes all points of the divisor $D_\alpha$.} \end{rem} \begin{prop} { If $\sum_{\pm v}{\rm ram}_H(v)\geq\epsilon \cdot \sharp (\pm v),$ then $|G:H| < 2^5\epsilon^{-2}[2^{72}\epsilon^{-42}]_p.$} \label{boundD} \end{prop} {\em Proof.} We can forget about $\pm$ signs in the above proposition. For any set $I$ of primitive vectors we define the {\it ramification mean}\/ of $I$ to be equal to $$(\sum_{v\in I}{\rm ram}_H(v))/|I|.$$ Clearly, the ramification mean never exceeds $1$. Among the subgroups of $V$ that are isomorphic to $({{\bf Z}/n{\bf Z}})^3$, we choose a subgroup $V_3$, such that the ramification mean of the set of primitive vectors that lie in it is maximum among all such subgroups. Any two primitive vectors are contained in the same number of subgroups that are isomorphic to $({{\bf Z}/n{\bf Z}})^3$, so the sum of the ramification means among these subgroups is at least $\epsilon$ times the number of subgroups. Hence, the ramification mean of $V_3$ is at least $\epsilon$. Analogously, we can choose the subgroup $V_2$ that has the maximum ramification mean among the subgroups with the properties (1) $V_2\simeq ({{\bf Z}/n{\bf Z}})^2$, (2) $V_2\subseteq V_3$, (3) $\langle,\rangle |_{V_2}=0$. Any two primitive vectors in $V_3$ are conjugates with respect to the stabilizer of $V_3$ and therefore are contained in the same number of subgroups $V_2$ that satisfy the above three properties. As a result, the ramification mean of the set of the primitive vectors that lie in $V_2$ is also at least $\epsilon$. The total number of primitive vectors $v$ in $V_2$ is $n^2(1-p^{-2}).$ One can easily show that at least $(\epsilon/2)n^2(1-p^{-2})$ of them have ${\rm ram}_H(v)$ bigger than $\epsilon/2$, because otherwise the ramification mean of $V_2$ would be less than $\epsilon.$ We call these vectors {\it good.} We may additionally assume without loss of generality that $V_2={}^t(*,*,0,0)$ and $V_3={}^t(*,*,*,0).$ If $v={}^t(x,y,0,0)$, then $r_{v,1}$ has the matrix $\pmatrix{{\bf 1}&B\cr {\bf 0}&{\bf 1}\cr}$, where $B=\pmatrix{x^2&xy\cr xy&y^2\cr}.$ Denote the group that consists of matrices $\pmatrix{{\bf 1}&B\cr {\bf 0}&{\bf 1}\cr}$ by $G_{V_2}.$ We can prove the following statement. \begin{lem} { $|G_{V_2}:(G_{V_2}\cap H)|\leq[\epsilon^{-9}2^{14}]_p.$} \label{V_2} \end{lem} {\em Proof of the lemma.} We assume that ${\rm ram}_H({}^t(1,0,0,0))\geq \epsilon/2.$ We can do it, because there is a primitive vector in $V_2$ with this property and we may transform it to ${}^t(1,0,0,0)$ by an element of $G$ that stabilizes $V_2.$ This transformation may not stabilize $V_3$, so we can not use this assumption in the proof of Proposition \ref{boundD}. At least $\epsilon n^2(1-p^{-2})/4$ good vectors ${}^t(x,y,0,0)$ satisfy ${\rm g.c.d.}(y,n)\leq [4/(\epsilon(1-p^{-2})]_p.$ Really, the number of vectors in $V_2$ that do not satisfy this condition is at most $\epsilon n^2(1-p^{-2})/4.$ We pick one such vector and call it ${}^t(x_1,y_1,0,0).$ Consider the set of vectors $v={}^t(x,y,0,0)$ that have the following properties (1) $v$ is good, (2) ${\rm g.c.d.}(y,n)\leq [4/(\epsilon(1-p^{-2}))]_p$, (3) ${\rm g.c.d.}(x_1y-y_1x,n)\leq [4/(\epsilon(1-p^{-2}))]_p$. There are at least $\epsilon n^2(1-p^{-2})/4$ vectors that satisfy the first two conditions and there are less than $\epsilon n^2(1-p^{-2})/4$ vectors that do not satisfy the third one. As a result, such a vector exists, and we denote it by $v={}^t(x_2,y_2,0,0)$. So $H$ contains three elements of $G_{V_2}$ with the matrices $$B=\pmatrix{\alpha^2&0\cr 0&0\cr},\pmatrix{\alpha^2x_1^2&\alpha^2x_1y_1\cr \alpha^2x_1y_1&\alpha^2y_1^2\cr},\pmatrix{\alpha^2x_2^2&\alpha^2x_2y_2\cr \alpha^2x_2y_2&\alpha^2y_2^2\cr},$$ where ${\rm g.c.d.}(\alpha,n)\leq [2/\epsilon]_p.$ They generate a subgroup of $G_{V_2}$ of index equal to the greatest common divisor of $n$ and the determinant of the corresponding three by three matrix. This is equal to $${\rm g.c.d.}(n,\alpha^6y_1y_2(x_1y_2-x_2y_1))\leq [(2/\epsilon)]_p^6[4/(\epsilon(1-p^{-2}))]^3 \leq [\epsilon^{-9}2^{14}]_p.$$ This proves the lemma. \hfill{$\Box$} We now recall that the ramification mean of the set of vectors $v=~{}^t(x,y,z,0)$ is at least $\epsilon$. It implies that there are at least $\epsilon n^3(1-p^{-3})/2$ of them that have ${\rm ram}_H(v)\geq \epsilon/2.$ There are at least $\epsilon n^3(1-p^{-3})/4$ of them that additionally satisfy ${\rm g.c.d.}(z,n)\leq 4/(\epsilon(1-p^{-3})).$ We abuse the notations and also call such vectors good. The operator $r_{v,\alpha}$ that corresponds to a vector $v\in V_3$ and a number $\alpha$ has the matrix $$\pmatrix{1+\alpha xz&0&-\alpha x^2&-\alpha xy\cr \alpha yz&1&- \alpha xy&-\alpha y^2\cr \alpha z^2&0&1-\alpha xz&-\alpha yz\cr 0&0&0&1\cr}.$$ All the elements we have described so far lie inside the group $$G_{V_3}=\pmatrix{a&0&b&m_3\cr m_1&1&m_2&m_4\cr c&0&d&m_5\cr 0&0&0&1\cr}({\rm mod}n),$$$$~ad-bc=1({\rm mod}n), ~bm_1+m_3=am_2({\rm mod}n),~dm_1+m_5=cm_2({\rm mod}n)\}.$$ This group has a natural projection $\lambda$ to the ${\rm Sl(2,{\bf Z}/n{\bf Z})}$ defined by the entries $a,b,c,d.$ Our next step is to show that the images of elements of $H$ generate a subgroup of ${\rm Sl(2,{\bf Z}/n{\bf Z})}$ of bounded index. We have at our disposal the matrices $\pmatrix{1+\alpha xz&-\alpha x^2\cr \alpha z^2&1-\alpha xz\cr}$, as well as $\pmatrix{1&\beta\cr 0&1\cr}$ with ${\rm g.c.d.}(\beta,n)\leq[\epsilon^{-9}2^{14}]_p.$ Here we use the estimate of $\beta$ that comes from the result of lemma \ref{V_2}. We fix $\alpha_0$ that satisfies ${\rm g.c.d.}(\alpha_0,n)=[2/\epsilon]_p.$ Notice that if $(1+\alpha_0 x_1z_1, \alpha_0 z_1^2)\neq (1+\alpha_0 x_2z_2,\alpha_0 z_2^2)$, then the matrices $$\pmatrix{1+\alpha_0 x_1z_1&-\alpha_0 x_1^2\cr \alpha_0 z_1^2& 1-\alpha_0x_1z_1\cr},\pmatrix{1+\alpha_0 x_2z_2&-\alpha_0 x_2^2\cr \alpha_0 z_2^2&1-\alpha_0 x_2z_2\cr}$$ lie in different cosets of the subgroup $\pmatrix{1&*\cr 0&1\cr}.$ Therefore, we can estimate the order of the subgroup generated by the elements that lie in $H$ simply by multiplying $n/[\epsilon^{-9}2^{14}]_p$ by the number of different pairs $(1+\alpha_0 xz, \alpha_0 z^2)$ that we are guaranteed to have. We have at least $\epsilon n^2(1-p^{-3})/4$ pairs $(x,z)$ that correspond to at least one good vector ${}^t(x,y,z,0)$ and thus give rise to an element in $H$ of the above form. We now need to estimate the number of pairs $(x,z)$ that can give the same $(1+\alpha_0 xz, \alpha_0 z^2).$ The number of different $z$ that have the same $\alpha_0 z^2$ is at most $4\cdot {\rm g.c.d.}(\alpha_0 z^2,n)$, which does not exceed $4\cdot [2/\epsilon]_p[4/(\epsilon(1-p^{-3}))]_p^2$. Once we know $z$, the number of $x$ that give the same $1+\alpha_0 xz$ is at most ${\rm g.c.d.}(\alpha_0 z,n)$, which is at most $[2/\epsilon]_p[4/\epsilon(1-p^{-3})]_p.$ So the total number of pairs $(1+\alpha_0 xz, \alpha_0 z^2)$ is at least $$(\epsilon n^2(1-p^{-3})/4)/(4[2/\epsilon]_p^2[4/(\epsilon(1-p^{-3}))]_p^3) \geq \epsilon n^2/(2^5[2^8\epsilon^{-5}]_p).$$ This implies that the images of elements that lie in $H$ generate a subgroup of ${\rm Sl(2,{\bf Z}/n{\bf Z})}$ of index at most $${{n^3(1-p^{-2})(1-p^{-1})} \over {(\epsilon n^2/(2^5[2^8\epsilon^{-5}]_p))\cdot (n/[\epsilon^{-9}2^{14}]_p)}}~\leq~[2^{22}\epsilon^{-14}]_p\epsilon^{-1}2^5.$$ On the other hand, let us estimate the index of $H\cap {\rm Ker}(\lambda)$ in ${\rm Ker}(\lambda)$. We use the formula $$ \pmatrix{1+\alpha xz&0&-\alpha x^2&-\alpha xy\cr \alpha yz&1&-\alpha xy&-\alpha y^2\cr \alpha z^2&0&1-\alpha xz &-\alpha yz\cr 0&0&0&1\cr}\cdot \pmatrix{1&0&0&b\cr 0&1&b&0\cr 0&0&1&0\cr 0&0&0&1\cr}\cdot $$$$ \pmatrix{1+\alpha xz&0&-\alpha x^2&-\alpha xy\cr \alpha yz&1&-\alpha xy&-\alpha y^2\cr \alpha z^2&0&1-\alpha xz&-\alpha yz\cr 0&0&0&1\cr}^{ \hspace{-2pt}-1} \hspace{-4pt}\cdot \pmatrix{1&0&0&-b-b\alpha xz\cr 0&1&-b-b\alpha xz&b\alpha z(-2y+ bz+b\alpha xz^2)\cr 0&0&1&0\cr 0&0&0&1\cr} $$ $$ =\pmatrix{1&0&0&0\cr -b\alpha z^2&1&0&0\cr 0&0&1&b\alpha z^2\cr 0&0&0&1} $$ to generate the subgroup of $\pmatrix{1&0&0&0\cr *&1&0&0\cr 0&0&1&*\cr 0&0&0&1}$ of index at most ${\rm g.c.d.}(\beta\alpha z^2,n),$ which we can estimate. $${\rm g.c.d.}(\beta\alpha z^2,n)\leq [\epsilon^{-9}2^{14}]_p[2\epsilon^{-1}]_p [4\epsilon^{-1}(1-p^{-3})]^2_p\leq [\epsilon^{-12}2^{20}]_p.$$ Because the kernel of $\lambda$ is a semidirect product of the above group and a subgroup of $G_{V_2}$, we have $$|{\rm Ker}\lambda:({\rm Ker}\lambda\cap H)|\leq [\epsilon^{-12}2^{20}]_p [\epsilon^{-9}2^{14}]_p\leq [\epsilon^{-21}2^{34}]_p$$ and $$|G_{V_3}:(G_{V_3}\cap H)|\leq [\epsilon^{-21}2^{34}]_p[\epsilon^{-14} 2^{22}]_p\epsilon^{-1}2^5 \leq [\epsilon^{-35}2^{56}]_p\epsilon^{-1}2^5.$$ There is only one more step necessary to prove this proposition. Because the ramification mean of $V$ is at least $\epsilon$, there are at least $\epsilon n^4(1-p^{-4})/4$ primitive vectors $v={}^t(x,y,z,t)$ that satisfy (1) ${\rm Ram}_H(v)\geq \epsilon/2$, (2) ${\rm g.c.d.}(t,n)\leq [4\epsilon^{-1}(1-p^{-4})^{-1}]_p.$ We continue to abuse the notations and call these vectors good. We use the number $\alpha_0$ defined earlier and consider elements $r_{v,\alpha_0}$ for all good vectors. They all lie in $H$, and the claim is that they lie in $\sim n^4$ different cosets of $G:G_{V_3}.$ Indeed, all elements of the group $G_{V_3}$ fix ${}^t(0,1,0,0)$, and $r_{v,\alpha_0}$ pushes ${}^t(0,1,0,0)$ to ${}^t(x\alpha_0 t, 1+y\alpha_0 t, z\alpha_0 t,\alpha_0 t^2).$ So if $${}^t(x\alpha_0 t, 1+y\alpha_0 t, z\alpha_0 t,\alpha_0 t^2)\neq {}^t(x_1\alpha_0 t_1, 1+y_1\alpha_0 t_1, z_1\alpha_0 t_1,\alpha_0 t_1^2),$$ then $r_{v,\alpha_0}$ and $r_{v_1,\alpha_0}$ lie in different cosets. We can estimate the number of vectors that can give the same fourtuple as follows. If we know $\alpha_0 t^2$, it leaves us with at most $$4\cdot {\rm g.c.d.}(\alpha_0 t^2,n)\leq 4[2\epsilon^{-1}]_p [4\epsilon^{-1}(1-p^{-4})^{-1}]_p^2\leq 4[2^6\epsilon^{-3}]_p$$ options for $t.$ Once we know $t$, we have at most $({\rm g.c.d.}(\alpha_0 t,n))^3$ choices for $(x,y,z).$ This gives us a total of at most $$ 4[2^6\epsilon^{-3}]_p \cdot ([2\epsilon^{-1}]_p [4\epsilon^{-1}(1-p^{-4})^{-1}]_p)^3\leq 4[2^{16}\epsilon^{-7}]_p$$ different good vectors $v$ that give $r_{v,\alpha_0}$ from the same coset. Therefore, we can estimate the number of different cosets that have representatives in $H$ by $$(\epsilon n^4(1-p^{-4})/4)/(4[2^{16}\epsilon^{-7}]_p) \geq \epsilon n^4/(2^5[2^{16}\epsilon^{-7}]_p).$$ Hence, we can estimate the order of $H$ by multiplying the estimate on the order of its intersection with $G_{V_3}$ by the number of cosets that it has representatives in, which gives $$|H|\geq {n^6(1-p^{-2})(1-p^{-1})\over 2^5\epsilon^{-1}[2^{56}\epsilon^{-35}] _p}\cdot{\epsilon n^4\over (2^5[2^{16}\epsilon^{-7}]_p)}\geq n^{10}\cdot{\epsilon^2 2^{-5}(1-p^{-2})(1-p^{-1})\over [2^{72}\epsilon^{-42}]_p}.$$ Therefore, $$|G:H|\leq n^{10}(1-p^{-4})(1-p^{-3})(1-p^{-2})(1-p^{-1}): (n^{10}\cdot{\epsilon^2 2^{-5}(1-p^{-2})(1-p^{-1})\over [2^{72}\epsilon^{-42}]_p}) $$$$ < 2^5\epsilon^{-2}[2^{72}\epsilon^{-42}]_p. $$ \hfill{$\Box$} \begin{rem} { The estimate of Proposition \ref{boundD} is probably far from optimum.} \label{remboundD} \end{rem} Now we consider subgroups that contain many elements that fix $E_i$ pointwise. \begin{dfn} { Let $(W_{\alpha1},W_{\alpha2})$ be a pair of complementary isotropic subgroups that corresponds to the divisor $E_\alpha$, as described in \ref{indexE}, and $\varphi_\alpha$ be the corresponding involution described in \ref{twodescrE}. We define ${\rm ram}_H(E_\alpha)$ to equal $1$ if $H\ni \varphi_\alpha$, and to equal $0$ otherwise. This definition makes sense because $H\ni\pm{\bf 1}$.} \end{dfn} \begin{rem} { We have shown already that $\varphi_\alpha$ fixes all points of $E_\alpha$.} \end{rem} \begin{prop} { If $\sum_\alpha {\rm ram}_H(E_\alpha) \geq \epsilon\sharp(\alpha),$ then $|G:H| < 2^7\epsilon^{-2}[2^{246}\epsilon^{-130}]_p$.} \label{boundE} \end{prop} {\em Proof.} For every set of indices $I$ we define the ramification mean of $I$ to be $\sum_{\alpha\in I}{\rm ram}_H(E_\alpha)/|I|$. For every primitive vector $v$ we consider the set $I_v$ of indices $\alpha$ such that $v$ is an eigenvector of $\varphi_\alpha$. Each index $\alpha$ belongs to the same number of sets $I_v$, therefore $$\sum_v {\rm ramif.mean}(I_v)\geq \epsilon \sharp(v).$$ Hence there are at least $(\epsilon/2)\cdot\sharp(v)$ vectors $v$ such that the ramification mean of $I_v$ is at least $\epsilon/2$. So now we try to estimate ${\rm ram}_H(v)$ for a vector $v$ with this property, and then use \ref{boundD}. We assume that $v={}^t(0,1,0,0).$ \begin{lem} { Involutions $\varphi_\alpha, \alpha \in I_v$ have matrices of the form $$\pmatrix{1&0&0&-2x\cr -2z&-1&2x&0\cr 0&0&1&-2z\cr 0&0&0&-1}.$$ The sign is chosen to satisfy $\varphi_\alpha v=-v$.} \end{lem} {\em Proof of the lemma.} Any involution of this kind is defined uniquely by the choice of $W_{\alpha_2}$. Because of $\langle W_{\alpha1},W_{\alpha2}\rangle =0$, the form $\langle,\rangle$ is unimodular on $W_{\alpha2}$. It implies that there is a basis of $W_{\alpha2}$ that consists of $v$ and ${}^t(x,0,z,1).$ The rest is just a calculation. \hfill$\Box$ We denote the involution with the matrix $$\pmatrix{1&0&0&-2x\cr -2z&-1&2x&0\cr 0&0&1&-2z\cr 0&0&0&-1}$$ by $\varphi_{x,z}$. We may assume without loss of generality that $\varphi_{0,0}\in H$. There are at least $\epsilon n^2/2$ pairs $(x,z)$ such that $\varphi_{x,z}\in H$. We call these pairs good. There are at least $\epsilon n^2/4$ good pairs that satisfy ${\rm g.c.d.}(z,n) \leq [4/\epsilon]_p$. We choose one of them and denote it by $(x_1,z_1)$. There is at least one good pair $(x,z)$ such that ${\rm g.c.d.}(xz_1-zx_1,n)\leq[2/\epsilon]\cdot {\rm g.c.d.}(z,n)$. Then $ {\rm g.c.d.}(xz_1-zx_1,n)\leq [8\epsilon^{-2}]_p.$ We denote this pair by $(x_2,z_2).$ It is a matter of calculation to check that $$(\varphi_{x_1,z_1}\varphi_{0,0}\varphi_{x_2,z_2})^2 = \pmatrix{1&0&0&0\cr 0&1&0&8(x_1z_2-x_2z_1)\cr 0&0&1&0\cr 0&0&0&1\cr}.$$ This element lies in $H$, therefore ${\rm ram}_H(v)\geq 1/[8\epsilon^{-2}]_p.$ Because we can prove the same estimate for every vector $v$ for which the ramification mean of $I_v$ is at least $\epsilon/2$, we get $$\sum_v{{\rm ram}_H(v)}\geq (8\epsilon^{-2})_p^{-1}\cdot(\epsilon/2).$$ Now we use Proposition \ref{boundD} to get $$|G:H| < 2^7\epsilon^{-2}[2^{246}\epsilon^{-130}]_p.$$ \hfill{$\Box$} Now let us consider subgroups that contain many elements that fix lines $D_i\cap D_j$ pointwise. \begin{dfn} { Every line $l_{\alpha\beta}=D_\alpha\cap D_\beta$ is a conjugate of the {\it standard}\/ line $l_0$, which is the intersection of the divisors that correspond to the $\pm$vectors $\pm{}^t(1,0,0,0)$, $\pm{}^t(0,1,0,0)$. We define ${\rm Ram}_G(l_0)$ to consist of matrices $$\pmatrix{1&0&*&0\cr 0&1&0&*\cr 0&0&1&0\cr 0&0&0&1\cr} ({\rm mod}n).$$ We then define ${\rm Ram}_G(g\cdot l_0)=g\cdot {\rm Ram}_G(l_0)\cdot g^{-1}.$ It can be defined invariantly as a subgroup of all matrices that fix both $v_\alpha$ and $v_\beta$, and also fix a pair of the isotropic subgroups $W_1\ni v_\alpha,W_2\ni v_\beta$ that correspond to a divisor $E_i$ that intersects $l_{\alpha\beta}$. It does not matter which $E_i$ we consider.} \end{dfn} \begin{rem} { We shall see later in Proposition \ref{jumpDD} that ${\rm Ram}_G(l_{\alpha\beta})$ consists of transformations that fix all points of the line $l_{\alpha\beta}$ and do not switch the divisors $D_\alpha$ and $D_\beta$.} \end{rem} \begin{dfn} { We define ${\rm Ram}_H(l_{\alpha\beta})=H\cap {\rm Ram}_G(l_{\alpha\beta}).$ We define ${\rm ram}_H(l_{\alpha\beta})$ to be the maximum order of the element of ${\rm Ram}_H(l_{\alpha\beta})$ divided by $n.$} \end{dfn} \begin{prop} { If $\sum_{\alpha\beta}{\rm ram}_H(l_{\alpha\beta})\geq \epsilon\sharp (\alpha\beta)$, then $$|G:H| \leq 2^{11}\epsilon^{-2}[2^{1020}\epsilon^{-350}]_p.$$} \label{boundDD} \end{prop} {\em Proof.} We will eventually use Proposition \ref{boundD}. We need another definition. \begin{lem} { Let $l_{\alpha\beta}$ be the line of the intersection of the divisors $D_\alpha$ and $D_\beta.$ Then ${\rm Ram}_G(l_{\alpha\beta})={\rm Ram}_G(v_\alpha)\oplus {\rm Ram}_G(v_\beta).$} \label{ramDD} \end{lem} {\em Proof of the lemma.} It is enough to consider the standard line, for which the statement follows from the explicit matrix representations of the three groups in question. \hfill$\Box$ \begin{dfn} { We define $${\rm ram}_H(l_{\alpha\beta}\subset D_\alpha)= {|{\rm Ram}_H(l_{\alpha\beta})|\over |{\rm Ram}_H(l_{\alpha\beta})\cap {\rm Ram}_G(v_\alpha)| \cdot n }$$ If $\alpha,\beta$ are standard, then ${\rm ram}_H(l_{\alpha\beta}\subset D_\alpha)$ is the inverse of the minimum ${\rm g.c.d.}(a,n)$ for $$\pmatrix{1&0&a&0\cr 0&1&0&c\cr 0&0&1&0\cr 0&0&0&1\cr}\in H.$$} \end{dfn} We notice that $${\rm ram}_H(l_{\alpha\beta})\leq {\rm max}\{{\rm ram}_H(l_{\alpha\beta}\subset D_\alpha),{\rm ram}_H(l_{\alpha\beta}\subset D_\beta)\}.$$ The usual argument shows that at least $(\epsilon/6)\cdot\sharp|D_\alpha|$ of divisors $D_\alpha$ obey the following property (1) at least $(\epsilon/6)\cdot\sharp (l_{\alpha\beta}\subset D_\alpha)$ of lines $l_{\alpha\beta}$ that are contained in it have ${\rm ram}_H(l_{\alpha\beta}\subset D_\alpha)\geq \epsilon/6.$ We call these divisors good. Now our goal is to prove that every good divisor $D_\alpha$ has sufficiently big ${\rm ram}_H(v_\alpha).$ We may assume without loss of generality that $D_\alpha=D_0$ is standard. We may also assume that the arrangement of lines in $D_0$ over the standard point on the Satake compactification contains at least $(\epsilon/6)\cdot n$ of the lines with ${\rm ram}_H(l_{0\beta}\subset D_0)\geq \epsilon/6.$ Divisors $D_\beta$ that intersect $D_0$ over the standard point of the Satake compactification correspond to $\pm$vectors of the form $\pm{}^t(1,b,0,0)$, see \ref{indexDD}. It implies, that there are at least $(\epsilon/6)\cdot n$ numbers $b$ such that $$ H\ni\pmatrix{1&0&a_0&ba_0\cr 0&1&ba_0&*\cr 0&0&1&0\cr 0&0&0&1\cr}$$ where ${\rm g.c.d.}(a_0,n) \leq [(\epsilon/6)^{-1}]_p$, and $*$ is an unknown number. We can choose $b_1$ and $b_2$ that give us the above elements in $H$ and additionally satisfy ${\rm g.c.d.}(b_1-b_2,n)\leq [6/\epsilon]_p.$ Then we can divide one such element by another to get $$H\ni\pmatrix{1&0&0&(b_1-b_2)a_0\cr 0&1&(b_1-b_2)a_0&*\cr 0&0&1&0\cr 0&0&0&1\cr}=\pmatrix{1&0&0&x\cr 0&1&x&*\cr 0&0&1&0\cr 0&0&0&1\cr}.$$ We can estimate ${\rm g.c.d.}(x,n)\leq [6^2\epsilon^{-2}]_p.$ We denote the above element by $\rho.$ Now we wander away from the standard point on the Satake compactification. All other divisors $D_\beta$ that intersect $D_0$ correspond to the $\pm$vectors $\pm{}^t(d,e,f,0)$ with $(d,f)\neq(0,0)(p)$. This also follows from Proposition \ref{indexDD}. At least $(\epsilon/6) \cdot n^3(1-p^{-2})$ of lines $l_{0\beta}$ satisfy ${\rm ram}_H(l_{0\beta}\subset D_0)\geq \epsilon/6.$ Therefore, at least one of them satisfies additionally ${\rm g.c.d.}(f,n)\leq [6\epsilon^{-1}(1-p^{-2})^{-1}]_p.$ It implies, that $H$ contains an element $\rho_1$ of the form $$\pmatrix{1+ dfa_0&0&-d^2a_0&-dea_0\cr efa_0&1&-dea_0&-ea_0^2+c\cr f^2a_0&0&1-dfa_0&-efa_0\cr 0&0&0&1\cr}$$ with ${\rm g.c.d.}(f,n)\leq [6\epsilon^{-1}(1-p^{-2})^{-1}]_p.$ One can calculate that $$\rho_1\rho\rho_1^{-1}\rho^{-1}\rho_1\rho^{-1}\rho_1^{-1}\rho= \pmatrix{1&0&0&0\cr 0&1&0&-2x^2f^2a\cr 0&0&1&0\cr 0&0&0&1},$$ which implies $${\rm ram}_H(v_0)\geq [2\cdot(6^2\epsilon^{-2})^2 \cdot(6\epsilon^{-1}(1-p^{-2})^{-1})^2\cdot(\epsilon/6)^{-1}]_p^{-1}\geq [2^{19}\epsilon^{-7}]_p^{-1}.$$ As a result, $\sum_{\pm v}{\rm ram}_H(v)\geq [2^{19}\epsilon^{-7}]_p^{-1} (\epsilon/6) \cdot \sharp (\pm v).$ By \ref{boundD}, it implies $$|G:H| \leq 2^{11}\epsilon^{-2} [2^{1020}\epsilon^{-350}]_p.$$ \hfill$\Box$ We also need to deal with subgroups that contain many elements that fix $F_i$ pointwise. \begin{dfn} { Let $\psi_\alpha$ be the involution that corresponds to the divisor $F_\alpha$ as described in \ref{indexinvF}. We define ${\rm ram}_H(F_\alpha)$ to equal $1$ if $H\ni \psi_\alpha$, and to equal $0$ otherwise.} \end{dfn} \begin{rem} { We have shown already that $\psi_\alpha$ fixes all points of $F_\alpha$.} \end{rem} \begin{prop} { If $\sum_\alpha {\rm ram}_H(F_\alpha) \geq \epsilon\sharp(\alpha),$ then $|G:H| \leq 2^{13}\epsilon^{-2}[2^{1722}\epsilon^{-702}]_p$.} \label{boundF} \end{prop} {\em Proof.} There are at least $(\epsilon/2)\sharp(\alpha\beta)$ lines $l_{\alpha\beta}$ such that at least $\epsilon n/2$ of divisors $F_\gamma$ that contain $l_{\alpha\beta}$ are ramification divisors. We call these lines good. Our goal is to estimate ${\rm ram}_H(l_{\alpha\beta})$ for a good line $l_{\alpha\beta}$. We may assume that $l_{\alpha\beta}$ is the standard line. If it is good, then the group $H$ contains at least $\epsilon n/2$ elements of the form $$\varphi_b=\pmatrix{0&1&0&b\cr 1&0&-b&0\cr 0&0&0&1\cr 0&0&1&0\cr}.$$ There are two elements $\varphi_{b_1}$ and $\varphi_{b_2}$ in $H$ such that ${\rm g.c.d.}(n,b_1-b_2)\leq [2\epsilon^{-1}]_p$. The matrix of the element $\varphi_{b_1}\varphi_{b_2}$ is equal to $$\pmatrix{1&0&b_1-b_2&0\cr 0&1&0&b_2-b_1\cr 0&0&1&0\cr 0&0&0&1}.$$ Therefore, ${\rm ram}_H(l_{\alpha\beta})\geq [2\epsilon^{-1}]_p^{-1}$. As a result, $$\sum_{\alpha\beta}{\rm ram}_H(l_{\alpha\beta})\geq \epsilon2^{-1}[2\epsilon^{-1}]_p^{-1}\sharp(\alpha\beta).$$ Proposition \ref{boundDD} gives $|G:H| \leq 2^{13}\epsilon^{-2} [2^{1722}\epsilon^{-702}]_p$. \hfill$\Box$ Finally, we will get an index estimate for subgroups such that their quotient varieties have bad singularities at the images of $D_i\cap D_j\cap D_k$. This is the most delicate calculation of the whole paper. We need some preliminary definitions. \begin{dfn} { Let $P_{\alpha\beta\gamma}$ be the point of the intersection of three infinity divisors $D_\alpha,D_\beta$, and $D_\gamma$. Define $${\rm Ram}_G(P_{\alpha\beta\gamma})={\rm Ram}_G(v_\alpha)\oplus {\rm Ram}_G(v_\beta)\oplus {\rm Ram}_G(v_\gamma).$$ If $P$ is the standard point, that is the one that corresponds to $v_\alpha={}^t(1,-1,0,0),v_\beta={}^t(1,0,0,0),v_\gamma={}^t(0,1,0,0),$ then this group consists of matrices $$\pmatrix{1&0&a&b\cr 0&1&b&c\cr 0&0&1&0\cr 0&0&0&1\cr}({\rm mod}n).$$ As usual, we define ${\rm Ram}_H(P_{\alpha\beta\gamma})=H\cap {\rm Ram}_G(P_{\alpha\beta\gamma})$.} \label{ramDDD} \end{dfn} \begin{dfn} { Consider the singularity at the image of $P_{\alpha\beta\gamma}$ in the quotient of a neighborhood of $P_{\alpha\beta\gamma}$ by the group ${\rm Ram}_H(P_{\alpha\beta\gamma})$. We define ${\rm mult}_H(P_{\alpha\beta\gamma})$ to be the multiplicity of this singular point.} \label{multDDD} \end{dfn} \begin{prop} { If $\,\sum\nolimits^* {\rm mult}_H{P_i}\geq \epsilon \sharp(i)$, where $\sum\nolimits^*$ means taking one point $P_{\alpha\beta\gamma}$ per orbit of the action of the group $H$, then $|G:H|\leq 2^{69}\epsilon^{-34}[2^{11170}\epsilon^{-5950}]_p$.} \label{boundDDD} \end{prop} {\em Proof.} For each point $P_{\alpha\beta\gamma}$ we define $\delta(H,P_{\alpha\beta\gamma})$ as a number $\delta$ defined in \ref{appdelta} for the group ${\rm Ram}_H(P_{\alpha\beta\gamma})$ acting in the tangent space at $P_{\alpha\beta\gamma}$. Notice that there is a natural choice of coordinates $(x_1,x_2,x_3)$ in a neighbourhood of $P_{\alpha\beta\gamma}$, such that the weights of an element $h\in {\rm Ram}_H(P_{\alpha\beta\gamma})$ are determined using ${\rm Ram}_G(P_{\alpha\beta\gamma})={\rm Ram}_G(v_\alpha)\oplus {\rm Ram}_G(v_\beta)\oplus {\rm Ram}_G(v_\gamma)$. Then $\delta(H)$ is defined as $(1/n) {\rm min}_{l\neq 0}(l_1+l_2+l_3)$, where minimum is taken over all $H$-invariant monomials $x_1^{l_1}x_2^{l_2}x_3^{l_3}$. First of all, we rewrite the condition of the proposition in terms of $\delta(H,P_{\alpha\beta\gamma})$. By \ref{appmult}, ${\rm mult}_H P_{\alpha\beta\gamma} \leq n^3\delta(H,P_{\alpha\beta\gamma}) /|{\rm Ram}_H(P_{\alpha\beta\gamma})|$. Therefore, $$\sum_{P_{\alpha\beta\gamma}}\delta(H,P_{\alpha\beta\gamma})\geq \sum_{P_{\alpha\beta\gamma}}n^{-3}|{\rm Ram}_H(P_{\alpha\beta\gamma})|{\rm mult}_H P_{\alpha\beta\gamma}$$ $$\geq \sum\nolimits^*(6n^3)^{-1}|H|{\rm mult}_H P_{\alpha\beta\gamma} \geq (6n^3)^{-1}\epsilon|H|\cdot|G:H|=\epsilon\sharp(P_{\alpha\beta\gamma}).$$ For every isotropic subgroup $V_2\simeq({{\bf Z}/n{\bf Z}})^2$ in $V$ we consider the set of the points $P_{\alpha\beta\gamma}$ with $v_\alpha,v_\beta, v_\gamma\in V_2$. Geometrically, these are the points that lie over certain cusp points of the Satake compactification, see \ref{indexptsSatake}. There are at least $(\epsilon/2)\sharp(V_2)$ of these subgroups that have $$\sum_{v_\alpha,v_\beta,v_\gamma\in V_2}\delta(H,P_{\alpha\beta\gamma}) \geq (\epsilon/2)\sharp(v_\alpha,v_\beta,v_\gamma\in V_2).$$ We call these subgroups good. We are going to prove that if $V_2$ is a good isotropic subgroup, then $$\sum_{v_\alpha,v_\beta\in V_2}{\rm ram}_H(l_{\alpha\beta})\geq \epsilon_1(\epsilon)\sharp(v_\alpha,v_\beta\in V_2),$$ and then use Proposition \ref{boundDD}. We assume without loss of generality that $V_2={}^t(*,*,0,0)$, and $\delta(H,P_0)\geq (\epsilon/2)$, where $P_0$ is the standard point. Notice that ${\rm Ram}_G(P_{\alpha\beta\gamma})$ and ${\rm Ram}_H(P_{\alpha\beta\gamma})$ do not depend on the point $P_{\alpha\beta\gamma}$, provided $v_\alpha,v_\beta,v_\gamma\in V_2$. We denote these groups by $G_1$ and $H_1$ respectively. The group $G_1$ is described in Definition \ref{ramDDD}. We are dealing with points $P_{\alpha\beta\gamma}$ obtained from the standard one by the action of elements of type $\pmatrix{A&0\cr 0&A\cr}$, where $A\in {\rm Gl(2,{\bf Z}/n{\bf Z})}$. Although the group $H_1$ is the same for all $P_{\alpha\beta\gamma}$, its action in the tangent spaces depends on $P_{\alpha\beta\gamma}$. It is the same as the action in the tangent space to the standard point $P_0$ of the group $AH{}^tA,~A\in{\rm Gl(2,{\bf Z}/n{\bf Z})}$, if we think of $G_1$ as the group of symmetric $2\times 2$ matrices. We define $\epsilon_1$ by the formula $$ \sum_{v_\alpha,v_\beta\in V_2}{\rm ram}_H(l_{\alpha\beta})= \epsilon_1\sharp(v_\alpha,v_\beta\in V_2).$$ There is a line $l_{\alpha\beta}$ such that ${\rm ram}_Hl_{\alpha\beta}\leq \epsilon_1$. It implies that the group $H_2=(H_1+[\epsilon_1^{-1}]_pG_1)/[\epsilon_1^{-1}]_pG_1$ is cyclic. When we pass from $H_1$ to $H_1+[\epsilon_1^{-1}]_pG_1$, the numbers $\delta$ do not decrease. Hence, $$\sum_{v_\alpha,v_\beta,v_\gamma\in V_2}\delta(H_1+[\epsilon_1^{-1}]_pG_1, P_{\alpha\beta\gamma})\geq (\epsilon/2)\sharp(v_\alpha,v_\beta,v_\gamma\in V_2).$$ This is equivalent to $$\sum_{A\in {\rm Gl(2,{\bf Z}/n{\bf Z})}}\delta(AH_1{}^tA+[\epsilon_1^{-1}]_pG_1,P_0) \geq (\epsilon/2)\sharp(A).$$ Let $H_2$ be generated by $B=\pmatrix{a&b\cr b&c\cr}$. One can show that $\delta(AH{}^tA+[\epsilon_1^{-1}]_pG_1,P_0)$ equals $\delta(({{\bf Z}/n{\bf Z}}) {\bar A}{\bar B}{}^t{\bar A},{\bar P}_0)$, where $n$ is replaced by $[\epsilon_1^{-1}]_p$ and bars means reduction ${\rm mod}[\epsilon_1^{-1}]_p$. Therefore, $$\sum_{C\in {\rm Gl(2,{\bf Z}/}[\epsilon_1^{-1}]_p{\bf Z})} \delta(({\bf Z}/[\epsilon_1^{-1}]_p{\bf Z})CB{}^tC,{\bar P_0})\geq (\epsilon/2)\sharp(C).$$ Because of the result of Proposition \ref{appfinmany}, there are at most $2^{10}\epsilon^{-8}[2^{12}\epsilon^{-5}]_p$ different matrices $CB{}^tC$ up to proportionality that give $\delta(({\bf Z}/[\epsilon_1^{-1}]_p{\bf Z}) CB{}^tC,{\bar P_0})\geq (\epsilon/4)\sharp(C)$. This implies that the orbit $CB{}^tC({\rm mod~proportionality})$ of the action of the group ${\rm Gl(2,{\bf Z}/}[\epsilon_1^{-1}]_p{\bf Z})$ has length at most $2^{12}\epsilon^{-9}[2^{12}\epsilon^{-5}]_p$. However, we can estimate this length by looking at matrices $C= \pmatrix{t&0\cr 0&1\cr}$. They give $CB{}^tC=\pmatrix{t^2a&tb\cr tb&c\cr}$, and so length of the orbit is at least $[\epsilon_1^{-1}]_p(1-p^{-1}) /{\rm g.c.d.}(bc,[\epsilon_1^{-1}]_p).$ Because we have assumed that $\delta ({\bf Z}/[\epsilon_1^{-1}]_p{\bf Z})B,{\bar P_0})\geq (\epsilon/2)$, we have ${\rm g.c.d.}(b,n)\leq[2\epsilon^{-1}]_p$ and ${\rm g.c.d.}(c,n)\leq [4\epsilon^{-1}]_p$. Really, the weights of $B$ are $(-b,a+b,c+b)$, and if ${\rm g.c.d.}[c,n]$ is greater than $[4\epsilon^{-1}]_p$, then we get $\delta\geq (\epsilon/2)$ because of the invariant monomial of the form $(x_1x_3)^{[\epsilon_1^{-1}]_p/[4\epsilon^{-1}]_p}$. As a result, the length of the orbit is at least $[\epsilon_1^{-1}]_p (1-p^{-1})/[8\epsilon^{-2}]_p$, and we have $[\epsilon_1^{-1}]_p(1-p^{-1})/[8\epsilon^{-2}]_p\leq 2^{12}\epsilon^{-9}[2^{12} \epsilon^{-5}]_p$ and $\epsilon_1\geq 2^{-28}\epsilon^{16}$. We now may use the result of Proposition \ref{boundDD} with $(2^{-28}\epsilon^{16})(\epsilon/2)$ in place of $\epsilon$. Thus, $|G:H|\leq 2^{69}\epsilon^{-34}[2^{11170}\epsilon^{-5950}]_p$. \hfill$\Box$ \section{Singularities of ${\cal H}/H$} It is easy to describe all elements of finite order in $\Gamma(2)$ by means of the following proposition. \begin{prop} { Any nonidentity element of finite order in $\Gamma(2)/\{\pm 1\}$ is conjugate in $\Gamma(1)/\{\pm 1\}$ to the element with the matrix $$\varphi_0=\pmatrix{1&0&0&0\cr0&-1&0&0\cr0&0&1&0\cr0&0&0&-1\cr}.$$} \label{einv} \end{prop} {\em Proof.} Denote the matrix of this element by $\varphi=\pmatrix {A&B\cr C&D\cr}.$ Because $\Gamma(4)$ is torsion free and $\varphi^2\in\Gamma(4)$, we obtain $\varphi^2=1$. Hence the following equalities hold $$A~{}^tB=B~{}^tA,~~C~{}^tD=D~{}^tC,~~A~{}^tD-B~{}^tC=1$$ $$A=~{}^tD,~~B=-{}^tB,~~C=-{}^tC.$$Really, the first three equalities hold for all symplectic matrices, and they imply $\varphi^{-1}=\pmatrix {{}^tD&-{}^tB\cr -{}^tC&{}^tA\cr}$, so $\varphi^{-1}=\varphi$ gives the last three ones. Six equalities together show that $$\varphi=\pmatrix{a_1&a_2&0&b\cr a_3&a_4&-b&0\cr0&c&a_1&a_3\cr -c&0&a_2&a_4\cr}$$ with $(a_1+a_4)b=(a_1+a_4)c=(a_1+a_4)a_2= (a_1+a_4)a_3=0,~a_1^2+a_2a_3-bc=a_4^2+a_2a_3-bc=1.$ Hence if $\varphi\neq1$, then $a_1+a_4=0$, so $$\varphi=\pmatrix{a_1&a_2&0&b\cr a_3&-a_1&-b&0\cr0&c&a_1&a_3\cr -c&0&a_2&-a_1\cr}$$ with $a_1^2+a_2a_3 -bc=1,~(a_1-1),a_2,a_3,b,c\equiv 0{\rm mod}(2)$. We need to prove that any matrix with these properties is conjugate to $\varphi_0$. The vector spaces ${\rm Ker}(\varphi-1)$ and ${\rm Ker}(\varphi+1)$ are orthogonal, so we should simply find four integer vectors $e_1,...,e_4$ that obey $\varphi(e_i)=(-1)^{i+1} e_i$ and $\langle e_2,e_4\rangle =\langle e_1,e_3\rangle =1$. Because of symmetry, it is enough to find $e_1$ and $e_3$. Let us denote $d={\rm g.c.d.}(b/2,a_3/2,(a_1-1)/2).$ There holds $\alpha b/2+\beta(a_1-1)/2+\gamma(-a_3/2)=d$ for some integers $\alpha,\beta,\gamma.$ Now we simply put $$e_1=\pmatrix{b/2d\cr 0\cr a_3/2d\cr(1-a_1)/2d\cr},~ e_3=\alpha\pmatrix{0\cr-b/2\cr(a_1+1)/2\cr a_2/2\cr}+\beta \pmatrix{a_2/2\cr(1-a_1)/2\cr c/2\cr 0\cr}+\gamma\pmatrix{(a_1+1) /2\cr a_3/2\cr0\cr-c/2\cr}$$ and check the required conditions by direct calculation. \hfill$\Box$ \begin{dfn} { Let $H$ be a subgroup of finite index in $\Gamma(1)$. We call $E_i$ or $F_j$ a {\em ramification divisor} iff $H$ contains the involution that fixes all points of the divisor. Because of the results of \ref{indexinvE} and \ref{indexinvF}, $E_\alpha$ is a ramification divisor iff ${\rm ram}_H(E_\alpha)=1$, and similarly for $F_\beta$.} \end{dfn} We are interested in singularities of ${\cal H}/H$. They occur at the images of the points of ${\cal H}$ that have nontrivial stabilizers in $H$. The goal of the rest of this section is to prove the following statement. \begin{prop} { Singularities of the images of the points $\xi\in{\cal H}$ that do not lie in ramification divisors $E_i$ or $F_j$ are canonical. Points that do lie in ramification divisors have solvable stabilizers of order at most $72$. We refer to \ref{defcan} for the definitions of canonical and terminal singularities. } \label{finsing} \end{prop} {\em Proof. }There are two possibilities: $\xi\in\cup E_i$ and $\xi\notin\cup E_i$. {\em Case 1. } $\xi\notin\cup E_i$. The stabilizer of $\xi$ in $\Gamma(2)$ equals $\{\pm 1\}$ because of Proposition \ref{einv} and the definition of $E_i$. We consider the quotient of ${\cal H}$ by the action of $\Gamma(2)$. It is the smooth part of the singular quartic $V$ defined by the equation $(\sum x_i^2)^2=4\sum x_i^4$ in coordinates $(x_1:...:x_6,~\sum x_i=0)$ of ${\bf P}^4$, see \cite{Geer}. The group $\Gamma(1)/\Gamma(2)\simeq \Sigma_6$ acts on $V$ by the permutations of the coordinates $x_i.$ The stabilizer $\xi$ in $\Gamma(1)$ equals that of the image of $\xi$ in $V$ in the group $\Sigma_6.$ Moreover, locally their actions are the same, so the resulting quotient singularities are isomorphic. Therefore, we need to study fixed points of $\Sigma_6$-action on $V$. \begin{lem} { A point $\xi\notin\cup E_i$ with a nontrivial stabilizer in $\Gamma(1)$ either lies in $\cup F_j$ or has the image in $V$ of type $\sigma(0:\theta:\theta^2:\theta^3:\theta^4:1),~\theta=\exp(2\pi i/5), ~\sigma\in\Sigma_6.$} \label{fixedsigma} \end{lem} {\em Proof of the lemma.} Denote by $x=(x_1:...:x_6)$ the image of $\xi$ in $V$. We may assume that the stabilizer of $x$ contains one of the permutations $$(1,2);(1,2)(3,4);(1,2)(3,4)(5,6);(1,2,3);(1,2,3)(4,5,6);(1,2,3,4,5).$$ Let us calculate the sets of fixed points of these permutations that lie in $V$. Case (1,2). We have $(x_1,x_2,...,x_6)=\lambda(x_2,x_1,...,x_6).$ If $\lambda=-1$, then $x=(-1:1:0:0:0:0)$, but this point does not lie in $V$. Hence $\lambda=1$. The set defined by "$x_1=x_2$" constitutes an irreducible divisor on $V$, so it is the closure of the image of some submanifold of dimension two in ${\cal H}$. Case (1,2)(3,4). We have $(x_1,x_2,x_3,x_4,x_5,x_6)=\lambda(x_2,x_1,x_4, x_3,x_5,x_6).$ If $\lambda=-1$, then $x_1=-x_2,~x_3=-x_4,~x_5=x_6=0$. The equality $(\sum x_i^2)^2=4\sum x_i^4$ implies that $x_1=x_3$ or $x_1=x_4$, so $x\in {\rm Sing}(V)$, see \cite{Geer}. If $\lambda=1$, then $x$ lies in the divisor "$x_1=x_2$". Case (1,2),(3,4),(5,6). We have $(x_1,x_2,x_3,x_4,x_5,x_6)=\lambda (x_2,x_1,x_4,x_3,x_6,x_5).$ If $\lambda=-1$, then $x_1=-x_2,~x_3=-x_4,~x_5=-x_6.$ Equality $(\sum x_i^2)^2=4\sum x_i^4$ leads to $(x_1+x_3+x_5)\cdot(x_1+x_3-x_5)\cdot(x_1-x_3+x_5)\cdot( -x_1+x_3+x_5)=0.$ Each of these linear equations implies that $x$ lies in the image of $\cup E_i$, see \cite{Geer}. If $\lambda=1$, then $x_1=x_2,~x_3=x_4,~x_5=x_6$ so $x\in {\rm Sing}(V)$. Case (1,2,3). We have $(x_1,x_2,x_3,x_4,x_5,x_6)=\lambda(x_2,x_3,x_1 ,x_4,x_5,x_6).$ If $\lambda\neq1$, then $x_1+x_2+x_3=0$, so $x$ lies in the image of $\cup E_i$. If $\lambda=1$, then $x_1=x_2=x_3$, and $x$ lies in the divisor "$x_1=x_2$". Case (1,2,3)(4,5,6). We have $(x_1,x_2,x_3,x_4,x_5,x_6)=\lambda(x_2, x_3,x_1,x_5,x_6,x_4).$ If $\lambda=1$, then $x=(1:1:1:-1:-1:-1)\notin V$. Otherwise, $x_1+x_2+x_3=0$, and $\xi\in\cup E_i$. Case (1,2,3,4,5). We have $(x_1,x_2,x_3,x_4,x_5,x_6)=\lambda(x_2,x_3 ,x_4,x_5,x_1,x_6).$ If $\lambda=1$, then $x=(1:1:1:1:1:-5)\notin V$. Otherwise $x=\sigma(0:\theta: \theta^2:\theta^3:\theta^4:1),~\theta=\exp(2\pi i/5),~\sigma\in\Sigma_6.$ The above calculation shows that there is only one up to $\Sigma_6$-action divisor on $V$ with a nontrivial stabilizer of a generic closed point. On the other hand, the images of $F_j$ on $V$ obey this condition. Therefore the images of $F_j$ are the conjugates of the divisor "$x_1=x_2$", which proves the lemma. \hfill$\Box$ \begin{rem} { As a corollary of this lemma, codimension one components of the ramification locus of the map from ${\cal H}/\Gamma_n$ to ${\cal H}/H$ can only be divisors $E_\alpha$ and $F_\beta$. Moreover, ramification occurs iff $E_\alpha(F_\beta)$ is a ramification divisor as defined above, and in this case the only nontrivial element that preserves all points of the divisor is the corresponding involution. Of course, when we consider the Igusa compactifications, we may have ramification at infinity divisors.} \label{ramfin} \end{rem} Let us come back to the proof of \ref{finsing}. We try to estimate the singularity at the image of the point $\xi\notin \cup E_i$ under the quotient map ${\cal H}\to{\cal H}/H.$ The group $\Gamma(2)/\{\pm 1\}$ acts freely on ${\cal H}-\cup E_i$, so we can work in terms of the image point $x\in V-{\rm Sing}(V)$ and the group ${\rm Stab}^H\xi\cdot\Gamma(2)/\Gamma(2)$, because these quotient singularities are isomorphic. There exists a useful criterion that enables one to find out whether the quotient singularity is canonical, see \cite{Reid}. In particular, it is always canonical, if the image of the group in ${\rm Gl}(T_x)$ lies in ${\rm Sl}(T_x)$. We use these facts extensively. First of all we consider the case $x=\sigma(0:\theta:...:1)$. Then either ${\rm Stab}^H\xi\cdot\Gamma(2)/\Gamma(2)=1$ or ${\rm Stab}^H\xi\cdot \Gamma(2)/\Gamma(2)={{\bf Z}/5{\bf Z}}.$ A direct calculation of the weights of the generator in the tangent space and the criterion of \cite{Reid} show that the quotient singularity is terminal, hence canonical. Now let us consider other points $x=(x_1:x_2:x_3:x_4:x_5:x_6)\in V-{\rm Sing}(V)$. The group $S={\rm Stab}^H\xi\cdot\Gamma(2)/\Gamma(2)$ contains no transpositions, because $\xi$ does not belong to any ramification divisors $F_i$. The proof of \ref{fixedsigma} shows that $S$ does not contain permutations of types $(*,*)(*,*)(*,*),(*,*,*)(*,*,*)$, and $(*,*,*,*,*)$. As a result, $S$ consists of permutations of types $(*,*) (*,*),(*,*,*),(*,*,*,*)$, and $(*,*)(*,*,*,*)$ only. Calculations similar to those of \ref{fixedsigma} show that if the group $S$ contains $(*,*)(*,*,*,*)$, then $\xi\in \cup E_i$. Moreover, if it contains a permutation of type $(*,*,*,*)$ and the proportionality coefficient $\lambda$ does not equal $1$, then $\xi\in \cup E_i$. Notice (see the proof of \ref{fixedsigma}) that the proportionality coefficients of elements of the group $S$ of types $(*,*)(*,*)$ and $(*,*,*)$ must also equal $1$. All these restrictions on the group $S$ imply that it consists of even permutations, and all proportionality coefficients are equal to $1$. Therefore, the group $S$ acts in the tangent space of $x$ by matrices from ${\rm Sl}$. The criterion of M. Reid shows that the quotient singularity is canonical. {\em Case 2.} $\xi\in \cup E_i$. all divisors $E_i$ are conjugates, so we may assume that $\xi$ is represented by a diagonal matrix. Different $E_i$ do not intersect, so the stabilizer $S$ of $\xi$ in $\Gamma(1)$ is a subgroup of $${\rm Stab}(\Delta)=\{\pmatrix{ a&0&b&0\cr0&a_1&0&b_1\cr c&0&d&0\cr0&c_1&0&d_1\cr}\cup\pmatrix{ 0&a_1&0&b_1\cr a&0&b&0\cr 0&c_1&0&d_1\cr c&0&d&0\cr},~ad-bc=a_1 b_1-c_1d_1=1\}.$$ Point $\xi$ may be transformed by the group ${\rm Stab}(\Delta)$ to the point $\pmatrix{x_0&0\cr0&z_0\cr}$ with $|{\rm Re}(x_0)|\leq 1/2,~|x_0|\geq 1,~|{\rm Re}(z_0)|\leq 1/2,~|z_0|\geq 1$. Without any loss of generality one may consider points of this type only. The stabilizer of the general such point in $\Gamma(1)/\{\pm 1\}$ equals ${{\bf Z}/2{\bf Z}}.$ It is generated by the involution of Proposition \ref{einv}. If this element is in $H$, then $\Delta$ is a ramification divisor by our definition. The order of the stabilizer can increase in the following curves and points (we have used the symmetry between $x$ and $z$) $$\pmatrix{x&0\cr0&x\cr},\pmatrix{i&0\cr0&x\cr},\pmatrix{\rho&0\cr0&x\cr} ,\pmatrix{i&0\cr0&\rho\cr},\pmatrix{i&0\cr0&i\cr}, \pmatrix{\rho&0\cr0&\rho\cr}.$$ Let us check all these cases. Case $\pmatrix{x&0\cr0&x\cr}$. Because $\Delta$ is not a ramification divisor, the order of ${\rm Stab}^H\xi$ is at most two, so the quotient singularity is canonical. Case $\pmatrix{i&0\cr0&x\cr}$. We get $| {\rm Stab}^H\xi|=1$ by the same argument. Case $\pmatrix{\rho&0\cr0&x\cr}$. In this case either $| {\rm Stab}^H\xi|=1$ or $ {\rm Stab}^H\xi$ is generated by the element of order $3$ whose action in the tangent space of $\xi$ has determinant $1$. Case $\pmatrix{i&0\cr0&\rho\cr}$. The argument is the same as in the previous case. Case $\pmatrix{i&0\cr0&i\cr}$. The stabilizer of $\xi$ in $\Gamma(1) /\{1,-1\}$ is generated by the images of elements of $\Gamma(1)$ with matrices $$\varphi=\pmatrix{0&1&0&0\cr 1&0&0&0\cr 0&0&0&1\cr 0&0&1&0\cr}, ~\alpha=\pmatrix{1&0&0&0\cr 0&0&0&1\cr 0&0&1&0\cr 0&-1&0&0\cr}, ~\beta=\pmatrix{0&0&1&0\cr 0&1&0&0\cr -1&0&0&0\cr 0&0&0&1\cr}.$$ The relations are $\alpha\beta=\beta\alpha,~\alpha^2=\beta^2,~ \varphi\alpha =\beta\varphi,~\varphi^2=\alpha^4=\beta^4=1.$ The order of the group is $16$. The point $\xi$ does not lie in the ramification divisors, so ${\rm Stab}^H\xi$ does not contain any conjugates of $\varphi$. As in the above cases, ${\rm Stab}^H\xi$ does not contain $\alpha^2$ either. We can also employ the following simple statement: if $s^2=1$ for all $s\in {\rm Stab}^H\xi$, then the quotient singularity is canonical. In our case it implies that if the quotient singularity is not canonical, then the group ${\rm Stab}^H\xi$ contains an element of order $4$. All these conditions on ${\rm Stab}^H\xi$ together hold iff this group is generated by a conjugate of $\varphi\beta.$ A direct calculation of the weights in the tangent space completes the argument. Case $\pmatrix{\rho&0\cr0&\rho\cr}.$ In this case it is possible to check that the condition "$\xi$ does not belong to any ramification divisors" implies that ${\rm Stab}^H\xi$ acts in the tangent space of $\xi$ by matrices with determinant $1$. To finish the proof of Theorem \ref{finsing}, we only need to check that stabilizers of all points of ${\cal H}$ are solvable groups whose orders are at most $72$. It can be done using the description of ${\cal H}/\Gamma(2)$ as the smooth part of the singular quartic. I skip the details, because this number is clearly bounded and only slightly affects the constant in the final result. \hfill$\Box$ \section{Finiteness Theorem for subgroups $H\supseteq \Gamma(p^t)$} We assume that $n=p^t$ throughout this section. We denote the subgroup $H\supseteq \Gamma(n)$ and the quotient $H/\Gamma(n)$ by the same letter, which should not lead to a confusion. The Igusa compactifications of ${\cal H} /\Gamma(n)$ and ${\cal H}/H$ are denoted by $X$ and $Y$. The quotient map $X\to X/H=Y$ is denoted by $\mu$. We start by pulling the problem from $Y$ to $X$. \begin{dfn} { Let $\pi:Z\to Y$ be a desingularization of $Y$. Denote by $-1+\delta$ the minimum discrepancy of $Y$, see \ref{discrep}. Because of \ref{qulog}, $\delta$ is a positive rational number.} \end{dfn} \begin{dfn} { Let $m$ be a sufficiently divisible number, so that $mK_Y$ is a Cartier divisor on $Y$. The vector space $H^0(Y,mK_Y-{\rm mlt})$ is defined as the space of global sections $s$ of the coherent subsheaf of ${\cal O}_Y(mK_Y)$ that consists of sections that lie in $m_y^{m(1-\delta)}({\cal O}_Y(mK_Y))_y$ for all noncanonical singular points $y\in Y$.} \end{dfn} \begin{rem} { We assume $m$ to be sufficiently divisible whenever it is necessary. We also omit ${\cal O}$ in the notations of the space of global sections, unless it can lead to a misunderstanding.} \end{rem} \begin{prop} { ${\rm dim}H^0(Z,mK_Z)\geq {\rm dim}H^0(Y,mK_Y-{\rm mlt}).$} \label{ZtoY} \end{prop} {\em Proof.} The pullbacks $\pi^*s$ vanish with the multiplicity at least $m(1-\delta)$ along exceptional divisors with negative discrepancies. Hence we can define an injective linear map from $H^0(Y,mK_Y-{\rm mlt})$ to $H^0(Z,mK_Z)$. \hfill$\Box$ \begin{dfn} { Denote by $H^0(Y,mK_Y-{\rm mlt}^0)$ the space of global sections $s\in H^0(Y,mK_Y)$ that satisfy $s\in m_y^{m(1-\delta)}({\cal O}_Y(mK_Y))_y$ for all noncanonical singular points of $Y$ except for the images of points $P_{\alpha\beta\gamma}$ that are triple intersections of infinity divisors on $X$.} \end{dfn} Clearly, $H^0(Y,mK_Y-{\rm mlt}^0))\supseteq H^0(Y,mK_Y-{\rm mlt})$. \begin{prop} { If $|G:H| > 2^{953}[2^{165870}]_p$, then ${\rm dim} H^0(Y,mK_Y-{\rm mlt}^0)-{\rm dim}H^0(Y,mK_Y-{\rm mlt}) \preceq_{m\to\infty} 2^{-8}3^{-6}5^{-1}m^3|G:H|.$} \label{excludeDDD} \end{prop} {\em Proof.} When $m\to\infty$, the codimension we are trying to estimate grows no faster than $(\sum_{Q\in Y}{\rm mult}_Q)(m^3/6)$, where $\sum_{Q\in Y}{\rm mult}_Q$ is the sum over all points $Q$ in the image of $\cup P_{\alpha\beta\gamma}$, and ${\rm mult}_Q$ is the multiplicity of the local ring of $Y$ at $Q$. We want to relate it to the statement of Proposition \ref{boundDDD}. We need an easy lemma. \begin{lem} { Let $P_{\alpha\beta\gamma}=D_\alpha\cap D_\beta \cap D_\gamma$ be a point on $X$, such that $\mu (P_{\alpha\beta\gamma})=Q$. Then ${\rm mult}_Q\leq 6^3 {\rm mult}_H(P_{\alpha\beta\gamma})$ with ${\rm mult}_H(P_{\alpha\beta\gamma})$ defined in \ref{multDDD}.} \end{lem} {\em Proof of the lemma.} Every element of $G$ that fixes $P_{\alpha\beta\gamma}$ permutes the triple of the $\pm$vectors $(\pm v_\alpha, \pm v_\beta, \pm v_\gamma)$. Hence the subgroup in ${\rm Stab}^H(P)$ of the elements that induce trivial permutations is a normal subgroup of order at most $6$. One can show that this subgroup coincides with ${\rm Ram}_H(P_{\alpha\beta\gamma})$ by the explicit matrix calculation for the standard triple $v_\alpha={}^t(0,1,0,0),v_\beta={}^t(-1,1,0,0), v_\gamma= {}^t(1,0,0,0)$. Therefore, the singularity of $Y$ at $Q$ can be obtained as the quotient of the singularity of $X/{\rm Ram}_H(P_{\alpha\beta\gamma})$ by the group of order at most $6$. Its multiplicity can be estimated by means of Proposition \ref{klem}. \hfill$\Box$ As a result of this lemma, the codimension we are trying to estimate grows no faster than $m^36^2\sum\nolimits^*{\rm mult}_H(P_{\alpha\beta\gamma})$, where one takes one point $P_{\alpha\beta\gamma}$ per orbit of $H$. By \ref{boundDDD} with $\epsilon=2^{-26}$, it grows no faster than $2^{-8} 3^{-6}5^{-1}m^3|G:H|$, if $|G:H| > 2^{953}[2^{165870}]_p$. \hfill$\Box$ \begin{prop} { Let $L_Y$ be a divisor of the modular form of weight $1$ on $Y$. Then ${\rm dim}H^0(Y,mL_Y)$ grows as $2^{-7}3^{-6}5^{-1}m^3|G:H|$.} \label{growthL} \end{prop} {\em Proof.} It can be derived, for instance, from the formula for ${\rm dim}H^0(X,mL_X)$ and $\oplus_mH^0(Y,mL_Y)=(\oplus_mH^0(X,mL_X))^H$. \hfill$\Box$ \begin{prop} { If ${\rm dim} H^0(Y,m(K_Y-L_Y)-{\rm mlt}^0)\neq 0$ for sufficiently big $m$ and $|G:H| > 2^{953}[2^{165870}]_p $, then the variety $Y$ is of general type.} \label{getridofDDD} \end{prop} {\em Proof.} We get $${\rm dim}H^0(Z,mK_Z)\geq {\rm dim}H^0(Y,mK_Y-{\rm mlt})$$ $$\geq {\rm dim}H^0(Y,mK_Y-{\rm mlt}^0)-2^{-8}3^{-6}5^{-1}|G:H|m^3$$ $$\succeq {\rm dim}H^0(Y,mL_Y)-2^{-8}3^{-6}5^{-1}|G:H|m^3\sim 2^{-8}3^{-6}5^{-1}|G:H|m^3.$$ \hfill$\Box$ We shall eventually prove that if $|G:H|$ is big, then ${\rm dim}H^0(Y,m(K_Y-L_Y)-{\rm mlt}^0)\neq 0$ for big $m$. \begin{dfn} { Let $R$ be the ramification divisor of the morphism $\mu$. We define $H^0(X,mK-mR-mL-{\rm mlt}^0)$ to be the space of global sections of ${\cal O}_X(m(K_X-R-L_X))$ that satisfy certain vanishing conditions. Namely, we require their germs to lie in $m_x^{m\cdot k({\rm Stab}^H(x))} {\cal O}_X(m(K_X-R-L_X))_x$ for all points $x\in X$ whose images in $Y$ have noncanonical singularities, except for $x=P_{\alpha\beta\gamma}.$ Here $k({\rm Stab}^H(x))$ is defined according to remark \ref{k}.} \end{dfn} \begin{prop} { If $|G:H|> 2^{953}[2^{165870}]_p$ and ${\rm dim}H^0(X,mK-mR-mL-{\rm mlt}^0)\neq 0$ for some $m>0$, then the variety $Y$ is of general type.} \label{alreadyonX} \end{prop} {\em Proof.} Because of \ref{klem}, all $H$-invariant elements of $H^0(X,mK-mR-mL-{\rm mlt}^0)$ can be pushed down to elements of $H^0(Y,m(K_Y-L_Y)-{\rm mlt}^0).$ Notice, that $\mu^*(m(K_Y-L_Y))=m(K_X-R-L_X)$, and $m\delta$ is dropped from the vanishing conditions to compensate for the constant $N$ from \ref{klem}. One can multiply the $H$-conjugates of any section to get an $H$-invariant one, so if ${\rm dim}H^0(X,mK-mR-mL-{\rm mlt}^0)\neq 0$ for some $m>0$, then ${\rm dim}H^0(Y,m(K_Y-L_Y)-{\rm mlt}^0)\neq 0$ for big $m$, and Proposition \ref{getridofDDD} finishes the proof.\hfill$\Box$ Let us now describe ramification divisors and points with noncanonical images. \begin{prop} { The ramification divisor $R$ equals $\sum_\alpha(n\cdot {\rm ram}_H(v_\alpha)-1)D_\alpha + \sum_\alpha {\rm ram}_H(E_\alpha)E_\alpha +\sum_\alpha {\rm ram}_H(F_\alpha)F_\alpha.$} \label{ramdiv} \end{prop} {\em Proof.} We know by \ref{ramfin} that the ramification divisor in the finite part is equal to $\sum_\alpha {\rm ram}_H(E_\alpha)E_\alpha+\sum_\alpha {\rm ram}_H(F_\alpha)F_\alpha$. We only need to show that the group of elements of $G$ that fix all points of the divisor $D_\alpha$ is exactly $\pm {\rm Ram}_G(v_\alpha)$. It can be done explicitly in coordinates for the standard divisor $D_0.$ \hfill$\Box$ \begin{prop} { If $x\in D_\alpha$, but $x\notin \cup (D_\alpha\cap D_\beta)$, then ${\rm Stab}^H(x)/(\pm {\rm Ram}_H(v_\alpha))$ is a group of order at most $6$.} \label{jumpD} \end{prop} {\em Proof.} We only need to consider the standard divisor $D_0$. It is the universal elliptic curve with level $n$ structure. It can be shown that the group ${\rm Stab}^G_{D_0}$ acts on it by a combination of modular transformations of the base, additions of points of order $n$ in the fibers, and the involution $a\to -a$ of the fibers. The order $6$ can be reached for the point $x$ on the curve with complex multiplication, such that $x$ satisfies $2n\cdot x=0$, and all other stabilizers are even smaller. I skip the details, because a different bound here would only slightly affect the final estimate. \hfill$\Box$ \begin{prop} { If $x\in l_{\alpha\beta}= D_\alpha\cap D_\beta$, but $x\notin \cup P_{\alpha\beta\gamma}$, then the order of the group ${\rm Stab}^H(x)/\pm {\rm Ram}_H(l_{\alpha\beta})$ is at most $4$.} \label{jumpDD} \end{prop} {\em Proof.} We may assume that $l_{\alpha,\beta}=l_0$ is the standard line. The group ${\rm Stab}^G(l_0)$ contains a subgroup of index $2$ of elements that preserve both $\pm{}^t(1,0,0,0)$ and $\pm{}^t(0,1,0,0).$ It in turn contains a subgroup of index $2$ that consists of matrices $\pm \pmatrix{{\bf 1}&B\cr {\bf 0}&{\bf 1}\cr}, ~B=\pmatrix{a&b\cr b&c\cr}.$ One can show using the explicit coordinate on $l_0$, that if $b$ is nonzero, then the action of this element has no fixed points on $l_0$, except for the points of triple intersection of the infinity divisors, which finishes the proof. \hfill$\Box$ \begin{prop} { If $\,|G:H|> 2^{953}[2^{165870}]_p$ and $${\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha -m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha$$ $$-m\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha -m\sum_{\alpha\beta}n\cdot {\rm ram}_H(l_{\alpha\beta}) 4 l_{\alpha\beta}) \neq 0$$ for some $m>0$, then the variety $Y$ is of general type.} \label{73} \end{prop} {\em Proof.} We know from Proposition \ref{finsing} that the points in the finite part, that do not lie in the ramification divisors $E_\alpha$ or $F_\beta$, do not contribute to ${\rm mlt}^0.$ Therefore, $$m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha+m\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha \geq {\rm mlt}+mR$$ in the finite part. This inequality, strictly speaking, is the inclusion of the sheaves of ideals. Analogously, Propositions \ref{jumpD} and \ref{jumpDD} show that $$m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha\geq mR_D+{\rm mlt}^0$$ on $D$ away from $\cup(D_\alpha\cap D_\beta)$, and $$m\sum_{\alpha\beta}n\cdot {\rm ram}_H(l_{\alpha\beta}) 4 l_{\alpha\beta} \geq {\rm mlt}^0$$ on $\cup(D_\alpha\cap D_\beta)$ away from points $P_{\alpha\beta\gamma}$. Then it remains to use Proposition \ref{alreadyonX}. \hfill$\Box$ \begin{prop} { If the variety $Y$ is not of general type, then at least one of the following inequalities holds true. (1) $$|G:H|\leq 2^{953}[2^{165870}]_p$$ (2) $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha)$$ $$\geq ((1/6)c_1(K-L)^3m^3)/5$$ (3) $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha)$$ $$\geq ((1/6)c_1(K-L)^3m^3)/5$$ (4) $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha)$$ $$\geq ((1/6)c_1(K-L)^3m^3)/5$$ (5) $$ {\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha -m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha$$ $$-m\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha) -{\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha$$ $$-m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha -m\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha$$ $$ -m\sum_{\alpha\beta}n\cdot {\rm ram}_H(l_{\alpha\beta}) 4 l_{\alpha\beta}) \geq ((1/6)c_1(K-L)^3m^3)/5$$} \label{splitcases} \end{prop} {\em Proof.} If (2),(3), and (4) are all false, then $${\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha -\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha$$ $$-\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha)\succeq (2/5)(1/6)c_1(K-L)^3m^3.$$ Really, ${\rm dim}H^0(m(K-L)$ grows like $(1/6)c_1(K-L)^3m^3$, because $K-L$ is ample for big $n$, and $E_\alpha, F_\beta, D_\gamma$ are different divisors. Hence, if (1) and (5) are also false, then Proposition \ref{73} proves that the variety $Y$ is of general type. \hfill$\Box$ Our next goal is to show that each of the statements (2)-(5) implies that $|G:H|$ is less than some constant. We use results of Yamazaki \cite{Yamazaki} and statements of Section 3. \begin{prop} { If $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha)$$ $$\geq ((1/6)c_1(K-L)^3m^3)/5,$$ then $|G:H|< 2^{41}[2^{828}]_p$.} \label{5.D} \end{prop} {\em Proof.} First of all, we get $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha)$$ $$\leq \sum_\alpha({\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-7mn\cdot {\rm ram}_H(v_\alpha) D_\alpha)).$$ The standard exact sequences associated to $D_\alpha\subset X$ allow us to estimate that $${\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-7mn\cdot {\rm ram}_H(v_\alpha) D_\alpha)$$ $$\leq \sum_{j=0}^{7mn\cdot {\rm ram}_H(v_\alpha)-1} {\rm dim}H^0(D_\alpha,m(K-L)-jD_\alpha)$$ $$=\sum_{j=0}^{7mn\cdot {\rm ram}_H(v_\alpha)-1} {\rm dim}H^0(D_\alpha,m(K-L)+(2j/n)(L+E)).$$ The divisor $L+E$ is nef on $X$, because $L$ is nef, divisors $E_i$ are disjoint, and $(L+E)E_i=0$. The divisor $K-L$ is ample on $X$, if $n$ is sufficiently big. Therefore, we may use the Riemann-Roch formula to calculate ${\rm dim}H^0(D_\alpha,m(K-L)+(2j/n)(L+E))$. Because we are only interested in the coefficient of $m^3$, as $m\to \infty$, we only need to take into account the term $(1/2)c_1(m(K-L)+(2j/n)(L+E))^2c_1(D_\alpha)$. When $j$ grows, this intersection number grows, therefore $${\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-7mn\cdot {\rm ram}_H(v_\alpha) D_\alpha)$$ $$\leq 7mn\cdot {\rm ram}_H(v_\alpha) (1/2) m^2c_1(K-L+14{\rm ram}_H(v_\alpha)(L+E))^2 c_1(D_\alpha)$$ $$\leq m^3 \sharp(v_\alpha)^{-1} {\rm ram}_H(v_\alpha) (7n/2)c_1(K-L+14 (L+E))^2c_1(D).$$ Hence, if the condition of the proposition is true, then $$\sharp(v_\alpha)^{-1}\sum_\alpha {\rm ram}_H(v_\alpha)\geq 105^{-1}c_1(K-L)^3/(c_1(K-L+14(L+E))^2c_1(nD)).$$ The right hand side can be calculated using the formulas of Yamazaki for the intersection numbers of the divisors $D,L,K$, and $E$. It is bigger than $2^{-18}$ if $n$ is sufficiently big, which we may assume without loss of generality. Therefore, by the result of Proposition \ref{boundD}, $|G:H|< 2^{41}[2^{828}]_p$.\hfill$\Box$ \begin{prop} { If $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha)$$ $$\geq ((1/6)c_1(K-L)^3m^3)/5,$$ then $|G:H|< 2^{53}[2^{3236}]_p$.} \label{5.E} \end{prop} {\em Proof.} Analogously to the proof of \ref{5.D}, we estimate $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha)$$ $$\leq \sum_\alpha {\rm ram}_H(E_\alpha) \sum_{j=0}^{73m-1} {\rm dim}H^0(E_\alpha,m(K-L)-jE_\alpha)$$ $$=\sum_\alpha {\rm ram}_H(E_\alpha) \sum_{j=0}^{73m-1} {\rm dim}H^0(E_\alpha,m(K-L)+jL)$$ $$\preceq \sharp(E_\alpha)^{-1}\sum_\alpha {\rm ram}_H(E_\alpha) (73/2)m^3 c_1(K+72L)^2c_1(E).$$ Therefore, $$\sharp(E_\alpha)^{-1}\sum_\alpha {\rm ram}_H(E_\alpha) \geq 73^{-1}15^{-1}c_1(K-L)^3/(c_1(K+72L)^2c_1(E))>2^{-23}.$$ Then Proposition \ref{boundE} tells us that $|G:H|< 2^{53}[2^{3236}]_p$.\hfill$\Box$ \begin{prop} { If $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha)$$ $$\geq ((1/6)c_1(K-L)^3m^3)/5,$$ then $|G:H|< 2^{73}[2^{22782}]_p$.} \label{5.F} \end{prop} {\em Proof.} As in the proof of \ref{5.E}, we estimate $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha)$$ $$\leq \sum_\alpha {\rm ram}_H(F_\alpha) \sum_{j=0}^{73m-1} {\rm dim}H^0(F_\alpha,m(K-L)-jF_\alpha).$$ Unfortunately, the geometry of $F$ is more complicated than that of $E$, and we do not have a nice formula like $(L+E_\alpha)E_\alpha=0$. We can get away with it by using the adjunction formula together with the Proposition \ref{gentypeF}. We can estimate $$\sum_\alpha {\rm ram}_H(F_\alpha) \sum_{j=0}^{73m-1} {\rm dim}H^0(F_\alpha,m(K-L)-jF_\alpha)$$ $$\leq \sum_\alpha {\rm ram}_H(F_\alpha) \sum_{j=0}^{73m-1} {\rm dim}H^0(F_\alpha,m(K-L)+jK-jK_{F_\alpha})$$ $$\leq \sum_\alpha {\rm ram}_H(F_\alpha) \sum_{j=0}^{73m-1} {\rm dim}H^0(F_\alpha,m(K-L)+jK)$$ $$\leq \sharp(F_\alpha)^{-1}\sum_\alpha {\rm ram}_H(F_\alpha) (73/2)m^3c_1(74K-L)^2c_1(F).$$ Therefore, $$\sharp(F_\alpha)^{-1}\sum_\alpha {\rm ram}_H(F_\alpha)\geq 73^{-1}15^{-1}c_1(K-L)^3/(c_1(74K-L)^2c_1(F)).$$ We need to have some upper bound on $c_1(74K-L)^2c_1(F).$ To do this, we recall the proof of Proposition \ref{finsing}, where we have shown that the images of the divisors $F_\alpha$ on the singular quartic $V$ have form $x_i=x_j$. The product $\prod_{i\neq j}(x_i-x_j)^2$ is invariant under the permutations of the coordinates, so it defines a modular form of weight $60$, that vanishes on $F$. Here we use the fact that the coordinates of ${\bf P}^4$ are given by the modular forms of weight $2$, see \cite{Geer}. As a result, $c_1(74K-L)^2c_1(F)\leq 60c_1(74K-L)^2c_1(L)$, and we can estimate $\sharp(F_\alpha)^{-1}\sum_\alpha {\rm ram}_H(F_\alpha)>2^{-30}$. Now Proposition \ref{boundF} implies that $|G:H|< 2^{73}[2^{22782}]_p$. \hfill$\Box$ \begin{prop} { If $$ {\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha -m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha$$ $$-m\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha) -{\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha$$ $$-m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha -m\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha$$ $$ -m\sum_{\alpha\beta}n\cdot {\rm ram}_H(l_{\alpha\beta}) 4 l_{\alpha\beta}) \geq ((1/6)c_1(K-L)^3m^3)/5,$$ then $|G:H|<2^{65}[2^{10470}]_p$.} \label{5.DD} \end{prop} {\em Proof.} Denote $$L_1=K-L-7\sum_\alpha n\cdot {\rm ram}(v_\alpha)D_\alpha - 73\sum_\alpha {\rm ram}_H(E_\alpha)E_\alpha-73\sum_\alpha {\rm ram}_H(F_\alpha)F_\alpha.$$ Then the left hand side of the proposition does not exceed the sum over all $l_{\alpha\beta}$ of $${\rm dim}H^0(mL_1)-{\rm dim}H^0(mL_1-4mn\cdot {\rm ram}_H(l_{\alpha\beta})l_{\alpha\beta}).$$ To estimate this codimension, we consider the blow-up of the variety $X$ along the line $l_{\alpha\beta}$, which we denote by $\pi:X_1\to X$. The normal bundle to $l_{\alpha\beta}$ is isomorphic to ${\cal O}(2)\oplus{\cal O}(2)$. This can be checked by direct calculation. Therefore, the exceptional divisor of $\pi$ is isomorphic to ${\bf P}^1\times {\bf P}^1$. We get $${\rm dim}H^0(mL_1)-{\rm dim}H^0(mL_1-4mn\cdot {\rm ram}_H(l_{\alpha\beta})l_{\alpha\beta})$$ $$={\rm dim}H^0(m\pi^*L_1)-{\rm dim}H^0(m\pi^*L_1-4mn\cdot {\rm ram}_H(l_{\alpha\beta})S)$$ $$\leq\sum_{j=0}^{4mn\cdot {\rm ram}(l_{\alpha\beta})-1}{\rm dim}H^0(S,m\pi^*L_1-jS).$$ We denote the fiber and the section of $S\to l_{\alpha\beta}$ by $f$ and $s$ respectively and get $(m\pi^*L_1-jS)S=m\cdot c_1(L_1)l_{\alpha\beta} \cdot f +j(2f+s)$. Hence, $H^0(S,m\pi^*L_1-jS)$ grows when $j$ grows, and we have $${\rm dim}H^0(mL_1)-{\rm dim}H^0(mL_1-4mn\cdot {\rm ram}_H(l_{\alpha\beta})l_{\alpha\beta})$$ $$\leq 4mn\cdot {\rm ram}_H(l_{\alpha\beta}){\rm dim}H^0(S,mc_1(L_1)l_{\alpha\beta}\cdot 2f+4mn\cdot {\rm ram}_H(l_{\alpha\beta})\cdot j )$$ $$\leq m^3{\rm ram}_H(l_{\alpha\beta})\cdot (8n\cdot {\rm ram}_H(l_{\alpha\beta}) +c_1(L_1)l_{\alpha\beta})\cdot4n\cdot {\rm ram}_H(l_{\alpha\beta})$$ $$\leq m^3{\rm ram}_H(l_{\alpha\beta})(128n^3+16n^2c_1(L_1)l_{\alpha\beta})$$ $$\preceq_{n\to\infty}~m^3{\rm ram}_H(l_{\alpha\beta})(128n^3+ 16n^2\cdot (7n\cdot 2\cdot 2+73\cdot 2\cdot n) $$ $$=m^3{\rm ram}_H(l_{\alpha\beta})\cdot 2912n^3.$$ The number of $l_{\alpha\beta}$ is equal to $2^{-3}n^7(1-p^{-4})(1-p^{-2})$, see \cite{Yamazaki}. Therefore, if the condition of the proposition is true, then $$(\sharp(l_{\alpha\beta}))^{-1}\sum_{\alpha\beta}{\rm ram}_H(l_{\alpha\beta}) \geq {{c_1(K-L)^3 }\over{ (30\cdot 2912\cdot 2^{-3}n^{10}(1-p^{-4})(1-p^{-2}))}}>2^{-27}.$$ Now the result of Proposition \ref{boundDD} gives $|G:H|<2^{65}[2^{10470}]_p$. \hfill$\Box$ We are now ready to prove the finiteness theorem for $H\supseteq \Gamma(p^t)$. \begin{prop} { If $|G:H|>2^{953}[2^{165870}]_p$, then the variety $Y$ is of general type.} \end{prop} {\em Proof.} We simply combine the results of Propositions \ref{5.D}, \ref{5.E}. \ref{5.F}, \ref{5.DD}, and \ref{splitcases}. \hfill$\Box$ \begin{prop} { {\bf Finiteness theorem for $H\supseteq \Gamma(p^t).$} There are only finitely many subgroups $H\subseteq {\rm Sp(4,{\bf Z})}$ of finite index that contain $\Gamma(p^t)$ for some $p$ and $t$, such that the variety ${\cal H}/H$ is not of general type.} \label{finthmprimary} \end{prop} {\em Proof.} It follows from the fact that $|G:H|$ is bounded. \hfill$\Box$ In particular, if $p$ is sufficiently big, then for any $H$, ${\rm Sp(4,{\bf Z})}\supset H\supseteq \Gamma(p^t)$ the variety $Y$ is of general type. \section{Finiteness Theorem, the general case} Now we no longer assume that $n$ is a power of a prime number. Our goal is to prove that the condition $n=p^t$ can be dropped from the statement of Proposition \ref{finthmprimary}. Our proof is the direct generalization of the argument of \cite{Thompson}. We first estimate prime factors of $n$. \begin{prop} { If $p>3$, and $$H\cdot\Gamma(p)=\Gamma(1),~ H\supseteq\Gamma(mp^\alpha),~{\rm g.c.d.}(m,p)=1,$$ then $H\supseteq\Gamma(m)$.} \label{bigprime} \end{prop} {\em Proof. } For any group $G$ we denote its image modulo $\Gamma(mp^\alpha)$ by $\hat G$. We have isomorphisms $$\hat\Gamma(1)\simeq \hat\Gamma(m)\times\hat\Gamma(p^\alpha),~\hat\Gamma(m)\simeq{\rm Sp(4,{\bf Z}/p^\alpha {\bf Z})},~\hat\Gamma(p^\alpha)\simeq{\rm Sp(4,{\bf Z}/m {\bf Z})}.$$ The group ${\rm PSp(4,{\bf Z}/p {\bf Z})}$ is simple for all $p\geq 3$. Because of $\hat H\cdot\hat\Gamma(p)/\hat\Gamma(p)\simeq {\rm Sp(4,{\bf Z}/p {\bf Z})}$, the group $\hat H$ has a section isomorphic to ${\rm PSp(4,{\bf Z}/p {\bf Z})}$. Consider the following normal subgroups of $\hat\Gamma(1)$. $$\hat\Gamma(1)\supset\hat\Gamma(m)\supset\hat\Gamma(mp)\supseteq\{e\}.$$ We easily get that $\hat H\cap\hat\Gamma(m)/\hat H\cap\hat\Gamma(mp)$ has a section isomorphic to ${\rm PSp(4,{\bf Z}/p{\bf Z})}$, so there holds $$(\hat H\cap\hat\Gamma(m))\cdot\hat\Gamma(mp)=\hat\Gamma(m)$$ Now it will suffice to prove that the last equality implies $\hat H\supseteq \hat\Gamma(m)$. Note that $\hat\Gamma(m)\simeq {\rm Sp(4,{\bf Z}/p^\alpha {\bf Z})}$ and $\hat\Gamma(mp)\simeq {\rm Ker}({\rm Sp(4,{\bf Z}/p^\alpha {\bf Z})} \to{\rm Sp(4,{\bf Z}/p {\bf Z}}))$. We denote by $K_i$ the kernels of ${\rm Sp(4 ,{\bf Z}/p^\alpha {\bf Z})}\to{\rm Sp(4,{\bf Z}/p^i {\bf Z})}$ for $i=1,...,\alpha$ and prove that $\hat H\supseteq K_i$ by the decreasing induction on $i$. For $i=\alpha$ there is nothing to prove. Besides we already have the last step of the induction. Suppose that $\hat H\supseteq K_i,~i>1$. To prove that $\hat H\supseteq K_{i-1}$ consider $h\in\hat H\cap\hat\Gamma(m)$ such that $$h\equiv \pmatrix{ 1&0&1&0\cr0&1&0&0\cr0&0&1&0\cr0&0&0&1}({\rm mod}~p).$$ Clearly, $h^{p^i}\in K_i$. Besides, a simple calculation shows that for $p\geq 5$ $$h^{p^{i-1}}\equiv\pmatrix{1&0&p^{i-1}&0\cr0&1&0&0\cr0&0&1&0 \cr0&0&0&1}({\rm mod}~p^i).$$ When the group $\hat\Gamma(m)$ acts on $K_{i-1}/K_i$ by conjugation, its subgroup $\hat\Gamma(mp)$ acts as identity. We have already known that $(\hat H\cap\hat\Gamma(m))\cdot \hat\Gamma(mp)=\hat\Gamma(m)$, so it is enough to show that conjugates of the element $h^{p^{i-1}}$ generate the whole group $K_{i-1}$ modulo $K_i$. This can be done by a direct calculation in the abelian group $K_{i-1}/K_i$. \hfill $\Box$ \begin{prop} { There exists a natural number $N$ such that if ${\cal H}/H$ is not of general type, then $$H\supseteq \Gamma(\prod_{p_i\leq N}p_i^{n_i})$$ for some natural numbers $n_i$.} \label{boundprime} \end{prop} {\em Proof.} Let $n$ be the minimum number such that $H\supseteq \Gamma(n)$. Because of the result of \ref{bigprime}, $H\cdot \Gamma(p)\neq \Gamma(1)$ for all prime factors of $p$ of $n$ bigger than $3$. If ${\cal H}/H$ is not of general type, then ${\cal H}/(H\cdot\Gamma_p)$ is not of general type either, see \ref{fincov}. Because of Proposition \ref{finthmprimary}, there are only finitely many choices for $p.$ \hfill$\Box$ We now prove the Finiteness Theorem in full generality. Define for any $H\subseteq\Gamma(1)$ and any prime $p$ the $p$-projection of $H$ as $H_p=\cap_1^\infty H\cdot\Gamma(p^j)$. Note that $H_p\supseteq H$ and $H_p\supseteq\Gamma(p^j)$ for some $j$. The following proposition allows us to work with $p$-projections only, after we have got an estimate on the primes. \begin{prop} { For any given set of subgroups $G_i\supseteq\Gamma(p_i^{n_i}), ~i=1,...,k$, there are only finitely many subgroups $H\supseteq\Gamma (p_1^{\alpha_1}\cdot...\cdot p_k^{\alpha_k})$ with $H_{p_i}=G_i$.} \label{splitprime} \end{prop} {\em Proof.} We can simply estimate the index of $H$ if we employ the fact that $\Gamma(p_i)$ are pro-$p_i$-groups.\hfill $\Box$ Now we can easily prove the Finiteness Theorem. \begin{prop} { {\bf Finiteness Theorem.} There are only finitely many subgroups $H\subseteq {\rm Sp(4,{\bf Z})}$ of finite index, such that ${\cal H}/H$ is not of general type.} \label{fintheorem} \end{prop} {\em Proof.} If ${\cal H}/H$ is not of general type, then ${\cal H}/H_p$ is not of general type either. Therefore, Proposition \ref{finthmprimary} tells us that there are only finitely many choices for $H_p$. By \ref{boundprime}, all prime factors of $n$ are bounded, so Proposition \ref{splitprime} finishes the proof. \hfill $\Box$ \section{Varieties of general type and singularities} We first recall some standard facts about varieties of general type and singularities. \begin{dfn} { A smooth compact algebraic variety X over ${\bf C}$ is called a variety of general type if there exists some constant $c>0$ such that ${\rm dim}H^0(X,{\cal O}_X(mK_X))>cm^{{\rm dim}X}$ for all sufficiently big (equivalent condition -- divisible by some integer $d$) positive integers $m$. Here $K_X$ is the canonical divisor of $X$.} \end{dfn} \begin{rem} { If $X$ and $Y$ are birational smooth compact algebraic varieties, then ${\rm dim}H^0(X,{\cal O}_X(mK_X))= {\rm dim}H^0(Y,{\cal O}_Y(mK_Y))$ for $m\geq0$. } \end{rem} \begin{dfn} { A field ${\cal K}\supset{\bf C}$ is called a field of general type if it is a field of the rational functions of a smooth compact algebraic variety of general type.} \end{dfn} \begin{dfn} { An algebraic variety over ${\bf C}$ is called a variety of general type if its field of functions is a field of general type.} \end {dfn} \begin{dfn} { A canonical divisor $K_X$ of a normal variety $X$ is a Weil divisor on $X$ that coincides with a canonical divisor on $X-{\rm Sing}(X)$. The variety $X$ is called ${\bf Q}$-$Gorenstein$ if $mK_X$ is a Cartier divisor for some integer $m$.} \end{dfn} \begin{rem} { If the variety $Y$ is normal ${\bf Q}$-Gorenstein but has singularities, then the condition "${\rm dim}H^0(Y,{\cal O}_Y(mK_Y))>cm^{{\rm dim}Y}$ for $m\to+\infty$" does not imply by itself that $Y$ is of general type. Really, if $\pi:Z\to Y$ is some desingularization, then there holds $$K_Z=\pi^*(K_Y)+\sum_i\alpha_iF_i,~~\alpha_i\in{\bf Q}$$ in the sense of equivalence of ${\bf Q}$-Cartier divisors, where $F_i$ are exceptional divisors of morphism $\pi$ and $\alpha_i$ are some rational numbers called {\it discrepancies}\/. If some $\alpha_i$ are negative, then ${\rm dim}H^0(Z,{\cal O}_Z(mK_Z))$ may be less than ${\rm dim}H^0(Y,{\cal O}_Y(mK_Y))$}. \label{discrep} \end{rem} \begin{dfn} { A normal ${\bf Q}$-Gorenstein variety $Y$ is said to have $log-terminal$ singularities if for some desingularization $\pi:Z\to Y$, such that the exceptional divisor $\sum F_i$ has simple normal crossings, all discrepancies are greater than $(-1)$. A singular point $y\in Y$ is called {\it canonical}\/ (resp. {\it terminal}\/) if the discrepancies $\alpha_i$ are nonnegative (resp. positive) for all $i$ such that $\pi(F_i)\ni y$. Once satisfied for some desingularization, whose exceptional locus is a divisor with simple normal crossings, these conditions are satisfied for any desingularization (see \cite{CKM}).} \label{defcan} \end{dfn} \begin{prop} { If $\mu:X\to Y$ is a finite morphism of algebraic varieties and $Y$ is of general type, then $X$ is also of general type.} \label{fincov} \end{prop} {\em Proof.} We find a surjective morphism $\hat\mu:\hat X\to\hat Y$, where $\hat X,\hat Y$ are smooth projective birational models of $X,Y$, and then pull back multicanonical forms. \hfill$\Box$ The following statement is well-known. \begin{prop} { {\rm (see \cite{CKM})} Let $X$ be a smooth projective algebraic variety over ${\bf C}$ with an action of a finite group $G$. Then the quotient variety $Y=X/G$ has log-terminal singularities.} \label{qulog} \end{prop} Now we shall prove a simple but important technical result about quotient singularities. Let $X$ be a projective algebraic variety with an action of a finite solvable group $H$. Let $x$ be a (closed) point of $X$, such that $hx=x$ for all $h\in H$. Suppose we have $\{e\}=H_0\subset H_1\subset ...\subset H_t=H$, where $H_{i-1}$ are normal subgroups of $H_i$ and $H_i/H_{i-1}$ are abelian groups with exponents $k_i$. Denote $k=k_1\cdot...\cdot k_t$. Denote the local ring of $x$ in $X$ by $(A,{\bf m})$. Then $(B,{\bf n})=(A^H,{\bf m}^H)$ is the local ring of the image of $x$ under the quotient morphism. \begin{prop} { In the above setup there exists a constant $N$, which depends only on $X$ and $H$ but not on $x$, such that there holds ${\bf m}^{kl+N}\cap B\subseteq {\bf n}^l$ for all $l\geq0$.} \label{klem} \end{prop} {\em Proof.} We do not suppose $X$ to be smooth, so it is enough to consider just the case of an abelian group $H$ with $kH=0$. There exists a linearized $H$-invariant very ample invertible sheaf ${\cal L}$ on $X$. Consider the corresponding closed embedding $X\to {\bf P}^{N_0}$. Because $H$ is abelian, there exists an open $H$-invariant affine neighborhood of $x$ with the ring $R$ equal to ${\bf C}[1,l_1/l_0,...,l_{N_0}/l_0]/I$ where $l_i\in H^0(X,{\cal L}),~ h(l_i)=\mu_i(h)\cdot l_i,~\forall h\in H$ and $I$ is some ideal. Moreover, we may assume that $f_i=l_i/l_0$ vanish at $x$, because of $Hx=x$. Hence the local ring $(A,{\bf m})$ is the localization of $R$ in $p=(f_1,...,f_{N_0})$. Because $H$ is finite, one can assume that all denominators are $H$-invariant. Therefore, the statement of the proposition is equivalent to $p^{kl+N}\cap R^H\subseteq (p^H)^l$. Each element of $p$ can be represented as a polynomial in $f_i$ with zero constant term. Therefore, each element of $p^{ kl+N}$ can be represented as a polynomial in $f_i$ with monomials of degree no less than $kl+N$. For any given $f\in p^{kl+N}\cap R^H$ consider such a representation with the minimum possible number of monomials. Then if for some monomial $g$ of this representation and some element $h\in H$ there holds $h(g)=w\cdot g,~w\neq1$, then the formula $f=f\cdot w/(w-1)-h(f)/(w-1)$ allows us to reduce the number of monomials. Hence every element $f\in p^{kl+N}\cap R^H$ is a sum of $H$-invariant monomials of degree at least $kl+N$. Now we only need to prove that any $H$-invariant monomial $g=f_1^{\alpha_1} \cdot...\cdot f_{N_0}^{\alpha_{N_0}}$ of degree at least $kl+N$ is a product of at least $l$ $H$-invariant monomials of positive degree. It is time to choose $N$, namely $N=k\cdot N_0$. Denote by $\gamma_i$ the maximum integers that do not exceed $\alpha_i/k$. Then $g=f_1^{k\gamma_1}\cdot...\cdot f_{N_0}^{k\gamma_{N_0}}\cdot g_1$ gives the required decomposition, because $\sum\gamma_i>\sum\alpha_i/k-N_0\geq l$.\hfill $\Box$ \begin{rem} { Due to the result of \cite{Hochster}, the above proposition holds for scheme points which correspond to the subvarieties that are pointwise $H$-invariant. I wish to thank Melvin Hochster for pointing out this reference.} \label{afterklem} \end{rem} \begin{rem} { In the rest of the paper $k(H)$ for a finite solvable group $H$ denotes the least possible value of $k$ that could be obtained in the above way.} \label{k} \end{rem} The rest of the section is devoted to multiplicities of certain toric singularities. Somewhat unnatural choice of notation is motivated by the notation of Section 3. \begin{dfn} { Let $G_1\simeq ({{\bf Z}/n{\bf Z}})^3$ act on ${\bf C}^3$ according to the formula $$(\xi_1,\xi_2,\xi_3)(x_1,x_2,x_3)=(e^{2\pi i\xi_1/n}x_1, e^{2\pi i\xi_2/n}x_2,e^{2\pi i\xi_3/n}x_3).$$ Let $H_1$ be a subgroup of $G_1$. Define $\delta(H_1)=(1/n) {\rm min}_{l\neq 0}(l_1+l_2+l_3)$, where the minimum is taken among all $H_1$-invariant monomials $x_1^{l_1}x_2^{l_2}x_3^{l_3}$.} \label{appdelta} \end{dfn} \begin{prop} { The multiplicity of the local ring of $C^3/H_1$ at zero is at most $n^3\delta(H_1)/|H_1|.$} \label{appmult} \end{prop} {\em Proof.} The exponents of the $H_1$-invariant monomials form a semigroup, which we denote by $K$. One can show that the multiplicity is equal to $vol({\bf R}^n_{>0}-conv(K-\{0\}))/|H_1|$, where the volume is normalized to be equal one on the basic tetrahedron. This result does not seem to be stated explicitly anywhere in the literature, but its proof is completely analogous to the calculation of \cite{Teissier} of multiplicities of the ideals in the polynomail ring that are generated by monomials. On the other hand, this set is contained in the set $$conv((l_1,l_2,l_3),(0,0,n),(0,n,0),(0,0,0))\cup...$$ $$...\cup conv((l_1,l_2,l_3),(0,n,0),(n,0,0),(0,0,0)),$$ which has volume $n^3\delta(H_1)$. \hfill$\Box$ \begin{rem} { Our results on the multiplicities of certain toric singularities can be generalized to arbitrary dimension, but we only need the case of dimension three.} \end{rem} Now we consider in detail the case when $n$ is a power of a prime number, and the group $H_1$ is cyclic. \begin{prop} { Let $K=K_{uvw}$ be a semigroup, defined by the conditions $\alpha u+ \beta v +\gamma w =0({\rm mod}{\em p^s})$ and $\alpha,\beta,\gamma\in {\bf Z}_{\geq 0},$ where $u$, $v$, and $w$ are some natural numbers. The number $\delta$ defined in \ref{appdelta} equals $p^{-s}{\rm min}_{K-\{0\}}(\alpha+\beta+\gamma).$ Then the number of homogeneous triples $(u:v:w)$ such that $\delta(u,v,w)\geq \epsilon$ is at most $2^2\epsilon^{-8}[4\epsilon^{-5}]_p$.} \label{appfinmany} \end{prop} {\em Proof.} Consider the intersection of $K$ and the coordinate plane $\alpha = 0$. It is the semigroup $K_1$ defined by the conditions $\beta,\gamma\in {\bf Z}_{\geq 0},~\beta v+\gamma w = 0({\rm mod}{\em p^s}).$ If $\delta(u,v,w)\geq \epsilon$, then $\beta+\gamma\geq \epsilon p^s$ for all nonzero $(\beta,\gamma)\in K_1$. Therefore, the area of ${\bf R}^2_{>0}-conv(K_1-\{0\})$ is at least $\epsilon^2p^{2s}$, if the area of the basic triangle in ${\bf Z}^2$ is equal to one. Because any triangle in ${\bf Z}^2$ with no lattice points inside and on the edges is basic, the number of points of $K_1$ that lie inside the positive quadrant and on the boundary of $conv(K_1-\{0\})$ is at least $-1+\epsilon^2p^{2s}/|{\bf Z}^2:span(K_1)|\geq -1+\epsilon^2p^s$. The function $\beta-\gamma$ is monotone on the boundary of $conv(K_1-\{0\})$, and changes by at most $2p^s$ inside the positive quadrant. Hence, there is a segment of this boundary, that is represented by the vector $(\beta_1,-\gamma_1)$ with $0<\beta_1,\gamma_1,~\beta_1 +\gamma_1\leq 2\epsilon^{-2}$. Hence there holds $v\beta_1=w\gamma_1({\rm mod}{\em p^s})$ with $0<\beta_1,\gamma_1,~\beta_1 +\gamma_1\leq 2\epsilon^{-2}$. Analogously, we have $u\alpha_2=w\gamma_2({\rm mod}{\em p^s})$ with $0<\alpha_2,\gamma_2,~\alpha_2+\gamma_2\leq 2\epsilon^{-2}$. Besides, ${\rm g.c.d.}(w,p^s)\leq[\epsilon^{-1}]_p$, because otherwise $(0,0,p^s/{\rm g.c.d.}(w,p^s))$ lies in $K$ and gives $\delta<\epsilon$. There are at most $[\epsilon^{-1}]_p$ choices of $w({\rm mod}{\em p^s})$ up to multiplication by $({\bf Z}/p^s{\bf Z})^*$. There are at most $2^2\epsilon^{-8}$ choices for the fourtuple $(\beta_1,\gamma_1, \alpha_2,\gamma_2)$. Once we know $(w,\beta_1,\gamma_1, \alpha_2,\gamma_2)$, there are at most $[2\epsilon^{-2}]_p$ for each of the numbers $u,v({\rm mod}{\em p^s})$. This proves the proposition. \hfill$\Box$ \bigskip

9505

alg-geom/9505020

en

https://arxiv.org/abs/alg-geom/9505020

alg-geom/9505020

Dan Abramovich

Dan Abramovich

Lang's conjectures, Conjecture H, and uniformity

7 pages. AMSLaTeX, dvi file available at http://math.bu.edu/INDIVIDUAL/abrmovic/conjh.dvi

The purpose of this note is to wish a happy birthday to Professor Lucia Caporaso.* We prove that Conjecture H of Caporaso et. al. ([CHarM], sec. 6) together with Lang's conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHarM]; a few applications in arithmetic and geometry are stated. Let X be a variety of general type defined over a number field K. It was conjectured by S. Lang that the set of rational points X(K) is not Zariski dense in X. In the paper [CHarM] of L. Caporaso, J. Harris and B. Mazur it is shown that the above conjecture of Lang implies the existence of a uniform bound on the number of K-rational points of all curves of fixed genus g over K. The paper [CHarM] has immediately created a chasm among arithmetic geometers. This chasm, which often runs right in the middle of the personalities involved, divides between loyal believers of Lang's conjecture, who marvel in this powerful implication, and the disbelievers, who try (so far in vain) to use this implication to derive counterexamples to the conjecture. In this paper we will attempt to deepen this chasm, using the techniques of [CHarM] and continuing [aleph], by proving more implications, some of which very strong, of various conjectures of Lang. Along the way we will often use a conjecture donned by Caporaso et. al. Conjecture H (see again [CHarM], sec. 6) about Higher dimensional varieties, which is regarded very plausible among experts of higher dimensional algebraic geometry. In particular, we will show

alg-geom

\section{Introduction} Let $X$ be a variety of general type defined over a number field $K$. It was conjectured by S. Lang that the set of rational points $X(K)$ is not Zariski dense in $X$. In the paper \cite{chm} of L. Caporaso, J. Harris and B. Mazur it is shown that the above conjecture of Lang implies the existence of a uniform bound on the number of $K$-rational points of all curves of fixed genus $g$ over $K$. The paper \cite{chm} has immediately created a chasm among arithmetic geometers. This chasm, which often runs right in the middle of the personalities involved, divides between loyal believers of Lang's conjecture, who marvel in this powerful implication, and the disbelievers, who try to use this implication to derive counterexamples to the conjecture. In this paper we will attempt to deepen this chasm, using the techniques of \cite{chm} and continuing \cite{abr}, by proving more implications, some of which very strong, of various conjectures of Lang. Along the way we will often use a conjecture donned by Caporaso et al. {\em Conjecture H} (see again \cite{chm}, \S 6) about {\em H}igher dimensional varieties, which is regarded very plausible among experts of higher dimensional algebraic geometry. Before we state any results, we need to specify various conjectures which we will apply. \subsection{A few conjectures of Lang} Let $X$ be a variety of general type over a field $K$ of characteristic 0. In view of Faltings's proof of Mordell's conjecture, Lang has stated the following conjectures: \begin{conj} \begin{enumerate} \item (Weak Lang conjecture) If $K$ is finitely generated over $\Bbb{Q}$ then the set of rational points $X(K)$ is not Zariski dense in $X$. \item (Weak Lang conjecture for function fields) If $k\subset K$ is a finitely generated regular extension in characteristic 0, and if $X(K)$ is Zariski dense in $X$, then $X$ is birational to a variety $X_0$ defined over $k$ and the {\em ``non-constant points''} $X(K)\setminus X_0(k)$ are not Zariski dense in $X$. \item (Geometric Lang's conjecture) Assuming only $Char(K) = 0$, there is a proper Zariski closed subset $Z(X) \subset X$, called in \cite{chm} the {\em Langian exceptional set}, which is the union of all positive dimensional subvarieties which are not of general type. \item (Strong Lang conjecture) If $K$ is finitely generated over $\Bbb{Q}$ then there is a Zariski closed subset $Z\subset X$ such that for any finitely generated field $L\supset K$ we have that $X(L)\setminus Z(L)$ is finite. \end{enumerate} \end{conj} These conjectures and the relationship between them are studied in \cite{langbul}, \cite{lang3} and in the introduction of \cite{chm}. For instance, it should be noted that the weak Lang conjecture together with the geometric conjecture imply the strong Lang conjecture. It should also be remarked that the analogous situation over fields of positive characteristic is subtle and interesting. See a recent survey by Voloch \cite{voloch}. \subsection{Conjecture H} An important tool used by Caporaso et al. in \cite{chm} is that of fibered powers. Let $X\rightarrow B$ be a morphism of varieties in characteristic 0, where the general fiber is a variety of general type. We denote by $X^n_B$ the $n$-th fibered power of $X$ over $B$. \begin{conj} (Conjecture H of \cite{chm}) For sufficiently large $n$, there exists a dominant rational map $h_n:X^n_B \dashrightarrow W_n$ where $W_n$ is a variety of general type, and where the restriction of $h_n$ to the general fiber $(X_b)^n$ is generically finite. \end{conj} This conjecture is known for curves and surfaces: \begin{thh} (Correlation theorem of \cite{chm}) Conjecture H holds when $X\rightarrow B$ is a family of curves of genus $>1$. \end{thh} \begin{thh} (Correlation theorem of \cite{hassett}) Conjecture H holds when $X\rightarrow B$ is a family of surfaces of general type . \end{thh} Using their Theorem H 1, and Lemma 1.1 of \cite{chm}, Caporaso et al. have shown that the weak Lang conjecture implies a uniform bound on the number of rational points on curves (Uniform bound theorem, \cite{chm} Theorem 1.1). It should be noted that the proofs of theorems H 1 and H 2 give a bit more: they describe a natural dominant rational map $X^n_B\rightarrow W$. For the case of curves, if $B_0$ is the image of $B$ in the moduli space, $ \mbox{ \bf M}_g$, then for sufficiently large $n$ the inverse image $B_n\subset {\mbox{ \bf M}}_{g,n}$ in the moduli space of $n$-pointed curves is a variety of general type. Therefore the moduli map $X^n_B \dashrightarrow B_n\subset \mbox{ \bf M}_{g,n}$ satisfies the requirements. A similar construction works for surfaces of general type, and one may ask whether this should hold in general. It is convenient to make the following definitions when discussing Lang's conjectures: \\ {\bf Definition:} {\em 1. A variety $X/K$ is said to be a {\bf Lang variety} if there is a dominant rational map $X_{\overline{K}} \dashrightarrow W$, where $W$ is a positive dimensional variety of general type. 2. A positive dimensional variety $X$ is said to be {\bf geometrically mordellic} (In short GeM) if $X_{\overline{K}}$ does not contain subvarieties which are not of general type.} In \cite{lang3}, in the course of stating even more far reaching conjectures, Lang defined by a notion of {\em algebraically hyperbolic} varieties which is very similar, and conjecturally the same as that of GeM varieties. I chose to use a different terminology here, to avoid confusion. Note that the weak Lang conjecture directly implies that the rational points on a Lang variety over a number field are not Zariski dense, and that there are only finitely many rational points over a number field on a GeM variety. \subsection{Summary of results} An indicated in \cite{chm} \S 6, Conjecture H together with Lang's conjectures should have very strong implications for counting rational points on varieties of general type, similar to the uniform bound theorem of \cite{chm} . Here we will prove the following basic result: \begin{th}\label{unif} Assume that the weak Lang conjecture as well as conjecture H hold. Let $X \rightarrow B$ be a family of GeM varieties over a number field $K$ (or any finitely generated field over $Q$). Then there is a uniform bound on $\sharp X_b(K)$. \end{th} One may refine this theorem for arbitrary families of varieties of general type, obtaining a bound on the number of points which do not lie in lang exceptional sets of fibers. If one assumes Lang's geometric conjecture, one obtains a closed subset $Z(X_b)$ for every $b\in B$. A natural question which arises in such a refinement is: how do these subsets fit together? An answer was given in \cite{chm} , Theorem 6.1, assuming conjecture H as well: the varieties $Z(X)$ are uniformly bounded. We will show that, using results of Viehweg, one does not need to assume conjecture H: \begin{th}\label{Z(X)}(Compare \cite{chm} , Theorem 6.1) Assume that the geometric Lang conjecture holds. Let $X\rightarrow B$ be a family of varieties of general type. Then there is a proper closed subvariety $\tilde{Z}\subset X$ such that for any $b\in B$ we have $Z(X_b)\subset \tilde{Z}$. \end{th} Using theorem \ref{Z(X)}, we can apply theorem \ref{unif} to any family $X\rightarrow B$ of varieties of general type, assuming that the geometric Lang conjecture holds: we can bound the rational points in the complement of $\tilde{Z}$. We will apply our theorem \ref{unif} in various natural cases. An immediate but rather surprising application is the following theorem: \begin{th}\label{unideg} Assume that the weak Lang conjecture as well as conjecture H hold. Let $X \rightarrow B$ be a family of GeM varieties over a field $K$ finitely generated over $Q$. Fix a number $d$. Then there is a uniform bound $N_d$ such that for any field extension $L$ of $K$ of degree $d$ and every $b\in B(L)$ we have $\sharp X_b(L)<N_d$. \end{th} As a corollary, we see that Lang's conjecture together with conjecture H imply the existence of a bound on the number of points on curves of fixed genus $g$ over a number field $K$ which depends only on the degree of the number field. These results have natural analogues for function fields. We will state a few of these, notably: \begin{th}\label{unigon} Assume that Lang's conjecture for function fields holds. Fix an integer $g>1$. Then there is an integer $N(g)$ such that for any generically smooth fibration of curves $C\rightarrow D$ where the fiber has genus $g$ and the base is hyperelliptic curve, there are at most $N$ non-constant sections $s:D\rightarrow C$. \end{th} We remind the reader that the {\em gonality } of a curve $D$ is the minimal degree of a nonconstant rational function on $D$ (so a curve of gonality 2 is hyperelliptic). One expects the above theorem to be generalized to the situation where ``hyperelliptic curve'' is replaced by ``curve of gonality $\leq d$'' for fixed $d$. \section{Proof of theorem \ref{unif}} \subsection{Preliminaries} Throughout this subsection {\bf we assume that conjecture H holds}, and the base field is algebraically closed. Observe that a positive dimensional subvariety of an GeM variety is GeM; and the normalization of an GeM variety is GeM. Note also that a variety dominating a Lang variety is a Lang variety as well. \begin{prp} Let $X \rightarrow B$ be a family of GeM varieties. Let $F \subset X$ be a reduced subscheme such that every component of $F$ dominating $B$ has positive fiber dimension. Then for $n$ sufficiently large, every component of the fibered power $F^n_B$ which dominates $B$ is a Lang variety. \end{prp} The proof will use the following lemmas: \begin{lem} Let $X\rightarrow B$ and $F$ be as above, and assume that the general fiber of $F\rightarrow B$ is irreducible. Then for $n$ sufficiently large, the dominant component of $F^n_B$ is a Lang variety. \end{lem} {\bf Proof.} Apply conjecture $H$ to $F \rightarrow B$, using the fact that the fibers of $F$ are of general type. \begin{lem} Let $X\rightarrow B$ and $F$ be as in the proposition, with $F$ irreducible. Then for $n$ sufficiently large, every component of the fibered power $F^n_B$ which dominates $B$ is a Lang variety. \end{lem} {\bf Proof.} Let $\tilde{F}$ be the normalization of $F$, and let $\tilde{F} \rightarrow \tilde{B} \rightarrow B$ be the Stein factorization. Denote by $c$ the degree of $\tilde{B}$ over $B$. Let $G \subset \tilde{F}^n_B$ be a dominant component. Then $G$ parametrizes $n$-tuples of points in the fibers of $\tilde{F}$ over $B$, and since $G$ is irreducible, there is a decomposition $\displaystyle \{1,\ldots,n\} = \cup_{i=1}^c J_i$ and $G$ surjects onto the dominant component of $\tilde{F}^{J_i}_{\tilde{B}}$. At least one of $J_i$ has at least $n/c$ elements. Using lemma 1 applied to $\tilde{F}\rightarrow \tilde{B}$, we see that for $n/c$ large enough $G$ is a Lang variety. {\bf Proof of proposition. } Let $F= F_1 \cup\ldots \cup F_m$ be the decomposition into irreducible components. Let $G$ be a dominant component of $F^n_B$. Then $G$ dominates $(F_1)^{n_1}_B \times_B \cdots\times_B(F_m)^{n_m}_B$. For at least one $i$ we have $n_i > n/m$, so applying the previous lemma we obtain that $G$ is a Lang variety. \subsection{Prolongable points} We return to the setup in theorem \ref{unif}. {\bf Definition.} 1. A point $x_n = (P_1,\ldots,P_n)\in X^n_B(K) $ is said to be off diagonal if for any $1\leq i< j\leq n$ we have $P_i\neq P_j$. We extend this for $n=0$ trivially by agreeing that any point of $B(K)$ is off diagonal. 2. Let $m>n$. An off diagonal point $x_n$ is said to be $m$-prolongable if there is an off-diagonal $x_m\in X^m_B(K)$ whose first $n$ coordinates give $x_n$. Let $E_n^{(m)}$ be the set of $m$-prolongable points on $X^n_B$, and let $F_n^{(m)}$ be the Zariski closure. Let $ F_n = \displaystyle\cap_{m>n} F_n^{(m)}$. By the Noetherian property of the Zariski topology we have $F_n = F_n^{(m)}$ for some $m$. All we need to show is $F_n = \emptyset$ for some $n$. \begin{lem} We have a surjection $F_{n+1} \rightarrow F_n$. \end{lem} {\bf Proof.} The set $E_{n+1}^{(m)}$ surjects to $E_n^{(m)}$ for any $m>n+1$. \begin{lem} Every fiber of $F_{n+1} \rightarrow F_n$ is positive dimensional. \end{lem} {\bf Proof.} Suppose that over an open set in $F_n$ the degree of the map is $d$. Then $E_n^{(n+d+1)}$ cannot be dense in $F_n$: if $(y_1,\ldots,y_{n+d+1})$ is an off diagonal prolongation of $(y_1,\ldots,y_{n+d+1})\in E_n^{(n+d+1)}$, then for $n+1\leq j\leq n+d+1$ we have that the points $(y_1,\ldots,y_{n}, y_j)\in E_{n+1}^{(n+d+1)}$ are distinct, therefore the degree of the map is at least $d+1$. \subsection{Proof of theorem.} We show by induction on $i$ that for any $n$ and $i$ the dimension of any fiber of $F_{n+1}\rightarrow F_n$ is at least $i+1$. Lemma 4 shows this for $i=0$. Assume it holds true for $i-1$, let $n\geq 0$ and let $G$ be a component of $F_{n}$, such that the fiber dimension of $F_{n+1}$ over $G$ is $i$. Applying the inductive assumption to each $F_{n+j+1}\rightarrow F_{n+j}$, we have that the dimension of every fiber of $F_{n+k}$ over $F_n$ is at least $ik$. On the other hand, $F_{n+k}$ is a subscheme of the fibered power $(F_{n+1})^k_{F_n}$, so over $G$ it has fiber dimension precisely $ik$. Therefore there exists a component $H_k$ of $F_{n+k}$ dominant over $G$ of fiber dimension $ik$, which is therefore identified as a dominant component of the fibered power $(F_{n+1})^k_{F_n}$. By proposition 1, for $k$ sufficiently large we have that $H_k$ is a Lang variety. Lang's conjecture implies that $H_k(K)$ is not dense in $K$, contradicting the definition of $F_{n+k}$. \qed \section{A few refinements and applications in arithmetic and geometry} \subsection{Proof of Theorem \ref{Z(X)}} Assume that $X\rightarrow B$ is a family of varieties of general type. By Hironaka's desingularization theorem, we may assume that $B$ is a smooth projective variety. Let $L$ be an ample line bundle on $B$, let $n>>0$ be a sufficiently large integer and let $H$ be a smooth divisor of $L^{\otimes n}$. Let $B_1\rightarrow B$ be the cyclic cover ramified to order $n$ along $H$. Then by adjunction, $B_1$ is a variety of general type. Let $X_1\rightarrow X$ be the pullback of $X$ to $B_1$. By the main theorem (Satz III) of \cite{viehweg}, the variety $X_1$ is of general type. Assuming the geometric Lang conjecture, Let $Z_1(X_1)$ be the Langian exceptional set. Let $\tilde{Z}$ be the image of $Z_1(X_1)$ in $X$. Then for any $b\in B$, we have by definition that $Z(X_b)\subset \tilde{Z}$. \qed It has been noted in \cite{chm} that Viehweg's work goes a long way towards proving conjecture H. It is therefore not surprising that it may be used on occasion to replace the assumption of conjecture H. \subsection{Uniformity in terms of the degree of an extension} Let $X \rightarrow B$ be a family of GeM varieties over $K$. Assuming the conjectures, theorem 1 gave us a uniform bound on the number of rational points over finite extension fields in the fibers. We will now see that this in fact implies a much stronger result, namely our theorem \ref{unideg}: the uniform bound only depends on the degree of the field extension. {\bf Proof of theorem \ref{unideg}:} for $n=1$ or $2$, Let $Y_n = \mbox{ \rm Sym}^d(X^n_B)$, and $Y_0 =\mbox{ \rm Sym}^d(B)$. Then we have natural maps $p_n:Y_n \rightarrow Y_{n-1}$. Let $\Gamma$ be the branch locus of the quotient map $X^d \rightarrow Y_1$, namely the set of points which are fixed by some permutation. If $P\not\in \Gamma$ then $p_2^{-1}(P)$ is a GeM variety, isomorphic over $\overline{K}$ to the product of $d$ fibers of $X$. Denote $Y_1' = Y_1\setminus \Gamma_1$, and $Y_2'=p_2^{-1}Y_1'$. Then $Y_2'\rightarrow Y_1'$ is a family of GeM varieties, and by Theorem 1 we have a bound on the cardinality of $(Y_2')_y(K)$ uniform over $y\in Y_1'(K)$. By induction, it suffices to bound the number of points in $X_b(L)$ over any field $L$ of degree $d$ over $K$, which are defined over $L$ but not over any intermediate field. If $\sigma_1,\ldots,\sigma_d$ are the distinct embeddings of $L$ in $\overline{K}$, and $P\in X_b(L)$ not defined over an intermediate field, then the points $\sigma_i(P)\in X_{\sigma_i(b)}(\sigma_i(K))\subset X(\overline{K})$ are distinct. If $(P_1,P_2)\in X_B^2(L)$ is a pair of such points, then the Galois orbit $\{\sigma_i(P_1,P_2), i=1,\ldots,d\}$ is Galois stable, therefore it gives rise to a point in $Y_2(K)$. This point has the further property that its image in $Y_1$ does not lie in $\Gamma_1$, so it gives rise to a point in $Y'_2(K)$. The previous paragraph shows that the number of points on a fiber is bounded. \qed Applying theorem 3 where $X\rightarrow B$ is the universal family over the Hilbert scheme of 3-canonical curves of genus $g$ (as in \cite{chm} , \S\S 1.2), we obtain the following: \begin{cor} Assume that the weak Lang conjecture as well as conjecture H hold. Fix integers $d, g>1$ and a number field $K$. Then there is a uniform bound $N_d$ such that for any field extension $L$ of $K$ of degree $d$ and every curve $C$ of genus $g$ over $L$ we have $\sharp C(L)<N_d$. \end{cor} We remark that in the cases of degrees $d\leq 3$ one does not need to assume conjecture H: this was proven in \cite{abr}, using the fact that conjecture H holds for families of curves or surfaces. A similar result is being worked out by P. Pacelli for arbitrary $d$. Here is a special case: let $f(x)\in {\Bbb Q}(x)$ be a polynomial of degree $>4$ with distinct complex roots. Then, assuming the weak Lang conjecture, the number of rational points over any quadratic field on the curve $C: y^2 = f(x)$ is bounded uniformly. We remark that, if $\deg f>6$, this in fact may be deduced using a combination of \cite{chm} and a theorem of Vojta \cite{vojta} which says that all but finitely many quadratic points on $C$ have rational $x$ coordinate. One then applies \cite{chm} which gives a uniform bound on the rational points on the twists $ty^2 = f(x)$. Following the suggestion of \cite{chm}, \S 6 one can apply Theorem 1 to symmetric powers of curves. Since conjecture H is known for surfaces, one obtains the following (stated without proof in \cite{chm}, Theorem 6.2): \begin{cor}(Compare \cite{chm} , Theorem 6.2) Assume that the weak Lang conjecture holds. Fix a number field $K$. Then there is a uniform bound $N$ for the number of quadratic points on any nonhyperelliptic, non-bielliptic curve $C$ of genus $g$ over $K$. \end{cor} Similarly, it was shown in \cite{ah}, lemma 1 that if the gonality of a curve $C$ is $>2d$ then $\mbox{ \rm Sym}^d(C)$ is GeM. Recall that a closed point $P$ on $C$ is said to be of degree $d$ over $K$ if $[K(P):K]=d$. We deduce the following: \begin{cor} Assume that the weak Lang conjecture holds. Fix a number field $K$ and an integer $d$. Then there is a uniform bound $N$ for the number of points of degree $d$ over $K$ on any curve $C$ of genus $g$ and gonality $>2d$ over $K$. \end{cor} \subsection{The geometric case} One can use the same methods using Lang's conjecture for function fields of characteristic 0, say over $\Bbb C$. Given a fibration $X\rightarrow B$ where the generic fiber is a variety of general type, a rational point $s\in X(K_B)$ over the function field of $B$ is called {\em constant} if $X$ is birational to a product $X_0\times B$ and $s$ corresponds to a point on $X_0$. Lang's conjecture for function fields says that the non-constant points are not Zariski dense. In this section we will restrict attention to the case where the base is the projective line ${\Bbb{P}^1}$. We will only assume the following statement: if $X$ is a variety of general type, then the rational curves in $X$ are not Zariski dense. It is easy to see that this statement in fact follows from the geometric Lang conjecture, as well as from Lang's conjecture for function fields. We would like to apply this conjecture to obtain geometric uniformity results. One has to be careful here, since the conjecture does not apply to Lang varieties, and one has to use a variety of general type directly. As stated in the introduction, if $X\rightarrow B$ is a family of curves of genus $>1$ the appropriate variety $W$ of general type dominated by $X^n_B$ is identified in \cite{chm} as the image $B_n\subset{\mbox{ \bf M}}_{g,n}$ of $X^n_B$ by the moduli map. We use this in the proof of the following proposition: \begin{prp}\label{unigeom} Assume that Lang's conjecture for function fields holds. Fix an integer $g>1$. Then there is a bound $N$ such that for any generically smooth family of curves $C\rightarrow \Bbb{P}^1$ of genus $g$ there are at most $N$ non-constant sections $s:\Bbb{P}^1\rightarrow C$. \end{prp} {\bf Proof.} First note that if $s:\Bbb{P}^1\rightarrow C$ is a nonconstant section whose image in $ \mbox{ \bf M}_{g,1}$ is a point, then $s$ becomes a constant section after a finite base change $D\rightarrow \Bbb{P}^1$. This implies that $s$ is fixed by a nontrivial automorphism of $C$, and the number of such points is bounded in terms of $g$. Therefore it suffices to bound the number of sections whose image in $\mbox{ \bf M}_{g,1}$ is non-constant. We will call such sections {\em strictly non-constant}. Let $B_0\subset \mbox{ \bf M}_g$ be a closed subvariety, and choose $n$ such that $B_n\subset \mbox{ \bf M}_{g,n}$ is of general type. If a family $C\rightarrow {\Bbb{P}^1}$ has moduli in $B_0$, then for any $n$-tuple of strictly non-constant sections $s_i:{\Bbb{P}^1}\rightarrow C$, we obtain a non-constant rational map ${\Bbb{P}^1} \rightarrow B_n$. Let ${F} \subset B_n$ be the Zariski closure of the images of the collection of non-constant rational maps obtained this way. Since $B_n$ is of general type, Lang's conjecture implies that $F\neq B_n$. Applying lemma 1.1 of \cite{chm} we obtain that there is an closed subset set $F_0\subset B_0$ and an integer $N$ such that, given a family of curve $C\rightarrow{\Bbb{P}^1}$ such that the rational image of ${\Bbb{P}^1}$ in $\mbox{ \bf M}_g$ lies in $B_0$ but not in $F_0$, there are at most $N$ non-constant sections of $C$. Noetherian induction gives the theorem. \qed Choosing a coordinate $t$ on $ {\Bbb P}^1$ we can pull back the curve $C$ along the map ${\Bbb{P}^1}\rightarrow {\Bbb{P}^1}$ obtained by taking $n$-th roots of $t$. Let ${\Bbb C}(t^{1/\infty}) = {\Bbb C}(\{t^{1/n}, n\geq 1\})$, the field obtained by adjoining all roots of $t$. If one restricts attention to non-isotrivial curves, one obtains the following amusing result (suggested to the author by Felipe Voloch): \begin{cor} Assume that the Lang conjecture for function fields holds. Fix an integer $g>1$. Then there is a bound $N$ such that for any smooth nonisotrivial curve $C$ over ${\Bbb C}(t)$ of genus $g$ there are at most $N$ points in $C({\Bbb C}(t^{1/\infty}))$. \end{cor} One can also try to prove uniformity results analogous to theorem 3. Using the results in \cite{abr} we can refine proposition \ref{unigeom} and obtain theorem \ref{unigon}. {\bf Proof of theorem \ref{unigon}.} The proof is a slight modification of the theorem of \cite{abr}, keeping track of the dominant map to a variety of general type. As in the proof of theorem 3, it suffices to look at sections $s:D\rightarrow C$ which are not pullbacks of sections of a family over ${\Bbb{P}^1}$. In an analogous way to the proof of theorem \ref{unif}, we say that an $n$-tuple of distinct, strictly non-constant sections is $m$-prolongable if it may be prolonged to an $m$-tuple of distinct, strictly non-constant sections, none of which being the pullback from a family over ${\Bbb{P}^1}$. Any $n$-tuple of distinct sections $s_i:D\rightarrow C$ over a hyperelliptic curve $D$ gives rise to a rational map ${\Bbb{P}^1}\rightarrow \mbox{ \rm Sym}^2(\mbox{ \bf M}_{g,n})$. We define $F_n^{(m)}$ to be the closure in $\mbox{ \rm Sym}^2(\mbox{ \bf M}_{g,n})$ of the images of $m$-prolongable sections, and $F_n = \cap_{m>n} F_n^{(m)}$. As in Lemma 1, we have that the relative dimension of any fiber of $F_{n+1}\rightarrow F_n$ is positive. We have two cases to consider: either for high $n$ there is a component of $F_{n+1}$ having fiber dimension 1 over $F_n$, or for all $n$ the fiber dimension is everywhere 2. In case the fiber dimension is 1, we will see that there is a component of $F_{n+k}$ which is a variety of general type. Assuming Lang's conjecture for function fields this contradicts the fact that the images of non-constant sections are dense. Fix a general fiber $f$ of $F_{n+1}$ over $F_{n}$. The curve $f$ lies inside a surface isomorphic to the product of two curves $C_{b_1} \times C_{b_2}$. By the definition of $m$-prolongable sections, and analogously to lemma 1, we obtain that there is a component $f'$ of $f$ which maps surjectively to both $C_{b_1}$ and $C_{b_2}$. Therefore as either $b_1$ or $b_2$ moves in $B_0$, the moduli of $f'$ move as well. Let $F'$ be a component of $F_{n+1}$ whose fibers have this property. If we follow the proof of proposition 1 and use the moduli description of the dominant map to a variety of general type $m:(F')^k_{F_n}\rightarrow W$, we see that if $E$ is a general curve in $(F')^k_{F_n}$ lying in a fiber of $m$, then $E$ projects to a point in $B_0$; moreover, by the definition of prolongable points, $E$ projects to an off diagonal point in some $(F')^l_{F_n}$. But the fibers over off-diagonal points are GeM varieties, therefore the general fiber of the map $m$ is of general type. By the main theorem of \cite{viehweg}, $(F')^k_{F_n}$ is itself a variety of general type, and therefore $F_{n+k}$ has a component of general type, contradicting Lang's conjecture. In case of fiber dimension 2, we use proposition 1 of \cite{abr}: let $B\subset \mbox{ \rm Sym}^2(\mbox{ \bf M}_g)$. Then for high $n$, the inverse image $B_n\subset \mbox{ \rm Sym}^2(\mbox{ \bf M}_{g,n})$ of $B$ is a variety of general type. Since the images of non-constant sections are dense in $F_n$, this again contradicts Lang's conjecture. \qed If one restricts attentions to trivial fibrations, one obtains as an immediate corollary: \begin{cor} Assume that the Lang conjecture for function fields holds. Fix an integer $g>1$. Then there is an integer $N$ such that for any curve $C$ of genus $g$ and any hyperelliptic curve $D$ there are at most $N$ non-trivial morphisms $f:D\rightarrow C$. \end{cor} It should be noted that the theory of Hilbert schemes gives the existence of a bound depending on the genus of $D$, which is clearly not as strong. As in the arithmetic case, I expect that work in progress of Pacelli should generalize these results to the case where $D$ is $d$-gonal, for fixed $d$.

9505

alg-geom/9505029

en

https://arxiv.org/abs/alg-geom/9505029

alg-geom/9505029

Teleman

Ch. Okonek and A. Teleman

Quaternionic Monopoles

LaTeX, 35 pages

10.1007/BF02099718

We present the simplest non-abelian version of Seiberg-Witten theory: Quaternionic monopoles. These monopoles are associated with Spin^h(4)-structures on 4-manifolds and form finite-dimensional moduli spaces. On a Kahler surface the quaternionic monopole equations decouple and lead to the projective vortex equation for holomorphic pairs. This vortex equation comes from a moment map and gives rise to a new complex-geometric stability concept. The moduli spaces of quaternionic monopoles on Kahler surfaces have two closed subspaces, both naturally isomorphic with moduli spaces of canonically stable holomorphic pairs. These components intersect along Donaldsons instanton space and can be compactified with Seiberg-Witten moduli spaces. This should provide a link between the two corresponding theories. Notes: To appear in CMP The revised version contains more details concerning the Uhlenbeck compactfication of the moduli space of quaternionic monopoles, and possible applications are discussed. Attention ! Due to an ununderstandable mistake, the duke server had replaced all the symbols "=" by "=3D" in the tex-file of the revised version we sent on February, the 2-nd. The command "\def{\ad}" had also been damaged !

alg-geom

\section{Introduction} Recently, Seiberg and Witten [W] introduced new 4-manifold invariants, essentially by counting solutions of the monopole equations. The new invariants have already found nice applications, like e.g. in the proof of the Thom conjecture [KM] or in a short proof of the Van de Ven conjecture [OT2]. In this paper we introduce and study the simplest and the most natural non-abelian version of the Seiberg-Witten monopoles, the quaternionic monopoles. Let $(X,g)$ be an oriented Riemannian manifold of dimension 4. The structure group $SO(4)$ has as natural extension the quaternionic spinor group $Spin^h(4):=Spin(4)\times_{{\Bbb Z}_2}Sp(1)$: $$1\longrightarrow Sp(1)\longrightarrow Spin^h(4)\longrightarrow SO(4)\longrightarrow 1 \ .$$ The projection onto the second factor $Sp(1)=SU(2)$ induces a "determinant map" $\delta:Spin^h(4)\longrightarrow PU(2)$. A $Spin^h(4)$-structure on $(X,g)$ consists of a $Spin^h(4)$-bundle over $X$ and an isomorphism of its $Sp(1)$-quotient with the (oriented) orthonormal frame bundle of $(X,g)$. Given a $Spin^h(4)$-structure on $X$, one has a one-one correspondence between $Spin^h$-connections projecting onto the Levi-Civita connection and $PU(2)$-connections in the associated "determinant" $PU(2)$-bundle. The quaternionic monopole equations are: $$\left\{\begin{array}{ccc}\hskip 4pt{\not}{D}_{A}\Psi&=&0\ \ \ \ \\ \Gamma(F_{A}^+)&=&(\Psi\bar\Psi)_0 \ \ , \end{array}\right.\eqno{ }$$ where $A$ is a $PU(2)$-connection in the "determinant" of the $Spin^h(4)$-structure and $\hskip 4pt{\not}{D}_A$ the induced Dirac operator; $\Psi$ is a positive quaternionic half-spinor. The Dirac operator satisfies the crucial Weitzenb\"ock formula : $$\hskip 4pt{\not}{D}_A^2=\nabla_{\hat A}^*\nabla_{\hat A}+\Gamma(F_A)+\frac{s}{4}{\rm id}$$ It can be used to show that the solutions of the quaternionic monopole equations are the absolute minima of a certain functional, just like in the $Spin^c(4)$-case [JPW]. The moduli space of quaternionic monopoles associated with a fixed $Spin^h(4)$-structure ${\fam\meuffam\tenmeuf h}$ is a real analytic space of virtual dimension $$m_{\fam\meuffam\tenmeuf h}=-\frac{1}{2}(3p_1+3e+4\sigma)\ .$$ Here $p_1$ is the first Pontrjagin class of the determinant, $e$ and $\sigma$ denote the Euler characteristic and the signature of $X$. Note that $m_{\fam\meuffam\tenmeuf h}$ is an even integer iff $X$ admits an almost complex structure. The moduli spaces of quaternionic monopoles contain the Donaldson instanton moduli spaces as well as the classical Seiberg-Witten moduli spaces, which suggests that they could provide a method of comparing the two theories. We study the analytic structure around the Donaldson moduli space. Much more can be said if the holonomy of $(X,g)$ reduces to $U(2)$, i.e. if $(X,g)$ is a K\"ahler surface. In this case we use the canonical $Spin^c(4)$-structure with $\Sigma^+=\Lambda^{00}\oplus\Lambda^{02}$ and $\Sigma^-=\Lambda^{01}$ as spinor bundles. The data of a $Spin^h(4)$-structure ${{\germ h}}$ in $(X,g)$ is then equivalent to the data of a Hermitian 2-bundle $E$ with $\det E=\Lambda^{02}$. The determinant $\delta({\germ h})$ coincides with the $PU(2)$-bundle $P(E)$ associated with $E$. A positive spinor can be written as $\Psi=\varphi+\alpha$, where $\varphi\in A^0(E^{\vee})$ and $\alpha\in A^{02}(E^{\vee})$ are $E^{\vee}$-valued forms. To give a $PU(2)$-connection in $P(E)$ means to give a $U(2)$-connection in $E$ inducing the Chern connection in $\Lambda^{02}$, or equivalently, a $U(2)$-connection $C$ in $E^{\vee}$ inducing the Chern connection in $K_X=\Lambda^{20}$. A pair $(C,\varphi+\alpha)$ solves the quaternionic monopole equation iff $C$ is a connection of type $(1,1)$, one of $\alpha$ or $\varphi$ vanishes while the other is $\bar\partial_C$-holomorphic, and a certain projective vortex equation is satisfied. This shows that in the K\"ahler case the moduli space decomposes as a union of two Zariski closed subspaces intersecting along the Donaldson locus. The two subspaces are interchanged by a natural real analytic involution, whose fixed point set is precisely the Donaldson moduli space. The projective vortex equation comes from a moment map which corresponds to a new stability concept for pairs $({\cal E},\varphi)$ consisting of a holomorphic bundle ${\cal E}$ with canonical determinant $\det{\cal E}={\cal K}_X$ and a holomorphic section $\varphi$. We call such a pair canonically stable iff either ${\cal E}$ is stable, or $\varphi\ne 0$ and the divisorial component $D_\varphi$ of the zero locus satisfies the inequality $$c_1\left({\cal O}_X(D_\varphi )^{\otimes2}\otimes K_X^{\vee}\right) \cup [\omega_g]<0 \ \ .$$ Our main result identifies the moduli spaces of irreducible quaternionic monopoles on a K\"ahler surface with the algebro-geometric moduli space of canonically stable pairs. In the algebraic case, moduli spaces of quaternionic monopoles can easily be computed using our main result (Theorem 7.3) and Lemma 5.5. The moduli spaces may have several components: Every component contains a Zariski open subset which is a holomorphic ${\Bbb C}^*$-bundle. For some components, this ${\Bbb C}^*$-bundle consists only of pairs $({\cal E},\varphi)$ with ${\cal E}$ stable as a bundle; components of this type can be obtained by compactifying the corresponding ${\Bbb C}^*$-bundle with a Donaldson moduli space at infinity. In the other direction, the component is not compact, but has a {\sl natural compactification} obtained by adding spaces associated with Seiberg-Witten moduli spaces. The other components can also be naturally compactified by using Seiberg-Witten moduli spaces in both directions. This compactification process, as well as the corresponding differential geometric interpretation will be the subject of a later paper. \footnote{ After having completed our results we received a manuscript by Labastida and Marino [LM] in which related ideas are proposed from a physical point of view, and physical implications are discussed } \section{$Spin^h$-structures} The quaternionic spinor group is defined as $$Spin^h:=Spin\times_{{\Bbb Z}/2}Sp(1)=Spin\times_{{\Bbb Z}/2}SU(2)\ ,$$ and fits in the exact sequences $$\begin{array}{c} 1\longrightarrow Sp(1)\longrightarrow Spin^h\stackrel{\pi}{\longrightarrow}SO\longrightarrow 1\\ 1\longrightarrow Spin\longrightarrow Spin^h\stackrel{\delta}{\longrightarrow}\ PU(2)\longrightarrow 1\end{array} \eqno{(1)}$$ These can be combined in the sequence $$1\longrightarrow{\Bbb Z}/2\longrightarrow Spin^h\textmap{(\pi,\delta)} SO\times PU(2)\longrightarrow 1\eqno{(2)}$$ In dimension 4, $Spin^h(4)$ has a simple description, coming from the splitting $Spin(4)=SU(2)\times SU(2)$: $$Spin^h(4)=\qmod{SU(2)\times SU(2) \times SU(2)}{{\Bbb Z}/2}$$ with ${\Bbb Z}/2=\langle(-{\rm id},-{\rm id},-{\rm id})\rangle$. There is another useful way to think of $Spin^h(4)$: let $G$ be the group $$G:=\{(a,b,c)\in U(2)\times U(2)\times U(2)|\ \det a=\det b= \det c\}\ .$$ One has an obvious isomorphism $Spin^h(4)=\qmod{G}{S^1}\ $ , and a commutative diagram with exact rows $$\matrix{1\rightarrow&{\Bbb Z}_2&\longrightarrow &SU(2)\times SU(2)\times SU(2)&\longrightarrow &Spin^h(4)&\rightarrow 1\cr &\downarrow& &\downarrow&&\parallel&\cr 1\rightarrow &S^1&\longrightarrow &G&\longrightarrow &Spin^h(4)&\rightarrow 1\cr }\eqno{(3)}$$ \begin{dt} Let $P$ be a principal $SO$-bundle over a space $X$. A $Spin^h$-structure in $P$ is a pair consisting of a $Spin^h$ bundle $P^h$ and an isomorphism $P\simeq P^h\times_\pi SO$. The $PU(2)$-bundle associated with a $Spin^h$-structure is the bundle $P^h\times_\delta PU(2)$. \end{dt} \begin{lm} A principal $SO$-bundle admits a $Spin^h$-structure iff there exists a $PU(2)$-bundle with the same second Stiefel-Whitney class. \end{lm} {\bf Proof: } This follows from the cohomology sequence $$\longrightarrow H^1(X,\underline{Spin^h})\longrightarrow H^1(X,\underline{{\phantom(}SO{\phantom)}}\times\underline{PU(2)})\textmap{\ beta} H^2(X,{\Bbb Z}/2)$$ associated to (2), since the connecting homomorphism $\beta$ is given by taking the sum of the second Stiefel-Whitney classes of the two factors. \hfill\vrule height6pt width6pt depth0pt \bigskip In this paper we will only use $Spin^h$-structures in $SO(4)$-bundles whose second Stiefel Whitney class admit {\sl integral} lifts. Then we have: \begin{lm} Let $P$ be a principal $SO(4)$-bundle whose second Stiefel-Whitney class $w_2(P)$ is the reduction of an integral class. Isomorphism classes of $Spin^h(4)$-structures in $P$ are in 1-1 correspondence with equivalence classes of triples consisting of a $Spin^c(4)$-structure $\qmod{P^c}{S^1}\simeq P$ in $P$, a $U(2)$-bundle $E$, and an isomorphism $\det P^c\simeq\det E$, where two triples are equivalent if they can be obtained from each other by tensoring with an $S^1$-bundle. \end{lm} {\bf Proof: } The cohomology sequence associated with the second row in (3) shows that $Spin^h$-structures in bundles whose second Stiefel-Whitney classes admit integral lifts are given by $G$-structures modulo tensoring with $S^1$-bundles. On the other hand, to give a $G$-structure in $P$ simply means to give a triple $(\Sigma^+,\Sigma^-,E)$ of $U(2)$-bundles together with isomorphisms $$\det\Sigma^+\simeq\det\Sigma^-\simeq\det E \ .$$ This is equivalent to giving a triple consisiting of a $Spin^c(4)$-structure $\qmod{P^c}{S^1}\simeq P$ in $P$, a $U(2)$-bundle, and an isomorphism $\det P^c\simeq\det E$. \hfill\vrule height6pt width6pt depth0pt \bigskip In the situation of this lemma, we get well defined vector bundles $${\cal H}^{\pm}:=\Sigma^{\pm}\otimes E^{\vee}\ $$ depending only on the $Spin^h$-structure and not on the chosen $G$-lifting. These spinor bundles have the following intrinsic interpretation: identify $SU(2)\times_{{\Bbb Z}/2} SU(2)$ with $SO(4)$, and denote by $$\pi_{ij}:Spin^h\longrightarrow SO(4)$$ the projections of $Spin^h(4)=\qmod{SU(2)\times SU(2)\times SU(2)}{{\Bbb Z}/2}$ onto the indicated factors ($\pi=\pi_{12}$). Using the inclusion $SO(4)\subset SU(4)$, we can form three $SU(4)$-vector bundles $P^h\times_{\pi_{ij}}{\Bbb C}^4$, $(i,j)\in\{(1,2),(1,3),(2,3)\}$. Under the conditions of the previous lemma we have $${\cal H}^+=P^h\times_{\pi_{13}}{\Bbb C}^4\ ,\ \ {\cal H}^-=P^h\times_{\pi_{23}}{\Bbb C}^4\ ,\ \ \Sigma^+\otimes(\Sigma^-)^{\vee}=P^h\times_{\pi}{\Bbb C}^4 \ .$$ The $PU(2)$-bundle $P^h\times_\delta PU(2)$ associated with the $Spin^h$-structure $\qmod{P^c}{S^1}\simeq P$ has in this case a very simple description: it is the projectivization $P(E)$ of the $U(2)$-bundle $E$. \section{ The quaternionic monopole equations} Let $(X,g)$ be an oriented Riemannian 4-manifold with orthonormal frame bundle $P$. The exact sequence (2) in the previous section shows two things: first, isomorphism classes of $PU(2)$-bundles with second Stiefel-Whitney class equal to $w_2(P)$ are in 1-1 correspondence with orbits of $Spin^h(4)$-structures in $P$ under the action of $H^1(X,{\Bbb Z} /2)$; second, $Spin^h(4)$-connections in a $Spin^h(4)$-bundle $P^h$ which induce the Levi-Civita connection in $P$ correspond bijectively to connections in the associated $PU(2)$-bundle $P^h\times_\delta PU(2)$. Now it is well known that $w_2(P)=w_2(X)$ is always the reduction of an integral class [HH], so that we can think of a $Spin^h$-structure in $P$ as a triple $(\Sigma^+,\Sigma^-,E)$ of $U(2)$-bundles with isomorphisms $\det\Sigma^+\simeq\det\Sigma^-\simeq\det E$ modulo tensoring with unitary line bundles. We denote the $Spin^h(4)$-connection corresponding to a connection $A\in{\cal A}(P(E))$ in the associated $PU(2)$-bundle by $\hat A$. \begin{re} Given a fixed $U(1)$-connection $c$ in $\det E$, the elements in ${\cal A}(P(E))$ can be identified with those $U(2)$-connections in $E$, which induce the fixed connection $c$. \end{re} Now view a $Spin^h(4)$-structure in $P$ as a $Spin^c(4)$-structure $\qmod{P^c}{S^1}\simeq P$ together with a $U(2)$-bundle $E$ and an isomorphism $\det P^c\simeq\det E$ . Recall that the choice of $\qmod{P^c}{S^1}\simeq P$ induces an isomorphism $$\gamma:\Lambda^1\otimes{\Bbb C}\longrightarrow(\Sigma^+)^{\vee}\otimes\Sigma^-$$ which extends to a homomorphism $$\Lambda^1\otimes{\Bbb C}\longrightarrow{\rm End}_0(\Sigma^+\oplus\Sigma^-)\ ,$$ mapping the bundle $\Lambda^1$ of real 1-forms into the bundle of trace-free skew-Hermitian endomorphisms. The induced homomorphism $$\Gamma:\Lambda^2\otimes{\Bbb C}\longrightarrow {\rm End}_0(\Sigma^+\oplus\Sigma^-)$$ maps the subbundles $\Lambda^2_{\pm}\otimes{\Bbb C}$ isomorphically onto the bundles ${\rm End}_0(\Sigma^{\pm})$, and identifies $\Lambda_{\pm}$ with the trace-free, skew-Hermitian endomorphisms ([H], [OT1]). \begin{dt} Let $P^h\times_{\pi}SO(4)\simeq P$ be a ${\rm Spin}^h(4)$-structure in $P$ with spinor bundle ${\cal H}:={\cal H}^+\oplus{\cal H}^-$ and associated $PU(2)$-bundle $P(E)$. Choose a connection $A\in{\cal A}(P(E))$, and let $\hat A$ be the corresponding $Spin^h(4)$-connection in $P^h$. The associated Dirac opearor is defined as the composition $$\hskip 4pt{\not}{D}_A:A^0({\cal H})\textmap{\nabla_{\hat A}} A^1({\cal H})\textmap{\gamma} A^0({\cal H})\ ,$$ where $\nabla_{\hat A}$ is the covariant derivative of $\hat A$ and $\gamma$ the Clifford multiplication. \end{dt} Note that the restricted operators $$\hskip 4pt{\not}{D}_A : A^0({\cal H}^{\pm})\longrightarrow A^0({\cal H}^{\mp})$$ interchange the positive and negative half-spinors. Let $s$ be the scalar curvature of $(X,g)$. \begin{pr} The Dirac operator $\hskip 4pt{\not}{D}_A:A^0({\cal H})\longrightarrow A^0({\cal H})$ is an elliptic, selfadjoint operator whose Laplacian satisfies the Weitzenb\"ock formula $$\hskip 4pt{\not}{D}_A^2=\nabla_{\hat A}^*\nabla_{\hat A}+\Gamma(F_A)+\frac{s}{4}id_{\cal H}\eqno{(4)}$$ \end{pr} {\bf Proof: } Choose a $Spin^c(4)$-structure $\qmod{P^c}{S^1}\simeq P$ and a $S^1$-connection $c$ in the unitary line bundle $\det P^c$. The connection $A\in {\cal A}(P(E))$lifts to a unique $U(2)$-connection $C$ in the bundle $E^{\vee}$ which induces the dual connection of $c$ in $\det E^{\vee}=(\det P^c)^{\vee}$. In [OT1] we introduced the Dirac operator $$\hskip 4pt{\not}{D}_{C,c}:A^0(\Sigma \otimes E^{\vee}) \longrightarrow A^0(\Sigma \otimes E^{\vee})\ ;$$ by construction it coincides with the operator $\hskip 4pt{\not}{D}_A:A^0({\cal H})\longrightarrow A^0({\cal H})$, and its Weitzenb\"ock formula reads $$\hskip 4pt{\not}{D}_{C,c}^2=\nabla_{\hat A}^*\nabla_{\hat A}+\Gamma(F_{C,c})+\frac{s}{4}id_{\cal H}\ ,$$ where $F_{C,c}=F_C+\frac{1}{2}F_c{\rm id}_{E^{\vee}}\in A^2( {\rm End} E^{\vee})$. Substituting $$F_C =\frac{1}{2}{\rm Tr}(F_C){\rm id}_{E^{\vee}}+F_A$$ and using $\frac{1}{2}{\rm Tr}(F_C)=-\frac{1}{2}F_c$ we get the Weitzenb\"ock formula (4) for $ \hskip 4pt{\not}{D}_A$. \hfill\vrule height6pt width6pt depth0pt \bigskip Consider now a section $ \Psi\in A^0({\cal H}^{\pm})$. We denote by $$(\Psi\bar\Psi)_0\in A^0({\rm End}_0 \Sigma^{\pm}\otimes{\rm End}_0 E^{\vee})$$ the projection of $\Psi\otimes\bar\Psi\in A^0({\rm End}{\cal H}^{\pm})$ onto the fourth summand in the decomposition $${\rm End}({\cal H}^{\pm})={\Bbb C}{\rm id}\oplus{\rm End}_0\Sigma^{\pm}\otimes{\rm End}_0 E^{\vee}\otimes({\rm End}_0\Sigma^{\pm}\otimes{\rm End}_0 E^{\vee})\ .$$ $(\Psi\bar\Psi)_0$ is a Hermitian endomorphism which is trace-free in both factors. \begin{dt}Choose a $Spin^h(4)$-structure in $P$ with spinor bundle ${\cal H}$ and associated $PU(2)$-bundle $P(E)$. The quaternionic monopole equations for the pair $(A,\Psi)\in{\cal A}(P(E))\times A^0({\cal H})$ are the following equations: $$\left\{\begin{array}{ccc}\hskip 4pt{\not}{D}_{A}\Psi&=&0\ \ \ \ \\ \Gamma(F_{A}^+)&=&(\Psi\bar\Psi)_0 \ .\ \ \end{array}\right.\eqno{(SW^h) }$$ \end{dt} The following result is the analog of Witten's formula in the quaternionic case (see [W], \S 3 ): \begin{pr} Let $\Psi\in A^0({\cal H}^+)$ be a positive half-spinor, $A\in {\cal A}(P(E))$ a connection in $P(E)$. Then we have $$\parallel\hskip 4pt{\not}{D}_{A}\Psi\parallel^2+ \frac{1}{2}\parallel\Gamma(F_{A}^+)-(\Psi\bar\Psi)_0\parallel^2=$$ $$= \parallel\nabla_{\hat A}\Psi\parallel^2+ \frac{1}{2}\parallel F_{A}^+\parallel^2+\frac{1}{2}\parallel(\Psi\bar\Psi)_0\parallel^2+ \frac{1}{4}\int\limits_X s|\Psi|^2. \eqno{(5)}$$ \end{pr} {\bf Proof: } The pointwise inner product $(\Gamma(F_A)\Psi,\Psi)$ for a positive half-spinor $\Psi$ simplifies: $(\Gamma(F_A)\Psi,\Psi)= (\Gamma(F_A^+)\Psi,\Psi)= (\Gamma(F_A^+),(\Psi\bar\Psi)_0)$, since $\Gamma(F_A^-)$ vanishes on $A^0({\cal H}^+)$, and since $ \Gamma(F_A^+)$ is trace-free in both arguments. Using the Weitzenb\"ock formula (5), we find $$(\hskip 4pt{\not}{D}_A\Psi,\Psi)=(\nabla_{\hat A}^*\nabla_{\hat A}\Psi,\Psi)+(\Gamma(F_A^+),(\Psi\bar\Psi)_0)+\frac{s}{4}|\Psi|^2\ ,\eqno{(6)}$$ which shows that $$(\hskip 4pt{\not}{D}_A^2\Psi,\Psi)+\frac{1}{2}|\Gamma(F_{A}^+)-(\Psi\bar\Psi)_0|^2= (\nabla_{\hat A}^*\nabla_{\hat A}\Psi,\Psi)+ \frac{1}{2}|F_{A}^+|^2+\frac{1}{2}|(\Psi\bar\Psi)_0|^2+\frac{s}{4}|\Psi|^2 $$ The identity (5) follows by integration over $X$. \section{Moduli spaces of quaternionic monopoles} Let $E$ be $U(2)$-bundle with $w_2(P)\equiv \overline{c_1(E)} $ (mod 2), and let $c$ be a fixed $S^1$-connection in $\det E^{\vee}$. We identify ${\cal A}(P(E))$ with the space ${\cal A}_c(E^{\vee})$ of $U(2)$-connections in $E^{\vee}$ which induce the fixed connection in $\det E^{\vee}$, and we set: $${\cal A}:={\cal A}_c(E^{\vee})\times A^0({\cal H}^+)$$ The natural gauge group is the group ${\cal G}$ consisting of unitary automorphisms in $E^{\vee}$ which induce the identity in $\det E^{\vee}$. ${\cal G}$ acts on ${\cal A}$ from the right in a natural way. Let ${\cal A}^*\subset {\cal A}$ be the open subset of ${\cal A}$ consisting of pairs $(C,\Psi)$ whose stabilizer ${\cal G}_{(C,\Psi)}$ is contained in the center ${\Bbb Z}/2=\{\pm{\rm id}_E\}$ of the gauge group. \begin{re} A pair $(C,\Psi)$ does not belong to ${\cal A}^*$ iff $\Psi=0$ and $C$ is a reducible connection. \end{re} Indeed, the isotropy group of ${\cal G}$ acting only on the first factor ${\cal A}_c(E^{\vee})$ is the centralizer of the holonomy of $C$ in $SU(2)$. The latter is $S^1$ if $C$ is reducible, and ${\Bbb Z}/2$ in the irreducible case. \hfill\vrule height6pt width6pt depth0pt \bigskip A pair belonging to ${\cal A}^*$ will be called irreducible. Note that the stabilizer of \underbar{any} pair with vanishing second componenent $\Psi$ contains ${\Bbb Z}/2$. {}From now on we also assume that ${\cal A}$ and ${\cal G}$ are completed with respect to suitable Sobolev norms $L^2_k$, such that ${\cal G}$ becomes a Hilbert Lie group acting smoothly on ${\cal A}$. Let ${\cal B}:=\qmod{{\cal A}}{{\cal G}}$ be the quotient, ${\cal B}^*:=\qmod{{\cal A}^*}{{\cal G}}$, and denote the orbit-map $[\ ]:{\cal A}\longrightarrow{\cal B}$ by $\pi$. An element in ${\cal A}^{*}$ will be called {\sl strongly irreducible} if its stabilizer is trivial. Let ${\cal A}^{**}\subset{\cal A}^*$ be the subset of strongly irreducible pairs, and put ${\cal B}^{**}:=\qmod{{\cal A}^{**}}{{\cal G}}$. \begin{pr} ${\cal B}$ is a Hausdorff space. ${\cal B}^{**}\subset{\cal B}$ is open and has the structure of a differentiable Hilbert manifold. The map ${\cal A}^{**}\longrightarrow{\cal B}^{**}$ is a differentiable principal ${\cal G}$-bundle. \end{pr} {\bf Proof: } Standard, cf. [DK], [FU]. \\ Fix a point $p=(C,\Psi)\in{\cal A}$. The differential of the map ${\cal G}\longrightarrow{\cal A}$ given by the action of ${\cal G}$ on $p$ is the map $$ \begin{array}{cccc}D^0_{p}&:A^0(su(E^{\vee}))&\longrightarrow& A^1(su(E^{\vee}))\oplus A^0(\Sigma^+\otimes E^{\vee})\\ &f&\longmapsto&(D_{C}(f),-f\Psi) \end{array}$$ Setting $$N_{p}(\varepsilon):=\{\beta\in A^1(su(E^{\vee}))\oplus A^0(\Sigma^+\otimes E^{\vee})|\ {D^0_{p}}^*\beta=0,\ ||\beta||<\varepsilon\}\ ,$$ for $\varepsilon>0$ sufficiently small, one obtains local slices for the action of ${\cal G}$ on ${\cal A}^{**}$ and charts $\pi|_{N_p(\varepsilon)}:N_p(\varepsilon)\longrightarrow {\cal B}^{**}$ for ${\cal B}^{**}$. \hfill\vrule height6pt width6pt depth0pt \bigskip Note that the curvature $F_A$ of a connection in $P(E)$ equals the trace-free part $F_C^0$ of the curvature of the corresponding connection $C\in{\cal A}_c(E^{\vee})$. Using the identification ${\cal A}(P(E))={\cal A}_c(E^{\vee})$, we can rewrite the quaternionic monopole equations in terms of pairs $(C,\Psi)\in{\cal A}$. Let ${\cal A}^{SW^h}\subset {\cal A}$ be the space of solutions. \begin{dt} Fix a $Spin^h$-structure in $P$. The moduli space of quaternionic monopoles is the quotient ${\cal M}:=\qmod{{\cal A}^{SW^h}}{{\cal G}}$. We denote by ${\cal M}^{**}:=\qmod{({\cal A}^{SW^h}\cap{\cal A}^{**})}{{\cal G}}$, ${\cal M}^{*}:=\qmod{({\cal A}^{SW^h}\cap{\cal A}^{*})}{{\cal G}}$ the subspaces of (strongly) irreducible monopoles. \end{dt} The tangent space to ${\cal A}^{SW^h}$ at $p=(C,\Psi)\in {\cal A}$ is the kernel of the operator $$ \begin{array}{c}D^1_{p} :A^1(su(E^{\vee}))\oplus A^0(\Sigma^+\otimes E^{\vee}) \longrightarrow A^0(su(\Sigma^+)\otimes su(E^{\vee}))\oplus A^0(\Sigma^-\otimes E^{\vee})\end{array}$$ defined by $$D^1_{p}((\alpha,\psi))= \left(\Gamma(D_C^+(\alpha))-[(\psi\bar\Psi)_0+ (\Psi\bar\psi)_0],\hskip 4pt{\not}{D}_{C,c}\psi +\gamma(\alpha)\Psi\right) \ , $$ where we consider $\gamma(\alpha)$ as map $\gamma(\alpha):\Sigma^+\longrightarrow \Sigma^-\otimes su(E^{\vee})$. Clearly $D^1_p\circ D_p^0=0$, since the monopole equations are gauge invariant. Using the isomorphism $\Gamma^{-1}: A^0(su(\Sigma^+))\longrightarrow A^2_+$, we can consider $D^1_p$ as an operator $D^1_p:A^1(su(E^{\vee}))\oplus A^0(\Sigma^+\otimes E^{\vee}) \longrightarrow A^2_+(su(E^{\vee}))\oplus A^0(\Sigma^-\otimes E^{\vee})$. Let $\sigma(X)$ and $e(X)$ be the signature and the topological Euler characteristic of the oriented manifold $X$. \begin{pr} For a solution $p=(C,\Psi)\in{\cal A}^{SW^h}$, the complex $$0\rightarrow A^0 su(E^{\vee}) \textmap{{D}^0_p} A^1 su(E^{\vee}) \oplus A^0 {\cal H}^+ \textmap{{D}^1_p}A^2_+ su(E^{\vee}) \oplus A^0 {\cal H}^- \rightarrow 0\eqno{({\cal C}_p)}$$ is elliptic and its index is $$\frac{3}{2}(4c_2(E^{\vee})-c_1(E^{\vee})^2)-\frac{1}{2}(3e(X)+4\sigma(X)) \ .\eqno{(7)}$$ \end{pr} {\bf Proof: } The complex ${\cal C}_p$ has the same symbol sequence as $$0\rightarrow A^0 su(E^{\vee}) \stackrel{(D_C,0)}{\longrightarrow} A^1 su(E^{\vee}) \oplus A^0 {\cal H}^+ \stackrel{(D_C^+,\hskip 4pt{\not}{D}_{C,c})}{\longrightarrow}A^2_+ su(E^{\vee}) \oplus A^0 {\cal H}^- \rightarrow 0 $$ which is an elliptic complex with index $$2(4c_2(E^{\vee})-c_1(E^{\vee})^2)-\frac{3}{2}(\sigma(X)+e(X))+index \hskip 4pt{\not}{D}_{C,c}\ .$$ The latter term is $$index\hskip 4pt{\not}{D}_{C,c}=[ch(E^{\vee})e^{\frac{1}{2} c_1(E^{\vee})}\hat A(X)]_4=-2c_2(E^{\vee})+\frac{1}{2}c_1(E^{\vee})^2-\frac{1}{2}\sigma(X)\ .$$ \hfill\vrule height6pt width6pt depth0pt \bigskip \begin{re} The integer in (7) is always an even number if $X$ admits almost complex structures. \end{re} Our next step is to endow the spaces ${\cal M}^{**}$ (${\cal M}^{*}$) with the structure of a real analytic space (orbifold). In the first case (compare with [FU], [DK], [OT1], [LT]), we have an analytic map $\sigma:{\cal A}\longrightarrow A^2_+(su(E^{\vee}))\oplus A^0( {\cal H}^-)$ defined by $$\sigma(C,\Psi)= \left((F_C^0)^+-\Gamma^{-1}(\Psi\bar\Psi)_0,\hskip 4pt{\not}{D}_{C,c}\Psi\right)$$ which gives rise to a section $\tilde\sigma$ in the bundle ${\cal A}^{**}\times_{{\cal G}} \left(A^2_+(su(E^{\vee}))\oplus A^0( {\cal H}^-) \right)$. We endow ${\cal M}^{**}$ with a real analytic structure by identifying it with the vanishing locus $Z(\tilde\sigma)$ of $\tilde\sigma$, regarded as a subspace of the Hilbert manifold ${\cal B}^{**}$ (in Douady's sense) ([M], [LT]). Now fix a point $p=(C,\Psi)\in{\cal A}^*$. We put $$S_p(\varepsilon):=\{p+\beta|\ \beta\in A^1su(E^{\vee}) \oplus A^0 {\cal H}^+ ,\ D^0_p{D^0_p}^* \beta +{D^1}^*_p\sigma(p+\beta)=0,\ ||\beta||<\varepsilon\}\ .$$ \begin{cl} For sufficiently small $\varepsilon>0$, $S_p(\varepsilon)$ is a finite dimensional submanifold of ${\cal A}$ which is contained in the slice $N_p(\varepsilon)$ and whose tangent space at $p$ is the first harmonic space ${\Bbb H}^1_p$ of the deformation complex ${\cal C}_p$. \end{cl} To prove this claim, we consider the map $$s_p:A^1(su(E^{\vee}))\oplus A^0({\cal H}^+)\longrightarrow {\rm im}( D^0_p)\oplus{\rm im} (D^1_p)^*$$ given by the left hand terms in the equations defining $S_p(\varepsilon)$. The derivative of $s_p$ at 0 is the first Laplacian $$\Delta^1_p:A^1(su(E^{\vee}))\oplus A^0({\cal H}^+)\longrightarrow {\rm im}( D^0_p)\oplus{\rm im} (D^1_p)^*$$ associated with the elliptic complex ${\cal C}_p$, hence $s_p$ is a submersion in 0. This proves the claim. \hfill\vrule height6pt width6pt depth0pt \bigskip The intersection ${\cal A}^{SW^h}\cap N_p(\varepsilon)=Z(\sigma)\cap N_p(\varepsilon)$ of the space of solutions with the standard slice through $p$ is contained in $S_p(\varepsilon)$ and can be identified with the finite dimensional model $$Z(\sigma)\cap N_p(\varepsilon)=Z(\sigma|_{S_p(\varepsilon)})\ .$$ If $p\in {\cal A}^{**}$ is strongly irreducible, then the map $$\pi|_{Z(\sigma|_{S_p(\varepsilon)})}:Z(\sigma|_{S_p(\varepsilon)})\longrightarrow {\cal M}^{**}$$ is a local parametrization of ${\cal M}^*$ at $p$, hence $Z(\sigma|_{S_p(\varepsilon)})$ is a local model for the moduli space around $p$. If $p\in{\cal A}^*\setminus{\cal A}^{**}$ is irreducible but not strongly irreducible, then necessarily $\Psi=0$, and the isotropy group ${\cal G}_p={\Bbb Z}/2$ acts on $S_p(\varepsilon)$. Since $\sigma$ is ${\Bbb Z}/2$-equivariant, we obtain an induced action on $Z(\sigma|_{S_p(\varepsilon)})$. In this case $\pi|_{Z(\sigma|_{S_p(\varepsilon)})}$ induces a homeomorphism of the quotient $\qmod{Z(\sigma|_{S_p(\varepsilon)})}{{\Bbb Z}/2}$ with an open neighbourhood of $p$ in ${\cal M}^*$, and ${\cal M}^*$ becomes an orbifold at $p$, if we use the map $$\pi|_{Z(\sigma|_{S_p})}:Z(\sigma|_{S_p(\varepsilon)})\longrightarrow {\cal M}^*$$ as an orbifold chart. \begin{re} Using a real analytic isomorphism which identifies the germ of $S_p(\varepsilon)$ at $p$ with the germ of ${\Bbb H}_p^1=T_p(S_p(\varepsilon))$ at 0, we obtain a local model of Kuranishi-type for ${\cal M}^*$ at $p$. \end{re} \begin{re} The points in ${\cal D}^*:={\cal M}^*\setminus{\cal M}^{**}$ have the form $[(C,0)]$, where $C$ is projectively anti-self-dual, i.e $(F_C^0)^+=0$. There is a natural finite map $${\cal D}^*\longrightarrow{\cal M}(P(E^{\vee}))$$ into the Donaldson moduli space of $PU(2)$-instantons in $P(E^{\vee})$, which maps ${\cal D}^*$ isomorphically onto ${\cal M}(P(E^{\vee})^*$ if $H^1(X,{\Bbb Z}/2)=0$. In general ${\cal D}^*$ and ${\cal M}(P(E^{\vee})^*$ cannot be identified. The difference comes from the fact that our gauge group is $SU(E^{\vee})$, whereas the $PU(2)$-instantons are classified modulo $PU(E^{\vee})$. \end{re} For simplicity we shall however refer to ${\cal D}^*$ as Donaldson instanton moduli space. Concluding, we get \begin{pr} ${\cal M}^{**}$ is a real analytic space. ${\cal M}^*$ is a real analytic orbi\-fold, and the points in ${\cal M}^*\setminus{\cal M}^{**}$ have neighbourhoods modeled on ${\Bbb Z}/2$-quotients. ${\cal M}^*\setminus{\cal M}^{**}$ can be identified as a set with the Donaldson moduli space ${\cal D}^*$ of irreducible projectively anti-self-dual connections in $E^{\vee}$ with fixed determinant $c$. \end{pr} The local structure of the moduli space ${\cal M}$ in reducible points, which correspond to pairs formed by a reducible instanton and a trivial spinor, can also be described using the method above (compare with [DK]). Let ${\cal M}^{SW}\subset{\cal M}$ be the subspace of ${\cal M}$ consisting of all orbits of the form $(C,\Psi)\cdot SU(E^{\vee})$, where $C$ is a reducible connection and $\Psi$ belongs to one of the summands. Let $L:=\det\Sigma^{\pm}=\det E$. It is easy to see that $${\cal M}^{SW}\simeq\mathop{\bigcup}\limits_{\matrix{^{S {\rm \ summand}}\cr ^{{\rm of} \ E^{\vee}}\cr}}{\cal M}^{SW}_{L \otimes S^{\otimes 2}} \ ,$$ where ${\cal M}^{SW}_M$ denotes the rank-1 Seiberg-Witten moduli space associated to a $Spin^c(4)$-structure of determinant $M$. The fact that the moduli spaces of quaternionic monopoles contain Donaldson moduli spaces as well of Seiberg-Witten moduli spaces suggests that they could provide a method for comparing the invariants given by the two theories. \section{Quaternionic monopoles on K\"ahler surfaces} Let $(X,g)$ be a K\"ahler surface with canonical $Spin^c(4)$-structure; in this case $\Sigma^+=\Lambda^{00}\oplus\Lambda^{02}$, and $\Sigma^-=\Lambda^{01}$. A $Spin^h(4)$-structure in the frame bundle is given by a unitary vector bundle $E$ together with an isomorphism $\det E\simeq \Lambda^{02}$. A $Spin^h(4)$-connection $\hat A$ corresponds to a $PU(2)$-connection $A$ in the associated bundle $P(E)$, or alternatively, to a unitary connection $C$ in $E^{\vee}$ which induces a fixed $S^1$-connection $c$ in $\Lambda^{20}$. Recall that the curvature $F_A$ of $A$ equals the trace-free component $F_C^0$ of $F_C$. If we choose $c$ to be the Chern connection in the canonical bundle $\Lambda^{20}=K_X$, then the $Spin^h(4)$-connection in ${\cal H}=\Sigma \otimes E^{\vee}$ is simply the tensor product of the canonical connection in $ \Sigma=\Sigma^{+}\oplus\Sigma^-$ and the connection $C$. A positive quaternionic spinor $\Psi\in A^0 ( {\cal H}^+) $ can be written as $\Psi=\varphi+\alpha$, with $\varphi\in A^0( E^{\vee}) $, and $\alpha\in A^{02}( E^{\vee})$. \begin{pr} Let $C$ be a unitary connection in $E^{\vee}$ inducing the Chern connection $c$ in $\det E^{\vee}=K_X$. A pair $(C,\varphi+\alpha)$ solves the quaternionic monopole equations if and only if $F_C$ is of type $(1,1)$ and one of the following conditions holds $$\matrix{1.\ \alpha=0\ ,\ \bar\partial_C\varphi=0\ and\ i\Lambda_g F_{C}^0 +\frac{1}{2} (\varphi\otimes\bar\varphi)_0\ =\ 0\ , \cr 2. \ \varphi=0\ ,\ \partial_C\alpha=0\ and\ i\Lambda_g F_{C}^0 -\frac{1}{2}* (\alpha\otimes\bar\alpha)_0=0\ .\cr}\eqno{(8)}$$ \end{pr} {\bf Proof: } Using the notation in the proof of the Weitzenb\"ock formula, we have $F_{C,c}=\frac{1}{2}({\rm Tr} F_C+F_c){\rm id}_{E^{\vee}}+ F_A=F_A=F_C^0\in A^2(su (E^{\vee}))$. By Proposition 2.6 of [OT1] the quaternionic Seiberg-Witten equations become $$\left\{ \begin{array}{lll} F_{A}^{20}&=&-\frac{1}{2}(\varphi\otimes\bar\alpha)_0\\ F_{A}^{02}&=&\frac{1}{2}(\alpha\otimes\bar\varphi)_0\\ i\Lambda_g F_{A}&=&-\frac{1}{2}\left[(\varphi\otimes\bar\varphi)_0- *(\alpha\otimes\bar\alpha)_0\right]\\ \bar\partial_C\varphi&=&i\Lambda_g\partial_C\alpha\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .\end{array}\right.$$ Note that the right-hand side of formula (5) is invariant under Witten's transformation $(C,\varphi+\alpha)\longmapsto (C, \varphi-\alpha)$. Therefore, every solution satisfies $F_A^{20}= F_A^{02}=0$, and $(\varphi\otimes\bar\alpha)_0=(\alpha\otimes\bar\varphi)_0=0$. Elementary computations show that this can happen only if $\varphi=0$ or $\alpha=0$. On the other hand, since the Chern connection in $K_X$ is integrable, we also get $F_C^{20}=F_C^{02}=0$. \hfill\vrule height6pt width6pt depth0pt \bigskip \begin{re} The second case in this proposition reduces to the first: in fact, if $\varphi=0$ and $\alpha\in A^{02}(E^{\vee})$ satisfies $i\Lambda_g\partial\alpha=0$, we set $\psi:=\bar\alpha\in A^{20}(\bar E^{\vee})=A^0(\Lambda^{20}\otimes E)=A^0(E^{\vee})$, and we obtain $\bar\partial_C\psi=\overline{\partial_C\bar\psi}=\overline{\partial_C\alpha }=0$. Here we used the fact that $\Lambda_g:\Lambda^{12}\longrightarrow \Lambda^{01}$ is an isomorphism, the adjoint of the Lefschetz isomorphism $\cdot\wedge\omega_g$ [LT]. A simple calculation in coordinates gives $-*(\alpha\otimes\bar\alpha)_0=(\bar\alpha\otimes\overline{\bar\alpha})_0= (\psi\otimes\bar\psi)_0$. \end{re} \section{Stability} Let $(X,g)$ be a compact K\"ahler manifold of arbitrary dimension, $E$ a differentiable vector bundle, and let ${\cal L}$ be a fixed holomorphic line bundle, whose underlying differentiable line bundle is $L:=\det E$. \begin{dt} A holomorphic pair of type $(E,{\cal L})$ is a pair $({\cal E},\varphi)$ consisting of a holomorphic bundle ${\cal E}$ and a section $\varphi\in H^0(X,{\cal E})$ such that the underlying differentiable bundle of ${\cal E}$ is $E$ and $\det{\cal E}={\cal L}$. \end{dt} Note that the determinant of the holomorphic bundle ${\cal E}$ is fixed, not only its isomorphism type. Two pairs $({\cal E}_i,\varphi_i)$, $i=1, 2$ of the same type are isomorphic if there exists an isomorphism $f:{\cal E}_1\longrightarrow{\cal E}_2$ with $f^*(\varphi_2)=\varphi_1$ and $\det f={\rm id}_{\cal L}$. In other words, $({\cal E}_i,\varphi_i)$ are isomorphic iff there exists a complex gauge transformation $f\in SL(E)$ with $f^*(\varphi_2)=\varphi_1$ such that $f$ is holomorphic as a map $f:{\cal E}_1\longrightarrow{\cal E}_2$. % \begin{dt} A holomorphic pair $({\cal E},\varphi)$ is simple if any automorphism of it is of the form $f=\varepsilon{\rm id}_{\cal E}$, where $\varepsilon^ {{\rm rk}{\cal E}}=1$. A pair $({\cal E},\varphi)$ is strongly simple if its only automorphism is ${\rm id}_{\cal E}$. \end{dt} Note that a simple pair $({\cal E},\varphi)$ with $\varphi\ne 0$ is stongly simple, whereas a pair $({\cal E},0)$ is simple iff ${\cal E}$ is a simple bundle. Note also that $({\cal E},\varphi)$ is simple iff any trace-free holomorphic endomorphism $f$ of ${\cal E}$ with $f(\varphi)=0$ vanishes. For a nontrivial torsion free sheaf ${\cal F}$ on $X$, we denote by $\mu_g({\cal F})$ its slope with respect to the K\"ahler metric $g$. Given a holomorphic bundle ${\cal E}$ over $X$ and a holomorphic section $\varphi\in H^0(X,{\cal E})$, we let ${\cal S}({\cal E})$ be the set of reflexive subsheaves ${\cal F}\subset{\cal E}$ with $0<{\rm rk}({\cal F})<{\rm rk}({\cal E})$, and we define $${\cal S}_\varphi({\cal E}):=\{{\cal F}\in{\cal S}({\cal E})|\ \varphi\in H^0(X,{\cal F})\} \ .$$ Recall the following stability concepts [B2]: \begin{dt}\hfill{\break} 1. ${\cal E}$ is $\varphi$-stable if $$\max\left(\mu_g({\cal E}),\sup\limits_{{\cal F}'\in{\cal S}({\cal E})} \mu_g({\cal F}')\right)< \inf\limits_{{\cal F}\in {\cal S}_\varphi({\cal E})}\mu_g(\qmod{{\cal E}}{{\cal F} })\ .$$ 2. Let $\lambda\in{\Bbb R}$ be a real parameter. The pair $({\cal E},\varphi)$ is $\lambda$-stable iff $$\max\left(\mu_g({\cal E}),\sup\limits_{{\cal F}'\in{\cal S}({\cal E})} \mu_g({\cal F}')\right)<\lambda< \inf\limits_{{\cal F}\in {\cal S}_\varphi({\cal E})}\mu_g(\qmod{{\cal E}}{{\cal F}})\ .$$ 3. $({\cal E},\varphi)$ is called $\lambda$-polystable if ${\cal E}$ splits holomorphically as ${\cal E}={\cal E}'\oplus{\cal E}''$, such that $\varphi\in H^0(X,{\cal E}')$, $({\cal E}',\varphi)$ is a $\lambda$-stable pair, and ${\cal E}''$ is a polystable vector bundle of slope $\lambda$. \end{dt} {}From now on we restrict ourselves to the case ${\rm rk}({\cal E})=2$. \begin{dt} \hfill{\break} 1. A holomorphic pair $({\cal E},\varphi)$ of type $({ E},{\cal L})$ is called stable if one of the following conditions is satisfied:\\ i) ${\cal E}$ is $\varphi$-stable.\\ ii) $\varphi\ne 0$ and ${\cal E}$ splits in direct sum of line bundle ${\cal E}={\cal E}'\oplus{\cal E}''$, such that $\varphi\in H^0({\cal E}')$ and the pair $({\cal E}',\varphi)$ is $\mu_g({ E})$-stable.\\ 2. A holomorphic pair $({\cal E},\varphi)$ of type $({ E},{\cal L})$ is called polystable if it is stable, or $\varphi=0$ and ${\cal E}$ is a polystable bundle. \end{dt} Note that there is \underbar{no} parameter $\lambda$ in the stability concept for holomorphic pairs of a fixed type. The conditions depend only on the metric $g$ and on the slope $\mu_g(E)$ of the underlying differentiable bundle $E$. \begin{lm} Let $({\cal E},\varphi)$ be a holomorphic pair of type $(E,{\cal L})$ with $\varphi\ne 0$. There exists a uniquely determined effective divisor $D=D_\varphi$ and a commutative diagram $$\begin{array}{cccc} 0\longrightarrow&{\cal O}_X(D)&\textmap{\hat\varphi}&{\cal E}\longrightarrow {\cal L}(-D)\otimes J_Z\longrightarrow 0\ ,\\ &{\scriptstyle D\cdot}\uparrow{\phantom i}&{\scriptstyle\varphi}\nearrow&\\ &\ {\cal O}_X&\end{array}\eqno{(9)}$$ with a local complete intersection $Z\subset X$ of codimension 2. The pair $({\cal E},\varphi)$ is stable if and only if $\mu_g({\cal O}_X(D))<\mu_g(E)$. \end{lm} {\bf Proof: } $D=D_\varphi$ is the divisorial component of the zero locus $Z(\varphi)$ of ${\cal E}$ which is defined by the ideal ${\rm im}(\varphi^{\vee}:{\cal E}^{\vee}\longrightarrow{\cal O}_X)$, and $\hat\varphi$ is the induced map. The set ${\cal S}_\varphi({\cal E})$ consists precisely of the line bundles ${\cal F}\subset{\cal O}_X({D})$, so that $$\inf\limits_{{\cal F}\in {\cal S}_\varphi({\cal E})}\mu_g(\qmod{{\cal E}}{{\cal F}})=2\mu_g(E)-\mu_g({\cal O}_X(D))\ .$$ Suppose $({\cal E},\varphi)$ is stable. If ${\cal E}$ is $\varphi$-stable, we have $\mu_g(E)<2\mu_g(E)-\mu_g({\cal O}_X(D))$, which gives the required inequality. If ${\cal E}$ is not $\varphi$-stable, then $Z=\emptyset$, the extension (9) splits, and the pair $({\cal O}_X(D),\varphi)$ is $\mu_g(E)$-stable, i.e. $\mu_g({\cal O}_X(D))<\mu_g(E)$. Conversely, suppose $\mu_g({\cal O}_X(D))<\mu_g(E)$, and assume first that the extension (9) does not split. In this case ${\cal E}$ is $\varphi$-stable: in fact, if ${\cal F}'\subset {\cal E}$ is an arbitrary line bundle, either ${\cal F}'\subset{\cal O}_X(D)$, or the induced map ${\cal F}'\subset{\cal E}\longrightarrow {\cal J}_Z\otimes{\cal L}(-D)$ is non-trivial. But then ${\cal F}'\simeq {\cal L}\otimes{\cal O}_X(-D-\Delta)$ for an effective divisor $\Delta$ containing $Z$, and we find $$\mu_g({\cal F}')=2\mu_g(E)-\mu_g(D)-\mu_g(\Delta)\leq 2\mu_g(E)-\mu_g({\cal O}_X(D)) \ . $$ Furthermore, strict inequality holds, unless $Z=\emptyset$ and the extension (9) splits, which it does not by assumption. In the case of a split extension, we only have to notice that a pair $({\cal E}',\varphi)$ is $\lambda$-stable for any parameter $\lambda>\mu_g({\cal E}') $ [B1]. \hfill\vrule height6pt width6pt depth0pt \bigskip \begin{re} Consider a pair $({\cal E},\varphi)$ of type $(E,{\cal L})$ with $\varphi\ne 0$ and associated extension (9). ${\cal E}$ is $\varphi$-stable iff $\mu_g({\cal O}_X(D))<\mu_g(E)$, and the extension does not split. \end{re} Indeed, if the extension splits, then ${\cal E}$ is not $\varphi$-stable, since $$\mu_g({\cal L}(-D))=\inf\limits_{{\cal F}\in {\cal S}_\varphi({\cal E})}\mu_g(\qmod{{\cal E}}{{\cal F}}) \ .$$ \section{The projective vortex equation} Let $E$ be a differentiable vector bundle over a compact K\"ahler manifold $(X,g)$. We fix a holomorphic line bundle ${\cal L}$ and a Hermitian metric $l$ in ${\cal L}$. Let $({\cal E},\varphi)$ be a holomorphic pair of type $(E,{\cal L})$. \begin{dt} A Hermitian metric in ${\cal E}$ with $\det h=l$ is a solution of the projective vortex equation iff the trace free part $F^0_h$ of the curvature $F_h$ satisfies the equation $$i\Lambda_g F_h^0 +\frac{1}{2}(\varphi\bar\varphi^h)_0=0\ .\eqno{(V)}$$ \end{dt} \begin{th} Let $({\cal E},\varphi)$ be a holomorphic pair of type $(E,{\cal L})$ with ${\rm rk}({\cal E})=2$. Fix a Hermitian metric $l$ in ${\cal L}$. The pair $({\cal E},\varphi)$ is polystable iff ${\cal E}$ admits a Hermitian metric $h$ with $\det h=l$ which is a solution of the projective vortex equation. If $({\cal E},\varphi)$ is stable, then the metric $h$ is unique. \end{th} {\bf Proof: } Suppose first that $h$ is a solution of the projective vortex equation $(V)$. Then we have $$i\Lambda F_h+\frac{1}{2}(\varphi\bar\varphi^h)=\frac{1}{2}(i\Lambda{\rm Tr} F_h+\frac{1}{2}|\varphi|^2){\rm id}_E \ , $$ i.e. $h$ satisfies the weak vortex equation $(V_t)$ associated to the real function $t:=\frac{1}{2}(2i\Lambda{\rm Tr} F_h+|\varphi|^2)$. Therefore, by [OT1], the pair $({\cal E},\varphi)$ is $\lambda$-polystable for the parameter $\lambda=\frac{(n-1)!}{4\pi}\int\limits_X t vol_g=\mu_g({\cal E})+\frac{(n-1)!}{8\pi}||\varphi||^2$. Let $A$ be the Chern connection of $h$, and denote by ${\cal E}'$ the minimal $A$-invariant subbundle which contains $\varphi$. If ${\cal E}'={\cal E}$, then ${\cal E}$ is $\varphi$-stable and the pair $({\cal E},\varphi)$ is stable. If ${\cal E}'=0$, hence $\varphi=0$, then $h$ is a weak Hermitian-Einstein metric, ${\cal E}$ is a polystable bundle, and the pair $({\cal E},\varphi)$ is polystable by definition. In the remaining case ${\cal E}'$ is a line bundle and $\varphi\ne 0$. Let ${\cal E}'':={\cal E}'^{\bot}$ be the orthogonal complement of ${\cal E}'$, and let $h'$ and $h''$ be the induced metrics in ${\cal E}'$ and ${\cal E}''$. We put $s:=i\Lambda_g{\rm Tr} F_h$. Then, since $h=h'\oplus h''$, the projective vortex equation can be rewritten as: $$ \left\{\begin{array}{ll} i\Lambda F_{h'}+\frac{1}{2}(\varphi\bar\varphi^{h'})=&\frac{1}{2}(s+ \frac{1}{2}|\varphi|_{h'}^2){\rm id}_{{\cal E}'}\\ i\Lambda F_{h''}=&\frac{1}{2}(s+\frac{1}{2}|\varphi|_{h'}^2) {\rm id}_{{\cal E}''} \ . \end{array}\right. $$ The first of these equations is equivalent to $$i\Lambda F_{h'}+\frac{1}{4}(\varphi\bar\varphi^{h'})=\frac{s}{2}{\rm id}_{{\cal E}'}\ ,$$ which implies that $({\cal E}',\frac{\varphi}{\sqrt 2})$ is $\mu_g({\cal E})$-stable by [OT1]. Conversely, suppose first that $({\cal E},\varphi)$ is stable. We have to consider two cases:\\ \underbar{Case 1}: ${\cal E}$ is $\varphi$ stable. Using Bradlow's existence theorem, we obtain Hermitian metrics in ${\cal E}$ satisfying the usual vortex equations associated with suitable chosen $\lambda$ , and, of course these metrics all satisfy the equation $(V)$. The problem is, however, to find a solution with an a priori given determinant $l$. In order to achieve this stronger result, Bradlow's proof has to modified slightly at some points: One starts by fixing a background metric $k$ such that $\det k=l$. Denote by $S_0(k)$ the space of {trace-free} $k$-Hermitian endomorphisms of $E$, and let ${\cal M} et(l)$ be the space of Hermitian metrics in $E$ with $\det h=l$. On $$ {\cal M} et(l)^p_2:=\{ke^s |\ s\in L^p_2(S_0(k))\}$$ we define the functional $M_\varphi:{\cal M} et(l)^p_2\longrightarrow{\Bbb R}$ by $$M_\varphi(h):=M_D(k,h)+||\varphi||^2_h-||\varphi||^2_k \ .$$ Here $M_D$ is the Donaldson functional, which is known to satisfy the identity $ \frac{d}{dt}M_D(k,h(t))=2\int\limits_X{\rm Tr}[ h^{-1}(t)\dot h(t) i\Lambda_g F_h ]\ $ for any smooth path of metrics $h(t)$ [Do], [Ko]. Since $h^{-1}(t)\dot h(t)$ is trace-free for a path in ${\cal M} et(l)$, we obtain $$\frac{d}{dt}|M_D(k,h(t))= 2\int\limits_X{\rm Tr}[ h^{-1}\dot h(t) i\Lambda_g F_h^0]\ .$$ Similarly, for a path of the form $h(t)=he^{ts}$, with $s \in S_0(h)$, we get $$\frac{d}{dt}_{|_{t=0}}||\varphi||^2_{h_t}=\frac{d}{dt}_{|_{t=0}}\langle e^{ts}\varphi,\varphi\rangle_h=\left\langle \frac{d}{dt}_{|_{t=0}}e^{ts}\varphi,\varphi\right\rangle_h= \left\langle s ,\varphi\bar\varphi^h\right\rangle_h=\int\limits_X{\rm Tr}[s (\varphi\bar\varphi^h)_0] \ .$$ This means that, putting $m_\varphi(h):=i\Lambda F_h^0+\frac{1}{2}(\varphi\bar\varphi^h)_0$, we always have $$\frac{d}{dt}_{|_{t=0}}M_\varphi(he^{ts})= 2\int\limits_X{\rm Tr}[ s\ m_\varphi(he^{ts})]\ ,\ $$ so that solving the projective vortex equation is equivalent to finding a critical point of the functional $M_\varphi$ (compare with Lemma 3.3 [B2]). \begin{cl} Suppose $({\cal E},\varphi)$ is simple. Choose $B>0$ and put $$ {\cal M} et(l)^p_2(B):=\{ h\in {\cal M} et(l)^p_2|\ ||m_\varphi(h)||_{L^p}\leq B\} \ .$$ Then any $h\in{\cal M} et(l)^p_2(B)$ which minimizes $M_\varphi$ on ${\cal M} et(l)^p_2(B)$ is a weak solution of the projective vortex equation. \end{cl} The essential point is the injectivity of the operator $s\longmapsto\Delta_h'(s)+\frac{1}{2}[(\varphi\bar\varphi) s]_0$ acting on $L^p_2 S_0(h)$. But from % $$\left\langle\Delta_h's+\frac{1}{2}[\varphi\bar\varphi^h) s]_0,\ s\right\rangle_h=||\bar\partial_h(s)||_{h}^2+||s\varphi||^2_{h} \ $$ we see that this operator is injective on trace-free endomorphisms if $({\cal E},\varphi)$ is simple. \hfill\vrule height6pt width6pt depth0pt \bigskip Now we can follow again Bradlow's proof : if ${\cal E}$ is $\varphi$-stable, then there exist positive constants $C_1$, $C_2$ such that for all $s\in L^p_2 S_0(k) $ with $ke^s\in{\cal M} et(l)^p_2(B)$ the following "main estimate" holds: $$\sup|s|\leq C_1 M_\varphi(ke^{s})+C_2\ .$$ This follows by applying Proposition 3.2 of [B2] to an arbitrary $\tau\in{\Bbb R}$ with $$\max\left(\mu_g({\cal E}),\sup\limits_{{\cal F}'\in{\cal S}({\cal E})}\mu_g({\cal F'})\right)<\frac{(n-1)!\tau Vol_g(X)}{4\pi}<\inf\limits_{{\cal F}\in{\cal S}_\varphi({\cal E})} \mu_g(\qmod{\cal E}{\cal F})\ , $$ since Bradlow's functional ${\cal M}_{\varphi,\tau}$ coincides on ${\cal M} et(l)$ with $M_{\varphi}$ . It remains to be shown that the existence of this main estimate implies the existence of a solution of the projective vortex equation. The main estimate implies that for any $c>0$, the set $$\{s\in\ L^p_2S_0(k) |\ ke^s\in {\cal M} et(l)^p_2(B) ,\ \ M_\varphi(ke^{s})<c\}$$ is bounded in $L^p_2$. Let $(s_i) $ be a sequence in $L^p_2 S_0(k) $ such that $ke^{s_i}\in {\cal M} et(l)^p_2(B)$ is a minimizing sequence for $M_\varphi$, and let $s$ be weak limit. Then $h:=ke^s$ is a weak solution of the projective vortex equation, which is smooth by elliptic regularity [B2]. Finally, we have to treat \underbar{Case 2}: $\varphi\ne 0$, ${\cal E}={\cal E}'\oplus{\cal E}''$, with $\varphi\in H^0({\cal E}')$, and the pair $({\cal E}',\varphi)$ is $\mu_g(E)$-stable.\\ We wish to find metrics $h'$ and $h''$ in ${\cal E}'$ and ${\cal E}''$, such that for $s:=i\Lambda F_l$ the following equations are satisfied: $$\left\{ \begin{array}{lll}h'\cdot h''&=&l\\ i\Lambda F_{h'}+\frac{1}{4}(\varphi\bar\varphi^{h'})&=&\frac{1}{2}s{\rm id}_{{\cal E}'} \\ i\Lambda F_{h''}&=&\frac{1}{2}(s+\frac{1}{2}|\varphi|_{h'}^2) {\rm id}_{{\cal E}''} \ . \end{array}\right.\ $$ Since the pair $({\cal E}',\frac{1}{\sqrt 2}\varphi)$ is $\mu_g(E)$-stable, there exists by [OT1] a unique Hermitian metric $h'$ in ${\cal E}'$ solving the second of these equations. With this solution the third equation can be rewritten as $$i\Lambda_g F_{h''}=s-i\Lambda_g F_{h'}\ . $$ Since $\int\limits_X(s-i\Lambda_g F_{h'})={\rm deg}({\cal E}'')$, we can solve this weak Hermitian-Einstein equation by a metric $h''$, which is unique up to constant rescaling. The pro\-duct $h'\cdot h''$ is a metric in ${\cal E}'\otimes{\cal E}''={\cal L}$ which has the same mean curvature $s$ as $l$, and therefore differs from $l$ by a constant factor. We can now simply rescale $h''$ by the inverse of this constant, and we get a pair of metrics satisfying the three equations above. \hfill\vrule height6pt width6pt depth0pt \bigskip \section{Moduli spaces of pairs} Let $E$ be a differentiable vector bundle of rank $r$ over a K\"ahler manifold $(X,g)$, and let ${\cal L}$ be a holomorphic line bundle whose underlying differentiable bundle is $L:=\det E$. \begin{pr} There exists a possibly non-Hausdorff complex analytic orbifold ${\cal M}^s(E,{\cal L} )$ parametrizing isomorphism classes of simple holomorphic pairs of type $(E,{\cal L})$. The open subset ${\cal M}^{ss}(E,{\cal L})\subset {\cal M}^{s}(E,{\cal L})$ consisting of strongly simple pairs is a complex analytic space, and the points in ${\cal M}^s(E,{\cal L})\setminus{\cal M}^{ss}(E,{\cal L})$ have neighbourhoods modeled on ${\Bbb Z}/r$-quotients. \end{pr} {\bf Proof: } Since we use the same method as in the proof of Proposition 3.9, we only sketch the main ideas. Let $\bar\lambda$ be the semiconnection defining the holomorphic structure of ${\cal L}$, and put $\bar{\cal A}:=\bar{\cal A}_{\bar\lambda}(E)\times A^0(E)$, where $\bar{\cal A}_{\bar\lambda}(E)$ denotes the affine space of semiconnections in $E$ inducing $\bar\lambda$ in $L=\det E$. The complex gauge group $SL(E)$ acts on $\bar{\cal A}$, and we write $\bar{\cal A}^s$ ($\bar{\cal A}^{ss})$ for the open subset of pairs whose stabilizer is contained in the center ${\Bbb Z}/r$ of $SL(E)$ ( is trivial). After suitable Sobolev completions, $\bar{\cal A}^{ss}$ becomes the total space of a holomorphic Hilbert principal $SL(E)$-bundle over $\bar{\cal B}^{ss}:=\qmod{\bar{\cal A}^{ss}}{SL(E)}$. A point $(\bar\delta,\varphi)\in \bar{\cal A}$ defines a pair of type $(E,{\cal L})$ iff it is integrable, i.e. iff it satisfies the following equations: $$\left\{\begin{array}{lll}F^{02}_{\bar\delta}&=&0\\ \bar\delta\varphi&=&0\ . \end{array}\right.\eqno{(10)}$$ Here $F^{02}_{\bar\delta}:=\bar\delta^2$ is a $(0,2)$-form with values in the bundle ${\rm End}_0(E)$ of trace-free endomorphisms. Moreover, isomorphy of pairs of type $(E,{\cal L})$ corresponds to equivalence modulo the action of the complex gauge group $SL(E)$. Let $\bar\sigma$ be the map ${\cal A}\longrightarrow A^{02}({\rm End}_0(E))\oplus A^{01}(E)$ sending a pair $(\bar\delta,\varphi)$ to the left hand sides of (10). We endow the sets ${\cal M}^{ss}_X(E,{\cal L})=\qmod{Z(\sigma)\cap\bar{\cal A}^{ss}}{SL(E)}$ ( ${\cal M}^{s }_X(E,{\cal L})=\qmod{Z(\sigma)\cap\bar{\cal A}^{s }}{SL(E)}$ ) with the structure of a complex analytic space (orbifold) as follows: ${\cal M}^{ss}_X(E,{\cal L})$ is defined to be the vanishing locus of the section $\tilde{\bar\sigma}$ in the Hilbert vector bundle $\bar{\cal A}^{ss}\times_{SL(E)}\left(A^{02}{\rm End}_0(E)\oplus A^{01}E\right)$ over $\bar{\cal B}^{ss}$ which is defined by $\bar\sigma$. To define the orbifold structure in ${\cal M}^{s }_X(E,{\cal L})$, we use local models derived from a deformation complex: Let $\bar p=(\bar\delta,\varphi)\in\bar{\cal A}$ an integrable point. The associated {deformation} {complex} $\bar{\cal D}_{\bar p}$ is the cone over the evaluation map $ev^*_\varphi$: $$ ev^q_\varphi:A^{0q}({\rm End}_0(E))\longrightarrow A^{0q}(E)\ ,$$ and has the form $$\matrix{0\rightarrow A^0 ({\rm End}_0(E))\textmap{\bar D^0_{\bar p}} A^{01}({\rm End}_0(E))\oplus A^0(E)\textmap{\bar D^1_{\bar p}}\cr\ \ \ \ \ \ \textmap{\bar D^1_{\bar p}} A^{02}({\rm End}_0(E))\oplus A^{01}(E)\textmap{\bar D^2_{\bar p}}\dots\cr}\eqno{(\bar{\cal D}_{\bar p})}$$ (compare with [OT1] \S 4). We define $$\bar S_{\bar p}(\varepsilon):=\{{\bar p}+\beta|\beta\in A^{01}{\rm End}_0(E)\oplus A^0E, \bar D^0_{\bar p}{\bar {D_{\bar p}^0}}^*(\beta)+{\bar {D_{\bar p}^1}}^*(\bar\sigma({\bar p}+\beta))=0, ||\beta||<\varepsilon\}.$$ The same arguments as in the proof of Proposition 3.9 show that for sufficiently small $\varepsilon>0$, $\bar S_{\bar p}(\varepsilon)$ is a submanifold of $\bar{\cal A}$, whose tangent space in $\bar p$ coincides with the first harmonic space $\bar{\Bbb H}^1_{\bar p}$ of the elliptic complex $(\bar{\cal D}_{\bar p})$. Therefore, we get a local finite dimensional model $Z(\bar\sigma|_{\bar S_{\bar p}(\varepsilon)}) $ for the intersection $Z(\bar\sigma)\cap \bar N_{\bar p}(\varepsilon)$ of the integrable locus with the standard slice $$ \bar N_{\bar p}(\varepsilon):=\{{\bar p} +\beta|\beta\in A^{01}({\rm End}_0(E))\oplus A^0(E), {\bar{ D_{\bar p}^0}}^*(\beta)=0,\ ||\beta||<\varepsilon\}$$ through $\bar p$. The restriction $$\bar\pi|_{Z(\bar\sigma|_{\bar S_{\bar p}(\varepsilon)})} : Z(\bar\sigma|_{\bar S_{\bar p}(\varepsilon)})\longrightarrow {\cal M}^{s}_X(E,{\cal L})$$ of the orbit map is \'etale if $[{\bar p}]\in {\cal M}^{ss}_X(E,{\cal L})$, and induces an open injection $$\qmod{ Z(\bar\sigma|_{\bar S_{\bar p}(\varepsilon)})}{ {\Bbb Z}/r }\longrightarrow {\cal M}^{s}_X(E,{\cal L})$$ if $[{\bar p}]\in {\cal M}^{s}_X(E,{\cal L})\setminus {\cal M}^{ss}_X(E,{\cal L})$. We define the orbifold structure of ${\cal M}^{s}_X(E,{\cal L})$ by taking the maps $\bar\pi|_{Z(\bar\sigma|_{\bar S_{\bar p}(\varepsilon)})}$ as orbifold-charts. \hfill\vrule height6pt width6pt depth0pt \bigskip Our next purpose is to compare the two types of moduli spaces constructed in this paper. Let $(X,g)$ be a K\"ahler surface endowed with the canonical $Spin^c$-structure ${\germ c} $. Let $E$ be a $U(2)$ bundle with $\det E=K_X$, and denote by ${\cal M}^*(E)$ the moduli space of irreducible quaternionic monopoles associated to the $Spin^h(4)$-structure defined by $({\germ c},E^{\vee})$ (Lemma 1.3) It follows from Proposition 4.1 that ${\cal M}^*(E)$ has a decomposition $${\cal M}^*(E)= {\cal M}^*(E)_{\alpha=0}\cup {\cal M}^*(E)_{\varphi=0}\ ,$$ where ${\cal M}^*(E)_{\alpha=0}$ ( ${\cal M}^*(E)_{\varphi=0}$ ) is the Zariski closed subspace of ${\cal M}^*(E)$ cut out by the equation $\alpha=0$ ( ${\varphi=0}$ ). The intersection $${\cal M}^*(E)_{\alpha=0}\cap {\cal M}^*(E)_{\varphi=0}$$ is the Donaldson instanton moduli space ${\cal D}^*$ of irreducible projectively anti-self-dual connections in $E$, inducing the Chern connection in ${\cal K}_X$. \begin{pr} The affine isomorphism ${\cal A}\ni(C,\varphi)\longmapsto(\bar\partial_C,\varphi)\in\bar{\cal A}$ induces a natural real analytic open embedding $$J:{\cal M}^*(E)_{\alpha=0}\hookrightarrow {\cal M}^s(E,{\cal K}_X)$$ whose image is the suborbifold of stable pairs of type $(E,{\cal K}_X)$. \end{pr} {\bf Proof: } Standard arguments (cf. [OT1]) show that $J$ is an \'etale map which induces natural identifications of the local models. A point $[(\bar\delta,\varphi)]$ lies in the image of $J$ iff the $SL(E)$-orbit of $(\bar\delta,\varphi)$ intersects the zero locus of the map $$m:\bar{\cal A}\longrightarrow A^0(su(E)), \ \ (\bar\partial_C,\varphi)\longmapsto \Lambda_g F_C^0-\frac{1}{2}(\varphi\bar\varphi)_0 \ .$$ Let $({\cal E},\varphi)$ be the holomorphic pair of type $(E,{\cal K}_X)$ defined by $(\bar\delta,\varphi)$. We can reformulate the condition above in the following way: $[({\cal E},\varphi)]$ lies in the image of $J$ iff there exists a Hermitian metric $h$ in ${\cal E}$ inducing the K\"ahler metric in ${\cal K}_X=\det{\cal E}$ which satisfies the projective vortex equation $(V)$. But we know already that this holds iff $({\cal E},\varphi)$ is stable. Moreover, the unicity of the solution of the projective vortex equation is equivalent to the injectivity of $J$. \hfill\vrule height6pt width6pt depth0pt \bigskip Using the remark after Proposition 4.1, we can now state the main result of this paper: \begin{th} Let $(X,g)$ be a K\"ahler surface with canonical bundle ${\cal K}_X$, and let $E$ be a $U(2)$-bundle with $\det E=K_X$. Consider the $Spin^h$-structure associated with the canonical $Spin^c(4)$-structure and the $U(2)$-bundle $E^{\vee}$. The corresponding moduli space of irreducible quaternionic monopoles is a union of two Zariski closed subspaces. Each of these subspaces is naturally isomorphic as a real analytic orbifold to the moduli space of stable pairs of type $(E,{\cal K}_X)$. There exists a real analytic involution on the quaternionic moduli space which interchanges these two closed subspaces. The fixed point set of this involution is the Donaldson moduli space of instantons in $E$ with fixed determinant, modulo the gauge group $SU(E)$. The closure of the complement of the Donaldson moduli space intersects the moduli space of instantons in the Brill-Noether locus. The union ${\cal M}^{SW}$ of all rank 1-Seiberg-Witten moduli spaces associated with splittings $E=E' \oplus E''$ corresponds to the subspace of stable pairs of type ii). \end{th} \newpage \centerline{\large{\bf References}} \vspace{10 mm} \parindent 0 cm [B1] Bradlow, S. B.: {\it Vortices in holomorphic line bundles over closed K\"ahler manifolds}, Comm. Math. Phys. 135, 1-17 (1990) [B2] Bradlow, S. B.: {\it Special metrics and stability for holomorphic bundles with global sections}, J. Diff. Geom. 33, 169-214 (1991) [D] Donaldson, S.: {\it Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles}, Proc. London Math. Soc. 3, 1-26 (1985) [DK] Donaldson, S.; Kronheimer, P.B.: {\it The Geometry of four-manifolds}, Oxford Science Publications (1990) [FU] Freed D. S. ; Uhlenbeck, K.: {\it Instantons and Four-Manifolds.} Springer-Verlag 1984. [HH] Hirzebruch, F.; Hopf H.: {\it Felder von Fl\"achenelementen in 4-dimensionalen 4-Mannigfaltigkeiten}, Math. Ann. 136 (1958) [H] Hitchin, N.: {\it Harmonic spinors}, Adv. in Math. 14, 1-55 (1974) [JPW] Jost, J.; Peng, X.; Wang, G. :{\it Variational aspects of the Seiberg-Witten functional}, Preprint, dg-ga/9504003, April (1995) [K] Kobayashi, S.: {\it Differential geometry of complex vector bundles}, Princeton University Press (1987) [KM] Kronheimer, P.; Mrowka, T.: {\it The genus of embedded surfaces in the projective plane}, Preprint (1994) [LM] Labastida, J. M. F.; Marino M.: {\it Non-abelian monopoles on four manifolds}, Preprint, Departamento de Fisica de Particulas, Santiago de Compostela, April (1995) [LT] L\"ubke, M.; Teleman, A.: {\it The Kobayashi-Hitchin correspondence}, World Scientific Publishing Co, to appear. [M] Miyajima K.: {\it Kuranishi families of vector bundles and algebraic description of the moduli space of Einstein-Hermitian connections}, Publ. R.I.M.S. Kyoto Univ. 25, 301-320 (1989) [OSS] Okonek, Ch.; Schneider, M.; Spindler, H: {\it Vector bundles on complex projective spaces}, Progress in Math. 3, Birkh\"auser, Boston (1980) [OT1] Okonek, Ch.; Teleman A.: {\it The Coupled Seiberg-Witten Equations, Vortices, and Moduli Spaces of Stable Pairs}, Preprint, Z\"urich, Jan. 13-th, (1995) [OT2] Okonek, Ch.; Teleman A.: {\it Les invariants de Seiberg-Witten et la conjecture de Van De Ven}, to appear in Comptes Rendus [OT3] Okonek, Ch.; Teleman A.: {\it Seiberg-Witten invariants and rationality of complex surfaces}, Preprint, Z\"urich, March (1995) [W] Witten, E.: {\it Monopoles and four-manifolds}, Mathematical Research Letters 1, 769-796 (1994) \vspace{0.4cm}\\ Authors addresses:\\ \\ Mathematisches Institut, Universit\"at Z\"urich,\\ Winterthurerstrasse 190, CH-8057 Z\"urich\\ e-mail: [email protected] \ ; \ [email protected] \end{document} --========================_16111448==_ Content-Type: text/plain; charset="us-ascii" Dr. Andrei Teleman (e-mail: [email protected]) Mathematisches Institut der Universitaet Zuerich Winterthurer Strasse 190, CH-8057 Zuerich-Irchel Tel.: (+411) 257 58 65; Fax 2575706 --========================_16111448==_--

9505

alg-geom/9505014

en

https://arxiv.org/abs/alg-geom/9505014

alg-geom/9505014

Teleman

Andrei Teleman and Christian Okonek

Seiberg-Witten Invariants and Rationality of Complex Surfaces

Duke preprint, LaTeX

The purpose of this paper is: 1) to explain the Seiberg-Witten invariants, 2) to show that - on a K\"ahler surface - the solutions of the monopole equations can be interpreted as algebraic objects, namely effective divisors, 3) to give - as an application - a short selfcontained proof for the fact that rationality of complex surfaces is a ${\cal C}^{\infty}$-property.

alg-geom

\section{Introduction} Recently, Seiberg and Witten introduced new differential invariants for 4-manifolds, which are defined by counting solutions of the so called {\sl monopole equations}, a system of non-linear differential equations of Yang-Mills-Higgs type [18]. The new invariants are expected to be equivalent to the Donaldson polynomial invariants, and they have already found important applications [15]. The purpose of this paper is:\\ - to explain the Seiberg-Witten invariants\\ - to show that --- on a K\"ahler surface --- the solutions of the monopole equations can be interpreted as algebraic objects, namely effective divisors\\ - to give --- as an application --- a short selfcontained proof for the fact that rationality of complex surfaces is a ${\cal C}^{\infty}$-property. \section{${\rm Spin}^c$-structures and the monopole equation} \begin{dt} {\rm [1], [11]} The group ${\rm Spin}^c(n):={\rm Spin}(n)\times_{{\Bbb Z}_2}S^1$ is called the complex spinor group. \end{dt} For the case $n=4$, there is a natural identification $${\rm Spin}^c(4)=\{(A,B)\in\U(2)\times\U(2)|\ {\rm det} A={\rm det} B\}\ .$$ The following diagram summarizes some of the basic relations of ${\rm Spin}^c(4)$ to other groups: $$\begin{array}{rrclcrl} &&&&\U(2)&&\\ &&&^l\swarrow& &\searrow^i\\ \ \ \ S^1&\longrightarrow&\ \ {\rm Spin}^c(4)\ &&\longrightarrow&&{\rm SO}(4)\\ (\cdot)^2\downarrow\ \ &{\scriptstyle{\rm det}}\swarrow&\ \downarrow&& \phantom{{\scriptstyle\lambda^+}}&&\ \ \ \ \ \downarrow {\scriptstyle(\lambda^+,\lambda^-)}\\ \ \ \ S^1&&\ \U(2)\times U(2)&&\textmap{{\rm ad}}&&{\rm SO}(3)\times SO(3) \ \ \ \ \end{array}$$ Here \ $l:\U(2)\longrightarrow{\rm Spin}^c(4)$ \ is the canonical lifting of the homomorphism \ $i\times{\rm det}:\U(2)\longrightarrow{\rm SO}(4)\times S^1$ [11], and acts by the formula\linebreak $\U(2)\ni a\longmapsto\left(\left(\matrix{{\rm id}&0\cr0&{\rm det} a\cr}\right),a\right)\in{\rm Spin}^c(4)$. $\lambda^{\pm}:{\rm SO}(4)\longrightarrow{\rm SO}(3)$ are the maps induced by the two projections of ${\rm Spin}(4)={\rm SU}(2)^+\times{\rm SU}(2)^-$ onto the factors. Let $X$ be a closed, oriented {\sl simply connected} 4-manifold, $\Lambda^p$ the bundle of $p$-forms on $X$, and $A^p:=A^0(X,\Lambda^p)$ the space of sections in this bundle. Let $g$ be a Riemannian metric on $X$, denote by $P$ the associated principal ${\rm SO}(4)$-bundle, and by $P^{\pm}$ the ${\rm SO}(3)$-bundles induced via the morphisms $\lambda^{\pm}$. The real 3-vector bundles $\Lambda^2_{\pm}:=P^{\pm}\times_{{\rm SO}(3)}{\Bbb R}^3$ can be identified with the bundles of (anti)self-dual 2-forms, hence there is an orthogonal splitting $\Lambda^2=\Lambda^2_+\oplus\Lambda^2_-$. \begin{lm}{\rm [10]} Given $c\in H^2(X,{\Bbb Z})$ with $w_2(X)\equiv\bar c$ (mod 2) there exists a unique ${\rm Spin}^c(4)$-bundle $\hat{P_c}$ with $P\simeq\qmod{\hat{P_c}}{S^1}$, and $c_1({\rm det}(\hat{P_c}))=c$. \end{lm} We denote by $\Sigma_c^{\pm}$ the induced $\U(2)$-vector bundles, and we put $\Sigma_c:=\Sigma^+_c\oplus\Sigma^-_c$. \begin{lm} {\rm [1], [11]} The choice of a ${\rm Spin}^c(4)$-lift $\hat{P_c}$ of $P$ induces an isomorphism $$\gamma_+:\Lambda^1\otimes{\Bbb C}\longrightarrow{\rm Hom}_{{\Bbb C}}(\Sigma^+_c,\Sigma^-_c)$$ satisfying the identity $\gamma_+(u)^*\gamma_+(v)+\gamma_+(v)^*\gamma_+(u)=2g(u,v){\rm id}_{\Sigma^+_c}$ for {\sl real} cotangent vectors $u,\ v\in\Lambda^1.$ \end{lm} We define the homomorphisms $\gamma:\Lambda^1\longrightarrow{\rm End}_0(\Sigma_c)$, $\Gamma:\Lambda^2\longrightarrow{\rm End}_0(\Sigma_c)$ by $$\gamma(u):=\left(\matrix{0&-\gamma_+(u)^*\cr\gamma_+(u)&0\cr}\right)$$ $$\Gamma(u\wedge v):=\frac{1}{2}[\gamma(u),\gamma(v)]\ ,$$ and we denote by the same symbols also their ${\Bbb C}$-linear extensions \linebreak$\Lambda^1\otimes{\Bbb C}\longrightarrow{\rm End}_0(\Sigma_c)$, and $\Lambda^2\otimes{\Bbb C}\longrightarrow{\rm End}_0(\Sigma_c)$. The homomorphism $\gamma$ defines a map $\Lambda^1\otimes\Sigma_c\longrightarrow\Sigma_c$\ , called the {\sl Clifford multiplication}. The map $\Gamma$ identifies the bundles $\Lambda^2_{\pm}$ with the bundles of {\sl trace free skew-Hermitian} endomorphisms of $\Sigma^{\pm}_c$. Fix a ${\rm Spin}^c(4)$-bundle $\hat{P_c}$ with $P\simeq\qmod{\hat{P_c}}{S^1}$, and let $L_c:={\rm det}(\hat{P_c})$ be the associated $S^1$-vector bundle. $L_c$ is the unique unitary line bundle with Chern class $c$. \begin{re} {\rm [11]} The choice of a $S^1$-connection $a$ in $L_c$ is equivalent to the choice of a ${\rm Spin}^c(4)$-connection $A$ in $\hat{P_c}$ projecting onto the Levi-Civita connection. \end{re} \begin{dt} The composition $\hskip 4pt{\not}{D}_a:A^0(\Sigma_c)\textmap{\nabla_A}A^1(\Sigma_c)\stackrel{\gamma}{\longrightarrow}A^ 0(\Sigma_c)$ is called the Dirac operator associated to the connection $a\in{\cal A}(L_c)$. \end{dt} {\bf Notation:} Let ${\cal A}(L_c)$ be the affine space of $S^1$-connections in $L_c$. For a connection $a\in{\cal A}(L_c)$, we denote by $F_a\in A^2({\rm ad}(L_c))= iA^2$ its curvature, and by $F_a^{\pm}\in i A^2_{\pm}$ the components of $F_a$ with respect to the orthogonal splitting $A^2=A^2_+\oplus A^2_-$. Every spinor $\Psi\in A^0(\Sigma^+_c)$ has a conjugate $\bar\Psi\in A^0(\bar\Sigma^+_c)$, and we can interpret $\Psi\otimes\bar\Psi$ as a Hermitian endomorphism of $\Sigma^+_c$. Let $(\Psi\otimes\bar\Psi)_0\in A^0({\rm End}_0(\Sigma^+_c))$ denote the trace-free component of it. \vspace{4mm}\\ The {\sl monopole equations} for a pair $(a,\Psi)\in {\cal A}(L_c)\times A^0(\Sigma^+_c)$ are the equations [18]: $$\left\{\begin{array}{ll} \hskip 4pt{\not}{D}_a\Psi=&0\\ \Gamma(F_a^+)=&2(\Psi\otimes\bar\Psi)_0 \end{array}\right.\eqno{(SW)}$$ \begin{pr} {\rm (The Weitzenb\"ock formula [11])}. Let $s$ be the scalar curvature of $(X,g)$. Fix a ${\rm Spin}^c(4)$-structure on $X$, and choose a $S^1$-connection $a\in{\cal A}(L_c)$. Then the following identity holds on $A^0(\Sigma_c)$ : $$\hskip 4pt{\not}{D}_a^2=\nabla_A^*\nabla_A+\frac{1}{2}\Gamma(F_a)+\frac{s}{4}{\rm id}_{\Sigma_c }\ .$$ \end{pr} \begin{co} Let $\Psi\in A^0(\Sigma^+_c)$. Then $$\parallel\hskip 4pt{\not}{D}_a\Psi\parallel^2+\frac{1}{2}\parallel\frac{1}{2}\Gamma(F_a^+)- (\Psi\bar\Psi)_0)\parallel^2=\parallel\nabla_A\Psi\parallel^2+ \frac{1}{8}\parallel F_a^+\parallel^2+\frac{1}{4}\parallel\Psi\parallel^4+ \frac{1}{4}\int\limits_X s|\Psi|^2.$$ \end{co} {\bf Proof: } By the Weitzenb\"ock formula we have $$(\hskip 4pt{\not}{D}^2_a\Psi,\Psi)=(\nabla_A^*\nabla_A\Psi,\Psi)+ \frac{1}{2}(\Gamma(F_a^+)(\Psi),\Psi)+\frac{s}{4}(\Psi,\Psi)\ ,$$ since $\Gamma(F_a^-)$ vanishes on $\Sigma_c^+$; integration over $X$ yields: $$\matrix{\parallel\hskip 4pt{\not}{D}_a\Psi\parallel^2+\frac{1}{2}\parallel\frac{1}{2} \Gamma(F_a^+)- (\Psi\bar\Psi)_0)\parallel^2=\int\limits_X (\hskip 4pt{\not}{D}^2_a\Psi,\Psi)+\frac{1}{2}\int\limits_X|\frac{1}{2}\Gamma(F_a^+)- (\Psi\bar\Psi)_0|^2=\cr =\parallel\nabla_A\Psi\parallel^2+ \frac{1}{2}\int\limits_X(\Gamma(F_a^+),(\Psi\bar\Psi)_0)+\frac{1}{4}\int\lim its_X s|\Psi|^2+\cr+\frac{1}{2}\int\limits_X\frac{1}{4}|\Gamma(F_a^+)|^2- \frac{1}{2}\int\limits_X(\Gamma(F_a^+),(\Psi\bar\Psi)_0)+ \frac{1}{4}\parallel\Psi\parallel^4\ .} $$ \begin{re} {\rm [18]} If $s\geq 0$ on $X$, then the only solutions $(a,\Psi)$ of $(SW)$ are pairs $(a,0)$ with $F_a^+=0$. \end{re} \section{Seiberg-Witten Invariants} The {\sl gauge group} ${\cal G}:={\cal C}^{\infty}(X,S^1)$ in the Seiberg-Witten theory {\sl is abelian} and acts on ${\cal A}(L_c)\times A^0(\Sigma^+_c)$ by $(a,\Psi)\cdot f:=(a+f^{-1}df,f^{-1}\Psi)$, letting invariant the set of solutions of the equations $(SW)$. We denote by ${\cal W}^g_X(c)$ the moduli space of solutions of the Seiberg-Witten equations, modulo gauge equivalence. A standard technique provides a natural structure of finite dimensional {\sl real analytic} space in ${\cal W}^g_X(c)$ [6], [5], [16]. The {\sl expected dimension} of this moduli space is $$w_c=\frac{1}{4}(c^2-2e(X)-3\sigma(X))\ ,$$ where $e(X)$ and $\sigma(X)$ stand for the Euler characteristic and the signature of the oriented manifold $X$. A solution $(a,\Psi)$ is reducible (has nontrivial stabilizer) if and only if $\Psi=0$, and then the connection $a$ must be anti-selfdual. We say that the metric $g$ is $c$-{\sl good} if the $g$-harmonic representative of the de Rham cohomology class $c_{\rm DR}$ is {\sl not} anti-selfdual. If $g$ is $c$-good, then ${\cal W}^g_X(c)$ consists only of irreducible orbits. Using the same technique as in Yang-Mills theory ([6], [5]), one defines a {\sl gauge invariant perturbation} of the Seiberg-Witten equations in order to get smooth moduli spaces of the expected dimension. For a selfdual form $\mu\in A^2_+$ we denote by ${\cal W}^{g,\mu}_X(c)$ the moduli space of solutions of the perturbed Seiberg-Witten equations $$\left\{\begin{array}{lll} \hskip 4pt{\not}{D}_a\Psi&=&0\\ \Gamma(F_a^+ +i\mu)&=&2(\Psi\otimes\bar\Psi)_0 \end{array}\right.\eqno{(SW_{\mu})}$$ We refer to [15] for the following \begin{lm}\hfill{\break} 1. For every $\mu\in A^2_+$, the moduli space ${\cal W}^{g,\mu}_X(c)$ is compact.\hfill{\break} 2. There is a dense, second category set of perturbations $\mu\in A^2_+$, for which the irreducible part ${\cal W}^{g,\mu}_X(c)^*$ of ${\cal W}^{g,\mu}_X(c)$ is smooth and has the expected dimension.\hfill{\break} 3. If $g$ is $c$-good, and $\mu$ is small enough in the $L^2$ topology, then ${\cal W}^{g,\mu}_X(c)$ consists only of irreducible orbits, i.e. ${\cal W}^{g,\mu}_X(c)$=${\cal W}^{g,\mu}_X(c)^*$ .\hfill{\break} 4. Let $g_0$ and $g_1$ be $c$-good metrics which can be connected by a smooth path of $c$-good metrics, and let $\varepsilon_i>0$ be small enough such that ${\cal W}^{g_i,\mu_i}_X(c)={\cal W}^{g_i,\mu_i}_X(c)^*$ for all perturbations $\mu_i$ with $\parallel\mu_i\parallel<\varepsilon_i$. Then any two moduli spaces ${\cal W}^{g_i,\mu_i}_X(c)\ ,\ \ i=0, 1$ , with $\parallel\mu_i\parallel<\varepsilon_i$, which are smooth and have the expected dimension, are cobordant. \end{lm} The first assertion is a simple consequence of the Weitzenb\"ock formula and of the Maximum Principle. The other three assertions follow as in Donaldson theory by the Sard theorem for smooth Fredholm maps, and by transversality arguments [5]. Note that in 4. we mean cobordism between {\sl non-oriented} compact smooth manifolds. A more delicate analysis of the monopole equations [18] shows that, in fact, the moduli spaces ${\cal W}^{g,\mu}_X(c)^*$ come with natural orientations, as soon as they are smooth and have the expected dimension, and that the conclusion in 4. holds for the oriented moduli spaces. \vspace{3mm}\\ The Seiberg-Witten theory provides strong differentiable invariants using only moduli spaces of dimension 0. Let $c$ be an integral lift of $w_2(X)$, with $w_c=0$, i.e. $$c^2=2e(X)+3\sigma(X)\ .$$ Such a lift is called an {\sl almost canonical class}, since the condition \hbox{$w_c=0$} is equivalent to the existence of an almost complex structure on $X$ with first Chern class $c$ [10], [16]. Now fix an almost canonical class $c$, choose a $c$-good metric $g$, and a small, sufficiently general perturbation $\mu$. Then ${\cal W}^{g,\mu}_X(c)={\cal W}^{g,\mu}_X(c)^*$ is compact, smooth of the expected dimension 0, and its bordism class is independent of $\mu$. Let $n_c^g:=|{\cal W}^{g,\mu}_X(c)|$ mod 2 be the number of points modulo 2 of this moduli space. Lemma 2.1 implies that $n_c^g$ is also independent of $g$ if any two $c$-good metrics can be connected by a smooth 1-parameter family of $c$-good metrics. The numbers $n_c:=n_c^g$ associated to such almost canonical classes are called the mod 2-Seiberg-Witten invariants, and the classes $c$ with $n_c\ne 0$ are then called mod 2-Seiberg-Witten classes of index 0. By definition they are differentiable invariants, in the following sense: If $f:X'\longrightarrow X$ is an orientation-preserving diffeomorphism, and for an almost canonical class $c$ of $X$ the Seiberg-Witten invariant $n_c$ is well defined, then $f^*(c)$ has the same property, and $n_{f^*(c)}=n_{c}$. \begin{re} Let $c$ be an almost canonical class of $X$. \hfill{\break} 1. If $c^2\geq 0$ and $c_{\rm DR}\ne 0$ , then {\sl any} Riemannian metric on $X$ is $c$-good.\hfill{\break} 2. If $b_2^+\geq 2$, then any two $c$-good metrics can be connected by a smooth path of $c$-good metrics. \end{re} Therefore, if one of the two conditions above is satisfied, then the \linebreak\hbox{mod 2-Seiberg-Witten} invariant $n_c$ is well-defined. \vspace{3mm} In the case $b_2^+=1$, invariants can still be defined, but the dependence of $n_c^g$ on the metric $g$ must be taken into account: In the real vector space $H^2_{\rm DR}(X)$ , consider the positive cone $${\cal K}=\{u\in H^2_{\rm DR}(X) \ | u^2>0\}\ .$$ Fix a {\sl non-vanishing} cohomology class $k\in H^2_{\rm DR}(X)$ with $k^2\geq 0$. The cone ${\cal K}$ splits as the disjoint union of its connected components ${\cal K}_{\pm}$, where $${\cal K}_{\pm}:=\{u\in{\cal K} \ |\ \pm u\cdot k>0\}\ .$$ If $c$ is an almost canonical class, let $c^{\bot}$ be the hyperplane $$c^{\bot}:=\{u\in H^2_{\rm DR}(X)\ |\ c\cdot u=0\}$$ If $c^{\bot}$ meets ${\cal K}_+$, then the intersection $c^{\bot}\cap{\cal K}_+$ is called the {\sl wall} of type $c$, and the two components of ${\cal K}_+\setminus c^{\bot}$ are called {\sl chambers} of type $c$. For every Riemannian metric $g$ on $X$, let $\omega_g$ be a generator of the real line of $g$-harmonic selfdual 2-forms, such that $[\omega_g]\in{\cal K}_+$ . Then the ray ${\Bbb R}_{>0}[\omega_g]\subset {\cal K}_+$ depends smoothly on the metric $g$. The property of a metric to be $c$-good has the following simple geometric interpretation: \begin{re}\hfill{\break} Suppose $b_2^+(X)=1$. Then:\hfill{\break} 1. The metric $g$ is $c$-good iff the ray ${\Bbb R}_{>0}[\omega_g]$ does not lie in the wall $c^{\bot}\cap{\cal K}_+$. \hfill{\break} 2. If $g_0$, and $g_1$ are $c$-good metrics, then $n_c^{g_0}=n_c^{g_1}$ iff the two rays ${\Bbb R}_{>0}[\omega_{g_i}]$ belong to the same chamber of type $c$. \end{re} The first assertion follows immediately from the definition. The second needs a careful analysis of a 1-parameter family of 0-dimensional smooth moduli spaces ${\cal W}^{g_t}_X(c)$ around the value of the parameter $t$ for which the ray ${\Bbb R}_{>0}[\omega_{g_t}]$ crosses the wall $c^{\bot}\cap{\cal K}_+$ (see [18], [15]). \section{Monopoles on K\"ahler surfaces} Let $(X,J,g)$ be an almost complex 4-manifold endowed with a Hermitian metric $g$. The almost complex structure $J$ defines a reduction of the structure group of the tangent bundle $T_X$ of $X$ from ${\rm SO}(4)$ to $\U(2)$. In particular, we get a {\sl canonical} ${\rm Spin}^c(4)$-structure on \ $X$ \ via \ the \ canonical \ lifting \ \ $l:\U(2)\longrightarrow{\rm Spin}^c(4)$ [11]. Let $\omega_g$ be the K\"ahler form of $g$. \begin{lm} {\rm [11]} The canonical ${\rm Spin}^c$-structure of an almost complex Hermitian 4-manifold has the following properties:\hfill{\break} 1. There are canonical identifications $\Sigma^+=\Lambda^{00}\oplus\Lambda^{02}$, $\Sigma^-=\Lambda^{01}$.\hfill{\break} 2. Via these identifications, the map $\Gamma:\Lambda^2_+\otimes{\Bbb C}\longrightarrow{\rm End}_0(\Sigma^+)$ is given by: $$\Lambda^{20}\oplus\Lambda^{02}\oplus\Lambda^{00}\omega_g\ni(\lambda^{20},\ lambda^{02}, f\omega_g)\textmap{\Gamma}2\left[\matrix{-if&-*(\lambda^{20}\wedge\cdot)\cr \lambda^{02}\wedge\cdot&if\cr}\right]\in{\rm End}_0(\Lambda^{00}\oplus\Lambda^{02 })\ .$$ \end{lm} Suppose now that $(X,J,g)$ is a K\"ahler surface. This means that $J$ is integrable, and $\omega_g$ is closed (or equivalently, $J$ is Levi-Civita parallel). In particular the holonomy group of the Levi-Civita connection also reduces to $\U(2)$, and the splittings $\Lambda^p\otimes{\Bbb C}=\bigoplus\limits_{i+j=p}\Lambda^{ij}$ are Levi-Civita parallel. We get a $\U(2)$-connection in the holomorphic tangent bundle ${\cal T}_X=T^{10}_X\simeq\Lambda^{01}$, which coincides with the {\sl Chern connection} of this bundle, i.e. with the unique connection compatible with the holomorphic structure and the Hermitian metric. The induced connection $c_0$ in the line bundle $K_X^{\vee}={\rm det}({\cal T}_X)\simeq\Lambda^{02}$ also coincides with the Chern connection of this Hermitian holomorphic line bundle. Every other ${\rm Spin}^c(4)$ structure $\hat{P_c}\longrightarrow P$ on $(X,g)$ has as spinor bundle $$\Sigma_c=\Sigma\otimes M\ ,$$ where $M$ is a differentiable $S^1$-bundle with $2c_1(M)+c_1(K_X^{\vee})=c$. (For a simply connected manifold $X$, $M$ is well defined up to isomorphy by this condition.) $S^1$-connections in ${\rm det}(\Sigma^{\pm}_c)=K_X^{\vee}\otimes M^{\otimes 2}$ correspond to $S^1$-connections in $M$. Given $b\in{\cal A}(M)$, the curvature of the corresponding connection $a\in{\cal A}(K_X^{\vee}\otimes M^{\otimes 2})$ is $F_a=F_{c_0}+2F_b$. A half-spinor $\Psi\in A^0(\Sigma^+\otimes M)$ can be written as $$\Psi=\varphi+\alpha\ ,\ \ \varphi\in A^0(M)\ ,\ \ \alpha\in A^{02}(M)\ .$$ We put $J(M):=c_1(\Sigma^+\otimes M)\cup[\omega_g]$. \begin{pr} Let $(X,g)$ be a K\"ahler surface with Chern connection $c_0$ in $K_X^{\vee}$, $M$ a differentiable $S^1$-bundle with $J(M)<0$. A pair $(b,\varphi+\alpha)\in {\cal A}(M)\times\left(A^0(M)\oplus A^{02}(M)\right)$ solves the monopole equations iff: $$ \begin{array}{l}F_b^{20}=F_b^{02}=0\ \\ \alpha=0\ ,\ \ \bar\partial_b(\varphi)=0 \ \\ i\Lambda F_b+\frac{1}{2}\varphi\bar\varphi+\frac{s}{2}=0\ .\end{array}\ \eqno{(*)}$$ \end{pr} {\bf Proof: } The pair $(b,\varphi+\alpha)$ solves the equations $(SW)$ iff the corresponding pair $(a,\varphi+\alpha)$ satisfies $$\begin{array}{ll}F_a^{20}&=-\varphi\otimes\bar\alpha\\ F_a^{02}&=\ \alpha\otimes\bar\varphi\\ \bar\partial_b(\varphi)&=\ i\Lambda\partial_b(\alpha) \\ i\Lambda F_a&=-\left(\varphi\bar\varphi-*(\alpha\wedge\bar\alpha)\right).\end{array}\ $$ By Corollary 1.7 it follows that $(b,\varphi+\alpha)$ solves $(SW)$ iff $(b,\varphi-\alpha)$ does (Witten's trick). Therefore $\varphi\otimes\bar\alpha=\alpha\otimes\bar\varphi=0$, hence $F_a^{20}=F_a^{02}=0$, and $\varphi$ or $\alpha$ must vanish. Integrating the equation $i\Lambda F_a=-\left(\varphi\bar\varphi- *(\alpha\wedge\bar\alpha)\right)$ over $X$, we find: $$J(M)=(2c_1(M)-c_1(K_X))\cup[\omega_g]=\int\limits_X\frac{i}{2\pi} F_a\wedge\omega_g= \frac{1}{8\pi}\int\limits_X(-|\varphi|^2+|\alpha|^2)\ ,$$ hence $\alpha=0$ if $J(M)<0$. \hfill\vrule height6pt width6pt depth0pt \bigskip The above proposition must be interpreted as follows: If $J(M)<0$, then the solutions of the monopole equations $(SW)$ are the pairs $(b,\varphi)\in{\cal A}(M)\times A^0(M)$, such that $b$ is the Chern connection of a holomorphic structure in $M$, $\varphi$ is a holomorphic section, and the mean curvature $i\Lambda F_b$ of $b$ satisfies the {\sl generalized vortex equation} [16], [3], [4], [9] $$i\Lambda F_b+\frac{1}{2}\varphi\bar\varphi+\frac{s}{2}=0\ .\eqno{(V_s)}$$ Moreover, every {\sl infinitesimal deformation} of a solution of the form $(b,\varphi)$, $\varphi\ne 0$ of the monopole equation still vanishes in the $\alpha$-direction. Therefore ${\cal W}_X^g(c)$ can be identified (as {\sl real analytic space}) with the moduli space of pairs $(b,\varphi)$ satisfying the above conditions, modulo the gauge group ${\cal C}^{\infty}(X,S^1)$ of unitary automorphisms of $M$. Under the assumption $J(M)<0$, the action of the gauge group is free on the space of solutions, because any solution $(b,\varphi)$ has a non-vanishing section $\varphi$. Alternatively, let ${\cal M}$ be a holomorphic line bundle with differentiable support $M$, and $\varphi$ a holomorphic section of ${\cal M}$. For a Hermitian metric $h$ in ${\cal M}$, we denote by $F_h$ the curvature of the associated Chern connection, and we consider the following equation for $h$: $$i\Lambda F_h+\frac{1}{2}\varphi\bar\varphi^h+\frac{s}{2}=0\ . \eqno{(V'_s)}$$ Standard arguments (see for instance [16], [9]) show that the problem of classifying the solutions $(b,\varphi)$ of $(*)$ modulo {\sl unitary automorphisms of $M$} is equivalent to the problem of classifying those pairs $({\cal M},\varphi)$ modulo {\sl holomorphic isomorphisms}, for which the equation $(V'_s)$ has a solution. \begin{pr} Let $(X,g)$ be a compact K\"ahler surface, $({\cal M},\varphi)$ a holomorphic line bundle with a {\sl non-vanishing} holomorphic section $\varphi\in H^0(X,{\cal M})$. ${\cal M}$ admits a metric $h$ satisfying the equation $(V'_s)$ iff $$c_1({\cal M})\cup[\omega_g]<\frac{1}{2}c_1(K_X)\cup[\omega_g]\ .$$ \end{pr} {\bf Proof: } (cf. [3]) Fix a background metric $h_0$; any other metric $h$ has the form $h=e^{2u}h_0$, with $u\in A^0$ a smooth function.The vortex equation $(V'_s)$ translates into $$\Delta u+\frac{1}{2}|\varphi|^2_{h_0} e^{2u}+(i\Lambda F_{h_0}+\frac{s}{2})=0\ .\eqno{(1)}$$ Set $q:=\int\limits_X(i\Lambda F_{h_0}+\frac{s}{2})=2\pi(c_1({\cal M})-\frac{1}{2}c_1(K_X)\cup[\omega_g]$, and choose $v\in A^0$ with $$-\Delta v=(i\Lambda F_{h_0}+\frac{s}{2})-q \ .$$ Define $w:=2(u-v)$. Then $(1)$ is equivalent to the following equation in $w$: $$\Delta w+(|\varphi|^2_{h_0} e^{2v})e^w +2q=0\ .\eqno{(2)}$$ Integrating over $X$, we see that if (2) has solutions, then $q$ must be negative. On the other hand, by a well known result of Kazdan and Warner [3], (2) has a unique solution if $q<0$. \hfill\vrule height6pt width6pt depth0pt \bigskip \begin{th} {\rm [18], [16]} Let $(X,g)$ be a simply connected K\"ahler surface, $c\in H^2(X,{\Bbb Z})$ with $c\equiv c_1(K_X)$ mod 2, and $\pm c\cup[\omega_g]<0$. \hfill{\break} 1. \ If $c\ \not\in\ {\rm NS}(X)$, then ${\cal W}_X^g(c)=\emptyset$ .\hfill{\break} 2. Suppose $c\in{\rm NS}(X)$. Then there is a natural real analytic isomorphism \linebreak ${\cal W}_X^g(c)\simeq{\Bbb P}(H^0(X,{\cal M}))$, where ${\cal M}$ is the (unique, up to isomorphy) holomorphic line bundle with $c_1(K_X^{\vee}\otimes{\cal M}^{\otimes 2})=\pm c$. \hfill{\break} 3. ${\cal W}^g_X(c)$ is always smooth.\ Let $D$ be the divisor of a nontrivial section in ${\cal M}$. Then ${\cal W}^g_X(c)$ has the expected dimension iff \hbox{$h^1({\cal O}_X(D)|_D)=0$}. \end{th} \section{Rationality of complex surfaces} A compact complex surface is {\sl rational} iff its field of meromorphic functions is isomorphic to ${\Bbb C}(u,v)$. Such a surface is always simply connected and has $b_2^+=1$ [2]. The following result has been has been announced by R. Friedman and Z. Qin [8]. Whereas their proof uses Donaldson theory and vector bundles techniques, our proof uses the new Seiberg-Witten invariants, and our interpretation of these invariants in terms of linear systems. \begin{th} {\rm [17]} A complex surface $X$ which is diffeomorphic to a rational surface is rational. \end{th} {\bf Proof: } The proof consists of the following three steps: 1.Any rational surface $X_0$ admits a {\sl Hitchin metric} [12], i.e. a K\"ahler metric $g_0$ with positive {\sl total scalar curvature}. This condition can be written as $c_1(K_{X_0})\cup[\omega_{g_0}]<0$. Let $c$ be any integral lift of $w_2(X_0)$, such that $g_0$ is $c$-good, i.e. such that the moduli space ${\cal W}^{g_0}_{X_0}(c)$ contains no reducible solutions. Since $p_g(X_0)=0$, $c$ has always type (1,1), and $g_0$ is $c$-good iff $c\cup[{\omega_{g_0}}]\ne 0$. We assert that ${\cal W}^{g_0}_{X_0}(c)$ is then empty, and in particular, all Seiberg-Witten invariants $n_c^{g_0}$ computed with respect to this metric vanish. Indeed, let ${\cal M}$ be the holomorphic line bundle defined in Theorem 3.4. If the moduli space ${\Bbb P}(H^0(X_0, {\cal M}))$ was not empty, then $$c_1({\cal M})\cup[\omega_{g_0}]\geq 0 \ .\eqno{(1)}$$ But we have $$0>\pm c\cup[\omega_{g_0}]=(2c_1({\cal M})-c_1(K_{X_0}))\cup[\omega_{g_0}]\ ,$$ hence, by (1) $$0\leq 2c_1({\cal M})\cup[\omega_{g_0}]<c_1(K_{X_0})\cup[\omega_{g_0}]\ ,$$ which contradicts the assumption on the total scalar curvature of $g_0$. \vspace{2mm}\\ 2. Let now $X$ be a simply connected projective surface with ${\rm kod}(X)> 0$. We may suppose that $X$ is the blow up in $k$ {\sl distinct} points of its minimal model $X_{\min}$. Denote by $\sigma:X\longrightarrow X_{\min}$ the contraction to the minimal model, and by $E=\sum\limits_{i=1}^k E_i$ the exceptional divisor. Fix an ample divisor $H_{\min}$ on $X_{\min}$, set $H_n:=\sigma^*(n H_{\min})-E$, and for $n\gg 0$ choose a K\"ahler metric $g_n$ on $X$ with $[\omega_{g_n}]=c_1(H_n)$. Given $I\subset\{1,\dots,k\}$, define $$\begin{array}{l}E_I:=\sum\limits_{i\in I} E_i\\ c_I:=2c_1(E_I)-c_1(K_X)\\ \bar I:=\{1,\dots,k\}\setminus I \ .\end{array}$$ Since $c_I$ is an almost canonical class, the expected dimension of the corresponding Seiberg-Witten moduli space is 0. For $n\gg 0$ we get $c_I\cup [\omega_{g_n}]<0$, and Theorem 3.4 gives $${\cal W}^{g_n}_X(c_I)\simeq\{E_I\}\ .$$ Therefore ${\cal W}^{g_n}_X(c_I)$ consists of a single smooth point, and $$n_{c_I}^{g_n}= 1\ {\rm mod}\ 2\ .\eqno{(2)}$$ 3. Suppose now that there is an orientation-preserving diffeomorphism $f:X\longrightarrow X_0$, where $X$ is projective surface with ${\rm kod} X\geq 0$. Since $X$ must have $p_g(X)=0$, and $\pi_1(X)=\{1\}$, it follows that, in fact, ${\rm kod} X>0$. Let $g=f^*(g_0)$ denote the pull-back of a Hitchin metric to $X$; clearly $$n_{c_I}^g=0 \eqno{(3)}$$ for all $I\subset\{1,\dots,k\}$ such that $g$ is $c_I$-good. We will now derive a contradiction in the following way: Using the Enriques-Kodaira classification of surfaces, it easy to see that the de Rham cohomology class $ k_{\min}:=\sigma^*(c_{1,{\rm DR}}(K_{\min}))$ is non-trivial and satisfies the condition $k^2_{\min}\geq 0$. Therefore we can consider the upper positive cone $${\cal K}_+:=\{u\in H^2_{\rm DR}(X)\ |\ u^2>0, \ u\cdot k_{\min}>0\}\ .$$ Clearly $[\omega_{g_n}]$ belongs to ${\cal K}_+$. We choose a harmonic $g$-selfdual form $\omega_g$, with $[\omega_g]\in{\cal K}_+$. \vspace{3mm}\\ {\bf Claim:} {\sl The rays ${\Bbb R}_{>0}[\omega_g]$ and ${\Bbb R}_{>0}[\omega_{g_n}]$ belong either to the same chamber of type $c_I$ or to the same chamber of type $c_{\bar I}$.} \vspace{1mm}\\ {\bf Proof: } If not, then, since $c_I\cup [\omega_{g_n}]<0$, we get $[\omega_g]\cdot c_I\geq0$ and $[\omega_g]\cdot c_{\bar I}\geq 0$. Write $$[\omega_g]=\sum\limits_{i=1}^k\lambda_i E_i+\sigma^*[\omega]\ ,$$ with $[\omega]\in H^2_{\rm DR}(X_{\min})$. Then $$\begin{array}{l} -\sum\limits_{i\in I}\lambda_i+\sum\limits_{j\in\bar I}\lambda_j-[\omega]\cdot [K_{\min}]\geq 0\\ -\sum\limits_{j\in\bar I}\lambda_j+\sum\limits_{i\in I}\lambda_i-[\omega]\cdot [K_{\min}]\geq 0\ .\end{array}$$ Adding these inequalities we find $[\omega]\cdot[K_{\min}]\leq 0$. But $[\omega]\cdot [K_{\min}]=[\omega_g]\cdot k_{\min}>0$, because $[\omega_g]\in{\cal K}_+$. This contradiction proves the claim. \vspace{3mm} It follows that either $g$ and $g_n$ are both $c_I$-good and $n_{c_I}^g=n_{c_I}^{g_n}$, or $g$ and $g_n$ are both $c_{\bar I}$-good and $n_{c_{\bar I}}^g=n_{c_{\bar I}}^{g_n}$. This gives now a contradiction with (2) and (3). \hfill\vrule height6pt width6pt depth0pt \bigskip Together with the results of Friedman and Morgan [7], we have: \begin{th}{\rm (The Van de Ven conjecture [19])} The Kodaira dimension of complex surfaces is a ${\cal C}^{\infty}$-invariant. \end{th} {\bf Remark:} It is possible to couple the Seiberg-Witten equations to connections in unitary bundles. The solutions of these coupled Seiberg-Witten equations over K\"ahler surfaces again have a purely complex-geometric interpretation [16]: The moduli space of solutions can be identified---via generalized vortex equations--- with moduli spaces of stable pairs [13], [4]. This construction could lead to new invariants which might be nontrivial for K\"ahler surfaces wit $p_g=0$. \vspace{0.8cm}\\ \parindent0cm \centerline {\Large {\bf Bibliography}} \vspace{0.5cm} 1. Atiyah M., Hitchin N. J., Singer I. M.: {\it Selfduality in four-dimensional Riemannian geometry}, Proc. R. Lond. A. 362, 425-461 (1978) 2. Barth, W., Peters, C., Van de Ven, A.: {\it Compact complex surfaces}, Springer Verlag (1984) 3. Bradlow, S. B.: {\it Vortices in holomorphic line bundles over closed K\"ahler manifolds}, Comm. Math. Phys. 135, 1-17 (1990) 4. Bradlow, S. B.: {\it Special metrics and stability for holomorphic bundles with global sections}, J. Diff. Geom. 33, 169-214 (1991) 5. Donaldson, S.; Kronheimer, P. B.: {\it The Geometry of four-manifolds}, Oxford Science Publications (1990) 6. Freed, D. S.; Uhlenbeck, K. K.: {\it Instantons and Four-Manifolds}, Springer Verlag (1984) 7. Friedman, R., Morgan, J.W.: {\it Smooth 4-manifolds and Complex Surfaces}, Springer Verlag 3. Folge, Band 27 (1994) 8. Friedman, R., Qin, Z.: {\it On complex surfaces diffeomorphic to rational surfaces}, Preprint (1994) 9. Garcia-Prada, O.: {\it Dimensional reduction of stable bundles, vortices and stable pairs}, Int. J. of Math. Vol. 5, No 1, 1-52 (1994) 10. Hirzebruch, F., Hopf H.: {\it Felder von Fl\"achenelementen in 4-dimensionalen 4-Mannigfaltigkeiten}, Math. Ann. 136, (1958) 11. Hitchin, N.: {\it Harmonic spinors}, Adv. in Math. 14, 1-55 (1974) 12. Hitchin, N.: {\it On the curvature of rational surfaces}, Proc. of Symp. in Pure Math., Stanford, Vol. 27 (1975) 13. Huybrechts, D.; Lehn, M.: {\it Stable pairs on curves and surfaces}, J. Alg. Geometry, (1995) 14. Kobayashi, S.: {\it Differential geometry of complex vector bundles}, Princeton University Press, (1987) 15. Kronheimer, P., Mrowka, T.: {\it The genus of embedded surfaces in the projective plane}, Preprint (1994) 16. Okonek, Ch.; Teleman A.: {\it The Coupled Seiberg-Witten Equations, Vortices, and Moduli Spaces of Stable Pairs}, Preprint, January, 13-th 1995 17. Okonek, Ch.; Teleman A.: {\it Seiberg-Witten invariants and the Van de Ven conjecture}, Preprint, February, 8-th 1995 18. Witten, E.: {\it Monopoles and four-manifolds}, Mathematical Research Letters 1, 769-796 (1994) 19. Van de Ven, A,: {\it On the differentiable structure of certain algebraic surfaces}, S\'em. Bourbaki ${\rm n}^o$ 667, Juin (1986) \vspace{0.5cm}\\ Authors addresses:\\ Mathematisches Institut, Universit\"at Z\"urich,\\ Winterthurerstrasse 190, CH-8057 Z\"urich\\ e-mail:[email protected] \ \ \ \ \ \ \ \ \ [email protected] \end{document} --========================_18689640==_ Content-Type: text/plain; charset="us-ascii" Dr. Andrei Teleman (e-mail: [email protected]) Mathematisches Institut der Universitaet Zuerich Winterthurer Strasse 190, CH-8057 Zuerich-Irchel Tel.: (+411) 257 58 65; Fax 2575706 --========================_18689640==_--

9505

alg-geom/9505038

en

https://arxiv.org/abs/alg-geom/9505038

alg-geom/9505038

Dan Abramovich

Dan Abramovich

Uniformity of stably integral points on elliptic curves

10 pages. Postscript file available at http://math.bu.edu/INDIVIDUAL/abrmovic/integral.ps, AMSLaTeX

A common practice in arithmetic geometry is that of generalizing rational points on projective varieties to integral points on quasi-projective varieties. Following this practice, we demonstrate an analogue of a result of L. Caporaso, J. Harris and B. Mazur, showing that the Lang - Vojta conjecture implies a uniform bound on the number of stably integral points on an elliptic curve over a number field, as well as the uniform boundedness conjecture (Merel's theorem).

alg-geom

\section{Introduction} Let $X$ be a variety of logarithmic general type, defined over a number field $K$. Let $S$ be a finite set of places in $K$ and let ${{\cal O}_{K,S}}$ be the ring of $S$-integers. Suppose that ${\cal{X}}$ is a model of $X$ over ${\mbox{Spec }}{{\cal O}_{K,S}}$. As a natural generalizasion of theorems of Siegel and Faltings, It was conjectured by S. Lang and P. Vojta (\cite{vojta}, conjecture 4.4) that the set of $S$-integral points ${\cal{X}}({{\cal O}_{K,S}})$ is not Zariski dense in ${\cal{X}}$. In case $X$ is projective, one may chose an arbitrary projective model ${\cal{X}}$ and then ${\cal{X}}({{\cal O}_{K,S}})$ is identified with $X(K)$. In such a case, one often refers to this conjecture of Lang and Vojta as just Lang's conjecture. L. Caporaso, J. Harris and B. Mazur \cite{chm} apply Lang's conjecture in the following way: Let $X\rightarrow B$ be a smooth family of curves of genus $g>1$. Let $X^n_B\rightarrow B$ be the $n$-th fibered power of $X$ over $B$. In \cite{chm} it is shown that for high enough $n$, the variety $X^n_B$ dominates a variety of general type. Assuming Lang's conjecture, they deduce the following remarkable result: the number of rational points on a curve of genus $g$ over a fixed number field is uniformly bounded. In this note we study an analogous implication for elliptic curves. Let $E/K$ be an elliptic curve over a number field, and let $P\in E(K)$. We say that $P$ is stably $S$-integral, denoted $P\in E(K,S)$, if $P$ is $S$-integral after semistable reduction (see \S\ref{stably}). Our main theorem states (see \S\ref{main}): \begin{th}(Main theorem in terms of points) Assume that the Lang - Vojta conjecture holds. Then for any number field $K$ and a finite set of places $S$, there is an integer $N$ such that for any elliptic curve $E/K$ we have $\#E(K,S)<N$. \end{th} Since the moduli space of elliptic curves is only one-dimensional, the computations and the proofs are a bit simpler than the higher genus cases. One can view the results in this paper as a simple application of the methods of \cite{chm}. \subsection{Overview} In section \ref{correlation} we prove a basic lemma analogous to lemma 1.1 \cite{chm} on uniformity of correlated points. In section \ref{level3} we study a particular pencil of elliptic curves which is the main building block for proving theorem 1. In section \ref{twist} we look at quadratic twists of an elliptic curve, motivating the study in section \ref{stably} of stably integral points. Section \ref{main} gives a proof of the main theorem. In section \ref{ubc} we will refine our methods and show that the Lang - Vojta conjecture implies the uniform boundedness conjecture for torsion on elliptic curves, thus giving a conditional (and therefore obsolete) proof of the following theorem of Merel: \begin{th}(Merel, \cite{merel}) For and integer $d$ there is an integer $N(d)$ such that given a number field $K$ with $[K:{\Bbb{Q}}]=d$, and given an elliptic curve $E/K$, then $\#E(K)_{tors}<N(d)$. \end{th} It should be noted that the methods introduced in section \ref{ubc} were essential for the developments in \cite{abr},\cite{abr1}. It would be interesting if one could apply the method to the study points on abelian varieties in general. \subsection{Acknowledgements} I am indebted to the ideas in the work \cite{chm} of Lucia Caporaso, Joe Harris and Barry Mazur and to conversations with them. Thanks to Joe Silverman and Sinnou David for discussions of points on elliptic curves and much encouragement. Much of this text was written during a pleasant visit with the group ``Problemes Diophantiens'' in Paris. It is also a pleasure to thank Henri Darmon, Gerhard Frey Brendan Hassett and Felipe Voloch for their suggestions. Special thanks are due to Frans Oort who read an earlier version of this paper and sent me many helpful comments\footnote{I take the opportunity to wish Professor Oort a happy 60th birthday.}. \subsection{The Lang - Vojta conjecture}\label{lv} A common practice in arithmetic geometry is that of generalizing rational points on projective varieties to integral points on quasi-projective varieties. We can summarize this in the following table, which will be explained below:\\ \begin{tabular}{l|l}\hline\hline Number field $K$ & Ring of $S$-integers $O_{K,S}$ \\[2mm] \hline Projective variety $X$ over $K$ & \parbox{3in}{\vspace*{1mm} Quasi projective variety $X$ and a model ${\cal{X}}$ over $O_{K,S}$ } \\ \hline \vspace*{1mm} Rational point $P\in X(K)$ & Integral point $P\in {\cal{X}}({{\cal O}_{K,S}})$ \\[2mm] \hline $X$ of general type & $X$ of log-general type \\ e.g.: $C$ a curve of genus $>1$ & \parbox{3in}{ e.g.: $E$ an elliptic curve with the origin removed } \\ \hline Faltings' theorem: $C(K)$ finite & Siegel's theorem: $E(O_{K,S})$ finite \\ \hline \vspace*{1mm} \parbox{3in}{\vspace*{1mm} Lang's conjecture: If $X$ is of general type then $X(K)$ not Zariski dense\\ } & \parbox{3in}{\vspace*{1mm} Lang-Vojta conjecture: If $X$ is of logarithmic general type then ${\cal{X}}({{\cal O}_{K,S}})$ not Zariski dense } \\ \hline \parbox{3in}{\cite{chm}: Lang's conjecture implies uniformity of $\#C(K)$} & \parbox{3in}{ \vspace*{1mm} ?? }\\ \hline\hline \end{tabular} \vspace*{2mm} We remind the reader of the definition of a variety of log general type: \begin{dfn} Let $X$ be a quasi-projective variety over ${\Bbb{C}}$. Let $f:Y \rightarrow X$ be a resolution of singularities, that is, a proper, birational morphism where $Y$ is a smooth variety. Let $Y\subset Y_1$ be a projective compactification, such that $Y_1$ is smooth and such that $D = Y_1\setminus Y$ is a divisor of normal crossings. Then $X$ is said to be of logarithmic general type if for some positive integer $m$, the rational map defined by the complete linear system $|m(K_{Y_1}+D)|$ is birational to the image. \end{dfn} Let $X$ be a quasi-projective variety of logarithmic general type, defined over a field $K$ which is finitely generated over ${\Bbb{Q}}$ (e.g., a number field). Let $R$ be a ring, finitely generated over ${\Bbb{Z}}$, whose fraction field is $K$ (e.g., the ring of $S$-integers in a number field). Choose a model ${\cal{X}}$ of $X$ over $R$. The following is a well-known conjecture of Lang and Vojta (\cite{vojta}, conjecture 4.4). {\bf Conjecture.} The set of integral points ${\cal{X}}(R)$ is not Zariski dense in ${\cal{X}}$. In case $X$ in the conjecture above is projective, then logarithmic general type means just general type; and integral points are just rational points. \subsection{What should the last entry in the table read?} We would like to fill in the question mark in the last entry in the table. One is tempted to ask: {\em Does the Lang - Vojta conjecture imply the uniformity of $\#E(O_{K,S})$?}\\ but one sees immediately that this cannot be true without some restrictions. Most importantly, one has to restrict the choice of the model $E$, as can be seen in the following example: Let $E$ be an elliptic curve over a number field $K$ such that $E(K) $ is infinite. Fix $P_1,\ldots ,P_n\in E(K)$. Choose an equation $$y^2 = x^3 + Ax+B$$ for $E$, where $A,B\in O_{K,S}$ for some finite $S$. Choose $c\in {{\cal O}_{K,S}}$ such that for each $P_i$ one has $c^2x(P_i),c^3 y(P_i)\in {{\cal O}_{K,S}}$. By changing coordinates $x_1 = c^2 x, y_1 = c^3y$, one obtains a new model $E_1$ given by the equation $y_1^2 = x_1^3 + c^4 A x_1 + c^6B$, on which all the points $P_i$ are integral. The problem with this new model arises because when one changes coordinates, one blows up the closed point corresponding to the origin at primes dividing $c$, so the resulting model has ``extraneous'' components over these primes. We are led to modify the statement: \begin{th}(Main theorem in terms of models, see \S\ref{stably}) Assume that the Lang-Vojta conjecture holds. Then for any number field $K$ and a finite set of places $S$ there is an integer $N$ such that for any stably minimal elliptic curve ${\cal{E}}$ over ${{\cal O}_{K,S}}$ we have $\#{\cal{E}}({{\cal O}_{K,S}})<N$. \end{th} It turns out that stably minimal models are very minimal indeed. In particular we will see that N\'eron models, the canonical models of elliptic curves over rings of integers, are not necessarily sufficiently minimal for the purpose of our methods. On the other hand we will see that semistable models are stably minimal. A precise definition of a stablly minimal model, and how to obtain a canonical one from the N\'eron model, will be given in \S\ref{stably}. In the case of semistable elliptic curves, it is worthwhile to state an immediate corollary of the theorem: \begin{cor} The Lang-Vojta conjecture implies that the number of integral points on semistable elliptic curves over ${\Bbb{Q}}$ is bounded.\end{cor} {\bf Remark.} Another conjecture of Lang (see \cite{lang1}) predicts that the number of all $S$-integral points on so called {\em quasi-minimal} elliptic curves should be bounded in terms of the rank and the number of elements in $S$. In view of the corollary, one is tempted to ask whether the rank of an elliptic curve can be bounded in terms of the places of additive reduction of the elliptic curve. \section{Boundedness of correlated points}\label{correlation} One of the main ideas in \cite{chm} is, that in order to bound the number of points on curves it is enough to show that they are {\em correlated}, that is, there is an algebraic relation between all $n$-tuples of these points. This is the content of the lemma below. First, some notation. Let $\pi:X\rightarrow B$ be a family of smooth irreducile curves over a field $K$. We denote by $\pi_n:X_B^n\rightarrow B$ the n-th fibered power of $X$ over $B$. Given a point $b\in B$ we denote by $X_b$ the fiber of $X$ over $b$; Similarly, given $Q=(P_1,\ldots,P_n)\in X^n_B$ we denote by $X_Q\subset X_B^{n+1}$ the fiber of $X_B^{n+1}$ over $Q$. Note that if $\pi_n(Q) = b$ then $X_Q \simeq X_b$. Denote by $p_n:X_n\rightarrow X_{n-1}$ the projection onto the first $n-1$ factors. Assume that we are given a subset ${\cal{P}}\subset X(K)$ (typical examples would be rational points, or integral points on some model of $X$). Again, we denote by ${\cal{P}}_B^n\subset X_B^n$ the fibered power of ${\cal{P}}$ over $B$ (namely the union of the $n$-tuples of points in ${\cal{P}}$ consisting of points in the same fiber), and by ${\cal{P}}_b$ the points of ${\cal{P}}$ lying over $b$. \begin{dfn} Assume that for some $n$ there is a proper closed subset $F_n\subset X^n_B$ such that ${\cal{P}}_B^n\subset F_n$. In such a case we say that the subset ${\cal{P}}$ is {\em $n$-correlated}. \end{dfn} For instance, a subset ${\cal{P}}$ is 1-correlated if and only if it is not Zariski dense; in which case it is easy to see that, over some open set in $B$, the number of points of ${\cal{P}}$ in each fiber is bounded. This is generalized by the following lemma: \begin{lem}(compare \cite{chm}, lemma 1.1) Let $X\rightarrow B$ be a family of smooth irreducible curves, and let ${\cal{P}}\subset X(K)$ be an $n$-correlated subset. Then there is a dense open set $U\subset B$ and an integer $N$ such that for every $b\in U$, we have $\# {\cal{P}}_b\leq N$. \end{lem} {\bf Proof:} Let $F_n = \overline{{\cal{P}}_B^n}$ be the Zariski closure, and $U_n= X_B^n\setminus F_n$ the complement. We now define be descending induction: $U_{i-1} = p_i (U_i)$ and $F_{i-1}=X^{i-1}_B\setminus U_{i-1} $ the complement. Notice that over $U_{i-1}$, the map $p_i$ restricts to a finite map on $F_i$: by definition if $x\in U_{i-1}$ then $p_i^{-1}(x)\not\subset F_i$, and $p_i^{-1}(x) $ is an irreducible curve. Therefore the number of points in the fibers of this map is bounded: if $x\in U_{i-1}$ then we can write $\#(p_i^{-1}(x)\cap F )\leq d_i$. Let $U=U_0\subset B$. We claim that over $U$, the number of points of ${\cal{P}}$ in each fiber is bounded. Consider a point $b\in U$. Case 1: ${\cal{P}}_b \subset F_1$. In this case, the number of points on ${\cal{P}}_b$ is bounded by $d_1$. Case 2: there is some $P\in {\cal{P}}_b, P\not\in F_1$, but $X_P\cap{\cal{P}}_b^2 \subset F_2$. In this case the number of points is bounded by $d_2$. Case $i$: $Q=(P_1,\ldots,P_{i-1})\in {\cal{P}}_b^{i-1}\setminus F_{i-1}$ but $X_Q\cap {\cal{P}}_b^i\subset F_i$. Here the number of points is bounded by $d_i$. Notice that in the case $i=n$ we have by definition $X_Q\cap {\cal{P}}_B^n\in F_n$, and the process stops. Therefore $N=\displaystyle \max_i d_i$ is a bound for the number of ${\cal{P}}$ points in each fiber over $U$. {\bf Example} (\cite{chm}): Let $X\rightarrow B$ be a family of smooth, irreducible curves of genus $>1$ over a number field $K$. Assume that Lang's conjecture holds true. Then in \cite{chm} it is shown that $X(K)$ is $n$-correlated, and the lemma above, with noetherian induction, is used to obtain the existence of a uniform bound on the number of rational points on such curves. {\bf Example:} Assume that $X_K\rightarrow B_K$ is a semistable family of curves of genus 1, together with a section $s:B_K\rightarrow X_K$, and assume that over an open set $B_0\subset B_K$ the restricted family $X_0\rightarrow B_0$ is smooth. Assume that the Lang - Vojta conjecture holds true. Given a semistable model $X$ of $X_K\setminus s(B_K)$ over ${{\cal O}_{K,S}}$, we will later show that $X({{\cal O}_{K,S}})\cap X_0$ is $n$-correlated for some integer $n$. We will deduce the existence of a uniform bound on the number of integral points on curves in this family. {\bf Example:} Let $S$ be a finite set of places in $K$. Assuming that the Lang - Vojta conjecture holds true, we will show that the set of stably $S$-integral points on any family of elliptic curves over a number field $K$ is $n$-correlated for some $n$. We will deduce the existence of a uniform bound on the number of stably $S$-integral points on an elliptic curve. \section{Moduli of elliptic curves with level 3 structure}\label{level3} We introduce here a building block in the proof of the main theorem. \subsection{The geometry} Let $E_1$ be the universal family of elliptic curves over ${\Bbb{C}}$ with full symplectic level 3 structure. The surface $E_1$ can be identified with the total space of the elliptic pencil written in bi-homogeneous coordinates as: $$(*)\quad \lambda(X^3+Y^3+Z^3) - 3\mu XYZ=0,$$ mapping to the moduli space ${\Bbb{P}}^1$ via $[\lambda:\mu]$. This equation gives a smooth model, which by abuse of notation we will also call $E_1$, of this space over ${\mbox{Spec }}{\Bbb{Z}}[1/3]$. We may choose the section $\Theta$ over the point $[X:Y:Z]=[1:-1:0]$ as the origin of the elliptic surface. Over ${\mbox{Spec }} {\Bbb{Z}}[1/3]$, the fibers of the elliptic pencil possess level 3 structure of type $\mu_3 \times {\Bbb{Z}}/3{\Bbb{Z}}$, in a way which is described precisely by Rubin and Silverberg in \cite{rs}; however in this section we will work over ${\Bbb{C}}$. The pencil $E_1\rightarrow {\Bbb{P}}^1$ is semistable, possessing four singular fibers, having 3 nodes each, over $\Sigma_0=\{0,1,\zeta_3,\zeta_3^2\}\subset {\Bbb{P}}^1$ where $\zeta_3$ is a primitive third root of 1. Let $L$ be the pullback of a line from the plane, and let $S_1,\ldots,S_9$ be the exceptional curves over the nine base points of the pencil fixing $S_1=\Theta$ to be the origin of the elliptic surface. Let $F$ be a fiber of the elliptic surface. We have the linear equivalence $-F\sim -3L +S_1+\cdots+S_9$. As a pencil of cubics with smooth total space, one easily calculates the relative dualizing sheaf, as follows: We know that $\omega_{{\Bbb{P}}^2} = {\cal{O}}_{{\Bbb{P}}^2}(-3)$. The canonical sheaf of the blown up surface is therefore ${\cal{O}}( -3L +S_1+\cdots+S_9)$. Therefore we have $\omega_{E_1} \simeq {\cal{O}}(-F)$, and $\omega_{E_1/{\Bbb{P}}^1} \simeq {\cal{O}}(F)$. Let $\pi_n:E_n\rightarrow {\Bbb{P}}^1$ be the $n$-th fibered power of $E_1$ over ${\Bbb{P}}^1$. Denote by $\pi_{n,i}:E_n\rightarrow E_1$ the projection onto the $i$-th factor. We have that $\omega_{E_n/{\Bbb{P}}^1} \simeq {\cal{O}}(nF)$, and therefore $\omega_{E_n} \simeq {\cal{O}}((n-2)F)$. We denote by $\Theta_n=\sum_{i=1}^n\pi_{n,i}^*\Theta$, the theta divisor. We denote by $\Sigma_n=\pi_n^{-1}\Sigma_0\subset E_n$ the locus of singular fibers, the inverse image of $\Sigma_0\subset {\Bbb{P}}^1$. It should be noted that $E_n$ is singular, but not too singular: \begin{lem} There is a desingularization $f_n:\tilde{E_n}\rightarrow{E_n}$ such that $\omega_{\tilde E_n} \simeq f_n^*\omega_{ E_n}(D)$, for some effective divisor $D$ such that $f_n(D)\subset \Sigma_n$, and such that $f_n^*\Theta_n$ is a reduced divisor of normal crossings. \end{lem} {\bf Proof:} The existence of a desingularization with $\omega_{\tilde E_n} \simeq f_n^*\omega_{ E_n}(D)$ follows from \cite{chm}, lemma 3.3, or lemma 3.6 of \cite{viehweg}. The desingularization is given by a succession of blowups along smooth centers. Since the singular locus of $E_n$ meets $\Theta_n$ transversally, the centers of the blowups can be taken to be transversal to $\Theta_n$, and therefore its inverse image is a divisor of normal crossings. This lemma shows that $E_n\setminus\Theta_n$ has log canonical singularities. This means that sections of powers of $\omega_{E_n}(\Theta_n)$ give regular sections of the logarithmic pluricanonical sheaves of $\tilde{E_n}$; therefore, in order to prove that $E_n$ is of logarithmic general type, there is no need to pass to a resolution of singularities - it suffices to show that $\omega^k_{E_n}(k\Theta_n)$ has many sections. \begin{lem} \begin{enumerate} \item The line bundle $\omega_{E_1/{\Bbb{P}}^1}(\Theta_1)$ is the pullback of an ample bundle along a birational morphism. \item Fix $n>2$. Then ${E_n}\setminus\Theta_n$ is of logarithmic general type. Moreover, the base locus of the logarithmic pluricanonical linear series is contained in $\Theta_n$. \end{enumerate} \end{lem} {\bf Proof: } Let $Y$ be the blowup of ${\Bbb{P}}^2$ at all the base points of our pencil except $[1,-1,0]$. The surface $Y$ is the same as $E_1$ blown down along $\Theta$. The line bundle $\omega_{E_1/{\Bbb{P}}^1}(\Theta_1)$ is the pullback of $M={\cal{O}}(3)\otimes{\cal{O}}(-(S_2+\cdots+S_9))$ from $Y$. On $Y$, $M$ is represented by the strict transform of one of the cubics of the pencil, hence it is a nef line bundle; it has self intersection number 1, therefore it is nef and big. In fact, it is easy to see by a dimension count that the complete linear system of sections of $M^{\otimes 3}$ gives a birational morphism of $Y$ to a surface in projective space, which blows down only the fibral components of $E_1$ which do not meet $S_1$. Part (2) follows by taking the products of sections pulled back along the projections: On $E_3$, let $p_{ij},p_k$ be the projections to the $i$-th and $j$-th factors, respectively $k$-th factor. We have the inclusion $p_{12}^*\omega_{E_2} \otimes p_3^*\omega_{E_1/{\Bbb{P}}^1}(\Theta_1) \subset\omega_{E_3}(\Theta_3)$. The sections of a power of this subsheaf give a map which generically separates between points whose third factors are different. Part 1 of this lemma implies that the base locus of these sections is contained in the theta divisors. By repeating this for the other two projections, we find that sections of powers of $\omega_{E_3}(\Theta_3)$ generically separate points; in particular $E_3\setminus\Theta_3$ is of logarithmic general type. Similarly, $E_n$ is of logarithmic general type for $n\geq 3$. \subsection{Boundedness of integral points on elliptic curves with level 3 structure} Let $K$ be a number field containing ${\Bbb{Q}}(\zeta_3)$, and let $E$ be an elliptic curve with full symplectic level 3 structure over $K$. Let $R={\cal O}_K[1/3]$. The curve $E$ occurs as a fiber in the surface over a point in ${\Bbb{P}}^1(K)$, which automatically has semistable reduction over $R$. Let ${\cal{E}}$ be the semistable model. Given any three $R$-integral points $P_i$ on ${\cal{E}}\setminus 0$, the point $(P_{1}, P_{2}, P_{3})$ gives rise to an $R$-integral point of the scheme $E_3\setminus\Theta_3$ (where by abuse of notation, we use the model of $E_3$ over $R$ which is the fibered cube of the given model $(*)$ of $E_1$). Assume that the Lang - Vojta conjecture holds for the variety $E_3\setminus \Theta_3$. Thus the Zariski closure $F$ of the set of integral points $(E_3\setminus\Theta_3)(R)$ is a proper subvariety of $E_3$; in other words, the set ${\cal{P}}=(E\setminus \Theta)(R)$ is 3-correlated. By lemma 1, there is a dense open set $U \in {\Bbb{P}}^1$ such that the number of integral points of fibers over $U(K)$ is bounded. The complement of $U$ is a finite number of points, therefore by Siegel's theorem there is a bound on the number of integral points on these curves as well. {\bf Open problem:} Show that the $S$-integral points on $E_3\setminus \Theta_3$ are not Zariski dense. \section{Quadratic twists of an elliptic curve}\label{twist} As a ``complementary case'' to the last section we will discuss here a typical case of isotrivial families of elliptic curves. This is in direct analogy with the exposition in \cite{chm}, \S\S 2.2. It will give us a good hint about the type of models of elliptic curves we need in order to obtain boundedness. A slightly more general version of the example here will be used in the proof of the main theorem. Let $E: y^2 = x^3 + Ax + B$ be a fixed elliptic curve. We denote $f(x) = x^3 + Ax + B$. We assume that $A$ and $B$ are relatively prime $S$-integers in a number field $L$, where $S$ is a finite set of places. All the quadratic twists of the curve over $L$ can be written in the form: $$E_t: ty^2 = f(x)$$ where $t$ may be chosen $S$-integral. We may form the family of Kummer surfaces associated to $E_t$: $$K_t: t^2 z^2 = f(x_1) f(x_2). $$ We have a morphism of $K_t$ to $K_1$ via $(x_1,x_2,z,t) \mapsto (x_1,x_2,tz)$. It can be easily verified that the affine surface $K_1$ is of logarithmic general type. Assume that the Lang - Vojta conjecture holds for $K_1$. It now follows from lemma 1, that there is a uniform bound on the number of integral points on $E_t$: the integral points on $K_1$ are not Zariski dense, therefore the integral points on $K_t$ are not Zariski dense, since they map to integral points on $K_1$. Lemma 1 says that there is an open set $U\subset {\Bbb A}_1$ such that there is a bound on the number of points on $E_t$ for an $S$-integer $t$ in $U$; for the remaining finitely many integers $t$ we can use Siegel's theorem. The same result can be obtained using any of the higher Kummer varieties $(E\times\cdots\times E)/(\pm 1)$. Note that the integral points on $E_t$ are not the same as the integral points on the N\'eron model. Suppose that $t$ is square free. Then a N\'eron integral point $P$ is integral on $E_t$, away from characteristic 2 and 3, if at a prime of additive reduction $P$ does not reduce to the component of the origin on the N\'eron model. In other words, even after semistable reduction (obtained by taking $y' = \sqrt{t} y$), $P$ remains integral on the N\'eron model. We call such points {\em stably integral}. {\bf Open problem:} Show that the $S$-integral points on $K_1$ are not Zariski dense. As a first step, describe the images of nontrivial morphisms ${\Bbb A}_1\setminus 0 \rightarrow K_1$. \section{Stably integral points}\label{stably} \begin{dfn} Let $E$ be an elliptic curve over a number field $K$, let $S$ be a finite set of places in $K$, and $P$ be a $K$-rational point on $E$. We say that $P$ is {\em stably $S$-integral}, written $P\in E(K,S)$ if the following holds: let $L$ be a finite extension of $K$, and let $T$ be the set of places above $S$, and assume that $E$ has semistable reduction ${\cal{E}}$ over ${\cal{O}}_{L,T}$; then $P\in ({\cal{E}}\setminus 0)({\cal{O}}_{L,T})$. In other words, $P$ is integral on the semistable model of $E\setminus 0$ over some finite field extension $L$ of $K$, where $T$ is the set of all places over $S$. \end{dfn} Stably integral points should be thought of as the rational points which are integral over the algebraic closure of the field. In this sense, they are a good analogue on elliptic curves, for rational points on curves of higher genus. It is important to note that stably integral points can be described as the integral points on a certain type of model of the curve. \begin{dfn} Let $E$ be an elliptic curve over a number field $K$, let $S$ be a set of places containing all places dividing 2 and 3, and let ${\cal{E}}$ be the N\'eron model over ${{\cal O}_{K,S}}$. Let $D_0$ be the zero section of ${\cal{E}}$. Let $S_a$ be the set of places of additive reduction, and for a place $v$ let ${\cal{E}}^0_v$ be the zero component. Let $D= D_0\cup \bigcup_{v\in S_a} {\cal{E}}^0_v$ and let ${\cal{E}}_0 = {\cal{E}}\setminus D$. We call ${\cal{E}}_0$ the {\em Stably minimal model} of $E$. \end{dfn} \begin{prp} Let $S$ be a set of places containing all places dividing 2 and 3. Then the $S$-integral points on the stably minimal model are precisely the stably $S$-integral points. \end{prp} {\bf Proof:} One can prove this proposition using the explicit list of possible reduction of the N\'eron model and their semistable reduction (Tate's algorithm). If one goes through this list, one sees that the kernel of the semistable reduction map away from characteristic 2 and 3 is precisely the additive components of the identity on the N\'eron model. A much more appealing proof follows directly from \cite{edix}, section 5 (especially remark 5.4.1): assume given a field extension $L\supset K$ which is tamely and totally ramified at a given prime $p$ (this can be assumed for a local field of semistable reduction of an elliptic curves once one avoids the primes dividing 2 and 3). Let ${\cal{E}}_K, {\cal{E}}_L$ be the N\'eron models of $E$ over $K$ and $L$ respectively. In \cite{edix} one obtains a description of the map induced on N\'eron models ${\cal{E}}_K\times{\mbox{Spec }}{\cal O}_L \rightarrow {\cal{E}}_L$, and one there sees that the group of components of the reduction $({\cal{E}}_K)_p$ maps isomorphically to the group of components of the fixed locus under the Galois action of $({\cal{E}}_L)_p$. Therefore the kernel of the map of N\'eron models is connected. {\bf Remark:} in order to include primes over 2 and 3 one simply needs to remove all the additive components which are in the kernel of the semistable reduction map. Since we are allowed to remove a finite number of places anyway, we can ignore this problem altogether. \section{The main theorem}\label{main} {\bf Proof of the main theorem:} We may assume that $S$ contains all places above 2 and 3, and that $K$ contains $\zeta_3$. Let $X \rightarrow U$ be the family of all elliptic curves given by the equation $$y^2 = x^3 + ax + b,$$ over an open set $U$ in $\Bbb{A}^2$, with parameters $a,b$. Let $B_0$ be any irreducible closed subset in $U$ and $B$ a compactification of $B_0$. We can add level 3 structure to produce a semistable family $X_1\rightarrow B_1$, over a Galois, generically finite cover $B_1$ of $B$. The Galois group of the cover is some $G_1\subset Sl_2({\Bbb{F}}_3) =G$. We have a natural map $X_1\rightarrow E_1$, coming from the moduli interpretation of $E_1$, which is $G_1$ equivariant. We have that $X^n_{B_0}$ maps to $((X_1)^n_{B_1})/G_1,$ which maps down to $E_n/G$. Since the family $E_1$ is semistable, an $n$-tuple of stably integral points on an elliptic curve gives rise to an integral point on $(E_n\setminus\Theta_n)/G$. Note that if $X_{B_0}\rightarrow B_0$ is not isotrivial, then $X_{B_0}^n$ dominates $E_n/G$. Otherwise, its image is isomorphic to $(E\setminus 0)^n/Aut\,E$ for some fixed elliptic curve $E$, where $Aut\, E$ acts diagonally. The following lemmas show that $(E_n\setminus\Theta_n)/G$ is of logarithmic general type for large $n$, and that for any fixed elliptic curve $E$, $(E\setminus 0)^n/Aut\,E$ is of logarithmic general type. Assuming the Lang-Vojta conjecture, the integral points on the variety $((X_1)^n_{B_1}\setminus \Theta)/G_1$ are not Zariski dense. By lemma 1, we obtain a uniform bound on the number of stably integral points on all elliptic curves away from a closed subset $B'$ of $B$. By Noetherian induction, we have a bound on all elliptic curves. This gives the theorem. \begin{lem}\label{flem}(Compare \cite{chm}, lemma 4.1) Let $X_0\subset X$ be an open inclusion of an irreducible variety $X_0$ in a smooth complex projective irreducible $X$ of dimension $n$, such that the complement $D = X \setminus X_0$ is a divisor of normal crossings. Let $G$ be a finite group acting on $ X,X_0,D$ compatibly. Let $\omega$ be a $G$- equivariant logarithmic $k-$canonical form on $X_0$. If at any point $x$ of $X_0$ which is fixed by some element in $G$, the form $\omega$ vanishes to order at least $C=k(|G|-1)$, then $\omega$ descends to a regular logarithmic $k$-canonical form on any desingularization of $X_0/G$ \end{lem} {\bf Proof:} let $Y_0$ be a desingularization of $X_0/G$ and let $Y$ be a regular compactification, mapping to $X/G$. Let $Z'$ be the graph of the rational map $X\rightarrow Y$. Let $Z$ be a $G$-equivariant desingularization of $Z'$, and $Z_0$ the inverse image of $X_0$. Let $S$ be the branch locus of $Z$ over $Y$. By a theorem of Hironaka, such desingularizations may be chosen such that $(Z\setminus Z_0)\cup S$ is a divisor of normal crossings. Let $F_1$ be the closed set in $Y$ where the fibers in $Z$ are positive dimensional. Let $D_Z$ be the inverse image of $D$ in $Z$, $D_Y$ its image in $Y$. Let $F_2\subset Y$ be the singular locus of $D_Y\cup S$. Clearly $F=F_1\cup F_2$ is of codimension at least 2 in $Y$. Note that away from $F_1$ the branch locus $S$ is of codimension 1, since $Y$ is smooth. Clearly $\omega$ descends to a logarithmic form on $Y\setminus S$. It is enough to show that it extends over $Y\setminus F$, since $F$ has codimension at least 2. Given a point $y\in S\setminus F$ let $z\in Z$ be a point mapping to it. We can choose formal coordinates $(z_1,z_2,\ldots,z_n)$ on $Z$ such that $(y_1,z_2,\ldots,z_n)$ are coordinates on $Y$, with $y_1 = z_1^m$. Since we removed the intersections of components of $D\cup S$, there are only two cases to consider: Case 1: $z\not\in D_Z$. We can write $\omega = f(z_1\ldots,z_m)z_1^C (dz_1\wedge\cdots\wedge dz_m)^k$. We have $dy_1 = m z_1^{m-1} dz_1$. Since $m<|G|$, we have that $\omega = f z_1^{C-k(m-1)} (dy_1\wedge dz_2\wedge\cdots\wedge dz_m)^k$, is regular, and since it is invariant it descends. Case 2: $z\in D_Z$ and $z_1=0$ is the equation of $D_Z$. We can write $\omega = f(z_1\ldots,z_m) (dz_1\wedge\cdots\wedge dz_m)^k/z_1^k$. Since $m dz_1/z_1 = dy_1/y_1,$ the invariance of $\omega$ means that $f$ descends to $Y$, and therefore $\omega$ descends. \begin{lem}\label{flem1}(Compare \cite{chm}, theorem 1.3) \begin{enumerate} \item There exists a positive integer $n$ such that $(E_n\setminus \Theta)/G$ ($G$ acting diagonally) is of logarithmic general type. \item For a fixed elliptic curve $E$, there is $n$ so that $(E\setminus 0)_n/Aut\,E$ ($Aut\,E$ acting diagonally) is of logarithmic general type. \end{enumerate} \end{lem} {\bf Proof:} Let $S\subset E_1$ be any divisor containing the locus of fixed points of elements of $G$, and let $F$ be a fiber. Then the fixed points in $E_n$ are contained in $S^n_{{\Bbb{P}}^1}$, the fibered product of $S$ with itself $n$ times. Recall that we have shown that $L=\omega_{E/{\Bbb{P}}^1}(\Theta) $ is big; therefore for some large $k$, the ${\Bbb{Q}}$-line bundle $L(-(S+2F)/k)$ is big. This means that for $n=k|G|$, on $E_n$ there are many sections of $\omega^m_{E_n}$ vanishing to order $m|G|$ on the fixed points in $E_n$. As in \cite{chm}, lemma 2.1, it follows that there are also many {\em invariant} sections vanishing to such order. The proof of part (2) is identical. \section{The uniform boundedness conjecture}\label{ubc} It is well known that torsion points of high order on an elliptic curve are integral; we will use this to study torsion points in terms of integral points. As quoted in the introduction, a long standing conjecture which was recently proved by Merel says that the order of a torsion point on an elliptic curve over a number field is bounded in terms of the degree of the field of definition only. We now indicate how Merel's theorem follows from the Lang - Vojta conjecture. We start with the basic proposition which makes things work (see the case of elliptic curves in \cite{oester}): \begin{prp} Let $P$ be a torsion point on an abelian variety $A$, both defined over a field $K$ of degree $d$. Denote by $n$ the order of $P$. Let $g$ be the dimension of $A$ and let $C$ be the order of the group $Sp_g(Z/5Z)$. Assume that either $n$ is not a prime power, or $p^k = n$, such that $p^k-p^{k-1} > Cd$. Then $P$ is stably integral on $A$, that is, its reduction at any place on the N\'eron model after semistable reduction is not the origin. \end{prp} {\bf Proof:} by adding level 3 structure we have (by a theorem of Raynaud) that there is a field of degree at most $Cd$ where $A$ has semistable reductions over all $p\neq 3$. Theorem IV.6.1 in \cite{silv} says that if $P$ is not integral, then it is not integral at a place $\frak{p}$ above some prime $p$ where $n=p^k$; and the valuation satisfies $v(p)>p^k-p^{k-1}$. Here by definition, $p=u\pi^{v(p)}$, where $u$ is a unit and $\pi$ a uniformizer of the valuation ring. But $v(p)$ is at most the degree of the field. We can similarly deal with primes over 3 by adding level 5 structure instead. \qed This following corollary is probably well known: torsion on abelian varieties is bounded in terms of the degree of the field, the dimension and a prime of potentially good reduction. \begin{cor} For any triple $(d,g,p)$ there is an (explicit) integer $N$ such that if $K$ is a number field of degree $d$, $A$ an abelian variety of dimension $g$ over $K$, and $p$ is a rational prime over which there is a place $\frak{p}$ of $K$ where $A$ has potentially good reduction, then $A(K)_{tors}<N$. \end{cor} {\bf Proof:} Let $L$ be a field of degree $\leq Cd$ over which $A$ has good reduction at some prime $\frak{p}$ over $p$. Let $A(L)_{tors}\rightarrow A_{\frak{p}}$ be the reduction map. By Weil's theorem, the image has cardinality $\leq (1+p^{Cd/2})^{2g}$. But by the proposition, any point in the kernel is of order $p^k$ satisfying $p^k-p^{k-1} < Cd$, which can be bounded as well.\qed We would like to apply the correlation method to torsion points of high order on elliptic curves, defined over all fields of degree $d$. By the proposition, we may use the fact that when $p$ is large these points are stably integral. Since we want to show that there is a bound for torsion over number fields depending only on the degree, we might as well assume that $E$ has level 3 structure: this has the effect of increasing the degree $d$ by a factor of 24. In \cite{kama}, Kamienny and Mazur show that it is enough to bound the order of prime torsion points. We will show the existence of a bound on prime order torsion points, assuming the Lang - Vojta conjecture. Let $P$ be a torsion point of large prime order $p$ on an elliptic curve $E$ which has level 3 structure, defined over some number field of degree $d$. The point $P$ gives a point on the surface $E_1$ introduced in \S\ref{level3}, defined over the same number field, and is in fact integral on $E_1\setminus \Theta$. The Galois orbit of $P$ gives a ${\Bbb{Q}}$-rational point on the $d-$th symmetric power of $E_1$. We can do a bit better: fix an integer $n$. Given $n$ torsion points on an elliptic curve defined over the same number field (e.g. multiples of a given torsion point) we in fact get a rational point on $Y_n={\mbox{Sym}}^d(E_n)$. In $Y_n$ there is a divisor $\Theta_{Y_n}$ which consists of those tuples of points such that at least one point is the origin, and the points thus obtained are in fact integral on the scheme $Y_n\setminus \Theta_{Y_n}$. Given an auxiliary integer $k$, let $F_{n,k}$ be the Zariski closure in $Y_n$ of the set of all points corresponding to Galois orbits of $n$-tuples of distinct torsion points of prime order {\em larger than $k$} defined over fields of degree exactly $d$. By definition, if $l>k$ then $F_{n,l}$ is contained in $F_{n,k}$. Let $F_n$ be the intersection of $F_{n,k}$ over all integers $k$. By the noetherian property of algebraic varieties, $F_n = F_{n,k}$ for some $k$. What we want to show is that $F_n$ is empty. We will assume the contrary and derive a contradiction. We have the natural symmetrization map $(E_n)^d \rightarrow Y_n$. Let $G_n$ be the inverse image of $F_n$ under this map. We denote by $\pi^d_i:(E_n)^d \rightarrow (E_1)^d$ the map induced from $\pi_i:E_n\rightarrow E_1$. The varieties $(E_n)^d$ can be viewed as compactified semiabelian schemes over the space ${\Bbb{P}}=({\Bbb{P}}^1)^d$. \begin{lem} Let $G$ be a component of $G_n$. There exists a closed subscheme $B\subset {\Bbb{P}}$, and subvarieties $A_i \subset E_1^d, 0< i<n+1$ mapping onto $B$, such that the general fiber of $A_i$ over $B$ is a finite union of abelian subvarieties, and $G$ is a component of the fibered product of $A_i$ over $B$. The varieties $A_i$ are not contained in any diagonal in $E_1^d$ or in the theta divisor, nor in the locus of singular fibers. \end{lem} Proof: if $P$ is a torsion point of some prime order $p$ defined over a number field, then any multiple of it $kP$, for $k$ prime to $p$, is also torsion defined over the same field. Fix an integer $1\leq i\leq n$. We look at the projection of $q_i:G\rightarrow E_{n-1}^d$ forgetting the $i$-th factor. It follows that each fiber of $G_n$ over $E_{n-1}^d$ is stable under multiplication by $k$ for any integer $k$. We now use the trick of Neeman and Hindry (see \cite{neeman} or \cite{hindry}), which tells us that a subvariety of an abelian variety which is stable under multiplication by all integers, is a union of abelian subvarieties. Let $G'\subset G_{n-1}$ be the image of $G$ under $q_i$. For each point $P\in E_{n-1}^d$ in $G'$ we have that $q_i^{-1}(P)\cap F_n$ is a union of finitely many abelian subvarieties of $E_1^d$. Since a subvariety of a constant abelian scheme is constant (say, by looking at torsion points), these abelian subvarieties depend only on the image of $P$ in ${\Bbb{P}}$. Therefore there is a subvariety $A_i$ as in the lemma such that $G$ is a component of $(\pi^d_i)^{-1}A_i\cap q_i^{-1}G'$. By induction we obtain the product structure. Since the variety $G$ was obtained from the closure of Galois orbits of points over fields of degree exactly $d$, none of them is fixed by any permutation, and none is in the theta divisor. Similarly, we see that they are not contained in the singular fibers of $E_n^d$. \qed We will now show that for high enough $n$, any candidate for a component of $F$ is of logarithmic general type. First note that, by noetherian induction, one may assume that the base $B$ of $A_i$ remains constant as $n$ grows. We will now see that if one of the $A_i$ appears many times in the product, then the image $F$ of $G$ is of logarithmic general type. \begin{lem} Let $B\subset {\Bbb{P}}$ be an irreducible closed subvariety. Let $A_i\subset (E_1)^d,\quad 1\leq i\leq m$ and $A_{m+1}$ be subschemes mapping to $B$ satisfying the conclusions in the previous lemma. There is an integer $k_0$ such that for any $k>k_0$ and any $l_i\geq 0$ the following holds: Let $G$ be a component of the scheme $$(A_1)^{l_1}_{B}\times_B\cdots\times_B (A_m)^{l_m}_B\times_B (A_{m+1})^k_B.$$ Let $F$ be the image of $G$ in $Y_n$, where $n=l_1+\cdots +l_n+k$. Let $F' = F\setminus\Theta_{Y_n}$. Then $F'$ is of logarithmic general type. \end{lem} {\bf Proof:} Let $M_i$ be the dimension of the fibers of $A_i$ over $B$. For each choice of a subset $J_i$ of $\{1,\ldots,d\}$ of size $M_i$, we have a variety $E_{J_i,B}$, the pullback of $(E_1)^{M_i}$ to $B$ along the projection $\pi_{J_i}$ to the factors in $J_i$. Since $A_i$ is not contained in the theta divisor of $E_1^d$, we have that $A_i$ surjects generically finitely onto $E_{J_i,B}$, whenever $J_i$ has size $M_i$. We treat $A_{m+1}$ a bit differently: using $d$ different generically finite surjections $A_{m+1}\rightarrow E_{J'_i,B}$ where $J'_i=\{i,\ldots,(i+M_{m+1} \mod d)\}\subset \{1,\ldots,d\}$, we can cook up a special generically finite surjection: write $k=qd+r$, then we map map $(A_{m+1})^k_B\rightarrow (E_{J_{m+1},B})^r_B \times_B (E_{qm})^d|_B $. In order to deal with the singularities, we desingularize the base: $B'\rightarrow B$. Now the pullback of the product of $E_{J_i,B}$ to $B'$ has semistable fibers, therefore has log canonical singularities as in lemma 2. Choose a canonical divisor $K_{B'}$. Choose an effective divisor $H\subset B$ such that the pull-back of $H$ to $B'$ is bigger than $-K_{B'}$. If $G'$ is a desingularization of $G$, it admits a generically finite surjection $$p:G\rightarrow V= (E_{J_1,B})^{l_1}_B \times_B\cdots\times_B (E_{J_m,B})^{l_m}_B\times_B (E_{J_{m+1},B})^r_B \times (E_{qm})^d|_B.$$ Notice that $V$ is the restriction to $B$ of a variety of the form $E_{r_1}\times \cdots \times E_{r_d}$, and if $k$ is large then {\em each of the $r_i$} is large as well. We can choose $G'$ so that it maps to $B'$. Let $V'$ be the pullback of $V$ to $B'$. We wish to use the sections of powers of the logarithmic relative dualizing sheaf of the product variety $V$ to construct differential forms on $F$. In view of lemmas \ref{flem},\ref{flem1} we need the sections to vanish sufficiently along the preimage of $H$, and their pullback to $G$ should vanish along the fixed points $\Delta$ of the symmetric group action to sufficiently high order. Since the relative dualizing sheaf $\omega_{E_1/{\Bbb{P}}^1}(\Theta)$ is nef and big, then the sheaf $\omega_{E_{J,B}/B}(\Theta)$ is nef, and the sheaf $\omega_{(E_{1})^d|_B/B}(\Theta)$ is nef and big. We have an injection $p^*(\omega^m_{V'/B'})(-mH+m\Theta) \rightarrow \omega^m_{G'}(m\Theta)$, since $V'\setminus \Theta$ has log canonical singularities. Since each of the $r_i$ in the description of $V$ can be made as large as we wish, the argument of lemma \ref{flem1} shows that $F$ is of logarithmic general type. \qed We can now show by induction on $M$ that the relative dimension of $G_{n+1}$ over $G_{n}$ is at least $M+1$, thus obtaining a contradiction. Clearly the relative dimension is at least 1. If for some $n$ the relative dimension is precisely $M$, then by induction, using the embedding $G_{n+k}\subset (G_{n+1})^k_{G_n}$, the relative dimension of $G_{n+k}$ over $G_n$ is $Mk$, and therefore there is a component $G$ of $G_{n+k}$ of relative dimension $Mk$. From lemma 6 it follows that $G$ is a component of a product variety of the form described in lemma 7, and therefore $F\setminus \Theta$ is of logarithmic general type. The Lang - Vojta conjecture implies that the integral points on $F$ are not dense, contradicting the definition of $F$. We arrived at a contradiction, therefore $F_n$ must be empty, and we conclude that there is a bound for torsion points of prime order. \qed

9505

alg-geom/9505023

en

https://arxiv.org/abs/alg-geom/9505023

alg-geom/9505023

Rahul Pandharipande

R. Pandharipande

A Note On Elliptic Plane Curves With Fixed j-Invariant

10 pages, AMSLatex

Let N_d be the number of degree d, nodal, rational plane curves through 3d-1 points in the complex projective plane. The number of degree d>=3, nodal, elliptic plane curves with a fixed (general) j-invariant through 3d-1 points is found to be {d-1 \choose 2}*N_d.

alg-geom

\section{Summary} Let $N_d$ be the number of irreducible, reduced, nodal, degree $d$ {\em rational plane curves} passing through $3d-1$ general points in the complex projective plane $\bold P^2$. The numbers $N_d$ satisfy a beautiful recursion relation ([K-M], [R-T]): $$N_1=1$$ $$\forall d>1, \ \ \ N_d= \sum_{i+j=d, \ i,j>0} N_iN_j \bigg( i^2j^2 {3d-4 \choose 3i-2} - i^3j {3d-4 \choose 3i-1} \bigg).$$ Let $E_{d,j}$ be the number of irreducible, reduced, nodal, degree $d$, {\em elliptic plane curves with fixed j-invariant j} passing through $3d-1$ general points $\bold P^2$. $E_{d,j}$ is defined for $d\geq 3$ and $\infty \neq j \in \overline{M}_{1,1}$. In this note, the following relations are established: \begin{eqnarray*} \forall j\neq 0,1728,\infty, & E_{d,j}= {d-1\choose 2}N_d, \\ j=0, & E_{d,0}= {1\over 3} {d-1 \choose 2} N_d, \\ j=1728, & \ E_{d,1728}= {1\over 2} {d-1\choose 2} N_d. \end{eqnarray*} If $d\equiv 0 \ mod \ 3$, then $ 3 \not\mid {d-1 \choose 2}$. Since $E_{3\hat{d},0}$ is an integer, $N_{3\hat{d}} \equiv 0 \ mod \ 3$ for $\hat{d}\geq 1$. In fact, a check of values in [D-I] shows $N_d \equiv 0 \ mod \ 3$ if and only if $d \equiv 0 \ mod \ 3$ for $3\leq d \leq 12$. P. Aluffi has calculated $E_{3,j}$ for $j<\infty$ in [A]. Aluffi's results agree with the above formulas. Thanks are due to Y. Ruan for discussions on Gromov-Witten invariants and quantum cohomology. The question of determining the numbers $E_{d,j}$ was first considered by the author in a conversation with Y. Ruan. \section{Kontsevich's Space of Stable Maps} \subsection{ The Quasi-Projective Subvarieties $U_C(\Gamma,c,\overline{w})$ , $U_{j=\infty}(\Lambda,\overline{w})$} \label{qpd} Fix $d\geq 3$ for the entire paper. Let $C$ be a nonsingular elliptic curve or an irreducible, nodal rational curve of arithmetic genus 1. Consider the coarse moduli space of $3d-1$-pointed stable maps from $C$ to $\bold P^2$ of degree $d\geq 3$, $\overline{M}_{C, 3d-1}(\bold P^2, d)$. For convenience, the notation $\barr{M}_C(d) = \overline{M}_{C,3d-1}(\bold P^2,d)$ will be used. Let $S_d= \{1,2,\ldots,3d-1\}$ be the marking set. Constructions of $\barr{M}_C(d)$ can be found in [Al], [K], [P]. Let $\Gamma$ be a tree consisting of a distinguished vertex $c$, $k\geq 0$ {\em other} vertices $v_1, \ldots, v_k$, and $3d-1$ marked legs. Let $0\leq e \leq d$. Weight the vertex $c$ by $e$. Let $w_1, \ldots, w_k$ be non-negative integral weights of the vertices $v_1, \ldots, v_k$ satisfying $$e+ w_1 + \cdots + w_k =d.$$ Denote the weighting by $\overline{w}=(e,w_1,\ldots, w_k)$. The marked, weighted tree with distinguished vertex $(\Gamma, c, \overline{w})$ is {stable} if the following implication holds for all $1\leq i\leq k$: $$w_i=0 \ \ \Rightarrow \ \ valence(v_i)\geq 3.$$ Two marked, weighted trees with distinguished vertex $(\Gamma, c,\overline{w})$ and $(\Gamma', c',\overline{w}')$ are isomorphic if there is an isomorphism of marked trees $\Gamma\rightarrow \Gamma'$ sending $c$ to $c'$ and respecting the weights. A quasi-projective subvariety $U_C(\Gamma,c, \overline{w})$ of $\barr{M}_C(d)$ is associated to each isomorphism class of stable, marked, weighted graph with distinguished vertex $(\Gamma, c,\overline{w})$. $U_C(\Gamma,c,\overline{w})$ consists of stable maps $\mu: (D, p_1,\ldots, p_{3d-1}) \rightarrow \bold P^2$ satisfying the following conditions. The domain $D$ is equal to a union: $$D=C \cup \bold P^1_1 \cup \cdots \cup \bold P^1_k.$$ The marked, weighted dual graph with distinguished vertex of the map $\mu$ is isomorphic to $(\Gamma, c, \overline{w})$. The distinguished vertex of the dual graph of $\mu$ corresponds to the (unique) component of $D$ isomorphic to $C$. Weights of the dual graph of $\mu$ are obtained by the degree of $\mu$ on the components. Note $U_C(\Gamma,c,\overline{w})=\emptyset$ if and only if $e=1$. Let $(\Gamma,c,\overline{w})$ be a stable, marked, weighted tree with distinguished vertex. Assume $e\neq 1$. The dimension of $U_{C}(\Gamma,c,\overline{w})$ is determined as follows. If $e\geq 2$, then $$dim \ U_{C}(\Gamma,c,\overline{w})= 6d-2-k.$$ If $e=0$, then $$dim \ U_{C}(\Gamma,c,\overline{w})= 6d-k$$ (where $k$ is the number of non-distinguished vertices of $\Gamma$). These calculations are straightforward. Let $C$ be a nonsingular elliptic curve. Every stable map in $\barr{M}_C(d)$ has domain obtained by attaching a finite number of marked trees to $C$. By the definition of tree and map stability: $$\bigcup_{(\Gamma,c, \overline{w})} U_C(\Gamma,c, \overline{w}) \ = \ \barr{M}_C(d).$$ Let $C$ be an irreducible, 1-nodal rational curve. The quasi-projective varieties $U_C(\Gamma,c,\overline{w})$ do not cover $\barr{M}_C(d)$. The curve $C$ can degenerate into a simple circuit of $\bold P^1$'s. Let $\Lambda$ be a graph with 1 circuit ($1^{st}$ Betti number equal to $1$, no self edges), $k\geq 1$ vertices $v_0,\ldots, v_k$, and $3d-1$ marked legs. Note the different vertex numbering convention. At least $2$ vertices are required to make a circuit, so $k\geq 1$. Let $w_0, w_1,\ldots,w_k$ be non-negative, integral weights summing to $d$. The marked, weighted graph with $1$ circuit $(\Lambda, \overline{w})$ is stable if each zero weighted vertex has valence at least 3. A quasi-projective subvariety $U_C(\Lambda, \overline{w})$ of $\barr{M}_C(d)$ is associated to each isomorphism class of stable, marked, weighted graph with $1$ circuit $(\Lambda, \overline{w})$. $U_C(\Lambda, \overline{w})$ consists of stable maps with marked, weighted dual graphs isomorphic to $(\Lambda, \overline{w})$. The union $$\bigcup_{(\Gamma,c, \overline{w})} U_C(\Gamma,c, \overline{w}) \ \ \cup \ \ \bigcup_{(\Lambda, \overline{w})} U_C(\Lambda, \overline{w})\ \ = \ \barr{M}_C(d)$$ holds by the definition of stability. Finally, the dimensions of the loci $U_C(\Lambda,\overline{w})$ will be required. Let $(\Lambda, \overline{w})$ be a stable, marked, weighted graph with $1$ circuit. Let $c_1,\ldots, c_l$ be the unique circuit of vertices of $\Lambda$. Let $e$ be the sum of the weights of the circuit vertices. $U_C(\Lambda, \overline{w})=\emptyset$ if and only if $e=1$. If $e\geq 2$, then $$dim\ U_C(\Lambda,\overline{w})= 6d-2-k.$$ If $e=0$, then $$dim\ U_C(\Lambda, \overline{w})= 6d-k$$ (where $k+1$ is the total number of vertices of $\Lambda$). Again, these results are straightforward. \subsection{The Component $\overline{W}_{1}(d)$} Let $\barr{M}_1(d) = \overline{M}_{1,3d-1}(\bold P^2,d)$ be Kontsevich's space of $3d-1$-pointed stable maps from genus $1$ curves to $\bold P^2$. There is canonical morphism $$\pi: \barr{M}_1(d) \rightarrow \overline{M}_{1,1}$$ obtained by forgetting the map and all the markings except $1\in S_d$ (the $3d-1$ possible choices of marking in $S_d$ all yield the same morphism $\pi$). Let $j\in \overline{M}_{1,1}$. By the universal properties of the moduli spaces, there is a canonical bijection $$\overline{M}_{C_j}(d) \rightarrow \pi^{-1}(j)$$ where $C_j$ is the elliptic curve (possibly nodal rational) with $j$-invariant $j$. When $j\in \overline{M}_{1,1}$ is automorphism-free, this bijection is an isomorphism. For $j=0,1728$, the scheme theoretic fiber $\pi^{-1}(j)$ is nonreduced. Define an open locus $W_{1}(d) \subset \barr{M}_1(d)$ by $[\mu: (D, p_1, \ldots, p_{3d-1}) \rightarrow \bold P^2]\in W_{1}(d)$ if and only if $D$ is irreducible. By considering the natural tautological spaces over the universal Picard variety of degree $d$ {\em line bundles} over $M_{1,3d-1}$, it is easily seen that $W_{1}(d)$ is a reduced, irreducible open set of dimension $6d-1$. Let $\overline{W}_1(d)$ be the closure of $W_{1}(d)$ in $\barr{M}_1(d)$. \section{A Deformation Result} Let $\Phi$ be a stable, marked, weighted tree with distinguished vertex $c$ determined by the data: $k=1$, $(e,w_1)=(0,d)$. There are $2^{3d-1}$ isomorphism classes of such $\Phi$ determined by the marking distribution. Let $j\in \overline{M}_{1,1}$. The dimension of $U_j(\Phi)$ is $6d-1$. A point $[\mu]\in U_j(\Phi)$ has domain $C_j \cup \bold P^1$. There are $3d-1$ dimensions of the map $\mu|_{\bold P^1}:\bold P^1\rightarrow \bold P^2$. The incidence point $p= C_j\cap \bold P^1$ moves in a $1$-dimensional family on $\bold P^1$. The remaining $3d-1$ markings move in $3d-1$ dimensions on $C_j$ and $\bold P^1$ (specified by the marking distribution). $6d-1=3d-1+1+3d-1$. A technical result is needed in the computation of the numbers $E_{d,j}$. \begin{lm} \label{aa} Let $I(\Phi,j)= \overline{W}_1(d) \cap U_j(\Phi) \subset \barr{M}_1(d)$. The dimension of $I(\Phi,j)$ is bounded by $dim\ I(\Phi,j) \leq 6d-3$. \end{lm} \begin{pf} Let $[\mu]\in I(\Phi,j)$ be a point. Let $D=C_j \cup \bold P^1$ be the domain of $\mu$ as above. The following condition will be shown to hold: the linear series on $\bold P^1$ determined by $\mu|_{\bold P^1}$ has vanishing sequence $\{0, \geq 2,*\}$ at the incidence point $p=C_j \cap \bold P^1$. The existence of a point with vanishing sequence $\{0,\geq 2,*\}$ is a $1$-dimensional condition on the linear series. The condition that the incidence point $p$ has this vanishing sequence is an additional $1$-dimensional constraint on $p$. Therefore, the dimension of $I(\Phi,j)$ is at most $6d-1-1-1=6d-3$. The vanishing sequence $\{0, \geq 2, *\}$ is equivalent to $d(\mu|_{\bold P^1})=0$ at $p$. It remains to establish the vanishing sequence $\{0,\geq 2,*\}$ at $p$. This result is easily seen in explicit holomorphic coordinates. Let $\bigtriangleup_t$ be a disk at the origin in $\Bbb{C}$ with coordinate $t$. Let $\eta: \cal{E} \rightarrow \bigtriangleup_t$ be a flat family of curves of arithmetic genus $1$ satisfying: \begin{enumerate} \item[(i.)] $\eta^{-1}(0)\stackrel{\sim}{=} C_j$. \item[(ii.)] $\eta^{-1}(t\neq 0)$ is irreducible, reduced, and (at worst) nodal. \end{enumerate} For each $1\leq i \leq d$, let $\cal{G}_i=\cal{H}_i\subset \cal{E}$ be the open subset of $\cal{E}$ on which the morphism $\eta$ is {\em smooth}. Consider the fiber product: $$X= \cal{G}_1 \times_{\bigtriangleup_t} \cdots \times_{\bigtriangleup_t} \cal{G}_d \times_{\bigtriangleup_t} \cal{H}_1 \times_{\bigtriangleup_t} \cdots \times_{\bigtriangleup_t} \cal{H}_d.$$ $X$ is a nonsingular open set of the $2d$-fold fiber product of $\cal{E}$ over $\bigtriangleup_t$. Let $Y\subset X$ be the subset of points $y=(g_1,\ldots, g_d,h_1, \ldots, h_d)$ where the two divisors $\sum g_i$ and $\sum h_i$ are linearly equivalent on the curve $\eta^{-1}(\eta(y))$. $Y$ is a nonsingular divisor in $X$. Let $p\in C_j=\eta^{-1}(0)$ be a nonsingular point of $C_j$. Certainly $p\in \cal{G}_i, \cal{H}_i$ for all $i$. Let $\gamma: \bigtriangleup_t \rightarrow \cal{E}$ be any local holomorphic section of $\eta$ such that $\gamma(0)=p$. Let $V$ be a local holomorphic field of vertical tangent vectors to $\cal{E}$ on an open set containing $p$. The section $\gamma$ and the vertical vector field $V$ together determine local holomorphic coordinates $(t,v)$ on $\cal{E}$ at $p$. Let $\phi_{V}:\cal{E}\times \Bbb{C} \rightarrow \cal{E}$ be the holomorphic flow of $V$ defined locally near $(p,0)\in \cal{E}\times \Bbb{C}$. The coordinate map $$\psi:(t,v) \rightarrow \cal{E}$$ is determined by $\psi(t,v)= \phi_V(\gamma(t),v)$. Local coordinates on $X$ near the point $x_p=(p,\ldots, p,p,\ldots, p)\in X$ are given by $$(t,v_1,\ldots, v_d, w_1, \ldots w_d).$$ The coordinate map is determined by: $$\psi_X(t,v_1,\ldots, v_d, w_1, \ldots w_d)= (\psi(t,v_1),\ldots, \psi(t,v_d), \psi(t,w_1), \ldots, \psi(t,w_d))\in X.$$ Note $x_p\in Y$. Let $f(t,v_1,\ldots, v_d, w_1, \ldots w_d)$ be a local equation of $Y$ at $x_p$. Since $f$ is identically $0$ on the line $(t,0,\ldots,0,0,\ldots,0)$, \begin{equation} \label{tvan} \forall k\geq 0, \ \ {\partial ^k f\over \partial t^k} |_{x_p} =0. \end{equation} The tangent directions in the plane $t=0$ correspond to divisors on the fixed curve $C_j$. Here, it is well know (up to $\Bbb{C}^*$- factor) \begin{equation} \label{lion} {\partial f\over \partial v_i}|_{x_p}=+1, \ \ {\partial f\over \partial w_i}|_{x_p}=-1. \end{equation} Equations (\ref{tvan}) and (\ref{lion}) are the only properties of $f$ that will be used. Let $\hat{\eta}:\cal{\hat{E}} \rightarrow \bigtriangleup_t$ be the family obtained by blowing-up $\cal{E}$ at $p$ and adding $3d-1$-marking. Let $\mu:\cal{\hat{E}}\rightarrow \bold P^2$ be a morphism. Let $\hat{\eta}^{-1}(0)=D= C_j \cup \bold P^1$. Assume the following conditions are satisfied: \begin{enumerate} \item[(i.)] $\mu$, $\hat{\eta}$, and the $3d-1$ markings determine a family of Kontsevich stable pointed maps to $\bold P^2$. \item[(ii.)] The markings of $D$ are distributed according to $\Phi$. \item[(iii.)] $deg(\mu|_{C_j})=0$, $deg(\mu|_{\bold P^1})=d$. \end{enumerate} Let $L_1$, $L_2$ be general divisors of $\mu^*({\cal{O}}_{\bold P^2}(1))$ that each intersect $\bold P^1$ transversely at $d$ distinct points. For $1\leq \alpha \leq 2$, $L_{\alpha}$ breaks into holomorphic sections $s_{\alpha,1}+\ldots+s_{\alpha,d}$ of $\hat{\eta}$ over a holomorphic disk at $0\in \bigtriangleup_t$. These sections $s_{\alpha,i}$ ($1\leq \alpha \leq 2$, $1\leq i \leq d$) determine a map $\lambda: \bigtriangleup_t \rightarrow Y$ locally at $0\in \bigtriangleup_t$. Let an affine coordinate on $\bold P^1$ be given by $\xi$ corresponding to the normal direction \begin{equation} \label{cord} {d\gamma\over dt}|_{t=0} + \xi \cdot V(p). \end{equation} Let $s_{1,i}(0)=\nu_i \in \Bbb{C} \subset \bold P^1$, $s_{2,i}(0)=\omega_i \in \Bbb{C} \subset \bold P^1$ be given in terms of the affine coordinate $\xi$. The map $\lambda$ has the form $$\lambda(t)= (t, \nu_1 t,\ldots, \nu_d t, \omega_1 t, \ldots, \omega_d t)$$ to first order in $t$ (written in the coordinates determined by $\psi_X$). Equations (\ref{tvan}), (\ref{lion}), and the condition $f(\lambda(t))=0$ implies \begin{equation} \label{cony} \sum_{i=1}^{d} \nu_i = \sum_{1}^{d} \omega_i. \end{equation} $L_1 \cap \bold P^1$ is a degree $d$ polynomial with roots at $\nu_i$. Condition (\ref{cony}) implies that the sums of the roots (in the coordinates (\ref{cord})) of general elements of the linear series $\mu|_{\bold P^1}$ are the same. Therefore, a constant $K$ exists with the following property. If $$\beta_0 + \beta_1 \xi+ \ldots + \beta_{d-1} \xi^{d-1} +\beta_{d} \xi^{d}$$ is an element of the linear series $\mu|_{\bold P^1}$, then $\beta_{d-1}+ K\cdot \beta_d=0$. The vanishing sequence at $\xi=\infty$ is therefore $\{0, \geq 2,*\}$. The point $\xi=\infty$ is the intersection $C_j \cap \bold P^1$. Suppose $\tilde{\eta}:\cal{\tilde{E}} \rightarrow \bigtriangleup_t$ is obtained from $\cal{E}$ by a sequence of $n$ blow-ups over $p$. The fiber $\tilde{\eta}^{-1}(0)$ is assumed to be $C_j$ union a chain of $\bold P^1$'s of length $n$. Each blow-up occurs in the exceptional divisor of the previous blow-up. Let $\bold P$ denote the extreme exceptional divisor. Let $\mu: \cal{\tilde{E}} \rightarrow \bold P^2$ be of degree $d$ on $\bold P$ and degree $0$ on the other components of the special fiber $\tilde{\eta}^{-1}(0)$. Let there be $3d-1$ markings as before. It must be again concluded that the linear series on $\bold P$ has vanishing sequence $\{0, \geq 2,*\}$ at the node. Let $\gamma$ be section of $\eta$ such that the lift of $\gamma$ to $\tilde{\eta}$ meets $P$. Let the coordinates $(t,v)$ on $\cal{E}$ be determined by this $\gamma$ (and any $V$). An affine coordinate $\xi$ is obtained on $\bold P$ in the follow manner. Let $\gamma_{\xi}$ be the section of $\eta$ determined in $(t,v)$ coordinates by $$\gamma_{\xi}(t)=(t,\xi t^n).$$ Let $\tilde{\gamma}_{\xi}$ be the lift of $\gamma_{\xi}$ to a section of $\tilde{\eta}$. The association $$\Bbb{C} \ni \xi \mapsto \tilde{\eta}(0)\in \bold P$$ is an affine coordinate on $\bold P$. Let $L_1, L_2$ be divisors in the linear series $\mu$ intersecting $\bold P$ transversely. As before, $L_{\alpha}$ breaks into holomorphic sections $s_{\alpha,1}$. Let $s_{1,i}=\nu_i \in \Bbb{C}\subset \bold P$, $s_{2,i}=\omega_i \in \Bbb{C} \subset \bold P$. As before, a map $\lambda:\bigtriangleup_t \rightarrow Y$ is obtained from the sections $s_{\alpha,i}$. In the coordinates determined by $\psi_X$, $$\lambda(t)=(t, \nu_1 t^n+O(t^{n+1}), \ldots, \nu_d t^n+ O(t^{n+1}), \omega_1 t^n+O(t^{n+1}), \ldots, \omega_d t^n+O(t^{n+1})).$$ As before $f(\lambda(t))=0$. The term of leading order in $t$ of $f(\lambda(t))$ is $$ (\sum_{i=1}^{d} \nu_i- \sum_{i=1}^{d}\omega_i) \cdot t^n.$$ This follows from equations (\ref{tvan}) and (\ref{lion}). The vanishing sequence $\{0,\geq 2,*\}$ is obtained as before. By definition, an element $[\mu]\in I(\Phi,j)$ can be obtained as the special fiber of family of Kontsevich stable maps where the domain is a smoothing of the node $p$. After resolving the singularity in the total space at the node $p$ by blowing-up, a family $\cal{\tilde{E}}$ is obtained. The above results show the linear series on $\bold P^1$ has vanishing sequence $\{0,\geq 2,*\}$ at $p$. \end{pf} The markings play no role in the preceding proof. An identical argument establishes the following: \begin{lm} \label{bb} Let $\Phi$ be a stable, marked, weighted tree with distinguished vertex satisfying $e=0$ and $w_i=d$ for some $i$. Let $k$ be the number of non-distinguished vertices of $\Phi$. Let $j\in \overline{M}_{1,1}$. Let $I(\Phi,j)= \overline{W}_1(d) \cap U_j(\Phi)$. The dimension of $I(\Phi,j)$ is bounded by $dim\ I(\Phi,j)\leq 6d-k-2$ \end{lm} \begin{lm} \label{cc} Let $\Omega$ be a stable, marked, weighted graph with $1$ circuit. Let $v_i$ be a non-circuit vertex with weight $w_i=d$ (this implies $e=0$). Let $k+1$ be the total number of vertices of $\Omega$. Let $I(\Omega,\infty)=\overline{W}_1(d) \cap U_{\infty}(\Omega)$. The dimension of $I(\Omega,\infty)$ is bounded by $dim \ I(\Omega,\infty)\leq 6d-k-2$. \end{lm} \noindent The vanishing sequence $\{0,\geq 2, *\}$ condition reduces the dimensions of $U_C(\Phi)$, $U_{\infty}(\Omega)$ by $2$. \section{The Numbers $E_{d,j}$} The space of maps $\barr{M}_1(d)$ is equipped with $3d-1$ evaluation maps corresponding to the marked points. For $i\in S_d$, let $e_i:\overline{W}_1(d)\rightarrow \bold P^2$ be the restriction of the $i^{th}$ evaluation map to $\overline{W}_1(d)$. let $\cal{L}_i= e_i^*({\cal{O}}_{\bold P^2})$ Let $$Z=c_1(\cal{L}_1)^2 \cap \ldots \cap c_1(\cal{L}_{3d-1})^2$$ Let $\pi_{\overline{W}}: \overline{W}_{1}(d) \rightarrow \overline{M}_{1,1}\cong \bold P^1$ be the restriction of $\pi$ to $\overline{W}_1(d)$. Let $$T=c_1(\pi_{\overline{W}}^*({\cal{O}}_{\bold P^1}(1))).$$ Note $\overline{W}_{1}(d)$ is an irreducible, projective scheme of dimension $6d-1$. The top intersection of line bundles on $\overline{W}_1(d)$, $Z\cap T$, is an integer. \begin{lm} \label{pal} \begin{eqnarray*} \forall j\neq 0,1728,\infty, & Z\cap T=E_{d,j}\ , \\ j=0, & \ Z\cap T = 3\cdot E_{d,0}\ , \\ j=1728, & \ \ Z\cap T= 2\cdot E_{d,1728}\ . \end{eqnarray*} \end{lm} \begin{pf} Via pull-back, lines in $\bold P^2$ yield representative classes of $c_1(\cal{L}_i)$. Therefore $3d-1$ general points in $\bold P^2$, $\overline{x}=(x_1, \ldots, x_{3d-1})$, determine a representative cycle $Z_{\overline{x}}$ of the the class $Z$. Let $\infty > j \in \overline{M}_{1,1}$. Let $\pi_W$ be the restriction of $\pi$ to $W_1(d)$. It is first established for a general representative $Z_{\overline{x}}$, \begin{equation} \label{erst} Z_{\overline{x}} \cap \pi_{\overline{W}}^{-1}(j) \subset \pi_W^{-1}(j). \end{equation} The statement (\ref{erst}) is proven by considering the quasi-projective strata of $\overline{M}_{C_j}(d)$. Note $\pi_W^{-1}(j)$ is the strata $U_{C_j}(\Gamma,c,\overline{w})$ where $(\Gamma,c,\overline{w})$ is the trivial, stable, marked, weighted tree with distinguished vertex. Assume now $(\Gamma,c,\overline{w})$ is not the trivial tree. By the equations for the dimension of $(\Gamma,c,\overline{w})$ $$dim U_{C_j}(\Gamma,c,\overline{w}) \leq 6d-3$$ unless $e=0$ and $k=1,2$. Since the linear series determined by the evaluation maps are base point free, the general intersection (\ref{erst}) will miss all loci of dimension less than $6d-2$. It remains to consider the trees $(\Gamma,c,\overline{w})$ where $e=0$ and $k=1,2$. If $k=1$, $(\Gamma,c,\overline{w})=\Phi$ satisfies the conditions of Lemma (\ref{aa}). By Lemma (\ref{aa}), $$dim\ I(\Phi,j) \leq 6d-3.$$ Hence, the general intersection (\ref{erst}) will miss all the loci $U_C(\Phi,c,(0,d))$. If $k=2$, there are two cases to consider. If there exists a vertex of weight $d$, then $(\Gamma,c,\overline{w})=\Phi$ satisfies the conditions of Lemma (\ref{bb}). By Lemma (\ref{bb}), $$dim\ I(\Phi,j) \leq 6d-4.$$ If $w_1+w_2=d$ is a positive partition, then the image of every map $[\mu]\in U_C(\Gamma,c,\overline{w})$ is the union of two rational curves of degrees $w_1$ and $w_2$. No such unions pass through $3d-1$ general points. The proof of claim (\ref{erst}) is complete. For $\infty>j\neq 0,1728$, $\pi_W^{-1}(j)$ is a reduced, irreducible divisor of $W_{1}(d)$. Since the linear series determined by the evaluation maps are base point free, the general intersection cycle \begin{equation} \label{al} Z_{\overline{x}} \cap \pi_W^{-1}(j) \end{equation} is a reduced collection of $Z\cap T$ points. The general intersection cycle $(\ref{al})$ also consists exactly of the reduced, nodal, degree $d$ elliptic plane curves with $j$-invariant $j$ passing through the points $\overline{x}$. The argument for $j=0,1728$ is identical except that $\pi_W^{-1}(0)$, and $\pi_W^{-1}(1728)$ are divisors in $W_1(d)$ with multiplicity $3$, $2$ respectively. These multiplicities arise from the extra automorphisms for $j=0,1728$. Therefore the cycle (\ref{al}) is a collection of $${1\over 3}\cdot Z\cap T,$$ $${1\over 2}\cdot Z\cap T$$ triple and double points respectively. \end{pf} It remains to evaluate $Z\cap T$. \begin{lm} $Z\cap T= {d-1\choose 2} N_d$. \end{lm} \begin{pf} It is first established for a general representative $Z_{\overline{x}}$, \begin{equation} \label{lrst} Z_{\overline{x}} \cap \pi_{\overline{W}}^{-1}(\infty) \subset \pi_W^{-1}(\infty). \end{equation} The statement (\ref{lrst}) is proven by considering the quasi-projective strata of $\overline{M}_{\infty}(d)$. By arguments of Lemma (\ref{pal}), all the loci $U_{\infty}(\Gamma,c,\overline{w})$ where $(\Gamma,c,\overline{w})$ is not the trivial tree are avoided in the general intersection (\ref{lrst}). Only the strata $U_{\infty}(\Lambda,\overline{w})$ remain to be considered. Let $k+1\geq 2$ be the total number of vertices of $\Lambda$. By the equations for the dimensions of $U_{\infty}(\Lambda,\overline{w})$, $$dim \ U_{\infty}(\Lambda, \overline{w}) \leq 6d-2-k\leq 6d-3$$ unless all the circuit vertices have weight zero. If all circuit vertices have weight zero, $k+1\geq 3$. Now $$dim \ U_{\infty}(\Lambda, \overline{w}) \leq 6d-k \leq 6d-3$$ unless $k=2$. Only one stable, marked, weighted, graph with $1$-circuit $(\Lambda, \overline{w})$ need be considered. Vertices $c_1, c_2$ form a weightless circuit. Vertex $v_3$ is connected to $c_2$ and $w_3=d$. $(\Lambda,\overline{w})= \Omega$ satisfies the conditions of Lemma (\ref{cc}). Therefore, $$dim\ I(\Omega, \infty) \leq 6d-4.$$ Claim (\ref{lrst}) is now proven. The divisor $\pi_W^{-1}(\infty)$ is reduced and irreducible in $W_{1}(d)$. As above, \begin{equation} \label{tal} Z_{\overline{x}} \cap \pi_W^{-1}(\infty) \end{equation} is a reduced collection of $Z\cap T$ points. The general intersection cycle $(\ref{tal})$ also consists exactly of degree $d$ {\em maps} of the $1$-nodal rational curve $C_{\infty}$ passing through $\overline{x}$. The image of such a map must be one of the $N_d$ degree $d$, nodal, rational plane curves passing through $\overline{x}$. The number of distinct birational maps (up to isomorphism) from $C_{\infty}$ to a ${d-1 \choose 2}$-nodal plane curve is exactly ${d-1 \choose 2}$. Therefore, $Z\cap T= {d-1\choose 2} N_d$. \end{pf}

9505

alg-geom/9505025

en

https://arxiv.org/abs/alg-geom/9505025

alg-geom/9505025

Tim Ford

Timothy J. Ford

Topological invariants of a fan associated to a toric variety

16 pages with 2 figures, Author-supplied DVI file available at ftp://ftp.math.fau.edu/pub/Ford/itv.dvi, Author-supplied PostScript file available at ftp://ftp.math.fau.edu/pub/Ford/itv.ps, AMSLaTeX v 1.2

Associated to a toric variety $X$ of dimension $r$ over a field $k$ is a fan $\Delta$ on $\Bbb R^r$. The fan $\Delta$ is a finite set of cones which are in one-to-one correspondence with the orbits of the torus action on $X$. The fan $\Delta$ inherits the Zariski topology from $X$. In this article some cohomological invariants of $X$ are studied in terms of whether or not they depend only on $\Delta$ and not $k$. Secondly some numerical invariants of $X$ are studied in terms of whether or not they are topological invariants of the fan $\Delta$. That is, whether or not they depend only on the finite topological space defined on $\Delta$. The invariants with which we are mostly concerned are the class group of Weil divisors, the Picard group, the Brauer group and the dimensions of the torsion free part of the \'etale cohomology groups with coefficients in the sheaf of units. The notion of an open neighborhood of a fan is introduced and examples are given for which the above invariants are sufficiently fine to give nontrivial stratifications of an open neighborhood of a fan all of whose maximal cones are nonsimplicial.

alg-geom

\section{Introduction} \label{sec1} Let $k$ be a field. Let $N = \Bbb Z^r$ and denote by $T_N$ the $k$-torus on $N$. Let $\Delta$ be a finite fan on $N \otimes \Bbb R$ and $X = T_N\emb(\Delta,k)$ the toric variety over $k$ associated to $\Delta$ \cite{D:Gtv}, \cite{F:ITV}, \cite{O:CBA}. This defines a functor $T_N\emb$ on the product category \begin{equation} \label{eq33} \begin{array}{ccc} (\text{finite fans on }N \otimes \Bbb R) \times (\text{fields}) & \stackrel{T_N\emb}{\longrightarrow} & (\text{toric varieties}) \\ (\Delta,k) & \mapsto & T_N\emb(\Delta,k) \end{array} \text{.} \end{equation} We define the topology on $\Delta$ as follows (cf. \cite[pp. 137--138]{DFM:CBg}). The orbit space $\tilde X$ of $X$ under the action of the torus $T_N$ is in one-to-one correspondence with the finite set of cones that belong to $\Delta$. There is a topology on $\tilde X$ inherited from $X$ by the continuous function $X \to \tilde{X}$. Identifying a cone $\sigma \in \Delta$ with the orbit $\operatorname{orb}{\sigma}$ in $\tilde X$, we see that the topology on $\tilde X$ corresponds to the topology on $\Delta$ under which the open sets are the subfans of $\Delta$. The fan $\Delta$ is now a two-faced beast. On the one hand $\Delta$ is an object in the category of fans on $N \otimes \Bbb R$. At the same time $\Delta$ is an object of the category of finite topological spaces. To distinguish between these roles played by $\Delta$, we denote by $\Delta_{fan}$ the object in the category of fans on $N \otimes \Bbb R$ and by $\Delta_{top}$ the object in the category of finite topological spaces. This defines a functor $\frak T$ (which factors via $T_N\emb$ through the category of toric varieties) \begin{equation} \label{eq26} \begin{array}{ccc} (\text{finite fans on }N \otimes \Bbb R) \times (\text{fields}) & \xrightarrow{\frak T} & (\text{finite top. spaces}) \\ (\Delta_{fan},k) & {\mapsto} & \Delta_{top} \end{array} \text{.} \end{equation} In Section~\ref{sec2} we consider some invariants of $\Delta_{fan}$ that are constant for all $k$. Suppose $\gamma(\Delta_{fan},k)$ is an invariant that is defined for any pair $(\Delta_{fan},k)$ (in this article $\gamma$ is usually an abelian group). We call $\gamma$ a {\em fan invariant} in case $\gamma(\Delta_{fan},k)$ depends only on $\Delta_{fan}$ and not on $k$ --- that is, given a fan $\Delta_{fan}$, $\gamma(\Delta_{fan},k_1) \cong \gamma(\Delta_{fan},k_2)$ for every pair of fields $k_1$, $k_2$. We show that the Brauer group $\operatorname{B}(~)$ is not a fan invariant for nonsingular fans. This is an observation based on a theorem of Hoobler and \cite[Theorem~1.1]{DF:Bgt}. In \cite[Theorem~1.1]{DF:Bgt} a complete computation of the Brauer group of a nonsingular toric variety $X=T_N\emb(\Delta)$ over an algebraically closed field $k$ was given in terms of the so-called invariant factors of the fan $\Delta$. In Theorem~\ref{th5} we give the Brauer group of $T_N\emb(\Delta,k)$ for any field $k$ in terms of the Brauer group and Galois group of $k$. The main result of Section~\ref{sec2} is Theorem~\ref{th6} in which it is stated that the class group, $\operatorname{Cl}(~)$, the Picard group, $\operatorname{Pic}(~)$, and the relative cohomology group, $H^2(K(~)/X(~)_{\operatorname{\acute{e}t}},\Bbb G_m)$ (where $K(X)$ is the function field of $X$), are fan invariants. The proof of Theorem~\ref{th6} follows from that of \cite[Theorem~1]{DFM:CBg} and is omitted. In Section~\ref{sec3} we consider some invariants of $\Delta_{fan}$ that are constant on fibers of the map $\frak T$ in \eqref{eq26}, hence depend only on $\Delta_{top}$. That is, suppose we have an invariant $\beta(\Delta_{fan})$ (usually a numerical invariant) associated to any fan $\Delta_{fan}$. If two fans $\Delta_1$, $\Delta_2$ have the same $\beta$-invariant whenever $(\Delta_1)_{top} \cong (\Delta_2)_{top}$, then we say $\beta$ is a {\em topological invariant of $\Delta_{fan}$}. We consider several invariants, all being cohomologically defined. The first sequence is defined by \[ \rho_0 = \operatorname{dim}_{\Bbb Q} \left[ H^0(X_{\operatorname{\acute{e}t}},\Bbb G_m) / k^* \otimes \Bbb Q \right] \text{,} \] and for $i \ge 1$, \[ \rho_i = \operatorname{dim}_{\Bbb Q} \left[ H^i(X_{\operatorname{\acute{e}t}},\Bbb G_m) \otimes \Bbb Q \right] \text{.} \] Also set \[ \rho_1' = \operatorname{dim}_{\Bbb Q} \left[ \operatorname{Cl}(X) \otimes \Bbb Q \right] \text{.} \] For $0 \le i \le 2$ these numbers are finite and are fan invariants. The first main result of Section~\ref{sec3} lists some facts about $\rho_0$ and $\rho_1'$. \medskip\noindent {\bf Theorem~\ref{th1}.} {\em Let $N = \Bbb Z^r$, $\Delta$ a fan on $N \otimes \Bbb R$, $X = T_N\emb(\Delta)$, and $s = \operatorname{dim}_{\Bbb R} \Bbb R|\Delta_{fan}|$ (that is, $s$ is the dimension of the $\Bbb R$-vector space spanned by the vectors in the support $|\Delta_{fan}|$). Then \begin{itemize} \item[(a)] $\rho_0 = r-s$, hence is a fan invariant, but not a topological invariant. \item[(b)] Suppose $\Delta_{fan}$ contains a cone $\sigma$ such that $\operatorname{dim}{\sigma} = r$. This is true for example if $\Delta_{fan}$ is a complete fan on $N \otimes \Bbb R$. Then $\rho_0 = 0$ and $\rho_0$ is a topological invariant of $\Delta_{fan}$. \item[(c)] $\rho_1'=\operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(X) \otimes \Bbb Q) = \#(\Delta(1)) -s$. If $\operatorname{dim}{\Delta_{top}} =r$, then $\rho_1'$ is a topological invariant of $\Delta_{fan}$. \item[(d)] The number $\rho_0-\rho_1'$ is a topological invariant of $\Delta_{fan}$. \end{itemize} } The second main result of Section~\ref{sec3} gives some results on $\rho_0$, $\rho_1$ and $\rho_2$ for simplicial fans. \medskip\noindent {\bf Theorem~\ref{th2}.~}{\em Let $N = \Bbb Z^r$. Let $\Delta$ be a simplicial fan on $N \otimes \Bbb R$ and $s = \operatorname{dim}_{\Bbb R}\Bbb R |\Delta_{fan}|$. Then \begin{itemize} \item[(a)] $\rho_1 = \#(\Delta(1))-s$. \item[(b)] $\rho_2 = 0$ hence is a topological invariant of $\Delta_{fan}$. \item[(c)] If $\operatorname{dim}{\Delta_{top}}=r$, then $\rho_1$ is a topological invariant of $\Delta_{fan}$. \item[(d)] $\rho_0-\rho_1+\rho_2$ is a topological invariant of $\Delta_{fan}$. \end{itemize}} The third main result of Section~\ref{sec3} gives some results on $\rho_0$, $\rho_1$ and $\rho_2$ for 3-dimensional fans. \medskip\noindent {\bf Theorem~\ref{th3}.~}{\em Let $\Delta$ be a fan on $N \otimes \Bbb R$. Let $\sigma_0, \dots, \sigma_w$ be the maximal cones in $\Delta$. Assume $\sigma_i \cap \sigma_j$ is simplicial for each $i \not = j$. These assumptions are satisfied for example if $\operatorname{dim}{\Delta_{top}} \le 3$. Then \begin{itemize} \item[(a)] \[ \rho_1 +s + \sum_{i=0}^w(\#(\Delta(\sigma_i)(1))-s_i) = \rho_2 + \#( \Delta(1)) \text{,} \] where we set $s_i = \operatorname{dim}{\sigma_i}$ for each $i = 0, \dots, w$ and $s = \operatorname{dim}_{\Bbb R} \Bbb R |\Delta_{fan}|$. \item[(b)] $\rho_0-\rho_1+\rho_2$ is a topological invariant of $\Delta_{fan}$. \end{itemize}} In Section~\ref{sec4} we introduce the notion of an {\em open neighborhood $B$ of a fan $\Delta$}. This is a subset of the fiber $\frak T^{-1}(\Delta_{top})$ that is parametrized by a dense subset of a real manifold. Let $\scr{S}\scr{F}$ denote the sheaf of $\Delta$-linear support functions on the topological space $\Delta_{top}$. It was shown in \cite{DFM:CBg} that the numbers $\rho_i$, $1\le i\le 2$, can be determined by the cohomology of the sheaf $\scr{S}\scr{F}$ on the finite topological space $\Delta_{top}$. Therefore we define another sequence of invariants by \[ \kappa_i = \operatorname{dim}_{\Bbb Q}\left[ H^i(\Delta_{top},\scr{S}\scr{F}) \otimes \Bbb Q \right] \] for $i \ge 0$. We consider the stratification of $B$ by the numerical invariant $\kappa_0$. Several examples are given for which the stratification of $B$ is nontrivial. We conjecture that $\kappa_0 = 3$ on a nonempty open subset of $B$ if $\Delta$ is a complete fan on $\Bbb R^3$ such that every maximal cone of $\Delta$ is nonsimplicial. Algorithm~\ref{alg1} is presented which computes an upper bound for $\kappa_0$. For complete 3-dimensional fans, this algorithm can be used to compute an upper bound for $\rho_1$ and $\rho_2$. For the benefit of the reader the following notation will be fixed throughout the rest of the paper. \begin{table}[htp] \label{tab1} \end{table} \begin{center} \begin{tabular}{|lp{14pc}|ll|}\hline $k$ & a field & $r$ & a positive integer \\ $N$ & $=\Bbb Z^r$ & $M $ & $=\operatorname{Hom}_{\Bbb Z}(N,\Bbb Z)$ \\ $\Delta$ & \raggedright a finite rational fan on $N\otimes \Bbb R$ & $X$ & $ =T_N\emb(\Delta,k)$ toric variety \\ $\Delta_{fan}$ & object in the category of fans & $\Delta_{top}$ & finite topological space \\ $\left|\Delta_{fan}\right|$ & support of the fan $\Delta$ & $\frak T$ & functor that maps $(\Delta_{fan},k)$ to $\Delta_{top}$ \\ $\operatorname{Cl}(X)$ &\raggedright class group of Weil divisor classes & $\operatorname{Pic}{X}$ & Picard group of invertible modules \\ $\operatorname{B}(X)$ & \raggedright Brauer group of Azumaya algebra classes & $\Bbb G_m$ & \'etale sheaf of units \\ $\rho_0 $ & $ = \operatorname{dim}_{\Bbb Q}\left[ H^0(X_{\operatorname{\acute{e}t}},\Bbb G_m)\otimes\Bbb Q\right]$ & $\rho_i $ & $ = \operatorname{dim}_{\Bbb Q}\left[ H^i(X_{\operatorname{\acute{e}t}},\Bbb G_m)\otimes\Bbb Q\right]$ (for $i>0$) \\ $\rho'_1$ & $ = \operatorname{dim}_{\Bbb Q}\left[ \operatorname{Cl}(X)\otimes\Bbb Q\right]$ & $\kappa_i$ & $=\operatorname{dim}_{\Bbb Q}\left[ H^i(\Delta_{top},\scr{S}\scr{F})\otimes\Bbb Q\right]$ (for $i\ge 0$) \\ $s $ & $=\operatorname{dim}_{\Bbb R}\Bbb R \left|\Delta_{fan}\right|$ & $\Delta(i)$ & $=\{\sigma\in\Delta | \operatorname{dim}{\sigma}=i\}$ \\ $K$ & $=K(X)$ the function field of $X$ & $\scr{S}\scr{F}$ & sheaf of $\Delta$-linear support functions \\ $\scr{W}$ & sheaf of Weil divisors & $\scr{P}$ & quotient sheaf $\scr{W} / \scr{S}\scr{F}$ \\ \hline \end{tabular}\end{center} \section{Fan Invariants} \label{sec2} In Theorem~\ref{th5} we determine the Brauer group of a nonsingular toric variety over $k$. This invariant depends on $k$. We then show in Theorem~\ref{th6} that $\operatorname{Cl}(X)$, $\operatorname{Pic}{X}$ and the relative cohomology group $H^2(K/X_{\operatorname{\acute{e}t}},\Bbb G_m)$ depend only on $\Delta_{fan}$, not on $k$. In order to determine the Brauer group of a nonsingular toric variety over $k$, we use the following theorem of Hoobler. \begin{theorem} \label{th4} Let $R = A[x_1,x_1^{-1}, \dots, x_r,x_r^{-1}]$, where $A$ is a connected, normal integral domain. Suppose $\nu$ is an integer relatively prime to the residue characteristics of $A$. Then \begin{equation} \label{eq8} H^1(R,\Bbb Z/\nu) = H^1(A,\Bbb Z/\nu) \oplus \left( \bigoplus^r \mu_\nu^{-1} \right) \text{ ,} \end{equation} and \begin{equation} \label{eq27} _\nu\operatorname{B}(R) = {_\nu\operatorname{B}(A)} \oplus \left( \bigoplus^r H^1(A, \Bbb Z/\nu) \right) \oplus \left( \bigoplus^{r(r-1)/2} \mu_\nu^{-1} \right) \text{ .} \end{equation} \end{theorem} \begin{pf} See \cite[Cor. 2.6]{H:Fgr}. \end{pf} Therefore the $\nu$-torsion of the Brauer group of $R=A[x_1,x_1^{-1}, \dots, x_r,x_r^{-1}]$ is generated by the Azumaya $A$-algebras and the classes of cyclic crossed product algebras of 2 types. For each cyclic Galois extension $C/A$ of degree $\nu$ with group $\langle \sigma \rangle$ and for each $1 \le i \le r$, there is the cyclic crossed product $(C/R,\langle \sigma \rangle, x_i)$ which is an Azumaya algebra over $R$. If there exists a primitive $\nu$-th root of unity $\zeta$ over $A$, then the symbol algebras $(x_i,x_j)_\nu$ are Azumaya algebras over $R$. \begin{example} \label{ex13} Let $R= \Bbb R [x_1,x_1^{-1},\dots, x_r,x_r^{-1}]$. Then by Theorem~\ref{th4} $\operatorname{B}(R)$ is an elementary 2-group and \begin{equation} \label{eq28} \begin{split} \operatorname{B}(R) & \cong {\operatorname{B}(\Bbb R)} \oplus \left( \bigoplus^r H^1(\Bbb R, \Bbb Z/2) \right) \oplus \left( \bigoplus^{r(r-1)/2} \mu_2^{-1} \right) \\ & \cong \left( \Bbb Z/2 \right)^{1+r+ r(r-1)/2} \end{split} \end{equation} \end{example} Define a sheaf ${\scr{S}\scr{F}}$ on $\Delta_{top} $ by assigning to each open set $\Delta ' \subseteq \Delta_{top}$ the abelian group $\operatorname{SF}(\Delta ')$ of support functions on $\Delta '$. Let $M = \operatorname{Hom}(N,{\Bbb Z})$ be the dual of $N$. There is a natural map $M \rightarrow \operatorname{SF}(\Delta ')$ which is locally surjective. If ${\scr{M}}$ denotes the constant sheaf of $M$ on $\Delta_{top}$, then there is an exact sequence of sheaves on $\Delta_{top}$: \begin{equation} \label{eq32} 0 \to {\scr{U}} \to {\scr{M}} \to {\scr{S}\scr{F}} \to 0 \end{equation} \noindent where ${\scr{U}}$ is defined by the sequence \eqref{eq32}. On any open $\Delta ' \subseteq \Delta_{top}$, ${\scr{U}} (\Delta ') = |\Delta '|^\perp \cap M = \{m \in M | \langle m,y\rangle = 0$ for all $y \in |\Delta '|\}$. Because ${\scr{M}}$ is flasque, $H^p(\Delta_{top} ,{\scr{M}} ) = 0$ for all $p \geq 1$, so $H^p(\Delta_{top} ,{\scr{S}\scr{F}} ) \cong H^{p+1}(\Delta_{top} ,{\scr{U}} )$ for all $p \geq 1.$ Let $k$ be a field and $X = T_N\emb(\Delta,k)$ a nonsingular toric variety over $k$. Let $N' = \langle \bigcup_{\sigma \in \Delta} \sigma \cap N \rangle$, let $\nu \ge 2$ be relatively prime to $\operatorname{char}{k}$, and let $M_\nu = \{ m \in M | \langle m,n' \rangle \equiv 0 \pmod \nu \text{ for all } n' \in N' \}$. The basis theorem for finitely generated abelian groups gives a basis $n_1, \dots, n_r$ of $N$ such that $N' = \Bbb Z a_1 n_1 \oplus \Bbb Z a_2 n_2 \oplus \dots \oplus \Bbb Z a_r n_r$ where the $a_i$ are nonnegative integers and $a_i | a_{i+1}$ for $1 \le i \le r-1$. As in \cite{DF:Bgt} call $a_1, \dots, a_r$ the set of invariant factors of $X$. \begin{theorem} \label{th5} In the above terminology, if $(\nu, a_i)$ is the greatest common divisor of $\nu$ and $a_i$, then \begin{equation} \label{eq29} \begin{split} H^1(X,\Bbb Z/\nu) &\cong H^1(k,\Bbb Z/\nu) \oplus \left( M_\nu/\nu M \otimes \mu_\nu ^{-1} \right) \\ &\cong H^1(k,\Bbb Z/\nu) \oplus \left( \bigoplus_{i=1}^r \Bbb Z/(\nu,a_i) \otimes \mu_\nu ^{-1} \right) \text{ .} \end{split} \end{equation} \begin{equation} \label{eq30} \begin{split} {_\nu \operatorname{B}(X)} & = {_\nu \operatorname{B}'(X)} \cong \\ & {_\nu \operatorname{B}(k)} \oplus \left( \bigoplus_{i=1}^r H^1(k,\Bbb Z/\nu) \otimes \Bbb Z/(\nu,a_i) \right) \oplus \left( \bigoplus_{i=1}^r \operatorname{Hom}(\Bbb Z/a_i \otimes \mu_\nu, \Bbb Q/ \Bbb Z )^{r-i} \right) \end{split} \end{equation} \end{theorem} \begin{pf} Follows from the proof of \cite[Theorem~1.1]{DF:Bgt} and Theorem~\ref{th4}. \end{pf} Therefore the $\nu$-torsion of the Brauer group of the nonsingular toric variety $X$ is generated by the classes of algebras from $k$ and cyclic crossed product algebras of 2 types. For each cyclic Galois extension $C/k$ of degree $\nu$ with group $\langle \sigma \rangle$ and for each $1 \le i \le r$, there is the cyclic crossed product $(C/k,\langle \sigma \rangle, x_i)$ which is an Azumaya algebra over the torus $T_N$. This algebra is unramified on $X$ if and only if the function $x_i$ corresponds to an element of $M_\nu$. If there exists a $\nu$-th root of unity $\zeta$ over $k$, then the symbol algebras $(x_i,x_j)_\nu$ are Azumaya algebras over $T_N$. Those symbols which are unramified on $X$ correspond to the last summand of \eqref{eq30}. \begin{example} \label{ex14} Let $k= \Bbb R$ and $X = T_N\emb(\Delta)$ a nonsingular toric variety over $\Bbb R$. Then by Theorem~\ref{th5} $\operatorname{B}(X)$ is an elementary 2-group. If $t = \left| \{ a_i | (2,a_i) \not = 1 \} \right|$, then \begin{equation} \label{eq31} \begin{split} \operatorname{B}(X) & \cong \operatorname{B}(\Bbb R) \oplus \left( \bigoplus_{i=1}^r \Bbb Z/(2,a_i) \right) \oplus \left( \bigoplus_{i=1}^r \operatorname{Hom}(\Bbb Z/a_i \otimes \mu_2, \Bbb Q/ \Bbb Z )^{r-i} \right) \\ & \cong \Bbb Z/2 \oplus (\Bbb Z/2)^t \oplus (\Bbb Z/2)^{t(t-1)/2} \end{split} \end{equation} \end{example} \begin{theorem} \label{th6} Let $k$ be a field and $X = T_N\emb(\Delta)$ a toric variety over $k$ with function field $K$. Then \begin{enumerate} \item $H^p(\Delta_{top},\scr{U}) \cong H^p(X_{\operatorname{Zar}},\scr{O}^*)$ for all $p \ge 1$ hence $H^p(X_{\operatorname{Zar}},\scr{O}^*)$ depends only on $\Delta_{fan}$, not $k$. In particular $\operatorname{Cl}(X)$ and $\operatorname{Pic}{X}$ depend only on $\Delta_{fan}$. \item $H^1(\Delta_{top},\scr{S}\scr{F}) \cong H^2(X_{\operatorname{Zar}},\scr{O}^*) \cong H^2(K/X_{\operatorname{\acute{e}t}},\Bbb G_m)$ hence $H^2(K/X_{\operatorname{\acute{e}t}},\Bbb G_m)$ depends only on $\Delta_{fan}$, not $k$. \item If $\tilde \Delta $ is a nonsingular subdivision of $\Delta $ and $\tilde X = T_N\emb(\tilde \Delta )$, then the sequence \[ 0 \to H^2(K/X_{\operatorname{\acute{e}t}},\Bbb G_m) \to H^2(X_{\operatorname{\acute{e}t}},\Bbb G_m) \to H^2(\tilde X_{\operatorname{\acute{e}t}},\Bbb G_m) \to 0 \] (with natural maps) is split-exact. \end{enumerate} \end{theorem} \begin{pf} The theorem follows from \cite{DFM:CBg}, noting that the proof of \cite[Theorem~1]{DFM:CBg} did not assume that $k$ is algebraically closed until the proof of Lemma~7 where it was not necessary anyway. \end{pf} \section{Topological Invariants} \label{sec3} The first invariants to be considered as candidates for topological invariants are the following. Let $\Delta$ and $X$ be as in the Introduction. For each $i \ge 0$ we define a positive integer $\rho_i$. Set \[ \rho_0 = \operatorname{dim}_{\Bbb Q} \left[ H^0(X_{\operatorname{\acute{e}t}},\Bbb G_m) / k^* \otimes \Bbb Q \right] \text{,} \] \[ \rho_1 = \operatorname{dim}_{\Bbb Q} \left[ H^1(X_{\operatorname{\acute{e}t}},\Bbb G_m) \otimes \Bbb Q \right] = \operatorname{dim}_{\Bbb Q} \left( \operatorname{Pic}{X} \otimes \Bbb Q \right) \] and for $i \ge 2$, \[ \rho_i = \operatorname{dim}_{\Bbb Q} \left[ H^i(X_{\operatorname{\acute{e}t}},\Bbb G_m) \otimes \Bbb Q \right] \text{.} \] The number $\rho_1$ is the traditional Picard number $\rho$ associated to $X$. Also set \[ \rho_1' = \operatorname{dim}_{\Bbb Q} \left[ \operatorname{Cl}(X) \otimes \Bbb Q \right] \text{.} \] It follows from Theorem~\ref{th1} below that $\rho_0$ is a fan invariant and from Theorem~\ref{th6} above that $\rho_1$, $\rho_1'$, and $\rho_2$ are fan invariants. Since $\Delta$ is finite, $\rho_0$, $\rho_1$, $\rho_1'$ and $\rho_2$ are finite. For $\rho_0$, $\rho_1$ and $\rho_1'$ see \cite{O:CBA} or \cite{F:ITV}. For $\rho_2$ this follows from \cite{DFM:CBg}. Examples where the number $\rho_2$ is computed seem to be somewhat scarce. Grothendieck \cite[II]{G:GB} and Childs \cite{C:Bgn} each give an example of a local ring $\scr{O}_x$ on a normal surface where $H^2((\scr{O}_x)_{\operatorname{\acute{e}t}},\Bbb G_m)$ is torsion free, but in each case $H^2((\scr{O}_x)_{\operatorname{\acute{e}t}},\Bbb G_m)$ is not finitely generated. \begin{remark}\label{re6} The dimension of the topological space $\Delta_{top}$ is defined to be the length of a maximal chain of irreducible closed subsets. One can check that this is equal to $\max{ \{ \operatorname{dim}{\sigma} |}$ ${ \sigma \in \Delta \} }$. Therefore $\operatorname{dim}{\Delta_{top}}$ is a topological invariant of $\Delta_{fan}$. \end{remark} \begin{remark}\label{re7} Define another sequence of invariants by \[ \kappa_i = \operatorname{dim}_{\Bbb Q}\left[ H^i(\Delta_{top},\scr{S}\scr{F}) \otimes \Bbb Q \right] \] for $i \ge 0$. It follows from Theorem~\ref{th6}~(2) that $\kappa_1 = \rho_2$. Let $\sigma_0, \dots, \sigma_m$ be the maximal cones in $\Delta$. From \cite[Lemma 8]{DFM:CBg} $\kappa_i$ can be computed from the $\operatorname{\check Cech}$ complex \begin{equation} \label{eq12} 0 \to \underset{i}{\oplus} \scr{S}\scr{F}(\sigma_i) \stackrel{\delta^0}{\rightarrow} \underset{i<j}{\oplus} \scr{S}\scr{F}(\sigma_{ij}) \stackrel{\delta^1}{\rightarrow} \underset{i<j<k}{\oplus} \scr{S}\scr{F}(\sigma_{ijk}) \to \dots \end{equation} For any cone $\tau \in \Delta$, $\operatorname{dim}_{\Bbb Q}(\scr{S}\scr{F}(\Delta(\tau)) \otimes \Bbb Q) = \operatorname{dim}{\tau}$. Therefore, if $C^i$ denotes the $i$-th group of $\operatorname{\check Cech}$ cochains in \eqref{eq12} and $c_i = \operatorname{dim}_{\Bbb Q}(C^i \otimes \Bbb Q)$, then the integer \begin{equation} \label{eq24} c_0 - c_1 + c_2 - \dots \end{equation} is a topological invariant of $\Delta_{fan}$. Note that there exists an integer $M$ such that $C^j = 0$ for all $j > M$. If $\operatorname{dim}(\Delta_{top}) = t$, then $\kappa_j = 0$ for all $j>t$. So the left hand side of \begin{equation} \label{eq25} \kappa_0 - \kappa_1 + \dots (-1)^t \kappa_t = c_0 - c_1 + c_2 - \dots (-1)^M c_M \end{equation} is a topological invariant of $\Delta_{fan}$. \end{remark} \begin{remark}\label{re5} Let $\Delta$ be a finite fan on $N \otimes \Bbb R$ where $N = \Bbb Z^r$. Setting $s = \operatorname{dim}_{\Bbb R} \Bbb R|\Delta_{fan}| $, we see that $s$ is not a topological invariant of $\Delta_{fan}$. Since $s \ge \operatorname{dim}{\Delta_{top}}$, if $\operatorname{dim}{\Delta_{top}} = r$, then $s=r$ so if $\Delta$ contains a cone $\sigma$ such that $\operatorname{dim}{\sigma}=r$, then $s=r$ and $s $ is a topological invariant of $\Delta_{fan}$. This condition is satisfied, for instance, if $\Delta$ is a complete fan on $N \otimes \Bbb R$. \end{remark} \begin{remark}\label{re2} Let $\sigma \in \Delta$ and let $\Delta(\sigma)$ denote the subfan of $\Delta$ consisting of the cone $\sigma$ and all of its faces. Then \[ \operatorname{dim}{\sigma} = \operatorname{dim}{\Delta(\sigma)_{top}} \text{,} \] so the dimensions of the cones in $\Delta$ depend only on $\Delta_{top}$. In particular, the number of 1-dimensional cones in $\Delta_{fan}$ is a topological invariant. \end{remark} \begin{remark}\label{re3} The fan $\Delta_{fan}$ is complete by definition if $|\Delta_{fan}| = \Bbb R^r$. This is true if and only if \begin{itemize} \item[(i)] $\Delta(r) \not = \emptyset$ and \item[(ii)] for each cone $\sigma \in \Delta(r)$ and every $r-1$-dimensional face $\tau$ of $\sigma$ there is a cone $\sigma_1 \in \Delta(r)$ such that $\tau = \sigma \cap \sigma_1$. \end{itemize} But these two conditions depend only on $\Delta_{top}$. That is, completeness can be thought of as a topological property of $\Delta_{fan}$. \end{remark} {F}rom the next theorem, which combines some results on $\rho_0$ and $\rho_1'$, we see that $\rho_0$ depends only on the dimension of the subspace spanned by $|\Delta_{fan}|$. \begin{theorem} \label{th1} Let $N = \Bbb Z^r$, $\Delta$ a fan on $N \otimes \Bbb R$, $X = T_N\emb(\Delta)$, and $s = \operatorname{dim}_{\Bbb R} \Bbb R|\Delta_{fan}|$ (that is, $s$ is the dimension of the $\Bbb R$-vector space spanned by the vectors in the support $|\Delta_{fan}|$). Then \begin{itemize} \item[(a)] $\rho_0 = r-s$, hence is a fan invariant, but not a topological invariant. \item[(b)] Suppose $\Delta_{fan}$ contains a cone $\sigma$ such that $\operatorname{dim}{\sigma} = r$. This is true for example if $\Delta_{fan}$ is a complete fan on $N \otimes \Bbb R$. Then $\rho_0 = 0$ and $\rho_0$ is a topological invariant of $\Delta_{fan}$. \item[(c)] $\rho_1'=\operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(X) \otimes \Bbb Q) = \#(\Delta(1)) -s$. If $\operatorname{dim}{\Delta_{top}} =r$, then $\rho_1'$ is a topological invariant of $\Delta_{fan}$. \item[(d)] The number $\rho_0-\rho_1'$ is a topological invariant of $\Delta_{fan}$. \end{itemize} \end{theorem} \begin{pf} (a) Let $N_1 = N \cap \Bbb R |\Delta_{fan}|$, $M_1 = \operatorname{Hom}_{\Bbb Z}(N_1,\Bbb Z)$, $M_2 = N_1^{\perp}$, $N_2 = \operatorname{Hom}_{\Bbb Z}(M_2,\Bbb Z)$. Then $M = M_1 \oplus M_2$ and $N = N_1 \oplus N_2$. Viewing $\Delta$ as a fan on the $s$-dimensional vector space $N_1 \otimes \Bbb R$, $X = T_{N_1}\operatorname{emb}(\Delta) \times T_{N_2}$ and $H^0(X,\Bbb G_m) / k^* \cong H^0(T_{N_2},\Bbb G_m) / k^* \cong \Bbb Z^{r-s}$. So $\rho_0 = r-s$. (b) In this case $s = \operatorname{dim}_{\Bbb R} \Bbb R|\Delta_{fan}| = \operatorname{dim}_{\Bbb R} (\Bbb R \sigma) = \operatorname{dim} \sigma = r$. (c) Let $N_0 = N \cap \Bbb R |\Delta_{fan}|$ be the set of lattice points in the subspace $\Bbb R |\Delta_{fan}|$ and $M_0 = \operatorname{Hom}_{\Bbb Z}(N_0,\Bbb Z)$. Then $\operatorname{dim}_{\Bbb Q}(M_0 \otimes \Bbb Q) = \operatorname{dim}_{\Bbb Q}(N_0 \otimes \Bbb Q) = \operatorname{dim}_{\Bbb R} \Bbb R |\Delta_{fan}| = s$. From \cite[Corollary~2.5]{O:CBA} there is a presentation of $\operatorname{Cl}(X)$ \begin{equation} \label{eq0} 0 \to M_0 \to \bigoplus_{\rho \in \Delta(1)} \Bbb Z \rho \to \operatorname{Cl}(X) \to 0 \text{.} \end{equation} So $\operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(X) \otimes \Bbb Q) = \#(\Delta(1)) -s$. In particular, if $\operatorname{dim}{\Delta_{top}} =r$, then $r=s$ so $\operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(X) \otimes \Bbb Q)$ is a topological invariant. (d) {F}rom (a) and (c), \[ \rho_0-\rho_1' = (r - s) - (\#(\Delta(1)) -s) = r - \#(\Delta(1)) \] which depends only on $\Delta_{top}$. \end{pf} \begin{remark} \label{re1} Let $N = \Bbb Z^r$, $\Delta$ a fan on $N \otimes \Bbb R$, $s = \operatorname{dim}_{\Bbb R} \Bbb R |\Delta_{fan}|$, $t = \underset{\sigma \in \Delta}{\max} \{ \operatorname{dim}{\sigma} \}$. It follows from \cite[Theorem~2.3]{DF:Bgt} that if $t \le 2$, then $\rho_1 = \#(\Delta(1)) - s$ and $\rho_2 = 0$. In this case $\Delta$ is a simplicial fan, so this is a special case of the following theorem. \end{remark} \begin{theorem} \label{th2} Let $N = \Bbb Z^r$. Let $\Delta$ be a simplicial fan on $N \otimes \Bbb R$ and $s = \operatorname{dim}_{\Bbb R}\Bbb R |\Delta_{fan}|$. Then \begin{itemize} \item[(a)] $\rho_1 = \#(\Delta(1))-s$. \item[(b)] $\rho_2 = 0$ hence is a topological invariant of $\Delta_{fan}$. \item[(c)] If $\operatorname{dim}{\Delta_{top}}=r$, then $\rho_1$ is a topological invariant of $\Delta_{fan}$. \item[(d)] $\rho_0-\rho_1+\rho_2$ is a topological invariant of $\Delta_{fan}$. \end{itemize} \end{theorem} \begin{pf} Suppose $\sigma \in \Delta$ is a simplicial cone and $\operatorname{dim}{\sigma} = d$. For any support function $h \in \scr{S}\scr{F}(\Delta)$, $h|_\sigma$ is linear and completely determined by its values on a spanning set $\{\eta_1, \dots, \eta_d \} \subseteq N$ for $\sigma$. Since $\operatorname{dim}{\sigma}=d$, $\sigma$ is spanned by $d$ lattice points. So $\scr{S}\scr{F}(\Delta(\sigma)) \otimes \Bbb Q \cong \Bbb Q^d$. If $\tau_0, \dots, \tau_n$ are the cones in $\Delta(1)$, and $\Gamma = \{0,\tau_0, \dots, \tau_n \}$, then $\Gamma_{top}$ is an open subset of the topological space $\Delta_{top}$. Define the sheaf $\scr{W}$ on $\Delta_{top}$ to be the direct image $i_*(\scr{S}\scr{F} |_{\Gamma_{top}})$. Since $\Gamma_{fan}$ is a nonsingular fan, $\scr{S}\scr{F} |_{\Gamma_{top}}$ is the sheaf defined by $\Xi \mapsto \Bbb Z^{\#(\Xi(1))}$ for each open $\Xi \subseteq \Delta_{top}$. It follows that $\scr{W}(\Xi) = \Bbb Z^{\# (\Xi(1))}$, hence $\scr{W}$ is a flasque sheaf. So there is an embedding $\scr{S}\scr{F} \to \scr{W}$ of sheaves on $\Delta_{top}$ and we define $\scr{P}$ by the exact sequence of sheaves \cite[(13), p. 149]{DFM:CBg} \begin{equation} \label{eq1} 0 \to \scr{S}\scr{F} \to \scr{W} \to \scr{P} \to 0 \text{.} \end{equation} Since $\Delta$ is simplicial, $ \scr{S}\scr{F}(\Delta(\sigma))$ and $\scr{W}(\Delta(\sigma))$ are free of the same rank $\operatorname{dim}{\sigma}$. Therefore, $\scr{P}$ is locally torsion, hence torsion. Because $\scr{W}$ is flasque, $H^1(\Delta_{top},\scr{W}) = 0$ and the long exact sequence associated to \eqref{eq1} becomes \begin{equation} \label{eq2} 0 \to H^0(\Delta_{top},\scr{S}\scr{F}) \to H^0(\Delta_{top},\scr{W}) \to H^0(\Delta_{top},\scr{P}) \to H^1(\Delta_{top}, \scr{S}\scr{F}) \to 0 \text{.} \end{equation} Because $\scr{P}$ is torsion, $H^0(\Delta_{top},\scr{P})$ is torsion. So $H^1(\Delta_{top}, \scr{S}\scr{F}) \otimes \Bbb Q = 0$. By \cite[Theorem~1]{DFM:CBg} $\rho_2 = \operatorname{dim}(H^1(\Delta_{top}, \scr{S}\scr{F}) \otimes \Bbb Q) = 0$. This proves (b). It also follows from \eqref{eq2} that we obtain the isomorphism of \cite[Proposition~2.1(v), p. 69]{O:CBA} \begin{equation} \label{eq10} H^0(\Delta_{top},\scr{S}\scr{F}) \otimes \Bbb Q \cong H^0(\Delta_{top},\scr{W}) \otimes \Bbb Q \text{.} \end{equation} By \cite[Lemma~8]{DFM:CBg} there is an exact sequence \begin{equation} \label{eq6} 0 \to M_0 \to \scr{S}\scr{F}(\Delta_{top}) \to \operatorname{Pic}{X} \to 0 \text{.} \end{equation} \noindent Combining \eqref{eq0} and \eqref{eq6}, we have a commutative diagram with exact rows and columns. \begin{equation} \label{eq9} \begin{CD} 0 @. 0 @. 0 \\ @VVV @VVV @VVV \\ M @>>> \scr{S}\scr{F}(\Delta_{top}) @>>> \operatorname{Pic}{X} @>>> 0 \\ @VV=V @VVV @VVV \\ M @>>> \scr{W}(\Delta_{top}) @>>> \operatorname{Cl}(X) @>>> 0 \end{CD} \end{equation} Because the center vertical arrow in \eqref{eq9} tensored with $\Bbb Q$ is the isomorphism \eqref{eq10}, from \eqref{eq9} it follows that \begin{equation} \label{eq11} \operatorname{Pic}(X) \otimes \Bbb Q \cong \operatorname{Cl}(X) \otimes \Bbb Q \text{.} \end{equation} It follows from Theorem~\ref{th1} that $\rho_1 = \#(\Delta(1))-s$. This proves (a). In case (c), $s=r$ so $\rho_1$ is a topological invariant. (d) {F}rom Theorem~\ref{th1} and parts (a) and (b), \[ \rho_0-\rho_1+\rho_2 = (r - s) - (\#(\Delta(1)) -s) + 0 = r - \#(\Delta(1)) \] which only depends on $\Delta_{top}$. \end{pf} \begin{lemma} \label{lem2} For any cone $\sigma \in \Delta$, let $\Delta(\sigma)$ denote the subfan of $\Delta$ consisting of $\sigma$ and all of its faces and $U_\sigma = T_N\emb(\Delta(\sigma))$. Then $H^0(\Delta(\sigma)_{top},\scr{P}) = \operatorname{Cl}(U_\sigma)$. \end{lemma} \begin{pf} For each $\sigma \in \Delta$ we have $H^1(\Delta(\sigma)_{top}, \scr{S}\scr{F}) = 0$ \cite[Lemma 2.a, p. 139]{DFM:CBg} so from \eqref{eq2} $$H^0(\Delta(\sigma)_{top},\scr{P}) = \scr{W}(\Delta(\sigma)_{top}) / \scr{S}\scr{F}(\Delta(\sigma)_{top}) \text{.}$$ Now $\scr{W}(\Delta(\sigma)_{top}) = \Bbb Z^{\#(\Delta(\sigma)(1))}$ and support functions are linear on a cone $\sigma$, so $$\scr{W}(\Delta(\sigma)_{top}) / \scr{S}\scr{F}(\Delta(\sigma)_{top}) \cong \Bbb Z^{\#(\Delta(\sigma)(1))}/ \operatorname{im}(M) = \operatorname{Cl}(U_\sigma) \text{.}$$ \end{pf} \begin{lemma} \label{lem3} Let $\sigma$ be a cone in $N \otimes \Bbb R$ and $s = \operatorname{dim}{\sigma}$. Then $\operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(U_\sigma) \otimes \Bbb Q) = \#(\Delta(\sigma)(1))-s$. Also $\sigma$ is simplicial if and only if $\operatorname{Cl}(U_\sigma) $ is torsion. \end{lemma} \begin{pf} Follows from Theorem~\ref{th1}. \end{pf} The following can be considered a theorem for 3-dimensional fans. \begin{theorem} \label{th3} Let $\Delta$ be a fan on $N \otimes \Bbb R$. Let $\sigma_0, \dots, \sigma_w$ be the maximal cones in $\Delta$. Assume $\sigma_i \cap \sigma_j$ is simplicial for each $i \not = j$. These assumptions are satisfied for example if $\operatorname{dim}{\Delta_{top}} \le 3$. Then \begin{itemize} \item[(a)] \[ \rho_1 +s + \sum_{i=0}^w(\#(\Delta(\sigma_i)(1))-s_i) = \rho_2 + \#( \Delta(1)) \text{,} \] where we set $s_i = \operatorname{dim}{\sigma_i}$ for each $i = 0, \dots, w$ and $s = \operatorname{dim}_{\Bbb R} \Bbb R |\Delta_{fan}|$. \item[(b)] $\rho_0-\rho_1+\rho_2$ is a topological invariant of $\Delta_{fan}$. \end{itemize} \end{theorem} \begin{pf} (a) The set $\{ \Delta(\sigma_i)_{top}\}_{i=0}^w$ is an open cover of $\Delta_{top}$ and the sequence \begin{equation} \label{eq3} 0 \to H^0(\Delta_{top},\scr{P}) \to \bigoplus_{i=0}^w H^0(\Delta(\sigma_i)_{top},\EuScript P) \to \bigoplus_{i=1}^w\bigoplus_{j=0}^{i-1}H^0(\Delta(\sigma_i \cap \sigma_j)_{top},\scr{P}) \end{equation} is exact since $\scr{P}$ is a sheaf. Applying Lemma~\ref{lem2}, the sequence \eqref{eq3} can be written \begin{equation} \label{eq4} 0 \to H^0(\Delta_{top},\scr{P}) \to \bigoplus_{i=0}^w \operatorname{Cl}(U_{\sigma_i}) \to \bigoplus_{i=1}^w\bigoplus_{j=0}^{i-1} \operatorname{Cl}(U_{\sigma_i \cap \sigma_j}) \text{.} \end{equation} By our assumption $\sigma_i \cap \sigma_j$ is simplicial. By Lemma~\ref{lem3}, $\operatorname{Cl}(U_{\sigma_i \cap \sigma_j})$ is torsion. By \cite[Theorem 1]{DFM:CBg} $H^1(\Delta_{top}, \scr{S}\scr{F}) \cong H^2(K/X_{\operatorname{\acute{e}t}},\Bbb G_m)$ and the torsion-free part of $H^2(X_{\operatorname{\acute{e}t}},\Bbb G_m)$ is equal to the torsion-free part of $H^2(K/X_{\operatorname{\acute{e}t}},\Bbb G_m)$. We compute the rank of the torsion-free part of $H^2(K/X_{\operatorname{\acute{e}t}},\Bbb G_m)$ from \eqref{eq2} tensored with $\Bbb Q$: \begin{equation} \label{eq5} 0 \to \scr{S}\scr{F}(\Delta_{top}) \otimes \Bbb Q \to \scr{W}(\Delta_{top}) \otimes \Bbb Q \to \scr{P}(\Delta_{top}) \otimes \Bbb Q \to H^2(K/X_{\operatorname{\acute{e}t}}, \Bbb G_m) \otimes \Bbb Q \to 0 \text{.} \end{equation} Tensoring \eqref{eq6} with $\Bbb Q$ and counting dimensions we find $\operatorname{dim}_{\Bbb Q}(\scr{S}\scr{F}(\Delta_{top}) \otimes \Bbb Q) =$ $ \operatorname{dim}_{\Bbb Q}(\operatorname{Pic}{X}$ $\otimes \Bbb Q) + s$ $ = \rho_1 +s$. By definition $\scr{W}(\Delta_{top}) \otimes \Bbb Q = \Bbb Q^{\#(\Delta(1))}$. {}From \eqref{eq4} $\scr{P}(\Delta_{top}) \otimes \Bbb Q = \bigoplus_{i=0}^w \operatorname{Cl}(U_{\sigma_i}) \otimes \Bbb Q$. {F}rom Lemma~\ref{lem3}, $\operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(U_{\sigma_i}) \otimes \Bbb Q) = \#(\Delta(\sigma_i)(1))-s_i$. Counting dimensions in \eqref{eq5}, we have the equation \begin{equation} \label{eq7} \rho_2 = \rho_1 +s + \sum_{i=0}^w(\#(\Delta(\sigma_i)(1))-s_i) - \#(\Delta(1)) \text{ .} \end{equation} (b) {F}rom (a) and Theorem~\ref{th1} we have \begin{align*} \rho_0-\rho_1+\rho_2 = & (r-s) +s + \sum_{i=0}^w(\#(\Delta(\sigma_i)(1))-s_i) - \# (\Delta(1)) \\ \mbox{} = & r+ \sum_{i=0}^w(\#(\Delta(\sigma_i)(1))-s_i) - \# (\Delta(1)) \end{align*} which depends only on $\Delta_{top}$. \end{pf} As the next example shows, $\rho_1$ and $\rho_2$ are not topological invariants of $\Delta_{fan}$ when $r \ge 3$ and $\Delta$ is not simplicial. \begin{example} \label{ex1} Let $\Delta$ be a fan on $\Bbb R^3$ and suppose $\Delta$ consists of three cones of dimension 3 and 6 cones of dimension 1 such that for each $\sigma_i \in \Delta(3)$, $\#(\Delta(\sigma_i)(1)) = 4$. Assume that the intersection of the fan $\Delta$ with the unit sphere traces a graph that looks like that shown in Figure~\ref{fig1}. \setlength{\unitlength}{.005in} \begin{figure} \center{ \begin{picture}(400,400) \thinlines \put(50,50){\line(1,0){300}} \put(50,50){\line(5,6){250}} \put(300,350){\line(1,-6){50}} \put(50,50){\line(2,1){100}} \put(150,100){\line(1,0){150}} \put(150,100){\line(1,1){100}} \put(300,100){\line(1,-1){50}} \put(250,200){\line(1,-2){50}} \put(250,200){\line(1,3){50}} \put(50,50){\circle*{5}} \put(150,100){\circle*{5}} \put(250,200){\circle*{5}} \put(300,100){\circle*{5}} \put(350,50){\circle*{5}} \put(300,350){\circle*{5}} \put(45,50){\makebox(0,0)[r]{\small{5}}} \put(355,50){\makebox(0,0)[l]{\small{4}}} \put(145,100){\makebox(0,0)[br]{\small{2}}} \put(305,100){\makebox(0,0)[bl]{\small{1}}} \put(245,200){\makebox(0,0)[br]{\small{0}}} \put(300,355){\makebox(0,0)[b]{\small{3}}} \put(180,180){\makebox(0,0)[l]{\small{$\sigma_0$}}} \put(240,75){\makebox(0,0)[r]{\small{$\sigma_2$}}} \put(310,180){\makebox(0,0)[r]{\small{$\sigma_1$}}} \end{picture} } \vspace{-0.35in} \caption{} \label{fig1} \end{figure} For any such fan $\Delta$, $\Delta_{top}$ is unique up to homeomorphism. We consider 2 such fans $\Delta$ and $\Delta'$ such that $\rho_1(\Delta) \not = \rho_1(\Delta')$ and $\rho_2(\Delta) \not = \rho_2(\Delta')$. For $\Delta$, take $\Delta(1)$ to be $\{ \Bbb R_{\ge} \eta_i | i=0..5\}$ where $\{\eta_0, \dots, \eta_5\}$ $ =$ \begin{equation} \label{eq23.5} \left\{ \begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix}, \begin{pmatrix} -1 \\ 2 \\ -2 \end{pmatrix}, \begin{pmatrix} -1 \\ -2 \\ -2 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}, \begin{pmatrix} -1 \\ 2 \\ 2 \end{pmatrix}, \begin{pmatrix} -1 \\ -2 \\ 2 \end{pmatrix} \right\} \text{.} \end{equation} Using the methods of \cite[Section 4]{F:Elt} we find that $\rho_1(\Delta) =1$ and $\rho_2(\Delta) =1$. For $\Delta'$, take $\Delta'(1)$ to be $\{ \Bbb R_{\ge} \eta_i | i=0..5\}$ where $\{ \eta_0, \dots, \eta_5\}$ $ =$ \[ \left\{ \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} -1 \\ 3 \\ 1\end{pmatrix}, \begin{pmatrix} -2 \\ -1 \\ 1\end{pmatrix}, \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix} \right\} \text{.} \] Using the methods of \cite[Section 4]{F:Elt} we find that $\rho_1(\Delta') =0$ and $\rho_2(\Delta') =0$. \end{example} \section{A Stratification of the fibers of $\frak T$} \label{sec4} Let $\Delta$ be a fan on $N \otimes \Bbb R= \Bbb R^r$ with $\Delta(1) = \{r_0, \dotsc, r_n \}$. The intersection of $\Delta(1)$ with the unit sphere $S$ in $\Bbb R^r$ is a finite set of points, say $\{ p_0, \dotsc, p_n\}$. About each $p_i$ we can find an open ball $B_i$ on $S$ such that if $p_i$ is parametrized by $B_i$, then each choice of $\vec{p} = (p_0, p_1, \dotsc, p_n)$ in $B_0 \times B_1 \times \dotsm \times B_n$ defines a fan $\Phi = \Phi(\vec{p})$ such that $\Phi_{top} \cong \Delta_{top}$. The manifold $\displaystyle{B = \prod_{i=0}^n B_i}$ parametrizes a subset of fans in the fiber $\frak T^{-1}(\Delta_{top})$. Call $B$ an {\em open neighborhood of $\Delta$}. If $\vec{p} \in B$, then the fan $\Phi = \Phi(\vec{p})$ is not necessarily rational. Sometimes it will be necessary to refer to points in $B$ that give rise to rational fans. In this case let \begin{multline} \label{eq35} B_{rat} = \{ (p_0, \dotsc, p_n) | \text{ for each } i \text{, } \\ p_i \text{ is the intersection of a rational 1-dimensional cone } r_i \text{ with } B_i \} \text{.} \end{multline} For the present section only we define the set of support functions on a fan to be a real vector space. If $\sigma$ is a cone, define $\scr{S}\scr{F}(\sigma) $ to be $\operatorname{Hom}_{\Bbb R}(\Bbb R \sigma, \Bbb R)$. Define $\scr{S}\scr{F}(\Delta)$ to be the kernel of $\delta^0$ in the $\operatorname{\check Cech}$ complex \begin{equation} \label{eq34} 0 \to \underset{i}{\oplus} \scr{S}\scr{F}(\sigma_i) \stackrel{\delta^0}{\rightarrow} \underset{i<j}{\oplus} \scr{S}\scr{F}(\sigma_{ij}) \stackrel{\delta^1}{\rightarrow} \underset{i<j<k}{\oplus} \scr{S}\scr{F}(\sigma_{ijk}) \to \dots \end{equation} where $\{ \sigma_0, \dotsc, \sigma_w \}$ is the set of maximal cones of $\Delta$. Define $\kappa_0(\Delta) = \operatorname{dim}_{\Bbb R} \scr{S}\scr{F}(\Delta)$. If $\Delta$ is a rational fan, then this definition of $\kappa_0$ agrees with the definition given in Remark~\ref{re7} of Section~\ref{sec3}. In this section we consider the stratification of the manifold $B$ by the invariant $\kappa_0$. \begin{example} \label{ex15} If $\Delta$ is simplicial, then $\kappa_0 = \#(\Delta(1)) = n+1$ so $B$ has only 1 stratum. As was suggested in Example~\ref{ex1}, we expect the stratification to be more interesting when $\Delta$ is nonsimplicial. \end{example} \begin{example} \label{ex16} Let $\Delta$ be the fan on $\Bbb R^3$ given in equation \eqref{eq23.5} of Example~\ref{ex1}. Let $\displaystyle{B = \prod_{i=0}^5 B_i}$ be an open neighborhood of $\Delta$. One can check that any support function $h \in \scr{S}\scr{F}(\Delta)$ is completely determined by its values on $r_0, r_1, r_2, r_3$ so $\kappa_0(\Delta) \le 4$. {F}rom Example~\ref{ex1} we know that $\kappa_0(\Delta) = 4$. It is possible to vary any one of the $r_i$ to achieve a fan $\Phi$ in $B$ with $\kappa = 3$. So $B$ has exactly 2 strata. We will see later that the stratum where $\kappa_0 = 4$ is a Zariski closed subset of $B$. \end{example} \begin{conjecture} \label{conj1} Let $\Delta$ be a complete fan on $\Bbb R^3$ such that for each cone $\sigma \in \Delta(3)$, $\sigma$ is nonsimplicial. Let $B$ be an open neighborhood of $\Delta$ as described above. Then for a general choice of $\vec{p} \in B$, if $\Phi = \Phi(\vec{p})$, then every $\Phi$-linear support function is linear. In particular for a general choice of $\vec{p} \in B_{rat}$, $\kappa_0(\Phi) = 3$ hence $\rho_1(\Phi) = 0$ and $\rho_2(\Phi)$ is a topological invariant. \end{conjecture} In Conjecture~\ref{conj1} by ``general choice'' of $\vec{p}$ we mean that there is a dense open subset $G \subseteq B$ and each fan in the set $\{ \Phi(\vec p) | \vec{p} \in G \}$ satisfies the conjecture. That is, if Conjecture~\ref{conj1} is true, a sufficiently general fan $\Delta'$ with $\Delta'_{top} \cong \Delta_{top}$ should satisfy $\kappa_0(\Delta') = 3$. As motivation for Conjecture~\ref{conj1}, consider the case where each $\sigma \in \Delta(3)$ has exactly 4 1-dimensional faces. Let $\Delta(3) = \{\sigma_0, \dots, \sigma_w \}$, $\Delta(2) = \{\tau_0, \dots, \tau_e \}$, $\Delta(1) = \{r_0, \dots, r_n \}$. The intersections of the cones in $\Delta(2)$ with the unit sphere $S$ in $\Bbb R^3$ trace out the edges of a graph on $S$. This graph has $e+1$ edges, $n+1$ vertices and $w+1$ regions. So $w+1 = (e+1)-(n+1)+2$. Each $\sigma_j$ has exactly 4 $\tau_i$'s and each $\tau_i$ is in exactly 2 $\sigma_j$'s, so $2(e+1)=4(w+1)$ or $e+1 = 2(w+1)$. Hence $w+1=n-1$. {F}rom Theorem~\ref{th1}~(c) $\rho'_1 = \operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(X) \otimes \Bbb Q) = (n+1)-3 = n-2$ and $\rho'_1(\Delta(\sigma_i)) = \operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(U_{\sigma_i}) \otimes \Bbb Q) = 4-3 = 1$. From \eqref{eq5} we have an exact sequence \begin{equation} \label{eq13} 0 \to \scr{S}\scr{F}(\Delta_{top}) \otimes \Bbb Q \to \scr{W}(\Delta_{top}) \otimes \Bbb Q \to \bigoplus_{i=0}^w (\operatorname{Cl}(U_{\sigma_i}) \otimes \Bbb Q) \end{equation} \noindent Since linear Weil divisors correspond to linear Cartier divisors \eqref{eq13} gives rise to \begin{equation} \label{eq14} 0 \to \operatorname{Pic}(X) \otimes \Bbb Q \to \operatorname{Cl}(X) \otimes \Bbb Q \to \bigoplus_{i=0}^w (\operatorname{Cl}(U_{\sigma_i}) \otimes \Bbb Q) \end{equation} \noindent In \eqref{eq14} the middle term has dimension $n-2$ and the third term dimension $n-1$. For each $i$ the map $\operatorname{Cl}(X) \to \operatorname{Cl}(U_{\sigma_i})$ is surjective. So we can view $ \operatorname{Pic}(X) \otimes \Bbb Q$ as the intersection of $n-1$ hyperplanes through $(0)$ in $\Bbb Q^{n-2}$. In general, this intersection should be $(0)$. If the conclusion of Conjecture~\ref{conj1} is satisfied, then from \eqref{eq7} it follows that $\rho_2 = 3 + (w+1)-(n+1)$ (for a general choice of $\Delta_{fan}$). Next we give an algorithm for computing an upper bound for $\kappa_0$ for a fan of arbitrary dimension. The algorithm can also be used to obtain an upper bound for $\rho_1$ and $\rho_2$ for complete rational 3-dimensional fans. If $\Delta$ is a fan on $\Bbb R^3$ which contains at least 1 cone of dimension 3, then $\rho_0 = 0$ and by Theorem~\ref{th3} we have $\rho_1 = \rho_2 + (\text{topological invariant})$. In this setting $\rho_1 = \kappa_0-3$ . \begin{algorithm} \label{alg1} Let $\Delta$ be a fan on $N \otimes \Bbb R$. The following is an algorithm for computing an upper bound for $\kappa_0$. \end{algorithm} The algorithm is based on the fact that the map $\scr{S}\scr{F}(\Delta) \to \Bbb Z^{\# (\Delta(1))}$ is injective. The algorithm finds a subset $G$ of $\Delta(1)$ such that any support function $h$ in $\scr{S}\scr{F}(\Delta) \otimes \Bbb Q$ is completely determined by its values on the 1-dimensional cones in $G$. If $\sigma$ is a maximal cone in $\Delta$ of dimension $d$, then a support function $h$ is determined by its values on any $d$ 1-dimensional faces of $\sigma$ that span a $d$-dimensional subspace of $N\otimes \Bbb R$. Pick $d$ such elements of $\sigma(1)$ and place them in a set called $G$. Place all other elements of $\sigma(1)$ in a set called $R$. Initially, $G$ and $R$ are both empty, and the starting cone $\sigma$ is chosen somewhat arbitrarily. The algorithm proceeds to branch from the starting cone $\sigma$ outward until all maximal cones of $\Delta$ have been visited and $\Delta(1)$ has been partitioned into $\Delta(1)=G\cup R$. The order in which the maximal cones are traversed is somewhat arbitrary and may affect both the resulting set $G$ and the resulting cardinality of $G$. \begin{itemize} \item[Step 0.] Set $B= \{ \sigma \in \Delta | \sigma \text{ is a maximal cone in } \Delta \}$. Set $G= \emptyset$ and $R= \emptyset$. Go to Step 3. \item[Step 1.] If there is a maximal cone $\sigma \in B$ such that $\sigma(1) \cap (G \cup R)$ contains a spanning set for $\Bbb R \sigma$, then add the remaining cones in $\sigma(1)-G-R$ to $R$. Remove $\sigma$ from $B$. repeat Step~1 until the condition is false. \item[Step 2.] If there is a maximal cone $\sigma \in B$ such that $\sigma(1) \cap (G \cup R) \not = \emptyset$, then pick $\sigma \in B$ such that \begin{itemize} \item[(i)] $e = \operatorname{dim}_{\Bbb R} \langle \sigma(1) \cap (G \cup R) \rangle$ is maximal and \item[(ii)] $d = \operatorname{dim}{\sigma}$ is maximal among all $\sigma \in B$ satisfying (i). \end{itemize} For any $\sigma$ satisfying (i) and (ii), pick $\tau_1, \dots, \tau_e$ in $\sigma(1) \cap (G \cup R)$ such that $\tau_1+ \dots+ \tau_e$ has dimension $e$. Choose $\tau_{e+1}, \dots, \tau_d$ in $\sigma(1)$ such that $\tau_1+ \dots+ \tau_d$ has dimension $d$. Add $\tau_{e+1}, \dots, \tau_d$ to $G$ and add the remaining elements $\sigma(1)-G-R-\{ \tau_{e+1}, \dots, \tau_d \}$ to $R$. Delete $\sigma$ from $B$. Go to Step 1. \item[Step 3.] If $B \not = \emptyset$, then pick $\sigma \in B$ such that $d = \operatorname{dim}{\sigma}$ is maximal. Pick $\tau_{1}, \dots, \tau_d$ in $\sigma(1)$ such that $\tau_1+ \dots+ \tau_d$ has dimension $d$. Add $\tau_{1}, \dots, \tau_d$ to $G$ and add the remaining cones in $\sigma(1)-\{ \tau_{1}, \dots, \tau_d \}$ to $R$. Delete $\sigma$ from $B$. Go to Step 1. \item[Step 4.] This point is reached only if $B = \emptyset$. Now $\Delta(1)$ is partitioned into 2 sets: $\Delta(1) = G \cup R$. Any support function $h$ in $\scr{S}\scr{F}(\Delta) \otimes \Bbb Q$ is determined completely by its values on $G$. So $\scr{S}\scr{F}(\Delta) \to \Bbb Z^{\#(G)}$ is injective. Therefore $\#(G)$ is an upper bound for $\kappa_0$. \end{itemize} \begin{example} \label{ex5} Let $\Delta$ be a fan on $\Bbb R^3$ that consists of three 3-dimensional cones and assume that the intersection of the fan $\Delta$ with the unit sphere $S$ traces a graph as shown in Figure~\ref{fig2}(a). In this example, we step through Algorithm~\ref{alg1} to see that $\kappa_0(\Delta) \le 4$. It is shown later in Example~\ref{ex2} that for this fan, $\kappa_0=4$. Initially, $B=\{\sigma_0,\sigma_1,\sigma_2\}$ and $G=R=\emptyset$. The algorithm proceeds to Step~3. Place $r_1$, $r_4$, $r_2$ from $\sigma_0(1)$ in $G$ and $r_0$ in $R$. Delete $\sigma_0$ from $B$. The condition in Step~1 is still false, so the algorithm goes to Step~2. For $\sigma_1$, $r_0$ and $r_2$ are both in $G\cup R$ and $r_0+r_2$ has dimension $e=2$. Place $r_5$ in $G$ and $r_3$ in $R$. Delete $\sigma_1$ from $B$. Go to Step~1. This time the set $G\cup R$ contains $\{ r_0, r_1, r_3\}$ which is a spanning set for $\Bbb R \sigma_2$. Therefore, remove $\sigma_2$ from $B$ and place $r_6$ in $R$. Any support function $h$ is completely determined by its values on $r_1$, $r_2$, $r_4$ and $r_5$, so $\kappa_0 \le 4$. \end{example} \setlength{\unitlength}{.005in} \begin{figure} \center{ \hfill (a) \begin{picture}(400,400) \thinlines \put(200,200){\line(0,1){150}} \put(75,125){\line(5,3){125}} \put(200,200){\line(5,-3){125}} \put(75,125){\line(0,1){150}} \put(75,125){\line(5,-3){125}} \put(75,275){\line(5,3){125}} \put(200,50){\line(5,3){125}} \put(200,350){\line(5,-3){125}} \put(325,125){\line(0,1){150}} \put(200,200){\circle*{5}} \put(200,350){\circle*{5}} \put(325,125){\circle*{5}} \put(75,125){\circle*{5}} \put(325,275){\circle*{5}} \put(200,50){\circle*{5}} \put(75,275){\circle*{5}} \put(195,205){\makebox(0,0)[br]{\small{$r_0$}}} \put(200,355){\makebox(0,0)[b]{\small{$r_1$}}} \put(330,125){\makebox(0,0)[l]{\small{$r_2$}}} \put(70,125){\makebox(0,0)[r]{\small{$r_3$}}} \put(330,275){\makebox(0,0)[l]{\small{$r_4$}}} \put(200,45){\makebox(0,0)[t]{\small{$r_5$}}} \put(70,275){\makebox(0,0)[r]{\small{$r_6$}}} \put(250,250){\makebox(0,0)[c]{\small{$\sigma_0$}}} \put(200,125){\makebox(0,0)[c]{\small{$\sigma_1$}}} \put(150,250){\makebox(0,0)[c]{\small{$\sigma_2$}}} \end{picture} \hfill (b) \setlength{\unitlength}{.005in} \begin{picture}(400,400) \thinlines \put(50,50){\line(1,0){300}} \put(50,50){\line(1,1){100}} \put(50,50){\line(0,1){300}} \put(50,350){\line(1,0){300}} \put(50,350){\line(1,-1){100}} \put(150,150){\line(1,0){100}} \put(150,150){\line(0,1){100}} \put(150,250){\line(1,0){100}} \put(250,150){\line(0,1){100}} \put(250,150){\line(1,-1){100}} \put(350,50){\line(0,1){300}} \put(250,250){\line(1,1){100}} \put(50,50){\circle*{5}} \put(50,350){\circle*{5}} \put(150,150){\circle*{5}} \put(150,250){\circle*{5}} \put(250,150){\circle*{5}} \put(250,250){\circle*{5}} \put(350,50){\circle*{5}} \put(350,350){\circle*{5}} \put(250,255){\makebox(0,0)[b]{\small{0}}} \put(250,145){\makebox(0,0)[t]{\small{1}}} \put(150,145){\makebox(0,0)[t]{\small{2}}} \put(150,255){\makebox(0,0)[b]{\small{3}}} \put(355,355){\makebox(0,0)[bl]{\small{4}}} \put(355,45){\makebox(0,0)[tl]{\small{5}}} \put(45,45){\makebox(0,0)[tr]{\small{6}}} \put(45,355){\makebox(0,0)[br]{\small{7}}} \put(200,200){\makebox(0,0)[c]{\small{$\sigma_0$}}} \put(300,200){\makebox(0,0)[c]{\small{$\sigma_1$}}} \put(200,100){\makebox(0,0)[c]{\small{$\sigma_2$}}} \put(100,200){\makebox(0,0)[c]{\small{$\sigma_3$}}} \put(200,300){\makebox(0,0)[c]{\small{$\sigma_4$}}} \put(10,200){\makebox(0,0)[c]{\small{$\sigma_5$}}} \end{picture} \hfill } \caption{} \label{fig2} \end{figure} \begin{example} \label{ex2} Let $\Delta$ be a complete fan on $\Bbb R^3$ and assume that the intersection of the fan $\Delta$ with the unit sphere $S$ traces a graph that corresponds to the edges of a cube as shown in Figure~\ref{fig2}(b). Applying Algorithm~\ref{alg1} to $\Delta$, we see that $\rho_1(\Delta) \le 1$ and $\rho_2(\Delta) \le 1$. Take $\Delta(1)$ to be $\{ \Bbb R_{\ge} \eta_i | i=0..7\}$ where $\{ \eta_0, \dots, \eta_7\}$ $ =$ \[ \left\{ \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ -1 \\ 1\end{pmatrix}, \begin{pmatrix} -1 \\ -1 \\ 1 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}, \begin{pmatrix} 1 \\ -1 \\ -1\end{pmatrix}, \begin{pmatrix} -1 \\ -1 \\ -1\end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ -1 \end{pmatrix} \right\} \text{.} \] Using the methods of \cite[Section 4]{F:Elt} we find that the upper bounds predicted by Algorithm~\ref{alg1} are reached: $\rho_1(\Delta) =1$ and $\rho_2(\Delta) =2$. Now change the fan so that $\Delta'(1)$ is no longer symmetrical about the origin. For example, take $\Delta'(1)$ to be $\{ \Bbb R_{\ge} \eta_i | i=0..7\}$ where $\{ \eta_0, \dots, \eta_7\}$ $ =$ \[ \left\{ \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ -1 \\ 1\end{pmatrix}, \begin{pmatrix} -1 \\ -1 \\ 1\end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}, \begin{pmatrix} 1 \\ -1 \\ -1\end{pmatrix}, \begin{pmatrix} -1 \\ -1 \\ -1\end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ -1 \end{pmatrix} \right\} \text{.} \] Using the methods of \cite[Section 4]{F:Elt} we find that the lower bounds predicted by Conjecture~\ref{conj1} are attained: $\rho_1(\Delta') =0$ and $\rho_2(\Delta') =1$. Now $\operatorname{Pic}{X'}$ is torsion-free for the complete toric variety $X' = T_N\emb{\Delta'}$. Since $\rho_1(\Delta')=0$ we see that $\operatorname{Pic}{X'}=0$. This proves that $X'$ is nonprojective (see Remark~\ref{re4}). If $B$ is an open neighborhood of $\Delta$, then the strata of $B$ are $\kappa_0 = 4$ and $\kappa_0 = 3$. We will show later that $\kappa_0 = 4$ corresponds to a Zariski closed subset of $B$. \end{example} \begin{remark} \label{re4} In general any toric variety satisfying Conjecture~\ref{conj1} is nonprojective. This is because a projective normal variety $X$ will always have a nonprincipal Cartier divisor corresponding to a hyperplane section. \samepage{ This follows from commutative diagram \eqref{eq15}. See \cite[Ex. 6.2, p. 146]{H:AG}. } \begin{equation} \label{eq15} \begin{CD} \operatorname{Pic}{ \Bbb P^N} = \operatorname{Cl}(\Bbb P^N) @>>> \operatorname{Pic}{X} @>>> \operatorname{Cl}(X) \\ @V{\operatorname{deg}}V{\cong}V @. @VV{\operatorname{deg}}V \\ \Bbb Z @>{\cdot(\operatorname{deg}{X})}>> \Bbb Z @= \Bbb Z \\ \end{CD} \end{equation} \end{remark} \begin{example} \label{ex3} We give an example to illustrate how \eqref{eq13} can be used to compute $\kappa_0$. Say $\Delta$ consists of three 3-dimensional cones as shown in Figure~\ref{fig2}(a). Then $\scr{W}(\Delta_{top}) = \Bbb Zr_0 \oplus \dots \oplus \Bbb Zr_6$ and $\scr{W}(\Delta(\sigma_0)) = \Bbb Zr_0 \oplus \Bbb Zr_1 \oplus \Bbb Zr_2 \oplus \Bbb Zr_4$. Let $\eta_i$ be a primitive lattice point on $r_i$, so that $r_i = \Bbb R_{\ge} \eta_i$ for $i=0..6$. The kernel of the surjection $\phi_0 : \scr{W}(\Delta) \to \operatorname{Cl}(U_{\sigma_0})$ is spanned by the vectors $ \begin{pmatrix} 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{pmatrix} ^\top$, $ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{pmatrix} ^\top$, $ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix} ^\top$, and the columns of \( \begin{pmatrix} \eta_0 & \eta_1 & \eta_2 & 0 & \eta_4 & 0 & 0 \end{pmatrix} ^\top \). So $\operatorname{ker}{\phi_0}$ is a subspace of codimension 1. Consider the matrix equation \begin{equation} \label{eq16} \begin{pmatrix} \eta_0 & \eta_1 & \eta_2 & \eta_4 \end{pmatrix} \overrightarrow{v_0} = 0 \text{.} \end{equation} Set $A = \begin{pmatrix} \eta_0 & \eta_1 & \eta_2 \end{pmatrix}$. Then \eqref{eq16} becomes \begin{equation} \label{eq16.6} \begin{pmatrix} I & A^{-1} \eta_4 \end{pmatrix} \overrightarrow{v_0} = 0 \text{.} \end{equation} Set $ A^{-1} \eta_4 = \begin{pmatrix} a_0 & b_0 & c_0 \end{pmatrix}^ \top $. So $\overrightarrow{v_0} = \begin{pmatrix} -a_0 z & -b_0 z & -c_0 z & z \end{pmatrix} ^ \top $. Since any 3 columns of the matrix in \eqref{eq16} are linearly independent, $\overrightarrow{v_0}$ has 4 nonzero entries, or $\overrightarrow{v_0}=0$. Normalize $\overrightarrow{v_0}$ by taking $z = -1$. Then $\operatorname{ker}{\phi_0}$ is the set of solutions to \begin{equation} \label{eq17} \begin{pmatrix} a_{0} & b_{0} & c_{0} & 0 & -1 & 0 & 0 \end{pmatrix} \vec{x} = 0 \text{.} \end{equation} Hence if $\phi : \scr{W}(\Delta) \to \operatorname{Cl}(U_{\sigma_0}) \oplus \operatorname{Cl}(U_{\sigma_1}) \oplus \operatorname{Cl}(U_{\sigma_2})$, then $\operatorname{ker}{\phi}$ is the set of solutions to \begin{equation} \label{eq18} \begin{pmatrix} a_{0} & b_{0} & c_{0} & 0 & -1 & 0 & 0 \\ a_{1} & 0 & b_{1} & c_{1} & 0 & -1 & 0 \\ a_{2} & b_{2} & 0 & c_{2} & 0 & 0 & -1 \end{pmatrix} \vec{x} = 0 \text{.} \end{equation} This coefficient matrix has rank 3 so $\operatorname{ker}{\phi}$ has rank 4. Therefore $\kappa_0 =4$ and for any open neighborhood $B$ of $\Delta$, $B$ has only 1 stratum. In this case, we see that $\kappa_0$ and hence $\rho_1$ and $\rho_2$ are topological invariants of the set of all (rational) fans that look like the one shown in Figure~\ref{fig2}(a). We could assume $\Delta$ has more than three (say $w+1$) 3-dimensional cones each with four 1-dimensional faces meeting around the common 1-dimensional face $r_0$. By a similar argument we see that $\kappa_0 = w+2$. \end{example} \begin{example} \label{ex4} Let $\Delta$ be a fan on $\Bbb R^3$ such that $\Delta_{top}$ is homeomorphic to the fan in Example~\ref{ex1}. Following the procedure of Example~\ref{ex3}, set up equations analogous to \eqref{eq16} \eqref{eq17} and \eqref{eq18}. Then $\operatorname{ker}{\phi}$ is the set of solutions to \begin{equation} \label{eq19} \begin{pmatrix} a_{0} & 0 & b_{0} & c_{0} & 0 & -1 \\ a_{1} & b_{1} & 0 & c_{1} & -1 & 0 \\ 0 & a_{2} & b_{2} & 0 & c_{2} & -1 \end{pmatrix} \vec{x} = 0 \text{.} \end{equation} The coefficient matrix in \eqref{eq19} clearly has rank 2 or more. This agrees with the upper bound 4 predicted for $\kappa_0$ by Algorithm~\ref{alg1}. The third, fourth and sixth columns of \eqref{eq19} are independent if and only if \begin{equation} \label{eq20} (b_2-b_0) c_1 \not = 0 \text{ .} \end{equation} This shows that on the complement of a Zariski open subset of $B$, $\kappa_0 = 3$. We check that \eqref{eq20} is satisfied on a nonempty subset of $B$. Note that \eqref{eq20} is satisfied if \begin{equation} \label{eq20.5} \text{the second row of } \begin{pmatrix} \eta_0 & \eta_2 & \eta_3 \end{pmatrix}^{-1} \eta_5 \not = \text{the second row of } \begin{pmatrix} \eta_1 & \eta_2 & \eta_4 \end{pmatrix}^{-1} \eta_5 \end{equation} which will be true for a sufficiently general choice of the fan. To see this, consider letting $p_0$ vary in $B_0$. Then in \eqref{eq20.5} the matrix $\begin{pmatrix} \eta_0 & \eta_2 & \eta_3 \end{pmatrix}^{-1} $ varies but the matrix $\begin{pmatrix} \eta_1 & \eta_2 & \eta_4 \end{pmatrix}^{-1}$ remains constant. \begin{comment} \setlength{\unitlength}{.006in} \begin{figure} \center{ \begin{picture}(400,400) \thinlines \put(100,100){\line(1,0){200}} \put(100,100){\line(2,5){100}} \put(200,350){\line(2,-5){100}} \put(100,100){\line(1,1){50}} \put(150,150){\line(1,0){100}} \put(250,150){\line(1,-1){50}} \put(150,150){\line(2,3){50}} \put(200,225){\line(2,-3){50}} \put(200,225){\line(0,1){125}} \put(100,100){\circle*{5}} \put(150,150){\circle*{5}} \put(200,225){\circle*{5}} \put(250,150){\circle*{5}} \put(300,100){\circle*{5}} \put(200,350){\circle*{5}} \put(200,200){\circle*{1}} \put(200,205){\circle*{1}} \multiput(190,210)(5,0){5}{\circle*{1}} \multiput(185,215)(5,0){7}{\circle*{1}} \multiput(180,220)(5,0){9}{\circle*{1}} \multiput(175,225)(5,0){11}{\circle*{1}} \multiput(170,230)(5,0){13}{\circle*{1}} \multiput(175,235)(5,0){11}{\circle*{1}} \multiput(175,240)(5,0){11}{\circle*{1}} \multiput(175,245)(5,0){11}{\circle*{1}} \multiput(175,250)(5,0){11}{\circle*{1}} \multiput(180,255)(5,0){9}{\circle*{1}} \multiput(180,260)(5,0){9}{\circle*{1}} \multiput(180,265)(5,0){9}{\circle*{1}} \multiput(180,270)(5,0){9}{\circle*{1}} \multiput(185,275)(5,0){7}{\circle*{1}} \multiput(185,280)(5,0){7}{\circle*{1}} \multiput(185,285)(5,0){7}{\circle*{1}} \multiput(185,290)(5,0){7}{\circle*{1}} \multiput(190,295)(5,0){5}{\circle*{1}} \multiput(190,300)(5,0){5}{\circle*{1}} \multiput(190,305)(5,0){5}{\circle*{1}} \multiput(190,310)(5,0){5}{\circle*{1}} \multiput(195,315)(5,0){3}{\circle*{1}} \multiput(195,320)(5,0){3}{\circle*{1}} \multiput(195,325)(5,0){3}{\circle*{1}} \multiput(195,330)(5,0){3}{\circle*{1}} \put(200,340){\circle*{1}} \multiput(250,150)(-5,5){25}{\circle*{1}} \multiput(150,150)(1.25,5){50}{\circle*{1}} \multiput(250,150)(-1.25,5){50}{\circle*{1}} \multiput(150,150)(5,5){25}{\circle*{1}} \put(95,100){\makebox(0,0)[r]{\small{$r_5$}}} \put(305,100){\makebox(0,0)[l]{\small{$r_4$}}} \put(145,150){\makebox(0,0)[br]{\small{$r_2$}}} \put(255,150){\makebox(0,0)[bl]{\small{$r_1$}}} \put(100,225){\makebox(0,0)[c]{\small{$r_0$}}} \put(107,225){\vector(1,0){85}} \put(195,350){\makebox(0,0)[r]{\small{$r_3$}}} \put(200,125){\makebox(0,0)[c]{\small{$\sigma_2$}}} \put(200,250){\makebox(0,0)[c]{$\bold G$}} \end{picture} } \vspace{-0.75in} \caption{} \label{fig4} \end{figure} \end{comment} \end{example} Consider \eqref{eq13} once again. Let $\Delta$ be a complete fan on $\Bbb R^3$. Let $B$ be an open neighborhood of $\Delta$. Proceed as in Examples \ref{ex3} and \ref{ex4}. Set up the matrix equation $\Phi \vec{x} = 0$ for $\operatorname{ker}{\phi}$. Since $M \to \operatorname{SF}(\Delta'_{top})$ is injective, $\operatorname{ker}{\phi}$ has rank at least 3. Consider an arbitrary $(n-2)- \text{by}-(n-2)$ submatrix $\Phi_0$ of $\Phi$. Then $\Phi_0$ has rank $n-2$ exactly when $\det(\Phi_0) \not = 0$. As in \eqref{eq19} and \eqref{eq20}, we can show that $\det(\Phi_0)=0$ is an equation in no more than $3(n-2)$ variables which are parametrized by points in $B$. The equation $\det(\Phi_0) =0$ defines a Zariski closed subset of $B$. On the complement of this closed set $\det(\Phi_0) \not = 0$, $\operatorname{rank}{\Phi} = n-2$ and $\operatorname{ker}{\phi}$ has rank 3. If there is at least one choice of $\Delta_{fan}$ for which $\det(\Phi_0) \not =0$, then the open set making up the complement of the determinant equations will be nonempty, hence the conclusion of Conjecture~\ref{conj1} will be satisfied. This shows for example that the general fan which is topologically homeomorphic to that of Figure~\ref{fig2}(b) satisfies the conclusion to Conjecture~\ref{conj1}, because in Example~\ref{ex2} an example is given which shows the determinants are nonzero on a nonempty Zariski open in $B$.

9505

alg-geom/9505018

en

https://arxiv.org/abs/alg-geom/9505018

alg-geom/9505018

Ron Stern

Ronald Fintushel and Ronald Stern

Rational Blowdowns of Smooth 4-Manifolds

34 pages with 14 figures (author-supplied), AMSLaTeX

In this paper we introduce a surgical procedure, called a rational blowdown, for a smooth 4-manifold X and determine how this procedure affects both the Donaldson and Seiberg-Witten invariants of X.

alg-geom

\section{Introduction\label{Intro}} The invariants of Donaldson and of Seiberg and Witten are powerful tools for studying smooth $4$-manifolds. A fundamental problem is to determine procedures which relate smooth $4$-manifolds in such a fashion that their effect on both the Donaldson and Seiberg-Witten invariants can be computed. The purpose of this paper is to initiate this study by introducing a surgical procedure, called rational blowdown, and to determine how this procedure affects these two sets of invariants. The technique of rationally blowing down and its effect on the the Donaldson invariant were first announced at the 1993 Georgia International Topology Conference and represents the bulk of the mathematics in this paper. We fell upon this surgical procedure while we were investigating the behavior of the Donaldson invariant in the presence of embedded spheres and while investigating methods for producing a topological logarithmic transform. As it turns out, this rational blowdown procedure allows for the full computation of the Donaldson series (and Seiberg-Witten invariants) of all elliptic surfaces with $p_g \ge 1$ with the only input being the Donaldson invariants of the Kummer surface; in particular this computation shows that the Donaldson series of elliptic surfaces is that conjectured by Kronheimer and Mrowka in \cite{KM}: \begin{th} Let $E(n;p,q)$ be the simply connected elliptic surface with $p_g=n-1$ and with multiple fibers of relatively prime orders $p,q\ge1$. Then \[{\bold{D}}_{E(n;p,q)}=\exp(Q/2){\sinh^n(f)\over\sinh(f_p)\sinh(f_q)}.\] \end{th} \noindent This theorem gives another, more topological, proof of the diffeomorphism classification of elliptic surfaces (\cite{Bauer,MorganMrowka,MorganOGrady,Fried1}). This procedure also goes further and routinely computes the Donaldson series (and Seiberg-Witten invariants) for many $4$-manifolds, some of which are complex surfaces, and for most of the currently known examples which are not even homotopy equivalent to complex surfaces. The ideas presented in this paper have led to rather easy proofs of the blowup formulas for the Donaldson invariants for arbitrary smooth $4$-manifolds \cite{FSblowup} and alternate proofs and generalizations \cite{FSstructure} of some of the results announced by Kronheimer and Mrowka (\cite{KM},\cite{KMbigpaper}). While we chose to first write up these later results, another major delay in the appearance of this paper was the introduction of the Seiberg-Witten invariants. { }From the beginning, Witten has conjectured how the Seiberg-Witten invariants and the Donaldson invariants determine each other (cf. \cite{Witten}). Some progress in proving this relationship has been announced by V. Pidstrigach and A. Tyurin. Our techniques verify Witten's conjecture for elliptic surfaces and for a large class of manifolds obtained from them by rational blowdowns. (See \S 8.) Here is an outline of the paper: In \S 2 we introduce the concept of a rational blowdown and discuss relevant topological issues. Our main analytical result, Theorem~\ref{basic}, gives a universal formula which relates the Donaldson invariants of a manifold with those of its rational blowdown. Three examples of the effect of a rational blowdown are given in \S 3 and these examples are used in subsequent sections to compute the universal quantities given in Theorem~\ref{basic}. In \S 4 we give the fundamental definitions of the Donaldson series, and \S 5 presents our key analytical results. Here we shall take advantage of our later results and techniques (\cite{FSblowup},\cite{FSstructure}) to streamline our earlier arguments. In particular, we will utilize the \lq\lq \ pullback --- pushforward " point of view introduced and developed by Cliff Taubes in \cite{Sxl,Reds,Circle,Holo} (or, alternatively the thesis of Wieczorek \cite{W}) to prove our basic universal formula (Theorem~\ref{basic}). Under the assumption of simple type, this universal formula takes on a particularly simple form (Theorem~\ref{BASIC}). Starting with the computations of the Donaldson series for elliptic surfaces without multiple fibers given in \cite{KM},\cite{FSstructure} and \cite{L}, we apply Theorem~\ref{BASIC} and some of the examples presented in \S 3 to compute the Donaldson series of the elliptic surfaces with multiple fibers in \S 6. Under the assumption of simple type and the additional assumption that the configuration of curves that is blown down is `taut', Theorem~\ref{BASIC} yields a very simple formula relating the basic classes of $X$ with those of its rational blowdowns (cf. Theorem~\ref{tautcalc}). This, as well as applications to the computations of the Donaldson series of other manifolds, is discussed in \S 7. Theorem~\ref{BASIC} has a straightforward analogue relating the Seiberg-Witten invariants of $X$ and those of its rational blowdowns. We conclude this paper with a statement and proof of this relationship in \S 8. \bigskip \section{The Topology of Rational Blowdowns\label{topology}} In this section we define what is meant by a rational blowdown. Let $C_p$ denote the simply-connected smooth $4$-manifold obtained by plumbing the $(p-1)$ disk bundles over the $2$-sphere according to the linear diagram \centerline{\unitlength 1cm \begin{picture}(5,2) \put(.9,.7){$\bullet$} \put(1,.8){\line(1,0){1.3}} \put(2.2,.7){$\bullet$} \put(2.3,.8){\line(1,0){.75}} \put(3.3,.8){.} \put(3.5,.8){.} \put(3.7,.8){.} \put(4,.8){\line(1,0){.75}} \put(4.65,.7){$\bullet$} \put(.35,1.1){$-(p+2)$} \put(2.1,1.1){$-2$} \put(4.55,1.1){$-2$} \put(.45,.4){$u_{p-1}$} \put(2.1,.4){$u_{p-2}$} \put(4.55,.4){$u_1$} \end{picture}} \noindent Here, each node denotes a disk bundle over $S^2$ with Euler class indicated by the label; an interval indicates that the endpoint disk bundles are plumbed, i.e identified fiber to base over the upper hemisphere of each $S^2$. Label the homology classes represented by the spheres in $C_p$ by $u_1,\dots,u_{p-1}$ so that the self-intersections are $u_{p-1}^2=-(p+2)$ and, for $j=1,\dots,p-2$, $u_j^2=-2$. Further, orient the spheres so that $u_j\cdot u_{j+1}=+1$. Then $C_p$ is a $4$-manifold with negative definite intersection form and with boundary the lens space $L(p^2,p-1)$. \begin{lem}\label{ratball} The lens space $L(p^2,p-1)=\partial C_p$ bounds a rational ball $B_p$ with \ $\pi_1(B_p)={\bold{Z}}_p$ and a surjective inclusion induced homomorphism \ $\pi_1(L(p^2,p-1)={{\bold{Z}}}_{p^2}\to \pi_1(B_p)$. \end{lem} \begin{pf} There are several constructions of $B_p$; we present three here. The first construction is perhaps amenable to showing that if the configuration of spheres $C_p$ are symplectically embedded in a symplectic $4$-manifold $X$, then the rational blowdown $X_p$ is also symplectic (cf. \cite{Gompf}). For this construction let ${\bold{F}}_{p-1}$, $p\ge 2$, be the simply connected ruled surface whose negative section $s_-$ has square $-(p-1)$. Let $s_+$ be a positive section (with square $(p-1)$) and $f$ a fiber. Then the homology classes $s_++f$ and $s_-$ are represented by embedded $2$-spheres which intersect each other once and have intersection matrix \[ \begin{pmatrix} p+1& 1\\ 1 & -(p-1) \end{pmatrix} \] It follows that the regular neighborhood of this pair of $2$-spheres has boundary $L(p^2,p-1)$. Its complement in ${\bold{F}}_{p-1}$ is the rational ball $B_p$. The second construction begins with the configuration of $(p-1)$ $2$-spheres \centerline{\unitlength 1cm \begin{picture}(5,2) \put(.9,.7){$\bullet$} \put(1,.8){\line(1,0){1.3}} \put(2.2,.7){$\bullet$} \put(2.3,.8){\line(1,0){.75}} \put(3.3,.8){.} \put(3.5,.8){.} \put(3.7,.8){.} \put(4,.8){\line(1,0){.75}} \put(4.65,.7){$\bullet$} \put(.6,1.1){p+2} \put(2.2,1.1){2} \put(4.65,1.1){2} \end{picture}} \noindent in $\#(p-1){\bold{CP}}^{\,2}$ where the spheres (from left to right) represent \[ 2h_1-h_2+\cdots-h_{p-1}, \ h_1+h_2, \ h_2+h_3, \dots , h_{p-2}+h_{p-1}\] where $h_i$ is the hyperplane class in the $i$\,th copy of ${\bold{CP}}^{\,2}$. The boundary of the regular neighborhood of the configuration is $L(p^2,p-1)$ and the classes of the configuration span $H_2({\bold{CP}}^{\,2};{\bold{Q}})$. The complement is the rational ball $B_p$. The third construction is due to Casson and Harer \cite{CH}. It utilizes the fact that any lens space is the double cover of $S^3$ branched over a 2-bridge knot. The 2-bridge knot $K((1-p)/p^2)$ corresponding to $L(p^2,1-p)$ is slice, and $B_p$ is the double cover of the $4$-ball branched over the slice disk. \end{pf} That all these constructions produce the same rational ball $B_p$ is an exercise in Kirby calculus. However, for the purposes of this paper, it is the third construction that is the most useful, since it allows us to quickly prove: \begin{cor} Each diffeomorphism of $L(p^2,1-p)$ extends over the rational ball $B_p$. \label{lensdiff}\end{cor} \begin{pf} It is a theorem of Bonahon \cite{Bonahon} that $\pi_0(\text{Diff}(L(p^2,1-p))=\bold{Z}_2$, and is generated by the deck transformation $\tau$ of the double branched cover of $K((1-p)/p^2)$. The extension of $\tau$ to $B_p$ is given by the deck transformation of the double cover of $B^4$ branched over the slice disk.\end{pf} Suppose that $C_p$ embeds in a closed smooth $4$-manifold $X$. Then let $X_p$ be the smooth $4$-manifold obtained by removing the interior of $C_p$ and replacing it with $B_p$. Corollary~\ref{lensdiff} implies that this construction is well-defined. We call this procedure a {\bf rational blowdown} and say that $X_p$ is obtained by {\bf rationally blowing down} $X$. Note that $b^+(X)=b^+(X_p)$ so that rationally blowing down increases the signature while keeping $b^+$ fixed. An algebro-geometric analogue of rationally blowing down is discussed in \cite{KSB}. With respect to the basis $\{u_1,\dots,u_{p-1}\}$ for $H_2(C_p)$, the plumbing matrix for $C_p$ is given by the symmetric $(p-1)\times(p-1)$ matrix \[ P= \begin{pmatrix} -2 & 1& & & & & \\ 1 & -2& 1& & &0 & \\ 0 & 1&-2 &1 & & & \\ & & & &\ddots & & \\ & 0& & & & -2& 1 \\ & & & & &1 & -(p+2) \end{pmatrix}\] with inverse given by $(P^{-1})_{i,j} =-j+{(ij)(p+1)\over p^2}$ for $j\le i$. Let $Q:H_2(C_p,\partial C_p;{{\bold{Z}}})\times H_2(C_p;{\bold{Z}}) \to {\bold{Z}}$ be the (relative) intersection form of $C_p$ and let $\{\gamma_1,\dots,\gamma_{p-1}\}$ be the basis of $H_2(C_p,\partial C_p;\bold{Z})$ dual to the basis $\{u_1,\dots, u_{p-1}\}$ of $H_2(C_p;{\bold{Z}})$ with respect to $Q$. I.e. $\gamma_k\cdot u_\ell=\delta_{k\ell}$. Let $i_\ast:H_2(C_p;{{\bold{Z}}}) \to H_2(C_p,\partial C_p;{\bold{Z}})$ be the inclusion induced homomorphism. Then the intersection form of $H_2(C_p,\partial C_p;{\bold{Q}})$ is defined by \[ \gamma_k\cdot\gamma_\ell={1\over p^2}\gamma_k \cdot {\gamma}'_\ell \] where ${\gamma}'_\ell\in H_2(C_p;{\bold{Z}})$ is chosen such that $i_\ast({\gamma}'_\ell)=p^2\gamma_\ell$. Since ${\gamma}'_\ell=p^2P^{-1}({\gamma}_\ell)$, the intersection matrix for $H_2(C_p,\partial C_p;{\bold{Q}})$ is $(\gamma_k\cdot\gamma_\ell)=P^{-1}$. Note also that using the sequence \[\begin{CD} 0 \to H_2(C_p;{\bold{Z}}) @>P>> H_2(C_p,\partial C_p;{\bold{Z}}) @>\partial>> H_1(L(p^2,1-p;{\bold{Z}}) \to 0 \end{CD}\] we may identify $H_1(L(p^2,1-p;{\bold{Z}})$ with ${{\bold{Z}}}_{p^2}$ so that $\partial$ is given by $\partial(\gamma_j)=j$. There is an alternative choice of dual bases for $H_2(C_p;{\bold{Z}})$ and $H_2(C_p,\partial C_p;{\bold{Z}})$ that we shall find useful because of its symmetry. Define the basis $\{v_i\}$ of $H_2(C_p;{\bold{Z}})$ by \[ v_i= u_{p-1}+\cdots + u_i, \hspace{.25in} u_j=v_j-v_{j+1} \] so $v_i^2=-(p+2)$ for each $i$, and if $i\ne j$ then $v_i\cdot v_j=-(p+1)$. The dual basis $\{\delta_i\}$ of $H_2(C_p,\partial C_p;{\bold{Z}})$ is given in terms of $\{\gamma_i\}$ by \begin{eqnarray*} \delta_i&=&\gamma_i-\gamma_{i-1}, \ \ i\ne 1\\ \delta_1&=&\gamma_1 \end{eqnarray*} Then \begin{eqnarray*} \delta_i\cdot\delta_j&=&{(p+1)\over p^2}, \ \ i\ne j\\ \delta_i^2&=&-{(p^2-p-1)\over p^2} \end{eqnarray*} and \[ \partial(\sum a_i\delta_i) = \sum a_i. \] Let the character variety of $SO(3)$ representations of $\pi_1(L(p^2,1-p))$ mod conjugacy be denoted by $\chi_{SO(3)}(L(p^2,1-p))$, and identify $\pi_1(L(p^2,1-p))$ with ${\bold{Z}}_{p^2}$ as above. Then we have an identification \[ \chi_{SO(3)}(L(p^2,1-p))\cong {{\bold{Z}}}_{p^2}/\{\pm1\}\cong H_1(L(p^2,1-p);{{\bold{Z}}})/\{\pm1\}. \] Let $\eta$ be the generator of $\chi_{SO(3)}(L(p^2,1-p))$ satisfying \[ \eta(1) =\begin{pmatrix} \cos({2\pi i/p^2}) &\sin({2\pi i/p^2})&0\\ -\sin({2\pi i/p^2})& \cos({2\pi i/p^2})&0\\ 0&0&1 \end{pmatrix} \] Let $e\in H_2(C_p,\partial C_p;{\bold{Z}})$; so $\partial e$ is some $n_e\in {\bold{Z}}_{p^2}$. Since $b^+(C_p)=0$, $e$ defines an anti-self-dual connection $A_e$ on the complex line bundle $L_e$ over $C_p$ whose first chern class is the Poincar\'e dual of $e$. Throughout this paper we shall identify $H_2(C_p,\partial C_p;{\bold{Z}})\equiv H^2(C_p;{\bold{Z}})$; so we may write $c_1(L_e)=e$. Consider $C_p$ with a metric which gives a collar $L(p^2,1-p)\times [ 0,\infty)$. The connection $A_e$ has an asymptotic value as $t\to\infty$, and this is a flat connection on $L(p^2,1-p)$. Dividing out by gauge equivalence, we obtain the element $\partial A_e=\eta^{n_e}\in\chi_{SO(3)}(L(p^2,1-p))$. For later use, we define \[ \partial':H_2(C_p,\partial C_p;{{\bold{Z}}})\to\chi_{SO(3)}(L(p^2,p-1)) ={{\bold{Z}}}_{p^2}/\{\pm1\}=\{0,1,\dots,[p/2]\} \] by $\partial'(e)=\bar{n}_e$, the equivalence class of $\partial e$. \bigskip \section{Examples of Rational Blowdowns\label{examples}} In this section we present four examples of the effect of rational blowdowns. These are essential for our later computations. \noindent {\bf Example 1.} Logarithmic transform as a rational blowdown This first example, whose discovery motivated our interest in this procedure, shows that a logarithmic transform of order $p$ can be obtained by a sequence of $(p-1)$ blowups (i.e. connect sum with $(p-1)$ copies of $\overline{\bold{CP}}^{\,2}$) and one rational blowdown of a natural embedding of the configuration $C_p$. First, some terminology. Recall that simply connected elliptic surfaces without multiple fibers are classified up to diffeomorphism by their holomorphic Euler characteristic $n=e(X)/12=p_g(X)+1$. The underlying smooth $4$-manifold is denoted $E(n)$. The tubular neighborhood of a torus fiber is a copy of $T^2\times D^2=S^1\times(S^1\times D^2)$. By a {\it log transform} on $E(n)$ we mean the result of removing this $T^2\times D^2$ from $E(n)$ and regluing it by a diffeomorphism $$\varphi: T^2\times\partial D^2\to T^2\times\partial D^2.$$ The {\it order} of the log transform is the absolute value of the degree of $${\text {pr}}_{\partial D^2}\circ\varphi:{\text {pt}}\times \partial D^2\to \partial D^2.$$ Let $E(n)_{\varphi}$ denote the result of this operation on $E(n)$. Note that multiplicity 0 is a possibility. It follows from Moishezon \cite{Moish} that if $\varphi$ and $\varphi'$ have the same order, there is a diffeomorphism, fixing the boundary, from $E(n)_{\varphi}$ to $E(n)_{\varphi'}$. What is needed here is the existence of a cusp neighborhood (cf. \cite{FScusp}). Let $E(n;p)$ denote any $E(n)_{\varphi}$ where the multiplicity of $\varphi$ is $p$. In $E(n;p)$ there is again a copy of the fiber $F$, but there is also a new torus fiber, the {\em multiple fiber}. Denote its homology class by $f_p$; so in $H_2(E(n;p);{\bold{Z}})$ we have $f=p\,f_p$. We can continue this process on other torus fibers; to insure that the resulting manifold is simply connected we can take at most two log-transforms with orders that are pairwise relatively prime. Let the orders be $p$ and $q$ and denote the result by $E(n;p,q)$. We sometimes write $E(n;p,q)$ in general, letting $p$ or $q$ equal $1$ if there are fewer than $2$ multiple fibers. Of course one can take arbitrarily many log transforms (which we shall sometimes do) and we denote the result of taking $r$ log transforms of orders $p_1,\dots,p_r$ by $E(n;p_1,\dots,p_r)$. The homology class $f$ of the fiber of $E(n)$ can be represented by an immersed sphere with one positive double point (a nodal fiber). Figure 1 represents a handlebody (Kirby calculus) picture for a cusp neighborhood $N$ which contains this nodal fiber. (See \cite{Kirby} for an explaination of such pictures and how to manipulate them.) Blow up this double point (i.e. take the proper transform of $f$) so that the class $f-2e_1$ (where $e_1$ is the homology class of the exceptional divisor) is represented by an embedded sphere with square $-4$ (cf. Figure 2). This is just the configuration $C_2$. Now the exceptional divisor intersects this sphere in two positive points. Blow up one of these points, i.e. again take a proper transform. One obtains the homology classes $u_2=f-2e_1-e_2$ and $u_1=e_1-e_2$ which form the configuration $C_3$. Continuing in this fashion, $C_p$ naturally embeds in $N\#_{p-1}{\overline{\bold{CP}}^{\,2}}\subset E(n)\#_{p-1}{\overline{\bold{CP}}^{\,2}}$ as in Figure 3. Our first important example of a rational blown down is: \begin{thm}\label{lgtr} The rational blowdown of the above configuration $C_p\subset E(n)\#(p-1)\overline{\bold{CP}}^{\,2}$ is diffeomorphic $E(n;p)$. \end{thm} \begin{pf} As proof, we offer a sequence of Kirby calculus moves in Figures 4 through 8. In Figure 4 we add to Figure 3 the handle (with framing $-1$) which has the property that when added to $\partial C_p$ one obtains $S^2\times S^1$ (so that when a further $3$ and $4$-handle are attached $B_p$ is obtained). Then we blow down the added handle, keeping track of the dual 2-handle (which is labelled in Figure 4 with 0-framing). In Figure 5 we blow down this added handle with framing $-1$ and rearrange to obtain Figure 6. Now slide $e_1$ over the handle with framing $+1$ and rearrange to obtain Figure 7. Blow down the $-1$ curve in Figure 7; so the $-2$ curve becomes a $-1$ curve. Continue this process $p-2$ times to obtain Figure 8. If in this final picture one replaces the handle with a dot on it by a 1-handle, there results the handlebody picture given by Gompf in \cite{nuc} for $N_p$, the order $p$ log-transformed cusp neighborhood. \end{pf} \noindent For the case $p=2$, this theorem was first observed by Gompf \cite{Gompf}. Here is a useful observation: To perform a log transform of order $pq$, first perform a log transform of order $p$ and then perform a log transform of order $q$ on the resulting multiple fiber $f_p$. This can also be obtained via a rational blowdown procedure. Figure 9 is a handlebody decomposition $N_p\#_{q-1}{\overline{\bold{CP}}^{\,2}}$ with an easily identified copy of $C_q$. The proof that the result of blowing down $C_q$ results in $E(n;pq)$ is to again follow through the steps of the proof of Theorem~\ref{lgtr}. \begin{prop}\label{ponq} Let $f_p$ be the multiple fiber in $E(n;p)$. Then there is an immersed (nodal) 2-sphere $S\subset E(n;p)$ representing the homology class of $f_q$. Let $q$ be a positive integer relatively prime to $p$. If the process of Theorem~\ref{lgtr} is applied to $S$, i.e. if $Y$ is the rational blowdown of the configuration $C_q$ in $E(n;p)\#(q-1)\overline{\bold{CP}}^{\,2}$ obtained from blowing up $S$, then $Y\cong E(n;pq)$, the result of a multiplicity $pq$ log transform on $E$.\end{prop} \noindent {\bf Example 2.} In $E(2)$ there is an embedded sphere with self-intersection $-4$ such that its blowdown is diffeomorphic to $3{\bold{CP}}^2\#18\overline{\bold{CP}}^{\,2}$. For this, any $-4$ curve suffices; however to verify that the rational blowdown decomposes requires more Kirby calculus manipulations. The Milnor fiber $M(2,3,5)$ for the Poincar\'e homology $3$-sphere $P=\Sigma(2,3,5)$ embeds in $E(2)$ so that $E(2)=M(2,3,5)\cup W$ for some $4$-manifold $W$ (cf. \cite{FScusp}). Now $\partial M(2,3,5)=P$ also bounds another negative definite $4$-manifold $S$ which is the trace of $-1$ surgery on the left handed trefoil. It is known that $S\cup W$ is diffeomorphic to $3{\bold{CP}}^2\#11\overline{\bold{CP}}^{\,2}$. Thus, to construct the example, it suffices find a $-4$ curve in $M(2,3,5)$ whose rational blowdown produces $S\#7\overline{\bold{CP}}^{\,2}$. Recall that $M(2,3,5)$ is just the $E_8$ plumbing manifold given in Figure 10. Slide the handle labeled $h$ over the handle labeled $k$ to obtain the $-4$ curve $h+k$ in Figure 11. Blow down this $-4$ curve to obtain Figure 12. Now slide the handle labeled $h'$ over the handle labeled $k'$ to obtain Figure 13. Now succesively blow down the $-1$ curves to obtain Figure 14. Cancelling the $1-$ handle with the $2-$handle with framing $-2$ yields $S\#7\overline{\bold{CP}}^{\,2}$. \noindent{\bf Example 3.} Given any smooth $4$-manifold $X$, there is an embedding of the configuration $C_p\subset X\#(p-1)\overline{\bold{CP}}^{\,2}=Y$ with $u_i=e_{p-(i+1)}-e_{p-i}$ for $i=1,\dots,p-2$, and $u_{p-1}=-2e_1-e_2-\cdots-e_{p-1}$ such that the rational blowdown $Y_p$ of $Y$ is diffeomorphic to $X\#H_p$ where $H_p$ is the homology $4$-sphere with $\pi_1={\bold{Z}}_p$ which is the double of the rational ball $B_p$. In fact $C_p\subset \#(p-1){\overline{\bold{CP}}^{\,2}}=Y$, and, from the proof of Lemma~\ref{ratball}, the result of blowing down this configuration is just the double of $B_p$. Note that Example 3 points out that although a smooth $4$-manifold $Y$ may have a symplectic structure, it need not be the case that a rational blowdown $Y_p$ of $Y$ also have a symplectic structure. For in this example $X\#H_p$ will never have a symplectic structure since its $p-$fold cover can be written as a connected sum of two $4$-manifolds with positive $b_+$ so has vanishing Seiberg-Witten invariants and hence, by Taubes \cite{TSymplectic1}, is not symplectic. Of course, in this example the configuration $C_p$ is not symplectically embedded. This brings up the possibility that any smooth $4$-manifold can be rationally blown up to a symplectic $4$-manifold. \bigskip \section{The Donaldson Series\label{def}} In this section we outline the definition of the Donaldson invariant. We refer the reader to \cite{Donpoly} and \cite{DK} for a more complete treatment. Given an oriented simply connected $4$-manifold with a generic Riemannian metric and an $SU(2)$ or $SO(3)$ bundle $P$ over $X$, the moduli space of gauge equivalence classes of anti-self-dual connections on $P$ is a manifold ${\cal M}_X(P)$ of dimension \[8\,c_2(P)-3\,(1+b_X^+)\] if $P$ is an $SU(2)$ bundle, and \[-2p_1(P)-3\,(1+b_X^+)\] if $P$ is an $SO(3)$ bundle. It will often be convenient to treat these two cases together by identifying ${\cal M}_X(P)$ and ${\cal M}_X(\text{ad}(P))$ for an $SU(2)$ bundle $P$. Over the product ${\cal M}_X(P) \times X$ there is a universal $SO(3)$ bundle ${\bold{P}}$ which gives rise to a homomorphism $\mu:H_i(X;\bold{R})\to H^{4-i}({\cal M}_X(P);\bold{R})$ obtained by decomposing the class $-{1\over 4}p_1({\bold{P}})\in H^4({\cal M}_X \times X)$. When either $w_2(P)\ne 0$ or when $w_2(P)= 0$, $d>\frac34(1+b_X^+)$, the Uhlenbeck compactification $\overline{{\cal M}}_X(P)$ carries a fundamental class. In practice, one is able to get around this latter restriction by blowing up $X$ and considering bundles over $X\#\overline{\bold{CP}}^{\,2}$ which are nontrivial when restricted to the exceptional divisor \cite{MMblowup}. In \cite{FMbook} it is shown that for $\alpha\in H_2(X;{{\bold{Z}}})$ the classes $\mu(\alpha)\in H^2({{\cal M}}_X(P))$ extend over $\overline{{\cal M}}_X(P)$. When $b_X^+$ is odd, $\dim {\cal M}_X(P)$ is even, say equal to $2d$. In fact, a class $c\in H_2(X;{{\bold{Z}}})$ and a nonnegative integer $d\equiv -c^2+\frac12(1+b^+)$ determine an $SO(3)$ bundle $P_{c,d}$ over $X$ with $w_2(P_{c,d})\equiv c$ (mod $2$) and formal dimension $\dim {\cal M}_X(P_{c,d}) = 2d$. For $\bar{\alpha}=(\alpha_1,\dots,\alpha_d)\in H_2(X;{{\bold{Z}}})^d$, write $\mu(\bar{\alpha})=\mu(\alpha_1)\cup\cdots\cup \mu(\alpha_d)$. Then one has \[ \langle \mu(\bar{\alpha}),[\overline{{\cal M}}_X(P_{c,d})]\rangle=\int_{\overline{{\cal M}}_X(P_{c,d})}\mu(\bar{\alpha}) \] when $\mu(\bar{\alpha})$ is viewed as a $2d$-form. If $[1]\in H_0(X;{{\bold{Z}}})$ is the generator, then $\nu=\mu([1])=-\frac14p_1(\beta)\in H^4({\cal M}_X(P))$ where $\beta$ is the basepoint fibration $\tilde{{\cal M}}_X(P)\to{\cal M}_X(P)$ with $\tilde{{\cal M}}_X(P)$ the manifold of anti-self-dual connections on $P$ modulo based gauge transformations, i.e. those that are the identity on the fiber over a fixed basepoint. The class $\nu$ extends over the Uhlenbeck compactification $\overline{{\cal M}}_X(P)$ if $w_2(P)\ne0$, and in case $P$ is an $SU(2)$ bundle, the class will extend under certain dimension restrictions. Once again, these restrictions can be done away with via the tricks mentioned above \cite{MMblowup}. Consider the graded algebra \[{\bold{A}}(X)=\text{Sym}_*(H_0(X)\oplus H_2(X))\] where $H_i(X)$ has degree $\frac12(4-i)$. The Donaldson invariant $D_c=D_{X,c}$ is then an element of the dual algebra ${\bold{A}}^*(X)$, i.e. a linear function \[ D_c:{\bold{A}}(X)\to {\bold{R}}. \] This is a homology orientation-preserving diffeomorphism invariant for manifolds $X$ satisfying $b_X^+\ge3$. Throughout this paper we assume $b_X^+\ge3$ and odd. We let $x\in H_0(X)$ be the generator $[1]$ corresponding to the orientation. In case $a+2b=d>\frac34(1+b^+_X)$ and $\alpha\in H_2(X)$, \[ D_c(\alpha^ax^b)=\langle\mu(\alpha)^a\nu^b,[{\overline{{\cal M}}_X(P_{c,d})}]\rangle\, . \] We may extend $\mu$ over ${\bold{A}}(X)$, and write for $z\in{\bold{A}}(X)$ of degree $d$, $D_c(z)=\langle\mu(z),[{\overline{{\cal M}}_X(P_{c,d})}]\rangle$. Since such moduli spaces ${\cal M}_X(P_{c,d})$ exist only for $d\equiv -c^2+\frac12(1+b^+_X)$ (mod 4), the Donaldson invariant $D_c$ is defined only on elements of ${\bold{A}}(X)$ whose total degree is congruent to $-c^2+\frac12(1+b^+_X)$ (mod $4$). By definition, $D_c$ is $0$ on all elements of other degrees. When $P$ is an $SU(2)$ bundle one simply writes $D$ or $D_X$. If $Y$ is a simply connected $4$-manifold with boundary, one can similarly construct relative Donaldson invariants. A good reference for this is \cite{MMR}. When the boundary is a lens space, the theory simplifies considerably, and we get relative Donaldson invariants \[D_{Y,c}[\lambda_i]:{\bold{A}}(Y)\to {\bold{R}}.\] Following \cite{KM}, one considers the invariant \[ \hat{D}_{X,c}:\text{Sym}_*(H_2(X))\to {\bold{R}} \] defined by $\hat{D}_{X,c}(u)=D_{X,c}((1+\frac{x}{2})u)$. Whereas $D_{X,c}$ can be nonzero only in degrees congruent to $-c^2+\frac12(1+b^+)$ (mod $4$), $\hat{D}_{X,c}$ can be nonzero in degrees congruent to $-c^2+\frac12(1+b^+)$ (mod 2). The {\em Donaldson series} ${\bold{D}}_c={\bold{D}}_{X,c}$ is defined by \[{\bold{D}}_{X,c}(\alpha)=\hat{D}_{X,c}(\exp(\alpha))=\sum_{d=0}^{\infty}{{\hat{D}_{X,c}}(\alpha^d)\over d!}\] for all $\alpha\in H_2(X)$. This is a formal power series on $H_2(X)$. A simply connected $4$-manifold $X$ is said to have {\em simple type} if the relation $D_{X,c}(x^2\,z)=4\,D_{X,c}(z)$ is satisfied by its Donaldson invariant for all $z \in {\bold{A}}(X)$ and for all $c\in H_2(X;{\bold{Z}})$. This important definition is due to Kronheimer and Mrowka \cite{KM} and was observed to hold for many $4$-manifolds \cite{KMbigpaper,FSstructure}. In terms of $\hat{D}_{X,c}$, the simple type condition is that $\hat{D}_{X,c}(zx)=2\hat{D}_{X,c}(z)$ for all $z\in {\bold{A}}(X)$ and for all $c\in H_2(X;{\bold{Z}})$. The assumption of simple type assures that for each $c$, the complete Donaldson invariant $D_{X,c}$ is determined by the Donaldson series ${\bold{D}}_{X,c}$. It is still an open question whether all $4$-manifolds are of simple type. The structure theorem is: \begin{thm}[Kronheimer and Mrowka \cite{KMbigpaper,FSstructure}]\label{KMstruct} Let $X$ be a simply connected 4-manifold of simple type. Then, there exist finitely many `basic' classes $\kappa_1$, \dots, $\kappa_p\in H_2(X,{\bold{Z}})$ and nonzero rational numbers $a_1$, \dots, $a_p$ such that \[{\bold{D}}_X\ =\ \exp(Q/2)\,\sum_{s=1}^p a_se^{\kappa_s}\] as analytic functions on $H_2(X)$. Each of the `basic classes' $\kappa_s$ is characteristic, i.e. $\kappa_s\cdot x \equiv x\cdot x$ (mod $2$)for all $x\in H_2(X;{\bold{Z}})$. Further, suppose $c\in H_2(X;{\bold{Z}})$. Then \[ {\bold{D}}_{X,c}\ =\ \exp(Q/2)\,\sum_{s=1}^p(-1)^{{c^2+\kappa_s\cdot c\over2}}a_se^{\kappa_s}\] \end{thm} \noindent Here the homology class $\kappa_s$ acts on an arbitrary homology class by intersection, i.e. $\kappa_s(u)=\kappa_s\cdot u$. The basic classes $\kappa_s$ satisfy certain inequalities analogous to the adjuction formula in a complex surface \cite{KMbigpaper,FSstructure}. We shall need \begin{thm}[\cite{FSstructure}]\label{FSadj} Let $X$ be a simply connected 4-manifold of simple type and let $\{\kappa_s\}$ be the set of basic classes as above. If $u\in H_2(X;{\bold{Z}})$ is represented by an immersed $2$-sphere with $p\ge 1$ positive double points, then for each $s$ \begin{equation} 2p-2\ \ge u^2 + |\kappa_s\cdot u|. \label{adjintro} \end{equation}\end{thm} \begin{thm}[\cite{FSstructure}]\label{FSadjspecial} Let $X$ be a simply connected 4-manifold of simple type with basic classes $\{\kappa_s\}$ as above. If the nontrivial class $u\in H_2(X;{\bold{Z}})$ is represented by an immersed $2$-sphere with no positive double points, then let \[ \{\kappa_s|\, s=1,\dots,2m\}\] be the collection of basic classes which violate the inequality (\ref{adjintro}). Then $\kappa_s\cdot u=\pm u^2$ for each such $\kappa_s$. Order these classes so that $\kappa_s\cdot u=-u^2\,(>0)$ for $s=1,\dots,m$. Then \[\sum_{s=1}^ma_se^{\kappa_s+u}-(-1)^{1+b_X^+\over2}\sum_{s=1}^ma_se^{-\kappa_s-u}=0.\] \end{thm} \bigskip \section{The Basic Computational Theorem\label{gauge}} Recall that for $y\in H_2(X)$ and $F\in {\bold{A}}(X)$, interior product \[ \iota_uF(v)= (\deg(v)+1) F(uv) \] defines a derivation which we denote by $\partial_u$ and call `partial derivation'. Our basic theorem is: \begin{thm}\label{basic} Let $X$ be a simply connected $4$-manifold of simple type containing the configuration $C_p$, and let $X_p$ be the result of rationally blowing down $C_p$. Then, restricted to $X^*=X_p \setminus B_p= X\setminus C_p$: \[{\bold{D}}_{X_p}=\sum_{i=1}^{m(p)}\alpha_i(p)\partial^{n_i(p)}{\bold{D}}_{X,c_i(p)} \] where $\alpha_i(p)\in {\bold{Q}}$, $c_i(p)\in H_2(C_p;{\bold{Z}})$, $\partial^{n_i(p)}$ is an $n_i$th order partial derivative with respect to classes in $H_2(C_p;{\bold{Z}})$, and these quantities depend only on $p$, not on $X$. \end{thm} As motivation, and for use in the next section, we begin with a `by hand' calculation. \begin{lem}\label{C2} Let $X$ be a simply connected $4$-manifold containing an embedded 2-sphere $\Sigma$ of square $-4$ representing the homology class $\sigma$. Let $X_2$ be the result of rationally blowing down $\Sigma$. Then \[ {{\bold{D}}_{X_2}|}_{X^*}={\bold{D}}_X-{\bold{D}}_{X,\sigma}. \]\end{lem} \begin{pf} Here we work with $SU(2)$ connections over $X_2$ and $X$. The conjugacy classes of $SU(2)$ representations of $L(4,-1)$ are $\{\pm1,i\}$. Since a multiple of any class in $H_2(X_2;{\bold{Z}})$ lives in $H_2(X^*;{\bold{Z}})$, it suffices to evaluate $D_{X_2}(z)$ for $z\in{\bold{A}}(X^*)$. The lemma is proved by a standard counting argument obtained by stretching the neck $\partial X^*\times {\bold{R}}$ in $X_2$. Doing this with nonempty moduli spaces leads to a sequence of anti-self-dual connections (with respect to a sequence of generic metrics on $X_2$) which limit to anti-self-dual connections $A^*$ over $X^*$, and $A_B$ over $B_2$ together perhaps with instantons on $X^*$ and $B_2$. Dimension counting shows that $A^*$ is irreducible, $A_B$ is reducible (hence flat), and that no instantons occur. (The key fact is that each representation of $L(4,-1)$ has a positive dimensional isotropy group.) The flat $SU(2)$ connections on $B_2$ are $\pm1$. Thus we have \[ D_{X_2}(z)=\pm D_{X^*}[1](z)\pm D_{X^*}[-1](z). \] The invariants $D_{X^*}[\pm1](z)$ are relative Donaldson invariants of $X^*$ with the given boundary values. We first claim that $D_{X^*}[1](z)=\pm D_X(z)$. This is almost obvious by applying an argument like the one above. We need to know that there are no nontrivial reducible connections on the neighborhood $C_2$ of $\Sigma$ with boundary value $1$ and in a moduli space of negative dimension. This follows simply from the fact that if $\lambda$ is the complex line bundle whose first chern class generates $H^2(C_2;{\bold{Z}})$, then the moduli space of anti-self-dual connections on $\lambda^m+\lambda^{-m}$ has dimension $4m-3$ (see \cite{FSstructure}). To compute $D_{X^*}[-1](z)$, note that the Poincar\'e dual of $\sigma$ in $H^2(X;{\bold{Z}}_2)$ is the unique nonzero class whose restrictions to $X^*$ and $C_2$ are both $0$. When passing to structure group $SO(3)$, the representation $-1$ becomes trivial, and thus extends over $C_2$ as the trivial $SO(3)$ connection. Now one can see that $D_{X^*}[-1](z)=\pm D_{X,\sigma}(z)$. Finally, we need to determine signs. A key point following from our discussion is that they are independent of $X$. Recall from Example 2 that there is a sphere $\Sigma$ of square $-4$ in the $K3$-surface $X$ which has a rational blowdown $X_2$ with ${\bold{D}}_{X_2}=0$. Since ${\bold{D}}_{X,\sigma}=\exp(Q/2)={\bold{D}}_X$, our formula must read \[ D_{X_2}(z)=\pm(D_X(z)-D_{X,\sigma}(z)). \] To compute the overall sign, we must compare the way that signs are attached to $A_0\#\Theta_{B_2}$, and $A_0\#\Theta_{C_2}$ where $A_0$ is an anti-self-dual connection on $X^*$ with boundary value $1$ and $\Theta_{B_2}$ and $\Theta_{C_2}$ are the trivial connections on $B_2$ and $C_2$. This is done in a way similar to the proof of \cite[Theorem 2.1]{FSblowup}, and the sign is easily seen to be `$+$'. \end{pf} We now proceed toward the proof of Theorem~\ref{basic}. The first step is to understand reducible connections over $C_p$. It will be convenient here to use the symmetric dual bases $\{v_i\}$ and $\{\delta_i\}$ of \S\ref{topology}. Using these coordinates, we express elements of $H_2(C_p,\partial C_p;{\bold{Z}})$ as \[ \beta=\sum t_i\delta_i = \langle t_1,\dots,t_{p-1}\rangle. \] Classes of the form $\langle t,\dots,t,s,\dots,s\rangle $ will play a special role. We shall use the abbreviation \[ \langle t,\dots,t,s,\dots,s \rangle =\langle t,s;b\rangle \] if the number of $s$'s is $1\le b\le p-1$. If $e\in H_2(C_p,\partial C_p;{\bold{Z}})$, write ${\cal M}_e$ for the $SO(3)$-moduli space of anti-self-dual connections on $C_p$ which contains the reducible connection in the bundle $L_e\oplus {\bold{R}}$ where $c_1(L_e)=e$, and which are asymptotically flat with boundary value $\partial' e\in \chi_{SO(3)}(L(p^2,1-p))$. Note that $\partial\langle t,t+1;b\rangle =(p-1)t+b$. \begin{lem}\label{dim} Let $e=\langle t,t+1;b\rangle$ with $0\le t\le p$. Then \ $\dim{{\cal M}}_e=2t-1$. \end{lem} \begin{pf} With respect to the basis $\{\delta_i\}$, the intersection form of $H_2(C_p,\partial)$ is \begin{equation}\label{Q} Q= -{(p^2-p-1)\over p^2}\sum x_i^2 + 2\ {p+1\over p^2}\sum_{i<j}x_ix_j\end{equation} and \begin{multline*} e^2=(b(t+1)^2+(p-b-1)t^2)(-{(p^2-p-1)\over p^2})\\ +2((p-b-1)bt(t+1)+ \binom{p-b-1}{2}t^2+\binom{b}{2}(t+1)^2){p+1\over p^2}\end{multline*} Hence \begin{equation}\label{esquare} e^2={1\over p^2}(b^2+b^2p-bp^2-2bt+t^2-pt^2). \end{equation} By hypothesis, $\partial e=(p-1)t+b\ne 0$. From \cite{Lawson} we have \[ {\rho\over 2}(\partial e)=-{1\over p^2}(-2b^2-2b^2p-p^2+2bp^2+4bt-2p^2t-2t^2+2pt^2) \] and by the index theorem \cite{APS}: \[ \dim {\cal M}_e= -2e^2-\frac32-\frac12(h+\rho)(\partial e)=-2e^2-2-{\rho\over 2}(\partial e)=2t-1.\] \end{pf} \begin{lem}\label{bvlem1} Let $e=\langle t,t+1;b\rangle $ with $t\ge 0$ and $(p-1)t+b\le p^2/2$. Suppose also that $e'=\langle\alpha_1,\dots,\alpha_{p-1}\rangle$ with $\sum\alpha_i=(p-1)t+b+rp^2$, $r\ne 0,-1$. Then \ $\dim{\cal M}_{e'}>\dim{\cal M}_e$.\end{lem} \begin{pf} Using \eqref{Q}, it follows from symmetry that for fixed $s=\sum x_i$, the minimum absolute value of $Q\langle x_1,\dots,x_{p-1}\rangle$ occurs at $\mu(s)=\langle s/(p+1),\dots,s/(p+1)\rangle$, and \[ \mu(s)^2= -{(p^2-p-1)\over p^2}(p-1){s^2\over (p-1)^2} + 2\ {p+1\over p^2}\binom{p-1}{2} {s^2\over (p-1)^2} ={s^2\over p^2-p^3}.\] On the other hand by \eqref{esquare}, $e^2={1\over p^2}(b^2+b^2p-bp^2-2bt+t^2-pt^2)$. Set $s= (p-1)t+b+rp^2$. Then \[ \mu(s)^2-e^2=-{1\over p-1}(b+b^2+2br+p^2r^2+2rt(p-1)-bp).\] Since $1\le b\le p-1$, we have $bp\le p^2-p\le p^2r^2$. So \[ \mu(s)^2-e^2\le -{1\over p-1}(b+b^2+2br+2rt(p-1))\] and if we assume $r\ge 1$, \ $\mu(s)^2< e^2 \text{(}<0\text{)}$. By the index theorem, \[\dim {\cal M}_{e'}= -2{e'}^2-\frac32-\frac12(h+\rho)(\partial e')\ge -2\mu(s)^2- \frac32-\frac12(h+\rho)(\partial e)\ge \dim {\cal M}_e \] since $(h+\rho)(\partial e')=(h+\rho)(\partial e)$. Notice that we have not yet used the hypothesis that $(p-1)t+b\le p^2/2$. If $r<-1$, set $\bar{e}=\langle t',t'+1;c\rangle$ with $t',c$ chosen such that \[ (p-1)t'+c=p^2-((p-1)t+b)\ge p^2/2.\] By Lemma \ref{dim}, $\dim{\cal M}_{\bar{e}}\ge\dim{\cal M}_e$ with equality only if $t'=t$. Note that $\dim{\cal M}_{-e'}=\dim{\cal M}_{e'}$, and $-\sum\alpha_i=(p-1)t'+c-(r+1)p^2$. Since $-(r+1)\ge 1$, the case we have already handled shows that $\dim{\cal M}_{-e'}\ge\dim{\cal M}_{\bar{e}}$. \end{pf} \begin{lem}\label{bvlem2} Let $e=\langle t,t+1;b\rangle $ with $t\ge 0$. Suppose that $e'=\langle\alpha_1,\dots,\alpha_{p-1}\rangle \ne e$ but $\sum\alpha_i=(p-1)t+b$. Then $\dim{\cal M}_{e'}>\dim{\cal M}_e$ unless $e'$ is a permutation of $e$.\end{lem} \begin{pf} It suffices to show that ${e'}^2<e^2$. Write $e'=e+\nu$ where \[\nu=\langle n_1,\dots,n_{p-b-1},n_{p-b},\dots,n_{p-1}\rangle.\] Since the sum of the coordinates of $e$ and $e'$ is the same, $\sum n_i=0$. Let \[ N_L=\sum_{i=1}^{p-b-1}n_i\hspace{.5in} N_R=\sum_{i=p-b}^{p-1}n_i.\] \begin{eqnarray*} {e'}^2&=&e^2+2(N_L((p-2)t+b)+N_R((p-2)t+b-1))({p+1\over p^2})\\ && \hspace{2in} - 2(N_Lt+N_R(t+1))({p^2-p-1\over p^2}) +\nu^2\\ &=&e^2-2N_R+\nu^2 \end{eqnarray*} since $N_L+N_R=0$. Hence $\frac12(\dim{\cal M}_{e'}-\dim{\cal M}_e)=e^2-{e'}^2=-\nu^2+2N_R$. However, if $y$ is the result of adding $+1$ to $x_{i_0}$ and $-1$ to $x_{i_1}$ in $x=\langle x_1,\dots,x_{p-1}\rangle$, then $y^2-x^2= 2(x_{i_1}-x_{i_0}-1)$. Starting with $x=\la0,\dots,0\rangle$ and making these $\pm1$ moves with constant sign in each coordinate until reaching $\nu$, we see that the minimum change in the square is $-2$. This is achieved only if each coordinate operated on is originally $0$. Thus, if $N_+$ is the sum of the positive coordinates $n_i$, we have $-\nu^2\ge 2N_+$. Equality occurs only if each $n_i$ is $\pm1$ or $0$. In this case there are $N_+$ such $-1$'s. If $|N_R|<N_+$ then $-\nu^2+2N_R\ge 2(N_+-|N_R|)>0$. If $|N_R|=N_+$ then each $-1$ occurs in a coordinate $n_i$, $i= p-b,\dots,p-1$, and so $e'$ is a permutation of $e$. If $-\nu^2>2N_+$ then since $|N_R|\le N_+$, we have $-\nu^2+2N_R>0$. \end{pf} \begin{prop}\label{bv} Let $e=\langle t,t+1;b\rangle $ with $t\ge 0$ and $(p-1)t+b\le p^2/2$. If $e'=\langle\alpha_1,\dots,\alpha_{p-1}\rangle$ with $e'\equiv e$ (mod $2$) and $\dim{\cal M}_{e'}\le\dim{\cal M}_e$, then $\partial e' \le \partial e$ as elements of ${\bold{Z}}_{p^2}$. \end{prop} \begin{pf} Let $\bar{e}=\langle s,s+1;c\rangle$ with $s\ge 0$, be the unique class of this form with $0\le\partial\bar{e}\le p^2/2$ satisfying $\partial e'=\partial\bar{e}$. Lemmas \ref{bvlem1} and \ref{bvlem2} imply that unless $-p^2/2\le\sum a_i<0$, we have $\dim{\cal M}_{\bar{e}}\le \dim{\cal M}_{e'}$; so $s\le t$. This holds in any case, since we can always work with $-e'$. If $s=t$ then $\bar{e}=e$ since no class $\langle t,t+1;b'\rangle$ with $b'\ne b$ is congruent to $e\pmod2$. This means that $\partial e' \le \partial e$. \end{pf} \begin{cor}\label{bddim} Let $e=\langle t,t+1;b\rangle $ with $t\ge 0$ and $(p-1)t+b\le p^2/2$. Suppose that $e'=\langle\alpha_1,\dots,\alpha_{p-1}\rangle$ with $\partial' e'=\partial' e\in\chi_{SO(3)}(L(p^2,1-p))$ and $e'\equiv e$ (mod $2$). Then $\dim{\cal M}_{e'}=\dim{\cal M}_e+4k$, $k\ge 0$.\end{cor} \begin{pf} As above, $\dim{\cal M}_e\le \dim{\cal M}_{e'}$. But $e'\equiv e$ (mod $2$) implies that ${e'}^2=e^2$ (mod $4$); so the corollary follows from the index theorem.\end{pf} We need one more simple fact. Let $\iota :(C_p,\emptyset)\to (C_p,\partial)$ be the inclusion. \begin{lem}\label{getc2} Let $e\in H_2(C_p,\partial;{\bold{Z}})$, and suppose that $\partial e\equiv 0$ (mod $2$) in case $p$ is even. Then there is a \ $c\in H_2(C_p;{\bold{Z}})$ such that $\iota_*(c)\equiv e$ (mod $2$).\end{lem} \begin{pf} This follows directly from the exact sequence \[ 0\to H_2(C_p;{\bold{Z}})\to H_2(C_p,\partial;{\bold{Z}})\to {\bold{Z}}_{p^2}\to 0\, .\] \end{pf} We now proceed toward the proof of Theorem \ref{basic}. We shall work always with structure group $SO(3)$ and identify $SU(2)$ connections with $SO(3)$ connections on $w_2=0$ bundles. We wish to calculate $D_{X_p}(z)$ for $z\in{\bold{A}}(X^*)$. If we blow up $X^*$ and evaluate $D_{X_p\#\overline{\bold{CP}}^{\,2},e}(ze)=D_{X_p}(z)$ where $e$ is the exceptional class \cite{MMblowup}, we can work under the assumption that there are no flat connections on the complement of $B_p$ with the same $w_2$ as our given bundle. Keeping this in mind, we may simplify notation without loss by making the same assumption for our given situation, $X_p=X^*\cup B_p$. Consider a sequence of generic metrics on $X_p$ which stretch a collar on $L(p^2,1-p)=\partial B_p$ to infinite length, giving the disjoint union of $X^*$ and $B_p$ with cylindrical ends as the limit. A sequence of anti-self-dual connections $\{ A_n\}$ with respect to these metrics, each of which also lies in the divisor $V_z$ corresponding to $z\in{\bold{A}}(X^*)$, must limit to $A_{X^*} \amalg A_{B_p}$. These are anti-self-dual connections over $X^*$ and $B_p$, and a counting argument shows that $A_{X^*}\in V_z$ and $A_{B_p}$ is reducible. (Our above assumption is helpful here.) Since the only reducible connections on the rational ball $B_p$ are flat, we get \begin{equation}\label{bareqn} D_{X_p}(z)=\sum_{n=0}^{[p/2]}\pm D_{X^*}[\eta^{np}](z). \end{equation} The notation $D_{X^*}[\eta^{np}]$ stands for the relative Donaldson invariant on $X^*$ constructed from the moduli space of anti-self-dual connections over $X^*$ (with a cylindrical end) which decay exponentially to a flat connection whose gauge equivalence class corresponds to the conjugacy class of the representation $\eta^{np}$. We need to calculate the summands of \eqref{bareqn}. We begin with $n=0$, i.e. $D_{X^*}[1](z)$. Consider $D_X(z)$. To calculate this, we use a neck-stretching argument as above. We see that on $C_p$ we must get a reducible anti-self-dual connection corresponding to chern class $e$ with $\dim{\cal M}_e<0$ and $e\equiv 0\pmod2$. This last condition means that $e$ cannot have the form $\langle 0,1;b\rangle$ (recall $1\le b\le p-1$); so by Lemma \ref{dim}, $e\ne \langle t,t+1;b\rangle$, $t\ge 0$. Now Proposition \ref{bv} implies that $e=0$. Thus \[ D_X(z)=\pm D_{X^*}[1](z) \] and the sign is independent of $X$. To calculate the other terms, we must utilize techniques of Taubes \cite{Sxl,Reds,Circle,Holo} or Wieczorek \cite{W} as in \cite[\S4]{FSstructure}. We shall quickly review the methods involved and refer the reader to \cite{FSstructure} and the references given there for more details. Our plan is to evaluate all the $D_{X^*}[\eta^m](z)$ inductively. (In case $p$ is even, we only need to calculate this for $m$ even.) We do this by computing $D_{X,c_m}(z\,w_m)$ where $c_m\in H_2(X;{\bold{Z}})$ is supported in $C_p$, $m=(p-1)t+b$, and $w_m\in \text{Sym}_t(H_2(C_p;{\bold{Z}}))$ depending only on $m$ and $p$. First we obtain $c_m$. Let $e_m\in H^2(C_p;{\bold{Z}})$ be the Poincar\'e dual of $\langle t,t+1;b\rangle$. By Lemma \ref{getc2} we can find $c_m\in H_2(C_p;{\bold{Z}})\subset H_2(X;{\bold{Z}})$ such that $\iota_*(c_m)\equiv \langle t,t+1;b\rangle\pmod2$. Thus the Poincar\'e dual of $c_m$ in $H^2(X;{\bold{Z}})$ restricts to $C_p$ congruent to $e_m\pmod2$ and restricts trivially to $X^*$. A dimension counting argument shows that in the formalism of Taubes \cite{Sxl}, $D_{X,c_m}(z\,w_m)$ is the sum of terms of the form \begin{equation}\label{terms} \int_{\tilde{{\cal M}}_{X^*}[\eta^j]\times_j\tilde{{\cal M}}_{C_p,\epsilon,\ell}} \tau\wedge\tilde{\mu}(z)\wedge\tilde{\mu}(w_m). \end{equation} In this formula, $\tilde{{\cal M}}_{C_p,\epsilon,\ell}$ is the based moduli space of exponentially decaying asymptotically flat anti-self-dual connections on the $SO(3)$ bundle $E_{\epsilon,\ell}$ which is obtained from the reducible bundle $L_\epsilon\oplus{\bold{R}}$ by grafting in $\ell$ instanton bundles. (The euler class of $L_\epsilon$ is $\epsilon$, $\partial\epsilon = j$, $\epsilon\equiv e_m\pmod2$, and $\dim{\cal M}_\epsilon+8\ell\le 2t-1$.) The notation `$\times_j$' in the formula denotes the fiber product with respect to the $SO(3)$-equivariant boundary value maps \[ \partial_{C_p,\epsilon,\ell}:\tilde{{\cal M}}_{C_p,\epsilon,\ell}\to G[j], \hspace{.25in} \partial_{X^*}[j]:\tilde{{\cal M}}_{X^*}[\eta^j]\to G[j] \] where $G[j]\subset SO(3)$ is the conjugacy class $\eta^j$ of representations of $\pi_1(L(p^2,1-p))$ to $SO(3)$. If $j\ne 0,p^2/2$ then $G[j]$ is a 2-sphere, $G[0]=\{ I\}$, and, in case $p$ is even, $G[p^2/2]\cong{\bold{RP}}^2$. Also, $\tau$ denotes a 3-form which integrates to 1 over the fibers of the basepoint fibration $\beta_{X^*,j}$ i.e. $\tilde{{\cal M}}_{X^*}[\eta^j]\to {\cal M}_{X^*}[\eta^j]$. The form $\tilde{\mu}(w_m)$ is supported near the orbit of the reducible connection corresponding to $\epsilon$. (If $\ell>0$, this reducible connection lies in the Uhlenbeck compactification of ${\cal M}_{C_p,\epsilon,\ell}$.) The principal $SO(3)$ bundle $\beta_{X^*,j}$ has a reduction to a bundle with structure group $S^1$. As in \cite[\S4]{FSstructure}, we let $\varepsilon\in H^2({\cal M}_{X^*}[\eta^j])$ denote the euler class of this $S^1$ bundle. The upshot of Taubes' work cited above is that there is a form $\tilde{\mu}(w_m)$ representing a class $\mu_{SO(3)}(w_m)$ in the $SO(3)$-equivariant cohomology of an enlargement of $\tilde{{\cal M}}_{C_p,\epsilon,\ell}$. The lift $\tilde{\mu}(z)$ defines an element of the equivariant cohomology $H^{2d}_{SO(3)}(\tilde{{\cal M}}_{X^*}[\eta^j])$. Furthermore, Taubes has shown that the push-forward $(\partial_{C_p,\epsilon,\ell})_*$ is well-defined, and \[ \int_{\tilde{{\cal M}}_{X^*}[\eta^j]\times_j\tilde{{\cal M}}_{C_p,\epsilon,\ell}} \tau\wedge\tilde{\mu}(z)\wedge\tilde{\mu}(w_m) = \int_{\tilde{{\cal M}}_{X^*}[\eta^j]}\tau\wedge\tilde{\mu}(z)\wedge (\partial_{X^*}[j])^*(\partial_{C_p,\epsilon,\ell})_*(\tilde{\mu}(w_m)) \] where $(\partial_{X^*}[j])^*$ denotes pullback in equivariant cohomology. For $j=0$, $\partial_{C_p,\epsilon,\ell}:\tilde{{\cal M}}_{C_p,\epsilon,\ell}\to \{1\}$, has fiber dimension equal to $\dim\tilde{{\cal M}}_{C_p,\epsilon,\ell}= 4k+8\ell$ for some $k\ge 0$. The cohomology class of $(\partial_{C_p,\epsilon,\ell})_*(\tilde{\mu}(w_m))$ lies in $H^{2t-4k-8\ell}_{SO(3)}(\{1\};{\bold{R}})=H^{2t-4k-8\ell}(BSO(3);{\bold{R}})$ which is a polynomial algebra on the 4-dimensional class $\wp$, which pulls back over $\tilde{{\cal M}}_{X^*}[\eta^j]$ as $p_1(\beta_{X^*,j})$. For $j\ne 0,p^2/2$, let $j=t_j(p-1)+b_j$ where $1\le b_j\le p-1$. Then $\partial_{C_p,\epsilon,\ell}:\tilde{{\cal M}}_{C_p,\epsilon,\ell}\to G[j]$ has fiber dimension $2t_j+2+8\ell+4k-2$ for some $k\ge 0$; so the cohomology class of $(\partial_{C_p,\epsilon,\ell})_*(\tilde{\mu}(w_m))$ lies in $H^{2(t-t_j)-8\ell-4k}_{SO(3)}(S^2;{\bold{R}})= H^{2(t-t_j)-8\ell-4k}({\bold{CP}}^\infty;{\bold{R}})$. Let $v$ be the 2-dimensional generator of $H^*({\bold{CP}}^\infty;{\bold{R}})$. The pullback $(\partial_{X^*}[j])^*(v)=\varepsilon$. Using the fact that $\varepsilon^2=p_1(\beta_{X^*,j})$, and arguing as in \cite[Prop.4.5,4.6]{FSstructure} we get \begin{equation}\label{expand} D_{X,c_m}(z\,w_m)=\sum_{t_j\equiv t\,(2)}\sum_q r_{m,j,q}D_{X^*}[\eta^j](zx^q) +\sum\begin{Sb} t_j\not\equiv t\,(2)\\j\ne 0\end{Sb} \sum_q r'_{m,j,q}D_{X^*}[\eta^j](zx^q\varepsilon). \end{equation} The notation $D_{X^*}[\eta^j](zx^q\varepsilon)$ is not standard, but its meaning is clear. It follows from Proposition \ref{bv} that the $\eta^j$ in \eqref{expand} have $j\le m$; so this bounds $j$ in both terms. We emphasize that in order to obtain $r_{m,j,q}$ or $r_{m,j,q}'\ne 0$ we must have an $\epsilon\in H^2(C_p;{\bold{Z}})$ satisfying $\partial'\epsilon = j$, $\epsilon\equiv e_m\pmod2$, and $\dim{\cal M}_\epsilon+8q\le 2t-1$. Assume inductively that: \begin{itemize} \item[a)]For each $j< m$ ($j\equiv 0\pmod2$ if $p$ is even) there are classes $w_{j,i}\in\text{Sym}_*(H_2(C_p;{\bold{Z}}))$ and rational numbers $a_{j,i}$ satisfying \begin{equation}\label{induct} D_{X^*}[\eta^j](z)=\sum_{i=1}^j a_{j,i}D_{X,c_i}(zw_{j,i}) \end{equation} \item[b)] For each $j$ with $t_j< t-1$ (and $j\equiv 0\pmod2$ if $p$ is even) there are classes $w'_{j,i}\in\text{Sym}_*(H_2(C_p;{\bold{Z}}))$ and rational numbers $a'_{j,i}$ satisfying \begin{equation}\label{inductepsilon} D_{X^*}[\eta^j](z\varepsilon)=\sum_{i=1}^j a_{j,i}D_{X,c_i}(zw'_{j,i}) \end{equation} \end{itemize} for all $z\in{\bold{A}}(X^*)$, and the coefficients $a_{j,i},a_{j,i}'$ are independent of $z$ and $X$. Recall that we are writing $m=(t-1)p+b$ with $1\le b\le p-1$, and let $e_m$ be the Poincar\'e dual of $\langle t,t+1;b\rangle=(t+1)\gamma_{p-1}-\gamma_{p-1-b}$. Also, we suppose that $m$ is even if $p$ is even. We set \[ w_m=(u_{p-1}-(t-1)u_{p-1-b})\cdot (u_{p-1})^{t-1}\in{\bold{A}}(C_p). \] We wish to calculate $D_{X,c_m}(z\,w_m)$ using \eqref{expand}. For $j=m$ in this formula, we need to compute $(\partial_{C_p,e_m,0})_*(\tilde{\mu}(w_m))\in H^0_{SO(3)}(G[m];{\bold{R}})={\bold{R}}$ since $t_m=t$. In fact, \begin{multline*} (\partial_{C_p,e_m,0})_*(\tilde{\mu}(w_m))=r_{m,m,0}\\ = -\frac12\langle u_{p-1}-(t-1)u_{p-1-b},e_m\rangle\,(-\frac12\langle u_{p-1},e_m\rangle)^{t-1} = (-\frac12)^t(2t)(t+1)^{t-1}\ne 0\end{multline*} (cf. \cite[p.187]{DK}). In \eqref{expand}, $ r_{m,m,0}D_{X^*}[\eta^m](z)$ is the only term which involves the boundary value $\eta^m$. If $j$ is the boundary value of an $\epsilon$ with $\epsilon\equiv e_m\pmod2$, and $\dim{\cal M}_\epsilon+8q\le 2t-1$, and if $t_j=t-1$, then by Corollary~\ref{bddim} and Lemma~\ref{bvlem2}, $\epsilon$ must be a permutation of $\langle t-1,t;p-1-b\rangle= t\gamma_{p-1}-\gamma_b$. In fact $\epsilon\equiv e_m\pmod2$ implies that $\epsilon = \langle t,t-1;b\rangle=(t-1)\gamma_{p-1}+\gamma_{p-1-b}$. So $j=(t-1)(p-1)+(p-1-b)$. Hence $\langle u_{p-1}-(t-1)u_{p-1-b},\epsilon\rangle =0$. Thus, no such $j$ occurs in the second sum of the expansion \eqref{expand} for $D_{X,c_m}(z\,w_m)$. (I.e. for such $j$, necessarily $q=0$ and $r'_{m,j,q}=0$.) Finally, if $p$ is even, then we are assuming that $m$ is also even. If $r_{m,i,q}$ or $r_{m,i,q}'\ne 0$ then as above there is an $\epsilon$ with $\partial\epsilon = i$ and $\epsilon\equiv e_m\pmod2$; so for \[ \partial_2:H_2(C_p,\partial;{\bold{Z}}_2)\to H_1(L(p^2,1-p);{\bold{Z}}_2)={\bold{Z}}_2 \] $j\equiv\partial_2(\epsilon)\equiv\partial_2 e_m\equiv m\pmod2$. Accordingly, all the other terms in \eqref{expand} are given inductively by \eqref{induct} and \eqref{inductepsilon}, and the powers of $x$ can be removed using the hypothesis that $X$ has simple type. Since the coefficient of $D_{X^*}[\eta^m](z)$ is nonzero, we may solve for it, completing the induction step for \eqref{induct}. For \eqref{inductepsilon}, we show how to compute $D_{X^*}[\eta^{m'}](z\varepsilon)$ for $m'=(t-1)(p-1)+(p-1-b)$ as required. Thus after completing the inductive step for each $t(p-1)+c$, $1\le c\le p-1$, we will have completed the calculation of $D_{X^*}[\eta^j](z\varepsilon)$ for all $j=(t-1)p+a$, $1\le a\le p-1$. So to calculate $D_{X^*}[\eta^{m'}](z\varepsilon)$ and thus complete the induction, we calculate $D_{X,c_m}(z\,w'_{m'})$ where $w'_{m'}=(u_{p-1}+(t+1)u_{p-1-b})\cdot(u_{p-1}+(t-1)u_{p-1-b})\cdot (u_{p-1})^{t-2}$. Using \eqref{expand} \begin{equation}\label{another} D_{X,c_m}(z\,w'_{m'})=\sum_{t_j\equiv t\,(2)}\sum_q s_{{m'},j,q}D_{X^*}[\eta^j](zx^q) +\sum\begin{Sb} t_j\not\equiv t\,(2)\\j\ne 0\end{Sb} \sum_q s'_{{m'},j,q}D_{X^*}[\eta^j](zx^q\varepsilon).\end{equation} Computing as above, we see that $s_{m',m',0}=0$. What we need to see is that $s'_{m',m',0}\ne 0$. By the argument of the above paragraph, $m'$ is the only possible boundary value not covered by the induction step. Let $\epsilon=(t-1)\gamma_{p-1}+\gamma_{p-1-b}$. This is the only euler class that can give boundary value $m'$ in \eqref{another}. Then \[ (\partial_{C_p,\epsilon,0})_*(\tilde{\mu}(w'_m))\in H^2_{SO(3)}(S^2;{\bold{R}})\cong H^2_{SO(3)}(G[m'];{\bold{R}})={\bold{R}} \] and $(\partial_{C_p,\epsilon,0})_*(\tilde{\mu}(w'_m))= (-\frac12)^{t-1}2t(2t-2)(t-1)^{t-2}v$ which pulls back over $\tilde{{\cal M}}_{X^*}[\eta^j]$ as $(-\frac12)^{t-3}t(t-1)(t-1)^{t-2}\varepsilon$. This means that we can solve \eqref{another} for $D_{X^*}[\eta^{m'}](z\varepsilon)$, completing the induction and the proof of Theorem \ref{basic}. The argument above shows that all of the relative invariants $D_{X^*}[\eta^{np}]$ can be expressed in terms of absolute invariants of $X$. Since we are assuming that $X$ has simple type, it follows that each of the relative invariants satisfies the formula \[ D_{X^*}[\eta^{np}](z\,x^2) = 4\,D_{X^*}[\eta^{np}](z). \] Hence it follows from \eqref{bareqn} that: \begin{cor}\label{st} Let $X_p$ be the result of rationally blowing down $C_p\subset X$. If $X$ has simple type, then so does $X_p$. \ \ \qed \end{cor} Now we shall make stronger use of the hypothesis that $X$ has simple type. By \cite{KM,FSstructure} we can write \begin{eqnarray*} {\bold{D}}_X&=&\exp(Q_X/2)\sum_{s=1}^na_se^{\kappa_s} \\ {\bold{D}}_{X,c}&=&\exp(Q_X/2)\sum_{s=1}^n(-1)^{\frac12(c^2+c\cdot\kappa_s)}a_se^{\kappa_s} \end{eqnarray*} for nonzero rational numbers $a_s$ and basic classes $\kappa_1,\dots,\kappa_n\in H_2(X;{\bold{Z}})$. Here $Q_X$ is the intersection form of $X$. Now \[ \partial_u(\exp(Q_X/2)e^{\kappa})=\exp(Q_X/2)(\tilde{u}+\kappa\cdot u)e^{\kappa} \] where $\tilde{u}: H_2(X)\to\bold{R}$ is $\tilde{u}(\alpha)=u\cdot\alpha$ and $\partial_v\tilde{u}=v\cdot u$. Apply Theorem \ref{basic}: since all derivatives are taken with respect to classes $u\in H_2(C_p;\bold{Z})$, after all derivatives are taken, the remaining $\tilde{u}$'s restricted to $X^*$ vanish. Hence, \begin{equation}\label{barseries0} {{\bold{D}}_{X_p}|_{X^*}}=\exp(Q_{X^*}/2)\sum_{s=1}^na_sb_se^{\kappa_s}|_{X^*}= \exp(Q_{X^*}/2)\sum_{s=1}^na_sb_se^{\kappa'_s} \end{equation} where $\kappa'_s=\kappa_s|_{X^*}=\text{PD}(i^*(\text{PD}(\kappa_s)))\in H_2(X^*,\partial;{\bold{Z}})$, where $\text{PD}$ denotes Poincar\'e duality, $i$ is the inclusion $X^*\subset X$, and $b_s$ depends only on the intersection numbers of $\kappa_s$ with the generators $u_i$ of $H_2(C_p;\bold{Z})$. \begin{lem}\label{theyextend} If $b_s\ne0$ in \eqref{barseries0} then \[ \partial\kappa'_s\in p{\bold{Z}}_{p^2}\subset H_1(L(p^2,1-p);{\bold{Z}})={\bold{Z}}_{p^2}. \] \end{lem} \begin{pf} Corollary~\ref{st} implies that $X_p$ has simple type. We thus have \begin{equation}\label{barseries} {\bold{D}}_{X_p}=\exp(Q_{X_p}/2)\sum_{r=1}^mc_re^{\lambda_r} \end{equation} where the basic classes of $X_p$ are $\lambda_1,\dots,\lambda_m$. Restrict ${\bold{D}}_{X_p}$ to $X^*$ and compare the restrictions of $\exp(Q_{X_p}/2)^{-1}{\bold{D}}_{X_p}$ in \eqref{barseries0} and \eqref{barseries}. Since for distinct $\alpha\in H_2(X^*,\partial;{\bold{Z}})$ the functions $e^\alpha:H_2(X^*)\to{\bold{R}}$ are linearly independent, it follows that if $b_s\ne 0$, then $\kappa'_s=\lambda_i|_{X^*}$ for some $i$. Thus $\kappa'_s$ extends over $B_p$, and hence $\partial \kappa'_s\in p{\bold{Z}}_{p^2}$. \end{pf} As a result, we have the following restatement of Theorem \ref{basic}. \begin{thm}\label{BASIC} Suppose that $X$ has simple type and \[{\bold{D}}_X=\exp(Q_X/2)\sum_{s=1}^na_se^{\kappa_s}.\] Let $C_p\subset X$ and let $X_p$ be its rational blowdown. Let $\{\kappa_t|t=1,\dots,m\}$ be the basic classes of $X$ which satisfy $\partial\kappa'_t\in p{\bold{Z}}_{p^2}$, and for each $t$, let $\bar{\k}_t$ be the unique extension of $\kappa'_t$. Then \[{\bold{D}}_{X_p}=\exp(Q_{X_p}/2)\sum_{t=1}^ma_tb_te^{\bar{\k}_t}\] where the $b_t$ depend only on the intersection numbers $u_i\cdot\kappa_t$, $i=1,\dots,p-1$. \ \ \qed \end{thm} \bigskip \section{The Donaldson Invariant of Elliptic Surfaces\label{ellipticcompute}} In this section we shall compute the result on the Donaldson series of performing log transforms. The Donaldson invariants of the elliptic surfaces $E(n), n\ge2$ without multiple fibers have been known for some time. There is a complete calculation in \cite{FSstructure}, for example. For $n\ge2$: \[ {\bold{D}}_{E(n)} = \exp(Q/2)\sinh^{n-2}(f) \] where $f$ is the class of a fiber. In this notation, the $K3$ surface is $E(2)$. As in Theorem~\ref{lgtr}, let $X=E(2)\#(p-1)\overline{\bold{CP}}^{\,2}$, and let $X_p$ be the rational blowdown of $C_p\subset X$, so that $X_p\cong E(2;p)$. Since ${\bold{D}}_{E(2)}=\exp(Q/2)$, the blowup formula \cite{FSblowup} yields \begin{equation}\label{blowup} {\bold{D}}_X={1\over 2^{p-1}}\exp(Q/2)\sum_J\exp(\sum_{i=1}^{p-1}\epsilon_{J,i}e_i) \end{equation} where the outer sum is taken over all $J=(\epsilon_{J,1},\dots,\epsilon_{J,p-1})\in\{\pm1\}^{p-1}$. The basic classes of $X$ are $\{\kappa_J=\sum\epsilon_{J,i}e_i\}$, and applying Theorem~\ref{BASIC} we get \begin{equation}\label{DXbar} {\bold{D}}_{X_p}={1\over2^{p-1}}\exp(Q_{X_p}/2)\sum_Jb_Je^{\bar{\k}_J} \end{equation} where $\bar{\k}_J\in H_2(X_p;{\bold{Z}})$ is the unique extension of ${\kappa_J|}_{X^*}$. Recall that the spheres of the configuration $C_p$ represent homology classes $u_i=e_{p-(i+1)}-e_{p-i}$ for $1\le i\le p-2$, and $u_{p-1}=f-2e_1-e_2-\cdots-e_{p-1}$. In $X_p$ we have the multiple fiber $f_p=f/p$. \begin{prop}\label{J} $\bar{\k}_J= |J|\cdot f_p$ where $|J|=\sum_{i=1}^{p-1}\epsilon_{J,i}$.\end{prop} \begin{pf} First we find a class $\zeta\in H_2(C_p;{\bold{Q}})$ so that $(\kappa_J+\zeta)\cdot u_i=0$ for each $i$. This means that $\kappa_J+\zeta\in H_2(X^*;{\bold{Q}})$, and as dual forms: $H_2(X^*;{\bold{Z}})\to{\bold{Z}}$, ${\kappa_J|}_{X^*}=\kappa_J+\zeta$. To find $\zeta$ we need to solve the linear system \[ (\kappa_J+\sum x_iu_i)\cdot u_j = 0,\ \ \ j=1,\dots,p-1. \] We begin by rewriting these equations. Let $\{\omega_i\}$ be a standard basis for ${\bold{Q}}^{p-1}$, and let $A$ be the $(p-1)\times(p-1)$ matrix whose $i$th row vector is \begin{eqnarray*} A_i&=& \omega_{p-(i+1)}-\omega_{p-i}, \ \ i=1,\dots,p-2\\ A_{p-1}&=&-2\omega_1-\omega_2-\dots-\omega_{p-1}. \end{eqnarray*} We have $u_i=A^t(\omega_i)\cdot{{\bold{e}}}$ and $u_{p-1}=f+A^t(\omega_{p-1})\cdot{\bold{e}}$ , where ${\bold{e}}=(e_1,\dots,e_{p-1})$. Our linear system is equivalent to \[ P{{\bold{x}}}=A\pmb{\epsilon}_J\] where ${{\bold{x}}}=(x_1,\dots,x_{p-1})$ and $\pmb{\epsilon}_J=(\epsilon_{J,1},\dots,\epsilon_{J,p-1})$. (The matrix $P$ is the plumbing matrix for $C_p$.) Hence ${\bold{x}}=P^{-1}A\pmb{\epsilon}_J$. We claim that $P(A^t)^{-1}=-A$. This can be checked on the basis \[ \{\omega_2-\omega_1,\dots,\omega_{p-1}-\omega_{p-2},\omega_{p-1} \}\] using \begin{eqnarray*} A(\omega_i)&=&-\omega_{p-1}-\omega_{p-(i+1)}+\omega_{p-i},\ 2\le i\le p-1 \ \ (\omega_0=0),\\ A(\omega_1)&=&-2\omega_{p-1}+\omega_{p-2},\\ P(\omega_i)&=&\omega_{i+1}-2\omega_i+\omega_{i-1},\ i\ne p-1,\\ P(\omega_{p-1})&=&-(p+2)\omega_{p-1}+\omega_{p-2}. \end{eqnarray*} It follows that $A^tP^{-1}A=-I$. Thus \[\kappa_J+\zeta=\kappa_J+\sum x_iu_i=({\bold{\epsilon}}_J+A^t{{\bold{x}}})\cdot{\bold{e}}+x_{p-1}f =(\pmb{\epsilon}_J-\pmb{\epsilon}_J)\cdot{\bold{e}}+x_{p-1}f=x_{p-1}f.\] To compute $x_{p-1}$ note that \[ A\pmb{\epsilon}_J=(\epsilon_{J,p-2}-\epsilon_{J,p-1},\epsilon_{J,p-3}-\epsilon_{J,p-2},\dots,\epsilon_{J,1}-\epsilon_{J,2}, -2\epsilon_{J,1}-\epsilon_{J,2}-\cdots-\epsilon_{J,p-1}) \] so that if $(P^{-1})_{p-1}$ denotes the bottom row of $P^{-1}$: \[x_{p-1}=(P^{-1})_{p-1}(A\pmb{\epsilon}_J)=-{1\over p^2}(1,2,\dots,p-1)\cdot(A{\bold{\epsilon}}_J) ={1\over p}\sum\epsilon_{J,i}={1\over p}|J|.\] Thus ${\kappa_J|}_{X^*}=\kappa_J+\zeta={1\over p}|J|f$ as forms: $H_2(X^*;{\bold{Z}})\to{\bold{Z}}$. The homology class $\kappa_J+\zeta$ is in fact an integral class $\bar{\kappa}_J=|J|f_p\in H_2(X_p;{\bold{Z}})$ which is the unique extension of ${\kappa_J|}_{X^*}$ \end{pf} In an arbitrary smooth $4$-manifold $X$, define a {\em nodal fiber} to be an immersed 2-sphere $S$ with one singularity, a positive double point, such that the regular neighborhood of $S$ is diffeomorphic to the regular neighborhood of a nodal fiber in an elliptic surface. (There need not be any associated ambient fibration of $X$.) Given such a nodal fiber $S$, one can perform a `log transform' of multiplicity $p$ by blowing up to get $C_p\subset X\#(p-1)\overline{\bold{CP}}^{\,2}$ with $u_{p-1}=S-2e_1-e_2-\cdots-e_{p-1}$, and then blowing down $C_p$. We denote the result of this process by $X_p$. Throughout, we use the following notation. If $X$ has simple type, and $${\bold{D}}_X=\exp(Q/2)\sum a_se^{\kappa_s},$$ then we write ${\bold{K}}_X=\sum a_se^{\kappa_s}$. \begin{prop}\label{formallog} Let $S$ be a nodal fiber which satisfies $S\cdot\lambda_j=0$ for each basic class $\lambda_j$ of $X$. Then \[{\bold{D}}_{X_p}=\begin{cases} \exp(Q_{X_p}/2){\bold{K}}_X\cdot(b_{p,0}+\sum\limits_{i=1}^{p-1\over2}b_{p,2i}(e^{2iS/p}+e^{-2iS/p})),\ &p\ \text{odd}\\ \exp(Q_{X_p}/2){\bold{K}}_X\cdot(\sum\limits_{i=1}^{p\over2}b_{p,2i-1}(e^{(2i-1)S/p}+e^{-(2i-1)S/p})), &p\ \text{even}\end{cases}\] where the coefficients $b_{p,j}$ depend only on $p$, not on $X$. \end{prop} \begin{pf} The Donaldson series of $X\#(p-1)\overline{\bold{CP}}^{\,2}$ is \[ {1\over2^{p-1}}{\bold{D}}_X\cdot\exp(Q_{(p-1)\overline{\bold{CP}}^{\,2}}/2)\sum_J\exp(\sum_{i=1}^{p-1}\epsilon_{J,i}e_i). \] Theorem~\ref{basic} states that ${\bold{D}}_{X_p}$ is obtained from this by applying a differential operator which by hypothesis evaluates trivially on ${\bold{D}}_X$. The proposition now follows from \eqref{DXbar} and Propostion~\ref{J} by the Leibniz rule. (That the coefficients of $e^{mp}$ and $e^{-mp}$ are equal follows from the fact that ${\bold{D}}_{E(2;p)}$ is an even function.) \end{pf} \begin{prop}\label{log2} The Donaldson series of the simply connected elliptic surface $E(n;2)$ with $p_g=n-1$ ($>0$) and one multiple fiber of multiplicity $2$ is \[ {\bold{D}}_{E(n;2)}=\exp(Q/2){\sinh^{n-1}(f)\over\sinh(f_2)}. \] \end{prop} \begin{pf} According to Theorem~\ref{lgtr}, we obtain $E(n;2)$ from $E(n)\#\overline{\bold{CP}}^{\,2}$ by blowing down the sphere of square $-4$ representing $f-2e$. We have ${\bold{D}}_{E(n)\#\overline{\bold{CP}}^{\,2}}=\exp(Q/2)\sinh^{n-2}(f)\cosh(e)$. Lemma~\ref{C2} gives \[ {{\bold{D}}_{E(n;2)}|}_{X^*} = ({\bold{D}}_{E(n)\#\overline{\bold{CP}}^{\,2}}-{\bold{D}}_{E(n)\#\overline{\bold{CP}}^{\,2},f-2e})|_{X^*}= 2\exp(Q/2)\sinh^{n-2}(f)\cosh(e)|_{X^*} \] (cf.\cite{KMbigpaper}, \cite[Thm.5.13]{FSstructure}). By Proposition~\ref{J} \[ {\bold{D}}_{E(n;2)}=2\exp(Q/2)\sinh^{n-2}(f)\cosh(f_2)=\exp(Q/2){\sinh^{n-1}(f)\over\sinh(f_2)}.\] \end{pf} Proposition~\ref{formallog} now implies: \begin{cor} If $S$ is a nodal fiber in $X$ orthogonal to all basic classes and $X_2$ is the multiplicity $2$ log transform of $X$ formed from $S$, then \[{\bold{D}}_{X_2}=\exp(Q_{X_2}/2){\bold{K}}_X\cdot(e^{S/2}+e^{-S/2}). \ \ \qed\] \end{cor} \begin{lem}\label{sum} Let $X$ contain a nodal fiber $S$ orthogonal to all basic classes. Then the sum of the coefficients $b_{p,j}$ in the expression for ${\bold{D}}_{X_p}$ in Proposition~\ref{formallog} is equal to $p$.\end{lem} \begin{pf} In Example 3 we showed that there is a configuration $C'_p\subset X\#(p-1)\overline{\bold{CP}}^{\,2}=Y$ where $u'_i=e_{p-(i+1)}-e_{p-i}$ for $i=1,\dots,p-2$, and $u'_{p-1}=-2e_1-e_2-\cdots-e_{p-1}$ such that the rational blowdown $Y_p=X\#H_p$ where $H_p$ is a homology $4$-sphere with $\pi_1={\bold{Z}}_p$. It follows easily that ${\bold{D}}_{Y_p}=p\cdot {\bold{D}}_X$. As above, we let $\kappa_J=\sum\epsilon_{J,i}e_i$, $J\in\{\pm1\}^{p-1}$; so \[ {\bold{D}}_Y={1\over2^{p-1}}{\bold{D}}_X\cdot\exp(Q_{(p-1)\overline{\bold{CP}}^{\,2}}/2)\sum_J e^{\kappa_J}. \] All partial derivatives of ${\bold{D}}_X$ with respect to classes in $H_2(C'_p)$ are trivial; so \[ p{\bold{D}}_X={\bold{D}}_{{Y}_p}={\bold{D}}_X\cdot\sum_Jb_Je^{\bar{\kappa}_J}. \] The proof of Proposition~\ref{J} shows that each $\bar{\kappa}_J=0$; so $\sum_Jb_J=p$. We can also form the configuration $C_p\subset Y$ whose blowdown is the $p$-log transform of the nodal fiber $S\subset X$. The configurations $C_p$, $C'_p$ agree, $u_i=u'_i$, except that $u_{p-1}=u'_{p-1}+S$. However, since $S$ is orthogonal to all the basic classes of $X$, for all $i$, all intersections of $u_i$ and $u'_i$ with all basic classes of $Y=X\#(p-1)\overline{\bold{CP}}^{\,2}$ agree. Thus, according to Theorem~\ref{BASIC}, the coefficients $b_J$ are the same coefficients that arise in the formula \[ {\bold{D}}_{X_p}=\exp(Q_{X_p}/2){\bold{K}}_X\sum_Jc_Je^{|J|S/p}. \] This means that the sum of the coefficients of the expression for ${\bold{D}}_{X_p}$ in Proposition~\ref{formallog} is $\sum_Jb_J=p$. \end{pf} We next invoke Proposition ~\ref{ponq} to see that if $p$ is any positive odd integer, then a multiplicity $2p$ log transform can be obtained as the result of either a multiplicity $p$ log transform on a nodal fiber of multiplicity 2, or by a multiplicity 2 log transform on a nodal fiber of multiplicity $p$. Thus \begin{eqnarray*} {\bold{D}}_{E(n;2p)}&=&\exp(Q/2)(e^{f_2}+e^{-f_2})(b_{p,0}+\sum_{i=1}^{(p-1)/2} b_{p,2i}(e^{2if_2/p}+e^{-2if_2/p}))\\ &=&\exp(Q/2)(b_{p,0}+\sum_{i=1}^{(p-1)/2}b_{p,2i}(e^{2if_p}+e^{-2if_p})) (e^{f_p/2}+e^{-f_p/2}) \end{eqnarray*} since we already know the formula for a log transform of multiplicity 2. We compare coefficients using $f_2=pf_{2p}$ and $f_p=2f_{2p}$. Assume for the sake of definiteness that $p\equiv 1\pmod4$ and let $r=(p-1)/4$. In the top expansion, the coefficient of $e^{\pm pf_{2p}}$ is $b_{p,0}$ and $b_{p,2j}$ is the coefficient of $e^{\pm (p+2j)f_{2p}}$ and $e^{\pm (p-2j)f_{2p}}$. In the second expansion, the coefficient of $e^{\pm f_{2p}}$ is $b_{p,0}$, and $b_{p,2j}$ is the coefficient of $e^{\pm (4j-1)f_{2p}}$ and $e^{\pm (4j+1)f_{2p}}$. To simplify notation, let $(m)_1$ be the coefficient of $e^{mf_{2p}}$ in the top expansion and $(m)_2$ its coefficient in the bottom expansion. Then, \begin{eqnarray*} b_{p,0}=(p)_1&=&(p)_2=b_{p,2r}=(p-2)_2=(p-2)_1=b_{p,2}=(p+2)_1=(p+2)_2 \\&=&b_{p,2(r+1)}= (p+4)_2=(p+4)_1=b_{p,4}=(p-4)_1=(p-4)_2\\&=&b_{p,2(r-1)}=(p-6)_2=(p-6)_1=b_{p,6}=\cdots \end{eqnarray*} and we see inductively that when $p$ is odd, all the $b_{p,2i}$ are equal. But by Lemma~\ref{sum} \[ b_{p,0}+2\sum_{i=1}^{(p-1)/2}b_{p,2i}=p.\] It follows that each $b_{p,2i}=1$, $i=0,\dots,(p-1)/2$. Similarly, if $p$ is even, let $q=p-1$. Expanding ${\bold{D}}_{E(n;pq)}$ we see that all $b_{p,2i-1}$, $i=1,\dots,p/2$ are equal; and so again each $b_{p,2i-1}=1$. \begin{thm}\label{toplt} Let $X$ be a $4$-manifold of simple type and suppose that $X$ contains a nodal fiber $S$ orthogonal to all its basic classes. Then \[ {\bold{D}}_{X_p}=\exp(Q_{X_p}/2){\bold{K}}_X\cdot{\sinh(S)\over\sinh(S/p)}. \] \end{thm} \begin{pf} If, e.g., $p$ is odd, \begin{eqnarray*} {\bold{D}}_{X_p}&=& \exp(Q_{X_p}/2){\bold{K}}_X\cdot(1+2\cosh(2S/p)+2\cosh(4S/p)+\cdots+2\cosh((p-1)S/p))\\&=& \exp(Q_{X_p}/2){\bold{K}}_X\cdot{\sinh(S)\over \sinh(S/p)}. \end{eqnarray*} \end{pf} As a result we have the calculation of the Donaldson series for all simply connected elliptic surfaces with $p_g\ge1$. \begin{thm}\label{ellformula} If $n\ge2$ and $p,q\ge1$ are relatively prime, \[{\bold{D}}_{E(n;p,q)}=\exp(Q/2){\sinh^n(f)\over\sinh(f_p)\sinh(f_q)}.\ \ \ \qed\]\end{thm} \noindent This formula was originally conjectured by Kronheimer and Mrowka \cite{KM}. As an example of Theorem~\ref{toplt} consider $E(n)$. It follows from \cite{GM1} and \cite{FScusp} that in $E(n)$ there are 3 pairs of disjoint nodal fibers such that the nodal fibers in each pair are homologous, but give three linearly independent homology classes. Form $E(n;p_1,q_1;p_2,q_2;p_3,q_3)$ by performing log transforms with each pair $\{ p_i,q_i\}$ relatively prime. The resulting manifold is simply connected and, \begin{prop}\label{noncomplex}\hspace{.1in}$\displaystyle {\bold{D}}_{E(n;p_1,q_1;p_2,q_2;p_3,q_3)}= \exp(Q/2){\sinh^{n+4}(f)\over\prod\limits_{i=1}^3\sinh(f_{p_i})\sinh(f_{q_i})}.\ \ \qed $\end{prop} \noindent Applying Theorem~\ref{ellformula} and Proposition~\ref{noncomplex} to the manifolds $E(n; p_1,q_1;p_2,q_2; p_3,q_3)$, we see that they do not admit complex structures with either orientation (cf.\cite{GM1},\cite[Theorem 8.3]{FScusp}). The manifolds $E(2;p_1,q_1;p_2,q_2;p_3,q_3)$ are the Gompf-Mrowka fake K3-surfaces \cite{GM1}. \bigskip \section{Tautly Embedded Configurations\label{taut}} Consider a $4$-manifold $X$ of simple type containing the configuration $C_p$. By Theorem~\ref{FSadj} for each 2-sphere $u_i$ in $C_p$ and each basic class $\kappa$ of $X$, we have \begin{equation}\label{tautly} -2\ge u_i^2+|u_i\cdot\kappa| \end{equation} except in the special case described in Theorem~\ref{FSadjspecial} where $0\ge u_i^2+|u_i\cdot\kappa|$. The only examples known where the special case arises are in blowups. This was the situation in the previous section where we studied log transforms. In this section, we assume that we are not in the special case. We say that a configuration is {\em tautly embedded} if \eqref{tautly} is satisfied for each $u_i$ of the configuration and each basic class $\kappa$ of $X$. Thus, if $C_p$ is tautly embedded, then for every basic class $\kappa$, $u_i\cdot\kappa=0$ for $i=1,\dots,p-2$ and $|u_{p-1}\cdot\kappa|\le p$. \begin{thm}\label{tautcalc} Suppose that $X$ is of simple type and contains the tautly embedded configuration $C_p$. If \[ {\bold{D}}_X=\exp(Q_X/2)\sum a_se^{\kappa_s} \] then the rational blowdown $X_p$ satisfies \[ {\bold{D}}_{{X_p}}=\exp(Q_{X_p}/2)\sum \bar{a}_se^{\bar{\kappa}_s} \] where \[\bar{a}_s=\begin{cases} 2^{p-1}a_s, \ \ \ &|u_{p-1}\cdot\kappa_s|=p\\ 0, &|u_{p-1}\cdot\kappa_s|<p \end{cases}\] Furthermore, if $|u_{p-1}\cdot\kappa_s|= p$, then $\bar{\kappa}_s^2={\kappa_s}^2+(p-1)$. \end{thm} \begin{pf} If $\kappa_s\cdot u_{p-1}\ne 0,\pm p$ then $\bar{a}_s=0$ by Lemma~\ref{theyextend}. For $\kappa_s\cdot u_{p-1}=0$, $\kappa_s\ne0$, note that since the $\kappa_s$ are characteristic, $p$ must be even. But then $\bar{\kappa}_s$ cannot even be characteristic in $X_p$, since $\bar{\kappa}_s^2=\kappa^2$ is not mod $4$ congruent to $(3\text{sign}+2e)(X_p)$. Thus, Theorem~\ref{BASIC} implies that $\bar{a}_s=0$. In case $\kappa_s\cdot u_{p-1}=\pm p$, we compare with the model for the order $p$ log transforms of $E(2)$; $C''_p\subset Y=E(2)\#(p-1)\overline{\bold{CP}}^{\,2}$ which is blown down to obtain $Y_p=E(2;p)$. Again let $\lambda_0=\pm(e_1+\cdots+e_{p-1})$; so by Lemma~\ref{J}, $\pm\lambda_0$ are the unique basic classes of $Y_p$ satisfying $\pm\bar{\lambda}_0=\pm(p-1)f_p\in H_2(Y_p;{\bold{Z}})$. Now \begin{eqnarray*} {\bold{D}}_{Y}&=&{1\over2^{p-1}}\exp(Q/2)\sum\exp(\pm e_1\pm\cdots\pm e_{p-1})= \exp(Q/2)\sum_J{1\over2^{p-1}}e^{\lambda_J}\\ {\bold{D}}_{Y_p}&=&\exp(Q/2)\sum\begin{Sb}|\ell|\le p-1\\\ell\equiv p\pmod2\end{Sb}e^{\ell f_p}= \exp(Q/2)\sum{1\over2^{p-1}}b_Je^{\lambda_J} \end{eqnarray*} Since $\pm\lambda_0$ are the unique $\lambda_J$ with $\bar{\lambda}_0=\pm (p-1)f_p$, the corresponding coefficient is $b_0=2^{p-1}$. We may now apply Theorem~\ref{BASIC} to obtain our result since $\kappa_s\cdot u_i=\lambda_0\cdot u''_i$ for each $i$. In order to compute $\bar{\kappa}_s^2$, we find $x_i\in{\bold{Q}}$, $i=1,\dots,p-1$, such that \[ \kappa_s+\zeta=\kappa_s+\sum_{i=1}^{p-1}x_iu_i \in H_2(X^*;{\bold{Q}}) \] as in the proof of Proposition~\ref{J}. We can solve for the $x_i$ using the model $C''_p\subset Y''$, and $\pmb{\epsilon}_J=\pm(1,\dots,1)$ in the proof of Proposition~\ref{J}. Referring there, we get \[ {\bold{x}}=P^{-1}A\pmb{\epsilon}_J=-(A^t)^{-1}\pmb{\epsilon}_J=\pm {1\over p}(1,2,\dots,p-1). \] So $\zeta=\pm\sum{i\over p}u_i$, and $\zeta^2={\bold{x}}\cdot P{\bold{x}}=1-p$. Hence \[ \bar{\kappa}_s^2 = (\kappa_s+\zeta)^2 = \kappa_s^2 + 2\kappa_s\cdot\zeta +\zeta^2= \kappa_s^2 +(p-1). \] \end{pf} Now consider the elliptic surface $E(1)$. It can be constructed by blowing up ${\bold{CP}}^2$ at the nine intersection points of a generic pencil of cubic curves. The fiber class of $E(1)$ is $f=3h-e_1-\cdots e_9$ where $3h$ is the class of the cubic in $H_2({\bold{CP}}^2;{\bold{Z}})$. The nine exceptional curves are disjoint sections of the elliptic fibration. The elliptic surface $E(n)$ can be obtained as the fiber sum of $n$ copies of $E(1)$, and these sums can be made so that the sections glue together to give nine disjoint sections of $E(n)$, each of square $-n$. In particular, consider $E(4)$ with 9 disjoint sections of square $-4$. The basic classes of $E(4)$ are $0$ and $2f$; so we see that each of the 9 sections gives us a tautly embedded configuration $C_2$. Let $W_n$ be the rational blowdown of $n$ of these sections, $1\le n\le 9$. For $n\le 8$, $W_n$ is simply connected. Gompf has shown that all these manifolds admit symplectic structures, and it is not hard to see that $W_2$ is the 2-fold branched cover of ${\bold{CP}}^2$ branched over the octic curve \cite[\S5.2]{Gompf}. \begin{prop} \hspace{.1in}$\displaystyle {\bold{D}}_{W_n}=2^{n-1}\exp(Q/2)\cosh(\kappa_n)$ \ where \ $\kappa_n^2=n$.\end{prop} \begin{pf} We have \[ {\bold{D}}_{E(4)}=\exp(Q/2)\sinh^2(f)=\exp(Q/2)(\frac12\cosh(2f)+\frac12). \] The basic classes $\pm 2f$ intersect each section twice; so Theorem~\ref{tautcalc} implies that each $X_n$ has only the basic classes, $\pm\kappa_n$, and that each blowdown multiplies its coefficient by 2 and increases its square by $1$. (We start with coefficient $\frac12$ and square $0$.) \end{pf} To further illustrate the utility of Theorem~\ref{tautcalc} we compute the Donaldson invariants of a family of Horikawa surfaces $\{ H(n)\}$ with $c_1(H(n))^2=2n-6$. To obtain $H(n)$, start with the simply connected ruled surface ${\bold{F}}_{n-3}$ whose negative section $s_-$ has square $-(n-3)$. We have seen in the proof of Lemma~\ref{ratball} that the classes $s_++f$ and $s_-$ form a configuration in ${\bold{F}}_{n-3}$ whose regular neighborhood $D_{n-2}$ has complement the rational ball $B_{n-2}$. The Horikawa surface $H(n)$ is defined to be the $2$-fold branched cover of ${\bold{F}}_{n-3}$ branched over a smoothing of $4(s_++f)+2s_-$. (Equivalently this is a smooth surface representing (6,$n+1$) in $S^2\times S^2$.) \begin{lem} For $n\ge 4$, the elliptic surface $E(n)$ contains a pair of disjoint configurations $C_{n-2}$ in which the spheres $u_{n-1}$ are sections of $E(n)$ and for $1\le j\le n-2$, $u_j\cdot f=0$. Furthermore, the rational blowdown of this pair of configurations is the Horikawa surface $H(n)$. \end{lem} \begin{pf} It follows from our description of $H(n)$ that there is a decomposition \[ H(n)= B_{n-2}\cup \tilde{D}_{n-2}\cup B_{n-2} \] where $\tilde{D}_{n-2}$ is the branched cover of $D_{n-2}$. Rationally blow up each $B_{n-2}$; this is then the $2-$fold branched cover of ${\bold{F}}_{n-3}$ with $B_{n-2}$ blown up. The result is the complex surface $ C_{n-2}\cup \tilde{D}_{n-2}\cup C_{n-2}$ which, by computing characteristic numbers, is just $E(n)$. \end{pf} \noindent The first case, $n=4$, gives the example $H(4)=W_2$ above. The Horikawa surfaces $H(n)$ lie on the Noether line $5c_1^2-c_2+36=0$, and of course the elliptic surfaces $E(n)$ lie on the line $c_1^2=0$ in the plane of coordinates $(c_1^2,c_2)$. Let $Y(n)$ be the simply connected $4$-manifold obtained from $E(n)$ by blowing down just one of the configurations $C_{n-2}$. Then $c_1(Y(n))^2=n-3$ and $c_2(Y(n))=11n+3$; so $Y(n)$ lies on the bisecting line $11c_1^2-c_2+36=0$. The calculation of Donaldson invariants of $Y(n)$ and $H(n)$ follows directly from Theorem~\ref{tautcalc}. \begin{prop} The Donaldson invariants of $Y(n)$ and $H(n)$ are: \begin{eqnarray*} {\bold{D}}_{Y(n)}&=&\begin{cases}\exp(Q/2)\sinh(\lambda_n),\ \ n\ \text{odd}\\ \exp(Q/2)\cosh(\lambda_n),\ \ n\ \text{even}\end{cases}\\ {\bold{D}}_{H(n)}&=&\begin{cases}2^{n-3}\exp(Q/2)\sinh(\kappa_n),\ \ n\ \text{odd}\\ 2^{n-3}\exp(Q/2)\cosh(\kappa_n),\ \ n\ \text{even}\end{cases} \end{eqnarray*} where $\lambda_n^2=n-3$ and $\kappa_n^2=2n-6$.\ \ \qed\end{prop} \begin{cor} The simply connected $4$-manifolds $Y(n)$ are not homotopy equivalent to any complex surface.\end{cor} \begin{pf} If $Y(n)$ were homeomorphic to a complex surface, this computation shows that it would have to be minimal, since the formula for ${\bold{D}}_{Y(n)}$ does not contain a factor $\cosh(e)$ where $e^2=-1$. Certainly the surface in question could not be elliptic since $c_1(Y(n))^2\ne 0$. Bt neither could the surface be of general type since $Y(n)$ violates the Noether inequality. Thus $Y(n)$ is not homeomorphic to any complex surface.\end{pf} D. Gomprecht \cite{Gomprecht} has computed the value of the Donaldson invariant $D_X(F^k)$ for any Horikawa surface $X$ and $k$ large, where $F$ is the branched cover of the fiber $f$ of $F_{n-3}$. \bigskip \section{Seiberg-Witten Invariants of Rational Blowdowns\label{SW}} Suppose we are given a spin$^{\text{c}}$ structure on an oriented closed Riemannian $4$-manifold $X$. Let $W^+$ and $W^-$ be the associated spin$^{\text{c}}$ bundles with $L=\det W^+=\det W^-$ the associated determinant line bundle. Since $c_1(L)\in H^2(X;{\bold{Z}})$ is a characteristic cohomology class, i.e. has mod 2 reduction equal to $w_2(X)\in H^2(X;{\bold{Z}}_2)$, we refer to $L$ as a characteristic line bundle. We will confuse a characteristic line bundle $L$ with its first Chern class $L \in H^2(X;{\bold{Z}})$. For simplicity we assume that $H^2(X;{\bold{Z}})$ has no $2$-torsion so that the set $Spin^{\text{c}}(X)$ of spin$^{\text{c}}$ structures on $X$ is precisely the set of characteristic line bundles on $X$. Clifford multiplication, $c$, maps $T^\ast X$ into the skew adjoint endomorphisms of $W^+\oplus W^-$ and is determined by the requirement that $c(v)^2$ is multiplication by $-|v|^2$. Thus $c$ induces a map $$c: T^\ast X \to {\text{Hom}}(W^+, W^-).$$ The $2$-forms $\Lambda^2=\Lambda^+\oplus\Lambda^-$ of $X$ then act on $W^+$ leading to a map $\rho:\Lambda^+\to {\text{su}}(W^+)$. A connection $A$ on $L$ together with the Levi-Civita connection on the tangent bundle of $X$ induces a connection $\nabla_A:\Gamma(W^+)\to \Gamma(T^\ast X\otimes W^+)$ on $W^+$. This connection, followed by Clifford multiplication, induces the Dirac operator $D_A:\Gamma(W^+)\to\Gamma(W^-)$. (Thus $D_A$ depends both on the connection $A$ and the Riemannian metric on $X$.) Given a pair $(A,\psi) \in {\cal{A}}_X(L)\times \Gamma(W^+)$, i.e. $A$ a connection in $L$ and $\psi$ a section of $W^+$, the monopole equations of Seiberg and Witten \cite{Witten} are \begin{eqnarray}\label{monopole} D_A\psi&=&0\\\rho(F_A^+)\notag &= &(\psi\otimes\psi^\ast)_o \end{eqnarray} where $(\psi\otimes\psi^\ast)_o$ is the trace-free part of the endomorphism $\psi\otimes\psi^\ast$. The gauge group $\text{Aut}(L)=\text{Map}(X,S^1)$ acts on the space of solutions, and its orbit space is the moduli space $M_X(L)$ whose formal dimension is \begin{equation} \dim M_X(L) = \frac14(c_1(L)^2-(3\,\text{sign}(X)+2\,e(X)). \label{dims} \end{equation} If this formal dimension is nonnegative and if $b^+>0$, then for a generic metric on $X$ the moduli space $M_X(L)$ contains no reducible solutions (solutions of the form $(A,0)$ where $A$ is an anti-self-dual connection on $L$), and for a generic perturbation of the second equation of \eqref{monopole} by the addition of a self-dual 2-form of $X$, the moduli space $M_X(L)$ is a compact manifold of the given dimension (\cite{Witten}). The {\em Seiberg-Witten invariant} for $X$ is the function $SW_X:Spin^{\text{c}}(X)\to {\bold{Z}}$ defined as follows. Let $L$ be a characteristic line bundle. If $\dim M_X(L)<0$ or is odd, then $SW_X(L)$ is defined to be $0$. If $\dim M_X(L)= 0$, the moduli space $M_X(L)$ consists of a finite collection of points and $SW_X(L)$ is defined to be the number of these points counted with signs. These signs are determined by an orientation on $M_X(L)$, which in turn is determined by an orientation on the determinant line $\det(H^0(X;{\bold{R}}))\otimes\det(H^1(X;{\bold{R}}))\otimes \det(H^2_+(X;{\bold{R}}))$. If $\dim M_X(L)>0$ then we consider the basepoint map $$ \tilde{M}_X(L)=\{\text{solutions}\, (A,\psi)\}/\text{Aut}^0(L)\to M_X(L) $$ where $\text{Aut}^0(L)$ consists of gauge transformations which are the identity on the fiber of $L$ over a fixed basepoint in $X$. If there are no reducible solutions, the basepoint map is an $S^1$ fibration, and we denote its euler class by $\beta\in H^2(M_X(L);\bold{Z})$. The moduli space $M_X(L)$ represents an integral cycle in the configuration space $B_X(L) =({\cal{A}}_X(L)\times \Gamma(W^+))/\text{Aut}(L)$, and if $\dim M_X(L)=2n$, the Seiberg-Witten invariant is defined to be the integer $$ SW_X(L)=\langle\beta^n,[M_X(L)]\rangle. $$ A fundamental result is that if $b^+(X)\ge2$, the map $$ SW_X: Spin^{\text{c}}(X) \to {\bold{Z}} $$ is a diffeomorphism invariant ({\cite{Witten}); i.e. $SW_X(L)$ does not depend on the (generic) choice of Riemannian metric on $X$ nor the choice of generic perturbation of the second equation of \eqref{monopole}. It is often convenient to observe that the space ${\cal{A}}_X(L)\times \Gamma(W^+)$ is contractible and $\text{Aut}(L)\cong\text{Map}(X,S^1)$ acts freely on ${\cal{A}}_X(L)\times (\Gamma(W^+)\setminus\{ 0 \})$. Since $S^1$ is a $K({\bold{Z}},1)$, if we further assume that $H^1(X;\bold{R})=0$, then the quotient $$ B^*_X(L)=\left({\cal{A}}_X(L)\times(\Gamma(W^+)\setminus\{ 0 \})\right)/S^1 $$ of this action is homotopy equivalent to ${\bold{CP}}^{\infty}$. So if there are no reducible solutions, we may view $M_X(L)\subset {\bold{CP}}^{\infty}$. Under these identifications, the class $\beta$ becomes the standard generator of $H^2({\bold{CP}}^{\infty};{\bold{Z}})$. Call a characteristic line bundle with nontrivial Seiberg-Witen invariant a {\it Seiberg-Witten class}. The assumption in Seiberg-Witten theory which is analogous to the assumption of simple type in Donaldson theory is \begin{enumerate}\item[(*)] For each Seiberg-Witten class $L$, $\dim M_L(X)=0$. \end{enumerate} If this condition is satisfied, $X$ is said to have {\it Seiberg-Witten simple type}. \begin{lem}\label{char} Let $C_p\subset X$ and let $X_p$ be its rational blowdown. Assume that $X_p$ is simply connected. \begin{enumerate} \item[(a)] A line bundle $L^*$ on $X^*$ extends over $X_p$ if and only if $c_1(L^*|_{L(p^2,1-p)})\in p{\bold{Z}}_{p^2}$. \item[(b)] If $\bar{L}$ is a characteristic line bundle on $X_p$, then there is a characteristic line bundle $L$ on $X$ such that $L|_{X^*}=\bar{L}|_{X^*}$. \item[(c)] If $L$ is a characteristic line bundle on $X$, then an extension $\bar{L}$ of $L|_{X^*}$ is characteristic on $X^*$ if and only if $\bar{L}|_{B_p}$ is characteristic. \end{enumerate}\end{lem} \begin{pf} (a) is obvious. For any simply connected manifold $Y=V\cup W$ where $\partial V=\partial W$ is a rational homology sphere, a class $c\in H^2(Y;\bold{Z})$ will be characteristic provided $\langle c,\alpha\rangle\equiv\alpha\cdot\alpha$ (mod 2) for all $\alpha\in H^2(Y;\bold{Z})$. Thus we need not worry about torsion classes; so a class is characteristic if and only if its restrictions to $V$ and $W$ are both characteristic. Applying this observation to $X_p=X^*\cup B_p$ proves (c). To prove (b), let $\bar{L}$ be a characteristic line bundle on $X_p$ and let $L^*=\bar{L}|_{X^*}$. By (a), $\delta c_1(L^*)=mp$ for some integer $m$. Suppose that $p$ is odd, then since $mp=(p+m)p\in\bold{Z}_{p^2}$, we may assume that $m$ is even. Let $L'$ be the line bundle on $C_p$ such that the Poincar\'e dual of $c_1(L')$ is $(m+1)\gamma_{p-1}+(m-p+1)\gamma_1$. Then $L'$ is characteristic on $C_p$ and $\delta c_1(L')=\delta c_1(L^*)$. It follows that $L^*$ extends to a characteristic line bundle on $X$ by our observation above. If $p$ is even, we may take $c_1(L')$ to be the Poincar\'e dual of $mp\gamma_1$ and get the extension of $L^*$ to a global line bundle $L$ on $X$ whose restriction to both $X^*$ and $C_p$ is characteristic. \end{pf} If $\bar{L}$ is a line bundle on $X_p$ and $L$ is a line bundle on $X$ satisfying $L|_{X^*}=\bar{L}|_{X^*}$, we say that $L$ is a {\em lift} of $\bar{L}$. \begin{thm}\label{swgen} Let $C_p\subset X$ and let $X_p$ be its rational blowdown. Let $\bar{L}$ be a characteristic line bundle on $X_p$ and let $L$ be any lift of $\bar{L}$ which is characteristic on $X$. Suppose that $\dim M_X(L)\equiv\dim M_{X_p}(\bar{L})\pmod 2$. Then \[ SW_{X_p}(\bar{L})=SW_X(L).\] \end{thm} \begin{pf} Since the rational ball $B_p$ embeds in the ruled surface ${\bold{F}}_{p-1}$ (see Lemma~\ref{ratball}), it admits a metric of positive scalar curvature. The gluing theory for solutions of the Seiberg-Witten equations follows the same pattern as for solutions of the anti-self-duality equations. Thus we study the solutions on $X_p$ for $\bar{L}$ by stretching the neck between $X^*$ and $B_p$. We may assume that there are positive scalar curvature metrics on both the neck $L(p^2,1-p)\times\bold{R}^+$ and on $B_p$. This means that the only solution to the Seiberg-Witten equations on $B_p$ with a cylindrical end is the reducible solution $(A',0)$, where $A'$ is an anti-self-dual connection on $L'=\bar{L}|_{B_p}$. Possible global solutions are constructed from asymptotically reducible solutions on $X^*$ glued to $(A',0)$. The formal dimension of $M_{B_p}(L')$ is odd and negative, and there is one gluing parameter (since the asymptotic value is reducible); so \[ \dim M_{X^*}(L^*)+1+\dim M_{B_p}(L') = \dim M_{X_p}(\bar{L})=2d_{\bar{L}}, \] say. (If $\dim M_{X_p}(\bar{L})$ is odd, there is nothing to prove.) Thus $\dim M_{X^*}(L^*)=2d_{L^*}$ where $d_{L^*}\ge d_{\bar{L}}$. This means that there is an obstruction to perturbing a glued-up $(A^*,\psi^*)\# (A',0)$ to a solution. As in Donaldson theory, there is an obstruction bundle $\xi$ over $M_{X^*}(L^*)$, and it is the complex vector bundle of rank $d_{L^*}-d_{\bar{L}}$ associated to the basepoint fibration. The zero set of a generic section of $\xi$ is homologous to $M_{X_p}$ in $B_{X_p}(\bar{L})$. Thus \[ SW_{X_p}(\bar{L})=\langle\beta^{d_{\bar{L}}},[M_{X_p}(\bar{L})]\rangle=\langle\beta^{d_{\bar{L}}}, \beta^{d_{L^*}-d_{\bar{L}}}\cap [M_{X^*}(L^*)]\rangle=\langle\beta^{d_{L^*}},[M_{X^*}(L^*)]\rangle. \] Let $L$ be a characteristic line bundle on $X$ which is a lift of $\bar{L}$, and let $\dim M_X(L)=2d_L$. The second construction of Lemma~\ref{ratball} shows that $C_p$ has a metric of positive scalar curvature. So the discussion of the last paragraph applies to show that \[ SW_X(L)=\langle\beta^{d_{L^*}},[M_{X^*}(L^*)]\rangle, \] completing the proof of the theorem. \end{pf} \begin{lem}\label{swdim} Suppose that $C_p\subset X$ with rational blowdown $X_p$. Let $L$ be a characteristic line bundle on $X$ such that $\langle c_1(L), u_i\rangle=0$ for $i=1,\dots,p-2$ and $\langle c_1(L), u_{p-1}\rangle=mp$ for some $m\in\bold{Z}$. Let $\bar{L}$ be a characteristic extension of $L|_{X^*}$ to all of $X_p$. Then $m$ is odd, and $\dim M_{\bar{L}}(X_p)=\dim M_L(X)+{m^2-1\over4}(p-1)$.\end{lem} \begin{pf} The proof of Theorem~\ref{tautcalc} shows that $m$ must be odd if $\bar{L}$ is to be characteristic and that $c_1(\bar{L})^2=c_1(L)^2+m^2(p-1)$. Since $3\,\text{sign}(X_p)+2\,e(X_p)=3\,\text{sign}(X)+2\,e(X)+(p-1)$, the lemma follows. \end{pf} \noindent Note that this shows that, unless $m=\pm1$, the dimensions of the moduli spaces will increase. We shall consider the two situations analogous to those studied in the previous sections: \smallskip \begin{itemize} \item[(i)] $C_p$ is embedded in $X=Y\#(p-1)\overline{\bold{CP}}^{\,2}$ so that $X_p$ is the result of an order $p$ log transform performed on a nodal fiber of $X$.\\ \vspace{-.1in}\item[(ii)] $C_p$ is tautly embedded in $X$ with respect to $L$, i.e. $\langle c_1(L),u_i\rangle=0$ for $i=1,\dots,p-2$, and $\langle c_1(L),u_{p-1}\rangle\le p$. \end{itemize} The next theorem follows directly from Theorem~\ref{swgen} and Lemma~\ref{swdim}. \begin{thm}\label{tautagain} Suppose that $X$ has Seiberg-Witten simple type and that $C_p\subset X$ with $X_p$ its rational blowdown. Assume that $X_p$ is simply connected and that $\bar{L}$ is a characteristic line bundle on $X_p$. Suppose further that $L$ is a characteristic lift of $\bar{L}$ and that $C_p$ is tautly embedded with respect to $L$. Then \[ SW_{X_p}({\bar{L}})=SW_X(L)\] and $c_1(\bar{L})^2=c_1(L)^2+(p-1)$. \ \ \qed \end{thm} Say that the configuration $C_p$ is {\it SW-tautly embedded} in $X$ if it is tautly embedded with respect to each Seiberg-Witten class. \begin{cor} Suppose that $X$ has Seiberg-Witten simple type and contains the SW-tautly embedded configuration $C_p$. Assume that the rational blowdown $X_p$ is simply connected. Then the Seiberg-Witten classes of $X_p$ are the characteristic line bundles $\bar{L}$ which have a lift to a Seiberg-Witten class $L$ of $X$, and $SW_{X_p}({\bar{L}})=SW_X(L)$. \ \ \qed\end{cor} In a fashion similar to the proof of Theorem~\ref{swgen}, one can prove a blowup formula for Seiberg-Witten invariants. The characteristic line bundles of $X\#\overline{\bold{CP}}^{\,2}$ are those of the form $L\otimes E^{2k+1}$ where $L$ is characteristic on $X$ and $c_1(E)=e$, and $\dim M_{L\otimes E^{2k+1}}(X\#\overline{\bold{CP}}^{\,2})=\dim M_L(X)-k(k+1)$. It is shown in \cite{Turkey} that $SW_{X\#\overline{\bold{CP}}^{\,2}}(L\otimes E^{2k+1})=SW_X(L)$ provided $\dim M_L(X)-k(k+1)\ge0$. It follows that if $X$ satisfies the Seiberg-Witten simple-type condition (*), then so does $X\#\overline{\bold{CP}}^{\,2}$. Suppose that $X$ contains the nodal fiber $S$, and $X_p$ is the result of performing an order $p$ log transform on $S$. The characteristic line bundles on $X_p$ are obtained from characteristic bundles $L\otimes E_1^{2k_1+1}\otimes\cdots\otimes E_{p-1}^{2k_{p-1}+1}$ on $Y=X\#(p-1)\overline{\bold{CP}}^{\,2}$ by restricting to $Y^*=Y\setminus C_p$ and then extending over $B_p$. If we assume that $\langle c_1(L),S\rangle=0$, then for each $L\otimes E_1^{\pm1}\otimes\cdots\otimes E_{p-1}^{\pm1}=L({\pmb\epsilon}_J)$ with $c_1(L({\pmb\epsilon}_J))= c_1(L)+\sum_J\epsilon_{J,i}e_i$, it follows from Proposition~\ref{J} that the unique extension $\bar{L}_J$ over $X_p$ has $c_1(\bar{L}_J)=c_1(L)+|J|\sigma_p$, where $\sigma_p$ is the Poincar\'e dual of $S/p$. (Note that when $p$ is even, $|J|$ must be odd; so the extension $\bar{L}_J$ is characteristic.) Hence \[ \dim M_{\bar{L}_J}(X_p) =\dim M_L(X),\] and Theorem~\ref{swgen} implies: \begin{thm}\label{SWlog} Suppose that $X$ has Seiberg-Witten simple type and contains the nodal fiber $S$. Let $L$ be a characteristic line bundle on $X$ with $\langle c_1(L),S\rangle=0$. Let $X_p$ be the result of performing an order $p$ log transform on $S$. For each $J\in \{\pm1\}^{p-1}$, we have $SW_{X_p}({\bar{L}_J})=SW_X(L)$. Suppose furthermore that $\langle c_1(L),S\rangle=0$ for each characteristic $L$ on $X$ with $SW_X(L)\ne0$. Then $X_p$ also has Seiberg-Witten simple type and each line bundle $\Lambda$ on $X_p$ with $SW_{X_p}(\Lambda)\ne 0$ is of the form $\Lambda=\bar{L}_J$.\ \ \qed \end{thm} By a the {\em nodal configuration} we shall mean a configuration $C_p\subset X\#(p-1)\overline{\bold{CP}}^{\,2}$ as above, obtained from a nodal fiber $S$ satisfying the condition $\langle c_1(L),S\rangle=0$ for each characteristic $L$ on $X$ with $SW_X(L)\ne0$. Witten \cite{Witten} has conjectured that (for manifolds with $b^+>1$) the Seiberg-Witten simple type condition is equivalent to the simple type condition of Kronheimer and Mrowka for Donaldson theory. Further, under this hypothess of simple type, Witten gives a precise conjecture for relating the Seiberg-Witten invariants and the Donaldson series, namely: \begin{conj}[Witten] The set of basic classes in the two theories are the same, and \[ {\bold{D}}_X= 2^{3\text{sign}+2e-({b^+-3\over 2})}\exp(Q/2)\sum SW_X(\kappa_s)e^{\kappa_s}.\] \end{conj} \begin{thm} Witten's conjecture is true for simply connected elliptic surfaces.\end{thm} \begin{pf} Witten has given a recipe for calculating $SW_X$ for all Kahler manifolds $X$. So one could prove this theorem simply by comparing the answer obtained with that of Theorem \ref{ellformula}. Alternatively, note that Witten's recipe gives the result that the nonzero Seiberg-Witten invariants of $E(n)$ are: \begin{equation}\label{SWEn} SW_{E(n)}((n-2-2r)f)= (-1)^r\binom{n-2}{r},\ \ r=0,\dots,n-2 \end{equation} (where $f$ is the fiber class). Suppose we define \[{\bold{W}}_X=2^{3\text{sign}+2e-({b^+-3\over 2})}\sum SW_X(\kappa_s)e^{\kappa_s}, \ \ {\bold{SW}}_X=\exp(Q_X/2){\bold{W}}_X\] Then \eqref{SWEn} shows that ${\bold{D}}_{E(n)}={\bold{SW}}_{E(n)}$. Suppose that $X_p$ is the result of an order $p$ log transform on a nodal fiber which is orthogonal to all classes in $H_2(X)$ with nontrivial Seiberg-Witten invariants. Then Theorem \ref{SWlog} implies that ${\bold{W}}_{X_p}={\bold{W}}_X\cdot(\sinh(f_p)/\sinh(f))$. It follows that ${\bold{SW}}_{E(n;p,q)}={\bold{D}}_{E(n;p,q)}.$\end{pf} Furthermore, we have \begin{thm} If $X$ satisfies the Witten conjecture, then so do all blowups and blowdowns and any rational blowdown $X_p$ of a nodal or taut configuration. \end{thm} \bigskip

9505

alg-geom/9505013

en

https://arxiv.org/abs/alg-geom/9505013

alg-geom/9505013

Teleman

Andrei Teleman and Christian Okonek

Seiberg-Witten Invariants and the Van De Ven Conjecture

Duke preprint, LATEX

The purpose of this note is to give a short, selfcontained proof of the following result: A complex surface which is diffeomeorphic to a rational surface is rational.

alg-geom

\section{Introduction} The purpose of this note is to give a short, selfcontained proof of the following result: \begin{th} A complex surface which is diffeomeorphic to a rational surface is rational. \end{th} This result has been announced by R. Friedman and Z. Qin [FQ]. Whereas their proof uses Donaldson theory and vector bundles techniques, our proof uses the new Seiberg-Witten invariants [W], and the interpretation of these invariants in terms of stable pairs [OT]. Combining the theorem above with the results of [FM], one obtains a proof of the Van de Ven conjecture [V]: \begin{co} The Kodaira dimension of a complex surface is a differential invariant. \end{co} {\bf Proof: } (of the Theorem) It suffices to prove the theorem for algebraic surfaces [BPV]. Let $X$ be an algebraic surface of non-negative Kodaira dimension, with $\pi_1(X)=\{1\}$ and $p_g(X)=0$. We may suppose that $X$ is the blow up in $k$ {\sl distinct} points of its minimal model $X_{\min}$. Denote the contraction to the minimal model by $\sigma:X\longrightarrow X_{\min}$, and the exceptional divisor by $E=\sum\limits_{i=1}^k E_i$. Fix an ample divisor $H_{\min}$ on $X_{\min}$, a sufficiently large integer $n$, and let $H_n:=\sigma^*(n H_{\min})-E$ be the associated polarization of $X$. For every subset $I\subset\{1,\dots,k\}$ we put $E_I:=\sum\limits_{i\in I} E_i$, and $L_I:=2[E_I]-[K_X]$, where $K_X$ is a canonical divisor. Clearly $L_I=[E_I]-[E_{\bar I}]-\sigma^*([K_{\min}])$, where $\bar I$ denotes the complement of $I$ in $\{1,\dots,k\}$. The cohomology classes $L_I$ are almost canonical classes in the sense of [OT]. Now choose a K\"ahler metric $g_n$ on $X$ with K\"ahler class $[\omega_{g_n}]=c_1({\cal O}_X(H_n))$. Since $[\omega_{g_n}]\cdot L_I<0$ for sufficiently large $n$, the main result of [OT] identifies the Seiberg-Witten moduli space ${\cal W}_X^{g_n}(L_I)$ with the union of all complete linear systems $|D|$ corresponding to effective divisors $D$ on $X$ with $c_1({\cal O}_X(2D-K_X))=L_I$. Since $H^2(X,{\Bbb Z})$ has no 2-torsion, and $q(X)=0$, there is only one such divisor, $D=E_I$. Furthermore, from $h^1({\cal O}_X(E_I)|_{E_I})=0$, and the smoothness criterion in [OT], we obtain: $${\cal W}_X^{g_n}(L_I)=\{E_I\}, $$ i.e. ${\cal W}_X^{g_n}(L_I)$ consists of a single smooth point. The corresponding Seiberg-Witten invariants are therefore odd: $n_{L_I}^{g_n}=\pm 1$. Consider now the positive cone ${\cal K}:=\{\eta\in H^2_{\rm DR}(X)|\ \eta^2>0\}$; using the Hodge index theorem, the fact that $K_{\min}$ is cohomologically nontrivial, and $K_{\min}^2\geq 0$, we see that ${\cal K}$ splits as a disjoint union of two components ${\cal K}_{\pm}:=\{\eta\in{\cal K}|\ \pm\eta\cdot\sigma^*(K_{\min})>0\}$. Clearly $[\omega_{g_n}]$ belongs to ${\cal K}_+$. Let $g$ be an {\sl arbitrary} Riemannian metric on $X$, and let $\omega_g$ be a $g$-selfdual closed 2-form on $X$ such that $[\omega_g]\in{\cal K}_+$. For a fixed $I\subset\{1,\dots,k\}$, we denote by $L_I^{\bot}\subset{\cal K}_+$ the wall associated with $L_I$, i.e. the subset of classes $\eta$ with $\eta\cdot L_I=0$.\\ \\ {\bf Claim:} The rays ${\Bbb R}_{>0}[\omega_g]$, ${\Bbb R}_{>0}[\omega_{g_n}]$ belong either to the same component of ${\cal K}_+\setminus L_I^{\bot}$ or to the same component of ${\cal K}_+\setminus L_{\bar I}^{\bot}$. \\ Indeed, since $[\omega_{g_n}]\cdot L_I<0$ and $[\omega_{g_n}]\cdot L_{\bar I}<0$, we just have to exclude that $$[\omega_{g}]\cdot L_I\geq 0 \ \ \ \ {\rm and}\ \ \ \ [\omega_{g}]\cdot L_{\bar I}\geq 0 .\eqno{(*)}$$ Write $[\omega_g]=\sum\limits_{i=1}^{k}\lambda_i[E_i]+\sigma^*[\omega]$, for some class $[\omega]\in H^2_{\rm DR}(X_{\min})$; then $[\omega]^2>\sum\limits_{i=1}^k\lambda_i^2$, and $[\omega]\cdot K_{\min}>0$, since $\omega_g$ was chosen such that its cohomology class belongs to ${\cal K}_+$. The inequalities $(*)$ can now be written as $$-\sum\limits_{i\in I}\lambda_i+\sum\limits_{j\in\bar I}\lambda_j- [\omega]\cdot K_{\min}\geq 0 \ \ \ {\rm and}\ \ \ -\sum\limits_{j\in\bar I}\lambda_j+\sum\limits_{i\in I}\lambda_i- [\omega]\cdot K_{\min}\geq 0,$$ and we obtain the contradiction $[\omega]\cdot K_{\min}\leq 0$. This proves the claim. \dfigure 80mm by 165mm (kegel scaled 700 offset 1mm:) We know already that the mod 2 Seiberg-Witten invariants $n^{g_n}_{L_I}$(mod 2) and $n^{g_n}_{L_{\bar I}}$(mod 2) are nontrivial for the special metric $g_n$. Since the invariants $n^{g}_{L_{I}}$(mod 2) and $n^{g}_{L_{\bar I}}$(mod 2) depend only on the chamber of the ray ${\Bbb R}_{>0}[\omega_g]$ with repect to the wall $L_{I}^{\bot}$, respectively $L_{\bar I}^{\bot}$ (see [W], [KM]), at least one of the invariants associated with the metric $g$ must be non-zero, too. But any rational surface admits a Hodge metric with positive total scalar curvature [H], and with respect to such a metric {\sl all} Seiberg-Witten invariants are trivial [OT]. \hfill\vrule height6pt width6pt depth0pt \bigskip \vspace{0.5cm}\\ \parindent0cm \centerline {\Large {\bf Bibliography}} \vspace{0.5cm} [BPV] Barth, W., Peters, C., Van de Ven, A.: {\it Compact complex surfaces}, Springer Verlag (1984) [FM] Friedman, R., Morgan, J.W.: {\it Smooth 4-manifolds and Complex Surfaces}, Springer Verlag 3. Folge, Band 27, (1994) [FQ] Friedman, R., Qin, Z.: {\it On complex surfaces diffeomorphic to rational surfaces}, Preprint (1994) [H] Hitchin, N.: {\it On the curvature of rational surfaces}, Proc. of Symp. in Pure Math., Stanford, Vol. 27 (1975) [KM] Kronheimer, P., Mrowka, T.: {\it The genus of embedded surfaces in the projective plane}, Preprint (1994) [OT] Okonek, Ch.; Teleman A.: {\it The Coupled Seiberg-Witten Equations, Vortices, and Moduli Spaces of Stable Pairs}, Preprint, January, 13-th 1995 [W] Witten, E.: {\it Monopoles and four-manifolds}, Mathematical Research Letters 1, 769-796 (1994) [V] Van de Ven, A,: {\it On the differentiable structure of certain algebraic surfaces}, S\'em. Bourbaki ${\rm n}^o$ 667, Juin (1986) \vspace{1cm}\\ Authors addresses:\\ \\ Mathematisches Institut, Universit\"at Z\"urich,\\ Winterthurerstrasse 190, CH-8057 Z\"urich\\ e-mail:[email protected] \ \ \ \ \ \ \ \ \ [email protected] \end{document}

9505

alg-geom/9505022

en

https://arxiv.org/abs/alg-geom/9505022

alg-geom/9505022

Rahul Pandharipande

R. Pandharipande

A Geometric Invariant Theory Compactification of M_{g,n} Via the Fulton-MacPherson Configuration Space

14 pages. AMSLatex

A compactification over $\overline{M}_g$ of $M_{g,n}$ is obtained by considering the relative Fulton-MacPherson configuration space of the universal curve. The resulting compactification differs from the Deligne-Mumford space $\overline{M}_{g,n}$. In case $n=2$, the compactification constructed here and the Deligne-Mumford compactification are essentially the distinct minimal resolutions of the fiber product over $\overline{M}_g$ of the universal curve with itself.

alg-geom

\section{Introduction} \label{int} In [F-M], a compactification of the configuration space of $n$ marked points on an algebraic variety is defined. For a nonsingular curve $X$ of genus $g \geq 2$, the Fulton-MacPherson configuration space of $X$ (quotiented by the automorphism group of $X$) is isomorphic to the (reduced) fiber of $\gamma:\overline{M}_{g,n} \rightarrow \overline{M}_{g}$ over $[X]\in M_g$. Since the Fulton-MacPherson configuration space is defined for singular varieties, it is natural to ask whether a compactification of $\gamma^{-1}(M_g)$ can be obtained over $\overline{M}_{g}$. First, we consider the Fulton-MacPherson configuration space for families of varieties. This relative construction is then applied to the universal curve over the Hilbert scheme of $10$-canonical, genus $g\geq 2$ curves. Following results of Gieseker, it is shown there exist linearizations of the natural ${SL}$-action on the relative configuration space of the universal curve that yield G.I.T. quotients compactifying $\gamma^{-1}(M_g)$. These new compactifications, $M_{g,n}^{c}$, are described. For $n=1$, $M_{g,1}^c$ and the Deligne-Mumford compactification $\overline{M}_{g,1}$ coincide. For $n=2$, $M_{g,2}^c$ and $\overline{M}_{g,2}$ are isomorphic on open sets with codimension $2$ complements. $M_{g,2}^c$ and $\overline{M}_{g,2}$ differ essentially by the birational modification corresponding to the two minimal resolutions of an ordinary threefold double point. For higher $n$, the compactifications $M_{g,n}^c$ and $\overline{M}_{g,n}$ differ more substantially. Thanks are due to J. Harris for mathematical guidance. The author has benefited from many discussions with him. \section{Relative Fulton-MacPherson Configuration Spaces} \label{rfm} \subsection{Terminology} \label{rmft} Let $\Bbb{C}$ be the ground field of complex numbers. A morphism $\mu: X \rightarrow Y$ is an {\it immersion} if $\mu$ is an isomorphism of $X$ onto an open subset of a closed subvariety of $Y$. A morphism $\gamma$ is {\it quasi-projective} if it factors as $\gamma =\rho \circ \mu$ where $\mu$ is an open immersion and $\rho$ is projective. The only smooth morphisms considered will be smooth morphisms of relative dimension $k$ between nonsingular varieties. \subsection{Definition} \label{rfmd} We carry out the construction of Fulton and MacPherson in the relative context. Suppose $\pi:{ F}\rightarrow{ B}$ is a (separated) morphism of algebraic varieties. Let $n$ be a positive integer. ${ N}= \{ 1,\ldots, n \} $. Wherever possible, products will be taken in the category of varieties over $B$. Define: $$ F_B^N=\prod_{N}F= \underbrace { F \times {_B} F \times {_B} \ldots \times {_B} F}_{n}\;\;. $$ And define: $$(F_B^N)_0= F_B^N \;\;\setminus \;\; (\bigcup \bigtriangleup_{\{a,b\}})$$ Where $\bigtriangleup_{\{a,b\}}$ denotes the large diagonal corresponding to the indices $a,b \in N$ and the union is over all pairs $\{ a,b \}$ of distinct element of $N$. For each subset $S$ of $N$ define $F_B^S=\prod_{S}F$. Following the notation of [F-M], let $Bl_\bigtriangleup (F_B^S)$ denote the blow-up of $F_B^S$ along the small diagonal. There exists a natural immersion: \begin{equation} (F_B^N)_0 \subset F_B^N \times \prod_{|S|\geq 2}{Bl_\bigtriangleup (F_B^S)}\;\; .\label{mouse} \end{equation} The relative Fulton-MacPherson configuration space of $n$ marked points of $F$ over $B$, $F_B[n]$, is defined to be the closure of $(F_B^N)_0$ in the above product. When $B$ is a point, this definition coincides with that of [F-M]. Consider the composition: \begin{equation} F_B[n] \subset F_B^N \times \prod_{|S|\geq 2}{Bl_\bigtriangleup (F_B^S)} \stackrel{\mu}{\rightarrow} F_B^N \times \prod_{|S|\geq 2}{(F_B^S)} \stackrel{\beta}{\rightarrow} F_B^N \end{equation} where $\mu$ is the natural blow-down morphism and $\beta$ is the projection on the first factor. Since $\mu$ is a projective morphism and $F_B[n]$ is a closed subvariety, $$\mu: F_B[n] \rightarrow \mu(F_B[n])$$ is also projective. Since $\beta: \mu(F_B[n]) \stackrel{\sim} {\rightarrow} F_B^N$ is an isomorphism, the morphism $\rho=\beta \circ \mu$ $$\rho: F_B[n] \rightarrow F_B^N$$ is projective. For our purposes, we shall only consider the case where $\pi:{F}\rightarrow{ B}$ is a quasi-projective morphism. Also, we will be mainly interested in the case where $F$ and $F^N_B$ are irreducible varieties. \subsection{The Blow-Up Construction} \label{rfmb} Consider again the birational projective morphism $$\rho :F_B[n]\rightarrow F_B^N$$ It is natural to inquire whether $\rho$ can be expressed as a composition of explicit blow-ups along canonical subvarieties. In [F-M], such a blow-up construction is given for the configuration space in case $B$ is a point. The blow-ups in [F-M] are canonical in the following sense: if $Y \rightarrow X$ is an immersion, the sequence of blow-ups resolving $Y[n] \rightarrow Y^N$ is the sequence of strict transformations of $Y^N$ under the blow-ups resolving $X[n] \rightarrow X^N$. The blow-up construction of Fulton and MacPherson is valid in the relative context. We now assume that $\pi: F\rightarrow B$ is a quasi-projective morphism. In this case, there exists a factorization: \begin{equation*} \begin{CD} F @>i>> {\bold P}^r \times B \\ @VV{\pi}V @VVV \\ B @= B \end{CD} \end{equation*} where $i$ is an immersion. We use the notation $\bold P^r \times B = \bold P_B^r$ and drop extra $B$ subscripts when the meaning is clear. For example, $(\bold P_B^r)^N$ instead of $(\bold P_B^r)_B^N$ . We have the following commutative diagram: \begin{equation} \begin{CD} (F_B^N)_0 @>>> F_B^N \times \prod_{|S|\geq 2}{Bl_\bigtriangleup (F_B^S)}\\ @VV{i^N}V @VV{i^{Bl}}V \\ (\bold P_B^r)^N_0 @>>> (\bold P_B^r)^N \times \prod_{|S|\geq 2}{Bl_\bigtriangleup ((\bold P_B^r)^S)} \label{snake} \end{CD} \end{equation} where $i^N$, $i^{Bl}$ are immersions. We conclude from diagram (\ref{snake}) that $F_B[n]$ is immersed in $\bold P_B^r[n]$. Hence: \begin{equation} \begin{CD} F_B[n] @>j>> \bold P_B^r[n] \\\ @VV{\rho}V @VV{\eta}V \\ F_B^N @>{i^N}>> (\bold P_B^r)^N \label{eagle} \end{CD} \end{equation} where $i^N$, $\;j$ are immersions. Since $\rho$ is a projective morphism, $j(F_B[n])$ is closed in $\eta^{-1}(i^N(F_B^N))$. $F_B[n]$ is therefore the strict transformation of $F_B^N$ under $\eta$. It is clear the following diagram holds: \begin{equation*} \begin{CD} \bold P_B^r[n] @= \bold P^r[n] \times B \\ @VV{\eta}V @VV{\gamma \times id}V \\ (\bold P_B^r)^N @=( \bold P^r)^N \times B \end{CD} \end{equation*} In [F-M], an explicit and canonical blow-up construction of $\gamma$ is given. By extending each exceptional locus over the base $B$, a blow-up construction of $\eta$ is obtained. We see from diagram (\ref{eagle}) that a blow-up construction of $F_B[n]$ exists by taking the strict transformation of $F_B^N$ at each blow-up of $(\bold P_B^r)^N$. \subsection{Comparing $F_b[n]$ and $F_B[n]_b$} \label{rfmc} For a given $b \in B$ let $F_b$ denote the fiber of $\pi$ over $b$. {}From equation (\ref{mouse}) and the definitions, it is clear there exists a natural closed immersion: $$F_b[n] \stackrel{i_b}{\hookrightarrow} F_B[n]_b .$$ It is possible for $i_b$ to be a proper inclusion. Examples of this behavior will be seen in section (\ref{disc}). We have the following: \begin{pr} \label{pfiber} If $B$ is irreducible and $\pi:F \rightarrow B$ is a smooth, quasi-projective morphism of nonsingular varieties, then for every $b \in B$, $i_b$ is an isomorphism. \end{pr} \begin{pf} Suppose $X$ is a fixed nonsingular algebraic variety. In [F-M], the canonical construction of $X[n]$ is given by a sequence of explicit blow-ups of $X^N$ along {\it nonsingular} centers. In the previous section, it was shown how the construction of [F-M] could be lifted to the relative context. Let $m$ be the number of blow-ups needed in the Fulton- MacPherson construction resolving $\rho: F_B[n] \rightarrow F_B^N$. Let $F_{B,j}^N$ for $0 \leq j \leq m$ denote the $j^{th}$ stage. $F_{B,0}^N= F_B^N$ and $F_{B,m}^N= F_B[n]$. Since the blow-up construction in [F-M] is canonical, for any variety $X$ similar definitions can be made. We show inductively, for each $b \in B$, the natural inclusion: \begin{equation} \label{duck} F_{b,j}^N \hookrightarrow (F_{B,j}^N)_b \end{equation} is an isomorphism. For $j=0$ the assertion is clear. The induction step rests on a simple {\bf Claim}: Suppose $S$ is an irreducible nonsingular variety, $ R \rightarrow S$ is a smooth morphism, and $T \hookrightarrow R$ is a closed immersion smooth over $S$ . Then, for any $s \in S$, the blow-up of $R_s$ along $T_s$ is naturally isomorphic to the fiber over $s$ of the blow-up of $R$ along $T$. Since all spaces are nonsingular, the assertion follows from examining normal directions of $T$ in $R$; the various smoothnesses imply all normal directions are represented in the fiber. Assume equation (\ref{duck}) is an isomorphism for all $b \in B$. Since $\pi$ is smooth of relative dimension $k$, $F_b$ and $F_{b,j}^N$ are nonsingular of pure dimensions $k$, $nk$. Hence, $(F_{B,j}^N)_b$ is nonsingular of pure dimension $nk$. Also, every irreducible component of $F_B^N$ (and hence $F_{B,j}^N$) is of relative dimension $nk$. The last two facts imply the morphism: $$\pi_j^N:F_{B,j}^N \rightarrow B$$ is smooth. Examination of the $(j+1)^{th}$ center is straightforward. Because of the assumed isomorphism (\ref{duck}) and the knowledge that the [F-M] construction of the configuration space of a nonsingular variety over a point only involves nonsingular centers, we see that the $(j+1)^{th}$ center is smooth over B. The above claim now proves the induction step. \end{pf} \subsection{Universal Families} \label{rfmu} Let $X$ be a nonsingular algebraic variety. Let $\overline{x}=(x_1, \ldots, x_n)$ be $n$ ordered points of $X$. A subset $S\subset N$ is said to be coincident at $z\in X$ if $|S| \geq 2$ and for all $i \in S$, $x_i=z$. Following [F-M], for every $S$ coincident at $z$, we define a {\it screen} of $S$ at $z$ to be an equivalence class of the data $(t_i)_{i\in S}$ where: \begin{enumerate} \item $t_i \in T_z$, the tangent space of $X$ at $z$. \item $\exists i,j \in S$ such that $t_i \neq t_j$. \end{enumerate} Two data sets $(t_i)_{i\in S}$ and $(t'_i)_{i\in S}$ are equivalent if there exists $\lambda \in C^*$ and $v \in T_z$ so that for all $i \in S$, $\lambda \cdot t_i + v= t'_i$. A screen shows the tangential separation of infinitely near points. An $n$-tuple $\overline{x}=(x_1, \ldots, x_n)$ together with a screen $Q_S$ for each coincident subset $S\subset N$ constitute an {\it n-pointed stable class} in $X$ if the screens are compatible in the following sense. Suppose $S_1\subset S_2$ are two subsets coincident at $z$ where $Q_{S_2}$ is represented by the data $(t_j)_{j \in S_2}$. If there exist $k,\hat{k} \in S_1$ so that $t_k\neq t_{\hat k}$, then $(t_j)_{j\in S_1}$ defines a screen for $S_1$. The compatibility condition requires that when this restriction of $Q_{S_2}$ is defined, it equals $Q_{S_1}$. For a nonsingular space $X$, $X[n]$ is the parameter space of $n$-pointed stable classes in $X$. Given an $n$-pointed stable class in $X$, an {\it $n$-pointed stable degeneration } of $X$ can be constructed (up to isomorphism) as follows. Let $z\in X$ occur with multiplicity in $\overline{x}$. Blow-up $X$ at $z$ and attach a $\bold P(T_z \oplus {\bold 1})$ in the natural way along the exceptional divisor at $z$. Note that $\bold P(T_z \oplus {\bold 1})$ minus the hyperplane at infinity, $\bold P(T_z)$, is naturally isomorphic to the affine space $T_z$. Let $S_z \subset N$ be the maximal subset coincident at $z$. The screen $Q_{S_z}$ associates (up to equivalence) points of $T_z$ to the indices that lie in $S_z$. Condition (2) of the screen data implies some separation of the marked points has occurred. The further screens specify in a natural way (up to equivalence of screens) the further blow-ups and markings required to separate the marked points. The final space obtained along with $n$ distinct marked points is the $n$-pointed stable degeneration associated to the given $n$-pointed stable class. See [F-M] for further details. It is shown in [F-M] that if $X$ is an nonsingular variety, there exists a universal family of $n$-pointed stable degenerations of $X$ over $X[n]$. Let $X[n]^+$ denote this universal family. $X[n]^+$ is equipped with the following maps: \begin{equation*} \begin{CD} X[n]^+ @>{\mu}>> X[n] \times X \\ @VV{\mu_p}V @VVV \\ X[n] @= X[n] \end{CD} \end{equation*} There are $n$ sections of ${\mu_p}$, $\{ \sigma_i \}_{i \in N}$ : $$X[n] \stackrel{\sigma_i}{\rightarrow} X[n]^+ . $$ For any $d\in X[n]$, the fiber $\mu^{-1}_p(d)$ along with the $n$-tuple $(\sigma_1(d), \ldots,\sigma_n(d))$ is the $n$-pointed stable degeneration of $X$ associated to the $n$-pointed stable class corresponding to $d$. We note here that if $C$ is a nonsingular automorphism-free curve, $n$-pointed stable classes in $C$ correspond bijectively to isomorphism classes of $n$-pointed Deligne-Mumford stable curves over $C$ . Moreover, the universal family over $C[n]$ defines a map to the reduced fiber $$\phi:C[n] \rightarrow \gamma^{-1}([C])$$ where $\gamma :\overline{M}_{g,n} \rightarrow \overline{M}_g$. Since $\phi$ is proper bijective and both spaces are normal, $\phi$ is an isomorphism. If $C$ has a finite automorphism group, $A$, we see $A$ acts on $C[n]$ and $\phi$ is $A$-invariant. Therefore $\phi$ descends to the quotient $$\phi (C[n]/A) \rightarrow \gamma^{-1}([C]).$$ It is not hard to see that this map is proper bijective and hence an isomorphism by normality. The map $\mu$ is a birational morphism and can be expressed as an explicit sequence of blow-ups of $X[n] \times X$ along canonical, nonsingular loci. Canonical here has the same meaning as in section (\ref{rfmb}) : if $Y \rightarrow X$ is an immersion of nonsingular varieties, the blow-up sequence resolving $Y[n]^+ \rightarrow Y[n]\times Y$ is the strict transform of of $Y[n] \times Y$ under the blow-up sequence resolving $X[n]^+ \rightarrow X[n] \times X$. Moreover, the sections of $Y[n]^+ \rightarrow Y[n]$ are restrictions of the sections of $X[n]^+ \rightarrow X[n]$. This canonical blow-up construction is given in [F-M]. \subsection{Relative Universal Families} \label{rfmru} Suppose $\pi:F \rightarrow B$ is a smooth, quasi-projective morphism of nonsingular varieties. In this case, the construction of the universal family that appears in [F-M] can be lifted to the relative context. Using the notation of section (\ref{rfmb}), we have an immersion: $$F_B[n] \times_B F \rightarrow \bold P_B^r[n] \times_B \bold P_B^r \;.$$ Consider the diagram: \begin{equation*} \begin{CD} \bold P_B^r[n]^+ @= \bold P^r[n]^+ \times B \\ @VV{\omega}V @VV{\mu \times id}V \\ \bold P_B^r[n] \times_B \bold P_B^r @= \bold P^r[n]\times \bold P^r \times B \end{CD} \end{equation*} For $\omega=\mu \times id$, the Fulton-MacPherson construction of the universal family can be carried out uniformly over the base by extending the centers of the blow-ups resolving $\mu$ trivially over $B$. Define $F_B[n]^+$ to be the proper transform of $ F_B[n] \times_B F $ under $\omega$. We have: $$\upsilon: F_B[n]^+ \rightarrow F_B[n] \times_B F$$ To show the space defined above, $F_B[n]^+$, has the desired geometrical properties, we argue as in the proof of Proposition 1. Let $ (F_B[n] \times_B F)_j $ denote the $j^{th}$ stage of the canonical sequence of blow-ups resolving $\upsilon$. Inductively, it is shown that for each $b \in B$ there is an isomorphism: \begin{equation} \label{cow} (F_b[n] \times F_b)_j \rightarrow (F_B[n] \times_B F)_{j,b}. \end{equation} The $j=0$ case is established by Proposition 1. The induction step follows from the the claim made in the proof of Proposition 1 and the fact that for a nonsingular variety $X$, the canonical Fulton-MacPherson resolution of $X[n]^+ \rightarrow X[n]\times X$ involves only nonsingular centers. We conclude that fiber $F_B[n]^+_b$ over $b \in B$ is naturally isomorphic to $F_b[n]^+$. It is clear that $n$ sections ${\sigma_i}$ exist for $$\omega_p: \bold P_B^r[n]^+ \rightarrow \bold P_B^r[n].$$ For each $b \in B$, these sections ${\sigma_i}$ are compatible with the $n$ natural sections of $F_b[n]^+ \rightarrow F_b[n].$ Therefore, via restriction, the ${\sigma_i}$ yield $n$ sections of $$\upsilon_p: F_B[n]^+ \rightarrow F_B[n].$$ The fiber of $F_B[n]^+_\xi$ over $\xi \in F_B[n]$ is a $n$-pointed stable degeneration of $F_{\pi(\xi)}$. We have: \begin{pr} Suppose $B$ is irreducible and $\pi: F \rightarrow B$ is a smooth, quasi-projective morphism of nonsingular varieties, then $F_B[n]^+$ along with $\upsilon$ and $\{\sigma_i\}_{i \in N}$ is a universal family of n-pointed stable degenerations of $F_B$ over $F_B[n]$. \end{pr} \subsection{Final Note} \label{rfmfn} Suppose $\pi:G \rightarrow B$ is a projective morphism, $G$ is nonsingular, irreducible, $B$ is nonsingular, $\pi$ is flat, and for every $b \in B$ the fiber $G_b$ is reduced. In this case, let $F \subset G$ be the open set where $\pi$ is smooth. Using flatness and a tangent space calculation, we see: $$ F=\{\xi \in G|\xi \hbox{ is a nonsingular point of } G_{\pi(\xi)} \} $$ and $\pi: F \rightarrow B$ is a smooth, surjective morphism of nonsingular varieties. We know the space $F_B[n]$ is equipped with a universal family $F_B[n]^+$ obtained from $F_B[n]\times_B F$ by a sequence of canonical blow-ups. The problem with this universal family is that its fibers over $F_B[n]$ are $n$-pointed stable degenerations of $F_B$ not $G_B$. This problem can easily be fixed. Note there is an open inclusion: $$F_B[n]\times_B F \subset F_B[n]\times_B G .$$ It is the case that the centers of the blow-ups resolving $$\upsilon:F_B[n]^+ \rightarrow F_B[n] \times_B F$$ are closed in $F_B[n]\times_B G$. Using the isomorphism (\ref{cow}) and the explicit description of the centers of blow-ups in [F-M], this closure is not hard to check. Hence, if the sequence of blow-ups is carried out over $F_B[n]\times_B G$ the desired family of $n$-pointed stable degenerations of $G_B$ is obtained over $F_B[n]$. An $n$-pointed stable degeneration of a fiber $G_b$ is as before with the additional condition that the marked points must lie over the smooth locus of $G_b$. \section{The Geometric Invariant Theory Set-Up} \label{git} \subsection{Notation} \label{gitn} Let ${\overline{ M}_g}$ denote the Deligne-Mumford compactification of the moduli space of nonsingular, genus $g$, projective curves, $M_g$. Let ${\overline{M}_{g,n}}$ denote the Deligne-Mumford compactification of the moduli space of genus $g$ curves with $n$ marked points. There exists a natural projective morphism $$\gamma : {\overline {M}_{g,n}} \rightarrow {\overline {M}_g}.$$ All these spaces are normal. \subsection{Gieseker's construction of $\overline{M}_g$} \label{gitg} Fix an integer $g \geq 2$. Define: $$d=10 \cdot (2g-2)$$ $$R=d-g.$$ Define the polynomial: $$f(m)= d \cdot m -g +1.$$ Note $f(m)$ is the Hilbert polynomial of a complete, genus $g$, $10$-canonical curve in $\bold P ^R$. Let $H_{f,R}$ denote the Hilbert scheme of the polynomial $f$ in $\bold P ^R$. If $X$ is a closed subscheme of $\bold P ^R$ with Hilbert polynomial $f$, we denote the point of $H_{f,R}$ corresponding to $X$ by $[X]$. It is well known that there exists an integer ${\widehat{m}}$ such that, for any $m \geq \widehat{m}$ and any closed subscheme $X \subset \bold P ^R$ corresponding to a point $ [X] \in H_{f,R}$, \begin{equation} \label{hippo} h^1(I_X(m),\bold P ^R) = 0 \end{equation} \begin{equation} \label{bear} h^0({\cal O}_X(m), X)= f(m) \; . \end{equation} Therefore, for any $m \geq \widehat{m}$, there is a natural map: $$i_m: H_{f,R} \rightarrow {\bold P}(\bigwedge ^{f(m)} H^0( {\cal O}_{{\bold P}^R}(m), \bold P ^R)^*).$$ Where $i_m$ is defined for each $[X] \in H_{f,R}$ as follows: by (\ref{hippo}), there is a natural surjection $$ H^0({\cal O}_{\bold P ^R}(m), \bold P ^R) \rightarrow H^0({\cal O}_X(m),X) $$ which yields, by (\ref{bear}), a surjection \begin{equation} \label{whale} \bigwedge ^{f(m)} H^0({\cal O}_{\bold P ^R}(m), \bold P ^R) \rightarrow \bigwedge ^{f(m)} H^0({\cal O}_X(m),X) \cong {\bold C}. \end{equation} The last surjection (\ref{whale}) is an element of ${\bold P}(\bigwedge ^{f(m)} H^0( {\cal O}_{{\bold P}^R}(m), {\bold P}^R)^*)$. The map $i_m$ is now defined on sets. That $i_m$ is an algebraic morphism of schemes can be seen by constructing (\ref{whale}) uniformly over $ H_{f,R}$ and using the universal property of ${\bold P}(\bigwedge ^{f(m)} H^0({\cal O}_{{\bold P}^R}(m), {\bold P}^R)^*)$. In fact, it can be shown there exists an integer $\overline{m}$ such that for every $m \geq \overline{m}$, $i_m$ is a closed immersion. {}From the universal property of the Hilbert scheme, we obtain a natural $SL_{R+1}$-action on $H_{f,R}$. For each $m \geq \overline{m}$, the closed immersion $i_m$ defines a linearization of the natural $SL_{R+1}$-action on $H_{f,R}$. Define the following locus $\overline{K}_g\subset H_{f,R}$: $[X] \in \overline{K}_g$ if and only if $X$ is a nondegenerate, 10-canonical, genus g, Deligne-Mumford stable curve in $\bold P^R$. $\overline{K}_g$ is a quasi-projective, $SL_{R+1}$-invariant subset. In [G], Gieseker shows a linearization $i_m$ can be chosen satisfying: \begin{enumerate} \item[(i)] The stable locus of the corresponding G.I.T. quotient contains $\overline{K}_g$. \item[(ii)] $\overline{K}_g$ is closed in the semistable locus. \end{enumerate} {}From (i), we see $\overline{K}_g/SL_{R+1}$ is a geometric quotient. By (ii), $\overline{K}_g/SL_{R+1}$ is a projective variety. Since $\overline{K}_g$ is a nonsingular variety ([G]), it follows that $\overline{K}_g/SL_{R+1}$ is normal. {}From the definition of $\overline{K}_g$, the universal family over $H_{f,R}$ restricted to $\overline{K}_g$ is a family of Deligne-Mumford stable curves. Therefore there exists a natural map $\mu: \overline{K}_g \rightarrow \overline{M}_g$. Since $\mu$ is $SL$-invariant, $\mu$ descends to a projective morphism from the quotient $\overline{K}_g/SL_{R+1}$ to $\overline{M}_g$. Since $\mu$ is one to one and $\overline{M}_g$ is normal, $\mu$ is an isomorphism. Note that since $\overline{M}_g$ is irreducible, $\overline{K}_g$ is also irreducible. \subsection{The Relative $n$-pointed Fulton-MacPherson Configuration Space of the Universal Curve} \label{gitrf} Let $\pi: U_{H} \rightarrow H_{f,R}$ be the universal family over the Hilbert scheme defined in section (\ref{gitg}) where $\pi$ is a flat, projective morphism. Let $\overline{K}_g \subset H_{f,R}$ be defined as above. Let $U_{\overline{K}_g}$ be the restriction of $U_H$ to $\overline{K}_g$. Following the notation of section (\ref{rfmd}), we define $U_{\overline {K}_g}[n]$ to be the relative Fulton- MacPherson space of $n$-marked points on $U_{\overline {K}_g}$ over ${\overline {K}_g}$. {}From section (\ref{rfmb}), we see the immersion $\zeta$: \begin{equation*} \begin{CD} U_{\overline {K}_g} @>{\zeta}>> \bold P ^R \times H_{f,R} \\ @VV{\pi}V @VV{\rho}V \\ {\overline {K}_g} @>>> H_{f,R} \end{CD} \end{equation*} yields another immersion $\zeta[n]$: \begin{equation*} \begin{CD} U_{\overline{K}_g}[n] @>{\zeta[n]}>> \bold P ^R[n] \times H_{f,R} \\ @VV{\pi[n]}V @VV{\rho[n]}V \\ {\overline{K}_g} @>>> H_{f,R} \end{CD} \end{equation*} There exists a natural $SL_{R+1}$-action on $ \bold P ^R[n]$ and therefore on $ \bold P ^R[n] \times H_{f,R}$. Since $U_{\overline {K}_g}$ is invariant under the natural $SL_{R+1}$-action, we see $U_{\overline{K}_g}[n]$ is also $SL_{R+1}$-invariant. Since $\pi$ is projective, $U_{\overline{K}_g}[n] \subset \rho[n]^{-1}(\overline {K}_g)$ is a closed subset. It follows from (i) and (ii) of section (\ref{gitg}) and Propositions (7.1.1) and (7.1.2) of [P] that there exist linearizations of the natural $SL_{R+1}$-action on $ \bold P ^R[n] \times H_{f,R}$ satisfying: \begin{enumerate} \item[(i)] $U_{\overline{K}_g}[n]$ is contained in the stable locus of the corresponding G.I.T. quotient. \item[(ii)] $(\rho[n]^{-1}(\overline{K}_g))^{SS}$ is closed in the semistable locus. \end{enumerate} {}From (i), (ii), and the fact that $U_{\overline{K}_g}[n]$ is closed in $\rho[n]^{-1}(\overline {K}_g)$, we see that $U_{\overline {K}_g}[n]/SL_{R+1}$ is a geometric quotient and a projective variety. Define: $$M_{g,n}^{c} = U_{\overline {K}_g}[n]/SL_{R+1} .$$ Note there is a natural projective morphism $$\rho : M_{g,n}^{c} \rightarrow {\overline {M}_{g}}$$ descending from the $SL_{R+1}$-invariant maps: $$U_{\overline {K}_g}[n] \rightarrow {\overline {K}_g} \rightarrow {\overline {M}_g}.$$ It follows easily that $M_{g,n}^{c}$ is a compactification of $\gamma^{-1}(M_g)$. To see this first make the definition: $${K_g} =\{[X] \in H_{f,R}|X \hbox{ is a nondegenerate, 10-canonical, nonsingular, genus g curve} \}. $$ $U_{K_g}[n]$ is a dense open $SL_{R+1}$-invariant subset of $U_{\overline{K}_g}[n]$. Since the morphism $\pi :U_{K_g} \rightarrow K_g$ is smooth, we see from section (\ref{rfmru}) that there exists a universal family of Deligne-Mumford stable $n$-pointed genus $g$ curves over $ U_{K_g}[n]$. This universal family yields a canonical morphism $$\mu: U_{K_g}[n] \rightarrow \gamma^{-1}(M_g).$$ It is easily checked that $\mu$ is $SL_{R+1}$-invariant. Therefore, $\mu$ descends to the open set, $\rho^{-1}(M_g)$, of $M_{g,n}^{c}$. One sees $$\mu_d:\rho^{-1}(M_g) \rightarrow \gamma^{-1}(M_g)$$ is a bijection by Proposition (\ref{pfiber}) and the fact that, for a smooth curve C, $$ (C[n]/\hbox{automorphisms}) \cong \gamma^{-1}([C])\subset \gamma^{-1}(M_g). $$ (See section (\ref{rfmu})). Since $\rho: \rho^{-1}(M_g) \rightarrow M_g$ is projective, $\gamma: \gamma^{-1}(M_{g}) \rightarrow M_g$ is separated, and $\rho= \gamma \circ \mu_d$, we conclude $\mu_d$ is projective. A bijective projective morphism onto a normal variety is an isomorphism. Since $\gamma^{-1}(M_g)$ is normal, $\mu_d$ is an isomorphism. \section{A Description Of $M_{g,n}^c$} \label{disc} \subsection{} \label{dis1} Let $\pi: U_{\overline{K}_g} \rightarrow \overline{K}_g$ be as above. Following section (\ref{rfmfn}), we define $F\subset U_{\overline{K}_g}$ to be the locus where $\pi$ is smooth. $F_{\overline{K}_g}[n] \subset U_{\overline{K}_g}[n]$ is an open $SL$-invariant subset. The points of $F_{\overline{K}_g}[n]$ parameterized $n$-pointed stable classes on the nonsingular locus of the fibers of $\pi$. There exists a universal family over $F_{\overline{K}_g}[n]$ which defines an $SL$-invariant morphism : \begin{equation*} \mu :F_{\overline{K}_g}[n] \rightarrow \overline{ M}_{g,n}^{s}. \end{equation*} Where $\overline{ M}_{g,n}^{s}$ parameterizes $n$-pointed, genus g, Deligne-Mumford stable curves with marked points lying over nonsingular points of the contracted stable model. Let $$F_{\overline{K}_g}[n]/SL_{R+1} = (M_{g,n}^c)^{s}.$$ $SL$-invariance implies $\mu$ descends to: \begin{equation} \label{quail} \mu_d: (M_{g,n}^c)^{s} \rightarrow \overline{ M}_{g,n}^{s}. \end{equation} {}From the arguments of section (\ref{rfmu}), we see $\mu_d$ is bijective. From the valuative criterion, it follows $\mu_d$ is proper. As before, by normality, it follows that $\mu_d$ is an isomorphism. \subsection{Points Of $M_{g,n}^c$ Over A Singular Point} \label{dissing} {}From section (\ref{dis1}), it is clear only the behavior of $U_{\overline {K}_g}[n]$ over a singular point of $U_{\overline{K}_g}$ remains to be investigated. Since this is a local question about the the smooth deformation of a node, it suffices to investigate the family: \begin{equation*} \begin{CD} G @>>> Spec(C[x,y]) \times Spec(C[t]) \\ @VV{\pi}V @VVV \\ Spec(C[t]) @= Spec(C[t]) \end{CD} \end{equation*} Where $G$ is defined by the equation $xy-t$. In the Fulton-MacPherson configuration space $Spec(C[x,y])[n]$, there is a closed subset $\cal {T}_n$ corresponding to the points lying over $(0,0)$. In the notation of section (\ref{rfmb}), $$ \cal{T}_n = \rho^{-1}(\underbrace{(0,0),(0,0), \ldots,(0,0)}_{n}).$$ Recall the notation of section (\ref{rfmd}). Let $B=Spec(C[t])$, $B^*=Spec(C[t])-(0)$, and $G^*=\pi^{-1}(B^*)$. We want to investigate the subset $\cal{W}_n \subset (\cal{T}_n,0)$ that lies in the closure of $G_{B^*}^{*}[n]$ in $Spec(C[x,y])[n]\times B$. Suppose $\kappa$ is a family in $(G_{B^*}^{*N})_0$ where all the marked points specialize to the node $\zeta$ of $G_0$. After a base change, $t \rightarrow t^r$, $\kappa$ can be defined by $n$ sections, $(\kappa_1, \ldots, \kappa_n)$, of $\pi$ in a neighborhood of $0 \in B$. The equation of $G$ after base change is now $G_r=xy-t^r$. Let us take $r=2$. The blow-up of $G_2$ at $\zeta$ is nonsingular and is defined in an open set by the equation $ab-1$ in $Spec(C[a,b]) \times Spec(C[t])$. The blow-down morphism is defined by the equations: $$x=at$$ $$y=bt.$$ Now assume that the strict transforms of the sections, $(\kappa_1, \ldots, \kappa_n)$, meet the exceptional curve $(ab=1,t=0)$ in distinct points $((a_1,a_1^{-1}), \ldots, (a_n,a_n^{-1}))$, $\forall i$ $a_i\neq 0$. Then it is clear that the $n$-pointed stable class in $\cal{T}_n$ that is the limit of $\kappa$ is the class in the tangent space of $C[x,y]$ at $(0,0)$ defined by the pairs of vectors: $$((a_1,a_1^{-1}), \ldots, (a_n,a_n^{-1}))$$ in the basis $(\frac{\partial}{\partial x}, \frac{\partial}{\partial y})$. We now define a map: \begin{equation*} \theta_n: (C^{*N})_0 \rightarrow \cal{T}_n \end{equation*} Where $\theta_n((a_1,\ldots,a_n))$ is the $n$-pointed stable class defined by the tangent vectors $((a_1,a_1^{-1}), \ldots, (a_n,a_n^{-1}))$. The preceding paragraph shows that $Image(\theta_n)\subset \cal{W}_n$. In fact, it is not hard to see that $Image(\theta_n)$ is dense in a component of $\cal{W}_n$. For $n=2$, $\cal{W}_2= \cal{T}_2$ where $\cal{T}_2$ is just the $\bold P^1$ of normal directions. Suppose $n \geq 3$. Let $\hat{a}=(a_1,\ldots,a_n)$ and $\hat{b}=(b_1,\ldots,b_n)$ be distinct points of $(C^{*N})_0$. Then, $\theta_n(\hat{a})=\theta_n(\hat{b})$ if and only if there exists a tangent vector $(v_1,v_2)$ and an element $\lambda \in C^*$ such that : $$ \forall i, \;\;\; \lambda \cdot a_i+v_1=b_i \hbox{ and } \lambda \cdot a_i^{-1}+v_2=b_i^{-1} .$$ These equations imply \begin{equation} \label{elk} \forall i,j,\; \;\;\; \lambda \cdot (a_i-a_j)=(b_i-b_j) \end{equation} \begin{equation} \label{rhino} \forall i,j,\; \;\;\; \lambda \cdot(a_i^{-1}-a_j^{-1})=(b_i^{-1}-b_j^{-1}) \end{equation} Dividing (\ref{elk}) by (\ref{rhino}) yields $a_i\cdot a_j=b_i\cdot b_j$. For $n\geq 3$, we easily obtain $\hat{a}=\pm\hat{b}$. Therefore, a component of $\cal{W}_n$ can be viewed as a compactification of $(C^{*N})_0/(\pm)$. We note that the dimension of $\cal{W}_n$ is $n$ for $n\geq 3$. \section{Comparison with $\overline {M}_{g,n}$ for $n=1,2$} \subsection{$n=1$} {}From the definitions, $M_{g,1}^c$ equals $U_{\overline {K}_g}/SL_{R+1}$. $\pi^*(U_{\overline{K}_g})$ is a family of 1-pointed Deligne-Mumford genus $g$ curves over $ U_{\overline{K}_g}$ via the natural diagonal section. This tautological family yields an $SL$-invariant morphism: $$\mu:U_{\overline{K}_g} \rightarrow \overline{M}_{g,1} $$ that descends to $$\mu_d: M_{g,1}^c \rightarrow \overline{M}_{g,1}.$$ Since $\mu_d$ is proper bijective and $\overline{M}_{g,1}$ is normal, $\mu_d$ is an isomorphism. \subsection{$n=2$} Consider the family: \begin{equation} U_{\overline {K}_g}^2=U_{\overline {K}_g} \times_{\overline {K}_g} U_{\overline {K}_g}\;. \label{mongoose} \end{equation} The singular locus of $U_{\overline {K}_g}^2$, $S$, is nonsingular of pure codimension $3$ and $SL_{R+1}$-invariant. The singular points are pairs $(\zeta,\zeta)$ where $\zeta \in U_{\overline {K}_g}$ is a node of a fiber. Moreover, the singularities of $ U_{\overline {K}_g}^2$ are \'etale-locally ordinary threefold double point singularities. That is, the singularities are of the form \begin{equation} \label{possum} W\times Spec(C[a,b,c,d]/(ab-cd)) \subset W\times Spec(C[a,b,c,d]) \end{equation} Where $W$ is nonsingular. These assertions about the singular locus follow from the deformation theory of a Deligne-Mumford stable curve and [G]. There are three standard resolutions of the ordinary double point singularity $Spec(C[a,b,c,d]/(ab-cd))$: \begin{enumerate} \item The blow-up along $(a,b,c,d)$. \item For any $\lambda \in C$, the blow-up along $(a-\lambda \cdot c, \lambda \cdot b-d)$. \item For any $\lambda \in C$, the blow-up along $(a-\lambda \cdot d, \lambda \cdot b-c)$. \end{enumerate} Methods (2) and (3) yield the distinct small resolutions. The local description (\ref{possum}) implies that the blow-up of $U_{\overline {K}_g}^2$ along $S$ is nonsingular with an exceptional divisor $E$ that is a $\bold P^1\times \bold P^1$- bundle over $S$. Using the techniques of section (\ref{gitrf}), it can be shown that the natural $SL_{R+1}$-action on the blow-up $Bl_{(S)}(U_{\overline {K}_g}^2)$ can be linearized so that all the points in question are stable and the quotient is projective. The diagonal embedding $$D:U_{\overline {K}_g} \hookrightarrow U_{\overline {K}_g}^2$$ is divisorial except along $S$ where it of the form of (2) and (3) in the local description (\ref{possum}). By definition, $$M_{g,2}^c= Bl_{(D)}(U_{\overline {K}_g}^2)\;\;/SL_{R+1}.$$ There is a natural blow-down map: $$\rho: Bl_{(S)}(U_{\overline {K}_g}^2)\rightarrow Bl_{(D)}(U_{\overline {K}_g}^2).$$ Another $SL_{R+1}$-invariant small resolution of $U_{\overline {K}_g}^2$ can be obtain by blowing-down uniformly along the opposite ruling of $E$ blown-down by $\rho$. Let $Y$ denote this other small resolution and let $$\overline{\rho}: Bl_{(S)}(U_{\overline {K}_g}^2) \rightarrow Y$$ be the blow-down. Linearizations can be chosen so that $$Y/SL_{R+1} \cong \overline{M}_{g,2}.$$ There are birational morphisms \begin{equation} \label{birad} M_{g,2}^c \ \leftarrow \ \ Bl_{(S)}(U_{\overline {K}_g}^2)\;\;/SL_{R+1} \ \ \rightarrow\ \overline{M}_{g,2}. \end{equation} Consider the open loci of $M^{c}_{g,2}$ and $\overline{M}_{g,2}$ where the underlying curve has no (nontrivial) automorphism. On the automorphism free loci the birational modification (\ref{birad}) is easy to describe. Let $F_1\subset M_{g,2}^c$ be the locus of of $2$-pointed stable classes that lie over a node in a Deligne-Mumford stable curve of genus g. Similarly, let $F_2\subset \overline{M}_{g,2}$ be the locus of $2$-pointed, genus g, Deligne-Mumford stable curves such that the marked points are coincident at a node in the stable contraction. On the automorphism free loci, $M_{g,2}^c$ and $\overline{M}_{g,2}$ are the distinct small resolution of the fiber product of the universal curve with itself. Hence, on the automorphism free loci, the blow-up of $M_{g,2}^2$ along $F_1$ is isomorphic to the blow-up of $\overline{M}_{g,2}$ along $F_2$. The modification (\ref{birad}) obtained by this isomorphism.

9505

alg-geom/9505009

en

https://arxiv.org/abs/alg-geom/9505009

alg-geom/9505009

Ron Donagi

Ron Donagi

Spectral Covers

This is a survey of various results about spectral covers and their relationship to Higgs bundles. To a G-principal Higgs bundle on a variety S corresponds a cameral cover \widetilde{S} of S (a W-Galois cover, where W is the Weyl group of G) together with a sheaf on \widetilde{S} which in simple cases is a line bundle, and is W-equivariant up to certain twists and shifts. Various other types of spectral covers, depending on the choice of a representation or weight of G, arise as associated objects of \widetilde{S}. We focus on the decomposition of the Picards of these spectral covers into Pryms (this includes various well-known Prym identities as special cases) and on the interpretation, in the spirit of Hitchin's abelianization program, of a distinguished Prym component as parameter space for higgs bundles.

alg-geom

\section{Introduction} \label{intro}\ \indent Spectral curves arose historically out of the study of differential equations of Lax type. Following Hitchin's work \cite{H1}, they have acquired a central role in understanding the moduli spaces of vector bundles and Higgs bundles on a curve. Simpson's work \cite{S} suggests a similar role for spectral covers $\widetilde{S}$ of higher dimensional varieties $S$ in moduli questions for bundles on $S$. The purpose of these notes is to combine and review various results about spectral covers, focusing on the decomposition of their Picards (and the resulting Prym identities) and the interpretation of a distinguished Prym component as parameter space for Higgs bundles. Much of this is modeled on Hitchin's system, which we recall in section \ref{Hitchin}, and on several other systems based on moduli of Higgs bundles, or vector bundles with twisted endomorphisms, on curves. By peeling off several layers of data which are not essential for our purpose, we arrive at the notions of an {\em abstract principal Higgs bundle} and a {\em cameral} (roughly, a principal spectral) cover. Following \cite{D3}, this leads to the statement of the main result, theorem \ref{main}, as an equivalence between these somewhat abstract `Higgs' and `spectral' data, valid over an arbitrary complex variety and for a reductive Lie group $G$. Several more familiar forms of the equivalence can then be derived in special cases by adding choices of representation, value bundle and twisted endomorphism. This endomorphism is required to be {\em regular}, but not semesimple. Thus the theory works well even for Higgs bundles which are everywhere nilpotent. After touching briefly on the symplectic side of the story In section \ref{symplectic}, we discuss some of the issues involved in removing the regularity assumption, as well as some applications and open problems, in sect! ion \ref{apps}. This survey is based on talks at the Vector Bundle Workshop at UCLA (October 92) and the Orsay meeting (July 92), and earlier talks at Penn, UCLA and MSRI. I would like to express my thanks to Rob Lazarsfeld and Arnaud Beauville for the invitations, and to them and Ching Li Chai, Phillip Griffiths, Nigel Hitchin, Vasil Kanev, Ludmil Katzarkov, Eyal Markman, Tony Pantev, Emma Previato and Ed Witten for stimulating and helpful conversations. We work throughout over $\bf C$ . The total space of a vector bundle (=locally free sheaf) $K$ is denoted $\Bbb{K}$. Some more notation: \begin{tabbing} bundles of algebras: \= \kill Groups: \>$G $ \=$B$ \=$T$ \=$N $ \=$C$\\ algebras: \>$\frak g$ \>$\frak b$ \>$\frak t$ \>$\frak n $ \>$\frak c $\\ Principal bundles: \>$\cal G$ \>$\cal B$ \>$\cal T$ \>$\cal N $ \>$\cal C$\\ bundles of algebras: \>\bdl{g} \>\bdl{b} \>\bdl{t} \>\bdl{n} \>\bdl{c} \end{tabbing} \section{Hitchin's system}\ \indent Let ${\cal M} := {\cal M} _{C}(n,d)$ be the moduli space of stable vector bundles of rank n and degree d on a smooth projective complex curve $C$. It is smooth and quasiprojective of dimension \begin{equation} \tilde{g} := n^{2} (g-1)+1. \end{equation} \noindent Its cotangent space at a point $ E \in {\cal M} $ is given by \begin{equation} T^*_{E}{\cal M} := H^{0} ( \End{E} \otimes \omega_{C} ) \end{equation} \noindent where $ \omega_{C} $ is the canonical bundle of $C$. Our starting point is: \begin{thm}[Hitchin\cite{H1}] \label{Hitchin} The cotangent bundle $T^*{\cal M}$ is an algebraically completely integrable Hamiltonian system. \end{thm} {\em Complete integrability} means that there is a map \[ h:T^*{\cal M}\longrightarrow B \] to a $\tilde{g}$-dimensional vector space $B$ which is Lagrangian with respect to the natural symplectic structure on $ T^*{\cal M}$ (i.e. the tangent spaces to a general fiber $h^{-1}(a)$ over $a \in B$ are maximal isotropic subspaces with respect to the symplectic form). In this situation one gets, by contraction with the symplectic form, a trivialization of the tangent bundle: \begin{equation} T_{h^{-1}(a)} \stackrel{ \approx}{ \longrightarrow} {\cal O}_{h^{-1}(a)} \otimes T^{*}_{a} B. \end{equation} In particular, this produces a family of ({\em `Hamiltonian'}) vector fields on $h^{-1}(a)$ which is parametrized by $T^{*}_{a} B$ , and the flows generated by these on $h^{-1}(a)$ all commute. {\em Algebraic complete integrability} means additionally that the fibers $h^{-1}(a)$ are Zariski open subsets of abelian varieties on which the Hamiltonian flows are linear, i.e. the vector fields are constant. We describe the idea of the proof in a slightly more general setting, following \cite{BNR}. Let $K$ be a line bundle on $C$, with total space $ \Bbb K$ . (In Hitchin's situation, $K$ is $ \omega_{C} $ and $ \Bbb K$ is $T^*C$ .) A {\em $K$-valued Higgs bundle} is a pair \[ (E \quad,\quad \phi :E \longrightarrow E \otimes K) \] consisting of a vector bundle $E$ on $C$ and a $K$-valued endomorphism. One imposes an appropriate stability condition, and obtains a good moduli space ${\cal M}_K$ parametrizing equivalence classes of $K$-valued semistable Higgs bundles, with an open subset ${\cal M}_K^s$ parametrizing isomorphism classes of stable ones, cf. \cite{S}. Let $B:=B_K$ be the vector space parametrizing polynomial maps \[ p_a : \Bbb K \longrightarrow \Bbb K^{n} \] of the form \[ p_{a} (x) = x^{n} + a_{1} x^{n-1} + \cdots + a_{n}, \qquad\qquad a_{i} \in H^{0}(K ^{\otimes i}). \] in other words, \begin{equation} B := \bigoplus_{i=1}^{n} H^{0}(K ^{\otimes i}). \end{equation} \noindent The assignment \begin{equation} (E, \phi) \longmapsto char(\phi):= \det{(xI-\phi}) \end{equation} \noindent gives a morphism \begin{equation} h_K:{\cal M}_{K}\longrightarrow B_{K}. \end{equation} In Hitchin's case, the desired map $h$ is the restriction of $h_{\omega _{C} }$ to $T^{*}{\cal M}$, which is an open subset of ${\cal M}_{\omega_C}^s$. Note that in this case $\dim{B}$ is, in Hitchin's words, `somewhat miraculously' equal to $\tilde{g} = \dim{{\cal M}}$. The {\em spectral curve} $\widetilde{C}:= \widetilde{C}_{a}$ defined by $a \in B_{K}$ is the inverse image in $\Bbb K$ of the $0$-section of $\Bbb K ^{\otimes n}$ under $ p_a : \Bbb K \longrightarrow \Bbb K^{n} $. It is finite over $C$ of degree $n$. The general fiber of $h_K$ is given by: \begin{prop} \cite{BNR} \label{prop:BNR} For $a \in B$ with {\em integral} spectral curve $\widetilde{C}_{a}$, there is a natural equivalence between isomorphism classes of: \begin{enumerate} \item Rank-$1$, torsion-free sheaves on $\widetilde{C}_{a}$. \item Pairs $(E \ , \ \phi:E \rightarrow E \otimes K)$ with $char(\phi)=a$. \end{enumerate} \end{prop} When $\widetilde{C}_{a}$ is non-singular, the fiber is thus $Jac(\widetilde{C}_{a})$, an abelian variety. In $T^*\cal M$ the fiber is an open subset of this abelian variety. One checks that the missing part has codimension $ \geq 2$, so the symplectic form, which is exact, must restrict to $0$ on the fibers, completing the proof. \section {Some related systems} \noindent \underline{\bf Polynomial matrices} \nopagebreak \noindent One of the earliest appearances of an ACIHS (algebraically completely integrable Hamiltonian system) was in Jacobi's work on the geodesic flow on an ellipsoid (or more generally, on a nonsingular quadric in ${\bf R}^k$). Jacobi discovered that this differential equation, taking place on the tangent (=cotangent!) bundle of the ellipsoid, can be integrated explicitly in terms of hyperelliptic theta functions. In our language, the total space of the flow is an ACIHS, fibered by (Zariski open subsets of) hyperelliptic Jacobians. We are essentially in the special case of Proposition \ref{prop:BNR} where \[ C={\bf P}^1, \quad n=2, \quad K= {\cal O}_{{\bf P}^1} (k). \] A variant of this system appeared in Mumford's solution \cite{Mu} of the Schottky problem for hyperelliptic curves. The extension to all values of n is studied in \cite{B} and, somewhat more analytically, in \cite{AHP} and \cite{AHH}. Beauville considers, for fixed $n$ and $k$, the space $B$ of polynomials: \begin{equation} p=y^n + a_1(x)y^{n-1}+\cdots +a_n(x), \quad, \deg{(a_i)}\leq ki \end{equation} \noindent in variables $x$ and $y$. Each $p$ determines an $n$-sheeted branched cover $$\widetilde{C}_p \rightarrow {\bf P}^1.$$ The total space is the space of polynomial matrices \begin{equation} M := H^0 ({\bf P}^1 , \End{{\cal O}^{\oplus n}} \otimes {\cal O}(d) ), \end{equation} \noindent the map $h:M \rightarrow B$ is the characteristic polynomial, and $M_p$ is the fiber over a given $p \in B$. The result is that for smooth spectral curves $\widetilde{C}_p$, $ {\bf P}GL(n)$ acts freely and properly on $M_p$; the quotient is isomorphic to $J(\widetilde{C}_p) \smallsetminus \Theta. $ (In order to obtain the entire $J(\widetilde{C}_p) ,$ one must allow all pairs $(E,\phi)$ with $E$ of given degree, say $0$. Among those, the ones with $E\approx {{\cal O}_{{\bf P}^1}}^{\oplus n}$ correspond to the open set $J(\widetilde{C}_p) \smallsetminus \Theta. $ ) This system is an ACIHS, in a slightly weaker sense than before: instead of a symplectic structure, it has a {\em Poisson structure}, i.e. a section $\beta$ of $\wedge^2 T$, such that the $\bf C$-linear sheaf map given by contraction with $\beta$ $$\begin{array}{ccc} {\cal O} & \rightarrow & {\cal T} \\ f & \mapsto & df \rfloor \beta \end{array}$$ sends the Poisson bracket of functions to the bracket of vector fields. Any Poisson manifold is naturally foliated, with (locally analytic) symplectic leaves. For a Poisson ACIHS, we want each leaf to inherit a (symplectic) ACIHS, so the symplectic foliation should be pulled back via $h$ from a foliation of the base $B$. The result of \cite{BNR} suggests that analogous systems should exist when ${\bf P}^1$ is replaced by an arbitrary base curve $C$. The main point is to construct the Poisson structure. This was achieved by Bottacin \cite{Bn} and Markman \cite{M1}, cf. section \ref{symplectic}. In the case of the polynomial matrices though, everything (the commuting vector fields, the Poisson structure, etc.) can be written very explicitly. What makes these explicit results possible is that every vector bundle over ${\bf P}^1$ splits. This of course fails in genus $>1$, but for elliptic curves the moduli space of vector bundles is still completely understood, so here too the system can be described explicitly: For simplicity, consider vector bundles with fixed determinant. When the degree is $0$, the moduli space is a projective space ${\bf P}^{n-1}$ (or more canonically, the fiber over $0$ of the Abel-Jacobi map $$C^{[n]} \longrightarrow J(C) = C.$$ The ACIHS which arises is essentially the Treibich-Verdier theory \cite{TV} of elliptic solitons. When, on the other hand, the degree is $1$ (or more generally, relatively prime to $n$), the moduli space is a single point; the corresponding system was studied explicitly in \cite{RS}. \\ \noindent \underline{\bf Reductive groups} \nopagebreak \noindent In another direction, one can replace the vector bundles by principal $G$-bundles ${\cal G}$ for any reductive group $G$. Again, there is a moduli space ${\cal M}_{G,K}$ parametrizing equivalence classes of semistable $K$-valued $G$-Higgs bundles, i.e. pairs $({\cal G}, \phi)$ with $\phi \in K \otimes \bdl{ad}(\cal G)$. The Hitchin map goes to $$B:=\oplus_{i} H^0(K^{\otimes d_i}),$$ where the $d_i$ are the degrees of the $f_i$, a basis for the $G$-invariant polynomials on the Lie algebra $\frak g$: \[ h: ({\cal G}, \phi) \longrightarrow (f_i (\phi))_{i}. \] When $K=\omega_C$, Hitchin showed \cite{H1} that one still gets a completely integrable system, and that it is algebraically completely integrable for the classical groups $GL(n), SL(n), SP(n), SO(n).$ The generic fibers are in each case (not quite canonically; one must choose various square roots! cf. sections \ref{reg.ss} and \ref{reg}) isomorphic to abelian varieties given in terms of the spectral curves $\widetilde{C}$: \begin{center} \begin{equation} \begin{array}{cl} \label{Pryms for groups} GL(n)& \widetilde{C} \mbox{ has degree n over C, the AV is Jac(} \widetilde{C}). \\ SL(n)& \widetilde{C} \mbox{ has degree n over C, the AV is Prym(} \widetilde{C} / C). \\ SP(n)& \widetilde{C} \mbox{ has degree 2n over C and an involution } x \mapsto -x. \\ & \mbox{ The map factors through the quotient } \overline{C}. \nonumber \\ & \mbox{ The AV is } Prym( \widetilde{C} / \overline{C}). \nonumber \\ SO(n)& \widetilde{C} \mbox{ has degree n and an involution , with: } \\ & \bullet \mbox{ a fixed component, when n is odd.} \\ & \bullet \mbox{ some fixed double points, when n is even.} \\ & \mbox{ One must desingularize } \widetilde{C} \mbox{ and the quotient } \overline{C}, \\ & \mbox{and ends up with the Prym of the} \\ & \mbox{desingularized double cover.} \ \end{array} \end{equation} \end{center} The algebraic complete integrability was verified in \cite{KP1} for the exceptional group $G_2$. A sketch of the argument for any reductive $G$ is in \cite{BK}, and a complete proof was given in \cite{F}. We will outline a proof in section \ref{abelianization} below. \\ \noindent \underline{\bf Higher dimensions} \nopagebreak \noindent Finally, a sweeping extension of the notion of Higgs bundle is suggested by the work of Simpson \cite{S}. To him, a Higgs bundle on a projective variety S is a vector bundle (or principal $G$-bundle \ldots) $E$ with a {\em symmetric}, $\Omega^1_S$-valued endomorphism \[ \phi : E \longrightarrow E \otimes \Omega^1_S. \] Here {\em symmetric} means the vanishing of: \[ \phi\wedge\phi : E \longrightarrow E \otimes \Omega^2_S, \] a condition which is obviously vacuous on curves. He constructs a moduli space for such Higgs bundles (satisfying appropriate stability conditions), and establishes diffeomorphisms to corresponding moduli spaces of representations of $\pi_1(S)$ and of connections. \section {Decomposition of spectral Picards} \subsection{The question}\ \indent Let $({\cal G},\phi)$ be a $K$-valued principal Higgs bundle on a complex variety $S$. Each representation \[ \rho : G \longrightarrow Aut(V) \] determines an associated $K$-valued Higgs bundle \[ ( {\cal V} := {\cal G} \times^{G} V, \qquad{\rho}(\phi)\ ), \] which in turn determines a spectral cover $\widetilde{S}_V \longrightarrow S$. The question, raised first in \cite{AvM} when $S={\bf P}^1$, is to relate the Picard varieties of the $\widetilde{S}_V$ as $V$ varies, and in particular to find pieces common to all of them. For Adler and van Moerbeke, the motivation was that many evolution DEs (of Lax type) can be {\em linearized} on the Jacobians of spectral curves. This means that the "Liouville tori", which live naturally in the complexified domain of the DE (and hence are independent of the representation $V$) are mapped isogenously to their image in $\mbox{Pic}(\widetilde{S}_V)$ for each nontrivial $V$ ; so one should be able to locate these tori among the pieces which occur in an isogeny decomposition of each of the $\mbox{Pic}(\widetilde{S}_V)$. There are many specific examples where a pair of abelian varieties constructed from related covers of curves are known to be isomorphic or isogenous, and some of these lead to important identities among theta functions. \begin{eg} \begin{em} Take $G=SL(4)$ . The standard representation $V$ gives a branched cover $\widetilde{S}_V \longrightarrow S$ of degree 4. On the other hand, the 6-dimensional representation $\wedge ^2 V$ (=the standard representation of the isogenous group $SO(6)$) gives a cover $ \stackrel{\approx}{S} \longrightarrow S$ of degree 6, which factors through an involution: \[ \stackrel{\approx}{S} \longrightarrow \overline{S} \longrightarrow S. \] One has the isogeny decompositions: \[ Pic \, (\widetilde{S}) \sim Prym(\widetilde{S} / S) \oplus Pic \,(S) \] \[ Pic \,(\stackrel{\approx}{S}) \sim Prym(\stackrel{\approx}{S} / \overline{S}) \oplus Prym(\overline{S} / S) \oplus Pic \,(S). \] It turns out that \[ Prym(\widetilde{S} / S) \sim Prym(\stackrel{\approx}{S} / \overline{S}) . \] For $S={\bf P}^1$, this is Recillas' {\em trigonal construction} \cite{R}. It says that every Jacobian of a trigonal curve is the Prym of a double cover of a tetragonal curve, and vice versa. \end{em} \end{eg} \begin{eg} \begin{em} Take $G=SO(8)$ with its standard 8-dimensional representation $V$. The spectral cover has degree 8 and factors through an involution, $ \stackrel{\approx}{S} \longrightarrow \overline{S} \longrightarrow S.$ The two half-spin representations $V_1, V_2$ yield similar covers \[ \stackrel{\approx}{S} _i \longrightarrow \overline{S} _i \longrightarrow S, \qquad i=1,2. \] The {\em tetragonal construction} \cite{D1} says that the three Pryms of the double covers are isomorphic. (These examples, as well as Pantazis' {\em bigonal construction} and constructions based on some exceptional groups, are discussed in the context of spectral covers in \cite{K} and \cite{D2}.) \end{em} \end{eg} It turns out in general that there is indeed a distinguished, Prym-like isogeny component common to all the spectral Picards, on which the solutions to Lax-type DEs evolve linearly. This was noticed in some cases already in \cite{AvM}, and was greatly extended by Kanev's construction of Prym-Tyurin varieties. (He still needs $S$ to be ${\bf P}^1$ and the spectral cover to have generic ramification; some of his results apply only to {\em minuscule representations}.) Various parts of the general story have been worked out recently by a number of authors, based on either of two approaches: one, pursued in \cite{D2,Me,MS}, is to decompose everything according to the action of the Weyl group $W$ and to look for common pieces; the other, used in \cite{BK,D3,F,Sc}, relies on the correspondence of spectral data and Higgs bundles . The group-theoretic approach is described in the rest of this section. We take up the second method, known as {\em abelianization}, in section~\ref{abelianization}. \subsection{Decomposition of spectral covers} \label{decomp covers}\ \indent The decomposition of spectral Picards arises from three sources. First, the spectral cover for a sum of representations is the union of the individual covers $\widetilde{S}_V$. Next, the cover $\widetilde{S}_V$ for an irreducible representation is still the union of subcovers $\widetilde{S}_{\lambda}$ indexed by weight orbits. And finally, the Picard of $\widetilde{S}_{\lambda}$ decomposes into Pryms. We start with a few observations about the dependence of the covers themselves on the representation. The decomposition of the Picards is taken up in the next subsection. \\ \noindent \underline{\bf Spectral covers} \nopagebreak \noindent There is an {\em infinite} collection (of irreducible representations $V := V_{\mu}$, hence) of spectral covers $\widetilde{S}_V$, which can be parametrized by their highest weights $\mu$ in the dominant Weyl chamber $\overline{C}$ , or equivalently by the $W$-orbit of extremal weights, in $\Lambda / W$. Here $T$ is a maximal torus in $G$, $\Lambda := Hom(T, {\bf C}^*)$ is the {\em weight lattice } (also called {\em character lattice }) for $G$, and $W$ is the Weyl group. Each of these $\widetilde{S}_V$ decomposes as the union of its subcovers $\widetilde{S}_{\lambda}$, parametrizing eigenvalues in a given $W$-orbit $W{\lambda}$ . ($\lambda$ runs over the weight-orbits in $V_{\mu}$.) \\ \noindent \underline{\bf Parabolic covers} \nopagebreak \noindent There is a {\em finite} collection of covers $\widetilde{S}_P$, parametrized by the conjugacy classes in $G$ of parabolic subgroups (or equivalently by arbitrary dimensional faces $F_P$ of the chamber $\overline{C}$) such that (for general $S$) each eigenvalue cover $\widetilde{S}_{\lambda}$ is birational to some parabolic cover $\widetilde{S}_{P}$, the one whose open face $F_P$ contains ${\lambda}$. \\ \noindent \underline{\bf The cameral cover} \nopagebreak \noindent There is a $W$-Galois cover $\widetilde{S} \longrightarrow S$ such that each $\widetilde{S}_{P}$ is isomorphic to $\widetilde{S} / W_P$, where $W_P$ is the Weyl subgroup of $W$ which stabilizes $F_P$. We call $\widetilde{S}$ the {\em cameral cover} , since, at least generically, it parametrizes the chambers determined by $\phi$ (in the duals of the Cartans), or equivalently the Borel subalgebras containing $\phi$. This is constructed as follows: There is a morphism ${\frak g}\longrightarrow {\frak t}/W$ sending $g \in {\frak g}$ to the conjugacy class of its semisimple part $g_{ss}$. (More precisely, this is $Spec$ of the composed ring homomorphism ${\bf C} [ {\frak t} ] ^{W} { \stackrel{\simeq}{\leftarrow}} {\bf C}[{\frak g}]^{G} \label{t/W} \hookrightarrow {\bf C}[{\frak g}]$.) Taking fiber product with the quotient map ${\frak t}\longrightarrow {\frak t}/W$, we get the cameral cover ${\tilde{\frak g}}$ of ${\frak g}$. The cameral cover $\widetilde{S} \longrightarrow S$ of a $K$-valued principal Higgs bundle on $S$ is glued from covers of open subsets in $S$ (on which $K$ and $\cal G$ are trivialized) which in turn are pullbacks by $\phi$ of ${\tilde{\frak g}} \longrightarrow {\frak g} $. \subsection{Decomposition of spectral Picards}\ \indent The decomposition of the Picard varieties of spectral covers can be described as follows:\\ \noindent \underline{\bf The cameral Picard} \nopagebreak \noindent From each isomorphism class of irreducible $W$-representations, choose an integral representative $\Lambda _i$. (This can always be done, for Weyl groups.) The group ring ${\bf Z} [W]$ which acts on $Pic(\widetilde{S}) $ has an isogeny decomposition: \begin{equation}\label{regular rep} {\bf Z} [W] \sim \oplus _i \Lambda _i \otimes_{\bf Z} \Lambda _i^{*}, \end{equation} \noindent which is just the decomposition of the regular representation. There is a corresponding isotypic decomposition: \begin{equation}\label{cameral Pic decomposition} Pic(\widetilde{S}) \sim \oplus _i \Lambda _i \otimes_{\bf Z} Prym_{\Lambda _i}(\widetilde{S}), \end{equation} \noindent where \begin{equation}\label{def of Prym_lambda} Prym_{\Lambda _i}(\widetilde{S} ):= Hom_W (\Lambda _i , Pic(\widetilde{S})). \end{equation}\\ \noindent \underline{\bf Parabolic Picards} \nopagebreak \noindent There are at least three reasonable ways of obtaining an isogeny decomposition of $Pic(\widetilde{S}_P) $, for a parabolic subgroup $P \subset G$: \begin{itemize} \item The `Hecke' ring $Corr_P$ of correspondences on $\widetilde{S}_P$ over $S$ acts on $Pic(\widetilde{S}_P) $, so every irreducible integral representation $M$ of $Corr_P$ determines a generalized Prym $$ Hom_{Corr_P} (M, Pic(\widetilde{S}_P)), $$ and we obtain an isotypic decomposition of $Pic(\widetilde{S}_P)$ as before. \item $Pic(\widetilde{S}_P)$ maps, with torsion kernel, to $Pic(\widetilde{S})$, so we obtain a decomposition of the former by intersecting its image with the isotypic components $\Lambda _i \otimes_{\bf Z} Prym_{\Lambda _i}(\widetilde{S})$ of the latter. \item Since $\widetilde{S}_P$ is the cover of $S$ {\em associated} to the $W$-cover $\widetilde{S}$ via the permutation representation ${\bf Z} [W_P \backslash W]$ of $W$, we get an isogeny decomposition of $Pic(\widetilde{S}_P)$ indexed by the irreducible representations in ${\bf Z} [W_P \backslash W]$. \end{itemize} It turns out (\cite{D2},section 6) that all three decompositions agree and can be given explicitly as \begin{equation} \label{multiplicity spaces} \oplus _i M _i \otimes Prym_{\Lambda _i}(\widetilde{S}) \subset \oplus _i \Lambda _i \otimes Prym_{\Lambda _i}(\widetilde{S}),\qquad M_i := (\Lambda_i)^{W_P}. \end{equation} \noindent \underline{\bf Spectral Picards} \nopagebreak \noindent To obtain the decomposition of the Picards of the original covers $\widetilde{S}_V$ or $\widetilde{S}_{\lambda}$, we need, in addition to the decomposition of $Pic(\widetilde{S}_P)$, some information on the singularities. These can arise from two separate sources: \begin{description} \item[Accidental singularities of the $\widetilde{S}_{\lambda}$. ] For a sufficiently general Higgs bundle, and for a weight $\lambda$ in the interior of the face $F_P$ of the Weyl chamber $\overline{C}$, the natural map: $$ i_{\lambda}: \widetilde{S}_P\longrightarrow \widetilde{S}_{\lambda} $$ is birational. For the {\em standard} representations of the classical groups of types $A_n, B_n$ or $C_n$, this {\em is} an isomorphism. But for general ${\lambda}$ it is {\em not}: In order for $i_{\lambda}$ to be an isomorphism, ${\lambda}$ must be a multiple of a fundamental weight, cf. \cite{D2}, lemma 4.2. In fact, the list of fundamental weights for which this happens is quite short; for the classical groups we have only: $\omega_1$ for $A_n, B_n$ and $C_n$, $\omega_n$ (the dual representation) for $A_n$, and $\omega_2$ for $B_2$. Note that for $D_n$ the list is {\em empty}. In particular, the covers produced by the standard representation of $SO(2n)$ are singular; this fact, noticed by Hitchin In \cite{H1}, explains the need for desingularization in his result~(\ref{Pryms for groups}). \item[Gluing the $\widetilde{S}_{V}$. ] In addition to the singularities of each $i_{\lambda}$, there are the singularities created by the gluing map $\amalg_{\lambda} \widetilde{S}_{\lambda} \longrightarrow \widetilde{S}_V$. This makes explicit formulas somewhat simpler in the case, studied by Kanev \cite{K}, of {\em minuscule} representations, i.e. representations whose weights form a single $W$-orbit. These singularities account, for instance, for the desingularization required in the $SO(2n+1)$ case in (\ref{Pryms for groups}). \end{description} \subsection{The distinguished Prym} \label{distinguished}\ \indent Combining much of the above, the Adler--van Moerbeke problem of finding a component common to the $Pic(\widetilde{S}_V)$ for all non-trivial $V$ translates into: \\ \begin{em} Find the irreducible representations $\Lambda_i $ of $W$ which occur in ${\bf Z} [W_P \backslash W] $ with positive multiplicity for all proper Weyl subgroups $W_P \subsetneqq W.$ \end{em} \\ By Frobenius reciprocity, or (\ref{multiplicity spaces}), this is equivalent to \\ \begin{em} Find the irreducible representations $\Lambda_i $ of W such that for every proper Weyl subgroup $W_P \subsetneqq W, $ the space of invariants $M_i := (\Lambda_i)^{W_P} $ is non-zero. \end{em} \\ One solution is now obvious: the {\em{reflection representation}} of $W$ acting on the weight lattice $\Lambda$ has this property. In fact, $\Lambda^{W_P}$ in this case is just the face $F_P$ of $\overline{C}$. The corresponding component $Prym_{\Lambda }(\widetilde{S})$ , is called {\em{the distinguished Prym}.} We will see in section \ref{abelianization} that its points correspond, modulo some corrections, to Higgs bundles. For the classical groups, this turns out to be the only common component. For $G_2$ and $E_6$ it turns out (\cite{D2}, section 6) that a second common component exists. The geometric significance of points in these components is not known. As far as I know, the only component other than the distinguished Prym which has arisen `in nature' is the one associated to the 1-dimensional sign representation of $W$, cf. section \ref{apps} and \cite{KP2}. \section {Abelianization}\label{abelianization} \subsection{Abstract vs. $K$-valued objects}\ \indent We want to describe the abelianization procedure in a somewhat abstract setting, as an equivalence between {\em{principal Higgs bundles}} and certain {\em spectral data}. Once we fix a {\em{values}} vector bundle $K$, we obtain an equivalence between {\em $K$-valued principal Higgs bundles} and {\em K-valued spectral data}. Similarly, the choice of a representation $V$ of $G$ will determine an equivalence of {\em $K$-valued Higgs bundles} (of a given representation type) with $K$-valued spectral data. As our model of a $W$-cover we take the natural quotient map $$G/T \longrightarrow G/N $$ and its partial compactification \begin{equation} \overline{G/T} \longrightarrow \overline{G/N}. \label{partial compactification} \end{equation} Here $T \subset G$ is a maximal torus, and $N$ is its normalizer in $G$. The quotient $G/N$ parametrizes maximal tori (=Cartan subalgebras) $\frak{t}$ in $\frak{g}$, while $G/T$ parametrizes pairs ${\frak t \subset \frak b}$ with ${\frak b \subset \frak g}$ a Borel subalgebra. An element $x \in {\frak g}$ is {\em regular} if the dimension of its centralizer ${\frak c \subset \frak g}$ equals $\dim{T}$ (=the rank of $\frak{g}$). The partial compactifications $ \overline{G/N}$ and $ \overline{G/T}$ parametrize regular centralizers ${\frak c }$ and pairs ${\frak c \subset \frak b}$, respectively. In constructing the cameral cover in section \ref{t/W}, we used the $W$-cover $\frak t \longrightarrow \frak t / W$ and its pullback cover ${ \widetilde{\frak g} \longrightarrow \frak g}$. Over the open subset $\frak g_{reg}$ of regular elements, the same cover is obtained by pulling back (\ref{partial compactification}) via the map $\alpha : \frak g_{reg} \longrightarrow \overline{G/N}$ sending an element to its centralizer: \begin{equation} \label{commutes} \begin{array}{lccccc} \frak t & \longleftarrow & \widetilde{\frak g}_{{reg}} & \longrightarrow & \overline{G/T} & \\ \downarrow & &\downarrow & & \downarrow & \\ \frak t /W & \longleftarrow & {\frak g}_{{reg}} & \stackrel{\alpha}{\longrightarrow} & \overline{G/N} &. \end{array} \end{equation} When working with $K$-valued objects, it is usually more convenient to work with the left hand side of (\ref{commutes}), i.e. with eigen{\em values}. When working with the abstract objects, this is unavailable, so we are forced to work with the eigen{\em vectors}, or the right hand side of (\ref{commutes}). Thus: \begin{defn} An abstract {\em cameral cover} of $S$ is a finite morphism $\widetilde{S} \longrightarrow S$ with $W$-action, which locally (etale) in $S$ is a pullback of (\ref{partial compactification}). \\ \end{defn} \begin{defn} A {\em $K$-valued cameral cover} ($K$ is a vector bundle on $S$) consists of a cameral cover $\pi : \widetilde{S} \longrightarrow S$ together with an $S$-morphism \begin{equation} \widetilde{S} \times \Lambda \longrightarrow \Bbb{K} \label{K-values} \end{equation} which is $W$-invariant ($W$ acts on $ \widetilde{S} , \Lambda,$ hence diagonally on $\widetilde{S} \times \Lambda $ ) and linear in $\Lambda$. \\ \end{defn} We note that a cameral cover $\widetilde{S}$ determines quotients $\widetilde{S}_P$ for parabolic subgroups $P \subset G$. A $K$-valued cameral cover determines additionally the $\widetilde{S}_{\lambda}$ for $\lambda \in \Lambda$, as images in $\Bbb{K}$ of $\widetilde{S} \times \{ \lambda \}$. The data of (\ref{K-values}) is equivalent to a $W$-equivariant map $\widetilde{S} \longrightarrow \frak{t}\otimes_{\bf C} K.$ \begin{defn} \label{princHiggs} A $G$-principal Higgs bundle on $S$ is a pair ($\cal{G}, \bdl{c})$ with $\cal{G}$ a principal $G$-bundle and $\bdl{c} \subset ad(\cal{G})$ a subbundle of regular centralizers. \\ \end{defn} \begin{defn} A $K$-valued $G$-principal Higgs bundle consists of $( \cal{G}, \bdl{c} )$ as above together with a section $\varphi$ of $\bdl{c} \otimes K$. \end{defn} A principal Higgs bundle $(\cal{G}, \bdl{c})$ determines a cameral cover $\widetilde{S}\longrightarrow S$ and a homomorphism $\Lambda \longrightarrow \mbox{Pic}(\widetilde{S}).$ Let $F$ be a parameter space for Higgs bundles with a given $\widetilde{S}$. Each non-zero $\lambda \in \Lambda$ gives a non-trivial map $F\longrightarrow \mbox{Pic}(\widetilde{S})$. For $\lambda$ in a face $F_P$ of $\overline{C}$, this factors through $\mbox{Pic}(\widetilde{S}_P)$. The discussion in section \ref{distinguished} now suggests that $F$ should be given roughly by the distinguished Prym, $$ Hom_W (\Lambda , \mbox{Pic}(\widetilde{S})). $$ It turns out that this guess needs two corrections. The first correction involves restricting to a coset of a subgroup; the need for this is visible even in the simplest case where $\widetilde{S}$ is etale over $S$, so $(\cal{G}, \bdl{c})$ is everywhere regular and semisimple (i.e. $ \bdl{c}$ is a bundle of Cartans.) The second correction involves a twist along the ramification of $\widetilde{S}$ over $S$. We explain these in the next two subsections. \subsection{The regular semisimple case: the shift} \label{reg.ss} \begin{eg} \label{unramified} \begin{em} Fix a smooth projective curve $C$ and a line bundle $K \in \mbox{Pic}(C)$ such that $K^{\otimes 2} \approx \cal{O}_C.$ This determines an etale double cover $\pi : \widetilde{C} \longrightarrow C$ with involution $i$, and homomorphisms \begin{center} $\begin{array}{cccccc} \pi^{*} &:& \mbox{Pic}(C) &\longrightarrow &\mbox{Pic}(\widetilde{C}) &, \\ \mbox{Nm} &:& \mbox{Pic}(\widetilde{C}) &\longrightarrow &\mbox{Pic}(C) &, \\ i^{*} &:& \mbox{Pic}(\widetilde{C}) &\longrightarrow &\mbox{Pic}(\widetilde{C}) &, \end{array}$ \end{center} satisfying $$ 1+i^{*} = \pi^* \circ \mbox{Nm}. $$ \begin{itemize} \item For $G = GL(2)$ we have $\Lambda = \bf{Z} \oplus \bf{Z}$, and $W = {\cal{S}}_{2}$ permutes the summands, so $$ Hom_W (\Lambda , \mbox{Pic}(\widetilde{C})) \approx \mbox{Pic}(\widetilde{C}). $$ And indeed, the Higgs bundles corresponding to $\widetilde{C}$ are parametrized by $\mbox{Pic}(\widetilde{C})$: send $L \in \mbox{Pic}(\widetilde{C})$ to $(\cal{G}, \bdl{c})$, where $\cal{G}$ has associated rank-2 vector bundle ${\cal V} := \pi_* L$, and $ \bdl{c} \subset \End{{\cal{V}}}$ is $\pi_* {\cal O}_{\widetilde{C}}.$ \item On the other hand, for $G=SL(2)$ we have $\Lambda=\bf{Z}$ and $W={\cal{S}}_2$ acts by $\pm 1$, so $$ Hom_W (\Lambda , \mbox{Pic}(\widetilde{S})) \approx \{L \in \mbox{Pic}(\widetilde{C})\ | \ i^*L \approx L^{-1} \} = \mbox{ker}(1+i^*). $$ This group has 4 connected components. The subgroup $\mbox{ker(Nm)}$ consists of 2 of these. The connected component of 0 is the classical Prym variety, cf. \cite{MuPrym}. Now the Higgs bundles correspond, via the above bijection $L\mapsto \pi_*L$, to $$\{L \in \mbox{Pic}(\widetilde{C}) \ |\ \det (\pi_*L) \approx {\cal O}_C \} = {\mbox{Nm}}^{-1}(K). $$ Thus they form the {\em non-zero} coset of the subgroup $\mbox{ker(Nm)}$. (If we return to a higher dimensional $S$, it is possible for $K$ not to be in the image of $\mbox{Nm}$, so there may be {\em no} $SL(2)$-Higgs bundles corresponding to such a cover.) \end{itemize} \end{em} \end{eg} This example generalizes to all $G$, as follows. The equivalence classes of extensions $$1 \longrightarrow T \longrightarrow N' \longrightarrow W \longrightarrow 1 $$ (in which the action of $W$ on $T$ is the standard one) are parametrized by the group cohomology $H^2(W,T)$. Here the 0 element corresponds to the semidirect product . The class $[N] \in H^2(W,T)$ of the normalizer $N$ of $T$ in $G$ may be 0, as it is for $G=GL(n) , {\bf P}GL(n) , SL(2n+1) $; or not, as for $G=SL(2n)$. Assume first, for simplicity, that $S,\widetilde{S}$ are connected and projective. There is then a natural group homomorphism \begin{equation} \label{c} c: Hom_W (\Lambda , \mbox{Pic}(\widetilde{S}))\longrightarrow H^2(W,T). \end{equation} Algebraically, this is an edge homomorphism for the Grothendieck spectral sequence of equivariant cohomology, which gives the exact sequence \begin{equation} \label{c-edge}\qquad 0 \longrightarrow H^1(W,T) \longrightarrow H^1(S,{\cal{C}}) \longrightarrow Hom_W (\Lambda , \mbox{Pic}(\widetilde{S})) \stackrel{c}{\longrightarrow} H^2(W,T). \end{equation} where ${\cal{C}} := \widetilde{S} \times _W T.$ Geometrically, this expresses a {\em Mumford group} construction: giving ${\cal{L}} \in \mbox{Hom}(\Lambda,\mbox{Pic}(\widetilde{S}))$ is equivalent to giving a principal $T$-bundle $\cal T$ over $\widetilde{S}$; for ${\cal{L}} \in \mbox{Hom}_W(\Lambda,\mbox{Pic}(\widetilde{S}))$, $c({\cal{L}})$ is the class in $H^2(W,T)$ of the group $N'$ of automorphisms of $\cal T$ which commute with the action on $\widetilde{S}$ of some $w \in W$. To remove the restriction on $S, \widetilde{S}$, we need to replace each occurrence of $T$ in (\ref{c}, \ref{c-edge}) by $\Gamma (\widetilde{S}, T)$, the global sections of the trivial bundle on $\widetilde{S}$ with fiber $T$. The natural map $H^2(W,T) \longrightarrow H^2(W,\Gamma (\widetilde{S}, T))$ allows us to think of $[N]$ as an element of $H^2(W,\Gamma (\widetilde{S}, T))$. \begin{prop} \cite{D3} Fix an etale $W$-cover $\pi: \widetilde{S}\longrightarrow S$. The following data are equivalent: \begin{enumerate} \item Principal $G$-Higgs bundles $(\cal{G}, \bdl{c})$ with cameral cover $\widetilde{S}$. \item Principal $N$-bundles $\cal N$ over $S$ whose quotient by $T$ is $\widetilde{S}.$ \item $W$-equivariant homomorphisms ${\cal{L}} : \Lambda \longrightarrow \mbox{Pic}(\widetilde{S})$ with $c({\cal L}) = [N] \in H^2(W,\Gamma (\widetilde{S}, T))$. \end{enumerate} \end{prop} We observe that while the shifted objects correspond to Higgs bundles, the unshifted objects $$ {\cal{L}} \in \mbox{Hom}_W(\Lambda,\mbox{Pic}(\widetilde{S})), \qquad c({\cal L})=0 $$ \noindent come from the $\cal C$-torsers in $H^1(S, {\cal C} ).$ \subsection{The regular case: the twist along the ramification} \label{reg} \begin{eg} \label{ramified} \begin{em} Modify example \ref{unramified} by letting $K \in \mbox{Pic}(C) $ be arbitrary, and choose a section $b$ of $K ^{\otimes 2}$ which vanishes on a simple divisor $B \subset C$. We get a double cover $\pi : \widetilde{C} \longrightarrow C$ branched along $B$, ramified along a divisor $$ R \subset \widetilde{C}, \quad \pi(R)=B. $$ Via $L\mapsto \pi_*L$, the Higgs bundles still correspond to $$\{L \in \mbox{Pic}(\widetilde{C}) \ |\ \det (\pi_*L) \approx {\cal O}_C \} = {\mbox{Nm}}^{-1}(K). $$ But this is no longer in $ Hom_W (\Lambda , \mbox{Pic}(\widetilde{S}))$; rather, the line bundles in question satisfy \begin{equation} \label{SL(2) twist} i^*L \approx L^{-1}(R). \end{equation} \end{em} \end{eg} For arbitrary $G$, let $\Phi$ denote the root system and $\Phi^+$ the set of positive roots. There is a decomposition $$ \overline{G/T} \ \smallsetminus \ G/T = \bigcup _{\alpha \in \Phi^+}R_{\alpha} $$ of the boundary into components, with $R_{\alpha}$ the fixed locus of the reflection $\sigma_{\alpha}$ in $\alpha$. (Via (\ref{commutes}), these correspond to the complexified walls in $\frak t$.) Thus each cameral cover $\widetilde{S} \longrightarrow S$ comes with a natural set of (Cartier) {\em ramification divisors}, which we still denote $R_{\alpha}, \quad \alpha \in \Phi^+.$ For $w \in W$, set $$ F_w := \left\{ \alpha \in \Phi^+ \ | \ w^{-1} \alpha \in \Phi^- \right\} = \Phi^+ \cap w \Phi^-, $$ and choose a $W$-invariant form $\langle , \rangle$ on $\Lambda$. We consider the variety $$ Hom_{W,R} (\Lambda , \mbox{Pic}(\widetilde{S})) $$ of $R$-twisted $W$-equivariant homomorphisms, i.e. homomorphisms $\cal L$ satisfying \begin{equation} \qquad \label{G twist} w^*{\cal L}(\lambda) \approx {\cal L}(w\lambda)\left( \sum_{\alpha \in F_w}{ {\langle-2\alpha,w\lambda \rangle \over \langle \alpha ,\alpha \rangle} R_{\alpha} } \right) , \qquad \lambda \in \Lambda, \quad w \in W. \end{equation} This turns out to be the correct analogue of (\ref{SL(2) twist}). (E.g. for a reflection $w=\sigma_{\alpha}$, \quad $F_w$ is $\left\{ \alpha \right\}$, so this gives $ w^*{\cal L}(\lambda) \approx {\cal L}(w\lambda)\left( {{\langle\alpha,2\lambda \rangle \over \langle \alpha,\alpha \rangle} R_{\alpha}} \right),$ which specializes to (\ref{SL(2) twist}).) As before, there is a class map \begin{equation} \label{c,R} c: Hom_{W,R} (\Lambda , \mbox{Pic}(\widetilde{S}))\longrightarrow H^2(W,\ \Gamma (\widetilde{S}, T)) \end{equation} \noindent which can be described via a Mumford-group construction. To understand this twist, consider the formal object \begin{center} $\begin{array}{cccc} {1 \over 2} \mbox{Ram}: & \Lambda & \longrightarrow & {\bf Q}\otimes \mbox{Pic}\widetilde{S}, \\ & \lambda & \longmapsto & \sum_{ ( \alpha \in {\Phi^+} ) }{{\langle\alpha,\lambda \rangle \over \langle \alpha,\alpha \rangle} R_{\alpha}}. \end{array}$ \end{center} In an obvious sense, a principal $T$-bundle $\cal T$ on $\widetilde{S}$ (or a homomorphism ${\cal L}: \Lambda \longrightarrow \mbox{Pic}(\widetilde{S})$) is $R$-twisted $W$-equivariant if and only if ${\cal T} (-{1 \over 2} Ram)$ is $W$-equivariant, i.e. if ${\cal T}$ and ${1 \over 2} Ram$ transform the same way under $W$. The problem with this is that ${1 \over 2} Ram$ itself does not make sense as a $T$-bundle, because the coefficients ${\langle\alpha,\lambda\rangle \over \langle\alpha,\alpha\rangle} $ are not integers. (This argument shows that if $Hom_{W,R} (\Lambda , \mbox{Pic}(\widetilde{S}))$ is non-empty, it is a torser over the untwisted $Hom_{W} (\Lambda , \mbox{Pic}(\widetilde{S}))$.) \begin{thm} \cite{D3} \label{main} For a cameral cover $\widetilde{S} \longrightarrow S$, the following data are equivalent: \\ (1) $G$-principal Higgs bundles with cameral cover $\widetilde{S}$. \\ (2) $R$-twisted $W$-equivariant homomorphisms ${\cal L} \in c^{-1}([N]).$ \end{thm} The theorem has an essentially local nature, as there is no requirement that $S$ be, say, projective. We also do not need the condition of generic behavior near the ramification, which appears in \cite{F, Me, Sc}. Thus we may consider an extreme case, where $\widetilde{S}$ is `everywhere ramified': \begin{eg}\begin{em} In example \ref{ramified}, take the section $b=0$. The resulting cover $\widetilde{C}$ is a `ribbon', or length-2 non-reduced structure on $C$: it is the length-2 neighborhood of $C$ in $\Bbb{K}$. The SL(2)-Higgs bundles $({\cal G},\bdl{c})$ for this $\widetilde{C}$ have an everywhere nilpotent $\bdl{c}$, so the vector bundle ${\cal V} := {\cal G} \times^{SL(2)} V \approx \pi_* L$ (where $V$ is the standard 2-dimensional representation) fits in an exact sequence $$ 0 \longrightarrow {\cal S} \longrightarrow {\cal V} \longrightarrow {\cal Q} \longrightarrow 0 $$ with ${\cal S} \otimes K \approx {\cal Q}.$ Such data are specified by the line bundle ${\cal Q}$, satisfying ${\cal Q}^{\otimes 2} \approx K$, and an extension class in $\mbox{Ext}^1({\cal Q}, {\cal S}) \approx H^1(K^{-1})$. The kernel of the restriction map $ \mbox{Pic}(\widetilde{C}) \longrightarrow \mbox{Pic}(C) $ is also given by $H^1(K^{-1})$ (use the exact sequence $0 \longrightarrow K^{-1} \longrightarrow \pi_*{\cal O}_{\widetilde{C}}^{\times} \longrightarrow {\cal O}_C^{\times} \longrightarrow 0$), and the $R$-twist produces the required square roots of $K$. (For more details on the nilpotent locus, cf. \cite{L} and \cite{DEL}.) \end{em}\end{eg} \subsection{Adding values and representations}\ \indent Fix a vector bundle $K$, and consider the moduli space $ {\cal M}_{S,G,K} $ of $K$-valued $G$-principal Higgs bundles on $S$. (It can be constructed as in Simpson's \cite{S}, even though the objects we need to parametrize are slightly different than his. In this subsection we outline a direct construction.) It comes with a Hitchin map: \begin{equation} \label{BigHitchin} h: {\cal M}_{S,G,K} \longrightarrow B_K \end{equation} \noindent where $B := B_K$ parametrizes all possible Hitchin data. Theorem \ref{main} gives a precise description of the fibers of this map, independent of the values bundle $K$. This leaves us with the relatively minor task of describing, for each $K$, the corresponding base, i.e. the closed subvariety $B_s$ of $B$ parametrizing {\em split} Hitchin data, or $K$-valued cameral covers. The point is that Higgs bundles satisfy a symmetry condition, which in Simpson's setup is $$ \varphi \wedge \varphi = 0, $$ and is built into our definition \ref{princHiggs} through the assumption that \bdl{c} is regular, hence abelian. Since commuting operators have common eigenvectors, this gives a splitness condition on the Hitchin data, which we describe below. (When $K$ is a line bundle, the condition is vacuous, $B_s = B$.) The upshot is: \begin{lem} \label{parametrization} The following data are equivalent: \\ (a) A $K$-valued cameral cover of $S$. \\ (b) A split, graded homomorphism $R{\bf \dot{\ }} \longrightarrow {Sym}{\bf \dot{\ }}K.$ \\ (c) A split Hitchin datum $b \in B_s$. \end{lem} Here $R{\bf \dot{\ }}$ is the graded ring of $W$-invariant polynomials on $\frak t$: \begin{equation} R{\bf \dot{\ }} := (\mbox{Sym}{\bf \dot{\ }} {\frak t}^*)^W \approx {\bf C}[\sigma_1,\ldots,\sigma_l], \qquad \deg (\sigma_i) = d_i \end{equation} \noindent where $l := \mbox{Rank}({\frak g})$ and the $\sigma_i$ form a basis for the $W$-invariant polynomials. The Hitchin base is the vector space $$ B := B_K := \oplus _{i=1}^l H^0(S, {Sym}^{d_i}K) \approx \mbox{Hom}(R{\bf \dot{\ }},\mbox{Sym}{\bf \dot{\ }}K). $$ \noindent For each $\lambda \in \Lambda$ (or $\lambda \in {\frak t}^*$, for that matter), the expression in an indeterminate $x$: \begin{equation} \label{rep poly} q_{\lambda}(x,t) := \prod_{w \in W}{(x-w\lambda(t))}, \qquad t \in {\frak t}, \end{equation} is $W$-invariant (as a function of $t$), so it defines an element $q_{\lambda}(x) \in R{\bf \dot{\ }}[x].$ A Hitchin datum $b \in B \approx \mbox{Hom}(R{\bf \dot{\ }},\mbox{Sym}{\bf \dot{\ }}K)$ sends this to $$ q_{\lambda,b}(x) \in \mbox{Sym}\dot{\ }(K)[x]. $$ We say that $b$ is {\em split} if, at each point of $S$ and for each $\lambda$, the polynomial $q_{\lambda,b}(x)$ factors completely, into terms linear in $x$. We note that, for $\lambda$ in the interior of $C$ (the positive Weyl chamber), $q_{\lambda,b}$ gives the equation in $\Bbb K$ of the spectral cover $\widetilde{S}_{\lambda}$ of section (\ref{decomp covers}): $q_{\lambda,b}$ gives a morphism $\Bbb K \longrightarrow \mbox{Sym}^N \Bbb K$, where $N:=\#W$, and $\widetilde{S}_{\lambda}$ is the invere image of the zero-section. (When $\lambda$ is in a face $F_P$ of $\overline{C}$, we define analogous polynomials $q_{\lambda}^P(x,t)$ and $q_{\lambda,b}^P(x)$ by taking the product in (\ref{rep poly}) to be over $w \in W_P \backslash W.$ These give the reduced equations in this case, and $q_{\lambda}$ is an appropriate power.) Over $B_s$ there is a universal $K$-valued cameral cover $$ \widetilde{\cal S} \longrightarrow B_s $$ with ramification divisor $R \subset \widetilde{\cal S}$. From the relative Picard, $$ \mbox{Pic}( \widetilde{\cal S} / B_s) $$ we concoct the relative $N$-shifted, $R$-twisted Prym $$ \mbox{Prym}_{\Lambda ,R}( \widetilde{\cal S} / B_s). $$ By Theorem \ref{main}, this can then be considered as a parameter space $ {\cal M}_{S,G,K} $ for all $K$-valued $G$-principal Higgs bundles on $S$. (Recall that our objects are assumed to be everywhere {\em regular}!) It comes with a `Hitchin map', namely the projection to $B_s$, and the fibers corresponding to smooth projective $\widetilde{S}$ are abelian varieties. When $S$ is a smooth, projective curve, we recover this way the algebraic complete integrability of Hitchin's system and its generalizations. \section {Symplectic and Poisson structures} \label{symplectic}\ \indent The total space of Hitchin's original system is a cotangent bundle, hence has a natural symplectic structure. For the polynomial matrix systems of \cite{B} and \cite{AHH} there is a natural Poisson structure which one writes down explicitly. In \cite{Bn} and \cite{M1}, this result is extended to the systems ${\cal M}_{C,K}$ of $K$-valued GL(n) Higgs bundles on $C$, when $K \approx \omega_C(D)$ for an effective divisor $D$ on $C$. There is a general-nonsense pairing on the cotangent spaces, so the point is to check that this pairing is `closed', i.e. satisfies the identity required for a Poisson structure. Bottacin does this by an explicit computation along the lines of \cite{B}. Markman's idea is to consider the moduli space ${\cal M}_D$ of stable vector bundles on $C$ with level-$D$ structure. He realizes an open subset ${\cal M}^0_{C,K}$ of ${\cal M}_{C,K}$, parametrizing Higgs bundles whose covers are nice, as a quotient (by an action of the level group) of $T^*{\cal M}_D$, so the natural symplectic form on $T^*{\cal M}_D$ descends to a Poisson structure on ${\cal M}^0_{C,K}$. This is identified with the general-nonsense form (wherever both exist), proving its closedness. In \cite{Muk}, Mukai constructs a symplectic structure on the moduli space of simple sheaves on a $K3$ surface $S$. Given a curve $C \subset S$, one can consider the moduli of sheaves having the numerical invariants of a line bundle on a curve in the linear system $ |nC| $ on $S$. This has a support map to the projective space $ |nC| $,which turns it into an ACIHS. This system specializes, by a `degeneration to the normal cone' argument, cf. \cite{DEL}, to Hitchin's, allowing translation of various results about Hitchin's system (such as Laumon's description of the nilpotent cone, cf. \cite{L} ) to Mukai's. In higher dimensions, the moduli space $\cal M$ of $\Omega^1$-valued Higgs bundles carries a natural symplectic structure \cite{S}. (Corlette points out in \cite{C} that certain components of an open subet in $\cal M$ can be described as cotangent bundles.) It is not clear at the moment exactly when one should expect to have an ACIHS, with symplectic, Poisson or quasi symplectic structure, on the moduli spaces of $K$-valued Higgs bundles for higher dimensional $S$, arbitrary $G$, and arbitrary vector bundle $K$. A beautiful new idea \cite{M2} is that Mukai's results extend to the moduli of those sheaves on a (symplectic, Poisson or quasi symplectic) variety $X$ whose support in $X$ is {\em Lagrangian.} Again, there is a general-nonsense pairing. At points where the support is non-singular projective, this can be identified with another, more geometric pairing, constructed using the {\em cubic condition} of \cite{DM1}, which is known to satisfy the closedness requirement. This approach is quite powerful, as it includes many non-linear examples such as Mukai's, in addition to the line-bundle valued spectral systems of \cite{Bn,M1} and also Simpson's $\Omega^1$-valued GL(n)-Higgs bundles: just take $X := T^*S \stackrel{\pi}{\rightarrow} S$, with its natural symplectic form, and the support in $X$ to be proper over $S$ of degree n; such sheaves correspond to Higgs bundles by $\pi_*$. The structure group $GL(n)$ can of course be replaced by an arbitrary reductive group $G$. Using Theorem \ref{main}, this yields (in the analogous cases) a Poisson structure on the Higgs moduli space ${\cal M}_{S,G,K}$ described at the end of the previous section. The fibers of the generalized Hitchin map are Lagrangian with respect to this structure. Along the lines of our general approach, the necessary modifications are clear: $\pi_*$ is replaced by the equivalence of Theorem \ref{main}. One thus considers only Lagrangian supports which retain a $W$-action, and only {\em equivariant} sheaves on them (with the numerical invariants of a line bundle). These two restrictions are symplecticly dual, so the moduli space of Lagrangian sheaves with these invariance properties is a symplectic (respectvely, Poisson) subspace of the total moduli space, and the fibers of the Hitchin map are Lagrangian as expected. A more detailed review of the ACIHS aspects of Higgs bundles will appear in \cite{DM2}. \section {Some applications and problems} \label{apps} \noindent \underline{\bf Some applications} \nopagebreak \noindent In \cite{H1}, Hitchin used his integrable system to compute several cohomology groups of the moduli space ${\cal SM}$ (of rank 2, fixed odd determinant vector bundles on a curve $C$) with coefficients in symmetric powers of its tangent sheaf ${\cal T}$. The point is that the symmetric algebra $Sym{\bf \dot{\ }} {\cal T}$ is the direct image of $ {\cal O}_{T^*{\cal SM}}$, and sections of the latter all pull back via the Hitchin map $h$ from functions on the base $B$, since the fibers of $h$ are open subsets in abelian varieties, and the missing locus has codimension $\geq 2$. Hitchin's system is used in \cite{BNR} to compute a couple of "Verlinde numbers" for GL(n), namely the dimensions $h^0({\cal M}, \Theta) = 1, \qquad h^0({\cal SM}, \Theta) = n^g$. These results are now subsumed in the general Verlinde formulas, cf. \cite{F2}, \cite{BL}, and other references therein. A pretty application of spectral covers was obtained by Katzarkov and Pantev \cite{KP2}. Let $S$ be a smooth, projective, complex variety, and $\rho : \pi_1(S)\longrightarrow G$ a Zariski dense representation into a simple $G$ (over $\bf{C}$). Assume That the $\Omega^1$-valued Higgs bundle $ ( {\cal V}, \phi) $ associated to $\rho$ by Simpson is (regular and) generically semisimple, so the cameral cover is reduced. Among other things, they show that $\rho$ factors through a representation of an orbicurve if and only if the non-standard component $Prym_{\epsilon}(\widetilde{S})$ is non zero, where $\epsilon $ is the one-dimensional sign representation of $W$. (In a sense, this is the opposite of $Prym_{\Lambda}(\widetilde{S})$: while $Prym_{\Lambda}(\widetilde{S})$ is common to $\mbox{Pic}(\widetilde{S}_P)$ for all proper Weyl subgroups, $Prym_{\epsilon}(\widetilde{S})$ occurs in none except for the full cameral Picard.) Another application is in \cite{KoP}: the moduli spaces of SL(n)- or GL(n)-stable bundles on a curve have certain obvious automorphisms, coming from tensoring with line bundles on the curve, from inversion, or from automorphisms of the curve. Kouvidakis and Pantev use the dominant direct-image maps from spectral Picards and Pryms to the moduli spaces to show that there are no further, unexpected automorphisms. This then leads to a `non-abelian Torelli theorem', stating that a curve is determined by the isomorphism class of the moduli space of bundles on it. \\ \mbox{}\\ \noindent \underline{\bf Compatibility?} \nopagebreak \noindent Hitchin's construction \cite {H2} of the projectively flat connection on the vector bundle of non-abelian theta functions over the moduli space of curves does not really use much about spectral covers. Nor do other constructions of Faltings \cite{F} and Witten et al \cite{APW}. Hitchin's work suggests that the `right' approach should be based on comparison of the non-abelian connection near a curve $C$ with the abelian connection for standard theta functions on spectral covers $\widetilde{C}$ of $C$. One conjecture concerning the possible relationship between these connections appears in \cite{A}, and some related versions have been attempted by several people, so far in vain. What's missing is a compatibility statement between the actions of the two connections on pulled-back sections. If the expected compatibility turns out to hold, it would give another proof of the projective flatness. It should also imply projective finiteness and projective unitarity of mo! nodromy for the non-abelian thetas , and may or may not bring us closer to a `finite-dimensional' proof of Faltings' theorem (=the former Verlinde conjecture).\\ \mbox{}\\ \noindent \underline{\bf {Irregulars?} } \nopagebreak \noindent The Higgs bundles we consider in this survey are assumed to be everywhere regular. This is a reasonable assumption for line-bundle valued Higgs bundles on a curve or surface, but {\em not} in $\dim \geq 3$. This is because the complement of ${\frak g}_{{reg}}$ has codimension 3 in ${\frak g}$. The source of the difficulty is that the analogue of (\ref{commutes}) fails over ${\frak g}$. There are two candidates for the universal cameral cover: $\widetilde{\frak g}$, defined by the left hand side of (\ref{commutes}), is finite over ${\frak g}$ with $W$ action, but does not have a family of line bundles parametrized by $\Lambda$. These live on $\stackrel{\approx}{\frak g}$, the object defined by the right hand side, which parametrizes pairs $(x,{\frak b}), \qquad x \in {\frak b} \subset {\frak g}$ . This suggests that the right way to analyze irregular Higgs bundles may involve spectral data consisting of a tower $$ \stackrel{\approx}{S} \stackrel{\sigma}{\longrightarrow} \widetilde{S} \longrightarrow S $$ together with a homomorphism $ {\cal L} : \Lambda \longrightarrow \mbox{Pic}(\stackrel{\approx}{S})$ such that the collection of sheaves $$ \sigma_*({\cal L}(\lambda)), \qquad \lambda \in \Lambda $$ on $\widetilde{S}$ is $R$-twisted $W$-equivariant in an appropriate sense. As a first step, one may wish to understand the direct images $ R^i \sigma_*({\cal L}(\lambda)) $ and in particular the cohomologies $H^i(F, {\cal L}(\lambda))$ where $F$, usually called a {\em Springer fiber}, is a fiber of $\sigma$. For regular $x$, this fiber is a single point. For $x=0$, the fiber is all of $G/B$, so the fiber cohomology is given by the Borel-Weil-Bott theorem. The question may thus be considered as a desired extension of BWB to general Springer fibers.

9505

alg-geom/9505012

en

https://arxiv.org/abs/alg-geom/9505012

alg-geom/9505012

Teleman

Ch. Okonek, and A. Teleman

The Coupled Seiberg-Witten Equations, vortices, and Moduli spaces of stable pairs

latex

We introduce coupled Seiberg-Witten equations, and we prove, using a generalized vortex equation, that, for Kaehler surfaces, the moduli space of solutions of these equations can be identified with a moduli space of holomorphic stable pairs. In the rank 1 case, one recovers Witten's result identifying the space of irreducible monopoles with a moduli space of divisors. As application, we give a short proof of the fact that a rational surface cannot be diffeomorphic to a minimal surface of general type.

alg-geom

\section{Introduction} Recently, Seiberg and Witten [W] introduced new invariants of 4-manifolds, which are defined by counting solutions of a certain non-linear differential equation. The new invariants are expected to be equivalent to Donaldson's polyno\-mial-invariants---at least for manifolds of simple type [KM 1]---and they have already found important applications, like e.g. in the proof of the Thom conjecture by Kronheimer and Mrowka [KM 2]. Nevertheless, the equations themselves remain somewhat mysterious, especially from a mathematical point of view. The present paper contains our attempt to understand and to generalize the Seiberg-Witten equations by coupling them to connections in unitary vector bundles, and to relate their solutions to more familiar objects, namely stable pairs. Fix a ${\rm Spin}^c$-structure on a Riemannian 4-manifold $X$, and denote by $\Sigma^{\pm}$ the associated spinor bundles. The equations which we will study are: $$ \left\{\begin{array}{lcc} \hskip 4pt{\not}{D}_{A,b}\Psi &=&0\\ \Gamma(F_{A,b}^+)&=&(\Psi\bar\Psi)_0\end{array}\right.$$ This is a system of equations for a pair $(A,\Psi)$ consisting of a unitary connection in a unitary bundle $E$ over $X$, and a positive spinor $\Psi\in A^0(\Sigma^+\otimes E)$. The symbol $b$ denotes a connection in the determinant line bundle of the spinor bundles $\Sigma^{\pm}$ and \hbox{$\hskip 4pt{\not}{D}_{A,b}:\Sigma^+\otimes E\longrightarrow\Sigma^-\otimes E$} is the Dirac operator obtained by coupling the connection in $\Sigma^+$ defined by $b$ (and by the Levi-Civita connection in the tangent bundle) with the variable connection $A$ in $E$. These equations specialize to the original Seiberg-Witten equations if $E$ is a line bundle. We show that the coupled equations can be interpreted as a differential version of the generalized vortex equations [JT]. Vortex equations over K\"ahler manifolds have been investigated by Bradlow [B1], [B2] and by Garcia-Prada [G1], [G2]: Given a pair $({\cal E},\varphi)$ consisting of a holomorphic vector bundle with a section, the vortex equations ask for a Hermitian metric $h$ in ${\cal E}$ with prescribed mean curvature: more precisely, the equations---which depend on a real parameter $\tau$---are $$i\Lambda F_h=\frac{1}{2}(\tau{\rm id}_{\cal E}-\varphi\otimes\varphi^*).$$ A solution exists if and only if the pair $({\cal E},\varphi)$ satisfies a certain stability condition ($\tau$-stability), and the moduli space of vortices can be identified with the moduli space of $\tau$-stable pairs. A GIT construction of the latter space has been given by Thaddeus [T] and Bertram [B] if the base manifold is a projective curve, and by Huybrechts and Lehn [HL1], [HL2] in the case of a projective variety. Other constructions have been given by Bradlow and Daskalopoulus [BD1], [BD2] in the case of a Riemann surface, and by Garcia-Prada for compact K\"ahler manifolds [G2]. In this connection also [BD2] is relevant. In this note we prove the following result: \begin{th} Let $(X,g)$ be a K\"ahler surface of total scalar curvature $\sigma_g$, and let $\Sigma$ be the canonical ${\rm Spin}^c$-structure with associated Chern connection $c$. Fix a unitary vector bundle $E$ of rank $r$ over $X$, and define $\mu_g(\Sigma^+\otimes E):=\frac{{\rm deg}_g(E)}{r} +\sigma_g$. Then for $\mu_g<0$, the space of solutions of the coupled Seiberg-Witten equation is isomorphic to the moduli space of stable pairs of topological type $E$, with parameter $\sigma_g$. \end{th} If the constant $\mu_g(\Sigma^+\otimes E)$ is positive, one simply replaces the bundle $E$ with $E^{\vee}\otimes K_X$, where $K_X$ denotes the canonical line bundle of $X$ (cf. Lemma 3.1). Note that the above theorem gives a complex geometric interpretation of the moduli space of solutions of the coupled Seiberg-Witten equation associated to \underbar{all} ${\rm Spin}^c$-structures on $X$: The change of the ${\rm Spin}^c$-structure is equivalent to tensoring $E$ with a line bundle. Notice also that in the special case $r=1$ one recovers Witten's result identifying the space of irreducible monopoles on a K\"ahler surface with the set of all divisors associated to line bundles of a fixed topological type; the stability condition which he mentions is the rank-1 version of the stable pair-condition. Having established this correspondence, we describe some of the basic properties of the moduli spaces, and give a first application: We show that minimal surfaces of general type cannot be diffeomorphic to rational ones. This provides a short proof of one of the essential steps in Friedman and Qin's proof of the Van de Ven conjecture [FQ]. More detailed investigations and applications will appear in a later paper. We like to thank A. Van de Ven for very helpful questions and remarks. \section{${\rm Spin}^c$-structures and almost canonical classes} The complex spinor group is defined as ${\rm Spin}^c:={\rm Spin}\times_{{\Bbb Z}_2}S^1$, and there are two non-split exact sequences $$\begin{array}{c} 1\longrightarrow S^1\longrightarrow{\rm Spin}^c\longrightarrow{\rm SO}\longrightarrow 1\\ 1\longrightarrow{\rm Spin}\longrightarrow{\rm Spin}^c\longrightarrow\ S^1\longrightarrow 1\end{array}$$ In dimension 4, ${\rm Spin}^c(4)$ can be identified with the subgroup of ${\rm U}(2)\times{\rm U}(2)$ consisting of pairs of unitary matrices with the same determinant, and one has two commutative diagrams: $$\begin{array}{ccllclll} & & 1 & & 1 & & & \\ & &\downarrow & &\downarrow& & & \\ 1&\longrightarrow& {\Bbb Z}_2 &\longrightarrow&{\rm Spin}(4) &\longrightarrow &{\rm SO}(4)\ \ \ &\longrightarrow 1\\ & &\downarrow & &\downarrow& & \ \ \parallel& \\ 1&\longrightarrow&S^1 &\longrightarrow&{\rm Spin}^c(4) &\longrightarrow &{\rm SO}(4) \ \ \ &\longrightarrow 1\\ & &\downarrow(\cdot)^2& &{\scriptstyle{\det}}\downarrow\ \ \ \ &\ \nwarrow{\scriptstyle\Delta}&\ \ \uparrow& \\ & &S^1 &= & S^1 &\longleftarrow&{\rm U}(2)\ \ \ & \\ & &\downarrow & &\downarrow&{\ }^{\det}&&\\ & &1 & &1 &&& \end{array} \eqno{(1)}$$ where $\Delta:{\rm U}(2)\longrightarrow{\rm Spin}^c(4)\subset{\rm U}(2)\times{\rm U}(2)$ acts by $a\longrightarrow\left(\left(\matrix{{\rm id}&0\cr 0&\det a\cr}\right),a\right)$, and $$ \matrix{ & & & & & & & & 1\ \ \ & &\cr & & & & & & & &\downarrow\ \ \ & &\cr & & & & 1 & & 1 & & {\Bbb Z}_2\ \ \ & &\cr & & & &\downarrow& &\downarrow& &\downarrow\ \ \ & &\cr & &1&\rightarrow&S^1 &\rightarrow&{\rm Spin}^c(4) &\rightarrow&{\rm SO}(4) &\rightarrow &1\cr & & & &\downarrow& &\downarrow& &\ \ \ \ \ \ \ \downarrow{\scriptstyle(\lambda^+,\lambda^-)}& &\cr & &1&\rightarrow&S^1\times S^1 &\rightarrow&{\rm U}(2)\times{\rm U}(2)&\stackrel{\rm ad}{\rightarrow}&{\rm SO}(3)\times{\rm SO}(3)\ &\rightarrow&1\cr & & & &\downarrow& &\downarrow& &\downarrow\ \ & &\cr 1&\rightarrow&{\Bbb Z}_2&\rightarrow&S^1&\stackrel{(\cdot)^2}{\rightarrow}&S^1& \rightarrow&1\ \ &\cr & & & &\downarrow& &\downarrow& & \ \ \ \ & &\cr & & & & 1 & & 1 & & & &\cr}$$ where $\lambda^{\pm}:{\rm SO}(4)\longrightarrow{\rm SO}(3)$ are induced by the two projections of ${\rm Spin}(4)={\rm SU}(2)^+\times{\rm SU}(2)^-$ [HH]. $\lambda^{\pm}$ can be also be seen as the representations of ${\rm SO}(4)$ in $\Lambda^2_{\pm}({\Bbb R}^4)\simeq{\Bbb R}^3$ induced by the canonical representation in ${\Bbb R}^4$. Let $X$ be a closed, oriented 4-manifold. Given any principal ${\rm SO}(4)$-bundle $P$ over $X$, we denote by $P^{\pm}$ the induced principal ${\rm SO}(3)$-bundles. If $\hat{P}$ is a ${\rm Spin}^c(4)$-bundle, we let $\Sigma^{\pm}$ be the associated ${\rm U}(2)$-vector bundles, and we set (via the vertical determinant-map in (1)) $\det(\hat{P})=L$, so that $\det(\Sigma^{\pm})=L$. \begin{lm} Let $P$ be a principal $SO(4)$-bundle over $X$ with characteristic classes $w_2(P)\in H^2(X,{\Bbb Z}_2)$, and $p_1(P), e(P)\in H^4(X,{\Bbb Z})$. Then\hfill{\break} i) $P$ lifts to a principal ${\rm Spin}^c(4)$-bundle $\hat{P}$ iff $w_2(P)$ lifts to an integral cohomology class.\hfill{\break} ii) Given a class $L\in H^2(X,{\Bbb Z})$ with $w_2(P)\equiv\bar L$(mod 2), the ${\rm Spin}^c(4)$-lifts $\hat{P}$ of $P$ with $\det\hat{P}=L$ are in 1-1 correspondence with the 2-torsion elements in $H^2(X,{\Bbb Z})$.\hfill{\break} iii) Let $\hat P$ be a ${\rm Spin}^c(4)$-principal bundle with $P\simeq\hat{P}/S^1$, and let $L=\det\hat{P}$. Then the Chern classes of $\Sigma^{\pm}$ are: $$\begin{array}{rl}c_1(\Sigma^{\pm})&=L\\ c_2(\Sigma^{\pm})&=\frac{1}{4}\left(L^2 -p_1(P)\mp 2e(P)\right)\end{array}$$ \end{lm} {\bf Proof: } [HH] and the diagrams above. \hfill\vrule height6pt width6pt depth0pt \bigskip Consider now a Riemannian metric $g$ on $X$, and let $P$ be the associated principal ${\rm SO}(4)$-bundle. In this case the real vector bundles associated to $P^{\pm}$ via the standard representations are the bundles $\Lambda^2_{\pm}$ of (anti-) self-dual 2-forms on $X$. The integral characteristic classes of $P$ are given by $p_1(P)=3\sigma$ and $e(P)=e$, where $\sigma$ and $e$ denote the signature and the Euler number of the oriented manifold $X$. Furthermore, $w_2(P)$ always lifts to an integral class, the lifts are precisely the characteristic elements in $H^2(X,{\Bbb Z})$, i.e. the classes $L$ with $x^2\equiv x\cdot L$ for every $x\in H^2(X,{\Bbb Z})$ [HH]. Let $T_X$ be the tangent bundle of $X$, and denote by $\Lambda^p$ the bundle of $p$-forms on $X$. The choice of a ${\rm Spin}^c(4)$-lift $\hat{P}$ of $P$ with associated ${\rm U}(2)$-vector bundles $\Sigma^{\pm}$ defines a vector bundle isomorphism $\gamma:\Lambda^1\otimes{\Bbb C}\longrightarrow{\rm Hom}_{{\Bbb C}}(\Sigma^+,\Sigma^-)$. There is also a ${\Bbb C}$-linear isomorphism $(\cdot)^{\#}:{\rm Hom}_{{\Bbb C}}(\Sigma^+,\Sigma^-)\longrightarrow {\rm Hom}_{{\Bbb C}}(\Sigma^-,\Sigma^+)$ which satisfies the identity: $$\gamma(u)^{\#}\gamma(v)+\gamma(v)^{\#}\gamma(u)=2g^{{\Bbb C}}(u,v){\rm id}_{\Sigma^+},$$ and $\gamma(u)^{\#}=\gamma(u)^*=g(u,u)\gamma(u)^{-1}$ for real non-vanishing cotangent vectors $u$. It is convenient to extend the homomorphisms $\gamma(u)$ to endomorphisms of the direct sum $\Sigma:=\Sigma^+\oplus\Sigma^-$. Putting $\gamma(u)|_{\Sigma^-}:=-\gamma(u)^{\#}$, we obtain a vector-bundle homomorphism $\gamma:\Lambda^1\otimes{{\Bbb C}}\longrightarrow{\rm End}_0(\Sigma)$, which maps the bundle $\Lambda^1$ of real 1-forms into the bundle of trace-free skew-Hermitian endomorphisms of $\Sigma$. With this convention, we get: $$\gamma(u)\circ\gamma(v)+\gamma(v)\circ\gamma(u)=-2g^{{\Bbb C}}(u,v){\rm id}_{\Sigma}.$$ Consider the induced homomorphism $$\Gamma:\Lambda^2\otimes{\Bbb C} \longrightarrow{\rm End}_0(\Sigma)$$ defined on decomposable elements by $$\Gamma(u\wedge v):=\frac{1}{2}[\gamma(u),\gamma(v)].$$ The restriction $\Gamma|_{\Lambda^2}$ identifies the bundle $\Lambda^2$ with the bundle ${\rm ad}_0(\hat{P})\simeq{\rm ad}(P)$ of skew-symmetric endomorphisms of the tangent bundle of $X$. $\Lambda^2$ splits as an orthogonal sum $\Lambda^2=\Lambda^2_+\oplus\Lambda^2_-$ and $\Gamma$ maps the bundle $\Lambda^2_{\pm}\otimes{\Bbb C}$ (respectively $\Lambda^2_{\pm}$) isomorphically onto the bundle ${\rm End}_0(\Sigma^{\pm})\subset{\rm End}(\Sigma)$ ($su(\Sigma^{\pm})\subset su(\Sigma)$) of trace-free (trace free skew-Hermitian) endomorphisms of $\Sigma^{\pm}$. We give now an explicit description of the two spinor bundles $\Sigma^{\pm}$ and of the map $\Gamma$ in the case of a ${\rm Spin}^c(4)$-structure coming from an almost Hermitian structure. \begin{dt} A characteristic element $K\in H^2(X,{\Bbb Z})$ is an almost canonical class if $K^2=3\sigma+2e$. \end{dt} Such classes exist on $X$ if and only if $X$ admits an almost complex structure. More precisely: \begin{pr} ({\rm Wu}) $K\in H^2(X,{\Bbb Z})$ is an almost canonical class if and only if there exists an almost complex structure $J$ on $X$ which is compatible with the orientation, such that $K=c_1(\Lambda^{10}_J)$. \end{pr} {\bf Proof: } [HH] \hfill\vrule height6pt width6pt depth0pt \bigskip Here we denote, as usual, by $\Lambda^{pq}_J$ the bundle of $(p,q)$-forms defined by the almost complex structure $J$. Notice that any almost complex structure $J$ on $X$ can be deformed into a $g$-orthogonal one, and that $J$ is $g$-orthogonal iff $g$ is $J$-Hermitian. The choice of a $g$-orthogonal almost complex structure $J$ on $X$ corresponds to to a reduction of the ${\rm SO}(4)$-bundle $P$ of $X$ to a $U(2)$-bundle via the inclusion ${\rm U}(2)\subset{\rm SO}(4)$; since the inclusion factors through the embedding $\Delta:{\rm U}(2)\longrightarrow{\rm Spin}^c(4)$ (see diagram (1)), this reduction defines a unique ${\rm Spin}^c(4)$-bundle $\hat{P_J}$ over $X$. By construction we have $\hat{P}_J/{S^1}\simeq P$, and $\det\hat{P}_J=-K$. \begin{pr} Let $J$ be a $g$-orthogonal almost complex structure on $X$, compatible with the orientation.\hfill{\break} i) The spinor bundles $\Sigma^{\pm}_J$ of $\hat{P}_J$ are: $$\Sigma^+_J\simeq\Lambda^{00}\oplus\Lambda^{02}_J,\ \ \Sigma^-_J\simeq\Lambda^{01}_J.$$ ii) The map $\Gamma:\Lambda_+^2\otimes{\Bbb C}\longrightarrow{\rm End}_0(\Sigma^+_J)$ is given by $$\Lambda^{20}_J\oplus\Lambda^{02}_J\oplus\Lambda^{00}\omega_g \ni(\lambda^{20},\lambda^{02},\omega_g)\stackrel{\Gamma}{\longmapsto} 2\left[\matrix{-i&-*(\lambda^{20}\wedge\cdot)\cr \lambda^{02}\wedge\cdot&i\cr}\right]\in{\rm End}_0(\Lambda^{00}\oplus\Lambda^{02}).$$ \end{pr} {\bf Proof: } i) $c_1(\Sigma^+_J)=c_1(\Sigma^-_J)=-K$, $c_2(\Sigma^+_J)=\frac{1}{4}[K^2-3\sigma-2e]$, $c_2(\Sigma^-_J)=\frac{1}{4}[K^2-3\sigma+2e]=c_2(\Sigma^+)+e$, and ${\rm U}(2)$-bundles on a 4-manifold are classified by their Chern classes.\hfill{\break} ii) With respect to a suitable choice of the isomorphisms i), the Clifford map $\gamma$ acts by $$\gamma(u)(\varphi+\alpha)=\sqrt{2}\left(\varphi u^{01}-i\Lambda_g u^{10}\wedge\alpha\right),$$ $$\gamma(u)^{\#}(\theta)=\sqrt{2}\left(i\Lambda_g(u^{10}\wedge\theta)-u^{01} \wedge \theta\right), \eqno{(3)}$$ where $\Lambda_g:\Lambda^{pq}_J\longrightarrow\Lambda^{p-1,q-1}_J$ is the adjoint of the map $\cdot\wedge\omega_g$ [H1]. \hfill\vrule height6pt width6pt depth0pt \bigskip \section{The coupled Seiberg-Witten equations} Let $P$ be the principal ${\rm SO}(4)$-bundle associated with the tangent bundle of the oriented, closed Riemannian 4-manifold $(X,g)$, and fix a ${\rm Spin}^c(4)$ structure $\hat{P}$ over $P$ with $L=\det(\hat{P})$. The choice of a ${\rm Spin}^c(4)$-connection in $\hat{P}$ projecting onto the Levi-Civita connection in $P$ is equivalent to the choice of a connection $b$ in the unitary line bundle $L$ [H1]. We denote by $B(b)$ the ${\rm Spin}^c(4)$-connection in $\hat{P}$ corresponding to $b$, and also the induced connection in the vector bundle $\Sigma=\Sigma^+\oplus\Sigma^-$. The curvature $F_{B(b)}$ of the connection $B(b)$ in $\Sigma$ has the form $$F_{B(b)}=\frac{1}{2}F_b{\rm id}_{\Sigma}+F_g= \left[\matrix{\frac{1}{2}F_b{\rm id}_{\Sigma^+}+F_g^+&0\cr 0&\frac{1}{2}F_b{\rm id}_{\Sigma^-}+F_g^-\cr}\right],$$ where $F_g$, and $F_g^{\pm}$ denote the Riemannian curvature operator, and its components with respect to the splitting ${\rm ad}(P)=\Lambda^2_+\oplus\Lambda^2_-$. Let now $E$ be an arbitrary Hermitian bundle of rank $r$ over $X$, and $A$ a connection in $E$. We denote by $A_b$ the induced connection in the tensor product $\Sigma\otimes E$, and by $\hskip 4pt{\not}{D}_{A,b}:A^0(\Sigma\otimes E)\longrightarrow A^0(\Sigma\otimes E)$ the associated Dirac operator. $\hskip 4pt{\not}{D}_{A,b}$ is defined as the composition: $$A^0(\Sigma\otimes E)\stackrel{\nabla_{A_b}}{\longrightarrow}A^1(\Sigma\otimes E)\stackrel{m}{\longrightarrow} A^0(\Sigma\otimes E)$$ where $m$ is the Clifford multiplication $m(u,\sigma\otimes e):=\gamma(u)(\sigma)\otimes e$. $\hskip 4pt{\not}{D}_{A,b}$ is an elliptic, self-adjoint operator and its square $\hskip 4pt{\not}{D}_{A,b}^2$ is related to the usual Laplacian $\nabla_{A_b}^*\nabla_{A_b}$ by the Weitzenb\"ock formula $$\hskip 4pt{\not}{D}_{A,b}^2=\nabla_{A_b}^*\nabla_{A_b}+\Gamma(F_{A_b}).$$ Here $\Gamma(F_{A_b})\in A^0({\rm End}(\Sigma\otimes E))$ is the Hermitian endomorphism defined as the composition $$A^0(\Sigma\otimes E)\textmap{F_{A_b}}A^0(\Lambda^2\otimes\Sigma\otimes E) \textmap{\Gamma} A^0({\rm End}_0(\Sigma)\otimes\Sigma\otimes E)\textmap{ev}A^0(\Sigma\otimes E).$$ We set $F_{A,b}:=F_A+\frac{1}{2}F_b{\rm id}_E\in A^0(\Lambda^2\otimes{\rm End}(E))$. \begin{pr} Let $s$ be the scalar curvature of the Riemannian 4-manifold $(X,g)$. Fix a ${\rm Spin}^c(4)$-structure on $X$ and choose connections $b$ and $A$ in $L$ and $E$ respectively. Then $$\hskip 4pt{\not}{D}_{A,b}^2=\nabla_{A_b}^*\nabla_{A_b}+\Gamma(F_{A,b})+\frac{s}{4} {\rm id}_{\Sigma\otimes E}.$$ \end{pr} {\bf Proof: } Since $\Gamma(F_g)=\frac{s}{4}{\rm id}_{\Sigma}$ [H1], and $F_{A_b}=F_{B(b)}\otimes {\rm id}_E+{\rm id}_{\Sigma}\otimes F_A=\frac{1}{2}F_b{\rm id}_{\Sigma}\otimes{\rm id}_E+F_g\otimes{\rm id}_E+{\rm id}_{\Sigma}\otimes F_A={\rm id}_{\Sigma}\otimes(F_A+\frac{1}{2}F_b{\rm id}_E)+F_g{\rm id}_E$, we find $\Gamma(F_{A_b})=\Gamma(F_{A,b})+\frac{s}{4}{\rm id}_{\Sigma\otimes E}$. \hfill\vrule height6pt width6pt depth0pt \bigskip \begin{re} One has a Bochner-type result for spinors $\Psi$ on which\linebreak \hbox{$\Gamma(F_{A,b})+\frac{s}{4}{\rm id}_{\Sigma\otimes E}$} is positive: Such a spinor is harmonic if and only if it is parallel [H1]. \end{re} Let $(\ ,\ )$ be the pointwise inner product on $\Sigma\otimes E$, $|\ |$ the associated pointwise norm, and $\parallel\ \parallel$ the corresponding $L^2$-norm. For a spinor $\Psi\in A^0(\Sigma^{\pm}\otimes E)$ we define $(\Psi\bar\Psi)_0\in A^0({\rm End}_0(\Sigma^{\pm}\otimes E))$ as the image of the Hermitian endomorphism $\Psi\otimes\bar\Psi\in A^0({\rm End}(\Sigma^{\pm}\otimes E))$ under the projection ${\rm End}(\Sigma^{\pm}\otimes E)\longrightarrow{\rm End}_0(\Sigma^{\pm})\otimes{\rm End}(E)$. \begin{co} With the notations above, we have $$(\hskip 4pt{\not}{D}_{A,b}^2\Psi,\Psi)=(\nabla_{A_b}^*\nabla_{A_b}\Psi,\Psi)+(\Gamma(F_{A, b}^+), (\Psi_+\bar\Psi_+)_0)+(\Gamma(F_{A,b}^-), (\Psi_-\bar\Psi_-)_0)+\frac{s}{4}|\Psi|^2,$$ where ($F_{A,b}^-$) $F_{A,b}^{+}$ is the (anti-)self-dual component of $F_{A,b}$. \end{co} {\bf Proof: } Indeed, since $\Gamma(F_{A,b}^{\pm})$ vanishes on $\Sigma^{\mp}$ and is trace free with respect to $\Sigma^{\pm}$, the inner product $(\Gamma(F_{A,b}),(\Psi\bar\Psi))$ in the Weitzenb\"ock formula simplifies for a spinor $\Psi\in A^0(\Sigma^{\pm}\otimes E) $: $$(\Gamma(F_{A,b}),(\Psi\bar\Psi))=(\Gamma(F_{A,b}^{\pm}),(\Psi\bar\Psi)_0)$$ \hfill\vrule height6pt width6pt depth0pt \bigskip For a positive spinor $\Psi\in A^0(E\otimes\Sigma^+)$, the following important identity follows immediately: $$(\hskip 4pt{\not}{D}_{A,b}^2\Psi,\Psi)+\frac{1}{2}|\Gamma(F_{A,b}^+)-(\Psi\bar\Psi)_0|^2= (\nabla_{A_b}^*\nabla_{A_b}\Psi,\Psi)+ \frac{1}{2}|F_{A,b}^+|^2+\frac{1}{2}|(\Psi\bar\Psi)_0|^2+\frac{s}{4}|\Psi|^2 \eqno{(4)}$$ If we integrate both sides of (4) over $X$, we get: \begin{pr} Let $(X,g)$ be an oriented, closed Riemannian 4-manifold with scalar curvature $s$, $E$ a Hermitian bundle over $X$. Choose a ${\rm Spin}^c(4)$-structure on $X$ and a connection $b$ in the determinant line bundle $L=\det(\Sigma^+)=\det(\Sigma^-)$. Let $A$ be a connection in $E$. For any $\Psi\in A^0(\Sigma^+\otimes E)$ we have: $$\parallel\hskip 4pt{\not}{D}_{A,b}\Psi\parallel^2+ \frac{1}{2}\parallel\Gamma(F_{A,b}^+)-(\Psi\bar\Psi)_0\parallel^2=$$ $$= \parallel\nabla_{A_b}\Psi\parallel^2+ \frac{1}{2}\parallel F_{A,b}^+\parallel^2+\frac{1}{2}\parallel(\Psi\bar\Psi)_0\parallel^2+ \frac{1}{4}\int\limits_X s|\Psi|^2.$$ \end{pr} We introduce now our coupled variant of the Seiberg-Witten equations. The unknown is a pair $(A,\Psi)$ consisting of a connection in the Hermitian bundle $E$ and a section $\Psi\in A^0(\Sigma^+\otimes E)$. The equations ask for the vanishing of the left-hand side in the above formula. $$\left\{\begin{array}{ccc}\hskip 4pt{\not}{D}_{A,b}\Psi&=&0\\ \Gamma(F_{A,b}^+)&=&(\Psi\bar\Psi)_0 \end{array}\right.\eqno{(SW)}$$ Proposition 2.4 and the inequality $|(\Psi\bar\Psi)_0|^2\geq\frac{1}{2}|\Psi|^4$ give immediately: \begin{re} If the scalar curvature $s$ is nonnegative on $X$, then the only solutions of the equations are the pairs $(A,0)$, with $F_{A,b}^+=0$. \end{re} If $L$ is the square of a line bundle $L^{\frac{1}{2}}$, and if we choose a connection $b^{\frac{1}{2}}$ in $L^{\frac{1}{2}}$ with square $b$, then $F_{A,b}$ is simply the curvature of the connection $A_{b^{\frac{1}{2}}}$ in $E\otimes L^{\frac{1}{2}}$. The solution of the coupled Seiberg-Witten equations on a manifold with $s\geq 0$ are in this case just ${\rm U}(r)$-instantons on $E\otimes L^{\frac{1}{2}}$. In the case of a K\"ahler surface $(X,g)$, the coupled Seiberg-Witten equation can be reformulated in terms of complex geometry. The point is that if we consider the canonical ${\rm Spin}^c(4)$-structure associated to the K\"ahler structure, the Dirac operator has a very simple form [H1]. The determinant of this ${\rm Spin}^c(4)$-structure is the anti-canonical bundle $K_X^{\vee}$ of the surface, which comes with a holomorphic structure and a natural metric inherited from the holomorphic tangent bundle. Let $c$ be the Chern connection in $K_X^{\vee}$. With this choice, the induced connection $B(c)$ in $\Sigma=\Lambda^{00}\oplus\Lambda^{02}\oplus\Lambda^{01}$ coincides with the connection defined by the Levi-Civita connection. Recall that on a K\"ahler manifold, the almost complex structure is parallel with respect to the Levi-Civita connection, so that the splitting of the exterior algebra $\bigoplus\limits_{p}\Lambda^p\otimes{\Bbb C}$ becomes parallel, too. \begin{pr} Let $(X,g)$ be a K\"ahler surface with Chern connection $c$ in $K_X^{\vee}$. Choose a connection $A$ in a Hermitian vector bundle $E$ over $X$ and a section $\Psi=\varphi+\alpha\in A^0(E)+A^0(\Lambda^{02}\otimes E)$. The pair $(A,\Psi)$ satisfies the Seiberg-Witten equations iff the following identities hold: $$ \begin{array}{lll} F_{A,c}^{20}&=&-\frac{1}{2}\varphi\otimes\bar\alpha\\ F_{A,c}^{02}&=&\frac{1}{2}\alpha\otimes\bar\varphi\\ i\Lambda_g F_{A,c}&=&-\frac{1}{2}\left(\varphi\otimes\bar\varphi- *(\alpha\otimes\bar\alpha)\right)\\ \bar\partial_A\varphi&=&i\Lambda_g\partial_A\alpha\end{array}$$ \end{pr} {\bf Proof: } The Dirac operator is in this case $\hskip 4pt{\not}{D}_{A,c}=\sqrt{2}(\bar\partial_A-i\Lambda_g\partial_A)$, and the endomorphism $(\Psi\bar\Psi)_0$ has the form: $$\left[\matrix{\frac{1}{2}(\varphi\otimes\bar\varphi- *(\alpha\otimes\bar\alpha))&*(\varphi\otimes\bar\alpha\wedge\cdot)\cr \alpha\otimes\bar\varphi&-\frac{1}{2}(\varphi\otimes\bar\varphi- *(\alpha\otimes\bar\alpha))\cr}\right].$$ Since $\Gamma(F_{A,c}^+)=\Gamma(F_{A,c}^{20}+F_{A,c}^{02}+\frac{1}{2}\Lambda_g F_{A,c}\cdot\omega_g)$ equals $$2\left[\matrix{-\frac{i}{2}\Lambda_gF_{A,c}&-*(F_{A,c}^{20}\wedge\cdot)\cr F_{A,c}^{20}\wedge\cdot&\frac{i}{2}\Lambda_gF_{A,c}\cr}\right],$$ the equivalence of the two systems of equations follows. \hfill\vrule height6pt width6pt depth0pt \bigskip \section{Monopoles on K\"ahler surfaces and the generalized vortex equation} Let $(X,g)$ be a K\"ahler surface with canonical ${\rm Spin}^c(4)$-structure, and Chern connection $c$ in the anti-canonical bundle $K_X^{\vee}$. We fix a unitary vector bundle $E$ of rank $r$ over $X$, and define $J(E):={\rm deg}_g(\Sigma^+\otimes E)$, i.e. $J(E)=2r(\mu_g(E)-\frac{1}{2}\mu_g(K_X))$, where $\mu_g$ denotes the slope with respect to $\omega_g$. Every spinor $\Psi\in A^0(\Sigma^+\otimes E)$ has the form $\Psi=\varphi+\alpha$ with $\varphi\in A^0(E)$ and $\alpha\in A^{0}(\Lambda^{02}\otimes E)$. We have seen that the coupled Seiberg-Witten equations are equivalent to the system: $$\left\{ \begin{array}{lll} F_{A,c}^{20}&=&-\frac{1}{2}\varphi\otimes\bar\alpha\\ F_{A,c}^{02}&=&\frac{1}{2}\alpha\otimes\bar\varphi\\ i\Lambda_g F_{A,c}&=&-\frac{1}{2}\left(\varphi\otimes\bar\varphi- *(\alpha\otimes\bar\alpha)\right)\\ \bar\partial_A\varphi&=&i\Lambda_g\partial_A\alpha\end{array}\right. \eqno{(SW^*)}$$ \begin{lm} \hfill{\break} A. Suppose $J(E)<0$: \hfill{\break} A pair $(A,\varphi+\alpha)$ is a solution of the system $(SW^*)$ if and only if \hfill{\break} i) $F_A^{20}=F_A^{02}=0$\hfill{\break} ii) $\alpha=0$, $\bar\partial_A\varphi=0$ \hfill{\break} iii) $i\Lambda_g F_A+\frac{1}{2}\varphi\otimes\bar\varphi+\frac{1}{2}s{\rm id}_E=0$. \hfill{\break} B. Suppose $J(E)>0$, and put $a:=\bar\alpha\in A^{20}(\bar E)=A^0(E^{\vee} \otimes K_X)$:\hfill{\break} A pair $(A,\varphi+\bar a)$ is a solution of the system $(SW^*)$ if and only if\hfill{\break} i) $F_A^{20}=F_A^{02}=0$\hfill{\break} ii) $\varphi=0$, $\bar\partial_A a=0$ \hfill{\break} iii) $i\Lambda_g F_A-\frac{1}{2}*(a\otimes\bar a)+\frac{1}{2}s{\rm id}_E=0$. \end{lm} {\bf Proof: } (cf. [W]) The splitting $\Sigma^+\otimes E=\Lambda^{00}\otimes E\oplus\Lambda^{02}\otimes E$ is parallel with respect to $\nabla_{A,c}$, so that, by Proposition 2.4 $$\parallel\hskip 4pt{\not}{D}_{A,c}\Psi\parallel^2+ \frac{1}{2}\parallel\Gamma(F_{A,c}^+)-(\Psi\bar\Psi)_0\parallel^2=$$ $$= \parallel\nabla_{A_c}\varphi\parallel^2+ \parallel\nabla_{A_c}\alpha\parallel^2+ \frac{1}{2}\parallel F_{A,c}^+\parallel^2+\frac{1}{2}\parallel(\Psi\bar\Psi)_0\parallel^2+ \frac{1}{4}\int\limits_X s(|\varphi|^2+|\alpha|^2).$$ The right-hand side is invariant under the transformation $(A,\varphi,\alpha)\longmapsto (A,\varphi,-\alpha)$, hence any solution $(A,\varphi+\alpha)$ must have $F_A^{20}=F_A^{02}=0$ and $\varphi\otimes\bar\alpha=\alpha\otimes\bar\varphi=0$; the latter implies obviously $\alpha=0$ or $\varphi=0$. Integrating the trace of the equation $i\Lambda F_{A,c}=-\frac{1}{2}\left(\varphi\otimes\bar\varphi- *(\alpha\otimes\bar\alpha)\right)$, we find: $$J(E)=c_1(\Sigma^+\otimes E)\cup[\omega_g]=(2c_1(E)-rc_1(K_X))\cup[\omega_g]=$$ $$= 2\int\limits_X\frac{i}{2\pi}{\rm Tr}(F_{A,c})\wedge\omega_g= \frac{1}{4\pi}\int\limits_X{\rm Tr}(i\Lambda F_{A,c})\omega_g^2=\frac{1}{8\pi}\int\limits_X{\rm Tr}(-\varphi\otimes\bar\varphi) +*(\alpha\otimes\bar\alpha))\omega_g^2$$ This equation shows that we must have $\alpha=0$, if $J(E)<0$, and $\varphi=0$, if $J(E)>0$. Notice that, replacing $E$ by $E^{\vee}\otimes K_X$, the second case reduces to the first one. The assertion follows now immediately from the identity $i\Lambda_g F_c=s$. \hfill\vrule height6pt width6pt depth0pt \bigskip Notice that the last equation $$i\Lambda_g F_A+\frac{1}{2}\varphi\otimes\bar\varphi+\frac{1}{2}s{\rm id}_E=0$$ has the form of a generalized vortex equation as studied by Bradlow [B1], [B2] and by Garcia-Prada [G2]; it is precisely the vortex equation with constant $\tau=-{s}$, if $(X,g)$ has constant scalar curvature. Let $s_m$ be the mean scalar curvature defined by $\int\limits_Xs\omega_g^2=s_m\int\limits_X\omega^2=2s_m{\rm Vol}_g(X)$. We are going to prove that the system $$\left\{\begin{array}{cl}\bar\partial_A^2&=0\\ \bar\partial_A\varphi&=0\\ i\Lambda_g F_A+\frac{1}{2}\varphi\otimes\bar\varphi+\frac{1}{2}s{\rm id}_E&=0 \end{array}\right.$$ for the pair $(A,\varphi)$ consisting of a unitary connection in $E$, and a section in $E$, is always equivalent to the vortex system with parameter $\tau=-s_m$, i.e. to the system obtained by replacing the third equation with $$i\Lambda_g F_A+\frac{1}{2}\varphi\otimes\bar\varphi+\frac{1}{2}s_m{\rm id}_E=0.$$ "Equivalent" means here that the corresponding moduli spaces of solutions are naturally isomorphic. Let generally $t$ be a smooth real function on $X$ with mean value $t_m$, and consider the following system of equations: $$\left\{\begin{array}{cl}\bar\partial_A^2&=0\\ \bar\partial_A\varphi&=0\\ i\Lambda_g F_A+\frac{1}{2}\varphi\otimes\bar\varphi-\frac{1}{2}t{\rm id}_E&=0 \end{array}\right.\eqno(V_t)$$ $(V_t)$ is defined on the space ${\cal A}(E)\times A^0(E)$, where ${\cal A}(E)$ is the space of unitary connections in $E$. The product ${\cal A}(E)\times A^0(E)$ (completed with respect to sufficiently large Sobolev indices) carries a natural $L^2$ K\"ahler metric $\tilde g$ and a natural right action of the gauge group $U(E)$: $(A,\varphi)^f:=(A^f,f^{-1}\varphi)$, where $d_{A^f}:=f^{-1}\circ d_A\circ f$. For every real function $t$ let $$m_t:{\cal A}(E)\times A^0(E)\longrightarrow A^0({\rm ad}(E))$$ be the map given by $m_t:=\Lambda_g F_A-\frac{i}{2}\varphi\otimes\bar\varphi+\frac{i}{2}t{\rm id}_E.$ \begin{pr} $m_t$ is a moment map for the action of $U(E)$ on \linebreak ${\cal A}(E)\times A^0(E)$. \end{pr} {\bf Proof: } Let $a^{\#}$ be the vector field on ${\cal A}(E)\times A^0(E)$ associated with the infinitesimal transformation $a\in A^0({\rm ad}(E))={\rm Lie }(U(E))$, and define the real function $m^a_t:{\cal A}(E)\times A^0(E)\longrightarrow{\Bbb R}$ by: $$ m^a_t(x)=\langle m_t(x),a\rangle_{L^2} .$$ We have to show that $m_t$ satisfies the identities: $$\iota_{a^{\#}}\omega_{\tilde g}=dm_t^a \ ,\ \ \ m_t^a\circ f=m^{{\rm ad}_f(a)}\ \ \ {\rm for\ all}\ \ a\in A^0({\rm ad}(E)), \ \ f\in U(E).$$ It is well known that, in general, a moment map for a group action in a symplectic manifold is well defined up to a constant central element in the Lie algebra of the group. In our case, the center of the Lie algebra $A^0({\rm ad}(E))$ of the gauge group is just $iA^0{\rm id}_E$, hence it suffices to show that $m_0$ is a moment map. This has already been noticed by Garcia-Prada [G1], [G2]. \hfill\vrule height6pt width6pt depth0pt \bigskip Note also that in our case every moment map has the form $m_t$ for some function $t$, which shows that from the point of view of symplectic geometry, the natural equations are the generalized vortex equations $(V_t)$. In order to show that Bradlow's main result [B2] also holds for the generalized system $(V_t)$, we have to recall some definitions. Let ${\cal E}$ be a holomorphic vector bundle of topological type $E$, and let $\varphi\in H^0({\cal E})$ be a holomorphic section. The pair $({\cal E},\varphi)$ is $\lambda$-\underbar{stable} with respect to a constant $\lambda\in{\Bbb R}$ iff the following conditions hold:\\ (1) $\mu_g({\cal E})<\lambda$ and $\mu_g({\cal F})<\lambda$ for all reflexive subsheaves ${\cal F}\subset{\cal E}$ with $0<{\rm rk}({\cal F})<r$.\\ (2) $\mu_g({\cal E}/{\cal F})>\lambda$ for all reflexive subsheaves ${\cal F}\subset{\cal E}$ with $0<{\rm rk}({\cal F})<r$ and $\varphi\in H^0({\cal F})$. \begin{th} Let $(X,g)$ be a closed K\"ahler manifold, $t\in A^0$ a real function, and $({\cal E},\varphi)$ a holomorphic pair over $X$. Set $\lambda:=\frac{1}{4\pi} t_m{\rm Vol}_g(X)$. ${\cal E}$ admits a Hermitian metric $h$ such that the associated Chern connection $A_h$ satisfies the vortex equation $$i\Lambda_g F_A+\frac{1}{2}\varphi\otimes\bar\varphi-\frac{1}{2}t{\rm id}_E=0$$ iff one of the following conditions holds:\\ (i) $({\cal E},\varphi)$ is $\lambda$-stable\\ (ii) ${\cal E}$ admits a splitting ${\cal E}={\cal E}'\oplus{\cal E}''$ with $\varphi\in H^0({\cal E}')$ such that $({\cal E}',\varphi)$ is $\lambda$-stable, and ${\cal E}''$ admits a weak Hermitian-Einstein metric with factor $\frac{t}{2}$. In particular ${\cal E}''$ is polystable of slope $\lambda$. \end{th} {\bf Proof: } In the case of a constant function $t=\tau$, the theorem was proved by Bradlow [B2], and his arguments work in the general context, too: The fact that the existence of a solution of the vortex equation implies $(i)$ or $(ii)$ follows by replacing the constant $\tau$ in [B2] everywhere with the function $t$. The difficult part consists in showing that every $\lambda$-stable pair $({\cal E},\varphi)$ admits a metric $h$ such that $(A_h,\varphi)$ satisfies the vortex equation $(V_t)$. To this end let $Met(E)$ be the space of Hermitian metrics in $E$, and fix a background metric $k\in Met(E)$. Bradlow constructs a functional $M_{\varphi,\tau}(\cdot,\cdot):Met(E)\times Met(E)\longrightarrow{\Bbb R}$, which is convex in the second argument, such that any critical point of $M_{\varphi,\tau}(k,\cdot)$ is a solution of the vortex equation; the point is then to find an absolute minimum of $M_{\varphi,\tau}(k,\cdot)$. The existence of an absolute minimum follows from the following basic ${\cal C}^0$ estimate: \begin{lm}{\rm (Bradlow)} Let $Met^p_2(E,B):=\{h=ke^a| a\in L^2_p({\rm End}(E)), a^{*k}=a, \parallel\mu_t(A_h,\varphi)\parallel_{L^p}\leq B\}$. If $({\cal E},\varphi)$ is $\frac{\tau}{4\pi}{\rm Vol}_g(X)$-stable, then there exist positive constants $C_1$, $C_2$ such that $$\sup|a|\leq C_1M_{\varphi,\tau}(k,ke^a)+ C_2,$$ for all $k$-Hermitian endomorphisms $a\in L^2_p({\rm End}(E))$. Moreover, any absolute minimum of $M_{\varphi,t}(k,\cdot)$ on $Met^p_2(E,B)$ is a critical point of $M_{\varphi,t}(k,\cdot)$, and gives a solution of the vortex equation $V_{\tau}$. \end{lm} Let now $t$ be a real function on $X$, and choose a solution $v$ of the Laplace equation $i\Lambda_g\bar\partial\partial v=\frac{1}{2}(t-t_m)$. If we make the substituion $h=h'e^v$, then $h$ solves the vortex equation $(V_t)$ iff $h'$ is a solution of $$i\Lambda_g F_{h'}+\frac{1}{2}e^v\varphi\otimes\bar\varphi^{h'}-\frac{1}{2}t_m{\rm id}_E=0.$$ Define $\mu_{t_m,v}(h'):=i\Lambda_g F_{h'}+\frac{1}{2}e^v\varphi\otimes\bar\varphi^{h'}-\frac{1}{2}t_m{\rm id}_E=0$, and $$M_{\varphi,t_m,v}(k,h):=M_D(k,h)+\parallel e^{\frac{v}{2}}\varphi\parallel^2_h-\parallel e^{\frac{v}{2}}\varphi\parallel^2_k-t_m\int\limits_X{\rm Tr}(\log(k^{-1}h)),$$ where $M_D$ is the Donaldson functional [D]. Then it is not difficult to show that all arguments of Bradlow remain correct after replacing $\mu_{t_m}$ and $M_{\varphi,t_m}$ with $\mu_{t_m,v}$ and $M_{\varphi,t_m,v}$ respectively. Indeed, let $l$ be a positive bound from below for the map $e^v$. Then $$\begin{array}{ll}M_{\varphi,t_m}(k,ke^{a+\log l})&\leq M_D(k,ke^{a})+M_D(ke^{a},lke^{a})+\parallel l\varphi\parallel^2_h-t_m\int\limits_X{\rm Tr}\log(lk^{-1}h)\cr &\leq M_{\varphi,t_m,v}(k,ke^{a})+\parallel e^{\frac{v}{2}}\varphi\parallel^2_k+2\log l{\rm deg}_g(E)-rt_m\log l{\rm Vol}_g(X)\cr &\leq M_{\varphi,t_m,v}(k,ke^{a})+C'(k,\varphi,v,l).\end{array}$$ Similarly, we get constants $n>0$, $C''$ and an inequality $$M_{\varphi,t_m,v}(k,ke^{a+\log n})\leq M_{\varphi,t_m}(k,ke^{a})+C'',$$ which shows that the basic ${\cal C}^0$ estimate in the Lemma above holds for $M_{\varphi,t_m,v}$ iff it holds for Bradlow's functional $M_{\varphi,t_m}$. \hfill\vrule height6pt width6pt depth0pt \bigskip \begin{re} In the special case of a rank-1 bundle $E$, a much more elementary proof based on [B1] is possible. \end{re} \section{Moduli spaces of monopoles, vortices, and stable pairs} Let $(X,g)$ be a closed K\"ahler manifold of arbitrary dimension, and fix a unitary vector bundle $E$ of rank $r$ over $X$. We denote by $\bar{{\cal A}}(E)$ the affine space of semiconnection of type $(0,1)$ in $E$. the complex gauge group $GL(E)$ acts on $\bar{{\cal A}}(E)\times A^0(E)$ from the right by $(\bar\partial_A,\varphi)^g:=(g^{-1}\circ\bar\partial_A\circ g, g^{-1}\varphi)$; this action becomes complex analytic after suitable Sobolev completions. We denote by $\bar{\cal S}(E)$ the set of pairs $(\bar\partial_A,\varphi)$ with trivial isotropy group. Notice that $\varphi\ne 0$ when $(\bar\partial_A,\varphi)\in\bar{\cal S}(E)$, and that $\bar{\cal S}(E)$ is an open subset of $\bar{{\cal A}}(E)\times A^0(E)$, by elliptic semi-continuity [K]. The action of $GL(E)$ on $\bar{\cal S}(E)$ is free, by definition, and we denote the Hilbert manifold $\qmod{\bar{\cal S}(E)}{GL(E)}$ by $\bar{{\cal B}}^s(E)$. The map $p:\bar{{\cal A}}(E)\times A^0(E)\longrightarrow A^{02}({\rm End}(E)\oplus A^{01}(E)$ defined by $p(\bar\partial_A,\varphi)=(F_A^{02},\bar\partial_A\varphi)$ is equivariant with respect to the natural actions of $GL(E)$, hence it gives rise to a section $\hat p$ in the associated vector bundle $\bar{\cal S}(E)\times_{GL(E)}\left(A^{02}({\rm End}(E)\oplus A^{01}(E)\right)$ over $\bar{{\cal B}}^s(E)$. We define the moduli space of \underbar{simple pairs} of type $E$ to be the zero-locus $Z(\hat{p})$ of this section. $Z(\hat{p})$ can be identified with the set of isomorphism classes consisting of a holomorphic bundle ${\cal E}$ of differentiable type $E$, and a holomorphic section $\varphi\ne 0$, such that the kernel of the evaluation map $ev(\varphi):H^0({\rm End}({\cal E}))\longrightarrow H^0({\cal E})$ is trivial. In a similar way we define the moduli space ${\cal V}^g_t$ of gauge-equivalence classes of irreducible solutions of the generalized vortex equation $V_t$: Let $B^+$ denote as usual the subbundle $\left((\Lambda^{02}+\Lambda^{20})\cap \Lambda^2\right)\oplus \Lambda^0\omega$ of the bundle $\Lambda^2$ of real 2-forms on $X$. We denote by ${\cal D}^*$ the open subset of the product ${\cal D}:={\cal A}(E)\times A^0(E)\simeq\bar{{\cal A}}(E)\times A^0(E)$ consisting of pairs with trivial isotropy group with respect to the action of the gauge group $U(E)$. The quotient ${\cal B}^*(E):=\qmod{{\cal D}^*(E)}{U(E)}$ comes with the structure of a real-analytic manifold. Let $v:{\cal D}(E)\longrightarrow A^0(B^+\otimes{\rm ad}(E))\oplus A^{01}(E)$ be the map given by: $$v(A,\varphi)=(F^{20}+F^{02},m_t(A,\varphi)\omega_g{\rm id}_E,\bar\partial_A\varphi).$$ Again $v$ is $U(E)$-equivariant, and the moduli space ${\cal V}^g_t$ of $t$-vortices is defined to be the zero-locus $Z(\hat{v})$ of the induced section $\hat v$ of \linebreak \hbox{${\cal D}^*(E)\times_{U(E)}A^0(B^+\otimes{\rm ad}(E))\oplus A^{01}(E)$} over ${\cal B}^*(E)$. Notice now that by Proposition 3.2, the second component $v^2$ of $v$ is a moment map for the $U(E)$ action. It is easy to see that (at least in a neighbourhood of $Z(v)\cap{\cal D}^*$) it has the general property of a moment map in the finite dimensional K\"ahler geometry: Its zero locus $Z(v^2)$ is smooth and intersects every $GL(E)$ orbit along a $U(E)$ orbit, and the intersection is transversal. This means that the natural map $A\longrightarrow\bar\partial_A$ defines in a neighbourhood of $Z(\hat{v})\cap{\cal B}^*(E)$ an open embedding $i:Z({\hat{v}^2})\longrightarrow\bar{\cal B}^s$ of smooth Hilbert manifolds. Regard now ${\cal V}^g_t$ as the subspace of $Z({\hat{v}^2})\subset{\cal B}^*(E)$ defined by the equation $(\hat{v}^1,\hat{v}^3)=0$. On the other hand, the pullback of the equation $\hat p=0$, cutting out the moduli space $Z(\hat{p})$ of simple holomorphic pairs, via the open embedding $i$ is precisely the equation $(\hat{v}^1,\hat{v}^3)=0$, cutting out ${\cal V}^g_t$. We get therefore an open embedding $i_0:{\cal V}^g_t\longrightarrow Z(\hat{p})$ of real analytic spaces induced by $i$, and by Theorem 3.3 the image of $i_0$ consists of the set of $\lambda$-stable pairs, with $\lambda:=\frac{1}{4\pi}t_m{\rm Vol}_g(X)$. Finally we denote by ${\cal M}_X^g(E,\lambda)\subset Z(\hat p)$ the open subspace of $\lambda$-stable pairs, with the induced complex space-structure. Putting everything together, we have: \begin{th} Let $(X,g)$ be a closed K\"ahler manifold, $t\in A^0$ a real function, and $\lambda: =\frac{1}{4\pi}t_m{\rm Vol}_g(X)$. Fix a unitary vector bundle $E$ of rank $r$ over $X$. There are natural real-analytic isomorphisms of moduli spaces $${\cal V}^g_t(E)\simeq{\cal V}^g_{t_m}(E)\simeq{\cal M}_X^g(E,\lambda).$$ \end{th} Let us come back now to the monopole equation $(SW^*)$ on a K\"ahler surface. In this case the function $t$ is the negative of the scalar curvature $s$, so that the corresponding constant $\lambda$ becomes: $$\lambda=\frac{-s_m}{4\pi}{\rm Vol}_g(X)=-\frac{1}{8\pi}\int\limits_Xs\omega^2= -\frac{1}{8\pi}\int\limits_X(i\Lambda F_c)\omega^2=-\frac{1}{4\pi}\int\limits_X i F_c\wedge\omega=\frac{1}{2}\mu_g(K).$$ This yields our main result: \begin{th} Let $(X,g)$ be a K\"ahler surface with canonical ${\rm Spin}^c(4)$-structure, and Chern connection $c$ in $K_X^{\vee}$. Fix a unitary vector bundle $E$ of rank $r$ over $X$, and suppose $J(E)={\rm deg}_g(\Sigma^+\otimes E)<0$. The moduli space of solutions of the coupled Seiberg-Witten equations is isomorphic to the moduli space ${\cal M}_X^g(E,\frac{1}{2}\mu_g(K))$ of $\frac{1}{2}\mu_g(K)$-stable pairs of topological type $E$. \end{th} At this point it is natural to study the properties of the moduli spaces ${\cal M}^g_X(E,\lambda)$. We do not want to go into details here, and we content ourselves by describing some of the basic steps. The infinitesimal structure of the moduli space around a point $[(A,\varphi)]$ is given by a deformation complex $(C_{\bar\partial_A,\varphi}^*, d_{A,\varphi}^*)$ which is the cone over the evaluation map $ev^*$, $ev^q(\varphi):A^{0q}({\rm End}(E))\longrightarrow A^{0q}(E)$. More precisely $C_{\bar\partial_A,\varphi}^q=A^{0q}({\rm End}(E))\oplus A^{0,q-1}(E)$ and the differential $d_{A,\varphi}^q$ is given by the matrix $$d_{A,\varphi}^q=\left[\matrix{-\bar D_A&0\cr ev(\varphi)&\bar\partial_A\cr}\right],$$ where $\bar\partial_A$ and $\bar D_A$ are the operators of the Dolbeault complexes $(A^{0*}(E),\bar\partial_A)$ and $(A^{0*}{\rm End}(E),\bar D_A)$ respectively. Associated to the morphism $ev^*(\varphi)$ is an exact sequence $$\dots\longrightarrow H^q({\rm End}({\cal E}_A))\textmap{ev^q(\varphi)}H^q({\cal E}_A)\longrightarrow H_{\bar\partial_A,\varphi}^{q+1}\longrightarrow H^{q+1}({\rm End}({\cal E}_A))\longrightarrow\dots $$ with finite dimensional vector spaces $$H^q_{\bar\partial_A,\varphi}= \ker(ev^q(\varphi))\oplus{\rm coker}(ev^{q-1}(\varphi)).$$ $H^0_{\bar\partial_A,\varphi}$ vanishes for a simple pair $(\bar\partial_A,\varphi)$, and $H^1_{\bar\partial_A,\varphi}$ is the Zariski tangent space of ${\cal M}^g_X(E,\lambda)$ at $[\bar\partial_A,\varphi]$. A Kuranishi type argument yields local models of the moduli space, which can be locally described as the zero loci of holomorphic map germs $$K_{[\bar\partial_A,\varphi]}:H^1_{\bar\partial_A,\varphi}\longrightarrow H^2_{\bar\partial_A,\varphi}$$ at the origin. One finds that $H^2_{\bar\partial_A,\varphi}=0$ is a sufficient smoothness criterion in the point $[\bar\partial_A,\varphi]$ of the moduli space, and that the expected dimension is \linebreak\hbox{$\chi(E)-\chi({\rm End}(E))$}. The necessary arguments are very similar to the ones in [BD1], [BD2]. The moduli spaces ${\cal M}^g(E,\lambda)$ will be quasi-projective varieties if the underlying manifold $(X,g)$ is Hodge, i.e. if $X$ admits a projective embedding such that a multiple of the K\"ahler class is a polarisation [G1]. A GIT construction for projective varieties of any dimension has been given in [HL2]. The spaces ${\cal M}^g_X(E,\lambda)$ vary with the parameter $\lambda$, and flip-phenomena occur just like in the case of curves [T]. \section{Applications} The equations considered by Seiberg and Witten are associated to a ${\rm Spin}^c(4)$-structure, and correspond to the case when (in our notations) the unitary bundle $E$ is the trivial line bundle. Alternatively, we can fix a ${\rm Spin}^c(4)$ structure ${\germ s}_0$ on $X$ , and regard the Seiberg-Witten equations corresponding to the other ${\rm Spin}^c(4)$-structures as {\sl coupled} Seiberg-Witten equations associated to ${\germ s}_0$ and to a unitary line bundle $E$. The ${\rm Spin}^c(4)$-structure we fix will always be the canonical structure defined by a K\"ahler metric. In the most interesting case of rank-1 bundles $E$ over K\"ahler surfaces the central result is: \begin{pr} Let $(X,g)$ be a K\"ahler surface with canonical class $K$, and let $L$ be a complex line bundle over $X$ with $L\equiv K$ (mod 2). Denote by ${\cal W}_X^g(L)$ the moduli space of solutions of the Seiberg-Witten equation for all ${\rm Spin}^c(4)$-structures with determinant $L$. Then\hfill{\break} i) If $\mu(L)<0$, ${\cal W}_X^g(L)$ is isomorphic to the space of all linear systems $|D|$, where $D$ is a divisor with $c_1({\cal O}_X(2D-K))=L$.\hfill{\break} ii) If $\mu(L)>0$, ${\cal W}_X^g(L)$ is isomorphic to the space of all linear systems $|D|$, where $D$ is a divisor with $c_1({\cal O}_X(2D-K))=-L$. \end{pr} {\bf Proof: } Use Theorem 4.2 and Bradlow's description of the moduli spaces of stable pairs in the case of line bundles [B1]. \hfill\vrule height6pt width6pt depth0pt \bigskip We have already noticed (Remark 2.5) that in the case of a Riemannian 4-manifold with nonnegative scalar curvature $s_g$, the Seiberg-Witten equations have only reducible solutions. In the K\"ahler case, the same result can be obtained under the weaker assumption $\sigma_g \geq 0$ on the total scalar curvature. \begin{co} Let $(X,g)$ be a K\"ahler surface with nonnegative total scalar curvature $\sigma_g$. Then all solutions of the Seiberg-Witten equations in rank 1 are reducible. If moreover the surface has $K^2>0$, then for every almost canonical class $L$, the corresponding Seiberg-Witten equations are incompatible. \end{co} {\bf Proof: } The first assertion follows directly from the theorem, since the condition $\sigma_g \geq 0$ is equivalent to $K\cup[\omega_g]\leq 0$. For the second assertion, note that if $L$ is an almost canonical class, then $L^2=K^2>0$, hence (regarded as line bundle) it cannot admit anti-selfdual connections. \hfill\vrule height6pt width6pt depth0pt \bigskip \begin{re} The Seiberg-Witten invariants associated to almost canonical classes are well-defined for oriented, closed 4-manifolds $X$ satisfying $3\sigma+2e>0$. \end{re} {\bf Proof: } Recall that if $L$ is an almost canonical class, then the expected dimension of the moduli space of solutions of the perturbed Seiberg-Witten equations [W, KM]corresponding to a ${\rm Spin}^c(4)$-structure of determinant $L$ is 0. Seiberg and Witten associate to every such class $L$ the number $n_L$ of points (counted with the correct signs [W]) of such a moduli space chosen to be smooth and of the expected dimension. In the case $b_+\geq 2$, using the same cobordism argument as in Donaldson theory, it follows that that these numbers are well-defined, i.e. independent of the metric, provided the moduli space has the expected dimension [KM]. The point is that the space of $L$-good metrics [KM] (i.e. metrics with the property that the space of harmonic anti-selfdual forms does not contain the harmonic representative of $c_1^{{\Bbb R}}(L)$) is in this case path-connected. On the other hand, under the assumption $3\sigma +2e>0$, it follows that $L^2>0$ for any almost canonical class $L$, hence all metrics are $L$-good. \hfill\vrule height6pt width6pt depth0pt \bigskip \begin{pr} Let $(X,H_0)$ be a polarised surface with $K$ nef and big, and choose a K\"ahler metric $g$ with K\"ahler class $[\omega_g]=H_0+nK=:H$ for some $n\geq KH_0$. Then ${\cal W}_X^g(L)$ is empty for all almost canonical classes, except for $L=\pm K$, when it consists of a simple point. \end{pr} {\bf Proof: } Let $L$ be an almost canonical class with $L H<0$. Suppose $D$ is an effective divisor with $c_1({\cal O}_X(2D-K))=L$, so that $D(D-K)=0$. Then $D^2=DK\geq 0$ since $K$ is nef. If $D^2$ were strictly positive, the Hodge index theorem would give $(D-K)^2\leq 0$, i.e. $K^2\leq D^2$. But from $LH<0$ we get $0>(2D-K)(H_0+nK)=(2D-K)H_0+n(2D^2-K^2)\geq (2D-K)H_0+n$, which leads to the contradiction $n<(K-2D)H_0\leq KH_0$. Therefore $D^2=DK=0$, so that, again by the Hodge index theorem, $D$ must be numerically zero. Since $D$ is effective, it must be empty, and $L=-K$. Replacing $L$ by $-L$ if $L$ is an almost canonical class with $LH>0$, we find $L=K$ in this case. The corresponding Seiberg-Witten moduli spaces are simple points in both cases, since $H^2_{\bar\partial_A,\varphi}=H^1({\cal O}(D)|_D)=0$. \hfill\vrule height6pt width6pt depth0pt \bigskip \begin{co} There exists no orientation-preserving diffeomorphism between a rational surface and a minimal surface of general type. \end{co} {\bf Proof: } Indeed, any rational surface $X$ admits a Hodge metric with positive total scalar curvature [H2]. If $X$ was orientation-preservingly diffeomorphic to a minimal surface of general type, then $K^2>0$, hence the Seiberg-Witten invariants are well defined (Remark 5.3), and vanish by Corollary 5.2. Proposition 5.4 shows, however, that the Seiberg-Witten invariants of a minimal surface of general type are non-trivial for two almost canonical classes. \hfill\vrule height6pt width6pt depth0pt \bigskip \vspace{3mm}\\ Witten has already proved [W] that for a minimal surface of general type with $p_g>0$ ($b_+\geq 2$), the only almost canonical classes which give non-trivial invariants are $K$ and $-K$. Their proof uses the moduli space of solutions of the perturbation of the Seiberg-Witten equation with a holomorphic form. Proposition 5.4 shows that a stronger result can be obtained with the non-perturbed equations by choosing the Hodge metric $H=H_0+nK,\ n\gg 0$. For the proof of Corollary 5.5, we need in fact only the mod. 2 version of the Seiberg-Witten invariants [KM2]. \newpage \parindent0cm \centerline {\Large {\bf Bibliography}} \vspace{1cm} [AHS] Atiyah M., Hitchin N. J., Singer I. M.: {\it Selfduality in four-dimensional Riemannian geometry}, Proc. R. Lond. A. 362, 425-461 (1978) [BPV] Barth, W., Peters, C., Van de Ven, A.: {\it Compact complex surfaces}, Springer Verlag (1984) [B] Bertram, A.: {\it Stable pairs and stable parabolic pairs}, J. Alg. Geometry 3, 703-724 (1994) [B1] Bradlow, S. B.: {\it Vortices in holomorphic line bundles over closed K\"ahler manifolds}, Comm. Math. Phys. 135, 1-17 (1990) [B2] Bradlow, S. B.: {\it Special metrics and stability for holomorphic bundles with global sections}, J. Diff. Geom. 33, 169-214 (1991) [BD1] Bradlow, S. B.; Daskalopoulos, G.: {\it Moduli of stable pairs for holomorphic bundles over Riemann surfaces I}, Intern. J. Math. 2, 477-513 (1991) [BD2] Bradlow, S. B.; Daskalopoulos, G.: {\it Moduli of stable pairs for holomorphic bundles over Riemann surfaces II}, Intern. J. Math. 4, 903-925 (1993) [D] Donaldson, S.: {\it Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles}, Proc. London Math. Soc. 3, 1-26 (1985) [DK] Donaldson, S.; Kronheimer, P.B.: {\it The Geometry of four-manifolds}, Oxford Science Publications (1990) [FM] Friedman, R.; Morgan, J.W.: {\it Smooth 4-manifolds and Complex Surfaces}, Springer Verlag 3. Folge, Band 27 (1994) [FQ] Friedman, R.; Qin, Z.: {\it On complex surfaces diffeomorphic to rational surfaces}, Preprint (1994) [G1] Garcia-Prada, O.: {\it Dimensional reduction of stable bundles, vortices and stable pairs}, Int. J. of Math. Vol. 5, No 1, 1-52 (1994) [G2] Garcia-Prada, O.: {\it A direct existence proof for the vortex equation over a compact Riemann surface}, Bull. London Math. Soc., 26, 88-96 (1994) [HH] Hirzebruch, F., Hopf H.: {\it Felder von Fl\"achenelementen in 4-dimensionalen 4-Mannigfaltigkeiten}, Math. Ann. 136 (1958) [H1] Hitchin, N.: {\it Harmonic spinors}, Adv. in Math. 14, 1-55 (1974) [H2] Hitchin, N.: {\it On the curvature of rational surfaces}, Proc. of Symp. in Pure Math., Stanford, Vol. 27 (1975) [HL1] Huybrechts, D.; Lehn, M.: {\it Stable pairs on curves and surfaces}, J. Alg. Geometry 4, 67-104 (1995) [HL2] Huybrechts, D.; Lehn, M.: {\it Framed modules and their moduli.} Int. J. Math. 6, 297-324 (1995) [JT] Jaffe, A., Taubes, C.: {\it Vortices and monopoles}, Boston, Birkh\"auser (1980) [K] Kobayashi, S.: {\it Differential geometry of complex vector bundles}, Princeton University Press (1987) [KM1] Kronheimer, P.; Mrowka, T.: {\it Recurrence relations and asymptotics for four-manifold invariants}, Bull. Amer. Math. Soc. 30, 215 (1994) [KM2] Kronheimer, P.; Mrowka, T.: {\it The genus of embedded surfaces in the projective plane}, Preprint (1994) [OSS] Okonek, Ch.; Schneider, M.; Spindler, H: {\it Vector bundles on complex projective spaces}, Progress in Math. 3, Birkh\"auser, Boston (1980) [Q] Qin, Z.: {\it Equivalence classes of polarizations and moduli spaces of stable locally free rank-2 sheaves}, J. Diff. Geom. 37, No 2 397-416 (1994) [S] Simpson, C. T.: {\it Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization}, J. Amer. Math. Soc. 1 867-918 (1989) [T] Thaddeus, M.: {\it Stable pairs, linear systems and the Verlinde formula}, Invent. math. 117, 181-205 (1994) [UY] Uhlenbeck, K. K.; Yau, S. T.: {\it On the existence of Hermitian Yang-Mills connections in stable vector bundles}, Comm. Pure App. Math. 3, 257-293 (1986) [W] Witten, E.: {\it Monopoles and four-manifolds}, Mathematical Research Letters 1, 769-796 (1994) \vspace{2cm}\\ Authors addresses:\\ \\ Mathematisches Institut, Universit\"at Z\"urich,\\ Winterthurerstrasse 190, CH-8057 Z\"urich\\ e-mail:[email protected] \ \ \ \ \ \ \ \ \ [email protected] \end{document}

9505

alg-geom/9505019

en

https://arxiv.org/abs/alg-geom/9505019

alg-geom/9505019

Carmen Schuhmann

Carmen Schuhmann

Mapping threefolds onto three-quadrics

13 pages, LaTeX v. 2.09

We prove that the degree of a nonconstant morphism from a smooth projective 3-fold $X$ with N\'{e}ron-Severi group ${\bf Z}$ to a smooth 3-dimensional quadric is bounded in terms of numerical invariants of $X$. In the special case where $X$ is a 3-dimensional cubic we show that there are no such morphisms. The main tool in the proof is Miyaoka's bound on the number of double points of a surface.

alg-geom

\section{Proof of Theorem 1} The proof of Theorem \ref{stelling} is based upon the following result of Miyaoka (see \cite{Mi}): \begin{theorem} \label{Miyaoka} Let $S$ be a complex projective surface with only ordinary double points and numerically effective dualizing sheaf $K_S$. Let $\tilde{S}$ be the minimal resolution of $S$. Then $$\#\mbox{\{double points\}} \leq \frac{2}{3}(c_2(\tilde{S}) - \frac{1}{3} K_S^2).$$ \end{theorem} Another important ingredient of the proof is the following lemma, which will be proven later on. Here the tangent hyperplane to a smooth quadric $Q$ of dimension 3 at a point $p \in Q$ is denoted by $T_pQ$. \begin{lemma} \label{lemma} Let $X$ be a smooth, projective variety of dimension 3 and $f$ a finite morphism from $X$ to $Q$. Then there is a dense open subset $U$ in $Q$ such that $f^*(T_pQ \cap Q)$ has no singularities away from $f^{-1}(p)$ for all $p\in U$. \end{lemma} Before starting the proof of Theorem \ref{stelling}, let us state the following lemma, which has an elementary proof. \begin{lemma} \label{NS} Let $X$, $Y$ be smooth projective threefolds and $f:X \rightarrow Y$ a morphism. Assume that $X$ has N\'{e}ron-Severi group ${\bf Z}$. Then the following two statements are equivalent: $i$) $f$ is nonconstant; $ii$) $f$ is finite. \end{lemma} {\it Proof of Theorem \ref{stelling}:} Suppose $f:X \rightarrow Q$ is a nonconstant morphism of generator degree d. By Lemma \ref{NS}, $f$ is finite. The degree of $f$ is equal to $H_X^3d^3/2$, where $H_X$ is the ample generator of the N\'{e}ron-Severi group of $X$. For every point $p$ on $Q$, the hyperplane section $H_p=T_pQ \cap Q$ is a quadric with one ordinary double point, namely $p$. So, if $p$ is not contained in the branch divisor of $f$, then the surface $f^*(H_p)$ contains $H_X^3d^3/2$ ordinary double points, which map to $p$ under $f$. {}From Lemma \ref{lemma} it follows that $f^*(H_p)$ has no other singularities for general $p$. Now fix a point $p$ such that $f^*(H_p)$ has exactly $H_X^3d^3/2$ ordinary double points and no other singularities and denote $f^*(H_p)$ by $S_p$. It will be shown that, if $d$ is large enough, Theorem \ref{Miyaoka} can be applied to $S_p$ and provides an upper bound on the number of ordinary double points of $S_p$ which is smaller than $H_X^3d^3/2$. In order to compute this bound, we have to compute $c_2(\tilde{S_p})$ and $K_{S_p}^2$, where $\tilde{S_p}$ is the minimal resolution of $S_p$. By Bertini's Theorem (see \cite{Jo}, Th\'{e}or\`{e}me 6.10), there is a hyperplane section $H$ of $Q$ such that the surface $f^*(H)$ is nonsingular. Denote this surface by $S$. The surfaces $S$ and $\tilde{S_p}$ are homeomorphic (see \cite{A}, Theorem 3), so $c_2(\tilde{S_p})=c_2(S)$ and $c_1^2(\tilde{S_p})=c_1^2(S)$. As $S_p$ has only ordinary double points, it follows that $K_{S_p}^2=K_{S}^2$. Using $K_X \equiv kH_X$, where k is the numerical index of $X$, the adjunction formula gives: $$K_{S} \cong (K_X + S) | _{S} \equiv (k+d)H_X | _{S},$$ so $$ K_{S}^2 = (k+d)^2H_X^2 | _{S} = (k+d)^2dH_X^3. $$ The second Chernclass of $S$ can easily be computed by means of adjunction: $$ c_2(S)=dc_2(X)H_X+d^2(d+k)H_X^3.$$ Thus the expression $\frac{2}{3}(c_2(\tilde{S_p}) - \frac{1}{3} K_{S_p}^2)$, which equals $\frac{2}{3}(c_2(S) - \frac{1}{3}K_S^2)$ by previous remarks, becomes the following polynomial expression in $d$: \begin{equation} \frac{4}{9}H_X^3d^3 + \frac{2}{9}kH_X^3d^2 + \frac{2}{3}(c_2(X)H_X - \frac{1}{3}k^2H_X^3)d. \label{eq:bound} \end{equation} We can apply Theorem \ref{Miyaoka} to $S_p$ if $S_p$ has only double points and $K_{S_p}$ is nef. As we remarked before, the first condition is certainly satisfied because of Lemma \ref{lemma}. As for the second one, using the adjunction formula, it follows that $K_{S_p}$ is linearly equivalent to $(K_{X}+S_p) |_{S_p}$. So $K_{S_p}$ is nef if and only if $(K_{X}+S_p) |_{S_p}$ is nef. As $(K_{X}+S_p) |_{S_p} \equiv (k+d)H_X |_{S_p}$, this is certainly true if $d \geq -k$. Notice that this condition is empty if $k \geq -1$. If $k=-2$, then the condition becomes $d \geq2$, which is also an empty condition as there are no morphisms of degree 1 between a variety of numerical index -2 and the quadric $Q$, which has numerical index -3. The only smooth threefolds with numerical index less than -2 are ${\bf P}^3$, which has numerical index -4 and $Q$ (see \cite{K-O}). So, the only cases in which we cannot apply Theorem \ref{Miyaoka} to $S_p$ are when $X$ is ${\bf P}^3$ and $d \leq 3$ or $X$ is $Q$ and $d \leq 2$. In the other cases, it tells us that the number of ordinary double points on $S_p$ is restricted by the expression (\ref{eq:bound}). However, we remarked that $S_p$ contains exactly $H_X^3d^3/2$ double points. As the leading term of (\ref{eq:bound}), $4H_X^3d^3/9$, is smaller than $H_X^3d^3/2$, the expression (\ref{eq:bound}) becomes smaller than $H_X^3d^3/2$ for large $d$. So we obtain a contradiction if $d$ is larger than the largest positive zero of the polynomial $H_X^3d^3/2-$(\ref{eq:bound}) (if $X$ is ${\bf P}^3$ resp. $Q$, then we moreover have to require $d \geq 4$ resp. $d \geq 3$). We conclude that the generator degree $d$ of $f$ and thus also the degree of $f$ is bounded in terms of the coefficients of this polynomial. Since $H_X=-c_1(X)/k$, the statement of the theorem follows.\\ {\it Proof of Lemma \ref{lemma}:} Denote the 4-dimensional dual projective space by ${{\bf P}^4} ^{\vee}$. Let ${Q}^{\vee}$ be the dual variety of $Q$. It is isomorphic to $Q$ via the following isomorphism: \[ \begin{array}{rcl} Q & \longrightarrow & {Q}^{\vee} \\ p & \longmapsto & T_p Q. \end{array} \] Given $T_p Q \in {Q}^{\vee}$, denote the surface $f^*(T_p Q \cap Q) \subset Y$ by $S_p$. We will study the singularities of $S_p$, using the following criterion: $$ {}~~~~~~~~~~~~~~~~~~~ x \in S_p \mbox{ is a singularity of } S_p \Leftrightarrow T_p Q \supset f_* T_x X .~~~~~~~~~~~~~~~~~~~~~~(*) $$ In order to describe $f_* T_x X$, we introduce some notation. For $i \in \{0, 1,2,3\}$, the set $$ X_i := \{x \in X \mid f \mbox{ has rank at most } i \mbox{ at }x\} $$ is an algebraic subset of $X$ of dimension $i$. So its image under the finite map $f$ is an algebraic subset of $Q$ of dimension $i$. Denote the surface $f(X_2)$ by $B$ (this is just the branch locus of $f$), the curve $f(X_1)$ by $C$ and the finite set $f(X_0)$ by $R$. Furthermore, denote the union of all irreducible curves along which $B$ is singular by $\Gamma$ and the set of isolated singularities of $B$ by $\Sigma$. Finally, the singular locus of the curve $C$ respectively $\Gamma$ is denoted by $Sing(C)$ respectively $Sing(\Gamma)$. Let us now examine when a point $x \in X$ is a singularity of $S_p$. If $x \in X_i \backslash X_{i+1}~(i \in \{1,2,3\})$ and $f(x_i)$ is a smooth point of $f(X_i)$, then $f_*T_xX=T_{f(x)}f(X_i)$. So, according to criterion $(*)$, $x$ is a singularity of $S_p$ if and only if $T_pQ$ contains $T_{f(x)}X_i$, in other words if and only if $T_pQ$ is tangent to $f(X_i)$ at the point $f(x)$. Especially, taking $i=3$, it follows that $x \in X_3 \backslash X_2$ is a singularity of $S_p$ if and only if $f(x)=p$. If $x \in X_2 \backslash X_1$ and $f(x) \in \Gamma \backslash Sing(\Gamma)$, then $f_*T_xX$ contains $T_{f(x)}\Gamma$. So in this case we see from $(*)$ that if $x$ is a singularity of $S_p$ then $T_pQ$ contains $T_{f(x)}\Gamma$, which means that $T_pQ$ is tangent to $\Gamma$ at $f(x)$. Finally, if $x \in X_0$, then $f_*T_xX=0$, so by $(*)$ the point $x$ is certainly a singularity of $S_p$. Combining these observations, it follows that $S_p$ has no singularities away from $f^{-1}(p)$ if the hyperplane $T_pQ$ lies in the intersection of the following 4 subsets of ${Q}^{\vee}$: \[ \begin{array}{lll} U_1 & := & \{H \in {Q}^{\vee} \mid H \cap (R \cup Sing(C) \cup Sing(\Gamma) \cup \Sigma) = \emptyset \}, \\ U_2 & := & \{H \in {Q}^{\vee} \mid H \mbox{ intersects } C \mbox{ transversally} \}, \\ U_3 & := & \{H \in {Q}^{\vee} \mid H \mbox{ intersects } \Gamma \mbox{ transversally} \}, \\ U_4 & := & \{H \in {Q}^{\vee} \mid H \cap B \backslash (\Gamma \cup \Sigma) \mbox{ is nonsingular} \}. \end{array} \] Thus, in order to prove the lemma, it is sufficient to show that the sets $U_i$ ($i \in \{1,2,3,4\}$) are Zariski-open in ${Q}^{\vee}$ and nonempty. As $R \cup Sing(C) \cup Sing(\Gamma) \cup \Sigma$ is a finite set, this is certainly true for $U_1$. As for $U_2$, notice that in order to show that a general hyperplane $H \in {Q}^{\vee}$ intersects $C$ transversally, it is sufficient to show that this is true for every irreducible component of $C$. So we can without loss of generality assume that $C$ is irreducible. Let $Z$ denote the following closed subscheme of $Q \times {Q}^{\vee}$: $$ Q \times {Q}^{\vee} \supset Z := \{(x,H) \in Q \times {Q}^{\vee} \mid x \in H \}. $$ Consider the scheme-theoretic intersection $(C \times {Q}^{\vee}) \cap Z$ and the restriction of the projections from $Q \times {Q}^{\vee}$ onto $Q$ respectively ${Q}^{\vee}$ to this subscheme of $Q \times {Q}^{\vee}$: \[ \begin{array}{l} q: (C \times {Q}^{\vee}) \cap Z \longrightarrow C,\\ r: (C \times {Q}^{\vee}) \cap Z \longrightarrow {Q}^{\vee}. \end{array} \] The fiber of $r$ over $H \in {Q}^{\vee}$ is the scheme-theoretic intersection of $C$ and $H$. So in order to show that $U_2$ is open and dense in ${Q}^{\vee}$, we have to prove that the general fiber of $r$ is reduced. Notice that all fibers of $q$ are singular quadric surfaces, so they have constant Hilbertpolynomial. It follows that $q$ is flat (see \cite{Ha}, Chapter III, Theorem 9.9). As all fibers of $q$ are reduced and $q$ itself is flat, we conclude that $U := q^{-1}(C \setminus Sing(C))$ is reduced. Now $U$ is a dense open subscheme of $(C \times {Q}^{\vee}) \cap Z$, so the general fiber of $r$, which is finite, is contained in $U$. Restricting to those fibers which are contained in the smooth part of $U$ and have empty intersection with the ramification locus of $r$, we see that the general fiber of $r$ is reduced. It follows that $U_2$ is Zariski-open in ${Q}^{\vee}$ and nonempty. Replacing $C$ by $\Gamma$ and repeating the above reasoning proves that $U_3$ is also Zariski-open in ${Q}^{\vee}$ and nonempty. In order to prove that $U_4$ is open and dense in ${Q}^{\vee}$, we will show that, for a general element $H \in {Q}^{\vee}$, the intersection of $H$ and $B$ is smooth away from the singular locus $\Gamma \cup \Sigma $ of $B$. As it is sufficient to show that this holds for every irreducible component of $B$, we may without loss of generality assume that $B$ is irreducible. Assume first that $B$ is a hyperplane section of $Q$, corresponding to an element $H_B$ in ${{\bf P}^4} ^{\vee}$. Then the hyperplane sections of $B$ correspond to the lines in ${{\bf P}^4} ^{\vee}$ through $H_B$. If $H_B$ is not contained in ${Q}^{\vee}$, then every line through $H_B$ intersects ${Q}^{\vee}$ in 2 (not necessarily distinct) points. So every hyperplane section of $B$ can be written as the intersection of $B$ and an element of ${Q}^{\vee}$. If $H_B$ is an element of ${Q}^{\vee}$, then every line through $H_B$ which is not contained in the tangent space to ${Q}^{\vee}$ at $H_B$ intersects ${Q}^{\vee}$ in exactly one point different from $H_B$. So in this case the elements of a dense open subset of the space of hyperplane sections of $B$ can uniquely be written as the intersection of $B$ and an element of ${Q}^{\vee}$. In both cases we conclude from Bertini's Theorem (see \cite{G-H}, page 137), applied to $B$, that the intersection of $B$ and a general element of ${Q}^{\vee}$ is smooth away from $\Gamma \cup \Sigma $. So $U_4$ is Zariski-open in ${Q}^{\vee}$ and nonempty. {}From now on, assume that $B$ is not a hyperplane section of $Q$ and interprete ${Q}^{\vee}$ as a quadric system contained in the space ${\bf P}(H^0(B,{\cal O}_B(1)))$ of all hyperplane sections of $B$. In order to prove that the intersection of $H$ and $B$ is smooth away from $\Gamma \cup \Sigma $ for general $H$ in ${Q}^{\vee}$, we will use the following special case of Bertini's Theorem (see \cite{G-H}, page 137). Here a one-dimensional linear system is called a pencil. \begin{lemma} \label{Bertini} Let $X \subset {\bf P}^N$ be a projective variety and $\Lambda \subset {\bf P}( H^0(X,{\cal O}_X(1)))$ a pencil. Then the general element of $\Lambda$ is smooth away from the base locus of $\Lambda$ and the singular locus of $X$. \end{lemma} We will show that ${Q}^{\vee}$ contains enough pencils to globalise Lemma \ref{Bertini}, which holds for any of these pencils, to a Bertini type theorem for the whole space ${Q}^{\vee}$. For every element $H$ of ${Q}^{\vee}$ there is a one-dimensional family of pencils in ${Q}^{\vee}$ containing $H$, parametrised by a plane quadric curve. By Lemma \ref{Bertini}, the general element of a pencil is smooth away from the base locus of the pencil and $\Gamma \cup \Sigma $. We will call such an element general for that pencil. Denote the base locus in $B$ of a pencil $\Lambda$ in ${Q}^{\vee}$ by $B_{\Lambda}$. Denote the isomorphism from $Q$ to ${Q}^{\vee}$, mapping $p \in Q$ to the hyperplane $T_p Q$, by $T$. An easy computation shows that \begin{equation} B_{\Lambda} = T^{-1}(\Lambda) \cap B. \label{eq:base locus} \end{equation} By a dimension argument, there is a Zariski-open subset $V$ of ${Q}^{\vee}$ such that every element $H$ of $V$ is general for almost all pencils containing $H$. We will show that $V \backslash (V \cap T(B))$ is contained in $U_4$. Let $T_p Q$ be an element of $V$. By (\ref{eq:base locus}) the intersection $B_{\Lambda} \cap B_{\Lambda'}$ of the base loci of any 2 pencils $\Lambda$ and $\Lambda'$ in ${Q}^{\vee}$ is empty if $p \not \in B$ and equals $p$ if $p \in B$. As $T_p Q$ is an element of $V$, it follows that the hyperplane section $T_p Q \cap B$ is smooth away from $\Gamma \cup \Sigma $ if $p \not \in B$. So the nonempty Zariski-open subset $V \backslash (V \cap T(B))$ of ${Q}^{\vee}$ is contained in $U_4$. We conclude that $U_4$ is open and dense in ${Q}^{\vee}$. \begin{remark} {\rm Notice that, if $X$ is ${\bf P}^3$ respectively a smooth threedimensional quadric $Q$, then the statement of Theorem \ref{stelling} follows also from Theorem \ref{Lazarsfeld} respectively the following theorem (see \cite{P-S}): \begin{theorem} \label{Pasri} Let $Y$ be a smooth quadric hypersurface of dimension at least 3, $X$ a smooth projective variety and $f:Y \rightarrow X$ a surjective morphism. Then either $f$ is an isomorphism or $X$ is isomorphic to a projective space. \end{theorem} Conversely, we conclude from Theorem \ref{stelling} that the degree of a nonconstant morphism from $Q$ to itself is bounded. So every such morphism must have degree 1, as otherwise we could produce nonconstant selfmaps of $Q$ of arbitrarily high degree by composition. More generally, let $X$ be a smooth threefold with N\'{e}ron-Severi group ${\bf Z}$. From Theorem \ref{stelling} it follows that, if $X$ does allow some nonconstant morphism to the quadric (for example if $X$ is a Fermat hypersurface of even degree in ${\bf P}^4$), then every nonconstant morphism from $X$ to itself has degree 1, so is an isomorphism.} \end{remark} \begin{remark}{\rm Replacing $Q$ by another smooth threefold with N\'{e}ron-Severi group ${\bf Z}$, I did not manage to prove an analogue of Theorem \ref{stelling}. For instance, let $Y$ be a smooth complete intersection of $N-3$ hyperplanes of degrees $m_1, \dots,m_{N-3}$ in ${\bf P}^N$. Let $X$ be as in Theorem \ref{stelling} and $f:X \rightarrow Y$ a finite morphism of generator degree $d$. Then the degree of $f$ equals $H_X^3d^3/(\prod_{i=1}^{N-3} m_i)$. So, if $H$ is a hyperplane section of $Y$ with $n$ ordinary double points, none of which lie in the branch locus of $f$, then $f^*(H)$ has $nH_X^3d^3/(\prod_{i=1}^{N-3} m_i)$ ordinary double points. As the leading term of the Miyaoka bound for $f^*(H)$ in Theorem \ref{Miyaoka} is equal to $\frac{4}{9}H_X^3$, it is clear that the idea of the proof of Theorem \ref{stelling} can only work in this case if \begin{equation} n \geq \frac{4}{9}\prod_{i=1}^{N-3} m_i. \label{eq:ci} \end{equation} The point is that, in order to apply Theorem \ref{Miyaoka}, it still has to be checked that for at least one such hyperplane section $H$ the surface $f^*(H)$ has only ordinary double points. If $Y$ is a cubic in ${\bf P}^4$ or the intersection of two quadrics in ${\bf P}^5$, it follows from (\ref{eq:ci}) that we have to study hyperplane sections of $Y$ with at least 2 ordinary double points. In both cases I did not succeed in proving an analogue of Lemma \ref{lemma}. If $Y$ is a cubic, the system of hyperplane sections with 2 ordinary double points has only dimension 2 and I couldn't apply any Bertini type argument; if $Y$ is an intersection of 2 quadrics, this system has dimension 3, but I could not get any grip on it. Another possibility is to try to prove something weaker than Lemma \ref{lemma}. If one can prove that $f^*(H)$ has only mild singularities apart from the ordinary double points lying over the double points of $H$, then one may try to apply Theorem \ref{Miyaoka} after blowing up these singularities. Of course this is only possible if one knows these singularities well. If one can prove that $f^*(H)$ contains only quotient singularities, then one may try to apply a more general version of Theorem \ref{Miyaoka}, valid for surfaces with only quotient singularities (see \cite{Mi}). However, these approaches seem quite hard and I did not try them seriously up to now.} \end{remark} \section{Proof of Theorem 2} Look at the special case where $X$ is a smooth hypersurface of degree $m$ in ${\bf P}^4$ and $f:X \rightarrow Q$ a nonconstant morphism of generator degree $d$. As $X$ has N\'{e}ron-Severi group ${\bf Z}$, we can apply Theorem \ref{stelling} which gives that the generator degree of $f$ is bounded. To estimate this bound, consider expression (\ref{eq:bound}). As \[ \begin{array}{lll} H_X^3 & = & m, \\ H_Xc_2(X) & = & m^3 - 5m^2 + 10m \end{array} \] and the index of $X$ is equal to $-5+m$, this expression becomes: \begin{equation} \frac{4}{9}md^3+(\frac{2}{9}m^2-\frac{10}{9}m)d^2+ (\frac{4}{9}m^3-\frac{10}{9}m^2+\frac{10}{9}m)d. \label{eq:boun} \end{equation} The degree of $f$ is equal to $md^3/2$. Thus, in this case, the upper bound on the generator degree of $f$ we obtained in the proof of Theorem \ref{stelling} is given by the maximal positive integer $d$ for which (\ref{eq:boun}) $ \geq md^3/2$. Denote this integer by $d_m$. A calculation shows that, for $m \gg 0$, the bound $d_m$ we obtain in this way on the generator degree of $f$ grows approximately linearly with $m$: $d_m \sim (2+2\surd 3)m+constant$. However, the generator degree of $f$ is also bounded from below. Choose coordinates $(x_0: \dots :x_4)$ on the projective space ${\bf P}^4$ containing $X$ and coordinates $(y_0: \dots :y_4)$ on the projective space ${\bf P}^4$ containing $Q$ such that $Q$ is given by the equation $\sum_{i=0}^{4} y_i^2 =0$. Then $f$ is given by \[ \begin{array}{rcl} f:X & \longrightarrow & Q, \\ (x_0: \dots :x_4) & \longmapsto & ({\phi}_0 (x_0: \dots :x_4): \dots : {\phi}_4 (x_0: \dots :x_4)) \end{array} \] where the ${\phi}_i$ are homogeneous polynomials of degree $d$, defined on $X$. As the natural map $H^0({\bf P}^4, {\cal O}_{{\bf P}^4}(d)) \rightarrow H^0(X, {\cal O}_X(d))$ is surjective, these polynomials can be extended to polynomials of degree $d$, defined on ${\bf P}^4$ (but not necessarily in a unique way). The extensions will also be denoted by ${\phi}_i$. Let $X \subset {\bf P}^4$ be given by the equation $F_X=0$, where $F_X$ is a homogeneous polynomial of degree $m$. As $\sum_{i=0}^{4} {\phi}_i^2 =0$ on $X$, it follows that \begin{equation} \sum_{i=0}^{4} {\phi}_i^2 = F_XG, \label{eq:vgl} \end{equation} where $G$ is a homogeneous polynomial of degree $2d-m$. Thus, the generator degree of $f$ must be larger than or equal to $m/2$. In fact, if $m$ is even, then there exist hypersurfaces $X$ of degree $m$ and morphisms of generator degree $m/2$ from $X$ to $Q$, for instance if $X$ is the Fermat hypersurface of degree $m$ in ${\bf P}^4$. From (\ref{eq:vgl}) it follows that, if $m$ is even, there is a morphism of generator degree $m/2$ from $X$ to $Q$ if and only if $F_X$ can be written as the sum of 5 squares of homogeneous polynomials of degree $m/2$, having no common zeroes on $X$. Now consider the case $m=3$, where $X$ is a smooth cubic. In order to prove Theorem \ref{kubiek}, we will use the following theorem of Lazarsfeld (see \cite{La}): \begin{theorem} \label{Lazarsfeld} Let $X$ be a smooth projective variety of dimension at least 1 and let $f:{\bf P}^n \rightarrow X$ be a surjective morphism. Then $X \cong {\bf P}^n$. \end{theorem} {\it Proof of Theorem \ref{kubiek}}: Let $f:X \rightarrow Q$ be a morphism of generator degree $d$. A computation shows that the upper bound $d_3$ on the generator degree of $f$, which was introduced above, is equal to 3. Morphisms of generator degree 3 cannot occur as the expression $3d^3/2$ which should be equal to the degree of such a morphism is not integer for $d=3$. So, all that remains to be proven is that there are no morphisms of generator degree 2 between cubics and quadrics. Assume $f$ has generator degree 2. As above, choose coordinates $(x_0: \dots :x_4)$ and $(y_0: \dots :y_4)$ on ${\bf P}^4$, and let ${\phi}_0, \dots, {\phi}_4$ be homogeneous polynomials of degree $2$, defining $f$. In this case we get, as in (\ref{eq:vgl}): \begin{equation} \sum_{i=0}^{4} {\phi}_i^2 = F_XL, \label{eq:kvgl} \end{equation} where $L$ is a homogeneous linear polynomial, defining a hyperplane in ${\bf P}^4$. This hyperplane will also be denoted by $L$, for convenience. We claim that the ${\phi}_i$ do not have any common zeroes on the hyperplane $L$. As the ${\phi}_i$ do not have any common zeroes on $X$, the claim follows for points in $X \cap L$. Now let $p$ be a point in $L \backslash (X \cap L)$. If ${\phi}_i(p)=0$ for all $i \in \{0, \dots,4 \}$, equation (\ref{eq:kvgl}) implies that $$ \frac{\partial F_XL}{\partial x_i}(p)=0,\mbox{ for all } i \in \{0, \dots,4 \}. $$ As $L(p)=0$ and $F_X(p)\not = 0$ by assumption, we get: $$ \frac{\partial L}{\partial x_i}(p)=0 \mbox{ for all } i \in \{0, \dots,4 \}. $$ But this is impossible because $L$, being a hyperplane, is nonsingular. This proves the claim. Thus, the ${\phi}_i$ define a morphism from $L$ to $Q$. Restricted to the surface $L \cap X$, this morphism equals $f | _{L \cap X}$, so it is not constant. This is a contradiction by Theorem \ref{Lazarsfeld}. \begin{remark} {\rm Notice that the argument in the proof of Theorem \ref{kubiek} works for every hypersurface $X$ of degree $m$ in ${\bf P}^4$ with a morphism $f:X \rightarrow Q$ of generator degree $d$ such that $2d-m=1$. So, if $m$ is odd, there are no morphisms of generator degree $(m+1)/2$ from $X$ to $Q$ (if $m \equiv 1mod4$, this is also clear from the fact that the expression $md^3/2$ which should be equal to the degree of such a morphism is not integer in this case).} \end{remark}

9505

alg-geom/9505024

en

https://arxiv.org/abs/alg-geom/9505024

alg-geom/9505024

Subhashis Nag

Indranil Biswas, Subhashis Nag, and Dennis Sullivan

Determinant Bundles, Quillen Metrics, and Mumford Isomorphisms Over the Universal Commensurability Teichm\"uller Space

ACTA MATHEMATICA (to appear); finalised version with a note of clarification regarding the connection of the commensurability modular group with the virtual automorphism group of the fundamental group of a closed Riemann surface; 25 pages. LATEX

IHES/M/95/43 (Paris)

There exists on each Teichm\"uller space $T_g$ (comprising compact Riemann surfaces of genus $g$), a natural sequence of determinant (of cohomology) line bundles, $DET_n$, related to each other via certain ``Mumford isomorphisms''. There is a remarkable connection, (Belavin-Knizhnik), between the Mumford isomorphisms and the existence of the Polyakov string measure on the Teichm\"uller space. This suggests the question of finding a genus-independent formulation of these bundles and their isomorphisms. In this paper we combine a Grothendieck-Riemann-Roch lemma with a new concept of $C^{*} \otimes Q$ bundles to construct such an universal version. Our universal objects exist over the universal space, $T_\infty$, which is the direct limit of the $T_g$ as the genus varies over the tower of all unbranched coverings of any base surface. The bundles and the connecting isomorphisms are equivariant with respect to the natural action of the universal commensurability modular group.

alg-geom

\section{Introduction} Let ${\cal T}_g$ denote the Teichm\"uller space comprising compact marked Riemann surfaces of genus $g$. Let $DET_n \longrightarrow {\cal T}_g$ be the determinant (of cohomology) line bundle on ${\cal T}_g$ arising from the $n$-th tensor power of the relative cotangent bundle on the universal family ${\cal C}_g$ over ${\cal T}_g$. The bundle $DET_0$ is called the Hodge line bundle. The bundle $DET_n$ is equipped with a hermitian structure which is obtained from the construction of Quillen of metrics on determinant bundles using the Poincar\'e metric on the relative tangent bundle of ${\cal C}_g$, [Q]. These natural line bundles over ${\cal T}_g$ carry liftings of the standard action of the mapping class group, $MC_g$, on ${\cal T}_g$. We shall think of them as $MC_g$-equivariant line bundles, and the isomorphisms we talk about will be $MC_g$-equivariant isomorphisms. By applying the Grothendieck-Riemann-Roch theorem, Mumford [Mum] had shown that $DET_n$ is a certain fixed (genus-independent) tensor power of the Hodge bundle. More precisely, $$ DET_{n} ~=~ DET^{\otimes (6n^2-6n+1)}_{0} \eqno{(1.1)} $$ this isomorphism of equivariant line bundles being ambiguous only up to multiplication by a non-zero scalar. (Any choice of such an isomorphism will be called a Mumford isomorphism in what follows.) There is a remarkable connection, discovered by Belavin and Knizhnik [BK], between the Mumford isomorphism above for the case $n=2$, [i.e., that $DET_{2}$ is the $13$-th tensor power of Hodge], and the existence of the Polyakov string measure on the moduli space ${\cal M}_g$. (See the discussion after Theorem 5.5 for more details.) This suggests the question of finding a genus-independent formulation of the Mumford isomorphisms over some ``universal'' parameter space of Riemann surfaces (of varying topology). In this paper we combine a Grothendieck-Riemann-Roch lemma (Theorem 2.9) with a new concept of ${{\CC}^{*}} \otimes {\QQ}$ bundles (Section 5), to construct a universal version of the determinant bundles and Mumford's isomorphism. Our objects exist over a universal base space ${\cal T}_{\infty} = {\cal T}_{\infty}(X)$, which is the infinite directed union of the complex manifolds that are the Teichm\"uller spaces of higher genus surfaces that are unbranched coverings of any (pointed) reference surface $X$. The bundles and the relating isomorphisms are equivariant with respect to the natural action of the universal commensurability group ${CM_{\infty}}$ -- which is defined (up to isomorphism) as the group of isotopy classes of unbranched self correspondences of the surface $X$ arising from pairs of non-isotopic pointed covering maps $X' {\rightarrow \atop \rightarrow } X$, (see below and in Section 5). In more detail, our universal objects are obtained by taking the direct limits using the following category ${\cal S}$: for each integer $g \geq 2$, there is one object in ${\cal S}$, an oriented closed pointed surface $X_g$ of genus $g$, and one morphism $X_{\tilde g} \rightarrow X_g$ for each based isotopy class of finite unbranched pointed covering map. For each morphism of ${\cal S}$ (say of degree $d$) we have an induced holomorphic injection of Teichm\"uller spaces arising from pullback of complex structure: $$ {\cal T}(\pi):~{\cal T}_g \rightarrow {\cal T}_{\tilde g} \eqno(1.2) $$ The GRR lemma provides a natural isomorphism of the line bundle $DET_{n,g}^{\otimes 12d}$ on ${\cal T}_g$ with the pullback line bundle ${{\cal T}(\pi)}^*DET_{n,{\tilde g}}^{\otimes 12}$. We may view this isomorphism, equivalently, as a degree $d$ homomorphism covering the injection ${\cal T}(\pi)$ between the principal $\C*$ bundles associated to $DET_{n,g}$ and $DET_{n,{\tilde g}}$, respectively. Each commutative triangle in ${\cal S}$ yields a commutative triangular prism whose top face is the following triangle of total spaces of principal $\C*$ bundles: $$ \matrix{ {DET}_{n,g}^{\otimes {12}} & &\mapright{} & &{DET}_{n,\tilde g}^{\otimes {12}} \cr & \searrow & &\swarrow & \cr && {DET}_{n,\bar g}^{\otimes {12}} && \cr} $$ and whose bottom face is the commuting triangle of base spaces for these bundles: $$ \matrix{ {\cal T}_g~~~~~ & &\mapright{} & &~~~~~{\cal T}_{\tilde g} \cr & \searrow & &\swarrow & \cr && {\cal T}_{\bar g} && \cr} \eqno(1.3) $$ Moreover, the canonical mappings above relating these $DET_n$ bundles over the various Teichm\"uller spaces preserve the Quillen hermitian structure of these bundles in the sense that unit circles are carried to unit circles. We explain in brief the commensurability Teichm\"uller space ${\cal T}_{\infty}$ and the large mapping class group ${CM_{\infty}}$ acting thereon. For each object $X$ in ${\cal S}$, consider the directed set $\{\alpha \}$ of all morphisms in ${\cal S}$ with range $X$. Then we form $$ {\cal T}_{\infty}(X) := dir. lim. {\cal T}_{g(\alpha )} \eqno(1.4) $$ where the limit is taken over $\{\alpha \}$, and $g(\alpha )$ is the genus for the domain of morphism $\alpha $. Each morphism $X_{g'} \rightarrow X_g$ induces a holomorphic {\it bijection} of the corresponding direct limits, and we denote any of these isomorphic ``ind-spaces'' (inductive limit of finite-dimensional complex spaces -- see [Sha]) by ${\cal T}_{\infty}$ -- the universal commensurability Teichm\"uller space. (Compare Section 2 and Example 4 on p. 547 of [S].) Notice that a pair of morphisms $X' {\rightarrow \atop \rightarrow } X$ determines an automorphism of ${\cal T}_{\infty}$; we call the group of automorphisms of ${\cal T}_{\infty}$ obtained this way the {\it commensurability modular group} ${CM_{\infty}}$. We now take the direct limit of the $\C*$ principal bundles associated to ${DET}_{n,g}^{\otimes 12}$ over ${\cal T}_g$ to obtain a new object -- a ${{\CC}^{*}} \otimes {\QQ}$ bundle over ${\cal T}_{\infty}$ -- denoted ${DET}(n,{\QQ})$. As sets the total space with action of the group ${{\CC}^{*}} \otimes {\QQ}$ is defined by the direct limit construction. Continuity and complex analyticity for maps into these sets are defined by the corresponding properties for factorings through the strata of the direct system. (Section 5.) There are the natural isomorphisms of Mumford, as stated in (1.1), at the finite-dimensional stratum levels. By our construction these isomorphisms are {\it rigidified} to be natural over the category ${\cal S}$. Therefore we have natural Mumford isomorphisms between the following ${{\CC}^{*}} \otimes {\QQ}$ bundles over the universal commensurability Teichm\"uller space ${\cal T}_{\infty}$: $$ {DET}(n,{\QQ})~~ {\rm and}~~ {{DET}(0,{\QQ})}^{\otimes (6n^2-6n+1)} \eqno(1.5) $$ We also show that the natural Quillen metrics of the $DET$ bundles fit together to define a natural analogue of Hermitian structure on these ${{\CC}^{*}} \otimes {\QQ}$ bundles; in fact, for all our canonical mappings in the direct system the unit circles are preserved. Note Theorem 5.5. In fact, the existence of the canonical relating morphism between determinant bundles (fixed $n$) in the fixed covering $\pi:X_{\tilde {g}} \rightarrow X_g$ situation was first conjectured and deduced by us utilizing the differential geometry of these Quillen metrics. Recall that the Teichm\"uller spaces ${\cal T}_g$ and ${\cal T}_{\tilde g}$ carry natural symplectic forms (defined using the Poincare metrics on the Riemann surfaces) -- the Weil-Petersson K\"ahler forms -- which are in fact the curvature forms of the natural Quillen metrics of these $DET$ bundles ([Wol], [ZT], [BGS]). If the covering $\pi$ is unbranched of degree $d$, a direct calculation shows that this natural WP form on ${\cal T}_{\tilde g}$ (appropriately renormalized by the degree) pulls back to the WP form of ${\cal T}_g$ by ${\cal T}(\pi)$ (the embedding of Teichm\"uller spaces induced by $\pi$). One expects therefore that if one raises the $DET_n$ bundle on ${\cal T}_g$ by the tensor power $d$, then it extends over the larger Teichm\"uller space ${\cal T}_{\tilde g}$ as the $DET_n$ bundle thereon. This intuition is, of course, what is fundamentally behind our direct limit constructions. Since it turns out to be technically somewhat difficult to actually prove that the relevant bundles are isomorphic using this differential geometric method, we have separated that aspect of our work into a different article [BN]. Can objects on ${\cal T}_{\infty}$ that are equivariant by the commensurability modular group ${CM_{\infty}}$ be viewed as objects on the quotient ${{\cal T}_{\infty}}/{{CM_{\infty}}}$ ? This quotient is problematical and interesting, so we work with the equivariant statement. We end the Introduction by mentioning some problems. The universal commensurability Teichm\"uller space, ${\cal T}_{\infty}$, is made up from embeddings ${\cal T}(\pi)$ that are isometric with respect to the natural Teichm\"uller metrics, so it carries a natural Teichm\"uller metric. Our theorems give us genus independent determinant line bundles ${DET}(n,{\QQ})$, Quillen metrics and Mumford isomorphisms over ${\cal T}_{\infty}$, all compatible with each other and the commensurability group ${CM_{\infty}}$. {\it Are the above structures uniformly continuous for this metric?} Then they would pass to the completion $\widetilde{T}_{\infty}$ of ${\cal T}_{\infty}$ for the Teichm\"uller metric. One knows that $\widetilde{T}_{\infty}$ is a separable complex Banach manifold which is the Teichm\"uller space of complex structures on the universal solenoidal surface $H_{\infty}=\limproj {\tilde X}$, where ${\tilde X}$ ranges, as above, over all finite covering surfaces of $X$. (See [S], [NS], for the Teichm\"uller theory of $H_{\infty}$.) We would conjecture that this continuity is true and that the ${{\CC}^{*}} \otimes {\QQ}$ bundles ${DET}(n,{\QQ})$, Quillen metrics, and Mumford isomorphisms can be defined over ${\cal T}(H_{\infty})=\widetilde{T}_{\infty}$ directly by looking at compact solenoidal Riemann surfaces themselves. Now we may consider square integrable holomorphic forms on the leaves of ${H}_{\infty}$ (which are uniformly distributed copies of the hyperbolic plane in ${H}_{\infty}$) regarded as modules over the $\CC*$ algebra of ${H}_{\infty}$ with chosen transversal. The measure of this transversal would become a real parameter extending the genus above. One expects that A. Connes' version of Grothendieck-Riemann-Roch would replace Deligne's functorial version which we are using here. Finally one would hope that the Polyakov measure (Section 5) on Teichm\"uller space, when viewed as a metric on the canonical bundle, would also make sense at infinity in the direct limit because this measure can be constructed by applying the $13$-th power Mumford isomorphism ((1.1) for $n=2$) to the $L^2$ inner product on the Hodge line bundle. That issue remains open. \noindent {\it Acknowledgements:} We would like express our gratitude to several mathematicians for their interest and discussions. In particular, we thank M.S.Narasimhan and E.Looijenga for helpful discussions in early stages of this work. Laurent Moret-Bailly deserves a very special place in our acknowledgements. In fact, he brought to our attention the Deligne pairing version of the Grothendieck-Riemann-Roch theorem that we use crucially here, and showed us Lemma 2.9, after seeing an earlier (less strong version) of the main theorem based on topology and the curvature calculations mentioned above. \section{A lemma on determinant bundles} Let $X$ be a compact Riemann surface, equivalently, an irreducible smooth projective curve over $\CC$. Let $L$ be a holomorphic line bundle on $X$. The determinant of $L$ is the defined to be the $1$-dimensional complex vector space $(\ext{top}H^0(X,L))\bigotimes (\ext{top}H^1(X,L)^*)$, and will be denoted by $det(L)$. Take a Riemannian metric $g$ on $X$ compatible with the conformal structure of $X$. Fix a hermitian metric $h$ on $L$. Using $g$ and $h$, a hermitian structure can be constructed on ${\Omega }^i(X,L)$, the space of $i$-forms on $X$ with values in $L$. Moreover the vector space $H^1(X,L)$ is isomorphic, in a natural way, with the space of harmonic $1$-forms with values in $L$. Consequently the vector spaces $H^0(X,L)$ and $H^1(X,L)$ are equipped with hermitian structures which in turn induce a hermitian structure on $det(L)$ -- this metric on $det(L)$ is usually called the $L^2$ metric. Let $\Delta := {\overline\partial}^*{\overline\partial}$ be the laplacian acting on the space of smooth sections of $L$. Let $\{{\lambda }_i\}_{i\geq 1}$ be the set of non-zero eigenvalues of $\Delta$; let $\zeta$ denote the analytic continuation of the function $s \longmapsto \sum_{i}1/{\lambda }^s_i$. The {\it Quillen metric} on $det(L)$ is defined to be the hermitian structure on $det(L)$ obtained by multiplying the $L^2$ metric with ${\rm exp}(-{\zeta}'(0))$, [Q]. To better suit our purposes, we will modify the above (usual) definition of the Quillen metric by a certain factor. Consider the real number $a(X)$ appearing in Th\'eor\`eme 11.4 of [D]. This number $a(X)$ depends only on the genus of $X$. The statement in Remark 11.5 of [D] -- to the effect that there is a constant $c$ such that $a(X) = c.\chi (X)$, where $\chi (X)$ is the Euler characteristic of $X$ -- has been established in [GS]. (The constant $c$ is related to the derivative at $(-1)$ of the zeta function for the trivial hermitian line bundle on $\CC P^1$ (4.1.7 of [GS]).) Let $H_Q(L; g,h)$ denote the Quillen metric on $det(L)$ defined above. Henceforth, by {\it Quillen metric} on $det(L)$ we will mean the hermitian metric $$ {\rm exp}(a(X)/12) H_Q(L; g, h) \eqno{(2.1)} $$ Next we will describe briefly some key properties of the determinant line and the Quillen metric. Let $\pi:{\cal X} \longrightarrow S$ be a family of compact Riemann surfaces parametrized by a base $S$. We can work with either holomorphic (Kodaira-Spencer) families over a complex-analytic variety $S$, or with algebraic families over complex algebraic varieties (or, more generally, over a scheme) $S$. In the algebraic category one means that $\pi$ is a proper smooth morphism of relative dimension one with geometrically connected fibers. In the analytic category, $\pi$ is a holomorphic submersion again with compact and connected fibers. Take a hermitian line bundle $L_S \longrightarrow {\cal X}$ with hermitian metric $h_S$. Fix a hermitian metric $g_S$ on the relative tangent bundle $T_{{\cal X}/S}$. For any point $s \in S$, the above construction gives a hermitian line $det(L_s)$ (the hermitian structure is given by the Quillen metric). The basic fact is that these lines fit together to give a line bundle on $S$ [KM], which is called the {\it determinant bundle} of $L_S$, and is denoted by $det(L_S)$. Moreover the function on the total space of $det(L_S)$ given by the norm with respect to the Quillen metric on each fiber is a $C^{\infty}$ function, and hence it induces a hermitian metric on $det(L_S)$ [Q]. This bundle will be denoted by $det(L_S)$. We shall make clear in Remark 2.13 below that this ``determinant of cohomology'' line bundle is also an algebraic or analytic bundle -- according to the category within which we work. The determinant bundle $det(L_S)$ is functorial with respect to base change. We describe what this means. For a morphism $\gamma : S' \longrightarrow S$ consider the bundle, $p^*_2L_S \longrightarrow S'\times_S{\cal X}$, on the fiber product, where $p_2 : S'\times_S{\cal X} \longrightarrow {\cal X}$ is the projection onto the second factor. The hermitian structure $h_S$ pulls back to a hermitian structure on $L_{S'} := p^*_2L_S$; and, similarly, the metric $g_S$ induces a hermitian structure on the relative tangent bundle of $S'\times_S{\cal X}$. ``Functorial with respect to base change'' now means that in the above situation there is a {\it canonical isometric isomorphism} $$ {\rho}_{S',S} ~:~ det(L_{S'})~\longrightarrow ~ {\gamma }^*det(L_S)$$ such that if $$S''~\mapright{\gamma '}~S'~\mapright{{\gamma }}~S$$ are two morphisms then the following diagram is commutative $$ \matrix{det(L_{S''})&\mapright{{\rho}_{S'',S'}}& {\gamma '}^*(det(L_{S'}))&\cr \mapdown{{\rho}_{S'',S}}&& \mapdown{{\gamma '}^*{\rho}_{S',S}}&\cr (\gamma \circ{\gamma }')^*det(L_S) &\mapright{id}& (\gamma \circ{\gamma }')^*det(L_S)&\cr} \eqno(2.2) $$ The determinant of cohomology construction $det(L_S)$ produces a bundle over the parameter space $S$ induced by the bundle over the total space ${\cal X}$; now, the Grothendieck-Riemann-Roch (GRR) theorem gives a canonical isomorphism of $det(L_S)$ with a combination of certain bundles obtained (on $S$) from the direct images of the bundle $L_S$ and the relative tangent bundle $T_{{\cal X}/S}$. In order to relate canonically the determinant bundle obtained from a given family ${\cal X} \rightarrow S$ (fibers of genus $g$, say) with the determinant arising from a covering family $\mbox{${\tilde{\cal X}}$}$ (having fibers of some higher genus ${\tilde g}$), we shall utilize the GRR theorem in a formulation due to Deligne, [D, Theorem 9.9(iii)]. In fact, Deligne introduces a ``bilinear pairing'' that associates a line bundle, denoted by $<L_S,M_S>$, over $S$ from any pair of line bundles $L_S$ and $M_S$ over the total space of the fibration ${\cal X} \rightarrow S$. If $L_S$ and $M_S$ carry hermitian metrics then a canonically determined hermitian structure gets induced on the Deligne pairing bundle $<L_S,M_S>$ as well. Denoting by ${\cal L} = L_S$ the given line bundle over ${\cal X}$, the GRR theorem in Deligne's formulation reads: $$ det({\cal L})^{\otimes 12}~=~ <T^*_{{\cal X} /S}, T^*_{{\cal X} /S}> \bigotimes <{{\cal L}} , {\cal L} {\otimes} T_{{\cal X} /S}>^{\otimes 6} \eqno(2.3) $$ Here $T^*_{{\cal X} /S}$ denotes the relative cotangent bundle over ${\cal X}$, and the equality asserts that there is a {\it canonical isomorphism, functorial with respect to base change,} between the bundles on the two sides. Furthermore, Th\'eor\`eme 11.4 of [D] says that the canonical identification in (2.3) is actually an {\it isometry} with the Quillen metric on the left side and the Deligne pairing metrics on the right. (The constant ${\rm exp}(a(X))$ in the statement of Th\'eor\`eme 11.4 of [D] has been absorbed in the definition (2.1).) We proceed to explain the Deligne pairing and the metric thereon in brief; details are to be found in sections 1.4 and 1.5 of [D]. Let $L$ and $M$ be two line bundles on a compact Riemann surface $X$. For a pair of meromorphic sections $l$ and $m$ of $L$ and $M$ respectively, with the divisor of $l$ being disjoint from the divisor of $m$, let $\CC <l,m>$ be the one dimensional vector space with the symbol $<l,m>$ as the generator. For two meromorphic functions $f$ and $g$ on $X$ such that $div(f)$ is disjoint from $div(m)$ and $div(g)$ is disjoint from $div(l)$, the following identifications of complex lines are to be made $$ \matrix{<fl,m> &= & f(div(m))<l,m>&\cr <l,gm>&=&g(div(l))<l,m>&} \eqno{(2.4)} $$ The Weil reciprocity law says that for any two meromorphic functions $f_1$ and $f_2$ on $X$ with disjoint divisors, $f_1(div(f_2)) = f_2(div(f_1))$ [GH, page 242]. So we have $$<fl,gm>~=~f(div(gm)).g(div(l))<l,m>~=~g(div(fl)).f(div(m))<l,m>.$$ {}From the above equality it follows that the identifications in (2.4) produce a complex one dimensional vector space, denoted by $<L,M>$, from the pair of line bundles $L$ and $M$. If $L$ and $M$ are both equipped with hermitian metrics then the hermitian metric on $\CC <l,m>$ defined by $$ log||<l,m>|| := {1\over{2{\pi}i}}\int_X\partial {\overline\partial}(log||l||.log||m||)~+\, log||l||(div(m)) \,+\,log||m||(div(l)) \eqno{(2.5)} $$ is compatible with the relations in (2.4) -- hence it gives a hermitian structure on $<L,M>$, see [D, 1.5.1]. Consider now a family of Riemann surfaces ${\cal X} \longrightarrow S$; let $L_S$ and $M_S$ be two line bundles on ${\cal X}$, equipped with hermitian structures. Over an open subset $U\subset S$, let $l_U$, $m_U$ be two meromorphic sections of $L_S$ and $M_S$ respectively, with finite supports over $U$ such that the support of $l_U$ is disjoint from the support of $m_U$. (Support of a section is the divisor of the section.) For another open set $V$ and two such sections $l_V$ and $m_V$, the relations in (2.4) give a function $$C_{U,V}~\in ~{{\cal O}}^*_{U\cap V}\,.$$ Using the Weil reciprocity law it can be shown that $\{C_{U,V}\}$ forms a 1-cocycle on $S$. In other words, we get a line bundle on $S$, which we will denote by $<L_S,M_S>$. The hermitian structure on $<L,M>$, described earlier, makes $<L_S,M_S>$ into a hermitian bundle. Given a meromorphic section $m$ of $M_S$, let $m^{\otimes n}$ be the meromorphic section of $M^n_S$ obtained by taking the $n$-th tensor power of $m$. Note that $div(m^{\otimes n}) = n.div(m)$. The map $<l,m^{\otimes n}> \longrightarrow <l,m>^{\otimes n}$ can be checked to be compatible with the relations (2.4) and hence it induces an isomorphism $$ <L_S,M^n_S> \longrightarrow <L_S,M_S>^n \eqno{(2.6)} $$ From the definition (2.5) we see that (2.6) is an isometry for the metric on $M^n_S$ induced by the metric on $M_S$. We shall now see how the critical formula (2.3) follows from the general GRR theorem of [D]. Indeed, let ${\cal L}$ denote any rank $n$ vector bundle on the total space of the family ${\cal X}$; we reproduce below the statement of Theorem 9.9(iii) of [D]: $$ det({\cal L} )^{\otimes 12}~=~ <T^*_{{\cal X} /S}, T^*_{{\cal X} /S}> \bigotimes <{\Lambda^{n}}({\cal L} ),{\Lambda^{n}}({\cal L} ) {\otimes} T_{{\cal X} /S}>^{\otimes 6} \bigotimes I_{{\cal X} /S}C^2({\cal L} )^{-12} \eqno{(GRR-D)} $$ Now, from the definition of $I_{{\cal X} /S}C^2$ in [D, 9.7.2] it follows that for a line bundle ${\cal L}$, the bundle $I_{{\cal X} /S}C^2({\cal L})$ is the trivial bundle on $S$, and the metric on it is the constant metric [D, Theorem 10.2(i)]. From Th\'eor\`eme 11.4 of [D] we conclude that that the canonical identification in the statement above is actually an isometric identification. (The factor ${\rm exp}(a(X))$ in Th\'eor\`eme 11.4 of [D] is taken care of by the definition (2.1).) Thus we have obtained the isometric isomorphism stated in (2.3). With this background behind us we can formulate our main lemma. Let ${\cal X}$ and $\mbox{${\tilde{\cal X}}$}$ be two families of Riemann surfaces over $S$ (say with fibers of genus $g$ and ${\tilde g}$, respectively), and $p: \mbox{${\tilde{\cal X}}$} \longrightarrow {\cal X}$ be an \'etale ({\it i.e.\/}\ unramified) covering of degree $d$, commuting with the projections onto $S$. In other words, the map $p$ fits into the following commutative diagram $$ \matrix{\mbox{${\tilde{\cal X}}$}& &\mapright{p}& & {\cal X} \cr &\searrow& & \swarrow & \cr & &S& & \cr} \eqno(2.7) $$ The situation implies that each fiber of the family $\mbox{${\tilde{\cal X}}$}$ is a degree $d=({\tilde g} - 1)/(g - 1)$ holomorphic covering over the corresponding fiber of the family ${\cal X}$. Fix also a hermitian metric $g$ on $T_{{\cal X} /S}$. Since $p$ is \'etale, $p^*T_{{\cal X} /S} = T_{\mbox{${\tilde{\cal X}}$} /S}$, and hence $g$ induces a hermitian metric $p^*g$ on $T_{{\mbox{${\tilde{\cal X}}$}}/S}$. Let ${{\cal X}}' \longrightarrow S$ be a third family of Riemann surface which is again an \'etale cover of $\mbox{${\tilde{\cal X}}$}$ and fits into the following commutative diagram $$ \matrix {{{\cal X}}' &\mapright{q} &\mbox{${\tilde{\cal X}}$} &\mapright{p}& {\cal X} \cr &\searrow& \mapdown{} &\swarrow & \cr & &S && \cr} \eqno(2.8) $$ We want to prove the following: \medskip \noindent{\bf Lemma 2.9.}\,\ {\it (i)~Let ${\cal L}$ be a hermitian line bundle on ${\cal X}$ and let $p^*{\cal L} \longrightarrow \mbox{${\tilde{\cal X}}$}$ be the pullback of ${\cal L}$ equipped with the pullback metric. Then there is a canonical isometric isomorphism $$det(p^*({\cal L} ))^{\otimes 12}~\cong ~ det({\cal L} )^{\otimes 12.{\rm deg}(p)}$$ of bundles on $S$. This isomorphism is functorial with respect to base change. \noindent (ii)~ Denoting the isometric isomorphism obtained in (i) by ${\Gamma }(p)$, and similarly defining ${\Gamma }(q)$ and ${\Gamma }(p\circ q)$, the following diagram commutes: $$\matrix{det((p\circ q)^*({\cal L} ))^{\otimes 12}&\mapright{{\Gamma }(q)}& det(p^*({\cal L} ))^{\otimes 12.{\rm deg}(q)}\cr \mapdown{{\Gamma }(p\circ q)}&& \mapdown{{\Gamma }(p)^{\otimes {\rm deg}(q) }}\cr det({\cal L} )^{\otimes 12.{\rm deg}(p\circ q)}&\mapright{id}& det({\cal L} )^{\otimes 12.{\rm deg}(p\circ q)}\cr} $$ where ${\Gamma }(p)^{\otimes {\rm deg}(q) }$ is the isomorphism on appropriate bundles, obtained by taking the ${\rm deg }(q)$-th tensor product of the isomorphism ${\Gamma }(p)$.} \medskip (The terminology ``functorial with respect to base change'' was explained earlier. We will use ``canonical'' to mean functorial with respect to base change.) \medskip \noindent {\bf Proof of Lemma 2.9.} The idea of the proof is to relate -- utilizing GRR in form (2.3) -- the determinant bundles, which are difficult to understand, with the more tractable ``Deligne pairings''. Let ${\cal M}$ be any line bundle on $\mbox{${\tilde{\cal X}}$}$ equipped with a hermitian structure. First we want to show that there is a canonical isometric isomorphism $$<p^*{\cal L} ,{\cal M} >~\longrightarrow ~<{\cal L} ,N({\cal M} )>\, , \eqno{(2.10)}$$ where $N({\cal M} ) \longrightarrow {\cal X}$ is the norm of ${\cal M}$. We recall the definition of $N({\cal M})$. The direct image $R^0p_*({\cal M} )$ is locally free on ${\cal X}$, and the bundle $R^0p_*({\cal M} )$ admits a natural reduction of structure group to the {\it monomial group} $G \subset GL({\rm deg}(p), \CC)$. (The group $G$ is the semi-direct product of permutation group, $P_{{\rm deg}(p)}$, with the invertible diagonal matrices defined using the permutation action of $P_{{\rm deg}(p)}$.) Mapping $g \in G$ to the permanent of $g$ (on $G$ it is simply the product of all non-zero entries) we get a homomorphism to ${\CC}^*$, which is denoted by $\mu$. Using this homomorphism $\mu$ we have a holomorphic line bundle on ${\cal X}$, associated to $R^0p_*({\cal M} )$, which is defined to be $N({\cal M})$. Clearly the fiber of $N({\cal M})$ over a point $x\in {\cal X}$ is the tensor product $$ N({\cal M} )_x ~ = ~ \bigotimes_{y\in p^{-1}(x)}{{\cal M}}_y\,, \eqno{(2.11)} $$ The hermitian metric on ${\cal M}$ gives a reduction of the structure group of $R^0p_*({\cal M} )$ to the maximal compact subgroup $G_U \subset G$. Since $\mu (G_U) = U(1)$, we have a hermitian metric on $N({\cal M})$. Note that the hermitian metric on $N({\cal M} )$ is such that the above equality (2.11) is actually an isometry. For a meromorphic section $m$ of $M$, the above identification of fibers gives a meromorphic section of $N(M)$ which is denoted by $N(m)$. Given sections $l$ and $m$ of ${\cal L}$ and ${\cal M}$ respectively, with finite support over $U\subset S$ (the support of $p^*l$ and $m$ being assumed disjoint) we map $<p^*l,m>$ to $<l,N(m)>$. It can be checked that this map is compatible with the relations in (2.4). Hence we get a homomorphism from the bundle $<p^*{\cal L} ,{\cal M}>$ to $<{\cal L} ,N({\cal M} )>$; this is our candidate for (2.10). To check that it is an isometry, we evaluate the (logarithms of) norms of the sections $<p^*l,m>$ and $<l,N(m)>$ given by definition (2.5). It is easy enough to see from (2.5) that the norms of these two sections coincide. Therefore for a hermitian line bundle ${{\cal L}}'$ on ${\cal X}$, the isomorphism (2.10) implies that $$<p^*{\cal L} ,p^*{{\cal L}}'> ~=~ <{\cal L} ,N(p^*{{\cal L}}')>$$ But $N(p^*{{\cal L}}') = {{\cal L}'}^d$, where $d :={\rm deg}(p)$, and moreover the hermitian metric on $N(p^*{{\cal L}}')$ coincides with that of ${{\cal L}'}^d$. Hence from the isometric isomorphism obtained in (2.6) we get the following identification of hermitian line bundles (the isomorphism so created being again functorial with respect to change of base space): $$ <p^*{\cal L},p^*{{\cal L}}'> ~=~<{\cal L} ,{{\cal L}}'>^d \eqno{(2.12)} $$ To prove part $(i)$ of the Lemma we apply the GRR isomorphism (2.3) to both ${\cal L}$ and $p^*{\cal L}$, and compare the Deligne pairing bundles appearing on the right hand sides using the result (2.12). To simplify notation set $\omega = T^*_{{\cal X} /S}$. By applying (2.3) to $p^*{\cal L}$, and noting that since the map $p$ is \'etale, the relative tangent bundle $T_{\mbox{${\tilde{\cal X}}$} /S} = p^*T_{{\cal X} /S}$, we deduce that $det(p^*{\cal L} )^{\otimes 12}$ is canonically isometrically isomorphic to $<p^*{\cal L} ,p^*({\cal L}\bigotimes{\omega }^{-1} )>^{\otimes 6} \bigotimes <p^*\omega ,p^*{\omega }>$. Taking ${{\cal L}}'$ to be ${{\cal L}}\bigotimes\omega $ in (2.12) we have $<{\cal L},{\cal L}\bigotimes{\omega }>^d=<p^*{\cal L},(p^*{\cal L}\bigotimes \omega )>$. Substituting $\omega $ in place on ${\cal L}$ and ${{\cal L}}'$ in (2.12) we have $<\omega ,\omega >^d=<p^*\omega ,p^*\omega >$. Therefore the bundle $<p^*{\cal L} ,p^*({\cal L}\bigotimes{\omega }^{-1})>^{\otimes 6}\bigotimes <p^*\omega ,p^*{\omega }>$ is isometrically isomorphic to $<{\cal L},{\cal L}\bigotimes{\omega }^{-1}>^{6d}\bigotimes <\omega ,\omega >^d$. But now applying (2.3) to ${\cal L}$ itself we see that this last bundle is isometrically isomorphic to $det({\cal L} )^{\otimes 12d}$. That completes the proof. Notice that since all isomorphisms used in the above proof were canonical (functorial with base change), the final isomorphism asserted in part $(i)$ is also canonical in the same sense. In order to prove part $(ii)$ of the Lemma, we first note that the isometric isomorphisms in (2.10) and (2.12) actually fit into the following commutative diagram $$ \matrix{<(p\circ q)^*{\cal L} ,{\cal M} >&\mapright{}&<p^*{\cal L} ,N({\cal M} )_q> \cr \mapdown{}&& \mapdown{}\cr <{\cal L} ,N({{\cal M}})>& = & <{\cal L} ,N({{\cal M}})>\cr} $$ where ${\cal L}$ is a hermitian line bundle on ${\cal X}$ and ${\cal M}$ is a hermitian line bundle on ${{\cal X}}'$, $N({\cal M} ) \longrightarrow {\cal X}$ is the norm of ${\cal M}$ for the covering $p\circ q$, and $N({\cal M} )_q \longrightarrow \mbox{${\tilde{\cal X}}$}$ is the norm of ${\cal M}$ for the covering $q$. Indeed, the commutativity of the above diagram is straightforward to deduce from the fact that the following two bundles on ${\cal X}$: namely, $N({\cal M} )$ and the norm of $N({\cal M} )_q$, are isometrically isomorphic. The isomorphism can be defined, for example, using (2.11). Now using (2.3), and repeatedly using the above commutative diagram, we obtain part $(ii)$. $\hfill{\Box}$ \medskip We will have occasion to use this general lemma in concrete situations. \medskip \noindent{\bf Remark 2.13.} In [KM] and in [D] the basic context is the algebraic families category, and the determinant of cohomology bundle as well as the Deligne pairing bundles are constructed in this category. However, since the constructions of the determinant bundles and of the Deligne pairing are {\it canonical} and {\it local}, they work equally well for holomorphic families of Riemann surfaces also. The point is that if ${\cal X} \rightarrow S$ is a holomorphic family of Riemann surfaces parametrized by a complex manifold $S$, and ${\cal L} \rightarrow {\cal X}$ is a holomorphic line bundle, then $det({\cal L}) \rightarrow S$ is a holomorphic line bundle which is functorial with respect to holomorphic base changes. And if ${\cal L}$ and ${\cal M}$ are two holomorphic line bundles on ${\cal X}$ then $<{\cal L},{\cal M}>$ is a holomorphic line bundle on $S$ -- again functorial with respect to holomorphic base changes. In fact, an analytic construction of the determinant bundle and the Quillen metric is to be found in [BGS]. Since the constructions of the Quillen metric and the metric on the Deligne pairing, (using (2.5)), also hold true for holomorphic families, consequently, Lemma 2.9 holds in the holomorphic category as well as in the algebraic category. \noindent{\bf Remark 2.14.} The statement that $det(p^*({\cal L}))^{\otimes 12}~\cong ~ det({\cal L} )^{\otimes 12{\rm deg}(p)}$ as line bundles actually holds for curves over any field. The statement about isometry makes sense only when we have Riemann surfaces. \section{Determinant bundles over Teichm\"uller spaces} Our aim in this section is to apply the Lemma 2.9 to the universal family of marked Riemann surfaces of genus $g$ over the Teichm\"uller space ${\cal T}_g$. The situation of Lemma 2.9 is precipitated by choosing any finite covering space over a topological surface of genus $g$. Let $\pi:{\tilde X} \longrightarrow X$ be an unramified covering map between two compact connected oriented two manifolds ${\tilde X}$ and $X$ of genera ${\tilde g}$ and $g$, respectively. Assume that $g \geq 2$. The degree of the covering $\pi$, which will play an important role, is the ratio of the respective Euler characteristics; namely, $deg(\pi)=({\tilde g}-1)/(g-1)$. We recall the basic deformation spaces of complex (conformal) structures on smooth closed oriented surfaces -- the Teichm\"uller spaces. Let $Conf(X)$ (respectively, $Conf({\tilde X})$) denote the space of all smooth conformal structures on $X$ (respectively, ${\tilde X}$). Define ${\rm Diff}^{+}(X)$ (respectively, ${\rm Diff}^+({\tilde X})$) to be the group of all orientation preserving diffeomorphisms of $X$ (respectively, ${\tilde X}$), and denote by ${\rm Diff}^{+}_0(X)$ (respectively, ${\rm Diff}^+_0({\tilde X})$) the subgroup of those that are homotopic to the identity. The group ${\rm Diff}^+(X)$ acts naturally on $Conf(X)$ by pullback of conformal structure. We define $$ {\cal T}(X)={\cal T}_g~:=~Conf(X)/{\rm Diff}^{+}_0(X) \eqno{(3.1)} $$ to be the Teichm\"uller space of genus $g$ (marked) Riemann surfaces. Similarly obtain ${\cal T}_{{\tilde g}} := Conf({\tilde X} )/{\rm Diff}^+_0({\tilde X} )$ -- the Teichm\"uller space for genus ${\tilde g}$. The Teichm\"uller space ${\cal T}_g$ carries naturally the structure of a $(3g-3)$ dimensional complex manifold which is embeddable as a contractible domain of holomorphy in the affine space $\CC^{3g-3}$. The mapping class group of the genus $g$ surface, namely the discrete group $MC_g := {\rm Diff}^+(X)/{\rm Diff}^+_0(X)$, acts properly discontinuously on ${\cal T}_g$ by holomorphic automorphisms, the quotient being the moduli space ${\cal M}_g$. For these basic facts see, for example, [N]. The Teichm\"uller spaces are fine moduli spaces. In fact, the total space $X\times {\cal T}_g$ admits a natural complex structure such that the projection to the second factor $$ {\psi}_g:{\cal C}_g:=X\times{\cal T}_g \longrightarrow {\cal T}_g \eqno{(3.2)} $$ gives the universal Riemann surface over ${\cal T}_g$. This means that for any $\eta \in {\cal T}_g$, the submanifold $X\times\eta$ is a complex submanifold of ${\cal C}_g$, and the complex structure on $X$ induced by this embedding is represented by $\eta$. As is well-known, (Chapter 5 in [N]), the family ${\cal C}_g \rightarrow {\cal T}_g$ is the {\it universal} object in the category of holomorphic families of genus $g$ marked Riemann surfaces. Given a complex structure on $X$, using $\pi$ we may pull back this to a complex structure on ${\tilde X}$. This gives an injective map $Conf(X)\longrightarrow Conf({{\tilde X}})$. Given an element of $f\in {\rm Diff}^+_0(X)$, from the homotopy lifting property, there is a unique diffeomorphism ${\tilde f}\in{\rm Diff}^{+}_0({{\tilde X}})$ such that $\tilde f$ is a lift of $f$. Mapping $f$ to $\tilde f$ defines an injective homomorphism of ${\rm Diff}^{+}_0(X)$ into ${\rm Diff}^{+}_0({{\tilde X}})$. We therefore obtain an injection $$ {\cal T}(\pi ):{\cal T}_g~\longrightarrow ~{\cal T}_{{\tilde g}} \eqno(3.3) $$ It is known that this map ${\cal T}(\pi )$ is a {\it proper holomorphic embedding} between these finite dimensional complex manifolds; ${\cal T}(\pi)$ respects the quasiconformal-distortion (=Teichm\"uller) metrics. {}From the definitions it is evident that this embedding between the Teichm\"uller spaces depends only on the (unbased) isotopy class of the covering $\pi$. \noindent{\bf Remark 3.4.} In fact, we see that $\cal T$ is thus a contravariant functor from the category of closed oriented topological surfaces, morphisms being covering maps, to the category of finite dimensional complex manifolds and holomorphic embeddings. We shall have more to say about this in Section 5. Over each genus Teichm\"uller space we have a sequence of natural determinant bundles arising from the powers of the relative (co-)tangent bundles along the fibers of the universal curve. Indeed, let ${\omega }_g \longrightarrow {\cal C}_g$ be the relative cotangent bundle for the projection ${\psi }_g$ in $(3.2)$. The determinant line bundle over ${\cal T}_g$ arising from its $n$-th tensor power is fundamental, and we shall denote it by: $$ DET_{n,g}:= det({\omega }^n_{g}) \longrightarrow {\cal T}_g, ~~n \in {\bf Z} \eqno(3.5) $$ Applying Serre duality shows that there is a canonical isomorphism $DET_{n,g} = DET_{1-n,g}$, for all $n$. $DET_{0,g} = DET_{1,g}$ is called the {\it Hodge} line bundle over ${\cal T}_g$. These holomorphic line bundles carry natural {\it Quillen hermitian structure} arising from the Poincar\'e metrics on the fibers of the universal curve. Recall that any Riemann surface $Y$ of genus $g\geq 2$ admits a unique conformal Riemannian metric of constant curvature $-1$, called the Poincar\'e metric of $Y$. This metric depends smoothly on the conformal structure, (because of the uniformization theorem with moduli parameters), and hence, for a family of Riemann surfaces of genus at least two, the Poincar\'e metric induces a hermitian metric on the relative tangent/cotangent bundle. We thus obtain Quillen metrics on each $DET_{n,g}$. The metric functorially assigned by the Quillen metric on any tensor power of $DET_{n,g}$ will also be referred to as the Quillen metric on that tensor power. Observe that by the naturality of the above constructions it follows that the action of $MC_g$ on ${\cal T}_g$ has a natural lifting as unitary automorphisms of these $DET$ bundles. We invoke back into play the unramified finite covering $\pi: {\tilde X} \rightarrow X$. Let $$ {\cal T}(\pi )^*{\cal C}_{{\tilde g}}~\longrightarrow ~{\cal T}_g \eqno{(3.6)} $$ be the pull-back to ${\cal T}_g$ of the universal family ${\cal C}_{{\tilde g}} \longrightarrow {\cal T}_{{\tilde g}}$ using the map ${\cal T}(\pi)$. Given the topological covering space $\pi$ we therefore obtain the following \'etale covering map between families of Riemann surfaces parametrized by ${\cal T}_g$: $$ {\pi}\times id ~:~ {\cal T}(\pi )^*{\cal C}_{{\tilde g}}~ \longrightarrow ~{\cal C}_g~:=~X\times {\cal T}_g $$ This is clearly a holomorphic map. In fact, we have the following commutative diagram $$ \matrix{{\cal T}(\pi )^*{\cal C}_{{\tilde g}}& &\mapright{\pi\times id}& & {\cal C}_g\cr &\searrow& & \swarrow &\cr & &{\cal T}_g& & \cr} \eqno{(3.7)} $$ exactly as in the general situation (2.7) above Lemma 2.9. Now let $$ id\times {\cal T}(\pi )~:~ {\cal T}(\pi )^*{\cal C}_{{\tilde g}} {}~\longrightarrow ~{\cal C}_{{\tilde g}} $$ denote the tautological lift of the map ${\cal T}(\pi)$. {}From the definition of the Poincar\'e metric it is clear that for an unramified covering of Riemann surfaces, ${\tilde Y} \longrightarrow Y$, the Poincar\'e metric on ${\tilde Y}$ is the pull-back of the Poincar\'e metric on $Y$. If ${\omega }_{{\tilde g}}$ is the relative cotangent bundle on ${\cal C}_{{\tilde g}}$ then this compatibility between Poincar\'e metrics implies that the two hermitian line bundles on ${\cal T}(\pi )^*{\cal C}_{{\tilde g}}$ namely, $(\pi\times id)^*{\omega }_g$ and $(id\times{\cal T}(\pi ))^*{\omega }_{{\tilde g}}$ are canonically isometric. But since the determinant bundle of a pullback family is the pullback of the determinant bundle, the holomorphic hermitian bundle ${\cal T}(\pi)^*(det({\omega }^n_{{\tilde g}})) \longrightarrow {\cal T}_g$ is canonically isometrically isomorphic to the determinant bundle of $(id\times {\cal T}(\pi))^*{\omega }^n_{{\tilde g}} \longrightarrow {\cal T}(\pi )^*{\cal C}_{{\tilde g}}$. Using this and simply applying Lemma 2.9 to the commutative diagram (3.7) we obtain the following theorem. (All the Quillen metrics are with respect to the Poincar\'e metric on fibers.) \medskip \noindent{\bf Theorem 3.8a.} {\it The two holomorphic hermitian line bundles $det({\omega }^n_g)^{12.{\rm deg}(\pi )}$ and\\ ${\cal T}(\pi )^*(det({\omega }^n_{{\tilde g}}))^{12}$ on ${\cal T}_g$ are canonically isometrically isomorphic for every integer $n$. In other words, there is a canonical isometrical line bundle morphism $\Gamma (\pi)$ lifting ${\cal T}(\pi)$ and making the following diagram commute: $$ \matrix{ {{DET}_{n,g}}^{\otimes {12.deg(\pi)}} &\mapright{{\Gamma }(\pi)} &{{DET}_{n,\tilde g}^{\otimes 12}} \cr \mapdown{} & &\mapdown{} \cr {\cal T}_g &\mapright{{\cal T}(\pi)} &{\cal T}_{\tilde g} \cr} $$ } \medskip \noindent {\bf Remark 3.9.} The bundle morphism $\Gamma (\pi)$ has been obtained from Riemann-Roch isomorphisms -- as evinced by the proof of Lemma 2.9. We shall therefore, in the sequel, refer to these canonical mappings as GRR morphisms. Tensor powers of the GRR morphisms will also be referred to as GRR morphisms. The functoriality of these morphisms is explained below in Theorem 3.8b. Let ${\bar X}~\mapright{\rho }~{\tilde X}~\mapright{\pi}~X$ be two unramified coverings between closed surfaces of respective genera ${{\bar g}}$, ${\tilde g}$ and $g$. By applying the Teichm\"uller functor we have the corresponding commuting triangle of embeddings between the Teichm\"uller spaces: $$ \matrix{ {\cal T}_g & &\mapright{{\cal T}(\pi)} & &{\cal T}_{\tilde g} \cr & \searrow & &\swarrow & \cr && {\cal T}_{\bar g} && \cr} \eqno(3.10) $$ Here the two slanting embeedings are, of course, ${\cal T}(\pi\circ\rho )$ and ${\cal T}(\rho )$. Applying Lemma 2.9(ii) we have \medskip \noindent{\bf Theorem 3.8b.}{\it The following triangle of GRR line bundle morphisms commutes: $$ \matrix{ {DET}_{n,g}^{\otimes {12.deg(\pi\circ\rho )}} & &\mapright{} & &{DET}_{n,\tilde g}^{\otimes {12.deg(\rho )}} \cr & \searrow & &\swarrow & \cr && {DET}_{n,\bar g}^{\otimes {12}} && \cr} \eqno(3.11) $$ All three maps in the diagram are obtained by applications of Theorem 3.8a, and raising to the appropriate tensor powers. The triangle above sits over the triangle of Teichm\"uller spaces (3.10), and the entire triangular prism is a commutative diagram.} \medskip \noindent {\bf Remark 3.12.}\,\ The nagging factor of $12$ in Theorems 3.8a and 3.8b can be dealt with as follows. The Teichm\"uller space being a contractible Stein domain, any two line bundles on ${\cal T}_g$ are isomorphic. Choose an isomorphism between ${\delta } : det({\omega }^n_g)^{{\rm deg}(\pi )} \longrightarrow {\cal T}(\pi )^*(det({\omega }^n_{{\tilde g}}))$. Hence $${\delta }^{\otimes 12} : det({\omega }^n_g)^{{12.\rm deg}(\pi )} \longrightarrow {\cal T}(\pi )^*(det({\omega }^n_{{\tilde g}}))^{12}$$ is an isomorphism. Let $${\tau} : det({\omega }^n_g)^{{12.\rm deg}(\pi )} \longrightarrow {\cal T}(\pi )^*(det({\omega }^n_{{\tilde g}}))^{12}$$ be the isomorphism given by the Theorem 3.8a. So $f := {\tau}\circ ({\delta }^{\otimes 12})^{-1}$ is a nowhere zero function on ${\cal T}_g$. Since ${\cal T}_g$ is simply connected, there is a function $h$ on ${\cal T}_g$ such that $h^{12} = f$. Any two such choices of $h$ will differ by a $12$-th root of unity. Consider the homomorphism ${\bar \tau} := h.\delta $. Clearly ${\bar \tau}^{\otimes 12} = \tau$. It is easy to see that for two different choices of the isomorphism $\delta $, the two ${\bar\tau}'$s differ by multiplication with a $12$-th root of unity. Moreover, if we consider a similar diagram to that in Theorem 3.8b with the factor 12 removed and all the homomorphisms being replaced by the corresponding analogues of $\bar\tau$, then the diagram commutes up to multiplication with a $12$-th root of unity. \noindent{\bf Remark 3.13.} Recall from above that the action of $MC_g$ in ${\cal T}_g$ lifts to the total space of $det({\omega }^n_g)$ as bundle automorphisms preserving the Quillen metric. There is no action, a priori, of $MC_g$ on the total space of the the pullback bundle ${\cal T}(\pi )^*(det({\omega }^n_{{\tilde g}}))$. However, from Theorem 3.8a the bundle ${\cal T}(\pi )^*(det({\omega }^n_{{\tilde g}}))^{12}$ gets an action of $MC_g$ which preserves the pulled back Quillen metric. Theorem 3.8b ensures the identity between the $MC_g$ actions obtained by different pullbacks. \medskip In [BN] we will consider two special classes of coverings, namely characteristic covers and cyclic covers. In such situations the map between Teichm\"uller spaces, induced by the covering, actually descends to a map between moduli spaces (possibly with level structure). As mentioned in the Introduction, in that context we were able to give a proof of the existence of the GRR morphism of Theorem 3.8a using Weil-Petersson geometry and topology. \section{Power law (Principal) bundle morphisms over Teichm\"uller spaces} We desire to obtain certain canonical geometric objects over the inductive limits of the finite dimensional Teichm\"uller spaces by coherently fitting together the determinant line bundles $DET_{n,g}$ thereon; the limit is taken as $g$ increases by running through a universal tower of covering maps. To this end it is necessary to find canonical mappings relating ${DET}_{n,g}$ to ${DET}_{n,\tilde g}$ where genus ${\tilde g}$ covers genus $g$. Now, given any complex line bundle $\lambda \rightarrow T$ over any base $T$, there is a certain canonical mapping of $\lambda $ to any positive integral ($d$-th) tensor power of itself, given by: $$ \omega _d: \lambda \longrightarrow {\lambda }^{\otimes d} \eqno(4.1) $$ where $\omega _d$ on any fiber of $\lambda $ is the map $z \mapsto z^{d}$. Observe that $\omega _d$ maps $\lambda $ minus its zero section to ${\lambda }^{\otimes d}$ minus its zero section by a map which is of degree $d$ on the $\CC^{*}$ fibers. We record the following properties of these maps: \noindent {\bf 4.1a.} The map $\omega _d$ is defined independent of any choices of basis, and it is evidently compatible with base change. [Namely, if we pull back both $\lambda $ and $\lambda ^{d}$ over some base $T_1 \rightarrow T$, then the connecting map $\omega _d$ (over $T$) also pulls back to the corresponding $\omega _d$ over $T_1$.] \noindent {\bf 4.1b.} The map $\omega _d$ is a homomorphism of the corresponding $\CC^{*}$ principal bundles. When $T$ is a complex manifold, and $\lambda $ is a line bundle in that category, then the map $\omega _d$ is a holomorphic morphism between the total spaces of the source and target bundles. \noindent {\bf 4.1c.} If $\lambda $ is equipped with a hermitian fiber metric, and its tensor powers are assigned the corresponding hermitian structures, the map $\omega _d$ carries the unit circles to unit circles. (The choice of a unit circle amongst the natural family of zero-centered circles in any complex line is clearly equivalent to specifying a hermitian norm. In this section we will think of hermitian structure on a line bundle as the choice of a smoothly varying family of unit circles in the fibers.) Thus, given a topological covering $\pi:{\tilde X} \rightarrow X$, as in the situation of Theorem 3.8a, we may define a canonical map $$ \Omega (\pi) := \Gamma (\pi) \circ \omega _{deg(\pi)}:{DET}_{n,g}^{\otimes 12} \longrightarrow {DET}_{n,\tilde g}^{\otimes 12} \eqno(4.2) $$ where $\Gamma (\pi)$ is the canonical GRR line bundle morphism found in Theorem 3.8a. Translating Theorems 3.8a and 3.8b in terms of these holomorphic maps $\Omega (\pi)$ of positive integral fiber degree, we get: \medskip \noindent{\bf Theorem 4.3a.} {\it For each integer $n$, there is a canonical isometrical holomorphic bundle morphism $\Omega (\pi)$ lifting ${\cal T}(\pi)$ and making the following diagram commute: $$ \matrix{ {{DET}_{n,g}^{\otimes 12}} &\mapright{{\Omega }(\pi)} &{{DET}_{n,\tilde g}^{\otimes 12}} \cr \mapdown{} & &\mapdown{} \cr {\cal T}_g &\mapright{{\cal T}(\pi)} &{\cal T}_{\tilde g} \cr} $$ By ``isometrical'' we mean that the unit circles of the Quillen hermitian structures are preserved by the $\Omega (\pi)$.} \smallskip \noindent{\bf 4.3b.} {\it Let $\pi$ and $\rho $ denote two composable covering spaces between surfaces, as in the situation of Theorem 3.8b. The following triangle of non-linear isometrical holomorphic bundle morphisms commutes: $$ \matrix{ {DET}_{n,g}^{\otimes {12}} & &\mapright{} & &{DET}_{n,\tilde g}^{\otimes {12}} \cr & \searrow & &\swarrow & \cr && {DET}_{n,\bar g}^{\otimes {12}} && \cr} $$ The horizontal map is $\Omega (\pi)$, and the two slanting maps are, (reading from left to right), $\Omega (\pi \circ \rho )$ and $\Omega (\rho )$. The triangle above sits over the triangle of Teichm\"uller spaces (3.10), and the entire triangular prism is a commutative diagram.} The canonical and functorial nature of these connecting maps, $\Omega (\pi)$, will now allow us to produce direct systems of line/principal bundles over direct systems of Teichm\"uller spaces. \section{Commensurability Teichm\"uller space and its Automorphism group} We construct a category ${\cal S}$ of certain topological objects and morphisms: the objects, $Ob({\cal S})$, are a set of compact oriented topological surfaces each equipped with a base point ($\star$), there being exactly one surface of each genus $g \geq 0$; let the object of genus $g$ be denoted by $X_g$. The morphisms are based isotopy classes of pointed covering mappings $$ \pi: (X_{\tilde g}, \star) \rightarrow (X_g, \star) $$ there being one arrow for each such isotopy class. Note that the monomorphism of fundamental groups induced by (any representative of the based isotopy class) $\pi$, is unambiguously defined. Fix a genus $g$ and let $X = X_g$. Observe that all the morphisms with the fixed target $X_g$: $$ M_g = \{\alpha \in Mor({\cal S}): Range(\alpha )=X_g \} $$ constitute a {\it directed set} under the partial ordering given by factorisation of covering maps. Thus if $\alpha $ and $\beta $ are two morphisms from the above set, then $\beta \succ \alpha $ if and only if the image of the monomorphism $\pi_1(\beta )$ is contained within the image of $\pi_1(\alpha )$; that happens if and only if there is a commuting triangle of morphisms of ${\cal S}$ as follows: $$ \matrix{ X_{g(\beta )} & &\mapright{\theta} & &X_{g(\alpha )} \cr & \searrow & &\swarrow & \cr && X_g && \cr} $$ Here $X_{g(\alpha )}$ denotes the domain surface for $\alpha $ (similarly $X_{g(\beta )}$), and the two slanting arrows are (reading from left to right), $\beta $ and $\alpha $. It is important to note that the factoring morphism $\theta$ is {\it uniquely} determined because we are working with base points. The directed property of $M_g$ follows by a simple fiber-product argument. [Remark: Notice that the object of genus $1$ in ${\cal S}$ only has morphisms to itself -- so that this object together with all its morphisms (to and from) form a subcategory.] Recall from Section 3 that each morphism of ${\cal S}$ induces a proper, holomorphic, Teichm\"uller-metric preserving embedding between the corresponding finite-dimensional Teichm\"uller spaces. We can thus create the natural {\it direct system of Teichm\"uller spaces} over the above directed set $M_g$, by associating to each $\alpha \in M_g$ the Teichm\"uller space ${\cal T}(X_{g(\alpha )})$, and for each $\beta \succ \alpha $ the corresponding holomorphic embedding ${\cal T}(\theta)$ (with $\theta$ as in the diagram above). Consequently, we may form the {\it direct limit Teichm\"uller space over $X=X_g$}: $$ {\cal T}_{\infty}(X_g) = {\cal T}_{\infty}(X) := ind. lim. {\cal T}(X_{g(\alpha )}) \eqno(5.1) $$ the inductive limit being taken over all $\alpha $ in the directed set $M_g$. This is our {\it commensurability Teichm\"uller space}. \noindent {\bf Remark:} Over the same directed set $M_g$ we may also define a natural {\it inverse system of surfaces}, by asscoiating to $\alpha \in M_g$ a certain copy, $S_{\alpha }$ of the pointed surface $X_{g(\alpha )}$. [Fix a universal covering over $X=X_g$. $S_{\alpha }$ can be taken to be the universal covering quotiented by the action of the subgroup $Im(\pi_1(\alpha )) \subset {\pi_1}(X,\star)$.] If $g \ge 2$, then the inverse limit of this system is the {\it universal solenoidal surface} $H_{\infty}$ whose Teichm\"uller theory was studied in [S], [NS]. The completion of ${\cal T}_{\infty}(X)$ in the Teichm\"uller metric is ${\cal T}(H_{\infty})$. A remarkable but obvious fact about this construction is that every morphism $\pi:Y \rightarrow X$ of ${\cal S}$ induces a natural Teichm\"uller metric preserving {\it homeomorphism} $$ {\cal T}_{\infty}(\pi): {\cal T}_{\infty}(Y) \longrightarrow {\cal T}_{\infty}(X) \eqno(5.2) $$ ${\cal T}_{\infty}(\pi)$ is invertible simply because the morphisms of ${\cal S}$ with target $Y$ are cofinal with those having target $X$. If we consider objects and maps to be continuous/holomorphic on the inductive limit spaces when they are continuous/holomorphic when restricted to the finite-dimensional strata, then it is clear that ${\cal T}_{\infty}(\pi)$ is a biholomorphic identification. (Note that ${\cal T}_{\infty}$ acts covariantly, since it is defined by a morphism of direct systems, although the Teichm\"uller functor ${\cal T}$ of (3.3) was contravariant.) It follows that each ${\cal T}_{\infty}(X)$, (and its completion ${\cal T}(H_{\infty})$), is equipped with a large {\it automorphism group} -- one from each (undirected) cycle of morphisms of ${\cal S}$ starting from $X$ and returning to $X$. By repeatedly using pull-back diagrams (i.e., by choosing the appropriate connected component of the fiber product of covering maps), it is easy to see that the automorphism arising from any (many arrows) cycle can be obtained simply from a two-arrow cycle ${\tilde X} {\rightarrow \atop \rightarrow } X$. Namely, whenever we have (the isotopy class of) a ``self-correspondence'' of $X$ given by two non-isotopic coverings, say $\alpha $ and $\beta $, $$ {\tilde X} {\rightarrow \atop \rightarrow } X \eqno(5.3) $$ we can create an automorphism of ${\cal T}_{\infty}(X)$ defined as the composition: ${{\cal T}_{\infty}(\beta )}\circ{({\cal T}_{\infty}(\alpha ))^{-1}}$. Therefore each of these automorphisms -- arising from any arbitrarily complicated cycle of coverings (starting and ending at $X$) -- is obtained as one of these simple ``two-arrow'' compositions. These automorphisms form a group that we shall call the {\it commensurability modular group}, $CM_{\infty}(X)$, acting on the universal commensurability Teichm\"uller space ${\cal T}_{\infty}(X)$. We make some further remarks regarding this large new mapping class group. Consider the abstract graph ($1$-complex), $\Gamma ({\cal S})$, obtained from the category ${\cal S}$ by looking at the objects as vertices and the (undirected) arrows as edges. It is clear from the definition above that the fundamental group of this graph, viz. $\pi_{1}(\Gamma ({\cal S}),X)$, is acting on ${\cal T}_{\infty}(X)$ as these automorphisms. In fact, we may fill in all the ``commuting triangles'' -- i.e., fill in the $2$-cells in this abstract graph whenever two morphisms (edges) compose to give a third edge; the thereby-reduced fundamental group of this $2$-complex produces on ${\cal T}_{\infty}(X)$ the action of $CM_{\infty}(X)$. It is interesting to observe that this new modular group $CM_{\infty}(X)$ of automorphisms on ${\cal T}_{\infty}(X)$ corresponds exactly to ``virtual automorphisms'' of the fundamental group $\pi_{1}(X)$, -- generalizing the classical situation where the usual automorphism group $Aut(\pi_{1}(X))$ appears as the action via modular automorphisms on ${\cal T}(X)$. Indeed, given any group $G$, one may define its associated group of ``virtual'' automorphisms; as opposed to usual automorphisms, for virtual automorphisms we demand that they be defined only on some finite index subgroup of $G$. To be precise, consider isomorphisms $\rho :H \rightarrow K$ where $H$ and $K$ are subgroups of finite index in $G$. Two such isomorphisms (say $\rho _1$ and $\rho _2$) are considered equivalent if there is a finite index subgroup (sitting in the intersection of the two domain groups) on which they coincide. The equivalence class $[\rho ]$ -- which is like the {\it germ} of the isomorphism $\rho $ -- is called a {\it virtual automorphism} of $G$; clearly the virtual automorphisms of $G$ constitute a group, $Vaut(G)$, under the obvious law of composition, (namely, compose after passing to deeper finite index subgroups, if necessary). We shall apply this concept to the fundamental group of a surface of genus $g$, ($g>1$). It is clear from definition that the group $Vaut(\pi_{1}(X_g))$ {\it is genus independent}, as is to be expected in our constructions. In fact, $Vaut$ presents us a neat way of formalizing the ``two-arrow cycles'' which we introduced to represent elements of ${CM_{\infty}}$. Letting $G = \pi_{1}(X)$, (recall that $X$ is already equipped with a base point), the two-arrow diagram (5.3) above corresponds to the following well-defined virtual automorphism of $G$: $$ [\rho ] = [{\beta }_{*}\circ{\alpha }_{*}^{-1}:{\alpha }_{*}(\pi_{1}({\tilde X})) \rightarrow {\beta }_{*}(\pi_{1}({\tilde X}))] $$ Here ${\alpha }_{*}$ denotes the monomorphism of the fundamental group $\pi_{1}({\tilde X})$ into $\pi_{1}(X) = G$, and similarly ${\beta }_{*}$. We let $Vaut^{+}({\pi}_{1}(X))$ denote the subgroup of $Vaut$ arising from pairs of orientation preserving coverings. The final upshot is that ${CM_{\infty}(X)}$ is {\it isomorphic} to $Vaut^{+}(\pi_{1}(X))$ and there is a natural surjective homomorphism: ${\pi}_{1}(\Gamma ({\cal S}),X) \rightarrow Vaut^{+}({\pi}_{1}(X))$ whose kernel is obtained by filling in all commuting triangles in $\Gamma ({\cal S})$. \noindent {\it Acknowledgement:} The concept of $Vaut$ has arisen in group theory papers -- for example [Ma],[MT]. We are grateful to Chris Odden for pointing out these references. \noindent {\bf Remark 5.4.} For the genus one object $X_1$ in ${\cal S}$, we know that the Teichm\"uller spaces for all unramified coverings are each a copy of the upper half-plane $H$. The maps ${\cal T}(\pi)$ are M\"obius identifications of copies of the half-plane with itself, and we easily see that the pair $({\cal T}_{\infty}(X_1),CM_{\infty}(X_1))$ is identifiable as $(H,PGL(2,\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}))$. In fact, $GL(2,\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}) \cong Vaut({\bf Z} \oplus {\bf Z})$, and $Vaut^{+}$ is precisely the subgroup of index $2$ therein, as expected. Notice that the action has dense orbits in the genus one case. On the other hand, if $X \in Ob({\cal S})$ is of any genus $g \geq 2$, then we get an infinite dimensional ``ind-space'' as ${\cal T}_{\infty}(X)$ with the action of ${\cal G}(X)$ on it as described. Since the tower of coverings over $X$ and $Y$ (both of genus higher than $1$) eventually become cofinal, it is clear that {\it for any choice of genus higher than one we get one isomorphism class of pairs} $({\cal T}_{\infty}, CM_{\infty})$. (It is not known whether the action has dense orbits in this situation; this matter is related to some old queries on coverings of Riemann surfaces.) We work now over the direct system of the higher genus example $(T_{\infty}, CM_{\infty})$ and obtain the main theorem. We will first explain some preliminary material on direct limits of holomorphic line bundles over a direct system of complex manifolds. Given a direct system $T_{\alpha }$ of complex manifolds, and line bundles $\xi_{\alpha }$ over these, suppose that there are power law maps as the $\Omega (\pi)$ above, between the corresponding principal $\C*$ bundles covering the mappings in the direct system of base manifolds. Let $N$ denote the directed set of positive integers ordered by divisibility. For each $\lambda \in N$ take a copy of $\C*$, call it $(\C*,\lambda )$ and form the direct system $\{ (\C* ,\lambda )\}$ where $(\C*,\lambda ) \rightarrow (\C* ,{\lambda }')$ is given by the power law map: $z \rightarrow z^d$ when ${\lambda }'=d{\lambda }$. These maps are homomorphisms of groups, and the direct limit over $N$ is canonically isomorphic to the group ${{\CC}^{*}} \otimes {\QQ} := \CC\otimes_{{\bf Z}}\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}$. [The isomorphism maps the equivalence class of the element $(z,\lambda ) \in (\C*,\lambda )$ to $z \otimes {1/\lambda } \in {{\CC}^{*}} \otimes {\QQ}$.] The direct limit object obtained from the power law connecting maps between the principal bundles associated to the ${DET}_{n}^{12}$ system over the Teichm\"uller spaces will give us a ${{\CC}^{*}} \otimes {\QQ}$ principal bundle over the universal commensurability Teichm\"uller space ${\cal T}_{\infty}$, at least at the level of sets. The topological and holomorphic structure on these sets is defined for maps into these objects which factor through the direct system by imposing these properties on the factorizations. Let us consider the direct limit bundles obtained from a family of such bundles $\xi_{\alpha }$, and from the family obtained by raising each $\xi_{\alpha }$ to the tensor power $d$. These are two ${{\CC}^{*}} \otimes {\QQ}$ bundles over the direct limit of the bases which may be thought to have the same total spaces (as sets) but the ${{\CC}^{*}} \otimes {\QQ}$ action on the second one is obtained by precomposing the original action by the automorphism of ${{\CC}^{*}} \otimes {\QQ}$ obtained from the homomorphism $z \mapsto z^d$ on $\C*$. \noindent {\bf Theorem 5.5.} {\it Fix any integer $n$. Starting from any base surface $X \in Ob({\cal S})$, we obtain a direct system of principal $\CC^*$ bundles ${\cal L}_n(Y) := DET_{n,g(Y)}^{\otimes 12}$ over the Teichm\"uller spaces ${\cal T}(Y)$ with holomorphic homomorphisms $\Omega (\pi)$ (see Theorem 4.3) between the total spaces; here $Y \mapright{\pi} X$ is an arbitrary morphism of ${\cal S}$ with target $X$. By passing to the direct limit, one therefore obtains over the universal commensurability Teichm\"uller space, ${\cal T}_{\infty}(X)$, a principal ${{\CC}^{*}} \otimes {\QQ}$ bundle: $$ {\cal L}_{n,\infty}(X) = ind. lim. {\cal L}_n(Y) $$ Since the maps $\Omega (\pi)$ preserved the Quillen unit circles, the limit object also inherits such a Quillen ``hermitian'' structure. The construction is functorial with respect to change of the base $X$ in the obvious sense that the directed systems and their limits are compatible with the biholomorphic identifications ${\cal T}_{\infty}(\pi)$ of equation (5.2). In particular, the commensurability modular group action $CM_{\infty}(X)$ on ${\cal T}_{\infty}(X)$ has a natural lifting to ${\cal L}_{n,\infty}(X)$ -- acting by unitary automorphisms. Finally, the Mumford isomorphisms persist: $$ {\cal L}_{n,\infty}(X) = {\cal L}_{0,\infty}(X)^{\otimes (6n^{2} - 6n + 1)} $$ Namely, if we change the action of ${{\CC}^{*}} \otimes {\QQ}$ on the ``Hodge'' bundle ${\cal L}_{0,\infty}$ by the ``raising to the $(6n^2-6n+1)$ power'' automorphism of ${{\CC}^{*}} \otimes {\QQ}$, then the principal ${{\CC}^{*}} \otimes {\QQ}$ bundles are canonically isomorphic. } \noindent {\bf Remark 5.6.} In other words, the Mumford isomorphism in the above theorem means that ${\cal L}_{n,\infty}$ and ${\cal L}_{0,\infty}$ are equivariantly isomorphic relative to the automorphism of ${{\CC}^{*}} \otimes {\QQ}$ induced by the homomorphism of $\C*$ that raises to the power exhibited. Also, we could have used the Quillen hermitian structure to reduce the structure group from $\CC^{*}$ to $U(1)$, and thus obtain direct systems of $U(1)$ bundles over the Teichm\"uller spaces. Passing to the direct limit would then produce $U(1) \otimes \rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$} :=$ {\it ``tiny circle''} bundles over ${\cal T}_{\infty}$, which can be tested for maps into these objects as above. \smallskip \noindent {\bf Rational line bundles on ind-spaces:} A line bundle on the inductive limit of an inductive system of varieties or spaces, is, by definition ([Sha]), a collection of line bundles on each stratum (i.e., each member of the inductive system of spaces) together with compatible line bundle (linear on fibers) morphisms. Such a direct system of line bundles determines an element of the inverse limit of the Picard groups of the stratifying spaces. See [KNR], [Sha]. (Recall: For any complex space $T$, $Pic(T)$ := the group (under $\otimes$) of isomorphism classes of line bundles on $T$. In the case of the Teichm\"uller spaces, we refer to the modular-group invariant bundles as constituting the relevant Picard group -- see [BN].) Now, utilising the GRR morphisms $\Gamma (\pi)$ themselves, (without involving the power law maps), we know from Section 3 that the ``$d$-th root'' of the bundle ${DET}_{n,\tilde g}$ fits together with the bundle ${DET}_{n,g}$ ($d=({\tilde g}-1)/(g-1)$). A ``rational'' line bundle over the inductive limit is defined to be an element of the inverse limit of the $Pic_{\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}} = Pic \otimes {\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}}$. Therefore we may also state a result about the existence of canonical elements of the inverse limit, $\lim_{\leftarrow}Pic({\cal T}_{g_i})_{\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}}$, by our construction. Indeed, in the notation of Section 3, by using the morphisms ${\Gamma (\pi)}\otimes {1/{deg(\pi)}}$ between ${DET}_{n,g}$ and ${{DET}_{n,\tilde g}}\otimes {1/{deg(\pi)}}$ to create a directed system, we obtain canonical elements representing the Hodge and higher $DET_n$ bundles, with respective Quillen metrics: $$ {\Lambda }_{m} \in \lim_{\leftarrow}Pic({\cal T}_{g_i})_{\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}}, ~~m=0,1,2,.. \eqno(5.7) $$ The pullback (i.e., restriction) of $\Lambda _m$ to each of the stratifying Teichm\"uller spaces ${\cal T}_{g_i}$ is $(n_i)^{-1}$ times the corresponding $DET_{m}$ bundle, (with $(n_i)^{-1}$ times its Quillen metric), over ${\cal T}_{g_i}$. Here $n_i$ is the degree of the covering of the surface of genus $g_i$ over the base surface. As rational hermitian line bundles the Mumford isomorphisms persist: $$ \Lambda _{m} = {\Lambda _{0}}^{\otimes (6m^{2} - 6m +1)} \eqno(5.8) $$ as desired. This statement is different from that of the Theorem. For further details see [BN]. \smallskip \noindent{\bf Polyakov measure on ${\cal M}_g$ and our constructions:} In his study of bosonic string theory, Polyakov constructed a measure on the moduli space ${\cal M}_g$ of curves of genus $g(\geq 2)$. Details can found, for example, in [Alv], [Nel]. Subsequently, Belavin and Knizhnik, [BK], showed that the Polyakov measure has the following elegant mathematical description. First note that a hermitian metric on the canonical bundle of a complex space gives a measure on that space. Fixing a volume form (up to scale) on a space therefore amounts to fixing a fiber metric (up to scale) on the canonical line bundle, $K$, over that space. But the Hodge bundle $\lambda $ has its natural Hodge metric (arising from the $L^2$ pairing of holomorphic 1-forms on Riemann surfaces). Therefore we may transport the corresponding metric on ${\lambda }^{13}$ to $K$ by Mumford's isomorphism, (as we know the choice of this isomorphism is unique up to scalar) -- thereby obtaining a volume form on ${\cal{M}}_g$. [BK] showed that this is none other than the Polyakov volume. Therefore, the presence of Mumford isomorphisms over the moduli space of genus $g$ Riemann surfaces describes the Polyakov measure structure thereon. Above we have succeeded in fitting together the Hodge and higher $DET$ bundles over the ind space ${\cal T}_{\infty}$, together with the relating Mumford isomorphisms. We thus have from our results a structure on ${\cal T}_{\infty}$ that suggests a genus-independent, universal, version of the Polyakov structure. We remark that since the genus is considered the perturbation parameter in the above formulation of the standard perturbative bosonic Polyakov string theory, our work can be considered as a contribution towards a {\it non-perturbative} formulation of that theory. \newpage

9505

alg-geom/9505037

en

https://arxiv.org/abs/alg-geom/9505037

alg-geom/9505037

Alice Silverberg

A. Silverberg and Yu. G. Zarhin

Hodge groups of abelian varieties with purely multiplicative reduction

This is an updated version of the paper. LaTeX2e or LaTeX2.09 or AMSLaTeX. Contact: [email protected]

10.1070/IM1996v060n02ABEH000074

The main result of the paper is that if $A$ is an abelian variety over a subfield $F$ of ${\bold C}$, and $A$ has purely multiplicative reduction at a discrete valuation of $F$, then the Hodge group of $A$ is semisimple. Further, we give necessary and sufficient conditions for the Hodge group to be semisimple. We obtain bounds on certain torsion subgroups for abelian varieties which do not have purely multiplicative reduction at a given discrete valuation, and therefore obtain bounds on torsion for abelian varieties, defined over number fields, whose Hodge groups are not semisimple.

alg-geom

\section{Introduction} We show that if $A$ is an abelian variety over a subfield $F$ of ${\bold C}$, and $A$ has purely multiplicative reduction at a discrete valuation of $F$, then the Hodge group of $A$ is semisimple (Theorem \ref{hodsemi}). Since the non-semisimplicity of the Hodge group of an abelian variety can be translated into a condition on the endomorphism algebra and its action on the tangent space (see Theorem \ref{notsemi}), this gives a useful criterion for determining when an abelian variety does not have purely multiplicative reduction. For abelian varieties over number fields, a result analogous to Theorem \ref{hodsemi} holds where the Hodge group is replaced by a certain linear algebraic group $H_\ell$ over ${\bold Q}_\ell$ arising from the image of the $\ell$-adic representation associated to $A$ (see Theorem \ref{notsemiGal}). The Mumford-Tate conjecture predicts that $H_\ell$ is the extension of scalars to ${\bold Q}_\ell$ of the Hodge group. Our result generalizes a result of Mustafin (Corollary after Theroem 3.2 of \cite{Mustafin}), which says that for a Hodge family of abelian varieties (as in \cite{MumP}) admitting a ``strong degeneration'', generically the fibers have semisimple Hodge group. The problem of describing the Hodge group of an abelian variety with purely multiplication reduction was posed by V.\ G.\ Drinfeld, in a conversation with Zarhin in the 1980's. In \S\ref{boundssect} we provide bounds on torsion for abelian varieties which do not have purely multiplicative reduction at a given discrete valuation. We apply this and Theorem \ref{hodsemi} to obtain bounds on torsion for abelian varieties whose Hodge groups are not semisimple. Silverberg would like to thank MSRI and IHES for their generous hospitality, and NSF for financial support. Zarhin would like to thank the Institute f\"ur Experimentelle Mathematik for its hospitality, and Gerhard Frey for his interest in the paper and useful discussions. \section{Definitions, notation, and lemmas} Suppose $A$ is an abelian variety defined over a field $F$ of characteristic zero, and $L$ is an algebraically closed field containing $F$. Write $\mbox{End}_F(A)$ for the set of endomorphisms of $A$ which are defined over $F$, let $\mbox{End}(A) = \mbox{End}_L(A)$, let $\mbox{End}^0(A) = \mbox{End}(A) \otimes_{\bold Z} {\bold Q}$, and let $\mbox{End}^0_F(A) = \mbox{End}_F(A) \otimes_{\bold Z} {\bold Q}$. Suppose $K$ is a field and $\iota : K \hookrightarrow \mbox{End}_F^0(A)$ is an embedding such that $\iota(1) = 1$. Let $\mbox{Lie}_F(A)$ be the tangent space of $A$ at the origin, an $F$-vector space. If $\sigma$ is an embedding of $K$ into $L$, let $$n_\sigma = \mbox{dim}_L\{t \in \mbox{Lie}_L(A) : \iota(\alpha)t = \sigma(\alpha)t {\text{ for all }} \alpha \in K\}.$$ Note that $n_\sigma$ is independent of the choice of an algebraically closed field $L$ containing $F$. Write ${\bar \sigma}$ for the composition of $\sigma$ with the involution complex conjugation of $K$. \begin{defn} If $A$ is an abelian variety over an algebraically closed field $L$ of characteristic zero, $K$ is a CM-field, and $\iota : K \hookrightarrow \mbox{End}^0(A)$ is an embedding such that $\iota(1) = 1$, we say $(A,K,\iota)$ is {\em of Weil type} if $n_\sigma = n_{\bar \sigma}$ for all embeddings $\sigma$ of $K$ into $L$. \end{defn} \begin{lem} \label{freeof} If $A$ is an abelian variety defined over a field $F$ of characteristic zero, $L$ is an algebraically closed field containing $F$, $K$ is a CM-field, and $\iota : K \hookrightarrow \mbox{End}_F^0(A) \subseteq \mbox{End}^0(A)$ is an embedding such that $\iota(1) = 1$, then the following statements are equivalent: \begin{enumerate} \item[\normalshape{(i)}] $(A,K,\iota)$ is of Weil type, \item[\normalshape{(ii)}] $\iota$ makes $\mbox{Lie}_{L}(A)$ into a free $(K \otimes_{\bold Q} L)$-module, \item[\normalshape{(iii)}] $\iota$ makes $\mbox{Lie}_F(A)$ into a free $(K \otimes_{\bold Q} F)$-module. \end{enumerate} \end{lem} \begin{pf} Let $\Sigma$ be the set of embeddings of $K$ into $L$, let $\psi_F : K \otimes_{\bold Q} F \to \mbox{End}(\mbox{Lie}_F(A))$ be the homomorphism induced by $\iota$, let $\psi : K \otimes_{\bold Q} L \to \mbox{End}(\mbox{Lie}_L(A))$ be the extension of scalars of $\psi_F$ to $K \otimes_{\bold Q} L$, let $m = 2\mbox{dim}(A)/[K : {\bold Q}]$, let $M_F = (K \otimes_{\bold Q} F)^m$, and let $M = (K \otimes_{\bold Q} L)^m = M_F \otimes_F L$. Let $\psi_F^\prime : K \otimes_{\bold Q} F \to \mbox{End}(M_F)$ and $\psi^\prime : K \otimes_{\bold Q} L \to \mbox{End}(M)$ be the natural homomorphisms. By \S 2.1 of \cite{Shimura}, for every $\sigma \in \Sigma$ we have $n_\sigma + n_{\bar \sigma} = m$. For $\alpha \in K$, taking the trace of $\psi(\alpha)$ gives $$\mbox{tr}(\psi(\alpha)) = \sum_{\sigma \in \Sigma}n_{\sigma}\sigma(\alpha).$$ The traces of $\psi$ and of $\psi^\prime$ coincide on $K$ if and only if $n_\sigma = n_{\bar \sigma} = m/2$ for every $\sigma \in \Sigma$. Since $K \otimes_{\bold Q} L$ is a semisimple ring, $\mbox{Lie}_L(A)$ and $M$ are semisimple $(K \otimes_{\bold Q} L)$-modules. Therefore, $\mbox{Lie}_L(A)$ is a free $(K \otimes_{\bold Q} L)$-module if and only if the traces of $\psi$ and of $\psi^\prime$ coincide on $K$. Therefore, $\mbox{Lie}_L(A)$ is a free $(K \otimes_{\bold Q} L)$-module if and only if $(A,K,\iota)$ is of Weil type. If $\mbox{Lie}_F(A)$ is a free $(K \otimes_{\bold Q} F)$-module, then clearly $\mbox{Lie}_L(A)$ ( $= \mbox{Lie}_F(A) \otimes_F L$) is a free $(K \otimes_{\bold Q} L)$-module. Conversely, if $\mbox{Lie}_L(A)$ is a free $(K \otimes_{\bold Q} L)$-module, then the traces of $\psi$ and of $\psi^\prime$ (and therefore of $\psi_F$ and of $\psi_F^\prime$) coincide on $K$. Since $K \otimes_{\bold Q} F$ is a semisimple ring, $\mbox{Lie}_F(A)$ and $M_F$ are semisimple $(K \otimes_{\bold Q} F)$-modules. Therefore $\mbox{Lie}_F(A)$ and $M_F$ are isomorphic as $(K \otimes_{\bold Q} F)$-modules, i.e., $\mbox{Lie}_F(A)$ is a free $(K \otimes_{\bold Q} F)$-module. \end{pf} See also p.~525 of \cite{Ribet} for the case where $K$ is an imaginary quadratic field. \begin{rem} If $A$ is an abelian variety defined over a field $F$ of characteristic zero, $K$ is a totally real number field, and $\iota : K \hookrightarrow \mbox{End}_F^0(A)$ is an embedding such that $\iota(1) = 1$, then $\iota$ makes $\mbox{Lie}_F(A)$ into a free $(K \otimes_{\bold Q} F)$-module. To see this, let $L$ be an algebraically closed field containing $F$, let $\Sigma$ be the set of embeddings of $K$ into $L$, and let $\psi : K \to \mbox{End}(\mbox{Lie}_L(A))$ be the homomorphism induced by $\iota$. Let $m^{\prime} = \mbox{dim}(A)/[K : {\bold Q}]$. We have $$\mbox{tr}(\psi(\alpha)) = m^{\prime}\sum_{\sigma \in \Sigma}\sigma(\alpha)$$ for every $\alpha \in K$, by \S 2.1 of \cite{Shimura}. Therefore, $\mbox{Lie}_L(A)$ is a free $(K \otimes_{\bold Q} L)$-module. As in the proof of Lemma \ref{freeof}, it follows that $\mbox{Lie}_F(A)$ is a free $(K \otimes_{\bold Q} F)$-module. \end{rem} Suppose $A$ is a complex abelian variety. Let $V = H_1(A,{\bold Q})$ and let ${\bold S} = \mbox{Res}_{{\bold C}/{\bold R}}{\bold G}_m$. The complex structure on $A$ gives rise to a rational Hodge structure on $V$ of weight $-1$, i.e., a homomorphism of algebraic groups $h : {\bold S} \to \mbox{GL}(V)_{\bold R}$. Let ${\bold T}$ be the kernel of the norm map ${\bold N} : {\bold S} \to {\bold G}_m$. Then ${\bold T}({\bold R}) = \{x \in {\bold C} : |x| = 1\}$. \begin{defn} If $A$ is an abelian variety over ${\bold C}$ and $V = H_1(A,{\bold Q})$, then the {\em Hodge group} $H$ is the smallest algebraic subgroup of $\mbox{GL}(V)$ defined over ${\bold Q}$ such that $H({\bold R})$ contains $h({\bold T}({\bold R}))$. Equivalently, $H$ is the largest algebraic subgroup of $\mbox{GL}(V)$ defined over ${\bold Q}$ such that all Hodge classes in $V^{\otimes p} \otimes (V^\ast)^{\otimes q}$, for all non-negative integers $p$ and $q$, are tensor invariants of $H$. I.e., $H$ is the largest algebraic subgroup of $\mbox{GL}(V)$ defined over ${\bold Q}$ which fixes all Hodge classes of all powers of $A$. \end{defn} It follows from the definition of $H$ that $\mbox{End}^0(A) = \mbox{End}_H(V)$. If now $F$ is a number field and $\ell$ is a prime number, let $T_\ell(A) = {\displaystyle \lim_\leftarrow A_{\ell^r}}$ (the Tate module), let $V_\ell(A) = T_\ell(A) \otimes_{{\bold Z}_\ell}{\bold Q}_\ell$, and let $\rho_{A,\ell}$ denote the $\ell$-adic representation $$\rho_{A,\ell} : \mbox{Gal}({\bar F}/F) \to \mbox{GL}(T_\ell(A)) \subseteq \mbox{GL}(V_\ell(A)).$$ Let $G_\ell$ denote the algebraic envelope of the image of $\rho_{A,\ell}$, i.e., the Zariski closure in $\mbox{GL}(V_\ell(A))$ of the image of $\rho_{A,\ell}$. By \cite{Faltings}, $G_\ell$ is a reductive algebraic group, and $\mbox{End}^0_F(A) \otimes_{\bold Q} {\bold Q}_\ell = \mbox{End}_{G_\ell}(V_\ell(A))$. Let $H_\ell$ be the identity connected component of $G_\ell \cap \mbox{SL}(V_\ell(A))$. Then $H_\ell$ is a connected reductive group and $\mbox{End}^0(A) \otimes_{\bold Q} {\bold Q}_\ell = \mbox{End}_{H_\ell}(V_\ell(A))$. We will repeatedly use the fact (see the first Theorem on p.~220 of \cite{Hum}) that if $G$ is a connected linear algebraic group over a field $F$ of characteristic zero, then $G(F)$ is Zariski-dense in $G$. \begin{lem} \label{semfin} If $G$ is a reductive linear algebraic group over a field $F$ of characteristic zero, and $Z$ is the center of $G$, then $G$ is semisimple if and only if $Z(F)$ is finite. \end{lem} \begin{pf} Let $Z^0$ denote the identity connected component of $Z$. Since $G$ is reductive, $G$ is semisimple if and only if $Z^0 = 1$ (see the lemma on p.~125 of \cite{Hum}). Since $Z^0(F)$ is Zariski-dense in $Z^0$, $Z^0 = 1$ if and only if $Z(F)$ is finite. \end{pf} \section{Semisimplicity criteria for the groups $H$ and $H_\ell$} If the center of $\mbox{End}^0(A)$ is a direct sum of totally real number fields, then it is well-known that the groups $H$ and $H_\ell$ are semisimple (see, for instance, Corollary 1 in \S 1.3.1 of \cite{Zarhintor} and Lemma 1.4 of \cite{Tankeev}). The following result follows easily from a result in \cite{moonzar}, and characterizes the endomorphism algebras of abelian varieties whose Hodge groups are not semisimple. \begin{thm} \label{notsemi} Suppose $A$ is an abelian variety defined over ${\bold C}$. Then the Hodge group of $A$ is not semisimple if and only if for some simple component $B$ of $A$, the center of $\mbox{End}^0(B)$ is a CM-field $K$ such that $(B,K,\mathrm{id})$ is not of Weil type, with $\mathrm{id}$ the identity embedding of $K$ in $\mbox{End}^0(B)$. \end{thm} \begin{pf} Let $V = H_1(A,{\bold Q})$. Fix a polarization on $A$. The polarization induces a non-degenerate alternating bilinear form $\varphi : V \times V \to {\bold Q}$ such that $H \subseteq \mbox{Sp}(V,\varphi)$ (see \cite{MumMatAnn}). Then $$H({\bold Q}) \subseteq \mbox{Sp}(V,\varphi)({\bold Q}) = \{ g \in \mbox{End}(V) : gg^\prime = 1 \},$$ where $g \mapsto g^\prime$ is the involution on $\mbox{End}(V)$ defined by $$\varphi(g(x),y) = \varphi(x,g^\prime (y)) \mbox{ for $x, y \in V$}.$$ The restriction of the involution ${}^\prime$ to $\mbox{End}^0(A)$ is the Rosati involution. Let $Z$ denote the center of $H$ and let $Z_{\mathrm{End}}$ denote the center of $\mbox{End}^0(A)$. If $\alpha \in Z({\bold Q})$, then $\alpha$ commutes with all elements of $H({\bold Q})$, so $\alpha \in \mbox{End}^0(A)$. Further, since $\alpha \in H({\bold Q})$, $\alpha$ commutes with all elements of $\mbox{End}^0(A)$, and therefore $\alpha \in Z_{\mathrm{End}}$. Therefore, \begin{equation} \label{Zs} Z({\bold Q}) \subseteq \{\alpha \in Z_{\mathrm{End}} : \alpha\alpha^{\prime} = 1\}. \end{equation} If $A$ is isogenous to a product of two abelian varieties, then the Hodge group $H$ of $A$ is a subgroup of the product of the Hodge groups $H_1$ and $H_2$ of the factors, in such a way that for $i = 1$ and $2$ the restriction to $H$ of the projection map from $H_1 \times H_2$ onto $H_i$ induces a surjective homomorphism from $H$ onto $H_i$ (see Proposition 1.6 of \cite{Hazama}). It follows easily that $H$ is semisimple if and only if both $H_1$ and $H_2$ are semisimple. We may therefore reduce to the case where $A$ is a simple abelian variety. Then the center $Z_{\mathrm{End}}$ of $\mbox{End}^0(A)$ is either a totally real number field or a CM-field. Suppose $Z_{\mathrm{End}}$ is totally real. Then all Rosati involutions are the identity when restricted to $Z_{\mathrm{End}}$. By (\ref{Zs}), $Z({\bold Q}) \subseteq \{\pm 1\}$. Therefore, $Z({\bold Q})$ is finite, so $H$ is semisimple by Lemma \ref{semfin}. Suppose $Z_{\mathrm{End}}$ is a CM-field $K$. Then every Rosati involution induces complex conjugation on $K$. Choose $\alpha \in K^\times$ such that ${\bar \alpha} = -\alpha$. Then there exists a unique $K$-Hermitian form $\psi : V \times V \to K$ such that $\varphi(x,y) = Tr_{K/{\bold Q}}(\alpha \psi(x,y))$ (see \cite{Shimura}). The unitary group $\mbox{U}(V,\psi)$ is an algebraic group over $K_0$, the maximal totally real subfield of $K$. Let $U = \mbox{Res}_{K_0/{\bold Q}} \mbox{U}(V,\psi)$, let $\mbox{SU}$ denote the kernel of the determinant homomorphism $\mbox{det}_K : \mbox{U} \to \mbox{Res}_{K/{\bold Q}} {\bold G}_m$, and let $\mbox{End}_K(V)$ denote the ring of $K$-linear endomorphisms of $V$. Then $$\mbox{U}({\bold Q}) = \{g \in \mbox{End}_K(V) : \psi(g(x),g(y)) = \psi(x,y) {\text{ for all }} x, y \in V\}$$ and $H \subseteq \mbox{U} \subseteq \mbox{Sp}(V,\varphi)$. By Lemma 2.8 of \cite{moonzar}, $H \subseteq \mbox{SU}$ if and only if $(A,K,\mathrm{id})$ is of Weil type. If $H$ is semisimple, then all homomorphisms from $H$ to commutative groups are trivial. Therefore $\mbox{det}_K(H) = 1$, so $H \subseteq \mbox{SU}$. Conversely, if $H \subseteq \mbox{SU}$, then $Z({\bold Q}) \subseteq \mbox{SU}({\bold Q}) \cap K$, the group of $(\mbox{dim}(V)/[K:{\bold Q}])$-th roots of unity in $K$. Therefore $Z({\bold Q})$ is finite and $H$ is semisimple. \end{pf} \begin{thm} \label{notsemiGal} Suppose $A$ is an abelian variety defined over a number field $F$. Then the following are equivalent: \begin{enumerate} \item[\normalshape{(i)}] $H$ is semisimple, \item[\normalshape{(ii)}] $H_\ell$ is semisimple, for one prime $\ell$, \item[\normalshape{(iii)}] $H_\ell$ is semisimple, for every prime $\ell$. \end{enumerate} \end{thm} \begin{pf} Let $\ell$ be a prime number and let $V_\ell = V_\ell(A)$. By Theorem \ref{notsemi}, it suffices to show that $H_\ell$ is not semisimple if and only if for some simple component $B$ of $A$, the center of $\mbox{End}^0(B)$ is a CM-field $K$ such that $(B,K,\mathrm{id})$ is not of Weil type, with $\mathrm{id}$ the identity embedding of $K$ in $\mbox{End}^0(B)$. Since $H_\ell$ is connected, it is invariant under finite extensions of the number field $F$. By replacing $F$ by a finite extension, we may suppose that $\mbox{End}^0(A) = \mbox{End}_F^0(A)$. We parallel the proof of Theorem \ref{notsemi}. Fix a polarization on $A$ defined over $F$. Let $V$ and $\varphi$ be as in the proof of Theorem \ref{notsemi}. Then $V_\ell = V \otimes_{\bold Q} {\bold Q}_\ell$. Let $\varphi_\ell : V_\ell \times V_\ell \to {\bold Q}_\ell$ be the ${\bold Q}_\ell$-linear extension of $\varphi$. It follows immediately from p.~516 of \cite{zarhin} and the definition of $H_\ell$ that $H_\ell \subseteq \mbox{Sp}(V_\ell,\varphi_\ell)$. Let $Z_\ell$ denote the center of $H_\ell$, let $Z_{\mathrm{End}}$ denote the center of $\mbox{End}^0(A)$, and let ${}^\prime$ denote the involution on $\mbox{End}(V_\ell)$ induced by $\varphi_\ell$. Following the proof of Theorem \ref{notsemi}, we conclude that $$Z_\ell({\bold Q}_\ell) \subseteq \{\alpha \in Z_{\mathrm{End}} \otimes_{\bold Q} {\bold Q}_\ell : \alpha\alpha^{\prime} = 1\}.$$ If $A$ is $F$-isogenous to a product of two abelian varieties, then $H_\ell$ is a subgroup of the product of the corresponding groups $H_{1,\ell}$ and $H_{2,\ell}$ for the factors, in such a way that for $i = 1$ and $2$ the restriction to $H_\ell$ of the projection map from $H_{1,\ell} \times H_{2,\ell}$ onto $H_{i,\ell}$ induces a surjective homomorphism from $H_\ell$ onto $H_{i,\ell}$. It follows that we may reduce to the case where $A$ is $F$-simple. If $Z_{\mathrm{End}}$ is totally real, we conclude that $H_\ell$ is semisimple as in the proof of Theorem \ref{notsemi}. Suppose $Z_{\mathrm{End}}$ is a CM-field $K$ and let $K_\ell = K \otimes_{\bold Q} {\bold Q}_\ell$. Let $\psi$ and $U$ be as in the proof of Theorem \ref{notsemi}, let $\psi_\ell : V_\ell \times V_\ell \to K_\ell$ denote the $K_\ell$-Hermitian form which extends the pairing $\psi$, let $\mbox{U}_\ell = \mbox{U} \times {\bold Q}_\ell$, let $\mbox{SU}_\ell$ denote the kernel of the determinant homomorphism $\mbox{det}_{K_\ell} : \mbox{U}_\ell \to \mbox{Res}_{K/{\bold Q}}{\bold G}_m \times {\bold Q}_\ell$, and let $\mbox{End}_{K_\ell}(V_\ell)$ denote the ring of $K_\ell$-linear endomorphisms of $V_\ell$. Then $$\mbox{U}_\ell({\bold Q}_\ell) = \{g \in \mbox{End}_{K_\ell}(V_\ell) : \psi_\ell(g(x),g(y)) = \psi_\ell(x,y) {\text{ for all }} x, y \in V_\ell\}$$ and $H_\ell \subseteq \mbox{U}_\ell \subseteq \mbox{Sp}(V_\ell,\varphi_\ell)$. By Lemma 2.8 of \cite{moonzar}, $H_\ell \subseteq \mbox{SU}_\ell$ if and only if $(A,K,\mathrm{id})$ is of Weil type. The group $\mbox{SU}_\ell({\bold Q}_\ell) \cap K_\ell$ is the finite group of $(\mbox{dim}_{{\bold Q}_\ell}(V_\ell)/\mbox{dim}_{{\bold Q}_\ell}(K_\ell))$-th roots of unity in the ring $K_\ell$. Paralleling the proof of Theorem \ref{notsemi}, $H_\ell \subseteq \mbox{SU}_\ell$ if and only if $H_\ell$ is semisimple. \end{pf} \begin{ex} If $A$ is odd-dimensional and the center of $\mbox{End}^0(A)$ is a CM-field $K$, then $H$ is not semisimple. To show this, note that $A$ is isogenous to a power of a simple odd-dimensional abelian variety $B$ such that $K$ is the center of $\mbox{End}^0(B)$. Then the Hodge groups of $A$ and of $B$ coincide, so we may reduce to the case where $A$ is simple. Let $d = \mbox{dim}(A)$ and use the notation of the proof of Lemma \ref{freeof}. Then $n_\sigma + n_{\bar \sigma} = 2d/[K : {\bold Q}]$. If $H$ were semisimple, then by Theorem \ref{notsemi}, $(A,K,\mathrm{id})$ would be of Weil type. We would therefore have $n_\sigma = n_{\bar \sigma}$, and so $2d/[K : {\bold Q}]$ would be even. However, $d$ is odd and $[K : {\bold Q}]$ is even, so this cannot happen. \end{ex} \section{Abelian varieties having purely multiplicative reduction} \begin{thm} \label{hodsemi} Suppose $A$ is an abelian variety over a subfield $F$ of ${\bold C}$, $v$ is a discrete valuation on $F$, and $A$ has purely multiplicative reduction at $v$. Then the Hodge group $H$ of $A$ is semisimple. \end{thm} \begin{pf} Since $H$ is semisimple if the Hodge groups of each of its $F$-simple components are, we may reduce to the case where $A$ is an $F$-simple abelian variety with purely multiplicative reduction at $v$. Since the properties of having semisimple Hodge group and having purely multiplicative reduction are invariant under finite extensions of the ground field, we may assume $\mbox{End}_F^0(A) = \mbox{End}^0(A)$. Suppose that $H$ is not semisimple. By Theorem \ref{notsemi}, the center of $\mbox{End}^0(A)$ is a CM-field $K$ such that $(A,K,\mathrm{id})$ is not of Weil type, with $\mathrm{id}$ the identity embedding of $K$ in $\mbox{End}^0(A)$. Let $L$ be a fixed algebraic closure of the completion $F_v$ of $F$ at $v$. Since $A$ has purely multiplicative reduction at $v$, $A$ admits a non-archimedean uniformation; i.e., (see \cite{Mumford} and \cite{Raynaud}) there are a discrete subgroup $\Gamma$ of ${\bold G}_m^d(F_v) = (F_v^{\times})^d$, isomorphic to ${\bold Z}^d$, and a $\mbox{Gal}(L/F_v)$-equivariant $v$-adically continuous isomorphism $(L^{\times})^d/\Gamma \cong A(L)$ which for some finite extension $M$ of $F_v$ induces an isomorphism $(M^{\times})^d/\Gamma \cong A(M)$ as $M$-Lie groups. Let ${\cal O}$ be the center of $\mbox{End}(A)$. Then ${\cal O}$ is an order in $K$. By Satz 6 of \cite{Gerritzen}, there is a homomorphism ${\cal O} \hookrightarrow \mbox{End}({\bold G}_m^d)$ which induces the inclusion ${\cal O} \subseteq \mbox{End}(A)$. Composing with the natural homomorphism $$\mbox{End}({\bold G}_m^d) \hookrightarrow \mbox{End}(\mbox{Hom}({\bold G}_m,{\bold G}_m^d))$$ and tensoring with ${\bold Q}$, we have $$K \hookrightarrow \mbox{End}(\mbox{Hom}({\bold G}_m,{\bold G}_m^d) \otimes {\bold Q}).$$ Therefore, the inclusion of $K$ in $\mbox{End}^0(A)$ induces a $K$-vector space structure on $\mbox{Hom}({\bold G}_m,{\bold G}_m^d) \otimes {\bold Q}$. Tensoring with $M$ makes $\mbox{Hom}({\bold G}_m,{\bold G}_m^d) \otimes_{\bold Z} M$ ( $ = M^d$) into a free $(K \otimes_{\bold Q} M)$-module. We can view $(M^\times)^d$ as a (non-archimedean) analytic variety over $M$. The tangent space to $(M^{\times})^d$ at $1$ is isomorphic to $M^d$. By \cite{Morik1} (see also Chapter 2 of \cite{Manin}), $(L^{\times})^d/\Gamma$ can be embedded, via theta functions, as an analytic subvariety of a projective space ${\bold P}^n(L)$, so that the image of $(M^{\times})^d/\Gamma$ is $A(M)$. Let $T$ denote the analytic tangent space at the origin of the analytic variety $A(M)$. The tangent map is an isomorphism $M^d \cong T$. The algebraic tangent space at the origin to the algebraic variety $A$ over $M$ is $\mbox{Lie}_M(A) = \mbox{Lie}_F(A) \otimes_F M$, and there is a canonical isomorphism between the analytic and algebraic tangent spaces to $A(M)$ (see subsection 3 of \S 2 of Chapter II of \cite{shaf}). Therefore, the identity embedding of $K$ into $\mbox{End}^0(A)$ makes $\mbox{Lie}_M(A)$ into a free $(K \otimes_{\bold Q} M)$-module. By Lemma \ref{freeof}, $(A,K,\mathrm{id})$ is of Weil type, contradicting our assumptions. \end{pf} Theorem \ref{hodsemi} remains true if we replace the assumption that $v$ is a discrete valuation by the assumption that $v$ is a valuation of rank $1$ and $A$ admits non-archimedean uniformization (Gerritzen's theorem remains true under these assumptions). \section {Bounds on torsion of abelian varieties which do not have purely multiplicative reduction} \label{boundssect} It is easy to find uniform bounds on orders of torsion points over number fields for abelian varieties with potential good reduction, or for elliptic curves which do not have multiplicative reduction, at a given discrete valuation (see \cite{contemp}, \cite{CM}, \cite{flexoest}). In this section we extend these results by finding bounds on torsion subgroups of abelian varieties which do not have purely multiplicative reduction at a given discrete valuation. Suppose $A$ is a $d$-dimensional abelian variety over a field $F$, $v$ is a discrete valuation on $F$ with finite residue field $k$ of order $q$, $n$ is a positive integer relatively prime to $q$, and $J$ is a non-zero subgroup of the group $A_n(F)$ of points in $A(F)$ of order dividing $n$. Let $A_v^0$ denote the connected component of the identity of the special fiber $A_v$ of the N\'eron minimal model of $A$ at $v$. Let $a$, $u$, and $t$ denote respectively the abelian, unipotent, and toric ranks of $A_v^0$. Then $d = a + t + u$. If $\lambda$ is a polarization on $A$ defined over an extension of $F$ which is unramified over $v$, define a skew-symmetric Galois-equivariant pairing $e_{\lambda,n}$ on $A_n$ by $e_{\lambda,n}(x,y) = e_n(x,\lambda(y))$, where $e_n$ is the Weil pairing. If $J$ is not isotropic with respect to $e_{\lambda,n}$, and $n$ is a prime number, then $\zeta_n \in F$, so the prime $n$ can be bounded independent of $A$ (with a bound depending on $F$). Therefore, the more interesting case is when $J$ is an isotropic subgroup of $A_n(F)$. If $J$ is a maximal isotropic subgroup of $A_n(F)$, and $A$ does not have semistable reduction at $v$, then $n \le 4$, by Theorem 6.2 of \cite{semistab}. The remaining case to consider is the case where $A$ has semistable reduction at $v$. Theorem \ref{bounds} below implies that in this case we can bound $n$ in terms of $q$ and $d$, as long as $A$ does not have purely multiplicative reduction at $v$. Note that if $P$ is a point of $A(F)$ of order $n$ which reduces to a point of $A_v^0$, then $n$ is bounded above by $\#A_v^0(k)$. Therefore even in the case of abelian varieties with purely multiplicative reduction one can easily bound, by a constant depending only on $d$ and $q$, the orders of torsion points whose reductions lie in $A_v^0$. As was the case for elliptic curves, the most difficult case is the case when the reduction is purely multiplicative and the reductions of the torsion points do not lie in the identity connected component of the special fiber of the N\'eron minimal model. \begin{lem}[Lemma 1 on pp.~494--495 of \cite{serretate}] \label{serretatelem} If $A$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$, and $n$ is a positive integer relatively prime to the residue characteristic of $v$, then $(A_v^0)_n$ is a free ${\bold Z}/n{\bold Z}$-module of rank $2a + t$. \end{lem} \begin{prop} \label{lessthan} Suppose $A$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$ with finite residue field $k$ of order $q$, $n$ is a positive integer relatively prime to $q$, and $J$ is a subgroup of $A_n(F)$. Suppose there is a positive constant $\epsilon$ such that $|J| \ge n^{t+2u+\epsilon}$. Then $n \le (\#(A_v^0)_n(k))^{1/\epsilon} \le (\#A_v^0(k))^{1/\epsilon}$. \end{prop} \begin{pf} Let $d = \mbox{dim}(A)$. Via the reduction map we may view $A_n(F)$, and therefore $J$, as a subgroup of $(A_v)_n$ (see \cite{serretate}). Therefore, $\#J \#(A_v^0)_n$ divides $n^{2d} \#(J \cap (A_v^0)_n)$. Thus by Lemma \ref{serretatelem}, $\#J$ divides $n^{t+2u}\#(J \cap (A_v^0)_n)$. Therefore, $$n^\epsilon \le \#(J \cap (A_v^0)_n) \le \#(A_v^0)_n(k) \le \#A_v^0(k).$$ \end{pf} \begin{thm} \label{bounds} Suppose $A$ is a $d$-dimensional abelian variety over a field $F$, $v$ is a discrete valuation on $F$ with finite residue field $k$ of order $q$, $n$ is a positive integer relatively prime to $q$, and $A_n(F)$ has a subgroup of order $n^d$. Suppose the reduction of $A$ at $v$ is semistable but not purely multiplicative. Then $$n \le (1 + \sqrt{q})^{2a}(1 + q)^t \le (1 + \sqrt{q})^{2d}.$$ \end{thm} \begin{pf} Since $A$ has semistable reduction at $v$, $u = 0$. Since the reduction of $A$ at $v$ is not purely multiplicative, $a \ge 1$. Applying Proposition \ref{lessthan} with $\epsilon = 1$, we have $n \le \#A_v^0(k)$. Since $A$ has semistable reduction at $v$, $A_v^0$ is an extension of an abelian variety $B$ by a torus $T$. We have the Weil bound $\# B(k) \le (1 + \sqrt{q})^{2a}$. Similarly, we have the bound $\# T(k) \le (1 + q)^t$, as follows. Let $X$ be the group of characters of $T \otimes {\bar k}$. The Frobenius element of $\mbox{Gal}({\bar k}/k)$ acts on $X$, say by $\varphi_0$. Since the torus $T$ splits over some finite extension of $k$, $\mbox{Gal}({\bar k}/k)$ acts on $X$ through a finite quotient, so all the eigenvalues of $\varphi_0$ have absolute value $1$. Therefore all eigenvalues of $q - \varphi_0$ are non-zero and have absolute value at most $1 + q$. We have (see Theorem 6.2 in \S 1 of Chapter VI of \cite{Vo}) $$\# T(k) = |\mbox{det}(q - \varphi_0)| \le (1 + q)^t.$$ Therefore, $$\#A_v^0(k) \le (1 + \sqrt{q})^{2a}(1 + q)^t = (1 + \sqrt{q})^{2a}(1 + q)^{d-a} \le (1 + \sqrt{q})^{2d}.$$ \end{pf} If $c$ and $d$ are positive integers, let $f(c,d)$ be the maximum of the orders of the elements of $GL_{2d}({\bold Z}/c{\bold Z})$. \begin{thm} \label{nrbounds} Suppose $A$ is a $d$-dimensional abelian variety over a number field $F$ of degree $m$, $v$ is a discrete valuation on $F$ at which $A$ does not have purely multiplicative reduction, $p$ is the residue characteristic of $v$, $n$ and $r$ are positive integers not divisible by $p$, $r \ge 3$, and $A_n(F)$ has a subgroup of order $n^d$. Then $$n \le (1 + p^{mf(r,d)/2})^{2d}.$$ \end{thm} \begin{pf} By a theorem of Raynaud (Proposition 4.7 of \cite{SGA}), $A$ has semistable reduction at the discrete valuations on the field $F(A_{r})$ of residue characteristic not dividing $r$. Let $v^\prime$ be a valuation on $F(A_r)$ extending $v$, and let $k$ be the corresponding residue field. Then $\# k$ divides $p^{mf}$, where $f$ is the order of Frobenius at $v^\prime$ in $\mbox{Gal}(F(A_r)/F)$. Since $\mbox{Gal}(F(A_r)/F)$ injects into $GL_{2d}({\bold Z}/r{\bold Z})$, $\# k$ divides $p^{mf(r,d)}$. The result now follows from Theorem \ref{bounds}. \end{pf} \begin{cor} \label{Hsemibds} Suppose $A$ is a $d$-dimensional abelian variety over a number field $F$ of degree $m$, and suppose the Hodge group of $A$ is not semisimple. Suppose $n$ is a positive integer and $A_n(F)$ has a subgroup of order $n^d$. Then $$n \le [(1 + 2^{mf(3,d)/2})(1 + 3^{mf(4,d)/2})]^{2d} < (1+10^{-11})\cdot (2^{3^{4d^2}}\cdot 3^{4^{4d^2}})^{md}.$$ \end{cor} \begin{pf} The result follows from Theorem \ref{hodsemi}, by applying Theorem \ref{nrbounds} with $p = 2$, $r = 3$ to bound the prime-to-two part of $n$, and with $p = 3$, $r = 4$ to bound the prime-to-three part of $n$. The final inequality follows from the bound $f(c,d) \le \# GL_{2d}({\bold Z}/c{\bold Z}) < c^{4d^2}$. \end{pf} The bounds on $n$ given in Corollary \ref{Hsemibds} were shown in Theorem 3.3 and Remark 2 of \cite{contemp} to be bounds on the orders of torsion subgroups of abelian varieties with potential good reduction at discrete valuations of residue characteristics $2$ and $3$. If we assume the existence of a polarization on $A$ of degree prime to $n$ (for example, a principal polarization) we obtain stronger bounds. The following results give bounds on torsion subgroups of order prime to the degree of a given polarization. \begin{thm} \label{lambdabounds} Suppose $A$ is a $d$-dimensional abelian variety over a number field $F$ of degree $m$, $v$ is a discrete valuation on $F$ at which $A$ does not have purely multiplicative reduction, $p$ is the residue characteristic of $v$, $\ell$ is a prime number, $\ell \ne p$, $J$ is a subgroup of $A_\ell(F)$ of order $\ell^d$, $\lambda$ is a polarization on $A$ defined over an extension of $F$ unramified at $v$, and $\ell$ does not divide the degree of $\lambda$. Then $$\ell \le (1 + p^{m/2})^{2d}.$$ \end{thm} \begin{pf} Since $\ell$ does not divide the degree of $\lambda$, the pairing $e_{\lambda,\ell}$ is nondegenerate. If $J$ is not isotropic with respect to $e_{\lambda,\ell}$, then $F$ contains a primitive $\ell$-th root of unity. Therefore $\ell - 1$ divides $[F : {\bold Q}]$, so $\ell \le 1 + m$. Suppose $J$ is isotropic with respect to $e_{\lambda,\ell}$. Then $J$ is a maximal isotropic subgroup of $A_\ell$ (since $\# J = \ell^d$ and $e_{\lambda,\ell}$ is nondegenerate). If $A$ does not have semistable reduction at $v$, then $\ell \le 3$ by Theorem 6.2 of \cite{semistab}. If $A$ has semistable reduction at $v$, then $\ell \le (1 + p^{m/2})^{2d}$ by Theorem \ref{bounds}. The result now follows since $(1 + p^{m/2})^{2d}$ is greater than $3$ and than $1 + m$. \end{pf} \begin{cor} \label{Hsemilambdabds} Suppose $(A,\lambda)$ is a $d$-dimensional polarized abelian variety over a number field $F$ of degree $m$, and suppose the Hodge group of $A$ is not semisimple. Suppose $\ell$ is a prime number which does not divide the degree of $\lambda$, and $J$ is a subgroup of $A_\ell(F)$ of order $\ell^d$. Then $$\ell \le (1 + 2^{m/2})^{2d}.$$ \end{cor} \begin{pf} By Theorem \ref{hodsemi}, $A$ does not have purely multiplicative reduction at any discrete valuations. Since $2 < (1 + 2^{m/2})^{2d}$, we may assume $\ell$ is an odd prime, and we obtain the result by applying Theorem \ref{lambdabounds} with $p = 2$. \end{pf} The proof of Theorem \ref{lambdabounds} shows that $\ell \le \max\{ 1+m, (1+\sqrt{q})^{2d}\}$, where $q$ is the order of the residue field of $v$. Therefore in Corollary \ref{Hsemilambdabds} we can conclude that $\ell \le \max\{ 1+m, (1+\sqrt{f})^{2d}\}$, where $f$ is the minimal order of the residue fields of the valuations on $F$ of residue characteristic $2$. \begin{cor} Suppose $(A,\lambda)$ is a $d$-dimensional polarized abelian variety over a number field $F$ of degree $m$, and suppose the Hodge group of $A$ is not semisimple. Suppose $n$ is a positive integer relatively prime to the degree of $\lambda$, and $J$ is a subgroup of $A_n(F)$ of order $n^d$ which is a maximal isotropic subgroup with respect to $e_{\lambda,n}$. Then $$n \le (1 + 2^{m/2})^{2d}(1 + 3^{m/2})^{2d}.$$ \end{cor} \begin{pf} By Theorem \ref{hodsemi}, $A$ does not have purely multiplicative reduction at any discrete valuations. The prime-to-$p$ part of $n$ is bounded above by $4$ if there is a valuation on $F$ of residue characteristic $p$ at which $A$ does not have semistable reduction (by Theorem 6.2 of \cite{semistab}), and otherwise is bounded above by $(1 + p^{m/2})^{2d}$ (by Theorem \ref{bounds}). Note that $4 < (1 + p^{m/2})^{2d}$. The result follows by letting $p = 2, 3$. \end{pf}

9505

alg-geom/9505016

en

https://arxiv.org/abs/alg-geom/9505016

alg-geom/9505016

Gerd Dethloff

Gerd Dethloff

Iitaka-Severi's Conjecture for Complex Threefolds

This paper has been withdrawn by the author. Withdrawn since its content had been subsumed (with improvement) in arXiv:alg-geom/9608033

We prove the following generalization of Severi's Theorem: Let $X$ be a fixed complex variety. Then there exist, up to birational equivalence, only finitely many complex varieties $Y$ of general type of dimension at most three which admit a dominant rational map $f$ from $X$ to Y$.

alg-geom

\section{Introduction} \noindent Let $X$ and $Y$ be algebraic varieties, i.e. complete integral schemes over a field of characteristic zero, and denote by $R(X,Y)$ the set of all dominant rational maps $f:X \rightarrow Y$. Moreover denote by ${\cal F} = {\cal F}(X)$ the set $\{ f: X \rightarrow Y|$ $f$ is a dominant rational map onto an algebraic variety $Y$ of general type$\}$ and by ${\cal F}_m = {\cal F}_m(X)$ the set $\{ f: X \rightarrow Y|$ $f$ is a dominant rational map and $Y$ is birationally equivalent to a nonsingular algebraic variety for which the $m$-th pluricanonical mapping is birational onto its image $\}$. We introduce an equivalence relation $\sim$ on the sets ${\cal F}$ and ${\cal F}_m$ as follows: $(f:X \rightarrow Y) \sim (f_1:X \rightarrow Y_1)$ iff there exists a birational map $b:Y \rightarrow Y_1$ such that $b \circ f = f_1$.\\ The classical theorem of Severi can be stated as follows (cf.\cite{Sa}): \begin{theo} \label{1.1} For a fixed algebraic variety $X$ there exist only finitely many hyperbolic Riemann surfaces $Y$ such that $R(X,Y)$ is nonempty. \end{theo} \noindent We may ask if a finiteness theorem of this kind also can be true in higher dimensions. This leads to the following: \begin{conj} \label{1.2} For a fixed variety $X$ there exist, up to birational equivalence, only finitely many varieties $Y$ of general type such that $R(X,Y)$ is nonempty. \\ Moreover, the set ${\cal F}/ \sim$ is a finite set. \end{conj} \noindent Maehara calls this conjecture Iitaka's Conjecture based on Severi's theorem (cf. \cite{Ma3}), and we abbreviate this as Iitaka-Severi's Conjecture. In \cite{Ma3} Maehara states the Conjecture more generally for algebraic varieties (over any field) and separable dominant rational maps. He also mentioned that K. Ueno proposed that a variety of general type could be replaced by a polarized non uniruled variety in this Conjecture.\\ Maehara proved in Proposition 6.5. in \cite{Ma2} that in characteristic zero the Conjecture is true if one restricts the image varieties $Y$ to such varieties that can be birationally embedded by the $m$-th pluricanonical map for any given $m$, i.e. ${\cal F}_m / \sim$ is finite for all $m$. This especially proves the Conjecture for surfaces $Y$ (take $m=5$). Furthermore Maehara shows that one can find a fixed $m$ such that for all {\bf smooth} varieties $Y$ which have nef and big canonical bundle the m-th pluricanonical map is a birational embedding, which proves the Conjecture also in this case. Earlier Deschamps and Menegaux \cite{DM2}, \cite{DM3} proved, in characteristic zero, the cases where the varieties $Y$ are surfaces which satisfy $q >0$ and $P_g \geq 2$, or where the maps $f:X \rightarrow Y$ are morphisms. In this direction Maehara \cite{Ma1} also showed finiteness of isomorphism classes of smooth varieties with ample canonical bundles which are dominated by surjective morphisms from a fixed variety.\\ There is a related classical result due to de Franchis \cite{Fr} which states that for any Riemann surface $X$ and any fixed hyperbolic Riemann surface $Y$ the set $R(X,Y)$ is finite. At the same time he gives an upper bound for $\#R(X,Y)$ only in terms of $X$. The generalization of this theorem to higher dimensions is not a conjecture any more: Kobayashi and Ochiai \cite{KO} proved that if $X$ is a Moisheson space and $Y$ a compact complex space of general type, then the set of surjective meromorphic maps from $X$ to $Y$ is finite. Deschamps and Menegaux \cite{DM1} proved that if $X$ and $Y$ are smooth projective varieties over a field of arbitrary characteristic, and $Y$ is of general type, then $\#R(X,Y)$ is finite (where one has additionally to assume that the dominant rational maps $f:X \rightarrow Y$ are separable). \\ {}From these results it follows that the second part of Conjecture \ref{1.2} is a consequence of the first part, hence we only have to deal with the first part.\\ Bandman \cite{Ba1}, \cite{Ba2} and Bandman and Markushevich \cite{BM} also generalized the second part of de Franchis' theorem, proving that for projective varieties $X$ and $Y$ with only canonical singularities and nef and big canonical line bundles $K_X$ and $K_Y$ the number $\#R(X,Y)$ can be bounded in terms of invariants of $X$ and the index of $Y$.\\ Another generalization of the (first part of) de Franchis' theorem was given by Noguchi \cite{No}, who proved that there are only finitely many surjective meromorphic mappings from a Zariski open subset $X$ of an irreducible compact complex space onto an irreducible compact hyperbolic complex space $Y$. Suzuki \cite{Su} generalized this result to the case where $X$ and $Y$ are Zariski open subsets of irreducible compact complex spaces $\overline{X}$ and $\overline{Y}$ and $Y$ is hyperbolically embedded in $\overline{Y}$. These results can be generalized to finiteness results for nontrivial sections in hyperbolic fiber spaces. But since a more precise discussion would lead us too far from the proper theme of this paper, we refer the interested reader to Noguchi \cite{No} and Suzuki \cite{Su}, or to the survey \cite{ZL} of Zaidenberg-Lin, where he also can find an overview for earlier results which generalized de Franchis' theorem.\\ It is a natural question if Conjecture \ref{1.2} can also be stated in terms of complex spaces. In \cite{No} Noguchi proposed the following: \begin{conj} \label{1.3} Let $X$ be a Zariski open subset of an irreducible compact complex space. Then the set of compact irreducible hyperbolic complex spaces $Y$ which admit a dominant meromorphic map $f:X \rightarrow Y$ is finite. \end{conj} Let us now return to Conjecture \ref{1.2}. In this paper we are only interested in the case of complex varieties. Since we want to prove finiteness only up to birational equivalence, we may assume without loss of generality that $X$ and all $Y$ in the Conjecture are nonsingular projective complex varieties, by virtue of Hironaka's resolution theorem \cite{Hi}, cf. also \cite{Ue}, p.73. Now fix a complex projective variety $X$. We define ${\cal G}_m:=\{$ a nonsingular complex projective variety $Y$ : the $m$-th pluricanonical map $\Phi_m: Y \rightarrow \Phi_m(Y)$ is birational onto its image and there exists a dominant rational map $f:X \rightarrow Y \} $. In order to show Conjecture \ref{1.2} it is sufficient, by Proposition 6.5. of Maehara \cite{Ma2}, to show the following \begin{conj} \label{1.4} There exists a natural number $m$ only depending on $X$ such that all smooth complex projective varieties $Y$ of general type which admit a dominant rational map $f:X \rightarrow Y$ belong to ${\cal G}_m$. \end{conj} \noindent We will prove that Conjecture \ref{1.4} is true for varieties $Y$ which are of dimension three, thus we prove Iitaka-Severi's Conjecture for complex 3-folds. Since for varieties $Y$ of dimension one resp. two we can take $m=3$ resp. $m=5$, our main theorem is: \begin{theo} \label{1.5} Let $X$ be a fixed complex variety. Then there exist, up to birational equivalence, only finitely many complex varieties $Y$ of general type of dimension at most three which admit a dominant rational map $f:X \rightarrow Y$.\\ Moreover the set ${\cal F}/ \sim$ is a finite set if one resticts to complex varieties $Y$ of dimension at most three . \end{theo} \noindent As Maehara \cite{Ma3}, p.167 pointed out already, in order to prove Conjecture \ref{1.4} it is enough to show that for all varieties $Y$ there exists a minimal model and the index of these minimal models can be uniformly bounded from above by a constant only depending on $X$. Since in dimension three minimal models and even canonical models do exist, the problem is reduces to the question how to bound the index.\\ But it turns out that one is running into problems if one directly tries to bound the indices of the canonical models $Y_c$ of threefolds $Y$, only using that they are all dominated by dominant rational maps from a fixed variety $X$. So we will proceed in a different way:\\ The first step of the proof is to show that the Euler characteristic $\chi (Y, {\cal O}_{Y})$ is uniformly bounded by an entire constant $C$ depending only on $X$ (Proposition 3.2), that is how we use the fact that all threefolds $Y$ are dominated by a fixed variety $X$. \\ In the second step of the proof, we show that we can choose another entire constant $R$, also only depending on $X$, such that for any threefold $Y$ of general type for which the Euler characteristic is bounded by $C$ the following holds (Proposition 3.3): Either the index of the canonical model $Y_c$ of $Y$ divides $R$ (first case) or the pluricanonical sheaf ${\cal O}_{Y_c}((13C)K_{Y_c})$ has two linearly independant sections on $Y_c$ (second case). In order to prove this Proposition, we use the Plurigenus Formula due to Barlow, Fletcher and Reid and estimates of some terms in this formula due to Fletcher. In the first case the index is bounded, and we are done (Proposition \ref{2.8}).\\ The third step of the proof deals with the second case. Here we remark that the two linearly independant sections on $Y_c$ can be lifted to sections in $H^0 (Y,{\cal O}_{Y}(mK_{Y}))$, and then we can apply a theorem of Kollar \cite{Ko} which states that now the $(11m+5)$-th pluricanonical map gives a birational embedding (Proposition 3.4), and we are also done in the second case. \\ Hence we do not prove directly that under our assumptions the index is uniformly bounded, we prove that if it is not, then there is some other way to show that some fixed pluricanonical map gives a birational embedding. The fact that the index actually has such a uniform bound then follows as a result of Theorem \ref{1.5}.\\ It finally might be worth while to point out that the second and the third step of our proof actually yield: \begin{theo} Let $C$ be a positive entire constant. Define $R={\rm lcm}(2,3,...$ $,26C-1)$ and $m={\rm lcm}(18R+1,143C+5)$. Then for all smooth projective 3-folds of general type for which the Euler characteristic is bounded above by $C$ the $m$-th pluricanonical map is birational onto its image. \end{theo} \noindent Despite the fact that our $m=m(C)$ is explicit, it is so huge that it is only of theoretical interest. For example for $C=1$ it is known by Fletcher \cite{Fl} that one can choose $m=269$, but for $C=1$ our $m$ is already for of the size $10^{12}$. Moreover J.P. Demailly recently told me that he conjectures that for 3-folds of general type any $m \geq 7$ should work, independantly of the size of the Euler characteristic.\\ The paper is organized as follows: In section 2 we collect, for the convenience of the reader and also for fixing the notations, the basic facts from canonical threefolds which we need. We try to give precise references to all these facts, but do not try to trace these facts back to the original papers. Where we could not find such references we give short proofs. However we expect that all these facts should be standard to specialists on threefolds. In section 3 we give the proof of Theorem \ref{1.5}.\\ The author would like to thank S.Kosarew (Grenoble) for pointing out Noguchi's Conjecture \ref{1.3} to him. This was his starting point for working on problems of this kind. He would also like to thank F.Catanese (Pisa) for pointing out Fletcher's paper \cite{Fl} to him, since this paper later gave him the motivation for the key step in the proof of Theorem \ref{1.5}. He finally would like to thank the Institut Fourier in Grenoble, the University of Pisa and the organizers of the conference Geometric Complex Analysis in Hayama for inviting him, since this gave him the possibility to discuss with many specialists. \section{Some Tools from the Theory of 3-folds} Let $Y$ be a normal complex variety of dimension $n$, $Y_{reg}$ the subspace of regular points of $Y$ and $j: Y_{reg} \hookrightarrow Y$ the inclusion map. Then the sheaves ${\cal O}_Y(mK_Y)$ are defined as $$ {\cal O}_Y(mK_Y) := j_*((\Omega^n_{Y_{reg}})^{\otimes m})$$ Equivalently ${\cal O}_Y(mK_Y)$ can be defined as the sheaf of $m$-fold tensor products of rational canonical differentials on $Y$ which are regular on $Y_{reg}$. The $mK_Y$ can be considered as Weil divisors. For this and the following definitions, cf. \cite{Re2} and \cite{Mo1}. \begin{defi} \label{2.1} $Y$ has only {\bf canonical} singularities if it satisfies the following two conditions:\\ i) for some integer $r \ge 1$, the Weil divisor $rK_Y$ is a Cartier divisor.\\ ii) if $f: \tilde{Y} \rightarrow Y $ is a resolution of $Y$ and $\{ E_i \}$ the family of all exceptional prime divisors of $f$, then $$ rK_{\tilde{Y}} = f^*(rK_Y) + \sum a_iE_i $$ with $a_i \geq 0$. \\ If $a_i >0$ for every exceptional divisor $E_i$, then $Y$ has only {\bf terminal} singularities.\\ The smallest integer $r$ for which the Weil divisor $rK_Y$ is Cartier is called the {\bf index} of $Y$. \end{defi} \begin{defi} \label{2.2} A complex projective algebraic variety $Y$ with only canonical (resp. terminal) singularities is called a {\bf canonical} (resp. {\bf minimal}) model if $K_Y$ is an ample (resp. a nef) $I \!\!\! Q$-divisor.\\ We say that a variety $Z$ has a {\bf canonical (resp. minimal) model} if there exists a canonical (resp. minimal) model which is birational to $Z$. \end{defi} Later on we will need the following theorem due to Elkik \cite{El} and Flenner \cite{Fle}, 1.3 (cf. \cite{Re2}, p.363): \begin{theo} \label{2.3} Canonical singularities are rational singularities. \end{theo} The first part of the following theorem, which is of high importance for the theory of 3-folds, was proved by Mori \cite{Mo2}, the second part follows from the first part by works of Fujita \cite{Fu}, Benveniste \cite{Be} and Kawamata \cite{Ka}: \begin{theo} \label{2.4} Let $Y$ be a non singular projective 3-fold of general type.\\ i) There exists a minimal model of $Y$.\\ ii) There exists a unique canonical model of $Y$, the canonical ring $R(Y,K_Y)$ of $Y$ is finitely generated, and the canonical model is just ${\rm Proj} R(Y,K_Y)$. \end{theo} We have the following Plurigenus Formula due to Barlow, Fletcher and Reid (cf. \cite{Fl}, \cite{Re2}, see also \cite{KM}, p.666 for the last part): \begin{theo} \label{2.5} Let $Y$ be a projective 3-fold with only canonical singularities. Then we have $$ \chi (Y, {\cal O}_Y(mK_Y)) = \frac{1}{12}(2m-1)m(m-1)K_Y^3 - (2m-1) \chi (Y, {\cal O}_Y) + \sum_Q l(Q,m)$$ with $$ l(Q,m) = \sum_{k=1}^{m-1} \frac{\overline{bk}(r- \overline{bk})}{2r}$$ Here the summation takes place over a basket of singularities $Q$ of type $\frac{1}{r}(a,-a,1)$ (see below for these notations). $\overline{j}$ denotes the smallest nonnegative residue of $j$ modulo $r$, and $b$ is chosen such that $\overline{ab}=1$.\\ Furthermore we have $$ {\rm index}(Y) = {\rm lcm}\{r=r(Q): \: Q\in {\rm basket} \} $$ \end{theo} \noindent A singularity of type $\frac{1}{r}(a,-a,1)$ is a cyclic quotient singularity $ I \!\!\!\! C^3 / \mu_r$, where $\mu_r$ denotes the cyclic group of $r$th roots of unity in $ I \!\!\!\! C$, and $\mu_r$ acts on $ I \!\!\!\! C^3$ via $$\mu_r \ni \epsilon: (z_1,z_2,z_3) \rightarrow (\epsilon^a z_1, \epsilon^{-a}z_2, \epsilon z_3)$$ Reid introduced the term `basket of singularities' in order to point out that the singularities $Q$ of the basket are not necessarily singularities of $Y$, but only `fictitious singularities'. However the singularities of $Y$ make the same contribution to $\chi (Y, {\cal O}_Y(mK_Y))$ as if they were those of the basket, hence we also can work with the singularities of the basket, which have the advantage that their contributions are usually easier than those of the original singularities. More precisely, one can pass from $Y$ to a variety where the singularities of the basket actually occur by a crepant partial resolution of singularities and then by a flat deformation. For the details cf. \cite{Re2}, p.404, 412. \\ \noindent For estimating from below the terms $l(Q,m)$ in the Plurigenus Formula, we will need two Propositions due to Fletcher \cite{Fl}. In those Propositions $[s]$ denotes the integral part of $s \in I \!\! R$. \begin{prop} \label{2.6} $$l( \frac{1}{r}(1,-1,1), m) = \frac{\overline{m} (\overline{m} -1)( 3r+1-2 \overline{m})}{12r} + \frac{r^2-1}{12} [\frac{m}{r}]$$ \end{prop} \begin{prop} \label{2.7} For $\alpha, \beta \in Z \!\!\! Z$ with $0 \leq \beta \leq \alpha$ and for all $m \leq [(\alpha +1)/2]$, we have: $$ l(\frac{1}{\alpha} (a,-a,1), m) \geq l(\frac{1}{\beta} (1,-1,1),m)$$ \end{prop} At last, we want to give a proof for Maehara's remark \cite{Ma3}, p.167 which we already mentioned, namely that it is enough to show that the index of the canonical models of the varieties $Y$ can be uniformly bounded from above by a constant only depending on $X$. We prove more precisely: \begin{prop} \label{2.8} Let $Y$ be a smooth projective 3-fold of general type and $l$ a natural number such that $l$ is an integer multiple of the index $r$ of the canonical model $Y_c$ of $Y$. Then the $(18l+1)$-th pluricanonical map is birational onto its image. \end{prop} {\bf Proof:} We could pass from the canonical model $Y_c$ of $Y$ to a minimal model $Y_m$ and then apply Corollary 4.6 of the preprint \cite{EKL} of Ein, K\"uchle and Lazarsfeld. But since it might even be easier we want to pass directly from $Y_c$ to $Y$, and then apply the corresponding result of Ein, K\"uchle and Lazarsfeld for smooth projective 3-folds, namely Corollary 3. of \cite{EKL}.\\ Since $l$ is a multiple of the index of $Y_c$, $lK_{Y_c}$ is an ample line bundle. Since we only are interested in $Y$ up to birational equivalence we may assume that $\pi : Y \rightarrow Y_c$ is a desingularization. Since the bundle $lK_{Y_c}$ on $Y_c$ is ample, the pulled back bundle $\pi^*(lK_{Y_c})$ on $Y$ is still nef and big. Hence we can apply Corollary 3. of \cite{EKL} to this bundle and get that the map obtained by the sections of the bundle $K_Y + 18\pi^*(lK_{Y_c})$ maps $Y$ birationally onto its image. But since by the Definition 2.1 of canonical singularities every section of the bundle $K_Y + 18\pi^*(lK_{Y_c})$ is also a section of $K_Y + 18lK_Y = (1+18l)K_Y$, the claim follows. \qed {\bf Remark:} Notice that for Proposition 2.8 we do not need the assumption that the smooth projective 3-folds $Y$ are dominated by a fixed complex variety $X$. This only will be needed to bound the indices of the canonical models $Y_c$ of the $Y$. \section{Bounding the Index of a Dominated Canonical 3-fold} In order to prove Theorem \ref{1.5}, it is enough to prove the following \begin{theo} \label{3.1} Let $X$ be a fixed smooth complex variety. Then there exists a natural number $m$ depending only on $X$ such that all smooth complex projective 3-folds $Y$ of general type which admit a dominant rational map $f:X \rightarrow Y$ belong to ${\cal G}_m$. \end{theo} \noindent Here, as in the introduction, ${\cal G}_m$ is defined as ${\cal G}_m:=\{$ a nonsingular complex projective variety $Y$ : the $m$-th pluricanonical map $\Phi_m: Y \rightarrow \Phi_m(Y)$ is birational onto its image and there exists a dominant rational map $f:X \rightarrow Y \} $. \\ The rest of this chapter is devoted to the proof of Theorem \ref{3.1}. We denote by $Y_c$ the canonical model of $Y$. Furthermore we may assume without loss of generality that $\pi : Y \rightarrow Y_c$ is a desingularization (since we only need to look at those smooth projective varieties $Y$ up to birational equivalence).\\ In the first step of the proof we show: \begin{prop} \label{3.2} Under the assumptions of Theorem \ref{3.1} there exists an entire constant $C \geq 1$ only depending on $X$, such that for all $Y$ we have $ \chi (Y, {\cal O}_Y) = \chi (Y_c, {\cal O}_{Y_c}) \leq C$. \end{prop} {\bf Proof:} First we get by Hodge theory on compact K\"ahler manifolds (cf. \cite{GH}, or \cite{Ii}, p.199): $$ h^i(Y, {\cal O}_Y) = h^0(Y, \Omega_Y^i), \:\: i=0,1,2,3$$ Now by Theorem 5.3. in Iitaka's book \cite{Ii}, p.198 we get that $$ h^0(Y, \Omega_Y^i) \leq h^0(X, \Omega_X^i), \:\: i=0,1,2,3$$ Hence by the triangle inequality we get a constant $C$, only depending on $X$, such that $$| \chi (Y, {\cal O}_Y) | \leq C$$ Now by the theorem of Elkik and Flenner (Theorem \ref{2.3}) $Y_c$ has only rational singularities, hence by degeneration of the Leray spectral sequence we get that $$ \chi (Y, {\cal O}_Y) = \chi (Y_c, {\cal O}_{Y_c})$$ This finishes the proof of Proposition \vspace{.5cm} \ref{3.2}. \qed In the second step of the proof of Theorem \ref{3.1} we show: \begin{prop} \label{3.3} Let $C\geq 1$ be an entire constant, $R := {\rm lcm}(2,3,..., 26C-1)$ and $m_1=18R+1$. Then for all smooth projective complex 3-folds $Y$ of general type with $\chi (Y, {\cal O}_Y) \leq C$ we have\\ either $Y \in {\cal G}_{m_1}$ or $h^0(Y_c, {\cal O}_{Y_c}((13C)K_{Y_c})) \geq 2$. \end{prop} {\bf Proof:} We distinguish between two cases: The first case is that the index of $Y_c$ divides $R$. Then applying Proposition \ref{2.8} we get that $Y \in {\cal G}_{m_1}$ and we are done. The second case is that the index does not divide $R$. Then in the Plurigenus Formula Theorem \ref{2.5} of Barlow, Fletcher and Reid we necessarily have at least one singularity $\tilde{Q}$ in the basket of singularities which is of the type $\frac{1}{r}(a,-a,1)$ with $r \geq 26C$. Now applying first a vanishing theorem for ample sheaves (cf. Theorem 4.1 in \cite{Fl}), the fact that $K^3_{Y_c} >0$ (since $K_{Y_c}$ is an ample $I \!\!\! Q$-divisor) and then the Propositions \ref{2.6} and \ref{2.7} due to Fletcher, we get: $$ h^0(Y_c, {\cal O}_{Y_c}((13C)K_{Y_c}))$$ $$ = \chi (Y_c, {\cal O}_{Y_c}((13C)K_{Y_c}))$$ $$ \geq (1-26C)\chi (Y_c, {\cal O}_{Y_c}) + \sum_{ Q \in {\rm basket}} l(Q,13C)$$ $$ \geq (1-26C)C + l(\tilde{Q},13C)$$ $$ \geq (1-26C)C + l(\frac{1}{26C}(1,-1,1),13C)$$ $$ = (1-26C)C + \frac{13C(13C-1)(78C+1-26C)}{312C}$$ $$ = \frac{312C^2 - 8112C^3 + 8788C^3 - 507C^2 -13C}{312C}$$ $$ = \frac{52C^2 - 15C - 1}{24}$$ $$ \geq \frac{36}{24} = 1.5 $$ \noindent The last inequality is true since $C\geq 1$. Since $ h^0(Y_c, {\cal O}_{Y_c}((13C)K_{Y_c}))$ is an entire, this finishes the proof of Proposition \vspace{.5cm} \ref{3.3}. \qed In the third step of the proof of Theorem \ref{3.1} we show: \begin{prop} \label{3.4} Assume that for a smooth projective complex 3-fold $Y$ of general type we have $h^0(Y_c, {\cal O}_{Y_c}((13C)K_{Y_c})) \geq 2$. Then $Y \in {\cal G}_{m_2}$ with $m_2=143C+5$. \end{prop} {\bf Proof:} Kollar proved that if $h^0(Y, {\cal O}_Y(lK_Y )) \geq 2$ then the $(11l+5)$-th pluricanonical map is birational onto its image (Corollary 4.8 in \cite{Ko}). So the only thing which remains to prove is that from $h^0(Y_c, {\cal O}_{Y_c}((13C)K_{Y_c})) \geq 2$ we get $h^0(Y, {\cal O}_{Y}((13C)K_{Y})) \geq 2$. This fact is standard for experts (cf. e.g. \cite{Re1}, p.277, or \cite{Fl}, p.225), but since we remarked in talks about this paper that this fact doesn't seem to be generally well known, we want to indicate how one can prove it:\\ What we have to prove is that taking linearly independant sections $s_1$, $s_2$ from $H^0(Y_c, {\cal O}_{Y_c}(lK_{Y_c}))$ we can get from them linearly independant sections $t_1$, $t_2$ from $H^0(Y, {\cal O}_Y(lK_Y))$. We mentioned at the beginning of section 1 that ${\cal O}_{Y_c}(lK_{Y_c})$ can also be defined as the sheaf of $l$-fold tensor products of rational canonical differentials on $Y_c$ which are regular on $(Y_c)_{reg}$. But since $Y$ and $Y_c$ are birationally equivalent, from this definition it is immediate that any linearly independant sections $s_1$, $s_2$ from $H^0(Y_c, {\cal O}_{Y_c}(lK_{Y_c}))$ can be lifted, namely as pull backs of (tensor products of rational) canonical differentials with the holomorphic map $\pi$, to linearly independant {\bf rational} sections $t_1$, $t_2$ of the bundle ${\cal O}_Y(lK_Y)$. These lifted sections are regular outside the family of the exceptional prime divisors $\{ E_i \}$ of the resolution $\pi : Y \rightarrow Y_c$. We have to show that $t_1$ and $t_2$ are regular everywhere. Since $Y$ is a manifold, by the First Riemann Extension Theorem it is sufficient to show that these sections are bounded near points of the $\{ E_i \}$. In order to show this, choose a natural number $p$, which now may depend on $Y$, such that index($Y_c$) divides $pl$. Then by the definition of canonical singularities (Definition \ref{2.1}) the sections $s_1^p$ and $s_2^p$ lift to {\bf regular} sections $t_1^p$ and $t_2^p$. Hence $t_1$ and $t_2$ have to be bounded near points of the $\{ E_i \}$, and we are \vspace{.5cm} done. \qed Now the proof of Theorem \ref{3.1} is immediate: If we take $m_0 := {\rm lcm}(m_1,m_2)$, then by Proposition \ref{3.3} and Proposition \ref{3.4} we have $Y \in {\cal G}_{m_0}$ for all $Y$ which occur in Theorem \vspace{0.5cm} \ref{3.1}. \qed

9505

alg-geom/9505033

en

https://arxiv.org/abs/alg-geom/9505033

alg-geom/9505033

Bruce Hunt

Bruce Hunt

Symmetric subgroups of rational groups of hermitian type

29 pages (11 pt), ps-file also available at the home page http://www.mathematik.uni-kl.de/~wwwagag, preprints. LaTeX v2.09

A rational group of hermitian type is an algebraic group over the rational numbers whose symmetric space is a hermitian symmetric space. We assume such a group $G$ to be given, which we assume is isotropic. Then, for any rational parabolic $P$ in the group $G$, we find a reductive rational subgroup $N$ closely related with $P$ by a relation we call incidence. This has implications to the geometry of arithmetic quotients of the symmetric space by arithmetic subgroups of $G$, in the sense that $N$ defines a subvariety on such an arithmetic quotient which has special behaviour at the cusp corresponding to the parabolic with which $N$ is incident.

alg-geom

\section{Real parabolics of hermitian type} \subsection{Notations}\label{section1.1} In this paper we will basically adhere to the notations of \cite{BB}. In the first two paragraphs $G$ will denote a real Lie group; later $G$ will be a $\fQ$-group of hermitian type. We assume $G$ is reductive, connected and with compact center; $K\subset} \def\nni{\supset} \def\und{\underline G$ will denote a maximal compact subgroup, ${\cal D} =G/K$ the corresponding symmetric space. Throughout this paper we will assume $G$ is of {\it hermitian type}, meaning that ${\cal D} $ is a hermitian symmetric space, hence a product ${\cal D} ={\cal D} _1\times \cdots \times {\cal D} _d$ of irreducible factors, each of which we assume is {\it non-compact}. Let $\Gg=\kk+\pp$ denote a Cartan decomposition of the Lie algebra of $G$, $\Gg_{\fC}=\kk_{\fC}+\pp^++\pp^-$ the decomposition of the complexified Lie algebra, with $\pp^{\pm}$ abelian subalgebras (and $\pp_{\fC}=\pp^+\oplus \pp^-$). Chooosing a Cartan subalgebra $\hh\subset} \def\nni{\supset} \def\und{\underline \Gg$, the set of roots of $\Gg_{\fC}$ with respect to $\hh_{\fC}$ is denoted $\Phi=\Phi(\hh_{\fC},\Gg_{\fC})$. As usual, we choose root vectors $E_{\ga}\in \Gg^{\ga}$ such that the relations $$[E_{\ga},E_{-\ga}]=H_{\ga}\in \hh_{\fC},\quad \ga(H_{\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda})=2{<\ga,\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda>\over <\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda,\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda>},\ \ \ga,\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda\in \Phi,$$ hold. Complex conjugation maps $\pp^+$ to $\pp^-$, in fact permuting $E_{\ga}$ and $E_{-\ga}$ for $E_{\ga}\in \pp^{\pm}$. Moreover, if $\gS^{\pm}:=\{\ga|E_{\ga} \in \pp^{\pm}\}$, then $$\pp^{\pm}=\span_{\fC}(E_{\ga}),\ \ga\in \gS^{\pm};\quad \pp=\span_{{\Bbb R}} \def\fH{{\Bbb H}}(X_{\ga},Y_{\ga}),\ \ga\in \gS^+,$$ where $X_{\ga}=E_{\ga}+E_{-\ga},\ Y_{\ga}=i(E_{\ga}-E_{-\ga})$ (twice the real and the (negative of the) imaginary parts, respectively). Let $\mu_1,\ldots,\mu_t$ denote a maximal set of strongly orthogonal roots, determined as in \cite{Helg}: $\mu_1$ is the smallest root in $\gS^+$, and $\mu_j$ is the smallest root in $\gS^+$ which is strongly orthogonal to $\mu_1,\ldots,\mu_{j-1}$. This set will be fixed once and for all. Once this set of strongly orthogonal roots has been chosen, a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus $A$ is uniquely determined by $Lie(A)=\aa=\span_{{\Bbb R}} \def\fH{{\Bbb H}}(X_{\mu_1},\ldots,X_{\mu_t})$. Then $\Phi_{{\Bbb R}} \def\fH{{\Bbb H}}=\Phi(\aa,\Gg)$ will denote the set of ${\Bbb R}} \def\fH{{\Bbb H}$-roots, and $\Gg$ has a decomposition $$\Gg=\frak z} \def\qq{\frak q(\aa)\oplus \sum_{\eta\in\Phi_{{\Bbb R}} \def\fH{{\Bbb H}}}\Gg^{\eta},$$ where $\Gg^{\eta}=\{x\in \Gg | \ad(s)x=\eta(s)x, \forall_{s\in A}\}$. For each irreducible component of $G$, the set of ${\Bbb R}} \def\fH{{\Bbb H}$-roots is either of type $\bf C_{\hbox{\scriptsize\bf t}}$ or $\bf BC_{\hbox{\scriptsize\bf t}}$, and of type $\bf C_{\hbox{\scriptsize\bf t}}$ $\iff$ the corresponding domain is a tube domain. If $\xi_i$ denote coordinates on $\aa$ dual to $X_{\mu_i}$, assuming for the moment ${\cal D} $ to be irreducible, the ${\Bbb R}} \def\fH{{\Bbb H}$-roots are explicitly \begin{equation}\label{e22b.2} \begin{array}{lcr}\Phi_{{\Bbb R}} \def\fH{{\Bbb H}}: & \pm(\xi_i\pm \xi_j),\ \pm 2 \xi_i \ (1\leq i\leq t,\ i<j) & \mbox{(Type $\bf C_{\hbox{\scriptsize\bf t}}$)} \\ & \pm(\xi_i\pm \xi_j),\ \pm 2 \xi_i, \ \pm\xi_i \ (1\leq i\leq t,\ i<j) & \mbox{(Type $\bf BC_{\hbox{\scriptsize\bf t}}$)} \\ \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{{\Bbb R}} \def\fH{{\Bbb H}}: & \eta_i=\xi_i-\xi_{i+1}, i=1,\ldots,t-1, \mbox{ and } \eta_t=2\xi_t \mbox{ (Type $\bf C_{\hbox{\scriptsize\bf t}}$) }, \eta_t=\xi_t & \mbox{ (Type $\bf BC_{\hbox{\scriptsize\bf t}}$).} \end{array} \end{equation} Here the simple roots $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{{\Bbb R}} \def\fH{{\Bbb H}}$ are with respect to the lexicographical order on the $\xi_i$. A general ${\Bbb R}} \def\fH{{\Bbb H}$-root system is a disjoint union of simple root systems. The choice of maximal set of strongly orthogonal roots determines an order on $\aa$ (the lexicographical order), which determines, on each simple ${\Bbb R}} \def\fH{{\Bbb H}$-root system, an order as above; this is called the {\it canonical order}. \subsection{Real parabolics} The maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split abelian subalgebra $\aa$, together with the order on it (induced by the choice of strongly orthogonal roots), determines a unique nilpotent Lie algebra of $\Gg$, $\frak n} \def\rr{\frak r=\sum_{\eta\in \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}^+}\Gg^{\eta}$. Set $A=\exp(\aa),\ N=\exp(\frak n} \def\rr{\frak r)$, and \begin{equation}\label{e2.1} B:={\cal Z} (A)\rtimes N; \end{equation} this is a minimal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic, the {\it standard} one, uniquely determined by the choice of strongly orthogonal roots. Every minimal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic of $G$ is conjugate to $B$. Note that, setting $M={\cal Z} (A)\cap K$, we have ${\cal Z} (A)=M\times A$, and the group $M$ is the {\it semisimple anisotropic kernel} of $G$. Assume again for the moment that ${\cal D} $ is irreducible, and let $\eta_i,\ i=1,\ldots,t$ denote the simple ${\Bbb R}} \def\fH{{\Bbb H}$-roots. Set $\aa_b:=\cap_{j\neq b} \Ker\eta_j,\ b=1,\ldots, t$, a one-dimensional subspace of $\aa$, and $A_b:=\exp(\aa_b)$, a one-dimensional ${\Bbb R}} \def\fH{{\Bbb H}$-split subtorus of $A$. Equivalently, $A_b=\left(\cap_{j\neq b}\Ker\eta_j\right)^0$, where $\eta_j$ is viewed as a character of $A$. The {\it standard maximal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic}, $P_b,\ b=1,\ldots, t$, is the group generated by ${\cal Z} (A_b)$ and $N$; equivalently it is the semidirect product (Levi decomposition) \begin{equation}\label{e2.2} P_b={\cal Z} (A_b)\rtimes U_b, \end{equation} where $U_b$ denotes the unipotent radical. The Lie algebra $\uu_b$ of $U_b$ is the direct sum of the $\Gg^{\eta},\ \eta\in \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}^+,\ \eta_{|\aa_b}\not\equiv 0$. The Lie algebra $\frak z} \def\qq{\frak q(\aa_b)$ of ${\cal Z} (A_b)$ has a decomposition: \begin{equation}\label{e2.3} \frak z} \def\qq{\frak q(\aa_b)=\mm_b\oplus \ll_b\oplus \ll_b'\oplus \aa_b,\quad \ll_b=\sum_{\eta\in [\eta_{b+1},\ldots,\eta_t]}\Gg^{\eta}+[\Gg^{\eta},\Gg^{-\eta}],\ \ \ll_b'=\sum_{\eta\in [\eta_{1},\ldots,\eta_{b-1}]}\Gg^{\eta}+[\Gg^{\eta},\Gg^{-\eta}], \end{equation} and $\mm_b$ is an ideal of $\mm$, the Lie algebra of the (semisimple) anisotropic kernel $M$. Both $\ll_b$ and $\ll_b'$ are simple, and the root system $[\eta_{b+1},\ldots,\eta_t]$ is of type ${\bf C}_{\hbox{\scriptsize\bf t-b}}$ or ${\bf BC}_{\hbox{\scriptsize\bf t-b}}$, while the root system $[\eta_1,\ldots, \eta_{b-1}]$ is of type ${\bf A}_{\hbox{\scriptsize\bf b-1}}$. Let $L_b,\ L_b'$ denote the analytic groups with Lie algebras $\ll_b$ and $\ll_b'$, respectively, and let ${\cal R} _b=L_b'A_b$, a reductive group (of type ${\bf A}_{\hbox{\scriptsize\bf b-1}}$). We call $L_b$ the {\it hermitian factor} of the Levi component and ${\cal R} _b$ the {\it reductive factor}. It is well known that $L_b$ defines the hermitian symmetric space which is the $b^{th}$ (standard) boundary component of ${\cal D} $. Indeed, letting $K_b\subset} \def\nni{\supset} \def\und{\underline L_b$ denote a maximal compact subgroup, ${\cal D} _b=L_b/K_b$ is hermitian symmetric, and naturally contained in ${\cal D} $ as a subdomain, ${\cal D} _b\subset} \def\nni{\supset} \def\und{\underline {\cal D} $. Let $\gz:{\cal D} \longrightarrow} \def\sura{\twoheadrightarrow \pp^+$ be the Harish-Chandra embedding, and let ${\bf D}=\gz({\cal D} ),\ {\bf D}_b=\gz({\cal D} _b)$ denote the images; ${\bf D}_b$ is a bounded symmetric domain contained in a linear subspace (which can be identified with $\pp_b^+=\ll_{b,\fC}\cap \pp^+$). Let $o_b=-(E_{\mu_1}+\cdots+E_{\mu_b}),\ 1\leq b\leq t$; as the elements $o_b$ are in $\pp^+$, one can consider the orbits $o_b\cdot G$ and $o_b\cdot L_b$. Since for $g\in L_b$ the action is described by $o_bg = o_b +\gz(g)$, one has $o_b\cdot L_b = o_b+\gz({\cal D} _b)=o_b+{\bf D}_b$, and this is the domain ${\bf D}_b$, translated into an affine subspace ($o_b+\pp_b^+$) of $\pp^+$. One denotes this domain by $F_b:=o_b\cdot L_b$, and this is the $b^{th}$ {\it standard boundary component} of ${\bf D}$. $G$ acts by translations on the various $F_b$, and the images are the {\it boundary components} of ${\bf D}$; one has $$\overline{\bf D}={\bf D}\cup \{\hbox{ boundary components }\}={\bf D} \cup \left(\cup_{b=1}^to_b\cdot G\right),$$ and $\overline{\bf D}\subset} \def\nni{\supset} \def\und{\underline \pp^+$ is the compactification of ${\bf D}$ in the Euclidean topology. For any boundary component $F$ one denotes by $N(F),\ Z(F)$ and $G(F)$ the normalizer, centralizer and automorphism group $G(F)=N(F)/Z(F)$, respectively. Then, letting $U(F)$ denote the unipotent radical of $N(F)$, \begin{equation}\label{e3.1} N(F_b)=P_b,\quad U(F_b)=U_b,\quad Z(F_b)=Z_b,\quad G(F_b)=L_b, \end{equation} where $Z_b$ is a closed normal subgroup of $P_b$ containing every normal subgroup of $P_b$ with Lie algebra $\frak z} \def\qq{\frak q_b=\mm_b\oplus\ll_b'\oplus \aa_b\oplus \uu_b$, which is an ideal in $\pp_b$. Now consider the general case, ${\cal D} ={\cal D} _1\times \cdots \times {\cal D} _d$, ${\cal D} _i$ irreducible. For each ${\cal D} _i$ we have ${\Bbb R}} \def\fH{{\Bbb H}$-roots $\Phi_{i,{\Bbb R}} \def\fH{{\Bbb H}}$, of ${\Bbb R}} \def\fH{{\Bbb H}$-ranks $t_i$ and simple ${\Bbb R}} \def\fH{{\Bbb H}$-roots $\{\eta_{i,1},\ldots,\eta_{i,t_i}\},\ i=1,\ldots, d$. For each factor we have standard parabolics $P_{i,b_i}$ $(1\leq b_i\leq t_i$) and standard boundary components $F_{i,b_i}$. The standard parabolics of $G$ and boundary components of ${\cal D} $ are then products \begin{equation}\label{e3.2} P_{\hbox{\scsi \bf b}}=P_{1,b_1}\times \cdots \times P_{d,b_d},\quad F_{\hbox{\scsi \bf b}}=F_{1,b_1}\times \cdots \times F_{d,b_d},\quad ({\bf b}=(b_1,\ldots, b_d)), \end{equation} and as above $P_{\hbox{\scsi \bf b}}=N(F_{\hbox{\scsi \bf b}})$. Furthermore, \begin{equation}\label{e3.3} G(F_{\hbox{\scsi \bf b}})=:L_{\hbox{\scsi \bf b}}=L_{1,b_1}\times \cdots\times L_{d,b_d}. \end{equation} As far as the domains are concerned, any of the boundary components $F_{i,b_i}$ may be the {\it improper} boundary component ${\cal D} _i$, which is indicated by setting $b_i=0$. Consequently, $P_{i,0}=L_{i,0}=G_i$ and in (\ref{e3.2}) and (\ref{e3.3}) any ${\bf b}=(b_1,\ldots,b_d),\ 0\leq b_i\leq t_i$ are admissible. \subsection{Fine structure of parabolics} For real parabolics of hermitian type one has a very useful refinement of (\ref{e2.2}). This is explained in detail in \cite{SC} and especially in \cite{S}, \S III.3-4. First we have the decomposition of ${\cal Z} (A_b)$ as described above, \begin{equation}\label{e3.4} {\cal Z} (A_b)=M_b\cdot L_b \cdot {\cal R} _b, \end{equation} where $M_b$ is compact, $L_b$ is the hermitian Levi factor, ${\cal R} _b$ is reductive (of type $\bf A_{\hbox{\scriptsize \bf b-1}}$), and the product is almost direct (i.e., the factors have finite intersection). Secondly, the unipotent radical decomposes, \begin{equation}\label{e3.5} U_b={\cal Z} _b\cdot V_b, \end{equation} which is a direct product, ${\cal Z} _b$ being the center of $U_b$. The action of ${\cal Z} (A_b)$ on $U_b$ can be explicitly described, and is the basis for the compactification theory of \cite{SC}. Before we recall this, let us note the notations used in \cite{SC} and \cite{S} for the decomposition. In \cite{SC}, we find \begin{equation}\label{e4.1} P(F)=(M(F)G_h(F)G_{\ell}(F))\rtimes U(F)\cdot V(F), \end{equation} and in \cite{S}, where the author uses Hermann homomorphisms $\gk:\frak s} \def\cc{\frak c\ll_2({\Bbb R}} \def\fH{{\Bbb H})\longrightarrow} \def\sura{\twoheadrightarrow \Gg$ to index the boundary component, \begin{equation}\label{e4.2} B_{\gk}=\left(G_{\gk}^{(1)}\cdot G_{\gk}^{(2)}\right) \rtimes U_{\gk}V_{\gk}. \end{equation} In (\ref{e4.1}), $M=M_b,\ G_h=L_b,\ G_{\ell}={\cal R} _b$, while in (\ref{e4.2}), $G_{\gk}^{(1)}=M_b\cdot L_b,\ G_{\gk}^{(2)}={\cal R} _b$ in our notations. The action can be described as follows (\cite{S}, III \S3-4). \begin{theorem}\label{t4.1} In the decomposition of the standard parabolic $P_b$ (see (\ref{e3.4}) and (\ref{e3.5})) $$P_b=(M_b\cdot L_b \cdot {\cal R} _b)\rtimes {\cal Z} _b\cdot V_b,$$ the following statements hold. \begin{itemize}\item[(i)] The action of $M_b\cdot L_b$ is trivial on ${\cal Z} _b$, while on $V_b$ it is by means of a symplectic representation $\gr:M_b\cdot L_b \longrightarrow} \def\sura{\twoheadrightarrow Sp(V_b,J_b)$, for a symplectic form $J_b$ on $V_b$. \item[(ii)] ${\cal R} _b$ acts transitively on ${\cal Z} _b$ and defines a homogenous self-dual (with respect to a bilinear form) cone $C_b\subset} \def\nni{\supset} \def\und{\underline {\cal Z} _b$, while on $V_b$ it acts by means of a representation $\gs:{\cal R} _b\longrightarrow} \def\sura{\twoheadrightarrow GL(V_b,I_b)$ for some complex structure $I_b$ on $V_b$. \end{itemize} Furthermore the representations $\gr$ and $\gs$ are compatible in a natural sense. \end{theorem} \section{Holomorphic symmetric embeddings of symmetric domains} \subsection{Symmetric subdomains} We continue with the notations of the previous paragraph. Hence $G$ is a real Lie group of hermitian type (reductive), ${\cal D} $ is the corresponding domain. We wish to consider reductive subgroups $N\subset} \def\nni{\supset} \def\und{\underline G$, also of hermitian type, defining domains ${\cal D} _N$, such that the inclusion $N\subset} \def\nni{\supset} \def\und{\underline G$ induces a holomorphic injection of the domains $i:{\cal D} _N\subset} \def\nni{\supset} \def\und{\underline {\cal D} $, and the $i({\cal D} _N)$ are totally geodesic. First of all we may assume that $K_N$, a maximal compact subgroup of $N$, is the intersection $K_N=K\cap N$; equivalently, letting $o\in {\cal D} $ and $o_N\in {\cal D} _N$ denote the base points, $i(o_N)=o$. Note that conjugating $N$ by an element of $K$ yields an isomorphic group $N'$ and subdomain $i':{\cal D} _{N'}\subset} \def\nni{\supset} \def\und{\underline {\cal D} $ such that $i'(o_{N'})=o$, and this defines an equivalence relation on the set of reductive subgroups $N\subset} \def\nni{\supset} \def\und{\underline G$ as described. For the irreducible hermitian symmetric domains, the equivalence classes of all such $N$ have been determined by Satake and Ihara (\cite{S1} for the cases of ${\cal D} $ of type $\bf I_{p,q},\ II_n, \ III_n$; \cite{I} for the other cases). Before quoting the results we will need, let us briefly remark on the mathematical formulation of the conditions. For this, let ${\cal D} ,\ {\cal D} '$ be hermitian symmetric domains, $G,\ G'$ the automorphism groups, $\Gg,\ \Gg'$ the Lie algebras, $\Gg=\kk\oplus\pp,\ \Gg'=\kk'\oplus\pp'$ the Cartan decompositions and $\gt,\ \gt'$ the Cartan involutions on $\Gg$ and $\Gg'$, respectively. To say that for an injection $i_{{\cal D} }:{\cal D} \hookrightarrow} \def\hla{\hookleftarrow {\cal D} '$ of symmetric spaces, $i_{{\cal D} }({\cal D} )$ is totally geodesic in ${\cal D} '$ is to say that $i_{{\cal D} }$ is induced by an injection $i:\Gg\hookrightarrow} \def\hla{\hookleftarrow \Gg'$ of the Lie algebras. If this holds, $i_{{\cal D} }$ is said to be {\it strongly equivariant}. Then, $\gt=\gt'_{|i(\Gg)}$, or $\kk=\Gg\cap \kk',\ \pp=\Gg\cap \pp'$. Since both ${\cal D} $ and ${\cal D} '$ are hermitian symmetric, there is an element $\xi$ in the center of $\kk$ (resp. $\xi'$ in the center of $\kk'$), such that $J=\ad(\xi)$ (resp. $J'=\ad(\xi')$) gives the complex structure. To say that the injection $i_{{\cal D} }:{\cal D} \hookrightarrow} \def\hla{\hookleftarrow {\cal D} '$ is holomorphic is the same as saying $i\circ J = J'\circ i$, or equivalently, \vspace*{.2cm} $\hbox{(H$_1$)}\hspace*{5.8cm} i\circ \ad(\xi) = \ad(\xi')\circ i.$ \vspace*{.2cm} \noindent This is the condition utilized by Satake and Ihara in their classifications. The condition (H$_1$) is clearly implied by \vspace*{.2cm}(H$_2$)\hspace*{6.5cm}$i(\xi)=\xi',$ \vspace*{.2cm} \noindent which however, if fulfilled, gives additional information. For example (\cite{S2}, Proposition 4) if ${\cal D} $ is a tube domain and $i$ satisfies (H$_2$), then ${\cal D} '$ is also a tube domain. Furthermore, (\cite{S}, Proposition II 8.1), if $i_{{\cal D} }:{\cal D} \longrightarrow} \def\sura{\twoheadrightarrow {\cal D} '$ is a holomorphic map which is strongly equivariant, the corresponding homomorphism $i$ fulfills (H$_1$), and, moreover, if ${\cal D} $ and ${\cal D} '$ are viewed as bounded symmetric domains ${\bf D}$, $\bf D'$ via the Harish-Chandra embeddings, then $i_{\hbf{D}}:{\bf D}\longrightarrow} \def\sura{\twoheadrightarrow {\bf D}'$ is the restriction of a $\fC$-linear map $i^+:\pp^+\longrightarrow} \def\sura{\twoheadrightarrow (\pp')^+$. If $i_{\fC}:\Gg_{\fC}\longrightarrow} \def\sura{\twoheadrightarrow \Gg_{\fC}'$ denotes the $\fC$-linear extension of $i$, and $\gs:\Gg_{\fC}\longrightarrow} \def\sura{\twoheadrightarrow \Gg_{\fC},\ \gs':\Gg_{\fC}'\longrightarrow} \def\sura{\twoheadrightarrow \Gg_{\fC}'$ denote the conjugations over $\Gg$ and $\Gg'$, respectively, then the condition $\gt=\gt'_{|i(\Gg)}$ is equivalent to the condition $i_{\fC}\circ \gs=\gs'\circ i_{\fC}$. This implies that $i:(\Gg,\xi)\longrightarrow} \def\sura{\twoheadrightarrow (\Gg',\xi')$ gives rise to a symmetric Lie algebra homomorphism $(\Gg_{\fC},\gs)\longrightarrow} \def\sura{\twoheadrightarrow (\Gg_{\fC}',\gs')$, and therefore, by \cite{S}, Proposition I 9.1, to a homomorphism of Jordan triple systems $i^+:\pp^+\longrightarrow} \def\sura{\twoheadrightarrow (\pp')^+$. It follows (\cite{S}, p.~85) that the following three categories are equivalent: \begin{tabular}{cc}$(\hbox{{\script S}} \hbox{{\script D}} )$ & \parbox[t]{15cm}{Category whose objects are symmetric domains $({\cal D} ,o)$ with base point $o$, whose morphisms $\gr_{{\cal D} }:({\cal D} ,o)\longrightarrow} \def\sura{\twoheadrightarrow ({\cal D} ',o')$ are strongly equivariant holomorphic maps $\gr_{{\cal D} }:{\cal D} \longrightarrow} \def\sura{\twoheadrightarrow {\cal D} '$ with $\gr_{{\cal D} }(o)=o'$.} \\ $(\hbox{{\script H}} \hbox{{\script L}} )$ & \parbox[t]{15cm}{Category whose objects are semisimple Lie algebras $(\Gg,\xi)$ of hermitian type (without compact factors), whose morphisms $\gr:(\Gg,\xi)\longrightarrow} \def\sura{\twoheadrightarrow (\Gg',\xi')$ are homomorphisms satisfying (H$_1$).} \\ $(\hbox{{\script H}} \hbox{{\script J}} )$ & \parbox[t]{15cm}{Category whose objects are positive definite hermitian Jordan triple systems $\pp^+$, whose morphisms $\gr_+:\pp^+\longrightarrow} \def\sura{\twoheadrightarrow (\pp')^+$ are $\fC$-linear homomorphisms of Jordan triple systems.} \end{tabular} \subsection{Positive-dimensional boundary components} We now quote some results which we will be using. First, assume we have fixed $A\subset} \def\nni{\supset} \def\und{\underline G$ as above, and let $F_b\subset} \def\nni{\supset} \def\und{\underline \overline{\bf D}$ be a standard boundary component of positive dimension, i.e., if ${\cal D} $ is irreducible, of rank $t$, then $b<t$; if ${\cal D} ={\cal D} _1\times \cdots\times {\cal D} _d$, then in the notations of (\ref{e3.2}), ${\bf b}=(b_1,\ldots, b_d)$, we have $b_i<t_i$ for {\it at least} one $i$. If ${\cal D} $ is irreducible, we list in Table \ref{T1} a positive-dimensional boundary component and a symmetric subdomain ${\cal D} _M\subset} \def\nni{\supset} \def\und{\underline {\cal D} $ with the property that ${\cal D} _N={\cal D} _F\times {\cal D} '$, where ${\cal D} _F$ is, as a hermitian symmetric space, isomorphic to the given boundary component. If ${\cal D} $ is reducible, ${\cal D} ={\cal D} _1\times\cdots\times{\cal D} _d$, and $F_{\hbox{\scsi \bf b}}\subset} \def\nni{\supset} \def\und{\underline \overline{\bf D}$ is a standard boundary component, we get a subdomain ${\cal D} _N={\cal D} _{N_1}\times \cdots \times {\cal D} _{N_d}$ such that ${\cal D} _{N_i}\subset} \def\nni{\supset} \def\und{\underline {\cal D} _i$ is of the type just mentioned with respect to the boundary component $F_{b_i}\subset} \def\nni{\supset} \def\und{\underline \overline{\bf D}_i$. \begin{table}\caption{\label{T1} Symmetric subdomains incident with positive-dimensional boundary components} $$\begin{array}{|c|c|c|c|}\hline {\cal D} & F_b,\ \ (b<t) & {\cal D} _N & \hbox{(H$_2$)} \\ \hline \hline \bf I_{\hbox{\scriptsize\bf p,q}} & \bf I_{\hbox{\scriptsize\bf p-b,q-b}} & \bf I_{\hbox{\scriptsize\bf p-b,q-b}}\times I_{\hbox{\scriptsize\bf b,b}} & p=q \\ \hline \bf II_{\hbox{\scriptsize\bf n}} & \bf II_{\hbox{\scriptsize\bf n-2b}} & \bf II_{\hbox{\scriptsize\bf n-2b}}\times II_{\hbox{\scriptsize\bf 2b}} & yes \\ \hline \bf III_{\hbox{\scriptsize\bf n}} & \bf III_{\hbox{\scriptsize\bf n-b}} & \bf III_{\hbox{\scriptsize\bf n-b}}\times III_{\hbox{\scriptsize\bf b}} & yes \\ \hline \bf IV_{\hbox{\scriptsize\bf n}} & \bf IV_{\hbox{\scriptsize\bf 1}} & \bf IV_{\hbox{\scriptsize\bf 1}}\times IV_{\hbox{\scriptsize\bf 1}} & yes \\ \hline \bf V & \bf I_{\hbox{\scriptsize\bf 5,1}} & \bf I_{\hbox{\scriptsize\bf 5,1}}\times I_{\hbox{\scriptsize\bf 1,1}} & yes \\ \hline \bf VI & \bf IV_{\hbox{\scriptsize\bf 10}} & \bf IV_{\hbox{\scriptsize\bf 10}}\times IV_{\hbox{\scriptsize\bf 1}} & yes \\ & \bf IV_{\hbox{\scriptsize\bf 1}} & \bf IV_{\hbox{\scriptsize\bf 1}}\times IV_{\hbox{\scriptsize\bf 10}} & yes \\ \hline \end{array}$$ \end{table} Next, choose $N\subset} \def\nni{\supset} \def\und{\underline G$ with ${\cal D} _N$ as in Table \ref{T1}, such that $A\subset} \def\nni{\supset} \def\und{\underline N$ is a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus in $N$, so that we can speak of standard boundary components of ${\cal D} _N$. Then the subdomains ${\cal D} _N$ listed in Table \ref{T1} have the following property. For simplicity we will assume from now on that $G$ is semisimple. \begin{proposition}\label{p7.1} Given $G$, simple of hermitian type with maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus $A$ and simple ${\Bbb R}} \def\fH{{\Bbb H}$-roots $\eta_i$ ($1\leq i\leq t=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G$), let $P_{b}$ and $F_{b}$ denote the standard maximal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic and standard boundary component determined by $\eta_b$ ($b<t$). Let $N\subset} \def\nni{\supset} \def\und{\underline G$ be a symmetric subgroup with $A\subset} \def\nni{\supset} \def\und{\underline N$, defining a subdomain ${\cal D} _N$ as in Table \ref{T1}, such that $N=N_{1}\times N_{2}$ and $N_{1}$ is a hermitian Levi factor of $P_b$. Let $P_0\times P_{{t_2}}$ be the standard maximal parabolic defined by the last simple ${\Bbb R}} \def\fH{{\Bbb H}$-root $\eta_{t_2}$ of the second factor in the decomposition $N=N_{1}\times N_{2}$. Then if $F:=F_0\times F_{{t_2}}$ ($\cong {\cal D} _{N_{1}}\times \{pt.\}$, $t_2=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}N_{2}$) denotes the corresponding standard boundary component, the equality $i_N(F)=F_{b}$ holds, where $i_N:{\cal D} _{N}\longrightarrow} \def\sura{\twoheadrightarrow {\cal D} $ denotes the injection. \end{proposition} {\bf Proof:} From construction, $F\cong{\cal D} _{N_{1}}\times \{pt\} \cong F_{b}$ as a hermitian symmetric space; to see that they coincide under $i_N$, recall from (\ref{e2.3}) the root space decomposition of the hermitian Levi component of $P_{b}$. Since $A\subset} \def\nni{\supset} \def\und{\underline N$ is also a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus of $N$, in the root system $\Phi(A,N)$ we have the subsystem $[\eta_{b+1},\ldots, \eta_t]$ giving rise, on the one hand to the hermitian Levi factor $\ll_b$ in $\pp_{b}$, on the other hand to the Lie algebra of the first factor $\frak n} \def\rr{\frak r_{1}$ of $N$. From this it follows that $P_{b}$ stabilizes $i_N(F)$, hence $i_N(F)=F_{b}$. \hfill $\Box$ \vskip0.25cm Before proceeding to the case of zero-dimensional boundary components, we briefly explain how the subgroups $N\subset} \def\nni{\supset} \def\und{\underline G$ (which are not unique, of course) arise in terms of $\pm$symmetric/hermitian forms, at least for the classical cases. For this, we note that $G$ can be described as follows (we describe here certain reductive groups; the simple groups are just the derived groups): \begin{equation}\label{e7.1}\begin{minipage}{14.5cm}\begin{itemize}\item[I] $\bf I_{\hbox{\scriptsize\bf p,q}}$: $G$ is the unitary group of a hermitian form on $\fC^{p+q}$ of signature $(p,q)$ ($p\geq q$). \item[II] $\bf II_{\hbox{\scriptsize\bf n}}$: $G$ is the unitary group of a skew-hermitian form on $\fH^n$. \item[III] $\bf III_{\hbox{\scsi \bf n}}$: $G$ is the unitary group of a skew-symmetric form on ${\Bbb R}} \def\fH{{\Bbb H}^{2n}$. \item[IV] $\bf IV_{\hbox{\scsi \bf n}}$: $G$ is the unitary group of a symmetric bilinear form on ${\Bbb R}} \def\fH{{\Bbb H}^{n+2}$ of signature $(n,2)$. \end{itemize}\end{minipage}\end{equation} Each of the $\pm$symmetric/hermitian forms is isotropic, and if $t=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G$, the maximal dimension of a totally isotropic subspace is $t=q,\ \left[{n\over 2}\right],\ n,\ 2$ in the cases I, II, III, and IV, respectively. Each maximal real parabolic is the stabilizer of a totally isotropic subspace, and using the canonical order on the ${\Bbb R}} \def\fH{{\Bbb H}$-roots as above, $P_{b}$ stabilizes a totally isotropic subspace of dimension $b$. Choosing a maximal torus $T$ (resp. a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus $A\subset} \def\nni{\supset} \def\und{\underline T$) amounts to choosing a basis of $V$ (resp. choosing a subset of this basis which spans a maximal totally isotropic subspace), and the standard parabolic is the stabilizer of a totally isotropic subspace spanned by some part of this basis. Now let $H\subset} \def\nni{\supset} \def\und{\underline V$ be a totally isotropic subspace with basis $h_1,\ldots, h_b$. Then there are elements $h_1',\ldots,h_b'$ of $V$ such that $h(h_i,h_j')=\gd_{ij},\ h(h_i,h_i)=h(h_i',h_i')=0$, and $h_1,\ldots, h_b,h_1',\ldots, h_b'$ span (over $D$) a vector subspace $W\subset} \def\nni{\supset} \def\und{\underline V$ on which $h$ restricts to a non-degenerate form. Let $W^{\perp}$ denote the orthogonal complement of $W$ in $V$, $W\oplus W^{\perp}=V$. Then \begin{equation}\label{e8.1} N:=U(W,W^{\perp};h):=\{g\in U(V,h) | g(W)\subset} \def\nni{\supset} \def\und{\underline W, g(W^{\perp})\subset} \def\nni{\supset} \def\und{\underline W^{\perp} \}\cong U(W,h_{|W})\times U(W^{\perp},h_{|{W^{\perp}}}). \end{equation} $N$ is a reductive subgroup of $G$, and as one easily sees, its symmetric space is just the domain denoted ${\cal D} _N$ in Table \ref{T1} above. The relation ``boundary component $\subset} \def\nni{\supset} \def\und{\underline $ symmetric subdomain'' translates into ``totally isotropic subspace $\subset} \def\nni{\supset} \def\und{\underline $ non-degenerate subspace'', $H\subset} \def\nni{\supset} \def\und{\underline W$, and {\it because} $h_{|W}$ is non-degenerate, any $g\in U(V,h)$ which stabilizes $W$ automatically stabilizes its orthogonal complement in $V$ as in (\ref{e8.1}). \subsection{Zero-dimensional boundary components} We now would like to consider the zero-dimensional boundary components, which correspond in the above picture to maximal totally isotropic subspaces. The construction above (\ref{e8.1}) doesn't necessarily work in this case, as $W^{\perp}$ may be $\{0\}$, and $N=G$. However, in terms of domains, given {\it any} subdomain ${\cal D} '\subset} \def\nni{\supset} \def\und{\underline {\cal D} $, it can be translated so as to contain a given zero-dimensional boundary component. We therefore place the following three conditions on such a subdomain: \begin{itemize}\item[1)] The subdomain ${\cal D} '$ has maximal rank ($\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G'=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G$). \item[2)] The subdomain ${\cal D} '$ is maximal and $G'$ is a maximal subgroup, or \item[2')] The subdomain ${\cal D} '$ is maximal of tube type and $G'$ is maximal with this property. \item[3')] The subdomain ${\cal D} '$ is maximal irreducible, and $F$ is a boundary component of ${\cal D} '$. \end{itemize} In Table \ref{T2} we list the subdomains (after \cite{I}) ${\cal D} '$ fulfilling 1), 2) and 3') in the column titled ``${\cal D} _N$''. We have listed also those subgroups fulfilling 1), 2') and 3') in the column titled ``maximal tube''. \begin{table}\caption{\label{T2} Symmetric subdomains incident with zero-dimensional boundary components} $$\begin{array}{|c|c|c|c|} \hline {\cal D} & {\cal D} _N & \hbox{(H$_2$)} & \hbox{maximal tube} \\ \hline \hline {\bf I_{\hbox{\scriptsize\bf p,q}}},\ p>q & \bf I_{\hbox{\scriptsize\bf p-1,q}} & no & \bf I_{\hbox{\scriptsize\bf q,q}} \\ \hline \bf I_{\hbox{\scriptsize\bf q,q}} & - & - & - \\ \hline \bf II_{\hbox{\scriptsize\bf n}},\ n\hbox{ even} & - & - & - \\ \hline \bf II_{\hbox{\scriptsize\bf n}},\ n\hbox{ odd} & \bf II_{\hbox{\scriptsize\bf n-1}} & yes & \bf II_{\hbox{\scriptsize\bf n-1}} \\ \hline \bf III_{\hbox{\scriptsize\bf n}} & - & - & - \\ \hline \bf IV_{\hbox{\scriptsize\bf n}} & \bf IV_{\hbox{\scriptsize\bf n-1}} & yes & \bf IV_{\hbox{\scriptsize\bf n-1}} \\ \hline \bf V & \bf I_{\hbox{\scriptsize\bf 2,4}},\ II_{\hbox{\scriptsize\bf 5}},\ IV_{\hbox{\scriptsize\bf 8}} & yes,\ no,\ no & \bf I_{\hbf{2,2}}, II_{\hbf{4}}, IV_{\hbox{\scriptsize\bf 8}} \\ \hline \bf VI & \bf I_{\hbox{\scriptsize\bf 3,3}},\ II_{\hbox{\scriptsize\bf 6}} & yes & \bf I_{\hbox{\scriptsize\bf 3,3}},\ II_{\hbox{\scriptsize\bf 6}} \\ \hline \end{array}$$ {\small In the column ``${\cal D} _N$'' the subgroups fulfilling 1), 2) and 3') are listed, and in the column ``maximal tube'' the subgroups fulfilling 1), 2') and 3') (i.e., not necessarily 2)) are listed.} \end{table} In Table \ref{T2}, if there is no entry in the column ``${\cal D} _N$'', no such subgroups exist. In these cases it is natural to take the polydisc ${\cal D} _{N_{\Psi}}$ defined by the maximal set of strongly orthogonal roots $\Psi=\{\pm\{\mu_1\},\ldots,\pm\{\mu_t\}\}$ as the subdomain ${\cal D} _N$, as there is no irreducible subdomain, and other products already occur in Table \ref{T1}. Hence for these cases we require the conditions 2'') and 3'') of the introduction. To sum up these facts we make the following definition. \begin{definition}\label{d9.1} Let $G$ be a simple real Lie group of hermitian type, $A$ a fixed maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus defined as above by a maximal set of strongly orthogonal roots, $\eta_i,\ i=1,\ldots, t$ the simple ${\Bbb R}} \def\fH{{\Bbb H}$-roots, $F_{b}$ a standard boundary component and $P_{b}$ the corresponding standard maximal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic. A reductive subgroup $N\subset} \def\nni{\supset} \def\und{\underline G$ (respectively the subdomain ${\cal D} _{N}\subset} \def\nni{\supset} \def\und{\underline {\cal D} $) will be called {\it incident} to $P_{b}$ (respectively to $F_{b}$), if ${\cal D} _N$ is isomorphic to the corresponding domain of Table \ref{T1} ($b<t$) or Table \ref{T2} ($b=t$), and if $N$ fulfills: \begin{itemize}\item $b<t$, then $N$ satisfies 1), 2), 3). \item $b=t,\ {\cal D} \not\in ({\cal E} {\cal D} )$, then $N$ satisfies 1), 2'), 3'). \item $b=t,\ {\cal D} \in ({\cal E} {\cal D} )$, then $N$ satisfies 1), 2''), 3''). \end{itemize} For reducible ${\cal D} ={\cal D} _1\times \cdots \times {\cal D} _d$, we have the product subgroups $N_{b_1,1}\times \cdots \times N_{b_d,d}$, where ${\cal D} _{N_{b_i,i}}$ is incident to the standard boundary component $F_{{b_i}}$ of ${\cal D} _i$ (and $N_{0,i}=G_i$). \end{definition} Next we briefly discuss uniqueness. We consider first the case of positive-dimensional boundary components. Let $P_{b}$, $1\leq b< t$ be a standard parabolic and let $L_b$ be the ``standard'' hermitian Levi factor, i.e., such that $Lie(L_b)=\ll_b$; then \begin{equation}\label{e10.1} N_b:= L_b\times {\cal Z} _G(L_b) \end{equation} is a subgroup having the properties of Proposition \ref{p7.1}, unique since $L_b$ is unique. We shall refer to this unique subgroup as the {\it standard} incident subgroup. The different Levi factors ${\cal L} $ in Levi decompositions $P_b={\cal L} \rtimes {\cal R} _u(P_b)$ are conjugate by elements of ${\cal R} _u(P_b)$, as is well known. This implies for the hermitian factors $L={\cal L} ^{herm}\subset} \def\nni{\supset} \def\und{\underline {\cal L} $ (which are uniquely determined by ${\cal L} $) by Theorem \ref{t4.1} the following. \begin{lemma}\label{l10A} Two hermitian Levi factors $L,\ L'\subset} \def\nni{\supset} \def\und{\underline P_b$ are conjugate by an element of $V_b\subset} \def\nni{\supset} \def\und{\underline P_b$. \end{lemma} It follows, since $g(L_b\times {\cal Z} _G(L_b))g^{-1}=gL_bg^{-1}\times {\cal Z} _G(gL_bg^{-1})$, that two subgroups $N,\ N'$, both incident with $P_b$, are conjugate by an element of $V_b$: \[\hbox{$N,\ N'$ incident to $P_b$ $\iff$ $N,\ N'$ conjugate (in $G$) by $g\in V_b$.}\] \begin{proposition} If $(N,P_b)$ are incident, there is $g\in V_b$ such that $N$ is conjugate by $g$ to the standard $N_b$ of (\ref{e10.1}). \end{proposition} {\bf Proof:} Since $N$ is incident, $N\cong N_1\times N_2$, where $N_1$ is a hermitian Levi factor of $N$. By Lemma \ref{l10A}, $N_1$ is conjugate by $g\in V_b$ to $L_b$, the hermitian Levi factor with Lie algebra $\ll_b$ in the notations of the last section. Hence $gNg^{-1}=g(N_1\times N_2)g^{-1} = gN_1g^{-1}\times gN_2g^{-1}=L_b\times N_{2,b}=N_b$, with $N_{2,b}={\cal Z} (L_b)$ (this follows from the maximality of $N_b$). Consequently, $N$ is conjugate by $g\in V_b$ to $N_b$. \hfill $\Box$ \vskip0.25cm The situation for zero-dimensional boundary components is more complicated, so we just observe the following. Suppose ${\cal D} \not\in ({\cal E} {\cal D} )$, and that ${\cal D} _N\subset} \def\nni{\supset} \def\und{\underline {\cal D} $ is incident to $F_t$, $F_t$=point. For any $g\in N(F_t)=P_t,\ g{\cal D} _N={\cal D} '\subset} \def\nni{\supset} \def\und{\underline {\cal D} $ is another subdomain, again incident to $F_t$. If $g\in P_t\cap N_t$, then $g{\cal D} '={\cal D} _N$. In this sense, letting $Q_t=P_t\cap N_t$, the coset space $P_t/Q_t$ is a parameter space of subdomains incident with $F_t$. Above we have defined the notion of symmetric subgroups incident with a standard parabolic. Any maximal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic is conjugate to one and only one standard maximal parabolic, $P=gP_bg^{-1}$ for some $b$. Let $N_b$ be any symmetric subgroup incident with $P_b$. Then just as one has the pair $(P_b,N_b)$ one has the pair $(P,N)$, \begin{equation}\label{e10.3} P=gP_bg^{-1},\quad N=g N_b g^{-1}. \end{equation} \begin{definition} \label{d10.1} A pair $(P,N)$ consisting of a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic $P$ and a symmetric subgroup $N$ is called {\it incident}, if the groups are conjugate by a common element $g$ as in (\ref{e10.3}) to a pair $(P_b,N_b)$ which is incident as in Definition \ref{d9.1}. \end{definition} \section{Rational parabolic and rational symmetric subgroups} \subsection{Notations} We now fix some notations to be in effect for the rest of the paper. We will be dealing with algebraic groups defined over $\fQ$, which give rise to hermitian symmetric spaces, groups of {\it hermitian type}, as we will say. As we are interested in the automorphism groups of domains, we may, without restricting generality, assume the group is {\it centerless}, and {\it simple} over $\fQ$. Henceforth $G$ will denote such an algebraic group. To avoid complications, we exclude in this paper the following case: \vspace*{.2cm}{\bf Exclude:}\hspace*{2cm} All non-compact real factors of $G({\Bbb R}} \def\fH{{\Bbb H})$ are of type $SL_2({\Bbb R}} \def\fH{{\Bbb H})$. \vspace*{.2cm} \noindent Finally, we shall only consider {\it isotropic} groups. This implies the hermitian symmetric space ${\cal D} $ has no compact factors. By our assumptions, then, we have \begin{itemize}\item[(i)] $G=Res_{k|\fQ}G'$, $k$ a totally real number field, $G'$ absolutely simple over $k$. \item[(ii)] ${\cal D} ={\cal D} _1\times \cdots \times {\cal D} _d$, each ${\cal D} _i$ a non-compact irreducible hermitian symmetric space, $d=[k:\fQ]$. \end{itemize} We now introduce a few notations concerning the root systems involved. Let $\gS_{\infty}$ denote the set of embeddings $\gs:k\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb R}} \def\fH{{\Bbb H}$; this set is in bijective correspondence with the set of infinite places of $k$. We denote the latter by $\nu$, and if necessary we denote the corresponding embedding by $\gs_{\nu}$. For each $\gs\in \gS_{\infty}$, the group $^{\gs}G'$ is the algebraic group defined over $\gs(k)$ by taking the set of elements $g^{\gs},\ g\in G'$. For each infinite prime $\nu$ we have $G_{k_{\nu}}\cong (^{\gs_{\nu}}G')_{{\Bbb R}} \def\fH{{\Bbb H}}$, and the decomposition of ${\cal D} $ above can be written $${\cal D} =\prod_{\gs\in \gS_{\infty}}{\cal D} _{\gs},\quad {\cal D} _{\gs}:=(^{\gs}G')_{{\Bbb R}} \def\fH{{\Bbb H}}/K_{(\gs)}=(^{\gs}G')_{{\Bbb R}} \def\fH{{\Bbb H}}^0/K_{(\gs)}^0.$$ Since $G'$ is isotropic, there is a positive-dimensional $k$-split torus $S'\subset} \def\nni{\supset} \def\und{\underline G'$, which we fix. Then ${^{\gs}S}'$ is a maximal $\gs(k)$-split torus of $^{\gs}G'$ and there is a canonical isomorphism $S'\ra {^{\gs}S}'$ inducing an isomorphism $\Phi_k=\Phi(S',G')\longrightarrow} \def\sura{\twoheadrightarrow \Phi_{\gs(k)}({^{\gs}S}',{^{\gs}G}')=:\Phi_{k,\gs}$. The torus $Res_{k|\fQ}S'$ is defined over $\fQ$ and contains $S$ as maximal $\fQ$-split torus; in fact $S\cong S'$, diagonally embedded in $Res_{k|\fQ}S'$. This yields an isomorphism $\Phi(S,G)\cong \Phi_k$, and the root systems $\Phi_{\fQ}=\Phi(S,G)$, $\Phi_k$ and $\Phi_{k,\gs}$ (for all $\gs\in \gS_{\infty}$) are identified by means of the isomorphisms. In each group $^{\gs}G'$ one chooses a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus $A_{\gs}\nni {^{\gs}S}'$, contained in a maximal torus defined over $\gs(k)$. Fixing an order on $X(S')$ induces one also on $X({^{\gs}S}')$ and $X(S)$. Then, for each $\gs$, one chooses an order on $X(A_{\gs})$ which is compatible with that on $X({^{\gs}S}')$, and $r:X(A_{\gs})\longrightarrow} \def\sura{\twoheadrightarrow X({^{\gs}S}')\cong X(S)$ denotes the restriction homomorphism. The canonical numbering on $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{{\Bbb R}} \def\fH{{\Bbb H},\gs}$ of simple ${\Bbb R}} \def\fH{{\Bbb H}$-roots of $G$ with respect to $A_{\gs}$ is compatible by restriction with the canonical numbering of $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{\fQ}$ (\cite{BB}, 2.8). Recall also that each $k$-root in $\Phi_k$ is the restriction of at most one simple ${\Bbb R}} \def\fH{{\Bbb H}$-root of $G'({\Bbb R}} \def\fH{{\Bbb H})$ (which is a simple Lie group). Let $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_k=\{\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_1,\ldots,\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_s\}$; for $1\leq i\leq s$ set $c(i,\gs)$:= index of the simple ${\Bbb R}} \def\fH{{\Bbb H}$-root of $^{\gs}G'$ restricting on $\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_i$. Then $i<j$ implies $c(i,\gs)<c(j,\gs)$ for all $\gs\in \gS$. Each simple $k$-root defines a unique standard boundary component: for $b\in \{1,\ldots,s\}$, \begin{equation}\label{e9.1} F_{\hbox{\scsi \bf b}}:=\prod_{\gs\in \gS_{\infty}}F_{c(b,\gs)}, \end{equation} which is the product of standard (with respect to $A_{\gs}$ and $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{{\Bbb R}} \def\fH{{\Bbb H},\gs}$) boundary components $F_{c(b,\gs)}$ of ${\cal D} _{\gs}$. It follows that $\overline{F}_{\hbox{\scriptsize\bf j}}\subset} \def\nni{\supset} \def\und{\underline \overline{F}_{\hbox{\scriptsize\bf i}}$ for $1\leq i\leq j\leq s$. Furthermore, setting $o_{\hbox{\scsi \bf b}}:=\prod o_{c(b,\gs)}$, then (\cite{BB}, p.~472) \begin{equation}\label{e9.2} F_{\hbox{\scsi \bf b}}=o_{\hbox{\scsi \bf b}}\cdot L_{\hbox{\scsi \bf b}}, \end{equation} where $L_{\hbox{\scsi \bf b}}$ denotes the hermitian Levi component (\ref{e3.3}) of the parabolic $P_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})=N(F_{\hbox{\scsi \bf b}})$. As these are the only boundary components of interest to us, we will henceforth refer to any conjugates of the $F_{\hbox{\scsi \bf b}}$ of (\ref{e9.1}) by elements of $G$ as {\it rational boundary components} (these should more precisely be called rational with respect to $G$), and to the conjugates of the parabolics $P_{\hbox{\scsi \bf b}}$ as the {\it rational parabolics}. \subsection{Rational parabolics} Let $G'$ be as above, $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_k=\{\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_1\ldots,\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_s\}$ the set of simple $k$-roots (having fixed a maximal $k$-split torus $S'$ and an order on $X(S')$). For $b\in \{1,\ldots,s\}$ we have the standard maximal $k$-parabolic $P_b'$ of $G'$, whose group of ${\Bbb R}} \def\fH{{\Bbb H}$-points is the normalizer of the standard rational boundary component $F_{c(b)}$ of the domain ${\cal D} '=G_{{\Bbb R}} \def\fH{{\Bbb H}}'/K'$, where $c(b)$ denotes the index of the simple ${\Bbb R}} \def\fH{{\Bbb H}$-root restricting to $\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_b$; since $G'$ is absolutely simple, $G_{{\Bbb R}} \def\fH{{\Bbb H}}'$ is simple and ${\cal D} '$ is irreducible. Hence Theorem \ref{t4.1} applies to $P_b'({\Bbb R}} \def\fH{{\Bbb H})$. Of the factors given there, the following are defined over $k$: the product $M_b'L_b'$ as well as $L_b'$ (but $M_b'$ is {\it not} defined over $k$, so the $k$-subgroups are (instead of $L_b'$ and $M_b'$) $L_b'$ and ${G_b'}^{(1)}:=M_b'L_b'$), ${\cal R} _b',{\cal Z} _b'$ and $V_b'$. As is well known, any maximal $k$-parabolic of $G'$ is conjugate to one and only one of the $P_b'$, and two parabolics are conjugate $\iff$ they are conjugate over $k$. There is a 1-1 correspondence between the set of $k$-parabolics of $G'$ and the set of $\fQ$-parabolics of $G$, given by $P'\mapsto Res_{k|\fQ}P'=:P$. The standard maximal $\fQ$-parabolic $P_{\hbox{\scriptsize\bf b}}$ of $G$ gives a $\fQ$-structure on the real parabolic $P_{\hbox{\scriptsize\bf b}}({\Bbb R}} \def\fH{{\Bbb H})$, which is the normalizer in ${\cal D} $ of the standard boundary component $F_{\hbox{\scsi \bf b}}$ as in (\ref{e9.1}) (see also (\ref{e3.2}) and (\ref{e3.3})), where ${\bf b}=(c(b,\gs_1),\ldots,c(b,\gs_d))$. In the decomposition of Theorem \ref{t4.1}, the factors $G_{\hbox{\scriptsize\bf b}}^{(1)}=M_{\hbox{\scriptsize\bf b}}L_{\hbox{\scriptsize\bf b}},\ {\cal R} _{\hbox{\scriptsize\bf b}}, {\cal Z} _{\hbox{\scriptsize\bf b}}$ and $V_{\hbox{\scriptsize\bf b}}$ are all defined over $\fQ$. In particular, for the factor $G_{\hbox{\scriptsize\bf b}}^{(1)}$, which we will call the $\fQ$-hermitian Levi factor (and similarly, we will call ${G_{b}'}^{(1)}$ the $k$-hermitian Levi factor of $P_{b}'$), we have \begin{equation}\label{e12.1} G_{\hbox{\scriptsize\bf b}}^{(1)}(\fQ)\cong \prod_{\gs}({{^{\gs}G}'}_b^{(1)})_{\gs(k)},\quad {\cal Z} _G(G_{\hbox{\scriptsize\bf b}}^{(1)})(\fQ)\cong \prod_{\gs}({\cal Z} _{(^{\gs}G'_{\gs(k)})}( ({{^{\gs}G}'_b}^{(1)})_{\gs(k)}). \end{equation} Furthermore, the hermitian Levi factor $L_{\hbox{\scsi \bf b}}$ is defined over $\fQ$, and \[L_{\hbox{\scsi \bf b}}(\fQ)=\prod_{\gs}({^{\gs}L}_b')_{\gs(k)}. \] We now make a few remarks about the factors of $G({\Bbb R}} \def\fH{{\Bbb H})$ and of $L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$. Since the map $G'\longrightarrow} \def\sura{\twoheadrightarrow {^{\gs}G}'$ is an isomorphism of a $k$-group onto a $\gs(k)$-group, the algebraic groups (over $\fC$) are isomorphic, hence the various ${^{\gs}G}'_{{\Bbb R}} \def\fH{{\Bbb H}}$ are all ${\Bbb R}} \def\fH{{\Bbb H}$-forms of some fixed algebraic group. Similarly, the factors of $L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ are all ${\Bbb R}} \def\fH{{\Bbb H}$-forms of a single $\fC$-group. However, they need not be isomorphic, unless the given $\fC$-group has a unique ${\Bbb R}} \def\fH{{\Bbb H}$-form of hermitian type (like $Sp(2n,\fC)$). Next we note the following. \begin{lemma}\label{L12a} $L_{\hbox{\scsi \bf b}}$ is anisotropic $\iff$ $b=s$. \end{lemma} {\bf Proof:} The group $L_{\hbox{\scsi \bf b}}$ is anisotropic precisely when the boundary component $F_{\hbox{\scsi \bf b}}$ defined by it contains no other boundary components $F_{\hbf{c}}^*\subset} \def\nni{\supset} \def\und{\underline F_{\hbox{\scsi \bf b}}^*$, which means $b\geq c$ for all $c$, or $b=s$. \hfill $\Box$ \vskip0.25cm In this case the group $L_{\hbox{\scsi \bf b}}$ does not fulfill the assumptions we have placed on $G$, and our results up to this point are not directly applicable to $L_{\hbox{\scsi \bf b}}$. Let us see how the phenomenon of compact factors of $L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ manifests itself in $F_{\hbox{\scsi \bf b}}=\prod_{\gs}F_{c(b,\gs)}$. Suppose some factor of $L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ is compact, say $L_{1,b}$. Then the symmetric space ${\cal D} _{b,\gs_1}$ of $L_{1,b}$ is compact, so it is not true that ${\cal D} _{b,\gs_1}\cong F_{c(b,\gs_1)}$, hence it is also not true that ${\cal D} _{\hbox{\scsi \bf b}}\cong F_{\hbox{\scsi \bf b}}$, where ${\cal D} _{\hbox{\scsi \bf b}}=\prod_{\gs}{\cal D} _{b,\gs}$ is the symmetric space of $L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$. However, letting ${\cal D} _{\hbox{\scsi \bf b}}'$ be the product of all compact factors, ${\cal D} _{\hbox{\scsi \bf b}}/{\cal D} _{\hbox{\scsi \bf b}}'$ is a symmetric space which is isomorphic to $F_{\hbox{\scsi \bf b}}$. What happens is that in the product $F_{\hbox{\scsi \bf b}}=\prod F_{c(b,\gs)}$, all factors $F_{c(b,\gs)}$ are {\it points} for which ${\cal D} _{b,\gs}$ is {\it compact}. Hence whether this occurs depends on whether any factors ${\cal D} _{\gs}$ have zero-dimensional (rational) boundary components or not. \subsection{Incidence} We keep the notations used above; $G$ is a simple $\fQ$-group of hermitian type. Our main definition gives a $\fQ$-form of Definition \ref{d10.1}, and is the following. \begin{definition}\label{d12.1} Let $P\subset} \def\nni{\supset} \def\und{\underline G$ be a maximal $\fQ$-parabolic, $N\subset} \def\nni{\supset} \def\und{\underline G$ a reductive $\fQ$-subgroup. Then we shall say that $(P,N)$ are {\it incident} (over $\fQ$), if $(P({\Bbb R}} \def\fH{{\Bbb H}),N({\Bbb R}} \def\fH{{\Bbb H}))$ are incident in the sense of Definition \ref{d10.1}. \end{definition} Note that in particular $N$ must itself be of hermitian type, and such that the Cartan involution of $G({\Bbb R}} \def\fH{{\Bbb H})$ restricts to the Cartan involution of $N({\Bbb R}} \def\fH{{\Bbb H})$. Furthermore, $N$ must be a $\fQ$-form of a product of groups, defining domains each of which is as in either Table \ref{T1} or Table \ref{T2}. The main result of this paper is the following existence result. \begin{theorem}\label{t12.1} Let $G$ be $\fQ$-simple of hermitian type subject to the restrictions above ($G$ is isotropic and $G({\Bbb R}} \def\fH{{\Bbb H})$ is not a product of $SL_2({\Bbb R}} \def\fH{{\Bbb H})$'s), $P\subset} \def\nni{\supset} \def\und{\underline G$ a $\fQ$-parabolic. Then there exists a reductive $\fQ$-subgroup $N\subset} \def\nni{\supset} \def\und{\underline G$ such that $(P,N)$ are incident over $\fQ$, with the exception of the indices $C^{(2)}_{2n,n}$ for the zero-dimensional boundary components. \end{theorem} We will give the proof in the following sections, where we consider separately different cases (of the $\fQ$-rank, the dimension of a maximal $\fQ$-split torus). But before we start, we note here that by definition, if the theorem holds for {\it standard} parabolics, then it holds for all parabolics, so it will suffice to consider only standard parabolics. The case that $G'$ has index $C^{(2)}_{2,1}$ was considered in \cite{hyp}; in that case there is a unique standard parabolic $P_1$, with zero-dimensional boundary component; the associated $N_1$ described in \cite{hyp} has domain ${\cal D} _{N_1}$ which is not a two-disc, but only a one-dimensional disc. \section{Split over ${\Bbb R}} \def\fH{{\Bbb H}$ case} In this paragraph we consider the easiest case. This could loosely be described as an ${\Bbb R}} \def\fH{{\Bbb H}$-Chevally form. \begin{definition}\label{d13.1} Let $G'$ be as in the last paragraph, absolutely simple over $k$, and let $\Phi_k$ be a root system (irreducible) for $G'$ with respect to a maximal $k$-split torus $S'\subset} \def\nni{\supset} \def\und{\underline G'$. Let $\Phi_{{\Bbb R}} \def\fH{{\Bbb H}}$ be the root system of $G'({\Bbb R}} \def\fH{{\Bbb H})$ with respect to a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus $A'$ of the real (simple) group $G'({\Bbb R}} \def\fH{{\Bbb H})$. We call $G'$ {\it split over ${\Bbb R}} \def\fH{{\Bbb H}$}, if $\Phi_k\cong \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}$ as root systems, and if the indices of $G'$ and $G'({\Bbb R}} \def\fH{{\Bbb H})$ coincide. \end{definition} Note that the indices are independent of the split tori used to form the root system, so there is no need to assume $S'\subset} \def\nni{\supset} \def\und{\underline A'$ in the above definition (the notion of isomorphism of indices is obvious). However, one can always find split tori $S', A'$ such that $S'\subset} \def\nni{\supset} \def\und{\underline A'$. From $\Phi_k\cong \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}$ it follows then that $S'=A'$, as both tori have the same dimension. \begin{lemma}\label{l13.1} Let $G$ be simple over $\fQ$ (=$Res_{k|\fQ}G'$), ${\cal D} =\prod_{\gs\in \gS_{\infty}}{\cal D} _{\gs}$ the domain defined by the real Lie group $G({\Bbb R}} \def\fH{{\Bbb H})\cong \prod_{\gs\in \gS_{\infty}}G_{\gs}=:\prod_{\gs}{^{\gs}G'_{{\Bbb R}} \def\fH{{\Bbb H}}}$. If $G'$ is split over ${\Bbb R}} \def\fH{{\Bbb H}$, then $G_{\gs}=G_{\tau}$ for all $\gs,\tau\in \gS_{\infty}$. \end{lemma} {\bf Proof:} For each $\gs$ we have $A_{\gs}\nni {^{\gs}S}'$, so by assumption $A_{\gs}\cong {^{\gs}S}'$, and for each $\gs$ the map $\phi:\Phi_k\longrightarrow} \def\sura{\twoheadrightarrow \Phi_{\gs(k)}(^{\gs}G')$ is an isomorphism, and since $\Phi_k\cong \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}$, $$\Phi_{\gs(k)}(^{\gs}G')\cong \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}(^{\gs}G_{{\Bbb R}} \def\fH{{\Bbb H}}').$$ It follows that $\Phi_{{\Bbb R}} \def\fH{{\Bbb H}}\cong \Phi_k\stackrel{\phi}{\cong} \Phi_{\gs(k)}(^{\gs}G')\cong \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}(^{\gs}G'_{{\Bbb R}} \def\fH{{\Bbb H}})\cong \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}$. Similarly, since the index of $G'$ is isomorphic to the index of $G'({\Bbb R}} \def\fH{{\Bbb H})$ (which determines the isomorphy class of $G'({\Bbb R}} \def\fH{{\Bbb H})$), the index of ${^{\gs}G}'$ is isomorphic to that of ${^{\gs}G}'({\Bbb R}} \def\fH{{\Bbb H})$. But the index of $G'({\Bbb R}} \def\fH{{\Bbb H})$ is the same as ${^{\gs}G}'({\Bbb R}} \def\fH{{\Bbb H})$, as an easy case by case check verifies. For example, for type (I), all factors have the same ${\Bbb R}} \def\fH{{\Bbb H}$-rank $q$, hence are all isomorphic to $SU(p,q)$. See Examples \ref{examples} below for the other cases. Hence ${^{\gs}G}'({\Bbb R}} \def\fH{{\Bbb H})\cong {^{\tau}G}'({\Bbb R}} \def\fH{{\Bbb H})$ for all $\gs, \tau$, as claimed. \hfill $\Box$ \vskip0.25cm {}From this it follows in particular that the (standard) boundary components are determined by $c(b,\gs)=b,\ \forall_{\gs},\ {\bf b}=(b,\ldots,b),\ 1\leq b\leq t=\rank_{\fQ}G=\rank_{k}G'=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G'({\Bbb R}} \def\fH{{\Bbb H})$. Hence they are of the form \begin{equation}\label{e13.1} F_{\hbox{\scsi \bf b}}=\prod_{\gs\in \gS_{\infty}}F_{b,\gs}, \end{equation} and $F_{b,\gs}$ is the standard rational boundary component of ${\cal D} _{\gs}$. \begin{examples}\label{examples} We now give examples of split over ${\Bbb R}} \def\fH{{\Bbb H}$ groups in each of the cases, and any such will be of one of the listed types. Let $k$ be a totally real number field. \begin{itemize}\item[I.] Let $K|k$ be imaginary quadratic, $V$ a $K$-vector space of dimension $n=p+q$, and $h$ a hermitian form on $V$ defined over $K$. Then the unitary group $U(V,h)$ is split over ${\Bbb R}} \def\fH{{\Bbb H}$ $\iff$ the hermitian form $h$ has Witt index $q$ and for all infinite primes, $h_{\nu}$ has signature $(p,q)$. \item[II.] Let $D|k$ be a totally definite quaternion algebra over $k$ (with the canonical involution), $V$ an $n$-dimensional right vector space over $D$, $h$ a skew-hermitian form on $V$ defined over $k$. Then the unitary group $U(V,h)$ is split over ${\Bbb R}} \def\fH{{\Bbb H}$ $\iff$ the skew-hermitian form has Witt index $[{n\over2}]$ ($n>4$). \item[III.] Take $G=Sp(2n,k)$. \item[IV.] Let $V$ be a $(n+2)$-dimensional $k$-vector space, $h$ a symmetric bilinear form defined over $k$ of Witt index 2. Then if $U(V,h)$ is of hermitian type, it is split over ${\Bbb R}} \def\fH{{\Bbb H}$. \item[V.] The Lie algebra in this case is of the form $\hbox{{\script L}} (\frak C_k,(J_1^b)_k)$, the Tits algebra, where $\frak C_k$ is an anisotropic octonion algebra and $(J_1^b)_k$ is the Jordan algebra $\BB^+$ for an associative algebra $\BB$ whose traceless elements with the Lie product form a Lie algebra of type $\frak s} \def\cc{\frak c\uu(2,1)$; since $G'$ is split over ${\Bbb R}} \def\fH{{\Bbb H}$, the algebra $\BB^-$ is the Lie algebra of a unitary group of a $K$-hermitian form ($K|k$ imaginary quadratic as in (I)) of Witt index 1. \item[VI.] The Lie algebra is isomorphic to $\hbox{{\script L}} (\AA_k,\JJ_k)$, the Tits algebra, where $\AA_k$ is a totally indefinite quaternion algebra over $k$ and $\JJ_k$ is a $k$-form of the exceptional Jordan algebra denoted $J^b$ by Tits. \end{itemize} \end{examples} \begin{lemma}\label{l14.1} In the notations above, let $N'({\Bbb R}} \def\fH{{\Bbb H})\subset} \def\nni{\supset} \def\und{\underline G'({\Bbb R}} \def\fH{{\Bbb H})$ be a subgroup such that the Lie algebra $\frak n} \def\rr{\frak r'\subset} \def\nni{\supset} \def\und{\underline \Gg'$ is a {\it regular} subalgebra, i.e., defined by a closed symmetric set of roots $\Psi$ of the (absolute) root system $\Phi$ of $G'$. Then $N'$ is defined over $k$, $N'\subset} \def\nni{\supset} \def\und{\underline G'$. \end{lemma} {\bf Proof:} From the isomorphism of the indices of $G'$ and $G'({\Bbb R}} \def\fH{{\Bbb H})$, it follows that any subalgebra $\Gg'\subset} \def\nni{\supset} \def\und{\underline \Gg$, such that for some subset $\Psi\subset} \def\nni{\supset} \def\und{\underline \Phi$, the subalgebra $\Gg'$ is given by $\Gg'=\tt+\sum_{\eta\in \Psi}\Gg^{\eta}$, is defined over ${\Bbb R}} \def\fH{{\Bbb H}$ $\iff$ it is defined over $k$. The regular subalgebra $\frak n} \def\rr{\frak r'$ is of this type, and it follows that $N'$ is defined over $k$. \hfill $\Box$ \vskip0.25cm \begin{corollary}\label{c14.1} Let $N'\subset} \def\nni{\supset} \def\und{\underline G'$ be as in Lemma \ref{l14.1}, $N=Res_{k|\fQ}N'\subset} \def\nni{\supset} \def\und{\underline G$. Then $N$ is defined over $\fQ$. \end{corollary} To apply Lemma \ref{l14.1} to ($k$-forms of) subgroups whose domains are listed in Tables \ref{T1} and \ref{T2}, we need to know which of the subgroups are defined by regular subalgebras. Ihara in \cite{I} considered this question, and the result is: all isomorphism classes of groups in Table \ref{T1} and all isomorphism classes of groups in Table \ref{T2}, with the exception of $SO(n-1,2)\subset} \def\nni{\supset} \def\und{\underline SO(n,2)$ for $n$ even, have representatives which are defined by (maximal) regular subalgebras. \begin{corollary}\label{c14.2} Let $G'$ be split over ${\Bbb R}} \def\fH{{\Bbb H}$. Then Theorem \ref{t12.1} holds for $G=Res_{k|\fQ}G'$. \end{corollary} {\bf Proof:} By Lemma \ref{l13.1}, $G({\Bbb R}} \def\fH{{\Bbb H})/K={\cal D} =\prod{\cal D} _{\gs}$, and all ${\cal D} _{\gs}$ are isomorphic to ${\cal D} '=G'({\Bbb R}} \def\fH{{\Bbb H})/K'$; the rational boundary components are as in (\ref{e13.1}), products of copies of $F_b'$, the standard boundary component of ${\cal D} '$, and each $\fQ$-parabolic of $G$ is conjugate to one of $P_b=Res_{k|\fQ}P_b'$, where $P_b'({\Bbb R}} \def\fH{{\Bbb H})=N(F_b')$ is the standard maximal real parabolic of $G'({\Bbb R}} \def\fH{{\Bbb H})$. Now locate $F_b'$ in Table \ref{T1} or \ref{T2} as the case may be; the corresponding group $N_b'$ is isomorphic to one defined by a regular subalgebra of $\Gg_{{\Bbb R}} \def\fH{{\Bbb H}}'$ with the one exception mentioned above. Then by Lemma \ref{l14.1}, $N_b'$ is defined over $k$, hence (Corollary) $N_b=Res_{k|\fQ}N_b'$ is defined over $\fQ$, and is incident with $P_b$. This takes care of all cases except the exception just mentioned, $\bf IV_{\hbox{\scriptsize\bf n-1}}\subset} \def\nni{\supset} \def\und{\underline IV_{\hbox{\scriptsize\bf n}}$, $n>3$ even. So let $V$ be a $k$-vector space of dimension $n+2$, $h$ a symmetric bilinear form on $V$. By assumption, $G'$ is split over ${\Bbb R}} \def\fH{{\Bbb H}$, so the Witt index of $h$ is 2. Let $H\subset} \def\nni{\supset} \def\und{\underline V$ be a maximal totally isotropic subspace (two-dimensional) defined over $k$, and $h_1,h_2$ a $k$-basis. Then there are $k$-vectors $h_i'$ such that $H_1:=<h_1,h_1'>$ and $H_2:=<h_2,h_2'>$ are hyperbolic planes; let $W=H_1\oplus H_2$ denote their direct sum. From $n>3$, $W$ has codimension $\geq1$ in $V$. Let $v\in W^{\perp}$ be a $k$-vector, and set: $$U:=v^{\perp}=\{w\in V|h(v,w)=0\}.$$ Then $W\subset} \def\nni{\supset} \def\und{\underline U$, the dimension of $U$ is $n+2-1=n+1$, and $h_{|U}$ still has Witt index 2. Hence $$N':=\{g\in U(V,h)|g(U)\subset} \def\nni{\supset} \def\und{\underline U\}$$ is a $k$-subgroup, and $N'({\Bbb R}} \def\fH{{\Bbb H})^0\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} SO(n-1,2)$. This is a group which is incident to a parabolic whose group of real points is the stabilizer of the zero-dimensional boundary component $F_{2}'$ of the domain ${\cal D} '$ of type $\bf IV_{\hbox{\scriptsize\bf n}}$. \hfill $\Box$ \vskip0.25cm This completes the discussion of the split over ${\Bbb R}} \def\fH{{\Bbb H}$ case. We just mention that, at least in the classical cases, we could have argued case for case with $\pm$symmetric/hermitian forms as in the proof of the exception above. Using the root systems simplified the discussion, and, in particular, gives the desired results for the exceptional groups without knowing their explicit construction. \section{Rank $\geq 2$} In this paragraph we assume $G$ in {\it not} split over ${\Bbb R}} \def\fH{{\Bbb H}$, but that $\rank_kG'=\rank_{\fQ}G\geq 2$. Under these circumstances, it is known precisely which $k$-indices are possible for $G'$ of hermitian type. \begin{proposition}\label{p16.1} Assume $\rank_{\fQ}G\geq 2$ and that $G'$ is not split over ${\Bbb R}} \def\fH{{\Bbb H}$. Then the $k$-index of $G'$ is one of the following: \begin{itemize}\item[(I)] ${^2A}^{(d)}_{n,s};\ s\geq2,\ d|n+1,\ 2sd\leq n+1$; if $d=1$, then $2s<n+1$. \item[(II)] ${^1D}^{(2)}_{n,s},\ s\geq2,\ s< \ell\ (n=2\ell);\quad {^2D}^{(2)}_{n,s},\ s\geq 2,\ s<\ell\ (n=2\ell+1)$. \item[(III)] $C^{(2)}_{n,s},\ s\geq 2,\ s< [{n\over 2}]$. \item[(IV)] none \item[(V)] none \item[(VI)] $E^{31}_{7,2}$. \end{itemize} \end{proposition} {\bf Proof:} All statements are self-evident from the description of the indices in \cite{T}; in the case (V) there are three possible indices, only one of which has rank $\geq 2$; this is the split over ${\Bbb R}} \def\fH{{\Bbb H}$ index. Similarly, in the case (IV), rank $\geq2$ implies split over ${\Bbb R}} \def\fH{{\Bbb H}$. For type (III), the indices $C^{(1)}$ are also split over ${\Bbb R}} \def\fH{{\Bbb H}$. \hfill $\Box$ \vskip0.25cm There is only one exceptional index to consider, so we start by dealing with this case. The index we must discuss is $$\begin{minipage}{16.5cm} \unitlength1.5cm \hspace*{2cm} \begin{picture}(9,1) \put(0,0){\circle*{.2}} \put(1,0){\circle{.2}} \put(2,0){\circle*{.2}} \put(3,0){\circle*{.2}} \put(4,0){\circle*{.2}} \put(5,0){\circle{.2}} \put(-1,0){$E^{31}_{7,2}$} \put(.1,0){\line(1,0){.8}} \put(1.1,0){\line(1,0){.8}} \put(2.1,0){\line(1,0){.8}} \put(3.1,0){\line(1,0){.8}} \put(4.1,0){\line(1,0){.8}} \put(3,.1){\line(0,1){.8}} \put(3,1){\circle*{.2}} \put(6,0){\parbox[b]{2cm}{with the \\ $k$-root \\ system:}} \put(7,0){\mbox{ \setlength{\unitlength}{0.0037500in}% \begin{picture}(216,60)(12,710) \thicklines \put(320,740){\circle{36}} \put(310,680){$\eta_1$} \put(160,740){\line( 1, 1){ 40}} \put(140,740){\circle{36}} \put(130,680){$\eta_2$} \put(160,740){\line( 1,-1){ 40}} \put(175,725){\line( 1, 0){135}} \put(175,755){\line( 1, 0){135}} \end{picture}}} \end{picture} \end{minipage}$$ There are two simple $k$-roots, $\eta_1$ and $\eta_2$; let $P_{b}'$ be the corresponding standard maximal $k$-parabolics, $F_{b}'$ the corresponding standard boundary components of the irreducible domain ${\cal D} '$. Then $F_{2}'$ is the one-dimensional boundary component, $F_{1}'$ is the ten-dimensional one. The $k$-root system is of type ${\bf BC}_2$ (since the highest simple ${\Bbb R}} \def\fH{{\Bbb H}$-root is anisotropic, see \cite{BB}, 2.9). Consider the decomposition of Theorem \ref{t4.1} for $P_{b}'({\Bbb R}} \def\fH{{\Bbb H})$; in both cases $L'_b$ is non-trivial, and, as mentioned above, $M_b'\cdot L_b'$ is defined over $k$. Here we have $b=1$ or 2. But for $E_7$, the compact factor $M_b'$ is in fact {\it absent}\footnote{see \cite{S}, p.~117}, and as $L_b'$ is defined over $k$, we can set $$N_b'=L_b'\times {\cal Z} _{G'}(L_b').$$ This is a $k$-subgroup which is a $k$-form of the corresponding ${\Bbb R}} \def\fH{{\Bbb H}$-subgroup whose domain is listed in Table \ref{T1}. Now consider $G=Res_{k|\fQ}G'$. It also has two standard maximal parabolics $P_{\hbox{\scriptsize\bf 1}}$ and $P_{\hbox{\scriptsize\bf 2}}$, and in each we have a non-trivial hermitian Levi factor\footnote{we note a change of notation here in that in (\ref{e3.3}), $L_{\hbox{\scsi \bf b}}$ denotes a real Lie group} $L_{\hbox{\scsi \bf b}}:= Res_{k|\fQ}L_b'$, such that $$L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})=\prod_{\gs\in \gS_{\infty}}{^{\gs}(L_b')}_{{\Bbb R}} \def\fH{{\Bbb H}}.$$ Also the symmetric subgroup $N_{\hbox{\scsi \bf b}}:=Res_{k|\fQ}N_b'$ is defined over $\fQ$ and satisfies $N_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})=\prod_{\gs\in \gS_{\infty}}{^{\gs}(N_b')}_{{\Bbb R}} \def\fH{{\Bbb H}}$. It follows that $(P_{\hbox{\scriptsize\bf b}},N_{\hbox{\scsi \bf b}})$ are incident: conditions 1) and 2) follow from the corresponding facts for $(P_b',N_b')$; we should check 3). But since it is obvious that ${^{\gs}(}L_b')_{{\Bbb R}} \def\fH{{\Bbb H}}\subset} \def\nni{\supset} \def\und{\underline {^{\gs}(}P_b')_{{\Bbb R}} \def\fH{{\Bbb H}}$ is a hermitian Levi factor, the same holds for $L_{\hbox{\scsi \bf b}}\subset} \def\nni{\supset} \def\und{\underline P_{\hbox{\scsi \bf b}}$; 3) is satisfied. This completes the proof of \begin{proposition}\label{p16.2} Theorem \ref{t12.1} is true for the exceptional groups in the rank$\geq 2$, not split over ${\Bbb R}} \def\fH{{\Bbb H}$ case. \end{proposition} We are left with the classical cases. Here we may use the interpretation of $G({\Bbb R}} \def\fH{{\Bbb H})$ as the unitary group of a $\pm$symmetric/hermitian form as in (\ref{e7.1}), and $G$ is a $\fQ$-form of this. The precise realisation of this is the interpretation in terms of central simple algebras with involution; this is discussed in \cite{W}. More precisely, the algebraic groups $G'$ which represent the indices of Proposition \ref{p16.1} are (here we describe reductive groups; the corresponding derived groups are the simple groups $G'$). \begin{itemize}\item[(I)] \begin{itemize}\item[$D$:] degree $d$ central simple division algebra over $K$, $K|k$ an imaginary quadratic extension, $D$ has a $K|k$-involution (involution of the second kind). \item[$V$:] right $D$-vector space, of dimension $m$ over $D$, $dm=n+1$. \item[$h$:] hermitian form $h:V\times V\longrightarrow} \def\sura{\twoheadrightarrow D$ of Witt index $s$, $2s\leq m$ ($2s<m$ if $d=1$), given by a matrix $H$. \item[$G'$:] unitary group $U(V,h)=\{g\in GL_D(V)|gHg^*=H\}$. \end{itemize} {\bf index:} ${^2A}^{(d)}_{n,s}$. \item[(II)] \begin{itemize}\item[$D$:] totally definite quaternion division algebra, central simple over $k$, with canonical involution. \item[$V$:] right $D$-vector space of dimension $m$ over $D$. \item[$h$:] skew-hermitian form $h$ of Witt index $s< [{m\over 2}]$, given by a matrix $H$. \item[$G'$:] unitary group $U(V,h)=\{g\in GL_D(V)|gHg^*=H\}$. \end{itemize} {\bf index:} $D^{(2)}_{m,s}$ ($m$ even), ${^2D}^{(2)}_{m,s}$ ($m$ odd). \item[(III)] \begin{itemize}\item[$D$:] totally indefinite quaternion division algebra, central simple over $k$, with the canonical involution. \item[$V$:] right $D$-vector space of dimension $m$. \item[$h$:] hermitian form $h:V\times V\longrightarrow} \def\sura{\twoheadrightarrow D$ of Witt index $s$, $2s\leq m$, given by a matrix $H$. \item[$G'$:] unitary group $U(V,h)=\{g\in GL_D(V)|gHg^*=H\}$. \end{itemize} {\bf index:} $C^{(2)}_{m,s}$. \end{itemize} Finally, we must consider the following ``mixed cases'', which still can give rise to groups of hermitian type: \begin{itemize}\item[(II-IV):]\begin{itemize}\item[$D$:] a quaternion division algebra over $k$, with $D_{\nu}$ definite for $\nu_1,\ldots,\nu_a$, $D_{\nu}$ indefinite for $\nu_{a+1},\ldots,\nu_f$, where $f=[k:\fQ]$. \item[$V$:] same as for (II) above. \item[$h$:] same as for (II) above, $h$ of Witt index $s$. \item[$G'$:] same as for (II) above. \end{itemize}\end{itemize} $G({\Bbb R}} \def\fH{{\Bbb H})$ is then a product $(SU(n,\fH))^a\times (SO(2n-2,2))^{f-a}$, where we have taken into account that we are assuming $G$ to be isotropic and of hermitian type. Note however, that since the factors $SO(2n-2,2)$ corresponding to the primes $\nu_{a+1},\ldots,\nu_f$ have ${\Bbb R}} \def\fH{{\Bbb H}$-split torus of dimension two, the $k$-rank of $G'$ must be $\leq2$. Hence the only indices where this can occur are: ${^iD}^{(2)}_{n,1}$ and ${^i}D^{(2)}_{n,2}$, $i=1,2$. In terms of the spaces $(V,h)$, the standard parabolics are stabilizers of totally isotropic subspaces $H_b\subset} \def\nni{\supset} \def\und{\underline V$, where $H_1$ is one-dimensional (over $D$), while $H_s$ is a maximal totally isotropic subspace. The latter case corresponds to zero-dimensional boundary components. We consider first the case $H_b,\ b<s$, of which at least $H_1$ exists, because of the assumption rank $\geq 2$. Fix a basis $h_1,\ldots, h_b$ of $H_b$ of isotropic vectors $h(h_i,h_i)=0$ for all $i=1,\ldots,b$. Then there exist, in $V$, elements $h_i',\ i=1,\ldots,b$ with $h(h_i,h_j')=\gd_{ij}$, and $h_1',\ldots,h_b'$ span a complementary totally isotropic subspace; denote it by $H_b'$. Then $H:=H_b\oplus H_b'$ is a {\it non-degenerate} space for $h$, $h_{|H}$ is a non-degenerate form. It follows that $\{g\in GL(V) | g(H)\subset} \def\nni{\supset} \def\und{\underline H\} = \{g\in GL(V) | g(H^{\perp})\subset} \def\nni{\supset} \def\und{\underline H^{\perp}\}$. In the following we will work in the (reductive) unitary group $G'=U(V,h)$; for any subgroup $H\subset} \def\nni{\supset} \def\und{\underline G'$ we can take the intersection $SL(V)\cap H\subset} \def\nni{\supset} \def\und{\underline SL(V)\cap G'$ to give subgroups of the simple group. Furthermore, up to Corollary \ref{c19.1} below, we omit the primes in the notations for the subgroups of $G'$. Set \begin{equation}\label{e18.1} N=U(H,H^{\perp};h)=\{g\in GL(V)|g(H)\subset} \def\nni{\supset} \def\und{\underline H,\ g(H^{\perp})\subset} \def\nni{\supset} \def\und{\underline H^{\perp}\}; \end{equation} then $N=U(H,h_{|H})\times U(H^{\perp},h_{|H^{\perp}}),$ and $U(H,h_{|H})\cong {\cal Z} _G(U(H^{\perp},h_{|H^{\perp}}))$. So setting $L=U(H^{\perp}, h_{|H^{\perp}})$, we have \begin{equation}\label{e18.2} N\cong L \times {\cal Z} _G(L). \end{equation} Next we note that the basis $h_1,\ldots, h_b$ of $H_b$ determines a unique ${\Bbb R}} \def\fH{{\Bbb H}$-split torus $A_b\subset} \def\nni{\supset} \def\und{\underline A$, where $A$ is the maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus defined by a basis $h_1,\ldots,h_s$ of a maximal totally isotropic subspace $H_s\nni H_b$, namely the scalars $\ga=\ga\cdot{\bf 1}\in GL(H_b)$, extended to $GL(V)$ by unity. Taking the centralizer of the torus $A_b$ gives a Levi factor of the parabolic $P_b={\cal N} _G(H_b),\ b<s$ (the normalizer in $G$ of $H_b$). \begin{lemma}\label{l18.1} $L=U(H^{\perp},h_{|H^{\perp}})$ is the $k$-hermitian factor $G_b^{(1)}=M_b\cdot L_b$ of $P_b$ in the decomposition of $P_b$ as in Theorem \ref{t4.1}. \end{lemma} {\bf Proof:} First observe that $L\subset} \def\nni{\supset} \def\und{\underline P_b$, as $H^{\perp}$ is orthogonal to the totally isotropic subspace, hence $L$ normalizes $H_b$. Since $L$ is reductive, there is a Levi decomposition of $P_b$ for which $L$ is contained in the Levi factor. It is clearly of hermitian type, and maximal with this property. We must explain why the Levi factor is the standard one ${\cal Z} (A_b)$. But this follows from the fact that $H_b$ is constructed by means of a basis, which in turn was determined by the choice of ${\Bbb R}} \def\fH{{\Bbb H}$-split torus $A_b$. It therefore suffices to explain the ``compact'' factor $M_b$. This factor occurs only in the cases $\bf I_{\bf\scriptstyle p,q}$ and $\bf IV_{\scriptstyle\bf n}$. We don't have to consider the latter case, as this is split over ${\Bbb R}} \def\fH{{\Bbb H}$ if rank $\geq 2$. So suppose $G\cong U(V,h)$, where $(V,h)$ is as in (I) above. We first determine the anisotropic kernel. Let $H_s$ be a maximal totally isotropic subspace, $S:=H_s\oplus H_s'$ as above. Then $U(S^{\perp}, h_{|S^{\perp}})$ is the anisotropic kernel, $U(S^{\perp},h_{|S^{\perp}})({\Bbb R}} \def\fH{{\Bbb H})\cong U(md-2sd)$. In particular, for $m=2s$, there is no anisotropic kernel. Now consider the group $L=U(H^{\perp},h_{|H^{\perp}})$. Clearly, for $b<s$, we have \[U(S^{\perp},h_{|S^{\perp}})\subset} \def\nni{\supset} \def\und{\underline U(H^{\perp},h_{|H^{\perp}})=L,\] so that $L$ contains the anisotropic kernel. Note that $SU(H^{\perp},h_{|H^{\perp}})({\Bbb R}} \def\fH{{\Bbb H})\cong L_b({\Bbb R}} \def\fH{{\Bbb H})$, while (if $H^{\perp}\neq \{0\}$) \[U(H^{\perp},h_{|H^{\perp}})({\Bbb R}} \def\fH{{\Bbb H})/SU(H^{\perp},h_{|H^{\perp}})({\Bbb R}} \def\fH{{\Bbb H})\cong M_b({\Bbb R}} \def\fH{{\Bbb H})\cong U(1).\] Here we have used that $U(H^{\perp},h_{|H^{\perp}})\subset} \def\nni{\supset} \def\und{\underline SU(V,h)$, as it is for the group $SU(V,h)$ (and not for $U(V,h)$) that $M_b({\Bbb R}} \def\fH{{\Bbb H})\cong U(1)$ (see \cite{S}, p.~115). This verifies the Lemma for the groups of type $\bf I$. \hfill $\Box$ \vskip0.25cm Now note that $L({\Bbb R}} \def\fH{{\Bbb H})\cong(M_b\cdot L_b)({\Bbb R}} \def\fH{{\Bbb H})=M_b({\Bbb R}} \def\fH{{\Bbb H})L_b({\Bbb R}} \def\fH{{\Bbb H})$, so for the domain defined by $L$ we have ${\cal D} _L=M_b({\Bbb R}} \def\fH{{\Bbb H})L_b({\Bbb R}} \def\fH{{\Bbb H})/M_b({\Bbb R}} \def\fH{{\Bbb H})K_b=L_b({\Bbb R}} \def\fH{{\Bbb H})/K_b\cong F_b$, hence ${\cal D} _N\cong {\cal D} _{N_b}$ as in Table \ref{T1}. Consider also ${\cal Z} _G(L_b)$ and ${\cal Z} _G(M_bL_b)$; both are defined over ${\Bbb R}} \def\fH{{\Bbb H}$, and clearly ${\cal Z} _{G({\Bbb R}} \def\fH{{\Bbb H})}(L_b({\Bbb R}} \def\fH{{\Bbb H}))/M_b({\Bbb R}} \def\fH{{\Bbb H})\cong {\cal Z} _{G({\Bbb R}} \def\fH{{\Bbb H})}(M_b({\Bbb R}} \def\fH{{\Bbb H})L_b({\Bbb R}} \def\fH{{\Bbb H}))$, so the group $L\times {\cal Z} _G(L)$ (both these factors being defined over $k$) is, over ${\Bbb R}} \def\fH{{\Bbb H}$, \begin{eqnarray}\label{e19.1} L({\Bbb R}} \def\fH{{\Bbb H})\times {\cal Z} _G(L)({\Bbb R}} \def\fH{{\Bbb H}) & \cong & M_b({\Bbb R}} \def\fH{{\Bbb H})\cdot L_b({\Bbb R}} \def\fH{{\Bbb H})\times {\cal Z} _{G({\Bbb R}} \def\fH{{\Bbb H})}(M_b({\Bbb R}} \def\fH{{\Bbb H})L_b({\Bbb R}} \def\fH{{\Bbb H})) \\ & \cong & M_b({\Bbb R}} \def\fH{{\Bbb H})\cdot L_b({\Bbb R}} \def\fH{{\Bbb H})\times {\cal Z} _{G({\Bbb R}} \def\fH{{\Bbb H})}(L_b({\Bbb R}} \def\fH{{\Bbb H}))/M_b({\Bbb R}} \def\fH{{\Bbb H}) \nonumber \\ & \cong & L_b({\Bbb R}} \def\fH{{\Bbb H})\times {\cal Z} _{G({\Bbb R}} \def\fH{{\Bbb H})}(L_b({\Bbb R}} \def\fH{{\Bbb H})).\nonumber \end{eqnarray} This completes the proof of \begin{proposition}\label{p19.1} The subgroup $N$ of (\ref{e18.1}) satisfies $N({\Bbb R}} \def\fH{{\Bbb H})\cong N_b({\Bbb R}} \def\fH{{\Bbb H})$, the latter group being the standard symmetric subgroup (\ref{e10.1}) standard incident to $P_b({\Bbb R}} \def\fH{{\Bbb H})$. \end{proposition} \begin{corollary}\label{c19.1} The parabolic $P_b$ and the symmetric subgroup $N$ of (\ref{e18.1}) are incident over $k$, i.e., $(P_b({\Bbb R}} \def\fH{{\Bbb H}),N({\Bbb R}} \def\fH{{\Bbb H}))$ are incident in the sense of Definition \ref{d9.1}. \end{corollary} Up to this point we have been working with the absolutely simple $k$-group; we now denote this situation by $G'$ as in section 3.1, and consider $G=Res_{k|\fQ}G'$. Let again primes in the notations denote subgroups of $G'$, the unprimed notations for subgroups of $G$. As above we set $P_{\hbox{\scsi \bf b}}:= Res_{k|\fQ}(P_b')$, and we denote the subgroup $N'$ of (\ref{e18.1}) henceforth by $N_b'$ and set: $N_{\hbox{\scsi \bf b}}:=Res_{k|\fQ}(N_b')$. Then Corollary \ref{c19.1} tells us that $(P_b'({\Bbb R}} \def\fH{{\Bbb H}),N'({\Bbb R}} \def\fH{{\Bbb H}))$ are incident. We now claim \begin{lemma}\label{l19.1} The $\fQ$-groups $(P_{\hbox{\scsi \bf b}},N_{\hbox{\scsi \bf b}})$ are incident. \end{lemma} {\bf Proof:} $P_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ is a product $P_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})\cong P_{b,1}({\Bbb R}} \def\fH{{\Bbb H})\times \cdots \times P_{b,d}({\Bbb R}} \def\fH{{\Bbb H})$ corresponding to (\ref{e9.1}); by assumption $F_{\hbox{\scsi \bf b}}$ is not zero-dimensional. Hence for at least one factor $P_{b,\gs}({\Bbb R}} \def\fH{{\Bbb H})$ the incident group ${^{\gs}N}'_b({\Bbb R}} \def\fH{{\Bbb H})\cong{^{\gs}L}'_b({\Bbb R}} \def\fH{{\Bbb H})\times {{\cal Z} }_{{^{\gs}G}'({\Bbb R}} \def\fH{{\Bbb H})}({^{\gs}L}'_b({\Bbb R}} \def\fH{{\Bbb H}))$ is defined. Consequently $N_{\hbox{\scsi \bf b}}$ is not trivial, and it is clearly a $\fQ$-form of $N_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})=\prod_{\gs}{^{\gs}N}'_b({\Bbb R}} \def\fH{{\Bbb H})$. \hfill $\Box$ \vskip0.25cm With Corollary \ref{c19.1} and Lemma \ref{l19.1}, we have just completed the proof of the following. \begin{proposition}\label{p19.2} To each standard maximal $\fQ$-parabolic $P_{\hbox{\scsi \bf b}}$ of $G$ with $b<s$, there is a symmetric $\fQ$-subgroup $N_{\hbox{\scsi \bf b}}\subset} \def\nni{\supset} \def\und{\underline G$ such that $(P_{\hbox{\scsi \bf b}},N_{\hbox{\scsi \bf b}})$ are incident. \end{proposition} Finally, we remark on what happens for the parabolic corresponding to the zero-dimensional boundary components. We have, in the notations above, $H=H_s\oplus H_s'$, and $H^{\perp}$ is anisotropic for $h$. It follows that the group $L$ of Lemma \ref{l18.1} is anisotropic; its semisimple part is the semisimple anisotropic kernel of $G'$. If $s={1\over 2}\dim V$, then $H=V$ already, $H^{\perp}=\{0\}$. Hence the group $N$ of (\ref{e18.2}) is the whole group ($L=1 \Ra {\cal Z} _G(L)=G$). Otherwise it is of the form $\{\hbox{anisotropic}\}\times \{\hbox{$k$-split}\}$. We list these in Table \ref{T3}. Note that the domains occuring have ${\Bbb R}} \def\fH{{\Bbb H}$-rank equal to the $\fQ$-rank of $G$, suggesting this as a possible modification of the definition of incident: \begin{itemize}\item[1')] $N$ has ${\Bbb R}} \def\fH{{\Bbb H}$-rank equal to the $\fQ$-rank of $G$. \end{itemize} Viewing things this way, we see that again indices $C^{(1)}$ represent an exception; for these 1) and 1') are equivalent. \begin{table}\caption{\label{T3} $k$-subgroups incident with zero-dimensional boundary components} $$\begin{array}{|c|c|c|c|c|} \hline \hbox{Index} & L & {\cal Z} _G(L) & \hbox{subdomains} & {\cal Z} _G(L)({\Bbb R}} \def\fH{{\Bbb H}) \\ \hline \hline {^2A}^{(d)}_{n,s} & {^2A}^{(d)}_{n-2ds,0} & {^2A}^{(d)}_{2ds-1,s} & \bf I_{\hbf{p-ds,q-ds}}\times I_{\hbf{ds,ds}} & SU(ds,ds) \\ \hline {^1D}^{(2)}_{n,s} & {^1D}^{(2)}_{n-2s,0} & {^1D}^{(2)}_{2s,s} & \bf II_{\hbf{n-s}}\times II_{\hbf{s}} & SU(2s,\fH)\ (\hbox{$n$ even}) \\ \hline {^2D}^{(2)}_{n,s} & {^2D}^{(2)}_{n-2s,0} & {^2D}^{(2)}_{2s,s} & \bf II_{\hbf{n-s}}\times II_{\hbf{ s}} & SU(2s,\fH)\ (\hbox{$n$ odd}) \\ \hline C^{(1)} & - & G & - & - \\ \hline C^{(2)}_{n,s} & C^{(2)}_{n-s,0} & C^{(2)}_{s,s} & \bf III_{\hbf{n-s}}\times III_{\hbox{\scriptsize\bf s}} & Sp(2s,{\Bbb R}} \def\fH{{\Bbb H}) \\ \hline \end{array}$$ \end{table} Let us now see which of the subgroups listed in Table \ref{T2} are defined over $k$. We use the notations $D,\ V, h$ and $G$ as described above in the cases (I)-(III). \begin{itemize}\item[(I)] Again $d$ denotes the degree of $D$. In $U(V,h)$ we have the subgroup $U(V',h_{|V'})$ for any codimension one subspace $V'\subset} \def\nni{\supset} \def\und{\underline V$. Let $W=(V')^{\perp}$ be the one-dimensional (over $D$) subspace orthogonal to $V'$. Then $U(W,h_{|W})$ is again a unitary group whose set of ${\Bbb R}} \def\fH{{\Bbb H}$-points is isomorphic to $U(p_W,q_W)$ for some $p_W,q_W$. Actually each $h_{\nu}$ for each infinite prime $\nu$ gives an ${\Bbb R}} \def\fH{{\Bbb H}$-group $U(p_{\scriptscriptstyle W,\nu},q_{\scriptscriptstyle W,\nu})$. Let $(p_{\nu},q_{\nu})$ be the signature of $h_{\nu}$ on $V_{\nu}$. Then $U(V'_{\nu},{h_{\nu}}_{|V'_{\nu}})\cong U(p_{\nu}-p_{\scriptscriptstyle W,\nu},q_{\nu}-q_{\scriptscriptstyle W,\nu})$. This gives rise to a product $N=\prod U(p_{\scriptscriptstyle W,\nu},q_{\scriptscriptstyle W,\nu})\times U(p_{\nu}-p_{\scriptscriptstyle W,\nu},q_{\nu}-q_{\scriptscriptstyle W,\nu})$, and the factors of the domain ${\cal D} _N$ are of type $\bf I_{\scriptstyle\bf p_{\hbox{$\scriptscriptstyle W$},\nu},q_ {\hbox{$\scriptscriptstyle W$},\nu}}\times I_{\scriptstyle\bf p_{\nu}-p_{\hbox{$\scriptscriptstyle W$},\nu},q_{\nu}-q_{\hbox{$\scriptscriptstyle W$},\nu}}$. In particular, for $p_{\scriptscriptstyle W,\nu}=0$, this is an irreducible group of type $\bf I_{\scriptstyle\bf p_{\nu},q_{\nu}-q_{\hbox{$\scriptscriptstyle W$},\nu}}$ and for $q_{\scriptscriptstyle W,\nu}=0$, of type ${\bf I}_{\scriptstyle\bf p_{\nu}-p_{\hbox{$\scriptscriptstyle W$},\nu},q_{\nu}}$. Now since $k$ is the degree of $D$, all of $p_{\nu}, q_{\nu}, p_{\hbox{$\scriptscriptstyle W$},\nu}, q_{\hbox{$\scriptscriptstyle W$},\nu}$ are divisible by $d$ and the net subdomains these subgroups (possibly) define are \begin{equation}\label{eZZ} \bf I_{\hbf{p-jd,q}},\quad I_{\hbf{p,q-jd}},\quad I_{\hbf{p-id,q-jd}} \subset} \def\nni{\supset} \def\und{\underline I_{\hbf{p,q}}, \ \ i,j=1,\ldots, s. \end{equation} \item[(II)] In $U(V,h)$ we have as above $U(V',h_{|V'})$; now if $h$ is non-degenerate on $V'$, then $U(V',h_{|V'})\cong U(n-1,D)$, giving subgroups of the real groups, defined over $k$, of type $U(n-1,\fH)\subset} \def\nni{\supset} \def\und{\underline U(n,\fH)$, with a corresponding subdomain of type $\bf II_{\scriptstyle\bf n-1}\subset} \def\nni{\supset} \def\und{\underline II_{\scriptstyle\bf n}$. This occurs at the primes for which $D$ is definite; at the others $SU(V',h_{|V'})\subset} \def\nni{\supset} \def\und{\underline SU(V,h)$ is of the type $SO(2n-4,2)\subset} \def\nni{\supset} \def\und{\underline SO(2n-2,2)$ (for $n$=dimension of $V$ over $D$). So we have maximal $k$-domains $$\bf II_{\hbf{n-1}}\subset} \def\nni{\supset} \def\und{\underline II_{\hbf{n}},\ (\nu \hbox{ definite}),\quad \quad IV_{\hbf{2n-4}}\subset} \def\nni{\supset} \def\und{\underline IV_{\hbf{2n-2}},\ (\nu \hbox{ indefinite}).$$ \item[(III)] The index is $C^{(2)}_{n,s}$; this case in considered in more detail below. \end{itemize} {}From this, we deduce \begin{proposition}\label{p23} Let $G'$ have $\rank_kG'=s\geq2$, not split over ${\Bbb R}} \def\fH{{\Bbb H}$, and let $P_s'$ be a standard $k$-parabolic defining a zero-dimensional boundary component, $P_s'({\Bbb R}} \def\fH{{\Bbb H})=N(F)$, and $\dim(F)=0$. Then there is a $k$-subgroup $N'$ incident with $P_s'$, with the following exception: Index $C^{(2)}_{2s,s}$. \end{proposition} {\bf Proof:} We first deduce for which of the indices listed in Proposition \ref{p16.1} zero-dimensional boundary components of ${\cal D} '$ are rational (this is necessary for the zero-dimensional boundary components of ${\cal D} $ to be rational). We need not consider exceptional cases or type $\bf IV_{\hbox{\scsi \bf n}}$. We first consider the groups of type ${^2A}$. \begin{Lemma}\label{L19A} For $G'$ with the index ${^2A}^{(d)}_{n,s}$, let $G'({\Bbb R}} \def\fH{{\Bbb H})\cong SU(p,q)$. Then the zero-dimensional boundary components are rational $\iff$ $sd=q$. \end{Lemma} {\bf Proof:} Let $H_s$ be an $s$-dimensional (over $D$) totally isotropic subspace, with basis $h_1,\ldots,h_s$. Let $h_i'\in V$ be vectors such that $h(h_i,h_j')=\gd_{ij}$, and set $H_s'=<h_1',\ldots,h_s'>$. Then $h$, restricted to $H:=H_s\oplus H_s'$, is non-degenerate, and $SU(H^{\perp},h_{|H^{\perp}})$ is the anisotropic kernel. The group $SU(H,h_{|H})({\Bbb R}} \def\fH{{\Bbb H}) \cong SU(sd,sd)$, while $SU(H^{\perp},h_{|H^{\perp}})({\Bbb R}} \def\fH{{\Bbb H})\cong SU(p-ds,q-ds)$. This defines the subdomain of type $\bf I_{\hbf{ds,ds}}\times I_{\hbf{p-ds,q-ds}}$ of Table \ref{T1}, hence the boundary component, which is the second factor, is zero-dimensional $\iff$ $q=ds$. \hfill $\Box$ \vskip0.25cm As to indices of type $D$ we observe the following. \begin{Lemma}\label{l5.8.1} $\dim(F)=0$ does not occur for the indices of type (II) in Proposition \ref{p16.1}. \end{Lemma} {\bf Proof:} Recall that $D$ is a quaternion division algebra, central simple over $k$, with the canonical involution, $V$ is an $n$-dimensional right $D$-vector space, and $h:V\times V\longrightarrow} \def\sura{\twoheadrightarrow D$ is a skew-hermitian form. Let $\nu_1,\ldots, \nu_a$ denote the infinite primes for which $D_{\nu}$ is definite, $\nu_{a+1},\ldots, \nu_d$ the primes at which $D_{\nu}$ is split. Then $G({\Bbb R}} \def\fH{{\Bbb H})$ is a product $$(SU(n,\fH))^a\times (SO(2n-2,2))^{d-a},$$ where we have taken into account that $G$ is assumed to be of hermitian type. At each of the first factors we have the Satake diagram \setlength{\unitlength}{0.005500in}% $$\begin{picture}(1202,145)(69,691) \thicklines \put(240,760){\circle*{22}} \put(160,760){\circle{22}} \put(315,760){\circle*{10}} \put(355,760){\circle*{10}} \put(400,760){\circle*{10}} \put(505,760){\circle*{22}} \put(579,701){\circle{22}} \put( 90,760){\line( 1, 0){ 60}} \put(170,760){\line( 1, 0){ 60}} \put(250,760){\line( 1, 0){ 45}} \put(425,760){\line( 1, 0){ 70}} \put(510,770){\line( 4, 3){ 60}} \put(511,752){\line( 4,-3){ 60}} \put(580,805){\vector( 0,-1){ 85}} \put(580,805){\vector( 0,1){ 0}} \put( 80,760){\circle*{22}} \put(1191,757){\line( 4,-3){ 60}} \put(580,820){\circle{22}} \put(1260,825){\circle{22}} \put(760,765){\circle*{22}} \put(920,765){\circle*{22}} \put(840,765){\circle{22}} \put(995,765){\circle*{10}} \put(1035,765){\circle*{10}} \put(1080,765){\circle*{10}} \put(1185,765){\circle{22}} \put(1259,706){\circle*{22}} \put(770,765){\line( 1, 0){ 60}} \put(850,765){\line( 1, 0){ 60}} \put(930,765){\line( 1, 0){ 45}} \put(1105,765){\line( 1, 0){ 70}} \put(1190,775){\line( 4, 3){ 60}} \end{picture}$$ \hspace*{3cm} for $n$ odd, \hspace*{6.5cm} for $n$ even. The corresponding ${\Bbb R}} \def\fH{{\Bbb H}$-root systems are then: \setlength{\unitlength}{0.005500in}% \begin{picture}(1108,70)(16,700) \thicklines \put(150,735){\circle{28}} \put(370,735){\circle{28}} \put(510,735){\circle{28}} \put(525,735){$\eta_t$} \put(235,735){\circle*{10}} \put(265,735){\circle*{10}} \put(290,735){\circle*{10}} \put( 45,735){\line( 1, 0){ 90}} \put(165,735){\line( 1, 0){ 50}} \put(310,735){\line( 1, 0){ 45}} \put(470,770){\line( 5,-6){ 25}} \put(495,730){\line(-5,-6){ 25}} \put(375,720){\line( 1, 0){110}} \put(375,750){\line( 1, 0){110}} \put( 30,735){\circle{28}} \put(630,735){\circle{28}} \put(1000,750){\line( 1, 0){110}} \put(750,735){\circle{28}} \put(970,735){\circle{28}} \put(1110,735){\circle{28}} \put(1125,735){$\eta_t$} \put(835,735){\circle*{10}} \put(865,735){\circle*{10}} \put(890,735){\circle*{10}} \put(645,735){\line( 1, 0){ 90}} \put(765,735){\line( 1, 0){ 50}} \put(910,735){\line( 1, 0){ 45}} \put(1000,720){\line( 1, 0){110}} \put(985,735){\line( 1,-1){ 35}} \put(985,735){\line( 1, 1){ 35}} \end{picture} In particular, the ${\Bbb R}} \def\fH{{\Bbb H}$-root corresponding to the parabolic $P_{t}$ with $\dim(F_{t})=0$ is the right-most one. On the other hand, the $k$-index is \begin{equation}\label{eZ1.1} \setlength{\unitlength}{0.006500in}% \begin{picture}(987,141)(14,670) \thicklines \put(25,680){$\underbrace{\hspace*{8.5cm}}_{\hbox{$2s$}}$} \put(435,740){\circle*{10}} \put(480,740){\circle*{10}} \put( 25,740){\circle*{22}} \put(185,740){\circle*{22}} \put(105,740){\circle{22}} \put(285,740){\circle{22}} \put(540,740){\circle{22}} \put(625,740){\circle*{22}} \put(740,740){\circle*{10}} \put(760,740){\circle*{10}} \put(720,740){\circle*{10}} \put(825,740){\circle*{22}} \put(395,740){\circle*{10}} \put(915,740){\circle*{22}} \put(921,732){\line( 4,-3){ 60}} \put(990,680){\circle*{20}} \put(990,800){\circle*{22}} \put( 35,740){\line( 1, 0){ 60}} \put(115,740){\line( 1, 0){ 60}} \put(195,740){\line( 1, 0){ 80}} \put(295,740){\line( 1, 0){ 75}} \put(500,740){\line( 1, 0){ 30}} \put(550,740){\line( 1, 0){ 65}} \put(635,740){\line( 1, 0){ 70}} \put(705,740){\line(-1, 0){ 5}} \put(785,740){\line( 1, 0){ 40}} \put(835,740){\line( 1, 0){ 70}} \put(920,750){\line( 4, 3){ 60}} \end{picture} \end{equation} with the $k$-root system \setlength{\unitlength}{0.00500in}% $$\begin{picture}(513,170)(51,610) \thicklines \put(185,745){\circle{28}} \put(405,745){\circle{28}} \put(545,745){\circle{28}} \put(540,780){$\eta_s$} \put(270,745){\circle*{10}} \put(300,745){\circle*{10}} \put(325,745){\circle*{10}} \put( 80,745){\line( 1, 0){ 90}} \put(200,745){\line( 1, 0){ 50}} \put(345,745){\line( 1, 0){ 45}} \put( 65,745){\circle{28}} \put( 60,780){$\eta_1$} \put(435,730){\line( 1, 0){110}} \put(415,660){\line( 1, 0){110}} \put(420,745){\line( 1,-1){ 35}} \put(420,745){\line( 1, 1){ 35}} \put(435,760){\line( 1, 0){110}} \put(200,645){(respectively)} \put(410,645){\circle{28}} \put(550,645){\circle{28}} \put(510,680){\line( 5,-6){ 25}} \put(535,640){\line(-5,-6){ 25}} \put(415,630){\line( 1, 0){110}} \end{picture}$$ from which it is evident that $P_{t}$ is defined over $k$ $\iff$ $s=t$ ($=[{n\over 2}]$). But this is the split over ${\Bbb R}} \def\fH{{\Bbb H}$ case. Consequently, $a=0$ and $D$ is totally indefinite. So we consider a prime $\nu$ where $D_{\nu}$ is split; the ${\Bbb R}} \def\fH{{\Bbb H}$-index is \setlength{\unitlength}{0.006500in}% $$\begin{picture}(522,171)(69,660) \thicklines \put(315,760){\circle*{10}} \put(355,760){\circle*{10}} \put(400,760){\circle*{10}} \put(505,760){\circle*{22}} \put(580,820){\circle*{22}} \put(579,701){\circle*{20}} \put( 80,760){\circle{22}} \put(160,760){\circle{22}} \put( 80,680){\circle{22}} \put(240,760){\circle*{22}} \put(160,680){\circle{22}} \put(120,655){\line( 4, 3){ 30}} \put(80,650){$\eta_2$} \put( 90,760){\line( 1, 0){ 60}} \put(170,760){\line( 1, 0){ 60}} \put(250,760){\line( 1, 0){ 45}} \put(425,760){\line( 1, 0){ 70}} \put(510,770){\line( 4, 3){ 60}} \put(511,752){\line( 4,-3){ 60}} \put( 85,690){\line( 1, 0){ 55}} \put( 85,670){\line( 1, 0){ 55}} \put(120,705){\line( 4,-3){ 30}} \put(160,650){$\eta_1$} \end{picture} $$ the ${\Bbb R}} \def\fH{{\Bbb H}$-root $\eta_2$ corresponding to the two-dimensional totally isotropic subspace and zero-dimensional boundary component. The $k$-index is as in (\ref{eZ1.1}), so $\eta_2$ is always anisotropic; the boundary components are actually one-dimensional. This verifies the statements of the lemma. \hfill $\Box$ \vskip0.25cm Note that this proves Proposition \ref{p23} for the indices of type (II). Now consider index $C^{(2)}_{n,s}$. The $k$-index is \setlength{\unitlength}{0.005500in}% $$\begin{picture}(962,22)(14,729) \thicklines \put(435,740){\circle*{10}} \put(480,740){\circle*{10}} \put( 25,740){\circle*{22}} \put( 25,720){$\underbrace{\hspace*{7.5cm}}_{\hbox{$2s$}}$} \put(185,740){\circle*{22}} \put(105,740){\circle{22}} \put(285,740){\circle{22}} \put(565,740){\circle{22}} \put(655,740){\circle*{22}} \put(750,740){\circle*{10}} \put(780,740){\circle*{10}} \put(810,740){\circle*{10}} \put(395,740){\circle*{10}} \put(900,740){\circle*{22}} \put(901,750){\line( 1, 0){ 60}} \put(965,740){\circle*{22}} \put( 35,740){\line( 1, 0){ 60}} \put(115,740){\line( 1, 0){ 60}} \put(195,740){\line( 1, 0){ 80}} \put(295,740){\line( 1, 0){ 75}} \put(500,740){\line( 1, 0){ 55}} \put(575,740){\line( 1, 0){ 70}} \put(665,740){\line( 1, 0){ 70}} \put(735,740){\line(-1, 0){ 5}} \put(830,740){\line( 1, 0){ 60}} \put(905,730){\line( 1, 0){ 55}} \end{picture}$$ and the $k$-root system is \setlength{\unitlength}{0.005500in}% $$\begin{picture}(508,70)(51,710) \thicklines \put(185,745){\circle{28}} \put(405,745){\circle{28}} \put(545,745){\circle{28}} \put(565,745){$\eta_s$} \put(270,745){\circle*{10}} \put(300,745){\circle*{10}} \put(325,745){\circle*{10}} \put( 65,745){\circle{28}} \put( 80,745){\line( 1, 0){ 90}} \put(435,760){\line( 1, 0){110}} \put(200,745){\line( 1, 0){ 50}} \put(345,745){\line( 1, 0){ 45}} \put(435,730){\line( 1, 0){110}} \put(420,745){\line( 1,-1){ 35}} \put(420,745){\line( 1, 1){ 35}} \end{picture}$$ The same reasoning as above shows that $F_{t}$ is rational $\iff$ $2s=t$, but that is only possible if the index is $C^{(2)}_{2n,n}$. Hence: \begin{Lemma}\label{l5.8.2} The only indices of Proposition \ref{p16.1}, case (III), for which zero-dimensional boundary components occur are $C^{(2)}_{2n,n}$. \end{Lemma} This index is that of the unitary group $U(V,h)$, where $V$ is a $2n$-dimensional vector space over $D$, and $h$ has Witt index $n$. We can find $n$ hyperbolic planes $V_i$ such that \[V=V_1\oplus\cdots \oplus V_n.\] This decomposition is defined over $k$, hence the subgroup \[N=U(V_1,h_{|V_1})\times \cdots \times U(V_n,h_{|V_n}),\] which is a product of groups with index $C^{(2)}_{2,1}$, is also defined over $k$. We have \begin{equation}\label{E20}N({\Bbb R}} \def\fH{{\Bbb H})\cong \underbrace{Sp(4,{\Bbb R}} \def\fH{{\Bbb H})\times \cdots \times Sp(4,{\Bbb R}} \def\fH{{\Bbb H})}_{n\ \hbox{\scriptsize times}} \end{equation} and the domain ${\cal D} _N$ is of type $\bf (III_{\hbox{\scriptsize\bf 2}})^n$. This is the exception in the statement of the main theorem. \vspace*{.3cm} \noindent{\bf Proof of Proposition \ref{p23}:} We have already completed the proof for (II) and (III), and as we mentioned above, the exceptional cases and (IV) need not be considered. It remains to show the existence of groups of the stated types for indices ${^2A}$. We explained above how one can find $k$-subgroups $N$ such that ${\cal D} _N$ has irreducible components of types $\bf I_{\hbox{\scriptsize\bf p-jd,q}}$ (see (\ref{eZZ})). Here we take a maximal totally isotropic subspace $H_s$, and $H:=H_s\oplus H_s'$ as described there. Let $H^{\perp}$ denote the orthogonal complement, so that $SU(H^{\perp},h_{|H^{\perp}})$ is the anisotropic kernel. Then, if $G'({\Bbb R}} \def\fH{{\Bbb H})=SU(p,q)$, we have \[SU(H,h_{|H})({\Bbb R}} \def\fH{{\Bbb H})\cong SU(sd,sd),\quad SU(H^{\perp},h_{|H^{\perp}})\cong SU(p-sd,q-sd).\] Therefore we get a subdomain of type \[\bf I_{\hbf{sd,sd}}\times I_{\hbf{p-sd,q-sd}},\] which is irreducible $\iff$ $sd=q$; Then $N=\{g\in G\Big| g(H)\subseteq H\}$ is a $k$-subgroup with $N({\Bbb R}} \def\fH{{\Bbb H})\sim SU(q,q)\times\{\hbox{compact}\}$, and $N$ then fulfills 1), 2') and 3'). By Lemma \ref{L19A}, this holds precisely when the boundary component $F_s$ is a point. This completes the proof if $p>q$. It remains to consider the case where ${\cal D} '$ is of type $\bf I_{\hbf{q,q}}$. In this case, $q=d\cdot j$ for some $j$, and the hermitian form $h:V\times V\longrightarrow} \def\sura{\twoheadrightarrow D$ has Witt index $j$. The vector space $V$ is then $2j$-dimensional, and it is the orthogonal direct sum of hyperbolic planes, $V=V_1\oplus \cdots \oplus V_j$, $\dim_DV_i=2$. Consider the $k$-subgroup \[ N=\{g\in GL_D(V) \Big| g(V_i)\subset} \def\nni{\supset} \def\und{\underline V_i, i=1,\ldots,j\}.\] Clearly $N\cong N_1\times \cdots \times N_j$, and each $N_i$ is a subgroup of rank one with index ${^2A}^{(d)}_{2d-1,1}$. As was shown in \cite{hyp}, in each $N_i$ we have a $k$-subgroup $N_i'\subset} \def\nni{\supset} \def\und{\underline N_i$, with ${\cal D} _{N_i'}$ of type $({\bf I_{\hbf{1,1}}})^d$. Then \[ N':=N_1'\times \cdots \times N_j'\] is a $k$-subgroup with ${\cal D} _{N'}$ of type $(({\bf I_{\hbf{1,1}}})^d)^j= ({\bf I_{\hbf{1,1}}})^{d\cdot j}= ({\bf I_{\hbf{1,1}}})^q$, which is a maximal polydisc, i.e., satisfies 1), 2'') and 3''). This completes the proof of Proposition \ref{p23} in this case also. \hfill $\Box$ \vskip0.25cm \section{Rank one} We now come to the most interesting and challenging case. In this last paragraph $G'$ will denote an absolutely simple $k$-group, $G$ the corresponding $\fQ$-simple group, both assumed to have rank one. There is only one standard maximal parabolic $P_1'\subset} \def\nni{\supset} \def\und{\underline G'$ in this case, so we may delete the subscript $_1$ in the notations. Let $P\subset} \def\nni{\supset} \def\und{\underline G$ be the corresponding $\fQ$-parabolic, so $P({\Bbb R}} \def\fH{{\Bbb H})=P_1({\Bbb R}} \def\fH{{\Bbb H})\times \cdots \times P_d({\Bbb R}} \def\fH{{\Bbb H})$, where $P_{\nu}({\Bbb R}} \def\fH{{\Bbb H})\subset} \def\nni{\supset} \def\und{\underline {^{\gs_{\nu}}G}'({\Bbb R}} \def\fH{{\Bbb H})$ is a standard maximal parabolic, say $P_{\nu}({\Bbb R}} \def\fH{{\Bbb H})=N(F_{b_{\nu}}),\ F_{b_{\nu}}\subset} \def\nni{\supset} \def\und{\underline \overline{{\cal D} }_{\gs_{\nu}}$. As we observed above, the $F_{b_{\nu}}$ are all hermitian spaces whose automorphism group is an ${\Bbb R}} \def\fH{{\Bbb H}$-form of some fixed algebraic group. As we are now assuming the rank to be one, it follows from Lemma \ref{L12a} that $L$ (=$L_{\hbox{\scsi \bf b}}$ in the notations above) is anisotropic. One way that this may occur was explained there, namely that if one of the factors $F_{b_{\nu}}$ is a {\it point}, in which case the symmetric space of $L({\Bbb R}} \def\fH{{\Bbb H})$ has a compact factor. Another possibility is that all $F_{b_{\nu}}$ are positive-dimensional, in which case $L$ is a ``genuine'' anisotropic group. The type of $F_{b_{\nu}}$ can be determined from the $k$-index of $G'$ and the ${\Bbb R}} \def\fH{{\Bbb H}$-index of ${^{\gs}_{\nu}G}'$. For example, for $G'$ of type ${^2A}$, these indices are: \vspace*{.5cm} \setlength{\unitlength}{0.004500in}% \begin{picture}(600,560)(90,235) \thicklines \put(280,780){\circle*{10}} \put(315,780){\circle*{10}} \put(105,780){\circle*{30}} \put(475,780){\circle*{30}} \put(120,780){\line( 1, 0){100}} \put(340,780){\line( 1, 0){120}} \put(845,780){\circle*{10}} \put(885,780){\circle*{10}} \put(920,780){\circle*{10}} \put(600,780){\circle{28}} \put(710,780){\circle*{28}} \put(1035,780){\circle*{30}} \put(490,780){\line( 1, 0){ 95}} \put(615,780){\line( 1, 0){ 80}} \put(720,780){\line( 1, 0){100}} \put(940,780){\line( 1, 0){100}} \put(240,620){\circle*{10}} \put(280,620){\circle*{10}} \put(315,620){\circle*{10}} \put(105,620){\circle*{30}} \put(475,620){\circle*{30}} \put(120,620){\line( 1, 0){100}} \put(340,620){\line( 1, 0){120}} \put(845,620){\circle*{10}} \put(885,620){\circle*{10}} \put(920,620){\circle*{10}} \put(600,620){\circle{28}} \put(710,620){\circle*{28}} \put(1035,620){\circle*{30}} \put(490,620){\line( 1, 0){ 95}} \put(615,620){\line( 1, 0){ 80}} \put(720,620){\line( 1, 0){100}} \put(940,620){\line( 1, 0){100}} \put(1135,705){\circle*{28}} \put(1035,780){\line( 4,-3){100}} \put(1035,620){\line( 6, 5){ 90}} \put(105,580){$\underbrace{\hspace*{4.4cm}}_{ \hbox{$d-1$ vertices}}$} \put(600,500){The $k$-index of $G'$} \end{picture} \setlength{\unitlength}{0.004500in}% \begin{picture}(1400,60)(90,235) \thicklines \put(240,250){\circle*{10}} \put(280,250){\circle*{10}} \put(315,250){\circle*{10}} \put(840,410){\circle*{10}} \put(880,410){\circle*{10}} \put(915,410){\circle*{10}} \put(840,250){\circle*{10}} \put(880,250){\circle*{10}} \put(240,780){\circle*{10}} \put(915,250){\circle*{10}} \put(340,250){\line( 1, 0){120}} \put(1215,410){\circle*{10}} \put(1255,410){\circle*{10}} \put(1290,410){\circle*{10}} \put(1215,250){\circle*{10}} \put(1255,250){\circle*{10}} \put(1290,250){\circle*{10}} \put(1505,335){\circle*{28}} \put(1405,410){\circle*{30}} \put(1405,250){\circle*{30}} \put(1405,410){\line( 4,-3){100}} \put(1405,250){\line( 6, 5){ 90}} \put(1310,410){\line( 1, 0){100}} \put(1310,250){\line( 1, 0){100}} \put(995,410){\circle{28}} \put(1120,410){\circle*{30}} \put(930,410){\line( 1, 0){ 55}} \put(1010,410){\line( 1, 0){ 95}} \put(1135,410){\line( 1, 0){ 60}} \put(1000,250){\circle{28}} \put(1125,250){\circle*{30}} \put(935,250){\line( 1, 0){ 55}} \put(1015,250){\line( 1, 0){ 95}} \put(1140,250){\line( 1, 0){ 60}} \put(240,410){\circle*{10}} \put(280,410){\circle*{10}} \put(315,410){\circle*{10}} \put(600,250){\circle{28}} \put(600,410){\circle{28}} \put(105,410){\circle{30}} \put(475,410){\circle{30}} \put(710,410){\circle{28}} \put(105,250){\circle{30}} \put(475,250){\circle{30}} \put(710,250){\circle{28}} \put(490,410){\line( 1, 0){ 95}} \put(615,410){\line( 1, 0){ 80}} \put(720,410){\line( 1, 0){100}} \put(120,410){\line( 1, 0){100}} \put(340,410){\line( 1, 0){120}} \put(490,250){\line( 1, 0){ 95}} \put(615,250){\line( 1, 0){ 80}} \put(720,250){\line( 1, 0){100}} \put(120,250){\line( 1, 0){100}} \put(105,230){$\underbrace{\hspace*{10.5cm}}_{\hbox{$q_{\nu}$ vertices}}$} \put(600,130){The ${\Bbb R}} \def\fH{{\Bbb H}$-index of ${^{\gs_{\nu}}G}'({\Bbb R}} \def\fH{{\Bbb H})$} \end{picture} \vspace*{1.5cm} From this we see that the boundary component is of type $\bf I_{\hbf{p$_{\nu}$-d,q$_{\nu}$-d}}$. There are basically two quite different cases at hand; the first is that the boundary components are positive-dimensional, the second occurs when the boundary components reduce to points. The former can be easily handled with the same methods as above, by splitting off orthogonal complements. The real interest is in the latter case, and here a basic role is played by the {\it hyperbolic planes}, which have been dealt with in detail in \cite{hyp}. We will essentially reduce the rank one case (at least for the classical groups) to the case of hyperbolic planes, then we explain how the results of \cite{hyp} apply to the situation here. \subsection{Positive-dimensional boundary components} Let $G', P'$ be as above, and consider the hermitian Levi factor ${G'}^{(1)}=M'L'$, which is defined over $k$. Over ${\Bbb R}} \def\fH{{\Bbb H}$ the factors $M'({\Bbb R}} \def\fH{{\Bbb H})$ and $L'({\Bbb R}} \def\fH{{\Bbb H})$ are defined. In this section we consider the situation that the boundary component $F'$ of ${\cal D} '$ defined by $P'$ (i.e., $P'({\Bbb R}} \def\fH{{\Bbb H})=N(F')$) is positive-dimensional, or equivalently, that the hermitian Levi factor $L'({\Bbb R}} \def\fH{{\Bbb H})$ is non-trivial. As above, we get the following $k$-group \begin{equation}\label{E23} N':={G'}^{(1)}\times {\cal Z} _{G'}({G'}^{(1)}). \end{equation} The same calculation as in ({\ref{e19.1}) shows that the domain ${\cal D} _{N'}$ defined by $N'$ is the same as that defined by $L'({\Bbb R}} \def\fH{{\Bbb H})\times {\cal Z} _{G'({\Bbb R}} \def\fH{{\Bbb H})}(L'({\Bbb R}} \def\fH{{\Bbb H}))$. Taking the subgroup $N=Res_{k|\fQ}N'$ defines a subdomain ${\cal D} _N\subset} \def\nni{\supset} \def\und{\underline {\cal D} $, which is a product ${\cal D} _N={\cal D} _{N,\gs_1}\times \cdots \times {\cal D} _{N,\gs_f}$. Each factor ${\cal D} _{N,\gs}$ is determined by the corresponding factor of ${^{\gs}L}'({\Bbb R}} \def\fH{{\Bbb H})$. The ${\Bbb R}} \def\fH{{\Bbb H}$-groups $N'({\Bbb R}} \def\fH{{\Bbb H})$ and $N({\Bbb R}} \def\fH{{\Bbb H})$ are determined in terms of the data $D,V,h$ as follows. \begin{itemize}\item[(I)] If $F'\cong {\bf I_{\hbox{\scriptsize\bf p-d,q-d}}}$, then ${\cal D} _{N'}\cong {\bf I_{\hbox{\scriptsize\bf p-d,q-d}}}\times {\bf I_{\hbox{\scriptsize\bf d,d}}}$. Note that in terms of the hermitian forms, this amounts to the following. Since $h$ has Witt index 1, the maximal totally isotropic subspaces are one-dimensional. Let $H_1=<v>$ be such a space; we can find a vector $v'\in V$ such that $H=<v,v'>$ is a hyperbolic plane, that is, $h_{|H}$ has Witt index 1. It follows that $h_{|H^{\perp}}$ is anisotropic. Consider the subgroup \begin{equation}\label{e22.0} N_k:=\{g\in U(V,h) | g(H)\subset} \def\nni{\supset} \def\und{\underline H\}. \end{equation} It is clear that for $g\in N_k$, it automatically holds that $g(H^{\perp})\subset} \def\nni{\supset} \def\und{\underline H^{\perp}$, hence \begin{equation}\label{e22.1} N_k\cong U(H,h_{|H})\times U(H^{\perp},h_{|H^{\perp}}). \end{equation} The first factor has ${\Bbb R}} \def\fH{{\Bbb H}$-points $U(H,h_{|H})({\Bbb R}} \def\fH{{\Bbb H})\cong U(d,d)$, while the second fulfills $U(H^{\perp},h_{|H^{\perp}})({\Bbb R}} \def\fH{{\Bbb H})\cong U(p-d,q-d)$. Thus $N_k\cong N'$ as in (\ref{E23}). At any rate, this gives us subdomains of type $${\bf I_{\hbox{\scriptsize\bf d,d}}\times I_{\hbox{\scriptsize\bf p-d,q-d}}\subset} \def\nni{\supset} \def\und{\underline } {\cal D} _{N'},$$ which, in case $d=p=q$ is the whole domain; in all other cases it is a genuine subdomain as listed in Table \ref{T1}, defined over $k$, and $(N',P')$ are incident. It follows from this that $(N,P)$ are incident over $\fQ$. The components $N_{\gs}({\Bbb R}} \def\fH{{\Bbb H})$ of $N_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ are determined as follows. Let $(p_{\nu},q_{\nu})$ be the signature of $h_{\nu}$ (so that $p_{\nu}+q_{\nu}=dm$ for all $\nu$). This implies \[ G({\Bbb R}} \def\fH{{\Bbb H})\cong \prod_{\nu}SU(p_{\nu},q_{\nu}).\] For each factor, we have the boundary component $F_{\gs}\cong SU(p_{\nu}-d, q_{\nu}-d)/K$, and for each factor for which $q_{\nu}>d$ this is positive-dimensional. As above, this leads to subdomains, in each factor, of type $\bf I_{\hbf{d,d}}\times I_{\hbf{p$_{\nu}$-d,q$_{\nu}$-d}}$, so that in sum \begin{equation}\label{E22a} {\cal D} _N\cong \prod_{\nu}{\cal D} _{\nu},\quad \hbox{${\cal D} _{\nu}$ of type $\bf I_{\hbf{d,d}}\times I_{\hbf{p$_{\nu}$-d,q$_{\nu}$-d}}$}. \end{equation} \item[(II)] Here rank 1 means we have the following $k$-index, $D^{(2)}_{n,1}$ \setlength{\unitlength}{0.004500in}% $$\begin{picture}(628,148)(26,666) \thicklines \put(300,740){\circle*{28}} \put(520,740){\circle*{28}} \put(640,800){\circle*{28}} \put(640,680){\circle*{28}} \put(440,740){\circle*{6}} \put(385,740){\circle*{6}} \put(410,740){\circle*{6}} \put( 40,740){\circle*{28}} \put(160,740){\circle{28}} \put(525,735){\line( 2,-1){114}} \put( 45,740){\line( 1, 0){100}} \put(175,740){\line( 1, 0){115}} \put(310,740){\line( 1, 0){ 55}} \put(460,740){\makebox(0.4444,0.6667){\circle{1}}} \put(460,740){\line( 1, 0){ 55}} \put(520,740){\line( 2, 1){120}} \end{picture}$$ In particular, the boundary component is of type $\bf II_{\hbox{\scriptsize\bf n-2}}$ if ${\cal D} '$ is of type $\bf II_{\hbox{\scriptsize\bf n}}$. This means also that the ``mixed cases'' only can occur if ${\cal D} '$ is of type $\bf II_{\hbox{\scriptsize\bf 4}}$, for then $\bf II_{\hbox{\scriptsize\bf 2}}\cong$ one-dimensional disc. Of course $\bf II_{\hbox{\scriptsize\bf 4}}\cong IV_{\hbox{\scriptsize\bf 6}}$ anyway, so we can conclude from this that mixed cases do not occur in the hermitian symmetric setting (for $\fQ$-simple $G$ of rank 1). The domain ${\cal D} _{N'}$ defined by $N'$ is of type $\bf II_{\hbox{\scriptsize\bf n-2}}\times II_{\hbox{\scriptsize\bf 2}}$. The components $N_{\gs}({\Bbb R}} \def\fH{{\Bbb H})$ of $N_{\hbf{1}}({\Bbb R}} \def\fH{{\Bbb H})$ are all of type $U(n-2,\fH)\times U(2,\fH)\subset} \def\nni{\supset} \def\und{\underline U(n,\fH)$, so the domain ${\cal D} _N$ is ot type \begin{equation}\label{E22a.1} {\bf (II_{\hbf{n-2}}\times II_{\hbf{2}})}^f. \end{equation} \item[(III)] Here rank 1 implies the index is one of $C^{(1)}_{1,1}$ (which we have excluded) or $C^{(2)}_{n,1}$. The corresponding boundary components in these cases are of type $\bf III_{\hbf{n-2}}$. The case $C^{(2)}_{2,1}$, for which the boundary component is a point, will be dealt with later, the others give rise to a subdomain of type $\bf III_{\hbox{\scriptsize\bf 2}}\times III_{\hbox{\scriptsize\bf n-2}}$. Consequently, ${\cal D} _N$ is of type ${\bf (III_{\hbf{n-2}}\times III_{\hbf{2}})}^f$, $f=[k:\fQ]$. \item[(IV)] Here we just have a symmetric bilinear form of Witt index 1. The $k$-index in this case is necessarily of the form \setlength{\unitlength}{0.004500in}% $$\begin{picture}(359,28)(86,766) \thicklines \put(240,780){\circle*{28}} \put(355,780){\circle*{10}} \put(100,780){\circle{28}} \put(400,780){\circle*{10}} \put(245,780){\line( 1, 0){ 85}} \put(440,780){\circle*{10}} \put(115,780){\line( 1, 0){115}} \end{picture}$$ The corresponding boundary component is a point, a case to be considered below. Splitting off an anisotropic vector (defined over $k$) in this case yields a codimension one subspace $H^{\perp}$ on which $h$ still has Witt index 1, hence the stabilizer $N'$ defines a subdomain ${\cal D} _{N'}$ of type $\bf IV_{\hbox{\scriptsize\bf n-1}}$. ${\cal D} _N$ is then of type ${(\bf IV_{\hbf{n-1}})}^f$. \item[(V)] The only index of rank 1 is $$ \unitlength1cm \begin{picture}(14,3) \put(1,2){$^2E_{6,1}^{28}$} \put(3,0){ \put(.1,2){\circle{0.2}} \multiput(.2,1.92)(.15,0){6}{-} \put(.1,2.3){$\gd$} \put(1.2,2){\circle{0.2}} \put(1.3,2){\line(1,0){0.9}} \put(1,2.3){$\ga_2$} \put(2.2,2.3){$\ga_4$} \put(2.3,2){\circle*{0.2}} \put(2.45,2){\line(1,1){1}} \put(2.45,2){\line(1,-1){1}} \put(2.8,3){$\ga_3$} \put(3.4,3){\circle*{0.2}} \put(3.4,2){\vector(0,1){.8}} \put(2.8,1){$\ga_5$} \put(3.4,1){\circle*{0.2}} \put(3.4,2){\vector(0,-1){.8}} \put(3.5,3){\line(1,0){1}} \put(3.5,1){\line(1,0){1}} \put(4.6,3){\circle*{0.2}} \put(4.6,2){\vector(0,1){.8}} \put(4.9,3){$\ga_1$} \put(4.6,1){\circle*{0.2}} \put(4.6,2){\vector(0,-1){.8}} \put(4.9,1){$\ga_6$} } \end{picture}$$ The vertex denoted $\ga_2$ gives rise to the five-dimensional boundary component. If $\gd$ denotes the lowest root, then, as is well known, $\gd$ is isotropic (does not map to zero in the $k$-root system), so the root $\gd$ defines a $k$-subalgebra $\frak n} \def\rr{\frak r_{\gd}:=\Gg^{\gd}+\Gg^{-\gd}+[\Gg^{\gd},\Gg^{-\gd}]\subset} \def\nni{\supset} \def\und{\underline \Gg'$ which is split over $k$. On the other hand the anisotropic kernel ${\cal K} $ is of type ${^2A}_5$, and ${\cal K} ({\Bbb R}} \def\fH{{\Bbb H})\cong U(5,1)$. Clearly ${\cal K} $ and the $k$-subgroup $N_{\gd}$ defined by $\frak n} \def\rr{\frak r_{\gd}$ are orthogonal, so we get a $k$-subgroup \[N'=N_{\gd}\times {\cal K} ,\] both factors being defined over $k$. The set of ${\Bbb R}} \def\fH{{\Bbb H}$-points is then of type $N'({\Bbb R}} \def\fH{{\Bbb H})\cong SL_2({\Bbb R}} \def\fH{{\Bbb H})\times SU(5,1)$, and the subdomain ${\cal D} _{N'}$ is \[{\cal D} _{N'}\cong {\bf I_{\hbox{\scriptsize\bf 1,1}}\times I_{\hbox{\scriptsize\bf 5,1}}}.\] This is one of the domains listed in Table \ref{T1}, incident to the five-dimensional boundary component. It follows that ${\cal D} _N$ is a product of factors of this type. \item[(VI)] There are no indices of hermitian type with rank one for $E_7$. \end{itemize} We sum up these results in the following. \begin{proposition}\label{p23.1} If the rational boundary components for $G'$ are positive-dimensional, then Theorem \ref{t12.1} holds for $G$. The subdomains defined by the symmetric subgroups $N'\subset} \def\nni{\supset} \def\und{\underline G'$ are: \begin{itemize}\item[(I)] $\bf I_{\hbox{\scriptsize\bf d,d}}\times I_{\hbox{\scriptsize\bf p-d,q-d}}$. \item[(II)] $\bf II_{\hbox{\scriptsize\bf n-2}}\times II_{\hbox{\scriptsize\bf 2}}$. \item[(III)] $\bf III_{\hbox{\scriptsize\bf n-2}}\times III_{\hbox{\scriptsize\bf 2}}$. \item[(IV)] $\bf IV_{\hbox{\scriptsize\bf n-1}}$ (here there are no positive-dimensional boundary components). \item[(V)] $\bf I_{\hbox{\scriptsize\bf 1,1}}\times I_{\hbox{\scriptsize\bf 5,1}}$. \end{itemize} Note here $\bf I_{\hbox{\scriptsize\bf 1,1}}\cong II_{\hbox{\scriptsize\bf 2}} \cong III_{\hbox{\scriptsize\bf 1}}\cong IV_{\hbox{\scriptsize\bf 1}}$. The corresponding domains ${\cal D} _N$ in ${\cal D} $ defined by the subgroups $N$ are products of domains of the types listed above. \end{proposition} \subsection{Zero-dimensional boundary components} The restrictions rank equal to one and zero-dimensional boundary componants are only possible for the domains of type $\bf I_{\hbox{\scriptsize\bf p,q}},\ III_{\hbox{\scriptsize\bf 2}}$ and $\bf IV_{\hbox{\scriptsize\bf n}}$ (see Lemmas \ref{l5.8.1} and \ref{l5.8.2}). Of these, the last case requires no further discussion: as above we find a codimension one $k$-subspace $V'\subset} \def\nni{\supset} \def\und{\underline V$, on which $h$ still is isotropic, and take its stabilizer as $N'$. This gives a $k$-subgroup $N'\subset} \def\nni{\supset} \def\und{\underline G'$, and defines a subdomain ${\cal D} _{N'}$ of type $\bf IV_{\hbox{\scriptsize\bf n-1}}$. In the ${^2A}^{(d)}$ case we may assume $d\geq 3$: the $d=1$ case is again easily dealt with as above. We have a $K$-vector space $V$ ($K|k$ imaginary quadratic) of dimension $p+q$ and a ($K$-valued) hermitian form $h$ of Witt index 1 on $V$. By taking a $K$-subspace $V'\subset} \def\nni{\supset} \def\und{\underline V$ of codimension one, such that $h_{|V'}$ still has Witt index 1, we get the $k$-subgroup $N'$ as the stabilizer of $V'$. Then the domain ${\cal D} _{N'}$ is either of type $\bf I_{\hbox{\scriptsize\bf p-1,q}}$ or $\bf I_{\hbox{\scriptsize\bf p,q-1}}$, and by judicious choice of $V'$ we can assume the first case, which is the domain listed in Table \ref{T2}. The $d=2$ case is ``lifted'' from the corresponding $d=2$ case with involution of the first kind: if $D$ is central simple of degree 2 over $K$ with a $K|k$-involution, then (\cite{A}, Thm.~10.21) $D=D'\otimes_kK$, where $D'$ is central simple of degree 2 over $k$ with the canonical involution. Consequently, $$U(V,h)=U(V'\otimes_kK,h'\otimes_kK)=U(V',h')_K,$$ the group is just the group $U(V',h')$ lifted to $K$. Since $U(V',h')$ has index $C^{(2)}_{n,1}$, while $U(V',h')_K$ has index $A^{(2)}_{2n-1,1}$, it follows that the boundary component is a point only if $n\leq 2$. This implies that if $d=2$, the index is $A^{(2)}_{3,1}$, the domain is $\bf I_{\hbf{2,2}}\cong IV_{\hbf{4}}$, so $U(V',h')_K$ is isomorphic to an orthogonal group over $k$ in six variables. As we just saw, in this case there is a subdomain defined over $k$ of type $\bf IV_{\hbf3}\subset} \def\nni{\supset} \def\und{\underline IV_{\hbf4}$. So we assume $d\geq 3$. Then, as we have seen, the boundary component $F'\cong \bf I_{\hbox{\scriptsize\bf p-d,q-d}}$ will be zero-dimensional $\iff$ $q=d$ (respectively $F\cong {\bf I_{\hbf{p$_1$-d,q$_1$-d}}\times \cdots \times I_{\hbf{p$_f$-d,q$_f$-d}}}$ will be zero-dimensional $\iff$ $q_{\nu}=d,\ \forall_{\nu}$. Here there are two possibilities: \begin{itemize}\item[1)] $p=q=d$, the group $N_k$ of (\ref{e22.0}) is $N_k\cong G'$. This is the case of {\it hyperbolic planes}. \item[2)] $p>q=d$, the group $N_k$ of (\ref{e22.0}) is over ${\Bbb R}} \def\fH{{\Bbb H}$ just $N_k({\Bbb R}} \def\fH{{\Bbb H}) = U(d,d)\times U(p-d)\subset} \def\nni{\supset} \def\und{\underline U(p,d)\cong G'({\Bbb R}} \def\fH{{\Bbb H})$. \end{itemize} Note that in the second case the domain ${\cal D} _{N_k}$ defined by $N_k$ is of type $\bf I_{\hbox{\scriptsize\bf d,d}}$, a maximal tube domain in $\bf I_{\hbox{\scriptsize\bf p,q}}$. So we are also finished in this case. For completeness, let us quickly go through the details to make sure nothing unexpected happens. \begin{proposition}\label{p24.1} Let $G'$ have index ${^2A}^{(d)}_{n,1},\ d=q,\ p>q$, $n+1=p+q$, and let $P'$ denote the corresponding standard parabolic and $N'= N_k$, where $N_k\subset} \def\nni{\supset} \def\und{\underline G'$ the symmetric subgroup defined in (\ref{e22.0}), where $H$ is the hyperbolic plane spanned by the vector which is stabilized by $P'$ and its ortho-complement ($v'$: $h(v,v')=1$). Then $(P',N')$ are incident, in fact standard incident. Consequently, $P=Res_{k|\fQ}P'$ and $N=Res_{k|\fQ}N'$ are incident over $\fQ$. \end{proposition} {\bf Proof:} We know that $N'({\Bbb R}} \def\fH{{\Bbb H})\cong U(q,q)\times U(p-q)$ which gives rise to the maximal tube subdomain $\bf I_{\hbox{\scriptsize\bf q,q}}\subset} \def\nni{\supset} \def\und{\underline I_{\hbox{\scriptsize\bf p,q}}$ of Table \ref{T2}. We need to check that the standard boundary component $F'$ stabilized by $P'({\Bbb R}} \def\fH{{\Bbb H})$ is also a standard boundary component of ${\cal D} _{N'}$; in particular we need the common maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus in $P'$ and $N'$. This is seen in (\ref{e22.0}), the ${\Bbb R}} \def\fH{{\Bbb H}$-split torus being contained in the hermitian Levi factor of $P'({\Bbb R}} \def\fH{{\Bbb H})$, which is contained in $N'({\Bbb R}} \def\fH{{\Bbb H})$. Consider the group $P'\cap N'$; this is nothing but the stabilizer of $v$ in $H$, which is a maximal standard parabolic in $N'$. Since $\nu$ determines the boundary component $F$, both in $G'({\Bbb R}} \def\fH{{\Bbb H})$ and in $N'({\Bbb R}} \def\fH{{\Bbb H})$, it is clear that $F$ is a boundary component of ${\cal D} _{N'}$. It follows that $(P',N')$ are incident, and this implies (see the discussion preceeding Proposition \ref{p19.2}) that $(P,N)\subset} \def\nni{\supset} \def\und{\underline G$ are incident. \hfill $\Box$ \vskip0.25cm We are left with the following cases: $\bf III_{\hbox{\scriptsize\bf 2}}$ with index $C^{(2)}_{2,1}$ and $\bf I_{\hbox{\scriptsize\bf q,q}}$ with index ${^2A}^{(d)}_{2d-1,1},\ d\geq3$. These indices are described in terms of hermitian forms as follows. Let $D$ be a central simple division algebra over $K$ ($K=k$ for $d=2$ and $K|k$ is imaginary quadratic if $d\geq3 $) and assume further that $D$ has a $K|k$-involution, $V$ is a two-dimensional right vector space over $D$ and $h:V\times V\longrightarrow} \def\sura{\twoheadrightarrow D$ is a hermitian form which is isotropic. Then $d=2$ gives groups with index $C^{(2)}_{2,1}$, and $d\geq 3$ gives groups with indices ${^2A}^{(d)}_{2d-1,1}$. \begin{lemma}\label{l25.1} There exists a basis $v_1,v_2$ of $V$ over $D$ such that the form $h$ is given by $h({\bf x},{\bf y})=x_1\overline{y}_2+x_2\overline{y}_1,\ {\bf x}=(x_1,x_2),\ {\bf y}=(y_1,y_2)$. \end{lemma} {\bf Proof:} Let $v$ be an isotropic vector, defined over $k$. Then there exists an isotropic vector $v'$, such that $h(v,v')=1$, hence also $h(v',v)=1$. Let ${v'}=({v'}_1,{v'}_2)$, and set $\gd={v'}_1\overline{v'}_2$, so that $h(v',v')=\gd+\overline{\gd}$. Then the matrix of $h$ with respect to the basis $v,v'$ is $H'={0\ 1\choose 1\ \ge}$, where $\ge=\gd+\overline{\gd}$. Now setting $$w=(w_1,w_2)=(-v_1\overline{\gd}+v_1',-v_2\overline{\gd}+v_2')$$ we can easily verify $h(w,w)=0,\ \ h(v,w)=h(w,v)=1$. Since the change of basis transformation is defined over $k$, the matrix of the hermitian form with respect to this $k$-basis $v,w$ is $H={0\ 1\choose 1\ 0}$. \hfill $\Box$ \vskip0.25cm So as far as the $\fQ$-groups are concerned, we may take the standard hyperbolic form given by the matrix $H$ as defining the hermitian form on $V$. We remark that the situation changes when one considers arithmetic groups, but that need not concern us here. At any rate, a two-dimensional right $D$-vector space $V$ with a hermitian form as in Lemma \ref{l25.1} is what we call a {\it hyperbolic plane}, and this case was studied in detail in \cite{hyp}. There it was determined exactly what kind of symmetric subgroups exist. These derive from the existence of splitting subfields $L\subset} \def\nni{\supset} \def\und{\underline D$, which may be taken to be cyclic of degree $d$ over $K$, if $D$ is central simple of degree $d$ over $K$. In fact, we have subgroups (\cite{hyp}, Proposition 2.4) $U(L^2,h)\subset} \def\nni{\supset} \def\und{\underline U(D^2,h)$, which give rise to the following subdomains: \begin{itemize} \item[1)] $d=2$; ${\cal D} _L\cong \left(\begin{array}{cc}\tau_1 & 0 \\ 0 & b^{\zeta_1}\tau_1 \end{array}\right)\times \cdots \times \left(\begin{array}{cc}\tau_1 & 0 \\ 0 & b^{\zeta_f}\tau_1 \end{array}\right)$, where $\zeta_i:k\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb R}$ denote the distinct real embeddings of $k$. \item[2)] $d\geq 3$; ${\cal D} _L\cong \left(\begin{array}{ccc}\tau_1 & & 0 \\ & \ddots & \\ 0 & & \tau_d\end{array}\right)^f$. \end{itemize} In other words, for hyperbolic planes we find subdomains of the following kinds \begin{equation}\label{E25} \bf III_{\hbox{\scriptsize\bf 1}}\subset} \def\nni{\supset} \def\und{\underline III_{\hbox{\scriptsize\bf 2}},\quad (I_{\hbox{\scriptsize\bf 1,1}})^d\subset} \def\nni{\supset} \def\und{\underline I_{\hbox{\scriptsize\bf d,d}}. \end{equation} The latter one is a polydisc, coming from a maximal set of strongly orthogonal roots, i.e., satisfying 1), 2'') and 3''). The first case is the only exception to the rule that we have symmetric subgroups $N'\subset} \def\nni{\supset} \def\und{\underline G'$ with $\rank_{{\Bbb R}} \def\fH{{\Bbb H}}N'=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G'$. \vspace*{.2cm} \noindent{\bf Proof of Theorem \ref{t12.1}:} We have split the set of cases up into the three considered in \S4, 5 and 6. Corollary \ref{c14.1} proves \ref{t12.1} for the split over ${\Bbb R}} \def\fH{{\Bbb H}$ case and Proposition \ref{p19.2} for the rank $\geq 2$ case and positive-dimensional boundary components. For $\rank \geq 2$ and zero-dimensional boundary components, Proposition \ref{p23} shows that with the exception given Theorem \ref{t12.1} holds in this case also. In the case of rank 1, Proposition \ref{p23.1} verifies \ref{t12.1} for the case that the boundary components are positive-dimensional, and Proposition \ref{p24.1} took care of the rest of the cases excepting hyperbolic planes. Then the results of \cite{hyp} verify \ref{t12.1} for hyperbolic planes, thus completing the proof. \hfill $\Box$ \vskip0.25cm \vspace*{.2cm} \noindent{\bf Proof of the Main Theorem:} The first statement is covered by Theorem \ref{t12.1}. The statements on the domains for the exceptions follow from (\ref{E20}) and (\ref{E25}). It remains to consider the condition 4). This is fulfilled for the groups $N$ utilized above by construction. For the exceptional cases this is immediate, as we took subgroups defined by symmetric closed sets of roots. Let us sketch this again for the classical cases, utilizing the description in terms of $\pm$symmetric/hermitian forms. The objects $D,\ V,\ h$ and $G'$ will have the meanings as above. Let $s=\rank_{k}G'$, and let $H_s$ be an $s$-dimensional (maximal) totally isotropic subspace in $V$, with basis $h_1,\ldots, h_s$. Let $h_i'\in V$ be vectors of $V$ with $h(h_i,h_j')=\gd_{ij}$, $H_s'=<h_1',\ldots, h_s'>$ and set $H=H_s\oplus H_s'$. Then $h_{|H}$ is non-degenerate of index $s$, and $H$ splits into a direct sum of hyperbolic planes, $H=V_1\oplus \cdots \oplus V_s$. The form $h$ restricted to $H^{\perp}$ is anisotropic; the semisimple anisotropic kernel is $SU(H^{\perp},h_{|H^{\perp}})$. Fixing the basis $h_1,\ldots, h_s,h_1',\ldots, h_s'$ for $H$ amounts to the choice of maximal $k$-split torus $S'$. For each real prime $\nu$, $(H_{\nu},h_{\nu})$ is a $2ds$-dimensional ${\Bbb R}} \def\fH{{\Bbb H}$-vecotr space with $\pm$symmetric/hermitian form. Choosing an ${\Bbb R}} \def\fH{{\Bbb H}$-basis of $H_{\nu}$ amounts to choosing a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus of $SU(H_{\nu},h_{\nu})$, and a choice of basis for a maximal set of hyperbolic planes (over ${\Bbb R}} \def\fH{{\Bbb H}$) amouts to the choice of maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus. Similarly, $(H_{\nu}^{\perp},{h_{\nu}}_{|H_{\nu}^{\perp}})$ is an ${\Bbb R}} \def\fH{{\Bbb H}$-vector space, $h_{|{H^{\perp}_{\nu}}}$ has some index $q_{\nu}$, and one can find a maximal set of hyperbolic planes $W_1,\ldots,W_r$, such that $H_{\nu}^{\perp}=(W_1)_{\nu}\oplus \cdots \oplus (W_r)_{\nu}\oplus W'$, where ${h_{\nu}}_{|W'}$ is anisotropic over ${\Bbb R}} \def\fH{{\Bbb H}$. A choice of basis of the $(W_i)_{\nu}$ amounts to the choice of maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus, and a choice of basis, over ${\Bbb R}} \def\fH{{\Bbb H}$, of $V_{\nu}$ amounts to the choice of maximal torus defined over ${\Bbb R}} \def\fH{{\Bbb H}$. From these descriptions we see that the polydisc group $N_{\Psi}$ defined by the maximal set of strongly orthogonal roots $\Psi$ splits into a component in $SU(H,h_{|H})$ and a component in $SU(H^{\perp},h_{|H^{\perp}})$, say $N_{\Psi}=N_{\Psi,1}\times N_{\Psi,2}$. Then $N_{\Psi,2}\subset} \def\nni{\supset} \def\und{\underline SU(H^{\perp},h_{|H^{\perp}})({\Bbb R}} \def\fH{{\Bbb H})$ and $N_{\Psi,1}\subset} \def\nni{\supset} \def\und{\underline SU(H,h_{|H})$. Since the subgroup $SU(H^{\perp},h_{|H^{\perp}})$ is contained in all the groups $N$ we have defined, we need only consider $N_{\Psi,1}$. $H$ is a direct sum of hyperbolic planes $V_i$, and the question is whether the corresponding polydisc group is contained in $SU(V_i,h_{|V_i})$. But this is what was studied in \cite{hyp}; the answer is affirmative. It follows that with the one exception stated, $C^{(2)}_{2,1}$, $N_{\Psi}\subset} \def\nni{\supset} \def\und{\underline N$. \hfill $\Box$ \vskip0.25cm

9505

alg-geom/9505001

en

https://arxiv.org/abs/alg-geom/9505001

alg-geom/9505001

Frank Sottile

Frank Sottile

Pieri's rule for flag manifolds and Schubert polynomials

21 pages with 1 figure. AMSLaTeX v 1.1

Ann. de l'Inst. Four., 46 (1996) 89-110

We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary symmetric polynomial or a complete homogeneous symmetric polynomial. Thus, we generalize the classical Pieri's rule for symmetric polynomials/Grassmann varieties to Schubert polynomials/flag manifolds. Our primary technique is an explicit geometric description of certain intersections of Schubert varieties. This method allows us to compute additional structure constants for the cohomology ring, which we express in terms of paths in the Bruhat order on the symmetric group.

alg-geom

\section{Introduction} Schubert polynomials had their origins in the study of the cohomology of flag manifolds by Bernstein-Gelfand-Gelfand~\cite{BGG} and Demazure~\cite{Demazure}. They were later defined by Lascoux and Sch\"utzenberger~\cite{Lascoux_Schutzenberger_polynomes_schubert}, who developed a purely combinatorial theory. For each permutation $w$ in the symmetric group $S_n$ there is a Schubert polynomial $\frak{S}_w$ in the variables $x_1,\ldots,x_{n-1}$. When evaluated at certain Chern classes, a Schubert polynomial gives the cohomology class of a Schubert subvariety of the manifold of complete flags in $\Bbb{C}\,^n$. In this way, the collection $\{\frak{S}_w\,|\, w\in S_n\}$ of Schubert polynomials determines an integral basis for the cohomology ring of the flag manifold. Thus there exist integer structure constants $c^u_{w\,v}$ such that $$ \frak{S}_w\cdot\frak{S}_v = \sum_u c^u_{w\,v}\frak{S}_u. $$ No formula is known, or even conjectured, for these constants. There are, however, a few special cases in which they are known. One important case is Monk's rule~\cite{Monk}, which characterizes the algebra of Schubert polynomials. While this is usually attributed to Monk, Chevalley simultaneously established the analogous formula for generalized flag manifolds in a manuscript that was only recently published~\cite{Chevalley91}. Let $t_{k\,k+1}$ be the transposition interchanging $k$ and $k+1$. Then $\frak{S}_{t_{k\,k+1}} = x_1{+}\cdots{+}x_k=s(x_1,\ldots,x_k)$, the first elementary symmetric polynomial. For any permutation $w\in S_n$, Monk's rule states $$ \frak{S}_w \cdot \frak{S}_{t_{k\,k+1}} \ =\ \frak{S}_w \cdot s_1(x_1,\ldots,x_k) \ =\ \sum \frak{S}_{w t_{a\,b}}, $$ where $t_{a\,b}$ is the transposition interchanging $a$ and $b$, and the sum is over all $a\leq k<b$ where $w(a)<w(b)$ and if $a<c<b$, then $w(c)$ is not between $w(a)$ and $w(b)$. \smallskip The classical Pieri's rule computes the product of a Schur polynomial by either a complete homogeneous symmetric polynomial or an elementary symmetric polynomial. Our main result is a formula for Schubert polynomials and the cohomology of flag manifolds which generalizes both Monk's rule and the classical Pieri's rule. Let $s_m(x_1,\ldots,x_k)$ and $s_{1^m}(x_1,\ldots,x_k)$ be respectively the complete homogeneous and elementary symmetric polynomials of degree $m$ in the variables $x_1,\ldots,x_k$ and let $\ell(w)$ be the length of a permutation $w$. These polynomials are the cohomology classes of special Schubert varieties. We will show \medskip \noindent{\bf Theorem~\ref{thm:main}.} {\em Let $k,m,n$ be positive integers, and let $w\in S_n$. \begin{enumerate} \item[I.] $\frak{S}_w\cdot s_m(x_1,\ldots,x_k) = \sum_{w'} \frak{S}_{w'}$, the sum over all $w' = w t_{a_1\,b_1}\cdots t_{a_m\,b_m}$, where $a_i\leq k < b_i$ and $\ell(w t_{a_1\,b_1}\cdots t_{a_i\,b_i}) = \ell(w) + i$ for $1\leq i\leq m$ with the integers $b_1,\ldots, b_m$ distinct. \item[II.] $\frak{S}_w\cdot s_{1^m}(x_1,\ldots,x_k) = \sum_{w'} \frak{S}_{w'}$, the sum over all $w'$ as in {\em I}, except that now the integers $a_1,\ldots,a_m$ are distinct. \end{enumerate} } \medskip Both $s_m(x_1,\ldots,x_k)$ and $s_{1^m}(x_1,\ldots,x_k)$ are Schubert polynomials, so Theorem~\ref{thm:main} computes some of the structure constants in the cohomology ring of the flag manifold. These formulas were stated in a different form by Lascoux and Sch\"utzenberger in~\cite{Lascoux_Schutzenberger_polynomes_schubert}, where an algebraic proof was suggested. They were later independently conjectured in yet another form by Bergeron and Billey~\cite{Bergeron_Billey}. Our formulation facilitates our proofs. Using geometry, we expose a surprising connection to the classical Pieri's rule, from which we deduce Theorem~\ref{thm:main}. These methods enable the determination of additional structure constants. We further generalize Theorem~\ref{thm:main} to give a formula for the multiplication of a Schubert polynomial by a hook Schur polynomial, indicating a relation between multiplication of Schubert polynomials and paths in the Bruhat order in $S_n$. This exposition is organized as follows: Section 2 contains preliminaries about Schubert polynomials while Section 3 is devoted to the flag manifold. In Section 4 we deduce our main results from a geometric lemma proven in Section~5. Two examples are described in Section~6, illustrating the geometry underlying the results of Section~5. We remark that while our results are stated in terms of the integral cohomology of the complex manifold of complete flags, our results and proofs are valid for the Chow rings of flag varieties defined over any field. We would like to thank Nantel Bergeron and Sara Billey for suggesting these problems and Jean-Yves Thibon for showing us the work of Lascoux and Sch\"utzenberger. \section{Schubert Polynomials} In~\cite{BGG,Demazure} cohomology classes of Schubert subvarieties of the flag manifold were obtained from the class of a point using repeated correspondences in $\Bbb{P}^1$-bundles, which may be described algebraically as ``divided differences.'' Subsequently, Lascoux and Sch\"utzenberger~\cite{Lascoux_Schutzenberger_polynomes_schubert} found explicit polynomial representatives for these classes. We outline Lascoux and Sch\"utzenberger's construction of Schubert polynomials. For a more complete account see~\cite{Macdonald_schubert}. For an integer $n>0$, let $S_n$ be the group of permutations of $[n] = \{1,2,\ldots,n\}$. Let $t_{a\, b}$ be the transposition interchanging $a <b$. Adjacent transpositions $s_i = t_{i\,i{+}1}$ generate $S_n$. The {\em length} $\ell(w)$ of a permutation $w$ is the minimal length of a factorization into adjacent transpositions. If $w = s_{a_1} s_{a_2}\cdots s_{a_m}$ is such a factorization, then the sequence $(a_1,\ldots,a_m)$ is a {\em reduced word} for $w$. The length of $w$ also counts the inversions of $w$, those pairs $i<j$ where $w(i)>w(j)$. It follows that $\ell(wt_{a\, b}) = \ell(w){+}1$ if and only if $w(a)<w(b)$ and whenever $a<c<b$, either $w(c)< w(a)$ or $w(b)<w(c)$. For each integer $n>1$, let $R_n = \Bbb{Z}[x_1,\ldots,x_n]$. The group $S_n$ acts on $R_n$ by permuting the variables. Let $f\in R_n$ and let $s_i$ be an adjacent transposition. The polynomial $f - s_i f$ is antisymmetric in $x_i$ and $x_{i+1}$, and so is divisible by $x_i - x_{i+1}$. Thus we may define the linear divided difference operator $$ \partial_i = (x_i-x_{i+1})^{-1} (1 - s_i). $$ If $f$ is symmetric in $x_i$ and $x_{i+1}$, then $\partial_i f$ is zero. Otherwise $\partial_i f$ is symmetric in $x_i$ and $x_{i+1}$. Divided differences satisfy \begin{eqnarray*} \partial_i\circ \partial_i & = & 0 \\ \partial_i \circ\partial_j &=& \partial_j \circ \partial_i \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ if } |i-j|\geq 2\\ \qquad\qquad\partial_{i+1}\circ\partial_i\circ\partial_{i+1} & = & \partial_i\circ\partial_{i+1}\circ\partial_i \end{eqnarray*} It follows that if $(a_1,\ldots,a_p)$ is a reduced word for a permutation $w$, the composition of divided differences $\partial_{a_1}\circ\cdots\circ\partial_{a_p}$ depends only upon $w$ and not upon the reduced word chosen. This defines an operator $\partial_{w}$ for each $w\in S_n$. Let $w_0$ be the longest permutation in $S_n$, that is $w_0(j) = n{+}1{-}j$. For $w \in S_n$, define the {\em Schubert polynomial} $\frak{S}_{w}$ by $$ \frak{S}_{w} = \partial_{w^{-1}w_0} \left( x_1^{n-1} x_2^{n-2}\cdots x_{n-1} \right). $$ The degree of $\partial_i$ is $-1$, so $\frak{S}_{w}$ is homogeneous of degree $ {n\choose 2} - \ell(w^{-1}w_0) = \ell(w)$. Let $\cal{S}\subset R_n$ be the ideal generated by the non-constant symmetric polynomials. The set $\{\frak{S}_{w}\,|\, w\in S_n\}$ of Schubert polynomials is a basis for $\Bbb{Z}\{ x_1^{i_1}\cdots x_{n-1}^{i_{n-1}}\,|\, i_j \leq n{-}j \}$, a transversal to $\cal{S}$ in $R_n$. Thus Schubert polynomials are explicit polynomial representatives of an integral basis for the ring $H_n = R_n/\cal{S}$. Courting ambiguity, we will use the same notation for Schubert polynomials in $R_n$ as for their images in the rings $H_n$. Recently, other descriptions have been discovered for Schubert polynomials~\cite{Bergeron,BJS,Fomin_Kirillov,Fomin_Stanley}. Combinatorists often define Schubert polynomials $\frak{S}_w$ for all $w\in S_\infty = \cup_{n=1}^\infty S_n$. One may show that our results are valid in this wider context. \smallskip A {\em partition} $\lambda$ is a decreasing sequence $\lambda_1 \geq \lambda_2\geq\cdots\geq \lambda_k$ of positive integers, called the {\em parts} of $\lambda$. Given a partition $\lambda$ with at most $k$ parts, one may define a Schur polynomial $s_\lambda = s_\lambda(x_1,\ldots,x_k)$, which is a symmetric polynomial in the variables $x_1,\ldots,x_k$. For a more complete treatment of symmetric polynomials and Schur polynomials, see~\cite{Macdonald_symmetric}. The collection of Schur polynomials forms an integral basis for the ring of symmetric polynomials, $\Bbb{Z}[x_1,\ldots,x_k]^{S_k}$. The Littlewood-Richardson rule is a formula for the structure constants $c^\lambda_{\mu\nu}$ of this ring, called {\em Littlewood-Richardson coefficients} and defined by $$ s_\mu \cdot s_\nu \ =\ \sum_\lambda \, c^\lambda_{\mu\nu}\, s_\lambda. $$ If $\lambda$ and $\mu$ are partitions satisfying $\lambda_i \geq \mu_i$ for all $i$, we write $\lambda \supset \mu$. This defines a partial order on the collection of partitions, called Young's lattice. Since $c^\lambda_{\mu\nu} = 0$ unless $\lambda \supset \mu$ and $\lambda\supset \nu$~(cf. \cite{Macdonald_symmetric}), we see that $\cal{I}_{n,k}=\{ s_\lambda\,|\, \lambda_1 \geq n-k\}$ is an ideal. Let $A_{n,k}$ be the quotient ring $\Bbb{Z}[x_1,\ldots,x_k]^{S_k}/\cal{I}_{n,k}$. To a partition $\lambda$ we may associate its Young diagram, also denoted $\lambda$, which is a left-justified array of boxes in the plane with $\lambda_i$ boxes in the $i$th row. If $\lambda \supset \mu$, then the Young diagram of $\mu$ is a subset of that of $\lambda$, and the skew diagram $\lambda/\mu$ is the set theoretic difference $\lambda-\mu$. If each column of $\lambda/\mu$ is either empty or a single box, then $\lambda/\mu$ is a {\em skew row} of {\em length} $m$, where $m$ is the number of boxes in $\lambda/\mu$. The transpose $\mu^t$ of a partition $\mu$ is the partition whose Young diagram is the transpose of that of $\mu$. We call the transpose of a skew row a {\em skew column}. The map defined by $s_\lambda \mapsto s_{\lambda^t}$ is a ring isomorphism $A_{n,k} \rightarrow A_{n,n-k}$. For example, let $\lambda = (5,2,1)$ and $\mu = (3,1)$ then $\lambda/\mu$ is a skew row of length 4 and $\mu^t = (2,1,1)$. The following are the Young diagrams of $\lambda$, $\mu$, $\lambda/\mu$, and $\mu^t$: \begin{picture}(400,60) \put(20,10){\begin{picture}(50,30) \thicklines \put(0, 0){\line(1,0){10}} \put(0,10){\line(1,0){20}} \put(0,20){\line(1,0){50}} \put(0,30){\line(1,0){50}} \put( 0, 0){\line(0,1){30}} \put(10, 0){\line(0,1){30}} \put(20,10){\line(0,1){20}} \put(30,20){\line(0,1){10}} \put(40,20){\line(0,1){10}} \put(50,20){\line(0,1){10}} \end{picture}} \put(140,10){\begin{picture}(50,30) \thicklines \put( 0,10){\line(1,0){10}} \put( 0,20){\line(1,0){30}} \put( 0,30){\line(1,0){30}} \put( 0,10){\line(0,1){20}} \put(10,10){\line(0,1){20}} \put(20,20){\line(0,1){10}} \put(30,20){\line(0,1){10}} \end{picture}} \put(235,10){\begin{picture}(50,30) \thicklines \put( 0, 0){\line(1,0){10}} \put( 0,10){\line(1,0){20}} \put(10,20){\line(1,0){10}} \put( 0, 0){\line(0,1){10}} \put(10, 0){\line(0,1){20}} \put(20,10){\line(0,1){10}} \put(30,20){\line(1,0){20}} \put(30,30){\line(1,0){20}} \put(30,20){\line(0,1){10}} \put(40,20){\line(0,1){10}} \put(50,20){\line(0,1){10}} \end{picture}} \put(360,10){\begin{picture}(40,30) \thicklines \put(0, 0){\line(1,0){10}} \put(0,10){\line(1,0){10}} \put(0,20){\line(1,0){20}} \put(0,30){\line(1,0){20}} \put( 0, 0){\line(0,1){30}} \put(10, 0){\line(0,1){30}} \put(20,20){\line(0,1){10}} \end{picture}} \end{picture} If $w$ has only one {\em descent} ($k$ such that $w(k) > w(k{+}1)$), then $w$ is said to be {\em Grassmannian} of descent $k$ and $\frak{S}_{w}$ is the Schur polynomial $s_{\lambda}(x_1,\ldots,x_k)$. Here $\lambda$ is the {\em shape} of $w$, the partition with $k$ parts where $\lambda_{k+1-j} = w(j){-}j$. For integers $k,m$ define $r[k,m]$ and $c[k,m]$ to be the Grassmannian permutations of descent $k$ with shapes $(m,0,\ldots,0) = m$ and $(1^m,0,\ldots,0) = 1^m$, respectively. These are the $m+1$-cycles \begin{eqnarray*} r[k,m] &=& (k{+}m\,\,\,\,k{+}m{-}1\,\ldots\,k{+}2\,\,\,\,k{+}1\,\,\,\,k)\\ c[k,m] &=& (k{-}m{+}1\,\,\,\,k{-}m{+}2\,\ldots\,k{-}1\,\,\,\,k\,\,\,\,k{+}1). \end{eqnarray*} \section{The Flag Manifold} Let $V$ be an $n$-dimensional complex vector space. A {\em flag} ${F\!_{\DOT}\,}$ in $V$ is a sequence $$ \{0\}\ =\ F_0 \subset F_1 \subset F_2\subset \cdots \subset F_{n-1} \subset F_n\ =\ V, $$ of linear subspaces with $\dim_{\Bbb{C}} F_i = i$. The set of all flags is a $\frac{1}{2}n(n-1)$ dimensional complex manifold, called the flag manifold and denoted $\Bbb{F}(V)$. Over $\Bbb{F}(V)$, there is a tautological flag ${\cal{F}\!_{\DOT}\,}$ of bundles whose fibre at a point ${F\!_{\DOT}\,}$ is the flag ${F\!_{\DOT}\,}$. Let $x_i$ be the Chern class of the line bundle $\cal{F}_i/\cal{F}_{i-1}$. Then the integral cohomology ring of $\Bbb{F}(V)$ is $H_n = \Bbb{Z}[x_1,\ldots,x_n]/\cal{S}$, where $\cal{S}$ is the ideal generated by those non-constant polynomials which are symmetric in $x_1,\ldots,x_n$. This description is due to Borel~\cite{Borel}. Given a subset $S \subset V$, let $\Span{S}$ be its linear span and for linear subspaces $W\subset U$ let $U-W$ be their set theoretic difference. An ordered basis $f_1,f_2,\ldots,f_n$ for $V$ determines a flag ${E_{\DOT}\,}$; set $E_i = \Span{f_1,\ldots,f_i}$ for $1\leq i \leq n$. In this case, write ${E_{\DOT}\,} = \Span{f_1,\ldots,f_n}$ and call $f_1,\ldots,f_n$ a {\em basis} for ${E_{\DOT}\,}$. A fixed flag ${F\!_{\DOT}\,}$ gives a decomposition due to Ehresmann~\cite{Ehresmann} of $\Bbb{F}(V)$ into affine cells indexed by permutations $w$ of $S_n$. The cell determined by $w$ is $$ X^{\circ}_w {F\!_{\DOT}\,} \ = \ \{ {E_{\DOT}\,}=\Span{f_1,\ldots,f_n}\,|\, f_i \in F_{n+1-w(i)}-F_{n-w(i)}, \,1\leq i\leq n\}. $$ The complex codimension of $X^{\circ}_w {F\!_{\DOT}\,}$ is $\ell(w)$ and its closure is the Schubert subvariety $X_w{F\!_{\DOT}\,}$. Thus the cohomology ring of $\Bbb{F}(V)$ has an integral basis given by the cohomology classes\footnote{Strictly speaking, we mean the classes Poincar\'e dual to the fundamental cycles in homology.} $[X_w{F\!_{\DOT}\,}]$ of the Schubert subvarieties. That is, $H^*\Bbb{F}(V) = \bigoplus_{w\in S_n}{\Bbb Z} [X_w{F\!_{\DOT}\,}]$. Independently, Bernstein-Gelfand-Gelfand~\cite{BGG} and Demazure~\cite{Demazure} related this description to Borel's, showing $[X_w{F\!_{\DOT}\,}] = \partial_{w^{-1}w_0}[\{{F\!_{\DOT}\,}\}]$. Later, Lascoux and Sch\"utzenberger~\cite{Lascoux_Schutzenberger_polynomes_schubert} obtained polynomial representatives $\frak{S}_w$ for $[X_w{F\!_{\DOT}\,}]$ by choosing $x_1^{n-1} x_2^{n-2}\cdots x_{n-1}$ for the representative of the class $[\{{F\!_{\DOT}\,}\}] = \frak{S}_{w_0}$ of a point. We use the term Schubert polynomial for both the polynomial and the associated cohomology class. This Schubert polynomial basis for cohomology diagonalizes the intersection pairing; If $\ell(w) + \ell(w') = \dim\Bbb{F}(V) = \frac{1}{2}n(n-1)$, then $$ \frak{S}_w\cdot \frak{S}_{w'} = \left\{ \begin{array}{ll} \frak{S}_{w_0} & \mbox{ if } w' = w_0 w\\ 0 & \mbox{ otherwise} \end{array} \right. $$ \smallskip For each $k\leq \dim V =n$, the set of all $k$-dimensional subspaces of $V$ is a $k(n{-}k)$ dimensional complex manifold, called the Grassmannian of $k$-planes in $V$, written $G_kV$. The cohomology ring of $G_kV$ is a quotient of the ring of symmetric polynomials in the Chern roots $x_1,\ldots,x_k$ of its tautological $k$-plane bundle. This identifies it with the ring $A_{n,k}$ of Section 2. A fixed flag ${F\!_{\DOT}\,}$ gives a decomposition of $G_kV$ into cells indexed by partitions $\lambda$ with $k$ parts, none exceeding $n{-}k$. The closure of such a cell is the Schubert variety $$ \Omega_\lambda {F\!_{\DOT}\,} = \{ H \in G_kV \,|\, \dim H\cap F_{n-k+j-\lambda_j} \geq j \mbox{ for } 1\leq j\leq k\}, $$ whose codimension is $\lambda_1{+}{\cdots}{+}\lambda_k = |\lambda|$. The classes $[\Omega_\lambda{F\!_{\DOT}\,}]$ form a basis for the cohomology ring of $G_kV$ and $[\Omega_\lambda{F\!_{\DOT}\,}]$ is the Schur polynomial $s_\lambda(x_1,\ldots,x_k)$. We use the term Schur polynomial for both the polynomial and its image in the cohomology ring of $G_kV$. The Schur polynomial $s_{m}$ is the complete homogeneous symmetric polynomial of degree $m$ in $x_1,\ldots,x_k$. The Schur polynomial $s_{1^m}$ is the $m$th elementary symmetric polynomial in $x_1,\ldots,x_k$. Pieri's rule is a formula for multiplying Schur polynomials by either $s_m$ or $s_{1^m}$. For $s_m$, it states $$ s_\mu \cdot s_m\ = \ \sum s_{\lambda}, $$ the sum over all partitions $\lambda$ with $n{-}k\geq\lambda_1\geq\mu_1\geq\cdots\geq\lambda_k\geq\mu_k$ and $|\lambda| = m{+} |\mu|$. That is, those partitions $\lambda\supset \mu$ with $\lambda/\mu$ a skew row of length $m$. To obtain the analogous formula for $s_{1^m}$, use the isomorphism $A_{n,k} \rightarrow A_{n,n-k}$ given by $s_\lambda \mapsto s_{\lambda^t}$. Doing so, we see that $$ s_\mu \cdot s_{1^m}\ = \ \sum s_\lambda, $$ the sum over all partitions $\lambda$ with $\lambda\supset \mu$ with $(\lambda/\mu)^t$ is a skew row of length $m$. That is, those $\lambda\supset \mu$ with $\lambda/\mu$ a skew column of length $m$. \smallskip If $Y\subset V$ has codimension $d$, then $G_kY \subset G_kV$ is a Schubert subvariety whose indexing partition is $d^k$, the partition with $k$ parts each equal to $d$. It follows that $\Omega_{(n{-}k)^k}{F\!_{\DOT}\,} = \{F_k\}$, so $s_{(n{-}k)^k}$ is the class of a point. The basis of Schur polynomials diagonalizes the intersection pairing; For a partition $\lambda$, let $\lambda^c$ be the partition $(n{-}k{-}\lambda_k,{\ldots},n{-}k{-}\lambda_1)$. If $|\mu| {+}|\lambda| = k(n{-}k)$, then $$ s_\lambda \cdot s_\mu = \left\{ \begin{array}{ll} s_{(n{-}k)^k}& \mbox{ if } \lambda^c = \mu\\ 0 & \mbox{ otherwise } \end{array} \right .. $$ We use this to reformulate Pieri's rule. Suppose $|\mu|+|\lambda|+m = k(n-k)$, then $$ s_\mu \cdot s_{\lambda^c}\cdot s_m = \left\{ \begin{array}{ll} s_{(n-k)^k} & \mbox{ if $\lambda/\mu$ is a skew row of length $m$}\\ 0 &\mbox{ otherwise}\end{array}\right. . $$ \smallskip For $k\leq n$, the association ${E_{\DOT}\,} \mapsto E_k$ defines a map $\pi :\Bbb{F}(V) \rightarrow G_kV$. The functorial map $\pi^*$ on cohomology is simply the inclusion into $H_n$ of polynomials symmetric in $x_1,\ldots,x_k$. That is, $A_{n,k} \hookrightarrow H_n$. If $\lambda$ is a partition with $k$ parts and $w$ the Grassmannian permutation of descent $k$ and shape $\lambda$, then $\pi^* s_\lambda = \frak{S}_w$. Under the Poincar\'e duality isomorphism between homology and cohomology groups, the functorial map $\pi_*$ on homology induces a a group homomorphism $\pi_*$ on cohomology. While $\pi_*$ is not a ring homomorphism, is does satisfy the projection formula (see Example 8.17 of~\cite{Fulton_intersection}): $$ \pi_*(\alpha\cdot \pi^* \beta) = (\pi_* \alpha)\cdot \beta, $$ where $\alpha$ is a cohomology class on $\Bbb{F}(V)$ and $\beta$ is a cohomology class on $G_kV$. \section{Pieri's Rule for Flag Manifolds} An open problem is to find the analog of the Littlewood-Richardson rule for Schubert polynomials. That is, determine the structure constants $c^u_{w\,v}$ for the Schubert basis of the cohomology of flag manifolds, which are defined by \begin{equation} \label{eq:structure} \frak{S}_w \cdot \frak{S}_v = \sum_u c^u_{w\,v} \frak{S}_u. \end{equation} These constants are positive integers as they count the points in a suitable triple intersection of Schubert subvarieties. They are are known only in some special cases. For example, if both $w$ and $v$ are Grassmannian permutations of descent $k$ so that $\frak{S}_w$ and $\frak{S}_v$ are symmetric polynomials in the variables $x_1,\ldots,x_k$, then (\ref{eq:structure}) is the classical Littlewood-Richardson rule. Another case is Monk's rule, which states: $$ \frak{S}_w\cdot \frak{S}_{t_{k\,k{+}1}} = \sum \frak{S}_{w t_{a\,b}}, $$ the sum over all $a\leq k <b$ with $\ell(w t_{a\,b})=\ell(w)+1$. The Schubert polynomial $\frak{S}_{t_{k\,k{+}1}}$ is $s_1(x_1,\ldots,x_k)$. We use geometry to generalize this formula, giving an analog of the classical Pieri's rule. \smallskip Let $w,w' \in S_n$. Write $w \stackrel{r[k,m]}{\relbar\joinrel\llra} w'$ if there exist integers $a_1,b_1,\ldots,a_m,b_m$ with \begin{enumerate} \item $a_i\leq k <b_i$ for $1\leq i\leq m$ and $w' = wt_{a_1\,b_1}\cdots t_{a_m\,b_m}$, \item $\ell(w t_{a_1\,b_1}\cdots t_{a_i\,b_i}) = \ell(w) +i$, and \item the integers $b_1, b_2,\ldots, b_m$ are distinct. \end{enumerate} Similarly, $w\stackrel{c[k,m]}{\relbar\joinrel\llra} w'$ if we have integers $a_1,\ldots,b_m$ as in (1) and (2) where now \begin{enumerate} \item[(3)$'$] the integers $a_1,a_2,\ldots, a_m$ are distinct. \end{enumerate} Our primary result is the following. \begin{thm} \label{thm:main} Let $w \in S_n$. Then \begin{enumerate} \item[I.] For all $k$ and $m$ with $k+m \leq n$, we have \ \ ${\displaystyle \frak{S}_{w}\cdot \frak{S}_{r[k,m]} = \sum_{w \stackrel{r[k,m]}{\relbar\joinrel\llra} w'} \frak{S}_{w'}}$. \smallskip \item[II.] For all $ m\leq k\leq n$, we have\rule{0pt}{20pt} \ \ ${\displaystyle \frak{S}_{w}\cdot \frak{S}_{c[k,m]} = \sum_{w \stackrel{c[k,m]}{\relbar\joinrel\llra} w'} \frak{S}_{w'}}$. \end{enumerate} \end{thm} Theorem~\ref{thm:main} may be alternatively stated in terms of the structure constants $c^u_{w\,v}$. \medskip \noindent{\bf Theorem 1$'\!$.} \ {\em Let $w, w' \in S_n$. Then \begin{enumerate} \item[I.] For all integers $k,m$ with $k+m\leq n$, \ \ ${\displaystyle c^{w'}_{w\, r[k,m]} = \left\{\begin{array}{ll} 1 &\mbox{ if } w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'\\ 0 & \mbox{ otherwise}\end{array}\right. }$. \item[ II.] For all integers $k,m$ with $m\leq k\leq n$,\rule{0pt}{28pt} \ \ ${\displaystyle c^{w'}_{w\, c[k,m]} = \left\{\begin{array}{ll} 1 &\mbox{ if } w\stackrel{c[k,m]}{\relbar\joinrel\llra} w'\\ 0 & \mbox{ otherwise}\end{array}\right. }$. \end{enumerate} } \bigskip We first show the equivalence of parts I and II and then establish part I. An order $<_k$ on $S_n$ is introduced, and we show that $c^{w'}_{w\, r[k,m]}$ is 0 unless $w<_k w'$. A geometric lemma enables us to compute $c^{w'}_{w\, r[k,m]}$ when $w<_k w'$. \begin{lemma}\label{lemma:equivalent} Let $w_0$ be the longest permutation in $S_n$, and $k{+}m \leq n$. Then \begin{enumerate} \item $w_0 r[k,m] w_0 = c[n{-}k,m]$. \item Let $w, w' \in S_n$. Then $w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'$ if and only if $\,\,\,w_0 w w_0\stackrel{c[n{-}k,m]}{\relbar\joinrel\relbar\joinrel\lllra} w_0w'w_0$. \item The map induced by $\frak{S}_w \mapsto \frak{S}_{w_0ww_0}$ is an automorphism of $H_n$. \item Statements {\em I} and {\em II} of Theorem~\ref{thm:main}$'$ are equivalent. \end{enumerate} \end{lemma} This automorphism $\frak{S}_w \mapsto \frak{S}_{w_0ww_0}$ is the Schubert polynomial analog of the map $s_\lambda(x_1,\ldots,x_k) \mapsto s_{\lambda^t}(x_1,\ldots,x_{n-k})$ for Schur polynomials. \medskip \noindent{\bf Proof:} Statements (1) and (2) are easily verified, as $w_0(j) = n+1-j$. Statement (3) is also immediate, as $\frak{S}_w \mapsto \frak{S}_{w_0ww_0}$ leaves Monk's rule invariant and Monk's rule characterizes the algebra of Schubert polynomials. For (4), suppose $k+m \leq n$ and $w, w' \in S_n$ and let $\overline{w}$ denote $w_0ww_0$. The isomorphism $\frak{S}_v \mapsto \frak{S}_{\overline{v}}$ of (3) shows $c^{w'}_{w\, r[k,m]}= c^{\overline{w'}}_{\overline{w}\,\overline{r[k,m]}}$. Part (1) shows $c^{\overline{w'}}_{\overline{w}\,\overline{r[k,m]}} = c^{\overline{w'}}_{\overline{w}\,c[n{-}k,m]}$. Then (2) shows the equality of the two statements of Theorem~\ref{thm:main}$'$. \QED Let $<_k$ be the transitive closure of the relation given by $w <_k w'$, whenever $w' = w t_{a\,b}$ with $a\leq k<b$ and $\ell(w\, t_{a\,b}) = \ell(w){+}1$. We call $<_k$ the {\em $k$-Bruhat order}, in~\cite{Lascoux_Schutzenberger_Symmetry} it is the $k$-colored Ehresmano\"edre. \begin{lemma}\label{lemma:order} If $\,c^{w'}_{w\, r[k,m]} \neq 0$, then $w<_k w'$ and $\ell(w') = \ell(w) + m$. \end{lemma} \noindent{\bf Proof:} By Monk's rule, $w<_kw'$ if and only if $\frak{S}_{w'}$ appears with a non-zero coefficient when $\frak{S}_w (\frak{S}_{t_{k\,k{+}1}})^{\ell(w')-\ell(w)}$ is written as a sum of Schubert polynomials. Since $r[k,m] = t_{k\,k{+}1} \cdot t_{k\,k{+}2}\cdots t_{k\,k{+}m}$, Monk's rule shows that $\frak{S}_{r[k,m]}$ is a summand of $(\frak{S}_{t_{k\,k+1}})^m$ with coefficient 1. Thus the coefficient of $\frak{S}_{w'}$ in the expansion of $\frak{S}_w \cdot (\frak{S}_{t_{k\,k+1}})^m$ exceeds the coefficient of $\frak{S}_{w'}$ in $\frak{S}_w \cdot \frak{S}_{r[k,m]}$. Hence $c_{w\,r[k,m]}^{w'} = 0$ unless $w<_k w'$ and $\ell(w') = \ell(w) +m$. \QED In Section 5 we use geometry to prove the following lemma. \begin{lemma}\label{lemma:pushforward} Let $w<_k w'$ be permutations in $S_n$. Suppose $w' = w t_{a_1\,b_1}\cdots t_{a_m\,b_m}$, where $a_i\leq k<b_i$, and $\ell(w t_{a_1\,b_1}\cdots t_{a_i\,b_i}) = \ell(w)+i$. Let $d = n-k-\#\{b_1,\ldots,b_m\}$. Then \begin{enumerate} \item There is a cohomology class $\delta$ on $G_kV$ such that $\pi_*(\frak{S}_w\cdot \frak{S}_{w_0w'}) = \delta \cdot s_{d^k}$. \item If $w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'$, then there are partitions $\lambda \supset \mu$ where $\lambda/\mu$ is a skew row of length $m$ whose $j$th row has length $\#\{i\,|\, a_i = j\}$ and $\pi_*(\frak{S}_w\cdot \frak{S}_{w_0w'}) = s_{\mu}\cdot s_{\lambda^c}$. \end{enumerate} \end{lemma} \noindent{\bf Proof of Theorem~\ref{thm:main}$'$:} By Lemma~\ref{lemma:order}, we need only show that if $w<_k w'$ and $\ell(w')-\ell(w) = m$, then $$ c^{w'}_{w\,r[k,m]} = \left\{ \begin{array}{ll} 1 &\mbox{ if } w \stackrel{r[k,m]}{\relbar\joinrel\llra} w'\\ 0 & \mbox{ otherwise} \end{array} \right. . $$ Begin by multiplying the identity $\frak{S}_w\cdot \frak{S}_{r[k,m]} = \sum_v\, c^v_{w\, r[k,m]}\, \frak{S}_v$ by $\frak{S}_{w_0\, w'}$ and use the intersection pairing to obtain $$ \frak{S}_w\cdot\frak{S}_{w_0\, w'}\cdot\frak{S}_{r[k,m]} \ =\ c^{w'}_{w\,r[k,m]}\, \frak{S}_{w_0}. $$ Recall that $\frak{S}_{r[k,m]} = \pi^* s_m(x_1,\ldots,x_k)$. As $\frak{S}_{w_0}$ and $s_{(n-k)^k}$ are the classes of points, $\pi_*\frak{S}_{w_0} = s_{(n-k)^k}$. Apply the map $\pi_*$ and then the projection formula to obtain: \begin{eqnarray*} \pi_*(\frak{S}_w\cdot\frak{S}_{w_0\, w'}\cdot \pi^* s_m) &=& c^{w'}_{w\,r[k,m]}\, \pi_*( \frak{S}_{w_0})\\ \pi_*(\frak{S}_w\cdot\frak{S}_{w_0\, w'}) \cdot s_m &=& c^{w'}_{w\,r[k,m]} \, s_{(n-k)^k}. \end{eqnarray*} By part (1) of Lemma~\ref{lemma:pushforward}, there is a cohomology class $\delta$ on $G_kV$ with $$ \pi_*(\frak{S}_w\cdot\frak{S}_{w_0\, w'}) \cdot s_m\ = \ \delta \cdot s_{d^k} \cdot s_m $$ But $s_{d^k} \cdot s_m = 0$ unless $d+m \leq n-k$. Since $d = n-k-\#\{b_1,\ldots,b_m\}\geq n-k-m$, we see that $ c^{w'}_{w\,r[k,m]} =0$ unless $m = \#\{b_1,\ldots,b_m\}$, which implies $w \stackrel{r[k,m]}{\relbar\joinrel\llra} w'$. To complete the proof of Theorem~\ref{thm:main}$'$, suppose that $w \stackrel{r[k,m]}{\relbar\joinrel\llra} w'$. By part (2) of Lemma~\ref{lemma:pushforward}, there are partitions $\lambda \supset \mu$ with $\lambda/\mu$ a skew row of length $m$ where we have $\pi_*(\frak{S}_w\cdot \frak{S}_{w_0w'}) = s_{\mu}\cdot s_{\lambda^c}$. Then $$ \pi_*(\frak{S}_w\cdot\frak{S}_{w_0\, w'}) \cdot s_m\ = \ s_{\mu}\cdot s_{\lambda^c} \cdot s_m \ = \ s_{(n-k)^k}, $$ by the ordinary Pieri's rule for Schur polynomials. So $c^{w'}_{w\, r[k,m]} = 1$. \QED Theorem \ref{thm:main}$'$ determines the structure constants $c^{w'}_{w\, r[k,m]}$ and $c^{w'}_{w\, c[k,m]}$. We compute more structure constants. For $\nu$ a partition with $k$ parts, let $w(\nu)$ be the Grassmannian permutation of descent $k$ and shape $\nu$. \begin{thm} Let $w, w'\in S_n$ and $k\leq n$ be an integer. Suppose $w\leq_k w'$ and $\ell(w') = \ell(w) +m$. Let $a_1,b_1,\ldots,a_m,b_m$ be such that $a_i\leq k <b_i$ where $w' = w t_{a_1\,b_1}\cdots t_{a_m\,b_m}$ and $\ell(w t_{a_1\,b_1}\cdots t_{a_i\,b_i}) = \ell(w) +i$. Let $\nu$ be a partition with $k$ parts. \begin{enumerate} \item If $\,w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'$, the structure constant $c^{w'}_{w\, w(\nu)}$ equals the Littlewood-Richardson coefficient $c^\lambda_{\mu\,\nu}$, where $\lambda/\mu$ is a skew row of length $m$ whose $j$th row has length $\#\{i \,|\, a_i = j\}$. \item If $\,w\stackrel{c[k,m]}{\relbar\joinrel\llra} w'$, the structure constant $c^{w'}_{w\, w(\nu)}$ equals the Littlewood-Richardson coefficient $c^\lambda_{\mu\,\nu}$, where $\lambda/\mu$ is a skew column of length $m$ whose $j$th column has length $\#\{i \,|\, b_i=j\}$. \end{enumerate} \end{thm} \noindent{\bf Proof:} Using the involution $\frak{S}_{w} \mapsto \frak{S}_{w_0ww_0}$, it suffices to prove part (1). We use part (2) of Lemma~\ref{lemma:pushforward} to evaluate $c^{w'}_{w\,w(\nu)}$. Recall that $\frak{S}_{w(\nu)} = \pi^*(s_\nu)$. Then \begin{eqnarray*} c^{w'}_{w\,w(\nu)}\, s_{(n-k)^k}\ =\ \pi_*(c^{w'}_{w\,w(\nu)}\, \frak{S}_{w_0}) &=& \pi_*(\frak{S}_w\cdot\frak{S}_{w_0w'}\cdot \frak{S}_{w(\nu)})\\ &=& \pi_*(\frak{S}_w\cdot\frak{S}_{w_0w'}) \cdot s_\nu\\ &=& s_\mu \cdot s_{\lambda^c}\cdot s_\nu\\ &=& c^{\lambda}_{\mu\nu}\, s_{(n-k)^k}. \ \ \ \ \QED \end{eqnarray*} The formulas of Theorem~\ref{thm:main} may be formulated as the sum over certain paths in the $k$-Bruhat order. We explain this formulation here. A (directed) path in the $k$-Bruhat order from $w$ to $w'$ is equivalent to a choice of integers $a_1,b_1,\ldots, a_m,b_m$ with $a_i\leq k < b_i$ and if $w^{(0)} = w$ and $w^{(i)} = w^{(i-1)}\cdot t_{a_i\,b_i}$, then $\ell(w^{(i)}) = \ell(w) + i$ and $w^{(m)} = w'$. In this case the path is $$ w = w^{(0)}<_k w^{(1)} <_k w^{(2)} <_k \cdots <_k w^{(m)} = w'. $$ \begin{lemma} Let $w, w' \in S_n$ and $k,m$ be positive integers. Then \begin{enumerate} \item $w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'$ if and only if there is a path in the $k$-Bruhat order of length $m$ such that $$ w^{(1)}(a_1) < w^{(2)}(a_2) < \cdots < w^{(m)}(a_m). $$ \item $w\stackrel{c[k,m]}{\relbar\joinrel\llra} w'$ if and only if there is a path in the $k$-Bruhat order of length $m$ such that $$ w^{(1)}(a_1) > w^{(2)}(a_2) > \cdots > w^{(m)}(a_m). $$ \end{enumerate} Furthermore, these paths are unique. \end{lemma} \noindent{\bf Proof:} If $w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'$, one may show that the set of values $\{ w^{(i)}(a_i)\}$ and the set of transpositions $\{t_{a_i\,b_i}\}$ depend only upon $w$ and $w'$, and not on the particular path chosen from $w$ to $w'$ in the $k$-Bruhat order. It is also the case that rearranging the set $\{ w^{(i)}(a_i)\}$ in order, as in (1), may be accomplished by interchanging transpositions $t_{a_i\,b_i}$ and $t_{a_j\,b_j}$ where $a_i\neq a_j$ (necessarily $b_i\neq b_j$). Both (1) and the uniqueness of this representation follow from these observations. Statement (2) follows for similar reasons. \QED For a path $\gamma$ in the $k$-Bruhat order, let $\mbox{end}(\gamma)$ be the endpoint of $\gamma$. We state a reformulation of Theorem 1. \begin{cor}[Path formulation of Theorem 1]\label{cor:pieri_paths} Let $w\in S_n$. \begin{enumerate} \item $ \frak{S}_w \cdot \frak{S}_{r[k,m]} \ =\ \sum_\gamma \frak{S}_{\mbox{\scriptsize end}(\gamma)}, $ the sum over all paths $\gamma$ in the $k$-Bruhat order which start at $w$ such that $$ w^{(1)}(a_1) < w^{(2)}(a_2) < \cdots < w^{(m)}(a_m), $$ where $\gamma$ is the path $w <_k w^{(1)} <_k w^{(2)} <_k \cdots <_k w^{(m)}$. Equivalently, $c^{w'}_{w\,r[k,m]}$ counts the number of paths $\gamma$ in the $k$-Bruhat order which start at $w$ such that $$ w^{(1)}(a_1) < w^{(2)}(a_2) < \cdots < w^{(m)}(a_m). $$ \item $ \frak{S}_w \cdot \frak{S}_{c[k,m]} \ =\ \sum_\gamma \frak{S}_{\mbox{\scriptsize end}(\gamma)}, $ the sum over all paths $\gamma$ in the $k$-Bruhat order which start at $w$ such that $$ w^{(1)}(a_1) > w^{(2)}(a_2) > \cdots > w^{(m)}(a_m), $$ where $\gamma$ is the path $w<_k w^{(1)} <_k w^{(2)} <_k \cdots <_k w^{(m)}$. Equivalently, $c^{w'}_{w\,r[k,m]}$ counts the number of paths $\gamma$ in the $k$-Bruhat order which start at $w$ such that $$ w^{(1)}(a_1) > w^{(2)}(a_2) > \cdots > w^{(m)}(a_m). $$ \end{enumerate} \end{cor} This is the form of the conjectures of Bergeron and Billey~\cite{Bergeron_Billey}, and it exposes a link between multiplying Schubert polynomials and paths in the Bruhat order. Such a link is not unexpected. The Littlewood-Richardson rule for multiplying Schur functions may be expressed as a sum over certain paths in Young's lattice of partitions. A connection between paths in the Bruhat order and the intersection theory of Schubert varieties is described in~\cite{Hiller_intersections}. We believe the eventual description of the structure constants $c^w_{uv}$ will be in terms of counting paths of certain types in the Bruhat order on $S_n$, and will yield new results about the Bruhat order on $S_n$. Corollary~\ref{cor:hook_enumeration} below is one such result. \smallskip Using multiset notation for partitions, $(p,1^{q-1})$ is the hook shape partition whose Young diagram is the union of a row of length $p$ and a column of length $q$. Define $h[k;\,p,q]$ to be the Grassmannian permutation of descent $k$ and shape $(p,1^{q-1})$. Then $\frak{S}_{h[k;\,p,q]} = \pi^* s_{(p,1^{q-1})}$. This permutation, $h[k;\,p,q]$, is the $p+q$-cycle $$ (k{-}q{+}1\,\,\,k{-}q{+}2\,\ldots\,k{-}1\,\,\, k\,\,\,k{+}p\,\,\,k{+}p{-}1\,\ldots\,k{+}1). $$ \begin{thm}\label{thm:hook_formula} Let $q\leq k$ and $k{+}p \leq n$ be integers. Set $m = p{+}q{-}1$. For $w\in S_n$, $$ \frak{S}_w \cdot \frak{S}_{h[k;\,p,q]} \ =\ \sum \frak{S}_{end(\gamma)}, $$ the sum over all paths $\gamma: w <_k w^{(1)} <_k w^{(2)} <_k \cdots <_k w^{(m)}$ in the $k$-Bruhat order with $$ w^{(1)}(a_1) < \cdots < w^{(p)}(a_p) \ \ \ \mbox{and}\ \ \ w^{(p)}(a_p) > w^{(p{+}1)}(a_{p{+}1})>\cdots > w^{(m)}(a_m). $$ Alternatively, those paths $\gamma$ with $$w^{(1)}(a_1) > \cdots > w^{(q)}(a_q) \ \ \ \mbox{and}\ \ \ w^{(q)}(a_q) <\cdots < w^{(m)}(a_m). $$ \end{thm} Setting either $p=1$ or $q=1$, we recover Theorem~\ref{thm:main}. If we consider the coefficient $c^{w'}_{w\,h[k;p,q]}$ of $\frak{S}_{w'}$ in the product $\frak{S}_w \cdot \frak{S}_{h[k;p,q]}$, we obtain: \begin{cor}\label{cor:hook_enumeration} Let $w, w' \in S_n$, and $p,q$ be positive integers where $\ell(w')-\ell(w) = p+q-1 = m$. Then the number of paths $w <_k w^{(1)} <_k w^{(2)} <_k \cdots <_k w^{(m)} = w'$ in the $k$-Bruhat order from $w$ to $w'$ with $$ w^{(1)}(a_1) < \cdots < w^{(p)}(a_p) \ \ \ \mbox{and}\ \ \ w^{(p)}(a_p) > w^{(p{+}1)}(a_{p{+}1})>\cdots > w^{(m)}(a_m) $$ equals the number of paths with $$w^{(1)}(a_1) > \cdots > w^{(q)}(a_q) \ \ \ \mbox{and}\ \ \ w^{(q)}(a_q) <\cdots < w^{(m)}(a_m). $$ \end{cor} \noindent{\bf Proof of Theorem~\ref{thm:hook_formula}:} By the classical Pieri's rule, $$ s_{p}\, \cdot \, s_{1^{(q-1)}} \ = \ s_{(p{+}1,1^{q-2})} \, + \, s_{(p,1^{q-1})}. $$ Expressing these as Schubert polynomials (applying $\pi^*$), we have: $$ \frak{S}_{r[k,p]}\, \cdot\, \frak{S}_{c[k,q{-}1]} \ =\ \frak{S}_{h[k;\,p{+}1,q{-}1]} + \frak{S}_{h[k;\,p,q]}. $$ Induction on either $p$ or $q$ (with $m$ fixed) and Corollary~\ref{cor:pieri_paths} completes the proof. \QED \section{Geometry of Intersections} We deduce Lemma~\ref{lemma:pushforward} by studying certain intersections of Schubert varieties. A key fact we use is that if $X_w{F\!_{\DOT}\,}$ and $X_v{G_{\DOT}}$ intersect generically transversally, then $$ [X_w{F\!_{\DOT}\,}\bigcap X_v{G_{\DOT}}] \ = \ [X_w{F\!_{\DOT}\,}]\cdot[X_v{G_{\DOT}}] \ = \ \frak{S}_w \cdot \frak{S}_v $$ in the cohomology ring. Flags ${F\!_{\DOT}\,}$ and ${G_{\DOT}}$ are {\em opposite} if for $1\leq i \leq n$, $F_i + G_{n-i} = V$. The set of pairs of opposite flags form the dense orbit of the general linear group $GL(V)$ acting on the space of all pairs of flags. Using this observation and Kleiman's Theorem concerning the transversality of a general translate~\cite{Kleiman}, we conclude that for any $w,v\in S_n$ and opposite flags ${F\!_{\DOT}\,}$ and ${G_{\DOT}}$, $X_w{F\!_{\DOT}\,}$ and $X_v{G_{\DOT}}$ intersect generically transversally. (One may also check this directly by examining the tangent spaces.) In this case the intersection is either empty or it is irreducible and contains a dense subset isomorphic to $(\Bbb{C}^\times\!)^m$, where $m+ \ell(w) + \ell(v)= \frac{1}{2}n(n-1)$ (cf.~\cite{Deodhar}). These facts hold for the Schubert subvarieties of $G_kV$ as well. Namely, if $\lambda$ and $\mu$ are any partitions and ${F\!_{\DOT}\,}$ and ${G_{\DOT}}$ are opposite flags, then $\Omega_\lambda{F\!_{\DOT}\,} \bigcap \Omega_\mu {G_{\DOT}}$ is either empty or it is an irreducible, generically transverse intersection containing a dense subset isomorphic to $(\Bbb{C}^\times\!)^m$, where $m+ |\lambda|+|\mu| = k(n-k)$. Let ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ be opposite flags in $V$. Let $e_1,\ldots,e_n$ be a basis for $V$ such that $e_i$ generates the one dimensional subspace $F_{n+1-i}\bigcap F'_i$. We deduce Lemma~\ref{lemma:pushforward} from the following two results of this section. \begin{lemma}\label{lemma:geometry_statementI} Let $w, w' \in S_n$ with $w <_k w'$ and $\ell(w')-\ell(w) =m$. Suppose that $w' = w t_{a_1\,b_1}\cdots t_{a_m\,b_m}$ with $a_i\leq k <b_i$ for $1\leq 1\leq m$ and $\ell(w t_{a_1\,b_1}\cdots t_{a_i\,b_i}) = \ell(w)+i$. Let $\pi : \Bbb{F}(V) \rightarrow G_kV$ be the canonical projection. Define $Y=\langle e_{w(j)}\,|\,j\leq k\,\mbox{ or }\,w(j)\neq w'(j)\rangle$. Then $Y$ has codimension $d=n- k - \#\{b_1,\ldots,b_m\}$ and $$ \pi ( X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,} ) \subset G_k Y. $$ Also, if ${E_{\DOT}\,}=\Span{f_1,\ldots,f_n} \in X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$, then we may assume that for $j>k$ with $w(j) = w'(j)$, we have $f_j = e_{w(j)}$. \end{lemma} \begin{lemma}\label{lemma:geometry_statementII} Let $w, w' \in S_n$ with $w \stackrel{r[k,m]}{\relbar\joinrel\llra} w'$ and let $a_1,\ldots,b_m$ be as in the statement of Lemma~\ref{lemma:geometry_statementI}. Then there exist opposite flags ${G_{\DOT}}$ and ${G_{\DOT}}\!'$ and partitions $\lambda\supset\mu$, with $\lambda/\mu$ a skew row of length $m$ whose $j$th row has length $\#\{i\,|\, a_i = j\}$ such that $$ \pi ( X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,} ) \quad =\quad \Omega_{\mu}{G_{\DOT}} \bigcap \Omega_{\lambda^c}{G_{\DOT}}\!', $$ and the map $\pi|_{X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}} : X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,} \rightarrow \Omega_{\mu}{G_{\DOT}} \bigcap \Omega_{\lambda^c}{G_{\DOT}}\!'$ has degree 1. \end{lemma} Lemma~\ref{lemma:geometry_statementII} is the surprising connection to the classical Pieri's rule that was mentioned in the Introduction. A typical geometric proof of Pieri's rule for Grassmannians (see~\cite{Griffiths_Harris,Hodge_Pedoe}) involves showing a triple intersection of Schubert varieties \begin{equation}\label{eq:triple_intersection} \Omega_{\lambda}{G_{\DOT}} \bigcap \Omega_{\mu^c}{G_{\DOT}}\!' \,\bigcap\, \Omega_m{G_{\DOT}}\!'' \end{equation} is transverse and consists of a single point, when ${G_{\DOT}}, {G_{\DOT}}\!'$, and ${G_{\DOT}}\!''$ are in suitably general position. We would like to construct a proof of Theorem~\ref{thm:main} along those lines, studying a triple intersection of Schubert subvarieties \begin{equation}\label{eq:triple_intersection_II} X_w {G_{\DOT}} \bigcap X_{w_0w'}{G_{\DOT}}\!' \,\bigcap\, X_{r[k,m]}{G_{\DOT}}\!'', \end{equation} where ${G_{\DOT}}, {G_{\DOT}}\!'$, and ${G_{\DOT}}\!''$ are in suitably general position. Doing so, one observes that the geometry of the intersection of~(\ref{eq:triple_intersection_II}) is governed entirely by the geometry of an intersection similar to that in~(\ref{eq:triple_intersection}). In part, that is because $ X_{r[k,m]}{G_{\DOT}}\!''=\pi^{-1}\Omega_m{G_{\DOT}}\!''$. This is the spirit of our method, which may be seen most vividly in Lemmas 14 and 15. \bigskip \noindent{\bf Proof of Lemma~\ref{lemma:pushforward}:} Since ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ are opposite flags, $X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$ is a generically transverse intersection, so in the cohomology ring $$ [X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}]\ = \ [X_w {F\!_{\DOT}\,}]\cdot [X_{w_0w'}{{F\!_{\DOT}}'\,}]\ =\ \frak{S}_w\cdot\frak{S}_{w_0w'}. $$ Let $Y$ be the subspace of Lemma~\ref{lemma:geometry_statementI}. Since $\pi ( X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,} ) \subset G_k Y$, the class $\pi_*(\frak{S}_w\cdot\frak{S}_{w_0w'})$ is a cohomology class on $G_kY$. However, all such classes are of the form $\delta \cdot [ G_k Y]$, for some cohomology class $\delta$ on $G_kV$. Since $d$ is the codimension of $Y$, we have $[G_k Y]= s_{d^k}$, establishing part (1) of Lemma~\ref{lemma:pushforward}. For part (2), suppose further that $w \stackrel{r[k,m]}{\relbar\joinrel\llra} w'$. If $\rho$ is the restriction of $\pi$ to $X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$, then $$ \pi_*(\frak{S}_w\cdot\frak{S}_{w_0w'}) \ = \ \pi_*([X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}]) \ = \ \deg \rho\cdot [\pi(X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,})]. $$ By Lemma~\ref{lemma:geometry_statementII}, $\deg \rho = 1$ and $\pi(X_w{F\!_{\DOT}\,} \bigcap x_{w_0w'}{{F\!_{\DOT}}'\,}) = \Omega_\mu {G_{\DOT}} \bigcap \Omega_{\lambda^c} {G_{\DOT}}\!'$. Since ${G_{\DOT}}$ and ${G_{\DOT}}\!'$ are opposite flags, we have $$ \pi_*(\frak{S}_w\cdot\frak{S}_{w_0w'}) \ = \ 1\cdot [ \Omega_{\lambda}{G_{\DOT}} \bigcap \Omega_{\mu^c}{G_{\DOT}}\!'] \ = \ [ \Omega_{\lambda}{G_{\DOT}} ] \cdot [\Omega_{\mu^c}{G_{\DOT}}\!'] \ =\ s_\lambda \cdot s_{\mu^c}, $$ completing the proof of Lemma~\ref{lemma:pushforward}.\QED \bigskip We deduce Lemma~\ref{lemma:geometry_statementI} from a series of lemmas. We first make a definition. Let $W\subsetneq V$ be a codimension 1 subspace and let $e \in V - W$ so that $V = \Span{W,e}$. For $1\leq p \leq n$, define an expanding map $\psi_p: \Bbb{F}(W) \rightarrow \Bbb{F}(V)$ as follows $$ (\psi_p {E_{\DOT}\,})_i = \left\{ \begin{array}{ll} E_i & \mbox{ if } i<p\\ \Span{E_{i-1},e} & \mbox{ if } i \geq p \end{array}\right. . $$ Note that if ${E_{\DOT}\,} = \Span{f_1,\ldots,f_{n-1}}$, then $\psi_p {E_{\DOT}\,}= \Span{f_1,\ldots,f_{p-1},e,f_p,\ldots,f_{n-1}}$. For $w\in S_n$ and $1\leq p \leq n$, define $w|_p \in S_{n-1}$ by $$ w|_p (j) = \left\{ \begin{array}{ll} w(j) & \mbox{ if } j<p \mbox{ and } w(j)<w(p)\\ w(j{+}1) & \mbox{ if } j\geq p \mbox{ and } w(j)<w(p)\\ w(j) - 1 & \mbox{ if } j<p \mbox{ and } w(j)>w(p)\\ w(j{+}1)-1 & \mbox{ if } j\geq p \mbox{ and } w(j)>w(p) \end{array} \right. . $$ If we represent permutations as matrices, $w|_p$ is obtained by crossing out the $p$th row and $w(p)$th column of the matrix for $w$. \begin{lemma}\label{lemma:expand} Let $W\subsetneq V$ and $e\in V - W$ with $V = \Span{W,e}$. Let ${G_{\DOT}}$ be a complete flag in $W$. For $1\leq p \leq n$ and $w\in S_n$, $$ \psi_p\left( X_{w|_p}{G_{\DOT}} \right) \subset X_w \left(\psi_{w_0w(p)}({G_{\DOT}})\right). $$ \end{lemma} \noindent{\bf Proof:} Let ${E_{\DOT}\,} \in X_{w|_p}{G_{\DOT}} $. Then $W$ has a basis $f_1,\ldots,f_{n-1}$ with ${E_{\DOT}\,} = \Span{f_1,\ldots,f_{n-1}}$ and for each $1\leq i \leq n-1$, $f_i \in G_{n-w|_p(i)}$. Then we necessarily have $ \psi_p({E_{\DOT}\,}) = \Span{\phi_1,\ldots,\phi_n} = \Span{f_1,\ldots,f_{p-1},e,f_p,\ldots,f_{n-1}}$. Noting $$ \left(\psi_{w_0w(p)}({G_{\DOT}})\right)_{n+1-j} \ = \ \left\{ \begin{array}{ll} G_{n+1-j} & \mbox{ if } j >w(p) \\ \Span{e,G_{n-j}} & \mbox{ if } j \leq w(p) \end{array}\right. , $$ we see that $\phi_i \in \left(\psi_{w_0w(p)}({G_{\DOT}})\right)_{n+1-w(i)}$. Thus $\psi_p\left( X_{w|_p}{G_{\DOT}} \right) \subset X_w \left(\psi_{w_0w(p)}({G_{\DOT}})\right)$. \QED \begin{lemma}\label{lemma:expanding} Let $W\subsetneq V$ and $e\in V - W$ with $V = \Span{W,e}$ and let ${G_{\DOT}}$ and ${G_{\DOT}}\!'$ be opposite flags in $W$. Suppose that $w<_kw'$ are permutations in $S_n$ and $p>k$ an integer such that $w(p) = w'(p)$. Let $w_0^{(j)}$ is the longest permutation in $S_j$. Then \begin{enumerate} \item $\ell(w'|_p)-\ell(w|_p)=\ell(w')-\ell(w)$ and $w|_p <_k w'|_p$. \item $\psi_p{\left(X_{w|_p}{G_{\DOT}} \bigcap X_{w_0^{(n-1)}(w'|_p)}{G_{\DOT}}\!'\right)} \, = \,X_w\left(\psi_{w_0^{(n)}w(p)}({G_{\DOT}}) \right) \bigcap X_{w_0^{(n)}w'}\left(\psi_{w'(p)}({G_{\DOT}}\!')\right)$. \item If ${E_{\DOT}\,} \in X_w\left(\psi_{w_0^{(n)}w(p)}({G_{\DOT}}) \right) \bigcap X_{w_0^{(n)}w'}\left(\psi_{w'(p)}({G_{\DOT}}\!')\right)$, then $E_p = \Span{E_{p-1},e}$. \item If ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ are opposite flags in $V$ and ${E_{\DOT}\,} \in X_w{F\!_{\DOT}\,}\bigcap X_{w_0^{(n)}w'}{{F\!_{\DOT}}'\,}$, then $E_k \in F_{n-w(p)}+F'_{w(p)-1}$. \end{enumerate} \end{lemma} \noindent{\bf Proof:} First recall that $\ell(v t_{a\,b}) = \ell(v)+1$ if and only if $v(a) < v(b)$ and if $a<j<b$, then $v(j)$ is not between $v(a)$ and $v(b)$. Thus if $\ell(v t_{a\,b}) = \ell(v)+1$ and $p\not\in \{a,b\}$, we have $\ell(v t_{a\,b}|_p) = \ell(v|_p)+1$. Statement (1) follows by induction on $\ell(w') - \ell(w)$. For (2), since $(w_0^{(n)}w')|_p = w_0^{(n-1)}(w'|_p)$ and $w_0^{(n)}w_0^{(n)}w'= w'$, Lemma~\ref{lemma:expand} shows $$ \psi_p\left(X_{w|_p}{G_{\DOT}} \bigcap X_{w_0^{(n-1)}(w'|_p)}{G_{\DOT}}\!'\right) \subset X_w\left(\psi_{w_0^{(n)}w(p)}({G_{\DOT}})\right) \bigcap X_{w_0^{(n)}w'}\left(\psi_{w'(p)}({G_{\DOT}}\!')\right). $$ The flags $\psi_{w_0^{(n)}w(p)}({G_{\DOT}})$ and $\psi_{w'(p)}({G_{\DOT}}\!')$ are opposite flags in $V$, since ${G_{\DOT}}$ and ${G_{\DOT}}\!'$ are opposite flags in $W$. Then part (1) shows both sides have the same dimension. Since $\psi_p$ is injective, they are equal. To show (3), let ${E_{\DOT}\,} \in X_w\left(\psi_{w_0^{(n)}w(p)}({G_{\DOT}}) \right) \bigcap X_{w_0^{(n)}w'}\left(\psi_{w'(p)}({G_{\DOT}}\!')\right)$. By (2), there is a flag ${E_{\DOT}\!'\,} \in X_{w|_p}{G_{\DOT}} \bigcap X_{w_0^{(n-1)}(w'|_p)}{G_{\DOT}}\!'$ with $\psi_p({E_{\DOT}\!'\,}) = {E_{\DOT}\,}$, so $E_p = \Span{E'_{p-1},e} = \Span{E_{p-1},e}$. For (4), let $W= F_{n-w(p)}+F'_{w'(p)-1}$ and $e$ any nonzero vector in the one dimensional space $F_{n+1-w(p)}\bigcap F'_{w'(p)}$. The distinct subspaces in ${F\!_{\DOT}\,}\bigcap W$ define a flag ${G_{\DOT}}$, and those in ${{F\!_{\DOT}}'\,} \bigcap W$ define a flag ${G_{\DOT}}\!'$. In fact, $\psi_{w_0^{(n)}w(p)}({G_{\DOT}})={F\!_{\DOT}\,}$ and $\psi_{w(p)}({G_{\DOT}}\!') = {{F\!_{\DOT}}'\,}$, and ${G_{\DOT}}$ and ${G_{\DOT}}\!'$ are opposite flags in $W$. By (2), $$ \psi_p\left(X_{w|_p}{G_{\DOT}} \bigcap X_{w_0^{(n-1)}(w'|_p)}{G_{\DOT}}\!'\right) = X_w{F\!_{\DOT}\,} \bigcap X_{w_0^{(n)}w'}{{F\!_{\DOT}}'\,}. $$ Thus flags in $X_w{F\!_{\DOT}\,} \bigcap X_{w_0^{(n)}w'}{{F\!_{\DOT}}'\,}$ are in the image of $\psi_p$. As $k < p$, $\left( \psi_p {E_{\DOT}\,}\right)_k = E_k\subset W$, establishing part (4). \QED \noindent{\bf Proof of Lemma~\ref{lemma:geometry_statementI}:} Let ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ be opposite flags in $V$, let $w<_k w'$ and let ${E_{\DOT}\,} \in X_w{F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$. Define a basis $e_1,\ldots,e_n$ for $V$ by $F_{n+1-j}\bigcap F'_j = \Span{e_j}$ for $1\leq j\leq n$. Suppose $w' = w t_{a_1\,b_1}\cdots t_{a_m\,b_m}$ with $a_i\leq k < b_i$. Let $\{p_1,\ldots,p_d\}$ be the complement of $\{b_1,\ldots,b_m\}$ in $\{k+1,\ldots,n\}$. For $1\leq i\leq d$, let $Y_i = \Span{e_1,\ldots,\widehat{e_{w(p_i)}},\ldots,e_n} = \Span{e_1,\ldots,e_{w(p_i)-1},e_{w(p_i)+1},\ldots,e_n}$. Since $w(p_i) = w'(p_i)$ and $k<p_i$, we see that $Y_i = F_{n-w(p_i)} + F'_{w(p_i)-1}$, so part (4) of Lemma~\ref{lemma:expanding} shows $E_k \subset Y_i$. Thus $$ E_k \in \bigcap_{i=1}^d Y_i \ = \ \Span{e_{w(j)}\,|\, j<k\mbox{ or }j=b_i} \ = \ Y. $$ Since $w(p_i) = w'(p_i)$ for $1\leq i \leq d$, we have $E_{p_i} = \Span{E_{p_i-1},e_{w(p_i)}}$, by part (3) of Lemma~\ref{lemma:expanding}. So if ${E_{\DOT}\,} = \Span{f_1,\ldots,f_n}$, we may assume that $f_{p_i} = e_{w(p_i)} \in F_{n+1-w(p_i)} \cap F'_{w'(p_i)}$ for $1\leq i \leq d$, completing the proof. \QED To prove Lemma~\ref{lemma:geometry_statementII}, we begin by describing an intersection in a Grassmannian. Recall that $\Omega_\lambda{F\!_{\DOT}\,} = \{H\in G_kV\,|\, \dim H\cap F_{k-j+\lambda_j} \geq j \mbox{ for } 1\leq j\leq k\}$. \begin{lemma}\label{lemma:grassmannian} Suppose that $L_1,\ldots,L_k,M \subset V$ with $V = M \bigoplus L_1\bigoplus\cdots\bigoplus L_k$. Let $r_j= \dim L_j -1$ and $m = r_1 + \cdots + r_k$. Then there are opposite flags ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ and partitions $\lambda\supset \mu$ with $\lambda_j - \mu_j = r_j$ and $\lambda/\mu$ a skew row of length $m$ such that in $G_kV$, $$ \Omega_\mu{F\!_{\DOT}\,} \bigcap \Omega_{\lambda^c}{{F\!_{\DOT}}'\,} = \{ H\in G_kV\,|\, \dim H\bigcap L_j = 1 \mbox{ for } 1\leq j \leq k\}. $$ \end{lemma} \noindent{\bf Proof:} Let $\mu_k =0$ and $\mu_j = r_{j+1}+\cdots+r_k$ for $1\leq j <k$ and $\lambda_j =r_j+\mu_j$ for $1\leq j\leq k$. Choose a basis $e_1,\ldots,e_n$ for $V$ such that \begin{eqnarray*} L_j & =& \Span{e_{k+1-j+\mu_j},e_{k+2-j+\mu_j},\ldots,e_{k+1+r_j-j+\mu_j} =e_{k+1-j+\lambda_j}}\\ M &=& \Span{e_{m+k+1},\ldots,e_n} \end{eqnarray*} Let ${F\!_{\DOT}\,} = \Span{e_n\ldots,e_1}$ and ${{F\!_{\DOT}}'\,}=\Span{e_1,\ldots,e_n}$. Then \begin{center} $F_{n-k+j-\mu_j}\quad =\quad M\bigoplus L_1 \bigoplus \cdots \bigoplus L_j \quad\quad$\\ $F'_{n-k+(k+1-j)-\lambda^c_{k+1-j}} \quad = \quad F'_{k+1-j+\lambda_j} \quad =\quad L_j\bigoplus \cdots \bigoplus L_k$. \end{center} If $H\in \Omega_\mu{F\!_{\DOT}\,} \bigcap \Omega_{\lambda^c}{{F\!_{\DOT}}'\,}$, then $\dim H\bigcap F_{n-k+j-\mu_j} \geq j$ for $1\leq j\leq k$ and $$ \dim H\bigcap F'_{n-k+(k+1-j)-\lambda^c_{k+1-j}}\geq k+1-j, $$ for $1\leq j\leq k$. Thus for $1\leq j \leq k$, $$ \dim H\bigcap F_{n-k+j-\mu_j} \bigcap F'_{n-k+(k+1-j)-\lambda^c_{k+1-j}} \geq 1. $$ But $F_{n-k+j-\mu_j} \bigcap F'_{n-k+(k+1-j)-\lambda^c_{k+1-j}} = L_j$, so $\dim H\bigcap L_j\geq 1$ for $1\leq j \leq k$. Since $L_j\bigcap L_i = \{0\}$ if $j\neq i$, we see that $\dim H\bigcap L_j =1$. Thus $$ \Omega_\mu{F\!_{\DOT}\,} \bigcap \Omega_{\lambda^c}{{F\!_{\DOT}}'\,} \subset \{ H\in G_kV\,|\, \dim H\bigcap L_j = 1 \mbox{ for } 1\leq j \leq k\}. $$ We show these varieties have the same dimension, establishing their equality: Since $|\lambda| = |\mu| +m$, and ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ are opposite flags, $\Omega_\mu{F\!_{\DOT}\,} \bigcap \Omega_{\lambda^c}{{F\!_{\DOT}}'\,}$ has dimension $m$. But the map $H \mapsto (H\bigcap L_1,\ldots,H\bigcap L_k)$ defines an isomorphism between $\{ H\in G_kV\,|\, \dim H\bigcap L_j = 1 \mbox{ for } 1\leq j \leq k\}$ and $\Bbb{P}L_1 \times \cdots \times \Bbb{P}L_k$, which has dimension $\sum_j (\dim L_j-1) = m$. Here, $\Bbb{P}L_j$ is the projective space of one dimensional subspaces of $L_j$. \QED We relate this to intersections of Schubert varieties in the flag manifold. \begin{lemma}\label{lemma:intersection_calculation} Suppose that $w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'$ and $w' = wt_{a_1\,b_1}\cdots t_{a_m\,b_m}$ with $a_i\leq k<b_i$ and $\ell(wt_{a_1\,b_1}\cdots t_{a_i\,b_i}) = \ell(w)+i$. Let ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ be opposite flags in $V$ and let $\Span{e_i} = F_{n+1-i}\bigcap F'_{i}$. Define \begin{eqnarray*} L_j & =& \Span{e_j, e_{w(b_i)}\,|\, a_i = j}\\ M & =&\Span{e_{w(p)}\,|\, k<p \mbox{ and } w(p) = w'(p)}. \end{eqnarray*} Then \begin{enumerate} \item $\dim L_j = 1 +\#\{i\,|\, a_i = j\}$ and $ V = M \bigoplus L_1\bigoplus \cdots \bigoplus L_k$. \item If ${E_{\DOT}\,} \in X_w{F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$, then $\dim E_k \bigcap L_j = 1$ for $1\leq j\leq k$. \item Let $\pi$ be the map induced by ${E_{\DOT}\,} \mapsto E_k$. Then $$ \pi : X_w{F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,} \rightarrow \{H\in G_kV\,|\, \dim H\bigcap L_j = 1 \mbox{ for } 1\leq j\leq k\} $$ is surjective and of degree 1. \end{enumerate} \end{lemma} \noindent{\bf Proof:} Part (1) is immediate. For (2) and (3), note that both $\{H\in G_kV\,|\, \dim H\bigcap L_j = 1 \mbox{ for } 1\leq j\leq k\}$ and $X_w{F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$ are irreducible and have dimension $m$. We exhibit an $m$ dimensional subset of each over which $\pi$ is an isomorphism. Let $\alpha = (\alpha_1,\ldots,\alpha_m) \in (\Bbb{C}^\times\!)^m$ be an $m$-tuple of nonzero complex numbers. We define a basis $f_1,\ldots,f_n$ of $V$ depending upon $\alpha$ as follows. $$ f_j = \left\{ \begin{array}{ll} e_{w(j)} + {\displaystyle \sum_{i : a_i=j} \alpha_i e_{w(b_i)}} & \mbox{ if } j\leq k \\ e_{w(j)} & \mbox{ if } j>k \mbox{ and } j\not\in\{b_1,\ldots,b_m\}\\ \rule{0pt}{22pt}{\displaystyle \sum_{\shortstack{\scriptsize $i: a_i = a_q$\\ \scriptsize $w(b_i)\geq w(j)$}} \alpha_i e_{w(b_i)}} & \mbox{ if } j = b_q > k \end{array} \right. . $$ Let $i_1<\cdots < i_s$ be those integers $i_l$ with $a_{i_l} = j$. Since $t_{a_i\,b_i}$ lengthens the permutation $wt_{a_1\,b_1}\cdots t_{a_{i-1}\,b_{i-1}}$, we see that $$ \begin{array}{ccccccc} w(j) & < & w(b_{i_1}) & < & \cdots & < & w(b_{i_s}) \\ \parallel & & \parallel & & & & \parallel \\ w'(b_{i_1}) & < & w'(b_{i_2}) & < & \cdots & < & w'(j) \end{array} $$ Thus the first term in $f_j$ is proportional to $e_{w(j)}$. Hence $f_j \in F_{n+1-w(j)} - F_{n-w(j)}$, and so $f_1,\ldots,f_n$ is a basis of $V$ and the flag ${E_{\DOT}\,}(\alpha) = \Span{f_1,\ldots,f_n}$ is in $X_w{F\!_{\DOT}\,}$. Note that $f'_1,\ldots,f'_n$ is also a basis for ${E_{\DOT}\,}(\alpha)$, where $f'_j$ is given by $$ f'_j = \left\{ \begin{array}{ll} f_j & \mbox{ if } j\leq k \\ f_j& \mbox{ if } j>k \mbox{ and } j\not\in\{b_1,\ldots,b_m\}\\ f_{a_q} - f_j & \mbox{ if } j = b_q > k \end{array} \right. . $$ Here, the last term in each $f'_j$ is proportional to $e_{w'(j)}$, so $f'_j \in F'_{w'(j)} = F'_{n+1-w_0w'(j)}$, showing that ${E_{\DOT}\,}(\alpha) \in X_{w_0w'}{{F\!_{\DOT}}'\,}$. Since $f_j \in L_j$ for $1\leq j\leq k$, we have $\dim {E_{\DOT}\,}(\alpha) \bigcap L_j = 1$ for $1\leq j\leq k$. As $\{{E_{\DOT}\,}(\alpha)\,|\, \alpha \in (\Bbb{C}^\times\!)^m\}$ is a subset of $ X_w{F\!_{\DOT}\,}\bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$ of dimension $m$, it is dense. Thus if ${E_{\DOT}\,} \in X_w{F\!_{\DOT}\,}\bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$, then $\dim E_k \bigcap L_j = 1$ for $1\leq j\leq k$. The set $\{({E_{\DOT}\,}(\alpha))_k\,|\, \alpha \in (\Bbb{C}^\times\!)^m\}$ is a dense subset of $$ \{H\in G_kV\,|\, \dim H\bigcap L_j = 1 \mbox{ for } 1\leq j\leq k\} \ \simeq \ \Bbb{P}L_1\times\cdots\times\Bbb{P}L_k. $$ Since $\pi$ is an isomorphism of this set with $\{{E_{\DOT}\,}(\alpha)\,|\, \alpha \in (\Bbb{C}^\times\!)^m\}$, the map $$ \pi: X_w{F\!_{\DOT}\,}\bigcap X_{w_0w'}{{F\!_{\DOT}}'\,} \rightarrow \{H\in G_kV\,|\, \dim H\bigcap L_j = 1 \mbox{ for } 1\leq j\leq k\} $$ is surjective of degree 1, proving the lemma. \QED We note that Lemma~\ref{lemma:geometry_statementII} is an immediate consequence of Lemmas~\ref{lemma:grassmannian} and~\ref{lemma:intersection_calculation}(3). \section{Examples} In this section we describe two examples, which should serve to illustrate the results of Section 5. This manuscript differs from the version we are submitting for publication only by the inclusion of this section, and its mention in the Introduction. \bigskip Fix a basis $e_1,\ldots,e_7$ for $\Bbb{C}\,^7$. This gives coordinates for vectors in $\Bbb{C}\,^7$, where $(v_1,\ldots,v_7)$ corresponds to $v_1e_1{+}\cdots{+}v_7e_7$. Define the opposite flags ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ by $$ {F\!_{\DOT}\,} = \langle e_7,e_6,e_5,e_4,e_3,e_2,e_1\rangle \ \ \mbox{and} \ \ {{F\!_{\DOT}}'\,} = \langle e_1,e_2,e_3,e_4,e_5,e_6,e_7\rangle. $$ For example, $F_3 = \langle e_7,e_6,e_5 \rangle$ and $F'_4 = \langle e_1,e_2,e_3,e_4 \rangle$. Let $w= 5412763$, $w' = 6524713$ and $w'' = 7431652$ be permutations in $S_7$. (We denote permutations by the sequence of their values.) Their lengths are 10, 14, and 14, respectively, and $w<_4 w'$ and $w<_3 w''$. We seek to describe the intersections $$ X_w{F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,} \ \ \ \ \mbox{and} \ \ \ \ X_w{F\!_{\DOT}\,} \bigcap X_{w_0w''}{{F\!_{\DOT}}'\,}. $$ Rather than describe each in full, we describe a dense subset of each which is isomorphic to the torus, $(\Bbb{C}^\times\!)^4$. This suffices for our purposes. Recall that the Schubert cell $X^\circ_w{F\!_{\DOT}\,}$ is defined to be $$ X^{\circ}_w {F\!_{\DOT}\,} = \{ {E_{\DOT}\,}=\Span{f_1,\ldots,f_7}\,|\, f_i \in F_{8-w(i)}-F_{7-w(i)}, \,1\leq i\leq 7\}. $$ Using the given coordinates of $\Bbb{C}\,^7$, we may write a typical element of $X^\circ_w{F\!_{\DOT}\,}$ in a unique manner. For each $f_i \in F_{8-w(i)}-F_{7-w(i)}$, the coordinate 7-tuple for $f_i$ has zeroes in the places $1,\ldots,w(i)-1$ and a nonzero coordinate in its $w(i)$th place, which we assume to be 1. We may also assume that the $w(j)$th coordinate of $f_i$ is zero for those $j<i$ with $w(j)> w(i)$, by subtracting a suitable multiple of $f_j$. Writing the coordinates of $f_1,\ldots,f_7$ as rows of an array, we conclude that a typical flag in $X^{\circ}_w {F\!_{\DOT}\,}$ has a unique representation of the following form: $$ \begin{array}{ccccccc} \cdot &\cdot&\cdot&\cdot& 1 & * & * \\ \cdot &\cdot&\cdot& 1 &\cdot& * & * \\ 1 & * & * &\cdot&\cdot& * & * \\ \cdot & 1 & * &\cdot&\cdot& * & * \\ \cdot &\cdot&\cdot&\cdot&\cdot&\cdot& 1 \\ \cdot &\cdot&\cdot&\cdot&\cdot& 1 &\cdot\\ \cdot &\cdot& 1 &\cdot&\cdot&\cdot& \cdot \end{array} $$ Here, the $i$th column contains the coefficients of $e_i$, the $\cdot$'s represent 0, and the $*$'s indicate some complex numbers, uniquely determined by the flag. Likewise, flags in $X^{\circ}_{w_0w'} {{F\!_{\DOT}}'\,}$ and $X^{\circ}_{w_0w''} {{F\!_{\DOT}}'\,}$ have unique bases of the forms: $$ \begin{array}{cccccccc} & * & * & * & * & * & 1 &\cdot\\ & * & * & * & * & 1 &\cdot&\cdot\\ & * & 1 &\cdot&\cdot&\cdot&\cdot&\cdot\\ & * &\cdot& * & 1 &\cdot&\cdot&\cdot\\ & * &\cdot& * &\cdot&\cdot&\cdot& 1 \\ & 1 &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ & \cdot&\cdot& 1 &\cdot&\cdot&\cdot& \cdot \end{array} \hspace{1in} \begin{array}{cccccccc} & * & * & * & * & * & * & 1 \\ & * & * & * & 1 &\cdot&\cdot&\cdot\\ & * & * & 1 &\cdot&\cdot&\cdot&\cdot\\ & 1 &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ & \cdot& * &\cdot&\cdot& * & 1 &\cdot\\ & \cdot& * &\cdot&\cdot& 1 &\cdot&\cdot\\ & \cdot& 1 &\cdot&\cdot&\cdot&\cdot& \cdot \end{array} $$ Let $\alpha,\beta,\gamma$ and $\delta$ be four nonzero complex numbers. Define bases $f_1,f_2,\ldots,f_7$ and $g_1,g_2,\ldots,g_7$ by the following arrays of coordinates. $$ \begin{array}{ccccccccc} f_1 &= &\cdot& \cdot &\cdot& \cdot & 1 & \alpha &\cdot\\ f_2 &= &\cdot& \cdot &\cdot& 1 & \beta & \cdot &\cdot\\ f_3 &= & 1 & \gamma &\cdot& \cdot & \cdot & \cdot &\cdot\\ f_4 &= &\cdot& 1 &\cdot& \delta & \cdot & \cdot &\cdot\\ f_5 &= &\cdot& \cdot &\cdot& \cdot & \cdot & \cdot & 1 \\ f_6 &= &\cdot& \cdot &\cdot& \cdot & \cdot & \alpha &\cdot\\ f_7 &= &\cdot& \cdot & 1 & \cdot & \cdot & \cdot &\cdot \end{array} \hspace{1in} \begin{array}{ccccccccc} g_1 & = &\cdot& \cdot & \cdot &\cdot& 1 & \alpha & \beta \\ g_2 & = &\cdot& \cdot & \cdot & 1 &\cdot& \cdot & \cdot \\ g_3 & = & 1 & \gamma & \delta &\cdot&\cdot& \cdot & \cdot \\ g_4 & = &\cdot& \gamma & \delta &\cdot&\cdot& \cdot & \cdot \\ g_5 & = &\cdot& \cdot & \cdot &\cdot&\cdot& \cdot & \beta \\ g_6 & = &\cdot& \cdot & \cdot &\cdot&\cdot& \alpha & \beta \\ g_7 & = &\cdot& \cdot & \delta &\cdot&\cdot& \cdot & \cdot \end{array} $$ Let ${E_{\DOT}\,} = \Span{f_1,f_2,\ldots,f_7}$ and ${E_{\DOT}\!'\,} = \Span{g_1,g_2,\ldots,g_7}$. Considering the left-most nonzero entry in each row, we see that both ${E_{\DOT}\,}$ and ${E_{\DOT}\!'\,}$ are in $X^\circ_w{F\!_{\DOT}\,}$. To see that ${E_{\DOT}\,} \in X^\circ_{w_0w'}{{F\!_{\DOT}}'\,}$ and ${E_{\DOT}\!'\,} \in X^\circ_{w_0w''}{{F\!_{\DOT}}'\,}$, note that we could choose $$ \begin{array}{ccccccccc} f_6' & = & 1 & \cdot &\cdot&\cdot& \cdot & \cdot&\cdot \end{array} \hspace{1in} \begin{array}{ccccccccc} g_4' & = & 1 & \cdot &\cdot&\cdot&\cdot& \cdot &\cdot\\ g_5' & = &\cdot& \cdot &\cdot&\cdot& 1 & \alpha &\cdot\\ g_6' & = &\cdot& \cdot &\cdot&\cdot& 1 & \cdot &\cdot\\ g_7' & = & 1 & \gamma &\cdot&\cdot&\cdot& \cdot &\cdot \end{array} $$ Replacing the unprimed vectors by the corresponding primed ones gives alternate bases for ${E_{\DOT}\,}$ and ${E_{\DOT}\!'\,}$. This shows ${E_{\DOT}\,} \in X^\circ_{w_0w'}{{F\!_{\DOT}}'\,}$ and ${E_{\DOT}\!'\,} \in X^\circ_{w_0w''}{{F\!_{\DOT}}'\,}$. We use this computation to illustrate Lemmas~\ref{lemma:geometry_statementI} and~\ref{lemma:geometry_statementII}. \begin{enumerate} \item[I.] First note that for ${E_{\DOT}\,} = \Span{f_1,f_2,f_3,f_4,f_5,f_6,f_7}$ as above, \begin{eqnarray*} E_3 & \subset & \Span{e_1,e_2,e_5,e_5,e_6}\\ & = & \Span{ e_{w(j)}\,|\, j\leq k\mbox{ or } w(j)\neq w'(j)}\\ & = & Y, \end{eqnarray*} the subspace of Lemma~\ref{lemma:geometry_statementI}. Since this holds for all ${E_{\DOT}\,}$ in a dense subset of $X_w{F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$, it holds for all ${E_{\DOT}\,}$ in that intersection. \item[II.] Recall that $w=5412763$ and note that $7431652 = w''= w\cdot t_{34}\cdot t_{16}\cdot t_{37}\cdot t_{15}$, so $w \stackrel{r[3,4]}{\relbar\joinrel\longrightarrow} w''$, and we are in the situation of Lemma~\ref{lemma:geometry_statementII}. Let $\mu = (2,2,0)$ and $\lambda = (4,2,2)$ be partitions. Then $\lambda^c = (2,2,0)$, and if ${E_{\DOT}\!'\,} = {E_{\DOT}\!'\,}(\alpha,\beta,\gamma,\delta)$ is a flag in the above form, then $$ E_3'(\alpha,\beta,\gamma,\delta) \in \Omega_{\mu}{F\!_{\DOT}\,} \bigcap \Omega_{\lambda^c} {{F\!_{\DOT}}'\,}, $$ since $$ \begin{array}{ccccl} f_1 &\in & F_3 & = & F_{7-3+1-\mu_1} \bigcap F'_{7-3+3-\lambda^c_3}\\ f_2 &\in & \Span{e_4} & =& F_{7-3+1-\mu_2} \bigcap F'_{7-3+3-\lambda^c_2}\\ f_3 &\in & F'_3 & = & F_{7-3+1-\mu_3} \bigcap F'_{7-3+3-\lambda^c_1}. \end{array} $$ Furthermore, the map $\pi: {E_{\DOT}\!'\,} \mapsto E_3'$ is injective for those ${E_{\DOT}\!'\,}(\alpha,\beta,\gamma,\delta)$ given above. Since that set is dense in $X_w{F\!_{\DOT}\,} \bigcap X_{w_0w''}{{F\!_{\DOT}}'\,}$, and the set of $E_3'(\alpha,\beta,\gamma,\delta)$ is dense in $\Omega_{\mu}{F\!_{\DOT}\,} \bigcap \Omega_{\lambda^c} {{F\!_{\DOT}}'\,}$, it follows that $$ \pi : X_w{F\!_{\DOT}\,} \bigcap X_{w_0w''}{{F\!_{\DOT}}'\,} \rightarrow \Omega_{\mu}{F\!_{\DOT}\,} \bigcap \Omega_{\lambda^c} {{F\!_{\DOT}}'\,} $$ is surjective and of degree 1. \end{enumerate} \smallskip Note that the description of $X_w{F\!_{\DOT}\,} \bigcap X_{w_0w''}{{F\!_{\DOT}}'\,}$ in II is consistent with that given for general $w\stackrel{r[k,m]}{\relbar\joinrel\llra} w''$ in the proof of Lemma~\ref{lemma:intersection_calculation}, part (2). This explicit description is the key to the understanding we gained while trying to establish Theorem~\ref{thm:main} \smallskip Also note that $w' = w \cdot t_{16}\cdot t_{26}\cdot t_{46}\cdot t_{36}$, thus $w \stackrel{c[4,4]}{\relbar\joinrel\llra} w'$. In I above, we give an explicit description of the intersection $X_w{F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$. This may be generalized to give a similar description whenever $w \stackrel{c[k,m]}{\relbar\joinrel\llra} w'$, and may be used to establish Theorem~\ref{thm:main} in much the same manner as we used the explicit description of intersections when $w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'$.

9505

alg-geom/9505021

en

https://arxiv.org/abs/alg-geom/9505021

alg-geom/9505021

Fabrizio Catanese and Klaus Hulek

Rational surfaces in P^4 containing a plane curve

25 pages, LaTeX2e

The families of smooth rational surfaces in $\PP^4$ have been classified in degree $\le 10$. All known rational surfaces in $\PP^4$ can be represented as blow-ups of the plane $\PP^2$. The fine classification of these surfaces consists of giving explicit open and closed conditions which determine the configurations of points corresponding to all surfaces in a given family. Using a restriction argument originally due independently to Alexander and Bauer we achieve the fine classification in two cases, namely non-special rational surfaces of degree 9 and special rational surfaces of degree 8. The first case completes the fine classification of all non-special rational surfaces. In the second case we obtain a description of the moduli space as the quotient of a rational variety by the symmetric group $S_5$. We also discuss in how far this method can be used to study other rational surfaces in $\PP^4$.

alg-geom

\section{Introduction}\label{sectionI} The families of smooth rational surfaces in ${\Bbb{P}}^4$ have been classified in degree $\leq10$ ( \cite{A1}, \cite{I1}, \cite{I2}, \cite{O1}, \cite{O2}, \cite{R1}, \cite{R2}, \cite{PR}). In this thesis Popescu \cite{P} constructed further examples of rational surfaces in degree $11$. The existence of these surfaces has been proved in various ways, using linear systems, vector bundles and sheaves or liaison arguments. All known rational surfaces can be represented as a blowing-up of ${\Bbb{P}}^2$. Although it would seem the most natural approach to prove directly that a given linear system is very ample, this turns out to be a very subtle problem in some cases, in particular when the surface $S$ in ${\Bbb{P}}^4$ is special (i.e.~$h^1({\cal O}_S(H))\neq0$). On the other hand, being able to handle the linear system often means that one knows the geometry of the surface very well. The starting point of our paper is the observation that every known rational surface in ${\Bbb{P}}^4$ contains a plane curve $C$. Using the hyperplanes through $C$ one can construct a residual linear system $|D|$. I.e., we can write $H\equiv C+D$ with $\dim |D|\geq1$. This situation was studied in particular by Alexander \cite{A1}, \cite{A2} and Bauer \cite{B}: if $|H|$ restricts to complete linear systems on $C$ and $D'$ where $D'$ varies in a $1$-dimensional linear subsystem of $|D|$, then $H$ is very ample on $S$ if and only if it is very ample on $C$ and the curves $D'$ (cf.~Theorem~(\ref{theo21})). In this way one can reduce the question of very ampleness of $H$ to the study of linear systems on curves. In \cite{CFHR} the following curve embedding theorem was proved which we shall state here only for the (special) case of curves contained in a smooth surface. \begin{theorem}\label{theo11} A divisor $H$ is very ample on $C$ if for every subcurve $Y$ of $C$ of arithmetic genus $p(Y)$ \noindent $\on{(i)}$ $H.Y\geq 2p(Y)+1$ or \noindent $\on{(ii)}$ $H.Y\geq 2p(Y)$ and there is no $2$-cycle $\xi$ of $Y$ such that $I_\xi{\cal O}_Y\cong \omega_Y(-H)$. More generally \noindent $\on{(iii)}$ If $\xi$ is an $r$-cycle of $C$, then $H^0(C,{\cal O}_C(H))$ surjects onto $H^0({\cal O}_C(H)\otimes{\cal O}_\xi)$ unless there is a subcurve $Y$ of $C$ and a morphism $\phi:I_\xi{\cal O}_Y\to \omega_Y(-H)$ which is "good" (i.e.~$\phi$ is injective with a cokernel of finite length) and which is not induced by a section of $H^0(Y,\omega_Y(-H))$. \end{theorem} The method described above was used in \cite{CF} to characterize exactly all configurations of points in ${\Bbb{P}}^2$ which define non-special rational surfaces of degree $\leq8$. In these cases $H.D\geq 2p(D)+1$. This left the case open of one non-special surface, namely the unique non-special surface of degree $9$. In this case one has a decomposition $H\equiv C+D$ where $C$ is a plane cubic, and $|D|$ is a pencil of curves of genus $p(D)=3$ and $H.D=6$. Section~\ref{sectionII} is devoted to this surface. In Theorem~(\ref{theo22}) we classify all configurations of points in the plane which lead to non-special surfaces of degree $9$ in ${\Bbb{P}}^4$. This completes the fine classification of non-special surfaces. In section~\ref{sectionIII} we show that this method can also be applied to study special surfaces. We treat the (unique) special surface of degree $8$. In this case there exists a decomposition $H\equiv C+D$ where $C$ is a conic and $|D|$ is a pencil of curves of genus $4$ with $H.D=6$. It turns out that for the general element $D'$ of $|D|$ (but not necessarily for all elements) $H$ is the canonical divisor on $D'$. In Theorem~(\ref{IIItheo14}) we give a characterization of these configurations of points which define smooth special surfaces of degree $8$ in ${\Bbb{P}}^4$. We then use this result to give an existence proof (in fact we construct the general element in the family) of these surfaces using only the linear system $|H|$ (Theorem~(\ref{IIItheo17})), and in particular to describe the moduli space of the above surfaces modulo projective equivalence (Theorem~(\ref{IIItheo20})). Finally in section~\ref{sectionIV} we discuss some posibilities how this method can be used to study other rational surfaces in ${\Bbb{P}}^4$, suggesting some explicit decompositions $H\equiv C+D$ of the hyperplane class as the sum of divisors. \medskip \noindent {\bf Acknowledgements.} The authors were partially supported by the DFG-Schwerpunktprogramm "Komplexe Mannigfaltigkeiten" under contract number Hu337/4-3, the EU HCM project AGE (Algebraic Geometry in Europe) contract number ERBCHRXCT 940557 and MURST $40\%$. The second author is also grateful to the Tata-Institute of Fundamental Research for their hospitality. The final version was written while the first author was "{\em Professore distaccato}" at the {\em Accademia dei Lincei}. \section{The non-special rational surface of degree $9$}\label{sectionII} In this section we want to give an application of Theorem (\ref{theo11}) to non-special rational surfaces. These surfaces have been classified by Alexander \cite{A1}. Catanese and Franciosi treated all non-special rational surfaces of degree $\leq8$ by studying suitable decompositions $H=C+D$ of the embedding linear systems. The crucial observation here is the following result, originally due to J.~Alexander and I.~Bauer \cite{B}. \begin{theorem}[Alexander-Bauer]\label{theo21} Let $X$ be a smooth projective variety and let $C,D$ be effective divisors with $\dim|D|\geq1$. Let $H$ be the divisor $H\equiv C+D$. If $\big|H\big||_C$ is very ample and for all $D'$ in a $1$-dimensional subsystem of $|D|$, $\big|H\big||_{D'}$ is very ample, then $|H|$ is very ample on $X$. \end{theorem} By Alexander's list there is only one non-special rational surface of degree bigger than $8$. This surface is a ${\Bbb{P}}^2$ blown up in 10 points $x_1,\ldots,x_{10}$ embedded by the linear system $|H|=|13L-4 \sum_{i=1}^{10}x_i|$. Alexander showed that for general position of the points $x_i$ the linear system $|H|$ embeds $S={\tilde{{\Bbb{P}}}}^2(x_1,\ldots,x_{10})$ into ${\Bbb{P}}^4$. Clearly the degree of $S$ is 9. Here we show that using Theorem (\ref{theo11}) one can also apply the decomposition method to this surface. In fact we obtain necessary and sufficient conditions for the position of the points $x_i$ for $|H|$ to be very ample. Our result is the following \begin{theorem}\label{theo22} The linear system $|H|= |13L-4\sum x_i|$ embeds the surface $S ={\tilde {\Bbb{P}}}^2(x_1,\ldots, x_{10})$ into ${\Bbb{P}}^4$ if and only if \noindent $\on{(0)}$ no $x_i$ is infinitely near, \noindent $\on{(1)}$ $|L-\sum\limits_{i\in\triangle}x_i|=\emptyset$ for $|\triangle|\geq4$, \noindent $\on{(2)}$ $|2L-\sum\limits_{i\in\triangle'}x_i|=\emptyset$ for $|\triangle'|\geq7$, \noindent $\on{(3)}$ $|3L-\sum\limits_ix_i|=\emptyset$, \noindent $\on{(3)}_{ij}'$ $|3L-\sum\limits_{k\neq i,j}x_k-2x_i|=\emptyset$ for all pairs $(i,j)$, \noindent $\on{(4)}_{ijk}$ $|4L-2x_i-2x_j-2x_k- \sum\limits_{l\neq i,j,k}x_l |=\emptyset$ for all triples $(i,j,k)$, \noindent $\on{(6)}_i$ $|6L-x_i-2\sum\limits_{j\neq i}x_j|=\emptyset$, \noindent $\on{(10)}_1$ If $D=10L-4x_1-3\sum\limits_{i\geq2}x_i$, then $\dim|D|=1$. \end{theorem} \begin{uremarks} (i) Clearly conditions (0) to (6) are open conditions. The expected dimension of $|D|$ is 1, hence this condition is also open. \noindent (ii) The last condition is asymmetrical. If $|H|$ is very ample condition $(10)_i$ is necessarily fulfilled for all $i$. On the other hand, our theorem shows that in order to prove very ampleness for $|H|$ it suffices to check only one of the conditions $(10)_i$. \end{uremarks} \begin{Proof} We shall first show that the conditions stated are necessary. Clearly (0) follows since $H.(x_i-x_j)=0$. Similarly the ampleness of $H$ immediately implies conditions (1) to (4). Assume the linear system $|6L-x_i-2\sum\limits_{j\neq i}x_j|$ contains some element $A$. Then $H.A=2$, and $p(A)=1$ which contradicts very ampleness of $H$. For (10) we consider $C\equiv H-D\equiv 3L-\sum\limits_{i\geq2}x_i$. Clearly $|C|$ is non empty. For $C'\in|C|$ we consider the exact sequence \setcounter{equation}{10} \begin{equation}\label{gl11} 0\longrightarrow{\cal O}_S(D)\longrightarrow{\cal O}_S(H)\longrightarrow {\cal O}_{C'}(H)\longrightarrow 0. \end{equation} If $h^0({\cal O}_S(D))\geq3$, then either $h^0({\cal O}_S(H)\geq6$ and $|H|$ does not embed $S$ into ${\Bbb{P}}^4$ or $|H|$ maps $C'$ to a line. But since $p(C)=1$ this means that $|H|$ cannot be very ample. Now assume that conditions (0) to $(10)_1$ hold. We shall first show \begin{align*} h^1({\cal O}_S(D))&=0\tag{I} \\ h^1({\cal O}_S(C))&=0\tag{II}\\ h^0({\cal O}_S(H))&=5 \tag{III} \end{align*} Ad (I): By condition $(10)_1$ we have $h^0({\cal O}_S(D))=2$. Clearly $h^2({\cal O}_S(D))=h^0({\cal O}_S(K-D))=0$. Hence the claim follows from Riemann-Roch, since $\chi({\cal O}_S(D))=2$. \noindent Ad (II): We consider $-K\equiv 3L-\sum\limits_ix_i\equiv C-x_1$. By condition (3) $h^0({\cal O}_S(-K))=0$. Clearly also $h^2({\cal O}_S(-K))=h^0({\cal O}_S(K))=0$. Hence by Riemann-Roch $h^1({\cal O}_S(-K))=-\chi ({\cal O}_S(-K))=0$. Now consider the exact sequence \begin{equation}\label{gl12} 0\longrightarrow{\cal O}_S(-K)\longrightarrow{\cal O}_S(C)\longrightarrow {\cal O}_S(C)|_{x_1}={\cal O}_{x_1}\longrightarrow 0. \end{equation} This shows $h^1({\cal O}_S(C))=0$. Note that this also implies (by Riemann-Roch) that $h^0({\cal O}_S(C))=1$, i.e.~the curve $C'$ is uniquely determined. \noindent Ad (III): In view of (I) and sequence (\ref{gl11}) it suffices to show that $h^0({\cal O}_{C'}(H))=3$. By Riemann-Roch on $C'$ this is equivalent to $h^1({\cal O}_{C'}(H))=0$. Since $K_{C'}$ is trivial this in term is equivalent to $h^0({\cal O}_{C'}(-H))=0$. By condition $(3)'$ the curve $C'$ contains no exceptional divisor. As a plane curve $C'$ can be irreducible or it can decompose into a conic and a line or three lines. In view of conditions (1) and (2), however, $C'$ cannot have multiple components and, moreover $H$ has positive degree on every component. This proves $h^0({\cal O}_{C'}(-H))=0$ and hence the claim. This shows that $|H|$ maps $S$ to ${\Bbb{P}}^4$ and that, moreover, $|H|$ restricts to complete linear systems on $C'$ and all curves $D'\in|D|$. We shall now show \begin{align*} \begin{split} &\ \ \ \,\text{ For every subcurve }A\leq C'\text{ we have } H.A\geq2p(A)+1 \end{split}\tag{IV} \end{align*} \begin{align*} \begin{split} &\text{For every proper subcurve }B'\subset D'\text{ of an element}\\ &D'\in |D|\text{ we have }H.B'\geq2p(B')+1 \end{split}\tag{V(i)} \end{align*} \begin{align*} \begin{split} &H\text{ does not restrict to a "$(2+K)$"-divisor on }D',\\ &\text{i.e. }{\cal O}_{D'}(H-K_{D'})\text{ does not have a good section}\\ &\text{defining a degree } 2\text{-cycle}. \end{split}\tag{V(ii)} \end{align*} It then follows from (IV) and \cite[Theorem 3.1]{CF} that $|H|$ is very ample on $C'$. Because of (V(i)) and (V(ii)) it follows from Theorem (\ref{theo11}) that $|H|$ is very ample on every element $D'$ of $|D|$. It then follows from Theorem (\ref{theo21}) that $|H|$ is very ample. \noindent Ad (V(ii)): Let $H_{D'}$ be the restriction of $H$ to $D'$, and denote the canonical bundle of $D'$ by $K_{D'}$. It suffices to show that $h^0({\cal O}_{D'}(H_{D'}-K_{D'}))=0$. Now \begin{align*} H_{D'}-K_{D'}&= (H-K-D)|_{D'}\\ &= (C-K)|_{D'}\\ &= (2C-x_1)|_{D'}. \end{align*} There is an exact sequence \begin{equation}\label{gl13} 0\longrightarrow{\cal O}_S(2C-x_1-D)\longrightarrow{\cal O}_S(2C-x_1) \longrightarrow {\cal O}_{D'}(H_{D'}-K_{D'})\longrightarrow 0. \end{equation} Since $$ 2C-x_1\equiv 6L-x_1-2\sum_{i=2}^{10}x_i $$ it follows from condition $(6)_1$ that $h^0({\cal O}_S(2C-x_1))=0$. Clearly $h^0({\cal O}_S(2C-x_1-D))=0$. Now $$ 2C-x_1-D\equiv -4L+3x_1+\sum_{i=2}^{10}x_i $$ resp. $$ K-(2C-x_1-D)\equiv L-2x_1. $$ Hence $h^2({\cal O}_S(2C-x_1-D))=h^0({\cal O}_S(K-(2C-x_1-D))=0$. Since moreover $\chi({\cal O}_S(2C-x_1-D))=0$ it follows that $h^1({\cal O}_S(2C-x_1-D))=0$. The assertion follows now from sequence (\ref{gl13}). \noindent Ad (IV) and (V(i)): We have to show that for all curves $A$ with $A\leq C'$, resp.~$A<D'$, $D'\in |D|$ the following holds \begin{equation}\label{gl14} H.A\geq2p(A)+1. \end{equation} We first notice that it is enough to prove (\ref{gl14}) for divisors $A$ with $p(A)\geq0$. Assume in fact we know this and that $p(A)<0$. Then $A$ is necessarily reducible. For every irreducible component $A'$ of $A$ we have $p(A')\geq0$ and hence $H.A'>0$. This shows $H.A>0$ and hence (\ref{gl14}). Clearly (\ref{gl14}) also holds for the lines $x_i$. Hence we can assume that $A$ is of the form \begin{equation}\label{gl15} A\equiv aL-\sum_ib_ix_i\quad\text{with }1\leq a\leq 10. \end{equation} Note that \begin{eqnarray} 2p(A)&=&a(a-3)-\sum_ib_i(b_i-1)+2\label{gl16}\\ H.A&=&13a-4\sum_ib_i.\label{gl17} \end{eqnarray} We proceed in several steps \medskip \noindent {\bf Claim 1 } Let $A$ be as in (\ref{gl15}) with $1\leq a\leq 3$. Assume that $p(A)\geq0$. Then (\ref{gl14}) is fulfilled. \medskip \noindent {\em Proof of Claim 1 } After possibly relabelling the $x_i$ we can assume that $b_1\geq b_2\geq\ldots\geq b_{10}$. If $a=1$ or 2 then $b_1\leq1$ and $b_2\geq0$. Moreover $p(A)=0$. If $H.A\leq2p(A)$ we get immediately a contradiction to conditions (1) or (2). If $a=3$ then we have two cases. Either $b_1\leq1$, $b_2\geq0$ as above and $p(A)=1$. Then $H.A\leq2p(A)$ violates condition (3). Or $b_1=2$ or $b_{10}=-1$ and the other $b_i$ are 0 or 1. Then $H.A\leq2p(A)$ is only possible for $b_1=2$, but this would violate condition $(3)'$. \medskip \noindent {\bf Claim 2 } $H$ is ample on $C$ and $D$, i.e.~for every irreducible component $A$ of $C'$, resp.~$D'$, $D'\in|D|$ we have $H.A>0$. \medskip \noindent {\em Proof of Claim 2 } Assume the claim is false. Let $A$ be an irreducible component with $H.A\leq0$. Since $A$ is irreducible, $p(A)\geq0$. By (\ref{gl16}), (\ref{gl17}) this leads to the two inequalities \begin{eqnarray} 13a&\leq&4\sum b_i\label{gl18}\\ \sum b_i(b_i-1)&\leq&a(a-3)+2.\label{gl19} \end{eqnarray} Multiplying (\ref{gl19}) by $13^2$ and using (\ref{gl18}) we obtain \begin{equation}\label{gl20} 169\left(\sum b_i^2-\sum b_i\right)\leq 16\left(\sum b_i\right)^2- 156\sum b_i+338. \end{equation} Now \begin{equation}\label{gl21} \left(\sum b_i\right)^2=10\sum b_i^2-\sum_{i<j}(b_i-b_j)^2 \end{equation} and using this (\ref{gl20}) becomes \begin{equation}\label{gl22} \sum_i(9b_i^2-13b_i)+16\sum_{i<j}(b_i-b_j)^2\leq 338. \end{equation} The function $f(b)=9b^2-13b$ for integers $b$ is non positive only for $b=0$ or 1. It is minimal for $b=1$. Since $f(1)=-4$ we derive from (\ref{gl22}) \begin{equation}\label{gl23} 16\sum_{i<j}(b_i-b_j)^2\leq 378 \end{equation} resp. \begin{equation}\label{gl24} \sum_{i<j}|b_i-b_j|^2\leq 23. \end{equation} At this point it is useful to introduce the following integer valued function $$ \delta=\delta(A)=\max_{i<j}|b_i-b_j|. $$ We have to distinguish several cases: \noindent $\delta\geq3$: Assume there is a pair $(i,j)$ with $|b_i-b_j|\geq3$. Then for all $k\neq i,j$: $$ |b_i-b_k|^2+|b_j-b_k|^2\geq5. $$ Hence $$ \sum_{i<j}|b_i-b_j|^2\geq9+5\cdot 8=49 $$ contradicting (\ref{gl24}). \noindent $\delta=2$: After possibly relabelling the $x_i$ we can assume that $b_2=b_1+2$ and $b_1\leq b_k\leq b_2$ for $k\geq3$. Then $$ |b_k-b_1|^2+|b_k-b_2|^2=\left\{\begin{array}{cl} 2&\text{if }b_k=b_1+1\\ 4&\text{if }b_k=b_1\text{ or }b_k=b_2. \end{array}\right. $$ Let $t$ be the number of $b_k$ which are either equal to $b_1$ or $b_2$. Then \begin{eqnarray*} \sum_{i<j}|b_i-b_j|^2&\geq&4+4t+2(8-t)+t(8-t)\\ &=&20+t(10-t). \end{eqnarray*} It follows from (\ref{gl24}) that $t=0$. But then (\ref{gl22}) gives $$ \sum(9b_i^2-13b_i)\leq18. $$ Looking at the values of $f(b)=9b^2-13b$ one sees immediately that this is only possible for $b_1=-1$ or $b_1=0$. In the first case it follows from (\ref{gl18}) that $a<0$ which is absurd. In the second case we obtain $a\leq 3$ and hence we are done by Claim 1. \noindent $\delta\leq1$: Here we can assume $$ b_1=\ldots=b_k=m,\quad b_{k+1}=\ldots=b_{10}=m+1. $$ Since $f(b)\geq42$ for $b\geq3$ it follows immediately from (\ref{gl22}) that $m\leq2$. If $m\leq0$ then (\ref{gl18}) gives $a\leq3$ and we are done by Claim 1. It remains to consider the subcases $m=1$ or 2. \noindent $m=2$: Since $f(2)=10$ and $f(3)=42$ formula (\ref{gl22}) implies $$ 10k+42(10-k)+16k(10-k)\leq338. $$ One checks easily that this is only possible for $k=9$ or 10. In this case (\ref{gl18}) gives $a\leq6$. If $k=9$ then (\ref{gl18}) gives $22\leq a(a-3)$, i.e.~$a\geq7$, a contradiction. If $k=10$, then (\ref{gl18}) implies $18\leq a(a-3)$. This is only possible for $a=6$. But now the existence of $A$ would contradict condition (6). \noindent $m=1$: Since $f(1)=-4$ and $f(2)=10$ formula (\ref{gl22}) reads $$ -4k+10(10-k)+16k(10-k)\leq338 $$ or equivalently $$ k(73-8k)\leq119. $$ It is straightforward to check that this implies $k\leq2$ or $k\geq7$. If $k\leq2$ then $\sum b_i(b_i-1)\geq16$ and (\ref{gl19}) shows that $a\geq6$. On the other hand $\sum b_i\leq19$ and this contradicts (\ref{gl18}). Now assume $k\geq7$. Then $\sum b_i\leq13$. It follows from (\ref{gl18}) that either $a\leq3$ -- and this case is dealt with by Claim 1 -- or $a=4$ and $\sum b_i=13$. Then $k=7$ and the existence of $A$ contradicts condition (4). \medskip \noindent {\em End of proof } It follows immediately from Claim 1 that (\ref{gl14}) holds for subcurves $A\leq C'$. It remains to consider subcurves $A< D'$, $D'\in|D|$. Since $H$ is ample on $D$ we have $H.A>0$, hence it suffices to consider curves with $p(A)\geq1$. Also by ampleness of $H$ on $D$ it follows that \begin{equation}\label{gl25} 1\leq H.A\leq5 \end{equation} since $H.D=6$. Also note that, as an immediate consequence of (\ref{gl17}): \begin{equation}\label{gl26} a\equiv H.A\on{mod}4. \end{equation} Finally we remark the following \medskip \noindent {\bf Observation:} If $A<D$ is not one of the exceptional lines $x_i$, then $H.A\leq4$ implies $b_i\geq0$ for all $i$. Otherwise at most one $b_i=-1$ and all other $b_i\geq0$. This follows from the ampleness of $H$ on $x_i$, since $H.x_i=4$. {}From now on we set \begin{equation}\label{gl27} B:=D-A. \end{equation} By adjunction \begin{equation}\label{gl28} p(A)+p(B)=p(D)+1-A.B=4-A.B. \end{equation} We write $$ B\equiv bL-\sum c_ix_i. $$ We shall now proceed by discussing the possible values of the coefficient $a$ of $A$ in decreasing order. \noindent $a=10$: Then $B=\sum c_ix_i$, $c_i\geq0$ and since $H.B\leq5$ we must have $B=x_i$. Then $A.B=4$ or 5 and $p(A)\leq0$ by (\ref{gl28}). \noindent $a=9$: By (\ref{gl25}), (\ref{gl26}) we have to consider two cases \begin{align*} H.A=5&, H.B=1 \tag{$\alpha$}\\ H.A=1&, H.B=5.\tag{$\beta$} \end{align*} Using our above observation for $B$ in case $(\alpha)$ we find that $$ B\equiv L-x_i-x_j-x_k. $$ But now $A.B\geq2$ and hence $p(A)\leq1$. Hence $H.A=5\geq2p(A)+1$. Using condition (1) we have to consider the following cases for $(\beta)$: \begin{eqnarray*} B&\equiv&L-x_i-x_j\\ B&\equiv&L-x_i-x_j-x_k+x_l. \end{eqnarray*} In the first case $A.B\geq4$ and $p(B)=0$, hence $p(A)\leq0$. In the second case $A.B\geq5$ and $p(B)=-1$, hence again $p(A)\leq0$. \noindent $a=8$: Here the only possibility is $$ H.A=4,\quad H.B=2. $$ Using our observation for $B$ we find that $$ B\equiv 2L-x_{i_1}-\ldots-x_{i_6}. $$ Either the $x_{i_j}$ are all different or we have 1 double point (and $B$ is a pair of lines) or 3 double points (and $B$ is a double line). Then $A.B\geq3$ (resp.~4, resp.~8) and $p(B)=0$ (resp.~$-1$, resp.~$-3$). In either case $p(A)\leq1$ and hence $H.A\geq2p(A)+1$. \noindent $a=7$: In this case $$ H.A=H.B=3. $$ All coefficients $b_i\geq0$. It is enough to consider divisors $A$ with $p(A)\geq2$. Together with $H.A=3$ this leads to the following conditions on the $b_i$: $$ \sum b_i=22,\quad \sum b_i(b_i-1)\leq26. $$ Let $\beta_i=\max(0,b_i-1)$. Then these conditions become $$ \sum\beta_i\geq12,\quad\sum(\beta_i+\beta_i^2)\leq26 $$ and it is easy to check that no solutions exist. \noindent $a=6$: We now have to consider $$ H.A=2,\quad H.B=4. $$ We have to consider divisors $A$ with $p(A)\geq1$. Arguing as in the case $a=7$ this leads to $$ \sum b_i=19,\quad \sum b_i(b_i-1)\leq18 $$ resp. $$ \sum\beta_i\geq9,\quad\sum(\beta_i+\beta_i^2)\leq18. $$ The only solution is $b_j=1$ for one $b_j$ and $b_i=2$ for $j\neq i$. But then $A\in|6L-x_j-2\sum\limits_{i\neq j}x_i|$ contradicting condition (6). \noindent $a=5$: Then we have two possible cases \begin{align*} H.A=5&, H.B=1 \tag{$\alpha$}\\ H.A=1&, H.B=5.\tag{$\beta$} \end{align*} We shall treat $(\alpha)$ first. Then by the ampleness of $H$ the curve $B$ must be irreducible. Set $$ B=5L-\sum c_ix_i,\quad c_i\geq0. $$ Then $H.B=1$ and irreducibility of $B$ gives: $$ \sum c_i=16,\quad \sum c_i(c_i-1)\leq12. $$ One easily checks that this is only possible if 6 of the $c_i$ are 2, and the others are 1. Hence $$ B\in|5L-2\sum_{i\in\triangle}x_i-\sum_{i\not\in\triangle}x_i|,\quad |\triangle|=6. $$ Then $p(B)=0$. Moreover $A.B\geq3$, hence $p(A)\leq1$ and hence $H.A\geq2p(A)+1$. In case $(\beta)$ we apply the above argument to $A$ and find $p(A)=0$, i.e.~again $H.A\geq2p(A)+1$. \noindent $a=4:$ Then $H.A=4$ and $H.B=2$. We are done if $p(A)\le 1$, and otherwise $H.A\ge 52-44=8$, a contradiction. \noindent $1\leq a\leq 3$: This follows immediately from Claim 1. \noindent $a=0$: The only possibility is $A=x_i$ when nothing is to show. This finishes the proof of the theorem. \end{Proof} \section{The special rational surface of degree $8$ in ${\Bbb{P}}^4$} \label{sectionIII} In this section we want to show how the decomposition method can be employed to obtain very precise geometric information also about special surfaces. We consider the rational surface in ${\Bbb{P}}^4$ of degree $8$, sectional genus $\pi=6$ and speciality $h=h^1({\cal O}_S(1))=1$. This surface was first constructed by Okonek \cite{O2} using reflexive sheaves. In geometric terms it is ${\Bbb{P}}^2$ blown-up in $16$ points embedded by a linear system of the form $$ |H|=|6L-2\sum_{i=1}^4x_i-\sum_{k=5}^{16}y_k|. $$ Our aim is to study the precise open and closed conditions which the points $x_i,y_k$ must fulfill for $|H|$ to be very ample. If $|H|$ is very ample, the exceptional lines $x_i$ are mapped to conics. Their residual intersection with the hyperplanes gives a {\em pencil} $|D_i|$. Hence we immediately obtain the (closed) necessary condition \begin{gather} |D_i|\equiv |6L-3x_i-2\sum_{j\neq i}x_j-\sum_{k=5}^{16}y_k|\text{ is a pencil} \tag{$D_i$} \end{gather} By Riemann-Roch this is equivalent to $h^1({\cal O}_S(D_i))=1$. We first want to study the linear system $|H|$ on the elements of the pencil $|D_i|$. Note that $$ p(D_i)=4,\ H.D_i=6. $$ If $D=A+B$ is a decomposition of some element $D\in |D_i|$, then \begin{gather} p(A)+p(B)+A.B=5\label{IIIgl1}\\ A.H+B.H=6.\label{IIIgl2} \end{gather} The first equality can be proved by adjunction, the second is obvious. \begin{lemma}\label{IIIlemma1} Assume $|H|$ is very ample. Then for every proper subcurve $Y$ of an element $D\in |D_i|$, $h^1({\cal O}_Y(H))\leq1$ and $p(Y)\leq3$. \end{lemma} \begin{Proof} Riemann-Roch on $Y$ gives \begin{gather} h^0({\cal O}_Y(H))=h^1({\cal O}_Y(H))+H.Y+1-p(Y).\label{IIIgl3} \end{gather} Consider the sequence \begin{gather} 0\longrightarrow{\cal O}_S(H-Y)\longrightarrow{\cal O}_S(H) \overset{\alpha}{\longrightarrow} {\cal O}_Y(H) \longrightarrow 0. \label{IIIgl4} \end{gather} Since $h^2({\cal O}_S(H-Y))=h^0({\cal O}_S(K-(H-Y)))=0$ and $h^1({\cal O}_S(H))=1$ we have $h^1({\cal O}_Y(H))\leq1$. We now consider the rank of the restriction map $H^0(\alpha)$. Since $Y$ is a curve contained in a hyperplane section $2\leq\on{rank}(\alpha)\leq4$. If $\on{rank}\alpha=2$, then $Y$ is a line, hence $p(Y)=0$. Next assume $\on{rank}(\alpha)=3$. In this case $Y$ is a plane curve of degree $d=Y.H$. Since $Y$ is a proper subcurve of $D$ which is not a line $2\leq d\leq 5$. Then $h^1({\cal O}_Y(H))=h^0({\cal O}_{{\Bbb{P}}^2}(d-4))$. Since $h^1({\cal O}_Y(H))\leq1$ this shows in fact $d\leq4$. But then $p(Y)\leq3$. Finally assume that $\on{rank}(\alpha)=4$, i.e.~$Y$ is a space curve. By (\ref{IIIgl3}) $$ p(Y)=h^1({\cal O}_Y(H))-h^0({\cal O}_Y(H))+H.Y+1\leq3 $$ since $H.Y\leq5$. \end{Proof} \begin{remark}\label{IIIrem2} Note that the above proof also shows the following: If $Y$ is a proper subcurve of $D$ with $p(Y)=3$, then $Y$ is a plane quartic with $H_Y=K_Y$ or $Y$ has degree $5$. \end{remark} Before proceeding we note the following result from \cite{CF} which we shall use frequently in the sequel. \begin{proposition}\label{IIIprop3} Let $Y$ be a curve contained in a smooth surface with $p(Y)\leq2$. If $H$ is very ample on $S$, then $H.Y\geq 2p(Y)+1$. \end{proposition} \begin{Proof} \cite[Prop.~5.2]{CF} \end{Proof} \begin{proposition}\label{IIIprop4} If $|H|$ is very ample, then every element $D\in |D_i|$ is $2$-connected. Moreover, either \noindent $\on{(i)}$ $D$ is $3$-connected or \noindent $\on{(ii)}$ Every decomposition of $D$ which contradicts $3$-connectedness is either of the form $D=A+B$ with $H.B=4$, $H_B=K_B$ or of the form $D=A+B$ with $H.B=5$. In the latter case $B=B'+B''$ with $H.B'=4$, $H_{B'}=K_{B'}$. \end{proposition} \begin{Proof} Let $D=A+B$. We first consider the case $p(A),p(B)>0$. Since $|H|$ is very ample, it follows that $H.A\geq3$, $H.B\geq3$. But then $H.A=H.B=3$ and hence $p(A)=p(B)=1$. By (\ref{IIIgl1}) this shows $A.B=3$. Now assume $p(A)\leq0$. Since $p(B)\leq3$ by Lemma~(\ref{IIIlemma1}) it follows from (\ref{IIIgl1}) that $A.B\geq2$. The only case where $A.B=2$ is possible is $p(A)=0$, $p(B)=3$. In this case $H.B\geq4$ since Riemann-Roch for $B$ gives $$ h^0({\cal O}_B(H))=h^1({\cal O}_B(H))+H.B-2 $$ and we know that $h^0({\cal O}_B(H))\geq3$. We first treat the case $H.B=4$. Then $h^1({\cal O}_B(H))=1$ and $h^0({\cal O}_B(H))=3$. In this case $B$ is a plane quartic and $H_B=K_B$. Now assume $H.B=5$. If $h^1({\cal O}_B(H))=0$ then $B$ is a plane quintic. But in this case $p(B)=6$, a contradiction. It remains to consider the case $h^1({\cal O}_B(H))=1$. By duality $h^0({\cal O}_B(K_B-H))=1$. Let $\sigma$ be a non-zero section of ${\cal O}_B(K_B-H)$. As usual we can write $B=Y+Z$ where $Z$ is the maximal subcurve where $\sigma$ vanishes. Note that $Z\neq\emptyset$, since $K_B-H$ has negative degree. Then $Y.(K_Y-H)\geq0$. By the very ampleness of $H$ this implies $p(Y)\geq3$ and hence $p(Y)=3$. Then we must have $H.Y=4$ and by the previous analysis $Y$ is a plane quartic with $H_Y=K_Y$. \end{Proof} At this point it is useful to introduce the following concept. \begin{definition} We say that an element $D\in |D_i|$ fulfills condition (C) if for every decomposition $D=A+B$: \noindent $\on{(i)}$ $p(A),p(B)\leq2$ \noindent $\on{(ii)}$ $H.A\geq 2p(A)+1$, $H.B\geq 2p(B)+1$. \end{definition} \begin{remark}\label{IIIrem5} It follows immediately from (\ref{IIIgl1}) that an element $D\in |D_i|$ which fulfills condition (C) is $3$-connected. \end{remark} For future use we also note \begin{lemma}\label{IIIlemma6} Let $D$ be a curve of genus $4$, and let $H$ be divisor on $D$ of degree $6$ with $h^0({\cal O}_D(H))\geq4$. Assume that for every proper subcurve $Y$ of $D$ we have $H.Y\geq 2p(Y)-1$. Then $H$ is the canonical divisor on $D$. \end{lemma} \begin{Proof} By Riemann-Roch and duality $h^0({\cal O}_D(K_D-H))\geq1$. Let $\sigma$ be a non-zero section of ${\cal O}_D(K_D-H)$. As usual this defines a decomposition $D=Y+Z$ where $Z$ is the maximal subcurve where $\sigma$ vanishes. If $Z=\emptyset$ the claim is obvious. Otherwise $(K_D-H).Y\geq Z.Y$ and by adjunction this gives $H.Y\leq 2p(Y)-2$, a contradiction. \end{Proof} Our next aim is to analyze the condition $h^0({\cal O}_S(H))=5$. For this we introduce the divisor $$ \Delta_i\equiv H-(L-x_i). $$ \begin{lemma}\label{IIIlemma7} The following conditions are equivalent: \noindent $\on{(i)}$ $h^0({\cal O}_S(H))=5$ (resp.~$h^1({\cal O}_S(H))=1$). \noindent $\on{(ii)}$ $h^0({\cal O}_D(H))=4$ (resp.~$h^1({\cal O}_D(H))=1$) for some (every) element $D\in |D_i|$. \noindent $\on{(iii)}$ $h^0({\cal O}_D(K_D-H))=1$ for some (every) element $D\in |D_i|$. \noindent Moreover assume that $D\in |D_i|$ fulfills condition $\on{(C)}$. Then the following conditions are equivalent to $\on{(i)}$-$\on{(iii)}$: \noindent $\on{(iv)}$ ${\cal O}_D(H)=K_D$. \noindent $\on{(v)}$ $\Delta_i|_D\equiv (2L-\sum x_i)|_D$. \end{lemma} \begin{Proof} Since $h^0({\cal O}_S(D_i))\geq1$ we have an exact sequence $$ 0\longrightarrow{\cal O}_S(x_i)\longrightarrow{\cal O}_S(H) \longrightarrow{\cal O}_D(H) \longrightarrow 0. $$ Since $h^1({\cal O}_S(x_i))=0$ the equivalence of (i) and (ii) follows. The equivalence of (ii) and (iii) is a consequence of Serre duality. It follows from Lemma~(\ref{IIIlemma6}) that (iii) implies (iv) if (C) holds. Conversely if ${\cal O}_D(H)=K_D$ then $h^0({\cal O}_D(K_D-H))=h^0({\cal O}_D)=1$, since $D$ is $3$-connected. To show the equivalence of (iv) and (v) note that by adjunction $$ K_D\equiv (K_S+D)|_D\equiv (3L-2x_i-\sum_{j\neq i}x_j)|_D. $$ Hence $K_D\equiv H|_D\equiv (\Delta_i+(L-x_i))|_D$ if and only if $\Delta_i|_D\equiv (K_D-(L-x_i))|_D\equiv (2L-\sum x_i)|_D$. \end{Proof} We want to discuss necessary open conditions which must be fulfilled if $|H|$ is ample. \begin{definition} We say that $|H|$ fulfills condition (P) if for every divisor $Y$ on $S$ with $Y.L\leq6$, $p(Y)\leq2$, $H.Y\leq2p(Y)$ the linear system $|Y|$ is empty. \end{definition} \begin{remark}\label{IIIrem8} (i) By Proposition (\ref{IIIprop3}) this condition is necessary for $|H|$ to be very ample. \noindent (ii) Note that in order to check (P) one only need check {\em finitely many} open conditions. \noindent (iii) For $Y.L=0$ condition (P) implies that the only points which can have infinitely near points are the $x_i$. The only possibility is that at most one of the points $y_k$ is infinitely near to some point $x_i$. \noindent (iv) If $Y.L=1$ then (P) implies $$ |L-\sum_{i\in\triangle}x_i-\sum_{k\in\triangle'}y_k|=\emptyset \text{ for }2|\triangle|+|\triangle'|\geq 6. $$ In particular no three of the points $x_i$ can lie on a line. \noindent (v) If $Y.L=6$ then (P) gives $$ |D_i-x_j|=\emptyset\ (j\neq i),\quad |D_i-y_k-y_l|=\emptyset\ (k\neq l). $$ \end{remark} There are, however, two more open conditions which are not as obvious to see. \begin{proposition}\label{IIIprop9} If $|H|$ embeds $S$ into ${\Bbb{P}}^4$ then the following open conditions hold: \begin{enumerate} \item[$\on{(Q)}$] $|D_i-2x_i|=\emptyset, \quad |D_i-x_i-y_k|=\emptyset, \quad |D_i-2y_k|=\emptyset$ \item[$\on{(R)}$] For any effective curve $C$ with $C\equiv L-x_i-x_j-y_k, C\equiv L-x_i-x_j$ or $ C\equiv y_k$ one has $\dim |D_i-C|\le 0$. Moreover $\dim |H-(L-x_i-x_j)|\le 1$. \end{enumerate} \end{proposition} \begin{Proof} We start with (R). We already know that $\dim |D_i|=1$. Hence we have to see that such a curve $C$ is not contained in the plane spanned by the conic $x_i$. But this would contradict very ampleness since $C.x_i=1$ or $0$. If $|H|$ is very ample then it embeds $\Lambda_{ij}=L-x_i-x_j$ as a plane conic (irreducible or reducible but reduced). The claim then follows from the exact sequence $$ 0\longrightarrow{\cal O}_S(H-(L-x_i-x_j))\longrightarrow {\cal O}_S(H) \longrightarrow {\cal O}_{\Lambda_{ij}}(H) \longrightarrow 0. $$ Next we consider the linear system $|D_i-2x_i|$. Assume there is a curve $B\in |D_i-2x_i|$. Then $p(B)=-3$. Since $H.B=2$ we have the following possibilities: $B$ is a reduced conic (either smooth or reducible). Then $p(B)=0$, a contradiction. If $B$ is the union of $2$ skew lines, then $p(B)=-1$ which is also not possible. Hence $B$ must be a non-reduced line. But this is not possible, since the class of $B$ on $S$ is not divisible by $2$. The crucial step is to prove the \begin{claim} Set $D=D_i$. If $|D|$ contains $y_k+B$, then $B$ is of the form $B=B'+(L-x_i-x_j-y_k)$ with $H_{B'}=K_{B'}$. \end{claim} It follows from Lemma~(\ref{IIIlemma7}) that there exists a non-zero section $0\neq\sigma\in H^0({\cal O}_D(K_D-H))$. As usual this defines a decomposition $D=Y+Z$. Since $(K_D-H).y_k=-1$ the curve $Z$ must contain the irreducible curve $y_k$. Moreover since $y_k.B=2$ and $(K_D-H).B=1$ it follows that $Z$ contains some further curve $Z'$ contained in $B$, i.e.~$B=B'+Z'$. Now as in proof of Lemma~(\ref{IIIlemma6}) $H.B'\leq2p(B')-2$ and very ampleness of $|H|$ together with (\ref{IIIlemma1}) implies $p(B')=3$. As in the proof of Proposition~(\ref{IIIprop4}) one concludes that $H.B'=4$, $H_{B'}=K_{B'}$. In particular $Z'$ is a line. Since $p(D_i-2y_k)=1$ it follows that $Z'\neq y_k$. First assume that $Z'.y_k=0$. Then $p(Z'+y_k)=-1$ and $B'.y_k=2$. It follows from (\ref{IIIgl1}) that $B'.Z'=1$. But now the decomposition $Z'+(B'+y_k)$ contradicts $2$-connectedness. Hence $Z'$ and $y_k$ are two lines meeting in a point. This gives $p(y_k+Z')=0$, $B'.(y_k+Z')=2$. We can write $$ Z'=aL-\beta_ix_i-\sum_{j\neq i}\beta_jx_j-y_k-\sum_{l\neq k}\alpha_ly_l. $$ If $a=0$ then $Z'=x_i-y_k$ or $Z'=x_j-y_k$, $j\neq i$. The first is impossible since $p(D_i-x_i)=1$ the second contradicts $|D_i-x_j|=\emptyset$. Hence $1\leq a\leq6$. Since $Z'$ is mapped to a line in ${\Bbb{P}}^4$ we find $Z'.y_l\leq1$, $Z'.x_j\leq2$, i.e. \begin{gather} 0\leq\alpha_l\leq1,\quad 0\leq\beta_i,\beta_j\leq2.\label{IIIgl5} \end{gather} It follows from (\ref{IIIgl5}) and from $p(Z')=0$ that $a\leq4$; moreover $p(Z')=0$, $p(B')=3$ and $p(B)=3$ imply $Z'.B'=1$. Using $0\leq\alpha_l\leq 1$ this gives \begin{gather} a(6-a)-\beta_i(3-\beta_i)-\sum_{j\neq i}\beta_j(2-\beta_j)=2. \label{IIIgl6} \end{gather} In view of (\ref{IIIgl5}) this shows $a(6-a)\leq7$ and since $a\leq4$ it follows that $a=1$. Then $\beta_i,\beta_j\leq1$. If $\beta_i=0$ then by (\ref{IIIgl6}) $\beta_j=1$ for $j\neq i$, but no three of the points $x_i$ can be collinear by (\ref{IIIgl6}). Hence $\beta_i=1$ and exactly one $\beta_j$ is $1$. Together with $H.Z'=1$ this gives $Z'=L-x_i-x_j-y_k$ as claimed. We are now in a position to prove that $|D_i-x_i-y_k|=\emptyset$ and $|D_i-2y_k|=\emptyset$. For this we have to show that $B'$ cannot contain $x_i$ or $y_k$. In the first case $B'=x_i+B''$. Then $H.x_i=2$ and $K_{B'}.x_i=1$ contradicting $H_{B'}=K_{B'}$. Similarly in the second case $B'=y_k+B''$ with $H.y_k=1$ and $K_{B'}.y_k=0$ giving the same contradiction. \end{Proof} Observe for future use that in the following proposition the assumption that $|H|$ is very ample is not made. \begin{proposition}\label{IIIprop10} Assume that the open conditions $\on{(P)}$ and $\on{(Q)}$ hold. Then an effective decomposition $D=A+B$ either fulfills condition $\on{(C)}$ and hence is not $3$-disconnecting or (after possibly interchanging $A$ and $B$) $A=y_k$, $L-x_i-x_j$ or $L-x_i-x_j-y_k$. \end{proposition} \begin{Proof} Let $D=A+B$. Clearly we can assume $A.L\leq3$. We shall first treat the case $A.L=0$, i.e.~$A$ is exceptional with respect to the blowing down map $S\to{\Bbb{P}}^2$. Then $p(A)\leq0$ and $A.H>0$ by (P). By conditions (Q) and (P) (cf. Remark (III.8)(v)) if $A.H=1$, then either $A=x_j-y_k$ or $A=x_i-y_k$ or $A=y_k$. In the first two cases $A.B\ge 3$ and $p(B)\le 2$, the third is one of the exceptions stated. If $A.H\ge 2$ then $p(B)\le 2$ and the claim follows from (P). Hence we can now write \begin{eqnarray*} A&\equiv&aL-\sum\alpha_jx_j-\sum a_ky_k\\ B&\equiv&bL-\sum\beta_jx_j-\sum b_ky_k \end{eqnarray*} with $a,b>0$. Using the open conditions from Remark~(\ref{IIIrem8})(v) (which are a consequence of (P)) and (Q) it follows that \begin{gather*} \begin{aligned} a_k,b_k &\geq -1,\\ \alpha_j,\beta_j &\geq 0,\\ \alpha_i,\beta_i &\geq -1, \end{aligned} \quad \begin{aligned} a_k+b_k &= 1\\ \alpha_j+\beta_j &= 2\\ \alpha_i+\beta_i &= 3 \end{aligned} \quad (j\neq i) \end{gather*} and moreover that at most one of the integers $a_k,b_k,\alpha_i,\beta_i$ can be negative. If $\beta_i=-1$ then $\alpha_i=4$. In this case $A$ cannot be effective since we have assumed $a\leq3$. If $\alpha_i=-1$ then $\beta_i=4$ and hence $b\geq4$. We have to consider the cases $a=1$ or $2$. In either case $p(A)\leq0$ and $H.A\geq2p(A)+1$ follows from (P). On the other hand \begin{eqnarray*} H.B-(2p(B)+1)&=&(9b-b^2+1)+\sum_{j\neq i}\beta_j(\beta_j-3)+\sum_k b_k(b_k-2)\\ &\geq&(9b-b^2+1)-6-12\geq3 \end{eqnarray*} since $b=4,5$. Hence we can now assume $\alpha_i,\beta_i\geq0$. \noindent $\mathbf{a=1}$. We first treat the case $a_k\geq0$ for all $k$. Then $$ A\equiv L-\sum_{j\in\triangle}x_j-\sum_{k\in\triangle'} y_k. $$ Clearly $p(A)\leq0$. Let $\delta_{i\triangle}=0$ (resp.~$1$) if $i\not\in\triangle$ (resp.~$i\in\triangle$). Then $$ p(B)=|\triangle|+\delta_{i\triangle}. $$ We only have to treat the cases where $p(B)\geq3$. Then either $\delta_{i\triangle}=0$, $|\triangle|\geq3$ or $\delta_{i\triangle}=1$, $|\triangle|\geq2$. In the first case $$ H.A=6-2|\triangle|-|\triangle'|\leq0 $$ contradicting (P) for $A$. In the second case the only possibilty is $|\triangle|=2$, $|\triangle'|\leq1$. But then $A=L-x_i-x_j$ or $L-x_i-x_j-y_k$. Now assume that one $a_k$ is negative. We can assume $a_{16}=-1$. Then $$ A\equiv L-\sum_{j\in\triangle}x_j-\sum_{k\in\triangle'}y_k+y_{16}. $$ In this case $p(A)=-1$ and $$ p(B)=|\triangle|+\delta_{i\triangle}-1. $$ Using the same arguments as before we find that $p(B)\leq2$ in all cases. \noindent $\mathbf{a=2}$. Again we first assume that all $a_k\geq0$. Then $$ A\equiv 2L-\sum_{j\in\triangle}x_j-\sum_{k\in\triangle'}2x_k- \sum_{l\in\triangle''}y_l-\sum_{m\in\triangle'''}2y_m. $$ Clearly $p(A)\leq0$. If $i\not\in\triangle\cup\triangle'$ then $p(B)\leq0$. If $i\in\triangle$ then $p(B)\leq2$. Now assume that $i\in\triangle'$. In this case $p(B)\leq2$ with one possible exception: $|\triangle|=3$ and $|\triangle'''|=0$. But then $$ A\equiv 2L-2x_i-x_j-x_k-x_l-\sum_{l\in\triangle''}y_l. $$ In this case $A$ splits into two lines meeting $x_i$. But then one of these lines must contain $3$ of the points $x_j$ contradicting condition (P). Finally let $a_{16}=-1$. The above arguments show that in this case $p(B)\leq2$. \noindent $\mathbf{a=3}$. Since in this case $p(A),p(B)\leq1$ condition (C) follows. \end{Proof} Propositions~(\ref{IIIprop4}) and (\ref{IIIprop10}) have provided us with a fairly good understanding of the behaviour of $H$ on the pencil $|D_i|$. \begin{cor}\label{IIIcor11} Assume $|H|$ embeds $S$ into ${\Bbb{P}}^4$. For every element $D\in |D_i|$ either: \noindent $\on{(i)}$ $D$ is $3$-connected and $H_D=K_D$ or \noindent $\on{(ii)}$ $D=B+(L-x_i-x_j)$ with $H_B=K_B$. \end{cor} \begin{remark}\label{IIIrem12} The conic $L-x_i-x_j$ can be irreducible or reducible in which case it splits as $(L-x_i-x_j-y_k)+y_k$. \end{remark} At this point we can also conclude our discussion about the linear system $|\Delta_i|=|H-(L-x_i)|$ (cf.~(\ref{IIIlemma7})). \begin{proposition}\label{IIIprop13} If $|H|$ embeds $S$ into ${\Bbb{P}}^4$, then $\dim|\Delta_i|=0$. \end{proposition} \begin{Proof} We first claim that the general element $D\in |D_i|$ is $3$-connected. Indeed if $D$ is not $3$-connected, then $D=B+(L-x_i-x_j)$. The conic $L-x_i-x_j$ spans a plane $E'$. If $E$ is the plane spanned by $x_i$ then $E\neq E'$ since $(L-x_i-x_j).x_i=1$. Hence $D$ is cut out by the hyperplane spanned by $E$ and $E'$. Varying the index $j$ there are at most $3$ such hyperplanes. Clearly $L-x_i$ is effective. Consider the exact sequence $$ 0\longrightarrow{\cal O}_S(\Delta_i)\longrightarrow{\cal O}_S(H) \longrightarrow {\cal O}_S(H)|_{L-x_i} \longrightarrow 0. $$ Since $H.(L-x_i)=4$ and $p(L-x_i)=0$ it follows that $|H|$ cannot map $L-x_i$ to a plane curve. This shows $h^0({\cal O}_S(\Delta_i))\leq1$. On the other hand choose an element $D\in|D_i|$ which is $3$-connected. We have an exact sequence $$ 0\longrightarrow{\cal O}_S(2x_i-L)\longrightarrow{\cal O}_S(\Delta_i) \longrightarrow {\cal O}_D(\Delta_i) \longrightarrow 0. $$ Now $h^0({\cal O}_S(2x_i-L))=h^2({\cal O}_S(2x_i-L))=0$ and hence $h^1({\cal O}_S(2x_i-L))=1$ by Riemann-Roch. Since $|H|$ is ample no $3$ of the points $x_i$ lie on a line. Hence $|2L-\sum x_i|$ is a base point free pencil. Since $|(2L-\sum x_i)-D|=\emptyset$ this shows that $|2L-\sum x_i|$ cuts out a base-point free pencil on $D$. Since $D$ is $3$-connected $(2L-\sum x_i)|_D\equiv\Delta_i|_D$ by Lemma~(\ref{IIIlemma7}) and hence $h^0({\cal O}_D(\Delta_i))\geq2$. By the above sequence this implies $h^0({\cal O}_S(\Delta_i))\geq 1$. \end{Proof} We are now ready to characterize very ample linear systems which embed $S$ into ${\Bbb{P}}^4$. \begin{theorem}\label{IIItheo14} The linear system $|H|$ embeds $S$ into ${\Bbb{P}}^4$ if and only if \noindent $\on{(i)}$ The open conditions $\on{(P)}$, $\on{(Q)}$ and $\on{(R)}$ hold. \noindent $\on{(ii)}$ The following closed conditions hold: \begin{enumerate} \item[$\on{(}D_i\on{)}$] $\dim |D_i|=1$ \item[$\on{(}\Delta_i\on{)}$] For a $3$-connected element $D\in|D_i|$ (whose existence follows from the above conditions) $\Delta_i.D\equiv (2L-\sum x_i).D$. \end{enumerate} \end{theorem} \begin{remark}\label{IIIrem15} As the proof will show it is enough to check the closed conditions ($D_i$), ($\Delta_i$) for one $i$. \end{remark} \begin{Proof} We have already seen that these conditions are necessary. Next we shall show that a $3$-connected element $D\in|D_i|$ exists if the open conditions and ($D_i$) are fulfilled. Assume that no element $D\in|D_i|$ is $3$-connected. Then by Proposition~(\ref{IIIprop10}) every element $D$ is of the form $D=B+C$ with $C=L-x_i-x_j$, $L-x_i-x_j-y_k$ or $y_k$. But by condition (R) there are only finitely many such elements in $|D_i|$. We shall now proceed in several steps. \noindent {\bf Step 1}: $h^0({\cal O}_S(H))=5$. We have seen in the proof of Lemma~(\ref{IIIlemma7}) that for a $3$-connected element $D$ the equality $\Delta_i.D\equiv (2L-\sum x_i).D$ implies $K_D=H_D$ and hence $h^0({\cal O}_D(K_D-H))=1$, resp.~$h^1({\cal O}_D(H))=1$. Now the claim follows from the equivalence of (i) and (ii) in Lemma~(\ref{IIIlemma7}). In order to prove very ampleness of $|H|$ we want to apply the Alexander-Bauer Lemma to the decomposition $$ H\equiv D_i+x_i. $$ We first have to show that $|H|$ cuts out complete linear systems on $x_i$ and $D\in|D_i|$. Recall that $x_i$ is either a ${\Bbb{P}}^1$ or consists of two ${\Bbb{P}}^1$'s meeting transversally (cf.~Remark~(\ref{IIIrem8})(iii)). Moreover $H.x_i=2$ and if $x_i$ is reducible then $H$ has degree $1$ on every component. Hence $h^0({\cal O}_{x_i}(H))=3$. The claim for $x_i$ then follows from the exact sequence $$ 0\longrightarrow{\cal O}_S(D_i)\longrightarrow{\cal O}_S(H) \longrightarrow {\cal O}_{x_i}(H)\longrightarrow 0. $$ and condition ($D_i$), i.e.~$h^0({\cal O}_S(D_i))=2$. The corresponding claim for $D$ follows from the sequence $$ 0\longrightarrow{\cal O}_S(x_i)\longrightarrow{\cal O}_S(H) \longrightarrow {\cal O}_S(H)|_D \longrightarrow 0. $$ Our above discussion also shows that $|H|$ embeds $x_i$ as a conic (which can be irreducible or consist of two different lines). \noindent {\bf Step 2}: If $D\in|D_i|$ is $3$-connected then $H_D=K_D$ and $|H|$ is very ample on $D$. We have already seen the first claim. We have to see that $K_D$ is very ample. For this we consider the pencils $|\Sigma_1|=|L-x_i|$, resp.~$|\Sigma_2|=|2L-\sum x_j|$. Clearly $|\Sigma_1|$ is base point free and the same is true for $|\Sigma_2|$ as no three of the points $x_i$ lie on a line (by (P)). Hence $$ |\Sigma_1+\Sigma_2|=|3L-2x_i-\sum_{j\neq i}x_j|=|D_i+K_S| $$ is base point free. By adjunction $(D_i+K_S)|_D\equiv K_D$ and the exact sequence $$ 0\longrightarrow{\cal O}_S(K_S)\longrightarrow{\cal O}_S(K_S+D_i) \longrightarrow {\cal O}_D(K_D) \longrightarrow 0 $$ shows that restriction defines an isomorphism $|\Sigma_1+\Sigma_2|\cong |K_D|$. Let $X$ be the blow-up of ${\Bbb{P}}^2$ in the points $x_j$ and $\pi:S\to X$ the map blowing down the exceptional curves $y_k$. The linear system $|\Sigma_1+\Sigma_2|$ defines a morphism $$ f=\phi_{|\Sigma_1+\Sigma_2|}:X\longrightarrow{\Bbb{P}}^3. $$ It is easy to understand the map $f$: Clearly $f$ contracts the three $(-1)$-curves $\Lambda_{ij}=L-x_i-x_j$, $j\neq i$. Let $\pi':X\to X'$ be the map which blows down the curves $\Lambda_{ij}$ (this makes also sense if $\Lambda_{ij}=(L-x_i-x_j-y_k)+y_k$ where we first contract $y_k$ and then $L-x_i-x_j-y_k$). Then $X'$ is a smooth surface and we have a commutative diagram $$ \unitlength1cm \begin{picture}(4,2) \put(0,1.5){$X$} \put(0.5,1.6){\vector(1,0){2.4}} \put(3.1,1.5){$f(X)$} \put(0.5,1.3){\vector(3,-2){1}} \put(1.5,0.1){$X'$} \put(2,0.62){\vector(3,2){1}} \put(0.5,0.7){{$\scriptstyle \pi'$}} \put(2.5,0.7){{$\scriptstyle f'$}} \put(1.5,1.7){{$\scriptstyle f$}} \end{picture} $$ where $f'$ maps $X'$ isomorphically onto a smooth quadric. This shows that $\phi_{|K_D|}:D\to{\Bbb{P}}^3$ is the composition of the blowing down maps $\pi:S\to X$ and $\pi':X\to X'={\Bbb{P}}^1\times{\Bbb{P}}^1$ followed by an embedding of $X'$. Now $D.y_k=1$, hence $\pi|_D$ can only fail to be an isomorphism if $D$ contains $y_k$. But this is impossible if $D$ is $3$-connected. Similarly $D.\Lambda_{ij}=1$ and $D$ cannot contain a component of $\Lambda_{ij}$. Hence we are done in this case. It remains to treat the case when $D$ is not $3$-connected. \noindent {\bf Step 3}: If $D$ is not $3$-connected, then $D=B+(L-x_i-x_j)$, $H_B=K_B$ and $|H|$ restricts onto $|K_B|$. We have already seen that $h^0({\cal O}_S(H))=5$ and hence $h^0({\cal O}_D(K_D-H))=1$. As usual a non-zero section $\sigma$ defines a decomposition $D=Y+Z$. Our first claim is that $Z$ is different from $0$. In fact if $Z=0$ then $K_D-H$ would be trivial on $D$. On the other hand $D$ is not $3$-connected, thus it splits as $D=A+B$ with $A$ as in Proposition~(\ref{IIIprop10}), in particular $p(A)=0$, $A.B=2$. Then $K_D.A=0$ contradicting $H.A>0$ which follows from (P). Thus $Z$ is different from $0$ and since the section $\sigma$ defines a good section $\sigma'$ of $H^0({\cal O}_Y(K_Y-H))$ it follows that $2p(Y)-2\ge H.Y$, and hence $p(Y)\ge 3, Y.Z\le 2$. Then Proposition (\ref{IIIprop10}) applies and $Z=y_k$ or $L-x_i-x_j-y_k$ or $L-x_i-x_j$. If $Z=y_k$ or $L-x_i-x_j$ then $(K_Y-H).Y=-1$, a contradiction. Hence $Z=L-x_i-x_j$ and $H_Y=K_Y$. We next claim that $B$ is $2$-connected. Assume we have a decomposition $B=B_1+B_2$ with $B_1.B_2\leq1$. Then $(B_1+B_2).(L-x_i-x_j)=2$, hence we can assume that $B_1.(L-x_i-x_j)\leq1$. But then $B_1.(B_2+L-x_i-x_j)\leq2$ contradicting Proposition~(\ref{IIIprop10}). This shows that $h^1({\cal O}_B(K_B))=1$ and $h^0({\cal O}_B(K_B))=3$. The claim then follows from the exact sequence $$ 0\longrightarrow{\cal O}_S(L-x_j)\longrightarrow{\cal O}_S(H) \longrightarrow {\cal O}_B(H) \longrightarrow 0. $$ \noindent {\bf Step 4}: $|H|$ embeds $D$. Our first claim is that $|H|$ embeds $B$ as a plane quartic. Since $B-y_k$ is not effective by condition (P) and $B.y_k=1$ it follows that the curve $B$ is mapped isomorphically onto its image under the blowing down map $\pi:S\to X$. On $X$ $$ B\equiv 5L-2x_i-x_j-2x_k-2x_l,\quad K_B\equiv (2L-x_i-x_k-x_l)|_B. $$ Thus $|K_B|$ is induced by a standard Cremona transformation centered at $x_j$, $x_k$ and $x_l$. Again by (P) it follows that $B-\Lambda_{ik}$ for $k\neq i$ and $B-\Lambda_{kl}$ for $k,l\neq i$ are not effective. Since $B.\Lambda_{ik}=B.\Lambda_{kl}=1$ it follows that $B$ is mapped isomorphically onto a plane quartic. It follows from condition (R) that $|H|$ embeds $\Lambda_{ij}$ as a plane conic $Q$. The planes containing $B$ and $Q$ intersect in a line and span a ${\Bbb{P}}^3$. The line of intersection cannot be a component of $Q$ since, by taking residual intersection with hyperplanes containing $B$, this would contradict $h^0({\cal O}_S(x_i+y_k))=1$, resp.~$h^0({\cal O}_S(L-x_j-y_k))=1$. Hence the schematic intersection of the embedded quartic $B$ and the conic $Q$ has length at most $2$. Let $D'$ be the schematic image of $D$. Then ${\cal O}_{D'}$ is contained in the direct image of ${\cal O}_D$. But the former has colength $\leq2$ in ${\cal O}_Q\oplus{\cal O}_B$, the latter has colength $2$, thus $D=D'$. \end{Proof} \begin{remark}\label{IIIrem16} We have already remarked that conditions (P) and (Q) lead to finitely many open conditions. Going through the proof of Proposition~(\ref{IIIprop10}) one sees that it is sufficient to check that no decomposition $A+B=D\in|D_i|$ exists where $A$ (or $B$) contradicts one of the following conditions below: Here $\triangle$ and $\triangle'$ are always disjoint subsets of $\{1,\ldots,4\}$ whereas $\triangle''$ is a subset of $\{5,\ldots,16\}$. We set $\delta_{i\triangle}=1$ (resp.~$0$) if $i\in\triangle$ (resp.~$i\not\in\triangle$). Similarly we define $\delta_{i\triangle'}$. Moreover $\delta_{m}=1$ for at most one $m\in\{5,\ldots,16\}$ and $\delta_{m}=0$ otherwise. If $\delta_{m}=1$ then ~$m\not\in\triangle''$. \begin{enumerate} \item[(0)] $|x_j-x_k|=\emptyset$ ($j\neq k$), $|y_k-y_l|=\emptyset$ ($k\neq l$), $|y_k-x_j|=\emptyset$, $|x_j-y_k-y_l|=\emptyset$. \item[(1)] $|L-\sum\limits_{j\in\triangle}x_j- \sum\limits_{k\in\triangle''}y_k|=\emptyset$ for $2|\triangle|+|\triangle'|\geq6$ \item[(2)] $|2L-\sum\limits_{j\in\triangle}x_j- \sum\limits_{k\in\triangle''}y_k|=\emptyset$ for $2|\triangle|+|\triangle'|\geq12$. \item[(3)] $|3L-2x_j-\sum\limits_{k\in\triangle}x_k- \sum\limits_{l\in\triangle''}y_l|=\emptyset$ for $2|\triangle|+|\triangle''|\geq14$\\ $|3L-\sum\limits_{j\in\triangle}x_j-2y_k- \sum\limits_{l\in\triangle''}y_l|=\emptyset$ for $2|\triangle|+|\triangle''|\geq16$\\ $|3L-\sum\limits_{j\in\triangle}x_j- \sum\limits_{k\in\triangle''}y_k|=\emptyset$ for $2|\triangle|+|\triangle''|\geq16$ \item[(4)] $|4L-(3-\delta_{i\triangle}-2\delta_{i\triangle'})x_i- \sum\limits_{j\neq i \atop j\in\triangle}x_j- 2\sum\limits_{k\neq i \atop k\not\in(\triangle\cup\triangle')}x_k- \sum\limits_{l\not\in\triangle''}y_l-\delta_my_m|=\emptyset$ for $|\triangle|+|\triangle'|+\delta_{i\triangle}+ 2\delta_{i\triangle'}-\delta_m\leq5$, $2|\triangle'|+|\triangle''|-2\delta_{i\triangle}-4\delta_{i\triangle'}+ \delta_m\leq0$, $2|\triangle|+4|\triangle'|+|\triangle''|\leq11$ \item[(5)] $|5L-(3-\delta_{i\triangle})x_i-\sum\limits_{j\neq i \atop j\in\triangle}x_j- 2\sum\limits_{k\neq i \atop k\not\in\triangle}x_k- \sum\limits_{l\not\in\triangle''}y_l-\delta_my_m|=\emptyset$ for $|\triangle|+\delta_{i\triangle}-\delta_m\leq2$, $|\triangle''|-2\delta_{i\triangle}+\delta_m\leq0$, $2|\triangle|+|\triangle''|\leq5$. \item[(6)] $|D_i-x_j|=\emptyset$ ($i\neq j$), $|D_i-2x_i|=\emptyset$, $|D_i-x_i-y_k|=\emptyset$, $|D_i-2y_k|=\emptyset$, $|D_i-y_k-y_l|=\emptyset$ ($k\neq l$). \end{enumerate} \end{remark} Now we want to show how Theorem (\ref{IIItheo14}) can be used to prove the existence of the special surfaces of degree 8 by explicitly constructing a very ample linear system $|H|$. Let $x_1,\ldots , x_4$ be points in general position in ${\Bbb{P}}^2$, and blow them up. The linear system $|5L-x_1-2\sum\limits_{j\ge 2} x_j|$ is 10-dimensional, its elements have arithmetic genus 3. Let $\Delta_1$ be a general (and hence smooth) element of the 10-dimensional linear system $|5L-x_1-2\sum\limits_{j\ge 2} x_j|$ on ${\hat{\Bbb{P}}}^2={\Bbb{P}}^2(x_1, \ldots , x_4)$. Note that the image of $\Delta_1$ in ${\Bbb{P}}^2$ is the image of the canonical model of $\Delta_1$ under a standard Cremona transformation. The linear system $|2L-\sum\limits_j x_j|$ cuts out a $g^1_3$ on $\Delta_1$, since $H^1({\hat {\Bbb{P}}}^2, {\cal O}_{{\hat {\Bbb{P}}}^2}(-3L+\sum x_j))=0$. The linear system $$ |L_0|:=|(6L-3x_1-2\sum\limits_{j\ge 2} x_j)|_{\Delta_1}-g^1_3|=|(4L-2x_1-\sum\limits_{j\ge 2} x_j)|_{\Delta_1}| $$ on $\Delta_1$ has degree 12 and dimension 9. The linear system $|4L-2x_1-\sum\limits_{j\ge 2} x_j|$ on ${\hat{\Bbb{P}}}^2$ cuts out a subsystem of codimension 1 in $|L_0|$. We consider the variety $$ {\cal M}:=\{(\Delta_1, \sum y_k);\ \Delta_1 \mbox{ smooth }, \sum y_k \in |L_0|\}. $$ ${\cal M}$ is rational of dimension 19. \begin{theorem}\label{IIItheo17} There is a non-empty open set ${\cal U}$ of the rational variety ${\cal M}$ for which the linear system $|H|$ embeds $S$ into ${\Bbb{P}}^4$. \end{theorem} \begin{Proof} We have to show that for a general choice of $\Delta_1$ and $\sum y_k \in |L_0|$ the linear system $|H|$ embeds $S$ into ${\Bbb{P}}^4$. We shall first treat the closed conditions. Since $\Delta_1$ is smooth we can identify it with its strict transform on $S$. Consider the exact sequence $$ 0\rightarrow {\cal O}_S(L-2x_1)\rightarrow {\cal O}_S(D_1)\rightarrow {\cal O}_{\Delta_1}(D_1)\rightarrow 0. $$ Since $\sum y_k\in |L_0|$ we have $$ (35) \quad 6L-3x_1-2\sum\limits_{j\ge 2} x_j-\sum y_k\equiv g^1_3 \mbox{ on } \Delta_1 $$ and hence $h^0({\cal O}_S(D_1))=h^0({\cal O}_{\Delta_1} (D_1))=2$. This is condition $\on {(D_1)}$. Condition $ (\Delta_1)$ holds by construction.\\ In order to treat the open conditions we will first consider special points in ${\cal M}$ which give us all open conditions but two. These we will then treat afterwards. The linear system $|4L-2x_1-\sum\limits_{j\ge 2} x_j|$ is free on ${\hat {\Bbb{P}}}^2$. Hence a general element $\Gamma$ is smooth and intersects $\Delta_1$ transversally in 12 points $y_k$ which neither lie on an exceptional line, nor on a line of the form $\Lambda_{kl}=L-x_k-x_l$. Moreover a general element $\Gamma$ is irreducible. This follows since $\Gamma^2=9$ and $|\Gamma|$ is not composed of a pencil, since the class of $\Gamma$ is not divisible by 3 on ${\hat {\Bbb{P}}}^2$. Let $\Gamma'$ be the smooth transform of $\Gamma$ on $S$. Since $\Gamma$ is smooth, $\Gamma'$ is isomorphic to $\Gamma$. \begin{claim} $|D_1|=\Gamma'+|2L-\sum\limits_j x_j|.$\\ This follows immediately since $D_1\equiv \Gamma'+(2L-\sum\limits_j x_j)$ and $\dim |D_1|=1=\dim(\Gamma'+|2L-\sum x_j|). $ \end{claim} The only curves contained in an element of $|D_1|$ are $\Gamma'$, conics $C\equiv 2L-\sum x_j$ and lines $\Lambda_{kl}=L-x_k-x_l$. The latter only happens for finitely many elements of $|D_1|$. This shows immediately that conditions (Q) and (R) are fulfilled with the possible exception that $\dim|H-\Lambda_{1j}|\ge 2$. To exclude this we consider w.l.o.g. the case $j=2$. Note that $H-\Lambda_{12}\equiv \Delta_2+x_1\equiv\Gamma ' + \Lambda_{34}+x_1$. Since $\Gamma '$ is smooth of genus 2 and $\Gamma '.(\Delta_2+x_1)=1$ it follows that $h^0({\cal O}_{\Gamma '}(\Delta_2+x_1))\le 1$. The claim now follows from the exact sequence $$ 0\rightarrow {\cal O}_S(\Lambda_{34}+x_1)\rightarrow {\cal O}_S(\Delta_2+x_1)\rightarrow {\cal O}_{\Gamma '}(\Delta_2+x_1)\rightarrow 0 $$ together with the fact that $h^0({\cal O}_S(\Lambda_{34}+x_1))=1$. It remains to consider (P). The curve $\Gamma'$ contradicts condition (P) since $p(\Gamma')=2, H.\Gamma'=4$. Similarly the decomposition $(\Gamma'+\Lambda_{ij})+\Lambda_{kl}$ contradicts (P) if $k, l\neq 1$. On the other hand the above construction shows that for one (and hence the general) pair $(\Delta_k, \sum y_k)$ all open conditions given by (P) are fulfilled for a decomposition $D=A+B$ of an element in $|D_1|$ with the possible exception of $|\Gamma'|\neq\emptyset$ or $|D_1-\Lambda_{kl}|\neq\emptyset$ for $k,l\neq 1$. The first case is easy, we can simply take an element $\sum y_k \in |L_0|$ which is not in the codimension 1 linear subsystem given by $|4L-2x_1-\sum\limits_{j\ge2} x_j|$ on ${\hat{\Bbb{P}}}^2$. Next we assume that there is an element $A\in |D_1-\Lambda_{kl}|$ where $k,l\neq 1$. Then $A.\Delta_1=2$. Since $\Delta_1$ cannot be a component of $A$ this means that $A$ intersects $\Delta_1$ in two points $Q_0, Q_1$. If $j$ is the remaining element of the set $\{1,\ldots,4\}$ then $L-x_1-x_j\equiv Q_0+Q_1$ on $\Delta_1$ The linear system $|L|$ cuts out a $g^2_5$ on $\Delta_1$ and is hence complete. Hence $Q_0 + Q_1$ is the intersection of $\Lambda_{1j}$ with $\Delta_1$. In particular $\Lambda_{1j}$ intersects $A$ in at least 2 points, namely $Q_0$ and $Q_1$. Since $A.\Lambda_{1j}=0$ this implies that $\Lambda_{1j}$ is a component of $A$ (we can assume that $\Lambda_{1j}$ is irreducible). Hence $A=A'+\Lambda_{1j}$ with $A'\in |D_1-\Lambda_{kl}-\Lambda_{1j}|=|\Gamma'|$ and we are reduced to the previous case. \end{Proof} \begin{remarks}\label{IIIrem18} (i) Originally Okonek \cite{O2} constructed surfaces of degree $8$ and sectional genus $6$ with the help of reflexive sheaves. \noindent (ii) According to \cite{DES} the rational surfaces of degree $8$ with $\pi=6$ arise as the locus where a general morphism $\phi:\Omega^3(3)\to{\cal O}(1)\oplus 4{\cal O}$ drops rank by $1$. The space of such maps has dimension $80$. Taking the obvious group actions into account we find that the moduli space has dimension $43=19+\dim\on{Aut}{\Bbb{P}}^4$. Moreover this description shows that the moduli space is irreducible and unirational. \noindent (iii) These surfaces are in $(3,4)$-liaison with the Veronese surface \cite{O2}. Counting parameters one finds again that they depend on $19$ parameters (modulo $\on{Aut}({\Bbb{P}}^4)$). \noindent (iv) It was pointed out to us by K.~Ranestad that Ellingsrud and Peskine (unpublished) also suggested a construction of these surfaces via linear systems. They start with a smooth quartic $K_4=\{f_4=0\}$ and a smooth quintic $K_5=\{f_5=0\}$ touching in $4$ points $x_1,\ldots,x_4$. Let $y_5,\ldots,y_{16}$ be the remaining points of intersection. Let $$ {\cal I}'={\cal O}_{{\Bbb{P}}^2}\left(-\sum x_i\right),\quad {\cal I}={\cal O}_{{\Bbb{P}}^2}\left(-2\sum x_i-\sum y_k\right). $$ Then we have an exact sequence $$ 0\longrightarrow {\cal I}'(-4)\longrightarrow {\cal I}\longrightarrow {\cal O}_{K_4}(-5) \longrightarrow 0. $$ Twisting this by ${\cal O}(6)$ and taking global section gives $$ 0\longrightarrow \Gamma({\cal I}'(2)) \longrightarrow \Gamma({\cal O}_S(H)) \longrightarrow \Gamma({\cal O}_{K_4}(1)) \longrightarrow 0. $$ Since $h^0({\cal I}'(2))=2$ and $h^0({\cal O}_{K_4}(1))=3$ this shows $h^0({\cal O}_S(H))=5$. One can easily see that $|\Delta_i|\neq\emptyset$ and $\dim|D_i|\geq1$ in this construction: counting parameters one shows that $\Delta_i=\{lf_4+f_5=0\}$ for some suitable linear form and that there is at least a $1$-dimensional family of curves in $|D_i|$ which are of the form $D=\{qf_4+lf_5\}$ where $q$ is of degree $2$ and $l$ is a linear form. This construction, too, depends on $19$ parameters. \end{remarks} Finally we want to discuss the moduli space of smooth special surfaces of degree 8 in ${\Bbb{P}}^4$ (modulo Aut ${\Bbb{P}}^4$). Recall the set ${\cal M}$ consisting of pairs $(\Delta_1, \sum y_k)$ where $\Delta_1 \in |H-(L-x_1)|$ is smooth and $\sum y_k \in |L_0|$. We have proved in Theorem (\ref{IIItheo17}) that for a general pair $(\Delta_1, \sum y_k)$ the linear system $|H|$ embeds $S$ into ${\Bbb{P}}^4$. Indeed in this way we obtain the general smooth surface of degree 8 in $ {\Bbb{P}}^4$. The surface $X={\hat {\Bbb{P}}}^2$, i.e. ${\Bbb{P}}^2$ blown up in $x_1,\ldots, x_4$ is the del Pezzo surface of degree 5. It is well known that Aut$X\cong S_5$ the symmetric group in 5 letters (Aut $X$ acts transitively on the 5 maximal sets of disjoint rational curves on $X$, see \cite [Chapter IV]{M}). \begin{proposition}\label{IIIprop19} For general $S$ the only lines contained in $S$ are the $y_k$'s. \end{proposition} \begin{Proof} Let $l$ be a line on S. The statement is clear if $l$ is $\pi$-exceptional as the $x_i$ are mapped to conics and since we can assume that there are no infinitesimally near points. If $l$ is not skew to the plane spanned by $x_i$ then $l$ is contained in a reducible member of $|D_i|$. But for general choice there is no decomposition A+B with A (or B) a line. Hence we can assume that $l.x_i=0$ for $i=1,\ldots, 4$ and $l.y_k\le 1$ for all $k$. Thus $l\equiv a L-\sum\limits_{k\in \triangle} y_k$ with $a\le 2$. Since $H.l=1$ we have either $a=1$ and $|\triangle|=5$ or $a=2$ and $|\triangle|=11$. In the first case 5 of the $y_k$ are collinear. But then it follows with the monodromy argument of \cite [p.111]{ACGH} that all the $y_k$'s are collinear which is absurd. In the same way the case $a=2$ would imply that all the $y_k$'s are on a conic which also contradicts very ampleness of $|H|$. \end{Proof} \begin{theorem}\label{IIItheo20} The moduli space of polarized rational surfaces (S,H) where $|H|$ embeds $S$ into ${\Bbb{P}}^4$ as a surface of degree 8, speciality 1 and sectional genus 6 is birationally equivalent to ${\cal M}/S_5$. \end{theorem} \begin{Proof} Let ${\cal V}$ be the open set of ${\cal M}$ where $|H|$ embeds $S$ into ${\Bbb{P}}^4$ and where all the $\Delta_i$'s are smooth. Let $(\Delta_1, \sum y_k)$ and $(\Delta_1 ', \sum y_k ')$ be two elements which give rise to surfaces $S, S'\subset {\Bbb{P}}^4$ for which a projective transformation ${\bar g}:S\rightarrow S'$ exists. Since obviously ${\bar g}$ carries lines to lines, it follows from Proposition (\ref{IIIprop19}) that ${\bar g}$ is induced by an automorphism $g:X\rightarrow X$ carrying the set $\{y_k\}$ to $\{y_k'\}$. Conversely, the group $S_5=\mbox { Aut } (X)$ acts on ${\cal V}$ as follows. Let $S$ correspond to $(\Delta_1 \sum y_k)$ and let $g\in \mbox { Aut } (X)$: Then, since $6L-2\sum x_j=-2K_X$ which is invariant under the action of $S_5$, we set $\{y_k'\}=g\{y_k\}, H'=-2K_X-\sum y_k'$. Then $H'$ embeds $S'={\tilde X}(y_1', \ldots, y_{12}')$ and we set $\Delta_1'$ to be the unique curve in $|H'-L+y_1|$. \end{Proof} \section{Further outlook}\label{sectionIV} In this section we want to discuss how this method can possibly be applied to other surfaces. For smooth surfaces of degree $\leq8$ it is rather straightforward to give a decomposition $H\equiv C+D$ which allows to apply the Alexander-Bauer lemma. This was done in \cite{B}, \cite{CF} and section~\ref{sectionIII} of this article. In degree $9$ there is one non-special surface, which was treated in section~\ref{sectionII} of this article, and a special surface with sectional genus $\pi=7$ which was found by Alexander \cite{A2}. Here $S$ is ${\Bbb{P}}^2$ blown up in 15 points $x_1,\ldots,x_{15}$ and $H\equiv 9L-3\sum\limits_{i=1}^6x_i- 2\sum\limits_{j=7}^9x_j-\sum\limits_{k=10}^{15}x_k$. As pointed out by Alexander one can take the decomposition $H\equiv C+D$ where $C\equiv 3L-\sum\limits_{i=1}^9x_i$ and $D\equiv H-C$. Then $C$ is a plane cubic and $|D|$ is a pencil of canonical curves of genus 4. Rational surfaces of degree 10 were treated by Ranestad \cite{R1}, \cite{R2}, Popescu and Ranestad \cite{PR} and Alexander \cite{A2}. There is one surface with $\pi=8$. In this case $S$ is ${\Bbb{P}}^2$ blown up in 13 points and $H\equiv 14L-6x_1-4\sum\limits_{i=2}^{10}x_i-2x_{11}-x_{12}-x_{13}$. Following Alexander \cite{A2} the curve $C\equiv 7L-3x_1-2\sum\limits_{i=2}^{10}x_i-\sum\limits_{j=11}^{13}x_j$ is a plane quartic and the residual pencil $|D|$ has $p(D)=3$ and degree 6. For sectional genus $\pi=9$ there are two possibilities. The first is ${\Bbb{P}}^2$ blown in 18 points with $H\equiv 8L-2\sum\limits_{i=1}^{12}x_i- \sum\limits_{j=13}^{18}x_j$. One can take $C\equiv 4L-\sum\limits_{i=1}^{16} x_i$ which becomes a plane quartic. For the residual intersection $|D|$ one finds $p(D)=3$, $H.D=6$. (For more details of this geometrically interesting situation see \cite[Proposition~2.2]{PR}. The second surface with $\pi=9$ is more difficult. Again we have ${\Bbb{P}}^2$ blown up in 18 points, but this time $H\equiv 9L-3\sum\limits_{i=1}^4x_i- 2\sum\limits_{j=5}^{11}x_j- \sum\limits_{k=12}^{18}x_k$. Clearly $S$ contains plane curves, e.g.~the conics $x_j$. But then for the residual pencil $|D|$ one has $p(D)=7$, $H.D=9$ and this case seems difficult to handle. Numerically it would be possible to have a decomposition $H\equiv C+D$ with $C\equiv 3L-\sum\limits_{i=1}^3x_i- \sum\limits_{j=5}^{11}x_j-x_{12}$ which would be a plane cubic. In this case $p(D)=4$, $H.D=6$. It might be interesting to check whether one can actually construct surfaces with such a decomposition. Of course, one can try and attempt to approach the problem of finding suitable decompositions $H\equiv C+D$ more systematically. Let us assume $S$ is a rational surface and $H\equiv C+D$ a decomposition to which the Alexander-Bauer lemma can be applied. Let $h=h^1({\cal O}_S(H))$ be the speciality of $S$. Since $C$ is mapped to a plane curve the exact sequence $$ 0\longrightarrow{\cal O}_S(D)\longrightarrow{\cal O}_S(H)\longrightarrow {\cal O}_C(H)\longrightarrow 0 $$ is exact on global sections, and hence $$ h=h^1(D)+\delta(C) $$ where $h^1(D)=h^1({\cal O}_S(D))$ and $\delta(C)=h^1({\cal O}_C(H))$. The analogous sequence for $D$ and the assumption that $|H|$ restricts to a complete system on the curves $D'\in|D|$ gives $$ h=h^1(C)+\delta(D) $$ where $h^1(C)$ and $\delta(D)$ are defined similarly. In general if $C$ is a curve of genus $(d-1)(d-2)/2$ and ${\cal O}_C(H)$ is a line bundle of degree $d$ it is difficult to show that $(C,{\cal O}_C(H))$ is a plane curve. Hence it is natural to assume $H.C\leq4$. In order to be able to control the linear system $|H|$ on the curves $D'\in|D|$ one is normally forced to assume that $H.D\geq 2p(D)-2$ and $H|_D=K_D$ in case of equality. Hence $\delta(D)=0$ if $H.D>2p(D)-2$ and $\delta(D)=1$ otherwise. Since $|H|$ is complete on $D$ we have $h^0({\cal O}_D(H))\leq4$. Now using our assumption that $H.D\geq 2p(D)-2$ and Riemann-Roch on $D$ we find $$ 2p(D)-2\leq H.D\leq p(D)+3+\delta(D) $$ and from this $$ p(D)\leq 5+\delta(D). $$ If $\delta(D)=0$ then $p(D)\leq5$. If $\delta(D)=1$ then $H|_D=K_D$ and $h^0({\cal O}_D(H))=p(D)$, i.e.~$p(D)\leq4$ in this case. But now $$ d=H.C+H.D\leq p(D)+7+\delta(D). $$ This shows that one can find such a decomposition only if the degree $d\leq12$. The case $d=12$ can only occur for $H.C=4$. Finally we want to discuss the case $d=11$. In his thesis Popescu \cite{P} gave three examples of rational surfaces of degree 11. In each case it is ${\Bbb{P}}^2$ blown up in 20 points. The linear systems are as follows: \begin{eqnarray} H&\equiv&10L-4x_1-3\sum_{i=2}^4x_i-2\sum_{j=5}^{14}x_j-\sum_{k=15}^{20}x_k \label{IVgl1}\\ H&\equiv&11L-5x_1-3\sum_{i=2}^7x_i-2\sum_{j=8}^{13}x_j-\sum_{k=14}^{20}x_k \label{IVgl2}\\ H&\equiv&13L-5x_1-4\sum_{i=2}^8x_i-2\sum_{j=9}^{11}x_j-\sum_{k=12}^{20}x_k \label{IVgl3} \end{eqnarray} In each of these cases $S$ contains a plane quintic. The residual intersection gives a pencil of rational (cases (\ref{IVgl1}) and (\ref{IVgl2})), resp.~elliptic (case (\ref{IVgl3})) sextics. Since the linear system $|H|$ is not complete on the curves of this linear system, one cannot immediately apply the Alexander-Bauer lemma to this decomposition. One can ask whether there are decompositions fulfilling the conditions given above. A candidate in case (\ref{IVgl1}) is given by $C\equiv 4L-x_1-\sum\limits_{i=2}^4x_i- \sum\limits_{j=5}^{14}x_j-\sum\limits_{k=15}^{17}x_k$ and $D\equiv H-C$. We do not know whether surfaces with such a decomposition actually occur. In the other cases one can show that no such decompositions exist. \bibliographystyle{alpha}

9505

alg-geom/9505003

en

https://arxiv.org/abs/alg-geom/9505003

alg-geom/9505003

Atsushi Moriwaki

Atsushi Moriwaki

Bogomolov conjecture over function fields for stable curves with only irreducible fibers (Version 2.0)

21 pages (with 1 figure), AMSLaTeX version 1.2 (In this version 2.0, the restriction of characteristic is removed.)

Let K be a function field and C a non-isotrivial curve of genus g >= 2 over K. In this paper, we will show that if C has a global stable model with only geometrically irreducible fibers, then Bogomolov conjecture over function fields holds.

alg-geom

\section{Introduction} Let $k$ be a field, $X$ a smooth projective surface over $k$, $Y$ a smooth projective curve over $k$, and $f : X \to Y$ a generically smooth semistable curve of genus $g \geq 2$ over $Y$. Let $K$ be the function field of $Y$, $\overline{K}$ the algebraic closure of $K$, and $C$ the generic fiber of $f$. For $D \in \operatorname{Pic}^1(C)(\overline{K})$, let $j_D : C_{\overline{K}} \to \operatorname{Pic}^0(C)_{\overline{K}}$ be an embedding defined by $j_D(x) = x - D$. Then, we have the following conjecture due to Bogomolov. \begin{Conjecture}[Bogomolov conjecture over function fields] \label{conj:Geometric:Bogomolov:Conjecture} If $f$ is non-isotrivial, then, for any embedding $j_D$, the image $j_D(C(\overline{K}))$ is discrete in terms of the semi-norm $\Vert \ \Vert_{NT}$ given by the Neron-Tate height pairing on $\operatorname{Pic}^0(C)(\overline{K})$, i.e., for any point $P \in \operatorname{Pic}^0(C)(\overline{K})$, there is a positive number $\epsilon$ such that the set \[ \left\{ x \in C(\overline{K}) \mid \Vert j_D(x) - P \Vert_{NT} \leq \epsilon \right\} \] is finite. \end{Conjecture} In this paper, we will prove the above conjecture under the assumption that the stable model of $f : X \to Y$ has only geometrically irreducible fibers. \begin{Theorem} \label{thm:conj:bogomolov} If the stable model of $f : X \to Y$ has only geometrically irreducible fibers, then Conjecture~\ref{conj:Geometric:Bogomolov:Conjecture} holds. More strongly, there is a positive number $A$ with the following properties. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item ${\displaystyle A \geq \sqrt{\frac{g-1}{12g(2g+1)}\delta}}$, where $\delta$ is the number of singularities in singular fibers of $f_{\bar{k}} : X_{\bar{k}} \to Y_{\bar{k}}$. \item For any small positive number $\epsilon$, the set \[ \left\{ x \in C(\overline{K}) \mid \Vert j_D(x) - P \Vert_{NT} \leq (1 - \epsilon)A \right\} \] is finite for any embedding $j_D$ and any point $P \in \operatorname{Pic}^0(C)(\overline{K})$. \end{enumerate} \end{Theorem} Our proof of Theorem~\ref{thm:conj:bogomolov} is based on the admissible pairing on semistable curves due to S. Zhang (cf. \S\ref{sec:metrized:graph:green:function:admissible:pairing}), Cornalba-Harris-Xiao's inequality over an arbitrary field (cf. Theorem~\ref{thm:Cornalba-Harris-Xiao:inequality}) and an exact calculation of a Green function on a certain metrized graph (cf. Lemma~\ref{lem:green:union:circle}). The estimation of a Green function also gives the following result, which strengthen S. Zhang's theorem \cite{Zh}. \begin{Theorem}[cf. Corollary~\ref{cor:lower:bound:w:w:not:smooth}] Let $K$ be a number field, $O_K$ the ring of integers, $f : X \to \operatorname{Spec}(O_K)$ a regular semistable arithmetic surface of genus $g \geq 2$ over $O_K$. If $f$ is not smooth, then \[ (\omega_{X/O_K}^{Ar} \cdot \omega_{X/O_K}^{Ar}) \geq \frac{\log 2}{6(g-1)}. \] \end{Theorem} \section{Metrized graph, Green function and admissible pairing} \label{sec:metrized:graph:green:function:admissible:pairing} In this section, we recall several facts of metrized graphs, Green functions and the admissible pairing on semistable curves. Details can be found in Zhang's paper \cite{Zh}. Let $G$ be a locally metrized and compact topological space. We say $G$ is a metrized graph if, for any $x \in G$, there is a positive number $\epsilon$, a positive integer $d = v(x)$ (which is called the valence at $x$), and an open neighborhood $U$ of $x$ such that $U$ is isometric to \[ \left\{ t e^{\frac{2 \pi \sqrt{-1} k}{d}} \in {\Bbb{C}} \mid 0 \leq t < \epsilon, k \in {\Bbb{Z}} \right\}. \] Let $\operatorname{Div}(G)$ be a free abelian group generated by points of $G$. An element of $\operatorname{Div}(G)$ is called a {\em divisor} on $G$. Let $F(G)$ be the set of all piecewisely smooth real valued functions on $G$. For $f \in F(G)$, we can define the Dirac function $\delta(f)$ associated with $f$ as follows. If $x \in G$ and $v(x) = n$, then $\delta(f)(x)$ is given by \[ (\delta(f)(x), g) = g(x) \sum_{i=1}^n \lim_{x_i \to 0} f'(x_i), \] where $g \in F(G)$ and $x_i$ is the arc-length parameter of one branch starting from $x$. The Laplacian $\Delta$ for $f \in F(G)$ is defined by \[ \Delta(f) = -f'' - \delta(f), \] where $f''$ is the second derivative of $f$ in the sense of distribution. Let $Q(G)$ be a subset of $F(G)$ consisting of piecewisely quadric polynomial functions. Let $V$ be a set of vertices of $G$ such that $G \setminus V$ is a disjoint union of open segments. Let $E$ be the collection of segments in $G \setminus V$. We denote by $Q(G, V)$ a subspace of $Q(G)$ consisting of functions whose restriction to each edge in $E$ are quadric polynomial functions, and by $M(G, V)$ a vector space of measures on $G$ generated by Dirac functions $\delta_v$ at $v \in V$ and by Lebesgue measures on edges $e \in E$ arising from the arc-length parameter. The fundamental theorem is the following existence of the admissible metric and the Green function. \begin{Theorem}[{\cite[Theorem 3.2]{Zh}}] \label{thm:existence:metric:green} Let $D = \sum_{x \in G} d_x x$ be a divisor on $G$ such that the support of $D$ is in $V$. If $G$ is connected and $\deg(D) \not= -2$, then there are a unique measure $\mu \in M(G, V)$ and a unique function $g_{\mu}$ on $G \times G$ with the following properties. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item ${\displaystyle \int_{G} \mu = 1}$. \item $g_{\mu}(x, y)$ is symmetric and continuous on $G \times G$. \item For a fixed $x \in G$, $g_{\mu}(x, y) \in Q(G)$. Moreover, if $x \in V$, then $g_{\mu}(x, y) \in Q(G, V)$. \item For a fixed $x \in G$, $\Delta_y(g_{\mu}(x, y)) = \delta_x - \mu$. \item For a fixed $x \in G$, ${\displaystyle \int_G g_{\mu} (x, y) \mu(y) = 0}$. \item $g_{\mu}(D, y) + g_{\mu}(y, y)$ is a constant for all $y \in G$, where $g_{\mu}(D, y) = \sum_{x \in G} d_x g_{\mu}(x, y)$. \end{enumerate} Further, if $d_x \geq v(x) - 2$ for all $x \in G$, then $\mu$ is positive. \end{Theorem} The measure $\mu$ in Theorem~\ref{thm:existence:metric:green} is called the {\em admissible metric} with respect to $D$ and $g_{\mu}$ is called the {\em Green function} with respect to $\mu$. The constant $g_{\mu}(D, y) + g_{\mu}(y, y)$ is denoted by $c(G, D)$. \bigskip Let $k$ be an algebraically closed field, $X$ a smooth projective surface over $k$, $Y$ a smooth projective curve over $k$, and $f : X \to Y$ a generically smooth semistable curve of genus $g \geq 1$ over $Y$. Let $\operatorname{CV}(f)$ be the set of all critical values of $f$, i.e., $y \in \operatorname{CV}(f)$ if and only if $f^{-1}(y)$ is singular. For $y \in \operatorname{CV}(f)$, let $G_y$ be the metrized graph of $f^{-1}(y)$ defined as follows. The set of vertices $V_y$ of $G_y$ is indexed by irreducible components of the fiber $f^{-1}(y)$ and singularities of $f^{-1}(y)$ correspond to edges of length $1$. We denote by $C_v$ the corresponding irreducible curve for a vertex $v$ in $V_y$. Let $K_y$ be a divisor on $G_y$ given by \[ K_y = \sum_{v \in V_y} (\omega_{X/Y} \cdot C_v) v. \] Let $\mu_y$ be the admissible metric with respect to $K_y$ and $g_{\mu_y}$ the Green function of $\mu_y$. The admissible dualizing sheaf $\omega_{X/Y}^a$ is defined by \[ \omega_{X/Y}^a = \omega_{X/Y} - \sum_{y \in \operatorname{CV}(f)} c(G_y, K_y) f^{-1}(y). \] Here we define a new pairing $(D \cdot E)_a$ for $D, E \in \operatorname{Div}(X) \otimes {\Bbb{R}}$ by \[ (D \cdot E)_a = (D \cdot E) + \sum_{y \in \operatorname{CV}(f)} \left\{ \sum_{v, v' \in V_y} (D \cdot C_v) g_{\mu_y}(v, v') (E \cdot C_{v'}) \right\}. \] This pairing is called the {\em admissible pairing}. It has lots of properties. For our purpose, the following are important. \medskip \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item (Adjunction formula) If $B$ is a section of $f$, then $(\omega_{X/Y}^a + B \cdot B)_a = 0$. \item (Intersection with a fiber) If $D$ is an ${\Bbb{R}}$-divisor with degree $0$ along general fibers, then $(D \cdot Z)_a = 0$ for all vertical curves $Z$. (cf. Proposition~\ref{prop:admissible:with:fiber}) \item (Compatibility with base changes) The admissible pairing is compatible with base changes. Namely, let $\pi : Y' \to Y$ be a finite morphism of smooth projective curves, and $X'$ the minimal resolution of the fiber product of $X \times_Y Y'$. We set the induced morphisms as follows. \[ \begin{CD} X @<{\pi'}<< X' \\ @V{f}VV @VV{f'}V \\ Y @<<{\pi}< Y' \end{CD} \] Then, for $D, E \in \operatorname{Div}(X) \otimes {\Bbb{R}}$, $({\pi'}^*(D) \cdot {\pi'}^*(E))_a = (\deg \pi)(D \cdot E)_a$. Moreover, we have ${\pi'}^*(\omega_{X/Y}^a) = \omega_{X'/Y'}^a$. Thus, $(\omega_{X'/Y'}^a \cdot \omega_{X'/Y'}^a)_a = (\deg \pi) (\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a$. \end{enumerate} \medskip\noindent Using the above properties, we can give the Neron-Tate height paring in terms of the admissible pairing. Let $C$ be the generic fiber of $f$, $K$ the function field of $Y$, and $L, M \in \operatorname{Pic}^0(C)(\overline{K})$. Then, there are a base change $Y' \to Y$, a semistable model $X'$ of $C$ over $Y'$, and line bundles ${\cal L}$ and ${\cal M}$ on $X'$ such that ${\cal L}_{\overline{K}} = L$ and ${\cal M}_{\overline{K}} = M$. Moreover, we can find vertical ${\Bbb{Q}}$-divisors $V$ and $V'$ on $X'$ such that $({\cal L} + V \cdot Z) = ({\cal M} + V' \cdot Z) = 0$ for all vertical curves $Z$ on $X'$. Then, it is easy to see that \[ \frac{-1}{[k(Y') : k(Y)]}({\cal L} + V \cdot {\cal M} + V') \] is well-defined. It is denoted by $(L \cdot M)_{NT}$ and is called the {\em Neron Tate height pairing}. Moreover, it is easy to see $(L \cdot L)_{NT} \geq 0$. So $\sqrt{(L \cdot L)_{NT}}$ is denoted by $\Vert L \Vert_{NT}$. On the other hand, by the definition of the admissible pairing, we have \[ ({\cal L} + V \cdot {\cal M} + V') = ({\cal L} + V \cdot {\cal M} + V')_a. \] Thus, using the second property of the above, we can see that \[ -[k(Y') : k(Y)] (L \cdot M)_{NT} = ({\cal L}\cdot {\cal M})_a, \] which means that the admissible pairing does not depend on the choice of the compactification of $L$ and $M$, and that of course \[ (L \cdot M)_{NT} = \frac{-({\cal L}\cdot {\cal M})_a}{[k(Y') : k(Y)]}. \] \medskip Next, let us consider a height function in terms of the admissible pairing. Let ${\cal L}$ be an ${\Bbb{R}}$-divisor on $X$ and $x \in C(\overline{K})$. Then, taking a suitable base change $\pi : Y' \to Y$, there is a semistable model $f' : X' \to Y'$ of $C$ such that $x$ is realized as a section $B_x$ of $f'$. We set \[ h^a_{{\cal L}}(x) = \frac{({\pi'}^*({\cal L}) \cdot B_x)_a}{\deg \pi}, \] where $\pi' : X' \to X$ is the induced morphism. One can easily see that $h^a_{{\cal L}}(x)$ is well-defined by the third property of the above. The following generic lower estimate of the height function is important for our purpose. \begin{Theorem}[{\cite[Theorem 5.3]{Zh}}] \label{thm:lower:estimate:height} If $\deg({\cal L}_K) > 0$ and ${\cal L}$ is $f$-nef, then, for any $\epsilon > 0$, there is a finite subset $S$ of $C(\overline{K})$ such that \[ h^a_{{\cal L}}(x) \geq \frac{({\cal L} \cdot {\cal L})_a}{2 \deg({\cal L}_K)} - \epsilon \] for all $x \in C(\overline{K}) \setminus S$. \end{Theorem} As corollary, we have the following. \begin{Corollary}[{\cite[Theorem 5.6]{Zh}}] \label{cor:lower:estimate:NT:metric} Let $D \in \operatorname{Pic}^1(C)(\overline{K})$. Then, for any $\epsilon > 0$, there is a finite subset $S$ of $C(\overline{K})$ such that \[ \Vert D - x \Vert_{NT}^2 \geq \frac{(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a}{4(g-1)} + \frac{\Vert \omega_C - (2g-2) D \Vert_{NT}^2}{4g(g-1)} - \epsilon \] for all $x \in C(\overline{K}) \setminus S$. \end{Corollary} {\sl Proof.}\quad Let $\pi_1 : Y_1 \to Y$ be a base change of $f : X \to Y$ such that $D$ is defined over the function field $k(Y_1)$ of $Y_1$. Let $f_1 : X_1 \to Y_1$ be the semistable model of $C$ over $Y_1$, $F$ a general fiber of $f_1$, and ${\cal D}$ a compactification of $D$ such that ${\cal D}$ is a ${\Bbb{Q}}$-divisor on $X_1$ and ${\cal D}$ is $f_1$-nef. Using adjuction formula and applying Theorem~\ref{thm:lower:estimate:height} to \[ {\cal L} = \omega_{X_1/Y_1}^a + 2 {\cal D} - ({\cal D} \cdot {\cal D})_a F, \] we have our corollary. \QED \section{Green function of a certain metrized graph} \label{sec:certain:graph} In this section, we will construct a Green function of a certain metrized graph. Let us begin with the following lemma. \begin{Lemma} \label{lem:laplacian:on:circle} Let $C$ be a circle with arc-length $l$. Fixing a point $O$ on $C$, let $t : C \to [0, l)$ be a coordinate of $C$ with $t(O) = 0$ coming from an arc-length parameterization of $C$. We set \[ \phi(t) = \frac{1}{2l} t^2 - \frac{1}{2} |t| \quad\text{and}\quad f(x, y) = \phi(t(x) - t(y)). \] Then, we have the following. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item $f(x, y)$ is symmetric and continuous on $C \times C$. \item $f(x, y)$ is smooth on the outside of the diagonal. \item For a fixed $x \in C$, ${\displaystyle \Delta_y(f(x, y)) = \delta_x - \frac{dt}{l}}$. \end{enumerate} \end{Lemma} {\sl Proof.}\quad We can check them by a straightforward calculation. \QED Let $C_1, \ldots, C_n$ be circles and $G$ a metrized graph constructed by joining $C_i$'s at a point $O$. Let $l_i$ be the arc-length of $C_i$ and $t_i : C_i \to [0, l_i)$ a coordinate of $C_i$ with $t_i(O) = 0$. \yes \par\bigskip \Draw \MoveTo(0,0) \MarkLoc(O) \Node(Q)(--$\bullet$--) \MoveTo(30,60) \MarkLoc(A) \MoveTo(-30,60) \MarkLoc(B) \MoveTo(-66.96,-4.02) \MarkLoc(C) \MoveTo(-36.96,-55.98) \MarkLoc(D) \MoveTo(36.96,-55.98) \MarkLoc(E) \MoveTo(66.96,-4.02) \MarkLoc(F) \Curve(O,A,B,O) \Curve(O,C,D,O) \Curve(O,E,F,O) \MoveTo(10,7) \Node(G)(--$O$--) \MoveTo(0,55) \Node(H)(--$C_1$--) \MoveTo(-47.63,-27.5) \Node(H)(--$C_2$--) \MoveTo(47.63,-27.5) \Node(H)(--$C_3$--) \EndDraw \bigskip \par\noindent \else \fi {}From now on, we will identify a point on $C_i$ with its coordinate. As in Lemma~\ref{lem:laplacian:on:circle}, for each $i$, we set \[ \phi_i(t) = \frac{1}{2l_i} t^2 - \frac{1}{2} |t|. \] We fix a positive integer $g$. Here we consider a measure $\mu$ and a divisor $K$ on $G$ defined by \[ \mu = \frac{g-n}{g} \delta_O + \sum_{i=1}^n \frac{d t_i}{gl_i} \quad\text{and}\quad K = (2g-2)O. \] Moreover, let us consider the following function $g_{\mu}$ on $G \times G$. \[ g_{\mu}(x, y) = \begin{cases} {\displaystyle \phi_i(x - y) - \frac{g-1}{g} \left(\phi_i(x) + \phi_i(y)\right) + \frac{L}{12g^2}} & \text{if $x, y \in C_i$} \\ {\displaystyle \frac{1}{g}\left( \phi_i(x) + \phi_j(y) \right) + \frac{L}{12g^2}} & \text{if $x \in C_i$, $y \in C_j$ and $i \not= j$} \end{cases} \] where $L = l_1 + \cdots + l_n$. Then, we can see the following. \begin{Lemma} \label{lem:green:union:circle} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item ${\displaystyle \int_G \mu = 1}$. \item $g_{\mu}(x, y)$ is symmetric and continuous on $G \times G$. \item For a fixed $x \in G$, $\Delta_y(g_{\mu}(x, y)) = \delta_x - \mu$. \item For a fixed $x \in G$, ${\displaystyle \int_G g_{\mu}(x, y) \mu(y) = 0}$. \item ${\displaystyle g_{\mu}(K, y) + g_{\mu}(y, y) = \frac{L(2g-1)}{12g^2}}$ for all $y \in G$. \end{enumerate} \end{Lemma} {\sl Proof.}\quad (1), (2) These are obvious. (3) We assume $x \in C_i$. By \cite[Lemma a.4, (a)]{Zh}, \[ \Delta_y(g_{\mu}(x, y)) = \sum_{j=1}^n \Delta_y(\rest{g_{\mu}(x, y)}{C_j}). \] Therefore, using Lemma~\ref{lem:laplacian:on:circle}, we get \begin{align*} \Delta_y(g_{\mu}(x, y)) & = \Delta_y(\rest{g_{\mu}(x, y)}{C_i}) + \sum_{j \not= i}^n \Delta_y(\rest{g_{\mu}(x, y)}{C_j}) \\ & = \left( \delta_x - \frac{dt_i}{l_i} - \frac{g-1}{g}\left(\delta_O - \frac{d t_i}{l_i}\right)\right) + \sum_{j \not= i}^n \frac{1}{g}\left( \delta_O - \frac{dt_j}{l_j} \right) \\ & = \delta_x - \mu. \end{align*} (4) We assume $x \in C_i$. Then, by a direct calculation, we can see \[ \int_{C_j} g_{\mu}(x, t_j) \frac{d t_j}{gl_j} = \begin{cases} {\displaystyle -\frac{g-1}{g^2}\phi_i(x) - \frac{l_i}{12g^2} + \frac{L}{12g^3}} & \text{if $j = i$} \\ {\displaystyle \frac{1}{g^2} \phi_i(x) - \frac{l_j}{12g^2} + \frac{L}{12g^3}} & \text{if $j \not= i$} \end{cases} \] Therefore, \[ \sum_{j=1}^n \int_{C_j} g_{\mu}(x, t_j) \frac{d t_j}{gl_j} = \frac{n-g}{g}\left( \frac{1}{g} \phi_i(x) + \frac{L}{12g^2} \right). \] Hence, \begin{align*} \int_G g_{\mu}(x, y) \mu(y) & = \frac{g-n}{g}g_{\mu}(x, 0) + \sum_{j=1}^n \int_{C_j} g_{\mu}(x, t_j) \frac{d t_j}{gl_j} \\ & = \frac{g-n}{g}\left( \frac{1}{g} \phi_i(x) + \frac{L}{12g^2} \right) + \frac{n-g}{g}\left( \frac{1}{g} \phi_i(x) + \frac{L}{12g^2} \right) \\ & = 0. \end{align*} (5) Since \[ g_{\mu}(O, x) = \frac{1}{g} \phi_i(x) + \frac{L}{12g^2} \quad\text{and}\quad g_{\mu}(x, x) = \frac{-2(g-1)}{g} \phi_i(x) + \frac{L}{12g^2}, \] (5) follows. \QED This lemma says us that $\mu$ is the admissible metric with respect to $K$, $g_{\mu}$ is the Green function of $\mu$, and ${\displaystyle c(G, K) = \frac{L(2g-1)}{12g^2}}$. \section{Cornalba-Harris-Xiao's inequality over an arbitrary field} In this section, we would like to generalize Cornalba-Harris-Xiao's inequality to fibered algebraic surfaces over an arbitrary field, namely, \begin{Theorem} \label{thm:Cornalba-Harris-Xiao:inequality} Let $k$ be a field, $X$ a smooth projective surface over $k$, $Y$ a smooth projective curve over $k$, and $f : X \to Y$ a generically smooth morphism with $f_*{\cal{O}}_X = {\cal{O}}_C$. If the genus $g$ of the generic fiber of $f$ is greater than or equal to $2$ and $\omega_{X/Y}$ is $f$-nef, then \[ (\omega_{X/Y} \cdot \omega_{X/Y}) \geq \frac{4(g-1)}{g} \deg(f_*(\omega_{X/Y})). \] \end{Theorem} The above was proved in \cite{CH} and \cite{Xi} under the assumption $\operatorname{char}(k) = 0$. Here we prove it using the following result of Bost. \begin{Theorem}[{\cite[Theorem III]{Bo}}] \label{thm:bost:ineq} Let $k$ be a field, $Y$ a smooth projective curve over $k$, and $E$ a vector bundle on $Y$. Let \[ \pi : P = \operatorname{Proj}\left( \bigoplus_{n=0}^{\infty} \operatorname{Sym}^n(E) \right) \longrightarrow Y \] be the projective bundle of $E$ and ${\cal{O}}_P(1)$ the tautological line bundle on $P$. If an effective cycle $Z$ of dimension $d \geq 1$ on $P$ is Chow semistable on the generic fiber of $\pi$, then \[ \frac{\left({\cal{O}}_P(1)^d \cdot Z \right)} {d \cdot \left( {\cal{O}}_P(1)^{d-1} \cdot Z \cdot F \right)} \geq \frac{\deg E}{\operatorname{rk} E}, \] where $F$ is a general fiber of $\pi$. \end{Theorem} \medskip First of all, let us begin with the following lemmas. \begin{Lemma} \label{lem:Chow:semistable:canonical} Let $K$ be a field, $C$ a smooth projective curve over $K$ of genus $g \geq 2$, and $\phi : C \to {\Bbb{P}}^{g -1}$ a morphism given by the complete linear system $|\omega_C|$. Then $\phi_*(C)$ is a Chow semistable cycle on ${\Bbb{P}}^{g-1}$. \end{Lemma} \proof Let $R$ be the image of $C$ by $\phi$ and $n$ an integer given by \[ n = \begin{cases} 1, & \text{if $C$ is non-hyperelliptic}, \\ 2, & \text{if $C$ is hyperelliptic}. \end{cases} \] Then, $\phi_*(C) = nR$. Thus a Chow form of $\phi_*(C)$ is the $n$-th power of a Chow form of $R$. Therefore, $\phi_*(C)$ is Chow semistable if and only if $R$ is Chow semistable. Moreover, Theorem~4.12 in \cite{Mu} says that Chow semistability of $R$ is derived from linear semistability of $R$ . Let $V$ be a subspace of $H^0(C, \omega_C)$, $p : {\Bbb{P}}^{g-1} \dashrightarrow {\Bbb{P}}^{\dim V - 1}$ the projection defined by the inclusion $V \hookrightarrow H^0(C, \omega_C)$, and $\phi' : C \to {\Bbb{P}}^{\dim V - 1}$ a morphism given by $V$. Then, $p \cdot \phi = \phi'$. We need to show that \addtocounter{Claim}{1} \begin{equation} \label{eqn:lem:stable:kernel:Chow:stable} \frac{2}{n} = \frac{\deg(R)}{g - 1} \leq \frac{\deg(p_*(R))}{\dim V - 1} \end{equation} to see linear semistability of $R$. Since $\deg({\phi'}^*({\cal{O}}(1))) = n \deg(p_*(R))$, (\ref{eqn:lem:stable:kernel:Chow:stable}) is equivalent to say \[ 2 \leq \frac{\deg({\phi'}^*({\cal{O}}(1)))}{\dim V - 1}. \] On the other hand, if we denote by $\omega^V_C$ the image of $V \otimes {\cal{O}}_C \to \omega_C$, then, by Clifford's lemma, we have \[ \dim V - 1 \leq \dim |\omega^V_C| \leq \frac{\deg(\omega^V_C)}{2}. \] Thus, we get (\ref{eqn:lem:stable:kernel:Chow:stable}) because $\deg({\phi'}^*({\cal{O}}(1))) = \deg(\omega^V_C)$. \QED \begin{Remark} By \cite[Proposition~4.2]{Bo}, $\phi_*(C)$ is actually Chow stable when $\operatorname{char}(K) = 0$. We don't know whether $\phi_*(C)$ is Chow stable if $\operatorname{char}(K) > 0$. Anyway, semistability is enough for our purpose. \end{Remark} \medskip Let us start the proof of Theorem~\ref{thm:Cornalba-Harris-Xiao:inequality}. Let \[ \phi : X \dashrightarrow P = \operatorname{Proj}\left( \bigoplus_{n=0}^{\infty} \operatorname{Sym}^n(f_*(\omega_{X/Y})) \right) \] be a rational map over $Y$ induced by $f^*f_*(\omega_{X/Y}) \to \omega_{X/Y}$. Here we take a birational morphism $\mu : X' \to X$ of smooth projective varieties such that $\phi' = \phi \cdot \mu : X' \to P$ is a morphism. \[ \begin{CD} X' @>{\mu}>> X \\ @V{\phi'}VV @VV{\phi}V \\ P @= P \end{CD} \] Then, there is an effective vertical divisor $D$ on $X'$ such that $\mu^*(\omega_{X/Y}) = {\phi'}^*({\cal{O}}_P(1)) + D$. Let $Z = {\phi'}_*(X')$. Then, by Lemma~\ref{lem:Chow:semistable:canonical}, $Z$ give a Chow semistable cycle on the generic fiber. Thus, by Theorem~\ref{thm:bost:ineq}, we have \[ \frac{\left({\phi'}^*({\cal{O}}_P(1)) \cdot {\phi'}^*({\cal{O}}_P(1))\right)}{4(g-1)} \geq \frac{\deg(f_*(\omega_{X/Y}))}{g}. \] On the other hand, since $\omega_{X/Y}$ is $f$-nef and $(D \cdot D) \leq 0$, \begin{align*} \left({\phi'}^*({\cal{O}}_P(1)) \cdot {\phi'}^*({\cal{O}}_P(1))\right) & = \left( \mu^*(\omega_{X/Y}) - D \cdot \mu^*(\omega_{X/Y}) - D \right) \\ & = \left( \omega_{X/Y} \cdot \omega_{X/Y} \right) -2 \left(\mu^*(\omega_{X/Y}) \cdot D \right) + (D \cdot D) \\ & \leq \left( \omega_{X/Y} \cdot \omega_{X/Y} \right). \end{align*} Therefore, we have our desired inequality. \QED \begin{Remark} \label{rem:semistable:kernel:another:proof} If $\operatorname{char}(k) = 0$, we can give another proof of Theorem~\ref{thm:Cornalba-Harris-Xiao:inequality} according to \cite{Mo2}. A rough idea is the following. Since the kernel $K$ of $f^*f_*(\omega_{X/Y}) \to \omega_{X/Y}$ is semistable on the generic fiber of $f$ by virtue of \cite{PR}, we can apply Bogomolov-Gieseker's inequality to $K$, which implies Cornalba-Harris-Xiao's inequality by easy calculations. \end{Remark} \section{Proof of Theorem~\ref{thm:conj:bogomolov}} In this section, we would like to give the proof of Theorem~\ref{thm:conj:bogomolov}. First of all, let us fix notations. Let $k$ be a field, $X$ a smooth projective surface over $k$, $Y$ a smooth projective curve over $k$, and $f : X \to Y$ a generically smooth semistable curve of genus $g \geq 2$ over $Y$. Let $K$ be the function field of $Y$, $\overline{K}$ the algebraic closure of $K$, and $C$ the generic fiber of $f$. We assume that $f$ is non-isotrivial and the stable model of $f : X \to Y$ has only geometrically irreducible fibers. Clearly, for the proof of Theorem~\ref{thm:conj:bogomolov}, we may assume that $k$ is algebraically closed. Then, we have the following lower estimate of $(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a$. \begin{Theorem} \label{thm:lower:bound:admissible:intersection} Under the above assumptions, $(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a$ is positive. Moreover, \[ (\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a \geq \frac{(g-1)^2}{3g(2g+1)} \delta, \] where $\delta$ is the number of singularities in singular fibers of $f$. \end{Theorem} {\sl Proof.}\quad Let $\operatorname{CV}(f)$ be the set of all critical values of $f$. For $y \in \operatorname{CV}(f)$, the number of singularities of $f^{-1}(y)$ is denoted by $\delta_y$. Let $G_y$ be the metrized graph of $f^{-1}(y)$ as in \S\ref{sec:metrized:graph:green:function:admissible:pairing}. Then, the total arc-length of $G_y$ is $\delta_y$. Let $K_y$ be the divisor on $G_y$ coming from $\omega_{X/Y}$ as in \S\ref{sec:metrized:graph:green:function:admissible:pairing}, $\mu_y$ the admissible metric of $K_y$, and $g_{\mu_y}$ the Green function of $\mu_y$. By the definition of $\omega_{X/Y}^a$ (see \S\ref{sec:metrized:graph:green:function:admissible:pairing}), we have \[ (\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a = (\omega_{X/Y} \cdot \omega_{X/Y}) + \sum_{ y \in \operatorname{CV}(f)} \left\{ g_{\mu_y}(K_y, K_y) - 2(2g-2) c(G_y, K_y) \right\}. \] On the other hand, $G_y$ is isometric to the graph treated in \S\ref{sec:certain:graph}. Thus, by Lemma~\ref{lem:green:union:circle}, \begin{align*} g_{\mu_y}(K_y, K_y) - 2(2g-2) c(G_y, K_y) & = (2g-2)^2 \frac{\delta_y}{12g^2} - 2(2g-2) \frac{(2g-1) \delta_y}{12g^2} \\ & = -\frac{g-1}{3g} \delta_y. \end{align*} Thus \[ (\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a = (\omega_{X/Y} \cdot \omega_{X/Y}) - \frac{g-1}{3g} \delta. \] By virtue of Theorem~\ref{thm:Cornalba-Harris-Xiao:inequality} and Noether formula \[ \deg(f_*(\omega_{X/Y})) = \frac{(\omega_{X/Y} \cdot \omega_{X/Y}) + \delta}{12}, \] we have \[ ( \omega_{X/Y} \cdot \omega_{X/Y} ) \geq \frac{g-1}{2g+1} \delta. \] Therefore, we get \[ (\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a \geq \frac{(g-1)^2}{3g(2g+1)} \delta. \] In particular, $(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a > 0$ if $f$ is not smooth. Further, if $f$ is smooth, then \[ (\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a = (\omega_{X/Y} \cdot \omega_{X/Y}) > 0 \] because $f$ is non-isotrivial. \QED \bigskip Let us start the proof of Theorem~\ref{thm:conj:bogomolov}. We set \[ A = \sqrt{\frac{(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a}{4(g-1)}}. \] Then, by Theorem~\ref{thm:lower:bound:admissible:intersection}, $A$ is positive and \[ A \geq \sqrt{\frac{g-1}{12g(2g+1)} \delta}. \] By virtue of Corollary~\ref{cor:lower:estimate:NT:metric}, for any $D \in \operatorname{Pic}^1(C)(\overline{K})$ and any $P \in \operatorname{Pic}^0(C)(\overline{K})$, there is a finite subset $S$ of $C(\overline{K})$ such that \[ \Vert x - D - P \Vert_{NT} > (1-\epsilon)A \] for all $x \in C(\overline{K}) \setminus S$. Therefore, we have \[ \left\{ x \in C(\overline{K}) \mid \Vert j_D(x) - P \Vert_{NT} \leq (1 - \epsilon)A \right\} \subset S. \] Thus, we get the second property of $A$. \QED \section{Effective lower bound of $(\omega \cdot \omega)$ for arithmetic surfaces} Let $K$ be a number field, $O_K$ the ring of integers, $f : X \to \operatorname{Spec}(O_K)$ a regular semistable arithmetic surface of genus $g \geq 2$ over $O_K$. In \cite{Mo3}, we proved the following. \begin{Theorem} If geometric fibers $X_{\overline{P}_1}, \ldots, X_{\overline{P}_n}$ of $X$ at $P_1, \ldots, P_n \in \operatorname{Spec}(O_K)$ are reducible, then \[ \left(\omega_{X/O_K}^{Ar} \cdot \omega_{X/O_K}^{Ar} \right) \geq \sum_{i=1}^n \frac{\log \#(O_K/P_i) }{6(g-1)}. \] \end{Theorem} Using Lemma~\ref{lem:green:union:circle}, we have the following exact lower estimate for stable curves with only irreducible fibers. \begin{Theorem} Assume that the stable model of $f : X \to \operatorname{Spec}(O_K)$ has only geometric irreducible fibers. If $\{ P_1, \ldots, P_n \}$ is the set of critical values of $f$, then \[ (\omega_{X/O_K}^{Ar} \cdot \omega_{X/O_K}^{Ar}) \geq \sum_{i=1}^n \frac{g-1}{3g} \delta_i \log\#(O_K/P_i), \] where $\delta_i$ is the number of singularities of the geometric fiber at $P_i$. Moreover, equality holds if and only if there is a sequence of distinct points $x_1, x_2, \ldots$ of $X(\overline{{\Bbb{Q}}})$ such that \[ \lim_{i \to \infty} \Vert (2g-2) x_i - \omega \Vert_{NT} = 0. \] \end{Theorem} {\sl Proof.}\quad By virtue of Lemma~\ref{lem:green:union:circle}, \[ (\omega_{X/O_K}^a \cdot \omega_{X/O_K}^a)_a = (\omega_{X/O_K}^{Ar} \cdot \omega_{X/O_K}^{Ar}) - \sum_{i=1}^n \frac{g-1}{3g} \delta_i \log \#(O_K/P_i). \] Therefore, our theorem follows from \cite[Corollary 5.7]{Zh}. \QED Combining the above two theorems, we have the following corollary, which is a stronger version of S. Zhang's result \cite{Zh}. \begin{Corollary} \label{cor:lower:bound:w:w:not:smooth} If $f : X \to \operatorname{Spec}(O_K)$ is not smooth, then \[ (\omega_{X/O_K}^{Ar} \cdot \omega_{X/O_K}^{Ar}) \geq \frac{\log 2}{6(g-1)}. \] \end{Corollary} \renewcommand{\thesection}{Appendix \Alph{section}} \renewcommand{\theTheorem}{\Alph{section}.\arabic{Theorem}} \renewcommand{\theClaim}{\Alph{section}.\arabic{Theorem}.\arabic{Claim}} \renewcommand{\theequation}{\Alph{section}.\arabic{Theorem}.\arabic{Claim}} \setcounter{section}{0} \section{Matrix representation of Laplacian} In this appendix, we will consider a matrix representation of the Laplacian and its easy application. Let $G$ be a metrized graph and $V$ a set of vertices of $G$ such that $G \setminus V$ is a disjoint union of open segments. Let $E$ be a set of edges of $G$ by $V$. The length of $e$ in $E$ is denoted by $l(e)$. Recall that $Q(G, V)$ is a set of continuous functions on $G$ whose restriction to each edge in $E$ are quadric polynomial functions, and $M(G, V)$ is a vector space of measures on $G$ generated by Dirac functions $\delta_v$ at $v \in V$ and by Lebesgue measures on edges $e \in E$ arising from the arc-length parameter. First, we define linear maps $p : Q(G, V) \to {\Bbb{R}}^{V}$ and $q : M(G, V) \to {\Bbb{R}}^{V}$ in the following ways. If $f \in Q(G, V)$, then $p(f)$ is the restriction to $V$. If $\delta_v$ is a Dirac function at $v \in V$, then \[ q(\delta_v)(v') = \begin{cases} 1 & \text{if $v' = v$} \\ 0 & \text{if $v' \not= v$} \end{cases} \] If $dt$ is a Lebesgue measure on a edge $e$ in $E$, then \[ q(dt)(v) = \begin{cases} l(e)/2 & \text{if $v$ is a vertex of $e$} \\ 0 & \text{otherwise} \end{cases} \] Next let us define a linear map $L : {\Bbb{R}}^{V} \to {\Bbb{R}}^{V}$. For distinct vertices $v, v'$ in $V$, let $E(v, v')$ be the set of edges in $E$ whose vertices are $v$ and $v'$. Here we set \[ a(v, v') = \begin{cases} 0 & \text{if $E(v, v') = \emptyset $} \\ {\displaystyle \sum_{e \in E(v, v')} \frac{1}{l(e)}} & \text{otherwise} \end{cases} \] for $v \not= v'$. Moreover, we set \[ a(v, v) = -\sum\begin{Sb} v' \in V \\ v' \not= v \end{Sb} a(v, v'). \] Let $L : {\Bbb{R}}^{V} \to {\Bbb{R}}^{V}$ be a linear map defined by a matrix $(-a(v, v'))_{v, v' \in V}$, i.e., if we denote $q(\delta_v)$ by $e_v$, then $L(e_v) = -\sum_{v' \in V} a(v, v') e_{v'}$. Thus, we have the following diagram: \[ \begin{CD} Q(G, V) @>{\Delta}>> M(G, V) \\ @V{p}VV @VV{q}V \\ {\Bbb{R}}^{V} @>>{L}> {\Bbb{R}}^{V} \end{CD} \] Then, we can see the following proposition as remarked in \cite[(5.3)]{BGS}. \begin{Proposition} \label{prop:commutativity:L:p:q:Delta} The above diagram is commutative, i.e., $L \circ p = q \circ \Delta$. \end{Proposition} {\sl Proof.}\quad Let $f \in Q(G, V)$. First, let us consider two special cases of $f$. Case 1 : A case where $f$ is a linear function on each edge in $E$. By the definition of $\Delta$, we can see that \[ \Delta(f) = - \sum_{v \in V}\left( \sum\begin{Sb} v' \in V \setminus \{ v \} \\ E(v, v') \not= \emptyset \end{Sb} \left( \sum_{e \in E(v, v')} \frac{f(v') - f(v)}{l(e)} \right) \right) \delta_v. \] On the other hand, by the definition of $a(v, v')$, \[ \sum\begin{Sb} v' \in V \setminus \{ v \} \\ E(v, v') \not= \emptyset \end{Sb} \left( \sum_{e \in E(v, v')} \frac{f(v') - f(v)}{l(e)} \right) = \sum_{v' \in V} a(v, v') f(v'). \] Therefore, we have \[ \Delta(f) = \sum_{v \in V} \left( \sum_{v' \in V} - a(v, v') f(v') \right) \delta_v, \] which shows us $q(L(f)) = L(p(f))$. \medskip Case 2 : A case where there is $e \in E$ such that $f \equiv 0$ on $G \setminus e$. Let $v, v'$ be vertices of $e$ and $\phi : [0, l(e)] \to e$ be the arc-length parameterization of $e$ with $\phi(0) = v$ and $\phi(l(e)) = v'$. Since $f(v) = f(v') = 0$, $f$ can be written in the form $f(t) = at(t-l(e))$, where $t$ is the arc-length parameter and $a$ is a constant. Thus, \[ \Delta(f) = al(e) \delta_v + al(e) \delta_{v'} - 2a dt. \] Therefore, $q(\Delta(f)) = 0$, which means that $q(\Delta(f)) = L(p(f))$. \medskip Let us consider a general case. Let $f_0$ be a continuous function on $G$ such that $f_0$ is a linear function on each $e \in E$ and $f_0(v) = f(v)$ for all $v \in V$. Then, $f - f_0$ can be written by a sum of functions $f_1, \ldots, f_k$ as in the case 2, i.e., \[ f = f_0 + f_1 + \cdots + f_k \] and $f_i$ ($1 \leq i \leq k$) is zero on the outside of some edge. By the previous observation, we know $q(\Delta(f_i)) = L(p(f_i))$ for all $i = 0, 1, \ldots, k$. Thus, using linearity of each map, we get our lemma. \QED As a corollary, we have the following. \begin{Corollary} Let $D = \sum_{v \in V} d_v v$ be a divisor on $G$, $\mu \in M(G, V)$, and $g \in Q(G, V)$ such that \[ \int_G \mu = 1 \quad\text{and}\quad \Delta(g) = \delta_D - (\deg D)\mu. \] Then, we have \[ d_v + \sum_{v' \in V} a(v, v')g(v') = (\deg D)q(\mu)(v) \] for all $v \in V$. \end{Corollary} {\sl Proof.}\quad Applying $q$ for $\Delta(g) = \delta_D - (\deg D)\mu$ and using Proposition~\ref{prop:commutativity:L:p:q:Delta}, we have \[ q(\delta_D) - L(p(g)) = (\deg D)q(\mu). \] Thus, by the definition of $L$, we get our corollary. \QED \bigskip Let $k$ be an algebraically closed field, $X$ a smooth projective surface over $k$, $Y$ a smooth projective curve over $k$, and $f : X \to Y$ a generically smooth semi-stable curve of genus $g \geq 1$ over $Y$. Let $\operatorname{CV}(f)$ be the set of all critical values of $f$ and $y \in \operatorname{CV}(f)$. Let $G_y$ be the metrized graph of $f^{-1}(y)$ as in \S\ref{sec:metrized:graph:green:function:admissible:pairing}. Let $V_y$ be a set of vertices coming from irreducible curves in $f^{-1}(y)$. For $v \in V_y$, the corresponding irreducible curve is denoted by $C_v$. Let $K_y$ be the divisor on $G_y$ defined by $K_y = \sum_{v \in V_y} (\omega_{X/Y} \cdot C_v) v$, $\mu_y$ the admissible metric of $K_y$, and $g_{\mu_y}$ the Green function of $\mu_y$. In this case, the map $L_y : {\Bbb{R}}^{V_y} \to {\Bbb{R}}^{V_y}$ defined in the above is given by a matrix $\left(-(C_v \cdot C_{v'})\right)_{v, v' \in V_y}$. Thus, the above corollary implies the following proposition. \begin{Proposition} \label{prop:admissible:with:fiber} Let $D$ be an ${\Bbb{R}}$-divisor on $X$ and $C_v$ the irreducible curve in $f^{-1}(y)$ corresponding to $v \in V_y$. Then, \[ (D \cdot C_v)_a = (D \cdot F) q(\mu_y)(v), \] where $F$ is a general fiber of $f$. In particular, $(D \cdot C_v)_a$ does not depend on the choice of compactification of $D$. \end{Proposition} \bigskip

9410

alg-geom/9410008

en

https://arxiv.org/abs/alg-geom/9410008

alg-geom/9410008

David B. Jaffe

Applications of iterated curve blowup to set-theoretic complete intersections in P3

57 pages, AMS-LaTeX

Let S, T be surfaces in P3. Suppose that S intersect T is set-theoretically a smooth curve C of degree d and genus g. Suppose that S and T have no common singular points. Then if C is not a complete intersection, then deg(S), deg(T) < 2d^4. Fixing (d,g), one can form a finite (shorter) list of all possible pairs (deg(S),deg(T)). For instance, when (d,g) = (4,0), and assuming for simplicity that deg(S) <= deg(T): (deg(S), deg(T)) \in {(3,4), (3,8), (4,4), (4,7), (6,26), (9,48), (10,28) (12,18), (13,16), (17,220), (18,118), (19,84), (20,67), (22,50), (28,33)}. Assume characteristic 0. [1] Suppose that S and T have non-overlapping rational singularities. Then d <= g+3. [2] Suppose that S is normal, and that d>deg(S). Then C is linearly normal (and so d <= g+3). [3] Suppose that S is a quartic surface having only rational singularities. Then C is linearly normal. Hard copy is available from the author. E-mail to [email protected].

alg-geom

\section{#1}} \def\abs#1{{\vert{#1}\vert}} \def\floor#1{\lfloor#1\rfloor} \def\makeaddress{ \vskip 0.15in \par\noindent {\footnotesize Department of Mathematics and Statistics, University of Nebraska} \par\noindent {\footnotesize Lincoln, NE 68588-0323, USA\ \ (jaffe{\kern0.5pt}@{\kern0.5pt}cpthree.unl.edu)}} \def \def\arabic{footnote}}\setcounter{footnote}{0{\fnsymbol{footnote}{ \def\arabic{footnote}}\setcounter{footnote}{0{\fnsymbol{footnote}} \par\noindent David B. Jaffe\protect\footnote{Partially supported by the National Science Foundation.} \makeaddress\def\arabic{footnote}}\setcounter{footnote}{0{\arabic{footnote}}\setcounter{footnote}{0}} \def\br#1{{[#1]}} \def\inn#1{\langle #1 \rangle} \newenvironment{proof}{\trivlist \item[\hskip \labelsep{\sc Proof.\kern1pt}]}{\endtrivlist \newenvironment{proofnodot}{\trivlist \item[\hskip \labelsep{\sc Proof}]}{\endtrivlist \newenvironment{sketch}{\trivlist \item[\hskip \labelsep{\sc Sketch.\kern1pt}]}{\endtrivlist \newenvironment{romanlist}{\begin{list}{(\roman{romanctr})}{\usecounter{romanctr}}}{\end{list} \newenvironment{arabiclist}{\begin{list}{(\arabic{arabicctr})}{\usecounter{arabicctr}}}{\end{list} \newenvironment{definition}{\trivlist \item[\hskip \labelsep{\bf Definition.\kern1pt}]}{\endtrivlist} \newenvironment{examples}{\trivlist \item[\hskip \labelsep{\bf Examples.\kern1pt}]}{\endtrivlist} \hfuzz 3pt \documentclass[12pt]{article}\usepackage{amssymb} \newtheorem{theorem}{Theorem}[section] \setlength{\parindent}{9mm} \setcounter{tocdepth}{3} \newtheorem{proposition}[theorem]{Proposition \newtheorem{lemma}[theorem]{Lemma \newtheorem{cor}[theorem]{Corollary \newtheorem{corollary}[theorem]{Corollary \newtheorem{prop}[theorem]{Proposition \newtheorem{claim}[theorem]{Claim \newtheorem{exampleth}[theorem]{Example} \newenvironment{example}{\begin{exampleth}\fontshape{n}\selectfont}{\end{exampleth}} \catcode`\@=11 \def\section*{Contents\markboth{CONTENTS}{CONTENTS}{\section*{Contents\markboth{CONTENTS}{CONTENTS}} \@starttoc{toc}} \def\l@part#1#2{\addpenalty{\@secpenalty} \addvspace{2.25em plus 1pt} \begingroup \@tempdima 3em \parindent \z@ \rightskip \@pnumwidth \parfillskip -\@pnumwidth {\footnotesize \rm \leavevmode #1\hfil \hbox to\@pnumwidth{\hss \ }}\par \nobreak \endgroup} \def\l@special{\@dottedtocline{1}{0.0em}{2.3em}} \def\l@section{\@dottedtocline{2}{1.5em}{2.3em}} \def\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}} \def\l@paragraph{\@dottedtocline{4}{7.0em}{4.1em}} \def\l@subparagraph{\@dottedtocline{5}{10em}{5em}} \def\listoffigures{\section*{List of Figures\markboth {LIST OF FIGURES}{LIST OF FIGURES}}\@starttoc{lof}} \def\l@figure{\@dottedtocline{1}{1.5em}{2.3em}} \def\listoftables{\section*{List of Tables\markboth {LIST OF TABLES}{LIST OF TABLES}}\@starttoc{lot}} \let\l@table\l@figure \catcode`\@=12 \begin{document} {\par\noindent\Large\bf Applications of iterated curve blowup to} \vskip 0.05in {\par\noindent\Large\bf set theoretic complete intersections in $\hbox{{\bbtwo P}}^3$} \vskip 0.15in \def\arabic{footnote}}\setcounter{footnote}{0{\fnsymbol{footnote} {\footnotesize\section*{Contents\markboth{CONTENTS}{CONTENTS}} \newpage \section*{Introduction} \addcontentsline{toc}{special}{Introduction} \def\arabic{footnote}}\setcounter{footnote}{0{\arabic{footnote}}\setcounter{footnote}{0} \indent We describe some new results on the set-theoretic complete intersection problem for projective space curves. Fix an algebraically closed ground field $k$. Let $S, T \subset \P3$ be surfaces. Suppose that $S \cap T$ is set-theoretically a smooth curve $C$ of degree $d$ and genus $g$. For purposes of the introduction, we label the main results as A, B, Q, X, I, II, and III.\footnote{The actual numbering in the text is A = \ref{thmA}, B = \ref{thmB}, Q = \ref{thmQ}, X = \ref{thmX}, I = \ref{thmI}, II = \ref{thmII}, III = \ref{thmIII}.} The results I, II, and III are more technical than A, B, Q, and X. Suppose that $S$ and $T$ have no common singular points. We discover that this requirement imposes severe limitations. Indeed, theorem (X) asserts that if $C$ is not a complete intersection, then $\deg(S), \deg(T) < 2d^4$. Fixing $(d,g)$, one can in fact form a finite list of all possible pairs $(\deg(S),\deg(T))$, which is much shorter than the list implied by theorem (X). For instance, when $(d,g) = (4,0)$, and assuming for simplicity that $\deg(S) \leq \deg(T)$, we find that $$(\deg(S), \deg(T)) \in \{ (3,4), (3,8), (4,4), (4,7), (6,26), (9,48), (10,28),$$ $$(12,18), (13,16), (17,220), (18,118), (19,84), (20,67), (22,50), (28,33) \}.$$ Very little is known about which of these degree pairs actually correspond to surface pairs $(S,T)$. Suppose that $S$ and $T$ have only rational singularities, and that the ground field $k$ has characteristic zero. We continue to assume that $S$ and $T$ have no common singular points. Under these conditions, we prove (A) that $d \leq g + 3$. (The actual statement is somewhat stronger.) Suppose that $S$ is normal, and that $d > \deg(S)$. Make no assumptions about how the singularities of $S$ and $T$ meet. Assume that $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = 0$. We show (Q) that $C$ is linearly normal. In particular, it follows by Riemann-Roch that $d \leq g+3$. Suppose that $S$ is a quartic surface having only rational singularities. Allow $T$ to be an arbitrary surface, and make no assumptions about how the singularities of $S$ and $T$ meet. Assume that $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = 0$. Under these conditions, we prove (B) that $C$ is linearly normal. In other papers\Lspace \Lcitemark 17\Rcitemark \Rspace{},\Lspace \Lcitemark 18\Rcitemark \Rspace{}, we have proved the following complementary results (in characteristic zero): if $S$ is has only ordinary nodes as singularities, or is a cone, or has degree $\leq 3$, then $d \leq g + 3$. It is conceivable (in characteristic zero) that this inequality is valid without any restrictions whatsoever on $S$ and $T$, or even that $C$ is always linearly normal. Examples of smooth set-theoretic complete intersection\ curves in ${\Bbb C}\kern1pt\P3$ have been constructed by Gallarati\Lspace \Lcitemark 8\Rcitemark \Rspace{}, Catanese\Lspace \Lcitemark 3\Rcitemark \Rspace{}, Rao (\Lcitemark 27\Rcitemark \ prop.\ 14), and the author\Lspace \Lcitemark 19\Rcitemark \Rspace{}. To explain the results (I), (II), and (III), and to describe the methods by which we prove (A), (B), and (X), there are two key ideas which must be discussed% .\footnote{We also give an alternate proof of the key ingredient of (X), which is independent of the main machine of this paper.} Both of these ideas have to do with the iterated blowing up of curves. The first idea has to do with certain invariants $p_i = p_i(S,C)$ $(i \in \xmode{\Bbb N})$ which we associate to a pair $(S,C)$ consisting of an abstract surface $S$ and a smooth curve $C$ on $S$ such that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(S)$. Let \mp[[ \pi || {\tilde{S}} || S ]] be the blowup along $C$. Then $p_1(S,C)$ is the sum of the multiplicities of the exceptional curves. (See \S\ref{measure} for details.) Moreover, $\pi$ admits a unique section ${\tilde{C}}$ over $C$, so we can define $p_2(S,C) = p_1({\tilde{S}},{\tilde{C}})$, $p_3(S,C) = p_2({\tilde{S}},{\tilde{C}})$, and so forth. We refer to the sequence $(p_1, p_2, \ldots)$ as the {\it type\/} of $(S,C)$. It is a sum of local contributions, one for each singular point of $S$ along $C$, and it is a rather mysterious measure of how singular $S$ is along $C$. The type depends not only on the particular species of singular points of $S$ which lie on $C$, but also on the way in which $C$ passes through those points. For example, the local contribution to the type coming from an $A_3$ singularity is either $(1,1,1,0,\ldots)$ or $(2,0,\ldots)$, depending on how $C$ passes through the singular point. The second idea is the following construction. For this we assume (as in the first paragraph) that $C = S \cap T$ (in $\P3$) and that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(S)$, $C \not\IN \mathop{\operatoratfont Sing}\nolimits(T)$. Other than this, no restrictions are necessary on the singularities of $S$ and $T$. Let $Y_1$ denote the blowup of $\P3$ along $C$. Let $S_1, T_1 \subset Y_1$ denote the strict transforms of $S$ and $T$ respectively. Let $E_1 \subset Y_1$ be the exceptional divisor, which is a ruled surface over ${\Bbb C}\kern1pt$. Then $S_1 \cap E_1$ is a curve $C_1$ (mapping isomorphically onto $C$), together with some rulings. The total number of rulings, counted with multiplicities, is $p_1(S,C)$. Now let $Y_2$ be the blowup of $Y_1$ along $C_1$. Let $S_2, T_2, E_2 \subset Y_2$ be as above. Then $S_2 \cap E_2$ is a curve $C_2$ plus $p_2(S,C)$ rulings. Iterate this construction $n$ times, where $n$ is the multiplicity of intersection of $S$ and $T$ along $C$. Then $S_n \cap T_n$ is a union of strict transforms of rulings. This fact leads us to theorems (I) and (II), which are statements about the numbers $p_i$. Theorem (III) is also such a statement, but it does not depend on the construction we have just described. We describe theorems (I), (II), and (III). These depend on the data $(s,t,d,g)$, where $s = \deg(S)$, $t = \deg(T)$. To make this description as simple as possible, we restrict our attention here to the special case where $(s,t,d,g) = (4,4,4,0)$. Theorem (I) has the hypothesis that $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$. Its conclusion (applied to our special case) is that: $$p_1 = p_2 = p_3 = 8.$% $Theorem (I) is used in the proofs of (A) and (X). Theorem (II) has no additional hypotheses. Its conclusion (applied to our special case) is that: \begin{eqnarray*} p_1 & \geq & 8; \\ 2p_1 + p_2 & \geq & 24; \\ 8p_1 + 3p_2 + p_3 & \geq & 96. \end{eqnarray*} Theorem (II) is not used in the proofs of (A) or (B). Theorem (III) has the hypotheses that $S$ has only rational singularities, and that $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = 0$. Its conclusion (applied to our special case) is that: $${1\over2} p_1 + {1\over6} p_2 + {1\over12}p_3 \many+ {1 \over k(k+1)}p_k + \cdots \geq 6.$% $Theorem (III), or actually a minor variant of it, is used in the proof of (B). We now mention some open problems and possible ways to improve upon the results in this paper. \par\noindent{\bf 1.} Let $(S,C)$ be the local scheme $S$ of a normal surface singularity, together with a smooth curve $C$ on $S$. There are three fundamental invariants of $(S,C)$ which are utilized in this paper. Firstly, there is the type of $(S,C)$. Secondly, there is the order of $(S,C)$, i.e.\ the smallest positive integer $n$ such that ${\cal O}_S(nC)$ is Cartier. Thirdly, there is $\Delta(S,C)$, which we describe in \S\ref{def-section}. What relationships exist between these three invariants? What is their relationship to the Milnor fiber? \par\noindent{\bf 2.} We suspect that (A), (B), and (III) are valid over an arbitrary algebraically closed field. There are significant difficulties in proving this which we have not explored fully. The proofs of (I) and (II) do not depend on the characteristic. \par\noindent{\bf 3.} The proofs of (A) and (B) use a bound \pref{bound-formula} on the number of exceptional curves in a minimal resolution for a surface $S \subset {\Bbb C}\kern1pt\P3$ having only rational singularities. Formulate and prove a suitable generalization for arbitrary normal surfaces. \par\noindent{\bf 4.} Construct examples of surfaces $S, T \subset {\Bbb C}\kern1pt\P3$, having only rational singularities, meeting set-theoretically along a smooth curve $C$, such that $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) \not= \varnothing$. The only example we know of is where $\deg(S) = \deg(T) = 2$, and $C$ is a line. \par\noindent{\bf 5.} The generic hypothesis that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(T)$ can probably be eliminated. \vspace*{0.1in} \par\noindent{\footnotesize{\it Acknowledgements.} I thank Dave Morrison for helpful comments, and Juan Migliore for raising the issue of linear normality of set-theoretic complete intersections.} \section*{Conventions} \addcontentsline{toc}{special}{Conventions} \begin{arabiclist} \item We fix an algebraically closed field $k$. \item A {\it curve\/} [resp.\ {\it surface}] [resp.\ {\it three-fold}] is an excellent $k$-scheme such that every maximal chain of irreducible proper closed subsets has length one [resp.\ two] [resp.\ three]. We make the following additional assumptions: \begin{itemize} \item all curves are reduced and irreducible; \item in part III and the introduction, all surfaces are reduced and irreducible. \end{itemize} \item A surface {\it embeds in codimension one\/} if it can be exhibited as an effective Cartier divisor on a regular three-fold. \item A {\it variety} is an integral separated scheme of finite-type over $k$. \item If $X$ and $Y$ are schemes, then the notation $X \subset Y$ carries the implicit assumption that $X$ is a {\it closed subscheme\/} of $Y$. \item In several situations, we use {\it bracketed exponents\/} to denote {\it repetition\/} in sequences, and we drop trailing zeros, where appropriate. For example, $$(2,1^\br{4}) = (2,1,1,1,1,0, \ldots)$% $and $$(3^\br{\infty}) = (3,3,\ldots).$$ \item We use the Grothendieck convention regarding projective space bundles. \item For any variety $V$, we let $A^k(V)$ denote the group of codimension $k$ cycles on $V$, modulo {\it algebraic\/} equivalence. When $d = \dim(V)$ and $V$ is complete, we identify $A^d(V)$ with $\xmode{\Bbb Z}$. \end{arabiclist} \vspace{0.25in} \part{Local geometry of smooth curves on singular surfaces} \block{Definitions}\label{def-section} \par\indent\indent We define the category of {\it surface-curve pairs}. (Sometimes, we use the shorthand term {\it pair\/} for a surface-curve pair.) An {\it object\/} $(S,C)$ in this category consists of a surface $S$, together with a curve $C \subset S$, such that $C$ is a regular scheme and $C \not\IN \mathop{\operatoratfont Sing}\nolimits(S)$. A {\it morphism\/} \mp[[ f || (S',C') || (S,C) ]] is a pair $(S' \mapE{} S, C' \mapE{} C)$ of morphisms of $k$-schemes, such that the diagram: \squareSE{C'}{C}{S'}{S% }commutes, and such that if \mp[[ \phi || C' || C \times_S S' ]] is the induced map, then $\phi \times_S \mathop{\operatoratfont Spec}\nolimits {\cal O}_{S,C}$ is an isomorphism. Such a morphism $f$ is {\it cartesian\/} if $\phi$ is an isomorphism. Most properties of morphisms of schemes also make sense as properties of morphisms in this category: the properties are to be interpreted as properties of the morphism \mapx[[ S' || S ]]. Let $(S,C)$ be a surface-curve pair. We say that: \begin{itemize} \item $(S,C)$ is {\it geometric\/} if $S$ is a variety; \item $(S,C)$ is {\it local\/} if $S$ is a local scheme; \item $(S,C)$ is {\it local-geometric\/} if $S$ is a local scheme, essentially of finite type over $k$. \end{itemize} To give a local surface-curve pair $(S,C)$ is equivalent to giving the data $(A,{\xmode{{\fraktur{\lowercase{P}}}}})$, consisting of an excellent local $k$-algebra $A$, of pure dimension two, together with a height one prime ${\xmode{{\fraktur{\lowercase{P}}}}} \subset A$ such that $A_{\xmode{{\fraktur{\lowercase{P}}}}}$ and $A/{\xmode{{\fraktur{\lowercase{P}}}}}$ are regular. We write $(S,C) = \mathop{\operatoratfont Spec}\nolimits(A,{\xmode{{\fraktur{\lowercase{P}}}}})$ to denote this correspondence. There are two operations on surface-curve pairs which we will be using. Firstly, if $(S,C)$ is a local surface-curve pair, then the {\it completion\/} $({\hat{S}},{\hat{C}})$ makes sense and is also a local surface-curve pair. Indeed, if $(S,C) = \mathop{\operatoratfont Spec}\nolimits(A,{\xmode{{\fraktur{\lowercase{P}}}}})$, then $A/{\xmode{{\fraktur{\lowercase{P}}}}}$ is regular, and so ${\hat{A}}/{\xmode{{\hat{\fraktur{\lowercase{P}}}}}}$ is regular, since it equals $\widehat{A/{\xmode{{\fraktur{\lowercase{P}}}}}}$, and the completion of a regular local ring is regular. The reader may also check easily that ${\hat{A}}_{\xmode{{\hat{\fraktur{\lowercase{P}}}}}}$ is regular. Moreover, ${\hat{A}}$ is excellent, since any noetherian complete local ring is excellent. Note also: there is a canonical morphism \mapx[[ (S,C) || ({\hat{S}},{\hat{C}}) ]]. Secondly, for any surface-curve pair $(S,C)$, one can define the {\it blowup\/} $({\tilde{S}},{\tilde{C}})$ of $(S,C)$. This is done by letting \mp[[ \pi || {\tilde{S}} || S ]] be the blowup of $S$ along $C$, and by letting ${\tilde{C}}$ be the unique section of $\pi$ over $C$, which exists e.g.\ by (\Lcitemark 11\Rcitemark \ 7.3.5). There is a canonical morphism \mapx[[ ({\tilde{S}}, {\tilde{C}}) || (S,C) ]]. Two local surface-curve pairs are {\it analytically isomorphic\/} if their completions are isomorphic. If $(S,C)$ is a surface-curve pair, and $p \in C$, we let $(S,C)_p$ denote the corresponding local surface-curve pair. A {\it configuration\/} is an element of the free abelian monoid on the set of analytic isomorphism classes of local-geometric pairs. Let $(S,C)$ be a geometric surface-curve pair. We may associate the configuration: $$\sum_{p \in \mathop{\operatoratfont Sing}\nolimits(S) \cap C} [(S,C)_p]$% $to $(S,C)$. On occasion, we shall identify $(S,C)$ with the associated configuration. We are interested in invariants of a geometric surface-curve pair $(S,C)$ which depend only on the associated configuration. There are four such invariants which we shall consider: \begin{arabiclist} \item The {\it order\/} of $(S,C)$ is the smallest $n \in \xmode{\Bbb N}$ such that ${\cal O}_S(nC)$ is Cartier, or else $\infty$ if ${\cal O}_S(nC)$ is not Cartier for all $n \in \xmode{\Bbb N}$. If $\mathop{\operatoratfont Sing}\nolimits(S) \cap C = \setof{\vec p1k}$, then $$\mathop{\operatoratfont order}\nolimits(S,C) = \mathop{\operatoratfont lcm}\nolimits\setof{\mathop{\operatoratfont order}\nolimits(S,C)_{p_1}, \ldots, \mathop{\operatoratfont order}\nolimits(S,C)_{p_k}},$% $so the computation of the order is a purely local problem. Moreover, at least if $S$ is normal, the order depends only on the associated configuration. Indeed, in that case, if $(S,C)$ is a local-geometric pair, then $S$ is excellent, so ${\hat{S}}$ is normal, and so by (\Lcitemark 6\Rcitemark \ 6.12) one knows that the canonical map \mapx[[ \mathop{\operatoratfont Cl}\nolimits(S) || \mathop{\operatoratfont Cl}\nolimits({\hat{S}}) ]] is injective. Hence $\mathop{\operatoratfont order}\nolimits(S,C) = \mathop{\operatoratfont order}\nolimits({\hat{S}},{\hat{C}})$. \item The {\it type\/} of $(S,C)$, which is the sequence $(p_i)_{i \in \xmode{\Bbb N}}$ discussed in the introduction, and studied in \S\ref{measure}. \item Assume that $S$ is normal. We define an invariant $\Delta(S,C) \in {\Bbb Q}\kern1pt$. Let \mp[[ \pi || {\tilde{S}} || S ]] be a minimal resolution, and let $\vec E1n \subset {\tilde{S}}$ be the exceptional curves. Let ${\tilde{C}} \subset {\tilde{S}}$ be the strict transform of $C$. According to (\Lcitemark 26\Rcitemark \ p.\ 241), there is a unique ${\Bbb Q}\kern1pt$-divisor $E = \sum a_i E_i$ such that $({\tilde{C}} + E) \cdot E_i = 0$ for all $i$. We define $\Delta(S,C) = -E^2$. Then $\Delta(S,C)$ is independent of $\pi$. If $S$ is projective, then $\Delta(S,C) = C^2 - {\tilde{C}}^2$, where $C^2$ is defined in (\Lcitemark 26\Rcitemark \ p.\ 241). \item Assume that $S$ has only rational double points along $C$. Let $\Sigma(S,C)$ equal the number of exceptional curves in the minimal resolution of those singularities of $S$ which lie on $C$. \end{arabiclist} \block{The type of a surface-curve pair}\label{measure} \par\indent\indent We define the {\it type\/} of a surface-curve pair, and show that it is an analytic invariant, at least when the surface embeds in codimension one. \begin{definition} Let $(S,C)$ be a surface-curve pair. Let $({\tilde{S}},{\tilde{C}})$ be the blowup of $(S,C)$. Let $\vec E1n \subset {\tilde{S}}$ be the (reduced) exceptional curves. We define numbers $p_i(S,C)$, for each $i \in \xmode{\Bbb N}$. Define: $$p_1(S,C) = \sum_{i=1}^n \mathop{\operatoratfont length}\nolimits {\cal O}_{\pi^{-1}(C), E_i},$% $where \mp[[ \pi || {\tilde{S}} || S ]] is the blowup map. For $i \geq 2$, recursively define $p_i(S,C)$ by: $$p_{i+1}(S,C) = p_i({\tilde{S}}, {\tilde{C}}).$% $The {\it type\/} of $(S,C)$ is the sequence $(p_1, p_2, \ldots)$. \end{definition} It is clear that the computation of the $p_i$ may be reduced to the computation of the $p_i$ when $(S,C)$ is a local pair. \begin{remark}\label{goober-peas} Let $(S,C)$ be a surface-curve pair, and assume that $S$ embeds in codimension one. Then we have $S \subset T$ for some smooth three-fold $T$. Let \mp[[ \pi_S || {\tilde{S}} || S ]] and \mp[[ \pi_T || {\tilde{T}} || T ]] be the blowups of $S$ and $T$ along $C$. Then $\pi_S^{-1}(C) \cong {\tilde{S}} \cap E$, where $E \subset {\tilde{T}}$ is the exceptional divisor. This fact plays an absolutely central role in our type computations. \end{remark} \begin{remark} We do not know for which $(S,C)$ we have $p_1(S,C) \geq p_2(S,C)$, and hence that $p_k(S,C) \geq p_{k+1}(S,C)$ for all $k \geq 1$. Conceivably, these inequalities may hold whenever $S$ embeds in codimension one, or even whenever $S$ is Cohen-Macaulay. By explicit calculation, we shall find in \pref{fantastico} that the inequalities hold if $S$ has only rational double points along $C$. However, as \pref{type-ex-2} shows, for some $(S,C)$ one has $p_1(S,C) < p_2(S,C)$. \end{remark} \begin{remark} We consider the following general question. Let $(S,C)$ be a local-geometric pair. Assume that $S$ is not smooth. Let $p \in S$ be the closed point. Let $({\tilde{S}},{\tilde{C}})$ be the blowup of $S$ along $C$. Let \mp[[ \pi || {\tilde{S}} || S ]] be the blowup map. What is the structure of $X = \RED{\pi^{-1}(p)}$? If $S$ embeds in codimension one, then $X$ will be a $\P1$. Weird things can happen if $S$ is not Cohen-Macaulay. For example, in \pref{type-ex-2}, $X$ is isomorphic to $\mathop{\operatoratfont Proj}\nolimits {\Bbb C}\kern1pt[s,t,u] / (s^3 - t^2u)$, which is a rational curve with a cusp. In \pref{type-ex-3}, $X$ is the disjoint union of a point and several copies of $\P1$, which do not meet ${\tilde{C}}$. The isolated point of $X$ is the unique point of ${\tilde{C}}$ lying over $p$. Assuming only that $S$ is Cohen-Macaulay, we do not know if $X$ is always isomorphic to $\P1$, or even if it is always connected. However: if $S$ embeds in codimension two, then $X$ embeds in $\P2$. \end{remark} We will prove \pref{analytic-invariant} that the type of a local-geometric pair $(S,C)$ is an analytic invariant, provided that $S$ embeds in codimension one. There are some preliminaries. \begin{lemma}\label{formal-woof-1} Let \mp[[ f || A || B ]] be a flat, formally smooth homomorphism of Artin local rings. Assume that $A$ contains a field. Then $\mathop{\operatoratfont length}\nolimits(A) = \mathop{\operatoratfont length}\nolimits(B)$. \end{lemma} \begin{proof} Let $K$ and $L$ be the residue fields of $A$ and $B$. Let \mp[[ i || K || L ]] be the induced map. Let ${\xmode{{\fraktur{\lowercase{M}}}}}$ be the maximal ideal of $A$. Then the map \mapx[[ K = A/{\xmode{{\fraktur{\lowercase{M}}}}} || B/{\xmode{{\fraktur{\lowercase{M}}}}} B ]] is formally smooth, so $B/{\xmode{{\fraktur{\lowercase{M}}}}} B = L$ and so $i$ is formally smooth. A theorem of Cohen (\Lcitemark 23\Rcitemark \ 28.J) implies that $A$ contains a coefficient field, which we also denote by $K$. Since $i$ is formally smooth, $L/K$ is a separable field extension. It follows by the cited theorem that we may find a coefficient field $L$ for $B$ which contains $f(K)$. Let ${\overline{A}} = A \o*_K L$. Then $f$ factors as: \diagramx{A&\mapE{h}&{\overline{A}}&\mapE{g}&B.% }Since $i$ is formally smooth, so is $h$. Since both $h$ and $g \circ h$ are formally smooth it follows by (\Lcitemark 12\Rcitemark \ 17.1.4) that $g$ is formally smooth. Clearly $B$ is a finite ${\overline{A}}$-module. In particular, $g$ is of finite-type, so $g$ is smooth. Since $g$ is smooth of relative dimension zero, $g$ is \'etale. By (\Lcitemark 12\Rcitemark \ 18.1.2), the obvious functor: \dfunx[[ \'etale ${\overline{A}}$-schemes || \'etale $L$-schemes ]]% is an equivalence of categories, so $g$ is an isomorphism. Hence $B \cong A \o*_K L$. Hence $\mathop{\operatoratfont length}\nolimits(A) = \mathop{\operatoratfont length}\nolimits(B)$. {\hfill$\square$} \end{proof} \begin{corollary}\label{formal-woof-2} Let \mp[[ f || X' || X ]] be a flat, formally smooth morphism of irreducible noetherian schemes. Assume that $X$ is defined over a field. Let $\eta$ and $\eta'$ be the generic points of $X$ and $X'$. Then: $$\mathop{\operatoratfont length}\nolimits {\cal O}_{X',\eta'} = \mathop{\operatoratfont length}\nolimits {\cal O}_{X,\eta}.$$ \end{corollary} \begin{remark} We do not know if \pref{formal-woof-1} and \pref{formal-woof-2} are true without the hypothesis of being ``defined over a field''. \end{remark} \begin{lemma}\label{flat-is-cartesian} Let \mp[[ f || (S_1,C_1) || (S_2,C_2) ]] be a flat morphism of surface-curve pairs. Then $f$ is cartesian. \end{lemma} \begin{proof} We must show that the induced map \mp[[ \phi || C_1 || C_2 \times_{S_1} S_2 ]] is an isomorphism. It suffices to show that $C_1 = C_2 \times_{S_1} S_2$ as closed subschemes of $S_2$. Let \mp[[ \pi || C_2 \times_{S_1} S_2 || C_2 ]] be the projection map. Because $\pi$ is flat, any irreducible component of $C_2 \times_{S_1} S_2$ must dominate $C_2$. (See e.g.{\ }\Lcitemark 14\Rcitemark \ III 9.7.) But $\phi \times_{S_2} \mathop{\operatoratfont Spec}\nolimits {\cal O}_{S_2,C_2}$ is an isomorphism, so it follows that $C_2 \times_{S_1} S_2$ is irreducible. Since $C_1 = C_2 \times_{S_1} S_2$ at their generic points, they are equal as closed subschemes of $S_2$. {\hfill$\square$} \end{proof} \begin{prop}\label{formal-woof-3} Let \mp[[ f || (S_1,C_1) || (S_2,C_2) ]] be a formally smooth, flat morphism of surface-curve pairs. Assume that the induced map \mapx[[ C_1 || C_2 ]] is bijective. Assume that $S_1$ and $S_2$ embed in codimension one. Then $(S_1,C_1)$ and $(S_2,C_2)$ have the same type. \end{prop} \begin{proof} The subscript $i$ will always vary through the set $\setof{1,2}$. Because of our hypothesis on the map \mapx[[ C_1 || C_2 ]], we may assume that $S_1, S_2$ are local schemes and that $f$ is a local morphism. Let \mp[[ \pi_i || ({\tilde{S}}_i, {\tilde{C}}_i) || (S_i, C_i) ]] be the blowup maps. By the universal property of blowing up, and because $f$ is cartesian by \pref{flat-is-cartesian}, we obtain a map \mp[[ {\tilde{\lowercase{F}}} || {\tilde{S}}_1 || {\tilde{S}}_2 ]] which makes the diagram: \diagramx{{\tilde{S}}_1&\mapE{{\tilde{\lowercase{F}}}}&{\tilde{S}}_2\cr \mapS{\pi_1}&&\mapS{\pi_2}\cr S_1&\mapE{f}&S_2\cr% }commute. Furthermore, using the flatness of $f$, we see that this diagram is cartesian and as a consequence that ${\tilde{\lowercase{F}}}$ is flat and formally smooth. Since $S_1$ and $S_2$ embed in codimension one, so do ${\tilde{S}}_1$ and ${\tilde{S}}_2$. Since $p_{k+1}(S_i,C_i) = p_k({\tilde{S}}_i,{\tilde{C}}_i)$ for all $k \geq 1$, the proof of the proposition will follow if we can show that $p_1(S_1,C_1) = p_1(S_2,C_2)$. Let $x_i \in S_i$ be the unique closed points. It is clear that $\pi_1^{-1}(x_1)$ maps onto $\pi_2^{-1}(x_2)$. Moreover, ${\tilde{\lowercase{F}}}({\tilde{C}}_1) = {\tilde{C}}_2$. Let ${\tilde{\lowercase{X}}}_i = \pi_i^{-1}(x_i) \cap {\tilde{C}}_i$. Then ${\tilde{\lowercase{F}}}({\tilde{\lowercase{X}}}_1) = {\tilde{\lowercase{X}}}_2$. Let $P_i = C_i \times_{S_i} {\tilde{S}}_i$. A little thought shows that there is a cartesian diagram: \squareSE{P_1}{P_2}{{\tilde{S}}_1}{{\tilde{S}}_2\makenull{.}% }Since $S_1$ and $S_2$ embed in codimension one, $P_i = {\tilde{C}}_i \cup E_i$, where $E_i \cong \P1$ and ${\tilde{C}}_i \cap E_i = {\tilde{\lowercase{X}}}_i$. The equality $p_1(S_1,C_1) = p_1(S_2,C_2)$ can then be deduced from \pref{formal-woof-2}. {\hfill$\square$} \end{proof} If $(S,C)$ is a local-geometric pair, then $S$ is excellent, so the completion map \mapx[[ {\hat{S}} || S ]] is formally smooth. Hence we have: \begin{corollary}\label{analytic-invariant} If two local-geometric surface-curve pairs embed in codimension one and are analytically isomorphic, then they have the same type. \end{corollary} \begin{remark} We do not know if \pref{formal-woof-3} and \pref{analytic-invariant} are true without the hypothesis of ``embedding in codimension one''. \end{remark} \block{Examples} \par\indent\indent We give three examples which illustrate type computations and pathological aspects of blowing up. Cf.\ \pref{fantastico}, where rational double points are dealt with. The first example illustrates a general conjecture which we cannot yet make precise: amongst surfaces of given degree in $\P3$, those which occur in positive characteristic can have ``larger'' type than those which occur in characteristic zero. Of course, the type also depends on the choice of a curve on the surface. More specifically, the example shows that in characteristic two, a quartic surface (together with a suitably chosen curve) can have $p_1 = p_2 = p_3 = 8$. We expect that this cannot happen in characteristic zero. If so, it would follow from (\ref{thmI} = ``I'') that a smooth quartic rational curve $C \subset {\Bbb C}\kern1pt\xmode{\Bbb P\kern1pt}^3$ cannot be the set-theoretic complete intersection\ of two quartic surfaces, unless $C$ is contained in the singular locus of one of the surfaces. On the other hand, Hartshorne \Lcitemark 15\Rcitemark \Rspace{} and Samuel\Lspace \Lcitemark 30\Rcitemark \Rspace{} have shown that in positive characteristic, the monomial rational quartic curve $C \subset \P3$ is a set-theoretic complete intersection. See \pref{examplex-char-two} for additional comments. The second two examples have to do with pairs $(S,C)$ in which $S$ is not Cohen-Macaulay. These seem to be of some intrinsic interest, but have no direct relevance to the problem of set-theoretic complete intersections\ in $\P3$. Example two might be viewed as a statement about the properties of the singularity at the vertex of the cone over a space curve. It would be very nice to understand better the connection between this singularity and the properties of the space curve. \begin{prop}\label{example-char-two} Let $k$ be an algebraically closed field of characteristic two. Let $S \subset \P3$ be the cuspidal cone given by $y^4 - x^3w = 0$. Let $C \subset S$ be the smooth rational quartic curve given by $$(s,t) \mapsto (x,y,z,w) = (s^4, s^3t, st^3, t^4).$% $Then $C$ meets $\mathop{\operatoratfont Sing}\nolimits(S)$ at the unique point $(0,0,0,1)$, and the type of $(S,C)$ is $(8,8,8)$. \end{prop} \begin{proof} We will calculate in the category of affine varieties, so we will replace $S$ by an affine variety, and when we refer to a {\it blowup}, we will actually mean a correctly chosen affine piece of the blowup. We let $(S_n, C_n)$ denote the \th{n} iterated blowup of $(S,C)$. The assertion that $C \cap \mathop{\operatoratfont Sing}\nolimits(S) = \setof{(0,0,0,1)}$ is easily checked. Taking the affine piece at $w = 1$, we find that $S$ is given by $y^4 = x^3$ and that $C$ is given by $x = yz$ and $y = z^3$. Making the change of variable $x \mapsto x + yz$, followed by $y \mapsto y+z^3$, we obtain the new equation: $$y^4 + z^{12} = (x+yz+z^4)^3$% $for $S$ and the equation $x = y = 0$ for $C$. Blow up $S$ along $C$, formally substituting $xy$ for $x$. Then $S_1$ is given by: $$y^3 = x^3y^2 + x^2y^2z + x^2yz^4 + xy^2z^2 + y^2z^3 + yz^6 + xz^8 + z^9.$% $Intersecting with the exceptional divisor, as in \pref{goober-peas}, corresponds to setting $y = 0$. We obtain $z^8(z + x) = 0$, which tells us that $p_1(S,C) = 8$, and that $C_1$ is given by $y = 0$, $z+x=0$. Making the change of variable $z \mapsto z - x$, we obtain the new equation: $$y^3 = y^2z^3 + yz^6 + x^4yz^2 + x^8z + z^9$% $for $S_1$, and the equation $y = z = 0$ for $C_1$. Blow up $S_1$ along $C_1$, formally substituting $zy$ for $z$. Then $S_2$ is given by: $$y^2 = y^4z^3 + y^6z^6 + x^4y^2z^2 + y^8z^9 + x^8z.$% $Setting $y = 0$, we obtain $x^8z = 0$, which tells us that $p_2(S,C) = 8$, and that $C_2$ is given by $y = z = 0$. Blow up $S_2$ along $C_2$, formally substituting $zy$ for $z$. Then the blown up surface $S_3$ is given by: $$y = y^6z^3 + y^{11}z^6 + x^4y^3z^2 + y^{16}z^9 + x^8z.$% $Setting $y = 0$, we obtain $x^8z = 0$, which tells us that $p_3(S,C) = 8$, and that the new curve $C_3$ is given by $y = z = 0$. One can check that $S_3$ is smooth along $C_3$, so $p_k(S,C) = 0$ for all $k > 3$. {\hfill$\square$} \end{proof} \begin{prop}\label{type-ex-2} Let $S \subset \xmode{\Bbb A\kern1pt}^4 = \mathop{\operatoratfont Spec}\nolimits {\Bbb C}\kern1pt[x,y,z,w]$ be the cone over the monomial quartic curve: $$(s,t) \mapsto (x,y,z,w) = (s^4, s^3t, st^3, t^4)$% $in $\P3$. Let $C \subset S$ be the ruling given by $y = z = w = 0$. Then $\mathop{\operatoratfont type}\nolimits(S,C) = (1,2)$. Moreover, if \mp[[ \pi || S_1 || S ]] denotes the blowup along $C$, and $p \in S$ denotes the unique singular point, then $\RED{\pi^{-1}(p)} \cong \mathop{\operatoratfont Proj}\nolimits {\Bbb C}\kern1pt[s,t,u](s^3-t^2u)$. \end{prop} \begin{proof} One sees that $S$ is given by the equations $yz = xw$, $x^2z = y^3$, $z^3 = yw^2$, and $y^2w = xz^2$. The blowup $S_1$ of $S$ along $C$ is obtained\footnote{In this situation, where $S$ does not embed in codimension one, it is apparently necessary to look at all of the affine pieces of the blowup. The details of this are left to the reader. These calculations are greatly facilitated by the use of a computer program such as Macaulay.} by formally substituting $z = sy$, $w = ty$. Then $S_1$ is given by $sy = tx$, $sx^2 = y^2$, $s^3 = t^2$, and $ty = s^2x$. Then: \disomorx[[ \pi_1^{-1}(C) || \mathop{\operatoratfont Spec}\nolimits {\Bbb C}\kern1pt[x,y,s,t]/(y,tx,sx^2,s^3-t^2,s^2x). ]]% Set-theoretically, $$\pi_1^{-1}(C) = V(s,t,y) \cup V(x,y, s^3-t^2).$% $We have $C_1 = V(s,t,y)$. Thus: $$p_1(S,C) = \mathop{\operatoratfont length}\nolimits {\Bbb C}\kern1pt[x,s,t]/(tx,sx^2,s^3-t^2,s^2x)_{(x,s^3-t^2)},$% $which equals one. The blowup $S_2$ of $S_1$ along $C_1$ is obtained by formally substituting $s = ay$, $t = by$. Then $S_2$ is given by $ax^2 = y$ and $b = a^2x$. Let \mp[[ \pi_2 || S_2 || S_1 ]] be the blowup map. Then: \begin{eqnarray*} \pi_2^{-1}(C_1) & \cong & \mathop{\operatoratfont Spec}\nolimits {\Bbb C}\kern1pt[a,b,x] / (ax^2,b-a^2x)\\ & \cong & \mathop{\operatoratfont Spec}\nolimits {\Bbb C}\kern1pt[a,x] / (ax^2). \end{eqnarray*} This implies that $p_2(S,C) = 2$. Since $S_2$ is smooth, we see that $\mathop{\operatoratfont type}\nolimits(S,C)$ is as claimed. \end{proof} \begin{example}\label{type-ex-3} Let $S$ be a smooth surface, which is a variety. Fix $n \geq 2$, and let $\vec p1n \in S$ be distinct (closed) points. Let \mp[[ \pi || S || {\overline{S}} ]] be the morphism which pinches $\vec p1n$ together, yielding $p \in {\overline{S}}$. Let $C \subset S$ be a smooth curve passing through $p_1$ but not through $\vec p2n$. Then ${\overline{C}} = \pi(C)$ is smooth. Let \mp[[ f || X || {\overline{S}} ]] be the blowup along ${\overline{C}}$. Then $X$ is obtained from $S$ by blowing up $\vec p2n$. Hence $f^{-1}({\overline{C}})$ is isomorphic to the disjoint union of ${\overline{C}}$ with $n-1$ copies of $\P1$. The type of $(S,C)$ is $(n-1)$. \end{example} \block{Classification of rational double point pairs}\label{class} \par\indent\indent In this section, we assume that $k$ has characteristic zero. We describe a classification of local-geometric pairs $(S,C)$, up to analytic isomorphism, where $S$ is the local scheme of a rational double point singularity. Let $(S,C)$ be a local-geometric pair corresponding to a rational double point. As is well-known, such objects $S$ are classified (up to analytic isomorphism) by A-D-E Dynkin diagrams. Let ${\tilde{S}}$ be the minimal resolution of $S$, and let ${\tilde{C}} \subset {\tilde{S}}$ be the strict transform of $C$. Let $\vec E1n \subset {\tilde{S}}$ be the exceptional curves, numbered as in (\Lcitemark 18\Rcitemark \ p.\ 167). Then ${\tilde{C}}$ meets a unique exceptional curve $E_k$, and we have ${\tilde{C}} \cdot E_k = 1$. Moreover, there are some restrictions on $k$, depending on $S$. (See\Lspace \Lcitemark 18\Rcitemark \Rspace{}\ 2.2.) In this way, we are able to define certain local-geometric pairs $A_{n,k}$, $D_{n,k}$, and $E_{n,k}$. In fact, one can show\Lspace \Lcitemark 20\Rcitemark \Rspace{} that these pairs are well-defined, up to analytic isomorphism. We have: \begin{theorem}\label{the-conjecture} Let $(S,C)$ be a local-geometric pair, where $S$ is the local scheme of a rational double point singularity. Then $(S,C)$ is analytically isomorphic to a unique member of the following list of local pairs: \begin{itemize} \item $A_{n,k}$ (for some positive integers $n,k$ with $k \leq (n+1)/2$); \item $D_{n,1}$ (for some integer $n \geq 4$); \item $D_{n,n}$ (for some integer $n \geq 5$); \item $E_{6,1}$; \item $E_{7,1}$. \end{itemize} \end{theorem} \begin{remark} Equations for these pairs may be found in the proof of \pref{fantastico}. \end{remark} \block{Invariants of rational double point configurations}\label{invrdp} \par\indent\indent In this section, we assume that $k$ has characteristic zero. We will calculate the type of $(S,C)$ in the case where $S$ is the local scheme of a rational double point singularity. This depends not only on $S$, but also on $C$. Note that if $S$ is the local scheme of any rational singularity, and $S$ embeds in a nonsingular three-fold, then $S$ ``is'' a rational double point. For each pair of positive integers $(n,k)$ with $k \leq n$, we define a sequence $\phi(n,k)$ of integers, via the following recursive definition: $$\phi(n,k) = \cases{ \phi(n,n-k+1),&if $k > {n+1\over2}$;\cr (k),&if $k = {n+1\over2}$;\cr (k,\phi(n-k,k)),&if $k < {n+1\over2}$.}$% $ \begin{examples} \ \begin{arabiclist} \item $\phi(n,1) = (1^\br{n})$ for all $n \geq 1$; \item $\phi(rk,k) = (k^\br{r-1},1^\br{k})$ for all $k \geq 1$, $r \geq 1$ (generalizing 1); \item $\phi(rk-1,k) = (k^\br{r-1})$ for all $r \geq 2$, $k \geq 1$ (also generalizing 1); \item $\phi(10,4) = (4,3,1^\br{3}).$ \end{arabiclist} \end{examples} Let $a, b \in \xmode{\Bbb N}$. For each integer $n \geq 0$, we define the \th{n} {\it iterated remainder\/} on division of $a$ by $b$, denoted $\mathop{\operatoratfont rem}\nolimits_n(a,b)$. Let $\mathop{\operatoratfont rem}\nolimits_0(a,b) = b$, and let $\mathop{\operatoratfont rem}\nolimits_1(a,b)$ be the usual remainder. For $n \geq 2$, define: $$\mathop{\operatoratfont rem}\nolimits_n(a,b) = \cases{ \mathop{\operatoratfont rem}\nolimits_1(\mathop{\operatoratfont rem}\nolimits_{n-2}(a,b), \mathop{\operatoratfont rem}\nolimits_{n-1}(a,b)), &if $\mathop{\operatoratfont rem}\nolimits_{n-1}(a,b) \not= 0$;\cr 0,&if $\mathop{\operatoratfont rem}\nolimits_{n-1}(a,b) = 0$.}$% $ \par Let $a,b \in \xmode{\Bbb N}$. For each integer $n \geq 1$, we define the \th{n} {\it iterated quotient\/} of $a$ by $b$, denoted $\mathop{\operatoratfont div}\nolimits_n(a,b)$. Let $\mathop{\operatoratfont div}\nolimits_1(a,b) = \floor{a/b}$. For $n \geq 2$, define: $$\mathop{\operatoratfont div}\nolimits_n(a,b) = \cases{ \mathop{\operatoratfont div}\nolimits_1(\mathop{\operatoratfont rem}\nolimits_{n-2}(a,b), \mathop{\operatoratfont rem}\nolimits_{n-1}(a,b)), &if $\mathop{\operatoratfont rem}\nolimits_{n-1}(a,b) \not= 0$;\cr 0,&if $\mathop{\operatoratfont rem}\nolimits_{n-1}(a,b) = 0$.}$% $ \begin{prop}\label{key-rem} Fix $k, n \in \xmode{\Bbb N}$ with $k \leq (n+1)/2$. Let $t$ be the largest integer such that $\mathop{\operatoratfont rem}\nolimits_t(n-k+1,k) \not= 0$. Let $r_i = \mathop{\operatoratfont rem}\nolimits_i(n-k+1,k)$, $d_i = \mathop{\operatoratfont div}\nolimits_i(n-k+1,k)$, for various $i$. Then: $$\phi(n,k) = (r_0^\br{d_1}, r_1^\br{d_2}, \ldots, r_t^\br{d_{t+1}}).$% $\end{prop} \begin{sketch} Define $r_{-1} = n-k+1$. One shows that for all $p \geq 0$, $$\phi(r_{p-1}+r_p-1, r_p) = \cases{(r_p^\br{d_{p+1}}, \phi(r_p + r_{p+1}-1, r_{p+1})), &if $r_{p+1} \not= 0$;\cr (r_p^\br{d_{p+1}}),&if $r_{p+1} = 0$.\cr}$% $The result then follows by induction. {\hfill$\square$} \end{sketch} \begin{prop}\label{fantastico} The type of $A_{n,k}$ is $\phi(n,k)$. The type of $D_{n,1}$ is $(2)$. We have: $${\begindiagram \mathop{\operatoratfont type}\nolimits(D_{n,n}) = \cases{({n\over2}),&if $n$ is even\kern1.5pt$;$\cr ({n-1\over2}, 1^\br{n-1}),& if $n$ is odd.\cr}}$% $The type of $E_{6,1}$ is $(2,2)$. The type of $E_{7,1}$ is $(3)$. \end{prop} \begin{proof} We let $(S,C)$ correspond to the given pair. The comments in the first paragraph of the proof of \pref{example-char-two} apply equally well here. We make use of the explicit resolutions of rational double points given in the appendix to\Lspace \Lcitemark 25\Rcitemark \Rspace{}. First we consider the $A_{n,k}$ case. (We allow $1 \leq k \leq n$.) Then $S$ is given by $xy - z^{n+1} = 0$, and $C$ is given parametrically by $x = u^k$, $y = u^{n-k+1}$, $z = u$. [In terms of the notation used in \Lcitemark 25\Rcitemark \Rspace{}, this may be seen as the image of $V(u_k=1) \subset W_k$.] After making the change of variable $x \mapsto x + z^k$ and $y \mapsto y + z^{n-k+1}$, we find that $S$ is given by: $$xy + yz^k + xz^{n-k+1} = 0,\eqno(*)$% $and that $C$ is given by $x = y = 0$. From now on, we assume that $k \leq {n+1\over2}$. Blow-up along $C$, formally substituting $yx$ for $y$. Then $S_1$ is given by: $$xy + yz^k + z^{n-k+1} = 0.$% $Intersecting with the exceptional divisor, as in \pref{goober-peas}, corresponds to setting $x = 0$. We obtain $z^k(y + z^{n-2k+1}) = 0$. This tells us that $p_1(S,C) = k$ and that $C_1$ is given by $x = 0$ and $y + z^{n-2k+1} = 0$. After making the change of variable $y \mapsto y - z^{n-2k+1}$, and thence $y \mapsto -y$, we obtain the equation: $$xy + yz^k + xz^{n-2k+1} = 0\eqno(**)$% $for $S_1$, and the equation $x = y = 0$ for $C_1$. If $n-2k+1=0$, then $S_1$ is smooth along $C_1$, and we are done. Otherwise, compare $(*)$ with $(**)$, to complete the $A_{n,k}$ case. Now we deal with the case $D_{n,1}$. In terms of the notation used in\Lspace \Lcitemark 25\Rcitemark \Rspace{}, $C$ is the image of $V(v_0 = 0) \subset W_0$. Following\Lspace \Lcitemark 25\Rcitemark \Rspace{}, we would have two cases ($n$ even, $n$ odd), but in fact these two cases are identical in this situation, after interchanging variables $(x \leftrightarrow y)$. We find that $S$ is given by: $$x^2z + y^2 - z^{n-1} = 0,$% $and that $C$ is given by $y = z = 0$. Blow up along $C$, formally substituting $zy$ for $z$. Then $S_1$ is given by: $$x^2z + y - y^{n-2}z^{n-1} = 0.$% $Setting $y = 0$, we obtain $x^2z = 0$. This tells us that $p_1(S,C) = 2$ and that $C_1$ is given by $y = z = 0$. An easy calculation shows that $S_1$ is smooth. The result for $D_{n,1}$ follows. Now we deal with the case $D_{n,n}$. In terms of the notation used in \Lcitemark 25\Rcitemark \Rspace{}, $C$ is the image of $V(u_n = 0) \subset W_n$. We may take the same equation for $S$ as we did in the case $D_{n,1}$. There are two cases: Case I: $n$ is even. Then $C$ is given parametrically by $x = u^{(n-2)/2}$, $y = 0$, $z = u$. After making the change of variable $x \mapsto x + z^{(n-2)/2}$, we find that $S$ is given by: $$x^2z + y^2 + 2xz^{n/2} = 0,$% $and that $C$ is given by $x = y = 0$. Blow up along $C$, formally substituting $xy$ for $x$. Then $S_1$ is given by: $$x^2yz + y + 2xz^{n/2} = 0.$% $Setting $y = 0$, we obtain $xz^{n/2} = 0$. This tells us that $p_1(S,C) = n/2$ and that $C_1$ is given by $x = y = 0$. On checks that $S_1$ is smooth. Case II: $n$ is odd. Then $C$ is given parametrically by $x = 0$, $y = u^{(n-1)/2}$, $z = u$. (In this case, $x$ and $y$ are interchanged from the notation in\Lspace \Lcitemark 25\Rcitemark \Rspace{}.) After making the change of variable $y \mapsto y + z^{(n-1)/2}$, we find that $S$ is given by: $$x^2z + y^2 + 2yz^{(n-1)/2} = 0,$% $and that $C$ is given by $x = y = 0$. Blow up along $C$, formally substituting $yx$ for $y$. Then $S_1$ is given by: $$xz + xy^2 + 2yz^{(n-1)/2} = 0.\eqno(*{*}*)$% $Setting $x = 0$, we obtain $yz^{(n-1)/2} = 0$. This tells us that: $$p_1(S,C) = (n-1)/2$% $and that $C_1$ is given by $x = y = 0$. Now blow-up along $C_1$, formally substituting $xy$ for $x$. Then the blow-up $S_2$ is given by: $$xz + xy^2 + 2z^{(n-1)/2} = 0.$% $Setting $y = 0$, we obtain $z(x + 2z^{(n-1)/2 - 1}) = 0$. This tells us that $p_2(S,C) = 1$ and that $C_2$ is given by ($y = 0$ and $x + 2z^{(n-1)/2 - 1} = 0$). After making the change of variable $x \mapsto x - 2z^{(n-1)/2 - 1}$, we find that $S_2$ is given by: $$xz + xy^2 - 2y^2z^{(n-1)/2 - 1} = 0,$% $and that $C_2$ is given by $x = y = 0$. Now blow up along $C_2$, formally substituting $xy$ for $x$. Then the blown up surface $S_3$ is given by: $$xz + xy^2 - 2yz^{(n-1)/2-1} = 0,$% $$C_3$ is given by $x = y = 0$, and $p_3(S,C) = 1$. Replacing $y$ by $-y$, we may assume that $S_3$ is given by: $$xz + xy^2 + 2yz^{(n-1)/2-1} = 0.$% $This looks like $(*{*}*)$, except that $n$ is now replaced by $n-2$. Note that if $n = 1$, then $(*{*}*)$ is smooth. A little thought shows that the asserted type of $D_{n,n}$ is correct. A posteriori, we see that $(S_1,C_1) = A_{n-1,1}$. A direct proof of this assertion would of course simplify the proof. For both $E_{6,1}$ and $E_{7,1}$, we may choose any smooth curve for $C$. For $E_{6,1}$, $S$ is given by $x^2 - y^3 - z^4 = 0$, and $C$ is given by $y = 0$, $x + z^2 = 0$. After making the change of variable $x \mapsto x - z^2$, we obtain the new equation $x^2 - 2xz^2 - y^3 = 0$ for $S$. Then $C$ is given by $x = y = 0$. Blow up along $C$, substituting $xy$ for $x$. The equation for $S_1$ is: $$x^2y - 2xz^2 - y^2 = 0.$% $Setting $y = 0$, we obtain $xz^2 = 0$. Hence $p_1(S,C) = 2$, and $C_1$ is given by $x = y = 0$. Blow up $S_1$ along $C_1$, substituting $xy$ for $x$. The equation for $S_2$ is $x^2y^2 - 2xz^2 - y = 0$. Substituting $y = 0$, we obtain $xz^2 = 0$. Hence $p_2(S,C) = 2$. One checks that $S_2$ is smooth, so $p_k(S,C) = 0$ for all $k > 2$. For $E_{7,1}$, $S$ is given by $x^2 + y^3 - yz^3 = 0$, and $C$ is given by $x = y = 0$. Blow up along $C$, substituting $yx$ for $y$. The equation for $S_1$ is $x + x^2y^3 - yz^3 = 0$. Setting $x = 0$, we obtain $yz^3 = 0$. Hence $p_1(S,C) = 3$. As $S_1$ is smooth, we see that the type of $E_{7,1}$ is as claimed. {\hfill$\square$} \end{proof} \begin{warning} Amongst the rational double point\ local-geometric pairs, those of the kind $(S,C) = D_{n,n}$ with $n$ odd ($n \geq 5$) are highly atypical. The following phenomena happen only for these special pairs: \begin{romanlist} \item $\Sigma(S,C) < \sum_{i=1}^\infty p_i(S,C)$; \item $p_r(S,C) \not= 0$ for some $r > \mathop{\operatoratfont order}\nolimits(S,C)$: see \pref{interesting}. \end{romanlist} \end{warning} The calculation in the proposition allows one to compute not just the type of a rational double point, but also the precise sequence of (analytic equivalence classes of) local surface-curve pairs which arise under successive blowups: \begin{itemize}\label{blowupAnk} \item $\mathop{\operatoratfont blowup}\nolimits(A_{n,k}) \cases{\hbox{is smooth},&if $k = {n+1\over2}$;\cr = A_{n-k,n-2k+1},&if ${n-k+1\over2} < k < {n+1\over2}$;\cr = A_{n-k,k},&if $k \leq {n-k+1\over2}$;}$ \item $\mathop{\operatoratfont blowup}\nolimits(D_{n,1})$ is smooth; \item $\mathop{\operatoratfont blowup}\nolimits(D_{n,n})$ is smooth, if $n$ is even; \item $\mathop{\operatoratfont blowup}\nolimits(D_{n,n}) = A_{n-1,1},$ if $n$ is odd; \item $\mathop{\operatoratfont blowup}\nolimits(E_{6,1}) = A_{3,2}$; \item $\mathop{\operatoratfont blowup}\nolimits(E_{7,1})$ is smooth. \end{itemize} We now calculate $\mathop{\operatoratfont order}\nolimits(S,C)$, where $S$ is the local scheme of a rational double point. \begin{prop}\label{order-calc} The order of $A_{n,k}$ is the order of ${\overline{\lowercase{K}}}$ in $\xmode{\Bbb Z}/(n+1)\xmode{\Bbb Z}$. The order of $D_{n,1}$ is $2$. The order of $D_{n,n}$ is $2$ if $n$ is even, and it is $4$ if $n$ is odd. The order of $E_{6,1}$ is $3$. The order of $E_{7,1}$ is $2$. \end{prop} \begin{proof} Let $(S,C)$ correspond to the given pair. Some of the orders ($E_{6,1}$, $E_{7,1}$, $D_{n,1}$ ($n$ even) and $D_{n,n}$ ($n$ even)) can be computed immediately if one knows the abstract group $\mathop{\operatoratfont Cl}\nolimits(S)$. A list of these groups may be found in (\Lcitemark 22\Rcitemark \ p.\ 258). We do not use this approach. Let ${\tilde{S}}$ be the minimal resolution of $S$. Let $\vec E1n \subset {\tilde{S}}$ be the exceptional curves. Let ${\tilde{C}} \subset {\tilde{S}}$ denote the strict transform of $C$. There is a unique ${\Bbb Q}\kern1pt$-divisor $E = a_1E_1 \many+ a_nE_n$ such that ${\tilde{C}} \cdot E_i = -E \cdot E_i$ for all $i$. (See\Lspace \Lcitemark 26\Rcitemark \Rspace{}.) The total transform ${\overline{C}} \subset {\tilde{S}}$ of $C$ is ${\tilde{C}} + E$; it is a ${\Bbb Q}\kern1pt$-divisor. According to (\Lcitemark 26\Rcitemark \ p.\ 242), ${\overline{C}}$ is integral (i.e.\ $E$ is integral, i.e.\ $\vec a1n \in \xmode{\Bbb Z}$) if and only if\ ``$C$ is locally analytically equivalent to zero''. Since $S$ is a rational double point, this is equivalent to $[C] = 0$ in $\mathop{\operatoratfont Cl}\nolimits(S)$. As this discussion applies not just to $C$, but also to positive integer multiples of $C$, we see that the order of $(S,C)$ is the least positive integer $N$ such that $N(\vec a1n) \in \xmode{\Bbb Z}^n$. Let $M$ be the inverse of the self-intersection matrix of the $E_i$. Then $(\vec a1n)$ is the \th{k} column of $M$. This may be computed from an explicit formula for $M$, which one may find in (\Lcitemark 18\Rcitemark \ p.\ 169). In case $(S,C) = A_{n,k}$, one finds that: $$a_i = \cases{-k(n-i+1)/(n+1),&if $i \geq k$;\cr ki/(n+1) - i,&if $i \leq k$.\cr}$% $From this we calculate that $a_1 = k/(n+1) - 1$. The proof for $A_{n,k}$ follows. In case $(S,C) = D_{n,1}$, one finds that: $$a_i = \cases{-1,&if $i \leq n-2$;\cr -1/2,&if $n-1 \leq i \leq n$.\cr}$% $Hence $\mathop{\operatoratfont order}\nolimits(D_{n,1}) = 2$. In case $(S,C) = D_{n,n}$, one finds that: $$a_i = \cases{-i/2,&if $i \leq n-2$;\cr -(n-2)/4,&if $i = n-1$;\cr -n/4,&if $i = n$.\cr}$% $Hence $\mathop{\operatoratfont order}\nolimits(D_{n,n})$ is as claimed. We now deal with the two exceptional cases. The inverses of the self-intersection matrices do not appear in \Lcitemark 18\Rcitemark \Rspace{}, and we omit them here for lack of space. In case $(S,C) = E_{6,1}$, one finds that: $$(\vec a1n) = (-{4\over3}, -{5\over3}, -2, -1, -{4\over3}, -{2\over3}),$% $and in case $(S,C) = E_{7,1}$, one finds that: \formulaqed{(\vec a1n) = (-{3\over2}, -2, -{5\over2}, -3, -{3\over2}, -2, -1).} \end{proof} It is interesting to note that for $D_{n,n}$ ($n$ odd), one has $p_r(S,C) \not= 0$ for some $r > \mathop{\operatoratfont order}\nolimits(S,C)$. This does not occur for the other rational double point\ pairs, as we shall see in \pref{interesting}. \begin{lemma}\label{gcdgcd} Let $k$ and $N$ be positive integers, with $k < N$. Assume that $k \nmid N$. Then: $$\floor{N/k} \leq (N-k) / \gcd(k,N).$$ \end{lemma} \begin{proof} First suppose that $k > N/2$. Then $\floor{N/k} = 1$, so we must show that $\gcd(k,N) \leq N-k$. Indeed, if $x | k$ and $x | N$, then $x | (N-k)$, so this is clear. Hence we may assume\ that $k \leq N/2$. Since $k \nmid N$, $\gcd(k,N) \leq k/2$. Therefore it suffices to show that $(k/2)(N/k) \leq N-k$. This follows from $k \leq N/2$. {\hfill$\square$} \end{proof} \begin{corollary}\label{orderbound} Let $k$ and $N$ be positive integers, with $k \leq N$. Let $t$ be the smallest positive integer such that $\mathop{\operatoratfont rem}\nolimits_t(N,k) = 0$. Let $d_i = \mathop{\operatoratfont div}\nolimits_i(N,k)$, for $i = 1, \ldots, t$. Then: $$\svec d1t \leq N/\gcd(k,N).$$ \end{corollary} \begin{proof} The case $k = N$ is clear, so we may assume\ that $k < N$. If $t = 1$, the result is clear. Let $r_1 = \mathop{\operatoratfont rem}\nolimits_1(N,k)$. We may assume that $r_1 \not= 0$. By induction on $t$, we may assume\ that: $$\svec d2t \leq k/\gcd(r_1,k).$% $Therefore it suffices to show that: $$d_1 + {k \over \gcd(r_1,k)} \leq {N \over \gcd(k,N)}.$% $One sees that $\gcd(r_1,k) = \gcd(k,N)$. Therefore it suffices to show that $d_1 \leq (N-k) / \gcd(k,N)$. This follows from \pref{gcdgcd}. {\hfill$\square$} \end{proof} \begin{prop}\label{interesting} Let $(S,C)$ be a local-geometric pair corresponding to a rational double point. Assume that $(S,C) \not= D_{n,n}$ for any odd integer $n \geq 5$. Then $p_r(S,C) = 0$ for all $r \geq \mathop{\operatoratfont order}\nolimits(S,C)$. \end{prop} \begin{proof} We utilize \pref{fantastico} and \pref{order-calc}. The only nontrivial case is $(S,C) = A_{n,k}$. We may assume that $k \leq (n+1)/2$. For any $a,b \in \xmode{\Bbb N}$, let $o(a,b)$ denote the order of ${\overline{\lowercase{A}}}$ in $\xmode{\Bbb Z}/b\xmode{\Bbb Z}$. In the notation of \pref{key-rem}, we must show that: $$\svec d1{t+1} < o(k,n+1).$% $Translating to the notation of \pref{orderbound} ($N = n+1$), both $d_1$ and $t$ change by $1$. The statement we need is: $$(d_1 - 1) + \svec d2t < o(k,N).$% $Since $o(k,N) = N/\gcd(k,N)$, this does follow from \pref{orderbound}. {\hfill$\square$} \end{proof} The content of the following proposition may be found in (\Lcitemark 18\Rcitemark \ proof of 2.3, pp.\ 169-170). \begin{prop}\label{delta-formulas} We have: \begin{eqnarray*} \Delta(A_{n,k}) & = & k(n-k+1)/(n+1)\\ \Delta(D_{n,1}) & = & 1\\ \Delta(D_{n,n}) & = & n/4\\ \Delta(E_{6,1}) & = & 4/3\\ \Delta(E_{7,1}) & = & 3/2. \end{eqnarray*} \end{prop} \block{Technical lemmas on rational double points} \par\indent\indent In this section we assume that $k$ has characteristic zero. We prove various technical relationships between the invariants of rational double point\ local-geometric pairs. We use these results in part III. The result \pref{rdp-one} appears to be of intrinsic interest. \begin{prop}\label{potato-1} Let $(S,C)$ be a local-geometric pair corresponding to a rational double point. Write: $$\mathop{\operatoratfont type}\nolimits(S,C) = (n_1^\br{k_1}, \ldots, n_r^\br{k_r})$% $with $n_1 > \cdots > n_r \geq 1$ and $k_i \geq 1$ for each $i$. Assume that $r > 1$. Then $k_r > 1$ and $n_r | n_{r-1}$. \end{prop} \begin{proof} We use \pref{fantastico}. The proposition is clear if $(S,C)$ is of species $D$ or $E$. Therefore we may assume that $(S,C)$ is of species $A$. In the notation of \pref{key-rem}, we may write: $$\mathop{\operatoratfont type}\nolimits(S,C) = (r_0^\br{d_1}, \ldots, r_t^\br{d_{t+1}}).$% $Since $r_{t+1} = 0$, we have $r_{t-1} = r_t d_{t+1}$. Hence $r_t | r_{t-1}$. Hence $n_r | n_{r-1}$. Since $r_{t-1} > r_t$, $d_{t+1} > 1$. Hence $k_r > 1$. {\hfill$\square$} \end{proof} \begin{prop}\label{potato-2} If $\mathop{\operatoratfont type}\nolimits(A_{n,k}) = \mathop{\operatoratfont type}\nolimits(A_{n',k'})$, where $k \leq (n+1)/2$ and $k' \leq (n'+1)/2$, then $n = n'$ and $k = k'$. \end{prop} \begin{proof} We use \pref{fantastico}. Write $\mathop{\operatoratfont type}\nolimits(A_{n,k}) = (r_0^\br{d_1}, \ldots, r_t^\br{d_{t+1}})$, as in \pref{key-rem}. Then $n = n' = r_0(d_1+1) + r_1 - 1$, and $k = k' = r_0$. {\hfill$\square$} \end{proof} \begin{lemma}\label{yechh} Fix positive integers $k$ and $N$ with $k \leq N/2$. Let $t$ be the smallest positive integer such that $\mathop{\operatoratfont rem}\nolimits_t(N,k) = 0$. Let $r_i = \mathop{\operatoratfont rem}\nolimits_i(N,k)$, $d_i = \mathop{\operatoratfont div}\nolimits_i(N,k)$, for various $i$. Then: \begin{romanlist} \item If $t = 1$, then $r_0d_1^{-1} = k^2/N$. \item If $t = 2$, then $(r_0-r_1)d_1^{-1} + r_1(d_1+d_2)^{-1} \leq k^2/N$. \item If $t \geq 3$, then: $$(r_0-r_1)d_1^{-1} + (r_1-r_2)(d_1+d_2)^{-1} + r_2(d_1+d_2+1)^{-1} \leq k^2/N.$% $\end{romanlist} \end{lemma} \begin{proof} First suppose that $t = 1$. Then $N = d_1 k$. Hence $k^2/N = k/d_1 = r_0d_1^{-1}$. This proves (i). Now suppose that $t = 2$. Then $N = r_0d_1 + r_1$ and $r_0 = r_1d_2$. We must show that: $${r_0 - r_1 \over d_1} + {r_1 \over d_1 + d_2} \leq {r_0^2\over r_0d_1+r_1}.$% $Substitute $r_0 = r_1d_2$, and cancel out $r_1$. We must show: $${d_2 - 1 \over d_1} + {1 \over d_1+d_2} \leq {d_2^2 \over d_1d_2 + 1}.$% $Eliminating denominators, we find that we must show: $$(d_1 - 1)(d_2 - 1) \geq 0,$% $which is certainly true. Finally, suppose that $t \geq 3$. Then $N = r_0d_1 + r_1$ and $r_0 = r_1d_2 + r_2$. We must show that: $${r_0-r_1 \over d_1} + {r_1-r_2 \over d_1+d_2} + {r_2 \over d_1+d_2+1} \leq {r_0^2 \over r_0d_1 + r_1}.$% $Substitute $r_0 = r_1d_2 + r_2$. We must show that: $${r_1d_2 + r_2 - r_1 \over d_1} + {r_1 - r_2 \over d_1 + d_2} + {r_2 \over d_1 + d_2 + 1} \leq {(r_1d_2 + r_2)^2 \over r_1d_1d_2 + r_2d_1 + r_1}.$% $Now cancel denominators. (This is best done with the aid of a computer.) We must show: \splitdiagram{d_2r_1^2 - d_1^2d_2r_1^2 - d_1d_2^2r_1^2 + d_1^2d_2^2r_1^2 - d_2^3r_1^2 + d_1d_2^3r_1^2 - d_1^2r_1r_2 - d_2r_1r_2% }{ - d_1d_2r_1r_2 + 2d_1^2d_2r_1r_2 - d_2^2r_1r_2 + d_1d_2^2r_1r_2 + d_1^2r_2^2 \geq 0.} Equivalently, we must show that: \splitdiagram{r_1^2d_2[ d_2^2(d_1-1) + d_1(d_1d_2 - d_1 - d_2)+1]% }{ + r_1r_2[ d_1^2(d_2-1) + d_2^2(d_1-1)+ d_2(d_1^2-d_1-1)] + d_1^2r_2^2 \geq 0.} If $d_1 \geq 2$ and $d_2 \geq 2$, this is clear. Since $k \leq N/2$, we have $d_1 \geq 2$. Suppose that $d_2 = 1$. Then the needed inequality simplifies to: $$r_1r_2(d_1^2 - 2) + d_1^2r_2^2 \geq 0,$% $which is true. {\hfill$\square$} \end{proof} {}From \pref{yechh}, we obtain the following weaker statement, which we use in \pref{rdp-one}: \begin{corollary}\label{weirdness} Fix positive integers $k$ and $N$, with $k \leq N/2$. Let $t$ be the smallest positive integer such that $rem_t(N,k) = 0$. Let $r_i = \mathop{\operatoratfont rem}\nolimits_i(N,k)$, $d_i = \mathop{\operatoratfont div}\nolimits_i(N,k)$, for various $i$. Then: $$(r_0-r_1)d_1^{-1} \many+ (r_{t-1}-r_t)(\svec d1t)^{-1} \leq k^2/N.$$ \end{corollary} \begin{warning} When we use \pref{weirdness}, the symbol $d_1$ will appear to have two different values, differing by $1$: in the application \pref{rdp-one}, $d_1$ will be smaller by $1$. \end{warning} \begin{prop}\label{rdp-one} Let $(S,C)$ correspond to a rational double point singularity. Let $p_k = p_k(S,C)$, for each $k \in \xmode{\Bbb N}$. Then: $$\sum_{k=1}^\infty{1 \over k(k+1)}p_k \geq \Delta(S,C).$$ \end{prop} \begin{proof} We use \pref{delta-formulas} and \pref{fantastico}. If $(S,C)$ is not of species $A$, then the proposition is proved by the following table: {\renewcommand{\arraystretch}{1.2} \vspace*{0.1in} \centerline{ \begin{tabular}{||c|c|c||} \hline singularity & $\sum_{k=1}^\infty {1\over k(k+1)} p_k$ & $\Delta(S,C)$\\ \hline $D_{n,1}$ & $1$ & $1$\\ \hline $D_{n,n}$ ($n$ even) & $n/4$ & $n/4$\\ \hline $D_{n,n}$ ($n$ odd) & ${n-1\over4}$ + $\sum_{k=2}^n {1\over k(k+1)}$ & $n/4$\\ \hline $E_{6,1}$ & $4/3$ & $4/3$\\ \hline $E_{7,1}$ & $3/2$ & $3/2$\\ \hline \end{tabular} } \vspace*{0.1in}} Suppose that $(S,C) = A_{n,k}$ for some $n, k$. We may assume that $k \leq (n+1)/2$. We must show that: $$\sum_{j=1}^\infty {1 \over j(j+1)}\phi(n,k)_j \geq k(n-k+1)/(n+1).\eqno(*)$% $Let $t$ be the largest integer such that $\mathop{\operatoratfont rem}\nolimits_t(n-k+1,k) \not= 0$. For $i = 0, \ldots, t$, let $r_i = \mathop{\operatoratfont rem}\nolimits_i(n-k+1,k)$, $d_{i+1} = \mathop{\operatoratfont div}\nolimits_{i+1}(n-k+1,k)$. By \pref{key-rem}, we see that $(*)$ is equivalent to: \splitdisplay{r_0 \sum_{j=1}^{d_1} {1 \over j(j+1)} + r_1 \sum_{j=d_1+1}^{d_1+d_2} {1 \over j(j+1)} \many+ r_t \sum_{j=\svec d1t + 1}^{\svec d1{t+1}} {1 \over j(j+1)}% }{\geq {k(n-k+1) \over n+1}.% }Note that for any $a,b \in \xmode{\Bbb N}$ with $a \leq b$, $$\sum_{j=a+1}^b {1 \over j(j+1)} = {b \over b+1} - {a \over a+1}.$% $Hence $(*)$ is equivalent to: \splitdisplay{\left(r_0-r_1\right)\left({d_1 \over d_1+1}\right) \many+ \left(r_{t-1}-r_t\right)\left({\svec d1t \over \svec d1t + 1}\right)% }{+ r_t\left({\svec d1{t+1} \over \svec d1{t+1} + 1}\right) \geq k(n-k+1)/(n+1).% }This is equivalent to: \splitdiagram{r_0 - (r_0 - r_1)(d_1 + 1)^{-1} - \cdots - (r_{t-1}-r_t) (\svec d1t + 1)^{-1}% }{- r_t(\svec d1{t+1} + 1)^{-1} \geq k(n-k+1)/(n+1).% }Since $r_0 = k$, this is equivalent to: \splitdiagram{(r_0 - r_1)(d_1 + 1)^{-1} \many+ (r_{t-1}-r_t) (\svec d1t + 1)^{-1}} { + r_t(\svec d1{t+1} + 1)^{-1} \leq k^2/(n+1).% }Let $N = n+1$. Then this follows from \pref{weirdness}, and thence completes the proof. {\hfill$\square$} \end{proof} \begin{definition} Let $(S,C)$ be a local-geometric pair corresponding to a rational double point. Then the {\it deficiency\/} of $(S,C)$ is: $$\mathop{\operatoratfont def \kern1pt}\nolimits(S,C) = \Sigma(S,C) - \sum_{i=1}^\infty p_i(S,C).$$ \end{definition} One always has $\mathop{\operatoratfont def \kern1pt}\nolimits(S,C) \geq 0$, except for $D_{n,n}$, with $n$ odd, $n \geq 5$. \part{Iterated curve blowups} \block{Intersection ring of a blow up}\label{oneblow} \par\indent\indent In this section we describe (without proof) the intersection ring of the blow-up of a nonsingular variety along a nonsingular subvariety, following the statements given in (\Lcitemark 7\Rcitemark \ 6.7, 8.3.9). There are two differences between the assertions we make and the assertions made in\Lspace \Lcitemark 7\Rcitemark \Rspace{}. Firstly, we work with cycles modulo algebraic equivalence, rather than modulo rational equivalence. Secondly, we have adjusted the signs to reflect our convention regarding projective space bundles. Let $X$ be a nonsingular closed subvariety of a nonsingular variety $Y$. Let $d = \mathop{\operatoratfont codim}\nolimits(X,Y)$, and assume that $d \geq 2$. Let $N$ be the normal bundle of $X$ in $Y$. Let ${\tilde{Y}}$ be the blow-up of $Y$ along $X$. The exceptional divisor is isomorphic to $\xmode{\Bbb P\kern1pt} N^*$. We use the following diagram to fix notation: \diagramx{\xmode{\Bbb P\kern1pt} N^*&\mapE{j}&{\tilde{Y}}\cr \mapS{g}&&\mapS{f}\cr X&\mapE{i}&Y.\cr% }Let $F = \mathop{\operatoratfont Ker}\nolimits[ g^*(N^*)\ \mapE{\rm{can}}\ {\cal O}_{\xmode{\Bbb P\kern1pt} N^*}(1)]$. For each $k$, there is a canonically split exact sequence: \sesmaps{A^{k-d}(X)}{\delta}{A^{k-1}(\xmode{\Bbb P\kern1pt} N^*) \o+ A^k(Y)}{\beta}{A^k({\tilde{Y}})% }of cycle groups modulo algebraic equivalence. The maps are given by: $$\delta(x) = (c_{d-1}(F) \cdot g^*(x), i_*(x))$% $and $$\beta({\tilde{\lowercase{X}}}, y) = j_*({\tilde{\lowercase{X}}}) + f^*(y).$% $ This describes $A^*({\tilde{Y}})$ as an abelian group. The ring structure is described by the following rules: \begin{eqnarray*} (f^*y) \cdot (f^*y') & = & f^*(y \cdot y') \\ (j_*{\tilde{\lowercase{X}}}) \cdot (j_*{\tilde{\lowercase{X}}}')&=&-j_*((c_1 {\cal O}_{\xmode{\Bbb P\kern1pt} N^*}(1))\cdot{\tilde{\lowercase{X}}} \cdot {\tilde{\lowercase{X}}}') \\ (f^*y) \cdot (j_*{\tilde{\lowercase{X}}}) & = & j_*((g^*i^*y) \cdot {\tilde{\lowercase{X}}}). \end{eqnarray*} \block{The intersection ring of an iterated curve blow-up}\label{iterate} \par\indent\indent The result of this section is: \begin{theorem}\label{iteration} Let $Y_0 = \P3$. Let $C_0 \subset Y_0$ be a nonsingular curve of degree $d$ and genus $g$. Let $Y_1$ be the blow-up of $Y_0$ along $C_0$. Choose a smooth curve $C_1$ which lies on the exceptional divisor $E_1 \subset Y_1$ and which meets each ruling on $E_1$ exactly once. Let $Y_2$ be the blow-up of $Y_1$ along $C_1$. Iterate this process: $Y_{k+1}$ is obtained by blowing up a smooth curve $C_k \subset E_k \subset Y_k$. We assume that $C_k$ meets each ruling on $E_k$ exactly once and that for all $k \geq 2$, $C_k \not= E_k \cap E_{k-1,k}$, where $E_{k-1,k} \subset Y_k$ denotes the strict transform of $E_{k-1}$. Let $H \subset Y_0$ be a plane. Let ${\mathbf{\lowercase{H}}} = [H] \in A^1(Y_0)$. Let ${\mathbf{\lowercase{E}}}_k = [E_k] \in A^1(Y_k)$. Let ${\mathbf{\lowercase{R}}}_k \in A^1(E_k)$ denote the class of a ruling, which we identify with its image in $A^2(Y_k)$. Identify ${\mathbf{\lowercase{H}}}$, ${\mathbf{\lowercase{E}}}_k$ and ${\mathbf{\lowercase{R}}}_k$ with their images in the intersection ring $A^*(Y_n)$ of the $\th{n}$ iterated blow-up $Y_n$. Then $A^k(Y_n)$ has as a basis: $$[Y_n]\ \ (k=0);\ {\mathbf{\lowercase{H}}}, \vec\lbE1n\ \ (k=1);\ {\mathbf{\lowercase{H}}}^2, \vec\lbR1n\ \ (k=2); \ 1\ \ (k=3).$% $This information, together with the following multiplication rules, completely describe $A^*(Y_n)$ as a graded ring: ${\mathbf{\lowercase{H}}}^3 = 1$, ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{R}}}_k = 0$, ${\mathbf{\lowercase{H}}}^2 \cdot {\mathbf{\lowercase{E}}}_k = 0$, ${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{R}}}_j = -\delta_{i,j}$, ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{E}}}_k = d{\mathbf{\lowercase{R}}}_k$, ${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{E}}}_j = -\beta_i{\mathbf{\lowercase{R}}}_j\ \ (\hbox{if } i < j)$, $${\mathbf{\lowercase{E}}}_k^2 = -d{\mathbf{\lowercase{H}}}^2 - \alpha_{k-1}{\mathbf{\lowercase{R}}}_k - \sum_{i=1}^{k-1} \beta_i{\mathbf{\lowercase{R}}}_i,$% $where $\alpha_k$ is determined by $[C_k] = c_1{\cal O}_{E_k}(1) - \alpha_k{\mathbf{\lowercase{R}}}_k$ in $A^1(E_k)$, for $k \geq 1$, $\alpha_0 = 2-2g-4d$, and $\beta_k = \alpha_{k-1} - \alpha_k$, for each $k \geq 1$. \end{theorem} We note the following generalization and conceptual reformulation of \pref{iteration}, whose proof is omitted. It will not be used again. \begin{theorem} Let $Y_0$ be a nonsingular complete three-fold. Let $C_0 \subset Y_0$ be a nonsingular curve. Let $Y_1$ be the blow-up of $Y_0$ along $C_0$. Choose a smooth curve $C_1$ which lies on the exceptional divisor $E_1 \subset Y_1$ and which meets each ruling on $E_1$ exactly once. Let $Y_2$ be the blow-up of $Y_1$ along $C_1$. Iterate this process: $Y_{k+1}$ is obtained by blowing up a smooth curve $C_k \subset E_k \subset Y_k$. We assume that $C_k$ meets each ruling on $E_k$ exactly once and that for all $k \geq 2$, $C_k \not= E_k \cap E_{k-1,k}$, where $E_{k-1,k} \subset Y_k$ denotes the strict transform of $E_{k-1}$. Let ${\mathbf{\lowercase{E}}}_k = [E_k] \in A^1(Y_k)$. Let ${\mathbf{\lowercase{R}}}_k \in A^1(E_k)$ denote the class of a ruling, which we identify with its image in $A^2(Y_k)$. Identify ${\mathbf{\lowercase{E}}}_k$ and ${\mathbf{\lowercase{R}}}_k$ with their images in the intersection ring $A^*(Y_n)$ of the $\th{n}$ iterated blow-up $Y_n$. Then $A^*(Y_n)$ is the graded $A^*(Y_0)$-algebra generated by $\vec\lbE1n$ (degree $1$) and $\vec\lbR1n$ (degree $2$), modulo the relations: $A^1(Y_0) \cdot {\mathbf{\lowercase{R}}}_k = 0$, $A^2(Y_0) \cdot {\mathbf{\lowercase{E}}}_k = 0$, ${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{R}}}_j = -\delta_{i,j}$, ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{E}}}_k = ({\mathbf{\lowercase{H}}} \cdot C_0){\mathbf{\lowercase{R}}}_k$\ (for all ${\mathbf{\lowercase{H}}} \in A^1(Y_0)$), ${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{E}}}_j = -\beta_i{\mathbf{\lowercase{R}}}_j\ \ (\hbox{if } i < j)$, $${\mathbf{\lowercase{E}}}_k^2 = - [C_0] - \alpha_{k-1}{\mathbf{\lowercase{R}}}_k - \sum_{i=1}^{k-1} \beta_i{\mathbf{\lowercase{R}}}_i,$% $where $\alpha_k$ is determined by $[C_k] = c_1{\cal O}_{E_k}(1) - \alpha_k{\mathbf{\lowercase{R}}}_k$ in $A^1(E_k)$, for $k \geq 1$, $\alpha_0 = \deg(N_{C_0}^*)$, and $\beta_k = \alpha_{k-1} - \alpha_k$, for each $k \geq 1$. \end{theorem} The remainder of this section breaks up into two parts. First we introduce various notations and conventions which we will use in the proof and in subsequent sections. Then we prove \pref{iteration}. There are group homomorphisms \mapx[[ A^i(E_k) || A^{i+1}(Y_k) ]] and injective ring homomorphisms: \diagramx{A^*(Y_0)&\mapE{}&A^*(Y_1)&\mapE{}&\cdots&\mapE{}&A^*(Y_n).% }We systematically identify various elements with their images, via these maps. Since the latter maps are ring homomorphisms, it is not necessary to distinguish between multiplication in $A^*(Y_i)$ and $A^*(Y_j)$, for any $i,j$. On the other hand, since the maps \mapx[[ A^i(E_k) || A^{i+1}(Y_k) ]] are not ring homomorphisms, it is necessary to distinguish between multiplication in $A^*(Y_k)$ and $A^*(E_k)$. We do this by using a dot ($\cdot$) to denote multiplication in $A^*(Y_k)$ and brackets ($\inn,$) to denote multiplication in $A^*(E_k)$. No problems are introduced by the fact that $k$ does not occur explicitly in the bracket notation. Let ${\mathbf{\lowercase{C}}}_k = [C_k]$. This is an element of $A^2(Y_k)$, and it is an element of $A^1(E_k)$ if $k \geq 1$. Let ${\mathbf{\lowercase{D}}}_k = c_1{\cal O}_{E_k}(1)$. It is an element of $A^1(E_k)$. For $k\leq n$, let $E_{k,n} \subset Y_n$ denote the strict transform of $E_k$. In $A^1(Y_n)$ we have ${\mathbf{\lowercase{E}}}_k = [E_{k,n}] \many+ [E_{n,n}]$. (This depends on our assumption that $C_k \not= E_k \cap E_{k-1,k}$.) {\it In particular, the reader should observe the following insidious source of error: ${\mathbf{\lowercase{E}}}_k \not= [E_{k,n}]$.} This same sort of error applies to other cycles which we shall discuss. In this section we do not fix a particular ruling $R_k \subset E_k$. We do so in the next section. Having made such a choice, one can then discuss the strict transform $R_{k,n} \subset Y_n$ of $R_k$. Let $N_k$ be the normal bundle of $C_k$ in $Y_k$. Then $E_k \cong \xmode{\Bbb P\kern1pt}(N_{k-1}^*)$. For each $k = 0, \ldots, n$, we let $\alpha'_k = \deg(N_k^*)$. For each $k = 1, \ldots, n$, we let $\beta'_k = \alpha'_{k-1} - \alpha_k$. (We will show that $\alpha_k = \alpha'_k$ and hence that $\beta_k = \beta'_k$.) \begin{proofnodot} (of \ref{iteration}). We make repeated use of the results of \S\ref{oneblow}, without explicitly referring to them. The abelian group structure of $A^*(Y_n)$ and the assertions that ${\mathbf{\lowercase{H}}}^3 = 1$, ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{R}}}_k = 0$, ${\mathbf{\lowercase{H}}}^2 \cdot {\mathbf{\lowercase{E}}}_k = 0$, and ${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{R}}}_j = -\delta_{i,j}$ are left to the reader. We compute ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{E}}}_i$. Let $\mu_i = {\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{C}}}_i$. Note that: $${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{E}}}_i\ =\ \mu_{i-1} {\mathbf{\lowercase{R}}}_i.$% $We show that $\mu_k$ is independent of $k$, and in fact equals $d$. First one checks that ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{C}}}_0 = d$. Now we have: \begin{eqnarray*} \mu_i & = & {\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{C}}}_i \\ & = & \inn{ ({\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{E}}}_i), {\mathbf{\lowercase{C}}}_i} \\ & = & \inn{\mu_{i-1} {\mathbf{\lowercase{R}}}_i, {\mathbf{\lowercase{C}}}_i} \\ & = & \inn{ \mu_{i-1} {\mathbf{\lowercase{R}}}_i, {\mathbf{\lowercase{D}}}_i - \alpha_i {\mathbf{\lowercase{R}}}_i } \\ & = & \mu_{i-1}. \end{eqnarray*} Hence $\mu_k = d$ for all $k$. Hence ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{C}}}_i\ =\ d$ and ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{E}}}_i\ =\ d{\mathbf{\lowercase{R}}}_i$. We now work on showing that $\alpha_k = \alpha'_k$. In the process we calculate ${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{C}}}_j$ for all $i \leq j$, a result we shall need later. We have: $$\inn{{\mathbf{\lowercase{D}}}_i,{\mathbf{\lowercase{D}}}_i} = c_1(N_{i-1}^*) = \alpha'_{i-1}.$% $Further: \begin{eqnarray*} {\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{C}}}_i & = & -\inn{{\mathbf{\lowercase{D}}}_i, {\mathbf{\lowercase{C}}}_i} \\ & = & -\inn{{\mathbf{\lowercase{D}}}_i, {\mathbf{\lowercase{D}}}_i - \alpha_i{\mathbf{\lowercase{R}}}_i} \\ & = & -(\alpha'_{i-1}-\alpha_i)\ =\ -\beta'_i. \end{eqnarray*} Using this we find: \begin{eqnarray*} {\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{C}}}_{i+1} & = & \inn{({\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{C}}}_i){\mathbf{\lowercase{R}}}_{i+1},{\mathbf{\lowercase{C}}}_{i+1}}\\ & = & {\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{C}}}_i. \end{eqnarray*} Continuing in this manner, the reader may verify that for $i \leq j$, ${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{C}}}_j = -\beta'_i$. The class of the canonical divisor on $Y_i$ is given by: $$[K_{Y_i}] = -4{\mathbf{\lowercase{H}}} + {\mathbf{\lowercase{E}}}_1 \many+ {\mathbf{\lowercase{E}}}_i.$% $(This may be computed from the formula for the canonical divisor of a blowup -- see\Lspace \Lcitemark 9\Rcitemark \Rspace{}\ p.\ 608.) For all $i \geq 0$, we have: \begin{eqnarray*}\label{quirkalpha} \alpha'_i & = & c_1 N_i^* \\ & = & -c_1 \det(N_i)\ =\ -[[K_{C_i}] - [K_{Y_i}]|_{C_i}] \\ & = & -[[K_{C_i}] - [K_{Y_i}] \ \cdot {\mathbf{\lowercase{C}}}_i] \\ & = & -[2g-2 - (-4{\mathbf{\lowercase{H}}} + {\mathbf{\lowercase{E}}}_1 \many+ {\mathbf{\lowercase{E}}}_i) \cdot {\mathbf{\lowercase{C}}}_i] \\ & = & -[2g-2 + 4d + \beta'_1 \many+ \beta'_i]. \end{eqnarray*} {}From this, and from the definition of the $\beta$'s and the $\alpha$'s, we conclude: $$\alpha'_k = \alpha_k\ \ \hbox{for all } k \geq 0.$% $ \par\indent For $i < j$, $${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{E}}}_j = ({\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{C}}}_{j-1}){\mathbf{\lowercase{R}}}_j,$% $so we obtain the formula ${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{E}}}_j = -\beta_i{\mathbf{\lowercase{R}}}_j$. We proceed to calculate ${\mathbf{\lowercase{E}}}_k^2$. By the definition of the map $\delta$ given in \S\ref{oneblow}, we have: \begin{eqnarray*} {\mathbf{\lowercase{D}}}_i & = & \alpha_{i-1}{\mathbf{\lowercase{R}}}_i + {\mathbf{\lowercase{C}}}_{i-1}\ \ \ (i \geq 1). \end{eqnarray*} Continuing to calculate, we find: \begin{eqnarray*}\label{quirkC} {\mathbf{\lowercase{C}}}_i & = & {\mathbf{\lowercase{D}}}_i - \alpha_i {\mathbf{\lowercase{R}}}_i\ \ \ (i \geq 1) \\ {\mathbf{\lowercase{D}}}_i & = & \alpha_{i-1}{\mathbf{\lowercase{R}}}_i + {\mathbf{\lowercase{D}}}_{i-1} - \alpha_{i-1}{\mathbf{\lowercase{R}}}_{i-1} \ \ \ (i \geq 2) \\ {\mathbf{\lowercase{D}}}_i - {\mathbf{\lowercase{D}}}_{i-1} & = & \alpha_{i-1}{\mathbf{\lowercase{R}}}_i - \alpha_{i-1}{\mathbf{\lowercase{R}}}_{i-1} \ \ \ (i \geq 2) \\ {\mathbf{\lowercase{D}}}_1 & = & \alpha_0 {\mathbf{\lowercase{R}}}_1 + d{\mathbf{\lowercase{H}}}^2 \\ {\mathbf{\lowercase{D}}}_k & = & d{\mathbf{\lowercase{H}}}^2 + \left( \sum_{i=1}^{k-1} (\alpha_{i-1} - \alpha_i) {\mathbf{\lowercase{R}}}_i \right) + \alpha_{k-1}{\mathbf{\lowercase{R}}}_k\ \ \ (k \geq 1) \\ {\mathbf{\lowercase{E}}}_k^2 & = & -{\mathbf{\lowercase{D}}}_k\ \ \ (k \geq 1) \\ & = & - \left[ d{\mathbf{\lowercase{H}}}^2 + \left( \sum_{i=1}^{k-1} \beta_i{\mathbf{\lowercase{R}}}_i \right) + \alpha_{k-1}{\mathbf{\lowercase{R}}}_k\right]. \end{eqnarray*} \vspace*{-0.25in} {\hfill$\square$} \end{proofnodot} \block{The strict transform of a ruling} \par\indent\indent The results of this section will be used in the proof of theorem II \pref{thmII}. The notations introduced in \S\ref{iterate} remain in effect in this section. Fix a particular ruling $R_k \subset E_k$, where $1 \leq k \leq n$. We compute the class of $R_{k,n}$ in $A^2(Y_n)$. A priori, this is a $\xmode{\Bbb Z}$-linear combination of ${\mathbf{\lowercase{H}}}^2, \vec\lbR1n$, which depends on the particular choice of $R_k$. We use the term {\it graph\/} to mean an undirected graph, which we shall formally view as a reflexive, symmetric relation. By an {\it augmented graph}, we shall mean a graph, together with a mapping from the set of vertices of that graph to $\xmode{\Bbb Z}$. If the augmentation map is injective, we shall refer to the graph as a {\it labeled graph}, with the obvious connotations. Let $\Gamma$ be a labeled graph, which we suppose has a maximum vertex $m$. We define various labeled graphs, coming from $\Gamma$, with maximum vertex $m+1$. First we define a labeled graph $\Gamma^+$ by $\mathop{\operatoratfont vertices}\nolimits(\Gamma^+) = \mathop{\operatoratfont vertices}\nolimits(\Gamma) \cup \setof{m+1}$ and $\mathop{\operatoratfont edges}\nolimits(\Gamma^+) = \mathop{\operatoratfont edges}\nolimits(\Gamma) \cup \setof{\mathop{\operatoratfont edge}\nolimits(m,m+1)}$. Now suppose that $\mathop{\operatoratfont edge}\nolimits(l,m) \in \Gamma$. We define a graph $\Gamma^l$ by $\mathop{\operatoratfont vertices}\nolimits(\Gamma^l) = \mathop{\operatoratfont vertices}\nolimits(\Gamma) \cup \setof{m+1}$ and $$\mathop{\operatoratfont edges}\nolimits(\Gamma^l) = \mathop{\operatoratfont edges}\nolimits(\Gamma) \cup \setof{\mathop{\operatoratfont edge}\nolimits(l, m+1), \mathop{\operatoratfont edge}\nolimits(m, m+1)} - \setof{\mathop{\operatoratfont edge}\nolimits(l,m)}.$% $Intuitively, this construction may be thought of as adding a vertex $(m+1)$ ``in the middle'' of the edge from $l$ to $m$. \begin{definition} A {\it standard operation\/} is an operation on a labeled graph of the form $\Gamma \mapsto \Gamma^+$ or $\Gamma \mapsto \Gamma^l$ for some $l$. A {\it standard labeled graph\/} is a labeled graph obtained from a one-vertex labeled graph by a finite sequence of standard operations. \end{definition} It is not hard to see that given a standard labeled graph, one may compute the last standard operation which was performed, and thence undo that operation. It follows that: \begin{prop}\label{unique-operations} Let $G$ be a standard labeled graph. Then there is a unique sequence of standard operations which gives rise to $G$. \end{prop} Fix integers $k$ and $m$ with $1 \leq k \leq m \leq n$. Let $R_k \subset E_k$ be a ruling. We will show how to associate a certain standard labeled graph $\Gamma_m(R_k)$ to $R_k$, in such a way that $[R_{k,m}] \in A^2(Y_m)$ depends only on $\Gamma_m(R_k)$. To do this, consider the set of all curves $H \subset Y_m$ which are the strict transforms of some ruling $R_l$ on $E_l$, for some $l$ with $k \leq l \leq m$. To each such $H$, we may associate an integer, namely $l$. It may be that $H \subset E_{l',m}$, for some $l'$ with $l' \not= l$ and $k \leq l' \leq m$, but this does not matter to us. The set of all such curves $H$ may be viewed as the vertices of a graph $\Gamma_m(k)$: two distinct vertices are connected by an edge if and only if\ the corresponding two curves on $Y_m$ meet. There is an augmentation on $\Gamma_m(k)$ given by $H \mapsto l$ as above. Define $\Gamma_m(R_k)$ to be the maximal connected subgraph of $\Gamma_m(k)$ which contains $R_{k,m}$. The augmentation on $\Gamma_m(k)$ induces an augmentation on $\Gamma_m(R_k)$. We shall prove shortly \pref{snooker} that $\Gamma_m(R_k)$ is a labeled graph, and that in fact it is a standard labeled graph. \begin{lemma}\label{lemma1} If two distinct curves $H_1, H_2 \in \Gamma_m(k)$ meet, then they meet at a unique point, and they meet transversally. \end{lemma} \begin{proof} What we need to show is that if $p \leq q$ are integers ($k \leq p,q \leq m$), and if $R_p \subset E_p$ and $R_q \subset E_q$ are rulings, and if $R_{p,m}$ meets $R_{q,m}$ (but $R_{p,m} \not= R_{q,m}$), then in fact $R_{p,m}$ meets $R_{q,m}$ at a unique point and they do so transversally. It suffices to show that $R_{p,q}$ meets $R_q$ in this way. We may assume that $p < q$. Indeed if $R_{p,q}$ met $R_q$ at more than one point, or if they did not meet transversally, then the image of $R_{p,q}$ under the map \mapx[[ Y_q || Y_p ]] would be singular, because this map contracts $R_q$. {\hfill$\square$} \end{proof} \begin{lemma}\label{morsel} No three distinct curves $H_1, H_2, H_3 \in \Gamma_m(k)$ meet at a common point. \end{lemma} \begin{proof} We may reduce to showing the following: if $p < q < r$ ($k \leq p,q,r \leq m$) and $R_p \subset E_p$, $R_q \subset E_q$, and $R_r \subset E_r$ are rulings, then $R_{p,r} \cap R_{q,r} \cap R_r = \varnothing$. We proceed by contradiction: let $x \in R_{p,r} \cap R_{q,r} \cap R_r$. We may assume that $r$ is minimal with respect to this assertion. Let $y$ be the image of $x$ under the map \mapx[[ Y_r || Y_{r-1} ]]. Then $y \in R_{p,r-1} \cap R_{q,r-1} \cap C_{r-1}$. If $q < r-1$, then for some ruling $R_{r-1} \subset E_{r-1}$, we have $y \in R_{p,r-1} \cap R_{q,r-1} \cap R_{r-1}$, thereby contradicting the minimality of $r$. Hence we may assume that $q = r-1$. To prove the lemma, it suffices to show that $\T_y(R_{p,r-1}) + \T_y(R_{q,r-1}) + \T_y(C_{r-1}) = \T_y(Y_{r-1})$. Since by \pref{lemma1} $R_{p,r-1}$ meets $R_{q,r-1}$ at a unique point, this will imply that $R_{p,r} \cap R_{q,r} = \varnothing$, thereby yielding a contradiction. Substituting $q = r-1$, we must show: $$\T_y(R_{p,r-1}) + \T_y(R_{r-1}) + \T_y(C_{r-1}) = \T_y(Y_{r-1}).\eqno(*)$% $The curves $R_{r-1}$ and $C_{r-1}$ meet transversally at $y$, tangentially spanning $\T_y(E_{r-1})$. Therefore, to prove $(*)$, and hence the lemma, it suffices to show that $R_{p,r-1}$ meets $E_{r-1}$ transversally. This may be deduced by repeated application of the following two facts, applied to integers $t$ with $p \leq t \leq r-2$: \begin{itemize} \item if $R_{p,t}$ meets $C_t$ transversally (on $Y_t$), then $R_{p,t+1}$ meets $E_{t+1}$ transversally (on $Y_{t+1}$); \item if $R_{p,t}$ meets $E_t$ transversally (on $Y_t$), then $R_{p,t}$ meets any smooth curve on $E_t$ transversally (if at all). {\hfill$\square$} \end{itemize} \end{proof} \begin{prop}\label{snooker} Let $k, m \in \xmode{\Bbb Z}$, with $1 \leq k \leq m \leq n$. Let $R_k \subset E_k$ be a ruling. Let $\Gamma = \Gamma_m(R_k)$. Then $\Gamma$ is a standard labeled graph with vertices $[k,m] \cap \xmode{\Bbb Z}$, and provided that $m < n$, $\Gamma_{m+1}(R_k)$ is obtained from $\Gamma$ by a single standard operation. \end{prop} \begin{proof} By induction, we may assume that $\Gamma$ is a standard labeled graph with vertices $[k,m] \cap \xmode{\Bbb Z}$. For each $q$ between $k$ and $m$, let let $R_q \subset E_q$ be the ruling corresponding to the vertex $q \in \Gamma$. First we show $(*)$ that if $l$ is such that $k \leq l < m$ and $R_{l,m}$ meets $C_m$, then in fact $R_{l,m}$, $R_m$, and $C_m$ meet at a common point. Suppose otherwise: $R_{l,m} \cap R_m \cap C_m = \varnothing$. We will obtain a contradiction. We may choose $m$ to be as small as possible. There are two cases. Case (a). We have $l = m-1$. Since $R_{m-1}$ meets $C_{m-1}$ transversally at a single point, $R_{m-1,m}$ meets $E_m$ at a single point. Since $\Gamma$ is a standard labeled graph, it is clear that $R_{m-1,m}$ meets $R_m$. Since $R_{m-1,m}$ meets $C_m$, we see that $R_{m-1,m}$ meets $E_m$ at two distinct points: contradiction. This proves case (a). Case (b). We have $l < m-1$. Since $R_{l,m}$ meets $C_m$ (and a fortiori $R_{l,m}$ meets $E_m$), it follows that $R_{l,m-1}$ meets $C_{m-1}$. By the minimality of $m$, $R_{l,m-1} \cap R_{m-1} \cap C_{m-1} \not= \varnothing$. It follows that $R_{l,m}$ and $R_{m-1,m}$ meet a common ruling on $E_m$. Since $R_{m-1,m}$ meets $R_m$, it is clear that this ruling must be $R_m$. Hence $R_{l,m}$ meets $R_m$. Thus $R_{l,m}$ meets both $R_m$ and $C_m$, but the three curves do not meet at a common point. Hence $R_{l,m}$ meets two distinct rulings on $E_m$. Hence $R_{l,m-1}$ meets $C_{m-1}$ at $\geq 2$ distinct points, so $R_l$ meets $C_l$ at $\geq 2$ distinct points: contradiction. This proves case (b), and hence $(*)$. We now proceed with the proof of the proposition. There are two cases. Case I. For no $l$ (with $k \leq l < m$) is it true that $R_{l,m}$, $R_m$ and $C_m$ have a point in common. We claim that $\Gamma_{m+1}(R_k) = \Gamma^+$. It suffices to show that $R_m$ is the unique curve in $\Gamma$ which meets $C_m$. This follows from $(*)$. Case II. For some $l$ (with $k \leq l < m$), $R_{l,m}$, $R_m$ and $C_m$ have a point (say $x$) in common. We claim that $\Gamma_{m+1}(R_k) = \Gamma^l$. To prove this, we need to prove two things: \begin{romanlist} \item for any $q$ such that $k \leq q < m$ and $q \not= l$, $R_{q,m}$ does not meet $C_m$; \item $\T_x(R_{l,m}) + \T_x(R_m) + \T_x(C_m) = \T_x(Y_m)$. \end{romanlist} The first assertion follows immediately from $(*)$ and from \pref{morsel}. The second assertion follows from the proof of \pref{morsel}. {\hfill$\square$} \end{proof} \begin{lemma}\label{pine-cone} Let $G$ be a standard labeled graph, constructed from the single vertex graph \setof{k} by a sequence of standard operations $+, k^\br{p},\vec o1r$, for some $p, r \geq 0$, such that $o_1 \not= k$. Then $G - \setof{k}$ is a standard labeled graph, which can be constructed from the single vertex graph \setof{k+1} by the sequence of standard operations: $$\cases{+^\br{p},\vec o1r,&if $p \geq 1;$\cr +, \vec o2r,&if $p=0$ and $r \geq 1;$\cr \varnothing,&if $p = r = 0$.}$$ \end{lemma} The proof of this lemma is left to the reader. Let $G$ be a standard labeled graph, with smallest vertex $k$, having at least two vertices. It is clear that there is a unique $r > k$ such that $\mathop{\operatoratfont edge}\nolimits(k,r)$ is in $G$. We define the {\it order\/} of $G$ to be $r-k$. Moreover, if $G$ has order $p$, then $G$ is constructed from the single vertex graph $\setof{k}$ by a sequence of operations which begins with $+, k^\br{p-1}$, and whose next operation (if any) is not $k$. Let $G$ be any standard labeled graph, with vertices $k, \ldots, m$. We associate a function \mp[[ \mu_G || G || \xmode{\Bbb N} ]], defined by inducting on $G$: if $G$ is a single vertex graph, then $\mu_G(k) = 1$. If $G$ is any standard labeled graph, then $\mu_{G^+}(j) = \mu_G(j)$ and $\mu_{G^l}(j) = \mu_G(j)$ for all $j$ with $k \leq j \leq m$, and $\mu_{G^+}(m+1) = \mu_G(m)$, $\mu_{G^l}(m+1) = \mu_G(m) + \mu_G(l)$. The fact that $\mu_G$ is well-defined follows from \pref{unique-operations}. \begin{prop}\label{spitup} Let $G$ be a standard labeled graph, with smallest vertex $k$, having at least two vertices. Then: $$\mu_G = \mu_{\setof{k}} + \sum_{i=1}^{\mathop{\operatoratfont ord}\nolimits(G)} \mu_{G - \setof{k, \ldots, k+i-1}},$% $where the functions on the right hand side\ are viewed as functions on $G$, via extension by zero. \end{prop} \begin{sketch} Use \pref{pine-cone}. If $p = \mathop{\operatoratfont ord}\nolimits(G)$, then $$G\ \longleftrightarrow\ +, k^\br{p-1}, *$% $where $*$ is a sequence of standard operations (possibly empty), not beginning with $k$. The case $p = 1$ is left to the reader. For $p \geq 2$: $$G - \setof{k}\ \longleftrightarrow\ +^\br{p-1}, *$$ $$G - \setof{k,k+1}\ \longleftrightarrow\ +^\br{p-2}, *$% $and so forth: $$G - \setof{k,\ldots,k+p-2}\ \longleftrightarrow\ +, *$% $$$G - \setof{k,\ldots,k+p-1}\ \longleftrightarrow\ *'$% $where $*'$ can be determined from \pref{pine-cone}. We compute $\mu$ in a special case, namely when $*$ is empty. Then: $$G\ \longleftrightarrow\ (1,1,2,3,\ldots,p)$$ $$G - \setof{k}\ \longleftrightarrow\ (0^\br{1}, 1^\br{p})$$ $$\cdots$$ $$G - \setof{k,\ldots,k+p-1}\ \longleftrightarrow\ (0^\br{p}, 1^\br{1}),$% $where the sequences on the right are $(\mu(k), \ldots, \mu(k+p))$. In this case the proposition is clear. The general case is left to the reader. {\hfill$\square$} \end{sketch} \begin{corollary} Fix an integer $k$ with $1 \leq k \leq n$. Let $R_k \subset E_k$ be a ruling. If $k = n$ then $[R_{k,n}] = {\mathbf{\lowercase{R}}}_n$, and if $k < n$, then there exists an integer $l$, with $k < l \leq n$, such that $$[R_{k,n}] = {\mathbf{\lowercase{R}}}_k - \sum_{i=k+1}^l {\mathbf{\lowercase{R}}}_i.$$ \end{corollary} \begin{sketch} For each integer $m$ with $k \leq m \leq n$, let $R_m \subset E_m$ be the ruling which enters into $\Gamma_n(R_k)$. For each integer $l$ with $k \leq l \leq n$, write: $$\mu_{\Gamma_n(R_l)} = (\vec bln).$% $By considering the scheme-theoretic inverse image of $R_l$ under the map \mapx[[ Y_n || Y_l ]], one can show that: $${\mathbf{\lowercase{R}}}_l = b_l [R_{l,n}] \many+ b_n [R_{n,n}].$% $The result then follows from \pref{spitup}. {\hfill$\square$} \end{sketch} \begin{corollary} Let $H \subset Y_n$ be a cycle which is a sum of strict transforms of rulings. Then $[H]$ is a positive $\xmode{\Bbb Z}$-linear combination of the classes: $${\mathbf{\lowercase{R}}}_n, ({\mathbf{\lowercase{R}}}_{n-1}-{\mathbf{\lowercase{R}}}_n), ({\mathbf{\lowercase{R}}}_{n-2}-{\mathbf{\lowercase{R}}}_{n-1}-{\mathbf{\lowercase{R}}}_n), \ldots, ({\mathbf{\lowercase{R}}}_1 - {\mathbf{\lowercase{R}}}_2 - \cdots - {\mathbf{\lowercase{R}}}_n).$$ \end{corollary} \begin{corollary}\label{snort-snort-snort} Let $H \subset Y_n$ be a cycle which is a sum of strict transforms of rulings. Then there exists integers $\vec a1n$ such that $[H] = a_1 {\mathbf{\lowercase{R}}}_1 \many+ a_n {\mathbf{\lowercase{R}}}_n$ and for each integer $k$ with $1 \leq k \leq n$, we have: $$\left(\sum_{i=1}^{k-1} 2^{k-i-1} a_i\right) + a_k \geq 0.$$ \end{corollary} \part{Application to set-theoretic complete intersections} \block{Theorems I and II}\label{section9} \par\indent\indent Let $S, T \subset \xmode{\Bbb P\kern1pt}^3$ be surfaces of degrees $s$ and $t$, respectively. Write $S_0 = S$, $T_0 = T$. Assume that $S \cap T$ is set-theoretically a smooth curve $C = C_0$. Let $d = \deg(C)$. Then $d | st$. Let $n = st/d$. Assume that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(S)$ and that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(T)$. Let $Y_0 = \xmode{\Bbb P\kern1pt}^3$. Let $Y_1$ be the blow-up of $Y_0$ along $C_0$. Let $S_1 \subset Y_1$ be the strict transform of $S$. There is a unique curve $C_1 \subset S_1$ which maps isomorphically onto $C_0$. Let $Y_2$ be the blow-up of $Y_1$ along $C_1$. Iterate this process. This puts us in the situation of \pref{iteration}. Let $p_i = p_i(S,C)$, for $i = 1, \ldots, n-1$. For each $k = 1, \ldots, n$, let $S_k$ and $T_k$ denote the strict transforms of $S$ and $T$ on $Y_k$. Since $C \not\IN \mathop{\operatoratfont Sing}\nolimits(S)$ and $C \not\IN \mathop{\operatoratfont Sing}\nolimits(T)$, it follows that for each $k$ with $0 \leq k \leq n$, $S_k$ meets $T_k$ along $C_k$ with multiplicity $n - k$. As consequences of this, we see that $S_n \cap T_n$ is a union of strict transforms of rulings, and that $[S_k] = s{\mathbf{\lowercase{H}}} - \sum_{i=1}^k {\mathbf{\lowercase{E}}}_i$, $[T_k] = t{\mathbf{\lowercase{H}}} - \sum_{i=1}^k {\mathbf{\lowercase{E}}}_i$. \label{turnipgreen} First we derive the formula: $$\beta_k = ds + (2-4d-2g) - p_k.\eqno(*)$% $We have $[S_k] \cdot {\mathbf{\lowercase{E}}}_k = {\mathbf{\lowercase{C}}}_k + p_k{\mathbf{\lowercase{R}}}_k$. (See \ref{goober-peas}.) Combining this with $[S_k] = s{\mathbf{\lowercase{H}}} - \sum_{i=1}^k {\mathbf{\lowercase{E}}}_i$, ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{E}}}_k = d{\mathbf{\lowercase{R}}}_k$ (from \ref{iteration}), and ${\mathbf{\lowercase{C}}}_k = {\mathbf{\lowercase{D}}}_k - \alpha_k{\mathbf{\lowercase{R}}}_k$ (from p.\ \pageref{quirkC}), we obtain: $$ds{\mathbf{\lowercase{R}}}_k - \sum_{i=1}^k ({\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{E}}}_k) = {\mathbf{\lowercase{D}}}_k - \alpha_k{\mathbf{\lowercase{R}}}_k + p_k{\mathbf{\lowercase{R}}}_k.$% $Combine this with ${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{E}}}_k = -\beta_i{\mathbf{\lowercase{R}}}_k$ (if $i < k$) (from \ref{iteration}) and ${\mathbf{\lowercase{E}}}_k^2 = -{\mathbf{\lowercase{D}}}_k$ (from p.\ \pageref{quirkC}) to obtain: $$ds{\mathbf{\lowercase{R}}}_k + (\svec\beta1{k-1}){\mathbf{\lowercase{R}}}_k = -\alpha_k{\mathbf{\lowercase{R}}}_k + p_k{\mathbf{\lowercase{R}}}_k.$% $Combine this with the formula: $$\alpha_k = (2-4d-2g) - (\svec\beta1k)\eqno(**)$% $from p.\ \pageref{quirkalpha}, to obtain $(*)$. Since $S_n \cap T_n$ is a union of strict transforms of rulings, it follows from \pref{snort-snort-snort} that: $$(s{\mathbf{\lowercase{H}}} - \sum_{i=1}^n {\mathbf{\lowercase{E}}}_i)(t{\mathbf{\lowercase{H}}} - \sum_{i=1}^n {\mathbf{\lowercase{E}}}_i) = \sum_{l=1}^n a_l {\mathbf{\lowercase{R}}}_l, \eqno(*{*}*)$% $for some integers $a_l$ such that for each $k$ with $1 \leq k \leq n$, we have: $$\left(\sum_{m=1}^{k-1} 2^{k-m-1} a_m\right) + a_k \geq 0.$% $ We proceed to analyze the consequences of this. The left hand side of $(*{*}*)$ equals: $$ st{\mathbf{\lowercase{H}}}^2 - d(s+t)(\sum_{i=1}^n {\mathbf{\lowercase{R}}}_i) - \sum_{1 \leq i < j \leq n} \beta_i {\mathbf{\lowercase{R}}}_j - \sum_{1 \leq j < i \leq n} \beta_j {\mathbf{\lowercase{R}}}_i$$ $$ - \sum_{k=1}^n \left[ d{\mathbf{\lowercase{H}}}^2 + \left( \sum_{i=1}^{k-1} \beta_i {\mathbf{\lowercase{R}}}_i \right) + \alpha_{k-1} {\mathbf{\lowercase{R}}}_k \right].$% $ Then for each $m$ with $1 \leq m \leq n$: $$-a_m = d(s+t) + 2\sum_{i=1}^{m-1} \beta_i + (n-m) \beta_m + \alpha_{m-1}.$% $Substituting $\alpha_k = (2-4d-2g) - (\beta_1 \many+ \beta_k)$, we obtain: $$-a_m = d(s+t) + \sum_{i=1}^{m-1} \beta_i + (n-m)\beta_m + (2-4d-2g).$% $ \par\indent In the special case where $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$, we have $a_m = 0$ for all $m$, with $1 \leq m \leq n$. Now substitute $\beta_i = ds + (2-4d-2g) - p_i$. We obtain: $$\left(\sum_{i=1}^{m-1} p_i\right) + (n-m)p_m = d[n(s-4) + t] + (2-2g)n.$% $This implies: \begin{theorem}\label{thmI} {\bf (``I'')} Let $C \subset \P3$ be a smooth curve of degree $d$ and genus $g$. Suppose that $C = S \cap T$, where $S$ and $T$ are surfaces of degree $s$ and $t$ respectively. Assume that $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$. Let $n = st/d$. Let $p_i = p_i(S,C)$, for each $i = 1, \ldots, n-1$, as defined in \S\ref{measure}. Then: $$p_1 = \cdots = p_{n-1} = {1 \over n-1} \left\{d[n(s-4)+t] + (2-2g)n\right\}.$$ \end{theorem} \begin{example}\label{examplex-char-two} If $s = t = 4$, $d = 4$, $g = 0$, we obtain $p_1 = p_2 = p_3 = 8$. This can occur in characteristic two, at least. Indeed, let $(S,C)$ be as in \pref{example-char-two}, and let $T$ be given by $z^4-xw^3=0$. \end{example} We now return to the general case. For each $k$ with $1 \leq k \leq n$, we have: \splitdisplay{\left(\sum_{m=1}^{k-1} 2^{k-m-1} [d(s+t) + \sum_{i=1}^{m-1} \beta_i + (n-m) \beta_m + (2-4d-2g)] \right)% }{+ [d(s+t) + \sum_{i=1}^{k-1} \beta_i + (n-k) \beta_k +(2-4d-2g)]\leq 0.% }A simplification yields: $$2^{k-1}[d(s+t-4)+2-2g] + \left( \sum_{i=1}^{k-1} 2^{k-i-1}(n-i+1) \beta_i \right) + (n-k) \beta_k \leq 0.$% $Note that: $$\sum_{i=1}^{k-1} 2^{k-i-1}(n-i+1) \ =\ (n-1)2^{k-1} + k - n.$% $Substitute $\beta_i = ds + (2-4d-2g) - p_i$. We obtain: \begin{theorem}\label{thmII} {\bf (``II'')} Let $C \subset \P3$ be a smooth curve of degree $d$ and genus $g$. Suppose that $C = S \cap T$, where $S$ and $T$ are surfaces of degree $s$ and $t$ respectively. Assume that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(S)$ and $C \not\IN \mathop{\operatoratfont Sing}\nolimits(T)$. Let $n = st/d$. Let $p_i = p_i(S,C)$, for each $i = 1, \ldots, n-1$, as defined in \S\ref{measure}. Then for each $k = 1, \ldots, n-1$, we have: $$\sum_{i=1}^{k-1} 2^{k-i-1}(n-i+1) p_i + (n-k)p_k \geq 2^{k-1}\left\{dt + n[d(s-4)+2-2g]\right\}.$$ \end{theorem} \begin{examples} \ \begin{itemize} \item $s = 2$, $t = 3$, $d = 3$, $g = 0$: then the theorem yields the single inequality $p_1 \geq 1$; \item $s = 2$, $t = 2$, $d = 1$, $g = 0$: as above the theorem yields $p_1 \geq 1$; \item $s = 4$, $t = 4$, $d = 4$, $g = 0$: the theorem yields three inequalities: \begin{romanlist} \item $p_1 \geq 8$; \item $2p_1 + p_2 \geq 24$; \item $8p_1 + 3p_2 + p_3 \geq 96$. \end{romanlist} \end{itemize} \end{examples} \block{Theorems III, Q, and B} \begin{theorem}\label{thmIII} {\bf (``III'')} Let $C \subset {\Bbb C}\kern1pt\P3$ be a smooth curve of degree $d$ and genus $g$ which is the set-theoretic complete intersection\ of two surfaces $S$, $T$ of degrees $s$, $t$, respectively. Let $n = st/d$. Let $p_k = p_k(S,C)$, for each $k \in \xmode{\Bbb N}$. Assume that $S$ has only rational singularities. Assume that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(T)$. Then: $$\sum_{k=1}^\infty{1 \over k(k+1)}p_k \geq {d^2 \over s} + d(s-4) + 2-2g.$$ \end{theorem} \begin{proof} By (\Lcitemark 18\Rcitemark \ 1.1), we know that $\Delta(S,C) = d^2/s + d(s-4) + 2-2g$. Apply \pref{rdp-one}. {\hfill$\square$} \end{proof} \begin{remark} In the statement of \pref{thmIII}, we do not know if $\sum_{k=1}^\infty$ can be replaced by $\sum_{k=1}^{n-1}$. From (\ref{interesting}), we see that this can at least be done if $(S,C)$ does not contain any singularities of type $D_{n,n}$ (with $n$ odd). \end{remark} \begin{lemma}\label{bound-formula} Let $S \subset {\Bbb C}\kern1pt\P3$ be a surface of degree $s$ having only rational singularities. Let \mp[[ \pi || {\tilde{S}} || S ]] be a minimal resolution. Let $N$ be the number of exceptional curves on ${\tilde{S}}$. Then: $$N \leq {s \over 3}(2s^2 - 6s + 7) - 1.$$ \end{lemma} \begin{proof} Clearly $N \leq \mathop{\operatoratfont rank}\nolimits \mathop{\operatoratfont NS}\nolimits({\tilde{S}}) - 1$. Also $\mathop{\operatoratfont rank}\nolimits \mathop{\operatoratfont NS}\nolimits({\tilde{S}}) \leq h^{1,1}({\tilde{S}})$, so it suffices to show that $h^{1,1}({\tilde{S}}) = {s \over 3}(2s^2-6s+7)$. By simultaneous resolution of rational double points \Lcitemark 2\Rcitemark \Rspace{}, and deformation invariance of Hodge numbers, we may reduce to showing that $h^{1,1}(S) = {s \over 3}(2s^2-6s+7)$ if $S$ is itself nonsingular. We have: \begin{eqnarray*} h^{1,1}(S) & = & h^2(S,{\Bbb Q}\kern1pt) - 2h^2(S,{\cal O}_S) \\ & = & [\chi_{\operatoratfont top}(S) + 4h^1(S,{\cal O}_S) - 2] - 2h^2(S,{\cal O}_S). \end{eqnarray*} Using the fact that the top Chern class of the tangent bundle equals the Euler characteristic (see e.g.{\ }\Lcitemark 1\Rcitemark \ 11.24, 20.10.6), and using Riemann-Roch, we find: $$\chi_{\operatoratfont top}(S) = c_2(S) = 12\chi(S) - c_1^2(S) = 12(1-h^1(S,{\cal O}_S)+h^2(S,{\cal O}_S)) - (4-s)^2s.$% $The formula for $h^{1,1}$ follows from $h^1(S,{\cal O}_S) = 0$ and \formulaqed{h^2(S,{\cal O}_S) = {s-1 \choose 3}.} \end{proof} \begin{remark} We do not know if the lemma remains valid if ${\Bbb C}\kern1pt$ is replaced by an algebraically closed field of positive characteristic. \end{remark} \begin{lemma}\label{bungobungo} Let $(p_k)_{k \in \xmode{\Bbb N}}$ be a sequence of nonnegative integers. Let $n$ be a nonnegative integer. Assume that: \begin{romanlist} \item $p_1 \leq 9 - {2\over5}n$ \item $p_1 \geq p_2 \geq p_3 \geq \cdots$ \item $\sum_{k=1}^\infty p_k \leq 19 - n$ \item $n/4 + \sum_{k=1}^\infty {1 \over k(k+1)} p_k \geq 6$. \end{romanlist} \par\noindent Then $n=0$ and $(p_k) \in \setof{ (9,8,2), (9,9), (9,9,1) }.$ \end{lemma} \begin{proof} Constraints (i), (iii), and (iv) imply that $$n/4 + \hbox{$1 \over 2$}\floor{9-\hbox{$2 \over 5$}n} + \hbox{$1 \over 6$}(19-n-\floor{9-\hbox{$2 \over 5$}n} ) \geq 6.$% $It follows that $n \in \setof{0,2}$. Suppose that $n=2$. Then the same constraints imply that $$1/2 + \hbox{$1 \over 2$}p_1 + \hbox{$1 \over 6$}(17-p_1) \geq 6,$% $so $p_1 = 8$. Now we see that the left hand side\ of (iv) is maximized when $(p_k) = (8,8,1)$. In that case, the left hand side\ of (iv) is $5{11\over12}$: contradiction. Hence $n=0$. Then $$\hbox{$1 \over 2$}p_1 + \hbox{$1 \over 6$}(19-p_1) \geq 6,$% $so $p_1 = 9$. If $p_2 \leq 7$, then the sum in (iv) is bounded by the sum obtained when $(p_k) = (9,7,3)$. This sum is $< 6$, so $p_2 \not\leq 7$. Hence $p_2 \in \setof{8,9}$. Etc. {\hfill$\square$} \end{proof} \begin{prop}\label{kformula} Let $C \subset {\Bbb C}\kern1pt\P3$ be a smooth curve of degree $d$ and genus $g$, which lies on a surface $S \subset {\Bbb C}\kern1pt\P3$ of degree $s$. Assume that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(S)$. Let $p_1 = p_1(S,C)$. Let $N$ be the normal bundle of $C$ in ${\Bbb C}\kern1pt{\Bbb P}^3$, and let $l$ be the maximum degree of a sub-line-bundle of $N$. Let $k = 3d + (2g-2) - l$. Then $p_1 \leq d(s-1) - k$. \end{prop} \begin{proof} We use the notation of \pref{iteration}. We also use various facts from \S\ref{section9}, which although apparently dependent on another surface $T$, actually make sense in this context. We have $\inn{{\mathbf{\lowercase{C}}}_1, {\mathbf{\lowercase{C}}}_1} \geq \deg(N) - 2l$. Since $\deg(N) = 4d+2g-2$ and $k = 3d + (2g-2) - l$, we have: $$\inn{{\mathbf{\lowercase{C}}}_1,{\mathbf{\lowercase{C}}}_1} \geq -2d+2-2g+2k.\eqno(\dag)$% $Now ${\mathbf{\lowercase{C}}}_1 = {\mathbf{\lowercase{D}}}_1 - \alpha_1{\mathbf{\lowercase{R}}}_1$ in $A^1(E_1)$, and $\inn{{\mathbf{\lowercase{D}}}_1,{\mathbf{\lowercase{D}}}_1} = 2-2g-4d$, so: $$\inn{{\mathbf{\lowercase{C}}}_1,{\mathbf{\lowercase{C}}}_1}\ =\ \inn{{\mathbf{\lowercase{D}}}_1,{\mathbf{\lowercase{D}}}_1} - 2\alpha_1\ = \ 2 - 2g - 4d - 2\alpha_1.$% $Combining this with $(\dag)$, we obtain $d + \alpha_1 \leq -k$. The formulas $(*)$ and $(**)$ from \S\ref{section9} imply that $\alpha_1 = p_1 - ds$. Hence $p_1 \leq d(s-1)-k$. {\hfill$\square$} \end{proof} \begin{remark} This result \pref{kformula} is a strengthening of the very elementary fact that: $$\abs{\mathop{\operatoratfont Sing}\nolimits(S) \cap C} \leq d(s-1).$$ \end{remark} \begin{theorem}\label{thmQ} {\bf (``Q'')} Let $C \subset {\Bbb C}\kern1pt{\Bbb P}^3$ be a curve. Assume that $C = S \cap T$ set-theoretically for some surfaces $S$ and $T$. Assume that $S$ is normal. Assume that $\deg(C) > \deg(S)$. Then $C$ is linearly normal. \end{theorem} \begin{proof} To any Weil divisor $E$ on a normal surface $S$, one can associate a reflexive ${\cal O}_S$-module ${\cal O}_S(E)$. We recall the following result of Sakai from \Lcitemark 29\Rcitemark \Rspace{}, which is a slightly less general version of theorem 5.1 of that paper: \begin{quote} {\it Let $S$ be a normal projective surface. Let $D$ be a nef Weil divisor on $S$ with $D^2 > 0$. Then $H^1(S, {\cal O}_S(-D)) = 0$.} \end{quote} Since the canonical map \mapx[[ H^0(\P3, {\cal O}_{\P3}(1)) || H^0(S, {\cal O}_S(1)) ]] is surjective, it suffices to show that the canonical map \mapx[[ H^0(S, {\cal O}_S(1)) || H^0(S, {\cal O}_C(1)) ]] is surjective. Let $H$ be a hyperplane section of $S$. From the long exact sequence coming from \sescomma{{\cal O}_S(H-C)}{{\cal O}_S(H)}{{\cal O}_C(H)% }we see that it is sufficient to show that $H^1(S, {\cal O}_S(H-C)) = 0$. Let $d = \deg(C)$. Let $s = \deg(S)$, $t = \deg(T)$, and let $n$ be the multiplicity of intersection of $S$ with $T$ along $C$, $n = st/d$. Since $d > s$, we have $t > n$. Hence $(t-n)H$ is a very ample Cartier divisor. Since $n(C-H) \sim (t-n)H$, the theorem follows from Sakai's result. {\hfill$\square$} \end{proof} \begin{corollary} Let $C \subset {\Bbb C}\kern1pt{\Bbb P}^3$ be a smooth curve. Assume that $C$ is the set theoretic complete intersection of two normal surfaces $S$ and $T$, with multiplicity $\leq 3$. Then $C$ is linearly normal. \end{corollary} \begin{proof} Using the notation of the proof of \pref{thmQ}, we are done if either $s$ or $t$ is bigger than $n$. Otherwise, $d \leq 3$, and so $C$ is linearly normal anyway. {\hfill$\square$} \end{proof} \begin{remark} For the case of multiplicity $4$, we must have $C$ linearly normal, except possibly for the case where $C$ is a rational quartic, which is the set-theoretic complete intersection\ of two normal quartic surfaces. It is not known if this is possible. \end{remark} \begin{theorem}\label{thmB} {\bf (``B'')} Let $S, T \subset {\Bbb C}\kern1pt\P3$ be surfaces. Assume that $S \cap T$ is set-theoretically a smooth curve. Assume that $\deg(S) = 4$ and that $S$ has only rational singularities. Then $C$ is linearly normal. \end{theorem} \begin{proof} Let $C$ have degree $d$ and genus $g$. By \pref{thmQ}, we may assume that\ $d=4$ and $g=0$. By\Lspace \Lcitemark 18\Rcitemark \Rspace{}, we may assume that\ $\deg(T) \geq 4$. Since $\deg(S) \leq \deg(T)$, we may assume that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(T)$, as follows. Suppose that $C \subset \mathop{\operatoratfont Sing}\nolimits(T)$. Write $S = V(f)$, $T = V(g)$. Choose $h$ so that $\deg(fh) = \deg(g)$, and so that $C \not\IN V(h)$. Then $C \not\IN \mathop{\operatoratfont Sing}\nolimits(V(fh + g))$. Hence we may replace $T$ by $V(fh + g)$. Write $(S,C) = (S',C') + (S'',C'')$, where $$(S'',C'') = D_{n_1,n_1} \many+ D_{n_r,n_r},$% $$\vec n1r$ are odd integers $\geq 5$, and $(S',C')$ is a configuration which does not involve any such singularities. Let $p_i = p_i(S',C')$. Let $n = \svec n1r$. We show that the hypotheses of \pref{bungobungo} are satisfied. We apply \pref{kformula}, using that fact\Lspace \Lcitemark 5\Rcitemark \Rspace{} that $l=7$, concluding that $p_1(S,C) \leq 9$. We have $p_1(\sum D_{n_i,n_i}) = \sum (n_i-1) / 2)$ by \pref{fantastico}, and $n_i - 1 \geq {4\over5} n_i$, so $p_1(\sum D_{n_i,n_i}) \geq {2\over5}n$. Thus hypothesis (i) is satisfied. Hypothesis (ii) holds. Hypothesis (iii) follows from \pref{bound-formula} and from the fact that $(S',C')$ contains no $D_{m,m}$ pairs with $m$ odd, $m \geq 5$, so that $\mathop{\operatoratfont def \kern1pt}\nolimits(S',C') \geq 0$. To prove hypothesis (iv), we would like to use (\ref{thmIII} = ``III''), but that is not quite good enough. By \pref{rdp-one}, $$\sum_{k=1}^\infty {1\over k(k+1)}p_k \geq \Delta(S',C').$% $Let $s = \deg(S) = 4$. By (\Lcitemark 18\Rcitemark \ 1.1), we know that: $$\Delta(S,C) = d^2/s + d(s-4) + 2 - 2g = 6.$% $Then $\Delta(S',C') = \Delta(S,C) - \Delta(S'',C'')$. By \pref{delta-formulas}, $\Delta(S'',C'') = n/4$. Hypothesis (iv) follows. By \pref{bungobungo}, we conclude that $n = 0$ and that: $$\mathop{\operatoratfont type}\nolimits(S,C) \in \setof{ (9,8,2), (9,9), (9,9,1) }.$% $We will use \pref{fantastico}, \pref{potato-1}, and \pref{potato-2}. Suppose that $\mathop{\operatoratfont type}\nolimits(S,C) = (9,9,1)$. Then for some $p \in C$, $\mathop{\operatoratfont type}\nolimits(S,C)_p = (r,1,1)$ for some $r \geq 1$. The case $r > 1 $ is impossible because $\mathop{\operatoratfont type}\nolimits(S,C) - \mathop{\operatoratfont type}\nolimits(S,C)_p = (9-r,9-1)$ and $9-r \geq 9-1$. Hence $\mathop{\operatoratfont type}\nolimits(S,C)_p = (1,1,1)$. Hence $(S,C)_p = A_{3,1}$. Since $\Sigma(S,C) \leq 19$ by \pref{bound-formula}, and since $9+9+1 = 19$, we have $\mathop{\operatoratfont def \kern1pt}\nolimits(S,C) = 0$. Therefore, since the other singularities of $S$ along $C$ have type $(k,k)$ for some $k \leq 8$, we see that the other singularities must be $A_{3k-1,k}$ for some $k \in \setof{1, \ldots, 8}$, depending on the singular point. Amongst these, only $A_{2,1}$ has deficiency zero. Hence $(S,C) = 8A_{2,1} + A_{3,1}$. Hence $\Delta(S,C) = 8({2\over3}) + {3\over4} \not= 6$: contradiction. Now suppose that $\mathop{\operatoratfont type}\nolimits(S,C) = (9,8,2)$. Then for some $p \in C$, $$\mathop{\operatoratfont type}\nolimits(S,C)_p \in \setof{ (1,1,1), (2,1,1), (2,2,2) }.$% $These types are realized by the singularities $A_{3,1}$, $A_{4,2}$, and $A_{7,2}$, respectively, and by no others. Since $A_{7,2}$ has nonzero deficiency, it can be excluded. Hence $(S,C)_p \in \setof{A_{3,1},A_{4,2}}$. If $(S,C)_p = A_{4,2}$, we find (by analogy with the $(9,9,1)$ case) that $(S,C) = 6A_{2,1} + A_{3,1} + A_{4,2}$. Hence $\Delta(S,C) = 6({2\over3}) + {3\over4} + {6\over5} \not= 6$: contradiction. If $(S,C)_p = A_{3,1}$, then we may assume that\ $(S,C) = 2A_{3,1} + \hbox{other}$, where the ``other'' part must have type $(7,6)$. The only zero-deficiency rational double point\ configuration which realizes this type is $A_{1,1} + 6A_{2,1}$. Hence $(S,C) = A_{1,1} + 6A_{2,1} + 2A_{3,1}$. By\Lspace \Lcitemark 24\Rcitemark \Rspace{}, the sum of the contributions of the singularities must not exceed $(2/3)\deg(S)(\deg(S)-1)^2 = 24$, where each singularity $p$ contributes $e(E) - 1/\abs{G}$, $e(E)$ is the topological Euler characteristic of the exceptional fiber in the minimal resolution of $p$, and $G$ is the order of the group which defines $p$ as a quotient singularity. In particular, an $A_n$ singularity contributes $(n+1) - (n+1)^{-1}$. Then the sum of the contributions is $25$: contradiction. Suppose that $\mathop{\operatoratfont type}\nolimits(S,C) = (9,9)$. Then each singularity of $S$ along $C$ must have type $(k,k)$ for some $k$, depending on the singular point. Hence $(S,C)$ must be built up from $E_{6,1}$ and $A_{3k-1,k}$ for various $k$. Since $\mathop{\operatoratfont def \kern1pt}\nolimits(E_{6,1}) = 2$, we may rule out that case. In fact, there are only two configurations with deficiency $\leq 1$: either $(S,C) = 9A_{2,1}$ or else $(S,C) = 7A_{2,1} + A_{5,2}$. In both cases, $\mathop{\operatoratfont order}\nolimits(S,C) = 3$. Hence we may assume that $\deg(T) = 3$. By \Lcitemark 18\Rcitemark \Rspace{}, we know that this is impossible. {\hfill$\square$} \end{proof} \block{Theorem A} \begin{lemma}\label{murky-algebra} Let $s,t,d,g \in \xmode{\Bbb Z}$. Assume that $t \geq s \geq 4$, $d \geq 1$, and $g \geq 0$. Assume that $d \vert st$. Let $n = st/d$. Assume that $n \geq 2$. Let $r = d[n(s-4)+t] + (2-2g)n$. Assume that $r \leq {s \over 3}(2s^2-6s+7)-1$. Then $d \leq g + 3$. \end{lemma} \begin{proof} We assume that $d \geq g + 4$, working toward a contradiction. We have $2 - 2g \geq 10 - 2d$, so: $$d[n(s-4) + t] + (10-2d)n \leq {s\over3}(2s^2-6s+7) - 1.\eqno(*)$% $ \par First suppose that $s = 4$. Then $t \geq 4$, $t \geq d/2$, and $dt + (10-2d)n \leq 19$. Substituting $n = st/d = 4t/d$ and simplifying, we obtain: $$(d^2 - 8d + 40)t \leq 19d.\eqno(\dag)$% $Since $t \geq d/2$, we have $(d^2 - 8d + 40)(1/2) \leq 19$. It follows that $d \leq 7$. Hence $d \in \setof{4,5,6,7}$. In each case, $(\dag)$ gives us an upper bound $t_{\rm max}$ for $t$: {\renewcommand{\arraystretch}{1.0} \vspace*{0.1in} \centerline{ \begin{tabular}{||c|c||} \hline $d$ & $t_{\rm max}$\\ \hline $4$ & $3$\\ \hline $5$ & $3$\\ \hline $6$ & $4$\\ \hline $7$ & $4$\\ \hline \end{tabular} } \vspace*{0.1in}} The cases $d \in \setof{4,5}$ contradict $t \geq 4$. In case $d \in \setof{6,7}$, we have $t = 4$, which contradicts our assumption that $d | st$. Hence $s > 4$. Now suppose that $s = 5$. Then $t \geq 5$, $t \geq 2d/5$, and $d(t+n) + (10-2d)n \leq 44$. Substituting $n = st/d = 5t/d$ and simplifying, we obtain: $$t \leq 44d / (d^2 - 5d + 50).$% $This implies that $t < 5$: contradiction. Hence $s \geq 6$. Since $n = st/d \geq s^2/d$, it follows from $(*)$ that: $$d[{s^2 \over d}(s-6) + s] + 10{s^2 \over d} \leq {s \over 3}(2s^2-6s+7).$% $This implies that: $$s(s-6) + d + 10{s \over d} \leq {1 \over 3}(2s^2 - 6s + 7)\eqno(**)$% $and in particular that: $$s(s-6) \leq {1 \over 3}(2s^2 - 6s + 7).$% $It follows that $s \leq 12$. Hence $6 \leq s \leq 12$. If $s = 12$, $(**)$ implies that $d + 120/d \leq 2{1\over3}$. This is absurd. In a similar manner, one may eliminate the cases where $6 \leq s \leq 11$. {\hfill$\square$} \end{proof} \begin{theorem}\label{thmA} {\bf (``A'')} Let $S, T \subset {\Bbb C}\kern1pt\xmode{\Bbb P\kern1pt}^3$ be surfaces. Assume that $S$ has only rational singularities. Assume that $\deg(S) \leq \deg(T)$. Assume that $S \cap T$ is set-theoretically a smooth curve $C$ of degree $d$ and genus $g$. Assume that $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$. Then $d \leq g + 3$. \end{theorem} \begin{proof} Let $s = \deg(S)$, $t = \deg(T)$, $n = st/d$. Let $p_i = p_i(S,C)$. By (\ref{thmI} = ``I''), we have: $$p_1 = \cdots = p_{n-1} = {1 \over n-1} \left\{d[n(s-4)+t] + (2-2g)n\right\}.$$ We show that $S$ has no singularities of type $D_{t,t}$ (with $t$ odd, $t \geq 5$), lying on $C$. There are two cases. If $n = 2$, then $\mathop{\operatoratfont order}\nolimits(S,C) | 2$. But by \pref{order-calc}, the order of $D_{t,t}$ (as above) is $4$. Hence $n > 2$. Hence $p_1 = p_2$. But $p_1(D_{t,t}) > p_2(D_{t,t})$ (by \ref{fantastico}), so ``$D_{t,t} \notin (S,C)$'' for $t$ odd, $t \geq 5$. {}From this, it follows that $\svec p1{n-1} \leq \Sigma(S,C)$. Let $r = d[n(s-4)+t] + (2-2g)n$. By \pref{bound-formula}, we conclude that $r \leq {s\over3}(2s^2-6s+7)-1$. The case $n = 1$ corresponds to a complete intersection, and the theorem is easily verified in this case. Therefore we may assume that\ $n \geq 2$. By\Lspace \Lcitemark 18\Rcitemark \Rspace{}, it follows that if $s \leq 3$, then $d \leq g+3$. Hence we may assume that $s \geq 4$. Therefore \pref{murky-algebra} applies, and we conclude that $d \leq g + 3$. {\hfill$\square$} \end{proof} \begin{corollary} Let $S, T \subset {\Bbb C}\kern1pt\xmode{\Bbb P\kern1pt}^3$ be surfaces. Assume that $S$ and $T$ have only rational singularities. Assume that $S \cap T$ is set-theoretically a smooth curve $C$ of degree $d$ and genus $g$. Assume that $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$. Then $d \leq g + 3$. \end{corollary} \block{Theorem X} \par\indent\indent As a corollary of theorem (I), we show: \begin{theorem}\label{thmX} {\bf (``X'')} Let $C \subset \P3$ be a smooth curve. Assume that $C$ is not a complete intersection. Suppose that $C = S \cap T$ as sets, where $S, T \subset \P3$ are surfaces. Assume that $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$. Then: $$\deg(S), \deg(T) < 2 \cdot \deg(C)^4.$$ \end{theorem} \par\noindent First we make a few remarks. \begin{arabiclist} \item The proof of theorem (X) depends primarily on the fact that the numbers $p_k$ in theorem (I) must be integers. \item The importance of theorem (X) is that an upper bound is given for the degrees of $S$ and $T$, that this bound is computable, and that this bound depends only on the degree of $C$. In the proof, we give the better bounds $\deg(S) < 2 \cdot \deg(C)^2$, $\deg(T) < 2 \cdot \deg(C)^4$, provided that $\deg(S) \leq \deg(T)$. \item Via the bounds in theorem (X), it becomes a computer triviality to find all possible degrees for $S$ and $T$ which are consistent with the integrality of the numbers $p_k$ in theorem (I). \item Doing this when $\deg(C) = 4$, $\mathop{\operatoratfont genus}\nolimits(C) = 0$, and assuming for efficiency that $\deg(S) \leq \deg(T)$, we find: $$(\deg(S), \deg(T)) \in \{ (3,4), (3,8), (4,4), (4,7), (6,26), (9,48), (10,28),$$ $$(12,18), (13,16), (17,220), (18,118), (19,84), (20,67), (22,50), (28,33) \}.$$ [We have excluded the cases where $\deg(S)$ is $1$ or $2$, which cannot occur.] \item We do not know which of these pairs of integers can be realized by pairs of surfaces, as in theorem (X). All that we know is that $(3,4)$ and $(3,8)$ cannot be realized in characteristic zero, and that $(4,4)$ can be realized in characteristic two. \item Theorem (X) is false without the hypothesis that $C$ is a complete intersection. Counterexample: for any $s \in \xmode{\Bbb N}$, one can find a smooth curve $D \subset \P2$ of degree $s$ and a line $L \subset \P2$ such that $D \cap L$ is a single point, set-theoretically. Let $S$ and $T$ be cones over $D$ and $L$, with the same vertex. Then $S \cap T$ is a line, set-theoretically. Theorem (X) is also false without the hypothesis that $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$. \end{arabiclist} Before proceeding with the proof of theorem (X), we need the following lemma, which was known in characteristic zero, and for the smooth case, was known in all characteristics. (See proof for references.) \begin{lemma}\label{torsion-free} Let $S \subset \P3$ be a normal surface. Then $\mathop{\operatoratfont Pic}\nolimits(S)/\mathop{\operatoratfont Pic}\nolimits(\P3)$ is torsion-free. \end{lemma} Before proceeding with the proof, we recall some standard material on differentials for which we do not have a good reference. First of all, for any scheme $X$, there is a map of sheaves of abelian groups: \dmap[[ \mathop{\operatoratfont dlog}\nolimits || {\cal O}_X^* || \Omega_X ]]% given by $f \mapsto df/f$. (All sheaves we shall discuss are sheaves on the Zariski site.) Now suppose that $X$ is a normal proper variety, defined over an algebraically closed field $k$ of positive characteristic $p$. Then we have an exact sequence: \diagramx{0&\mapE{}&{\cal O}_X^*&\mapE{F}&{\cal O}_X^*&\mapE{\mathop{\operatoratfont dlog}\nolimits}&\Omega_X\cr% }of sheaves of abelian groups on $X$, where $F$ denotes the Frobenius map. The exactness in the middle depends on normality, and may be deduced e.g.\ from (\Lcitemark 16\Rcitemark \ I\ 4.2). Let ${\cal{D}}$ be the image of $\mathop{\operatoratfont dlog}\nolimits$. Since $X$ is proper, $H^0(X, {\cal O}_X^*) = k^*$, so $H^0(X, F)$ is an isomorphism, and we obtain an isomorphism $H^0(X, {\cal{D}}) \cong \ker H^1(X, F)$. We have $\ker H^1(X, F) \cong {}_p\mathop{\operatoratfont Pic}\nolimits(X)$. Composing with the canonical injection \mapx[[ H^0(X, {\cal{D}}) || H^0(X, \Omega_X) ]], we obtain an injective group homomorphism: \dmap[[ \psi_X || {}_p\mathop{\operatoratfont Pic}\nolimits(X) || H^0(X, \Omega_X). ]]% \begin{proofnodot} (of \ref{torsion-free}). First we show $(*)$ that $\mathop{\operatoratfont Pic}\nolimits(S)/\mathop{\operatoratfont Pic}\nolimits(\P3)$ has no torsion, except possibly for $p$-torsion, when the ground field has positive characteristic $p$. These arguments are very similar to those given by Lang\Lspace \Lcitemark 21\Rcitemark \Rspace{}. The methods were invented by Grothendieck (\Lcitemark 10\Rcitemark \ Expos\'e XI), and further studied by Hartshorne (\Lcitemark 13\Rcitemark \ \S4.3). We refer the reader to\Lspace \Lcitemark 21\Rcitemark \Rspace{} or\Lspace \Lcitemark 13\Rcitemark \Rspace{} for details. Let $S_n$ be the \th{n} infinitesimal neighborhood of $S$ in $\P3$. Then: $$\mathop{\operatoratfont Pic}\nolimits(\P3) \cong \displaystyle{\lim_{\overleftarrow{\hphantom{\lim}}}} \mathop{\operatoratfont Pic}\nolimits(S_n).$% $Moreover, for each $n$ there is an exact sequence of abelian groups: \les{\mathop{\operatoratfont Pic}\nolimits(S_{n+1})}{\mathop{\operatoratfont Pic}\nolimits(S_n)}{H^2(S, {\cal{J}}^n/{\cal{J}}^{n+1}),% }where ${\cal{J}}$ is the ideal sheaf of $S$ in $\P3$. Since the $H^2$ term is a vector space, $(*)$ follows. {}From now on we may assume that the ground field has positive characteristic $p$. A standard calculation shows that $H^0(S, \Omega_S) = 0$. Up to now, we have not used the hypothesis that $S$ is normal. We now use this hypothesis. Via the map $\psi_S$, defined immediately above this proof, we see that ${}_p\mathop{\operatoratfont Pic}\nolimits(S) = 0$. (This argument is essentially that used in\Lspace \Lcitemark 21\Rcitemark \Rspace{}.) Finally, to complete the proof, we must show that $[O_S(1)]$ does not have a \th{p} root in $\mathop{\operatoratfont Pic}\nolimits(S)$. The argument given here is essentially the argument given in (\Lcitemark 4\Rcitemark \ 1.8). For any variety $X$, there is a natural group homomorphism \mp[[ H^1(\mathop{\operatoratfont dlog}\nolimits) || \mathop{\operatoratfont Pic}\nolimits(X) || H^1(X, \Omega_X) ]]. Consider this map when $X = S$ and when $X = \P3$. A standard calculation shows that the map \mapx[[ H^1(\P3, \Omega_{\P3}) || H^1(S, \Omega_S) ]] is injective. Moreover, one knows that the image of $[{\cal O}_{\P3}(1)]$ in $H^1(\P3, \Omega_{\P3})$ is not zero. (See e.g.{\ }\Lcitemark 14\Rcitemark \ Chapter 3, exercise 7.4.) Hence the image of $[{\cal O}_S(1)]$ in $H^1(S, \Omega_S)$ is not zero. Hence $[O_S(1)]$ does not have a \th{p} root in $\mathop{\operatoratfont Pic}\nolimits(S)$. {\hfill$\square$} \end{proofnodot} \begin{remark} Over an algebraically closed field of positive characteristic, let $S \subset \P3$ be a surface, not necessarily normal. We do not know if $\mathop{\operatoratfont Pic}\nolimits(S)/\mathop{\operatoratfont Pic}\nolimits(\P3)$ is torsion-free, or even if $\mathop{\operatoratfont Pic}\nolimits(S)$ is torsion-free. Answers to these questions might be obtained from a general structure theorem for \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm\hbox{$\mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(S)\ \mapE{}\ \mathop{\operatoratfont Pic}\nolimits(\nor{S})]$}, where $S$ is an arbitrary projective variety. \end{remark} \begin{corollary}\label{smooth-stci} In $\P3$, suppose that $C = S \cap T$ as sets, where $C$ is a curve, and $S, T$ are surfaces. Assume that $C$ does not meet $\mathop{\operatoratfont Sing}\nolimits(S)$. Then there exists a surface $T' \subset \P3$ such that $C = S \cap T'$, scheme-theoretically. \end{corollary} \begin{proof} Since $T \cap \mathop{\operatoratfont Sing}\nolimits(S) = \varnothing$, $S$ is normal. By \pref{torsion-free}, $\mathop{\operatoratfont Pic}\nolimits(S)/\mathop{\operatoratfont Pic}\nolimits(\P3)$ is torsion-free. Hence $[{\cal O}_S(C)] = 0$ in $\mathop{\operatoratfont Pic}\nolimits(S)/\mathop{\operatoratfont Pic}\nolimits(\P3)$. Hence ${\cal O}_S(C) \cong {\cal O}_S(t)$ for some $t \in \xmode{\Bbb N}$. Since the canonical map \mapx[[ H^0(\P3, {\cal O}_{\P3}(t)) || H^0(S, {\cal O}_S(t)) ]] is surjective, it follows that there exists a surface $T'$ of degree $t$ as claimed. {\hfill$\square$} \end{proof} \begin{remark} Robbiano\Lspace \Lcitemark 28\Rcitemark \Rspace{} proved this in the case where $S$ is smooth and the ground field has characteristic zero. \end{remark} \begin{proofnodot} (of theorem X). Let $d = \deg(C)$, $g = \mathop{\operatoratfont genus}\nolimits(C)$, $s = \deg(S)$, $t = \deg(T)$. We may assume that $s \leq t$. We show that $s < 2d^2$ and $t < 2d^4$. Let $n = st/d$. By theorem (I), we know that: $$(n-1) \kern3pt | \kern3pt \setof{ d [ n (s-4) + t ] + (2-2g)n }.$% $A proof of this fact, independent of (I), is given at the end of this paper. By \pref{smooth-stci}, we know that $C$ meets $\mathop{\operatoratfont Sing}\nolimits(S)$. Hence $p_1(S,C) > 0$. Hence the right hand side\ is positive. Write $d = d_s d_t$, where $d_s, d_t \in \xmode{\Bbb N}$, $d_s|s$, and $d_t|t$. Let $s_1 = s/d_s$, $t_1 = t/d_t$. Then $n = s_1t_1$, so: $$(s_1t_1 - 1) | \setof{ d [ s_1t_1 (d_s s_1 - 4) + d_t t_1 ] + (2-2g) s_1 t_1 }.$% $The right hand side\ is divisible by $t_1$, and $\gcd(s_1t_1 - 1, t_1) = 1$, so: $$(s_1t_1 - 1) | \setof{ d [ s_1 (d_s s_1 - 4) + d_t ] + (2-2g) s_1}.\eqno(*)$% $Thus for some $k \in \xmode{\Bbb N}$, we have: $$(s_1t_1 - 1)k = d [ s_1 (d_s s_1 - 4) + d_t ] + (2-2g) s_1.$% $Reorganizing, we find: $$(s_1t_1 - 1)k = (d d_s)s_1^2 + (2-2g-4d)s_1 + d d_t. \eqno(**)$% $Now we have $t \geq s$, so $t_1 \geq (d_s/d_t)s_1$. Hence: $$\left[s_1^2\left({d_s \over d_t}\right) - 1\right]k \leq (d d_s)s_1^2 + (2-2g-4d)s_1 + d d_t.$% $It is conceivable that the left hand side\ of this inequality is negative. This will not effect the following argument. Suppose that $k \geq d d_t$. After a short calculation, one finds that $s_1 \leq d d_t / (2d+g-1)$, and hence that $s \leq d^2/(2d+g-1)$. This implies that $s < 2d^2$. Hence, in order to prove our assertion that $s < 2d^2$, we may assume that $k < d d_t$. \par\noindent From $(**)$ we obtain: $$(d d_s)s_1^2 + (2 - 2g - 4d - t_1 k)s_1 + (k + d d_t) = 0.$% $Hence $s_1 | (k + d d_t)$. Hence $s_1 \leq k + d d_t$. Hence $s_1 < 2d d_t$. Hence $s < 2d^2$. To complete the proof, we must show that $t < 2d^4$. The right hand side\ of $(*)$ is nonzero, so: $$s_1 t_1 - 1 \leq d[s_1(d_s s_1 - 4) + d_t] + (2 - 2g)s_1.$% $Dividing by $s_1$ and isolating $t_1$, we find: $$t_1 \leq d [ d_s s_1 - 4 + d_t s_1^{-1} ] + 2 - 2g + s_1^{-1}.$% $Taking account of $t_1 = t d_t^{-1}$ and $s_1 = s d_s^{-1}$, we obtain: $$t \leq d_t \setof{ d [ s - 4 + ds^{-1} ] + 2 - 2g } + ds^{-1}.$% $Since $s < 2d^2$, it follows (with a little work) that $t < 2d^4$. {\hfill$\square$} \end{proofnodot} \begin{remark} We give here an alternate proof of the main ingredient of the proof of (X), namely that $$(n-1) \kern3pt | \kern3pt \setof{ d [ n (s-4) + t ] + (2-2g)n }. \eqno(\dag)$% $Let ${\tilde{C}}$ be the scheme-theoretic complete intersection of $S$ and $T$. Let ${\cal{J}}$ be the ideal sheaf of $C$ in ${\tilde{C}}$. Let $p$ be a closed point of $C$. If ${\cal O}_{S,p}$ is regular, then near $p$, ${\tilde{C}}$ and $C$ are Cartier divisors on $S$, with ${\tilde{C}} = nC$. Choose an isomorphism ${\cal O}_{S,p} \cong k[[x,y]]$, such that $C$ corresponds to $V(x)$. Then ${\tilde{C}}$ corresponds to $V(x^n)$. Therefore the algebra of conormal invariants $${\cal{A}}\ =\ {\cal O}_{{\tilde{C}}}/{\cal{J}} \o+ {\cal{J}}/{\cal{J}}^2 \o+ {\cal{J}}^2/{\cal{J}}^3 \o+ \cdots$% $is a locally free ${\cal O}_C$-module near $p$. But we similarly get the same conclusion if ${\cal O}_{T,p}$ is regular, so in fact ${\cal{A}}$ is locally free since $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$. Moreover, there is a line bundle ${\cal{L}}$ on $C$ such that ${\cal{A}} \cong {\cal O}_C \o+ {\cal{L}} \o+ {\cal{L}}^2 \manyo+ {\cal{L}}^{n-1}$. Hence $\chi({\cal{A}}) = n(1-g) + {n \choose 2}\deg({\cal{L}})$. On the other hand, $\chi({\cal{A}}) = \chi({\cal O}_{{\tilde{C}}})$, which (via ${\tilde{C}} = S \cap T$) is easily computed to be $st(4-s-t)/2$. Hence $$n(1-g) + {n \choose 2}\deg({\cal{L}}) = {st(4-s-t) \over 2}.$% $Hence $${n \choose 2} \kern3pt \left| \kern3pt {st(4-s-t) \over 2} - n(1-g).\right.$% $It is not difficult to verify that this is equivalent to $(\dag)$. \end{remark} \vspace{0.25in} \section*{References} \addtocontents{toc}{\protect\vspace*{2.25em}} \addcontentsline{toc}{special}{References} \ \par\noindent\vspace*{-0.25in} \hfuzz 5pt \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{1}% \def\Atest{ }\def\Astr{Bott\Revcomma R\Initper % \Aand L\Initper \Initgap W\Initper Tu}% \def\Ttest{ }\def\Tstr{Differential Forms in Algebraic Topology}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Dtest{ }\def\Dstr{1982}% \def\Qtest{ }\def\Qstr{access via "bott tu"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{2}% \def\Atest{ }\def\Astr{Brieskorn\Revcomma E\Initper }% \def\Ttest{ }\def\Tstr{Die Aufl\"osung der rationalen Singularit\"aten holomorpher Abbildungen}% \def\Jtest{ }\def\Jstr{Math. Ann.}% \def\Vtest{ }\def\Vstr{178}% \def\Dtest{ }\def\Dstr{1968}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{255--270}% \def\Qtest{ }\def\Qstr{access via "brieskorn simultaneous resolution annalen"}% \def\Xtest{ }\def\Xstr{Not on file. The author studies simultaneous resolution of rational singularities.}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{3}% \def\Atest{ }\def\Astr{Catanese\Revcomma F\Initper }% \def\Ttest{ }\def\Tstr{Babbage's conjecture, contact of surfaces, symmetric determinantal varieties and applications}% \def\Jtest{ }\def\Jstr{Invent. Math.}% \def\Vtest{ }\def\Vstr{63}% \def\Dtest{ }\def\Dstr{1981}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{433--465}% \def\Qtest{ }\def\Qstr{access via "catanese contact"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{4}% \def\Atest{ }\def\Astr{Deligne\Revcomma P\Initper }% \def\Ttest{ }\def\Tstr{{\rm\tolerance=1000 Cohomologie des intersections compl\`etes, expos\'e\ XI in {\itS\'em\-in\-aire de G\'eom\'e\-trie Al\-g\'e\-bri\-que} (SGA 7)}}% \def\Stest{ }\def\Sstr{Lecture Notes in Mathematics}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Vtest{ }\def\Vstr{340}% \def\Dtest{ }\def\Dstr{1973}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{39--61}% \def\Qtest{ }\def\Qstr{access via "deligne intersections"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{5}% \def\Ven{Van de Ven}{}% \def\Atest{ }\def\Astr{Eisenbud\Revcomma D\Initper % \Aand A\Initper \Ven}% \def\Ttest{ }\def\Tstr{On the normal bundles of smooth rational space curves}% \def\Jtest{ }\def\Jstr{Math. Ann.}% \def\Vtest{ }\def\Vstr{256}% \def\Dtest{ }\def\Dstr{1981}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{453--463}% \def\Qtest{ }\def\Qstr{access via "eisenbud normal bundles"}% \def\Xtest{ }\def\Xstr{Not on file.}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{6}% \def\Atest{ }\def\Astr{Fossum\Revcomma R\Initper \Initgap M\Initper }% \def\Ttest{ }\def\Tstr{The Divisor Class Group of a Krull Domain}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Dtest{ }\def\Dstr{1973}% \def\Qtest{ }\def\Qstr{access via "fossum"}% \def\Xtest{ }\def\Xstr{I don't have this.}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{7}% \def\Atest{ }\def\Astr{Fulton\Revcomma W\Initper }% \def\Ttest{ }\def\Tstr{Intersection Theory}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Dtest{ }\def\Dstr{1984}% \def\Qtest{ }\def\Qstr{access via "fulton intersection theory"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{8}% \def\Atest{ }\def\Astr{Gallarati\Revcomma D\Initper }% \def\Ttest{ }\def\Tstr{Ricerche sul contatto di superfiche algebriche lungo curve}% \def\Jtest{ }\def\Jstr{Acad\'emie royale de Belgique, Classe des Sciences, M\'emoires, Collection in-$8^0$}% \def\Vtest{ }\def\Vstr{32}% \def\Dtest{ }\def\Dstr{1960}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{1--78}% \def\Qtest{ }\def\Qstr{access via "gallarati belgique"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{9}% \def\Atest{ }\def\Astr{Griffiths\Revcomma P\Initper % \Aand J\Initper Harris}% \def\Ttest{ }\def\Tstr{Principles of Algebraic Geometry}% \def\Itest{ }\def\Istr{John Wiley \& Sons}% \def\Ctest{ }\def\Cstr{New York}% \def\Dtest{ }\def\Dstr{1978}% \def\Qtest{ }\def\Qstr{access via "griffiths harris principles" (was "griffiths harris")}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{10}% \def\Atest{ }\def\Astr{Grothendieck\Revcomma A\Initper }% \def\Ttest{ }\def\Tstr{{\rm Cohomologie Locale des Faisceaux Coherents et Theorems de Lefschetz Locaux et Globaux, in {\it S\'em\-in\-aire de G\'eom\'e\-trie Al\-g\'e\-bri\-que} (SGA 2)}}% \def\Itest{ }\def\Istr{North-Holland}% \def\Dtest{ }\def\Dstr{1968}% \def\Qtest{ }\def\Qstr{access via "grothendieck lefschetz"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{11}% \def\Atest{ }\def\Astr{Grothendieck\Revcomma A\Initper % \Aand J\Initper \Initgap A\Initper Dieudonn\'e}% \def\Ttest{ }\def\Tstr{El\'ements de g\'eom\'etrie\ alg\'e\-brique II}% \def\Jtest{ }\def\Jstr{Inst. Hautes \'Etudes Sci. Publ. Math.}% \def\Vtest{ }\def\Vstr{8}% \def\Dtest{ }\def\Dstr{1961}% \def\Qtest{ }\def\Qstr{access via "EGA2"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{12}% \def\Atest{ }\def\Astr{Grothendieck\Revcomma A\Initper % \Aand J\Initper \Initgap A\Initper Dieudonn\'e}% \def\Ttest{ }\def\Tstr{El\'ements de g\'eom\'etrie\ alg\'e\-brique IV (part four)}% \def\Jtest{ }\def\Jstr{Inst. Hautes \'Etudes Sci. Publ. Math.}% \def\Vtest{ }\def\Vstr{32}% \def\Dtest{ }\def\Dstr{1967}% \def\Qtest{ }\def\Qstr{access via "EGA4-4"}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{13}% \def\Atest{ }\def\Astr{Hartshorne\Revcomma R\Initper }% \def\Ttest{ }\def\Tstr{Ample Subvarieties of Algebraic Varieties}% \def\Stest{ }\def\Sstr{Lecture \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm Notes in Mathematics}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Vtest{ }\def\Vstr{156}% \def\Dtest{ }\def\Dstr{1970}% \def\Qtest{ }\def\Qstr{access via "hartshorne ample subvarieties"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{14}% \def\Atest{ }\def\Astr{Hartshorne\Revcomma R\Initper }% \def\Ttest{ }\def\Tstr{Algebraic Geometry}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Dtest{ }\def\Dstr{1977}% \def\Qtest{ }\def\Qstr{access via "hartshorne algebraic geometry"}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{15}% \def\Atest{ }\def\Astr{Hartshorne\Revcomma R\Initper }% \def\Ttest{ }\def\Tstr{Complete intersections in characteristic $p > 0$}% \def\Jtest{ }\def\Jstr{Amer. J. Math.}% \def\Vtest{ }\def\Vstr{101}% \def\Dtest{ }\def\Dstr{1979}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{380--383}% \def\Qtest{ }\def\Qstr{access via "hartshorne complete intersections characteristic"}% \def\Xtest{ }\def\Xstr{had: (Based on a talk given at the 1964 Woods Hole conference)}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{16}% \def\Atest{ }\def\Astr{Iversen\Revcomma B\Initper }% \def\Ttest{ }\def\Tstr{Generic Local Structure in Commutative Algebra}% \def\Stest{ }\def\Sstr{Lecture Notes in Mathematics}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Vtest{ }\def\Vstr{310}% \def\Dtest{ }\def\Dstr{1973}% \def\Qtest{ }\def\Qstr{access via "iversen generic local structure"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{17}% \def\Atest{ }\def\Astr{Jaffe\Revcomma D\Initper \Initgap B\Initper }% \def\Ttest{ }\def\Tstr{Space curves which are the intersection of a cone with another surface}% \def\Jtest{ }\def\Jstr{Duke Math. J.}% \def\Vtest{ }\def\Vstr{57}% \def\Dtest{ }\def\Dstr{1988}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{859--876}% \def\Qtest{ }\def\Qstr{access via "jaffe another"}% \def\Htest{ }\def\Hstr{1}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{18}% \def\Atest{ }\def\Astr{Jaffe\Revcomma D\Initper \Initgap B\Initper }% \def\Ttest{ }\def\Tstr{On set theoretic complete intersections in $\P3$}% \def\Jtest{ }\def\Jstr{Math. Ann.}% \def\Vtest{ }\def\Vstr{285}% \def\Dtest{ }\def\Dstr{1989}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{165--173, 175, 174, 176}% \def\Qtest{ }\def\Qstr{access via "jaffe on set theoretic annalen"}% \def\Htest{ }\def\Hstr{2}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{19}% \def\Atest{ }\def\Astr{Jaffe\Revcomma D\Initper \Initgap B\Initper }% \def\Ttest{ }\def\Tstr{Smooth curves on a cone which pass through its vertex}% \def\Jtest{ }\def\Jstr{Manu\-scripta Math.}% \def\Vtest{ }\def\Vstr{73}% \def\Dtest{ }\def\Dstr{1991}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{187--205}% \def\Qtest{ }\def\Qstr{access via "jaffe vertex"}% \def\Htest{ }\def\Hstr{5}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{20}% \def\Atest{ }\def\Astr{Jaffe\Revcomma D\Initper \Initgap B\Initper }% \def\Ttest{ }\def\Tstr{Local geometry of smooth curves passing through rational double points}% \def\Jtest{ }\def\Jstr{Math. Ann.}% \def\Vtest{ }\def\Vstr{294}% \def\Dtest{ }\def\Dstr{1992}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{645--660}% \def\Qtest{ }\def\Qstr{access via "jaffe local geometry"}% \def\Htest{ }\def\Hstr{6}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{21}% \def\Atest{ }\def\Astr{Lang\Revcomma W\Initper \Initgap E\Initper }% \def\Ttest{ }\def\Tstr{Remarks on $p$-torsion of algebraic surfaces}% \def\Jtest{ }\def\Jstr{Compositio Math.}% \def\Vtest{ }\def\Vstr{52}% \def\Dtest{ }\def\Dstr{1984}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{197--202}% \def\Qtest{ }\def\Qstr{access via "william lang torsion"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{22}% \def\Atest{ }\def\Astr{Lipman\Revcomma J\Initper }% \def\Ttest{ }\def\Tstr{Rational singularities, with applications to algebraic surfaces and unique factorization}% \def\Jtest{ }\def\Jstr{Inst. Hautes \'Etudes Sci. Publ. Math.}% \def\Vtest{ }\def\Vstr{36}% \def\Dtest{ }\def\Dstr{1969}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{195--279}% \def\Qtest{ }\def\Qstr{access via "lipman rational"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{23}% \def\Atest{ }\def\Astr{Matsumura\Revcomma H\Initper }% \def\Ttest{ }\def\Tstr{Commutative Algebra, second edition}% \def\Itest{ }\def\Istr{Benjamin/Cum\-mings}% \def\Ctest{ }\def\Cstr{Reading, Massachusetts}% \def\Dtest{ }\def\Dstr{1980}% \def\Qtest{ }\def\Qstr{access via "matsumura commutative algebra"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{24}% \def\Atest{ }\def\Astr{Miyaoka\Revcomma Y\Initper }% \def\Ttest{ }\def\Tstr{The maximal number of quotient singularities on surfaces with given numerical invariants}% \def\Jtest{ }\def\Jstr{Math. Ann.}% \def\Vtest{ }\def\Vstr{268}% \def\Dtest{ }\def\Dstr{1984}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{159--171}% \def\Qtest{ }\def\Qstr{access via "miyaoka quotient"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{25}% \def\Atest{ }\def\Astr{Morrison\Revcomma D\Initper \Initgap R\Initper }% \def\Ttest{ }\def\Tstr{The birational geometry of surfaces with rational double points}% \def\Jtest{ }\def\Jstr{Math. Ann.}% \def\Vtest{ }\def\Vstr{271}% \def\Dtest{ }\def\Dstr{1985}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{415--438}% \def\Qtest{ }\def\Qstr{access via "morrison rational double points"}% \def\Xtest{ }\def\Xstr{MR 87e:14011. \par The author considers the problem of when a curve on a surface can be contracted to a surface having only rational double points. Has appendix on rational double points.}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{26}% \def\Atest{ }\def\Astr{Mumford\Revcomma D\Initper }% \def\Ttest{ }\def\Tstr{The topology of normal singularities of an algebraic surface and a criterion for simplicity}% \def\Jtest{ }\def\Jstr{Inst. Hautes \'Etudes Sci. Publ. Math.}% \def\Vtest{ }\def\Vstr{9}% \def\Dtest{ }\def\Dstr{1961}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{229--246}% \def\Qtest{ }\def\Qstr{access via "mumford topology of normal"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{27}% \def\Atest{ }\def\Astr{Rao\Revcomma A\Initper \Initgap P\Initper }% \def\Ttest{ }\def\Tstr{On self-linked curves}% \def\Jtest{ }\def\Jstr{Duke Math. J.}% \def\Vtest{ }\def\Vstr{49}% \def\Dtest{ }\def\Dstr{1982}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{251--273}% \def\Qtest{ }\def\Qstr{access via "rao self linked"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{28}% \def\Atest{ }\def\Astr{Robbiano\Revcomma L\Initper }% \def\Ttest{ }\def\Tstr{A problem of complete intersections}% \def\Jtest{ }\def\Jstr{Nagoya Math. J.}% \def\Vtest{ }\def\Vstr{52}% \def\Dtest{ }\def\Dstr{1973}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{129--132}% \def\Qtest{ }\def\Qstr{access via "robbiano complete intersections nagoya"}% \def\Xtest{ }\def\Xstr{Not on file.}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{29}% \def\Atest{ }\def\Astr{Sakai\Revcomma F\Initper }% \def\Ttest{ }\def\Tstr{Weil divisors on normal surfaces}% \def\Jtest{ }\def\Jstr{Duke Math. J.}% \def\Vtest{ }\def\Vstr{51}% \def\Dtest{ }\def\Dstr{1984}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{877--887}% \def\Qtest{ }\def\Qstr{access via "sakai weil divisors"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{30}% \def\Atest{ }\def\Astr{Samuel\Revcomma M\Initper \Initgap P\Initper }% \def\Ttest{ }\def\Tstr{Handwritten notes (one page)}% \def\Dtest{ }\def\Dstr{July 1964}% \def\Qtest{ }\def\Qstr{access via "samuel unpublished"}% \Refformat\egroup% \end{document}

9410

alg-geom/9410002

en

https://arxiv.org/abs/alg-geom/9410002

alg-geom/9410002

David Reed

David Reed

Topology of Conjugate Varieties

LATEX, 27 pages

Serre and Abelson have produced examples of non-homeomorphic conjugate varieties. We show that if the field of definition of a polarized projective variety coincides with its field of moduli then all of its conjugates have the same topological type. This extends the class of varieties known to posses conjugacy invariant to canonically embedded varieties. We also show that normal complete interswections in homogeneous varieties have this property.

alg-geom

\section{#1}\setcounter{equation}{0} } \newcommand{\bitt}[1]{\subsection{#1} } \renewcommand{\theequation}{\thesubsection .\arabic{equation}} \newcommand{\re}[1]{{\bf (\ref{#1})}} \catcode`\@=\active \catcode`\@=11 \def\@eqnnum{\hbox to .01pt{}\rlap{\bf \hskip -\displaywidth(\theequation)}} \catcode`\@=12 \newenvironment{s}[1] { \addvspace{12pt} \def\smallskipamount{6pt} \refstepcounter{equation} \noindent {\bf (\theequation) #1.} \begin{em}} {\end{em} \par \addvspace{12pt} \def\smallskipamount{6pt} } \newenvironment{r}[1] { \addvspace{12pt} \def\smallskipamount{6pt} \refstepcounter{equation} \noindent {\bf (\theequation) #1.} } {\par \addvspace{12pt} \def\smallskipamount{6pt} } \begin{document} \catcode`\@=\active \catcode`\@=11 \newcommand{\newcommand}{\newcommand} \newcommand{\vars}[2] {{\mathchoice{\mbox{#1}}{\mbox{#1}}{\mbox{#2}}{\mbox{#2}}}} \newcommand{\Aff}{\vars{\bf} \def\blb{\small \bf} \def\frak{ A}{\blb A}} \newcommand{\C}{\vars{\bf} \def\blb{\small \bf} \def\frak{ C}{\blb C}} \newcommand{\G}{\vars{\bf} \def\blb{\small \bf} \def\frak{ G}{\blb G}} \newcommand{\Hyp}{\vars{\bf} \def\blb{\small \bf} \def\frak{ H}{\blb H}} \newcommand{\N}{\vars{\bf} \def\blb{\small \bf} \def\frak{ N}{\blb N}} \newcommand{\Pj}{\vars{\bf} \def\blb{\small \bf} \def\frak{ P}{\blb P}} \newcommand{\Q}{\vars{\bf} \def\blb{\small \bf} \def\frak{ Q}{\blb Q}} \newcommand{\R}{\vars{\bf} \def\blb{\small \bf} \def\frak{ R}{\blb R}} \newcommand{\V}{\vars{\bf} \def\blb{\small \bf} \def\frak{ V}{\blb V}} \newcommand{\Z}{\vars{\bf} \def\blb{\small \bf} \def\frak{ Z}{\blb Z}} \newcommand{\oper}[1]{\mathop{\mathchoice{\rm #1}{\rm #1} {\scriptstyle \rm #1}{\scriptstyle \rm #1}}\nolimits} \newcommand{\Aut}{\oper{Aut}} \newcommand{\Def}{\oper{Def}} \newcommand{\End}{\oper{End}} \newcommand{\Hilb}{\oper{Hilb}} \newcommand{\Hom}{\oper{Hom}} \newcommand{\diag}{\oper{diag}} \newcommand{\Fl}{\oper{Fl}} \newcommand{\Gr}{\oper{Gr}} \newcommand{\NS}{\oper{NS}} \newcommand{\Par}{\oper{Par}} \newcommand{\Pic}{\oper{Pic}} \newcommand{\Proj}{\oper{Proj}} \newcommand{\Quot}{\oper{Quot}} \newcommand{\Spec}{\oper{Spec}} \newcommand{\GL}[1]{{\rm GL(#1)}} \newcommand{\PSL}[1]{{\rm PSL(#1)}} \newcommand{\PGL}[1]{{\rm PGL(#1)}} \newcommand{\SL}[1]{{\rm SL(#1)}} \def\commrect#1#2#3#4#5#6#7#8{% \begin{center}% \begin{picture}(130,90)% \put(120,70){\vector( 0,-1){50}}% \put(10,80){\vector( 1, 0){100}}% \put(0,70){\vector( 0,-1){50}}% \put(10,10){\vector( 1, 0){100}}% \put(115,80){\makebox(0,0)[l]{#2}}% \put(5,80){\makebox(0,0)[r]{#1}}% \put(115,10){\makebox(0,0)[l]{#4}}% \put(5,10){\makebox(0,0)[r]{#3}}% \put(-3,50){\makebox(0,0)[r]{#5}} \put(123,50){\makebox(0,0)[l]{#6}} \put(60,3){\makebox(0,0)[c]{#8}} \put(60,88){\makebox(0,0)[c]{#7}} \end{picture}\end{center}} \def\commtriang#1#2#3#4#5#6{% \begin{center}\begin{picture}(120,80)% \put(55,70){\vector(-1,-2){30}} \put(65,70){\vector(1,-2){30}} \put(30,5){\vector(1,0){60}} \put(60,75){\makebox(0,0)[c]{#1}} \put(25,5){\makebox(0,0)[r]{#2}} \put(95,5){\makebox(0,0)[l]{#3}} \put(60,0){\makebox(0,0)[c]{#6}} \put(37,43){\makebox(0,0)[r]{#4}} \put(83,43){\makebox(0,0)[l]{#5}} \end{picture}\end{center}} \newcommand{\down}{\Big\downarrow} \newcommand{\beqas}{\begin{eqnarray*}} \newcommand{\beqa}{\begin{eqnarray}} \newcommand{\beq}{\begin{equation}} \newcommand{\bl}{\vskip 1.2ex } \newcommand{\eeqas}{\end{eqnarray*}} \newcommand{\eeqa}{\end{eqnarray}} \newcommand{\eeq}{\end{equation}} \newcommand{\fp}{\mbox{ $\Box$}} \newcommand{\half}{\frac{\scriptstyle 1}{\scriptstyle 2}} \newcommand{\m}{{\bf m}} \newcommand{\mod}{/ \! \! /} \newcommand{\pf}{{\em Proof}} \newcommand{\sans}{\backslash} \newcommand{\st}{\, | \,} \newcommand{\cC}{{\cal C}} \newcommand{\hcC}{{\hat {\cal C}}} \catcode`\@=12 \newpage \title {The Topology of Conjugate Varieties} \author {David Reed\\ Mathematical Institute\\ 24 - 29 St Giles'\\ Oxford OX1 3LB\\ UK\\} \date{October 1994} \maketitle \begin{abstract} Serre \cite{Se:64} and Abelson \cite{Ab:74} have produced examples of conjugate algebraic varieties which are not homeomorphic. We show that if the field of definition of a polarized projective variety coincides with its field of moduli then all of its conjugates have the same topological type. This immediately extends the class of varietie s known to possess invariant topological type to all canonically embedded varieties. We also show that (normal) complete intersections in projective space and, more generally in homogeneous varieties, satisfy the condition. \end{abstract} \bit{Introduction} If $V$ is an algebraic variety defined over $k$, a finitely generated extension of $\Q$, for each embedding $\sigma :k \rightarrow \C$ we can extend scalars to form the complex algebraic variety $V_{\sigma}$ defined by the following cartesian square (base change or extension of scalars) \[ \begin{array}{ccc} V_{\sigma} := V \times _{\Q} \C & \rightarrow & V\\ \downarrow & & \downarrow \\ \Spec \C & \stackrel{\sigma}{\rightarrow} & \Spec k \end{array} \] The complex points of this variety $V_{\sigma}(\C)$ form a topological space and the topological type of two such spaces, $V_{\sigma}(\C)$ and $V_{\tau}(\C)$ for two different embeddings $k \rightarrow \C$ can be compared. We will refer to varieties $V_{\sigma}$ and $V_{\tau}$ obtained in this manner as {\em conjugate varieties}. Serre \cite{Se:64} and Abelson \cite{Ab:74} have produced examples of conjugate varieties whose complex points constitute non-homeomorphic topological spaces. Since the publication of these papers there has been little published work in this area. The principal result of the research reported upon here is a sufficient condition for the topological spaces of complex points of conjugate varieties to be homeomorphic. We begin by recalling definitions due to Matsusaka-Shimura-Koizumi \cite{Ko:72}. \begin{r}{Definition} For a divisor $X$ on a projective variety $V$ define the class ${\cal P}(X)$ to be the class of all divisors $X'$ on $V$ such that there are integers $m, n$ with $mX \equiv nX'$ (algebraic equivalence). If ${\cal P}(X)$ contains an ample divisor then it is called a {\em polarization} (this term is also applied to the ample divisor in the class). An isomorphism of projective varieties $f:V \rightarrow W$ is said to be a isomorphism of polarized projective varieties if there are polarizations ${\cal P}$, ${\cal P}'$ on $V$, and $W$ respectively such that the map on divisors induced by $f$ takes ${\cal P}$ to ${\cal P}'$ \end{r} \begin{r}{Definition} The field of moduli for polarized projective variety $(V,{\cal P})$ is the field $k$ such that for $\sigma \in \Aut(\C)$, $\sigma$ is in fact in $\Aut(\C /k)$ if and only if $V^{\sigma} \simeq V$ as polarized varieties.\end{r} Compare this field to the {\em field of definition} of the variety which is the field $K$ such that $\sigma \in \Aut(\C)$ is in fact in $\Aut(\C /k)$ iff and only if $V^{\sigma}=V$. \bl A discussion of when fields of moduli exist in general for varieties can be found in \cite{Ko:72}. An example of a variety (a hyperelliptic curve) whose field of moduli differs from its field of definition can be found in \cite{Sh:72}. \begin{s}{Theorem} \label{big'} If $V$ is a polarized projective variety defined over $k$, a finitely generated extension of $\Q$, and if the field of moduli for $V$ coincides with $k$, then the topological type of $V_{\sigma}(\C)$ is independent of $\sigma$.\end{s} The proof of this Theorem relies on a strengthened form of Thom's stratified Isotopy Theorem which is given in \S 2 below. The Theorem itself is proved in \S3. It was previously known that certain types of varieties whose topology is rather easily described, such as non-singular curves, abelian varieties, $K-3$ surfaces and simply connected non-singular surfaces of general type, have topological types which do n ot vary under conjugation of their fields of definition. The above Theorem extends this list to include all canonically embedded varieties. In \S 4, the criterion given above will be used to show that (normal) complete intersections in projective space, and, more generally, in homogeneous varieties, also belong on this list by showing that they satisfy the condition of the Theorem as well. In \S 5 we point out a few of the many questions that remain open in this area. \bit{Stratified Isotopy Theorems and the Topology of Conjugate Varieties} \bitt{Preliminaries from Algebraic Geometry} Our intention is to review the proof of the ``stratified isotopy Theorem'' as it applies to algebraic varieties with a view to establishing that the stratification described by the Theorem can be defined without extension of the base fields of the varieti es involved. This material is essentially contained in \cite{Ve:76}. In the course of the analysis we will frequently rely on two sets of well-known and indeed basic facts from algebraic geometry which are stated here with an emphasis on the relevant fields of definition. The first set of facts deals with the singular locus of a variety defined over a field $k$ of characteristic 0 (here we do not need any restriction on the nature of the extension $k/\Q$). Let $X$ be a variety of dimension $n$ over a characteristic zero field $k$, then for any point $x \in X$ the following are equivalent: \begin{enumerate} \item $\Omega^1_{X,x}$ (the module of differentials at $x$) is a free module of rank $n$ over the local ring ${\cal O}_{X,x}$ of $X$ at $x$; \item ${\cal O}_{X,x}$ is a regular local ring. \end{enumerate} and, if either of these conditions obtains at $x$ we say $X$ is {\em smooth} at $x$. \begin{s}{Fact} \label{sm} There is an everywhere dense Zariski open set $U \subset X$ which is smooth and the Zariski closed set $X-U$ is defined over $k$. \end{s} {\bf Remark}: For a discussion of fields of definition for arbitrary subsets of schemes see [{\bf EGA IV}, \S 4.8]. The second set of facts is just Hironaka's well-known resolution of singularities. Once again our only restriction is that we work over characteristic 0 fields. Let $X$ be variety defined over $k$ and ${\cal J}$ a coherent sheaf of ideals defining a closed sub-scheme $D$ then we make the usual: \begin{r}{Definition} A blow-up of $X$ at $D$, otherwise known as a monoidal transformation of $X$ with center $D$, is a pair $(P, f)$ consisting of a variety $P$ and a morphism $f:P \rightarrow X$ such that $f^{-1}({\cal J})$ is an invertible sheaf on $P$ and, for any other p air $(P',f')$ with $f':P' \rightarrow X$ and $f'^{-1}({\cal J})$ an invertible sheaf on $P'$, there is a unique morphism $g:P' \rightarrow P$ such that \commtriang{$P'$}{$P$}{$X$}{$g$}{$f$}{$f'$} commutes.\end{r} A general procedure for constructing $P$ is to define \[ P:=\Proj(\oplus_{d=0}^{\infty} {\cal J}^d) \] where we set ${\cal J}^0={\cal O}_X$. There is a natural map $P \rightarrow X$ (given by ${\cal O}_X \rightarrow \oplus {\cal J}^d$) and the universal property is proved in \cite{H:77} Ch II, \S 7. In particular the field of definition for $P$ is just t he field of definition of $D$ or, equivalently, of ${\cal J}$. \bl Hironaka has shown, \begin{s}{Theorem} \label{m1} Let $X$ be a variety defined over $k$, characteristic 0, then there is a closed subscheme $D$ of $X$ such that: \begin{enumerate} \item the set of closed points of $D$ is the singular locus of $X$; and \item if $f:{\tilde X} \rightarrow X$ is the monoidal transform of $X$ at $D$ then $\tilde X$ is smooth. \end{enumerate} \end{s} \pf. \cite{Hi:64} (Main Theorem 1) \fp For our purposes we note in particular that since $D$ is defined over $k$ by Fact \re{sm} above we have that $\tilde X$ and $f$ are defined over $k$ as well. \begin{r}{Definition} A divisor with normal crossings $D$ in a smooth variety $X$ is a divisor such that for any $x \in D \subset X$ with local ring ${\cal O}_{X,x}$ and maximal ideal ${\bf m}_{X,x}=(z_1, \dots , z_n)$, each component of $D$ passing through $x$ is described b y precisely one ideal $(z_i)$. \end{r} \begin{s}{Theorem} \label{ms} Let $X$ be a smooth variety defined over $k$, $W$ a nowhere dense sub-scheme of $X$, then there exists a finite set of monoidal transforms \[ f_i:X_{i+1} \rightarrow X_i \] with smooth centers $D_i$, for $0 \leq i < r$ and $X_0=X$ such that \begin{enumerate} \item $X_r$ is smooth; \item if $\bar{f_i}$ is the composition of the $f_j$ for $0 \leq j < i$, then $D_i \subset {\bar {f_i}}(W)$ for all $i$; and \item ${\bar{f_r}}^{-1}(W)$ is an invertible sheaf whose support is a divisor with normal crossings. \end{enumerate} \end{s} \pf. \cite{Hi:64} (Cor 3, to Main Theorem II). \fp Since we know that the singular locus of a variety $X$ over $k$ is nowhere dense and hence its inverse image under the monoidal transform $f$ from Theorem \re{m1} is nowhere dense, we can summarize the above by saying that Hironaka's resolution of singula rities starts with an arbitrary variety $X$, defined over $k$ characteristic 0, and produces a smooth variety $X'$ and a morphism $f:X' \rightarrow X$ such that the inverse image of the singular locus of $X$ becomes a divisor with normal crossings in $X'$ and $X'$, $D$ and $f:X \rightarrow X$ are defined over $k$ as well. \bitt{Stratifications and stratified isotopy in the Real Analytic Category} The most natural setting for the study of stratifications of singular spaces is the category of real analytic subspaces of smooth (real analytic) manifolds and proper maps between them. A brief sketch of the aspects of the theory used below is give n here. The application to complex algebraic varieties follows. The best current reference is \cite{G-M:88}. Let $M$ be a real analytic manifold, $Z \subset M$ a closed subset and \[ Z = \bigcup_{i\in S}S_i \] ($S$ a partially ordered set) a decomposition of $Z$ as a union of a locally finite collection of disjoint locally closed ``pieces'' or ``strata'' satisfying the {\em boundary} consition \[ S_i \cap {\bar{S_j}} \neq \emptyset} \def\dasharrow{\to \Leftrightarrow S_i \subset {\bar {S_j}}\Leftrightarrow i=j\;\;or\;\;i<j \] (in the last case we also write $S_i < S_j$). Such a decomposition is called a {\em Whitney Stratification} if and only if it also satisfies: \begin{enumerate} \item each $S_i$ is smooth (not necessarily connected), and \item each pair $(S_i,S_j)$ satisfies the {\em a} and {\em b} conditions, namely, if we have a collection of points $\{x_i\} \subset S_i$ such that $\{x_i\} \rightarrow y \in S_j$ and another set of points $\{y_i\} \subset S_j$ with $\{y_i\} \rightarrow y $ such that the secant lines ${\overline{x_iy_i}}\rightarrow l$ and the tangent planes $T_{x_i}S_i \rightarrow \tau$ then we have \begin{itemize} \item {\em a}: $T_yS_j \subset \tau$; and \item {\em b}: $l \subset \tau$ \end{itemize} \end{enumerate} These conditions ensure that the pieces $S_i$ ``fit together'' well at an infinitesimal level (see \cite{B-C-R:87} for examples). The conditions are local and can be tested by taking local coordinates in $M$ about $y$. The validity of the conditions is independent of the choice of coordinate system. It is a theorem (Hironaka-Hardt) that any subanalytic manifold admits such a stratification. If a map behaves well with respect to stratifications we say it is a {\em stratified} map. Specifically, let $Y_1 \subset M_1, \;\; Y_2 \subset M_2$ be Whitney stratified subsets of manifolds $M_1, \; M_2$, and let $f:M_1 \rightarrow M_2$ be a real analy tic map such that $f\!\mid Y_1$ is proper and $f(Y_1) \subset Y_2$, then $f$ is {\em stratified} if for each stratum $A\subset Y_2$ we have $f^{-1}(A)$ a union of connected components of strata of $Y_1$, say $f^{-1}(A) = \cup S_i$ and $f$ takes each $S_i$ submersively to $A$ (surjection on tangent spaces). There are two key results on stratified maps which are often referred to as the $1^{st}$ and $2^{nd}$ (stratified) Isotopy Theorems. \begin{s}{Theorem} For $Z \subset M$ a Whitney stratified subset of a real analytic manifold, $f:Z \rightarrow \R^n$ proper and such that the restriction to each stratum \newline $f\!\mid A : A \rightarrow \R^n$ is a submersion, then there is a stratum preserving homeomorphism $h:Z \rightarrow \R^n \times (f^{-1}(0) \cap Z)$ such that \commrect{$Z$}{$\R^n\times(f^{-1}(0)\cap Z)$}{$\R^n$}{$\R^n$}{$f$}{$pr_1$}{$h$}{$id$} \noindent commutes. In particular, the fibers of $f\!\mid Z$ are homeomorphic by a stratum preserving homeomorphism. \end{s} \begin{s}{Theorem} $A \subset M$, $B \subset N$ subanalytic subsets of real analytic manifolds, $F: A \rightarrow B$ a proper subanalytic map. Then there exist stratifications $S$, $T$ of $A$, $B$ into smooth subanalytic manifolds such that $f$ is stratified with respect t o $S$ and $T$. Furthermore, given any locally finite collection of subanalytic subsets $\cal C$ of $A$ (resp. ${\cal D}$ of $B$ we can choose $S$ (resp. $T$) such that each elements of ${\cal C}$ (resp. ${\cal D}$) is a union of strata of $S$ (resp. $T$) {}. \end{s} By the $1^{st}$ isotopy Theorem one obtains local topological triviality of the $f$ along connected components of strata of $B$. For further discussion and guidance to the literature see \cite{G-M:88} Part I, Chapter 1, pp. 36 - 44. We wish to employ this theory in the context of complex algebraic varieties and to take our stratifications to be constructible sets whose fields of definition we can control. Following a suggestion of Bernstein, Beilinson, Deligne \cite{BBD:81} Chapter 6, we find that such a version of stratified isotopy theory has been given by Verdier. \bitt{Whitney Stratifications \`{a} la Verdier} We now define a notion of Whitney stratification which is adapted to algebraic varieties. The properties {\em a} and {\em b} above will be replaced by a single property {\em w} which is also local in nature. Hence we will always assume that {\em smooth} complex algebraic varieties have been equipped with coordinate charts given by their underlying real analytic manifold structure. Nothing will depend on the choice of coordinates (see comments below). We use the following notion of distnace between sub-vector spaces in a finite dimensional Euclidean space $E$, ${\delta}(F,G)$ defined by \[{\delta}(F,G) := \sup_{\small{\begin{array}{ccc} x\! & \in & F \\ \parallel x \parallel & = & 1 \end{array}}}\!dist (x,G)\] In particular, $\delta(F,G)=0 \Rightarrow F \subset G$. \begin{r}{Definition} A Verdier-Whitney stratification of (the complex points of) a complex algebraic variety $V$ where $V$ is a $k$-variety of finite type, $k$ a finitely generated extension of $\Q$, is a finite disjoint partition of $V$ by smooth constructible sets $A_i$ \[V=\bigcup_{{i}=1}^n A_{i}\] such that \begin{enumerate} \item the ``boundary property'' holds, namely ${\overline{A_{i}}} \cap A_{j} \neq \emptyset} \def\dasharrow{\to$ implies ${\overline {A_{i}}} \supset A_{j}$ and \item if ${\overline {A_{i}}} \supset A_{\j}$ with $i \neq j$ then the pair $(A_{i},A_{j})$ satisfies the following property ``{\em w}" at every point $y \in A_{i}$: Consider $A_{\alpha}$ and $A_{\beta}$ as real analytic manifolds and take coordinate patches around $y$ to some Euclidean space $E$, then there exists a neighborhood $U \subset E$ of (the image of) $y$ and a positive real number $C$ such that $\forall x \in U \cap A_{i}$ and $y' \in U \cap A_{j}$ (here $A_i$ and $A_j$ are taken to mean the images of some small open subsets around $y$ in $E$) we have \[{\delta}(T_{y'}{A_{j}},T_x{A_{i}}) \leq C \parallel x - y' \parallel\] where $T_x{A_{\alpha}}$ is the tangent plane to $A_{\alpha}$ at $x$ and $\delta$ is as defined above \end{enumerate} \end{r} A number of remarks are called for here. As we are restricting ourselves to algebraic varieties, it is sufficient to consider stratifications with finite collections of subsets. The analytic cases require infinite collections of subsets. This permits a certain amount of simplification in the definition and the subsequent arguments. It also permits us to speak of the (common) field of definition of the stratification as being the smallest field containing the fields of definition of the $A_{\alpha}$. On the other hand the condition {\em w} (so-called by Verdier) replaces the more familiar conditions `{\em a} and `{\em b} above. Condition {\em w} implies condition {\em a} simply because it is a uniform version of it but {\em w} does not imply {\em b} in general (consider the logarithmic spiral at 0). The key fact however is that this implication does hold when $A_{\alpha}$ is a smooth subanalytic subspace of a real analytic space and $A_{\beta}$ is a smooth analytic subspace of $\overline {A_{\alpha}}$ (Kuo). Verification of property {\em w} does not depend on the choice of coordinates (for this it is important that the strata $A_i$ are required to be smooth). The next key fact is that using resolution of singularities, we can stratify arbitrary complex algeb raic varieties in much the same way as real analytic manifolds. The first Theorem we will require is, \begin{s}{Theorem} \label {3b} If $V$ is a complex algebraic variety as above and $V_{\beta}$ a finite family of constructible subsets of $V$ then there is a Verdier-Whitney stratification of $V$ such that each $V_{\beta}$ is obtained as the union of strata. The stratification is defined over the (common) field of definition of $V$ and the $V_{\beta}$.\end{s} The proof of this is based on \begin{s}{Theorem} \label {3a} $V$ as above, $M$, $M'$ smooth, connected, locally closed subsets such that $M \cap M' = \emptyset} \def\dasharrow{\to$, $M' \subset {\overline M}$ and ${\overline M} - M'$ is closed (all for the Zariski topology), then there is a Zariski open $Y \subset M'$ containing all of the points $y \in M'$ such that $(M, M')$ has the property w at $y$ and $M' - V$ is Zariski closed. $Y$ is defined over the (common) field of definition of $V$, $M$ and $M'$. \end{s} \pf\quad of \re{3a}. When $M$, $M'$ are locally closed smooth {\em subanalytic subspaces of a second countable real analytic space} $X$ a corresponding Theorem is proved by \begin{enumerate} \item defining a subset $V \subset M'$ by removing ``bad points'' to arrive at an open subanalytic subset of $X$ which is dense in $M'$, and then \item showing that $(M, M')$ has property {\em w} at all points of $V$ by taking coordinate charts in which $M'$ is an open subset of an affine space $F$. There is an affine space $G$ such that locally $F\oplus G=X$, there is a ``blow-up'' $W$ of $X$ in w hich $M'$ is described as a divisor with normal crossings $\prod_1^qz^{n_i}_i$ and there is a map $\pi:W \rightarrow X=F\oplus G$. Additional work involving an analysis of the matrix representation of $d\pi$ then gives the desired result. In what follows we show how $V$ is defined in the complex algebraic case, show that $V$ is a Zariski open and describe its field of definition. We do not review the proof that $(M, M')$ has the property {\em w} at all points of $V$ since we are principal ly interested in showing that the stratifications can be taken to be algebraic and have the right fields of definition. The reference for all omitted parts of proofs is \cite{Ve:76}. We use resolution of singularities to find a smooth complex algebraic variety $W$ and a proper morphism $\pi :W \rightarrow V$ with $\pi(W)={\overline M}$ and $\pi^{-1}(M')=D \subset W$ is a divisor whose singularities are at worst normal crossings. By \re{m1} and \re{ms} resolution of singularities takes place without extension of the field of definition so that both $W$ and $D$ are defined over the same field as $V$. The next step is to produce the desired open set by removing ``bad points'', in this case, the normal crossings singularities. Consider the subset $D_q \subset D$ of points of $D$ where at least $q$ irreducible local components of $D$ meet (this set is empty for $q \gg 0$) let ${\tilde {D_q}}$ be the normalization of $D_q$ (separating the points lying on the various components) and let $i:{\tilde{D_q}}\rightarrow D_q$ be the normalization map. ${\tilde{D_q}}$ is a smooth algebraic variety defined over the same field as $D$ so we can consider \[d({\pi}\circ i_q):\; \Omega^1_{D_q} \rightarrow (\pi \circ i_q)^*\Omega^1_{\overline M}\] where $\Omega^1$ is the sheaf of differentials. We now have one more correction to make, namely we must consider the points where this map is not surjective so that we do not have a submersion. These form a Zariski closed subset $R_q \subset D_q$ [EGA IV, \S. 17.15.13] defined by a Jacobian condition and hence this singular locus is defined over the same field as $D_q$. $i_q(R_q)$ is closed since $i_q$ is finite and hence a closed map (the ``going-down" Theorem) and the collection of $i_q(R_q)$ is finite so that $S:=\pi(\bigcup_q i_q(R_q)) \subset {\overline M}$ is Zariski closed. We thus have that \[Y:=M' \cap (M - S) \subset {\overline M}\] is a Zariski open subset of $M$ defined without extension of the base field such that $Y$ is dense in $M'$ and it is smooth by construction. The proof of \re{3a} now proceeds by considering the local analysis of the smooth real analytic varieties undrlying $M$, $M'$ and $Y$ as very briefly described above.\fp \pf\quad of Theorem \re{3b}. We need two Lemmas. \begin{s}{Lemma} \label{311} Let $X$ be an algebraic variety, $Y_{\beta}$ a finite family of constructible subsets of $X$, then there is another finite family of subsets of $X$, $B_{\alpha}$ satisfying: \begin{enumerate} \item for all $\alpha$, ${\overline B}_{\alpha}$ and ${\overline B}_{\alpha}-B_{\alpha}$ are Zariski closed and the $B_{\alpha}$ are connected and smooth; \item the $B_{\alpha}$ partition $X$ and each $Y_{\beta}$ is the union of a collection of $B_{\alpha}$; \item ${\overline B}_{\alpha} \cap B_{\beta} \neq \emptyset} \def\dasharrow{\to \Rightarrow B_{\beta} \subset {\overline B}_{\alpha}$ \end{enumerate} and the $B_{\alpha}$ can be defined without extending the (common) field of definition of the $Y_{\beta}$. \end{s} \pf. Since the $Y_{\beta}$ are locally closed we have that the sets ${\overline Y}_{\beta}$ and ${\overline Y}_{\beta} - Y_{\beta}$ are Zariski closed and we replace the family $Y_{\beta}$ with the family of Zariski closed sets $\{X,{\overline Y}_{\beta },{\overline Y}_{\beta}-Y_{\beta}\}$ which we continue to refer to as $Y_{\beta}$. Let $\cal F$ be the largest collection of Zariski closed subsets of $S$ which is such that: \begin{itemize} \item for all $\beta$, $Y_{\beta} \in {\cal F}$; \item ${\cal F}$ is closed under intersection; \item for all $Z \in {\cal F}$ the irreducible components of $Z$ are in ${\cal F}$; and \item $Z \in {\cal F}$ implies that the set of singular points of $Z$, $sing\; Z \in {\cal F}$. \end{itemize} We can get at least one such collection with these properties by taking the family of sets consisting of the $Y_{\beta}$ and their singular loci $(Y_{\beta})_{sing}$ and then closing this collection under taking of irreducible components (defined over the field of definition of the $Y_{\beta}$) and intersections. In particular $\cal F$ can be constructed without extending the field of definition of the $Y_{\beta}$. Then we define \[B_{\alpha}=Z_{\alpha} - \bigcup_{\stackrel{Z_{\beta}\subset Z_{\alpha}}{Z_{\beta} \neq Z_{\alpha}}}Z_{\beta}\] where the $Z_{\gamma}$ are the irreducible sets in $\cal F$. It is clear that the $B_{\alpha}$ have the properties set out in the Lemma.\fp \begin{s}{Lemma} \label{3l2} Let $V \subset X$ be a constructible set which is connected and smooth and let $Z \subset {\overline V} - V$ be Zariski closed, then there is a (finite) partition $B_{\alpha}$ of $Z$ such that such that for all $\alpha$, ${\overline B}_{\alpha}$, ${\overline B}_{\alpha} - B_{\alpha}$ are Zariski closed in $X$, the $B_{\alpha}$ are smooth and connected and the pairs $(V, B_{\alpha})$ have property {\em w}. The $B_{\alpha}$ are defined over the (common) field of definition of $X$, $V$ and $Z$ \end{s} \pf. By induction. The statement is true for $Z= \emptyset} \def\dasharrow{\to$. Assume $Z \neq \emptyset} \def\dasharrow{\to$ so we apply Lemma \re{311} to get a finite collection of smooth connected subsets $U_{\alpha} \subset Z$, $U_{\alpha} \cap U_{\beta} = \emptyset} \def\dasharrow{\to$ defined over the field of definition of $Z$such that the ${\overline U}_{\alpha}$, ${\overline U}_{\alpha} -U_{\alpha}$, $Z-U_{\alpha}$ are all Zariski closed and $Z-\bigcup_{\alpha}U_{\alpha}$ is a Zariski closed set of lower dimension. By Theorem \re{3a} there is an open dense $W_{\alpha} \subset U_{\alpha}$ such that the ${\overline U}_{\alpha} - W_{\alpha}$ are Zariski closed and the pairs $(V, W_{\alpha})$ have the property {\em w}. Now $Z_1:=Z - \cup W_{\alpha}$ is Zariski closed with dimension lower than $Z$ so we can apply the induction hypothesis to it. Combine the $B_{\alpha}$ thus obtained with the $W_{\alpha}$ to get a new collection of $B_{\alpha}$. Since the $W_{\alpha}$ provided by Theorem \re{3a} are constructed without extending fields of definition we are done.\fp Returning to the Proof of Theorem \re{3b} we start by restricting to the case of $V$ irreducible and proceed once again by induction. For $V= \emptyset} \def\dasharrow{\to$ the Theorem is trivially true so suppose $V \neq \emptyset} \def\dasharrow{\to$ and replace the $V_{\beta}$ in the statement of the Theorem with a the family $\{V, {\overline V}_{\beta}, V - V_{\beta}, {\overline V}_{\beta} - V_{\beta}\}$ as in the proof of Lemma \re{311} so that this new family (which we still call $Y_{\beta}$) is made up of Zariski closed sets. Since $V$ is irreducible there is a ${\beta}_0$ such that $Y_{{\beta}_0}$ is open and dense in $V$. Hence $V_1:=V-Y_{{\beta}_0}$ is closed and is of lower dimension than $V$. We also clearly have that $\beta \neq {\beta}_0 \Rightarrow Y_{\beta} \subset X_1$. Let $B_{\alpha}$ be a partition of $V_1$ coming from Lemma \re{3l2} and apply Lemma \re{311} to the $Y_{\beta}$ and $B_{\alpha}$ together. This produces a common refinement $\{C_{\gamma}\}$ which still has property {\em w} because, in general, if $M$, $M'$ are locally closed subsets of an algebraic variety $X$ which are smooth with $M' \subset \overline M$ and $M' \cap M = \emptyset} \def\dasharrow{\to$, then if there is a locally closed and smooth $M'' \subset M'$, the pair $(M, M'')$ will have the property {\em w} if the pair $(M,M')$ does. Now assume for the moment the following \newline {\bf Claim}: If $V = V_{\alpha}$ (finite union) and the $V_{\alpha}$ are Zariski closed, then if Theorem \re{3b} is true for the $V_{\alpha}$ it is true for $V$. $V_1$ is a finite union of irreducibles of dimension lower than $V$ so apply the induction hypothesis to $V_1$ and the $C_{\gamma}$ to get a Whitney stratification of $V_1$ and add $Y_{{\beta_0}}$ If we can now prove the {\bf Claim} we have just made we will both complete the proof for the irreducible case and for the general case as well. So let $V=Y \cup Z$ be a union of irreducibles. Apply the result in the irreducible case to $Z$, $Y_{\beta} \cap Z$, $Y \cap Z$ to get a Whitney stratification of $Z$ such that $Y_{\beta} \cap Z$ and $Y \cap Z$ are unions of strata. Now apply the result again to $Y$, $Y_{\beta} \cap Y$ and the $A_{\alpha}$ such that $A_{\alpha} \subset Y \cap Z$ to get a Whitney stratification of $Y$ such that the $A_{\alpha}$ and $Y_{\beta} \cap Y$ are unions of strata. Take $B_{\beta}$ and those $A_{\alpha}$ such that $A_{\alpha} \subset Z - (Y \cap Z)$. Note that property {\em w} still holds by the remark made above. Continue by induction. Nothing in any of these procedures requires an extension of fields of definition.\fp \bitt{Stratified Morphisms} The next ingredient is the demonstration that quite general algebraic-geometric morphisms behave well with respect to stratifications. Recall that a morphism of stratified spaces $f:X \rightarrow Y$ is called a {\em stratified} morphism if it is proper and if the inverse image of a stratum of $Y$ under $f$ is a union of strata of $X$ and each component of these strata is mapped sunmersiv ely (as a real analytic manifold) to $Y$. We now have a version of the $2^{nd}$ isotopy theorem for complex algebraic varieties. \begin{s}{Theorem}(Verdier) \label{3c} If $f:X \rightarrow Y$ is a morphism of complex algebraic varieties and $f$ is proper then there are Verdier-Whitney stratifications $S$ and $T$ of $X$ and $Y$, defined over the (common) field of definition of $X$,$Y$ and $f$, such that $f$ is transverse to $S$ and $T$ \end{s} \pf. {\em Step 1} We first show that if $X \rightarrow Y$ is proper and $S$ a stratification of $X$ such as given in Theorem \re{3b}, then there is a Zariski open $U \subset f(X) \subset Y$ which is smooth in $Y$ and (Zariski) dense in $f(x)$ such that $f\!\m id f^{-1}(U) \rightarrow U$ takes the connected components of$S \cap f^{-1}(U)$ submersively to $U$. Set \[X_q=\bigcup_{dim\:S_{\alpha} \leq q}S_{\alpha}\] this is closed, smooth, constructible subset of $X$ and we write $f_q$ for $f\mid X_q:X_q \rightarrow Y$. Once again, let $R_q$ be the set of points in $X_q$ where \[df_q: \Omega^1_{X_q} \rightarrow f^*_q(\Omega^1_Y)\] is not onto and set \[U_q=Y-Y_{reg}\cap (\bigcup_{q'\leq q}f_{q'}(R_{q'}))\] where $Y_{reg}$ is the set of regular points of $Y$ (as noted above this set is defined without extension of the field of definition in characteristic 0). The proof now proceeds by induction on $q$ to show that \[f_q\mid U_q:f^{-1}_q(U_q) \rightarrow U_q\] maps the connected components of to $S\cap f^{-1}_q(U_q)$ to $f_q(U_q)$ where we always have ${U_q}$ Zariski dense in $Y$. Note that $Y-U_q$ is automatically a Zariski closed subset since the $f_{q'}(R_{q'})$ are. So let $S_{\alpha} \subset X_{q}$ be a stratum. If dim $S_{\alpha} < q$ then there is a $q'<q$ such that $f_{q'}$ maps the connected components of $S \cap f^{-1}_{q'}(U_{q'})$ to $U_{q'}$. Thus we can assume dim$S_{\alpha}=q$ so $S_{\alpha} \subset X_{ q}$ is Zariski open and smooth and is a connected component of $W=X_q - X_{q-1}$ which is also open and smooth. Now $R_q \cap f^{-1}(U_{q-1}) \subset W$ since $\forall q' < q$ \[coker\:(df_{q'}) \rightarrow coker\:(df_q)\] is surjective over points of $X_{q'}$ as $f_q$ agrees with $f_{q'}$ on $X_{q'}$. $f_q(R_q) \cap U_{q-1}$ is a Zariski closed subset and $U_{q-1} - (f_q(R_q) \cap U_{q-1})$ is dense in $U_{q-1}$ and $f_q \mid U_q$ takes $S \cap f^{-1}_q(U_q)$ submersively to $f_q(U_q)$. But $f:X \rightarrow Y$ is proper so $f_q(R_q)$ is empty for $ q \gg 0$ and we can form $U:= \bigcap_q U_q$. This is dense in $Y$ with $Y - U$ an algebraic Zariski closed set and by construction $f$ takes the connected components of $S\cap f^{-1}(U)$ submersively to $U$. {\em Step 2} Now take $f:X \rightarrow Y$ with stratifications $S$ and $T$ respectively as guaranteed by Theorem \re{3b}. By step 1 we find a Zariski open $U \subset f(X)$ smooth in $Y$ and dense in $f(X)$ with $f\!\mid f^{-1}(U)$ submersive on connecte d components. We now re-stratify $Y$ per Theorem \re{3b} with $U$ a union of strata. Take $f^{-1}$ of these strata and restratify $X$. $f$ now behaves as we want on $f^{-1}(U)$. Consider $f\!\mid X-f^{-1}(U)$, find a $U' \subset f(X-f^{-1}(U))=f(X)-f( U) \subset Y$ which is Zariski open, smooth in $Y$ and dense in $f(X)-f(U)$ and repeat the above restratification process. Since $dim\: U' < dim \:U$ the procedure terminates after a finite number of steps. Using Theorem \re{3b} we will always have that the $U$, $U'$, $\dots$ will be unions of strata and similarly for the $f^{-1}(U)$, $f^{-1}(U')$, $\dots$ and $f$ will clearly take each connected component of the stratifications of the $f^{-1}(U)$, $f^{-1}(U')$, $\dots$ to $f(U)$, $f(U')$, $\dots$ \fp \bitt{Stratified Isotopy} Lastly we state a version of the $1^{st}$ isotopy Theorem: \begin{s}{Theorem}(Thom) \label{Th} $X$, $Y$ are real analytic spaces, $S$ and $T$ stratifications, $f:X \rightarrow Y$ proper and submersive on the connected components of the strata of $X$. Set $y_0 \in Y$. Write $X_0=f^{-1}(y_0)$, $S_0=X_0 \cap S$. Then there is an open neighborhood (in the complex topology) $y_0 \in V \subset Y$ and a homeomorphism $\phi:(f^{-1}(V),\;S \cap f^{-1}(V)) \rightarrow (X_0 \times V,\;S_0 \times V)$ preserving the stratifications and compatible with projections to $V$.\end{s} \pf. Classically this is proved using techniques from differential topology and is quite difficult. Using the condition {\em w} in place of the more standard {\em a} and {\em b} conditions Verdier is able to give a fairly self-contained proof in a few p ages \cite{Ve:76}. Nonetheless we will pass this over in silence since our objective is not to see how the result is proved but rather to show how it can be applied to give useful transversality properties for the algebraically defined stratifications de scribed above. See \cite{Ve:76} for the missing details. \fp When combined with Verdier's results this gives: \begin{s}{Corollary} \label{313} Let $X \stackrel{f}{\rightarrow} Y$ be a proper morphism of algebraic varieties, then the topological type of the fibers of $f$ over a connected component of a stratum of $T$ is constant. \end{s} \pf. By Theorem \re{3c} there are stratifications $S$ and $T$ of $X$ and $Y$ respectively, defined over the common fields of definition of $X$, $Y$ and $f$, such that the inverse image of any stratum of $T$ is a union of strata of $S$ $f^{-1}(T_{\alpha}) = \cup S_{\beta}$ and each connected component of a stratum of $S$ is mapped submersively onto a stratum of $T$. Thus only the last statement requires discussion. Consider a connected component of a stratum of $T$, and call it $W$. Partition $W$ into subs ets such that the topological type of the fibers of $f$ are constant on each member of the partition. The sets partitioning $W$ are then open by Theorem \re{Th} and they are disjoint by construction. Since $W$ is connected only one of the sets in the pa rtition is non-empty. \fp \bit{Principal Results} \bitt {A Sufficient Condition for Topological Stability Under Conjugation} The results described above can be applied to give a sufficient criterion for an algebraic variety and its conjugates to have the same topological type. \begin{s}{Theorem} \label{main} Let $V$ be a $k$-variety, $k$ a finitely generated extension of $\Q$ and suppose there exists a family $f:Y \rightarrow B$, that is, a proper morphism of complex algebraic varieties such that: \begin{enumerate} \item all of the conjugate complex algebraic varieties $V_{\sigma}$ are isomorphic as $k$-varieties to fibers of $f$ (in other words for each $\sigma : k \rightarrow \C$ there is a point $b_{\sigma} \in B$ and $k$-isomorphism $V_{\sigma} \simeq f^{-1}(b_{\sigma})$); and \item $f: Y \rightarrow B$ arises by base extension from $\Q$, that is, there are $\Q$ varieties $Y_{/{\Q}}$ and $B_{/{\Q}}$ and a morphism $f_{/{\Q}}: Y_{/{\Q}} \rightarrow B_{/{\Q}}$ such that \commrect{$Y \simeq Y_{/{\Q}} \times \C$}{$Y_{/{\Q}}$}{$B \simeq B_{/{\Q}} \times \C$}{$B_{/{\Q}}$}{$f$}{$f_{/{\Q}}$}{${\beta}_Y$}{${\beta}_B$} commutes, \end{enumerate} then the topological type of $V_{\sigma}(\C)$ is independent of $\sigma$. \end{s} \pf. By Corollary \re{313} we can stratify $Y$ and $B$ with stratifications $S$ and $T$ defined over $\Q$ so that the map $f$ is topologically locally trivial over each connected component of the strata. We need only show therefore that the points $b_{\sigma}$,$b_{\tau}$ corresponding to conjugate varieties $V_{\sigma}$, $V_{\tau}$ must lie in a single connected component of a stratum of $S$. Since $v_{\sigma}:=f^{-1}(b_{\sigma})$, $v_{\tau}:=f^{-1}(b_{\tau})$ are $k$-isomorphic to varieties which differ only by conjugation of their field of definition these fibers of $f$ are mapped to the same subscheme $v_{/{\Q}}$ of $Y_{/{\Q}}$ by ${\beta}_Y$. This is in turn mapped to a subscheme of $B_{/{\Q}}$; call it $b_{/{\Q}}$. Now we claim $v_{/{\Q}}$ and hence $b_{/{\Q}}$ are irreducible. If not, the $v_{\sigma}$ divide up into subsets which are interchanged by some $\phi \in \Aut(\C)$ (each $v_{\sigma}$ is irreducibl e since it is isomorphic to a variety $V_{\sigma}$. Thus the largest irreducible closed subschemes of $Y_{\Q}$ containing the images of these subsets are not equal to $Y_{\Q}$. But $Y_{\Q}$ is irreducible since $Y$ is, so we have a contradiction. Thus $ b_{/{\Q}}$ is defined by a sheaf of prime ideals ${\cal P}$ with local ring ${\cal O}_{\cal P}$ and residue field ${\bf k}_{\cal P}$. Take the inverse image of $b_{/{\Q}}$ under ${\beta}_B$ in $B \simeq B_{/{\Q}} \times \C$. Call this $b_{/{\C}}$. The points $b_{\sigma}$ and $b_{\tau}$ lie in $b_{/{\C}}$ by commutativity of the diagram in Theorem \re{main} and call the residue fields of these points ${\bf k}_{\sigma}$ and ${\bf k}_{\tau}$. Now ${\beta}_B$ is an open map so that we have ${\bf k}_{\sigma} \simeq {\bf k}_{\cal P} \otimes \C$ and similarly for ${\bf k}_{\tau}$. The stratification $T$ of $B$ is defined over $\Q$ and we claim that points of a subscheme of $B$, defined over $\Q$ and with residue fields isomorphic to $k_{\cal P}$ over $\C$ must lie in a single irreducible component of a stratum of $T$. Assume not. By Verdier's results $b_{/{\C}}(\C)$, the set of complex points of $b_{/{\C}}$ is a union of strata $ \cup T_i$ and each $T_i$ is defined over $\Q$. Assume that we have $b_{\sigma} \in T_1$ and $b_{\tau} \in T_2$. We may assume $T_1 \cup T_2$ is all of $b_{/{\C}}(\C)$. By the definition of a stratification, if ${\overline {T_1}} \cap T_2 \neq \emptyset} \def\dasharrow{\to$ then ${\overline {T_1}} \supset T_2$. This implies that one of the points $b_{\sigma}$ or $b_{\tau}$ lies in a zariski closed subset of $b_{\C}$ and hence does not have residue field isomorphic to $k_{\cal P}$ over $\C$. So we must have ${\overline {T_1}} \cap T_2 = \emptyset} \def\dasharrow{\to$ and similarly ${\overline {T_2}} \cap T_1 = \emptyset} \def\dasharrow{\to$. But then $b_{/{\Q}}$ is not irreducible. Thus the points $b_{\sigma}$ and $b_{\tau}$ lie in a single irreducible component of the stratification of $B$. Finally, over $\C$, irreducibility in the zariski topology implies connectedness in the complex topology so $b _{\sigma}$ and $b_{\tau}$ lie in a single connected component of the stratification and hence the topological types of $f^{-1}(b_{\sigma})$ and $f^{-1}(b_{\tau})$ are the same.\fp {\em Remark}: Shimura has used the existence of non-homeomorphic conjugate varieties to show that the irreducible components of Chow varieties are not necessarily defined over $\Q$ \cite{Sh:68}. I am grateful to Professor J-P. Serre for providing this r eference. \bitt{Corollaries} Following, for example \cite{BBD:81}, it is easy to see that any complex projective variety $V$ (say defined over a field $k$) can be embedded in a family defined over $\Q$. One merely considers the coefficients $c_{\alpha \beta}$ of the homogeneous ideal defining $V$ as indeterminates. This produces a family $f: Y \rightarrow S$ over an affine base $S$ with the original variety $V$ isomorphic to the fiber of $f$ over the point of $S$ corresponding to the $c_{\alpha \beta}$. This family clearly contains all of the conjugates of $V$, since these are obtained by conjugating the coefficients in its homogeneous ideal. Furthermore, since $S$ is affi ne this family is actually ``arises via base extension from $\Q$" in the sense of Theorem \re{main}. Fortunately we cannot use this technique to show that all varieties have invariant topological type under conjugation due to the fact that the families we obtain in this way may not be irreducible. Consider as an example the hyperelliptic curve ${\cal C }$ constructed by Shimura \cite{Sh:72} \[ y^2=a_0x^m + \sum^m_{r=1}(a_rx^{m+r} + (-1)^r{\bar {a_r}}x^{m-r}) \] If we treat the $a_r$ and $\bar{a_r}$ as independent indeterminates $a_r$, $b_r$ we get a family of curves $F$ and $\cal C$ lies over a point $p_0$ in the locus where $a_r=\bar{b_r}$. The family has two components which are interchanged by complex conjug ation. We cannot use it in Theorem \re{main} therefore to show that the conjugates of $\cal C$ are homeomorphic (although, of course, this can be shown in other ways). If the field of moduli of $V$ coincides with its field of definition however, we can use the BBD type family in Theorem \re{main} by virtue of, \begin{s}{Proposition} Let $V$ be a projective variety whose field of definition $k$, a finitely generated extension of $\Q$, coincides with its field of definition. Define the family $f: Y \rightarrow S$ as above, by letting the coefficients $\{c_{\alpha \beta}\}$ of the homo geneous ideal of $V$ vary. Then all of the conjugates of $V$ are isomorphic to fibers $f^{-1}(b_i)$ where the $b_i$ lie in a single connected component of a Verdier-Whitney stratification of $S$ and hence have the same topological type .\end{s} \pf. Examining the proof of Theorem \re{main} we see that we did not need there the full assumption that $Y$ is irreducible, but merely that the conjugates of $V$ do not lie in separate irreducible components of $Y$. Consider the action of $\sigma \in \Aut(\C)$ on $f:Y \rightarrow S$. Since the field of moduli of $V$ coincides with its field of definition $k$ we have that $Y \rightarrow S$ is stable under $\sigma$ if and only if $\sigma \in \Aut (\C /k)$ (that is, th e field of definition of $f:Y \rightarrow S$ is $k$). Let $Y=\cup Y_i$ be a decomposition of $Y$ into irreducible components with fields of definition $k_i$. If $Y_i \neq Y$ then $k \subset k_i$ and $k \neq k_i$. Let the fibers of $f$ corresponding to t he conjugates $V_{\sigma}$ be $v_{\sigma}$ and suppose, for example, that $v_{\sigma}$ and $v_{\tau}$ were in $Y_i$ and $Y_j$ respectively. Then there is some $\phi \in \Aut(\C/ k)$ taking $Y_i$ to $Y_j$. but $\phi$ fixes $v_{\sigma}$ and $v_{\tau}$ so b oth must be in $Y_i \cap Y_j$. This argument applies to all of the $v_{\sigma}$ and all components $Y_i$ hence all of the conjugates of $V$ can be identified with fibers of $f$ lying in a single component $Y_i$ (we may pick any component). We now replac e $Y$ with $Y_i$, which is irreducible, and apply the theorem. \fp \begin{s}{Corollary} The topological type of a canonically embedded variety is invariant under conjugation.\end{s} \pf. The canonical embedding is given over the field of definition and hence the field of definition coincides with the field of moduli. \fp This enlarges the class of varieties previously known to have conjugate invariant topological type. Further examples are given in \S 4 where we show that complete intersections in homogeneous varieties have fields of moduli which coincide with the fields of definition. For a proof that Serre's original examples of non-homeomorphic conjugate varieties do not have fields of definition which coincide with their fields of moduli see \cite{Re:94} \bit{Complete Intersection Type Varieties} \bitt{Generalities} We can use Theorem \re{main} to exhibit further classes of varieties which have topological type invariant under conjugation. These classes of varieties, which include (normal) complete intersection varieties in projective space, may be parametrized by vector spaces of sections of vector bundles and this linear structure provides a natural method of descending from $\C$ to $\Q$. The Deformation Theory of these varieties has been studied by a number of authors including \cite{K-S:58}, \cite{S:75}, \cite{B:83} and \cite{W:84} in a series of papers with results extending from smooth hypersurfaces through to the more general cases. The most general result along these lines is: \begin{s}{Theorem} (Wehler) \label{weh} Let $Z=G/H$ be a non-singular homogeneous complex variety, quotient of a simple, simply connected Lie group $G$ by a parabolic subgroup $H$, $E=\oplus_{j=1}^r{\cal O}_Z(d_j)$ a vector bundle, $s \in H^0(Z,E)$ a section and $X$ a complex variety described by the zero locus of $s$ such that codim$X=r$ and $X$ is not a $K$-3 surface. Then the vector space $H^0(Z,E)$ parametrizes a complete set of small deformations of $X$ and these deformations are given by the family \[ Y :=\{(z,s) \mid z\in Z, s\in H^0(Z,E), s(z)=0\} \rightarrow H^0(Z,E) \] \end{s} If we are to apply the Theorem to this case we must show that the family $Y \rightarrow H^0(Z, E)$ satisfies the conditions of Theorem \re{main}. To do this we reprove the theorem using algebraic geometric techniques to obtain a family over $\Q$ which will have the required properties. The proofs given here therefore are modelled on Wehler's but adapted to algebraic rather than analytic geometry. The key in both the analytic and algebraic approach is to establish a close connection between the deformation theory of the objects and their Hilbert schemes as will be explained shortly. For another approach to establishing the algebraic deformation theory of complete intersections see \cite{Ma:68}. \bitt{Comparison of Hilbert Scheme and Deformation Functors} Let $Z$ be an arbitrary non-singular projective variety over a field $k$ and $E \rightarrow Z$ an algebraic $k$ vector bundle over $Z$. Consider the scheme $X \subset Z $ defined by a global section $s_0 \in H^0(Z,E)$. We construct this scheme as follows: the section $s$ defines a map of sheaves ${\cal O}_Z \rightarrow E$ sending the section ${\bf 1}$ of ${\cal O}_Z$ to the section $s$ (we are abusing notation here by not distinguishing between the vector bundle and its associated locally free sheaf). There is a dual map $\check s:\check E \rightarrow {\cal O}_Z$ and $X$ is said to be defined by $s$ if ${\cal O}_X$ fits into an exact sequence of sheaves \[\check E \stackrel{\check s}{\rightarrow} {\cal O}_Z \rightarrow {\cal O}_X \rightarrow 0 \] $X$ is sometimes referred to as the zero scheme of $s$. \begin{r}{Definition} The Hilbert functor of $X$ (in $Z$), is the functor $\Hilb$ from $\cal C$, the category of local artin rings with residue field $k$ to the category ${\em Sets}$ which assigns to each object $A$ in $\cal C$ the set of schemes $Y$ which fit into the following diagram \commrect{$Z \supset X\simeq Y \times_k A$}{$Y \subset Z \times \Spec A$}{$\Spec k$}{$\Spec A$}{}{}{}{} with $Y$ flat over $\Spec A$.\end{r} \begin{r}{Definition} The (affine) projective cone (hereinafter simply ``the cone'') $C_X$ of (or on) a projective variety $X \subset \Pj^n$ is given by \[ C_X:= \Spec ({\cal O}_X \oplus {\cal O}_X(1) \oplus {\cal O}_X(2) \oplus \dots ) \] The vertex $p$ of $C_X$ is the (closed) subscheme defined by the augmentation ideal, $ker\:{\epsilon}$, where \[ \epsilon: ({\cal O}_X \oplus {\cal O}_X(1) \oplus {\cal O}_X(2) \oplus \dots) \rightarrow {\cal O}_X \] $\Spec ({\epsilon})$ thus defines a map $X \rightarrow C_X$ whose image is $p$.\end{r} For future use we recall that the cone with vertex removed (\'{e}point\'{e}) \[C_X - p \simeq \V({\cal O}_X(-1))\simeq \Spec (\oplus {\cal O}_X(n))\] where $\V$ denotes the operation of taking the vector bundle associated to a locally free sheaf. There is an action of $\G_m$ on $C_X$ (and on $C_X-p$) with integral weights coming from the action on each tensor power ${\cal O}(n)$. For details see [{\bf {EGA II}} \S 8.4 - 8.6]. Next we define a deformation functor $\Def_{C_X}$ as the functor which assigns to each object $A$ in $\cal C$ the set of deformations ${\cal O}_{C_X,p}(A)$ of the $k$-algebra ${\cal O}_{C_X,p}$ which is the local ring of $C_X$ at $p$. There is a natural morphism $h$ from the Hilbert functor to this deformation functor obtained by assigning to a scheme $Y$ as above the local ring of the vertex of the projective cone on $Y$. This is naturally a deformation of the local ring of the vertex of the projective cone on $X$. The morphism thus consists of ``forgetting the embedding in $\Pj^n$ or, in terms of the underlying rings, forgetting the gradings. Comparison Theorems between Hilbert functors and Deformation functors go back (at least) to Schlessinger \cite{Sch:71} and can be found in \cite{P:74}, \cite{K:79} and \cite{Wa:92}. We will use a version due to Kleppe which employs Andr\'{e}-Quillen cohomology to avoid unnecessary smoothness conditions. Since $X$ is given as the zero-scheme of a section of a vector bundle we want to describe its Hilbert scheme in the same way. So define functors $F_{s_0}$ from the category ${\cal C}$ to {\em Sets} which, for any given section $s_0 \in H^0(Z,E)$, assign to an object $A$ of ${\cal C}$ the set of zero schemes in $Z \times \Spec A$ of sections $s_A \in H^0(Z \times\Spec A, E \times \Spec A)$ which reduce to $s_0$ over the closed point $k$ under the map $\Spec k \rightarrow \Spec A$. If we further assume that codim$X$=rank$E$ we have that $X$ is (at least) a local complete intersection. Moreover, such zero schemes $s_A$ are flat over $\Spec A$. By [{\bf EGA IV} \S 19.3.8] the zero scheme defined by any of the $s_A$ is a local complete intersection as well. Thus we obtain elements of ${\Hilb}_X(A)$ as sections of vector bundles $E \times \Spec A$ and a morphism of functors $F_{s_0} \stackrel{f}{\rightarrow} \Hilb_X$. We now study this morphism of functors. \begin{s}{Proposition} \label{surj} Let $Z$, $E$ and $A$ be as above and let $X$ be the zero set of a section $s_0$ with codim $X$=rank$E$, write ${\cal I}_X \subset {\cal O}_Z$ for the ideal sheaf of $X$ and suppose that \[H^1(Z,E \otimes{\cal I}_X)=0\] then the above morphism of functors $f: F \rightarrow \Hilb$ is surjective on tangent spaces.\end{s} \pf. The tangent space to $H$ at $h_0$, is $\Hilb_{s_0}(k[{\epsilon}])$ and it is standard that \[T(H,h_0)=Hom_{{\cal O}_Z}({\cal I}_X, {\cal O}_X)=Hom_{{\cal O}_X}({\cal I}_X/{\cal I}^2_X, {\cal O}_X)\] (for the first isomorphism see \cite {G:61}, for the second see \cite{H:77} Ch. II, \S 8). We also have \[Hom_{{\cal O}_X}({\cal I}_X/{\cal I}^2_X, {\cal O}_X)\simeq H^0(X,N_{X/Z}) \simeq H^0(X,E\!\mid_X)\] where $N$ is isomorphic to the tangent space to $\Hilb$. But the tangent space to $F:=F(k[{\epsilon}])$ is the vector space of sections of $E \times \Spec(k[{\epsilon}]) \rightarrow Z \times \Spec(k[{\epsilon}]) $ which reduce to $s_0$ over $k$ and this is just $H^0(Z,E)$. Consider the standard short exact sequence \[ 0 \rightarrow {\cal I}_X \rightarrow {\cal O}_Z \rightarrow {\cal O}_X \rightarrow 0 \] tensor it with $E$ and take cohomology to get \[0 \rightarrow H^0(Z,E\otimes {\cal I}_X) \rightarrow H^0(Z,E) \rightarrow H^0(X,E\!\mid_X)\] \[ \rightarrow H^1(Z,E\otimes {\cal I}_X) \rightarrow \dots \hfill \] Hence if $H^1(Z,E\otimes{\cal I}_X)=0$ we have the desired surjectivity. \fp So far we have developed portions of a triangle of functors \commtriang{$F_{s_0}$}{$\Hilb_{s_0}$}{$\Def_{C_X}$}{$f$}{?}{h} To fill in the morphism marked with ``?'' note that the local ring of the vertex of the cone of the zero scheme of $s_A$ is naturally a deformation of the local ring of the vertex of the cone of the zero scheme of $s_0$. It is clear that the triangle commutes because $g$ is just the composition of $f$ and $h$. We now invoke the comparison Theorem relating the Hilbert functor and the deformation functor \begin{s}{Theorem} Suppose that $V$ is projectively normal and that $T^1(V)$,the tangent space to $\Def_{C_V}$, is negatively graded (in a sense to be made precise below), then the natural morphism of functors \[h: \Hilb_{V} \rightarrow \Def_{C_V}\] is smooth. \end{s} \pf. See \cite{K:79} \fp As we will see in the case of the zero schemes of sections of certain vector bundles, this result together with Proposition \re{surj} will enable us to establish the existence of families of the desired type. The codim$X$=rank$E$ condition that we have been placing on our zero schemes of sections of $E$ ensures projective normality (assuming that the varieties are normal in the first place) by a straightforward adaptation of the proof for complete intersections in projective space. It is the grading condition in the comparison theorem which is the more difficult of the two to verify and we will reduce it to a condition on vanishing of cohomology. Recall that $\Def_{C_X}$ is the deformation functor of the vertex on the projective cone $C_X$ over $X$ and that there is an action by $\G_m (k)$ on $C_X$ with weights ranging over the integers. $T^1(X):={\Def}_{C_X}(k[{\epsilon}])$ is a vector space and the action of $\G_m$ on on $C_X$ becomes an action on $T^1(X)$, so that we get a decomposition $T^1(X)=\oplus_{\nu = -\infty}^{\infty}T^1(\nu )$ and $T^1(X)$ becomes a graded vector space (as $p$ is an isolated singularity in $C_X$, in fact $T^1(\nu)=0$ for $\nu \gg 0$ and for $\nu \ll 0$ but we will not need this). We say that $T^1(X)$ is {\em negatively graded} if $T^1(X)(\nu )=0$ for $\nu > 0$. To demonstrate that this condition obtains we show that $\Hilb (k[{\epsilon}])$ too has a grading, which is negative in this sense and that the map $Th$ is surjective and respects both gradings. \begin{s}{Proposition} Let $X$ be the zero scheme of the section $s_0$ of $E \rightarrow Z$ and assume that $X$ projectvely normal (e.g. codim $X$ = rank $E$). Suppose the conclusion of Proposition \re{surj} holds, that is, the morphism of functors $F_{s_0} \rightarrow \Hilb_X$ is surjective on tangent spaces. Let $g: F_{s_0} \rightarrow Def_{C_X}$ be the natural morphism taking an element of $F_{s_0}(A)$ to an element of $Def_{C_x}(A)$. Denote the map on tangent spaces by $H^0(Z,E) \stackrel{Tg}{\rightarrow} T^1(X)$. If this map is surjective then $T^1(X)$ is negatively graded and the morphism of functors $h$ is smooth. \end{s} \pf. We have a triangle of tangent maps \commtriang{$H^0(Z,E)$}{$\Hilb(k[{\epsilon}])$}{$\Def{C_X}(k[{\epsilon}])$}{$Tf$}{$Tg$}{$Th$} which commutes because the triangle of underlying maps does and we are assuming that $Tf$ is onto. If we now further assume that $H^0(Z,E) \rightarrow T^1(X)$ is onto then $\Hilb(k[{\epsilon}]) \rightarrow T^1(X)$ must also be onto as well. $\Hilb(k[{\epsilon}])\simeq H^0(X,N_{X/Z})$ as noted above and $H^0(X,N_{X/Z}) \simeq H^0(C_X,N_{C_X})$ where $C_X$ is the projective cone over $X$. By projective normality the singularity at the vertex $p$ of $C_X$ has depth=2 so global sections of $C_X-p$ extend to global sections of $C_X$. Since $X$ is a local complete intersection, this is true of all coherent sheaves on $C_X$ by local duality, and so, in particular \[H^0(C_X,N_{C_X}) \simeq H^0({C_X}-p,N_{C_X}) \simeq H^0(C_X-p,N_{C_X-p})\] But \[C_X-p \simeq \V({\cal O}_X(-1)) - {\em zero section}\] so there is a natural affine map $\pi:C_X-p \rightarrow X$ and $N_{C_C-p} \simeq \pi^*N_X$ and we can compute \[H^0(C_X-p,N_{C_X-p}) \simeq H^0(C_X-p,\pi^*N_X) \simeq H^0(X,\pi_*\pi^*N_X) \simeq \oplus_{\nu = -\infty}^0 H^0(X,N_X(\nu))\] where $N_X(\nu):= N_X \otimes {\cal O}(\nu)$ as usual. The second isomorphism comes from the fact that $\pi$ is affine and the conclusion that $H^0(X,N_X(\nu))=0$ for $\nu > 0$ comes from the fact that the sections must extend over all of $C_X$ and hence cannot have any poles at $p$. The grading thus produced on $\Hilb (k[{\epsilon}])$ arises from the action of $\G_m$ on $C_X$ and the morphism $\Hilb \rightarrow \Def$ is contructed precisely by passing to the projective cone to go from the varieties in projective space to abstract varieties. Hence the action of $\G_m$ on $T^1(X)$ and $\Hilb(k[{\epsilon}])$ is compatible with this morphism. The result is that $T^1(X)$ must also be negatively graded and hence we can apply the comparison Theorem to get that this morphism is smooth. \fp Finally it is not difficult to reduce the requirement ``$H^0(Z,E) \rightarrow T^1(X)$ surjective'' to a statement of vanishing of cohomology. The tangent sheaf to a variety $V$ is defined by $\Theta_V := Hom_{{\cal O}_V}(\Omega_V, {\cal O}_V)$, where $\Omega_V$ is the sheaf of differentials of $V$. Since $C_X \subset \Aff^{n+1}$ and $C_X -p$ is smooth we have that \[ 0 \rightarrow \Theta_{C_X}\!\mid_{C_X-p}\rightarrow \Theta_{\Aff ^{n+1}}\!\mid_{C_X-p} \rightarrow {\cal N}_{C_X}\!\mid_{C_X-p} \rightarrow 0 \] is exact. Note that $C-p$ is not affine so there is a longer exact cohomology sequence \[ 0 \rightarrow H^0(C_X-p, \Theta_{C_X}\!\mid_{C_X-p}) \rightarrow H^0(C_X-p,\Theta_{\Aff^{n+1}\!\mid_{C_X-p}}) \rightarrow\] \[H^0(C_X-p,{\cal N}_{C_X}\!\mid_{C_X-P}) \rightarrow H^1(C_X-p, \Theta_{C_X}\!\mid_{C_X-p}) \rightarrow H^1(C_X-p, \Theta_{\Aff^{n+1}\!\mid_{C_X-p}}) \rightarrow \] The theory of the ``cotangent complex'' \cite{Li-Sch 67} gives us the following exact sequence \[ 0 \rightarrow H^0(C,\Theta_{C_X}) \rightarrow H^0(C,\Theta_{\Aff^{n+1}}\!\mid_{C_X}) \rightarrow H^0(C_X,{\cal N}_{C_X})\] \[ \rightarrow T^1(X) \rightarrow H^1(X,\Theta_Z\!\mid_X) \rightarrow \dots \hfill \] \noindent Once again since $X$ is projectively normal, sections over $C_X-p$ extend to $C_X$ so that \[H^0(C_X,\Theta_{C_X}) \simeq H^0(C_X-p,\Theta_{C_X}\!\mid_{C_X-p}) \] \[ H^0(C_X,\Theta_{\Aff^{n+1}}\!\mid_{C_X}) \simeq H^0(C_X-p,\Theta_{\Aff^{n+1}}\!\mid_{{C_X}-p})\] and \[H^0(C_X,{\cal N}_{C_X}) \simeq H^0(C_x-p,{\cal N}_{C_X-p})\] Putting all of this together we see that if \[ H^1(C_x-p,\Theta_{\Aff^{n+1}}\!\mid_{C_X-p})\simeq H^1(X,\Theta_Z\!\mid_X)=\emptyset} \def\dasharrow{\to \] then \[ T^1(X) \hookrightarrow H^1(C_X-p,\Theta_{C_X}\!\mid_{C_X-p})=0 \] To summarize, \begin{s}{Proposition} For $X$ the zero scheme of a section $S$ bundle $E$ over a non-singular projective variety $Z$ with codim $X$ = rank $E$ where \[H^1(Z, E \otimes {\cal I}_X)=H^1(X,\Theta_Z\!\mid_X)=0\] and $X$ is normal we have the commutative triangle \commtriang{$F_{s_0}$}{$\Hilb_X$}{$\Def_{C_X}$}{}{}{} where all of the arrows are smooth. \end{s} \pf. We only need to show that $F_{s_0} \stackrel{f}{\rightarrow} \Hilb_X$ is smooth since then the third side of the triangle will be smooth by \cite{Sch:67}. We know that this arrow is surjective on tangent spaces hence we only need to show that $F_{s_0}$ is less obstructed than $\Hilb_X$. In other words, if $B \stackrel{\phi}{\rightarrow} A$ is a surjection in $\cal C$ with $ker(\phi)^2=0$ and $({\bf m}_B)ker(\phi)=0$ (so $ker(\phi)$ is a $k$-vector space) and if for $\xi_0 \in \Hilb_X(A)$ which is in the image of $f$ so, $\xi_0=f(\zeta_0)$, $\zeta_0 \in F_{s_0}(A)$, there is a $\xi \in \Hilb_X(B)$ such that $\Hilb_X(\phi)(\xi)=\xi_0$ then there is a $\zeta \in F_{s_0}(B)$ such that $F_{s_0}(\phi)(\zeta)=\zeta_0$. The obstruction to the existence of such $\zeta$ lies in $Ext^2(L_{X/Z},{\cal O}_X)$ where $L_{X/Z}$ is the cotangent complex of $X$. This is a two term complex \[ 0 \rightarrow {\cal I}_X/({\cal I}_X)^2 \rightarrow i^*(\Omega^1_Z) \rightarrow 0 \] since $X$ is a local complete intersection \cite{I:71} Ch. III, \S 3.2 and the $Ext^2$ terms vanish since $H^1(X,\Theta_Z\!\mid_X)=0$. \fp \bl Since we know that the formal scheme prorepresenting $\Hilb_X$ is algebraizable \cite{G:61}, we now have the same conclusion for $\Def_{C_X}$. All infinitesimal deformations of $X$ come from small deformations and all small deformations lie in $Z$. Indeed, by versality $H^0(Z,E)$ is a complete deformation space and the family \[ Y:=\{(z,s)\mid z \in Z, s \in H^0(Z,E), s(z)=0\} \subset Z \times H^0(Z,E) \rightarrow H^0(Z,E) \] is a universal family both in the sense of the deformation functor and the Hilbert functor. These results are valid for $X$ defined over any field. \bitt{Application of the theory} We now apply this theory to our problem. If a normal $k$-variety $V$ ($k$ a finitely generated extension of $\Q$) is given as the zero scheme of a section $s\in H^0(Z,E)$ (everything defined over $k$) and if we have \begin{itemize} \item codim$V$=rank$E$ \item $H^1(Z,E\otimes {\cal I}_V)=H^1(V,\Theta_V)=0$ \end{itemize} then we know that the $k$-vector space $H^0(Z,E)$ parametrizes a complete family of deformations of $V$. There is a map \[ H^0(Z,E) \rightarrow H^0(Z,E)_{/{\Q}} \otimes k \] giving a $\Q$ structure to $H^0(Z,E)$. This induces a morphism of functors $F^k_{s} \rightarrow F^{\Q}_{s}$. We observe that this morphism is smooth since first, if $A' \rightarrow A$ is a surjection in $\cal C$ then \[ F^k(A') \rightarrow F^k(A) \times_{F^{\Q}(A)} F^{\Q}(A') \] must also be onto. To see this note that the arrow $\beta$ in \[ \begin{array}{ccc} F^k(A) \times_{F^{\Q}(A)} F^{\Q}(A') & \stackrel{{\beta}'}{\rightarrow} & F^{\Q}(A')\\ \downarrow & & \downarrow\\ F^k(A) & \stackrel{\beta}{\rightarrow} & F^{\Q}(A) \end{array} \] is onto and hence ${\beta}'$ is also and that $F^k(A') \rightarrow F^{\Q}(A')$ is onto as well. Secondly the morphism also induces a bijection on tangent spaces since $H^0(Z,E)_k$ and $H^0(Z,E)_{/{\Q}}$ are vector spaces of the same dimension over fields of the same cardinality. Thus the projection $H^0(Z,E) \rightarrow H^0(Z,E)_{/{\Q}}$ induces a map of deformation spaces between the functors $\Def^k$ and ${\Def}^{\Q}$ and $\Hilb_X^k$ and $\Hilb_X^{\Q}$. We have a universal family $Y_{/{\Q}} \rightarrow H^0(Z,E)_{/{\Q}}$ and the above ensures that we recover the corresponding universal family over $k$ by extension of scalars \[ \begin{array}{ccc} Y_k & \rightarrow & Y_{/{\Q}}\\ \downarrow & & \downarrow \\ H^0(Z,E)_k & \rightarrow & H^0(Z,E)_{/{\Q}} \end{array} \] But there is a unique extension of scalars from $\Q$ to $\C$ and this gives a diagram of the sort described at the beginning of this \S \commrect{$Y_{/{\C}}\simeq Y_{/{\Q}} \times \C$}{$Y_{/{\Q}}$}{$H^0(Z,E)_{/{\C}}\simeq H^0(Z,E)_{/{\Q}}\times \C$}{$H^0(Z,E)_{/{\Q}}$}{$f$}{$f_{/{\Q}}$}{$\beta_Y$}{$\beta$} Since we can now consider $V$, $E$, $Z$, and $s$ as objects defined over the complex numbers it is clear that all of the conjugate varieties $V_{\sigma}$ can be obtained by conjugating the section $s$, that is, $V_{\sigma}$ is the zero scheme of $s_{\sigma}:=\sigma (s)$. Hence all of the conjugates $V$ are in the family $Y$ if $V$ is. Thus the conditions of Theorem \re{main} are satisfied and the independence of the topological type from variation with $\sigma$ is ensured. It only remains to spell out some types of varieties which are defined as the zero sections of vector bundles satisfying our conditions. If we assume that $Z$ is homogeneous, that is \[Z \simeq G/H\] where $G$ is a simple, simply connected, split algebraic group over $\Q$, and $H$ a parabolic subgroup, then by algebraic versions of Bott's vanishing Theorems \cite{Dm:76} we get that $H^1(Z, \Theta_Z)=0$ for all $i > 0$. If we further suppose that $E\simeq \oplus_{j=1}^r{\cal O}_Z(d_j)$ with $d_j > 0$ and $X$ defined by a section $s$ such that \begin{itemize} \item codim $X$=$r$ \item $X$ is not a $K$-3 surface \item dim $Z \geq 3$ \end{itemize} then a reworking of calculations of Borcea using an algebraic version of the Kodaira-Nakano-Akizuki vanishing theorem shows that $H^1(Z,E\otimes{\cal I}_X)=0$. For details see \cite{Re:94} If we now assume that $X$ is normal and defined over $k$ that is, the section $s$ satisfies \[ s \in H^0(Z,E)_{/{\Q}} \otimes k \] \noindent then we have a class of $k$ varieties whose topology is independent of the embedding of $k$. This class includes complete intersections in projective space. A somewhat different proof is available for the special case of complete intersections in projective space which does not require normality \cite{Re:94}. We note finally that while the case of $K$-3 surfaces must be treated separately, the result is the same since all $K$-3's are homeomorphic (indeed diffeomorphic) and conjugates of $K$-3 varieties are $K$-3. \bit{Comments and Open Questions} The above discussion can be used to shed a bit of light on the nature of the Serre-Abelson examples. Both authors construct their varieties as quotients using finite group actions. In both cases the action is varied under conjugation. The difference lies in the part of the homotopy type which is affected by conjugation. In Serre, the variety acted upon is a product of a diagonal hypersurface by an abelian variety and the group action on the abelian variety makes the $\pi_1$ of the variety into a module in demonstrably different ways under conjugation. In Abelson's example the group acts on complete intersection which is constructed via a representation of the group and all of this varies under conjugation. Special choices of the group allow one to demonstrate variation in the Postnikov tower. It is not difficult to see, in both cases that the field of moduli of the varieties thus constructed is smaller than their field of definition. The group action creates some ``symmetries'' which produce this result. This suggests the following vague question: A) Is it possible to produce examples of non-homeomorphic conjugate varieties without using (finite) group actions? This question can be made more specific in a number of ways. Because of the use of group actions the examples of Serre and Abelson are rather rigid. A small deformation of one of their examples no longer maintains the structure required to compare it with its conjugates. One approach might therefore be to ask, A1) Is it possible to construct examples of non-homemorphic conjugate varieties which are stable under small deformations? This seems unlikely. Another way to cut out finite group actions is to ask for simply-connected examples, A2) Are there examples of simply connected non-homemorphic conjugate varieties? A useful source of new examples may be provided by Shimura varieties. Note: It may be useful to employ Kollar's notions of ''essentially large'' fundamental groups here instead. Finally, along these lines one has the fundamental question, A4) Are the simply connected covering spaces of conjugate algebraic varieties analytically isomorphic? The criterion developed in this paper does not provide an indication of the minimum '``necessary'' conditions under which the topology of varieties remains stable under conjugation. Neither does it give any indication of the ``part'' (if any) of the topological type which are conjugations invariant (over and above the \'{e}tale homotopy type which is clearly invariant). A Theorem of Deligne \cite{D e:87} shows that the nilpotent completion of the fundamental group of an algebraic variety is algebraically determined. One is led to ask, B) Is the entire rational homotopy type a conjugation invariant? One might also pose the following question which seems to lie somewhere between A) and B), C) Is simple connectivity a conjugation invariant? \newpage

9410

alg-geom/9410005

en

https://arxiv.org/abs/alg-geom/9410005

alg-geom/9410005

Lothar Goettsche

Geir Ellingsrud and Lothar G\"ottsche

Variation of moduli spaces and Donaldson invariants under change of polarization

44 pages, amslatex, no figures

The paper determines the change of moduli spaces of rank $2$ sheaves on surfaces with $p_g=0$ under change of polarization and the corresponding change of the Donaldson invariants. In this revised version we have made some minor stylistic changes in the previous text. In addition we have added a final chapter of about 20 pages (announced in the previous version), in which the six lowest order terms (three of them non-zero) of the change are computed explicitely using computations in the cohomology of Hilbert schemes of points.

alg-geom

P(n){P(n)} \def{\text{\rom{\bf b}}}{{\text{\rom{\bf b}}}} \defS^{(\bb)}{S^{({\text{\rom{\bf b}}})}} \def{\text{\rom{\bf E}}}{{\text{\rom{\bf E}}}} \def\Inc#1{Z_{#1}} \def\inc#1{\zeta_{#1}} \def\boh#1{pt_{#1} \def\Boh#1{Pt_{#1}} \def\mah#1{al_{#1} \def\Mah#1{Al_{#1}} \def\pr#1{W_{#1}} \def\alpha{\alpha} \def\Gamma{\Gamma} \def\bar \Gamma{\bar \Gamma} \def\gamma{\gamma} \def\bar \eta{\bar \eta} \def{\cal Z}{{\cal Z}} \def{pt}{{pt}} \def\alpha{\alpha} \def{S^{(n)}}{{S^{(n)}}} \def{S^{(m)}}{{S^{(m)}}} \def\<{\langle} \def\>{\rangle} \def{\text{\rom{Hilb}}}{{\text{\rom{Hilb}}}} \def{\Cal W}{{\cal W}} \def{\hbox{\rom{Tor}}}{{\hbox{\rom{Tor}}}} \def{\hbox{\rom{Ext}}}{{\hbox{\rom{Ext}}}} \def\stil#1{\tilde S^#1} \def\pi{\pi} \defg{g} \def\phi{\varphi} \def\tilde{\tilde} \def{\bar\al}{{\bar\alpha}} \begin{document} \title[Variation of Donaldson invariants]{Variation of moduli spaces and Donaldson invariants under change of polarization} \author{Geir Ellingsrud} \address{Mathematical Institute\\University of Oslo\{\Bbb P}.~O.~Box~1053\\ N--0316 Oslo, Norway} \email{ellingsr@@math.uio.no} \keywords{Moduli spaces, Donalson invariants, Hilbert scheme of points} \author{Lothar G\"ottsche} \address{Max--Planck--Institut f\"ur Mathematik\\Gottfried--Claren--Stra\ss e 26\\ D-53225 Bonn, Germany} \email{lothar@@mpim-bonn.mpg.de} \maketitle\ \section{Introduction} Let $S$ be a smooth projective surface over the complex numbers and let $c_1\in H^2(S,{\Bbb Z})$ and $c_2\in H^4(S,{\Bbb Z})$ be two classes. For an ample divisor $H$ on $S$, one can study the moduli space $M_H(c_1,c_2)$ of $H$-semistable torsion-free sheaves $E$ on $S$ of rank $2$ with $c_1(E)=c_1$ and $c_2(E)=c_2$. We want to study the change of $M_H(c_1,c_2)$ under variation of $H$. It is known that the ample cone of $S$ has a chamber structure, and that $M_H(c_1,c_2)$ depends only on the chamber containing $H$. In this article we will try to understand how $M_H(c_1,c_2)$ changes, when $H$ passes through a wall separating two chambers. The set-theoretic changes of the subspace consisting of locally free sheaves and of $M_H(c_1,c_2)$ have been treated in \cite{Q1} and \cite{Go1} respectively. We show that the change of the moduli space when $H$ passes through a wall, can be expressed as a sequence of operations similar to a flip. In fact the moduli spaces at each step can be identified as moduli spaces of torsion-free sheaves with a suitable parabolic structure of length $1$. We assume that either the geometric genus $p_g(S)$ is $0$ or that $K_S$ is trivial. We shall also make suitable hypotheses on the wall, and walls fulfilling this condition we call good. This assumption is reasonably weak if the Kodaira dimension of $S$ is at most $0$, but gets stronger if e.g., $S$ is of general type. When the polarization passes through a good wall, each of the steps above is realized by a smooth blow-up along a projective bundle over a product of Hilbert schemes of points on $S$, followed by a smooth blow-down of the exceptional divisor in another direction. The change of moduli spaces can be viewed as a change of GIT quotients, treated in \cite{Th2} and \cite{D-H}. These results could in principle be applicable, although it would still take quite some work to do so. We have however chosen a more direct approach via elementary transforms of universal families, which is more in the spirit of \cite{Th1}, and which also immediately gives the change of the universal sheaves needed for the computation of Donaldson invariants. In the case that $K_S$ is trivial, i.e., $S$ is an abelian or a $K3$ surface, we see that the change of $M_H(c_1,c_2)$, when $H$ passes through a wall, is given by elementary transformations of symplectic varieties. In the case that $p_g(S)=q(S)=0$ we use these results in order to compute the change of the Donaldson polynomials under change of polarisation. The Donaldson polynomials of a $C^\infty$-manifold $M$ of dimension $4$ are defined using a Riemannian metric on $S$, but in case $b^+(M)>1$ they are known to be independent of the metric, as long as it is generic. In case $b_+(M)=1$, (which for an algebraic surface $S$ corresponds to $p_g(S)=0$), the invariants have been introduced and studied by Kotschick in \cite{Ko}. In \cite{K-M} Kotschick and Morgan show that the invariants only depend on the chamber of the period point of the metric in the positive cone of $H^2(M,{\Bbb R})$. They also compute the lowest order term of the change and conjecture the shape of a formula for the change. The case we are studying corresponds to $M$ being an algebraic surface $S$ with $p_g(S)=q(S)=0$ and a wall lying in the ample cone, in addition we assume that the wall is good. In a first step we compute the change of the Donaldson invariants in terms of natural cohomology classes on Hilbert schemes of points on $S$ and then we use some computations in the cohomology rings of these Hilbert schemes to determine the six lowest order terms of the change of the Donaldson invariants explicitly. The results are compatible with the conjecture of \cite{K-M} (which in particular predicts that three of the terms above are zero. Parallelly and independently similar results to ours have been obtained by other authors. Matsuki and Wentworth show in \cite{M-W} that the change of moduli spaces of torsion-free sheaves of arbitrary rank on a projective variety under change of polarisation can be described as a sequence of flips. In \cite{F-Q} Friedman and Qin obtain very similar results to ours. \section{Background material} In this paper let $S$ be a projective surface over ${\Bbb C}$. By the Neron-Severi group $NS(S)$ of $S$ we mean the group of divisors modulo homological equivalence, i.e., the image of $Div(S)$ in $H^2(S,{\Bbb Z})$ under the map sending the class of a divisor $D$ to its fundamental cycle $[D]$. Let $Div^0(S)$ be its kernel. Let $c_1\in H^2(S,{\Bbb Z})$ and $c_2\in H^4(S,{\Bbb Z})={\Bbb Z}$ be elements which will be fixed throughout the paper. Let $H$ be a polarization of $S$. As we mostly shall consider stability and semistability in the sense of Gieseker and Maruyama we shall write $H$-stable (resp. $H$-semistable) instead of Gieseker stable (resp. semistable) with respect to $H$ and $H$-slope stable (resp. $H$-slope semistable) instead of stable (resp. semistable) with respect to $H$ in the sense of Mumford-Takemoto. Denote by $M_H(c_1,c_2)$ the moduli space of H-semistable torsion-free sheaves $E$ on $S$ of rank $2$ with $c_1(E)=c_1$ and $c_2(E)=c_2$ and $M_H^s(c_1,c_2)$ the open subscheme of $M_H^s(c_1,c_2)$ of stable sheaves. Let $Spl(c_1,c_2)$ be the moduli space of simple torsion-free sheaves with $c_1(E)=c_1$ and $c_2(E)=c_2$ (see \cite{A-K}). \begin{nota} For a sheaf ${\cal F}$ on a scheme $X$ and a divisor $D$ let ${\cal F}(D):={\cal F}\otimes {\cal O}_X(D)$ . Many of our arguments will take place over products $S\times X$, where $X$ is a scheme. We shall denote by $p:S\times X\longrightarrow S$ and $q_X:S\times X\longrightarrow X$ the two projections and if there is no danger of confusion, we will drop the index $X$. For a divisor $D$ on $X$ we denote $D_S:=q_X^*(D)$. For a sheaf ${\cal F}$ on $S\times X$ and a divisor or divisor class, $D$ on $S$ we denote by ${\cal F}(D)$ the sheaf ${\cal F}(p^*(D))$. If $X$ is a smooth variety of dimension $n$, we denote the cup product of two elements $\alpha$ and $\beta$ in $H^*(X,{\Bbb Z})$ by $\alpha\cdot\beta$ and the degree of a class $\alpha\in H^{2n}(X,{\Bbb Z})$ by $\int_X\alpha$. For $\alpha,\beta\in H^2(S,{\Bbb Z})$ let $\<\alpha\cdot\beta\>:=\int_S\alpha\cdot\beta$. We write $\alpha^2$ for $\<\alpha\cdot\alpha\>$ and, for $\gamma\in H^2(S,{\Bbb Z})$, we put $\<\alpha,\gamma\>:=\<\alpha\cdot \check\gamma\>$, where $\check\gamma$ is the Poincar\'e dual of $\gamma$. \end{nota} \begin{conve} \label{convent} If $Y,X$ are schemes and there is a "canonical" map $f:X\longrightarrow Y$, then for a cohomology class $\alpha\in H^*(Y,{\Bbb Z})$ (resp. for a vector bundle $E$ on $Y$) we will very often also denote the pull-back via $f$ by $\alpha$ (resp. $E$). \end{conve} \begin{defn} \cite{OG2}\label{defpseudofam} Let $B$ be a scheme. A family of sheaves, ${\cal F}$, on $S$ parametrized by $B$ is a $B$-flat sheaf on $S\times B$. Two families of sheaves ${\cal F}$ and ${\cal G}$ on $S$ parametrized by $B$ are called equivalent if there exists an isomorphism ${\cal F}\simeq {\cal G}\otimes q_B^*M$, for some line bundle $M$ on $B$. Let $(B_j)_{j\in J}$ be an \'etale cover of $B$ by schemes. Assume that for each $j\in J$ there is a family ${\cal F}_j$ of sheaves on $X$ parametrized by $B_j$, and that for each pair $k,l\in J$ the pullbacks of ${\cal F}_k $ and ${\cal F}_l$ to $B_k\times_B B_l$ are equivalent. Then we will say that the above data defines a pseudo-family of sheaves on $S$ parametrized by $B$. We will denote it by ${\cal F}$. It is clear what is meant by a map of pseudo-families and by two pseudo-families being equivalent. \end{defn} The main reason to introduce pseudo-families is that the moduli space $M^s_H(c_1,c_2)$ does not always carry a universal family of sheaves, but there will always be a universal pseudo-family. By the universal property of $M^s_H(c_1,c_2)$ a pseudo family of $H$-stable torsion-free sheaves $E$ on $S$ with $c_1(E)=c_1$, $c_2(E)=c_2$ parametrized by $B$ gives rise to a morphism $B\longrightarrow M^s_H(c_1,c_2)$. \bigskip {\bf Walls and chambers for torsion-free sheaves} We now recall some results about walls and chambers from \cite{Q1}, \cite{Q2} and \cite{Go1}. \begin{defn}\label{defwall}(for the first part see \cite{Q1} Def I.2.1.5) Let $C_S$ be the ample cone in $NS(S)\otimes {\Bbb R}$. For $\xi\in NS(S)$ let $$W^\xi:=C_S\cap\big \{ x\in NS(S)\otimes{\Bbb R} \bigm| \<x\cdot\xi\>=0\big\}.$$ We shall call $W^\xi$ a wall of type $(c_1,c_2)$, and say that it is defined by $\xi$ if the following conditions are satisfied: \begin{enumerate} \item $\xi+c_1$ is divisible by $2$ in $NS(S)$, \item $c_1^2-4c_2\le \xi^2<0$, \item there is a polarisation $H$ with $\< H\cdot \xi\>=0$. \end{enumerate} In particular $d_\xi:= (4c_2-c_1^2+\xi^2)/4$ is a nonnegative integer. An ample divisor $H$ is said to lie in the wall $W$ if $[H]\in W$. If $D$ is a divisor with $[D]=\xi$, we will also say that $D$ defines the wall $W$. A {\it chamber} of type $(c_1,c_2)$ or simply a chamber, is a connected component of the complement of the union of all the walls of type $(c_1,c_2)$. Two different chambers will be said to be {\it neighbouring chambers} if the intersection of their closures contains a nonempty open subset of a wall. We will call a wall $W$ {\it good}, if $D+K_S$ is not effective for any divisor $D$ defining the wall $W$. \end{defn} If $D$ defines a wall, then neither $D$ nor $-D$ can be effective because $D$ is orthogonal to an ample divisor. In particular every wall will be good if $-K_S$ is effective or if $[K_S]$ is a torsion class. More generally, a wall $W$ will be good if there exists an ample divisor $H$ in $W$ with $\<K_S \cdot H\>\le 0$. \begin{defn}\label{defenm} Let ${\text{\rom{Hilb}}}^l(S)$ be the Hilbert scheme of subschemes of length $l$ on $S$. For $\alpha\in NS(S)$ and $l\in {\Bbb Z}$, let $M(1,\alpha,l)$ be the moduli space of rank $1$ torsion-free sheaves ${\cal I}_{Z}(F)$ on $S$ with $c_1({\cal I}_{Z}(F))=[F]=\alpha$, $c_2(F)=length(Z)=l$. Let $$T^{n,m}_\xi:=\coprod_{2\alpha=c_1+\xi} M(1,\alpha,n)\times M(1,c_1-\alpha,m).$$ Let $N_2 \subset NS(S)$ be the subgroup of $2$-torsion elements. There is a (noncanonical) isomorphism $$T^{n,m}_\xi\simeq N_2\times{\text{\rom{Hilb}}}^n S\times Div^0(S)\times {\text{\rom{Hilb}}}^mS\times Div^0(S),$$ which depends on the choice of an $\alpha\in NS(S)$ with $2\alpha=c_1+\xi$ and on a representative $F$ in $Div(S)$ for $\alpha$. For any extension \begin{equation*}\tag{$\epsilon$}0\longrightarrow A_1\longrightarrow E\longrightarrow A_2\longrightarrow 0\end{equation*} where $A_1$ and $A_2$ are torsion-free rank one sheaves, we define $\Delta(\epsilon):=\chi(A_1)-\chi(A_2)$. Then if $\alpha=c_1(A_1)-c_1(A_2)$, the Riemann-Roch theorem gives $\Delta(\epsilon)=1/2\<(c_1(E)-K_S)\cdot\alpha\>+c_2(A_2)-c_2(A_1)$. Furthermore for any divisor $D$ we have $\Delta(\epsilon(D))=\Delta(\epsilon)+\<\alpha\cdot D\>$, where $ \epsilon(D)$ denotes the extension $\epsilon$ twisted by the line bundle ${\cal O}(D)$. This follows immediately from the fact that $c_1(E(D))=c_1(E)+2[D]$. Assume that $\xi$ defines a wall of type $(c_1,c_2)$, and that $n$ and $m$ are nonnegative integers with $n+m=d_\xi=c_2-(c_1^2-\xi^2)/4$. Let ${\text{\rom{\bf E}}}^{n,m}_\xi$ be the set of sheaves lying in nontrivial extensions \begin{eqnarray}\label{splitting} &&0\longrightarrow {\cal I}_{Z_1}(F_1)\longrightarrow E\longrightarrow {\cal I}_{Z_2}(F_2)\longrightarrow 0\end{eqnarray} where $({\cal I}_{Z_1}(F_1),{\cal I}_{Z_2}(F_2))$ runs through $T^{n,m}_\xi$. It is easy to see that every sheaf $E\in {{\text{\rom{\bf E}}}}^{n,m}_\xi$ is simple (\cite{Go1}, lemma 2.3). Let $$V^{n,m}_\xi:={\text{\rom{\bf E}}}^{n,m}_\xi\setminus\Big(\bigcup_{l,s} {\text{\rom{\bf E}}}_{-\xi}^{l,s}\Big).$$ \end{defn} \begin{nota}\label{defaplus} Assume that $H_+$ and $H_-$ are ample divisors lying in neighbouring chambers separated by the wall $W$. Then we define $$A^+(W):=\Big\{\xi\in NS(S)\Bigm| \xi \text{ defines } W \text{ and } \ \<\xi\cdot H_+\> >0\Big\}$$ and $A^-(W):=-A^+(W)$.\end{nota} The following proposition mostly comprizes some of the results of \cite{Go1}, that are generalizations of the corresponding results of \cite{Q1}, \cite{Q2} and will be important for the rest of the paper. Note that unlike \cite{Go1} we assume walls to be defined by classes in $NS(S)$ and not by numerical equivalence classes, and that we look at moduli spaces with fixed first Chern class and not with fixed determinant. The proofs in \cite{Go1} stay however valid with very few changes. \begin{prop} \label{wall} \begin{enumerate} \item For $H$ not lying on a wall, $M_H(c_1,c_2)\setminus M_H^s(c_1,c_2)$ is independent of $H$ and $M_H(c_1,c_2)$ depends only on the chamber of $H$. \noindent For the rest of the proposition we assume that we are in the situation of \ref{defaplus} and that $\xi\in A^+(W)$. \item Every $E\in {\text{\rom{\bf E}}}^{n,m}_\xi$ is $H_+$ slope-unstable and the sequence (\ref{splitting}) is its Harder-Narasimhan filtration with respect to $H_+$. \item ${\hbox{\rom{Hom}}}({\cal I}_{Z_1}(F),E)={\Bbb C}$. Thus, for $E\in {\text{\rom{\bf E}}}^{n,m}_\xi$, the sequence \ref{splitting} is the unique extension $$0\longrightarrow {\cal I}_{W_1}(F_1)\longrightarrow E\longrightarrow {\cal I}_{W_2}(G_2) \longrightarrow 0$$ with $\<(2F_1-c_1)\cdot H_+\> >0$. \item In particular we see that, for $\xi,\eta\in A^+(W)$, the subsets ${\text{\rom{\bf E}}}_\xi^{n,m}$, ${\text{\rom{\bf E}}}_{\eta}^{k,l}$ of $Spl(c_1,c_2)$ do not intersect, unless $\xi=\eta$ and $(n,m)=(k,l)$. \item If $E\in {\text{\rom{\bf E}}}_\xi^{n,m}$ then $E$ is $H_-$-slope stable if and only if $E\in V_\xi^{n,m}$ and $H_-$-slope unstable otherwise. \item On the other hand let $E$ be a torsion-free sheaf with $c_1(E)=c_1$ and $c_2(E)=c_2$, which is $H_-$-semistable and $H_+$-unstable. Then $E$ is $H_-$-slope stable and $E\in {{\text{\rom{\bf E}}}}^{n,m}_\xi$ for suitable numbers $n$ and $m$ and $\xi\in A^+(W)$. \end{enumerate} \end{prop} \begin{pf} (1) is (\cite{Go1}, theorem 2.9(1)). (2) is easy. (3) follows from (2) and (\cite{Go1}, lemma 2.3). (4) follows from (2). (5) is (\cite{Go1}, prop 2.5). (6) is (\cite{Go1}, lemma 2.2). \end{pf} \section{Parabolic structures and the passage through a wall} As mentioned in the previous section, $M_H(c_1,c_2)$ depends only on the chamber to which $H$ belongs. If $H'$ lies in a neighbouring chamber to $H$ the moduli space $M_{H'}(c_1,c_2)$ will in most cases be birational to $M_H(c_1,c_2)$, although new components do occur in some cases. If the wall separating the two chambers is good, we will describe the birational transformation in detail by giving an explicit sequence of blow-ups and blow-downs with smooth centers which are known. If the wall is good, but the transformation is not birational, our arguments give a description of the components which are added to or deleted from the moduli space. For the rest of the paper we will assume that $H_+$ and $H_-$ are ample divisors lying in neighbouring chambers separated by the wall $W$, and that $H$ is an ample divisor in the wall $W$ which lies in the closure of the chambers containing $H_-$ and $H_+$ respectively and which does not lie in any other wall. Furthermore we shall assume that $M=H_+-H_-$ is effective. By replacing $H_+$ by a high multiple if necessary, we can always achieve this. Our aim is to divide the passage through a wall into a number of smaller steps. To this purpose we will introduce a finer notion of stability. The starting point is the observation that unlike slope stability, Gieseker stability is not invariant under tensorization by a line bundle. \begin{lem}\label{tensor} There is a positive integer $n_0$ such that for all $l\ge n_0$ and all torsion-free rank $2$ sheaves $E$ on $S$ with $c_1(E)=c_1$, $c_2(E)=c_2$ \begin{enumerate} \item $E$ is $H_-$-stable (resp. semistable) if and only if $E(-lM)$ is $H$-stable (resp. semistable). \item $E$ is $H_+$-stable (resp. semistable) if and only if $E(lM)$ is $H$-stable (resp. semistable). \end{enumerate}\end{lem} \begin{pf} It will be enough to show (1). As $H_-$ does not lie on a wall, it is easy to see that $E$ is $H_-$-(semi)stable if and only if $E(M)$ is. Also there are only finitely many $\xi\in NS(S)$ defining the wall $W$. Therefore lemma \ref{tensor} follows immediately from lemma \ref{tensor1} and lemma \ref{tensor2} below.\end{pf} \begin{lem}\label{tensor1} \begin{enumerate} \item Assume $E$ is $H_-$-semistable but $H$-unstable. Then $E\in {{\text{\rom{\bf E}}}}^{n,m}_\xi$ for suitable $n,m$ and $\xi\in A^+(W)$. \item Assume $E$ is $H_-$-unstable but $H$-semistable. Then $E\in {{\text{\rom{\bf E}}}}^{n,m}_{-\xi}$ for suitable $n,m$ and $\xi\in A^+(W)$. \end{enumerate}\end{lem} \begin{pf} We just prove (1), the proof of (2) being analoguous. By assumption there is an extension \begin{equation*}\tag{$\epsilon$}0\longrightarrow {\cal I}_{Z_1}(F_1)\longrightarrow E\longrightarrow {\cal I}_{Z_2}(F_2)\longrightarrow 0\end{equation*} with $\Delta(\epsilon(lH_-))\le 0$ and $\Delta(\epsilon(lH))>0$ for $l>>0$. In particular we have $\<\eta \cdot H_-\>\le 0\le\< \eta \cdot H\>$ where $\eta:=2[F_1]-c_1$. If $0<\< \eta\cdot H\>$, there would be a wall separating $H_-$ and $H$. So $\< \eta\cdot H\>=0$, and unless $\eta$ is a torsion class, it defines a wall in which $H$ lies. As $H$ lies in a unique wall it must be $W$. Hence $\eta\in A^+(W)$, and $E\in {{\text{\rom{\bf E}}}}^{n,m}_{\eta}$. Assume that $\eta$ is a torsion class. Then $F_1$ and $F_2$ are numerically equivalent, and it is easily verified that \begin{equation*} \Delta(\epsilon(lH_-))=\Delta(\epsilon(lH)) \end{equation*} which is a contradicition.\end{pf} \begin{lem}\label{tensor2} Given $n,m,\xi$. Then there exists an integer $k_0$ such that for all $k>k_0$ and all $E\in {{\text{\rom{\bf E}}}}^{n,m}_\xi$, the sheaf $E(-k M)$ is $H$-stable if and only $E$ is $H_-$-slope stable. Otherwise $E$ is both $H_-$-slope unstable and $H$-unstable. \end{lem} \begin{pf} Let $E\in E^{n,m}_\xi$. Then there is an extension \begin{equation*}\label{split2}\tag{$\epsilon_1$}0\longrightarrow {\cal I}_{Z_1}(F_1)\longrightarrow E\longrightarrow {\cal I}_{Z_2}(F_2)\longrightarrow 0\end{equation*} with $\xi=2[F_1]-c_1$. Assume first that $E$ is $H_-$-slope-stable and let \begin{equation*}\tag{$\epsilon_2$}0\longrightarrow {\cal I}_{Y_1}(G_1)\longrightarrow E\longrightarrow {\cal I}_{Y_2}(G_2)\longrightarrow 0\end{equation*} be another extension. Put $\eta:=2[G_1]-c_1$. As $E$ is $H_-$-slope stable we have $\<\eta\cdot H_-\>< 0$, and because there is no wall between $H_-$ and $H$, we know that $\<\eta\cdot H\>\le 0$. For any integers $k$ and $l$ $$\Delta(\epsilon_2(-kM+lH))=\Delta(\epsilon_2)-k\<\eta\cdot M\>+l\<\eta\cdot H\>.$$ Hence if $\<\eta\cdot H\><0$ for all extensions $\epsilon_2$ above, then $E(-k M)$ will be $H$-stable for any $k$. Assume that $\<\eta\cdot H\>=0$. By assumption $H$ is contained in a single wall $W$, so necessarily $\eta\in A^+(W)$. Hence by proposition \ref{wall}(3), we get ${\cal I}_{Z_1}(F_1)={\cal I}_{Y_1}(G_1)$. Therefore it suffices to see that for $k>>0$ and any $l$ we have the inequality $$\Delta(\epsilon_1(-kM+lH))<0.$$ Now $$\Delta(\epsilon_1(-kM+lH))=\Delta(\epsilon_1)-k\<\eta\cdot M\>,$$ which is negative for $k>>0$ as $\<\eta\cdot M\>>0$. To prove the converse, assume that $E$ is not $H_-$-slope stable. Then by proposition \ref{wall}(5) there is an extension \begin{equation*}\tag{$\epsilon_3$}0\longrightarrow {\cal I}_{Y_1}(F_2)\longrightarrow E\longrightarrow {\cal I}_{Y_2}(F_1)\longrightarrow 0.\end{equation*} Because $2[F_2]-c_1=-\xi$ we have $$\Delta(\epsilon_3(-kM+lH))=\Delta(\epsilon_3)+\<-\xi\cdot (-kM+lH)\>= \Delta(\epsilon_3)+k\<\xi\cdot M\>>0$$ for $k>>0$\end{pf} {}From now on until the end of this section we fix $n_0$ as in lemma \ref{tensor}, and we put $C:= (n_0+1)M$. \begin{defn}\label{defalstable} Let $a$ be a real number between $0$ and $1$. For any torsion-free sheaf $E$ we define $$P_a(E)=((1-a)\chi(E(-C))+a\chi(E(C)))/rk(E).$$ A torsion-free sheaf $E$ on $S$ is called $a$-semistable if and only if every subsheaf $E'\subset E$ satisfies $P_a(E'(lH))\le P_a(E(lH))$ for all $l>>0,$ and it is called $a$-stable if strict inequality holds. \end{defn} In particular, by lemma \ref{tensor}, $E$ is $0$-semistable if and only if it is $H_-$-semistable, and it is $1$-semistable if and only if it is $H_+$-semistable. For any extension \begin{equation*}\tag{$\epsilon$}0\longrightarrow A_1\longrightarrow E\longrightarrow A_2\longrightarrow 0\end{equation*} we define $\Delta_a(\epsilon):=P_a(A_1)-P_a(A_2)$. Then $\Delta_a(\epsilon)=\Delta(\epsilon)+(2a-1)\<C\cdot\alpha\>$ where $\alpha=c_1(A_1)-c_1(A_2)$. Clearly $\Delta_a(\epsilon(D))=\Delta_a(\epsilon)+\<D\cdot\alpha\>$ for any divisor $D$. A sheaf $E$ is $a$-stable (resp. $a$-semistable) if $\Delta_a(\epsilon(lH))<0$ (resp. $\le0$) for all $l>>0$ and for any extension $\epsilon$ whose middle term is $E$. \begin{rem} It is easy to see that $P_a(E(lH))$ is the parabolic Hilbert polynomial of the parabolic bundle $(E(C),E(-C),a)$, (i.e. with a filtration of length $1$). Therefore $E$ is $a$-semistable if and only if $(E(C),E(-C),a)$ is semistable. In \cite{Ma-Yo} a coarse quasiprojective moduli space of stable parabolic sheaves with fixed Hilbert polynomial is constructed, and by \cite{Yo} there exists a projective coarse moduli space for $S$-equivalence classes of semistable parabolic sheaves. In particular there exists a coarse moduli space $M_{a}(c_1,c_2)$ for $a$-semistable sheaves $E$ on $S$ with $c_1(E)=c_1$ and $c_2E=c_2$. We denote by $M_{a}^s(c_1,c_2)$ its open subscheme of stable sheaves. \end{rem} \begin{rem} We see that $M_{H_-}(c_1,c_2)$ and $M_0(c_1,c_2)$ respectively $M_{H_+}(c_1,c_2)$ and $M_1(c_1,c_2)$ are coarse moduli schemes for the same functor and therefore they are isomorphic. \end{rem} \begin{rem} The same proof as in the case of $H$-stable sheaves shows that $M_{a}^s(c_1,c_2)$ carries a universal pseudofamily. One checks easily that every $E\in M^s_{a}(c_1,c_2)$ is simple. As $M^s_{a}(c_1,c_2)$ and $Spl(c_1,c_2)$ both carry universal pseudofamilies, ${\cal V}$ and ${\Cal W}$ respectively, there exists a morphism $f:M^s_{a}(c_1,c_2)\to Spl(c_1,c_2)$ such that $({\hbox{\rom{id}}}_S\times f)^*({\Cal W})={\cal V}$. Let $M$ be its image. By the same argument there exists a map $g:M\to M^s_{a}(c_1,c_2)$, with $({\hbox{\rom{id}}}_S\times f)^*({\hbox{\rom{id}}}_S\times g)^*({\cal V})={\cal V}$. Hence $f$ is an open embedding. In particular and what is the most important thing for us, the tangent space to $M^s_{a}(c_1,c_2)$ at a point $E$ is ${\hbox{\rom{Ext}}}^1(E,E)$. \end{rem} \begin{defn}\label{defminiwall} For all $a\in [0,1]$ let $A^+(a)$ be the set of $(\xi,n,m)\in A^+(W)\times {\Bbb Z}_{\ge 0}^2$ satisfying \begin{eqnarray}\label{minicond} n+m&=& c_2-(c_1^2-\xi^2)/4,\\ n-m&=& \<\xi\cdot (c_1-K_S)\>/2 +(2a-1)\<\xi\cdot[C]\>.\end{eqnarray} A number $a$ is called a {\it miniwall} if $A^+(a)\ne \emptyset$. A {\it minichamber} is a connected component of the complement of the set of all miniwalls in $[0,1]$. It is clear that there are finitely many minichambers. Two minichambers are called neighbouring minichambers if their closures intersect. \end{defn} \begin{rem}Note that $A^+(a)$ is the set of all $\xi,n,m$ with $\xi\in A^+(W)$ for which there exists a (possibly split) extension \begin{equation*}\tag{$\epsilon$}0\longrightarrow A_1\longrightarrow E\longrightarrow A_2\longrightarrow 0\end{equation*} with $\xi=c_1(A_1)-c_1(A_2)$, $n=c_2(A_1)$, $m=c_2(A_2)$ and $\Delta_a(\epsilon)=0$. \end{rem} \begin{lem} Let $0\le a_-< a_+\le 1$ and assume that neither $a_-$ nor $a_+$ is a miniwall. Let $E$ be $a_-$-semistable and $a_+$-unstable. Then there exists a miniwall $a$ between $a_-$ and $a_+$ and an element $(\xi,n,m)\in A^+(a)$, such that $E\in {{\text{\rom{\bf E}}}}_{\xi}^{n,m}$. \end{lem} \begin{pf} By assumption $E$ is $a_+$-unstable. Hence there is an extension \begin{equation*}\tag{$\epsilon$}0\longrightarrow A_1\longrightarrow E\longrightarrow A_2\longrightarrow 0\end{equation*} such that for all $l>>0$ we have $\Delta_{a_+}(\epsilon(lH))>0$. Putting $\xi:=c_1(A_1)-c_1(A_2)$ and using that $E$ is $a_-$-semistable, we obtain the following inequalities valid for all $l>>0$ $$\Delta_{a_-}(\epsilon(lH))=\Delta_{a_-}(\epsilon)+l\<H\cdot\xi\>\le0< \Delta_{a_+}(\epsilon(lH))=\Delta_{a_+}(\epsilon)+l\<H\cdot\xi\>.$$ In particular $\<H\cdot\xi\>=0$ and $\Delta_{a_-}(\epsilon)<0<\Delta_{a_+}(\epsilon)$. Furthermore $\xi$ is not a torsion class and $\xi$ defines a wall on which $H$ is lying, which therefore must be $W$. There clearly is an $a$ such that $\Delta_a(\epsilon)=0$. \end{pf} \begin{lem} \label{stab} Let $a_-<a_+$ be in neighbouring minichambers separated by the miniwall $a$. Let $(\xi,n,m)\in A^+(a)$. \begin{enumerate} \item Any $E\in {{\text{\rom{\bf E}}}}_\xi^{n,m}$ is $a_-$-stable, strictly $a$-semistable and $b$-unstable for all $b>a$. \item Any $E\in {{\text{\rom{\bf E}}}}_{-\xi}^{m,n}$ is $a_+$-stable, strictly $a$-semistable and $b$-unstable for all $b<a$.\end{enumerate} \end{lem} \begin{pf} By symmetry it is enough to show (1). Let $E\in {{\text{\rom{\bf E}}}}_\xi^{n,m}$. Then $E$ is given by an extension \begin{equation*}\tag{$\epsilon$}0\longrightarrow {\cal I}_{Z_1}(F_1)\longrightarrow E\longrightarrow {\cal I}_{Z_2}(F_2)\longrightarrow 0\end{equation*} with $\xi=2[F_1]-c_1$ and $length(Z_1)=n$, $length(Z_2)=m$. Now if $b>a$ we have $\Delta_b(\epsilon(lH))=\Delta_a(\epsilon(lH))+2(b-a)\<C\cdot\xi\>=2(b-a)\<C \cdot\xi\>>0$ since $\Delta_a(\epsilon)=0$ and $\<C\cdot\xi\>>0$. Thus $E$ is $b$-unstable. Assume that $E$ is not $a_-$-stable. Then it lies in an extension \begin{equation*}\tag{$\epsilon_1$}0\longrightarrow {\cal I}_{Y_1}(G_1)\longrightarrow E\longrightarrow {\cal I}_{Y_2}(G_2)\longrightarrow 0,\end{equation*} for which $\Delta_{a_-}(\epsilon_1(lH))\ge 0>\Delta_{a}(\epsilon_1(lH))$ for $l>>0$. Hence we obtain $\<(2G_1-c_1)\cdot H\>\ge \<(2F_1-c_1)\cdot H\>$ and $P_{a_-}({\cal I}_{Z_1}(F_1+lH))<P_{a_-}({\cal I}_{Y_1}(G_1+lH))$ and thus $\chi({\cal I}_{Z_1}(F_1+lH-C))<\chi({\cal I}_{Y_1}(G_1+lH-C))$ or $\chi({\cal I}_{Z_1}(F_1+lH+C))<\chi({\cal I}_{Y_1}(G_1+lH+C))$. Consequently ${\hbox{\rom{Hom}}}({\cal I}_{Y_1}(G_1),{\cal I}_{Z_1}(F_1))=0$ and the obvious map ${\cal I}_{Y_1}(G_1)\longrightarrow {\cal I}_{Z_2}(F_2)$ is an injection. Hence $F_2-G_1$ is effective. If $F_2\ne G_1$, we would have $\<(G_1-F_2)\cdot H\><0$, and, by $\<\xi \cdot H\>=0$, we would get the contradiction $\<(2G_1-c_1)\cdot H\><0$. So $G_1=F_2$. By the injectivity of ${\cal I}_{Y_1}(G_1)\longrightarrow {\cal I}_{Z_2}(F_2)$ and the fact that \ref{splitting} is not split, we get $length(Z_2)<length(Y_1)$ which shoes that $E$ is $a_-$-stable. A similar argument shows that $E$ is strictly $a$-semistable. \end{pf} \begin{rem}\label{stab1} We can also easily see from the above arguments that in the situation of \ref{stab} any sheaf $E\in M_{a_-}(c_1,c_2)$, which does not lie in any ${\text{\rom{\bf E}}}_{\xi}^{n,m}$ for $(\xi,n,m)\in A^+(a)$ is $a_-$-stable (resp. semistable) if and only if it is $a$-stable (resp. semistable). \end{rem} \begin{rem}\label{fine} \begin{enumerate} \item Looking at the proof of \cite{Ma2} for the sufficient criterion for the existence of a universal family on $M_H(c_1,c_2)$, we see that the same proof also works for $M_{a}(c_1,c_2)$ and we get the same criterion, i.e. if $c_1$ is not divisible by $2$ in $NS(S)$ or otherwise $4c_2-c_2^2$ is not divisible by $8$ and $M_a(c_1,c_2)= M_a^s(c_1,c_2)$, then $M^s_a(c_1,c_2)$ carries a universal family. \item From the results obtained so far it follows easily that, under the above conditions for the Chern classes, $M_a(c_1,c_2)=M^s_a(c_1,c_2)$ if and only if $a$ is not a miniwall. \end{enumerate} \end{rem} \begin{prop}\label{flip} \begin{enumerate} \item $M_0(c_1,c_2)=M_{H_-}(c_1,c_2)$ and $M_1(c_1,c_2)=M_{H_+}(c_1,c_2)$. \item If $b\in [0,1]$ is not on a miniwall, the moduli space $M_b(c_1,c_2)$ depends only on the minichamber in which $b$ is lying, and $M_b(c_1,c_2)\setminus M^s_b(c_1,c_2)$ is independent of $b$. \item Let $a_-<a_+$ be in neighbouring minichambers separated by the miniwall $a$. Then we have a set-theoretical decomposition $$M_{a_+}(c_1,c_2)=\left(M_{a_-}(c_1,c_2)\setminus \coprod_{(\xi,n,m)\in A^+(a)} {\text{\rom{\bf E}}}^{n,m}_{\xi} \right){\sqcup} \left(\coprod_{(\xi,n,m)\in A^+(a)} {\text{\rom{\bf E}}}^{m,n}_{-\xi}\right),$$ and there are morphisms $$\matrix M_{a_-}(c_1,c_2)&&&&M_{a_+}(c_1,c_2)\cr &\mapse{\psi_-}&&\mapsw{\psi_+}\cr &&M_{a}(c_1,c_2)\cr\endmatrix$$ which are open embeddings over $$M_{a_-}(c_1,c_2)\setminus \coprod_{(\xi,n,m)\in A^+(a)} {\text{\rom{\bf E}}}^{n,m}_{\xi} \text{ and }\ \ M_{a_+}(c_1,c_2)\setminus \coprod_{(\xi,n,m)\in A^+(a)} {\text{\rom{\bf E}}}^{m,n}_{-\xi}.$$ \end{enumerate}\end{prop} \begin{pf} (1), (2), (3) follow by putting together the results of this section. By lemma \ref{stab} all the points of $M_{a_-}(c_1,c_2)$ and $M_{a_+}(c_1,c_2)$ are $a$-semistable and hence we get the morphisms $\psi_-$ and $\psi_+$. The statement that they be open embeddings over the indicated open subsets, follows from remark \ref{stab1}.\end{pf} \section{The normal bundles of the exceptional sets} Our aim in this and the next chapter is to describe the passage through a miniwall which corresponds to a good wall. We keep the assumptions from the beginning of the previous section. In addition to those we assume that either $p_g(S)=0$ or $K_S$ is trivial, and that the wall $W$ is good. Let $a$ define a miniwall and let $(\xi,n,m)\in A^+(a)$. Let $a_-<a_+$ lie in neighbouring minichambers separated by $a$. For simplicity of notation we shall assume that $A^+(a)=\{(\xi,n,m)\}$. Because, for $(\xi,n_1,m_1)$, $ (\xi_2,n_2,m_2)$ distinct elements of $A^+(a)$, the sets ${\text{\rom{\bf E}}}_{\xi_1}^{n_1,m_1}$ and ${\text{\rom{\bf E}}}_{\xi_2}^{n_2,m_2}$ are disjoint by proposition \ref{wall} and our arguments are local in a neighbourhood of each ${\text{\rom{\bf E}}}_{\eta}^{l,s}$, this assumption can be made without loss of generality. Furthermore we assume for simplicity of notation that $NS(S)$ has no $2$-torsion. Then the classes $(c_1+\xi)/2$, $(c_1-\xi)/2\in NS(S)$ are well-defined and $T_\xi^{n,m}=M(1,(c_1+\xi)/2,n)\times M(1,(c_1-\xi)/2,m)$. Again this assumption is not important, as otherwise the components of $E_{\xi}^{n,m}$ and $E_{-\xi}^{m,n}$ are disjoint. \begin{nota} \label{notsec4} We shall write $M_-:=M_{a_-}(c_1,c_2)$, $M_+:=M_{a_+}(c_1,c_2)$, $M^s_-:=M^s_{a_-}(c_1,c_2)$, $M^s_+:=M^s_{a_+}(c_1,c_2)$ and put ${\text{\rom{\bf E}}}_-:={\text{\rom{\bf E}}}_\xi^{n,m}$ and ${\text{\rom{\bf E}}}_+:={\text{\rom{\bf E}}}^{n,m}_{-\xi}$. \end{nota} \begin{defn} Let $ {\cal F}_1'$ (resp. $ {\cal F}_2'$) be the pull-back of a universal sheaf over $S\times M(1,(c_1+\xi)/2,n)$ (resp. $S\times M(1,(c_1-\xi)/2,m)$) to $S\times T$, where $T:= M(1,(c_1+\xi)/2,n)\times M(1,(c_1-\xi)/2,m)$. Let $q=q_T:S\times T\to T$ be the projection. Let ${\cal A}_{-}':={\hbox{\rom{Ext}}}_{q}^1({\cal F}_2',{\cal F}_1')$ and ${\cal A}_{+}':={\hbox{\rom{Ext}}}_{q}^1({\cal F}_1',{\cal F}_2')$ and ${\Bbb P}_{-}:={\Bbb P}({\cal A}_{-}'),$ ${\Bbb P}_{+}:={\Bbb P}({\cal A}_{+}').$ Let $\pi_-$ (resp. $\pi_+$) be the projections of ${\Bbb P}_-$ (resp. ${\Bbb P}_+$) to $T$ and $\tau_-$ (resp. $\tau_+$) the tautological sublinebundles of ${\cal A}_-:=\pi_-^*({\cal A}_-')$ (resp. ${\cal A}_+:=\pi_+^*({\cal A}_+')$). Let ${\cal F}_1:=({\hbox{\rom{id}}}_S\times\pi_-)^*{\cal F}_1'$ and ${\cal F}_2:=({\hbox{\rom{id}}}_S\times \pi_-)^*{\cal F}_2'$. \end{defn} \begin{lem}\label{enm} \begin{enumerate} \item $ {\cal A}_-'$ is locally free of rank $-\xi(\xi-K_S)/2 +n+m-\chi({\cal O}_S)$ and its formation commutes with arbitrary base change. \item There is an isomorphism ${\hbox{\rom{Ext}}}^1({\cal F}_2',{\cal F}_1')\longrightarrow H^0(T,{\cal A}_-')$, hence over $S\times {\Bbb P}_-$ there is a tautological extension \begin{eqnarray}\label{globext}&& 0\longrightarrow {\cal F}_1\longrightarrow {{\cal E}}\to {\cal F}_2(\tau_-)\longrightarrow 0.\end{eqnarray} There is a morphism $i_-:{\Bbb P}_-\longrightarrow M_-$ with image ${\text{\rom{\bf E}}}_-$. \end{enumerate} \end{lem} \begin{pf}As $\xi$ defines a wall, ${\hbox{\rom{Hom}}}_{q}({\cal F}_2,{\cal F}_1)$ is fibrewise $0$, and, as the wall is good, $F_1-F_2+K_S$ is not effective for $(F_1,F_2)\in T$, therefore by Serre duality for the extension groups \cite{Mu2} also ${\hbox{\rom{Ext}}}^2_{q}({\cal F}_2,{\cal F}_1)=0$. So (1) follows by Riemann-Roch for the extension groups \cite{Mu2}. Now we apply \cite{La}. \end{pf} \begin{prop}\label{normal} \begin{enumerate} \item If $p_g(S)=0$ or if $K_S$ is trivial, then $i_-:{\Bbb P}_-\longrightarrow M_-$ is a closed embedding and $M_-$ is smooth along ${\text{\rom{\bf E}}}_-$. The irreducible component of $M_-$ containing ${\text{\rom{\bf E}}}_-$ has the expected dimension. \item If $p_g(S)=0$, then the normal bundle $N_{{\text{\rom{\bf E}}}_-/M_-}$ of ${\text{\rom{\bf E}}}_-$ in $M_-$ is equal to ${\cal A}_+(\tau_-)$. \item If $K_S$ is trivial, then $N_{{\text{\rom{\bf E}}}_-/M_-}=Q^\vee(\tau_-)$, where $Q$ is the universal quotient bundle on ${\Bbb P}_-={\Bbb P}({\cal A}_-)$. \end{enumerate} \end{prop} \begin{pf} By proposition \ref{wall}(3) and lemma \ref{enm} the map ${\Bbb P}_-\longrightarrow M_-$ is injective with image ${\text{\rom{\bf E}}}_-$. We also see by proposition \ref{wall} that ${\text{\rom{\bf E}}}_-\subset M_-^s$. In case $K_S$ is trivial, $Spl(c_1,c_2)$ and thus also the open subscheme $M_-^s$ are smooth by \cite{Mu1}. In order to see that $M_-$ is smooth along ${\text{\rom{\bf E}}}_-$ in the case $p_g(S)=0$, we have to show that ${\hbox{\rom{Ext}}}^2(E,E)=0$ for any $E\in {{\text{\rom{\bf E}}}}_-$. So let $E\in {{\text{\rom{\bf E}}}}_-$ be given by a nontrivial extension (\ref{splitting}) \begin{equation*} \tag{$\epsilon$} 0\longrightarrow {\cal I}_{Z_1}(F_1)\longrightarrow E\longrightarrow {\cal I}_{Z_2}(F_2)\longrightarrow 0.\end{equation*} As the wall $W$ is good, we obtain by Serre duality and the fact that $p_g(S)=0$ that ${\hbox{\rom{Ext}}}^2({\cal I}_{Z_i}(F_i),{\cal I}_{Z_j}(F_j))=0$ for $i=1,2$ and $j=1,2$. Hence applying ${\hbox{\rom{Ext}}}^2({\cal I}_{Z_i}(F_i),\cdot)$ to $(\epsilon)$ we get ${\hbox{\rom{Ext}}}^2({\cal I}_{Z_i}(F_i),E)=0$ for $i=1,2$ and this in turn shows that ${\hbox{\rom{Ext}}}^2(E,E)=0$. We now want to compute the normal bundle to ${\text{\rom{\bf E}}}_-$. {\it First Case: }$p_g(S)=0$. Applying ${\hbox{\rom{Hom}}}_{q}(\cdot,\cdot)$ on both sides of the sequence (\ref{globext}) and denoting by $\pi_i$ the composition of $\pi_-$ with the projection to the $i^{th}$ factor we get the following exact diagram of locally free sheaves on ${\Bbb P}_-$ \begin{eqnarray}\label{globdiag}&&\matrix &&0&&0&&0\cr &&\mapd{}&&\mapd{}&&\mapd{}\cr 0&\mapr{}&T_{{\Bbb P}_-/T}&\mapr{}&{\hbox{\rom{Ext}}}^1_{q}({\cal F}_2(\tau_-),{\cal E})&\mapr{}& \pi_2^*T_{M(1,(c_1-\xi)/2,m)}&\longrightarrow& 0\cr &&\mapd{}&&\mapd{\psi}&&\mapd{}\cr 0&\mapr{}&{\hbox{\rom{Ext}}}^1_{q}({\cal E},{\cal F}_1)&\mapr{\varphi}&i_-^*(T_{M_-})&\mapr{}& {\hbox{\rom{Ext}}}^1_{q}({\cal E},{\cal F}_2)(\tau_-)&\longrightarrow& 0\cr &&\mapd{}&&\mapd{}&&\mapd{}\cr 0&\mapr{}&\pi_1^*T_{M(1,(c_1+\xi)/2,n)}&\mapr{} &{\hbox{\rom{Ext}}}^1_{q}({\cal F}_1,{\cal E})&\mapr{}& {\cal A}_+(\tau_-)&\longrightarrow& 0\cr &&\mapd{}&&\mapd{}&&\mapd{}\cr &&0&&0&&0\cr \endmatrix\end{eqnarray} To identify the entries in this diagram we have used the following facts. \begin{enumerate} \item ${\hbox{\rom{Hom}}}_q({\cal F}_1,{\cal F}_1)={\hbox{\rom{Hom}}}_q({\cal F}_2,{\cal F}_2)={\cal O}_{{\Bbb P}_-}$. \item If $Q$ is the universal quotient on ${\Bbb P}({\cal A}_-)$, then the relative tangent bundle is $T_{{\Bbb P}_-/T}=Q(-\tau_-)$, i.e. the cokernel of the natural map ${\cal O}_{{\Bbb P}_-}={\hbox{\rom{Hom}}}_q({\cal F}_1,{\cal F}_1)\longrightarrow {\hbox{\rom{Ext}}}^1_q({\cal F}_2(\tau_-),{\cal F}_1)$. \item $\pi_2^\ast(T_{M(1,(c_1-\xi)/2,m)})={\hbox{\rom{Ext}}}^1_{q}({\cal F}_2,{\cal F}_2)$ and $\pi_2^\ast T_{M(1,(c_1+\xi)/2,n)}={\hbox{\rom{Ext}}}^1_{q}({\cal F}_1,{\cal F}_1)$. \item By Mukai's sheafified Kodaira-Spencer map \cite{Mu1} we have $i_-^*T_{M_-}={\hbox{\rom{Ext}}}^1_{q}({\cal E},{\cal E}).$ Mukai shows the result only if $S$ is an abelian or K3-surface, but in his proof he only uses that $Spl(c_1,c_2)$ is smooth in a neighbourhood of ${\text{\rom{\bf E}}}_-$, (which we have just seen) and ${\hbox{\rom{Ext}}}^1_{q}({\cal E},{\cal E})$ is locally free and compatible with base change. \end{enumerate} To show that the sequences in the diagram are exact we just use standard techniques. It is enough to check the exactness fibrewise. One has repeatedly to make use of the fact that $\xi$ defines a good wall, i.e. if $E\in {{\text{\rom{\bf E}}}}_-$ is given by (\ref{splitting}), then $F_1-F_2$, $F_2-F_1$, $F_1-F_2+K_S$, $F_2-F_1+K_S$ are not effective, which implies that ${\hbox{\rom{Hom}}}_q({\cal F}_1,{\cal F}_2(\tau_-))= {\hbox{\rom{Hom}}}_q({\cal F}_2(\tau_-),{\cal F}_1)= {\hbox{\rom{Ext}}}^2_q({\cal F}_1,{\cal F}_2(\tau_-))= {\hbox{\rom{Ext}}}^2_q({\cal F}_2(\tau_-),{\cal F}_1)=0$. In addition we use that all $E\in {{\text{\rom{\bf E}}}}_-$ are simple and that ${\hbox{\rom{Ext}}}^2_{q}({\cal F}_2,{\cal F}_2)={\hbox{\rom{Ext}}}^2_{q}({\cal F}_1,{\cal F}_1)=0$. We also use the vanishings from the proof of the smoothness of $M_-$ along ${\text{\rom{\bf E}}}_-$. \noindent{\it Second Case: } $K_S$ is trivial. We apply essentially the same arguments as in the first case. Now however we have ${\hbox{\rom{Ext}}}^2_{q}({\cal E},{\cal F}_1)={\hbox{\rom{Ext}}}^2_{q}({\cal F}_2(\tau_-),{\cal E})={\hbox{\rom{Ext}}}^2_{q}({\cal E},{\cal E})= {\hbox{\rom{Ext}}}^2_{q}({\cal F}_1,{\cal F}_1)={\hbox{\rom{Ext}}}^2_{q}({\cal F}_2,{\cal F}_2)={\cal O}_{{\Bbb P}_-}$, which follows easily from Mukai's results \cite{Mu1}. We also notice that by Serre-duality ${\cal A}_+$ is canonically dual to ${\cal A}_-$. Using all this we again get the diagram (\ref{globdiag}) with the entry ${\cal A}_+(\tau_-)$ in the lower right corner replaced by the kernel of the natural map ${\cal A}_+(\tau_-)\to {\cal O}_{{\Bbb P}_-}$, i.e. $Q^\vee(\tau_-)$. \noindent{\it Claim: } The image of the Kodaira-Spencer map $\kappa:T_{{\Bbb P}_-}\to {\hbox{\rom{Ext}}}^1_{q}({\cal E},{\cal E})$ is $Im(\varphi)$+$Im(\psi)$ (see (\ref{globdiag})). Note that, by what we have shown so far, the claim implies the theorem. \noindent{\it Proof of the Claim.} For dimension reasons it is enough to show that $Im(\varphi)$ and $Im(\psi)$ both are contained in the image of $\kappa$. We show it for $Im(\varphi)$. It is enough to show this fibrewise. Let $F_1\in M(1,(c_1+\xi)/2,n)$ and let $({\Bbb P}_-)_{F_1}$ be the fibre of the projection $\pi_1:{\Bbb P}_- \longrightarrow M(1,(c_1+\xi)/2,n)$ over $F_1$. Then $({\Bbb P}_-)_{F_1}$ is the space of extensions $$0\to F_1\to E\to G\to 0$$ with $G$ running through $M(1,(c_1-\xi)/2,m)$. Let $x\in ({{\Bbb P}}_-)_{F_1}$ be given by an extension \begin{equation}\tag{$\lambda_x$} 0\to F_1\to E\to G_1\to 0.\end{equation} We will want to show that $\kappa(T_{({\Bbb P}_-)_{F_1}}(x))=\varphi({\hbox{\rom{Ext}}}^1(G_1,E))$. The tangent space to $({\Bbb P}_-)_{F_1}$ at $x$ is the space of first order deformations of $E$ together with an injection $F_1\to E$. For $t\in T_{({\Bbb P}_-)_{F_1}}(x)$ we get therefore the diagram $$\matrix &&0&&0&&0\cr &&\mapd{}&&\mapd{}&&\mapd{}\cr 0&\mapr{}&F_1&\mapr{}&E&\mapr{}&G_1&\mapr{}&0\cr &&\mapd{}&&\mapd{}&&\mapd{}\cr 0&\mapr{}&F_1\oplus F_1&\mapr{}&\widetilde E&\mapr{}&\widetilde G&\mapr{}&0\cr &&\mapd{}&\mapne{\gamma}&\mapd{}&&\mapd{}\cr 0&\mapr{}&F_1&\mapr{}&E&\mapr{}&G_1&\mapr{}&0\cr &&\mapd{}&&\mapd{}&&\mapd{}\cr &&0&&0&&0\cr \endmatrix \eqno (*)$$ and we see that $T_{({\Bbb P}_-)_{F_1}}(x)$ can be identified with the space of diagrams $(*)$. Furthermore $\kappa(t)$ is the extension class of the middle column of $(*)$. {}From $(*)$ we also get a sequence $0\longrightarrow E\longrightarrow \widetilde E/\gamma(F_1)\longrightarrow G_1\to 0$ such that $\widetilde E$ is defined by pull-back $$\matrix \widetilde E/\gamma(F_1)&\mapr{}&G_1\cr \mapu{}&&\mapu{}\cr \widetilde E.&\mapr{}&E\cr \endmatrix$$ This gives a map $\theta:T_{({\Bbb P}_-)_{F_1}}(x)\longrightarrow Ext^1(G_1,E)$, such that the restriction of $\kappa$ to $T_{({\Bbb P}_-)_{F_1}}(x)$ is $\varphi\circ\theta$. To finish the proof we have to see that $\theta$ is an isomorphism. We give an inverse. Let $$0\longrightarrow E\longrightarrow W\longrightarrow G_1\longrightarrow 0$$ be an extension. We define $\widetilde E$ as the fibre product $$\matrix W&\mapr{}&G_1\cr \mapu{}&&\mapu{}\cr \widetilde E,&\mapr{}&E\cr \endmatrix$$ and we see that it lies in a diagram $(*)$. \end{pf} \begin{rem}\label{newcomp} Assume $p_g(S)=0$. From lemma \ref{normal} it follows that the dimension of ${\text{\rom{\bf E}}}_-$ is at most the expected dimension $N=(4c_2-c_1^2)-3\chi({\cal O}_S)+q(S)$. We have to distinguish two cases. \begin{enumerate} \item $dim({\text{\rom{\bf E}}}_-)<N$ and $dim({\text{\rom{\bf E}}}_+)<N$. Then the change from $M_-$ to $M_+$ is a birational transformation. \item $dim({\text{\rom{\bf E}}}_-)=N$ or $dim({\text{\rom{\bf E}}}_+)=N$. We can assume that $dim({\text{\rom{\bf E}}}_-)=N$. Then by lemma \ref{normal} ${\text{\rom{\bf E}}}_-$ is a smooth connected component of $M_-$, which is isomorphic to ${\Bbb P}_-$. And, ${\cal A}_+(\tau_-)$ being the normal bundle to ${\text{\rom{\bf E}}}_-$, we have ${\cal A}_+=0$ and therefore ${\text{\rom{\bf E}}}_+=\emptyset$. This happens if and only if $\<\xi\cdot(\xi-K_S)\>/2+d_\xi=\chi({\cal O}_S)$. If we allow $NS(S)$ to contain $2$-torsion, we see that all the connected components of ${\text{\rom{\bf E}}}_-$ are connected components of $M_-$. \end{enumerate} \end{rem} Assume for the following definition and corollary that we are in case (1) of \ref{newcomp}, i.e. that the change from $M_-$ to $M_+$ is birational. \begin{defn} Let $\widetilde M_-$ be the blow-up of $M_-$ along ${\text{\rom{\bf E}}}_-$ and $D$ the exceptional divisor. Similarly let $\widetilde M_+$ be the blow up of $M_+$ along ${\text{\rom{\bf E}}}_+$. Let $\pi_D$, $\pi_{D-}$, $\pi_{D+}$ be the projections from $D$ to $T$, ${\Bbb P}_-$, ${\Bbb P}_+$ respectively. \end{defn} \begin{cor}\begin{enumerate} \item If $p_g(S)=0$ then $D$ is isomorphic to ${\Bbb P}_-\times_T {\Bbb P}_+$ and with this identification ${\cal O}(D)|_D={\cal O}(\tau_-+\tau_+)$. \item If $K_S$ is trivial, then ${\cal A}_-$ and ${\cal A}_+$ are canonically dual and $D$ is the incidence correspondence $\{(l,H)\in {\Bbb P}({\cal A}_-)\times_T{\Bbb P}^{\vee}({\cal A}_-) \,|\, l\subset H\}$ and ${\cal O}(D)|_D$ is the restriction of ${\cal O}(\tau_-+\tau_+)$. \end{enumerate} \end{cor} \section{Blow-up construction} We keep the assumptions and notations of the last section. In addition we assume that we in case (1) of \ref{newcomp}, i.e. the map $\widetilde M_-\longrightarrow M_-$ is birational. In this section we want to show that $\widetilde M_-$ and $\widetilde M_+$ are isomorphic. We shall construct a morphism $\varphi_+:\widetilde M_- \to M_+$, which we shall show is the blow-up of $M_+$ along ${\text{\rom{\bf E}}}_+$. Let $\varphi_-:\widetilde M_-\to M_-$ be the blow-up map and $j:D\to \widetilde M_-$ be the embedding. We denote $\widetilde M_-^s:=\varphi_-^{-1}M_-^s$. Let ${\cal U}_-$ be a universal pseudo-family on $S\times M^s_-$ and ${\cal V}_-:=({\hbox{\rom{id}}}_S\times \varphi_-)^*{\cal U}_-$. We want to make an elementary transform of ${\cal V}_-$ along $D_S:=S\times D$ to obtain a pseudo-family ${\cal V}_+$ of $a_+$-stable sheaves on $\widetilde M_-^s$ and thus the desired map $\varphi_+$. If ${\cal U}_-$ is a universal family, then also ${\cal V}_+$ will be one. \begin{nota} For a sheaf ${\cal H}$ on $S\times{\Bbb P}_-$ (resp. $S\times{\Bbb P}_+$) we will write ${\cal H}_D$ for $({\hbox{\rom{id}}}_S\times \pi_{D-})^*{\cal H}$ (resp. $({\hbox{\rom{id}}}_S\times \pi_{D+})^*{\cal H}$). We also write ${\cal F}_{1D}$ and ${\cal F}_{2D}$ instead of $({\cal F}_{1})_{D}$ and $({\cal F}_{2})_{D}$. \end{nota} \begin{defn} By the universal property of $M_-$ and lemma \ref{enm} there is a line bundle $\lambda$ on $D$ such that there is an exact sequence \begin{eqnarray} \label{restrsec}&&0\to {\cal F}_{1D}(\lambda)\longrightarrow {\cal V}_-|_{D_S}\longrightarrow {\cal F}_{2D}(\tau_-+\lambda)\longrightarrow 0, \end{eqnarray} indeed there is already a sequence like this on ${\text{\rom{\bf E}}}_-$. Let $\gamma$ be the composition ${\cal V}_-\longrightarrow {\cal V}_-|_{D_S}\longrightarrow {\cal F}_{2D}(\tau_-+\lambda).$ Then we put ${\cal V}_+:=ker\gamma$. Because ${\cal V}_-$ is flat on $S\times \widetilde M^s_-$, and ${\cal F}_{2D}(\lambda+\tau_-)$ is flat on the Cartier divisor $S\times D$, ${\cal V}_+$ is flat over $S\times \widetilde M^s_-$. The restrictions of ${\cal V}_+$ and ${\cal V}_-$ to $S\times \widetilde M^s_-\setminus D$ are naturally isomorphic. There are diagrams of sheaves on $S\times \widetilde M_-^s$ \begin{eqnarray}\label{dia1} &&{\matrix &&&&0&&0\cr &&&&\mapd{}&&\mapd{}\cr 0&\mapr{}&{\cal V}_-(-D_S)&\mapr{}&{\cal V}_+&\mapr{}&{\cal F}_{1D}(\lambda)&\mapr{}&0\cr &&\Big|\Big|&&\mapd{}&&\mapd{}\cr 0&\mapr{}&{\cal V}_-(-D_S)&\mapr{}&{\cal V}_-&\mapr{}&{\cal V}_-|_{D_S}&\mapr{}&0\cr &&&&\mapd{}&&\mapd{}\cr &&&&{\cal F}_{2D}(\tau_-+\lambda)&\relgl\joinrel\relgl\joinrel\relgl&{\cal F}_{2D}(\tau_-+\lambda) &\mapr{}&0\cr &&&&\mapd{}&&\mapd{}\cr &&&&0&&0\cr\endmatrix}\end{eqnarray} \begin{eqnarray}\label{dia2}&&{\matrix &&&&0&&0\cr &&&&\mapd{}&&\mapd{}\cr 0&\mapr{}&{\cal V}_+(-D_S)&\mapr{}&{\cal V}_-(-D_S)&\mapr{} &{\cal F}_{2D}(\lambda-\tau_+)&\mapr{}&0\cr &&\Big|\Big|&&\mapd{}&&\mapd{}\cr 0&\mapr{}&{\cal V}_+(-D_S)&\mapr{}&{\cal V}_+&\mapr{}&{\cal V}_+|_{D_S}&\mapr{}&0\cr &&&&\mapd{}&&\mapd{}\cr &&&&{\cal F}_{1D}(\lambda)&\relgl\joinrel\relgl\joinrel\relgl&{\cal F}_{1D}(\lambda)&\mapr{}&0\cr &&&&\mapd{}&&\mapd{}\cr &&&&0&&0\cr\endmatrix} \end{eqnarray} By the rightmost column of (\ref{dia2}), $({\cal V}_+)_x\in {\text{\rom{\bf E}}}_+$ for all $x\in D$. Therefore by proposition \ref{flip} ${\cal V}_+$ is a pseudo-family of $a_+$-stable sheaves over $\widetilde M^s_-$ and defines a morphism $\varphi_+:\widetilde M^s_- \to M^s_+$. We see from the definitions that the restriction of $\varphi_+$ to $\widetilde M^s_-\setminus D$ is an isomorphism to $M^s_+\setminus {\text{\rom{\bf E}}}_+$, which coincides with the natural identification $\widetilde M^s_-\setminus D\simeq M^s_-\setminus {\text{\rom{\bf E}}}_-\simeq M^s_+\setminus {\text{\rom{\bf E}}}_+$. As ${\text{\rom{\bf E}}}_-\subset M_-^s$ and ${\text{\rom{\bf E}}}_+\subset M_+^s$, we see that $\varphi_+$ extends to a morphism $\widetilde M_-\longrightarrow M_+$, which we still denote by $\varphi_+$. \end{defn} \begin{thm} $\varphi_+:\widetilde M_-\longrightarrow M_+$ is the blow up of $M_+$ along ${\text{\rom{\bf E}}}_+$. \end{thm} \begin{pf} By the above $\varphi_+(D)\subset {\text{\rom{\bf E}}}_+$. We want to show that $\varphi_+|_D$ is the projection $\pi_{D+}:D\longrightarrow {\text{\rom{\bf E}}}_+$. For this we have to show that the extension $$0\longrightarrow{\cal F}_{2D}(\lambda-\tau_+)\longrightarrow {\cal V}_+|_{S\times D} \longrightarrow {\cal F}_{1D}(\lambda)\mapr{} 0$$ from the rightmost column of (\ref{dia2}) is the pull-back via $\pi_{D+}$ of the tautological extension on ${\Bbb P}_+$ (defined analogously to \ref{globext}) tensorized with ${\cal O}_D(\lambda-\tau_+)$. It is enough to show this fibrewise. Let $x=(x_-,x_+)\in D\subset {\Bbb P}_-\times_T {\Bbb P}_+$ and let $V_-:=({\cal V}_-)_x$ and $V_+:=({\cal V}_+)_x$ be given by extensions \begin{eqnarray}\label{seq1} &&0\longrightarrow F_1\longrightarrow V_-\longrightarrow F_2\longrightarrow 0,\\ \label{seq2} &&0\longrightarrow F_2\longrightarrow V_+\longrightarrow F_1\longrightarrow 0. \end{eqnarray} Then $\pi_D(x)$ is the point $(F_1,F_2)\in T$ and $x_-\in ({\Bbb P}_-)_{(F_1,F_2)}= {\Bbb P}({\hbox{\rom{Ext}}}^1(F_2,F_1))$ is the extension class of (\ref{seq1}). Then we have to show that $x_+\in ({\Bbb P}_+)_{(F_1,F_2)}={\Bbb P}({\hbox{\rom{Ext}}}^1(F_1,F_2))$ is the extension class of (\ref{seq2}). Let $R:={\hbox{\rom{Spec}\,}}{\Bbb C}[\epsilon]/(\epsilon^2)$ and let $t:R\to \widetilde M_-$ be a tangent vector to $\widetilde M_-$ at $x$, which is not tangent to $D$. Then $t$ factors through $2D$ (i.e. the subscheme defined by ${\cal I}_D^2$). If we restrict the diagrams (\ref{dia1}), (\ref{dia2}) to $2D_S$, we see that the image of the map ${\cal V}_-(-D_S)|_{2D_S}\longrightarrow {\cal V}_+|_{2D_S}$ is ${\cal I}_{D_S}{\cal V}_-/{\cal I}^2_{D_S}{\cal V}_-$ and the image of the composition ${\cal V}_+(-D_S)|_{2D_S}\longrightarrow{\cal V}_-(-D_S)|_{2D_S}\longrightarrow V_+|_{2D_S}$ is ${\cal I}_{D_S}{\cal F}_{1D}(\lambda)\cdot/{\cal I}_{D_S}^2{\cal F}_{1D}(\lambda)$. Therefore, by pulling back the diagrams (\ref{dia1}), (\ref{dia2}) to $S\times R$ via $({\hbox{\rom{id}}}_S\times t)$ and pushing down with the projection $p:S\times R\to S$, we get the diagrams \begin{eqnarray}\label{dia3}\matrix &&&&0&&0\cr &&&&\mapd{}&&\mapd{}\cr 0&\mapr{}&V_-&\mapr{}&\widetilde V_+&\mapr{}&F_1&\mapr{}&0\cr &&\Big|\Big|&&\mapd{}&&\mapd{}\cr 0&\mapr{}&V_-&\mapr{}&\overline V_-&\mapr{}&V_-&\mapr{}&0\cr &&&&\mapd{}&&\mapd{}\cr &&&&F_2&\relgl\joinrel\relgl\joinrel\relgl&F_2&\mapr{}&0\cr &&&&\mapd{}&&\mapd{}\cr &&&&0&&0\cr\endmatrix\end{eqnarray} \begin{eqnarray} \label{dia4} \matrix &&&&0&&0\cr &&&&\mapd{}&&\mapd{}\cr 0&\mapr{}&F_1&\mapr{}& V_-&\mapr{}&F_2&\mapr{}&0\cr &&\Big|\Big|&&\mapd{}&&\mapd{}\cr 0&\mapr{}&F_1&\mapr{}&\widetilde V_+&\mapr{}&V_+&\mapr{}&0\cr &&&&\mapd{}&&\mapd{}\cr &&&&F_1&\relgl\joinrel\relgl\joinrel\relgl&F_1&\mapr{}&0\cr &&&&\mapd{}&&\mapd{}\cr &&&&0&&0\cr\endmatrix \end{eqnarray} The extension class $\delta\in {\Bbb P}({\hbox{\rom{Ext}}}^1(V_-,V_-))$ of the middle row of (\ref{dia3}) is the class of the image of $t$ under $d\varphi_-:T_{\widetilde M_-}(x)\longrightarrow T_{ M_-}(\varphi_-(x))={\hbox{\rom{Ext}}}^1(V_-,V_-)$. The image of the composition $$T_{ M_-}(x)\longrightarrow \varphi_-^*(T_{M_-}(x))\,\mapr{\rho} \,\varphi_-^*(N_{{\text{\rom{\bf E}}}_-/M_-}(x))$$ is the tautological subline-bundle of $\varphi_-^*(N_{{\text{\rom{\bf E}}}_-/M_-}(x))={\hbox{\rom{Ext}}}^1(F_1,F_2)$ and the kernel is $T_{D}(x)$. Therefore the image of $\rho(\delta)$ in $({\Bbb P}_+)_{(F_1,F_2)}={\Bbb P}({\hbox{\rom{Ext}}}^1(F_1,F_2))$ is $x_+$. By (\ref{globdiag}) the map $\rho$ is the composition $$ {\hbox{\rom{Ext}}}^1(V_-,V_-)\,\mapr{\rho_1}\,{\hbox{\rom{Ext}}}^1(F_1,V_-)\,\mapr{\rho_2}\, {\hbox{\rom{Ext}}}^1(F_1,F_2)$$ given by applying ${\hbox{\rom{Hom}}}(\cdot,\cdot)$ on both sides to the sequence $0\to F_1\to V_-\to F_2\to 0$. By (\ref{dia3}) $\rho_1(\delta)$ is the extension class of the first row of (\ref{dia3}) giving $\widetilde V_+$, and then, by (\ref{dia4}), $\rho(\delta)$ is the extension class of (\ref{seq2}). So we see that $\varphi_+|_D$ is the projection to ${\text{\rom{\bf E}}}_+$. If for the moment we call $\overline \varphi_+:\widetilde M_+\to M_+$ the blow-up of $M_+$ along ${\text{\rom{\bf E}}}_+$ and $\overline D$ the exceptional divisor, we get analogously that $\overline D\simeq {\Bbb P}_-\times_T {\Bbb P}_+$, (or the incidence correspondence in ${\Bbb P}_-\times_T {\Bbb P}_+$ in case $K_S$ is trivial). In the same way as above we can construct a morphism $\overline \varphi_-:\widetilde M_+\longrightarrow M_-$ such that $\overline \varphi_-|_{\overline D}$ is the projection to ${\text{\rom{\bf E}}}_-$ and $\overline \varphi_-|_{\widetilde M_+\setminus \overline D}$ is just the natural identification $\widetilde M_+\setminus \overline D\simeq M_+\setminus {\text{\rom{\bf E}}}_+\simeq M_-\setminus {\text{\rom{\bf E}}}_-$. Therefore we have morphisms $\varphi_-\times \varphi_+:\widetilde M_-\to M_-\times M_+$, $\overline \varphi_-\times \overline \varphi_+:\widetilde M_+\to M_-\times M_+$, which by the above are injective and easily seen to be injective on tangent vectors. Furthermore $(\varphi_-\times \varphi_+)(\widetilde M_-\setminus D)= (\overline \varphi_-\times \overline \varphi_+) (\widetilde M_+\setminus \overline D)$. Therefore $\widetilde M_-$ and $\widetilde M_+$ are isomorphic and in fact both isomorphic to the closure of the graph of the obvious rational map $M_-\to M_+$. \end{pf} In the following theorem we put together the main results we have obtained so far. \begin{thm}\label{zblowup} Let $S$ be a surface with either $p_g(S)=0$ or $K_S$ trivial. Let $c_1\in NS(S)$, $c_2\in {\Bbb Z}$ and put $N:=4c_2-c_1^2-3\chi({\cal O}_S)+q(S)$. Let $W$ be a good wall of type $(c_1,c_2)$ and let $H_-$, $H_+$ be ample divisors on $S$ in neighbouring chambers separated by $W$. Then for all $a\in [0,1]$ there exist spaces $M_a(c_1,c_2)$ and a finite set of miniwalls dividing $[0,1]$ into finitely many minichambers such that the following holds: \begin{enumerate} \item $M_0(c_1,c_2)=M_{H_-}(c_1,c_2)$, $M_1(c_1,c_2)=M_{H_+}(c_1,c_2)$. \item If $a_1$, $a_2$ are in the same minichamber then $M_{a_1}(c_1,c_2)=M_{a_2}(c_1,c_2)$. \item If $a_-<a<a_+$ and $a$ is the unique miniwall between $a_-$ and $a_+$ then $M_{a_+}(c_1,c_2)$ is obtained from $M_{a_-}(c_1,c_2)$ as follows: We blow up $M_{a_-}(c_1,c_2)$ along the disjoint smooth subvarieties ${\text{\rom{\bf E}}}_{\xi}^{n,m}$, with $(\xi,n,m)\in A^+(a)$ (see \ref{defminiwall}) which fulfill $0\le dim({\text{\rom{\bf E}}}_{\xi}^{n,m})< N$ and blow-down the exceptional divisors to ${\text{\rom{\bf E}}}_{-\xi}^{m,n}$ respectively. Then we remove the ${\text{\rom{\bf E}}}_{\xi}^{n,m}$ with $(\xi,n,m)\in{\cal A}^+(a)$ and $dim({\text{\rom{\bf E}}}_{\xi}^{n,m})= N$ (which are unions of connected components of $M_{a_-}(c_1,c_2)$) and take the disjoint union with all ${\text{\rom{\bf E}}}_{-\xi}^{m,n}$ with $(\xi,n,m)\in{\cal A}^+(a)$ and ${\text{\rom{\bf E}}}_{\xi}^{n,m}= \emptyset$. \item If $H$ is an ample divisor on $W$ which lies in the closure of both of the chambers containing $H_-$ and $H_+$, then, for all $b\in [0,1]$, the space $M_b(c_1,c_2)$ is a moduli space of H-semistable sheaves on $S$ with a suitable parabolic structure. \end{enumerate} \end{thm} In \cite{Mu1} Mukai defines elementary transforms of a symplectic variety $X$ as follows. Assume $X$ contains a subvariety $P$, which has codimension $n$ and is a ${\Bbb P}_n$-bundle over a variety $Y$. Let $\widetilde X$ be the blow-up of $X$ along $P$. Then the exceptional divisor $E$ is isomorphic to the incidence correspondence in $P\times_Y P'$, where $P'$ is the dual projective bundle to $P$. One can then blow down $E$ to $P'$ to obtain a smooth symplectic variety $X'$. We will for the moment call $Y$ the center of such an elementary transformation. So by the above we obtain the following: \begin{cor} Let $S$ be a K3-surface or an abelian surface. Let $H_-,H_+$ be polarisations which both do not lie on a wall. Then $M_{H_+}(c_1,c_2)$ is obtained from $M_{H_-}(c_1,c_2)$ by a series of elementary transforms, whose centers are of the form $M(1,(c_1+\xi)/2,n)\times M(1,(c_1-\xi)/2,m)$ for $\xi$ defining a wall between $H_-$ and $H_+$ and $(n,m)$ running through the nonnegative integers with $n+m=(4c_2-c_1^2+\xi^2)/4$. \end{cor} \begin{rem} If $q(S)\ne 0$ we can also, for $A\in Pic(S)$, $c_2\in {\Bbb Z}$ and an ample divisor $H$, study the moduli space $\widetilde M_H(A,c_2)$ of rank $2$ torsion-free sheaves $E$ on $S$ with $det(E)=A$ and $c_2(E)=c_2$. Then there is a morphism $M_H(c_1,c_2)\longrightarrow Pic^0(S)$, whose fibres are the various $\widetilde M_H(A,c_2)$ for $A$ with $c_1(A)=c_1$. Then, by restricting our arguments to the fibres, we get that theorem \ref{zblowup} also holds with the obvious changes for $\widetilde M_H(A,c_2)$. \end{rem} \section{The change of the Donaldson invariants in terms of Hilbert schemes} In this section we assume that $q(S)=0$. Let $\gamma_{c_1,c_2,g}$ be the Donaldson polynomial with respect to a Riemannian metric $g$ associated to the principal $SO(3)$-bundle $P$ on $S$ whose second Stiefel-Whitney class $w_2(P)$ is the reduction of $c_1$ mod $2$ and whose first Pontrjagin class is $p_1(P)=(c_1^2-4c_2)$. Then $\gamma_{c_1,c_2,g}$ is a homogeneous polynomial on $H_*(S,{\Bbb Q})$ of weight $2N=2(4c_2-c_1^2-3\chi({\cal O}_S))$, where the elements of $H_i(S,{\Bbb Q})$ have weight $4-i$. In case $p_g(S)>0$ it is known that $\gamma_{c_1,c_2,g}$ does not depend on the metric (as long as it is generic). In [Ko] the invariant has been introduced for $4$-manifolds $M$ with $b_+(M)=1$. In [K-M] it has been shown that in case $b_+(M)=1$, $b_1(M)=0$ it depends only on the chamber of the period point of the metric in the positive cone of $H^2(M,{\Bbb R})$. The algebro-geometric analogues of the Donaldson polynomials are defined as follows: \begin{defn} \label{algana} (\cite{OG1}, \cite{OG2}) Assume that $M_H(c_1,c_2)$ is a fine moduli space, i.e. $M_H(c_1,c_2)=M_H^s(c_1,c_2)$, and there is a universal sheaf ${\cal U}$ on $S\times M_H(c_1,c_2)$. We define a linear map $$\nu_{c_1,c_2,H}:H_i(S,{\Bbb Q})\to H^{4-i}(M_H(c_1,c_2),{\Bbb Q}); \ \ \nu_{c_1,c_2,H}(\alpha):=(c_2({\cal U})-{1\over 4}c_1^2({\cal U}))/\alpha,$$ where $/$ denotes the slant product. We assume furthermore that $M_H(c_1,c_2)$ is of the expected dimension $N:=4c_2-c_1^2-3\chi({\cal O}_S)$. Given classes $\alpha_{s}\in H_{2j_s}(S,{\Bbb Q})$, for $s=1,\ldots,k$ with $2k-\sum_s j_s=N$, we set $$\Phi_{c_1,c_2,H}(\alpha_1,\ldots,\alpha_k):=\int\limits_{M_H(c_1,c_2)} \nu_{c_1,c_2,H}(\alpha_1)\cdot \ldots \cdot\nu_{c_1,c_2,H}(\alpha_k).$$ As $c_1,c_2$ are fixed in our paper, we will write $\nu_H=\nu_{c_1,c_2,H}$ and $\Phi_{H}=\Phi_{c_1,c_2,H}$. Let ${pt}\in H_0(S,{\Bbb Z})$ be the class of a point in $S$. Knowing $\Phi_{H}$ is equivalent to knowing the numbers $$\Phi_{H,l,r}(\alpha) :=\int\limits_{M_H(c_1,c_2)}\nu_H(\alpha)^l\cdot\nu_H({pt})^r.$$ for all $l,r$ with $l+2r=N$ and all $\alpha\in H_2(S,{\Bbb Q})$. \end{defn} \begin{defn} Following \cite{OG2}, we call $M_H(c_1,c_2)$ admissible if the following holds: \begin{enumerate} \item $H$ does not lie on a wall of type $(c_1,c_2)$; \item $dim(M_H(c_1,c_2))=N$, \item if $c_1$ is divisible by $2$ in $NS(S)$, then $N> (4c_2-c_1^2)/2$; \item $dim(M_H(c_1,k))+2(c_2-k)<N$ for all $k<c_2$. \end{enumerate} \end{defn} For admissible $M_H(c_1,c_2)$ the results of \cite{Mo} and \cite{Li} give $$\Phi_{H}|_{H^2(S,{\Bbb Q})}=(-1)^{(c_1^2+\<c_1\cdot K_S\>)} \gamma_{c_1,c_2,g(H)}|_{H^2(S,{\Bbb Q})},$$ where $g(H)$ is the Fubini-Study metric associated to $H$. Furthermore if $c_2>>0$, then $\Phi_{H}=(-1)^{(c_1^2+\<c_1\cdot K_S \>)}\gamma_{c_1,c_2,g(H)}$. We now want to determine how $\Phi_{H}$ changes, when $H$ passes through a wall. We assume that if $c_1$ is divisible by $2$ in $NS(S)$ then $(4c_2-c_1^2)$ is not divisible by $8$. Then, by the criterion of \cite{Ma2}, $M_H(c_1,c_2)$ is a fine moduli space, unless $H$ lies on a wall. Now we assume that we are in the situation of section 3, i.e. $H_-$ and $H_+$ are ample divisors lying in neighbouring chambers separated by $W$, and $H$ a polarization on the wall $W$ not lying on any other wall and lying in the closure of both the chambers containing $H_-$ and $H_+$. We assume furthermore that $W$ is a good wall. For $b\in [0,1]$ we have $M_b(c_1,c_2)$ as in section 3. \begin{defn} By remark \ref{fine} we see that, for $b$ not on a miniwall, $M_b(c_1,c_2)=M^s_b(c_1,c_2)$ and there is a universal sheaf on $M_b(c_1,c_2)$. Assume that $b\in [0,1]$ does not lie on a miniwall. Then analoguosly to the definition of $\Phi_{H}$ and $\Phi_{H,l,r}$ in \ref{algana}, we may define $\Phi_{b}$ and $\Phi_{b,l,r}$ by always replacing $M_H(c_1,c_2)$ by $M_b(c_1,c_2)$. \end{defn} We notice that $\Phi_{H_-}=\Phi_{0}$ and $\Phi_{H_+}=\Phi_{1}$ and it is obvious that $\Phi_{b}$ only depends on the minichamber containing $b$. We therefore have to determine the change of $\Phi_b$ when $b$ passes through a miniwall. We will make the same assumptions as in section 4, i.e. let $a$ be a miniwall and let $(\xi,n,m)\in A^+(a)$. Let $a_-<a_+$ lie in neighbouring minichambers separated by $a$. To simplify the notation we will for the moment assume that $A^+(a)=\{ (\xi,n,m)\}$ and that $H^2(S,{\Bbb Z})$ contains no $2$-torsion. We also assume that either $p_g(S)=0$ or $K_S$ is trivial. \begin{nota}\label{notado} We use the notations and definitions of sections 4 and 5. If the change is birational, i.e. we are not in case (1) of \ref{newcomp}, we shall write $\widetilde M$ instead of $\widetilde M_-$. Let $d:=d_\xi=n+m$, $e_-=rk ({\cal A}_-)$, $e_+=rk ({\cal A}_+)$, then $N=2d+e_-+e_+-1$ if $p_g(S)=0$ and $N=2d+e_-+e_+-2$ if $K_S$ is trivial. We put $\nu_+:=\nu_{a_+}$, $\nu_-:=\nu_{a_-}$, $\Phi_+:=\Phi_{a_+}$, $\Phi_-:=\Phi_{a_-}$, $\Phi_{+,l,r}:=\Phi_{a_+,l,r}$ and $\Phi_{-,l,r}:=\Phi_{a_-,l,r}$. Note that the condition $q(S)=0$ implies $Pic(S)\simeq NS(S)$. For $\beta\in NS(S)$ we may therefore denote by ${\cal O}_S(\beta)$ the corresponding line bundle. Let $q_1,q_2$ be the two projections of $T={\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)$. \end{nota} \begin{rem}\begin{enumerate} \item If the change is birational, then by the projection formula $\Phi_+,$ $\Phi_{+,l,r}$ (resp. $\Phi_-$, $\Phi_{-,l,r}$) coincide with the numbers which are defined analogously by replacing $M_{a}(c_1,c_2)$ by $\widetilde M$ and the universal sheaf by ${\cal V}_+$ (resp. ${\cal V}_-$). \item Assume $p_g(S)=0$ and say ${\text{\rom{\bf E}}}_+=\emptyset$. Let ${\cal E}$ be the universal sheaf on ${\text{\rom{\bf E}}}_-$ from (\ref{globext}), then we can define $\sigma_-:H_i(S,{\Bbb Q})\longrightarrow H^{4-i}({\text{\rom{\bf E}}}_-,{\Bbb Q})$ and $\delta_-$ and $\delta_{-,l,r}$ in the same way as $\nu_-$ and $\Phi_-$ and $\Phi_{-,l,r}$ by replacing $M_-$ by ${\text{\rom{\bf E}}}_-$ and the universal sheaf on $M_-$ by ${\cal E}$. Then $\Phi_+-\Phi_-=-\delta_-$. \end{enumerate} \end{rem} \begin{defn}\label{hilbdef} Let $Z_n(S)\subset S\times {\text{\rom{Hilb}}}^n(S)$ be the universal subscheme. In $S\times {\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)$, we put ${\cal Z}_1:=({\hbox{\rom{id}}}_S\times q_1)^{-1}(Z_n(S))$, ${\cal Z}_2:=({\hbox{\rom{id}}}_S\times q_2)^{-1}(Z_m(S))$ and denote by ${\cal I}_{{\cal Z}_1}$, ${\cal I}_{{\cal Z}_2}$ the corresponding idealsheaves. Let $F_1:={\cal O}_S((c_1+\xi)/2)$, $F_2:={\cal O}_S((c_1-\xi)/2)$. By our assumptions $T={\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)$ and ${\cal F}_1'={\cal I}_{{\cal Z}_1}(F_1)$, $ {\cal F}_2'={\cal I}_{{\cal Z}_2}(F_2)$. Let $h_n:{\text{\rom{Hilb}}}^n(S)\longrightarrow S^{(n)}$ be the Hilbert-Chow morphism \cite{Fo}, where $S^{(n)}$ is the $n$-fold symmetric power of $S$ with the quotient map $\varphi_n:S^n\longrightarrow S^{(n)}$. For $i=1,\ldots,n$ we denote by $p_i:S^n\to S$ the projection to the $i^{th}$ factor. We denote $\Delta_{i}:=\big\{(x,x_1,\ldots,x_n)\in S\times S^n\bigm| x=x_i\big\}$ and $Y_n:=({\hbox{\rom{id}}}_S\times \varphi_n)(\Delta_{1}).$ We have linear maps \begin{eqnarray*} \iota_n:H_i(S,{\Bbb Q})\longrightarrow H^{4-i}({\text{\rom{Hilb}}}^n(S),{\Bbb Q});&& \iota_{n}(\alpha)= [Z_n(S)]/\alpha\ \ \text{ and }\\ \bar\iota_n:H_i(S,{\Bbb Q})\longrightarrow H^{4-i}S^{(n)},{\Bbb Q});&& \bar\iota_{n}(\alpha)= [Y_n]/\alpha.\end{eqnarray*} For $\alpha\in H^i(S,{\Bbb Q})$ put $\alpha_{n,m}:= [{\cal Z}_1]/\alpha+[{\cal Z}_2]/\alpha=q_1^*(\iota_n(\alpha))+q_2^*(\iota_n(\alpha)) \in H^{4-i}(T,{\Bbb Q})$. \end{defn} The map $\iota_{n}$ is in fact easy to describe: \begin{lem} \label{hilbkuenn} \begin{enumerate} \item $[Z_n(S)]=({\hbox{\rom{id}}}_S\times h_n)^*([Y_n])$. \item $({\hbox{\rom{id}}}_S\times\varphi_n)^*([Y_n])=\sum_i[\Delta_i]$ \item For $\alpha\in H^i(S,{\Bbb Q})$ we have $\iota_n(\alpha)=h_n^*(\bar\iota_n(\alpha))$ and $\varphi_n^*(\bar\iota_n(\alpha))=\sum_{i=1}^n p_i^*(\check \alpha)$, where $\check \alpha$ is the Poincar\'e dual of $\alpha$. \end{enumerate} \end{lem} \begin{pf} (1). Out of codimension $3$ on $S\times {\text{\rom{Hilb}}}^n(S)$ we have ${\cal O}_{Z_n(S)}= ({\hbox{\rom{id}}}_S\times h_n)^*({\cal O}_{Y_n})$. So we get $[Z_n(S)]=({\hbox{\rom{id}}}_S\times h_n)^*([Y_n]).$ Out of codimension $3$ we also have $({\hbox{\rom{id}}}_S\times \varphi_n)^*({\cal O}_{Y_n})= \bigoplus_i{\cal O}_{\Delta_{i}}.$ Therefore (2) follows in the same way as (1). (3) follows immediately from (1) and (2).\end{pf} \begin{rem}\label{restch} For the total Chern classes we have $c(({\hbox{\rom{id}}}_S\times j)^*{\cal V}_-)=({\hbox{\rom{id}}}_S\times j)^*c({\cal V}_-)$ and $c(({\hbox{\rom{id}}}_S\times j)^*{\cal V}_+)=({\hbox{\rom{id}}}_S\times j)^*c({\cal V}_+)$, where, as above, $j:D\longrightarrow \widetilde M$ is the embedding of the exceptional divisor. \end{rem} \begin{pf} We have to see that ${\hbox{\rom{Tor}}}_k({\cal V}_-,{\cal O}_{S\times D})=0$ for all $k>0$ (and similarly for ${\cal V}_+$). This follows however easily from the flatness of ${\cal V}_-$ over $\widetilde M_-$. \end{pf} \begin{lem} \label{nu}\begin{enumerate} \item Assume that we are in case (1) of \ref{newcomp}, i.e. the change of moduli is birational. Then, for $\alpha\in H_2(S,{\Bbb Q})$, we have \begin{eqnarray*}\nu_+(\alpha)-\nu_-(\alpha) &=&-{1\over 2}\<\xi,\alpha\>[D],\\ \nu_{+}(pt)-\nu_{-}({pt})&=&{1\over 4}j_*([\tau_-]-[\tau_+]). \end{eqnarray*} \item If ${\text{\rom{\bf E}}}_+=\emptyset$ then \begin{eqnarray*} \sigma_-(\alpha)&=& {1\over 2}\<\xi,\alpha\>[\tau_-],\\ \sigma_{-}({pt})&=&-{1\over 4}[\tau_-]^2. \end{eqnarray*} \end{enumerate}\end{lem} \begin{pf}By (\ref{dia1}) we have the sequence $$0\longrightarrow {\cal V}_-\longrightarrow {\cal V}_+(D_S)\longrightarrow {\cal F}_{1D}(\lambda+\tau_-+\tau_+)\longrightarrow 0.$$ Using Riemann-Roch without denominators \cite{Jo} we get \begin{eqnarray*} c_1({\cal F}_{1D}(\lambda+\tau_-+\tau_+))&=&[D_S]\\ c_2({\cal F}_{1D}(\lambda+\tau_-+\tau_+))&=& -c_1({\cal F}_{1D}(\lambda)), \end{eqnarray*} and thus \begin{eqnarray*} c_1({\cal V}_+(D_S))&=&c_1({\cal V}_-)+[D_S],\\ c_2({\cal V}_+(D_S))&=&c_2(V_-)+[D_S]\cdot c_1(V_-)-({\hbox{\rom{id}}}_S\times j)_*(c_1({\cal F}_{1D}(\lambda)))\\ 4c_2({\cal V}_+)-c_1({\cal V}_+)^2&=&4c_2({\cal V}_+(D_S))-c_1({\cal V}_+(D_S))^2\\ &=& 4c_2({\cal V}_-)-c_1({\cal V}_-)^2+2[D_S]\cdot c_1({\cal V}_-) -[D_S]^2-4({\hbox{\rom{id}}}_S\times j)_*(c_1({\cal F}_{1D}(\lambda))). \end{eqnarray*} Let $\alpha\in H_2(S,{\Bbb Q})$. As $[D_S]$ is the pull-back of $[D]$ from $\widetilde M$, we have $$([D_S]\cdot c_1({\cal V}_-))/\alpha=[D](c_1({\cal V}_-)/\alpha)=\<c_1,\alpha\>[D].$$ Furthermore $({\hbox{\rom{id}}}_S\times j)_*c_1({\cal F}_{1D}(\lambda))/\alpha= j_*(c_1({\cal F}_1(\lambda))/\alpha)$, where the second slant product is taken on $S\times D$ and $c_1({\cal F}_1)=\pi_D^*(c_1({\cal F}_1'))=p^*([F_1])$. So we get $({\hbox{\rom{id}}}_S\times j)_\ast (c_1({\cal F}_1)/\alpha)=\<F_1\cdot \alpha\>[D]$. As $\lambda$ is the pull-back of a divisor on $D$, we have $({\hbox{\rom{id}}}_S\times j)_*c_1(\lambda)/\alpha= 0$ and similarly $[D_S]^2/\alpha=0$. So we get $\nu_+(\alpha)-\nu_-(\alpha)=-{1\over 2}\<\xi,\alpha\>[D]$ By $c_1({\cal F}_1)=p^*([F_1])$, $c_1({\cal F}_2)=p^*([F_2])$, we get $c_1({\cal F}_1')/{pt}=c_1({\cal F}_2')/{pt}=0.$ Then the sequence $$0\longrightarrow {\cal F}_{1D}(\lambda)\longrightarrow {\cal V}_-|_D\longrightarrow {\cal F}_{2D}(\tau_-+\lambda)\longrightarrow 0,$$ and remark \ref{restch} give \begin{eqnarray*} (c_1({\cal V}_-)\cdot [D_S])/{pt} &=& ({\hbox{\rom{id}}}_S\times j)_*(c_1({\cal V}_-|_D))/{pt}=j_*([\tau_-]+2[\lambda]),\\ (c_1({\cal F}_{1D}(\lambda)))/{pt} &=& j_*(c_1({\cal F}_1(\lambda))/{pt})= j_*([\lambda]),\\{} [D_S]^2/{pt} &=& [D]^2. \end{eqnarray*} So we get \begin{eqnarray*}\nu_+({pt})-\nu_-({pt})&=&{1\over 4}([D]^2 +j_*([2\tau_-+4\lambda])-4j_*([\lambda])) \\ &=& {1\over 4}j_*([\tau_-]-[\tau_+]).\end{eqnarray*} (2) can be shown using essentially the same arguments. \end{pf} \begin{lem} \label{change1} Let $l+2r=N$. \begin{enumerate} \item If we are in case (1) of \ref{newcomp}, then \begin{eqnarray*} &&\Phi_{+,l,r}(\alpha)-\Phi_{-,l,r}(\alpha)\\&& \quad=\sum_{b=0}^{l}\sum_{c=0}^r (-1)^{r-c+1}2^{b+2c-N}{l\choose b} {r\choose c}\<\xi,\alpha\>^{l-b}\int\limits_D \left(\alpha_{n,m}^{b}{pt}_{n,m}^{c}\sum_{s+t=N-b-2c-1} (-\tau_+)^s\tau_-^t\right)\end{eqnarray*} \item ${\text{\rom{\bf E}}}_+=\emptyset$, then \begin{eqnarray*} &&\Phi_{+,l,r}(\alpha)-\Phi_{-,l,r}(\alpha)\\&& \quad=\sum_{b=0}^{l}\sum_{c=0}^r (-1)^{r-c+1}2^{b+2c-N}{l\choose b} {r\choose c}\<\xi,\alpha\>^{l-b}\int\limits_{{\text{\rom{\bf E}}}_-} \left(\alpha_{n,m}^{b}{pt}_{n,m}^{c}\tau_-^{N-b-2c}\right)\end{eqnarray*} \end{enumerate}\end{lem} \begin{pf} (1) By remark \ref{restch} we get for $\alpha\in H_{i}(S,{\Bbb Q})$ that $[D]\cdot \nu_+(\alpha)= j_*((4c_2({\cal V}_+|_D)-c_1({\cal V}_+|_D)^2)/4\alpha$ (and similar for $\nu_-$). By the sequences \begin{eqnarray*}&&0\longrightarrow {\cal F}_{2D}(-\tau_++\lambda)\longrightarrow {\cal V}_+|_{D_S} \longrightarrow {\cal F}_{1D}(\lambda)\longrightarrow 0\\ &&0\longrightarrow {\cal F}_{1D}(\lambda)\longrightarrow {\cal V}_-|_{D_S}\longrightarrow {\cal F}_{2D}(\tau_-+\lambda)\longrightarrow 0\end{eqnarray*} we get \begin{eqnarray*}&&4c_2({\cal V}_+|_D)-c_1({\cal V}_+|_D)^2 =4(c_2({\cal F}_{1D})+c_2({\cal F}_{2D}))-(c_1({\cal F}_{2D})-c_1({\cal F}_{1D})-[\tau_+])^2\\ &&4c_2({\cal V}_-|D)-c_1({\cal V}_-|D)^2 =4(c_2({\cal F}_{1D})+c_2({\cal F}_{2D}))-(c_1({\cal F}_{1D})-c_1({\cal F}_{2D})-[\tau_-])^2 \end{eqnarray*} By the above we have $c_1({\cal F}_{1D})=p^*([F_1])$, $c_1({\cal F}_{2D})=p^*([F_2])$, $c_2({\cal F}_{1D})=({\hbox{\rom{id}}}_S\times \pi_D)^*(c_2({\cal I}_{{\cal Z}_1}))=({\hbox{\rom{id}}}_S\times \pi_D)^*([{\cal Z}_1])$ and $c_2({\cal F}_{2D})=({\hbox{\rom{id}}}_S\times \pi_D)^*([{\cal Z}_2])$, where, as above, $\pi_D:D\longrightarrow T$ is the projection. So we have \begin{eqnarray*}4c_2({\cal V}_+|_D)-c_1({\cal V}_+|_D)^2 &=&4({\hbox{\rom{id}}}_S\times \pi_D)^*([{\cal Z}_1]+[{\cal Z}_2])-(p^*(\xi)+[\tau_+])^2,\\ 4c_2({\cal V}_-|_D)-c_1({\cal V}_-|_D)^2 &=&4({\hbox{\rom{id}}}_S\times \pi_D)^*([{\cal Z}_1]+[{\cal Z}_2])-(p^*(\xi)-[\tau_-])^2, \end{eqnarray*} and thus for $\alpha\in H_2(S,{\Bbb Q})$: \begin{eqnarray*} j^*(\nu_+(\alpha))&=&\alpha_{n,m}+{1\over 2}\<\xi,\alpha\>[-\tau_+]\\ j^*(\nu_-(\alpha))&=&\alpha_{n,m}+{1\over 2}\<\xi,\alpha\>[\tau_-]\\ j^*(\nu_+({pt}))&=&{pt}_{n,m}-{1\over 4}[\tau_+]^2\\ j^*(\nu_-({pt}))&=&{pt}_{n,m}-{1\over 4}[\tau_-]^2 \end{eqnarray*} We write \begin{eqnarray*} &&\Phi_{+,l,r}(\alpha)-\Phi_{-,l,r}(\alpha)=\\ &&\qquad=\int\limits_{\widetilde M} \left(\nu_+(\alpha)^{l}(\nu_+({pt})^r-\nu_-({pt})^r)+ \nu_+({pt})^r(\nu_+(\alpha)^{l}-\nu_-(\alpha)^{l})\right)\\ &&\qquad=\int\limits_D\left({1\over 4}([-\tau_+]+[\tau_-]) j^*\left(\sum_{s+t=r-1}\nu_+({pt})^s\nu_-({pt})^t \nu_+(\alpha)^l\right) \right.\\ &&\qquad\qquad-\left. {1\over 2}\<\xi,\alpha\>\ j^*\left(\sum_{s+t=l-1}\nu_+(\alpha)^s \nu_-(\alpha)^t \nu_-({pt})^r\right) \right).\end{eqnarray*} Now the claim follows after a straightforward computation. (2) follows easily from lemma \ref{nu}(2).\end{pf} \begin{prop}\label{donmin} \begin{enumerate} \item If $S$ is a $K3$ surface and $N>0$, then $\Phi_+=\Phi_-$. \item If $p_g(S)=0$, then for $\alpha\in H_2(S,{\Bbb Q})$ and $l,r$ with $l+2r=N$ we have \begin{eqnarray*} &&\Phi_{+,l,r}(\alpha)-\Phi_{-,l,r}(\alpha)\\ &&\quad=\sum_{b=0}^{l}\sum_{c=0}^r (-1)^{r-c+e_-}2^{b+2c-N}{l\choose b} {r\choose c}\<\xi,\alpha\>^{l-b}\int\limits_{T} \left(\alpha_{n,m}^{b} {pt}_{n,m}^{c} s_{2d-b-2c}({\cal A}_+'\oplus{{\cal A}_-'}^{\vee})\right) \end{eqnarray*} \end{enumerate} \end{prop} \begin{pf} (1) It easy to show using Riemann-Roch, that the condition $N>0$ implies $e_->1$ and $e_+>1$. Therefore, as $\alpha_{n,m}$ and ${pt}_{n,m}$ are pull-backs from $T$, it is enough to show that for $k\le e_-+e_+-2$ we have $$(\pi_D)_*\left(\sum_{s+t=k}(-\tau_+)^s \tau_-^t\right)=0.$$ Now $D$ is the projectivisation ${\Bbb P}(Q)$ where $Q={\cal A}_-/\tau_-$ over ${\Bbb P}_{-}={\Bbb P}({\cal A}_-')$. Therefore \begin{eqnarray*} (\pi_D)_*\left(\sum_{s+t=k}(-\tau_+)^s\tau_-^t\right)&=& (\pi_{-})_*\left(\sum_{s+t=k}s_{s-e_++2}(Q)\tau_-^t\right)\\ &=&(\pi_{-})_*(s_{k-e_++2}({\cal A}_-))\\ &=&(\pi_{-})_*\pi_-^*(s_{k-e_++2}({\cal A}_-'))=0.\end{eqnarray*} Here $\pi_{-}:{\Bbb P}_-\longrightarrow T$ is the projection. (2) We just note that $\pi_{+})_*((-\tau_+)^k)=s_{k-e_++1}({\cal A}_+')$ and $(\pi_{-})_*(\tau_-^k)=(-1)^{e_-+1}s_{k-e_-+1}({{\cal A}_-'}^{\vee})$. Then the result follows immediately from the definitions and lemma \ref{change1}. \end{pf} For the rest of the chapter we assume that $p_g(S)=q(S)=0$. On the other hand we allow $NS(S)=H^2(S,{\Bbb Z})$ to contain torsion. \begin{defn}\label{defchange} Let $\xi\in H^2(S,{\Bbb Z})$ be a class defining a good wall of type $(c_1,c_2)$. Let $d_\xi:=(4c_1-c_1^2+\xi^2)/4$, $e_{\xi}:=-\<\xi\cdot(\xi-K_S)\>/2+d_\xi+1$ and $$T_\xi:={\text{\rom{Hilb}}}^{d_\xi}(S\sqcup S)= \coprod_{n+m=d_\xi}{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S).$$ Let $q_\xi:S\times T_\xi\longrightarrow T_\xi$ be the projection. Let $V_\xi$ be the sheaf $p^*({\cal O}_S(-\xi)\oplus{\cal O}_S(-\xi+K_S))$ on $S\times T_\xi$. Let ${\cal Z}^\xi_{1}$ (resp.${\cal Z}^\xi_{2}$) be the subscheme of $S\times T_\xi$ which restricted to each component $S\times{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)$ is the subscheme ${\cal Z}_1$ (resp. ${\cal Z}_2$) from \ref{hilbdef}. Let ${\cal I}_{{\cal Z}^\xi_{1}}$, ${\cal I}_{{\cal Z}^\xi_{2}}$ be the corresponding ideal sheaves. For $\alpha\in H_i(S,{\Bbb Q})$ let $\widetilde\alpha\in H^{4-i}(T_\xi,{\Bbb Q})$ be the class whose restriction to each component ${\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)$ of $T_\xi$ is $\alpha_{n,m}$. Then for all $l,r$ with $l+2r=N$ we define a map $H_2(S,{\Bbb Q})\longrightarrow {\Bbb Q}$ by \begin{eqnarray*} \delta_{\xi,l,r}(\alpha)&:=& \sum_{b=0}^{l}\sum_{c=0}^r (-1)^{r-c+e_\xi}2^{b+2c-N}{l\choose b} {r\choose c}\<\xi,\alpha\>^{l-b}\\ &&\int\limits_{T_\xi} \left(\widetilde\alpha^{b} \widetilde{pt}^{c}s_{2d_\xi-2c-b} ({\hbox{\rom{Ext}}}^1_{q}({\cal I}_{{\cal Z}^\xi_{1}}, {\cal I}_{{\cal Z}^\xi_{2}}\otimes V_\xi)\right)\end{eqnarray*} \end{defn} \begin{thm}\label{donch1} Let $S$ be a surface with $p_g(S)=q(S)=0$. Let $c_1\in H^2(S,{\Bbb Z})$ and $c_2\in {\Bbb Z}$. Assume that, if $c_1$ is divisible by $2$ in $H^2(S,{\Bbb Z})$ then $(4c_2-c_1^2)$ is not divisible by $8$. Let $W$ be a good wall of type $(c_1,c_2)$ and let $H_-$ and $H_+$ be ample divisors on $S$ lying in neighbouring chambers separated by $W$. Let $n_2$ be the number of $2$-torsion points in $H^2(S,{\Bbb Z})$. Then for all $l,r$ with $l+2r=N=(4c_2-c_1^2)-3$ we have $$\Phi_{H_+,l,r}-\Phi_{H_-,l,r}=n_2\sum_{\xi\in A^+(W)} \delta_{\xi,l,r}.$$ Here, as above, $$A^+(W)=\big\{ \xi\in H^2(S,{\Bbb Z})\bigm | Z\hbox{ defines the wall W and } \<\xi\cdot H_+\> >0 \}.$$ Therefore we get for a class $\alpha\in H_2(S,{\Bbb Q})$ \begin{eqnarray*} (\gamma_{c_1,c_2,g(H_+)}- \gamma_{c_1,c_2,g(H_-)}) (\underbrace{{pt},\ldots,{pt}}_r, \underbrace{\alpha,\ldots,\alpha}_l) =(-1)^{(c_1^2+\<c_1\cdot K_S \>)} n_2\sum_{\xi\in A^+(W)}\delta_{\xi,l,r}(\alpha). \end{eqnarray*} \end{thm} \begin{pf} If $H^2(S,{\Bbb Z})$ contains no $2$-torsion, and $a_-<a_+$ are in neighbouring minichambers separated by a miniwall $a$ with $A^+(a)=\{(\xi,n,m)\}$, then proposition \ref{donmin} computes $\Phi_{a_+,l,r}-\Phi_{a_-,l,r}$. By Serre duality and the definitions we see that in the notations of proposition \ref{donmin} ${\cal A}_+'\oplus{{\cal A}_-'}^\vee={\hbox{\rom{Ext}}}^1_{q}({\cal I}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes V_\xi)$. Thus, if for all miniwalls $a$ the set $A^+(a)$ consists of only one element, the theorem follows. If $N_2\subset H^2(S,{\Bbb Z})$ is the subgroup of $2$-torsion, then $T_\xi^{n,m}\simeq N_2\times{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)$. So the exceptional divisor in $\widetilde M$ has $n_2$ isomorphic components (or we add $n_2$ isomorphic connected components to $\widetilde M$ or subtract them), and each component gives the same contribution to $\Phi_{a_+,l,r}-\Phi_{a_-,l,r}$. Assume that $A^+(a)=\{(\xi_1,n_1,m_1),\ldots,(\xi_2,n_2,m_2)\}.$ Then, as we have seen above, the ${\text{\rom{\bf E}}}_{\xi_i}^{n_i,m_i}$ are disjoint, and, as the change $\Phi_{a_+,l,r}-\Phi_{a_-,l,r}$ can be computed on the exceptional divisor (or the added components), it is just the sum of the contributions for all $(\xi_i,n_i,m_i)$. The result now follows by adding up the contributions of all the miniwalls. \end{pf} By the results we have obtained so far, in order to compute explicitly the change of the Donaldson invariants, when the polarisation passes through a good wall $W=W^\xi$, we have first to determine the Chern classes of the bundles ${\hbox{\rom{Ext}}}^1_{q_\xi}({\cal I}_{{\cal Z}^\xi_{1}},{\cal I}_{{\cal Z}^\xi_{2}}\otimes V_\xi)$ on $T_\xi$, and then make explicit computations in the cohomology ring of ${\text{\rom{Hilb}}}^d(S\sqcup S)$. In the rest of this section we will again use the assumptions and notations from \ref{notado}, and will adress the first question, i.e. we express the Chern classes of the vector bundles ${\hbox{\rom{Ext}}}^1_{q}({\cal I}_{{\cal Z}_{1}},{\cal I}_{{\cal Z}_{2}}\otimes V)$ on $T={\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)$, (where we have written $V:=V_\xi$) in terms of those of ``standard bundles''. \begin{defn} Using the projections $p:S\times T \longrightarrow S$ and $q:S\times T\longrightarrow T$ we associate to a vector bundle $U$ of rank $r$ on $S$ the vector bundles $[U]_1:=q_*({\cal O}_{{\cal Z}_2}\otimes p^*(U))$ and $[U]_2:= q_*({\cal O}_{{\cal Z}_1}\otimes p^*(U))$ of ranks $rn$ (resp. $rm$) on $T$. \end{defn} For a Cohen-Macaulay scheme $Z$, we denote by $\omega_Z$ its dualizing sheaf. \begin{lem}\label{extlem} $${\hbox{\rom{Ext}}}^2_q({\cal O}_{{\cal Z}_1},{\cal O}_{{\cal Z}_2}\otimes p^*V)= q_*(\omega_{{\cal Z}_1}\otimes \omega_T^{-1}\otimes{\cal O}_{{\cal Z}_2}\otimes p^*V)$$ and ${\hbox{\rom{Ext}}}^i_q({\cal O}_{{\cal Z}_1},{\cal O}_{{\cal Z}_2}\otimes p^*V)=0$ for $i\ne 2$. \end{lem} \begin{pf} Let \begin{eqnarray}\label{resol}&&0\longrightarrow B_2\longrightarrow B_1 \longrightarrow {\cal O}_{S\times T}\longrightarrow {\cal O}_{{\cal Z}_1}\longrightarrow 0\end{eqnarray} be a locally free resolution on $S\times T$. We apply ${\cal Hom}(\cdot,{\cal O}_{{\cal Z}_2}\otimes p^*V)$ to obtain the complex $$0\longrightarrow {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow B_1^*\otimes {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow B_2^*\otimes {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow 0,$$ whose cohomologies are the ${\cal Ext}^i({\cal O}_{{\cal Z}_1},{\cal O}_{{\cal Z}_2}\otimes p^*V)$. We can arrive at this complex differently, namely by first dualizing and then tensorizing by ${\cal O}_{{\cal Z}_2}\otimes p^*V$. By dualizing and using that ${\cal Z}_1$ is Cohen-Macauley we obtain $$0\longrightarrow {\cal O}_{S\times T}\longrightarrow B_1^*\longrightarrow B_2^*\longrightarrow \omega_{{\cal Z}_1}\otimes \omega_T^{-1}\longrightarrow 0.$$ Tensorizing by ${\cal O}_{{\cal Z}_2}\otimes p^*V$ gives the sequence $$0\longrightarrow {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow B_1^*\otimes {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow B_2^*\otimes {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow \omega_{{\cal Z}_1}\otimes \omega_T^{-1}\otimes{\cal O}_{{\cal Z}_2}\otimes p^*V \longrightarrow 0,$$ which is exact by the corollaire on p. V.20 in \cite{Se} because ${\cal Z}_1$ and ${\cal Z}_2$ are Cohen-Macaulay and intersect properly. Hence ${\cal Ext}^2({\cal O}_{{\cal Z}_1},{\cal O}_{{\cal Z}_2}\otimes p^*V)= \omega_{{\cal Z}_1}\otimes \omega_T^{-1}\otimes{\cal O}_{{\cal Z}_2}\otimes p^*V$ and ${\cal Ext}^i({\cal O}_{{\cal Z}_1},{\cal O}_{{\cal Z}_2}\otimes p^*V)=0$ for $i<2$. As ${\cal Z}_2$ and ${\cal Z}_1$ are flat of dimension $0$ over $T$, the result follows by applying $q_*$. \end{pf} \begin{prop}\label{grot} In the Grothendieck ring of sheaves on $T$ we have the equality \begin{eqnarray*}{\hbox{\rom{Ext}}}^1_q({\cal I}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V)&=& [V]_2+([V^{\vee}(K_S)]_1)^{\vee} +(H^1(S,{\cal O}_S(-\xi))\oplus H^1(S,{\cal O}_S(-\xi+K_S)))\otimes{\cal O}_T\\ &&\qquad - q_*(\omega_{{\cal Z}_1}\otimes\omega_T^{-1}\otimes{\cal O}_{{\cal Z}_2}\otimes p^*V). \end{eqnarray*} \end{prop} \begin{pf}{\it Case n=0:} We will use repeatedly that $\xi$ defines a good wall, so in particular $q_*(p^*V)=R^2q_*(p^*V)=0$. We apply ${\hbox{\rom{Hom}}}_q({\cal O}_{S\times T},\cdot)$ to the sequence $$0\longrightarrow {\cal I}_{{\cal Z}_2}\otimes p^*V\longrightarrow p^*V\longrightarrow {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow 0$$ to obtain \begin{eqnarray}\label{ex1} 0\longrightarrow [V]_2\longrightarrow {\hbox{\rom{Ext}}}^1_q({\cal O}_{S\times T},{\cal I}_{{\cal Z}_2}\otimes p^*V) \longrightarrow R^1q_*(p^*V)\longrightarrow 0. \end{eqnarray} The surjectivity follows as ${\cal Z}_2$ is flat of dimension $0$ over $T$ and the injectivity by $q_*p^*V=0$. \noindent {\it General case:} We apply ${\hbox{\rom{Hom}}}_q(\cdot,{\cal I}_{{\cal Z}_2}\otimes p^*V)$ to the sequence $0\longrightarrow {\cal I}_{{\cal Z}_1}\longrightarrow{\cal O}_{S\times T}\longrightarrow {\cal O}_{{\cal Z}_1}\longrightarrow 0$ to get \begin{eqnarray}\label{ex2} \qquad\qquad 0\to {\hbox{\rom{Ext}}}^1_q({\cal O}_{S\times T},{\cal I}_{{\cal Z}_2}\otimes p^*V)\to {\hbox{\rom{Ext}}}^1_q({\cal I}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V)\to {\hbox{\rom{Ext}}}^2_q({\cal O}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V)\to 0. \end{eqnarray} The exactness on the left follows from the fact that $q_*({\cal I}_{{\cal Z}_2}\otimes p^*V)=0$ and so ${\hbox{\rom{Ext}}}^1_q({\cal O}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V)$ is torsion-free being a subsheaf of the locally free sheaf $R^1q_*({\cal I}_{{\cal Z}_2}\otimes p^*V)$. Its support is contained in $q({\cal Z}_1\cap{\cal Z}_2)$ and thus it is the zero sheaf. We apply ${\hbox{\rom{Hom}}}_q({\cal O}_{{\cal Z}_1},\cdot)$ to $0\longrightarrow {\cal I}_{{\cal Z}_2}\otimes p^*V\longrightarrow p^*V\longrightarrow {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow 0$ and use lemma \ref{extlem} to obtain \begin{eqnarray}\label{ex3} &&\qquad 0\longrightarrow {\hbox{\rom{Ext}}}^2_q({\cal O}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V)\longrightarrow {\hbox{\rom{Ext}}}^2_q({\cal O}_{{\cal Z}_1},p^*V) \longrightarrow q_*(\omega_{{\cal Z}_1}\otimes\omega_T^{-1}\otimes{\cal O}_{{\cal Z}_2}\otimes p^*V) \longrightarrow 0. \end{eqnarray} By duality ${\hbox{\rom{Ext}}}^2_q({\cal O}_{{\cal Z}_1},p^*V)=q_*({\cal O}_{{\cal Z}_1}\otimes p^*(V^\vee(K_S))^\vee=[V^\vee(K_S)]_1^\vee$. Thus the result follows by putting \ref{ex1} to \ref{ex3} together. \end{pf} \def\int\limits{\int\limits} \def{\cal U}{{\cal U}} \def{\hbox{\rom{Spec}\,}}{{\hbox{\rom{Spec}\,}}} \def{\Bbb P}_2^*{{\Bbb P}_2^*} \def{\Bbb P}_2{{\Bbb P}_2} \def{\cal M}{{\cal M}} \def{m}{{m}} \def{pt}{{\Bbb P}_2} \def{\cal S}{{\cal S}} \def\varepsilon{\varepsilon} \def{\Bbb R}{{\Bbb R}} \def{\cal P}{{\cal P}} \def{\cal A}{{\cal A}} \def{\cal G}{{\cal G}} \def{\cal V}{{\cal V}} \def{\Bbb Q}{{\Bbb Q}} \def{\Bbb Z}{{\Bbb Z}} \def{\Bbb N}{{\Bbb N}} \def{\Bbb P}{{\Bbb P}} \def{\Bbb C}{{\Bbb C}} \def{\cal I}{{\cal I}} \def{\tilde L}{{\tilde L}} \def\X#1#2{X(#1,#2)} \def{\hbox{\rom{id}}}{{\hbox{\rom{id}}}} \def{\cal Coh}{{\cal Coh}} \def{\cal I}{{\cal I}} \def{\cal F}{{\cal F}} \def{\cal E}{{\cal E}} \def{\cal O}{{\cal O}} \def{\cal O}{{\cal O}} \def{\cal P}{{\cal P}} \def\varepsilon{\varepsilon} \def{\hbox{\rom{Ext}}}{{\hbox{\rom{Ext}}}} \def{\hbox{\rom{Hom}}}{{\hbox{\rom{Hom}}}} \def{\cal Ext}{{\cal Ext}} \def{\cal Hom}{{\cal Hom}} \def{X^{[n]}}{{X^{[n]}}} \def{S^{[n]}}{{S^{[n]}}} \def{X(n)}{{X(n)}} \def{\widetilde \xn}{{\widetilde {X(n)}}} \def\uberschrift#1{\bigskip\bigskip {\bf #1}\medskip} \def\Theor#1#2{\medskip \noindent{{\bf Theorem~#1.~~}}{\it #2}\medskip} \def\Thm#1#2{\medskip \noindent{{\bf Theorem~#1.~~}}{\it #2}\medskip} \def\Prop#1#2{\medskip \noindent{{\bf Proposition~#1.~~}}{\it #2}\medskip} \def\Cor#1#2{\medskip \noindent{{\bf Corollary~#1.~~}}{\it #2}\medskip} \def\Lemma#1#2{\medskip \noindent{{\bf Lemma~#1.~~}}{\it #2}\medskip} \def\Def#1{\medskip \noindent{{\bf Definition~#1.~~}}} \def\Not#1{\medskip \noindent{{\bf Notation~#1.~~}}} \def\Rmk#1{\medskip \noindent{{\bf Remark~#1.~~}}} \def\Rem#1{\Rmk{#1}} \def\Ex#1{\medskip \noindent{{\bf Example~#1.~~}}} \def\Parag#1#2{\bigskip \noindent {\S}{\bf #1.~#2.} \medskip} \def\demo{Proof}{\demo{Proof}} \def\vskip 1pt \noindent{\vskip 1pt \noindent} \def\whsq{\vbox to 5.8pt {\offinterlineskip\hrule \hbox to 5.8pt{\vrule height 5.1pt\hss\vrule height 5.1pt}\hrule}} \def\vrule height 8pt width 6pt{\vrule height 8pt width 6pt} \def{\hfill {\whsq}}\enddemo{{\hfill {\whsq}}\enddemo} \def\Qed{{\hfill {\whsq}}\enddemo} \def{\Bbb P}{{\Bbb P}} \def\longrightarrow{\longrightarrow} \def{\cal O}{{\cal O}} \def{\cal I}{{\cal I}} \def\diagramm#1{ \def\baselineskip18pt\lineskip7pt\lineskiplimit7pt{\baselineskip18pt\lineskip7pt\lineskiplimit7pt} \matrix{#1}} \def\mapr#1{\smash{ \mathop{\longrightarrow}\limits^{#1}}} \def\mapsr#1{{ \mathop{\longrightarrow}\limits^{#1}}} \def\mapinto#1{\smash{ \mathop{\longrighthook}\limits^{#1}}} \def\mapl#1{\smash{ \mathop{\longleftarrow}\limits^{#1}}} \def\mapd#1{\Big\downarrow \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} \def\mapu#1{\Big\uparrow \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} \def\mapsec#1#2{\mapdownl#1\mapup#2} \def\mapne#1{\nearrow \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} \def\mapse#1{\searrow \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} \def\mapnw#1{\nwarrow \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} \def\mapsw#1{\swarrow \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} \def\mathrel{\smash=}{\mathrel{\smash=}} \def\relgl\joinrel\relgl\joinrel\relgl{\mathrel{\smash=}\joinrel\mathrel{\smash=}\joinrel\mathrel{\smash=}} \def{\text{\rom{Hilb}}}^n(S){{\text{\rom{Hilb}}}^n(S)} \def{S^{[n-1,n]}}{{S^{[n-1,n]}}} \defS^{(n)}{S^{(n)}} \def{\cal H}{{\cal H}} \def{\Bbb C}{{\Bbb C}} \def{\Bbb R}{{\Bbb R}} \def{\Bbb P}{{\Bbb P}} \def{\Bbb N}{{\Bbb N}} \def{\cal P}{{\cal P}} \def{\Bbb Z}{{\Bbb Z}} \def{\Bbb Q}{{\Bbb Q}} \def{\cal T}{{\cal T}} \def\varepsilon{\varepsilon} \def\textstyle\bigoplus\limits{\textstyle\bigoplus\limits} \def\varphi{\varphi} \defC^{\infty}{C^{\infty}} \def\frak S{\frak S} \def\({\left(} \def\){\right)} \def{\cal B}{{\cal B}} \def{\cal B}^\prime{{\cal B}^\prime} \def\hbox{Supp}\,{\hbox{Supp}\,} \defP(n){P(n)} \def{\text{\rom{\bf b}}}{{\text{\rom{\bf b}}}} \defS^{(\bb)}{S^{({\text{\rom{\bf b}}})}} \def{\text{\rom{\bf E}}}{{\text{\rom{\bf E}}}} \def\alpha{\alpha} \def\Gamma{\Gamma} \def\bar \Gamma{\bar \Gamma} \def\gamma{\gamma} \def\bar \eta{\bar \eta} \def{\cal Z}{{\cal Z}} \def{pt}{{pt}} \def\alpha{\alpha} \def{S^{(n)}}{{S^{(n)}}} \def{S^{(m)}}{{S^{(m)}}} \def\<{\langle} \def\>{\rangle} \def{\text{\rom{Hilb}}}{{\text{\rom{Hilb}}}} \def{\Cal W}{{\cal W}} \def{\hbox{\rom{Tor}}}{{\hbox{\rom{Tor}}}} \def{\hbox{\rom{Ext}}}{{\hbox{\rom{Ext}}}} \def\stil#1{\tilde S^#1} \def\pi{\pi} \def{{\hbox{$*$}}}{{{\hbox{$*$}}}} \defg{g} \def\phi{\varphi} \def\tilde{\tilde} \def{\bar\al}{{\bar\alpha}} \section{Explicit computations on Hilbert schemes of points} The aim of this section is to make theorem \ref{donch1} more explicit. We want to compute the contributions $\delta_{\xi}$ to the change of the Donaldson invariants for a class $\xi$ defining a good wall, in terms of cohomology classes and intersection numbers on $S$. We do not succeed in determining $\delta_{\xi}$ completely. It turns however out that $\delta_{\xi}$ can be developed in terms of powers of $\xi$ and we will compute the six lowest order terms (as predicted by the conjecture of Kotschick and Morgan half of them are zero). \begin{nota} \label{xino} In this section we fix a class $\xi\in H^2(S,{\Bbb Z})$ which defines a good wall of type $(c_1,c_2)$ and will therefore drop $\xi$ in our notation. In particular we write $d:=d_\xi$, $e:=e_\xi$ and $T:={\text{\rom{Hilb}}}^{d}(S\sqcup S)$. As usual let $p$ and $q$ be the projections of $S\times T$ to $S$ and $T$ respectively. We write $V:={\cal O}_S(-\xi)\oplus{\cal O}_S(-\xi+K_S)$, ${\cal Z}_1:={\cal Z}_1^\xi$, ${\cal Z}_2:={\cal Z}_2^\xi$ and $\delta_{l,r}:=\delta_{\xi,l,r}$. We put $\Gamma:=q_*(\omega_{{\cal Z}_1}\otimes\omega_{S\times T}^{-1}\otimes {\cal O}_{{\cal Z}_2}\otimes p^*V)$. \end{nota} We see by theorem \ref{donch1} that, in order to compute the change $\delta_{l,r}$, it is enough to compute $\int_T s({\hbox{\rom{Ext}}}^1_q({\cal I}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V))\cdot \gamma$ for all classes $\gamma\in H^*(T,{\Bbb Q})$ which are pull-backs from $S^{(d)}$ via the natural map ${\text{\rom{Hilb}}}^d(S\sqcup S)\longrightarrow (S\sqcup S)^{(d)}\longrightarrow S^{(d)}$. By proposition \ref{grot} we have \begin{eqnarray*} \int_T s({\hbox{\rom{Ext}}}^1_q({\cal I}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V))\cdot\gamma&=& \int_T s([V^\vee(K_S)]_1^\vee \oplus [V]_2)\cdot \gamma\\ &&\quad +\int_T (c(\Gamma)-1)s([V^\vee(K_S)]_1^\vee \oplus [V]_2)\cdot\gamma.\\ \end{eqnarray*} In the first part of this section we compute the first of these two integrals. As said in the beginning of this section, we only want to compute the terms of lowest order of the change of the Donaldson invariants. This corresponds to restricting our attention to a big open subset of the Hilbert scheme of points. \begin{nota} A point $\sigma\in {S^{(n)}}$ is a formal linear combination $\sum_i m_i x_i$ of points on $S$ with positive integer coefficients and $\sum_i m_i=n$. The support $supp(\sigma)$ is the set of points $x_i$. For all $i\le n$ let $$S^{(n)}_i:=\big\{\sigma\in S^{(n)}\bigm| \# supp(\sigma)\ge n-i+1\big\}.$$ Furthermore, for any variety $X$ with a canonical morphism $f:X\longrightarrow S^{(n)}$, we denote $f^{-1}S^{(n)}_i$ by $X_i$. For the universal family $Z_n(S)\subset S\times {\text{\rom{Hilb}}}^n(S)$ we denote by $Z_n(S)_i$ the preimage of ${\text{\rom{Hilb}}}^n(S)_i$. \end{nota} In order to compute the first integral we will use an inductive approach, which is based on results of \cite{E1},\cite{F-G} and which is similar to computations in \cite{Go2} on the Hilbert scheme of $3$ points. \begin{defn}(\cite{E1},\cite{F-G}) Let $S^{[n-1,n]}\longrightarrow S\times {\text{\rom{Hilb}}}^{n-1}(S)$ be the blow-up along the universal family $Z_{n-1}(S)$, and let $F_n$ the exceptional divisor. Contrary to our conventions in the previous section for any vector bundle $E$ on $S$ we will denote by $E[n]$ the vector bundle $q_*({\cal O}_{Z_n(S)}\otimes p^* E)$ on ${\text{\rom{Hilb}}}^n(S)$. \end{defn} \begin{thm} \label{snn}(\cite{E1}) $S^{[n-1,n]}$ is smooth. There is a natural morphism $S^{[n-1,n]}\longrightarrow {\text{\rom{Hilb}}}^n(S)$, and on $S^{[n-1,n]}$ we have an exact sequence \begin{eqnarray}\label{hilbseq} 0\longrightarrow V(-F_n)\longrightarrow V[n]\longrightarrow V[{n-1}]\longrightarrow 0, \end{eqnarray} where we have used convention \ref{convent}. \end{thm} It is easy to see that the induced map $S^{[n-1,n]}\longrightarrow S\times {\text{\rom{Hilb}}}^n(S)$ factors through $Z_n(S)\subset S\times {\text{\rom{Hilb}}}^n(S)$, and that the map $S^{[n-1,n]}\longrightarrow Z_n(S)$ is an isomorphism over the open set $Z_n(S)_1$. We denote by $S^{[n-1,n]}_i$ the preimage of $Z_n(S)_i$. \begin{lem}\label{hilbnorm} Let $N^\vee_n$ be the conormal sheaf of $Z_n(S)$ in $S\times {\text{\rom{Hilb}}}^n(S)$. Then we have an exact sequence on $S^{[n-1,n]}_2$ \begin{eqnarray*}\label{hilbnseq} 0 \longrightarrow N^\vee_n\longrightarrow T_S^\vee\longrightarrow {\cal O}_{F_n}(-F_n)\longrightarrow 0. \end{eqnarray*} Here we have used the convention \ref{convent}. In particular on $S^{[n-1,n]}_2$ we get $$s(N^\vee_n)=s(T^\vee_S){1-F_n\over 1-2F_n}.$$ \end{lem} \begin{pf} It is easy to see that $S^{[n-1,n]}_2\longrightarrow {\text{\rom{Hilb}}}^n(S)_2$ is a branched $n$-fold cover, \'etale out of $F_n$ and with ramification of order $1$ along $F_n$. So the result follows in the same way as in the proof of (\cite{F-G}, lemma 2.10). \end{pf} \begin{lem} \label{discard} Let $i$ be a positive integer and assume that $\alpha_1,\alpha_2\in A^{*}({\text{\rom{Hilb}}}^n(S))$ have the same pull-back to $ {\text{\rom{Hilb}}}^n(S)_i$. Then $$\int\limits_{{\text{\rom{Hilb}}}^n(S)}\alpha_1\cdot \beta= \int\limits_{{\text{\rom{Hilb}}}^n(S)}\alpha_2\cdot \beta$$ for all $\beta\in H^{4n-4i-2}(S^{(n)},{\Bbb Q})$. The same result holds if we replace ${\text{\rom{Hilb}}}^n(S)_i$ by $S^{[n-1,n]}_i$. \end{lem} \begin{pf} Let $j:{\text{\rom{Hilb}}}^n(S)\setminus {\text{\rom{Hilb}}}^n(S)_i\longrightarrow Hilb^n(S)$ be the inclusion. We get $\alpha_1=\alpha_2+j_{*}(\mu)$ for a class $\mu\in A^*({\text{\rom{Hilb}}}^n(S)\setminus {\text{\rom{Hilb}}}^n(S)_i).$ As the codimension of the complement of $S^{(n)}_i$ in $S^{(n)}$ is $2i$, the result follows by the projection formula. \end{pf} \begin{nota} \label{tplusminus} For all $l\ge 1$ we denote by $\Delta_l$ the "small" diagonal $\{(x,\ldots,x)\ | \ x\in S\}$ and by $[\Delta_l]$ its cohomology class. We define classes $t_{1-},t_{2-},t_{3-}\in H^*(S,{\Bbb Q})$ by \begin{eqnarray*} t_{1-}&:=&1+(2\xi-K_S)+(3\xi^2-3\xi K_S+K_S^2),\\ t_{2-}&:=&3+(18\xi-13K_S)+(63\xi^2-91\xi K_S+33K_S^2+5s_2(S)),\\ t_{3-}&:=&27+(270\xi-237K_S). \end{eqnarray*} Here $s_i(S):=s_i(T_S)$ is the $i^{th}$ Segre class of $S$. We define $t_{1+}$, $t_{2+}$, $t_{3+}$ by replacing $K_S$ by $(-K_S)$ in the definition of $t_{1-}$, $t_{2-}$, $t_{3-}$ respectively and put $t_i:=t_{i-}+t_{i+},$ i.e. \begin{eqnarray*} t_1&=&2+4\xi+6\xi^2+2K_S^2,\\ t_2&=&6+36\xi+126\xi^2+66K_S^2+10s_2(S),\\ t_3&=&54+540\xi. \end{eqnarray*} \end{nota} \begin{lem}\label{hilbind} Let $\gamma\in H^{4n-2k}(S^{(n)},{\Bbb Q})$ with $k\le 5$. Then \begin{eqnarray*} n\int\limits_{{\text{\rom{Hilb}}}^n(S)}s(V[n])\cdot\gamma =\sum_{l=1}^3\int\limits_{S^l\times {\text{\rom{Hilb}}}^{n-l}(S)} (-1)^{l-1} [\Delta_l]p_1^*t_{l-}\cdot s(V[n-l])\cdot \gamma, \end{eqnarray*} where $p_1:S^l\longrightarrow S$ is the projection to the first factor. \end{lem} \begin{pf} By theorem \ref{snn} we have the identity $s(V[n])=s(V(-F_n))s(V[n-1])$ on $S^{[n-1,n]}$ and furthermore $$s(V(-F_n))=\sum_{i,j\ge 0}{i+j+1\choose i+1} s_i(V) F_n^j.$$ So we get \begin{eqnarray} n\int_{{\text{\rom{Hilb}}}^n(S)}s(V[n])\cdot\gamma &=&\int\limits_{S\times{\text{\rom{Hilb}}}^{n-1}(S)}s(V)s(V[{n-1}])\cdot\gamma\label{ha}\\ \label{secint} +\sum_{i,j\ge 0}&&\int\limits_{{S^{[n-1,n]}}}F_n {i+j+2\choose i+1} s_i(V)F_n^j s(V[{n-1}])\cdot\gamma.\label{hb} \end{eqnarray} By using $V={\cal O}_S(-\xi)\oplus{\cal O}_S(-\xi+K_S)$, we see immediately that $s(V)=t_{1-}$. We denote for all $i$ by $f_i$ the composition $$S^{[n-i,n-i+1]}\to S\times {\text{\rom{Hilb}}}^{n-i}(S)\to S\times S^{(n-i)} \to S^i\times S^{(n-i)} \to S^{(n)},$$ where the second map is induced by the diagonal map $S\longrightarrow S^i$ and put we $\gamma_i:=f_i^*(\gamma)$. The integral (\ref{secint}) can be expressed as an integral over $F_n$. We push it forward to $Z_{n-1}(S)\subset S\times {\text{\rom{Hilb}}}^{n-1}(S)$ and pull back to $S^{[n-2,n-1]}$. Note that $f_2$ maps $S^{[n-2,n-1]}_i$ to $S^{(n)}_{i+1}$. So we get, in view of lemma \ref{hilbnorm} and lemma \ref{discard}, \begin{eqnarray}(\ref{hb})=-\int\limits_{S^{[n-2,n-1]}}\sum_{i,j>0} {i+j+3\choose i+1} s_i(V) s_j(N^\vee_{n-1})s(V[{n-1}])\cdot \gamma_2. \label{hc} \end{eqnarray} We now again use the identity $s(V[{n-1}])=s(V(-F_{n-1}))s(V[{n-2}])$ on $S^{[n-2,n-1]}$ and obtain \begin{eqnarray} (\ref{hc})=\!\!\!\!\! \int\limits_{S\times {\text{\rom{Hilb}}}^{n-2}(S)} \sum_{i+j+l\le 2}{i+l+3\choose i+1} s_i(V) s_j(V)s_l(T^\vee_S) s(V[{n-2}])\cdot \gamma_2\label{hd}\\ \ \ \ \ \ \ \ \ +\!\!\!\!\!\!\!\int\limits_{S^{[n-2,n-1]}}\!\!\sum_{i,j,l}\!\!\!\!{i+l+3\choose i+1} s_i(V) s(V[{n-2}]) \!\left(s_j(V(F_{n-1}))s_l(N^\vee_{n-1})-s_j(V)s_l(T^\vee_S) \right)\!\cdot\! \gamma_2\label{he} \end{eqnarray} By explicit calculation and the definition of $V$, we get for the first integral \begin{eqnarray*}&&\sum_{i+j+l\le 2}{i+l+3\choose i+1} s_i(V) s_j(V)s_l(T^\vee_S)\\ &&\quad= 3+9s_1(V)-4K_S+13s_2(V)+6s_1(V)^2-14s_1(V)K_S+5s_2(S)\\ &&\quad=t_{2-}. \end{eqnarray*} Now we compute the integral (\ref{he}). We use the formula $$s(N^\vee_{n-1})=s(T^\vee_S){1-F_{n-1}\over 1-2F_{n-1}}$$ and the notation $$2^{[l]}=\begin{cases} 1&l<0;\\ 2^l&l\ge 0.\end{cases}$$ to obtain \begin{eqnarray*}\qquad (\ref{he})&=&- \int\limits_{S^{[n-2,n-1]}}\sum_{i,j_1,j_2,k_1,k_2} F_{n-1}{i+k_1+k_2+3\choose i+1}{j_1+j_2+1\choose j_1+1} 2^{[k_2-1]} \cdot\\ &&\qquad\qquad\qquad\qquad\qquad\qquad \cdot s_i(V)s_{j_1}(V)s_{k_1}(T^\vee_S)F_{n-1}^{j_2+k_2-1} s(V[{n-2}])\cdot\gamma_2. \end{eqnarray*} This can again be expressed as an integral over $F_{n-1}$. We push forward to $Z_{n-2}(S)\subset S\times{\text{\rom{Hilb}}}^{n-2}(S)$ and then pull back to $S^{[n-3,n-2]}$. Note that $f_3$ maps $S^{[n-3,n-2]}_i$ to $S^{(n)}_{i+2}$. Therefore using lemma \ref{hilbnorm} and lemma \ref{discard} to see that we can replace the push-forward of $F_{n-1}^l$ by the pull-back of $s_{l-2}(T^\vee_S)$ via the projection $S^{[n-3,n-2]}\longrightarrow S\times{\text{\rom{Hilb}}}^{n-3}(S)$. We then push forward to $S\times {\text{\rom{Hilb}}}^{n-3}(S)$ and notice that by theorem \ref{snn} and lemma \ref{discard} we can replace the push-forward of $s(V[{n-2}])$ by $s(V)s(V[{n-3}])$. Putting all this together we obtain \begin{eqnarray*} (\ref{he})&=& \int\limits_{S\times Hilb^{n-3}(S)}\sum_{i+j_1+j_2+k_1+k_2+l\le 3} {i+k_1+k_2+3\choose i+1}{j_1+j_2+1\choose j_1+1} 2^{[k_2-1]} \cdot\\ &&\qquad\qquad\cdot s_i(V)s_{j_1}(V)s_l(V)s_{k_1}(T^\vee_S)s_{j_2+k_2-2}(T^\vee_S) s(V[{n-3}])\cdot\gamma_3. \end{eqnarray*} We obtain, again by direct calculation, \begin{eqnarray*} \sum_{i+j_1+j_2+k_1+k_2+l\le 3} &&{i+k_1+k_2+3\choose i+1}{j_1+j_2+1\choose j_1+1} 2^{[k_2-1]} s_i(V)s_{j_1}(V)s_l(V)s_{k_1}(T^\vee_S)s_{j_2+k_2-2}(T^\vee_S)\\&& = 27+135s_1(V)-102K_S=27+270\xi-237 K_S. \end{eqnarray*} This completes the proof. \end{pf} \begin{rem}\label{hilbind1} Let $\gamma\in H^{4n-2k}(S^{(n)},{\Bbb Q})$ with $k\le 5$. Then the same proof shows \begin{eqnarray*} n\int\limits_{{\text{\rom{Hilb}}}^n(S)}s((V^\vee(K_S)[n])^\vee)\cdot\gamma =\sum_{l=1}^3\int\limits_{S^l\times {\text{\rom{Hilb}}}^{n-l}(S)} (-1)^{l-1} [\Delta_l]p_1^*t_{l+}\cdot s((V^\vee(K_S)[n-l])^\vee)\cdot \gamma. \end{eqnarray*} \end{rem} We will now introduce a compact notation for some symmetric cohomology classes on $S^n$ that will also help us in organizing our combinatorical calculations. \begin{defn}\label{polnota} We denote by $\frak S_{n}$ the symmetric group on $n$ letters, which acts on $S^n$ by permuting the factors. For $\alpha\in H^{2*}(S^k,{\Bbb Q})$ and $\beta\in H^{2*}(S^l,{\Bbb Q})$ we define $\alpha{{\hbox{$*$}}} \beta\in H^{2*}(S^{k+l},{\Bbb Q})^{\frak S_{k+l}}$ by putting $$\alpha{{\hbox{$*$}}} \beta:={1\over (k+l)!} \sum_{\sigma\in \frak S_{k+l}} (p_{\sigma(1)}\times\ldots\times p_{\sigma(k)})^{{\hbox{$*$}}} \alpha \cdot (p_{\sigma(k+1)}\times\ldots\times p_{\sigma(k+l)})^*\beta.$$ It is easy to see that $*$ is a commutative and associative operation. We will denote $$\alpha^{{{\hbox{$*$}}} k}:=\underbrace{\alpha{{\hbox{$*$}}} \alpha{{\hbox{$*$}}} \ldots{{\hbox{$*$}}} \alpha}_k.$$ \end{defn} \begin{rem} \label{starrem} The following elementary properties of $*$ will be very important for our further computations: \begin{enumerate} \item For $\alpha\in H^{2*}(S^k,{\Bbb Q})$, $\beta\in H^{2*}(S^l,{\Bbb Q})$ and $w\in H^*(S^{k+l},{\Bbb Q})^{\frak S_{k+l}}$ it follows immediately from the symmetry of $w$ that \begin{eqnarray*}\int_{S^{k+l}}(\alpha{{\hbox{$*$}}} \beta)\cdot w &=&\int_{S^{k+l}}(p_{1}\times\ldots \times p_{k})^*\alpha \cdot (p_{k+1}\times\ldots \times p_{k+l})^*\beta\cdot w\\ &=&\sum_{(w_1,w_2)}\int_{S^{k}}\alpha w_1 \cdot \int_{S^{l}}\beta w_2. \end{eqnarray*} Here $w=\sum_{(w_1,w_2)} w_1\cdot w_2$ is the K\"unneth decomposition. Analogous results hold if more then two factors are multiplied via $*$. \item Let $1$ denote the neutral element of the ring $H^*(S,{\Bbb Q})$. Then $1^{{{\hbox{$*$}}} k}$ is the neutral element of $H^*(S^k,{\Bbb Q})$. \item It is also easy to see from the definitions that $*$ fulfills the distributive law $\alpha {{\hbox{$*$}}} (\beta_1+\beta_2)=\alpha {{\hbox{$*$}}} \beta_1+\alpha {{\hbox{$*$}}} \beta_1$. In fact $+$ and ${{\hbox{$*$}}} $ make $\bigoplus_{n\ge 0}H^{2*}(S^n,{\Bbb Q})^{\frak S_n}$ a commutative ring. \item In particular the binomial formula holds: $$\sum_{k+l=n} {n\choose k}\alpha^{{{\hbox{$*$}}} k}{{\hbox{$*$}}}\beta^{{{\hbox{$*$}}} l} =(\alpha+\beta)^{{{\hbox{$*$}}} n}.$$ \end{enumerate} \end{rem} \begin{nota}\label{starnota} For a class $\alpha\in H^*(S,{\Bbb Q})$ and a postive integer $i$ we denote by $(\alpha)_i:=[\Delta_i]p_1^*\alpha\in H^*(S^i,{\Bbb Q})^{\frak S_i}$, where $[\Delta_i]$ is the (small) diagonal $\{(x,\ldots,x) \ | \ x\in S\}$ in $S^i$. In particular $(\alpha)_1=\alpha$. We will in the future write $(\alpha)_i(\beta)_j$ instead of $(\alpha)_i{{\hbox{$*$}}} (\beta)_j$ and $(\alpha)_i^{m}$ instead of $(\alpha)_i^{{{\hbox{$*$}}} m}$. Furthermore we write $\alpha^{{{\hbox{$*$}}} m}\beta^{{{\hbox{$*$}}} l}$ and $\alpha^{{{\hbox{$*$}}} m}(\beta)_i$ instead of $\alpha^{{{\hbox{$*$}}} m}{{\hbox{$*$}}} \beta^{{{\hbox{$*$}}} l}$ and $\alpha^{{{\hbox{$*$}}} m}{{\hbox{$*$}}} (\beta)_i$. \end{nota} \begin{prop}\label{erstint} Let $\gamma\in H^{4d-2k}S^{(d)},{\Bbb Q})$ with $k\le 5$ and $w\in H^{4d-2k}(S^d,{\Bbb Q})$ its pull-back to $S^d$. Then \begin{eqnarray*} &&d!\int\limits_{{\text{\rom{Hilb}}}^d(S\sqcup S)} s([V^\vee(K_S)]^\vee_1\oplus [V]_2)\cdot \gamma\\ &&\qquad\qquad = \int\limits_{S^d}\left(t_1^{{{\hbox{$*$}}} d}-{d\choose 2}(t_2)_2 t_1^{{{\hbox{$*$}}} (d-2)} +2{d\choose 3}(t_3)_3t_1^{{{\hbox{$*$}}} (d-3)} +3{d\choose 4}(t_2)_2^{2}t_1^{{{\hbox{$*$}}} (d-4)} \right)\cdot w. \end{eqnarray*} \end{prop} \begin{pf} Let $n,m$ be nonnegative integers with $n+m=d$. Let $\gamma_1\cdot\gamma_2\in H^*(S^{(n)})\times S^{(m)},{\Bbb Q})\setminus 0$ be a K\"unneth component of the pull-back of $\gamma$ via $S^{(n)}\times S^{(m)} \longrightarrow S^{(d)}$. Let $w_1\cdot w_2\in H^{4n-2l}(S^n,{\Bbb Q})^{\frak S_n}\times H^{4n-2r}(S^m,{\Bbb Q})^{\frak S_m}$ be the pull-back of $\gamma_1\cdot\gamma_2$. Then $0\le l,r\le 5$. By an easy induction using lemma \ref{hilbind}, remark \ref{hilbind1} and remark \ref{starrem} and ignoring all terms of codimension $\ge 6$ we get \begin{eqnarray*} n!\int\limits_{{\text{\rom{Hilb}}}^n(S)} s((V^\vee(K_S)[n])^\vee)\cdot \gamma_1&=& \int_{S^n} P_{n}\cdot w_1 \\ {} m!\int\limits_{{\text{\rom{Hilb}}}^m(S)} s(V[m])\cdot \gamma_2&=& \int_{S^m}Q_m\cdot {w_2}, \end{eqnarray*} where \begin{eqnarray*} P_{n}&=& t_{1+}^{{{\hbox{$*$}}} n}-\sum_{i=2}^n(i-1)(t_{2+})_2t_{1+}^{{{\hbox{$*$}}} (n-2)}+ \sum_{i=2}^{n-2}\sum_{j=i+2}^n(i-1)(j-1)(t_{2+})_2^{2}t_{1+}^{{{\hbox{$*$}}} (n-4)}\\ &&\qquad +\sum_{i=3}^n(i-1)(i-2)(t_{3+})_3 t_{1+}^{{{\hbox{$*$}}} (n-3)}. \end{eqnarray*} and $Q_m$ is defined analogously to $P_n$ replacing $n,$ $ t_{1+},$ $t_{2+}$ and $t_{3+}$ by $m,$ $ t_{1-},$ $t_{2-}$ and $t_{3-}$ respectively. Applying again remark \ref{starrem} we obtain $$n!m!\int\limits_{{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)} s((V^\vee(K_S)[n])^\vee)s(V[m])\cdot \gamma= \int\limits_{S^d}(P_n{{\hbox{$*$}}} Q_m)\cdot w, $$ and thus $$ d!\int\limits_{{\text{\rom{Hilb}}}^d(S\sqcup S)} s([V^\vee(K_S)]^\vee_1)s([V]_2)\cdot \gamma\\ = \sum_{n+m=d} {d\choose n} \int\limits_{S^d} (P_n{{\hbox{$*$}}} Q_m)\cdot w$$ Finally we have \begin{eqnarray*} &&\sum_{n+m=d} {d\choose n} P_n{{\hbox{$*$}}} Q_m\\ &&\quad =\sum_{n+m=d} {d\choose n} \Bigg(t_{1+}^{{{\hbox{$*$}}} n} t_{1-}^{{{\hbox{$*$}}} m}-{n\choose 2} (t_{2+})_2t_{1+}^{{{\hbox{$*$}}}(n-2)}{{\hbox{$*$}}} t_{1-}^{{{\hbox{$*$}}} m} -{m\choose 2}(t_{2-})_2t_{1+}^{*n} t_{1-}^{{{\hbox{$*$}}} (m-2)}\\ &&\quad\quad + 3{n\choose 4}(t_{2+})_2^{ 2} t_{1+}^{{{\hbox{$*$}}}(n-4)}t_{1-}^{{{\hbox{$*$}}} m}+ {n\choose 2}{m\choose 2}(t_{2+})_2 (t_{2-})_2 t_{1+}^{{{\hbox{$*$}}}(n-2)}t_{1-}^{{{\hbox{$*$}}}(m-2)} + 3{m\choose 4}(t_{2-})_2^{ 2} t_{1+}^{{{\hbox{$*$}}} n} t_{1-}^{{{\hbox{$*$}}}(m-4)}\\ &&\quad\quad +2{n\choose 3}(t_{3+})_3 t_{1+}^{{{\hbox{$*$}}} (n-3)} t_{1-}^{{{\hbox{$*$}}} m} +2{m\choose 3}(t_{3-})_3 t_{1+}^{{{\hbox{$*$}}} n} t_{1-}^{{{\hbox{$*$}}}(m-3)}\Bigg)\\ &&=\quad t_1^{{{\hbox{$*$}}} d}-{d\choose 2}(t_2)_2 t_1^{{{\hbox{$*$}}} (d-2)} +3{d\choose 4}(t_2)_2^{ 2} t_1^{{{\hbox{$*$}}} (d-4)} +2{d\choose 3}(t_3)_3 t_1^{{{\hbox{$*$}}} (d-3)}. \end{eqnarray*} \end{pf} \def{pt}{{pt}} \def\alpha{\alpha} \def{S^{(n)}}{{S^{(n)}}} \def{S^{(m)}}{{S^{(m)}}} \def{S^{(n)}}{{S^{(n)}}} \def{S^{(m)}}{{S^{(m)}}} \def\<{\langle} \def\>{\rangle} \def{\text{\rom{Hilb}}}{{\text{\rom{Hilb}}}} \def{\Cal W}{{\Cal W}} \def{\hbox{\rom{Tor}}}{{\hbox{\rom{Tor}}}} \def\Hilb^{n} (S){{\text{\rom{Hilb}}}^{n} (S)} \def\Hilb^{m} (S){{\text{\rom{Hilb}}}^{m} (S)} \def\tilde S^{n}{\tilde S^{n}} \def\Hilb^{m} (S){{\text{\rom{Hilb}}}^{m} (S)} \def\tilde S^{m}{\tilde S^{m}} \def\pi{\pi} \defg{g} \def\phi{\varphi} \def\tilde{\tilde} \def{\bar\al}{{\bar\alpha}} \def\Star#1{{(#1)}_*} \def{\ti{\cal Z_2}}{{\tilde{\cal Z_2}}} \def\Delta^0{\Delta^0} \def{\cal E}#1#2{E_{#1#2}} \def\Ez#1#2{E^0_{#1#2}} \def\bE#1#2{\bar E_{#1#2}} \def\bEz#1#2{\bar E^0_{#1#2}} \def\bar F{\bar F} Now we want to compute the second integral $$\int_T(c(\Gamma)-1)s([V^\vee(K_S)]_1^\vee)s([V]_2)\cdot\gamma$$ for $\gamma\in H^{4d-2k}(S^{(d)},{\Bbb Q})$ with $k\le 5$, The conventions of \ref{convent} stay in effect. \begin{defn}\label{diazwei} Let $n,m$ be nonnegative integers with $n+m=d$. We consider the following diagram $$\CD S\times \Hilb^{n} (S) \times\Hilb^{m} (S) & @>q >> &\Hilb^{n} (S) \times \Hilb^{m} (S) &@>g>> &{S^{(n)}}\times {S^{(m)}}\cr @AA{\tilde\phi}A&&@A{\phi}AA&&@A\phi'AA\cr S\times \tilde S^{n} \times\tilde S^{m} & @>\tilde q>> &\tilde S^{n} \times \tilde S^{m}&@>\tig>> &S^{n}\times S^{m}\cr \endCD$$ Here, as above, $q$ and $\tilde q$ are the projections, $g:\Hilb^{n} (S) \times \Hilb^{m} (S) \longrightarrow {S^{(n)}}\times {S^{(m)}}$ is the product of the Hilbert-Chow morphisms and $\varphi':S^{n}\times S^{m} \longrightarrow {S^{(n)}}\times {S^{(m)}}$ is the product of the quotient maps, and all the other varieties and maps are defined via pull-back. For $i=1,2$ we put $\widetilde Z_i:=\tilde\varphi^{-1}({\cal Z}_i)$ and $\widetilde Z_{1,2}:=\widetilde Z_1\cap \widetilde Z_2$, (i.e. the scheme-theoretic intersection). \end{defn} \begin{nota} We denote by \begin{eqnarray*}&&({S^{(n)}}\times {S^{(m)}})_{*} :=\Big\{ (\sigma_+,\sigma_-)\in {S^{(n)}}\times{S^{(m)}} \Bigm|\\ &&\qquad\qquad\qquad\qquad\qquad\# supp(\sigma_+)\ge n-1,\, \# supp(\sigma_-)\ge m-1,\, \# supp(\sigma_++\sigma_-)\ge d-2\Big\}. \end{eqnarray*} Furthermore for all $X$ with a natural morphism $f:X\longrightarrow {S^{(n)}}\times {S^{(m)}}$ we denote $X_{*}:=f^{-1}({S^{(n)}}\times {S^{(m)}})_{*}$. We put $$\Gamma_{n,m}:=\varphi^*(\Gamma|_{(\Hilb^{n} (S)\times \Hilb^{m} (S))_{*}})$$ (see 7.1 for the definition of $\Gamma$). For $1\le i\le n$ (resp. $1\le j\le m$) we denote by $p_{i+}$ (resp. $p_{j-}$) the projection from $S^{n}\times S^{m}$ onto the $i^{th}$ factor of $S^{n}$ (resp. the $j^{th}$ factor of $S^{m}$). For $\epsilon =+,-$, $\eta =+,-$ we put \begin{eqnarray*}\Delta_{0,i}^{\epsilon }&:=&\Bigm\{(x,x_1^+,\ldots x_{n}^+, x_1^-,\ldots x_{m}^-)\in S\times S^{n}\times S^{m}\Bigm| x=x_i^\epsilon \Big\}\\ \Delta_{i,j}^{\epsilon \eta }&:=&\Big\{(x_1^+,\ldots x_{n}^+, x_1^-,\ldots x_{m}^-)\in S^{n}\times S^{m}\Bigm| x_i^\epsilon =x_j^\eta \Big\}\\ \Delta_{0,i,j}^{\epsilon \eta }&:=&\Big\{(x,x_1^+,\ldots x_{n}^+, x_1^-,\ldots x_{m}^-)\in S\times S^{n}\times S^{m}\Bigm| x=x_i^\epsilon =x_j^\eta \Big\}\\ \end{eqnarray*} We will also denote by $\Delta_{i,j}^{+-}$, $\Delta_{0,i,j}^{+-}$, $\Delta^{\epsilon }_{0,i}$ the pull-backs $\tilde g^{-1}(\Delta_{i,j}^{+- })$, $({\hbox{\rom{id}}}_S\times \tilde g)^{-1}(\Delta_{0,i,j}^{+-})$, $({\hbox{\rom{id}}}_S\times \tilde g)^{-1}(\Delta^{\epsilon }_{0,i})$. We denote $D_{i,j}:= \tilde g^{-1}(\Delta^{++}_{i,j})$ and $E_{i,j}:= \tilde g^{-1}(\Delta^{--}_{i,j})$. $D_{i,j}$ and $E_{i,j}$ are divisors (see below), we denote $F_{i}:=\sum_{j<i}D_{i,j}$ and $G_{i}:=\sum_{j<i}E_{i,j}$. \end{nota} \begin{rem}\label{hilbfacts} The following easy facts will be used throughout the computation. \begin{enumerate} \item It is well known that $({\text{\rom{Hilb}}}^{n}(S)\times {\text{\rom{Hilb}}}^{m}(S))_{*}$ is obtained from $(S^{n}\times S^{m})_{*}$ by blowing up all the $\Delta^{++}_{i,j}$ and $\Delta^{--}_{i,j}$ and taking the quotient by the action of the product of the symmetric groups $\frak S_{n}\times \frak S_{m}$. It follows that in fact $(\tilde S^{n}\times \tilde S^{m})_{*}$ is just the blow up of $(S^{n}\times S^{m})_{*}$ along the (disjoint) smooth subvarieties $(\Delta^{++}_{i,j})_*$ and $(\Delta^{--}_{i,j})_*$ and the $(D_{i,j})_*$ and $(E_{i,j})_*$ are the exceptional divisors. \item It is also easy to see that $(\widetilde Z_1)_{*}=\bigcup_{i=1}^{n} (\Delta_{0,i}^{+})_*$, $(\widetilde Z_2)_{*}=\bigcup_{j=1}^{m} (\Delta_{0,j}^{-})_*$ and therefore $$(\widetilde Z_{1,2})_{*}=\bigcup_{i=1}^{n}\bigcup_{j=1}^{m} (\Delta_{0,i,j}^{+-})_{*}.$$ (We mean here the scheme theoretic union, i.e. the scheme defined by the intersection of the ideals). \item For $i\ne j$ we have (scheme-theoretically) \begin{eqnarray*} (\Delta_{0,i}^{+})_{*}\cap \Delta_{0,j}^{+}&=& (\Delta_{0,i}^{+})_{*}\cap D_{i,j},\\ (\Delta_{0,i}^{-})_{*}\cap \Delta_{0,j}^{-}&=& (\Delta_{0,i}^{-})_{*}\cap E_{i,j}. \end{eqnarray*} \end{enumerate} \end{rem} \begin{lem}\label{comalg} \begin{enumerate} \item Let $X$ be a smooth variety, and let $Y$ and $Z$ be Cohen-Macauley subschemes of $X$ such that the ideal ${\cal I}_{Z/(Y\cup Z)}$ of $Z$ in $Y\cup Z$ is ${\cal O}_Y(-D)$ for a divisor $D$ on $Y$. Then in the Grothendieck ring of $X$ we have \begin{eqnarray*}{\cal O}_{Y\cup Z}&=&{\cal O}_Y(-D)+{\cal O}_Z\ \hbox{ and }\\ \omega_X^{-1}\otimes \omega_{Y\cup Z}&=&\omega_X^{-1}\otimes\omega_Y(D) +\omega_X^{-1} \otimes\omega_Z.\end{eqnarray*} \item Let $f:X\longrightarrow Y$ be a morphism between smooth varieties. Let $Z\subset Y$ be a Cohen-Macauley subscheme of codimension $2$ and assume $W:=f^{-1}(Z)$ has pure codimension $2$ in $X$. Then $$f^{*}(\omega_Y^{-1}\otimes \omega_Z)=\omega_X^{-1}\otimes \omega_W.$$ \item Let $X$ be a smooth variety and $Y$ and $Z$ Cohen-Macauley subschemes of codimension $2$ intersecting properly. Then in the Grothendieck ring of $X$ we have $${\cal O}_{Y}\otimes{\cal O}_{Z}={\cal O}_{Y\cap Z}.$$ \end{enumerate} \end{lem} \begin{pf} (1) The first identity follows from the standard exact sequence $$0\longrightarrow {\cal O}_Y(-D)\longrightarrow {\cal O}_{Y\cup Z}\longrightarrow {\cal O}_Z\longrightarrow 0.\eqno (*)$$ Now we dualize $(*)$ and use that for a two codimensional Cohen-Macauley subscheme $W\subset X$ we have $${\cal Ext}^i({\cal O}_W,{\cal O}_X)=\begin{cases} 0& i<2,\\ \omega_X^{-1}\otimes \omega_X& i=2\end{cases}$$ to obtain the sequence $$0\longrightarrow \omega_X^{-1}\otimes \omega_Z\longrightarrow \omega_X^{-1}\otimes \omega_{Y\cup Z}\longrightarrow \omega_X^{-1}\otimes\omega_Y(F)\longrightarrow 0$$ and thus the second identity. (2) We take a locally free resolution $$0\longrightarrow B\longrightarrow A\longrightarrow {\cal O}_Y\longrightarrow {\cal O}_Z\longrightarrow 0.$$ Pulling it back we obtain the sequence $$0\longrightarrow f^*B\longrightarrow f^*A\longrightarrow {\cal O}_X\longrightarrow {\cal O}_W\longrightarrow 0,$$ which stays exact by the Hilbert-Birch theorem (see e.g. \cite{P-S} lemma 3.1). Dualizing we obtain the exact sequence $$0\longrightarrow {\cal O}_X\longrightarrow f^*A\longrightarrow f^*B\longrightarrow \omega_X^{-1} \otimes\omega_Z\longrightarrow 0.$$ We can also arrive at this sequence differently, by first dualizing and then pulling back. This way we obtain the sequence $$0\longrightarrow {\cal O}_X\longrightarrow f^*A\longrightarrow f^*B\longrightarrow f^*(\omega_X^{-1} \otimes\omega_Z)\longrightarrow 0,$$ and (2) follows. (3) By the corollaire on p. 20 in \cite{Se} we have ${\hbox{\rom{Tor}}}_i({\cal O}_Y,{\cal O}_Z)=0$ for $i>0$, and (3) follows. \end{pf} \begin{lem}\label{grot2} In the Grothendieck ring of $(\widetilde S^{n}\times \widetilde S^{m})_{*}$ we have the equality $$\varphi^*(\Gamma_{n,m})=\sum_{i=1}^{n}\sum_{j=1}^{m}\Big( {\cal O}_{\Delta_{i,j}^{+-}}(F_i-G_j-p_{i+}^*\xi)+ {\cal O}_{\Delta_{i,j}^{+-}}(F_i-G_j-p_{i+}^*(\xi+K_S))\Big).$$ \end{lem} \begin{pf} Using remark \ref{hilbfacts}(2) and remark \ref{hilbfacts}(3) and applying lemma \ref{comalg}(1) inductively we obtain in the Grothendieck ring of $S\times(\widetilde S^n\times \widetilde S^m)_*$ the equalities \begin{eqnarray} \label{gr1}{\cal O}_{\widetilde Z_1}&=&\sum_{i=1}^n {\cal O}_{\Delta_{0,i}^+}(-F_i)\\ \label{gr2}{\cal O}_{\widetilde Z_2}&=&\sum_{j=1}^m {\cal O}_{\Delta_{0,j}^-}(-G_j)\ \hbox{ and} \end{eqnarray} \begin{eqnarray*} \widetilde\varphi^*(\omega_{S\times{\text{\rom{Hilb}}}^n(S)\times{\text{\rom{Hilb}}}^m(S)}^{-1}\otimes \omega_{{\cal Z}_1})&=& \omega_{S\times\widetilde S^n\times \widetilde S^m}^{-1} \otimes\omega_{\widetilde Z_1}\\ &=&\sum_{i=1}^n \omega_{S\times\widetilde S^n\times \widetilde S^m}^{-1} \omega_{\Delta_{0,i}^+}(F_i)\\ &=&\sum_{i=1}^n {\cal O}_{\Delta_{0,i}^+}(-p_{i+}^*K_S+F_i), \end{eqnarray*} where in the third and the last line we have used lemma \ref{comalg}(2). Now using lemma \ref{comalg}(3) and tenzorizing by $p^*V$ we obtain in the Grothendieck ring of $S\times(\widetilde S^n\times \widetilde S^m)_*$ the equality $$({\hbox{\rom{id}}}_S\times \varphi)^*(\omega_T^{-1}\otimes \omega_{{\cal Z}_1}\otimes {\cal O}_{{\cal Z}_2}\otimes p^*V) =\sum_{i=1}^{n}\sum_{j=1}^{m}\Big({\cal O}_{\Delta_{0,i,j}^{+-}}(F_i-G_j -p_{i+}^*\xi)+ {\cal O}_{\Delta_{0,i,j}^{+-}}(F_i-G_j -p_{i+}^*(\xi+K_S)\Big) .$$ The morphism $\varphi:(\widetilde S^{n}\times \widetilde S^{m})_{*} \longrightarrow {\text{\rom{Hilb}}}^{n}(S)\times {\text{\rom{Hilb}}}^{m}(S)$ is flat. Therefore we get by (\cite{Ha} prop.III.9.3) \begin{eqnarray*} &&\varphi^*q_*(\omega_T^{-1}\otimes \omega_{Z_1}\otimes {\cal O}_{Z_2}\otimes p^*V)\\ &&\qquad = \tilde q_*({\hbox{\rom{id}}}_S\times \varphi)^*(\omega_T^{-1}\otimes \omega_{Z_1}\otimes {\cal O}_{Z_2}\otimes p^*V)\\ &&\qquad=\sum_{i=1}^{n}\sum_{j=1}^{m}\tilde q_*( ({\cal O}_{\Delta_{0,i,j}^{+-}}(F_i-G_j -p_{i+}^*\xi)+ {\cal O}_{\Delta_{0,i,j}^{+-}}(F_i-G_j -p_{i+}^*(\xi+K_S))\\ &&\qquad =\sum_{i=1}^{n}\sum_{j=1}^{m}({\cal O}_{\Delta_{i,j}^{+-}}(F_i-G_j -p_{i+}^*\xi)+ {\cal O}_{\Delta_{i,j}^{+-}}(F_i-G_j -p_{i+}^*(\xi+K_S)), \end{eqnarray*} in the Grothendieck ring of $(\widetilde S^{n}\times \widetilde S^{m})_{*}$. The last identity follows from the fact that the projection $\tilde q|_{\Delta_{0,i,j}^{+-}}:{\Delta_{0,i,j}^{+-}}\longrightarrow \Delta_{i,j}^{+-} $ is an isomorphism. \end{pf} \begin{lem}\label{rrsd} Let $X$ be a smooth variety and let $i:Y\longrightarrow X$ be the closed embedding of a smooth subvariety of codimension $2$ with conormal bundle $N^\vee$. Let $D$ be a divisor on $Y$. Then $$c(i_*({\cal O}_Y(-D)))=1-i_*\Big(\sum_{k,l\ge 0} {k+l+1\choose l} D^l s_k(N^\vee)\Big).$$ \end{lem} \begin{pf} This is a straightforward application of Riemann-Roch without denominators \cite{Jo}. \end{pf} \begin{lem}\label{cgamma} Let $1\le k\le 5$. Then $c_k(\Gamma_{n,m}|_{(\widetilde S^n\times \widetilde S^m)_*})$ is the part of degree $k$ of $$\matrix \displaystyle -\sum_{(i,j)} [\Delta_{i,j}^{+-}]\Big(2+4p_{i+}^*\xi+4(G_j-F_i) +p_{i+}^*(6\xi^2+3s_2(S)-K_S^2) +12p_{i+}^*\xi(G_j-F_i)\cr \displaystyle+6(F_i^2+G_j^2)+p_{i+}^*(24\xi^2+12s_2(S)-4K_S^2) (G_j-F_i) +24p_{i+}^*\xi(G_j^2+F_i^2)+8(G_j^3-F_i^3)\Big)\cr +\displaystyle \sum_{(i,j)\ne (i_1,j_1)} [\Delta_{i,j}^{+-}][\Delta_{i_1,j_1}^{+-}](4+8p_{i+}^*\xi+8p_{i_1+}^*\xi) \endmatrix\eqno (7.20.1)$$ Here $(i,j)$ and $(i_1,j_1)$ run through $\{1,\ldots, n\}\times \{1,\ldots, m\}$. \end{lem} \begin{pf} We compute on $(\widetilde S^n\times \widetilde S^m)_*$. We notice that $[\Delta_{i,j}^{+-}]$ is just the pull-back of the corresponding class in $S^{n}\times S^{m}$ via $\tilde g$ and the conormal bundle of $\Delta_{i,j}^{+-}$ is just the pull-back of the conormal bundle, i.e. $p_{i+}^*(T^\vee_S)$. Furthermore we note that on $(\widetilde S^{n}\times \widetilde S^{m})_{*}$ we have $[\Delta_{i,j}^{+-}]\cdot F_i\cdot G_j=0$. Therefore we obtain by lemma \ref{rrsd} after some calculation that for $1\le k\le 5$ the Chern class $c_k({\cal O}_{\Delta_{i,j}^{+-}}(-p_{i+}^*\xi +F_i-G_j))$ is the part of degree $k$ of \begin{eqnarray*}&&-[\Delta_{i,j}^{+-}]\Big(1+p_{i+}^*(2\xi-K_S)+2(G_j-F_i) +p_{i+}^*(3\xi^2-3\xi K_S+s_2(S)) \\ &&\qquad+p_{i+}^*(6\xi-3K_S)(G_j-F_i)+3(G_j^2+F_i^2)\\ &&\qquad +p_{i+}^*(12\xi^2-12K_S\xi+4s_2(S))(G_j-F_i) +p_{i+}^*(12\xi-6K_S)(G_j^2+F_i^2)+4(G_j^3-F_i^3)\Big).\\ \end{eqnarray*} Analogously we obtain that $c_k({\cal O}_{\Delta_{i,j}^{+-}} (-p_{i+}^*(\xi+K_S) +F_i-G_j))$ is the part of degree $k$ of \begin{eqnarray*} &&1-[\Delta_{i,j}^{+-}]\Big(1+p_{i+}^*(2\xi+K_S)+2(G_j-F_i) +p_{i+}^*(3\xi^2+3\xi K_S+s_2(S))\\ &&\qquad +p_{i+}^*(6\xi+3K_S)(G_j-F_i)+3(G_j^2+F_i^2)\\ &&\qquad +p_{i+}^*(12\xi^2+12K_S\xi+4s_2(S))(G_j-F_i) +p_{i+}^*(12\xi+6K_S)(G_j^2+F_i^2)+4(G_j^3-F_i^3)\Big).\\ \end{eqnarray*} We notice that $[\widetilde\Delta_{i,j}^{+-}]^2=[\Delta_{i,j}^{+-}]p_{i+}^*(c_2(S))$. Thus, by multiplying out, we get that $c_k({\cal O}_{\Delta_{i,j}^{+-}}(-p_{i+}^*\xi -F_i+G_j)\oplus {\cal O}_{\Delta_{i,j}^{+-}} (-p_{i+}^*(\xi+K_S) -F_i+G_j))$ is the part of degree $k$ of \begin{eqnarray*} &&1-[\Delta_{i,j}^{+-}]\Big(2+4p_{i+}^*\xi+4(G_j-F_i)\\ &&\qquad +p_{i+}^*(6\xi^2+3s_2(S)-K_S^2)+ 12p_{i+}^*\xi(G_j-F_i)+6(G_j^2+F_i^2)\\ &&\qquad +p_{i+}^*(24\xi^2+12s_2(S)-4K_S^2)(G_j-F_i) +24p_{i+}^*\xi(G_j^2+F_i^2)+8(G_j^3-F_i^3)\Big).\\ \end{eqnarray*} Now we take the product over all $i,j$. We use that on $(\tilde S^{n}\times\tilde S^{m})_{*}$ we have $[\Delta_{i_1,j_1}^{+-}]\cdot [\Delta_{i_2,j_2}^{+-}]\cdot F_i=[\Delta_{i_1,j_1}^{+-}]\cdot [\Delta_{i_2,j_2}^{+-}]\cdot G_j=0$ unless $\{i_1,j_1\}=\{i_2,j_2\}$, and obtain the result. \end{pf} \begin{rem} \label{cstand} \begin{enumerate} \item In the Grothendieck ring of $(\widetilde S^{n}\times \widetilde S^{m})_{*}$ we have \begin{eqnarray*} \varphi^*([V]_2|_{{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)}) &=& \sum_{j=1}^{m} p_{j-}^*V(-G_j),\\ \varphi^*([V^\vee(K_S)]^\vee|_{{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)}) &=& \sum_{i=1}^{n} p_{i+}^*(V(-K_S))(F_i).\\ \end{eqnarray*} \item Therefore, for $l\le 3$, $s_l(\varphi^*([V]_2|_{{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)}))$ is the part of degree $l$ of $$\matrix\displaystyle\prod_{j=1}^{m}p_{j-}^*t_{1-} +\sum_{1\le j\le j_1\le m}\Big( +2E_{j,j_1} +p_{j-}^*(10\xi-5K_S)E_{j,j_1}+3E_{j,j_1}^2)\cr +p_{j-}^*(30\xi^2-30\xi K_S+9K_S^2)E_{j,j_1} +p_{j-}^*(18\xi-9K_S)E_{j,j_1}^2+4E_{j,j_1}^3 \Big) \displaystyle\prod_{j_2\not\in \{j,j_1\}} p_{j_2-}^*t_{1-}, \endmatrix\eqno (7.21.1)$$ and $s_l(\varphi^*([V^\vee(K_S)])^\vee|_{{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)} )$ is the part of degree $l$ of $$\matrix \displaystyle\prod_{i=1}^{n}p_{i+}^*t_{1+} + \sum_{1\le i\le i_1\le n} \Big( -2D_{i,i_1} -p_{i+}^*(10\xi+5K_S)D_{i,i_1}+3D_{i,i_1}^2 \cr -p_{i+}^*(30\xi^2+30\xi K_S+9K_S^2)D_{i,i_1} +p_{i+}^*(18\xi+9K_S)D_{i,i_1}^2-4D_{i,i_1}^3\Big) \displaystyle\prod_{i_2\not\in \{i,i_1\}} p_{i_2+}^*t_{1+}. \endmatrix \eqno (7.21.2)$$ \end{enumerate} \end{rem} \begin{pf} (1) follows from the formulas $\ref{gr1}$, $\ref{gr2}$ by tensorizing with $p^*V$ (resp. $p^*(V^\vee(K_S))$) and pushing down via $\tilde q_*$. (2) is just a straightforward computation using that $E_{i,j}\cdot E_{k,l}=D_{i,j}\cdot D_{k,l}=0$ for $\{i,j\}\ne \{k,l\}$. \end{pf} \begin{rem}\label{discard2} Let $k\le 5$ and $\gamma\in H^{4d-2k}(\tilde S^{n}\times \tilde S^{m},{\Bbb Q})$ and assume that $\alpha_1,\alpha_2\in A^*(\tilde S^{n}\times \tilde S^{m})$ have the same pull-back to $(\tilde S^{n}\times \tilde S^{m})_{*}$. Then, for all $i\le n$, $j\le m$, we get analogously to lemma \ref{discard} $$\int\limits_{\tilde S^{n}\times \tilde S^{m}}\Delta_{i,j}^{+-}\cdot(\alpha_1-\alpha_2)\cdot\gamma=0.$$ \end{rem} \begin{prop}\label{intzwei} Let $\gamma\in H^{4d-2k}(S^{(d)},{\Bbb Q})$ with $k\le 5$, and let $w\in H^{4d-2k}(S^d,{\Bbb Q})$ be the pull-back of $\gamma$ to $S^d$. Then \begin{eqnarray*} &&d!\int\limits_{{\text{\rom{Hilb}}}^d(S\sqcup S)} (c(\Gamma)-1)s([V^\vee(K_S)]_1^\vee)s([V]_2)\cdot\gamma\\ &&\qquad = \int_{S^d}\Big( -d(d-1)(2+12\xi+42\xi^2+3s_2(S)+K_S^2)_2 t_1^{{{\hbox{$*$}}} (d-2)}\\ &&\qquad\quad\qquad\quad +d(d-1)(d-2)(30+260\xi)_3 t_1^{{{\hbox{$*$}}} (d-3)}\\ &&\qquad\quad\qquad\quad +2d(d-1)(d-2)(d-3)(2+12\xi)_2^{ 2} t_1^{{{\hbox{$*$}}} (d-4)}\Big)\cdot w, \end{eqnarray*} and, with \begin{eqnarray*} R_d&:=&t_1^{{{\hbox{$*$}}} d}-d(d-1)(5+30\xi+105\xi^2+8s_2(S)+34K_S^2)_2 t_1^{{{\hbox{$*$}}} (d-2)}\\ &&\quad+d(d-1)(d-2)(48+440\xi)_3 t_1^{{{\hbox{$*$}}} (d-3)}\\ &&\quad +{d(d-1)(d-2)(d-3)\over 2} (5+30\xi)_2^{ 2} t_1^{{{\hbox{$*$}}} (d-4)},\end{eqnarray*} we get \begin{eqnarray*} d!\int\limits_{{\text{\rom{Hilb}}}^d(S\sqcup S)} c(\Gamma)s([V^\vee(K_S)]_1^\vee)s([V]_2)\cdot\gamma =\int\limits_{S^d}R_d\cdot w. \end{eqnarray*} \end{prop} \begin{pf} We fix $n$ and $m$ with $n+m=d$ and start by computing on $(\widetilde S^{n}\times \widetilde S^{m})$. Using remark \ref{discard2} we can restrict our attention to $(\widetilde S^{n}\times \widetilde S^{m})_{*}$. We multiply out the formulas (7.21.1),(7.21.2) and (7.20.1) and push down to $S^{n}\times S^{m}$. We shall use the following facts: On $(\tilde S^{n}\times \tilde S^{m})_{*}$ any of $D_i$ and $E_j$, gives zero when multiplied by $[\Delta_{i,j}^{+-}][\Delta_{i_1,j_1}^{+-}]$. Furthermore $\tilde g_*(D_{i,j})=\tilde g_*E_{i,j}=0$, $\tilde g_*(D_{i,i_1}^2)=-[\Delta_{i,i_1}^{++}]$, $\tilde g_*(E_{j,j_1})^2=-[\Delta_{j,j_1}^{--}]$, $\tilde g_*(D_{i,i_1}^3)=[\Delta_{i,i_1}^{++}]p_{i+}^*(K_S)$, $\tilde g_*(E_{j,j_1}^3)=[\Delta_{j,j_1}^{--}]]p_{j-}^*(K_S)$. Below we collect the result of the push-down in ten terms according to the factors that they contain {\it before} the push-down. All the summands contain at least one diagonal factor $[\Delta_{i,j}^{+-}]$ and at most two diagonal factors $[\Delta_{i,j}^{+-}]$, $[\Delta_{i_1,j_1}^{+-}]$. The first seven terms come from summands containing precisely one factor $[\Delta_{i,j}^{+-}]$. So to define these summandss we can fix $i$ and $j$. The first term corresponds to summands not containing any exceptional divisor $D_{i,i_1}$ or $E_{j,j_1}$. The second to seventh summands correspond in that order to the push-downs of the terms containing only powers of $D_{i,i_1}$ with $i_1<i$, $E_{j,j_1}$ with $j_1<j$, $D_{i,i_1}$ with $i_1>i$, $E_{j,j_1}$ with $j_1>j$, $D_{i_1,i_2}$ with $i\not\in \{i_1,i_2\}$ and $E_{j_1,j_2}$ with $j\not\in \{j_1,j_2\}$. Notice that on $(\widetilde S^{n}\times \widetilde S^{m})_{*}$ the class $[\Delta_{i,j}^{+-}]D_{i_1,i_2}E_{j_1,j_2}$ is zero for all $i_1,i_2,j_1,j_2$ and $[\Delta_{i,j}^{+-}]D_{i_1,i_2}D_{i_3,i_4} =[\Delta_{i,j}^{+-}]E_{j_1,j_2}E_{j_3,j_4}=0$ unless $\{i_1,i_2\}=\{i_3,i_4\}$ (resp. $\{j_1,j_2\}=\{j_3,j_4\}$). The last three summands correspond to terms containing two diagonal factors $[\Delta_{i,j}^{+-}][\Delta_{i_1,j_1}^{+-}]$. In that order they correspond to the possibilties that $j=j_1$, that $i=i_1$ and finally that $i\ne i_1$ and $j\ne j_1$. After a long but elementary computation we get that, if $k\le 5$, $\tilde g_*(\varphi^*(c_k(\Gamma_{n,m})-1)s([V^\vee(K_S)[n]^\vee)s(V[m]))$ is the part of degree $k$ of \begin{eqnarray*} &&\sum_{(i,j)}\Bigg( - [\Delta_{i,j}^{+-}] p_{i+}^*(2+12\xi+42\xi^2+3s_2(S)+K_S^2) \prod_{i_1\ne i}p_{i_1+}^*(t_{1+}) \prod_{j_1\ne j} p_{j_1-}^*(t_{1-})\\ &&\qquad\quad +\sum_{i_1<i} [\Delta_{i,j}^{+-}][\Delta_{i,i_1}^{++}] p_{i+}^*(20+160\xi+90K_S) \prod_{i_2\not \in \{i,i_1\}}p_{i_2+}^*t_{1+} \prod_{j_1\ne j} p_{j_1-}^*t_{1-}\\ &&\qquad\quad +\sum_{j_1<j}[\Delta_{i,j}^{+-}][\Delta_{j,j_1}^{--}] p_{i+}^*(20+160 \xi-90K_S) \prod_{i_1\ne i }p_{i_1+}^*t_{1+} \prod_{j_2\not \in \{j,j_1\}} p_{j_2-}^*t_{1-} \\ &&\qquad\quad +\sum_{i_1>i} [\Delta_{i,j}^{+-}][\Delta_{i,i_1}^{++}] p_{i+}^*(6+60\xi+20K_S) \prod_{i_2\not \in \{i,i_1\}}p_{i_2+}^*t_{1+} \prod_{j_1\ne j} p_{j_1-}^*t_{1-}\\ &&\qquad\quad +\sum_{j_1>j}[\Delta_{i,j}^{+-}][\Delta_{j,j_1}^{--}] p_{j-}^*(6+60 \xi-20K_S) \prod_{i_1\ne i }p_{i_1+}^*t_{1+} \prod_{j_2\not \in \{j,j_1\}} p_{j_2-}^*t_{1-}\\ &&\qquad\quad +\sum_{i_1\ne i}\sum_{i_2\ne i,i_2< i_1}\Bigg( [\Delta_{i,j}^{+-}][\Delta_{i_1,i_2}^{++}] p_{i+}^*(2+12 \xi) p_{i_1+}^*(3+18\xi+13K_S)\\ &&\qquad\quad\cdot \prod_{i_3\not\in\{ i,i_1,i_2\} }p_{i_3+}^*t_{1+} \prod_{j_1\ne j} p_{j_1-}^*t_{1-}\Bigg) +\sum_{j_1\ne j}\sum_{j_2\ne j,j_2< j_1}\Bigg( [\Delta_{i,j}^{+-}][\Delta_{j_1,j_2}^{--}] p_{j-}^*(2+12 \xi)\\ &&\qquad\quad\cdot p_{j_1-}^*(3+18\xi-13K_S) \prod_{i_1\ne j}p_{i_1+}^*t_{1+} \prod_{j_3\not\in\{ j,j_1,j_2\} }p_{j_3-}^*t_{1-}\Bigg) \\ &&\qquad\quad +\sum_{i_1<i} [\Delta_{i,j}^{+-}][\Delta_{i_1,j}^{+-}] p_{i+}^*(4+40\xi+4K_S) \prod_{i_2\not \in \{i,i_1\}}p_{i_2+}^*t_{1+} \prod_{j_1\ne j} p_{j_1-}^*t_{1-}\\ &&\qquad\quad +\sum_{j_1<j}[\Delta_{i,j}^{+-}][\Delta_{i,j_1}^{+-}] p_{j-}^*(4+40 \xi-4K_S) \prod_{i_1\ne i }p_{i_1+}^*t_{1+} \prod_{j_2\not \in \{j,j_1\}} p_{j_2-}^*t_{1-}\\ &&\qquad\quad +\sum_{i_1<i}\sum_{j_1\ne j} [\Delta_{i,j}^{+-}][\Delta_{i_1,j_1}^{+-}] p_{i+}^*(2+12 \xi) p_{i_1+}^*(2+12\xi) \prod_{i_2\not\in\{ i,i_1\} }p_{i_2+}^*t_{1+} \prod_{j_2\not \in \{j,j_1\}} p_{j_2-}^*t_{1-}\Bigg). \end{eqnarray*} Now we want to translate this result into the notation \ref{starnota}. Using remark \ref{starrem} and notation \ref{starnota} we see that for $w\in H^{4d-2k}(S^d,{\Bbb Q})^{\frak S_{d}}$ and $a\in H^*(S,{\Bbb Q})$ we have \begin{eqnarray*} \int_{S^d}[\Delta^{+-}_{i,j}] p_{i+}^*a\prod_{i_1\ne i}p_{i_1+}^*t_{1+} \prod_{j_1\ne i}p_{j_1-}^*t_{1-}\cdot w &=&\int_{S^d}(a)_2 t_{1+}^{{{\hbox{$*$}}} (n-1)} t_{1-}^{{{\hbox{$*$}}} (m-1)} \cdot w. \end{eqnarray*} Now assume $j\ne j_1$. Then \begin{eqnarray*} \int_{S^d}[\Delta^{+-}_{i,j}][\Delta^{+-}_{i,j_1}]p_{i+}^*a \prod_{i_1\ne i} p_{i_1+}^*t_{1+} \prod_{j_2\not\in \{j,j_1\}}p_{j_2-}^*t_{1-}\cdot w &=& \int_{S^d}(a)_3 t_{1+}^{{{\hbox{$*$}}} (n-1)} t_{1-}^{{{\hbox{$*$}}} (m-2)}\cdot w. \end{eqnarray*} We also see that $[\Delta^{+-}_{i,j}][\Delta^{+-}_{i,j_1}] =[\Delta^{+-}_{i,j}][\Delta^{--}_{j,j_1}]$ and $[\Delta^{+-}_{i,j}][\Delta^{+-}_{i_1,j}] =[\Delta^{+-}_{i,j}][\Delta^{++}_{i,i_1}]$. If $i\ne i_1$ and $j\ne j_1$ we get similarly \begin{eqnarray*} \int_{S^d}[\Delta^{+-}_{i,j}][\Delta^{+-}_{i_1,j_1}] p_{i+}^*a_1 p_{i_1+}^*a_2\! \!\!\!\!\!\prod_{i_2\not\in \{i,i_1\}}\!\!\!\!\! p_{i_2+}^*t_{1+} \!\!\!\!\!\prod_{j_2\not\in \{j,j_1\}}\!\!\!\!\!p_{j_2-}^*t_{1-}\cdot w \!\!&=&\! \int_{S^d}\!\!(a_1)_2(a_2)_2t_{1+}^{{{\hbox{$*$}}} (n-2)} t_{1-}^{{{\hbox{$*$}}} (m-2)}\cdot w. \end{eqnarray*} We can translate our result into this notation and simplify it by collecting the terms number $2,4,8$ and the terms $3,5,9$ respectively. So we get for $w\in H^{4d-2k}(S^d,{\Bbb Q})^{\frak S_{d}}$ with $k\le 5$: \begin{eqnarray*} &&\int_{S^d}\tilde g_*\big(\varphi^*\big((c(\Gamma)-1) s([V^\vee(K_S)[n]^\vee)s(V[m])\big)\big)\cdot w\\ &&\quad =\int_{S^d}\Bigg(-nm (2+12\xi+42\xi^2+3s_2(S)+K_S^2)_2 t_{1+}^{{{\hbox{$*$}}} (n-1)} t_{1-}^{{{\hbox{$*$}}} (m-1)}\\ &&\qquad\quad +{n\choose 2}m (30+260\xi+114K_S)_3 t_{1+}^{{{\hbox{$*$}}} (n-2)} t_{1-}^{{{\hbox{$*$}}} (m-1)}\\ &&\qquad\quad +{m\choose 2}n(30+260\xi-114K_S)_3 t_{1+}^{{{\hbox{$*$}}} (n-1)} t_{1-}^{{{\hbox{$*$}}} (m-2)}\\ &&\qquad\quad +2{n\choose 2}{m\choose 2} (2+12 \xi)_2^{ 2} t_{1+}^{{{\hbox{$*$}}} (n-2)} t_{1-}^{{{\hbox{$*$}}} (m-2)}\\ &&\qquad\quad +mn{n-1\choose 2} (2+12 \xi)_2 (3+18\xi+13K_S)_2 t_{1+}^{{{\hbox{$*$}}} (n-3)} t_{1-}^{{{\hbox{$*$}}} (m-1)}\\ &&\qquad\quad +nm{m-1\choose 2} (2+12 \xi)_2 (3+18\xi-13K_S)_2 t_{1-}^{{{\hbox{$*$}}} (m-3)} t_{1+}^{{{\hbox{$*$}}} (n-1)}\Bigg)\cdot w \end{eqnarray*} Now we sum over all $m,n$ and keep in mind that the map $\varphi:(\widetilde S^{n}\times \widetilde S^{m})_{*}\longrightarrow {\text{\rom{Hilb}}}^{n}(S)\times {\text{\rom{Hilb}}}^{m}(S)$ has degree $m! n!$. So we obtain \begin{eqnarray*} &&d!\int_{{\text{\rom{Hilb}}}^d(S\sqcup S)} (c(\Gamma)-1)s([V^\vee(K_S)]_1^\vee)s([V]_2)\cdot\gamma\\ &&\qquad =\sum_{n+m=d} {d\choose n} \int_{S^d}\tilde g_*\big(\varphi^*\big((c(\Gamma)-1)s([V^\vee(K_S)[n]^\vee)s(V[m])\big)\big) \cdot w\\ &&\qquad =\sum_{n+m=d} \int_{S^d}\Bigg(- d(d-1) {d-2\choose n-1} (2+12\xi+42\xi^2+3s_2(S)+K_S^2)_2 t_{1+}^{{{\hbox{$*$}}} (n-1)} t_{1-}^{{{\hbox{$*$}}} (m-1)}\\ &&\qquad\quad +d(d-1)(d-2) {d-3\choose n-2}(30+260\xi)_3 t_{1+}^{{{\hbox{$*$}}} (n-2)} t_{1-}^{{{\hbox{$*$}}} (m-1)}\\ &&\qquad\quad +2{d(d-1)(d-2)(d-3)}{d-4\choose n-2} (2+12 \xi)_2^{ 2} t_{1+}^{{{\hbox{$*$}}} (n-2)} t_{1-}^{{{\hbox{$*$}}} (m-2)}\Bigg)\cdot w\\ &&\qquad =\int_{S^d}\Big(-d(d-1)(2+12\xi+42\xi^2+3s_2(S)+K_S^2)_2 t_1^{{{\hbox{$*$}}} (d-2)}\\ &&\qquad\quad +d(d-1)(d-2)(30+260\xi)_3 t_1^{{{\hbox{$*$}}} (d-3)}\\ &&\qquad\quad +2{d(d-1)(d-2)(d-3)}(2+12 \xi)_2^{ 2} t_1^{{{\hbox{$*$}}} (d-4)}\Big)\cdot w. \end{eqnarray*} This shows the first formula. The second follows by combining this formula with proposition \ref{erstint}. \end{pf} Now we have described the intersection numbers $\int_T s({\hbox{\rom{Ext}}}^1_q({\cal I}_{Z_1},{\cal I}_{Z_2}\otimes p^*V)\cdot\gamma$, and are in a position to finish our computation of the leading terms of the change of the Donaldson invariants $\delta_{l,r}(\alpha)$. We first want to compute a formula for the change of $\delta_{N,0}(\alpha)$ and then compute how one has to modify this formula to get $\delta_{l,r}(\alpha)$. The reason that the computation of $\delta_{N,0}(\alpha)$ is easier, is the following fact: \begin{rem} \label{pteinszwei} Let $l,j,k$ be positive integers, $\alpha\in H^2(S,{\Bbb Q})$, $\beta\in H^{2i}(S,{\Bbb Q})$ and $\gamma\in H^*(S^k,{\Bbb Q})^{\frak S_{k}}$. Then we get \begin{eqnarray}\label{pteinszw1} \qquad\qquad\int_{S^{k+j}} (\beta)_j{{\hbox{$*$}}} \gamma\cdot (p_1^*\alpha+\ldots +p_{k+j}^*\alpha)^l= j^{2-i}\int_{S^{k+j}} \beta{{\hbox{$*$}}} pt^{{{\hbox{$*$}}} (j-1)}{{\hbox{$*$}}} \gamma\cdot (p_1^*\alpha+\ldots +p_{k+j}^*\alpha)^l \end{eqnarray} \end{rem} \begin{pf} For the diagonal $\Delta_{j}\subset S^j$ and a class $\alpha\in H^2(S,{\Bbb Q})$, we have $(p_1^*\alpha+\ldots +p_j^*\alpha)\cdot [\Delta_{j}]=jp_1^*(\alpha)[\Delta_{j}]$. By remark \ref{starrem} the left hand side of (\ref{pteinszw1}) is equal to \begin{eqnarray*} \left(\int_{S^{k}} \Delta_{j}p_1^*\beta\cdot (p_1^*\alpha+\ldots +p_{k}^*\alpha)^{2-i}\right)\left(\int_{S^{j}}\gamma\cdot (p_1^*\alpha+\ldots +p_{j}^*\alpha)^{l+i-2}\right). \end{eqnarray*} So the result follows. \end{pf} \begin{nota} We denote by $q_S$ the quadratic form on $H_2(S,{\Bbb Z})$ and, for $\gamma\in H^2(S,{\Bbb Q})$, we let $L_{\gamma}$ be the linear form on $H_2(S,{\Bbb Q})$ given by $\alpha\mapsto \<\gamma,\alpha\>$. For a class $\beta\in H_i(S,{\Bbb Q})$ we denote $\bar \beta:=p_1^*\check \beta+ \ldots +p_d^*\check \beta\in H^{4-i}(S^d,{\Bbb Q})$, where as above, $\check\beta$ is the Poincar\'e dual of $\beta$. Note that by lemma \ref{hilbkuenn} and definition \ref{defchange} $\varphi^*(\widetilde\beta|_{{\text{\rom{Hilb}}}^{n}(S)\times {\text{\rom{Hilb}}}^{m}(S)})$ is the pullback of $\bar \beta$. Let $N=4c_2-c_1^2-3$ again be the expected dimension of $M_H(c_1,c_2)$. \end{nota} \begin{lem}\label{formel1} For all $x,y\ge 0$ and all $\alpha\in H_2(S,{\Bbb Q})$ we have \begin{eqnarray*}\int_{S^d}\xi^{{{\hbox{$*$}}} x} {pt}^{{{\hbox{$*$}}} y} 1^{{{\hbox{$*$}}} (d-x-y)} \cdot \bar\alpha^{2d-x-2y}={(2d-x-2y)!\over 2^{d-x-y}} q_S(\alpha)^{d-x-y}\<\xi,\alpha\>^x.\end{eqnarray*} \end{lem} \begin{pf} By remark \ref{starrem} we have \begin{eqnarray*} \int_{S^d}\!\!\!\! \xi^{{{\hbox{$*$}}} x} {pt}^{{{\hbox{$*$}}} y}1^{{{\hbox{$*$}}}(d-x-y)} \cdot{\bar\alpha^{N-m}}= \int_{S^d} \!\!\! p_1^{*}\xi\cdot \ldots \cdot p_x^{*}\xi \cdot p_{x+1}^{*}{pt}\cdot \ldots\cdot p_{x+y}^{*}{pt} \cdot (p_1^*\check\alpha +\ldots +p_d^*\check\alpha)^{N-m}, \end{eqnarray*} and it is easy to see that this is just ${(2d-x-2y)!\over 2^{d-x-y}} q_S(\alpha)^{d-x-y}\<\xi,\alpha\>^x.$ \end{pf} \begin{thm} \label{chthm1} In the polynomial ring on $H^*(S,{\Bbb Q})$ we have $$\delta_{\xi,N,0}\equiv(-1)^{e_\xi}\sum_{k=0}^2 {N!\over (N-2d+2k)!(d-k)!}Q_{k}(N,d,K_S^2) L_{\xi/2}^{N-2d+2k}q_S^{d-k}\ \hbox{ modulo } L_\xi^{N-2d+6},$$ where, by convention ${1\over m!}=0$ for $m<0$ and \begin{eqnarray*} Q_{0}(N,d,K_S^2)&=& 1\\ Q_{1}(N,d,K_S^2)&=& 2N+2K_S^2-2d+8\\ Q_{2}(N,d,K_S^2)&=& 2N^2-4dN+4NK_S^2+21N+2d^2 -4dK_S^2 -18d+2(K_S^2)^2+18K_S^2+49. \end{eqnarray*} \end{thm} \begin{pf} Let $R_d\in H^*(S^d,{\Bbb Q})^{\frak S_{d}}$ be the class from proposition \ref{intzwei} with $$d!\int\limits_{{\text{\rom{Hilb}}}^d(S\sqcup S)} c(\Gamma)s([V^\vee(K_S)]_1^\vee)s([V]_2)\cdot \gamma=\int\limits_{S^d} R_d\cdot {w}.$$ By remark \ref{pteinszwei} there is a class $U'_d$ which is a linear combination of classes of the form $\xi^{{{\hbox{$*$}}} x}{pt}^{{{\hbox{$*$}}} y} 1^{{{\hbox{$*$}}} (d-x-y)}$ with $\int_{S^d} R_d\cdot\bar \alpha^b=\int_{S^d} U'_d\cdot {\bar \alpha^b}$ for all $\alpha\in H_2(S,{\Bbb Q})$. We write $U'_d:=\sum_{x,y\ge 0}u_{x,y}\xi^{{{\hbox{$*$}}} x}{pt}^{{{\hbox{$*$}}} y} 1^{{{\hbox{$*$}}} (d-x-y)}$ and $U_d:=\sum_{x+y\le 2}u_{x,y}\xi^{{{\hbox{$*$}}} x}{pt}^{{{\hbox{$*$}}} y} 1^{{{\hbox{$*$}}} (d-x-y)}$ By definition \ref{defchange} and theorem \ref{donch1} we see that \begin{eqnarray*} \delta_{\xi,N,0}(\alpha)=\sum_{i=0}^{2d}A_i\cdot\<\xi,\alpha\>\int_{S^d} \{U'_d\}_i\bar\alpha^{2d-i}, \end{eqnarray*} where $\{\ \}_i$ denotes the part of degree $i$, and the $A_i$ are suitable rational numbers. Thus $\delta_{\xi,N,0}$ modulo $L_\xi^{n-2d+6}$ is already determined by $U_d$. As $S$ is a surface with $p_g(S)=q(S)=0$, we have $12=12\chi({\cal O}_S)=K_S^2+c_2(S)$ and thus we can replace $s_2(S)$ by $2K_S^2-12$. So, using proposition \ref{intzwei}, we obtain after a short calculation that \begin{eqnarray*} U_d&=& 2^{d}1^{{{\hbox{$*$}}} d}+2^{d+1}d \cdot1^{{{\hbox{$*$}}} (d-1)}{{\hbox{$*$}}}\xi +2^{d+1}d(d-1)1^{{{\hbox{$*$}}} (d-2)} \xi^{{{\hbox{$*$}}} 2}\\ &&\quad+2^d d(3\xi^2+K_S^2-5d+5)1^{{{\hbox{$*$}}} (d-1)} {{\hbox{$*$}}} {pt}\\ &&\quad+2^d d(d-1)(6\xi^2+2K_S^2-10d+5)1^{{{\hbox{$*$}}} (d-2)}{{\hbox{$*$}}} \xi{{\hbox{$*$}}} {pt}\\ &&\quad+2^{d-2} d(d-1)(18(\xi^2)^{2}+12\xi^2K_S^2+2(K_S^2)^{ 2} -60d\xi^2-20dK_S^2+50d^2\\ &&\quad +15\xi^2-10K_S^2-34d-36){pt}^{{{\hbox{$*$}}} 2} 1^{{{\hbox{$*$}}} (d-2)}, \end{eqnarray*} where we view $\xi^2$ and $K_S^2$ as integers and not as cohomology classes. Now we apply definition \ref{defchange} and lemma \ref{formel1}. Then, after some computation, we get the result with $Q_0(N,d,K_S^2)$, $Q_1(N,d,K_S^2)$, $Q_2(N,d,K_S^2)$ replaced by \begin{eqnarray*} P_0(N,d,K_S^2,\xi^2)&=&1\\ P_1(N,d,K_S^2,\xi^2)&=&8N-26d+6\xi^2+2K_S^2+26\\ P_2(N,d,K_S^2,\xi^2)&=&18(\xi^2)^2+12(\xi^2)(K_S^2)+2(K_S^2)^2+48N\xi^2 -156d\xi^2\\ &&\quad-52dK_S^2+338d^2 +16K_S^2N+32N^2-208d N+207\xi^2\\ &&\quad+54K_S^2+264N -882d+508. \end{eqnarray*} We notice that by definition $d=(4c_2-c_1^2+\xi^2)/4$ and $N=4c_2-c_1^2-3$ and thus $\xi^2=4d-N-3$. Substituting this into the $P_i(N,d,K_S^2,\xi^2)$ we obtain the result. \end{pf} We see that the result is compatible with the conjecture of Kotschick and Morgan. In fact it suggests a slightly sharper statement. \begin{conj} In the polynomial ring on $H^2(S,{\Bbb Q})$ we have $$\delta_{\xi,N,0}=(-1)^{e_\xi}\sum_{k=0}^d {N!\over (N-2d+2k)!(d-k)!}Q_{k}(N,d,K_S^2) L_{\xi/2}^{N-2d+2k}q_S^{d-k},$$ where $Q_{k}(N,d,K_S^2)$ is a polynomial of degree $k$ in $N,d,K_S^2$, which is independent of $S$ and $\xi$. \end{conj} Now we want to compute $\delta_{l,r}$ in general. We shall see that there is reasonably simple relationship between the formula for $\delta_{N,0}$ and that for $\delta_{l,r}$ (with $l+2r=N$), which is however obscured by the existence of a correction term coming from the failure of remark \ref{pteinszwei} for classes of the form $\bar\alpha^{k-2}\bar {pt}$ (instead of $\bar\alpha^{k}$). \begin{lem}\label{formel2} \begin{enumerate} \item For all $x,y\ge 0$, all $c\le r$ and all $\alpha\in H_2(S,{\Bbb Q})$ we have with $m:=2d-2c-x-2y$: \begin{eqnarray*} \int_{S^d} \xi^{{{\hbox{$*$}}} x} {pt}^{{{\hbox{$*$}}} y}1^{{{\hbox{$*$}}} (d-x-y)}\cdot {\bar{pt}^c\bar\alpha^{m}}={(d-x-y)!\over (d-x-y-c)!}{m!\over 2^{d-x-y-c}}q_S(\alpha)^{d-x-y-c} \<\xi,\alpha\>^{x} \end{eqnarray*} \item \begin{eqnarray*} \int_{S^d}(1)_2 1^{{{\hbox{$*$}}} (d-2)}\cdot {\bar{pt}\, \bar\alpha^{2d-4}} ={4d-6\over d-1}\int_{S^d}{pt}{{\hbox{$*$}}} 1^{{{\hbox{$*$}}} (d-1)}\cdot {\bar{pt}\, \bar\alpha^{2d-4}} \end{eqnarray*} \end{enumerate} \end{lem} \begin{pf} (1) By remark \ref{starrem} we have \begin{eqnarray*} &&\int_{S^d}\xi^{{{\hbox{$*$}}} x} {pt}^{{{\hbox{$*$}}} y}\cdot{\bar{pt}^c\bar\alpha^{m}} = \int_{S^d} p_1^{*}\xi\ldots p_x^{*}\xi \cdot p_{x+1}^{*}{pt}\ldots p_{x+y}^{*}{pt} \cdot (p_1^*\check\alpha +\ldots +p_d^*\check\alpha)^{m} \cdot (p_1^*\check{pt} +\ldots +p_d^*\check{pt})^{c} \end{eqnarray*} and it is elementary to show that this is just $$ {(d-x-y)!\over (d-x-y-c)!}{m!\over 2^{d-x-y-c}}q_S(\alpha)^{d-x-y-c} \<\xi,\alpha\>^{x}. $$ (2) By remark \ref{starrem} and remark \ref{pteinszwei} we have \begin{eqnarray*}&&\int_{S^d} (1)_2 1^{{{\hbox{$*$}}} (d-2)}\cdot{\bar{pt}\, \bar\alpha^{2d-4}}\\ &&\qquad =\int_{S^d} [\Delta_{1,2}] \cdot (p_1^* \check\alpha +\ldots +p_d^*\check\alpha)^{2d-4} \cdot (p_1^*\check{pt} +\ldots +p_d^*\check{pt})\\ &&\qquad=2\int_{S^{d-2}} (p_1^*\check\alpha +\ldots +p_{d-2}^*\check\alpha)^{2d-4}+(d-2)\int_{S^{d-1}} [\Delta_{1,2}] \cdot (p_1^*\check\alpha +\ldots +p_d^*\check\alpha)^{2d-4}\\ &&\qquad=(4d-6)\int_{S^{d-2}}\bar\alpha^{2d-4}\\ &&\qquad=(4d-6)\int_{S^{d-1}}{pt}*1^{{{\hbox{$*$}}} (d-2)}\bar\alpha^{2d-4}\\ &&\qquad={4d-6\over d-1}\int_{S^d}{pt}{{\hbox{$*$}}} 1^{ {{\hbox{$*$}}} (d-1)}\cdot{\bar{pt}\,\bar\alpha^{2d-4}} \end{eqnarray*} \end{pf} \begin{thm} \label{chthm2} Let $l,r$ be nonnegative integers with $l+2r=N$. Then in the polynomial ring on $H^*(S,{\Bbb Q})$ we get $$ \delta_{\xi,l,r}\equiv\sum_{c=0}^2{(-1)^{r-c+{e_\xi}}\over 2^{-3c+2r}} {r\choose c}\sum_{k=c}^{2} {l!\over (l-2d+2k)!(d-k)!} Q_{k-c,c}(l,d,K_S^2,\xi^2) L_{\xi/2}^{l-2d+2k}q_S^{d-k}$$ modulo $ \xi^{N-2d+6},$ where \begin{eqnarray*} Q_{m,c}(l,d,K_S^2,\xi^2)&=& P_{m}(l,d,K_S^2,\xi^2)+21mc\ \hbox{ for } m+c\le 2 \end{eqnarray*} Here the $P_{i}(N,d,K_S^2,\xi^2)$ are the polynomials from the proof of theorem \ref{chthm1}. \end{thm} \begin{pf} For $i\le r$ and a class $\gamma\in H^*(S^d,{\Bbb Q})^{\frak S_{d}}$ we denote by $W_{l,r,c}(\gamma)$ the map that associates to $\alpha\in H_2(S,{\Bbb Q})$ the number \begin{eqnarray*} \sum_{b=0}^{l}(-1)^{r-c+e_\xi}2^{b+2c-N}{l\choose b}{r\choose c}\<\xi,\alpha\>^{l-b} \int_{S^d}\gamma\cdot{\bar\alpha^b\bar{pt}^c}. \end{eqnarray*} Let $R_d$, $U_d\in H^*(S^d,{\Bbb Q})^{\frak S_d}$ be the classes from the proof of \ref{chthm1}. By thm \ref{donch1} and proposition \ref{intzwei} we get $$\delta_{l,r}\equiv\sum_{c=0}^r W_{l,r,c}(R_d).$$ By lemma \ref{pteinszwei} we see that $W_{l,r,0}(R_d)\equiv W_{l,r,0}(U_d)$ modulo $L_\xi^{N-2d+6}$. Furthermore we get modulo $L_\xi^{N-2d+6}$ \begin{eqnarray*} W_{l,r,k}(R_d)&\equiv& 0 \ \hbox{ for } k>2\\ W_{l,r,2}(R_d)&\equiv & W_{l,r,2}(2^d 1^{{{\hbox{$*$}}} d})\\ W_{l,r,1}(R_d)&\equiv& W_{l,r,1}(\bar R_d), \end{eqnarray*} where $\bar R_d=t_1^{{{\hbox{$*$}}} d}-5\cdot 2^{d-2}d(d-1)(1)_21^{{{\hbox{$*$}}} (d-2)}$. By lemma \ref{formel2}(2) $W_{l,r,1}(\bar R_d)\equiv W_{l,r,1}(\bar U_d)$ where $$\bar U_d= t_1^{{{\hbox{$*$}}} d}-5\cdot 2^{d-2}d(4d-6){pt}{{\hbox{$*$}}} 1^{{{\hbox{$*$}}}(d-1)}.$$ Now the result follows by applying lemma \ref{formel2}(1) and some computation. \end{pf} \begin{rem} Using $\xi^2=4d-2r-l-3$ we get equivalently \begin{eqnarray*} Q_{0,0}(l,d,K_S^2,\xi^2)&=&Q_{0,1}(l,d,K_S^2,\xi^2)= Q_{0,2}(l,d,K_S^2,\xi^2)=1\\ Q_{1,0}(l,d,K_S^2,\xi^2)&=&2l-2d-12r+2K_S^2+8\\ Q_{2,0}(l,d,K_S^2,\xi^2)&=&72r^2-24rl+24dr-24K_S^2r+2l^2-4dl+4K_S^2l\\ &&\quad +2d^2-4dK_S^2+2(K_S^2)^2-198r+21l-18d+18K_S^2+49\\ Q_{1,1}(l,d,K_S^2,\xi^2)&=&2 l - 2 d - 12 r +2K_S^2+ 29.\\ \end{eqnarray*} \end{rem} \begin{rem} We see that our results contain as a special case the formulas for the change for $d\le 2$. In the case that $d=3$ we notice that the spaces $X_*$ and $Y_{*}$ (for $X$ and $Y$ schemes with a natural morphism to $S^{(d)}$ and $S^{(n)}\times S^{(m)}$ respectively) just coincide with $X$ respectively $Y$. Therefore our computations are valid on the whole of ${\text{\rom{Hilb}}}^d(S\sqcup S)$ and our methods will also give complete formulas for the change of the Donaldson invariants in case $d=3$. We however do not carry out the elementary but long computations here. \end{rem}

9410

alg-geom/9410001

en

https://arxiv.org/abs/alg-geom/9410001

alg-geom/9410001

Victor Batyrev

Victor V. Batyrev and Dimitrios I. Dais

Strong McKay Correspondence, String-theoretic Hodge Numbers and Mirror Symmetry

42 pages, Latex

In the revised version of the paper, we correct misprints and add some new statements.

alg-geom

\section{Introduction} Throughout this paper by an {\em algebraic variety} (or simply {\em variety}) we mean an integral, separated algebraic scheme over ${\bf C}$. By a {\em compact algebraic variety} we mean the representative of a complete variety within the analytic category. The {\em singular} {\em locus} of an algebraic variety $X$ is denoted by ${\rm Sing}\,X$. The words {\em smooth variety} and {\em manifold} are used interchangeably. By the word {\em singularity} we sometimes intimate a singular point and sometimes the underlying space of a neighbourhood or the germ of a singular point, but its meaning will be always clear from the context. Following Danilov \cite{danilov}, \S 13.3, we shall say that an $x \in X$ is a {\em toroidal singularity} of $X$, if there is an analytic isomorphism between the germ $(X, x)$ and the germ corresponding to the toric singularity $({\bf A}_{\sigma}, p_{\sigma})$ (see also \S 4). Our main tool will be certain algebraic varieties with special Gorenstein singularities, primarily having in mind the Calabi-Yau varieties. A {\em Calabi-Yau variety} is defined to be a normal projective algebraic variety $X$ with trivial canonical sheaf ${\omega}_X$ and $H^i(X, {\cal O}_X) = 0$, $0 < i < {\rm dim}_{\bf C}\,X$, which, in addition, can have at most {\em canonical Gorenstein singularities}. (For the notion of {\em canonical singularity} we refer to \cite{reid1}.) If ${\rm Sing}\,X = \emptyset$, then $X$ is called, as usual, {\em Calabi-Yau manifold}. In this paper we shall attempt to realize some Hodge-theoretical invariants used by physicists for singular varieties being related to the mirror symmetry phenomenon. The necessity of working with singular varieties becomes unavoidable from the fact that, in many examples of pairs $X$, $X^*$ of mirror symmetric Calabi-Yau manifolds, at least one of the two manifolds $X$ or $X^*$ is obtained as {\em a crepant desingularization} of a singular Calabi-Yau variety \cite{batyrev1,morrison}. Here, by a crepant desingularization of a Gorenstein variety $Z$, we mean a birational morphism $\pi\,: \, Z' \rightarrow Z$, such that $\pi^*(\omega_Z) \cong \omega_{Z'}$, where $\omega_Z$ and $\omega_{Z'}$ denote the canonical sheaves on $Z$ and $Z'$ respectively. 3-dimensional Gorenstein quotient singularities and their crepant desingularizations have been studied in \cite{bertin,ito1,ito2,markushevich,markushevich2,roan0,roan1,roan2,roan3,roan-y au,reid2,yau}. The most known physical cohomological invariant of singular varieties obtained as quotient-spaces of certain compact manifolds by actions of finite groups is the so called {\em physicists Euler number} \cite{dixon}. It has been investigated by several mathematicians in \cite{atiyah,got2,hirzebruch,roan,roan1,reid2}. Let $X$ be a smooth simplectic manifold over ${\bf C}$ having an action of a finite group $G$ such that the simplectic volume form $\omega$ is $G$-invariant. For any $g \in G$, we set $X^g : = \{ x \in X \mid g(x) = x \}$. Physicists have proposed the following formula for computing the {\em orbifold Euler number} \cite{dixon}: \begin{equation} e(X,G) = \frac{1}{\mid G \mid} \sum_{gh = hg} e(X^g \cap X^h). \label{euler.phys} \end{equation} It is expected that $e(X,G)$ coincides with the usual Euler number $e(\widehat{X/G})$ of a crepant desingularization $\widehat{X/G}$ of the quotient space $X/G$ provided such a desingularization exists. For a volume-invariant linear action on ${\bf C}^n$ of a finite group $G$, the corresponding conjectural local properties of crepant desingularizations were formulated by M. Reid \cite{reid2}: \begin{conj} {\rm (generalized McKay correspondence)} Let $X = {\bf C}^n$, $G$ an arbitrary finite subgroup in $SL(n, {\bf C})$. Assume that $Y = X/G$ admits a crepant desingularization $\pi \,: \, \hat{Y} \rightarrow Y$. Then $H^*(\pi^{-1}(0), {\bf C})$ has a basis consisting of classes of algebraic cycles $Z_c \subset \pi^{-1}(0)$ which are in $1$-to-$1$ correspondence with conjugacy classes $c$ of $G$. In particular, we obtain for the Euler number \[ e(\hat{Y}) = e(\pi^{-1}(0)) = \# \{ \mbox{\rm conjugacy classes in $G$} \}. \] \label{general} \end{conj} \begin{rem} {\rm For $n =2$ an one-to-one correspondence between the nontrivial irreducible representations of a subgroup $G \subset SL(2, {\bf C})$ and the irreducible components of $\pi^{-1}(0)$ was discovered by McKay \cite{mckay} and investigated in \cite{gonzalez,knorrer,sandro-infirri}. } \end{rem} Our first purpose is to use some stronger version of Conjecture \ref{general} in order to give an analogous interpretation for the {\em physicists Hodge numbers} $h^{p,q}(X,G)$ of orbifolds considered by C. Vafa \cite{vafa} and E. Zaslow \cite{zaslow}. Let $X$ be a smooth compact K\"ahler manifold of dimension $n$ over ${\bf C}$ being equipped with an action of a finite group $G$, such that $X$ has a $G$-invariant volume form. Let $C(g) : = \{ h \in G \mid hg = gh\}$. Then the action of $C(g)$ on $X$ can be restricted on $X^g$. For any point $x \in X^g$, the eigenvalues of $g$ in the holomorphic tangent space $T_x$ are roots of unity: \[ e^{2\pi \sqrt{-1} \alpha_1}, \ldots, e^{2\pi \sqrt{-1} \alpha_d} \] where $0 \leq \alpha_j <1 $ $(j =1, \ldots, d)$ are locally constant functions on $X^g$ with values in ${\bf Q}$. We write $X^g = X_1(g) \cup \cdots \cup X_{r_g}(g)$, where $X_1(g), \ldots, X_{r_g}(g)$ are the smooth connected components of $X^g$. For each $ i \in \{1, \ldots, r_{g} \}$, the {\em fermion shift number} $F_i(g)$ is defined to be equal to the value of $\sum_{1 \leq j \leq n} {\alpha_j}$ on the connected component $X_i(g)$. We denote by $h^{p,q}_{C(g)}(X_i(g))$ the dimension of the subspace of $C(g)$-invariant elements in $H^{p,q}(X_i(g))$. We set \[ h^{p,q}_g(X,G) : = \sum_{i =1}^{r_g} h^{p - F_i(g),q - F_i(g)}_{C(g)}(X_i(g)).\] The {\em orbifold Hodge numbers} of $X/G$ are defined by the formula (3.21) in \cite{zaslow}: \begin{equation} h^{p,q}(X,G): = \sum_{\{ g \}} h^{p,q}_g(X,G) \label{phys.form} \end{equation} where $\{g\}$ runs over the conjugacy classes of $G$, so that $g$ represents $\{ g \}$. As we shall see in Corollary \ref{c.des}, these numbers coincide with the {\em usual} Hodge numbers of a crepant desingularization of $X/G$. One of our next intentions is to convince the reader of the existence of some {\em new cohomology theory} $H^*_{\rm st}(X)$ of more general algebraic varieties $X$ with mild Gorenstein singularities. Since this cohomology is inspired from the string theory, we call $H^*_{\rm st}(X)$ the {\em string cohomology of} $X$. For compact varieties $X$, we expect that the string cohomology groups $H^*_{\rm st}(X)$ will satisfy the Poincar\'{e} duality and will be endowed with a pure Hodge structure. The role of crepant resolutions for the string-cohomology $H^*_{\rm st}(X)$ is analogous to that one of small resolutions for the intersection cohomology $IH^*(X)$ with middle perversity. Physicists compute orbifold Hodge numbers without using crepant desingularizations. From mathematical point of view, however, crepant desingularizations seem to be rather helpful, although they have some disadvantages. Firstly, they might not exist (at least in dimension $\geq 4$) and ,secondly, even if they exist, they might be not unique. The consistency of the physical approach naturally suggests the formulation of the following conjecture (which can be verified for the toric case by Theorem \ref{invariants}): \begin{conj} Hodge numbers of smooth crepant resolutions do not depend on the choice of such a resolution. \end{conj} Let us briefly review the rest of the paper. In Section 2, we consider an example showing the importance of the ``physical Hodge numbers'' in connection with the mirror duality. In Section 3, we remind basic properties of $E$-polynomials. In Section 4, we study the Hodge structure of the exceptional loci of local crepant toric resolutions. In Section 5, we formulate the conjecture concerning the strong McKay correspondence and we prove that it is true for $2$- and $3$-dimensional Gorenstein quotient singularities, as well as for abelian Gorenstein quotient singularities of arbitrary dimension. This correspondence will be used in Section 6 in order to give the formal definition of the string-theoretic Hodge numbers and to study their main properties. In Section 7, we give some applications relating to the mirror symmetry and formulate the string-theoretic Hodge diamond-mirror conjecture for Calabi-Yau complete intersections in $d$-dimensional toric Fano varieties. This conjecture will be proved in Section 8 for the case of $\Delta$-regular hypersurfaces in toric Fano varieties ${\bf P}_{\Delta}$ which are defined by $d$-dimensional reflexive simplices $\Delta$ (for arbitrary $d$); it gives the mirror duality {\em for all string-theoretic Hodge numbers} $h^{p,q}_{\rm st}$ of abelian quotients of Calabi-Yau Fermat-type hypersurfaces which are embedded in $d$-dimensional weighted projective spaces. This duality agrees with the mirror construction proposed by Greene and Plesser \cite{greene0,greene1,greene} and the polar duality of reflexive polyhedra proposed in \cite{batyrev1}. \bigskip {\bf Acknowledgements.} We would like to express our thanks to D. Cox, A. Dimca, H. Esnault, L. G\"ottsche, Yu. Ito, D. Kazhdan, M. Kontsevich, D. Markushevich, Yu. Manin, K. Oguiso, M. Reid, A. V. Sardo-Infirri, D. van Straten and E. Viehweg for fruitful discussions, suggestions and remarks. \section{Hodge numbers and mirror symmetry} At the beginning we shall state some introductory questions which could be considered also as another motivation for the paper. These questions are related to singular varieties of dimension $\geq 4$ which arose as examples of the mirror duality \cite{batyrev1,candelas,greene,schimmrigk1,schimmrigk2,schimmrigk3}. If two $d$-dimensional Calabi-Yau manifolds $X$ and $Y$ form a mirror pair, then for all $0 \leq p, q \leq d$ their Hodge numbers must satisfy the relation \begin{equation} h^{p,q}(X) = h^{d-p,q}(Y). \label{duality} \end{equation} However, it might happen that a mirror pair consists of two $d$-dimensional Calabi-Yau varieties $X$ and $Y$ having singularities. In this case, the duality (\ref{duality}) is expected to take place not for $X$ and $Y$ themselves, but for their crepant desingularizations $\hat{X}$ and $\hat{Y}$, if such desingularizations exist. Using the existence of smooth crepant desigularizations of Gorenstein toroidal singularities in dimension $\leq 3$, one can check the relations (\ref{duality}) for many examples of $3$-dimensional mirror pairs \cite{batyrev1,roan0}. But there are difficulties to prove (\ref{duality}) {\em for all} $p,q$ and $d \geq 4$, even if one heuristically knows a mirror pair of singular Calabi-Yau varieties, for instance, as an orbifold. The main problem in dimension $d \geq 4$ is due to the existence of many {\em terminal} Gorenstein quotient singularities, i.e., to singularities which obviously do not admit any crepant resolution. In \cite{batyrev1}, the first author constructed the so called {\em maximal projective crepant partial desingularizations} (MPCP-desingularizations) of singular Calabi-Yau hypersurfaces in toric varieties. Using MPCP-desingularizations, the relation (\ref{duality}) was proved for $h^{1,1}$ and $h^{d-1,1}$ in \cite{batyrev1}. We shall show later that MPCP-desingularizations are sufficient to establish (\ref{duality}) for $q=1$ and arbitrary $p$ in the case of $d$-dimensional Calabi-Yau hypersurfaces in toric varieties (see \ref{p1}, \ref{p1cor}). Although MPCP-desingularizations always exist, it is important to stress that they are not sufficient to prove (\ref{duality}) for all $p,q$, and $d \geq 4$, because of the following two properties which can be easily illustrated by means of various examples: \begin{itemize} \item In general, a MPCP-desingularization of a Gorenstein toroidal singularity is not a manifold, but a variety with Gorenstein terminal abelian quotient singularities. \item Cohomology and Hodge numbers of different MPCP-desingularizations might be different. \end{itemize} \bigskip It turns out that, even for $3$-dimensional Calabi-Yau manifolds, the mirror construction inspired from the superconformal field theory demands consideration of higher dimensional manifolds with singularities \cite{batyrev-borisov,candelas,schimmrigk1,schimmrigk2,schimmrigk3}. In this case, we again meet difficulties if we wish to obtain analogues of the duality in (\ref{duality}). Let us explain them for the example which was discussed in \cite{candelas}. Let $E_0$ be the unique elliptic curve having an authomorphism of order $3$ with $3$ fixed points $p_0, p_1, p_2 \in E_0$. We consider the natural diagonal action of $G \cong {\bf Z}/3{\bf Z}$ on $Z = E_0\times E_0 \times E_0$. The quotient $X = Z/G$ is a singular Calabi-Yau variety whose smooth crepant resolution $\hat{X}$ has Hodge numbers \[ h^{1,1}(\hat{X}) = 36,\;\; h^{2,1} (\hat{X}) = 0. \] As the mirror partner of $X$, it has been proposed the $7$-dimensional orbifold $Y$ obtained from the quotient of the Fermat-cubic $(W\, :\, z_0^3 + \cdots z_8^3 = 0)$ in ${\bf P}^8$ by the order 3 cyclic group action defined by the matrix \[ g = {\rm diag}( 1,1,1,e^{2\pi \sqrt{-1}/3},e^{2\pi \sqrt{-1}/3} ,e^{2\pi \sqrt{-1}/3}, e^{-2\pi \sqrt{-1}/3},e^{-2\pi \sqrt{-1}/3},e^{-2\pi \sqrt{-1}/3}). \] By standard methods, counting $G$-invariant monomials in the Jacobian ring, one immediately verifies that $h^{4,3}(Y) = 30$. One could expect that a crepant resolution of singularities of $Y$ along the $3$ elliptic curves \[ C_0 = \{ z_3 = \cdots = z_8 = 0 \} \cap Y, \] \[ C_1 = \{ z_0 = z_1= z_2 = z_6 = z_7 = z_8 = 0 \} \cap Y, \] \[ C_2 = \{ z_0 = \cdots = z_5 = 0 \} \cap Y \] would give the missing $6$ dimensions to $h^{4,3}(Y)$ in order to obtain $36$ (this would be the analogue of (\ref{duality})). But also this hope must be given up because of a very simple reason: all singularities along $C_0, C_1, C_2$ are terminal, i.e., they do not admit any smooth crepant resolution. \begin{ques} What could be that suitable mathematical reasoning which would give back the missing $6$ in the above example? \end{ques} {}From the viewpoint of physicists, one should consider $Y$ as an orbifold quotient of $W$ by $G = \{ e, g, g^{-1} \}$. By physicists' formula (\ref{phys.form}), \[ h^{4,3}(W,G) = h^{4,3}_e(W,G) + h^{4,3}_g(W,G) + h^{4,3}_{g^{-1}}(W,G). \] It is clear that $h^{4,3}_g(W,G) = h^{4,3}_{g^{-1}}(W,G)$ and $h^{4,3}_e(W,G) = h^{4,3}(Y) = 30$. So, it remains to compute $h^{4,3}_g(W,G)$. Notice that $W^g = C_0 \cup C_1 \cup C_2$; i.e., $W_i(g) = C_i$ $( i =0,1,2)$. Moreover, $g$ acts on the tangent space $T_w$ of a point $w \in W^g$ by the matrix \[ {\rm diag}( 1,e^{2\pi \sqrt{-1}/3},e^{2\pi \sqrt{-1}/3} ,e^{2\pi \sqrt{-1}/3}, e^{-2\pi \sqrt{-1}/3},e^{-2\pi \sqrt{-1}/3},e^{-2\pi \sqrt{-1}/3}). \] Therefore, $F_i(g) = 3$ $({\rm for}\; i=0,1,2)$. So $h^{4,3}_g(W,G) = \sum_{i =0}^2 h^{4-F_i(g),3-F_i(g)}(C_i) = 3$ and the required $6$ is indeed present! \begin{ques} Is there a local version of the formula {\rm (\ref{phys.form})} for the underlying space of a quotient singularity extending that of \ref{general}? \end{ques} We shall answer both questions in Sections 5 and 6. \section{$E$-polynomials of algebraic varieties} In this section we recall some basic properties of the {\em $E$-polynomials} of (not necessarily smooth or compact) {\em algebraic varieties}. $E$-polynomials are defined by means of the mixed Hodge structure (MHS) of rational cohomology groups with compact supports \cite{dan.hov}. As we shall see below, these polynomials obey to similar additive and multiplicative laws as those of the {\em usual} Euler characteristic, which enables us to compute all the Hodge numbers coming into question in a very convenient way. As Deligne shows in \cite{deligne}, the cohomology groups $H^k(X, {\bf Q})$ of a (not necessarily smooth or compact) algebraic variety $X$ carry a natural MHS. By similar methods, one can determine a canonical MHS by considering $H^k_c(X, {\bf Q})$, i.e., the cohomology groups {\em with compact supports}. Compared with $H^k(X, {\bf Q})$, the MHS on $H^k_c(X, {\bf Q})$ presents some additional technical advantages. One of them is the existence of the following exact sequence: \begin{prop} Let $X$ be an algebraic variety and $Y \subset X$ a closed subvariety. Then there is an exact sequence \[ \ldots \rightarrow H^k_c(X \setminus Y, {\bf Q}) \rightarrow H^k_c(X, {\bf Q}) \rightarrow H^k_c(Y, {\bf Q}) \rightarrow \cdots \] consisting of $MHS$-morphisms. \label{exact-s} \end{prop} \begin{dfn} {\rm Let $X$ be an algebraic variety over ${\bf C}$ which is not necessarily compact or smooth. Denote by $h^{p,q}(H^k_c(X, {\bf C}))$ the dimension of the $(p,q)$-Hodge component of the $k$-th cohomology with compact supports. We define: \[e^{p,q}(X) := \sum_{k \geq 0} (-1)^k h^{p,q}(H_c^k(X, {\bf C})). \] The polynomial \[ E(X; u,v) : = \sum_{p,q} e^{p,q}(X) u^p v^q \] is called the {\em E-polynomial} of $X$}. \label{e-poly} \end{dfn} \begin{rem} {\rm If the Hodge structure of $X$ in \ref{e-poly} is {\em pure}, then the coefficients $e^{p,q}(X)$ of the E-polynomial of $X$ are related to the usual Hodge numbers by $e^{p,q}(X) = (-1)^{p+q}h^{p,q}(X)$. In fact, the E-polynomial (in the general case) can be regarded as a notional refinement of the {\em virtual Poincar\'{e} polynomial} $E(X; -u,-u)$ and, of course, of the {\em Euler characteristic with compact supports} $e_c(X): = E(X, -1,-1)$. It should be also mentioned, that $e_c(X) = e(X)$, i.e., that $e_c(X)$ is equal to the {\em usual} Euler characteristic of $X$ (cf. \cite{fulton}, pp. 141-142). } \end{rem} Using Proposition \ref{exact-s}, one obtains: \begin{prop} Let $X$ be a disjoint union of locally closed subvarieties $X_i$ $(i \in I)$. Then \[E(X;u,v) = \sum_{i \in I} E(X_i;u,v). \] \label{proper1} \end{prop} \begin{dfn} {\rm Let $X$ be a disjoint union of locally closed subvarieties $X_i$ $(i \in I)$. We shall write $X_{i'} < X_i$, if $X_{i'} \neq X_i$ and $X_{i'}$ is contained in the Zariski closure $\overline{X}_i$ of $X_i$.} \end{dfn} \begin{prop} For any $i_0 \in I$, one has \[ E(X_{i_0}; u,v) = \sum_{k \geq 0} (-1)^k \sum_{X_{i_k} < \cdots < X_{i_1} < X_{i_0} } E(\overline{X}_{i_k}; u,v). \] \label{stra1} \end{prop} \noindent {\em Proof.} By \ref{proper1}, we get \[ E(X_{i_0}; u,v) = E(\overline{X}_{i_0}; u,v) - E(\overline{X}_{i_0} \setminus X_{i_0}; u,v). \] Moreover, \[ E(\overline{X}_{i_0} \setminus X_{i_0}; u,v) = \sum_{X_{i_1} < X_{i_0}} E(X_{i_1}; u,v). \] Repeating the same procedure for $i_1 \in I$, we obtain: \[ E(X_{i_1}; u,v) = E(\overline{X}_{i_1}; u,v) - E(\overline{X}_{i_1} \setminus X_{i_1}; u,v), \] \[ E(\overline{X}_{i_1} \setminus X_{i_1}; u,v) = \sum_{X_{i_2} < X_{i_1}} E(X_{i_2}; u,v), \;\; \; \mbox{\rm etc. } \dots \] This leads to the claimed formula. \hfill $\Box$ \bigskip Applying the K\"unneth formula, we get: \begin{prop} Let $\pi\,: \, X \rightarrow Y$ be a locally trivial fibering in Zariski topology. Denote by $F$ the fiber over a closed point in $Y$. Then \[ E(X;u,v) = E(Y;u,v) \cdot E(F;u,v). \] \label{proper2} \end{prop} We shall use \ref{proper1} and \ref{proper2} in the following situation. Let $\pi\,:\, {X}' \rightarrow X$ be a proper birational morphism of algebraic varieties ${X}'$ and $X$. Let us further assume that ${X}'$ is smooth and $X$ has a stratification by locally closed subvarieties $X_i$ $(i \in I)$, such that each $X_i$ is smooth and the restriction of $\pi$ on $\pi^{-1}(X_i)$ is a locally trivial fibering over $X_i$ in Zariski topology. Using \ref{proper1} and \ref{proper2}, we can compute all Hodge numbers of ${X}'$ as follows: \begin{prop} Let $F_i$ $(i \in I)$ denote the fiber over a closed point of $X_i$. Then \[ E({X}'; u,v) = \sum_{i \in I} E(X_i; u,v) \cdot E(F_i;u,v). \] \label{formula} \end{prop} We shall next deal with the case in which $\pi\,:\, \tilde{X} \rightarrow X$ represents a crepant resolution of singularities of an algebraic variety $X$ having only Gorenstein singularities. The problem of main interest is to characterize the $E$-polynomials $E(F_i; u,v)$ in terms of singularities of $X$ along the $X_i$'s. This problem will be solved in the case when $X$ has Gorenstein toroidal or quotient singularities. \section{Local crepant toric resolutions} We shall compute here the $E$-polynomials of the fibers of crepant toric resolution mappings of Gorenstein toric singularities by using their combinatorial description in terms of convex cones. It is assumed that the reader is familiar with the theory of toric varieties as it is presented, for instance, in the expository article of Danilov \cite{danilov}, or in the books of Oda \cite{oda} and Fulton \cite{fulton}. Let $M$, $N$ be two free abelian groups of rank $d$, which are dual to each other, and let $M_{\bf R}$ and $N_{\bf R}$ be their real scalar extensions. The type of every $d$-dimensional Gorenstein toroidal singularity can be described combinatorially by a $d$-dimensional cone $\sigma = \sigma_{\Delta} \subset N_{\bf R}$ which supports a $(d-1)$-dimensional lattice polyhedron $\Delta \subset N_{\bf R}$ \cite{reid1}. This lattice polyhedron $\Delta$ can be defined as $\{ x \in \sigma \mid \langle x, m_{\sigma} \rangle = 1 \}$ for some uniquely determined element $m_{\sigma} \in M$. Let $\check{\sigma} \subset M_{\bf R}$ be dual to $\sigma$ and set ${\bf A}_{\sigma} := {\rm Spec}\, {\bf C} \lbrack \check{\sigma} \cap M \rbrack$. Then ${\bf A}_{\sigma}$ is a $d$-dimensional affine toric variety with only Gorenstein singularities. We denote by $p= p_{\sigma}$ the unique torus invariant closed point in ${\bf A}_{\sigma}$. \begin{dfn} {\rm A finite collection ${\cal T} = \{ \theta \}$ of simplices with vertices in $\Delta \cap N$ is called a {\em triangulation} of $\Delta$ if the following properties are satisfied: (i) if $\theta'$ is a face of $\theta \in {\cal T}$, then $\theta' \in {\cal T}$; (ii) the intersection of any two simplices $\theta_1', \theta_2' \in {\cal T}$ is either empty, or a common face of both of them; (iii) $\Delta = \bigcup_{\theta \in {\cal T}} \theta$. } \end{dfn} Every triangulation ${\cal T}$ of $\Delta$ gives rise to a partial crepant toric desingularization $\pi_{\cal T}\, : \, X_{\cal T} \rightarrow {\bf A}_{\sigma}$ of ${\bf A}_{\sigma}$, so that $X_{\cal T}$ has at most abelian quotient Gorenstein singularities. \begin{dfn} {\rm A simplex $\theta \subset \Delta \subset \{ x \in N_{\bf R} \mid \langle x, m_{\sigma} \rangle =1 \}$ is called {\em regular} if its vertices form a part of a ${\bf Z}$-basis of $N$. } \end{dfn} \noindent It is known (see, for instance, \cite{oda}, Thm. 1.10, p.15) that $X_{\cal T}$ is smooth if and only if all simplices in ${\cal T}$ are regular. \begin{theo} Assume that $\Delta$ admits a triangulation ${\cal T}$ into regular simplices; i.e., that the corresponding toric variety $X_{\cal T}$ in the crepant resolution \[ \pi_{\cal T}\, : \, X_{\cal T} \rightarrow {\bf A}_{\sigma} \] is smooth. Then $F = \pi_{\cal T}^{-1}(p)$ can be stratified by affine spaces. \label{stratification} \end{theo} \noindent {\em Proof.} Let $\theta_0$ be an arbitrary $(d-1)$-dimensional simplex in ${\cal T}$ with vertices $ e_1, \ldots, e_d$. Choose an 1-parameter multiplicative subgroup $G_{\omega} \subset ({\bf C}^*)^d$ whose action on ${\bf A}_{\sigma}$ is defined by a weight-vector $\omega \in \sigma \cap N$, so that $\omega = \omega_1 e_1 + \cdots + \omega_d e_d$, where $\omega_1, \ldots, \omega_d$ are positive integers. The action of $G_{\omega}$ on ${\bf A}_{\sigma}$ extends naturally to an action on $X_{\cal T}$. If $\{ \theta_0, \theta_1, \ldots, \theta_s \}$ denotes the set of all $(d-1)$-dimensional simplices in ${\cal T}$, then $\sigma = \bigcup_{i =0}^s \sigma_{\theta_i}$, and $X_{\cal T}$ is canonically covered by the corresponding $G_{\omega}$-invariant open subsets $U_0, \ldots, U_s$, so that $U_i \cong {\bf C}^d$. Denote by $p_i$ $(i = 0,1, \ldots, s)$ the unique torus invariant point in $U_i$. We assume that $\omega$ has been already chosen in such a way, that $p_i$ is the unique $G_{\omega}$-invariant point in $U_i$. We consider a multiplicative parameter $t$ on $G_{\omega}$ for which the action of $G_{\omega}$ on $U_0$ is defined as follows: \[ t \cdot (x_1, \dots, x_d): = (t^{\omega_1}x_1, \ldots, t^{\omega_d}x_d). \] Furthermore, we set: \[ X_i : = \{ x = (x_1, \ldots, x_d) \in X_{\cal T} \mid \lim_{t \rightarrow \infty} t(x) = p_i \}. \] Since $\pi_{\cal T}(p_i) = p$, we have $X_i \subset F$. By compactness of $F$, for every point $x \in F$, there exists $\lim_{t \rightarrow \infty} t(x)$ which is a $G_{\omega}$-invariant point; i.e., $\lim_{t \rightarrow \infty} t(x) = p_i$ for some $i$ $(0 \leq i \leq s)$. So $\bigcup_{i =0}^s X_i = F$. Obviously, $X_i \subset U_i$. Moreover, $X_i \cap X_j = \emptyset$ for $i \neq j$. If we now choose appropriate torus coordinates $y_1, \ldots, y_d$ on $U_i$, so that $G_{\omega}$ acts by \[ t\cdot (y_1, \ldots, y_k, y_{k+1}, \ldots, y_d) = (t^{\lambda_1} y_1, \ldots, t^{\lambda_k} y_k, t^{\lambda_{k+1}}y_{k+1}, \ldots, t^{\lambda_d}y_d ) \] with $\lambda_1, \ldots, \lambda_k$ positive and $\lambda_{k+1}, \ldots, \lambda_d$ negative, $X_i$ is defined by the equations $y_1 = \ldots = y_k = 0$. Therefore, $X_i$ is isomorphic to an affine space. \hfill $\Box$ \medskip Let $l(k\Delta)$ denote the number of lattice points of $k\Delta$. Then the {\em Ehrhart power series} \[ P_{\Delta}(t): = \sum_{k \geq 0} l(k\Delta) t^k \] can be considered as a numerical characteristic of the toric singularity at $p_{\sigma}$. It is well-known (see, for instance, \cite{batyrev1}, Thm 2.11, p.356) that $P_{\Delta}(t)$ can be always written in the form: \[ P_{\Delta}(t) = \frac{\psi_0(\Delta) + \psi_1(\Delta)t + \cdots + \psi_{d-1}(\Delta)t^{d-1}}{(1-t)^d}, \] where $\psi_0(\Delta) = 1$ and $\psi_1(\Delta), \ldots, \psi_{d-1}(\Delta)$ are certain nonnegative integers. \begin{theo} Let $\Delta$ be as in {\rm \ref{stratification}}, and $F = \pi_{\cal T}^{-1}(p)$. Then the cohomology groups $ H^{2i}(F, {\bf C}), \;\; i = 0, \ldots, d-1$ are generated by the $(i,i)$-classes of algebraic cycles, and $H^{j}_c(F, {\bf C}) = 0$ for odd values of $j$ . Moreover, $h^{i,i}(F) = \psi_i(\Delta) \;\; i = 0, \ldots, d-1$. In particular, the dimensions $h^{i,i}(F) = {\rm dim}\, H^{2i}(F, {\bf C})$ $( 0 \leq i \leq d-1)$ do not depend on the choice of the triangulation ${\cal T}$. \label{invariants} \end{theo} \noindent {\em Proof.} The first statement follows immediately from Theorem \ref{stratification}. Since $F$ is compact, we have $H^i(F,{\bf C}) = H^i_c(F, {\bf C})$. Therefore, it is sufficient to compute the $E$-polynomial \[ E(F; u,v) = \sum_{p,q} e^{p,q}(F) u^p v^q. \] Since $X_{\cal T}$ is a toric variety, it admits a natural stratification by strata which are isomorphic to algebraic tori $T_{\theta}$ corresponding to regular subsimplices $\theta \in {\cal T}$, such that \[ \mbox{\rm dim}\, T_{\theta} + \mbox{\rm dim}\, \theta = d-1. \] The natural stratification of $X_{\cal T}$ induces a stratification of $F$. Notice that $\pi_{\cal T} (T_{\theta}) = p_{\sigma}$ (i.e., $T_{\theta} \in F$) if and only if $\theta$ does not belong to the boundary of $\Delta$. If $a_i$ denotes the number of $i$-dimensional regular simplices of ${\cal T}$ which do not belong to the boundary of $\Delta$, then $a_i$ can be identified with the number of $(d-1-i)$-dimensional tori in the natural stratification of $\pi_{\cal T}^{-1}(p)$. By \ref{proper1}, we get: \[ E(F; u,v) = \sum_{\pi_{\cal T} (T_{\theta}) = p} E(T_{\theta}; u, v). \] Since $E(({\bf C}^*)^k; u,v) = (uv -1 )^k$, we obtain \[ E(F; u,v) = a_0(u v -1 )^{d-1} + a_1(u v -1 )^{d-2} + \cdots + a_{d-1}. \] Now we compute $P_{\Delta}(t)$ by using the numbers $a_i$. If $\theta \in {\cal T}$ is a $i$-dimensional regular simplex, then \[ l(k\theta) = {k+i \choose k}; \;\;\;\;\;\; {\rm i.e.,}\; \;\;\;\;\; P_{\theta}(t) = \frac{1}{(1-t)^{i+1}}. \] Applying the usual inclusion-exclusion principle for the counting of lattice points of $k\Delta$, we obtain: \[ l(k\Delta) = \sum_{i =0}^{d-1} \sum_{{\rm dim}\, \theta=d-1-i} (-1)^i l(k\theta), \] where $\theta$ runs over all regular simplices in ${\cal T}$ which do not belong to the boundary of $\Delta$. Thus, \[ P_{\Delta}(t) = \frac{a_{d-1}}{(1-t)^d} - \frac{a_{d-2}}{(1 - t)^{d-1}} + \cdots + (-1)^{d-1} \frac{a_0}{(1 - t)} \] and the polynomial \[ \psi_0(\Delta) + \psi_1(\Delta)t + \cdots \psi_{d-1}(\Delta)t^{d-1} = P_{\Delta}(t) (1-t)^d \] is equal to \[ a_{d-1} + a_{d-2}(t -1) + \cdots + a_0(t-1)^{d-1}. \] The latter coincides with the $E$-polynomial $E(F; u,v)$ after making the substitution $t = uv$. Hence, $\psi_i(\Delta) = e^{i,i}(F)$ ($0 \leq i \leq d-1$). \hfill $\Box$ \begin{dfn} {\rm Let $\Delta$ be a $(d-1)$-dimensional lattice polyhedron defining a $d$-dimensional Gorenstein toric singularity $p \in {\bf A}_{\sigma}$. Then \[ S(\Delta;uv): = \psi_0(\Delta) + \psi_1(\Delta)uv + \cdots + \psi_{d-1}(\Delta)(uv)^{d-1} \] will be called the {\em $S$-polynomial} of the Gorenstein toric singularity at $p$. } \end{dfn} \begin{coro} The Euler number $e(F)$ equals $S(\Delta, 1) = (d-1)! {\rm vol}(\Delta)$. \label{eu.number} \end{coro} \noindent {\em Proof.} By definition of $P_{\Delta}(t)$, \[ \psi_0(\Delta) + \psi_1(\Delta) + \cdots + \psi_{d-1}(\Delta) = (d-1)! {\rm vol}(\Delta). \] Obviously, the left hand side equals $e(F)$. \hfill $\Box$ \begin{rem} {\rm It is known that the coefficient $\psi_{d-1}(\Delta)$ equals $l^*(\Delta)$, i.e., the number of rational points in the interior of $\Delta$ (see \cite{dan.hov}, pp. 292-293). } \label{lead} \end{rem} \section{Gorenstein quotient singularities} Let $G$ be a finite subgroup of $SL(d, {\bf C})$. We shall use the fact that any element $ g \in G$ is obviously conjugate to a diagonal matrix. \begin{dfn} {\rm If an element $g \in G$ is conjugate to \[ {\rm diag}( e^{2\pi \sqrt{-1}\alpha_1}, \ldots, e^{2\pi \sqrt{-1}\alpha_d} ) \] with $\alpha_i \in {\bf Q} \cap [0,1)$, then the sum \[ wt(g): = \alpha_1 + \cdots + \alpha_d \] will be called the {\em weight} of the element $g \in G$. The number $ht(g): = {\rm rk} (g - e)$ will be called the {\em height} of $g$. } \end{dfn} \begin{prop} For any $g \in G$, one has \[ wt(g) + wt(g^{-1}) = ht(g) = ht(g^{-1}). \] \label{dualit} \end{prop} \noindent {\em Proof.} Let $g = {\rm diag}( e^{2\pi \sqrt{-1}\alpha_1}, \ldots, e^{2\pi \sqrt{-1}\alpha_d} )$, $g^{-1} = {\rm diag}( e^{2\pi \sqrt{-1}\beta_1}, \ldots, e^{2\pi \sqrt{-1}\beta_d} )$. Then $ht(g)$ equals the number of nonzero elements in $\{ \alpha_1, \ldots, \alpha_d \}$. On the other hand, $\alpha_i + \beta_i = 1$ if $\alpha_i \neq 0$, and $\alpha_i + \beta_i = 0$ otherwise. Hence $\sum_{i =1}^d (\alpha_i + \beta_i) = ht(g)$. \hfill $\Box$ \begin{conj} {\rm (strong McKay correspondence)} Let $G \subset SL(d, {\bf C})$ be a finite group. Assume that $X = {\bf C}^d/G$ admits a smooth crepant desingularization $\pi \,: \, \hat{X} \rightarrow X$ and $F:= \pi^{-1}(0)$. Then $H^*(F, {\bf C})$ has a basis consisting of classes of algebraic cycles $Z_{\{g\}} \subset F$ which are in $1$-to-$1$ correspondence with the conjugacy classes $\{g\}$ of $G$, so that \[ {\rm dim}\, H^{2i}(F, {\bf C}) = \# \{ \mbox{\rm conjugacy classes $\{g\} \subset G$, such that $wt(g) = i$} \}. \] \label{strong1} \end{conj} Now we give several evidences in support of Conjecture \ref{strong1}. \begin{theo} Let $G \subset SL(d, {\bf C})$ be a finite abelian group. Suppose that $X = {\bf C}^d/G$ admits a smooth crepant toric desingularization $\pi \,: \, \hat{X} \rightarrow X$ and $F: = \pi^{-1}(0)$. Then $H^*(F, {\bf C})$ has a basis consisting of classes of algebraic cycles $Z_g \subset F$ which are in $1$-to-$1$ correspondence with the elements $g$ of $G$, so that \[ {\rm dim}\, H^{2i}(F, {\bf C}) = \# \{ \mbox{\rm elements $g \in G$, such that $wt(g) = i$} \}. \] In particular, the Euler number of $F$ equals $\mid G \mid$. \label{strong} \end{theo} \noindent {\em Proof.} Let $N \subset {\bf R}^d$ be the free abelian group generated by ${\bf Z}^d \subset {\bf R}^d$ and all vectors $(\alpha_1, \ldots, \alpha_d)$ where $g = {\rm diag}( e^{2\pi \sqrt{-1}\alpha_1}, \ldots, e^{2\pi \sqrt{-1}\alpha_d} )$ runs over all the elements of $G$. Then $N$ is a full sublattice of ${\bf R}^d = {\bf N}_{\bf R}$, ${\bf Z}^d$ is a subgroup of finite index in $N$, and $N/{\bf Z}^d$ is canonically isomorphic to $G$. Let $M = {\rm Hom}(N, {\bf Z})$. We identify ${\bf Z}^d$ with ${\rm Hom}({\bf Z}^d, {\bf Z})$ by using the dual basis. $M$ is a canonical sublattice of ${\bf Z}^d$ and can be identified with the set of all Laurent monomials in variables $t_1, \ldots, t_d$ which are $G$-invariant. Therefore, the cone $\sigma$ defining the affine toric variety $X = {\bf A}_{\sigma}$ is the positive $d$-dimensional octant ${\bf R}_{\geq 0}^d \subset {\bf R}^d = N_{\bf R}$. Furthermore, the element $m_{\sigma} \in M$, which was mentioned at the beginning of the previous section, equals $(1,\ldots, 1) \in {\bf Z}^d$. Now if $S: = {\bf C} \lbrack \sigma \cap N \rbrack $ and if for any $x \in \sigma \cap N$, we define a {\em degree} ${\rm deg}\, x : = \langle x, m_{\sigma} \rangle$, $S$ becomes a graded ring, so that \[ n_1 := (1, 0, \ldots ,0), \ldots, n_d : = (0,\ldots, 0,1) \] form a regular sequence of elements of degree $1$ in $S$. This means that $S/(n_1, \ldots, n_d)$ has a monomial basis corresponding to those elements of $N$ which are not in ${\bf Z}^d$. The element $(\alpha_1, \ldots, \alpha_d) \in N$ corresponds precisely to the element $g = {\rm diag}( e^{2\pi \sqrt{-1}\alpha_1}, \ldots, e^{2\pi \sqrt{-1}\alpha_d} ) \in G$. Moreover, \[ \langle (\alpha_1, \ldots, \alpha_d), m_{\sigma} \rangle = w(g). \] Thus, the Poincar\'{e} series of the quotient ring $S/(n_1, \ldots, n_d)$ equals \[ \psi_0(\Delta) + \psi_1(\Delta)t + \cdots + \psi_{d-1}(\Delta)t^{d-1} \] with coefficients \[ \psi_i(\Delta) = \# \{ \mbox{\rm elements $g \in G$ such that $wt(g) = i$} \} \] and $\sigma = \sigma_{\Delta}$ as in \S 4. The proof is completed after making use of Theorem \ref{invariants} and Corollary \ref{eu.number}. \hfill $\Box$ \begin{exam} {\rm For an abelian finite group $G \subset SL(3, {\bf C})$, the quotient $X = {\bf C}^3 /G$ admits always smooth crepant toric desingularizations coming from the full triangulations of the corresponding triangle $\Delta$ which is determined by $n_1, n_2, n_3$. All these triangulations contain only regular simplices and each of them differs from another one by finitely many elementary transformations (cf. \cite{oda}, Prop. 1.30 (ii)). In particular, if $G$ is a cyclic group generated by \[ {\rm diag}( e^{\frac{2\pi \sqrt{-1}\lambda_1}{|G|} }, e^{\frac{2\pi \sqrt{-1} \lambda_2}{|G|}}, e^{\frac{2\pi \sqrt{-1}\lambda_3}{|G|} }) \] with \[ 0 < \lambda_1, \lambda_2, \lambda_3 < | G |, \; \lambda_1 + \lambda_2 + \lambda_3 = | G |,\; {\rm gcd}(\lambda_1, \lambda_2, \lambda_3) =1, \] then: \[ {\rm dim}\, H^0(F, {\bf C}) = 1, \;{\rm dim}\, H^1(F, {\bf C}) = {\rm dim}\, H^3(F, {\bf C}) = {\rm dim}\, H^5(F, {\bf C}) = 0, \] \[ {\rm dim}\, H^2(F, {\bf C}) = \frac{1}{2} \left( |G| + \sum_{i =1}^3 {\rm gcd}(\lambda_i, |G|) \right) - 2 \] and \[ {\rm dim}\, H^4(F, {\bf C}) = \frac{1}{2} \left( |G| - \sum_{i =1}^3 {\rm gcd}(\lambda_i, |G|) \right) +1. \]} \end{exam} \begin{prop} The Conjecture \ref{strong1} is true for $d \leq 3$. \end{prop} \noindent {\em Proof.} If $d = 2$, then $wt(g) = 1$ unless $g = e$. The number of the conjugacy classes with weight $1$ is equal to the number of nontrivial irreducible representations of $G$. Since the exceptional locus $F$ of a crepant resolution is a tree of rational curves, ${\rm dim}\, H^0(F, {\bf C})$= 1, and ${\rm dim}\, H^2(F, {\bf C})$ is the number of irreducible components of $F$. By the classical McKay correspondence \cite{gonzalez,knorrer,mckay}, we obtain the statement \ref{strong1}. If $d=3$, we use the result of Roan \cite{roan3} about the existence of crepant resolutions and the Euler number of the exceptional locus. Let $F$ be the exceptional locus over $0$ of a crepant resolution $\pi \, : \, \hat{X} \rightarrow X$. Then $F$ is a strong deformation retract of $\hat{X}$; i.e., $H^i(F, {\bf C}) = H^i(\hat{X}, {\bf C})$. On the other hand, $H^4(\hat{X}, {\bf C})$ is Poincar\'{e} dual to $H_c^2(\hat{X}, {\bf C})$. Note that ${\rm dim}\, H^4(F, {\bf C})$ is nothing but the number of irreducible $2$-dimensional components of $F$. Since $H^2(\hat{X}, {\bf Z})$ is isomorphic to the Picard group of $\hat{X}$, ${\rm dim}\, H^2(\hat{X}, {\bf C})$ is equal to the number of $\pi$-exceptional divisors. Moreover, the subspace $H^2_c(\hat{X}, {\bf C}) \subset H^2(\hat{X}, {\bf C})$ is spanned exactly by the classes of those exceptional divisors whose image under $\pi$ is $0$. Therefore, \[ {\rm dim}\, H^2(\hat{X}, {\bf C}) - {\rm dim}\, H^4(\hat{X}, {\bf C}) = \] \[ = \# \{ \mbox{\rm exceptional divisors $E \subset \hat{X}$, such that $ \pi (E) $ is a curve on $X$} \}. \] By the classical McKay correspondence in dimension $2$, \[ {\rm dim}\, H^2(\hat{X}, {\bf C}) - {\rm dim}\, H^4(\hat{X}, {\bf C}) = \] \[ = \# \{ \mbox{\rm conjugacy classes $\{g\} \subset G$, such that $wt(g) = 1$ and $ht(g) = 2$} \}. \] By \cite{roan3}, \begin{equation} 1 + {\rm dim}\, H^2(\hat{X}, {\bf C}) + {\rm dim}\, H^4(\hat{X}, {\bf C}) = \# \{ \mbox{\rm all conjugacy classes $\{g\} \subset G$} \}. \label{euler3} \end{equation} By \ref{dualit}, \[ \# \{ \mbox{\rm conjugacy classes $\{g\} \subset G$, with $wt(g) = 1$ and $ht(g) = 3$} \} = \] \[ = \# \{ \mbox{\rm conjugacy classes $\{g\} \subset G$, with $wt(g) = 2$ and $ht(g) = 3$} \}. \] Hence, \[ \# \{ \mbox{\rm conjugacy classes $\{g\} \subset G$, with $wt(g) = 2$ and $ht(g) = 3$ } \} = {\rm dim}\, H^4(\hat{X}, {\bf C}). \] Notice that if $wt(g)= 2$, then the height of $g$ must be equal to $3$. Thus, \[ {\rm dim}\, H^4(F, {\bf C}) = \# \{ \mbox{\rm conjugacy classes $\{g\} \subset G$, such that $wt(g) = 2$ } \} . \] Finally, \[ {\rm dim}\, H^2(F, {\bf C}) = \# \{ \mbox{\rm conjugacy classes $\{g\} \subset G$, such that $wt(g) = 1$ } \} \] follows immediately from (\ref{euler3}). \hfill $\Box$ \bigskip \begin{dfn} {\rm Let $G$ be a finite subgroup of $SL(d, {\bf C})$ and $0 \in {\bf C}^d/G$ the corresponding $d$-dimensional Gorenstein toric singularity. If we denote by $\psi_i(G)$ the number of the conjugacy classes of $G$ having the weight $i$, then \[ S(G;uv): = \psi_0(G) + \psi_1(G)uv + \cdots + \psi_{d-1}(G)(uv)^{d-1} \] will be called the {\em $S$-polynomial} of the regarded Gorenstein quotient singularity at $0$. } \end{dfn} \begin{dfn} {\rm Let $G$ be a finite subgroup of $SL(d, {\bf C})$ and $0 \in {\bf C}^d/G$ the corresponding $d$-dimensional Gorenstein toric singularity. If we denote by $\tilde{\psi}_i(G)$ the number of the conjugacy classes of $G$ having the weight $i$ and the height $d$, then \[ \tilde{S}(G;uv): = \tilde{\psi}_0(G) + \tilde{\psi}_1(G)uv + \cdots + \tilde{\psi}_{d-1}(G)(uv)^{d-1} \] will be called the {\em $\tilde{S}$-polynomial} of the Gorenstein quotient singularity at $0$. } \end{dfn} By \ref{dualit}, we easily obtain: \begin{prop} The $\tilde{S}$-polynomial satisfies the following reciprocity relation: \[ \tilde{S}(G; uv) = (uv)^d \tilde{S}(G; (uv)^{-1}). \] \label{dualit1} \end{prop} \section{String-theoretic Hodge numbers} Let $X$ be a compact $d$-dimensional Gorenstein variety with ${\rm Sing}\,X$ consisting of at most toroidal or quotient singularities. \begin{dfn} {\rm Let $x \in {\rm Sing}\,X$. We say that the $d$-dimensional singularity at $x$ has the {\em splitting codimension $k$}, if $k$ is the maximal number for which the analytic germ at $x$ is locally isomorphic to the product of ${\bf C}^{d-k}$ and a $k$-dimensional toric singularity defined by a $(k-1)$-dimensional lattice polyhedron $\Delta'$ or, correspondingly, to the product of ${\bf C}^{d-k}$ and the underlying space ${\bf C}^k/G'$ of a $k$-dimensional quotient singularity defined by a finite subgroup $G' \subset SL(k, {\bf C})$. For simplicity, we also say that the singularity at $x$ is defined by $\Delta'$, or by $G'$.} \end{dfn} Using standard arguments, we can easily show that $X$ is always stratified by locally closed subvarieties $X_i$ $(i \in I)$, such that the germs of the singularities of $X$ along $X_i$ are analytically isomorphic to that of a Gorenstein toric singularity defined by means of a $(k-1)$-dimensional lattice polytope $\Delta_i$ or to that of a quotient singularity defined by means of a finite subgroup $G_i$ of $SL(k, {\bf C})$, respectively, where $k$ denotes the splitting codimension of singularities on $X_i$. \begin{dfn} We denote by $S(X_i; uv)$ the $S$-polynomial $S(\Delta_i; uv)$ or $S(G_i; uv)$. Analogously, $\tilde{S}(X_i; uv)$ will denote the $\tilde{S}$-polynomial $\tilde{S}(G_i; uv)$ if $X_i$ has only Gorenstein quotient singularities. \end{dfn} \begin{dfn} {\rm Suppose that $X$ has at most quotient Gorenstein singularities. A stratification $X = \bigcup_{i \in I} X_i$, as above, is called {\em canonical}, if for every $i \in I$ and every $x \in X_i$, there exists an open subset $U \cong {\bf C}^d/G_i$ in $X$ and an element $g \in G_i$, such that $\overline{X_i} \cap U = ({\bf C}^d)^g/C(g)$, where $({\bf C}^d)^g$ is the set of $g$-invariant points of ${\bf C}^d$.} \label{stratif} \end{dfn} \begin{rem} {\rm An algebraic variety is called {\em V-variety} if it has at most quotient singularities. A {\em Gorenstein $V$-variety} (abbreviated {\em $GV$-variety}) is a $V$-variety having at most Gorenstein quotient singularities. The notion of $V$-variety (or $V$-manifold) was first introduced by Satake \cite{satake}. The existence and the uniqueness of the canonical stratification for a $V$-variety was proved by Kawasaki in \cite{kawasaki}. ( Note that our {\em canonical} stratification in \ref{stratif} is not the first, but the second stratification of $X$ defined by Kawasaki in \cite{kawasaki}, p. 77.) } \end{rem} \begin{prop} Suppose that $X$ is a $GV$-variety and $X = \bigcup_{i \in I} X_i$ is its canonical stratification. Then for any $i_0 \in I$, one has: \[ S(X_{i_0}; uv) = \tilde{S}(X_{i_0}; uv) + \sum_{X_{i_0} < X_{i_1}} \tilde{S}(X_{i_1}; uv). \] \label{can.strat} \end{prop} \noindent {\em Proof}. It is sufficient to prove the corresponding local statement; i.e., we can assume, without loss of generality, that $X_{i_0} = {\bf C}^k/G_{i_0}$. For simplicity, we set $Y = {\bf C}^k$, $Z = X_{i_0}$. Denote by $\pi$ the natural mapping $Y \rightarrow Z$. For $g \in G_{i_0}$, the image $Z(g) : = \pi(Y^g) \subset Z$ depends only on the conjugacy class of $g$. Since $ht(g)$ equals the codimension of $Z(g)$ in $Z$, we obtain \[ S(X_{i_0}; uv) = \tilde{S}(X_{i_0}; uv) + \sum_{X_{i_0} < X_{i_1}} \tilde{S}(X_{i_1}; uv). \] \hfill $\Box$ \begin{coro} Suppose that $X$ is a $GV$-variety and $X = \bigcup_{i \in I} X_i$ is its canonical stratification. Then for any $i_0 \in I$ one has: \[ \tilde{S}(X_{i_0}; uv) = \sum_{k \geq 0} (-1)^k \sum_{X_{i_0} < \cdots < X_{i_k}} {S}(X_{i_k}; uv). \] \label{stra2} \end{coro} \noindent {\em Proof.} By \ref{can.strat}, we have \[ \tilde{S}(X_{i_0}; uv) = S(X_{i_0}; uv) - \sum_{X_{i_0} < X_{i_1}} \tilde{S}(X_{i_1}; uv). \] After that we apply \ref{can.strat} to $X_{i_1}$: \[ \tilde{S}(X_{i_1}; uv) = S(X_{i_1}; uv) - \sum_{X_{i_1} < X_{i_2}} \tilde{S}(X_{i_2}; uv), \;\;\; {\rm etc} \ldots \] The repetition of this procedure completes the proof of the assertion. \hfill $\Box$ \begin{dfn} {\rm Let $X$ be a stratified variety with at most Gorenstein toroidal or quotient singularities. We shall call the polynomial \[ E_{\rm st}(X; u,v):= \sum_{i \in I} E(X_i; u,v) \cdot S(X_i; uv) \] the {\em string-theoretic $E$-polynomial of $X$}. Let us write $E_{\rm st}(X;u,v)$ in the following expanded form: \[ E_{\rm st}(X; u,v) = \sum_{p,q} a_{p,q} u^p v^q. \] The numbers $h^{p,q}_{\rm st}(X): = (-1)^{p+q} a_{p,q}$ will be called the {\em string-theoretic Hodge numbers of $X$}. Correspondingly, \[ e_{\rm st}(X): = E_{\rm st}(X; -1,-1) = \sum_{p,q} (-1)^{p+q} h^{p,q}_{\rm st}(X) \] will be called the {\em string-theoretic Euler number of $X$}. } \end{dfn} \begin{rem} {\rm If $X$ admits a smooth crepant toroidal desingularization $\pi\, : \, \hat{X} \rightarrow X$, then, by \ref{formula} and \ref{invariants}, the $E$-polynomial of $\hat{X}$ equals \[ E(\hat{X}; u, v) = \sum_{i \in I} E(X_i; u, v)\cdot E(F_i; u, v) \] where $F_i$ denotes a the special fiber $\pi^{-1}(x)$ over a point $x \in X_i$.} \label{crep1} \end{rem} By \ref{crep1}, we obtain: \begin{theo} If $X$ admits a smooth crepant toroidal desingularization $\hat{X}$, then the string-theoretic Hodge numbers $h^{p,q}_{\rm st}(X)$ coincide with the ordinary Hodge numbers $h^{p,q}(\hat{X})$. In particular, the numbers $h^{p,q}_{\rm st}(X)$ are nonnegative and satisfy the Poincar\'{e} duality $h^{p,q}_{\rm st}(X) = h^{d-p,d-q}_{\rm st}(X)$. \end{theo} The next theorem will play an important role in the forthcoming statements: \begin{theo} Suppose that $X$ is a $GV$-variety and $X = \bigcup_{i \in I} X_i$ denotes its canonical stratification. Then \[ E_{\rm st}(X; u,v) = \sum_{i \in I} E(\overline{X}_i; u,v) \cdot \tilde{S}(X_i; uv). \] \label{second.f} \end{theo} \noindent {\em Proof.} By \ref{stra1}, we get \[ E(X_{i_0}; u,v) = \sum_{k \geq 0} (-1)^k \sum_{X_{i_k} < \cdots < X_{i_1} < X_{i_0} } E(\overline{X}_{i_k}; u,v). \] Therefore, \[E_{\rm st}(X; u,v) = \sum_{i_0 \in I} \left( \sum_{k \geq 0} (-1)^k \sum_{X_{i_k} < \cdots < X_{i_1} < X_{i_0} } E(\overline{X}_{i_k}; u,v) \right) \cdot S(X_{i_0}; uv) = \] \[ = \sum_{i_k \in I} E(\overline{X}_{i_k}; u,v) \cdot \left( \sum_{k \geq 0} (-1)^k \sum_{X_{i_k} < \cdots < X_{i_1} < X_{i_0} } S(X_{i_0}; uv) \right). \] By \ref{stra2}, we have \[ \tilde{S}(X_{i_k}; uv) = \sum_{k \geq 0} (-1)^k \sum_{X_{i_k} < \cdots < X_{i_1} < X_{i_0} } S(X_{i_0}; uv). \] This implies the required formula. \hfill $\Box$ \begin{coro} Suppose that $X$ is a $GV$-variety. Then the numbers $h^{p,q}_{\rm st}(X)$ are nonnegative and satisfy the Poincar\'{e} duality $h^{p,q}_{\rm st}(X) = h^{d-p,d-q}_{\rm st}(X)$. \label{main.prop} \end{coro} \noindent {\em Proof.} Since $\overline{X}_i$ itself is a $V$-variety, one has $h^{p,q}(\overline{X}_i) \geq 0$, as well as the Poincar\'{e} duality \[ E(\overline{X}_i; u,v) = (uv)^{{\rm dim}\,\overline{X}_i} E(\overline{X}_i; u^{-1},v^{-1}). \] On the other hand, by \ref{dualit1}, we obtain \[ \tilde{S}(X_i; uv) = (uv)^{{\rm dim}\,\overline{X}_i} \tilde{S}(X_i; (uv)^{-1}). \] This implies \[ E_{\rm st}(X; u,v) = (uv)^{{\rm dim}\,X} E_{\rm st}(X; u^{-1},v^{-1}). \] Since $\tilde{S}(X_i; uv)$ is a polynomial of $uv$ with nonnegative coefficients, we conclude that $h^{p,q}_{\rm st}(X) \geq 0$. \hfill $\Box$ \begin{theo} Suppose that $X$ has at most toroidal Gorenstein singularities. Let $\pi \, : \, \hat{X} \rightarrow X$ be a $MPCP$-desingularization of $X$. Then \[ E_{\rm st}(X;, u,v) = E_{\rm st}(\hat{X}; u,v). \] Moreover, \[ h^{p,1}_{\rm st}(X) = h^{p,1}(\hat{X}), \;\;\;\; \mbox{\rm for all $p$}. \] \label{MPCP-desing} \end{theo} \noindent {\em Proof. } Let $X = \bigcup_{i \in I} X_i$ be a stratification of $X$, such that \[ E_{\rm st}(X; u,v) = \sum_{i \in I} E(X_i; u,v) \cdot S(\Delta_i; uv) \] and $\pi\, : \, \hat{X} \rightarrow X$ be a MPCP-desingularization of $X$. We set $\hat{X}_i : = \pi^{-1}(X_i)$. Then $\hat{X}_i$ has the natural stratification by products $X_i \times ({\bf C}^*)^{{\rm codim}\, \theta}$ induced by the triangulation \[ \Delta_i = \bigcup_{\theta \in {\cal T}_i} \theta. \] Thus, \[ E_{\rm st}(\hat{X}; u,v) = \sum_{i \in I} \left(\sum_{\theta \in {\cal T}_i} (uv-1)^{{\rm codim}\, \theta} E(X_i; u,v) \cdot S(\theta; uv) \right). \] By counting lattice points in $k\Delta_i$, we obtain \[ S(\Delta_i; uv) = \sum_{\theta \in {\cal T}_i} (uv-1)^{{\rm codim}\, \theta} S(\theta; uv). \] Hence, \[ E_{\rm st}(X;, u,v) = E_{\rm st}(\hat{X}; u,v). \] Since $\hat{X}$ has only terminal ${\bf Q}$-factorial singularities, for any $\theta \in {\cal T}_i$ we obtain \[ \psi_1(\theta) = 0;\;\; \mbox{\rm i.e.,} \;\; S(\theta; uv) = 1 + \psi_2(\theta) (uv)^2 + \cdots . \] Therefore, the coefficient of $u^p v$ in $E_{\rm st}(\hat{X}; u,v)$ coincides with the coefficient of $u^p v$ in the usual $E$-polynomial $E(\hat{X}; u,v)$. As $\hat{X}$ is a $V$-variety, the Hodge structure in $H^*(\hat{X}, {\bf C})$ is pure, and \[ h^{p,1}_{\rm st}(X) = h^{p,1}(\hat{X}),\; \mbox{\rm for all $p$}. \] \hfill $\Box$ \begin{coro} Suppose that $X$ has at most toroidal Gorenstein singularities. Then the numbers $h^{p,q}_{\rm st}(X)$ are nonnegative and satisfy the Poincar\'{e} duality $h^{p,q}_{\rm st}(X) = h^{d-p,d-q}_{\rm st}(X)$. \end{coro} \noindent {\em Proof.} By \ref{MPCP-desing}, it is sufficient to prove the statement for a $MPCP$-desingularization $\hat{X}$ of $X$. The latter follows from \ref{main.prop}. \hfill $\Box$ \begin{theo} Let $X$ be a smooth compact K\"ahler manifold of dimension $n$ over ${\bf C}$ being equipped with an action of a finite group $G$, such that $X$ has a $G$-invariant volume form. Then the orbifold Hodge numbers $h^{p,q}(X,G)$ which were defined in the introduction coincide with the string-theoretic Hodge numbers $h^{p,q}_{\rm st}(X/G)$. Moreover, \[ e(X,G) = e_{\rm st}(X/G). \] \label{str-eul} \end{theo} \noindent {\em Proof.} We use the canonical stratification of $Y= X/G$: \[ Y = \bigcup_{i \in I} Y_i. \] For every stratum $Y_i$, there exists an element $g_i \in G$, such that $\overline{Y}_i = X^{g_i}/C(g_i)$. We note that \[ E(\overline{Y}_i; u,v) = \sum_{p,q} (-1)^{p+q} {\rm dim} H^{p,q}(X^{g_i})^{C(g_i)} u^pv^q. \] Now the equality \[ h^{p,q}_{\rm st}(X/G) = h^{p,q}(X,G) \] follows from Theorem \ref{second.f}. \newline In order to get $e(X,G) = e_{\rm st}(X/G)$, it remains to prove the equality \[ e(X,G) = \sum_{p,q} (-1)^{p+q} h^{p,q}(X,G). \] We shall make use of the notation which was introduced in \S 1. Since $\{ g \}$ expresses a system of representatives for $G/C(g)$ and the number of conjugacy classes of $G$ equals \[ \frac{1}{\mid G \mid} \sum_{g \in G} \mid C(g) \mid, \] one can rewrite the physicists Euler number (\ref{euler.phys}) as \[ e(X, G) = \frac{1}{\mid G \mid} \sum_{g} | C(g) | \cdot e(X^g/C(g)) = \sum_{\{ g \}} e(X^g/C(g)), \] where $\{g\}$ runs over all conjugacy classes of $G$ with $g$ representing $\{ g \}$. We show that \[ \sum_{p,q} (-1)^{p+q} h^{p,q}_g(X,G) = e(X^g/C(g)). \] This follows from the equalities \[ \sum_{p,q} (-1)^{p+q} h^{p,q}_g(X,G) = \sum_{i =1}^{r_g} \sum_{p,q} (-1)^{p+q - 2 F_i(g)} h^{p - F_i(g),q - F_i(g)}_{C(g)}(X_i(g)) = \] \[ = \sum_{p,q} (-1)^{p+q} h^{p,q}_{C(g)}(X^g) = e(X^g/C(g)). \] \hfill $\Box$ \bigskip \begin{coro} Suppose that $X/G$ has a crepant desingularization $\widehat{X/G}$ and that the strong McKay correspondence $($Conjecture \ref{strong1}$)$ holds true for the singularities occuring along every stratum of $X/G$. Then \[ h^{p,q}(\widehat{X/G}) = h^{p,q}_{\rm st}(X/G). \] \label{c.des} \end{coro} \begin{exam} {\rm Let us first give a $3$-dimensional example of an orbit space (with a simple acting group) containing both abelian and non-abelian quotient singularities, and which was proposed by F. Hirzebruch. We consider the Fermat quintic $X = \{ [z_1, \ldots, z_5] \in {\bf P}^4 \mid \sum_{i =1}^5 z_i^5 = 0 \}$ and let the alternating group ${\cal A}_5$ act on it coordinatewise. The group ${\cal A}_5$ has five conjugacy classes: the trivial, one consisting of all $20$ $3$-cycles, one consisting of the $15$ products of disjoint transpositions, and two more conjugacy classes of $5$-cycles, each of which has $12$ elements. Note that the action of the elements of these last two conjugacy classes is fixed point free. Each of the $20$ $3$-cycles fixes a plane quintic and two additional points. Correspondingly, each of the $15$ products of disjoint traspositions fixes a plane quintic and a projective line (without common points). As $X/{\cal A}_5$ is a Calabi-Yau variety, the generic points of the $1$-dimensional components of ${\rm Sing}\, X/{\cal A}_5$ are compound Du Val points \cite{reid1}. Up to the above mentioned $40$ additional points coming from the $3$-cycles and having isotropy groups $\cong {\bf Z}/3{\bf Z}$, there exist $175$ more fixed points on $X$ creating (after appropriate group identifications) {\em dissident} points on $X/{\cal A}_5$ (we follow here the terminology of M. Reid). Namely, the $25$ points of the intersection locus of the $20$ plane quintics (with isotropy groups $\cong {\cal A}_4$), further $125$ points lying in the intersection locus of the $15$ plane quintics (with isotropy groups $\cong {\cal S}_3$), as well as $15 + 10 = 25$ points coming from the intersection of the projective lines (with isotropy groups isomorphic to the Kleinian four-group and to ${\cal S}_3$ respectively). Using Ito's results \cite{ito1,ito2}, we can construct global crepant desingularizations $\pi\, : \, \widehat{X/{\cal A}_5} \rightarrow X/{\cal A}_5$. By \ref{c.des}, $h^{p,q}(\widehat{X/{\cal A}_5}) = h^{p,q}_{\rm st}({X/{\cal A}_5})$. Thus, for the computation of $h^{p,q}(\widehat{X/{\cal A}_5})$, we just need to choose two representatives, say $(123)$ and $(12)(34)$, of the two non-freely acting conjugacy classes. We have: \begin{itemize} \item $h^{p,q}({X/{\cal A}_5}) = h^{p,q}_{\{1\}}({X,{\cal A}_5})$ equals $\delta_{p,q}$ ( $=$ Kronecker symbol) for $p + q \neq 3$, $h^{p,q}_{\{1\}}({X,{\cal A}_5}) = 1$ for $(p,q) \in \{ (3,0), (0,3) \}$ and $h^{p,q}_{\{1\}}({X,{\cal A}_5}) = 5$ for $(p,q) \in \{ (2,1), (1,2) \}$; \item $h^{p,q}_{\{(123)\}} (X, {\cal A}_5)$ equals $2$ for $(p,q) \in \{ (1,1), (2,2) \}$, $h^{p,q}_{\{(123)\}} (X, {\cal A}_5) =6$ for $(p,q) \in \{ (2,1), (1,2) \}$, $h^{p,q}_{\{(123)\}} (X, {\cal A}_5) = 0$ otherwise; \item $h^{p,q}_{\{(12)(34)\}} (X, {\cal A}_5)$ equals $2$ for $1 \leq p,q \leq 2$ and $0$ otherwise. \end{itemize} Thus, we get: \[ h^{2,1}_{\rm st}(X/{\cal A}_5) = h^{1,2}_{\rm st}(X/{\cal A}_5) = 13, \] \[ h^{1,1}_{\rm st}(X/{\cal A}_5) = h^{2,2}_{\rm st}(X/{\cal A}_5) = 5. \] In particular, $e(\widehat{X/{\cal A}_5}) = e_{\rm st}(X/{\cal A}_5) = -16$, in agreement with the calculations of physicists (cf. \cite{KS}, p. 57). } \end{exam} \begin{exam} {\rm Let $X^{(n)} : = X^n/{\cal S}_n$ be the $n$-th symmetric power of a smooth projective surface $X$. As it is known (see, for instance, \cite{got2}, p.54 or \cite{hirzebruch}, p.258), $X^{(n)}$ is endowed with a canonical crepant desingularization $X^{[n]}: = {\rm Hilb}^n(X) \rightarrow X^{(n)}$ given by the Hilbert scheme of finite subschemes of length $n$. In \cite{got1,got2}, G\"ottsche computed the Poincar\'{e} polynomial of $X^{[n]}$. In particular, his formula for the Euler number gives: \[ \sum_{n =0}^{\infty} e(X^{[n]}) t^n = \prod_{k =1}^{\infty}(1-t^k)^{-e(X)}. \] Using power series comparison and the above formula, Hirzebruch and H\"ofer gave in \cite{hirzebruch} a formal proof of the equality $e(X^{[n]}) = e(X^{(n)}, {\cal S}_n)$. In fact, for the proof of the validity of {\em orbifold Euler formulae} of this kind, it is enough to check locally that the Conjecture \ref{general} of M. Reid is true (cf. \cite{roan3}, Lemma 1). Our results \ref{str-eul} and \ref{c.des} say more: in order to obtain the equality $ h^{p,q}(X^{[n]}) = h^{p,q}(X^{(n)},G)$ it is sufficient to verify locally our ``strong'' McKay correspondence. The latter has been checked by G\"ottsche in \cite{got3}. The numbers $h^{p,q}(X^{[n]})$ can be computed by means of the Hodge polynomial $h(X^{[n]}; u,v) := E(X^{[n]}; -u,-v)$. If $\Pi(n)$ denotes the set of all finite series $(\alpha) = ( \alpha_1, \alpha_2, \ldots )$ of nonnegative integers with $\sum_i i \alpha_i = n$, then the conjugacy class of a permutation $\sigma \in {\cal S}_n$ is determined by its type $(\alpha) = ( \alpha_1, \alpha_2, \ldots ) \in \Pi(n)$, where $\alpha_i$ expresses the number of cycles of length $i$ in $\sigma$. G\"ottsche and Soergel \cite{GS,got2} proved that \[ h(X^{[n]}; u,v) = \sum_{(\alpha) \in \Pi(n)} (uv)^{n- \mid \alpha \mid} \prod_{k =1}^{\infty} h(X^{(\alpha_k)}; u,v), \] where $\mid \alpha \mid : = \alpha_1 + \alpha_2 + \cdots $ denotes the sum of the members of $(\alpha) \in \Pi(n)$. (Similar formulae can be obtained for the even-dimensional Kummer varieties of higher order, cf. \cite{got2,got3,GS}.) } \end{exam} \section{Applications to quantum cohomology $\;\;\;\;\;\;\;\;\;\;\; \;\;\; $ and mirror symmetry} {}From now on, and throughout this section, we use the notion of {\em reflexive polyhedron} being introduced in \cite{batyrev1}. \begin{prop} Let $\Delta$ be a reflexive polyhedron of dimension $d$. Then \[ S(\Delta,t) = (t -1)^d + \sum_{\begin{array}{c} {\scriptstyle 0 \leq {\rm dim}\,\theta \leq d-1} \\ {\scriptstyle \theta \subset \Delta} \end{array}} S(\theta,t) \cdot (t -1)^{{\rm dim}\, \theta^*}. \] \label{relation} \end{prop} \noindent {\em Proof. } Denote by $\partial \Delta$ the $(d-1)$-dimensional boundary of $\Delta$ which is homeomorphic to $(d-1)$-dimensional sphere. Let $l(k \cdot \partial \Delta)$ be the number of lattice points belonging to the boundary of $k\Delta$. The reflexivity of $\Delta$ implies: \[ l(k \cdot \partial \Delta) = \sum_{0 \leq {\rm dim}\, \theta \leq d-1} (-1)^{d-1 - {\rm dim}\, \theta} l(k\theta), \; \; \; \mbox{\rm for $k > 0$}. \] Since the Euler number of a $(d-1)$-dimensional sphere is $1 + (-1)^{d-1}$, we obtain \[ (-1)^{d-1} + (1-t)P_{\Delta}(t)\;\; = \;\; (-1)^{d-1} \sum_{0 \leq {\rm dim}\, \theta \leq d-1} (-1)^{{\rm dim}\, \theta} P_{\theta}(t), \] i.e., \[ (-1)^{d-1} + \frac{S(\Delta;t)}{(1-t)^d}\;\; = \;\; (-1)^{d-1} \sum_{0 \leq {\rm dim}\, \theta \leq d-1} (-1)^{{\rm dim}\, \theta} \frac{S(\theta;t)}{(1-t)^{{\rm dim}\, \theta +1}}. \] This implies the required equality. \hfill $\Box$ \medskip We prove the following relation between the polar duality of lattice polyhedra and string-theoretic cohomology: \begin{theo} Let ${\bf P}_{\Delta}$ be a $d$-dimensional Gorenstein toric Fano variety corresponding to a $d$-dimensional reflexive polyhedron $\Delta$. Then \[ E_{\rm st}({\bf P}_{\Delta}; u, v) = (1- uv)^{d+1} P_{\Delta^*} (uv) \] where $\Delta^*$ is the dual reflexive polyhedron. \end{theo} \noindent {\em Proof.} ${\bf P}_{\Delta}$ has a natural stratification being defined by the strata $T_{\theta}$, where $\theta$ runs over all the faces of $\Delta$. On the other hand, the Gorenstein singularities along $T_{\theta}$ are determined by the dual face $\theta^*$ of the dual polyhedron $\Delta^*$ (cf. \cite{batyrev1}, 4.2.4). We set $S(\theta^*, uv) =1$ if $\theta = \Delta$. Then \[ E_{\rm st}({\bf P}_{\Delta}; u, v) = \sum_{\theta \subset \Delta} E(T_{\theta}; u,v) \cdot S(\theta^*; uv). \] Note that \[ E(T_{\theta}; u,v) = (uv -1)^{{\rm dim}\, \theta}, \] and that, for ${\rm dim}\, \theta < d$, one has by definition: \[ S(\theta^*; uv) = (1 - uv)^{{\rm dim}\, \theta^* +1} P_{\theta^*}(uv). \] If we apply Proposition \ref{relation} to the dual reflexive polyhedron $\Delta^*$, then, using ${\rm dim}\, \theta + {\rm dim}\, \theta^* = d-1$, we obtain the desired formula for $E_{\rm st}({\bf P}_{\Delta}; u, v)$. \hfill $\Box$ \bigskip \begin{coro} The string-theoretic Euler number of ${\bf P}_{\Delta}$ is equal to $d!({\rm vol}\, \Delta^*)$. \end{coro} \begin{rem} {\rm The quantum cohomology ring of a smooth toric variety was described in \cite{batyrev00}. It was proved that the usual cohomology of a smooth toric manifold can be obtained as a limit of the quantum cohomology ring. On the other hand, one can immediately extend the description of the quantum cohomology ring to arbitrary (possibly singular) toric variety (cf. \cite{batyrev00}, 5.1). In particular, one can easily show that ${\rm dim}\, QH^*_{\varphi}({\bf P}_{\Delta}, {\bf C}) = d! ({\rm vol}\, \Delta^*)$, for any $d$-dimensional reflexive polyhedron. Comparing dimensions, we see that, for singular toric Fano varieties ${\bf P}_{\Delta}$, the limit of the quantum cohomology ring is not the usual cohomology ring, but rather the cohomology of a smooth crepant desingularization, if such a desingularization exists (cf. \cite{batyrev00}, 6.5). By our general philosophy, we should consider the string-theoretic Hodge numbers $h^{p,p}_{\rm st}({\bf P}_{\Delta})$ as the Betti numbers of a limit of the quantum cohomology ring $QH^*_{\varphi}({\bf P}_{\Delta}, {\bf C})$. } \end{rem} \bigskip Let $\overline{Z}_f: = \overline{Z}_{f_1} \cap \cdots \cap \overline{Z}_{f_r}$ be a generic $(d-r)$-dimensional Calabi-Yau complete intersection variety, which is embedded in a Gorenstein toric Fano variety ${\bf P}_{\Delta}$ corresponding to a $d$-dimensional reflexive polyhedron $\Delta = \Delta_1 + \cdots \Delta_r$, where $\Delta_i$ is the Newton polyhedron of $f_i$ $(i = 1, \ldots, r)$. Assume that the lattice polyhedra $\Delta_1, \ldots, \Delta_r$ are defined by a {\em nef-partition} of vertices of the dual reflexive polyhedron $\Delta^* = {\rm Conv}\{ \nabla_1, \ldots , \nabla_r\}$. (For definitions and notations the reader is referred to \cite{batyrev-borisov,borisov}.) Denote by $\overline{Z}_g : = \overline{Z}_{g_1} \cap \cdots \cap \overline{Z}_{g_r}$ a generic Calabi-Yau complete intersection variety in the Gorenstein toric Fano variety ${\bf P}_{\nabla^*}$, which is defined by the reflexive polyhedron $\nabla^* = {\rm Conv}\{ \Delta_1, \ldots , \Delta_r \}$, where $\nabla_i$ is the Newton polyhedron of $g_i$ $(i = 1, \ldots, r)$. \begin{conj} {\rm (Mirror duality of string-theoretic Hodge numbers)} The string-theoretic $E$-polynomials of $\overline{Z}_f$ and $\overline{Z}_g$ obey to the following reciprocity law: \[ E_{\rm st}(\overline{Z}_f; u,v) = (-u)^{d-r}E_{\rm st}(\overline{Z}_g;u^{-1},v). \] Equivalently, the string-theoretic Hodge numbers of $\overline{Z}_f$ and $\overline{Z}_g$ are related to each other by: \[ h^{p,q}_{\rm st}(\overline{Z}_f) = h^{d-r-p,q}_{\rm st}(\overline{Z}_g), \;\; \mbox{ {\rm for all $p,q$}}. \] \label{symmetry} \end{conj} \noindent We want to show some evidences in support of Conjecture \ref{symmetry} for Calabi-Yau hypersurfaces ($r =1$). \begin{theo} Let $\overline{Z}_f$ be a $\Delta$-regular Calabi-Yau hypersurface in ${\bf P}_{\Delta}$. Then \[ E_{\rm st}(\overline{Z}_f; 1,v) = \frac{S(\Delta^*;v)}{v} + (-1)^{d-1} \frac{S(\Delta;v)}{v} + \] \[ + \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\ {\scriptstyle \theta \subset \Delta} \end{array}} \frac{(-1)^{{\rm dim}\, \theta-1}}{v} \left( S(\theta; v) \cdot S(\theta^*; v) \right) - \] \[ - \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\,\theta = d-1} \\ {\scriptstyle \theta \subset \Delta} \end{array}} (-1)^{d-1} \frac{S(\theta,v)}{v} - \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\,\theta^* = d-1} \\ {\scriptstyle \theta^* \subset \Delta^*} \end{array}} \frac{S(\theta^*,v)}{v}. \] \label{formul0} \end{theo} \begin{coro} \[ E_{\rm st}(\overline{Z}_f; 1,v) =(-1)^{d-1} E_{\rm st}(\overline{Z}_g; 1,v). \] \end{coro} At first we need the following formula: \begin{prop} Let $\theta$ be a face of $\Delta$ and ${\rm dim}\, \theta \geq 1$. Then \[ E(Z_{f,\theta}; 1,v) = \frac{ (v-1)^{{\rm dim}\, \theta}}{v} + (-1)^{{\rm dim}\, \theta-1} \frac{S(\theta,v)}{v} . \] \label{e-f} \end{prop} \noindent {\em Proof. } It follows from the formula of Danilov and Khovanski\^i (\cite{dan.hov}, Remark 4.6): \[ (-1)^{{\rm dim}\, \theta -1} \sum_p e^{p,q}(Z_{f,\theta}) = (-1)^q { n \choose q + 1 } + \psi_{q+1}(\theta). \] \hfill $\Box$ \noindent {\bf Proof of Theorem \ref{formul0}}. By definition, \[ E_{\rm st}(\overline{Z}_f; 1,v) \; = \; E(Z_{f, \Delta}; 1,v) \; + \; \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\,\theta = d-1} \\ {\scriptstyle \theta \subset \Delta} \end{array}} E(Z_{f, \theta}; 1,v) \;\; + \] \[ + \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2 } \\ {\scriptstyle \theta \subset \Delta} \end{array}} E(Z_{f, \theta}; 1,v) \cdot S(\theta^*; v). \] Substituting the expressions which follow from \ref{e-f}, we get: \[ E(\overline{Z}_{f, \Delta}; 1,v) = \frac{(v-1)^{d}}{v} + (-1)^{d-1} \frac{S(\Delta,v)}{v} + \] \[ + \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\,\theta = d-1} \\ {\scriptstyle \theta \subset \Delta} \end{array} } \left( (-1)^{d-2} \frac{S(\theta,v)}{v} + \frac{(v-1)^{d-1}}{v} \right) + \] \[ + \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\ {\scriptstyle \theta \subset \Delta} \end{array}} \left( (-1)^{{\rm dim}\, \theta - 1} \frac{S(\theta,v)}{v} + \frac{(v-1)^{{\rm dim}\, \theta}}{v} \right) \cdot S(\theta^*; uv). \] It remains to use \ref{e-f} and \ref{relation}. \hfill $\Box$ \bigskip \begin{dfn} {\rm For a face $\theta$ of $\Delta$, we denote by ${\bf v}(\theta)$ the normalized volume of $\theta$: $({\rm dim}\,\theta)! {\rm vol}(\theta)$. } \end{dfn} \begin{coro} Let $\Delta$ be a $d$-dimensional reflexive polyhedron. Then \[ e_{\rm st} (\overline{Z}_f) = \sum_{i =1}^{d-2} \sum_{{\rm dim}\, \theta = i} (-1)^i {\bf v}(\theta)\cdot {\bf v}(\theta^*). \] In particular, \[ e_{\rm st} (\overline{Z}_f) = (-1)^{d-1} e_{\rm st} (\overline{Z}_g). \] \end{coro} We remark that \ref{symmetry} is evident if $q =0$, because $h^{p,0}_{\rm st} (\overline{Z}_f) = 1$, for $q = 0, d-1$ and $h^{p,0} _{\rm st} (\overline{Z}_f) = 0$ otherwise. For $q = 1$ $(r=1)$, and $p \in \{ 1,d-2\}$, Conjecture \ref{symmetry} is proved by Theorem \ref{MPCP-desing} combined with Thm. 4.4.3 from \cite{batyrev1}. We generalize this for arbitrary values of $p$. \begin{theo} For a face $\theta$ of $\Delta$, we denote by $l^*(\theta)$ the number of lattice points in the relative interior of $\theta$. Assume that $d \geq 5$. Then for $ 2 \leq p \leq d-3$ one has \[ h^{p,1}_{\rm st}(\overline{Z}_f) = \sum_{{\rm codim}\, \theta = p } l^*(\theta)\cdot l^*(\theta^*). \] By the duality among faces, one has \[ h^{p,1}_{\rm st}(\overline{Z}_f) = h^{d-1-p,1}_{\rm st}(\overline{Z}_g). \] \label{p1} \end{theo} \noindent {\em Proof.} By the Poincar\'{e} duality, it is enough to compute $h^{d-1-p,d-2}_{\rm st}(\overline{Z}_f) = h^{p,1}_{\rm st}(\overline{Z}_f)$. We use \[ E_{\rm st} (\overline{Z}_f;u,v) = \sum_{\theta \subset \Delta} E(Z_{f,\theta};u,v)\cdot S(\theta^*; uv). \] By \ref{lead}, \[ S(\theta^*; uv) = l^*(\theta^*)(uv)^{{\rm dim}\, \theta^*} + \mbox{\rm $\{$lower order terms in $uv$$\}$}. \] On the other hand, by \cite{dan.hov}, Prop. 3.9, \[e^{p,q}(Z_{f,\theta}) =0\;\; \mbox{if $p + q > {\rm dim}\, \theta - 1 = {\rm dim}\, Z_{f,\theta}$ and $p \neq q$}. \] Hence, the only possible case in which we can meet the monomial of type $u^{d-1-p}v^{d-2}$ within the product $ E(Z_{f,\theta};u,v)\cdot S(\theta^*; u,v)$ is that occuring by consideration of the product of the term $l^*(\theta^*)(uv)^{{\rm dim}\, \theta^*}$ from $S(\theta^*; uv)$ and the term \[ e^{0, {\rm dim}\, \theta -1}(Z_{f,\theta}) v^{{\rm dim}\, \theta -1}, \] where ${\rm dim}\, \theta^* = d - 1 - p$. As it is known (cf. \cite{dan.hov}, Prop. 5.8.): \[ e^{0, {\rm dim}\, \theta -1}(Z_{f,\theta}) = (-1)^{{\rm dim}\, \theta -1}l^*(\theta). \] Therefore, \[ h^{d-1-p,d-2}_{\rm st}(\overline{Z}_f) = l^*(\theta) \cdot l^*(\theta^*). \] \hfill $\Box$ \begin{coro} Let $\hat{Z}_f$ be a MPCP-desingularization of $\overline{Z}_f$. Assume that $d \geq 5$. Then, for $ 2 \leq p \leq d-3$, one has \[ h^{p,1}(\hat{Z}_f) = \sum_{{\rm codim}\, \theta = p } l^*(\theta)\cdot l^*(\theta^*). \] \label{p1cor} \end{coro} \noindent {\em Proof.} It follows from Theorem \ref{p1} and Theorem \ref{MPCP-desing}. \hfill $\Box$ \section{Duality of string-theoretic Hodge numbers for the Greene-Plesser construction} In \cite{greene0,greene1} B. Greene and R. Plesser proposed an explicit construction of mirror pairs of Calabi-Yau orbifolds which are obtained as abelian quotients of Fermat hypersurfaces in weighted projective spaces. As it was shown in \cite{batyrev1}, 5.5, the Greene-Plesser construction can be interpreted in terms of the polar duality of {\em reflexive simplices}. The main purpose of this section is to verify the mirror duality of all string-theoretic Hodge numbers for this construction. {}From now on, we assume that $\Delta$ and $\Delta^*$ are $d$-dimensional reflexive simplices. We shall prove Conjecture \ref{symmetry} for $\Delta$-regular Calabi-Yau hypersurfaces in ${\bf P}_{\Delta}$ and ${\bf P}_{\Delta^*}$. (We remind that, for this kind of hypersurfaces and for $d = 4$, Conjecture \ref{symmetry} was proved in \cite{roan0,batyrev1}.) \begin{dfn} {\rm Let $\Theta$ be a $k$-dimensional lattice simplex. We denote by $\tilde{S}(\Theta; uv)$ the $\tilde{S}$-polynomial of the $(k+1)$-dimensional abelian quotient singularity defined by $\Theta$. We denote the corresponding finite abelian subgroup of $SL(k+1,{\bf C})$ by $G_{\Theta}$ (in the sence of \S 4,5).} \end{dfn} Our main statement is an immediate consequence of the following: \begin{theo} Let $\overline{Z}_f$ be a $\Delta$-regular Calabi-Yau hypersurface in ${\bf P}_{\Delta}$. Then \[ E_{\rm st}(\overline{Z}_f; u,v) = \frac{1}{uv}\tilde{S}(\Delta^*;uv) + (-1)^{d-1} \frac{u^{d}}{v} \tilde{S}(\Delta; u^{-1}v) + \] \[ + \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\ {\scriptstyle \theta \subset \Delta} \end{array}} (-1)^{{\rm dim}\, \theta-1} \left( \frac{u^{{\rm dim}\, \theta}}{v} \tilde{S}(\theta; u^{-1}v) \cdot \tilde{S}( \theta^*; uv) \right). \] \label{formul} \end{theo} \noindent Indeed, if we apply Theorem \ref{formul} to the dual polyhedron $\Delta^*$, then we get \[ E_{\rm st}(\overline{Z}_g; u,v) = \frac{1}{uv}\tilde{S}(\Delta;uv) + (-1)^{d-1} \frac{u^d}{v} \tilde{S}(\Delta^*; u^{-1}v) + \] \[ + \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta^* \leq d-2} \\ {\scriptstyle \theta^* \subset \Delta^*} \end{array}} (-1)^{{\rm dim}\, \theta^* -1} \left( \frac{u^{{\rm dim}\, \theta^*}}{v} \tilde{S}(\theta^*; u^{-1}v) \cdot \tilde{S}( \theta; uv) \right). \] Now the required equality \[ E_{\rm st}(\overline{Z}_f; u,v) =(-u)^{d-1} E_{\rm st}(\overline{Z}_g; u^{-1},v) \] follows evidently from the $1$-to-$1$ correspondence $\theta \leftrightarrow \theta^*$ $( 1 \leq {\rm dim}\, \theta,\, {\rm dim}\, \theta^* \leq d-1)$ and from the property: ${\rm dim}\, \theta + {\rm dim}\, \theta^* = d-1$. \bigskip For the proof of Theorem \ref{formul}, we need some preliminary facts. \begin{prop} Let $\theta$ be a face of $\Delta$ and ${\rm dim}\, \theta \geq 1$. Then \[ E(Z_{f,\theta}; u,v) = \frac{ (uv-1)^{{\rm dim}\, \theta} - (-1)^{{\rm dim}\, \theta}}{uv} + (-1)^{{\rm dim}\, \theta-1} \left( \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1} \\ {\scriptstyle \tau \subset \theta} \end{array}} \frac{u^{{\rm dim}\, \tau}}{v} \tilde{S}(\tau; u^{-1}v) \right) . \] \label{e-ff} \end{prop} \noindent {\em Proof. } By \cite{dan.hov}, Prop. 3.9, the natural mapping \[ H^i_c(Z_{f, \theta}) \rightarrow H^{i+1}_c(T_{\theta}) \] is an isomorphism if $i > {\rm dim}\, \theta -1$ and surjective if $i = {\rm dim}\, \theta -1$. Moreover, $H^i_c(Z_{f, \theta}) = 0$ if $i < {\rm dim}\, \theta -1$. In order to compute the mixed Hodge structure in $H^{{\rm dim}\, \theta -1}_c(Z_{f, \theta})$, we use the explicit description of the weight filtration in $H^{{\rm dim}\, \theta -1}_c(Z_{f, \theta})$ (see \cite{batyrev0}). Note that if we choose a $\theta$-regular Laurent polynomial $f$ containing only ${\rm dim}\, \theta + 1$ monomials associated with vertices of $\theta$ (such a polynomial $f$ defines a Fermat-type hypersurface $\overline{Z}_f$ in ${\bf P}_{\theta}$), then the corresponding Jacobian ring $R_f$ has a monomial basis. Thus, the weight filtration on $R_f$ can be described in terms of the partition of monomials in $R_f$ which is defined by the faces $\tau \subset \theta$. To get the claimed formula, it suffices to identify the partition of monomials in $R_f$ with the height-partition of elements of the finite abelian group $G_{\theta} \subset SL({\rm dim}\, \theta +1, {\bf C})$ and its subroups $G_{\tau} \subset G_{\theta}$. Another way to obtain the same result is to use the formulae of Danilov and Khovanski\^i (cf. \cite{dan.hov}, \S 5.6,5.7) which are valid for an arbitrary simple polyhedron $\Delta$. \hfill $\Box$ \begin{prop} Let $\theta$ be a face of $\Delta$ and ${\rm dim}\, \theta \geq 1$. Then \[ S(\theta; t) = 1 + \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \eta \geq 1} \\ {\scriptstyle \eta \subset \theta} \end{array} } \tilde{S}( \eta; t). \] \label{can} \end{prop} \noindent {\em Proof. } It is similar to that of \ref{can.strat}. \hfill $\Box$ \begin{prop} We fix a face $\tau \subset \Delta$ and a face $\eta \subset \Delta^*$, such that: $\tau$ is a face of $\eta^*$. Then \[ \sum_{\theta,\; \tau \subset \theta \subset \eta^*} (-1)^{{\rm dim}\, \theta} = (-1)^{{\rm dim}\, \tau}\;\; \mbox{if $ \tau = \eta^*$ } \] and \[ \sum_{\theta,\; \tau \subset \theta \subset \eta^*} (-1)^{{\rm dim}\, \theta} = 0\;\; \mbox{if $ \tau \neq \eta^*$. } \] \label{sum1} \end{prop} \noindent {\em Proof.} If $\eta^* = \tau$, this is obvious. For ${\rm dim}\, \eta^* > {\rm dim}\, \tau $, the number of faces $\theta \subset \Delta$, for which $\tau \subset \theta \subset \eta^*$, is equal to ${ {\rm dim}\, \eta^* - {\rm dim}\, \tau \choose {\rm dim}\, \theta - {\rm dim}\, \tau }$. It remains to use the equality \[ \sum_{\theta, \tau \subset \theta \eta^*} (-1)^{ {\rm dim}\, \theta } = (-1)^{{\rm dim}\, \tau } \left( \sum_{ i =0}^{{\rm dim}\, \eta^* - {\rm dim}\, \tau} (-1)^i { {\rm dim}\, \eta^* - {\rm dim}\, \tau \choose i} \right) = 0. \] \hfill $\Box$ \begin{prop} \[ \frac{1}{uv} \tilde{S}(\Delta; uv) = \frac{(uv)^d -1}{uv -1}\;\;\; + \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\tau \leq d-2} \\ {\scriptstyle \tau \subset \Delta} \end{array}} \left( \frac{(uv)^{{\rm dim}\, \tau^*} -1}{uv -1} \right) \cdot \tilde{S}( \eta; uv). \] \label{tilde-s} \end{prop} \noindent {\em Proof.} By Proposition \ref{relation}, we have \[ (-1)^{d-1} + \frac{S(\Delta;t)}{(1-t)^d}\;\; = \;\; (-1)^{d-1} \sum_{0 \leq {\rm dim}\, \theta \leq d-1} (-1)^{{\rm dim}\, \theta} \frac{S(\theta;t)}{(1-t)^{{\rm dim}\, \theta +1}}. \] Applying Proposition \ref{can} to both sides of this equality, we get \[ (-1)^{d-1}\;\; + \;\; \frac{1}{(1-t)^d}\;\; + \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1} \\ {\scriptstyle \tau \subset \Delta} \end{array}} \frac{\tilde{S}(\tau; t)}{(1-t)^d} \;\; = \] \[ = \;\; (-1)^{d-1} \sum_{0 \leq {\rm dim}\, \theta \leq d-1} \frac{(-1)^{{\rm dim}\, \theta}}{( 1- t)^{{\rm dim}\, \theta + 1}} \;\; + \] \[ + \;\; (-1)^{d-1} \sum_{0 \leq {\rm dim}\, \theta \leq d-1} (-1)^{{\rm dim}\, \theta} \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1} \\ {\scriptstyle \tau \subset \theta} \end{array}} \frac{\tilde{S}(\tau; t)}{(1-t)^{{\rm dim}\, \theta +1}}. \] As the number of $k$-dimensional faces of $\Delta$ equals ${ d+1 \choose k+1 }$, we have \[ - \;\; (-1)^{d-1} \; - \; \frac{1}{(1-t)^d} + (-1)^{d-1} \sum_{0 \leq \theta \leq d-1} \frac{(-1)^{{\rm dim}\, \theta}}{( 1- t)^{{\rm dim}\, \theta + 1}} \; = \] \[ -\;\; (-1)^{d-1} \; - \; \frac{1}{(1-t)^d} \;\; + \;\; \sum_{k =0}^{d-1} \frac{(-1)^k}{(1-t)^{k+1}} { d+1 \choose k+1 } \; = \; (-1)^d \frac{t^{d+1} - t}{(t-1)^{d+1}} \] and we can deduce that: \[ \frac{\tilde{S}(\Delta, t)}{(1-t)^d} \; + \sum_{{\rm dim}\, \tau = d-1} \frac{\tilde{S}(\tau, t)}{(1-t)^d} \; + \sum_{1 \leq {\rm dim}\, \tau \leq d-2} \frac{\tilde{S}(\tau, t)}{(1-t)^d} \; = \] \[ = (-1)^d \frac{t^{d+1} - t}{(t-1)^{d+1}}\;\; + \sum_{{\rm dim}\, \tau = d-1} \frac{\tilde{S}(\tau, t)}{(1-t)^d} \;\; + \] \[ + \sum_{{\rm dim}\, \theta = d-1} \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\tau \leq d-2} \\ {\scriptstyle \tau \subset \theta} \end{array}} \frac{\tilde{S}(\tau, t)}{(1-t)^d} \;\; + \] \[ + \;\; (-1)^{d-1} \sum_{ 1 \leq {\rm dim}\, \theta \leq d-2} (-1)^{{\rm dim}\, \theta} \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1} \\ {\scriptstyle\tau \subset \theta} \end{array} } \frac{\tilde{S}(\tau, t)}{(1- t)^{{\rm dim}\, \theta + 1}}. \] The terms containing $\tilde{S}(\tau, t)$, with ${\rm dim}\, \tau = d-1$, have the same contribution to the right and left hand sides. The coefficient of $\tilde{S}(\tau, t)$ $( 1 \leq {\rm dim}\, \tau \leq d-2 )$ in the right hand side of the last equality is \[ (-1)^{d-1} \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \theta \leq d-2} \\ {\scriptstyle \tau \subset \theta} \end{array}} (-1)^{{\rm dim}\, \theta} \frac{1}{(1-t)^{{\rm dim}\, \theta + 1}} \;\; = \] \[ = \;\; \frac{(-1)^d}{(t-1)^{d+1}} \left( t^{d- {\rm dim}\, \tau} - 1 - (d - {\rm dim}\,\tau ) (t-1) \right). \] Correspondingly, the coefficient of $\tilde{S}(\tau, t)$ $(1 \leq {\rm dim}\, \tau \leq d-2)$ in the left hand side equals \[ \frac{d - 1 - {\rm dim}\, \tau}{(1-t)^d}. \] Finally, using ${\rm dim}\, \tau + {\rm dim}\, \tau^* = d -1$, we obtain: \[ \frac{\tilde{S}(\Delta, t)}{(1-t)^d} \; = \; (-1)^d \frac{t^{d+1} - t}{(t-1)^{d+1}} \; + \; (-1)^d \sum_{1 \leq \tau \leq d-2} \tilde{S}(\tau, t) \frac{(t^{{\rm dim}\, \tau^* + 1} - t)}{ (t - 1)^{d + 1} }. \] \hfill $\Box$ \noindent {\bf Proof of Theorem \ref{formul}}. By definition, \[ E_{\rm st}(\overline{Z}_f; u,v) \; = \; E(Z_{f, \Delta}; u,v) \; + \; \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\,\theta = d-1} \\ {\scriptstyle \theta \subset \Delta} \end{array}} E(Z_{f, \theta}; u,v) \;\; + \] \[ + \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2 } \\ {\scriptstyle \theta \subset \Delta} \end{array}} E(Z_{f, \theta}; u,v) \cdot S(\theta^*; uv). \] Substituting the expressions which were found out in \ref{e-ff} for the $E$-polynomials of the above three summands, we get: \[ E(Z_{f, \Delta}; u,v) = \frac{ (uv-1)^{d} - (-1)^{d}}{uv} + (-1)^{d-1} \left( \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1} \\ {\scriptstyle \tau \subset \Delta} \end{array}} \frac{u^{{\rm dim}\, \tau}}{v} \tilde{S}(\tau; u^{-1}v) \right) , \] \newline \[ \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\,\theta = d-1} \\ {\scriptstyle \theta \subset \Delta} \end{array} } E(Z_{f, \theta}; u,v) \;\; = \;\; \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\,\theta = d-1} \\ {\scriptstyle \theta \subset \Delta} \end{array}} \frac{ (uv-1)^{{\rm dim}\, \theta} - (-1)^{{\rm dim}\, \theta}}{uv} \;\; + \] \[ + \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\,\theta = d-1} \\ {\scriptstyle \theta \subset \Delta} \end{array}} (-1)^{{\rm dim}\, \theta-1} \left( \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1} \\ {\scriptstyle \tau \subset \theta} \end{array}} \frac{u^{{\rm dim}\, \tau}}{v} \tilde{S}(\tau; u^{-1}v) \right), \] and \[ \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\ {\scriptstyle \theta \subset \Delta} \end{array}} E(Z_{f, \theta}; u,v) \cdot S(\theta^*; uv) \;\; = \;\; \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\ {\scriptstyle \theta \subset \Delta} \end{array}} \frac{ (uv-1)^{{\rm dim}\, \theta} - (-1)^{{\rm dim}\, \theta}}{uv} \;\; + \] \[ + \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\ {\scriptstyle \theta \subset \Delta} \end{array}} (-1)^{{\rm dim}\, \theta-1} \left( \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1} \\ {\scriptstyle \tau \subset \theta} \end{array}} \frac{u^{{\rm dim}\, \tau}}{v} \tilde{S}(\tau; u^{-1}v) \right) \cdot \left( 1 + \sum_{ \begin{array}{c} {\scriptstyle {\rm dim}\, \eta \geq 1} \\ {\scriptstyle \eta \subset \theta^*} \end{array}} \tilde{S}( \eta; uv) \right). \] Hence, $ E_{\rm st}(\overline{Z}_f; u,v)$ can be written as the sum of the following $4$ terms $E_i$ $( i =1,2,3,4)$: \newline \[ E_1 = \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\, \theta} \\ {\scriptstyle \theta \subset \Delta} \end{array}} \frac{ (uv-1)^{{\rm dim}\, \theta} - (-1)^{{\rm dim}\, \theta}}{uv}, \] \newline \[ E_2 = \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\, \theta} \\ {\scriptstyle \theta \subset \Delta} \end{array}} (-1)^{{\rm dim}\, \theta-1} \left( \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1} \\ {\scriptstyle \tau \subset \theta} \end{array}} \frac{u^{{\rm dim}\, \tau}}{v} \tilde{S}(\tau; u^{-1}v) \right) ,\] \[ E_3 = \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\ {\scriptstyle \theta \subset \Delta} \end{array}} \left( \frac{ (uv-1)^{{\rm dim}\, \theta} - (-1)^{{\rm dim}\, \theta}}{uv} \right) \cdot \left( \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \eta \geq 1} \\ {\scriptstyle \eta \subset \theta^*} \end{array}} \tilde{S}( \eta; uv) \right), \] and \[ E_4 = \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\ {\scriptstyle \theta \subset \Delta} \end{array}} (-1)^{{\rm dim}\, \theta-1} \left( \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1} \\ {\scriptstyle \tau \subset \theta} \end{array}} \frac{u^{{\rm dim}\, \tau}}{v} \tilde{S}(\tau; u^{-1}v) \right) \cdot \left( \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \eta \geq 1} \\ {\scriptstyle \eta \subset \theta^*}\end{array}} \tilde{S}( \eta; uv) \right). \] By \ref{sum1}, we can simplify the multiple summation into a single sum: \[ E_4 = \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\ {\scriptstyle \theta \subset \Delta} \end{array}} (-1)^{{\rm dim}\, \theta-1} \left( \frac{u^{{\rm dim}\, \theta}}{v} \tilde{S}(\theta; u^{-1}v) \cdot \tilde{S}( \theta^*; uv) \right). \] If we make use of the combinatorial identity \[ \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\, \theta} \\ {\scriptstyle \theta \subset \Delta} \end{array}} a^{{\rm dim}\, \theta} = \sum_{k =2}^{d+1} { d+1 \choose k } a^{k-1} = a^{-1} \left( (a+1)^{d+1} - 1 - (d+1) a \right), \] we obtain: \[ E_1 = \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\, \theta} \\ {\scriptstyle \theta \subset \Delta}\end{array}} \frac{ (uv-1)^{{\rm dim}\, \theta} - (-1)^{{\rm dim}\, \theta}}{uv} = \] \[ = [uv(uv - 1)]^{-1} \left( (uv)^{d+1} - 1 - (d+1)(uv -1) \right) + d (uv)^{-1} = \frac{(uv)^d -1}{uv -1}. \] By \ref{sum1}, we get \[ E_2 = \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\, \theta} \\ {\scriptstyle \theta \subset \Delta} \end{array}} (-1)^{{\rm dim}\, \theta-1} \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1} \\ {\scriptstyle \tau \subset \theta} \end{array}} \frac{u^{{\rm dim}\, \tau}}{v} \tilde{S}(\tau; u^{-1}v) = (-1)^{d-1} \frac{u^d}{v} \tilde{S}(\Delta; u^{-1}v). \] \noindent It remains to compute $E_3$. As above for $E_1$, we have \[ \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\, \theta} \\ {\scriptstyle \theta \subset \eta^*} \end{array}} \frac{ (uv-1)^{{\rm dim}\, \theta} - (-1)^{{\rm dim}\, \theta}}{uv} = \frac{(uv)^{{\rm dim}\, \eta^*} -1}{uv -1}. \] Hence, by \ref{tilde-s}, \[ E_3 = \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\eta \leq d-2} \\ {\scriptstyle \eta \subset \Delta^*} \end{array}} \left( \frac{(uv)^{{\rm dim}\, \eta^*} -1}{uv -1} \right) \cdot \tilde{S}( \eta; uv) = \frac{1}{uv}\tilde{S}(\Delta^*;uv) - \frac{(uv)^d -1}{uv -1}. \] Finally, we get altogether \[ E_{\rm st}(\overline{Z}_f; u,v) = \frac{1}{uv}\tilde{S}(\Delta^*;uv) + (-1)^{d-1} \frac{u^d}{v} \tilde{S}(\Delta; u^{-1}v) + \] \[ + \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\ {\scriptstyle \theta \subset \Delta} \end{array}} (-1)^{{\rm dim}\, \theta-1} \left( \frac{u^{{\rm dim}\, \theta}}{v} \tilde{S}(\theta; u^{-1}v) \cdot \tilde{S}( \theta^*; uv) \right). \] \hfill $\Box$ \begin{exam} {\rm The polar duality between reflexive simplices shows (cf. \cite{batyrev1}, Thm. 5.1.1.) that the family of all smooth Calabi-Yau hypersurfaces $X_{d+1}$ of degree $d+1$ in ${\bf P}^d$ has as its mirror partner the one-parameter family $\{ Q_{d+1}(\lambda)/G_{d+1} \}$, where \[ Q_{d+1}(\lambda) := \{ [z_0, \ldots, z_d ] \in {\bf P}^d \mid \sum_{i=0}^d z_i^{d+1} - (d+1)\lambda\prod_{ i=0}^d z_i = 0 \} \] denotes the so called {\em Dwork pencil} and $G_{d+1}$ the acting finite abelian group \[ G_{d+1} := \{ (\alpha_0, \ldots, \alpha_d) \in ({\bf Z}/(d+1){\bf Z})^{d+1} \mid \prod_{i =0}^d \alpha_i = 1 \} / \{\rm scalars \}, \] which is abstractly isomorphic to $({\bf Z}/(d+1){\bf Z})^{d-1}$. The moduli space ${\bf P}^1 \setminus \{ 0,1, \infty \}$ of $\{ Q_{d+1}(\lambda)/G_{d+1} \}_{\lambda}$ can be described by means of the parameter $\lambda^{d+1}$ (cf. \cite{greene}, \S 3.1, \cite{morrison}, \S 5, and \cite{morrison1} \S 11). Since Conjecture \ref{symmetry} is true for the case being under consideration, the quotient $Q_{d+1}(\lambda)/G_{d+1}$ has the following string-theoretic Hodge numbers: \[ h^{p,q}_{\rm st} (Q_{d+1}(\lambda)/G_{d+1}) = h^{p,q}(Q_{d+1}(\lambda),G_{d+1}) = h^{d-1-p,q}(X_{d+1}) = \delta_{d-1-p,q}, \;\;\; \mbox{\rm for $p \neq q$}; \] \[ h^{p,p}_{\rm st} (Q_{d+1}(\lambda)/G_{d+1}) = h^{p,p}(Q_{d+1}(\lambda),G_{d+1}) = h^{d-1-p,p}(X_{d+1}) \;= \] \[ = \; \sum_{i =0}^p (-1)^i { d+1 \choose i } { (p+1 -i)d + p \choose d } + \delta_{2p,d-1} . \] In particular, the string-theoretic Euler number is given by: \[ e_{\rm st} (Q_{d+1}(\lambda)/G_{d+1}) = e(Q_{d+1}(\lambda),G_{d+1}) = - e(X_{d+1})\; = \] \[ = \; \frac{1}{d+1} \left( (-1)^{d+2} \cdot d^{d+1} + 1 \right) - d - 1. \] The first two equalities follow from Lefschetz hyperplane section theorem and from the ``four-term formula'' (cf. \cite{hirzebruch1}, \S 2.2 ). The third one can be obtained directly by computing the $(d-1)$-th Chern class of $X_{d+1}$. } \end{exam}

9410

alg-geom/9410006

en

https://arxiv.org/abs/alg-geom/9410006

alg-geom/9410006

Barbara Fantechi, Rita Pardini

Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers

30 pages, LaTeX

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties. In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group $G$ there exist infinitely many components $M$ of the moduli of varieties with ample canonical class such that the generic automorphism group $G_M$ is equal to $G$. In order to construct the examples, we use abelian covers, i.e. Galois cover whose Galois group is finite and abelian. We prove two results about abelian covers: first, that if the building data are sufficiently ample, then the natural deformations surject on the Kuranishi family of $X$; second, that if the building data are sufficiently ample and generic, then $Aut(X)=G$.

alg-geom

\section{Introduction} Coverings of algebraic varieties are a classical theme in algebraic geometry, since Riemann's description of curves as branched covers of the projective line. Double covers were used by the Italian school to construct examples that shed light on the theory of surfaces and to describe special classes of surfaces, as in the case of Enriques surfaces. More recently, cyclic coverings have been extensively applied by several authors to the study of surfaces of general type; it will be enough to recall the work of Horikawa, Persson and Xiao Gang. Abelian covers have been used by Hirzebruch to give examples of surfaces of general type on and near the line $c_1^2=3c_2$; Catanese and Manetti have used bidouble and iterated double covers, respectively, of $\P^1\times \P^1$ to construct explicitly connected components of the moduli space of surfaces of general type. In \cite{Pa1}, the second author has given a complete description of abelian covers of algebraic varieties in terms of the so-called building data, namely of certain line bundles and divisors on the base of the covering, satisfying suitable compatibility relations. Natural deformations of an abelian cover $f:X\to Y$ are also introduced there and it is shown that they are complete, if $Y$ is rigid, regular and of dimension $\ge 2$, and if the building data are sufficiently ample. (Natural deformations are obtained by modifying the equations defining $X$ inside the total space of the bundle $f_*\O_X$). In this paper we study natural deformations of an abelian cover $f:X\to Y$ and prove that they are complete for varieties of dimension at least two if the branch divisors are sufficiently ample. The result requires no assumption on $Y$, and in particular also holds when the cover has obstructed deformations; this is a key technical step towards the moduli space constructions described below. We then turn to the study of the automorphism group of the cover. Since the automorphism group of a variety of general type is finite, one would expect that in the case of a Galois cover it coincides with the Galois group, at least if the cover is generic. Our main theorem \ref{mainthm} shows that this is indeed the case for an abelian cover, if the branch divisors are generic and sufficiently ample. We construct explicitly coarse moduli spaces of abelian covers and complete families of natural deformations for a fixed base of the cover $Y$; this is useful if one wants to investigate the birational structure of the components of the moduli obtained by the methods of this paper. The main application of the results described so far is the study of moduli of varieties with ample canonical class. Recently Viehweg proved the existence of a coarse moduli space for varieties with ample canonical class of arbitrary dimension, generalizing Gieseker's result for surfaces. Given an irreducible component $M$ of the moduli space of varieties with ample canonical class, the automorphism group $G_M$ of a generic variety in $M$ is well-defined. In contrast with the case of curves (where this group is trivial for $g\ge 3$), it was already known in the case of surfaces that there exist infinitely many components $M$ of the moduli with nontrivial automorphism group $G_M$; it is easy to construct examples such that $G_M$ contains an involution, and Catanese gave examples where $G_M$ contains a subgroup isomorphic to ${\bf Z}_2\times {\bf Z}_2$. There are also, of course, easy examples of components $M$ where $G_M$ is trivial (for instance the hypersurfaces of degree $d\ge 5$ in $\P^3$). As a first application of theorem \ref{mainthm} we prove that for any finite abelian group $G$ there are infinitely many irreducible components $M$ of the moduli of varieties with ample canonical class such that $G_M=G$; notice that we precisely determine $G_M$ instead of just bounding it from below. We also prove that there are varieties with ample canonical class lying on arbitrarily many irreducible components of the moduli. We distinguish these components by means of their generic automorphism group; there are examples both in the equidimensional and in the non-equidimensional case. In the surface case, this answers a question raised by Catanese in \cite{Ca2}. Let $S$ be a surface of general type; Xiao has given explicit upper bounds both for the cardinality of $Aut(S)$ and of an abelian subgroup of $Aut(S)$, in terms of the invariants of $S$ (\cite{Xi1}, \cite{Xi2}). Some upper bounds are also known for a higher-dimensional variety $X$ with ample canonical class, although sharp bounds are still lacking. It seems interesting to ask whether these bounds can be improved by considering instead of $Aut(X)$ the group $Aut_{\rm gen}(X)$, namely the intersection in $Aut(X)$ of the images of the generic automorphism groups $G_M$ of all irreducible components $M$ of the moduli space containg $X$ (in particular, if $X$ lies in a unique component $M$, then $Aut_{\rm gen}(X)=G_M$). As a first step towards the computation of a sharp bound for $\#Aut_{\hbox{\rm gen}}(S)$, we show that such a bound cannot be ``too small''; in fact we give a sequence of surfaces $S_n$ of general type, whose Chern numbers tend to infinity with $n$, and such that $\#Aut_{\hbox{\rm gen}}(S_n)\ge 2^{-4}K_{S_n}^2$. The paper goes as follows: in section 2 we collect some results from the literature and set up the notation. In section 3 we prove that, if the branch divisors are sufficiently ample, then infinitesimal natural deformations are complete. In section 4 we prove (theorem \ref{mainthm}) that the automorphism group of an abelian cover coincides with the Galois group if the building data are sufficiently ample and generic. To do this, we prove some results on extensions of automorphisms, which we believe should be of independent interest. The proof of \ref{mainthm} is based on a degeneration argument and requires an explicit partial desingularization, contained in section 7. Section 5 contains the construction of a coarse moduli space for abelian covers of a given variety $Y$ and of a complete family of natural deformations. Finally, in section 6 we apply the results of sections 3 and 4 to the study of moduli spaces of varieties with ample canonical class, as stated above. \smallskip \noindent{\em Acknowledgements}. This work was supported by the italian MURST 60\% funds. The first author would also like to thank the Max-Planck-Institut f\"ur Mathematik (Bonn) for hospitality and the italian CNR for support. \section{Notation and conventions} All varieties will be complex, and smooth and projective unless the contrary is explicitly stated. For a projective morphism of schemes $Y\to S$, $Hilb_S(Y)$ will be the relative Hilbert scheme (see \cite{FGA}, expos\'e 221). When $Y$ is smooth over $S$, $Hilb^{\rm div}_S(Y)$ will be the (open and closed) subscheme of $Hilb_S(Y)$ parametrizing divisors (see \cite{Fo} for a proof of this). When $S$ is a point, it will be omitted from the notation. For $Y$ a smooth projective variety, let $c_1:Pic(Y)\to H^2(Y,{\bf Z})$ be the map associating to a line bundle its first Chern class; let $NS(Y)$ be the image in $H^2(Y,{\bf Z})$ of $Pic(Y)$, and $Pic^\xi(Y)$ the inverse image of $\xi\in NS(Y)$. Let $q(Y)=\dim H^1(Y,\O_Y)$ be the dimension of $Pic^0(Y)$. Let ${\cal X}\to B$ be any flat family, with integral fibres. Then there are open subschemes $Aut_{{\cal X}/B}$ and $Bir_{{\cal X}/B}$ of the relative Hil\-bert sche\-me \hbox{$Hilb_B({\cal X}\times_B{\cal X})$} parametrizing fibrewise the (graphs of) automorphisms and birational automorphisms of the fibre (\cite{FGA}, \cite{Ha}). We denote the cardinality of a (finite) set $S$ by $\#S$; for each integer $m\ge 2$, let $\zeta_m=e^{2\pi i/m}$. \smallskip \noindent{\em Notation for abelian covers}. The following notation will be used freely throughout the paper: we collect it here for the reader's convenience. $G$ will be a finite abelian group, $G^*$ its dual; the order of an element $g$ will be denoted by $\ord{g}$. Let $I_G$ be the set of all pairs $(H,\psi)$ where $H$ is a cyclic subgroup of $G$ with at least two elements and $\psi$ is a generator of $H^*$. There is a bijection between $I_G$ and $G\setminus 0$ given by $(H,\psi)\mapsto g$ where $g\in H$ is such that $\psi(g)=\zeta_{\#H}$. For $\chi\in G^*$, $i=(H_i,\psi_i)\in I_G$, let $\re^i_\chi$ be the unique integer such that $0\le \re^i_\chi<m_i$ (where $m_i=\#H_i$) and $\chi_{|H_i}=\psi_i^{\re^i_\chi}$ (cfr.\ \cite{Pa1}, remark 1.1 on p.~195, where $\re^i_\chi$ is denoted by $f_{H,\psi}(\chi)$). Let $\eps^i_{\chi,\chi'}=[(\re^i_\chi+\re^i_{\chi'})/m_i]$, where $[r]$ is the integral part of a real number $r$; note that $\eps^i_{\chi,\chi'}$ is either $0$ or $1$. A basis of $G$ will be a sequence of elements of $G$, $(e_1,\dots,e_s)$, such that $G$ is the direct sum of the (cyclic) subgroups generated by the $e_j$'s, and such that $\ord{e_j}$ divides $\ord{e_{j+1}}$ for each $j=1,\ldots,s-1$. Given a basis $(e_1,\dots,e_s)$ of $G$, we will call dual basis of $G^*$ the $s$-tuple $(\chi_1,\ldots,\chi_s)$, where $\chi_j(e_i)=1$ if $i\ne j$ and $\chi_i(e_i)=\zeta_{\ord{e_i}}$. We will write $\re^i_j$ instead of $\re^i_{\chi_j}$, for all $j=1,\ldots,s$; for $\chi=\chi_1^{\alpha_1}\cdots\chi_s^{\alpha_s}$, let $$q^i_\chi=\left[\sum_{j=1}^s\frac{\alpha_i\re^i_j}{m_i}\right].$$ Note that, unlike $\re^i_\chi$, $q^i_\chi$ depends on the choice of the basis and not only on $\chi$ and $i$. \begin{lem} Let $G$ be as above, and let $I\subset I_G$ be a subset with $k$ elements {\rm(}which we denote by $1,\ldots,k${\rm)} such that the natural map $H_1\oplus \ldots \oplus H_k\to G$ is surjective. Then the $k\times s$ matrix $(\re^i_j)$ has rank $s$ over ${\bf Q}$. \end{lem} \noindent{\sc Proof.~} Let $g_i$ be the element corresponding to $(H_i,\psi_i)$ via the bijection $I_G\leftrightarrow G\setminus 0$ described above. Then, for any $i=1,\ldots,k$ and for any $j=1,\ldots,s$, one has $\re^i_j/m_i=\lambda_{ij}/n_j$, where $n_j=\ord{e_j}$ and $g_i=\sum \lambda_{ij}e_j$, with $0\le \lambda_{ij}<n_j$ and $\lambda_{ij}\in{\bf Z}$ . So the matrix $(\re^i_j)$ has the same rank over ${\bf Q}$ as the matrix $\lambda_{ij}$. On the other hand $\lambda_{ij}$ is the matrix associated to the natural map $H_1\oplus \ldots \oplus H_k\to G$, which is surjective. Let $p$ be a prime factor of $n_1$, hence of all of the $n_j$'s. Then the map ${\bf Z}_p^k\to {\bf Z}_p^s$ represented by the matrix $(\lambda_{ij})\ \hbox{\rm mod}\; p$ is also surjective, hence the matrix $(\lambda_{ij})$ has an $s\times s$ minor whose determinant is nonzero modulo $p$. This implies that the determinant is nonzero, hence the result. \ $\Box$\par\smallskip \smallskip Let $X$ be any projective variety. A {\sl deformation} of $X$ over a pointed analytic space $(T,o)$ will be a flat, proper map ${\cal X}\to T$, together with an isomorphism of the special fibre ${\cal X}_o$ with $X$. Deformations modulo isomorphism are a contravariant functor $Def_X$ from the category $\hbox{\sl Ansp}_0$ of pointed analytic spaces to the category $\hbox{\sl Sets}$, where the functoriality is given by pullback. More generally, given a contravariant functor $F:\hbox{\sl Ansp}_0\to \hbox{\sl Sets}$, we will use the same letter $F$ to denote the induced functor on the categories $\hbox{\sl Germs}$ of germs of analytic spaces and $\hbox{\sl Art}^*$ of finite length spaces supported in a point (i.e. $Spec$'s of local Artinian ${\bf C}$-algebras). For the properties of functors on $\hbox{\sl Art}^*$, we refer the reader to \cite{schl}. \smallskip Let $M$ be an irreducible component of the moduli space of (projective) manifolds with ample canonical class. As the automorphism group is semicontinuous (see corollary 4.5), it makes sense to speak of the automorphism group of a generic manifold in $M$; we will denote it by $G_M$. Note that for any $X$ such that $[X]\in M$, there is a natural identification of $G_M$ with a subgroup of $Aut(X)$. If $X$ is a minimal surface of general type, we denote the intersection in $Aut(X)$ of $G_M$ for all components $M$ containing $[X]$ by $Aut_{\hbox{\rm gen}}(X)$; it is the largest subgroup $H$ of $Aut(X)$ such that the action of $H$ extends to any small deformation of $X$. \section{Deformations of abelian covers} In this section we introduce natural deformations of a smooth abelian cover and prove that infinitesimal natural deformations are complete, if the branch divisors are sufficiently ample and the dimension is at least two. We start by recalling from \cite{Pa1} some fundamental results on abelian covers; the reader will find there a more detailed exposition and proofs of the following statements. Let $G$ be a finite abelian group and let $I$ be a subset of $I_G$: we will use freely throughout the paper the notation introduced in section 2. Let $Y$ be a smooth projective variety: a $(G,I)$-cover of $Y$ is a normal variety $X$ and a Galois cover $f:X\to Y$ with Galois group $G$ and branch divisors $D_i$ (for $i\in I)$ having $(H_i,\psi_i)$ as inertia group and induced character (see \cite{Pa1} for details). $X$ is smooth if and only if the $D_i$'s are smooth, their union is a normal crossing divisor, and, whenever $D_{i_1},\ldots,D_{i_k}$ have a common point, the natural map $H_{i_1}\oplus\ldots\oplus H_{i_k}\to G$ is injective. The cover is said to be {\em totally ramified} if the natural map $\bigoplus_{i\in I}H_i\to G$ is surjective. Note that each abelian cover can be factored as the composition of a totally ramified with an unramified cover. Let $M_i=\O_Y(D_i)$. The vector bundle $f_*\O_X$ on $Y$ splits naturally as sum of eigensheaves $L_\chi^{-1}$ for $\chi\in G^*$, and multiplication in the $\O_Y$-algebra $f_*\O_X$ induces isomorphisms \begin{equation} \label{bdata}L_\chi\otimes L_{\chi'}=L_{\chi\chi'}\otimes\Bigotimes_{i\in I} M_i^{\otimes \eps^i_{\chi,\chi'}}\qquad\quad \hbox{for all $\chi,\chi'\in G^*\setminus 1$}. \end{equation} Denote $L_{\chi_j}$ by $L_j$, and let $n_j=\ord{\chi_j}$. The isomorphisms above induce isomorphisms \begin{equation} \label{rbdata} L_j^{\otimes n_j} =\Bigotimes_{i\in I}M_i^{\otimes\reb^i_j}\qquad\quad \hbox{for all $j=1,\ldots,s$}. \end{equation} The $(D_i,L_\chi)$ are the {\em building data} of the cover; the $(D_i,L_j)$ are the {\em reduced building data}. The sheaves $L_\chi$ can be recovered from the reduced building data by setting, for $\chi=\chi_1^{\alpha_1}\cdots\chi_s^{\alpha_s}$, \begin{equation} \label{chidarbd} L_\chi=\Bigotimes_{j=1}^s L_j^{\alpha_j}\otimes\Bigotimes_{i\in I}M_i^{-q^i_\chi}. \end{equation} Conversely, for each choice of $(D_i,L_\chi)$ (resp.\ $(D_i,L_j)$) satisfying equation (\ref{bdata}) (resp.\ (\ref{rbdata})), there exists a unique cover having these as (reduced) building data. Note that equations (\ref{rbdata}) have a solution in $Pic(Y)$ (viewing the line bundles $M_i$'s as parameters and the $L_j$'s as variables) if and only if their images via $c_1$ have a solution in $\hbox{\sl NS}(Y)$. \begin{assu}\label{totram} In this paper all $(G,I)$-covers will be totally ramified. Unless otherwise stated, $f:X\to Y$ will be a $(G,I)$-cover, with reduced building data $(D_i,L_j)$. We will also assume that $X$ and $Y$ are smooth, of dimension $\ge 2$, and that $X$ has ample canonical class. \end{assu} We say that a property holds whenever a line bundle $L$ (or a divisor $D$) is sufficiently ample if it holds whenever $c_1(L)$ (or $c_1(D)$) belongs to a (given) suitable translate of the ample cone. It is easy to see that assumption \ref{totram} implies the following: if all of the $D_i$'s are sufficiently ample then so is $L_\chi$ for any $\chi\ne 1$. Moreover, if $V$ is a vector bundle, $V\otimes L$ is ample for any sufficiently ample $L$. Let $S=\{({i,\chi})\in I\times G^*|\chi_{|H_i}\ne\psi_i^{-1}\}$. Given a $(G,I)$-cover $X\to Y$ as above, together with sections $s_{i,\chi}$ of $H^0(M_i\otimes L_\chi^{-1})$ for all $({i,\chi})\in S$, a natural deformation of $X$ was defined in \cite{Pa1}, \S 5. We now give a functorial (and more general) version of that definition in order to be able to apply standard techniques from deformation theory. \begin{defn}\label{natdef}{\rm A {\em natural deformation of the reduced building data} of $f:X\to Y$ over $(T,o)\in\hbox{\sl Ansp}_0$ is $({\cal Y},{\cal M}_i,\L_j,s_{{i,\chi}},\phi_j)$ where:\begin{enumerate} \item $i\in I$, $j=1,\ldots,r$, and $({i,\chi})\in S$; \item ${\cal Y}\to T$ is a deformation of $Y$ over $T$; \item $\L_j$ and ${\cal M}_i$ are line bundles on ${\cal Y}$ such that $\L_j$ restricts to $L_j$ and ${\cal M}_i$ to $M_i$ over $o$; \item $ \phi_j:\L_j^{\otimes n_j}\to \bigotimes {\cal M}_i^{\otimes \reb^i_j}$ is an isomorphism whose restriction to ${\cal Y}_o$ coincides with the isomorphism $L_j^{\otimes n_j}\to \bigotimes M_i^{\otimes \reb^i_j}$ given by multiplication; \item $s_{{i,\chi}}$ is a section of $\L_\chi^{-1}\otimes{\cal M}_i$, where $ \L_\chi=\Bigotimes_{j=1}^s \L_j^{\alpha_j}\otimes\Bigotimes_{i\in I}{\cal M}_i^{-q^i_\chi}$; \item $s_{{i,\chi}}$ restricts over ${\cal Y}_o$ to $s_{i,\chi}^0$, where $s_{i,\chi}^0=0$ if $\chi\ne 1$, and $s^0_{i,1}$ is a section of $M_i$ defining $D_i$. \end{enumerate} We will say that a deformation is {\em Galois} if $s_{i,\chi}=0$ for $\chi\ne 1$.} \end{defn} Natural deformations modulo isomorphism define a contravariant functor $\hbox{\rm Dnat}_X:\hbox{\sl Ansp}_0\to \hbox{\sl Sets}$, and Galois deformations are a subfunctor $\hbox{\rm Dgal}_X$. Note that the inclusion $\hbox{\rm Dgal}_X\hookrightarrow \hbox{\rm Dnat}_X$ is naturally split. We now extend formulas in \S 5 of \cite{Pa1} to define a natural transformation of functors $\hbox{\rm Dnat}_X\to \defor X$. \begin{defn}\label{trasnat}{\rm Let $T$ be a germ of an analytic space, and let $$({\cal Y},\L_j,{\cal M}_i,\phi_j,s_{i,\chi})\in\hbox{\rm Dnat}_X(T).$$ Let $V$ be the total space of the vector bundles $\bigoplus_{\chi\in G^*}\L_\chi$, and let $\pi:V\to {\cal Y}$ be the natural projection. For a line bundle $\L$ on ${\cal Y}$, denote its pullback to $V$ by $\bar \L$, and analogously for sections and isomorphisms. Each of the line bundles $\bar\L_\chi$ has a tautological section $\sigma_\chi$. For each pair $(\chi,\chi')\in G^*\times G^*$, the isomorphisms $\phi_j$ induce isomorphisms $$\phi_{\chi,\chi'}:\L_\chi\otimes\L_{\chi'} \to \L_{\chi\chi'}\otimes\Bigotimes {\cal M}_i^{\eps^i_{\chi,\chi'}}.$$ Let $\tau_i\in H^0(V,\bar{\cal M}_i)$ be defined by $$ \tau_i=\sum_{\{\chi|({i,\chi})\in S\}}\bar s_{i,\chi}\sigma_\chi.$$ Define a section $\rho_{\chi,\chi'}$ of $\bar\L_\chi\otimes\bar\L_{\chi'}$ by $$\rho_{\chi,\chi'}=\sigma_\chi\sigma_{\chi'}- \bar\phi_{\chi,\chi'}^*(\sigma_{\chi\chi'}{\textstyle\prod}\tau_i^{\eps^i_{\chi,\chi'}}).$$ Then the zero locus of all the $\rho_{\chi,\chi'}$ is naturally a deformation ${\cal X}\to T$ of $X$ over $T$ (in particular $X$ can be naturally identified with the fibre of ${\cal X}\to T$ over the closed point). This is proven in \cite{Pa1} in the case where the deformation of $Y$, $L_j$ and $M_i$ is the trivial one, but it is easy to see that the same proof works in our generalized setting. The deformation ${\cal X}\to T$ so obtained is called the {\em natural deformation of $X$} associated to the given natural deformation of the reduced building data.} \end{defn} It is now clear why Galois deformations were called that way: \begin{rem} Let ${\cal X}\to T$ be a deformation of $X$ induced by a Galois deformation $({\cal Y},\ldots)$ of the reduced building data; ${\cal X}$ has a canonical structure of $(G,I)$-cover of ${\cal Y}$, induced by the action of $G$ on the total space of the line bundle $\L_\chi$ given by the character $\chi$. \end{rem} The restrictions to the category $\hbox{\sl Art}^*$ of the functors $\hbox{\rm Dnat}_X $ and $\hbox{\rm Dgal}_X$ satisfy Schlessinger's conditions for the existence of a projective hull (see \cite{schl}); in fact, they can be described (as usual in deformation theory) in terms of tangent and obstruction spaces. If $F:\hbox{\sl Art}^*\to \hbox{\sl Sets}$ is a contravariant functor, then we denote its tangent (resp.~obstruction) space by $T^1(F)$ (resp.~$T^2(F)$), when this makes sense. \setcounter{equation}{0} \begin{lem}\label{tdefnat} There is a natural action of $G$ on $\hbox{\rm Dnat}_X$, whose invariant locus is $\hbox{\rm Dgal}_X $; the decomposition of $T^l(\hbox{\rm Dnat}_X )$ according to characters, for $l=1,2$, is the following: \begin{eqnarray} &&T^l(\hbox{\rm Dgal}_X )=T^l(\hbox{\rm Dnat}_X )^{\rm inv}=H^l(Y,T_Y(-\log{\textstyle\sum} D_i));\label{tgal}\\ &&T^l(\hbox{\rm Dnat}_X )^{\chi}=\Bigoplus_{i\in S_\chi} H^{l-1}(Y,\O_Y(D_i)\otimes L_\chi^{-1})\qquad\hbox{for $\chi\ne1$;} \end{eqnarray} where $S_\chi=\{i\in I|({i,\chi})\in S\}$. \end{lem} \noindent{\sc Proof.~} An element $g\in G$ acts by $$({\cal Y},{\cal M}_i,\L_j,s_{i,\chi},\phi_j) \mapsto ({\cal Y},{\cal M}_i,\L_j,\chi(g)s_{i,\chi},\phi_j).$$ It is clear that $\hbox{\rm Dgal}_X$ is contained in the invariant locus. It is not difficult to show the other inclusion using the fact that the cover is totally ramified. We now study separately tangent and obstructions spaces corresponding to the different characters. For the trivial character, i.e. $\hbox{\rm Dgal}_X$, the functor is isomorphic to the deformation functor of the data $(Y,M_i,s_i)$; (\ref{tgal}) is then well known (see \cite{We}). Fix a nontrivial character $\chi$. Then the problem reduces to studying the deformations of the zero section of a line bundle, given a deformation of the base and of the bundle. The statement can then be proven by applying the following lemma. \ $\Box$\par\smallskip \begin{lem} Let $o\in B'\subset B\in\hbox{\sl Art}^*$ be schemes of length $1,n,n+1$ respectively for some $n$; for schemes, etc.~over $B$ denote the restriction to $B'$ by a prime and the restriction to $o$ by ${}_o$. Let ${\cal Y}\to B$ be a smooth projective morphism, $\L$ a line bundle on ${\cal Y}$; let $s'$ be a section of $\L'$, such that $s'_o=0$. Then the obstruction to lifting $s'$ to a section $s$ of $\L$ lies in $H^1({\cal Y}_o,\L_o)$, and two liftings differ by an element of $H^0({\cal Y}_o,\L_o)$. \end{lem} \noindent{\sc Proof.~} Let $\{U_\alpha\}$ be an affine open cover of $Y={\cal Y}_o$ such that $L$ is trivial on each $U_\alpha$. Let $U_{\al\be}$ be $U_\alpha\cap U_\beta\subset U_\alpha$. As $Y$ is smooth, we have that ${\cal Y}$ is covered by open subsets $V_\alpha$ isomorphic to $U_\alpha\times B$, glued via $B$-isomorphisms $\phi_{{\al\be}}:U_{\al\be}\times B\to U_{\be\al}\times B$ satisfying the cocycle condition and restricting to the identity over $o$. Let $g_{{\al\be}}$ be transition functions for $\L$ with respect to the open cover $V_\alpha$. The section $s'$ can be described by functions $s'_\alpha$ on $U_\alpha\times B'$ such that, on $U_{\al\be}\times B'$, $$ s'_\alpha=g'_{\al\be}(s'_\beta\circ\phi_{\al\be}).$$ Extend $s'_\alpha$ arbitrarily to a function $s_\alpha$ on $U_\alpha\times B$; any other extension is of the form $s_\alpha+\varepsilon\sigma_\alpha$, where $\varepsilon=0$ is an equation of $B'$ in $B$ and $\sigma_\alpha$ is a function on $U_\alpha$ (as $\varepsilon f=0$ for any function $f$ in the ideal of $o$ in $B$). If an extension $s$ of $s'$ exists, then there must be functions $\sigma_\alpha$ on $U_\alpha$ such that, on $U_{\al\be}\times B$, $$ s_\alpha+\varepsilon\sigma_\alpha=g_{\al\be}((s_\beta+\varepsilon\sigma_\beta)\circ\phi_{\al\be}).$$ Let $u_{\al\be}=s_\alpha-g_{\al\be}(s_\beta\circ\phi_{\al\be})$. The restriction of $u_{\al\be}$ to $U_{\al\be}\times B'$ is zero, hence $u_{\al\be}$ is divisible by $\varepsilon$: let $u_{\al\be}=\varepsilon v_{\al\be}$. One can verify, using the fact that $s_o=0$, that $v_{\al\be}$ is a cocycle in $H^1(Y,\L_o)$: it is enough to check that $$ u_{\al\be}+g_{\al\be}(u_{\be\gamma}\circ\phi_{\al\be})=u_{\al\gamma} $$ on $U_{{\al\be}\gamma}$, for all triples $\alpha,\beta,\gamma$ of indices of the cover. It is then immediate to verify that $v_{\al\be}$ is the obstruction to lifting $s'$ to ${\cal Y}$, and the statement about the difference of two liftings can be proven in a similar way. \ $\Box$\par\smallskip We now recall some properties of $\defor X$. Let $\defg X:\hbox{\sl Ansp}_0\to \hbox{\sl Sets}$ be the functor of deformations of $X$ together with the $G$ action. \begin{lem} There is a natural action of $G$ on $\defor X$, whose invariant locus is $\defg X$. \end{lem} \noindent{\sc Proof.~} Let ${\cal X}\to T$ be a deformation of $X$ over $(T,o)$; there is a given isomorphism $i:X\to {\cal X}_o$. The action of an element $g\in G$ is given by replacing $i$ with $i\circ \phi(g)$, where $\phi:G\to Aut(X)$ is the natural action. It is clear that if $G$ acts on a deformation ${\cal X}\to T$, then this belongs to $\defg X$. The other implication follows from \cite{Ca2}, \S 7 or directly from the fact that the automorphisms of $X$ and of its deformations are rigid. \ $\Box$\par\smallskip Note that, as $X$ is of general type, the $G$-action on $\defor X$ induces an action on the Kuranishi family ${\cal X}\to B$ of $X$; the restriction of the Kuranishi family to the fixed locus $B^G$ is universal for the functor $\defg X$ (compare (\cite{Pi}, (2.8) p.~19, \cite{Ca2}, \S 7). Recall the following result from \cite{Pa1}. \setcounter{equation}{0} \begin{lem} Let $X$ be a smooth $(G,I)$-cover of $Y$ with building data $(D_i,L_\chi)$. Then the decomposition according to characters of $H^l(X,T_X)$ is as follows: \begin{eqnarray} &&H^l(X,T_X)^{\rm inv}=H^l(T_Y(-\log \sum_{i\in I} D_i))\\ &&H^l(X,T_X)^\chi=H^l(T_Y(-\log \!\!\!\sum_{i\in S_\chi}\!\!\! D_i)\otimes L_\chi^{-1})\qquad \hbox{\ if $\chi\ne1$} \end{eqnarray} where $S_\chi$ is the same as in lemma \rm{\ref{tdefnat}}. \end{lem} \noindent{\sc Proof.~} This follows immediately from proposition 4.1. in \cite{Pa1}. \ $\Box$\par\smallskip \setcounter{equation}{0} \begin{cor}\label{tdef} Assume that, for all $\chi\in G^*\setminus 1$, the bundles $L_\chi$ and $\Omega_Y^1\otimes L_\chi$ are ample. Then there are natural exact sequences, for all $\chi\in G^*\setminus 1$: \begin{eqnarray} &&0\to \textstyle\bigoplus\limits_{i\in S_\chi} H^0(Y,\O(D_i)\otimes L_\chi^{-1})\to H^1(X,T_X)^\chi\to 0.\\ &&0\to \textstyle\bigoplus\limits_{i\in S_\chi} H^1(Y,\O(D_i)\otimes L_\chi^{-1})\to H^2(X,T_X)^\chi. \end{eqnarray} \end{cor} \noindent{\sc Proof.~} Fix $\chi\ne 1$, let $D=\sum_{i\in S_\chi}D_i$, and consider the following diagram of sheaves with exact rows and columns: $$ \begin{array}{ccccccccc} &&&& 0 && 0 \\ &&&& \downarrow && \downarrow \\ &&&& \Bigoplus_{i\in S_\chi}^{\phantom{S_\chi}}\O_Y & = & \Bigoplus_{i\in S_\chi}\O_Y\\ &&&& \downarrow && \downarrow \\ 0 & \longrightarrow & T_Y(-\log D) & \longrightarrow &{\cal P}^*&\longrightarrow& \Bigoplus_{i\in S_\chi}^{\phantom{S_\chi}}\O_Y(D_i)&\longrightarrow&0\\ &&\Vert&& \downarrow && \downarrow \\ 0 & \longrightarrow & T_Y(-\log D) & \longrightarrow &T_Y&\longrightarrow& \Bigoplus_{i\in S_\chi}^{\phantom{S_\chi}}\O_{D_i}(D_i)&\longrightarrow&0\\ &&&& \downarrow && \downarrow \\ &&&& 0 && 0 \end{array}$$ where ${\cal P}$ is the prolongation bundle associated to the normal crossing divisor $D$. By the previous lemma, it is enough to prove that the first two cohomology groups of ${\cal P}^*\otimes L_\chi^{-1}$ vanish; this follows from the corresponding vanishing for $L_\chi^{-1}$ and $T_Y\otimes L_\chi^{-1}$, and the latter is just Kodaira vanishing (it is here that one needs the assumption $\dim Y\ge 2$). \ $\Box$\par\smallskip The natural transformation of functors $\hbox{\rm Dnat}_X\to \defor X$ defined in \ref{trasnat} is equivariant with respect to the natural actions of $G$ on these functors. Therefore, there is a commutative diagram $$\begin{array}{ccc} \hbox{\rm Dgal}_X & \longrightarrow & \defg X \\ \downarrow & &\downarrow \\ \hbox{\rm Dnat}_X & \longrightarrow & \defor X \end{array} $$ where the vertical arrows are injections. The following theorem shows that the horizontal arrows are smooth morphisms of functors when the branch divisors are sufficiently ample. This was proven in \cite{Pa1} under the hypothesis that $Y$ be rigid and regular; in this case natural deformations are unobstructed, and it is enough to check the surjectivity of the Kodaira-Spencer map. In the general case one has to take into account the obstructions as well. \begin{thm}\label{complete} Let $f:X\to Y$ be a totally ramified $(G,I)$-cover with building data $D_i$, $L_\chi$, such that $X$ and $Y$ are smooth of dimension $\ge 2$ and that $X$ is of general type. Assume that for all $\chi\in G^*\setminus 1$ the bundles $L_\chi$ and $\Omega_Y^1\otimes L_\chi$ are ample. Then the natural map of functors (from $\hbox{\sl Art}^*$ to $\hbox{\sl Sets}$) $\hbox{\rm Dnat}_X \to \defor X$ is smooth, and so is the induced map $\hbox{\rm Dgal}_X $ and $\defg X$. \end{thm} \noindent{\sc Proof.~} By a well-known criterion, smoothness of a natural transformation of functors is implied by surjectivity of the induced map on tangent spaces, and injectivity on obstruction spaces. This is immediate by lemma \ref{tdefnat} and corollary \ref{tdef}, and by the fact that the map between tangent (obstruction) spaces induced by the map of functors is the natural one. \ $\Box$\par\smallskip \section{Main theorem} In this section we will prove that the automorphism group of an abelian cover is precisely the Galois group, provided that the branch divisors are sufficiently ample and generic. The proof depends on the construction of an explicit partial resolution of some singular covers, which will be given in section 7. Although the result is in some sense expected, the proof is rather involved and the techniques applied are, we believe, of independent interest. The following lemma is inspired by a similar result of McKernan (\cite{McK}). \begin{lem} \label{mac} Let $\Delta$ be the unit disc in ${\bf C}$, $\Delta^*=\Delta\setminus\{0\}$. Let $p:{\cal X}\to \Delta$ be a flat map, smooth over $\Delta^*$, whose fibres are integral projective varieties of non negative Kodaira dimension. Assume we are given a section $\sigma$ of $Aut_{{\cal X}/\Delta^*}$. If there exists a resolution of singularities $\varepsilon:\tilde {\cal X}\to{\cal X}$ such that each divisorial component of the exceptional locus has Kodaira dimension $-\infty$, then $\sigma$ can be (uniquely) extended to a section of $Bir_{{\cal X}/\Delta}$. \end{lem} \noindent{\sc Proof.~} The section $\sigma$ induces a birational map $\phi:{\cal X}\hbox{\rm-}\,\hbox{\rm-}\,\hbox{\rm-}\!\!\!\!\!> {\cal X}$ over $\Delta$; the uniqueness of the extension follows from this. Let $\tilde\phi:\tilde {\cal X}\hbox{\rm-}\,\hbox{\rm-}\,\hbox{\rm-}\!\!\!\!\!>\tilde {\cal X}$ be the induced birational map, and let $\Gamma$ be a resolution of the closure of the graph of $\tilde\phi$; let $p_1$, $p_2$ be the natural projections of $\Gamma$ on $\tilde{\cal X}$ (such that $p_2=\tilde\phi\circ p_1$), and let $q_i=\varepsilon\circ p_i$. The strict transform ${\cal X}_0'$ of ${\cal X}_0$ in $\Gamma$ via $q_1^{-1}$ has positive Kodaira dimension, hence it cannot be contracted by $p_2$, which is a birational morphism with smooth image. Therefore the restriction of $p_2$ to ${\cal X}_0'$ is birational (because ${\cal X}_0'$ is not contained in the exceptional locus of $p_2$) onto some irreducible divisor ${\cal X}_0''$ in ${\cal X}$. As ${\cal X}''_0$ is birational to ${\cal X}_0$ it cannot be of Kodaira dimension $-\infty$; hence it is not contained in the exceptional locus of $\varepsilon$. Therefore $\varepsilon({\cal X}_0'')$ is a divisor contained in ${\cal X}_0'$, hence it is ${\cal X}_0'$ by irreducibility, and the map $\varepsilon:{\cal X}_0''\to{\cal X}_0$ is birational. So the birational map $\phi$ can be extended to ${\cal X}_0$ by the birational map $q_2\circ \left(q_{1|{\cal X}_0'}\right)^{-1}$. \ $\Box$\par\smallskip \begin{lem} \label{trick}In the same hypotheses of lemma {\rm \ref{mac}}, assume moreover that there is a line bundle $L$ on ${\cal X}$, flat over $\Delta$, whose restriction to ${\cal X}_t$ is very ample for all $t$, and such that $h^0({\cal X}_t,L_{|{\cal X}_t})$ is constant in $t$. If the action of $\sigma$ can be lifted to an action on $L$, then $\sigma$ can be uniquely extended to a section of $Aut_{{\cal X}/\Delta}$. \end{lem} \noindent{\sc Proof.~} Let $N$ be the rank of the vector bundle $p_*L$ on $\Delta$; choosing a trivializing basis yields an embedding ${\cal X}\hookrightarrow\P^{N-1}\times \Delta$. The automorphisms $\phi_t$ of ${\cal X}_t$ are restrictions to ${\cal X}_t$ of nondegenerate projectivities of $\P^{N-1}$; their limit, as $t\to 0$, is a well-defined, possibly degenerate projectivity $\phi_0$. This gives an extension of $\phi$ to an open set of ${\cal X}_0$; this must now be birational by the previous lemma, which in turn implies that $\phi_0$ is nondegenerate (as ${\cal X}_0$ is not contained in a hyperplane), and therefore that $\phi_0$ is a morphism. Applying the same argument to $\phi^{-1}$ concludes the proof. \ $\Box$\par\smallskip \begin{rem} The hypothesis that $\sigma$ acts on $L$ is obviously verified if $L_{|{\cal X}_t}$ is a pluricanonical bundle for all $t\ne 0$. \end{rem} \begin{prop}\label{prop.aut Let $p:{\cal X}\to \Delta$ be a flat family of integral projective varieties of general type, smooth over $\Delta^*$. Assume that there is a line bundle $L$ on ${\cal X}$, flat over $\Delta$, with $L_t:=L_{|{\cal X}_t}$ ample on ${\cal X}_t$, and $Aut({\cal X}_t)$ acts on $L_t$ for $t\ne 0$. Assume moreover that for any $m$-th root base change $\rho_m:\Delta\to \Delta$ the pullback $\rho_m^*{\cal X}$ admits a resolution having only divisors of negative Kodaira dimension in the exceptional locus. Then $Aut_{X/\Delta}$ is proper over $\Delta$, and the cardinality of the fibre is an upper semi-continuous function. \end{prop} \noindent{\sc Proof.~} After replacing $L$ with a suitable multiple and maybe shrinking $\Delta$, we can assume that $L_t$ is very ample on ${\cal X}_t$, and that $h^0({\cal X}_t,L_t)$ is constant in $t$. The map $Aut_{{\cal X}/\Delta}\to \Delta$ is obviously quasi-finite (because the fibres are of general type) and the fibres are reduced (because automorphism groups are always reduced in char.~$0$). It is enough to prove that given a map of a pointed curve $(C,P)$ to $\Delta$ and a lifting of the map to $Aut_{{\cal X}/B}$ out of $P$, the lifting can be extended to $P$. Via restriction to an open set we can assume that $C$ is the unit disc $\Delta$, $P$ is the origin and $\Delta\to \Delta$ is the map $z\to z^m$; we can then apply lemma \ref{trick} to conclude the proof. \ $\Box$\par\smallskip \begin{cor}\label{gen.aut} Let ${\cal X}\to B$ be a smooth family of varieties having ample canonical bundle. Then the scheme $Aut_{X/B}$ is proper over $B$, and the cardinality of the fibre is an upper semi-continuous function. \end{cor} \noindent{\sc Proof.~} We can apply the previous proposition with $L=K_{{\cal X}/\Delta}$. \ $\Box$\par\smallskip \begin{thm} \label{mainthm} Let $Y$ be a smooth projective variety, and $X$ a smooth $(G,I)$-cover with ample canonical bundle, with covering data $L_\chi$, $D_i$. Let $H=\O_Y(1)$ for some embedding of $Y$ in $\P^{N-2}$; assume that the linear system $$|D_1-m_1NH|$$ is base-point-free. Assume also that the ${\bf Q}$-divisor $$ M=K_Y-(m_1-1)NH+\sum_{i\in I}\frac{(m_i-1) }{m_i}D_i$$ is ample on $Y$. Then, for a generic choice of $D_1$ in its linear system, $X$ has automorphism group isomorphic to $G$. \end{thm} \newcommand{1}{1} \noindent{\sc Proof.~} Let $d$ be the number of automorphisms of a generic cover with the given covering data (cfr.~corollary \ref{gen.aut}). It is enough to show that $d\le \#G$, the other inequality being obvious. Let $H$ be as in the statement of the theorem, and let ${\cal H}\subset |H|$ be the (not necessarily complete) linear system giving the embedding; let $H_1,\ldots,H_N$ be $N$ projectively independent divisors in ${\cal H}$. Assume that the $H_i$'s are generic, in particular that they are smooth and that their union with all of the $D_i$'s has normal crossings. Let $m=m_1$, $D=D_1$. The strategy of the proof is the following: start from a generic cover $X$ of $Y$, and construct a sequence of manifolds $X_1,\ldots,X_N$ and of subgroups $G_k$ of $Aut(X_k)$ such that $$ \#Aut(X)\le \#G_1\le\ldots\le \#G_N\qquad{\rm and}\qquad G_N=G.$$ In fact, $X_k$ will be a $(G,I)$-cover of $Y$ with covering data $D^{(k)},D_2,\ldots$, $L_\chi^{(k)}$, where $L_\chi^{(k)}=L_\chi-k\re^1_\chi H$ and $D^{(k)}$ is a generic divisor in $|D-kmH|$ (recall that $\re^i_\chi$ was defined as the unique integer $a$ satisfying $0\le a\le m_i-1$ and $\chi_{|H_i}=\psi_i^a$). We let $G_k$ be the group of automorphisms of $X_k$ preserving the inverse images of the curves $H_1,\ldots,H_k$ in $Y$. We therefore want to prove the following:\begin{enumerate} \item $\#Aut(X)\le \#G_1$; \item $\#G_k\le \#G_{k+1}$; \item $G_N=G$. \end{enumerate} \smallskip \noindent {\sc First step:} $\#Aut(X)\le \#G_1$. Let $D^{(1)}$ be a generic divisor in $|D-m H|$, and choose equations $f_1$, $g$ and $h_1$ for $H_1$, $D$ and $D^{(1)}$ respectively. Define divisors ${\cal D}_i$ on $Y\times {\bf C}$ by ${\cal D}_i=D_i\times {\bf C}$ for $i\ne 1$, ${\cal D}_1=\{(1-t)f_1^mh_1+tg=0\}$; let ${\cal X}^1$ be the corresponding abelian cover. ${\cal X}^1_0$ is a singular variety (singular along the inverse image of the curve $H_1$ in $Y$), with smooth normalization $X_1$ (see \cite{Pa1}, step 1 of normalization algorithm of p.~203). Note that $X_1$ is of general type by the ampleness assumption on $M$. By proposition \ref{reslemma}, the family ${\cal X}^1$ and each $n$-th root base change of ${\cal X}^1$ admit a resolution with only divisors of Kodaira dimension $-\infty$ in the exceptional locus. Moreover, the pull-back of $(\#G)M$ restricts to the $\#G$-canonical bundle on the smooth fibres of ${\cal X}^1$ (cfr the proof of prop.~4.2 in \cite{Pa1}, p.~208). Applying proposition \ref{prop.aut} gives that $Aut_{{\cal X}^1/{\bf C}}$ is proper over ${\bf C}$, and hence that $\#Aut(X)\le Aut({\cal X}^1_0)$ (as we assumed $X$ to be generic). On the other hand it is clear that each automorphism of ${\cal X}^1_0$ lifts to the normalization $X_1$, yielding an automorphism which maps to itself the inverse image of the singular locus, i.e., the inverse image of the curve $H_1$. \smallskip \noindent {\sc Second step:} $\#G_{k-1}\le \#G_{k}$. We use a similar construction; let $X_{k-1}$ be as above, let $h_{k-1}$ be an equation of $D^{(k-1)}$, $f_k$ an equation of $H_k$, and $h_k$ an equation of $D^{(k)}$. Define a $(G,I)$-cover ${\cal X}^k$ of $Y\times {\bf C}$ branched over $D_i\times {\bf C}$ for $i\ne 1$, and over ${\cal D}_1^{(k)}=\{(1-t)f_k^mh_k+th_{k-1}=0\}$; ${\cal X}^k_0$ is singular along the inverse image $C_k$ of $H_k$, and its normalization is $X_k$; again $X_k$ is of general type. Again by proposition \ref{reslemma} the family ${\cal X}^{(k)}$ and all its $n$-th root base changes have a resolution with only uniruled components in the exceptional locus; the same argument as before proves the result. \smallskip\noindent {\sc Final step:} $G_N=G$. Let $\pi:X_N\to Y$ be the covering map: $G_N$ is the group of automorphisms of $Y$ fixing the inverse images of the curves $H_1,\ldots,H_N$. Every element of $G_N$ preserves $\pi^*\left({\cal H}\right)$, hence induces an automorphism of $Y$; this automorphism must be the identity as it induces the identity on ${\cal H}$. Therefore $G_N$ must coincide with $G$. \ $\Box$\par\smallskip \begin{rem} In theorem {\rm \ref{mainthm}} we can replace the assumption that the linear system $|D_1-m_1NH|$ be base point free by asking that for each $i\in I$ $$ |D_i-m_iN_iH|$$ be base point free, with $N_i$ nonnegative integers with sum $N${\rm ;} we then get that, for a generic choice of the $D_i$'s such that $N_i\ne 0$, $Aut(X)=G$. \end{rem} \begin{ex}{\rm One might wonder whether it is always true that a generic abe\-lian cover of general type has no ``extra automorphisms". Here is an easy example where this is not the case. Consider a ${\bf Z}_3$-cover of $\P^1$, branched over two pairs of distinct points, with opposite characters. A generic such cover is a smooth genus $2$ curve, hence its automorphism group cannot be ${\bf Z}_3$. } \end{ex} \begin{ex}{\rm Here is a slightly more complicated example of extra automorphisms, which works in any dimension. Let $Y$ be a principally polarized abelian variety, and let $L$ be a principal polarization; assume that $L$ is symmetric, i.e. invariant under the natural involution $\sigma(y)=-y$ on $Y$. The sections of $L^{\otimes 2}$ are all symmetric, and the associated linear system has no base points. Let $G={\bf Z}_2^s$, with the canonical basis $e_1,\ldots,e_s$. Choose $I=\{1,\ldots,s\}$, and let $H_i$ be the subgroup generated by $e_i$, for $i=1,\ldots,s$. \par The equations for the reduced building data become $L_j^{\otimes 2}=\O_Y(D_j)$; we choose the solution $L_j=L$, $M_i=L^{\otimes 2}$ for all $i,j$. We are in fact constructing a fibred product of double covers. Choose the $D_i$'s to be generic divisors in the linear system $|L^{\otimes 2}|$. Each of them must be symmetric; this implies that the involution $\sigma$ can be lifted to an involution of $X$, which is an automorphism not contained in the Galois group of the cover. \par Note that in this case the total branch divisor can become arbitrarily large, still all $(G,I)$-covers have an automorphism group bigger than $G$. } \end{ex} \section{Moduli spaces of abelian covers and global constructions} In this section we will explicitly construct a coarse moduli space for abelian covers of a smooth variety $Y$ and a complete space of natural deformations. Although some of the material in this section is implicit in \cite{Pa1}, we find it important to state it in a precise and explicit way. In particular we will apply theorem \ref{complete} to construct (under suitable ampleness assumptions) a family of natural deformations which maps dominantly to the moduli (theorem 5.12). Let $Y$ be a smooth, projective variety, $G$ an abelian group, $I$ a subset of $I_G$. A {\em family of smooth $(G,I)$-covers} of $Y$ over a base scheme $T$ is a smooth, proper map ${\cal X}\to T$ and an action of $G$ on ${\cal X}$ compatible with the projection on $T$, together with a $T$-isomorphism of the quotient ${\cal X}/G$ with $Y\times T$, such that for each $t\in T$ the induced cover ${\cal X}_t\to Y$ is a $(G,I)$-cover. Two families over $T$ are {\em {\rm (}strictly\/{\rm )} isomorphic} if there is a $G$-equivariant isomorphism inducing on the quotient $Y\times T$ the identity map. A (coarse) moduli space $\ZZ$ for smooth $(G,I)$-covers of $Y$ is a scheme structure on the set of smooth $(G,I)$-covers modulo isomorphisms, such that for any family of $(G,I)$-covers of $Y$ with base $T$ the induced map $T\to \ZZ$ is a morphism. \begin{thm} There is a coarse moduli space of $(G,I)$-covers of $Y$, which is a Zariski open set $\ZZ=\ZZ(Y,G,I)$ in the closed subvariety of $$\prod_{\chi\in G^*\setminus 1}Pic(Y)\times \prod_{i\in I} Hilb^{\rm div}(Y)$$ of all the $(L_\chi,D_i)$ satifying the relations {\rm (\ref{bdata})}. The open set $\ZZ$ is the set of $(L_\chi,D_i)$'s which satisfy the additional conditions: \begin{enumerate} \item each $D_i$ is smooth and the union of the $D_i$'s is a divisor with normal crossings; \item whenever $D_{i_1}, \ldots, D_{i_k}$ meet, the natural map $H_{i_1}\oplus\cdots\oplus H_{i_k}\to G$ is injective. \end{enumerate} \end{thm} \noindent{\sc Proof.~} The set $\ZZ$ parametrizes the smooth abelian covers of $Y$ by \cite{Pa1}, theorem 2.1. The fact that the induced maps from a family of abelian covers to $\ZZ$ are morphisms follows from the corresponding property of the Hilbert schemes and Picard groups. \ $\Box$\par\smallskip Proposition 2.1 of \cite{Pa1} implies: \begin{rem} For any basis $\chi_1,\ldots,\chi_s$ of $G^*$, the natural map $$\ZZ\to\prod_{j=1}^s Pic(Y)\times\prod_{i\in I} Hilb^{\rm div}(Y)$$ induced by projection is an isomorphism with its image. \end{rem} $\ZZ$ decomposes as the disjoint union of infinitely many quasiprojective varieties $Z(\xi_i,\eta_\chi)=Z(\xi_i,\eta_\chi)(Y,G,I)$, where $\eta_\chi$, $\xi_i$ are the Chern classes of $L_\chi$ and $\O(D_i)$, respectively. We now give an explicit description of $Z(\xi_i,\eta_\chi)$ under the assumption that the $\xi_i$'s are sufficiently ample. \setcounter{equation}{0} \begin{prop}\label{moduli} Let $\xi_i$, $\eta_\chi$ be cohomology classes satisfying the following relations {\rm(}compare {\rm(\ref{bdata})):} \begin{equation} \label{Chbdata}\eta_\chi+\eta_{\chi'}=\eta_{\chi\chi'}+\sum_{i\in I} \eps^i_{\chi,\chi'}\xi_i\qquad\quad \hbox{for all $\chi,\chi'\in G^*\setminus 1$}. \end{equation} Assume moreover that $\xi_i-c_1(K_Y)$ is the class of an ample line bundle for all $i\in I$. Then $Z(\xi_i,\eta_\chi)$ is an open set in a smooth fibration (with fibre a product of projective spaces) over an abelian variety $A(\xi_i,\eta_\chi)$ isogenous to $Pic^0(Y)^{\#I}$. $Z(\xi_i, \eta_\chi)$ is nonempty iff there are smooth effective divisors $D_i$, with $c_1(D_i)=\xi_i$, such that their union has normal crossings. \end{prop} \noindent{\sc Proof.~} Let $A=A(\xi_i,\eta_\chi)\subset\prod_{i\in I}Pic^{\xi_i}(Y)\times \prod_{\chi\in G^*\setminus 1}Pic^\chi(Y)$ be the image of $Z(\xi_i,\eta_\chi)$; by equations (\ref{rbdata}) the natural map $A\to \prod_{i\in I}Pic^{\xi_i}(Y)$ is a finite \'etale cover of degree $(2q)^{\#G}$, where $q$ is the irregularity of $Y$. So each connected component of $A$ is an abelian variety, isogenous to $Pic^0(Y)^{\#H}$. The fact that $A$ is connected is a consequence of the covering being totally ramified. In fact, choose a basis $\chi_1,\ldots,\chi_s$ of $G^*$, and consider the diagram $$ \begin{array}{ccc} A&\longrightarrow&\prod_{i\in I}Pic^{\xi_i}(Y)\\ \downarrow& &\downarrow\\ \prod_{j=1}^sPic^{\eta_j}(Y)&\longrightarrow&\prod_{j=1}^sPic^{\ord{\chi_j} \eta_j}\\ \end{array}$$ with maps given by $$ \begin{array}{ccc} (M_i,L_j)&\mapsto&(M_i)\\ \downarrow& & \downarrow\\ (L_j)&\mapsto&(L_j^{\otimes n_j}=\otimes M_i^ {\reb^i_{j}}).\\ \end{array}$$ The diagram is a fibre product of (connected) abelian varieties; to prove that $A$ is connected is equivalent to proving that $\pi_1(\prod_{i\in I}Pic^{\xi_i}(Y))$ surjects on $$\pi_1(\prod_{j=1}^sPic^{\ord{\chi_j}\eta_j})/ \pi_1(\prod_{j=1}^sPic^{\eta_j}(Y));$$ this is in turn equivalent to proving that $G^*$ injects in $\oplus_{i\in I}H_i^*$, which follows by dualizing from assumption \ref{totram}. Let ${\cal P}_i$ on $A\times Y$ be the pullback of the Poincar\'e line bundles from $Pic^{\xi_i}(Y)\times Y$; the pushforward of ${\cal P}_i$ to $A$ is a vector bundle $E_i$ because of the ampleness condition (the rank of $E_i$ can be computed by Riemann-Roch). The moduli space $Z(\xi_i,\eta_\chi)$ is an open set of the fibred product of the $\P(E_i)$. \ $\Box$\par\smallskip \begin{rem} {\rm If $q(Y)$ is not zero, then the components $Z(\xi_i,\eta_\chi)$ are uniruled, but not unirational.} \end{rem} \begin{rem}{\rm In general $Z(\xi_i,\eta_\chi)$ is a coarse but not a fine moduli space, i.e., it does not carry a universal family. Keeping the notation of proposition \ref{moduli}, let ${\cal V}$ be the total space of the fibred product of the $E_i$'s, and let ${\cal V}^o$ the inverse image of $Z(\xi_i,\eta_\chi)$; we have a natural abelian cover of $Y\times {\cal V}^o$, which is a complete family of smooth covers of $Y$ with the given data.} \end{rem} There is a natural action of $Aut(Y)$ on the moduli space of $(G,I)$-covers $\ZZ$, given by $$\phi(D_i,L_\chi)=(\phi(D_i),(\phi^{-1})^*L_\chi)\qquad\qquad\hbox{for $\phi\in Aut(Y)$.}$$ The automorphism group of $G$ acts naturally on $G^*$ (by $\Phi(\chi)=\chi\circ \Phi^{-1}$) and on $I_G$ (by $\Phi (H,\psi)=(\Phi(H),\psi\circ \Phi^{-1})$); given a subset $I$ of $I_G$, let $Aut_I(G)$ be the set of automorphisms of $G$ preserving $I$. There is a natural action of $Aut_I(G)$ on $\ZZ$, induced by the natural action of this group on the indexing sets $G^*\setminus 1$ and $I$. \begin{prop} If the classes $\xi_i$'s are ample enough {\rm (}so that theorem {\rm \ref{mainthm}} applies to some cover in $Z(\xi_i,\eta_j)${\rm),} then the quotient of $Z(\xi_i,\eta_j)$ by the natural action of $Aut(Y)\times Aut_I(G)$ maps birationally to its image in the moduli of manifolds with ample canonical class. \end{prop} \noindent{\sc Proof.~} That the natural map to the moduli factors via this action is clear. Viceversa, given a generic cover $X$ in $Z(\xi_i,\eta_j)$, by theorem \ref{mainthm} its automorphism group is isomorphic to $G$; so it can be identified uniquely as a $(G,I)$-cover up to isomorphisms of $G$ and of $Y$. \ $\Box$\par\smallskip \begin{defn}{\rm Let ${\cal Y}\to T$ be a deformation of $Y$ over a simply connected pointed analytic space $(T,o)$. As $T$ is simply connected, the cohomology of every fibre ${\cal Y}_t$ is canonically isomorphic with that of $Y$. Then the varieties $Z(\xi_i,\eta_\chi)({\cal Y}_t,G,I)$ (resp.\ $A(\xi_i,\eta_\chi)({\cal Y}_t,G,I)$) for $t\in T$ glue to a global variety $\ZZ_T(\xi_i,\eta_\chi)=\ZZ_T(\xi_i,\eta_\chi)({\cal Y},G,I)$ (resp.\ ${\cal A}_T(\xi_i,\eta_\chi)$), surjecting on the locus on $T$ where the classes $\xi_i$ {\rm(}and hence also the $\eta_\chi${\rm)} stay of type $(1,1)$. The global varieties are constructed by replacing the Hilbert and Picard schemes in the construction of $Z(\xi_i,\eta_\chi)$ and $A(\xi_i,\eta_\chi)$ with their relative versions. The previous results can all be extended to this relative setting. } \end{defn} For each smooth $(G,I)$-cover $f:X\to Y$, the natural deformations of the reduced building data such that the induced deformations of $(Y,L_j,M_i)$ is trivial are parametrized naturally by $\prod_{({i,\chi})\in S}H^0(Y,M_i\otimes L_\chi^{-1})$, as in \S 5 of \cite{Pa1}. \begin{thm} Let ${\cal Y}\to T$ be a deformation of $Y$ over a germ $(T,o)$, and assume that the $\xi_i$'s stay of type $(1,1)$ on $T$. Then there is a quasiprojective morphism ${\cal W}_T(\xi_i,\eta_\chi)\to {\cal A}_T(\xi_i,\eta_\chi)$ whose fibre over a point parametrizing line bundles $(L_j,M_i)$ on ${\cal Y}_t$ is canonically isomorphic to $\prod_{({i,\chi})\in S}H^0(Y,M_i\otimes L_\chi^{-1})$. \end{thm} \noindent{\sc Proof.~} The theorem follows, by taking suitable fibre products, from the following two lemmas. \ $\Box$\par\smallskip \begin{lem} Let $Y$ be a smooth projective variety, and $\xi\in NS(Y)$. Then there exists a morphism of schemes $\pi:W^\xi(Y)\to Pic^\xi(Y)$ such that the fibre over a point $[L]$ is naturally isomorphic to the vector space $H^0(Y,L)$. For any choice of the Poincar\'e line bundle ${\cal P}$ on $Y\times Pic^\xi(Y)$, there exists such a $W^\xi(Y)$ with the property that the line bundle $\pi^*{\cal P}$ on $Y\times W^\xi(Y)$ has a tautological section. \end{lem} Let ${\cal P}$ be the Poincar\'e line bundle on $Y\times Pic^\xi(Y)$, and let $p:Y\times Pic^\xi(Y)\to Pic^\xi(Y)$ and $q:Y\times Pic^\xi(Y)\to Y$ be the projections; if $p_*({\cal P})$ is a vector bundle, it is enough to take $W$ to be the total space of this vector bundle. It is also clear that if $\xi-c_1(K_Y)$ is an ample class, then $p_*({\cal P})$ is indeed a vector bundle. For the general case, let $A$ be a line bundle on $Y$ such that $c_1(A)+\xi-c_1(K_Y)$ is ample, and such that there exists an $s\in H^0(Y,A)$ defining an effective, smooth divisor $D$. Let $\pi:V\to Pic^\xi(Y)$ be the total space of the vector bundle $p_*({\cal P}\otimes q^*A)$, and let $\sigma:\O_{Y\times V}\to \pi^*({\cal P}\otimes q^*A)$ be the tautological section. For every $y\in D$, let $\sigma_y$ be the induced section of $\pi^*({\cal P}\otimes q^*A)|_{\{y\}\times V}$; let $W_y\subset V$ be the divisor defined by $\sigma_y$. Let $W=W^\xi(Y)$ be the intersection of all $W_y$'s for $y\in D$: then $\sigma/s$ is regular on $W$, and defines the required tautological section. \ $\Box$\par\smallskip \begin{lem}{Let ${\cal Y}\to T$ be a deformation of $Y$ over a germ of analytic space $T$, and assume that $\xi$ stays of type $(1,1)$ over $T$. Then, after maybe replacing $T$ with a Zariski-open subset, the spaces $W^\xi({\cal Y}_t)$ glue together to a quasiprojective morphism $W^\xi_T({\cal Y})\to Pic^\xi_T({\cal Y})$.} \end{lem} \noindent{\sc Proof.~} After possibly restricting $T$, we can extend $A$ to a line bundle ${\cal A}$ over ${\cal Y}$, and $s$ to a section of ${\cal A}$. The rest of the proof remains the same, using the fact that the relative Picard scheme exists and carries a Poincar\'e line bundle. \ $\Box$\par\smallskip We now want to describe explicitly $W^\xi(Y)$ in the case $\xi=0$, which we will use repeatedly later. \begin{rem}{\rm For any deformation ${\cal Y}\to T$ over a germ of analytic space, $W^0_T({\cal Y})$ is naturally isomorphic to the union in $Pic^0_T({\cal Y})\times {\bf C}$ of $j(T)\times{\bf C}$ and $Pic^0_T({\cal Y})\times \{0\}$, where $j:T\to Pic^0_T({\cal Y})$ is the zero section.} \end{rem} In particular $W^0(Y)$ is reducible when $q(Y)\ne 0$; this reflects the fact that the deformations, as pair (line bundle, section), of $(\O_Y,0)$ are obstructed; one can either deform the line bundle or the section, but not both at the same time. This remark will be used to construct examples of manifolds lying in several components of the moduli in section 6. \begin{thm} {\rm (i)} Let $Y$ be a smooth projective variety, and let $X\to Y$ be a smooth $(G,I)$-cover such that theorem {\rm \ref{complete}} holds. Then there exists a pointed analytic space $({\cal W},w)$ and a natural deformation of the reduced building data of $X$ over ${\cal W}$ such that the induced map of germs from $({\cal W},w)$ to the Kuranishi family of $X$ {\rm(}defined as in {\rm 3.3}{\rm)} is surjective. \par\noindent {\rm (ii)} One can choose ${\cal W}$ to be a quasi-projective scheme, and then the induced rational map from ${\cal W}$ to the moduli of manifolds with ample canonical class is dominant onto each component of the moduli containing $[X]$. \end{thm} \noindent{\sc Proof.~} (i) Let ${\cal Y}\to T$ be the restriction of the Kuranishi family of $Y$ to the locus where all the $\xi_i$'s stay of type $(1,1)$. Let ${\cal W}={\cal W}_T(\xi_i,\eta_\chi)$, ${\cal Y}_{\cal W}={\cal Y}\times_T{\cal W}$. Over ${\cal Y}_{\cal W}$ there are tautological line bundles $\L_j$, ${\cal M}_i$ and tautological sections $s_{{i,\chi}}$ of ${\cal M}_i\otimes\L_\chi^{-1}$ (where $\L_\chi$ is defined as in \ref{natdef}); moreover $\L_j^{\otimes n_j}$ is isomorphic to $\Bigotimes {\cal M}_i^{\reb^i_j}$. ${\cal W}$ parametrizes data $({\cal Y}_t,L_j,M_i,s_{{i,\chi}})$ such that $t\in T$, $L_j$ and $M_i$ are line bundles on ${\cal Y}_t$ satisfying (\ref{rbdata}) and having Chern classes $\eta_j,\xi_i$, and $s_{i,\chi}$ are sections of $L_\chi\otimes M_i^{-1}$. Let $w\in {\cal W}$ be a point corresponding to the reduced building data of $X$: that is, assume that $w$ corresponds to the data $({\cal Y}_o,L_j,M_i,s_{{i,\chi}})$, where $s_{i,\chi}=0$ for all $\chi\ne 1$, $o$ is the chosen point in $T$, and the sections $s_{i,0}$ define divisors $D_i$ such that $(L_j,D_i)$ are the reduced building data of $X$. Choose arbitrarily isomorphisms $\Phi_j:\L_j^{\otimes n_j}\to \Bigotimes {\cal M}_i^{\otimes\reb^i_j}$, extending the isomorphism over $w$ induced by multiplication in $\O_X$. By theorem \ref{complete}, together with Artin's results on approximation of analytic mappings (see \cite{Ar}), it is enough to show that every natural deformation of the reduced building data of $X$ over a germ of analytic space can be obtained as pullback from $({\cal W},w)$. It is clear that all small deformations of the data $(Y,L_j,M_i,s_{i,\chi})$ can be obtained as pullback from $W$. So it is enough to prove that, up to isomorphism of natural deformations, we can choose the $\phi_j$'s arbitrarily. This is proven in lemma \ref{lautnonconta}. \noindent (ii) Start by noting that one can construct a deformation ${\cal Y}\to B$ of $Y$ over a pointed quasi-projective variety $(B,o)$, such that the germ of $B$ at $o$ maps surjectively to the locus in the Kuranishi family of $Y$ where the classes $\xi_i$'s stay of type $(1,1)$. In fact, choose any $\chi\in G^*\setminus 1$, and let $L$ be a sufficiently big multiple of $L_\chi$; assume in particular that $L$ is very ample and that all its higher cohomology groups vanish. Let $N=\dim H^0(Y,L)-1$; choosing a basis of $ H^0(Y,L)$ gives an embedding of $Y$ in $\P^N$. Take the union of the irreducible components of the Hilbert scheme of $\P^N$ containing $b=[Y]$, and consider inside it the open locus $B'$ of points corresponding to smooth subvarieties. Then the natural map from the germ of $B'$ at $b$ to the Kuranishi family of $X$ surjects on the locus where $\eta_\chi$ stays of type $(1,1)$. Let $B$ be the closed subscheme of $B'$ where also the classes $\xi_i$ stay of type $(1,1)$. Let ${\cal Y}\to B$ be the universal family; by replacing $B$ with an \'etale open subset we can assume that ${\cal Y}\to B$ has a section. Then (compare for instance \cite{Mu}, p.~20) there exists a global projective morphism ${\cal A}\to B$ and line bundles ${\cal M}_i$, $\L_j$ on ${\cal Y}\times_B{\cal A}$, such that ${\cal A}_b$ parametrizes line bundles $(M_i,L_j)$ on ${\cal Y}_b$ such that firstly, they satisfy the usual compatibility conditions, and secondly, the Chern classes of $(M_i,L_j)$ lie in the orbit of $(\xi_i,\eta_j)$ via the monodromy action of $\pi_1(B,b)$. Mimicking the proof in the germ case, and replacing $B$ by an \'etale open subset if necessary, one can find a quasi-projective morphism ${\cal W}\to {\cal A}$ whose fibre over a point corresponding to line bundles $(M_i,L_j)$ on ${\cal Y}_b$ is isomorphic to $\prod H^0({\cal Y}_b,M_i\otimes L_\chi^{-1})$ for $({i,\chi})\in S$, together with tautological sections $\sigma_{{i,\chi}}$ of the pullbacks to ${\cal Y}\times_B{\cal W}$ of ${\cal M}_i\otimes\L_\chi^{-1}$. Let $w\in {\cal W}$ be a point corresponding to the building data of $X$ as before. Again (possibly passing to an \'etale open subset) one can extend the multiplications isomorphisms $\phi_j$ to isomorphisms $\Phi_j:\L_j^{\otimes n_j}\to \Bigotimes {\cal M}_i^{\otimes\reb^i_j}$. Putting everything together, we have a natural deformation of the building data of $X$ over $({\cal W},w)$; this induces by (3.3) a rational map to the moduli of manifolds with ample canonical class, which is a morphism on the open subset of ${\cal W}$ where the natural deformation of $X$ is smooth. Applying the same methods as in (i) implies that the map from ${\cal W}$ to the moduli is dominant on each irreducible component containing $[X]$. \ $\Box$\par\smallskip \begin{lem}\label{lautnonconta} Let $T$ be a germ of analytic space. For any $({\cal Y},{\cal M}_i,\L_j,s_{{i,\chi}},\phi_j)\in\hbox{\rm Dnat}_X(T)$, and for any other admissible choice of isomorphisms $\phi_j':\L_j^{\otimes n_j}\to\bigotimes M_i^{\otimes \re^i_j}$, there exist sections $s'_{i,\chi}$ such that $ ({\cal Y},{\cal M}_i,\L_j,s_{{i,\chi}},\phi_j)$ is isomorphic to $({\cal Y},{\cal M}_i,\L_j,s'_{{i,\chi}},\phi'_j)$. \end{lem} \noindent{\sc Proof.~} It is enough to show that there are automorphisms $\psi_i$ of ${\cal M}_i$ such that the composition $(\bigotimes \psi_i^{\otimes \reb^i_j})\circ \phi_j$ equals $\phi_j'$; in fact in this case one can choose $s'_{i,\chi}=\psi_i^*(s_{i,\chi})$, for all $({i,\chi})\in S$. As both $\phi_j$ and $\phi_j'$ are isomorphisms, $\phi_j=f_j\phi_j'$, where $f_j$ is an invertible function on ${\cal Y}$ restricting to $1$ on the central fibre. Finding the $\psi_i$'s is equivalent to finding functions $g_i$'s on ${\cal Y}$ such that $g_i$ restricts to $1$ on the central fibre and $f_j=\prod g_i^{\reb^i_j}$, for all $j=1,\ldots,s$. The existence of such $g_i$'s follows from the fact that the matrix $\re^i_j$ has rank equal to $s$, which in turn is implied by the cover being totally ramified (see lemma 2.1). \ $\Box$\par\smallskip \setcounter{equation}{0} \begin{rem}{\rm There is a natural action of $(C^*)^{\#I}$ on the functor of natural deformations, which is the identity on $({\cal Y},\L_j,{\cal M}_i,\phi_j)$ and acts on $\sigma_{i,\chi}$ by \begin{equation}\label{C*action} (\lambda_i)_{i\in I}(\sigma_{j,\chi})=\prod_{i\in I}\lambda_i^{\delta_{ij}m_i-\re^i_\chi}\cdot \sigma_{j,\chi};\end{equation} This action has the property that the induced flat maps ${\cal X}\to T$ are invariant under it; in particular the natural map from ${\cal W}(\xi_i,\eta_j)$ to the moduli factors through the corresponding action. } \end{rem} \section{Applications to moduli} In this chapter we want to apply the results on deformation theory together with theorem \ref{mainthm} to study the generic automorphism group of some components of the moduli spaces of manifolds with ample canonical class, components containing suitable abelian covers with sufficiently ample branch divisors. To begin with, we study the case of simple cyclic covers (i.e., those for which the Galois group $G$ is cyclic and there is only one irreducible branch divisor). \begin{prop}\label{cyclic} Let $f:X\to Y$ be a smooth simple cyclic cover, with Galois group ${\bf Z}_m$, and reduced building data $D$ and $L$ (where $D$ is a smooth divisor and $L$ is a line bundle satisfying $mL\equiv D$). Assume that $D$ is sufficiently ample. Let $M$ be an irreducible component of the moduli space of surfaces of general type containing $X$. Then $G_M$ is trivial if $m\ge 3$, and $G_M=G$ if $m=2$. \end{prop} \noindent{\sc Proof.~} In case $m=2$, it is easy to check that $H^i(X,T_X)$ is $G$-invariant for $i=1,2$; hence the natural map $\hbox{\rm Dgal}_X \to \hbox{\rm Dnat}_X$ is surjective, and all deformations are Galois. By theorem \ref{mainthm}, $Aut(X)=G$ for a generic choice of $D$ in its linear system. If $m\ge 3$, assume without loss of generality that $D$ is generic in its linear system. Let $(G,\chi)$ be the element of $I_G$ corresponding to the only nonempty branch divisor. Then the natural deformations of $X$ such that $Y$ and $\O(D)$ are fixed are parametrized by $$ \Bigoplus_{i=0}^{m-2} H^0(Y,L^{-i}(D))=\Bigoplus_{i=0}^{m-2} H^0(Y,L^{m-i});$$ in particular they are unobstructed. Moreover, given any nontrivial element $g$ of the Galois group $G$, it acts on the (necessarily nonzero) summand $H^0(Y,L^{m-1})$ as multiplication by $\chi(g)$, hence nontrivially; therefore $g$ does not extend to the generic deformation. By genericity however $Aut(X)=G$, hence by semicontinuity of the automorphism group the proof is complete. \ $\Box$\par\smallskip Hence, to get nontrivial examples, and to prove the results on the moduli claimed in the introduction, it is necessary to study more general abelian covers. \begin{constr}\label{construction} {\rm Let $s$ be an integer $\ge2$. Let $d_1,\ldots,d_s$ be integers $\ge 2$, such that $d_i|d_{i+1}$ for $i\le s-1$; let $d_0=d_s$, and define integers $b_i$ by requiring that $b_id_i=d_0$, for all $i=1,\ldots,s$. Let $G={\bf Z}_{d_1}\times\cdots\times {\bf Z}_{d_s}$, and let $e_1,\ldots,e_s$ be the canonical basis of $G$; let $\chi_1,\ldots,\chi_s$ be the dual basis of $G^*$. \par Let $e_0:=-(e_1+\ldots+e_s)$, and let $H_i$ be the subgroup generated by $e_i$; for $i=0,\ldots,s$, let $\psi_i\in H_i^*$ be the unique character such that $\psi_i(e_i)=\zeta_{d_i}$; note that, for each $j=1,\ldots,s$ and $i\ne 0$ we have $\re^i_j=\delta_{ij}$, while $\re^0_j=b_j(d_j-1)$. Moreover $\ord{e_i}=d_i$, for $i=0,\ldots,s$. Let $I=\{0,\ldots,s\}$; identify $I$ with a subset of $I_G$ via $i\mapsto (H_i,\psi_i)$. \par Fix a smooth projective variety $Y$ of dimension $d$, and assume that $s\ge d\ge 2$. Let $f:X\to Y$ be a $(G,I)$-cover of $Y$, with branch divisors $D_i$. Equations (\ref{rbdata}) become $$ L_j^{\otimes d_j}=M_j\otimes M_0^{\otimes (d_j-1)}$$ for all $j=1,\ldots,s$, hence they can be solved by letting $L_j=M_0\otimes F_j$, $M_j=M_0\otimes F_j^{\otimes d_j}$, for all $j=1,\ldots ,s$. \par We compute explicitly $L_\chi$ for $\chi\in G^*$, using equation (\ref{chidarbd}). Let $\chi\in G^*$, and write $\chi=\chi_1^{\alpha_1}\cdots\chi_s^{\alpha_s}$, with $0\le \alpha_j< d_j$. One gets $$ L_{\chi}=\Bigotimes_{j=1}^s F_j^{\otimes \alpha_j}\otimes M_0^{\otimes N_\chi},$$ where $N_\chi=-[(-\alpha_1b_1-\ldots-\alpha_sb_s)/d_0]$. In particular $N_\chi$ is an integer $\ge 0$; $N_\chi=0$ if and only if $\chi=0$, $N_\chi=1$ if and only if $\sum(\alpha_ib_i)\le d_0$. In the following we will always assume that $c_1(F_j)=0$, for $j=1,\ldots,s$; let $\xi=c_1(M_0)$. Assume also that $X$ is a smooth $(G,I)$-cover, that is that the divisors $D_i$ are smooth and their union has normal crossings.\par \setcounter{equation}{0} In the surface case, one can compute the Chern invariants of the cover $X$: \begin{eqnarray*} K_X^2/\#G&=&\left(K_Y+(s-(d_0^{-1}+\ldots+d_s^{-1}))\xi\right)^2\\ c_2(X)/\#G&=&c_2(Y)-((s+1)-(d_0^{-1}+\ldots+d_s^{-1}))\xi K_Y+\\ &&\left({s+2\choose 2}+ \sum_{i=0}^sd_i^{-1}+\sum_{0\le i<j\le s}d_i^{-1}d_j^{-1}\right)\xi^2. \end{eqnarray*} The first equality follows from \cite{Pa1}, proposition 4.2; the second from the additivity of the Euler characteristic, by decomposing $Y$ in locally closed subsets according to whether a point lies in $2$, $1$ or no branch divisor. Note that no other possibilities can occur, as we assume that the union of the branch divisors has normal crossings. The second equality could also be derived by Noether's formula and proposition 4.2 in \cite{Pa1}. } \end{constr} \begin{lem}\label{keylemma} Let $f:X\to Y$ be a $(G,I)$-cover as in construction \ref{construction}. Assume that $q(Y)$ is nonzero, that $\xi\in NS(Y)$ is sufficiently ample, that $F_j=\O_Y$ (for $j=1,\ldots,s$), and that $D_i\in |M_i|=|M_0|$ is generic {\rm(}for $i=0,\ldots,s${\rm)}. Then, for each $k=0,\ldots, s$, there exists a component $M_k$ of the moduli of manifolds with ample canonical class, containing $X$, such that the generic automorphism group $G_{M_k}\subset G$ is equal to $G_k={\bf Z}_{d_{1}}\times\ldots\times{\bf Z}_{d_k}$. \end{lem} \noindent{\sc Proof.~} By assumption $X$ has ample canonical class, $Aut(X)=G$ and the natural deformations of $X$ are complete. Assume first that $Y$ is rigid. Let $\chi\in G^*$ be such that $N_\chi=1$, and let $({i,\chi})\in S$; these are the only values of ${i,\chi}$ (with $\chi$ nontrivial) for which $M_i\otimes L_\chi^{-1}$ can have sections, i.e.~can contribute to non-Galois deformations. In fact $c_1(M_i\otimes L_\chi^{-1})=0$, hence it has sections if and only if it is trivial (compare remark 5.11). The condition that the line bundle $M_i\otimes L_\chi^{-1}$ be trivial can be expressed, in terms of the $F_j$'s, as \begin{equation}\label{trivial} \sum_j \alpha_j F_j= d_i F_i.\end{equation} Let $T_k\subset Pic^0(Y)^s$ be the locus where $F_i=0$ for all $i>k$. Note that $F_i=0$ for all $i>k$ implies that $M_i=M_0$ for all $i>k$, and that $M_i\otimes L_\chi^{-1}$ is trivial for any $({i,\chi})$ such that $N_\chi=1$, $i>k$ and $\chi$ restricted to $G_k$ is trivial. For a generic choice of $(F_j)\in T_k$, the line bundles $M_i\otimes L_\chi^{-1}$ are nontrivial for each $\chi$ such that $\chi_{|G_k}\ne 1$; in fact, for any such $\chi$ there exists $j_0\le k$ such that $\alpha_{j_0}> 0$, hence the coefficient of $F_{j_0}$ in (\ref{trivial}) is nonzero (being either $\alpha_{j_0}>0$ or $\alpha_{j_0}-d_{j_0}<0$). On the other hand, for each $j>k$, one has $(0,\chi_j)\in S$ and $M_0\otimes L_j^{-1}$ is trivial (in fact one has to exclude here the case where $d_0$ is equal to $2$, and hence all $d_i$'s are; this case needs a slightly different analysis, see below). Hence for every $g\in G\setminus G_k$, and for any $(G,I)$-cover with building data in $T_k$, there are natural deformations of the cover to which the action of $g$ does not extend. Therefore the $(G,I)$-covers whose building data are in $T_k$, together with their natural deformations such that $s_{{i,\chi}}=0$ for all $\chi$ acting nontrivially on $G_k$, form an irreducible component of the Kuranishi family of $X$; in fact they are parametrized by an irreducible variety, and at some point they are complete (at least at all points corresponding to $(G,I)$-covers with a generic choice of the $F_j$'s for $j\le k$). The generic element of this component has therefore automorphism group $G_k$. In the case where $d_0=2$, $(0,\chi_j)\notin S$; however, if $k\ne s-1$, we can consider $M_{j'}\otimes L_j^{-1}$ instead of $M_0\otimes L_j^{-1}$, where $j'$ is any index $>k$ and different from $j$. If $k=s-1$, let $\chi=\chi_1+\chi_s$; then $N_\chi=1$ (as $s\ge 2$), and $(0,\chi)\in S$. As $\chi(e_s)\ne 0$, there are natural deformations to which the action of $G$ does not extend. The same argument applies if $Y$ is non-rigid, by replacing $Pic^0(Y)^s$ with $Pic^0_T({\cal Y})^s$, where ${\cal Y}\to T$ is the restriction of the Kuranishi family of $Y$ to the locus where $\xi$ stays of type $(1,1)$. \ $\Box$\par\smallskip \begin{rem} {\rm We can find a $Y$ of arbitrary dimension and an ample class $\xi$ such that deformations of $Y$ for which $\xi$ stays of type $(1,1)$ are unobstructed; for instance, by taking $Y$ a product of curves of genus at least two and $\xi$ the canonical class.} \end{rem} \begin{thm} Let $d\ge 2$ be an integer. Given any integer $N$, there exists a point in the moduli space of manifolds of dimension $d$ with ample canonical class which is contained in at least $N$ distinct irreducible components. \end{thm} \setcounter{equation}{0} \noindent{\sc Proof.~} Without loss of generality, assume that $N\ge d$. Choose arbitrarily integers $d_1,\ldots,d_N$, each of them $\ge2$ and such that $d_i|d_{i+1}$. Let $(Y,L)$ be as in lemma \ref{keylemma}; then for each $k=1,\ldots,N$ there exists a component of the moduli containing $X$ and having generic automorphism group isomorphic to ${\bf Z}_{d_1}\times\ldots\times{\bf Z}_{d_k}$. Hence $X$ lies in at least $N$ different irreducible components of the moduli. \ $\Box$\par\smallskip In the case of surfaces, this result gives a strong negative answer to the open problem (ii) on page 485 of \cite{Ca1}. \begin{thm} Let $G$ be a finite abelian group, and $d\ge 2$ an integer. Then there exist infinitely many components $M$ of the moduli space of manifolds of dimension $d$ with ample canonical class such that $G_M=G$. \end{thm} \noindent{\sc Proof.~} Write $G$ as ${\bf Z}_{d_1}\times\ldots\times{\bf Z}_{d_k}$, with $d_i|d_{i+1}$. If $k\ge d$, let $s=k$; if $k<d$, let $s=d$ and let $d_{k+1}=\ldots=d_s=d_k$. Choose $(Y,\xi)$ as in lemma \ref{keylemma}. Applying the lemma to $(Y,i\xi)$ for $i\ge 1$ gives the claimed result. \ $\Box$\par\smallskip In the case of surfaces, another natural question concerns the cardinality of the automorphism group. Xiao proved in \cite{Xi1} that if $X$ is a minimal surface of general type, $\#G\le 52K_X^2+32$ for all abelian subgroups $G$ of $Aut(X)$; it is not known whether this bound is sharp, but he gives examples to the effect that any better bound must still be linear in $K_X^2$. It seems natural to ask if there is a smaller bound if one replaces $Aut(X)$ by $Aut_{\hbox{\rm gen}}(X)$, the intersection in $Aut(X)$ of $G_M$, for each irreducible component $M$ containg $X$. Notice that in Xiao's examples the generic automorphism group is obviously smaller, so a better bound should be possible. We prove here that such a bound cannot be less than linear in $K_X^2$. \begin{prop} There exists a sequence $S_n$ of minimal surfaces of general type such that \begin{enumerate} \item $k_n=K_{S_n}^2$ tends to infinity with $n$; \item $S_n$ lies on a unique irreducible component, $M_n$; \item $\#G_{M_n}> 2^{-4}k_n$. \end{enumerate} \end{prop} \noindent{\sc Proof.~} Let $n\ge 2$ be an integer. Apply contruction \ref{construction} with $s=2$, $d_1=d_2=n$, $Y$ a principally polarized abelian surface with $NS(Y)= {\bf Z}$ and $\xi$ equal to the double of the class of the principal polarization. Choose $S_n$ to be a cover branched over divisors $D_i$ whose linear equivalence classes are generic; then all infinitesimal deformations must be Galois, and the Kuranishi family of $S_n$ is smooth. So $G_{M_n}$ must contain ${\bf Z}_n^2$, hence $\#G_{M_n}\ge n^2$. On the other hand, $k_n=16(n-1)^2$. Note that as we only want to bound $G_M$ from below, we don't need to apply theorem \ref{mainthm}, which would have forced us to choose as class $\xi$ a higher multiple of the principal polarization.\ $\Box$\par\smallskip \begin{rem} {\rm Using the computation of Chern numbers for construction \ref{construction}, one can determine where the examples constructed so far lie in the geography of surfaces of general type. For instance by setting all $d_i$'s equal to $m$ and letting $s$ and $m$ go to infinity, one gets a sequence of examples where $K^2/c_2$ tends to $2$ from below.} \end{rem} \section{Resolution of singularities} \begin{rem} \label{resnc} Let $\pi:X\to Y$ be a $(G,I)$-cover with $Y$ smooth and branch locus with normal crossings. Let $Z\to X$ be a resolution of singularities; then the exceptional locus of $Z$ has uniruled divisorial components. \end{rem} \noindent{\sc Proof.~} The question is local on $Y$, so we can assume that $Y$ is affine and that the line bundles $L_\chi$ and $\O(D_i)$ are trivial. Let $G'$ be the abelian group with $\#I$ generators $e_1,\ldots,e_s$, and relations $m_ie_i=0$ (where $m_i=\#H_i$). There exists a smooth $G'$-cover $X'$ of $Y$ branched over the $D_i$ such that the inertia subgroup of $D_i$ is generated by $e_i$, and such that the map $V\to Y$ factors via $X$. Let $Z'$ be a resolution of singularities of the fibre product $Z\times_X X'$; we have a commutative diagram $$ \begin{array}{ccc} Z' & \to & X'\\ \downarrow & & \downarrow\\ Z&\to &X\\ \end{array} $$ Let $E$ be an irreducible divisorial component of the exceptional locus of $Z\to X$; its strict transform $E'$ in $Z'$ must be contracted in $X'$ as $X'\to X$ is finite. As $X'$ is smooth and $Z'\to X'$ is birational, $E'$ must be ruled by \cite{Ab}, therefore $E$ must be uniruled. \ $\Box$\par\smallskip \begin{lem} Let ${\cal Y}\to \Delta$ be a family of smooth manifolds, ${\cal X}\to {\cal Y}$ an abelian cover branched on divisors which are all smooth except $D$, of branching order $n$, which has local equation $f^nh+tg=0$ with $f$, $t$, $h$, $g$ local coordinates on $Y$ (and $t$ coordinate on $\Delta$). Then there exists a morphism $\tilde{\cal Y}\to {\cal Y}$ such that: \begin{enumerate} \item $\tilde{\cal Y}\to {\cal Y}$ is a composition of blowups with smooth center; \item the normalization $\tilde {\cal X}$ of the induced cover of $\tilde {\cal Y}$ is an abelian cover of $\tilde {\cal Y}$ branched over a normal crossing divisor; \item the exceptional divisors of $\tilde {\cal X}\to {\cal X}$ have Kodaira dimension $-\infty$. \end{enumerate} \end{lem} \noindent{\sc Proof.~} We will construct $\tilde{\cal Y}$ by successive blowups; a local coordinate and its strict transform after the blowup will be denoted by the same letter. At each blowing-up step one checks that the normalization of the last introduced exceptional divisor has Kodaira dimension $-\infty$ (further blowups change the situation only up to birational maps). The strategy of the proof is as follows; each blowup introduces a divisor which is a $\P^r$ bundle (for $r=1,2$), and we prove that the induced cover of the generic $\P^r$ has Kodaira dimension $-\infty$. We can assume that the Galois group coincides with the inertia subgroup $H$ of $D$; if this is not the case, consider the factorization ${\cal X}\to {\cal X}/H\to Y$, and note that the map ${\cal X}/H\to Y$ is unramified near generic points of $D$, hence after blowing up the inverse image of the generic $\P^r$ is an unramified cover, which is therefore a disjoint union of copies of $\P^r$. We first prove the result on the locus where $h\ne 0$ (this is all one needs if ${\cal Y}$ is a threefold). By changing local coordinates one can assume $h=1$. Let $n$ be the order of $H$. We distinguish two cases: $n$ even and $n$ odd. Let $E_1,E_2,\ldots$ be the subsequent exceptional divisors. \smallskip \noindent{\sc Case of $n$ even.} Blow up at each step the singular locus $t=f=g=0$ and look at the $f$ chart. At the first step one obtains $$ z^n=f^2(f^{n-2}+tg)$$ and the total transform of the branch locus $D$ is $D+2E_1$. The covering restricted to $E_1$ is the composition of a totally ramified cover of degree $n/2$ and of a double cover ramified over $D\cap E_1$ which is (on each $\P^2$ in $E_1$) a (possibly reducible) conic. Hence the cover of $E_1$ is fibered in two-dimensional quadrics (maybe singular). At the $k$-th step ($1<k\le n/2$) we have $$ z^n=f^{2k}(f^{n-2k}+tg)$$ and the total transform of $D$ is $$ D+2E_1+\ldots+2kE_k.$$ Again $D$ cuts out a (possibly reducible) conic on the $\P^2$ fibration of $E_k$; moreover, $E_k\cap E_i=\emptyset$ if $i<k-1$, and $E_k\cap E_{k-1}$ is (fibrewise) a line which is not contained in $D$. If $\xi$ is a generator of the group $H$, the induced cover of $E_k$ is the composite of a totally ramified cover and of a cyclic cover of degree $r$, where $r$ is the cardinality of $H/\< \xi^{2k}\>$; the cover is ramified on each $\P^2$ on a conic and on a line. The pairs (inertia group, character) for the branch divisors correspond, via the bijection defined in \S 2, to $\xi$ for the conic and to $\xi^{-2}$ for the line. The canonical bundle of the cover is (fibrewise) the pullback of a multiple of a line in $\P^2$, the multiple being $$ -3+2\left(\frac{r-1}{r}\right) +\left(\frac{r/2-1}{r/2}\right)<0 $$ if $r$ is even and $$ -3+2\left(\frac{r-1}{r}\right)+\left(\frac{r-1}{r}\right)<0$$ if $r$ is odd; in both cases the anticanonical bundle of the cover is ample and the surface must be of Kodaira dimension $-\infty$. \smallskip \noindent{\sc Case of $n$ odd.} Start by blowing up the singular locus $t=f=g=0$. At the first step the total transform of $D$ is $D+2E_1$ and the cover of $E_1$ is totally ramified (as $2$ is prime with $n$), hence the cover is again $E_1$. If $2k<n$ the same formulas as before hold; we can repeat the previous argument where $r$ is necessarily odd. Look now at the $k=(n-1)/2$ case. The total transform of $D$ is $$ D+2E_1+\ldots+(n-1)E_{(n-1)/2}.$$ The strict transform of $D$ is now smooth; $E_{(n-1)/2}\cap D$ is fibered in singular conics, and we blow up the singular locus. The center of this blowup does not meet $E_k$ for $k<(n-1)/2$, and $D$ and $E_{(n-1)/2}$ have the same tangent space there. Therefore after blowing one gets an exceptional divisor $E_{(n+1)/2}$ intersecting both $E_{(n-1)/2}$ and $D$ in the same line. The equation (in the $g$ chart) becomes $$ z^n=f^{n-1}g^n(f+tg).$$ The cover of $E_{(n+1)/2}$ is a $\P^1$-bundle ramified on a generic $\P^1$ with opposite characters on the same divisor, hence when normalizing it splits completely. The components of the total transform of $D$ are smooth, but they meet non-transversally along the $\P^1$-bundle $f=g=0$. We now blow up the locus $f=g=0$ and call the exceptional divisor $F$; the total transform of $D$ is $$ D+2E_1+\ldots+(n-1)E_{(n-1)/2}+nE_{(n+1)/2}+2nF,$$ and $F$ is a $\P^1$-bundle over a $\P^1$-bundle. The covering of the generic $\P^1$-fibre of $F$ is ramified of degree $n$ over two points (corresponding to $F\cap D$ and $F\cap E_{(n+1)/2}$) with opposite characters, hence is again isomorphic to $\P^1$. \smallskip In both cases the fact that the divisors are smooth and transversal can be checked at each step out of the center of the next blowup. We now work in the neighborhood of a point where $h=0$. If $n$ is even, one can perform the same blowups as in the previous case and check that the same arguments work. If $n$ is odd, one can perform the first $(n-1)/2$ blowups as before. After them, the total transform of $D$ has equation $f^{n-1}(fh+tg)$. In particular (the strict transform of) $D$ is not smooth any more; we blow up its singular locus, and get a smooth exceptional divisor $\bar E$. The total transform of $D$ is $$ D+2E_1+\ldots+(n+1)\bar E$$ and is given (in local equations in the $h$ chart) by $$ f^{n-1}h^{n+1}(f+tg).$$ Let $\xi$ be a generator of $H$; the induced cover of $\bar E$ is cyclic with group $H/\<\xi^{n+1}\>$, hence it is totally ramified and therefore of Kodaira dimension $-\infty$, being a $\P^2$-bundle. We are not done because the divisors $D$ and $E_{(n-1)/2}$ are not transversal along $f=g=t=0$; but now we can apply the previous blowup procedure again. \ $\Box$\par\smallskip \begin{prop}\label{reslemma} Let ${\cal X}\to {\cal Y}\to \Delta$ be an abelian cover, branched over all smooth divisors except one, which has local equation $f^mh+tg$, where $f,t,g$ are coordinates and $m$ is the order of branching (where $t$ is the coordinate on $\Delta$). Then ${\cal X}$ and all its transforms via an $n$-th root base change admit a resolution of singularities such that the divisorial components of the exceptional divisor all have Kodaira dimension $-\infty$. \end{prop} \noindent{\sc Proof.~} The statement without the base change has already been proved; let $\tilde {\cal X}$ be such a resolution. By Hironaka's resolution of singularities (\cite{Hi}, p.~113, lines 8--4 from the bottom) we can assume that $\tilde{\cal X}_0$ is a normal crossing divisor. Let now $\rho_n:\Delta\to \Delta$ be the map $t\mapsto t^n$. There is a natural birational mapping $\rho_n^*\tilde {\cal X}\to \rho_n^*{\cal X}$; moreover $\rho_n^*\tilde {\cal X}$ is a cyclic cover of the manifold $\tilde {\cal X}$ ramified over $\tilde{\cal X}_0$, which has normal crossings, hence by remark \ref{resnc} $\rho_n^*\tilde {\cal X}$ has a resolution such that the divisorial components of the exceptional divisor all have Kodaira dimension $-\infty$. \ $\Box$\par\smallskip

9410

alg-geom/9410010

en

https://arxiv.org/abs/alg-geom/9410010

alg-geom/9410010

David B. Jaffe

Functorial structure of units in a tensor product

23 pages, AMS-LaTeX. Hard copy is available from the author. E-mail to [email protected]

We study the units in a tensor product of rings. For example, let k be an algebraically closed field. Let A and B be reduced rings containing k, having connected spectra. Let u \in A tensor_k B be a unit. Then u = a tensor_k b for some units a \in A and b \in B. Here is a deeper result, stated for simplicity in the affine case only. Let k be a field, and let f: R --> S be a homomorphism of f.g. k-algebras such that Spec(f) is dominant. Assume that every irreducible component of Spec(R_red) or Spec(S_red) is geometrically integral and has a rational point. Let B --> C be a faithfully flat homomorphism of reduced k-algebras. For A a k-algebra, define Q(A) to be (S tensor_k A)^*/(R tensor_k A)^*. Then Q satisfies the following sheaf property: the sequence 0 --> Q(B) --> Q(C) --> Q(C tensor_B C) is exact. This and another result are used in the proof of the following statement from "The kernel of the map on Picard groups induced by a faithfully flat homomorphism" by R. Guralnick, D. Jaffe, W. Raskind, R. Wiegand: Let K/k be an algebraic field extension and let A be a f.g. k-algebra. Assume resolution of singularities. Then there is a finite extension E/k contained in K/k such that Pic(A tensor_k E) --> Pic(A tensor_k K) is injective.

alg-geom

\section{#1}} \def\et#1{#1_{\hbox{\footnotesize\'et}}} \def\Ext_{\op fpqc}{\mathop{\operatoratfont Ext}\nolimits_{\operatoratfont fpqc}} \def\makeaddress{ \vskip 0.15in \par\noindent {\footnotesize Department of Mathematics and Statistics, University of Nebraska} \par\noindent {\footnotesize Lincoln, NE 68588-0323, USA\ \ (jaffe{\kern0.5pt}@{\kern0.5pt}cpthree.unl.edu)}} \def \def\arabic{footnote}}\setcounter{footnote}{0}{\fnsymbol{footnote}{ \def\arabic{footnote}}\setcounter{footnote}{0}{\fnsymbol{footnote}} \par\noindent David B. Jaffe\protect\footnote{Partially supported by the National Science Foundation.} \makeaddress\def\arabic{footnote}}\setcounter{footnote}{0}{\arabic{footnote}}\setcounter{footnote}{0}} \newenvironment{proof}{\trivlist \item[\hskip \labelsep{\sc Proof.\kern1pt}]}{\endtrivlist \newenvironment{proofnodot}{\trivlist \item[\hskip \labelsep{\sc Proof}]}{\endtrivlist \newenvironment{sketch}{\trivlist \item[\hskip \labelsep{\sc Sketch.\kern1pt}]}{\endtrivlist \newenvironment{alphalist}{\begin{list}{(\alph{alphactr})}{\usecounter{alphactr}}}{\end{list} \newenvironment{romanlist}{\begin{list}{(\roman{romanctr})}{\usecounter{romanctr}}}{\end{list} \newenvironment{definition}{\trivlist \item[\hskip \labelsep{\bf Definition.\kern1pt}]}{\endtrivlist} \newenvironment{remarks}{\trivlist \item[\hskip \labelsep{\bf Remarks.\kern1pt}]}{\endtrivlist} \newenvironment{problem}{\trivlist \item[\hskip \labelsep{\bf Problem.\kern1pt}]}{\endtrivlist} \def\arrow(#1,#2){\ncline[nodesep=5pt]{->}{#1}{#2}} \def\dottedarrow(#1,#2){\ncline[linestyle=dashed,nodesep=5pt]{->}{#1}{#2}} \def\vspec#1{\special{ps:#1}} \def\circno#1{{\bf [#1]}} \def\smallcat#1{\cat{\kern3pt\fontsize{8}{10pt}\selectfont #1\kern3pt}} \def\Morkschemes{\mathop{\operatoratfont Mor}\nolimits_{\cat{\kern3pt\fontsize{8}{10pt}\selectfont $k$-schemes\kern3pt}}} \hfuzz 3pt \documentclass[12pt]{article}\usepackage{amssymb} \newtheorem{theorem}{Theorem}[section] \setlength{\parindent}{9mm} \setcounter{tocdepth}{3} \newtheorem{fact}[theorem]{Fact \newtheorem{proposition}[theorem]{Proposition \newtheorem{lemma}[theorem]{Lemma \newtheorem{conjecture}[theorem]{Conjecture \newtheorem{cor}[theorem]{Corollary \newtheorem{corollary}[theorem]{Corollary \newtheorem{prop}[theorem]{Proposition \newtheorem{claim}[theorem]{Claim \newtheorem{exampleth}[theorem]{Example} \newenvironment{example}{\begin{exampleth}\fontshape{n}\selectfont}{\end{exampleth}} \begin{document} \vskip 0.15in \def\cat{(abelian group)-valued $k$-functors}{\cat{(abelian group)-valued $k$-functors}} \par\noindent{\Large\bf Functorial structure of units in a tensor product} \vspace{0.15in} \def\arabic{footnote}}\setcounter{footnote}{0}{\fnsymbol{footnote} \vspace{0.1in} \block{Introduction} Let $k$ be a field, and let $A$ be a finitely generated\ $k$-algebra.\footnote{All rings in this paper are commutative.} We explore the structure of the functor from \cat{$k$-algebras} to \cat{abelian groups} given by $B \mapsto (A \o*_k B)^*$. More generally, if $S$ is a $k$-scheme of finite type, not necessarily affine, we study the functor $\mu(S)$ given by $B \mapsto (\Gamma(S,{\cal O}_S) \o*_k B)^*$. This was done in (\Lcitemark 8\Rcitemark \ 4.5) for the case where $k$ is algebraically closed and $S$ is a variety. We make the assumption that every irreducible component of $\RED{S}$ is geometrically integral and has a rational point. We summarize these properties by saying that $S$ is {\it geometrically stable}. If $S$ is any $k$-scheme of finite type, we can always find a finite extension $k'$ of $k$ such that $S \times_k \mathop{\operatoratfont Spec}\nolimits(k')$ is geometrically stable as a $k'$-scheme. With the assumption that $S$ is geometrically stable, we find that $\mu(S)$ fits into an exact sequence \ses{{\Bbb G}_m^r \times U \times \xmode{\Bbb Z}^n}{\mu(S)}{I% }in which $I$ is a sheaf (for the fpqc topology), $I(B) = 0$ for every reduced $k$-algebra $B$, and $U$ admits a finite filtration with successive quotients isomorphic to ${\Bbb G}_a^{\kern1pt\beta}$, for various $\beta \in \xmode{\Bbb N} \cup \setof{\infty}$. We summarize these properties by saying that $I$ is {\it nilpotent\/} and $U$ is {\it additive}. In the sequence, $\xmode{\Bbb Z}^n$ denotes the constant sheaf associated to the abelian group $\xmode{\Bbb Z}^n$, or equivalently, the functor which represents the constant group scheme associated to the abelian group $\xmode{\Bbb Z}^n$. Moreover, suppose we have a dominant morphism \hbox{\mp[[ f || S || T ]],} in which both $S$ and $T$ are geometrically stable. There is an induced morphism of functors \mp[[ \mu(f) || \mu(T) || \mu(S) ]]. Let $Q = \mathop{\operatoratfont Coker}\nolimits[\mu(f)]$. We find that $Q$ also fits into an exact sequence as shown above, except that ${\Bbb G}_m^r \times U \times \xmode{\Bbb Z}^n$ is replaced by an extension of a finitely generated\ abelian group (i.e.\ the associated constant sheaf) by ${\Bbb G}_m^r \times U$, $U$ is pseudoadditive (see p.\ \pageref{pseudoadditive-def}), and we do not know if $I$ is a sheaf. Correspondingly, we do not know if $Q$ is a sheaf, but we do know at least that $Q|_{\smallcat{reduced $k$-algebras}}$ is a sheaf and moreover that the canonical map \mapx[[ Q || Q^+ ]] is a monomorphism. Specializing to the affine case, we see for example that if $A$ is a subalgebra of a $k$-algebra $C$ (and $\mathop{\operatoratfont Spec}\nolimits(A)$, $\mathop{\operatoratfont Spec}\nolimits(C)$ are geometrically stable), then the functor given by $B \mapsto (C \o* B)^*/(A \o* B)^*$ fits into such an exact sequence. We have thus far described the content of the first theorem \pref{tori-result-generalized} of this paper. Now we describe the second theorem \pref{kernel-pic-nilimmersion}, which is an application of the first. Let $X$ be a geometrically stable $k$-scheme. Let \hbox{\mp[[ i || X_0 || X ]]} be a nilimmersion, such that the ideal sheaf ${\cal{N}}$ of $X_0$ in $X$ has square zero. Let $P$ be the functor from \cat{$k$-algebras} to \cat{abelian groups} given by $$P(B) = \mathop{\operatoratfont Ker}\nolimits[ \mathop{\operatoratfont Pic}\nolimits(X \times_k \mathop{\operatoratfont Spec}\nolimits(B))\ \rightarrow \ \mathop{\operatoratfont Pic}\nolimits(X_0 \times_k \mathop{\operatoratfont Spec}\nolimits(B))].$% $Of course, if $X$ is affine, $P = 0$, but in general $P$ is not zero. We find that $P$ fits into an exact sequence \sescomma{D \o+ I}{U}{P% }in which $I$ is nilpotent (except possibly not a sheaf), $U$ is pseudoadditive, and $D$ is the constant sheaf associated to a finitely generated\ abelian group. Although this theorem does not imply that $P$ is a sheaf, it does imply that if \mp[[ f || B || C ]] is a faithfully flat homomorphism of reduced $k$-algebras, then $P(f)$ is injective \pref{sheaf-kernel-pic-nilimmersion}. In fact, this holds even if ${\cal{N}}^2 \not= 0$. We indicate the idea of the proof of the second theorem. We have an exact sequence \splitdiagram{H^0(X,{\cal O}_X^*)&\mapE{}&H^0(X_0,{\cal O}_{X_0}^*)% }{\mapE{}&H^1(X,{\cal{N}})&\mapE{}& \mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(X)\ \rightarrow\ \mathop{\operatoratfont Pic}\nolimits(X_0)]&\mapE{}&0.% }Functorializing this yields an exact sequence: \sescomma{\mathop{\operatoratfont Coker}\nolimits[\mu(i)]}{{\Bbb G}_a^{\kern1pt\beta}}{P% }in which $\beta = h^1(X,{\cal{N}})$. The first theorem tells us what $\mathop{\operatoratfont Coker}\nolimits[\mu(i)]$ is like. The second theorem is deduced from this. Finally, we describe a theorem about the Picard group, whose proof in \Lcitemark 7\Rcitemark \Rspace{} uses both theorems of this paper. Let $k$ be a field, and let $X$ be a separated $k$-scheme of finite type. Then there exists a finite field extension $k^+$ of $k$ such that for every algebraic extension $L$ of $k^+$, the canonical map \mapx[[ \mathop{\operatoratfont Pic}\nolimits(X_L) || \mathop{\operatoratfont Pic}\nolimits(X_{L^a}) ]] is injective. \vspace{0.1in} \par\noindent{{\bf Acknowledgements.}\ Bob Guralnick supplied the neat proof of \pref{unit-lemma-generalized}. Faltings kindly provided example \pref{Faltings}, thereby correcting an error. \vspace{0.1in} \par\noindent{\bf Conventions.} \begin{alphalist} \item A {\it $k$-functor\/} is a functor from \cat{$k$-algebras} to \cat{sets}. (The usage of the term {\it $k$-functor\/} here is slightly different from the usage in\Lspace \Lcitemark 8\Rcitemark \Rspace{}.) If $V$ is a $k$-scheme, then we also let $V$ denote the representable $k$-functor given by $V(B) = \Morkschemes(\mathop{\operatoratfont Spec}\nolimits(B),V)$. \item A $k$-functor $F$ is a {\it sheaf\/} (by which we mean {\it sheaf for the fpqc topology}) if for every faithfully flat homomorphism \mp[[ p || B || C ]], the canonical map\label{Psi-place} \dmap[[ \Psi_{F,p} || F(B) || \setof{x \in F(C): F(i_1)(x) = F(i_2)(x)} ]]% is bijective, where \mp[[ i_2, i_2 || C || C \o*_B C ]] are given by $c \mapsto c \o* 1$ and $c \mapsto 1 \o* c$, respectively. \item The superscript $+$ is used to denote {\it associated sheaf}. \item If $k$ is a field, $X$ is a $k$-scheme, and $L$ is a field extension of $k$, we let $X_L$ denote $X \times_k \mathop{\operatoratfont Spec}\nolimits(L)$. We let $k^a$ denote an algebraic closure of $k$. \item If $X$ is a scheme, we let $\Gamma(X)$ denote $\Gamma(X,{\cal O}_X)$, and we let $\Gamma^*(X)$ denote $\Gamma(X,{\cal O}_X)^*$. \item If $B$ is a ring, $\mathop{\operatoratfont Nil}\nolimits(B)$ denotes its nilradical. \item $k$-functors are said to be {\it (abelian group)-valued\/} if they take values in \cat{abelian groups} rather than \cat{sets}. \end{alphalist} We give some definitions which are adapted from \Lcitemark 8\Rcitemark \Rspace{}\ pp.\ 173, 180. If $k$ is a field and $X$, $Y$ are $k$-schemes, then $\mathop{\mathbf{Hom}}\nolimits(X,Y)$ denotes the $k$-functor given by $$B \mapsto \Morkschemes(X \times_k \mathop{\operatoratfont Spec}\nolimits(B), Y).$% $ An (abelian group)-valued $k$-functor $F$ is {\it nilpotent\/} if it is a sheaf and $F(B) = 0$ for every reduced $k$-algebra $B$. We say that $F$ is {\it subnilpotent\/} if it can be embedded as a subsheaf of a nilpotent $k$-functor. An (abelian group)-valued $k$-functor is {\it discrete\/} if it is a constant sheaf. We also say that such a functor is (for example) {\it discrete and finitely generated}, if it is the constant sheaf associated to a finitely generated\ abelian group. \block{Additive $k$-functors} Let $k$ be a field. We need to consider a countably-infinite-dimensional analog of unipotent group schemes over $k$. Actually, what we will be considering is more restrictive, as we shall only be considering the analog of unipotent group schemes over $k$ which are smooth, connected, and moreover which are $k$-solvable. (See\Lspace \Lcitemark 10\Rcitemark \Rspace{}\ \S5.1.) The simplest infinite dimensional example is the (abelian group)-valued $k$-functor ${\Bbb G}_a^\infty$ given by $B \mapsto \o+_{i=1}^\infty B$. However, this is not good enough for our purposes, since in positive characteristic one can have nontrivial extensions of ${\Bbb G}_a$ by ${\Bbb G}_a$. (See e.g.{\ }\Lcitemark 13\Rcitemark \ p.\ 67, exercise 8 or \Lcitemark 12\Rcitemark \Rspace{}\ VII\ \S2.) We want to define a class of objects which is closed under extension. \begin{definition} Let $k$ be a field, and let $F$ be an (abelian group)-valued $k$-functor. Then $F$ is {\it strictly additive\/} if it is isomorphic to ${\Bbb G}_a^\alpha$ for some $\alpha \in \setof{0,1,\ldots,\infty}$, and $F$ is {\it additive\/} if it admits a filtration: $$0\ =\ F_0\ \subset F_1\ \subset \cdots \subset F_n\ =\ F,$% $whose successive quotients are strictly additive. \end{definition} This terminology is not perfect, but it is at least consistent with the usage of the word {\it additive\/} in\Lspace \Lcitemark 8\Rcitemark \Rspace{}: by (\ref{ext-results}\ref{additive-additive-char0}), it will follow that if $k$ has characteristic zero, then additive $\Longrightarrow$ strictly additive. We define the {\it dimension\/} of an additive $k$-functor $F$ to be the sum of the dimensions of the successive quotients in a filtration of $F$, as in the definition of additive. Thus $\dim(F) \in \setof{0,1,\ldots,\infty}$. If $F$ is additive, we define its {\it period\/} to be the smallest $n$ for which there exists a filtration as in the definition of additive. A direct sum of countably many additive $k$-functors need not be additive, even if the summands are finite-dimensional. Also, we shall not concern ourselves with uncountable direct sums (e.g.\ of ${\Bbb G}_a$), as they seem not to arise in practice. The following two statements are easily checked: \begin{prop}\label{extension-of-additive} Let $k$ be a field. Let \ses{F'}{F}{F''% }be an exact sequence of (abelian group)-valued $k$-functors, in which $F'$ and $F''$ are additive. Then $F$ is additive. \end{prop} \begin{prop}\label{quotient-of-additive} Let $k$ be a field of characteristic zero. Let \ses{F'}{F}{F''% }be an exact sequence of (abelian group)-valued $k$-functors, in which $F'$, $F$ are additive. Then $F''$ is additive. \end{prop} \begin{prop}\label{lemma-two} Let $k$ be a field. Let \ses{F'}{F}{F''% }be an exact sequence of (abelian group)-valued $k$-functors, in which $F'$, $F$ are additive and finite-dimensional. Then $F''$ is additive. \end{prop} \begin{proof} By (\Lcitemark 3\Rcitemark \ 11.17), $(F'')^+$ is representable. Let \mp[[ p || F || (F'')^+ ]] be the canonical map, which is fpqc-surjective. By (\Lcitemark 11\Rcitemark \ Theorem 10), there exists a morphism \mp[[ \sigma || (F'')^+ || F ]] of $k$-functors such that $p \circ \sigma = 1_{(F'')^+}$. Hence $F'' = (F'')^+$. Let $X'$, $X$, and $X''$ be the group schemes which represent $F'$, $F$, and $F''$, respectively. Then we have an exact sequence \ses{X'}{X}{X''% }in \cat{commutative $k$-group schemes}. Since $F$ is additive and finite-dimensional, $X$ admits a series whose factors are copies of the group scheme ${\Bbb G}_a$. Hence $X''$ admits a series whose factors are group scheme quotients of ${\Bbb G}_a$. But any quotient of ${\Bbb G}_a$ is $0$ or ${\Bbb G}_a$ (\Lcitemark 10\Rcitemark \ 2.3), so $X''$ admits a series whose factors are the group scheme ${\Bbb G}_a$. From the argument at the beginning of the proof (showing that under certain circumstances fpqc-surjective $\Longrightarrow$ surjective), we see that $F''$ admits a series whose factors are the (abelian group)-valued $k$-functor ${\Bbb G}_a$. Hence $F''$ is additive. {\hfill$\square$} \end{proof} Unfortunately, \pref{quotient-of-additive} fails in positive characteristic. We will give an example of this, but there are a couple of preliminaries: \begin{lemma}\label{strictly-additive-splitting} If $F$ and $G$ are strictly additive and \mp[[ \pi || F || G ]] is an epimorphism in \cat{(abelian group)-valued $k$-functors}, then $\pi$ splits. \end{lemma} \begin{sketch} The lemma is clear if $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = 0$, so we may assume that\ $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = p > 0$. We do the case where $F = G = {\Bbb G}_a^\infty$; the proof in the other cases is the same. Let $e_i$ denote the element of ${\Bbb G}_a^\infty(k)$ which has $1$ in the \th{i} spot and $0$'s elsewhere. Let $A = k[t]$. For each $i$, choose $f_i \in {\Bbb G}_a^\infty(A)$ such that $\pi(f_i) = te_i$. Let $g_i$ be the part of $f_i$ involving only the monomials $t,t^p,t^{p^2},\ldots$. Since the monomials which appear in an expression for $\pi$ also have this form (with various $t$), it follows that $\pi(g_i) = te_i$. Regard $g_i$ as a function of $t$. Define \mp[[ \sigma || G || F ]] by $\sigma(be_i) = g_i(b)$, where $B$ is a $k$-algebra and $b \in B$. Then $\sigma$ splits $\pi$. {\hfill$\square$} \end{sketch} \begin{corollary}\label{strictly-additive-quotient} If $F$ is strictly additive and $G$ is additive, and \hbox{\mp[[ \pi || F || G ]]} is an epimorphism in \cat{(abelian group)-valued $k$-functors}, then $G$ is strictly additive. \end{corollary} \begin{proof} Induct on the period of $G$. If $\mathop{\operatoratfont period}\nolimits(G) \leq 1$ we are done. Otherwise, we can find $G' \subset G$ and an exact sequence \sesmaps{G'}{}{G}{p}{H% }in which $H$ is strictly additive, $G'$ is additive, and $\mathop{\operatoratfont period}\nolimits(G') < \mathop{\operatoratfont period}\nolimits(G)$. By \pref{strictly-additive-splitting}, $p \circ \pi$ splits. Hence $p$ splits. Hence $\mathop{\operatoratfont period}\nolimits(G) = \mathop{\operatoratfont period}\nolimits(G')$: contradiction. {\hfill$\square$} \end{proof} \begin{example}\label{Faltings} (provided by G.\ Faltings) \par\noindent Let \mp[[ f || {\Bbb G}_a^\infty || {\Bbb G}_a^\infty ]] be given by $(x_1,x_2,x_3,\ldots) \mapsto (x_1, x_2 - x_1^p, x_3 - x_2^p, \ldots)$. Then $f$ is a monomorphism. Let $F'' = \mathop{\operatoratfont Coker}\nolimits(f)$. If $F''$ were additive, then by \pref{strictly-additive-quotient} $F''$ would be strictly additive, and so by \pref{strictly-additive-splitting} $f$ would split. However, this is clearly not the case. Hence $F''$ is not additive. \end{example} The following generalization of {\it additive\/} allows us to work around the behavior illustrated by the example: \begin{definition}\label{pseudoadditive-def} An (abelian group)-valued $k$-functor $P$ is {\it pseudoadditive\/} if for some $n \in \xmode{\Bbb N}$ there exists an exact sequence \Rowseven{0}{U_1}{U_2}{\cdots}{U_n}{P}{0% }in \cat{(abelian group)-valued $k$-functors}\ in which $\vec U1n$ are additive. \end{definition} By example \pref{Faltings}, one cannot always take $n=1$, i.e.\ additive $\not=$ pseudoadditive. We do not know if one can always take $n=2$. \begin{prop}\label{pseudoadditive-is-sheaf} If $P$ is pseudoadditive, then $P$ is a sheaf. \end{prop} \begin{proof} Let ${\cal{C}}$ be the class of (abelian group)-valued $k$-functors $F$ with the property that for any faithfully flat homomorphism \mapx[[ B || C ]] of $k$-algebras, the usual \v Cech complex \sRowsix{0}{F(B)}{F(C)}{F(C \o*_B C)}{F(C \o*_B C \o*_B C)}{\cdots% }is exact. Then ${\Bbb G}_a^\alpha \in {\cal{C}}$ for all $\alpha$. If \ses{F'}{F}{F''% }is an exact sequence and any two of $F', F, F''$ are in ${\cal{C}}$, then so is the third. Hence $P$ is in ${\cal{C}}$, so $P$ is a sheaf. {\hfill$\square$} \end{proof} \block{Extensions in \cat{(abelian group)-valued $k$-functors}} For any objects $F_1, F_2$ in an abelian category, one can define an abelian group $\mathop{\operatoratfont Ext}\nolimits^1(F_1, F_2)$, whose elements are isomorphism classes of extensions \sesdot{F_2}{F}{F_1% }(The general theory is described in \Lcitemark 9\Rcitemark \Rspace{}\ Ch.\ VII, among other places.) Also, we will refer to such an exact sequence as defining an {\it extension of $F_1$ by $F_2$}. In particular, the theory applies to \cat{(abelian group)-valued $k$-functors}. We shall say that an exact sequence \sesmaps{F_2}{}{F}{\pi}{F_1% }in this category is {\it set-theoretically split\/} if there exists a morphism of $k$-functors \mp[[ \sigma || F_1 || F ]] such that $\pi \circ \sigma = 1_{F_1}$. We also refer to {\it set-theoretically split extensions}. Let $\mathop{\operatoratfont Ext}\nolimits^1_s(F_1,F_2)$ denote the subgroup of elements of $\mathop{\operatoratfont Ext}\nolimits^1(F_1,F_2)$ which correspond to set-theoretically split extensions. To compute $\mathop{\operatoratfont Ext}\nolimits^1_s(F_1,F_2)$, we copy (with appropriate but minor changes) some definitions which may be found in (\Lcitemark 12\Rcitemark \ VII\ \S4). For this discussion, fix $F_1$ and $F_2$. A {\it symmetric factor system\/} is a morphism \mp[[ f || F_1 \times F_1 || F_2 ]] of $k$-functors such that \begin{eqnarray*} 0 & = & f(y,z) - f(x+y,z) + f(x,y+z) - f(x,y)\\ f(x,y) & = & f(y,x) \end{eqnarray*} for all $k$-algebras $B$ and all $x,y,z \in F_1(B)$. If \mp[[ g || F_1 || F_2 ]] is a morphism of $k$-functors, then there is a symmetric factor system $\delta g$ defined by $$\delta g(x,y) = g(x+y) - g(x) - g(y);$% $such a system is called {\it trivial}. The group structure on $F_2$ makes the set of symmetric factor systems into a group. Then by standard arguments, $\mathop{\operatoratfont Ext}\nolimits^1_s(F_1,F_2)$ is isomorphic to the group of symmetric factor systems, modulo the subgroup of trivial factor systems. If $F_1$ and $F_2$ are sheaves, then one can also compute the group $\mathop{\operatoratfont Ext}\nolimits^1_{\operatoratfont fpqc}(F_1,F_2)$, i.e.\ the group of isomorphism classes of extensions of $F_1$ by $F_2$ in \cat{(abelian group)-valued $k$-functors which are sheaves}. If moreover $F_1$ and $F_2$ are represented by commutative group schemes $X_1$ and $X_2$ of finite type over $k$, then [see\Lspace \Lcitemark 4\Rcitemark \Rspace{}\ 5.4 and \Lcitemark 2\Rcitemark \Rspace{}\ 3.5, 7.3(ii)] $\mathop{\operatoratfont Ext}\nolimits^1_{\operatoratfont fpqc}(F_1,F_2) = \mathop{\operatoratfont Ext}\nolimits^1(X_1,X_2)$, where the latter $\mathop{\operatoratfont Ext}\nolimits$ group is computed relative to the abelian category \cat{commutative group schemes of finite type over $k$}. For arbitrary sheaves $F_1$, $F_2$, we have $\mathop{\operatoratfont Ext}\nolimits^1(F_1,F_2) \subset \mathop{\operatoratfont Ext}\nolimits^1_{\operatoratfont fpqc}(F_1,F_2)$, but not equality in general, as may be seen e.g.\ from the exact sequence \sesmaps{\xmode{\Bbb Z}/p\xmode{\Bbb Z}}{}{{\Bbb G}_a}{t\ \mapsto\ t^p - t}{{\Bbb G}_a% }in the group scheme category, where $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = p > 0$. \begin{prop}\label{ext-results} Let $k$ be a field. We consider objects and morphisms in \cat{(abelian group)-valued $k$-functors}. Then: \begin{alphalist} \item If \mapx[[ W || G ]] is an epimorphism, and $V$ is additive, then the induced map \mapx[[ \mathop{\operatoratfont Mor}\nolimits_{\smallcat{$k$-functors}}(V,W) || \mathop{\operatoratfont Mor}\nolimits_{\smallcat{$k$-functors}}(V,G) ]] is surjective. \item\label{additive-any} If $V$ is additive, then $\mathop{\operatoratfont Ext}\nolimits^1_s(V,F) = \mathop{\operatoratfont Ext}\nolimits^1(V,F)$ for all $F$. \item\label{Z-sheaf} If $F$ is a sheaf then $\mathop{\operatoratfont Ext}\nolimits^1(\xmode{\Bbb Z},F) = 0$. \item\label{Gm-Ga} If we have an exact sequence \ses{U_1}{U_2}{P% }in which $U_1$ and $U_2$ are additive, then $\mathop{\operatoratfont Ext}\nolimits^1({\Bbb G}_m, P) = 0$. \item\label{nilpotent-discrete} $\mathop{\operatoratfont Ext}\nolimits^1(I,D) = 0$ if $I$ is subnilpotent and $D$ is discrete. \item\label{additive-additive-char0} $\mathop{\operatoratfont Ext}\nolimits^1(U,V) = 0$ if $U$ and $V$ are additive and $k$ has characteristic zero. \end{alphalist} \end{prop} \begin{proof} {\bf (a):\ } First we prove this when $V = {\Bbb G}_a^\alpha$ for some $\alpha$. If $\alpha < \infty$, the claim is immediate. Otherwise, the essential point is that in a commutative diagram we can (exercise) fill in a dotted arrow as shown. Now suppose that $V$ is arbitrary. From what we have just shown, it follows that $\mathop{\operatoratfont Ext}\nolimits^1_s({\Bbb G}_a^\alpha,F) = \mathop{\operatoratfont Ext}\nolimits^1({\Bbb G}_a^\alpha,F)$ for all $F$. In turn, this implies that $V \cong {\Bbb G}_a^{\kern1pt\beta}$ in \cat{$k$-functors}, for some $\beta$. Hence (a) holds when $V$ is arbitrary. \vspace{0.1in} \par\noindent{\bf (b):\ } follows immediately from (a). \vspace{0.1in} \par\noindent{\bf (c):\ } We have to show that if \mp[[ \pi || H || \xmode{\Bbb Z} ]] is an epimorphism in \cat{(abelian group)-valued $k$-functors}, and $H$ is a sheaf, then $\pi$ splits. For each $n \in \xmode{\Bbb Z}$, let $y_n \in \xmode{\Bbb Z}(k)$ correspond to the constant map \mapx[[ \mathop{\operatoratfont Spec}\nolimits(k) || \xmode{\Bbb Z} ]] of topological spaces with value $n$, and choose $x_1 \in H(k)$ such that $\pi(x_1) = y_1$. For each $n \in \xmode{\Bbb Z}$, define $x_n \in H(k)$ to be $n x_1$. Define \mp[[ \sigma || \xmode{\Bbb Z} || H ]] as follows. For any ring $B$, an element $\lambda \in \xmode{\Bbb Z}(B)$ corresponds to a locally constant map \mapx[[ \mathop{\operatoratfont Spec}\nolimits(B) || \xmode{\Bbb Z} ]] of topological spaces, and therefore we may write $B = B_1 \times \cdots \times B_n$ in such a way that $\lambda$ is induced by $(y_{r_1},\ldots,y_{r_n})$ for suitable $\vec r1n \in \xmode{\Bbb Z}$. Since $H$ is a sheaf for the Zariski topology, there is a unique element $x_{r_1,\ldots,r_n} \in H(k^n)$ whose image in $H(k)$ under the \th{i} projection map is $x_{r_i}$. Now set $\sigma(\lambda)$ equal to the image of $x_{r_1,\ldots,r_n}$ under the canonical map \mapx[[ H(k^n) || H(B_1 \times \cdots \times B_n) ]]. This defines $\sigma$, and thus proves that $\pi$ splits. \vspace{0.1in} \par\noindent{\bf (d):\ } First suppose that $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = 0$. Then $P \cong {\Bbb G}_a^\alpha$ for some $\alpha$. If $\alpha < \infty$, the statement follows from [\Lcitemark 10\Rcitemark \ 5.1.1(i)]. We have $\mathop{\operatoratfont Ext}\nolimits^1({\Bbb G}_m,{\Bbb G}_a^\infty) = \mathop{\operatoratfont Ext}\nolimits^1_s({\Bbb G}_m,{\Bbb G}_a^\infty)$. Therefore an extension of ${\Bbb G}_m$ by ${\Bbb G}_a^\infty$ corresponds to a symmetric factor system \mp[[ f || {\Bbb G}_m \times {\Bbb G}_m || {\Bbb G}_a^\infty ]]. For $n \gg 0$, we can find a morphism \mp[[ f_n || {\Bbb G}_m \times {\Bbb G}_m || {\Bbb G}_a^n ]] through which $f$ factors. But then $f_n$ is a symmetric factor system, and so $f_n$ is trivial, since we already know that $\mathop{\operatoratfont Ext}\nolimits^1({\Bbb G}_m,{\Bbb G}_a^n) = 0$. Hence $f$ is trivial. Hence $\mathop{\operatoratfont Ext}\nolimits^1({\Bbb G}_m,{\Bbb G}_a^\infty) = 0$. Now suppose that $k$ has characteristic $p > 0$. Since ${\Bbb G}_m$ and $P$ are sheaves, it suffices to show that $\Ext_{\op fpqc}^1({\Bbb G}_m,P) = 0$. For $n$ sufficiently large, multiplication by $p^n$ gives a zero map from $P$ to $P$. It follows from the fpqc-exact sequence \sesmapsone{\mu_{p^n}}{}{{\Bbb G}_m}{{p^n}}{{\Bbb G}_m% }that it is enough to show $\mathop{\operatoratfont Hom}\nolimits(\mu_{p^n}, P) = 0$. Let \mp[[ f || \mu_{p^n} || P ]] be a morphism. Let $H$ be the fiber product of $\mu_{p^n}$ and $U_2$ over $P$. Then we have an exact sequence: \diagramno{(*)}{\rowfive{0}{U_1}{H}{\mu_{p^n}}{1.}% }We will show that this sequence splits. We can do this by showing that $\mathop{\operatoratfont Ext}\nolimits^1(\mu_{p^n},U_1) = 0$, but by the definition of additive, it is clearly enough to show that $\mathop{\operatoratfont Ext}\nolimits^1(\mu_{p^n}, {\Bbb G}_a^\alpha) = 0$ for all $\alpha$. Arguing as in the characteristic zero case, one sees further that it is further enough to show that $\mathop{\operatoratfont Ext}\nolimits^1(\mu_{p^n}, {\Bbb G}_a) = 0$. This is a special case of \Lcitemark 10\Rcitemark \Rspace{}\ 5.1.1(d). Hence $(*)$ splits. Hence there exists a morphism \mp[[ \sigma || \mu_{p^n} || U_2 ]] such that $\pi \circ \sigma = f$, where \mp[[ \pi || U_2 || P ]] is the given map. Now I claim that $\sigma = 0$. For this (arguing as above), it is enough to show that $\mathop{\operatoratfont Hom}\nolimits(\mu_{p^n}, {\Bbb G}_a) = 0$. It is enough to do this when $k = k^a$, and then the statement is well-known. Hence $f = 0$. Hence $\mathop{\operatoratfont Hom}\nolimits(\mu_{p^n}, P) = 0$, which completes the proof. \vspace{0.1in} \par\noindent{\bf (e):\ } Let \ses{D}{L}{I% }be an exact sequence of (abelian group)-valued $k$-functors. Define a $k$-functor $I'$ by $I'(B) = \mathop{\operatoratfont Ker}\nolimits[L(B)\ \rightarrow\ L(\RED{B})]$. Then $I'$ defines a splitting of the sequence. \vspace{0.1in} \par\noindent{\bf (f):\ } It suffices to show that $\mathop{\operatoratfont Ext}\nolimits^1({\Bbb G}_a^\alpha, {\Bbb G}_a^{\kern1pt\beta}) = 0$ for all $\alpha, \beta$. Moreover, since $\mathop{\operatoratfont Ext}\nolimits^1$ converts a coproduct in the first variable into a product, we may assume that\ $\alpha = 1$. By (b), it suffices to show that $\mathop{\operatoratfont Ext}\nolimits^1_s({\Bbb G}_a, {\Bbb G}_a^{\kern1pt\beta}) = 0$. Arguing as in (d), it suffices to show that $\mathop{\operatoratfont Ext}\nolimits^1_s({\Bbb G}_a,{\Bbb G}_a^n) = 0$, and moreover we may as well take $n=1$. Suppose we have an exact sequence \sesdot{{\Bbb G}_a}{X}{{\Bbb G}_a% }By (\Lcitemark 10\Rcitemark \ 3.9 ter.), $X \cong {\Bbb G}_a^2$. But (in characteristic zero) morphisms from ${\Bbb G}_a^n$ to ${\Bbb G}_a^m$ are in bijective correspondence with vector space homomorphisms from $k^n$ to $k^m$, so the sequence splits. {\hfill$\square$} \end{proof} \block{Functorial structure of units in a tensor product} The main purpose of this section is to prove \pref{tori-result-generalized}, which generalizes (\Lcitemark 8\Rcitemark \ 4.5). The preparatory lemmas are similar to those in (\Lcitemark 8\Rcitemark \ \S4), and we shall omit their proofs if the proofs of the corresponding statements in\Lspace \Lcitemark 8\Rcitemark \Rspace{} carry over with minor changes. \begin{lemma}\label{unit-lemma-generalized} Let $k$ be an algebraically closed field. Let $A$ and $B$ be reduced rings containing $k$, having connected spectra. Let $u \in A \o*_k B$ be a unit. Then $u = a \o* b$ for some units $a \in A$ and $b \in B$. \end{lemma} \begin{proof} The statement generalizes (\Lcitemark 8\Rcitemark \ 4.2), but we give a new and simpler proof, due to Guralnick. Let $X$ be the set of maximal ideals of $A$, and let $Y$ be the set of maximal ideals of $B$. Let $x_0 \in X$, $y_0 \in Y$. We will prove the lemma by showing that for all $x \in X$, $y \in Y$, we have: $$u(x,y)\ =\ {u(x,y_0) u(x_0,y) \over u(x_0, y_0)}.\eqno(*)$% $For this we may suppose that $k$ is uncountable. By a variant of a result of Roquette (see \Lcitemark 7\Rcitemark \Rspace{}\ 1.5) the group $B^*/k^*$ is finitely generated, so $F = \setof{f \in B^*: f(y_0) = 1}$ is countable. For each $f \in F$, let $$Q(f)\ =\ \setof{x \in X: u(x,y) = u(x,y_0)f(y)\hbox{\ for all\ } y \in Y}.$% $Then $Q(f)$ is a closed subset of $X$. For any given $x \in X$, the function on $Y$ given by $y \mapsto u(x,y)/u(x,y_0)$ sends $y_0$ to $1$ and so lies in $F$. Hence $X = \cup_{f \in F}Q(f)$. Since $k$ is uncountable, it follows that if $I$ is an irreducible component of $X$, then $I \subset Q(f)$ for some $f \in F$. Hence for any fixed $y \in Y$, the function \mp[[ g_y || X || k ]] given by $x \mapsto u(x,y)/u(x,y_0)$ is constant on each irreducible component of $X$. Since $X$ is connected, $g_y$ is constant. Then $u(x,y) = u(x,y_0)g_y(x_0)$, which proves $(*)$. {\hfill$\square$} \end{proof} \begin{corollary}\label{second-generalized} Let $k$ be an algebraically closed field. Let $A$ and $B$ be rings containing $k$, having connected spectra. Assume that $A$ is reduced. Let ${\xmode{{\fraktur{\lowercase{M}}}}} \subset A$ be a maximal ideal such that $A/{\xmode{{\fraktur{\lowercase{M}}}}} = k$. Then $(A \o*_k B)^*$ is the direct sum of the two subgroups $A^*B^*$ and $1 + ({\xmode{{\fraktur{\lowercase{M}}}}} \o* \mathop{\operatoratfont Nil}\nolimits(B))$% .\footnote{Statements 4.3 and 4.4 from\Lspace \Lcitemark 8\Rcitemark \Rspace{} should also have the hypothesis that $A/{\xmode{{\fraktur{\lowercase{M}}}}} = k$.} \end{corollary} \begin{corollary}\label{third-generalized} Let $k$ be an algebraically closed field. Let $A$ and $B$ be rings containing $k$. Assume that $A$ is reduced and has a connected spectrum. Let ${\xmode{{\fraktur{\lowercase{M}}}}} \subset A$ be a maximal ideal such that $A/{\xmode{{\fraktur{\lowercase{M}}}}} = k$. \begin{itemize} \item For any decomposition $B = B_1 \times \cdots \times B_n$, there is a subgroup: $$\mu(\vec B1n) = \oplus_{i=1}^n [A^*B_i^* \o+ (1 + ({\xmode{{\fraktur{\lowercase{M}}}}} \o* \mathop{\operatoratfont Nil}\nolimits(B_i))]$% $of $(A \o*_k B)^*$. \item For any $x \in (A \o*_k B)^*$, there exists a decomposition $B = B_1 \times \cdots \times B_n$ such that $x \in \mu(\vec B1n)$. \item If $B$ has only finitely many idempotent elements (e.g.\ if $B$ is noetherian), we can write $B = B_1 \times \cdots \times B_n$ for rings $B_i$ having connected spectra. Then $\mu(\vec B1n) = (A \o*_k B)^*$. \end{itemize} \end{corollary} Let $k$ be a field, and let $S$ be a $k$-scheme of finite type. Let $F = \mathop{\mathbf{Hom}}\nolimits(S,{\Bbb G}_m)$. Then $$F(B)\ =\ \Gamma(S \times \mathop{\operatoratfont Spec}\nolimits(B), {\cal O}^*_{S \times \mathop{\operatoratfont Spec}\nolimits(B)}) \ =\ [\Gamma(S,{\cal O}_S) \o*_k B]^*,$% $by (\Lcitemark 6\Rcitemark \ 9.3.13 (i)). In particular, if $S = \mathop{\operatoratfont Spec}\nolimits(A)$, then $F(B) = (A \o*_k B)^*$. The next theorem gives an abstract description of $F$, and thus (in effect) a description of how units in a tensor product $A \o*_k B$ vary as $B$ varies. First, for convenience, we encapsulate the following definition: \begin{definition} Let $k$ be a field. A $k$-scheme $S$ is {\it geometrically stable\/} if $(1)$ it is of finite type, and $(2)$ every irreducible component of $\RED{S}$ is geometrically integral and has a rational point. \end{definition} \begin{theorem}\label{tori-result-generalized} Let $k$ be a field. Define an (abelian group)-valued $k$-functor $F$ to be of type $(*)$ if there exist exact sequences \seslabcomma{R}{F}{I}{\dag% }\seslab{{\Bbb G}_m^r \times U}{R}{L}{\dag\dag% }in \cat{(abelian group)-valued $k$-functors}, in which $r \geq 0$, $U$ is pseudoadditive, $I$ is subnilpotent, and $L$ is discrete and finitely generated. \par\noindent{\bf\rm (a):\ } Let $S$ be a geometrically stable $k$-scheme. Then $\mathop{\mathbf{Hom}}\nolimits(S,{\Bbb G}_m)$ is of type $(*)$ and we have $r =$ the number of connected components of $S$. Also $U$ is additive of dimension $\dim_k \mathop{\operatoratfont Nil}\nolimits[\Gamma(S,{\cal O}_S)]$. Also, $I$ is nilpotent, $L$ is free, and $(\dag\dag)$ splits. \par\noindent{\bf\rm (b):\ } Let $S$ and $T$ be geometrically stable $k$-schemes, and let \mp[[ f || S || T ]] be a dominant morphism of $k$-schemes. Then the cokernel of $\mathop{\mathbf{Hom}}\nolimits(f,{\Bbb G}_m)$ is of type $(*)$. Moreover, $r$ equals the number of connected components of $S$ minus the number of connected components of $T$. \end{theorem} \begin{corollary}\label{tori-quotient-sheaf} Let $S$ and $T$ be geometrically stable $k$-schemes, and let \mp[[ f || S || T ]] be a dominant morphism of $k$-schemes. Let $Q$ be the cokernel of $\mathop{\mathbf{Hom}}\nolimits(f,{\Bbb G}_m)$. Then the canonical map \mapx[[ Q || Q^+ ]] is a monomorphism, and $Q|_{\smallcat{reduced $k$-algebras}}$ is a sheaf, in the sense that if \mp[[ p || B || C ]] is a faithfully flat homomorphism of reduced $k$-algebras, then $\psi_{Q,p}$ (see p.\ \pageref{Psi-place}) is bijective. \end{corollary} \begin{remarks} \ \begin{romanlist} \item In part (a) of the theorem, one can choose $(\dag)$ so that it splits if $S$ is reduced, but probably not in general. \item In part (b), the sequence $(\dag\dag)$ does not always split. For an example, take $k$ to be an imperfect field of characteristic $p$, let $u \in k - k^p$, and let $f$ be $\mathop{\operatoratfont Spec}\nolimits$ of the ring map \mapx[[ k[t,t^{-1}] || k[x,x^{-1}] \times k ]], given by $t \mapsto (x^p,u)$. \item The hypothesis that the schemes in the theorem be geometrically stable can be weakened slightly, as is indicated in the proof. They presumably can be weakened further, but we do not know what is possible in this direction. \item\label{tori-result-remark-three} We suspect that $I$ in part (b) of the theorem is a sheaf (and thus satisfies the definition of {\it nilpotent}). If true, this would imply (in the corollary) that the cokernel of $\mathop{\mathbf{Hom}}\nolimits(f,{\Bbb G}_m)$ is a sheaf. To prove that $I$ is a sheaf, it would be sufficient (at least in the case where $S$ and $T$ are connected) to show that if $C$ is a subalgebra of a reduced $k$-algebra $A$, then the (abelian group)-valued $k$-functor given by $$B\ \mapsto\ {1 + A \o* \mathop{\operatoratfont Nil}\nolimits(B) \over 1 + C \o* \mathop{\operatoratfont Nil}\nolimits(B)}$% $ is a sheaf. \item In part (b), we have $U$ pseudoadditive with $n=2$, as in the definition of pseudoadditive. However, it is conceivable that $U$ is always additive. \end{romanlist} \end{remarks} \begin{proofnodot} (of \ref{tori-result-generalized}.) The hypothesis that $S$ be geometrically stable is chosen for simplicity and we note here some consequences which are in fact sufficient to prove the theorem: \vspace{0.05in} \par\circno{A}:\ every connected component of $S$ has a rational point; \vspace{0.05in} \par\circno{B}:\ $\RED{S}$ is geometrically reduced [by\Lspace \Lcitemark 5\Rcitemark \Rspace{}\ 4.6.1(e)]. \vspace{0.05in} \par\noindent Moreover, \circno{A}\ and \circno{B}\ also imply: \vspace{0.05in} \par\circno{C}:\ every connected component of $S$ is geometrically connected [by\Lspace \Lcitemark 5\Rcitemark \Rspace{}\ 4.5.14]; \vspace{0.05in} \par\circno{D}:\ if $Q$ is a connected component of $S$, then $k$ is integrally closed in $\Gamma(\RED{Q})$. \par\noindent All of these comments apply equally to $T$. Now we want to reduce to the affine case. This is not literally possible, because $\Gamma(S)$ need not be finitely generated\ as a $k$-algebra. What we can do is reformulate the theorem in terms of a certain class of $k$-algebras. This class is chosen simply to serve the needs of the proof: a $k$-algebra is {\it good\/} if it is of the form $\Gamma(S)/N$, where $S$ is a geometrically stable $k$-scheme and $N \subset \Gamma(S)$ is a nilpotent ideal. Here is a reformulation of the theorem in terms of good $k$-algebras: \begin{quote} \par\noindent{\bf\rm (a):\ } Let $A$ be a good $k$-algebra. Then $B \mapsto (A \o* B)^*$ is of type $(*)$ and we have $r =$ the number of connected components of $\mathop{\operatoratfont Spec}\nolimits(A)$, $\dim(U) = \dim_k \mathop{\operatoratfont Nil}\nolimits(A)$. Also, $L$ is free and $I$ is nilpotent. \par\noindent{\bf\rm (b):\ } Let \mp[[ \phi || C || A ]] be a homomorphism of good $k$-algebras. Assume that $\mathop{\operatoratfont Ker}\nolimits(\phi)$ is nilpotent. Then the cokernel of the morphism from $B \mapsto (C \o* B)^*$ to $B \mapsto (A \o* B)^*$ is of type $(*)$. Moreover, $r$ equals the number of connected components of $\mathop{\operatoratfont Spec}\nolimits(A)$ minus the number of connected components of $\mathop{\operatoratfont Spec}\nolimits(C)$. \end{quote} Since the map \mapx[[ (C \o* B)^* || (C/\mathop{\operatoratfont Ker}\nolimits(\phi) \o* B)^* ]] is surjective, we may reduce to the case where $\phi$ is {\it injective}. It was the need for this reduction which lead to the introduction of good $k$-algebras in the proof. We proceed to build a diagram involving (abelian group)-valued $k$-functors, which we associate to $A$, and which is functorial in $A$. Let $G_A$ be given by $G_A(B) = (\RED{A} \o* B)^*$. Write $\RED{A} = A_1 \times \cdots \times A_r$, where $\vec A1r$ have connected spectra. We can identify $A \o* B$ with $(A_1 \o* B) \times \cdots \times (A_r \o* B)$. Let $F_A$ be the sheaf associated to the subfunctor of $G_A$ given by $B \mapsto \setof{(a_1 \o* b_1, \ldots, a_r \o* b_r): a_i \in A_i^*, b_i \in B^*}$. Let $E_A$ be the subfunctor of $F_A$ given by $E_A(B) = \setof{(\vec b1r): \vec b1r \in B^*}$. Let $D_A = F_A/E_A$. Let $I_A = G_A/F_A$. Define $H_A$ by $H_A(B) = (A \o* B)^*$. Let \mp[[ p || H_A || G_A ]] be the canonical map, and let $U_A$ be its kernel. We have $U_A(B) = 1 + \mathop{\operatoratfont Nil}\nolimits(A) \o* B$. For each $n \in \xmode{\Bbb N}$, let $U_A^n$ be given by $U_A^n(B) = 1 + \mathop{\operatoratfont Nil}\nolimits(A)^n \o* B$. Here is the diagram of (abelian group)-valued $k$-functors which we have built: \diagramx{&&&&0\cr &&&&\mapS{}\cr && 0 &&U_A & = \kern10pt \hbox to 0pt{$U_A^1 \kern10pt \supset \kern10pt U_A^2 \kern10pt \supset \kern10pt \cdots$}\cr && \mapS{} && \mapS{}\cr && E_A && H_A \cr && \mapS{} && \mapS{}\cr \rowfive{0}{F_A}{G_A}{I_A}{0}\cr && \mapS{} && \mapS{} && \vbox to 0pt{\box5}\cr && D_A && 0\cr && \mapS{}\cr && 0} We proceed to analyze the various components of this diagram. In particular, we will show that \circno1\ $G_A \cong F_A \times I_A$ and $I_A$ is nilpotent, \circno2\ $E_A \cong {\Bbb G}_m^r$, where $r$ is the number of connected components of $\mathop{\operatoratfont Spec}\nolimits(A)$, \circno3\ $D_A$ is represented by a constant group scheme (corresponding to a finitely generated\ free abelian group), and that \circno4\ $U_A$ is additive. For these purposes, we may assume that\ $\mathop{\operatoratfont Spec}\nolimits(A)$ is connected. Then $E_A(B) = B^*$. \vspace{0.05in} \par\noindent\circno1\ Let ${\xmode{{\fraktur{\lowercase{M}}}}}_A \subset \RED{A}$ be a maximal ideal such that $A/{\xmode{{\fraktur{\lowercase{M}}}}}_A = k$. (This is possible by \circno{A}.) Define a subfunctor $I'_A$ of $G_A$ by $I'_A(B) = 1 + {\xmode{{\fraktur{\lowercase{M}}}}}_A \o* \mathop{\operatoratfont Nil}\nolimits(B)$. Let \mp[[ \psi || F_A \o+ I'_A || G_A ]] be the canonical map. We will show that $\psi$ is an isomorphism. If $k$ is algebraically closed, this follows from \pref{third-generalized}. But both the source and the target of $\psi$ are sheaves, so it follows (using \circno{B}\ and \circno{C}) that $\psi$ is an isomorphism for any $k$. Thus $I'_A \cong I_A$, so $I_A$ is nilpotent. \vspace{0.05in} \par\noindent\circno2\ We have $E_A \cong {\Bbb G}_m$. \vspace{0.05in} \par\noindent\circno3\ Write $A = \Gamma(S)/N$, as in the definition of good. Let ${\cal{N}}$ be the nilradical of $S$. Then we have an exact sequence: \les{H^0(S,{\cal{N}})}{H^0(S,{\cal O}_S)}{H^0(\RED{S},{\cal O}_{\RED{S}})% }and so $\RED{A}$ is a subring of $H^0(\RED{S},{\cal O}_{\RED{S}})$. Since $k$ is integrally closed in $H^0(\RED{S},{\cal O}_{\RED{S}})$ by \circno{D}, it follows by (\Lcitemark 7\Rcitemark \ 1.5),that $D_A(k)$ is free abelian of finite rank. In fact, $D_A$ is represented by the corresponding constant group scheme. \vspace{0.05in} \par\noindent\circno4\ For each $n$ we have $${U_A^n \over U_A^{n+1}}(B) \ =\ {1 + \mathop{\operatoratfont Nil}\nolimits(A)^n \o* B \over 1 + \mathop{\operatoratfont Nil}\nolimits(A)^{n+1} \o* B} \ \cong\ {\mathop{\operatoratfont Nil}\nolimits(A)^n \over \mathop{\operatoratfont Nil}\nolimits(A)^{n+1}} \o* B$% $as (abelian group)-valued $k$-functors. Hence $U_A$ is additive. Now we describe $H_A$, making a number of non-canonical choices. We have $F_A \cong E_A \times D_A$ non-canonically, e.g.\ by (\ref{ext-results}\ref{Z-sheaf}), but it is easily proved directly. Hence $F_A \cong {\Bbb G}_m^r \times \xmode{\Bbb Z}^n$ for some $n$. Also, we have shown that $G_A \cong F_A \times I_A$. Therefore we have an exact sequence \sesmapsdot{U_A}{}{H_A}{q}{{\Bbb G}_m^r \times \xmode{\Bbb Z}^n \times I_A% }Let $M = q^{-1}({\Bbb G}_m^r \times \xmode{\Bbb Z}^n)$. Then we have an exact sequence \sesdot{U_A}{M}{{\Bbb G}_m^r \times \xmode{\Bbb Z}^n% }By (\ref{ext-results}\ref{Z-sheaf}\ref{Gm-Ga}) this sequence splits. This proves (a). Now, to prepare for proving (b), we analyze the functorial behavior of each basic component of the big diagram shown above. Let \mp[[ \phi || C || A ]] be an injective homomorphism of good $k$-algebras. First we analyze $E_\phi$. It is a monomorphism, corresponding to a map \mapx[[ {\Bbb G}_m^{r_1} || {\Bbb G}_m^{r_2} ]], for some $r_1$ and $r_2$, which is given by an $r_2 \times r_1$ matrix of $0$'s and $1$'s. The cokernel of $E_\phi$ is isomorphic to ${\Bbb G}_m^{r_2 - r_1}$. Now we analyze $D_\phi$. Let us show that $D_\phi$ is a monomorphism. Since its source and target are constant sheaves, it suffices to show that $D_\phi(k^a)$ is injective. The assertion then boils down to showing that if one has a dominant morphism \mp[[ \psi || V || W ]] of reduced schemes of finite type over an algebraically closed field $k$, and $W$ is connected, and \mp[[ g || W(k) || k ]] is a non-constant regular function, then $g \circ \psi(k)$ is not constant on each connected component of $V$. This is clear, so $D_\phi$ is a monomorphism. The cokernel of $D_\phi$ is the constant sheaf associated to a finitely generated\ abelian group. We show that $I_\phi$ is a monomorphism. In the process of doing so, we justify remark \pref{tori-result-remark-three} from p.\ \pageref{tori-result-remark-three}. Also, once we know that $I_\phi$ is a monomorphism, it will follow immediately that $\mathop{\operatoratfont Coker}\nolimits(I_\phi)$ is subnilpotent. We may assume that $\mathop{\operatoratfont Spec}\nolimits(C)$ is connected. We have a canonical map \mapx[[ 1 + \RED{C} \o* \mathop{\operatoratfont Nil}\nolimits(B) || I_C(B) ]], and likewise for $A$. From our discussion of $I'$, it is clear that these maps are surjective. Letting $X_C(B)$ and $X_A(B)$ denote their kernels, we have a commutative diagram with exact rows: \diagramx{\sesonerow{X_C(B)}{1 + \RED{C} \o* \mathop{\operatoratfont Nil}\nolimits(B)}{I_C(B)}\cr && \mapS{} && \mapS{} && \mapS{}\cr \sesonerowdot{X_A(B)}{1 + \RED{A} \o* \mathop{\operatoratfont Nil}\nolimits(B)}{I_A(B)}} We describe $X_C(B)$. Let $x \in X_C(B)$. Locally (for the fpqc topology) on $B$, we may write $x = c \o* b = 1 + \sum_{i=1}^s c_i \o* b_i$, where $c \in \RED{C}^*$, $b \in B^*$, $c_i \in \RED{C}$, and $b_i \in \mathop{\operatoratfont Nil}\nolimits(B)$. It follows that $b$ must lie in the $k$-linear span of $1$ and the $b_i$, and in particular that we may write $b = 1 + n$, where $n \in \mathop{\operatoratfont Nil}\nolimits(B)$. Passing to $\RED{C} \o* \RED{B}$, we see then that $c = 1$. Hence $x = 1 + n$. From this it follows that $X_C(B) = 1 + \mathop{\operatoratfont Nil}\nolimits(B)$. Similarly, we have $X_A(B) = (1+\mathop{\operatoratfont Nil}\nolimits(B)) \times \cdots \times (1+\mathop{\operatoratfont Nil}\nolimits(B))$, with one copy for each connected component of $\mathop{\operatoratfont Spec}\nolimits(A)$. It follows (details omitted) that the canonical map \dmapx[[ {X_A(B) \over X_C(B)} || {1 + \RED{A} \o* \mathop{\operatoratfont Nil}\nolimits(B) \over 1 + \RED{C} \o* \mathop{\operatoratfont Nil}\nolimits(B)} ]]% is injective, and hence that $I_\phi$ is a monomorphism. Assume now that $\mathop{\operatoratfont Spec}\nolimits(A)$ and $\mathop{\operatoratfont Spec}\nolimits(C)$ are connected. Let ${\xmode{{\fraktur{\lowercase{M}}}}}_C$ be the preimage of ${\xmode{{\fraktur{\lowercase{M}}}}}_A$ under the map \mapx[[ \RED{C} || \RED{A} ]] induced by $\phi$. From our discussion of $I'$, it is clear that $\mathop{\operatoratfont Coker}\nolimits(I_\phi)$ is isomorphic to the cokernel of the morphism given at $B$ by \mapx[[ 1 + {\xmode{{\fraktur{\lowercase{M}}}}}_C \o* \mathop{\operatoratfont Nil}\nolimits(B) || 1 + {\xmode{{\fraktur{\lowercase{M}}}}}_A \o* \mathop{\operatoratfont Nil}\nolimits(B) ]]. In turn this implies remark (iii). We have exact sequences \sescomma{\mathop{\operatoratfont Coker}\nolimits(U_\phi)}{\mathop{\operatoratfont Coker}\nolimits(H_\phi)}{\mathop{\operatoratfont Coker}\nolimits(G_\phi)% }\sescomma{\mathop{\operatoratfont Coker}\nolimits(F_\phi)}{\mathop{\operatoratfont Coker}\nolimits(G_\phi)}{\mathop{\operatoratfont Coker}\nolimits(I_\phi)% }\sesdot{\mathop{\operatoratfont Coker}\nolimits(E_\phi)}{\mathop{\operatoratfont Coker}\nolimits(F_\phi)}{\mathop{\operatoratfont Coker}\nolimits(D_\phi)% }Since $\mathop{\operatoratfont Coker}\nolimits(U_\phi)$ is clearly pseudoadditive, part (b) of the theorem follows from these sequences and (\ref{ext-results}\ref{Gm-Ga}). {\hfill$\square$} \end{proofnodot} \block{Line bundles becoming trivial on pullback by a nilimmersion} If $X$ is a $k$-scheme, we let $\mathop{\mathbf{Pic}}\nolimits(X)$ denote the $k$-functor given by $B \mapsto \mathop{\operatoratfont Pic}\nolimits(X \times_k \mathop{\operatoratfont Spec}\nolimits(B))/\mathop{\operatoratfont Pic}\nolimits(B)$. Then $\mathop{\mathbf{Pic}}\nolimits$ itself defines a functor whose source is \opcat{$k$-schemes}. If \mp[[ f || X || Y ]] is a morphism of $k$-schemes of finite type, such that $X$ and $Y$ each have a rational point, then $\mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(f)]$ is isomorphic to the $k$-functor given by $$B \mapsto \mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(Y \times_k \mathop{\operatoratfont Spec}\nolimits(B)) \ \rightarrow\ \mathop{\operatoratfont Pic}\nolimits(X \times_k \mathop{\operatoratfont Spec}\nolimits(B))].$$ \begin{theorem}\label{kernel-pic-nilimmersion} Let $k$ be a field, and let $X$ be a geometrically stable $k$-scheme. Let \mp[[ i || X_0 || X ]] be a nilimmersion, such that the ideal sheaf ${\cal{N}}$ of $X_0$ in $X$ has square zero. Then there is an exact sequence of (abelian group)-valued $k$-functors \sescomma{D \o+ I}{P}{\mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(i)]% }in which $D$ is discrete and finitely generated, $I$ is subnilpotent, and $P$ is pseudoadditive. \end{theorem} \begin{remarks} \ \begin{alphalist} \item If $X$ is affine, $\mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(i)] = 0$. \item If $X$ is proper over $k$, $D = 0$ and $I = 0$, so $\mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(i)] \cong P$. Also, $P$ is additive and finite-dimensional. One way to get examples is to take $k$ to be algebraically closed, $Y$ to be a projective variety over $k$, and ${\cal{M}}$ to be a coherent ${\cal O}_Y$-module with $H^1(Y,{\cal{M}}) \not= 0$. Make ${\cal O}_Y \o+ {\cal{M}}$ into a coherent ${\cal O}_Y$-algebra via the rule ${\cal{M}}^2 = 0$. Let $X = \mathop{\mathbf{Spec}}\nolimits({\cal O}_Y \o+ {\cal{M}})$, and let \mp[[ i || \RED{X} || X ]] be the inclusion. Then $\mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(i)] \cong {\Bbb G}_a^\alpha$, where $\alpha = h^1(Y,{\cal{M}})$. \item In the non-affine, non-proper case, we have not determined exactly what can happen. In particular, we do not know if $D$ can be nonzero. If $k$ has characteristic zero, then $D \cong \xmode{\Bbb Z}^n$ for some $n$. If $k$ has characteristic $p > 0$, then $D \cong \xmode{\Bbb Z}^n \o+ (\xmode{\Bbb Z}/p\xmode{\Bbb Z})^m$ for some $n$ and some $m$. \item Conceivably the theorem holds without the assumption that ${\cal{N}}^2 = 0$. To prove this, one would at least have to understand $\mathop{\operatoratfont Coker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(i)]$ in the case where ${\cal{N}}^2 = 0$, which we do not. \item We will find an exact sequence \Rowsix{0}{U_1}{U_2}{{\Bbb G}_a^{\kern1pt\beta}}{P}{0% }in which $U_1$ and $U_2$ are additive, for some $\beta$. This is stronger than saying that $P$ is pseudoadditive. Perhaps $P$ is always additive. \end{alphalist} \end{remarks} \begin{proof} We have an exact sequence of sheaves of abelian groups on $X$ \Rowfive{0}{{\cal{N}}}{{\cal O}_X^*}{{\cal O}_{X_0}^*}{1,% }and thus an exact sequence of abelian groups \splitdiagram{H^0(X,{\cal O}_X^*)&\mapE{}&H^0(X_0,{\cal O}_{X_0}^*)% }{\mapE{}&H^1(X,{\cal{N}})&\mapE{}& \mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(X)\ \rightarrow\ \mathop{\operatoratfont Pic}\nolimits(X_0)]&\mapE{}&0.% }In fact, everything is functorial in $k$, and we thus obtain an exact sequence of (abelian group)-valued $k$-functors \splitdiagram{\mathop{\mathbf{Hom}}\nolimits(X,{\Bbb G}_m)&\mapE{}&\mathop{\mathbf{Hom}}\nolimits(X_0,{\Bbb G}_m)&\mapE{}& {\Bbb G}_a^{\kern1pt\beta}}{\mapE{}& \mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(X)\ \rightarrow\ \mathop{\mathbf{Pic}}\nolimits(X_0)]&\mapE{}&0,% }where $\beta = \dim_k[H^1(X,{\cal{N}})]$. Let $K = \mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(X)\ \rightarrow\ \mathop{\mathbf{Pic}}\nolimits(X_0)]$, $L = \mathop{\operatoratfont Coker}\nolimits[\mathop{\mathbf{Hom}}\nolimits(X,{\Bbb G}_m)\ \rightarrow\ \mathop{\mathbf{Hom}}\nolimits(X_0,{\Bbb G}_m)]$. We have an exact sequence \sesdot{L}{{\Bbb G}_a^{\kern1pt\beta}}{K% }According to (\ref{tori-result-generalized}b), there are exact sequences \sescomma{P'}{R}{Q% }\sescomma{R}{L}{I% }in which $Q$ is discrete and finitely generated, $P'$ is additive, and $I$ is subnilpotent. By definition, $\mathop{\operatoratfont Coker}\nolimits[ P'\ \rightarrow\ {\Bbb G}_a^{\kern1pt\beta}]$ is pseudoadditive. Hence by (\ref{ext-results}\ref{nilpotent-discrete}), we have an exact sequence \sescomma{Q \times I}{P}{K% }in which $P$ is pseudoadditive. {\hfill$\square$} \end{proof} \begin{corollary}\label{sheaf-kernel-pic-nilimmersion} Let $k$ be a field, and let $X$ be a geometrically stable $k$-scheme. Let \mp[[ i || X_0 || X ]] be a nilimmersion. Let $F = \mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(i)]$. Let \mp[[ p || B || C ]] be a faithfully flat ring homomorphism. \begin{alphalist} \item If $B$ and $C$ are reduced, then the canonical map \mapx[[ F(B) || F(C) ]] is injective. \item Assume that $k$ has characteristic zero and that the ideal sheaf of $X_0$ in $X$ has square zero. Assume that $B$ is normal and that $C$ is \'etale over $B$. Then $\Psi_{F,p}$ (see p.\ \pageref{Psi-place}) is bijective. \end{alphalist} \end{corollary} \begin{proof} Let ${\cal{N}}$ be the ideal sheaf of $X_0$ in $X$. First suppose that ${\cal{N}}^2 = 0$. Then (a) follows from \pref{kernel-pic-nilimmersion}. For (b), let $S = \mathop{\operatoratfont Spec}\nolimits(B)$. Since $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = 0$, $D$ is torsion-free. Therefore it suffices to show that in the category of (abelian group)-valued $k$-functors which are sheaves for the \'etale topology, the sequence \ses{\xmode{\Bbb Z}^n}{{\Bbb G}_a^\alpha}{({\Bbb G}_a^\alpha/\xmode{\Bbb Z}^n)^+% }is exact when evaluated at $B$. (Here we let $({\Bbb G}_a^\alpha/\xmode{\Bbb Z}^n)^+$ denote the quotient in this category.) Since $H^1(\et{S}, \xmode{\Bbb Z}) = 0$ by (\Lcitemark 1\Rcitemark \ 3.6(ii)), we are done. Now we prove the general case of (a). For each $m$, let $X_m$ be the closed subscheme of $X$ defined by ${\cal{N}}^m$. Choose $n \in \xmode{\Bbb N}$ so that ${\cal{N}}^n = 0$. Let $K_m = \mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(X_{m+1})\ \rightarrow\ \mathop{\mathbf{Pic}}\nolimits(X_m)]$. Let $F_m = \mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(X_m)\ \mapE{}\ \mathop{\mathbf{Pic}}\nolimits(X_0)]$. We have an exact sequence \Rowfour{0}{K_m}{F_{m+1}}{F_m.% }Then \mapx[[ K_m(B) || K_m(C) ]] is injective by \pref{kernel-pic-nilimmersion}. By induction on $m$, it follows that \mapx[[ F_m(B) || F_m(C) ]] is injective for all $m$. Taking $m = n$, we get (a). {\hfill$\square$} \end{proof} \begin{problem} Is the functor $F$ a sheaf? \end{problem} \section*{References} \addcontentsline{toc}{section}{References} \ \par\noindent\vspace*{-0.25in} \hfuzz 5pt \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{1}% \def\Atest{ }\def\Astr{Artin\Revcomma M\Initper }% \def\Ttest{ }\def\Tstr{Faisceaux constructibles, cohomologie d'un courbe alg\`ebrique, expos\'e\ IX in {\itS\'em\-in\-aire de G\'eom\'e\-trie Al\-g\'e\-bri\-que} (SGA 4)}% \def\Stest{ }\def\Sstr{Lecture Notes in Mathematics}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Vtest{ }\def\Vstr{305}% \def\Dtest{ }\def\Dstr{1973}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{1--42}% \def\Qtest{ }\def\Qstr{access via "artin constructible sheaves"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{2}% \def\Atest{ }\def\Astr{Artin\Revcomma M\Initper }% \def\Ttest{ }\def\Tstr{The implicit function theorem in algebraic geometry}% \def\Btest{ }\def\Bstr{Algebraic Geometry (Bombay Colloquium, 1968)}% \def\Itest{ }\def\Istr{Oxford Univ. Press}% \def\Dtest{ }\def\Dstr{1969}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{13--34}% \def\Xtest{ }\def\Xstr{Theorem: $G/H$ is representable by an algebraic space. repr-quot: section 7.}% \def\Qtest{ }\def\Qstr{access via "artin implicit function theorem"}% \def\Xtest{ }\def\Xstr{descent: p. 31}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{3}% \def\Atest{ }\def\Astr{Bertin\Revcomma J\Initper \Initgap E\Initper }% \def\Ttest{ }\def\Tstr{{\rm\tolerance=1000 Generalites sur les preschemas en groupes, expos\'e\ ${\rm VI}_{\rm B}$ in {\itS\'em\-in\-aire de G\'eom\'e\-trie Al\-g\'e\-bri\-que} (SGA 3)}}% \def\Stest{ }\def\Sstr{Lecture Notes in Mathematics}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Vtest{ }\def\Vstr{151}% \def\Dtest{ }\def\Dstr{1970}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{318--410}% \def\Qtest{ }\def\Qstr{access via "bertin preschemas"}% \def\Xtest{ }\def\Xstr{This includes the following result, announced by Raynaud, but apparently not proved in this volume: Theorem (11.11.1) Let $S$ be a regular noetherian scheme of dimension $/leq 1$. Let \map(\pi,G,S) be an $S$-group scheme of finite-type. Assume that $\pi$ is a flat, affine morphism. Then $G$ is isomorphic to a closed subgroup of $\AUT(V)$, where $V$ is a vector bundle on $S$.}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{4}% \def\Atest{ }\def\Astr{Gabriel\Revcomma P\Initper }% \def\Ttest{ }\def\Tstr{{\rm\tolerance=1000 Generalites sur les groupes algebriques, expos\'e\ ${\rm VI}_{\rm A}$ in {\itS\'em\-in\-aire de G\'eom\'e\-trie Al\-g\'e\-bri\-que} (SGA 3)}}% \def\Stest{ }\def\Sstr{Lecture Notes in Mathematics}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Vtest{ }\def\Vstr{151}% \def\Dtest{ }\def\Dstr{1970}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{287--317}% \def\Qtest{ }\def\Qstr{access via "gabriel generalites"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{5}% \def\Atest{ }\def\Astr{Grothendieck\Revcomma A\Initper % \Aand J\Initper \Initgap A\Initper Dieudonn\'e}% \def\Ttest{ }\def\Tstr{El\'ements de g\'eom\'etrie\ alg\'e\-brique IV (part two)}% \def\Jtest{ }\def\Jstr{Inst. 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9410

alg-geom/9410007

en

https://arxiv.org/abs/alg-geom/9410007

alg-geom/9410007

Zhenbo Qin

Robert Friedman and Zhenbo Qin

Flips of moduli spaces and transition formulas for Donaldson polynomial invariants of rational surfaces

56 pages, amstex

OSU Math 1995-11

We study the change of moduli spaces of Gieseker-semistable torsion free rank-$2$ sheaves on algebraic surfaces as we vary the polarizations. When the surfaces are rational with an effective anti-canonical divisor, the moduli spaces are linked by a series of flips (blowups and blowdowns). Using these results, we compute the transition formulas for Donaldson polynomial invariants of rational surfaces. Part of the work is also obtained independently by Matsuki-Wentworth and Ellingsrud-G{\" o}ttsche.

alg-geom

\section{1. Introduction.} In \cite{7}, Donaldson has defined polynomial invariants for smooth simply connected 4-manifolds with $b_2^+\geq 3$. These invariants have also been defined for 4-manifolds with $b_2^+=1$ in \cite{24, 17, 18}, along lines suggested by the work of Donaldson in \cite{5}. In this case, however, they depend on an additional piece of information, namely a chamber defined on the positive cone of $H^2(X; \Ar)$ by a certain locally finite set of walls. Explicitly, let $X$ be a simply connected, oriented, and closed smooth $4$-manifold with $b_2^+ = 1$ where $b_2^+$ is the number of positive eigenvalues of the quadratic form $q_X$ when diagonalized over $\Ar$. Let $$\Omega_X = \{\, x \in H^2(X, \Ar)\mid x^2 > 0 \,\}$$ be the positive cone. Fix a class $\Delta$ in $H^2(X, \Zee)$ and an integer $c$ such that $d = 4c - \Delta ^2 - 3$ is nonnegative. A {\sl wall of type $(\Delta, c)$\/} is a nonempty hyperplane: $$W^\zeta = \{\, x \in \Omega_X\mid x \cdot \zeta = 0 \,\}$$ in $\Omega_X$ for some class $\zeta \in H^2(X, \Zee)$ with $\zeta \equiv \Delta \pmod 2$ and $\Delta ^2 - 4c \le \zeta^2 < 0$. The connected components of the complement in $\Omega_X$ of the walls of type $(\Delta, c)$ are the {\sl chambers of type $(\Delta, c)$}. Then the Donaldson polynomial invariants of $X$ associated to $\Delta$ and $c$ are defined with respect to chambers of type $(\Delta, c)$. The invariants only depend on the class $w=\Delta \bmod 2\in H^2(X; \Zee/2\Zee)$ and the integer $p = \Delta ^2 - 4c$, and we shall often refer to walls and chambers of type $(w,p)$ as well. We shall write $D^X _{w,p}(\Cal C)$ for the Donaldson polynomial corresponding to the $SO(3)$ bundle $P$ with invariants $w_2(P) = w$ and $p_1(P) = p$, depending on the chamber $\Cal C$. A basic question is then the following: Suppose that $\Cal C_+$ and $\Cal C_-$ are separated by a single wall $W^\zeta$. Here there may be more than one class $\zeta$ of type $(\Delta, c)$ defining $W^\zeta$. Then find a formula for the difference $$\delta ^X_{w,p}(\Cal C_+, \Cal C_-) = D^X _{w,p}(\Cal C_+) - D^X _{w,p}(\Cal C_-).$$ We shall refer to such a difference as a {\sl transition formula}. There has been considerable interest in the above problem. The first result in this direction is due to Donaldson in \cite{5}, who gave a formula in case $\Delta = 0$ and $c = 1$. Kotschick \cite{17} showed that, on the part of the symmetric algebra generated by $2$-dimensional classes, $\delta ^X_{w,p}(\Cal C_+, \Cal C_-) = \pm \zeta^d$ for $\zeta^2 = -(4c - \Delta^2)=p$, and that $\delta ^X_{w,p}(\Cal C_+, \Cal C_-)$ is in fact always divisible by $\zeta$, except when $p=-5$ and $\zeta ^2 = -1$ (cf\. also Mong \cite{24} for some partial results along these lines). For a rational ruled surface $X$, all the transition formulas for $\Delta = 0$ and $2 \le c \le 4$ have been determined in \cite{24, 33, 22}. Using a gauge-theoretic approach, Yang \cite{35} settled the problem for $\Delta = 0$ and $c = 2$, and computed the degree $5$ Donaldson polynomials for rational surfaces. The known examples and the work of Kotschick and Morgan \cite{18} raise the following rather natural conjecture: \medskip\noindent {\bf Conjecture.} {\it The transition formula $\delta ^X_{w,p}(\Cal C_+, \Cal C_-)$ is a homotopy invariant of the pair $(X, \zeta)$; more precisely, if $\phi$ is an oriented homotopy equivalence from $X'$ to $X$, then $$\delta^{X'}_{\phi ^*w,p}(\phi ^*(\Cal C_+), \phi ^*(\Cal C_-))=\phi ^*\delta ^{X}_{w,p}(\Cal C_+, \Cal C_-).$$} \medskip We remark that this conjecture is essentially equivalent to the following statement: the transition formula $\delta ^X_{w,p}(\Cal C_+, \Cal C_-)$ is a polynomial in $\zeta$ and the quadratic form $q_X$ with coefficients involving only $\zeta^2$, homotopy invariants of $X$ (i.e\. $b^-_2(X)$), and universal constants. Our goal in this paper is to study the corresponding problem in algebraic geometry. More precisely, let $X$ be an algebraic surface (not necessarily with $b_2^+(X)= 1$) and let $L$ be an ample line bundle on $X$. We can then identify the moduli space of $L$-stable rank two bundles $V$ on $X$ with $c_1(V)=\Delta$ and $c_2(V)=c$ with the moduli space of equivalence classes of ASD connections on $X$ with respect to a Hodge metric on $X$ corresponding to $L$. Let $\frak M_L(\Delta, c)$ be the Gieseker compactification of this moduli space. It is known that $\frak M_L(\Delta, c)$ changes as we change $L$, and that $\frak M_L(\Delta, c)$ is constant on a set of chambers for the ample cone of $X$ which are defined in a way analogous to the definition of chambers for $\Omega _X$ given above. Using the recent result of Morgan \cite{25} and Li \cite{21} that the Donaldson polynomial of an algebraic surface can be evaluated using the Gieseker compactification ${\frak M}_L(\Delta, c)$ of the moduli space of stable bundles, we shall work on ${\frak M}_L(\Delta, c)$ for suitable choices of $L$ and in particular analyze the change in ${\frak M}_L(\Delta, c)$ for $L\in \Cal C_+$ or $L\in \Cal C_-$, where $\Cal C_\pm$ are two adjacent chambers. It turns out that we can obtain $\frak M _{L_+}(\Delta, c)$ from $\frak M _{L_-}(\Delta, c)$ by a series of blowups and blowdowns (flips). Our results are thus very similar to those of Thaddeus in \cite{31}. Thaddeus \cite{32} and also Dolgachev-Hu \cite{3} have developed a general picture for the variation of GIT quotients after a change of polarization, and although our methods are somewhat different it seems quite possible that they fit into their general framework. We have also found it convenient to borrow some of Thaddeus' notation. Next we shall apply our results on the change in the moduli spaces to determine the transition formula for Donaldson polynomials in case $X$ is a rational surface with $-K_X$ effective. We shall give explicit formulas for $\delta ^X_{w,p}(\Cal C_+, \Cal C_-)$ in case the nonnegative integer $\ell _\zeta = (\zeta ^2 - p)/4 \leq 2$. These formulas are in agreement with the above conjecture, in the sense that the transition formula is indeed a polynomial in $\zeta$ and $q_X$ with coefficients involving only $\zeta^2$, $K_X^2$, and universal constants. We shall also give a formula in principle for $\delta ^X_{w,p}(\Cal C_+, \Cal C_-)$ in general (see Theorem 5.4), but to make this formula explicit involves more knowledge of the enumerative geometry of $\Hilb^nX$ than seems to be available at present. In case $-K_X$ is effective, the moduli spaces are (essentially) smooth and the centers of the blowup are smooth as well; in fact they are $\Pee ^k$-bundles over $\Hilb ^{n_1}X\times \Hilb ^{n_2}X$ for appropriate $k$, $n_1$ and $n_2$. In this way, we obtain general formulas which can be made explicit for low values of $n$. For instance, we show the following (see Theorem 6.4 for details): \theorem{} Assume that the wall $W^\zeta$ is defined only by $\pm \zeta$ with $\ell_\zeta = 1$ and that $\Cal C_\pm$ lies on the $\pm$-side of $W^\zeta$. Then, on the subspace of the symmetric algebra generated by $H_2(X)$, $\delta^X_{w, p}(\Cal C_-, \Cal C_+)$ is equal to $$(-1)^{{(\Delta \cdot K_X + \Delta^2) + (\zeta \cdot K_X - \zeta^2)} \over 2} \cdot \left \{ d(d - 1) \cdot \left(\zeta \over 2 \right)^{d - 2} \cdot q_X + (2K_X^2 + 2d + 6) \cdot \left ({\zeta \over 2} \right )^d \right \}.$$ \endstatement Along the direction of the work of Kronheimer and Mrowka \cite{19, 20}, we also consider the difference of Donaldson polynomial invariants involving the natural generator $x \in H_0(X; \Zee)$. More precisely, let $\nu$ be the corresponding $4$-dimensional class in the instanton moduli space. For $\alpha \in H_2(X; \Zee)$, we give a formula for the difference $\delta^X_{w, p}(\Cal C_-, \Cal C_+)(\alpha^{d - 2}, \nu)$ in Theorem 5.5. It is worth to point out that the similarity between Theorem 5.4 and Theorem 5.5 may indicate that there exists a deep relation between $\delta^X_{w, p}(\Cal C_-, \Cal C_+)(\alpha^d)$ and $\delta^X_{w, p}(\Cal C_-, \Cal C_+)(\alpha^{d - 2}, \nu)$, and suggest a way to generalize the notion of {\sl simple type} in \cite{19, 20} from the case of $b_2^+ > 1$ to the case of $b_2^+ = 1$. For instance, modulo some lower degree terms, $\delta^X_{w, p}(\Cal C_-, \Cal C_+)(\alpha^{d - 2}, \nu)$ can be obtained from $(-1/4) \cdot \delta^X_{w, p}(\Cal C_-, \Cal C_+)(\alpha^d)$ by replacing $d$ by $(d - 2)$ (see Theorem 5.13 and Theorem 5.14). In fact, based on some heuristic arguments, it seems reasonable to conjecture that $\delta^X_{w, p}(\Cal C_-, \Cal C_+)(\alpha^{d - 2}, \nu)$ is a combination of $\delta^X_{w, p'}(\Cal C_-, \Cal C_+)(\alpha^{d - 4k})$ for various nonnegative integers $k$ if the degrees are properly arranged. We hope to return to this issue in future. Our paper is organized as follows. In section 2, we study rank two torsion free sheaves which are semistable with respect to ample divisors in $\Cal C_-$ but not semistable with respect to ample divisors in $\Cal C_+$. When the surface $X$ is rational with $-K_X$ effective, these sheaves are parametrized by an open subset of a union of projective bundles over the product of two Hilbert schemes of points in $X$. More precisely, if $\zeta$ defines the wall separating $\Cal C_-$ from $\Cal C_+$, define $E_\zeta^{n_1, n_2}$ be the set of all isomorphism classes of nonsplit extensions of the form $$0 \to \scrO _X(F )\otimes I_{Z_1} \to V \to \scrO _X(\Delta -F ) \otimes I_{Z_2} \to 0,$$ where $F$ is a divisor class such that $2F-\Delta \equiv \zeta$ and $Z_1$ and $Z_2$ are two zero-dimensional subschemes of $X$ with $\ell (Z_i) = n_i$ such that $n_1 + n_2 = \ell _\zeta$. In case $X$ is rational, $E_\zeta^{n_1, n_2}$ is a $\Pee ^N$ bundle over $\Hilb ^{n_1}X\times \Hilb ^{n_2}X$, and the set of points of $E_\zeta^{n_1, n_2}$ lying in $\frak M_{L_-}(\Delta, c)$ but not in $\frak M_{L_+}(\Delta, c)$ is a Zariski open subset of $E_\zeta^{n_1, n_2}$. The main technical difficulty is that it is hard to control the rational map from $E_\zeta^{n_1, n_2}$ to $\frak M_{L_-}(\Delta, c)$, and in particular this map is not a morphism. The general picture that we establish is the following: first, the map $E_\zeta^{0, \ell_\zeta}\dasharrow\frak M_{L_-}(\Delta, c)$ is a morphism, and it is possible to make an elementary transformation, or {\sl flip\/}, along its image. The result is a new space for which the rational map $E_\zeta^{1, \ell_\zeta - 1}\dasharrow\frak M_{L_-}(\Delta, c)$ becomes a morphism, and it is possible to make a flip along {\sl its\/} image. We continue in this way until we reach $\frak M_{L_+}(\Delta, c)$. It seems rather difficult to see that the above picture holds directly. Instead we shall proceed as follows. We define abstractly a sequence of moduli spaces, indexed by an integer $k$ with $0\leq k \leq \ell _\zeta+1$, such that the moduli space for $k=0$ is $\frak M_{L_-}(\Delta, c)$, the moduli space for $k=\ell _\zeta+1$ is $\frak M_{L_+}(\Delta, c)$, and moreover the $k^{\text{th}}$ moduli space contains an embedded copy of $E_\zeta^{k, \ell _\zeta -k}$ such that the flip along this copy yields the $(k+1)^{\text{st}}$ moduli space. Thus the picture is very similar to that developed independently by Thaddeus in \cite{31}. To define our sequence of moduli spaces, we define $(L_0, \boldsymbol \zeta, \bold k)$-semistability in section 3 for rank two torsion free sheaves, where $L_0$ is any ample divisor contained in the common face of $\Cal C_+$ and $\Cal C_-$, $\boldsymbol \zeta$ is the set of classes of type $(\Delta, c)$ defining the common wall of $\Cal C_+$ and $\Cal C_-$, and $\bold k$ is a set of integers. We show that $\frak M _{L_-}(\Delta, c)$ and $\frak M _{L_+}(\Delta, c)$ are linked by the moduli spaces $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ where the data $\bold k$ is allowed to vary. When the surface $X$ is rational with $-K_X$ effective, we can obtain $\frak M _{L_+}(\Delta, c)$ from $\frak M _{L_-}(\Delta, c)$ by a series of flips. The fact that all $(L_0, \boldsymbol \zeta, \bold k)$-semistable rank two torsion free sheaves do form a moduli space $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ in the usual sense is proved in section 4 where we introduce an equivalent notion of stability called {\sl mixed stability}. Our method follows Gieseker's GIT argument in \cite{13}. Roughly speaking, the goal of mixed stability is to define stability for a sheaf of the form $V\otimes \Xi$, where $V$ is a torsion free sheaf but $\Xi$ is just a $\Bbb Q$-divisor. To make this idea precise, given actual divisors $H_1$ and $H_2$ and positive weights $a_1$ and $a_2$, we shall define a notion of stability which ``mixes" stability for $V\otimes H_1$ with stability for $V\otimes H_2$, together with weightings of the stability condition for $V\otimes H_i$. The effect of this definition will be formally the same as if we had defined stability of $V\otimes \Xi$, where $\Xi$ is the $\Bbb Q$-divisor $$\frac{a_1}{a_1+a_2}H_1 + \frac{a_2}{a_1+a_2}H_2.$$ In section 5, using our results on flips of moduli spaces, we give a formula for the transition formula of Donaldson polynomials when $X$ is rational with $-K_X$ effective, and compute the leading term in the transition formula. In section 6, we obtain explicit transition formulas when $\ell_\zeta \le 2$. Some of the material in our section 2 has been worked out independently by Hu and Li \cite{16} and G\"ottsche \cite{14}. Moreover Ellingsrud and G\"ottsche \cite{8} have recently studied the change in the moduli space by similar methods and have obtained results very similar to ours. Using very different methods, the results in Section 4 have also been obtained by Matsuki and Wentworth \cite{23}, who also consider the case of higher rank. They use branched covers of the surface $X$ to study the change in the moduli space. We expect that a minor modification of the arguments in Section 4 of this paper will also handle the case of higher rank. \section{Conventions and notations} We fix some conventions and notations for the rest of this paper. Let $X$ be a smooth algebraic surface. We shall be primarily interested in the case where $X$ is simply connected and $-K_X$ is effective and nonzero. Thus necessarily $X$ is a rational surface. However much of the discussion in sections 1--4 will also apply to the general case. Stability and semistability with respect to an ample line bundle $L$ will always be understood to mean Gieseker stability or semistability unless otherwise noted. We shall not mention the choice of $L$ explicitly if it is clear from the context. Recall that a torsion free sheaf $V$ of rank two is Gieseker $L$-stable if and only if, for every rank one subsheaf $W$ of $V$, either $\mu _L(W) < \mu _L(V)$ or $\mu _L(W) = \mu _L(V)$ and $2\chi(W) < \chi(V)$, where $\mu _L$ is the normalized degree with respect to $L$. Semistability is similarly defined, where the second inequality is also allowed to be an equality. For a torsion free sheaf $V$, we use $V\ddual$ to stand for its double dual. For two divisors $D_1$ and $D_2$ on $X$, the notation $D_1 \equiv D_2$ means that $D_1$ and $D_2$ are numerically equivalent, that is, $D_1 \cdot D = D_2 \cdot D$ for any divisor $D$. For a locally free sheaf (or equivalently a vector bundle) $\Cal E$ over a smooth variety $Y$, we use $\Pee(\Cal E)$ to denote the associated projective space bundle, that is, $\Pee(\Cal E)$ is the {\bf Proj} of $\oplus_{d \ge 0} S^d(\Cal E)$. Fix a divisor $\Delta$ and an integer $c$. Let $\Cal C_-$ and $\Cal C_+$ be two adjacent chambers of type $(\Delta, c)$ separated by the wall $W^\zeta$. We assume that $\zeta \cdot \Cal C_- < 0<\zeta \cdot \Cal C_+$. Let $L_\pm\in \Cal C_\pm$ be an ample line bundle, so that $L_- \cdot \zeta < 0 < L_+\cdot \zeta$, and denote by $\frak M_\pm$ the moduli space $\frak M _{L_\pm}(\Delta, c)$ of rank two Gieseker semistable torsion free sheaves $V$ with $c_1(V) = \Delta$ and $c_2(V) = c$. Let $L_0$ be any ample divisor contained in the interior of the intersection of $W^\zeta$ and the closures of $\Cal C_\pm$. Let $\zeta = \zeta _1, \dots, \zeta _n$ be all the positive rational multiples of $\zeta$ such that $\zeta_i$ is an integral class of type $(w,p)$ which also defines the wall $W^\zeta$. In sections 5--6, we will assume that $n = 1$ for notational simplicity. Finally, we point out that our $\mu$-map is half of the $\mu$-map used in \cite{17, 18} (see (viii) and (ix) in Notation 5.1). Thus our transition formula differs from the one defined in \cite{18} by a universal constant. \medskip\noindent {\it Acknowledgements.} We would like to thank Hong-Jie Yang for invaluable access to his calculations, which helped to keep us on the right track. The second author would like to thank Wei-ping Li and Yun-Gang Ye for helpful discussions, and the Institute for Advanced Study at Princeton for its hospitality and financial support through NSF grant DMS-9100383 during the academic year 1992--1993 when part of this work was done. \section{2. Preliminaries on the moduli space.} In this section, we study rank two torsion free sheaves which are related to walls. These sheaves arise naturally from the comparison of $L_-$-semistability and $L_+$-semistability. We will show that when the surface $X$ is rational with $-K_X$ effective, the moduli spaces $\frak M_\pm$ are smooth at the points corresponding to these sheaves. We start with the following lemma, which for simplicity is just stated for $L_-$-stability. \lemma{2.1} Let $V$ be a rank two torsion free sheaf on $X$ with $c_1(V) = \Delta$ and $c_2(V) = c$. If $V$ is $L_-$-semistable, then exactly one of the following holds: \roster \item"{(i)}" Both $V$ and $V\ddual$ are $L_-$-stable and Mumford $L_-$-stable. \item"{(ii)}" $V$ sits in an exact sequence $$0 \to \scrO_X(F_1)\otimes I_{Z_1} \to V \to \scrO_X(F_2) \otimes I_{Z_2} \to 0$$ where $2F_1 \equiv \Delta \equiv 2F_2$, and $Z_1$ and $Z_2$ are zero-dimensional subschemes of $X$ such that $\ell (Z_1) \geq \ell(Z_2)$. Moreover in this case $V$ is $L$-semistable for every choice of an ample line bundle $L$ and $V$ is strictly $L_\pm$-semistable if and only if $\ell(Z_1) = \ell (Z_2)$. \endroster \endstatement \proof Suppose that $V$ is (Gieseker) $L_-$-semistable. The vector bundle $V\ddual$ satisfies $c_1(V\ddual ) = \Delta$ and $c_2(V\ddual) \leq c$. Standard arguments \cite{10} show that $V\ddual$ is Mumford $L_-$-semistable. If $V\ddual$ is strictly Mumford $L_-$-semistable, then by \cite{10, 30}, either $L_-$ must lie on a wall of type $(\Delta, c)$ or if $\scrO_X(F_1)$ is a destabilizing sub-line bundle then $\Delta \equiv 2F_1$. Since by assumption $L_-$ does not lie on a wall of type $(\Delta, c)$, either $V\ddual$ is Mumford $L_-$-stable or there is an exact sequence $$0 \to \scrO_X(F_1) \to V\ddual \to \scrO_X(F_2) \otimes I_Z \to 0,$$ where $F_2 = \Delta - F_1 \equiv F_1$ and $Z$ is a zero-dimensional subscheme of $X$. If $V\ddual$ is Mumford $L_-$-stable, then $V$ is Mumford $L_-$-stable and therefore $L_-$-stable. Thus case (i) holds. Otherwise $\scrO_X(F_1) \cap V$ is of the form $\scrO_X(F_1)\otimes I_{Z_1}$ for some $Z_1$ and $V/\scrO_X(F_1)\otimes I_{Z_1}$ is a subsheaf of $\scrO_X(F_2) \otimes I_Z$ and thus of the form $\scrO_X(F_2) \otimes I_{Z_2}$ for some $Z_2$. Thus we are in case (ii) of the lemma. Since $\mu(\scrO_X(F_1)\otimes I_{Z_1}) = \mu (V)$ and $V$ is semistable, we have $$2\chi(\scrO_X(F_1)\otimes I_{Z_1}) \leq \chi (V)= \chi(\scrO_X(F_1) \otimes I_{Z_1})+ \chi(\scrO_X(F_2)\otimes I_{Z_2}).$$ Hence $\chi(\scrO_X(F_2)\otimes I_{Z_2}) - \chi(\scrO_X(F_1)\otimes I_{Z_1}) \geq 0$. As $F_1 \equiv F_2$ and $\chi(\scrO_X(F_i)\otimes I_{Z_i}) = \chi(\scrO_X(F_i)) - \ell(Z_i)$, we must then have $\ell(Z_1) - \ell (Z_2) \geq 0$. The last sentence of (ii) is a straightforward argument left to the reader. \endproof If $V$ satisfies the conclusions of (2.1)(ii), we shall call $V$ {\sl universally semistable}. Next we shall compare stability for $L_-$ and $L_+$. \lemma{2.2} Let $V$ be a torsion free rank two sheaf with $c_1(V) = \Delta$ and $c_2(V) = c$. \roster \item"{(i)}" If $V$ is $L_-$-stable but $L_+$-unstable, then there exist a divisor class $F$ and two zero-dimensional subschemes $Z_-$ and $Z_+$ of $X$ and an exact sequence $$0 \to \scrO _X(F )\otimes I_{Z_-} \to V \to \scrO _X(\Delta -F ) \otimes I_{Z_+} \to 0,$$ where $L_-\cdot (2F-\Delta) < 0 < L_+\cdot (2F-\Delta) $. Moreover the divisor $F$, the schemes $Z_-$ and $Z_+$, and the map $F \otimes I_{Z_-} \to V$ are unique mod scalars, and $\zeta = 2F - \Delta$ defines a wall of type $(\Delta, c)$. \item"{(ii)}" Conversely, suppose that there is a nonsplit exact sequence as above. Then $V$ is simple. Moreover, $V$ is not $L_-$-stable if and only if it is $L_-$-unstable if and only if there exist subschemes $Z'$ and $Z''$ and an exact sequence $$0 \to \scrO _X(\Delta - F )\otimes I_{Z'} \to V \to \scrO _X(F ) \otimes I_{Z''} \to 0,$$ if and only if $V\ddual$ is a direct sum $\scrO _X(F) \oplus \scrO _X(\Delta -F )$. In this case the scheme $Z'$ strictly contains the scheme $Z_+$, $\ell (Z') > \ell (Z_+)$ and $\ell (Z')+\ell (Z'') = \ell (Z_-)+\ell (Z_+)$. Finally, if $Z_-=\emptyset$ then $V$ is always $L_-$-stable. \endroster \endstatement \proof We first show (i). Suppose that $V$ is $L_-$-stable but $L_+$-unstable. Then by (2.1) $V\ddual$ is also $L_-$-stable and $L_+$-unstable. By \cite{30}, there is a uniquely determined line bundle $\scrO_X(F)$ and a map $\scrO _X(F) \to V\ddual$ with torsion free quotient such that $L_-\cdot (2F-\Delta) < 0 < L_+\cdot (2F-\Delta) $. Moreover $\zeta = 2F - \Delta$ defines a wall of type $(\Delta, c)$. The subsheaf $\scrO_X(F) \cap V$ of $V\ddual$ is a subsheaf of $\scrO_X(F)$ and agrees with it away from finitely many points. Thus $\scrO_X(F) \cap V = \scrO _X(F )\otimes I_{Z_-}$ for some well-defined subscheme $Z_-$. Moreover the quotient $V\Big/[\scrO _X(F )\otimes I_{Z_-}]$ is a subsheaf of $\scrO _X(\Delta -F ) \otimes I_Z$ for some zero-dimensional subscheme $Z$, and agrees with $\scrO _X(\Delta -F ) $ away from finitely many points. Thus the quotient is of the form $\scrO _X(\Delta -F ) \otimes I_{Z_+}$ for some zero-dimensional subscheme $Z_+$. The uniqueness is clear. To see (ii), suppose that $V$ is given as a nonsplit exact sequence $$0 \to \scrO _X(F )\otimes I_{Z_-} \to V \to \scrO _X(\Delta -F ) \otimes I_{Z_+} \to 0$$ as above, where $L_-\cdot (2F-\Delta) < 0 < L_+\cdot (2F-\Delta) $. Again by (2.1), $V$ is $L_-$-semistable if and only if it is $L_-$-stable if and only if $V\ddual$ is $L_-$-stable. Now taking double duals of the above exact sequence, there is an exact sequence $$0 \to \scrO _X(F ) \to V\ddual \to \scrO _X(\Delta -F ) \otimes I_Z \to 0$$ for some zero-dimensional scheme $Z$. Moreover, by \cite{30}, $V\ddual$ is $L_-$-unstable if and only if the above exact sequence splits, and in particular if and only if $Z=\emptyset$ and $V\ddual = \scrO _X(F)\oplus \scrO _X(\Delta -F )$. In this case, the map $\scrO _X(\Delta -F ) \to V\ddual$ induces a map $\scrO _X(\Delta -F )\otimes I_{Z'} \to V$ for some ideal sheaf $I_{Z'}$. We may clearly assume that the quotient is torsion free, in which case it is necessarily of the form $\scrO _X(F ) \otimes I_{Z''}$ with $\ell (Z')+\ell (Z'') = \ell (Z_-)+\ell (Z_+)$. Using the nonzero map $\scrO _X(\Delta -F )\otimes I_{Z'} \to \scrO _X(\Delta -F )\otimes I_{Z_+}$, we see that there is an inclusion $I_{Z'} \subseteq I_{Z_+}$; moreover this inclusion must be strict since the defining exact sequence for $V$ is nonsplit. Thus $Z'$ strictly contains $Z_+$ and in particular $\ell (Z') > \ell (Z_+)$. Conversely, if there exists a nonzero map $\scrO _X(\Delta - F )\otimes I_{Z'} \to V$, then there is a nonzero map $\scrO _X(\Delta - F ) \to V\ddual$ and thus $V\ddual $ is the split extension. We next show that $V$ is simple. If $V$ is stable then it is simple. If $V$ is not stable, then $V\ddual = \scrO _X(F)\oplus \scrO _X(\Delta -F )$. There is an inclusion $\Hom (V, V) \subseteq \Hom (V\ddual, V\ddual)$. If $V\ddual$ is split, then $\Hom (V\ddual, V\ddual) = \Cee \oplus \Cee$. In this case, using a nonscalar endomorphism of $V$, it is easy to see that we can split the exact sequence defining $V$. Finally suppose that $Z_- = \emptyset$ in the notation of (2.2). If $V$ is $L_-$-unstable, then we can find $Z'$ with $\ell (Z') > \ell (Z_+)$ and a subscheme $Z''$ such that $\ell (Z') + \ell (Z'') = \ell (Z_+)$. Thus $\ell (Z') \leq \ell (Z_+)$, a contradiction. It follows that $V$ is $L_-$-stable. \endproof For the rest of this section, we shall assume that $-K_X$ is effective and nonzero and that $q(X)=0$. Thus $X$ is a rational surface. \lemma{2.3} Suppose that $\frak M_\pm$ is nonempty. Suppose that $(w,p)\neq (0,0)$, or equivalently that $\frak M_\pm$ does not consist of a single point corresponding to a twist of the trivial vector bundle. Then the open subset of $\frak M_\pm$ corresponding to Mumford stable rank two vector bundles is nonempty and dense. Every component of $\frak M_\pm$ has dimension $4c-\Delta ^2 -3=-p-3$. The points of $\frak M_\pm$ corresponding to $L_\pm$-stable sheaves $V$ are smooth points. \endstatement \proof Suppose that $\frak M_\pm$ is nonempty, and let $V$ correspond to a point of $\frak M_\pm$. Then by general theory (e.g\. Chapter 7 of \cite{10}), $\frak M_\pm$ is smooth of dimension $4c-\Delta ^2 -3=-p-3$ at $V$ if $V$ is stable and $\Ext^2(V, V) = 0$, since $h^2(X; \scrO _X) = 0$. Moreover, setting $W = V\ddual$, there is a surjection from $H^2(X; Hom(W, W))$ to $\Ext^2(V, V)$. Thus to show that $\Ext^2(V, V) =0$ it suffices to show that $H^2(X; Hom(W, W))=0$. Now $H^2(X; Hom(W, W))$ is dual to $H^0(X; Hom (W,W) \otimes K_X)$. Since $-K_X$ is effective, there is an inclusion of $H^0(X; Hom (W,W) \otimes K_X)$ in $H^0(X; Hom (W,W))$. If $W$ is stable, then $H^0(X; Hom (W,W)) \cong \Cee$ and $H^0(X; Hom (W,W) \otimes K_X) = 0$. Thus $\frak M_\pm$ is smooth at $V$. Standard theory \cite{1, 10} also shows that every torsion free sheaf $V$ for which $V\ddual$ is stable is smoothable. Thus the set of locally free sheaves is nonempty and dense in the component containing $V$ in this case. Now consider a $V$ such that $W=V\ddual$ is not stable. Using the exact sequence $$0 \to \scrO_X(F) \to W \to \scrO_X(F) \otimes I_Z \to 0$$ for $W$ which was given in the course of the proof of (2.1), it is easy to check that there is an exact sequence $$0 \to \Hom (I_Z, W\otimes \scrO_X(-F)\otimes K_X) \to \Hom (W,W \otimes K_X) \to H^0(W\otimes \scrO_X(-F)\otimes K_X).$$ Since $-K_X$ is effective and nonzero, $H^0(W\otimes \scrO_X(-F)\otimes K_X)=\Hom (I_Z, W\otimes \scrO_X(-F)\otimes K_X) =0$. Thus $\Hom (W,W \otimes K_X)=0$ as well. Once again $V$ is smoothable. Now we claim that a general smoothing $V'$ of $V$ is Mumford stable. For otherwise by the proof of (2.1) there is an exact sequence $$0 \to \scrO_X(F) \to V' \to \scrO_X(F) \otimes I_Z \to 0$$ as above, with $\ell (Z) \leq \ell (\emptyset) =0$. In this case $V'$ is an extension of $\scrO_X(F)$ by $\scrO_X(F)$, forcing $w=p=0$ and (since $h^1(\scrO_X)= 0$) $V' = \scrO_X(F) \oplus \scrO_X(F) $. \endproof It is natural to make the following conjecture, which is true for geometrically ruled $X$ by \cite{29} and is verified in certain other cases by \cite{34}. \proclaim{ Conjecture 2.4} If $X$ is a rational surface with $-K_X$ effective, then for every choice of $L$, $\Delta$ and $c$, ${\frak M}_L(\Delta, c)$ is either empty or irreducible. \endstatement Let us fix some notations for the rest of this paper. \definition{Definition 2.5} Let $X$ be an algebraic surface (not necessarily rational), and let $\zeta$ be a fixed numerical equivalence class defining a wall of type $(\Delta, c)$. Set $\ell _\zeta = (4c-\Delta ^2 +\zeta ^2)/4 = (\zeta ^2-p)/4$. Choose two nonnegative integers $n_-$ and $n_+$ with $n_-+ n_+ = \ell _\zeta$, and let $E_\zeta^{n_-, n_+}$ be the set of all isomorphism classes of nonsplit extensions of the form $$0 \to \scrO _X(F )\otimes I_{Z_-} \to V \to \scrO _X(\Delta -F ) \otimes I_{Z_+} \to 0$$ with $\zeta \equiv 2F -\Delta$ and $\ell (Z_\pm) = n_\pm$. \enddefinition \medskip We remark that since $\zeta \equiv \Delta \pmod 2$ and $\Delta ^2 - 4c \le \zeta^2 < 0$, $\ell _\zeta$ is a nonnegative integer. If $V$ corresponds to a point of $E_\zeta^{n_-, n_+}$, then $V$ is $L_+$-unstable since $L_+\cdot \zeta > 0$. By (2.2)(ii), $V$ is simple, and if it is $L_-$-semistable then it is actually stable. By (2.3), if $X$ is a rational surface with $-K_X$ effective, then $\frak M_-$ is smooth in a neighborhood of a point corresponding to a sheaf $V$ lying in $E_\zeta^{n_-, n_+}$ for some $\zeta, n_-, n_+$. We shall now study $E_\zeta^{n_-, n_+}$ in more detail for rational surfaces. \lemma{2.6} Suppose that $-K_X$ is effective and that $q(X) =0$. For $Z_-$ and $Z_+$ two fixed zero-dimensional subschemes of $X$ of lengths $n_-$ and $n_+$ respectively, $$\dim \Ext^1(\scrO _X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-}) =n_-+ n_+ + h(\zeta)= \ell _\zeta + h(\zeta),$$ where $$h(\zeta ) = h^1(X; \scrO _X(2F-\Delta)) = \frac{(\zeta \cdot K_X)}{2} - \frac{\zeta ^2}{2} -1.$$ \endstatement \proof Note that $\Hom(\scrO _X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})\subseteq H^0(\scrO_X(2F-\Delta)) = 0$, since $L_- \cdot (2F-\Delta ) < 0$. Likewise $\Ext^2(\scrO _X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})$ is Serre dual to $\Hom (\scrO _X(F )\otimes I_{Z_-} , \scrO _X(\Delta -F ) \otimes I_{Z_+}\otimes K_X) \subseteq H^0(\scrO_X(\Delta -2F) \otimes K_X) \subseteq H^0(\scrO_X(\Delta -2F))$, since $-K_X$ is effective. Thus as $L_+ \cdot (\Delta - 2F) < 0$, $\Ext^2(\scrO _X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})=0$ as well. If we set $\chi (\scrO _X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-}) = \sum _i (-1)^i\dim \Ext^i(\scrO _X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})$, then $\chi (\scrO _X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-}) = -\dim \Ext^1(\scrO _X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})$. Now a standard argument \cite{27} shows that $$\gather \chi (\scrO_X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})\\ = \int_X\ch(\scrO _X(\Delta -F ) \otimes I_{Z_+})\spcheck\cdot \ch(\scrO _X(F)\otimes I_{Z_-})\cdot \Todd_X. \endgather$$ Here given a class $a = \sum a_i \in \bigoplus _iA^i(X)$, we denote by $a\spcheck$ the class $\sum _i (-1)^ia_i$. An easy computation gives $$\gather \int _X\ch(\scrO_X(\Delta -F ) \otimes I_{Z_+})\spcheck\cdot \ch(\scrO _X(F )\otimes I_{Z_-})\cdot \Todd_X\\ =\int _X\ch(\scrO_X(\Delta -F )\spcheck\cdot \ch(\scrO _X(F )\cdot\Todd _X - \ell (Z_-) - \ell(Z_+). \endgather$$ Reversing the above argument, we see that $$\align \int _X\ch(\scrO_X(\Delta -F )\spcheck\cdot \ch(\scrO _X(F )\cdot\Todd _X &= \chi (\scrO_X(2F-\Delta))\\ =- h^1(X; \scrO _X(2F-\Delta)) &= \frac{\zeta ^2}{2}-\frac{(\zeta \cdot K_X)}{2} +1 = -h(\zeta ). \endalign$$ Putting these together we see that $\dim \Ext^1(\scrO _X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})$ is equal to $n_-+ n_+ + h(\zeta)$. \endproof Let us describe the scheme structure on $E_\zeta^{n_-, n_+}$ more carefully. For $Z_-$ and $Z_+$ fixed, the set of extensions in $E_\zeta^{n_-, n_+}$ corresponding to $Z_-$, $Z_+$, is equal to $\Pee \Ext^1(\scrO _X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})$. To make a universal construction, let $H_{n_\pm} = \Hilb ^{n_\pm}X$. Let $\Cal Z_{n_\pm}$ be the universal codimension two subscheme of $X\times H_{n_\pm}$. Let $\pi _1, \pi _2$ be the projections of $X\times H_{n_-}\times H_{n_+}$ to $X$, $H_{n_-}\times H_{n_+}$ respectively, and let $\pi _{1,2}$, $\pi _{1,3}$ be the projections of $X\times H_{n_-}\times H_{n_+}$ to $X\times H_{n_-}$, $X\times H_{n_+}$ respectively. Define $$\Cal E_\zeta ^{n_-, n_+} = Ext ^1_{\pi _2}(\pi _1^*\scrO _X(\Delta -F ) \otimes \pi _{1,3}^*I_{\Cal Z_{n_+}}, \pi _1^*\scrO _X(F )\otimes \pi _{1,2}^*I_{\Cal Z_{n_-}}).$$ The previous lemma and standard base change results show that $\Cal E_\zeta ^{n_-, n_+}$ is locally free of rank $h(\zeta)+\ell _\zeta$ over $H_{n_-}\times H_{n_+}$. We set $E_\zeta^{n_-, n_+} = \Pee ((\Cal E_\zeta ^{n_-, n_+})\spcheck)$, if $h(\zeta)+\ell _\zeta > 0$. Moreover by standard facts about relative Ext sheaves there is an exact sequence $$\gather 0 \to R^1\pi _2{}_*Hom\Big(\pi _1^*\scrO _X(\Delta -F ) \otimes \pi _{1,3}^*I_{\Cal Z_{n_+}}, \pi _1^*\scrO _X(F )\otimes \pi _{1,2}^*I_{\Cal Z_{n_-}}\Big) \to \Cal E_\zeta ^{n_-, n_+} \to \\ \to \pi _2{}_*Ext ^1\Big(\pi _1^*\scrO _X(\Delta -F ) \otimes \pi _{1,3}^*I_{\Cal Z_{n_+}}, \pi _1^*\scrO _X(F )\otimes \pi _{1,2}^*I_{\Cal Z_{n_-}}\Big) \to 0. \endgather$$ \corollary{2.7} With $X$ as in \rom{(2.6)}, if $h(\zeta) +\ell _\zeta= h^1(X; \scrO _X(2F-\Delta)+\ell_\zeta \neq 0$, $E_\zeta^{n_-, n_+}$ is a $\Pee ^{N_\zeta}$-bundle over $H_{n_-}\times H_{n_+}$, where $N_\zeta = \dim \Ext^1 - 1 = h(\zeta)+\ell _\zeta -1$. Thus if $h(\zeta)+\ell _\zeta\neq 0$, then $\dim E_\zeta^{n_-, n_+} = 3\ell _\zeta + h(\zeta)-1$. Moreover in this case $E_{-\zeta}^{n_+, n_-}$ is a $\Pee ^{N_{-\zeta}}$-bundle over $H_{n_+}\times H_{n_-}$, and $N_\zeta + N_{-\zeta} + 2\ell _\zeta = -p-4$. If $h(\zeta) +\ell _\zeta =0$, then $E_\zeta^{0, 0} = \emptyset$ and $E_{-\zeta}^{0, 0} = \Pee ^{-p-3}$ is a component of $\frak M_+$. Finally this last case arises if and only if $\zeta ^2 = p$ and $\zeta \cdot K_X = \zeta ^2+2 = p+2$. \endstatement \proof Note that $N_\zeta \geq 0$ unless $h(\zeta) +\ell _\zeta =0$. Under this assumption, we have $$N_\zeta + N_{-\zeta} + 2\ell _\zeta = 4\ell _\zeta - \zeta ^2 -4= -p-4.$$ The case where $h(\zeta) +\ell _\zeta =0$ is similar. Moreover if $h(\zeta) +\ell_\zeta =0$, then it follows from (2.2)(ii) that all of the sheaves $V$ corresponding to points of $E_{-\zeta}^{0, 0}$ are $L_+$-stable. By (2.2)(i) the map $E_{-\zeta}^{0, 0}\to \frak M_+$ is one-to-one. Since $\frak M_+$ is of dimension $-p-3$ and smooth at points corresponding to the sheaves in $E_{-\zeta}^{0, 0}\to \frak M_+$, the map $E_{-\zeta}^{0, 0}\to \frak M_+$ must be an embedding onto a component of $\frak M_+$. The final statement follows from the formulas $\zeta ^2 = 4\ell_\zeta +p$ and $\dsize h(\zeta)= \frac{(\zeta \cdot K_X)}{2} -\frac{\zeta ^2}{2} -1$. \endproof If $h(\zeta)+\ell_\zeta\neq 0$, then by Lemma 2.2 there is a rational map from $E_\zeta^{n_-, n_+}$ to the moduli space $\frak M_-$ which is birational onto its image. However this map will not in general be a morphism if $n_->0$ (see \cite{16}). We shall study this more carefully in the next sections. Let us also remark that standard theory gives a universal sheaf $\Cal V$ over $E_\zeta^{n_-, n_+}$: \proposition{2.8} Let $\rho \: X\times E_\zeta^{n_-, n_+} \to X\times H_{n_-}\times H_{n_+}$ be the natural projection, and let $\pi_2\: X\times E_\zeta^{n_-, n_+} \to E_\zeta^{n_-, n_+}$ be the projection. Then there is a coherent sheaf $\Cal V$ over $X\times E_\zeta^{n_-, n_+}$ and an exact sequence $$\gather 0 \to \rho ^*\left(\pi _1^*\scrO _X(F )\otimes \pi _{1,2}^*I_{\Cal Z_{n_-}}\right)\otimes \pi_2^*\scrO_{E_\zeta^{n_-, n_+}}(1)\\ \to \Cal V \to \rho ^*\left(\pi _1^*\scrO _X(\Delta -F ) \otimes \pi _{1,3}^*I_{\Cal Z_{n_+}}\right) \to 0.\qed \endgather$$ \endstatement \noindent {\bf Remark 2.9.} Very similar results hold in the case where $-K_X$ is effective and nonzero (corresponding to certain elliptic ruled surfaces) or $K_X=0$ (corresponding to $K3$ or abelian surfaces). For example, in the case of a $K3$ surface $X$, the moduli space is smooth of dimension $-p-6$ away from the sheaves which are strictly semistable for every ample divisor (although there exist components consisting entirely of non-locally free sheaves for small values of $-p$). In this case however $h(\zeta ) = -\zeta ^2/2 -2$ and $N_\zeta + N_{-\zeta} + 2\ell _\zeta = -p-6$, which is equal to the dimension $d$ of the moduli space instead of to $d-1$. For example, if $\ell _\zeta = 0$, then $N_\zeta = N_{-\zeta} = d/2$. In this case $E_\zeta ^{0,0} \cong \Pee ^{d/2}$ is a maximal isotropic submanifold of the symplectic manifold $\frak M_-$. In other words, the natural holomorphic $2$-form $\omega$ on $\frak M_-$ vanishes on $E_\zeta ^{0,0}$ and identifies the normal bundle of $E_\zeta ^{0,0}$ in $\frak M_-$ with the cotangent bundle of $E_\zeta ^{0,0}$. \section{3. Flips of moduli spaces.} In this section, we begin by assuming again that $X$ is an arbitrary algebraic surface. Let $\zeta = \zeta _1, \dots, \zeta _n$ be the positive rational multiples of $\zeta$ such that $\zeta_i$ is an integral class also defining the wall $W^\zeta$. Our goal in this section is to deal with the problem that there is only a rational map in general from $E_{\zeta_i}^{n_-, n_+}$ to $\frak M_-$. We shall do so by finding a sequence of spaces between $\frak M_-$ and $\frak M_+$, each one given by blowing up and down the previous one, such that for an appropriate member of the sequence the rational map $E_{\zeta_i}^{n_-, n_+}\dasharrow \frak M_-$ becomes a morphism (and a smooth embedding in the case of rational surfaces). Throughout the rest of this paper, $L_0$ shall denote any ample divisor contained in the interior of the intersection of $W^\zeta$ and the closures of $\Cal C_\pm$. Recall that we have defined universal semistability after the proof of (2.1). \definition{Definition 3.1} Let $k$ be an integer. A rank two torsion free sheaf $V$ with $c_1(V) = \Delta$ and $\Delta ^2 - 4c_2(V) = p$ is {\sl $(L_0, \zeta, k)$-semistable\/} if $V$ is Mumford $L_0$-semistable and if it is strictly Mumford semistable, then either it is universally semistable or, for all divisors $F$ such that $2F-\Delta \equiv \zeta$, we have the following: \roster \item"{(i)}" If there exists an exact sequence $$0 \to \scrO _X(F )\otimes I_{Z_1} \to V \to \scrO _X(\Delta -F ) \otimes I_{Z_2} \to 0,$$ then $\ell (Z_2) \leq k$ and thus $\ell (Z_1) \geq \ell _\zeta - k$. \item"{(ii)}" If there exists an exact sequence $$0 \to \scrO _X(\Delta - F )\otimes I_{Z_1} \to V \to \scrO _X(F ) \otimes I_{Z_2} \to 0,$$ then $\ell (Z_1) \geq k+1$ and thus $\ell (Z_2) \leq \ell _\zeta - k-1$. \endroster Likewise, setting $\boldsymbol \zeta = (\zeta _1, \dots,\zeta _n)$ and $\bold k = (k_1, \dots, k_n)$, we say that $V$ is {\sl $(L_0, \boldsymbol \zeta, \bold k)$-semistable\/} if $V$ is $(L_0, \zeta _i, k_i)$-semistable for every $i$. Let $\frak M_0^{(\boldsymbol \zeta, \bold k)}$ denote the set of isomorphism classes of $(L_0, \boldsymbol \zeta, \bold k)$-semistable rank two sheaves $V$ with $c_1(V) = \Delta$ and $\Delta ^2 - 4c_2(V) = p$. \enddefinition Next we give some easy properties of $(L_0, \boldsymbol \zeta, \bold k)$-semistability. \lemma{3.2} \roster \item"{(i)}" If $k_i \geq \ell _{\zeta_i}$ for all $i$, and $V$ is not universally semistable, then $V$ is $(L_0, \boldsymbol \zeta, \bold k)$-semistable if and only if it is $L_-$-stable. Likewise if $k_i \leq -1$ for all $i$ and $V$ is not universally semistable, then $V$ is $(L_0, \boldsymbol \zeta, \bold k)$-semistable if and only if it is$L_+$-stable. \item"{(ii)}" If $k_i \geq \ell _{\zeta_i}$ for all $i$, then $\frak M_0^{(\boldsymbol \zeta, \bold k)} = \frak M_-$. Likewise if $k_i \leq -1$ for all $i$, then $\frak M_0^{(\boldsymbol \zeta, \bold k)} = \frak M_+$. \item"{(iii)}" For $n_2 > k_i$, $\frak M_0 ^{(\boldsymbol \zeta, \bold k)} \cap E_{\zeta_i}^{n_1, n_2}= \emptyset$. \item"{(iv)}" There is an injection $E_{\zeta_i}^{\ell_{\zeta_i}-k_i, k_i} \to \frak M_0 ^{(\boldsymbol \zeta, \bold k)}$. Likewise there is an injection $E_{-\zeta_i}^{k_i+1, \ell _{\zeta_i} - k_i-1} \to \frak M_0 ^{(\boldsymbol \zeta, \bold k)}$. Finally, the images of $E_{\zeta_i}^{\ell _{\zeta_i} -k_i, k_i}$ and $E_{\zeta_j}^{\ell _{\zeta_j}-k_j, k_j}$ are disjoint if $i\neq j$. \endroster \endstatement \proof If $k_i\geq \ell _{\zeta_i}$ for all $i$, then the condition that $\ell(Z_2) \leq \ell _{\zeta_i}$ and $\ell (Z_1) \geq 0$ are trivially always satisfied and the conditions $\ell(Z_2) \leq -1$ and $\ell (Z_1) \geq \ell _{\zeta_i} +1$ are vacuous. A similar argument handles the case $k_i \leq -1$ for all $i$. It is easy to see that this implies (i). Statement (ii) follows from (i), and (iii) follows from the definitions. As for (iv), let $V\in E_{\zeta_i}^{\ell _{\zeta_i} -k_i, k_i}$. To decide if $V$ is in $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$, we look for potentially destabilizing subsheaves with torsion free quotient. Similar arguments as in \cite{30} show that the only potentially destabilizing subsheaves with torsion free quotient must be either $\scrO _X(F )\otimes I_{Z_1}$ or $\scrO _X(\Delta - F )\otimes I_{Z}$. By hypothesis, there is a unique subsheaf of $V$ of the form $\scrO _X(F )\otimes I_{Z_1}$, and it is not destabilizing. If there is a subsheaf of the form $\scrO _X(\Delta - F )\otimes I_{Z}$ with torsion free quotient, then by Lemma 2.2 we have $\ell (Z) > \ell (Z_2) = k_i$ and so $\ell (Z) \geq k_i + 1$. Hence such a subsheaf is also not destabilizing. Thus by Definition 3.1 $V$ is $(L_0, \boldsymbol \zeta, \bold k)$-semistable. The fact that the map $E_{\zeta_i}^{\ell _{\zeta_i} -k_i, k_i} \to \frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ is one-to-one and that $E_{\zeta_i}^{\ell _{\zeta_i} -k_i, k_i}$ and $E_{\zeta_j}^{\ell _{\zeta_j} -k_j, k_j}$ are disjoint if $i\neq j$ also follow from similar arguments in \cite{30}. The statement about $E_{-\zeta_i}^{k_i+1, \ell _{\zeta_i} - k_i-1}$ is similar. \endproof Next suppose that we are given two integral vectors $\bold k$ and $\bold k'$ and a subset $I$ of $\{1 \dots, n\}$ such that $k_i' = k_i$ if $i\notin I$ and $k_i' = k_i-1$ if $i\in I$. We investigate the change as we pass from $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ to $\frak M_0 ^{(\boldsymbol \zeta, \bold k')}$. \lemma{3.3} The set of sheaves $V$ in $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ which are not $(L_0, \boldsymbol \zeta, \bold k')$-semistable is exactly the image of $\bigcup _{i\in I}E_{\zeta_i}^{\ell _{\zeta_i}-k_i, k_i}$. Likewise the set of $V\in \frak M_0 ^{(\boldsymbol \zeta, \bold k')}$ which are not $(L_0, \boldsymbol \zeta, \bold k)$-semistable is exactly the image of $\bigcup _{i\in I}E_{-\zeta_i}^{k_i,\ell _{\zeta_i} -k_i}$. \endstatement \proof If $V$ is $(L_0, \boldsymbol \zeta, \bold k)$-semistable but not $(L_0, \boldsymbol \zeta, \bold k')$-semistable, then $V$ must be Mumford strictly $L_0$-semistable. Suppose that the $(L_0, \boldsymbol \zeta, \bold k')$-destabilizing subsheaf is of the form $\scrO _X(F )\otimes I_{Z_1}$, where $F$ corresponds to $\zeta _i$ for some $i\in I$. Then $\ell (Z_2) \leq k_i$ (since $V\in \frak M_0 ^{(\boldsymbol \zeta, \bold k)}$) but $\ell (Z_2) \geq k_i$ (since the subsheaf is $(L_0, \boldsymbol \zeta, \bold k')$-destabilizing, for $k_i' = k_i-1$) so that $\ell (Z_2) = k_i$. Thus $V\in E_\zeta^{\ell _{\zeta_i} -k_i, k_i}$. The other possibility is that the destabilizing subsheaf is of the form $\scrO _X(\Delta - F )\otimes I_{Z_1}$. Here we need $\ell (Z_1) \geq k_i+1$ but $\ell (Z_1) < k_i$ and there are no such sheaves. The statement about $\frak M_0 ^{(\boldsymbol \zeta, \bold k')}$ follows by symmetry. \endproof We shall now describe a sequence of actual moduli spaces $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ for which the integral vector $\bold k$ change in the way described before the statement of (3.3). \definition{Definition 3.4} Suppose that $\zeta _i = r_i\zeta _1$, where $r_i$ is a positive rational number. Given $t\in \Bbb Q$, let $t_i = r_it$, so that $t_1=t$. Suppose that $\dsize \frac{\ell _{\zeta _i} + t_i}2$ is not an integer for any $i$. In this case, define $$k_i(t) = \fracwithdelims[]{\ell _{\zeta _i} + t_i}{2},$$ where $[x]$ is the greatest integer function, and define $\bold k(t)$ to be the vector formed by the $k_i(t)$. A rational number $t$ is {\sl $\zeta _i$-critical\/} if $\dsize \frac{\ell_{\zeta _i} + t_i}2 \in \Zee$ and $-1 \leq \dsize \frac{\ell _{\zeta _i} + t_i}2 \leq \ell _{\zeta _i}$. We shall also say that $t_i$ is {\sl $\zeta _i$-critical\/}. Finally $t$ is {\sl $\boldsymbol \zeta$-critical\/} if it is $\zeta _i$-critical for some $i$. Note that there are only finitely many such $t$. \enddefinition \medskip Given $t \in \Bbb Q$, let $I(t) = \{\, i: t {\text{ is $\zeta _i$-critical}}\,\}$. Suppose that $\varepsilon$ is chosen so that, for every $i$, either there is no $\zeta _i$-critical rational number in $[t_i - r_i\varepsilon, t_i + r_i\varepsilon]$ or $t_i$ is the unique $\zeta _i$-critical rational number in $[t_i - r_i\varepsilon, t_i + r_i\varepsilon]$. Equivalently either there is no $\boldsymbol \zeta$-critical number in $[t-\varepsilon, t+\varepsilon]$ or $t$ is the unique $\boldsymbol \zeta$-critical number in $[t-\varepsilon, t+\varepsilon]$. Then we clearly have: $$k_i(t - \varepsilon) = \cases k_i(t+ \varepsilon) , &\text{if $i\notin I(t)$}\\ k_i(t + \varepsilon) -1, &\text{if $i\in I(t)$.} \endcases$$ In particular if there is no $\boldsymbol \zeta$-critical number in $[t-\varepsilon, t+\varepsilon]$, so that $I(t)= \emptyset$, then $k_i(t - \varepsilon) = k_i(t + \varepsilon)$ for every $i$. Further note that if $t\gg 0$, then $k_i(t) > \ell _{\zeta _i}$ for every $i$, and if $t\ll 0$, then $k_i(t) <-1$ for every $i$. We then have the following theorem, whose proof will be given in the next section: \theorem{3.5} For all $t\in \Bbb Q$ which are not $\boldsymbol \zeta$-critical, there exists a natural structure of a projective scheme on $\frak M_0 ^{(\boldsymbol \zeta, \bold k(t))}$ for which it is a coarse moduli space. \endstatement \medskip The proof of (3.5) will also show that $\frak M_0 ^{(\boldsymbol \zeta, \bold k(t))}$ has the usual properties of a coarse moduli space: all sheaves corresponding to points of $\frak M_0 ^{(\boldsymbol \zeta, \bold k(t))}$ will turn out to be simple (as they will turn out to be stable for an appropriate notion of stability), a classical or formal neighborhood of a point of $\frak M_0 ^{(\boldsymbol \zeta, \bold k(t))}$ may be identified with the universal deformation space of the corresponding sheaf, and there exists a universal sheaf locally in the classical or \'etale topology around every point of $\frak M_0 ^{(\boldsymbol \zeta, \bold k(t))}$. For the rest of this section, we shall again restrict to the case where $X$ is a rational surface with $-K_X$ effective, unless otherwise noted. Let $\zeta = \zeta _i$ for some $i$ and let $\bold k=\bold k(t)$ for some $t$ which is not $\boldsymbol \zeta$-critical. The first step is to make some infinitesimal calculations concerning the differential of the map $E_\zeta^{\ell _\zeta -k, k}\to \frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ and the normal bundle to its image. \proposition{3.6} The map $E_\zeta^{\ell _\zeta -k, k}\to \frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ is an immersion. The normal bundle $\Cal N_\zeta ^{\ell _\zeta -k, k}$ to $E_\zeta^{\ell _\zeta -k, k}$ in $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ is exactly $\rho ^*\Cal E _{-\zeta}^{k, \ell _\zeta -k}\otimes \scrO_{E_\zeta ^{\ell _\zeta -k, k}}(-1)$, in the notation of the previous section. \endstatement \proof Since every sheaf in $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ is actually stable and therefore simple (which was also proved in (2.2)) we may identify an analytic neighborhood of $V\in \frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ with the germ of the universal deformation space for $V$, i\.e\. with $\Ext ^1(V,V)$. Let us now calculate the tangent space to $E_\zeta ^{\ell _\zeta -k, k}$ at $V$: suppose that $\xi \in \Ext^1(\scrO _X(\Delta -F ) \otimes I_{Z_2} , \scrO _X(F)\otimes I_{Z_1}) =\Ext^1$ is a nonzero extension class corresponding to $V$, where $\ell (Z_1) = \ell _\zeta -k$ and $\ell (Z_2) = k$. Let $H_{\ell _\zeta -k} = \Hilb ^{\ell _\zeta -k}X$ and $H_k = \Hilb ^kX$. Then there is the following exact sequence for the tangent space to $E_\zeta ^{\ell _\zeta -k, k}$ at $\xi$: $$0\to \Ext ^1/\Cee \cdot \xi \to T_\xi E_\zeta ^{\ell _\zeta -k, k}\to T_{Z_1}H_{\ell _\zeta -k} \oplus T_{Z_2}H_k \to 0.$$ Note further that the tangent space to $\Hilb ^nX$ at $Z$ is equal to $\Hom(I_Z,\scrO_Z)$, which we may further canonically identify with $\Ext^1(I_Z, I_Z)$ since $X$ is rational and by a local calculation. We then have the following: \proposition{3.7} For all nonzero $\xi \in \Ext^1$, the natural map from a neighborhood of $\xi$ in $E_\zeta ^{\ell _\zeta -k, k}$ to $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ is an immersion at $\xi$. The image of $T_\xi E_\zeta ^{\ell _\zeta -k, k}$ in $\Ext ^1(V,V)$ is exactly the kernel of the natural map $\Ext ^1(V,V)\to \Ext ^1(\scrO _X(F)\otimes I_{Z_1}, \scrO _X(\Delta -F ) \otimes I_{Z_2})$, and the normal space to $E_\zeta ^{\ell _\zeta -k, k}$ at $\xi$ in $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ may be canonically identified with $\Ext ^1 (\scrO _X(F)\otimes I_{Z_1}, \scrO _X(\Delta -F ) \otimes I_{Z_2})$. \endstatement \noindent {\it Proof.} Consider the natural map from $\Ext ^1(V,V)$ to $\Ext ^1 (\scrO _X(F)\otimes I_{Z_1}, \scrO _X(\Delta -F ) \otimes I_{Z_2})$. We claim that this map is surjective and will describe its kernel in more detail. The map factors into two maps: $$\gather \Ext ^1(V,V) \to \Ext^1(V, \scrO _X(\Delta -F ) \otimes I_{Z_2}) \\ \Ext^1(V, \scrO _X(\Delta -F ) \otimes I_{Z_2}) \to \Ext ^1 (\scrO _X(F)\otimes I_{Z_1}, \scrO _X(\Delta -F ) \otimes I_{Z_2}). \endgather$$ The cokernel of the first map is contained in $\Ext^2(V, \scrO _X(F)\otimes I_{Z_1})$. To see that this group is zero, apply Serre duality: it suffices to show that $\Hom(\scrO _X(F)\otimes I_{Z_1}, V\otimes K_X) =0$. From the defining exact sequence for $V$, we have an exact sequence $$\gather 0 \to \Hom(\scrO _X(F)\otimes I_{Z_1},\scrO _X(F)\otimes I_{Z_1}\otimes K_X) \to \Hom(\scrO _X(F)\otimes I_{Z_1}, V\otimes K_X)\\ \to \Hom(\scrO _X(F)\otimes I_{Z_1},\scrO _X(\Delta -F ) \otimes I_{Z_2}). \endgather$$ The first term is just $H^0(K_X) = 0$ and the third is contained in $H^0(\scrO_X(\Delta -2F) \otimes K_X) =0$. Thus $\Hom(\scrO _X(F)\otimes I_{Z_1}, V\otimes K_X) =0$. The vanishing of the cokernel of the second map, namely $\Ext ^2(\scrO _X(\Delta -F ) \otimes I_{Z_2}, \scrO _X(\Delta -F ) \otimes I_{Z_2})$, is similar. Thus $\Ext ^1(V,V)\to \Ext ^1 (\scrO _X(F)\otimes I_{Z_1}, \scrO _X(\Delta -F ) \otimes I_{Z_2})$ is onto. If $K$ is the kernel, then arguments as above show that there is an exact sequence $$0 \to \Ext^1(V, \scrO _X(F)\otimes I_{Z_1}) \to K \to \Ext^1(\scrO _X(\Delta -F ) \otimes I_{Z_2}, \scrO _X(\Delta -F ) \otimes I_{Z_2})\to 0.$$ Here $\Ext^1(\scrO _X(\Delta -F ) \otimes I_{Z_2}, \scrO _X(\Delta -F ) \otimes I_{Z_2}) = \Ext^1(I_{Z_2}, I_{Z_2})$ is the tangent space to $H_k$. Moreover, there is an exact sequence $$\gather \Hom(\scrO _X(F)\otimes I_{Z_1}, \scrO _X(F)\otimes I_{Z_1}) \to \Ext^1(\scrO _X(\Delta -F ) \otimes I_{Z_2},\scrO _X(F)\otimes I_{Z_1}) \to \\ \to \Ext^1(V, \scrO _X(F)\otimes I_{Z_1}) \to \Ext^1(\scrO _X(F)\otimes I_{Z_1}, \scrO _X(F)\otimes I_{Z_1})\to 0. \endgather$$ The last term is $\Ext^1(I_{Z_1}, I_{Z_1})$ which is the tangent space to $H_{\ell _\zeta -k}$ at $Z_1$, and the first two terms combine to give $\Ext^1/\Cee \cdot \xi$. Thus the kernel $K$ looks very much like the tangent space to $E_\zeta ^{\ell _\zeta -k, k}$ at $\xi$ and both spaces have the same dimension. Let us describe the tangent space to $E_\zeta ^{\ell _\zeta -k, k}$ at $\xi$ and the differential of the map $E_\zeta ^{\ell _\zeta -k, k}$ to $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ in more intrinsic terms. It is easy to see that a $\Spec \Cee[\epsilon]$-valued point of $E_\zeta ^{\ell _\zeta -k, k}$ which restricts to $\xi$ defines two codimension two subschemes $\Cal Z_1\subseteq X\times \Spec\Cee[\epsilon]$, $\Cal Z_2\subseteq X\times \Spec\Cee[\epsilon]$, flat over $\Spec\Cee[\epsilon]$, restricting to $Z_i$ over $X$, and an extension $\Cal V$ over $X\times \Spec\Cee[\epsilon]$ of the form $$0 \to \pi _1^* \scrO _X(F)\otimes I_{\Cal Z_1} \otimes \to \Cal V \to \pi _1^*\scrO _X(\Delta -F ) \otimes I_{\Cal Z_2}\to 0.$$ Conversely such a choice of $\Cal Z_1$, $\Cal Z_2$ and $\Cal V$ define a $\Spec \Cee[\epsilon]$-valued point of $E_\zeta ^{\ell _\zeta -k, k}$. Thus there is a commutative diagram with exact rows and columns: $$\CD @. 0 @. 0 @. 0 @.\\ @. @VVV @VVV @VVV @. \\ 0 @>>> \scrO _X(F)\otimes I_{Z_1} @>>> V @>>> \scrO _X(\Delta -F ) \otimes I_{Z_2} @>>> 0\\ @. @VVV @VVV @VVV @. \\ 0 @>>> \pi _1^* \scrO _X(F)\otimes I_{\Cal Z_1} @>>> \Cal V @>>> \pi _1^*\scrO _X(\Delta -F ) \otimes I_{\Cal Z_2}@>>> 0\\ @. @VVV @VVV @VVV @. \\ 0 @>>> \scrO _X(F)\otimes I_{Z_1} @>>> V @>>> \scrO _X(\Delta -F ) \otimes I_{Z_2} @>>> 0\\ @. @VVV @VVV @VVV @. \\ @. 0 @. 0 @. 0 @. \endCD$$ Here the extension $\Cal V$ of $V$ by $V$, viewed as a point of $\Ext ^1(V,V)$, corresponds to the Kodaira-Spencer map of the deformation $\Cal V$ of $V$. Likewise the left and right hand columns give classes in $\Ext ^1(I_{Z_1}, I_{Z_1})$ and $\Ext ^1(I_{Z_2}, I_{Z_2})$ corresponding to $\Cal Z_1$ and $\Cal Z_2$. A straightforward diagram chase shows that if $\Cal V$ fits into this commutative diagram then the image of the extension class $\xi \in \Ext ^1(V,V)$ corresponding to $\Cal V$ in $\Ext ^1 (\scrO _X(F)\otimes I_{Z_1}, \scrO _X(\Delta -F ) \otimes I_{Z_2})$ is zero. To see the converse, that every element in the kernel $K$ of the map $\Ext ^1(V,V) \to \Ext ^1 (\scrO _X(F)\otimes I_{Z_1}, \scrO _X(\Delta -F ) \otimes I_{Z_2})$ is the image of a tangent vector to $E_\zeta ^{\ell _\zeta -k, k}$ at $\xi$, use the arguments above which show that there is a surjection from $K$ to $$\Ext^1(\scrO _X(\Delta -F ) \otimes I_{Z_2}, \scrO _X(\Delta -F ) \otimes I_{Z_2}) = \Ext ^1(I_{Z_2}, I_{Z_2}).$$ Thus there is an induced extension of $\scrO _X(\Delta -F ) \otimes I_{Z_2}$ by $\scrO _X(\Delta -F ) \otimes I_{Z_2}$, necessarily of the form $\scrO _X(\Delta -F ) \otimes I_{\Cal Z_2}$, and a map from $\Cal V$ to $\scrO _X(\Delta -F ) \otimes I_{\Cal Z_2}$, necessarily a surjection. The kernel of this surjection then defines an extension $\scrO _X(F)\otimes I_{\Cal Z_1}$ of $\scrO _X(F)\otimes I_{Z_1}$ by $\scrO _X(F)\otimes I_{Z_1}$. It follows that $K$ is in the image of the tangent space to $E_\zeta ^{\ell _\zeta -k, k}$ at $\xi$. By counting dimensions the map on tangent spaces from $T_\xi E_\zeta ^{\ell _\zeta -k, k}$ to $\Ext ^1(V,V)$ is injective, showing that the map from $E_\zeta ^{\ell _\zeta -k, k}$ to $\frak M_0^{(\boldsymbol \zeta, \bold k)}$ is an immersion and identifying the normal space at $\xi$. \qed Let us continue the proof of Proposition 3.6. To give a global description of the normal bundle to $E_\zeta ^{\ell _\zeta -k, k}$ in $\frak M_0^{(\boldsymbol \zeta, \bold k)}$, recall by standard deformation theory \cite{10} that the pullback of the tangent bundle of $\frak M_0^{(\boldsymbol \zeta, \bold k)}$ to $E_\zeta ^{\ell _\zeta -k, k}$ is just $Ext ^1_{\pi_2}(\Cal V, \Cal V)$, where $\Cal V$ is the universal sheaf over $X\times E_\zeta ^{\ell _\zeta -k, k}$ described in (2.8) and $\pi_2\: X \times E_\zeta ^{\ell _\zeta -k, k} \to E_\zeta ^{\ell _\zeta -k, k}$ is the second projection. Moreover the calculations above globalize to show that the normal bundle is exactly $$Ext ^1_{\pi_2}(\rho ^*\left(\pi _1^*\scrO _X(F )\otimes \pi _{1,2}^*I_{\Cal Z_1}\right)\otimes \pi_2^*\scrO_{E_\zeta^{\ell _\zeta -k, k}}(1), \rho ^*\left(\pi _1^*\scrO _X(\Delta -F ) \otimes \pi _{1,3}^*I_{\Cal Z_2}\right)),$$ where $\rho \: X\times E_\zeta^{\ell _\zeta -k, k} \to X \times H_{\ell _\zeta -k}\times H_k$ is the natural projection. Using standard base change results and the projection formula, we see that this sheaf is equal to $$\rho ^*Ext ^1_{\pi _2}(\pi _1^*\scrO _X(F )\otimes \pi _{1,2}^*I_{\Cal Z_1}, \pi _1^*\scrO _X(\Delta -F ) \otimes \pi _{1,3}^*I_{\Cal Z_2})\otimes \scrO_{E_\zeta^{\ell _\zeta -k, k}}(-1),$$ which is the same as $\rho ^*\Cal E _{-\zeta}^{k,\ell _\zeta -k}\otimes \scrO_{E_\zeta ^{\ell _\zeta -k, k}}(-1)$. \endproof Finally, to compare the moduli space $\frak M_0 ^{(\boldsymbol \zeta, \bold k(t+\varepsilon))}$ with $\frak M_0 ^{(\boldsymbol \zeta, \bold k(t-\varepsilon))}$, where $t$ is the unique $\boldsymbol \zeta$-critical point in $[t-\varepsilon, t +\varepsilon]$, we shall need the following result which is a straightforward generalization of (A.2) of \cite{11}. \proposition{3.8} Let $X$ be a smooth projective scheme or compact complex manifold, and let $T$ be smooth. Suppose that $\Cal V$ is a rank two reflexive sheaf over $X\times T$, flat over $T$. Let $D$ be a reduced divisor on $T$, not necessarily smooth and let $i\: D \to T$ be the inclusion. Suppose that $L$ is a line bundle on $X$ and that $\Cal Z$ is a codimension two subscheme of $X\times D$, flat over $D$. Suppose further that $\Cal V \to i_*\pi _1^*L\otimes I_{\Cal Z}$ is a surjection, and let $\Cal V'$ be its kernel: $$0 \to \Cal V' \to \Cal V \to i_*\pi _1^*L\otimes I_{\Cal Z} \to 0.$$ Then there is a line bundle $M$ on $X$ and a subscheme $\Cal Z'$ of $X\times D$ codimension at least two, flat over $D$, with the following properties: \roster \item"{(i)}" $\Cal V'$ is reflexive and flat over $T$. \item"{(ii)}" There are exact sequences $$\align 0 \to \pi _1^*M \otimes I_{\Cal Z'} \to &\Cal V|D \to \pi _1^*L\otimes I_{\Cal Z} \to 0 ;\\ 0\to \pi _1^*L\otimes I_{\Cal Z}\otimes \scrO_D(-D) \to &\Cal V'|D \to \pi _1^*M \otimes I_{\Cal Z'} \to 0, \endalign$$ which restrict for each $t\in D$ to give exact sequences $$\align 0 \to M\otimes I_{Z'} \to &V_t \to L\otimes I_Z \to 0;\\ 0\to L\otimes I_Z \to &(V_t)' \to M\otimes I_{Z'} \to 0. \endalign$$ Here $Z$ is the subscheme of $X$ defined by $\Cal Z$ for the slice $X\times \{t\}$ and $Z'_t$ is likewise defined by $\Cal Z'$. \item"{(iii)}" If $D$ is smooth, then the extension class corresponding to $(V_t)'$ in $\Ext ^1(M\otimes I_W, L\otimes I_Z)$ is defined by the image of the normal vector to $D$ at $t$ under the composition of the Kodaira-Spencer map from the tangent space of $T$ at $t$ to $\Ext ^1(V_t, V_t)$, followed by the natural map $\Ext ^1(V_t, V_t) \to \Ext ^1(M\otimes I_{Z'}, L\otimes I_Z)$. \endroster \endstatement \medskip Here $\Cal V'$ is called the {\sl elementary modification\/} of $\Cal V$ along $D$. This construction has the following symmetry: if we make the elementary modification of $\Cal V'$ along $D$ corresponding to the surjection $\Cal V' \to i_*\bigl(\pi _1^*M \otimes I_{\Cal Z'}\bigr)$, then the result is $\Cal V \otimes \scrO_{X\times T}(-(X\times D))$. Here is the typical way that we will apply the above: given $X$, let $M$ be a smooth manifold and $Y$ a submanifold of $M$. Let $T$ be the blowup of $M$ along $Y$ and let $D$ be the exceptional divisor. Let $\pi \: T \to M$ be the natural map. Then, given $\xi \in D$, the image in the normal space to $\pi(\xi)$ of the normal direction at $\xi$ to $D$ under $\pi _*$ may be identified with the line in the normal space corresponding to $\xi$. We can now state the main result as follows: \theorem{3.9} Suppose that $t$ is the unique $\boldsymbol \zeta$-critical point in $[t-\varepsilon, t +\varepsilon]$. If $h(\pm\zeta_i) + \ell _{\pm\zeta_i} \neq 0$ for every $i$, then the rational map $\frak M_0 ^{(\boldsymbol \zeta, \bold k(t+\varepsilon))} \dasharrow \frak M_0 ^{(\boldsymbol \zeta, \bold k(t-\varepsilon))}$ is obtained as follows. For every $i$, fixing $\zeta _i =\zeta$ and $k_i(t+\varepsilon) = k$, blow up $E_\zeta^{\ell _\zeta -k, k}$ in $\frak M_0 ^{(\boldsymbol \zeta, \bold k(t+\varepsilon))}$. Then the exceptional divisor $D$ is a $\Pee ^{N_\zeta}\times \Pee ^{N_{-\zeta}}$-bundle over $\Hilb ^{\ell _\zeta -k}X\times \Hilb ^kX$. Moreover this divisor can be contracted in two different ways. Contracting the $\Pee ^{N_{-\zeta}}$ fibers for all possible $\zeta$ gives $\frak M_0 ^{(\boldsymbol \zeta, \bold k(t+\varepsilon))}$. Contracting the $\Pee ^{N_{\zeta}}$ fibers for all possible $\zeta$ gives $\frak M_0 ^{(\boldsymbol \zeta, \bold k(t-\varepsilon))}$. Moreover the morphism from the blowup to $\frak M_0 ^{(\boldsymbol \zeta, \bold k(t-\varepsilon))}$ is induced by an elementary modification as in \rom{(3.8)}, and the image of the the component of the exceptional divisor which is the blowup of $E_\zeta^{\ell _\zeta -k, k}$ is $E_{-\zeta}^{k, \ell _\zeta -k}$. Finally the construction is symmetric. Similar statements hold if $h(\pm\zeta_i) + \ell _{\pm\zeta_i}=0$ for some $i$, where we must also add in or delete an extra component coming from $\pm\zeta _i$. \endstatement \proof Begin by blowing up $E_\zeta^{\ell _\zeta -k, k}$ in $\frak M_0 ^{(\boldsymbol \zeta, \bold k(t+\varepsilon))}$ for all possible $\zeta$. For simplicity we shall just write down the argument in case there is only one $\zeta$; the general case is just additional notation. Let $\widetilde{\frak M}_0^{(\boldsymbol \zeta, \bold k(\bold t+\varepsilon))}$ denote the blowup and $D$ the exceptional divisor. Note that the normal bundle $\Cal N_\zeta ^{\ell _\zeta -k, k}$ to $E_\zeta^{\ell _\zeta -k, k}$ in $\frak M_0 ^{(\boldsymbol \zeta, \bold k(\bold t+\varepsilon))}$ is $\rho ^*\Cal E _{-\zeta}^{k, \ell _\zeta -k}\otimes \scrO_{E_\zeta ^{\ell _\zeta -k, k}}(-1)$, where $\rho\: E_\zeta ^{\ell _\zeta -k, k} \to \Hilb ^{\ell _\zeta -k}X\times \Hilb ^kX$ is the projection. In particular $\Cal N_\zeta ^{\ell _\zeta -k, k}$ restricts to each fiber $\Pee ^{N_\zeta}$ to a bundle of the form $\left[\scrO_{\Pee ^{N_\zeta}}^N\right]\otimes \scrO_{\Pee ^{N_\zeta}}(-1)$, and an easy calculation using (2.7) shows that $N= N_{-\zeta}+1$. It follows that the fibers of the induced map from $D$ to $\Hilb ^{\ell _\zeta -k}X\times \Hilb ^kX$ are naturally $\Pee ^{N_\zeta}\times \Pee ^{N_{-\zeta}}$. Moreover it is easy to see that $\scrO(D)|\Pee ^{N_\zeta}= \scrO_{\Pee ^{N_\zeta}}(-1)$, using for example the fact that $\scrO(D)|\Pee ^{N_\zeta}\times \Pee ^{N_{-\zeta}}= \scrO(a,-1)$ for some $a$ and the fact that $$\Cal N_\zeta ^{\ell _\zeta -k, k}|\Pee ^{N_\zeta} = R^0\pi _1{}_*[\scrO(-D)|\Pee ^{N_\zeta}\times \Pee ^{N_{-\zeta}}]= \left[\scrO_{\Pee ^{N_\zeta}}^{N_{-\zeta}+1}\right]\otimes \scrO_{\Pee ^{N_\zeta}}(-a).$$ For the rest of the argument, we assume that there exists a universal family on $X\times \frak M_0^{(\boldsymbol \zeta, \bold k(\bold t+\varepsilon))}$. Of course, such a family will only exist locally in the classical or \'etale topology, but this will suffice for the argument. Let $\Cal U$ be the pullback of the universal family to $X\times \widetilde{\frak M}_0^{(\boldsymbol \zeta, \bold k(\bold t+\varepsilon))}$. Locally again we may assume that the restriction of $\Cal U$ to $X\times D$ is the pullback of the universal extension $\Cal V$ of (2.8): $$\gather 0 \to \rho ^*\left(\pi _1^*\scrO _X(F )\otimes \pi _{1,2}^*I_{\Cal Z_{n_-}}\right)\otimes \pi_2^*\scrO_{E_\zeta^{n_-, n_+}}(1)\\ \to \Cal V \to \rho ^*\left(\pi _1^*\scrO _X(\Delta -F ) \otimes \pi _{1,3}^*I_{\Cal Z_{n_+}}\right) \to 0. \endgather$$ Now consider the effect of making an elementary transformation of $\Cal U$ on $X\times \widetilde{\frak M}_0^{(\boldsymbol \zeta, \bold k(\bold t+\varepsilon))}$ along the divisor $D$, using the morphism from $\Cal U$ to the pullback of $\rho ^*\left(\pi _1^*\scrO _X(\Delta -F ) \otimes \pi _{1,3}^*I_{\Cal Z_k}\right)$ given by considering the pullback of the universal extension. Applying (3.8) to the elementary transformation $\Cal U'$, we see that the fiber of $\Cal U'$ at a point of the fiber $\Pee ^{N_\zeta}\times \Pee ^{N_{-\zeta}}$ lying over a point $(Z_1, Z_2)\in \Hilb ^{\ell _\zeta -k}X\times \Hilb ^kX$ is given by a nonsplit extension of the form $$0 \to \scrO_X(\Delta - F)\otimes I_{Z_2} \to U \to \scrO_X(F)\otimes I_{Z_1} \to 0.$$ Moreover the extension class corresponding to $U$ is given by the projectivized normal vector in $\Pee ^{N_{-\zeta}}$. Thus it is independent of the first factor $\Pee ^{N_\zeta}$ and the set of all possible such classes is parametrized by the second factor $\Pee ^{N_{-\zeta}}$. There is then an induced morphism from $\widetilde{\frak M}_0^{(\boldsymbol \zeta, \bold k(t+\varepsilon))}$ to $\frak M_0^{(\boldsymbol \zeta, \bold k(t-\varepsilon))}$ and clearly it has the effect of contracting $D$ along its first ruling and has the property that the image of $D$ is exactly $E_{-\zeta}^{k, \ell _\zeta -k}$. We leave the symmetry of the construction to the reader. This concludes the proof of (3.9). \endproof \noindent {\bf Remark 3.10.} In the $K3$ or abelian case, the arguments of this section show that the rational map $\frak M_0 ^{(\boldsymbol \zeta, \bold k(\bold t+\varepsilon))} \dasharrow \frak M_0 ^{(\boldsymbol \zeta, \bold k(t-\varepsilon))}$ is a Mukai elementary transformation \cite{26, 28}. \medskip We can also use (3.8) to analyze the rational map from $E_\zeta ^{n_-, n_+}$ to $\frak M_-$, in the case where it is not a morphism. For simplicity we shall only consider the case of $E_\zeta ^{1,0}$, i\.e\. $\ell _\zeta = 1$. In this case $Z_- = p\in X$ and $I_{Z_-} = \frak m_p$ is the maximal ideal sheaf of $p$. Moreover $\Ext ^1(\scrO _X(\Delta -F ), \scrO _X(F)\otimes \frak m_p) = H^1(\scrO _X(2F-\Delta)\otimes \frak m_p)$. There is an exact sequence $$0 \to H^0(\Cee _p) \to H^1(\scrO _X(2F-\Delta)\otimes \frak m_p) \to H^1(\scrO _X(2F-\Delta)) \to 0.$$ Moreover, for $p$ fixed, the extensions $V$ corresponding to a split extension for $V\ddual$ are exactly the kernel of the map from $H^1(\scrO _X(2F-\Delta)\otimes \frak m_p)$ to $H^1(\scrO _X(2F-\Delta))$, i\.e\. the image of $H^0(\Cee _p)$. The normal space is thus identified with $H^1(\scrO _X(2F-\Delta))$. Now if the extension for $V\ddual$ is split, then there is a map $\scrO _X(\Delta -F )\otimes \frak m_p \to V$ with quotient $\scrO _X(F)$. This way of realizing $V$ as an extension gives a surjection $\Ext ^1(V,V) \to \Ext ^1(\scrO _X(\Delta -F )\otimes \frak m_p, \scrO _X(F))$, and we must look at the image of the normal space $H^1(\scrO _X(2F-\Delta))$ in this extension group. On the other hand, we have an exact sequence $$0\to H^1(\scrO _X(2F-\Delta)) \to \Ext ^1(\scrO _X(\Delta -F )\otimes \frak m_p, \scrO _X(F)) \to H^0(\Cee _p) \to 0$$ coming from the long exact Ext sequence, and it is an easy diagram chase to see that the induced map $\Ext ^1(\scrO _X(\Delta -F ), \scrO _X(F)\otimes \frak m_p) \to \Ext ^1(\scrO _X(\Delta -F )\otimes \frak m_p, \scrO _X(F))$ factors through the map $\Ext ^1(\scrO _X(\Delta -F ), \scrO _X(F)\otimes \frak m_p) \to H^1(\scrO _X(2F-\Delta))$ and that the image is exactly the natural subgroup $H^1(\scrO _X(2F-\Delta))$ of $\Ext ^1(\scrO _X(\Delta -F )\otimes \frak m_p, \scrO _X(F))$. The above has the following geometric interpretation: the locus $U$ in $E_\zeta ^{1,0}$ of $L_-$-unstable sheaves is in fact a section of $E_\zeta ^{1,0}$. If we blow up this section and then make the elementary transformation, the result is exactly the set of elements of $E_\zeta ^{0,1}$ corresponding to nonlocally free sheaves. This set is already a divisor in $E_\zeta ^{0,1}$. There is thus a morphism from the blowup of $E_\zeta ^{1,0}$ along $U$ to $\frak M_-$ which is an embedding into $\frak M_-$. Its image $(E_\zeta ^{1,0})'$ in $\frak M_-$ meets $E_\zeta ^{0,1}$ exactly along the divisor in $E_\zeta ^{0,1}$ of nonlocally free sheaves. We can now give a picture of the birational map from $\frak M_-$ to $\frak M_+$ in this case. Begin with the subvariety $E_\zeta ^{0,1}$ in $\frak M_-$ and blow it up. Let $D^{0,1}$ be the exceptional divisor, ruled in two different ways. As $E_\zeta ^{0,1}$ meets $(E_\zeta ^{1,0})'$ along a divisor, the proper transform of $(E_\zeta ^{1,0})'$ in the blowup is again $(E_\zeta ^{1,0})'$. Making the elementary modification along $D^{0,1}$, we then blow down $D^{0,1}$ to get a new moduli space. This moduli space then contains $E_\zeta ^{1,0}$. At this point we can then blow up $E_\zeta ^{1,0}$ and contract the new exceptional divisor $D^{1,0}$ to obtain $\frak M_+$ (a few extra details need to be checked here concerning the Kodaira-Spencer class). Note again the symmetry of the situation. In principle we could hope to carry through this analysis to the case where $\ell _\zeta >1$ as well, but we run into trouble with the birational geometry of $\Hilb ^nX$. Somehow the construction of our auxiliary sequence of moduli spaces has eliminated the necessity for understanding this birational geometry in detail. \section{4. Mixed stability and mixed moduli spaces.} Our goal in this section is to give a proof of Theorem 3.5 (for an arbitrary algebraic surface $X$). By way of motivation for our construction, let us analyze Gieseker semistability more closely. In the notation of the last section, we suppose that $L_0$ is an ample line bundle lying on a unique wall $W$ of type $(w,p)$, and let $\zeta _1, \dots, \zeta _n$ be the integral classes of type $(w,p)$ defining $W$. Let $V$ be an $L_0$-semistable rank two sheaf. Thus either $V$ is Mumford $L_0$-stable or it is Mumford strictly semistable. In the second case, let $\scrO_X(F)\otimes I_{Z_1}$ be a destabilizing subsheaf and suppose that there is an exact sequence $$0 \to \scrO_X(F)\otimes I_{Z_1} \to V \to \scrO_X(\Delta -F)\otimes I_{Z_2}\to 0.$$ Let $\zeta = 2F-\Delta$. We shall assume that $\zeta =\zeta _i$ for some $i$, or equivalently that $\zeta$ is not numerically equivalent to zero (i.e\., $V$ is not universally semistable). By assumption $\mu _{L_0}(V) \geq \mu _{L_0}(\scrO_X(F)\otimes I_{Z_1})$, and so $\chi(V) \geq 2\chi (\scrO_X(F)\otimes I_{Z_1})$. Since $\chi (V) = \chi (\scrO_X(F)\otimes I_{Z_1}) + \chi (\scrO_X(\Delta -F)\otimes I_{Z_2})$, we may rewrite this last condition as $$\chi(\scrO_X(\Delta -F)\otimes I_{Z_2}) - \chi (\scrO_X(F)\otimes I_{Z_1}) \geq 0.$$ Now from the exact sequence $$0 \to \scrO_X(F)\otimes I_{Z_1} \to \scrO_X(F) \to \scrO_{Z_1} \to 0,$$ we see that $\chi(\scrO_X(F)\otimes I_{Z_1}) = \chi(\scrO_X(F)) - \ell (Z_1)$, and similarly $\chi(\scrO_X(\Delta -F)\otimes I_{Z_2}) = \chi(\scrO_X(\Delta -F)) - \ell(Z_2)$. By Riemann-Roch, $$\align \chi(\scrO_X(\Delta -F)) - \chi(\scrO_X(F)) &= \frac12((\Delta -F)^2 - (\Delta -F)\cdot K_X - F^2 + F\cdot K_X)\\ &= \frac12(\Delta ^2 - 2\Delta \cdot F + \zeta \cdot K_X) \\ &= \frac12\zeta\cdot(K_X-\Delta)=t. \endalign$$ Thus we have the following conditions on $Z_1$ and $Z_2$: $$\align \ell (Z_2) - \ell (Z_1) &\leq t;\\ \ell (Z_2) + \ell (Z_1) &= \ell _\zeta, \endalign$$ and so $2\ell (Z_2) \leq \ell _\zeta +t$. Setting $k= \dsize\fracwithdelims[]{\ell _\zeta +t}2$, we have $\ell (Z_2) \leq k$. Applying a similar analysis to a subsheaf of the form $\scrO_X(\Delta -F) \otimes I_{Z_1}$ shows that, if there is such a subsheaf, with a torsion free quotient $\scrO_X(F)\otimes I_{Z_2}$, then $$\ell (Z_2)\leq \frac{\ell _\zeta - t}2 = \ell _\zeta - \frac{\ell _\zeta +t}2.$$ In particular, if $\dsize\frac{\ell _\zeta +t}2$ is not an integer, then this condition becomes $\ell (Z_2) \leq \ell _\zeta - k-1$. Thus, provided $\dsize\frac{\ell _\zeta +t}2$ is not an integer for every $\zeta$ defining the wall $W$ (i.e\. $t$ is not $\zeta$-critical for every $\zeta$), $V$ is $(L_0, \zeta, k)$-semistable for $k =\dsize\fracwithdelims[]{\ell _\zeta +t}2$ and indeed $V$ is $(L_0, \boldsymbol\zeta, \bold k)$-semistable, where $\bold k$ is defined in the obvious way. Conversely, assuming that $t$ is not $\zeta$-critical for every $\zeta$, $V$ is Gieseker $L_0$-semistable, indeed Gieseker $L_0$-stable, if it is $(L_0, \boldsymbol\zeta, \bold k)$-semistable for $\bold k$ as above. We would like to produce a similar condition where $t$ is allowed to be any rational number which is not $\zeta$-critical. One way to think of this problem is to consider the analogous problem where we replace $\Delta$ by $\Delta + 2 \Xi$ and make the corresponding change in $c$, so that $\Delta$ and $p$ remain the same. This corresponds to twisting $V$ by $\scrO_X(\Xi)$, and $t$ is replaced by $t - \zeta \cdot \Xi$. In particular, we see that the notion of Gieseker stability is rather sensitive to twisting by a line bundle. Moreover if $W$ is defined by exactly one $\zeta$ such that there exists a divisor $\Xi$ with $\zeta \cdot \Xi=1$, for example if $\zeta$ is primitive and $p_g(X)=0$, it is easy to see that we can construct the appropriate moduli spaces as Gieseker moduli spaces corresponding to twists of $V$ by various multiples of $\Xi$. In general however we will need to consider a problem which is roughly analogous to allowing twists of $V$ by a $\Bbb Q$-divisor $\Xi$. This is the goal of the following definition of mixed stability: \definition{Definition 4.1} Let $X$ be an algebraic surface and let $L_0$ be an ample line bundle on $X$. Fix line bundles $H_1$ and $H_2$ on $X$ and positive integers $a_1$ and $a_2$. For every torsion free sheaf $V$ on $X$ of rank $r$, define $$p_{V; H_1, H_2, a_1, a_2}(n)= \frac{a_1}{r}\chi(V\otimes H_1 \otimes L_0^n) + \frac{a_2}{r}\chi(V\otimes H_2 \otimes L_0^n).$$ A torsion free sheaf $V$ is {\sl$(H_1, H_2, a_1, a_2)$ $L_0$-stable\/} if, for all subsheaves $W$ of $V$ with $0< \rank W < \rank V$ and for all $n \gg 0$, $$p_{V; H_1, H_2, a_1, a_2}(n) > p_{W; H_1, H_2, a_1, a_2}(n).$$ $(H_1, H_2, a_1, a_2)$ $L_0$-semistable and unstable are defined similarly. \enddefinition The usual arguments show the following: \lemma{4.2} If $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-stable, then it is simple. \qed \endstatement In the case of rank two on a surface $X$ (which is the only case which shall concern us), $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-stable if and only if, for all rank one subsheaves $W$, and for all $n\gg 0$, we have $$a_1(\chi(V\otimes H_1 \otimes L_0^n) - 2\chi(W\otimes H_1 \otimes L_0^n))+ a_2(\chi(V\otimes H_2 \otimes L_0^n) - 2\chi(W\otimes H_2 \otimes L_0^n)) >0.$$ In particular, if $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-stable then either $V\otimes H_1$ or $V\otimes H_2$ is stable, and a similar statement holds for semistability. A short calculation shows that the coefficient of $n$ in the above expression (which is a degree two polynomial in $n$) is $(a_1+a_2)(L_0\cdot (c_1(V) - 2F))$ and that the constant term is $$(a_1+a_2)(\chi(V) - 2\chi(W)) + a_1H_1\cdot (c_1(V) - 2F) + a_2H_2\cdot (c_1(V) - 2F).$$ Thus $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-stable (resp\. semistable) if and only if it is either Mumford $L_0$-stable or Mumford strictly semistable and the above constant term is positive (resp\. nonnegative). It is easy to see, comparing this with the discussion at the beginning of this section, that formally this is the same as requiring that $V\otimes \Xi$ is (Gieseker) $L_0$-stable or semistable, where $\Xi$ is the $\Bbb Q$-divisor $$\frac{a_1}{a_1+a_2}H_1 + \frac{a_2}{a_1+a_2}H_2.$$ Thus for example taking $H_2=0$ and replacing $H_1$ by a positive integer multiple we see that we can take for $\Xi$ an arbitrary $\Bbb Q$-divisor. Let us explicitly relate mixed stability to our previous notion of $(L_0, \boldsymbol \zeta, \bold k)$-semistability: \lemma{4.3} Given $\Delta$ and $c$ and the corresponding $w$ and $p$, let $L_0$ be an ample divisor lying on a unique wall of type $(w,p)$ and let $V$ be a rank two torsion free sheaf with $c_1(V) = \Delta$ and $c_2(V)=c$. Let $\Xi$ be the $\Bbb Q$-divisor $\dsize \frac{a_1}{a_1+a_2}H_1 + \frac{a_2}{a_1+a_2}H_2$ and suppose that the rational number $t_i = \frac12\zeta_i\cdot(K_X-\Delta) - \zeta_i \cdot \Xi$ is not $\zeta_i$-critical for every $\zeta_i$ of type $(w,p)$ defining $W$. Then, with $t=t_1$, $V$ is $(L_0, \boldsymbol \zeta, \bold k(t))$-semistable if and only if it is $(H_1, H_2, a_1, a_2)$ $L_0$-semistable if and only if it is $(H_1, H_2, a_1, a_2)$ $L_0$-stable. \endstatement \proof Using the additivity of the polynomials $p_{V; H_1, H_2, a_1, a_2}$ over exact sequences, it is easy to check that $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-semistable if and only if it is Mumford $L_0$-semistable, and for every Mumford destabilizing subsheaf of the form $\scrO_X(F)\otimes I_{Z_1}$, either $V$ is universally semistable or we have $$\chi (V) - 2\chi (\scrO_X(F)\otimes I_{Z_1}) - \zeta _i\cdot \Xi >0,$$ where $\zeta_i = 2F-\Delta$. Using our calculations above, this works out to $$\ell (Z_2) - \ell (Z_1) \leq \frac12\zeta_i\cdot(K_X-\Delta) - \zeta_i \cdot \Xi =t_i.$$ Equivalently since $\ell (Z_1) + \ell (Z_2) = \ell _{\zeta _i}$, this becomes $\ell (Z_2) \leq \dsize \fracwithdelims[]{\ell _{\zeta _i}+t_i}2$. Thus $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-semistable if and only if it is $(L_0, \boldsymbol \zeta, \bold k(t))$-semistable. Moreover, since $t$ is not $\zeta_i$-critical, the inequalities are automatically strict, so that $V$ is also $(H_1, H_2, a_1, a_2)$ $L_0$-stable. \endproof Now choosing a $\Xi_0$ such that $\zeta _1\cdot \Xi _0 \neq 0$, every rational number $t$ is of the form $\frac12\zeta_1\cdot(K_X-\Delta) - \zeta_1 \cdot r\Xi_0$ for some rational number $r$. Thus Theorem 3.5 will follow from Lemma 4.3 and from the more general result below: \theorem{4.4} Let $X$ be an algebraic surface $X$ and let $L_0$ be an ample line bundle on $X$. Given a divisor $\Delta$ and an integer $c$, line bundles $H_1$ and $H_2$ on $X$ and positive integers $a_1$ and $a_2$, suppose that every rank two torsion free sheaf $V$ with $c_1(V) = \Delta$, $c_2(V) = c$ which is $(H_1, H_2, a_1, a_2)$ $L_0$-semistable is actually $(H_1, H_2, a_1, a_2)$ $L_0$-stable. Then there exists a projective coarse moduli space $\frak M _{L_0}(\Delta, c;H_1, H_2, a_1, a_2)$ of isomorphism classes of rank two torsion free sheaves $V$ with $c_1(V) = \Delta$, $c_2(V) = c$, which are $(H_1, H_2, a_1, a_2)$ $L_0$-semistable. \endstatement \proof The argument will follow the arguments in \cite{13} as closely as possible, and we shall assume a familiarity with that paper. Suppose that $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-semistable. Then either $V\otimes H_1$ or $V\otimes H_2$ is $L_0$-semistable, and thus by \cite{13}, Lemma 1.3 the set of all such $V$ is bounded. We may thus choose an $n$ such that, for all $V$ which are $(H_1, H_2, a_1, a_2)$ $L_0$-semistable, $V\otimes H_i \otimes L_0^n$ is generated by its global sections and has no higher cohomology, for $i=1,2$. Fix such an $n$ for the moment, and let $d_i = h^0(V\otimes H_i \otimes L_0^n)$. Then $d_i$ is independent of $V$ and $V$ is a quotient of $(H_i^{-1}\otimes L_0^{-n})^{\oplus d_i}$. Let $Q_i$ be the open subset of the corresponding Quot scheme associated to $(H_i^{-1}\otimes L_0^{-n})^{\oplus d_i}$ consisting of quotients which are rank two torsion free sheaves $V_i$ with $c_1(V_i) = \Delta$ and $c_2(V_i) = c$, and such that $V_i\otimes H_i \otimes L_0^n$ is generated by its global sections and has no higher cohomology. We will write a point of $Q_i$ as $V_i$, suppressing the surjection $(H_i^{-1}\otimes L_0^{-n})^{\oplus d_i} \to V_i$. Inside $Q_1\times Q_2$, we have the closed subscheme $I_0$ consisting of quotients $V_1$ and $V_2$ such that $\dim \Hom (V_1, V_2) \geq 1$. There is also the open subvariety $I_0'$ of $I_0$ consisting of $(V_1, V_2)$ with $\dim \Hom (V_1, V_2) =1$. Using the universal sheaves $\Cal U_i$ over $X\times Q_i$, we can construct a $\Cee ^*$ bundle $I$ over $I_0'$ whose points are $(V_1, V_2, \varphi)$, where $\varphi\: V_1 \to V_2$ is a nonzero homomorphism, unique up to scalars. For $i=1,2$, let $E_i$ be a fixed vector space of dimension equal to $d_i = h^0(V\otimes H_i \otimes L_0^n)$. Fix once and for all an isomorphism $(H_i^{-1}\otimes L_0^{-n})^{\oplus d_i} \cong (H_i^{-1}\otimes L_0^{-n})\otimes E_i$. A surjection $(H_i^{-1}\otimes L_0^{-n})^{\oplus d_i}\to V_i$ then gives a map $E_i \to H^0(V_i\otimes H_i \otimes L_0^n)$ and via such a surjection a basis $v_1,\dots, v_{d_1}$ of $E_1$ gives $d_1$ sections of $V_1 \otimes H_1 \otimes L_0^n$ and similarly for a basis $w_1, \dots, w_{d_2}$ of $E_2$. Moreover $GL(d_i)$ acts on $(H_i^{-1}\otimes L_0^{-n})^{\oplus d_i}$ and on $Q_i$. By the universal property of the Quot scheme, this action extends to a $GL(d_i)$-linearization of the universal sheaf $\Cal U_i$ over $X\times Q_i$. Thus there is a right action of $GL(d_1) \times GL(d_2)$ on $I$, and it is easy to see that the elements $(\lambda\Id, \lambda\Id)$ act trivially. Let $F_i$ be the fixed vector space $H^0(\Delta \otimes H_i^2 \otimes L_0^{2n})$, and $F$ the fixed vector space $H^0(\Delta \otimes H_1 \otimes H_2 \otimes L_0^{2n})$. Let $$U= \Hom(\bigwedge ^2E_1, F_1) \oplus \Hom(\bigwedge ^2E_2, F_2)\oplus \Hom (E_1\otimes E_2, F).$$ (The factor $\Hom (E_1\otimes E_2, F)$ is there to make sure that the destabilizing subsheaves for $V\otimes H_1$ and $V\otimes H_2$ are in fact the same.) Note that $GL(d_1) \times GL(d_2)$ operates on the right on $U$ and $\Pee U$. For example, the pair $(\lambda\Id, \mu\Id)$ acts on the triple $(T_1, T_2, T) \in U$ via $(T_1, T_2, T) \mapsto (\lambda ^2T_1, \mu ^2T_2, \lambda\mu T)$. Thus $(A_1, A_2)$ acts trivially on $\Pee U$ if and only if $(A_1, A_2) = (\lambda\Id, \lambda \Id)$. Given a quintuple $\underline{V} = (V_1, V_2,\psi _1, \psi_2, \varphi)$, where $V_i \in Q_i$, $\psi _i\: E_i \to H^0(V_i \otimes H_i \otimes L_0^n)$ is an isomorphism, and $\varphi\: V_1 \to V_2$ is a nonzero map, we will define a point $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V})) \in \Pee U$. To do so, fix an isomorphism $\alpha _2 \: \det V_2 \to \scrO_X(\Delta)$, and set $\alpha _1 = \alpha _2 \circ \det\varphi$. (Thus $\alpha_1 = 0$ if $\varphi$ is not an isomorphism.) Given $v,v' \in E_1$ and $w, w' \in E_2$, identify $v,v'$ with their images in $H^0(V_i \otimes H_1 \otimes L_0^n)$ and similarly for $w,w'$, and let $$\align T_1(\underline{V})(v\wedge v') &= \alpha _1(v\wedge v') =\alpha _2\circ \det\varphi (v\wedge v') \in H^0(\Delta \otimes H_1^2\otimes L_0^{2n});\\ T_2(\underline{V})(w\wedge w') &= \alpha _2(w\wedge w') \in H^0(\Delta \otimes H_2^2\otimes L_0^{2n});\\ T(\underline{V})(v\otimes w) &= \alpha _2(\varphi(v)\wedge w) \in H^0(\Delta \otimes H_1 \otimes H_2 \otimes L_0^{2n}). \endalign$$ Changing $\alpha _2$ by a nonzero scalar $\lambda$ multiplies $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ by $\lambda$, so that the induced element of $\Pee U$ is well defined. Similarly, if we replace $\varphi$ by $\lambda\varphi$, then $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is replaced by $(\lambda ^2T_1(\underline{V}), T_2(\underline{V}), \lambda T(\underline{V}))$. It is easy to check that the map $\underline{V} \mapsto T(\underline{V})$ induces a morphism from $I$ to $\Pee U$ which is $GL(d_1) \times GL(d_2)$-equivariant. Further note that we can define $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ more generally if we are given the data $\underline{V}$ of two rank two torsion free sheaves $V_1$ and $V_2$ with $\det V_i = \Delta$, a morphism $\varphi\: V_1 \to V_2$, and linear maps $\psi _i\: E_i \to H^0(V_i \otimes H_i \otimes L_0^n)$, not necessarily isomorphisms, although it is possible for $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ to be zero in this case. We have not yet introduced the extra parameters $a_1$ and $a_2$. To do so, define $G(a_1, a_2) \subset GL(d_1) \times GL(d_2)$ as follows: $$G(a_1, a_2) = \{\, (A_1,A_2)\mid \det A_1^{a_1}\det A_2^{a_2} = \Id\,\}.$$ Thus unlike Thaddeus we don't change the polarization or the linearization but the actual group which we use to determine stability; still our construction could probably be interpreted in his general framework. Fixing $a_1$ and $a_2$ for the rest of the discussion, we shall denote $G(a_1, a_2)$ by $G$. Since $a_1$ and $a_2$ are positive, the matrix $(\lambda\Id, \lambda\Id)$ lies in $G$ if and only if $\lambda$ is an $m^{\text{th}}$ root of unity, where $m = a_1d_1+a_2d_2$. Thus a quotient of $G$ by a finite group acts faithfully on $\Pee U$. Moreover, the problem of finding a good quotient of $\Pee U$ (for an appropriate open subset of $\Pee U$) for $G$ is the same as that of finding a good quotient of $\Pee U$ for $GL(d_1) \times GL(d_2)$, since $$G\cdot \Cee ^*(\Id, \Id) = GL(d_1) \times GL(d_2).$$ This last statement follows since $G$ clearly contains $SL(d_1) \times SL(d_2)$ and since $\Cee ^*\times \Cee ^*$ is generated by its diagonal subgroup and by the subgroup $$\{\,(\lambda, \mu): \lambda ^{a_1d_1}\mu ^{a_2d_2} = 1\,\}.$$ We may thus apply the general machinery of GIT to the group $G$ acting on $\Pee U$. A one parameter subgroup of $G$ is given by a basis $\{v_i\}$ of $E_1$, a basis $\{w_k\}$ of $E_2$ and weights $n_i$, $m_k\in \Zee$, such that $v_i^\lambda = \lambda ^{n_i}v_i$, $w_k ^\lambda = \lambda ^{m_k}w_k$, and $$a_1\sum _in_i + a_2 \sum _km_k = 0.$$ We shall always arrange our choice of basis so that $n_1 \leq n_2 \leq \dots \leq n_{d_1}$ and $m_1 \leq m_2 \leq \dots \leq m_{d_2}$. Given $(T_1, T_2, T) \in U$ and a one parameter subgroup of $G$ as above, we see that $\lim _{\lambda \to 0}(T_1, T_2, T)^\lambda =0$ if and only if $T_1(v_i\wedge v_j) = 0$ for every pair of indices $i,j$ such that $n_i+n_j \leq 0$, $T_2(w_k\wedge w_\ell) = 0$ for every pair of indices $k, \ell$ such that $m_k+m_\ell \leq 0$, and $T(v_i\otimes w_j) = 0$ for every pair $i,k$ such that $n_i+m_k\leq 0$. Likewise the condition that $\lim _{\lambda \to 0}(T_1, T_2, T)^\lambda$ exists is similar, replacing the $\leq$ by strict inequality. Finally note that if $n_i+n_j \leq 0$, then $n_1+n_j \leq 0$, if $m_k+m_\ell \leq 0$ then $m_1+m_\ell \leq 0$, and if $n_i+m_k\leq 0$ then $n_1+m_k\leq 0$ and $n_i+m_1\leq 0$. We then have the following: \lemma{4.5} \roster \item"{(i)}" Suppose that we are given the data $\underline{V}$ of two rank two torsion free sheaves $V_1$ and $V_2$ with $\det V_i = \Delta$, a morphism $\varphi\: V_1 \to V_2$, and a linear map $E_i \to H^0(V_i \otimes H_i \otimes L_0^n)$, not necessarily an isomorphism. If $E_i \to H^0(V_i \otimes H_i \otimes L_0^n)$ is not injective for some $i$ or if $\varphi$ is not an isomorphism, then $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is either zero or $G$-unstable. \item"{(ii)}" For $n$ sufficiently large depending only on $\Delta$ and $c$ and for $V$ a rank two torsion free sheaf with $\det V = \Delta$ and $c_2(V) = c$, $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-unstable if and only if $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-unstable for all choices of data $\underline{V}$ such that $E_i \to H^0(V_i \otimes H_i \otimes L_0^n)$ is injective and $\varphi\: V_1 \to V_2\cong V$ is an isomorphism, and $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-strictly semistable if and only if $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-strictly semistable for all such $\underline{V}$. Thus $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-stable if and only if $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-stable for all such $\underline{V}$. \endroster \endstatement \proof First let us prove (i). We may assume that $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))\neq 0$. Suppose for example that $v_1 \in E_1 \mapsto 0 \in H^0(V_1\otimes H_1\otimes L_0^n)$. Complete $v_1$ to a basis of $E_1$ and choose a basis $\{w_k\}$ for $E_2$. Then $T_1(\underline{V})(v_1\wedge v_i) = 0$ for all $i$ and $T(\underline{V})(v_1\otimes w_k) =0$ for all $k$. Define a one parameter subgroup of $G$ as follows: let $v_1^\lambda = \lambda ^{-N}v_1$, $v_i^\lambda = \lambda ^av_i$ for $i>1$, and $w_k^\lambda = \lambda ^bw_k$ for all $k$. Clearly $\lim _{\lambda \to 0}(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))^\lambda =0$ provided that $a$ and $b$ are positive, so that $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-unstable provided that the one parameter subgroup so constructed lies in $G$, or on other words provided that $$a_1(-N + a(d_1-1))+ a_2bd_2 =0.$$ It thus suffices to take $a$ an arbitrary positive integer, $b=a_1$, and $N= a(d_1-1) + a_2d_2$. The argument in case $\varphi$ has a kernel is similar: in this case let $v_1 \in \Ker \varphi$. Then $T_1(\underline{V})=0$ and $T(\underline{V})(v_1\otimes w_k) =0$ for all $k$, so that the previous argument handles this case also. Next we show (ii). Let $p_{V\otimes H_i}$ be the usual normalized Hilbert polynomial of $V\otimes H_i$, and similarly for $p_{W\otimes H_i}$, where $W$ is a rank one subsheaf of $V$. Thus $p_{V\otimes H_i}$ and $p_{W\otimes H_i}$ have the same leading term. Given a polynomial $p$, let $\Delta p$ denote the difference polynomial. In our case, all of the polynomials $p$ that occur are quadratic polynomials with the same fixed degree two term. Thus if $p_1$ and $p_2$ are two such polynomials, then $p_1(n) > p_2(n)$ for all $n\gg 0$ if and only if the linear term of $p_1$ is greater than or equal to the linear term of $p_2$, and if the linear terms are equal then the constant term of $p_1$ is greater than the constant term of $p_2$. In this last case, where the linear terms are also equal, we see that $p_1(n) > p_2(n)$ for all $n\gg 0$ if and only if $p_1(n) > p_2(n)$ for some $n$. Finally the linear term of $p_1$ is greater than or equal to the linear term of $p_2$ if and only if $\Delta p_1(n) \geq \Delta p_2(n)$ for all $n$, which we shall write as $\Delta p_1\geq \Delta p_2$. Thus if $\Delta p_1 \geq \Delta p_2$ and $p_1(n) > p_2(n)$ for some $n$, then $p_1(n) > p_2(n)$ for all $n\gg 0$. If $\Delta p_1 = \Delta p_2$, then $p_1(n) > p_2(n)$ for some $n$ if and only if $p_1(n) > p_2(n)$ for all $n$. We shall show that, for sufficiently large $n$, if $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-semistable and $\underline{V}$ corresponds to data where $E_i \to H^0(V\otimes H_i\otimes L_0^n)$ is injective and $\varphi$ is an isomorphism, then $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-semistable. Note that $V$ is Mumford semistable. First we may choose $n$ so that $V\otimes H_i$ is generated by its global sections and has no higher cohomology, and so $\chi (V\otimes H_i \otimes L_0^n) = h^0(V\otimes H_i \otimes L_0^n) = d_i$. Hence, since $E_i \to H^0(V\otimes H_i\otimes L_0^n)$ is injective, it is an isomorphism. Let $W$ be a rank one subsheaf of $V$. Since $V\otimes H_i$ is Mumford semistable, $\Delta p_{W\otimes H_i} \leq \Delta p_{V\otimes H_i}$. Now the proof of (3) of Lemma 1.2 in \cite{13} shows that there exists an $N$ so that, for all $n\geq N$, with $d_i$ as above, if $W$ is a rank one subsheaf of $V$ and such that $h^0(W\otimes H_i \otimes L_0^n ) \geq d_i/2$ for at least one $i$ ($i=1,2$), then in fact $\Delta p_{V\otimes H_i} = \Delta p_{W\otimes H_i}$ for all such $W$, and thus $\mu _{L_0}(V) = \mu _{L_0}(W)$. It is then easy to see that there is a twist $W\otimes H_i\otimes L_0^{-k}$, depending only on $L_0$ and $\Delta$, such that $h^0((V/W)\otimes H_i\otimes L_0^{-k}) = 0$. The proof of Proposition 3.1 in \cite{13} shows that in this case $h^1(W\otimes H_i \otimes L_0^{-k})$ is bounded by $Q$, where $Q$ is some universal bound for the numbers $h^1(V\otimes H_i\otimes L_0^{-k})$ as $V\otimes H_i$ ranges over the appropriate set of $L_0$-semistable sheaves Thus by (4) of Lemma 1.2 in \cite{13}, the $W$ satisfying the condition that $h^0(W\otimes H_i \otimes L_0^n ) \geq d_i/2$ for at least one $i$ form a bounded family, and we may choose $n$ so large, depending only on $L_0$, $\Delta$, $c$, such that $h^j(W\otimes H_i \otimes L_0^n)=0$ for $j\geq 1$ and $i=1,2$. Now suppose that $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-unstable. Then there exists a one parameter subgroup of $G$ as above such that $\lim_{\lambda \to 0} (T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))^\lambda =0$. Let $$\align s_1&= \#\{\, j: T_1(\underline{V})(v_1\wedge v_j) =0\,\} \geq \max \{\, j: n_1 + n_j \leq 0\,\};\\ s_2 &= \#\{\, j: T_2(\underline{V})(w_1\wedge w_\ell) =0\,\} \geq \max \{\, \ell: m_1 + m_\ell \leq 0\,\}. \endalign$$ Since $a_1\sum _in_i + a_2 \sum _km_k =0$, at least one of $n_1, m_1$ is negative. By symmetry we may assume that $n_1$ is negative, and that $n_1 \leq m_1$. Since for $j\leq s_1$, $v_1\wedge v_j$ is zero as a section of $\det (V\otimes H_1\otimes L_0^n)$, the sections corresponding to $v_j$, $1\leq j \leq s_1$, are all sections of a rank one subsheaf $W_1$ of $V$. Likewise the sections $w_\ell$, $1\leq \ell \leq s_2$, if there are any such, are all sections of a rank one subsheaf $W_2$ of $V$. The condition that $T(\underline{V})(v_1\otimes w_1) =0$ insures that $W_1$ and $W_2$ are contained in a saturated rank one subsheaf $W$, if $s_2\neq 0$, otherwise we shall just take for $W$ the saturated rank one subsheaf containing $W_1$. Moreover $h^0(W\otimes H_1\otimes L_0^n) \geq s_1$ and $h^0(W\otimes H_2\otimes L_0^n) \geq s_2$. Suppose that we show that $$a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2) <0.$$ Thus in particular $s_i\geq d_i/2$ for at least one $i$. By our choice of $n$ and the previous paragraph, if $s_i\geq d_i/2$ for at least one $i$, then $h^0(W\otimes H_i\otimes L_0^n) = \chi (W\otimes H_i\otimes L_0^n)$ and furthermore $\mu _{L_0}(V) = \mu _{L_0}(W)$. Thus $$h^0(W\otimes H_i \otimes L_0^n) = \chi (W\otimes H_i \otimes L_0^n) \geq s_i$$ for $i=1,2$ and so $p_{V; H_1, H_2, a_1, a_2}(n) < p_{W; H_1, H_2, a_1, a_2}(n)$. On the other hand, $p_{V; H_1, H_2, a_1, a_2}$ and $p_{W; H_1, H_2, a_1, a_2}$ are two quadratic polynomials with the same linear and quadratic terms (since $\mu _{L_0}(V) = \mu _{L_0}(W)$), and $p_{V; H_1, H_2, a_1, a_2}(n) < p_{W; H_1, H_2, a_1, a_2}(n)$ for one value of $n$. Thus the constant term of $p_{W; H_1, H_2, a_1, a_2}$ must be larger than that of $p_{V; H_1, H_2, a_1, a_2}$. This contradicts the $(H_1, H_2, a_1, a_2)$ $L_0$-semistability of $V$. To see that $a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2) <0$, let $$t_1 = \#\{\, j: n_j +m_1 \leq 0\,\} \leq s_1.$$ Here $t_1 \leq s_1$ since $T(\underline{V})(v_j \otimes w_1) = 0$ implies that $v_j$ and $w_1$ are contained in a rank one subsheaf of $V$, necessarily $W$, and thus that $v_1\wedge v_j = 0$. Let $$t_2 = \#\{\, \ell: n_1 +m_\ell \leq 0\,\} \leq s_2.$$ We have assumed that $n_1 \leq m_1$. Then consider the expression $$a_1\sum _j (n_1+ n_j) + a_2\sum _\ell (n_1 + m_\ell).$$ On the one hand from the definition of the one parameter subgroup we have $$a_1\sum _j (n_1+ n_j) + a_2\sum _\ell (n_1 + m_\ell) = a_1d_1n_1 + a_2d_2n_1.$$ On the other hand, to estimate $\sum _j (n_1+ n_j)$, we can ignore the positive terms where $n_1+n_j \geq 0$ and each of the $s_1$ negative terms are at least $n_1 + n_1 \geq 2n_1$. Thus $\sum _j (n_1+ n_j) \geq 2s_1n_1$. Since $n_1 <0$, this term is $\geq 2s_1n_1$. Also this inequality is strict or $n_1 + n_i \leq 0$ for every $i$, which would say that every section of $V\otimes H_1 \otimes L_0^n$ is really a section of $W\otimes H_1 \otimes L_0^n$ contradicting the fact that $V\otimes H_1 \otimes L_0^n$ is generated by global sections. So $\sum _j (n_1+ n_j) < 2s_1n_1$. Likewise we claim that $\sum _\ell (n_1 + m_\ell) \geq 2s_2n_1$. Here, to estimate $\sum _\ell (n_1 + m_\ell)$, we may ignore the terms with $n_1 + m_\ell$ positive, leaving $t_2$ terms $n_1+m_\ell$ which are $\leq 0$, and moreover each such term is at least $n_1+m_1 \geq 2n_1$. Thus $\sum _\ell (n_1 + m_\ell) \geq 2t_2n_1$, and since $t_2 \leq s_2$ and $n_1 <0$, we have $2t_2n_1 \geq 2s_2n_1$. Putting this together we have $$\align a_1d_1n_1 + a_2d_2n_1 &= a_1\sum _j (n_1+ n_j) + a_2\sum _\ell (n_1 + m_\ell) \\ &> a_1(2s_1n_1)+ a_2(2s_2n_1), \endalign$$ so that $$a_1(d_1 - 2s_1)n_1 + a_2(d_2 - 2s_2)n_1 >0.$$ As $n_1 <0$, we must have $a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2) <0$, as desired. We have thus shown that, if $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-unstable, then $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-unstable. A very similar argument handles the $G$-strictly semi\-stable case. Now we turn to the converse statement, that if $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-unstable then $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-unstable. Suppose instead that $$(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$$ is $G$-semistable. Let $W$ be a rank one subsheaf of $V$ such that $p_{W; H_1, H_2, a_1, a_2}(m) > p_{V; H_1, H_2, a_1, a_2}(m)$ for all $m\gg 0$. We may assume that the quotient $W' = V/W$ is torsion free. Thus $p_{W'; H_1, H_2, a_1, a_2}(m) < p_{V; H_1, H_2, a_1, a_2}(m)$ for all $m\gg 0$, and so $\Delta p_{W'\otimes H_i} \leq \Delta p_{V\otimes H_i}$. Now we have the map $E_i \to H^0(V\otimes H_i \otimes L_0^n)$. Consider $E_i \cap H^0(W\otimes H_i \otimes L_0^n)\subseteq E_i$. Let $\dim E_i \cap H^0(W\otimes H_i \otimes L_0^n) = s_i$. Suppose first that $a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2) <0$. We claim that in this case $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-unstable, a contradiction. To see this, choose a basis $v_1, \dots, v_{d_1}$ for $E_1$ such that $$v_i \in E_1\cap H^0(W\otimes H_1 \otimes L_0^n)$$ for $i\leq s_1$, and similarly choose a basis $w_1, \dots, w_{d_2}$ for $E_2$ such that $w_k \in E_2\cap H^0(W\otimes H_2 \otimes L_0^n)$ for $i\leq s_2$. Thus, if $i,j\leq s_1$ then $T_1(\underline{V})(v_i\wedge v_j) =0$; if $k, \ell \leq s_2$ then $T_2( \underline{V})(w_k\wedge w_\ell) = 0$; if $i\leq s_1$ and $k\leq s_2$ then $T(\underline{V})(v_i\otimes w_k) =0$. We will try to find a one parameter subgroup of $G$ of the form $$v_i^\lambda = \cases \lambda ^{-m}v_i, &\text{for $i \leq s_1$;} \\ \lambda ^nv_i , &\text{for $i > s_1$,} \endcases$$ and similarly $$w_k^\lambda = \cases \lambda ^{-m}w_k, &\text{for $i \leq s_2$;} \\ \lambda ^nw_k , &\text{for $i > s_2$.} \endcases$$ It is easy to check that $\lim _{\lambda \to 0}(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))^\lambda=0$ if and only if $n>m$. What we must arrange is the condition $$a_1(-ms_1 + n(d_1-s_1)) + a_2(-ms_2 +n(d_2-s_2)) = 0.$$ Now consider the linear function with rational coefficients $$f(t) = a_1(-s_1 + t(d_1-s_1)) + a_2(-s_2 +t(d_2-s_2)).$$ Since the coefficient of $t$ is strictly positive $f(t)$ is increasing, and $$\align f(1) &= a_1(-s_1 + (d_1-s_1)) + a_2(-s_2 +(d_2-s_2))\\ &= a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2) <0. \endalign$$ Thus there is a rational number $t= n/m>1$ such that $f(t) = 0$, and this gives the desired choice of $n$ and $m$. Thus if $a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2) <0$, then $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-unstable, contradicting our hypothesis. The other possibility is that $a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2) \geq 0$. In this case $d_i \geq 2 s_i$ for at least one $i$. Recalling that we have the quotient $W'$ of $V$ by $W$, it then follows that for such an $i$ the image of $E_i$ in $H^0(W' \otimes H_i \otimes L_0^n)$ must have dimension at least $d_i/2$. Arguing as in Proposition 3.2 of \cite{13}, it then follows from Lemma 1.2 of \cite{13} that $\Delta p_{W'\otimes H_i} = \Delta p_{V\otimes H_i}$ and so that $V$ is Mumford $L_0$-semistable and $\mu _{L_0}(V) = \mu _{L_0}(W)$. Moreover, after enlarging $n$ if necessary (independently of $V$) we may assume that $h^j(V\otimes H_i \otimes L_0^n) = 0$ for $j>0$. In particular, $d_i = \dim H^0(V\otimes H_i \otimes L_0^n)$ for $i=1,2$, and $E_i \to H^0(V \otimes H_i \otimes L_0^n)$ is an isomorphism; so $s_i = h^0(W \otimes H_i \otimes L_0^n)$. As $\mu _{L_0}(V) = \mu _{L_0}(W)$, the polynomials $p_{W; H_1, H_2, a_1, a_2}$ and $p_{V; H_1, H_2, a_1, a_2}$ have the same terms in degree one and two, and thus since $p_{W; H_1, H_2, a_1, a_2}(m) > p_{V; H_1, H_2, a_1, a_2}(m)$ for some $m$ the same is true for all $m$, in particular for $m=n$. Moreover, for a general choice of a smooth curve $C$ in the linear system corresponding to $L_0$, there is a fixed bound on the line bundle $W\otimes H_i|C$. A standard argument as in the proof of (2) of Lemma 1.2 of \cite{13} shows that, for $n$ sufficiently large but independent of $V$, we have $H^2(W\otimes H_i\otimes L_0^n) =0$. Thus $s_i = h^0(W \otimes H_i \otimes L_0^n) \geq p_{W\otimes H_i}(n)$. It follows that $$\align a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2)&\leq a_1(d_1 - 2p_{W\otimes H_1}(n)) + a_2(d_2 - 2p_{W\otimes H_2}(n))\\ &= 2(p_{V; H_1, H_2, a_1, a_2}(n) - p_{W; H_1, H_2, a_1, a_2}(n)) <0. \endalign$$ This contradicts the assumption that $a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2) \geq 0$. It then follows that $(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-unstable. The strictly semistable case is similar. \endproof We may now finish the proof of Theorem 4.4. Let $\Pee U_{\text{ss}}$ be the set of $G$-semistable points of $\Pee U$. Let $I_{\text{ss}}$ be the inverse image of $\Pee U_{\text{ss}}$ under the morphism $I \to \Pee U$. Since every semistable sheaf is stable, $I_{\text{ss}}$ is a $\Cee ^*$-bundle over its image in $Q_1\times Q_2$. Moreover the representable functor corresponding to $I_{\text{ss}}$ is easily seen to be formally smooth over the moduli functor. Arguments very similar to those for Lemma 4.3 and 4.5 of \cite{13} show that the morphism $I_{\text{ss}} \to \Pee U_{\text{ss}}$ is one-to-one and proper, and thus in particular finite. Thus we may construct a quotient $\frak M _{L_0}(\Delta, c;H_1, H_2, a_1, a_2)$ of $I_{\text{ss}}$ by $G$. This quotient maps in a one-to-one and proper way to the GIT quotient of $\Pee U_{\text{ss}}$ and is therefore projective. By the discussion at the beginning of the proof of Theorem 4.4 the points of $\frak M _{L_0}(\Delta, c;H_1, H_2, a_1, a_2)$ may be identified with isomorphism classes of $(H_1, H_2, a_1, a_2)$ $L_0$-semistable rank two sheaves. Standard arguments then show that $\frak M _{L_0}(\Delta, c;H_1, H_2, a_1, a_2)$ has the usual properties of a coarse moduli space. \endproof \section{5. The transition formula for Donaldson polynomial invariants.} From now on, we will assume that the surface $X$ is rational with $-K_X$ effective, and will study the transition formula of Donaldson polynomial invariants: $$\delta ^X_{w,p}(\Cal C_+, \Cal C_-) = D^X _{w,p}(\Cal C_+) - D^X _{w,p}(\Cal C_-)$$ where $\Cal C_-$ and $\Cal C_+$ are two adjacent chambers separated by a single wall $W^{\zeta}$ of type $(w, p)$ or equivalently of type $(\Delta, c)$. For simplicity, we assume that the wall $W^{\zeta}$ is only represented by $\pm \zeta$ since the general case just involves additional notation. We use $\frak M_0^{(k)}$ to stand for the moduli space $\frak M_0^{(\boldsymbol \zeta, k)}$. When $\ell_\zeta = 0$, we also assume that $$h(\zeta) = h^1(X; \scrO _X(2F-\Delta) \neq 0$$ (see Corollary 2.7). The special case when $\ell_\zeta = h(\zeta) = 0$ will be treated in Theorem 6.1. By Theorem 3.9 and Lemma 3.2 (ii), we have the following diagram: $$\matrix & &{\widetilde{\frak M}}_0^{(\ell_\zeta)}&&&&\ldots&&&&{\widetilde{\frak M}}_0^{(0)}&&\\ &\swarrow&&\searrow&&\swarrow&&\searrow&&\swarrow&&\searrow\\ \frak M_0^{(\ell_\zeta)} & & && \frak M_0^{(\ell_\zeta - 1)} & &&& \frak M_0^{(0)} & &&& \frak M_0^{(-1)}\\ \Vert \quad&&&&&&&&&&&& \Vert \quad\\ \frak M_- &&&&&&&&&&&&\frak M_+\\ \endmatrix $$ where the morphism ${\widetilde{\frak M}}_0^{(k)} \to \frak M_0^{(k)}$ is the blowup of $\frak M_0^{(k)}$ at $E_\zeta^{\ell_\zeta -k, k}$, and the morphism ${\widetilde{\frak M}}_0^{(k)} \to \frak M_0^{(k - 1)}$ is the blowup of $\frak M_0^{(k - 1)}$ at $E_{-\zeta}^{k, \ell_\zeta -k}$. Next, we collect and establish some notations. Recall that in section 2 we have constructed the bundle $\Cal E_\zeta^{\ell_\zeta -k, k}$ over $ H_{\ell_\zeta - k} \times H_{k}$, where $H_k = \Hilb^k X$. \medskip \noindent {\bf Notation 5.1}. Let $\zeta$ define a wall of type $(w, p)$. \roster \item"{(i)}" $\lambda_k$ is the tautological line bundle over $E_\zeta^{\ell_\zeta - k, k} = \Pee((\Cal E_\zeta^{\ell_\zeta -k, k})\spcheck)$; for simplicity, we also use $\lambda_k$ to denote its first Chern class; \item"{(ii)}" $\rho_k: X \times E_\zeta^{\ell_\zeta - k, k} \to X \times H_{\ell_\zeta - k} \times H_{k}$ is the natural projection; \item"{(iii)}" $p_k: \widetilde{\frak M}_0^{(k)} \to \frak M_0^{(k)}$ is the blowup of $\frak M_0^{(k)}$ at $E_\zeta^{\ell_\zeta -k, k}$; \item"{(iv)}" $q_{k - 1}: \widetilde{\frak M}_0^{(k)} \to \frak M_0^{(k-1)}$ is the contraction of $\widetilde{\frak M}_0^{(k)}$ to $\frak M_0^{(k-1)}$; \item"{(v)}" $\Cal N_k$ is the normal bundle of $E_\zeta^{\ell_\zeta -k, k}$ in $\frak M_0^{(k)}$; by Proposition 3.7, we have $$\Cal N_k = \rho_k^*\Cal E_{-\zeta}^{k, \ell_\zeta -k} \otimes \lambda_k^{-1};$$ \item"{(vi)}" $D_k = \Pee(\Cal N_k\spcheck)$ is the exceptional divisor in $\widetilde{\frak M}_0^{(k)}$; \item"{(vii)}" $\xi_k = \Cal O_{\widetilde{\frak M}_0^{(k)}}(-D_k)|D_k$ is the tautological line bundle on $D_k$; again, for simplicity, we also use $\xi_k$ to denote its first Chern class; \item"{(viii)}" $\mu^{(k)}(\alpha) = -{1 \over 4} p_1(\Cal U^{(k)})/\alpha$ where $\alpha \in H_2(X; \Zee)$ and $\Cal U^{(k)}$ is a universal sheaf over $X \times \frak M_0^{(k)}$. Let $\mu^{(\ell_\zeta)}(\alpha) = \mu_-(\alpha)$ and that $\mu^{(-1)}(\alpha) = \mu_+(\alpha)$. \item"{(ix)}" $\nu^{(k)} = -{1 \over 4} p_1(\Cal U^{(k)})/x$ where $x \in H_0(X; \Zee)$ is the natural generator. Let $\nu^{(\ell_\zeta)}= \nu_-$ and that $\nu^{(-1)} = \nu_+$. \endroster Note that, in (viii) and (ix) above, the sheaf $\Cal U^{(k)}$ is only defined locally in the classical topology. However, since it is defined on the level of the Quot scheme a straightforward argument shows that $p_1(\Cal U^{(k)})$ is a well-defined element in the rational cohomology of $X \times \frak M_0^{(k)}$, at least in the complement of the universally semistable sheaves. In case there are universally semistable sheaves, then the work of Li \cite{21} extends the $\mu$-map to $\frak M_0^{(k)}$, at least for the two-dimensional algebraic classes. We can then extend the $\mu$-map to the 4-dimensional class via a blowup formula due to O'Grady (unpublished). Moreover, there is a universal sheaf $\Cal V_k$ over $X \times E_\zeta^{\ell_\zeta - k, k}$. In what follows, we shall work as if there were a universal sheaf $\Cal U^{(k)}$, and leave it to the reader to check that our final Chern class calculations can be verified directly even when no universal sheaf exists. In the following lemma, we study the restrictions of $p_k^*\mu^{(k)}(\alpha)$ and $p_k^*\nu^{(k)}$ to $D_k$. \lemma{5.2} Let $\alpha \in H_2(X; \Zee)$ and $a = (\zeta \cdot \alpha)/2$. Let $\tau_1$ and $\tau_2$ be the projections of $E_\zeta^{\ell_\zeta - k, k}$ to $H_{\ell_\zeta - k}$ and $H_k$ respectively. Then, $$\align &(\operatorname{Id} \times p_{k})^*c_1(\Cal U^{(k)})|(X \times D_k) = \pi _1^*\Delta + (p_k|D_k)^*\lambda_k\\ &p_k^*\mu^{(k)}(\alpha)|D_k = (p_k|D_k)^*\left[\tau_1^*([{\Cal Z_{\ell_\zeta - k}}]/\alpha) + \tau_2^*([{\Cal Z_{k}}]/\alpha) - a \lambda_k\right]\\ &p_k^*\nu^{(k)}|D_k = {1 \over 4} (p_k|D_k)^* \left [ 4 \tau_1^*([{\Cal Z_{\ell_\zeta - k}}]/x) + 4 \tau_2^*([{\Cal Z_{k}}]/x) - \lambda_k^2 \right ]. \\ \endalign$$ \endstatement \proof Note that $\Cal U^{(k)}|X \times E_\zeta^{\ell_\zeta - k, k} = \Cal V_k$, where the sheaf $\Cal V_k$ is constructed by Proposition 2.8 and sits in the exact sequence: $$0 \to \pi _1^*\scrO _X(F )\otimes \rho_k ^*\pi _{1,2}^*I_{\Cal Z_{\ell_\zeta - k}} \otimes \pi_2^*\lambda_k \to \Cal V_k \to \pi _1^*\scrO _X(\Delta -F ) \otimes \rho_k ^*\pi _{1,3}^*I_{\Cal Z_{k}} \to 0.$$ Thus, $c_1(\Cal V_k) = \pi _1^*\Delta + \pi_2^*\lambda_k$ and $(\operatorname{Id} \times p_{k})^*c_1(\Cal U^{(k)})|(X \times D_k) = \pi _1^*\Delta + (p_k|D_k)^*\lambda_k$. Moreover, $c_2(\Cal V_k) = \rho_k ^*\pi_{1,2}^*[{\Cal Z_{\ell_\zeta - k}}] + \rho_k ^*\pi_{1,3}^*[{\Cal Z_{k}}] + (\pi _1^*F + \pi_2^*\lambda_k) \cdot \pi _1^*(\Delta - F)$. Since $p_k^*\mu^{(k)}(\alpha)|D_k = (p_k|D_k)^*[\mu^{(k)}(\alpha)|E_\zeta^{\ell_\zeta - k, k}] = (p_k|D_k)^*[-{1 \over 4} p_1(\Cal V_k)/\alpha]$, we have $$p_k^*\mu^{(k)}(\alpha)|D_k = (p_k|D_k)^*\left[\tau_1^*([{\Cal Z_{\ell_\zeta - k}}]/\alpha) + \tau_2^*([{\Cal Z_{k}}]/\alpha) - a \lambda_k\right].$$ Similarly, $p_k^*\nu^{(k)}|D_k = {1 \over 4} (p_k|D_k)^* \left [ 4 \tau_1^*([{\Cal Z_{\ell_\zeta - k}}]/x) + 4 \tau_2^*([{\Cal Z_{k}}]/x) - \lambda_k^2 \right ]$. \endproof It follows from the work of Morgan \cite{25} and Li \cite{21}, together with unpublished work of Morgan, that $D^X _{w,p}(\Cal C_\pm)(\alpha^d) = \delta(\Delta) \cdot \mu_\pm(\alpha)^d$ and $$D^X _{w,p}(\Cal C_\pm)(\alpha^{d - 2}, x) = \delta(\Delta) \cdot \mu_\pm(\alpha)^{d - 2} \cdot \nu_\pm$$ where $d = -p - 3$, $\delta(\Delta) = (-1)^{{{(\Delta^2 + \Delta \cdot K_X)}/2}}$ is the difference between the complex orientation and the standard orientation on the instanton moduli space (see \cite{6}), and $x \in H_0(X; \Zee)$ is the natural generator. Strictly speaking, their methods only handle the case of $D^X _{w,p}(\Cal C_\pm)(\alpha^d)$. To handle the case of $D^X _{w,p}(\Cal C_\pm)(\alpha^{d - 2}, x) $, one needs a blowup formula in algebraic geometry, which has been established by O'Grady (unpublished). To compute the differences $$\mu_+(\alpha)^d - \mu_-(\alpha)^d \quad \text{and} \quad \mu_+(\alpha)^{d - 2} \cdot \nu_+ - \mu_-(\alpha)^{d - 2} \cdot \nu_-,$$ we need to know how $\mu^{(k)}(\alpha)$ and $\mu^{(k - 1)}(\alpha)$ are related, and also how $\nu^{(k)}$ and $\nu^{(k - 1)}$ are related. The following lemma handles this problem. \lemma{5.3} For $\alpha \in H_2(X; \Zee)$ and the natural generator $x \in H_0(X; \Zee)$, we have $$\align &q_{k - 1}^*\mu^{(k-1)}(\alpha) = p_k^*\mu^{(k)}(\alpha) - aD_k\\ &q_{k - 1}^*\nu^{(k - 1)} = p_k^*\nu^{(k)} - {1 \over 4} [D_k^2 + 2(p_k|D_k)^*\lambda_k].\\ \endalign$$ \endstatement \proof From the construction, the sheaf $(\operatorname{Id} \times q_{k - 1})^*\Cal U^{(k - 1)}$ on $X \times \widetilde{\frak M}_0^{(k)}$ is the elementary modification of $(\operatorname{Id} \times p_{k})^*\Cal U^{(k)}$ along the divisor $X \times D_k$, using the surjection from $(\operatorname{Id} \times p_{k})^*\Cal U^{(k)}$ to the pullback of $\rho_k^*(\pi_1^*\Cal O_X(\Delta - F) \otimes \pi_{1, 3}^*I_{\Cal Z_k})$: $$0 \to (\operatorname{Id} \times q_{k - 1})^*\Cal U^{(k - 1)} \to (\operatorname{Id} \times p_{k})^*\Cal U^{(k)}$$ $$\to (\operatorname{Id} \times p_{k}|D_k)^* \rho_k^*(\pi_1^*\Cal O_X(\Delta - F) \otimes \pi_{1, 3}^*I_{\Cal Z_k}) \to 0$$ where $(2F - \Delta) = \zeta$ and $\pi_1$ is the natural projection $X \times H_{\ell_\zeta - k} \times H_{k} \to X$. Note that $(\operatorname{Id} \times p_{k}|D_k)^* \rho_k^*(\pi_1^*\Cal O_X(\Delta - F) \otimes \pi_{1, 3}^*I_{\Cal Z_k})$ is a sheaf supported on $X \times D_k$, and that its first and second Chern classes are equal to $(X \times D_k)$ and $(X \times D_k^2) - \pi_1^*(\Delta - F) \cdot (X \times D_k)$ respectively. It follows that $$\align &(\operatorname{Id} \times q_{k - 1})^* c_1(\Cal U^{(k - 1)}) = (\operatorname{Id} \times p_{k})^* c_1(\Cal U^{(k)}) - (X \times D_k)\\ &(\operatorname{Id} \times q_{k - 1})^* c_2(\Cal U^{(k - 1)}) = (\operatorname{Id} \times p_{k})^* c_2(\Cal U^{(k)}) - (\operatorname{Id} \times p_{k})^* c_1(\Cal U^{(k)}) \cdot (X \times D_k)\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad + \pi_1^*(\Delta - F) \cdot (X \times D_k).\\ \endalign$$ By Lemma 5.2, $(\operatorname{Id} \times p_{k})^*c_1(\Cal U^{(k)}) \cdot (X \times D_k) = (\Delta \times D_k) + (X \times (p_k|D_k)^*\lambda_k)$. Thus, $$\align (\operatorname{Id} \times q_{k - 1})^* p_1(\Cal U^{(k - 1)}) &= (\operatorname{Id} \times p_{k})^* p_1(\Cal U^{(k)}) + (X \times D_k^2) - 4 (\Delta - F) \times D_k\\ &\qquad\qquad + 2 (\operatorname{Id} \times p_{k})^*c_1(\Cal U^{(k)}) \cdot (X \times D_k)\\ &= (\operatorname{Id} \times p_{k})^* p_1(\Cal U^{(k)}) + 2(2F - \Delta) \times D_k\\ &\qquad\qquad + X \times [D_k^2 + 2(p_k|D_k)^*\lambda_k].\\ \endalign$$ Now the conclusions follow from some straightforward calculations. \endproof In the next two theorems, we will give formulas for the differences $[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d$ and $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ in terms of the intersections in $H_{\ell_\zeta - k} \times H_k$ and the Segre classes of the vector bundles $\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck$ on $H_{\ell_\zeta - k} \times H_k$, where $k = 0, 1, \ldots, \ell_\zeta$. The arguments are a little complicated, but the idea is that we are trying to get rid of the exceptional divisors $D_k$ as well as the Chern classes of the tautological line bundles $\xi_k$ and $\lambda_k$. \theorem{5.4} Let $\zeta$ define a wall of type $(w, p)$, and $d = (-p - 3)$. For $\alpha \in H_2(X; \Zee)$, put $a = (\zeta \cdot \alpha)/2$. Then, $[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d$ is equal to $$\sum_{j = 0}^{2\ell_\zeta}~ {d \choose j} \cdot (-1)^{h(\zeta) + \ell_\zeta + j} \cdot a^{d - j} \cdot \sum_{k = 0}^{\ell_\zeta}~ ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot s_{2\ell_\zeta - j}(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck).$$ \endstatement \proof By Lemma 5.3, we have $q_{k - 1}^*\mu^{(k-1)}(\alpha) = p_k^*\mu^{(k)}(\alpha) - aD_k$. Since $p_k$ and $q_{k - 1}$ are birational morphisms, $[p_k^*\mu^{(k)}(\alpha)]^d = [\mu^{(k)}(\alpha)]^d$ and $[q_{k - 1}^*\mu^{(k-1)}(\alpha)]^d = [\mu^{(k-1)}(\alpha)]^d$. Thus, $[\mu^{(k - 1)}(\alpha)]^d - [\mu^{(k)}(\alpha)]^d$ is equal to $$\align &\quad \sum_{i = 1}^d~ {d \choose i} \cdot [p_k^*\mu^{(k)}(\alpha)|D_k]^{d - i} \cdot (-D_k|D_k)^{i - 1} \cdot (-a^i) \\ &= \sum_{i = 1}^d~ {d \choose i} \cdot [p_k^*\mu^{(k)}(\alpha)|D_k]^{d - i} \cdot \xi_k^{i - 1} \cdot (-a^i). \\ \endalign$$ By Lemma 5.2, $p_k^*\mu^{(k)}(\alpha)|D_k = (p_k|D_k)^*([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha - a \lambda_k)$. So we have $$\align &\quad [\mu^{(k - 1)}(\alpha)]^d - [\mu^{(k)}(\alpha)]^d \\ &= \sum_{i = 1}^d~ {d \choose i} \cdot \sum_{j = 0}^{2\ell_\zeta}~ {{d - i} \choose j} \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot (-a\lambda_k)^{d - i - j} \cdot \xi_k^{i - 1} \cdot (-a^i)\\ &= \sum_{j = 0}^{2\ell_\zeta}~ \sum_{i = 1}^{d - j}~ {d \choose j} \cdot {{d - j} \choose i} \cdot (-a^{d - j}) \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \xi_k^{i - 1} \cdot (-\lambda_k)^{d - i - j}\\ &= \sum_{j = 0}^{2\ell_\zeta}~ {d \choose j} \cdot (-a^{d - j}) \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \sum_{i = 1}^{d - j}~ {{d - j} \choose i} \cdot \xi_k^{i - 1} \cdot (-\lambda_k)^{d - i - j}\\ &= \sum_{j = 0}^{2\ell_\zeta}~ {d \choose j} \cdot (-a^{d - j}) \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \sum_{i = 0}^{d - 1 - j}~ {{d - j} \choose {i + 1}} \cdot \xi_k^{i} \cdot (-\lambda_k)^{d - 1 - j - i}\\ \endalign$$ Now, our formula follows from the following claim by summing $k$ from $0$ to $\ell_\zeta$. \claim{} $$([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \sum_{i = 0}^{d - 1 - j}~ {{d - j} \choose {i + 1}} \cdot \xi_k^{i} \cdot (-\lambda_k)^{d - 1 - j - i}$$ $$= ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot (-1)^{h(\zeta) + \ell_\zeta + j - 1} \cdot s_{2\ell_\zeta - j}(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck).$$ \endstatement \par\noindent {\it Proof.} For simplicity, on the exceptional divisor $D_k$, we put $$\sigma_s = ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \sum_{i = 0}^s {{s + 1} \choose {i + 1}} \cdot \xi_k^i \cdot (-\lambda_k)^{s - i}.$$ So we must compute $\sigma_{d - 1 - j}$. Notice the relation $$\sigma_s + \lambda_k \cdot \sigma_{s - 1} = ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot (\xi_k - \lambda_k)^s.$$ Thus for $0 \le t \le s$, we have $$\sigma_s = (-\lambda_k)^t \cdot \sigma_{s - t} + ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \sum_{i = 0}^{t - 1} (\xi_k - \lambda_k)^{s - i} \cdot (-\lambda_k)^i.$$ Put $s = d - 1 - j$ and $t = s - {N_{-\zeta}} = d - 1 - j - {N_{-\zeta}}$, where ${N_{-\zeta}} = \ell_{-\zeta} + h(-\zeta) - 1 = \ell_{\zeta} + h(-\zeta) - 1$ as defined in Corollary 2.7. Then, $\sigma_{d - 1 - j}$ is equal to $$(-\lambda_k)^{d - 1 - j - {N_{-\zeta}}} \cdot \sigma_{N_{-\zeta}} + ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \sum_{i = 0}^{d - 2 - j - {N_{-\zeta}}} (\xi_k - \lambda_k)^{(d - 1 - j) - i} \cdot (-\lambda_k)^i.$$ Since $\dim E_\zeta^{\ell_\zeta - k, k} = d - 1 - {N_{-\zeta}}$, we see that $(-\lambda_k)^{d - 1 - j - {N_{-\zeta}}} \cdot \sigma_{N_{-\zeta}}$ is equal to $$\align &\quad (-\lambda_k)^{d - 1 - j - {N_{-\zeta}}} \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \sum_{i = 0}^{N_{-\zeta}} {{{N_{-\zeta}} + 1} \choose {i + 1}} \cdot \xi_k^i \cdot (-\lambda_k)^{{N_{-\zeta}} - i}\\ &= (-\lambda_k)^{d - 1 - j - {N_{-\zeta}}} \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \xi_k^{N_{-\zeta}}\\ &= ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot (-\lambda_k)^{d - 1 - j - {N_{-\zeta}}} \cdot (\xi_k - \lambda_k)^{N_{-\zeta}}\\ \endalign$$ since the restriction of $\xi_k$ to a fiber of $D_k \to E_\zeta^{\ell_\zeta - k, k}$ is a hyperplane. Therefore, $$\sigma_{d - 1 - j} = ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \sum_{i = 0}^{d - 1 - j - {N_{-\zeta}}} (\xi_k - \lambda_k)^{(d - 1 - j) - i} \cdot (-\lambda_k)^i.$$ Now, we shall simplify $(\xi_k - \lambda_k)^{(d - 1 - j) - i}$. Since $\xi_k$ is the tautological line bundle on $D_k = \Pee(\Cal N_k\spcheck)$, the line bundle $(\xi_k \otimes \lambda_k^{-1})$ is the tautological line bundle on $$\Pee(\Cal N_k\spcheck \otimes \lambda_k^{-1}) = \Pee[((\rho_k|E_\zeta^{k, \ell_\zeta - k})\spcheck\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck].$$ Since ${N_{-\zeta}} + 1$ is the rank of $\Cal E_{-\zeta}^{k, \ell_\zeta -k}$, it follows that $$(\xi_k - \lambda_k)^{1 + {N_{-\zeta}}} = -\sum_{j = 1}^{1 + {N_{-\zeta}}} c_j(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot (\xi_k - \lambda_k)^{1 + {N_{-\zeta}} - j}.$$ One verifies that in general, for $u' \ge {N_{-\zeta}}$, one has $$(\xi_k - \lambda_k)^{u'} = s_{u' - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot (\xi_k - \lambda_k)^{{N_{-\zeta}}} + O\left((\xi_k - \lambda_k)^{{N_{-\zeta}} - 1}\right)$$ where $s_i(\Cal E_{-\zeta}^{k, \ell_\zeta -k})$ is the $i^{\text{th}}$ Segre class of $\Cal E_{-\zeta}^{k, \ell_\zeta -k}$. Therefore, since $(d - 1 - j) - i \ge {N_{-\zeta}}$, we see that $(\xi_k - \lambda_k)^{(d - 1 - j) - i}$ is equal to $$s_{d - 1 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot (\xi_k - \lambda_k)^{{N_{-\zeta}}} + O\left((\xi_k - \lambda_k)^{{N_{-\zeta}} - 1}\right)$$ and that $([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot (\xi_k - \lambda_k)^{(d - 1 - j) - i} \cdot (-\lambda_k)^i$ is equal to $$([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \left[s_{d - 1 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot (\xi_k - \lambda_k)^{{N_{-\zeta}}}\right] \cdot (-\lambda_k)^i$$ $$= ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot s_{d - 1 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot (-\lambda_k)^i.$$ Next, we note that $([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot s_{d - 1 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k})$ is a cycle on $E_{\zeta}^{\ell_\zeta -k, k}$ pulled-back from $H_{\ell_\zeta -k} \times H_k$. So this term is zero unless $d - 1 - i - {N_{-\zeta}} \le 2\ell_\zeta$, that is, $i \ge d - 1 - {N_{-\zeta}} - 2\ell_\zeta$. Note that by Corollary 2.7, $d - 1 - {N_{-\zeta}} - 2\ell_\zeta = {N_\zeta}$ and ${N_\zeta} + 1 = h(\zeta) + \ell_\zeta$ is the rank of $\Cal E_{\zeta}^{\ell_\zeta -k, k}$. Since $\lambda_k$ is the tautological line bundle on $E_{\zeta}^{\ell_\zeta -k, k} = \Pee((\Cal E_{\zeta}^{\ell_\zeta -k, k})\spcheck)$, we see as before that $$\lambda_k^i = s_{i - {N_\zeta}}(\Cal E_{\zeta}^{\ell_\zeta -k, k}) \cdot \lambda_k^{N_\zeta} + O\left(\lambda_k^{{N_\zeta} - 1}\right).$$ Putting all these together, we conclude that $\sigma_{d - 1 - j}$ is equal to $$\align &\quad ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \sum_{i = {N_\zeta}}^{d - 1 - j - {N_{-\zeta}}}~ s_{d - 1 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot (-1)^i \cdot s_{i - {N_\zeta}}(\Cal E_{\zeta}^{\ell_\zeta -k, k})\\ &= ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \sum_{i = 0}^{2\ell_\zeta - j}~ (-1)^{i + {N_\zeta}} \cdot s_{(2\ell_\zeta - j) - i}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot s_{i}(\Cal E_{\zeta}^{\ell_\zeta -k, k})\\ &= ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot (-1)^{j + {N_\zeta}} \cdot\sum_{i = 0}^{2\ell_\zeta - j}~ s_{(2\ell_\zeta - j) - i}((\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck) \cdot s_{i}(\Cal E_{\zeta}^{\ell_\zeta -k, k})\\ &= ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot (-1)^{j + {N_\zeta}} \cdot s_{2\ell_\zeta - j}(\Cal E_{\zeta}^{\ell_\zeta -k, k} \oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck) \qed\\ \endalign$$ This completes the proof of the Theorem. \endproof For the difference $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$, we have the following. \theorem{5.5} Let $\zeta$ define a wall of type $(w, p)$, and $d = -p - 3$. For $\alpha \in H_2(X; \Zee)$, put $a = (\zeta \cdot \alpha)/2$. Then, $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to $${1 \over 4} \cdot \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot (-1)^{h(\zeta) + \ell_\zeta - 1 + j} \cdot a^{d - 2 - j} \cdot$$ $$\sum_{k = 0}^{\ell_\zeta} ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \left [ s_{2 \ell_\zeta - j} - 4 ([{\Cal Z_{\ell_\zeta - k}}] + [{\Cal Z_{k}}])/x \cdot s_{2 \ell_\zeta - 2 - j} \right ]$$ where $s_i$ stands for the $i^{\text{th}}$ Segre class of $\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck$. \endstatement \proof By Lemma 5.3, we have $q_{k - 1}^*\mu^{(k-1)}(\alpha) = p_k^*\mu^{(k)}(\alpha) - aD_k$ and $$q_{k - 1}^*\nu^{(k - 1)} = p_k^*\nu^{(k)} - {1 \over 4} [D_k^2 + 2 (p_k|D_k)^*\lambda_k].$$ It follows that $[\mu^{(k-1)}(\alpha)]^{d - 2} \cdot \nu^{(k-1)} - [\mu^{(k)}(\alpha)]^{d - 2} \cdot \nu^{(k)} = I_1 + I_2$ where $$\align &I_1 = [\mu^{(k)}(\alpha) - aD_k]^{d - 2} \cdot {1 \over 4} [-D_k^2 - 2 (p_k|D_k)^*\lambda_k] \\ &\quad = [\mu^{(k)}(\alpha)|D_k + a\xi_k]^{d - 2} \cdot {1 \over 4} (\xi_k - 2 \lambda_k) \\ &I_2 = \sum_{i = 1}^{d - 2} {{d - 2} \choose i} \cdot \mu^{(k)}(\alpha)^{d - 2 - i} \cdot (-aD_k)^i \cdot \nu^{(k)} \\ &\quad = \sum_{i = 1}^{d - 2} {{d - 2} \choose i} \cdot [\mu^{(k)}(\alpha)|D_k]^{d - 2 - i} \cdot \xi_k^{i - 1} \cdot (-a^i) \cdot (\nu^{(k)}|D_k).\\ \endalign$$ First of all, since $\mu^{(k)}(\alpha)|D_k = ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha - a \lambda_k)$, we see that $$\align I_1 &= \left [([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha) + a (\xi_k - \lambda_k) \right]^{d - 2} \cdot {1 \over 4} (\xi_k - 2 \lambda_k)\\ &= {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot a^{d - 2 - j} \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot (\xi_k - \lambda_k)^{d - 2 - j} \cdot (\xi_k - 2 \lambda_k)\\ &= {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot a^{d - 2 - j} \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \\ &\quad\quad\quad\quad \cdot \left [ (\xi_k - \lambda_k)^{d - 1 - j} - \lambda_k \cdot (\xi_k - \lambda_k)^{d - 2 - j} \right ]\\ &= {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot a^{d - 2 - j} \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \\ &\quad\quad\quad\quad \cdot \left [ s_{d - 1 - j - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) - \lambda_k \cdot s_{d - 2 - j - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \right ].\\ \endalign$$ Next, by Lemma 5.2, we have $\nu^{(k)}|D_k = {1 \over 4} \left [ 4 [{\Cal Z_{\ell_\zeta - k}}]/x + 4 [{\Cal Z_{k}}]/x - \lambda_k^2 \right ]$. Thus, as in the proof of Theorem 5.4, we can verify that $I_2$ is equal to $$\align &\quad {1 \over 4} \left [ 4 [{\Cal Z_{\ell_\zeta - k}}]/x + 4 [{\Cal Z_{k}}]/x - \lambda_k^2 \right ] \cdot \sum_{i = 1}^{d - 2} {{d - 2} \choose i} \cdot [\mu^{(k)}(\alpha)|D_k]^{d - 2 - i} \cdot \xi_k^{i - 1} \cdot (-a^i) \\ &= {1 \over 4} \left [ 4 [{\Cal Z_{\ell_\zeta - k}}]/x + 4 [{\Cal Z_{k}}]/x - \lambda_k^2 \right ] \cdot \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot (-a^{d - 2 - j}) \cdot \\ &\quad\quad\quad \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \sum_{i = 0}^{2 \ell_\zeta + {N_\zeta} - 2 - j} s_{d - 3 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot (- \lambda_k)^i\\ &= {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot (-a^{d - 2 - j}) \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \\ &\cdot \left [ 4 ([{\Cal Z_{\ell_\zeta - k}}] + [{\Cal Z_{k}}])/x \cdot (-1)^{j + {N_\zeta}} \cdot s' - \sum_{i = 0}^{2 \ell_\zeta + {N_\zeta} - 2 - j} s_{d - 3 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot (- \lambda_k)^{i + 2} \right ] \\ &= {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot a^{d - 2 - j} \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \\ &\cdot \left [ \sum_{i = 0}^{2 \ell_\zeta + {N_\zeta} - 2 - j} s_{d - 3 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot (- \lambda_k)^{i + 2} - 4 ([{\Cal Z_{\ell_\zeta - k}}] + [{\Cal Z_{k}}])/x \cdot (-1)^{j + {N_\zeta}} \cdot s' \right ] \\ \endalign$$ where $s'$ stands for $s_{2 \ell_\zeta - 2 - j}(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck)$. Thus, $I_1 + I_2$ is equal to $$\align &\quad {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot a^{d - 2 - j} \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \\ &\cdot \left [ \sum_{i = -2}^{2 \ell_\zeta + {N_\zeta} - 2 - j} s_{d - 3 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot (- \lambda_k)^{i + 2} - 4 ([{\Cal Z_{\ell_\zeta - k}}] + [{\Cal Z_{k}}])/x \cdot (-1)^{j + {N_\zeta}} \cdot s' \right ] \\ &= {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot a^{d - 2 - j} \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \\ &\quad \cdot \left [ (-1)^{j + {N_\zeta}} \cdot s'' - 4 ([{\Cal Z_{\ell_\zeta - k}}] + [{\Cal Z_{k}}])/x \cdot (-1)^{j + {N_\zeta}} \cdot s' \right ] \\ &= {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot (-1)^{h(\zeta) + \ell_\zeta - 1 + j} \cdot a^{d - 2 - j} \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \\ &\quad \cdot \left [ s'' - 4 ([{\Cal Z_{\ell_\zeta - k}}] + [{\Cal Z_{k}}])/x \cdot s' \right ] \\ \endalign$$ since ${N_\zeta} = h(\zeta) + \ell_\zeta - 1$, where $s''$ stands for $s_{2 \ell_\zeta - j}(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck)$. Letting $k$ run from $0$ to $\ell_\zeta$, we obtain the desired formula. \endproof \par\noindent {\bf Remark 5.6.} For the sake of convenience, we record here the following relation among the Chern classes and the Segre classes of a vector bundle: $$s_n = -c_1 \cdot s_{n - 1} - c_2 \cdot s_{n - 2} - \ldots - c_n$$ with the convention that $s_0 = 1$. We refer to \cite{12} for details. \medskip In the next section, using Theorem 5.4 and Theorem 5.5, we shall compute $[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d$ and $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ explicitly when $0 \le \ell_\zeta \le 2$. In principle, Theorem 5.4 and Theorem 5.5 give formulas for these differences in terms of certain intersections in $H_{\ell_\zeta -k} \times H_k$. However, it is difficult to evaluate these intersection numbers in general. In the following, we shall compute the term $$S_j = \sum_{k = 0}^{\ell_\zeta}~ ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot s_{2\ell_\zeta - j}(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck) \eqno (5.7)$$ in the special cases when $j = 2\ell_\zeta$ and $2\ell_\zeta - 1$. We start with a simple lemma. \lemma{5.8} Let $\alpha, \beta \in H_2(X; \Zee)$. Then $$\align &([\Cal Z_k]/\alpha)^{2k} = {{(2k)!} \over {2^k \cdot k!}} \cdot (\alpha^2)^k\\ &([\Cal Z_k]/\alpha)^{2k - 1} \cdot ([\Cal Z_k]/\beta) = {{(2k)!} \over {2^k \cdot k!}} \cdot (\alpha^2)^{k - 1} \cdot (\alpha \cdot \beta)\\ &([\Cal Z_k]/\alpha)^{2k - 2} \cdot ([\Cal Z_k]/\beta)^2 = \\ &{{(2k - 2)!} \over {2^{k - 1} \cdot (k - 1)!}} \cdot (\alpha^2)^{k - 1} \cdot \beta^2 + {{(2k - 2)!} \over {2^{k - 2} \cdot (k - 2)!}} \cdot (\alpha^2)^{k - 2} \cdot (\alpha \cdot \beta)^2.\\ \endalign$$ \endstatement \proof The first equality is well-known (see \cite{28} for instance). The other statements follow from the first one by considering $$ ([\Cal Z_k]/\alpha + [\Cal Z_k]/\beta)^{2k}= {{(2k)!} \over {2^k \cdot k!}} \cdot ((\alpha + \beta)^2)^k,$$ and formally equating the terms involving $(2k-1)$ copies of $\alpha$ and one $\beta$ or $(2k-2)$ copies of $\alpha$ and two copies of $\beta$. \endproof The next result computes the term (5.7) when $j = 2\ell_\zeta$. \proposition{5.9} Let $\zeta$ define a wall of type $(w, p)$, and $\alpha \in H_2(X; \Zee)$. Then, $$S_{2\ell_\zeta} = \sum_{k = 0}^{\ell_\zeta}~ ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^{2\ell_\zeta} = {{(2\ell_\zeta)!} \over {\ell_\zeta !}} \cdot (\alpha^2)^{\ell_\zeta}.$$ \endstatement \par\noindent {\it Proof.} This follows in a straightforward way from Lemma 5.8 (i): $$\align &\quad \sum_{k = 0}^{\ell_\zeta}~ ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^{2\ell_\zeta}\\ &= \sum_{k = 0}^{\ell_\zeta}~{{2\ell_\zeta} \choose {2k}} \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha)^{2(\ell_\zeta - k)} \cdot ([{\Cal Z_{k}}]/\alpha)^{2k} \\ &= \sum_{k = 0}^{\ell_\zeta}~{{2\ell_\zeta} \choose {2k}} \cdot \left [{{(2\ell_\zeta - 2k)!} \over {2^{\ell_\zeta - k} \cdot (\ell_\zeta - k)!}} \cdot (\alpha^2)^{\ell_\zeta - k} \right ] \cdot \left [{{(2k)!} \over {2^k \cdot k!}} \cdot (\alpha^2)^k \right ]\\ &= \sum_{k = 0}^{\ell_\zeta}~ {\ell_\zeta \choose k} \cdot {{(2\ell_\zeta)!} \over {2^{\ell_\zeta} \cdot \ell_\zeta !}} \cdot (\alpha^2)^{\ell_\zeta} \\ &= {{(2\ell_\zeta)!} \over {\ell_\zeta !}} \cdot (\alpha^2)^{\ell_\zeta}\qed\\ \endalign$$ To compute the term (5.7) when $j = (2\ell_\zeta - 1)$, we study $\Cal E_{-\zeta}^{k, \ell_\zeta -k}$ and $\Cal E_{\zeta}^{\ell_\zeta -k, k}$, and evaluate their first Chern classes. We begin with a general lemma. \lemma{5.10} Let $Z, W$ be codimension $2$ cycles in a smooth variety $Y$. \roster \item"{(i)}" If $Z \subseteq W$, then $Hom(I_W, I_Z) = \Cal O_Y$; \item"{(ii)}" If $(Z - Z \cap W)$ is open and dense in $Z$, then $Hom(I_W, I_Z) = I_Z$; \item"{(iii)}" If $Z$ and $W$ are local complete intersections meeting properly, then there is an exact sequence: $$0 \to Ext ^1(I_W, I_Z) \to \Cal O _W \otimes \det N_W \to \Cal O _{W \cap Z} \otimes \det N_W \to 0$$ where $N_W$ is the normal bundle of $W$ in $Y$; \item"{(iv)}" Assume that $Z \cap W$ is nowhere dense in $W$ and that $W$ is smooth at a generic point. Then, as a sheaf on $W$, $Ext^1(I_W, I_Z)$ is of rank $1$; thus, $$c_0(Ext^1(I_W, I_Z)) = c_1(Ext^1(I_W, I_Z)) = 0, \quad c_2(Ext^1(I_W, I_Z)) = -[W].$$ \endroster \endstatement \proof (i) Applying the functor $Hom(I_W, \cdot)$ to the exact sequence $$0 \to I_Z \to \Cal O_Y \to \Cal O_Z \to 0,$$ we obtain $0 \to Hom(I_W, I_Z) \to Hom(I_W, \Cal O_Y) = \Cal O_Y$. Thus, $Hom(I_W, I_Z) = I_U$ for some closed subscheme $U$ of $Y$. On the other hand, since $Z \subseteq W$, $$H^0(Y; Hom(I_W, I_Z)) = \Hom(I_W, I_Z) \ne 0.$$ Thus, $U$ must be empty, and $Hom(I_W, I_Z) = \Cal O_Y$. (ii) As in the proof of (i), $Hom(I_W, I_Z) = I_U$ for some closed subscheme $U$ of $Y$. Applying the functor $Hom(\cdot, I_Z)$ to the exact sequence $$0 \to I_W \to \Cal O_Y \to \Cal O_W \to 0,$$ we get $0 \to I_Z \to Hom(I_W, I_Z) = I_U \to Ext^1(\Cal O_W, I_Z)$. Thus, $U \subseteq Z$; moreover, since $Ext^1(\Cal O_W, I_Z) = 0$ on $(X - W)$, we have $(Z - Z \cap W) = (U - U \cap W)$. So $$(Z - Z \cap W) \subseteq U \subseteq Z.$$ Since $(Z - Z \cap W)$ is open and dense in $Z$, it follows that $U = Z$. (iii) We begin with the local identification: let $R$ be a regular local ring, and let $Z$ and $W$ be two codimension $2$ local complete intersection subschemes of $R$ meeting properly. Applying the functor $Hom_R(\cdot, I_Z)$ to the Koszul resolution of $W$ $$0 \to R \to R \oplus R \to I_W \to 0$$ gives $I_Z \oplus I_Z \to I_Z \to Ext_R^1(I_W, I_Z) \to 0$. It follows that $Ext_R^1(I_W, I_Z) = I_Z/(I_Z \cdot I_W)$. Since $Z$ and $W$ are codimension $2$ local complete intersections meeting properly, we have $I_Z \cdot I_W = I_Z \cap I_W$. Thus, $Ext^1_R(I_W, I_Z) \cong I_Z/(I_Z \cap I_W)$, and we can fit it into an exact sequence $$0 \to Ext^1_R(I_W, I_Z) \to R/I_W \to R/(I_W + I_Z) \to 0.$$ Here $(I_W + I_Z)$ corresponds to the intersection $W \cap Z$. The identification of $Ext_R^1(I_W, I_Z)$ and $I_Z/(I_Z \cap I_W)$ is not canonical. Globally we must correct by $\det N_W$. Thus globally we have an exact sequence: $$0 \to Ext ^1(I_W, I_Z) \to \Cal O _W \otimes \det N_W \to \Cal O _{W \cap Z} \otimes \det N_W \to 0.$$ (iv) It is clear that $Ext^1(I_W, I_Z)$ is a sheaf supported on $W$. To show that it has rank $1$ as a sheaf on $W$, it suffices to verify that it has rank $1$ at a generic point $w$ of $W$. Since $Z \cap W$ is nowhere dense in $W$ and $W$ is smooth at a generic point, we may assume that $w \not \in Z$ and that $w$ is a smooth point of $W$. Then it follows from (iii) that $Ext^1(I_W, I_Z)$ is of rank $1$ at $w$. \endproof \lemma{5.11} Let $Hom = Hom(I_{\Cal Z_k}, I_{\Cal Z_{\ell_\zeta - k}})$, $Ext^1 = Ext^1(I_{\Cal Z_k}, I_{\Cal Z_{\ell_\zeta - k}})$, $\pi_1$ and $\pi_2$ be the projections from $X \times (H_{\ell_\zeta - k} \times H_k)$ to $X$ and $(H_{\ell_\zeta - k} \times H_k)$ respectively. \roster \item"{(i)}" There exist a row exact sequence and a column exact sequence: $$\matrix &0&\\ &\downarrow&\\ &\pi_{2*}(\pi_1^*\Cal O_X(\zeta) \otimes \Cal O_{\Cal Z_{\ell_\zeta - k}}) &\\ &\downarrow&\\ 0 \to & R^1\pi_{2*} \left (\pi_1^*\Cal O_X(\zeta) \otimes Hom \right ) & \to \Cal E_{\zeta}^{\ell_\zeta -k, k} \to \pi_{2*} \left (\pi_1^*\Cal O_X(\zeta) \otimes Ext^1 \right ) \to 0; \\ &\downarrow&\\ &[\Cal O_{H_{\ell_\zeta - k} \times H_k}]^{\oplus~ h(\zeta)} &\\ &\downarrow&\\ &0&\\ \endmatrix$$ \item"{(ii)}" $c_1 \left(R^1\pi_{2*} (\pi_1^*\Cal O_X(\zeta) \otimes Hom)\right) = [\Cal Z_{\ell_\zeta - k}]/(\zeta - K_X/2) + \pi_{2*}[c_3(\Cal O_{\Cal Z_{\ell_\zeta - k}})]/2$; \item"{(iii)}" $c_1 \left(\pi_{2*} (\pi_1^*\Cal O_X(\zeta) \otimes Ext^1 )\right ) = [\Cal Z_k]/(\zeta - K_X/2) + \pi_{2*}[c_3(Ext^1)]/2$. \endroster \endstatement \proof (i) Note that the bundle $\Cal E_\zeta ^{\ell_\zeta -k, k}$ is defined as $$Ext^1_{\pi _2}(\pi_1^*\scrO _X(\Delta - F) \otimes I_{\Cal Z_k}, \pi _1^*\scrO _X(F)\otimes I_{\Cal Z_{\ell_\zeta -k}}) = Ext^1_{\pi _2}(I_{\Cal Z_k}, \pi _1^*\scrO _X(\zeta) \otimes I_{\Cal Z_{\ell_\zeta -k}}).$$ Since $R^2\pi_{2*}(\pi_1^*\Cal O_X(\zeta) \otimes Hom) = 0$, the row exact sequence follows from standard facts about relative Ext sheaves. To see the column exact sequence, we use Lemma 5.10 (ii) and apply the functor $\pi_{2*}$ to the exact sequence $$0 \to \pi _1^*\scrO _X(\zeta) \otimes I_{\Cal Z_{\ell_\zeta -k}} \to \pi _1^*\scrO _X(\zeta) \to \pi _1^*\scrO _X(\zeta) \otimes \Cal O_{\Cal Z_{\ell_\zeta -k}} \to 0.$$ (ii) Note that $Hom = I_{\Cal Z_{\ell_\zeta -k}}$ and that $R^i\pi_{2*}(\pi_1^*\Cal O_X(\zeta) \otimes Hom) = 0$ for $i = 0, 2$. By the Grothendieck-Riemann-Roch Theorem, we have $$\align &\quad -\ch \left (R^1\pi_{2*}(\pi_1^*\Cal O_X(\zeta) \otimes Hom)\right )\\ &= \pi_{2*}\left (\ch (\pi_1^*\Cal O_X(\zeta) \otimes I_{\Cal Z_{\ell_\zeta -k}}) \cdot \pi_1^*\Todd (T_X) \right )\\ &= \pi_{2*}\left (\pi_1^*\ch (\Cal O_X(\zeta)) \cdot \ch (I_{\Cal Z_{\ell_\zeta -k}}) \cdot \pi_1^*\Todd (T_X) \right ).\\ \endalign$$ Now, the conclusion follows by comparing the degree $1$ terms and by the fact that $$\ch (I_{\Cal Z_{\ell_\zeta -k}}) = 1 - \ch (\Cal O_{\Cal Z_{\ell_\zeta - k}}) = 1 - [\Cal Z_{\ell_\zeta - k}] - {c_3(\Cal O_{\Cal Z_{\ell_\zeta - k}}) \over 2} + (\text{terms with degree} \ge 4).$$ (iii) We have $R^i\pi_{2*}(\pi_1^*\Cal O_X(\zeta) \otimes Ext^1) = 0$ for $i = 1, 2$. By Lemma 5.10 (iv), $$\ch (Ext^1) = [\Cal Z_k] + {c_3(Ext^1) \over 2} + (\text{terms with degree} \ge 4).$$ Again, using the Grothendieck-Riemann-Roch Theorem, we obtain $$\align &\quad \ch \left (\pi_{2*}(\pi_1^*\Cal O_X(\zeta) \otimes Ext^1)\right )\\ &= \pi_{2*}\left (\ch (\pi_1^*\Cal O_X(\zeta) \otimes Ext^1) \cdot \pi_1^*\Todd (T_X) \right )\\ &= \pi_{2*}\left (\pi_1^*\ch (\Cal O_X(\zeta)) \cdot \ch (Ext^1) \cdot \pi_1^*\Todd (T_X) \right ).\\ \endalign$$ Then, our conclusion follows by comparing the degree $1$ terms. \endproof Now, we can compute the term (5.7) for $j = 2\ell_\zeta - 1$. \proposition{5.12} Let $\alpha \in H_2(X; \Zee)$ and $a = (\zeta \cdot \alpha)/2$. Then, $$\align S_{2\ell_\zeta - 1} &= \sum_{k = 0}^{\ell_\zeta}~ ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^{2\ell_\zeta - 1} \cdot s_1(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck) \\ &= (-4) \cdot {{(2\ell_\zeta)!} \over {\ell_\zeta !}} \cdot (\alpha^2)^{\ell_\zeta - 1} \cdot a.\\ \endalign$$ \endstatement \par\noindent {\it Proof.} By the symmetry between $k$ and $(\ell_\zeta - k)$, we see that $S_{2\ell_\zeta - 1}$ is equal to $$\sum_{k = 0}^{\ell_\zeta}~ ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^{2\ell_\zeta - 1} \cdot {{s_1(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck) + s_1(\Cal E_{\zeta}^{k, \ell_\zeta -k} \oplus (\Cal E_{-\zeta}^{\ell_\zeta - k, k})\spcheck)} \over 2}.$$ From Lemma 5.11, we conclude that $c_1(\Cal E_{\zeta}^{\ell_\zeta - k, k})$ is equal to $$([\Cal Z_{\ell_\zeta - k}] + [\Cal Z_k])/(\zeta - K_X/2) + {{\pi_{2*}[c_3(\Cal O_{\Cal Z_{\ell_\zeta - k}}) + c_3(Ext^1(I_{\Cal Z_k}, I_{\Cal Z_{\ell_\zeta - k}}))]} \over 2}.$$ Since $s_1(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck) = c_1(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) - c_1(\Cal E_{\zeta}^{\ell_\zeta - k, k})$, we see that $${{s_1(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck) + s_1(\Cal E_{\zeta}^{k, \ell_\zeta -k} \oplus (\Cal E_{-\zeta}^{\ell_\zeta - k, k})\spcheck)} \over 2} = (-2) \cdot ([\Cal Z_{\ell_\zeta - k}] + [\Cal Z_k])/\zeta$$ where the $c_3$'s are cancelled out. Therefore, by Lemma 5.8, $$\align S_{2\ell_\zeta - 1}&= \sum_{k = 0}^{\ell_\zeta}~ ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^{2\ell_\zeta - 1} \cdot (-2) \cdot ([\Cal Z_{\ell_\zeta - k}]/\zeta + [\Cal Z_k]/\zeta)\\ &= (-2) \cdot \sum_{k = 0}^{\ell_\zeta}~ [ {{2\ell_\zeta - 1} \choose {2k}} \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha)^{2\ell_\zeta - 2k - 1} \cdot ([{\Cal Z_{k}}]/\alpha)^{2k} \cdot [\Cal Z_{\ell_\zeta - k}]/\zeta\\ &\quad\quad\quad\quad + {{2\ell_\zeta - 1} \choose {2k - 1}} \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha)^{2\ell_\zeta - 2k} \cdot ([{\Cal Z_{k}}]/\alpha)^{2k - 1} \cdot [\Cal Z_k]/\zeta]\\ &= (-4) \cdot \sum_{k = 1}^{\ell_\zeta}~ {{2\ell_\zeta - 1} \choose {2k - 1}} \cdot ([{\Cal Z_{\ell_\zeta - k}}]/\alpha)^{2\ell_\zeta - 2k} \cdot ([{\Cal Z_{k}}]/\alpha)^{2k - 1} \cdot [\Cal Z_k]/\zeta\\ &= (-4) \cdot {{(2\ell_\zeta)!} \over {\ell_\zeta !}} \cdot (\alpha^2)^{\ell_\zeta - 1} \cdot a \qed\\ \endalign$$ It is possible, but far more complicated, to compute (5.7) for $j = 2\ell_\zeta - 2$. Next, we shall draw some consequences from our previous computations. Recall that $q_X$ denotes the intersection form of $X$, and that $$\delta(\Delta) = (-1)^{{{\Delta^2 + \Delta \cdot K_X} \over 2}}$$ is the difference between the complex orientation and the standard orientation on the instanton moduli space (see \cite{6}). Theorem 5.13 below has already been obtained by Kotschick and Morgan \cite{18} for any smooth $4$-manifold with $b_2^+ = 1$. \theorem{5.13} Let $\zeta$ define a wall of type $(w, p)$, and $d = -p - 3$. Then, $$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d \equiv (-1)^{h(\zeta) + \ell_\zeta} \cdot {{d!} \over {\ell_\zeta! \cdot (d - 2\ell_\zeta)!}} \cdot a^{d - 2\ell_\zeta} \cdot (\alpha^2)^{\ell_\zeta} \pmod {a^{d - 2\ell_\zeta + 2}}$$ for $\alpha \in H_2(X; \Zee)$, where $a = (\zeta \cdot \alpha)/2$. In other words, $$\delta^X_{w, p}(\Cal C_-, \Cal C_+) \equiv \delta(\Delta) \cdot (-1)^{h(\zeta) + \ell_\zeta} \cdot {{d!} \over {\ell_\zeta! \cdot (d - 2\ell_\zeta)!}} \cdot \left ( {\zeta\over 2} \right )^{d - 2\ell_\zeta} \cdot q_X^{\ell_\zeta} \pmod {\zeta^{d - 2\ell_\zeta + 2}}.$$ \endstatement \par\noindent {\it Proof.} By Theorem 5.4 and our notation (5.7), we have $$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d \equiv \sum_{j = 2\ell_\zeta - 1}^{2\ell_\zeta}~ {d \choose j} \cdot (-1)^{h(\zeta) + \ell_\zeta + j} \cdot a^{d - j} \cdot S_j \pmod {a^{d - 2\ell_\zeta + 2}}.$$ By Proposition 5.12, $S_{2\ell_\zeta - 1}$ is divisible by $a$. Therefore, $$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d \equiv {d \choose {2\ell_\zeta}} \cdot (-1)^{h(\zeta) + \ell_\zeta} \cdot a^{d - 2\ell_\zeta} \cdot S_{2\ell_\zeta} \pmod {a^{d - 2\ell_\zeta + 2}}.$$ Now, our conclusion follows from Proposition 5.9 and the fact that $$\gamma_{\pm}(\alpha^d) = \delta(\Delta) \cdot \mu_{\pm}(\alpha)^d.\qed$$ The following is proved by using a similar method. \theorem{5.14} Let $\zeta$ define a wall of type $(w, p)$. For $\alpha \in H_2(X; \Zee)$, let $a = (\zeta \cdot \alpha)/2$. Then, modulo $a^{d - 2\ell_\zeta}$, $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to $${1 \over 4} \cdot (-1)^{h(\zeta) + \ell_\zeta - 1} \cdot {{(d - 2)!} \over {\ell_\zeta! \cdot (d - 2 - 2\ell_\zeta)!}} \cdot a^{d - 2 - 2\ell_\zeta} \cdot (\alpha^2)^{\ell_\zeta}.$$ \endstatement \par\noindent {\it Proof.} By Theorem 5.5, $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to $${1 \over 4} \cdot \sum_{j = 2 \ell_\zeta - 1}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot (-1)^{h(\zeta) + \ell_\zeta - 1 + j} \cdot a^{d - 2 - j} \cdot S_j$$ modulo $a^{d - 2\ell_\zeta}$, where $S_j$ is the notation introduced in (5.7). By Proposition 5.12, $S_{2 \ell_\zeta - 1}$ is divisible by $a$; by Proposition 5.9, we have $$S_{2\ell_\zeta} = {{(2\ell_\zeta)!} \over {\ell_\zeta !}} \cdot (\alpha^2)^{\ell_\zeta}.$$ Therefore, modulo $a^{d - 2\ell_\zeta}$, $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to $${1 \over 4} \cdot (-1)^{h(\zeta) + \ell_\zeta - 1} \cdot {{(d - 2)!} \over {\ell_\zeta! \cdot (d - 2 - 2\ell_\zeta)!}} \cdot a^{d - 2 - 2\ell_\zeta} \cdot (\alpha^2)^{\ell_\zeta}. \qed$$ \section{6. The formulas when $\ell_\zeta = 0, 1, 2$.} In this section, we shall compute $[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d$ and $$[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$$ by assuming that $\ell_\zeta = 0, 1, 2$. Our first result, Theorem 6.1 below, was first obtained by Mong and Kotschick \cite{17}. \theorem{6.1} Let $\zeta$ define a wall of type $(w, p)$ with $\ell_\zeta = 0$. Then, $$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d = (-1)^{h(\zeta)} \cdot \left ({{\zeta \cdot \alpha} \over 2} \right )^d$$ for $\alpha \in H_2(X; \Zee)$. In other words, $\delta^X_{w, p}(\Cal C_-, \Cal C_+) = \delta(\Delta) \cdot (-1)^{h(\zeta)} \cdot ({\zeta}/2)^d$. \endstatement \par\noindent {\it Proof.} There are two cases: $h(\zeta)> 0$ and $h(\zeta)= 0$. In the first case when $h(\zeta)> 0$, the formula follows immediately from Theorem 5.4. In the second case when $h(\zeta)= 0$, we must have $\zeta^2 = p$ and $\zeta \cdot K_X = \zeta^2 + 2 = p + 2$ by Corollary 2.7. Then $\frak M_+$ consists of $\frak M_-$ and an additional connected component $E_{-\zeta}^{0, 0} \cong \Pee^{-p - 3}$. We have constructed a universal sheaf $\Cal U$ over $X \times E_{-\zeta}^{0, 0}$: $$0 \to \pi_1^*\Cal O_X(\Delta - F) \otimes \pi_2^*\lambda \to \Cal U \to \pi_1^*\Cal O_X(F) \to 0$$ where $F$ is the unique divisor satisfying $(2F - \Delta) = \zeta$, $\lambda$ is the line bundle corresponding to a hyperplane in $E_{-\zeta}^{0, 0} \cong \Pee^{-p - 3}$, and $\pi_1$ and $\pi_2$ are the natural projections of $X \times E_{-\zeta}^{0, 0}$. Thus for $\alpha \in H_2(X; \Zee)$, we have $$\mu_+(\alpha) = \mu_-(\alpha) - {1 \over 4} \cdot p_1(\Cal U)/\alpha = \mu_-(\alpha) + a \lambda$$ where $a = ({\zeta \cdot \alpha})/2$. Since $h(\zeta) = 0$, we conclude that $$\mu_+(\alpha)^d = \mu_-(\alpha)^d + \left ({{\zeta \cdot \alpha} \over 2} \right )^d = \mu_-(\alpha)^d + (-1)^{h(\zeta)} \cdot \left ({{\zeta \cdot \alpha} \over 2} \right )^d. \qed$$ The proof of the next result is similar to the proof of Theorem 6.1. \theorem{6.2} Let $\zeta$ define a wall of type $(w, p)$ with $\ell_\zeta = 0$, let $d = -p - 3$. Then, for $\alpha \in H_2(X; \Zee)$, we have $$[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_- = {1 \over 4} \cdot (-1)^{h(\zeta) - 1} \cdot \left ({{\zeta \cdot \alpha} \over 2} \right)^{d - 2}. \qed$$ \endstatement Next, we shall study the difference $\delta^X_{w, p}(\Cal C_-, \Cal C_+)$ when $\ell_\zeta = 1$. In this case, we have to know (5.7) for $j = 2, 1, 0$. In view of Propositions 5.9 and 5.12, it suffices to calculate (5.7) for $j = 0$. The following lemma deals with this. \lemma{6.3} Let $\zeta$ define a wall of type $(w, p)$ with $\ell_\zeta = 1$. Then $$S_0 = \sum_{k = 0}^1~ s_2(\Cal E_{\zeta}^{1 - k, k} \oplus (\Cal E_{-\zeta}^{k, 1 - k})\spcheck) = (6 \zeta^2 + 2K_X^2).$$ \endstatement \par\noindent {\it Proof.} First, we compute the Chern classes of $\Cal E_{\zeta}^{1, 0}$. Let notations be as in Lemma 5.11, and set $\ell_\zeta = 1$ and $k = 0$ in Lemma 5.11. Then $Ext^1 = 0$. Since $(H_{\ell_\zeta - k} \times H_k) = X$, the codimension $2$ cycle $\Cal Z_1$ is exactly the diagonal in $X \times (H_{\ell_\zeta - k} \times H_k) = X \times X$. Thus, $\pi_{2*}(\pi_1^*\Cal O_X(\zeta) \otimes \Cal O_{\Cal Z_{\ell_\zeta - k}}) = \Cal O_X(\zeta)$. By Lemma 5.11 (i), the bundle $\Cal E_{\zeta}^{1, 0}$ sits in an exact sequence: $$0 \to \Cal O_X(\zeta) \to \Cal E_{\zeta}^{1, 0} \cong R^1\pi_{2*} \left (\pi_1^*\Cal O_X(\zeta) \otimes Hom \right ) \to \Cal O_X^{\oplus~ h(\zeta)} \to 0.$$ Thus, $c_1(\Cal E_{\zeta}^{1, 0}) = \zeta$ and $c_2(\Cal E_{\zeta}^{1, 0}) = 0$. Next, we compute the Chern classes of $\Cal E_{\zeta}^{0, 1}$. Let $\ell_\zeta = 1$ and $k = 1$ in Lemma 5.11. Then, $Ext^1 = \det (N)$ where $N$ is the normal bundle of $\Cal Z_1$ in $X \times X$. Thus, $$\pi_{2*} \left (\pi_1^*\Cal O_X(\zeta) \otimes Ext^1 \right ) = \Cal O_X(\zeta - K_X).$$ By Lemma 5.11 (i), the bundle $\Cal E_{\zeta}^{0, 1}$ sits in an exact sequence: $$0 \to \Cal O_X^{\oplus~ h(\zeta)} \to \Cal E_{\zeta}^{0, 1} \to \Cal O_X(\zeta - K_X) \to 0.$$ Thus, $c_1(\Cal E_{\zeta}^{0, 1}) = \zeta - K_X$ and $c_2(\Cal E_{\zeta}^{0, 1}) = 0$. Replacing $\zeta$ by $-\zeta$ gives $c_1(\Cal E_{-\zeta}^{0, 1}) = -\zeta - K_X$ and $c_2(\Cal E_{-\zeta}^{0, 1}) = 0$. It follows that $c_1(\Cal E_{\zeta}^{1, 0} \oplus (\Cal E_{-\zeta}^{0, 1})\spcheck) = 2\zeta + K_X$ and that $$c_2(\Cal E_{\zeta}^{1, 0} \oplus (\Cal E_{-\zeta}^{0, 1})\spcheck) = \zeta \cdot (\zeta + K_X) = \zeta^2 + \zeta \cdot K_X.$$ So we conclude that the Segre class $s_2(\Cal E_{\zeta}^{1, 0} \oplus (\Cal E_{-\zeta}^{0, 1})\spcheck)$ is equal to $$c_1(\Cal E_{\zeta}^{1, 0} \oplus (\Cal E_{-\zeta}^{0, 1})\spcheck)^2 - c_2(\Cal E_{\zeta}^{1, 0} \oplus (\Cal E_{-\zeta}^{0, 1})\spcheck) = 3\zeta^2 + 3 \zeta \cdot K_X + K_X^2.$$ Replacing $\zeta$ by $-\zeta$ gives $s_2(\Cal E_{-\zeta}^{1, 0} \oplus (\Cal E_{\zeta}^{0, 1})\spcheck) = 3\zeta^2 - 3 \zeta \cdot K_X + K_X^2$. Therefore, $$\align S_0 &= \sum_{k = 0}^1~ s_2(\Cal E_{\zeta}^{1 - k, k} \oplus (\Cal E_{-\zeta}^{k, 1 - k})\spcheck)\\ &= s_2(\Cal E_{\zeta}^{1, 0} \oplus (\Cal E_{-\zeta}^{0, 1})\spcheck) + s_2(\Cal E_{\zeta}^{0, 1} \oplus (\Cal E_{-\zeta}^{1, 0})\spcheck)\\ &= s_2(\Cal E_{\zeta}^{1, 0} \oplus (\Cal E_{-\zeta}^{0, 1})\spcheck) + s_2((\Cal E_{\zeta}^{0, 1})\spcheck \oplus \Cal E_{-\zeta}^{1, 0})\\ &= (3\zeta^2 + 3 \zeta \cdot K_X + K_X^2) + (3\zeta^2 - 3 \zeta \cdot K_X + K_X^2)\\ &= 6 \zeta^2 + 2K_X^2.\qed\\ \endalign$$ Now we can compute the difference $\delta^X_{w, p}(\Cal C_-, \Cal C_+)$ when $\ell_\zeta = 1$. \theorem{6.4} Let $\zeta$ define a wall of type $(w, p)$ with $\ell_\zeta = 1$. Then, $$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d = (-1)^{h(\zeta) + 1} \cdot \left \{ d(d - 1) \cdot a^{d - 2} \cdot \alpha^2 + (2K_X^2 + 2d + 6) \cdot a^d \right \}$$ for $\alpha \in H_2(X; \Zee)$, where $a = (\zeta \cdot \alpha)/2$. In other words, $\delta^X_{w, p}(\Cal C_-, \Cal C_+)$ is equal to $$\delta(\Delta) \cdot (-1)^{h(\zeta) + 1} \cdot \left \{ d(d - 1) \cdot \left(\zeta \over 2 \right)^{d - 2} \cdot q_X + (2K_X^2 + 2d + 6) \cdot \left ({\zeta \over 2} \right )^d \right \}.$$ \endstatement \par\noindent {\it Proof.} From 5.4, 5.9, 5.12, and 6.3, we conclude that $$\align &\quad [\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d\\ &= (-1)^{h(\zeta) + 1} \cdot d(d - 1) \cdot a^{d - 2} \cdot \alpha^2 + (-1)^{h(\zeta) + 1} \cdot 8d \cdot a^d\\ &\quad\quad\quad\quad\quad\quad\quad\quad + (-1)^{h(\zeta) + 1} \cdot a^d \cdot (6 \zeta^2 + 2K_X^2)\\ &= (-1)^{h(\zeta) + 1} \cdot \left \{ d(d - 1) \cdot a^{d - 2} \cdot \alpha^2 + (2K_X^2 + 2d + 6) \cdot a^d \right \}. \qed\\ \endalign$$ For $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$, we have the following. \theorem{6.5} Let $\zeta$ define a wall of type $(w, p)$ with $\ell_\zeta = 1$, let $d = -p - 3$. For $\alpha \in H_2(X; \Zee)$, let $a = (\zeta \cdot \alpha)/2$. Then, $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to $${1 \over 4} \cdot (-1)^{h(\zeta)} \cdot \left [ (d - 2)(d - 3) \cdot a^{d - 4} \cdot \alpha^2 + (2K_X^2 + 2d - 18) \cdot a^{d - 2} \right ].$$ \endstatement \par\noindent {\it Proof.} By Theorem 5.5, $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to $$\align &\quad {1 \over 4} \cdot \sum_{j = 0}^{2} {{d - 2} \choose j} \cdot (-1)^{h(\zeta) + j} \cdot a^{d - 2 - j} \cdot S_j - (-1)^{h(\zeta)} \cdot a^{d - 2} \cdot \sum_{k = 0}^1 ([{\Cal Z_{1 - k}}] + [{\Cal Z_{k}}])/x \\ &= {1 \over 4} \cdot \sum_{j = 0}^{2} {{d - 2} \choose j} \cdot (-1)^{h(\zeta) + j} \cdot a^{d - 2 - j} \cdot S_j - (-1)^{h(\zeta)} \cdot 2a^{d - 2}.\\ \endalign$$ By Proposition 5.9, Proposition 5.12, and Lemma 6.3, we have $$S_2 = 2 \alpha^2, S_1 = -8a, S_0 = 6 \zeta^2 + 2K_X^2.$$ Therefore, we conclude that $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to $${1 \over 4} \cdot (-1)^{h(\zeta)} \cdot \left [ (d - 2)(d - 3) \cdot a^{d - 4} \cdot \alpha^2 + (2K_X^2 + 2d - 18) \cdot a^{d - 2} \right ]. \qed$$ In the rest of this section, we assume that $\ell_\zeta = 2$. The following standard facts about double coverings can be found in \cite{2, 10}. \lemma{6.6} Let $\phi: Y_1 \to Y_2$ be a double covering between two smooth projective varieties with $\phi_*\Cal O_{Y_1} = \Cal O_{Y_2} \oplus L^{-1}$ where $L$ is a line bundle on $Y_2$. \roster \item"{(i)}" $K_{Y_1} = \phi^*(K_{Y_2} \otimes L)$ and $L^{\otimes 2} = \Cal O_{Y_2}(B)$ where $B$ is the branch locus in $Y_2$ and is the image of the fixed set of the involution $\iota$ on $Y_1$; \item"{(ii)}" If $D$ is a divisor on $Y_1$, then $\phi_*(\Cal O_{Y_1}(D))$ is a rank $2$ bundle on $Y_2$ with $c_1(\phi_*(\Cal O_{Y_1}(D))) = \phi_*D - L$ and $$c_2(\phi_*(\Cal O_{Y_1}(D))) = {1 \over 2} \cdot \left [(\phi_*D)^2 - \phi_*(D^2) - \phi_*D \cdot L \right].$$ \endroster \endstatement Next, we recall some standard facts about the Hilbert scheme $H_2 = \Hilb ^2(X)$. Let $\Delta_0 \subset X \times X$ be the diagonal, and let $\iota$ be the obvious involution on $\tilde H_2 = \operatorname{Bl}_{\Delta_0} (X\times X)$, the blowup of $X\times X$ along $\Delta_0$. Let $E$ be the exceptional divisor of the blowup in $\tilde H_2$. Then, $H_2 = \tilde H_2/\iota$ and the branch locus lies under $E$. Let $\tilde \Cal Z_2 \subset X \times \tilde H_2$ be the pullback of the codimension $2$ cycle $\Cal Z_2 \subset X \times H_2$. Then, $\tilde \Cal Z_2$ splits into a union of two cycles $\tilde H_{12}$ and $\tilde H_{13}$ in $X \times \tilde H_2$, which are the proper transforms in $X \times \tilde H_2$ of the two morphisms of $X \times X$ into $X \times (X \times X)$: the first maps the first factor in $X \times X$ diagonally into $X \times X$ which is the product of the first and second factors in $X \times (X \times X)$, while the second maps the first factor in $X \times X$ diagonally into $X \times X$ which is the product of the first and third factors in $X \times (X \times X)$. Thus each $\tilde H_{1j}$ is isomorphic to $\operatorname{Bl}_{\Delta_0} (X\times X)$, and the projection of each to $\tilde H_2$ is an isomorphism. If $\alpha \in H_2(X; \Zee)$, then $$[\tilde \Cal Z_2]/\alpha = \alpha \otimes 1 + 1 \otimes \alpha = \alpha \otimes 1 + \iota^*(\alpha \otimes 1) \eqno(6.7)$$ where $\alpha \otimes 1$ and $1 \otimes \alpha$ are the pull-backs of $\alpha$ by the two projections of $\tilde H_2$ to $X$. Fix $x \in X$. Let $\tilde X_x$ be the pull-back of $X \times x \subset X \times X$ to $\tilde H_2$. Then, $\tilde X_x$ is isomorphic to the blow-up of $X$ at $p$ with the exceptional divisor $(\tilde X_x \cap E)$; moreover, $$[\tilde \Cal Z_2]/x = \tilde X_x + \iota^* \tilde X_x. \eqno(6.8)$$ It is known (see p. 685 in \cite{9}) that $\Cal Z_2$ is smooth. Let $B$ be the branch locus of the natural double covering from $\Cal Z_2$ to $H_2$. Then, $B \sim 2L$ for some divisor $L$ on $H_2$, and the pull-back of $B \subset H_2$ to $\tilde H_2$ is $2E$. Let $i: \Cal Z_2 \to X \times H_2$ be the embedding, and $\pi_1$ and $\pi_2$ be the natural projections of $X \times H_2$ to $X$ and $H_2$ respectively. In the following, we compute the Chern and Segre classes of $\Cal E_{\zeta}^{2 - k, k}$ for $k = 0, 1, 2$. The method is to use Lemma 5.11 together with Lemma 6.6. We start with $\Cal E_{\zeta}^{2, 0}$. \lemma{6.9} $c_3(\Cal E_{\zeta}^{2, 0}) = c_4(\Cal E_{\zeta}^{2, 0}) = 0$, $c_1(\Cal E_{\zeta}^{2, 0}) = [\Cal Z_2]/\zeta - L$, and $$c_2(\Cal E_{\zeta}^{2, 0}) = {1 \over 2} \left [([\Cal Z_2]/\zeta)^2 - \zeta^2 \cdot X_x - [\Cal Z_2]/\zeta \cdot L \right]$$ where $x$ is any point on $X$, and $X_x$ stands for $[\Cal Z_2]/x$. \endstatement \proof Let notations be as in Lemma 5.11, and let $\ell_{\zeta} = 2$ and $k = 0$. Then, $ Ext^1 = 0$. By Lemma 5.11 (i), $\Cal E_{\zeta}^{2, 0}$ sits in an exact sequence $$0 \to (\pi_2 \cdot i)_*(\pi_1 \cdot i)^*\Cal O_X(\zeta) \to \Cal E_{\zeta}^{2, 0} \to [\Cal O_{H_2}]^{\oplus~ h(\zeta)} \to 0.$$ Since $(\pi_2 \cdot i)_*(\pi_1 \cdot i)^*\Cal O_X(\zeta)$ has rank $2$, $c_3(\Cal E_{\zeta}^{2, 0}) = c_4(\Cal E_{\zeta}^{2, 0}) = 0$. By Lemma 6.6 (ii), $$c_1(\Cal E_{\zeta}^{2, 0}) = (\pi_2 \cdot i)_*(\pi_1 \cdot i)^*\zeta - L = [\Cal Z_2]/\zeta - L$$ since $(\pi_2 \cdot i)_*(\pi_1 \cdot i)^*\zeta = [\Cal Z_2]/\zeta$; moreover, we have $$\align c_2(\Cal E_{\zeta}^{2, 0}) &= {1 \over 2} \left [((\pi_2 \cdot i)_*(\pi_1 \cdot i)^*\zeta)^2 - (\pi_2 \cdot i)_*((\pi_1 \cdot i)^*\zeta)^2 - (\pi_2 \cdot i)_*(\pi_1 \cdot i)^*\zeta \cdot L \right]\\ &= {1 \over 2} \left [([\Cal Z_2]/\zeta)^2 - \zeta^2 \cdot X_x - [\Cal Z_2]/\zeta \cdot L \right]\\ \endalign$$ since $(\pi_2 \cdot i)_*((\pi_1 \cdot i)^*\zeta)^2 = \zeta^2 \cdot (\pi_2 \cdot i)_*(\pi_1 \cdot i)^*x = \zeta^2 \cdot [\Cal Z_2]/x = \zeta^2 \cdot X_x$. \endproof The following follows from Lemma 6.9 and Remark 5.6. \corollary{6.10} The Segre classes of the bundle $\Cal E_{\zeta}^{2, 0}$ are given by $$\align &s_1(\Cal E_{\zeta}^{2, 0}) = L - [\Cal Z_2]/\zeta\\ &s_2(\Cal E_{\zeta}^{2, 0}) = {1 \over 2} \left[[[\Cal Z_2]/\zeta]^2 - 3 [\Cal Z_2]/\zeta \cdot L + 2L^2 + \zeta^2 \cdot X_x \right]\\ &s_3(\Cal E_{\zeta}^{2, 0}) = [\Cal Z_2]/\zeta]^2 \cdot L - 2 [\Cal Z_2]/\zeta \cdot L^2 + L^3 - \zeta^2 \cdot X_x \cdot [\Cal Z_2]/\zeta + \zeta^2 \cdot X_x \cdot L\\ &s_4(\Cal E_{\zeta}^{2, 0}) = {(\zeta^2)^2 \over 2} - 5\zeta^2 - {5 \over 2} \zeta \cdot K_X + (6\chi(\Cal O_X) - K_X^2).\\ \endalign$$ Here we have identified degree $4$ classes with the corresponding integers. \endstatement \par\noindent {\it Proof.} Since the computation is straightforward, we only calculate $s_4(\Cal E_{\zeta}^{2, 0})$. For simplicity, let $c_i$ denote the $i^{\text{th}}$ Chern class of $\Cal E_{\zeta}^{2, 0}$. Note that $c_3 = c_4 = 0$ by Lemma 6.9. Thus, $s_4(\Cal E_{\zeta}^{2, 0}) = c_1^4 - 3 c_1^2 c_2 + c_2^2$ by Remark 5.6. Therefore, $$\align s_4(\Cal E_{\zeta}^{2, 0}) &= ([\Cal Z_2]/\zeta - L)^4 - 3 ([\Cal Z_2]/\zeta - L)^2 \cdot {1 \over 2} \left [([\Cal Z_2]/\zeta)^2 - \zeta^2 \cdot X_x - [\Cal Z_2]/\zeta \cdot L \right]\\ &\quad\quad\quad\quad\quad\quad + {1 \over 4} \left [([\Cal Z_2]/\zeta)^2 - \zeta^2 \cdot X_x - [\Cal Z_2]/\zeta \cdot L \right]^2\\ &= L^4 - {5 \over 2} \cdot [\Cal Z_2]/\zeta \cdot L^3 + {7 \over 4} \cdot ([\Cal Z_2]/\zeta)^2 \cdot L^2 + {3 \over 2}\zeta^2 \cdot X_x \cdot L^2\\ &\quad\quad\quad\quad\quad\quad - {1 \over 4} ([\Cal Z_2]/\zeta)^4 + {1 \over 4}(\zeta^2)^2 \cdot X_x^2 + \zeta^2 \cdot ([\Cal Z_2]/\zeta)^2 \cdot X_x\\ \endalign$$ since $([\Cal Z_2]/\zeta)^3 \cdot L = 0 = [\Cal Z_2]/\zeta \cdot L \cdot X_x$. Now, we need a claim. \claim{} Let $\alpha, \beta \in H_2(X; \Zee)$. Then, we have the following: \roster \item"{(i)}" $[\Cal Z_2]/\alpha \cdot [\Cal Z_2]/\beta \cdot X_x = \alpha \cdot \beta$; \item"{(ii)}" $X_x^2 = 1$; \item"{(iii)}" $X_x \cdot L^2 = -1$; \item"{(iv)}" $L^4 = 6\chi(\Cal O_X) - K_X^2$; \item"{(v)}" $[\Cal Z_2]/\alpha \cdot L^3 = \alpha \cdot K_X$; \item"{(vi)}" $[\Cal Z_2]/\alpha \cdot [\Cal Z_2]/\beta \cdot L^2 = - 2(\alpha \cdot \beta)$. \endroster \endstatement \par\noindent {\it Proof.} Let $\pi: \tilde H_2 \to H_2 = \tilde H_2/\iota$ be the quotient map. By (6.8), we have $$\pi^*X_x = \pi^*([\Cal Z_2]/x) = [\tilde \Cal Z_2]/x = (\tilde X_x + \iota^* \tilde X_x).$$ (i) Recall from (6.7) that $\pi^*([\Cal Z_2]/\alpha) = [\tilde \Cal Z_2]/\alpha = \alpha \otimes 1 + 1 \otimes \alpha$. Thus, $$\align [\Cal Z_2]/\alpha \cdot [\Cal Z_2]/\beta \cdot X_x &= {1 \over 2} \cdot \pi^*([\Cal Z_2]/\alpha) \cdot \pi^*([\Cal Z_2]/\beta) \cdot \pi^*X_x \\ &= {1 \over 2} \cdot (\alpha \otimes 1 + 1 \otimes \alpha) \cdot (\beta \otimes 1 + 1 \otimes \beta) \cdot (\tilde X_x + \iota^* \tilde X_x)\\ &= \alpha \cdot \beta.\\ \endalign$$ (ii) Let $x_1 \in X$ be a point different from $x$. Then, $$\align X_x^2 &= X_x \cdot X_{x_1} = {1 \over 2} \cdot \pi^*(X_x) \cdot \pi^*(X_{x_1}) \\ &= {1 \over 2} \cdot (\tilde X_x + \iota^* \tilde X_x) \cdot (\tilde X_{x_1} + \iota^* \tilde X_{x_1}) \\ &= 1.\\ \endalign$$ (iii) Since $B \sim 2L$ and $\pi^*(B) = 2E$, $\pi^*(L) \sim E$. Thus, $$X_x \cdot L^2 = {1 \over 2} \cdot (\tilde X_x + \iota^* \tilde X_x) \cdot E^2 = \tilde X_x \cdot E^2 = (\tilde X_x \cdot E)^2 = -1.$$ (iv) Since $E = \Pee(N\spcheck)$ where $N$ is the normal bundle of $\Delta_0$ in $X \times X$, $-E|E = \xi$ is the tautological line bundle on $E$. Since $N = T_{\Delta_0}$, $$\xi^2 = -(\pi|E)^*c_1(N) \cdot \xi - c_2(N) = (\pi|E)^*K_{\Delta_0} \cdot \xi + (K_X^2 - 12 \chi(\Cal O_X)).$$ It follows that $\xi^3 = (2K_X^2 - 12 \chi(\Cal O_X)) \cdot \xi$. Therefore, $$L^4 = {1 \over 2} \cdot E^4 = -{1 \over 2} \cdot \xi^3 = 6\chi(\Cal O_X) - K_X^2.$$ (v) Note that $(\alpha \otimes 1)|E = (\pi|E)^*\alpha$ since ${\Delta_0} \cong X$. Thus, $$[\Cal Z_2]/\alpha \cdot L^3 = {1 \over 2} \cdot (\alpha \otimes 1 + 1 \otimes \alpha) \cdot E^3 = (\alpha \otimes 1) \cdot E^3 = (\pi|E)^*\alpha \cdot \xi^2 = \alpha \cdot K_X.$$ (vi) Again since $(\alpha \otimes 1)|E = (\pi|E)^*\alpha = (1 \otimes \alpha)|E$, we have $$\align [\Cal Z_2]/\alpha \cdot [\Cal Z_2]/\beta \cdot L^2 &= {1 \over 2} \cdot (\alpha \otimes 1 + 1 \otimes \alpha) \cdot (\beta \otimes 1 + 1 \otimes \beta) \cdot E^2\\ &= -2 \cdot (\pi|E)^*\alpha \cdot (\pi|E)^*\beta \cdot \xi \\ &= -2 (\alpha \cdot \beta).\qed \\ \endalign$$ We continue the calculation of $s_4(\Cal E_{\zeta}^{2, 0})$. By Lemma 5.8 (i), $([\Cal Z_2]/\zeta)^4 = 3 (\zeta^2)^2$. It follows from the above Claim with a straightforward computation that $$s_4(\Cal E_{\zeta}^{2, 0}) = {(\zeta^2)^2 \over 2} - 5\zeta^2 - {5 \over 2} \zeta \cdot K_X + (6\chi(\Cal O_X) - K_X^2). \qed$$ Next, we compute the Chern and Segre classes of $\Cal E_{\zeta}^{0, 2}$ on $H_2$. \lemma{6.11} $c_3(\Cal E_{\zeta}^{0, 2}) = c_4(\Cal E_{\zeta}^{0, 2}) = 0$, $c_1(\Cal E_{\zeta}^{0, 2}) = [\Cal Z_2]/(\zeta - K_X) + L$, and $$c_2(\Cal E_{\zeta}^{0, 2}) = {1 \over 2} \left [ L \cdot [\Cal Z_2]/(\zeta - K_X) + [[\Cal Z_2]/(\zeta - K_X)]^2 - (\zeta - K_X)^2 \cdot X_x \right].$$ \endstatement \proof Let $\ell_{\zeta} = 2$ and $k = 2$ in Lemma 5.11. By Lemma 6.6 (i), $$(\det T_{\Cal Z_2})^{-1} = \Cal O_{\Cal Z_2}(K_{\Cal Z_2}) = (\pi_2 \cdot i)^*\Cal O_{H_2}(K_{H_2} + L).$$ Let $N_{\Cal Z_2}$ be the normal bundle of $\Cal Z_2$ in $X \times H_2$. Since $\Cal Z_2$ is smooth and has codimension $2$ in $X \times H_2$, $Ext^1 = Ext^1(I_{\Cal Z_2}, \Cal O_{X \times H_2})$ is isomorphic to $$\det N_{\Cal Z_2} = i^*\det T_{X \times H_2} \otimes (\det T_{\Cal Z_2})^{-1} = \Cal O_{\Cal Z_2}((\pi_2 \cdot i)^*L - (\pi_1 \cdot i)^*K_X).$$ By Lemma 5.11 (i), $\Cal E_{\zeta}^{0, 2}$ sits in an exact sequence $$0 \to [\Cal O_{H_2}]^{\oplus~ h(\zeta)} \to \Cal E_{\zeta}^{0, 2} \to (\pi_2 \cdot i)_*\Cal O_{\Cal Z_2}((\pi_2 \cdot i)^*L + (\pi_1 \cdot i)^*(\zeta - K_X)) \to 0.$$ Note that $(\pi_2 \cdot i)_*(\pi_2 \cdot i)^*L = 2L$. Thus, by Lemma 6.6 (ii), $$c_1(\Cal E_{\zeta}^{0, 2}) = (\pi_2 \cdot i)_*\left[(\pi_2 \cdot i)^*L + (\pi_1 \cdot i)^*(\zeta - K_X)\right ] - L = [\Cal Z_2]/(\zeta - K_X) + L.$$ Also, Lemma 6.6 (ii) together with a straightforward calculation gives $$c_2(\Cal E_{\zeta}^{0, 2}) = {1 \over 2} \left [ L \cdot [\Cal Z_2]/(\zeta - K_X) + [[\Cal Z_2]/(\zeta - K_X)]^2 - (\zeta - K_X)^2 \cdot X_x \right ]$$ where we have used the projection formula $$(\pi_2 \cdot i)_*[(\pi_2 \cdot i)^*L \cdot (\pi_1 \cdot i)^*(\zeta - K_X)] = L \cdot (\pi_2 \cdot i)_*(\pi_1 \cdot i)^*(\zeta - K_X)$$ and the fact that $(\pi_2 \cdot i)_*(\pi_2 \cdot i)^*L^2 = 2L^2$. \endproof The following follows from Lemma 6.11 and Remark 5.6. \corollary{6.12} The Segre classes of $\Cal E_{\zeta}^{0, 2}$ are given by $$\align &s_1(\Cal E_{\zeta}^{0, 2}) = [\Cal Z_2]/(K_X - \zeta) - L\\ &s_2(\Cal E_{\zeta}^{0, 2}) = {1 \over 2} \left[[[\Cal Z_2]/(\zeta - K_X)]^2 + 3 [\Cal Z_2]/(\zeta - K_X) \cdot L + 2L^2 + (\zeta - K_X)^2 \cdot X_x \right]\\ &s_3(\Cal E_{\zeta}^{0, 2}) = -[[\Cal Z_2]/(\zeta - K_X)]^2 \cdot L - 2 [\Cal Z_2]/(\zeta - K_X) \cdot L^2 - L^3\\ &\quad\quad\quad\quad\quad\quad - (\zeta - K_X)^2 \cdot X_x \cdot [\Cal Z_2]/(\zeta - K_X) - (\zeta - K_X)^2 \cdot X_x \cdot L\\ &s_4(\Cal E_{\zeta}^{0, 2}) = {((K_X - \zeta)^2)^2 \over 2} - 5(K_X - \zeta)^2 - {5 \over 2} (K_X - \zeta) \cdot K_X + (6\chi(\Cal O_X) - K_X^2).\\ \endalign$$ \endstatement \proof The calculation of $s_4(\Cal E_{\zeta}^{0, 2})$ is similar to that of $s_4(\Cal E_{\zeta}^{2, 0})$ in Corollary 6.10. \endproof Note that $s_4(\Cal E_{\zeta}^{0, 2})$ may be obtained from $s_4(\Cal E_{\zeta}^{2, 0})$ by replacing $\zeta$ by $K_X - \zeta$, and indeed this holds more generally for $s_i$ when we add the sign $(-1)^i$. Now we compute the Chern and Segre classes of $\Cal E_{\zeta}^{1, 1}$ on $X \times X$. \lemma{6.13} Let $\tau_1$ and $\tau_2$ be the two natural projections of $X \times X$ to $X$, let ${\Delta_0}$ be the diagonal in $X \times X$, and let $j\: \Delta _0 \to X\times X$ be the inclusion. Then $$\align &c_1(\Cal E_{\zeta}^{1, 1}) = \tau_1^*\zeta + \tau_2^*(\zeta - K_X) \\ &c_2(\Cal E_{\zeta}^{1, 1}) = \tau_1^*\zeta \cdot \tau_2^*(\zeta - K_X) + {\Delta_0} \\ &c_3(\Cal E_{\zeta}^{1, 1}) = \tau_1^*\zeta \cdot {\Delta_0} - \tau_2^*(\zeta - K_X) \cdot {\Delta_0} - j_*K_{\Delta_0} \\ &c_4(\Cal E_{\zeta}^{1, 1}) = -{{K_{\Delta_0}^2} \over 2}.\\ \endalign$$ \endstatement \proof Let $\ell_{\zeta} = 2$ and $k = 1$ in Lemma 5.11. Recall that $\pi_1$ and $\pi_2$ are the natural projections of $X \times (X \times X)$ to $X$ and $(X \times X)$ respectively. \claim{1} $\pi_{2*}\left (\pi_1^*\Cal O_X(\zeta) \otimes Ext^1 \right) \cong \tau_2^*\Cal O_X(\zeta - K_X) \otimes I_{\Delta_0}$. \endstatement \par\noindent {\it Proof.} Let $\Delta_{12}$ be the diagonal in $X \times X$ which is formed by the first and second factors in $X \times (X \times X)$, and let $\Delta_{13}$ be the diagonal in $X \times X$ which is formed by the first and third factors in $X \times (X \times X)$. Then, $\Delta_{12} \times X$ and $\Delta_{13} \times X$ are smooth codimension $2$ subvarieties in $X \times (X \times X)$. Here it is understood that the factor $X$ in $\Delta_{13} \times X$ is embedded as the second factor in $X \times (X \times X)$. Moreover, $\Delta_{12} \times X$ and $\Delta_{13} \times X$ intersect properly along the diagonal $\Delta_{123}$ in $X \times X \times X$. Thus, from Lemma 5.10 (iii), we conclude that $$Ext^1 = Ext^1(I_{\Delta_{13} \times X}, I_{\Delta_{12} \times X}) \cong I \otimes \det N$$ where $N$ is the normal bundle $\Delta_{13} \times X$ in $X \times (X \times X)$, and $I$ is the ideal sheaf of $\Delta_{123}$ in $\Delta_{13} \times X$. Now, the restriction of $\pi_2$ to $\Delta_{13} \times X$ gives an isomorphism from $\Delta_{13} \times X$ to $X \times X$. Via this isomorphism, $\Delta_{123}$ in $\Delta_{13} \times X$ is identified with the diagonal ${\Delta_0}$ in $X \times X$, $\det N$ is identified with $\tau_2^*(-K_X)$, and the restriction $\pi_1^*\Cal O_X(\zeta)|(\Delta_{13} \times X)$ is identified with $\tau_2^*(\zeta)$. Therefore, $$\pi_{2*}\left (\pi_1^*\Cal O_X(\zeta) \otimes Ext^1 \right) \cong \pi_{2*}\left (\pi_1^*\Cal O_X(\zeta) \otimes I \otimes \det N \right) = \tau_2^*\Cal O_X(\zeta - K_X) \otimes I_{\Delta_0}. \qed$$ Note that $\pi_{2*}(\pi_1^*\Cal O_X(\zeta) \otimes \Cal O_{\Delta_{12} \times X}) = \tau_1^*(\zeta)$. Thus by Lemma 5.11 (i) and Claim 1, we have a row exact sequence and a column exact sequence $$\matrix &0&\\ &\downarrow&\\ &\tau_1^*(\zeta) &\\ &\downarrow&\\ 0 \to & R^1\pi_{2*} \left (\pi_1^*\Cal O_X(\zeta) \otimes Hom \right ) & \to \Cal E_{\zeta}^{1, 1} \to \tau_2^*\Cal O_X(\zeta - K_X) \otimes I_{\Delta_0} \to 0. \\ &\downarrow&\\ &[\Cal O_{X \times X}]^{\oplus~ h(\zeta)} &\\ &\downarrow&\\ &0&\\ \endmatrix \eqno (6.14)$$ In the next claim, we compute the Chern classes of $I_{\Delta_0}$. Clearly, $c_0(I_{\Delta_0}) = 1$. \claim{2} $c_1(I_{\Delta_0}) = 0$, $c_2(I_{\Delta_0}) = {\Delta_0}$, $c_3(I_{\Delta_0}) = -j_*K_{\Delta_0}$, $c_4(I_{\Delta_0}) = {K_{\Delta_0}^2}/2$. \endstatement \proof Note that $\Todd (N_{\Delta_0})^{-1} = 1 + K_{\Delta_0}/2 + (K_{\Delta_0}^2/4 - \chi (\Cal O_{\Delta_0}))$. By a formula on p.288 of \cite{12} (a special case of the Grothendieck-Riemann-Roch Theorem), $$\align \ch (j!\Cal O_{\Delta_0}) &= j_*(\Todd (N_{\Delta_0})^{-1} \cdot \ch (\Cal O_{\Delta_0})) = j_*(\Todd (N_{\Delta_0})^{-1})\\ &= {\Delta_0} + {{j_*K_{\Delta_0}} \over 2} + j_*\left({{K_{\Delta_0}^2} \over 4} - \chi (\Cal O_{\Delta_0}) \right).\\ \endalign$$ Since $\ch (j!\Cal O_{\Delta_0})$ is just equal to $\ch (j_*\Cal O_{\Delta_0})$, we obtain $$\ch (I_{\Delta_0}) = \ch (\Cal O_{X \times X}) - \ch (j_*\Cal O_{\Delta_0}) = 1 - {\Delta_0} - {{j_*K_{\Delta_0}} \over 2} - j_*\left({{K_{\Delta_0}^2} \over 4} - \chi (\Cal O_{\Delta_0}) \right).$$ From this, the Chern classes of $I_{\Delta_0}$ follows immediately. In particular, $$c_4(I_{\Delta_0}) = {{{\Delta_0}^2} \over 2} + j_*\left({{3K_{\Delta_0}^2} \over 2} - 6\chi (\Cal O_{\Delta_0}) \right) = {K_{\Delta_0}^2 \over 2}$$ since ${\Delta_0}^2 = c_2(T_X) = 12 \chi (\Cal O_X) - K_X^2$ (see the Example 8.1.12 in \cite{12}). \endproof Now the calculation of the Chern classes of $\Cal E_{\zeta}^{1, 1}$ follows from (6.14) and Claim 2. In particular, $$\align c_4(\Cal E_{\zeta}^{1, 1}) &= -\tau_1^*\zeta \cdot \tau_2^*(\zeta - K_X) \cdot {\Delta_0} - \tau_1^*\zeta \cdot j_*K_{\Delta_0} + \tau_2^*(\zeta - K_X)^2 \cdot {\Delta_0}\\ &\quad\quad\quad\quad + 2 \tau_2^*(\zeta - K_X) \cdot j_*K_{\Delta_0} + {{j_*K_{\Delta_0}^2} \over 2}= -{{K_{\Delta_0}^2} \over 2}\\ \endalign$$ since $\tau_1^*\zeta \cdot \tau_2^*(\zeta - K_X) \cdot {\Delta_0} = \zeta \cdot (\zeta - K_X)$ and $\tau_1^*\zeta \cdot j_*K_{\Delta_0} = \zeta \cdot K_X$. \endproof The next result follows immediately from Lemma 6.13 and Remark 5.6. \corollary{6.15} Let notations be the same as in Lemma 6.13. Then $$\align &s_1(\Cal E_{\zeta}^{1, 1}) = -\tau_1^*\zeta - \tau_2^*(\zeta - K_X) \\ &s_2(\Cal E_{\zeta}^{1, 1}) = \tau_1^*\zeta^2 + \tau_1^*\zeta \cdot \tau_2^*(\zeta - K_X) + \tau_2^*(\zeta - K_X)^2 - {\Delta_0} \\ &s_3(\Cal E_{\zeta}^{1, 1}) = -\tau_1^*\zeta^2 \cdot \tau_2^*(\zeta - K_X) - \tau_1^*\zeta \cdot \tau_2^*(\zeta - K_X)^2\\ &\quad\quad\quad\quad\quad + \tau_1^*\zeta \cdot {\Delta_0} + 3 \tau_2^*(\zeta - K_X) \cdot {\Delta_0} + j_*K_{\Delta_0}\\ &s_4(\Cal E_{\zeta}^{1, 1}) = (12 \zeta \cdot K_X - 12 \zeta^2 - 3K_X^2).\qed \endalign$$ \endstatement We can now work out (5.7) explicitly for $\ell_\zeta = 2$ and $j = 2, 1, 0$. For simplicity, let $$S_j = \sum_{k = 0}^2~ S_{j, k} = \sum_{k = 0}^2~ ([{\Cal Z_{2 - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot s_{4 - j}(\Cal E_{\zeta}^{2 - k, k} \oplus (\Cal E_{-\zeta}^{k, 2 - k})\spcheck). \eqno (6.16)$$ \lemma{6.17} $S_2 = 64a^2 + (12 \zeta^2 + 4K_X^2 - 20)\alpha^2$ where $a = (\zeta \cdot \alpha)/2$. \endstatement \proof Note that $s_i(\Cal E_{-\zeta}^{k, 2 - k})$ (respectively, $S_{j, k}$) can be obtained from $s_i(\Cal E_{\zeta}^{k, 2 - k})$ (respectively, $(-1)^j \cdot S_{j, 2 - k}$) by replacing $\zeta$ by $-\zeta$. Also, $S_{2, 2}$ is equal to $$([{\Cal Z_2}]/\alpha)^2 \cdot s_2(\Cal E_{\zeta}^{0, 2} \oplus (\Cal E_{-\zeta}^{2, 0})\spcheck) = ([{\Cal Z_2}]/\alpha)^2 \cdot \left [ s_2(\Cal E_{\zeta}^{0, 2}) - s_1(\Cal E_{\zeta}^{0, 2}) \cdot s_1(\Cal E_{-\zeta}^{2, 0}) + s_2(\Cal E_{-\zeta}^{2, 0}) \right].$$ Therefore, by Corollary 6.10 and Corollary 6.12, we obtain $$S_{2, 2} + S_{2, 0} = 32a^2 + (6 \zeta^2 + 2K_X^2 - 12)\alpha^2 + 2(\alpha \cdot K_X)^2.$$ Let $\tau_1$ and $\tau_2$ be the projections of $X \times X$ to $X$. Then, by Corollary 6.15, $$\align S_{2, 1} &= (\tau_1^*\alpha + \tau_2^*\alpha)^2 \cdot s_2(\Cal E_{\zeta}^{1, 1} \oplus (\Cal E_{-\zeta}^{1, 1})\spcheck)\\ &= (\tau_1^*\alpha + \tau_2^*\alpha)^2 \cdot \left[ s_2(\Cal E_{\zeta}^{1, 1}) - s_1(\Cal E_{\zeta}^{1, 1}) \cdot s_1(\Cal E_{-\zeta}^{1, 1}) + s_2(\Cal E_{-\zeta}^{1, 1}) \right]\\ &= 32a^2 + (6 \zeta^2 + 2K_X^2 - 8)\alpha^2 - 2(\alpha \cdot K_X)^2.\\ \endalign$$ It follows that $S_2 = (S_{2, 2} + S_{2, 0}) + S_{2, 1} = 64a^2 + (12 \zeta^2 + 4K_X^2 - 20)\alpha^2$. \endproof Next, adopting the same method as in the proof of Lemma 6.17, we compute the values of $S_1$ and $S_0$ in the next two lemmas respectively. \lemma{6.18} $S_1 = -(48 \zeta^2 + 16K_X^2 - 120) a$ where $a = (\zeta \cdot \alpha)/2$. \endstatement \proof In view of (6.16), we have to compute $S_{1, 2}, S_{1, 1}$, and $S_{1, 0}$. Note that $S_{1, 0}$ can be obtained from $-S_{1, 2}$ by replacing $\zeta$ by $-\zeta$. Using Corollary 6.10 and Corollary 6.12, we see that $(S_{1, 2} + S_{1, 0}) = -(24 \zeta^2 + 8K_X^2 - 72) a - 6(\zeta \cdot K_X) (\alpha \cdot K_X)$. Let $\tau_1$ and $\tau_2$ be the projections of $X \times X$ to $X$. Then, by Corollary 6.15, $$S_{1, 1} = (\tau_1^*\alpha + \tau_2^*\alpha) \cdot s_3(\Cal E_{\zeta}^{1, 1} \oplus (\Cal E_{-\zeta}^{1, 1})\spcheck) = -(24 \zeta^2 + 8K_X^2 - 48) a + 6(\zeta \cdot K_X) (\alpha \cdot K_X).$$ It follows that $S_1 = (S_{1, 2} + S_{1, 0}) + S_{1, 1} = - (48 \zeta^2 + 16K_X^2 - 120) a$. \endproof \lemma{6.19} $S_0 = 18 (\zeta^2)^2 + (14K_X^2 - 105) \zeta^2 + [2(K_X^2)^2 - 50K_X^2 + 96]$. \endstatement \par\noindent {\it Proof.} We need to compute $S_{0, 2}, S_{0, 1}$, and $S_{0, 0}$. Again, $S_{0, 0}$ can be obtained from $S_{0, 2}$ by replacing $\zeta$ by $-\zeta$. Using Corollary 6.10 and Corollary 6.12, we see that $$(S_{0, 2} + S_{0, 0}) = 9 (\zeta^2)^2 + (8K_X^2 - 63) \zeta^2 + [(K_X^2)^2 - 43K_X^2 + 60].$$ By Corollary 6.15, $S_{0, 1} = 9 (\zeta^2)^2 + (6K_X^2 - 42) \zeta^2 + [(K_X^2)^2 - 7K_X^2 + 36]$. Therefore, $$S_0 = (S_{0, 2} + S_{0, 0}) + S_{0, 1} = 18 (\zeta^2)^2 + (14K_X^2 - 105) \zeta^2 + [2(K_X^2)^2 - 50K_X^2 + 96]. \qed$$ Now we can calculate the difference $\delta^X_{w, p}(\Cal C_-, \Cal C_+)$ when $\ell_\zeta = 2$. \theorem{6.20} Let $\zeta$ define a wall of type $(w, p)$ with $\ell_\zeta = 2$. Then $$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d = (-1)^{h(\zeta)} \cdot \left \{ g_0 \cdot a^d + g_1 \cdot a^{d - 2} \cdot \alpha^2 + g_2 \cdot a^{d - 4} \cdot (\alpha^2)^2 \right \}$$ for $\alpha \in H_2(X; \Zee)$, where $a$ stands for $(\zeta \cdot \alpha)/2$ and $$\align &g_2 = {{d!} \over {2! \cdot (d - 4)!}}\\ &g_1 = {d \choose 2} \cdot (4K_X^2 + 4d + 8)\\ &g_0 = 2d^2 + 2d \cdot K_X^2 + 2 (K_X^2)^2 + 13d + 20K_X^2 + 21.\\ \endalign$$ In other words, the difference $\delta^X_{w, p}(\Cal C_-, \Cal C_+)$ is equal to $$\delta(\Delta) \cdot (-1)^{h(\zeta)} \cdot \left \{ g_0 \cdot \left({\zeta \over 2} \right)^d + g_1 \cdot \left({\zeta \over 2} \right)^{d - 2} \cdot q_X + g_2 \cdot \left({\zeta \over 2} \right)^{d - 4} \cdot q_X^2 \right \}.$$ \endstatement \par\noindent {\it Proof.} In view of Theorem 5.4 and the notation (6.16), we have $$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d = \sum_{j = 0}^{4}~ {d \choose j} \cdot (-1)^{h(\zeta) + j} \cdot a^{d - j} \cdot S_j.$$ Now, $S_4$ and $S_3$ are given by Proposition 5.9 and Proposition 5.12 respectively; $S_2, S_1$, and $S_0$ are computed in the previous three lemmas. So it follows that the coefficient of $(-1)^{h(\zeta)} \cdot a^{d - 4} \cdot (\alpha^2)^2$ is equal to $$g_2 = {{d!} \over {2! \cdot (d - 4)!}}.$$ Similarly, also keeping in mind that $\zeta^2 = (p + 8) = (5 - d)$, we have $$\align &g_1 = {d \choose 2} \cdot (12 \zeta^2 + 4K_X^2 + 16d - 52) = {d \choose 2} \cdot (4K_X^2 + 4d + 8)\\ &g_0 = 64 \cdot {d \choose 2} + (48 \zeta^2 + 16K_X^2 - 120) \cdot d~ + \\ &\quad\quad\quad + \left [18(\zeta^2)^2 + 14 \cdot \zeta^2 \cdot K_X^2 + 2 (K_X^2)^2 - 105 \zeta^2 - 50 K_X^2 + 96 \right ]\\ &\quad= 2d^2 + 2d \cdot K_X^2 + 2 (K_X^2)^2 + 13d + 20K_X^2 + 21. \qed \\ \endalign$$ \corollary{6.21} Let $\zeta$ define a wall of type $(w, p)$ with $\ell_\zeta \le 2$. Then, the difference $\delta^X_{w, p}(\Cal C_-, \Cal C_+)$ of Donaldson polynomial invariants is a polynomial in $\zeta$ and $q_X$ with coefficients involving only $\zeta^2$, homotopy invariants of $X$, and universal constants. \endstatement \proof Follows from Theorems 6.1, 6.4, and 6.20. \endproof Finally, we compute the difference $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ for $\ell_\zeta = 2$. \theorem{6.22} Let $\zeta$ define a wall of type $(w, p)$ with $\ell_\zeta = 2$, and let $d = -p - 3$. Then, $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to $${1 \over 4} \cdot (-1)^{h(\zeta) + 1} \cdot \left \{ {\tilde g}_0 \cdot a^{d - 2} + {\tilde g}_1 \cdot a^{d - 4} \cdot \alpha^2 + {\tilde g}_2 \cdot a^{d - 6} \cdot (\alpha^2)^2 \right \}$$ for $\alpha \in H_2(X; \Zee)$, where $a$ stands for $(\zeta \cdot \alpha)/2$ and $$\align &{\tilde g}_2 = {{(d - 2)!} \over {2! \cdot (d - 6)!}}\\ &{\tilde g}_1 = {{d - 2} \choose 2} \cdot (4K_X^2 + 4d - 40)\\ &{\tilde g}_0 = 2d^2 + 2d \cdot K_X^2 + 2 (K_X^2)^2 - 35d - 28K_X^2 - 99.\\ \endalign$$ \endstatement \proof By Theorem 5.5, $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to $${1 \over 4} \cdot \sum_{j = 0}^4 {{d - 2} \choose j} \cdot (-1)^{h(\zeta) + 1 + j} \cdot a^{d - 2 - j} \cdot S_j - \sum_{j = 0}^2 {{d - 2} \choose j} \cdot (-1)^{h(\zeta) + 1 + j} \cdot a^{d - 2 - j} \cdot T_j$$ where for simplicity we have defined $T_j$ as $$T_j = \sum_{k = 0}^2 T_{j, k} = \sum_{k = 0}^2 ([{\Cal Z_{2 - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot ([{\Cal Z_{2 - k}}] + [{\Cal Z_{k}}])/x \cdot s_{2 - j}(\Cal E_{\zeta}^{2 - k, k} \oplus (\Cal E_{-\zeta}^{k, 2 -k})\spcheck).$$ Next, we compute $T_0$. Using Corollary 6.10 and Corollary 6.12, we obtain $$\align T_{0, 0} &= X_x \cdot s_2(\Cal E_{\zeta}^{2, 0} \oplus (\Cal E_{-\zeta}^{0, 2})\spcheck) \\ &= X_x \cdot \left [s_2(\Cal E_{\zeta}^{2, 0}) - s_1(\Cal E_{\zeta}^{2, 0}) \cdot s_1(\Cal E_{-\zeta}^{0, 2}) + s_2(\Cal E_{-\zeta}^{0, 2}) \right ]\\ &= (3\zeta^2 + 3 \zeta \cdot K_X + K_X^2 - 3).\\ \endalign$$ Note that $T_{0, 2}$ can be obtained from $T_{0, 0}$ by replacing $\zeta$ by $-\zeta$. Thus, $$T_{0, 2} = (3\zeta^2 - 3 \zeta \cdot K_X + K_X^2 - 3).$$ Similarly, using Corollary 6.15, we get $T_{0, 1} = (6\zeta^2 + 2K_X^2 - 4)$. Therefore, $$T_0 = \sum_{k = 0}^2 T_{0, k} = (12\zeta^2 + 4K_X^2 - 10).$$ By similar but much simpler arguments, we conclude that $T_1 = -16a$ and $T_2 = 4 \alpha^2$. From (5.9), (5.12), (6.17), (6.18), and (6.19), we have $$\align &S_4 = 12 (\alpha^2)^2, \\ &S_3 = -48 a \cdot \alpha^2,\\ &S_2 = 64a^2 + (12 \zeta^2 + 4K_X^2 - 20)\alpha^2, \\ &S_1 = -(48 \zeta^2 + 16K_X^2 - 120) a, \\ &S_0 = 18 (\zeta^2)^2 + (14K_X^2 - 105) \zeta^2 + [2(K_X^2)^2 - 50K_X^2 + 96].\\ \endalign$$ Putting all these together, we see that $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to $${1 \over 4} \cdot (-1)^{h(\zeta) + 1} \cdot \left \{ {\tilde g}_0 \cdot a^{d - 2} + {\tilde g}_1 \cdot a^{d - 4} \cdot \alpha^2 + {\tilde g}_2 \cdot a^{d - 6} \cdot (\alpha^2)^2 \right \}$$ where ${\tilde g}_0, {\tilde g}_1$, and ${\tilde g}_2$ are as defined in the statement of Theorem 6.22 above. \endproof \Refs \ref \no 1 \by I.V. Artamkin \paper Deforming torsion-free sheaves on an algebraic surface \jour Math. USSR Izv. \vol 36 \pages 449-485 \yr 1991 \endref \ref \no 2 \by W. Barth, C. Peters, A. 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9410

alg-geom/9410028

en

https://arxiv.org/abs/alg-geom/9410028

alg-geom/9410028

Rick Miranda

Bruce Crauder and Rick Miranda

Quantum Cohomology of Rational Surfaces

31 pages, AMS-LaTeX Version 1.1

In this article formulas for the quantum product of a rational surface are given, and used to give an algebro-geometric proof of the associativity of the quantum product for strict Del Pezzo surfaces, those for which $-K$ is very ample. An argument for the associativity in general is proposed, which also avoids resorting to the symplectic category. The enumerative predictions of Kontsevich and Manin concerning the degree of the rational curve locus in a linear system are recovered. The associativity of the quantum product for the cubic surface is shown to be essentially equivalent to the classical enumerative facts concerning lines: there are $27$ of them, each meeting $10$ others.

alg-geom

\section*{Introduction} The purpose of this article is to give an algebro-geometric description of the quantum cohomology ring for a general rational surface $X$. By a ``general'' rational surface we mean one in which all linear systems have the expected dimension, and in which the locus of rational curves in each linear system also has the expected dimension. We understand that it is not known whether the general blowup of the plane in ten or more general points is general in this sense; however we proceed anyway, developing the quantum cohomology ring based on the expected linear systems. In this sense the theory is entirely a numerical one. We view the genericity assumption on the surface $X$ as capable of replacing the genericity assumptions for the symplectic geometry on which the quantum theory is usually based (see for example \cite{ruan-tian,siebert-tian} and a forthcoming paper of Grassi \cite{grassi} which also addresses aspects of the quantum cohomology of rational surfaces). It seems to us desirable to have a description of the quantum product which is based on algebraic geometry rather than symplectic geometry, and this is what we have tried to offer in the case of rational surfaces. In Section \ref{sectionX3class} we explain how classes in the triple product $X^3$ are used to define classes on $X$ itself, via the K\"unneth formula; this is formal. In Section \ref{sectionTPclass} we introduce the ``triple-point'' classes, or Gromov-Witten classes, from which the quantum product is defined. Gromov-Witten classes were first rigorously defined by Ruan \cite{ruan} using symplectic deformations. We have found it convenient to define these classes (which measure rational curves on $X$ with $3$ marked points satisfying certain geometric conditions) in terms of the locus of such curves inside linear systems on $X$, rather than using the space of maps from ${\Bbb P}^1$ to $X$. Other definitions (see \cite{kontsevich-manin,ruan,siebert-tian,witten1,witten2}) use the space of maps instead; for rational surfaces this seems unnecessary, at least for the definitions, and slightly less transparent for our construction. In Section \ref{sectiondefqntm} we give the definition of the quantum cohomology ring. As noted above, the quantum product is defined in terms of classes of loci of rational curves on $X$; not all such loci or cohomology classes appear in the definition, and we call those that do ``relevant'' classes. In Section \ref{strictrelevantsection} we enumerate the relevant classes for the general strict Del Pezzo surfaces, that is, for the general blow-up of the plane at $6$ or fewer points. (These are the surfaces which are Fano surfaces in the strictest sense, namely that $-K$ is very ample.) In Section \ref{sectionformula} we give explicit formulas for the quantum product in terms of the relevant classes on $X$, and using these formulas in Section \ref{sectionordinary} we show that there is a natural map from the quantum cohomology ring to the ordinary cohomology ring. In Section \ref{sectionexamples} we explicitly compute the quantum cohomology rings using the formulas for the minimal rational strict Del Pezzo surfaces, namely ${\Bbb P}^2$, ${\Bbb F}_0 = {\Bbb P}^1\times{\Bbb P}^1$, and ${\Bbb F}_1$. Quantum cohomology seems to be an admixture of homology and cohomology; as such, a functoriality property seems elusive (e.g., should it be contravariant or covariant?). We show in Section \ref{sectionfunctor} that there is at least some degree of functoriality for a blowup. We use this to give a new proof of the associativity of the quantum product for the general strict Del Pezzo surface in Section \ref{sectionSDPassoc}; the functoriality property is strong enough so that it suffices to check associativity on the general $6$-fold blowup of the plane. Moreover the results hold without the genericity assumptions on the $6$-fold blowup mentioned above, although it is known in this case that the general $n$-fold blowup of the plane is general in the above sense for $n \leq 9$. We view this section as the primary technical contribution of the paper, namely the verification of the associativity of the quantum product using algebro-geometric techniques rather than methods from symplectic geometry. Associativity in general is a tricky business; the original heuristic arguments from physics have been made precise only using symplectic geometry (\cite{mcduff-salamon,ruan-tian}). We have given in Section \ref{sectionassoc} an alternate approach using algebraic geometry. The sketch which we offer here is related to the argument given in \cite{witten2}. Finally in Section \ref{sectionenum} we draw some of the enumerative consequences of the associativity of the quantum product. These lead in particular to the formulas appearing in \cite{kontsevich-manin}, interpreted properly. As a small application we show that associativity for the general cubic surface is essentially equivalent to the standard enumerative facts concerning lines: there are $27$ of them and each meets $10$ others. The authors would like to thank Igor Dolgachev, who inspired the second author to think about these questions with an excellent lecture on the subject. We also profited greatly from conversations with Antonella Grassi, Sheldon Katz, and David Morrison. \section{Classes on the triple product} \label{sectionX3class} Let $X$ be a general rational surface; by this we mean firstly that if $X$ is obtained by blowing up $n$ points $p_i$, creating exceptional curves $E_i$, then for every $d$ and $m_i$, the linear system $|dH - \sum_{i=1}^n m_i E_i|$ has the expected dimension (where $H$ is the pullback from the plane of the line class); this expected dimension is ${1\over 2}[d^2+3d - \sum_i m_i(m_i+1)]$ unless this is negative. Secondly we also assume that the locus of irreducible rational curves in this system has codimension equal to the arithmetic genus of the general curve in the system, which is the expected codimension. The ordinary integral cohomology $H^*(X) = H^*(X,{\Bbb Z})$ is \begin{eqnarray*} H^0(X) &=& {\Bbb Z}[X], \\ H^1(X) &=& \{0\}, \\ H^2(X) &\cong& {\Bbb Z}^\rho, \\ H^3(X) &=& \{0\}, \\ H^4(X) &=& {\Bbb Z}[p], \text{ and } \\ H^i(X) &=& \{0\} \text{ for } i \geq 5, \end{eqnarray*} where here $[X]$ is the fundamental class of the surface $X$ itself, and $[p]$ is the class of a single point. The Picard number $\rho$ is the rank of the $H^2$ group. The triple product $X^3 = X \times X \times X$ is then an algebraic six-fold. Its cohomology is (by the K\"unneth Theorem) the triple tensor product of the cohomology of $X$, and is therefore generated over ${\Bbb Z}$ by tensors of the form $\alpha \otimes \beta \otimes \gamma$, where $\alpha$, $\beta$, and $\gamma$ are either $[X]$, $[p]$, or generators for $H^2(X,{\Bbb Z})$. In particular all cohomology is even-dimensional. Suppose that $[A]$ is a cohomology class in the triple product, so that $[A] \in H^{2d}(X^3)$, where $d$ is the complex codimension of the class. Suppose that we choose $\alpha$ and $\beta$ to be homogeneous elements of $H^*(X)$, of degrees $2a$ and $2b$, such that \begin{equation} \label{dimcondition} 4 \leq a + b + d \leq 6. \end{equation} In this case if we let $c = 6 - a - b - d$, then for any class $\gamma \in H^{2c}(X)$, the class $(\alpha\otimes\beta\otimes\gamma)$ will have the complementary dimension to the class $[A]$, and therefore the intersection product \[ [A] \cdot (\alpha \otimes \beta \otimes \gamma) \in {\Bbb Z} \] will be defined. We therefore obtain a linear functional \[ \Phi_{[A]}(\alpha,\beta): H^{2c}(X) \to {\Bbb Z} \] which by duality must be represented by intersection with a cohomology class in $H^{4-2c}(X)$. Call this cohomology class $\phi_{[A]}(\alpha,\beta)$; in this case we have by definition that for all $\gamma \in H^{2c}(X)$, \[ [A] \cdot (\alpha \otimes \beta \otimes \gamma) = \phi_{[A]}(\alpha,\beta) \cdot \gamma, \] and indeed by duality this characterizes the class $\phi_{[A]}(\alpha,\beta)$. Note that the intersection on the left side of this formula is intersection in the cohomology of the triple product $X^3$, while the intersection on the right side is intersection in the cohomology of $X$. The element $\phi_{[A]}(\alpha,\beta)$, by definition, is linear in both $\alpha$ and $\beta$. For notational sanity we declare $\phi_{[A]}(\alpha,\beta) = 0$ unless we have the dimension condition that $4 \leq a + b + d \leq 6$. \begin{example} \label{phiDelta} Let $[\Delta]$ be the class of the diagonal $\Delta \subset X \times X \times X$. Since $\Delta$ has complex codimension $d = 4$, $[\Delta] \in H^8(X^3)$. Let $a$, $b$, and $c$ be non-negative integers such that $a+b \leq 2$ (this is (\ref{dimcondition})) and $c = 2 - a - b$. In this case if $\alpha$, $\beta$, and $\gamma$ are classes in $H^*(X)$ of degrees $a$, $b$, and $c$ respectively, the intersection product $\alpha \cdot \beta \cdot \gamma$ is defined in ${\Bbb Z}$; this is just cup product to $H^4(X)$, then taking the degree. In particular it is equal to $(\alpha \cup \beta) \cdot \gamma$. Moreover $[\Delta] \cdot \alpha\otimes\beta\otimes\gamma$ is equal to this triple intersection. Hence we see that $\phi_{[\Delta]}(\alpha,\beta) = \alpha\cup\beta$. \end{example} \begin{example} Let $[A]$ be the fundamental class of $X^3$; this has codimension $d = 0$, and lies in $H^0(X^3)$. Indeed, it is equal to $[X]\otimes[X]\otimes[X]$. Suppose that $a$, $b$, and $c=6-a-b$ are possible complex codimensions in $X$, satisfying (\ref{dimcondition}), which is that $4 \leq a+b \leq 6$; since $X$ is a surface, we must have $a = b = c = 2$. If $\alpha$, $\beta$, and $\gamma$ are classes in $H^4(X)$, by linearity for the computation we may take all three classes equal to the class $[p]$ of a point. In this case we obviously have $[A] \cdot [p]\otimes[p]\otimes[p] = 1$. Therefore $\phi_{[A]}([p],[p]) = [X]$, and unless both $\alpha$ and $\beta$ lie in $H^4(X)$, $\phi_{[A]}(\alpha,\beta) = 0$. \end{example} \begin{example} \label{exCCC} Let $C$ be an divisor on $X$, and let $[A] = [C] \otimes [C] \otimes [C] \in H^6(X^3)$. Suppose that $a$, $b$, and $c=6-a-b$ are possible complex codimensions in $X$, satisfying (\ref{dimcondition}), which is that $1 \leq a+b \leq 3$, and $\alpha$, $\beta$, and $\gamma$ are classes in $H^*(X)$ of degrees $a$, $b$, and $c$ respectively. Then $[A] \cdot \alpha\otimes\beta\otimes\gamma = ([C]\cdot \alpha)([C]\cdot \beta)([C]\cdot \gamma)$, which is zero unless $a=b=c=1$, and $\alpha$, $\beta$, $\gamma$ are divisor classes. Therefore in this case $\phi_{[A]}(\alpha,\beta) = ([C]\cdot \alpha)([C]\cdot \beta)[C]$ for divisor classes $\alpha$ and $\beta$, and is zero otherwise. \end{example} \begin{example} Let $C$ be an divisor on $X$, and let $[A] = [C] \otimes [C] \otimes [X] \in H^4(X^3)$. Suppose that $\alpha$ and $\beta$ are divisor classes in $H^2(X)$. Then $[A] \cdot \alpha\otimes\beta\otimes[p] = ([C]\cdot \alpha)([C]\cdot \beta)$. Therefore in this case $\phi_{[A]}(\alpha,\beta) = ([C]\cdot \alpha)([C]\cdot \beta)[X]$ for divisor classes $\alpha$ and $\beta$, and is zero otherwise. \end{example} \begin{example} Let $C$ be an divisor on $X$, and instead let $[A] = [C] \otimes [X] \otimes [C] \in H^4(X^3)$. Suppose that $\alpha$ and $\gamma$ are divisor classes in $H^2(X)$. Then $[A] \cdot \alpha\otimes[p]\otimes\gamma = ([C]\cdot \alpha)([C]\cdot \gamma)$. Therefore in this case $\phi_{[A]}(\alpha,[p]) = ([C]\cdot \alpha)[C]$ for a divisor class $\alpha$ (and is zero otherwise). \end{example} \begin{example} Let $C$ be an divisor on $X$, and let $[A] = [C] \otimes [X] \otimes [X] \in H^2(X^3)$. Suppose that $\alpha$ is a divisor classes in $H^2(X)$. Then $[A] \cdot \alpha\otimes[p]\otimes[p] = ([C]\cdot \alpha)$. Therefore in this case $\phi_{[A]}(\alpha,[p]) = ([C]\cdot \alpha)[X]$ for a divisor class $\alpha$ (and is zero otherwise). \end{example} \begin{example} Let $C$ be an divisor on $X$, and instead let $[A] = [X] \otimes [X] \otimes [C] \in H^2(X^3)$. Then for any divisor class $\gamma$ in $H^2(X)$, we have $[A] \cdot [p]\otimes[p]\otimes\gamma = ([C]\cdot \gamma)$. Therefore in this case $\phi_{[A]}([p],[p]) = [C]$. \end{example} \section{Three-point classes on the triple product} \label{sectionTPclass} Fix a divisor class $[C] \in H^2(X)$, such that $|C|$ has no fixed components and is non-empty. Define the locus ${\cal R}_{[C]} \subset |C|$ representing irreducible rational curves with only nodes as singularities. Inside the product ${\cal R}_{[C]} \times X^3$ consider the incidence correspondence \[ {\cal S}_{[C]} = \{(C,p_1,p_2,p_3)\;|\; \text{ the points }p_i \text{ are distinct smooth points on the curve }C\}. \] Let $\overline{{\cal S}}_{[C]}$ be the closure of this subvariety inside $|C|\times X^3$. The second projection gives a regular map $\pi:\overline{{\cal S}}_{[C]} \to X^3$. We define the {\em three-point class} $[A_{[C]}]$ to be $\pi_*[\overline{{\cal S}}_{[C]}]$, the image of the fundamental class. (This is a priori in homology, but we consider it in cohomology via duality.) Note that $\dim {\cal S}_{[C]} = 3 + \dim {\cal R}_{[C]}$. A degenerate version of this locus is obtained when we allow the cohomology class $[C]\in H^2(X)$ to be trivial; we declare in this case that the three-point class $[A_{[0]}]$ is the class of the diagonal $\Delta$. It is an elementary matter to compute the dimensions of these three-point classes, in terms of the numerical characters of the class $[C]$. Since $X$ is a general rational surface, the general member of the linear system $|C|$ is smooth, and the system has the expected dimension, which is $(C\cdot C) +1-p_a(C)$; here $p_a(C)$ is the arithmetic genus, and equals $p_a(C) = 1 + (C \cdot C + K_{X})/2$ by Riemann-Roch. Imposing a node on a member of $|C|$ is one condition; hence the locus of nodal rational curves ${\cal R}_{[C]}$ has dimension $\dim |C| - p_a(C) = (C \cdot C) + 1 - 2p_a(C)$ by our general assumption on $X$. Therefore the dimension of the incidence locus ${\cal S}_{[C]}$ is $(C \cdot C) + 4 - 2p_a(C)$, which we may re-write as \[ \dim {\cal S}_{[C]} = 2 - (C \cdot K_{X}). \] To be more explicit, suppose that $[C] = dH - \sum_{i=1}^n m_i E_i$, where $H$ is the pullback of the line class from ${\Bbb P}^2$ and $E_i$ is the class of the exceptional curve over the blown up point $p_i$. Then $K_{X} = -3H + \sum_i E_i$, so that $(C \cdot K_{X}) = -3d + \sum_i m_i$. Hence we have \[ \dim {\cal S}_{[C]} = 3d +2 - \sum_i m_i. \] The only classes of curves on $X$ which are not of this form are the classes of the exceptional curves $E_i$ themselves. Here $|E_i|$ is a single point (the only member is $E_i$ itself) and $E_i$ is a smooth rational curve; so $\dim {\cal R}_{[E_i]} = 0$ and $\dim {\cal S}_{[E_i]} = 3$. (Actually the formula holds in this case also, with $d=0$, $m_i = -1$, and $m_j = 0$ for $j \neq i$.) \begin{definition} A class $[C] \in H^2(X)$ is {\em relevant} (for quantum cohomology) if either $[C] = 0$ or ${\cal R}_{[C]} \neq \emptyset$ and $\dim {\cal S}_{[C]} \leq 6$. \end{definition} If $\dim {\cal S}_{[C]} > 6$, then the image of the fundamental class of its closure in the six-dimensional variety $X^3$ is trivial. Therefore all non-relevant classes induce a trivial three-point class $[A_{[C]}]$. Given the class $[A_{[C]}]$, they induce as noted in the previous section classes $\phi_{[A_{[C]}]}(\alpha,\beta)$ in the cohomology of $X$. We will abbreviate the notation for these classes and write simply $\phi_{[C]}(\alpha,\beta)$. We note that there is an obvious $S_3$-symmetry to the three-point classes, in the sense that \[ [A_{[C]}] \cdot \alpha_1 \otimes \alpha_2 \otimes \alpha_3 = [A_{[C]}] \cdot \alpha_{\sigma(1)} \otimes \alpha_{\sigma(1)} \otimes \alpha_{\sigma(1)} \] for any permutation $\sigma \in S_3$. This is simply because the locus ${\cal S}_{[C]}$ is $S_3$-invariant. As a consequence of this, we see that the $\phi$-classes are symmetric: \[ \phi_{[C]}(\alpha,\beta) = \phi_{[C]}(\beta,\alpha) \] and of course they are bilinear in $\alpha$ and $\beta$. \begin{example} \label{phi0} If we start with the trivial class $[C] = 0$, then the three-point class $[A_{[0]}]$ is the class of the diagonal. Hence as we have noted above, for any classes $\alpha$ and $\beta$ in $H^*(X)$, \[ \phi_0(\alpha,\beta) = \alpha \cup \beta. \] \end{example} \begin{example} Suppose that $E$ is a $(-1)$-curve on $X$, that is, a smooth rational curve with $(E \cdot E) = -1$. The only member of the linear system $|E|$ is the curve $E$ itself; the locus ${\cal R}_{[E]}$ of nodal rational curves in $|E|$ is the whole system $|E| = \{E\}$. The incidence locus ${\cal S}_{[E]} = \{E\} \times E \times E \times E$ with the large diagonal removed; its closure $\overline{{\cal S}}_{[E]} = \{E\} \times E \times E \times E$. Hence the image under the projection to $X^3$ is $E\times E\times E$, and the class $[A_{[E]}] = [E]\otimes[E]\otimes[E]$. Hence by the computation in Example \ref{exCCC}, we have \[ \phi_{[E]}(\alpha,\beta) = ([E]\cdot \alpha)([E]\cdot \beta)[E] \] for divisor classes $\alpha$ and $\beta$, and is zero unless $\alpha$ and $\beta$ are both in $H^2(M)$. \end{example} \begin{example} More generally suppose that $E$ is an irreducible curve on $X$ whose class is relevant, and $(E\cdot E) - 2p_a(E) = -1$, so that $\dim {\cal R}_{[E]} = 0$ and is therefore a finite set; say it has $d$ members $E_1,\dots,E_d$. The incidence locus ${\cal S}_{[E]} = \bigcup_{i=1}^d \{E_i\} \times E_i \times E_i \times E_i$ with the large diagonal removed; its closure $\overline{{\cal S}}_{[E]} = \bigcup_{i=1}^d \{E_i\} \times E_i \times E_i \times E_i$. Hence the image under the projection to $X^3$ is $\bigcup_{i=1}^d E_i\times E_i\times E_i$, and the class $[A_{[E]}] = d [E]\otimes[E]\otimes[E]$. Hence by the computation in Example \ref{exCCC}, we have \[ \phi_{[E]}(\alpha,\beta) = d ([E]\cdot \alpha)([E]\cdot \beta)[E] \] for divisor classes $\alpha$ and $\beta$, and is zero unless $\alpha$ and $\beta$ are both in $H^2(M)$. This generalizes the previous example, where $d=1$. \end{example} \begin{example} Suppose that $F$ is a fiber in a ruling on $X$, that is, a smooth rational curve with $(F \cdot F) = 0$. The linear system $|F|$ is a pencil; the locus ${\cal R}_{[F]}$ of irreducible nodal rational curves in $|F|$ is an open dense subset of the whole system $|F|$ (it is the set of smooth members of $|F|$). The incidence locus ${\cal S}_{[F]}$ has complex dimension $4$; an element is obtained by choosing a member of $|F|$, then three points on this member. The complementary classes in $H^*(X^3)$ have complex codimension $4$, that is, they are the classes in $H^8(X^3)$. This group is generated by the classes $[p]\otimes[p]\otimes[X]$, $\alpha\otimes\beta\otimes[p]$ for divisor classes $\alpha$ and $\beta$, and the associated classes obtained by symmetry. The intersection product $[A_{[F]}] \cdot [p]\otimes[p]\otimes[X] = 0$, since there is a no curve in the system through two general points $p$. The intersection product $[A_{[F]}] \cdot \alpha\otimes\beta\otimes[p] = (F\cdot \alpha)(F \cdot \beta)$; forcing the curve to pass through the general point $p$ gives a unique curve in the system, and the choice of the other two points, which must lie in the divisor $\alpha$ and $\beta$ respectively, gives the result above. Therefore we have \[ \phi_{[F]}(\alpha,\beta) = ([F]\cdot \alpha)([F]\cdot \beta)[X] \] for divisor classes $\alpha$ and $\beta$. Moreover by the symmetry we also have $[A_{[F]}] \cdot \alpha\otimes[p]\otimes\gamma = (F\cdot \alpha)(F \cdot \gamma)$, so that (after taking symmetry into account) \[ \phi_{[F]}([p],\alpha) = \phi_{[F]}(\alpha,[p]) = ([F]\cdot \alpha)[F] \] for a divisor class $\alpha$. All other $\phi$-classes are zero. \end{example} \begin{example} Suppose that $F$ gives a relevant class on $X$ with $(F\cdot F) - 2p_a(F) = 0$; then the locus ${\cal R}_{[F]}$ of nodal rational curves in $|F|$ forms a curve. Denote by $d$ the degree of this curve in the projective space $|F|$. The incidence locus ${\cal S}_{[F]}$ has complex dimension $4$; an element is obtained by choosing a member of ${\cal R}_{[F]}$, then three points on this member. The complementary classes are the classes in $H^8(X^3)$; This group as above is generated by the classes $[p]\otimes[p]\otimes[X]$, $\alpha\otimes\beta\otimes[p]$ for divisor classes $\alpha$ and $\beta$, and the associated classes obtained by symmetry. The intersection product $[A_{[F]}] \cdot [p]\otimes[p]\otimes[X] = 0$, since there is a no curve in the system through two general points $p$. The intersection product $[A_{[F]}] \cdot \alpha\otimes\beta\otimes[p] = d (F\cdot \alpha)(F \cdot \beta)$; forcing the curve to pass through the general point $p$ gives $d$ curves in the system, and the choice of the other two points, which must lie in the divisor $\alpha$ and $\beta$ respectively, contributes $(F\cdot \alpha)$ and $(F\cdot \beta)$ respectively to the number of choices. Therefore we have \[ \phi_{[F]}(\alpha,\beta) = d ([F]\cdot \alpha)([F]\cdot \beta)[X] \] for divisor classes $\alpha$ and $\beta$. Moreover by the symmetry we also have $[A_{[F]}] \cdot \alpha\otimes[p]\otimes\gamma = d (F\cdot \alpha)(F \cdot \gamma)$, so that (after taking symmetry into account) \[ \phi_{[F]}([p],\alpha) = \phi_{[F]}(\alpha,[p]) = d([F]\cdot \alpha)[F] \] for a divisor class $\alpha$. All other $\phi$-classes are zero. This generalizes the previous example, where $d=1$. \end{example} \begin{example} Suppose that $L$ is a smooth rational curve on $X$ with $(L \cdot L) = 1$. The linear system $|L|$ is a net; the locus ${\cal R}_{[L]}$ of nodal (i.e. smooth) rational curves in $|L|$ is an open dense subset of $|L|$. The incidence locus ${\cal S}_{[L]}$ has complex dimension $5$; an element is obtained by choosing a smooth member of $|L|$, then three points on this member. The complementary classes in $H^*(X^3)$ have complex codimension $5$, that is, they are the classes in $H^{10}(X^3)$. This group is generated by the classes $[p]\otimes[p]\otimes\alpha$, for a divisor classes $\alpha$, and the associated classes obtained by symmetry. The intersection product $[A_{[L]}] \cdot [p]\otimes[p]\otimes\alpha = (L \cdot \alpha)$, since through two general points there is a unique member $L_0$ of $|L|$, whose third point can be any of the points where $L_0$ meets the divisor $\alpha$. Therefore we have \[ \phi_{[L]}(\alpha,[p]) = (L\cdot \alpha)[X] \] for a divisor classes $\alpha$, and \[ \phi_{[L]}([p],[p]) = [L]. \] All other $\phi$-classes are zero. \end{example} \begin{example} Again we may generalize the above in case $L$ induces any relevant class with $(L \cdot L) -2 p_a(L) = 1$. The locus ${\cal R}_{[L]}$ of nodal (i.e. smooth) rational curves in $|L|$ is a surface inside the projective space $|L|$; let $d$ be the degree of this surface. The incidence locus ${\cal S}_{[L]}$ has complex dimension $5$. The complementary classes in $H^*(X^3)$ are the classes in $H^{10}(X^3)$, which is generated by the classes $[p]\otimes[p]\otimes\alpha$, for a divisor classes $\alpha$, and the associated classes obtained by symmetry. The intersection product $[A_{[L]}] \cdot [p]\otimes[p]\otimes\alpha = d (L \cdot \alpha)$, since through two general points there are $d$ members of ${\cal R}_{[L]}$, whose third point can be any of the points where the member meets the divisor $\alpha$. Therefore we have \[ \phi_{[L]}(\alpha,[p]) = d (L\cdot \alpha)[X] \] for a divisor classes $\alpha$, and \[ \phi_{[L]}([p],[p]) = d [L]. \] All other $\phi$-classes are zero. This generalizes the previous example, where $d=1$. \end{example} \begin{example} Suppose that $C$ is a smooth rational curve on $X$ with $(C \cdot C) = 2$. The linear system $|C|$ is a web (that is, it is $3$-dimensional); the locus ${\cal R}_{[C]}$ of nodal (i.e. smooth) rational curves in $|C|$ is again an open dense subset of $|C|$. The incidence locus ${\cal S}_{[C]}$ has complex dimension $6$; an element is obtained by choosing a smooth member of $|C|$, then three points on this member. The complementary classes in $H^*(X^3)$ have complex codimension $6$, that is, they are the classes in $H^{12}(X^3)$. This group has rank one, and is generated by the class $[p]\otimes[p]\otimes[p]$. The intersection product $[A_{[C]}] \cdot [p]\otimes[p]\otimes[p] = 1$, since through three general points there is a unique member of $|C|$. Indeed, we have that $[A_{[C]}] = [X]\otimes[X]\otimes[X]$. Therefore we have \[ \phi_{[C]}([p],[p]) = [X]. \] All other $\phi$-classes are zero. \end{example} \begin{example} \label{phiC2=2} Again if $C$ is a relevant class with $(C\cdot C) - 2p_a(C) = 2$, then the locus ${\cal R}_{[C]}$ of nodal rational curves in $|C|$ is a threefold of $|C|$; let $d$ be the degree of this threefold. The incidence locus ${\cal S}_{[C]}$ has complex dimension $6$; and the complementary classes in $H^*(X^3)$ have complex codimension $6$, that is, they are the classes in $H^{12}(X^3)$, which is generated by the class $[p]\otimes[p]\otimes[p]$. The intersection product $[A_{[C]}] \cdot [p]\otimes[p]\otimes[p] = d$, since through three general points there is are $d$ members of ${\cal R}_{[C]}$. Indeed, we have that $[A_{[C]}] = d [X]\otimes[X]\otimes[X]$. Therefore we have \[ \phi_{[C]}([p],[p]) = d [X]. \] All other $\phi$-classes are zero. This generalizes the previous example, where $d=1$. \end{example} We offer the following example which shows that the above phenomenon occurs, namely that there are relevant classes which come from singular rational curves. \begin{example} Let $X$ be the blow-up of the plane ${\Bbb P}^2$ at $5$ general points $p_1,\dots p_5$. Let $H$ denote the line class on $X$ and $E_i$ denote the exceptional curve lying over $p_i$. Consider the anti-canonical class $C = -K_{X} = 3H - \sum_{i=1}^5 E_i$. This is the linear system of cubics passing through the $5$ points $p_i$. Note that $(C\cdot C) = 4$ and $p_a(C) = 1$ so that $(C\cdot C) - 2p_a(C) = 2$. We have $\dim {\cal S}_{[C]} = 2 - (C \cdot K_{X}) = 6$, so $[C]$ is a relevant class. The map \[ \pi_2:\overline{{\cal S}_{[C]}} \to X^3 \] has as its general fiber over a triple $(q_1,q_2,q_3)$ those nodal rational curves in the linear system $|C|$ through the three points $q_1$, $q_2$, and $q_3$. This is exactly the set of nodal rational cubics in the plane passing through the eight points $p_1,\dots p_5$, $q_1,\dots q_3$. The system of cubics through these general eight points forms a pencil of genus one curves, which has exactly $12$ singular members. Hence the map $\pi_2:\overline{{\cal S}_{[C]}} \to X^3$ is generically finite of degree $12$, and so pushing down the fundamental class we see that \[ [A_{[C]}] = 12[X]\otimes[X]\otimes[X]. \] Hence \[ [A_{[C]}] \cdot [p]\otimes[p]\otimes[p] = 12 \text{ and } \phi_{[C]}([p],[p]) = 12[X]. \] All other $\phi$-classes are zero. \end{example} \section{Definition of quantum cohomology} \label{sectiondefqntm} We denote by $\operatorname{Eff}(X)$ the cone of effective divisor classes in $H^2(X)$. It forms a semigroup under addition. Let $Q = {\Bbb Z}[[\operatorname{Eff}(X)]]$ be the completion of the integral group ring over $\operatorname{Eff}(X)$; $Q$ is a ${\Bbb Z}$-algebra. It is customary to formally introduce a multi-variable $q$ and to write the module generators of $Q$ as elements $q^{[D]}$, where $[D]$ is an effective cohomology class in $\operatorname{Eff}(X)$. With this notation, every element of $Q$ can be written as a formal series \[ \sum_{[D]\in\operatorname{Eff}(X)} n_{[D]} q^{[D]} \] with integral coefficients $n_{[D]}$, and divisor class exponents $[D] \in \operatorname{Eff}(X)$. In this way multiplication in the ring $Q$ is induced by the relations that \[ q^{[D_1]}q^{[D_2]} = q^{[D_1+D_2]} \] for divisors $D_1$ and $D_2$ on $X$, and the ordinary distributive and associative laws. Define the {\em quantum cohomology ring} of $X$ to be \[ H^*_Q(X) = H^*(X) \otimes_{\Bbb Z} Q \] as a free abelian group. Moreover it is also a $Q$-module, with the obvious structure. The multiplication $\operatorname{\ast_Q}$ on $H^*_Q(X)$, called the {\em quantum product}, is determined by knowing the products of (homogeneous) elements from $H^*(X)$, since the rest comes from linearity and the $Q$-module structure. For two homogeneous elements $\alpha$ and $\beta$ in $H^*(X)$ define \[ \alpha \operatorname{\ast_Q} \beta = \sum_{[C]} \phi_{[C]}(\alpha,\beta) q^{[C]}, \] the sum begin taken over the relevant cohomology classes in $H^2(X)$. We remark that if there are only finitely many relevant classes in $H^2(X)$, then the quantum cohomology ring may be formulated as a polynomial ring instead of a power series ring; in other words, one may take $Q = {\Bbb Z}[\operatorname{Eff}(X)]$ to be simply the integral semigroup ring instead of its completion. This is the case for a Del Pezzo surface $X$. We have an immediate identification for the identity of the quantum product: \begin{lemma} The fundamental class $[X]$ is the identity for the quantum product. In other words, for every class $\alpha \in H^*(X)$, \[ \phi_0([X],\alpha) = \alpha \] and if $[C] \neq 0$, then \[ \phi_{[C]}([X],\alpha) = 0. \] \end{lemma} \begin{pf} The $[C]= 0$ statement follows from the computation in Example \ref{phiDelta}; we have \[ \phi_0([X],\alpha) = \phi_{[\Delta]}([X],\alpha) = [X]\cup\alpha = \alpha \] since $[X]$ is the identity for the ordinary cup product. If $[C] \neq 0$, then $\dim {\cal S}_{[C]} = 3 + \dim {\cal R}_{[C]} \geq 3$ if $[C]$ is relevant. Therefore the three-point class $[A_{[C]}]$ lies in $H^{2k}(X^3)$ for $k \leq 3$. Any complementary class of the form $[X]\otimes\beta\otimes\gamma$ must have $\deg(\beta)+\deg(\gamma) = 12-2k \leq 6$. Suppose first that $\dim {\cal S}_{[C]} = 3$, so that ${\cal R}_{[C]}$ is a finite set and the three-point class $[A_{[C]}]$ lies in $H^{6}(X^3)$. The only complementary classes involving the fundamental class $[X]$ have the form $[X]\otimes\beta\otimes[p]$ for some divisor class $\beta$. But $[A_{[C]}] \cdot [X]\otimes\beta\otimes[p]$ counts the number of curves in ${\cal S}_{[C]}$ whose second point lies in the divisor $\beta$ and whose third point equals $p$; since $p$ is a general point and ${\cal R}_{[C]}$ is a finite set, there are no curves in ${\cal R}_{[C]}$ through $p$ and this intersection number is zero. Hence $\phi_{[C]}([X],\alpha) = 0$ for any $\alpha$. Suppose next that $\dim {\cal S}_{[C]} = 4$, so that the locus ${\cal R}_{[C]}$ is one-dimensional and the three-point class $[A_{[C]}]$ lies in $H^{4}(X^3)$. The only complementary classes involving the fundamental class $[X]$ have the form $[X]\otimes[p]\otimes[p]$. The intersection product $[A_{[C]}] \cdot [X]\otimes[p]\otimes[p]$ counts the number of curves in ${\cal S}_{[C]}$ whose second and third point are specified general points; since ${\cal R}_{[C]}$ is one-dimensional, there are no curves in ${\cal R}_{[C]}$ through two specified general points, and this intersection number is zero. Hence again $\phi_{[C]}([X],\alpha) = 0$ for any $\alpha$. Finally if $\dim {\cal S}_{[C]} \geq 5$, there are no complementary classes in the cohomology of $X^3$ of the form $[X]\otimes\beta\otimes\gamma$ at all. \end{pf} \section{Relevant classes on strict Del Pezzo surfaces} \label{strictrelevantsection} Let $X$ be a general strict Del Pezzo surface, that is, $X = {\Bbb F}_0 \cong {\Bbb P}^1\times {\Bbb P}^1$ or a general blowup of the plane such that $-K_{X}$ is very ample. This amounts to having $X \cong {\Bbb F}_0$ or $X$ being a blowup of the plane at $n \leq 6$ general points. It is an elementary matter to compute the relevant classes on such a strict Del Pezzo surface $X$. The results are shown in the tables below. In the first few columns are the numerical characters of the class: the bidegree in the case of ${\Bbb F}_0$ and the integers $d$ and $m_i$ for a class of the form $dH - \sum_i m_i E_i$ on a blowup of the plane; here $d$ is the degree and $m_i$ is the multiplicity of the curves at the associated blown up point. The final three columns contain the quantity $C^2-2p_a(C)$ (on which relevance is based), the arithmetic genus $p_a(C)$, and the number of such classes up to permutations of the $E_i$'s. \begin{center} \begin{tabular}{c|c|c|c} \multicolumn{4}{c}{Relevant nonzero classes on ${\Bbb P}^2$} \\ \hline degree & $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\ 1 & 1 & 0 & 1 \\ \end{tabular} \end{center} \begin{center} \begin{tabular}{c|c|c|c} \multicolumn{4}{c}{Relevant classes on ${\Bbb F}_0$} \\ \hline bidegree & $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\ (0,1) & 0 & 0 & 1 \\ (1,0) & 0 & 0 & 1 \\ (1,1) & 2 & 0 & 1 \\ \end{tabular} \end{center} Denote by $X_n$ a general blowup of ${\Bbb P}^2$ at $n$ points. \begin{center} \begin{tabular}{c|c|c|c|c} \multicolumn{5}{c}{Relevant classes on $X_1$} \\ \hline degree & $m_1$ & $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\ 0 & -1 & -1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ \end{tabular} \end{center} \begin{center} \begin{tabular}{c|cc|c|c|c} \multicolumn{6}{c}{Relevant nonzero classes on $X_2$} \\ \hline degree & $m_1$ & $m_2$ & $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\ 0 & -1 & 0 & -1 & 0 & 2 \\ 1 & 1 & 1 & -1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 & 2 \\ 1 & 0 & 0 & 1 & 0 & 1 \\ 2 & 1 & 1 & 2 & 0 & 1 \\ \end{tabular} \end{center} \begin{center} \begin{tabular}{c|ccc|c|c|c} \multicolumn{7}{c}{Relevant classes on $X_3$} \\ \hline degree & $m_1$ & $m_2$ & $m_3$ & $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\ 0 & -1 & 0 & 0 & -1 & 0 & 3 \\ 1 & 1 & 1 & 0 & -1 & 0 & 3 \\ 1 & 1 & 0 & 0 & 0 & 0 & 3 \\ 1 & 0 & 0 & 0 & 1 & 0 & 1 \\ 2 & 1 & 1 & 1 & 1 & 0 & 1 \\ 2 & 1 & 1 & 0 & 2 & 0 & 3 \\ \end{tabular} \end{center} \begin{center} \begin{tabular}{c|cccc|c|c|c} \multicolumn{8}{c}{Relevant nonzero classes on $X_4$} \\ \hline degree & $m_1$ & $m_2$ & $m_3$ & $m_4$ & $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\ 0 & -1 & 0 & 0 & 0 & -1 & 0 & 4 \\ 1 & 1 & 1 & 0 & 0 & -1 & 0 & 6 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 4 \\ 2 & 1 & 1 & 1 & 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\ 2 & 1 & 1 & 1 & 0 & 1 & 0 & 4 \\ 2 & 1 & 1 & 0 & 0 & 2 & 0 & 6 \\ 3 & 2 & 1 & 1 & 1 & 2 & 0 & 4 \\ \end{tabular} \end{center} \begin{center} \begin{tabular}{c|ccccc|c|c|c} \multicolumn{9}{c}{Relevant classes on $X_5$} \\ \hline degree & $m_1$ & $m_2$ & $m_3$ & $m_4$ & $m_5$ & $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\ 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 5 \\ 1 & 1 & 1 & 0 & 0 & 0 & -1 & 0 & 10 \\ 2 & 1 & 1 & 1 & 1 & 1 & -1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 5 \\ 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 5 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\ 2 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 10 \\ 3 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 5 \\ 2 & 1 & 1 & 0 & 0 & 0 & 2 & 0 & 10 \\ 3 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 \\ 3 & 2 & 1 & 1 & 1 & 0 & 2 & 0 & 20 \\ 4 & 2 & 2 & 2 & 1 & 1 & 2 & 0 & 10 \\ \end{tabular} \end{center} \begin{center} \begin{tabular}{c|cccccc|c|c|c} \multicolumn{10}{c}{Relevant classes on $X_6$} \\ \hline degree & $m_1$ & $m_2$ & $m_3$ & $m_4$ & $m_5$ & $m_6$ & $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 6 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 15 \\ 2 & 1 & 1 & 1 & 1 & 1 & 0 & -1 & 0 & 6 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 \\ 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 15 \\ 3 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 6 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\ 2 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 3 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 3 & 2 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 30 \\ 4 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 0 & 20 \\ 5 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 0 & 1 \\ 2 & 1 & 1 & 0 & 0 & 0 & 0 & 2 & 0 & 15 \\ 3 & 1 & 1 & 1 & 1 & 1 & 0 & 2 & 1 & 6 \\ 3 & 2 & 1 & 1 & 1 & 0 & 0 & 2 & 0 & 60 \\ 4 & 2 & 2 & 1 & 1 & 1 & 1 & 2 & 1 & 15 \\ 4 & 2 & 2 & 2 & 1 & 1 & 0 & 2 & 0 & 60 \\ 4 & 3 & 1 & 1 & 1 & 1 & 1 & 2 & 0 & 6 \\ 5 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 1 & 6 \\ 5 & 3 & 2 & 2 & 2 & 1 & 1 & 2 & 0 & 60 \\ 6 & 3 & 3 & 2 & 2 & 2 & 2 & 2 & 0 & 15 \\ \end{tabular} \end{center} The table of relevant classes on $X_6$ has some features which will be useful below. Let us collect them in the following lemma. Note that the anti-canonical class $-K$ on the surface has $d = 3$ and $m_i = 1$ for each $i = 1,\ldots,6$. \begin{lemma} \label{X6relevantlemma} Let $X_6$ denote a general $6$-fold blowup of the plane (that is, a general cubic surface in ${\Bbb P}^3$). \begin{enumerate} \item All relevant classes $[C]$ on $X_6$ have arithmetic genus $p_a(C) \leq 1$. \item The anti-canonical class $-K$ is the unique relevant class $[C]$ on $X_6$ with $C^2 - 2p_a(C) = 1$ and $p_a(C) = 1$. \item There are exactly $27$ relevant classes $[E]$ on $X_6$ with $E^2 - 2p_a(E) = -1$; all have $p_a(E) = 0$. These are the classes of the $27$ lines on the cubic surface. \item There are exactly $27$ relevant classes $[F]$ on $X_6$ with $F^2 - 2p_a(F) = 0$; all have $p_a(F) = 0$. Each such class $F$ is obtained from a unique relevant class $E$ with $E^2 - 2p_a(E) = -1$ by subtracting $E$ from the anticanonical class: $F = -K - E$. Any two such classes $F$, $G$ satisfy $0 \leq (F \cdot G) \leq 2$; $(F\cdot G) = 0$ if and only if $F = G$. If $(F \cdot G) = 1$ then $C = F+G$ is a relevant class with $C^2 - 2p_a(C) = 2$ and $p_a(C) = 0$. If $(F \cdot G) = 2$ then $C = F+G$ is a relevant class with $C^2 - 2p_a(C) = 2$ and $p_a(C) = 1$. \item There are exactly $72$ relevant classes $[L]$ on $X_6$ with $L^2 - 2p_a(L) = 1$ and $p_a(L) = 0$. For each such class $[L]$ and each relevant class $E$ with $E^2 - 2p_a(E) = -1$ we have $0 \leq (L \cdot E) \leq 2$. If $(L\cdot E) = 1$ then $C = L+E$ is a relevant class with $C^2 - 2p_a(C) = 2$ and $p_a(C) = 0$. If $(L \cdot E) = 2$ then $C = L+E$ is a relevant class with $C^2 - 2p_a(C) = 2$ and $p_a(C) = 1$. \item There are exactly $216$ relevant classes $[C]$ on $X_6$ with $C^2 - 2p_a(C) = 2$ and $p_a(C) = 0$. Each such class $C$ can be written uniquely (up to order) as $C = F + G$, where $F$ and $G$ are classes with $F^2 - 2p_a(F) = G^2 - 2p_a(G) = 0$. \item There are exactly $27$ relevant classes $[C]$ on $X_6$ with $C^2 - 2p_a(C) = 2$ and $p_a(C) = 1$. Each such class $C$ is obtained from a unique relevant class $E$ with $E^2 - 2p_a(E) = -1$ by adding $E$ to the anticanonical class: $C = -K + E$. \end{enumerate} \end{lemma} The proof of the above lemma is left to the reader. All of the statements can be easily shown by careful examination of the table of relevant classes on $X_6$; many of the statements are also elementary consequences of intersection theory on rational surfaces. \section{A formula for the quantum product} \label{sectionformula} It is obvious that the computation of the quantum product depends on knowing the intersection numbers \[ [A] \cdot (\alpha \otimes \beta \otimes \gamma) \in {\Bbb Z} \] for the relevant three-point loci $A$, and for generators $\alpha$, $\beta$, and $\gamma$ of $H^*(X)$. {}From the computations made in Examples \ref{phi0}-\ref{phiC2=2}, the degree of the closure of the locus ${\cal R}_{[C]}$ is an important number for the quantum product. We will denote this degree by $d_{[C]}$: \[ d_{[C]} = \text{degree of }\overline{{\cal R}_{[C]}} \text{ in the projective space }|C|. \] The following lemma is then immediate from the computations made in Examples \ref{phi0}-\ref{phiC2=2}. \begin{lemma} Let $X$ be a Del Pezzo surface, and suppose that $C$ gives a relevant class on $X$ (or $C = 0$). We denote by $(C \cdot C)$ the self-intersection of $C$ and by $p_a(C)$ its arithmetic genus. Let $d_{[C]}$ denote the degree of the closure of the locus ${\cal R}_{[C]}$ in the projective space $|C|$. Then: \begin{itemize} \item in case $C=0$, the three-point class $[A_0]$ is the class of the small diagonal and has real codimension $8$ in $X^3$. The classes $[X]\otimes[X]\otimes[p]$ and $[D_1]\otimes[D_2]\otimes[X]$ (for divisors $D_i$) generate the complementary space $H^4(X^3)$ (together with the classes obtained from these by permutations). We have: \begin{itemize} \item[] $[A_0] \cdot ([X]\otimes[X]\otimes[p]) = 1$ and \item[] $[A_0] \cdot ([D_1]\otimes[D_2]\otimes[X]) = (D_1\cdot D_2)$. \end{itemize} \item in case $(C\cdot C) -2p_a(C)= -1$, the image of ${\cal S}_{[C]}$ has real codimension $6$ in $X^3$. The classes $[X]\otimes[D]\otimes[p]$ and $[D_1]\otimes[D_2]\otimes[D_3]$ (for divisors $D$, $D_i$) generate the complementary space $H^6(X^3)$ (together with the classes obtained from these by permutations). We have: \begin{itemize} \item[] $[A_{[C]}] \cdot ([X]\otimes[D]\otimes[p]) = 0$ and \item[] $[A_{[C]}] \cdot ([D_1]\otimes[D_2]\otimes[D_3]) = d_{[C]} (C \cdot D_1)(C \cdot D_2)(C \cdot D_3)$. \end{itemize} In this case $[A_{[C]}] = d_{[C]} [C]\otimes[C]\otimes[C] \in H^6(X^3)$. \item in case $(C\cdot C) -2p_a(C) = 0$, the image of ${\cal S}_{[C]}$ has real codimension $4$ in $X^3$. The classes $[X]\otimes[p]\otimes[p]$ and $[D_1]\otimes[D_2]\otimes[p]$ (for divisors $D_i$) generate the complementary space $H^8(X^3)$ (together with the classes obtained from these by permutations). We have: \begin{itemize} \item[] $[A_{[C]}] \cdot ([X]\otimes[p]\otimes[p]) = 0$ and \item[] $[A_{[C]}] \cdot ([D_1]\otimes[D_2]\otimes[p]) = d_{[C]} (C \cdot D_1)(C \cdot D_2)$. \end{itemize} In this case $[A_{[C]}] = d_{[C]} [C]\otimes[C]\otimes[X] + d_{[C]} [C]\otimes[X]\otimes[C] + d_{[C]} [X]\otimes[C]\otimes[C] \in H^4(X^3)$. \item in case $(C\cdot C) -2p_a(C) = 1$, the image of ${\cal S}_{[C]}$ has real codimension $2$ in $X^3$. The classes $[D]\otimes[p]\otimes[p]$ (for divisors $D$) generate the complementary space $H^{10}(X^3)$ (together with the classes obtained from these by permutations). We have: \begin{itemize} \item[] $[A_{[C]}] \cdot ([D]\otimes[p]\otimes[p]) = d_{[C]} (C \cdot D)$. \end{itemize} In this case $[A_{[C]}] = d_{[C]} [C]\otimes[X]\otimes[X] + d_{[C]} [X]\otimes[C]\otimes[X] + d_{[C]} [X]\otimes[X]\otimes[C] \in H^2(X^3)$. \item in case $(C\cdot C) = 2$, the image of ${\cal S}_{[C]}$ is all of $X^3$ (and therefore has codimension zero). The class $[p]\otimes[p]\otimes[p]$ generates the complementary space $H^{12}(X^3)$. We have: \begin{itemize} \item[] $[A_{[C]}] \cdot ([p]\otimes[p]\otimes[p]) = d_{[C]}$ \end{itemize} and $[A_{[C]}] = d_{[C]} [X]\otimes[X]\otimes[X] \in H^0(X^3)$. \end{itemize} \end{lemma} This gives the following descriptions of the classes $\phi_{[C]}(\alpha,\beta)$: \begin{corollary} Let $X$ be a Del Pezzo surface, and suppose that $C$ gives a relevant class on $X$ (or $C = 0$). We denote by $(C \cdot C)$ the self-intersection of $C$ and by $p_a(C)$ the arithmetic genus. Let $d_{[C]}$ denote the degree of the closure of the locus ${\cal R}_{[C]}$ in the projective space $|C|$. Then: \begin{itemize} \item in case $C=0$, \begin{itemize} \item[] $\phi_0([X],[X]) = [X]$, \item[] $\phi_0([X],[p]) = [p]$, \item[] $\phi_0([X],[D]) = [D]$ for a divisor $D$, and \item[] $\phi_0([D_1],[D_2]) = (D_1\cdot D_2)[p]$ for divisors $D_i$. \item[] The classes $\phi_{[0]}([D],[p]) = \phi_{[0]}([p],[p]) = 0$. \end{itemize} \item in case $(C\cdot C) -2p_a(C) = -1$, \begin{itemize} \item[] $\phi_{[C]}([D_1],[D_2]) = d_{[C]} (C \cdot D_1)(C \cdot D_2) [C]$. \item[] If $\alpha$ and $\beta$ are homogeneous elements of $H^*(X)$, $\phi_{[C]}(\alpha,\beta) = 0$ unless both $\alpha$ and $\beta$ lie in $H^2(X)$. \end{itemize} \item in case $(C\cdot C) -2p_a(C) = 0$, \begin{itemize} \item[] $\phi_{[C]}([D_1],[D_2]) = d_{[C]} (C \cdot D_1)(C \cdot D_2)[X]$, and \item[] $\phi_{[C]}([D],[p]) = d_{[C]} (C \cdot D) [C]$. \item[] $\phi_{[C]}([p],[p]) = 0$ and $\phi_{[C]}([X],\beta) = 0$ for all $\beta$. \end{itemize} \item in case $(C\cdot C) -2p_a(C) = 1$, \begin{itemize} \item[] $\phi_{[C]}([p],[p]) = d_{[C]} [C]$ and \item[] $\phi_{[C]}([D],[p]) = d_{[C]} (C \cdot D) [X]$. \item[] $\phi_{[C]}([D_1],[D_2]) = 0$ and $\phi_{[C]}([X],\beta) = 0$ for all $\beta$. \end{itemize} \item in case $(C\cdot C) -2p_a(C) = 2$, \begin{itemize} \item[] $\phi_{[C]}([p],[p]) = d_{[C]} [X]$. \item[] If $\alpha$ and $\beta$ are homogeneous elements of $H^*(X)$, $\phi_{[C]}(\alpha,\beta) = 0$ unless both $\alpha$ and $\beta$ lie in $H^4(X)$. \end{itemize} \end{itemize} \end{corollary} Finally we deduce the formulas for the quantum product. \begin{proposition} \label{quantumproductformulas} Let $X$ be a Del Pezzo surface. \begin{enumerate} \item The class $[X]$ is an identity for the quantum product. \item For two divisors $D_1$ and $D_2$, \[ [D_1] \operatorname{\ast_Q} [D_2] = (D_1 \cdot D_2)[p] q^{[0]} + \sum\begin{Sb} E \text{ relevant}\\{E^2-2p_a(E) = -1}\end{Sb} d_{[E]}(E \cdot D_1)(E \cdot D_2) [E] q^{[E]} \]\[ + \sum\begin{Sb} F \text{ relevant}\\{F^2-2p_a(F) = 0}\end{Sb} d_{[F]} (F \cdot D_1)(F \cdot D_2) [X] q^{[F]} \] where the sum is taken over the linear systems (not over the curves actually). \item For a divisor $D$, \[ [D] \operatorname{\ast_Q} [p] = \sum\begin{Sb} F \text{ relevant}\\{F^2-2p_a(F) = 0}\end{Sb} d_{[F]} (F \cdot D) [F] q^{[F]} + \sum\begin{Sb} L \text{ relevant}\\{L^2-2p_a(L) = 1}\end{Sb} d_{[L]} (L \cdot D) [X] q^{[L]} \] where the sum is again taken over the linear systems. \item \[ [p] \operatorname{\ast_Q} [p] = \sum\begin{Sb} L \text{ relevant}\\{L^2-2p_a(L) = 1}\end{Sb} d_{[L]} [L] q^{[L]} + \sum\begin{Sb} C \text{ relevant}\\{C^2-2p_a(C) = 2}\end{Sb} d_{[C]} [X] q^{[C]} \] where the sum is again taken over the linear systems. \end{enumerate} \end{proposition} \section{The relationship with ordinary cohomology} \label{sectionordinary} The effective cone $\operatorname{Eff}(X)$ in $H^2(X)$ is a proper cone, in the sense that it contains no subgroups of $H^2(X)$. Hence there is an ``augmentation'' ring homomorphism \[ G: H^*_Q(X) \to H^*(X) \] defined by sending a quantum cohomology class $\sum_{[D]} \alpha_D q^{[D]}$ to the coefficient $\alpha_0$ of the $q^{[0]}$ term. \begin{proposition} Let $X$ be a Del Pezzo surface. Then the map $G$ is a ring homomorphism from the quantum cohomology ring $H^*_Q(X)$ (with the quantum product) to the integral cohomology ring $H^*(X)$ (with the cup product). \end{proposition} This is clear from the formulas for the quantum product given in Proposition \ref{quantumproductformulas}. \section{Examples} \label{sectionexamples} \begin{example} Let $X = {\Bbb P}^2$, the complex projective plane. \end{example} Then $H^2(X) = {\Bbb Z}[L]$, where $[L]$ is the class of a line. The quantum products determining the multiplication are \[ [L] \operatorname{\ast_Q} [L] = [p]q^{[0]}, \] \[ [L] \operatorname{\ast_Q} [p] = [X]q^{[L]}, \text{ and } \] \[ [p] \operatorname{\ast_Q} [p] = [L]q^{[L]}. \] We may identify $q^{[0]}$ and $[X]$ with $1$ and $q^{[L]}$ with $q$; if we write $\ell$ for the class $[L]$, the above relations are that $\ell^2 = [p]$, $\ell^3 = q$, and $\ell^4 = \ell q$. Hence the quantum cohomology ring is isomorphic to \[ H^*_Q({\Bbb P}^2) = {\Bbb Z}[\ell,q]/(\ell^3 - q). \] \begin{example} Let $X = {\Bbb P}^1 \times {\Bbb P}^1$, the smooth quadric surface. \end{example} Then $H^2(X) = {\Bbb Z}[F_1] \oplus {\Bbb Z}[F_2]$, where the classes $[F_i]$ are those of the two rulings on $X$. The quantum products determining the multiplication are \[ [F_1] \operatorname{\ast_Q} [F_1] = [X]q^{[F_2]}, \] \[ [F_1] \operatorname{\ast_Q} [F_2] = [p]q^{[0]}, \] \[ [F_2] \operatorname{\ast_Q} [F_2] = [X]q^{[F_1]}, \] \[ [F_1] \operatorname{\ast_Q} [p] = [F_2]q^{[F_2]}, \] \[ [F_2] \operatorname{\ast_Q} [p] = [F_1]q^{[F_1]}, \text{ and } \] \[ [p] \operatorname{\ast_Q} [p] = [X]q^{[F_1+F_2]}. \] Denote $[F_i]$ by $f_i$, and $q^{[F_i]}$ by $q_i$. These relations then become $f_1^2 = q_2$, $f_1f_2 = [p]$, $f_2^2 = q_1$, $f_1[p] = f_2q^2$, $f_2[p] = f_1q_1$, and $[p]^2 = q_1q_2$. Hence the quantum cohomology ring is isomorphic to \[ H^*_Q({\Bbb P}^1 \times {\Bbb P}^1) = {\Bbb Z}[f_1,f_2,q_1,q_2]/(f_1^2-q_2,f_2^2-q_1). \] \begin{example} Let $X = {\Bbb F}_1$, the blowup of the plane at one point. \end{example} Then $H^2(X) = {\Bbb Z}[E] \oplus {\Bbb Z}[F]$, where $[E]$ is the class of the exceptional curve and $[F]$ is the class of the fiber. The only other class with a smooth rational curve of self-intersection at most $2$ is the class $[L] = [E]+[F]$; it has self-intersection $1$. The quantum products determining the multiplication are \[ [E] \operatorname{\ast_Q} [E] = -[p]q^{[0]} + [E]q^{[E]} + [X]q^{[F]}, \] \[ [E] \operatorname{\ast_Q} [F] = [p]q^{[0]} - [E]q^{[E]}, \] \[ [F] \operatorname{\ast_Q} [F] = [E]q^{[E]}, \] \[ [E] \operatorname{\ast_Q} [p] = [F]q^{[F]}, \] \[ [F] \operatorname{\ast_Q} [p] = [X]q^{[E]+[F]}, \text{ and } \] \[ [p] \operatorname{\ast_Q} [p] = [L]q^{[E]+[F]}. \] Denote $[E]$ by $e$, $q^{[E]}$ by $q$, $[F]$ by $f$, $q^{[F]}$ by $r$, and $[p]$ by $p$. These relations then become $e^2 = -p+eq+r$, $ef = p-eq$, $f^2 = eq$, $ep = fr$, $fp = qr$, and $p^2 = (e+f)qr$. We may eliminate $p$ from the generators since $p = ef+eq$; after doing so, the first and third relations become $e^2 = r - ef$ and $f^2 = eq$, and the other relations formally follow from these two. Hence the quantum cohomology ring is isomorphic to \[ H^*_Q({\Bbb F}_1) = {\Bbb Z}[e,f,q,r]/ (e^2+ef-r,f^2-eq). \] \section{A functoriality property} \label{sectionfunctor} Let $\pi:X \to Y$ be a general blowup of a Del Pezzo surface $Y$ at a single point $p$, with an exceptional curve $E$. Then we have $\pi\times\pi\times\pi: X^3 \to Y^3$. Suppose that $C$ is an irreducible curve in $Y$, such that its cohomology class $[C]$ is relevant for the quantum cohomology of $Y$. We may assume that $p$ is not on $C$. Then $[\pi^{-1}(C)] = \pi^*[C]$ as a class on $X$, and has the same self-intersection and arithmetic genus as does $[C]$; therefore it is a relevant class for the quantum cohomology for $X$. In other words, the three-point classes $[A_{[C]}]$ on $Y^3$ and $[A_{\pi^*[C]}]$ on $X^3$ are both defined. \begin{lemma} With the above notation and conditions, if $[C] \neq 0$, then \[ [A_{\pi^*[C]}] = {(\pi\times\pi\times\pi)}^*([A_{[C]}]). \] \end{lemma} \begin{pf} What is clear is that the nodal rational curve loci ${\cal R}_{[C]}$ and ${\cal R}_{\pi^*[C]}$ are birational, so that $\pi\times\pi\times\pi$ induces a birational map \[ {\cal S}_{\pi^*[C]} \to {\cal S}_{[C]}. \] since we are blowing up a point which is not on the general member of ${\cal R}_{[C]}$. This implies immediately that \[ \pi\times\pi\times\pi([A_{\pi^*[C]}]) = [A_{[C]}] \] as classes on $Y^3$. We want to investigate the pull-back, not the image; however this at least says that $[A_{\pi^*[C]}]$ and ${(\pi\times\pi\times\pi)}^*([A_{[C]}])$ will differ only on the exceptional part of the map $\pi\times\pi\times\pi$, i.e., only over the fundamental locus ${\cal E} = (\{p\}\times Y \times Y) \cup (Y\times \{p\} \times Y) \cup (Y\times Y \times \{p\})$. We can take the different cases up one by one. If $(C \cdot C) - 2p_a(C) = -1$, then the rational curve locus ${\cal R}_{[C]}$ is a finite set, and by assumption the point $p$ is not on any member; hence the loci in question are disjoint from the fundamental loci for $\pi\times\pi\times\pi$, and there is nothing to prove. Similarly if $(C\cdot C) -2p_a(C)= 2$, then $[A_{[C]}] = d_{[C]}[Y^3]$ and $[A_{\pi^*[C]}] = d_{\pi^*[C]}[X^3]$; since $d_{[C]} = d_{\pi^*[C]}$, the result follows in this case. Suppose that $(C \cdot C) -2p_a(C)= 0$, so that ${\cal R}_{[C]}$ is a curve. Let $C_1, \dots C_d$ be the members of ${\cal R}_{[C]}$ through $p$ (here $d = d_{[C]}$). Then $[A_{[C]}]$ intersects the fundamental locus ${\cal E}$ exactly in the union of the loci $(\{p\} \times C_j \times C_j) \cup (C_j\times \{p\} \times C_j) \cup (C_j\times C_j \times \{p\})$; over this locus in $X^3$ is the union of the loci $(E \times (E + \overline{C_j}) \times (E + \overline{C_j})) \cup ((E + \overline{C_j})\times \{p\} \times (E + \overline{C_j})) \cup ((E + \overline{C_j})\times (E + \overline{C_j}) \times \{p\})$ (where $(E + \overline{C_j})$ means the union of the exceptional curve $E$ with the proper transform of $C_j$). Note that this is a union of three-folds in $X^3$; since in this case $[A_{\pi^*[C]}]$ is the class of a four-fold, we have no extra contribution to the pull-back class. The final case of $(C \cdot C) -2p_a(C) = 1$ is similar; here the locus $[A_{[C]}]$ intersects the fundamental locus ${\cal E}$ in a three-fold, over which lies a four-fold in $X^3$; since $[A_{\pi^*[C]}]$ is the class of a five-fold, again we have no extra contribution to the pull-back class. \end{pf} This implies the following. \begin{corollary} \label{pi*phi} With the above notations, if $[C] \neq 0$, then for any homogeneous classes $\alpha$ and $\beta$ in $H^*(Y)$, we have \[ \pi^*(\phi_{[C]}(\alpha,\beta)) = \phi_{[\pi^*C]}(\pi^*\alpha,\pi^*\beta). \] \end{corollary} \begin{pf} Choose a class $\gamma$ of the correct dimension on $Y$, and compute \begin{eqnarray*} \phi_{[\pi^*C]}(\pi^*\alpha,\pi^*\beta) \cdot \pi^*\gamma &=& [A_{\pi^*[C]}] \cdot \pi^*\alpha\otimes \pi^*\beta \otimes \pi^*\gamma \\ &=& {(\pi\times\pi\times\pi)}^*[A_{[C]}] \cdot {(\pi\times\pi\times\pi)}^*(\alpha\otimes\beta\otimes\gamma \\ &=& [A_{[C]}] \cdot (\alpha\otimes\beta\otimes\gamma \\ &=& \phi_{[C]}(\alpha,\beta) \cdot \gamma. \end{eqnarray*} Moreover if $\phi_{[C]}(\alpha,\beta)$ is a class in $H^2(Y)$, and $E$ is the exceptional curve for the map $\pi$, then \begin{eqnarray*} \phi_{[\pi^*C]}(\pi^*\alpha,\pi^*\beta) \cdot E &=& [A_{\pi^*[C]}] \cdot \pi^*\alpha\otimes \pi^*\beta \otimes E \\ &=& {(\pi\times\pi\times\pi)}^*[A_{[C]}] \cdot \pi^*\alpha\otimes \pi^*\beta \otimes E \\ &=& 0. \end{eqnarray*} Hence as far as intersections go, the class $\phi_{[\pi^*C]}(\pi^*\alpha,\pi^*\beta)$ is behaving exactly like the class $\pi^*(\phi_{[C]}(\alpha,\beta))$. However this class is defined in terms of its intersection behaviour, and so the equality as claimed holds. \end{pf} Note that we in any case have the formula \[ \pi^*(\phi_{0}(\alpha,\beta)) = \phi_{0}(\pi^*\alpha,\pi^*\beta) \] since $\phi_{0}(\pi^*\alpha,\pi^*\beta) = \pi^*\alpha \cup \pi^*\beta$ (this is cup product on $X$) which is in turn equal to $\pi^*(\alpha \cup \beta)$ since $\pi^*$ is a ring homomorphism on ordinary cohomology. Since $\pi^*(0) = 0$, we view this as the ``$[C] = 0$'' case of Corollary \ref{pi*phi}. Putting this together we derive the following version of functoriality for the quantum ring: \begin{corollary} \label{functoriality} Let $\alpha$ and $\beta$ be ordinary cohomology classes in $H^*(Y)$. Then for any relevant class $[C]$ in $H^2(Y)$, the $q^{\pi^*[C]}$-term of $\pi^*\alpha \operatorname{\ast_Q} \pi^*\beta$ is equal to $\pi^*$ of the $q^{[C]}$-term of $\alpha\operatorname{\ast_Q}\beta$. \end{corollary} Another way of saying this is to define the quantum pullback \[ \pi_Q^*:H^*_Q(Y) \to H^*_Q(X) \] by setting \[ \pi^*_Q(\sum_{[D]} c_{[D]} q^{[D]} ) = \sum_{[D]} \pi^*(c_{[D]}) q^{\pi^*[D]}. \] This is NOT in general a ring homomorphism. But the above corollary says that for classes $\alpha$ and $\beta$ in $H^*(Y)$, the two quantum cohomology classes \[ \pi^*(\alpha) \operatorname{\ast_Q} \pi^*(\beta) \text{ and } \pi^*_Q(\alpha \operatorname{\ast_Q} \beta) \] differ only in the $q^{[D]}$ terms for those classes $[D]$ on $X$ which are NOT pullbacks from $Y$. That is, they agree on all the $q^{[\pi^*C]}$ terms, for any effective classes $C$ on $Y$. \section{Associativity of the quantum product for strict Del Pezzo surfaces} \label{sectionSDPassoc} In this section we will use the formulas of Proposition \ref{quantumproductformulas} to check the associative law for the quantum product for general strict Del Pezzo surfaces. These are the surfaces ${\Bbb P}^2$, ${\Bbb F}_0 = {\Bbb P}^1 \times {\Bbb P}^1$, and $X_n$ (the $n$-fold general blowup of ${\Bbb P}^2$) for $n \leq 6$. The first reduction is to note that it suffices to prove the associative law for the general surface $X_6$, the six-fold blowup of the plane. (This is the general cubic surface in ${\Bbb P}^3$.) This is due to the functoriality property stated in Corollary \ref{functoriality}; if associativity holds on a blowup $X$ of a surface $Y$, then in fact it must hold on $Y$. By the tables of relevant classes on these surfaces given in Section \ref{strictrelevantsection}, we note that all relevant classes have arithmetic genus at most one. (This is no longer true if one blows up $7$ general points in the plane; the class of quartics double at one point and passing through $6$ others is relevant on $X_7$, and has arithmetic genus $2$.) Hence for the strict Del Pezzo surface case, the following lemma suffices to give us all the relevant $d_{[C]}$ numbers. \begin{lemma} Suppose that $X$ is a general rational surface, and $[C]$ is a relevant class on $X$. Then \[ d_{[C]} = \left\{\begin{array}{cl} 1 & \text{ if }\;\; p_a(C) = 0, \text{ and } \\ 12 & \text{ if }\;\; p_a(C) = 1. \end{array}\right. \] \end{lemma} \begin{pf} If $p_a(C) = 0$, then the locus ${\cal R}_{[C]}$ of irreducible rational curves in the linear system $|C|$ is an open subset of $|C|$ (it is the subset parametrizing the smooth curves of $|C|$). Hence its closure is the entire linear system $|C|$, and therefore has degree one. If $p_a(C) = 1$, then the locus ${\cal R}_{[C]}$ of irreducible rational curves in the linear system $|C|$ is an open subset of the discriminant locus of $|C|$. Its degree is the number of irreducible nodal rational curves in a general pencil of curves in $|C|$. Such a general pencil, after blowing up the base points, will give a fibration of elliptic curves on a rational surface; the degree of ${\cal R}_{[C]}$ is the number of singular fibers of this fibration. This is $12$, by standard Euler number considerations. \end{pf} Let us begin with checking associativity for a triple product of the form $[p] \operatorname{\ast_Q} [p] \operatorname{\ast_Q} [D]$ for a divisor $D$. For notational convenience let us define \[ s(C) = (C \cdot C) - 2 p_a(C) \] for an irreducible curve $C$ in a relevant class $[C]$. We have \begin{eqnarray*} ([p] \operatorname{\ast_Q} [p]) \operatorname{\ast_Q} [D] &=& (\sum_{s(L)= 1} d_{[L]}[L] q^{[L]} + \sum_{s(C)= 2} d_{[C]}[X] q^{[C]}) \operatorname{\ast_Q} [D] \\ &=& \sum_{s(L) = 1} d_{[L]}([L]\operatorname{\ast_Q}[D]) q^{[L]} + \sum_{s(C)= 2} d_{[C]}[D] q^{[C]} \\ &=& \sum_{s(C)= 2} d_{[C]}[D] q^{[C]} + \sum_{s(L)= 1} d_{[L]}(L \cdot D)[p] q^{[L]} + \\ && + \sum_{s(L)= 1}\sum_{s(E)= -1} d_{[L]} d_{[E]} (E \cdot L)(E \cdot D) [E] q^{[E+L]}\\ && + \sum_{s(L) = 1}\sum_{s(F)= 0} d_{[L]} d_{[F]} (F \cdot L)(F \cdot D) [X] q^{[F+L]}) \end{eqnarray*} while \begin{eqnarray*} [p] \operatorname{\ast_Q} ([p] \operatorname{\ast_Q} [D]) &=& [p]\operatorname{\ast_Q} (\sum_{s(F)= 0} d_{[F]}(F \cdot D) [F] q^{[F]} + \sum_{s(L)= 1} d_{[L]} (L \cdot D) [X] q^{[L]}) \\ &=& \sum_{s(F)= 0} d_{[F]}(F \cdot D) ([p]\operatorname{\ast_Q} [F]) q^{[F]} + \sum_{s(L)= 1} d_{[L]} (L \cdot D) ([p]\operatorname{\ast_Q} [X]) q^{[L]} \\ &=& \sum_{s(L)= 1} d_{[L]} (L \cdot D) [p] q^{[L]} +\\ &&+\sum_{s(F)= 0}\sum_{s(G)= 0} d_{[F]}d_{[G]}(F \cdot D)(G \cdot F)[G] q^{[F+G]} + \\ && + \sum_{s(F)= 0}\sum_{s(L)= 1} d_{[F]}d_{[L]}(F \cdot D)(L \cdot F)[X] q^{[F+L]}. \end{eqnarray*} Comparing terms in the above two expressions we see that this particular triple product is associative if and only if \begin{eqnarray} \label{ppD} \sum_{s(C)= 2} d_{[C]} [D] q^{[C]} &+ & \sum_{s(L)= 1}\sum_{s(E)= -1} d_{[L]} d_{[E]} (E \cdot L)(E \cdot D) [E] q^{[E+L]} \\ &=& \sum_{s(F)= 0}\sum_{s(G)= 0} d_{[F]}d_{[G]}(F \cdot D)(G \cdot F)[G] q^{[F+G]}. \nonumber \end{eqnarray} Of course only relevant classes are included in the above sums. By Lemma \ref{X6relevantlemma}, if $L$ and $E$ are relevant classes with $s(L) = 1$ and $s(E) = -1$, then $0 \leq (L\cdot E) \leq 2$; if $(L \cdot E) = 0$ then there is no contribution in (\ref{ppD}). If $(L\cdot E) \neq 0$ then $L+E$ is a relevant class with $s(L+E) = 2$; if $L \neq -K$ then $p_a(L+E) = (L\cdot E) - 1$, and if $L = -K$ then $(L \cdot E) = 1$ and $p_a(L+E) = 1$. Similarly if $F$ and $G$ are relevant classes with $s(F) = s(G) = 0$, then $0 \leq (F\cdot G) \leq 2$; if $(F \cdot G) = 0$ then there is no contribution in (\ref{ppD}). If $(F\cdot G) \neq 0$ then $F+G$ is a relevant class with $s(F+G) = 2$ and $p_a(F+G) = (F \cdot G) - 1$. Therefore the only terms which can appear in the associativity formula (\ref{ppD}) are those $q^{[C]}$ terms for relevant classes $[C]$ with $s(C) = 2$ (and $p_a(C) \leq 1$). In fact we have the following. \begin{lemma} The associativity of the triple product $p \operatorname{\ast_Q} p \operatorname{\ast_Q} D$ on $X_6$ is equivalent to the following two formulas: \begin{itemize} \item[(a)] For every relevant class $C$ with $s(C) = 2$ and $p_a(C) = 0$, and every divisor $D$, \[ [D] + \sum\begin{Sb} E \\ s(E) = -1 \\ (C \cdot E) = 0 \end{Sb} (D \cdot E) [E] = \sum\begin{Sb} (F,G) \\ s(F) = s(G) = 0 \\ F+G = C \end{Sb} (D \cdot F) [G]. \] \item[(b)] For every relevant class $C$ with $s(C) = 2$ and $p_a(C) = 1$, \[ 6[D] + 6(K+C \cdot D) [K+C] + \sum\begin{Sb} E \\ s(E) = -1 \\(C \cdot E) = 1 \end{Sb} (D \cdot E) [E] = \sum\begin{Sb} (F,G) \\ s(F) = s(G) = 0 \\ F+G = C\end{Sb} (D \cdot F) [G]. \] \end{itemize} \end{lemma} \begin{pf} As noted above, we may decompose (\ref{ppD}) into the $q^{[C]}$ terms fixing a relevant class $C$ with $s(C) = 2$. The two cases of the lemma correspond to the two possibilities for $p_a(C)$. If $p_a(C) = 0$, then the only pairs $(L,E)$ with $s(L) = 1$ and $s(E) = -1$ having $L + E = C$ must have $L \neq -K$ (and therefore $d_{[L]} = 1$). Moreover $(L \cdot E) = 1$ (else $p_a(C) = 1$ by Lemma \ref{X6relevantlemma}). Hence $(C \cdot E) = (L + E \cdot E) = 1 - 1 = 0$. Conversely for any class $E$ with $(C \cdot E) = 0$, the class $L = C-E$ has $s(L) = 1$ and occurs in the sum. Therefore this $(L,E)$ sum with $L+E = C$ is a sum over those $E$'s with $(C \cdot E) = 0$. In this case $d_{[E]} = (L \cdot E) = 1$ also, so these contributions can be ignored. Similarly, if $p_a(C) = 0$, then if $C = F+G$ with $s(F) = s(G) = 0$, then $d_{[F]} = d_{[G]} = (F\cdot G) = 1$. This then produces the equation of part (a). Suppose then that $p_a(C) = 1$. Then $d_{[C]} = 12$, and in the $(L,E)$ sum, $L = -K$ is a possibility. The $E$ that pairs with $L = -K$ is of course $E = K+C$, and has $(E \cdot L) = (E \cdot -K) = 1$. This gives a term $12(K+C\cdot D)[K+C]$ to the $(L,E)$ sum. If $L \neq -K$ and $L+E = C$, then by Lemma \ref{X6relevantlemma} we have $(L\cdot E) = 2$, or, equivalently, $(C \cdot E) = 1$. Conversely, any class $E$ with $s(E) = -1$ and $(C \cdot E) = 1$ occurs, and is paired with the class $L = C-E$. Therefore again this sum can be written as a sum over such $E$'s, each $E$ giving the term $2(E\cdot D)[E]$ (since $d_{[L]} = d_{[E]} = 1$ and $(L\cdot E) = 2$). Finally, in the $(F,G)$ sum, for two such classes to sum to a $C$ with $p_a(C) = 1$, we must have $(F \cdot G) = 2$ by Lemma \ref{X6relevantlemma}; since $d_{[F]}=d_{[G]}=1$, each such pair $(F,G)$ contributes a term of the form $2 (F\cdot D)[G]$. Dividing all terms by two produces the equation of part (b). \end{pf} It remains to prove these two formulas. We begin with (a). \begin{lemma} Let $C$ be a relevant class on $X_6$ with $s(C) = 2$ and $p_a(C) = 0$. Then for any divisor $D$ on $X_6$, \[ [D] + \sum\begin{Sb} E \\ s(E) = -1 \\ (C \cdot E) = 0 \end{Sb} (D \cdot E) [E] = \sum\begin{Sb} (F,G) \\ s(F) = s(G) = 0 \\ F+G = C \end{Sb} (D \cdot F) [G]. \] \end{lemma} \begin{pf} By Lemma \ref{X6relevantlemma}, the class $C$ can be written uniquely (up to order) as $C = F+G$, with $s(G) = s(G) = 0$; if we do so, we see that the right-hand side of the above equation consists of only the two terms $(D \cdot F) [G] + (D \cdot G) [F]$. Therefore we must actually show that \[ [D] = (D \cdot F) [G] + (D \cdot G) [F] - \sum\begin{Sb} E \\ s(E) = -1 \\ (C \cdot E) = 0 \end{Sb} (D \cdot E) [E]. \] The two pencils $|F|$ and $|G|$ on $X_6$ give a birational map $\pi:X_6 \to {\Bbb F}_0$, realizing $X_6$ as a general five-fold blowup of ${\Bbb F}_0$. The only curves $E$ on $X_6$ with $s(E) = -1$ which do not meet $C = F+G$ are the five exceptional curves for this blowup; call these five curves $E_1,\ldots,E_5$. Note that the seven classes $[F],[G],[E_1],\ldots,[E_5]$ generate the Picard group over ${\Bbb Z}$; the intersection matrix is unimodular. Now the above formula is exactly the writing of the class $[D]$ in terms of these generators. \end{pf} Finally we address the equation (b). \begin{lemma} Let $C$ be a relevant class on $X_6$ with $s(C) = 2$ and $p_a(C) = 1$. Then for any divisor $D$ on $X_6$, \[ 6[D] + 6(K+C \cdot D) [K+C] + \sum\begin{Sb} E \\ s(E) = -1 \\(C \cdot E) = 1 \end{Sb} (D \cdot E) [E] = \sum\begin{Sb} (F,G) \\ s(F) = s(G) = 0 \\ F+G = C\end{Sb} (D \cdot F) [G]. \] \end{lemma} \begin{pf} Let $\hat{E}$ denote the class $K+C$; we have $s(\hat{E}) = -1$ as noted above. Any class $E$ with $s(E) = -1$ and $(C \cdot E) = 1$ must therefore have $(E \cdot \hat{E}) = 0$ and conversely; therefore the $E$ sum above is a sum over those $E$'s with $(E \cdot \hat{E}) = 0$. On the other side, suppose that $F+G=C$ with $s(F)=s(G)=0$. By Lemma \ref{X6relevantlemma}, we must have $(F \cdot G) = 2$, and so $(F\cdot C) = (G \cdot C) = 2$. Then $(F \cdot \hat{E}) = (F \cdot K+C) = (F \cdot K) + 2 = 0$ since $(F \cdot K) = -2$ by the genus formula. Similarly $(G \cdot \hat{E}) = 0$. Therefore in the two pencils $|F|$ and $|G|$, the curve $\hat{E}$ occurs in a singular fiber of each. Hence there are unique curves $E_F$ and $E_G$ with $s(E_F) = s(E_G) = -1$ such that $F = \hat{E} + E_F$ and $G = \hat{E} + E_G$. Moreover $(E_F \cdot E_G) = 1$ since $(F \cdot G) = 2$. Note that the three curves $\hat{E}$, $E_F$, and $E_G$ form a triangle on $X_6$ (considered as a cubic surface). Conversely, given a triangle of curves $\hat{E}$, $E_F$, and $E_G$ with $s(E_F) = s(E_G)= -1$, we obtain a unique pair $(F,G)$ with $s(F) = s(G) = 0$ and $F+G = C$ by setting $G = \hat{E}+E_G$ and $F = \hat{E} + E_G$. Therefore the $(F,G)$ sum above can be made into a sum over such triangles; for a fixed $\hat{E}$ there are five such \cite{beauville,reid}. Therefore the equation in question may be written as \[ 6[D] + 6(\hat{E} \cdot D) [\hat{E}] + \sum\begin{Sb} E \\ s(E) = -1 \\(\hat{E} \cdot E) = 0 \end{Sb} (D \cdot E) [E] = \sum\begin{Sb} \text{five triangles}\\ \hat{E}+E_F+E_G\end{Sb} (D \cdot \hat{E}+E_F) [\hat{E}+E_G] + (D \cdot \hat{E}+E_G) [\hat{E}+E_F]. \] We first claim that the above equation holds when $D = \hat{E}$. In this case the first two terms on the left side cancel, while each term in the two sums are clearly zero. Next we claim that the equation holds when $D = \hat{E} + E'$, for any curve $E'$ with $s(E') = -1$ and $(\hat{E}\cdot E') = 1$. In this case $(\hat{E} \cdot D) = 0$ so the second term of the equation drops out. Of those $E$'s which satisfy $(\hat{E} \cdot E) = 0$, there are exactly $8$ which meet $E'$ and contribute to the sum on the left-hand side; these are exactly the other curves in the four triangles containing $E'$ which do not involve $\hat{E}$. These come in pairs, and if $E_1$ and $E_2$ form a pair, then they contribute $(D\cdot E_1)[E_1] + (D\cdot E_2)[E_2] = [E_1+E_2]$. However each triangle is equivalent to $-K$, so this pair's contribution may be written as $[-K - E']$; hence this sum reduces to $-4[K+E']$. Hence the entire left-hand side is equal to $6[\hat{E} + E'] -4[K+E'] = 6[\hat{E}] - 4[K] + 2[E']$. On the right-hand side, if we have a triangle $E+E_F+E_G$, $(D \cdot \hat{E}+E_F) = (E' \cdot \hat{E}+E_F) = 1 + (E'\cdot E_F)$ and similarly for the $E_G$ term. Now $E'$ is part of a triangle containing $\hat{E}$, say $\hat{E} + E' + E''$; the other four triangles are disjoint from $E'$ (except for the curve $\hat{E}$). For one of these four triangles, we obtain a contribution of $[\hat{E}+E_G] + [\hat{E}+E_F] = [\hat{E} - K]$. For the triangle with $E'$ and $E''$, we have a contribution of $2[\hat{E}+E']$. Thus the right-hand sum reduces to $6[\hat{E}] - 4[K] + 2[E']$. As noted above, this is equal to the left-hand side; therefore the equation holds for this $D$. The proof now finishes by remarking that the Picard group of $X_6$ is generated rationally by $\hat{E}$ and the classes $\hat{E}+E'$ considered above. Since the equation is linear in $D$, and is true for these generators, it is true for all divisors $D$. \end{pf} Let us now address the associativity for a triple product of the form $[p]\operatorname{\ast_Q}[D_1]\operatorname{\ast_Q}[D_2]$ for divisors $D_i$. We have \[ [p]\operatorname{\ast_Q}([D_1]\operatorname{\ast_Q}[D_2]) = \] \[ {\renewcommand{\arraystretch}{1.5} \begin{array}{ll} = & [p]\operatorname{\ast_Q}((D_1 \cdot D_2)[p] q^{[0]} + \sum_{s(E) = -1} d_{[E]}(E \cdot D_1)(E \cdot D_2) [E] q^{[E]} + \sum_{s(F) = 0} d_{[F]}(F \cdot D_1)(F \cdot D_2) [X] q^{[F]}) \\ = & (D_1 \cdot D_2)[p]\operatorname{\ast_Q}[p] q^{[0]} + \sum_{s(E) = -1} d_{[E]}(E \cdot D_1)(E \cdot D_2) [p]\operatorname{\ast_Q}[E] q^{[E]} \\ & + \sum_{s(F) = 0} d_{[F]}(F \cdot D_1)(F \cdot D_2) [p]\operatorname{\ast_Q}[X] q^{[F]} \\ = & (D_1 \cdot D_2)(\sum_{s(L) = 1}d_{[L]}[L] q^{[L]} + \sum_{s(C) = 2} d_{[C]}[X] q^{[C]}) \\ & + \sum_{s(E) = -1} d_{[E]}(E \cdot D_1)(E \cdot D_2) (\sum_{s(F) = 0} d_{[F]}(F \cdot E) [F] q^{[F]} +\sum_{s(L) = 1} d_{[L]}(L \cdot E) [X] q^{[L]}) q^{[E]} \\ & + \sum_{s(F) = 0} d_{[F]}(F \cdot D_1)(F \cdot D_2) [p]q^{[F]} \\ = & \sum_{s(L) = 1} d_{[L]}(D_1 \cdot D_2) [L] q^{[L]} + \sum_{s(C) = 2} d_{[C]}(D_1 \cdot D_2) [X] q^{[C]} \\ & + \sum_{s(E) = -1}\sum_{F^2 = 0} d_{[E]}d_{[F]} (E \cdot D_1)(E \cdot D_2)(F \cdot E) [F] q^{[E]+[F]} \\ & + \sum_{s(E) = -1}\sum_{L^2 = 1} d_{[E]}d_{[L]}(E \cdot D_1)(E \cdot D_2)(L \cdot E) [X] q^{[E]+[L]} \\ & + \sum_{s(F) = 0} d_{[F]}(F \cdot D_1)(F \cdot D_2) [p]q^{[F]} \end{array} } \] while \[ ([p]\operatorname{\ast_Q}[D_1])\operatorname{\ast_Q}[D_2]) = \] \[ {\renewcommand{\arraystretch}{1.5} \begin{array}{ll} = & (\sum_{s(F) = 0} d_{[F]}(F \cdot D_1) [F] q^{[F]} + \sum_{s(L) = 1} d_{[L]}(L \cdot D_1) [X] q^{[L]}) \operatorname{\ast_Q} [D_2] \\ = & \sum_{s(F) = 0} d_{[F]}(F \cdot D_1) ([F]\operatorname{\ast_Q}[D_2]) q^{[F]} + \sum_{s(L) = 1} d_{[L]}(L \cdot D_1) [D_2] q^{[L]} \\ = & \sum_{s(F) = 0} d_{[F]}(F \cdot D_1) ( (F \cdot D_2)[p] q^{[0]} + \sum_{s(E) = -1} d_{[E]}(E \cdot F)(E \cdot D_2) [E] q^{[E]} \\ & + \sum_{s(G) = 0} d_{[G]}(G \cdot F)(G \cdot D_2) [X] q^{[F]} ) q^{[F]} \\ & + \sum_{s(L) = 1} d_{[L]}(L \cdot D_1) [D_2] q^{[L]} \\ = & \sum_{s(F) = 0} d_{[F]}(F \cdot D_1) (F \cdot D_2)[p] q^{[F]} \\ & + \sum_{s(F) = 0} \sum_{s(E) = -1} d_{[F]}d_{[E]}(F \cdot D_1)(E \cdot F)(E \cdot D_2) [E] q^{[E+F]} \\ & + \sum_{s(F) = 0} \sum_{s(G) = 0} d_{[F]}d_{[G]}(F \cdot D_1)(G \cdot F)(G \cdot D_2) [X] q^{[F+G]} \\ & + \sum_{s(L) = 1} d_{[L]}(L \cdot D_1) [D_2] q^{[L]} . \\ \end{array} } \] Therefore associativity of this triple product is equivalent to the identity \[ {\renewcommand{\arraystretch}{1.5} \begin{array}{ll} & \sum_{s(L) = 1} d_{[L]}(D_1 \cdot D_2) [L] q^{[L]} + \sum_{s(C) = 2} d_{[C]}(D_1 \cdot D_2) [X] q^{[C]} \\ & + \sum_{s(E) = -1}\sum_{s(F) = 0} d_{[F]}d_{[E]}(E \cdot D_1)(E \cdot D_2)(F \cdot E) [F] q^{[E+F]} \\ & + \sum_{s(E) = -1}\sum_{s(L) = 1} d_{[L]}d_{[E]}(E \cdot D_1)(E \cdot D_2)(L \cdot E) [X] q^{[E+L]} \\ =&\\ & \sum_{s(F) = 0} \sum_{s(E) = -1} d_{[F]}d_{[E]}(F \cdot D_1)(E \cdot F)(E \cdot D_2) [E] q^{[E+F]} \\ & + \sum_{s(F) = 0} \sum_{s(G) = 0} d_{[F]}d_{[G]}(F \cdot D_1)(G \cdot F)(G \cdot D_2) [X] q^{[F+G]} \\ & + \sum_{s(L) = 1} d_{[L]}(L \cdot D_1) [D_2] q^{[L]} . \end{array} } \] Comparing those terms with coefficients in $H^0(X)$ and those in $H^2(X)$, we see that $ p \operatorname{\ast_Q} (D_1 \operatorname{\ast_Q} D_2) = (p \operatorname{\ast_Q} D_1) \operatorname{\ast_Q} D_2$ if and only if the following two equations hold: \begin{equation} \label{pDD1} {\renewcommand{\arraystretch}{1.5} \begin{array}{ll} & \sum_{s(L) = 1} d_{[L]}(D_1 \cdot D_2) [L] q^{[L]} \\ & + \sum_{s(E) = -1}\sum_{s(F) = 0} d_{[F]}d_{[E]}(E \cdot D_1)(E \cdot D_2)(F \cdot E) [F] q^{[E+F]} \\ =&\\ & \sum_{s(L) = 1} d_{[L]}(L \cdot D_1) [D_2] q^{[L]} \\ & + \sum_{s(F) = 0} \sum_{s(E) = -1} d_{[F]}d_{[E]}(F \cdot D_1)(E \cdot F)(E \cdot D_2) [E] q^{[E+F]} \end{array} } \end{equation} and \begin{equation} \label{pDD2} {\renewcommand{\arraystretch}{1.5} \begin{array}{ll} & \sum_{s(C) = 2} d_{[C]}(D_1 \cdot D_2) q^{[C]} \\ & + \sum_{s(E) = -1}\sum_{s(L) = 1} d_{[L]}d_{[E]}(E \cdot D_1)(E \cdot D_2)(L \cdot E) q^{[E+L]} \\ =&\\ & \sum_{s(F) = 0} \sum_{s(G) = 0} d_{[F]}d_{[G]}(F \cdot D_1)(G \cdot F)(G \cdot D_2) q^{[F+G]} .\\ \end{array} } \end{equation} \begin{lemma} Equation (\ref{pDD2}) follows from (\ref{ppD}). \end{lemma} \begin{pf} In fact, it is obtained from (\ref{ppD}) by setting $D = D_1$ and dotting with $D_2$. \end{pf} The proof of associativity in the $p \operatorname{\ast_Q} D_1 \operatorname{\ast_Q} D_2$ case now follows from the lemma below. \begin{lemma} Equation (\ref{pDD1}) holds for $X_6$. \end{lemma} \begin{pf} We will show that (\ref{pDD1}) follows from two types of relations, one of which holds generally for generic rational surfaces and the other of which is special to the cubic surface. Note that every relevant class $F$ with $s(F) = 0$ can be written uniquely as $F = -K - E_F$, where $E_F$ is a relevant class with $s(E_F) = -1$ (see Lemma \ref{X6relevantlemma}). Therefore for any relevant class $E$ with $s(E) = -1$, we must have $0 \leq (F \cdot E) \leq 2$; moreover $(F \cdot E) = 2$ if and only if $E = E_F$, and $(F \cdot E) = 1$ if and only if $(E \cdot E_F) = 1$. In this latter case $F+E$ is a relevant class with $s(F+E) = 1$ and $p_a(F+E) = 0$. Therefore (\ref{pDD1}) can be analyzed by considering only these types of $q$-terms. We begin by considering a term of the form $q^{[L]}$, where $L$ is a relevant class with $s(L) = 1$ and $p_a(L) = 0$. Note that in this case if $L=E+F$, then $(E \cdot F) = 1$, $(E \cdot L) = 0$ and $(F\cdot L) = 1$. Moreover all classes contributing to this term have $d=1$. Thus considering the coefficent of $q^{[L]}$ in (\ref{pDD1}) gives the equation \[ (D_1 \cdot D_2)[L] + \sum\begin{Sb} s(E) = -1 \\ s(F) = 0 \\ E+F=L \end{Sb} (E \cdot D_1)(E \cdot D_2)[F] = (L \cdot D_1)[D_2] + \sum\begin{Sb} s(E) = -1 \\ s(F) = 0 \\ E+F=L \end{Sb} (E \cdot D_2)(F \cdot D_1)[E] \] Conversely we note that if $(L \cdot E) = 0$ for $E$ an exceptional curve, then by Riemann-Roch, $L\equiv E+F$ for some $F$. Also if $E+F\equiv E'+F'$, then $E\equiv E'$ and $F\equiv F'$ or $E$ and $E'$ are disjoint; this follows from the fact that $0 = (E' \cdot L) = (E' \cdot E) + (E' \cdot F) \geq (E' \cdot E)$ since $F$ moves in a pencil. For each such class $L$, it is easy to see that there are exactly $6$ classes $E$ with $s(E) = -1$ and $(E \cdot L) = 0$. Therefore there is a disjoint basis of $\operatorname{Pic}(X)$, $[L], [E_1], \ldots, [E_6]$ where $E_1, \dots , E_6$ are exceptional curves, and since $(L \cdot E_i) = 0$, $L$ can be written as $F_i + E_i$ for each $i$. In this basis, the equation above is equivalent to: \[ (D_1 \cdot D_2)[L] - (L \cdot D_1)[D_2] = \sum_i ((E_i \cdot D_2)(F_i \cdot D_1)[E_i] - (E_i \cdot D_1)(E_i\cdot D_2)[F_i]) \] and the right-hand side of this equation is equal to \begin{eqnarray*} &=& \sum_i ((E_i \cdot D_2)(F_i \cdot D_1)[E_i] - (E_i \cdot D_1)(E_i\cdot D_2)[F_i]) \\ &=& \sum_i (E_i \cdot D_2) ((L \cdot D_1)-(E_i \cdot D_1)) [E_i] - (E_i \cdot D_1) [L-E_i]) \\ &=& \sum_i (E_i \cdot D_2) (- (E_i \cdot D_1)[L] + ((L \cdot D_1)-(E_i \cdot D_1) + (E_i \cdot D_1)) [E_i]) \\ &=& (L \cdot D_1) \sum_i (E_i \cdot D_2)[E_i] - \sum_i (E_i \cdot D_2)(E_i \cdot D_1)[L] \end{eqnarray*} and so we must show that \[ (D_1 \cdot D_2)[L] - (L \cdot D_1)[D_2] = (L \cdot D_1) \sum_i (E_i \cdot D_2)[E_i] - \sum_i (E_i \cdot D_2)(E_i \cdot D_1)[L]. \] On the other hand $[D_2] = (L \cdot D_2)[L] - \sum_i(E_i \cdot D_2)[E_i]$. Plugging this expression into the above equation, we obtain an expression all in terms of the basis $[L],[E_1],\ldots,[E_6]$. The coefficients of $[E_i]$ on the two sides are obviously equal, to $(L \cdot D_1)(E_i \cdot D_2)$. Hence we must only check the coefficient of $[L]$, and so the above equation follows from the identity \[ (D_1 \cdot D_2) = (L \cdot D_1)(L \cdot D_2) - \sum_i(E_i \cdot D_1)(E_i \cdot D_2) \] which is immediate from writing $[D_1]$ and $[D_2]$ in terms of the basis $[L],[E_1],\ldots,[E_6]$. To complete the proof of the lemma, we need to consider the coefficient of $q^{[-K]}$ in (\ref{pDD1}); $L = -K$ is the unique relevant class with $s=1$ and $p_a = 1$. In the $E,F$ sums, we may sum over the $E$'s only, setting $F = -K-E$; noting that in this case $(E \cdot F) = 2$, equating the coefficients of $q^{[-K]}$ in (\ref{pDD1}) and dividing by two gives \begin{eqnarray} 6 (K \cdot D_1)[D_2] - 6(D_1 \cdot D_2)[K] &= & \sum_{s(E)= -1} ((-K-E)\cdot D_1)(E \cdot D_2)[E] - (E \cdot D_2)(E \cdot D_1)[-K-E]\nonumber \\ &=& \sum_{s(E)= -1} (-K \cdot D_1)(E \cdot D_2)[E] - (E \cdot D_2)(E \cdot D_1)[-K]. \label{pDD-K} \end{eqnarray} As we saw in the proof above for the associativity of the triple product $p \operatorname{\ast_Q} p \operatorname{\ast_Q} D$, $-K$ is linearly equivalent to any triangle of exceptional curves; moreover precisely ten exceptional curves meet any given exceptional curve. Thus for all exceptional curves $E'$, \begin{eqnarray*} 5[-K] &=& 5[E'] + \sum_{(E \cdot E') = 1} [E] \\ &=& 6[E'] + \sum_E (E \cdot E')[E]. \end{eqnarray*} Thus for all exceptional curves $E'$ and $E''$, \[ 6[E'] + \sum_E (E \cdot E')[E] = 6[E''] + \sum_E (E \cdot E'')[E] \] and so \[ 6[E'] = 6[E''] + \sum_E (E \cdot E'')[E] - \sum_E (E \cdot E')[E]. \] Intersecting with $D_2$ and noting that $(-K \cdot E') = 1$ for all $E'$, we have \[ 6(E' \cdot D_2) = ( 6(E'' \cdot D_2) + \sum_E (E \cdot D_2)(E \cdot E'')) (-K \cdot E') - \sum_E (E \cdot D_2)(E \cdot E'). \] Here this equation holds for all exceptional curves $E'$, which generate $\operatorname{Pic}(X_6)$, so \[ 6[D_2] = (6(E'' \cdot D_2) + \sum_E(E \cdot D_2)(E \cdot E''))[-K] - \sum_E (E \cdot D_2)[E]. \] Intersecting now with $D_1$ and again noting that $(-K \cdot E'')=1$, we have \[ (6(D_1 \cdot D_2)+ \sum_E (E \cdot D_2)(E \cdot D_1))(-K \cdot E'') = (-K \cdot D_1)(6(E'' \cdot D_2) + \sum_E(E \cdot D_2)(E \cdot E'')). \] As this is true for all $E''$, which generate $\operatorname{Pic}(X_6)$, we see that \[ (6(D_1 \cdot D_2)+ \sum_E (E \cdot D_2)(E \cdot D_1))[-K] = (-K \cdot D_1)(6[D_2] + \sum_E(E \cdot D_2)[E]). \] This can be re-written as \begin{eqnarray*} 6(D_1 \cdot D_2)[-K] - 6(-K \cdot D_1)[D_2] &=& \sum_E (-K \cdot D_1)(E \cdot D_2)[E] - (E \cdot D_1)(E \cdot D_2)[-K]\\ &=& \sum_E (E \cdot D_2)((-K \cdot D_1)[E] - (E \cdot D_1)[-K]). \end{eqnarray*} This is exactly the desired equation (\ref{pDD-K}). \end{pf} To conclude our proof of associativity for the quantum product on $X_6$, we must deal with triple quantum products of the form $D_1 \operatorname{\ast_Q} D_2 \operatorname{\ast_Q} D_3$ for divisors $D_i$. Note that $d_{[C]} = 1$ whenever $[C]$ is a relevant class on $X_6$ with $s(c) \leq 0$; we then compute: \begin{eqnarray*} D_1 \operatorname{\ast_Q} (D_2 \operatorname{\ast_Q} D_3) & =& D_1 \operatorname{\ast_Q} ( (D_2 \cdot D_3)[p]q^0 + \!\!\sum_{s(E)= -1} (E \cdot D_2)(E \cdot D_3)[E]q^E + \!\!\sum_{s(F)= 0} (F \cdot D_2)(F \cdot D_3)[X]q^F) \\ & = & (D_2 \cdot D_3)(D_1 \operatorname{\ast_Q} [p])q^0 + \sum_{s(E)= -1} (E \cdot D_2)(E \cdot D_3)(D_1 \operatorname{\ast_Q} [E])q^E \\ &&+ \sum_{s(F)= 0} (F \cdot D_2)(F \cdot D_3)[D_1]q^F \\ & = & \sum_{s(F)= 0} (D_2 \cdot D_3)(F \cdot D_1)[F]q^F + \sum_{s(L)= 1} d_{[L]}(D_2 \cdot D_3)(L \cdot D_1)[X]q^L \\ &&+ \sum_{s(F)= 0} (F \cdot D_2)(F \cdot D_3)[D_1]q^F \\ &&+ \sum_{s(E)= -1} (E \cdot D_2)(E \cdot D_3) ( (E \cdot D_1)[p]q^E + \sum_{s(E')= -1} (E' \cdot D_1)(E \cdot E')[E']q^{E+E'} \\ &&+ \sum_{s(F)= 0} (F \cdot D_1)(F \cdot E)[X]q^{E+F} ) \\ & = & \text{(dimension zero terms:)} \sum_{s(E)= -1} (E \cdot D_1)(E \cdot D_2)(E \cdot D_3)[p]q^E \\ & + & \text{(dimension two terms:)} \sum_{s(F)= 0} ( (D_2 \cdot D_3)(F \cdot D_1)[F] + (F \cdot D_2)(F \cdot D_3)[D_1] ) q^F \\ && + \sum_{s(E)=s(E')= -1} (E \cdot D_2)(E \cdot D_3)(E' \cdot D_1)(E \cdot E')[E']q^{E+E'} \\ & + & \text{(dimension four terms:)} \sum_{s(L)= 1} d_{[L]}(D_2 \cdot D_3)(L \cdot D_1)[X]q^L \\ && + \sum\begin{Sb} s(E)= -1 \\ s(F)= 0 \end{Sb} (E \cdot D_2)(E \cdot D_3)(F \cdot D_1)(F \cdot E)[X]q^{E+F} \\ \end{eqnarray*} On the other hand \begin{eqnarray*} (D_1 \operatorname{\ast_Q} D_2) \operatorname{\ast_Q} D_3 & =& D_3 \operatorname{\ast_Q} (D_1 \operatorname{\ast_Q} D_2) \\ &=& \text{(dimension zero terms:)} \sum_{s(E)= -1} (E \cdot D_3)(E \cdot D_1)(E \cdot D_2)[p]q^E \\ & + & \text{(dimension two terms:)} \sum_{s(F)= 0} ( (D_1 \cdot D_2)(F \cdot D_3)[F] + (F \cdot D_1)(F \cdot D_2)[D_3] ) q^F \\ && + \sum_{s(E)=s(E')= -1} (E \cdot D_1)(E \cdot D_2)(E' \cdot D_3)(E \cdot E')[E']q^{E+E'} \\ & + & \text{(dimension four terms:)} \sum_{s(L)= 1} d_{[L]}(D_1 \cdot D_2)(L \cdot D_3)[X]q^L \\ && + \sum\begin{Sb} s(E)= -1 \\ s(F)= 0 \end{Sb} (E \cdot D_1)(E \cdot D_2)(F \cdot D_3)(F \cdot E)[X]q^{E+F} \\ \end{eqnarray*} (This is obtained from the previous by permuting indices.) Comparing terms, we see that the dimension zero terms are identical and that the dimension four terms follow from Equation (\ref{pDD1}) (obtained by considering the $p \operatorname{\ast_Q} D_1 \operatorname{\ast_Q} D_2$ product) intersected with $D_3$. Comparing terms in dimension two, we need to show that \begin{equation} \label{DDD} {\renewcommand{\arraystretch}{1.5} \begin{array}{ll} & \sum_{s(F)= 0} ( (D_1 \cdot D_2)(F \cdot D_3)[F] + (F \cdot D_1)(F \cdot D_2)[D_3] )q^F \\ & + \sum_{s(E)=s(E')= -1} (E \cdot D_1)(E \cdot D_2)(E' \cdot D_3)(E' \cdot E) [E'] q^{E+E'} \\ =&\\ & \sum_{s(F)= 0} ( (D_2 \cdot D_3)(F \cdot D_1)[F] + (F \cdot D_2)(F \cdot D_3)[D_1] ) q^F \\ & + \sum_{s(E)=s(E')= -1} (E' \cdot D_1)(E \cdot D_2)(E \cdot D_3)(E \cdot E')[E']q^{E+E'}. \end{array} } \end{equation} Now if $E$ and $E'$ are disjoint, then there is no contribution to either side. If $E = E'$, then the coefficients of $q^{E+E'}$ are seen to be equal. If $E$ meets $E'$, then $(E \cdot E')=1$ and $E+E' \equiv F$ for some $F$. Hence the equality above follows from the equality of the coefficients of $q^F$ for a particular class $[F]$ with $s(F) = 0$. Hence associativity is implied by the following lemma. \begin{lemma} For all relevant classes $[F]$ with $s(F)=0$, \begin{eqnarray*} &&(D_1 \cdot D_2)(F \cdot D_3)[F] + (F \cdot D_1)(F \cdot D_2)[D_3] -(D_2 \cdot D_3)(F \cdot D_1)[F] - (F \cdot D_2)(F \cdot D_3)[D_1] \\ &=& \sum_{E+E' \equiv F}((E' \cdot D_1)(E \cdot D_2)(E \cdot D_3) - (E \cdot D_1)(E \cdot D_2)(E' \cdot D_3))[E']. \end{eqnarray*} \end{lemma} \begin{pf} Our fixed curve $F$ gives $X$ a structure of ruled surface with $F$ as general fiber. Each $E+E'$ summing to $F$ is a reducible fiber with respect to that ruling (there are five such reducible fibers). Note that there is a basis, $[F],[G],[E_1],\dots,[E_5]$ of $\operatorname{Pic}(X_6)$ such that $G$ has self-intersection zero, the $E_i$'s have self-intersection -1, $(F \cdot G) =1$, and all other intersections are zero (this is equivalent to $X_6$ having ${\Bbb F}_0 = {\Bbb P}^1\times {\Bbb P}^1$ as a minimal ruled model). We now write the right hand side of the above equation in terms of the basis. Note that $E'_i \equiv F - E_i$ and that the role of $E'$ is played by both $E'$ and $E$. We have \begin{eqnarray*} && \sum_{E+E' \equiv F}((E' \cdot D_1)(E \cdot D_2)(E \cdot D_3) - (E \cdot D_1)(E \cdot D_2)(E' \cdot D_3))[E'] \\ &=& \sum_i ((F-E_i) \cdot D_1)(E_i \cdot D_2)(E_i \cdot D_3) - (E_i \cdot D_1)(E_i \cdot D_2)((F-E_i) \cdot D_3))[F-E_i] \\ && + \sum_i (E_i \cdot D_1)((F-E_i) \cdot D_2)((F-E_i) \cdot D_3) - ((F-E_i) \cdot D_1)((F-E_i) \cdot D_2)(E_i \cdot D_3))[E_i]. \end{eqnarray*} The coefficient of each $[E_i]$ in this expression contains a sum of twelve products, all of which cancel except for $(E_i \cdot D_1)(F \cdot D_2)(F \cdot D_3) - (F \cdot D_1)(F \cdot D_2)(E_i \cdot D_3)$. The coefficient of the $[F]$ term is seen to be $\sum_i ( (F \cdot D_1)(E_i \cdot D_2)(E_i \cdot D_3)- (E_i \cdot D_1)(E_i \cdot D_2)(F \cdot D_3) )$. Thus we may re-write the equation of the lemma as \begin{eqnarray*} &&(D_1 \cdot D_2)(F \cdot D_3)[F] + (F \cdot D_1)(F \cdot D_2)[D_3] -(D_2 \cdot D_3)(F \cdot D_1)[F] - (F \cdot D_2)(F \cdot D_3)[D_1] \\ &=& \\ && [F]\sum_i ( (F \cdot D_1)(E_i \cdot D_2)(E_i \cdot D_3)- (E_i \cdot D_1)(E_i \cdot D_2)(F \cdot D_3) ) \\ && + \sum_i ((E_i \cdot D_1)(F \cdot D_2)(F \cdot D_3) - (F \cdot D_1)(F \cdot D_2)(E_i \cdot D_3)) [E_i]. \end{eqnarray*} To show that this linear equivalence is true, it suffices to show equality when dotted with a basis of $\operatorname{Pic}(X)$. It is easy to see that equality holds when the expression above is dotted with $F$ or any $E_i$. Dotting with $G$, and recalling that $(F\cdot G) = 1$ and $(G \cdot E_i)= 0$ yields the following expression: \begin{eqnarray*} \label{mess} &&(D_1 \cdot D_2)(F \cdot D_3) + (F \cdot D_1)(F \cdot D_2)(G\cdot D_3) -(D_2 \cdot D_3)(F \cdot D_1) - (F \cdot D_2)(F \cdot D_3)(G \cdot D_1) \\ &=& \\ && \sum_i ( (F \cdot D_1)(E_i \cdot D_2)(E_i \cdot D_3)- (E_i \cdot D_1)(E_i \cdot D_2)(F \cdot D_3) ). \end{eqnarray*} On the other hand, the divisors $D_j$, $j=1,2,3$ are written in terms of the basis as \[ [D_j] \equiv (D_j \cdot G)[F] + (D_j \cdot F)[G] - \sum_i (D_j \cdot E_i)[E_i]. \] Substituting these expressions into $(D_1 \cdot D_2)$ and $(D_2 \cdot D_3)$ of (\ref{mess}) and collecting terms proves the result. \end{pf} This completes our analysis of the associativity of the quantum product for $X_6$, and therefore for all general strict Del Pezzo surfaces. We have proved the following. \begin{theorem} The quantum product $\operatorname{\ast_Q}$ is associative for ${\Bbb P}^2$, ${\Bbb F}_0$, and $X_1,\ldots,X_6$. \end{theorem} \section{Associativity in general} \label{sectionassoc} In this section we offer an algebro-geometric approach to proving the associativity of the quantum product for a general rational surface. This approach avoids the reliance on perturbing to a non-integrable almost complex structure (see \cite{mcduff-salamon,ruan-tian}); we work with the existing complex/algebraic structure. The associativity of the quantum product is implied by checking associativity for triple products of homogeneous generators for $H^*(X)$. In other words, we must check that if $\alpha$, $\beta$, and $\gamma$ are homogeneous classes in $H^*(X)$, then \begin{equation} \label{assoc1} \alpha \operatorname{\ast_Q} (\beta \operatorname{\ast_Q} \gamma) = (\alpha \operatorname{\ast_Q} \beta) \operatorname{\ast_Q} \gamma. \end{equation} We need not check the formula when one of the constituents is the identity $[X]$, or when they are all equal. \begin{lemma} \label{assoclemma} The associativity of the quantum product is equivalent to the following identity: \[ \sum_{([C_1],[C_2]):[C_1+C_2]=[D]} \phi_{[C_2]}(\alpha,\delta) \cdot \phi_{[C_1]}(\beta,\gamma) = \sum_{([C_1],[C_2]):[C_1+C_2]=[D]} \phi_{[C_2]}(\gamma,\delta) \cdot \phi_{[C_1]}(\alpha,\beta). \] This identity must hold for all divisor classes $[D]$ and all homogeneous classes $\alpha$, $\beta$, $\gamma$, and $\delta$ in $H^*(X)$. \end{lemma} \begin{pf} Expanding the two sides of (\ref{assoc1}), we have \begin{eqnarray*} \alpha \operatorname{\ast_Q} (\beta \operatorname{\ast_Q} \gamma) &=& \alpha \operatorname{\ast_Q} ( \sum_{[C_1]} \phi_{[C_1]}(\beta,\gamma) q^{[C_1]}) \\ &=& \sum_{[C_1]} \sum_{[C_2]} \phi_{[C_2]}( \alpha,\phi_{[C_1]}(\beta,\gamma) ) q^{[C_1+C_2]} \end{eqnarray*} while \begin{eqnarray*} (\alpha \operatorname{\ast_Q} \beta) \operatorname{\ast_Q} \gamma &=& ( \sum_{[C_1]} \phi_{[C_1]}(\alpha,\beta) q^{[C_1]} ) \operatorname{\ast_Q} \gamma \\ &=& \sum_{[C_1]} \sum_{[C_2]} \phi_{[C_2]}(\phi_{[C_1]}(\alpha,\beta),\gamma ) q^{[C_1+C_2]}. \end{eqnarray*} For these to be equal, they must have equal coefficients for all terms $q^{[D]}$. Therefore associativity of the quantum product is equivalent to having \[ \sum_{([C_1],[C_2]):[C_1+C_2]=[D]} \phi_{[C_2]}( \alpha,\phi_{[C_1]}(\beta,\gamma) ) = \sum_{([C_1],[C_2]):[C_1+C_2]=[D]} \phi_{[C_2]}(\phi_{[C_1]}(\alpha,\beta),\gamma ) \] for all homogeneous classes $\alpha$, $\beta$, and $\gamma$ in $H^*(X)$ and all divisor classes $[D]$. The equality is equivalent to knowing that for all homogeneous $\delta \in H^*(X)$, the intersection products with $\delta$ are equal, i.e., \[ \sum_{([C_1],[C_2]):[C_1+C_2]=[D]} \phi_{[C_2]}( \alpha,\phi_{[C_1]}(\beta,\gamma) ) \cdot \delta = \sum_{([C_1],[C_2]):[C_1+C_2]=[D]} \phi_{[C_2]}(\phi_{[C_1]}(\alpha,\beta),\gamma ) \cdot \delta. \] (Here a dot product is taken to be zero unless the codimensions of the classes are complementary.) These intersection products can be computed on $X^3$, and we then are requiring that \[ \sum_{([C_1],[C_2]):[C_1+C_2]=[D]} [A_{[C_2]}] \cdot (\alpha\otimes\phi_{[C_1]}(\beta,\gamma)\otimes\delta) = \sum_{([C_1],[C_2]):[C_1+C_2]=[D]} [A_{[C_2]}] \cdot (\phi_{[C_1]}(\alpha,\beta) \otimes\gamma \otimes \delta). \] By the symmetry of the $[A]$-classes we may rewrite this as \[ \sum_{([C_1],[C_2]):[C_1+C_2]=[D]} [A_{[C_2]}] \cdot (\alpha\otimes\delta\otimes\phi_{[C_1]}(\beta,\gamma)) = \sum_{([C_1],[C_2]):[C_1+C_2]=[D]} [A_{[C_2]}] \cdot ( \gamma \otimes \delta\otimes\phi_{[C_1]}(\alpha,\beta)), \] which we may then reformulate as \[ \sum_{([C_1],[C_2]):[C_1+C_2]=[D]} \phi_{[C_2]}(\alpha,\delta) \cdot \phi_{[C_1]}(\beta,\gamma) = \sum_{([C_1],[C_2]):[C_1+C_2]=[D]} \phi_{[C_2]}(\gamma,\delta) \cdot \phi_{[C_1]}(\alpha,\beta). \] This is the desired identity. \end{pf} Let $[C]$ be a relevant class on $X$, and let ${\cal R}_{[C]}$ denote the locus of irreducible nodal rational curves in the linear system $|C|$. Recall that $d_{[C]}$ is the degree of the closure $\overline{{\cal R}_{[C]}}$. Suppose that $[C]$ decomposes as $[C] = [C_1] +[C_2]$ where $[C_1]$ and $[C_2]$ are relevant classes. We may form the following locus \begin{eqnarray*} {\cal S}_{[C_1],[C_2]} &= \{& (C_1,C_2,x_1,y_1,x_2,y_2,z) \in {\cal R}_{[C_1]} \times {\cal R}_{[C_2]} \times X^5 \;|\; \\ && C_1 \text{ and } C_2 \text{ meet transversally at } z, \\ && x_1 \text{ and } y_1 \text{ are smooth points of }C_1, \text{ and } \\ && x_2 \text{ and } y_2 \text{ are smooth points of }C_2 \}. \end{eqnarray*} There is a natural map \[ {\cal S}_{[C_1],[C_2]} \to X^6 \] sending $(C_1,C_2,x_1,y_1,x_2,y_2,z)$ to $(x_1,y_1,z,x_2,y_2,z)$. Call the image of the fundamental class $[A^6_{[C_1],[C_2]}]$. Related to this is the map \[ {\cal S}_{[C_1]} \times {\cal S}_{[C_2]} \to X^6 \] sending a pair $((C_1,x_1,y_1,z_1),(C_2,x_2,y_2,z_2))$ to $(x_1,y_1,z_1,x_2,y_2,z_2)$. The image of the fundamental class of this map is clearly $[A_{[C_1]}] \otimes [A_{[C_2]}]$. If we denote by $\pi_{36}:X^6 \to X^2$ the projection onto the third and sixth coordinates, we see that \[ [A^6_{[C_1],[C_2]}] = ([A_{[C_1]}] \otimes [A_{[C_2]}]) \cup \pi_{36}^*([\Delta]) \] where $\Delta \subset X^2$ is the diagonal. Finally consider the natural map \[ {\cal S}_{[C_1],[C_2]} \to X^4 \] sending $(C_1,C_2,x_1,y_1,x_2,y_2,z)$ to $(x_1,y_1,x_2,y_2)$. Call the image of the fundamental class $[A^4_{[C_1],[C_2]}]$. \begin{lemma} With the above notation, \[ \phi_{[C_1]}(\alpha,\beta) \cdot \phi_{[C_2]}(\gamma,\delta) = [A^4_{[C_1],[C_2]}] \cdot \alpha\otimes\beta\otimes\gamma\otimes\delta. \] \end{lemma} \begin{pf} Write $[\Delta] = \sum_i u_i \otimes v_i$ in $H^4(X^2)$, where the $u_i$ and $v_i$ are classes in $H^*(X)$. Denote by $\pi_{1245}:X^6 \to X^4$ the projection onto the first, second, fourth, and fifth factors. Then \begin{eqnarray*} \phi_{[C_1]}(\alpha,\beta) \cdot \phi_{[C_2]}(\gamma,\delta) &=& [\phi_{[C_1]}(\alpha,\beta) \otimes \phi_{[C_2]}(\gamma,\delta)] \cdot [\Delta] \\ &=& \sum_i [\phi_{[C_1]}(\alpha,\beta) \otimes \phi_{[C_2]}(\gamma,\delta)] \cdot[u_i\otimes v_i] \\ &=& \sum_i (\phi_{[C_1]}(\alpha,\beta) \cdot u_i) (\phi_{[C_2]}(\gamma,\delta) \cdot v_i) \\ &=& \sum_i([A_{[C_1]}] \cdot \alpha\otimes\beta\otimes u_i) ([A_{[C_2]}] \cdot \gamma\otimes\delta\otimes v_i) \\ &=& \sum_i ([A_{[C_1]}] \otimes [A_{[C_2]}])\cdot (\alpha\otimes\beta\otimes u_i \otimes \gamma\otimes\delta\otimes v_i) \\ &=& ([A_{[C_1]}] \otimes [A_{[C_2]}])\cup (\alpha\otimes\beta\otimes X \otimes \gamma\otimes\delta\otimes X) \cup \pi_{36}^*([\Delta]) \\ &=& ([A_{[C_1]}] \otimes [A_{[C_2]}])\cup \pi_{1245}^*(\alpha\otimes\beta\otimes \gamma\otimes\delta) \cup \pi_{36}^*([\Delta]) \\ &=& [A^6_{[C_1],[C_2]}] \cdot \pi_{1245}^*(\alpha\otimes\beta\otimes \gamma\otimes\delta) \\ &=& [A^4_{[C_1],[C_2]}] \cdot (\alpha\otimes\beta\otimes \gamma\otimes\delta) \end{eqnarray*} as claimed. \end{pf} Next we introduce the space \begin{eqnarray*} {\cal S}^4_{[C]} &= \{& (C,x_1,x_2,x_3,x_4) \in {\cal R}_{[C]} \times X^4 \;|\; \\ && x_i \text{ are smooth points of }C \}. \end{eqnarray*} There is a natural projection to $X^4$; we denote the image of the fundamental class by $[A^4_{[C]}]$. We note that we can consider the spaces ${\cal S}_{[C_1],[C_2]}$ as lying in the closure of the space ${\cal S}^4_{[C_1+C_2]}$, via the natural map \[ \psi:{\cal S}_{[C_1],[C_2]} \to \overline{ {\cal S}^4_{[C_1+C_2]} } \] defined by sending $(C_1,C_2,x_1,y_1,x_2,y_2,z)$ to $(C_1+C_2,x_1,y_1,x_2,y_2)$. Actually, this map $\psi$ may be finite-to-one, if $(C_1\cdot C_2) \geq 2$. The image points represent the addition of an extra node to a curve which already has arithmetic genus zero; this then breaks the curve into two components. One expects that the boundary of ${\cal S}^4_{[C]}$ in its closure will have exactly the images of these loci $\psi({\cal S}_{[C_1],[C_2]})$ with $[C_1]+[C_2] = [C]$ as components. Now the space ${\cal S}^4_{[C]}$ has a cross-ratio function on it, \[ \operatorname{CR}:{\cal S}^4_{[C]} \to {\Bbb P}^1, \] defined by sending $(C,x_1,x_2,x_3,x_4)$ to the cross-ratio $(x_1-x_3)(x_2-x_4)/(x_1-x_4)(x_2-x_3)$. (Here a coordinate is chosen on the normalization of $C$.) Again one expects that this cross-ratio function will extend to the closure $\overline{ {\cal S}^4_{[C_1+C_2]} }$, or at least to a model of the closure which is birational on the boundary divisors $\psi({\cal S}_{[C_1],[C_2]})$ with $[C_1]+[C_2] = [C]$. For distinct points, the cross-ratio takes values in ${\Bbb P}^1 - \{0,1,\infty\}$. For the general coalescence of two of the four points, the cross-ratio takes value $0$ when $x_1 = x_3$, value $1$ when $x_1 = x_2$, and value $\infty$ when $x_1 = x_4$. Therefore $\operatorname{CR}^{-1}(1)$ on the closure should be the space where the first two points $x_1$ and $x_2$ come together; when this happens, the curve will split, with $x_1$ and $x_2$ moving to points on one curve and $x_3$ and $x_4$ lying on the other. Therefore if we denote by $[A^4_{[C]}(\lambda)]$ the class of the image of $\operatorname{CR}^{-1}(\lambda)$ in $X^4$, we have that \[ [A^4_{[C]}(1)] = \sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb} [A^4_{[C_1],[C_2]}] \] Therefore \begin{eqnarray*} \sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb} \phi_{[C_2]}(\alpha,\delta) \cdot \phi_{[C_1]}(\beta,\gamma) &=& \sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb} [A^4_{[C_1],[C_2]}] \cdot \beta\otimes\gamma\otimes\alpha\otimes\delta \\ &=& [A^4_{[C]}(1)] \cdot \beta\otimes\gamma\otimes\alpha\otimes\delta. \end{eqnarray*} Now as $\lambda$ varies, the classes $[A^4_{[C]}(\lambda)]$ are rationally equivalent; hence the intersection product is the same. Therefore \[ [A^4_{[C]}(1)] \cdot \beta\otimes\gamma\otimes\alpha\otimes\delta = [A^4_{[C]}(0)] \cdot \beta\otimes\gamma\otimes\alpha\otimes\delta. \] This class $[A^4_{[C]}(0)]$ can be analyzed by studying the maps \[ \tilde{\psi}:{\cal S}_{[C_1],[C_2]} \to \overline{ {\cal S}^4_{[C_1+C_2]} } \] defined by sending $(C_1,C_2,x_1,y_1,x_2,y_2,z)$ to $(C_1+C_2,x_1,x_2,y_1,y_2)$. This is just the map $\psi$ above, followed by a permutation of the four points; but we see that if we denote the image of these classes by $[\tilde{A}^4_{[C_1],[C_2]}]$, then \[ [A^4_{[C]}(0)] = \sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb} [\tilde{A}^4_{[C_1],[C_2]}] \] since the cross-ratio being zero represents when the first and third points come together, and this should be modelled by having them split off to one of the component curves (in this case $C_1$). This class $[\tilde{A}^4_{[C_1],[C_2]}]$ is related to the original version $[A^4_{[C_1],[C_2]}]$ by the relation that \[ [A^4_{[C_1],[C_2]}] \cdot \beta\otimes\alpha\otimes\gamma\otimes\delta = [\tilde{A}^4_{[C_1],[C_2]}] \cdot \beta\otimes\gamma\otimes\alpha\otimes\delta \] because of the permutation which relates the maps $\psi$ and $\tilde{\psi}$. Hence \begin{eqnarray*} [A^4_{[C]}(0)] \cdot \beta\otimes\gamma\otimes\alpha\otimes\delta &=& \sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb} [\tilde{A}^4_{[C_1],[C_2]}] \cdot \beta\otimes\gamma\otimes\alpha\otimes\delta \\ &=& \sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb} [A^4_{[C_1],[C_2]}] \cdot \beta\otimes\alpha\otimes\gamma\otimes\delta \\ &=& \sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb} \phi_{[C_1]}(\beta,\alpha) \cdot \phi_{[C_2]}(\gamma,\delta) \end{eqnarray*} We conclude that \[ \sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb} \phi_{[C_2]}(\alpha,\delta) \cdot \phi_{[C_1]}(\beta,\gamma) = \sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb} \phi_{[C_1]}(\beta,\alpha) \cdot \phi_{[C_2]}(\gamma,\delta) \] which is equivalent to the associative law for the quantum product by Lemma \ref{assoclemma}, using the symmetry of the $\phi$-classes. The reader will note that the approach given above to the proof of associativity relies on the existence of a model of the closure of the spaces ${\cal S}^4_{[C]}$, which has rather nice properties: the boundary is well-understood in terms of splittings of $C$ as $C = C_1 + C_2$, and the cross-ratio function extends nicely to it. The existence of such a model we only conjecture, and have not attempted to construct it in this paper. \section{Enumerative consequences of associativity} \label{sectionenum} We will now extract several enumerative consequences from the associative law for the quantum product which have been noted by Kontsevich and Manin in \cite{kontsevich-manin}. Let us change notation somewhat and introduce the integer \[ k(C) = k([C]) =\begin{cases} (-K\cdot C) & \text{ if the class $[C]$ is relevant on $X$}\\ 0 & \text{if $[C]$ is not relevant.} \end{cases} \] We note that by the adjunction formula, if $[C]$ is a relevant class, then $k(C) = s(C) + 2$ (recall that $s(C) = (C\cdot C) - 2p_a(C)$). Moreover $k(-)$ is linear in relevant classes: \[ k(C_1+C_2) = k(C_1)+k(C_2). \] Also, the subscript notation for the degree $d_{[C]}$ of the rational curve locus ${\cal R}_{[C]}$ in the linear system $|C|$ is too cumbersome; we will switch notation and call this degree $N(C)$ (following the notation in \cite{kontsevich-manin}). The equations which were seen in Section \ref{sectionSDPassoc} to be equivalent to the associative law for the quantum product for strict Del Pezzos are actually equivalent to associativity for any rational surface $X$. These were (\ref{ppD}), (\ref{pDD1}), and (\ref{DDD}). With the above notation, they can be written as \begin{eqnarray} \label{assocc1} \sum_{k(C)= 4} N(C) [D] q^{[C]} &+ & \sum_{k(L)= 3}\sum_{k(E)= 1} N(L) N(E) (E \cdot L)(E \cdot D) [E] q^{[E+L]} \\ &=& \sum_{k(F)= 2}\sum_{k(G)= 2} N(F)N(G)(F \cdot D)(G \cdot F)[G] q^{[F+G]} \nonumber \end{eqnarray} for any divisor class $[D]$, \begin{equation} \label{assoc2} {\renewcommand{\arraystretch}{1.5} \begin{array}{ll} & \sum_{k(L) = 3} N(L)(D_1 \cdot D_2) [L] q^{[L]} \\ & + \sum_{k(E) = 1}\sum_{k(F) = 2} N(F)N(E)(E \cdot D_1)(E \cdot D_2)(F \cdot E) [F] q^{[E+F]} \\ =&\\ & \sum_{k(L) = 3} N(L)(L \cdot D_1) [D_2] q^{[L]} \\ & + \sum_{k(F) = 2} \sum_{k(E) = 1} N(F)N(E)(F \cdot D_1)(E \cdot F)(E \cdot D_2) [E] q^{[E+F]} \end{array} } \end{equation} for any divisor classes $[D_1]$ and $[D_2]$, and \begin{equation} \label{assoc3} {\renewcommand{\arraystretch}{1.5} \begin{array}{ll} & \sum_{k(F)= 2} N(F) ( (D_1 \cdot D_2)(F \cdot D_3)[F] + (F \cdot D_1)(F \cdot D_2)[D_3] )q^F \\ & + \sum_{k(E)=k(E')= 1} N(E)N(E') (E \cdot D_1)(E \cdot D_2)(E' \cdot D_3)(E' \cdot E) [E'] q^{E+E'} \\ =&\\ & \sum_{k(F)= 2}N(F) ( (D_2 \cdot D_3)(F \cdot D_1)[F] + (F \cdot D_2)(F \cdot D_3)[D_1] ) q^F \\ & + \sum_{k(E)=k(E')= 1} N(E)N(E') (E' \cdot D_1)(E \cdot D_2)(E \cdot D_3)(E \cdot E')[E']q^{E+E'}. \end{array} } \end{equation} for any divisor classes $[D_1]$, $[D_2]$, and $[D_3]$. (Note that (\ref{DDD}) had no $N(-)$ numbers; all were equal to one in the case of $X_6$, but in general they must be included of course.) Suppose that $X = X_n$, the general $n$-fold blowup of the plane. We take as a basis for $\operatorname{Pic}(X)$ the classes $[H]$, $[E_1], \dots, [E_n]$, where $[H]$ is the class of the pullback of a line from ${\Bbb P}^2$, and $E_i$ is the exceptional curve over the $i$-th point $p_i$ which is blown up. In this case every divisor class can be written as \[ [D] = d[H] - \sum_{i=1}^n m_i [E_i] \] which we will abbreviate to $[D] = (d;m_1,\dots,m_n)$. Note then that the anticanonical class $[-K] = (3;1^n)$ where we use the exponential notation for repeated $m_i$'s, as is rather standard. Hence if $[D] = (d;m_1,\dots,m_n)$ then $(-K\cdot D) = 3d-\sum_i m_i$. With this notation we see that quantum cohomology is a completely numerically based theory. A class $[D] = (d;m_1,\dots,m_n)$ is relevant if and only if $d^2 + 3d \geq \sum_i m_i^2 + \sum_i m_i$ and $k(D) \leq 4$. (The first condition is that the expected dimension of $|D|$ is non-negative, so that there will be curves in $|D|$; the second is the relevance condition, that the dimension of the locus of rational curves in $|D|$ is not more than $3$.) Suppose that we take the associativity condition (\ref{assocc1}), set $D = H$, and dot with $H$: we obtain \begin{eqnarray} \label{assoc1D=HdotH} \sum_{k(C)= 4} N(C) q^{[C]} &= & \sum_{k(F)= 2}\sum_{k(G)= 2} N(F)N(G)(F \cdot H)(G \cdot F)(G \cdot H) q^{[F+G]} \\ && - \sum_{k(L)= 3}\sum_{k(E)= 1} N(L) N(E) (E \cdot L){(E \cdot H)}^2q^{[E+L]} \nonumber \end{eqnarray} We want to use this to develop a recursive formula for the degrees $N(C)$ if possible. The $q^{[C]}$ terms of the (\ref{assoc1D=HdotH}) are \begin{eqnarray} \label{assoc1qC} N(C) &= & \sum\begin{Sb} (F,G) \\ k(F)=k(G)=2 \\ F+G \equiv C \end{Sb} N(F)N(G)(F \cdot H)(G \cdot F)(G \cdot H) \\ && - \sum\begin{Sb} (E,L) \\ k(E)=1, k(L) = 3 \\ E+L \equiv C \end{Sb} N(L) N(E) (E \cdot L){(E \cdot H)}^2 \nonumber \end{eqnarray} To relate this recursive formula to those of Kontsevich and Manin (Claims 5.2.1 and 5.2.3b of \cite{kontsevich-manin}), we write this as \begin{eqnarray} \label{assoc2qC} & & \\ N(C) &= & \sum\begin{Sb} (C_1,C_2) \\ C_1+C_2 \equiv C \end{Sb} N(C_1)N(C_2)(C_1 \cdot C_2)(H \cdot C_1) ( (H \cdot C_2)\delta_{k(C_1)-2} - (H \cdot C_1)\delta_{k(C_1)-1} ) \nonumber \end{eqnarray} where $\delta_n = \begin{cases} 1 & \text{ if $n=0$ } \\ 0 & \text{ if $n\ne 0$ } \end{cases}$. This expression is equivalent to Claim 5.2.3b of \cite{kontsevich-manin} in the case for which $k(C) = 4$, assuming that their convention for $\left( \begin{array}{c} 0 \\ n \end{array} \right)$ is that $\left( \begin{array}{c} 0 \\ n \end{array} \right) = \delta_n$. Note that \ref{assoc2qC} is not valid when $k(C) \ne 4$, as may be seen when $C \equiv -K_X$ on $X=X_6$. It is unclear what 5.2.3b of \cite{kontsevich-manin} means in this case, since $(-K \cdot C) = 3$ and so 5.2.3b involves terms of the form $\left( \begin{array}{c} -1 \\ n \end{array} \right)$ where $n \le 0$. Note also that the same $C$ is indecomposible in the semi-group of numerically effective curves and yet has $N(C)=12$, contrary to the expectation expressed in 5.2.3b of \cite{kontsevich-manin}. Our next goal is to compute $N(d)$, the degree of the locus of rational curves of degree $d$ in the plane. This is not a relevant class on the plane, unless $d = 1$. Since forcing a curve to pass through a generically chosen point is a linear condition on the linear system, we have \[ N(d;m_1,\dots,m_n,1) = N(d;m_1,\dots,m_n). \] Hence by induction we have that \[ N(d) = N(d;1^{3d-4}) \] in particular. Now on the surface $X_{3d-4}$, the class $C = (d;1^{3d-4})$ is a relevant class; in fact $k(d;1^{3d-4}) = 4$ and so \ref{assoc1qC} may be used to compute $N(d)$. We now need to understand those $k(E) = 1$ and $k(L) = 3$ classes which sum to $C$, and those $k(F) = k(G) = 2$ classes which sum to $C$. First consider the class $E_i$ itself, which has $k(E) = 1$. However $(E_i \cdot H) = 0$, so these $k(E)=1$ classes do not contribute to the recursive formula of (\ref{assoc1qC}). Hence we may assume $E$ is a relevant class with all $m_i$'s non-negative. In this case since $C = (d;1^{3d-4})$, all $m_i$'s for $E$ (and for the complementary $k=3$ class $L$) must be $0$ or $1$. Therefore $E = (e;1^{3e-1})$ for some $e$ with $1 \leq e \leq d-1$, where this notation means that $3e-1$ of the $m_i$'s are $1$, and all the others are zero. (There are {\small $\left(\begin{array}{c} 3d-4 \\ 3e-1 \end{array}\right)$} such classes.) The complementary class $L$ is of the form $L = (d-e;1^{3d-3e-3})$ where the $1$'s occur in the complementary positions. Note that with this notation $(E\cdot L) = e(d-e)$ and $(E \cdot H) = e$, with $N(E) = N(e)$ and $N(L) = N(d-e)$; so the second sum above reduces to \[ \sum_{E+L\equiv C} N(L) N(E) (E \cdot L){(E \cdot H)}^2 = \sum_{e=1}^{d-1} \left(\begin{array}{c} 3d-4 \\ 3e-1 \end{array}\right) N(e) N(d-e) (d-e) e^3. \] Now suppose that $F$ is a $k=2$ class; again all its multiplicity numbers $m_i$ must be zero or one, and so $F$ must have the form $F = (e;1^{3e-2})$ for some $e$ with $1 \leq e \leq d-1$; there are {\small $\left(\begin{array}{c} 3d-4 \\ 3e-2 \end{array}\right)$} such classes. The complementary class $G$ is $G = (d-e;1^{3d-3e-2})$, where again the $1$'s occur in the complementary positions. Note that with this notation $(F\cdot G) = e(d-e)$, $(F \cdot H) = e$, and $(G \cdot H) = d-e$; also $N(F) = N(e)$ and $N(G) = N(d-e)$. Hence the first sum above reduces to \[ \sum_{F+G\equiv C} N(F)N(G)(F \cdot H)(G \cdot F)(G \cdot H) = \sum_{e=1}^{d-1} \left(\begin{array}{c} 3d-4 \\ 3e-2 \end{array}\right) N(e) N(d-e) e^2{(d-e)}^2. \] Collecting terms gives the following recursion relation for the degrees $N(d)$: \begin{equation} \label{km5.2.1} N(d) = \sum_{e=1}^{d-1} e^2(d-e) \left[ (d-e) \left(\begin{array}{c} 3d-4 \\ 3e-2 \end{array}\right) - e \left(\begin{array}{c} 3d-4 \\ 3e-1 \end{array}\right) \right] N(e) N(d-e). \end{equation} This is exactly the enumerative prediction made by Kontsevich and Manin (Claim 5.2.1 of \cite{kontsevich-manin}). We note here that from our point of view this prediction follows from the associativity of the quantum product for arbitrarily large blowups of the plane; it is not enough to know it just for the plane. To further illustrate the geometric and enumerative significance of associativity of quantum cohomology, we return to $X_6$. \begin{proposition} Associativity of quantum cohomology on strict Del Pezzo surfaces is equivalent to the fact that there are 27 exceptional curves on $X_6$, each of which meets precisely 10 others. \end{proposition} \begin{pf} In Section \ref{sectionSDPassoc}, associativity for $X_6$ (and so all strict Del Pezzos) was shown using the fact that the number and mutual disposition of exceptional curves are as above. To complete our proof, it suffices to show that if $\hat E$ is an exceptional curve on $X_6$, $m$ is the number of other exceptional curves meeting $\hat E$ and $e$ is the total number of exceptional curves, then $m=10$ and $e=27$. Consider the associativity relation \ref{assoc2} with $ L \equiv D_1 \equiv D_2 \equiv -K $. Allowing ourselves the knowledge that $N(F)=N(E)=1$ for all cases for which $F+E \equiv -K$ and collecting terms, \ref{assoc2} implies that \begin{eqnarray} 0 & = & \sum\begin{Sb} (E,F) \\ k(F)=2, k(E)=1 \\ E+F \equiv -K \end{Sb} (E \cdot -K)( (F \cdot -K)[E] - (E \cdot -K)[F] ) \nonumber \\ & = & \sum_{k(E)=1} 2[E] - [-K-E] \nonumber \\ & = & \sum_{k(E)=1} ( 3[E] + [K] ) \nonumber \end{eqnarray} Intersecting with our fixed exceptional curve $\hat E$, we have $ 0 = \sum_E ( 3 (E \cdot \hat E) - 1)$ and so $0 = -4 + 2m - (e- (m+1))$ and thus $e = 3m - 3$. On the other hand, applying the associativity relation \ref{assocc1} with $C \equiv -K + \hat E$, we get for all divisors $D$ \[ 12[D] = \sum_{F+G=C} 1 \cdot 1 \cdot (F \cdot D) \cdot 2 [G] - \sum\begin{Sb} L+E \equiv -K+\hat E \\ p_a(L) = 0 \end{Sb} 1 \cdot 1 \cdot 2 \cdot (E \cdot D)[E] - 12 (\hat E \cdot D)[\hat E] \] which is equivalent, after noting that $F+G \equiv C$ if and only if $F \equiv \hat E + E_F$ for some exceptional curve $E_F$, to \[ 6( [D] + (\hat E \cdot D)[\hat E] ) = \sum\begin{Sb} \hat E + E_F \\ \hat E + E_G \end{Sb} ( (\hat E + E_F) \cdot D)[\hat E + E_G] - \sum_{(E \cdot \hat E) = 0} (E \cdot D)[E]. \] Letting $D \equiv -K$ and intersecting with $-K$ yields $24 = 4m - (e-(m+1))$ and so $e = 5m -23$, which combined with our previous equation yields $e=27$ and $m=10$. \end{pf} Note finally that $N(d;m_1, \dots , m_n)$ is invariant under permutations and Cremona transformations. The former is obvious and the latter follows from the fact that $k$ and the decomposition of a curve into sums of curves are invariant under Cremona transformations. Also of course the arithmetic genus of a class is invariant under symmetries and Cremona transformations. It is tempting to conjecture that the number $N(d;m_1, \dots , m_n)$ depends only on the genus. However a recent computation of A. Grassi \cite{grassi} shows that this is not the case in general for classes with arithmetic genus at least $2$.

9410

alg-geom/9410027

en

https://arxiv.org/abs/alg-geom/9410027

alg-geom/9410027

Heath Martin

Heath Martin and Juan Migliore

Submodules of the deficiency modules and an extension of Dubreil's Theorem

18 pages, LaTeX, version 2.09

In its most basic form, Dubreil's Theorem states that for an ideal $I$ defining a codimension $2$, arithmetically Cohen--Macaulay subscheme of projective $n$-space, the number of generators of $I$ is bounded above by the minimal degree of a minimal generator plus $1$. By introducing a new ideal $J$ which is the complete intersection of $n-1$ general linear forms, we are able to extend Dubreil's Theorem to an ideal $I$ defining a locally Cohen--Macaulay subscheme $V$ of any codimension. Our new bound involves the lengths of the Koszul homologies of the cohomology modules of $V$, with respect to the ideal $J$, and depends on a careful identification of the module $(I \cap J)/IJ$ in terms of the maps in the free resolution of $J$. As a corollary to this identification, we also give a new proof of a theorem of Serre which gives a necessary and sufficient condition to have the equality $I \cap J = IJ$ in the case where $I$ and $J$ define disjoint schemes in projective space.

alg-geom

\section{When does $I \cap J = IJ$?} Let $S= k[x_0, \dots, x_n]$ be a polynomial ring over the algebraically closed field $k$. Let $I$ and $J$ be ideals defining subschemes $V$ and $Y$, respectively, of the projective space $\Bbb P^n_k = {\Bbb P^n}$ over $k$. In particular, both $I$ and $J$ are homogeneous, saturated ideals. In this section, we will derive a relationship between the quotient module $(I \cap J)/IJ$ and the cohomology of $V$, when $V$ and $Y$ meet in the expected dimension. In general, if $V$ is a subscheme of ${\Bbb P^n}$, with saturated homogeneous defining ideal $I = I_V$, the cohomology modules of $V$ (or, less precisely, of $I$) are defined, for $i=0, \dots, n-1$, by $$ H^i_*({\cal I}_V) = H^i_*(V) = \bigoplus_j H^i({\Bbb P^n}, {\cal I}_V(j)), $$ where ${\cal I}_V = \widetilde{I_V}$ is the ideal sheaf of $V$. These are all graded $S$-modules. Moreover, $H^0_*(V) = I_V$ and $H^i_*(V) = 0$ for $i > \dim V + 1$. Usually, when $i = 1, \dots, \dim V$, we will call $H^i_*(V)$ a deficiency module. This name comes from the fact that the $H^i_*(V)$, $i = 1, \dots, \dim V$, measure the failure of $V$ to be an arithmetically Cohen--Macaulay subscheme, since they vanish whenever $V$ is aCM. We will also have need to use the cohomology of modules. If $M$ is a (finitely generated) $S$-module, let $\widetilde{M}$ be its sheafification. Then, exactly as in the case of ideal sheaves, we define the cohomology module of $\widetilde{M}$ to be $$ H^i_*(\widetilde{M}) = \bigoplus_j H^i({\Bbb P^n}, \widetilde{M}(j)). $$ These are again graded $S$-modules. We note here for future reference that the cohomology modules of $M$ are related to the local cohomology modules $H^i_{\goth m}(M)$ of $M$ with respect to the homogeneous maximal ideal ${\goth m}$ as follows: \begin{equation}\label{local-coh} 0 \rightarrow H^0_{\goth m}(M) \rightarrow M \rightarrow H^0_*(\widetilde{M}) \rightarrow H^1_{\goth m}(M) \rightarrow 0 \end{equation} $$ H^i_{\goth m}(M) \cong H^i_*(\widetilde{M})\mbox{\quad\quad for $i>1$.} $$ See \cite[Chapter 0]{SV:buchsbaum} for a good discussion of graded and local cohomology. In this paper, we will sometimes require subschemes of ${\Bbb P^n}$ to be locally Cohen--Macaulay and equidimensional. This is equivalent to saying that all the cohomology modules have finite length, except of course for the top cohomology $H^{d+1}_*(V)$, $d = \dim V$. By Serre's vanishing theorem, this is again equivalent to having $[H^i_*(V)]_j = 0$ for $j \ll 0$, and $1 \le i \le d$, since in any case the cohomology modules vanish in high degrees. Now, let $I$ and $J$ be as above, let $s = \mathop{\rm pd\,} J$ be the projective dimension of $J$, and write a minimal graded free resolution of $J$ as follows: \begin{equation} \begin{array}{ccccccccccccccccc} 0 & \rightarrow & F_s & \buildrel {\phi_s} \over \longrightarrow & \dots & {\buildrel {\phi_2} \over \rightarrow} & F_1& \buildrel {\phi_1} \over \longrightarrow & F_0 & \rightarrow & J & \rightarrow & 0. \end{array} \end{equation} where $F_j = \bigoplus_i S(-a_{ji})$ are free modules. For each $j = 1, \dots, s$, let $K_j$ be the $j$-th syzygy module, so that there are short exact sequences $$ 0 \rightarrow K_{j+1} \buildrel {\psi_{j+1}} \over \longrightarrow F_j \buildrel {\eta_j} \over \longrightarrow K_j \rightarrow 0, $$ where the maps $\psi_j$ and $\eta_j$ are the canonical inclusions and projections, respectively. Note that for $j = s$, we have $K_s = F_s$, $\psi_s = \phi_s$ and $\eta_s = id$. For $S$-modules $M$ and $N$, and a map $f : M \to N$, we denote by $f^i : H^i_*(\widetilde{M \otimes I}) \to H^i_*(\widetilde{N \otimes I})$ the map induced on cohomology by $f \otimes id : M \otimes I \to N \otimes I$. Our main technical result for this paper is the following Theorem. \begin{thm}\label{main:technical} Suppose the ideals $I$ and $J$ as above define disjoint subschemes $V$ and $Y$, respectively. Then for each $i \ge 1$, there are isomorphisms $$ \ker \psi_i^1 \cong \mathop{\rm Tor}\nolimits_i^S(S/I, S/J) $$ and, for each $i,j \ge 1$, a long exact sequence \begin{equation}\label{main:sequence} 0 \rightarrow \mathop{\rm im\,} \psi_{i+1}^j \rightarrow \ker \phi_i^j \rightarrow \ker \psi_i^j \rightarrow \ker \psi_{i+1}^{j+1} \rightarrow {\mathop{\rm coker\,}} \phi_i^j \rightarrow {\mathop{\rm coker\,}} \psi_i^j \rightarrow 0. \nonumber \end{equation} \end{thm} \begin{proof} We remark that for $i > s$, both statements are trivial, since then $\phi_i = \psi_i$ is the zero map. Also, if $j > \dim S/I$, then because $H^j_*(V) = 0$ again the second statement is trivial. Now, let $\mu : I \otimes J \to IJ$ be the natural surjection, and note that $\ker \mu \cong \mathop{\rm Tor}\nolimits_2^S(S/I, S/J)$. This follows, for instance, by tensoring $$ 0 \rightarrow I \rightarrow S \rightarrow S/I \rightarrow 0 $$ with $J$, comparing the resulting sequence with $$ 0 \rightarrow IJ \rightarrow J \rightarrow J/IJ \rightarrow 0 $$ via the multiplication map, and using that $\mathop{\rm Tor}\nolimits_1^S(S/I, J) \cong \mathop{\rm Tor}\nolimits_2^S(S/I, S/J)$. Note especially that $\ker \mu$ has finite length since it is annihilated by $I+J$. In particular, by sheafifying and taking cohomology of the short exact sequence $$ 0 \rightarrow \ker \mu \rightarrow I \otimes J \rightarrow IJ \rightarrow 0, $$ we see that $H^i_*(\widetilde{I \otimes J}) \cong H^i_*(\widetilde{IJ})$ for all $i \ge 0$. Next, using the functorial map $M \to H^0_*(\widetilde{M})$ for any $S$-module $M$, we get a commutative diagram \begin{equation}\label{eq:1} \begin{array}{ccccccccc} &&&& 0 && 0 \\ &&&& \downarrow && \downarrow \\ &&&& H^0_{\goth m}(I \otimes J) & \rightarrow & H^0_{\goth m}(IJ) \\ &&&& \downarrow && \downarrow \\ 0 &\rightarrow & \mathop{\rm Tor}\nolimits_2^S(S/I, S/J) & \rightarrow & I \otimes J & {\buildrel {\mu} \over {\longrightarrow}} & IJ & \rightarrow & 0 \\ &&&& \downarrow && \downarrow \\ && 0 & \rightarrow & H^0_*(\widetilde{I \otimes J}) &\rightarrow & H^0_*({\widetilde{IJ}}) & \rightarrow & 0 \\ &&&& \downarrow && \downarrow \\ &&&& H^1_{\goth m}(I \otimes J) & \rightarrow &H^1_{\goth m}(IJ) \\ &&&& \downarrow && \downarrow \\ &&&& 0 && 0 \end{array} \end{equation} But $IJ$ is an ideal, so $H^0_{\goth m}(IJ) = 0$. Hence the kernel of the map $I \otimes J \to H^0_*(\widetilde{I \otimes J})$ is $\mathop{\rm Tor}\nolimits_2^S(S/I, S/J)$. Now, with these preliminaries out of the way, we prove the isomorphisms by induction on $i$. For $i = 1$, tensor the exact sequence $$ 0 \rightarrow K_1 {\buildrel {\psi_1} \over \longrightarrow } F_0 \rightarrow J \rightarrow 0 $$ by $I$. This yields an exact sequence $$ 0 \rightarrow \mathop{\rm Tor}\nolimits_1^S(I, J) \rightarrow K_1 \otimes I {\buildrel {\psi_1 \otimes 1} \over {\hbox{$\hbox to .35in{\rightarrowfill}$} }} F_0 \otimes I \rightarrow J \otimes I \rightarrow 0. $$ Since $\mathop{\rm Tor}\nolimits_1^S(I, J) \cong \mathop{\rm Tor}\nolimits_3^S(S/I, S/J)$ is annihilated by $I+J$, in particular it has finite length. Thus, taking cohomology and comparing with the original sequence gives a diagram \begin{equation}\nonumber \setlength{\arraycolsep}{1pt} \begin{array}{ccccccccccccc} 0 & \rightarrow & \mathop{\rm Tor}\nolimits_3^S(S/I, S/J) & \rightarrow & K_1 \otimes I & \rightarrow & F_0 \otimes I & \rightarrow & J \otimes I & \rightarrow & 0 \\ &&&& \downarrow &&\downarrow&&\downarrow \\ && 0 & \rightarrow & H^0_*(\widetilde{K_1 \otimes I}) & \rightarrow & H^0_*({\widetilde{F_0 \otimes I}}) & \rightarrow & H^0_*(\widetilde{J \otimes I}) & \rightarrow & \ker \psi_1^1 & \rightarrow & 0. \end{array} \end{equation} Here, the middle vertical map is an isomorphism, since $I$ is saturated. Thus the snake lemma shows that there are exact sequences $$ 0 \rightarrow \mathop{\rm Tor}\nolimits_3^S(S/I, S/J) \rightarrow K_1 \otimes I \rightarrow H^0_*(\widetilde{K_1 \otimes I}) \rightarrow \mathop{\rm Tor}\nolimits_2^S(S/I, S/J) \rightarrow 0, $$ and $$ 0 \rightarrow \mathop{\rm Tor}\nolimits_2^S(S/I, S/J) \rightarrow J \otimes I \rightarrow H^0_*(\widetilde{J \otimes I}) \rightarrow \ker \psi_1^1 \rightarrow 0. $$ But from the above discussion, the last sequence implies the short exact sequence $$ 0 \rightarrow IJ \rightarrow H^0_*(\widetilde{IJ}) \rightarrow \ker \psi_1^1 \rightarrow 0. $$ Since $I$ and $J$ define disjoint varieties, we have $H^0_*(\widetilde{IJ}) = I \cap J$. Thus, the above sequence shows that $$ \ker \psi_1^1 \cong {{I \cap J} \over {IJ}} \cong \mathop{\rm Tor}\nolimits_1^S(S/I, S/J). $$ By induction, we may assume that $\ker \psi_i^1 \cong \mathop{\rm Tor}\nolimits_i^S(S/I, S/J)$, and that there is an exact sequence $$ 0 \rightarrow \mathop{\rm Tor}\nolimits_{i+2}^S(S/I, S/J) \rightarrow K_i \otimes I \rightarrow H^0_*(\widetilde{K_i \otimes I}) \rightarrow \mathop{\rm Tor}\nolimits_{i+1}^S(S/I, S/J) \rightarrow 0. $$ Tensoring the exact sequence $$ 0 \rightarrow K_{i+1} {\buildrel {\psi_{i+1}} \over \hbox{$\hbox to .35in{\rightarrowfill}$} } F_i \rightarrow K_i \rightarrow 0 $$ with $I$ yields $$ 0 \rightarrow \mathop{\rm Tor}\nolimits_1^S(K_i, I) \rightarrow K_{i+1} \otimes I \rightarrow F_i \otimes I \rightarrow K_i \otimes I \rightarrow 0. $$ Here, $\mathop{\rm Tor}\nolimits_1^S(K_i, I) \cong \mathop{\rm Tor}\nolimits_{i+1}^S(I, J) \cong \mathop{\rm Tor}\nolimits_{i+3}^S(S/I, S/J)$. In particular, it is finite length. Hence, taking cohomology and comparing yields a diagram \begin{equation}\nonumber \setlength{\arraycolsep}{1pt} \begin{array}{ccccccccccccc} 0 & \rightarrow & \mathop{\rm Tor}\nolimits_{i+3}^S(S/I, S/J) & \rightarrow & K_{i+1} \otimes I & \rightarrow & F_i \otimes I & \rightarrow & K_i \otimes I & \rightarrow & 0 \\ &&&& \downarrow &&\downarrow&&\downarrow \\ && 0 & \rightarrow & H^0_*(\widetilde{K_{i+1} \otimes I}) & \rightarrow & H^0_*({\widetilde{F_i \otimes I}}) & \rightarrow & H^0_*(\widetilde{K_i \otimes I}) & \rightarrow & \ker \psi_{i+1}^1 & \rightarrow & 0. \end{array} \end{equation} But by the inductive hypothesis, we know the kernel and cokernel of the right-hand vertical map. Thus the snake lemma implies that $$ \ker \psi_{i+1}^1 \cong \mathop{\rm Tor}\nolimits_{i+1}^S(S/I, S/J), $$ and that there is a long exact sequence $$ 0 \rightarrow \mathop{\rm Tor}\nolimits_{i+3}^S(S/I, S/J) \rightarrow K_{i+1} \otimes I \rightarrow H^0_*(\widetilde{K_{i+1} \otimes I}) \rightarrow \mathop{\rm Tor}\nolimits_{i+2}^S(S/I, S/J) \rightarrow 0, $$ which finishes the proof of the isomorphisms. Next, we show that the long exact sequence exists. Fix an $i \ge 1$. Thus there is an exact sequence $$ 0 \rightarrow K_{i+1} {\buildrel {\psi_{i+1}} \over \hbox{$\hbox to .35in{\rightarrowfill}$} } F_i {\buildrel {\eta_{i}} \over \hbox{$\hbox to .35in{\rightarrowfill}$} } K_i \rightarrow 0. $$ Tensor this sequence with $I$, to obtain $$ 0 \rightarrow \mathop{\rm Tor}\nolimits_1^S(K_i, I) \rightarrow K_{i+1} \otimes I {\buildrel {\psi_{i+1} \otimes 1} \over \hbox{$\hbox to .35in{\rightarrowfill}$} } F_i \otimes I {\buildrel {\eta_{i} \otimes 1} \over \hbox{$\hbox to .35in{\rightarrowfill}$} } K_i \otimes I \rightarrow 0, $$ and note that $\mathop{\rm Tor}\nolimits_1^S(K_i , I) = \mathop{\rm Tor}\nolimits_{i+3}^S(S/I, S/J)$, has finite length. Thus, after sheafifying and taking cohomology, at the $j$-th stage this yields isomorphisms \begin{eqnarray*} \ker \eta_i^j &\cong& \mathop{\rm im\,} \psi_{i+1}^j \\ {\mathop{\rm coker\,}} \eta_i^j &\cong& \ker \psi_{i+1}^{j+1}. \end{eqnarray*} Now, using the functoriality of tensor products and of cohomology, we obtain a commutative square \begin{equation} \begin{array}{ccc} H^j_*(\widetilde{F_i \otimes I}) & \widetilde{\hbox{$\hbox to .35in{\rightarrowfill}$} } & H^j_*(\widetilde{F_i \otimes I}) \\ \mapdown{\eta_i^j} && \mapdown{\phi_i^j} \\ H^j_*(\widetilde{K_i \otimes I}) & {\buildrel {\psi_i^j} \over {\longrightarrow}} & H^j_*(\widetilde{F_{i-1} \otimes I}). \end{array} \end{equation} Applying the snake lemma to the columns, and using the two isomorphisms above shows that there is a sequence $$ 0 \rightarrow \mathop{\rm im\,} \psi_{i+1}^j \rightarrow \ker \phi_i^j \rightarrow \ker \psi_i^j \rightarrow \ker \psi_{i+1}^{j+1} \rightarrow {\mathop{\rm coker\,}} \phi_i^j \rightarrow {\mathop{\rm coker\,}} \psi_i^j \rightarrow 0, $$ which is what we claimed. \end{proof} We note that this greatly extends the arguments in \cite[Section 1]{Mig:submodules}. The situation there was much simpler in that it only considered the case that $J$ was codimension $2$ and arithmetically Cohen--Macaulay (so most of the terms in the sequence~(\ref{main:sequence}) vanish), and only the case $i=j=1$ was studied, so it focused on $\ker \psi_1^1 = (I \cap J)/IJ$. Our extension makes no assumptions on the Cohen--Macaulayness of $J$, nor on its codimension. Of course, our conclusion is much more complicated, reflecting the fact that so much information is encoded in the free resolution of $J$. As an application of this technical result, in the next theorem we give a proof of a statement due to Serre on when there is an equality $I \cap J = IJ$. \begin{thm}{\rm \cite[Corollaire, p. 143]{Serre}} Suppose the ideals $I$ and $J$ define disjoint subschemes of ${\Bbb P}^n$. Then $IJ = I \cap J$ if and only if $\dim S/I + \dim S/J = \dim S$ and both $S/I$ and $S/J$ are Cohen--Macaulay. \end{thm} \begin{proof} Suppose first that $S/I$ and $S/J$ are Cohen--Macaulay with $\dim S/I + \dim S/J = \dim S$. Then by the Auslander--Buchsbaum formula, $s = \mathop{\rm pd\,} J = \dim S/I - 1$, and moreover $H^i_*(\widetilde{I}) = 0$ for $i = 1, \dots, s$. In particular, $\ker \phi_i^i = 0 = {\mathop{\rm coker\,}} \phi_i^i$ for $i = 1, \dots, s$. Since $\psi_s = \phi_s$, by reverse induction the sequence (\ref{main:sequence}) with $j=i$ shows that $\ker \psi_i^i = 0$ for $i = 1, \dots, s$. Thus $(I \cap J)/IJ = \ker \psi_1^1 = 0$. Conversely, since the subschemes defined by $I$ and $J$ are disjoint, we have $\dim S/I + \dim S/J \le \dim S$ Hence \begin{equation}\label{ineqs} \dim S/I \le \dim S - \dim S/J \leq \dim S - \mathop{\rm depth\,} S/J = s+1 \end{equation} where the last equality is by the Auslander--Buchsbaum formula. Now, if $IJ = I \cap J$, then $\mathop{\rm Tor}\nolimits_1^S(S/I, S/J) = 0$, and so by rigidity, $\mathop{\rm Tor}\nolimits_i^S(S/I, S/J) = 0$ for $i \ge 1$. Hence the isomorphisms of Theorem~\ref{main:technical} show that $\ker \phi_s^1 = 0$. But this implies that $H^1_*(I) = 0$. Thus also $\ker \phi_i^1 = 0 = {\mathop{\rm coker\,}} \phi_i^1$ for all $i = 1, \dots, s$, and since $\ker \psi_1^1 = (I \cap J)/IJ = 0$, the exact sequence (\ref{main:sequence}) with $j = 1$, $i = 1, \dots, s$ implies that $\ker \psi^2_i = 0$, for $i = 2, \dots, s$. Since $\psi_s = \phi_s$, this shows $\ker \psi_s^2 = 0$ and hence also $H^2_*(\widetilde{I}) = 0$. Continuing inductively, we see that $\ker \psi_i^j = 0$ for all $i$ and $j$ with $i \ge j$. In particular, since $\psi_s = \phi_s$ we get that $H^j_*(\widetilde{I}) = 0$ for $j = 1, \dots, s$. Let $d = \dim S/I$. We have seen that $d-1 \le s$. If this inequality were strict, then in particular $H^d_*(\widetilde{I}) = 0$, which is impossible. Hence we have $d-1 = s$ and $H^j_*(\widetilde{I}) = 0$ for $j = 1, \dots, d-1$; that is $S/I$ is Cohen--Macaulay. But furthermore, each of the inequalities in (\ref{ineqs}) is actually in equality. This shows both that $\dim S/J = \mathop{\rm depth\,} S/J$, i.e., $S/J$ is Cohen--Macaulay, and that $\dim S/I + \dim S/J = \dim S$, which finishes the proof. \end{proof} \section{An Extension of Dubreil's Theorem} In this section, we wish to use the results of Section 1 to extend a theorem of Dubreil on the number of generators of certain ideals. Let $\nu(I)$ denote the minimal number of generators of $I$, and let $\alpha(I)$ denote the least degree of a minimal generator. In its most basic form, Dubreil's Theorem states: \begin{thm} Let $I$ be a homogeneous ideal of $k[x,y]$. Then $\nu(I) \le \alpha(I) + 1$. \end{thm} See \cite{DGM:Dubreil} for a proof of this; note however that it is essentially a consequence of the Hilbert--Burch theorem. Dubreil's theorem is easily extended to the case that $I$ is a codimension $2$ arithmetically Cohen--Macaulay ideal in any polynomial ring $k[x_0, \dots, x_n]$; again, see \cite{DGM:Dubreil} for the details. On the other hand, when $I$ is not arithmetically Cohen--Macaulay, or when $I$ is not codimension $2$, not much is known in this direction. However, in the case of an ideal defining a subscheme of $\Bbb P^3$, the following theorem of Migliore shows that the general case will involve the cohomology of the subscheme. \begin{thm}{\rm \cite[Corollary 3.3]{Mig:submodules}} Suppose $I$ defines a subscheme $V$ of $\Bbb P^3$, of codimension at least $2$. Let $A = (L_1, L_2)$ be the complete intersection of two general linear forms, and let $K_A$ denote the submodule of $H^1_*(V)$ annihilated by $A$. Then $$ \nu(I) \le \alpha(I) + 1 + \nu(K_A). $$ \end{thm} We note in particular that this formula is valid both for the case that $V$ is codimension $2$, not necessarily arithmetically Cohen--Macaulay, and the case that $V$ is codimension $3$. In the latter case, even though $H^1_*(V)$ is not finitely generated, we still have that at least $K_A$ is finitely generated (see \cite[Theorem 2.1]{Mig:submodules} or our Lemma~\ref{finite-length}), so the theorem still has useful content. Furthermore, in case $I$ defines an arithmetically Buchsbaum curve, so that $H^1_*(V)$ is a $k$-vector space, $K_A = H^1_*(V)$ and $\nu(K_A) = \dim_k H^1_*(V)$. Thus the Buchsbaum case is particularly easy to calculate in examples. In this special case, the bound can be obtained from \cite{Amasaki:structure}. In this section, we will give a generalization of Dubreil's Theorem to ideals defining subschemes of ${\Bbb P^n}$ of arbitrary codimension. As an easy consequence, we recover by our methods the above two theorems, and also part of a result of Chang, \cite{Chang:charac}, on the number of generators of an ideal defining a Buchsbaum codimension $2$ subscheme of ${\Bbb P^n}$, which again seems to be the best understood case. Our generalization is a corollary to the technical statement Theorem~\ref{main:technical} in Section~1, underscoring the usefulness of identifying the difference between intersections and products. Our generalization will be based on the Koszul homologies of the cohomology modules of an ideal $I$ defining a subscheme of projective space. As such, we will make some general remarks concerning Koszul homology. These comments are basic, and can be found, for instance, in \cite{Mat}. We first set the notation. If $R$ is a ring, and $y_1, \dots, y_s$ elements of $R$, we let ${\Bbb K}((y_1, \dots, y_s);R)$ denote the Koszul complex with respect to $y_1, \dots, y_s$. If $M$ is an $R$-module, put ${\Bbb K}((y_1, \dots, y_s); M) = {\Bbb K}((y_1, \dots, y_s);R) \otimes M$, the Koszul complex on $M$ with respect to $y_1, \dots, y_s$. Set ${\Bbb H}_i((y_1, \dots, y_s); M)$ to be the $i$-th homology module of ${\Bbb K}((y_1, \dots, y_s); M)$; this is the Koszul homology on $M$ with respect to $y_1, \dots, y_s$. We will need the following facts: \begin{remark}\label{Koszul} \begin{enumerate} \item If $y_1, \dots, y_n$ forms a regular sequence on $M$, then ${\Bbb K}((y_1, \dots, y_n); M)$ is acyclic. \item Let $J = (y_1, \dots, y_s)$. Then for each $i = 0, \dots, s$, $J \subseteq \ann {\Bbb H}_i((y_1, \dots, y_s);M)$. \item Suppose there is a short exact sequence of $R$-modules $$ 0 \rightarrow M_1 \rightarrow M_2 \rightarrow M_3 \rightarrow 0. $$ Then there is a long exact sequence on Koszul homology $$ \begin{array}{c} \cdots \rightarrow {\Bbb H}_{i+1}((y_1, \dots, y_n); M_3) \rightarrow {\Bbb H}_i((y_1, \dots, y_n); M_1) \hspace{1.5in} \\ \hspace{1.5in} \rightarrow {\Bbb H}_i((y_1, \dots, y_n); M_2) \rightarrow {\Bbb H}_i((y_1, \dots, y_n); M_3) \rightarrow \cdots. \end{array} $$ \item For each $i = 0, \dots, s$, there is an isomorphism $$ {\Bbb H}_i((0, y_2, \dots, y_s); M) \cong {\Bbb H}_i((y_2, \dots, y_s); M) \oplus {\Bbb H}_{i-1}((y_2, \dots, y_s); M). $$ \end{enumerate} \end{remark} Throughout this section, let $I$ be the saturated defining ideal of a locally Cohen--Macaulay, equidimensional subscheme $V$ of ${\Bbb P}^n$; put $d=\dim V$. Let $J = (L_1, \dots, L_{n-1})$ be the complete intersection of $n-1$ general linear forms. In particular, ${\Bbb K}((L_1, \dots, L_{n-1}); R)$ is a free resolution of $S/J$. Recall that the highest non-zero cohomology module $H^{d+1}_*(\widetilde{I})$ is never finitely generated, when $d \ge 0$. In the next result, we show that nonetheless, most of its Koszul homologies are finitely generated. As general notation, for a module $M$ and an ideal $A$, let $M_A$ denote the submodule of $M$ which is annihilated by $A$; that is, $M_A = (0 :_M A)$. \begin{prop}\label{finite-length} Let $I$ and $J = (L_1, \dots, L_{n-1})$ be as above. Then the Koszul homology ${\Bbb H}_i((L_1, \dots, L_{n-1});H^{d+1}_*(\widetilde{I}))$ is finitely generated for each $i \ge d+2$. In particular, if $d \ge 0$, the Koszul homology has finite length. \end{prop} \begin{proof} By changing coordinates if necessary, we may, without loss of generality, assume that $L_i = x_{i-1}.$ We will prove the proposition by using induction on $d$. If $d = -1$, i.e., $I$ defines the empty subscheme, then $H^0_*(\widetilde{I}) = I$ is already finitely generated, so each of its Koszul homologies is also finitely generated. Suppose $d \ge 0$. The exact sequence $$ 0 \rightarrow I {\buildrel {x_0} \over \longrightarrow} I \rightarrow I/x_0 I \rightarrow 0 $$ induces the long exact sequence on cohomology \begin{equation}\label{endcohomology} \begin{array}{rcl} 0 \rightarrow A \rightarrow H^{d}_*(\widetilde{I/x_0 I}) & \hbox{$\hbox to .45in{\rightarrowfill}$} & H^{d+1}_*(\tilde{I}) {\buildrel {x_0} \over \longrightarrow} H^{d+1}_*(\tilde{I}) \rightarrow 0 \\ & \searrow \hfill \nearrow \\ & H^{d+1}_*(\tilde{I})_{(x_0)} \\ & \nearrow \hfill \searrow \\ 0 && 0 \end{array} \end{equation} where $A = H^{d}_*(\widetilde{I}) / x_0 H^{d}_*(\widetilde{I})$. Note in particular that $A$ is finitely generated, since it is the quotient of two finitely generated modules. {}From the right-hand part of this sequence, we obtain a long exact sequence of Koszul homology (see Remark~\ref{Koszul}) $$ \begin{array}{c} \cdots {\buildrel {x_0} \over \longrightarrow} {\Bbb H}_{i+1}((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})) \rightarrow {\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)}) \hspace*{1in}\\ \hspace*{1in}\rightarrow {\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})) {\buildrel {x_0} \over \longrightarrow} {\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})) \rightarrow \cdots. \end{array} $$ But for any module $M$, ${\Bbb H}_i((x_0, \dots, x_{n-2}); M)$ is annihilated by $x_0$, so the long exact sequence breaks into short exact sequences \begin{center} \makebox[\textwidth][l]{$\qquad 0 \longrightarrow {\Bbb H}_{i+1}((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})) \longrightarrow {\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)}) \longrightarrow$} \makebox[\textwidth][r]{${\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})) \longrightarrow 0.\qquad$} \end{center} Thus to show that ${\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I}))$ is finitely generated for $i \ge d+2$, it will suffice to show that ${\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)})$ is finitely generated for $i \ge d+2$. However, $x_0$ kills $H^{d+1}_*(\widetilde{I})_{(x_0)}$, and so {\setlength{\arraycolsep}{0pt} \begin{eqnarray*} {\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)}) =&& {\Bbb H}_i((0, x_1, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)}) \\ = {\Bbb H}_i((x_1, \dots, &&x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)}) \oplus {\Bbb H}_{i-1}((x_1, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)}), \end{eqnarray*}} and we can calculate this over $R = S/(x_0) = k[x_1, \dots, x_n]$. Now, the left-hand part of (\ref{endcohomology}) yields a long exact sequence of Koszul homology \begin{equation}\label{sequence1} \begin{array}{c} \cdots \rightarrow {\Bbb H}_{j+1}((x_1, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)}) \rightarrow {\Bbb H}_j((x_1, \dots, x_{n-2}); A) \rightarrow \hspace*{1in} \\ \hspace*{1in} {\Bbb H}_j((x_1, \dots, x_{n-2}); H^d_*(\widetilde{I/x_0 I})) \rightarrow {\Bbb H}_j((x_1, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)}) \rightarrow\cdots. \end{array} \end{equation} Here, the saturation of $I/x_0 I \subseteq R$ defines a subscheme $\overline{V}$ of ${\Bbb P}^{n-1}$, with $\dim \overline{V} = d-1$. Thus, by the induction hypothesis, ${\Bbb H}_j(H^d_*(\widetilde{I/ x_0 I}))$ is finitely generated for each $j \ge d+1$. In particular, since $i \ge d+2$, this is true for $j = i, i-1$. Also, since $A$ is finitely generated, all of its Koszul homologies are also finitely generated. But then (\ref{sequence1}) shows that both ${\Bbb H}_i((x_1, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)})$ and ${\Bbb H}_{i-1}((x_1, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)})$ are finitely generated. This implies that ${\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I}))$ is finitely generated, which finishes the proof of the first statement. For the second statement, recall that the Serre vanishing theorem says that $H^{d+1}(\widetilde{I}(t))$ vanishes for large $t$, and hence the Koszul homology ${\Bbb H}_i((L_1, \dots, L_{n-1}); H^{d+1}_*(\widetilde{I}))$ also vanishes in high degrees. But since it is finitely generated, it must also vanish in low degrees, and we can conclude that it must have finite length. \end{proof} \begin{thm}\label{extended-Dubreil} Suppose $I$ defines a locally Cohen--Macaulay, equidimensional subscheme $V$ of dimension $d$ of ${\Bbb P^n}$. Let $J = (L_1, \dots, L_{n-1})$ be generated by $n-1$ general linear forms, and let $\Bbb H_i((L_1, \dots, L_{n-1}); H^j_*(\widetilde{I}))$ be the Koszul homologies of $H^j_*(\widetilde{I})$ with respect to $J$. Then \begin{equation}\label{dubreil:formula} \nu(I) \le \alpha(I) + 1 + \sum_{i=1}^{n-2} \dim \Bbb H_{i+1}((L_1, \dots, L_{n-1});H^i_*(V)). \end{equation} \end{thm} \begin{proof} Note that we have an exact sequence $$ 0 \rightarrow {{I \cap J} \over {IJ}} \rightarrow {I \over {IJ}} \rightarrow {{I+J} \over J} \rightarrow 0. $$ Hence, $\nu(I) = \nu(I/IJ) \le \nu({{I+J}\over J}) + \nu({{I \cap J} \over IJ})$. Now, since $(I+J)/J$ is an ideal in $S/J$, which is a polynomial ring in two variables, Dubreil's Theorem applies, and says that $\nu((I+J)/J) \le \alpha((I+J)/J) + 1$. Since $J$ is generated by general linear forms, $\alpha((I+J)/J) = \alpha(I)$, and so $\nu((I+J)/J) \le \alpha(I) + 1$. Thus it only remains to estimate $\nu((I \cap J)/IJ)$. Note that $I$ and $J$ define disjoint schemes, since $J$ is generated by general linear forms, and that the Koszul complex ${\Bbb K}((L_1, \dots, L_{n-1});S)$ is a free resolution of $S/J$. In particular, the isomorphisms and exact sequences of Theorem~\ref{main:technical} hold. For each $i = 1, \dots, n-1$, let $P_i = \ker \phi_i^i/\mathop{\rm im\,} \psi_{i+1}^i$. Note then, that $\Bbb H_{i+1}(H^i_*(V))$ naturally maps surjectively onto $P_i$, since $\mathop{\rm im\,} \psi_{i+1}^i$ contains $\mathop{\rm im\,} \phi_{i+1}^i$. Thus, in particular, $\dim P_i \le \dim \Bbb H_{i+1}(H^i_*(V))$. Hence it follows from Theorem~\ref{main:technical} that \begin{eqnarray} \nu((I \cap J) / IJ) \le \dim ((I \cap J)/IJ) &\le& \dim \Bbb H_2(H^1_*(V)) + \dim \ker\psi_2^2 \nonumber \\ &\le& \dim \Bbb H_2(H^1_*(V)) + \dim \Bbb H_3(H^2_*(V)) + \dim \ker \psi_3^3 \nonumber \\ &\vdots & \nonumber\\ &\le& \sum_{i=1}^{n-2} \dim \Bbb H_{i+1}(H^i_*(V)). \nonumber \end{eqnarray} \end{proof} \begin{remark} Since $H^i_*(V) = 0$ whenever $i > \dim V + 1$, many of the terms in the formula~(\ref{dubreil:formula}) vanish. For instance, if $V$ is a curve in ${\Bbb P}^5$, there are only two terms coming from the cohomology of $V$. \end{remark} \begin{remark} In general, for a finite length graded module, $\nu(M)$ is much less than $\dim M$, and so we would like to be able to replace ``$\dim$'' by ``$\nu$'' throughout in the above formula. However, counting minimal generators is much more difficult in general than counting vector space dimensions. \end{remark} One important case in which we can replace ``$\dim$'' by ``$\nu$'' is when all but the top cohomology of $V$ is annihilated by the maximal ideal. Recall the definition: \begin{definition} A subscheme $V$ of ${\Bbb P^n}$ of dimension $d$ is said to be {\em arithmetically Buchsbaum} if $H^i_*(V)$ is annihilated by the maximal ideal for each $i=1, \dots, d$, and if for each general linear subspace $Y$ of ${\Bbb P^n}$, the cohomology $H^i_*(V \cap Y)$ is annihilated by the maximal ideal for $i = 1, \dots, \dim V \cap Y$. \end{definition} Note that the condition on linear subspaces of $V$ is required, for there are examples of subschemes $V$ whose cohomology is annihilated by the maximal ideal which have hypersurface sections whose cohomology is not annihilated by the maximal ideal; see for instance, \cite{Miy:graded}. In general, we will not require the full strength of this definition, only that the cohomologies are annihilated by the maximal ideal. Such subschemes are called {\em quasi--Buchsbaum}. The next corollary was obtained for the case $n=3$ in \cite{Mig:submodules}, and a somewhat better bound is stated in \cite{Chang:charac} for arithmetically Buchsbaum subschemes of ${\Bbb P}^n$. \begin{cor} Let $I$ define a codimension $2$ subscheme of ${\Bbb P^n}$ which is quasi-Buchsbaum. Then $$ \nu(I) \le \alpha(I) + 1 + \sum_{i=1}^{n-2} {{n-1} \choose {i+1}} \dim_k H^i_*(V). $$ \end{cor} \begin{proof} We only have to note that since $H^i_*(V)$ is annihilated by the maximal ideal for $i = 1, \dots, n-1$, and since every non-zero entry in a matrix representation for $\phi_i$ is a linear form, then for each $i = 1, \dots, n-1$, $\Bbb H_{i+1}(H^i_*(V)) = \ker \phi_i^i$ is a direct sum of ${{n-1} \choose {i+1}}$ copies of (twists of) $H^i_*(V)$. \end{proof} More generally, we can apply the same kind of analysis to quasi-Buchsbaum subschemes of arbitrary codimension, except that we now have to consider the top cohomology as well. \begin{cor}\label{number} Let $V$ be a $d$-dimensional quasi-Buchsbaum subscheme of ${\Bbb P^n}$, defined by the saturated ideal $I$. Let $J = (L_1, \dots, L_{n-1})$ be generated by $n-1$ general linear forms, and let $\Bbb H_i(H^j_*(V))$ be the Koszul homologies of $H^j_*(V)$ with respect to $J$. Then $$ \nu(I) \le \alpha(I) + 1 + \sum_{i=1}^{d} {{n-1} \choose {i+1}} \dim_k H^i_*(V) + \dim \Bbb H_{d+2}(H^{d+1}_*(V)). $$ \end{cor} \begin{proof} Again, since $H^i_*(V)$ is annihilated by every linear form for $i = 1, \dots, d$, then $\Bbb H_{i+1}(H^i_*(V))$ is just a direct sum of ${{n-1} \choose {i+1}}$ copies of $H^i_*(V)$. \end{proof} \begin{remark} By Lemma~\ref{finite-length}, even though $H^{d+1}_*(V)$ does not have finite length, the Koszul homology $\Bbb H_{d+2}(H^{d+1}_*(V))$ does have finite length, and so this corollary really does give a finite bound on the number of generators of $I$. \end{remark} \begin{example} Unfortunately, the bound in Theorem~\ref{extended-Dubreil} does not seem to be very sharp. For example, in ${\Bbb P}^4$, let $V$ be the union of a conic and a line not meeting the plane of the conic. Then $V$ is arithmetically Buchsbaum, and $\dim_k H^1_*(V) = 1$. Also, $\alpha(I_V) = 2$, and $\nu(I_V) = 7$. However, if $J$ is generated by three general linear forms, then $H^2_*(V)_J$ is at least $2$, as can be seen, for instance, by a calculation using the inductive procedure in Proposition~\ref{finite-length}. So $\alpha(I_V) + 1 + 3\dim_k H^1_*(V) + \dim H^2_*(V)_J \geq 8$, but $\nu(I_V) = 7$. \end{example} \begin{remark} During the final preparation of this paper, the authors received the preprint \cite{Hoa} of Hoa, which contains bounds on the number of generators of an ideal based in part on the cohomology of the ideal, but involving different invariants of the ideal than our bounds. Neither Hoa's bounds nor our bounds seem to be particularly sharp in general. \end{remark} \section{On the Least Degree of Surfaces Containing Certain Buchsbaum Subschemes} In this section, we want to use the bound given in Section~2 to extend a result of Amasaki on the minimal degree of the minimal generators of an ideal $I$ defining a codimension $2$ Buchsbaum subscheme of ${\Bbb P^n}$. In \cite{Amasaki:structure}, Amasaki showed that if $C$ is a Buchsbaum curve in $\Bbb P^3$, and if $N = \dim_k H^1_*(C)$ is the Buchsbaum invariant of $C$, then $\alpha(I) \ge 2N$. A different proof was subsequently given in \cite{GM:generators} based on combining the upper bound estimate for $\nu(I)$ of Corollary~\ref{number} in the case of curves in $\Bbb P^3$ together with a lower bound estimate coming from a determination of the free resolution of the ideal $I$ from a free resolution of $H^1_*(C)$. Also, Chang extended Amasaki's bound to a codimension $2$ Buchsbaum subscheme of any ${\Bbb P^n}$ in \cite{Chang:charac}, based on a structure theorem for the locally free resolution of the ideal sheaf associated to the subscheme. Here, we would like to use our methods to give a different proof of Amasaki's bound for certain codimension $2$ subschemes of ${\Bbb P^n}$. Specifically, we will give a lower bound for $\alpha(I)$ in terms of $H^1_*(V)$ for a codimension $2$ subscheme of $V$ for which $H^1_*(V)$ is annihilated by the maximal ideal, and $H^i_*(V) = 0$ for $i = 2, \dots, \dim V$. Note that these quasi-Buchsbaum schemes are in fact Buchsbaum, since if $H$ is a general hyperplane defined by a linear form $L$, the from the standard exact sequence $$ 0 \rightarrow I {\buildrel {\times L} \over {\longrightarrow}} I \rightarrow I/LI \rightarrow 0, $$ it is easy to see that $H^i_*(V \cap H) \cong H^i_*(V)$ for $1 \le i \le \dim V - 1$. Our method of proof will be to follow the lines of \cite{GM:generators}. That is, we will combine our upper bound estimate of $\nu(I)$ with a lower bound estimate for $\nu(I)$ based on the free resolution of $I$. We begin with an extension of a result in \cite{Rao}. \begin{prop} Suppose $V$ is a codimension $2$ subscheme of ${\Bbb P^n}$, such that $H^i_*(V) = 0$ for all $i = 2, \dots, \dim V$. Let $$ 0 \rightarrow L_{n+1} {\buildrel {\sigma_{n+1}} \over \longrightarrow} L_n {\buildrel {\sigma_n} \over \longrightarrow} L_{n-1} \rightarrow \dots {\buildrel {\sigma_1} \over \longrightarrow} L_0 \rightarrow H^1_*(V) \rightarrow 0 $$ be the minimal free resolution of the finite length module $H^1_*(V)$. Then the saturated defining ideal $I = I_V$ of $V$ has a minimal free resolution $$ 0 \rightarrow L_{n+1} {\buildrel {\sigma_{n+1}} \over \longrightarrow} L_n {\buildrel {\sigma_{n}} \over \longrightarrow} L_{n-1} {\buildrel_{\sigma_{n-1}} \over \longrightarrow} \dots {\buildrel {\sigma_4} \over \longrightarrow} L_4 {\buildrel {{[\sigma_3\; 0]}} \over \hbox{$\hbox to .35in{\rightarrowfill}$} } L_3 \oplus \bigoplus_{i=1}^r S(-b_i) \rightarrow \oplus_{j=1}^p S(-a_j) \rightarrow I \rightarrow 0, $$ for some $r \ge 0$, and $p = \nu(I)$. \end{prop} \begin{proof} Write a minimal free resolution of $S/I$ as follows: $$ 0 \rightarrow F_n {\buildrel {\phi_n} \over \longrightarrow} F_{n-1} {\buildrel {\phi_{n-1}} \over \longrightarrow} \dots {\buildrel {\phi_{2}} \over \longrightarrow} F_1 \rightarrow S \rightarrow S/I \rightarrow 0, $$ and let $E_i$ be the $i$-th syzygy module. Sheafifying and dualizing the short exact sequence $$ 0 \rightarrow F_n {\buildrel {\phi_{n-1}} \over \longrightarrow} F_{n-1} \rightarrow E_{n-1} \rightarrow 0 $$ yields the exact sequence $$ 0 \rightarrow {\cal E}_{n-1}^\vee \rightarrow {\cal F}_{n-1}^\vee {\buildrel {\phi_{n-1}^\vee} \over \hbox{$\hbox to .35in{\rightarrowfill}$} } {\cal F}_n^\vee \rightarrow 0. $$ Taking cohomology then gives a sequence $$ 0 \rightarrow H^0_*({\cal E}_{n-1}^\vee) \rightarrow F_{n-1}^\vee {\buildrel {\phi_{n-1}^\vee} \over \longrightarrow} F_n^\vee \rightarrow H^1_*({\cal E}_{n-1}^\vee) \rightarrow 0. $$ Note, though, that $H^1_*({\cal E}_{n-1}^\vee) \cong \mathop{\rm Ext}\nolimits_S^{n+1}(H^1_*(V), S))$. Next, for each $i = 2, \dots, n-2$, consider the sequence $$ 0 \rightarrow E_{i+1} \rightarrow F_{i} \rightarrow E_{i} \rightarrow 0. $$ Sheafifying, dualizing and taking cohomology gives an exact sequence $$ 0 \rightarrow H^0_*({\cal E}_i^\vee) \rightarrow F_{i}^\vee \rightarrow H^0({\cal E}_{i+1}^\vee) \rightarrow H^1_*({\cal E}_i^\vee) \rightarrow 0. $$ But $H^1_*({\cal E}_i^\vee) = \mathop{\rm Ext}\nolimits_S^{n+1}(H^{n-i}_*(V), S) = 0$, by assumption. Hence, we can paste together all these exact sequences to get a long exact sequence $$ F_2^\vee {\buildrel {\phi_{3}^\vee} \over \hbox{$\hbox to .35in{\rightarrowfill}$} } F_3^\vee {\buildrel {\phi_{4}^\vee} \over \hbox{$\hbox to .35in{\rightarrowfill}$} } \dots {\buildrel {\phi_{n}^\vee} \over \hbox{$\hbox to .35in{\rightarrowfill}$} } \mathop{\rm Ext}\nolimits_S^{n+1}(H^1_*(V), S) \rightarrow 0. $$ However, a minimal free resolution of $\mathop{\rm Ext}\nolimits_S^{n+1}(H^1_*(V), S)$ is given by just dualizing the resolution of $H^1_*(V)$, and so we see that $F_i = L_{i+1}$ and $\phi_i = \sigma_{i+1}$ for $i = 3, \dots, n$, and $F_2 = L_3 \oplus \bigoplus_{i=1}^r S$ and $\phi_2 = [\sigma_3\; 0]$, for some $r \ge 0$. This finishes the proof. \end{proof} \begin{cor} Let $V$ be as in the previous proposition, and let $I$ be its saturated defining ideal. Then $\nu(I) \ge 1 + \sum_{i=3}^{n+1} (-1)^i \mathop{\rm rank\,} L_i$. \end{cor} \begin{proof} With the notation as in the statement of the Proposition, we have $$ \nu(I) = p = 1 + \mathop{\rm rank\,} L_3 + r -\mathop{\rm rank\,} L_4 + \mathop{\rm rank\,} L_5 + \dots \ge 1 + \sum_{i=3}^{n+1} (-1)^i \mathop{\rm rank\,} L_i. $$ \end{proof} \begin{cor} In addition to the assumptions of the Proposition, suppose that $H^1_*(V)$ is annihilated by the maximal ideal. Let $N = \dim_k H^1_*(V)$. Then $\alpha(I) \ge (n-2)N$. \end{cor} \begin{proof} A minimal free resolution of $H^1_*(V)$ is just a direct sum of $N$ copies of the Koszul complex resolving $k = S/{\goth m}$. Thus, $\mathop{\rm rank\,} L_i = N{{n+1} \choose i}$. By the previous Corollary and Corollary~\ref{number}, we have $$ 1 + \sum_{i=3}^{n+1} (-1)^i N{{n+1} \choose i} \le 1 + \alpha(I) + (n+1)N. $$ A simple arithmetic calculation then reduces this to $(n-2)N \le \alpha(I)$, as claimed. \end{proof}

9410

alg-geom/9410031

en

https://arxiv.org/abs/alg-geom/9410031

alg-geom/9410031

Robert Guralnick, David Jaffe, Wayne Raskind, and Roger Wiegand

On the Picard group: torsion and the kernel

27 pages, AMS-LaTeX

For a homomorphism f: A --> B of commutative rings, let D(A,B) denote Ker[Pic(A) --> Pic(B)]. Let k be a field and assume that A is a f.g. k-algebra. We prove a number of finiteness results for D(A,B). Here are four of them. 1: Suppose B is a f.g. and faithfully flat A-algebra which is geometrically integral over k. If k is perfect, we find that D(A,B) is f.g. (In positive characteristic, we need resolution of singularities to prove this.) For an arbitrary field k of positive characteristic p, we find that modulo p-power torsion, D(A,B) is f.g. 2: Suppose B = A tensor k^sep. We find that D(A,B) is finite. 3: Suppose B = A tensor L, where L is a finite, purely inseparable extension. We give examples to show that D(A,B) may be infinite. 4: Assuming resolution of singularities, we show that if K/k is any algebraic extension, there is a finite extension E/k contained in K/k such that D(A tensor E,A tensor K) is trivial. The remaining results are absolute finiteness results for Pic(A). 5: For every n prime to char(k), Pic(A) has only finitely many elements of order n. 6: Structure theorems are given for Pic(A), in the case where k is absolutely f.g. All of these results are proved in a more general form, valid for schemes. Hard copy is available from the authors.

alg-geom

\section{The kernel under a separable extension} We start by recalling some material on Galois actions, leading up to an application of the Hochschild-Serre spectral sequence to the computation of $\operatorname{Pic}$. This material is more or less standard, but it is not available in the literature in quite the form we need. In particular, we want the statement of \pref{the-kernel} to be free of noetherian hypotheses. Later (see \ref{belch}) this will be important, because a noetherian scheme $X$ may have $\Gamma(X)$ non-noetherian, and our proofs about $\operatorname{Pic}(X)$ depend on understanding $\operatorname{Pic} \Gamma(X)$. If $Y$ is a scheme, and $G$ is a group (or just a set), we let $Y \times G$ denote the scheme which is a disjoint union of copies of $Y$, one for each $g \in G$. \begin{definition}\label{Galois-defn} Let $X$ be a scheme, and let $Y$ be a finite \'etale $X$-scheme. Suppose a finite group $G$ acts on the right of $Y$ as an $X$-scheme.\footnote{Hereafter we say simply that {\it $G$ acts on $Y/X$.}} Then this action is {\it Galois\/} if the map $Y \times G \to Y \times_X Y$ given by $(y,g) \mapsto (y, yg)$ is an isomorphism of schemes.% \footnote{If $y \in Y$, then $yg \in Y$, and $g$ induces an isomorphism of $k(y)$ with $k(yg)$. Therefore we get maps \mp[[ \sigma_1, \sigma_2 || \operatorname{Spec}(k(y)) || Y ]], such that $\pi \circ \sigma_1 = \pi \circ \sigma_2$, where \mp[[ \pi || Y || X ]] is the structure map. By the universal property of the fiber product, we obtain a morphism \mapx[[ \operatorname{Spec}(k(y)) || Y \times_X Y ]], whose image is by definition the point $(y,yg)$.} \end{definition} It is important to note that $G$ cannot be recovered from $Y \rightarrow X$ in general: for instance this is the case if $Y$ consists of a disjoint union of copies of $X$. If we have a Galois action of $G$ on $Y/X$, and $X' \to X$ is any morphism, we get a Galois action of $G$ on $Y \times_X X'$ as an $X'$-scheme. Let $A \subset B$ be rings, and let a finite group $G$ act on $B/A$, meaning that $G$ acts on (the left of) $B$ as an $A$-algebra. Then we shall call this action {\it Galois\/} if the action of $G$ on $\operatorname{Spec}(B)/\operatorname{Spec}(A)$ is Galois with respect to definition \pref{Galois-defn}. For some definitions, stated directly for rings, see \cite{KO2, Chapter II, \S5}. In the case where $A$ and $B$ are fields, the action of $G$ on $B/A$ is Galois if and only if $B/A$ is a Galois extension (in the usual sense), and $G = \operatorname{Aut}_A(B)$. We now recall the {\it Hochschild-Serre spectral sequence}. This may be found in Milne \cite{Mi1, p.\ 105}. Although Milne refers only to locally noetherian schemes, the argument he presents is valid for any scheme. For clarity, however, we note that $X_{\hbox{\footnotesize\'et}}$ as used here means the (small) \'etale site (on an arbitrary scheme $X$), as defined in \cite{G2}. Let $X$ be a scheme, let $Y$ be a finite \'etale $X$-scheme, and let a Galois action of a finite group $G$ on $Y/X$ be given. Let ${\cal{F}}$ be a sheaf (of abelian groups) for the \'etale topology on $X$. Then the Hochschild-Serre spectral sequence is: $$E^{p,q}_2 = H^p(G, H^q(Y_{\hbox{\footnotesize\'et}}, \pi^*{\cal{F}})) \ \Longrightarrow \ H^{p+q}(X_{\hbox{\footnotesize\'et}}, {\cal{F}}),$$% where $\pi: Y \to X$ is the structure morphism. Apply this with ${\cal{F}} = {\Bbb G}_{\op m}$. One has an exact sequence: $$0 \to E^{1,0}_2 \to H^1(X_{\hbox{\footnotesize\'et}},{\Bbb G}_{\op m}) \to E^{0,1}_2.$$% The first term is $H^1(G,\Gamma(Y)^*)$. The middle term is $\operatorname{Pic}(X)$. The last term is $H^0(G,\operatorname{Pic}(Y))$, which embeds in $\operatorname{Pic}(Y)$. Hence: \begin{proposition}\label{the-kernel} Let \mp[[ f || Y || X ]] be a finite \'etale morphism of schemes, and suppose we have a Galois action of a finite group $G$ on $Y/X$. Then $$\operatorname{Ker}[\operatorname{Pic}(f)]\ \cong\ H^1(G,\Gamma(Y)^*).$$ \end{proposition} In particular \cite{Sw2, (4.2)}, if $A \subset B$ are rings, and we are given a Galois action of a finite group $G$ on $B/A$, then $\operatorname{D}(A,B) \cong \operatorname{H}^1(G,B^*)$. The following example shows that one has to be a bit careful about generalizing the proposition to the non-Galois case. \begin{example} Let $A$ be a complete discrete valuation ring with fraction field $K$, and let $L$ be a finite Galois extension of $K$ with Galois group $G$. Denote the integral closure of $A$ in $L$ by $B$. We would like there to be an exact sequence: $$0\to H^1(G,B^*)\to \operatorname{Pic}(A)\to \operatorname{Pic}(B),$$ and hence conclude that $H^1(G,B^*) = 0$. But this is not the case in general: $H^1(G,B^*)$ has order equal to the ramification index of $B/A$. \end{example} We recall some easy facts about cohomology of groups: \begin{proposition}\label{group-cohomology-basics} Let $G$ be a finite group and $M$ a left $\Bbb Z G$-module. \begin{enumerate} \item If $M$ is finitely generated, then $\operatorname{H}^n(G,M)$ is finite for every $n > 0$. \item If $M$ has trivial $G$-action then $H^1(G,M) \cong \operatorname{Hom}_{\smallcat{groups}}(G,M)$. \end{enumerate} \end{proposition} \begin{proof} (1) Since $\operatorname{H}^n(G,M) = \operatorname{Ext}^n_{\Bbb Z G}(\Bbb Z,M)$ and $\Bbb Z G$ is left Noetherian it is clear that all the cohomology groups are finitely generated modules (over $\Bbb Z G$ or, equivalently, over $\Bbb Z$). Moreover, they are annihilated by $|G|$ by \cite{Br, Chap.\ III, (10.2)}. (2) is clear from the representation of $\operatorname{H}^1(G,M)$ in terms of crossed homomorphisms (or, see \cite{Bab, \S23}). \qed \end{proof} \begin{proposition}\label{90} Let a finite group $G$ act on a field $K$. Then $H^1(G,K^*)$ is finite, and (Hilbert's Theorem 90) it is $0$ if the group action is faithful. \end{proposition} \begin{proof} Let $H$ be the subgroup of $G$ which acts trivially on $K$, and let $\overline{G} = G/H$. Let $k = K^{\overline{G}}$. Then the extension $K/k$ is Galois, with Galois group $\overline{G}$. By \pref{the-kernel}, $H^1(\overline{G},K^*) \cong \operatorname{Ker}[\operatorname{Pic}(k) \to \operatorname{Pic}(K)]$, which is $0$. We are done if $G$ acts faithfully on $K$. We have the inflation-restriction exact sequence $$0 \to \operatorname{H}^1(\overline{G}, K^*) \to \operatorname{H}^1(G,K^*) \to \operatorname{H}^1(H,K^*),$$ so it is enough to show that $\operatorname{H}^1(H,K^*)$ is finite, which is clear from \pref{group-cohomology-basics}(2). \qed \end{proof} The next result is a variant of a well-known result due to Roquette \cite{Ro}: \begin{proposition}\label{roquette} Let $K$ be a field, $X$ a $K$-scheme of finite type, and $\Lambda$ the integral closure of $K$ in $A:=\Gamma(X)_{\op{red}}$. Then $\Lambda$ is finite-dimensional as a $K$-vector space, and $A^*/\Lambda^*$ is a finitely generated\ free abelian group. \end{proposition} \begin{proof} Let $\{U_1,\dots,U_m\}$ be an affine open cover of $X$, and set $R_i = \Gamma(U_i)_{\op{red}}$. Each $R_i$ is a a reduced $K$-algebra of finite type. We have an embedding $A \to B := R_1\times\dots \times R_m$. Let $P_1,\dots,P_n$ be the minimal prime ideals of $B$, and let $C_j$ be the normalization of the domain $B/P_j$. Then each $C_j$ is a normal domain of finite type over $K$, and $A \subset C_1\times\dots\times C_n$. Let $\Delta_j$ be the integral closure of $K$ in $C_j$. By the usual formulation of Roquette's theorem (see \cite{L, Chapter 2, (7.3)} or \cite{Kr, (1.4)}) $C_j^*/\Delta_j^*$ is finitely generated\ for each $j$. We have $\Lambda = A \cap (\prod_j\Delta_j)$. Therefore $A^*/\Lambda^*$ embeds in the finitely generated group $\prod_j(C_j^*/\Delta_j^*)$. Obviously $A^*/\Lambda^*$ is torsion-free, and since it is finitely generated, it is free. The fraction field $K_j$ of $C_j$ is a finitely generated\ field extension of $K$. Each $\Delta_j$, being $0$-dimensional, reduced and connected, is a field algebraic over $K$. Since $K_j/K$ is finitely generated, so is $\Delta_j/K$. Therefore $\prod_j\Delta_j$ is a finite-dimensional $K$-algebra, and hence so is its subalgebra $\Lambda$. \qed \end{proof} \begin{theorem}\label{kernel2b} Let $X$ be a scheme of finite type over a field $k$. Let $f: Y \to X$ be a finite, \'etale, surjective morphism of schemes. If $k$ has positive characteristic, assume that $X$ is reduced. Then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is finite. \end{theorem} \begin{remark} If $X$ is affine, $\operatorname{Pic}(X) = \operatorname{Pic}(X_{\op{red}})$, and so the assumption about $X$ being reduced (in positive characteristic) is not needed. In general it is: see \pref{finite-etale-infinite}. \end{remark} \begin{proofnodot} (of \ref{kernel2b}.) We may assume that $X$ is connected. Let $Y_0$ be a connected component of $Y$. Then $f|_{Y_0}$ is finite and \'etale, and since $X$ is connected, it is surjective. Therefore we may reduce to the case where $Y$ is connected. By \cite{Mur, (4.4.1.8)}, there exists a scheme $Y'$ over $Y$ such that $Y' \to X$ is finite \'etale\ surjective and the action of the finite group $\operatorname{Aut}(Y'/X)$ on $Y'/X$ is Galois. Therefore we may assume that\ in fact there is a Galois action of a finite group $G$ on $Y/X$. By \pref{the-kernel}, $\operatorname{Ker}[\operatorname{Pic}(f)] \cong H^1(G,\Gamma(Y)^*)$. We will complete the proof by showing that $H^1(G,\Gamma(Y)^*)$ is finite. This will depend only on the fact that we have a finite group $G$ acting on an algebraic scheme $Y$, which is reduced if the characteristic is positive. Let $B = \Gamma(Y)$. Let $\Lambda$ be the integral closure of $k$ in $B_{\op{red}}$. By \pref{roquette}, $\Lambda$ is a finite-dimensional $k$-algebra and $B_{\op{red}}^*/\Lambda^*$ is finitely generated. By \pref{group-cohomology-basics}(1) we know that $H^1(G,B_{\op{red}}^*/\Lambda^*)$ is finite. We show that $H^1(G,\Lambda^*)$ is finite. Write $\Lambda = \prod_{j\in J}F_j$, where each $F_j$ is a finite-dimensional field extension of $k$ (and $J$ is a finite index set). Since the action of $G$ preserves idempotents, we can define an action of $G$ on $J$ by the rule $F_{gj} = gF_j$. Let $I$ be any orbit of $G$ on $J$, and look at $\Upsilon := \prod_{i\in I}F_i$. Since $\Lambda^*$ is the direct sum of the groups $\Upsilon^*$ (over the various orbits of $G$ on $J$), it is enough to show that $\operatorname{H}^1(G,\Upsilon^*)$ is finite. Fix $i \in I$, let $H$ be the isotropy subgroup of $i$, and put $F = F_i$. It follows from \cite{Br, Chap. III, (5.3), (5.9), (6.2)} that $\operatorname{H}^1(G,\Upsilon^*) \cong \operatorname{H}^1(H,F^*)$, which is finite by \pref{90}. Hence $H^1(G,\Lambda^*)$ is finite. Running the long exact sequence of cohomology coming from the exact sequence $$1 \to \Lambda^* \to B_{\op{red}}^* \to B_{\op{red}}^*/\Lambda^* \to 1,$$ we conclude that $H^1(G,B_{\op{red}}^*)$ is finite. Thus we are done if $X$ is reduced. We may assume that $\mathop{\operator@font char \kern1pt}\nolimits(k) = 0$. Let $N$ be a $G$-stable nilpotent ideal of $B$. (For instance, we might take $N$ to be the nilradical of $B$.) It is enough to show that the canonical map \mapx[[ H^1(G,B^*) || H^1(G,(B/N)^*) ]] is injective. If $N^r = 0$, note that we can factor the map \mapx[[ B || B/N ]] as \diagramx{B & \mapE{} & B/N^{r-1} & \mapE{} & B/N^{r-2} & \mapE{} & \cdots & \mapE{} & B/N,}% and $G$ acts on everything in the sequence, so we can reduce to the case where $N$ has square zero. We have an exact sequence \diagramx{0&\mapE{} & N & \mapE{} & B^* & \mapE{} & (B/N)^* & \mapE{} & 1,}% and it is enough to show that $H^1(G,N) = 0$. On the one hand, $H^1(G,N)$ is annihilated by $|G|$, and so is torsion. On the other hand, $H^1(G,N)$ is an $B[G]$-module, thus a $\Bbb Q$-vector space, and so is torsion-free. Hence $H^1(G,N) = 0$. \qed \end{proofnodot} \begin{remark}\label{belch} In the theorem, if $Y = X_L$ for some finite separable field extension $L/k$, we will show that there is no need to assume $X$ is reduced, even in positive characteristic. Indeed in that case, $\Gamma(X_L) = \Gamma(X)_L$, so we have a Galois action of $G$ on $\operatorname{Spec} \Gamma(Y) / \operatorname{Spec} \Gamma(X)$. Applying \pref{the-kernel} twice, we conclude that $\operatorname{Ker}[\operatorname{Pic}(f)] = \operatorname{Ker}[\operatorname{Pic}(\Gamma(f))]$. But $\operatorname{Pic}$ of a ring is the same as $\operatorname{Pic}$ of its reduction, so if $C = \Gamma(X)$, we conclude that $\operatorname{Ker}[\operatorname{Pic}(f)] = \operatorname{Ker}[\operatorname{Pic}(C_{\op{red}}) \to \operatorname{Pic}((C_{\op{red}})_L)]$. Applying \pref{the-kernel} once again, we see that $\operatorname{Ker}[\operatorname{Pic}(f)] = H^1(G, (C_{\op{red}})_L) = H^1(G,B_{\op{red}})$. Now the remainder of the proof of the theorem goes through. \end{remark} \begin{problem}\label{fidofido} The proof of the theorem shows that if $k$ is a field, $A$ is a finitely generated\ $k$-algebra (reduced if $\mathop{\operator@font char \kern1pt}\nolimits(k)\not=0$), and a finite group $G$ acts on $A$, then $H^1(G,A^*)$ is finite. Under the same hypotheses, for which $n \in \cal N$ is the set $H^1(G,\operatorname{GL}_n(A))$ finite? \end{problem} \begin{theorem}\label{makes-injective} Let $k$ be a field, and let $K/k$ be a Galois extension of fields, not necessarily finite. Let $S$ be a $k$-scheme of finite type, and assume that each connected component of $S$ is geometrically connected. Let $\Lambda$ be the integral closure of $K$ in $\Gamma(S_K)_{\op{red}}$. Assume that $\Gamma(S)_{\op{red}}^* \Lambda^* = \Gamma(S_K)_{\op{red}}^*$. Then the canonical map $\operatorname{Pic}(S) \to \operatorname{Pic}(S_K)$ is injective. \end{theorem} \begin{proof} It is enough to show that for each finite Galois extension $L/k$ with $k \subset L \subset K$, the map $\operatorname{Pic}(S) \to \operatorname{Pic}(S_L)$ is injective. Let $G = \operatorname{Gal}(L/k)$. Let $B_0 = \Gamma(S_L)_{\op{red}}$. Let $\Lambda_0$ be the integral closure of $L$ in $B_0$. The condition of the theorem implies that $\Gamma(S)_{\op{red}}^* \Lambda_0^* = B_0^*$. Therefore the action of $G$ on $B_0^*/\Lambda_0^*$ is trivial. Hence $H^1(G,B_0^*/\Lambda_0^*) \cong \operatorname{Hom}_{\smallcat{groups}}(G, B_0^*/\Lambda_0^*)$ by \pref{group-cohomology-basics}(2). Since $G$ is finite and $B_0^*/\Lambda_0^*$ is torsion-free [by \pref{roquette}], $H^1(G,B_0^*/\Lambda_0^*) = 0$. Since $S$ is geometrically connected, $S_L$ is connected, and so $\Lambda_0$ is a field. Since $G$ acts faithfully on $\Lambda_0$, $H^1(G,\Lambda_0^*) = 0$ by \pref{90}. Hence $H^1(G,B_0^*) = 0$. Arguing as in \pref{belch}, one obtains the theorem. \qed \end{proof} \begin{theorem}\label{separable-extension} Let $k$ be a field and $S$ a scheme of finite type over $k$. Let $K/k$ be a separable algebraic field extension. Then the kernel of the map $\operatorname{Pic}(S) \to \operatorname{Pic}(S_K)$ (induced by the projection $\pi: S_K \to S$) is a finite group. \end{theorem} \vspace{0.08in} \par\noindent{\bf Affine version of Theorem \ref{separable-extension}.} \ {\it Let $k$ be a field, and let $A$ be a finitely generated\ $k$-algebra. Let $K/k$ be a separable algebraic field extension. Then the kernel of the induced map $\operatorname{Pic}(A) \to \operatorname{Pic}(A_K)$ is finite.} \vspace{0.1in} \begin{proofnodot} (of \ref{separable-extension}). By passing to the normal closure, we may assume that $K/k$ is Galois (possibly infinite). Let $B = \Gamma(S_K)_{\op{red}}$, and let $\Lambda$ be the integral closure of $K$ in $B$. By \pref{roquette}, $B^*/\Lambda^*$ is finitely generated. Therefore we can find a finite Galois extension $L$ of $k$ (with $L \subset K$) such that the canonical map $(\Gamma(S_L)_{\op{red}})^* \to B^*/\Lambda^*$ is surjective. We can also choose $L$ so that each connected component of $S_L$ is geometrically connected. By \pref{kernel2b} and \pref{belch}, the kernel of $\operatorname{Pic}(S) \to \operatorname{Pic}(S_L)$ is finite, and by \pref{makes-injective}, the map $\operatorname{Pic}(S_L) \to \operatorname{Pic}(S_K)$ is injective, so we are done. \qed \end{proofnodot} We will see in \S\ref{examples-section} that \pref{separable-extension} can fail if $K/k$ is not assumed to be separable. On the other hand, assuming resolution of singularities, we will show in \S4 that for any separated scheme $S$ of finite type over $k$, and any algebraic field extension $K/k$, there is an intermediate field $E/k$ of finite degree over $k$ such that $\operatorname{Pic}(S_E) \to \operatorname{Pic}(S_K)$ is one-to-one. We conclude this section by showing that the kernel is not always trivial. In fact, any finite abelian group can occur. If, in the following construction, one takes $K/k = \Bbb C/\Bbb R$, one obtains the familiar example $A = \Bbb R[X,Y]/(X^2+Y^2-1)$. The Picard group of $A$ has order two (generated by the M\"obius band), whereas $A\otimes_{\Bbb R}\Bbb C \cong \Bbb C[U,U^{-1}]$, which has trivial Picard group. \begin{example} Let $K/k$ be a finite Galois extension with Galois group $G$. There is a domain $A$ of finite type over $k$ such that $\operatorname{Pic}(A) \cong G/[G,G]$ but $\operatorname{Pic}(A\otimes_kK)$ is trivial. \end{example} \begin{proof} Make $\Bbb Z G$ into a $G$-module via the left regular action. We have an exact sequence of $G$-modules $$0 \to \Bbb Z \mapE{\sigma} \Bbb Z G \to L \to 0,$$% where $G$ acts trivially on $\Bbb Z$, $\sigma$ takes $1$ to $\sum_{g\in G}g$, and $L$ is defined by the sequence. Let $F = \operatorname{Im}(\sigma)$, which is the set of fixed points of $\Bbb Z G$ under the action of $G$. Clearly $L$ is a free $\Bbb Z$-module of rank $|G| - 1$. Form the group ring $B = K[L]$. Since $L$ is a free abelian group, $B$ is isomorphic to the Laurent polynomial ring in $|G|-1$ variables and hence has trivial Picard group. Note that $G$ acts on $B$ by acting as the Galois group on $K$ and by the $G$-module structure on $L$. Let $A = B^G$. Then $A\otimes_kK \cong B$ by \cite{Sw1, (2.5)}, and the action of $G$ on $B/A$ is Galois. Using (\ref{the-kernel}) we get $\operatorname{Pic}(A) = \operatorname{D}(A,B) = \operatorname{H}^1(G,B^*)$. But $B^* \cong K^* \oplus L$, so $\operatorname{Pic}(A) \cong \operatorname{H}^1(G,K^*) \oplus \operatorname{H}^1(G,L) \cong \operatorname{H}^1(G,L)$, since $G$ acts faithfully on $K^*$ and so $H^1(G,K^*) = 0$ by \pref{90}. We will show that $\operatorname{H}^1(G,L) \cong G/[G,G]$. We know \cite{Br, Chap. III. (6.6)} that $\operatorname{H}^i(G,\Bbb Z G) = 0$ for $i>0$. Therefore, by the long exact sequence of cohomology we have $\operatorname{H}^1(G,L) \cong \operatorname{H}^2(G,\Bbb Z)$. But $\operatorname{H}^2(G,\Bbb Z) \cong \operatorname{H}^1(G,\Bbb Q/\Bbb Z) \cong \operatorname{Hom}_{\smallcat{groups}}(G, \Bbb Q/\Bbb Z) \cong G/[G,G]$. (See \cite{Bab, \S23}.) \qed \end{proof} \section{Torsion in Picard groups}\label{torsion-section} We note that $_n\operatorname{Pic}(R)$ can be infinite even for $R$ an domain finitely generated\ over a field. For example, take $R = k[T^2,T^3]$, where $k$ is an infinite field of characteristic $p > 0$. Then $\operatorname{Pic}(R)$ is isomorphic to to the additive group of $k$ and is therefore an infinite group of exponent $p$. As long as we avoid the characteristic, however, this cannot happen. First we need the following lemma (cf.\ \cite{Bas1, IX, (4.7)}): \begin{lemma}\label{kernel-is-torsion} Let \mp[[ f || X || S ]] be a finite flat morphism of schemes, of constant degree $d > 0$. Then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is $d$-torsion. \end{lemma} \begin{proof} Let $\cal M \in \operatorname{Pic}(S)$. Then $f_*f^*\cal M \cong \cal M \o* f_* \cal O_X$ as $\cal O_S$-modules. If moreover $\cal M \in \operatorname{Ker}[\operatorname{Pic}(f)]$, then $f_* f^* \cal M \cong f_* \cal O_X$. Hence $\cal M \o* f_* \cal O_X \cong f_* \cal O_X$. Apply $\wedge^d$, yielding $\cal M^{\o* d} \o* \wedge^d(f_*\cal O_X) \cong \wedge^d(f_*\cal O_X)$, and hence $\cal M^{\o* d} \cong \cal O_S$. \qed \end{proof} Now let us generalize to the proper case. First we need: \begin{lemma}\label{is-locally-free} Let \mp[[ f || X || S ]] be a proper flat morphism of noetherian schemes. Then $f_*\cal O_X$ is a locally free $\cal O_S$-module. \end{lemma} \begin{proof} We may assume that $S$ is affine. Let $W = \mathop{\mathbf{Spec}}\nolimits(f_*\cal O_X)$ (cf.\ \cite{Ha, II, exercise 5.17}), and label morphisms \diagramx{X & \mapE{\varphi} & W & \mapE{h} & S.}% Since $f$ is proper, $f_*\cal O_X$ is coherent \cite{EGA$3_1$, 3.2.1}, and so it is enough to show that $h$ is flat. Therefore it is enough to show that for any injection $i$ of coherent $\cal O_S$-modules, $h^*(i)$ is also injective. By construction, $\varphi_* \cal O_X = \cal O_W$, and so $\varphi_* \varphi^* h^*(i) = h^*(i)$ by the projection formula \cite{Ha, II, exercise 5.1d}. Since $f$ is flat, $f^*(i)$ [which equals $\varphi^* h^*(i)$] is injective. Since $\varphi_*$ is left exact, $\varphi_* \varphi^* h^*(i)$ is also injective. \qed \end{proof} \begin{corollary}\label{proper-bounded} Let \mp[[ f || X || S ]] be a surjective proper flat morphism of noetherian schemes. Then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is a bounded torsion group. \end{corollary} \begin{proof} We may assume that $S$ is connected. By \pref{is-locally-free}, $f_*\cal O_X$ is a locally free $\cal O_S$-module. Factor $f$ as in the proof of \pref{is-locally-free}. By the projection formula, $\varphi_* \varphi^*$ is the identity, so $\operatorname{Pic}(\varphi)$ is injective. Apply \pref{kernel-is-torsion}. \qed \end{proof} \begin{theorem}\label{finitely-many} Let $S$ be a scheme of finite type over a field $k$ and let $n \in \Bbb N$ be prime to the characteristic of $k$. Then $_n\operatorname{Pic}(S)$ is finite. \end{theorem} \vspace{0.03in} \par\noindent{\bf Affine version of Theorem \ref{finitely-many}.} \ {\it Let $k$ be a field, and let $A$ be a finitely generated\ $k$-algebra. Let $n \in \Bbb N$ be prime to the characteristic of $k$. Then $_n\operatorname{Pic}(A)$ is finite.} \vspace{0.1in} \begin{proofnodot} (of \ref{finitely-many}). Suppose first that $k$ is separably closed. The argument in this case seems to be fairly well known and was pointed out to us several years ago by David Saltman and Tim Ford. We consider the Kummer sequence \cite{SGA4$1\over2$, p.\ 21, (2.5)}: \diagramx{1 & \mapE{} & \mu_n & \mapE{} & {\Bbb G}_{\op m} & \mapE{n} & {\Bbb G}_{\op m} & \mapE{} & 1}% This is an exact sequence of sheaves for the \'etale topology on $S$. Taking \'etale cohomology, we get an exact sequence \diagramno{(*)}{\operatorname{H}^1(S_{\hbox{\footnotesize\'et}},\mu_n) & \mapE{} & \operatorname{H}^1(S_{\hbox{\footnotesize\'et}},{\Bbb G}_{\op m}) & \mapE{n} & \operatorname{H}^1(S_{\hbox{\footnotesize\'et}},{\Bbb G}_{\op m}).}% Now $\operatorname{H}^1(S_{\hbox{\footnotesize\'et}},{\Bbb G}_{\op m})\cong\operatorname{Pic}(S)$ by \cite{SGA4$1\over2$, p.\ 20, (2.3)}. Since $k$ is separably closed, $\mu_n$ is isomorphic to the constant sheaf $\Bbb Z/n\Bbb Z$. (See \cite{SGA4$1\over2$, p.\ 21, (2.4)}.) Therefore $\mu_n$ is constructible \cite{SGA4$1\over2$, p.\ 43, (3.2)}. By \cite{SGA4$1\over2$, p.\ 236, (1.10)} $\operatorname{H}^1(S_{\hbox{\footnotesize\'et}},\mu_n)$ is finitely generated. Hence the kernel of the map $n$ in $(*)$ is finite, so the theorem holds when $k$ is separably closed. Apply \pref{separable-extension}. \qed \end{proofnodot} As a consequence of \pref{proper-bounded} and \pref{finitely-many}, we have: \begin{corollary}\label{proper-Pic-finite} Let $S$ be a scheme of finite type over a field $k$. Let \mp[[ f || X || S ]] be a proper flat surjective morphism of schemes. Assume that over each connected component of $S$, the rank of the locally free sheaf $f_*\cal O_X$ is invertible in $k$. Then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is finite. \end{corollary} We can use (\ref{finitely-many}) to answer a question of S.\ Montgomery about outer automorphism groups of Azumaya algebras: \begin{corollary} Let $k$ be a field. Let $R$ be a finitely generated\ $k$-algebra. Let $A$ be a (not necessarily commutative) Azumaya algebra over $R$ of degree $d$. Then $\operatorname{Out}_R(A)$ is finite whenever $d$ is not a multiple of the characteristic of $k$. \end{corollary} \begin{proof} There is an embedding (see e.g.\ \cite{DI}) of $\operatorname{Out}_R(A)$ in $\operatorname{Pic}(R)$, and in fact the image of $\operatorname{Out}_R(A)$ is contained in $_d \operatorname{Pic}(R)$. (See \cite{KO1}.) \qed \end{proof} In fact, one can show that the finiteness of $_{d^e}\operatorname{Pic}(R)$ for all $e$ is equivalent to the finiteness of $\operatorname{Out}_R(M_{d^e}(R))$ for all $e$ (see \cite{BG}). Finally, we consider $n$-torsion in a normal algebraic scheme. It turns out that this is finite, even if $n$ is not prime to the characteristic, at least assuming that resolution of singularities holds. To prove this, we need to know what happens to $\operatorname{Pic}$ of a normal scheme when its singular locus is deleted. The kernel is described by the following ``folklore'' lemma, which we state in greater generality for later application \pref{seminormal-S2}: \begin{lemma}\label{S2} Let $X$ be a noetherian $S_2$ scheme% \footnote{Recall that a noetherian scheme $X$ is by definition $S_2$ if for every $x \in X$, $\operatorname{depth} \cal O_{X,x} \geq \min\setof{\operatorname{dim} \cal O_{X,x},2}$.}, and let $C \subset X$ be a closed subset of codimension $\geq 2$. Let $U = X - C$. Then the canonical map $\operatorname{Pic}(X) \to \operatorname{Pic}(U)$ is injective. \end{lemma} \begin{proof} Let $\cal L$ be a line bundle on $X$ which becomes trivial on $U$. For any line bundle $\cal M$ on $X$, the long exact sequence of local cohomology gives us $$\operatorname{H}^0_C(X,\cal M) \to \operatorname{H}^0(X,\cal M) \mapE{\rho_{\cal M}} \operatorname{H}^0(U,\cal M) \to \operatorname{H}^1_C(X,\cal M).$$% Since $X$ is $S_2$, the end terms vanish \cite{G1, (1.4), (3.7), (3.8)}. Let $\phi: \cal O_U \to \cal L|_U$ be an isomorphism. Since $\rho_{\cal L}$ is an isomorphism, we can lift $\phi$ to a morphism $\psi: \cal O_X \to \cal L$. Since $\rho_{\cal L^{-1}}$ is an isomorphism, we can lift $\phi^{-1}$ to a morphism $\psi': \cal L \to \cal O_X$. Since $\rho_{\cal O_X}$ and $\rho_{\cal L}$ are isomorphisms, $\psi' \circ \psi$ and $\psi \circ \psi'$ are the identity maps. \qed \end{proof} \begin{lemma}\label{alg-closed-regular} Let $k$ be an algebraically closed field. Assume that resolutions of singularities exist for varieties over $k$. Let $X$ be a normal $k$-scheme of finite type. Let $n \in {\Bbb N}$. Then $_n \operatorname{Pic}(X)$ is finite. \end{lemma} \begin{sketch} We may assume that $X$ is connected. By \pref{S2}, we may replace $X$ by $X_{\operatoratfont reg}$ and so assume that $X$ is regular. If we further replace $X$ by a nonempty open subscheme, we kill a finitely generated\ subgroup of $\operatorname{Pic}(X)$. In this way we may reduce to the case where $X$ is affine. Since we have resolution of singularities, we can embed $X$ as an open subscheme of a regular $k$-scheme ${\overline{X}}$. Then $\operatorname{Pic}(X)$ is the quotient of $\operatorname{Pic}({\overline{X}})$ by a finitely generated\ subgroup. Now $\operatorname{Pic}^0({\overline{X}})$ is the group $A(k)$ of $k$-valued points of an abelian variety $A$ over $k$, and $\operatorname{Pic}({\overline{X}})/\operatorname{Pic}^0({\overline{X}})$ is finitely generated\ (see e.g.\ \cite{K, (5.1)}). Therefore it suffices to show that $_n A$ is finite. This is well-known \cite{Mum, p.\ 39}. \qed \end{sketch} \section{Faithfully flat extensions} The main results of this section are \pref{faithfully-flat-fg}, \pref{faithfully-flat-fg-imperfect}, and \pref{open-cover}, which give information about the kernel of the map on Picard groups induced by a faithfully flat morphism of algebraic schemes. First we consider the case where the target of the morphism is normal, in which case we can weaken the hypothesis of faithful flatness. \begin{theorem}\label{normal-faithfully-flat} Let $k$ be a field. Let $X$ be a normal $k$-scheme of finite type. Let \mp[[ f || Y || X ]] be a dominant morphism of finite type. Then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is finitely generated\ (if $\mathop{\operator@font char \kern1pt}\nolimits(k) = 0$) and is the direct sum of a finitely generated\ group and a bounded $p$-group (if $\mathop{\operator@font char \kern1pt}\nolimits(k) = p > 0$). If $k$ is algebraically closed, and resolution of singularities holds, then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is finitely generated. \end{theorem} \begin{proof} We may assume that $X$ is connected. By \pref{S2}, we may assume that\ $X$ is regular. Then we may replace $X$ by any nonempty open subscheme. In particular, we may assume that\ $X$ is affine. Moreover, by replacing $Y$ by a suitable open subscheme, we may assume that\ $Y$ is affine too. We may assume that $Y$ is a regular integral scheme. We may embed $Y$ as an open subscheme of an $X$-scheme ${\overline{Y}}$ which is projective over $X$ and is an integral scheme as well. Replace ${\overline{Y}}$ be its normalization. Now by again replacing $X$ by a nonempty open subscheme, we may assume (by generic flatness) that the morphism \mapx[[ {\overline{Y}} || X ]] is flat; certainly we may assume that\ it is surjective. Call this morphism $\varphi$. By \pref{proper-bounded}, $\operatorname{Ker}[\operatorname{Pic}(\varphi)]$ is a bounded torsion group. Hence by \pref{finitely-many}, $\operatorname{Ker}[\operatorname{Pic}(\varphi)]$ is finite (if $\mathop{\operator@font char \kern1pt}\nolimits(k) = 0$) and is the direct sum of a finite group and a bounded $p$-group (if $\mathop{\operator@font char \kern1pt}\nolimits(k) = p > 0$). [By \pref{alg-closed-regular}, if $k$ is algebraically closed, then $\operatorname{Ker}[\operatorname{Pic}(\varphi)]$ is always finite.] By \pref{S2} and \cite{Ha, II, (6.5c), (6.16)}, $\operatorname{Ker}[\operatorname{Pic}({\overline{Y}}) \rightarrow \operatorname{Pic}(Y)]$ is finitely generated. The theorem follows. \qed \end{proof} Now we want to see what happens when we consider a faithfully flat morphism \mapx[[ Y || X ]], where $X$ is not necessarily normal. The issue is complicated by nilpotents, even in the affine case. The problem [see examples \pref{reduction-not-faithfully-flat}, \pref{rnff2} below] is that one can have a domain $A$, and a finitely generated\ faithfully flat $A$ algebra $B$, such that $B_{\op{red}}$ is not flat over $A$. \begin{lemma}\label{B-cap-K-if-flat} Let $A$ be a reduced ring, with total ring of fractions $K$. Let $B$ be a faithfully flat $A$-algebra. Then (inside $B \o*_A K$) $B \cap K = A$. \end{lemma} \begin{proof} Let $b \in B \cap K$, so we have an equation of the form $bu = v$, for some $u, v \in A$ with $u$ a non-zero-divisor. By the equational criterion for faithful flatness \cite{Bour, Ch.\ I, \S3, $\operatoratfont{n}^\circ$ 7, Prop.\ 13}, $b \in A$. \qed \end{proof} \begin{example}\label{reduction-not-faithfully-flat} Let $k$ be a field of characteristic $2$. Let $A = k[s,t]/(s^2-t^3)$. Let $B = A[x,y]/(x^2-s,y^2-t)$. Then $B$ is a faithfully flat $A$-algebra. Now $B \cong k[x,y]/(x^4-y^6)$, so the nilradical of $B$ is generated by $(x^2-y^3)$. Hence $B_{\op{red}} = A[x,y]/(x^2-s,y^2-t,x^2-y^3)$. Since $s=ty$ in $B_{\op{red}}$, we have $y \in B_{\op{red}} \cap A_{\op{nor}}$ and $y \notin A$, so it follows from \pref{B-cap-K-if-flat} that $B_{\op{red}}$ is not flat over $A$. One can also prove this directly by taking $M = A/(t)$ and checking that the map \mapx[[ M || M \o*_A B_{\op{red}} ]] is not injective. \end{example} \begin{example}[shown to us by Bill Heinzer and Sam Huckaba]\label{rnff2} It is known that there is a smooth curve $C \subset {\Bbb C}\kern1pt{\Bbb P}^3$ of degree $8$ and genus $5$ which is set-theoretically the intersection of two surfaces $S$, $T$, but which is not arithmetically Cohen-Macaulay (see \cite{Bar}, \cite{Hu}). Let $\tilde C = S \cap T$, scheme-theoretically. Choose lines $L, L'$ which are noncoplanar and do not meet $C$. Then projection from $L$ onto $L'$ defines a nonconstant morphism \mapx[[ \tilde C || \Bbb P^1 ]]. Let $A$ be the homogeneous coordinate ring of $\Bbb P^1$, and let $B$ be the homogeneous coordinate ring of $\tilde C$. Then $A = \Bbb C[x,y]$, $B$ is Cohen-Macaulay, $A \subset B$, and $B$ is module-finite over $A$, so $B$ is faithfully flat over $A$. (See \cite{Ma, p.\ 140}.) On the other hand, $B_{\op{red}}$ is not Cohen-Macaulay, so $B_{\op{red}}$ is {\it not\/} flat over $A$. \end{example} It is not clear if the behavior illustrated by the characteristic $p$ example can be mimicked in characteristic zero: \begin{problem}\label{furry-friend} Let $k$ be an algebraically closed field of characteristic zero. Let $A$ be a finitely generated, reduced $k$-algebra. Let $B$ be a faithfully flat and finitely generated\ $A$-algebra. Do we have $B_{\op{red}} \cap A_{\op{nor}} = A$? \end{problem} \begin{theorem}\label{faithfully-flat-fg} Let $k$ be a perfect field. Assume that resolutions of singularities exist for varieties over $k^a$. Let $X$ and $Y$ be geometrically integral $k$-schemes of finite type. Let \mp[[ f || Y || X ]] be a faithfully flat morphism of $k$-schemes. Then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is finitely generated. \end{theorem} We are not sure to what extent the hypothesis ``geometrically integral'' can be relaxed. Certainly if $X$ is not affine, nonreduced, and $Y$ is disconnected, one can have trouble \pref{nonreduced-cover-example}. In positive characteristic, the assumption that $Y$ is reduced is needed \pref{random-rabbits}. In characteristic zero, we do not know if it is necessary to assume that $Y$ is reduced. [If the answer to \pref{furry-friend} is yes, then we do not need to assume $Y$ reduced.] We do not know if it is necessary to assume that $X$ and $Y$ are geometrically irreducible. Cf.\ \pref{open-cover}. \vspace{0.1in} \par\noindent{\bf Affine version of Theorem \ref{faithfully-flat-fg}.} \ {\it Let $k$ be a perfect field. Assume that resolutions of singularities exist for varieties over $k^a$. Let $A$ be a finitely generated\ $k$-algebra. Let $B$ be a finitely generated\ and faithfully flat $A$-algebra, which is geometrically integral over $k$. Then $\operatorname{Ker}[\operatorname{Pic}(A) \rightarrow \operatorname{Pic}(B)]$ is finitely generated.} \vspace{0.1in} \begin{proofnodot} (of \ref{faithfully-flat-fg}). By \pref{separable-extension}, $\operatorname{Ker}[\operatorname{Pic}(X) \rightarrow \operatorname{Pic}(X^{\operatoratfont a})]$ is finite, so we may assume that\ $k$ is algebraically closed. By \pref{normal-faithfully-flat}, the theorem holds when $X$ is normal. To complete the proof, we need to show that $\operatorname{Ker}[\operatorname{Pic}(f)] \cap \operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(X_{\op{nor}})]$ is finitely generated. If we relax the assumptions on $Y$, by assuming only that each connected component of $Y$ is geometrically integral, it is enough to do the two cases: \begin{romanlist} \item $Y$ is an ``open cover'' of $X$; \item $X$, $Y$ are both affine. \end{romanlist} Let $P$ be the fiber product of $X_{\op{nor}}$ and $Y$ over $X$. Then $P$ is reduced. Let \mp[[ \pi || X_{\op{nor}} || X ]] and \mp[[ \tau || P || Y ]] be the canonical maps. Let ${\cal{C}}$ be the quotient sheaf $(\pi_*\cal O_{X_{\op{nor}}}^*)/\cal O_X^*$, and similarly let ${\cal{D}} = (\tau_*\cal O_P^*)/\cal O_{Y}^*$. (Note that the canonical map \mapx[[ \cal O_{Y} || \tau_*\cal O_P ]] is injective.) We have a commutative diagram with exact rows (and some maps labelled): \diagramx{1 & \mapE{} & {\Gamma(X_{\op{nor}})^*/\Gamma(X)^*} & \mapE{i} & \Gamma({\cal{C}}) & \mapE{} & \operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(X_{\op{nor}})] & \mapE{} & 1\cr && \mapS{\lambda} && \mapS{\delta} && \mapS{}\cr 1 & \mapE{} & {\Gamma(P)^*/\Gamma(Y)^*} & \mapE{} & \Gamma({\cal{D}}) & \mapE{} & \operatorname{Ker}[\operatorname{Pic}(Y) \to \operatorname{Pic}(P)] & \mapE{} & 1\makenull{.}}% We will be done with the proof if we can show that $\delta$ is injective and that $\operatorname{Coker}(\lambda)$ is finitely generated. We show that $\delta$ is injective. In case (i) this is clear, since ${\cal{C}}$ is a sheaf. So we may assume that\ $X$, $Y$ are both affine. Let $A = \Gamma(X)$, $B = \Gamma(Y)$. Let $\alpha \in \Gamma({\cal{C}})$. Then there exist elements $\vec f1n \in A$ with $(\vec f1n) = (1)$ and elements $\alpha_i \in (A_{\op{nor}})_{f_i}^*$ $(i=1,\ldots,n)$, $\beta_{ij} \in A_{f_i f_j}^*$ such that $\alpha_i = \beta_{ij}\alpha_j$ in $(A_{\op{nor}})_{f_i f_j}^*$ and $\alpha$ induces $\vec \alpha1n$. Suppose $\alpha$ maps to $1 \in \Gamma({\cal{D}})$. Then for each $i$ we have elements $b_i \in B_{f_i}^*$ such that $b_i = \alpha_i$ in $B_{f_i} \o*_A A_{\op{nor}}$ for each $i$. By \pref{B-cap-K-if-flat}, applied with $A_{f_i}$ substituted for $A$, we see that $\alpha_i \in A_{f_i}$. Hence $\alpha = 1$. Hence $\delta$ is injective. To complete the proof, we need to show that $$\operatorname{Coker}(\lambda) = {\Gamma(P)^* \over \Gamma(Y)^* \Gamma(X_{\op{nor}})^*}$$% is finitely generated. For this, we may assume that $Y$ is connected. By \pref{roquette}, it is enough to show that the map \mapx[[ P || X_{\op{nor}} ]] is bijective on connected components. This is true since $X$ and $Y$ are integral schemes. \qed \end{proofnodot} We note that the proof breaks down if we do not assume that $Y$ is geometrically irreducible. Indeed, without this hypothesis, the map \mapx[[ P || X_{\op{nor}} ]] may not be bijective on connected components: \begin{example} Let $k$ be a field of characteristic $\not= 2$. Let $A$ be the subring $k[t^2, t+t^{-1}]$ of $k[t,t^{-1}]$. Let $B = A[x]/(x^2-t^2)$. Then $B$ is a faithfully flat $A$-algebra. We will see that (i) $\operatorname{Spec}(B)$ is connected, but (ii) $\operatorname{Spec}(B \o*_A A_{\op{nor}})$ is not connected. First note that $t = (t^2+1)/(t+t^{-1})$, from which it follows that $A_{\op{nor}} = k[t,t^{-1}]$. Then $B \o*_A A_{\op{nor}} \cong A_{\op{nor}} \times A_{\op{nor}}$, so (ii) holds. Let $z = t^2 + 1$, $y = t+t^{-1}$. Then $A \cong k[y,z]/(y^2+z^2-y^2z)$, so $$B \cong k[x,y,z]/(x^2-z+1, y^2+z^2-y^2z) \cong k[x,y]/((x^2+1)^2-x^2y^2).$$% {}From this one sees that $\operatorname{Spec}(B)$ has two smooth components, meeting at the two points $(\pm i,0)$. Hence $\operatorname{Spec}(B)$ is connected. \end{example} \begin{theorem}\label{faithfully-flat-fg-imperfect} Let $k$ be a field of characteristic $p > 0$. Let $X$ and $Y$ be geometrically irreducible $k$-schemes of finite type. Let \mp[[ f || Y || X ]] be a faithfully flat morphism of $k$-schemes. Then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is the direct sum of a finitely generated\ group and a $p$-group. \end{theorem} \vspace{0.05in} \par\noindent{\bf Affine version of Theorem \ref{faithfully-flat-fg-imperfect}.} \ {\it Let $k$ be a field of characteristic $p > 0$. Let $A$ be a finitely generated\ $k$-algebra. Let $B$ be a finitely generated\ and faithfully flat $A$-algebra such that $\operatorname{Spec}(A \o*_k k^a)$ is irreducible. Then the group $\operatorname{Ker}[\operatorname{Pic}(A) \rightarrow \operatorname{Pic}(B)]$ is the direct sum of a finitely generated\ group and a $p$-group.} \vspace{0.1in} \begin{proofnodot} (of \ref{faithfully-flat-fg-imperfect}). It follows from \pref{kernel-is-torsion} and \pref{finitely-many} that $\operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(X^a)]$ is the direct sum of a finitely generated\ group and a $p$-group, so we may assume that\ $k$ is algebraically closed. Since $\operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(X_{\op{red}})]$ is a $p$-group, we may assume that\ $X$ is reduced. If $Y$ is reduced, we are done by \pref{faithfully-flat-fg}. In the general case, we modify the argument of the proof of \pref{faithfully-flat-fg}. Since the scheme $P$ in that argument may be nonreduced, we do not get that $\operatorname{Coker}(\lambda)$ is finitely generated, but rather $(*)$ it is finitely generated\ mod $p$-power torsion. We still get that $$M := {\Gamma(P)_{\op{red}}^* \over \Gamma(Y)_{\op{red}}^* \Gamma(X_{\op{nor}})^*}$$% is finitely generated. Let $N$ be the nilradical of $\Gamma(P)$. Then the kernel of the canonical map \mapx[[ \operatorname{Coker}(\lambda) || M ]] is a quotient of $1+N$, and hence is a bounded $p$-group, so $(*)$ follows. \qed \end{proofnodot} \begin{remark} By \pref{bounded}, it will follow that the $p$-group of the theorem is actually a bounded $p$-group, provided that resolution of singularities is valid. \end{remark} \begin{theorem}\label{open-cover} Let $X$ be a reduced scheme of finite type over a perfect field. Let $\vec U1n$ be an open cover of $X$. Then the canonical map \dmapx[[ \operatorname{Pic}(X) || \operatorname{Pic}(U_1) \times \cdots \times \operatorname{Pic}(U_n) ]] has finitely generated\ kernel. \end{theorem} For the case where $X$ is normal (in fact, any noetherian normal scheme), this is easy, following roughly from \pref{S2}. For the general case, one can follow the proof of \pref{faithfully-flat-fg}, taking $Y$ to be the disjoint union of the $U_i$'s; one need only adjust the last sentence. \section{Examples where the kernel is not finitely generated}% \label{examples-section} Let \mp[[ f || Y || X ]] be a faithfully flat morphism of noetherian schemes. The results \ref{kernel2b}, \ref{separable-extension}, \ref{proper-Pic-finite}, \ref{normal-faithfully-flat}, \ref{faithfully-flat-fg}, and \ref{open-cover} all give conditions under which $\operatorname{Ker}[\operatorname{Pic}(f)]$ is finitely generated. While it is reasonable to think that there are unifying and more general results with the same conclusion, we do not know what form such results should take. With this in mind, we give in this section a varied collection of examples in which $\operatorname{Ker}[\operatorname{Pic}(f)]$ is {\it not\/} finitely generated. We mention an obviously related question, about which we know very little. For which faithfully flat proper morphisms \mp[[ f || Y || X ]] is it the case that the map \dmapx[[ \setofh{iso.\ classes of vector bundles on $X$} || \setofh{iso.\ classes of vector bundles on $Y$} ]]% is finite-to-one? Cf.\ \pref{fidofido}, \pref{woofwoof}. Returning to $\operatorname{Pic}$, first we see that there are examples with $X$ algebraic of positive characteristic, and $f$ a finite \'etale\ morphism. For these, by \pref{kernel2b}, $X$ must be nonreduced. \begin{example}\label{finite-etale-infinite} Let $k$ be an algebraically closed field of characteristic $p > 0$. Let \mp[[ \lambda || F || E ]] be a finite \'etale\ morphism of varieties over $k$, of degree $p$, and suppose we have a Galois action of a finite group $G$ on $F/E$. Let $R = k[\epsilon]/(\epsilon^p)$. Let $X = E \times_k R$, $Y = F \times_k R$, and let \mp[[ f || Y || X ]] be the induced morphism. Then $\operatorname{Ker}[\operatorname{Pic}(f)] \cong H^1(G, \Gamma(Y)^*)$ by \pref{the-kernel}. Since $\Gamma(Y)$ is just $R$, and $G$ acts trivially on it, we have by \pref{group-cohomology-basics}(2) that $\operatorname{Ker}[\operatorname{Pic}(f)] \cong \operatorname{Hom}(\Bbb Z/p\Bbb Z, R^*)$, which is not finitely generated, since $(1+c\epsilon)^p = 1$ in $R$ for each $c \in k$. To get examples of such morphisms $\lambda$, let $E$ be an elliptic curve over $k$ which is not supersingular. Then the Tate module $T_p(E)$ is isomorphic to the $p$-adic integers $\Bbb Z_p$. According to a theorem of Serre-Lang \cite{SGA1, XI, (2.1)}, the $p$-primary part of the algebraic fundamental group $\pi_1(E)$ is $T_p(E)$. Therefore there exists a surjective homomorphism \mapx[[ \pi_1(E) || \Bbb Z/p\Bbb Z ]], and so there exists a variety $F$ and a morphism $\lambda$ as indicated. More concretely, this may be seen as follows. The multiplication by $p$ map $E\ \mapE{p}\ E$ factors through $E^{(p)}$, the scheme defined in essence by raising the coefficients in the equation defining $E$ to the \th{p} power. The induced map \mapx[[ E^{(p)} || E ]], called the {\it Verschiebung}, is exactly $\lambda$. \end{example} Now we see that there are examples with $X$ reduced and algebraic, even over an algebraically closed field (of positive characteristic), and $f$ a finite flat morphism: \begin{example}\label{random-rabbits} Take example \pref{reduction-not-faithfully-flat}, and use $X = \operatorname{Spec}(A)$, $Y = \operatorname{Spec}(B)$. Since the map \mapx[[ A || B_{\op{red}} ]] factors through $A_{\op{nor}}$, it follows that $\operatorname{Ker}[\operatorname{Pic}(f)] = k$, which is not finitely generated\ if $k$ is infinite. \end{example} Now we see that there are examples with $X$ algebraic and $Y$ \'etale, even in characteristic zero. Indeed, one may take $Y$ to be an ``affine open cover'' of $X$: \begin{example}\label{nonreduced-cover-example} Let $T$ be a projective variety over an algebraically closed field $k$, and let ${\cal{F}}$ be a coherent sheaf on $T$ with $H^1(T,{\cal{F}}) \not= 0$. Make ${\cal{A}} := \cal O_T \oplus {\cal{F}}$ into an $\cal O_T$-algebra by forcing ${\cal{F}} \cdot {\cal{F}} = 0$. Let $X = \mathop{\mathbf{Spec}}\nolimits({\cal{A}})$. Let $Y$ be an affine open cover of $X$, i.e.\ the disjoint union of the schemes in such a cover. Then $\operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(X_{\op{red}})] \subset \operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(Y)]$, and $\operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(X_{\op{red}})]$ is a nonzero vector space (over $k$), so $\operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(Y)]$ is not finitely generated. \end{example} Now we give families of examples in which $X$ and $Y$ can be chosen to be reduced noetherian schemes of characteristic zero, but the morphism is not of finite type: \begin{example} Given any ring $A$ there exists a faithfully flat extension $B$ with $\operatorname{Pic}(B) =1$: take $B=A[x]$ localized at the set of primitive polynomials (i.e.\ polynomials such that the coefficients generate the unit ideal). For the fact that $\operatorname{Pic}(B)=1$, see \cite{EG, (5.4), (3.5), (2.6 with $R=S$)}. \end{example} \begin{example} Let $k$ be a field and let $X = \operatorname{Spec}(k[t^2,t^3])$. Let $$Y = \operatorname{Spec}(k[t,t^{-1}]) \times k[t^2,t^3]_{(t^2,t^3)}),$$% which is the disjoint union of $X_{\op{reg}}$ and $\operatorname{Spec} \cal O_{X,x}$, where $x$ is the singular point of $X$. Then $\operatorname{Ker}[\operatorname{Pic}(f)] = k$. \end{example} Now we will give examples based on purely inseparable base extension. For purely inseparable extensions we cannot use Galois cohomology to control the Picard group. Instead, we use differentials, following the ideas in Samuel's notes on unique factorization domains \cite{Sam}. We build a purely inseparable form of the affine line whose Picard group is infinite. We need the following version of a result of Samuel \cite{Sam, 2.1, p.\ 62}: \begin{lemma}\label{son-of-samuel} Let $B$ be a domain of characteristic $p > 0$ with fraction field $Q(B)$. Let $\mathop{\delta}$ be a $\Bbb Z$-linear derivation of $B$ with $A$ the subring of invariants of $\mathop{\delta}$ (i.e.\ the elements with $\mathop{\delta}(a)=0$). Let $\mathop{\Delta}$ be the logarithmic derivative of $\mathop{\delta}$ defined on $Q(B)^*$ (i.e. $\mathop{\Delta}(b)=\mathop{\delta}(b)/b)$. If $M_1, M_2$ are invertible ideals of $A$ with $M_iB = b_iB$ for each $i$ ($b_1,b_2 \in B$), then $M_1 \cong M_2$ if and only if $\Delta(b_1)-\Delta(b_2) \in \Delta(B^*)$. \end{lemma} \begin{proof} If $M_1 \cong M_2$, then $b_1=aub_2$ for some $a$ in the quotient field of $A$ and $u \in B^*$. Thus, $\Delta(b_1)=\Delta(b_2) + \Delta(u)$. Conversely, if $\Delta(b_1)-\Delta(b_2) = \Delta(u)$ for some $u \in B^*$, then replacing $b_1$ by $b_1u$ allows us to assume that $\Delta(b_1)=\Delta(b_2)$ whence $\delta(b_1^{-1}b_2 )=0$. Set $a=b_1^{-1}b_2$. Thus, $a$ is in the quotient field of $A$. Replacing $M_1$ by $aM_1$ allows us to assume that $M_1B=M_2B$. We claim that this implies $M_1=M_2$. It suffices to check this locally and so we may assume that each $M_i$ is principal. Let $a_i$ be a generator for $M_i$. It follows that $a_1/a_2$ is a unit in $B$. We have $\delta(a_1/a_2) = 0$ and $a_1/a_2 \in B$, so $a_1/a_2 \in A$. Similarly, $a_2/a_1 \in A$, so $a_1/a_2 \in A^*$, and hence $M_1 = M_2$. \qed \end{proof} \begin{example}\label{deranged-derivations} Let $k$ be a separably closed imperfect field of characteristic $p$. Let $\alpha \in k - k^p$. Let $q$ be a power of $p$ with $q > 2$, and let $K = k(\alpha^{1/q})$. Let $A = k[X,Y]/(X^q-X-\alpha Y^q)$. Set $r=q/p$. Then $A$ is a Dedekind domain and $P=\operatorname{Pic}(A)$ is a group of exponent $q$ with $P/{_r}P$ infinite. Also, $A_K \cong K[Z]$. Hence the kernel of the map $\operatorname{Pic}(A) \rightarrow \operatorname{Pic}(A_K)$ is infinite of exponent $q$. \end{example} \begin{proof} Let $\beta = \alpha^{1/q}$. In the polynomial ring $B = K[Z]$, put $x = Z^q$ and $y = \beta^{-1}(Z^q - Z)$. Note that $Z = x-\beta y$, so that $K[Z] = K[x,y]$. Moreover, we can identify $A$ with the subring $k[x,y]$ of $B$. Then $A_K = B$. Thus $A$ is a Dedekind domain and $\operatorname{Pic}(A_K)$ is trivial. It follows that $\operatorname{Pic}(A)$ has exponent dividing $q$. Let $V=\{a \in k : a^q -a \in \alpha k^q\}$. Since $k$ is separably closed, for every $b \in k$ there is an element $a\in V$ such that $$a^q - a -\alpha b^q = 0.$$ Thus $V$ is infinite. Given $a \in V$, define $b \in k$ by the above equation. Let $M(a)$ be the maximal ideal $(x-a,y-b)$ of $A$. Set $c= a- \beta b=a^{1/q}$. Note that $M(a)B=(Z-c)B$. Let $a_1$ and $a_2$ be distinct elements of $V$. Let $b_i$ and $c_i$ be the corresponding elements defined above. It suffices to show that $M(a_1)^r$ and $M({a_2})^r$ are nonisomorphic. Let $\gamma = \alpha^{1/p}$. Define a derivation $\mathop{\delta}$ on $k[\gamma]$ with $\mathop{\delta}(k)=0$ and $\mathop{\delta}(\gamma)= \gamma$. Extend this derivation to $A_0=A[\gamma]$ (and to its quotient field) by taking $\mathop{\delta}$ trivial on $A$. Set $W=Z^r$. Since $Z=x - \beta y$, $W=x^r -\gamma y^r$. Thus, $\mathop{\delta}(W)= -\gamma y^r = W-W^q$. Let $\mathop{\Delta}$ denote the logarithmic derivative of $\mathop{\delta}$. The following observation will be useful: $\mathop{\delta}(c_i^r)=-\gamma b_i^r= c_i^r -c_i^{rq}$. Thus $$\mathop{\Delta}(W-c_i^r) ={{W-W^q-c_i^r+c_i^{rq}}\over{W-c_i^r}}= 1-(W-c_i^r)^{q-1}.$$ Thus, $$\Delta(W-c_1^r) - \Delta(W-c_2^r) = (W - c_2^r)^{q-1} - (W - c_1^r)^{q-1}$$ is a polynomial in $Z$ of degree $r(q-2) > 0$ as long as $ q > 2$. Let $J=(W-c_i^r) A_0$ and $I=M(a_i)^r A_0$. Then $I^pA_0=J^pA_0$, since after extension to B, they become equal. Now $A_0$ is a normal domain, so its group of invertible fractional ideals is torsion-free. Hence $I A_0 = J A_0$, i.e.\ $M(a_i)^rA_0=(W-c_i^r)A_0$. Therefore, it suffices to show (by \ref{son-of-samuel}) that $\Delta(W-c_1^r) - \Delta(W-c_2^r) \ne \Delta(u)$ for any $u \in A_0^*$. Since $\Delta(u)$ is a constant in $Z$, the result follows by the previous paragraph. \qed \end{proof} Now we give an alternate (more geometric) explanation of the preceeding result. It will show that the kernel of \mapx[[ \operatorname{Pic}(A) || \operatorname{Pic}(A_K) ]] is infinite and $q$-torsion, but not that the kernel has exponent $q$. Let $U = \operatorname{Spec}(A)$, and let $V = \operatorname{Proj}(k[X,Y,T]/(X^q-X T^{q-1}-\alpha Y^q))$. Then $U$ is an open subscheme of $V$. We have a commutative diagram \squareSE{\operatorname{Pic}(V)}{\operatorname{Pic}(U)}{\operatorname{Pic}(V_K)}{\operatorname{Pic}(U_K)\makenull{.}}% Of course $\operatorname{Pic}(U_K) = 0$ since $A_K$ is a polynomial ring. Since $U$ is obtained from $V$ by deleting the single regular point $[X,Y,T] = [1,0,0]$, \mapx[[ \operatorname{Pic}(V) || \operatorname{Pic}(U) ]] is surjective with cyclic kernel, and in fact one sees that the kernel is infinite cyclic. On the other hand, the map \mapx[[ \operatorname{Pic}(V) || \operatorname{Pic}(V_K) ]] is injective, as is well-known.% \footnote{More generally, it is even true that if $K/k$ is any field extension, $V$ is a projective $k$-scheme, and one has two coherent $\cal O_V$-modules which become isomorphic over $V_K$, then they were already isomorphic over $V$ -- see \cite{Wi, (2.3)}.} Hence the kernel of \mapx[[ \operatorname{Pic}(U) || \operatorname{Pic}(U_K) ]] is exactly $\operatorname{Pic}^0(V)$. By results of Grothendieck \cite{FGA, (2.1), (3.1)}, there exists a commutative group scheme $P$ of finite type over $k$ such that $\operatorname{Pic}^0(V) = P(k)$. Since $\operatorname{dim}(V) = 1$, $H^2(V,\cal O_V) = 0$, so $P$ is smooth and $\operatorname{dim}(P) = h^1(V,\cal O_V)$ by \cite{FGA, \#236, 2.10(ii, iii)}. Now $h^1(V,\cal O_V)$ is just the arithmetic genus of a plane curve of degree $q$, which is $(q-1)(q-2)/2$. In particular, since $q \geq 3$, we have $h^1(V,\cal O_V) > 0$. Hence $P$ is positive-dimensional. Now since $P$ is a geometrically integral scheme of finite type over a separably closed field $k$, $P(k)$ is Zariski dense in $P(k^a)$ -- see \cite{Se, discussion on p.\ 107}. In particular, since $P$ is positive-dimensional, it follows that $P(k)$ is infinite. Hence $\operatorname{Pic}^0(V)$ is infinite. Hence the kernel of the map \mapx[[ \operatorname{Pic}(U) || \operatorname{Pic}(U_K) ]] is infinite; it is $q$-torsion by \pref{kernel-is-torsion}. This completes the alternate proof of \pref{deranged-derivations}, except that we have not shown that the kernel has exponent $q$. \qed \vspace{0.03in} We will see in the next section (at least assuming resolution of singularities) that for a separated scheme $X$ of finite type over a field $k$, the kernel of the map $\operatorname{Pic}(X) \to \operatorname{Pic}(X_{k^{\text a}})$ is always bounded (i.e., torsion with finite exponent). The Picard group itself, however, can have infinite exponent. For example, if $X$ is any smooth affine or projective curve of positive genus over an algebraically closed field, then the torsion subgroup of $\operatorname{Pic}(X)$ is unbounded, because then $\operatorname{Pic}(X)$ is the quotient of an abelian variety by a finitely generated\ subgroup. \section{Eventual vanishing of the kernel of Pic}\label{eventual-section} Let $k$ be a field. The main theorem of this section \pref{eventually-injective} asserts that if $X$ is a separated $k$-scheme of finite type, then there exists a finite field extension $k^+$ of $k$, such that for every algebraic field extension $L$ of $k^+$, the canonical map $\operatorname{Pic}(X_L) \to \operatorname{Pic}(X_{L^{\text a}})$ is injective. This statement has a sheaf-theoretic formulation, which we consider, in part because it figures in the proof. Let $F$ be an (abelian group)-valued $k$-functor, meaning a functor from $\cat{$k$-algebras}$ to $\cat{abelian groups}$. Let $p: B \to C$ be a faithfully flat homomorphism of $k$-algebras. There are maps $i_1,i_2: C \to C\otimes_BC$, given by $c \mapsto c \o* 1$ and $c \mapsto 1 \o* c$, respectively. If the sequence \diagramno{(*)}{0 & \mapE{} & F(B) & \mapE{F(p)} & F(C) & \mapE{F(i_1)-F(i_2)} & F(C\o*_BC)}% is exact for all $p$, then one says that $F$ is a {\it sheaf\/} (for the fpqc [faithfully flat quasi-compact] topology). This is often too much to ask, and so one may look only at certain maps $p$, or ask only that $F(p)$ be injective. To relate this to the theorem \pref{eventually-injective}, let $F$ be given by $B \mapsto \operatorname{Pic}(X_B)/\operatorname{Pic}(B)$. Of course if $B$ is a field, we have $F(B) = \operatorname{Pic}(X_B)$. What the theorem says is that if we enlarge $k$ sufficiently (replacing it by a finite extension), then $F(L \to L^{\text a})$ is injective, for all algebraic extensions $L$ of $k$. In this sense, $F$ becomes close to being a sheaf, if we allow for the enlargement of $k$. However, in general, one cannot get $(*)$ exact in an analogous manner. More precisely, for suitable $k$ and $X$, one cannot find a finite extension $k^+$ of $k$ such that for every algebraic extension $L$ of $k^+$, if $p:L \to L^{\text a}$ is the canonical map, then $(*)$ is exact. As an example, let $k$ be a separable closure of $\Bbb F_q(t)$, for some prime $q$, and let $X = \operatorname{Spec}(k[x,y]/(y^2-x^3))$. Then $\operatorname{Pic}(X_L) = L$ for every extension field $L$ of $k$. For any finite extension $k^+$ of $k$, there is some $a \in k^+ - (k^+)^q$. If $L = k^+$, one finds that $a^{1/q}$ lies in the kernel of $F(i_1)-F(i_2)$, but not in the image of $F(p)$. \begin{definition} Let $k$ be a field. A $k$-scheme $S$ is {\it geometrically stable\/} if (1) it is of finite type, and (2) every irreducible component of $S_{\op{red}}$ is geometrically integral and has a rational point. \end{definition} \begin{theorem}\label{eventually-injective} Let $k$ be a field; assume that resolutions of singularities exist for varieties over $k^a$. Let $X$ be a separated $k$-scheme of finite type. Then there exists a finite field extension $k^+$ of $k$, such that for every algebraic field extension $L$ of $k^+$, the canonical map $\operatorname{Pic}(X_L) \to \operatorname{Pic}(X_{L^{\text a}})$ is injective. \end{theorem} \vspace{0.1in} \par\noindent{\bf Proof of (\ref{eventually-injective})} In the course of the proof, we will refer to {\it enlarging\/} $k$, by which we mean that $k$ is to be replaced by a suitably large finite field extension, contained in $k^{\text a}$. This is done only finitely many times. Then, at the end of the proof, the $k$ we have is really the $k^+$ of which the theorem speaks. We may treat $L$ as an extension of $k$ which is contained in $k^{\text a}$. By induction on $\operatorname{dim}(X)$, we may assume that the theorem holds when $X$ is replaced by a scheme of strictly smaller dimension. (The case of dimension zero is trivial.) \vspace{0.1in} \par\noindent{\bf Step 1. The case of a geometrically normal scheme} \vspace{0.1in} If $T$ is a $k$-scheme, we have let $T^{\text a}$ denote $T_{k^{\text a}}$. However, in some places in the next paragraph we shall define a $k^{\text a}$-scheme $T^{\text a}$, even though $T$ has not yet been defined; we will then proceed to construct a $k$-scheme $T$ such that $T^{\text a} = T_{k^{\text a}}$. Suppose $X$ is geometrically normal. By \cite{N}, there is a proper $k^{\text a} $-scheme ${\overline{X}}^{\text a}$ which contains $X^{\text a} $ as a dense open subscheme. (Note that ${\overline{X}}$ does not yet been defined.) After normalizing ${\overline{X}}^{\text a}$ we may assume that\ ${\overline{X}}^{\text a}$ is normal. Let $\pi^{\text a} : {\overline{Y}}^{\text a} \to {\overline{X}}^{\text a}$ be a {\it resolution of singularities}, by which we mean that ${\overline{Y}}^{\text a}$ is regular, $\pi^{\text a}$ is a proper morphism, and $\pi^{\text a}$ is an isomorphism over ${\overline{X}}^{\text a} - \operatorname{Sing}({\overline{X}}^{\text a})$. (Note that ${\overline{Y}}$ and $\pi$ have not yet been defined.) By looking at the equations defining ${\overline{X}}^{\text a}$, ${\overline{Y}}^{\text a}$, and $\pi^{\text a}$, we can (after enlarging $k$ if necessary) find a $k$-scheme ${\overline{X}}$ (containing $X$ as a dense open subscheme), a $k$-scheme ${\overline{Y}}$, and a morphism $\pi: {\overline{Y}} \to {\overline{X}}$ such that $\pi \times_k k^{\text a} = \pi^{\text a}$. Then ${\overline{X}}$ is geometrically normal, ${\overline{Y}}$ is geometrically regular, and (by faithfully flat descent \cite{EGA4, (2.7.1)(vii)}) $\pi$ is proper. Let $Y = \pi^{-1}(X)$. By enlarging $k$ if necessary, we may assume that if $C_1,\dots,C_n$ are the irreducible codimension one components of ${\overline{Y}}-Y$, and if $p: {\overline{Y}}^{\text a} \to {\overline{Y}}$ is the natural map, then $p^{-1}(C_1), \dots, p^{-1}(C_n)$ are the irreducible codimension one components of ${\overline{Y}}^{\text a} - Y^{\text a} $. Let $d : \operatorname{Pic}({\overline{Y}}^{\text a} ) \to \operatorname{Pic}(Y^{\text a} )$, $e : \operatorname{Pic}({\overline{Y}}) \to \operatorname{Pic}({\overline{Y}}^{\text a} )$, $f : \operatorname{Pic}({\overline{Y}}) \to \operatorname{Pic}(Y)$ and $h : \operatorname{Pic}(Y) \to \operatorname{Pic}(Y^{\text a} ) $ be the canonical maps. For any normal proper $k$-scheme $V$ of finite type, the canonical map $\operatorname{Pic}(V) \to \operatorname{Pic}(V^{\text a} ) $ is injective. (See \cite{Mi3, (6.2)}.) In particular, $e$ is injective. Since $$[p^{-1}(C_1)],\ldots,[p^{-1}(C_n)]$$% generate $\operatorname{Ker}(d)$, it follows that $[C_1], \ldots, [C_n]$ generate $\operatorname{Ker}(de)$. Since $f$ is surjective, it follows that $h$ is injective. A slight modification of the argument above shows that the canonical map $h_L : \operatorname{Pic}(Y_L) \to \operatorname{Pic}(Y^{\text a} ) $ is injective for every algebraic extension $L/k$. Let $\cal L$ be a line bundle on $X_L$ that becomes trivial on $X^{\text a} $. Since $h_L$ is injective, $\cal L$ becomes trivial on pullback to $Y_L$. Let $r : X^{\text a} \to X_L $ be the canonical map. Then the restriction of $\cal L$ to $X_L - r(\operatorname{Sing}(X^{\text a} ))$ is trivial. By (\ref{S2}) $\cal L$ is trivial. This completes the proof when $X$ is geometrically normal. \vspace{0.1in} \par\noindent{\bf Step 2. The case of a geometrically reduced scheme} \vspace{0.1in} If $X$ is geometrically reduced we may assume, by enlarging $k$ if need be, that the normalization $X_{\op{nor}}$ is geometrically normal. Let $\pi : X_{\op{nor}} \to X $ be the canonical map. Let $\cal I = [\cal O_X : \pi_*\cal O_{X_{\op{nor}}}]$ be the conductor of $X_{\op{nor}} $ into $X$. This is a coherent sheaf of ideals in $\cal O_X$. Let $X/\cal I$ denote $\hbox{\bf Spec}(\cal O_X/\cal I)$, and let $X_{\op{nor}}/\cal I$ denote $(X/\cal I) \times_X X_{\op{nor}} $. By further enlarging $k$, we may assume that the pullback of $\cal I$ to $X^{\text a} $ is the conductor of $(X_{\op{nor}} )^{\text a} $ into $X^{\text a} $. Enlarging $k$ still further, we may assume that $X_{\op{nor}} $, $X/\cal I$, and $X_{\op{nor}} /\cal I$ are all geometrically stable. Let $F_1$ and $F_2$ be the (abelian group)-valued $k$-functors defined by $$ F_1(B) = \frac{\Gamma((X_{\op{nor}} )_B)^*}{\Gamma(X_B)^*} \text{\ \ and\ \ } F_2(B) = \frac{\Gamma((X_{\op{nor}} /\cal I)_B)^*}{ \Gamma((X/\cal I)_B)^*}. $$ Let $G = F_2/F_1$, the quotient in $\cat{(abelian group)-valued $k$-functors}$. We have a commutative diagram $$ \begin{CD} {} @. 0 @. 0 @. {} @. {}\\ @. @VVV @VVV @. @. \\ 0 @>>> F_1(L) @>>> F_2(L) @>>> G(L) @>>> 0 \\ @. @VVV @VVV @VVV @. \\ 0 @>>> F_1(L^{\text a}) @>>> F_2(L^{\text a}) @>>> G(L^{\text a}) @>>> 0 \\ @. @VVV @VVV @VVV @. \\ 0 @>>> F_1(L^{\text a}\o*_LL^{\text a}) @>>> F_2(L^{\text a}\o*_LL^{\text a}) @>>> G(L^{\text a}\o*_LL^{\text a}) @>>> 0 \end{CD} $$ with exact rows. Now we use \cite{J2, (4.5)}: \vspace{0.1in} \par\noindent{\bf Theorem}\ \ {\it Let $S$ and $T$ be geometrically stable $k$-schemes, and let $f: S \to T$ be a dominant morphism of $k$-schemes. Let $Q$ be the (abelian group)-valued $k$-functor given by $Q(A) = \Gamma(S_A)^* / \Gamma(T_A)^*$. Let $p: B \to C$ be a faithfully flat homomorphism of reduced $k$-algebras. Then the sequence $$0 \to Q(B) \to Q(C) \to Q(C \otimes_B C)$$% is exact.} \vspace{0.1in} This implies that the first two columns are exact. It follows that the canonical map $G(L) \to G(L^{\text a})$ is injective. We need a scheme-theoretic version of Milnor's Mayer-Vietoris sequence \cite{Bas1, Chap.\ IX, (5.3)}, which may be found in \cite{We, (7.8)(i)}, and which implies that there is an exact sequence $$0 \to F_1(L) \to F_2(L) \to \operatorname{Pic}(X_L) \to \operatorname{Pic}((X_{\op{nor}})_L) \times \operatorname{Pic}((X/\cal I)_L) \eqno(\diamondsuit)$$\label{diamond}% for each algebraic extension $L/k$. By the induction hypothesis announced near the beginning of the proof, we may assume that\ $\operatorname{Pic}((X/\cal I)_L) \to \operatorname{Pic}((X/\cal I)^{\text a} )$ is injective. Also, by Step 1, $\operatorname{Pic}((X_{\op{nor}})_L) \to \operatorname{Pic}((X_{\op{nor}})^{\text a})$ is injective. Since $G(L) \to G(L^{\text a})$ is injective, it follows that $\operatorname{Pic}(X_L) \to \operatorname{Pic}(X^{\text a})$ is injective. Thus the theorem is true for geometrically reduced schemes. \vspace{0.1in} \par\noindent{\bf Step 3. Deal with the nonreduced case} \vspace{0.1in} This step may of course be ignored if $X$ is affine. Otherwise, we use the following result \cite{J2,(5.2)(a)}: \vspace{0.1in} \par\noindent{\bf Theorem}\ \ {\it Let $k$ be a field and $X$ a geometrically stable $k$-scheme. Let $i: X_0 \to X$ be a nilimmersion. For any $k$-algebra $A$, let $\kappa(A)$ be the kernel of the natural map $\operatorname{Pic}(X\times_kA) \to \operatorname{Pic}(X_0\times_kA)$. Let $A \to B$ be a faithfully flat homomorphism of reduced $k$-algebras. Then the induced map $\kappa(A) \to \kappa(B)$ is injective.} \vspace{0.1in} To complete the proof of (\ref{eventually-injective}), we may assume, by enlarging $k$ if need be, that $X$ is geometrically stable and that $k$ is big enough so that the conclusion is valid for the (geometrically reduced) scheme $X_{\op{red}}$. We have the following commutative diagram, for any algebraic extension $L/k$: $$ \begin{CD} \operatorname{Pic}(X_L) @>\alpha>> \operatorname{Pic}((X_{\op{red}})_L)\\ @V\gamma VV @V\delta VV\\ \operatorname{Pic}(X^{\text a}) @>\beta>> \operatorname{Pic}(X_{\op{red}}^{\text a}) \end{CD} $$ Taking $(A \to B) = (L \to k^{\text a})$ in \cite{J2,(5.2)(a)}, cited above, we see that $\gamma$ is one-to-one on $\operatorname{Ker}(\alpha)$. Since $\delta$ is injective, so is $\gamma$. \qed \begin{remark}\label{woofwoof} Let $F$ be a finite field of characteristic different from $5$ and containing a primitive fifth root of unity $\zeta$. Let $Y\subset {\Bbb P}^3_F$ be the Fermat quintic given by the equation: $$x^5+y^5+z^5+t^5=0.$$% Then the group of fifth roots of unity acts on $Y$ by sending $(x,y,z,t)$ to $(x,\zeta y,\zeta ^2z,\zeta^3t)$. This action has no fixed points, and the quotient $X$ is a smooth projective surface which is called the {\it Godeaux surface}. For any finite extension $E/F$, $\operatorname{CH}^2(X_E)_{\operatoratfont tors}=\Bbb Z/5\Bbb Z$ \cite{KS1, Proposition 9}. On the other hand, over an algebraic closure ${\overline F}$, we have $\operatorname{CH}^2(X_{\overline F})_{\operatoratfont tors})=0$ \cite{Mi2}. Thus for each $E$ there exists a finite extension $H$ over which the $\Bbb Z/5\Bbb Z$ dies, but a new one comes to take its place. Now for any smooth surface over a field, the natural map $K_0(X)\to \operatorname{CH}^*(X)$ is an isomorphism between the Grothendieck ring and the Chow ring. Hence we produce an element of $K_0(X)$ with similar properties. Thus the theorem does not hold with $K_0$ in place of $\operatorname{Pic}$. \end{remark} \begin{corollary}\label{bounded} Let $k$ be a field; assume that resolutions of singularities exist for varieties over $k^a$. Let $X$ be a $k$-scheme of finite type. Then the kernel of the map $\operatorname{Pic}(X) \to \operatorname{Pic}(X^a)$ is a bounded torsion group. \end{corollary} \begin{proof} One can (details omitted) use \pref{open-cover} to reduce to the case where $X$ is separated. Let $K = k^{\text a}$. Let $k^+$ be as in \pref{eventually-injective}. Then $\operatorname{Pic}(X) \to \operatorname{Pic}(X_K)$ and $\operatorname{Pic}(X) \to \operatorname{Pic}(X_{k^+})$ have the same kernel. By (\ref{kernel-is-torsion}) the kernel has exponent dividing $[k^+ : k]$. \qed \end{proof} \section{Finitely Generated Fields}\label{abs-section} We say a field $k$ is {\it absolutely finitely generated\/} if it is finitely generated over its prime subfield. In this section we will study the structure of $\operatorname{Pic}(X)$, where $X$ is a scheme of finite type over an absolutely finitely generated field. We begin with a result that is presumably well known, but for which we have found no reference. \begin{proposition}\label{normal-over} Let $X$ be a normal scheme of finite type over $\Bbb Z$ or over an absolutely finitely generated field $k$. Then $\operatorname{Pic}(X)$ is finitely generated. \end{proposition} \begin{proof} A theorem due to Roquette \cite{Ro}, \cite{L, Chap. 2, (7.6)} handles the case of schemes of finite type over $\Bbb Z$. Suppose now that $X$ is of finite type over the absolutely finitely generated field $k$. By \pref{S2} the map $\operatorname{Pic}(X) \to \operatorname{Pic}(X - \operatorname{Sing}(X))$ is injective. Therefore we may assume that\ $X$ is regular. There exists a finitely generated $\Bbb Z$-algebra $A \subset k$ and an $A$-scheme $X_0$ of finite type such that $X\cong X_0\times_A k$. We have $X \subset X_0$. Since $X$ is regular, $\cal O_{X_0,x}$ is regular for every $x\in X$. Since $X_0$ is excellent, its regular locus is open, so there exists an open subscheme of $X_0$ which is regular and contains $X$. By replacing $X_0$ by this subscheme, we may assume that $X_0$ is regular. The map on divisor class groups $\operatorname{Cl}(X_0) \to \operatorname{Cl}(X)$ is certainly surjective, and since both $X_0$ and $X$ are regular, the map $\operatorname{Pic}(X_0) \to \operatorname{Pic}(X)$ is surjective. Since $\operatorname{Pic}(X_0)$ is finitely generated (by Roquette's theorem), so is $\operatorname{Pic}(X)$. \qed \end{proof} The following examples show, in contrast, that the torsion subgroup of $\operatorname{Pic}(X)$ need not be finite if $X$ is not normal. \begin{example} Let $B = \Bbb Q[x,x^{-1}]$, and put $A = \Bbb Q+(x-1)^2B$. Then $A$ is a one-dimensional domain, finitely generated as a $\Bbb Q$-algebra, and $\operatorname{Pic}(A) \cong \Bbb Q/\Bbb Z$. In particular, $\operatorname{Pic}(A)$ is an unbounded torsion group. \end{example} \begin{proof} We note that $I:= (x-1)^2B$ is the conductor of $B$ into $A$. Therefore by Milnor's Mayer-Vietoris exact sequence \cite{Bas1, Chap. IX, (5.3)} [or see ($\diamondsuit$, p.\ \pageref{diamond}) for the scheme-theoretic version], we have $\operatorname{Pic}(A) \cong (B/I)^*/((A/I)^*U)$, where $U$ is the image of $B^*$ in $(B/I)^*$. But $(A/I)^* = \Bbb Q^*$, so $\operatorname{Pic}(A) \cong (B/I)^*/U$. Now $$B^* = \{sx^j:s\in\Bbb Q^*,j\in \Bbb Z\}\cong \Bbb Q^*\oplus \Bbb Z,$$% and $(B/I)^* \cong \Bbb Q^*\oplus W$, where $W =\{1+s(x-1):s\in\Bbb Q\} \cong \Bbb Q$. By keeping track of these identifications, one easily gets $\operatorname{Pic}(A) \cong \Bbb Q/\Bbb Z$. \qed \end{proof} By a slight modification we get an example of finite type over $\Bbb Z$: \begin{example} Fix a positive integer $m$, put $B = \Bbb Z[x,x^{-1},\frac{1}{m}]$, and let $A = \Bbb Z[\frac{1}{m}]+(x-1)^2B$. Then $A$ is a two-dimensional domain finitely generated as a $\Bbb Z$-algebra, and $\operatorname{Pic}(A) \cong \bigoplus_{p|m}\Bbb Z_{p^{\infty}}$. \end{example} The pathology in the examples above stems from the fact that $B/I$ is not reduced. Before stating our main finiteness theorems [\pref{seminormal-S2} and \pref{absolutely-p}] we record the following result from \cite{CGW, (7.4)}: \begin{theorem}\label{CGW} Let $k$ be an absolutely finitely generated\ field and let $\Lambda$ be a finite-dimensional reduced $k$-algebra. Let $E_1$ and $E_2$ be intermediate subalgebras of $\Lambda/k$. \begin{enumerate} \item If $k$ has positive characteristic $p$, then $\Lambda^*/E_1^*E_2^*$ is a direct sum of a countably generated free abelian group, a finite group, and a bounded $p$-group. \item If $\Lambda/k$ is separable, then $\Lambda^*/E_1^*E_2^*$ is a direct sum of a countably generated free abelian group and a finite group. \end{enumerate} \end{theorem} \begin{theorem}\label{seminormal-S2} Let $k$ be a field finitely generated over $\Bbb Q$ and let $X$ be a reduced $k$-scheme of finite type which is seminormal and $S_2$. Then $\operatorname{Pic}(X)$ is isomorphic to the direct sum of a free abelian group and a finite abelian group. \end{theorem} \begin{proof} Let $\cal I$ be the conductor of $X_{\op{nor}}$ into $X$ (see \S\ref{eventual-section}, Step 2). Let $X/\cal I := \hbox{\bf Spec}(\cal O_X/\cal I)$ and $X_{\op{nor}}/\cal I := (X/\cal I) \times_X X_{\op{nor}}$ denote the corresponding closed subschemes. Since $X$ is seminormal, it follows \cite{T, (1.3)} that $X/\cal I$ is reduced. Let $Q$ be the non-normal locus of $X/\cal I$, which has codimension $\geq 2$ in $X$. By \pref{S2}, the canonical map $\operatorname{Pic}(X) \to \operatorname{Pic}(X-Q)$ is injective. Therefore we may replace $X$ by $X-Q$ and start the proof over, with the added assumption that $X/\cal I$ is normal. Let $D$ be the kernel of the map $\phi: \operatorname{Pic}(X) \to \operatorname{Pic}(X/\cal I) \times \operatorname{Pic}(X_{\op{nor}})$. By (\ref{normal-over}), the target of this morphism is finitely generated. Let $\Lambda$ be the integral closure of $k$ in $\Gamma(X_{\op{nor}}/\cal I)$, $E_1$ the integral closure of $k$ in $\Gamma(X/\cal I)$, and $E_2$ the image in $\Lambda$ of the integral closure of $k$ in $\Gamma(X_{\op{nor}})$. Using \pref{roquette} and the exact sequence ($\diamondsuit$, p.\ \pageref{diamond}) with $L = k$, we see that there is an exact sequence $$\hbox{finitely generated} \to \Lambda^*/E_1^*E_2^* \to D \to \hbox{finitely generated}.$$% By (\ref{CGW})(2), $\Lambda^*/E_1^*E_2^*$ is free $\oplus$ finite. It follows that $D$ and thence $\operatorname{Pic}(X)$ is free $\oplus$ finite. \qed \end{proof} \begin{theorem}\label{absolutely-p} Let $k$ be an absolutely finitely generated field of positive characteristic $p$, and let $X$ be a $k$-scheme of finite type. Then $\operatorname{Pic}(X)$ has the form $$\hbox{(countably generated free abelian group)} \oplus \hbox{(bounded $p$-group)} \oplus \hbox{(finite group)}.$$ \end{theorem} \begin{proof} Let $\cal C$ be the class of abelian groups having the form ascribed to $\operatorname{Pic}(X)$ in the theorem. We leave to the reader to verify that $\cal C$ is closed under formation of subgroups and extensions. Induct on $\operatorname{dim}(X)$; the case where $\operatorname{dim}(X) = 0$ is trivial. We will reduce to the case where $X$ is reduced. For this, it is enough to show $(*)$ that if $\cal J \subset \cal O_X$ is a square-zero ideal, $X_0 = \hbox{\bf Spec}(\cal O_X/\cal J)$, and $\operatorname{Pic}(X_0) \in \cal C$, then $\operatorname{Pic}(X) \in \cal C$. The standard exact sequence of sheaves on $X$ $$\begin{CD} 0 @>>> \cal J @>{a\kern2pt \mapsto 1+a}>> \cal O_X^* @>>> (\cal O_X)/\cal J @>>> 1 \end{CD}$$ yields on taking cohomology an exact sequence $$H^1(X,\cal J) \to H^1(X,\cal O_X^*) \to H^1(X,(\cal O_X/\cal J)^*),$$% from which $(*)$ follows, since $H^1(X,\cal J)$ is an $\Bbb F_p$-vector space. Therefore we may assume that $X$ is reduced. Let $\cal I, D, \phi$, etc.\ be as in the proof of \pref{seminormal-S2}. (Here we do not know that $X/\cal I$ is normal.) By induction and \pref{normal-over}, the target of $\phi$ is in $\cal C$, and therefore $\operatorname{Im}(\phi)$ is also. Since $\cal C$ is closed under extensions, it suffices to show that $D \in \cal C$. Arguing as in the proof of \pref{seminormal-S2}, with \pref{CGW}(2) replaced by \pref{CGW}(1), we see that this is the case. \qed \end{proof} \begin{remark} For $k = \Bbb F_p$, it was shown in \cite{J1, (10.11)} that $\operatorname{Pic}(X)$ has the form $$\left( \oplus_{n=1}^\infty F \right) \oplus \hbox{(finitely generated\ abelian group)},$$ where $F$ is a finite $p$-group. \end{remark} \section{Complements on $K_0(X)$} Let $k$ be a field, and let $X$ be a $k$-scheme of finite type. Let $K_0(X)$ denote the Grothendieck group of vector bundles on $X$. In this section, which is purely expository, we consider the analog for $K_0(X)$ of the absolute finiteness results for $\operatorname{Pic}(X)$ proved in sections \ref{torsion-section} and \ref{abs-section}. Consider the following table: \vspace{0.15in} \vspace{0.05in} \begin{verbatim} k alg. k abs. f.g. k finite k abs f.g. closed of char. p>0 of char. 0 X arbitrary [1] finite [3] finite [4] f.g. n-torsion for torsion mod mod p^n-torsion all n p^n-torsion, for some n invertible in k for some n X regular [2] finite n-torsion [5] f i n i t e l y g e n e r a t e d for all n \end{verbatim} If we view this as a collection of statements about $\operatorname{Pic}(X)$, then all five statements are true\footnote{For statement \circno2, we have used resolution of singularities.}, as we have seen in the preceeding sections. {\bf From now on, regard the table as a table of five conjectures about $K_0(X)$.} The numbering of these conjectures is not related to the numbering of results in the introduction. It would be surprising if all five of these conjectures held. There is no field over which any of them are known to hold, even if one restricts attention to smooth projective or smooth affine schemes $X$. Conjecture \circno5\ would follow from a conjecture of Bass \cite{Bas2, \S9.1} to the effect that $K_i(X)$ is finitely generated\ for all $i \geq 0$ and all regular schemes $X$ which are of finite type over $\Bbb Z$. Over an arbitrary field $k$, it is not clear what sort of finiteness statement might hold for $K_0(X)$: there are examples of infinite $n$-torsion in the Chow groups of smooth projective varieties over a field of characteristic zero \cite{KM}. If $X$ is smooth of dimension $n$, then the operation of taking Chern classes defines a homomorphism of graded rings \mapx[[ \operatorname{Gr}[K_0(X)] || \operatorname{CH}^*(X) ]], which becomes an isomorphism after tensoring by $\Bbb Z[1 / (n-1)!]$. (See \cite{F, (15.3.6)}.) It follows that for any almost any question about $K_0(X)$, there is a parallel question about the groups $\operatorname{CH}^q(X)$, $q = 1, \ldots, n$. Note that when $q = 1$, $\operatorname{CH}^1(X) = \operatorname{Pic}(X)$. For the remainder of this section, suppose that $X$ is smooth, projective, and of dimension $n$; we give a partial discussion of results and conjectures about $\operatorname{CH}^q(X)$, for $q \geq 2$. First suppose that $k$ is algebraically closed. Some things are known when $q \in \setof{2,n}$: (i) The group ${}_m \operatorname{CH}^2(X)$ is finite if $m$ is invertible in $k$ \cite{Ra, (3.1)}; (ii) the group ${}_m \operatorname{CH}^n(X)$ is finite for every $m$. This follows from Roitman's theorem (See e.g.{\ } \cite{Ra, (3.2)}.) Hence \circno2\ holds for smooth projective surfaces. Now suppose that $k$ is a number field. Bloch has conjectured that $\operatorname{CH}^q(X)$ is finitely generated\ for all $q$. All results in this direction assume at least that $H^2(X,\cal O_X) = 0$. With this hypothesis, it has been shown that the torsion subgroup of $\operatorname{CH}^2(X)$ is finite \cite{CR}. Moreover, if $X$ is a surface which is not of general type, it is known \cite{Sal}, \cite{CR} that $\operatorname{CH}^2(X)$ is finitely generated. Hence conjecture \circno5\ holds for a smooth projective surface over a number field which is not of general type. Finally, suppose that $k$ is a finite field. Again, it is conjectured that $\operatorname{CH}^q(X)$ is finitely generated\ for all $q$. What is known is that $\operatorname{CH}^n(X)$ is finitely generated\ (in fact $\operatorname{CH}_0(X)$ is finitely generated\ for any scheme $X$ of finite type over $\Bbb Z$ \cite{KS2}), and that the torsion subgroup of $\operatorname{CH}^2(X)$ is finite \cite{CSS}, \ cf.{\ }\cite{CR, (3.7)}. The first assertion implies that conjecture \circno5\ holds for any smooth projective surface over a finite field.

9410

alg-geom/9410029

en

https://arxiv.org/abs/alg-geom/9410029

alg-geom/9410029

Marco Andreatta

M. Andreatta, M. Mella

Contractions on a manifold polarized by an ample vector bundle

18 pages, LateX

A complex manifold $X$ of dimension $n$ together with an ample vector bundle $E$ on it will be called a {\sf generalized polarized variety}. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the positivity (the nefness or ampleness) of the adjoint bundle in the case $r := rank (E) = (n-2)$. If $r\geq (n-1)$ this was previously done in a series of paper by Ye-Zhang, Fujita, Andreatta-Ballico-Wisniewski. If $K_X+detE$ is nef, then by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map $\pi :X \longrightarrow W$ from $X$ onto a normal projective variety $W$ with connected fiber and such that $K_X + det(E) = \pi^*H$, for some ample line bundle $H$ on $W$. We describe those contractions for which $dimF \leq (r-1)$. We extend this result to the case in which $X$ has log terminal singualarities. In particular this gives the Mukai's conjecture1 for singular varieties. We consider also the case in which $dimF = r$ for every fibers and $\pi$ is birational. Hard copies of the paper are available.

alg-geom

\section*{Introduction} An algebraic variety $X$ of dimension $n$ (over the complex field) together with an ample vector bundle $E$ on it will be called a {\sf generalized polarized variety}. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. Problems concerning adjoint bundles have drawn a lot of attention to algebraic geometer: the classical case is when $E$ is a (direct sum of) line bundle (polarized variety), while the generalized case was motivated by the solution of Hartshorne-Frankel conjecture by Mori ( \cite{Mo}) and by consequent conjectures of Mukai (\cite{Mu}). \par A first point of view is to study the positivity (the nefness or ampleness) of the adjoint line bundle in the case $r = rank (E)$ is about $n = dim X$. This was done in a sequel of papers for $r\geq (n-1)$ and for smooth manifold $X$ ([Ye-Zhang], [Fujita], [Andreatta-Ballico-Wisniewski]). In this paper we want to discuss the next case, namely when $rank (E) = (n-2)$, with $X$ smooth; we obtain a complete answer which is described in the theorem (4.1). This is divided in three cases, namely when $K_X + det(E)$ is not nef, when it is nef and not big and finally when it is nef and big but not ample. If $n=3$ a complete picture is already contained in the famous paper of Mori (\cite{Mo1}), while the particular case in which $E = \oplus^{(n-2)} (L)$ with $L$ a line bundle was also studied (\cite{Fu1}, \cite{So}; in the singular case see \cite{An}). The part 1 of the theorem was proved (in a slightly weaker form) by Zhang (\cite{Zh}) and, in the case $E$ is spanned by global sections, by Wisniewski (\cite{Wi2}). \par Another point of view can be the following: let $(X,E)$ be a generalized polarized variety with $X$ smooth and $rankE=r$. If $K_X + det(E)$ is nef, then by the Kawamata-Shokurov base point free theorem it supports a contraction (see (1.2)); i.e. there exists a map $\pi :X \rightarrow W$ from $X$ onto a normal projective variety $W$ with connected fiber and such that $K_X + det(E) = \pi^*H$ for some ample line bundle $H$ on $W$. It is not difficult to see that, for every fiber $F$ of $\pi$, we have $dimF \geq (r-1)$, equality holds only if $dimX > dimW$. In the paper we study the "border" cases: we assume that $dimF = (r-1)$ for every fibers and we prove that $X$ has a ${\bf P}^r$-bundle structure given by $\pi$ (theorem (3.2)). We consider also the case in which $dimF = r$ for every fibers and $\pi$ is birational, proving that $W$ is smooth and that $\pi$ is a blow-up of a smooth subvariety (theorem (3.1)). This point of view was discussed in the case $E = \oplus^r L$ in the paper [A-W]. \par Finally in the section (4) we extend the theorem (3.2) to the singular case, namely for projective variety $X$ with log-terminal singularities. In particular this gives the Mukai's conjecture1 for singular varieties. \bigskip \section{Notations and generalities} \addtocounter{subsection}{1 We use the standard notations from algebraic geometry. Our language is compatible with that of [K-M-M] to which we refer constantly. We just explain some special definitions and propositions used frequently. In particular in this paper $X$ will always stand for a smooth complex projective variety of dimension $n$. Let $Div(X)$ the group of Cartier divisors on $X$; denote by $K_X$ the {\sf canonical divisor} of $X$, an element of $Div(X)$ such that ${\cal O}_{X}(K_X) = \Omega^n_{X}$. Let $N_1(X)=\frac{\{1-cycles\}}{\equiv}\otimes {\bf R}$, $N^1(X)= \frac{\{divisors\}}{\equiv}\otimes {\bf R}$ and $\overline {<NE(X)>}=\overline{\{\mbox{effective 1-cycles}\}}$; the last is a closed cone in $N_1(X)$. Let also $\rho(X)=dim_{{\bf R}}N^1(X)<\infty$. \medskip Suppose that $K_X$ is not nef, that is there exists an effective curve $C$ such that $K_X\cdot C<0$. \begin{Theorem}\cite{KMM} Let $X$ as above and $H$ a nef Cartier divisor such that $F:= H^{\bot} \cap \overline {<NE(X)>} \setminus \{0\}$ is entirely contained in the set $\{Z\in N_1(X) :K_X\cdot Z<0\}$, where $H^{\bot} = \{Z:H\cdot Z=0\}$. Then there exists a projective morphism $\varphi:X\rightarrow W$ from $X$ onto a normal variety $W$ with the following properties: \begin{itemize} \item[{i})] For an irreducible curve $C$ in $X$, $\varphi(C)$ is a point if and only if $H.C = 0$, if and only if $cl(C) \in F$. \item[{ii})] $\varphi$ has only connected fibers \item[{iii})] $H = \varphi^*(A)$ for some ample divisor $A$ on $W$. \item[{iv})] The image $\varphi^* :Pic(W) \rightarrow Pic(X)$ coincides with $\{D \in Pic(X): D.C = 0 \mbox{ \rm for all } C \in F\}.$ \end{itemize} \label{contractionth} \end{Theorem} \begin{Definition} The following terminology is mostly used (\cite{KMM}, definition 3-2-3). Referring to the above theorem, the map $\varphi$ is called a {\sf contraction} (or an {\sf extremal contraction}); the set $F$ is an {\sf extremal face}, while the Cartier divisor $H$ is a {\sf supporting divisor} for the map $\varphi$ (or the face $F$). If $dim_{{\bf R}}F = 1$ the face $F$ is called an {\sf extremal ray}, while $\varphi$ is called an {\sf elementary contraction}. \end{Definition} \begin{remark} We have also (\cite{Mo1}) that if $X$ has an extremal ray $R$ then there exists a rational curve $C$ on $X$ such that $0< -K_X \cdot C\leq n+1$ and $R=R[C]:=\{D\in <NE(X)>: D\equiv \lambda C, \lambda\in {\bf R}^+\}$. Such a curve is called an {\sf extremal curve}. \end{remark} \begin{remark}\label{biraz} Let $\pi:X\rightarrow V$ denote a contraction of an extremal face $F$, supported by $H=\pi^*A$([iii]\ref{contractionth}) . Let $R$ be an extremal ray in $F$ and $\rho:X\rightarrow W$ the contraction of $R$. Since $\pi^*A\cdot R=0$, $\pi^*A$ comes from $Pic (W)$ ([iv]\ref{contractionth}). Thus $\pi$ factors trough $\rho$. \end{remark} \begin{Definition} To an extremal ray $R$ we can associate: \begin{itemize} \item[{i})] its {\sf length} $l(R):=min\{ -K_X\cdot C;$ for $C$ rational curve and $C\in R\}$ \item[{ii})] the {\sf locus} $E(R):=\{$the locus of the curves whose numerical classes are in $R\}\subset X$. \end{itemize} \end{Definition} \begin{Definition} It is usual to divide the elementary contractions associated to an extremal ray $R$ in three types according to the dimension of $E(R)$: more precisely we say that $\varphi$ is of {\sf fiber type}, respectively {\sf divisorial type}, resp. {\sf flipping type}, if $dim E(R) = n$, resp. $n-1$, resp. $< n-1$. Moreover an extremal ray is said not nef if there exists an effective $D\in Div(X)$ such that $D\cdot C<0$. \end{Definition} The following very useful inequality was proved in \cite{Io} and \cite{Wi3}. \begin{Proposition} Let $\varphi$ the contraction of an extremal ray $R$, $E^{\prime}(R)$ be any irreducible component of the exceptional locus and $d$ the dimension of a fiber of the contraction restricted to $E^{\prime}(R)$. Then $$ dim E^{\prime}(R)+d\geq n+l(R)-1.$$ \label{diswis} \end{Proposition} \addtocounter{subsection}{1 Actually it is very useful to understand when a contraction is elementary or in other words when the locus of two distinct extremal rays are disjoint. For this we will use in this paper the following results. \begin{Proposition}\cite[Corollary 0.6.1]{BS} Let $R_1$ and $R_2$ two distinct not nef extremal rays such that $l(R_1)+l(R_2)>n$. Then $E(R_1)$ and $E(R_2)$ are disjoint. \label{birelementare} \end{Proposition} Something can be said also if $l(R_1)+l(R_2)=n$: \begin{Proposition}\cite[Theorem 2.4]{Fu3} Let $\pi:X\rightarrow V$ as above and suppose $n\geq 4$ and $l(R_i)\geq n-2$. Then the exceptional loci corresponding to different extremal rays, are disjoint with each other. \label{n=4} \end{Proposition} \begin{Proposition}\cite{ABW1} Let $\pi:X\rightarrow W$ be a contraction of a face such that $dimX > dim W$. Suppose that for every rational curve $C$ in a general fiber of $\pi$ we have $-K_X\cdot C\geq (n+1)/2$ Then $\pi$ is an elementary contraction except if \begin{itemize} \item[a)] $-K_X\cdot C=(n+2)/2$ for some rational curve $C$ on $X$, $W$ is a point, $X$ is a Fano manifold of pseudoindex $(n+2)/2$ and $\rho(X)=2$ \item[b)] $-K_X\cdot C=(n+1)/2$ for some rational curve $C$, and $dim$W$\leq 1$ \end{itemize} \label{fibelementare} \end{Proposition} The following definition is used in the theorem: \begin{Definition} Let $L$ be an an ample line bundle on $X$. The pair $(X,L)$ is called a scroll (respectively a quadric fibration, respectively a del Pezzo fibration) over a normal variety $Y$ of dimension $m$ if there exists a surjective morphism with connected fibers $\phi: X \rightarrow Y$ such that $$K_X+(n-m+1)L \approx p^*{\cal L}$$ (respectively $K_X+(n-m)L \approx p^*{\cal L}$; respectively $K_X+(n-m-1)L \approx p^*{\cal L}$) for some ample line bundle ${\cal L}$ on $Y$. $X$ is called a classical scroll (respectively quadric bundle) over a projective variety $Y$ of dimension $r$ if there exists a surjective morphism $\phi : X\rightarrow Y$ such that every fiber is isomorphic to ${\bf P}^{n-r}$ (respectively to a quadric in ${\bf P}^{(n-r+1)}$) and if there exists a vector bundle $E$ of rank $(n-r+1)$ (respectively of rank $n-r+2$) on $Y$ such that $X\simeq {\bf P}(E)$ (respectively exists an embedding of $X$ as a subvariety of ${\bf P}(E)$). \end{Definition} \section{A technical construction} \label{tech} Let $E$ be a vector bundle of rank $r$ on $X$ and assume that $E$ is ample, in the sense of Hartshorne. \begin{remark} Let $f:{\bf P}^1\rightarrow X$ be a non constant map, and $C=f({\bf P}^1)$, \label{mha} then $detE\cdot C\geq r$. \par In particular if there exists a curve $C$ such that $(K_X+detE).C \leq 0$ (for instance if $(K_X+detE)$ is not nef) then there exists an extremal ray $R$ such that $l(R) \geq r$. \end{remark} \addtocounter{subsection}{1 \label{sopra} Let $Y={\bf P}(E)$ be the associated projective space bundle, $p:Y \rightarrow X$ the natural map onto $X$ and ${\xi}_E$ the tautological bundle of $Y$. Then we have the formula for the canonical bundle $K_Y=p^*(K_X+detE)-r{\xi}_E$. Note that $p$ is an elementary contraction; let $R$ be the associated extremal ray. Assume that $K_X+detE$ is nef but not ample and that it is the supporting divisor of an elementary contraction $\pi:X\rightarrow W$. Then $\rho(Y/W) = 2$ and $-K_Y$ is $\pi\circ p$-ample. By the relative Mori theory over $W$ we have that there exists a ray on $NE(Y/W)$, say $R_1$, of length $\geq r$, not contracted by $p$, and a relative elementary contraction $\varphi:Y\rightarrow V$. We have thus the following commutative diagram. \begin{equation} \label{dia1} \matrix{{\bf P}(E)=Y&\mapright\varphi&V\cr \mapdown{p}&&\mapdown\psi\cr X&\mapright\pi&W} \end{equation} where $\varphi$ and $\psi$ are elementary contractions. Let $w\in W$ and let $F(\pi)_w$ be an irreducible component of $\pi^{-1}(w)$; choose also $v$ in $\psi^{-1}(w)$ and let $F(\varphi)_v$ be an irreducible component of $\varphi ^{-1}(v)$ such that $p(F(\varphi)_v) \cap F(\pi)_w \not= \emptyset$; then $p(F(\varphi)_v) \subset F(\pi)_w$. This is true by the commutativity of the diagram. Since $p$ and $\varphi$ are elementary contractions of different extremal rays we have that $dim(F(\varphi)\cap F(p))=0$, that is curve contracted by $\varphi$ cannot be contracted by $p$. In particular this implies that $dim p(F(\varphi)_v) = dim F(\varphi)_v$; therefore $$dimF(\varphi)_v\leq dimF(\pi)_w.$$ \begin{remark} If $dimF(\varphi)_v=dimF(\pi)_w$, then $dim F(\psi)_w:=dim(\psi^{-1}(w)) = r-1$; if this holds for every $w \in W$ then $\psi$ is equidimensional. \end{remark} \noindent{\bf Proof. } Let $Y_w$ be an irreducible component of $p^{-1}\pi^{-1}(w)$ such that $\varphi (Y_w) = F(\psi)_w$. Then $dim F(\psi)_w = dim Y_w - dimF(\varphi)_v = dim Y_w - dimF(\pi)_w = dimF(p) = (r-1)$. \par\hfill $\Box$\par \addtocounter{subsection}{1 {\bf Slicing techniques} \label{adj} Let $H = \varphi ^*(A)$ be a supporting divisor for $\varphi$ such that the linear system $|H|$ is base point free. We assume as in (\ref{sopra}) that $( K_X + detE)$ is nef and we refer to the diagram (\ref{dia1}). The divisor $K_Y+r{\xi}_E =p^*( K_X + detE)$ is nef on $Y$ and therefore $m(K_Y+r{\xi}_E+aH)$, for $m\gg 0$, $a\in{\bf N}$, is also a good supporting divisor for $\varphi$. Let $Z$ be a smooth n-fold obtained by intersecting $r-1$ general divisor from the linear system H, i.e. $Z = H_1\cap \dots \cap H_{r-1}$ (this is what we call a {\sf slicing}); let $H_i = \varphi^{-1} A_i$. Note that the map ${\varphi}^{\prime}=\varphi_{|Z}$ is supported by $m|(K_Y+r{\xi}_E+a\varphi^*A)_{|Z}|$, hence, by adjunction, it is supported by $K_Z+rL$, where $L={{\xi}_E}_{|Z}$. Let $p^{\prime}=p_{|Z}$; by construction $p^{\prime}$ is finite. If $T$ is (the normalization of) $\varphi (Z)$ and $\psi^{\prime} :T \rightarrow W$ is the map obtained restricting $\psi$ then we have from (\ref{dia1}) the following diagram \begin{equation} \label{dia2} \matrix{Z&\mapright{\varphi\prime}&T\cr \mapdown{p\prime}&&\mapdown{\psi\prime}\cr X&\mapright\pi&W} \end{equation} In general one has a good comprehension of the map $\varphi^{\prime}$ (for instance in the case $r = (n-2)$ see the results in \cite{Fu1} or in \cite{An}). The goal is to "transfer" the information that we have on $\varphi^{\prime}$ to the map $\pi$. The following proposition is the major step in this program. \begin{Proposition} Assume that $\psi$ is equidimensional (in particular this is the case if for every non trivial fiber we have $dimF(\varphi)=dimF(\pi)$). Then $W$ has the same singularities of $T$. \label{fujita} \end{Proposition} \begin{proof} By hypothesis any irreducible reduced component $F_i$ of a non trivial fiber $F(\psi)$ is of dimension $r-1$; this implies also that $F_i=\varphi(F(p))$ for some fiber of $p$. Now, let us follow an argument as in \cite[Lemma 2.12]{Fu1}. We can assume that the divisor $A$ is very ample; we will choose $r-1$ divisors $A_i \in |A|$ as above such that, if $T = {\bigcap_{i}} A_i$, then $T \cap \psi^{-1}(w)_{red}= N$ is a reduced 0-cycle and $Z = H_1\cap \dots \cap H_{r-1}$ is a smooth n-fold, where $H_i = \varphi^{-1} A_i$. This can be done by Bertini theorem. Moreover the number of points in $N$ is given by $A^{r-1}\cdot \psi^{-1}(w)_{red}=\sum_i A^{r-1}\cdot F_i=\sum_i d_i$. Note that, by projection formula, we have $A^{r-1}\cdot F_i= \varphi^*A^{r-1}\cdot F(p)$; moreover, since $p$ is a projective bundle, the last number is constant i.e. $\varphi^*A^{r-1}\cdot F(p) = d$ for all fiber $F(p)$, that is the $d_i$'s are constant. Now take a small enough neighborhood $U$ of $w$, in the metric topology, such that any connected component $U_{\lambda}$ of $\psi^{-1}(U)\cap T$ meets $\psi^{-1}(w)$ in a single point. This is possible because $\psi^{\prime}:=\psi_{|T}: T \rightarrow W$ is proper and finite over $w$. Let $\psi_{\lambda}$ the restriction of $\psi$ at $U_{\lambda}$ and $m_{\lambda}$ its degree. Then $deg{\psi}^{\prime}=\sum m_{\lambda}\geq \sum_i d_i = \sum_i d$ and equality holds if and only if $\psi$ is not ramified at $w$ (remember that $\sum_i d_i$ is the number of $U_{\lambda}$). The generic $F(\psi)_w$ is irreducible and generically reduced. Note that we can choose $\tilde{w}\in W$ such that $\psi^{-1}(\tilde{w})= \varphi(F(p))$ and $deg{\psi}^{\prime}=A^{r-1}\cdot\psi^{-1}(\tilde{w})$, the latter is possible by the choice of generic sections of $|A|$. Hence, by projection formula $deg\psi^{\prime}= A^{r-1}\cdot \psi^{-1}(\tilde{w})= \varphi^*A^{r-1}\cdot F(p)=d$, that is $m_{\lambda}=1$ and the fibers are irreducible. Since $W$ is normal we can conclude, by Zarisky's Main theorem, that $W$ has the same singularity as $T$. \par\hfill $\Box$\par \section{Some general applications} As an application of the above construction we will prove the following proposition; the case $r = (n-1)$ was proved in \cite {ABW2}. \begin{Proposition} Let $X$ be a smooth projective complex variety and $E$ be an ample vector bundle of rank $r$ on $X$. Assume that $K_X+detE$ is nef and big but not ample and let $\pi:X\rightarrow W$ be the contraction supported by $K_X+detE$. Assume also that $\pi$ is a divisorial elementary contraction, with exceptional divisor $D$, and that $dim F\leq r$ for all fibers $F$. Then $W$ is smooth, $\pi$ is the blow up of a smooth subvariety $B: = \pi (D)$ and $E =\pi^*E^{\prime}\otimes[-D]$, for some ample $E^{\prime}$ on $W$. \label{bd} \end{Proposition} \noindent{\bf Proof. } Let $R$ be the extremal ray contracted by $\pi$ and $F:=F(\pi)$ a fiber. Then $l(R)\geq r$ and thus $dimF\geq r$ by proposition (\ref{diswis}). Hence all the fibers of $\pi$ have dimension $r$. Consider the commutative diagram (\ref{dia1}); let $R_1$ be the ray contracted by $\varphi$. Since $l(R_1)\geq r$, again by proposition (\ref{diswis}), we have that $dimF(\varphi) \geq r$ (note that $R_1$ is not nef). Therefore, since $dimF(\varphi) \leq dimF$, we have that $dimF(\varphi) = dimF = r$, $l(R)=l(R_1)=r$ and ${\xi}_E\cdot C_1=1$, where $C_1$ is a (minimal) curve in the ray $R_1$. Via slicing we obtain the map $\varphi^{\prime}:Z\rightarrow T$ which is supported by $K_Z+r{\xi}_E{}_{|Z}$. This last map is very well understood: namely by \cite[Th 4.1 (iii)]{AW} it follows that $T$ is smooth and $\varphi^{\prime}$ is a blow up along a smooth subvariety. By proposition (\ref{fujita}) also $W$ is smooth. Therefore $\pi$ is a birational morphism between smooth varieties with exceptional locus a prime divisor and with equidimensional non trivial fibers; by \cite[Corollary 4.11]{AW} this implies that $\pi$ is a blow up of a smooth subvariety in $W$. We want to show that $E =\pi^*E^{\prime}\otimes[-D]$. Let $D_1$ be the exceptional divisor of $\varphi$; first we claim that ${\xi}_E+D_1$ is a good supporting divisor for $\varphi$. To see this observe that $({\xi}_E+D_1)\cdot C_1=0$, while $({\xi}_E+D_1)\cdot C>0$ for any curve $C$ with $\varphi(C)\not= pt$ (in fact ${\xi}_E$ is ample and $D_1\cdot C\geq 0$ for such a curve). Thus ${\xi}_E+D_1=\varphi^*A$ for some ample $A\in Pic(V)$; moreover by projection formula $A\cdot l=1$, for any line $l$ in the fiber of $\psi$. Hence by Grauert theorem $V={\bf P}(E^{\prime})$ for some ample vector bundle $E^{\prime}$ on $W$. This yields, by the commutativity of diagram (1), to $E\otimes D=p_*({\xi}_E+D_1)=p_*\varphi^*A=\pi^*\psi_*A=\pi^*E^{\prime}$. \par\hfill $\Box$\par We now want to give a similar proposition for the fiber type case. \begin{Theorem} Let $X$ be a smooth projective complex variety and $E$ be an ample vector bundle of rank $r$ on $X$. Assume that $K_X+detE$ is nef and let $\pi:X\rightarrow W$ be the contraction supported by $K_X+detE$. Assume that $r\geq (n+1)/2$ and $dim F\leq r-1$ for any fiber $F$ of $\pi$. Then $W$ is smooth, for any fiber $F\simeq {\bf P}^{r-1}$ and $E_{|F}=\oplus^r{\cal O}(1)$. \label{relscroll} \end{Theorem} \noindent{\bf Proof. } Note that by proposition (\ref{diswis}) $\pi$ is a contraction of fiber type and all the fibers have dimension $r-1$. Moreover the contraction is elementar, as it follows from proposition (\ref{fibelementare}). We want to use an inductive argument to prove the thesis. If $dim W=0$ then this is Mukai's conjecture1; it was proved by Peternell, Koll\'ar, Ye-Zhang (see for instance \cite{YZ}). Let the claim be true for dimension $m-1$. Note that the locus over which the fiber is not ${\bf P}^{r-1}$ is discrete and $W$ has isolated singularities. In fact take a general hyperplane section $A$ of $W$, and $X^{\prime}=\pi^{-1}(A)$ then $\pi_{|X^{\prime}}:X^{\prime}\rightarrow A$ is again a contraction supported by $K_{X^{\prime}}+det E_{|X^{\prime}}$, such that $r\geq ((n-1)+1)/2$. Thus by induction $A$ is smooth, hence $W$ has isolated singularities. \par Let $U$ be an open disk in the complex topology, such that $U\cap SingW=\{0\}$. Then by lemma below \ref{scroll} we have locally, in the complex topology, a $\pi$-ample line bundle $L$ such that restricted to the general fiber is ${\cal O}(1)$. As in \cite[Prop. 2.12]{Fu1} we can prove that $U$ is smooth and all the fibers are ${\bf P}^{r-1}$. \par\hfill $\Box$\par \begin{Lemma} \label{scroll} Let $X$ be a complex manifold and $(W,0)$ an analityc germ such that $W\setminus \{0\}\simeq \Delta^m\setminus \{0\}$. Assume we have an holomorphic map $\pi:X\rightarrow W$ with $-K_X$ $\pi$-ample; assume also that $F\simeq {\bf P}^r$ for all fibers of $\pi$, $F\not= F_0=\pi^{-1}(0)$, and that $codim F_0\geq 2$. Then there exists a line bundle $L$ on $X$ such that $L$ is $\pi$-ample and $L_{|F}={\cal O}(1)$. \end{Lemma} \noindent{\bf Proof. } (see also \cite[pag 338, 339]{ABW2}) Let $W^*=W\setminus \{0\}$ and $X^*=X\setminus F_0$. By abuse of notation call $\pi=\pi_{|X^*}:X^*\rightarrow W^*$; it follows immediately that $R^1\pi_*{\bf Z}_{X^*}=0$ and $R^2\pi_*{\bf Z}_{X^*}={\bf Z}$. If we look at Leray spectral sequence, we have that: $$ E^{0,2}_2= {\bf Z}\mbox{ and } E^{p,1}_2= 0 \mbox{ for any p.}$$ Therefore $d_2:E^{0,2}_2\rightarrow E^{2,1}_2$ is the zero map and moreover we have the following exact sequence $$0\rightarrow E^{0,2}_{\infty}\rightarrow E^{0,2}_2\stackrel{d_3}{\rightarrow} E^{3,0}_2,$$ since the only non zero map from $ E^{0,2}_2$ is $d_3$ and hence $E^{0,2}_{\infty}=kerd_3$. On the other hand we have also, in a natural way, a surjective map $H^2(X^*,{\bf Z})\rightarrow E^{0,2}_{\infty}\rightarrow 0$. Thus we get the following exact sequence $$ H^2(X^*,{\bf Z})\stackrel{\alpha}{\rightarrow} E^{0,2}_2\rightarrow E^{3,0}_2=H^3(W^*,{\bf Z}).$$ We want to show that $\alpha$ is surjective. If $dimW := w\geq 3$ then $H^3(W^*,{\bf Z})=0$ and we have done. Suppose $w=2$ then $H^3(W^*,{\bf Z})={\bf Z}$; note that the restriction of $-K_X$ gives a non zero class (in fact it is $r+1$ times the generator) in $E^{0,2}_2$ and is mapped to zero in $E^{0,3}_2$ thus the mapping $E^{0,2}_2\rightarrow E^{3,0}_2$ is the zero map and $\alpha$ is surjective. Since $F_0$ is of codimension at least 2 in $X$ the restriction map $H^2(X,{\bf Z})\rightarrow H^2(X^*,{\bf Z})$ is a bijection. By the vanishing of $R_i\pi_*{\cal O}_X$ we get $H^2(X,{\cal O}_X)=H^2(W,{\cal O}_W)=0$ hence also $Pic(X)\rightarrow H^2(X,{\bf Z})$ is surjective. Let $L\in Pic(X)$ be a preimage of a generator of $E^{0,2}_2$. By construction $L_t$ is ${\cal O}(1)$, for $t\in W^*$. Moreover $(r+1)L=-K_X$ on $X^*$ thus, again by the codimension of $X^*$, this is true on $X$ and $L$ is $\pi$-ample. \par\hfill $\Box$\par \section{An approach to the singular case} The following theorem arose during a discussion between us and J.A. Wisniewski; we would like to thank him. The idea to investigate this argument came from a preprint of Zhang [Zh2] where he proves the following result under the assumption that $E$ is spanned by global sections. For the definition of log-terminal singularity we refer to \cite{KMM}. \begin{Theorem} Let $X$ be an n-dimensional log-terminal projective variety and $E$ an ample vector bundle of rank $n+1$, such that $c_1(E)=c_1(X)$. Then $(X,E)=({\bf P}^n,\oplus^{n+1}{\cal O}_{{\bf P}^n}(1))$. \end{Theorem} \noindent{\bf Proof. } We will prove that $X$ is smooth, then we can apply proposition (\ref{relscroll}). We consider also in this case the associated projective space bundle $Y$ and the commutative diagram \begin{equation} \matrix{{\bf P}(E)=Y&\mapright\varphi&V\cr \mapdown{p}&&\mapdown\psi\cr X&\mapright\pi&pt} \end{equation} as in (\ref{dia1}); it is immediate that $Y$ is a weak Fano variety (i.e. $Y$ is Gorenstein, log-terminal and $-K_Y$ is ample; in particular it has Cohen-Macaulay singularities); moreover, as in (3.1), $dimF(\varphi) \leq dimF(\pi) = n$ and the map $\varphi$ is supported by $K_Y+(n+1)H$, where $H ={\xi}_E + A$, with ${\xi}_E$ the tautological line bundle and $A$ a pull back of a ample line bundle from $V$. It is known that a contraction supported by $K_Y+rH$ on a log terminal variety has to have fibers of dimension $\geq (r-1)$ and of dimension $\geq r$ in the birational case (\cite [remark 3.1.2]{AW}). Therefore in our case $\varphi$ can not be birational and all fibers have dimension $n$; moreover, by the Kobayashi-Ochiai criterion the general fiber is $F\simeq {\bf P}^n$. We want to adapt the proof of \cite[Prop 1.4]{BS}; to this end we have only to show that there are no fibers of $\varphi$ entirely contained in $Sing(Y)$. Note that, by construction, $Sing(Y)\subset p^{-1}(Sing X)$ hence no fibers $F$ of $\varphi$ can be contained in $Sing(Y)$. Hence the same proof of \cite[Prop 1.4]{BS} applies and we can prove that $V$ is nonsingular and $\varphi:Y\rightarrow V$ is a classical scroll. In particular $Y$ is nonsingular and therefore also $X$ is nonsingular. \par\hfill $\Box$\par More generally we can prove the following. \begin{Theorem} Let $X$ be an n-dimensional log-terminal projective variety and $E$ be an ample vector bundle of rank $r$. Assume that $K_X+det E$ is nef and let $\pi:X\rightarrow W$ be the contraction supported by $K_X+det E$. Assume also that for any fiber $F$ of $\pi$ $dimF\leq r-1$, that $r\geq (n+1)/2$ and $codim Sing(X)>dim W$. Then $X$ is smooth and for any fiber $F\simeq {\bf P}^{r-1}$. \end{Theorem} \noindent{\bf Proof. } The proof that $X$ is smooth is as in the theorem above and then we use proposition (\ref{relscroll}) \par\hfill $\Box$\par \section{Main theorem} This section is devoted to the proof of the following theorem. \begin{Theorem} Let $X$ be a smooth projective variety over the complex field of dimension $n \geq 3$ and $E$ an ample vector bundle on $X$ of rank $r= (n-2)$. Then we have \begin{itemize} \item[1)] $K_X + det(E)$ is nef unless $(X,E)$ is one of the following: \begin{itemize} \item[{i})] there exist a smooth $n$-fold, $W$, and a morphism $\phi : X \rightarrow W$ expressing $X$ as a blow up of a finite set $B$ of points and an ample vector bundle $E'$ on $W$ such that $E = \phi^*E'\otimes[-\phi^{-1}(B)]$. \par\noindent Assume from now on that $(X,E)$ is not as in (i) above (that is eventually consider the new pair $(W,E')$ coming from (i)). \item[{ii})] $X = {\bf P}^n$ and $E =\oplus^{(n-2)}{\cal O}(1)$ or $\oplus^{2}{\cal O}(2)\oplus^{(n-4)}{\cal O}(1)$ or ${\cal O}(2)\oplus^{(n-3)}{\cal O}(1)$ or ${\cal O}(3)\oplus^{(n-3)}{\cal O}(1)$. \item[{iii})] $X = {\bf Q}^n$ and $E =\oplus^{(n-2)}{\cal O}(1)$ or ${\cal O}(2)\oplus^{(n-3)}{\cal O}(1)$ or ${\bf E}(2)$ with ${\bf E}$ a spinor bundle on ${\bf Q}^n$. \item[{iv})] $X = {\bf P}^2 \times {\bf P}^2$ and $E = \oplus^2{\cal O}(1,1)$ \item[{v})] $X$ is a del Pezzo manifold with $b_2 = 1$, i.e. $Pic(X)$ is generated by an ample line bundle ${\cal O}(1)$ such that ${\cal O}(n-1) = {\cal O}(-K_X)$ and $E = \oplus^{(n-1)}{\cal O}(1)$. \item[{vi})] $X$ is a classical scroll or a quadric bundle over a smooth curve $Y$. \par \item[{vii})] $X$ is a fibration over a smooth surface $Y$ with all fibers isomorphic to ${\bf P}^{(n-2)}$. \end{itemize} \item[2)] If $K_X + det(E)$ is nef then it is big unless there exists a morphism $\phi : X \rightarrow W$ onto a normal variety $W$ supported by (a large multiple of) $K_X + det(E)$ and $dim(W) \leq 3$; let $F$ be a general fiber of $\phi$ and $E^{\prime}=E_{|F}$. We have the following according to $s = dim W$: \begin{itemize} \item[{i})] If $s = 0$ then $X$ is a Fano manifold and $K_X + det(E) = 0$. If $n\geq 6$ then $b_2(X) = 1$ except if $X={\bf P}^3\times{\bf P}^3$ and $E=\oplus^4{\cal O}(1,1)$. \item[{ii})] If $s = 1$ then $W$ is a smooth curve and $\phi$ is a flat (equidimensional) map. Then $(F,E')$ is one of the pair described in \cite{PSW}, in particular $F$ is either ${\bf P}^n$ or a quadric or a del Pezzo variety. If $n \geq 6$ then $\pi$ is an elementary contraction. If the general fiber is $P^{n-1}$ then $X$ is a classical scroll while if the general fiber is ${\bf Q}^{n-1}$ then $X$ is a quadric bundle. \item[{iii})] If $s = 2$ and $n \geq 5$ then $W$ is a smooth surface, $\phi$ is a flat map and $(F,E^{\prime})$ is one of the pair described in the Main Theorem of \cite{Fu2}. If the general fiber is ${\bf P}^{n-2}$ all the fibers are ${\bf P}^{n-2}$. \item[{iv})] If $s = 3$ and $n \geq 5$ then $W$ is a smooth 3-fold and all fibers are isomorphic to ${\bf P}^{n-3}$. \end{itemize} \end{itemize} \item[3)] Assume finally that $K_X + det(E)$ is nef and big but not ample. Then a high multiple of $K_X + det(E)$ defines a birational map, $\varphi :X \rightarrow X'$, which contracts an "extremal face" (see section 2). Let $R_i$, for $i$ in a finite set of index, the extremal rays spanning this face; call $\rho_i: X \rightarrow W$ the contraction associated to one of the $R_i$. Then we have that each $\rho_i$ is birational and divisorial; if $D$ is one of the exceptional divisors (we drop the index) and $Z = \rho (D)$ we have that $dim(Z) \leq 1$ and the following possibilities occur: \begin{itemize} \item[{i})] $dimZ = 0$, $D = {\bf P}^{(n-1)}$ and $D_{|D} = {\cal O}(-2)$ or ${\cal O}(-1)$; moreover, respectively, $E_{|D} =\oplus^{n-2}{\cal O}(1)$ or $E_{|D} =\oplus^{n-1}{\cal O}(1)\oplus {\cal O}(2)$. \item[{ii})] $dimZ = 0$, $D$ is a (possible singular) quadric, ${\bf Q}^{(n-1)}$, and $D_{|D} = {\cal O}(-1)$; moreover $E_{|D} =\oplus^{n-2}{\cal O}(1)$. \item[{iii})] $dimZ = 1$, $W$ and $Z$ are smooth projective varieties and $\rho$ is the blow-up of $W$ along $Z$. Moreover $E_{|F} =\oplus^{n-2}{\cal O}(1)$. \end{itemize} If $n > 3$ then $\varphi$ is a composition of "disjoint" extremal contractions as in i), ii) or iii). \label{main} \end{Theorem} \noindent{\bf Proof. } Proof of part 1) of the theorem Let $(X,E)$ be a generalized polarized variety and assume that $K_X + det(E)$ is not nef. Then there exist on $X$ a finite number of extremal rays, $R_1, \dots , R_s$, such that $(K_X + det(E))^.R_i < 0$ and therefore, by the remark in section (2), $l(R_i) \geq (n-1)$. Consider one of this extremal rays, $R = R_i$, and let $\rho : X \rightarrow Y$ be its associated elementary contraction. Then $L := -(K_X+det(E))$ is $\rho$-ample and also the vector bundle $E_1 := E \oplus L$ is $\rho$-ample; moreover $K_X + det(E_1) = {\cal O}_X$ relative to $\rho$. We can apply the theorem in \cite{ABW2} which study the positivity of the adjoint bundle in the case of $rank E_1 = (n-1)$. More precisely we need a relative version of this theorem, i.e. we do not assume that $E_1$ is ample but that it is $\rho$-ample (or equivalently a local statement in a neighborhood of the exceptional locus of the extremal ray $R$). We just notice that the theorem in \cite{ABW2} is true also in the relative case and can be proved exactly with the same proof using the relative minimal model theory (see [K-M-M]; see also the section 2 of the paper \cite{AW} for a discussion of the local set up). Assume first that $\rho$ is birational, then $K_X + det(E_1)$ is $\rho$-nef and $\rho$-big; note also that, since $l(R_i) \geq (n-1)$, $\rho$ is divisorial. Therefore we are in the (relative) case C of the theorem in \cite{ABW2} (see also the proposition \ref{bd} with $r = (n-1)$); this implies that $Y$ is smooth and $\rho$ is the blow up of a point in $Y$. Since $l(R_i) \geq (n-1)$, the exceptional loci of the birational rays are pairwise disjoint by proposition (\ref{birelementare}). This part give the point {\sf (i)} of the theorem \ref{main}; i.e. the birational extremal rays have disjoint exceptional loci which are divisors isomorphic to ${\bf P}^{(n-1)}$ and which contract simultaneously to smooth distinct points on a $n$-fold $W$. The description of $E$ follows trivially (see also \cite{ABW2}). If $\rho$ is not birational then we are in the case B of the theorem in \cite{ABW2}; from this we obtain similarly as above the other cases of the theorem \ref{main}, with some trivial computations needed to recover $E$ from $E_1$. \par\hfill $\Box$\par \bigskip Proof of the part 2) of the theorem Let $K_X+detE$ be nef but not big; then it is the supporting divisor of a face $F = (K_X+detE)^{\bot}$. In particular we can apply the theorems of section (\ref{tech}): therefore there exist a map $\pi:X\rightarrow W$ which is given by a high multiple of $K_X+detE$ and which contracts the curves in the face. Since $K_X+detE$ is not big we have that $dimW<dimX$. Moreover for every rational curve $C$ in a general fiber of $\pi$ we have $-K_X\cdot C \geq (n-2)$ by the remark in section (\ref{tech}). We apply proposition (\ref{fibelementare}), which, together with the above inequality on $-K_X\cdot C$, says that $\pi$ is an elementary contraction if $n\geq 5$ unless either $n=6$, $W$ is a point and $X$ is a Fano manifold of pseudoindex $4$ and $\rho(X) = 2$ or $n = 5$ and $dimW \leq 1$. By proposition (\ref{diswis}) we have the inequality $$n+dimF\geq n+n-2-1;$$ in particular it follows that $dim W\leq 3$. \addtocounter{subsection}{1 Let $dimW=0$, that is $K_X+detE=0$ and therefore $X$ is a Fano manifold. By what just said above we have that $b_2(X)=1$ if $n \geq 6$ with an exception which will be treated in the following lemma. \begin{Lemma} Let $X$ be a $6$ dimensional projective manifold, $E$ is an ample vector bundle on $X$ of rank $4$ such that $K_X+detE=0$. Assume moreover that $b_2 \geq 2$. Then $X={\bf P}^3\times{\bf P}^3$ and $E=\oplus^4{\cal O}(1,1)$. \label{slice} \end{Lemma} \noindent{\bf Proof. } The lemma is a slight generalizzation of \cite[Prop B]{Wi1} for dimension $6$; the poof is similar and we refer to this paper. In particular as in \cite {Wi1} we can see that $X$ has two extremal rays whose contractions, $\pi_i$,$i =1,2$, are of fiber type with equidimensional fibers onto 3-folds $W_i$ and with general fiber $F_i\simeq {\bf P}^3$. We claim that the $W_i$ are smooth and thus $W_i\simeq {\bf P}^3$. First of all note that $W_i$ can have only isolated singularity and only isolated points over which the fiber is not ${\bf P}^{n-3}$; in fact let $S$ be a general hyperplane section of $W_i$ and $T_i=\pi_i^*(S)$, then $(\pi_i)_{|T_i}$ is an extremal contraction, by proposition \ref{fibelementare}; hence by \cite[Prop 1.4.1]{ABW2} $S$ is smooth; moreover the contraction is supported by $K_{T_i}+det E_{T_i}$ hence all fibers are ${\bf P}^3$ by the main theorem of \cite{ABW2}. Now we are (locally) in the hypothesis of lemma \ref{scroll} so we get, locally in the complex topology, a tautological bundle and we can conclude, by \cite[Prop 2.12]{Fu1}, that $W_i$ is smooth. Let $T = H_1 \cap H_2$, where $H_i$ are two general elements of $\pi_1^*({\cal O}(1)$. $T$ is smooth, we claim that $T\simeq {\bf P}^1\times {\bf P}^3$. In fact $\pi_{1 _{|T}}$ makes $T$ a projective bundle over a line (since $H^2({\bf P}^1,{\cal O}^*)=0$), that is $T={\bf P}({\cal F})$. Moreover $\pi_{2_{|T}}$ is onto ${\bf P}^3$, therefore the claim follows. Therefore we conclude that $\pi_i^*{\cal O}_{{\bf P}^3}(1)_{|F_i}\simeq {\cal O}_{{\bf P}^3}(1)$ for $i=1,2$. This implies by Grauert Theorem that the two fibrations are classical scroll, that is $X={\bf P}({\cal F}_i)$, for $i=1,2$; moreover computing the canonical class of $X$ the ${\cal F}_i$ are ample and the lemma easily follows. \end{proof} \addtocounter{subsection}{1 Let $dimW=1$. Then $W$ is a smooth curve and $\pi$ is a flat map. Let $F$ be a general fiber, then $F$ is a smooth Fano manifold and $E_{|F}$ is an ample vector bundle on $F$ of rank $(n-2) = dimF - 1$ such that $-K_F = det(E_{|F})$. These pairs $(F, E_{|F})$ are classified in the Main Theorem of \cite{PSW}; in particular if $dimF \geq 5$ $F$ is either ${\bf P}^{(n-1)}$ or ${\bf Q}^{(n-1)}$ or a del Pezzo manifold with $b_2(F) = 1$. Moreover if $n \geq 6$ then $\pi$ is an elementary contraction by proposition (\ref{fibelementare}). \noindent{\bf Claim } Let $n\geq 6$ and assume that the general fiber is ${\bf P}^{n-1}$, then $X$ is a classical scroll and $E_{|F}$ is the same for all $F$. (See also \cite {Fu2}) Let $S= W\setminus U$ be the locus of points over which the fiber is not ${\bf P}^{n-1}$. Over $U$ we have a projective fiber bundle. Since $H^2(U,{\cal O}^*)=0$ we can associate this ${\bf P}$-bundle to a vector bundle ${\cal F}$ over $U$. Let $Y={\bf P}({\cal F})$ and $H$ the tautological bundle; by abuse of language let $H$ the extension of $H$ to $X$. Since $\pi$ is elementary $H$ is an ample line bundle on $X$. Therefore by semicontinuity $\Delta(F,H_F)\geq \Delta(G,H_G)$, for any fiber $G$, where $\Delta(X,L)$ is Fujita delta-genus. In our case this yields $0=\Delta(F,H_F)\geq \Delta(G,H_G)\geq 0$. Moreover by flatness $(H_G)^{n-1}=(H_F)^{n-1}=1$ and Fujita classification allows to conclude. The possible vector bundle restricted to the fibers are all decomposables, hence they are rigid, that is $H^1(End(E))=\oplus_i H^1(End({\cal O}(a_i))= \oplus_i H^1({\cal O}(-a_i))=0$. Hence the decomposition is the same along all fibers of $\pi$. \noindent{\bf Claim } Let $n\geq 6$ and assume that the general fiber is ${\bf Q}^{n-1}$. Then $X$ is a quadric bundle. Let as above $S=W\setminus U$ be the locus of points over which the fiber is not a smooth quadric. Let $X^*=\pi^{-1}(U)$ then we can embed $X^*$ in a fiber bundle of projective spaces over $U$, since it is locally trivial. Associate this $P$-bundle over $U$ to a projective bundle and argue as before. \par\hfill $\Box$\par \addtocounter{subsection}{1 Let now $dimW=2$ and assume that $n\geq 5$; then $\pi$ is an elementary contraction. This implies first, by \cite[Prop. 1.4.1]{ABW2}, that $W$ is smooth; secondly that $\pi$ is equidimensional, hence flat and the general fiber is ${\bf P}^{n-2}$ or ${\bf Q}^{n-2}$, see \cite{Fu2}. \noindent{\bf Claim } Let $n\geq 5$ and the general fiber is ${\bf P}^{n-2}$ then for any fiber $F\simeq {\bf P}^{n-2}$ and $E_{|F}$ is the same for all $F$. Let $S\subset W$ be the locus of singular fibers, then $dimS\leq 0$ since $W$ is normal. Let $U\subset W$ be an open set, in the complex topology, with $U\cap S=\{0\}$ and let $V\subset X$ such that $V=\pi^{-1}(U)$. We are in the hypothesis of lemma \ref{scroll} thus we get a "tautological" line bundle $H$ on $V$ and we conclude by \cite[Prop. 2.12]{Fu1}. There are two possible restriction of $E$ to the fiber, namely $E_{|F}\simeq {\cal O}(2)\oplus(\oplus^{n-1}{\cal O}(1))$ or $E_{|F}$ is the tangent bundle. As observed by Fujita in \cite{Fu2} this two restrictions have a different behavior in the diagram (\ref{dia1}), in the former $\varphi$ is birational while in the latter it is of fiber type. Hence the restriction has to be constant along all the fibers. \par\hfill $\Box$\par \addtocounter{subsection}{1 Let finally $dimW=3$; the general fiber is ${\bf P}^{n-3}$ (see for instance \cite{Fu2}). Assume that $n\geq 5$, therefore $\pi$ is elementary; we claim that all fibers are ${\bf P}^{n-3}$. Since $\pi$ is elementary any fiber $G$ has $cod G\geq 2$. Let $S\subset W$ be the locus of point over which the fiber is not ${\bf P}^{n-3}$; $dimS\leq 0$ since a generic linear space section can not intersect $S$, by the above. Let $U\subset W$ be an open set, in the complex topology, with $U\cap S=\{0\}$ and let $V\subset X$ such that $\pi(V)=U$. Then by lemma \ref{scroll} we get a "tautological" line bundle $H$ on $V$; $\pi: V\rightarrow U$ is supported by $K_V+(n-2)H$. Thus by \cite[Th 4.1]{AW} $U$ is smooth and all the fibers are ${\bf P}^{n-3}$ ( we use that $n\geq 5$). \par\hfill $\Box$\par Proof of the part 3) of the theorem In the last part of the theorem we assume that $K_X+detE$ is nef and big but not ample. Then $K_X+detE$ is a supporting divisor of an extremal face, $F$; let $R_i$ the extremal rays spanning this face. Fix one of this ray, say $R = R_i$ and let $\pi:X\rightarrow W$ be the elementary contraction associated to $R$. We have $l(R)\geq n-2$; this implies first that the exceptional loci are disjoint if $n > 3$, proposition (\ref{n=4}). Secondly, by the inequality (\ref{diswis}), we have $$dimE(R)+dimF(R)\geq 2n - 3.$$ Therefore $dimE(R)=n-1$ and either $dimF(R) = n-1$ or $dimF(R) = n-2$; if $Z := \rho (E)$ and $D=E(R)$ this implies that either $dimZ = 0$ or $1$. If $dimZ = 1$ then $dim F(\pi) = n-2$ for all fibers (note that since the contraction $\pi$ is elementary there cannot be fiber of dimension $(n-1)$); thus we can apply proposition (\ref{bd}) with $r = (n-2)$. This will give the case 3-(iii) of the theorem. Consider again the construction in section (\ref{tech}), in particular we refer to the diagram (\ref{dia1}). Let $S$ be the extremal ray contracted by $\varphi$; note that $l(S)\geq n-2$ and that the inequality (\ref{diswis}) gives $$dimE(S)+dimF(S)\geq 3n - 6;$$ in particular, since $dim F(S) \leq dim F(R) $, we have two cases, namely $dimE(S) = 2n-5$ and $dimF(S) = (n-1)$ or $dimE(S) = 2n-4$ and $dimF(S) = (n-1)$ or $(n-2)$. The case in which $dimE(S) = 2n-5$ will not occur. In fact, after "slicing", (see \ref{adj}), we would obtain a map $\varphi^{\prime}=\varphi_{|Z}$ which would be a small contraction supported by a divisor of the type $K_Z+(n-2)L$ but this is impossible by the classification of \cite[Th 4]{Fu1} (see also \cite{An}). \medskip Hence $dimE(S)=2n-4$, that is also $\varphi$ is divisorial. Suppose that the general fiber of $\varphi$, $F(S)$, has dimension $(n-2)$. After slicing we obtain a map ${\varphi}^{\prime}=\varphi_{|Z}: Z \rightarrow T$ supported by $K_Z+(n-2)L$, where $L={{\xi}_E}_{|Z}$. This map contracts divisors $D$ in $Z$ to curves; by (\cite[Th 4]{Fu1}) we know that every fiber $F$ of this map is ${\bf P}^{(n-2)}$ and that $D_{|F} = {\cal O}(-1)$ (actually this map is a blow up of a smooth curve in a smooth variety). In particular there are curves in $Y$, call them $C$, such that $-E(S).C = 1$. We will discuss this case in a while. Suppose then the general fiber of $\varphi$, $F(S)$, has dimension $(n-1)$; therefore all fibers have dimension $(n-1)$. Slicing we obtain a map ${\varphi}^{\prime}=\varphi_{|Z}: Z \rightarrow T$ supported by $K_Z+(n-2)L$, where $L={{\xi}_E}_{|Z}$. This map contracts divisors $D$ in $Z$ to points; by (\cite{Fu1}) we know that these divisors are either ${\bf P}^{(n-1)}$ with normal bundle ${\cal O}(-2)$ or ${\bf Q}^{(n-1)}\subset {\bf P}^n$ with normal bundle ${\cal O}(-1)$. In the latter case we have as above that there are curves $C$ in $Y$, such that $-E(S).C = 1$. In these cases observe that $E(S)\cdot \tilde{C}=0$, where $\tilde{C}$ is a curve in the fiber of $p$. Hence $E(S)=p^*(-M)$ for some $M\in Div(X)$. Let $l$ be an extremal curve of $E(S)$. Then, by projection formula, we have $-1=E(S)\cdot l=-M\cdot mC$ and thus $M$ generates $Im[Pic(X)\rightarrow Pic(D)]$, hence $M$ is $\pi$-ample; note that in general it does not generate $Pic(D)$. We study now the Hilbert polynomial of $M_{|D}$ to show that $\Delta(D,M_{|D})=0$, where $\Delta(X,L)$ is Fujita delta genus. Let ${\cal O}_D(-K_X) \simeq{\cal O}_D(pM)$, where $p=l(R)\geq n-2$, and ${\cal O}_D(-D)\simeq{\cal O}_D(qM)$ for some $p,q\in {\bf N}$. By adjunction formula $\omega_D\simeq {\cal O}_D(-(p+q)M)$. By \cite[Lemma 2.2]{Ando} or \cite[pag 179]{BS}, Serre duality and relative vanishing we obtain that $q\leq 2$, the Hilbert polynomial is $$P(D,M_{|D})= \frac{a}{(n-1)!}(t+1)\cdots(t+(n-2))(t+c)$$ and the only possibilities are $a=1, c=n-1, q=1 or 2$ and $a=2, c=(n-1)/2, q=1$. In particular $\Delta(D,M_{|D})=0$ and, by Fujita classification, $D$ is equal to ${\bf P}^{(n-1)}$ or to ${\bf Q}^{(n-1)}\subset {\bf P}^n$. Now the rest of the claim in 3) i) and ii) follows easily. It remains the case in which ${\varphi}^{\prime}=\varphi_{|Z}: Z \rightarrow T$ contracts divisors $D= {\bf P}^{(n-1)}$ with normal bundle ${\cal O}(-2)$ to points. We can apply the above proposition (\ref{fujita}) and show that the singularities of $W$ are the same as those of $T$. Then, as in (\cite{Mo1}), this means that we can factorize $\pi$ with the blow up of the singular point. Let $X^{\prime}=Bl_{w}(W)$, then we have a birational map $g:X\rightarrow X^{\prime}$. Note that $X^{\prime}$ is smooth and that $g$ is finite. Actually it is an isomorphism outside $D$ and cannot contract any curve of $D$. Assume to the contrary that $g$ contracts a curve $B\subset D$; let $N\in Pic(X^{\prime})$ be an ample divisor then we have $g^*N\cdot B=0$ while $g^*N\cdot C\not=0$ contradiction. Thus by Zarisky's main theorem $g$ is an isomorphism. This gives a case in 3)i). \small

9410

alg-geom/9410030

en

https://arxiv.org/abs/alg-geom/9410030

alg-geom/9410030

Oleg Viro

Oleg Viro

Self-linking number of a real algebraic link

7 pages, AMS-LaTeX with 6 pictures in the format of *.EPS files

For a nonsingular real algebraic curve in the 3-dimensional projective space and sphere a new numeric characteristic is introduced. It takes integer values, is invariant under rigid isotopy, multiplied by -1 under mirror reflection. In a sense it is a Vassiliev invariant of degree 1 and a counter-part of a link diagram writhe.

alg-geom

\section{Introduction}\label{sI} In the classical knot theory by a link one means a smooth closed 1-dimensional submanifold of the 3-dimensional sphere $S^3$, i.~e. several disjoint circles smoothly embedded into $S^3$. A classical link may emerge as the set of real points of a real algebraic curve. First, it gives rise to questions about relations between invariants of the same curve which are provided by link theory and algebraic geometry. Second, it suggests to develop a theory parallel to the classical link theory, but taking into account the algebraic nature of objects. From this viewpoint it is more natural to consider real algebraic links up to isotopy consisting of real algebraic links, which belong to the same continuous family of algebraic curves, rather than up to smooth isotopy in the class of classical links. I call an isotopy of the former kind a {\it rigid isotopy\/} following a terminology established by Rokhlin \cite{R} in a similar study of real algebraic plane projective curves and extended later to various other situations (see, e.g., \cite{Viro New pr.}). Of course, there is a forgetting functor: any real algebraic link can be considered as a classical link and a rigid isotopy as a smooth isotopy. It is interesting, how much is lost under this transition. In this note I point out a characteristic of a real algebraic link which is lost. It is unexpectedly simple. In an obvious sense it is a nontrivial Vassiliev invariant of degree 1 on the class of real algebraic knots.\footnote{Recall that a knot is a link consisting of one component.} In the classical knot theory the lowest degree of a nontrivial Vassiliev knot invariant is 2. Thus there is an essential difference between classical knot theory and the theory of real algebraic knots. The characteristic of real algebraic links which is defined below is very similar to self-linking number of framed knots. I call it also {\it self-linking number.\/} Its definition looks like a refinement of an elementary definition of the writhe of a knot diagram, but taking into consideration the imaginary part of the knot. \section{Self-linking of a nonalgebraic knot}\label{slknonalg}In the classical theory, a self-linking number of a knot is defined only if the knot is equipped with an additional structure like framing or just a vector field nowhere tangent to the knot.\footnote{A framing is a pair of orthogonal to each other normal vector fields on a knot. There is an obvious construction which makes a framing from a nontangent vector field and establishes one to one correspondence between homotopy classes of framings and nontangent vector fields. The vector fields are more flexible and relevant to the case.} The self-linking number is the linking number of the knot oriented somehow and its copy obtained by a small shift in the direction specified by the vector field. It does not depend on the choice of orientation, since reversing the orientation of the knot is compensated by reversing the induced orientation of its shifted copy. Of course, it depends on the homotopy class of the vector field. A knot has no natural preferable homotopy class of framings which would allow to speak about a self-linking number of the knot without a special care on choice of a framing.\footnote{Moreover, the self-linking number is used to define a natural class of framings: namely, framings with the self-linking number zero.} Some framings appear naturally in geometric situations. For example, if one fixes a generic projection of a knot to a plane, the vector field of directions of the projection appears. The corresponding self-linking number is called the {\it writhe\/} of the knot. However, it depends on the choice of projection and changes under isotopy. The linking number is a Vassiliev invariant of order 1 of two-component oriented links. That means that it changes by a constant (in fact, by 2) when the link experiences a homotopy with a generic appearance of an intersection point of the components. Whether the linking number increases or decreases depends only on the local picture of orientations near the double point: when it passes from $\vcenter{\epsffile{f01.eps}}$ through $\vcenter{\epsffile{f02.eps}}$ to $\vcenter{\epsffile{f03.eps}}$, the linking number increases by 2. Generalities on Vassiliev invariants see, e.~g., in \cite{V} In a sense the linking number is the only Vassiliev invariant of degree 1 of two-component oriented links: any Vassiliev invariant of degree 1 of two-component oriented links is a linear function of the linking number. Similarly, the self-linking number is a Vassiliev invariant of degree 1 of framed knots (it changes by 2 when the knot experiences a homotopy with a generic appearance of a self-intersection point) and it is the only Vassiliev of degree 1 of framed knots in the same sense. Necessity of framing for definition of self-linking number can be formulated now more rigorously: only constants are Vassiliev invariants of degree 1 of (nonframed) knots. The definition of the writhe, which is mimicked below, runs as follows: for each crossing point of the knot projection one defines a {\it local writhe\/} equal to $+1$ if near the point the knot diagram looks like $\vcenter{\epsffile{f03.eps}}$ and $-1$ if it looks like $\vcenter{\epsffile{f01.eps}}$. Then one sums up the local writhes over all double points of the projection. The sum is the writhe. A continuous change of the projection may cause vanishing of a crossing point. It happens under the first Reidemeister move shown in the left hand half of Figure \ref{f1}. This move changes the writhe by $\pm 1$. \section{How algebraicity enhances self-linking number}\label{Genrem}If a link is algebraic then its projection to a plane is algebraic, too. A generic projection has only ordinary double points. The total number of complex double points is constant. The number of real double points can change, but only by an even number. A real double point cannot turn alone into imaginary one, as it seems happen under the first Reidemeister move. Under the algebraic version of the first Reidemeister move the double point stays in the real domain, but becomes solitary, like the only real point of the curve $x^2+y^2=0$. The algebraic version of the first Reidemeister move is shown in the right hand half of Figure \ref{f1}. It is not difficult to prove that the corresponding family of plane curves can be transformed by a local diffeomorphism to the family of real rational cubic curves $y^2=x^2(t-x)$ with $t\in \Bbb R$. \begin{figure}[h] \centerline{\epsffile{f1.eps}} \caption{Topological and real algebraic versions of the first Reidemeister move} \label{f1} \end{figure} A solitary double point of the projection is not image of any real point of the link. It is the image of two imaginary complex conjugate points of the complexification of the link. The preimage of the point in the 3-space under the projection is a real line. It is disjoint from the real part of the link, but intersects its complexification in a couple of complex conjugate imaginary points. In the next section with any solitary double point of the projection, a local writhe equal to $\pm1$ is associated. It is done in such a way that the local writhe of the crossing point vanishing in the first Reidemeister move is equal to the local writhe of the borning solitary double point. In the case of an algebraic knot the sum of local writhes of all double points, both solitary and crossings, does not depend on the choice of projection and is invariant under rigid isotopy. This sum is the self-linking number. There are two types of generic deformations of an algebraic link changing the rigid isotopy type. One of them is exactly as in the category of classical links: two pieces of the set of real points come to each other and pass through each other. A generic projection of the link experiences an isotopy. No events happen besides that one crossing point becomes for a moment the image of a double point of the link and then turns back into a crossing point, but with the opposite writhe. Another type has no counter-part in the topological context. Two complex conjugate imaginary branches pass through each other. At the moment of passing they intersect in a real isolated double point. A generic projection of the link experiences an isotopy. No events happen besides that one solitary double point becomes for a moment the image of an isolated double point of the link and then turns back into a usual solitary double point, but with the opposite writhe. It is clear that the self-linking number of an algebraic knot changes under both modifications by $\pm2$ with the sign depending only on the local structure of the modification near the double point. It means that the self-linking number is the Vassiliev invariant of degree 1. A construction similar to the construction of the self-linking number of an algebraic knot can be applied to algebraic {\it links.\/} However in this case it is necessary either to orient the link or to exclude from the sum the crossings where the branches belong to distinct components of the set of real points. In fact, the local writhe depends on the orientations of the branches, but if the branches belong to the same component orientations of the branches can be induced from the same orientation of the component. It is easy to see that the result does not depend on the choice of orientation of the component. In the case of knots, self-linking number defines a natural class of framings, since for knots homotopy classes of framings are enumerated by their self-linking numbers and we can choose the framing having the self-linking number equal to the algebraic self-linking number constructed here. I do not know any direct construction of this framing. Moreover, there seems to be a reason for absence of such a construction. In the case of links the construction above gives a single number, while framings are enumerated by sequences of numbers with entries corresponding to components. The construction of this paper can be applied to algebraic links in the sphere $S^3$. Although from the viewpoint of knot theory this is the most classical case, from the viewpoint of algebraic geometry the case of curves in the projective space is simpler, and I will start from it. The case of spherical links is postponed to the Section \ref{srS}. \section{Real algebraic projective links}\label{s0}Let $A$ be a nonsingular real algebraic curve in the 3-dimensional projective space. Then the set $\Bbb R A$ of its real points is a smooth closed 1-dimensional submanifold of $\Bbb R P^3$, i.~e. a smooth projective link. The set $\Bbb C A$ of its complex points is a smooth complex 1-dimensional submanifold of $\Bbb C P^3$. Let $c$ be a point of $\Bbb R P^3$. Consider the projection $p_c:\Bbb C P^3\smallsetminus c\to \Bbb C P^2$ from $c$. Assume that $c$ is such that the restriction to $\Bbb C A$ of $p_c$ is generic. This means that it is an immersion without triple points and at each double point the images of the branches have distinct tangent lines. As it follows from well-known theorems, those $c$'s for which this is the case form an open dense subset of $\Bbb R P^3$ (in fact, it is the complement of a 2-dimensional subvariety). The real part $p_c(\Bbb C A)\cap\Bbb R P^2$ of the image consists of the image $p_c(\Bbb R A)$ of the real part and, maybe, several solitary points, which are double points of $p_c(\Bbb C A)$. There is a purely topological construction which assigns a local writhe equal to $\pm1$ to a crossing belonging to the image of only one component of $\Bbb R A$. This construction is well-known in the case of classical knots. Here is its projective version. I borrow it from Drobotukhina's paper \cite{Dr} on generalization of Kauffman brackets to links in the projective space. Let $K$ be a smooth connected one-dimensional submanifold of $\Bbb R P^3$, and $c$ be a point of $\Bbb R P^3\smallsetminus K$. Let $x$ be a generic double point of the projection $p_c(K)\subset \Bbb R P^2$ and $L\subset \Bbb R P^3$ be the line which is the preimage of $x$ under the projection. Denote by $a$ and $b$ the points of $L\cap \Bbb R P^3$. \begin{figure}[t] \centerline{\epsffile{f2.eps}} \caption{Construction of the frame $v$, $l$, $w'$.} \label{f2} \end{figure} The points $a$ and $b$ divides the line $L$ into two segments. Choose one of them and denote it by $S$. Choose an orientation of $K$. Let $v$ and $w$ be tangent vectors of $K$ at $a$ and $b$ respectively directed along the selected orientation of $K$. Let $l$ be a vector tangent to $L$ at $a$ and directed inside $S$. Let $w'$ be a vector at $a$ such that it is tangent to the plane containing $L$ and $w$ and is directed to the same side of $S$ as $w$ (in an affine part of the plane containing $S$ and $w$). See Figure \ref{f2}. The triple $v$, $l$, $w'$ is a base of the tangent space $T_a\Bbb R P^3$. The value taken by the orientation of $\Bbb R P^3$ on this frame is the local writhe of $x$. Its definition involves several choices. However it is easy to prove that the result does not depend on them. Let $A$, $c$ and $p_c$ be as in the beginning of this Section and let $s\in\Bbb R P^2$ be a solitary double point of $p_c$. Here is a construction assigning $\pm1$ to $s$. I will call the result also a {\it local writhe\/} at $s$. Denote the preimage of $s$ under $p_c$ by $L$. This is a real line in $\Bbb R P^3$ connecting $c$ and $s$. It intersects $\Bbb C A$ in two imaginary complex conjugate points, say, $a$ and $b$. Since $a$ and $b$ are conjugate they belong to different components of $\Bbb C L\smallsetminus\Bbb R L$. Choose one of the common points of $\Bbb C A$ and $\Bbb C L$, say, $a$. The natural orientation of the component of $\Bbb C L\smallsetminus\Bbb R L$ defined by the complex structure of $\Bbb C L$ induces orientation on $\Bbb R L$ as on the boundary of its closure. The image under $p_c$ of the local branch of $\Bbb C A$ passing through $a$ intersects the plane of the projection $\Bbb R P^2$ transversally at $s$. Take the local orientation of the plane of projection such that the local intersection number of the plane and the image of the branch of $\Bbb C A$ is $+1$. Thus the choice of one of two points of $\Bbb C A\cap\Bbb C L$ defines an orientation of $\Bbb R L$ and a local orientation of the plane of projection $\Bbb R P^2$ (we can speak only on a local orientation of $\Bbb R P^2$, since the whole $\Bbb R P^2$ is not orientable). The plane of projection intersects\footnote{We may think on the plane of projection as embedded into $\Bbb R P^3$. If you would like to think on it as on the set of lines of $\Bbb R P^3$ passing through $c$, please, identify it in a natural way with any real projective plane contained in $\Bbb R P^3$ and disjoint from $c$. All such embeddings $\Bbb R P^2\to\Bbb R P^3$ are isotopic.} transversally $\Bbb R L$ in $s$. The local orientation of the plane, orientation of $\Bbb R L$ and the orientation of the ambient $\Bbb R P^3$ determine the intersection number. This is the local writhe. It does not depend on the choice of $a$. Indeed, if one chose $b$ instead, then both the orientation of $\Bbb R L$ and the local orientation of $\Bbb R P^2$ would be reversed. The orientation of $\Bbb R L$ would be reversed, because $\Bbb R L$ receives opposite orientations from different halves of $\Bbb C L\smallsetminus\Bbb R L$. The local orientation of $\Bbb R P^2$ would be reversed, because the complex conjugation involution $%\operatorname{conj: \Bbb C P^2\to\Bbb C P^2$ preserves the complex orientation of $\Bbb C P^2$, preserves $\Bbb R P^2$ (point-wise) and maps one of the branches of $p_c(\Bbb C A)$ at $s$ to the other reversing its complex orientation. Now for any real algebraic projective link $A$ choose a point $c\in\Bbb R P^3$ such that the projection of $A$ from $c$ is generic and sum up writhes at all crossing points of the projection belonging to image of only one component of $\Bbb R A$ and writhes of all solitary double points. The sum is called the {\it self-linking number of $A$.\/} It does not depend on the choice of projection. Moreover it is invariant under {\it rigid isotopy\/} of $A$. By rigid isotopy we mean an isotopy made of nonsingular real algebraic curves. The effect of a movement of $c$ on the projection can be achieved by a rigid isotopy defined by a path in the group of projective transformations of $\Bbb R P^3$. Therefore the following theorem implies both independence of the self-linking number on the choice of projection and its invariance under rigid isotopy. \begin{thm}\label{mainth} For any two rigidly isotopic real algebraic projective links $A_1$ and $A_2$ such that their projections from the same point $c\in\Bbb R P^3$ are generic, the self-linking numbers of $A_1$ and $A_2$ defined via $c$ are equal.\end{thm} To prove this statement, first replace any rigid isotopy by a generic one. As in purely topological situation of classical links, any generic rigid isotopy may be decomposed to a composition of rigid isotopies, each of which makes a local standard move of the projection. There are 5 local standard moves. They are similar to the Reidemeister moves. The first of these 5 moves is shown in the right hand half of Figure \ref{f1}. The next two coincide with the second and third Reidemeister moves. The fourth move is similar to the second Reidemeister move: also two double points of projection come to each other and disappear. However the double points are solitary. The fifth move is similar to the third Reidemeister move: also a triple point appears for a moment. But at this triple point only one branch is real, the other two are imaginary conjugate to each other. In this move a solitary double point traverses a real branch. Only in the first, fourth and fifth moves solitary double points are involved. The invariance under the second and the third move follows from well-known fact of knot theory that the topological writhe is invariant under the second and third Reidemeister moves. Thus we have to prove that:\begin{enumerate} \item in the first move the writhe of vanishing crossing point is equal to the writhe of the borning solitary point, \item in the fourth move the writhes of the vanishing solitary points are opposite and \item in the fifth move the writhe of the solitary point does not change. \end{enumerate} The proof is not complicated, but would take room inappropriate in this short note. The same construction may be applied to real algebraic curves in $\Bbb R P^3$ having singular imaginary points, but no real singularities. In the construction we can avoid usage of projections from the points such that some singular point is projected from it to a real point. Indeed, for any imaginary point there exists only one real line passing through it (the line connecting the point with its complex conjugate), thus we have to exclude a finite number of real lines. \section{Real algebraic links in sphere}\label{srS} The three-dimensional sphere $S^3$ is a real algebraic variety. It is a quadric in the four-dimensional real affine space. A stereographic projection is a birational isomorphism of $S^3$ onto $\Bbb R P^3$. It defines a diffeomorphism between the complement of the center of projection in $S^3$ and a real affine space. Given a real algebraic link in $S^3$, one may choose a real point of $S^3$ from the complement of the link and project the link from this point to an affine space. Then include the affine space into the projective space and apply the construction above. The image has no real singular points, therefore we can use the remark from the end of the previous section. \section{Other generalizations}\label{sGeneralizations} It is difficult to survey all possible generalizations. Here I indicate only two directions. First, consider the most straightforward generalization. Let $L$ be a nonsingular real algebraic $(2k-1)$-dimensional subvariety in the projective space of dimension $4k-1$. Its generic projection to $\Bbb R P^{4k-2}$ has only ordinary double points. At each double point either both branches of image are real or they are imaginary complex conjugate. If set of real points is orientable then one can repeat everything from Section \ref{s0} with obvious changes and obtain a definition of a numeric invariant generalizing the self-linking number defined in Section \ref{s0}. Let $M$ be a nonsingular three-dimensional real algebraic variety with oriented set of real points equipped with a real algebraic fibration over a real algebraic surface $F$ with fiber a projective line. There is a construction which assigns to a real algebraic link (i.~e., a nonsingular real algebraic curve in $M$) with a generic projection to $F$ an integer, which is invariant under rigid isotopy, multiplied by $-1$ under reversing of the orientation of $M$ and is a Vassiliev invariant of degree 1. This construction is similar to that of Section \ref{s0}, but uses, instead of projection to $\Bbb R P^2$, an algebraic version of Turaev's shadow descriptions of links \cite{T}.

9410

alg-geom/9410019

en

https://arxiv.org/abs/alg-geom/9410019

alg-geom/9410019

Bernd Siebert

B.Siebert and G.Tian

Recursive relations for the cohomology ring of moduli spaces of stable bundles

15 pages, Latex

The authors learnt that similar results have been independently found by D.Zagier, V.Baranovsky and V.Balaji/A.King/P.Newstead. The corresponding references have been added (and some typos corrected).

alg-geom

\section{Method and notation} Let us now fix a Riemann surface $\Sigma$ of genus $g$ and a line bundle $L$ on $\Sigma$ of odd degree. We write ${\calm_g}={{\cal M}}_g(\Sigma,L)$. Generators for $H^*({\calm_g},{\Bbb Q})$ occur as coefficients in the K\"unneth decomposition of a characteristic class associated to the universal bundle ${\cal U}$ over ${\calm_g}\times\Sigma$ \[ c_2({\mbox{\rm End}\skp\skp}{\cal U})\ =\ -\beta+4\psi+2\alpha\otimes\omega\,. \] Here $\omega\in H^2(\Sigma,{\Bbb Z})$ is the normalized volume form and $\psi$ is the part of type $(3,1)$. Choosing a standard basis $e_i$ for $H^1(\Sigma,{\Bbb Z})$, $i=1,\ldots,2g$ (s.th.\ $e_i e_j=0$ unless $i\equiv j(g)$ and $e_i e_{i+g}[\Sigma]=1$), $\psi$ decomposes further $\psi=\sum_{i=1}^{2g} \psi_i\otimes e_i$. The classes $\alpha\in H^2({\calm_g},{\Bbb Q})$, $\psi_i \in H^3({\calm_g},{\Bbb Q})$ and $\beta\in H^4({\calm_g},{\Bbb Q})$ are actually integral and generate the cohomology ring \cite{newstead1}, \cite{atiyah-bott}. There is an interesting subring of $H^*({{\cal N}}_g,{\Bbb Q})$ to which the intersection pairing may easily be reduced by geometric arguments as noted by Thaddeus \cite{thaddeus}. Namely, any orientation preserving diffeomorphism of $\Sigma$ induces a diffeomorphism of ${{\cal N}}_g$ by acting on $\pi_1$. The corresponding action on $H^*({\calm_g},{\Bbb Q})$ leaves $c_2({\mbox{\rm End}\skp\skp}{\cal U})$ fixed. Thus $\alpha$ and $\beta$ are invariant and the $\psi_i$ transform dually to the $e_i$. It is not hard to show that the ring of such transformations is precisely the subring generated by $\alpha$, $\beta$ and a newly defined class $\gamma:=2\sum_{i=1}^{2g}\psi_i\psi_{i+g}$, or more intrinsically $\psi^2=\gamma\omega$ (in view of the functional equation for the generating function $\Phi$ it might be more natural to take twice this class, but to be consistent with the work of Newstead and Thaddeus we keep this definition). On the other hand, there is a method introduced by Mumford to construct relations among the generators, cf.\ \cite{atiyah-bott}: Letting $L$ vary among the line bundles of fixed (odd) degree $D$ (the space of which we denote by ${\mbox{\rm Pic}\skp}^D(\Sigma)$), one gets a moduli space $\tilde{{\cal M}}(\Sigma)$ and a fibration $\tilde{{\cal M}}(\Sigma)\rightarrow{\mbox{\rm Pic}\skp}^D(\Sigma)$ with fibre ${\cal N}_g$. In rational cohomology this fibration is trivial, i.e.\ $H^*(\tilde{{\cal M}}(\Sigma),{\Bbb Q})\simeq H^*({\cal N}_g,{\Bbb Q})\otimes_{\Bbb Q} H^*({\mbox{\rm Pic}\skp}^D(\Sigma),{\Bbb Q})$. But ${\mbox{\rm Pic}\skp}^D(\Sigma)\simeq{\mbox{\rm Pic}\skp}^0(\Sigma)$ and $H^*({\mbox{\rm Pic}\skp}^0(\Sigma),{\Bbb Q})$ is a free alternating algebra in $2g$ generators ${\hat e}_i$ of degree 1 (the push-forwards of $e_i$ under the Jacobi map $\Sigma\rightarrow{\mbox{\rm Pic}\skp}^0(\Sigma)$). Now let $\tilde{\cal U}$ be the universal bundle over ${\tilde{\cal M}}_g\times\Sigma$ and let $\tilde\pi:{\tilde{\cal M}}_g \times\Sigma\rightarrow{\tilde{\cal M}}_g$ be the projection. If $D=4g-3$, $R^1\tilde\pi_*\tilde{\cal U}=0$ and $\tilde\pi_*\tilde{\cal U}$ is locally free of rank $2g-1$. By Grothendieck-Riemann-Roch then the Chern classes of $\tilde\pi_*\tilde{\cal U}=\tilde\pi_!\,\tilde{\cal U}$ are expressed as polynomials in $\alpha$, $\beta$, $\psi_i$ and $\hat e_i$. Equating to zero the coefficients of $\hat e_{i_1}\ldots\hat e_{i_\nu}$ in $c_r(\tilde\pi_*\tilde{\cal U})$, $r\ge 2g$ (which vanish for rank reasons) gives a number of relations among the generators $\alpha$, $\beta$ and $\psi_i$. The point of letting $L$ vary is of course to lower the degrees of the relations by up to $2g$. The smallest degree of a relation we thus obtain is $4g-2g=2g$. The conjecture of Mumford which Kirwan recently succeeded to prove as remarked in the introduction, is that this set of relations is complete. But in view of the explicit formula for the intersection pairing of Thaddeus the authors could not believe in this end of the story. In fact, computer evidence (up to $g=18$ using ``Macaulay'' \cite{mcly}) showed that for the subring generated by $\alpha$, $\beta$, $\gamma$ there should be only three independent relations of degrees $g$, $g+1$, $g+2$ respectively, coming from the lowest degree equation $c_{2g}=0$. Unfortunately, explicit computations are rather arduous, e.g.\ due to the presence of the odd degree classes $\hat e_i$. To find relations of low degrees without bringing ${\mbox{\rm Pic}\skp}(\Sigma)$ into the game we first remark that because the ${{\cal M}}(\Sigma,L)$ are all diffeomorphic as long as we fix the genus of $\Sigma$ and the degree of $L$ modulo 2, we may restrict ourselves to a hyperelliptic curve $\Sigma$. In this case, there is a closed embedding $\varphi:{\calm_g}\hookrightarrow G(g+3,2g+2)$ into a Grassmannian as follows \cite{ramanan}: Let $p:\Sigma\rightarrow{\rm I \! P}^1$ represent $\Sigma$ as two-fold covering of ${\rm I \! P}^1$, $\iota:\Sigma\rightarrow\Sigma$ the corresponding hyperelliptic involution (s.th.\ $p\circ\iota=p$) and $B\subset{\rm I \! P}^1$ the branch locus of $p$ ($\sharp B=2g+2$). Now let $L$ be a line bundle of degree $2g+1$ (as opposed to $d=4g-3$ in Mumford's method) and let $E$ be a stable 2-bundle over $\Sigma$ with determinant $L$. Applying $\iota^*$ to $E\otimes\iota^* E$ and switching factors induces an involution $J: E\otimes\iota^*E\rightarrow E\otimes\iota^* E$. Denote by $p_*(E\otimes \iota^*E)^\natural$ the $J$-anti-invariant subsheaf of $p_*(E\otimes\iota^* E)$. One shows $h^0({\rm I \! P}^1,p_*(E\otimes\iota^* E)^\natural)=g+3$ (loc.\ cit., Prop.~2.2). In a branch point $t\in B$ we have the identification \[ p_*(E\otimes\iota^* E)^\natural_t\ \simeq\ p_*(E\wedge E)_t \ =\ (p_*L)_t\,. \] The map $\varphi:{\calm_g}\rightarrow G(g+3,2g+2)$ is then defined as \[ E\longmapsto\left(H^0\Big({\rm I \! P}^1,p_*(E\otimes\iota^* E)^\natural \Big)\Big|_B\subset H^0({\rm I \! P}^1,p_* L|_B)\simeq\cz^{2g+2}\right)\,. \] Now let $S$ and $Q$ be the universal bundle and the universal quotient bundle on $G(g+3,2g+2)$ respectively. The key observation is that $Q$ has rank $g-1$! Similarly to Mumford's method we ``just'' have to express the Chern classes of $\varphi^* Q$ in terms of $\alpha$, $\beta$, $\gamma$ (essentially by Grothendieck-Riemann-Roch of course) to get relations $c_r(\varphi^* Q)=0$, $r=g,g+1,g+2$. The Chern class computations are based on the following exact sequence. \begin{lemma}\label{exact} On ${\calm_g}\times\Sigma$, there is an exact sequence \[ 0\longrightarrow\hat p^*\hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural \longrightarrow{\cal U}\otimes\iota^*{\cal U}\longrightarrow S^2{\cal U}|_{{\calm_g}\times p^{-1}(B)}\longrightarrow 0\,, \] where $\hat p={\mbox{\rm Id}\skp}\times p:{\calm_g}\times\Sigma\rightarrow{\calm_g}\times{\rm I \! P}^1$. \end{lemma} {\em Proof. } The restriction map $\hat p^*\hat p_* E\rightarrow E$ is injective for any locally free sheaf $E$ by flatness of $p$. Composing this map with the inclusion $\hat p^*\hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural\hookrightarrow \hat p^*\hat p_*({\cal U}\otimes\iota^*{\cal U})$ yields exactness at the first place. Outside of the branch points this map is obviously an isomorphism (by anti-invariance elements of $\hat p^*\hat p_*({\cal U}\otimes \iota^*{\cal U})^\natural$ are uniquely determined by their behaviour on one branch). It is then a matter of linear algebra to check that at a branch point $y\in\Sigma$ the cokernel is given by mapping a germ of sections of ${\cal U} \otimes\iota^*{\cal U}$ to $(s+J(s))(y)\in{\cal U}\otimes\iota^*{\cal U} |_{{\calm_g}\times p^{-1}(B)}\simeq S^2{\cal U}|_{{\calm_g}\times p^{-1}(B)}$. {\hfill$\Diamond$}\vspace{1.5ex} \section{Computations of Chern classes} This section contains the computational heart of the paper, summarized in the following proposition. We adopt the convention that in writing Chern classes or Chern characters as analytic functions in certain cohomology classes we understand to evaluate on these classes the corresponding power series expansion. \begin{prop}\label{chernquot} Denote by $c(\varphi^*Q)=\sum_{i\ge0}c_i(\varphi^*Q)$ the total Chern class of $\varphi^*Q$. Then \[ c(\varphi^*Q)=(1-\beta)^{-1/2}\exp\bigg[\alpha+\Big(\alpha+ \frac{2\gamma}{\beta}\Big)\sum_{m\ge1}\frac{\beta^m}{2m+1}\bigg]\,. \] \end{prop} Note that the $c_i(\varphi^*(Q))$ are really {\em polynomials} in $\alpha$, $\beta$, $\gamma$, the denominator $\beta$ cancels. Also, if one prefers, one could write $({\mbox{\rm arctanh}}(\sqrt{\beta})/\sqrt{\beta})-1$ (with $\sqrt{\beta}$ formally adjoint to $H^*({\calm_g})$) instead of the infinite sum. Before turning to the proof we will need some preparations. Letting $\pi:{\calm_g}\times{\rm I \! P}^1\rightarrow{\calm_g}$ and $\tilde\pi=\pi\circ\hat p:{\calm_g}\times \Sigma\rightarrow{\calm_g}$ be the projections, we know $\varphi^*S={\tilde\pi}_*( {\cal U}\otimes\iota^*{\cal U})^\natural\otimes H^{-1}$ with $H\in{\mbox{\rm Pic}\skp}({\calm_g}) \simeq{\Bbb Z}$ the ample generator and ${\tilde\pi}_*( {\cal U}\otimes\iota^*{\cal U})^\natural:=\pi_*(\hat p_* ({\cal U}\otimes\iota^*{\cal U})^\natural)$ \cite{ramanan}. To apply Grothendieck-Riemann-Roch to this sheaf we first need a closed formula for the Chern character of ${\cal U}\otimes\iota^*{\cal U}$. We will use a compuational trick which the authors learned from \cite[...]{kirwan} to represent most of the terms in exponential form. This will be convenient later when we transform back to Chern classes. For that fix a large number $N$ such that $\beta^N=0$ (e.g.\ for dimension reasons) and let $\mu_1,\ldots,\mu_N\in\cz$ be such that \[ N_k(\mu_1,\ldots,\mu_N)\ :=\ \sum_{\nu=1}^N\mu_\nu^k\ =\ \frac{1}{k+1} \ \ \ \mbox{for } 1\le k\le N. \] The existence of $\mu_1,\ldots,\mu_N$ is clear either by an elementary argument or from the finiteness of the map $(N_1,\ldots,N_N):\cz^N\rightarrow\cz^N$. \begin{lemma} Formally adjoining $\sqrt{\beta}$ and $\alpha':=\alpha+2\gamma/\beta$ (both of real degree 2) to $H^{2*}({\calm_g},{\Bbb Q})$ the following holds: \begin{eqnarray*} {\mbox{\rm ch}\skp}({\cal U}\otimes\iota^*{\cal U})&=&e^\alpha\bigg[\Big( 2+e^{\sqrt{\beta}}+e^{-\sqrt{\beta}}\Big)(1+D\omega)-2\alpha\omega\\ &&-\,\alpha'\omega\sum_\nu \Big(e^{\mu_\nu\sqrt{\beta}}+e^{-\mu_\nu\sqrt{\beta}}-2\Big)\bigg]\,. \end{eqnarray*} \end{lemma} {\em Proof. } We have $c({\cal U}^*\otimes{\cal U})=1+c_2({\mbox{\rm End}\skp\skp}{\cal U})=1-\beta+4\psi+2\alpha\omega$ with $\psi=\sum_i\psi_i\otimes e_i$. Note that $\iota^*:H^1(\Sigma)\rightarrow H^1(\Sigma)$ is just multiplication by $-1$. In fact, if $\delta$ is a closed $1$-form on $\Sigma$, then $\delta+\iota^*\delta$ is closed and $\iota^*$-invariant, hence $\delta+\iota^*\delta=p^*df=dp^*f$ for $f\in{\cal C}^\infty({\rm I \! P}^1)$ since $H^1({\rm I \! P}^1)=0$. But any orientation preserving diffeomorphism leaves $c({\mbox{\rm End}\skp\skp}{\cal U})$ unchanged, hence $\iota^*\psi_i=-\psi_i$. Now it is almost clear and can be easily verified by a standard Chern class computation that $c_2({\cal U}^*\otimes\iota^*{\cal U})=-\beta+2\alpha\omega$, i.e.\ that the factor $4\psi$ drops out. Instead, there is a non-trivial $c_4$, namely $c_4({\cal U}^*\otimes\iota^*{\cal U})=4\psi^2=4\gamma\omega$. Summarizing, we have \[ c({\cal U}^*\otimes\iota^*{\cal U})\ =\ 1+(-\beta+2\alpha\omega) +4\gamma\omega\,. \] Next, for any bundle $E$ with only $c_2$ and $c_4$ non-vanishing \[ {\mbox{\rm ch}\skp}_{2k}(E)\ =\ 2(-1)^k c_2(E)^{k-2}[c_2(E)^2-kc_4(E)],\ \ \ {\mbox{\rm ch}\skp}_{2k+1}(E)=0 \] (induction on $k$). Thus for $k\ge2$ ($k=1$: ${\mbox{\rm ch}\skp}_2({\cal U}^*\otimes \iota^*{\cal U} )=-2c_2=2\beta-4\alpha\omega$) \begin{eqnarray*} {\mbox{\rm ch}\skp}_{2k}({\cal U}^*\otimes\iota^*{\cal U})&=&2(-1)^k\left((-\beta)^{k-2} +(k-2)(-\beta)^{k-3}2\alpha\omega\right)(\beta^2-4\alpha\omega-4k \gamma\omega)\\ &=&2\beta^k-4k(\alpha\beta+2\gamma)\beta^{k-2}\omega\,. \end{eqnarray*} Formally adjoining $\sqrt{\beta}$ and $\alpha'=\alpha+2\gamma/\beta$ ($\beta$ in the denominator always cancels in the following) we get \begin{eqnarray*} {\mbox{\rm ch}\skp}({\cal U}^*\otimes\iota^*{\cal U})&=&2+2\sum_{k\ge0}\frac{\beta^k}{(2k)!} -2(\alpha\beta+2\gamma)\omega\sum_{k\ge2}2k\frac{\beta^{k-2}}{(2k)!} -\frac{4\alpha\omega}{2}\\ &=&2+e^{\sqrt{\beta}}+e^{-\sqrt{\beta}}-2\alpha\omega-2\alpha'\omega \sum_{k\ge1}\frac{1}{2k+1}\frac{\beta^k}{(2k)!}\,. \end{eqnarray*} The computational trick consists in writing \[ 2\sum_{k\ge1}\frac{1}{2k+1}\frac{\beta^k}{(2k)!}\ =\ \sum_{\nu=1}^N\Big(e^{\mu_\nu\sqrt{\beta}} +e^{-\mu_\nu\sqrt{\beta}}-2\Big)\,. \] Finally using $c_1({\cal U})=\alpha+D\omega$ ($D=2g+1$) together with the isomorphim ${\cal U}\simeq{\cal U}^*\otimes\det{\cal U}$ and the multiplicativity of the Chern character we deduce ${\mbox{\rm ch}\skp}({\cal U}\otimes\iota^*{\cal U})=e^\alpha {\mbox{\rm ch}\skp}({\cal U}^*\otimes\iota^*{\cal U})$ which upon inserting the previous computations gives the stated formula. {\hfill$\Diamond$}\vspace{1.5ex} \vspace{10pt} Pushing-forward the exact sequence from Lemma~\ref{exact} we get \[ 0\longrightarrow \hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural \otimes\hat p_*{\cal O}\longrightarrow \hat p_*({\cal U}\otimes\iota^*{\cal U}) \longrightarrow\hat p_*\left(S^2{\cal U}|_{{\calm_g}\times p^{-1}(B)}\right) \longrightarrow0 \] (for the first term apply the projection formula). \begin{lemma} Denoting $\bar\omega$ the normalized volume form on ${\rm I \! P}^1$ the following holds \begin{eqnarray*} {\mbox{\rm ch}\skp}\left(\hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural\right) &=&e^\alpha\bigg[(2+e^{\sqrt{\beta}}+e^{-\sqrt{\beta}}) (1-\frac{\bar\omega}{2})\\ &&+\,\bar\omega\Big(-\alpha-\frac{\alpha'}{2}\sum_\nu\big( e^{\mu_\nu\sqrt{\beta}}+e^{-\mu_\nu\sqrt{\beta}}-2\big) +(g+1)\Big)\bigg]\,. \end{eqnarray*} \end{lemma} {\em Proof. } {}From the above exact sequence, we see \[ {\mbox{\rm ch}\skp}\left(\hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural\right) \ =\ \left({\mbox{\rm ch}\skp} \hat p_*({\cal U}\otimes\iota^*{\cal U})-{\mbox{\rm ch}\skp} \hat p_*(S^2{\cal U}|_{{\calm_g}\times p^{-1}(B)})\right)/{\mbox{\rm ch}\skp}(\hat p_*{\cal O}) \] which by Grothendieck-Riemann-Roch applied to $\hat p$ and writing $\bar g=g-1$ and ${\mbox{\rm ch}\skp}({\cal U}\otimes\iota^*{\cal U})-{\mbox{\rm ch}\skp}(S^2 {\cal U}|_{{\calm_g}\times p^{-1}(B)})=A+B\omega$, equals \begin{eqnarray*} \hat p_*\left[(A+B\omega)(1-\bar g\omega)\right]/ \hat p_*(1-\bar g\omega) \ =\ \hat p_*\Big(A+(B-\bar g A)\omega\Big)/(2-\bar g\bar\omega)\\ =\ \Big(2A+(B-\bar g A)\bar\omega\Big)\frac{1}{2}(1+ \frac{\bar g}{2}\bar\omega)\ =\ A+B\frac{\bar\omega}{2}\,. \end{eqnarray*} To compute the Chern character of $S^2{\cal U}|_{{\calm_g}\times p^{-1}(B)}$ we restrict the exact sequence of Lemma~\ref{exact} to ${\calm_g}\times p^{-1}(B)$. Then since $\hat p^*\hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural|_{{\calm_g}\times p^{-1} (B)}$ $\simeq\ \det{\cal U}|_{{\calm_g}\times p^{-1}(B)}$ we get \[ {\mbox{\rm ch}\skp}(S^2{\cal U}|_{{\calm_g}\times p^{-1}(B)})\ =\ \left[{\mbox{\rm ch}\skp}({\cal U}\otimes\iota^* {\cal U})-{\mbox{\rm ch}\skp}(\det{\cal U})\right](2g+2)\omega\,. \] Thus \begin{eqnarray*} A+B\omega&=&{\mbox{\rm ch}\skp}({\cal U}\otimes\iota^*{\cal U})\Big(1-(2g+2)\omega\Big) +{\mbox{\rm ch}\skp}(\det{\cal U})(2g+2)\omega\\ &=&e^\alpha\bigg[(2+e^{\sqrt{\beta}}+e^{-\sqrt{\beta}}) (1-\omega)\\ &&+\,\omega\Big(-\alpha-\frac{\alpha'}{2}\sum_\nu\big(e^{\mu_\nu \sqrt{\beta}}+e^{-\mu_\nu\sqrt{\beta}}-2\big)+(g+1)\Big)\bigg]\,, \end{eqnarray*} hence the claim. {\hfill$\Diamond$}\vspace{1.5ex} \vspace{10pt} \noindent {\em Proof of Proposition~\ref{chernquot}}: It follows from Proposition~2.2 of \cite{ramanan} applied to ${\cal U}\otimes\iota^*{\cal U}$ that $R^1\pi_*(\hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural)=0$. The Grothendieck-Riemann-Roch theorem for pushing-forward $\hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural$ by $\pi:{\calm_g}\times{\rm I \! P}^1\rightarrow{\calm_g}$ thus reads \begin{eqnarray*} \lefteqn{{\mbox{\rm ch}\skp}(\tilde\pi_*({\cal U}\otimes\iota^*{\cal U})^\natural)\ =\ \pi_*\left({\mbox{\rm ch}\skp}(\hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural)\cdot (1+\bar\omega)\right)}\hspace{1cm}\\ &=&e^\alpha\bigg[\frac{1}{2}(2+e^{\sqrt{\beta}}+e^{-\sqrt{\beta}})-\alpha -\frac{\alpha'}{2}\sum_\nu\Big(e^{\mu_\nu\sqrt{\beta}}+ e^{-\mu_\nu\sqrt{\beta}}-2\Big)+(g+1)\bigg]\,. \end{eqnarray*} Plugging in the relation expressing the pull-back of S, i.e.\ $\varphi^*S=\tilde\pi_*({\cal U}\otimes\iota^*{\cal U})^\natural\otimes H^{-1}$, we get \begin{eqnarray*} {\mbox{\rm ch}\skp}\varphi^*Q&=&(2g+2)-{\mbox{\rm ch}\skp}\varphi^*S\ =\ (2g+2)-{\mbox{\rm ch}\skp}\tilde\pi_*({\cal U}\otimes\iota^*{\cal U})^\natural/{\mbox{\rm ch}\skp}(H)\\ &=&(g-1)-\frac{1}{2}\Big(e^{\sqrt{\beta}}+e^{-\sqrt{\beta}}\Big)+\alpha +\frac{\alpha'}{2}\sum_\nu\Big(e^{\mu_\nu\sqrt{\beta}} +e^{-\mu_\nu\sqrt{\beta}}-2\Big)\,. \end{eqnarray*} Next we need to make use of the computational trick of Kirwan again: Assume $M\gg 0$ s.th.\ $\alpha^M=\alpha'^M=0$ and find $\lambda_\kappa\in\cz$ with $N_1(\lambda_1,\ldots,\lambda_M)=1$, $N_k(\lambda_1,\ldots,\lambda_M)=0$ for $2\le k\le M$. Then $\alpha=\sum_\kappa(e^{\lambda_\kappa\alpha}-1)$, $\alpha'=\sum_\kappa(e^{\lambda_\kappa\alpha'}-1)$. Inserting we get \begin{eqnarray*} {\mbox{\rm ch}\skp}\varphi^*Q&=&(g-1)-\frac{1}{2}\Big(e^{\sqrt{\beta}}+e^{-\sqrt{\beta}}\Big) +\sum_\kappa(e^{\lambda_\kappa\alpha}-1)\\ &&+\frac{1}{2}\sum_{\kappa,\nu}\left(e^{\lambda_\kappa\alpha'+\mu_\nu \sqrt{\beta}}+e^{\lambda_\kappa\alpha'-\mu_\nu\sqrt{\beta}} -e^{\mu_\nu\sqrt{\beta}}-e^{-\mu_\nu\sqrt{\beta}}-2(e^{\lambda_\kappa \alpha'}-1)\right)\,. \end{eqnarray*} This is a sum of exponentials and as such easily transformed into the corresponding total Chern class: \begin{eqnarray*} c(\varphi^*Q)&=&\Big[\Big(1+\sqrt{\beta}\Big)\Big(1-\sqrt{\beta}\Big) \Big]^{-1/2} \prod_\kappa(1+\lambda_\kappa\alpha)\\ &&\cdot\,\bigg[\prod_{\kappa,\nu} \frac{1+\mu_\nu\sqrt{\beta}+\lambda_\kappa\alpha'}{1+\mu_\nu\sqrt{\beta}} \cdot \frac{1-\mu_\nu\sqrt{\beta}+\lambda_\kappa\alpha'}{1-\mu_\nu\sqrt{\beta}} \cdot\frac{1}{(1+\lambda_\kappa\alpha')^2}\bigg]^{1/2}\!\!. \end{eqnarray*} To get rid of the $\lambda_\kappa$ we observe that $\sigma_k(\lambda_1, \ldots,\lambda_M)=1/k!$ \cite[p.862]{kirwan}. The product over $\kappa$ can thus be carried out, e.g. \begin{eqnarray*} \lefteqn{\prod_{\kappa,\nu}\frac{1+\mu_\nu\sqrt{\beta}+\lambda_\kappa \alpha'}{1+\mu_\nu\sqrt{\beta}}\ =\ \prod_{\kappa,\nu}\bigg(1+\lambda_\kappa\frac{\alpha'}{1+\mu_\nu\sqrt{\beta}} \bigg)\ =\ \prod_\nu\exp\frac{\alpha'}{1+\mu_\nu\sqrt{\beta}}}\hspace{4cm}\\ &=&\exp\alpha'\sum_\nu\sum_{l\ge0}\Big(-\mu_\nu\sqrt{\beta}\Big)^l\ =\ \exp\alpha'\sum_{l\ge0}\frac{(-\sqrt{\beta})^l}{l+1}\,. \end{eqnarray*} Inserting into our last formula we thus find (the term $(1+\lambda_\kappa \alpha')^{-2}$ cancels the summand for $l=0$) \begin{eqnarray*} c(\varphi^*Q)&=&(1-\beta)^{-1/2}\exp\bigg(\alpha+\frac{\alpha'}{2}\sum_{l\ge1} \frac{(-\sqrt{\beta})^l+(\sqrt{\beta})^l}{l+1}\bigg)\\ &=&(1-\beta)^{-1/2}\exp\bigg(\alpha+\alpha'\sum_{m\ge1}\frac{1}{2m+1} \beta^m\bigg) \end{eqnarray*} as claimed. {\hfill$\Diamond$}\vspace{1.5ex} \vspace{10pt} It is convenient to gather the relations in a generating function. \begin{defi}\label{genfct} We define $\Phi\in{\Bbb Q}[\alpha,\beta,\gamma][\![t]\!]$ by \[ \Phi(t)\ :=\ (1-\beta t^2)^{-1/2} \exp\bigg[\alpha t+\Big(\alpha+\frac{2\gamma}{\beta} \Big)t\sum_{m\ge 1}\frac{\beta^m t^{2m}}{2m+1}\bigg]\,. \] \end{defi} For later use let us also state a functional equation that $\Phi$ obeys. This equation is actually equivalent to the recursion formula to be proved below (Proposition~\ref{recursion}). \begin{prop}\label{fctleqn} $\Phi$ obeys the following differential equation: \[ \Phi'(t)\ =\ \frac{\alpha+\beta t+2\gamma t^2}{1-\beta t^2} \cdot\Phi(t)\,. \] \end{prop} {\em Proof. } Direct computation. {\hfill$\Diamond$}\vspace{1.5ex} \vspace{10pt} Let us add that with the same methods, it is not hard to deduce also a closed formula for the Chern classes of ${\cal N}_g$. The result is: \[ c({\cal N}_g)\ =\ (1-\beta)^g \exp\Big(\frac{-8\gamma}{1-\beta}\Big)\cdot c(\varphi^*Q)^2\,. \] Note the simple dependence on the genus! \section{A minimal set of relations} We have remarked in the introduction that the three generating relations $f_1^g$, $f_2^g$, $f_3^g$ of degrees $g$, $g+1$ and $g+2$ are uniquely determined by their initial terms $\alpha^g$, $\alpha^{g-1}\beta$ and $\alpha^{g-1}\gamma$ respectively (w.r.t.\ the reverse lexicographic order; we will prove this as an easy consequence of the recursion relations, see Proposition~\ref{inf}). It is then an exercise in calculus to find the following \begin{defi} Writing $\Phi^{(r)}=\displaystyle\frac{d^r\Phi}{dt^r}(0)$ we define for $g\ge1$ \begin{eqnarray*} f_1^g&:=&\Phi^{(g)}\\ f_2^g&:=&\frac{1}{g^2}\left(\Phi^{(g+1)}-\alpha\Phi^{(g)}\right)\\ f_3^g&:=&\frac{1}{2g(g+1)}\left(\Phi^{(g+2)}-\alpha\Phi^{(g+1)} -(g+1)^2\beta\Phi^{(g)}\right)\,. \end{eqnarray*} \end{defi} \vspace{10pt} We are now in a position to prove the recursion relations. \begin{prop}\label{recursion} $(f_1^1,f_2^1,f_3^1)=(\alpha,\beta,\gamma)$ and inductively for $g\ge1$ \begin{eqnarray*} f_1^{g+1}&=&\alpha f_1^g+g^2 f_2^g\\ f_2^{g+1}&=&\beta f_1^g+\frac{2g}{g+1}f_3^g\\ f_3^{g+1}&=&\gamma f_1^g\,. \end{eqnarray*} \end{prop} {\em Proof. } The first claim is by direct check. Next, the recursion relations for $f_1^{g+1}$ and $f_2^{g+1}$ are immediate consequences of their definition. All the work is thus shifted to the innocuous looking formula for $f_3^{g+1}$. What we have to show is the vanishing of \begin{eqnarray*} \lefteqn{2(g+1)(g+2)\left(f_3^{g+1}-\gamma f_1^g\right)}\hspace{1cm}\\ &=&\Phi^{(g+3)}-\alpha\Phi^{(g+2)}-(g+2)^2\beta\Phi^{(g+1)}-2(g+1)(g+2) \gamma\Phi^{(g)}\\ &=&(g+2)!\left[(g+3)\varphi_{g+3}-\alpha\varphi_{g+2}-(g+2)\beta\varphi_{g+1} -2\gamma\varphi_g\right] \end{eqnarray*} with $\varphi_k$ the $k$-th Taylor coefficient of $\Phi$ at $t=0$. Multiplying with $t^{g+3}$ and taking the sum this will follow from \[ \sum_{g\ge1}(g+3)\varphi_{g+3}t^{g+3}\ =\ \alpha t\sum_{g\ge1}\varphi_{g+2}t^{g+2} +\beta t^2\sum_{g\ge1}(g+2)\varphi_{g+1}t^{g+1}+2\gamma t^3\sum_{g\ge1}\varphi_g t^g \] which is the part of order larger 3 of \[ t\cdot\Phi'\ =\ \alpha t\Phi+\beta t^2(\Phi\cdot t)'+2\gamma t^3\Phi \ =\ (\alpha t+\beta t^2+2\gamma t^3)\Phi+\beta t^3\Phi'\,, \] the functional equation for $\Phi$ (Proposition~\ref{fctleqn}). {\hfill$\Diamond$}\vspace{1.5ex} \section{The Leitideal} The decisive step in the proof of completeness of our relations is that the Leitideal (initial ideal) can be computed completely and has a particularly simple form. In all that follows we use the (graded) reverse lexicographic order in ${\Bbb Q}[\alpha,\beta,\gamma]$ and write ${\mbox{\rm In}\skp}(f)$ (${\mbox{\rm In}\skp}({\cal I})$) for the initial term of $f\in{\Bbb Q}[\alpha,\beta,\gamma]$ (resp.\ the initial ideal of an ideal ${\cal I}\subset{\Bbb Q}[\alpha,\beta,\gamma]$). As warm-up let us check that the initial terms of the $f_i^g$ are as promised in the last chapter: \begin{prop}\label{inf} In the reverse lexicographic order ${\mbox{\rm In}\skp}(f_1^g)=\alpha^g$, ${\mbox{\rm In}\skp}(f_2^g)=\alpha^{g-1}\beta$, ${\mbox{\rm In}\skp}(f_3^g)=\alpha^{g-1}\gamma$. \end{prop} {\em Proof. } By induction on $g$. $g=1$ is clear by the first line of Proposition~\ref{recursion}. Applying our recursion relations and the induction hypothesis, we get $f_1^{g+1}=\alpha^{g+1}+\alpha^g\beta+\ldots\,$, $f_2^{g+1}=\alpha^g\beta+\frac{2g}{g+1}\alpha^{g-1}\gamma+\ldots\,$, $f_3^{g+1}=\alpha^g\gamma+\ldots\,$, where $\ldots$ mean terms of lower order. {\hfill$\Diamond$}\vspace{1.5ex} Now setting ${\cal I}_g:=(f_1^g,f_2^g,f_3^g)\subset {\Bbb Q}[\alpha,\beta,\gamma]$, the ideal spanned by $f_i^g$, $i=1,2,3$, then \begin{prop}\label{inideal} ${\mbox{\rm In}\skp}({\cal I}_g)=(\alpha^a\beta^b\gamma^c,a+b+c\ge g)$. \end{prop} {\em Proof. } By induction on $g$, $g=1$ being trivially true. From \begin{eqnarray*} \gamma f_1^g&=&f_3^{g+1}\\ g^2\gamma f_2^g&=&\gamma f_1^{g+1}-\alpha\gamma f_1^g \ =\ \gamma f_1^{g+1}-\alpha f_3^{g+1}\\ \frac{4g}{g+1}\gamma f_3^g&=&\gamma f_2^{g+1}-\beta\gamma f_1^g \ =\ \gamma f_2^{g+1}-\beta f_3^{g+1} \end{eqnarray*} we see that $\gamma{\cal I}_g\subset{\cal I}_{g+1}$ (this is also clear from the observation that $\gamma\in H^*({\cal M}_{g+1})$ is Poincar\'e dual to $2g$ copies of ${\calm_g}$, cf.\ below). By induction hypothesis the claim is thus true for $c>0$. But in any homogeneous expression (with $\alpha$, $\beta$, $\gamma$ having weights $1$, $2$, $3$ respectively) the monomials containing $\gamma$ have lower order than those without. Therefore, we can reduce modulo $\gamma$ (indicated by a bar) and have only to show ${\mbox{\rm In}\skp}(\bar{{\cal I}_g})=(\bar\alpha^a\bar\beta^b,a+b\ge g)$. Modulo $\gamma$ the recursion relations read \[ \bar{f}_1^{g+1}=\bar\alpha\bar{f}_1^g+g^2\bar{f}_2^g,\ \ \ \bar{f}_2^{g+1}=\bar\beta\bar{f}_1^g\,. \] Now we are able to repeat the argument from above with $\bar\beta$ instead of $\gamma$, because \begin{eqnarray*} \bar\beta\bar{f}_1^g&=&\bar{f}_2^{g+1}\\ g^2\bar\beta\bar{f}_2^g&=&\bar\beta\bar{f}_1^{g+1}-\bar\alpha\bar\beta \bar{f}_1^g\ =\ \bar\beta\bar{f}_1^{g+1}-\bar\alpha\bar{f}_2^{g+1}\,. \end{eqnarray*} This leaves us with the case $b=0$, $c=0$, which is clearly true since $\alpha^g={\mbox{\rm In}\skp}(f_1^g)$ is the smallest power of $\alpha$ contained in ${\cal I}_g$ (for $\deg f_i^g\le g$, $i=1,2,3$). {\hfill$\Diamond$}\vspace{1.5ex} \section{Completeness} The strategy of showing that ${\cal I}_g=(f_1^g,f_2^g,f_3^g)\subset \cz[\alpha,\beta,\gamma]$ is really the ideal of relations between $\alpha$, $\beta$, $\gamma$ is a simple dimension count. But although the subring $\langle\alpha,\beta,\gamma\rangle\subset H^*({\calm_g},{\Bbb Q})$ generated by $\alpha$, $\beta$, $\gamma$ is the invariant ring under the action of the orientation preserving diffeomorphisms, the authors do not know a direct way to compute $\dim_{\Bbb Q}\langle\alpha,\beta,\gamma\rangle$. Instead we are viewing the even cohomology $H^{2*}({\calm_g},{\Bbb Q})$ as module over $\cz[\alpha,\beta,\gamma]/ {\cal I}_g$ and check injectivity of the structure map $\cz[\alpha,\beta,\gamma]/ {\cal I}_g\rightarrow H^{2*}({\calm_g},{\Bbb Q})$ by refining the basis $\{\alpha^a \beta^b\gamma^c\mid a+b+c<g\}$ of $\cz[\alpha,\beta,\gamma]/{\cal I}_g$ to a basis of $H^{2*}({\calm_g},{\Bbb Q})$. As a by-product we will actually find an explicit basis of the latter, which in a sense explains the inductive formulas for the even Betti numbers found by Newstead \cite{newstead2}. \begin{prop}\label{generators} $H^{2*}({\calm_g},{\Bbb Q})$ is generated by elements of the form\\[5pt] \hspace*{2cm} $\alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_{2l}}$ with $a+b+2l<g-1$, \\ \hspace*{1cm}and $\alpha^a\beta^b\gamma^k\psi_{i_1}\ldots\psi_{i_{2l}}$ with $a+b+k+2l=g-1$, $k\ge0$,\\[5pt] where $1\le i_1<\ldots<i_{2l}\le2g$. \end{prop} As an intermediate notion between the $\psi_i$ and $\gamma$ let us introduce the classes $\gamma_j:=\psi_j\psi_{j+g}$, $j=1,\ldots,g$ (then $\gamma=2\sum_j\gamma_j$). Each of the $\gamma_j$ is Poincar\'e dual to a diffeomorphic image $N_j$ of ${\cal M}_{g-1}$ (by ``contracting a handle'', cf.\ no.26 in \cite{thaddeus}). Moreover, ${\cal U}|N_j$ is topologically a universal bundle on ${\cal M}_{g-1}$, so $\alpha$, $\beta$, $\gamma$ restrict to generators $\hat\alpha$, $\hat\beta$, $\hat\psi_i$ ($i\neq j,j+g$) of $H^*({\cal M}_{g-1},{\Bbb Q})$ ($\psi_j|N_j=0=\psi_{j+g}|N_j$ since $\psi_j\gamma_j=0 =\psi_{j+g}\gamma_j$ trivially). We will also use the fact that intersection proucts $\alpha^a\beta^b\psi_{i_1}\ldots\psi_{2l}[{\calm_g}]$ ($a+2b+3l=3g-3$) are zero unless $\{i_1,\ldots,i_{2l}\}=\{j_1,j_1+g,\ldots,j_l,j_l+g\}$ in which case \[ \alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_{2l}}[{\calm_g}] =\frac{1}{g!}\alpha^a\beta^b\gamma^l[{\calm_g}]\,, \] i.e.\ only depending on the length of the sequence $(j_1,\ldots,j_l)$, cf.\ \cite{thaddeus}. \vspace{15pt} \noindent {\em Proof of proposition.} We want to refine the result of Proposition~\ref{inideal} that a monomial $\alpha^a\beta^b\gamma^c$ is equivalent (= may be reduced modulo ${\cal I}_g$) to a polynomial of lower order. For this we will use the reverse lexicographic order $\alpha>\beta>\psi_1>\ldots>\psi_{2g}>\gamma_1>\ldots>\gamma_{2g}> \gamma$. Let $1\le i_1<\ldots<i_k\le2g$ and $1\le j_1<\ldots<j_k\le g$ with $\{i_1,\ldots,i_k\}\cap\{j_1,j_1+g,\ldots,j_l,j_l+g\}=\emptyset$. \vspace{5pt} \noindent {\em Claim:} If $a+b+k+l\ge g$ then $\alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_k} \gamma_{j_1}\ldots\gamma_{j_l}$ is equivalent to a polynomial of lower order modulo ${\cal I}_g$, which can be taken of the form $F(\alpha,\beta)\, \psi_{i_1}\ldots\psi_{i_k}\gamma_{j_1}\ldots\gamma_{j_l}$. \vspace{-7pt} \noindent The claim certainly holds if $k+l=0$ by Proposition~\ref{inideal}. For $l>0$ let $\iota:{\cal M}_{g-1}\hookrightarrow{\calm_g}$ have image $N_{j_k}$ (Poincar\'e dual to $\gamma_{j_k}$) and use a hat to denote pull-back by $\iota$. By descending induction on $g$ then ($l<g$ because $\gamma_1\ldots\gamma_g=0$ for dimension reasons), \begin{eqnarray*} \lefteqn{\iota^*\left(\alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_k} \gamma_{j_1}\ldots\gamma_{j_{l-1}}\right) \ =\ {\hat\alpha}^a{\hat\beta}^b{\hat\psi_{i_1}}\ldots{\hat\psi_{i_k}} {\hat\gamma_{j_1}}\ldots{\hat\gamma_{j_{l-1}}} }\\ &=&F(\hat\alpha,\hat\beta)\,{\hat\psi_{i_1}}\ldots{\hat\psi_{i_k}} {\hat\gamma_{j_1}}\ldots{\hat\gamma_{j_{l-1}}} \ =\ \iota^*\left(F(\alpha,\beta)\,\psi_{i_1}\ldots\psi_{i_k} \gamma_{j_1}\ldots\gamma_{j_{l-1}}\right) \end{eqnarray*} with $\mbox{order}(F)<a+b$. This means $\alpha^a\beta^b\psi_{i_1}\ldots \psi_{i_k}\gamma_{j_1}\ldots\gamma_{j_l}=F(\alpha,\beta)\,\psi_{i_1} \ldots\psi_{i_k}\gamma_{j_1}$ $\ldots\gamma_{j_l}$ as wanted. Finally the case $l=0$, $k>0$: Set $\bar\imath_k=i_k+g$ if $i_k\le g$ and $\bar\imath_k=i_k-g$ if $i_k>g$. By the previous case, we get $\alpha^a\beta^b\psi_{i_1}\ldots \psi_{i_k}\psi_{\bar\imath_k}=F(\alpha,\beta)\psi_{i_1}\ldots \psi_{i_k}\psi_{\bar\imath_k}$. Then the above remarks on the intersection product show \[ \left(\alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_k} -F(\alpha,\beta)\,\psi_{i_1}\ldots\psi_{i_k}\right)\cdot A[{\calm_g}]\ =\ 0 \] for all $A\in H^*({\calm_g})$, i.e.\ $\alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_k} =F(\alpha,\beta)\psi_{i_1}\ldots\psi_{i_k}$. This proves the claim.\\ The proposition is now clearly reduced to a second \vspace{5pt} \noindent {\em Claim:} Let $M=\alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_k} \gamma_{j_1}\ldots\gamma_{j_l}$ with $a+b+k+l=g-2$. Then for all $1\le i,j \le g$, $M\gamma_i-M\gamma_j$ may be reduced to lower order modulo ${\cal I}_g$. \vspace{5pt} \noindent In fact, from the above we know already $M\gamma_i\gamma_j=F\gamma_i\gamma_j$ in $H^*({\calm_g})$ with $\mbox{order}(F)<g-2$. $F\gamma_i-F\gamma_j$ is our candidate for the lower order term. If $\{i,i+g,j,j+g\}\cap\{i_1,\ldots,i_k\} =\emptyset$ and $A=\alpha^{a'}\beta^{b'}\psi_{i_1}\ldots\psi_{i_k}$ then \[ \left(M\gamma_i-M\gamma_j\right)\cdot A[{\calm_g}]\ =\ 0\ =\ \left(F\gamma_i-F\gamma_j\right)\cdot A[{\calm_g}] \] again by the symmetry in the $\gamma_i$ of the intersection pairing. On the other hand \begin{eqnarray*} (M\gamma_i-M\gamma_j)\gamma_j[{\calm_g}] &=&M\gamma_i\gamma_j A[{\calm_g}] \ =\ F\gamma_i\gamma_j A[{\calm_g}]\\ &=& (F\gamma_i-\gamma_j)\gamma_j[{\calm_g}] \end{eqnarray*} and analogously with $\gamma_i$. Thus $M\gamma_i=M\gamma_j+ F(\gamma_i-\gamma_j)$ modulo ${\cal I}_g$ as claimed.\\ (Note: This argument fails for the $\psi_i$ because of the presence of $\psi_{i+g}$ respectively $\psi_{i-g}$.) {\hfill$\Diamond$}\vspace{1.5ex} The only thing we finally need to do is to count the number of generators and compare with the inductive formula for the Betti numbers found by Newstead. \begin{prop}\label{dimcount} The generators for $H^{2*}({\calm_g},{\Bbb Q})$ in Proposition~\ref{generators} are linearly independent up to the middle dimension. \end{prop} {\em Proof. } Let $G_r$ be the set of generators of (real) degree $r$ from Proposition~\ref{generators}, $g_r:=\sharp G_r$. We will show that for $s\displaystyle\le\left[\frac{3g-1}{2}\right]$ \[ g_{2s+4}\ =\ g_{2s}+\sum_{l=s-g+1}^{[s/3]}{2g\choose 2l} \] which together with $g_0=1$ and $g_2=1$ is exactly Newstead's formula for the even Betti numbers \cite{newstead1}. Define a map $\varphi:G_{2s}\rightarrow G_{2s+4}$ by \[ \alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_{2l}}\longmapsto \alpha^a\beta^{b+1}\psi_{i_1}\ldots\psi_{i_{2l}}\ \ \ \mbox{for }a+b+2l<g-1 \] and \begin{eqnarray*} \alpha^a\beta^b\gamma^k\psi_{i_1}\ldots\psi_{i_{2l}}\longmapsto \alpha^{a-1}\beta^b\gamma^{k+1}\psi_{i_1}\ldots\psi_{i_{2l}} &&\mbox{ for }a+b+k+2l=g-1, \end{eqnarray*} with $k\ge0$ and $a>0$. Note that the case $a=k=0$ does not occur (then $b+2l=g-1$ and $2b+3l=s$ imply $s-2g+2=-l\le0$ which contradicts $s\le(3g-1)/2$). Now $G_{2s+4}\setminus{\mbox{\rm im}\skp\skp}\varphi=\{\alpha^a\psi_{i_1}\ldots \psi_{i_{2l}}\mid a+2l\le g-1, a+3l=s\}$ s.th.\ $l$ runs from $s-g+1$ to $[s/3]$ ($a$ is determined through $a+3l=s$). The contribution for $l$ fixed is then precisely $2g\choose2l$. {\hfill$\Diamond$}\vspace{1.5ex}

9410

alg-geom/9410020

en

https://arxiv.org/abs/alg-geom/9410020

alg-geom/9410020

Bas Edixhoven

Bas Edixhoven

On the prime-to-$p$ part of the groups of connected components of N\'eron models

21 pages, LaTeX, 94-23

This article improves certain results of Dino Lorenzini concerning the groups of connected components of special fibres of N\'eron models of abelian varieties. Lorenzini has shown the existence of a four step filtration on the prime-to-$p$ part ($p$ is the residue characteristic), and proved certain bounds for the successive quotients. We improve these bounds and show that our bounds are sharp. As an application, we give a complete classification of the groups that can arise as the prime-to-$p$ part of the group of connected components of the Neron model of an abelian variety with given dimensions for the abelian, toric and unipotent part. Hard copies of this preprint are available.

alg-geom

\section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus -.2ex}{2.3ex plus .2ex}{\large\bf}} \makeatother \makeatletter \newenvironment{eqn}{\refstepcounter{subsection} $$}{\leqno{\rm(\thesubsection)}$$\global\@ignoretrue} \makeatother \makeatletter \newenvironment{subeqn}{\refstepcounter{subsubsection} $$}{\leqno{\rm(\thesubsubsection)}$$\global\@ignoretrue} \makeatother \makeatletter \newenvironment{subeqarray}{\renewcommand{\theequation}{\thesubsubsection} \let\c@equation\c@subsubsection\let\cl@equation\cl@subsubsection \begin{eqnarray}}{\end{eqnarray}} \makeatother \makeatletter \def\@rmkcounter#1{\noexpand\arabic{#1}} \def\@rmkcountersep{.} \def\@beginremark#1#2{\trivlist \item[\hskip \labelsep{\bf #2\ #1.}]} \def\@opargbeginremark#1#2#3{\trivlist \item[\hskip \labelsep{\bf #2\ #1\ (#3).}]} \def\@endremarkwithsquare{~\hspace{\fill}~$\square$\endtrivlist} \makeatother \newenvironment{prf}[1]{\trivlist \item[\hskip \labelsep{\it #1.\hspace*{.3em}}]}{~\hspace{\fill}~$\square$\endtrivlist} \newenvironment{proof}{\begin{prf}{\bf Proof}}{\end{prf}} \let\tempcirc=\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}} \def\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}{\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}} \def\ac{\nobreak\hskip.1111em\mathpunct{}\nonscript\mkern-\thinmuskip{:}\hskip .3333emplus.0555em\relax} \setlength{\textheight}{250mm} \setlength{\textwidth}{170mm} \setlength{\evensidemargin}{0in} \setlength{\oddsidemargin}{0in} \setlength{\topmargin}{-2cm} \setcounter{tocdepth}{1} \title{On the prime-to-$p$ part of the groups of connected components of N\'eron models.} \author{Bas Edixhoven} \newtheorem{theorem}[subsection]{Theorem.} \newtheorem{proposition}[subsection]{Proposition.} \newtheorem{lemma}[subsection]{Lemma.} \newtheorem{corollary}[subsection]{Corollary.} \newtheorem{tabel}[subsection]{Table.} \newtheorem{conjecture}[subsection]{Conjecture.} \newtheorem{definition}[subsection]{Definition.} \newremark{rmk}[subsection]{Remark} \newremark{subrmk}[subsubsection]{Remark} \newremark{notation}[subsection]{Notation} \newremark{example}[subsection]{Example} \renewcommand{\baselinestretch}{1.3} \newcommand{\bf}{\bf} \newcommand{\varepsilon}{\varepsilon} \newcommand{{\rm Spec}}{{\rm Spec}} \newcommand{{\msy Z}}{{\bf Z}} \newcommand{{\msy Q}}{{\bf Q}} \newcommand{{\msy C}}{{\bf C}} \newcommand{{\msy R}}{{\bf R}} \newcommand{\langle}{\langle} \newcommand{\rangle}{\rangle} \newcommand{{\msy F}}{{\bf F}} \newcommand{{\rm Gal}}{{\rm Gal}} \newcommand{{\rm Pic}}{{\rm Pic}} \newcommand{{{\rm G}_m}}{{{\rm G}_m}} \newcommand{{\rm Ext}}{{\rm Ext}} \newcommand{{\rm im}}{{\rm im}} \newcommand{{K^{\rm s}}}{{K^{\rm s}}} \newcommand{{D^{\rm s}}}{{D^{\rm s}}} \newcommand{{\overline{k}}}{{\overline{k}}} \newcommand{{\rm T}}{{\rm T}} \newcommand{{\rm H}}{{\rm H}} \newcommand{{\rm tors}}{{\rm tors}} \newcommand{{I_{\rm t}}}{{I_{\rm t}}} \newcommand{{\widetilde{V}}}{{\widetilde{V}}} \newcommand{{\widetilde{W}}}{{\widetilde{W}}} \newcommand{{\tilde{t}}}{{\tilde{t}}} \newcommand{{\tilde{a}}}{{\tilde{a}}} \newcommand{{\widetilde{K}}}{{\widetilde{K}}} \newcommand{{K^{\rm t}}}{{K^{\rm t}}} \newcommand{{t_{\rm t}}}{{t_{\rm t}}} \newcommand{{a_{\rm t}}}{{a_{\rm t}}} \newcommand{{u_{\rm t}}}{{u_{\rm t}}} \newcommand{{D^{\rm t}}}{{D^{\rm t}}} \newcommand{{\rm rank}}{{\rm rank}} \newcommand{{\widetilde{I}}}{{\widetilde{I}}} \newcommand{{\widetilde{\phi}}}{{\widetilde{\phi}}} \newcommand{{\overline{\phi}}}{{\overline{\phi}}} \newcommand{{\rm a}}{{\rm a}} \newcommand{{\rm t}}{{\rm t}} \begin{document} \maketitle \tableofcontents \section{Introduction.} The aim of this article was originally to improve certain results of Dino Lorenzini concerning the groups of connected components of special fibres of N\'eron models of abelian varieties. Let $D$ be a strictly henselian discrete valuation ring, $K$ its field of fractions, $k$ its residue field and $A_K$ an abelian variety over $K$ with N\'eron model $A$ over $D$. Let $p\geq0$ be the characteristic of $k$ and let $\Phi_{(p)}$ denote the prime-to-$p$ part of the group of connected components $\Phi$ of $A_k$. In \cite{Dino1} Lorenzini shows the existence of a functorial four step filtration on $\Phi_{(p)}$ and he proves certain properties satisfied by this filtration. In particular, he gives bounds for the successive quotients. These bounds are of the following type. For a prime $l$ and a finite abelian group $G$ of $l$-power order, say $G\cong\oplus_{i\geq1}{\msy Z}/l^{a_i}{\msy Z}$ with $a_1\geq a_2\geq\cdots$, he defines $\delta_l'(G):=l^{a_1}-1+(l-1)\sum_{i\geq2}a_i$. Then he gives bounds for the $\delta_l'$ of certain successive quotients in terms of the dimensions of the toric and abelian variety parts of the special fibres of N\'eron models of $A_K$ over various extensions of~$D$. In \cite[Remark~2.16]{Dino1} he remarks that the bounds might possibly be improved by replacing $\delta_l'$ by an other invariant $\delta_l$ defined as follows: for $G$ as above one has $\delta_l(G)=\sum_{i\geq1}(l^{a_i}-1)$. This improvement is exactly what we do in this article. The results can be found in \S\ref{section3}. Needless to say, we follow very much the approach of \cite{Dino1} in order to prove these sharper bounds. In fact, only Lemma~2.13 of \cite{Dino1} has to be changed, so the proof we give is rather short. We have taken this opportunity to weaken slightly the hypotheses of Lorenzini's results (he supposes $D$ to be complete and $k$ to be algebraically closed). In \S\ref{section2} we recall Lorenzini's filtration. In \S\ref{section3} we state and prove the bounds on the $\delta_l$ of the $l$-parts of certain successive quotients and in \S\ref{section5} we show by some examples that the bounds of \S\ref{section3} are sharp; \S\ref{section4} is used to show some results on finite abelian groups that are needed in the other sections. After all this work it turned out that a complete classification of the possible $\Phi_{(p)}$ for abelian varieties whose reduction has toric part, abelian variety part and unipotent part of fixed dimensions was in reach. The result, which is surprisingly simple to state, can be found in Thm.~\ref{thm61}. In this article we will frequently speak of the abelian variety part, the toric part and the unipotent part of the fibre over $k$ of a N\'eron model over $D$. Since we are only interested in the characteristic polynomials of certain endomorphisms on the toric and abelian variety part, it suffices to define these parts after base change to an algebraic closure of $k$, and up to isogeny. Over the algebraic closure of $k$ we can apply Chevalley's theorem; in \cite[Thm.~9.2.1, Thm.~9.2.2]{BLR} one finds what is needed, and even more. I would like to thank Xavier Xarles for indicating a mistake in an earlier version of this article, and Rutger Noot for his help concerning the proofs of Lemma's~\ref{lemma410} and~\ref{lemma411}. \section{Lorenzini's filtration.} \label{section2} Let $D$ be a discrete valuation ring, let $K$ be its field of fractions and $k$ its residue field. Let $A_K$ be an abelian variety over $K$, $A$ its N\'eron model over $D$ and $\Phi:=A_k/A_k^0$ the finite \'etale group scheme over $k$ of connected components of the special fibre $A_k$. Let $p\geq0$ be the characteristic of $k$ and let $\Phi_{(p)}$ be the prime-to-$p$ part of $\Phi$; if $p=0$ we define $\Phi_{(p)}$ to be equal to $\Phi$. In this section we will briefly recall the construction in \cite{Dino1} of a descending filtration \begin{eqn} \label{eqn21} \Phi_{(p)} = \Phi_{(p)}^0 \supset \Phi_{(p)}^1 \supset \Phi_{(p)}^2 \supset \Phi_{(p)}^3 \supset \Phi_{(p)}^4 = 0 \end{eqn} which is functorial in $A_K$ and invariant under base change by automorphisms of $D$. Since $\Phi_{(p)}$ is the direct sum of its $l$-parts $\Phi_l$, with $l$ ranging through the primes different from $p$, it suffices to describe the filtration on each $\Phi_l$. We replace $D$ by its strict henselization and view the group scheme $\Phi$ over the separably closed field $k$ as just a group. Let $l\neq p$ be a prime number. Let $K\to{K^{\rm s}}$ be a separable closure, let ${D^{\rm s}}$ be the integral closure of $D$ in ${K^{\rm s}}$ and let ${\overline{k}}$ be the residue field of ${D^{\rm s}}$; note that ${\overline{k}}$ is an algebraic closure of $k$ and that $k\to{\overline{k}}$ is purely inseparable. The first step in the construction of the filtration is the description of $\Phi_l$ in terms of the Tate module $U_l:={\rm T}_l(A({K^{\rm s}}))$ with its action by $I:={\rm Gal}({K^{\rm s}}/K)$ given in Prop.~11.2 of \cite{Grothendieck1}: \begin{eqn} \label{eqn22} \Phi_l = \left(U_l\otimes{\msy Q}/{\msy Z}\right)^I/ \left(U_l^I\otimes{\msy Q}/{\msy Z}\right) \end{eqn} The long exact cohomology sequence of the short exact sequence $$ 0\to U_l\to U_l\otimes{\msy Q}\to U_l\otimes{\msy Q}/{\msy Z}\to 0 $$ of continuous $I$-modules gives a canonical isomorphism (see \cite[(11.3.8)]{Grothendieck1}) \begin{eqn} \label{eqn23} \Phi_l = {\rm tors}({\rm H}^1(I,U_l)) \end{eqn} where for $M$ any abelian group, ${\rm tors}(M)$ denotes the subgroup of torsion elements. Let ${I_{\rm t}}$ be the quotient of $I$ corresponding to the maximal tamely ramified extension of $D$, and let $P$ be the kernel of $I\to{I_{\rm t}}$. Then ${I_{\rm t}}$ is canonically isomorphic to $\prod_{q\neq p}{\rm T}_q({{\rm G}_m}(k))=\prod_{q\neq p}{\msy Z}_q(1)$ and $P$ is a pro-$p$ group. The Hochschild-Serre spectral sequence shows that \begin{eqn}\label{eqn24} \Phi_l = {\rm tors}({\rm H}^1(I,U_l)) = {\rm tors}({\rm H}^1({I_{\rm t}},U_l^P)) = {\rm tors}((U_l^P)_{I_{\rm t}})(-1) = {\rm tors}((U_l)_I)(-1) \end{eqn} with the lower indices ${I_{\rm t}}$ and $I$ denoting coinvariants and ``$(-1)$'' a Tate twist. Let $N_l$ be the submodule of $U_l$ which is generated by the elements $\sigma(x)-x$ with $\sigma$ in $I$ and $x$ in $U_l$. Then by definition we have $(U_l)_I=U_l/N_l$. As in \cite[\S2.5]{Grothendieck1}, we define $V_l:=U_l^I$. Then $V_l$, which is called the fixed part of $U_l$, is canonically isomorphic to ${\rm T}_l(A_k(k))$. Let $A_K'$ be the dual of $A_K$, i.e., $A_K'={\rm Pic}^0_{A_K/K}$. We will denote by $A'$ the N\'eron model over $D$ of $A_K'$, by $\Phi'$ its group of connected components, etc. Let $\langle{\cdot},{\cdot}\rangle\colon U_l\times U_l'\to{\msy Z}_l(1)$ be the Weil pairing. For any $y$ in $V'_l$, $\sigma$ in $I$ and $x$ in $U_l$ we have $\langle\sigma(x)-x,y\rangle = \langle\sigma(x),y\rangle-\langle x,y\rangle = \sigma(\langle x,\sigma^{-1}(y)\rangle) - \langle x,y\rangle = 0$. It follows that $N_l$ is contained in the orthogonal ${V_l'}^\perp$ of $V_l'$ in $U_l$. Since $U_l/{V_l'}^\perp$ is torsion free, we conclude that \begin{eqn} \label{eqn25} \Phi_l = {\rm tors}\left({V_l'}^\perp/N_l\right)(-1) \end{eqn} \begin{rmk} \label{rmk26} In the proof of Thm.~\ref{thm33} we will see that ${V_l'}^\perp/N_l$ is in fact a finite group, hence we have $\Phi_l = ({V_l'}^\perp/N_l)(-1)$. \end{rmk} Now it is clear that any filtration on ${V_l'}^\perp$ induces a filtration on $\Phi_l$. As in \cite[\S2.5]{Grothendieck1}, we define $W_l\subset V_l$ to be the submodule corresponding to the maximal torus in $A_k$. Let ${\widetilde{W}}_l\subset{\widetilde{V}}_l\subset V_l$ be the submodules called the essentially toric part and the essentially fixed part in \cite[\S4.1]{Grothendieck1}; if $G/k'$ is the connected component of the special fibre of a semi-stable N\'eron model of $A_K$ over a suitable sub-extension of $K\to{K^{\rm s}}$ then ${\widetilde{V}}_l$ corresponds to ${\rm T}_l(G({\overline{k}}))$ and ${\widetilde{W}}_l$ to the Tate module of the maximal torus in $G$. We denote by $t$, $a$ and $u$ the dimensions of the toric part, the abelian variety part and the unipotent part of $A^0_{\overline{k}}$; we denote by ${\tilde{t}}$ and ${\tilde{a}}$ the analogous dimensions of any semi-stable reduction of $A_K$. Note that $t+a+u={\tilde{t}}+{\tilde{a}}=\dim(A_K)$. An easy application of the Igusa-Grothendieck orthogonality theorem (which states that $W_l=V_l\cap{V_l'}^\perp$, see \cite[Thm.~2.4]{Grothendieck1}, or \cite[Thm.~3.1]{Oort1}), gives us the following filtration of ${V_l'}^\perp$, in which the successive quotients are torsion free and of the indicated rank: \begin{eqn} \label{eqn27} {V_l'}^\perp \;\;\stackrel{{\tilde{t}}-t}{\supset}\;\; {\widetilde{V}}_l\cap{V_l'}^\perp \;\;\stackrel{2({\tilde{a}}-a)}{\supset}\;\; {\widetilde{W}}_l \;\;\stackrel{{\tilde{t}}-t}{\supset}\;\; W_l \;\;\stackrel{t}{\supset}\;\; 0 \end{eqn} Lorenzini's filtration (\ref{eqn21}) on $\Phi_l$ is the filtration induced by (\ref{eqn25}) and (\ref{eqn27}). Note that in fact any finite sub-extension of $K\to{K^{\rm s}}$ induces a filtration on $\Phi_l$ as above; see \cite[Thm.~3.1]{Dino1} for results concerning those filtrations. The reason we only consider the filtration coming from extensions over which $A_K$ has semi-stable reduction is that only that filtration matters for the bounds on $\Phi_{(p)}$ of the next section. \section{Bounds on $\Phi_{(p)}$.} \label{section3} We keep the notation of the previous section. Recall that $K$ is strictly henselian. First we define some invariants of finite abelian groups and fix some notation needed to state our results. \begin{definition} \label{def31} For $l$ a prime number and $a=(a_1,a_2,\ldots)$ a sequence of integers $a_i\geq0$ with $a_i=0$ for $i$ big enough, let $\delta_l(a):=\sum_i(l^{a_i}-1)$. For $l$ a prime number and $G\cong\oplus_i{\msy Z}/l^{a_i}{\msy Z}$ a finite abelian group of $l$-power order let $\delta_l(G):=\delta_l(a)$, where $a:=(a_1,a_2,\ldots)$. For $G$ a finite abelian group let $\delta(G):=\sum_l\delta_l(G_l)$, where $G=\oplus_l G_l$ is the decomposition of $G$ into groups of prime power order. \end{definition} \begin{notation} \label{notation32} Let ${\widetilde{K}}$ be the minimal sub-extension of ${K^{\rm s}}$ over which $A_K$ has semi-stable reduction; it corresponds to the kernel of $I$ acting on ${\widetilde{V}}$, see \cite[\S4.1]{Grothendieck1}. We define ${K^{\rm t}}$ to be the maximal tame extension in ${\widetilde{K}}$, and for all $l\neq p$ we let $K_l$ denote the maximal sub-extension of ${\widetilde{K}}$ whose degree over $K$ is a power of $l$. We denote by ${\tilde{t}}$, ${\tilde{a}}$, ${t_{\rm t}}$, ${a_{\rm t}}$, ${u_{\rm t}}$, $t_l$, $a_l$ and $u_l$ the dimensions of the toric parts, the abelian variety parts and the unipotent parts of the corresponding N\'eron models of $A_K$. For each prime $l\neq p$ we let $I_{(l)}$ be the subgroup of $I$ such that $I/I_{(l)}$ is the quotient ${\msy Z}_l(1)$ of ${I_{\rm t}}$. Let $A^{\rm t}$ be the N\'eron model of $A_K$ over the ring of integers ${D^{\rm t}}$ of ${K^{\rm t}}$. Then ${\rm Gal}({K^{\rm t}}/K)$ acts (from the right) on $A^{\rm t}$, compatibly with its right-action on ${\rm Spec}({D^{\rm t}})$. This action induces an action of ${\rm Gal}({K^{\rm t}}/K)$ on the special fibre $A^{\rm t}_k$. Let $\sigma$ be a generator of the cyclic group ${\rm Gal}({K^{\rm t}}/K)$. Let $l\neq p$ be a prime number and $i\geq1$ an integer. Let $f_{l,i}$ denote the cyclotomic polynomial whose roots are the roots of unity of order $l^i$. We define $m_{{\rm a},l,i}$ and $m_{{\rm t},l,i}$ to be the multiplicities of $f_{l,i}$ in the characteristic polynomials of $\sigma$ on the abelian variety part and on the toric part, respectively, of $A^{\rm t}_k$ (say one lets $\sigma$ act on ${\rm T}_l(A^{\rm t}_k({\overline{k}}))\otimes{\msy Q}$). Let $m_{l,i}:=m_{{\rm a},l,i}+m_{{\rm t},l,i}$. Finally, for $j\geq1$ we define $p_{{\rm a},l,j}:=|\{i\geq1\;|\;m_{{\rm a},l,i}\geq j\}|$, $p_{{\rm t},l,j}:=|\{i\geq1\;|\;m_{{\rm t},l,i}\geq j\}|$ and $p_{l,j}:=|\{i\geq1\;|\;m_{l,i}\geq j\}|$. For an interpretation of $p_{{\rm a},l}=(p_{{\rm a},l,1},p_{{\rm a},l,2},\ldots)$ in terms of $m_{{\rm a},l}=(m_{{\rm a},l,1},m_{{\rm a},l,2},\ldots)$ etc. using partitions, see the beginning of the proof of Lemma~\ref{lemma45}. \end{notation} \begin{theorem} \label{thm33} Let $l\neq p$ be a prime number and consider the filtration (\ref{eqn21}) on $\Phi_l$. With the notations above, we have: \begin{enumerate} \item The group $\Phi_l^3$ can be generated by $t$ elements. \item $\delta_l(\Phi_l^2/\Phi_l^3) \leq \delta_l(p_{{\rm t},l}) \leq t_l-t$. \item $\delta_l(\Phi_l^1/\Phi_l^2) \leq \delta_l(p_{{\rm a},l}) \leq 2(a_l-a)$. \item $\delta_l(\Phi_l/\Phi_l^1) \leq \delta_l(p_{{\rm t},l}) \leq t_l-t$. \item $\delta_l(\Phi_l/\Phi_l^2) \leq \delta_l(p_l) \leq (t_l-t)+2(a_l-a)$. \item $\delta_l(\Phi_l^1/\Phi_l^3) \leq \delta_l(p_l) \leq (t_l-t)+2(a_l-a)$. \end{enumerate} \end{theorem} \begin{corollary} \label{cor34} \begin{enumerate} \item The group $\Phi_{(p)}^3$ can be generated by $t$ elements. \item $\delta(\Phi_{(p)}^2/\Phi_{(p)}^3) \leq {t_{\rm t}}-t.$ \item $\delta(\Phi_{(p)}^1/\Phi_{(p)}^2) \leq 2({a_{\rm t}}-a).$ \item $\delta(\Phi_{(p)}/\Phi_{(p)}^1) \leq {t_{\rm t}}-t.$ \item $\delta(\Phi_{(p)}/\Phi_{(p)}^2) \leq ({t_{\rm t}}-t)+2({a_{\rm t}}-a).$ \item $\delta(\Phi_{(p)}^1/\Phi_{(p)}^3) \leq ({t_{\rm t}}-t)+2({a_{\rm t}}-a).$ \end{enumerate} \end{corollary} \begin{prf}{{\bf Proof} {\rm (of Thm.~\ref{thm33})}} We begin with some generalities. We always have $M_I=(M_{I_{(l)}})_{{\msy Z}_l(1)}$. The functors $M\mapsto M_{I_{(l)}}$ and $M\mapsto M^{I_{(l)}}$ are exact on the category of finitely generated ${\msy Z}_l$-modules with continuous $I_{(l)}$-action, and, for such modules, the canonical map $M^{I_{(l)}}\to M_{I_{(l)}}$ is an isomorphism, hence $M_{I_{(l)}}$ is torsion free if $M$ is torsion free. For $M$ a finitely generated ${\msy Z}_l$-module with continuous ${\msy Z}_l(1)$-action we have $M_{{\msy Z}_l(1)}=M/(\sigma-1)M$ and $M^{{\msy Z}_l(1)}=M[\sigma-1]$, where $\sigma$ is any topological generator of ${\msy Z}_l(1)$. Next we recall some general facts on the action of $I$ on $U_l$. Let ${\widetilde{I}}$ denote the subgroup ${\rm Gal}({K^{\rm s}}/{\widetilde{K}})$ of $I$. Then ${\widetilde{I}}$ acts trivially on ${\widetilde{V}}_l=U_l^{\widetilde{I}}$ and on $U_l/{\widetilde{V}}_l$; the action of ${\widetilde{I}}$ on $U_l$ factors through the biggest pro-$l$ quotient ${\msy Z}_l(1)$ of ${\widetilde{I}}$ and is given by an isogeny $U_l/{\widetilde{V}}_l\to{\widetilde{W}}_l(-1)$ (see \cite[\S9.2, Thm.~10.4]{Grothendieck1}). It follows that $N_l\cap{\widetilde{W}}_l$ is open in ${\widetilde{W}}_l$. The group $I$ acts on ${\widetilde{V}}_l$ via its finite quotient ${\rm Gal}({\widetilde{K}}/K)=I/{\widetilde{I}}$; this action can be described in terms of an action of $I/{\widetilde{I}}$ on the special fibre of the N\'eron model of $A_{\widetilde{K}}$ (see \cite[\S4.2]{Grothendieck1}). Dually, $I$ acts with finite image on $U_l/{\widetilde{W}}_l$. As promised in Remark.~\ref{rmk26} we will show that $\Phi_l={V_l'}^\perp/N_l$. It suffices to show that ${V_l'}^\perp$ and $N_l$ have the same rank. We have ${\rm rank}(U_l/{V_l'}^\perp)=t+2a$. From the generalities at the beginning of the proof it follows that $U_l/N_l=(U_l)_I=((U_l)_{I_{(l)}})_{{\msy Z}_l(1)}=(U_l^{I_{(l)}})_{{\msy Z}_l(1)}$. Let $\sigma$ be a topological generator of ${\msy Z}_l(1)$. The exact sequence \begin{eqn} \label{eqn35} 0 \longrightarrow U_l^I \longrightarrow U_l^{I_{(l)}} \;\;\stackrel{\sigma-1}{\longrightarrow} \;\; U_l^{I_{(l)}} \longrightarrow \left(U_l^{I_{(l)}}\right)_{{\msy Z}_l(1)} \longrightarrow 0 \end{eqn} shows that ${\rm rank}((U_l)_I)={\rm rank}(U_l^I)={\rm rank}(V_l)=t+2a$. In order to prove Thm.~\ref{thm33} we may neglect the Tate twist in~(\ref{eqn25}). By definition, we have $\Phi_l^3=W_l/N_l\cap W_l$. Since $W_l$ is a free ${\msy Z}_l$ module of rank~$t$, $\Phi_l^3$ can be generated by $t$ elements. Let us now consider $\Phi_l^2/\Phi_l^3$. Since $\Phi_l^2={\widetilde{W}}_l/{\widetilde{W}}_l\cap N_l$, the group $\Phi_l^2/\Phi_l^3$ is a quotient of $({\widetilde{W}}_l/W_l)_I=(({\widetilde{W}}_l/W_l)_{I_{(l)}})_{{\msy Z}_l(1)}$. Lemma~\ref{lemma44} implies that $\delta_l(\Phi_l^2/\Phi_l^3)\leq \delta_l((({\widetilde{W}}_l/W_l)_{I_{(l)}})_{{\msy Z}_l(1)})$. By the generalities above, $({\widetilde{W}}_l/W_l)_{I_{(l)}}$ is isomorphic as ${\msy Z}_l(1)$-module to ${\widetilde{W}}_l^{{\rm Gal}({\widetilde{K}}/K_l)}/W_l$. Note that ${\widetilde{W}}_l^{{\rm Gal}({\widetilde{K}}/K_l)}$ is the Tate module of the toric part of the special fibre of the N\'eron model of $A_K$ over the ring of integers of $K_l$, and that $W_l={\widetilde{W}}_l^{{\rm Gal}({\widetilde{K}}/K)}$. It follows that for all $i\geq1$ the multiplicity of $f_{l,i}$ in the characteristic polynomial of a generator $\sigma$ of ${\rm Gal}({K^{\rm t}}/K)$ on $({\widetilde{W}}_l/W_l)_{I_{(l)}}$ is $m_{{\rm t},l,i}$ and that $1$ is not a root of this characteristic polynomial. Applying Lemma~\ref{lemma45} and Cor.~\ref{cor46} gives the second part of the theorem. The proof of parts 3, 4, 5 and 6 of the theorem follows the same lines. For example, $\Phi_l^1/\Phi_l^2$ is a quotient of $(({\widetilde{V}}_l\cap{V_l'}^\perp)/{\widetilde{W}}_l)_I$. The group $I$ acts with finite image on $({\widetilde{V}}_l\cap{V_l'}^\perp)/{\widetilde{W}}_l$. The Grothendieck-Igusa orthogonality theorem \cite[Thm.~2.4]{Grothendieck1} shows that $({\widetilde{V}}_l\cap{V'_l}^\perp/{\widetilde{W}}_l)_{I_{(l)}}$ has rank $2(a_l-a)$. We have \begin{eqn} \label{eqn36} \left(\frac{{\widetilde{V}}_l\cap{V_l'}^\perp}{{\widetilde{W}}_l}\right)_I\otimes{\msy Q} = \frac{({\widetilde{V}}_l\cap{V_l'}^\perp)^I}{{\widetilde{W}}_l^I}\otimes{\msy Q} = \frac{V_l\cap{V_l'}^\perp}{W_l}\otimes{\msy Q} = \frac{W_l}{W_l}\otimes{\msy Q} = 0 \end{eqn} which shows that the hypotheses of Lemma~\ref{lemma45} are satisfied. Since ${\widetilde{V}}_l\cap{V_l'}^\perp/{\widetilde{W}}_l$ is isogenous to ${\widetilde{V}}_l/V_l$, the multiplicities of the $f_{l,i}$ in the characteristic polynomial of a generator $\sigma$ of ${\rm Gal}({K^{\rm t}}/K)$ on $({\widetilde{V}}_l\cap{V_l'}^\perp/{\widetilde{W}}_l)_{I_{(l})}$ are precisely the $m_{{\rm a},l,i}$. The proof of part 6 is entirely similar to the proofs of parts~2 and 3. For parts~4 and 5 one notes that ${V'_l}^\perp/{\widetilde{V}}_l\cap {V'_l}^\perp$ is dual to ${\widetilde{W}}_l'/W_l'$, that ${V'_l}^\perp/{\widetilde{W}}_l$ is dual to ${\widetilde{V}}_l'/V_l'$ and one uses that $A_K$ and $A_K'$ are isogenous. \end{prf} \begin{prf}{{\bf Proof} {\rm (of Cor.~\ref{cor34})}} One just considers the factorization into irreducible factors of the characteristic polynomial of a generator $\sigma$ of ${\rm Gal}({K^{\rm t}}/K)$ acting on the semi-abelian variety part of~$A^{\rm t}_k$. \end{prf} \section{Some abelian group theory.} \label{section4} In this section we prove some results needed in the proof of Thm.~\ref{thm33}. We fix a prime number $l$ and consider finite ${\msy Z}_l$-modules, i.e., finite abelian groups of $l$-power order. Recall that there is a bijection between the set of isomorphism classes of finite ${\msy Z}_l$-modules and the set of partitions (i.e., sequences $m=(m_1,m_2,\ldots)$ of non-negative integers such that $m_1\geq m_2\geq\cdots$ and $m_i=0$ for $i$ big enough): a finite ${\msy Z}_l$-module $M$ corresponds to the partition $m=(m_1,m_2,\ldots)$ which satisfies $M\cong\oplus_{i\geq1}{\msy Z}/l^{m_i}{\msy Z}$. To any partition $m$ we attach the number $\delta_l(m):=\sum_{i\geq1}(l^{m_i}-1)$. Note that with these definitions, we have $\delta_l(M)=\delta_l(m)$, with $\delta_l(M)$ as in Def.~\ref{def31}. \begin{lemma} \label{lemma41} Let $0\to B\to E\to A\to 0$ be an extension of finite ${\msy Z}_l$-modules. Let $b=(b_1,b_2,\ldots)$, $e$ and $a$ denote their invariants. Define $n_i:=a_i+b_i$ and $n=(n_1,n_2,\ldots)$. Let $m=(m_1,m_2,\ldots)$ be the invariant of $A\oplus B$, i.e., $m$ is the sequence obtained by reordering $(a_1,b_1,a_2,b_2,\ldots)$. Then we have $m\leq e\leq n$, with ``$\leq$'' the lexicographical ordering. \end{lemma} \begin{proof} Let us first prove that $e\leq n$. We use induction on $|E|$. We have $e_1\leq a_1+b_1=n_1$ since $l^{a_1+b_1}$ kills $E$. If $e_1<n_1$ there is nothing to prove, so we suppose that $e_1=n_1$. Choose any element $x$ in $E$ of order $l^{e_1}$ and consider the subgroup it generates. We get a diagram \begin{subeqn} \label{eqn411} \begin{array}{ccccccccc} & & 0 & & 0 & & 0 & & \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 & \to & {\msy Z}/l^{b_1}{\msy Z} & \to & {\msy Z}/l^{e_1}{\msy Z} & \to & {\msy Z}/l^{a_1}{\msy Z} & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 & \to & B & \to & E & \to & A & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 & \to & B' & \to & E' & \to & A' & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ & & 0 & & 0 & & 0 & & \end{array} \end{subeqn} in which the rows and columns are exact. Now the columns are split, since $l^{b_1}$ is the exponent of $B$, etc. Hence $b':=(b_2,b_3,\ldots)$, $e':=(e_2,e_3,\ldots)$ and $a':=(a_2,a_3,\ldots)$ are the invariants of $B'$, $E'$ and $A'$, respectively. The proof is finished by induction. Let us now prove that $e\geq m$. By passing to Pontrjagin duals, if necessary, we may assume that $b_1\geq a_1$. Then $m_1=b_1$. If $e_1>m_1$ there is nothing to prove, hence we suppose that $e_1=m_1=b_1$. We choose any element $x$ in $B$ of order $l^{b_1}$. Just as above we find a diagram \begin{subeqn} \label{eqn412} \begin{array}{ccccccccc} & & 0 & & 0 & & & & \\ & & \downarrow & & \downarrow & & & & \\ & & {\msy Z}/l^{b_1}{\msy Z} & \stackrel{\rm id}{\to} & {\msy Z}/l^{b_1}{\msy Z} & & & & \\ & & \downarrow & & \downarrow & & & & \\ 0 & \to & B & \to & E & \to & A & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 & \to & B' & \to & E' & \to & A & \to & 0 \\ & & \downarrow & & \downarrow & & & & \\ & & 0 & & 0 & & & & \end{array} \end{subeqn} in which the columns are split. Induction finishes the proof. \end{proof} \begin{rmk} \label{rmk42} It would be nice to have a complete description of the possible invariants of extensions $E$ of finite ${\msy Z}_l$-modules $A$ by $B$ in terms of the invariants of $A$ and $B$. In particular, are there more restrictions than the following: those in Lemma~\ref{lemma41}, $e_i\geq a_i$ and $e_i\geq b_i$ for all~$i$, and the minimal number of generators for $E$ does not exceed the sum of those numbers for $A$ and $B$? As Hendrik Lenstra pointed out to me, the problem can be phrased in terms of Hall polynomials, see for example \cite{MacDonald1}. \end{rmk} \begin{lemma} \label{lemma43} Suppose that $a=(a_1,a_2,\ldots)$ and $b=(b_1,b_2,\ldots)$ are partitions of $N$ (i.e., $\sum_{i\geq1}a_i=N=\sum_{i\geq1}b_i$) and that $a\geq b$ in the lexicographical ordering. Then $\delta_l(a)\geq\delta_l(b)$, with equality if and only if $a=b$. \end{lemma} \begin{proof} Consider the set $X$ of all partitions of $N$ with its lexicographical ordering. From the inequality $$ (l^{n+1}-1) + (l^{m-1}-1) > (l^n-1) + (l^m-1) $$ satisfied for any integers $n\geq m$ it follows that $\delta_l\colon X\to{\msy Z}$ is strictly increasing. \end{proof} \begin{lemma} \label{lemma44} \begin{enumerate} \item For $M$ a finite ${\msy Z}_l$-module we have $\delta_l(M)\geq0$, with equality if and only if $M=0$. \item Let $0\to M'\to M\to M''\to 0$ be a short exact sequence of finite ${\msy Z}_l$-modules. Then $\delta_l(M)\geq\delta_l(M')+\delta_l(M'')$, with equality if and only if the sequence is split. \item Let $0\to M'\to M\to M''\to 0$ be a short exact sequence of finite ${\msy Z}_l$-modules. Suppose that $M$ is killed by $l^a$ and that $|M'|=l^{b}$. Then $\delta_l(M)\leq \delta_l(M'')+b(l^a-l^{a-1})$. \end{enumerate} \end{lemma} \begin{proof} This follows directly from Lemmas~\ref{lemma41} and \ref{lemma43}. \end{proof} \begin{lemma} \label{lemma45} Let $M$ be a finitely generated free ${\msy Z}_l$-module with an automorphism $\sigma$ of finite order. Suppose that $M/(\sigma-1)M$ is finite, or, equivalently, that the automorphism $\sigma\otimes1$ of the ${\msy Q}_l$-vector space $M\otimes{\msy Q}$ does not have $1$ as eigenvalue. For $i\geq1$ let $m_i$ be the multiplicity, in the characteristic polynomial of $\sigma$, of the cyclotomic polynomial $f_i$ whose roots are the roots of unity of order $l^i$. For each $j\geq1$, let $p_j:=|\{i\geq1\;|\; m_i\geq j\}|$. Then $\delta_l(M/(\sigma-1)M)\leq \sum_{i\geq1}(l^{p_i}-1)$. \end{lemma} \begin{proof} Let $q=(q_1,q_2,\ldots)$ be the partition obtained by reordering $(m_1,m_2,\ldots)$. Then $p:=(p_1,p_2,\ldots)$ is what is usually called the conjugate of $q$: when viewing partitions as Young diagrams, $p$ and $q$ are obtained from each other by interchanging rows and columns. In particular, we have $\sum_{i\geq1}p_i=\sum_{i\geq1}m_i$. Let $n$ be the order of $\sigma$. Then $M$ becomes a module over the ring ${\msy Z}_l[x]/(x^n-1)$. Let us write $n=l^rn'$ with $n'$ not divisible by $l$. Then ${\msy Z}_l[x]/(x^n-1)$ is the product of the ring ${\msy Z}_l[x]/(x^{l^r}-1)$ by another ring $R$ and $x-1$ is invertible in $R$. This implies that $M$ is the direct sum of two modules, one over ${\msy Z}_l[x]/(x^{l^r}-1)$ and the other over $R$, and that the module over $R$ does not contribute to $M/(\sigma-1)M$. Hence we have reduced the problem to the case where the order of $\sigma$ is $l^r$. Let $i_1<i_2<\cdots<i_{p_1}$ denote the integers $i\geq1$ such that $m_i>0$. For $1\leq j\leq p_1$, let $F_j:=f_{i_j}$ be the corresponding cyclotomic polynomials, and let $F:=F_1{\cdot}F_2\cdots F_{p_1}$. Since $1$ is not an eigenvalue of $\sigma$ on $M\otimes{\msy Q}$, and $M$ is torsion free as ${\msy Z}_l$-module, $M$ is a module over the ring $A:={\msy Z}_l[x]/(F)$. For any $A$-module $N$, we define $\overline{N}:=N/(x-1)N$. Let us first note that for all $j$ we have $F_j(1)=l$. It follows that $\overline{A}={\msy Z}/l^{p_1}{\msy Z}$. For $N$ an $A$-module, $\overline{N}$ is an $\overline{A}$-module, hence $l^{p_1}$ annihilates $\overline{N}$. We claim that $|\overline{M}|=l^{\sum_{i\geq1}p_i}$. To prove this, note that $|\overline{M}|=|\det(\sigma-1)|_l^{-1}$, with $|\cdot|_l$ the $l$-adic absolute value on ${\msy Q}_l$, normalized by $|l|_l=1/l$. So in order to compute $|\overline{M}|$ we may replace $M$ by any $\sigma$-stable lattice $M'$ in $M\otimes{\msy Q}\cong\oplus_{i\geq1}({\msy Q}_l[x]/(f_i))^{m_i}$. Taking $M':=\oplus_{i\geq1}({\msy Z}_l[x]/(f_i))^{m_i}$ and noting that $f_i(1)=1$ gives the result. Let $a=(a_1,a_2,\ldots)$ be the invariant of $\overline{M}$, i.e., $\overline{M}\cong\oplus_{i\geq1}{\msy Z}/l^{a_i}{\msy Z}$ and $a_1\geq a_2\geq\cdots$. Note that $a$ and $p$ are partitions of the same number, hence in view of Lemma~\ref{lemma43}, it suffices to show that $a\leq p$ in the lexicographical ordering. Since $l^{p_1}$ annihilates $\overline{M}$, we have $a_1\leq p_1$. If $a_1<p_1$ there is nothing to prove, so we assume that $a_1=p_1$. Let $y$ be in $M$ such that its image $\overline{y}$ in $\overline{M}$ corresponds to $(1,0,0,\ldots)$. Let $A'$ denote the submodule $Ay$ of $M$. Since $M$ is free as a ${\msy Z}_l$-module, $A'$ is free as a ${\msy Z}_l$-module, and we have $A'={\msy Z}_l[x]/(G)$, with $G$ dividing $F$. Let $0\leq s\leq p_1$ be the number of irreducible factors of $G$. Then we have $\overline{A'}={\msy Z}/l^s{\msy Z}$. We have a short exact sequence \begin{subeqn} \label{eqn451} 0 \to A' \to M \to M' \to 0 \end{subeqn} of $A$-modules, with $M'$ not necessarily free as ${\msy Z}_l$-module. Multiplication by $x-1$ on this sequence induces an exact sequence \begin{subeqn} \label{eqn452} 0 \to M'[x-1] \to \overline{A'} \to \overline{M} \to \overline{M'} \to 0 \end{subeqn} The element $\overline{1}$ of $\overline{A'}$, which is annihilated by $l^s$, is mapped to $\overline{y}$ which has annihilator $l^{p_1}$. It follows that $s=p_1$, that $G=F$ and that $M'[x-1]=0$. Let us now consider the finite $A$-module ${\rm tors}(M')$. Multiplication by $x-1$ acts injectively, hence bijectively. Since $x-1$ is in the maximal ideal of $A$, it follows that ${\rm tors}(M')=0$, hence that $M'$ is free as ${\msy Z}_l$-module. The proof is now finished by induction on ${\rm rank}(M)$, since $\overline{M'}\cong\oplus_{i\geq2}{\msy Z}/l^{a_i}{\msy Z}$ and the partition $p'$ obtained from $M'$ is $(p_2,p_3,\ldots)$. \end{proof} \begin{rmk}\label{rmk46} Lemma~\ref{lemma45} can be seen as a bound on the cohomology group ${\rm H}^1({\msy Z}/n{\msy Z},M)$, where $1$ in ${\msy Z}/n{\msy Z}$ acts on $M$ via $\sigma$. It is an interesting question, raised by Xavier Xarles, to obtain similar bounds for non-cyclic groups. \end{rmk} \begin{corollary} \label{cor46} Let $M$ be a finitely generated free ${\msy Z}_l$-module with an automorphism $\sigma$ of finite order. Suppose that $M/(\sigma-1)M$ is finite. Then $\delta_l(M/(\sigma-1)M)\leq{\rm rank}(M)$. \end{corollary} \begin{proof} We use the notation of the beginning of the proof of Lemma~\ref{lemma45}. Then one has: \begin{subeqn} \label{eqn461} {\rm rank}(M)\geq \sum_{i\geq1}m_i\phi(l^i) \geq \sum_{i\geq1}q_i\phi(l^i) = \sum_{i\geq1}\sum_{j=1}^{p_i}\phi(l^j) = \sum_{i\geq1}(l^{p_i}-1) \end{subeqn} The proof is finished by applying Lemma~\ref{lemma45}. \end{proof} \begin{lemma} \label{lemma47} Let $M$ be a finitely generated free ${\msy Z}_l$-module with an automorphism $\sigma$ of finite order. Suppose that $M/(\sigma-1)M$ is finite and that $\delta_l(M/(\sigma-1)M)={\rm rank}(M)$. Then $M$ is a direct sum of ${\msy Z}_l$-modules of the type ${\msy Z}_l[x]/(f_1{\cdot}f_2\cdots f_r)$ with $\sigma$ acting as multiplication by $x$ and where $f_i$ denotes the cyclotomic polynomial whose roots are the roots of unity of order $l^i$. \end{lemma} \begin{proof} The proof is by induction on ${\rm rank}(M)$ and consists of an inspection of the proofs of Lemma~\ref{lemma45} and Cor.~\ref{cor46}. First of all we must have that $n'=1$. Secondly, we note that $m_i=q_i$ for all $i\geq1$ since the inequalities in (\ref{eqn461}) are equalities (here we use that $\sum_{i\geq1}m_i=\sum_{i\geq1}q_i$ and that $\phi(l^i)<\phi(l^j)$ if $1\leq i<j$). So $m$ is the conjugate partition of $p$, hence $A={\msy Z}_l[x]/(F)$ with $F=f_1{\cdot}f_2\cdots f_{p_1}$. The formula for the number of elements of $|\overline{M}|$ in the proof of Lemma~\ref{lemma45} shows that $a$ and $p$ are partitions of the same number. By the hypotheses of the lemma we are proving, we have $\delta_l(a)=\delta_l(p)$. Lemma~\ref{lemma43} implies that $a=p$. The end of the proof of Lemma~\ref{lemma45} shows that $A'=A$ and that $M'$ is free as ${\msy Z}_l$-module. By induction on ${\rm rank}(M)$, we know that $M'$ is of the indicated type. It remains to show that the short exact sequence (\ref{eqn451}) splits. To do that, it is sufficient to show that ${\rm Ext}^1_A(A_i,A)=0$, where $A_i={\msy Z}_l[x]/(f_1\cdots f_i)$ with $i\leq p_1$. This ${\rm Ext}^1$ is easily computed using the projective resolution \begin{subeqn}\label{eqn471} \cdots \longrightarrow A \;\;\stackrel{f}{\longrightarrow}\;\;A \;\;\stackrel{g}{\longrightarrow}\;\;A \;\;\stackrel{f}{\longrightarrow}\;\;A \longrightarrow A_i \longrightarrow 0 \end{subeqn} with $f=f_1\cdots f_i$ and $g=f_{i+1}\cdots f_{p_1}$. \end{proof} The following lemmas will be used in \S\ref{section5} and \S\ref{section6}. \begin{lemma} \label{lemma48} Let $M$ be a finite ${\msy Z}_l$-module and let $$ M=M^0\supset M^1\supset M^2\supset\cdots\supset M^r=0 $$ be a strictly descending filtration. Suppose that for all $i$ with $0\leq i\leq r-2$ the group $M^i/M^{i+2}$ is cyclic. Then $M$ is cyclic. \end{lemma} \begin{proof} For $r\leq2$ there is nothing to prove. If we know the result for $r=3$, the general case follows by induction since then $M^0/M^3$ is cyclic and the filtration $M^0\supset M^2\supset M^3\supset\cdots\supset M^r$ has length $r-1$. So assume now that $r=3$. Let $x$ be an element of $M^0$ such that its image in $M^0/M^2$ is a generator. Then a certain multiple $ax$ of $x$ gives a generator of $M^1/M^2$. Since $M^1$ is cyclic, and $M^1/M^2$ a non-trivial quotient, $ax$ is a generator of $M^1$. The subgroup of $M^0$ generated by $x$ contains $M^1$ and its quotient by $M^1$ is $M^0/M^1$. We conclude that $x$ generates~$M^0$. \end{proof} \begin{rmk}\label{remark49} The proof of Lemma~\ref{lemma48} generalizes immediately to a proof of the following assertion: let $A$ be a local ring and $M$ an $A$-module with a finite strictly descending filtration $M^i$ such that the $M^i/M^{i+2}$ are cyclic, then $M$ is cyclic. \end{rmk} \begin{lemma}\label{lemma410} Let $0\to B\to E\to A\to 0$ be a short exact sequence of finite ${\msy Z}_l$-modules with invariants $b$, $e$ and $a$. Let $t\geq0$ be an integer and suppose that $B$ is generated by $t$ elements. Then for all $i\geq1$ we have $a_i\geq e_{i+t}$. \end{lemma} \begin{proof} For a partition $p$, let $p'$ denote its conjugate. Then for all $i\geq1$ we have $l^{a_i'}=|A[l^i]A[l^{i-1}]|$. Let $d$ be the endomorphism of the set of partitions defined by: $d(p)_i=p_{i+1}$ for all $i\geq1$. Let $d'$ be the conjugate of $d$: $d'(p)=d(p')'$. Then $d'(p)_i=\max(0,p_i-1)$. When viewing a partition $p$ as a Young diagram in which the $p_i$ are the lengths of the columns, $d$ and $d'$ remove the longest column and row, respectively. Note that for a finite ${\msy Z}_l$-module $M$ with invariant $m$, the submodule $lM$ has invariant $d'(m)$. Note that $d$ and $d'$ commute. In the rest of this proof we will consider the partial ordering on the set of partitions in which $p\leq q$ if and only if for all $i\geq1$: $p_i\leq q_i$. Note that $p\leq q$ is equivalent to $p'\leq q'$. Below we will use that $p\geq q$ if and only if: $p_1'\geq q_1'$ and $d'(p)\geq d'(q)$. We will also use that if $N$ and $M$ are finite ${\msy Z}_l$-modules with invariants $n$ and $m$ such that $N$ is a subquotient of $M$, then $n\leq m$. The proof of the lemma is by induction on $|E|$. What we have to prove is that $a\geq d^t(e)$. The exact sequence $0\to B[l]\to E[l]\to A[l]$ shows that $a_1'\geq e_1'-t$. Note that $d^t(e)'_1=\max(0,e_1'-t)$, hence we have $a_1'\geq d^t(e)'_1$. The exact sequence $0\mapsto B\cap lE\to lE\to lA$ shows (induction hypothesis) that $d'(a)\geq d^t(d'(e))=d'(d^t(e))$. The two inequalities just proved imply that $a\geq d^t(e)$. \end{proof} \begin{lemma}\label{lemma411} Let $l$ be a prime. Let $0\to B\to E\to A$ be a short exact sequence of finite ${\msy Z}_l$-modules with invariants $b$, $e$ and $a$, respectively. Then \begin{subeqn}\label{eqn4111} \delta_l(b) + \delta_l(a) \geq \sum_{i\geq1}\left( \frac{l^{\lfloor e_i/2\rfloor}+l^{\lceil e_i/2\rceil}}{2}-1\right) \end{subeqn} where for any real number $x$, $\lfloor x\rfloor$ and $\lceil x\rceil$ denote the largest (resp. smallest) integer $\leq x$ (resp. $\geq x$). \end{lemma} \begin{proof} We have $\sum_i(a_i+b_i)=\sum_i e_i$. Lemma~\ref{lemma41} asserts that $a+b\geq e$ in the lexicographical ordering. Consider the set $S$ of all pairs $(r,s)$ of partitions, such that $r+s\geq e$ and $\sum_i(r_i+s_i)=\sum_i e_i$. Let $f\colon S\to {\msy Z}$ be the map which sends $(r,s)$ to $\delta_l(r)+\delta_l(s)$. We will show that $f$ achieves its minimum at all $(r,s)$ in $S$ with the property that, for all $i\geq1$, one has $\{r_i,s_i\}=\{\lfloor e_i/2\rfloor,\lceil e_i/2\rceil\}$. Suppose now that $(r,s)$ is an element of $S$ where $f$ has a minimum. We have to show that $|r_i-s_i|\leq1$ for all $i\geq1$. Suppose that this is not the case. Let $j\geq1$ be minimal for the property that $|r_j-s_j|>1$ and $|r_i-s_i|\leq1$ for all $i<j$. We may and do suppose that $r_j-s_j>1$. Note that if $j>1$ we have $s_{j-1}>s_j$. We define $r'$ and $s'$ as follows: $(r'_i,s'_i)=(r_i-1,s_i+1)$ if $i\geq j$ and $r_i=r_j$; in all other cases $(r'_i,s'_i)=(r_i,s_i)$. Note that $r'$ and $s'$ are partitions, that $\sum_i(r'_i+s'_i)$ is equal to $\sum_i e_i$ and that $f(r',s')$ is strictly smaller than $f(r,s)$. \end{proof} \section{Examples.} \label{section5} The aim of this section is to give examples that show that the bounds in Thm.~\ref{thm33} and Cor.~\ref{cor34} are sharp, in a sense that will become clear in the examples. The examples we construct here will play an important role in \S\ref{section6}. We give our examples over the field $K:={\msy C}((q))$ of formal Laurent series over the complex numbers with its usual valuation, but it is easy to get similar examples in mixed characteristic, or equal characteristic $p>0$. The building stones of our examples are the following. For each integer $n\geq1$ we let $E_n$ be the so-called Tate elliptic curve ``${{\rm G}_m}/q^{n{\msy Z}}$'' over $K$ as described in \cite[\S VII]{DeligneRapoport} or in \cite[\S6]{Mumford} ($E_n$ is obtained from the analytic family of elliptic curves over the punctured unit disc with coordinate $q$ whose fibres are the ${\msy C}^*/q^{n{\msy Z}}$, by base change from the field of finite tailed convergent Laurent series to $K$). It is well known that the special fibre of the N\'eron model of $E_n$ over $D:={\msy C}[[q]]$ is an extension of ${\msy Z}/n{\msy Z}$ by the multiplicative group. For each prime $l$ and integer $r\geq0$ we define the ring $\Lambda_{l,r}:={\msy Z}[x]/(f_{l,1}\cdots f_{l,r})$, where as before $f_{l,i}$ is the polynomial whose roots are the roots of unity of order $l^i$. When $l>2$, we let $A_{l,r}$ be an abelian variety over ${\msy C}$ obtained as follows: we choose an isomorphism of ${\msy R}$-algebras between $\Lambda_{l,r}\otimes{\msy R}$ and a product of a number of copies of ${\msy C}$ and define $A_{l,r}:=(\Lambda_{l,r}\otimes{\msy R})/\Lambda_{l,r}$ (it is well known that the trace form on $\Lambda_{l,r}$ implies the existence of a polarization). The first three examples will be isogenous to twists of products of copies of $E_{n}$ and of $A_{l,r,K}$. Of course Lemma~\ref{lemma47} tells us how to cook up the required examples. \begin{example}\label{example51} Let $d\geq0$ and let $G$ be any finite abelian group that can be generated by $d$ elements. Then $G\cong\oplus_{i=1}^d{\msy Z}/n_i{\msy Z}$, say. For $A_K:=\prod_{i=1}^d E_{n_i}$ one has $\Phi=\Phi^3=\oplus_{i=1}^t{\msy Z}/n_i{\msy Z}$ and we have $d=t=\dim(A_K)$. \end{example} \begin{example}\label{example52} Now consider parts~2 and 4 of Thm.~\ref{thm33}. Let $l$ be a prime. For $i\geq1$ we let $B_{l,i}$ be the abelian variety $E_1\otimes\Lambda_{l,i}$ over ${\msy C}$, i.e., $B_{l,i}$ is a direct sum of copies of $E_1$, indexed by some ${\msy Z}$-basis of $\Lambda_{l,i}$, and $\Lambda_{l,i}$ acts on $B_{l,i}$ according to its action on itself. In particular, multiplication by $x$ in $\Lambda_{l,i}$ induces an automorphism $\sigma$ of $B_{l,i}$. Note that $\sigma$ has order $l^i$. Let $C_{l,i}$ be the twist of $B_{l,i,K}$ over $K(q^{1/l^i})$ by $\sigma$, i.e., $C_{l,i}$ is the quotient of the $K$-scheme $B_{l,i,K}\times_{{\rm Spec}(K)}{\rm Spec}(K(q^{1/l^i}))$ by the group ${\rm Gal}(K(q^{1/l^i})/K)={\msy Z}/l^i{\msy Z}$ (here we choose a root of unity of order $l^i$) which acts by $a\mapsto\sigma^a$ on the first factor and via its natural action on the second factor. We will now compute the group of connected components $\Psi$ of the N\'eron model of $C_{l,i}$ over $D$, using (\ref{eqn24}). First of all we have ${\rm T}_l(E_1({K^{\rm s}}))={\msy Z}_l(1)\oplus{\msy Z}_l$, with $I$ acting via its quotient ${\msy Z}_l(1)$ in the following way: an element of $I$ with image $a$ in ${\msy Z}_l(1)$ acts as multiplication by the matrix $({1\atop0}{a\atop1})$. By construction, ${\rm T}_l(C_{l,i}({K^{\rm s}}))={\rm T}_l(E_1({K^{\rm s}}))\otimes\Lambda_{l,i}$ and an element in $I$ with image $a$ in ${\msy Z}_l(1)$ acts as $({1\atop0}{a\atop1})\otimes x^a$. Since $C_{l,i}$ has ${\tilde{a}}=t=0$, we have $\Psi_l^1=\Psi_l^2$ and $\Psi_l^3=0$. The filtration ${\msy Z}_l(1)\subset{\rm T}_l(E_1({K^{\rm s}})$ induces the filtration ${\widetilde{W}}_l\subset{\rm T}_l(C_{l,i}({K^{\rm s}}))$. It follows that $\Psi_l$ is the cokernel of $({x-1\atop0}{x\atop x-1})$ and that $\Psi_l/\Psi_l^1$ and $\Psi_l^2/\Psi_l^3$ are both isomorphic to $\Lambda_{l,i}/(x-1)={\msy Z}/l^i{\msy Z}$. An analogous computation shows that $\Psi=\Psi_l$. One can show that $\Psi$ is isomorphic to ${\msy Z}/l^i{\msy Z}\oplus{\msy Z}/l^i{\msy Z}$ if $l>2$ and to ${\msy Z}/2^{i+1}{\msy Z}\oplus{\msy Z}/2^{i-1}{\msy Z}$ if $l=2$. Let $G$ be a finite abelian group of $l$-power order, say with invariant $a=(a_1,a_2,\ldots)$. Then for $A_K:=\prod_{i\geq1}C_{l,a_i}$ we have $\Phi_l/\Phi_l^1\cong\Phi_l^2/\Phi_l^3\cong G$ and $\delta_l(G)=t_l=\dim(A_K)$. We remark that abelian varieties over $K$ that are isogeneous to $A_K$ provide examples with $\Phi_l/\Phi_l^1$ not isomorphic to $\Phi_l^2/\Phi_l^3$. \end{example} \begin{example}\label{example53} For $l>2$ prime and $i\geq0$ we let $D_{l,i}$ be the abelian variety over $K$ obtained by twisting $A_{l,i,K}$ over $K(q^{1/l^i})$ by the automorphism $\sigma$ of $A_{l,i}$ which is induced from the multiplication by $x$ in $\Lambda_{l,i}$. Then we have ${\rm T}_l(D_{l,i}({K^{\rm s}}))={\rm T}_l(A_{l,i}({\msy C}))=\Lambda_{l,i}\otimes{\msy Z}_l$, and an element in $I$ with image $a$ in ${\msy Z}_l(1)$ acts as $x^a$. Let $\Psi_l$ denote the group of connected components of attached to $D_{l,i}$. In this case we have ${\tilde{t}}=a=0$, hence $\Psi_l=\Psi_l^1$ and $\Psi_l^2=0$. By (\ref{eqn24}) we have $\Psi_l=\Lambda_{l,i}/(x-1)={\msy Z}/l^i{\msy Z}$. Suppose now that $l\neq2$. Let $G$ be a finite abelian group of $l$-power order, say with invariant $a=(a_1,a_2,\ldots)$. Then for $A_K:=\prod_{i\geq1}D_{l,a_i}$ we have $\Phi_l=\Phi_l^1$, $\Phi_l^2=0$, $\Phi_l^1/\Phi_l^2\cong G$ and $\delta_l(G)=2a_l=2\dim(A_K)$. The case $l=2$ is a little bit different because $m_{{\rm a},2,1}$ is always even. \end{example} \begin{example} \label{example54} Let $l$ be prime and let $r>0$ and $s>0$ be positive integers. We will construct an abelian variety $A_K$ with $t=a=0$, ${\tilde{t}}=l^r-1$, ${\tilde{a}}=(l^{r+s}-l^r)/2$ and $\Phi=\Phi_l\cong{\msy Z}/l^{2r+s}{\msy Z}$. It follows from Thm.~\ref{thm33} that in such an example $\Phi_l/\Phi_l^1$ and $\Phi_l^2$ are cyclic of order $l^r$, that $\Phi_l^1/\Phi_l^2$ is cyclic of order $l^s$ and that $\Phi_l/\Phi_l^2$ and $\Phi_l^1$ are cyclic of order $l^{r+s}$. Hence this example shows that, as far as the exponent is concerned, the two-fold extension $\Phi_l/\Phi_l^3$ can be arbitrary. As in the previous examples, $f_{l,i}$ will denote the polynomial whose roots are the roots of unity of order $l^i$, and $\Lambda_{l,r}$ is the ring ${\msy Z}[x]/(f_{l,1}\cdots f_{l,r})$. Let $\Lambda_{l,r,s}:={\msy Z}[x]/(f_{l,r+1}\cdots f_{l,r+s})$. Let $D_{l,r,s}$ be an abelian variety over $K$ obtained by replacing $\Lambda_{l,i}$ by $\Lambda_{l,r,s}$ and $q^{1/l^i}$ by $q^{1/l^{r+s}}$ in the construction of $D_{l,i}$ in Example~\ref{example53}. Let $C_{l,r}$ be as in Example~\ref{example52}. Our example $A_K$ will be isogeneous to $C_{l,r}\times D_{l,r,s}$. Let $V:={\rm T}_l((C_{l,r}\times D_{l,r,s})({K^{\rm s}}))\otimes{\msy Q}$. Then $V$ is a ${\msy Q}_l$-vector space with an action of $I={\rm Gal}({K^{\rm s}}/K)$. We have an isomorphism of ${\msy Q}_l$-vector spaces with $I$-action \begin{subeqn}\label{eqn541} \Lambda_{l,r}\otimes{\msy Q}_l \;\oplus\; \Lambda_{l,r,s}\otimes{\msy Q}_l \;\oplus\; \Lambda_{l,r}\otimes{\msy Q}_l \;\;\tilde{\longrightarrow}\;\; V \end{subeqn} such that an element of $I$ with image $a$ in ${\msy Z}_l(1)$ acts via \begin{subeqn}\label{eqn542} \left(\begin{array}{ccc}x^a&0&ax^a\\0&x^a&0\\0&0&x^a\end{array}\right) \end{subeqn} Let \begin{subeqn}\label{eqn543} V=V^0\supset V^1\supset V^2\supset V^3=0 \end{subeqn} be the filtration (\ref{eqn27}) on $V$. Then $V^2$ is simply the first term in (\ref{eqn541}) and $V^1$ is the sum of the first two terms. For any ${\msy Z}_l$-lattice $M$ in $V$ let $M^i:=M\cap V^i$. To get our example $A_K$, it suffices to find an $I$-invariant ${\msy Z}_l$-lattice $M$ in $V$ such that $M^1$ and $M/M^2$ are isomorphic, as ${\msy Z}_l[I]$-modules, to $\Lambda_{l,r+s}\otimes{\msy Z}_l$, where an element of $I$ with image $a$ in ${\msy Z}_l(1)$ acts on $\Lambda_{l,r+s}\otimes{\msy Z}_l$ as $x^a$. Namely, since $M$ is $I$-invariant, $M$ is the $l$-adic Tate module of an abelian variety $A_K$ which is isogeneous to $C_{l,r}\times D_{l,r,s}$; for $A_K$ one has $\Phi_l/\Phi_l^2$ and $\Phi_l^1$ cyclic of order $l^{r+s}$, hence $\Phi_l$ cyclic of order $l^{2r+s}$ by Lemma~\ref{lemma48}. Let us now try to find such a $M$. Note that we have canonical projections $\Lambda_{l,r+s}\to\Lambda_{l,r}$ and $\Lambda_{l,r+s}\to\Lambda_{l,r,s}$ which induce an embedding $\Lambda_{l,r+s}\otimes{\msy Z}_l\subset V^1$. We will first show that we only have to look among the sublattices $M$ with $M^1=\Lambda_{l,r+s}\otimes{\msy Z}_l$. Namely, if $M$ is an $I$-invariant ${\msy Z}_l$-lattice in $V$ of the type we are looking for, then for a suitable element of the form $v=(a,b,a)$ of $V^*$ (here we consider $V$ as a ${\msy Q}_l$-algebra and $V^*$ denotes the group of units of $V$) $vM$ is isomorphic to $M$ as ${\msy Z}_l[I]$-module and has $(vM)^1=\Lambda_{l,r+s}\otimes{\msy Z}_l$. Since the ${\msy Z}_l[I]$-module structure determines the filtration, we also have $(vM)/(vM)^2\cong M/M^2$. \par From now on we only consider $M$ with $M^1=\Lambda_{l,r+s}\otimes{\msy Z}_l$. Such $M$ are determined by their image in $V/M^1$. So we look for an $I$-invariant torsion free ${\msy Z}_l$-submodule $N$ of $V/M^1=V^1/M^1\oplus\Lambda_{l,r}\otimes{\msy Q}_l$ whose image in $\Lambda_{l,r}\otimes{\msy Q}_l$ is a lattice and for whose associated $M$ we have $M/M^2\cong\Lambda_{l,r+s}\otimes{\msy Z}_l$. It follows that such a $N$ is isomorphic, via the canonical projection, to its image in $\Lambda_{l,r}\otimes{\msy Q}_l$. Lemma~\ref{lemma47} implies that this image is isomorphic to $\Lambda_{l,r}\otimes{\msy Z}_l$, hence of the form $z{\cdot}\Lambda_{l,r}\otimes{\msy Z}_l$ for some $z$ in $(\Lambda_{l,r}\otimes{\msy Q}_l)^*$. We conclude that $N$ is of the form ${\rm im}(\alpha)$, where \begin{subeqn}\label{eqn544} \alpha\colon \Lambda_{l,r}\otimes{\msy Z}_l \longrightarrow V^1/M^1\;\oplus\;\Lambda_{l,r}\otimes{\msy Q}_l, \quad a\mapsto(\phi(a),za) \end{subeqn} with $\phi\colon\Lambda_{l,r}\otimes{\msy Z}_l\to V^1/M^1$ a morphism of ${\msy Z}_l$-modules, and $z\in(\Lambda_{l,r}\otimes{\msy Q}_l)^*$. For a given pair $(\phi,z)$, let $N_{\phi,z}$ denote the image of the corresponding $\alpha$. Let us first study what it means for $(\phi,z)$ that $N_{\phi,z}$ is $I$-invariant. Using that $N_{\phi,z}$ is $I$-invariant if and only if it is invariant under the matrix in (\ref{eqn542}) with $a$ replaced by $1$, one easily sees that $N_{\phi,z}$ is $I$-invariant if and only if \begin{subeqn}\label{eqn545} \forall a\in\Lambda_{l,r}\otimes{\msy Z}_l\colon \quad \phi(xa)=x\phi(a)+\overline{(xza,0)} \end{subeqn} To find out which $(\phi,z)$ satisfy (\ref{eqn545}), we write out everything in terms of the ${\msy Z}_l$-basis $(1,x,\ldots,x^{l^r-2})$ of $\Lambda_{l,r}\otimes{\msy Z}_l$. Let $y=(y_1,y_2)$ be in $V^1$ such that $\phi(1)=\overline{y}$. One then checks that \begin{subeqn}\label{eqn546} \phi(x^i) = x^i\overline{y} + ix^i\,\overline{(z,0)}, \quad \mbox{for $0\leq i\leq l^r-2.$} \end{subeqn} Applying (\ref{eqn545}) with $a=x^{l^r-2}$, and using that $\sum_{i=0}^{l^r-1}x^i=0$ in $\Lambda_{l,r}$, gives \begin{subeqn}\label{eqn547} (xg_r'(x)z,g_r(x)y_2) \in \Lambda_{l,r+s}\otimes{\msy Z}_l \end{subeqn} where $g_r=f_{l,1}\cdots f_{l,r}$ and $g_r'$ is the derivative of $g_r$. The conclusion is that $N_{\phi,z}$ is $I$-invariant if and only if $\phi$ is given by (\ref{eqn546}) and $(y,z)$ satisfies (\ref{eqn547}). For a given such pair $(y,z)$, let $M_{y,z}$ denote the lattice $M$ in $V$ corresponding to $N_{\phi,z}$. It remains now to be seen that there exist $(y,z)$ satisfying (\ref{eqn547}) such that $M_{y,z}/M_{y,z}^2$ is isomorphic to $\Lambda_{l,r+s}\otimes{\msy Z}_l$, or, equivalently, such that $(M_{y,z}/M_{y,z}^2)_I$ is cyclic. In order to have a useful description of $M_{y,z}$, we lift $\phi$ to $V^1$ as follows: let ${\widetilde{\phi}}\colon\Lambda_{l,r}\otimes{\msy Z}_l\to V^1$ be the morphism of ${\msy Z}_l$-modules such that \begin{subeqn}\label{eqn548} {\widetilde{\phi}}\colon x^i\mapsto x^iy + ix^i(z,0), \quad\mbox{for $0\leq i\leq l^r-2$} \end{subeqn} Then we have an isomorphism of ${\msy Z}_l$-modules: \begin{subeqn}\label{eqn549} \beta\colon \Lambda_{l,r+s}\otimes{\msy Z}_l\;\oplus\;\Lambda_{l,r}\otimes{\msy Z}_l \;\;\tilde{\longrightarrow}\;\; M\subset V,\quad (a,b)\mapsto (a,0)+({\widetilde{\phi}}(b),zb) \end{subeqn} Let $\tau$ be an element of $I$ with image $1$ in ${\msy Z}_l(1)$. Then $\tau$ acts on $V$ by the matrix in (\ref{eqn542}) with $a=1$. One computes that in order to make $\beta$ invariant under $I$, one must let $\tau$ act on the source of $\beta$ in (\ref{eqn549}) by \begin{subeqn}\label{eqn5410} \tau\colon (a,b)\mapsto \left(xa+(xzb,0)+x{\widetilde{\phi}}(b)-{\widetilde{\phi}}(xb),xb\right) \end{subeqn} Using this formula, we can study $M_{y,z}/M_{y,z}^2$. Recall that $\Lambda_{l,r+s}\otimes{\msy Z}_l$ is the image in $V^1$ of the sum of the two canonical projections from $\Lambda_{l,r+s}\otimes{\msy Z}_l$ to $\Lambda_{l,r}\otimes{\msy Z}_l$ and $\Lambda_{l,r,s}\otimes{\msy Z}_l$. It follows that \begin{subeqn}\label{eqn5411} M_{y,z}/M_{y,z}^2 \cong N:= \Lambda_{l,r,s}\otimes{\msy Z}_l \;\oplus\; \Lambda_{l,r}\otimes{\msy Z}_l \end{subeqn} with $\tau$ acting on $N$ by \begin{subeqn}\label{eqn5412} \tau\colon (a,b)\mapsto (xa+x{\overline{\phi}}(b)-{\overline{\phi}}(xb),xb) \end{subeqn} where ${\overline{\phi}}\colon\Lambda_{l,r}\otimes{\msy Z}_l\to V^1/V^2=\Lambda_{l,r,s}\otimes{\msy Q}_l$ is ${\widetilde{\phi}}$ composed with the projection $V^1\to V^1/V^2$; we have ${\overline{\phi}}(x^i)=x^iy_2$ for $0\leq i\leq l^r-2$. Note that $\Phi_l/\Phi_l^2\cong N/(\tau-1)N$. Hence $\Phi_l/\Phi_l^2$ is cyclic if and only if the endomorphism $\tau-1$ of the ${\msy F}_l$-vector space $N\otimes{\msy F}_l$ has corank $1$. Now $N\otimes{\msy F}_l$ is the direct sum of ${\msy F}_l[\varepsilon]/(\varepsilon^{l^{r+s}-l^r})$ and ${\msy F}_l[\varepsilon]/(\varepsilon^{l^r-1})$, with $\varepsilon=x-1$. The matrix of $\tau-1$ with respect to the direct sum of the bases $(1,\varepsilon,\ldots,\varepsilon^{l^{r+s}-l^r-1})$ and $(1,\varepsilon,\ldots,\varepsilon^{l^r-2})$ is of the form \begin{subeqn}\label{eqn5413} \renewcommand{\baselinestretch}{1} \left( \begin{array}{cccc|cccc} 0& & & & & & & \\ 1&0& & & &A& & \\ &\ddots&\ddots& & & & & \\ & &1&0& & & & \\ \hline & & & &0& & & \\ & & & &1&0& & \\ & & & & &\ddots&\ddots& \\ & & & & & &1&0 \end{array} \right) \end{subeqn} It follows that $\tau-1$ has corank $1$ if and only if the upper right coefficient of $A$ is not zero, or, equivalently, if and only if there exists $b$ in $\Lambda_{l,r}\otimes{\msy Z}_l$ such that $x{\overline{\phi}}(b)-{\overline{\phi}}(xb)$ is a unit in $\Lambda_{l,r,s}\otimes{\msy Z}_l$. A computation shows that $x{\overline{\phi}}(x^{l^r-2})-{\overline{\phi}}(x^{l^r-1})=g_r(x)y_2$. Now recall that we are free to choose $y=(y_1,y_2)$ in $V^1$ and $z$ in $(\Lambda_{l,r}\otimes{\msy Q}_l)^*$ as long as $(y,z)$ satisfies (\ref{eqn547}). Note that $g_r(x)$ and $g_r'(x)$ are units in $\Lambda_{l,r,s}\otimes{\msy Q}_l$ and $\Lambda_{l,r}\otimes{\msy Q}_l$, respectively. Hence we can choose $y_2=g_r(x)^{-1}$ and $z=x^{-1}g_r'(x)^{-1}$. \end{example} \begin{example}\label{example55} Our final example is the analog of Example~\ref{example54} in the case ${\tilde{a}}=0$. More precisely, let $l$ be a prime and $r\geq0$ an integer. Then there exists an abelian variety $A_K$ with $t={\tilde{a}}=0$, ${\tilde{t}}=l^r-1$ and $\Phi=\Phi_l\cong{\msy Z}/l^{2r}{\msy Z}$. Let $C_{l,r}$ be as in Example~\ref{example52}. The abelian variety $A_K$ can be found in the isogeny class of $C_{l,r}$ in the same way as used in Example~\ref{example54}. In this case the construction is somewhat easier, since the filtration on $V$ has only two steps ($V^1=V^2$), so we leave the details to the reader. Let us just mention that all formulas up to (\ref{eqn5410}) remain valid (in adapted form), and after (\ref{eqn5410}) one shows that $M/(\tau-1)M$ can be cyclic with the same method as used to show that $N/(\tau-1)N$ can be cyclic. \end{example} \section{Classification of the $\Phi_{(p)}$.} \label{section6} The aim of this section is to prove the following theorem. \begin{theorem}\label{thm61} Let $D$ be a strictly henselian discrete valuation ring of residue characteristic $p\geq0$. Let $G$ be a finite commutative group of order not divisible by $p$. For each prime $l\neq p$, let $m_l:=(m_{l,1},m_{l,2},\ldots)$ be the partition corresponding to the $l$-part $G_l$ of $G$ (i.e., $G_l\cong\oplus_{i\geq1}{\msy Z}/l^{m_{l,i}}{\msy Z}$ and $m_{l,1}\geq m_{l,2}\cdots$). Let $d$, $t$, $a$ and $u$ be non-negative integers such that $d=t+a+u$. Then there exists an abelian variety over the field of fractions of $D$, of dimension $d$, toric rank $t$, abelian rank $a$ and unipotent rank $u$ which has $\Phi_{(p)}\cong G$, if and only if \begin{subeqn}\label{eqn611} u \geq \sum_{l\neq p}\sum_{i\geq t+1}\left( \frac{l^{\lfloor m_{l,i}/2\rfloor}+l^{\lceil m_{l,i}/2\rceil}}{2}-1\right) \end{subeqn} where for any real number $x$, $\lfloor x\rfloor$ and $\lceil x\rceil$ denote the largest (resp. smallest) integer $\leq x$ (resp. $\geq x$). \end{theorem} \begin{proof} We will start by showing that if $A_K$ is as indicated in the theorem, then (\ref{eqn611}) holds. Let $l\neq p$ be a prime. Let $f_l$ be the map from the set of partitions to ${\msy R}$ defined by \begin{subeqn}\label{eqn612} f_l(m) = \sum_{i\geq1} \left(\frac{l^{\lfloor m_i/2\rfloor}+ l^{\lceil m_i/2\rceil}}{2}-1\right) \end{subeqn} Then $f_l$ is strictly increasing for the partial ordering in which $a\geq b$ if and only if $a_i\geq b_i$ for all~$i$. One easily sees that $f_l$ is increasing for the lexicographical ordering on the set of partitions of a fixed number, but we won't use that. Consider the filtration \begin{subeqn}\label{eqn613} \Phi_l \supset \Phi_l^1 \supset \Phi_l^3 \supset 0 \end{subeqn} induced by (\ref{eqn21}). Theorem~\ref{thm33} shows that \begin{subeqn}\label{eqn614} 2(t_l-t + a_l-a) \geq \delta_l(\Phi_l/\Phi_l^1) + \delta_l(\Phi_l^1/\Phi_l^3) \end{subeqn} Let $n_l$ be the invariant of $\Phi_l/\Phi_l^3$. Lemma~\ref{lemma410} shows that for all $i\geq1$ we have $n_{l,i}\geq m_{l,i+t}$, or, in the terminology of the proof of that lemma, that $n_l\geq d^t(m_l)$ in the partial ordering. Lemma~\ref{lemma411} says that \begin{subeqn}\label{eqn615} \delta_l(\Phi_l/\Phi_l^1)+\delta_l(\Phi_l^1/\Phi_l^3)\geq 2f_l(n_l) \end{subeqn} It follows that \begin{subeqn}\label{eqn616} 2(t_l-t+a_l-a) \geq 2f_l(n_l) \end{subeqn} Summing over all $l\neq p$ and dividing by $2$ gives (\ref{eqn611}). It remains to show that all groups $G$ satisfying (\ref{eqn611}) can occur as the $\Phi_{(p)}$ of an abelian variety $A_K$ over the field of fractions $K$ of $D$ of dimension $d$, toric rank $t$, abelian rank $a$ and unipotent rank $u$. It is sufficient to show that all groups $G$ satisfying \begin{subeqn}\label{eqn617} u = \lceil\left(\sum_{l\neq p}\sum_{i\geq t+1}\left( \frac{l^{\lfloor m_{l,i}/2\rfloor}+l^{\lceil m_{l,i}/2\rceil}}{2}-1\right) \right)\rceil \end{subeqn} occur in such a way, since one can replace $A_K$ by the product of $A_K$ with an abelian variety $B_K$ which has unipotent reduction and trivial group of connected components. Let us first suppose that $K={\msy C}((q))$. Let $d$, $t$, $a$, $u$ and $G$ be as in the theorem, and suppose that they satisfy (\ref{eqn617}). We have $G\cong \oplus_{i\geq1}{\msy Z}/n_i{\msy Z}$ with $n_i\geq1$ and $n_{i+1}|n_i$ for all $i$. Let $B_K$ be of the type described in Example~\ref{example51}: it has dimension $t$, completely toric reduction and group of connected components ${\msy Z}/n_1{\msy Z}\oplus\cdots\oplus{\msy Z}/n_t{\msy Z}$. The abelian variety $A_K$ we are constructing will be of the form \begin{subeqn}\label{eqn618} A_K=B_K\times\prod_{l\neq p}C_{K,l}, \qquad{\rm with}\quad \dim(C_{K,l})=\lceil\left(\sum_{i\geq t+1}\left( \frac{l^{\lfloor m_{l,i}/2\rfloor}+l^{\lceil m_{l,i}/2\rceil}}{2}-1\right) \right)\rceil \end{subeqn} and such that all $C_{K,l}$ have unipotent reduction. Note that in fact such an $A_K$ has unipotent rank $u$, since for $l\neq2$ the function $f_l$ defined above has integer values. For $l\neq2$ we define \begin{subeqn}\label{eqn619} C_{K,l} = \prod_{i>t} C_{K,l,i} \end{subeqn} where $C_{K,l,i}$ is the abelian variety constructed in Example~\ref{example54} with $r=(m_{l,i}-1)/2$ and $s=1$ if $m_{l,i}\neq1$ is odd, where $C_{K,l,i}$ is the abelian variety constructed in Example~\ref{example53} with $i=1$ if $m_{l,i}=1$, and $C_{K,l,i}$ is the abelian variety constructed in Example~\ref{example55} with $r=m_{l,i}/2$ if $m_{l,i}$ is even. Note that the group of connected components of the reduction of $C_{K,l,i}$ is cyclic of order~$l^{m_{l,i}}$. For $l=2$ the construction of $C_{K,l}$ is a bit different. Let $r\geq0$ be maximal such that $m_{2,i}>1$ for all $i\leq r$. Then $C_{K,2}$ will be of the form \begin{subeqn}\label{eqn6110} C_{K,2} = D_{K,2} \times \prod_{t<i\leq r}C_{K,2,i} \end{subeqn} where $C_{K,2,i}$ is defined as $C_{K,l,i}$ but with $l$ replaced by $2$, and where $D_{K,2}$ is as follows. Let $v$ be the number of $i>t$ such that $m_{2,i}=1$. If $v$ is even we let $D_{K,2}$ be the product of $v/2$ elliptic curves which have unipotent reduction and group of connected components isomorphic to ${\msy Z}/2{\msy Z}\times{\msy Z}/2{\msy Z}$. If $v$ is odd we let $D_{K,2}$ be the product of $(v-1)/2$ elliptic curves with unipotent reduction and group of connected components isomorphic to ${\msy Z}/2{\msy Z}\times{\msy Z}/2{\msy Z}$ and one elliptic curve with unipotent reduction and group of connected components cyclic of order~$2$. One verifies easily that $A_K$ has all the desired properties. To finish the proof of the theorem, we have to show that similar examples exist over any strictly henselian discrete valuation ring $D$ with residue characteristic $p$. Since our examples are products of the examples of \S\ref{section5}, it suffices to show that the examples in \S\ref{section5} exist over $D$. Since we do not suppose $D$ complete, we cannot use a Tate curve ``${{\rm G}_m}/q^{\msy Z}$'' with $q$ a uniformizer of~$D$. Instead we can use any elliptic curve $E$ over $K$ which has toric reduction and trivial group of connected components. Then $I$ acts on the Tate module ${\rm T}_l(E({K^{\rm s}}))$ through its quotient ${\msy Z}_l(1)$ and for a suitable choice of a ${\msy Z}_l$-basis of ${\rm T}_l(E({K^{\rm s}}))$, an element of $I$ with image $a$ in ${\msy Z}_l(1)$ acts as $({1\atop0}{a\atop1})$. It follows that Examples~\ref{example51} and \ref{example52} with $E_1$ replaced by $E$ still work. To make Example~\ref{example53} work over $D$, it is enough to show that for all $l\neq p$ and $r>0$ such that $l^r>2$, there exists an abelian scheme over $D$ of relative dimension $l^{r-1}(l-1)/2$ and with an action by ${\msy Z}[x]/(f_{l,r})$. Once one has these abelian schemes, the constructions of \S\ref{section5} can be carried out over~$D$. The fact that such abelian schemes exist is a consequene of the theory of abelian varieties of ``CM-type''. Fix an $l$ and $r$ as above. The moduli scheme over ${\msy Z}[1/l]$ of abelian schemes of the desired type, with a suitable polarization and $l$-power level structure, is finite etale and not empty. Another way to prove the desired existence is to consider isogeny factors over ${\msy Q}(\zeta_{l^r})$ of the jacobian of the Fermat curve of degree $l^r$. \end{proof} \section{Further remarks and questions.} \label{section7} Although Theorem~\ref{thm61} gives a complete classification of the prime-to-$p$ parts of the groups of connected components of special fibres of N\'eron models with some fixed invariants, there are still questions left. For example, it is clear that the groups of connected components $\Phi$ have functorial additional structure coming from the fact that the category of abelian varieties has an involution: every abelian variety has its dual. More precisely, suppose that $\Phi$ comes from the abelian variety $A_K$. Let $A'_K$ be the dual of $A_K$ and denote its group of connected components by $\Phi'$. Then there are several pairings with values in ${\msy Q}/{\msy Z}$, conjecturally perfect and the same, between $\Phi$ and $\Phi'$; see \cite[\S\S1.2, 1.3, 11.2]{Grothendieck1}, \cite{Milne1}, \cite[\S3]{Dino1}, \cite{Moret-Bailly1} and \cite[Prop.~3.3]{Oort1}. Let us note by the way that the last reference is clearly wrong since it says that the pairing has values in $({\msy Q}/{\msy Z})(1)$; the mistake in the proof is that the direct sum decomposition in the unique displayed formula in it is not unique. Anyway, for each of the remaining pairings we get a filtration \begin{eqn}\label{eqn71} \Phi_{(p)} = {\Phi'}_{(p)}^{4,\perp} \supset {\Phi'}_{(p)}^{3,\perp} \supset {\Phi'}_{(p)}^{2,\perp} \supset {\Phi'}_{(p)}^{1,\perp} \supset {\Phi'}_{(p)}^{0,\perp} = 0 \end{eqn} It would be interesting to know the common refinement of this filtration with (\ref{eqn21}). Also, it would be of interest to prove that the various pairings are the same up to a determined sign. Some relations between the two filtrations (\ref{eqn21}) and (\ref{eqn71}) on the $l$-part for $l\neq p$ can be found in \cite[Thm.~3.21]{Dino1}, under the hypothesis that $A_K$ has a polarization of degree prime to~$l$. Let us consider the functor from the category of abelian varieties over $K$ to the category of finite abelian groups which associates to each abelian variety the group of connected components of the special fibre of its N\'eron model. A rather vague question one can ask is through what categories of abelian groups endowed with some extra structure this functor factors. We have seen for example that there is a filtration of four steps on the prime-to-$p$ part, but as we have just remarked that is certainly not all there is. Lorenzini has shown \cite[Thm.~3.22]{Dino1}, under the hypothesis that there is a polarization of degree prime to $l$, that ${\Phi'}_l^{2,\perp}$ is the prime-to-$p$ part of the kernel of the map from $\Phi$ to the group of connected components of $A_L$, where $K\to L$ is any extension over which $A_K$ has semi-stable reduction. It would be interesting to generalize this. Even in the case in which $A_K$ acquires semi-stable reduction after a tamely ramified extension $K\to L$, when the theory of \cite{Edixhoven1} applies, I have not been able to give a description of the filtration (\ref{eqn21}) in terms of the special fibre of the N\'eron model of $A_L$ with its action of ${\rm Gal}(K/L)$. The $p$-part of $\Phi$ remains difficult. For example, one expects a bound for its order in terms of the dimension of $A_K$ if the toric part of the reduction is zero, but even in the case of potentially good reduction I don't know of any such bound (of course, if $A_K$ is the jacobian of a curve with a rational point, the usual bound, i.e., the bound we have when $k$ is of characteristic zero, holds, since one can apply Winters's theorem \cite{Winters1}). In a forthcoming article \cite{EdiLiuLor} one can find a generalization of a result of McCallum (unpublished) which says that in the case of potentially good reduction the $p$-part is annihilated by the degree of any extension after which one obtains semi-stable reduction, but not in general by the exponent of the Galois group of such an extension. Work in progress by Bosch and Xarles, using a rigid analytic uniformization of N\'eron models, seems to imply that there is a four-step functorial filtration on the whole of~$\Phi$, for which three of the four successive quotients can be described in terms of the ineria group acting on the character group of the toric part of the semi-stable reduction. The remaining successive quotient comes from an abelian variety, obtained by Raynaud's extension, which has potentially good reduction. This part is still a mystery.

9410

alg-geom/9410015

en

https://arxiv.org/abs/alg-geom/9410015

alg-geom/9410015

Bert van Geemen

Bert van Geemen and Emma Previato

On the Hitchin System

23 pages, LaTeX Version 2.09 <7 Dec 1989>

The Hitchin system is a completely integrable hamiltonian system (CIHS) on the cotangent space to the moduli space of semi-stable vector bundles over a curve. We consider the case of rank-two vector bundles with trivial determinant. Such a bundle $E$ defines a divisor $D_E$ in the Jacobian of the curve and for any smooth point of $D_E$ we define a cotangent vector (a Higgs field). The Hitchin map on these Higgs fields is then determined in terms of the Gauss map on the divisor $D_E$. We apply the results to the $g=2$ case and show how Hitchin's system is related to classical line geometry in $\PP^3$.

alg-geom

\section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus -.2ex}{1.5ex plus .2ex}{\large\bf} \def\secdef\empsubsection{\emppsubsection*}{\@startsection{subsection}{2}{\z@}{-3.25ex plus -1ex minus -.2ex}{-1em}{\normalsize\bf}} \let\emppsubsection\secdef\empsubsection{\emppsubsection*} \def\empsubsection[#1]#2{\emppsubsection[#1]{#2\unskip}} \def\secdef\empsubsection{\emppsubsection*}{\secdef\empsubsection{\emppsubsection*}} \def\subsection{\secdef\empsubsection{\emppsubsection*}} \makeatother \newcommand{\newsubsubsection}% {{\bf\refstepcounter{subsubsection}\thesubsubsection\ \ } \renewcommand{\Huge}{\huge \renewcommand{\theequation}{\thesubsubsection} \renewcommand{\equation}{$$ \refstepcounter {subsubsection}} \addtolength{\textheight}{2cm} \addtolength{\topmargin}{-2cm} \newcommand{\bf}{\bf} \newcommand{\vspace{\baselineskip}\noindent{\bf Proof.}$\;\;$}{\vspace{\baselineskip}\noindent{\bf Proof.}$\;\;$} \newcommand{{\Bbb A}}{{\bf A}} \newcommand{{\Bbb B}}{{\bf B}} \newcommand{{\Bbb C}}{{\bf C}} \newcommand{{\Bbb E}}{{\bf E}} \newcommand{{\Bbb F}}{{\bf F}} \newcommand{{\Bbb H}}{{\bf H}} \newcommand{{\Bbb N}}{{\bf N}} \newcommand{{\Bbb P}}{{\bf P}} \newcommand{{\Bbb Q}}{{\bf Q}} \newcommand{{\Bbb R}}{{\bf R}} \newcommand{{\Bbb S}}{{\bf S}} \newcommand{{\Bbb Z}}{{\bf Z}} \newcommand{\alpha}{\alpha} \newcommand{\epsilon}{\epsilon} \newcommand{\Gamma}{\Gamma} \newcommand{\lambda}{\lambda} \newcommand{\omega}{\omega} \newcommand{\sigma}{\sigma} \newcommand{\theta}{\theta} \newcommand{\Theta}{\Theta} \newcommand{\Omega}{\Omega} \newcommand{{\cal O}}{{\cal O}} \newcommand{{\cal P}}{{\cal P}} \newcommand{{\overline{Q}}}{{\overline{Q}}} \newcommand{{\tilde{C}}}{{\tilde{C}}} \newcommand{{\unskip\nobreak\hfill\hbox{ $\Box$}\par}}{{\unskip\nobreak\hfill\hbox{ $\Box$}\par}} \title{On the Hitchin System} \author{{\sc Bert van Geemen}\\ University of Utrecht, The Netherlands\\ {\sc Emma Previato}\\ Boston University, USA \thanks {Research partially supported by NSF Grant DMS-9105221 at Boston University and DMS-9022140 at MSRI.} } \date{} \begin{document} \maketitle \section{Introduction} \secdef\empsubsection{\emppsubsection*}{} What is known as the Hitchin system is a completely integrable hamiltonian system (CIHS) involving vector bundles over algebraic curves, identified by Hitchin in (\cite{H1}, \cite{H2}). It was recently generalized by Faltings \cite{F}. In this paper we only consider the case of rank-two vector bundles with trivial determinant. In that case the Hitchin system corresponding to a curve $C$ of genus $g$ is obtained as follows. Let $$ {\cal M}:=\{ E\rightarrow C:\;E\;\mbox{a semi-stable rank two bundle},\; \wedge^2E\cong {\cal O}\;\}/\sim_{\rm S} $$ be the moduli space of (S-equivalence classes of) semi-stable rank-two vector bundles on $C$. Then ${\cal M}$ is a projective variety (singular if $g>2$) of dimension $3g-3$. The locus of stable bundles ${\cal M}^s$ is the set of smooth points of ${\cal M}$ for $g>2$. The cotangent space of ${\cal M}$ at a stable bundle $E$ is : $$ T^*_E{\cal M}=Hom_0(E,E\otimes K),\mbox{with}\quad Hom_0(E,E\otimes K):=H^0(C,{\cal E}nd_0(E)\otimes K) $$ where ${\cal E}nd_0(E)$ is the sheaf of endomorphisms of $E$ with trace zero and $K$ is the canonical bundle on $C$. A $\Phi\in Hom_0(E,E\otimes K)$ is called a Higgs field. The determinant of a Higgs field $det(\Phi)\in Hom(\wedge^2E,\wedge^2(E\otimes K))=H^0(C,2K)$ gives a map $$ det:T^*_E{\cal M}=Hom_0(E,E\otimes K)\longrightarrow H^0(C,2K), $$ which globalizes to a map on $T^*{\cal M}^s$. Hitchin considered the map: $$ H:T^*{\cal M}^s\longrightarrow H^0(C,2K),\quad \Phi\mapsto det(\Phi) $$ and showed that it is a CIHS in the following sense: the functions on $T^*{\cal M}^s$ that one obtains by choosing any basis in $H^0(C,2K)$ are a complete set of hamiltonians in involution (with respect to the natural symplectic structure on a cotangent bundle). Since $det$ is homogeneous of degree two in the fibre variables ($det(t\Phi)=t^2det(\Phi))$, one can define a (rational) projective Hitchin map $$ \overline{H}:{\Bbb P} T^*{\cal M}^s\longrightarrow{\Bbb P} H^0(C,2K)=|2K| $$ and it is in fact this map that we consider. \secdef\empsubsection{\emppsubsection*}{} In the first two sections of this paper we define and study a set of Higgs fields associated to any semi-stable bundle $E$. These results are then applied to the $g=2$ case; they may also be of independent interest for studying moduli spaces of Higgs bundles, which are pairs $(E,\Phi)$ as above, with certain restrictions. To study ${\cal M}$ as a projective variety (see \cite{NR}, \cite{B1}, \cite{B2}) one associates to any $E\in {\cal M}$ a divisor $D_E$ in the Jacobian of $C$ using which, questions on rank two bundles are rephrased in terms of line bundles (and extensions). We exhibit a natural map $$ \phi_E:D^{sm}_E\longrightarrow {\Bbb P} Hom_0(E,E\otimes K),\qquad \xi\mapsto \Phi_\xi $$ (with $D_E^{sm}$ the smooth points of $D_E$) and we are able to compute $det(\Phi_\xi)$. The result is best stated in a diagram (the stable case of Proposition \ref{detPhi}): For any stable bundle $E$ the following diagram commutes: \begin{equation}\label{diagram} \begin{array}{rcccl} &&{\Bbb P} T_E^*{\cal M}&&\\ &\phi_E\nearrow&&\searrow\bar{H}&\\ D_E^{sm} &&&&{\Bbb P} H^0(C,2K)\\ &\psi_E\searrow&&\nearrow Sq&\\ &&{\Bbb P} H^0(K)&& \end{array} \end{equation} here $\psi_E$ is just the Gauss map of the divisor $D_E$ inside the Jacobian and $Sq(\omega)=\omega\otimes \omega$. Thus the divisor $D_E$ (rather, its image in ${\Bbb P} T_E^*{\cal M}$) plays an important role in the study of the fibers of $\bar{H}$ over the quadratic differentials which are squares of one forms. However, our fiberwise approach (for the map $T^*{\cal M}\rightarrow {\cal M}$) is, in a sense, perpendicular to Hitchin's approach which studies the fibers of $H:T^*{\cal M}\rightarrow H^0(C,2K)$. That approach establishes that such a fiber, over a (general) quadratic differential $\eta$ is the Prym variety associated with a `spectral' double cover $C_\eta\rightarrow C$ defined by $\eta$ (\cite{H2},\,\cite{BNR}). It would be interesting to relate our results to a study of the fibers of $H$. \secdef\empsubsection{\emppsubsection*}{} In the remaining sections we apply these results to investigate the case $g=2$. Then the space ${\cal M}$ is isomorphic to ${\Bbb P}^3$ (\cite{NR}), so that we look for a CIHS on $T^*{\Bbb P}^3$ (and also on the open subset $T^*{\Bbb C}^3$). Using information on $\phi_E$ from $\S$\ref{phiE} we work out the maps of the diagram (1.2.1) in the genus two case in $\S$\ref{g=2}. Here we encounter some classical algebraic geometry of curves of genus two and three. It turns out that finding $H$ explicitly involves a problem in line geometry in ${\Bbb P}^3$ (a sketch of the solution in fact appears in in J.H. Grace's article ``Line Geometry'' in the Encyclopaedia Britannica, 1911). We can thus make an educated guess as to what the explicit hamiltonians should be. A computer calculation (using the {\it Mathematica} system) showed that our candidates actually define a CIHS. We are not able to show that our hamiltonians define the Hitchin map, but we can prove that Hitchin's hamiltonians and ours differ by multiplication by functions from the base (an open set in ${\Bbb P}^3$). For a more precise result we would have to extend the results of section \ref{g=2} to enlarge the open set in the base were those results hold, or we would need further information on Hitchin's system. \secdef\empsubsection{\emppsubsection*}{Acknowledgements.} The first named author wishes to thank: the University of Pavia for a six month stay where much of the work on this paper was done, the NSF for supporting visits to Boston University under Grant DMS-9105221, and J. de Jong for helpful discussions. The second named author wishes to acknowledge: Carolyn Gordon's invitation to MSRI for two weeks of the special year in Differential Geometry 1993/94 (research at MSRI supported in part by NSF grant \# DMS 9022140), and participation in the LMS/Europroj Workshop ``Vector bundles in algebraic geometry'' (Durham, 1993; organizers N. Hitchin, P. Newstead and W.M. Oxbury), on which occasion N. Hitchin provided generous insight. \section{Higgs fields} \secdef\empsubsection{\emppsubsection*}{} We fix some notations and recall some basic facts. In this paper $C$ will be a smooth, irreducible projective curve of genus $g>1$ over ${\Bbb C}$ and $E$ will be a rank two semi-stable bundle on $C$ with trivial determinant. Since $\wedge^2E\cong {\cal O}$, we have: $$ E\wedge E={\cal O}_C,\qquad {\rm so}\quad E\cong E^*:={\cal H}om(E,{\cal O}_C),\quad e\mapsto [f\mapsto e\wedge f] $$ thus $E$ is self-dual. This gives isomorphisms: $$ E\otimes E\cong E^*\otimes E={\cal E}nd(E),\qquad S^2E\cong {\cal E}nd_0(E) $$ with ${\cal E}nd_0(E)\subset {\cal E}nd(E)$ the sheaf of endomorphisms of trace zero. We recall that $End_0(E):=H^0(C,{\cal E}nd_0(E))=0$ for a stable bundle $E$, the only endomorphisms of $E$ being scalar multiples of the identity. Thus: $ H^0(C,S^2E)=0$. For a vector space $V$, we let ${\Bbb P} V$ be the space of one dimensional linear subspaces of $V$. \secdef\empsubsection{\emppsubsection*}{}\label{inv} We will construct Higgs fields by relating $E$ to line bundles. Such a connection is provided by the following results. Let $E$ be a semi-stable rank two bundle on $C$ with $det(E)={\cal O}$. Associated to $E$ is a divisor (\cite{B1}, 2.2): $$ D_E:=\{\xi\in Pic^{g-1}(C):\;\dim H^0(\xi\otimes E)>0\;\}. $$ With its natural scheme structure, $D_E$ is linearly equivalent to $2\Theta$. Here $\Theta$ is the natural theta divisor: $$ \Theta:=\{\xi\in Pic^{g-1}(C):\;\dim H^0(C,\xi)>0\;\},\qquad D_E\in |2\Theta|. $$ On $Pic^{g-1}(C)$ there is a natural involution: $$ \iota:Pic^{g-1}(C)\longrightarrow Pic^{g-1}(C),\qquad \xi\mapsto K\otimes \xi^{-1}. $$ All divisors in $|2\Theta|$ are invariant under the involution $\iota$. For a $D_E$ that is easy to check since by Riemann-Roch and Serre duality: $$ \dim H^0(\xi\otimes E)=\dim H^1(\xi\otimes E)= \dim H^0(\xi^{-1}\otimes K\otimes E). $$ Note that $H^0(\xi^{-1}\otimes K\otimes E)=Hom(\xi,E\otimes K)$ and that, since $E$ is self-dual, $H^0(\xi\otimes E)=Hom(E,\xi)$. Thus we have: $$ \xi\in D_E\Longleftrightarrow Hom(E,\xi)\neq 0 \Longleftrightarrow Hom(\xi,E\otimes K)\neq 0. $$ \secdef\empsubsection{\emppsubsection*}{} For any semi-stable $E$ (cf.\ \cite{L}, Cor. V.6): $$ \xi\in D^{sm}_E\;\Longrightarrow\; \dim H^0(\xi\otimes E)=1. $$ Thus for $\xi\in D^{sm}_E$ there are unique (up to scalar multiple) maps: $$ \pi:E\longrightarrow \xi,\qquad \tau: \xi\longrightarrow E\otimes K. $$ The composition $$ \tau\circ \pi:\;E\longrightarrow E\otimes K $$ is an element of $Hom(E,E\otimes K)$, defined (up to scalar multiple) by $\xi$. \secdef\empsubsection{\emppsubsection*}{Definitions.} \label{defpp} Let $E$ be a semi-stable rank-two bundle on $C$ with $det(E)={\cal O}$. We define rational maps: $$ \phi_E:\; D_E^{sm}\longrightarrow {\Bbb P} Hom_0(E,E\otimes K), $$ $$ \xi\mapsto \Phi_\xi:= \tau\circ \pi-(1/2)(id_E\otimes tr(\tau\circ \pi)): E\otimes{\cal O}\longrightarrow E\otimes K $$ and $$ \psi_E:\; D_E^{sm}\longrightarrow {\Bbb P} H^0(C,K)\;={\Bbb P} Hom(\xi,\xi\otimes K), $$ $$ \psi_E(\xi)=(\pi\otimes id_K)\circ\tau:\; \xi\stackrel{\tau}{\longrightarrow} E\otimes K \stackrel{\pi\otimes 1}{\longrightarrow}\xi\otimes K . $$ \secdef\empsubsection{\emppsubsection*}{} Now we have a large supply of Higgs fields, the $\Phi_\xi$'s. It is surprisingly easy to determine $\psi_E$. In Proposition \ref{detPhi} we will see how that already determines the Hitchin map to a large extent. Recall that the cotangent bundle to $Pic^{g-1}(C)$ is trivial: $$ T^*Pic^{g-1}(C)\cong Pic^{g-1}(C)\times H^1(Pic^0(C),{\cal O})^*. $$ For a smooth point $\xi$ in a divisor $D\subset Pic^{g-1}(C)$ the tangent space to $D$ at $\xi$ is then defined by an element of $H^1(Pic^0(C),{\cal O})^*$, unique up to scalar multiple. The corresponding morphism $D^{sm}\rightarrow {\Bbb P} H^1(Pic^0(C),{\cal O})^*$ is called the Gauss map. \secdef\empsubsection{\emppsubsection*}{Proposition.}\label{psig} The map $\psi_E$ is the Gauss map on $D_E\subset Pic^{g-1}(C)$. $$ \psi_E:D_E^{sm}\longrightarrow {\Bbb P} H^1(Pic^0(C),{\cal O})^*={\Bbb P} H^0(C,K). $$ In particular, $\psi_E$ is a morphism. \vspace{\baselineskip}\noindent{\bf Proof.}$\;\;$ By \cite{L}, Prop.\ V.2, we know that for $\xi\in D_E^{sm}$ the space $$ T_\xi D_E\subset T_\xi Pic^{g-1}(C)= H^1(Pic^0(C),{\cal O})=H^1(C,{\cal O}) $$ is defined by the image of the cup-product map $$ H^0(C,E\otimes\xi)\otimes H^0(C,E\otimes \iota(\xi))\longrightarrow H^0(C,K)\cong H^1(C,{\cal O})^*. $$ This map coincides with the composition: $$ Hom(E\otimes K,\xi\otimes K)\otimes Hom(\xi,E\otimes K) \longrightarrow Hom(\xi,\xi\otimes K)\cong H^0(K), $$ and in our case we recover the definition of $\psi_E$: $$ (\pi\otimes 1)\otimes \tau\mapsto (\pi\otimes 1)\circ\tau =\psi_E(\xi). $$ (One may in fact also consider $Hom(E,\xi_\eta)$ where $\xi_\eta$ is a deformation of $\xi$ given by $\eta\in H^1(C,{\cal O})$. Then $\eta\in T_\xi D_E$ iff $\pi\in Hom(E,\xi)$ lifts to $Hom(E,\xi_\eta)$ iff $\pi\cup \eta=0\in Ext^1(E,\xi)=H^1(E\otimes\xi)$, which gives the statement above. The justification for this argument is given in \cite{L}, II.) {\unskip\nobreak\hfill\hbox{ $\Box$}\par} \secdef\empsubsection{\emppsubsection*}{} We are interested in computing the determinant of the Higgs field $\Phi_\xi$. Since the maps in \ref{defpp} are only defined up to scalar multiple, we consider $$ det: Hom_0(E,E\otimes K) \longrightarrow {\Bbb P} H^0(C, 2K), \qquad \Phi\mapsto \langle det(\Phi)\rangle. $$ Let $$ Sq: H^0(C,K)\longrightarrow H^0(C, 2K),\qquad \omega\mapsto\omega^{\otimes 2}. $$ \secdef\empsubsection{\emppsubsection*}{Proposition.}\label{detPhi} For a semi-stable $E$ and $\xi\in D_E^{sm}$ we have: $$ det(\Phi_\xi)=\psi_E(\xi)^{\otimes 2}\qquad (\in {\Bbb P} H^0(C, 2K)). $$ Thus, the compositions $det\circ\phi_E$ and $Sq\circ\psi_E$ coincide: $$ \begin{array}{cccccc} det\circ \phi_E: & D_E^{sm}&\stackrel{\phi_E}{\longrightarrow}& {\Bbb P} Hom_0(E,E\otimes K)& \stackrel{det}{\longrightarrow}& {\Bbb P} H^0(C, 2K),\\ Sq\circ\psi_E:& D_E^{sm}&\stackrel{\psi_E}{\longrightarrow} &{\Bbb P} H^0(C,K)& \stackrel{Sq}{\longrightarrow} &{\Bbb P} H^0(C, 2K). \end{array} $$ \vspace{\baselineskip}\noindent{\bf Proof.}$\;\;$ Since $\psi_E:D_E^{sm}\rightarrow {\Bbb P} H^0(C,K)$ is a morphism, $\psi_E(\xi)$ is (represented by) a non-zero differential form for each $\xi\in D_E^{sm}$. Define a canonical divisor on $C$ by: $$ K_\xi:=(\psi_E(\xi)),\qquad {\rm let}\quad U:=C-Support(K_\xi), $$ On the open set $U$, the map (defined by) $\psi_E(\xi):\xi\rightarrow \xi\otimes K$ is an isomorphism. Its inverse, composed with $\tau:\xi\rightarrow E\otimes K$, gives a map $\xi\otimes K\rightarrow E\otimes K$ which splits the map $\pi\otimes1:E\otimes K\rightarrow \xi\otimes K$. $$ \begin{array}{ccc} \xi & {}_{\psi_E(\xi)}& \\ \tau\Big\downarrow\phantom{\tau}&\searrow &\\ E\otimes K &\stackrel{\pi\otimes 1}{\longrightarrow}& \xi\otimes K \end{array} $$ Thus over $U$, the bundle $E$ splits: $$ {E}_{|U}\cong L\oplus\xi_{|U},\qquad{\rm with}\quad L:=\ker(\pi_U:E_{|U}\longrightarrow \xi_{|U}) $$ and $L$ is a line bundle on $U$. Since $\Phi_\xi:=\tau\pi-(1/2)tr(\tau\pi)$, we get $\Phi_{\xi|U}(L)=0$ so that: $$ \Phi_{\xi|U}=\left(\begin{array}{cc} -(1/2)\psi_E(\xi)&\ast\\0&(1/2)\psi_E(\xi) \end{array}\right). $$ Then $det(\Phi_{\xi{|U}})=-(1/4)\psi_E(\xi)^{\otimes 2}$, which does not vanish at any point of $U$. If $D_E$ is irreducible, the image of the Gauss map on $D^{sm}_E$ contains an open subset of ${\Bbb P} H^0(C,K)$. Thus, for general $\xi$ on such a $D_E$, $2K_\xi$ is the only divisor in ${\Bbb P} H^0(C,2K)$ with support in $C-U$. Since $det(\Phi_\xi)\in H^0(C,2K)$ and since its divisor must have support in $C-U$, we conclude that $$ (det(\Phi_\xi))=2K_\xi. $$ For general $C$, the rank of the N\' eron-Severi group of $Pic^{g-1}(C)$ is one, and then the reducible divisors in $|2\Theta|$ are a subvariety of dimension $g$ (they are unions of two translates of $\Theta$). Since $\Delta({\cal M})$ (see \ref{delta}) has dimension $3g-3$ (\cite{B1}), the divisor $D_E$ is irreducible for general $C$. Thus if we work in a family of $D_E$'s over a general family of curves containing the given $D_E$, the maps $\xi\mapsto det(\Phi_\xi)$ and $\xi\mapsto \psi_E(\xi)^{\otimes 2}$ agree on a non-empty open subset, hence must agree everywhere. {\unskip\nobreak\hfill\hbox{ $\Box$}\par} \section{The map $\phi_E$} \label{phiE} \secdef\empsubsection{\emppsubsection*}{}\label{delta} In this section we consider only stable bundles $E$ and we study the map $\phi_E:D_E^{sm}\rightarrow {\Bbb P} T^*_E{\cal M}$. We use the codifferential of the map: $$ \Delta: {\cal M}\longrightarrow |2\Theta|,\qquad E\mapsto D_E $$ to relate the cotangent bundles of ${\cal M}$ and of the projective space $|2\Theta|$. First of all we recall some facts on the cotangent bundle to a projective space and on the dual of $|2\Theta|$ (following \cite{NR2}, $\S$ 3). \secdef\empsubsection{\emppsubsection*}{} \label{dual} Let $V$ be a vector space. The dual of the Euler sequence on ${\Bbb P} V$ gives: $$ 0\longrightarrow T^*{\Bbb P} V\longrightarrow V^*\otimes{\cal O}(-1)\longrightarrow {\cal O} \longrightarrow 0, $$ the last non-trivial map is given by $(\ldots,s_i,\ldots)\mapsto\ldots+x_is_i+\ldots$ over $(\ldots :x_i:\ldots)\in{\Bbb P} V$. Taking the associated projective bundles we have an isomorphism: $$ {\Bbb P} T^*{\Bbb P} V\cong I:=\{\;(x,h)\in {\Bbb P} V\times {\Bbb P} V^*:\; x\in h\;\}, $$ the variety $I$ is called the incidence bundle. \secdef\empsubsection{\emppsubsection*}{}\label{delta2} To identify the dual of $|2\Theta|$ we use the map: $$ \delta:Pic^{g-1}(C)\longrightarrow |2\Theta_0|,\qquad \xi\mapsto D_\xi:=L^*_\xi\Theta +L^*_{\iota(\xi)}\Theta, $$ here $\Theta_0$ is (any) symmetric theta divisor in $Pic^0(C)$ (the linear equivalence class of $2\Theta_0$ is independent of the choice) and $$ L_\alpha:Pic(C)\longrightarrow Pic(C),\qquad \beta\mapsto \alpha\otimes\beta $$ is translation by $\alpha$ in $Pic(C)$. Pulling back the linear forms on $|2\Theta_0|$ gives an isomorphism $$ \delta^*:H^0(|2\Theta_0|,{\cal O}(1))=H^0(Pic^0(C),2\Theta_0)^* \stackrel{\cong}{\longrightarrow} H^0(Pic^{g-1}(C),2\Theta). $$ Projectivizing gives $\delta^*:|2\Theta_0|^*\stackrel{\cong}{\longrightarrow} |2\Theta|$. In fact, there is a commutative diagram: $$ \begin{array}{ccc} & & |2\Theta|^*\\ &{}^\nu\nearrow\phantom{{}^\nu}& \\ Pic^{g-1}(C)& &\;\;\downarrow (\delta^*)^*\\ & {}_\delta\searrow\phantom{{}_\delta}&\\ & &|2\Theta_0| \end{array} $$ where $\nu$ is the natural map (see \cite{NR2} and \cite{B2} \S 2 for a variant). From now on, $I$ will be the incidence bundle: $$ I:={\Bbb P} T^*|2\Theta|\subset |2\Theta|\times |2\Theta_0|. $$ We will denote by $(d\Delta)^*$ the projectivized codifferential of $\Delta$ (see \ref{delta}): $$ (d\Delta)^*:{\Bbb P} T^*|2\Theta|= I\longrightarrow {\Bbb P} T^*{\cal M}. $$ \secdef\empsubsection{\emppsubsection*}{Lemma.}\label{dd} Let $D\in |2\Theta|$ and let $\xi\in Pic^{g-1}(C)$. Then $$ (D,D_\xi)\in I\;\Longleftrightarrow\; \xi\in D. $$ \vspace{\baselineskip} \noindent {\bf Proof.}$\;\;$ For $\xi\in Pic^{g-1}(C)$, $\nu(\xi)\in |2\Theta|^*$ is the hyperplane in $|2\Theta|$ consisting of the divisors passing through $\xi$. Thus $D\in |2\Theta|$ and $\xi\in |2\Theta|^*$ are incident iff $\xi\in D$. The dual of the isomorphism $\delta^*$ maps $\nu(\xi)$ to $D_\xi$ so the result follows. {\unskip\nobreak\hfill\hbox{ $\Box$}\par} \secdef\empsubsection{\emppsubsection*}{} We recall that for non-hyperelliptic curves, the map $\Delta$ has degree one over its image \cite{B1} (so it is locally an isomorphism with its image for generic $E$) and is an embedding for the general curve as recently announced by Y. Laszlo and also by S. Brivio and A. Verra jointly. In case $g=2$ the map is an isomorphism \cite{NR} but for hyperelliptic curves of genus greater than two the map is 2:1 and `ramifies' along a subvariety of dimension $2g-1$ \cite{B1}. \secdef\empsubsection{\emppsubsection*}{Proposition.}\label{glue} Let $E$ be a stable bundle such that the map $\Delta$ is locally at $E$ an isomorphism with its image. Then the rational map $$ \phi_E:D_E\longrightarrow {\Bbb P} T^*_E{\cal M},\qquad \xi\mapsto \Phi_{\xi}, $$ is the left-hand column in the diagram: $$ \begin{array}{ccc} D_E&\hookrightarrow&Pic^{g-1}(C)\\ \Bigg\downarrow&&\Bigg\downarrow\delta\\ {\Bbb P} T^*_{D_E}|2\Theta| &\hookrightarrow& |2\Theta_0|\\ (d\Delta_E)^*\Bigg\downarrow\phantom{(d\Delta_E)^*} & &\\ {\Bbb P} T^*_E {\cal M}&&\\ \end{array} $$ where the last vertical arrow is a linear projection given by the dual of the differential of $\Delta$ at $E\in{\cal M}$. \vspace{\baselineskip} \noindent {\bf Proof.}$\;\;$ Let $\xi\in D_E$. Then $\delta(\xi)=D_\xi\in|2\Theta_0|=|2\Theta|^*$ corresponds to a hyperplane $H_\xi\subset |2\Theta|$. By Lemma \ref{dd}, $\xi\in D_E$ implies $D_E\in H_\xi\subset |2\Theta|$. This says that $H_\xi$ passes through $D_E=\Delta(E)\in\Delta({\cal M})\subset |2\Theta|$ and (by the assumption on local isomorphism) defines a codimension $\leq 1$ subspace in $T_E{\cal M}$. We must show that $\Phi_\xi\;(\in T^*_E{\cal M})$ is the defining equation for this subspace. We first determine $H_\xi\cap \Delta({\cal M})$; the pull-back $\Delta^*H_\xi$ will be the divisor $\tilde{D}_\xi$ defined below; in particular, the subspace of $T_E{\cal M}$ defined by $H_\xi$ is $T_E\tilde{D}_\xi$. We recall from \cite{B1} that $$ Pic({\cal M})\cong{\Bbb Z},\qquad{\rm and}\quad {\cal L}:=\Delta^*({\cal O}(1)) $$ is the ample generator of this group. Moreover, the natural map $$ {\cal M}\longrightarrow {\Bbb P} H^0({\cal M},{\cal L})^* $$ actually coincides with $\Delta$. Define for $\xi\in Pic^{g-1}(C)$: $$ \tilde{D}_\xi:=\{E\in{\cal M}:\; H^0(C,E\otimes\xi)\neq 0\;\}. $$ This divisor, with its natural scheme structure, is defined by a section of $H^0({\cal M},{\cal L})$. Restriction to the Kummer variety of $Pic^0(C)$ (= locus of non-stable bundles) in ${\cal M}$ induces the isomorphism (\cite{B1}): $$ {\Bbb P} H^0({\cal M},{\cal L})\stackrel{\cong}{\longrightarrow} |2\Theta_0|, \qquad {\rm and}\quad \tilde{D}_\xi\mapsto D_\xi $$ (indeed, for $L\in Pic^0(C)$ one has $H^0((L\oplus L^{-1})\otimes\xi)>0$ iff $L\in L_\xi^*\Theta$ or $L\in L_{\iota(\xi)}^*\Theta$ iff $L\in D_\xi$). Now, by definition, the hyperplane $H_\xi$ intersects $Pic^0(C)$ in $D_\xi$ so that $H_\xi$ intersects ${\cal M}$ in $\tilde{D}_\xi$, as desired. We must now show that $\Phi_\xi$ defines the subspace $T_E \tilde{D}_\xi$ of $T_E{\cal M}$. The divisor $\tilde{D}_\xi$ is (the closure of) the image of the (rational) map: $$ \rho:{\Bbb P} H^1(\xi^{-2})\longrightarrow {\cal M},\qquad \epsilon\mapsto [E_\epsilon] $$ where $E_\epsilon$ is the extension defined by $\epsilon\in Ext^1(\xi,\xi^{-1})=H^1(\xi^{-2})$, \begin{equation} \label{ext} 0\longrightarrow \xi^{-1}\longrightarrow E_\epsilon\stackrel{\pi}{\longrightarrow} \xi \longrightarrow 0. \end{equation} These maps were studied in detail by Bertram in \cite{B}. We will now assume $\pi:E\rightarrow \xi$ to be surjective, so $\xi^{-1}$ is a subbundle of $E$. By specialization the result follows for all $\xi\in D_E$, all $E$. We tensor the sequence \ref{ext} by $E$, obtaining the following sequence for $S^2E$: $$ 0\longrightarrow \xi^{-2}\longrightarrow S^2E \longrightarrow \xi\otimes E \longrightarrow 0. $$ Since ${\cal E}nd_0(E)\cong S^2E$, the differential of $\rho$ is the natural map: $$ d\rho:H^1(\xi^{-2})/H^0(\xi\otimes E)\longrightarrow H^1(S^2E)=T_E{\cal M}, \qquad {\rm so}\quad T_E \tilde{D}_\xi=Im(d\rho). $$ Dualizing the sequence above and tensoring it by $K$ we get: $$ 0\longrightarrow \xi^{-1}\otimes E\otimes K\longrightarrow S^2E\otimes K \longrightarrow \xi^{ 2}\otimes K \longrightarrow 0. $$ Now we have: $$ \begin{array}{rcl} \langle\Phi_\xi\rangle &=& Im(H^0(\xi^{-1}\otimes E\otimes K)=\langle\tau\rangle \longrightarrow H^0(S^2E\otimes K))\\ &=& \ker(H^1(E\otimes\xi)\longleftarrow H^1(S^2E))^*\\ &=& Im(H^1(S^2E)\longleftarrow H^1(\xi^{-2}))^*. \end{array} $$ Therefore we have indeed, as claimed: $$ T_E \tilde{D}_\xi=\ker(\Phi_\xi). $$ {\unskip\nobreak\hfill\hbox{ $\Box$}\par} \secdef\empsubsection{\emppsubsection*}{}\label{covD} The previous proposition shows that the line bundle ${\cal O}_{Pic^{g-1}(C)}(2\Theta)_{|D_E}\cong {\cal O}_{D_E}(D_E)$ plays an essential role as regards the map $\phi_E$. Recall that the divisor $D_E$ is invariant under the involution $\iota$ (see \ref{inv}). The following lemma shows how $\iota$ acts on the global sections of this line bundle. \secdef\empsubsection{\emppsubsection*}{Lemma.}\label{split} The involution $\iota$ gives a splitting in an invariant and an anti-invariant part: $$ \begin{array}{rcccc} H^0(D_E,{\cal O}_{D_E}(D_E))&=& H^0(D_E,{\cal O}_{D_E}(D_E))_+&\oplus& H^0(D_E,{\cal O}_{D_E}(D_E))_-\\ &\cong&H^0(Pic^{g-1}(C),{\cal O}(D_E))/\langle s_E\rangle & \oplus& H^1(Pic^{g-1}(C),{\cal O}). \end{array} $$ Here $s_E\in H^0(Pic^{g-1}(C),{\cal O}(D_E))$ is a section with divisor $(s_E)=D_E$. The map $\phi_E$ factors over the natural map $$ D_E\longrightarrow {\Bbb P} H^0(D_E,{\cal O}_{D_E}(D_E))_+^*\;(\cong {\Bbb P} T^*_{D_E}|2\Theta|). $$ The map $\psi_E$ (the Gauss map) is the natural rational map: $$ D_E\longrightarrow {\Bbb P} H^0(D_E,{\cal O}_{D_E}(D_E))_-^*\cong {\Bbb P} H^1(Pic^{g-1}(C),{\cal O})^*. $$ Therefore both $\phi_E$ and $\psi_E$ factor over $\bar{D}_E$. \vspace{\baselineskip}\noindent{\bf Proof.}$\;\;$ The exact sequence of sheaves on $Pic^{g-1}(C)$: $$ 0\longrightarrow {\cal O}\stackrel{\cdot s_E}{\longrightarrow} {\cal O}(D_E)\longrightarrow {\cal O}_{D_E}(D_E)\longrightarrow 0 $$ gives the cohomology sequence: $$ 0\longrightarrow H^0(Pic^{g-1}(C),{\cal O}(D_E))/\langle s_E \rangle \longrightarrow H^0(D_E,{\cal O}_{D_E}(D_E)) \longrightarrow H^1(Pic^{g-1}(C),{\cal O})\longrightarrow 0. $$ It is well known that $\iota^*$ acts as the identity on $H^0(Pic^{g-1}(C),{\cal O}(D_E))$ and as minus the identity on $H^1(Pic^{g-1}(C),{\cal O})$. The remaining assertions are standard. {\unskip\nobreak\hfill\hbox{ $\Box$}\par} \section{The genus two case}\label{g=2} \secdef\empsubsection{\emppsubsection*}{} To determine the Hitchin map in the genus two case, we study first the divisors $D_E$ for general stable $E$ and we study three of the four maps from the diagram \ref{diagram} (see \ref{lower2}, \ref{upper}). This leads to quite classical geometry involving for instance \' etale double covers and tangent conics. Then we can easily determine the fourth map, which is Hitchin's map (projectivized and restricted to ${\Bbb P} T^*_E{\cal M}$). We then `rigidify' our construction using the classical Proposition \ref{prym}. From now on, $C$ will be a genus two curve. \secdef\empsubsection{\emppsubsection*}{} The main result of \cite{NR} is that the map $$ \Delta:{\cal M}\stackrel{\cong}{\longrightarrow} |2\Theta|\cong{\Bbb P}^3,\qquad E\mapsto D_E $$ is an isomorphism, so ${\cal M}\cong {\Bbb P}^3$. In particular, any element in $|2\Theta|$ is a $D_E$ for some $E$. Since this linear system is base-point free, the divisors, now in fact curves, $D_E$ are smooth and have genus 5 for general stable $E$ (a description of the singular curves in $|2\Theta|$ can be found in \cite{Verra}). \secdef\empsubsection{\emppsubsection*}{}\label{cov} Each divisor in $|2\Theta|$ is fixed by $\iota$ and for general $E$, the involution $\iota$ restricted to $D_E$ is a fixed-point free involution on a smooth curve. The induced covering $$ \pi_E:D_E\longrightarrow \bar{D}_E:=D_E/\iota $$ is an \' etale 2:1 covering (the associated Prym variety is $Jac(C)$, see for example \cite{Verra}, p.\ 438). In particular, for general $E$, $\bar{D}_E$ is a smooth genus three curve. From now on we will consider only such $E$. \secdef\empsubsection{\emppsubsection*}{}\label{bunde} The kernel of the map $\pi_E^*:Pic(\bar{D}_E)\rightarrow Pic(D_E)$ is generated by a point $\alpha$ of order two. One has: $$ \pi_E^* K_{\bar{D}_E}\cong K_{D_E},\quad \pi_{E*}{\cal O}_{D_E}\cong{\cal O}_{\bar{D}_E}\oplus\alpha. $$ The adjunction formula on $Pic^{g-1}(C)$ shows ${\cal O}_{D_E}(D_E)\cong K_{D_E}$. The involution $\iota$ gives a splitting in an invariant and an anti-invariant part: $$ H^0(D_E,K_{D_E})\cong H^0(\bar{D}_E,\pi_* K_{D_E})\cong\; H^0(\bar{D}_E,K_{\bar{D}_E})\oplus H^0(\bar{D}_E,K_{\bar{D}_E}\otimes\alpha), $$ (projection formula) which coincides with the splitting given in Lemma \ref{split}. In particular, since $H^1(Pic^{g-1}(C),{\cal O})\cong H^1(C,{\cal O})\cong H^0(C,K)^*$, we have a natural identification $$ H^0(C,K)\cong H^0(\bar{D}_E,K_{\bar{D}_E}\otimes\alpha)^*. $$ \secdef\empsubsection{\emppsubsection*}{} \label{lower2} We will now write $C_3$ for $\bar{D}_E$, $K_3$ for $K_{\bar{D}_E}$. The Gauss map on $D_E$ then factors over $C_3$, and on $C_3$ coincides with the natural map $$ C_3\longrightarrow {\Bbb P} H^0(C_3,K_3\otimes\alpha)^*\cong {\Bbb P} H^1(Pic^{g-1}(C),{\cal O})^* \cong {\Bbb P}^1 $$ which is therefore also essentially $\psi_E$. The map $Sq:|K|\rightarrow |2K|$ from diagram \ref{diagram} corresponds to the second Veronese map which embeds ${\Bbb P}^1$ as a conic in ${\Bbb P}^2$. The (three dimensional) space $S^2H^0(C_3,K_3\otimes\alpha)^*$ may be identified with a quotient of the (six dimensional) $H^0(C_3,2K_3)^*$ (note $2(K_3\otimes\alpha)\equiv 2K_3$), thus we have a diagram (where the last map is a linear projection): $$ \begin{array}{ccccccccc} D_E& &\stackrel{\psi_E}{\longrightarrow}& &|K|&\stackrel{Sq}{\longrightarrow}&|2K|& & \\ \Big|\,\!\Big|&& && \;\;\Big\downarrow \cong && \;\Big\downarrow\cong& & \\ D_E&\stackrel{\pi_E}{\longrightarrow}\!&C_3&\!\rightarrow&\!{\Bbb P} H^0(C_3,K_3\otimes\alpha)^*&\longrightarrow&\!{\Bbb P} S^2H^0(C_3,K_3\otimes\alpha)^*& \leftarrow &\!{\Bbb P} H^0(C_3,2K_3)^*. \end{array} $$ \secdef\empsubsection{\emppsubsection*}{}\label{upper} As $\Delta: {\cal M}{\rightarrow} |2\Theta|={\Bbb P}^3$, is an isomorphism, the cotangent bundle to ${\cal M}$ is the incidence bundle. The map $(d\Delta)^*$ induces an isomorphism. $$ (d\Delta)^*:I=\{(x,h)\in {\Bbb P}^3\times({\Bbb P}^3)^*:\;x\in h\;\} \stackrel{\cong}{\longrightarrow} {\Bbb P} T^*{\cal M} \quad {\rm and}\quad {\Bbb P} T_{D_E}^*|2\Theta|\cong {\Bbb P} T^*_E{\cal M}. $$ From Lemma \ref{split} and \ref{bunde} we get: $$ \phi_E:D_E\stackrel{\pi_E}{\longrightarrow} C_3 \stackrel{\kappa}{\longrightarrow} {\Bbb P} H^0(C_3, K_3)^*\cong {\Bbb P} T_E^*{\cal M} $$ where $\kappa$ is just the canonical map. \secdef\empsubsection{\emppsubsection*}{}\label{kum} We show how the various $C_3$'s ($=\bar{D}_E$'s) fit together as $E$ moves over ${\Bbb P}^3\;(={\cal M})$. The image $S$ of the map $$ \delta:Pic^{g-1}(C)\longrightarrow S\subset |2\Theta_0| = {\Bbb P} H^0(Pic^{g-1}(C),2\Theta)^*=({\Bbb P}^3)^*. $$ (see \ref{delta2}) is the Kummer surface of $Pic^{g-1}(C)$: $$ S\cong Pic^{g-1}(C)/\iota,\qquad \iota:L\mapsto K\otimes L^{-1}. $$ The surface $S$ is a quartic surface and its singular locus consists of the 16 fixed points of $\iota$ (which are the theta characteristics on $C$). Moreover, we can view ${\Bbb P} T_E^*{\cal M}$ as a plane in $({\Bbb P}^3)^*$: $$ {\Bbb P} T_E^*{\cal M}=\{h\in({\Bbb P}^3)^*:\; E\in h\;\}. $$ Proposition \ref{glue} shows: $$ {\Bbb P} T_E^*{\cal M}\cap S\,=\; \phi_E(D_E),\qquad ({\rm with}\quad \phi_E:D_E\longrightarrow {\Bbb P} T_E^*{\cal M}). $$ Thus $\phi_E(D_E)$ is a hyperplane section of the Kummer surface $S$, hence a quartic plane curve. For general $E$, this curve will be smooth (i.e.\ the curve $\bar{D}_E$ is non-hyperelliptic). We consider only these $E$. \secdef\empsubsection{\emppsubsection*}{}\label{hf3} The curve $C_3$ is now non-hyperelliptic by assumption, so the canonical map $\kappa$ is an embedding, and the image of $C_3$ is a smooth quartic in ${\Bbb P}^2$. Pull-back along $\kappa$ gives an isomorphism $H^0(C_3,2K_3)\cong H^0({\Bbb P}^2,{\cal O}(2))$. Let $s,\,t$ be a basis of $H^0(C_3,K_3\otimes\alpha)$. We define conics $Q_i$ in ${\Bbb P}^2$ by: $$ s\otimes s=Q_1,\quad s\otimes t=Q_2,\quad t\otimes t=Q_3 $$ and the essential part (that is, on $D_E/\iota=C_3$) of $Sq\circ\psi_E$ is now: $$ C_3\longrightarrow {\Bbb P} S^2H^0(C_3,K_3\otimes\alpha)^*\cong {\Bbb P}^2, $$ $$ x\mapsto (s^2(x):st(x):t^2(x))=(Q_1(\kappa(x)): Q_2(\kappa(x)):Q_3(\kappa(x))). $$ Since $Sq\circ \psi_E=\bar{H}\circ\phi_E$, we conclude that the Hitchin map is given by: $$ \bar{H}:{\Bbb P} T_E^*{\cal M} \longrightarrow {\Bbb P} H^0(2K)=|2K|,\qquad p\mapsto (Q_1(p):Q_2(p):Q_3(p)) $$ (since $\kappa(C_3)=\phi_E(D_E)$ has degree 4, spans ${\Bbb P} T_E^*{\cal M}$ and $\bar{H}$ has quadratic coordinate functions, $\bar{H}$ is determined by its restriction to $\phi_E(D_E)$). Note that the inverse image in ${\Bbb P} T_E^*{\cal M}$ of the conic $Sq|K|\subset |2K|$ under $\bar{H}$ is a quartic curve containing $\phi_E(D_E)$ and thus is equal to $\phi_E(D_E)$. \secdef\empsubsection{\emppsubsection*}{}\label{bit} We study the construction above a little more closely and exhibit natural (up to scalar multiple) subsets of $H^0(C_3,K_3\otimes\alpha)^*$ and $H^0(K)$ which correspond under the isomorphism $H^0(C_3,K_3\otimes\alpha)^*\cong H^ 0(K)$ that we found in \ref{bunde}. For any $a,\,b\in{\Bbb C}$ we have a section $as+bt\in H^0(C_3,K_3\otimes\alpha)$, let: $$ Q_{(a:b)}:=S^2(as+bt)=a^2Q_1+2abQ_2+b^2Q_3. $$ The $Q_{(a:b)}$ are a quadratic system of conics, each of which is tangent to $C_3$ (that is, has even intersection multiplicity at each intersection point) because $Q_{(a:b)}$ cuts out twice the divisor of the section $as+bt\in H^0(K_3\otimes \alpha)$. There are 6 conics $H_i$, $i\in\{1,\ldots ,6\}$, in the quadratic system of tangent conics which split as pairs of bitangents. They correspond to the six points $$ \langle s_i\rangle=\langle a_is+b_it\rangle\in{\Bbb P}^1= {\Bbb P} H^0(K_3\otimes\alpha),\qquad{\rm with}\quad det(a_i^2Q_1+2a_ib_iQ_2+b_i^2Q_3)=0 $$ (where we now view the $Q_i$ as $3\times 3$ matrices). In this way obtain 12 bitangents of $C_3$. The other 16 bitangents are best seen by identifying $C_3$ with a hyperplane section of the Kummer surface $S$ (\ref{kum}). In fact the divisor $\Theta$ and its translates by points of order two map to conics in $S$, the plane through such a conic intersects $S$ in a double conic, and thus intersects the plane in which $\bar{D}_E$ lies in a bitangent of $\bar{D}_E$. The following classical result relates these six points $\langle s_i\rangle$ to the Weierstrass points of the curve $C$. \secdef\empsubsection{\emppsubsection*}{Proposition.}\label{prym} Under the natural isomorphism from \ref{bunde}: $$\ {\Bbb P} H^0(C_3,K_3\otimes\alpha)\stackrel{\cong}{\longrightarrow} {\Bbb P} H^0(K)^*, $$ the six points which correspond to pairs of bitangents are mapped to the six linear maps corresponding to the Weierstrass points $p_i$ of $C$: $$ \langle s_i\rangle\mapsto \langle [\omega\mapsto \omega(p_i)]\rangle \qquad\qquad (\omega\in H^0(K)). $$ \vspace{\baselineskip}\noindent{\bf Proof.}$\;\;$ We recall the way $D_E$ and $C$ can be recovered from $H^0(C_3,K_3\otimes\alpha)$. It will be shown that the double cover of ${\Bbb P} H^0(C_3,K_3\otimes\alpha)$ branched over the 6 points corresponding to pairs of bitangents is isomorphic to $C$, which proves the Proposition. Since we have $(st)^{ 2}=(s^ {2})( t^{2})$ on $C_3$, the quartic equation of $C_3$ must be: $$ C_3:\quad Q_1Q_3-Q_2^2=0. $$ The curve $C_5$ defined by: $$ s^2=Q_1,\quad st=Q_2,\quad t^2=Q_3 \qquad(\subset{\Bbb P}^4) $$ is a canonically embedded genus 5 curve in ${\Bbb P}^4$ (with coordinates $x,y,z,s,t$ and where $x,y,z$ are a basis of $H^0(C_3,K_3)$). Projection onto ${\Bbb P}^2$ defines a 2:1 unramified covering $\pi:C_5\rightarrow C_3$ and clearly $\pi^*( K_{3}\otimes\alpha)\cong K_{C_5}$, so $\pi$ is defined by $\alpha$ and $C_5\cong D_E$. From the theory of Prym varieties we have (cf.\ \cite{Mumford}, $\S$ 6): $$ Nm^{-1}(K_3)=P^+\cup P^- \qquad{\rm with}\quad Nm:\; Pic^4(C_5)\longrightarrow Pic^4(C_3) $$ and $P^+,\,P^-$ are both isomorphic to $J(C)$, the Prym variety of the cover $C_5\rightarrow C_3$ (\cite{Verra}, p.\ 438). Here we have: $$ P^+:=\{ L\in Pic^4(C_5):\; Nm(L)=K_3,\;\; \dim H^0(C_5,L)\equiv 0\;{\rm mod}\;2\},\quad{\rm and}\quad \tilde{\Theta}\cap P^+=\Xi, $$ where $\tilde{\Theta}$ is the theta divisor in $Pic^4(C_5)$ and where $\Xi$ is the theta divisor of the Prym variety (actually the intersection has multiplicity 2), so in our case $\Xi=C$. A point of $C$ thus corresponds to a $g^1_4$ on $C_5$ with norm $K_3$. The $g^1_4$'s on $C_5$ are cut out by rulings of quadrics in the ideal of $C_5$ of rank $\leq 4$. The hyperelliptic involution on $C$ corresponds to the permutation of the rulings in the rank 4 quadrics, so the Weierstrass points correspond to the $g^1_4$'s from rank 3 quadrics. To a section $as+bt\in H^0(K_3\otimes\alpha)$ corresponds a quadric of rank $\leq 4$ in the ideal of $C_5$ given by: $$ (as+bt)^2=a^2Q^2+2abQ_2+b^2Q_3,\quad{\rm so}\quad (as+bt)^2=Q_{(a:b)}. $$ Such a quadric has rank 3 iff $det(Q_{(a:b)})=0$. Thus these rank 4 quadrics are parametrized by $|K_3\otimes\alpha|$ and there are 6 rank 3 quadrics that correspond to the pairs of bitangents. Each quadric is a cone over a 2:1 cover of the plane $s=t=0$ branched along the conic $Q_{(a:b)}$. The rulings of a rank four quadric are the two irreducible components in the inverse image of lines tangent to the conic $Q_{(a:b)}$; they are interchanged by the covering involution ($s,\,t\mapsto -s,\,-t$). Any ${\Bbb P}^2$ in such a rank 4 quadric thus projects to a line tangent to the conic $Q_{(a:b)}$ and the divisor cut out by the ${\Bbb P}^2$ on $C_5$ maps onto the divisor cut out on $C_3$ by that tangent line. Hence the norm of the $g^1_4$'s obtained from these quadrics is $K_3$. This shows that $C$ is indeed the double cover of $|K_3\otimes\alpha|$ branched over the six points corresponding to pairs of bitangents. {\unskip\nobreak\hfill\hbox{ $\Box$}\par} \secdef\empsubsection{\emppsubsection*}{} The proposition allows us to make `consistent' choices for the coordinate functions of the Hitchin map as $E$ varies. Let $s_i\in H^0(C_3,K_3\otimes\alpha)$ be the six sections which correspond to bitangents. Then $H_i$ restricts to $s_i^2$ on $C_3$, so if we put: $$ \bar{H}:{\Bbb P} T^*_E{\cal M} \longrightarrow {\Bbb P}^2,\qquad p\mapsto (H_1(p):H_2(p):H_3(p)) $$ then we have a choice of coordinate functions for $\bar{H}$ which makes sense for any (general) $E$. The only remaining problem is that we can can still multiply each $H_i$ by a function on ${\cal M}\cong{\Bbb P}^3$ which has poles and zeros in the locus where the map $D_E\rightarrow \bar{D}_E$ is not an \' etale 2:1 map of smooth curves. \section{Computing the Hitchin map}\label{compute} \secdef\empsubsection{\emppsubsection*}{} In the previous section we saw that the polynomials $H_i$ on ${\Bbb P}^3\times ({\Bbb P}^3)^*$ defining the Hitchin map have the property: for any general $q\in{\Bbb P}^3$, $$ (H_i=0)\cap {\Bbb P} T^*_q{\Bbb P}^3=l_i\cup l_i'\qquad ({\Bbb P} T^*_q{\Bbb P}^3\subset{\Bbb P} T^*{\Bbb P}^3=I=\{(x,h)\in{\Bbb P}^3\times({\Bbb P}^3)^*:\;x\in h\;\}), $$ where $l_i$ and $l_i'$ form the pair of bitangents to the smooth curve $$ C_3:=S\cap {\Bbb P} T^*_q{\Bbb P}^3 $$ (see \ref{kum}) corresponding to $s_i\in H^0(C_3,K_3\otimes\alpha)$ (i.e.\ $(s_i^2)=C_3\cap (l_i\cup l_i'$)). Here $S\subset ({\Bbb P}^3)^*$ is the Kummer surface of $Pic^{g-1}(C)$ and $\alpha$ is the bundle of order two defined by the \' etale double cover of $C_3$ obtained by pull-back from the map $Pic^{g-1}(C)\rightarrow S$. We now consider the problem of finding such polynomials. \secdef\empsubsection{\emppsubsection*}{} This problem was actually solved a century ago using the relation between Kummer surfaces and Quadratic line complexes. The classical solution is as follows. A line in $({\Bbb P}^3)^*$ with Klein coordinates (see \ref{linec}) $(x_1:\ldots:x_6)\in{\Bbb P}^5$ is a bitangent to the Kummer surface $S$ occurring in one of the six pairs iff there is an $i\in\{1,\ldots ,6\}$ such that the following two equations are satisfied (see Proposition \ref{bip}): $$ x_i=0,\qquad \sum_{j\neq i} \frac{x_j^2}{\lambda_i-\lambda_j}=0 \qquad\qquad(j\in\{1,\ldots ,6\}), $$ where the $\lambda_i$ correspond to the Weierstrass points of the curve $C$: $$ C:\quad y^2=(x-\lambda_1)\ldots (x-\lambda_6). $$ We will show in \ref{biteq} how to derive the $H_i$ by `restricting' these two equations to the incidence bundle $I$. \secdef\empsubsection{\emppsubsection*}{}\label{linec} We start with some definitions from line geometry. The Pl\"ucker coordinates of the line $l=\langle (Z_0:\ldots: Z_3),\;(W_0:\ldots :W_3)\rangle\subset({\Bbb P}^3)^*$ are: $$ p_{ij}:=Z_iW_j-W_iZ_j\quad{\rm and}\quad G:\;\;p_{01}p_{23}-p_{02}p_{13}+p_{03}p_{12}=0 $$ is the equation of the Grassmannian of lines, embedded in ${\Bbb P}^5$. The Klein coordinates of a line are: $$ \begin{array}{lll} X_1=p_{01}+p_{23},\quad& X_3=i(p_{02}+p_{13}),\quad&X_5=p_{03}+p_{12}\\ X_2=i(p_{01}-p_{23}),&X_4=p_{02}-p_{13},&X_6=i(p_{03}-p_{12}). \end{array} $$ Note that each $X_i$ corresponds to a non-degenerate alternating bilinear form in the $Z_i,\,W_i$. These six bilinear forms give sections $\Phi_i$ of the bundle projection ${\Bbb P} T^*{\Bbb P}^3\longrightarrow {\Bbb P}^3$: $$ \Phi_i:{\Bbb P}^3\longrightarrow {\Bbb P} T^*{\Bbb P}^3=I\subset{\Bbb P}^3\times({\Bbb P}^3)^*, \quad q\mapsto (q,\epsilon_i(q)):=(q,X_i(q,-)). $$ That $\Phi_i(q)\in I$ follows from the fact that $X_i$ is alternating: $X_i(q,q)=0$. Explicitly, if $q=(x:y:z:t)\in{\Bbb P}^3$, then the $\epsilon_i=\epsilon_i(q)\in{\Bbb P}^{3*}$ have the dual coordinates: $$ \begin{array}{lll} \epsilon_1=(y:-x:t:-z),\quad& \epsilon_3=(z:t:-x:-y),\quad&\epsilon_5=(t:z:-y:-x)\\ \epsilon_2=(y:-x:-t:z),&\epsilon_4=(z:-t:-x:y),&\epsilon_6=(t:-z:y:-x). \end{array} $$ \secdef\empsubsection{\emppsubsection*}{}\label{biteq} We show how to take care of the first equation. Let $q=(x:y:z:t)\in{\Bbb P}^3$. As the incidence bundle is the cotangent bundle we have: $$ T^*_q{\Bbb P}^3=\{(u:v:w:s)\in ({\Bbb P}^3)^*:\; xu+yv+zw+ts=0\;\}. $$ Note that we can rewrite the equation to obtain: $$ T^*_q{\Bbb P}^3=\{p\in({\Bbb P}^3)^*:\;X_i(\epsilon_i(q),p)=0\;\}. $$ This implies that the lines in $T^*_q{\Bbb P}^3$ with $X_i=0$ (which form a linear line complex) are exactly the lines passing through the point $\epsilon_i(q)$ (cf.\ \cite{GH}, p. 759-760). In particular, if $T^*_q{\Bbb P}^3\cap S$ is a smooth quartic curve, and $l_i,\;l'_i$ are a pair of bitangents as before, then both lines have $X_i=0$ and thus they must intersect in $\epsilon_i(q)$. Let now $p\in T^*_q{\Bbb P}^3,\; p\neq \epsilon_i(q)$. The condition that $p\in l_i\cup l'_i$ is equivalent to demanding that the line $\langle\epsilon_i(q),p\rangle$ is one of these two bitangents. The $i$-th Klein coordinate of this line is zero because it passes through $\epsilon_i(q)$. Thus for these lines the first equation is verified. We conclude: $$ p\in l_i\cup l_i'\subset {\Bbb P} T^*_q{\Bbb P}^3\; \Longleftrightarrow\; H_i(p,q):=\sum_{j\neq i} \frac{x_j^2}{\lambda_i-\lambda_j}=0,\quad {\rm with}\quad x_j:=X_j(\langle\epsilon_i(q),p\rangle), $$ the Klein coordinates of the line $\langle\epsilon_i(q),p\rangle\subset ({\Bbb P}^3)^*$. The coordinates of $\epsilon_i(q)$ are linear in those of $q$ and so the Pl\"ucker coordinates of $\langle\epsilon_i(q),p\rangle$ are homogeneous of bidegree (1,1) in those of $q$ and $p$. Thus $H_i$ is given by a homogeneous polynomial of bidegree (2,2) on ${\Bbb P}^3\times ({\Bbb P}^3)^*$. \secdef\empsubsection{\emppsubsection*}{} On the open subset $T^*{\Bbb C}^3={\Bbb C}^3\times ({\Bbb C}^3)^*$ of $T^*{\Bbb P}^3$ one can obtain a CIHS from the polynomials $H_i$ as follows. Let $(x,y,z)$ be coordinates on ${\Bbb C}^3$ and let $(u,v,w)$ be the dual coordinates on ${\Bbb C}^{3*}$. Then, the inclusion of cotangent bundles followed by the (rational) projectivization map is given by $$ T^*{\Bbb C}^3 \longrightarrow T^*{\Bbb P}^3\longrightarrow {\Bbb P} T^*{\Bbb P}^3 $$ $$ (q,p):=((x,y,z),(u,v,w))\mapsto (\tilde{q},\tilde{p}):=((x:y:z:1),(u:v:w:-(xu+yv+zw))). $$ (Note that the last coordinate is obtained from the incidence condition.) Now we define: $$ H^a_i(p,q)=\sum_{j\neq i} \frac{X_j(\langle \epsilon_i(\tilde{q})\,,\,(u:v:w:-(xu+yv+zw))\rangle)^2}{\lambda_i-\lambda_j}. $$ The $H_i$ are homogeneous of degree (2,2), and the last coordinate has degree one in $x,\,y,\,z$ so the $H^a_i$ will have degree $\leq 4$ in the $x,\,y,\,z$ (and need not be homogeneous in these variables), but they are still homogeneous of degree $2$ in the $u,\,v,\,w$. With the help of a computer, one can explicitly write down the polynomials $H^a_i$ (the expressions are rather long though). To verify that these polynomials actually Poisson commute (with respect to the standard two form $dx\wedge du+dy\wedge dv+dz\wedge dw$) we again used the computer (after normalizing three of the $\lambda_i$'s by a linear fractional transformation). This then allows us to conclude that the map $H^a:T^*{\Bbb C}^3\rightarrow {\Bbb C}^3$ (whose coordinate functions are any three of the six $H_i^a$'s) is a CIHS. It seems reasonable to expect that the CIHS defined by these $H^a_i$ is actually Hitchin's system, but we could not establish that. \section{Quadratic Line Complexes}\label{Qcomp} \secdef\empsubsection{\emppsubsection*}{} In this section we recall how the equations for the bitangents are determined. We summarize the results we need from \cite{GH}, Chapter 6 and follow \cite{Hu}. Let $G\subset {\Bbb P}^5$ be the Grassmannian of lines in ${\Bbb P}^3$, so $G$ is viewed as a quadric in ${\Bbb P}^5$. For $x\in G$ we denote by $l_x$ the corresponding line in ${\Bbb P}^3$. For $p\in{\Bbb P}^3$ and $h\subset {\Bbb P}^3$ a plane we define $$ \sigma(p):=\{x\in G:\; p\in l_x\},\qquad \sigma(h):=\{x\in G:\; l_x\subset h\}. $$ Both $\sigma(p)$ and $\sigma(h)$ are isomorphic to ${\Bbb P}^2$, in fact any (linear) ${\Bbb P}^2$ in $G$ is either a $\sigma(p)$ or a $\sigma(h)$. Let $L$ be a line in $G$, then $L$ is the intersection of a (unique) $\sigma(p)$ with a (unique) $\sigma(h)$: $$ L=\sigma(p)\cap\sigma(h)=\{x\in G:\; p\in l_x\subset h\}. $$ We will sometimes write $h=h_L$, $p=p_L$ and $L=L_{p,h}$. Thus the points on the line $L$ (in $G$) correspond to the lines (in ${\Bbb P}^3$) in a pencil in $h$ with `focus' $p$. \secdef\empsubsection{\emppsubsection*}{} A quadratic line complex $X$ is the intersection of $G$ with another quadric $F$; we assume $X$ to be smooth. $$ X:=G\cap F. $$ For any $p\in{\Bbb P}^3$, the intersection of $\sigma(p)={\Bbb P}^2\subset G$ with the quadric $F$ is a conic in $\sigma(p)$. Let $$ S:=\{p\in{\Bbb P}^3:\; \sigma(p)\cap F\;{\rm is}\;{\rm singular}\}, $$ then $S$ is a Kummer surface. \secdef\empsubsection{\emppsubsection*}{} If $\sigma(p)\cap F$ is singular, it is the union of two lines $L,\,L'$ or it is a double line. The double lines correspond to the 16 singular points of $S$. The points $x\in L$ correspond to the lines in a plane $h_L$ passing through $p$, similarly the points in $L'$ correspond to lines in a plane $h_{L'}$ passing through $p$. These pencils are called confocal pencils (having the same focus $p$). Note that the line $l=h_L\cap h_{L'}$ lies in both these pencils; it is called a singular line of the complex $X$. This line $l$ corresponds to the intersection of $L$ and $L'$ in $G$: $[l]=L\cap L'$. The singular lines of $X$ form a smooth surface $\Sigma$ in $G$. $$ \Sigma:=\{x\in X:\;l_x\;\mbox{is a singular line in}\;X\}. $$ The set $\Sigma$ is determined in \cite{GH}, p.\ 767-769: $$ x\in\Sigma\; \Longleftrightarrow\; T_xF=T_{x'}G\quad \mbox{for some}\;x'\in G. $$ \secdef\empsubsection{\emppsubsection*}{}\label{kc} In Klein coordinates $X_i$ the relation between the points $x$ and $x'$ above assumes a very simple form. Any quadratic line complex $X$ can be given by (\cite{GH},p.\ 789): $$ G:\;\; X_1^2+X_2^2+\dots +X_6^2=0,\quad F:\;\;\lambda_1X_1^2+\lambda_2X_2^2+\ldots +\lambda_6X_6^2=0, \quad X=G\cap F. $$ Then $S$ is the Kummer variety associated with the genus two curve $$ C:\quad y^2=(x-\lambda_1)\ldots (x-\lambda_6). $$ The equations defining the surface $\Sigma$ are then (\cite{GH}, p.\ 769) $$ \Sigma=G\cap F\cap F_2,\qquad F_2:\quad \lambda_1^2X_1^2+\dots +\lambda_6^2X_6^2=0. $$ Let now $$ x=(x_1:\ldots:x_6)\in \Sigma\;\Longrightarrow\;\; T_xF:\quad\lambda_1x_1X_1+\ldots+\lambda_6x_6X_6=0. $$ Defining $$ x':=(\lambda_1x_1:\ldots:\lambda_6x_6),\;\Longrightarrow\;\; \quad x'\in G,\quad T_{x'}G:\quad\lambda_1x_1X_1+\ldots+\lambda_6x_6X_6=0, $$ so $T_xF=T_{x'}G$ and $x'$ satisfies the required condition. \secdef\empsubsection{\emppsubsection*}{Lemma.}\label{lembit} Let $x\in \Sigma\subset G$ and let $x'\in G$ as above. Define a line: $$ L:=\langle x,\,x'\rangle\;\subset {\Bbb P}^5. $$ Then we have $L\subset G$ and $$ L=L_{p,h}\qquad {\rm with}\quad p\in S,\quad h=T_pS, $$ so that the points $y\in L$ correspond to the lines $l_y\subset {\Bbb P}^3$ with $p\in l_y\subset T_pS$. For $i\in\{1,\ldots ,6\}$ let $$ \{[l_i]\}:=\;(X_i=0) \cap L\qquad(\in G\subset {\Bbb P}^5). $$ Then $l_i$ is a bitangent line to $S$. Moreover, if $p$ is a general point of $S$ then any bitangent to $S$ passing through $p$ is one of the six $l_i$'s. \vspace{\baselineskip}\noindent{\bf Proof.}$\;\;$ Since $x\in T_{x'}G$ we have $L\subset G$ (this is also easily verified using the three equations defining $\Sigma$). Then $L=L_{p,h}$ with $p=l_x\cap l_{x'}$. We claim that $p\in S$. Since $\sigma(p)$ is a linear subspace in $G$ passing through $x'$ we have $\sigma(p)\subset T_{x'}G$, and thus also $\sigma(p)\subset T_xF(=T_{x'}G)$. Thus $\sigma(p)$ is tangent to $F$ at $x$, so $\sigma(p)\cap F$ is singular in $x$. Therefore $p\in S$ (and $l_x$ is the singular line of $X$ passing through $p$). Any point on $L$, distinct from $x$, can be written as: $$ x_\lambda:=\lambda x+x'=(\ldots:(\lambda +\lambda_i)x_i:\ldots)\qquad (\lambda\in {{\Bbb C}}). $$ It is easy to check by substitution that $x\in \Sigma\Rightarrow x_\lambda\in\Sigma_\lambda$ with: $$ \Sigma_\lambda:= G\cap F^{(\lambda)}\cap F^{(\lambda)}_2,\quad(\lambda\neq -\lambda_i) $$ and where we define: $$ F^{(\lambda)}:\;(\lambda+\lambda_1)^{-1}X_1^2+\ldots+(\lambda+\lambda_6)^{-1}X_6^2=0,\quad F^{(\lambda)}_2:\;(\lambda+\lambda_1)^{-2}X_1^2+\ldots+(\lambda+\lambda_6)^{-2}X_6^2=0. $$ Thus $x_\lambda$ corresponds to a singular line for the quadratic complex $X_\lambda:=G\cap F^{(\lambda)}$. As above, there exists thus a point $x'_\lambda\in G$ with: $$ T_{x_\lambda}F^{(\lambda)}=T_{x'_\lambda}G,\qquad x'_\lambda:=(\ldots:x_i:\ldots)=x\in G. $$ Therefore $x_\lambda$ and $x'_\lambda$ lie on the line $L\subset G$ and thus the point $p$ is a point of $S_\lambda$, the Kummer surface associated to the quadratic line complex $X_\lambda$. This holds for all singular lines $x$ of $X$ (and thus for all points $p\in S$), therefore we conclude: $$ S=S_\lambda. $$ In particular, there is a one-dimensional family of quadratic line complexes $X_\lambda$ which give rise to the same Kummer surface, the so called Klein variety (see \cite{NR} for a modern treatment). Now we can determine $h$. Each $x_\lambda\in L$ is a singular line for a quadratic line complex defining $S$. Then the line in ${\Bbb P}^3$ corresponding to it is tangent to $S$ at the (unique; cf. the verification on p.\ 767 of \cite{GH}) point $p_\lambda\in S$ with $x_\lambda=Sing(\sigma(p_\lambda)\cap F)$ (cf.\ \cite{GH}, p.\ 764-765, p. 791). In our case, $p_\lambda=p$ for all $\lambda$, so we conclude that $L$ is the pencil of lines in ${\Bbb P}^3$ that are tangent to $S$ at $p$, which implies $h=T_pS$. The lines from $L$ that are bitangent to $S$ are thus the bitangents of the curve $T_pS\cap S$. This is, in general, a plane quartic curve with a node at $p$, so there are six bitangents to $S$ in the pencil $L$. These must then correspond to the values $\lambda=-\lambda_i$, since for other values the lines $x_\lambda$ are singular lines of a smooth quadratic line complex and cannot be tangent to $S$ at other points. Thus the bitangent lines in $T_pS$ passing through $p$ correspond to the points on $L$ with exactly one Klein coordinate equal to zero. {\unskip\nobreak\hfill\hbox{ $\Box$}\par} \secdef\empsubsection{\emppsubsection*}{Remarks.} Viewing $S$ as $Pic^{g-1}(C)/\iota$, the divisors $T_pS\cap S$ (for $p$ smooth) correspond to the divisors $$ D_\beta:=L_\beta^*\Theta+L_{-\beta}^*\Theta\in |2\Theta|,\qquad( \beta \in Pic^0(C)) $$ with $2\beta\neq{\cal O}$. These are the union of two copies of $C\,(=\Theta)$ meeting in two points. These two points, and the two copies, are interchanged by $\iota$, the quotient is a nodal curve isomorphic to $C$ with the two points identified. The normalization of $T_pS\cap S$ is thus isomorphic with $C$. The six bitangents to $C$ correspond to the lines spanned by $p$ and (the image in $T_pS\cap S$) of a Weierstrass point of $C$. This, once again, establishes the connection between bitangents to $S$ (and its plane sections) and Weierstrass points on $C$. The 16 non-reduced divisors $L_\beta^*(2\Theta)\in |2\Theta|$ (so $2\beta={\cal O}$) map to double conics. A point on a double conic is `exceptional' for the Lemma since any line tangent to $S$ at a point of the conic lies in the plane of the conic and is thus a bitangent to $S$. These (double) conics are called the tropes of the Kummer surface. The `self'-duality of the Kummer surface $S=\delta(Pic^{g-1}(C))$ fits in nicely with the map $\delta:Pic^{g-1}(C)\rightarrow |2\Theta|_0$ from \ref{delta}, the duality between $|2\Theta_0|$ and $|2\Theta|$, and the map $\delta':Pic^0(C)\rightarrow |2\Theta|,\; \beta\mapsto D_\beta$. In fact, it identifies the tangent planes to points of $S$ (which cut out $D_\beta$) with the points $\pm \beta \in Pic^0(C)/\pm 1 \cong \delta'(Pic^0(C))$ which is the Kummer surface of $Pic^0(C)$. This surface is isomorphic to $S=Pic^{g-1}(C)/\iota$, but the `self'-duality is however not an isomorphism; it is a birational isomorphism which blows up double points and blows down tropes. The special case that the line $L$ in the Lemma actually lies in $X$ is studied in \cite{GH}, p.\ 791-796 (note that they fix the quadratic line complex whereas in the proof of the Lemma we consider a family of complexes). \secdef\empsubsection{\emppsubsection*}{Proposition.}\label{bip} For $x\in G$ the line $l_x$ is a bitangent to a general point of $S$ iff for some $i\in\{1,\ldots ,6\}$ one has: $$ x_i=0,\qquad \sum_{j\neq i} \frac{x_j^2}{\lambda_i-\lambda_j} =0. $$ \vspace{\baselineskip} \noindent{\bf Proof.}$\;\;$ In Lemma \ref{lembit} we saw that any such bitangent $l_z$ of $S$ has one Klein coordinate equal to zero, we will assume it is the first one. Then $$ z=(0:(-\lambda_1+\lambda_2)x_2:\ldots :(-\lambda_1+\lambda_6)x_6)\quad {\rm with}\quad x=(x_1:x_2:\ldots:x_6)\in\Sigma, $$ in particular $x\in X=G\cap F$. Substituting the coordinates of $z$ in the second equation we get: $$ (\lambda_1-\lambda_2)x^2_2+\ldots +(\lambda_1-\lambda_6)x_6^2=\lambda_1(x_1^2+\ldots +x_6^2)- (\lambda_1x_1^2+\ldots+\lambda_6x_6^2)=0. $$ Conversely, let $z=(0:z_2:\ldots :z_6)\in G$ satisfy also the second equation above, so: $$ z_2^2+\ldots +z_6^2=0,\quad (\lambda_1-\lambda_2)^{-1}z^2_2+\ldots +(\lambda_1-\lambda_6)^{-1}z_6^2=0. $$ Then we define: $$ x_i:=(\lambda_1-\lambda_2)^{-1}z_i,\quad(2\leq i\leq 6),\qquad x_1:=\sqrt{-(x_2^2+\ldots +x_6^2)}, $$ here the choice of the square root does not matter. Define $$ x:=(x_1:x_2:\ldots :x_6),\quad{\rm so}\quad x_1^2+\ldots +x_6^2=0 $$ and we have $x\in G$ (the quadric defined by $X_1^2+\ldots +X_2^2$.) We claim that $l_x$ is a singular line of $S$. For this we verify that $x$ satisfies the other two quadratic equations defining $\Sigma$. First of all: $$ \begin{array}{rcl} \lambda_1x_1^2+\lambda_2x_2^2+\ldots+\lambda_6x_6^2&=& -\lambda_1(x_2^2+\ldots +x_6^2)+\lambda_2x_2^2+\ldots+\lambda_6x_6^2\\ &=&(\lambda_1-\lambda_2)x_2^2+\ldots+(\lambda_1-\lambda_6)x_6^2\\ &=&(\lambda_1-\lambda_2)^{-1}z^2_2+\ldots +(\lambda_1-\lambda_6)^{-1}z_6^2\\ &=&0, \end{array} $$ so $x\in F$. We use these two relations on the $x_i$'s to obtain the third: $$ \begin{array}{rcl} \lambda_1^2x_1^2+\ldots+\lambda_6^2x_6^2&=& (\lambda_1^2x_1^2+\ldots+\lambda_6^2x_6^2)-2\lambda_1(\lambda_1x_1^2+\ldots+\lambda_6x_6^2) +\lambda_1^2(x_1^2+\ldots+x_6^2)\\ &=& (\lambda_1-\lambda_2)^2x_2^2+\ldots +(\lambda_1-\lambda_6)^2x_6^2\\ &=& z_2^2+\ldots +z_6^2\\ &=&0, \end{array} $$ since $z=(0:z_2:\ldots:z_6)\in G$. Thus $x\in F_2$ and we conclude $x\in\Sigma$, so we verified that $l_x$ is a singular line. Note that (with notation from \ref{kc}): $$ x'=(\lambda_1x_1:\ldots:\lambda_6x_6),\quad {\rm so}\;\; z=-\lambda_1x+x' $$ and thus $l_z$ is indeed a bitangent to $S$ (see the proof of the Lemma). {\unskip\nobreak\hfill\hbox{ $\Box$}\par}

9410

alg-geom/9410003

en

https://arxiv.org/abs/alg-geom/9410003

alg-geom/9410003

Valery Alexeev

Valery Alexeev

Moduli spaces $M_{g,n}(W)$ for surfaces

21 pages, written in AMS-LaTeX

We construct and prove the projectiveness of the moduli spaces which are natural generalizations to the case of surfaces of the following: 1) $M_{g,n}$, the moduli space of $n$-marked stable curves, 2) $M_{g,n}(W)$, the moduli space of $n$-marked stable maps to a variety $W$.

alg-geom

\section{Introduction} \label{sec:introduction} \begin{say} \label{say:moduli spaces for curves} We recall the following coarse moduli spaces in the case of curves: \begin{enumerate} \item $M_{g}$, parameterizing nonsingular curves of genus $g\ge2$ and its compactification $\overline{M_{g}}$, parameterizing Mumford-Deligne moduli-stable curves, see Mumford \cite{Mumford77}, \item spaces $M_{g,n}$, $2g-2+n>0$, for stable $n$-pointed curves, see Knudsen \cite{Knudsen83}, \item a moduli space $M_{g,n}(W)$ of stable maps from reduced curves to a variety $W$, see Kontsevich \cite{Kontsevich94}. \end{enumerate} It is well known that $\overline{M_{g}}$ and $M_{g,n}$ are projective, $M_{g}$ is quasi-projective. \end{say} \begin{say} For surfaces, Gieseker \cite{Gieseker77} established the existence of a quasi-projective scheme parameterizing surfaces with at worst Du Val singularities, ample canonical class $K$ and fixed $K^{2}$, this is a straightforward analog of $M_{g}$ and we will denote it by $M_{K^{2}}$. A geometrically meaningful compactification of this space, $\overline{M^{sm}_{K^{2}}}$, was constructed by Koll\'ar and Shepherd-Barron in \cite{KollarShepherdBarron88} as a separated algebraic space. It is a moduli space of smoothable stable (not in the G.I.T. sense) surfaces of general type. In \cite{Kollar90} Koll\'ar has shown that if the class of smoothable stable surfaces with a fixed $K^2$ is bounded then $\overline{M^{sm}_{K^{2}}}$ is in fact a projective scheme (Corollary 5.6). Finally, the boundedness was proved by the author in \cite{Alexeev94b}. \end{say} \begin{say} The main purpose of this paper is to construct analogs of $M_{g,n}$ and $M_{g,n}(W)$ in the case of surfaces, and to prove their projectiveness. After this is done, we touch on a connection between our moduli spaces and the standard moduli spaces of K3 and Abelian surfaces. \end{say} \begin{say} An idea of ``$M_{g,n}$ for surfaces'' occured to me when I mentioned that my boundedness theorem 9.2 \cite{Alexeev94b} is strictly stronger than what was used for $\overline{M^{sm}_{K^{2}}}$. Then, looking at the definition of $M_{g,n}(W)$ in \cite{Kontsevich94} I realized that this is simply a relative version of the same scheme, and can be done for surfaces too. \end{say} \begin{say} The basic construction of a moduli space as an algebraic space used here is the same as in \cite{Mumford82}, \cite{Kollar85}, \cite{Viehweg94} and elsewhere. For the hardest question involved, proof of local closedness, we refer to a result of Koll\'ar \cite{Kollar94}. For proving that our moduli spaces are projective schemes, rather than mere algebraic spaces, we use Koll\'ar's Ampleness Lemma 3.9 \cite{Kollar90}, which can be applied in a straightforward way to a variety of complete moduli problems. \end{say} \begin{say} Kontsevich and Manin \cite{KontsevichManin94} use the moduli spaces $M_{g,n}(W)$ to define Gromov-Witten classes of varieties in the ``quantum cohomology'' theory. Hence one distant application of ``$M_{g,n}(W)$ for surfaces'' might be ``higher'' GW-classes of schemes. \end{say} \begin{notation} All schemes are assumed to be at least Noetherian and defined over a fixed algebraically closed field $k$ of characteristic zero. Obstacles to extending the results to positive characteristic are discussed in the last section. In most cases, we prefer the additive notation to the multiplicative one, for divisors and line bundles. All moduli spaces in this paper are coarse. \end{notation} \begin{ack} It is a pleasure to acknowledge useful discussions that I had with F.Campana, Y.Kawamata, J.Koll\'ar, Yu.Manin, E.Sernesi and V.V.Shokurov while working on this paper. \end{ack} \section{Overview} \label{sec:overview} \begin{say} One possible approach to solving an algebro-geometric moduli problem goes through the following steps: \begin{enumerate} \item defining the objects that we are trying to parameterize, \item giving the right definition for a moduli functor, \item establishing properties of this functor, \item constructing a moduli space in some fashion, \item proving finer facts about this space. \end{enumerate} In our treatment, we will follow two guiding principles well understood in the field: \begin{principle} Most moduli spaces exist in the category of algebraic spaces. \end{principle} \begin{principle} Most complete and separated moduli spaces are projective. \end{principle} \end{say} \begin{say} Moduli spaces of nonsingular curves $M_g$ and of Deligne-Mumford stable curves $\overline{M_{g}}$ of genus $g$ provide a textbook illustration of how this works in practice. Since nonsingular curves can degenerate into singular ones in an uncorrectable way, $M_g$ is not complete. There are many different ways to compactify it but the one we are interested in here is adding more curves and trying to solve an enlarged moduli problem. It turns out that the curves one has to add are Deligne-Mumford moduli-stable curves which are defined as connected and complete reduced curves with ordinary nodes only such that every smooth rational irreducible component intersects others in at least 3 points and every irreducible component of arithmetical genus one intersects the rest in at least 1 point. The latter condition has two equivalent meanings: \begin{enumerate} \label{enu:properties of deligne-mumford curves} \item the automorphism group $\operatorname{Aut} (X)$ is finite (and this property is a must for the Geometric Invariant Theory), \item the dualizing sheaf $\omega_X$ is ample. \end{enumerate} To arrive at this answer, one can look at the way the good limits are obtained. One considers a family ${\cal X}$ of curves over a marked curve, or the specter of a DVR ring, $({\cal S},0)$ with a nonsingular general fibre and a degenerate special fibre. Then by the Semistable Reduction Theorem, after making a finite base change ${\cal S}'\to{\cal S}$ and resolving the singularities of ${\cal X}'$, the central fibre will be a reduced curve with simple nodes. Following (1) above one should contract all the rational curves $E$ in the central fiber that intersect the rest only at 1 or 2 points. These have self-intersection numbers $E^2=-1$ and $E^2=-2$ respectfully. Contracting $(-1)$-curves leaves the ambient space, which is a surface, nonsingular. Contracting $(-2)$-curves introduces very simple surface singularities, called Du Val or rational double. The central fiber has nodes only. \end{say} \begin{say} One can recognize that the above is a field of the Minimal Model Program. In fact, we have just constructed the canonical model, in dimension 2, of ${\cal X}'$ over ${\cal S}'$. So, to generalize $\overline{M_{g}}$ to the surfaces of general type we have to apply the Minimal Model Program in dimension 3. This was done by Koll\'ar and Shepherd-Barron in \cite{KollarShepherdBarron88}. By that time, the end of 1980-s, all the necessary for this construction tools from MMP in dimension 3 were already available. The new objects that one has to add are defined as connected reduced surfaces with semi-log canonical singularities and ample tensor power of the dualizing sheaf $(\omega_X^N)^{**}$, where $^{**}$ means taking the self-dual. Using the additive notation, we say that a $\Bbb Q$-Cartier divisor $K_X$ is ample. \end{say} \begin{say} At the present time, the {\em log} Minimal Model Program in dimension 3 is in a pretty good shape. Let us see what kind of statements we can get using its principles. Keeping in line with what we did before, we now consider pairs $(X,B)$ of surfaces $X$ and divisors $B=\sum_{j=1}^{n} B_j$ with ample $K_X+B$. A construction very similar to the one above, with Semistable Reduction Theorem and, this time {\em log} canonical model, shows that we again get a complete moduli functor. What about singularities of the pair $(X,B)$? Why, they ought to be semi-log canonical, of course! What is the analog of this in dimension one? That is easy to answer and we get something very familiar. The divisor $B=\sum_{j=1}^{n} B_j$ becomes a set of marked points. Semi-log canonical means that the curve has nodes only, and that marked points are distinct and lie in the nonsingular part. These are exactly the $n$-marked semistable curves of Knudsen \cite{Knudsen83}. \end{say} \begin{say} Another possible generalization would be looking not at absolute curves (or surfaces) $X$ (or pairs $(X,B)$) but doing it in the relative setting. In other words, let us consider maps $X\to W$ to a fixed projective scheme $W$ with $K_X$ (resp. $K_X+B$) {\em relatively\/} ample. The only modification in the above construction will be that we have to apply the relative version of the (log) Minimal Model Program over ${\cal S}'\times W$ instead of over ${\cal S}'$. What we get for curves is the moduli space $M_{g,n}(W)$ introduced by Kontsevich in \cite{Kontsevich94}. \end{say} \begin{say} Now that we have outlined the objects we will be dealing with, let us return back to the basic example of $\overline{M_g}$. We recall two different approaches to constructing it. \end{say} \begin{approach}[G.I.T.] One first proves that moduli-stable curves are asymptotically Hilbert-stable, \cite{Mumford77}. Then the standard G.I.T. machinery produces a quasi-projective moduli space. Since it is complete, it is actually projective. \end{approach} \begin{approach} Using a fairly general argument (\cite{Mumford82}, p.172) one proves the existence of a moduli space in the category of algebraic spaces. To a family of curves $\pi:{\cal X}\to{\cal S}$ one can in a natural way associate line bundles on ${\cal S}$ which are defined as $\det(\pi_*\omega^k)$. They descend to ($\Bbb Q$-)line bundles $\lambda_k$ on $\overline{M_g}$ and one can further show that $\lambda_k$ are ample for $k\ge1$. \end{approach} \begin{say} As mentioned in \cite{ShepherdBarron83} and \cite{Kollar90}, for surfaces of general type the first approach fails. By \cite{Mumford77} 3.19 in order to be asymptotically Chow- or Hilbert-stable a surface has to have singularities of multiplicities at most 7. On the other hand, the semi-stable limits described above have semi-log canonical singularities and it looks like they must be included in any reasonable complete moduli problem. These semi-log canonical singularities include all quotient singularities, for example, and can have arbitrarily high multiplicities. \end{say} \begin{say} The second approach is what we will be using here. After establishing the existence as an algebraic space, we will use Koll\'ar's Ampleness Lemma \cite{Kollar90} to prove that it is projective. The Ampleness Lemma is a general scheme that shows projectiveness once some good properties of the moduli functor are established: local closedness, completeness, separateness, semipositiveness, finite reduced automorphism groups, and, crucially, boundedness. The projectiveness will be the only ``finer'' property of the obtained moduli spaces that we will consider. \end{say} \section{The objects} \label{sec:the objects} \begin{say} The main objects into the consideration will be {\em stable maps of pairs\/} $g:(X,B)\to W$, where \begin{enumerate} \item $W\subset\Bbb P$ is a fixed projective scheme, \item $X$ is a connected projective surface, \item $B=\sum_{j=1}^n B_j$ is a divisor on $X$, $B_j$ are reduced but not necessarily irreducible, \item the pair $(X,B)$ has semi-log canonical singularities, \item the divisor $K_X+B$ is relatively $g$-ample. \end{enumerate} The precise definitions follow. \end{say} \begin{say} For a normal variety $X$, $K_{X}$ or simply $K$ will always denote the class of linear equivalence of the canonical Weil divisor. The corresponding reflexive sheaf ${\cal O}_X(K_X)$ is defined as $i_*(\Omega_U^{\dim X})$, where $i:U\to X$ is the embedding of the nonsingular part of $X$. \end{say} \begin{defn} An {\em $\Bbb R$-divisor\/} $D=\sum d_{j}D_{j}$ is a linear combination of prime Weil divisors with real coefficients, i.e.\ an element of $N^{1}\otimes \Bbb R$. An $\Bbb R$-divisor is said to be $\Bbb R$-Cartier if it is a combination of Cartier divisors with real coefficients, i.e.\ if it belongs to the image of $Div(X)\otimes\Bbb R \to N^{1}(X)\otimes\Bbb R$ (this map is of course injective for normal varieties). The $\Bbb Q$-divisors and $\Bbb Q$-Cartier divisors are defined in a similar fashion. \end{defn} \begin{defn} Consider an $\Bbb R$-divisor $K+B=K_{X}+\sum b_{j}B_{j}$ and assume that \begin{enumerate} \item $K+B$ is $\Bbb R$-Cartier, \item $0\le b_{j}\le1$. \end{enumerate} For any resolution $f:Y\to X$ look at the natural formula \begin{eqnarray} \label{defn:codiscrepancies} K_{Y}+B^{Y}= f^{*}(K_{X}+\sum b_{j}B_{j})= K_{Y}+\sum b_{j}f^{-1}B_{j} + \sum b_{i}F_{i} \end{eqnarray} or, equivalently, \begin{eqnarray} \label{defn:log discrepancies} K_{Y}+\sum b_{j}f^{-1}B_{j} + \sum F_{i}= f^{*}(K_{X}+\sum b_{j}B_{j}) + \sum a_{i}F_{i} \end{eqnarray} Here $f^{-1}B_{j}$ are the proper preimages of $B_{j}$ and $F_{i}$ are the exceptional divisors of $f:Y\to X$. The coefficients $b_{i},b_{j}$ are called codiscrepancies, the coefficients $a_{i}=1-b_{i},a_{j}=1-b_{j}$ -- log discrepancies. \end{defn} \begin{rem} In fact, $K+B$ is not a usual $\Bbb R$-divisor but rather a special gadget consisting of a linear class of a Weil divisor $K$ (or a corresponding reflexive sheaf) and an honest $\Bbb R$-divisor $B$. This, however, does not cause any confusion. \end{rem} \begin{defn} A pair $(X,B)$ (or a divisor $K+B$) is said to be \begin{enumerate} \item log canonical, if the log discrepancies $f_{k}\ge0$ \item Kawamata log terminal, if $f_{k}>0$ \item canonical, if $f_{k}\ge1$ \item terminal, if $f_{k}>1$ \end{enumerate} for every resolution $f:Y\to X$, $\{k\}=\{i\} \cup \{j\}$. \end{defn} \begin{say} The notion of {\em semi-log canonical\/} is a generalization of {\em log canonical\/} to the case of varieties that are singular in codimension 1. The basic observation here is that for a curve with a simple node the definition of the log discrepancies still makes sense and gives $a_1=a_2=0$, so it can also be considered to be (semi-)log canonical. No new Kawamata semi-log terminal singularities appear, however. Recall that according to Serre's criterion normal is equivalent to Serre's condition $S_2$ and regularity in codimension 1. So, if we do allow singularities in codimension 1, $S_2$ will be exactly what we will need to keep. \end{say} \begin{defn} \label{defn:semi-log canonical} Let $X$ be a reduced (but not necessarily irreducible) equidimensional scheme which satisfies Serre's condition $S_{2}$ and is Gorenstein in codimension 1. Let $B=\sum b_{j}B_{j}$, $0\le b_{j}\le1$ be a linear combination with real coefficients of codimension 1 subvarieties none of irreducible components of which is contained in the singular locus of $X$. Denote by ${\cal O}(K_X)$ the reflexive sheaf $i_*(\omega_U)$, where $i:U\to X$ is the open subset of Gorenstein points of $X$ and $\omega_U$ is the dualizing sheaf of $U$. We can again consider a formal combination of $K_X$ and an $\Bbb R$-divisor $B$, and there is a good definition for $K_X+\sum b_jB_j$ to be $\Bbb R$-Cartier. It means that in a neighborhood of any point $P\in X$ we can choose a section $s$ of ${\cal O}(K_X)$ such that the divisor $(s)+\sum b_jB_j$ is a formal combination with real coefficients of Cartier divisors with no components entirely in the singular set. A pair $(X,B)$ (or a divisor $K_{X}+B$) is said to be semi-log canonical if, similar to the above, \begin{enumerate} \item $K_{X}+B$ is $\Bbb R$-Cartier, \item for any morphism $f:Y\to X$ which is birational on every irreducible component, and with a nonsingular $Y$, in the natural formula \begin{displaymath} f^*(K_{X}+B)=K_{Y}+f^{-1}B+ \sum b_{i}F_{i} \end{displaymath} with $F_{i}$ being irreducible components of the exceptional set, all $b_{i}\le 1$ (resp.~ $a_{i}=1-b_{i}\ge0$). \end{enumerate} As before, the coefficients $b_i,b_j$ are called codiscrepancies, the coefficients $a_i,a_j$ -- the log discrepancies. \end{defn} \begin{rem} In the case when $(X,B)$ has a good semi-resolution (for example, for surfaces) this definition is equivalent to that of \cite{KollarShepherdBarron88}, \cite{FAAT} chapter 12. In our opinion, it is more natural to give a definition which is independent of the existence of a semi-resolution. \end{rem} \begin{rem} For surfaces the condition $S_2$ is of course equivalent to Cohen-Macaulay. \end{rem} \begin{defn} By the Kleiman's criterion, the ampleness for proper schemes is a numerical condition, hence it extends to $\Bbb R$-Cartier divisors. If coefficients of $B$ are rational, $K_{X}+B$ is $g$-ample iff for some positive integer $n$ the divisor $n(K_X+B)$ is Cartier and $g$-ample in the usual sense. \end{defn} \begin{rem} Below we will only consider the case when $B$ is reduced, i.e.\ all the coefficients $b_j=1$. See the last section for the discussion on non-integral coefficients. \end{rem} \begin{exmp} If $X$ is a curve then $(X,B)$ is semi-log canonical iff the only singularities of $X$ are simple nodes and $B$ consists of distinct points lying in the nonsingular part of $X$. $K_X+B$ is relatively ample iff every smooth rational component of $X$ mapping to a point on $W$ has at least 3 points of intersection with the rest of $X$, or points in $B_j$, and every component of arithmetical genus 1 has at least 1 such point. In the absolute case, i.e. when $W$ is a point, this is the usual definition of a stable curve with marked points. Every $B_j$ can also be considered as a group of unordered points. \end{exmp} \begin{exmp} The only codimension 1 semi-log canonical singularities are normal crossings. \end{exmp} \begin{exmp} If $X$ is a nonsingular surface then $(X,B)$ is semi-log canonical iff $B$ has only normal intersections. \end{exmp} \begin{exmp} For the case when $X$ is a surface and $B$ is empty the semi-log canonical singularities over $\Bbb C$ were classified in \cite{KollarShepherdBarron88}. They are (modulo analytic isomorphism): nonsingular points, Du Val singularities, cones over nonsingular elliptic curves, cusps or degenerate cusps (which are similar to cones over singular curves of arithmetical genus 1), double normal crossing points $xy=0$, pinch points $x^{2}=y^{2}z$, and all cyclic quotients of the above. If $B$ is nonempty then the singularities of $X$ are from the same list and, in addition, there are different ways $B$ can pass through them. For normal $X$ the list could be found in \cite{Alexeev92} for example. \end{exmp} \begin{say} The following describes an easy reduction of semi-log canonical singularities to log canonical, cf. \cite{FAAT} 12.2.4. \end{say} \begin{lem} \label{lem:reduction of semi-log canonical to log canonical} Let $(X,B)$ be as in the definition \ref{defn:semi-log canonical} and denote by $\nu:X^{\nu}\to X$ its normalization. Assume that $K_X+B$ is $\Bbb R$-Cartier. Then $(X,B)$ is semi-log canonical iff $(X^{\nu},\nu^{-1}B+cond(\nu))$ is log canonical, and they have the same log discrepancies. \end{lem} \begin{pf} Clear from the definition. \end{pf} \begin{say} The next theorem explains how semi-log canonical surfaces appear in families (cf. \cite{KollarShepherdBarron88} 5.1). But first we will need an auxiliary definition. \end{say} \begin{defn} Let $f:({\cal X},{\cal B})\to {\cal S}$ be a 3-dimensional one-parameter family. Let ${\cal B}=\sum b_{j}{\cal B}_{j}$ with $0\le b_{j} \le1$ be an $\Bbb R$-divisor and assume that ${\cal X}$ and all ${\cal B}_j$ are flat over ${\cal S}$ and that $K_{{\cal X}}+{\cal B}$ is $\Bbb R$-Cartier. We will say that the pair $({\cal X},{\cal B})$ (or the divisor $K_{{\cal X}}+{\cal B}$) is {\em $f$-canonical} if in the definition of log discrepancies for all exceptional divisors with $f(F_i)$ a closed point on ${\cal S}$ one has for the corresponding log discrepancy $a(F_i)\ge1$ (resp. $b(F_i)\le0$). This condition does not say anything about log discrepancies of divisors that map surjectively onto ${\cal S}$. \end{defn} \begin{thm} \label{thm:family is good iff central fiber is good} Let $f:({\cal X},{\cal B})\to {\cal S}$ be a 3-dimensional one-parameter family over a pointed curve or a specter of a DVR (a discrete valuation ring). Let ${\cal B}=\sum b_{j}{\cal B}_{j}$ with $0\le b_{j} \le1$ be an $\Bbb R$-divisor and assume that ${\cal X}$ is irreducible, ${\cal X}$ and all ${\cal B}_j$ are flat over ${\cal S}$ and that the fibers satisfy Serre's condition $S_2$ and are Gorenstein in codimension 1 (note that this implies that ${\cal X}$ itself is Cohen-Macaulay and is Gorenstein in codimension 1). Further assume that $K_{{\cal X}}+{\cal B}$ is $\Bbb R$-Cartier. Then the following is true: \begin{enumerate} \item If $K_{{\cal X}_{0}}+{\cal B}_{0}$ is semi-log canonical then $K_{{\cal X}}+{\cal B}$ is log canonical and $f$-canonical. \item Under assumptions of (1), the general fiber is also semi-log canonical. \item Suppose that there exists a birational morphism $\mu: {\cal Y} \to {\cal X}$ with a nonsingular ${\cal Y}$ such that all exceptional divisors of $\mu$ and strict preimages of ${\cal B}_{i}$ have normal crossings and such that the central fiber is reduced. Then the opposite to (1) is true. \end{enumerate} \end{thm} \begin{pf} The proof is an application of the adjunction formula. (1) The log adjunction theorem \cite{FAAT} 17.12 and \ref{lem:reduction of semi-log canonical to log canonical} imply that $K_{{\cal X}_0}+{\cal B}_{0}$ is semi-log canonical iff $K_{{\cal X}}+{\cal B}+{\cal X}_0$ is. Now, the connection between the log discrepancies of the divisors $K_{{\cal X}}+{\cal B}$ and $K_{{\cal X}}+{\cal B}+{\cal X}_0$ is clear. For a divisor $E$ mapping onto ${\cal S}$ the log discrepancies are the same. For $E$ mapping to a central point of ${\cal S}$ the difference is the coefficient of $E$ in the central fiber of ${\cal Y}\to{\cal S}$, and so is at least 1. (3) Here the differences between the log discrepancies over the central fiber are all equal to 1. (2) is by adjunction. \end{pf} \begin{say} Finally, we show how to pass from a relatively ample $K+B$ to an ample divisor. \end{say} \begin{lem} \label{lem:absolute ampleness} Let $g:(X,B)\to W\subset \Bbb P$ be a map, where $X$ is a projective surface and $B=\sum b_jB_j$ is an $\Bbb R$-divisor on $X$. Assume that $K_X+B$ is semi-log canonical and is relatively $g$-ample. Then $K_X+B+4H$ is ample, where $H=g^*{\cal O}(1)$. \end{lem} \begin{pf} It is enough to prove that the restriction on the normalization of $X$ is ample, therefore by \ref{lem:reduction of semi-log canonical to log canonical} we can assume that $X$ is normal and that $(X,B)$ is log canonical. We show that $K_X+B+3H$ is nef (numerically effective) and this implies the statement. Indeed, $K_X+B+MH$ is ample for $M\gg0$ and $K_X+B+4H$ is a weighted average of the above two divisors. Assume that $K_X+B+3H$ is not nef. Then the Cone Theorem, which holds for arbitrary normal surfaces, tells us that there exists an irreducible curve $C$ generating an extremal ray and such that \begin{displaymath} (K_X+B)C<0 \end{displaymath} This is possible only if $C$ does not map to a point. But then $C\cdot3H\ge3$ and $(K_X+B)C\ge-3$ by a theorem on the length of extremal curves, see \cite{MiyaokaMori86}. In dimension 2 the latter statement is very elementary. Let $f:Y\to X$ be a minimal resolution of singularities of $X$. Then, if $X\ne\Bbb P^2$, one necessarily has $(f^{-1}C)^2\le0$ and \begin{eqnarray*} (K_X+B)C \ge (K_Y+f^{-1}B)f^{-1}C \ge \\ (K_Y+f^{-1}C)f^{-1}C = 2p_a(f^{-1}C)-2 \ge -2. \end{eqnarray*} And the case of $X=\Bbb P^2$ is clear. \end{pf} \section{Definition and properties of a moduli functor} \label{sec:definition and properties of a moduli functor} \begin{say} Below we give a few general definitions for moduli functors. They are fairly standard (see e.g. \cite{Viehweg94}, \cite{Kollar90}) but we need to make slight modifications to adapt them to our situation. \end{say} \begin{say} The moduli functor for a moduli problem of polarized schemes is normally constructed in the following way. One fixes a class ${\cal C}$ of schemes $X/k$ with a polarization, i.e. an ample line bundle, $L$ and some extra structure and subject to some conditions. Then for an arbitrary scheme ${\cal S}/k$ one defines ${{\cM\cC}} ({\cal S})$ as the set of all (relatively) polarized flat families over ${\cal S}$ with all geometric fibers from ${\cal C}$ and, possibly, subject to more conditions. The families are considered modulo an equivalence relation. Usually it is an isomorphism between ${\cal X}_1/{\cal S}$ and ${\cal X}_2/{\cal S}$ with whatever extra structure they have and a fiber-wise linear equivalence between ${\cal L}_1$ and ${\cal L}_2$. In other cases it is an algebraic or a numerical, or a numerical up to a scalar equivalence, instead of linear. Sometimes, it is also useful considering a $\Bbb Q$-polarization $L$ on $X$, i.e a reflexive sheaf such that $(L^{\otimes N})^{**}$ is an ample line bundle. \end{say} \begin{say} The above definition is intentionally vague since extra structures and conditions vary greatly from one moduli problem to another. Instead of trying to cover all future generalizations, we will formulate general principles and, when nontrivial, say exactly how they specialize to our situation. \end{say} \begin{defn} The class ${\cal S}$ is said to be {\em bounded\/} if there exists a scheme $({\cal X},{\cal L})$ with an extra structure, and a morphism $F:{\cal X}\to{\cal S}$ to a scheme ${\cal S}$ of finite type such that all elements of ${\cal C}$ appear as geometric fibers of $F$, not necessarily in a one-to-one way. There are two important variations of this definition. There is the {\em polarized\/} boundedness, when one requires $F$ to be projective and ${\cal L}$ to restrict to the given polarization $L$ on a fiber, versus {\em non-polarized\/}. One can also consider boundedness {\em in the narrow sense\/}, requiring that all fibers of $F$ belong to ${\cal C}$, or {\em in the wide sense\/}, asking only for some of the fibers to be from ${\cal C}$. Here we make the choice of the polarized boundedness in the wide sense. \end{defn} \begin{defn} A moduli functor ${{\cM\cC}} $ is said to be {\em separated\/} if every one-parameter family in ${{\cal M}{\cal C}({\cal S}_{gen})}$, where ${\cal S}_{gen}$ is a generic point of a DVR, has at most one extension to ${\cal S}$. \end{defn} \begin{defn} A moduli functor ${{\cM\cC}} $ is said to be {\em complete\/} if every one-parameter family in ${{\cal M}{\cal C}({\cal S}_{gen})}$, where ${\cal S}_{gen}$ is a generic point of a DVR, has at least one extension after a finite cover ${{\cal S}'\to{\cal S}}$. \end{defn} \begin{defn} \label{defn:local closedness} A class ${\cal C}$ is said to be {\em locally closed\/} if for every flat family $F:({\cal X},{\cal L})\to{\cal S}$ with an extra structure there exist locally closed subschemes ${\cal S}_l\subset{\cal S}$ with the following universal property: \begin{itemize} \item A morphism of schemes ${\cal T}\to{\cal S}$ factors through $\coprod S_l$ iff $({\cal X},{\cal L})\underset{{\cal S}}{\times}{\cal T} \to{\cal T}$ belongs to ${{\cM\cC}} ({\cal T})$. \end{itemize} \end{defn} \begin{defn} The class ${\cal C}$ is said to {\em have finite reduced automorphisms\/} if every object in ${\cal C}$ has a finite and reduced (the latter is automatic in characteristic 0) group of automorphisms. \end{defn} \begin{defn} A moduli functor ${{\cM\cC}} $ is said to be {\em functorially polarizable\/} if for every family $({\cal X},{\cal L})$ in {$\cM\cC(\cS)$} there exists an equivalent family $({\cal X},{\cal L}^c)$ such that \begin{enumerate} \item if $({\cal X}_1,{\cal L}_1)$ and $({\cal X}_2,{\cal L}_2)$ are equivalent, then $({\cal X}_1,{\cal L}^c_1)$ and $({\cal X}_2,{\cal L}^c_2)$ are isomorphic, \item for any base chance $h:{{\cal S}'\to{\cal S}}$, $({\cal X}',{{\cal L}'}^c)$ and $({\cal X}',h^*({\cal L}^c))$ are isomorphic. \end{enumerate} The main example of a functorial polarization is delivered by the polarization $\omega_{{\cal X}/{\cal S}}$ for canonically polarized manifolds. \end{defn} \begin{defn} A functorial polarization ${\cal L}^c$ is said to be {\em semipositive\/} if there exists a fixed $k_{0}$ such that whenever ${\cal S}$ is a complete smooth curve and $f:({\cal X},{\cal L})\to{\cal S}$ an element in {$\cM\cC(\cS)$} , then for all $k\ge k_{0}$ the vector bundles $f_*(k{\cal L}^c)$ are semipositive, i.e.\ all their quotients have nonnegative degrees. This definition will be slightly modified for our purposes, we will also require semipositiveness of restrictions of ${\cal L}^c$ to certain divisors ${\cal B}_j$ on ${\cal X}$. \end{defn} \begin{say} The following is the class that we will be considering from now on. \end{say} \begin{defn} \label{defn:the class} The elements of the class ${\cal C}^N={\cal C}^N_{(K+B)^2,(K+B)H,H^2}$ are {\em stable maps of pairs\/} $g:(X,B,L_N)\to W$, where \begin{enumerate} \item $W\subset\Bbb P$ is a fixed projective scheme, \item $X$ is a connected projective surface, \item $B=\sum_{j=1}^n B_j$ is a divisor on $X$, $B_j$ are reduced but not necessarily irreducible, \item the pair $(X,B)$ has semi-log canonical singularities, \item the divisor $K_X+B$ is relatively $g$-ample, \item $(K_X+B)^2=C_1, (K_X+B)H=C_2, H^2=C_3$ are fixed,. \item $L_N={\cal O}(N(K_X+B+5H))$, where $H=g^*{\cal O}_W(1)$. Here $N$ is a positive integer such that for every map as above $L_N$ is a line bundle. For example, we can choose $N$ to be the minimal positive integer satisfying this condition. The existence of such an $N$ will be proved in \ref{thm:boundedness of maps}, and it is ample by \ref{lem:absolute ampleness}. \end{enumerate} \end{defn} \begin{say} The classes ${\cal C}^N$ and ${\cal C}^M$ for different $N,M$ are in a one-to-one correspondence between each other, and the only difference is the polarizations. As a consequence, the polarization in our functor plays a secondary role. We will switch from a polarization $L_N$ to its multiple $L_M$ when it will be convenient. \end{say} \begin{thm} \label{thm:boundedness of maps} For some $M>0$ the class ${\cal C}^M$ is bounded. \end{thm} \begin{pf} We start with the boundedness theorem which gives what we want in the absolute case. \begin{thm}[\cite{Alexeev94b}, 9.2] \label{thm:absolute boundedness} Fix a constant $C$ and a set ${\cal A}$ satisfying the descending chain condition. Consider all surfaces $X$ with an $\Bbb R$-divisor $B=\sum b_jB_j$ such that the pair $(X,B)$ is semi-log canonical, $K_X+B$ is ample, $b_j\in {\cal A}$ and $(K_X+B)^2=C$. Then the class $\{(X,\sum b_jB_j)\}$ is bounded. \end{thm} Apply this theorem with the set ${\cal A}=\{1\}$ to $K_X+B+D$, where $D$ is a general member of the linear system $|4H|$. Since this linear system is base point free, the pair $(X,B+D)$ also has semi-log canonical singularities. Therefore, all pairs $(X,B)$ satisfying the conditions of the theorem can be embedded by a linear system $|M(K_X+B+4L)|$ for a fixed large divisible $M$ in a fixed projective space $\Bbb P^{d_1}$. Every map $g:X\to W$ is defined by its graph $\Gamma_g$. Consider a Veronese embedding of $W$ by $|{\cal O}_W(M)|$ in some $\Bbb P^{d_2}$ and then look at the graphs $\Gamma_g$ in a Segre embedding $\Bbb P^{d_1}\times\Bbb P^{d_2} \subset \Bbb P^{d_3}$. Note that ${\cal O}_{\Bbb P^{d_3}}(1)$ restricted on $X\simeq\Gamma_g$ is $L_M=M(K_X+B+5H)$. $L^2$ is fixed, hence by the boundedness theorem \ref{thm:boundedness of maps} above there are only finitely many possibilities for Hilbert polynomials $\chi({\cal O}_{\Gamma_g}(t))$. By the same theorem, there are also only finitely many possibilities for Hilbert polynomials $\chi({\cal O}_{B_j}(t))$. Therefore, all elements of our class $g:(X,B)\to W$ are parameterized by finitely many products of Hilbert schemes. In each product, we have to extract a subscheme parameterizing subschemes of $\Bbb P^{d_1}\times W$ and with fixed ${\cal O}_{\Bbb P^{d_1}}(1)^2$, ${\cal O}_{\Bbb P^{d_1}}(1)\cdot{\cal O}_{\Bbb P^{d_2}}(1)$ and ${\cal O}_{\Bbb P^{d_2}}(1)^2$, and these are obviously closed algebraic conditions. We also need to extract the graphs, i.e subschemes mapping isomorphically to $\Bbb P^{d_1}$, and this is an open condition. The resulting scheme will parameterize the maps, including all maps from the class ${\cal C}^M$. This proves the theorem. \end{pf} \begin{say} We won't need the boundedness of the class ${\cal C}^N$ itself, although it will follow from the proof of the local closedness \ref{thm:local closedness}. \end{say} \begin{defn} \label{defn:moduli functor 1} There are several ways to define the moduli functor for our class. The one we use here is the most straightforward one (cf. \cite{KollarShepherdBarron88}, \cite{Viehweg94} in the absolute case with $B=\emptyset$). For any scheme ${\cal S}/k$, ${{\cM\cC}} ^N={{\cM\cC}} ^N_{(K+B)^2,(K+B)H,H^2}$ is given by \begin{displaymath} {{{\cM\cC}} ^N({\cal S})}= \left\{ \begin{aligned} & \text{all families } f:({\cal X},{\cal L})\to{\cal S} \text{ with a divisor } {\cal B}=\sum_{j=1}^N {\cal B}_j \text{ on } X, \\ & \text{a map } g:{\cal X}\to W \text{ and a line bundle } {\cal L} \text{ such that every }\\ & \text{geometric fiber belongs to } {\cal C}, X \text{ and all } {\cal B}_j \text{ are flat over } {\cal S} \\ \end{aligned} \right\} \end{displaymath} Two families over ${\cal S}$ are equivalent if they are isomorphic fiber-wise. In this functor we consider a sub-functor ${{\cM\cC}} {'}^N$, requiring in addition that for each $s$ there exists a 1-dimensional family from ${{\cM\cC}} ^N$ with the central fiber ${\cal X}_s$ and an irreducible general fiber ${\cal X}_g$ such that: \begin{enumerate} \item ${\cal X}_g$ is irreducible, \item the pair $({\cal X}_g,0)$ is (Kawamata) log terminal. \end{enumerate} This is similar to the smoothability condition for $\overline{M_{K^2}^{sm}}\subset\overline{M_{K^2}}$ (see \cite{Kollar90}) and is necessary due to the technical reasons. Consider a one parameter family of maps. Then we would like the ambient 3-fold to be irreducible since MMP is not developed for non-irreducible varieties yet. We would also want the 3-fold to have log terminal singularities because they are Cohen-Macaulay in characteristic 0. \end{defn} \begin{say} A little disadvantage of the above definition is that even though ${{\cM\cC}} ^{N,irr}$ and, say, ${{\cM\cC}} ^{2N,irr}$ are the same on the closed points, the corresponding moduli spaces can potentially have different scheme structures, the second one could be strictly larger. So, in fact, we have not one but infinitely many moduli spaces. It would be better if we had a formula for the minimal $N$ in terms of $(K+B)^2,(K+B)H,H^2$. We know, however, only that such an $N$ exists. \end{say} \begin{say} A different solution was suggested (again, in the absolute case with $B=\emptyset$) by Koll\'ar in \cite{Kollar90},\cite{Kollar94}. In a sense, it produces a moduli space with the ``minimal'' scheme structure. We introduce some necessary notation first. \end{say} \begin{defn} Let $F:{\cal X}\to{\cal S}$ be a projective family of graphs of maps $(X,B)\to W$. Assume that every fiber is Gorenstein in codimension 1 and satisfies Serre's condition $S_2$. Denote by $i:{\cal U}\hookrightarrow{\cal X}$ the open subset where $f$ is Gorenstein and the divisors ${\cal B}_j$ are Cartier. Note that on every fiber one has $\operatorname{codim} _{{\cal X}_s}({\cal X}_s-{\cal U}_s)\ge2$. Define the sheaves ${\cal L}_{{\cal U},k}$ and ${\cal L}_k$ by \begin{displaymath} {\cal L}_{{\cal U},k}={\cal O}_{{\cal U}}(k(K_{{\cal U}/{\cal S}}+{\cal B}+g^*{\cal O}_W(5)) \end{displaymath} \begin{displaymath} {\cal L}_k=i_*{\cal L}_{{\cal U},k} \end{displaymath} It follows that the sheaves ${\cal L}_k$ on ${\cal X}$ are coherent. \end{defn} \begin{notationnum} Let $f:{\cal X}\to{\cal S}$ be a morphism of schemes, $i:{\cal U}\hookrightarrow{\cal X}$ be the immersion of an open set and ${\cal F}$ be a coherent sheaf on ${\cal U}$ which is flat over ${\cal S}$. For a base change $h:{\cal S}'\to{\cal S}$ we obtain ${\cal X}^h:={\cal X}\underset{{\cal S}}{\times}{\cal S}'$, ${\cal U}^h:={\cal U}\underset{{\cal S}}{\times}{\cal S}'$ etc. Denote the induced morphism ${\cal U}^h\to {\cal U}$ by $h_{{\cal U}}$ and set ${\cal F}^h:=h_{{\cal U}}^*{\cal F}$. The induced morphism ${\cal X}^h\to{\cal X}$ is denoted by $h_X$ One says that {\em the push forward of ${\cal F}$ commutes with a base change\/} $h:{\cal S}'\to{\cal S}$ if the natural map $h_X^*(i_*{\cal F})\to i^h_*{\cal F}^h$ is an isomorphism. \end{notationnum} \begin{defn} Define ${{\cM\cC}} ^{all}={{\cM\cC}} ^{all}_{(K+B)^2,(K+B)H,H^2}$ by \begin{displaymath} {{{\cM\cC}} ^{all}({\cal S})}= \left\{ \begin{aligned} & \text{all families } f:{\cal X}\to{\cal S} \text{ with a divisor } {\cal B}=\sum_{j=1}^N {\cal B}_j \text{ on } {\cal X} \text{ and} \\ & \text{a map } g:{\cal X}\to W \text{ such that every geometric fiber belongs to } {\cal C}, \\ & {\cal X} \text{ and all } {\cal B}_j \text{ are flat over } {\cal S}, \text{ and for each } k \\ & i_*{\cal L}_{{\cal U},k} \text{ commutes with arbitrary base changes} \end{aligned} \right\} \end{displaymath} As above, one can consider a sub-functor ${{\cM\cC}} {'}^{all}\subset{{\cM\cC}} ^{all}$. We will not go into detailed discussion of this functor. \end{defn} \begin{say} One can see that if we require that $i_*{\cal L}_{{\cal U},k}$ commutes with arbitrary base changes only for $k=N$ instead of all positive $k$, then we get the previous definition of the moduli functor. Indeed, if a line bundle ${\cal L}$ exists, then ${\cal L}_N={\cal L}+f^*{\cal E}$ for some invertible sheaf ${\cal E}$ on ${\cal S}$. Then for every $h:{\cal S}'\to{\cal S}$ the two sheaves $i^h_*{\cal L}_{{\cal U},N}^h$ and $h_X^*(i_*{\cal L}_{{\cal U},N})=h_X^*({\cal L}_N)$ on ${\cal X}'$ are both reflexive and coincide on $h_X^{-1}({\cal U})$, hence everywhere. Vice versa, if $i_*{\cal L}_{{\cal U},N}$ commutes with base changes, then ${\cal L}_N$ is flat and for every closed point $s\in{\cal S}$ \begin{displaymath} {\cal L}_N\big|_{{\cal X}_s}={\cal O}_{{\cal X}_s}(N(K+B+g^*{\cal O}_W(5))) \end{displaymath} Since the latter restriction is locally free for every $s$ and the sheaves ${\cal O}_{{\cal X}}$, ${\cal L}_N$ are coherent and flat over ${\cal S}$, it follows by \cite{Matsumura86} 22.5, 22.3 that ${\cal L}_N$ is locally free. \end{say} \begin{say} Now let us show that our moduli functor ${{\cM\cC}} {'}^N$ has all the good properties listed above. We start with the local closedness. The main technical result we will be using is the following theorem. \end{say} \begin{thm}[Koll\'ar \cite{Kollar94}] \label{kollar's flattening decomposition} With the above notations, assume that $f:{\cal X}\to{\cal S}$ is projective, $i_*{\cal F}$ is coherent and that for every point $s\in{\cal S}$ the sheaf ${\cal F}_s$ on the fiber ${\cal X}_s$ satisfies Serre's condition $S_2$. Then there exist locally closed subschemes ${\cal S}_l\subset{\cal S}$ such that for any morphism $h:{\cal T}\to{\cal S}$ the following are equivalent: \begin{enumerate} \item $h$ factors through ${\cal T}\to\coprod{\cal S}_l\to{\cal S}$, \item $i^h_*{\cal F}^h$ commutes with all future base changes. \end{enumerate} \end{thm} \begin{thm} \label{thm:local closedness} The functors ${{\cM\cC}} ^N$ and ${{\cM\cC}} {'}^N$ are locally closed. \end{thm} \begin{pf} Let $F:{\cal X}\to{\cal S}$ be an arbitrary projective family of graphs of maps $(X,B)\to W$. First, after the flattening decomposition (see \cite{Mumford66} lecture 8) of ${\cal S}$ into locally closed subschemes, we can assume that ${\cal X}$ and ${\cal B}_j$ are flat over ${\cal S}$ if they are not already. Consider a one-parameter sub-family ${\cal X}_{{\cal R}}\to{\cal R}$ and a point $P$ on the central fiber ${\cal X}_0$. Then ${\cal X}_0$ is Cohen-Macaulay at $P$ iff the 3-fold ${\cal X}_{{\cal R}}$ is. The property of a local ring to be Cohen-Macaulay is open (\cite{Matsumura86} 24.5) and the morphism $F$ is projective. Therefore, if ${\cal X}_0$ is Cohen-Macaulay then there exists an open neighborhood of ${\cal R}$, and also of ${\cal S}$, that contains exactly the points over which the fibers are Cohen-Macaulay. The property of a local ring to be Gorenstein is also open (\cite{Matsumura86} 24.6) and by the same argument there exists a closed subset $Z$ of non-Gorenstein points in ${\cal X}$. Give it the structure of a reduced scheme. Then we have to throw away all fibers on which the Hilbert polynomial of ${\cal O}_Z\underset{{\cal O}_{{\cal S}}}{\otimes}k(s)$ has degree $\ge1$. There are only finitely many possible Hilbert polynomials and the condition on the degree is obviously closed. At this point we use the previous theorem \ref{kollar's flattening decomposition} to the sheaf ${\cal L}_{{\cal U},N}$ to conclude that there exist locally closed subschemes ${\cal S}_l\subset{\cal S}$ such that every map $h:{\cal T}\to{\cal S}$ with ${\cal X}\underset{{\cal S}}{\times}{{\cal T}}\in{{\cM\cC}} ({\cal T})$ factors through $\coprod{\cal S}_l$. ${\cal S}_l$ are disjoint, so we can concentrate on one of them. If $P$ is a point of ${\cal S}$ and some $h$ as in the definition does not factor through ${\cal S}-P$, then the fiber of $F$ over $P$ has to be a pair $(X,B)$ from our class. The sheaf ${\cal L}_N$ on $X\underset{{\cal S}}{\times}S_l$ is flat over $S_l$ and its restriction to the fiber over $P$ is locally free. Hence, it has to be locally free in a neighborhood of the fiber. Therefore, for each $S_l$ if we denote by $U_l\subset S_l$ the open set over which ${\cal L}_N$ is locally free, then $h:{\cal T}\to{\cal S}$ has to factor through $\coprod{\cal U}_l$. Now we can apply \ref{thm:family is good iff central fiber is good}(2) to conclude that there exist open subsets ${\cal V}_l\subset{\cal U}_l$ containing all the points over which the fibers have semi-log canonical singularities. Also, ${{\cM\cC}} {'}^N\subset{{\cM\cC}} ^N$ is evidently closed and we end up with a disjoint union of locally closed subschemes. There is one more thing one has to take care of: the polarization ${\cal O}_{\Bbb P^{d_3}}(1)$ on the fibers has to coincide with ${\cal L}_N$ or its fixed multiple ${\cal L}_M$. Standard semi-continuity theorems for $h^0$ in flat families show that there exists a closed subset where the two sheaves are the same. One can also define the scheme structure on it, see lemma 1.26 \cite{Viehweg94}. \end{pf} \begin{lem} For the functors ${{\cM\cC}} ^N$ and ${{\cM\cC}} {'}^N$ the polarization ${\cal L}_N$ is functorial. \end{lem} \begin{pf} $K_{{\cal U}/{\cal S}}$ of a flat family commutes with base changes, and so do ${\cal O}({\cal B}_j)$ and $g^*{\cal O}_W(1)$. Therefore, ${\cal L}_{{\cal U},k}$ are functorial. By the definitions of the functor ${{\cM\cC}} $ the same is true for ${\cal L}_k$ (resp. ${\cal L}_N$). \end{pf} \begin{thm} ${{\cM\cC}} {'}^N$ is \begin{enumerate} \item separated, \item complete, \item have finite and reduced automorphisms. \end{enumerate} \end{thm} \begin{pf} The first two properties have code names in the Minimal Model Program: ``uniqueness and existence of the log canonical model''. It is enough to check them in the case when the general fiber is irreducible and has log terminal singularities. (1) Let ${\cal S}$ be a specter of a DVR or a pointed curve. Two families in ${{\cM\cC}} ({\cal S})$ that coincide outside of $0$ are birationally isomorphic. \ref{thm:family is good iff central fiber is good}(1) implies that they are both log canonical and both are relative log canonical models over ${\cal S}\times W$ for the same divisor, hence isomorphic. If ${\cal Y}\to{\cal S}$ is a common resolution then the divisor is \begin{displaymath} K_{{\cal Y}}+f^{-1}{\cal B} + \sum{\cal E}_i \end{displaymath} where ${\cal E}_i$ are exceptional divisors that do not map to a central point $0\in{\cal S}$. (2) If there is a family over ${\cal S}-0$, we can complete it over $0$ somehow. Then by a variant of the Semistable Reduction Theorem, after a finite base change, there is a resolution ${\cal Y}$ of singularities such that the central fiber is reduced and all exceptional divisors and ${\cal B}_j$ have normal crossings. Consider the log canonical model for the same divisor as above, relative over ${\cal S}\times W$. It exists by \cite{KeelMcKernanMatsuki93} for example. This log canonical model has the same fibers as $(X,B)$ outside $0$. It has log terminal singularities only, which are Cohen-Macaulay in dimension 3 and characteristic 0. Therefore, the central fiber is also Cohen-Macaulay and it is from our class ${\cal C}$ by \ref{thm:family is good iff central fiber is good}(3). We also have to show that the sheaf ${\cal L}_N$ for this family is locally free. It amounts to proving that the Hilbert polynomials $h_1(t)$ of the sheaf $L_{N,0}$ on the special fiber, and $h_2(t)$ of the sheaf $L_{N,g}$ of the general fiber coincide. Both sheaves are locally free. But the log canonical model is constructed by applying the Base Point Freeness theorem, and by the very construction we have that some ${\cal L}_M$ for a large divisible $M$ is locally free on ${\cal X}$. Therefore the polynomials $h_1(M/Nt)$ and $h_2(M/Nt)$ are the same, and that means that $h_1(t)$ and $h_2(t)$ are also the same. (3) In the absolute case, the fact that $K+B$ is ample and log canonical implies that the automorphism group is finite by \cite{Iitaka82}. In the relative case we apply the same theorem to $K_X+B+D$, $D\in|4H|$ general, which is ample by lemma \ref{lem:absolute ampleness}. We are working in characteristic 0 and so the group scheme $\operatorname{Aut} X$ is reduced. \end{pf} \begin{thm} \label{thm:semipositiveness} The functors ${{\cM\cC}} ^N$ and ${{\cM\cC}} {'}^N$ are semipositive. \end{thm} \begin{pf} One has the following \begin{thm}[Koll\'ar \cite{Kollar90} 4.12] \label{thm:kollar's semipositiveness} Let $Z$ be a complete variety over a field of characteristic zero. Assume that $Z$ satisfies Serre's condition $S_2$ and that it is Gorenstein in codimension one. Let $Z\to C$ be a map onto a smooth curve. Assume that the general fiber of $f$ has only semi-log canonical singularities, and further that $K$ of the general fiber is ample. Then $f_*{\cal O}(kK_{Z/C})$ is semipositive for $k\ge1$. \end{thm} For the sheaves ${\cal L}_N= O_{{\cal X}}(N(K_{{\cal X}/{\cal S}}+{{\cal B}}))$ with empty ${\cal B}$ in the absolute case this is exactly what we need. Analyzing the proof of \ref{thm:kollar's semipositiveness} shows that it works with very minor changes in the case of a non-empty reduced ${\cal B}$. In the relative case instead of $K_{{\cal X}/{\cal S}}+{{\cal B}}$ we consider $K_{{\cal X}/{\cal S}}+{{\cal B}}+5H$, $H=g^*{\cal O}_W(1)$. We can think of $5H$ simply as of an additional component of the boundary ${\cal B}$. If a member of the linear system $|5H|$ is chosen generically, on the general fiber of $f$ the pair $(X,B+5H)$ will still be semi-log canonical. For the positiveness of the sheaves ${\cal L}_N\Big|_{{\cal B}_j}$ we use the log adjunction formula, see \cite{Shokurov91} or \cite{FAAT} chapter 16. We get the following semi-log canonical divisors on ${\cal B}_j$: \begin{displaymath} K_{{\cal X}}+{\cal B}\Big|_{{\cal B}_j}=K_{{\cal B}_j}+\sum(1-1/m_{k}){\cal M}_{k} \end{displaymath} for some Weil divisors ${\cal M}_{k}$ on ${\cal B}_j$ and $m_k\in\Bbb N\cup\{\infty\}$. So, here we need a more general semipositiveness theorem, with nonempty ${\cal B}$ that has fractional coefficients. The situation is saved by the fact that the relative dimension of ${\cal B}_j$ over ${\cal S}$ equals 1, and the semipositiveness for this case is proved in \cite{Kollar90} 4.7. \end{pf} \section{Existence and projectivity of a moduli space} \label{sec:existence and projectivity of a moduli space} \begin{thm} \label{thm:existence as an algebraic space} The functor ${{\cM\cC}} ={{\cM\cC}} {'}^N$ is coarsely represented by a proper separated algebraic space of finite type ${{\bold M \bold C}} ={{\bold M \bold C}} {'}^N$. \end{thm} \begin{pf} The proof is essentially the same as in \cite{Mumford82}, p.172. We remind that we are working in characteristic zero, and over $\Bbb C$ the argument is easier. The class ${\cal C}^M$ is bounded, and we can embed all graphs $\Gamma_g$ of the maps $g$ by a linear system $|M(K_X+B+5H)|$ in $\Bbb P^{d_1}\times\Bbb P^{d_2} \subset \Bbb P^{d_3}$ as in \ref{thm:boundedness of maps} for a large divisible $M$. By taking $M$ even larger we can assume that all $X=\Gamma_g$ and all $B_j\subset X$ are projectively normal, $h^0(M(K_X+B+5H))$ is locally constant and there are no higher cohomologies. $(\Gamma_g,B)$ are parameterized, not in a one-to-one way, by some scheme that we will denote by ${\cal H}$. For any family in ${{\cM\cC}} ^N({\cal T})$, the embedding by a relatively very ample linear system $|M(K_X+B+4H)|$ defines a non-unique map ${\cal T}\to{\cal H}$. By \ref{thm:local closedness} there exists a disjoint union of locally closed subschemes ${\cal S}=\coprod{\cal S}_l\hookrightarrow{\cal H}$ with a universal property, and ${\cal T}\to{\cal H}$ factors through ${\cal S}$. We conclude that the coarse moduli space ${{\bold M \bold C}} $ is a categorial quotient of ${\cal S}$ by an equivalence relation $R$, described as follows. $R$ is a set of pairs $(h,G)$, where $h\in{\cal S}$ and $G$ corresponds to a different embedding of $X$ in $\Bbb P^{d_1}$, i.e. $G$ varies in a group $PGL(d_1+1)$. There is a natural map $F:R\to{\cal S}\times{\cal S}$. Every fiber of $\pi_1\circ F$ is isomorphic to $PGL(d_1+1)$ and this map is obviously smooth. The map $F$ is quasi-finite and unramified because its fibers are automorphism groups of objects in ${\cal C}$, and these are finite reduced. The fact that ${{\cM\cC}} $ is also proper implies that $F$ is finite. The rest of the proof is the same as in \cite{Mumford82}, p.172 verbatim. By taking the transversal sections locally the question is reduced to the case of a finite equivalence relation dominated by a map $F':R\to{\cal H}'\times{\cal H}'$ with $\pi_{1}\circ F'$ \'etale, and then the quotient is easily constructed as an algebraic space. Finally, since ${{\cM\cC}} $ is proper, so is ${{\bold M \bold C}} $. \end{pf} \begin{thm} \label{thm:projectiveness} The moduli space ${{\bold M \bold C}} ={{\bold M \bold C}} {'}^N$ is projective. \end{thm} \begin{pf} The proof follows the general scheme of \cite{Kollar90}. By the very construction of ${{\bold M \bold C}} $, there exists a subscheme ${\cal S}\subset{\cal H}$ of a product of Hilbert schemes, with the corresponding universal family $V_{{\cal S}}\to{\cal S}$, that maps to ${{\bold M \bold C}} $. One starts by constructing a {\em finite\/} morphism from a scheme $Y\to{{\bold M \bold C}} $ with a universal family $f:V_Y\to Y$. This is done locally by cutting ${\cal S}\to{{\bold M \bold C}} $ transversally, then adding more copies of these sections, so that the automorphisms do not obstruct gluing the local pieces together, see \cite{Kollar90} 2.7. The only properties of the class ${\cal C}$ used in this construction are boundedness and finiteness of automorphisms, which we have. Next step is to consider the line bundles \begin{displaymath} \lambda_M=\det(f_*{\cal L}_M\oplus f_*{\cal L}_M\big|_{{\cal B}_j}) \end{displaymath} on $Y$ for $M$ large divisible, where \begin{displaymath} {\cal L}_M={\cal O}_V(M(K_{V/Y}+{\cal B}+g^*{\cal O}_W(5))). \end{displaymath} These line bundles do not descend to ${{\bold M \bold C}} $ because of automorphisms, but since the objects of ${\cal C}$ have finite groups of automorphisms and ${\cal C}$ is bounded, for every $M$ there is a finite power of $\lambda_M$ that does come from a line bundle on ${{\bold M \bold C}} $. To prove that ${{\bold M \bold C}} $ is projective it is enough to show that one of $\lambda_M$ is ample, which is achieved by the following theorem. For simplicity we formulate it only in characteristic 0. \begin{notationnum} Let $Y$ be a scheme and let $W$ be a vector bundle of rank $w$ with structure group $\rho:G\to GL_w$. Let $q:W\to Q$ be a quotient vector bundle of rank $k$. Let $Gr(w,k)/G$ denote the set of $G$-orbits on the $k$-dimensional quotients of a $w$-dimensional vector space. The natural map of sets \begin{displaymath} u_{Gr}:\{\text{closed points of }X\}\to Gr(w,k)/G \end{displaymath} is called the {\em classifying map}. One says that the classifying map is {\em finite\/} if \begin{enumerate} \item every fiber of $u_{Gr}$ is finite, and \item for every $y\in Y$ only finitely many elements of $G$ leave $\ker q_y$ invariant. \end{enumerate} \end{notationnum} \begin{thm}[Koll\'ar's Ampleness Lemma, \cite{Kollar90} 3.9] Let $Y$ be a proper algebraic space and let $W$ be a semipositive vector bundle with structure group $G$. Let $Q$ be a quotient vector bundle of $W$. Assume that \begin{enumerate} \item $G$ is reductive, \item the classifying map is finite. \end{enumerate} Then $\det Q$ is ample. In particular, $Y$ is projective. \end{thm} This is what it translates to in our situation. The sheaves are \begin{displaymath} W=\operatorname{Sym} ^j(f_*{\cal L}_M)\oplus\operatorname{Sym} ^j(f_*{\cal L}_M\big|_{{\cal B}_j}) \end{displaymath} and \begin{displaymath} Q=f_*L_{jM}\oplus f_*L_{jM}\big|_{{\cal B}_j}, \end{displaymath} $q$ is the multiplication map. By \ref{thm:semipositiveness} we already know that $Q$ is semipositive, and so is $W$ since symmetric powers of a semipositive sheaf are semipositive. Recall that the universal family $U_Y$ over $Y$ is embedded into a product of $Y$ and \begin{displaymath} \Bbb P^{d_1}\times W \subset \Bbb P^{d_1}\times\Bbb P^{d_2} \subset \Bbb P^{d_3} \end{displaymath} and that the sheaf $L_{M}$ is the restriction of ${\cal O}_{P^{d_3}}(1)$ in this embedding. The group $G$ acting on $W$ is $GL_{d_1+1}\times GL_1$. If every fiber ${\cal X}={\cal\Gamma}_g$ together with all $B_j$ can be uniquely reconstructed from the map $W_s\to Q_s$, then the fibers of $u_{Gr}$ will be exactly the same as fibers of $Y\to{{\bold M \bold C}} $, hence finite. For this to be true we need the following: \begin{enumerate} \item every fiber in $\Bbb P^{d_3}$ is set-theoretically defined by degree $\le j$ equations, \item the multiplication maps $\operatorname{Sym} ^j(f_*{\cal L}_M)\to f_*L_{jM}$ and $\operatorname{Sym} ^j(f_*{\cal L}_M\big|_{{\cal B}_J})\to f_*L_{jM}\big|_{{\cal B}_J}$ are surjective. \end{enumerate} (1) holds if $j$ is large enough. (2) is satisfied because we have chosen $M$ so large that all $X$ and $B_j$ are projectively normal in $\Bbb P^{d_3}$. Finally, the second condition in the definition of finiteness of the classifying map is satisfied because all graphs $(\Gamma_g,B)=(X,B)$ in $\Bbb P^{d_3}$ have finite groups of automorphisms. \end{pf} \section{Related questions} \label{sec:related questions} \begin{say} Let us see how our moduli spaces are related to some others. For example, consider the moduli space ${\cal M}_{L^2}$ of K3 surfaces $X$ with a polarization $L$ with a fixed square. Compare it with ${\cal M}_{(K+B)^2}$, where $W=pt$, $B=B_1$ is one reduced divisor and $(K+B)^2=L^2$ is the same number. ${\cal M}_{(K+B)^2}$ contains an open subset $U$ parameterizing K3 surfaces with reduced divisors having normal intersections only, and we have a map $F:U\to{\cal M}_{H^2}$. A well-known result (Saint-Donat \cite{SaintDonat74}) says that every ample linear system $|L|$ on a K3 surface contains at least one reduced divisor with normal intersections, therefore $F$ is surjective. In fact, ${\cal M}_{H^2}$ is a quotient of $U$ modulo an obvious equivalence relation $R$: $(X_1,B_1)\underset{R}{\sim}(X_2,B_2)$ iff $X_1$, $X_2$ are isomorphic and $B_1$, $B_2$ are linearly equivalent. There is a natural map $G:R\to U\times U$. $\pi_1\circ G$ is smooth and its fibers are open subsets in $\Bbb P^{h^0(H)-1}$. The situation is very similar to what we had in theorem \ref{thm:existence as an algebraic space}, except this time the quotient $U/R$ is not proper. The obvious way to try to obtain a compactification of ${\cal M}_{H^2}$ is to consider the closure $\overline{U}$ of $U$ in ${\cal M}_{(K+B)^2}$, then somehow define the closure $\overline{R}$ of $R$, and ask if it has good enough properties enabling one to construct $\overline{U}/\overline{R}$ and to prove that it is projective. Alternatively, one can ask if the closure of $\overline{G}(R)$ in $\overline{U}\times\overline{U}$ has good properties. The situation resembles what happens for elliptic curves. The natural compactification of the moduli space ${\cal M}_1=\Bbb A^1_k$ is $\Bbb P_k^1$, and the infinite point corresponds not to one but to many degenerations: wheels of rational curves of lengths $1\dots n$ if we consider ${\cal M}_1$ as a factor of ${\cal M}_{1,n}$. Similarly, the boundary points of ${\cal M}_{H^2}$ should correspond to many different degenerations of smooth K3 surfaces with geometric divisors, properly identified. The first thing to ask on this way is: \begin{question} Is it possible to define an equivalence relation $\overline{G}:\overline{R}\to\overline{U}\times\overline{U}$, so that the morphism $\pi_1\circ\overline{G}$ is smooth or at least flat? \end{question} Even if this is done, there are problems with taking the quotient. There does not seem to exist in the literature a ready-to-use method that would cover our situation. There is, on one hand, a theorem of M.Artin (see \cite{Artin69} 7.1, \cite{Artin74b} 6.3) that shows that if $\overline{G}:\overline{R}\to\overline{U}\times\overline{U}$ were a monomorphism (which it is not) with flat projections, then the quotient would be defined as an algebraic space. In this case it would also easily follow that the quotient is actually projective. On the other hand, there is the method of \cite{Mumford82}, p.172 that we used in the previous section, in which the equivalence relation is smooth, and the map $\overline{G}$ is finite. Natural degenerations of K3 surfaces can have infinite groups of automorphisms, however. I think that the question deserves a more detailed consideration. \end{say} \begin{say} Similarly to K3 surfaces, for any principally polarized Abelian variety $A$ with a theta divisor $\Theta$ the pair $(A,\Theta)$ has log canonical singularities, see \cite{Kollar93}. So. the previous discussion applies to principally polarized Abelian surfaces too. One can also ask what happens if the polarization is not principal. \end{say} \begin{say} It goes without saying that the projectivity theorem \ref{thm:projectiveness} applies in the case of curves, with significant simplifications. Therefore, the moduli spaces $M_{g,n}(W)$ of \cite{Kontsevich94} are also projective. \end{say} \begin{say} Most ${{\bold M \bold C}} _{(K+B)^2,(K+B)H,H^2}$ are definitely not irreducible and not even connected. They are subdivided according to various invariants, such as the numerical or homological type of $g(X)$ and $g(B_j)$, intersection numbers $(K+B)B_j$ etc. One can also get by fixing only one number, $(K+B+4H)^2$. Then there are only finitely many possibilities for other invariants. \end{say} \begin{say} The boundedness theorem \ref{thm:absolute boundedness} is in fact even stronger than what we used here: it applies to the case when the coefficients $b_j$ belong to an arbitrary set $\cal A$ that satisfies the descending chain condition. One, perhaps, would want to define even more general moduli spaces. There are two obstacles, however. First, the semipositiveness theorem \ref{thm:kollar's semipositiveness} for the case of fractional coefficients seems to be quite hard to prove, but probably still possible. The second obstacle is a fundamental one: for proving the semipositiveness theorems for ${\cal L}_k|_{{\cal B}_j}$ we used the log adjunction formula. It basically just says $K+B|_B=K_B$, and here the coefficient 1 of $B$ is important. \end{say} \begin{say} The places where assumption about the characteristic 0 was used: \begin{enumerate} \item MMP in dimension 3. This is not serious since we worked in the situation of the relative dimension 2. For surfaces log MMP is characteristic free, and perhaps it is true for families of surfaces in generality needed. For the case $B=\emptyset$ see \cite{Kawamata91} \item The semipositiveness theorem \ref{thm:kollar's semipositiveness} requires characteristic 0. Since we are dealing with a case of relative dimension 2 only, this also probably can be dealt with. \item A group scheme in characteristic 0 is reduced, hence smooth. This was used in the proof of \ref{thm:existence as an algebraic space}. Perhaps, the argument could be strengthened. \item The argument of \cite{Mumford82} p.172 is a whole lot more complicated in characteristic $p>0$. \end{enumerate} \end{say} \begin{say} It should be possible to prove the semipositiveness theorems and the Ampleness Lemma, as well as the \cite{Mumford82} p.172 argument, entirely in the relative situation $/W$, without appealing to absolutely ample divisors. The moduli spaces obtained should be then projective over $W$. \end{say} \begin{say} One can see that most of the theorems that we proved for the functor ${{\cM\cC}} {'}^N$ apply to the functor ${{\cM\cC}} {'}^{all}$ as well. \end{say} \makeatletter \renewcommand{\@biblabel}[1]{\hfill#1.}\makeatother

9410

alg-geom/9410026

en

https://arxiv.org/abs/alg-geom/9410026

alg-geom/9410026

Severinas Zube

Exceptional vector bundle on Enriques surfaces

12 pages, LATEX

The main purpose in this paper is to study exceptional vector bundles on Enriques surfaces.

alg-geom

\subsection{ Introduction} The purpose of this note is to study exceptional\ vector\ bundles on Enriques surfaces. Exceptional bundle E on a surface with irregularity $q=h^1(O_S)$ and geometric genus $p_g=h^2(O_S)$ is the bundle with the following properties:\\ $Ext^0(E,E)=\bbbc ,Ext^1(E,E)=q, Ext^2(E,E)=p_g$. On an Enriques surface Kim and Naie in [Ki1],[Ki2],[N] have been studied extremal bundles which are very similar to exceptional\ \ bundles. Extremal bundle on Enriques surface by definition is a simple $Ext^0(E,E)=\bbbc$, rigid $Ext^1(E,E)=\bbbc$ with the following condition:$Ext^2(E,E)=\bbbc$. From the Riemann-Roch theorem easily follows that any exceptional\ bundle has odd rank and any extremal even rank. In [Ki2], Kim characterized extremal bundles on Enriques surfaces. They exist only on nodal surfaces and satisfy $c_1^2=4n-2,c_2=n$ for $n>4$. They have also geometrical meaning. It turns out that the existence of extremal bundles is closely related to the embedding a general Enriques surface in the Grassmannian G(2,n+1).\\ The main result in this paper is to give the necessary and sufficient conditions for the existence exceptional\ bundles on Enriques surfaces. The statement is similar to the theorem 4 in [Ki2]. The proof fill a gap in the proof of this theorem 4. Also I give constructions of them by using some constructions of Enriques surfaces and by using modular operations (reflections) which is described in the last section. I think the reflection is very useful to construct and study moduli of sheaves on Enriques surfaces. \subsection{Enriques surfaces} A smooth irreducible surface $S$, such that $h^1 (O_S )=h^2 (O_S )=0 $ and $2K_S \sim O_S $, is called a Enriques surface. Recall that a divisor $D$ on a smooth surface $X$ is said to be $nef$ if $DC \geq 0$ for every curve $C$ on $X$ . The following useful properties will be used throughout, sometimes without explicit mention: (A)([C,D] Corollary 3.1.3) If $D$ is a $nef$ divisor and $D^2 > 0 $ , then $H^1 (O_S (-D)) = 0$ and $\chi(O(D)) -1 =dim|D| = \frac{D^2 }{2}$. \label{222} (B)([C,D] Proposition 3.1.4) If $ \mid D \mid $ has no fixed components, then one of the following holds: \nonumber \\ (i) $D^2 > 0$ and there exist an irreducible curve $C$ in $ \mid D \mid $. \nonumber \\ (ii) $D^2 =0$ and there exist a genus 1 pencil $ \mid P \mid $ such that $D \sim kP$ for some $k \geq 1 $. \label{333} (C)([C,D]Chapter 4, appendix , corollary 1. and corollary 2.) If $D^2 \geq 6$ and $D$ is $nef$ then $D$ is ample , $2D$ is generated by its global sections, $3D$ is very ample. The Enriques surface $S$ is called nodal (resp. unnodal) if there are (resp. not) a smooth (-2)-curve contained in $S$. A general Enriques surface is an unnodal. Every Enriques surface admits an elliptic fibration over ${\rm I\!P} ^1$ with exactly two multiple fibers F, F' and an elliptic pencil $\mid 2F\mid =\mid 2F'\mid$ with $K_S=F-F'$. On a general Enriques surface there are ten different elliptic pencils $\mid 2F_1\mid ,\mid 2F_2\mid,..., \mid 2F_{10}\mid $ (see [CD]). \addtocounter{subsection}{1} \subsection{Mukai lattice} It is convenient to describe discrete invariants of sheaves and bundles on a K3 or an Enriques surface $X$ in the form of vectors in the algebraic Mukai lattice $$M(X) = H^0 (X,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) \oplus Pic X \oplus H^4 (X,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) = {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} \oplus Pic X \oplus {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} \ni v=(r,D,s)$$ with inner product $< , >$ $$<(r,D,s), (r',D',s')> = rs'+s'r-D.D'$$ To each sheaf $E$ on $X$ with $c_1(E) = D, c_2(E) \in H^4 (X, {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) = {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}$ we associate the vector $$v(E) = \left( rk(E),D, \frac{1}{2}D^2 - c_2 + rk(E) \frac{\chi(O_X )}{2}\right) , $$ where $\chi(O_X )=1-q+p_g $ is equal to 2 for a K3 surface and 1 for an Enriques surface. This formula,the Riemann-Roch theorem yield the equalities: \begin{eqnarray} <v(F),v(E)>=<v(E),v(F)>&=&\chi(E,F)\nonumber\\ = dim Ext^0 (E,F) - dim Ext^1 (E,F)&+& dim Ext^2 (E,F) \end{eqnarray} Because $K_X $ is numerically equal to zero we have that $\chi(E,F) = \chi (F,E)$ on a K3 or an Enriques surface. For the short exact sequence $$0 \to F \to E \to G \to 0$$ we have the following equalities $$v(E) = v(F) + v(G) = (r(F)+r(G), c_1 (F) +c_1 (G),s(F) + s(G)).$$ \addtocounter{section}{1} \subsection{Exceptional bundles} {\bf Definition:} E is an exceptional sheaf on surface S if:\\ ${~~~~~~~~~~~~~~}dim Ext^0 (E,E) = 1, dim Ext^1 (E,E) = q , dim Ext^2 (E,E) = p_g$.\newline From the Riemann-Roch theorem we have: $$\chi(E,E) = r^2 \chi(O_S )+(r-1)c_1^2 -2rc_2.$$ Since $H^2 (S,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} )$ is even lattice and $\chi(O_S)=1$ for an Enriques surface S we see that E has odd rank if E is an exceptional bundle. For the description of exceptional vector bundles we need the following result of Kuleshov. \newtheorem{ttt}{Theorem [Ku]}[subsection] \begin{ttt} Let X be a smooth a K3 surface and let $H$ be an arbitrary ample divisor on X, and $v=(r,D,s), r > 0$ is an exceptional vector (i.e $v^2 =2$) belonging to the Mukai lattice on X. Then there exist a simple , $\mu_H $-semi-stable bundle E which realize the vector v (i.e.v=v(E)). \end{ttt} I wish to start from some useful facts. The first one is about torsion free sheaves with some homological condition. \newtheorem{Mukai}{Proposition}[subsection] \begin{Mukai} Let {\it E} be a torsion free sheaf on a smooth surface S and $dim Ext^1 ({\it E,E}) =1$ or 0. Then {\it E} is locally free. \end{Mukai} {\bf Proof:} \ We have the following exact sequence: $$0 \to {\it E} \to {\it E}^{**} \to M \to 0, $$ where ${\it E}^{**}$ is double dual of {\it E} and cokernel M is of finite length. Now I use Mukai [M] result ( see Corollary 2.11 and 2.12) and obtain the following inequality: $$ dim Ext^1 ({\it E}^{**},{\it E}^{**}) + dim Ext^1 (M,M) \leq dim Ext^1 ({\it E},{\it E}) $$ Because $v^2 (M)=0$ we have that $ dim Ext^1 (M,M)$ is equal to $2dim End_{O_S} (M)$.\\ Since $Ext^1 ({\it E},{\it E})=0$ we obtain that M=0 and ${\it E} = {\it E}^{**}$. And hence ${\it E}$ is locally free the statement follows. $\odot$ Now I wish formulate the main result. \newtheorem{exc}[ttt]{Theorem \begin{exc} Let S be a smooth Enriques surface S, $v=(r,D,s) \in M(S)\\ ( r > 0 )$ and $v^2=1$ then: \\ (i)\ There is an ample divisor H such that $D\cdot H$ and r have not common divisor greater than 1 (i.e. $(D\cdot H,r) = 1$). \\ (ii)\ For any ample divisor H with condition $(D\cdot H,r)=1$ exist an exceptional vector bundle E and only one such that v(E)=v and E is H-stable. \end{exc} {\bf Proof:}\ \ (i) Because $v^2=1$ we have $2rs-D^2=1$. This means that $D \not\subset r'\cdot H^2 (S,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} ) $ for any $r'$ such that $( r',r ) > 1 $ (here ( , ) means the greatest common divisor). Since our lattice $H^2 (S,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} ) $ is unimodular there is $X \in H^2 (S,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} ) $ such that $(X\cdot D,r)=1$. Hence $H_k =X + krH$ will be very ample for any ample divisor H and $k\gg 0$ . And $H_k$ satisfy our condition $(H_k \cdot D,r)=1$. This prove the first statement.\\ (ii) Let a universal covering space of S be X which is a K3 surface and let $\pi$ be the quotient map. Consider the vector $\hat v =(r,D',s')=\pi ^*(v)$ on K3 surface X. It turns out that $\hat v ^2=2$ so by theorem of Kuleshov there is an exceptional vector bundle F on X such that $\hat v (F)=\hat v $ and F is $\hat H = \pi ^*(H)$-semi-stable. In fact F is $\hat H $-stable. Indeed, we have $(\hat H \cdot D',r) =(2H\cdot D,r) = 1$ (recall that r is even number as we notice above) and for any subsheaf W with $0 < rank(W) < rank(F) $ the following: $$\frac{c_1 (W)\cdot \hat H }{rank(W)}<\frac{D'\cdot \hat H }{rank(F)},$$ therefore F is $\hat H$ -stable . So F and $\sigma ^* (F)$ both are $\hat H$ -stable, where $\sigma$ is the free involution on X such that $X/\sigma = S.$ It is easy to see that $\chi(F, \sigma ^* (F)) = 2$. Hence there is non zero homomorphism from F to $\sigma ^* (F)$ which should be isomorphism because both vector bundles have the same determinant and are $\hat H$ -stable. This isomorphism means that there is a vector E on the surface S such that $\pi ^*(E)=F$. Of course, the vector bundle E is H-stable and v(E)=v. Assume that there is another H-stable vector bundle $\hat E$ and $v(E)=v(\hat E )$. Then $\chi(E,\hat E )=1$, therefore there is non trivial map from $\phi:E\to \hat E $ or by Serre duality $\rho:\hat E \to E\otimes K$. In both cases it should be isomorphisms by stability assumption. But then $E=\hat E$ or $det\rho$ gives non zero element of canonical class. This contradiction prove that H-stable bundle is unique.$\odot$ \\ {\bf Notice} that from condition $Ext^1(E,E)=0$ we get only that moduli space of E contain only discrete set of bundles. But by (ii) such E is only one so moduli space consist of only one point. \paragraph{Remark}: It is not clear (to my ) whether an exceptional H-stable vector bundle E is G-stable for any another ample divisor G. By theorem it can happen only if $(G\cdot c_1 (E),rk(E)) > 1$. \addtocounter{section}{1} \subsection{Examples} I wish to give some explicit examples of exceptional\ \ vector\ \ bundles. The one way to construct them is as in the theorem above to find a stable, invariant, exceptional\ vector\ bundle on a K3 surface. Consider Horikawa's representation of Enriques surface (see for details [BPV]). We introduce coordinates $(z_0:z_1:z_2:z_3)$ on ${{\rm I\!P} ^3}$ such that a quadric $Q={\rm I\!P} ^1 \times {\rm I\!P} ^1$ is embedded by $$z_0=x_0y_0,\ \ z_1=x_1y_1,\ \ z_2=x_0y_1,\ \ z_3=x_1y_0.$$ If we define the involution $\tau$ on ${{\rm I\!P} ^3}$ by $\tau (z_0:z_1:z_2:z_3)=(z_0:z_1:-z_2:-z_3)$, then Q is $\tau$ invariant with $\tau$ acting on Q by $$\tau ((x_0:x_1)(y_0:y_1)=(x_0:-x_1)(y_0:-y_1).$$ On Q the involution $\tau$ has the four fixed points $$(x_0:x_1)(y_0:y_1)=(1:0)(1:0) , (1:0)(0:1),(0:1)(0:1) , (0:1)(1:0).$$ Take a polynomial of bidegree (4,4) which define a $\tau$ invariant curve B. Assume that B have not any of fixed points. Consider a surface X which is a double cover of Q ramified over B. It turns out that X is a K3 surface and the involution $\tau$ induce the involution $\sigma$ on X which is without fixed points. Hence $X/\sigma =S$ is an Enriques surface. Notice that general Enriques surface can be obtain by this construction. Let $\pi :X \to S$ be factorization map and $\phi :X \to Q$ the double cover ramified over B. If E is exceptional\ vector\ bundle on Q then $\phi ^* (E)=\hat E $ is an exceptional\ . Indeed , by projection formula we have $H^i (End_{O_X}(\hat E ) = H^i (End_{O_Q}(E) \oplus H^i (End_{O_Q}(E) \otimes (-K_Q))$ (recall that $B=-2K_Q$). Since, by the Serre duality, we have $H^i (End_{O_Q}(E) \otimes (-K_Q))=H^{2-i}(End_{O_Q}(E))$ therefore $\hat E$ is an exceptional\ \ vector\ \ bundle on the K3 surface X. This bundle $\hat E$ is $\sigma$ invariant because E is $\tau$ invariant on quadric Q. (Indeed, E is rigid, therefore E is $PGL(2)\times PGL(2)$-homogeneous.) Because $\hat E$ is $\sigma$ invariant, there is a vector\ \ bundle F on S such that $\hat E =\pi ^* (F)$. It is easy to see that $v^2 (F)=1$ and F is rigid, therefore F is exceptional\ \ vector\ \ bundle on Enriques surface S. I am not able to say anything about the stability of $\hat E$ and F. This produce a lot of exceptional\ \ vector\ bundles because we know how to construct all exceptional\ \ vector\ \ bundles on smooth quadric Q. In the similar way we can consider quartic X in ${\rm I\!P} ^3$ which is defined by the equation $z_0^4+z_1^4-z_2^4-z_3^4$. This is a smooth K3 surface (see for details in [GH]). Let T be an automorphism on ${\rm I\!P} ^3$ defined as follows: $$T: (z_0,z_1,z_2,z_3) \to (z_0,\sqrt{-1}z_1,-z_2,-\sqrt{-1}z_3).$$ This automorphism T has 4 fixed points on ${\rm I\!P} ^3$ no one of which lay on the surface X , $T^2$ has 2 fixed lines: $$l_1=(z_0=z_2=0), \ l_2=(z_1=z_3=0). $$ These lines intersect the surface X in 8 points $p_1,....,p_8.$ Consider blow-up of X in these 8 points ${\bar X} \to X$. Let ${\bar T}$ denote induced automorphism on ${\bar X}$. It turns out that $X'={\bar X}/\{ {\bar T}^{2n} \} $ is a K3 surface and $\bar T$ acts on $X'$ as an involution without fixed points. So $X'/\bar T$ is an Enriques surface S. Now we can get an exceptional\ \ vector\ \ bundle on S from any exceptional\ \ vector\ bundle on ${\rm I\!P} ^3$ because each exceptional\ \ vector\ \ bundle on ${\rm I\!P} ^3$ is a homogeneous, therefore it is T invariant. Of course, we get the vector\ bundle on $X'$ which is an exceptional\ and the bundle descend to S also as an exceptional\ . This procedure gives us a lot exceptional\ \ vector\ \ bundles on S and we can describe it because we know constructions of exceptional\ bundles on ${\rm I\!P} ^3$. Now I wish discuss about the ability to construct exceptional collections on Enriques surface. Recall that by definition ${E_1,E_2,...,E_n}$ is an exceptional\ \ collection if $Ext^i (E_k,E_j)=0$ for any i and $k>j.$ In particular, we have that $\chi (E_k,E_j)=0$. But on an Enriques surface we have $\chi (E,F)=\chi (F,E)$ for any sheaves E and F. Hence for an exceptional\ \ collection on Enriques surface should be true the following: $$ Ext^i (E_k,E_j)=0 , \forall \ i \ and\ k>j;\ \ \ \ \chi (E_a,E_b)=0\ if\ a\not= b.$$ On a general Enriques surface exist exceptional\ \ colection with ten bundles. Indeed, it is well known that on general Enriques surface there are ten different elliptic pencils say $\mid 2F_1 \mid ,...,\mid 2F_{10} \mid $ (see [CD]). It is easy to see that $Ext^i (F_k,F_j)=0 , \forall \ i \ and\ k\not= j,$ therefore ${F_1,F_2,...,F_{10}}$ is an exceptional\ \ collection. It will be very interesting to describe the orthogonal category in the derived category D(S) (which is finite ) of all sheaves on Enriques surface S. This orthogonal category should have only two independent elements. \addtocounter{section}{1} \subsection{ Modular operations } There are some natural modular operations from one moduli space to another which gives an isomorphism of tangent bundles of moduli spaces.For example $E\longleftrightarrow E^*, E \longleftrightarrow E\otimes D$, where D is a line bundle. On an Enriques surface we have the very interesting modular operation which I call a reflection. This operation is similar to the reflection on a K3 surface (see [T] 4.10,4.11). I wish to describe it. First of all this reflection acts on Mukai lattice in the following way: $$v=(r,D,s) \longleftrightarrow R(v)=\hat v =\left( 2s,D+\left( s+\frac{r}{2}\right) K_S,\frac{r}{2}\right).$$ Notice that $v^2 = 2rs-D^2 = \hat v ^2 $.\\ Now I describe it on the level of sheaves. Assume that a torsion free sheaf E is generated by its section,v(E)=v=(r,D,s) and $\chi(E\otimes K)=h^0 (E\otimes K),\\ h^1 (E\otimes K)= h^1 (E) = 0$. Notice that from this we have $h^2 (E\otimes K)=\\ h^2 (E) = 0, h^0 (E)=\chi (E)=\chi (E\otimes K)$. For example, if we twist any stable bundle by sufficiently large very ample divisor then our condition will be satisfied. Consider the following exact sequence : \begin{equation} 0 \to \bar E^* \to H^0 (E)\otimes O_S\stackrel{ev}{ \to } E \to 0 , \label{ebundle} \end{equation} where $ev: H^0 (E)\otimes O_S \to E \to 0$ is the canonical evaluation map (surjective by assumption). For the convenience denote $H=H^0 (E),\ h=dim H$ and consider the dual sequence: \begin{equation} 0 \to E^* \to H^* \otimes O_S \to \bar E \to 0. \label{edual} \end{equation} By our conditions and Serre duality we have that $$h^1 (\bar E )=h^2 (E^* )=h^0 (E \otimes K) =\chi (E\otimes K)=h.$$ Consider the following sequence: \begin{equation} 0\to H^1(\bar E ) \otimes K_S \to \hat E \to \bar E \to 0, \label{extention} \end{equation} where $\hat E$ is given by universal extension element $id \in Ext^1 (\bar E, H^1(\bar E )) = End(H^1(\bar E )).$ Denote $R(E)=\hat E $. As an easy consequence of two sequences (\ref{ebundle}) and (\ref{extention}) our assumptions and Serre duality is the following \newtheorem{PP}{Proposition}[subsection] \begin{PP} Assume E is the sheaf as above then sheaves $\bar E$ and $\hat E $ from sequences (\ref{ebundle}) and (\ref{extention}) satisfies the following properties: \\ 1. ${\bar E}$ is globally generated by sections.\\ 2. $\chi(\bar E)=0,\ h^0 (\bar E)=h=h^1 (\bar E), \ h^2 (\bar E)=0.$\\ 3. $\chi(\bar E\otimes K)=0,\ h^i (\bar E\otimes K_S)=0, \ for \ \forall i>0$.\\ 4. $v(\hat E )=\hat v =R(v)= \left( 2s,D+\left( s+\frac{r}{2}\right) K_S,\frac{r}{2}\right) \ and\\ h^0 ( \hat E )= h^0 ( \hat E \otimes K)=h,\ h^i ( \hat E )= h^i ( \hat E \otimes K)=0 \ for \ \forall i>0$.\\ 5.$Hom(E,E)=Hom(\bar E,\bar E)=Hom(\hat E ,\hat E ),Ext^2(\bar E,\bar E)=0.$\\ 6. $rank(E)=rank(\hat E)\pmod{2}$. Moreover, if $rank(E)=2k+1$ and both E and $\hat E$ are H-stable then $Ext^2(\hat E ,\hat E )=Ext^2(E,E)=0,Ext^1(\hat E ,\hat E )=Ext^1(E,E)$. \label{PP} \end{PP} {\bf Proof:} \ \ 1. The sheaf $H^* \otimes O_S$ in the middle of the sequence (\ref{edual}) is globally generated by sections so $\bar E$ is too.\\ \noindent 2.The corresponding long in cohomology to the sequence (\ref{edual}) gives us $h^0 (\bar E)=h$ because by Serre duality $h^0 (E^*)=h^2 (E\otimes K)=0$ and $h^1 (E^*)=h^1 (E\otimes K)=0$. Also $h^2(\bar E)=0$ and $h^1 (\bar E)=h$ as we already noticed. Hence $\chi(\bar E)=0.$\\ \noindent 3.Consider the sequence (\ref{edual}) twisted by K: \begin{equation} 0 \to E^*\otimes K \to H^* \otimes K_S \to \bar E\otimes K \to 0. \label{etwistK} \end{equation} In the same way the corresponding long exact sequence in cohomology gives: $h^0(\bar E \otimes K)=h^1( E^* \otimes K)=h^1( E) $ and $\ h^1(\bar E \otimes K)=0$ because $h^2(E^* \otimes K)=h^0(E)=h=h^2(H^*\otimes K);\ h^2(\bar E\otimes K) =h^0(\bar E^*)$ (by the sequence (\ref{ebundle})).\\ \noindent 4. An easily calculation shows that $v(\hat E )=\hat v =\left( 2s,c_1(E)+\left( s+\frac{r}{2}\right) K_S,\frac{r}{2}\right).$ Since the sequence (\ref{extention}) is the universal extension we have $h^1( \hat E )=h^2( \hat E )=0$, therefore, by the Riemann-Roch theorem, we obtain that $h^0( \hat E )=h$. If we twist (\ref{extention}) by K and then use properties of $\bar E\otimes K$ we easily get that\\ $h^0( \hat E \otimes K)=h,\ h^1(\hat E \otimes K)=h^2(\hat E \otimes K)=0$. \\ \noindent 5. Applying $Hom(\bar E,*)$ to (\ref{edual}) we get the long exact sequence: \begin{eqnarray*} 0\to Ext^0(\bar E,E^*)\to & Ext^0(\bar E,H^*\otimes O_S) & \to Ext^0(\bar E,\bar E) \to \\ \to Ext^1(\bar E,E^*)\to & Ext^1(\bar E,H^*\otimes O_S) & \to Ext^1(\bar E,\bar E) \to \\ \to Ext^2(\bar E,E^*)\to & Ext^2(\bar E,H^*\otimes O_S) & \to Ext^2(\bar E,\bar E) \to0 \end{eqnarray*} By Serre duality and the statement 3 Ext groups in the middle are\\ $H^i(\bar E\otimes K)\otimes H^* =0$. Hence we have $ Ext^1(\bar E,E^*)= Ext^0(\bar E,\bar E) ,\\ Ext^2(\bar E,\bar E) =0.$ Also applying $Hom(*,E^*) $ to (\ref{edual}) we get: \begin{eqnarray*} 0\to Ext^0(\bar E,E^*)\to & Ext^0(H^*\otimes O_S,E^*) & \to Ext^0(E^*,E^*) \to \\ \to Ext^1(\bar E,E^*)\to & Ext^1(H^*\otimes O_S,E^*) & \to Ext^1(E^*,E^*) \to \end{eqnarray*} By our assumption and Serre duality the middle Ext groups are \\ $Ext^0(H^*\otimes O_S,E^*)=H^2(E\otimes K) = 0 = Ext^1(H^*\otimes O_S,E^*)=H^1(E\otimes K)$.\\ Hence $Hom(E^*,E^*)=Hom(E,E)=Hom(\bar E,\bar E).$ Now applying $Hom(\hat E ,*)$ to (\ref{extention}) we get: \begin{eqnarray*} 0\to Ext^0(\hat E ,H\otimes K)\to & Ext^0(\hat E ,\hat E ) & \to Ext^0(\hat E ,\bar E) \to \\ 0\to Ext^1(\hat E ,H\otimes K)\to & Ext^1(\hat E ,\hat E ) & \to Ext^1(\hat E ,\bar E) \to \end{eqnarray*} Since $Ext^0(\hat E ,H\otimes K)=H^2(\hat E )\otimes H=0$ we obtain that $Ext^0(\hat E ,\hat E )= Ext^0(\hat E ,\bar E)$. And, in the similar way, after applying $Hom(*,\bar E)$, we get that $Ext^0(\hat E ,\hat E )= Ext^0(\bar E,\bar E)$. 6.If rank(E) is odd then $s(E)\in \frac{1}{2}{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}$ but $s(E)\not\subset {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} $, therefore $rank(\hat E)\\ = 2s(E)$ is odd number too. If rank(E) is even then $s(E)\in {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}$, therefore $rank(\hat E) = 2s(E)$ is even.Consider any non zero element $\phi$ in $Ext^2(E,E)=\\Ext^0(E,E\otimes K)^*$ by the stability assumption this should be an isomorphism but then $det\phi$ is non zero element of canonical class. This contradiction shows that $Ext^2(E,E)=0$ and $Ext^2(\hat E ,\hat E )=0$. Because $v^2=\hat v ^2$ we have $Ext^1(E,E)=Ext^1(\hat E ,\hat E )$.$\odot$\\ I am able to reverse this operation in the following situation. Consider a sheaf F and the following exact sequence: \begin{equation} 0\to H^0( F\otimes K_S ) \otimes K_S \stackrel{ev}{\to} F \to \bar F \to 0, \label{fx} \end{equation} where $ ev:H^0( F\otimes K_S ) \otimes K_S \to F$ is the canonical evaluation map. Assume that $\bar F$ is globally generated a torsion free sheaf then we have the following exact sequence: \begin{equation} 0 \to \hat F ^* \to H^0 (\bar F) \otimes O_S \stackrel{ev}{\to} \bar F \to 0. \label{ff} \end{equation} Denote $R(F)=\hat F$. Under these assumptions we can prove the similar result: \newtheorem{P'}[PP]{Proposition \begin{P'} Assume F is the sheaf as above then sheaves $\bar F$ and $\hat F$ from sequences (\ref{ff}) and (\ref{fx}) satisfies the following properties: \\ 1. $\chi(\bar F)=0,\ h^0 (\bar F)=h=h^1 (\bar F), \ h^2 (\bar F)=0.$\\ 2. $\chi(\bar F\otimes K)=0,\ h^i (\bar F\otimes K_S)=0, \ for \ \forall i>0$.\\ 3. $v(\hat F )=\hat v =\left( 2s,D+\left( s+\frac{r}{2}\right) K_S,\frac{r}{2}\right) \ and\ h^0 ( \hat F )= h^0 ( \hat F \otimes K)=h,\\ \ \ \ h^i ( \hat F )= {~~}h^i ( \hat F \otimes K)=0 \ for \ \forall i>0$.\\ 4.$Hom(F,F)=Hom(\bar F,\bar F)=Hom(\hat F ,\hat F ),Ext^2(\bar F,\bar F)=0.$\\ 5. $rank(F)=rank(\hat F)\pmod{2}$. Moreover, if $rank(F)=2k+1$ and both F and $\hat F$ are H-stable then $Ext^2(\hat F ,\hat F )=Ext^2(F,F)=0,Ext^1(\hat F ,\hat F )=Ext^1(F,F)$. \label{P'} \end{P'} Notice that from both propositions follows that R(R(E))=E. \paragraph{Remark:} The reflection always exist in the derived category of sheaves on surface S. It does not matter whether the evaluation map $ H^0( F ) \otimes O_S \stackrel{ev}{\to} F$ is a surjective or an injective map.\\ Now I wish to give a few examples of reflections.\\ 1. Assume we have a smooth curve C on S, A a globally generated divisor on the curve C with the properties $h^1 (O_C(A))=h^1(O_C(A\otimes K_S))=0$. We consider the following exact sequence: \begin{equation} 0 \to E(C,A)^* \to H^{0}(O_C(A)) \otimes O_S\stackrel{ev}{\to} O_{C}(A) \to 0 \label{E*} \end{equation} The dual sequence to (\ref{E*}) is : \begin{equation} 0 \to H^{0}(A)^{*} \otimes O_S \to E(C,A) \to O_{C}(C) \otimes A^* \to 0 \end{equation} By our assumption, this sequence and Serre duality on the curve C, we obtain that $h^1(E(C,A))=h^1(O_C(C-A))=h^0(O_C(A)$ and get the following sequence: \begin{equation} 0 \to H^1(E(C,A))\otimes K_S \to \hat E \to E(C,A) \to 0 \end{equation} So we have $R(O_C(A))=\hat E $. \\ If C is (-2)-curve and $A=O_C$, we see that $R(O_C)=F$ is an extremal rank 2 vector bundle (i.e. $Ext^0(F,F)=Ext^2(F,F)=\bbbc, Ext^1(F,F)=0.)$ 2.If $\mid 2F\mid,\ \mid 2G\mid$ are two elliptic pencils on S then a pencil $\mid F+G\mid$ has two different base points x and y. From the standard sequence: \begin{equation} 0 \to J_{x+y} (F+G) \to O(F+G)\to O_{x+y}(F+G)\to 0, \label{jj} \end{equation} we see that $h^1(J_{x+y} (F+G))=2$. This gives us the following bundle E, defined by the universal extension element $id\in End(H^1(J_{x+y} (F+G))$: \begin{equation} 0 \to H^1(J_{x+y} (F+G)) \otimes K_S \to E\to J_{x+y} (F+G) \to 0, \label{ej} \end{equation} By lemmas 1.1,1.2 in [T], E is a simple bundle. An easily calculation shows that E is an exceptional\ \ bundle. Also by \ref{P'}, we see that $R(E)=O(F+G)$. Notice that $O(F+G)$ is not globally generated by section, therefore we cannot use \ref{PP} to this bundle to produce $R(O(F+G))$, but we can do this in the derived category. Since $R(E)=O(F+G)$, we also obtain that $R(O(F+G))=E$. 3.If we consider a divisor $O(aF+bG)$ for $a\geq b\geq 2$ then this divisor will be an ample on a general Enriques surface. Hence $R(O(aF+bG))$ is an exceptional\ \ bundle of rank $2ab+1$. \paragraph{References} : \newline [BPV] W.Bart, C.Peter, A.Van de Ven. Compact complex surfaces. Berlin, Heidelberg,New York :Springer 1984. \newline [CD] F. Cossec ; I. Dolgachev "Enriques Surfaces 1", Birkh\"{a}user 1989. \newline [GH] Griffiths,Ph., Harris J.,Principles of algebraic geometry, New York\newline (1978) \newline [Ki1] Kim,Hoil.:Exceptional bundles on nodal Enriques surfaces,\newline Bayreuth preprint(1991). \newline [Ki2] Kim,Hoil.:Exceptional bundles and moduli spaces of stable vector \newline bundles on Enriques surfaces,Bayreuth preprint(1991). \newline [Ku] Kuleshov,S.A:An existence theorem for exceptional\ \ bundles on K3 \newline surfaces,Math.USSR Izvestia,vol.34,373-388.(1990). \newline [M] Mukai,S.On the moduli space of bundles on K3 surfaces I, in Vector \newline Bundles, ed. Atiay et all, Oxford University Press, Bombay, 341-413(1986). \newline [N] Daniel Naie. Special rank two vector bundles over Enriques surfaces, \newline preprint. \newline [T] A.N. Tyurin "Cycles, curves and vector bundles on algebraic surfaces." Duke Math.J. 54, 1-26,(1987). \newline Department of geometry and topology, Faculty of mathematics, Vilnius university, Naugarduko g.24, 2009 Vilnius, Lithuania. e-mail:[email protected] \end{document}

9410

alg-geom/9410014

en

https://arxiv.org/abs/alg-geom/9410014

alg-geom/9410014

Dmitri Zaitsev

On the linearization of the automorphism groups of algebraic domains

10 pages, LaTeX

Let $D$ be a domain in $C^n$ and $G$ a topological group which acts effectively on $D$ by holomorphic automorphisms. In this paper we are interested in projective linearizations of the action of $G$, i.e. a linear representation of $G$ in some $C^{N+1}$ and an equivariant imbedding of $D$ into $\P^N$ with respect to this representation. The domains we discuss here are open connected sets defined by finitely many real polynomial inequalities or connected finite unions of such sets. Assume that the group $G$ acts by birational automorphisms. Our main result is the equivalence of the following conditions: 1) there exists a projective linearization, i.e. a linear representation of $G$ in some $\C^{N+1}$ and a biregular imbedding $i\colon \P^n \hookrightarrow \P^N$ such that the restriction $i|_D$ is $G$-equivariant. 2) $G$ is a subgroup of a Lie group $\hat G$ of birational automorphisms of $D$ which extends the action of $G$ and has finitely many connected components; 3) $G$ is a subgroup of a Nash group $\hat G$ of birational automorphisms of $D$ which extends the action of $G$ to a Nash action $\hat G\times D\to D$; 4) $G$ is a subgroup of a Nash group $\hat G$ such that the action $G\times D\to D$ extends to a Nash action $\hat G\times D\to D$; 5) the degree of the automorphism $\phi_g\colon D\to D$

alg-geom

\section{Linearization Theorem and applications} Let $D$ be a domain in $\C^n$ and $G$ a topological group which acts effectively on $D$ by holomorphic automorphisms. In this paper we are interested in projective linearizations of the action of $G$, i.e. a linear representation of $G$ in some $\C^{N+1}$ and an equivariant imbedding of $D$ into $\P^N$ with respect to this representation. Since $G$ acts effectively, the representation in $\C^{N+1}$ must be faithful. In our previous paper~\cite{Z}, however, we considered an example of a bounded domain $D\subset\C^2$ with an effective action of a finite covering $G$ of the group $SL_2(\R)$. In this case the group $G$ doesn't admit a faithful representation. The example shows that a linearization in the above sense doesn't exist in general. In the present paper we give a criterion for the existence of the projective linearization for birational automorphisms. The domains we discuss here are open connected sets defined by finitely many real polynomial inequalities or connected finite unions of such sets. These domains are called {\it algebraic}. For instance, in the above example the domain $D$ is algebraic. \begin{Def}\label{Nash} \begin{enumerate} \item A {\bf Nash map} is a real analytic map $$f=(f_1,\ldots,f_m)\colon U\to \R^m$$ (where $U\subset\R^n$ is open) such that for each of the components $f_k$ there is a non-trivial polynomial $P_k$ with $$P_k(x_1,\ldots,x_n,f_k(x_1,\ldots,x_n))=0$$ for all $(x_1,\ldots,x_n)\in U$. \item A {\bf Nash manifold} $M$ is a real analytic manifold with finitely many coordinate charts $\phi_i\colon U_i\to V_i$ such that $V_i\subset\R^n$ is Nash for all $i$ and the transition functions are Nash (a Nash atlas). \item A {\bf Nash group} is a Nash manifold with a group operation $(x,y)\mapsto xy^{-1}$ which is Nash with respect to all Nash coordinate charts. \end{enumerate} \end{Def} In the above example the group $SL_2(\R)$ and its finite covering $G$ are Nash groups (The universal covering of $SL_2(\R)$ is a so-called locally Nash group). Moreover, the action $G\times D\to D$ is also Nash. Since the linearization doesn't exist here, we need a stronger condition on the action of $G$. A topological group $G$ is said to be a {\it group of birational automorphisms} of a domain $D\subset\C^n$ if we are given an effective (continuous) action $G\times D\to D$ such that every element $g\in G$ defines an automorphism of $D$ which extends to a birational automorphism of $\C^n$. By the {\it degree} of a Nash map $f$ we mean the minimal natural number $d$ such that all polynomials $P_k$ in Definition~\ref{Nash} can be chosen such that their degrees don't exceed $d$. Finally, under a {\it biregular} map between two algebraic varieties we understand an isomorphism in sense of algebraic geometry. Our main result is the following linearization criterion. It will be proved in section~\ref{proof}. \begin{Th}\label{main} Let $D\subset\C^n$ be a algebraic domain and $G$ a group of birational automorphisms of $D$. The following properties are equivalent: \begin{enumerate} \item $G$ is a subgroup of a Lie group $\hat G$ of birational automorphisms of $D$ which extends the action of $G$ and has finitely many connected components; \item $G$ is a subgroup of a Nash group $\hat G$ of birational automorphisms of $D$ which extends the action of $G$ to a Nash action $\hat G\times D\to D$; \item $G$ is a subgroup of a Nash group $\hat G$ such that the action $G\times D\to D$ extends to a Nash action $\hat G\times D\to D$; \item the degree of the automorphism $\phi_g\colon D\to D$ defined by $g\in G$ is bounded; \item there exists a projective linearization, i.e. a linear representation of $G$ in some $\C^{N+1}$ and a biregular imbedding $i\colon \P^n \hookrightarrow \P^N$ such that the restriction $i|_D$ is $G$-equivariant. \end{enumerate} \end{Th} We finish this section with applications of Theorem~\ref{main}. In the previous paper~(\cite{Z}) we gave sufficient conditions on $D$ and $G$ such that $G$ is a Nash group and the action $G\times D\to D$ is Nash. The condition on $D$ is to be bounded and to have a non-degenerate boundary in the following sense. \begin{Def}\label{deg} A boundary of a domain $D\subset\C^n$ is called {\bf non-degenerate} if it contains a smooth point where the Levi-form is non-degenerate. \end{Def} The group $G$ is taken to be the group $Aut_a(D)$ of all holomorphic Nash (algebraic) automorphisms of $D$. We proved in~\cite{Z} that, if $D$ is a algebraic bounded domain with non-degenerate boundary, the group $Aut_a(D)$ is closed in $Aut(D)$ and carries a unique structure of a Nash group such that the action $Aut_a(D)\times D\to D$ is Nash with respect to this structure. Now let $G=Aut_b(D)\subset Aut_a(D)$ be the group of all birational automorphisms of $D$. Then $G$ satisfies the property~3 in Theorem~\ref{main} with $\hat G = Aut_a(D)$. By the property~2, $G$ is a subgroup of a Nash group of birational automorphisms of $D$. Since $G$ contains all the birational automorphisms of $D$, $G$ is itself a Nash group with the Nash action on $D$. We obtain the following corollary. \begin{Cor}\label{aut-b} Let $D\subset\subset\C^n$ be a bounded algebraic domain with non-degenerate boundary. Then the group $Aut_b(D)$ of all birational automorphisms of $D$ is Nash with the Nash action on $D$ which admits a projective linearization, i.e. there exist a representation of $Aut_b(D)$ in some $\C^{N+1}$ and a biregular imbedding $i\colon \P^n \hookrightarrow \P^N$ such that the restriction $i|_D$ is $Aut_b(D)$-equivariant. \end{Cor} Furthermore, S.~Webster (see \cite{W}) established the following sufficient conditions on $D$ which make its automorphisms birational. Let $D$ be a algebraic domain. The theory of semialgebraic sets (see Benedetti-Risler~\cite{BR}) implies that the boundary $\partial D$ is contained in finitely many irreducible real hypersurfaces. Several of them, let say $M_1,\ldots,M_k$, have generically non-degenerate Levi forms. If $\partial D$ is non-degenerate in sense of Definition~\ref{deg}, such hypersurfaces exist. The complexifications ${\cal M}_i$'s of $M_i$'s are defined to be their complex Zariski closures in $\C^n\times\overline{\C^n}$ where $M_i$'s are totally real imbedded via the diagonal map $z\mapsto (z,\bar z)$. It follows that ${\cal M}_i$'s are the irreducible complex hypersurfaces. Furthermore, the so-called Segre varieties $Q_{iw}$'s, $w\in \C^n$ are defined by $$ Q_{iw} := \{z\in \C^n \mid (z, \bar w) \in {\cal M}_i \}.$$ The complexifications and Segre varieties are the important biholomorphic invariants of a domain $D$ and play a decisive role in the reflection principle. Now we are ready to formulate the conditions of S.~Webster. \begin{Def} A algebraic domain is said to satisfy the condition $(W)$ if for all $i$ the Segre varieties $Q_{iw}$ uniquely determine $z\in\C^n$ and $Q_{iw}$ is an irreducible hypersurface in $\C^n$ for all $z$ off a proper subvariety $V_i\subset\C^n$. \end{Def} The Theorem of S.~Webster (see \cite{W}, Theorem~3.5) can be formulated in the following form: \begin{Th}\label{bir} Let $D\subset\C^n$ be a algebraic domain with non-degenerate boundary which satisfies the condition $(W)$. Further, let $f\in Aut(D)$ be an automorphism which is holomorphically extendible to a smooth boundary point with non-degenerate Levi-form. Then $f$ is birationally extendible to the whole $\C^n$. \end{Th} Since every Nash automorphism $f\in Aut_a(D)$ extends holomorphically to generic boundary points, we obtain the following Corollary. \begin{Cor} Let $D\subset\C^n$ be a bounded algebraic domain which satisfies the condition $(W)$. Then the whole group $Aut_a(D)$ is projective linearizable, i.e. there exist a representation of $Aut_a(D)$ in some $\C^{N+1}$ and a biregular imbedding $i\colon \P^n \hookrightarrow \P^N$ such that the restriction $i|_D$ is $Aut_a(D)$-equivariant. \end{Cor} To obtain the extendibility of the whole group $Aut(D)$ of holomorphic automorphisms, we consider the {\it algebraic} domains in sense of Diederich-Forn\ae ss (see \cite{DF}). \begin{Def} A domain $D\subset\subset C^n$ is called {\bf algebraic} if there exists a real polynomial $r(z,\bar z)$ such that $D$ is a connected component of the set $$\{z\in\C^n \mid r(z,\bar z)<0 \}$$ and $dr(z)\ne 0$ for $z\in\partial D$. \end{Def} The following fundamental result for such domains is due to K.~Diederich and J.~E.~Forn\ae ss (see \cite{DF}). \begin{Th}\label{hol} Let $D\subset\subset\C^n$ be an algebraic domain. Then $Aut_a(D)=Aut(D)$. \end{Th} Thus we obtain the linearization of the whole automorphism group $Aut(D)$. \begin{Th}\label{alg} Let $D\subset\subset \C^n$ be an algebraic domain which satisfies the condition $(W)$. Then the group $Aut(D)$ is projective linearizable, i.e. there exist a representation of $Aut(D)$ in some $\C^{N+1}$ and a biregular imbedding $i\colon \P^n \hookrightarrow \P^N$ such that the restriction $i|_D$ is $Aut(D)$-equivariant. \end{Th} Further corollaries are devoted to the constructions of complexifications. \section{Complexifications} Using the linearization criterion we establish here existences of complexifications. To every real Lie group $G$ one can associate its complexification (see Hochschild~\cite{Ho}) defined as follows. \begin{Def}\label{cpx} Let $G$ be a real Lie group. A complex Lie group $G^{\C}$ together with a Lie homomorphism $\imath\colon G \to G^{\C}$ is called a {\bf complexification} of $G$ if for a given Lie homomorphism $\phi$ from $G$ into a complex Lie group $H$, there exists exactly one holomorphic Lie homomorphism $\phi^{\C}\colon G^{\C} \to H$ such that $\phi=\phi^{\C}\circ\imath$. A real Lie group $G$ is called {\bf holomorphically extendible} if the map $\imath\colon G\to G^{\C}$ is injective. \end{Def} A complexification always exists and is unique up to biholomorphisms (see Hochschild~\cite{Ho} and Heinzner~\cite{He} and \cite{He1}). Further, one defines the complexification of an action (see Heinzner~\cite{He}). \begin{Def}\label{G-cpx} Let a real Lie group $G$ act on a complex space $X$ by holomorphic automorphisms. A complex space $X^{\C}$ together with a holomorphic action of $G^{\C}$ and a $G$-equivariant map $\imath\colon X\to X^{\C}$ is called a $G$-{\bf complexification} of $X$ if to every holomorphic $G$-equivariant map $\phi\colon X\to Y$ into another complex space $Y$ with a holomorphic action of $G^{\C}$ there exists exactly one holomorphic $G^{\C}$-equivariant map $\phi^{\C}$ such that $\phi=\phi^{\C}\circ\imath$. \end{Def} A $G$-complexification is unique up to biholomorphic $G^{\C}$-equivariant maps provided it exists. P.~Heinzner proved in \cite{He} the existence of a $G$-complexification of $X$ with properties that $\imath\colon X\to X^{\C}$ is an open imbedding and $X^{\C}$ is Stein in case $G$ is compact and $X$ is a Stein space. Now the projective linearization in Theorem~\ref{main} implies the existence of complexifications in our situation. \begin{Cor}\label{main1} Let $D\subset\C^n$ be a algebraic domain and $G$ a Lie group of birational automorphisms of $D$ which satisfies one of the equivalent properties in Theorem~\ref{main}. Then the group $G$ is holomorphically extendible and there exists a smooth $G$-complexification $D^{\C}$ of $D$ such the map $\imath\colon D\to D^{\C}$ is an open imbedding. \end{Cor} {\bf Proof.} By property~5 in Theorem~\ref{main}, $G$ is a subgroup of the complex Lie group $GL_N(\C)$. By Definition~\ref{cpx}, $G$ is holomorphically extendible. Let $i$ be the embedding of $\C^n\supset D$, given by Theorem~\ref{main}. Since the stability group $H\subset GL_N(\C)$ of the complex projective variety $X:=i(\P^n)$ is a complex Lie group and $G\subset H$, the Definition~\ref{cpx} yields a holomorphic action of $G^{\C}$ on $X$. We claim that $D^{\C}:=G^{\C}\cdot D\subset X$ is the required $G$-complexification of $D$. Indeed, let $\phi\colon D\to Y$ be a $G$-equivariant holomorphic map into another complex space $Y$ with a holomorphic action of $G^{\C}$. To define the required in Definition~\ref{G-cpx} map $\phi^{\C}$ we take a point $z\in D^{\C}$ which is always of the form $z=Ax$ with $A\in G^{\C}$ and $x\in D$. Then we set $\phi^{\C}(z):=A\phi(x)$. Why is $\phi^{\C}(z)$ independent of the representation $z=Ax$? Because the holomorphic map $A\mapsto A\phi(x)$ is determined by values on the maximal totally real subgroup $G$: for $A\in Aut_b(D)$ one has $A\phi(x)=\phi(Ax)$. We obtain a well-defined $G^{\C}$-equivariant map $\phi^{\C}(z)\colon D\to D^{\C}$ with the property $\phi=\phi^{\C}\circ\imath$ (because for $z\in D$ one can choose $A=1$). The holomorphicity of $\phi^{\C}$ is obtained by fixing $A$ in the formula $\phi^{\C}(z):=A\phi(x)$. \hfill $\square$ For the algebraic domains we obtain the following Corollaries. \begin{Cor} Let $D\subset\subset\C^n$ be a algebraic domain with non-degenerate boundary. Then the group $Aut_b(D)$ is holomorphically extendible and there exists an $Aut_b(D)$-complexification of $D$. \end{Cor} \begin{Cor} Let $D\subset\subset\C^n$ be a algebraic domain with non-degenerate boundary which satisfies the condition $(W)$. Then the group $Aut_a(D)$ is holomorphically extendible and there exists an $Aut_a(D)$-complexification of $D$. \end{Cor} \begin{Cor} Let $D\subset\subset \C^n$ be an algebraic domain which satisfies the condition $(W)$. Then the group $Aut(D)$ is holomorphically extendible and there exists an $Aut(D)$-complexification of $D$. \end{Cor} \section{Proof of the main Theorem}\label{proof} Let $D$ be a algebraic domain and $G$ a group of birational automorphisms of $D$. We prove the equivalence of the properties in Theorem~\ref{main} in the direction of the following two chains: $2\Longrightarrow 3\Longrightarrow 4\Longrightarrow 5\Longrightarrow 2$ and $2\Longrightarrow 1\Longrightarrow 4$. \fbox{$2\Longrightarrow 3$.} The proof is trivial. \hfill $\square$ \bigskip \fbox{$3\Longrightarrow 4$.} Let $G$ be a subgroup of a Nash group $\hat G$ such that the action $G\times D\to D$ extends to a Nash action $\hat G\times D\to D$. We prove the statement for arbitrary Nash manifold $\hat G$ and Nash map $\hat G\times D\to D$ by induction on $\dim G$. It is obvious for $\dim G=0$. Let $U\subset \hat G$ be a Nash coordinate chart and $\phi_i(g)\colon D\to \R$ be the $i$th coordinate of $\phi_g\colon D\to D$ for $g\in U$. Since the map $\phi_i\colon U\times D\to \R$ is Nash, it satisfies a polynomial equation $P(g,x,\phi_i(g,x))\equiv 0$. This yields polynomial equations of the same degree for all $g\in U$ outside a proper Nash submanifold. This submanifold has lower dimension and the statement is true for it by induction. In summary, we obtain the boundness of the degree for the whole neighborhood $U$ and, since the Nash atlas is finite, for $G$. \hfill$\square$ \bigskip \fbox{$4\Longrightarrow 5$.} Here is a sketch of the proof. The idea is to imbed the group $G$ into a complex algebraic variety so that the action on $D$ is given by a rational mapping. Using this mapping we construct a collection of homogeneous polynomials on $\C^{n+1}$ which generate a finite-dimensional linear subspace, invariant with respect to the action of $G$. These polynomials yield the required projective linearization. The imbedding of $G$ is obtained by associating to every element $g\in G$ the Chow coordinates of the complex Zariski closure of the graph of the automorphism defined by $g$ (see Shafarevich, \cite{S} ,page 65). The main problem here is that the Chow scheme $C$ has infinitely many disjoint components parameterized by dimensions and degrees of subvarieties. In order to concern finitely many components of $C$ we have required the degree of automorphisms $\phi_g\colon D\to D$ to be bounded. To every $g\in G$ one associates the $n$-dimensional (complex) Zariski closure $\tilde\Gamma_g\subset \P^n\times\P^n$ of the graph $\Gamma_g\subset D\times D$ of the automorphism defined by $g$. To regard $\tilde\Gamma_g$ as a subvariety of some $P^N$ let us consider the Segre imbedding: \begin{equation}\label{segre} v([z_0,...,z_n],[w_0,...,w_n])=[{z_iw_j}]_ {0 \le i \le n, 0 \le j \le n}, \end{equation} $$v\colon \P^n\times \P^n \to \P^{n^2+2n}.$$ We set $N=n^2+2n$ and obtain a family of subvarieties $\rho(g):=v(\tilde\Gamma_g)\subset \P^N$ parameterized by $g\in G$. The family $V$ of all algebraic subvarieties of $\P^n$ is parameterized by the Chow scheme $C$ (see Shafarevich, \cite{S} ,page 65). Different automorphisms $g\in G$ define different subvarieties $v(\tilde\Gamma_g)\subset \P^N$ and one obtains an imbedding $\rho$of $G$ in the Chow scheme $C$. The (complex) dimension of the subvarieties $v(\tilde\Gamma_g)$ is $n$. The degree of $v(\tilde\Gamma_g)\subset\P^N$ is the intersection number with $N-n$ generic linear hyperplanes $\{L_1=0\},\ldots,\{L_{N-n}=0\}$. It is equal to the intersection number of $\tilde\Gamma_g\subset\P^N\times\P^N$ with divisors $v^*L_1,\ldots,v^*L_{N-n}$. By the Bezout theorem this intersection number is bounded. Thus, $G$ lies in fact in finitely many components of the Chow scheme $C$. Let $C_0$ denote the union of these components and $V_0$ the corresponding family of subvarieties of $P^N$. We obtain an imbedding $\rho$ of $G$ in a complex projective variety $C_0$. \begin{Lemma} The imbedding $\rho\colon G\to C_0$ is continuous. \end{Lemma} {\bf Proof.} Assume the contrary. Then there exists a sequence $g_n\to g$ in $G$ such that no subsequence $\rho(g_{n(k)})$ converges to $\rho(g)$. On the other hand, since the degree of $\rho(g_n)$ is bounded, there exists a subsequence $\rho(g_{n(k)})$ which converges in $C_0$. This follows from the Theorem of Bishop (see e.g. F.~Campana, \cite{Cam}). Let $A\in C_0$ be the limes cycle of this subsequence. Our cycles lie in $v(\P^n\times\P^n)$ and we identify them with the preimages in $\P^n\times\P^n$. Since the action $G\times D\to D$ is continuous, the cycle $A$ contains the graph of $\phi_g$ and therefore its Zariski closure $\rho(g)$. We claim that $A=\rho(g)$. This yields a contradiction with the choice of $\rho(g_{n(k)})$. Indeed, otherwise there exists a horizontal of vertical $n$-dimensional projective subspace $H\subset\P^n\times\P^n$ ($H=\{z\}\times\P^n$ or $H=\P^n\times\{w\}$) such that the intersection number of $A$ and $H$ is more than one. Then the intersection number of $\rho(g_{n(k)})$ and $H$ is also more than one which contradicts to the birationality of $\phi_{g_{n(k)}}$. \hfill $\square$ Further let $\tilde G$ be the complex Zariski closure of $G$ in $C_0$. \begin{Lemma} The action $\phi\colon G\times D\to D$ extends to a rational map $\tilde\phi \colon \tilde G\times \P^n \to \P^n$. \end{Lemma} {\bf Proof.} We begin with the construction of the graph $\Gamma_{\tilde\phi} \subset \tilde G\times \P^n \to \P^n$ of $\tilde\phi$. For this we regard $\P^n \times \P^n$ as a subset of $P^N$ (via the Segre imbedding $v$ in (\ref{segre})). We then define $\Phi = \Gamma_{\tilde\phi}$ to be the intersection of the Chow family $V_0$ with $\tilde G\times \P^n \to \P^n$. This is a complex algebraic variety. Moreover, for $g\in G$ and $x\in D$ the fibre $\Phi_{(g,x)}\subset\P^n$ consists of the single point $g(x)$. Since the set $G\times D$ is Zariski dense in $\tilde G\times \P^n$, this is true for every generic fibre of $\Phi$. This means that $\Phi$ is the graph of a rational map $\tilde\phi \colon \tilde G\times \P^n \to \P^n$. \hfill $\square$ The projective variety $\tilde G$ is imbedded in a projective space $\P^m$. The map $\tilde\phi$ can be extended to a rational map from $\P^m \times \P^n$ into $\P^n$. Such map is given by $n+1$ polynomials $P_1(x,y),\dots,P_{n+1}(x,y)$, homogeneous separately in $x\in\C^{m+1}$ and $y\in\C^{n+1}$. Let $h$ be a fixed homogeneous polynomial on $\C^{n+1}$. Then the function $$(x,y)\mapsto h(P_1(x,y),\ldots,P_{n+1}(x,y))$$ is a separately homogeneous polynomial on $\C^{m+1} \times \C^{n+1}$. The algebra $\C_h[x,y]$ of such polynomials is equal to the tensor product $\C_h[x]\otimes\C_h[y]$. Therefore there exist polynomials $\varphi_i\in \C_h[x]$, $\psi_i\in \C_h[y]$, $i=1,\ldots,l$ such that $$h(P_1(x,y),\ldots,P_{n+1}(x,y)) = \sum_{i=1}^l \varphi_i(x) \psi_i(y).$$ For $x=g\in G$ fixed we obtain $$\alpha_*(f^{-1}) h = \sum_{i=1}^l c_i \psi_i(y),$$ where $\alpha_*$ denotes the associated action of $G$ on homogeneous polynomials. In other words, the orbit of $h$ via the action of $G$ is contained in the finite-dimensional subspace $<\psi_1,\ldots,\psi_l> \subset \C_h[y]$. The linear hull of this orbit is a finite-dimensional $G$-invariant subspace containing $h$. We choose mow sufficiently many polynomials $h_j$, $j=1,\ldots,s$ which separate the points of $\C^{n+1}$ and such that neither $h_j$ nor the differentials $dh_j$ nowhere vanish simultaneously. They lie in a finite-dimensional $G$-invariant subspace $L\subset \C_h[y]$. Let $(p_1,\ldots,p_{N+1})$ be a collection of homogeneous polynomials which yields a basis of $L$. The required representation of $G$ is the action on $L$ and the polynomial map $(p_1,\ldots,p_{u+1})\colon \C^{n+1}\to \P^N$ defines the required projective linearization. \hfill $\square$ \bigskip \fbox{$5\Longrightarrow 2$.} Assume we are given a projective linearization of the action of $G$ on $D$. It follows that the given representation of $G$ is faithful and we identify $G$ with its image in $GL_{N+1}(\C)$. We define now the group $\hat G\supset G$ to be the subgroup of all $g\in GL_{N+1}$ such that $g(i(D))=i(D)$. It follows that $G\subset\hat G$. We wish to prove that $\hat G$ is a Nash subgroup of $GL_{N+1}(\C)$. For the proof we use the technique of semialgebraic sets and maps which are closely related to the Nash manifolds and maps. The semialgebraic subsets of $\R^n$ are the sets of the form $\{P_1=\cdots=P_k, Q_1<0,\ldots,Q<s\}$ and finite unions of them where $P_1,\ldots,P_k$ and $Q_1,\ldots,Q_s$ are real polynomials on $\R^n$. More generally, the semialgebraic subsets of a Nash manifold $M$ are the subsets which have semialgebraic intersections with every Nash coordinate chart. The semialgebraic maps between semialgebraic sets are any maps with semialgebraic graphs. The Nash submanifolds of $\R^n$ are exactly semialgebraic real analytic submanifolds and the Nash maps are semialgebraic real analytic maps. Now the graph $\Gamma\subset Gl_{N+1}\times i(D) \to \P^N$ of the restriction to $i(D)$ of the linear action of $GL_{N+1}$ on $\P^N$ is a semialgebraic subset. The condition $g(i(D))=i(D)$ on $g$ defines a semialgebraic subset $\hat G\subset Gl_{N+1}$. We proved this in the previous paper (see \cite{Z}, Lemma~6.2). Since $\hat G$ is a subgroup, it is Nash. Thus, $G$ in a subgroup of the Nash group $\hat G$ of birational automorphisms of $i(D)\cong D$ with required properties.\hfill $\square$ \bigskip \fbox{$2\Longrightarrow 1$.} A Nash open subset of $\R^n$ is semialgebraic and has therefore finitely many connected components (see Benedetti-Risler, \cite{BR}, Theorem~2.2.1). The Nash group $\hat G$ admits a finite Nash atlas and has also finitely many components.\hfill $\square$ \bigskip \fbox{$1\Longrightarrow 4$.} Assume $G$ is a subgroup of a Lie group $\hat G$ of birational automorphisms of $D$ with finitely many connected components. Consider the complex coordinates $\phi_i\colon \hat G\times D\to \C$ of the action of $\hat G$. For fixed $g\in \hat G$ the map $\phi_i(g)\colon D\to \C$ extends to a rational map $\tilde\phi_i(g)\colon \C^n\to \C$. These extensions define a map $\tilde\phi_i\colon \hat G\times\C^n\to \C$. A priori we don't know whether this new map is real analytic or even continuous. To prove this we use the following result of Kazaryan (\cite{Ka}): \begin{Prop}\label{Kaz} Let $D'$ be a domain in $\C^n$ and let $E\subset D'$ be a nonpluripolar\footnote{a subset $E\subset D'$ is called {\it nonpluripolar} if there are no plurisubharmonic functions $f\colon D'\to \R\cup\{-\infty\}$ such that $f|_E\equiv -\infty$} subset. Let $D''$ be an open set in a complex manifold $X$. If $f$ is a meromorphic function on $D'\times D''$ such that $f(g,\cdot)$ extends to a meromorphic function on $X$ for all $g\in E$, then $f$ extends to a meromorphic function in a neighborhood of $E\times X\subset D'\times X$. \end{Prop} \begin{Lemma} The map $\tilde\phi_i\colon \hat G\times\C^n\to \C$ is real analytic. \end{Lemma} {\bf Proof.} The question of real analyticity of $\tilde\phi_i$ is local with respect to $\hat G$ so we can take a real analytic coordinate neighborhood $E$ in $\hat G$, regarded as an open subset of $\R$ . The map $\phi_i$ is real analytic in $E\times D$ and extends therefore to a holomorphic function in a neighborhood $D'\times D''$ of $E\times D''$ in the complex manifold $\C\times X$. Here we must replace $D$ by a bit smaller neighborhood $D''\subset D$. The set $E$, being an open subset of $\R$ , is nonpluripolar. By Proposition~\ref{Kaz}, $\phi_i$ extends to a meromorphic functions in a neighborhood of $E\times X$. The restriction $\tilde\phi_i$ is therefore real analytic. \hfill $\square$ According to the construction of Chow scheme (see Shafarevich, \cite{S},p.65) every graph $\Gamma_{\tilde\phi_i}\subset\C^{2n}\subset\P^{2n}$ has its Chow coordinate in the Chow scheme C. The Chow coordinates yield a continuous mapping $f\colon \hat G \to C$. This is the universal property of the Chow scheme. It follows from the Theorems of D. Barlet on universality of the Barlet space and on the equivalence of the latter to the Chow scheme in case of projective space (see Barlet, \cite{B}). Since $\hat G$ has finitely many components, the image in $C$ is has also this property. But the degree of variety is constant on the components of $C$. This implies that the degree of the variety in $\P^{2n}$ associated to $g\in \hat G$ is bounded. This implies that the degrees of defining polynomials are bounded and the statement is proven. \hfill $\square$

9410

alg-geom/9410009

en

https://arxiv.org/abs/alg-geom/9410009

alg-geom/9410009

David B. Jaffe

Coherent functors, with application to torsion in the Picard group

46 pages, AMS-LaTeX

Let A be a commutative noetherian ring. Call a functor <<commutative A-algebras>> --> <<sets>> coherent if it can be built up (via iterated finite limits) from functors of the form B \mapsto M tensor_A B, where M is a f.g. A-module. When such a functor F in fact takes its values in <<abelian groups>>, we show that there are only finitely many prime numbers p such that _p F(A) is infinite, and that none of these primes are invertible in A. This (and related statements) yield information about torsion in Pic(A). For example, if A is of finite type over Z, we prove that the torsion in Pic(A) is supported at a finite set of primes, and if _p Pic(A) is infinite, then the prime p is not invertible in A. These results use the (already known) fact that if such an A is normal, then Pic(A) is finitely generated. We obtain a parallel result for a reduced scheme X of finite type over Z. We show that the groups which can occur as the Picard group of a scheme of finite type over a finite field all have the form (finitely generated) + sum_{n=1}^infty F, where F is a finite p-group. Hard copy is available from the author. E-mail to [email protected].

alg-geom

\section{#1}} \def\abs#1{{\vert{#1}\vert}} \def\makeaddress{ \vskip 0.15in \par\noindent {\footnotesize Department of Mathematics and Statistics, University of Nebraska} \par\noindent {\footnotesize Lincoln, NE 68588-0323, USA\ \ (jaffe{\kern0.5pt}@{\kern0.5pt}cpthree.unl.edu)}} \def \par\noindent David B.\ Jaffe \makeaddress{ \par\noindent David B.\ Jaffe \makeaddress} \newenvironment{proof}{\trivlist \item[\hskip \labelsep{\sc Proof.\kern1pt}]}{\endtrivlist \newenvironment{proofnodot}{\trivlist \item[\hskip \labelsep{\sc Proof}]}{\endtrivlist \newenvironment{sketch}{\trivlist \item[\hskip \labelsep{\sc Sketch.\kern1pt}]}{\endtrivlist \newenvironment{alphalist}{\begin{list}{(\alph{alphactr})}{\usecounter{alphactr}}}{\end{list} \newenvironment{arabiclist}{\begin{list}{(\arabic{arabicctr})}{\usecounter{arabicctr}}}{\end{list} \defbounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded{bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded} \def{\lambda \in \Lambda}{{\lambda \in \Lambda}} \def\underline{\underline} \def\overline{\overline} \newenvironment{definition}{\trivlist \item[\hskip \labelsep{\bf Definition.\kern1pt}]}{\endtrivlist} \newenvironment{remarks}{\trivlist \item[\hskip \labelsep{\bf Remarks.\kern1pt}]}{\endtrivlist} \newenvironment{problem}{\trivlist \item[\hskip \labelsep{\bf Problem.\kern1pt}]}{\endtrivlist} \hfuzz 3pt \documentclass{article}\usepackage{amssymb} \input xypic.sty \textwidth 6.5in \hoffset=-0.88in \newtheorem{theorem}{Theorem}[section] \setlength{\parindent}{9mm} \setcounter{tocdepth}{3} \newtheorem{proposition}[theorem]{Proposition \newtheorem{lemma}[theorem]{Lemma \newtheorem{conjecture}[theorem]{Conjecture \newtheorem{cor}[theorem]{Corollary \newtheorem{corollary}[theorem]{Corollary \newtheorem{prop}[theorem]{Proposition \newtheorem{claim}[theorem]{Claim \newtheorem{problemx}[theorem]{Problem \newtheorem{exampleth}[theorem]{Example} \newenvironment{example}{\begin{exampleth}\fontshape{n}\selectfont}{\end{exampleth}} \def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax} \begin{document} \vskip 0.15in \def\nullfoot#1{} \def\arabic{footnote}}\setcounter{footnote}{0{\nullfoot{footnote}} \footnote{{\it 1991 Mathematics Subject Classification.} Primary: 14C22, 18A25, 14K30, 18A40.} \footnote{{\it Key words and phrases.\/} Coherent functor, representable functor, Picard group.} \footnote{Partially supported by the National Science Foundation.} \def\arabic{footnote}}\setcounter{footnote}{0{\fnsymbol{footnote}}\setcounter{footnote}{0} \par\noindent{\bf\LARGE Coherent functors, with application to torsion in the Picard group} \def\arabic{footnote}}\setcounter{footnote}{0{\arabic{footnote}}\setcounter{footnote}{0} \vspace*{15pt} \par\noindent David B.\ Jaffe \makeaddress \vspace*{0.2in} \medskip \par\noindent{\bf\large Abstract} \medskip Let $A$ be a commutative noetherian ring. We investigate a class of functors from \cat{commutative $A$-algebras} to \cat{sets}, which we call {\it coherent}. When such a functor $F$ in fact takes its values in \cat{abelian groups}, we show that there are only finitely many prime numbers $p$ such that ${}_p F(A)$ is infinite, and that none of these primes are invertible in $A$. This (and related statements) yield information about torsion in $\mathop{\operatoratfont Pic}\nolimits(A)$. For example, if $A$ is of finite type over $\xmode{\Bbb Z}$, we prove that the torsion in $\mathop{\operatoratfont Pic}\nolimits(A)$ is supported at a finite set of primes, and if ${}_p \mathop{\operatoratfont Pic}\nolimits(A)$ is infinite, then the prime $p$ is not invertible in $A$. These results use the (already known) fact that if such an $A$ is normal, then $\mathop{\operatoratfont Pic}\nolimits(A)$ is finitely generated. We obtain a parallel result for a reduced scheme $X$ of finite type over $\xmode{\Bbb Z}$. We classify the groups which can occur as the Picard group of a scheme of finite type over a finite field. \medskip \par\noindent{\bf\large Coherent functors (introductory remarks)} \medskip Let us say that an {\it $A$-functor\/} is a functor from the category of commutative $A$-algebras to \cat{sets}. Some such $A$-functors have additional structure: they are actually functors from \cat{commutative $A$-algebras} to \cat{groups}. We refer to such functors as {\it group-valued\/} $A$-functors. We will also consider $A$-functors $F$ such that $F(B)$ is a $B$-module for every $B$; these {\it module-valued\/} $A$-functors are discussed later in the introduction. For now, all $A$-functors which we consider will be treated as set-valued functors. An $A$-functor is {\it coherent\/} if it may be built up as an iterated finite limit of functors of the form ${\underline{M}}$, given by ${\underline{M}}(B) = M \o*_A B$, where $M$ is a finitely generated\ $A$-module. We do not know if every coherent functor may be expressed as a finite limit of such functors ${\underline{M}}$. However, the analogous question regarding module-valued\ functors is answered affirmatively below. The idea of {\it coherent functor\/} was originally devised by Auslander \Lcitemark 5\Rcitemark \Rspace{}, in a somewhat different setting; his notion of coherence applied to functors from an abelian category to \cat{abelian groups}. Later Artin\Lspace \Lcitemark 2\Rcitemark \Rspace{} transposed Auslander's notion to a setting closer to that given here. Artin also raised a question about coherence of higher direct images as functors. This question is considered in \S\ref{higher-section}. If an $A$-functor is representable by a commutative $A$-algebra of finite type, then it is coherent. There are many examples of non-representable $A$-functors which are coherent. For example, if $M$ is a finitely generated\ $A$-module, then $B \mapsto \mathop{\operatoratfont Aut}\nolimits_{B-{\operatoratfont mod}}(M \o*_A B)$ defines a coherent $A$-functor. More examples may be found in \S\ref{examples-section}. A {\it module-valued\ $A$-functor\/} is an (abelian group)-valued $A$-functor $F$, together with the following additional structure: for each commutative $A$-algebra $B$, $F(B)$ has the structure of a $B$-module, such that for any homomorphism \mapx[[ B_1 || B_2 ]] of commutative $A$-algebras, the induced map \mapx[[ F(B_1) || F(B_2) ]] is a homomorphism of $B_1$-modules. The module-valued\ $A$-functors form an abelian category. A module-valued\ $A$-functor $F$ is {\it module-coherent\/} if there exists a homomorphism \mp[[ f || M || N ]] of finitely generated\ $A$-modules such that $F$ is isomorphic to the module-valued\ $A$-functor given by $B \mapsto \ker(f \o*_A B)$. The module-coherent\ $A$-functors form a full subcategory of the category of module-valued\ $A$-functors. Most examples of module-valued\ $A$-functors are induced naturally by functors from \cat{$A$-modules} to \cat{$A$-modules}. Certainly for many purposes it makes more sense to study the latter sort of functor. On the other hand, (as pointed out by Artin\Lspace \Lcitemark 2\Rcitemark \Rspace{}) one can set up a correspondence between module-coherent\ $A$-functors and functors from \cat{$A$-modules} to \cat{$A$-modules} which satisfy an analogous coherence axiom. This creates a bridge to the ideas of Auslander\Lspace \Lcitemark 5\Rcitemark \Rspace{} and Grothendieck (\Lcitemark 18\Rcitemark \ \S7). We have not exploited this point of view. A key result is that if \mp[[ \sigma || F || G ]] is a morphism of module-coherent\ $A$-functors, then $\ker(\sigma)$ and $\mathop{\operatoratfont coker}\nolimits(\sigma)$ are module-coherent. (These are to be computed in \cat{module-valued\ $A$-functors}.) One deduces easily from this that any finite limit or finite colimit of module-coherent\ $A$-functors is module-coherent. In particular, the iterated finite limit construction which we used in the definition of {\it coherent\/} $A$-functor is not necessary here, although in the body of the paper we find it convenient to begin with a definition of module-coherent\ which uses iterated finite limits. Let $X$ be a noetherian scheme. An {\it $X$-functor\/} is a functor from \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm\hbox{\opcat{$X$-schemes}} to \cat{sets}. We define the notion of coherent $X$-functor by analogy with the definition for $A$-functors. When $X = \mathop{\operatoratfont Spec}\nolimits(A)$, the theory of coherent $X$-functors is identical to the theory of coherent $A$-functors. Similarly, we define module-coherent\ $X$-functors. We show that the property of being a module-coherent\ $X$-functor is local on $X$, assuming that $X$ is separated. We conjecture that the property of being a coherent $X$-functor is local on $X$. \medskip \par\noindent{\bf\large Finiteness theorems (introductory remarks)} \medskip If $F$ is an (abelian group)-valued coherent $X$-functor, we prove that there are only finitely many primes $p$ such that ${}_p F(X)$ is infinite, and that none of these primes are invertible in $\Gamma(X,{\cal O}_X)$. A stronger form of this statement holds if $X$ is essentially of finite type over $\xmode{\Bbb Z}$ or over $\xmode{\Bbb Z}_p$ for some prime $p$. For example, if $X$ is of finite type over ${\Bbb Q}\kern1pt$ or over ${\Bbb Q}\kern1pt_p$, then the torsion subgroup of $F(X)$ is finite. Assuming that $X$ is reduced and that the canonical map \mapx[[ \nor{X} || X ]] is finite, consider the quotient sheaf $F = {\cal O}_{\nor{X}}^*/{\cal O}_X^*$ on $X$. We extend $F$ to an (abelian group)-valued $X$-functor, also denoted here by $F$. We do not know if $F$ is coherent, but we are able (more or less) to find an (abelian group)-valued coherent $X$-functor $G$, and a morphism \mp[[ \psi || F || G ]] such that $\psi(X)$ is injective. We say ``more or less'' because the actual proof works via a sequence of partial normalizations. The end result however is the same: there are only finitely many primes $p$ such that ${}_p F(X)$ is infinite, and none of these primes are invertible in $\Gamma(X,{\cal O}_X)$. There is also a stronger form for certain $X$ as discussed in the previous paragraph. It follows that if the group $Q = \Gamma({\cal O}_{\nor{X}}^*)/\Gamma({\cal O}_X^*)$ is finitely generated, and if $K = \mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(X)\ \mapE{}\ \mathop{\operatoratfont Pic}\nolimits(\nor{X})]$, then there are only finitely many primes $p$ such that ${}_p K$ is infinite, and none of these primes are invertible in $\Gamma(X,{\cal O}_X)$. This holds for instance if $X$ is of finite type over $\xmode{\Bbb Z}$, thereby yielding one of the results stated in the summary. When $X$ is of finite type over a field $k$, it has been shown \Lcitemark 21\Rcitemark \Rspace{} that ${}_n \mathop{\operatoratfont Pic}\nolimits(X)$ is finite for every $n$ which is invertible in $k$. For a finite field $k$ of characteristic $p$, we prove a strengthened form of this statement: modulo $p$-power torsion, $\mathop{\operatoratfont Pic}\nolimits(X)$ is finitely generated. We completely describe the structure of $\mathop{\operatoratfont Pic}\nolimits(X)$ as an abstract abelian group. The following related result is relevant. Claborn\Lspace \Lcitemark 12\Rcitemark \Rspace{} has shown that every abelian group occurs as the Picard group of some Dedekind domain over ${\Bbb Q}\kern1pt$. (See also\Lspace \Lcitemark 15\Rcitemark \Rspace{}\ \S14.) In particular, for suitable $X$, $\mathop{\operatoratfont Pic}\nolimits(X)$ itself has infinite $n$-torsion, for every $n$. It would be interesting to know to what extent the results of this paper on $\mathop{\operatoratfont Pic}\nolimits(X)$ can be obtained via \'etale cohomology. \vspace{0.1in} \par\noindent{\footnotesize{\it Acknowledgements.} I thank Deligne and Ogus for much help on \S\ref{higher-section}.} \vspace*{0.1in} \par\noindent{\bf Conventions} \ \begin{itemize} \item $A$ denotes an arbitrary commutative noetherian ring (unless specified otherwise); $B$ usually denotes an arbitrary commutative $A$-algebra; \item $X$ denotes an arbitrary noetherian scheme; \item By a {\it Zariski sheaf}, we mean a sheaf for the Zariski topology. \item If $S$, $T$ are sets, and $M$ is an abelian group, by a {\it left exact sequence} \diagramx{S&\rightarrowtail&T&\mapE{f}&M\cr% }we mean that the map from $S$ to $T$ is injective, and that $S$ is the kernel of the map from $T$ to $M$, meaning that $S = \setof{t \in T: f(t) = 0}$. Similar language applies when $S$, $T$, and $M$ are functors. \end{itemize} \block{Coherent functors} In this section we develop the basic theory of coherent functors. We have already defined the notion of {\it $A$-functor\/} in the introduction. These form a category \cat{$A$-functors} whose morphisms are natural transformations. If $M$ is an $A$-module, then there is an $A$-functor ${\underline{M}}$ given by ${\underline{M}}(B) = M \o*_A B$. If a given $A$-functor $F$ is isomorphic to ${\underline{M}}$ for some finitely generated\ $A$-module $M$, we shall say that $F$ is {\it strictly coherent}. \begin{definition} Let ${\cal{C}}$ be a category. Let $S$ be a collection of objects in ${\cal{C}}$. Let $S_0 = S$, and for each $n \geq 0$, let $S_{n+1}$ be the collection of all objects of ${\cal{C}}$, which may be obtained as limits (in ${\cal{C}}$) of diagrams involving finitely many objects in $S_n$ and finitely many morphisms. Let $S_\infty = \cup_{n=0}^\infty S_n$. Then we say that the objects in $S_\infty$ are {\it iterated finite limits\/} of objects in $S$. \end{definition} \begin{definition} An $A$-functor is {\it coherent\/} if it may be obtained as an iterated finite limit of strictly coherent $A$-functors, where the limits are taken in \cat{$A$-functors}. \end{definition} We may define $\cat{coherent $A$-functors}$: it is a full subcategory of \cat{$A$-functors}, which may be thought of as the {\it finite completion\/} of the subcategory \cat{strictly coherent $A$-functors} of \cat{$A$-functors}. Let ${\cal{C}}_A^0$ denote the collection of strictly coherent $A$-functors. For each $n \geq 0$, let ${\cal{C}}_A^{n+1}$ denote the collection of $A$-functors which may be obtained as finite limits (inside \cat{$A$-functors}) of objects in ${\cal{C}}_A^n$. We have: $${\cal{C}}_A^0 \subset {\cal{C}}_A^1 \subset {\cal{C}}_A^2 \subset \cdots.$% $Let ${\cal{C}}_A = \cup_{n=0}^\infty {\cal{C}}_A^n$. Then the objects in ${\cal{C}}_A$ are exactly the coherent $A$-functors. \begin{definition} Let $F$ be a coherent $A$-functor. Then the {\it level\/} of $F$ is the smallest integer $n$ such that $F \in {\cal{C}}_A^n$. \end{definition} In some proofs, we will need to induct on the level of a given coherent $A$-functor. This process will be facilitated by the following lemma, whose proof is left to the reader: \begin{lemma}\label{lesx-exists} Let $F$ be a coherent $A$-functor of level $n \geq 1$. Then there exists a left exact sequence: \lesx{F}{G}{{\underline{M}}% }in which $G$ is a coherent $A$-functor of level $n-1$ and $M$ is a finitely generated\ $A$-module. Moreover, $F$ may be embedded as a subfunctor of a strictly coherent $A$-functor. \end{lemma} If $k$ is a field, then every coherent $k$-functor is representable, and from this one deduces easily that every coherent $k$-functor has level $\leq 1$. Later \pref{main-theorem} we shall prove that a large class of coherent $k$-functors have level $\leq 1$. However, we do not know the answer to the following basic question: \begin{problemx} Does every coherent $A$-functor have level $\leq 1$? \end{problemx} Note that for a given $A$-functor $F$, this is the case if and only if\ there exist finitely generated\ $A$-modules $M$ and $N$, together with a morphism \mp[[ \phi || {\underline{M}} || {\underline{N}} ]] of $A$-functors such that $F$ is the ``kernel'' of $\phi$, meaning that $$F(B) = \setof{x \in M \o*_A B: \phi(x) = 0}.$% $The maps $\phi(B)$ need not be homomorphisms of $B$-modules. In some situations, for a given coherent $A$-functor $F$, it will be necessary to consider the set ${\cal{S}} = \setof{\vec M1n}$ of all $A$-modules which enter into its construction. This set is not uniquely determined by $F$. Also it does not carry information about multiplicity: ${\cal{S}}$ might consist of a single module $M$, but many copies of $M$ might enter into the construction of $F$. If $F$ has level $0$, we can choose ${\cal{S}}$ to have one element. If $F$ has level $1$, then we may view $F$ as a limit of strictly coherent $A$-functors, and thus we may choose ${\cal{S}}$ to consist of the corresponding modules. If $F$ has level $2$, then $F$ is a limit of level $1$ coherent $A$-functors $\vec F1k$, and we may choose ${\cal{S}}$ to be the union of the sets corresponding (as just considered) to $\vec F1k$. In any case, we shall say that $F$ is {\it built up from\/} $\vec M1n$. Conversely, given an arbitrary class ${\cal{S}}$ of finitely generated\ $A$-modules, we may speak of coherent $A$-functors which are {\it built up from\/} ${\cal{S}}$, meaning that such $A$-functors are built up from finite subsets of ${\cal{S}}$. \block{Module-coherent functors} In this section we develop the basic theory of module-coherent functors. The definition given initially will not be the same as that given in the introduction. It is only after considerable work that we will find \pref{main-theorem} that the two definitions agree. We have already defined the notion of {\it module-valued\/} $A$-functor in the introduction. If $F$ and $G$ are module-valued\ $A$-functors, then a {\it morphism\/} \mp[[ \sigma || F || G ]] is a natural transformation of functors, in the following sense. It is a system of homomorphisms \mp[[ \sigma(B) || F(B) || G(B) ]] of $B$-modules, for each commutative $A$-algebra $B$, such that for any $A$-algebra homomorphism \mp[[ f || B_1 || B_2 ]], the diagram: \diagramx{F(B_1)&\mapE{\sigma(B_1)}&G(B_1)\cr \mapS{F(f)}&&\mapS{G(f)}\cr F(B_2)&\mapE{\sigma(B_2)}&G(B_2)\cr% }commutes. With this definition of morphism, the module-valued\ $A$-functors form a category, which is abelian. Kernels and cokernels are computed in the obvious way; if \mp[[ \sigma || F || G ]] is a morphism, we have: $$[\ker(\sigma)](B)\ =\ \ker(\sigma(B)) \ =\ \setof{x \in F(B): \sigma(x) = 0},$% $$$[\mathop{\operatoratfont coker}\nolimits(\sigma)](B)\ =\ \mathop{\operatoratfont coker}\nolimits(\sigma(B))\ =\ G(B)/\sigma(F(B)).$% $One sees that $\sigma$ is a {\it monomorphism\/} if and only if\ $\sigma(B)$ is injective for every $B$, and $\sigma$ is an {\it epimorphism\/} if and only if\ $\sigma(B)$ is surjective for every $B$. Evidently, any module-valued\ $A$-functor may be viewed also as an $A$-functor. In some situations we shall want to consider \mp[[ \phi || F || G ]] in which $F$ and $G$ are module-valued\ $A$-functors but $\phi$ is a morphism of $A$-functors, not necessarily preserving the module structure. For clarity, we may say that $\phi$ is {\it linear}, if we wish to assume that it is a morphism of module-valued\ $A$-functors. In this section, all morphisms are linear. If $M$ is an $A$-module, then ${\underline{M}}$ is a module-valued\ $A$-functor. If a given module-valued\ $A$-functor $F$ is isomorphic to ${\underline{M}}$ for some finitely generated\ $A$-module $M$, we shall say that $F$ is {\it strictly module-coherent}. \begin{definition} A module-valued\ $A$-functor is {\it module-coherent\/} if it may be obtained as an iterated finite limit of strictly module-coherent\ $A$-functors. These limits are all taken in \cat{module-valued\ $A$-functors}. \end{definition} We will show \pref{main-theorem}, that in fact this definition is equivalent to the (much simpler) definition of module-coherent\ given in the introduction. \begin{problemx} If a module-valued\ $A$-functor $F$ is coherent (when thought of simply as an $A$-functor), does it follow that $F$ is module-coherent? \end{problemx} We may define $\cat{module-coherent\ $A$-functors}$: it is a full subcategory of \cat{module-valued\ $A$-functors}, which may be thought of as the {\it finite completion\/} of the subcategory \cat{strictly module-coherent\ $A$-functors} of \cat{module-valued\ $A$-functors}. Let ${\cal{M}}{\cal{C}}_A^0$ denote the collection of strictly module-coherent\ $A$-functors. For each $n \geq 0$, let ${\cal{M}}{\cal{C}}_A^{n+1}$ denote the collection of module-valued\ $A$-functors which may be obtained as finite limits (inside \cat{module-valued\ $A$-functors}) of objects in ${\cal{M}}{\cal{C}}_A^n$. We have: $${\cal{M}}{\cal{C}}_A^0 \subset {\cal{M}}{\cal{C}}_A^1 \subset {\cal{M}}{\cal{C}}_A^2 \subset \cdots.$% $Let ${\cal{M}}{\cal{C}}_A = \cup_{n=0}^\infty {\cal{M}}{\cal{C}}_A^n$. Then the objects in ${\cal{M}}{\cal{C}}_A$ are exactly the module-coherent\ $A$-functors. \begin{definition} Let $F$ be a module-coherent\ $A$-functor. Then the {\it level\/} of $F$ is the smallest integer $n$ such that $F \in {\cal{M}}{\cal{C}}_A^n$. \end{definition} To show that our definition of module-coherent\ is equivalent to the definition given in the introduction, we will show \pref{main-theorem} that the level of $F$ is always $\leq 1$. In the meantime, however, we will employ induction on the level of a given module-coherent\ $A$-functor. For this we use the following analog of \pref{lesx-exists}, whose proof is left to the reader: \begin{lemma}\label{les-exists} Let $F$ be a module-coherent\ $A$-functor of level $n \geq 1$. Then there exists a left exact sequence: \les{F}{G}{{\underline{M}}% }in which $G$ is a module-coherent\ $A$-functor of level $n-1$ and $M$ is a finitely generated\ $A$-module. Moreover, $F$ may be embedded as a sub-module-valued functor of a strictly module-coherent\ $A$-functor. \end{lemma} There is a functor $i_A$ from \cat{finitely generated\ $A$-modules} to \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm\hbox{\cat{module-coherent\ $A$-functors}}, given by $M \mapsto {\underline{M}}$. It is easily seen that $i_A$ is fully faithful, so we may view module-coherent $A$-functors as a sort of generalization of finitely generated\ $A$-modules. The functor $i_A$ is cocontinuous: it preserves colimits. However, $i_A$ does not carry monomorphisms to monomorphisms and is not continuous: it does not preserve limits. For example, if $J \subset A$ is an ideal, then $J$ is the kernel of the canonical map \mapx[[ A || A/J ]] of modules, but ${\underline{J}}$ is not the kernel of the induced map \mp[[ \phi || {\underline{A}} || \underline{A/J} ]]. The kernel of $\phi$ is instead given by $B \mapsto JB$. Let $H$ be a module-coherent\ $A$-functor. It would be very convenient if there existed an epimorphism \mapx[[ \underline{A^n} || H ]] for some $n$. Unfortunately, this is not always the case. For example, it is not the case if $H(B) = \mathop{\operatoratfont Ann}\nolimits_B(x)$, where $A = {\Bbb C}\kern1pt[x]$. (The module-coherence of this functor follows from example \pref{annihilator} of \S\ref{examples-section}.) As a compromise, we are lead to the following notion: \begin{definition} A module-valued\ $A$-functor $F$ is {\it linearly representable\/} if there exists a left exact sequence: \les{F}{\underline{A^n}}{\underline{A^k}% }in \cat{module-valued\ $A$-functors}, for some $n, k \geq 0$. \end{definition} If $F$ is linearly representable, it is module-coherent, and it is representable by an $A$-algebra of the form $$A[\vec x1n]/(\vec f1k),$% $where $\vec f1k$ are linear and homogeneous in $\vec x1n$. The following proposition is a basic tool, because it exhibits any module-coherent\ $A$-functor as a quotient of ``something simple''. \begin{prop}\label{dog-eats-dog} Let $F$ be a module-coherent\ $A$-functor. Then there exists a linearly representable $A$-functor $R$ and an epimorphism \mapx[[ R || F ]]. \end{prop} There are some preliminaries. \begin{lemma}\label{gerbil} Let $F$ be a linearly representable $A$-functor, represented by $C = A[\vec x1n]/(\vec f1k)$, where $\vec f1k$ are linear and homogeneous. We assume that the module structure on $F$ is the canonical one, induced from the embedding in $\underline{A^n}$ defined by $\vec x1n$. Let $C_1$ denote the degree $1$ part of $C$. Let $N$ be an $A$-module. Then morphisms from $F$ to ${\underline{N}}$ are in bijective correspondence with elements of $N \o*_A C_1$. \end{lemma} \begin{proof} We can think of $F$ and ${\underline{N}}$ as functors from the category of commutative $A$-algebras to \cat{sets}. If we take this point of view, then some morphisms (i.e.\ natural transformations) from $F$ to ${\underline{N}}$ define morphisms in \cat{module-valued $A$-functors}, and some do not. Those which do will be called {\it linear}, for purposes of this proof. The linear morphisms are those which preserve the module structure. If we take this point of view, then morphisms from $F$ to ${\underline{N}}$ are in bijective correspondence with elements of ${\underline{N}}(C) = N \o*_A C$, and it is clear that the elements of $N \o*_A C_1$ define linear morphisms. To complete the proof, we must show that if an element $\eta \in N \o*_A C$ corresponds to a linear morphism, then $\eta \in N \o*_A C_1$. The grading of $C$ induces a grading of $N \o*_A C$. Let $\eta_1 \in N \o*_A C_1$ denote the degree $1$ part of $\eta$. Let $\eta_0 = \eta - \eta_1$. Then the degree $1$ part of $\eta_0$ is $0$ and $\eta_0$ defines a linear morphism \mp[[ \psi || F || {\underline{N}} ]]. We must show that $\psi = 0$. Let $B$ be a commutative $A$-algebra. Let $D = B[t]$. Then $\psi(dx) = d\psi(x)$ for all $d \in D$ and all $x \in F(D)$. In particular, $\psi(tx) = t\psi(x)$ for all $x \in F(B)$. We have $\psi(x) \in N \o*_A B$, $\psi(tx) \in N \o*_A D = (N \o*_A B)[t]$. Since $\psi(tx) = t\psi(x)$, $\psi(tx)$ is a homogeneous linear polynomial in $t$. The element $tx \in F(D)$ defines a ring homomorphism \mp[[ \rho || C || D ]], which maps each generator $x_i$ of $C$ to a homogeneous linear polynomial in $t$. The map $\rho$ induces a map \mapx[[ N \o*_A C || N \o*_A D ]], which sends $\eta_0$ to $\psi(tx)$. But $\eta_0$ has no linear part, so it follows that $\psi(tx)$ has no linear part. Hence $\psi(tx) = 0$. Hence $t\psi(x) = 0$, so $\psi(x) = 0$. Hence $\psi = 0$. {\hfill$\square$} \end{proof} \begin{prop}\label{make-arrow} Let $P$ be an $A$-module. Suppose given a diagram: \diagramx{\rowfour{0}{F}{\underline{A^n}}{\underline{A^k}}\cr &&\mapS{\phi}\cr &&{\underline{P}}\cr% }in \cat{module-valued\ $A$-functors}, with the row exact. Then there exists a morphism \mp[[ h || \underline{A^n} || {\underline{P}} ]] which makes the diagram commute. \end{prop} \begin{proof} Let $C = A[\vec x1n]$, $${\overline{C}} = A[\vec x1n]/(\vec f1k),$% $where $\vec f1k$ are homogeneous linear elements determined by the given map from $\underline{A^n}$ to $\underline{A^k}$. Then $F$ represents ${\overline{C}}$. According to \pref{gerbil}, $\phi$ corresponds to an element of $P \o*_A {\overline{C}}_1$. The canonical map \mapx[[ P \o*_A C_1 || P \o*_A {\overline{C}}_1 ]] is surjective, so $h$ exists. {\hfill$\square$} \end{proof} \begin{corollary}\label{limit-of-linearly-rep} Let ${\cal{D}}$ be a finite diagram in \cat{module-valued $A$-functors}, in which the objects are linearly representable. Then the limit of ${\cal{D}}$ is linearly representable. \end{corollary} \begin{proof} It is clear that a product of finitely many linearly representable $A$-functors is linearly representable. Since any linearly representable $A$-functor embeds in $\underline{A^r}$ for some $r$, we may reduce to showing that if $F$ is linearly representable and \mp[[ \varphi || F || \underline{A^r} ]] is a morphism, then $\ker(\varphi)$ is linearly representable. Let \lesmaps{F}{}{\underline{A^n}}{h}{\underline{A^k}% }be as in the definition of {\it linearly representable}. By \pref{make-arrow}, $\varphi$ extends to a morphism \mp[[ \psi || \underline{A^n} || \underline{A^r} ]]. Hence $\ker(\varphi) = \ker(\psi) \cap \ker(h)$, so $\ker(\varphi)$ is linearly representable. {\hfill$\square$} \end{proof} \begin{proofnodot} (of \ref{dog-eats-dog}.) For purposes of the proof, let us say that a module-valued\ $A$-functor $G$ {\it dominates\/} a module-valued\ $A$-functor $F$ if there exists an epimorphism \mapx[[ G || F ]], and that a module-valued\ $A$-functor $F$ is {\it linearly-affine-dominated\/} if it is dominated by a linearly representable $A$-functor. Let $n$ be the level of $F$. The case $n = 0$ is clear -- in that case $F$ is dominated by $\underline{A^r}$ for some $r$. Suppose that $n \geq 1$. By \pref{les-exists}, we may find a left exact sequence: \les{F}{G}{{\underline{M}}% }in which $G$ is a module-coherent\ $A$-functor of level $n-1$ and $M$ is a finitely generated\ $A$-module. By induction on $n$, we may assume that\ there exists a linearly representable $A$-functor $H$ and an epimorphism \mapx[[ H || G ]]. Let $P$ be the fiber product of $H$ with $F$ over $G$. Then $P$ dominates $F$, so it suffices to show that $P$ is linearly-affine-dominated. We have a left exact sequence: \lesdot{P}{H}{{\underline{M}}% }Choose an epimorphism \mp[[ h || A^r || M ]]. Let $L$ be the fiber product of $H$ with $\underline{A^r}$ over ${\underline{M}}$. Let $Q$ be the fiber product of $L$ with $P$ over $H$. We have a diagram with cartesian squares, in which some maps are labelled: \diagramx{Q&\rightarrowtail&L&\mapE{\sigma}&\underline{A^r}\cr \mapS{}&&\mapS{}&&\mapS{\pi}\cr P&\rightarrowtail&H&\mapE{f}&{\underline{M}}\makenull{.}\cr% }The bottom row (but not the top) is exact. The vertical arrows are all epimorphisms. Since $Q$ dominates $P$, it suffices to show that $Q$ is linearly-affine-dominated. Let $K = \ker(\pi)$. Then $Q = \sigma^{-1}(K)$, so we have a cartesian diagram \squareSE{Q}{K}{L}{\underline{A^r}\makenull{.}% }We will show that $K$ and $L$ are linearly-affine-dominated. It will follow (by taking suitable fiber products) that $Q$ is dominated by a fiber product of linearly-representable $A$-functors. By \pref{limit-of-linearly-rep}, it will follow that $Q$ is linearly-affine-dominated. There is a canonical epimorphism \mapx[[ \underline{\ker(h)} || K ]]. Choose an epimorphism \mapx[[ A^s || \ker(h) ]]. Then $K$ is dominated by $\underline{A^s}$. To complete the proof, we will show that $L$ is linearly-affine-dominated. By \pref{gerbil}, it follows that $f$ factors through $\pi$; let \mp[[ g || H || \underline{A^r} ]] be such that $f = \pi \circ g$. Then \begin{eqnarray*} L(B) & = & \setof{(x,y) \in H(B) \times \underline{A^r}(B): f(x) = \pi(y)}\\ & = & \setof{(x,y) \in H(B) \times \underline{A^r}(B): \pi(g(x) - y) = 0}. \end{eqnarray*} Hence the morphism of module-valued\ $A$-functors \mapx[[ H \times K || L ]] given by $$(x,y) \mapsto (x,g(x)+y)$% $is an isomorphism. Since $K$ is linearly-affine-dominated, and $H$ is linearly representable, it follows that $L$ is linearly-affine-dominated. {\hfill$\square$} \end{proofnodot} \begin{theorem}\label{coherence-of-cokernel} Let \mp[[ \varphi || F || G ]] be a morphism of module-coherent\ $A$-funct\-ors. Then $\mathop{\operatoratfont Coker}\nolimits(\varphi)$ is module-coherent. \end{theorem} \begin{proof} Let $n$ be the level of $G$. First suppose that $n = 0$, so we may assume that\ $G = {\underline{M}}$ for some finitely generated\ $A$-module $M$. Choose an epimorphism \hbox{\mapx[[ A^m || M ]]} of $A$-modules and thus an epimorphism \mapx[[ \underline{A^m} || {\underline{M}} ]]. Let $F'$ be the fiber product of $F$ with $\underline{A^m}$ over ${\underline{M}}$. Let $C = \mathop{\operatoratfont Coker}\nolimits(\varphi)$. Then we have a right exact sequence: \resmapsdot{F'}{f}{\underline{A^m}}{}{C% }By \pref{dog-eats-dog}, we may assume that\ $F'$ is linearly representable. Choose a left exact sequence: \lesdot{F'}{\underline{A^n}}{\underline{A^k}% }By \pref{make-arrow}, there exists a morphism \mapx[[ \underline{A^n} || \underline{A^m} ]] which makes the following diagram commute: $$\diagram F' \rto^f \dto & \underline{A^m} \rto & C \rto & 0 \\ \underline{A^n} \dto \urto \\ \underline{A^k}\makenull{.} \enddiagram$% $Let $D$ be the co-fiber product of $A^m$ and $A^k$ over $A^n$, computed in the category of $A$-modules. Then in fact ${\underline{D}}$ is the co-fiber product of $\underline{A^m}$ and $\underline{A^k}$ over $\underline{A^n}$. Let \mp[[ g || \underline{A^m} || {\underline{D}} ]] be the canonical map. Then $\ker(g) = \mathop{\operatoratfont Im}\nolimits(f)$, so $C$ is isomorphic to $\mathop{\operatoratfont Im}\nolimits(g)$, which is module-coherent by example \pref{image} from \S\ref{examples-section}. This completes the case $n = 0$. Now suppose that $n \geq 1$. By \pref{les-exists}, we may choose a left exact sequence: \les{G}{H}{{\underline{M}}% }in which $H$ is a module-coherent\ $A$-functor of level $n-1$ and $M$ is a finitely generated\ $A$-module. Abusing notation slightly, we have a left exact sequence: \lesdot{G/F}{H/F}{{\underline{M}}% }By induction on $n$, we may assume that\ $H/F$ is module-coherent. But then $\mathop{\operatoratfont Coker}\nolimits(\varphi) = G/F$ is exhibited as the kernel of a morphism of module-coherent\ $A$-functors, so it too is module-coherent. {\hfill$\square$} \end{proof} \begin{corollary}\label{result-A} If \mp[[ \phi || F || G ]] is a morphism of module-coherent\ $A$-functors, then $\mathop{\operatoratfont Im}\nolimits(\phi)$ is module-coherent. \end{corollary} \begin{proof} Let $K = \mathop{\operatoratfont Ker}\nolimits(\phi)$. Then $K$ is module-coherent. Hence \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm $\mathop{\operatoratfont Coker}\nolimits[K \mapE{} F]$ is module-coherent\ by \pref{coherence-of-cokernel}, but this equals $\mathop{\operatoratfont Im}\nolimits(\phi)$. {\hfill$\square$} \end{proof} The next result says that the definition of module-coherent\ given in this section coincides with the simpler definition given in the introduction. \begin{corollary}\label{main-theorem} Let $F$ be a module-coherent\ $A$-functor. Then $F$ has level $\leq 1$. \end{corollary} \begin{proof} By \pref{les-exists}, we may embed $F$ as a subfunctor of ${\underline{M}}$, for some finitely generated\ $A$-module $M$. Let $Q = {\underline{M}}/F$. By \pref{coherence-of-cokernel}, $Q$ is module-coherent. By \pref{les-exists}, we may embed $Q$ as a subfunctor of ${\underline{N}}$, for some finitely generated\ $A$-module $N$. Hence $F$ is the kernel of a morphism from ${\underline{M}}$ to ${\underline{N}}$. {\hfill$\square$} \end{proof} \block{Quasi-coherent functors} In this section we sketch a theory (parallel to the last two sections) of quasi-coherent\ and module-quasi-coherent\ $A$-functors. The results of this section will be used in \S\ref{global-section}. In particular, it is the case that quasi-coherent\ $A$-functors (which are not coherent) are useful in the study of coherent $A$-functors. However, the reader interested only in the Picard group results may ignore this section and everything from \pref{locally-mc} to the end of \S\ref{global-section}. The reason for this is explained in the paragraph preceding \pref{locally-mc}. If a given $A$-functor is isomorphic to ${\underline{M}}$ for some $A$-module $M$, we shall say that $F$ is {\it strictly quasi-coherent}. Similarly, if a given module-valued\ $A$-functor $F$ is isomorphic to ${\underline{M}}$ for some $A$-module $M$, we shall say that $F$ is {\it strictly module-quasi-coherent}. An $A$-functor is {\it quasi-coherent\/} if it may be obtained as an iterated finite limit of strictly coherent $A$-functors. These limits are all taken in \cat{$A$-functors}. We shall not have much more to say about quasi-coherent\ $A$-functors per se in this paper. A module-valued\ $A$-functor is {\it module-quasi-coherent\/} if it may be obtained as an iterated finite limit of strictly module-quasi-coherent\ $A$-functors. These limits are all taken in \cat{module-valued\ $A$-functors}. The rest of this section is about module-quasi-coherent\/ $A$-functors. All morphisms will be linear. We may define the {\it level\/} of a module-quasi-coherent\ $A$-functor, as we have done for module-coherent\ $A$-functors. As we shall see \pref{main-theorem-mqc}, any module-quasi-coherent\ $A$-functor has level $\leq 1$. The analog of \pref{les-exists} for module-quasi-coherent\ $A$-functors is: \begin{lemma}\label{les-exists-qc} Let $F$ be a module-quasi-coherent\ $A$-functor of level $n \geq 1$. Then there exists a left exact sequence: \les{F}{G}{{\underline{M}}% }in which $G$ is a module-quasi-coherent\ $A$-functor of level $n-1$ and $M$ is an $A$-module. Moreover, $F$ may be embedded as a sub-module-valued functor of a strictly module-quasi-coherent\ $A$-functor. \end{lemma} We make the following convention: if $N$ is a set, then $A^N$ denotes a {\it direct sum\/} of copies of $A$, one for each element of $N$. If $S$ is a subset of $N$, then we may view $A^S$ as a submodule of $A^N$, and thence we may view $\underline{A^S}$ as a subfunctor of $\underline{A^N}$. \begin{prop}\label{make-arrow-generalized} Let $P$ be an $A$-module. Let $N$ and $K$ be sets. Suppose given a diagram: \diagramx{0&\mapE{}&F&\mapE{}&\underline{A^N}&\mapE{g}&\underline{A^K}\cr &&\mapS{\phi}\cr &&{\underline{P}}\cr% }in \cat{module-valued $A$-functors}, with the row exact. Then there exists a morphism \mp[[ h || \underline{A^N} || {\underline{P}} ]] which makes the diagram commute. \end{prop} \begin{proof} Let $\mathop{\operatoratfont fin}\nolimits(N)$ denote the collection of finite subsets of $N$. Then $$\setof{\underline{A^S}}_{S \in \mathop{\operatoratfont fin}\nolimits(N)}$% $forms a directed system of subfunctors of $\underline{A^N}$, whose union is $\underline{A^N}$. Let $F_S = F \cap \underline{A^S}$ for each $S$. Then $\setof{F_S}_{S \in \mathop{\operatoratfont fin}\nolimits(N)}$ forms a directed system of subfunctors of $F$, whose union is $F$. It is clear that $g|_{\underline{A^S}}$ factors through the subfunctor $\underline{A^{S^*}}$ of $\underline{A^K}$, for some finite subset $S^*$ of $K$. It follows that $F_S$ is linearly representable by a ring $\overline{C_S} = A[\sets xsS]/I_S$, where $I_S$ is generated by linear homogeneous elements. Let $C_S$ be the polynomial ring $A[\sets xsS]$. By \pref{gerbil}, $\phi|_{F_S}$ corresponds to an element of $P \o*_A (\overline{C_S})_1$. By lifting this element to an element of $P \o*_A (C_S)_1$, we see that $\phi|_{F_S}$ can be extended to a morphism \mp[[ h_S || \underline{A^S} || {\underline{P}} ]]. As $S$ varies, we have to choose these extensions $h_S$ so that they are compatible with each other. To do this is equivalent to showing that the canonical map: \dmapx[[ \invlim{S \in \mathop{\operatoratfont fin}\nolimits(N)} P \o*_A (C_S)_1 || \invlim{S \in \mathop{\operatoratfont fin}\nolimits(N)} P \o*_A (\overline{C_S})_1 ]]% is surjective. In general, it is not true that an inverse limit of surjective module maps is surjective, but (\Lcitemark 4\Rcitemark \ 10.2) it is the case if the transition maps in the system of kernels are surjective. To show this, it suffices to show that if $S, S' \in \mathop{\operatoratfont fin}\nolimits(N)$, with $S \subset S'$, then the canonical map \mapx[[ P \o*_A (I_{S'})_1 || P \o*_A (I_S)_1 ]] is surjective. This follows from the fact that the canonical map \mapx[[ I_{S'} || I_S ]] is surjective. {\hfill$\square$} \end{proof} \begin{definition} A module-valued $A$-functor $F$ is {\it linearly quasi-representable\/} if there exist sets $N$ and $K$ and a left exact sequence: \les{F}{\underline{A^N}}{\underline{A^K}% }in \cat{module-valued\ $A$-functors}. \end{definition} Using \pref{make-arrow-generalized} one may prove the following analog of \pref{limit-of-linearly-rep}: \begin{corollary}\label{limit-of-linearly-rep-gen} Let ${\cal{D}}$ be a finite diagram in \cat{module-valued\ $A$-functors}, in which the objects are linearly quasi-representable. Then the limit of ${\cal{D}}$ is linearly quasi-representable. \end{corollary} Evidently, any linear quasi-representable $A$-functor is module-quasi-coherent. \begin{prop}\label{rat-eats-rat} Let $F$ be a module-quasi-coherent\ $A$-functor. Then there exists a linearly quasi-representable $A$-functor $R$ and an epimorphism \mapx[[ R || F ]]. \end{prop} \begin{sketch} Take the proof of \pref{dog-eats-dog}, and modify it in the following ways. In the various places where $A^r$ is written, one has to allow $r$ to be an arbitrary set. Do the same with $A^s$. Change each reference to {\it module-coherent\/} to {\it module-quasi-coherent}. Drop the assumption that $M$ is finitely generated. The construction of $g$ requires the use of \pref{make-arrow-generalized}. Use \pref{limit-of-linearly-rep-gen} instead of \pref{limit-of-linearly-rep}. Use \pref{les-exists-qc} instead of \pref{les-exists}. {\hfill$\square$} \end{sketch} \begin{prop}\label{quasi-coherence-of-cokernel} Let \mp[[ \varphi || F || G ]] be a morphism of module-quasi-coherent\ $A$-functors. Then $\mathop{\operatoratfont Coker}\nolimits(\varphi)$ is module-quasi-coherent. \end{prop} \begin{sketch} Take the proof of \pref{coherence-of-cokernel}, and modify it in the following ways. Change each reference to {\it module-coherent\/} to {\it module-quasi-coherent}. Drop the assumption that $M$ is finitely generated. In the notations $A^m$, $A^n$, and $A^k$, one must allow $m$, $n$, and $k$ to be arbitrary sets. Use \pref{make-arrow-generalized} instead of \pref{make-arrow}. Use \pref{rat-eats-rat} instead of \pref{dog-eats-dog}. Use \pref{les-exists-qc} instead of \pref{les-exists}. Modify reference to example \pref{image} from \S\ref{examples-section} appropriately. {\hfill$\square$} \end{sketch} \begin{corollary}\label{result-A-MQC} Let \mp[[ \phi || F || G ]] be a morphism of module-quasi-coherent\ $A$-functors. Then $\mathop{\operatoratfont Im}\nolimits(\phi)$ is module-quasi-coherent. \end{corollary} \begin{corollary}\label{main-theorem-mqc} Let $F$ be a module-quasi-coherent\ $A$-functor. Then $F$ has level $\leq 1$. \end{corollary} \begin{lemma}\label{noetherian} Let $F$ be a module-coherent\ $A$-functor. Let $\sets F\lambda\Lambda$ be a system of subfunctors of $F$, whose union is $F$. Then $F = F_{\lambda_0}$ for some $\lambda_0 \in \Lambda$. \end{lemma} \begin{proof} By \pref{dog-eats-dog}, we may find a linearly-representable $A$-functor $H$, and an epimorphism \mp[[ \pi || H || F ]]. Let $H_\lambda = \pi^{-1}(F_\lambda)$, for each ${\lambda \in \Lambda}$. (That is, $H_\lambda$ is the fiber product of $F_\lambda$ with $H$ over $F$.) Then $H$ is the union of the $H_\lambda$. Identify $H$ with the functor representing some $A$-algebra $C$: $H(B) = \mathop{\operatoratfont Hom}\nolimits_{A-{\operatoratfont alg}}(C,B)$. Choose $\lambda_0 \in \Lambda$ such that $1_C \in H_{\lambda_0}(C)$. It follows that there exists a natural transformation of functors \mp[[ s || H || H_{\lambda_0} ]] (possibly not preserving module structures) such that $i \circ s = 1_H$, where \mp[[ i || H_{\lambda_0} || H ]] is the inclusion. Hence $i$ is an isomorphism. Hence $H_{\lambda_0} = H$. Since $\pi$ is an epimorphism, it follows that $F_{\lambda_0} = F$. {\hfill$\square$} \end{proof} \begin{remark}\label{not-coh-example} If $F$ is module-coherent\ and $G$ is module-quasi-coherent, and \mp[[ \varphi || F || G ]] is a morphism, then $\ker(\varphi)$ need not be module-coherent. For an example, let $A = \xmode{\Bbb Z}$, $F = \underline{\xmode{\Bbb Z}}$, $G = \underline{{\Bbb Q}\kern1pt}$, and let $\varphi$ be the map given by $n \mapsto n$. Indeed, $$\ker(\varphi)(B)\ =\ \setofh{$b \in B: nb = 0$ for some $n \in \xmode{\Bbb N}$}.$% $Then $\ker(\varphi)$ is the direct limit of its subfunctors $F_n = \setof{b \in B: nb = 0}$, for $n \geq 1$, and since $\ker(\varphi) \not= F_n$ for all $n$, it follows from \pref{noetherian} that $\ker(\varphi)$ is not module-coherent. \end{remark} \block{Examples}\label{examples-section} First we give some examples, all of which are easily seen to satisfy the simple definition of module-coherent\ which we gave in the introduction. \begin{arabiclist} \item $B \mapsto M \o*_A B$, where $M$ is a finitely generated $A$-module; \item\label{kernel} $B \mapsto \mathop{\operatoratfont Ker}\nolimits(f \o*_A B)$, where \mp[[ f || M || N ]] is a homomorphism of finitely generated\ $A$-modules \par[We shall denote this $A$-functor by $\mathop{\underline{\operatoratfont Ker}}\nolimits(f)$.] \item $B \mapsto IB$, where $I$ is an ideal of $A$ \par[Consider the kernel of the map \mapx[[ {\underline{A}} || \underline{A/I} ]].] \item\label{image} $B \mapsto \mathop{\operatoratfont Im}\nolimits(f \o*_A B)$, where \mp[[ f || M || N ]] is a homomorphism of finitely generated $A$-modules. \par[We shall denote this functor by $\mathop{\underline{\operatoratfont Im}}\nolimits(f)$. Let \mp[[ g || N || N/\mathop{\operatoratfont Im}\nolimits(f) ]] be the canonical map. Then $\mathop{\underline{\operatoratfont Im}}\nolimits(f) = \mathop{\underline{\operatoratfont Ker}}\nolimits(g)$, so $\mathop{\underline{\operatoratfont Im}}\nolimits(f)$ is module-coherent.] \item\label{hom-example} $B \mapsto \mathop{\operatoratfont Hom}\nolimits_{B-{\operatoratfont mod}}(M \o*_A B, N \o*_A B)$, where $M$ and $N$ are finitely generated $A$-modules \par[We shall denote this functor by $\mathop{\underline{\operatoratfont Hom}}\nolimits(M, N)$. To see why it is module-coherent, choose a presentation $A^k \mapE{} A^n \mapE{} M \mapE{} 0$ for $M$. Consider the induced map \mp[[ f || \mathop{\operatoratfont Hom}\nolimits(A^n, N) || \mathop{\operatoratfont Hom}\nolimits(A^k, N) ]]. Then $\mathop{\underline{\operatoratfont Hom}}\nolimits(M, N)$ is isomorphic to $\mathop{\underline{\operatoratfont Ker}}\nolimits(f)$.] \item\label{end-example} $B \mapsto \mathop{\operatoratfont End}\nolimits_{B-{\operatoratfont mod}}(M \o*_A B)$, where $M$ is a finitely generated\ $A$-module \par[We shall denote this functor by $\mathop{\underline{\operatoratfont End}}\nolimits(M)$.] \item\label{annihilator} $B \mapsto \mathop{\operatoratfont Ann}\nolimits_B(I)$, where $I \subset A$ is a fixed ideal \par[We shall denote this functor by $\mathop{\underline{\operatoratfont Ann}}\nolimits(I)$. The point is that $\mathop{\operatoratfont Ann}\nolimits_B(I) = \mathop{\operatoratfont Hom}\nolimits_{B-{\operatoratfont mod}}(B/I,B)$; use example \pref{hom-example}.] \end{arabiclist} \par\noindent Now we give some simple examples of coherent $A$-functors. \begin{arabiclist} \setcounter{arabicctr}{7} \item\label{representable-example} $B \mapsto \mathop{\operatoratfont Hom}\nolimits_{A-{\operatoratfont alg}}(C,B)$, where $C$ is a commutative $A$-algebra of finite type \par[Choose a presentation $C = A[\vec x1n]/(\vec f1k)$. Then $\vec f1k$ define a morphism of $A$-functors \mapx[[ \underline{A^n} || \underline{A^k} ]], whose kernel is the given functor.] \item $B \mapsto \setof{x \in B: x^2 \in IB}$ \par[Consider the kernel of the map \mapx[[ {\underline{A}} || \underline{A/I} ]] given by $x \mapsto x^2$.]; \item $B \mapsto \setof{a \in A^n: f(a) = 0}$, where $f \in M[\vec x1n]$ \par[Consider kernels of maps \mapx[[ \underline{A^n} || {\underline{M}} ]]; this generalizes the preceding example.] \end{arabiclist} The coherence of the remaining examples of this section follows without great difficulty from the tools developed so far. However, the remaining examples seem to be deeper, in the sense that their coherence cannot be deduced directly from the definitions. Presumably, all of the usual linear algebra operations ($\mathop{\operatoratfont Hom}\nolimits$, $\o*$, $\Lambda^n$, $\ldots$) have analogs for module-valued\ $A$-functors. A thorough study (which we do not give) would include definitions of these operations and an analysis of which preserve module-coherence. We restrict our attention to some special cases. Let $F$ and $G$ be module-valued\ $A$-functors. Then there is a module-valued\ $A$-functor $F \o* G$, given by $B \mapsto F(B) \o*_B G(B)$. If $F$ and $G$ are module-coherent, one can ask if $F \o* G$ is module-coherent. It turns out \pref{tensor-mc} that this is not the case. However, there is the following special case: \begin{prop}\label{tensor-basic} Let $F$ be a module-coherent\ $A$-functor, and let $M$ be a finitely generated\ $A$-module. Then $F \o* {\underline{M}}$ is module-coherent. \end{prop} \begin{proof} Choose a right exact sequence: \res{A^k}{A^n}{M% }of $A$-modules. We obtain a right exact sequence: \res{F \o* \underline{A^k}}{F \o* \underline{A^n}}{F \o* {\underline{M}}% }of module-valued\ $A$-functors, and thence a right exact sequence: \resdot{F^k}{F^n}{F \o* {\underline{M}}% }Since $F^k$ and $F^n$ are module-coherent, it follows by \pref{coherence-of-cokernel} that $F \o* {\underline{M}}$ is module-coherent. {\hfill$\square$} \end{proof} To show that tensor products do not (in general) preserve module-coherence, we need the following lemma, which will also be used in a counterexample presented in \S\ref{higher-section}. \begin{lemma}\label{ding-dong} Let $A$ be a noetherian local ring of dimension $d$, having maximal ideal ${\xmode{{\fraktur{\lowercase{M}}}}}$. Let $F$ be a module-coherent\ $A$-functor. Then there exists a constant $c$, such that for each $n \in \xmode{\Bbb N}$, and every ideal $I$ of $A$ with ${\xmode{{\fraktur{\lowercase{M}}}}}^n \subset I$, we have $\mu[F(A/I)] \leq c n^d$, where $\mu$ gives the minimal number of generators of an $A$-module. \end{lemma} \begin{remark} Perhaps the bound $c n^d$ can be replaced by $c n^{d-1}$. \end{remark} \begin{proofnodot} (of \ref{ding-dong}.) We may assume that $F = \mathop{\underline{\operatoratfont Ker}}\nolimits(f)$, for some homomorphism \mp[[ f || M || N ]] of finitely generated\ $A$-modules. Let $\lambda$ denote {\it length}. Choose a surjection \mapx[[ A^s || M ]]. Since we have $${\xmode{{\fraktur{\lowercase{M}}}}}^{n+1}M\ \subset\ {\xmode{{\fraktur{\lowercase{M}}}}} f^{-1}(IN)\ \subset\ f^{-1}(IN) \subset\ M,$% $it follows that: \begin{eqnarray*} \mu[\mathop{\operatoratfont Ker}\nolimits(f \o*_A A/I)] & = & \mu[f^{-1}(IN)/IM]\ \leq\ \mu[f^{-1}(IN)]\\ & & =\ \mu[f^{-1}(IN)/{\xmode{{\fraktur{\lowercase{M}}}}} f^{-1}(IN)]\\ & & =\ \lambda[f^{-1}(IN)/{\xmode{{\fraktur{\lowercase{M}}}}} f^{-1}(IN)]\\ & & \leq\ \lambda(M/{\xmode{{\fraktur{\lowercase{M}}}}}^{n+1}M) \ \leq\ \lambda(A^s/{\xmode{{\fraktur{\lowercase{M}}}}}^{n+1} A^s)\\ & & =\ s\lambda(A/{\xmode{{\fraktur{\lowercase{M}}}}}^{n+1}). \end{eqnarray*} The lemma follows from the theory of the Hilbert-Samuel polynomial. {\hfill$\square$} \end{proofnodot} Now we show that the tensor product of two module-coherent\ $A$-functors need not be module-coherent: \begin{prop}\label{tensor-mc} Let $A = {\Bbb C}\kern1pt[[s,t,u]]$, and let $I$ be the ideal $(s)$ of $A$. Then the module-valued\ $A$-functor $\mathop{\underline{\operatoratfont Ann}}\nolimits(I) \o* \mathop{\underline{\operatoratfont Ann}}\nolimits(I)$ is not module-coherent. \end{prop} \begin{proof} Let $F$ denote the given functor. Fix $n \in \xmode{\Bbb N}$, and let $B = A/(s,t,u)^n$. Then a minimal generating set for $\mathop{\operatoratfont Ann}\nolimits_B(s)$ is $$\setof{s^i t^j u^k}_{0 \leq i,j,k \leq n-1,\ i+j+k = n-1},$% $which has cardinality $n(n+1)/2$. Then $\mu[F(B)] = \mu[\mathop{\operatoratfont Ann}\nolimits_B(s)]^2 = [n(n+1)/2]^2$. By \pref{ding-dong}, $F$ is not module-coherent. {\hfill$\square$} \end{proof} Let $M$ and $N$ be finitely generated\ $A$-modules. For $n \geq 0$, one can ask if the functor $\mathop{\underline{\operatoratfont Tor}}\nolimits_{\kern1pt n}(M,N)$ given by $B \mapsto \mathop{\operatoratfont Tor}\nolimits_n^B(M \o*_A B, N \o*_A B)$ is module-coherent. This seems unlikely for $n \geq 2$ (but we do not have a counterexample). For $n = 0$, \pref{tensor-basic} applies. For $n = 1$, we have: \begin{prop} Let $M$ and $N$ be finitely generated\ $A$-modules. Then $\mathop{\underline{\operatoratfont Tor}}\nolimits_1(M,N)$ is module-coherent. \end{prop} \begin{proof} Choose an epimorphism \mapx[[ A^n || M ]] and thence a short exact sequence: \ses{K}{\underline{A^n}}{{\underline{M}}% }in which $K$ is module-coherent. One obtains a left exact sequence: \lesdot{\mathop{\underline{\operatoratfont Tor}}\nolimits_1(M,N)}{K \o* {\underline{N}}}{\underline{A^n} \o* {\underline{N}}% }The corollary follows now from \pref{tensor-basic}. {\hfill$\square$} \end{proof} If $F$ and $G$ are module-valued\ $A$-functors, we let $\mathop{\mathbf{Hom}}\nolimits(F,G)$ denote the module-valued\ $A$-functor given by $$B \mapsto \mathop{\operatoratfont Hom}\nolimits_{\kern2pt{\operatoratfont module-valued}\ B-{\operatoratfont functors}} (F, G),$% $where $F$ and $G$ may be viewed as module-valued\ $B$-functors because any $B$-algebra is an $A$-algebra. In a natural way, $\mathop{\mathbf{Hom}}\nolimits(F,G)$ is itself a module-valued\ $A$-functor. We let $\mathop{\mathbf{End}}\nolimits(F)$ denote $\mathop{\mathbf{Hom}}\nolimits(F,F)$. \begin{example}\label{uHomHOM} Let $M$ and $N$ be finitely generated\ $A$-modules. Then there is a canonical isomorphism of module-valued\ $A$-functors \dmapx[[ \mathop{\underline{\operatoratfont Hom}}\nolimits(M,N) || \mathop{\mathbf{Hom}}\nolimits({\underline{M}},{\underline{N}}). ]] \end{example} \begin{prop}\label{HOM-coherent} Let $F$ and $G$ be module-coherent\ $A$-functors. Then \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm $\mathop{\mathbf{Hom}}\nolimits(F,G)$ is module-coherent. \end{prop} \begin{proof} First observe that the bifunctor \dmap[[ \mathop{\mathbf{Hom}}\nolimits || \opcat{module-valued $A$-functors} \times \cat{module-valued $A$-functors} || \cat{module-valued $A$-functors} ]]% is left exact in both variables. By \pref{main-theorem}, there is a left exact sequence: \les{G}{\underline{M_1}}{\underline{M_2}% }for some finitely generated\ $A$-modules $M_1$ and $M_2$. This yields a left exact sequence: \lesdot{\mathop{\mathbf{Hom}}\nolimits(F,G)}{\mathop{\mathbf{Hom}}\nolimits(F,\underline{M_1})}{\mathop{\mathbf{Hom}}\nolimits(F,\underline{M_2})% }Thus it suffices to show that $\mathop{\mathbf{Hom}}\nolimits(F,\underline{M_i})$ is module-coherent\ for each $i$. Let $M = M_i$. It follows from \pref{dog-eats-dog} that there exists a right exact sequence \res{L_2}{L_1}{F% }in which $L_1$ and $L_2$ are linearly representable. We obtain a left exact sequence: \lesdot{\mathop{\mathbf{Hom}}\nolimits(F,{\underline{M}})}{\mathop{\mathbf{Hom}}\nolimits(L_1,{\underline{M}})}{\mathop{\mathbf{Hom}}\nolimits(L_2,{\underline{M}})% }Therefore we may reduce to proving that $\mathop{\mathbf{Hom}}\nolimits(L,{\underline{M}})$ is module-coherent\ for any linearly representable $A$-functor $L$. The $A$-functor $L$ is representable by a ring $C = A[\vec x1n]/(\vec f1k)$, where $\vec f1k$ are homogeneous linear polynomials in $\vec x1n$. From \pref{gerbil}, it follows that $\mathop{\mathbf{Hom}}\nolimits(L,{\underline{M}}) \cong \underline{M \o*_A C_1}$, so $\mathop{\mathbf{Hom}}\nolimits(L,{\underline{M}})$ is module-coherent. {\hfill$\square$} \end{proof} \begin{corollary} For any module-coherent\ $A$-functor $F$, $\mathop{\mathbf{End}}\nolimits(F)$ is module-coherent. \end{corollary} If $F$ is a module-valued\ $A$-functor, we let $\mathop{\mathbf{Aut}}\nolimits(F)$ denote the $A$-functor given by $$B \mapsto \mathop{\operatoratfont Aut}\nolimits_{\kern2pt{\operatoratfont module-valued} \ B-{\operatoratfont functors}} (F),$% $where $F$ is viewed as a $B$-functor. Then $\mathop{\mathbf{Aut}}\nolimits(F)$ is an $A$-functor. \begin{corollary} For any module-coherent\ $A$-functor $F$, $\mathop{\mathbf{Aut}}\nolimits(F)$ is coherent. \end{corollary} \begin{proof} Take the kernel of the map \dmapx[[ \mathop{\mathbf{End}}\nolimits(F) \times \mathop{\mathbf{End}}\nolimits(F) || \mathop{\mathbf{End}}\nolimits(F) \times \mathop{\mathbf{End}}\nolimits(F) ]]% given by $(\alpha,\beta) \mapsto (\alpha \circ \beta - \mathop{\operatoratfont id}\nolimits, \beta \circ \alpha - \mathop{\operatoratfont id}\nolimits)$. {\hfill$\square$} \end{proof} By an {\it algebra-valued\/} $A$-functor $F$, we shall mean a module-valued\ $A$-functor $F$ which has the additional structure of a $B$-algebra on $F(B)$, for each $B$. We do not assume that these algebras $F(B)$ are commutative. If $F$, $G$, and $H$ are module-valued\ $A$-functors, then $\mathop{\mathbf{Bil}}\nolimits(F \times G, H)$ will denote the functor of bilinear maps from $F \times G$ to $H$, which sends $B$ to the $B$-module consisting of all morphisms of $B$-functors from $F \times G$ to $H$ which are bilinear. Then $\mathop{\mathbf{Bil}}\nolimits(F \times G, H)$ is a module-valued\ $A$-functor. \begin{corollary} Let $F$, $G$, and $H$ be module-coherent\ $A$-functors. Then \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm\hbox{$\mathop{\mathbf{Bil}}\nolimits(F \times G, H)$} is module-coherent. \end{corollary} \begin{proof} We may identify $\mathop{\mathbf{Bil}}\nolimits(F \times G, H)$ with $\mathop{\mathbf{Hom}}\nolimits(F,\mathop{\mathbf{Hom}}\nolimits(G,H))$. {\hfill$\square$} \end{proof} If $F$ and $G$ are algebra-valued $A$-functors, we let $\mathop{{\mathbf{Hom}}_{\kern1pt\operatoratfont alg}(F,G)$ denote the $A$-functor given by $$B \mapsto \mathop{\operatoratfont Hom}\nolimits_{\kern2pt{\operatoratfont algebra-valued} \ B-{\operatoratfont functors}} (F, G),$% $where $F$ and $G$ are viewed as algebra-valued $B$-functors. Similarly, we may define $\mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(F)$. \begin{corollary} Let $R$ and $S$ be module-coherent, algebra-valued $A$-functors. Then $\mathop{{\mathbf{Hom}}_{\kern1pt\operatoratfont alg}(R,S)$ is coherent. \end{corollary} \begin{proof} Let \mp[[ \mu_R || R \times R || R ]] and \mp[[ \mu_S || S \times S || S ]] denote the multiplication maps. Then $\mathop{{\mathbf{Hom}}_{\kern1pt\operatoratfont alg}(R,S)$ is the kernel of the map: \dmapx[[ \mathop{\mathbf{Hom}}\nolimits(R,S) || \mathop{\mathbf{Bil}}\nolimits(R \times R, S) ]]% given by $f \mapsto [f \circ \mu_R] - [\mu_S \circ (f \times f)]$. {\hfill$\square$} \end{proof} \begin{example} Let $R$ and $S$ be module-finite $A$-algebras (not necessarily commutative). Then the $A$-functor defined by $$B \mapsto \mathop{\operatoratfont Hom}\nolimits_{B-{\operatoratfont alg}}(R \o*_A B, S \o*_A B)$% $is coherent. \end{example} \begin{corollary}\label{autalg-is-coherent} For any module-coherent, algebra-valued $A$-functor $R$, \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm $\mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(R)$ is coherent. \end{corollary} \begin{example} Let $R$ be a module-finite $A$-algebra (not necessarily commutative). Then the $A$-functor given by $B \mapsto \mathop{\operatoratfont Aut}\nolimits_{B-{\operatoratfont alg}}(R \o*_A B)$ is coherent. \end{example} \block{The global case}\label{global-section} We have defined the notion of $X$-functor in the introduction; these form a category \cat{$X$-functors}. If $X = \mathop{\operatoratfont Spec}\nolimits(A)$, then there is a canonical equivalence of categories between \cat{$A$-functors which are Zariski sheaves} and \cat{$X$-functors which are Zariski sheaves}; we can pass back and forth freely between these two categories. Similarly, for an arbitrary noetherian scheme $X$, we may (instead of looking at $X$-functors which are Zariski sheaves) look at functors which are Zariski sheaves and whose source is $$\opcat{$X$-schemes which are quasi-compact}.$$ Let \mp[[ \phi || X || Y ]] be a morphism of noetherian schemes. We consider pull-back and push-forward of functors: \begin{itemize} \item Let $F$ be a $Y$-functor. Then there is an $X$-functor $\phi^*F$, given by $(\phi^*F)(T) = F(T)$ for all $X$-schemes $T$. Sometimes we will write $\phi|_X$ instead of $\phi^*F$, and refer to the {\it restriction\/} of $F$ to $X$. \item Let $G$ be an $X$-functor. Then there is a $Y$-functor $\phi_*G$, given by $(\phi_*G)(S) = F(X \times_Y S)$, for all $Y$-schemes $S$. \item Let $F$ be a $Y$-functor. Then the $Y$-functor $\phi_*\phi^*(F)$ is given by $S \mapsto F(X \times_Y S)$, for all $Y$-schemes $S$. Instead of writing $\phi_*\phi^*(F)$, we may refer to ``$F|_X$, viewed as a $Y$-functor.'' \end{itemize} \par\noindent Note that both $\phi^*$ and $\phi_*$ are {\it exact\/} functors. If $Y$ is an $X$-scheme, \mp[[ \pi || Y || X ]] is a morphism of schemes, and ${\cal{M}}$ is a quasi-coherent\ ${\cal O}_X$-module, we let ${\cal{M}}_Y$ denote $\pi^*{\cal{M}}$. For any such ${\cal{M}}$, there is an $X$-functor ${\underline{\cal M}}$ given by ${\underline{\cal M}}(Y) = \Gamma(Y,{\cal{M}}_Y)$. An $X$-functor is {\it strictly coherent\/} if it is isomorphic to ${\underline{\cal M}}$ for some coherent ${\cal O}_X$-module ${\cal{M}}$. An $X$-functor is {\it coherent\/} if it is an iterated finite limit of strictly coherent $X$-functors, where the limits are taken in \cat{$X$-functors}. Similarly, one may define {\it quasi-coherent\/} $X$-functors, by allowing ${\cal{M}}$ to be quasi-coherent. We may define the {\it level\/} of a coherent $X$-functor, exactly as we have done for coherent $A$-functors. We may also define the {\it level\/} of a quasi-coherent\ $X$-functor, and it is distantly conceivable that there exists a coherent $X$-functor whose level is lower when viewed as a quasi-coherent\ $X$-functor. It is very important to note that if $X = \mathop{\operatoratfont Spec}\nolimits(A)$, then coherent $X$-functors are essentially the same as coherent $A$-functors. This follows from the fact that coherent $X$-functors are sheaves for the Zariski topology. Indeed we have: \begin{prop}\label{sheaf} Let $F$ be a quasi-coherent\ $X$-functor. Then $F$ is a sheaf for the fpqc topology. \end{prop} \begin{proof} If $F = {\underline{\cal M}}$, for some quasi-coherent\ ${\cal O}_X$-module ${\cal{M}}$, then the statement is true. The proposition follows because any limit of sheaves is a sheaf. {\hfill$\square$} \end{proof} \begin{definition} A {\it module-valued\ $X$-functor\/} is an (abelian group)-valued $X$-functor $F$, together with the structure of a $\Gamma(Y,{\cal O}_Y)$-module on each set $F(Y)$, with the property that for each map of $X$-schemes \mapx[[ Y_1 || Y_2 ]], the induced map \mapx[[ F(Y_2) || F(Y_1) ]] is a homomorphism of $\Gamma(Y_2, {\cal O}_{Y_2})$-modules. \end{definition} The module-valued\ $X$-functors form an abelian category. When we have $X = \mathop{\operatoratfont Spec}\nolimits(A)$, there is a canonical equivalence of categories between: $$\cat{module-valued\ $A$-functors which are Zariski sheaves}$% $and $$\cat{module-valued\ $X$-functors which are Zariski sheaves}.$% $ If ${\cal{M}}$ is a quasi-coherent\ ${\cal O}_X$-module, we let ${\underline{\cal M}}$ denote the module-valued\ $X$-functor given by $Y \mapsto \Gamma(Y,{\cal{M}}_Y)$. A module-valued\ $X$-functor $F$ is {\it strictly module-coherent\/} if there exists a coherent ${\cal O}_X$-module ${\cal{M}}$ such that $F \cong {\underline{\cal M}}$. \begin{definition} A module-valued\ $X$-functor is {\it module-coherent\/} if it may be obtained as an iterated finite limit of strictly module-coherent\ $X$-functors. These limits are all taken in \cat{module-valued\ $X$-functors}. \end{definition} In a similar way, one may define module-quasi-coherent\ $X$-functors. If $X = \mathop{\operatoratfont Spec}\nolimits(A)$, then module-coherent\ $X$-functors are essentially the same as module-coherent\ $A$-functors. For arbitrary $X$, the theory of module-coherent\ $X$-functors runs parallel to the theory of module-coherent\ $A$-functors, but there is one difference. When one takes the cokernel of a morphism of module-coherent\ $X$-functors, it is necessary to take the associated sheaf (with respect to\ the Zariski topology), in order to obtain a module-coherent\ $X$-functor. The {\it level\/} of a module-coherent\ (or module-quasi-coherent) $X$-functor is defined analogously to the definition of level for a module-coherent\ $A$-functor. As in that case, we will find ultimately that the level is always $\leq 1$. An $X$-functor $F$ is {\it locally coherent\/} if $F$ is a Zariski sheaf and if there exists an open cover $\vec U1n$ of $X$ such that $F|_{U_i}$ is a coherent $U_i$-functor for each $i$. Similarly, a module-valued\ $X$-functor $F$ is {\it locally module-coherent\/} if $F$ is a Zariski sheaf and if there exists an open cover $\vec U1n$ of $X$ such that $F|_{U_i}$ is a module-coherent\ $U_i$-functor for each $i$. We will show shortly that any locally module-coherent\ $X$-functor is module-coherent, assuming that $X$ is separated. (One can also define {\it locally quasi-coherent\/} and {\it locally module-quasi-coherent\/} $X$-functors.) We do not know the answer to the analogous question for locally coherent $X$-functors: \begin{conjecture}\label{locally-coherent-conjecture} Every locally coherent $X$-functor is coherent. \end{conjecture} If the conjecture were true, it would follow immediately that if an $X$-functor $F$ represents an affine $X$-scheme of finite type, then $F$ is coherent. (This is true if $X$ is affine: see example \pref{representable-example} from \S\ref{examples-section}.) More generally, one could ask: \begin{problemx} Let \mp[[ \phi || Y || X ]] be a faithfully flat morphism of noetherian schemes. Let $F$ be an $X$-functor, which is a sheaf for the fpqc topology. Assume that $\phi^*F$ is a coherent $Y$-functor. Does it follow that $F$ is a coherent $X$-functor? \end{problemx} We now consider push-forward and pull-back of coherent and quasi-coherent $X$-functors. These operations also make sense for module-valued\ $X$-functors. \begin{prop}\label{pull} Let \mp[[ \phi || X || Y ]] be a morphism of noetherian schemes. Let $F$ be a coherent $Y$-functor. Then $\phi^*F$ is coherent. Similarly, if $F$ is a module-coherent\ $Y$-functor, then $\phi^*F$ is module-coherent. \end{prop} \begin{proof} Suppose that $F$ is a coherent $Y$-functor. (The parallel case for module-coherent\ $Y$-functors is left to the reader.) Let $n$ be the level of $F$. First suppose that $n = 0$, so $F \cong {\underline{\cal M}}$ for some coherent ${\cal O}_Y$-module ${\cal{M}}$. But then $\phi^*F \cong \underline{\phi^*{\cal{M}}}$, so $\phi^*F$ is coherent. Now suppose that $n \geq 1$. By an unstated analog of \pref{lesx-exists}, there is a left exact sequence: \lesx{F}{G}{{\underline{\cal M}}% }of $X$-functors in which $G$ is coherent of level $n-1$ and ${\cal{M}}$ is a coherent ${\cal O}_X$-module. By induction on $n$, we may assume that\ $\phi^*G$ is coherent. Since $\phi^*$ is an exact functor, we have a left exact sequence: \lesxdot{\phi^*F}{\phi^*G}{\phi^*{\underline{\cal M}}% }Hence $\phi^*F$ is coherent. {\hfill$\square$} \end{proof} \begin{prop}\label{push} Let \mp[[ \phi || X || Y ]] be a morphism of noetherian schemes. Let $F$ be an $X$-functor. \begin{alphalist} \item If $F$ is quasi-coherent\ and $\phi$ is affine, then $\phi_*F$ is quasi-coherent. \item If $F$ is coherent and $\phi$ is finite, then $\phi_*F$ is coherent. \item Parallel statements apply if $F$ is a module-valued\ $X$-functor. \end{alphalist} \end{prop} \begin{proof} (a): Let $n$ be the level of $F$. If $n = 0$, $F = {\underline{\cal M}}$ for some quasi-coherent\ ${\cal O}_X$-module ${\cal{M}}$, so $(\phi_*F)(T) = \Gamma(X \times_Y T, {\cal{M}}_{X \times_Y T})$ for any $Y$-scheme $T$. By (\Lcitemark 20\Rcitemark \ I:9.1.1), it follows that $(\phi_*F)(T) \cong \Gamma(T, (\phi_*{\cal{M}})_T)$. Hence $\phi_*F$ is coherent. Now suppose that $n \geq 1$. By the (unstated) analog of \pref{lesx-exists} for quasi-coherent\ $X$-functors, there is a left exact sequence: \lesx{F}{G}{{\underline{\cal M}}% }in which $G$ is a quasi-coherent\ $X$-functor of level $n-1$ and ${\cal{M}}$ is a quasi-coherent\ ${\cal O}_X$-module. Then we have a left exact sequence: \lesx{\phi_*F}{\phi_*G}{\phi_*{\underline{\cal M}}% }By induction on $n$, we may assume that\ $\phi_*G$ is quasi-coherent. By the $n=0$ case, we may identify $\phi_*{\underline{\cal M}}$ with $\underline{\phi_*{\cal{M}}}$. Hence $\phi_*F$ is quasi-coherent. Parts (b) and (c) are left to the reader. {\hfill$\square$} \end{proof} The next result is key, since it permits us to reduce to the affine case, and thereby obtain the analogs of the results for module-valued\ $A$-functors. In particular, it will follow that many examples of $X$-functors are coherent. However, the reader interested only in the Picard group results may ignore the next result and its corollaries, since for purposes of the finiteness result \pref{coherent-implies-linear}, it is sufficient to know that a given $X$-functor is {\it locally coherent}. \begin{theorem}\label{locally-mc} Assume that $X$ is separated. Let $F$ be a locally module-quasi-coherent\ [resp.\ locally module-coherent\/] $X$-functor. Then $F$ is module-quasi-coherent\ [resp.\ module-coherent]. \end{theorem} \begin{proof} In the course of the proof we refer to sheaves, which shall always mean sheaves for the {\it Zariski topology}. We work not with $X$-functors, but with functors whose source is \opcat{$X$-schemes which are quasi-compact}, as discussed briefly at the beginning of this section. Let $\vec U1n$ be as in the definition of {\it locally module-quasi-coherent\/} (or {\it locally module-coherent}). By \pref{pull}, we may assume that\ each $U_i$ is affine. Since $X$ is separated, it follows that the open subschemes $U_i \cap U_j$ are affine and that the inclusions of $U_i$ in $X$ and of $U_i \cap U_j$ in $X$ are affine morphisms. First we prove the module-quasi-coherent\ case. (This will be needed for the module-coherent\ case.) Regard $F|_{U_i}$ and $F|_{U_i \cap U_j}$ as module-valued\ $X$-functors. It follows from \pref{push} and \pref{pull} that these are module-quasi-coherent. Because $F$ is a sheaf, we have a left exact sequence: \les{F}{\prod_{i=1}^n F|_{U_i}}{\prod_{1 \leq i < j \leq n} F|_{U_i \cap U_j}% }of module-valued\ $X$-functors. Hence $F$ is module-quasi-coherent. Now we show that if $F$ is a module-quasi-coherent\ $X$-functor, then there exists a morphism \mp[[ \phi || {\cal{M}} || {\cal{N}} ]] of quasi-coherent\ ${\cal O}_X$-modules such that $F \cong \mathop{\underline{\operatoratfont Ker}}\nolimits(\phi)$. By an unstated analog of \pref{les-exists}, we may embed $F$ as a sub-module-valued-functor of ${\underline{\cal M}}$ for some quasi-coherent\ ${\cal O}_X$-module ${\cal{M}}$. Let $G = ({\underline{\cal M}}/F)^*$, where the superscript $*$ denotes sheafification. By \pref{quasi-coherence-of-cokernel}, it follows that $G$ is locally module-quasi-coherent, so (by the first part of the proof) $G$ is module-quasi-coherent. Embed $G$ as a sub-module-valued-functor of ${\underline{\cal N}}$ for some quasi-coherent\ ${\cal O}_X$-module ${\cal{N}}$. Let \mp[[ \phi || {\cal{M}} || {\cal{N}} ]] be the induced map. Then $F = \mathop{\underline{\operatoratfont Ker}}\nolimits(\phi)$, as required. Now we begin the proof of the module-coherent\ case. By what we have already shown, we may assume that\ there exists a morphism \mp[[ \phi || {\cal{M}} || {\cal{N}} ]] of quasi-coherent\ ${\cal O}_X$-modules such that $F = \mathop{\underline{\operatoratfont Ker}}\nolimits(\phi)$. We will show that there exist coherent sub-${\cal O}_X$-modules ${\cal{M}}_0 \subset {\cal{M}}_1 \subset {\cal{M}}$ and ${\cal{N}}_0 \subset {\cal{N}}$ such that $\phi({\cal{M}}_0) \subset {\cal{N}}_0$ and such that in the induced diagram: \diagramx{\underline{{\cal{M}}_1}\cr \mapN{g}\cr \underline{{\cal{M}}_0}&\mapE{f}&\underline{{\cal{N}}_0}\cr% }we have $F \cong g[\ker(f)]^*$. Let us verify that the construction of this data will complete the proof. We must show that $g[\ker(f)]^*$ is module-coherent. Let ${\cal{P}}$ denote the co-fiber product of ${\cal{M}}_1$ with ${\cal{N}}_0$ over ${\cal{M}}_0$, taken in the category of quasi-coherent\ ${\cal O}_X$-modules. Then in fact the induced diagram \diagramx{\underline{{\cal{M}}_1}&\mapE{h}&{\underline{\cal P}}\cr \mapN{g}&&\mapN{}\cr \underline{{\cal{M}}_0}&\mapE{f}&\underline{{\cal{N}}_0}\cr% }is cocartesian, if it is viewed as a diagram in $$\cat{module-valued\ $X$-functors which are sheaves}.$% $Since we have $\ker(h) \cong g[\ker(f)]^*$, the theorem will follow. It remains to construct the data. Certainly, for any choice of data, there is a canonical morphism \dmap[[ \psi || g[\ker(f)]^* || F ]]% of module-valued\ $X$-functors. We work on choosing ${\cal{M}}_0$. Let $\sets {\cal{M}}\lambda\Lambda$ be the coherent sub-${\cal O}_X$-modules of ${\cal{M}}$. Let $H_\lambda$ be the sheafified image of the map \mapx[[ \underline{{\cal{M}}_\lambda} || {\underline{\cal M}} ]]. Then the $H_\lambda$ form a directed system of module-valued\ subfunctors of ${\underline{\cal M}}$ (which are sheaves), whose union is ${\underline{\cal M}}$. (The validity of the last assertion depends on the simplifying assumption made in the first paragraph of this proof, to the effect that we work only with quasi-compact $X$-schemes.) Let $F_\lambda = F \cap H_\lambda$. Then the $F_\lambda$ form a directed system of module-valued\ subfunctors of $F$ (which are sheaves), whose union is $F$. Then it follows from \pref{noetherian} that $F = F_\lambda$ for some ${\lambda \in \Lambda}$. Let ${\cal{M}}_0 = {\cal{M}}_\lambda$. Then $F$ is contained in the sheafified image of the map \mapx[[ \underline{{\cal{M}}_0} || {\underline{\cal M}} ]]. Now we work on choosing ${\cal{N}}_0$. Let $\sets {\cal{N}}\lambda\Lambda$ be the coherent sub-${\cal O}_X$-modules of ${\cal{N}}$ which contain $\phi({\cal{M}}_0)$. Let $I_\lambda$ be the sheafified image of the map \mapx[[ \ker[\underline{{\cal{M}}_0}\ \mapE{}\ \underline{{\cal{N}}_\lambda}] || {\underline{\cal M}} ]]. Then the $I_\lambda$ form a directed system of module-valued\ subfunctors of $F$ (which are sheaves). We will show that the union of the $I_\lambda$ is $F$. Let $J_\lambda = \ker[\underline{{\cal{M}}_0}\ \mapE{}\ \underline{{\cal{N}}_\lambda}]$. It suffices to show that $\ker[\underline{{\cal{M}}_0}\ \mapE{}\ {\underline{\cal N}}]$ is the union of the $J_\lambda$. Let $Y$ be a quasi-compact $X$-scheme. Let $\alpha \in \Gamma[({\cal{M}}_0)_Y]$, and assume that $\alpha \mapsto 0$ in $\Gamma({\cal{N}}_Y)$. We must show that there exists some ${\cal{N}}_\lambda$ such that $\alpha \mapsto 0$ in $\Gamma[({\cal{N}}_\lambda)_Y]$. By (\Lcitemark 20\Rcitemark \ I.6.9.9), we know that ${\cal{N}}$ is the direct limit of the ${\cal{N}}_\lambda$. It follows that ${\cal{N}}_Y$ is the direct limit of the $({\cal{N}}_\lambda)_Y$. This implies the statement about $\alpha \mapsto 0$, and hence that the union of the $I_\lambda$ is $F$. Arguing as in the preceding paragraph, we see that for some ${\lambda \in \Lambda}$, we have $I_\lambda = F$. Let ${\cal{N}}_0 = {\cal{N}}_\lambda$. It follows now that now matter how we choose ${\cal{M}}_1$, the map $\psi$ will be an epimorphism, when viewed as a morphism in the category of module-valued\ $X$-functors which are sheaves. Now we work on choosing ${\cal{M}}_1$. Let $G = \ker(f)$. Then $G$ is module-coherent. Let $K = \ker[G\ \mapE{}\ {\underline{\cal M}} ]$. Then $K = \ker[G\ \mapE{}\ F ]$, so $K$ is locally module-coherent. Let $\sets {\cal{M}}\lambda\Lambda$ be the coherent sub-${\cal O}_X$-modules of ${\cal{M}}$ which contain ${\cal{M}}_0$. (These ${\cal{M}}_\lambda$ are not the same as those defined earlier.) Let $K_\lambda = \ker[ G\ \mapE{}\ \underline{{\cal{M}}_\lambda} ]$. Then the $K_\lambda$ form a directed system of module-valued\ subfunctors of $K$ (which are sheaves). Arguing as in the preceding paragraph, we see that the union of the $K_\lambda$ is $K$. But $K$ is locally module-coherent, so it follows from \pref{noetherian} that $K_\lambda = K$ for some ${\lambda \in \Lambda}$. Let ${\cal{M}}_1 = {\cal{M}}_\lambda$. Then $\psi$ is a monomorphism. Hence $\psi$ is an isomorphism. {\hfill$\square$} \end{proof} \begin{corollary}\label{mump} Assume that $X$ is separated. Let \mp[[ \phi || F || G ]] be a morphism of module-coherent\ $X$-funct\-ors. Then the Zariski sheaf associated to $\mathop{\operatoratfont Coker}\nolimits(\phi)$ is module-coherent. \end{corollary} \begin{corollary} Assume that $X$ is separated. Then the category of module-coherent\ $X$-functors is abelian. \end{corollary} \begin{corollary} Assume that $X$ is separated. Let $F$ be a module-coherent\ $X$-functor. Then $F$ has level $\leq 1$. That is, there exists a left exact sequence: \les{F}{{\underline{\cal M}}}{{\underline{\cal N}}% }in which ${\cal{M}}$ and ${\cal{N}}$ are coherent ${\cal O}_X$-modules. \end{corollary} The constructions $\mathop{\mathbf{Hom}}\nolimits$, $\mathop{\mathbf{Aut}}\nolimits$ and so forth which we defined in \S\ref{examples-section} make sense for $X$-functors. For example, if $F$ and $G$ are module-valued $X$-functors, we let $\mathop{\mathbf{Hom}}\nolimits(F,G)$ denote the $X$-functor given by $$Y \mapsto \mathop{\operatoratfont Hom}\nolimits_{\kern2pt{\operatoratfont module-valued}\ Y-{\operatoratfont functors}} (F|_Y, G|_Y).$% $ \begin{prop} Assume that $X$ is separated. Let $F$ and $G$ be module-coherent\ $X$-functors. Then \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm $\mathop{\mathbf{Hom}}\nolimits(F,G)$ is module-coherent. \end{prop} \begin{proof} By \pref{HOM-coherent} and \pref{locally-mc}, it suffices to show that $\mathop{\mathbf{Hom}}\nolimits(F,G)$ is a Zariski sheaf. This can be directly checked from the definition. {\hfill$\square$} \end{proof} Similarly, we have: \begin{corollary}\label{lump} Assume that $X$ is separated. Then for any module-coherent\ $X$-functor $F$, $\mathop{\mathbf{End}}\nolimits(F)$ is module-coherent, and $\mathop{\mathbf{Aut}}\nolimits(F)$ is coherent. Let $R$ and $S$ be module-coherent, algebra-valued $X$-functors. Then $\mathop{{\mathbf{Hom}}_{\kern1pt\operatoratfont alg}(R,S)$ and $\mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(R)$ are coherent. \end{corollary} Now we consider the extent to which a module-quasi-coherent\ $A$-functor can be viewed as a direct limit of module-coherent\ $A$-functors. These considerations will enter into an analysis of extensions of module-coherent\ $X$-functors, which will be the last topic discussed in this section. Unfortunately, it is not the case that every module-quasi-coherent\ $A$-functor $H$ is the direct limit of its module-coherent\ subfunctors. For an example, let $H = \mathop{\operatoratfont Im}\nolimits(\phi)$, where $\phi$ is as in remark \pref{not-coh-example}. If there existed a directed system $\sets H\lambda\Lambda$ of module-coherent\ subfunctors of $H$, with union $H$, it would follow by \pref{noetherian}, applied with $F = \underline{\xmode{\Bbb Z}}$, $F_\lambda = \phi^{-1}(H_\lambda)$, that $\phi$ factors through a module-coherent\ subfunctor of $\underline{{\Bbb Q}\kern1pt}$, and hence that $\ker(\phi)$ is module-coherent, which is not the case. \begin{definition} A module-valued\ $A$-functor $C$ is {\it bar-module-coherent\/} if it is module-quasi-coherent\ and if there exists a module-coherent\ $A$-functor $F$, together with an epimorphism \mapx[[ F || C ]]. \end{definition} A bar-module-coherent\ $A$-functor need not be module-coherent. For an example, let $H = \mathop{\operatoratfont Im}\nolimits(\phi)$, where $\phi$ is as in remark \pref{not-coh-example}. Then $H$ is bar-module-coherent, but not module-coherent, since otherwise $\ker(\phi)$ would be module-coherent. \begin{lemma}\label{directed-system-of-images} Let $F$ be a module-quasi-coherent\ $A$-functor. Then there exists a directed system $\sets F\lambda\Lambda$ of bar-module-coherent\ subfunctors of $F$, with union $F$. \end{lemma} \begin{proof} Choose $A$-modules $M$, $N$ and a left exact sequence: \lesmaps{F}{}{{\underline{M}}}{h}{{\underline{N}}% }of module-valued $A$-functors, in which $h$ is induced by a homomorphism \mp[[ \phi || M || N ]] of $A$-modules. Let ${\cal{S}}$ denote the collection $\setof{(M_\lambda,N_\lambda)}_{{\lambda \in \Lambda}}$ consisting of all pairs \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm\hbox{$(M_\lambda,N_\lambda)$} in which $M_\lambda$ is a finitely generated\ submodule of $M$, $N_\lambda$ is a finitely generated\ submodule of $N$, and $\phi(M_\lambda) \subset N_\lambda$. For each ${\lambda \in \Lambda}$, let \mp[[ h_\lambda || \underline{M_\lambda} || \underline{N_{\lambda}} ]] be the induced morphism of $A$-functors. Let $K_\lambda = \ker(h_\lambda)$. Then $K_\lambda$ is module-coherent. There is a canonical map \mp[[ f_\lambda || K_\lambda || F ]]. Let $F_\lambda = \mathop{\operatoratfont Im}\nolimits(f_\lambda)$. Then $F_\lambda$ is bar-module-coherent, and the $F_\lambda$ form a directed system of subfunctors of $F$. Let $B$ be a commutative $A$-algebra, and let $c \in F(B)$. Then $c \in M_B$. Choose a finitely generated\ submodule $M_\lambda \subset M$ and an element $c_\lambda \in (M_\lambda)_B$ such that $c_\lambda \mapsto c$. There exists a finitely generated\ submodule $N_\lambda$ of $N$ such that $\phi(M_\lambda) \subset N_\lambda$ and such that $c_\lambda \mapsto 0$ in $(N_\lambda)_B$. It follows that $\cup_{{\lambda \in \Lambda}}F_\lambda = F$. {\hfill$\square$} \end{proof} We close this section with some questions and a result about extensions: \begin{problemx} Let \Rowfive{1}{F'}{F}{F''}{1% }be a short exact sequence of group-valued $X$-functors. Assume that $F'$ and $F''$ are coherent. Is $F$ coherent? \end{problemx} \begin{problemx} Let \ses{F'}{F}{F''% }be a short exact sequence of module-valued\ $X$-functors. Assume that $F'$ and $F''$ are module-coherent. Is $F$ module-coherent? \end{problemx} We can prove this if we assume that $F$ is module-quasi-coherent\ and that $X$ is separated: \begin{prop}\label{extension-is-mc} Assume that $X$ is separated. Let \ses{F'}{F}{F''% }be a short exact sequence of module-valued\ $X$-functors. Assume that $F'$ and $F''$ are module-coherent. Assume that $F$ is module-quasi-coherent. Then $F$ is module-coherent. \end{prop} \begin{proof% }By \pref{locally-mc}, we may reduce to working with $A$-functors. By \pref{directed-system-of-images}, $F$ is the direct limit of its bar-module-coherent\ subfunctors. By \pref{noetherian}, it follows that there exists a bar-module-coherent\ subfunctor $B$ of $F$ such that $B$ maps onto $F''$. Since $F'$ is module-coherent, we see that $F$ is itself bar-module-coherent. Choose a module-coherent\ $A$-functor $C$ and an epimorphism \mp[[ \beta || C || F ]]. Let $P$ be the fiber product of $F'$ and $C$ over $F$. Then we have a commutative diagram with exact rows: \diagramx{\rowfive{0}{P}{C}{F''}{0}\cr &&\mapS{\alpha}&&\mapS{\beta}&&\mapS{=}\cr \rowfive{0}{F'}{F}{F''}{0}\makenull{.}% }Hence $\ker(\alpha) \cong \ker(\beta)$. Since $C$ and $F''$ are module-coherent, so is $P$. Since $F'$ and $P$ are module-coherent, so is $\ker(\alpha)$. Hence $\ker(\beta)$ is module-coherent. Since $\ker(\beta)$ and $C$ are module-coherent, it follows by \pref{coherence-of-cokernel} that $F$ is also module-coherent. {\hfill$\square$} \end{proof} This fact will be used in the proof of \pref{mulch-material}. \block{Continuity}\label{continuity-section} We consider the extent to which quasi-coherent\ $A$-functors preserve limits, and briefly, the extent to which they preserve colimits. We prove that a module-quasi-coherent\ $A$-functor which preserves products is module-coherent. Although these topics do not play much of a role in the subsequent parts of this paper, they are very natural. Some important examples of limit and colimit preservation which have arisen previously are Grothendieck's theorem on formal functions (see e.g.{\ }\Lcitemark 22\Rcitemark \ III:11), and Grothendieck's notion of functors which are locally of finite presentation (\Lcitemark 1\Rcitemark \ 1.5), which enters into Artin's criterion for representability (\Lcitemark 3\Rcitemark \ 3.4). We begin by recalling some definitions about continuity of functors. Let ${\cal{C}}$ and ${\cal{A}}$ be complete (meaning small-complete) categories, and let \mp[[ F || {\cal{C}} || {\cal{A}} ]] be any functor. Then $F$ is {\it continuous\/} if it {\it preserves limits}, i.e.\ if for every small category ${\cal{D}}$, and every functor \mp[[ H || {\cal{D}} || {\cal{C}} ]], the canonical map \dmapx[[ F \left( \displaystyle{\lim_{\overleftarrow{\hphantom{\lim}}}}\ H \right) || \displaystyle{\lim_{\overleftarrow{\hphantom{\lim}}}}(F \circ H) ]]% is an isomorphism. (See e.g.{\ }\Lcitemark 27\Rcitemark \ V.4.) It is also of interest to know if $F$ preserves more restricted sorts of limits, e.g.\ does it preserve arbitrary products, or does it preserve equalizers. These conditions correspond to placing appropriate restrictions on ${\cal{D}}$. For particular sorts of limits, one can usually rephrase the continuity condition in a simpler way. For example, $F$ preserves products if and only if\ for every set \sets XiI of objects in ${\cal{C}}$, the canonical map \dmapx[[ F \left( \prod_{i \in I} X_i \right) || \prod_{i \in I} F(X_i) ]]% is an isomorphism. Just as one can check a category for completeness by checking if it has products and equalizers, so one can check a functor for continuity by checking if it preserves products and equalizers. We will study the continuity properties of $A$-functors. There are two general observations to be made. The first observation is that an (abelian group)-valued $A$-functor or a module-valued\ $A$-functor is continuous (or preserves a particular kind of limit) if and only if\ the same statement holds for the underlying functor from \cat{commutative $A$-algebras} to \cat{sets}. The second observation is the following lemma, whose proof is left to the reader: \begin{lemma}\label{kernel-preserves} Let $M$ be an $A$-module and let \lesx{F}{G}{{\underline{M}}% }be a left exact sequence of $A$-functors. If ${\underline{M}}$ and $G$ preserve a particular type of limit, then so does $F$. \end{lemma} The phrase ``particular type of limit'' is to be construed as referring to a class of limits which is constrained by some restriction on the categories ${\cal{D}}$ and/or the functors \mp[[ H || {\cal{D}} || \cat{commutative $A$-algebras} ]] which are allowed. We proceed to investigate the extent to which quasi-coherent\ $A$-functors preserve various types of limits. First we consider finite limits, that is limits in which the category ${\cal{D}}$ has only finitely many objects and morphisms. The most important examples are finite products (including terminal objects), and equalizers. Also, if a functor preserves finite products and equalizers, then it preserves all finite limits. As for finite products, one sees easily (using \ref{kernel-preserves}) that: \begin{prop}\label{number-1} Let $F$ be a quasi-coherent\ $A$-functor. Then $F$ preserves finite products. \end{prop} Now we consider equalizers. An $A$-module $M$ is flat if and only if\ the functor \fun[[ \o*_A M || $A$-modules || $A$-modules ]] preserves equalizers, so the analogous fact for $A$-functors is hardly surprising: \begin{prop}\label{commutes-eq-iff-flat} Let $M$ be an $A$-module. Then the $A$-functor ${\underline{M}}$ preserves equalizers if and only if\ $M$ is flat. \end{prop} \begin{proof} The case where $M$ is flat is left to the reader. So suppose that $M$ is not flat. Then (see e.g.{\ }\Lcitemark 28\Rcitemark \ 3.53) there exists an ideal $I \subset A$ such that the induced map \mp[[ \phi || M \o*_A I || M ]] is not injective. Let $y \in \ker(\phi) - \setof{0}$. Let $\vec a1n$ be generators for $I$. Choose $\vec m1n \in M$ such that $y = \sum_{i=1}^n m_i \o* a_i$. Let $B = A[\vec x1n]/(\setof{x_i x_j}_{1 \leq i,j \leq n})$. Let \mp[[ f,g || B || A ]] be the $A$-algebra maps given by $f(x_i) = a_i$ and $g(x_i) = 0$ for each $i$. Let $p \in M \o*_A B$ be $\sum_{i=1}^n m_i \o* x_i$. Then $(f \o*_A M)(p) = (g \o*_A M)(p)$. Let $\mathop{\operatoratfont Eq}\nolimits(f,g)$ be the equalizer of $f$ and $g$. Since $y \not= 0$, it follows (after a little work) that $p$ does not lie in the image of the canonical map \mp[[ \lambda || \mathop{\operatoratfont Eq}\nolimits(f,g) \o*_A M || \mathop{\operatoratfont Eq}\nolimits(f \o*_A M, g \o*_A M) ]], and hence that $\lambda$ is not an isomorphism. Hence ${\underline{M}}$ does not preserve the equalizer of $f$ and $g$. {\hfill$\square$} \end{proof} It follows that ${\underline{M}}$ preserves finite limits if and only if\ $M$ is flat. In addition, \pref{kernel-preserves} yields the following corollary: \begin{corollary}\label{number-2} Any quasi-coherent\ $A$-functor which is built up from flat modules will preserve finite limits. \end{corollary} It is worth noting that if $F$ is an $A$-functor which preserves finite limits, then $F$ is a sheaf with respect to\ the fppf topology; indeed the sheaf axioms may be viewed as a statement about continuity. While quasi-coherent\ $A$-functors do not always preserve finite limits, we do know that they are sheaves with respect to\ the fppf topology. \begin{remark}\label{number-4} It is not difficult to see that any coherent $A$-functor preserves products. Any coherent $A$-functor which is built up from finitely generated\ projective $A$-modules will also preserve equalizers, and hence all limits. Of course, the $A$-functors which are built up in this way are exactly the $A$-functors which are representable by an $A$-algebra of finite type. \end{remark} In general, coherent $A$-functors do not preserve inverse limits. For example, if $A = \xmode{\Bbb Z}$, then the $A$-functor $\underline{\xmode{\Bbb Z}/2\xmode{\Bbb Z}}$ does not preserve the limit of \diagramx{\xmode{\Bbb Z}[x]/(x^2) & \mapW{x\kern2pt\mapsto\kern2pt 3x} & \xmode{\Bbb Z}[x]/(x^2) & \mapW{x\kern2pt\mapsto\kern2pt 3x} & \cdots.% }The transition maps here are not surjective. One might hope that the limit would be preserved if the transition maps were surjective, or at least if the system satisfied the Mittag-Leffler condition (\Lcitemark 17\Rcitemark \ 0:13.1). Unfortunately this is not the case: \begin{example} Let $A = \xmode{\Bbb Z}$. Let $$B_n = \xmode{\Bbb Z}[x_1, x_2,\ldots, y, z_1,\ldots,z_{n-1}] / (2x_1,2x_2,\ldots,Q),$% $where $Q$ denotes the set of homogeneous quadratic polynomials in all of the given variables. Form an inverse system of commutative $A$-algebras \diagramx{B_1 & \mapW{} & B_2 & \mapW{} & B_3 & \mapW{} & \cdots% }in which (for each $n > 1$) the transition map \mapx[[ B_n || B_{n-1} ]] is given by $x_k \mapsto x_{k+1}$, $y \mapsto x_1 + y$, $z_1 \mapsto x_1$, $z_k \mapsto z_{k-1}$ for $k \geq 2$. The transition maps of this system are surjective. The elements $2y, 2y, \ldots$ form a coherent sequence. Each element in this sequence is divisible by $2$, but there is no coherent sequence which when multiplied by $2$ yields $2y, 2y, \ldots$. Hence the $A$-functor $\underline{\xmode{\Bbb Z}/2\xmode{\Bbb Z}}$ does not preserve the limit of this system. \end{example} {}From (\Lcitemark 4\Rcitemark \ 10.13) and \pref{kernel-preserves} it follows that: \begin{prop}\label{number-5} Let $F$ be a coherent $A$-functor. Let $B$ be a commutative noetherian $A$-algebra. Let $I \subset B$ be an ideal. Then $F$ preserves the limit of \diagramx{B/I & \mapW{} & B/I^2 & \mapW{} & B/I^3 & \cdots.} \end{prop} There may well be interesting situations in which coherent functors preserve inverse limits, other than those given in \pref{number-5} and \pref{number-4}. Our next objective is to show that a module-quasi-coherent\ $A$-functor which preserves products is module-coherent. This as well as \pref{commutes-eq-iff-flat} allow one to use knowledge about continuity to deduce some information about how a quasi-coherent\ $A$-functor is built up. \begin{lemma}\label{fg-sufficient} Let $L_1$ and $L_2$ be module-quasi-coherent\ subfunctors of a module-quasi-coherent\ $A$-functor $H$. If $L_1(B) \subset L_2(B)$ for every finitely generated\ commutative $A$-algebra $B$, then $L_1 \subset L_2$. \end{lemma} \begin{proof} Let $G = L_1 / (L_1 \cap L_2)$. Since $G$ is module-quasi-coherent, we have $G \cong \mathop{\underline{\operatoratfont Ker}}\nolimits(f)$ for some homomorphism \mp[[ f || M || N ]] of $A$-modules. Then $\mathop{\operatoratfont Ker}\nolimits(f \o*_A B) = 0$ for every finitely generated\ commutative $A$-algebra $B$, from which it follows that $\mathop{\operatoratfont Ker}\nolimits(f \o*_A B) = 0$ for {\it every\/} commutative $A$-algebra $B$. Hence $G = 0$. {\hfill$\square$} \end{proof} \begin{lemma}\label{factor-fg} Let \mp[[ \phi || H || N ]] be a homomorphism of $A$-modules. Assume that $H$ is finitely generated. Let $B$ be a commutative $A$-algebra. Let $h \in \mathop{\operatoratfont Ker}\nolimits(\phi \o*_A B)$. Then there exists a finitely generated\ submodule $N_0$ of $N$ such that $\phi$ factors through $N_0$ and such that $h \mapsto 0$ in $N_0 \o*_A B$. \end{lemma} \begin{proof} Certainly $N$ is the direct limit of its finitely generated\ submodules which contain $\phi(H)$. The statement follows from the fact that tensor products commute with direct limits. {\hfill$\square$} \end{proof} \begin{prop}\label{product-preserving} Let $F$ be a module-quasi-coherent\ $A$-functor which preserves products. Then $F$ is module-coherent. \end{prop} \begin{proof} All tensor products in this proof are over $A$. We may assume that $F = \mathop{\underline{\operatoratfont Ker}}\nolimits(f)$ for some homomorphism \mp[[ f || M || N ]] of $A$-modules. We will show that there exists a finitely generated\ submodule $L \subset M$ such that if \mp[[ i || L || M ]] is the inclusion, then $\mathop{\underline{\operatoratfont Ker}}\nolimits(f) \subset \mathop{\underline{\operatoratfont Im}}\nolimits(i)$. Choose a complete set of isomorphism class representatives $\sets B\lambda\Lambda$ for the finitely generated\ commutative $A$-algebras. Form the disjoint union $T = \coprod_{{\lambda \in \Lambda}} \mathop{\operatoratfont Ker}\nolimits(f \o* B_\lambda)$. For any $t \in T$, let $\lambda(t)$ denote the corresponding element of $\Lambda$. Let $B = \prod_{t \in T} B_{\lambda(t)}$. The elements $t \in T$ define an element $x \in \prod_{t \in T} \mathop{\operatoratfont Ker}\nolimits(f \o* B_{\lambda(t)})$. Since $F$ preserves products, $x \in \mathop{\operatoratfont Ker}\nolimits(f \o* B)$. Write $x = \sum_{j=1}^r m_j \o* b_j$, where $\vec m1r \in M$, $\vec b1r \in B$. Let $L$ be the submodule of $M$ generated by $\vec m1r$. The expansion of $x$ defines an element ${\tilde{\lowercase{X}}} \in L \o* B$ with the property that ${\tilde{\lowercase{X}}} \mapsto x$. From this it follows that $\mathop{\operatoratfont Ker}\nolimits(f \o* B_\lambda) \subset \mathop{\operatoratfont Im}\nolimits(i \o* B_\lambda)$ for every ${\lambda \in \Lambda}$. By \pref{fg-sufficient}, we have $\mathop{\underline{\operatoratfont Ker}}\nolimits(f) \subset \mathop{\underline{\operatoratfont Im}}\nolimits(i)$. Since ${\tilde{\lowercase{X}}} \mapsto 0$ in $N \o* B$, it follows from \pref{factor-fg} that there exists a finitely generated\ submodule $N_0$ of $N$ such that \mapx[[ L || N ]] factors through $N_0$ and such that ${\tilde{\lowercase{X}}} \mapsto 0$ in $N_0 \o* B$. For any $t \in T$, let $\lambda = \lambda(t)$, and let ${\tilde{\lowercase{T}}} \in L \o* B_\lambda$ be the image of ${\tilde{\lowercase{X}}}$ under the canonical map \mp[[ \pi_t || L \o* B || L \o* B_\lambda ]] which projects onto the \th{t} factor. Then $(i \o* B_\lambda)({\tilde{\lowercase{T}}}) = t$, and ${\tilde{\lowercase{T}}} \mapsto 0$ in $N_0 \o* B_\lambda$. Let ${\underline{K}} = \mathop{\underline{\operatoratfont Ker}}\nolimits(L\ \mapE{}\ N_0)$. Then ${\underline{K}}(B_\lambda)$ maps {\it onto\/} $F(B_\lambda)$. By \pref{fg-sufficient}, the map \mp[[ \psi || {\underline{K}} || F ]] is an epimorphism. In particular, $F$ is the image of a module-coherent\ $A$-functor. Let $Q = \mathop{\operatoratfont Ker}\nolimits(\psi)$. Since ${\underline{K}}$ is module-coherent, it preserves products. Since $F$ also preserves products, it follows by \pref{kernel-preserves} that $Q$ preserves products. Replaying the first part of the proof, with $F$ replaced by $Q$, we see that $Q$ is also the image of a module-coherent\ $A$-functor. Hence $F$ is the cokernel of a map of module-coherent\ $A$-functors, so $F$ is module-coherent. {\hfill$\square$} \end{proof} The last objective of this section is to consider (briefly) the extent to which coherent functors preserve colimits. Whether or not coherent functors preserve finite colimits and coproducts does not seem to be an interesting question. One reason for this is that the forgetful functor from \cat{groups} to \cat{sets} does not preserve coproducts or finite colimits. (One can substitute various other categories for \cat{groups} with the same outcome.) Therefore a functor from \cat{commutative $A$-algebras} to \cat{groups} might preserve such colimits, but the induced functor from \cat{commutative $A$-algebras} to \cat{sets} might not. With either interpretation, preservation of coproducts or finite colimits seems like a bizarre requirement. On the other hand, the forgetful functor \funx[[ groups || sets ]] does preserve direct limits\footnote{The reader is reminded that direct limits are a type of {\it colimit}.}, and the same statement is valid with various other categories substituted for \cat{groups}. Therefore, the situation for direct limits is just like the situation which holds for all limits: an (abelian group)-valued $A$-functor or a module-valued\ $A$-functor preserves direct limits if and only if\ the same statement holds for the underlying functor from \cat{commutative $A$-algebras} to \cat{sets}. The analog of \pref{kernel-preserves} for direct limits is valid, and since tensor products commute with direct limits, it follows that any quasi-coherent\ $A$-functor preserves direct limits. Artin remarks that nearly all $A$-functors which occur in practice do this; in Artin's terminology an $A$-functor which preserves direct limits is said to be {\it locally of finite presentation} (\Lcitemark 1\Rcitemark \ 1.5). This condition enters into his criterion for representability (\Lcitemark 3\Rcitemark \ 3.4). \block{Coherence of higher direct images as functors}\label{higher-section} In this section we consider a question which was posed (in an equivalent form) by Artin\Lspace \Lcitemark 2\Rcitemark \Rspace{}: \begin{problem} Let $X$ be a proper $A$-scheme, let ${\cal{F}}$ be a coherent sheaf on $X$, and fix $n \geq 0$. Is the $A$-functor $H = H^n_{\cal{F}}$ given by $B \mapsto H^n(X_B, {\cal{F}}_B)$ module-coherent? \end{problem} Taking the \u Cech resolution of ${\cal{F}}$ relative to some affine open cover of $X$ yields a complex $K$ of $A$-modules, and by (\Lcitemark 17\Rcitemark \ 1.4.1) we have $H^n(X_B, {\cal{F}}_B) \cong H^n(K \o* B)$ for all commutative $A$-algebras $B$. It follows that at least $H$ is {\it module-quasi-coherent}. The issue of whether $H$ is module-coherent\ is quite subtle. We will show that if ${\cal{F}}$ is $A$-flat, or $A$ is a Dedekind domain, then $H$ is module-coherent. By \pref{product-preserving}, we know also that $H$ is module-coherent\ if and only if\ $H$ preserves products, but it is not clear how to use this statement. We will give an example which shows that in general $H$ is not module-coherent. Let us say that a module-valued\ $A$-functor $F$ is {\it upper semicontinuous\/} if for every commutative $A$-algebra $B$, and every ${\xmode{{\fraktur{\lowercase{P}}}}} \in \mathop{\operatoratfont Spec}\nolimits(B)$, there is a neighborhood $U$ of ${\xmode{{\fraktur{\lowercase{P}}}}}$ such that $\dim_{k({\xmode{{\fraktur{\lowercase{Q}}}}})} F(k({\xmode{{\fraktur{\lowercase{Q}}}}})) \leq \dim_{k({\xmode{{\fraktur{\lowercase{P}}}}})} F(k({\xmode{{\fraktur{\lowercase{P}}}}}))$ for all ${\xmode{{\fraktur{\lowercase{Q}}}}} \in U$. Similarly, one defines {\it lower semicontinuous\/} by reversing the inequality. If $M$ is a finitely generated\ $A$-module, then it follows by Nakayama's lemma that ${\underline{M}}$ is upper semicontinuous. If \mp[[ f || P_1 || P_2 ]] is a map of finitely generated\ projective $A$-modules, then $\mathop{\underline{\operatoratfont Ker}}\nolimits(f)$ is upper semicontinuous, whereas $\mathop{\underline{\operatoratfont Im}}\nolimits(f)$ is lower semicontinuous. For $I \subset A$ an ideal, $\mathop{\underline{\operatoratfont Ker}}\nolimits(A\ \mapE{}\ A/I)$ is in general not upper semicontinuous. (But it is lower semicontinuous.) If the numerator of a quotient is upper semicontinuous and the denominator is lower semicontinuous, then the quotient is itself upper semicontinuous. It follows that if $K_0$ is a complex of finitely generated\ free $A$-modules, and $n \in \xmode{\Bbb Z}$, then the module-valued\ $A$-functor given by $B \mapsto H^n(K_0 \o* B)$ is upper semicontinuous. It is also module-coherent. \begin{prop}\label{flat-implies-cohomologically-coherent} If $X$ is proper over $A$ and ${\cal{F}}$ is a coherent sheaf on $X$ which is $A$-flat, then the module-valued\ $A$-functor $H^n_{\cal{F}}$ given by $B \mapsto H^n(X_B, {\cal{F}}_B)$ is module-coherent\ and upper semicontinuous. \end{prop} \begin{proofnodot} (pointed out to me by Deligne and Ogus; cf.{\ }\Lcitemark 23\Rcitemark .) Let $K$ be the \u Cech complex discussed above. It is bounded and flat. Since $X$ is proper over $A$, the complex $K$ has finitely generated\ cohomology modules. It follows that there exists a complex $K_0$ of finitely generated\ free $A$-modules which is bounded above and a quasi-isomorphism \mp[[ \phi || K_0 || K ]]. Since $K$ is flat and bounded above, there is a spectral sequence $$E_{p,q}^2\ =\ \mathop{\operatoratfont Tor}\nolimits_p(H^{-q}(K), B)\ \Longrightarrow\ H^{-p-q}(K \o* B).$% $(See e.g.{\ }\Lcitemark 28\Rcitemark \ 11.34.) Similarly, one has such a spectral sequence for $K_0$. Moreover, $\phi$ induces a morphism from the spectral sequence for $K_0$ to the spectral sequence for $K$. Since $\phi$ is a quasi-isomorphism, the induced maps \mapx[[ H^{-q}(K_0) || H^{-q}(K) ]] are isomorphisms, and so the induced maps \mapx[[ \mathop{\operatoratfont Tor}\nolimits_p(H^{-q}(K_0), B) || \mathop{\operatoratfont Tor}\nolimits_p(H^{-q}(K), B) ]] are isomorphisms. Hence $H^n(\phi)$ is an isomorphism. Hence the module-valued\ $A$-functor $B \mapsto H^n(X_B, {\cal{F}}_B)$ is isomorphic to the module-valued\ $A$-functor $B \mapsto H^n(K_0 \o* B)$. {\hfill$\square$} \end{proofnodot} The {\it upper semicontinuity\/} part of the proposition is of course the usual theorem on upper semicontinuity of cohomology (see e.g.{\ }\Lcitemark 22\Rcitemark \ III\ 12.8). \begin{theorem}\label{mulch-material} If $X$ is proper over a Dedekind domain $A$ and ${\cal{F}}$ is a coherent sheaf on $X$, then the module-valued\ $A$-functor $H^n_{\cal{F}}$ given by $B \mapsto H^n(X_B, {\cal{F}}_B)$ is module-coherent. \end{theorem} Before proving this, there are some preliminaries. By a {\it truncated discrete valuation ring}, we shall mean a ring $A$ of the form $R/I$ where $R$ is a discrete valuation ring\ and $I$ is a proper nonzero ideal. By a {\it uniformizing parameter\/} for $A$, we shall mean the image in $A$ of a uniformizing parameter for $R$. \begin{lemma}\label{mulch-material-lemma} Let $A$ be a truncated discrete valuation ring\ with uniformizing parameter $t$. Let $C$ be a bounded complex of $A$-modules. Assume that $H^n(C \o* A/(t^l))$ is finitely generated\ for all $n$ and all $l$. Then for each $n$, the $A$-functor $F$ given by $B \mapsto H^n(C \o* B)$ is module-coherent. \end{lemma} \begin{proof} Any module $M$ over $A$ is a direct sum of cyclic modules. (See e.g.{\ }\Lcitemark 11\Rcitemark \ Ch.\ VII\ \S2\ exercise 12(b).) It follows that if $M_0$ is a finitely generated\ submodule of $M$, then there exists a finitely generated\ direct summand $M_1$ of $M$ with $M_0 \subset M_1$. For each $l$, let $A_l$ denote $A/(t^l)$. By working from low indices to high indices, one can construct a subcomplex $C_0$ of $C$ with the properties that for each $n$: \begin{alphalist} \item for each $l$, $\mathop{\operatoratfont Ker}\nolimits[ d^n \o* A_l ] \subset C_0^n \o* A_l + \mathop{\operatoratfont Im}\nolimits[ d^{n-1} \o* A_l ]$ and \item $C_0^n$ is finitely generated\ and is a direct summand of $C^n$. \end{alphalist} Property (b) comes from the first paragraph. Let ${\overline{C}} = C/C_0$. It follows from (b) that the sequence \ses{C_0}{C}{{\overline{C}}% }is universally exact. Since any $A$-module is a direct sum of cyclic modules, it follows from (a) that the induced map \mapx[[ H^n(C_0 \o* B) || H^n(C \o* B) ]] is surjective for every $n$ and every $B$, and hence that we have short exact sequences: \sesdot{H^{n-1}({\overline{C}} \o* B)}{H^n(C_0 \o* B)}{H^n(C \o* B)% }Now construct a subcomplex ${\overline{C}}_0 \subset {\overline{C}}$ in the same way that we constructed $C_0 \subset C$. It follows that $F$ is expressible as the cokernel of a map of module-coherent\ functors, and hence that $F$ is itself module-coherent. {\hfill$\square$} \end{proof} Now we prove \pref{mulch-material}, using arguments provided by Deligne. \begin{proof} There is an exact sequence of coherent sheaves on $X$: \ses{{\cal{F}}'}{{\cal{F}}}{{\cal{F}}''% }in which ${\cal{F}}'$ is $A$-torsion and ${\cal{F}}''$ is $A$-torsion-free. Since $A$ is a Dedekind domain, ${\cal{F}}''$ is $A$-flat. Hence this sequence remains exact after tensoring over $A$ by anything, so by the long exact sequence of cohomology and \pref{extension-is-mc}, it suffices to prove the theorem when ${\cal{F}}$ is either $A$-torsion or $A$-flat. The second case is taken care of by \pref{flat-implies-cohomologically-coherent}. Therefore we may assume that\ ${\cal{F}}$ is $A$-torsion. Then in fact $a{\cal{F}} = 0$ for some nonzero $a \in A$. We may then reduce to the case where $A$ is a truncated discrete valuation ring. Apply \pref{mulch-material-lemma} to the \u Cech complex of ${\cal{F}}$. {\hfill$\square$} \end{proof} Finally we give a counterexample which shows that in general, $H^n_{\cal{F}}$ is not module-coherent. The counterexample to be given here is based on examples constructed by H.\ Cohen\Lspace \Lcitemark 13\Rcitemark \Rspace{}. The same examples appear in his thesis\Lspace \Lcitemark 14\Rcitemark \Rspace{}, where Cohen remarks briefly that the techniques therein yield a counterexample to Artin's problem. However, he says nothing more about the matter. It seems likely that Cohen and/or Verdier did construct a counterexample, but it has now (apparently) been lost. Let $k$ be a field, let $A = k[[s,t]]$, and let $X = \P3_A$, with coordinates $x,y,z,w$. Let ${\cal{F}}$ be the cokernel of the map \mapx[[ {\cal O}_X || {\cal O}_X(1) ]] given by multiplication by $sx-ty$% .\footnote{In terms of Cohen's construction, we have $r = 3$, $c_0 = s$, $c_1 = -t$, and $m = 0$.} Let $n = 1$. We will show that $H$ is not module-coherent, using \pref{ding-dong}. In his proposition 2, Cohen shows (in effect) that if $B$ is a commutative $A$-algebra, and $R = B[x^{\pm 1}, y^{\pm 1}, z^{\pm 1}, w^{\pm 1}]$, then $H(B)$ is isomorphic to the degree $0$ part of the quotient of $$\left\{ {f \over x^a y^b z^c w^d} \in R: f \in B[x,y,z,w],\ a,b,c,d \in \xmode{\Bbb N}, \hbox{\ and\ } (sx-ty)f = 0 \right\}$% $by the sub-$B[x,y,z,w]$-module generated by $$\left\{ {f \over yzw}, {f \over xzw}, {f \over xyw}, {f \over xyz} \in R: f \in B[x,y,z,w] \hbox{\ and\ } (sx-ty)f = 0 \right\}.$% $ Let $B = A/(s^k, t^k)$, for some $k \in \xmode{\Bbb N}$. We proceed to compute $H(B)$. \begin{lemma} The $B[x,y,z,w]$-module $\mathop{\operatoratfont Ann}\nolimits_{B[x,y,z,w]}(sx-ty)$ is generated by $$(st)^{k-1},\ (st)^{k-2} \sum_{i=0}^1 (sx)^i (ty)^{1-i},\ \ldots, \ (st)^0 \sum_{i=0}^{k-1} (sx)^i (ty)^{k-1-i}.$$ \end{lemma} \begin{sketch} Let $C = k[s,t,x,y]/(s^k,t^k)$. Since $B[x,y,z,w]$ is a flat $C$-algebra, it suffices to show that the $C$-module $\mathop{\operatoratfont Ann}\nolimits_C(sx-ty)$ admits the given generators. Let $f \in \mathop{\operatoratfont Ann}\nolimits_C(sx-ty)$. Write $$f = \sum_{0 \leq i,j \leq k-1} f_{ij} s^i t^j,$% $where $f_{ij} \in k[x,y]$. The following assertions are easily checked: \begin{itemize} \item $f_{i,0} = 0$ if $0 \leq i \leq k-2$; \item $f_{0,j} = 0$ if $0 \leq j \leq k-2$; \item $f_{i,j-1} = (x/y) f_{i-1,j}$ if $1 \leq i,j \leq k-1$. \end{itemize} Hence $f$ is completely determined by $f_{0,k-1}, \ldots, f_{k-1,k-1}$. The lemma follows. {\hfill$\square$} \end{sketch} {}From this lemma it follows that $H(B)$ is isomorphic to the sub-$B$-module of $R$ generated by $$\bigcup_{j=5}^k \left\{ {(st)^{k-j} \sum_{i=0}^{j-1} (sx)^i (ty)^{j-1-i} \over x^a y^b z^c w^d}: a,b,c,d \in \xmode{\Bbb N} \hbox{\ and\ } a+b+c+d = j-1 \right\}.$% $Since this is a minimal generating set, we have: $$\mu[H(B)]\ =\ \sum_{j=5}^k {j-2 \choose 3}\ \geq\ O(k^3).$% $Since $(s,t)^{2k-1} \subset (s^k, t^k)$, it follows from \pref{ding-dong} that $H$ is not module-coherent. \block{Global sections% }\label{torsion-section} Let us say that a group is {\it linear\/} if it may be embedded as a subgroup of $\mathop{\operatoratfont GL}\nolimits_n(k_1) \times \cdots \times \mathop{\operatoratfont GL}\nolimits_n(k_r)$ for some $n$ and some fields $\vec k1r$. We shall want to have some control over the fields: a group is {\it $X$-linear\/} if there exist points $\vec x1r \in X$ (not necessarily closed) and finitely generated\ field extensions $k_i$ of $k(x_i)$ (for each $i$) such that the group may be embedded as a subgroup of $\mathop{\operatoratfont GL}\nolimits_n(k_1) \times \cdots \times \mathop{\operatoratfont GL}\nolimits_n(k_r)$ for some $n$. The purpose of this section is two-fold. The first purpose is to develop a tool (the next theorem) for proving that groups are $X$-linear. The second purpose is to study the torsion in $X$-linear groups. Our result is \pref{linear-implies-finite-torsion}, or in a slightly different form \pref{linear-bound}. \begin{theorem}\label{coherent-implies-linear} Let $G$ be a group-valued locally coherent $X$-functor. Then $G(X)$ is $X$-linear. \end{theorem} Before proving this theorem, we will study the torsion in $X$-linear groups. For any abelian group $H$, one can try to determine for which $n \in \xmode{\Bbb N}$ one has $\abs{{}_nH} < \infty$. If $n = p_1^{k_1} \cdots p_r^{k_r}$, where $\vec p1r$ are prime numbers, then $\abs{{}_nH} < \infty$ if and only if\ $\abs{{}_{p_i} H} < \infty$ for each $i$. Therefore we may as well restrict to the problem of determining when $\abs{{}_pH} < \infty$, where $p$ is prime. If $C$ is a commutative ring, let us say that a morphism \mp[[ \pi || X || \mathop{\operatoratfont Spec}\nolimits(C) ]] is {\it essentially of finite type\/} if there exists a commutative ring $D$, a homomorphism \mp[[ \phi || C || D ]] which is essentially of finite type, and a morphism of finite type \mp[[ \pi_0 || X || \mathop{\operatoratfont Spec}\nolimits(D) ]], such that $\pi = \mathop{\operatoratfont Spec}\nolimits(\phi) \circ \pi_0$. Note that if $X$ is essentially of finite type over $\xmode{\Bbb Z}$, and $x \in X$, then $k(x)$ is a finitely generated\ field extension of its prime subfield. \begin{prop}\label{linear-implies-finite-torsion} Let $H$ be an $X$-linear abelian group. \begin{alphalist} \item There are only finitely many prime numbers $p$ such that $H$ has infinite $p$-torsion. Moreover, such a $p$ cannot be invertible in $\Gamma(X,{\cal O}_X)$. \item If $X$ is essentially of finite type over $\xmode{\Bbb Z}$ or over $\xmode{\Bbb Z}_p$ (for some prime number $p$), then there exist prime numbers $\vec p1n$, none of which are invertible in $\Gamma(X,{\cal O}_X)$, such that the subgroup of $H$ consisting of torsion prime to $p_1 \cdot \ldots \cdot p_n$ is finite. \end{alphalist} \end{prop} Clearly, if in the above proposition, $X$ is essentially of finite type over $\xmode{\Bbb Z}_p$, then the list of primes $\vec p1n$ may be taken to be the single prime $p$. Also we have: \begin{corollary} Let $H$ be an $X$-linear abelian group. If $X$ is essentially of finite type over ${\Bbb Q}\kern1pt$ or over ${\Bbb Q}\kern1pt_p$ (for some prime number $p$), then the torsion subgroup of $H$ is finite. \end{corollary} We now state a generalization of the proposition to the non-abelian case: \begin{prop}\label{linear-bound} Let $H$ be an $X$-linear group. \begin{alphalist} \item Let $n \in \xmode{\Bbb N}$ be invertible in $\Gamma(X,{\cal O}_X)$. Then there exists some $N \in \xmode{\Bbb N}$, such that whenever $K$ is an $n$-torsion abelian subgroup of $H$, we have $\abs{K} \leq N$. \item If $X$ is essentially of finite type over $\xmode{\Bbb Z}$ or over $\xmode{\Bbb Z}_p$ (for some prime number $p$), then there exist prime numbers $\vec p1n$, none of which are invertible in $\Gamma(X,{\cal O}_X)$, and some $N \in \xmode{\Bbb N}$, such that if $K$ is an abelian subgroup of $H$, and every element of $K$ is torsion prime to $p_1 \cdot \ldots \cdot p_n$, then $\abs{K} \leq N$. \end{alphalist} \end{prop} \begin{lemma}\label{injective-on-roots} Let $(A,{\xmode{{\fraktur{\lowercase{M}}}}},k)$ be a local ring. Let $n \in \xmode{\Bbb N}$, and assume that $n$ is invertible in $A$. Then the canonical map: \dmapx[[ \setofh{\th{n} roots of unity in $A$} || \setofh{\th{n} roots of unity in $k$} ]]% is injective. \end{lemma} \begin{proof} Let $n \in \xmode{\Bbb N}$, $x \in A$, $a \in {\xmode{{\fraktur{\lowercase{M}}}}}$, and suppose that $x^n = (x+a)^n = 1$. Then $0 = [(x+a)^n - x^n] = a[nx^{n-1} + c]$, for some $c \in {\xmode{{\fraktur{\lowercase{M}}}}}$. But $nx^{n-1}$ is a unit, so $nx^{n-1}+c$ is a unit, so $a = 0$. {\hfill$\square$} \end{proof} It is known (\Lcitemark 11\Rcitemark \ \S14, \#7, Cor.\ 2 to Prop.\ 17) that a field finitely generated\ over its prime subfield (as a field extension) contains only finitely many roots of unity. This also holds for a field finitely generated\ over ${\Bbb Q}\kern1pt_p$. We will need a modest generalization of these statements: \begin{lemma}\label{root-bound} Let $K$ be a finitely generated\ field extension of $F$, where $F$ is ${\Bbb Q}\kern1pt$, or ${\Bbb F}\kern1pt_p$, or ${\Bbb Q}\kern1pt_p$, for some prime number $p$. Then there exists a constant $c$ such that for every $m \in \xmode{\Bbb N}$, and every finite field extension $L$ of $K$ with $[L:K] \leq m$, the number of roots of unity in $L$ is $\leq c m$ (if $F = {\Bbb Q}\kern1pt$), and is $\leq c^m$ if $F$ is ${\Bbb F}\kern1pt_p$ or ${\Bbb Q}\kern1pt_p$. \end{lemma} \begin{proof} Let $\vec x1r$ be a transcendence basis for $K$ over $F$. Let $s = [K:F(\vec x1r)]$. If $F = {\Bbb Q}\kern1pt$, let $c = 2s$. If $F = {\Bbb F}\kern1pt_p$, let $c = p^s$. If $F = {\Bbb Q}\kern1pt_p$, let $c = 2sp^s$. Let $L_0$ be the subfield of $L$ consisting of elements algebraic over $F$. Then $[L:F(\vec x1r)] = s[L:K] \leq sm$, so $[L_0:F] \leq sm$. If $F = {\Bbb Q}\kern1pt$ or $F = {\Bbb F}\kern1pt_p$, it is clear that the given $c$ works. Suppose that $F = {\Bbb Q}\kern1pt_p$. Extend the standard absolute value on ${\Bbb Q}\kern1pt_p$ to $L_0$. Let $(A,{\xmode{{\fraktur{\lowercase{M}}}}},k)$ be the valuation ring of $L_0$. If $x \in L_0$ is a root of unity, $\abs{x} = 1$, so $x \in A$. Since $k$ is an extension of ${\Bbb F}\kern1pt_p$ of degree $\leq sm$, the number of elements in $k$ is bounded by $p^{sm}$. By \pref{injective-on-roots}, it follows that for any $r \in \xmode{\Bbb N}$ which is prime to $p$ (and hence invertible in $A$), the number of \th{r} roots of unity in $A$ is $\leq p^{sm}$. For $p \not= 2$, ${\Bbb Q}\kern1pt_p$ has no \th{p} roots of unity other than $1$ (see \Lcitemark 24\Rcitemark \Rspace{}\ p.\ 20\ exercise 14). For $p = 2$, ${\Bbb Q}\kern1pt_p$ contains no square root of $-1$. For any field $M$, let $M'$ denote its subfield generated by \setof{x \in M: x^{(p^n)} = 1 \hbox{\ for some\ } n \in \xmode{\Bbb N}}. Then ${\Bbb Q}\kern1pt_p' = {\Bbb Q}\kern1pt$. Hence $[L_0':{\Bbb Q}\kern1pt] \leq sm$. Hence $$\abs{\setof{x \in L_0: x^{(p^n)} = 1 \hbox{\ for some\ } n \in \xmode{\Bbb N}}} \ \leq\ 2sm.$% $Hence the number of roots of unity in $L_0$ is $\leq (2sm)p^{sm} \leq c^m$. {\hfill$\square$} \end{proof} \begin{proofnodot} (of \ref{linear-bound}) For part (a), we may assume that\ $H \subset \mathop{\operatoratfont GL}\nolimits_r(k)$ for some field $k$, where $n$ is invertible in $k$. Let $\vec g1l \in \mathop{\operatoratfont GL}\nolimits_r(k)$ be distinct commuting elements with $g_i^n = 1$ for each $i$. We need to prove that there is some $N \in \xmode{\Bbb N}$ (independent of $\vec g1l$) such that $l \leq N$. Let $C$ be the subalgebra of $\mathop{\operatoratfont Mat}\nolimits_{r \times r}(k)$ generated by $\vec g1l$. Then $C$ is a commutative, artinian $k$-algebra, and $\mathop{\operatoratfont Spec}\nolimits(C)$ has at most $r$ components, since $\mathop{\operatoratfont Mat}\nolimits_{r \times r}(k)$ has at most $r$ distinct nonzero orthogonal idempotents. It follows from \pref{injective-on-roots} that the equation $x^n = 1$ has at most $n^r$ solutions in $C$. Let $N = n^r$. For part (b), the field $k$ will be a finitely generated\ field extension of ${\Bbb Q}\kern1pt$, ${\Bbb F}\kern1pt_p$, or ${\Bbb Q}\kern1pt_p$, for some prime number $p$. Construct $C$ as in the preceding paragraph. We have to bound $l$ in terms of $r$ alone, and not in terms of $n$. We may assume that $C$ is local. The residue field $L$ of $C$ is a finite extension of $k$, and $[L:k] \leq r^2$. Apply \pref{root-bound} and \pref{injective-on-roots}. {\hfill$\square$} \end{proofnodot} We now work towards a proof of \pref{coherent-implies-linear}. \begin{lemma}\label{artinian-exists} Let $M$ be a finitely generated\ $A$-module. Then there exists a commutative artinian $A$-algebra $B$, such that $B$ is essentially of finite type over $A$, and such that the canonical map \mapx[[ M || M \o*_A B ]] is injective. \end{lemma} \begin{proof} Let us say that an ideal $I \subset A$ is {\it good\/} if there exists a commutative artinian $(A/I)$-algebra $B_{[I]}$ which is essentially of finite type over $A/I$ such that the canonical map \dmapx[[ M \o*_A (A/I) || M \o*_A B_{[I]} ]]% is injective. Let $I \subset A$ be an ideal, and suppose that every ideal properly containing $I$ is good. To prove the lemma, it suffices (by a sort of noetherian induction) to show that $I$ is good. Replacing $A$ by $A/I$, we may assume that\ $I = 0$. Choose a primary decomposition $0 = Q_1 \cap \cdots \cap Q_r$ of $0$ in $M$. Then the map \mapx[[ M || M/Q_1 \times \cdots \times M/Q_r ]] is injective and each module $M/Q_i$ has a unique associated prime ${\xmode{{\fraktur{\lowercase{P}}}}}_i$. Since each module $M/Q_i$ admits a filtration with quotients isomorphic to $A/{\xmode{{\fraktur{\lowercase{P}}}}}_i$, it follows that there exists an integer $N$ such that the maps \mapx[[ M/Q_i || (M/Q_i) \o*_A A/{\xmode{{\fraktur{\lowercase{P}}}}}_i^N ]] are injective. Note also that the maps \mapx[[ M/Q_i || (M/Q_i) \o*_A A_{{\xmode{{\fraktur{\lowercase{P}}}}}_i} ]] are injective. We may assume that $\vec \lfP1r$ are arranged so that $\vec \lfP1k$ are minimal primes of $A$ and $\vec {\xmode{{\fraktur{\lowercase{P}}}}}{k+1}r$ are not. Then ${\xmode{{\fraktur{\lowercase{P}}}}}_{k+1}^N,\ldots,{\xmode{{\fraktur{\lowercase{P}}}}}_r^N$ are nonzero. Let \formulaqed{B = A_{{\xmode{{\fraktur{\lowercase{P}}}}}_1} \times A_{{\xmode{{\fraktur{\lowercase{P}}}}}_k} \times B_{[{\xmode{{\fraktur{\lowercase{P}}}}}_{k+1}^N]} \times B_{[{\xmode{{\fraktur{\lowercase{P}}}}}_r^N]}.} \end{proof} We use Witt rings in this and the next section. The reader may find treatments of the subject in (\Lcitemark 26\Rcitemark \ VIII\ exercises\ 42--44), (\Lcitemark 29\Rcitemark \ II\ \S5,\ \S6), and\Lspace \Lcitemark 9\Rcitemark \Rspace{}. Some remarks about notation are in order. There is a version of the Witt ring which does {\it not\/} depend on the choice of a prime number $p$, as discussed for example in (\Lcitemark 26\Rcitemark \ VIII\ exercise\ 42). It seems reasonable to denote this version of the Witt ring by $W(A)$. There is a second version of the Witt ring which does depend on the choice of a prime number $p$, as discussed for example in (\Lcitemark 26\Rcitemark \ VIII\ exercise\ 43). To avoid confusion, we will denote this version of the Witt ring by $W^p(A)$, as is done in (\Lcitemark 9\Rcitemark \ p.\ 179). However, the ring we denote by $W^p(A)$ is the same as the ring denoted $W(A)$ in\Lspace \Lcitemark 29\Rcitemark \Rspace{}. We will be using the ring $W^p(A)$, as well as the truncated version $W^p_n(A)$. \begin{lemma}\label{witt-coherent} Let $k$ be a perfect field of positive characteristic $p$. Fix $n \in \xmode{\Bbb N}$, and let $A = W^p_n(k)$. Let $F$ be a coherent $A$-functor. Define a $k$-functor ${\tilde{F}}$ by ${\tilde{F}}(B) = F(W^p_n(B))$, for all (commutative) $k$-algebras $B$. Then ${\tilde{F}}$ is coherent. \end{lemma} \begin{proof} We let\ $\tilde{}$\ denote the operation which is in effect defined in the statement. Let $r$ be the level of $F$. Suppose that $r = 0$, so we may assume that\ $F = {\underline{M}}$ for some finitely generated\ $A$-module $M$. Since $A = W^p(k)/(p^n)$, and $W^p(k)$ is a complete discrete valuation ring\ with uniformizing parameter $p$, it follows that $M$ may be expressed as a direct sum of modules of the form $A/(p^i)$, where $i \in \setof{1,\ldots,n}$. We may assume that in fact $M = A/(p^i)$. Then: $${\tilde{F}}(B)\ =\ A/(p^i) \o*_A W_n^p(B)\ =\ W_i^p(B),$% $which may be identified (as a set) with $B^i$. Hence ${\tilde{F}}$ is coherent. Now suppose that $r \geq 1$. By \pref{lesx-exists}, we may choose a left exact sequence: \lesx{F}{G}{{\underline{N}}% }of $A$-functors where $G$ is a coherent $A$-functor of level $r-1$ and $N$ is a finitely generated\ $A$-module. We obtain a left exact sequence: \lesx{{\tilde{F}}}{{\tilde{G}}}{\tilde{{\underline{N}}}% }of $k$-functors. By induction on $r$, we may assume that\ ${\tilde{G}}$ is coherent, and by the case $r = 0$, $\tilde{{\underline{N}}}$ is coherent. Hence ${\tilde{F}}$ is coherent. {\hfill$\square$} \end{proof} \begin{proofnodot} (of \ref{coherent-implies-linear}) Since $G$ is locally coherent, it is a sheaf with respect to\ the Zariski topology, so we may immediately reduce to the case where $X$ is affine, say $X = \mathop{\operatoratfont Spec}\nolimits(A)$. We work with $A$-functors. By \pref{lesx-exists}, there exists a finitely generated\ $A$-module $M$ and an embedding $G\ \includeE{}\ {\underline{M}}$ of $A$-functors. (Note that this map need not preserve the group structure; otherwise the proof would be much easier!) By \pref{artinian-exists}, we can choose an artinian $A$-algebra $B$ which is essentially of finite type over $A$ such that the map \mapx[[ {\underline{M}}(A) || {\underline{M}}(B) ]] is injective. Let $G|_B$ denote the $B$-functor given by $G|_B(C) = G(C)$. Then $G|_B$ is coherent by \pref{pull}. Therefore it suffices to show that $G|_B$ has the desired property. Replacing $A$ by $B$, we may reduce to the case where $A$ is artinian. Since $A$ is a product of Artin local rings, we may in fact reduce to the case where $A$ is an Artin local ring. First suppose that $A$ is a field. Then $A$ is a coherent $A$-functor, so $G$ is representable by an affine group scheme of finite type over $A$. Hence (\Lcitemark 10\Rcitemark \ 11.11) $G(A)$ embeds in $\mathop{\operatoratfont GL}\nolimits_n(A)$ for some $n$. Now suppose that $A$ contains a field. Then from the Cohen structure theorem for complete local rings, we know that $A$ contains a coefficient field $k$. Since $A$ is artinian, it follows that $A$ is module-finite over $k$. By \pref{push}, we may reduce to the case $A = k$. Finally, suppose that $A$ is an Artin local ring which does not contain a field. Then $A$ has mixed characteristic. Let ${\xmode{{\fraktur{\lowercase{M}}}}}$ be its maximal ideal, and let $k$ be its residue field. By (\Lcitemark 20\Rcitemark \ 0.6.8.3), there exists a (commutative) faithfully flat noetherian local $A$-algebra $({\tilde{A}},{\xmode{{\tilde{\fraktur{\lowercase{M}}}}}}, {\tilde{\lowercase{K}}})$ with ${\tilde{\lowercase{K}}}$ being an algebraic closure of $k$, such that ${\xmode{{\fraktur{\lowercase{M}}}}}{\tilde{A}} = {\xmode{{\tilde{\fraktur{\lowercase{M}}}}}}$. {}From the latter fact, it follows that ${\tilde{A}}$ is artinian. Since $G$ is coherent, by \pref{sheaf} it is a sheaf for the fpqc topology, so \mapx[[ F(A) || F({\tilde{A}}) ]] is injective. Hence we may assume that\ $A = {\tilde{A}}$ and so $k$ is algebraically closed. By the Cohen structure theorem, $A \cong W^p_n(k)[[\vec x1r]]/I$ for some $r,n \in \xmode{\Bbb N}$ and some ideal $I$. Since $W^p_n(k)$ maps onto the residue field of $A$, and since $A$ is artinian, it follows that $A$ is module-finite over $W^p_n(k)$. By \pref{push}, we may reduce to the case $A = W^p_n(k)$. Apply \pref{witt-coherent} to reduce to the case $A = k$. {\hfill$\square$} \end{proofnodot} \block{Global sections -- arithmetic case} In this section we refine the results of the last section, in the special case where $X$ is of finite type over $\xmode{\Bbb Z}$. In particular, (\ref{linear-implies-finite-torsion}b) is supplanted by \pref{arith-linear-structure}. Let us say that a group is {\it arithmetically linear\/} if it may be embedded as a subgroup of $\mathop{\operatoratfont GL}\nolimits_n(C)$, for some $n$ and some finitely generated\ commutative $\xmode{\Bbb Z}$-algebra $C$. We will prove: \begin{theorem}\label{coherent-implies-arithmetically-linear} Assume that $X$ is of finite type over $\xmode{\Bbb Z}$. Let $G$ be a group-valued locally coherent $X$-functor. Then $G(X)$ is arithmetically linear. Moreover, if $n$ is invertible in $\Gamma(X,{\cal O}_X)$, then the $n$-torsion in $G(X)$ is finite. \end{theorem} First we analyze the structure of arithmetically linear abelian groups. Recall that an abelian group $H$ is {\it bounded\/} if $nH = 0$ for some $n \in \xmode{\Bbb N}$. It is known [see\Lspace \Lcitemark 16\Rcitemark \Rspace{}\ 11.2 or \Lcitemark 11\Rcitemark \Rspace{}\ Ch.\ VII\ \S2\ exercise 12(b)] that any bounded abelian group is a direct sum of cyclic groups. Thus one may characterize the bounded abelian groups as those which can be expressed as direct sums of cyclic groups, in which the orders of the summands are bounded. For purposes of this paper, let us say that an abelian group $H$ is {\it cobounded\/} if it may be embedded as a subgroup of a direct sum of (possibly infinitely many) copies of $\xmode{\Bbb Z}[1/n]$ for some $n \in \xmode{\Bbb N}$. This is equivalent to saying that $H$ is torsion-free and that $H \o*_\xmode{\Bbb Z} \xmode{\Bbb Z}[1/n]$ is a free $\xmode{\Bbb Z}[1/n]$-module for some $n$. \begin{remark} We do not know of a structure theorem for abelian groups which are countable and cobounded. Certainly such groups can be rather complicated. For example, not every subgroup $H$ of $\o+_{k=1}^\infty \xmode{\Bbb Z}[1/2]$ can be expressed as a direct sum of copies of $\xmode{\Bbb Z}$ and copies of $\xmode{\Bbb Z}[1/2]$; consider: $$H\ = \ \Bigl\{ a \in \o+_{k=1}^\infty \xmode{\Bbb Z}[1/2]: \sum_{k=1}^\infty {a_k \over k} \in \xmode{\Bbb Z} \Bigr\}.$% $Let $L$ be the maximal $2$-divisible subgroup of $H$. Then $H/L \cong {\Bbb Q}\kern1pt$. \end{remark} Let us say that an abelian group $H$ is {\itbounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded\/} if $H \cong B \times C$ for some bounded group $B$ and some cobounded group $C$. We shall see \pref{arith-linear-structure} that arithmetically linear abelian groups are the same as countable (bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded) abelian groups. \begin{lemma}\label{bxc} \ \begin{alphalist} \item Let $M$ be a bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded\ abelian group, and let $H$ be a subgroup of $M$. Then $H$ is bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded. \item Let \ses{M'}{M}{M''% }be a short exact sequence of abelian groups, in which $M'$ and $M''$ are bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded. Then $M$ is bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded. \end{alphalist} \end{lemma} \begin{proof} Part (a) follows from (\Lcitemark 16\Rcitemark \ 50.3): if the torsion subgroup of an abelian group is bounded, then it is a direct summand. For part (b), write $M' = B' \times C'$ and $M'' = B'' \times C''$ where $B'$, $B''$ are bounded and $C'$, $C''$ are cobounded. Let $M_{\operatoratfont tor}$ be the torsion subgroup of $M$. We have a left exact sequence: \les{B'}{M_{\operatoratfont tor}}{B''% }from which it follows that $M_{\operatoratfont tor}$ is bounded, and hence that $M_{\operatoratfont tor}$ is a direct summand of $M$. Therefore it suffices to show that $M/M_{\operatoratfont tor}$ is cobounded. We have an exact sequence: \ses{C'}{M/M_{\operatoratfont tor}}{\overline{B''} \times C''% }in which $\overline{B''}$ is a quotient of $B''$ and hence is bounded. Choose $n \in \xmode{\Bbb N}$ such that $n\overline{B''} = 0$, and such that $n$ satisfies the property of $n$ in the definition of cobounded, for both $C'$ and $C''$. Tensoring by $\xmode{\Bbb Z}[1/n]$ yields an exact sequence: \sesdot{C'[1/n]}{(M/M_{\operatoratfont tor})[1/n]}{C''[1/n]% }By construction, $C'[1/n]$ and $C''[1/n]$ are submodules of free $\xmode{\Bbb Z}[1/n]$-modules and hence are themselves free. Hence $(M/M_{\operatoratfont tor})[1/n]$ is free so $M/M_{\operatoratfont tor}$ is cobounded. {\hfill$\square$} \end{proof} \begin{lemma}\label{pork} Let $A$ be a commutative $\xmode{\Bbb Z}$-algebra of finite type. Let $M$ be a finitely generated\ $A$-module. Then the abelian group $M$ is bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded. \end{lemma} \begin{proof} As an abelian group, we may embed $M$ as a subgroup of the additive group of the symmetric algebra of $M$, which is a finitely generated\ $A$-algebra. Therefore we may reduce to the case $M = A$. Take a primary decomposition $0 = {\xmode{{\fraktur{\lowercase{Q}}}}}_1 \cap \cdots \cap {\xmode{{\fraktur{\lowercase{Q}}}}}_m$ of $0$ in $A$. Then the canonical map \mapx[[ A || \prod_{i=1}^n A/{\xmode{{\fraktur{\lowercase{Q}}}}}_i ]] is injective, so we may reduce to the case where $A$ has a unique associated prime. Let $q$ be the characteristic of $A$. First suppose that $q > 0$. Then $qA = 0$ so $A$ is bounded. Now suppose that $q = 0$. By the Noether normalization lemma, there exist elements $\vec x1r \in A \o*_\xmode{\Bbb Z} {\Bbb Q}\kern1pt$ which are algebraically independent over ${\Bbb Q}\kern1pt$ and such that $A \o*_\xmode{\Bbb Z} {\Bbb Q}\kern1pt$ is module-finite over ${\Bbb Q}\kern1pt[\vec x1r]$. We may assume that $\vec x1r \in A$. Let $S = \mathop{\operatoratfont Spec}\nolimits(\xmode{\Bbb Z})$, $X = \mathop{\operatoratfont Spec}\nolimits(A)$, $Y = \mathop{\operatoratfont Spec}\nolimits(\xmode{\Bbb Z}[\vec x1r])$, so we have a morphism \mp[[ \phi || X || Y ]] of $S$-schemes. If $\eta \in S$ is the generic point, then $\phi_\eta$ is finite, so it follows from (\Lcitemark 19\Rcitemark \ 8.1.2(a), 8.10.5(xii), 8.11.1, 9.6.1(vii)) that for some $n \in \xmode{\Bbb N}$, $\phi \o*_S \mathop{\operatoratfont Spec}\nolimits(\xmode{\Bbb Z}[1/n])$ is finite, i.e.\ that $A_n = A \o*_\xmode{\Bbb Z} \xmode{\Bbb Z}[1/n]$ is a module-finite $\xmode{\Bbb Z}[1/n, \vec x1r]$-algebra. Since $A$ has characteristic zero and has a unique associated prime, it follows that $A_n$ is a torsion-free $\xmode{\Bbb Z}[1/n, \vec x1r]$-module and that the canonical map \mapx[[ A || A_n ]] is injective. Hence $A$ embeds (as an abelian group) in $(\xmode{\Bbb Z}[1/n, \vec x1r])^k$ for some $k$, so $A$ is cobounded. {\hfill$\square$} \end{proof} \begin{prop}\label{arith-linear-structure} Let $H$ be an abelian group. Then $H$ is arithmetically linear if and only if\ it is countable and bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded. \end{prop} \begin{proof} First suppose that $H$ is countable and bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded. For some $n,k \in \xmode{\Bbb N}$ and some prime numbers $\vec p1k$, we may embed $H$ as a subgroup of a countable direct sum $K$ of copies of $$\xmode{\Bbb Z}[1/n] \o+ \left( \o+_{i=1}^k \xmode{\Bbb Z}/p_i^n\xmode{\Bbb Z} \right).$% $Then $K$ is the additive group of the ring $$A = \xmode{\Bbb Z}[1/n,t] \times (\xmode{\Bbb Z}/p_1^n\xmode{\Bbb Z})[t] \times \cdots \times (\xmode{\Bbb Z}/p_k^n\xmode{\Bbb Z})[t],$% $so $K$ may be embedded as a subgroup of $\mathop{\operatoratfont GL}\nolimits_2(A)$. Hence $H$ is arithmetically linear. Now suppose that $H$ is arithmetically linear. The countability of $H$ is clear. Embed $H$ as a subgroup of $\mathop{\operatoratfont GL}\nolimits_r(A)$, for some finitely generated\ commutative $\xmode{\Bbb Z}$-algebra $A$. Let $R$ be the sub-$A$-algebra of $\mathop{\operatoratfont Mat}\nolimits_{r \times r}(A)$ generated by $H$. Then $R$ is a finite $A$-algebra, and $H$ is a subgroup of $R^*$. By (\ref{bxc}a), it suffices to show that $R^*$ is bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded. We may as well view $R$ as an arbitrary finitely generated\ commutative $\xmode{\Bbb Z}$-algebra. Let $J$ be the nilradical of $R$. For each $n \in \xmode{\Bbb N}$, there is a short exact sequence of abelian groups: \diagramx{0&\mapE{}&J^n/J^{n+1}&\mapE{}&(R/J^{n+1})^*&\mapE{}&(R/J^n)^*& \mapE{}&1.% }The group $(R/J)^*$ is finitely generated\ (see \ref{units-over-Z}), and hence is bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded. Hence by (\ref{bxc}b), it suffices to show that $J^n/J^{n+1}$ is bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded. Apply \pref{pork}. {\hfill$\square$} \end{proof} We now work towards a proof of \pref{coherent-implies-arithmetically-linear}. \def\overline{S}^{\kern2pt\lower3pt\hbox{$\scriptstyle -1$}}{\overline{S}^{\kern2pt\lower3pt\hbox{$\scriptstyle -1$}}} \begin{lemma}\label{localize-now} Fix $n \in \xmode{\Bbb N}$ and a prime number $p$. Let $S \subset W_n^p(A)$ be a multiplicatively closed set. Let \mp[[ \mu || W_n^p(A) || A ]] be the canonical map. Let ${\overline{S}} = \mu(S)$. Then the canonical map \mp[[ i || W_n^p(A) || W_n^p(\overline{S}^{\kern2pt\lower3pt\hbox{$\scriptstyle -1$}} A) ]] factors through $S^{-1}W_n^p(A)$. \end{lemma} \begin{proof} Let \mp[[ \nu || W_n^p(\overline{S}^{\kern2pt\lower3pt\hbox{$\scriptstyle -1$}} A) || \overline{S}^{\kern2pt\lower3pt\hbox{$\scriptstyle -1$}} A ]] be the canonical map. Let $f \in S$. Then the image of $\mu(f)$ in $\overline{S}^{\kern2pt\lower3pt\hbox{$\scriptstyle -1$}} A$ is invertible, so $\nu(i(f))$ is invertible. But $\nu$ is surjective and has nilpotent kernel, so $i(f)$ is invertible. {\hfill$\square$} \end{proof} Note that if $A$ is any ${\Bbb F}\kern1pt_p$-algebra, there is a canonical map \mapx[[ W_n^p({\Bbb F}\kern1pt_p) || W_n^p(A) ]], and since $W_n^p({\Bbb F}\kern1pt_p) = \xmode{\Bbb Z}/p^n\xmode{\Bbb Z}$, we see that $p^n = 0$ in $W_n^p(A)$. \begin{prop}\label{witt-ring-of-poly-ring} For any $n, k \in \xmode{\Bbb N}$, and any prime number $p$, the ring $$W_n^p({\Bbb F}\kern1pt_p[\vec t1k])$% $is isomorphic to the subring of $(\xmode{\Bbb Z}/p^n\xmode{\Bbb Z})[t_1^{1/p^{n-1}},\ldots,t_k^{1/p^{n-1}}]$ generated by the elements $p^r t_i^{j/p^r}$ for $0 \leq r \leq n-1$, $1 \leq i \leq k$, and $1 \leq j \leq p-1$. \end{prop} \begin{proof} Let $R = {\Bbb F}\kern1pt_p[t_1^{p^{-\infty}},\ldots,t_k^{p^{-\infty}}]$, and let $V = W_n^p(R)$. Let $$W = (\xmode{\Bbb Z}/p^n\xmode{\Bbb Z})[t_1^{p^{-\infty}},\ldots,t_k^{p^{-\infty}}].$% $It is easily seen that there exists a unique ring homomorphism \mp[[ \eta || W || V ]] with the property that $\eta(t_i^{p^{-m}}) = (t_i^{p^{-m}},0,\ldots,0)$ for each $i$ and each $m \geq 0$. We show that $\eta$ is surjective. Let \mp[[ \mu || V || R ]] be the canonical map. Since $R$ is perfect, $\mathop{\operatoratfont Ker}\nolimits(\mu) = (p)$. Because of this, because $p^n = 0$ in $V$, and because $t_i^{1/p^r}$ (for various $i$, $r$) generate $R$ as a $\xmode{\Bbb Z}$-algebra, it follows that the $\eta(t_i^{1/p^r})$ generate $V$ as a $\xmode{\Bbb Z}$-algebra. Hence $\eta$ is surjective. We show that $\eta$ is injective. Suppose otherwise. Let $x \in \mathop{\operatoratfont Ker}\nolimits(\eta)$, $x \not= 0$. We may assume that $px = 0$. Then $x = p^{n-1}y$ for some $y \in W$; we may assume that\ the coefficients which appear in $y$ lie in the set \setof{1,\ldots,p-1}. It follows that the $\th{0}$ component of $\eta(y)$ is nonzero. Hence $p^{n-1} \eta(y) \not= 0$, so $\eta(x) \not= 0$: contradiction. Hence $\eta$ is injective. We return to the proof of the proposition. Let $A_n = W_n^p({\Bbb F}\kern1pt_p[\vec t1k])$. From what we have just done, it follows that $A_n$ may be identified with a subring of $W$. Since $\eta(p^r t_i^{j/p^r}) = (0,\ldots,0,t_i^j,0,\ldots,0)$, where $t_i^j$ appears in the \th{r} spot, it follows that $p^r t_i^{j/p^r} \in A_n$, for each $i$, $r$, and $j$. To complete the proof, we must show that these elements generate $A_n$. Let $A_n'$ be the subring of $A_n$ generated by the elements $p^r t_i^{j/p^r}$. Consider the canonical map \mp[[ \tau || A_n || A_{n-1} ]]. By induction on $n$ we may assume that\ the elements $p^r t_i^{j/p^r}$ generate $A_{n-1}$, and so that $\tau(A_n') = A_{n-1}$. Let $f \in {\Bbb F}\kern1pt_p[\vec t1k]$. Write $f = \sum_I a_I t^I$, where $I$ is a multi-index. Since $a_I \in \setof{0,\ldots,p-1}$, we may view $a_I$ as an element of $\xmode{\Bbb Z}/p^n\xmode{\Bbb Z}$. Let ${\tilde{\lowercase{F}}} = p^{n-1} \sum_I a_I (t^I)^{1/p^{n-1}} \in W$. Then ${\tilde{\lowercase{F}}}$ corresponds to the element $(0,\ldots,0,f)$ of $A_n$. Hence $\mathop{\operatoratfont Ker}\nolimits(\tau) \subset A_n'$. Since $\tau(A_n') = A_{n-1}$, it follows that $A_n' = A_n$. {\hfill$\square$} \end{proof} \begin{corollary}\label{witt-finite} Fix $n \in \xmode{\Bbb N}$ and a prime number $p$. Let $A$ be an ${\Bbb F}\kern1pt_p$-algebra of finite type. Then $W_n^p(A)$ is a $\xmode{\Bbb Z}$-algebra of finite type. \end{corollary} \begin{proof} Choose a surjection \mp[[ \pi || {\Bbb F}\kern1pt_p[\vec x1k] || A ]]. Then $W_n^p(\pi)$ is a surjection, and since \pref{witt-ring-of-poly-ring} $W_n^p({\Bbb F}\kern1pt_p[\vec x1k])$ is of finite type over $\xmode{\Bbb Z}$, so is $W_n^p(A)$. {\hfill$\square$} \end{proof} The following result is a variant of \pref{witt-coherent}. \begin{lemma}\label{sneezewort} Fix $n \in \xmode{\Bbb N}$ and a prime number $p$. Assume that $A$ is an ${\Bbb F}\kern1pt_p$-algebra of finite type. Let $C = W_n^p(A)$. (By \pref{witt-finite} $C$ is noetherian.) Let $F$ be a coherent $C$-functor, built up from $\setof{C/(p^k)}_{1 \leq k \leq n}$. Let $G$ be the $A$-functor given by $G(B) = F(W_n^p(B))$. Let ${\tilde{G}}$ be the sheaf associated to $G$ for the ffqc topology. Then ${\tilde{G}}$ is representable. \end{lemma} \begin{remarks} The functor $G$ is not in general coherent, as it is not in general an ffqc sheaf. For an arbitrary coherent $C$-functor $F$, with no restrictions on how it is built up, it may be that the corresponding functor ${\tilde{G}}$ is always representable. \end{remarks} \begin{proofnodot} (of \ref{sneezewort}.) Since any limit of finitely many representable functors is representable, we may assume that\ $F = \underline{C/(p^k)}$ for some $k$. Then we may describe $G$ by: $$G(B)\ =\ {\setof{(\vec b0{n-1}): b_i \in B \hbox{\ for all\ } i} \over \setof{(0,\ldots,0,d_k^{p^k},\ldots,d_{n-1}^{p^k}): d_i \in B \hbox{\ for all\ } i}},$% $where it is to be understood that this quotient of abelian groups takes place with respect to\ the abelian group structure on $W_n^p(B)$. For for any $b \in B$, there exists a faithfully flat ring extension \mp[[ \phi || B || B' ]] such that $\phi(b)$ is a \th{(p^k)} power. Therefore to complete the proof, it suffices to show that the $A$-functor $H$ given by $$H(B)\ =\ {\setof{(\vec b0{n-1}): b_i \in B \hbox{\ for all\ } i} \over \setof{(0,\ldots,0,e_k,\ldots,e_{n-1}): e_i \in B \hbox{\ for all\ } i}}$% $is representable. (Then we will have $H = {\tilde{G}}$.) But \begin{eqnarray*} H(B) & \cong & W_n^p(B) / \mathop{\operatoratfont Ker}\nolimits[W_n^p(B)\ \mapE{}\ W_k^p(B)]\\ & \cong & W_k^p(B)\ \cong\ B^k, \end{eqnarray*} so $H$ is representable. {\hfill$\square$} \end{proofnodot} Let $A = {\Bbb F}\kern1pt_p[\vec t1k]$. Let $C = (\xmode{\Bbb Z}/p^n\xmode{\Bbb Z})[\vec t1k]$. There is a canonical map \mp[[ \phi || C || W_n^p(A) ]] given by setting $\phi(t_i) = (t_i,0,\ldots,0)$ for each $i$. \begin{prop}\label{bindweed} Let $A = {\Bbb F}\kern1pt_p[\vec t1k]$. Let $C = (\xmode{\Bbb Z}/p^n\xmode{\Bbb Z})[\vec t1k]$. Let $S$ be a multiplicatively closed subset of $C$. Let \mp[[ \pi || C || A ]] be the canonical map. Let $F$ be a coherent $S^{-1}C$-functor. Assume that $F$ is built up from $\setof{S^{-1}C/(p^r)}_{1 \leq r \leq n}$. Let ${\overline{S}} = \pi(S)$. Let $G$ be the ffqc sheaf associated to the ${\overline{S}}^{-1}A$-functor given by $B\mapsto F(W_n^p(B))$. (This makes sense by \ref{localize-now}.) Then: \begin{alphalist} \item $G$ is representable; \item the canonical map \mp[[ i || F(S^{-1}C) || G({\overline{S}}^{-1}A) ]] is injective. \end{alphalist} \end{prop} \begin{proof} {\bf (a):} Let $D = W_n^p({\overline{S}}^{-1}A)$. Let \mp[[ \phi || \mathop{\operatoratfont Spec}\nolimits(D) || \mathop{\operatoratfont Spec}\nolimits(S^{-1}C) ]] be the canonical map. Then $\phi^*F$ is a $D$-functor, which is coherent (see \ref{pull}), and is in fact built up from $\setof{D/(p^r)}_{1 \leq r \leq n}$. Apply \pref{sneezewort}. {\bf (b):} The construction is functorial in $F$, so we may reduce to the case where $F = \underline{S^{-1}C/(p^r)}$. In that case, one sees that $i$ is isomorphic to the canonical map \mapx[[ S^{-1}(\xmode{\Bbb Z}/p^r\xmode{\Bbb Z})[\vec t1k] || W_r^p({\overline{S}}^{-1}{\Bbb F}\kern1pt_p[\vec t1k]) ]]. We may reduce to showing that the canonical map \mp[[ j || (\xmode{\Bbb Z}/p^r\xmode{\Bbb Z})[\vec t1k] || W_r^p({\Bbb F}\kern1pt_p[\vec t1k]) ]] is injective. This follows from the proof of \pref{witt-ring-of-poly-ring}. {\hfill$\square$} \end{proof} \begin{lemma}\label{generic-representability} Assume that $A$ is a domain. Let $F$ be a coherent $A$-functor. Then for some $f \in A - \setof{0}$, the pullback [along \mapx[[ \mathop{\operatoratfont Spec}\nolimits(A_f) || \mathop{\operatoratfont Spec}\nolimits(A) ]]{}] of $F$ to $A_f$ is representable by an $A$-algebra of finite type. \end{lemma} \begin{sketch} The functor $F$ is built up from finitely many $A$-modules, each finitely generated. Pick some $f \in A - \setof{0}$ such that the localization of each such module at $f$ is free. {\hfill$\square$} \end{sketch} \begin{lemma}\label{generic-linearity} Assume that $X$ is integral. Let $G$ be an affine group scheme of finite type over $X$. Then there exists some $n \in \xmode{\Bbb N}$, a nonempty open subscheme $U \subset X$, and a closed immersion \mapx[[ G_U || \mathop{\operatoratfont GL}\nolimits_n(U) ]] of $U$-schemes which is also a homomorphism. \end{lemma} \begin{sketch} Let $\eta$ be the generic point of $X$. For some $n \in \xmode{\Bbb N}$, we have a closed immersion (and a homomorphism) \mp[[ h || G_\eta || \mathop{\operatoratfont GL}\nolimits_n(X)_\eta ]] of $\mathop{\operatoratfont Spec}\nolimits k(\eta)$-schemes. There is a nonempty open subscheme $V \subset X$ and a closed immersion \mapx[[ G_V || \mathop{\operatoratfont GL}\nolimits_n(V) ]] of $V$-schemes which induces $h$. Replacing $V$ by a sufficiently small nonempty open subscheme $U$, we obtain (by restriction) a homomorphism (and a closed immersion) \mapx[[ G_U || \mathop{\operatoratfont GL}\nolimits_n(U) ]] of $U$-schemes. {\hfill$\square$} \end{sketch} \begin{lemma}\label{soup-bone} Let $A = (\xmode{\Bbb Z}/p^n\xmode{\Bbb Z})[\vec t1k]$. Let $M$ be a finitely generated\ $A$-module. Then there exists a non-zero-divisor $f \in A$ and positive integers $\vec l1r$ such that $M_f \cong A_f/(p^{l_1}) \manyo+ A_f/(p^{l_r})$ as $A_f$-modules. \end{lemma} \begin{sketch} Let $S$ be the set of non-zero-divisors of $A$. In $S^{-1}A$, the only elements (up to associates) are $1,p,\ldots,p^{n-1},0$. Hence every ideal of $S^{-1}A$ is principal, so every finitely generated\ $S^{-1}A$-module is a direct sum of cyclic modules, necessarily of the form $S^{-1}A/(p^j)$ for various $j$. The lemma follows. {\hfill$\square$} \end{sketch} \begin{proofnodot} (of \ref{coherent-implies-arithmetically-linear}.) The comment about what happens when $n$ is invertible in $\Gamma(X,{\cal O}_X)$ follows from (\ref{linear-implies-finite-torsion}b), so a direct proof is omitted. Some of the steps here follow the proof of \pref{coherent-implies-linear}. We may assume that $X$ is affine, $X = \mathop{\operatoratfont Spec}\nolimits(A)$. Choose $M$ and $B$ as in the proof of \pref{coherent-implies-linear}. Write $B = S^{-1}C$ for some finitely generated\ $A$-algebra $C$ and some multiplicatively closed subset $S \subset C$. Certainly we may replace $A$ by $C$, so $B = S^{-1}A$. Replacing $A$ by $A_g$ for some suitably chosen $g \in S$, we may assume that\ the connected components of $\mathop{\operatoratfont Spec}\nolimits(A)$ are irreducible and that they correspond bijectively with the points of $\mathop{\operatoratfont Spec}\nolimits(S^{-1}A)$. Write $A = A_1 \times \cdots \times A_m$, where $\mathop{\operatoratfont Spec}\nolimits(A_i)$ is irreducible for each $i$. Localizing further if necessary, we may assume that\ $A_i$ has a unique associated prime for each $i$. We may reduce to the following situation: $A$ has a unique associated prime, $S^{-1}A$ is an Artin local ring. It suffices to show that for some $f \in S$, $G(A_f)$ is arithmetically linear. Let $r$ be the characteristic of $A$. Since $A$ has a unique associated prime and $S^{-1}A$ is an Artin local ring, every non-nilpotent element of $A$ lies in $S$. First suppose that $r = 0$. By Noether normalization, we may find algebraically independent elements $\vec x1s \in A$ such that $A \o*_\xmode{\Bbb Z} {\Bbb Q}\kern1pt$ is module-finite over ${\Bbb Q}\kern1pt[\vec x1s]$. As in the proof of \pref{pork}, there is some $n \in \xmode{\Bbb N}$ such that $A \o*_\xmode{\Bbb Z} \xmode{\Bbb Z}[1/n]$ is a module-finite $\xmode{\Bbb Z}[1/n, \vec x1s]$-algebra. Every nonzero element of $\xmode{\Bbb Z}[\vec x1s]$ lies in $S$. We may replace $A$ by $A \o*_\xmode{\Bbb Z} \xmode{\Bbb Z}[1/n]$. By (\ref{push}b), the pushforward of $G$ to $\xmode{\Bbb Z}[1/n, \vec x1s]$ is coherent. By \pref{generic-representability}, there is some $f \in \xmode{\Bbb Z}[1/n,\vec x1s] - \setof{0}$ such that the pullback of $G$ to $\xmode{\Bbb Z}[1/n,\vec x1s,f^{-1}]$ is representable by an affine $\xmode{\Bbb Z}[1/n,\vec x1s,f^{-1}]$-algebra of finite type. Apply \pref{generic-linearity}. Now suppose that $r > 0$. Then $r = p^m$ for some prime $p$ and some $m$. By the Noether normalization theorem, we may find algebraically independent elements $\vec x1s \in A$ such that $A \o*_\xmode{\Bbb Z} {\Bbb F}\kern1pt_p$ is module-finite over ${\Bbb F}\kern1pt_p[\vec x1s]$. It follows that $A$ is module-finite over $E = (\xmode{\Bbb Z}/p^m\xmode{\Bbb Z})[\vec x1s]$. By (\ref{push}b), the pushforward $H$ of $G$ to $E$ is coherent, so we may reduce to the case where $A = (\xmode{\Bbb Z}/p^m\xmode{\Bbb Z})[\vec x1s]$. Let $\vec M1d$ be $A$-modules from which $G$ can be built up. By \pref{soup-bone}, there is some $f \in S$ such that each $(M_i)_f$ is a direct sum of modules of the form $A_f/(p^r)$, for various $r$. Let $G_f$ be the pullback of $G$ to $A_f$. Then $G_f$ is built up from $\setof{A_f/(p^r)}_{1 \leq r \leq n}$. The theorem follows now by applying \pref{bindweed} and \pref{generic-linearity}. {\hfill$\square$} \end{proofnodot} \begin{problemx} Which arithmetically linear groups arise as $G(X)$, for some scheme $X$ of finite type over $\xmode{\Bbb Z}$ and some group-valued coherent $X$-functor $G$? \end{problemx} Certain groups can be shown to be quotients of arithmetically linear abelian groups by finitely generated\ subgroups. For example, in the next section we shall see that this is the case for $\mathop{\operatoratfont Pic}\nolimits(X)$, where $X$ is a reduced scheme of finite type over $\xmode{\Bbb Z}$. Although it may in fact be the case that $\mathop{\operatoratfont Pic}\nolimits(X)$ is itself arithmetically linear, we have not been able to show this, so we are lead to the following lemma: \begin{lemma}\label{arith-linear-over-finite} Let $G$ be an arithmetically linear abelian group. Let $H$ be a finitely generated\ subgroup of $G$. Then the torsion subgroup of $G/H$ is supported at a finite set of primes. \end{lemma} \begin{proof} Write $G = B \times C$, where $B$ is bounded and $C$ is cobounded. For any abelian group $M$, let $M[1/n]$ denote $M \o*_\xmode{\Bbb Z} \xmode{\Bbb Z}[1/n]$. Choose $n \in \xmode{\Bbb N}$ such that $nB = 0$ and such that $C[1/n]$ is a free $\xmode{\Bbb Z}[1/n]$-module. Let $Q = G/H$. It suffices to show that the torsion subgroup of $Q[1/n]$ is supported at a finite set of primes. We have an exact sequence: \ses{H[1/n]}{C[1/n]}{Q[1/n]% }of $\xmode{\Bbb Z}[1/n]$-modules. Since $H[1/n]$ is contained in a finitely generated\ direct summand of $C[1/n]$, it suffices to show that for any finitely generated\ $\xmode{\Bbb Z}[1/n]$-module $M$, the torsion subgroup of $M$ is supported at a finite set of primes. This is easily checked. {\hfill$\square$} \end{proof} \block{Application to the Picard group} Let $M$ be a finitely generated\ $A$-module. Let $\mathop{\underline{\operatoratfont PGL}}\nolimits(M)$ denote the Zariski sheaf associated to the $A$-functor given by $$B \mapsto {\mathop{\operatoratfont Aut}\nolimits_B(M \o*_A B) \over B^*}.$% $More generally, let ${\cal{M}}$ be a coherent ${\cal O}_X$-module. Let $\mathop{\underline{\operatoratfont PGL}}\nolimits({\cal{M}})$ denote the Zariski sheaf associated to the $X$-functor given by $$Y\ \mapsto\ {\mathop{\operatoratfont Aut}\nolimits_Y({\cal{M}}_Y) \over \Gamma({\cal O}_Y^*)}.$% $Since $\mathop{\underline{\operatoratfont Aut}}\nolimits({\cal{M}})$ acts by conjugation on $\mathop{\underline{\operatoratfont End}}\nolimits({\cal{M}})$, we obtain a canonical morphism of group-valued $X$-functors: \dmap[[ \psi || \mathop{\underline{\operatoratfont PGL}}\nolimits({\cal{M}}) || \mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(\mathop{\underline{\operatoratfont End}}\nolimits({\cal{M}})).\footnote{For the actual arguments which we use, one could substitute the simpler functor $\mathop{\mathbf{Aut}}\nolimits(\mathop{\underline{\operatoratfont End}}\nolimits({\cal{M}}))$, but we use $\mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(\mathop{\underline{\operatoratfont End}}\nolimits({\cal{M}}))$ instead for asthetic reasons because it makes $\psi$ closer to an isomorphism.} ]]% The $X$-functor $\mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(\mathop{\underline{\operatoratfont End}}\nolimits({\cal{M}}))$ is locally coherent by \pref{autalg-is-coherent} and example \pref{end-example} from \S\ref{examples-section}; it is coherent, at least assuming that $X$ is separated, by \pref{lump}. We would like to show that $\mathop{\underline{\operatoratfont PGL}}\nolimits({\cal{M}})$ is coherent. We could do this by showing that $\psi$ is an isomorphism. However, other than the case where ${\cal{M}}$ is locally free, we do not know if $\psi$ is an isomorphism. Therefore, we settle for showing that (under certain special circumstances) $\psi(X)$ is injective. This is a weak substitute for showing that $\mathop{\underline{\operatoratfont PGL}}\nolimits({\cal{M}})$ is coherent.\footnote{In trying to show that $\psi$ is a monomorphism, one comes to the following question: Let $M$ be a finitely generated\ $A$-module. Let $\sigma$ be an automorphism of $M$ as an $A$-module. Assume that for every commutative $A$-algebra $B$, $\sigma \o*_A B$ lies in the center of $\mathop{\operatoratfont End}\nolimits_B(M \o*_A B)$. Does it follow that $\sigma$ is a homothety, i.e.\ that $\sigma$ is given by multiplication by an element of $A$?} First we prove a lemma, then a corollary which says something directly about $\psi$. \begin{lemma}\label{homothety} Assume that $A$ is reduced. Let $A'$ be a ring, with $A \subset A' \subset \nor{A}$. Assume that the ideal $[A:A']$ of $A$ is prime. Let $\sigma$ be an endomorphism of $A'$ as an $A$-module. Assume that for every ${\xmode{{\fraktur{\lowercase{P}}}}} \in \mathop{\operatoratfont Spec}\nolimits(A)$, $\sigma \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}})$ is a homothety of $A' \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}})$ as a $k({\xmode{{\fraktur{\lowercase{P}}}}})$-module. Then $\sigma$ is a homothety of $A'$ as an $A$-module. \end{lemma} \begin{proof} First we show that $\sigma$ is given by multiplication by $\sigma(1)$. By subtracting the endomorphism of $A'$ given by multiplication by $\sigma(1)$, we may reduce to showing that if $\sigma(1) = 0$, then $\sigma = 0$. Let $\vec\lfP1r$ be the minimal primes of $A$. Using the fact that $\sigma \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}}_i)$ is a homothety, we conclude that $\sigma \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}}_i) = 0$. Let $x \in A'$. Then $\sigma(x) \mapsto 0$ in $A' \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}}_i)$. Since the map \dmapx[[ A' || \oplus_{i=1}^r A' \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}}_i) ]]% is injective, it follows that $\sigma(x) = 0$, and hence that $\sigma = 0$. Hence (reverting to the original problem), we see that $\sigma$ is given by multiplication by $\sigma(1)$. Let $Q = A'/A$, which is an $A$-module. Let ${\xmode{{\fraktur{\lowercase{P}}}}} = [A:A']$. Let $\overline{\sigma(1)}$ denote the image of $\sigma(1)$ in $Q$. It follows that $\overline{\sigma(1)} \mapsto 0$ in $Q \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}})$. But by the construction of ${\xmode{{\fraktur{\lowercase{P}}}}}$, the canonical map \mapx[[ Q || Q \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}}) ]] is injective. Hence $\overline{\sigma(1)} = 0$. Hence $\sigma(1) \in A$. {\hfill$\square$} \end{proof} \begin{corollary}\label{key-embedding} Assume that $A$ is reduced. Let $A'$ be a ring, with $A \subset A' \subset \nor{A}$. Assume that the ideal $[A:A']$ of $A$ is prime. Assume that $A'$ is a finitely generated\ $A$-module. Let \dmap[[ \psi || \mathop{\underline{\operatoratfont PGL}}\nolimits(A') || \mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(\mathop{\underline{\operatoratfont End}}\nolimits(A')) ]]% be the canonical (conjugation) morphism of group-valued $A$-functors. Then $\psi(A)$ is injective. \end{corollary} \begin{proof} Consider the morphism \mp[[ \psi_0 || \mathop{\underline{\operatoratfont Aut}}\nolimits(A') || \mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(\mathop{\underline{\operatoratfont End}}\nolimits(A')) ]] which induces $\psi$. Then $\ker(\psi_0(A))$ consists of those automorphisms $\sigma$ of $A'$ as an $A$-module with the property that $\sigma \o*_A B \in Z[\mathop{\operatoratfont End}\nolimits_B(A' \o*_A B)]$ for every commutative $A$-algebra $B$. If $B$ is a field, it follows that for such a $\sigma$, $\sigma \o*_A B$ is given by multiplication by an element of $B$. By \pref{homothety}, such a $\sigma$ is itself a homothety. Hence $\ker(\psi_0(A)) = A^*$. Let $f \in A - \setof{0}$. Then the ideal $[A_f:A'_f]$ of $A_f$ is prime and it follows that $\ker(\psi_0(A_f)) = A_f^*$. Hence the map \dmapx[[ {\mathop{\operatoratfont Aut}\nolimits_{A_f}(A' \o*_A A_f) \over A_f^*} || \mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(\mathop{\underline{\operatoratfont End}}\nolimits(A'))(A_f) ]]% is injective. Considering the sheafification which occurs in the definition of $\mathop{\underline{\operatoratfont PGL}}\nolimits(A')$, we see that $\psi(A)$ is injective. {\hfill$\square$} \end{proof} \begin{lemma}\label{filtration-of-normalization} Let $C$ be an overring of $A$, with $C$ finitely generated\ as an $A$-module. Then there exists a chain: $$A\ =\ A_0\ \subset\ A_1\ \subset\ \cdots\ \subset\ A_n\ =\ C$% $of rings such that for each $k = 1, \ldots, n$, the ideal $[A_{k-1}:A_k]$ of $A_{k-1}$ is prime. \end{lemma} \begin{proof} We may assume that $C \not= A$. For each $y \in C - A$, let $I_y = [A:A[y]]$. (All conductors are to be computed as ideals in $A$.) Choose $y \in C - A$ so that $I_y$ is maximal amongst all such ideals. We will show that $I_y$ is prime. This will complete the proof. Pick $a \in A$ such that $[I_y:a]$ is prime. Choose $n$ so that $y$ satisfies a monic polynomial of degree $n$ with coefficients in $A$. Then \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm\hbox{$I_y = [A: y,y^2,\ldots,y^{n-1}]$}, and $[I_y:a] = [A: ay,ay^2,\ldots,ay^{n-1}]$. First we show that $ay \notin A$. Suppose otherwise. Since $[I_y:a] \not= A$, $a \notin I_y$. Hence $ay^k \notin A$, for some $k$ with $2 \leq k \leq n-1$. Consider all pairs $(r,s) \in \xmode{\Bbb N}^2$, $s < n$, such that $a^r y^s \notin A$. Choose such a pair $(r_0,s_0)$ such that the ratio $s_0/r_0$ is as small as possible. (This is possible because $ay \in A$, and hence $a^r y^s \in A$ for all $r \geq s$.) Let $(r,s) \in \xmode{\Bbb N}^2$ be such that $a^r y^s \notin A$. We will show that $s_0/r_0 \leq s/r$. Suppose otherwise: $s_0/r_0 > s/r$. We may assume that $s \geq n$. We have: $$a^r y^s = \sum_{i = 0}^{n-1} c_i a^r y^i$% $for suitable $c_i \in A$. For $i$ in the given range, $s/r > i/r$, so $s_0/r_0 > i/r$. Hence $a^r y^i \in A$. Hence $a^r y^s \in A$: contradiction. Hence $s_0/r_0 \leq s/r$. Let $z = a^{r_0} y^{s_0}$. We have $I_y \subset I_z$, so by the maximality of $I_y$, we have $I_y = I_z$. For any $k \in \xmode{\Bbb N}$, $az^k = a^{r_0 k+1} y^{s_0 k}$, and $${s_0 k \over r_0 k + 1} < {s_0 \over r_0}.$% $Hence $az^k \in A$. Hence $a \in I_z$. Hence $a \in I_y$: contradiction. Hence $ay \notin A$. Note that $ay$ satisfies a monic polynomial of degree $n$ with coefficients in $A$. We have: $$I_y\ \subset\ [I_y:a]\ \subset\ [A:ay,(ay)^2,\ldots,(ay)^{n-1}]\ =\ I_{ay},$% $so by the maximality of $I_y$ we must have $I_y = I_{ay}$ and hence $I_y$ is prime. {\hfill$\square$} \end{proof} Let us say that an abelian group $G$ is {\it pseudo-$X$-linear\/} if there exists a filtration $$0 = G_0 \subset G_1 \subset \cdots \subset G_m = G,$% $with the property that $G_i/G_{i-1}$ is $X$-linear for each $i$. It is conceivable that every such group $G$ is $X$-linear. \begin{theorem}\label{filtration} Assume that $X$ is reduced, and that the canonical map \mp[[ \pi || \nor{X} || X ]] is finite. Let $C$ be the quotient sheaf $\pi_*{\cal O}_{\nor{X}}^*/{\cal O}_X^*$. Then the group $C(X)$ is pseudo-$X$-linear. If $X$ is of finite type over $\xmode{\Bbb Z}$, then $C(X)$ is arithmetically linear. \end{theorem} \begin{proof} The comment about what happens when $X$ is of finite type over $\xmode{\Bbb Z}$ is left to the reader; the proof given below works with appropriate changes, provided that one uses in addition (\ref{bxc}b) and \pref{arith-linear-structure}. One uses \pref{coherent-implies-arithmetically-linear} instead of \pref{coherent-implies-linear}. The following two facts are easily verified: any subgroup of a pseudo-$X$-linear abelian group is pseudo-$X$-linear, and any product of finitely many pseudo-$X$-linear abelian groups is pseudo-$X$-linear. It follows that we may reduce to the case where $X$ is affine. Then by \pref{filtration-of-normalization}, we may reduce to the following situation: $A$ and $A'$ are reduced noetherian rings, with $A \subset A' \subset \nor{A}$ (and $\nor{A}$ is module-finite over $A$), and the ideal $[A:A']$ of $A$ is prime. We must show that if $F$ is the Zariski sheaf associated to the $A$-functor given by $B \mapsto (A' \o*_A B)^*/B^*$, then $F(A)$ is $X$-linear. But $F(A) \cong [\mathop{\underline{\operatoratfont PGL}}\nolimits(A')](A)$, so by \pref{key-embedding}, it suffices to show that if $G = \mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(\mathop{\underline{\operatoratfont End}}\nolimits(A'))$, then $G(A)$ is $X$-linear. But $G$ is coherent by \pref{autalg-is-coherent}, so the theorem follows from \pref{coherent-implies-linear}. {\hfill$\square$} \end{proof} One can check that \pref{linear-implies-finite-torsion} applies to a pseudo-$X$-linear abelian group, so one obtains the following corollary: \begin{corollary}\label{finiteness-of-C} Assume that $X$ is reduced, and that the canonical map \mp[[ \pi || \nor{X} || X ]] is finite. Let $C$ be the quotient sheaf $\pi_*{\cal O}_{\nor{X}}^*/{\cal O}_X^*$. Then: \begin{alphalist} \item There are only finitely many prime numbers $p$ such that ${}_p C(X)$ is infinite. Moreover, such a $p$ cannot be invertible in $\Gamma(X,{\cal O}_X)$. \item If $X$ is essentially of finite type over $\xmode{\Bbb Z}$ or $\xmode{\Bbb Z}_p$ (for some prime number $p$), then there exist prime numbers $\vec p1n$, none of which are invertible in $\Gamma(X,{\cal O}_X)$, such that the subgroup of $C(X)$ consisting of torsion prime to $p_1 \cdot \ldots \cdot p_n$ is finite. \end{alphalist} \end{corollary} \begin{corollary}\label{pic-main-theorem} Assume that $X$ is reduced, and that the canonical map \mp[[ \pi || \nor{X} || X ]] is finite. Let $Q = \Gamma({\cal O}_{\nor{X}}^*)/\Gamma({\cal O}_X^*)$. Let $K$ be the kernel of the canonical map \mapx[[ \mathop{\operatoratfont Pic}\nolimits(X) || \mathop{\operatoratfont Pic}\nolimits(\nor{X}) ]]. \begin{alphalist} \item Fix $n \in \xmode{\Bbb N}$, and assume that $n$ is invertible in $\Gamma(X,{\cal O}_X)$. If $Q$ is finitely generated, or more generally if it admits an $n$-divisible subgroup with finitely generated\ cokernel, then ${}_n K$ is finite. \item If $Q$ admits a divisible subgroup with finitely generated\ cokernel, then there are only finitely many prime numbers $p$ such that ${}_p K$ is infinite. \end{alphalist} \end{corollary} \begin{proof} Any finitely generated\ projective module of constant rank over a semilocal ring is free, so any line bundle on $\nor{X}$ may be trivialized by an open cover pulled back from $X$. (This argument was shown to me by R.\ Wiegand.) Hence $R^1\pi_*({\cal O}_{\nor{X}}^*) = 0$, and so by the Leray spectral sequence we see that $H^1(X,\pi_* {\cal O}_{\nor{X}}^*) \cong \mathop{\operatoratfont Pic}\nolimits(\nor{X})$. Run the long exact sequence of cohomology on \sesdot{{\cal O}_X^*}{\pi_*{\cal O}_{\nor{X}}^*}{C% }We obtain a short exact sequence: \sesdot{Q}{C(X)}{K% }By \pref{finiteness-of-C}, it suffices to show that if \ses{M'}{M}{M''% }is a short exact sequence of abelian groups, $\abs{{}_n M} < \infty$, and there exists an $n$-divisible subgroup $H \subset M'$ such that $M'/H$ is finitely generated, then $\abs{{}_n M''} < \infty$. This is easily proved -- see \Lcitemark 21\Rcitemark \Rspace{}. {\hfill$\square$} \end{proof} \begin{example} Let $X = \mathop{\operatoratfont Spec}\nolimits {\Bbb Q}\kern1pt[x,y]/(x^2 - 2y^2)$. Then $\Gamma({\cal O}_{\nor{X}}^*) / \Gamma({\cal O}_X^*) \cong {\Bbb Q}\kern1pt[\sqrt{2}]^*/{\Bbb Q}\kern1pt^*$, which is not finitely generated. \end{example} We consider what happens when $X$ is of finite type over $\xmode{\Bbb Z}$. We need the following well-known result, which is apparently due to Roquette. A proof of the key case ($X$ integral, affine) may be found in (\Lcitemark 8\Rcitemark \ p.\ 39). \begin{prop}\label{units-over-Z} Let $X$ be a reduced scheme of finite type over $\xmode{\Bbb Z}$. Then the group $\Gamma(X,{\cal O}_X)^*$ is finitely generated. \end{prop} \begin{theorem}\label{rabbit-food} Let $X$ be a reduced scheme of finite type over $\xmode{\Bbb Z}$. Then the torsion subgroup of $\mathop{\operatoratfont Pic}\nolimits(X)$ is supported at a finite set of primes, and if ${}_p \mathop{\operatoratfont Pic}\nolimits(X)$ is infinite, then the prime $p$ is not invertible in $\Gamma(X,{\cal O}_X)$. \end{theorem} \begin{proof} By (\Lcitemark 25\Rcitemark \ 2.7.6) we know that $\mathop{\operatoratfont Pic}\nolimits(\nor{X})$ is finitely generated. Therefore it suffices to show that if $K = \mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(X)\ \mapE{}\ \mathop{\operatoratfont Pic}\nolimits(\nor{X})]$, then $K$ is supported at a finite set of primes, and if ${}_p K$ is infinite, then the prime $p$ is not invertible in $\Gamma(X,{\cal O}_X)$. By \pref{filtration}, \pref{units-over-Z} and the argument of \pref{pic-main-theorem}, one sees that $K$ is isomorphic to the quotient of an arithmetically linear group by a finitely generated\ subgroup. Hence \pref{arith-linear-over-finite} tells us that the torsion subgroup of $\mathop{\operatoratfont Pic}\nolimits(X)$ is supported at a finite set of primes. The last assertion of the theorem follows from (\ref{pic-main-theorem}a). {\hfill$\square$} \end{proof} For $X$ a non-reduced scheme of finite type over $\xmode{\Bbb Z}$, we do not know if the torsion subgroup of $\mathop{\operatoratfont Pic}\nolimits(X)$ is supported at a finite set of primes. However, for any commutative noetherian ring $A$, the canonical map \mapx[[ \mathop{\operatoratfont Pic}\nolimits(A) || \mathop{\operatoratfont Pic}\nolimits(\RED{A}) ]] is an isomorphism, so we have: \begin{corollary}\label{yyyyy} Let $A$ be a finitely generated\ commutative $\xmode{\Bbb Z}$-algebra. Then the torsion subgroup of $\mathop{\operatoratfont Pic}\nolimits(A)$ is supported at a finite set of primes, and if ${}_p \mathop{\operatoratfont Pic}\nolimits(A)$ is infinite, then the prime $p$ is not invertible in $A$. \end{corollary} Let $A$ be a finitely generated\ commutative $\xmode{\Bbb Z}$-algebra. If is natural to ask if there exist prime numbers $\vec p1n$, none of which are invertible in $A$, such that the subgroup of $\mathop{\operatoratfont Pic}\nolimits(A)$ consisting of torsion prime to $\vec p1n$ is finite. Unfortunately, the answer is no. For a counterexample, see (\Lcitemark 21\Rcitemark \ 6.3). Now we consider what happens when $X$ is of finite type over ${\Bbb F}\kern1pt_p$. \begin{theorem}\label{F-sub-p} Let $X$ be a scheme of finite type over ${\Bbb F}\kern1pt_p$. Then there exists a finitely generated\ abelian group $H$ and a finite $p$-group $F$ such that $$\mathop{\operatoratfont Pic}\nolimits(X)\ \cong\ H\ \o+ \ [\o+_{n=1}^\infty F].$$ \end{theorem} \begin{sketch} Consider the class ${\cal{C}}$ of groups of the form ascribed to $\mathop{\operatoratfont Pic}\nolimits(X)$ in the theorem. The groups in ${\cal{C}}$ are all bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded. One can check without great difficulty that ${\cal{C}}$ is closed under formation of subgroups, quotient groups, and extensions. First suppose that $X$ is reduced. Since $\mathop{\operatoratfont Pic}\nolimits(\nor{X})$ is finitely generated, it suffices to show that $K = \mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(X)\ \mapE{}\ \mathop{\operatoratfont Pic}\nolimits(\nor{X})]$ is in ${\cal{C}}$. We may assume that $X$ is affine. In fact, it suffices to show that if $X' = \mathop{\operatoratfont Spec}\nolimits(A')$ is a partial normalization of $X = \mathop{\operatoratfont Spec}\nolimits(A)$, and if $[A':A]$ is prime, then $K' = \mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(X)\ \mapE{}\ \mathop{\operatoratfont Pic}\nolimits(X')]$ is in ${\cal{C}}$. The proof of \pref{rabbit-food} shows that $K'$ is a quotient of an arithmetically linear group. In fact, the arguments used to arrive at this result show that there exists an ${\Bbb F}\kern1pt_p$-algebra $A$ of finite type such that $K'$ is a quotient of an abelian subgroup $H \subset \mathop{\operatoratfont GL}\nolimits_n(A)$. Therefore there exists an ${\Bbb F}\kern1pt_p$-algebra $C$ of finite type such that $K'$ is a quotient of a subgroup of $C^*$. The usual methods show that $C^*$ may be built up via extensions from a finitely generated\ abelian group and some ${\Bbb F}\kern1pt_p$-vector spaces. It follows that $C^* \in {\cal{C}}$, and hence that any quotient of $C^*$ is in ${\cal{C}}$. Now suppose that $X$ is arbitrary, not necessarily reduced. Let ${\cal{J}}$ be the nilradical of $X$. There is an exact sequence: \Rowfive{0}{{\cal{J}}^n/{\cal{J}}^{n+1}}{({\cal O}_X/{\cal{J}}^{n+1})^*}{({\cal O}_X/{\cal{J}}^n)^*}{1% }of sheaves of abelian groups on $X$. Let $X_n$ be the closed subscheme of $X$ corresponding to the ideal ${\cal{J}}^n$. By taking cohomology, one sees that $$\mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(X_{n+1}) \ \mapE{}\ \mathop{\operatoratfont Pic}\nolimits(X_n)]$% $is $p$-torsion. The theorem follows. {\hfill$\square$} \end{sketch} Finally, we compute the Picard group of a few simple examples. The calculations are an easy consequence of an exact sequence of Milnor (\Lcitemark 6\Rcitemark \ IX\ 5.3) applied to the cartesian square \squareSE{A}{\nor{A}}{A/{\xmode{{\fraktur{\lowercase{C}}}}}}{\nor{A}/{\xmode{{\fraktur{\lowercase{C}}}}}\makenull{,}% }where ${\xmode{{\fraktur{\lowercase{C}}}}}$ is the conductor of $\nor{A}$ into $A$. The exact sequence is: \splitdiagram{A^*&\mapE{}&\nor{A}^* \times (A/{\xmode{{\fraktur{\lowercase{C}}}}})^*&\mapE{}% &(\nor{A}/{\xmode{{\fraktur{\lowercase{C}}}}})^*}{\mapE{}&\mathop{\operatoratfont Pic}\nolimits(A)&\mapE{}% &\mathop{\operatoratfont Pic}\nolimits(\nor{A}) \times \mathop{\operatoratfont Pic}\nolimits(A/{\xmode{{\fraktur{\lowercase{C}}}}})&\mapE{}&\mathop{\operatoratfont Pic}\nolimits(\nor{A}/{\xmode{{\fraktur{\lowercase{C}}}}}).% }Let $p$ be a prime number. The first four rings given below may be viewed as subrings of $\xmode{\Bbb Z}[t,x]$. \vspace*{0.1in} \begin{center} \renewcommand{\arraystretch}{1.2} \begin{tabular}{||c|c||} \hline ring $A$ & structure of $\mathop{\operatoratfont Pic}\nolimits(A)$ \\ \hline\hline $\xmode{\Bbb Z}[t^2,t^3]$ & $\xmode{\Bbb Z}$ \\ \hline $\xmode{\Bbb Z}[t^2, t^3, x]$ & free abelian of countably infinite rank \\ \hline $\xmode{\Bbb Z}[pt, t^2, t^3]$ & ${\Bbb F}\kern1pt_p$ \\ \hline $\xmode{\Bbb Z}[pt, t^2, t^3, x]$ & ${\Bbb F}\kern1pt_p$-vector space of countably infinite rank \\ \hline $\xmode{\Bbb Z}[1/p, t^2, t^3]$ & $\xmode{\Bbb Z}[1/p]$ \\ \hline ${\Bbb F}\kern1pt_p[t^2, t^3, x]$ & ${\Bbb F}\kern1pt_p$-vector space of countably infinite rank \\ \hline \end{tabular} \end{center} \section*{References} \addcontentsline{toc}{section}{References} \ \par\noindent\vspace*{-0.25in} \hfuzz 5pt \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{1}% \def\Atest{ }\def\Astr{Artin\Revcomma M\Initper }% \def\Ttest{ }\def\Tstr{Algebraic approximation of structures over complete local rings}% \def\Jtest{ }\def\Jstr{Inst. Hautes \'Etudes Sci. Publ. Math.}% \def\Vtest{ }\def\Vstr{36}% \def\Dtest{ }\def\Dstr{1969}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{23--58}% \def\Qtest{ }\def\Qstr{access via "artin algebraic approximation"}% \def\Xtest{ }\def\Xstr{In bound volume.}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{2}% \def\Atest{ }\def\Astr{Artin\Revcomma M\Initper }% \def\Ttest{ }\def\Tstr{Letter to Grothendieck}% \def\Dtest{ }\def\Dstr{Nov.\ 5, 1968}% \def\Qtest{ }\def\Qstr{access via "artin letter grothendieck"}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{3}% \def\Atest{ }\def\Astr{Artin\Revcomma M\Initper }% \def\Ttest{ }\def\Tstr{Algebraization of formal moduli: I}% \def\Btest{ }\def\Bstr{Global Analysis: Papers in Honor of K.\ Kodaira}% \def\Itest{ }\def\Istr{Princeton Univ. Press}% \def\Dtest{ }\def\Dstr{1969}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{21--71}% \def\Qtest{ }\def\Qstr{access via "artin formal moduli one"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{4}% \def\Atest{ }\def\Astr{Atiyah\Revcomma M\Initper \Initgap F\Initper % \Aand I\Initper \Initgap G\Initper Macdonald}% \def\Ttest{ }\def\Tstr{Introduction to Commutative Algebra}% \def\Itest{ }\def\Istr{Addison-Wesley}% \def\Ctest{ }\def\Cstr{Reading, Mass.}% \def\Dtest{ }\def\Dstr{1969}% \def\Qtest{ }\def\Qstr{access via "atiyah macdonald"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{4}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{5}% \def\Atest{ }\def\Astr{Auslander\Revcomma M\Initper }% \def\Ttest{ }\def\Tstr{Coherent functors}% \def\Btest{ }\def\Bstr{Proceedings of the Conference on Categorical Algebra (La Jolla, 1965)}% \def\Etest{ }\def\Estr{S\Initper Eilenberg% \Ecomma D\Initper \Initgap K\Initper Harrison% \Ecomma S\Initper MacLane% \Eandd H\Initper R\"ohrl}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{189--231}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Qtest{ }\def\Qstr{access via "auslander coherent functors"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{6}% \def\Atest{ }\def\Astr{Bass\Revcomma H\Initper }% \def\Ttest{ }\def\Tstr{Algebraic K-Theory}% \def\Itest{ }\def\Istr{W.\ A.\ Benjamin}% \def\Ctest{ }\def\Cstr{New York}% \def\Dtest{ }\def\Dstr{1968}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{7}% \def\Atest{ }\def\Astr{Bass\Revcomma H\Initper }% \def\Ttest{ }\def\Tstr{Some problems in ``classical'' algebraic K-theory}% \def\Stest{ }\def\Sstr{Lecture Notes in Mathematics}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Vtest{ }\def\Vstr{342}% \def\Dtest{ }\def\Dstr{1973}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{3--73}% \def\Qtest{ }\def\Qstr{access via "bass classical problems"}% \def\Xtest{ }\def\Xstr{I don't have this.}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{8}% \def\Atest{ }\def\Astr{Bass\Revcomma H\Initper }% \def\Ttest{ }\def\Tstr{Introduction to some methods of algebraic K-theory}% \def\Jtest{ }\def\Jstr{Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics}% \def\Vtest{ }\def\Vstr{20}% \def\Dtest{ }\def\Dstr{1974}% \def\Qtest{ }\def\Qstr{access via "bass introduction methods"}% \def\Xtest{ }\def\Xstr{MR 50:441. (pamphlet on shelf) \par Let $R$ be either $\xmode{\Bbb Z}$ or $F_q[t]$. Let $A = R[\vec t2d]$, where $d \geq 1$. Then $\mathop{\operatoratfont GL}\nolimits_n(A)$ is a finitely generated group for $n \geq d+2$. Also, let $A$ be any semilocal ring or $\xmode{\Bbb Z}$ or $k[t]$, where $k$ is any field. Then $\SL_n(A)$ is generated by elementary matrices for any $n \geq 1$. (See 2.6, 2.8.)}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{9}% \def\Atest{ }\def\Astr{Bergman\Revcomma G\Initper \Initgap M\Initper }% \def\Ttest{ }\def\Tstr{Ring schemes; the Witt scheme}% \def\Btest{ }\def\Bstr{Lectures on Curves on an Algebraic Surface {\rm by David Mumford}}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{171---187}% \def\Itest{ }\def\Istr{Princeton Univ.\ Press}% \def\Dtest{ }\def\Dstr{1966}% \def\Qtest{ }\def\Qstr{access via "bergman witt scheme"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{10}% \def\Atest{ }\def\Astr{Bertin\Revcomma J\Initper \Initgap E\Initper }% \def\Ttest{ }\def\Tstr{{\rm\tolerance=1000 Generalites sur les preschemas en groupes, expos\'e\ ${\rm VI}_{\rm B}$ in {\itS\'em\-in\-aire de G\'eom\'e\-trie Al\-g\'e\-bri\-que} (SGA 3)}}% \def\Stest{ }\def\Sstr{Lecture Notes in Mathematics}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Vtest{ }\def\Vstr{151}% \def\Dtest{ }\def\Dstr{1970}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{318--410}% \def\Qtest{ }\def\Qstr{access via "bertin preschemas"}% \def\Xtest{ }\def\Xstr{This includes the following result, announced by Raynaud, but apparently not proved in this volume: Theorem (11.11.1) Let $S$ be a regular noetherian scheme of dimension $/leq 1$. Let \map(\pi,G,S) be an $S$-group scheme of finite-type. Assume that $\pi$ is a flat, affine morphism. Then $G$ is isomorphic to a closed subgroup of $\mathop{\mathbf{Aut}}\nolimits(V)$, where $V$ is a vector bundle on $S$.}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{11}% \def\Atest{ }\def\Astr{Bourbaki\Revcomma N\Initper }% \def\Ttest{ }\def\Tstr{Elements of Mathematics (Algebra II, Chapters 4--7)}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Dtest{ }\def\Dstr{1990}% \def\Qtest{ }\def\Qstr{access via "bourbaki algebra part two"}% \def\Xtest{ }\def\Xstr{I don't have this.}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{12}% \def\Atest{ }\def\Astr{Claborn\Revcomma L\Initper }% \def\Ttest{ }\def\Tstr{Every abelian group is a class group}% \def\Jtest{ }\def\Jstr{Pacific J. Math.}% \def\Vtest{ }\def\Vstr{18}% \def\Dtest{ }\def\Dstr{1966}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{219--222}% \def\Qtest{ }\def\Qstr{access via "claborn"}% \def\Xtest{ }\def\Xstr{I don't have this.}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{13}% \def\Atest{ }\def\Astr{Cohen\Revcomma H\Initper }% \def\Ttest{ }\def\Tstr{Un faisceau qui ne peut pas \^etre d\'etordu universellement}% \def\Jtest{ }\def\Jstr{C. R. Acad. Sci. Paris S\'er. I Math.}% \def\Vtest{ }\def\Vstr{272}% \def\Dtest{ }\def\Dstr{1971}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{799--802}% \def\Qtest{ }\def\Qstr{access via "cohen faisceau article"}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{14}% \def\Atest{ }\def\Astr{Cohen\Revcomma H\Initper }% \def\Ttest{ }\def\Tstr{Detorsion universelle de faisceaux coherents}% \def\otest{ }\def\ostr{thesis (Docteur $3^\circ$ Cycle)}% \def\Dtest{ }\def\Dstr{1972}% \def\Itest{ }\def\Istr{Universit\'e de Paris}% \def\Ctest{ }\def\Cstr{Orsay}% \def\Qtest{ }\def\Qstr{access via "cohen thesis"}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{15}% \def\Atest{ }\def\Astr{Fossum\Revcomma R\Initper \Initgap M\Initper }% \def\Ttest{ }\def\Tstr{The Divisor Class Group of a Krull Domain}% \def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}% \def\Ctest{ }\def\Cstr{New York}% \def\Dtest{ }\def\Dstr{1973}% \def\Qtest{ }\def\Qstr{access via "fossum"}% \def\Xtest{ }\def\Xstr{I don't have this.}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{16}% \def\Atest{ }\def\Astr{Fuchs\Revcomma L\Initper }% \def\Ttest{ }\def\Tstr{Abelian Groups}% \def\Itest{ }\def\Istr{Hungarian Academy of Sciences}% \def\Ctest{ }\def\Cstr{Budapest}% \def\Dtest{ }\def\Dstr{1958}% \def\Qtest{ }\def\Qstr{access via "fuchs abelian groups 1958"}% \def\Xtest{ }\def\Xstr{I don't have this.}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{17}% \def\Atest{ }\def\Astr{Grothendieck\Revcomma A\Initper % \Aand J\Initper \Initgap A\Initper Dieudonn\'e}% \def\Ttest{ }\def\Tstr{El\'ements de g\'eom\'etrie\ alg\'e\-brique III (part one)}% \def\Jtest{ }\def\Jstr{Inst. 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9410

alg-geom/9410025

en

https://arxiv.org/abs/alg-geom/9410025

alg-geom/9410025

Dmitri Zaitsev

On the automorphism groups of algebraic bounded domains

29 pages, LaTeX, Mathematischen Annalen, to appear

Let $D$ be a bounded domain in $C^n$. By the theorem of H.~Cartan, the group $Aut(D)$ of all biholomorphic automorphisms of $D$ has a unique structure of a real Lie group such that the action $Aut(D)\times D\to D$ is real analytic. This structure is defined by the embedding $C_v\colon Aut(D)\hookrightarrow D\times Gl_n(C)$, $f\mapsto (f(v), f_{*v})$, where $v\in D$ is arbitrary. Here we restrict our attention to the class of domains $D$ defined by finitely many polynomial inequalities. The appropriate category for studying automorphism of such domains is the Nash category. Therefore we consider the subgroup $Aut_a(D)\subset Aut(D)$ of all algebraic biholomorphic automorphisms which in many cases coincides with $Aut(D)$. Assume that $n>1$ and $D$ has a boundary point where the Levi form is non-degenerate. Our main result is theat the group $Aut_a(D)$ carries a unique structure of an affine Nash group such that the action $Aut_a(D)\times D\to D$ is Nash. This structure is defined by the embedding $C_v\colon Aut_a(D)\hookrightarrow D\times Gl_n(C)$ and is independent of the choice of $v\in D$.

alg-geom

\section{Introduction}\label{in} Let $D$ be a domain in $\C^n$ and $Aut(D)$ be the group of all biholomorphic automorphisms of $D$. Let $v\in D$ be fixed and define the map $C_v\colon Aut(D) \to D\times Gl(n)$ by $f\mapsto (f(v),f_{*v})$. The theorem of H.~Cartan (see Narasimhan, \cite{N}, p.~169) can be stated as follows. \begin{samepage} \begin{Th} Let $D$ be bounded. Then: \begin{enumerate} \item The group $Aut(D)$ possesses a natural Lie group structure compatible with the compact-open topology such that the action $Aut(D)\times D\to D$ is real analytic. \item For all $v\in D$ the map $C_v$ is a real-analytic homeomorphism onto its image. \end{enumerate} \end{Th} \end{samepage} The domains we discuss here are open connected sets defined by finitely many real polynomial inequalities or connected finite unions of such sets. These are the domains in the so-called ``semi-algebraic category'' defined below (Definition~\ref{def-s-a}). For this reason we call them ``semi-algebraic domains''. In this paper we are interested in the algebraic nature of the image of $Aut(D)$ and its subgroups in $D\times Gl(n)$ under the map $C_v$. {\bf Example.1.} The simplest example of a semi-algebraic domain is the unit disk $D=\{|z|<1\}$. For this domain $$Aut(D)=PGL_2(\R)_+:= \{A\in PGL_2(\R) \mid \det A > 0 \}.$$ We see that $Aut(D)$, as a subgroup of $PGL_2(\R)$, is defined by an inequality and therefore is not an algebraic subgroup. In fact, the group $Aut(D)$ here does not admit algebraic structure as a Lie group. To show this, assume that there is a Lie isomorphism $\varphi\colon PGL_2(\R)_+\to G$, where $G$ is a real algebraic group. It continues to an isomorphism of complexifications $\varphi^{\C}\colon PGL_2(\C) \to G^{\C}$. The latter, being a Lie isomorphism between semi-simple complex algebraic groups, is algebraic. Since $G\subset G^{\C}$ is real algebraic, so is its preimage $(\varphi^{\C})^{-1}(G)=PGL_2(\R)_+$. On the other hand, $PGL_2(\R)_+\subset PGL_2(\C)$ is not real Zariski closed. This is a contradiction. \par\hfill {\bf Q. E. D.} {\bf Example.2.} More generally let $D$ be a bounded homogeneous domain in $\C^n$. By the classification theorem of Vinberg, Gindikin and Pyatetskii-Shapiro (see \cite{VGP}, Theorem~6, p.~434), $D$ is biholomorphic to a homogeneous Siegel domain of the 1st or the 2nd kind. Such a domain is defined algebraically in terms of a homogeneous convex cone (\cite{VGP}) . Rothaus (\cite{R}) gave a procedure for constructing all homogeneous convex cones. The construction implies that all homogeneous convex cones, and therefore all homogeneous Siegel domains of the 1st and 2nd kind, are defined by finitely many polynomial inequalities. The Siegel domains are unbounded but they are birationally equivalent to bounded domains which are also defined by finitely many polynomial inequalities and therefore are semi-algebraic. Thus, $D$ is biholomorphic to a bounded semi-algebaic domain. The automorphism group $Aut(D)$ of a bounded homogeneous domain was discussed by Kaneyuki (see \cite{K}) where he proved in particular that the identity component $Aut(D)^0$ is isomorphic to an identity component of a real algebraic group (\cite{K}, Theorem.3.2., p.106). Let $x_0\in D$ be a fixed point. Since $D=Aut(D)/Iso(x_0)$ and the isotropy group $Iso(x_0)$ is compact, the automorphism group $Aut(D)$ has finitely many components. Together with the result of Kaneyuki this implies that $Aut(D)$ is isomorphic to an open subgroup of a real algebraic group. In the above examples the automorphism group $Aut(D)$ is isomorphic to an open subgroup of a real algebraic group. Therefore, it admits a faithful representation. For general domains however the automorphism group does not admit a faithfull representation. {\bf Example.3.} Let $$D:=\{(z,w)\in\C^2 \mid |z|^2+|w|^2<1, w\ne 0 \} $$ be the unit ball in $\C^2$ with a unit disk removed. We consider $D$ as a subset of $\P^2$ with homogeneous coordinates $\xi_0,\xi_1,\xi_2$, $z=\xi_1/\xi_0$, $w=\xi_2/\xi_0$. The automorphism group of $D$ is $Aut(D)=SU(1,1)\times S^1$ with the action on $D$ given by $$ (A,\tau)(\xi_0,\xi_1,\xi_2) = \pmatrix{ A & 0 \atop 0 \cr 0 \, 0 & \tau \cr} \pmatrix{ \xi_0 \cr \xi_1 \cr \xi_2 \cr },\quad A\in SU(1,1),\quad \tau \in S^1. $$ The isomorphism $K\colon C\mapsto JCJ^{-1}$ between $SL_2(\R)$ and $SU(1,1)$ ($J=\pmatrix{-i&1\cr i&1\cr}$) yelds an effective action of $SL_2(\R)$ on $D$. Furthermore, each map $j_x\colon SL_2(\R)\to D$, $j_x(C):=Cx$ induces an isomorphism between fundamental groups $\pi_1(SL_2(\R))=\pi_1(D)=\Bbb Z$. This implies that the induced action of a finite covering of $SL_2(\R)$ on $D$ lifts to an effective action on the finite covering $D'$ of $D$ of the same degree. But no covering of $SL_2(\R)$ admits a faithfull representation (because every such representation factorizes through a representation of $SL_2(\C)$). On the other hand, the map $(z,w)\mapsto (z,\root d \of w)$ defines an isomorphism between a finite covering of $D$ of degree $d$ and a bounded semi-algebraic domain $\tilde D\subset\C^2$. In this example the group $Aut(D)$ is not isomorphic to an open subset of an algebraic group. Moreover, even in case it is, the action $Aut(D)\times D\to D$ can be ``far from algebraic''. This phenomena is shown in the following example. {\bf Example.4.} Let $F=\C/\Lambda$ be a complex elliptic curve and ${\cal P}\colon F\to \P^1$ the Weierstra\ss\ ${\cal P}$-function which defines a $2-1$ ramified covering over $P^1$. The strip $\{z\in\C \mid c-\epsilon<{\rm Im}z<c+\epsilon\}$ covers a ``circle strip'' $\tilde D\subset \C/\Lambda$. Let $D$ be the projection of $\tilde D$ on $P^1$. If the constant $c$ is generic and $\epsilon$ is small enough, the projection of $\tilde D$ is biholomorphic. The real algebraic group $S^1\subset \C/\Lambda$ acts by translations on $\tilde D$ which yields an effective action on $D$. Since the domain $D$ is bounded by real elliptic curves, it is therefore semi-algebraic. However the action of $S^1$ is expressed in terms of the Weierstra\ss\ ${\cal P}$-function and is not algebraic. In fact there is no homomorphism of $S^1$ in a real algebraic group $G$ such that the action $S^1\times D\to D$ is given by restrictions of polynomials on $G$. This follows from the classification of 1-dimensional Nash (semi-algebraic) groups given by Madden and Stanton (see \cite{MS}). We see therefore that even in simple cases the class of real algebraic groups and their subgroups is not large enough to describe the group $Aut(D)$ and its action on $D$. Consequently, we consider a larger class of groups where {\em defining inequalities} are allowed. \begin{Def} \begin{enumerate} \item A {\bf Nash function} is a real analytic function $f=(f_1,\ldots,f_m)\colon U\to \R^m$ (where $U$ is an open semi-algebraic subset of $\R^n$) such that for each of the components $f_k$ there is a nontrivial polynomial $P$ with $P(x_1,\ldots,x_n,f_k(x_1,\ldots,x_n))=0$ for all $(x_1,\ldots,x_n)\in U$. \item A {\bf Nash manifold} $M$ is a real analytic manifold with finitely many coordinate charts $\phi_i\colon U_i\to V_i$ such that $V_i\subset\R^n$ is semi-algebraic for all $i$ and the transition functions are Nash (a Nash atlas). \item A Nash manifold is called {\bf affine} if it can be Nash (locally closed) imbedded in $\R^N$ for some $N$. \item A {\bf Nash group} is a Nash manifold with a group operation $(x,y)\to xy^{-1}$ which is Nash with respect to every Nash coordinate chart. \end{enumerate} \end{Def} \begin{Rem} The simplest example of a Nash manifold which is not affine is the quotient $\R/\Bbb Z$ with the Nash structure inherited from the standard Nash structure on $\R$. For the classification of such groups in the one-dimensional case see J.~J.~Madden and C.~M.~Stanton in \cite{MS}. \end{Rem} Roughly speaking, the goal of this paper is to prove that the automorphism group $Aut(D)$ of a semi-algebraic domain $D$ has a natural Nash group structure such that the action $Aut(D)\times D\to D$ is also Nash. For this we need a certain non-degeneracy condition on the boundary of $D$. To give the reader a flavour of the main result, we first mention an application for the {\it algebraic} domains introduced by Diederich and Forn\ae ss (\cite{DF}), for which this condition is automatically satisfied. \begin{Def}(see Diederich-Forn\ae ss, \cite{DF}) A domain $D\subset\subset C^n$ is called {\bf algebraic} if there exists a real polynomial $r(z,\bar z)$ such that $D$ is a connected component of the set $$\{z\in\C^n \mid r(z,\bar z)<0 \}$$ and $dr(z)\ne 0$ for $z\in\partial D$. \end{Def} \begin{Th}\label{cor} Let $D\subset\subset\C^n$, $n>1$, be an algebraic domain. The group $Aut(D)$ possesses a unique structure of an affine Nash group so that the action $Aut(D)\times D\to D$ is Nash. For all $v\in D$, $C_v\colon Aut(D)\to D\times Gl(n)$ is a Nash isomorphism onto its image. \end{Th} \begin{Cor} Let $D$ be as in Theorem~\ref{cor}. Then the group $Aut(D)$ has finitely many connected components. \end{Cor} \begin{Rem} In general the number of components of $Aut(D)$ can be infinite. For example, let $H:=\{z\in\C \mid {\rm Im}z=0\}$ be the upper half-plane and $$\hat H:=H\cup \bigcup_{n\in\Bbb Z} B_{\epsilon}(n),$$ where $B_{\epsilon}(n)$ is the ball with centre $n$ and radius $\epsilon<1/2$. Let $D\subset\C^2$ be the union of $H\times H$ and $(H+i)\times \hat H$. Then $D$ is biholomorphic to a simply connected bounded domain. The flat pieces of the boundary of $D$ admit canonical foliations $z=\rm const$ and $w=\rm const$. The latters induce foliations of $D$ of the same form which are preserved by the automorphisms (see Remmert and Stein \cite{RS}). By this argument one shows that $Aut(D)=\R\oplus\Bbb Z$. \end{Rem} For the formulation of our main result we need the following condition on $D$, which is automatically satisfied for all bounded domains with smooth boundary, in particular, for all algebraic domains. \begin{Def}\label{L} \begin{enumerate} \item A domain $D\subset\C^n$ or its boundary is called {\bf Levi-non-degenerate}, if there exists a point $x_0\in\partial D$ and a neighborhood $U\subset\C^n$ of $x_0$ such that $$D\cap U=\{z\in U \mid \varphi(z)<0 \}$$ for a $C^2$-function $\varphi$ with $d\varphi\ne 0$ and such that the Levi form in $x_0$ $$L_r(x_0):=\sum_{k,l=1}^n {\partial^2 r \over \partial z_i \, \partial \bar z_j} dz_k \otimes d\bar z_l$$ restricted to the holomorphic tangent space of $\partial D$ in $x_0$ is non-degenerate; \item A domain $D\subset\C^n$ or its boundary is called {\bf completely Levi-non-degenerate}, if every boundary point outside a real analytic subset of dimension $2n-2$ is non-degenerate in the above sense. \end{enumerate} \end{Def} Now let $D$ be semi-algebraic and consider the subset $Aut_a(D)\subset Aut(D)$ of all (biholomorphic) automorphisms which are Nash. In the following sense Nash automorphisms are branches of algebraic maps (see Proposition~\ref{bra}). \begin{Def}\label{br} Let $D$ be a domain in $\C^n$. A holomorphic map $f\in Aut(D)$ is a {\bf branch of an algebraic map} if there exists a complex $n$-dimensional algebraic subvariety $G\subset\C^n\times\C^n$ which contains the graph of $f$. \end{Def} The following is the main result of this paper. \begin{Th}\label{main} Let $D\subset\subset\C^n$, $n>1$, be a semi-algebraic Levi-non-degenerate domain. Then \begin{enumerate} \item $Aut_a(D)$ is a (closed) Lie subgroup of $Aut(D)$, \item $Aut_a(D)$ possesses a unique structure of an affine Nash group so that the action $Aut_a(D)\times D\to D$ is Nash. \item For all $v\in D$, $C_v\colon Aut_a(D)\to D\times Gl(n)$ is a Nash isomorphism onto its image. \end{enumerate} \end{Th} Theorem~\ref{cor} is now a corollary of Theorem~\ref{main}. This is a consequence of the following result of K.~Diederich and J.~E.~Forn\ae ss (\cite{DF}). {\bf Theorem~1.4} {\it Let $D\subset\subset\C^n$ be an algebraic domain. Then $Aut_a(D)=Aut(D)$.} {\bf Remark.} The proof of Theorem~1.4 makes use of the reflection principle (see S.~Pin\v cuk, \cite{P}) and basic methods of S.~Webster (\cite{W}) which are also fundamental in the present paper. Due to the results of K.~Diederich and S.~Webster (\cite{DW}) and of S.~Webster (\cite{W}) on the continuation of automorphisms, Theorem~\ref{main} can be applied to every situation where the automorphisms can be $C^\infty$ extended to the boundary. K.~Diederich has informed the author that he and S.~Pin\v cuk recently proved that, under natural non-degeneracy conditions on the boundary, automorphisms of domains are always almost everywhere continuously extendable. Applying this along with the reflection method, one would expect $Aut(D)=Aut_a(D)$ for $D$ a semi-algebraic domain with $\partial D$ completely Levi-non-degenerate. Thus Theorem~\ref{main} applied to this situation would show that $Aut(D)$ is an affine Nash group acting semi-algebraically on $D$. {\bf Acknowledgement.} On this occasion I would like to thank my teacher A.~T.~Huckleberry for formulating the problem, for calling my attention to the relevant literature and for numerous very useful discussions. \section{Real semi-algebraic sets}\label{sa} Here we present some basic properties of semi-algebraic sets which will be used in the proof of Theorem~\ref{main}. For the proofs we refer to Benedetti-Risler \cite{BR}. \begin{samepage} \begin{Def}\label{def-s-a} A subset $V$ of \/ $\R^n$ is called {\bf semi-algebraic} if it admits some representation of the form $$V = \bigcup_{i=1}^s \bigcap_{j=1}^{r_i} V_{ij}$$ where, for each $i=1,\ldots,s$ and $j=1,\ldots,r_i$, $V_{ij}$ is either $\{ x\in\R^n \mid P_{ij}(x)<0 \}$ or $\{ x\in\R^n \mid P_{ij}(x)=0 \}$ for a real polynomial $P_{ij}$. \end{Def} \end{samepage} As a consequence of the definition it follows that finite unions and intersections of semi-algebraic sets are always semi-algebraic. Moreover, closures, boundaries, interiors (see Proposition~\ref{cl}) and connected components of semi-algebraic sets are semi-algebraic. Further, the number of connected components is finite (see Corollary~\ref{con}). Finally, any semi-algebraic set admits a finite semi-algebraic stratification (see Definition~\ref{str} and Proposition~\ref{strat}). The natural morphisms in the category of semi-algebraic set are {\bf semi-algebraic maps}: \begin{Def}\label{map} Let $X\subset\R^n$ and $Y\subset\R^n$ be semi-algebraic sets. A map $f\colon X\to Y$ is called {\bf semi-algebraic} if the graph of $f$ is a semi-algebraic set in $\R^{m+n}$. \end{Def} \begin{samepage} \begin{Def}\label{str} A {\bf stratification} of a subset $E$ of $\R^n$ is a partition $\{A_i\}_{i\in I}$ of $E$ such that \begin{enumerate} \item each $A_i$ (called a {\bf stratum}) is a real analytic locally closed submanifold of $\R^n$; \item if $\overline{A_i} \cap A_j \ne \emptyset$, then $\overline{A_i} \supset A_j$ and $\dim A_j < \dim A_i$ (frontier condition). \end{enumerate} A stratification is said to be {\bf finite} if there is a finite number of strata and to be {\bf semi-algebraic} if furthermore each stratum is also a semi-algebraic set. \end{Def} \end{samepage} \begin{Prop}\label{strat} Every semi-algebraic set $E\subset\R^n$ admits a semi-algebraic stratification. \end{Prop} \begin{Cor}\label{con} Every semi-algebraic set has a finite number of connected components and each such component is semi-algebraic. \end{Cor} \begin{Prop}\label{cl} Let $X$ be a semi-algebraic set in $\R^m$. Then the closure $\bar X$, its interior $\mathop{X}\limits^0$, and its boundary $\partial X$ are semi-algebraic sets. \end{Prop} Using Proposition~\ref{strat}, dimension of a semi-algebraic set is defined to be the maximal dimension of its stratum. This is independent of the choice of a finite stratification. \begin{Prop}\label{as} Let $Y\subset\R^m$ be a semi-algebraic set of $\dim Y\le k$. Then it is contained in some real algebraic set $Z$ with ${\rm dim}Z\le k$. \end{Prop} The following results on images and local triviality of semi-algebraic maps will play an important role in the present paper. \begin{Th}[Tarski-Seidenberg]\label{TS} Let $f\colon X \to Y$ be a semi-algebraic map. Then the image $f(X)\subset Y$ is semi-algebraic set. \end{Th} Further we need the Theorem on local triviality (see Benedetti-Risler \cite{BR}, Theorem~2.7.1, p.~98). \begin{samepage} \begin{Th}\label{tr} Let $X$ and $Y$ be semi-algebraic sets and let $f\colon X\to Y$ be a continuous semi-algebraic map. Fix a finite semi-algebraic partition of $X$, $\{X_1,\ldots,X_h\}$. Then there exists \begin{enumerate} \item a finite semi-algebraic stratification $\{Y_1,\ldots,Y_k\}$ of $Y$; \item a collection of semi-algebraic sets $\{F_1,\ldots,F_k\}$ and, for every $i=1,\ldots,k$, a finite semi-algebraic partition $\{ F_{i1},\ldots,F_{ir} \}$ of $F_i$ (typical fibres); \item a collection of semi-algebraic homeomorphisms $$ g_i\colon f^{-1}(Y_i) \to Y_i\times F_i, \quad i=1,\ldots,k$$ such that \begin{enumerate} \item the diagram $$\def\baselineskip20pt\lineskip3pt\lineskiplimit3pt{\baselineskip20pt\lineskip3pt\lineskiplimit3pt} \def\mapright#1{{\buildrel #1 \over \longrightarrow}} \def\mapdown#1{\Big\downarrow \rlap{$\scriptstyle#1$}} \matrix{ f^{-1}(Y_i) & \mapright{g_i} & Y_i\times F_i \cr \mapdown{f} & &\mapdown{p} \cr Y_i & \mapright{id} & Y_i \cr }$$ is commutative ($p\colon Y_i\times F_i \to Y_i$ is the natural projection); \item for every $i=1,\ldots,k$ and every $h=1,\ldots,r$, $$ g_i(f^{-1}(Y_i)\cap X_h) = Y_i \times F_{ih}. $$ \end{enumerate} \end{enumerate} \end{Th} \end{samepage} \begin{Rem} In the above setting one says that $f$ is a trivial semi-algebraic map over $Y_i$ with typical fibre $F_i$ and structure homeomorphisms $g_i$. \end{Rem} Further, we shall use the following proposition from real algebraic geometry (see \cite{BR}, Proposition~3.2.4). \begin{Prop}\label{ri} Let $V\subset\R^n$ be a real algebraic variety. The set of singular points is an algebraic set properly contained in $V$. \end{Prop} The following Proposition provides a motivation for Definition~\ref{br}. \begin{Prop}\label{bra} The Nash automorphisms $f\in Aut_a(D)$ are branches of algebraic maps. \end{Prop} {\bf Proof.} Let $f\in Aut_a(D)$ be a Nash automorphism. Then $f$ is holomorphic and semi-algebraic. Every coordinate $f_j\colon D\to \C$ is also holomorphic and, by Theorem~\ref{TS}, semi-algebraic. By Proposition~\ref{as}, there exist real algebraic sets $Z_j$ of (real) codimension $2$ with $\Gamma_{f_j}\subset Z_j$. By Proposition~\ref{ri}, there exists a regular point $(w_0,f(w_0))\in \Gamma_{f_j}$ where $\Gamma_{f_j}$ is locally given by two real polynomials $P_1(z,\bar z)$, $P_2(z,\bar z)$ or by a complex one $P(z,\bar z):=P_1(z,\bar z)+iP_2(z,\bar z)$, such that $dP\ne 0$. The latter property implies that either $\partial P\ne 0$ or $\bar\partial P\ne 0$. We can assume that $\partial P\ne 0$, otherwise $P$ can be replaced with $\bar P$. Let $P(z,\bar z)=P'(z)+P''(z,\bar z)$ be a decomposition of $P$ in the holomorphic part $P'$ and the remainder $P''$ which consists only of terms with non-trivial powers of $\bar z$. The part $P'$ is not zero because $dP'=\partial P\ne 0$. We wish to prove that $\Gamma_{f_j}$ is locally defined by the holomorphic polynomial $P'(z)$. The identity $$P(w,f_j(w),\bar w, \overline{f_j(w)}) \equiv 0$$ for all $w$ near $w_0$ implies in particular the vanishing of the Taylor coefficients of $w^k$ for all multi-indices $k$. But these coefficients are just $${1\over |k|!}{\partial^{|k|} P'(w,f_j(w)) \over \partial w^k}$$ and their vanishing means the vanishing of $P'(z,f_j(z))$. Thus the graph $\Gamma_{f_j}$ is locally defined by a holomorphic polynomial $P_j$ and the polynomials $P_1,\ldots,P_n$ define the $n$-dimensional algebraic variety required in the definition of a branch of an algebraic map. \nopagebreak\par\hfill {\bf Q. E. D.} We shall also make frequent use of Chevalley's theorem on constructible sets (see Mumford \cite{M}, p.~72). \begin{Def}\label{cons} A subset $A\in\C^n$ is called {\bf constructible} if it is a finite union of locally closed complex algebraic subvarieties. \end{Def} \begin{Rem} Constructible sets are semi-algebraic. \end{Rem} \begin{Th}[Chevalley]\label{Chev} Let $X$ and $Y$ be affine varieties and $f\colon X\to Y$ any morphism. Then $f$ maps constructible sets in $X$ to constructible sets in $Y$. \end{Th} \section{A scheme of the proof} The proof of Theorem~\ref{main} can be divided in two steps. The essential ingredient for the first step is the method of S.~Webster (see \cite{W}) based on the reflection principle (see S.~Pin\v cuk, \cite{P}). We use it to construct an appropriate {\it family of graphs} of automorphisms from $Aut_a(D)$. In fact we construct a constructible family $F$ which fibres contain automorphisms from $Aut_a(D)$. This is carried out in sections~\ref{wm} and \ref{we}. In the second step we show using the family constructed in the first step that the set $C_v(Aut_a(D))$ is Nash. Taking a neighborhood where $C_v(Aut_a(D))$ is not empty and closed and taking its pullback in $Aut(D)$ we obtain a neighborhood where $Aut_a(D)$ is closed, which implies statement~1. in Theorem~\ref{main}. In fact we prove that the {\it exact} family of graphs $$\Gamma:=\{(C_v(f),w,f(w) \mid f\in Aut_a(D), \, w\in\C^n \}$$ is Nash. If we identify $Aut_a(D)$ with $C_v(Aut_a(D))$, $\Gamma$ is the graph of the action $Aut_a(D)\times D\to D$. This proves statement~3. To obtain statement~2., we observe that group operation can be defined {\em semi-algebraically} in terms of $\Gamma$. Here we use the theory of semi-algebraic sets and their morphisms. \section{Reflection principle}\label{wm} Let $D\subset\subset\C^n$ be a semi-algebraic Levi-non-degenerate domain. By Proposition~\ref{cl}, the boundary $\partial D$ is semi-algebraic. By Proposition~\ref{as}, there exists a real algebraic set $H$ of dimension $2n-1$ which contains $\partial D$. Let $H_i$ be irreducible components of $H$ of dimension $2n-1$. By Definition~\ref{L}, the Levi form of some component of $H$, let say of $H_1$, is not everywhere degenerate. To every irreducible hypersurface $H_i$ we associate a real Zariski open set $U_i\subset\C^n$ and a real polynomial $r_i(z,\bar z)$ with $H_i\cap U_i= \{r_i=0\} \cap U_i$ and $dr\ne 0$ on $U_i$. By Proposition~\ref{ri}, such $U_i$'s and $r_i$'s exist. Let $f\in Aut_a(D)$ be any fixed map which is, by Proposition~\ref{bra} a branch of an algebraic map and $V\subset\C^n\times\C^n$ be the corresponding $n$-dimensional algebraic subvariety which contains the graph of $f$. We wish to extend $f$ in a neighborhood of a boundary point $x\in H_1$. Outside a proper complex algebraic subvariety $V'\subset V$ the variety $V$ defines $f$ and $f^{-1}$ as possibly multiple-valued algebraic maps. Since $\dim_{\C} V=n$, $\dim_{\C} V'\le n-1$. Since $\dim_{\R} H_1=2n-1$, there exists a point $x\in H_1\cap U_1$ and a neighborhood $U\subset U_1$ of $x$ such that $V$ is a trivial covering over $U$ and all sections of this covering define biholomorphic maps onto their images. One of these maps coincides with $f$ over $D\cap U$. This map yields the desired extension of $f$. Since $f$ is an automorphism of $D$, it maps $H_1\cap U$ into $\partial D$. {\bf Notation.} Let $i=i(f)$ be such that $f(H_1\cap U)\subset H_i$. Let $w_0\in H_1\cap U$ be an arbitrary point such that for $w'_0=f(w_0)$ one has $dr_i(w'_0)\ne 0$. We use the notation $$ r:=r_1, r':=r_i, H:=H_1 \hbox{, and } H':=H_i.$$ Then $f(H\cap U)\subset H'$ and we have a relation \begin{equation}\label{pz} r'(f(z),\bar f(\bar z)) = g(z,\bar z) r(z,\bar z), \end{equation} where $g(z,\bar z)$ is real analytic. Let $z=x+iy$, where $x$ and $y$ are real coordinate vectors. Since the functions in (\ref{pz}) are given by power series in $(x,y)$, they are still defined for complex vectors $x$ and $y$ near $w_0$. This is equivalent to varying $z$ and $\bar z$ independently. The relation (\ref{pz}) persists: \begin{equation}\label{pzw} r'(f(z),\bar f(\bar w)) = g(z,\bar w) r(z,\bar w). \end{equation} Now we consider the spaces $Z:=\C^n$, $Z':=\C^n$, $W:=\C^n$ and $W':=\C^n$ and define the complexifications ${\cal H}\subset Z\times\bar W$ and ${\cal H'}\subset Z'\times\bar W'$ by \begin{equation}\label{wz} r(z, \bar w) = 0, \quad r'(z', \bar w') = 0. \end{equation} The so-called {\em Segre complex varieties} associated to the points $w\in W$ are defined by $$ Q_w = \{ z\in Z \mid r(z,\bar w)=0 \}. $$ Since $g(z,\bar w)$ is holomorphic for $z$ and $w$ near $w_0$, we see from (\ref{pzw}) that the map $f^{\C}:=f\times\bar f$ takes $\{z\} \times Q_z$ into $\{z'\} \times Q'_{z'}$, where $z'=f(z)$. Hence, the family of complex hypersurfaces $\{z\} \times Q_z$ is invariantly related to $H$. Since $r(z,\bar z)$ is real, we have $$ r(z,\bar w) = \bar r(\bar w, z) = \overline{r(w,\bar z)}, $$ so that $z\in Q_w \iff w\in Q_z$. Also $z\in Q_z \iff z\in H$. Since $r$ is real and \newpage \noindent $dr=\partial r + \bar\partial r$ does not vanish at $w_0$ (resp. $r'$ is real and $dr'=\partial r' + \bar\partial r'$ does not vanish at $w'_0$), we have \begin{equation}\label{dr} \partial r(z,\bar w)\ne 0, \quad \partial r'(z',\bar w')\ne 0 \end{equation} for $z$ and $w$ near $w_0$ and for $z'$ and $w'$ near $w'_0$. The relation (\ref{dr}) implies that $Q_w$ is non-singular in $z$ if $(z,\bar w)$ is near $(w_0,\bar w_0)$. Let $\pi_z(\bar w)$ denote the complex tangent space $T_z Q_w$ as an element in the grassmanian $G_{n,n-1}$. It follows that $\pi_z$ is an antiholomorphic map from $\{z\}\times Q_z$ to $G_{n,n-1}$. S.~Webster proves the following fact (\cite{W}, p.~55, Lemma~1.1): \begin{Lemma}\label{inv} The antiholomorphic map $\pi_z(\bar w)$ is locally invertible near the points of $H$ where the Levi form is non-degenerate. \end{Lemma} Since the Levi form of $U\cap H$ is non-degenerate, and a biholomorphic map preserves this property, the Levi form of $H'$ is non-degenerate around $w'_0=f(w_0)\in H'$. By Lemma~\ref{inv}, $\pi_z$ and $\pi'_{z'}$ are locally invertible for $(z,\bar w)$ near $(w_0,\bar w_0)$ and $(z',\bar w')$ near $(w'_0,\bar w'_0)$. The first step of Webster's method is to describe the map $f$ between $Q_z$ and $Q'_{z'}$. We have seen that $f$ takes $Q_w$ into $Q'_{w'}$, $w'=f(w)$. We take $w\in Q_z$. Then all $Q_w$'s pass through the point $z$ and all $Q'_{w'}$'s through $z'$. Therefore the differential $f_{*z}\in Gl(n)$ takes $T_zQ_w$ into $T_{z'}Q'_{w'}$, i.e. $\pi_z(\bar w)$ into $\pi_{z'}(\bar w')$. These considerations mean that the restriction $f|_{Q_z}$ can be decomposed as follows (\cite{W}, p.~56): \begin{equation}\label{dec} w \stackrel{\vphantom{\pi_{z'}^{-1}}\pi_z}{\longmapsto} T_zQ_w \stackrel{\vphantom{\pi_{z'}^{-1}}J_{n-1}(f_{*z})}{\longmapsto} T_{z'}Q'_{w'} \stackrel{\pi_{z'}^{-1}}{\longmapsto} w', \end{equation} where the second map $J_{n-1}(f_{*z})\colon G_{n,n-1}\to G'_{n,n-1}$ is the natural map between grassmanians which is induced by the differential $f_{*z}\colon T_zZ \to T_{z'}Z'$. The decomposition (\ref{dec}) implies that for $w\in Q_z$ and $w'=f(w)$, $z'=f(z)$ and $q=f_{*z}\in Gl(n)$ the following relation is satisfied: \begin{equation}\label{wqw} \pi'_{z'}(\bar w') = J_{n-1}(q)(\pi_z(\bar w)). \end{equation} This relation expresses the restriction $\phi:=f|_{Q_z}$ in terms of parameters $(z,z',q)\in Z\times Z'\times Gl(n)$. This is of great importance for our parametrization of the graphs of elements of $Aut_a(D)$. Hence we underline this fact by introducing the notation \begin{equation}\label{phi} \phi(\bar z,\bar z',\bar q)\colon Q_z \to Q'_{z'} \end{equation} for $\phi:=f|_{Q_z}$. We write the conjugate variables for arguments of $\phi$ in order to emphasize that $\phi$ depends holomorphically on them. The idea of the second step is to express the map $f$ in terms of restrictions $f|_{Q_z}$. This is done separately for $n=2$ and $n\ge 3$. \subsection{The case $n\ge 3$} Let $z_0\in U\cap Q_{w_0}$ be any point. Lemma~\ref{inv} gives points $v_1,\ldots,v_n\in Q_{z_0}$ near $w_0$ such that all $Q_{v_j}$ are non-singular and transverse in $z_0$ and the algebraic curve $\gamma$, defined by \begin{equation}\label{zv0} r(z,\bar v_1) = \cdots = r(z,\bar v_{n-1}) = 0, \end{equation} is transverse to $Q_{w_0}$. Every $w$ near $w_0$ lies in some $Q_z,z\in\gamma$, namely for $z\in \gamma\cap Q_w$. Therefore, to describe $f(w)$ we need only to consider restrictions $f|_{Q_z},z\in \gamma$. Given the values $z'=f(z)$ and differentials $q=f_{*z}\in Gl(n)$, the map $f|_{Q_z}$ is determined by (\ref{wqw}). Since $\gamma\subset Q_{v_1}$, the values $z'=f(z)$ along $\gamma$ are determined, in turn, due to (\ref{wqw}) by parameters $(v_1,v'_1,l_1)=(v_1,f(v_1),f_{*v_1})$. Namely, we use the map $\phi$ in (\ref{phi}) and set \begin{equation}\label{zz'} z'=\phi(\bar v_1,\bar v'_1,\bar l_1)(z). \end{equation} Further, the differentials $f_{*z}$ along $\gamma$ can be expressed in terms of parameters $(v_1,v'_1,l_1)=(v_1,f(v_1),f_{*v_1})$. Consider the differential 1-forms $$\theta_\alpha=\partial r(z,\bar v_\alpha).$$ They define a frame in the cotangent spaces. Let $$\{Y_j=Y_j(z,\bar v_1,\ldots,\bar v_{n-1},\bar w),j=1,\ldots,n\}$$ be the dual vector field frame. This frame has rational coefficients in the variables $(z,\bar v_1,\ldots,\bar v_{n-1},\bar w)$ and satisfies the conditions \begin{equation}\label{Y} \left.\matrix{ \hbox{$Y_1$ is transverse to $Q_{v_1}$ and tangent to $Q_{v_2}$,} \cr \hbox{$Y_2$ is transverse to $Q_{v_2}$ and tangent to $Q_{v_1}$,} \cr \hbox{$Y_3,\ldots,Y_n$ are tangent to $Q_{v_1} \cap Q_{v_2}$.}\cr }\right\rbrace \end{equation} Similar differential $1$-forms $\theta'_\alpha$ and frame vector fields $Y'_j$ are constructed for $H'$. Relative to these two frame fields $$ f_{*z}Y_l = \sum q_{lj}Y'_j, $$ where \begin{equation}\label{mat} [q_{lj}]= \left[ \matrix{ q_{11} & 0 & 0 \cr 0 & q_{22} & 0 \cr q_{1\beta} & q_{2\beta} & q_{\alpha\beta} \cr } \right], \end{equation} $\alpha,\beta=3,\ldots,n$. The functions $q_{11}$, $q_{1\beta}$, $q_{\alpha\beta}$ are determined by values of $f$ along $Q_{v_1}$, i.e. by $\phi(v_1,v'_1,l_1)$ (where $v'_1=f(v_1)$ and $l_1=f_{*v_1}$). Similarly, $q_{22}$ and $q_{2\beta}$ are determined by $\phi(v_2,v'_2,l_2)$. These dependencies can be expressed by relations \begin{equation}\label{q} \left.\matrix{ q_{11} & = &\theta'_1(\phi_{*z}(v_1,v'_1,l_1)Y_1), \cr q_{1\beta} & = &\theta'_1(\phi_{*z}(v_1,v'_1,l_1)Y_\beta), \cr q_{\alpha\beta}& = &\theta'_\alpha(\phi_{*z}(v_1,v'_1,l_1)Y_\beta), \cr q_{22} & = &\theta'_2(\phi_{*z}(v_2,v'_2,l_2)Y_2), \cr q_{2\beta} & = &\theta'_2(\phi_{*z}(v_2,v'_2,l_2)Y_\beta), \cr q_{12} & = & q_{21} = q_{\alpha 1} = q_{\alpha 2} = 0. \cr }\right\rbrace. \end{equation} Thus, the map $f$ is completely determined by parameters $v_j\in V_j:=\C^n$, $v'_j\in V'_j:=\C^n$ and $l_i\in L_j:=Gl(n)$, $j=1,\ldots,n$. \subsection{The case $n=2$} In case $n=2$ there are no frames with properties~(\ref{Y}) and another construction (\cite{W},~p.~58) is needed. By Lemma~\ref{inv}, two points $\zeta_1,\zeta_2\in Q_{w_0}$ can be chosen such that $Q_{\zeta_1}$ and $Q_{\zeta_2}$ are non-singular and transverse in $w_0$. Then choose $v_1\in Q_{\zeta_1}$ and $v_2\in Q_{\zeta_2}$ such that each $Q_{v_j}$ is non-singular in $\zeta_j$ and transverse to $Q_{w_0}$ there. Now fix $v_1$ and $v_2$ and let $z_1$ and $z_2$ move along $Q_{v_1}$ and $Q_{v_2}$ respectively. For $z_1$ and $z_2$ near $\zeta_1$ and $\zeta_2$, it follows that $Q_{z_1}$ and $Q_{z_2}$ are still transverse near $w_0$ and intersect each other in a single point $w$ there. Conversely, for given $w$ near $w_0$, $Q_w$ intersects each $Q_{v_j}$ transversely in a point $z_j$ near $\zeta_j$. In this way a local biholomorphic correspondence between $w\in W(:=\C^n)$ and $(z_1,z_2)\in Q_{v_1}\times Q_{v_2}$ is obtained. It is defined by relations \begin{equation}\label{etaw} r(w,\bar{z}_j) = 0 \quad (\iff z_j\in Q_w), \quad j=1,2; \end{equation} \begin{equation}\label{etav} r(v_j,\bar{z}_j) = 0 \quad (\iff z_j\in Q_{v_j}). \end{equation} Further, set $w'_0:=f(w_0)$, $\zeta'_j:=f(\zeta_j)$, $v'_j:=f(v_j)$. All transverse properties are preserved by the biholomorphic map $f$. Again, one obtains a local biholomorphic correspondence between $w'\in W'$ and $(z'_1,z'_2)\in Q_{v'_1}\times Q_{v'_2}$, which is defined by \begin{equation}\label{etaw'} r'(w',\bar{z}'_j) = 0 \quad (\iff w'\in Q'_{z'_j}); \end{equation} \begin{equation}\label{etav'} r'(v'_j,\bar{z}'_j) = 0 \quad (\iff z'_j\in Q'_{v'_j}). \end{equation} Since the Segre varieties $Q_z$ are invariant with respect to $f$, if $z'_j:=f(z_j)$, one obtains the corresponding point $w'=f(w)$. Thus, $f$ can be decomposed in the following way: \begin{equation}\label{dec1} W \longrightarrow Q_{v_1}\times Q_{v_2} \stackrel{f\times f} {\longrightarrow} Q'_{v'_1}\times Q'_{v'_2} \longrightarrow W'. \end{equation} The middle map here is in fact $f|_{Q_{v_1}} \times f|_{Q_{v_2}}$ which is equal to $\phi(\bar v_1,\bar v'_1,\bar l_1) \times \phi(\bar v_2,\bar v'_2,\bar l_2)$, where $\phi$ is the map (\ref{phi}) and $l_j:=f_{*v_j}$. In other words we have a relation between $z_j$ and $z'_j$: \begin{equation}\label{zz'1} z'_j = \phi(\bar v_j,\bar v'_j,\bar l_j)(z_j) \end{equation} Thus, $f$ is completely determined by parameters $v_j\in V_j:=\C^n$, $v'_j\in V'_j:=\C^n$ and $l_i\in L_j:=Gl(n)$, $j=1,2$. \subsection{Reflection principle with parameters}\label{we} The local construction recalled in previous paragraph is in fact global because of its algebraic nature. The map $f$ was locally expressed in terms of parameters $(v,v',l)=(v,f(v),f_{*v})\in P$, where $v:=(v_1,\ldots,v_n)$ and $$P:=\prod_{j=1}^n (V_j\times V'_j\times L_j).$$ Using the same algebraic relations globally, we shall obtain a constructible family $F\subset P\times W\times W'$ such that the graph $\Gamma_f$ is an open subset of the closure of the fibre $\overline{F_p}$ for generic $v$ and $p=(v,f(v),f_{*v})$ (this will be made precise below). We start with construction of a constructible family for the map $$\phi(z,z',q)\colon Q_z \to Q'_{z'}$$ in (\ref{phi}). For this we consider the constructible subset $$\Phi \subset \overline{Gl(n)} \times \bar Z \times W \times \bar Z' \times W'$$ defined by relations (\ref{wz}), (\ref{dr}), and (\ref{wqw}). The relations (\ref{dr}) provide the existence of $\pi_z(\bar w)$ and $\pi'_{z'}(\bar w')$ respectively. Furthermore, we have seen that $$(\overline{f_{*z}},\bar z,w,\overline{f(z)},f(w))\in \Phi$$ for $(z,\bar w)\in \cal H$ near $(w_0,\bar w_0)$, and $\pi_z$ and $\pi_{z'}$ are local invertible there (Lemma~\ref{inv}). To provide this local invertibility ``globally'', we assume, changing if necessary to a smaller constructible subset of $\Phi$, that \begin{equation}\label{dpi} \det {\partial \pi_z(\bar w) \over \partial \bar w} \ne 0, \quad \det {\partial \pi'_{z'}(\bar w') \over \partial \bar w'} \ne 0. \end{equation} The set $\Phi$ defines now a family of possibly multiple-valued maps $$\phi(\bar z,\bar z',\bar q)\colon Q_z \to Q'_{z'}.$$ Consider the complexifications ${\cal H}\subset Z\times \bar W$ and ${\cal H'}\subset Z'\times \bar W'$. Then $\Phi$ is a subset in $\overline{Gl(n)} \times \bar{\cal H} \times \bar{\cal H}'$. \begin{Lemma}\label{f1} The projection $\delta\colon \Phi \to \overline{Gl(n)} \times \bar Z' \times \bar {\cal H}$ has finite fibres and is locally biholomorphic. \end{Lemma} {\bf Proof.} We need to prove that $w'\in\delta^{-1}(\bar q,\bar z',\bar z,w)$ depends locally holomorphically on $(\bar q,\bar z',\bar z,w)$. For this it is enough to observe, that, by (\ref{dpi}), $\pi_{z'}(\bar w)$ is locally invertible and, by (\ref{wqw}), $\bar w'=(\pi'_{z'})^{-1}(J_{n-1}(q)(\pi_z(\bar w)))$. Since the above fibres are constructible, they are finite. \par\hfill {\bf Q. E. D.} In the following let $\phi$ denote the multiple-valued map defined by $\Phi$. Since every value of $\phi$ is, by Lemma~\ref{f1}, locally holomorphic in $w$, we can discuss its differential $\phi_*=\phi(\bar q,\bar z',\bar z,w)_*$ which is also possibly multiple-valued. For the construction of the required family we need to consider auxiliary parameter spaces $A:=Z\times Z'\times Gl(n)$ for $n>2$ and $A:=(Z\times Z')^2$ for $n=2$. Let $F\subset \bar A \times P \times W\times W'$ be the constructible subset defined by relations (\ref{wz}), (\ref{wqw}), (\ref{zv0}), (\ref{zz'}) and (\ref{q}) in case $n>2$ and by (\ref{etaw}), (\ref{etav}), (\ref{etaw'}), (\ref{etav'}) and (\ref{zz'1}) in case $n=2$. Passing if necessary to a constructible subset, we can require that in case $n>3$ all $Q_{v_j}$'s and $Q_w$ are transverse in $z$ and all $Q'_{v'_j}$'s and $Q'_{w'}$ are transverse in $z'$. In case $n=2$ we require that each $Q_{v_j}$ is transverse to $Q_w$ in $z_j$ and $Q_{z_j}$'s are transverse in $w$ and, similarly, each $Q'_{v'_j}$ is transverse to $Q'_{w'}$ in $z'_j$ and $Q'_{z'_j}$'s are transverse in $w'$. Further, by Theorem~\ref{Chev} of Chevalley, the projection $\pi(F)$ of $F$ on $P\times W\times W'$ is also constructible. We don't have in general a local biholomorphic property as in Lemma~\ref{f1} for $\pi(F)$, but we still can prove the finiteness: \begin{Lemma}\label{f2} The projection $\sigma\colon \pi(F) \to P\times W$ has finite fibres. \end{Lemma} {\bf Proof.} Let fix $(p,w)\in P\times W$. Let $(p,w,w')\in F$ be any point. By the construction of $\pi(F)$, there exist points $a\in A$ such that $(\bar a,p,w,w')\in F$. {\em Case $n>2$} Let $a=(z,z',q)$. We constructed $F$ such that $Q_{v_1},\ldots,Q_{v_{n-1}}$ and $Q_w$ are transversal in $z$. Then, by (\ref{wz}) and (\ref{zv0}), the set of possible $z\in Z$ is discrete and therefore finite. Further, by (\ref{zz'}) and (\ref{q}), only finitely many $z'$'s and $q$'s are possible. Here we use Lemma~\ref{f1}. Now $w'\in W'$ is determined by (\ref{wqw}), which implies finiteness of the set of $w'$'s. {\em Case $n=2$} Let $a=(z_1,z'_1,z_2,z'_2)$. By definition of $F_2$, $Q_w$ and $Q_{v_j}$ are transverse in $z_j$. Then there are only finitely many possible intersections $z_j$. By (\ref{zz'1}), the number of possible $z'_j$'s is also finite. Finally, since $Q'_{z'_j}$ are transverse in $w'$, the number of possible $w'$'s is also finite. \par\hfill {\bf Q. E. D.} Now let $f\in Aut_a(D)$ be fixed and $H=H_1$, $H'=H_i$ for $i=i(f)$ (we defined $i(f)$ by the condition $f(H_1\cap U)\subset H_i$ for some open $U\subset\C^n$ with $H_1\cap U\ne\emptyset$). By our construction, $(v,f(v),f_{*v},w,f(w))\in \pi(F)$ for all $(v,w)$ in some open subset $U_1\subset V\times W$. Here we wish to point out that the family $\pi(F)$ depends on the index $i=i(f)$. To include all automorphisms $f\in Aut_a(D)$, we just consider the finite union of $\pi(F)_i\subset P\times W\times W'$ for all possible $i=i(f)$, $f\in Aut_a(D)$ and denote it again by $F$. The set $F$ is constructible, i.e. a finite union of locally closed algebraic subvarieties. It follows that the set $$E(f):= \{(v,w)\in D^{n+1} \mid (v,f(v),f_{*v},w,f(w))\notin F \} $$ is analytically constructible, i.e. a finite union of locally closed analytic subvarieties. If $(v,w)$ is outside the closure of $E(f)$, we have $(p,w,f(w))\in F$ for $p=(v,f(v),f_{*v})\in P$. This means that the graph $\Gamma_f:=\{(w,f(w)) \mid w\in D \}$ lies in the closure of the fibre $F_p$. Now we wish to prove main result of this section. \begin{samepage} \begin{Prop}\label{par} Let $P$, $W$ and $W'$ be as above. There exist constructible subsets $F\subset P\times W\times W'$ and $E\subset P\times V\times W$ such that \begin{enumerate} \item The projection $\sigma\colon F\to P\times W$ has finite fibres; \item For every fixed $f\in Aut_a(D)$ there exists a proper subset $E(f)\subset D^{n+1}$, such that for all $(v,w)\in (D^{n+1})\backslash E(f)$ and $p=(v,f(v),f_{*v})\in P$ one has $(p,w,f(w))\in F$, the graph $\Gamma_f$ is a subset of the closure of the fibre $F_p$ and $E(f)\subset E_p$; \item $E_p$ is of complex codimension at least $1$ in $V\times W$. \end{enumerate} \end{Prop} \end{samepage} We need the following lemma. \begin{Lemma}\label{clos} Let $A$, $B$ and $C\subset A\times B$ be constructible subsets of arbitrary algebraic varieties. Then the {\bf fibrewise} closure of $C$ in $A\times B$, i.e. the union of closures of the fibres $C_a := (\{a\} \times B) \cap C$, $a\in A$ is constructible. \end{Lemma} The proof is based on the following fact (see Mumford, \cite{M}, Corollary~1, p.71). Recall that a morphism is {\it dominating} if its image is dense. \begin{Prop}\label{mor} Let $X$ and $Y$ be two complex algebraic varieties, $f\colon X\to Y$ be a dominating morphism and $r=\dim X-\dim Y$. Then there is a nonempty open set $U\subset Y$ such that, for all $y\in U$, $f^{-1}(y)$ is a nonempty ``pure'' $r$-dimensional set, i.e. all its components have dimension $r$. \end{Prop} {\bf Proof of Lemma~\ref{clos}.} We first observe that given two constructible subsets $C_1,C_2\subset A\times B$ which have constructible fibrewise closures, the union $C=C_1\cup C_2$ has also this property. Changing to locally closed irreducible components, we can assume that $A$, $B$ and $C$ are irreducible algebraic varieties. Now we prove the statement by induction on dimension of $A$. In case $\dim A=0$ the fibrewise closure of $C$ is just the closure of $C$ which is constructible. Let $\pi\colon \bar C\to A$ denote the projection of the closure $\bar C$ on $A$. We can assume $\pi$ to be dominant, otherwise $A$ is replaced by the closure of $\pi(\bar C)$ which has a smaller dimension. Now we apply Proposition~\ref{mor} to the projection $\pi$ and obtain an open subset $U\subset A$, such that the fibre's over $U$ have pure dimension $\dim C-\dim A$. We have a partition $A=U\cup (A\backslash U)$ of $A$ and the corresponding partition $C=(C_1\cup C_2)$ ($C_1:=C\cap (U\times B)$, $C_2:=C\cap ((A\backslash U)\times B)$). By the above observation, it is enough to prove the statement for $C_1$ and $C_2$ separately. The statement for $C_2$ follows by induction, because $\dim (A\backslash U)< \dim A$. Therefore we can assume $A=U$. Now we consider the irreducible components $C_i$ of $\bar C \backslash C$, $\dim C_i<\dim C$. If $S_i:=\overline{\pi(C_i)}\ne A$ for some $i$, then we replace $A$ by $A\backslash S_i$ and correspondingly $C$ by $C\cap\pi^{-1}(A\backslash S_i)$. Thus we may assume that $\pi\colon C_i\to A$ is dominaiting for all $i$. Then we can apply Proposition~\ref{mor} to every $C_i$ and obtain a number of open sets $U_i\subset A$. Let $U$ be the intersection of all $U_i$'s. Since $A$ is irreducible, $U$ is not empty. Again, proceeding by induction, we can reduce the statement to the case $A=U$. But in this case the fibres of $\bar C$ are of pure dimension $\dim C-\dim A$ and the fibres of $\bar C \backslash C$ have smaller dimension. This implies that the fibrewise closure of $C$ coincides with the usual closure $\bar C$ which is constructible. \par\hfill {\bf Q. E. D.} {\bf Proof of Proposition~\ref{par}.} Statement~1 follows from Lemma~\ref{f2}. It follows from the local Webster's construction (section~\ref{wm}) that $(p,w,f(w))\in F$ ($p=(v,f(v),f_{*v})$) for all $(v,w)$ in an open subset $U\subset D^{n+1}$. This means that $U$ lies in the complement of the `` exceptional set'' $E(f)$. Let $\Omega(f):=D^{n+1}\backslash E(f)$, i.e. $$\Omega(f)= \{(v,w)\in D^{n+1} \mid (v,f(v),f_{*v},w,f(w))\in F \}. $$ The set $E$ must be globally defined independently of any automorphism $f\in Aut_a(D)$. For this it is necessary to define $\Omega(f)$ in another way. Changing if necessary to a constructible subset of $F$, we may assume that the projection $\sigma_p \colon F_p\to W$ is locally biholomorphic and $\Omega(f)$ still contains an open subset $U\subset D^{n+1}$. Then the differentials $\partial w' \over \partial w$ are certainly defined. We now define the family $F'\subset F\times Gl(n)$ of differentials by adding values of $\partial w' \over \partial w$: $$F':= \{ (p,w,w',q)\in F\times Gl(n) \mid q={ \partial w' \over \partial w} \}$$ This is a constructible set and we have $(v,f(v),f_{*v},w,f(w),f_{*w})\in F'$ for $(v,w)$ in some open set $U\subset D^{n+1}$. Now we write the definition of $\Omega(f)$ in the form: $$\Omega(f):= \{ (v,w) \in V\times W \mid \hbox{ for } v'=f(v),l=f_{*v},w'=f(w): $$ $$ (v,v',l,w,w')\in F \}. $$ Now we define a set $\Omega$ which contains $\Omega(f)$ for all $f\in Aut_a(D)$: \begin{samepage} $$\Omega:= \{ (p,v,w,v',w',l) \in P\times V\times W\times V'\times W'\times Gl(n) \mid $$ $$\forall j: (p,v_j,v'_j,l_j)\in F' \land (p,w,w')\in F \land (v,v',l,w,w')\in F \}. $$ \end{samepage} For $f\in Aut_a(D)$, $(v,w)\in U$, and $p=(v,f(v),f_*v)$, we have $$(v,w,f(v),f(w),f_{*v})\in \Omega_p.$$ Let $\Omega'$ be the fibrewise closure of $\Omega$, i.e. the union of all closures of $\Omega_p$, $p\in P$. By Lemma~\ref{clos}, $\Omega'$ is constructible. Finally, we define $E\subset P\times V\times W$ to be the projection of $\Omega'\backslash \Omega$ on $P\times V\times W$. By Theorem~\ref{Chev} of Chevalley, $E$ is constructible. We now wish to prove that every fibre $E_p$ is of codimension at least $1$. For this we note that for $p\in P$ fixed the projection of $\Omega_p$ on $V\times W$ has finite fibres, i.e. $\dim \Omega_p\le \dim (V\times W)$. This implies $\dim (\Omega'\backslash \Omega)_p < \dim (V\times W)$ and $E_p$ is of codimension at least $1$ as required in statement~3. We take now any $(v,w)\in D^{n+1}$ outside $E(f)$ and set $p=(v,f(v),f_{*v})$. For the proof of statement~2, consider $f\in Aut_a(D)$, take $(v,w)\in D^{n+1}\backslash E(f)$, and set $p=(v,f(v),f_*v)$. We shall prove that $E(f)\subset E_p$. Let $(v_0,w_0)\in E(f)$ be any point. If $(v,w)\in U$, we have $$(v,f(v),f_{*v},w,f(w),f_{*w})\in F',$$ which implies $$(p, v, w, f(v), f(w), f_{*v}) \in \Omega.$$ Here $(v,w)\in U$ is arbitrary. Since $f$ is holomorphic, we have this property globally for all $(v,w)\in D^{n+1}$ if we replace $\Omega$ with its fibrewise closure $\Omega'$. In particular, we have $$(v_0,w_0,f(v_0),f(w_0),f_{*v_0}) \in \Omega'_p.$$ Since $(v_0,w_0)\in E(f)$, the point $$(v_0,f(v_0),f_{*v_0},w_0,f(w_0))$$ does not lie in $F$. This implies that $$(v_0,w_0,f(v_0),f(w_0),f_{*v_0})$$ does not lie in $\Omega_p$ and then it is in $\overline{\Omega_p}\backslash \Omega_p$. This means $(v_0,w_0)\in E_p$, which is required. The proof of Proposition~\ref{par} is finished. \par\hfill {\bf Q. E. D.} \section{The choice of parameters} In the previous section we proved the existence of a constructible algebraic family $F\subset P\times W\times W'$ with the property that for all $f\in Aut_a(D)$ there exists a point $p\in P$ such that \begin{equation}\label{subs} \Gamma_f\subset \overline{F_p}. \end{equation} The goal of this section is to choose for every $f$ appropriate parameter $p$ with this property and obtain a map $\imath$ from $Aut_a(D)$ in the corresponding parameter space $P$. The first idea is to take some generic $v\in V(:=V_1\times\cdots\times V_n=\C^{n^2})$ and to define $p=(v,f(v),f_{*v})$. If $(v,w)\notin E(f)$, Proposition~\ref{par} yields the required property (\ref{subs}). However, if we wish to define a global map $Aut_a(D)\to P$, the condition $(v,w)\notin E(f)$ must be satisfied for all $f\in Aut_a(D)$. Unfortunately, this is not true in general. It is therefore necessary to take sufficiently many points $(v_\mu,w_\mu) \in V\times W$ instead of one $(v,w)$, such that $(v_\mu,w_\mu) \notin E(f)$ is always true at least for one $\mu$. In fact, we prove the following Proposition. \begin{samepage} \begin{Prop}\label{par1} There exists a natural number $N$, a constructible subset $F\subset P^N\times W\times W'$ and a collection of points $v_1,\ldots,v_m\in D$, $m=nN$, such that \begin{enumerate} \item the projection $\sigma \colon F\to P^N\times W$ has finite fibres, \item for all $f\in Aut_a(D)$ the graph $\Gamma_f$ is a subset of the closure $\overline{F_{\imath(f)}}$, where the map $\imath\colon Aut(D)\to P^N$ is given by $\imath(f) = (v_1,f(v_1),f_{*v_1},\ldots,v_m,f(v_m),f_{*v_m})$. \end{enumerate} \end{Prop} \end{samepage} \begin{Rem}\label{any-v} Once the set $v_1,\ldots,v_m$ is chosen, we can add to it finitely many other $v$'s and not change the statement of Proposition~\ref{par1}. \end{Rem} Before we start with the proof we need a technical lemma. \begin{Lemma}\label{pts} Let $A$, $B$, $C\subset A\times B$ be constructible sets and every fibre $C_a:=\{b\in B\mid (a,b)\in C \}$ be of codimension at least one. Then there exists a finite number of points $b_\mu\in B$, $\mu=1,\ldots,s$ such that for every $a\in A$ there is a point $b_\mu\notin C_a$. \end{Lemma} {\bf Proof.} We first prove the statement for $A$ a locally closed irreducible subvariety by induction on dimension of $A$. If $\dim A=0$, the statement is obvious. Assume it to be proven for $\dim A < d$. By definition of constructible sets, $C$ is a finite union of locally closed subvarieties $C_\alpha=U_\alpha \cap F_\alpha$ where $U_\alpha$ are Zariski open and $F_\alpha$ are closed subvarieties. The subvarieties $F_\alpha$ are not open, otherwise a fibre $C_a$ would contain an open subset. So we can choose a point $$(a_0,b_0)\in (\cap_\alpha U_\alpha) \backslash (\cup_\alpha F_\alpha)$$ The set $A'$ of points $a\in A$, such that $b_0\in C_a$, is the projection on $A$ of the intersection $(A\times \{b_0\})\cap C$, which is constructible. There is an entire neighborhood of $a_0$ in the complement and, hence, $A'$ has lower dimension than $A$. Now we use induction for all irreducible components of the closure $\overline{A'}$. This yields a number of points $b_\mu$. These points together with $b_0$ satisfy the required condition. To prove the statement in case $A$ is constructible we note, that $A$ is by Definition~\ref{cons} a finite union of locally closed $A_\alpha$'s. For every $A_\alpha$ with $C_\alpha:=(A_\alpha \times B) \cap C$ the statement of Lemma gives a finite set of points $b_\mu$. The union of these finite sets for all $\alpha$ satisfies the required property. \par\hfill {\bf Q. E. D.} {\bf Proof of Proposition~\ref{par1}.} Now we apply Lemma~\ref{pts} to our situation. Let $P'$ be the constructible subset of all parameters $p\in P$ such that $E_p\subset V\times W$ is of codimension at least $1$. Then we set in Lemma~\ref{pts} $A:=P'$, $B:=V\times W$ and $C:=E \cap (P'\times V\times W)$. The statement of Lemma yields a number of points $$(v^{(\mu)},w_\mu) \in D^{n+1},\quad \mu=1,\ldots,N.$$ For every $f\in Aut_a(D)$ and $(v,w)\notin E(f)$ we have by condition~3 in Proposition~\ref{par}, $E(f)\subset E_p$ for $p=(v,f(v),f_{*v})$. Then for some $\mu=1,\ldots,N$ we have $(v^{(\mu)},w_\mu)\notin E_p$, i.e. $(v^{(\mu)},w_\mu)\notin E(f)$ and, by condition~2 in Proposition~\ref{par}, $(p,w,f(w))\in F$. We obtain $m=Nn$ points $v_1,\ldots,v_m$. Now we construct the required family $F$ to be the union of the sets $F_\mu$ defined by \begin{equation}\label{fnu} F_\mu := \{ (p_1,\ldots,p_N,w,w')\in P^N\times W\times W' \mid (p_\mu,w,w')\in F \}, \end{equation} Statement~1 in Proposition~\ref{par1} follows from condition~1 in Proposition~\ref{par}. Let $\imath\colon Aut(D) \to P^N$ be the map defined by \begin{equation}\label{imath} \imath(f) := (v_1,f(v_1),f_{*v_1},\ldots,v_m,f(v_m),f_{*v_m}). \end{equation} It is in fact a product of Cartan maps $C_v\colon f\mapsto (f(v),f_{*v})$ and is therefore a homeomorphism onto its image. Statement~2 in Proposition~\ref{par1} follows now from the above choice of $v_j$'s. \par\hfill {\bf Q. E. D.} \section{Defining conditions for $Aut_a(D)$} In Proposition~\ref{par1} we constructed a map $\imath\colon Aut(D) \to P^N$. Our goal here is to give semi-algebraic description of the image $\imath(Aut_a(D))$ and to prove the following Proposition. \begin{Prop}\label{Ga} The image $\imath(Aut_a(D))$ and the set of all graphs $$ \Gamma := \{(\imath(f),w,f(w)) \mid f\in Aut_a(D) \land w\in D \} $$ are semi-algebraic. \end{Prop} \subsection{Reduction to a fixed pattern.} In Proposition~\ref{par1} we obtained a constructible family $F\subset P^N\times W\times W'$. Our goal now is to find a stratification of $P^N\times W$ such that $F$ has a simplier form over each stratum. This is done by applying Theorem~\ref{tr} on local triviality of semi-algebraic morphisms. To simplify the notation we shall write $P$ for $P^N$. We first consider the projection $\sigma\colon F\to P\times W$. Since we are interested only in points over $P\times D\subset P\times W$, we write $F\subset P\times D\times W'$ for the intersection with $P\times D\times W'$. Since $D$ is semi-algebraic, $F$ is semi-algebraic. The projection $\sigma\colon F\to P\times D$ is a continuous semi-algebraic map (see Definition~\ref{map}) and we can apply Theorem~\ref{tr} on local triviality. Theorem~\ref{tr} yields a finite semi-algebraic stratification $\{Y_1,\ldots,Y_h\}$ of $P\times D$ (see Definition~\ref{str}), a collection of semi-algebraic typical fibres $\{E_1,\ldots,E_h\}$ and a collection of semi-algebraic structural homeomorphisms \begin{equation}\label{E} \tilde e_i\colon Y_i\times E_i \to \sigma^{-1}(Y_i), \quad i=1,\ldots,h \end{equation} (the $\tilde e_i$'s here are the inverses of the $g_i$'s in Theorem~\ref{tr}). By statement~1 in Proposition~\ref{par1}, every typical fibre $E_i$ is finite. The semi-algebraic stratification $\{Y_i\}$ of the product $P\times D$ defines a stratification of every fibre $\{p\}\times D$. This stratification depends on $p\in P$ and the qualitative picture (e.g. the number of open strata) can also depend on $p$. However, by changing to a partition of $P$ we reduce this general case to the case of fixed stratification of $\{p\}\times D$, a fixed {\em pattern}. For this we apply Theorem~\ref{tr} again to the projection $\rho\colon P\times D \to P$ and partition $\{Y_1,\ldots,Y_h\}$ of $P\times D$. We obtain a finite semi-algebraic stratification $\{P_1,\ldots,P_r\}$ of $P$, a collection of semi-algebraic typical fibres $\{G_1,\ldots,G_r\}$, for every $l=1,\ldots,r$ a finite semi-algebraic partition $\{G_{l1},\ldots,G_{lh} \}$ of $G_l$ and a collection of semi-algebraic structural homeomorphisms \begin{equation}\label{G} g_l\colon P_l\times G_l \to \rho^{-1}(P_l), \quad l=1,\ldots,r, \end{equation} such that \begin{equation}\label{Gi} g_l(P_l\times G_{li}) = (\rho^{-1}(P_l)) \cap Y_i, \quad l=1,\ldots,r, \quad i=1,\ldots,h. \end{equation} \subsection{The set of all automorphisms.} Here we discuss the set of all automorphisms of $D$ the graphs of which are contained in the closures of fibres of our family $F$. By condition~1 in Proposition~\ref{par1}, only finitely many automorphisms can be contained in the closure of a fixed fibre. On the other hand, a fixed automorphism can be contained in closures of a multitude of fibres. Without loss of generality we assume, that $\{G_{l1},\ldots,G_{lh} \}$ is a finite semi-algebraic stratification of $G_l$ and $G_{li}$ are connected (see Proposition~\ref{strat} and Corollary~\ref{con}). Now for fixed $p\in P_l$ we wish to determine if the fibre $F_p\subset D\times W'$ is related to some $f\in Aut(D)$. Our procedure for doing this is semi-algebraic: over the fixed decomposition $D=\sqcup_i G_{il}$ (in fact only over open $G_{il}$'S) we consider the pieces of $F_p$, determined by the trivialization of it with typical fibres $E_i$. The condition that certain of these pieces fit together to form a graph of an automorphism proves to be semi-algebraic. Among the strata $G_{li}$, $i=1,\ldots,h$ we choose the open one's, which are assumed to be $G_{li}$, $i=1,\ldots,t$, $t\le h$. By Proposition~\ref{par1}, the projection $\sigma \colon F\to P\times D$ has finite fibres so the typical fibres $E_i$ are all finite. Let us fix a $t$-tuple $e=(e_1,\ldots,e_t)\in E:=E_1\times\cdots\times E_t$. The number of possible $t$-tuples is finite. Further, we define the maps $\xi_{e,p}$ over each open $(Y_i)_p := \{w\in D \mid (p,w)\in Y_i\}\subset D$ by \begin{equation}\label{xi} (p,w,\xi_{e,p}(w)) = \tilde e_i(p,w,e_i), \quad i=1,\ldots,t, \end{equation} where $\tilde e_i\colon Y_i\times E_i \to \sigma^{-1}(Y_i)$ are the trivialization morphisms in~(\ref{E}). \begin{Prop}\label{pl} Let $P_l$ and $e\in E$ be fixed. The set $P_{e,l}$ of all parameters $p\in P_l$ such that the map $\xi_{e,p}$ extends to a biholomorphic automorphism from $Aut_a(D)$ is semi-algebraic. \end{Prop} We begin with three lemmas. The first one is a semi-algebraic version of Lemma~\ref{clos} on constructible sets. \begin{Lemma}\label{clos-s-a} Let $A$, $B$ and $C\subset A\times B$ be semi-algebraic sets. Then the ``fibrewise'' closure of $C$, i.e. the union of closures of the fibres $C_a :=(\{a\} \times B) \cap C$, $a\in A$ is semi-algebraic. \end{Lemma} {\bf Proof.} Let $X:=A\times B$ and consider the partition $X_1:=C$, $X_2:=(A\times B)\backslash C$ of $X$. Apply Theorem~\ref{tr} to the projection of $X$ on $A$. To obtain the closures of fibres we take the closures of typical fibres $F_{i1}$ in $F_i$ and their images in $X$ under the structural trivializing homeomorphisms. The images are semi-algebraic by Theorem~\ref{TS}. The union of these images for all $i$ is semi-algebraic and is exactly the ``fibrewise'' closure of $C$. \nopagebreak\par\hfill {\bf Q. E. D.} \begin{Lemma}\label{sub} Let $A$, $B$ and $C,D\subset A\times B$ be semi-algebraic sets. Then the set of $a\in A$ such that $C_a \subset D_a$ is semi-algebraic. \end{Lemma} {\bf Proof.} The complement of the required set in $A$ coincides with the projection on $A$ of the difference $C\backslash D$. The difference of semi-algebraic set is semi-algebraic, the projection is semi-algebraic by the Tarski-Seidenberg theorem (Theorem~\ref{TS}). \par\hfill {\bf Q. E. D.} \begin{Lemma}\label{1} Let $A$, $B$, $C$, $E\subset A\times B$ and $G\subset E\times C$ be semi-algebraic sets. Then the set of all $a\in A$, such that for all $b\in E_a$ the fibre $G_{(a,b)}$ consists of exactly one point, is also semi-algebraic. \end{Lemma} {\bf Proof.} We apply the Theorem~\ref{tr} on local trivialization to the projection of $G$ on $E$. This yields a partition $\{Y_i\}$ of $E$. The set $E'\subset E$ of one-point fibres $G_{(a,b)}$ is then the finite union of $Y_i$'s such that the corresponding typical fibres $F_i$ consist of one point. It follows that $E'$ is semi-algebraic. The required set in $A$ coincides with the set of $a\in A$ such that $C_a\subset E'_a$. The latter set is semi-algebraic by Lemma~\ref{sub}. \par\hfill {\bf Q. E. D.} {\bf Proof of Proposition~\ref{pl}.} We first consider the condition that $\xi_{e,p}$ extends to a well-defined continuous map on $D$. This means that for any point $w\in \overline{(Y_i)_p} \cap \overline{(Y_j)_p}$, $i,j=1,\ldots,t$, the limits of graphs of $\xi_{e,p}$ over $(Y_i)_p$ and $(Y_j)_p$ coincide over $w$ and consist of one point. For every stratum $G_{ls}$, $G_{li}$, $G_{lj}$, $s=1,\ldots,h$, $i,j=1,\ldots,t$, with $$G_{ls}\subset \overline{(Y_i)_p} \cap \overline{(Y_j)_p}$$ we write these conditions in a form \begin{equation}\label{cont} \left. \matrix{ i) & \overline{\Gamma_i(p)} \cap (B(p)\times W') = \overline{\Gamma_j(p)} \cap (B(p)\times W'), \cr\cr ii) & \forall w\in B(p) : \#(\overline{\Gamma_i(p)} \cap (\{w\}\times W')) = 1 } \right\rbrace, \end{equation} where $$B(p) := g_l(\{p\} \times G_{ls}) \subset \{p\} \times D$$ and $\Gamma_i(p) := \tilde e_i((Y_i)_p\times \{e_i\})$ is the graph of $\xi_{e,p}$ over $(Y_i)_p$. Now, by Lemmas~\ref{clos-s-a} and \ref{sub}, the set $\{p\in P_l\mid i)$ in (\ref{cont}) is satisfied $\}$ is a semi-algebraic subset of $P_l$. For the condition ii) in (\ref{cont}) we set in Lemma~\ref{1} $A:=P'$, $B:=D$, $C:= W'$, $$E:=\{(p,w)\in P'\times D \mid w\in B(p) \}$$ and $$G:=\{(p,w,w')\in E\times W' \mid w\in \overline{\Gamma_i(p)} \cap (B(p)\times W')\}.$$ Then, by Lemma~\ref{1}, the set $\{p\in P_l\mid ii)$ in (\ref{cont}) is satisfied $\}$ is a semi-algebraic subset of $P_l$. Without loss of generality, $i)$ and $ii)$ are satisfied for $p\in P_l$. Thus, the closures of graphs of $\xi_{e,p}$ over $\{p\} \times D$ yield well-defined maps $\xi_p\colon D\to W'$ (We do not know yet, whether or not these maps are continuous). The next condition on $\xi_{e,p}$ is \begin{equation}\label{D} \xi_{e,p}(D) = D, \end{equation} which is, by Lemma~\ref{sub}, a semi-algebraic condition. Now, if conditions (\ref{cont}) and (\ref{D}) are satisfied, we can prove that $\xi_{e,p}$ is continuous. For this let $U_p\subset D$ denote the union of all $(Y_i)_p$'s, $i=1,\ldots,t$. This is an open dense subset of $D$ where $\xi_{e,p}$ is continuous. Fix a point $w_0\in D$. By (\ref{cont}), $\xi_{e,p}(w_0)$ is the only limit value of $\xi_{e,p}(w)$ for $w\in U_p$. Since $\xi_{e,p}$ is bounded, we have \begin{equation}\label{lim} \xi_{e,p}(w_0) = \lim_{{w\to w_0 \atop w\in U}} \xi_{e,p}(w), \end{equation} which means $\xi_{e,p}$ is continuous. Thus, we obtained a family of continuous maps $\xi_{e,p}$ from $D$ onto $D$, which are holomorphic outside some real analytic locally closed subvariety of codimension~$1$. By the theorem on removable singularities, $\xi_{e,p}$ is holomorphic on $D$. Further, by the theorem of Osgood (see \cite{N}, Theorem~5, Chapter~5) $\xi_{e,p}$ is biholomorphic if and only if it is injective. This is the condition on fibres: \begin{equation}\label{inj} \#(\xi_{e,p}^{-1}(y))=1,\, y\in D. \end{equation} The set $\{p\in P_l\mid$ (\ref{inj}) is satisfied $\}$ is, by Lemma~\ref{1}, semi-algebraic (we set $A:=P'$, $B:= W'$, $C:=D$, $E:=P'\times D\subset A\times B$ and $G$ is the family of graphs of $\xi_{e,p}$). This finishes the proof of Proposition~\ref{pl}. \par\hfill {\bf Q. E. D.} \subsection{The image $\imath(Aut_a(D))$ and the set of associated graphs.} Here we wish to prove Proposition~\ref{Ga}. Let $P_{e,l}$ be the semi-algebraic subsets from Proposition~\ref{pl}. We obtain a diagram: \begin{equation}\label{di} \def\baselineskip20pt\lineskip3pt\lineskiplimit3pt{\baselineskip20pt\lineskip3pt\lineskiplimit3pt} \def\mapup#1{\Big\uparrow \rlap{$\scriptstyle#1$}} \matrix{ & & P \cr & \nearrow & \mapup{\imath} \cr P_{e,l} & \to & Aut_a(D) \cr p & \mapsto & \xi_{e,p} \cr }, \end{equation} where the map from $P_{e,l}$ into $P$ is the usual inclusion. We define $P'_{e,l}\subset P_{e,l}$ to be the subset of all points $p\in P_{e,l}$, for which the diagram is commutative. This condition means $p=(v,v',l)=(v,\xi_{e,p}(v),(\xi_{e,p})_{*v})$ and is therefore semi-algebraic. Therefore, $P'_{e,l}$ is semi-algebraic. The semi-algebraic property of $\imath(Aut_a(D))$ is a consequence of the following observation. \begin{Lemma} $$\imath(Aut_a(D))=\bigcup_{{e\in E \atop l=1,\ldots,r}} P'_{e,l}.$$ \end{Lemma} {\bf Proof.} Let $p\in \imath(Aut_a(D))$, i.e. $p=\imath(f)$ for some $f\in Aut_a(D)$. Then, by Proposition~\ref{par1}, $\Gamma_f\subset \overline{F_p}$. We have $p\in P_l$ for some $l=1,\ldots,r$. The graph $\Gamma_f$ defines sections in $F$ over every connected open stratum $(Y_i)_p$, $i=1,\ldots,t$. This means that for some choice $e\in E$ we have $f=\xi_{e,p}$. Then $\xi_{e,p}\in Aut_a(D)$, which implies $p\in P_{e,l}$. Further, the equality $f=\xi_{e,p}$ means that diagram~(\ref{di}) is commutative for $p$. Then $p\in P'_{e,l}$ and the inclusion in one direction is proven. Conversely, let $e\in E$ be fixed and $p\in P'_{e,l}$. Since $p\in P_{e,l}$, $f:=\xi_{e,p}$ is an automorphism in $Aut_a(D)$. The commutativity of diagram~(\ref{di}) means $p=\imath(f)$. This implies $p\in \imath(Aut_a(D))$, which proves the inclusion in other direction. \nopagebreak\par\hfill {\bf Q. E. D.} {\bf Proof of Proposition~\ref{Ga}.} The family $$\Gamma:=\{(\imath(f),w,f(w)) \mid f\in Aut_a(D) \land w\in D \}$$ over $P'_{e,l}$ coincides now with the family of graphs of $\xi_{e,p}$. The latter is, by construction, semi-algebraic and Proposition~\ref{Ga} is proven. \nopagebreak\par\hfill {\bf Q. E. D.} \section{Semi-algebraic structures on $Aut_a(D)$} Here we finish the proof of Theorem~\ref{main}. In previous section we considered imbeddings $\imath\colon Aut(D)\to P^N$. Here we wish to change to Cartan imbeddings $C_v(f):=(f(v),f_{*v})$. By Proposition~\ref{par1}, $\imath$ is given by $\imath(f)=(v,f(v),f_{*v})$, where $v=(v_1,\ldots,v_m)\in D^m$. The image $C_{v_j}(Aut_a(D))$ is equal to the projection of $\imath(Aut_a(D))$ on $V'_j\times L_j$ (we use our notations $v'_j=f(v_j)\in V'_j$, $l_j=f_{*v_j}\in L_j$). By Theorem~\ref{TS} of Tarski-Seidenberg, $C_{v_j}(Aut_a(D))$ is semi-algebraic. Further, it is semi-algebraically isomorphic to $\imath(Aut_a(D))$. By Remark~\ref{any-v}, any $v$ and $v'$ can be among $v_j$'s. Thus we obtain the following result. \begin{Prop}\label{stru} Let $v\in D$ be any point. The image $C_v(Aut_a(D))$ is semi-algebraic and this semi-algebraic structure is independent of $v\in D$. \end{Prop} Now we fix some $v\in D$ and denote by $K$ the image $C_v(Aut_a(D))\subset D\times Gl(n)$. The family $\Gamma'\subset C_v(Aut_a(D))\times D^2$ of graphs over $C_v(Aut_a(D))$ is a projection of $\Gamma$ and is therefore semi-algebraic. To simplify our notation we set $P:=D\times Gl(n)$, and $\Gamma:=\Gamma'$. Statements~2 and 3 in Theorem~\ref{main} can now be formulated as follows: \begin{Lemma}\label{s-a} With respect to the group operation of $Aut_a(D)$, $K$ is a Nash group and the action on $D$ is Nash. \end{Lemma} {\bf Proof.} We consider the graph of the operation $(x,y)\mapsto xy^{-1}$ in $K^3$. For this, start with the family $\Gamma\subset K\times D\times D$ and define a new family $\Gamma_1\subset K^3\times D^3$ by \begin{equation}\label{G1} \Gamma_1 := \{ (x,y,z,w,w',w'')\in K^3\times D^3 \mid (y,w',w)\in \Gamma \land (x,w',w'')\in \Gamma \}. \end{equation} The conditions in (\ref{G1}) express the fact that $y^{-1}\in K$ transforms $w$ in $w'$ and $x\in K$ transforms $w'$ in $w''$. The projection $\Gamma_2$ of $\Gamma_1$ on $K^3\times D^2$ (with coordinates $(x,y,z,w,w'')$) is, by the Theorem of Tarski-Seidenberg, semi-algebraic. Now the condition $z=xy^{-1}$ means that the graphs of $z$ and $xy^{-1}$ coincide, i.e. the fibres $(\Gamma_2)_{(x,y,z)}$ and $(\Gamma_3)_{(x,y,z)}$ coincide, where $\Gamma_3 := \{(x,y,z,w,w'') \mid (z,w,w'')\in \Gamma \}$ is an extension of $\Gamma$. By Lemma~\ref{sub}, the coincidence of fibres is a semi-algebraic condition on $(x,y,z)\in K^3$. This proves that the graph of the correspondence $(x,y)\mapsto xy^{-1}$ is semi-algebraic, which means that the group operation is semi-algebraic. Since the latter is also real analytic by Theorem of Cartan, $K$ is an affine Nash group. Furthermore, the graph $\Gamma$ of the action of $K$ on $D$ is semi-algebraic and real analytic and therefore Nash. \par\hfill {\bf Q. E. D.} It remains to prove statement~1 in Theorem~\ref{main} which asserts that $Aut_a(D)$ is a Lie subgroup of $Aut(D)$. {\bf Proof of statement~1. } We begin with the semi-algebraic set $K$. By Proposition~\ref{strat}, it admits a finite semi-algebraic stratification. Let $x\in K$ be a point in a stratum of maximal dimension. Then there is a neighborhood $U_x\subset P$ of $x$, such that $K\cap U_x$ is a closed real analytic submanifold of $U_x$. The preimage $K':=C_v^{-1}(K\cap U_x)$ is a closed real analytic submanifold in the neighborhood $U_f:=C_v^{-1}(U_x)$ of $f:=C_v^{-1}(x)$. Since $Aut_a(D)$ is a subgroup of $Aut(D)$, we see that $Aut_a(D)\cap (f^{-1} \cdot U_f) = f^{-1} \cdot K'$ is a closed real analytic submanifold in the neighborhood $f^{-1} \cdot U_f$ of the unit $id\in Aut(D)$. This implies that $Aut_a(D)$ is a real analytic subgroup of $Aut(D)$. \hfill {\bf Q. E. D.}

9710

alg-geom/9710020

en

https://arxiv.org/abs/alg-geom/9710020

alg-geom/9710020

Victor V. Batyrev

Birational Calabi--Yau n-folds have equal Betti numbers

AMS-LaTeX, 11 pages, to appear in in Proc. European Algebraic Geometry Conference (Warwick, 1996)

Let X and Y be two smooth projective n-dimensional algebraic varieties X and Y over C with trivial canonical line bundles. We use methods of p-adic analysis on algebraic varieties over local number fields to prove that if X and Y are birational, they have the same Betti numbers.

alg-geom

\section{Introduction} The purpose of this note is to show how to use the elementary theory of $p$-adic integrals on algebraic varieties to prove cohomological properties of birational algebraic varieties over $\C$. We prove the following theorem, which was used by Beauville in his recent explanation of a Yau--Zaslow formula for the number of rational curves on a K3 surface \cite{Beauville} (see also \cite{FGS,YZ}): \begin{thm} \label{main-th} Let $X$ and $Y$ be smooth $n$-dimensional irreducible projective algebraic varieties over $\C$. Assume that the canonical line bundles $\Om^n_X$ and $\Om^n_Y$ are trivial and that $X$ and $Y$ are birational. Then $X$ and $Y$ have the same Betti numbers, that is, \[ H^i(X,\C) \cong H^i(Y,\C) \quad \text{for all $i\ge0$.} \] \end{thm} Note that Theorem~\ref{main-th} is obvious for $n =1$, and for $n=2$, it follows from the uniqueness of minimal models of surfaces with $\kappa\ge0$, that is, from the property that any birational map between two such minimal models extends to an isomorphism \cite{KMM}. Although $n$-folds with $\kappa\ge0$ no longer have a unique minimal models for $n\ge3$, Theorem~\ref{main-th} can be proved for $n=3$ using a result of Kawamata (\cite{kawamata}, \S6): he showed that any two birational minimal models of $3$-folds can be connected by a sequence of flops (see also \cite{kollar}), and simple topological arguments show that if two projective $3$-folds with at worst $\Q$-factorial terminal singularities are birational via a flop, then their singular Betti numbers are equal. Since one still knows very little about flops in dimension $n\ge4$, it seems unlikely that a consideration of flops could help to prove Theorem~\ref{main-th} in dimension $n\ge4$. Moreover, Theorem~\ref{main-th} is false in general for projective algebraic varieties with at worst $\Q$-factorial Gorenstein terminal singularities of dimension $n\ge4$. For this reason, the condition in Theorem~\ref{main-th} that $X$ and $Y$ are smooth becomes very important in the case $n\ge4$. We remark that in the case of holomorphic symplectic manifolds some stronger result is obtained in \cite{H}. \bigskip \noindent {\bf Acknowledgements:} The author would like to thank Professors A. Beauville, B. Fantechi, L. G\"ottsche, K. Hulek, Y. Kawamata, M. Kontsevich, S. Mori, M. Reid and D. van Straten for fruitful discussions and stimulating e-mails. \section{Gauge forms and $p$-adic measures} Let $F$ be a local number field, that is, a finite extension of the $p$-adic field $\Q_p$ for some prime $p\in\Z$. Let $R\subset F$ be the maximal compact subring, $\mathfrak q\subset R$ the maximal ideal, $F_{\mathfrak q}=R/\mathfrak q$ the residue field with $|F_{\mathfrak q}|=q=p^r$. We write \[ N_{F/{\Q}_p}\colon F\to\Q_p \] for the standard norm, and $\| \cdot \|\colon F \to \R_{\ge 0}$ for the multiplicative $p$-adic norm \[ a \mapsto \|a\|=p^{-\operatorname{Ord}(N_{F/\Q_p}(a))}. \] Here $\operatorname{Ord}$ is the $p$-adic valuation. \begin{dfn} Let $\mathcal X$ be an arbitrary flat reduced algebraic $S$-scheme, where $S=\operatorname{Spec} R$. We denote by $\mathcal X(R)$ the set of $S$-morphisms $S\to\mathcal X$ (or sections of $\mathcal X\to S$). We call $\mathcal X(R)$ the set of $R$-{\em integral points} in $\mathcal X$. The set of sections of the morphism $\mathcal X\times_S\operatorname{Spec} F\to\operatorname{Spec} F$ is denoted by $\mathcal X(F)$ and called the set of $F$-{\em rational points} in $\mathcal X$. \end{dfn} \begin{rem}\label{point} \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item If $\mathcal X$ is an affine $S$-scheme, then one can identify $\mathcal X(R)$ with the subset \[ \bigl\{x\in\mathcal X(F) \bigm| \text{$f(x)\in R$ for all $f\in\Ga(\mathcal X,\mathcal O_{\mathcal X})$}\bigr\} \subset \mathcal X(F). \] \item If $\mathcal X$ is a projective (or proper) $S$-scheme, then $\mathcal X(R)=\mathcal X(F)$. \end{enumerate} \end{rem} Now let $X$ be a smooth $n$-dimensional algebraic variety over $F$. We assume that $X$ admits an extension $\mathcal X$ to a regular $S$-scheme. Denote by $\Om^n_X$ the canonical line bundle of $X$ and by $\Om^n_{\mathcal X/S}$ the relative dualizing sheaf on $\mathcal X$. Recall the following definition introduced by Weil \cite{weil}: \begin{dfn} A global section $\om\in\Ga(\mathcal X,\Om^n_{\mathcal X/S})$ is called a {\em gauge form} if it has no zeros in $\mathcal X$. By definition, a gauge form $\om$ defines an isomorphism $\mathcal O_\mathcal X\cong\Om^n_{\mathcal X/S}$, sending $1$ to $\om$. Clearly, a gauge form exists if and only if the line bundle $\Om^n_{\mathcal X/S}$ is trivial. \end{dfn} Weil observed that a gauge form $\om$ determines a canonical $p$-adic measure $\dd\mu_\om$ on the locally compact $p$-adic topological space $\mathcal X(F)$ of $F$-rational points in $\mathcal X$. The $p$-adic measure $\dd\mu_\om$ is defined as follows: Let $x\in\mathcal X(F)$ be an $F$-point, $t_1, \dots, t_n$ local $p$-adic analytic parameters at $x$. Then $t_1, \dots , t_n$ define a $p$-adic homeomorphism $\theta \colon U \to\A^n(F)$ of an open subset $\mathcal U\subset\mathcal X(F)$ containing $x$ with an open subset $\theta(\mathcal U)\subset\A^n(F)$. We stress that the subsets $\mathcal U\subset\mathcal X(F)$ and $\theta(\mathcal U)\subset\A^n(F)$ are considered to be open in the $p$-adic topology, not in the Zariski topology. We write \[ \om=\theta^*\left(g \dd t_1 \wedge \cdots \wedge \dd t_n\right), \] where $g=g(t)$ is a $p$-adic analytic function on $\theta(\mathcal U)$ having no zeros. Then the $p$-adic measure $\dd\mu_\om$ on $\mathcal U$ is defined to be the pullback with respect to $\theta$ of the $p$-adic measure $\|g(t)\|\bdt$ on $\theta(\mathcal U)$, where $\bdt$ is the standard $p$-adic Haar measure on $\A^n(F)$ normalized by the condition \[\int_{\A^n(R)} \bdt =1. \] It is a standard exercise using the Jacobian to check that two $p$-adic measures $\dd\mu_\om',\dd\mu_\om''$ constructed as above on any two open subsets $\mathcal U',\mathcal U''\subset\mathcal X(F)$ coincide on the intersection $\mathcal U'\cap\,\mathcal U''$. \begin{dfn} The measure $\dd\mu_\om$ on $\mathcal X(F)$ constructed above is called the {\em Weil $p$-adic measure} associated with the gauge form $\om$. \end{dfn} \begin{thm}[\cite{weil}, Theorem~2.2.5] Let $\mathcal X$ be a regular $S$-scheme, $\om$ a gauge form on $\mathcal X$, and $\dd\mu_\om$ the corresponding Weil $p$-adic measure on $\mathcal X(F)$. Then \[ \int_{\mathcal X(R)} \dd\mu_\om=\frac{|\mathcal X(F_\mathfrak q)|}{q^n}, \] where $\mathcal X({F_\mathfrak q})$ is the set of closed points of $\mathcal X$ over the finite residue field $F_\mathfrak q$. \label{weil1} \end{thm} \begin{pf} Let \[ \fie\colon \mathcal X(R)\to\mathcal X({F_\mathfrak q}) \quad\text{given by}\quad x \mapsto \xbar\in\mathcal X({F_\mathfrak q}) \] be the natural surjective mapping. The proof is based on the idea that if $\xbar\in\mathcal X({F_\mathfrak q})$ is a closed ${F_\mathfrak q}$-point of $\mathcal X$ and $g_1,\dots,g_n$ are generators of the maximal ideal of $\xbar$ in $\mathcal O_{\mathcal X,\xbar}$ modulo the ideal $\mathfrak q$, then the elements $g_1,\dots,g_n$ define a $p$-adic analytic homeomorphism \[ \ga\colon\fie^{-1}(\xbar)\to\A^n(\mathfrak q), \] where $\fie^{-1}(\xbar)$ is the fiber of $\fie$ over $\xbar$ and $\A^n(\mathfrak q)$ is the set of all $R$-integral points of $\A^n$ whose coordinates belong to the ideal $\mathfrak q \subset R$. Moreover, the $p$-adic norm of the Jacobian of $\ga$ is identically equal to $1$ on the whole fiber $\fie^{-1}(\xbar)$. In order to see the latter we remark that the elements define an \'etale morphism $g\colon V \to \A^n$ of some Zariski open neighbourhood $V$ of $\xbar\in\mathcal X$. Since $\fie^{-1}(\xbar) \subset V(R)$ and $g^*(\dd t_1,\wedge \cdots \wedge \dd t_n)=h \om$, where $h$ is invertible in $V$, we obtain that $h$ has $p$-adic norm $1$ on $\fie^{-1}(\xbar)$. So, using the $p$-adic analytic homeomorphism $\ga$, we obtain \[ \int_{\fie^{-1}(\xbar)} \dd\mu_\om=\int_{\A^n(\mathfrak q)}\bdt =\frac{1}{q^n} \] for each $\xbar\in\mathcal X({F_\mathfrak q})$. \ifhmode\unskip\nobreak\fi\quad $\square$\end{pf} Now we consider a slightly more general situation. We assume only that $\mathcal X$ is a regular scheme over ${S}$, but do not assume the existence of a gauge form on $\mathcal X$ (that is, of an isomorphism $\mathcal O_\mathcal X\cong\Om^n_{\mathcal X/S}$). Nevertheless under these weaker assumptions we can define a unique natural $p$-adic measure $\dd\mu$ at least on the compact $\mathcal X(R)\subset\mathcal X(F)$ -- although possibly not on the whole $p$-adic topological space $\mathcal X(F)$! Let $\mathcal U_1,\dots,\mathcal U_k$ be a finite covering of $\mathcal X$ by Zariski open $S$-subschemes such that the restriction of $\Om^n_{\mathcal X/S}$ on each $\mathcal U_i$ is isomorphic to $\mathcal O_{\mathcal U_i}$. Then each $\mathcal U_i$ admits a gauge form $\om_i$ and we define a $p$-adic measure $\dd\mu_i$ on each compact $\mathcal U_i(R)$ as the restriction of the Weil $p$-adic measure $\dd\mu_{\om_i}$ associated with $\om_i$ on $\mathcal U_i(F)$. We note that the gauge forms $\om_i$ are defined uniquely up to elements $s_i\in\Ga(\mathcal U_i,\mathcal O^*_\mathcal X)$. On the other hand, the $p$-adic norm $\|s_i(x)\|$ equals $1$ for any element $s_i\in\Ga( \mathcal U_i, \mathcal O^*_\mathcal X)$ and any $R$-rational point $x\in\mathcal U_i(R)$. Therefore, the $p$-adic measure on $\mathcal U_i(R)$ that we constructed does not depend on the choice of a gauge form $\om_i$. Moreover, the $p$-adic measures $\dd\mu_i$ on $\mathcal U_i(R)$ glue together to a $p$-adic measure $\dd\mu$ on the whole compact $\mathcal X(R)$, since one has \[ \mathcal U_i(R)\cap\,\mathcal U_j(R)=(\mathcal U_i\cap\,\mathcal U_j)(R) \quad\text{for $i,j=1,\dots,k$} \] and \[ \mathcal U_1(R)\cup\cdots\cup\,\mathcal U_k(R)= (\mathcal U_1\cup\cdots\cup\,\mathcal U_k)(R)=\mathcal X(R). \] \begin{dfn} \label{can-m} The $p$-adic measure constructed above defined on the set $\mathcal X(R)$ of $R$-integral points of a $S$-scheme $\mathcal X$ is called the {\em canonical $p$-adic measure}. \end{dfn} For the canonical $p$-adic measure $\dd\mu$, we obtain the same property as for the Weil $p$-adic measure $\dd\mu_\om$: \begin{thm} \label{integ2} \[ \int_{\mathcal X(R)} \dd\mu=\frac{|\mathcal X({F_\mathfrak q})|}{q^n}. \] \end{thm} \begin{pf} Using a covering of $\mathcal X$ by some Zariski open subsets $\mathcal U_1,\dots, \mathcal U_k$, we obtain \[ \int\limits_{\mathcal X(R)} \dd\mu=\sum_{i_1}\int\limits_{\mathcal U_{i_1}(R)} \kern-2mm\dd\mu -\sum_{i_1<i_2}\kern2mm\int\limits_{(\mathcal U_{i_1}\cap\,\mathcal U_{i_2})(R)} \kern-4mm\dd\mu \kern2mm+\cdots+\kern2mm (-1)^k \kern-4mm\int\limits_{(\mathcal U_1\cap\cdots\cap\,\mathcal U_{k})(R)} \kern-4mm\dd\mu \] and \begin{multline*} \bigl|\mathcal X({F_\mathfrak q})\bigr|= \sum_{i_1}\bigl|\mathcal U_{i_1}({F_\mathfrak q})\bigr|-\sum_{i_1<i_2}\bigl| (\mathcal U_{i_1}\cap\,\mathcal U_{i_2})({F_\mathfrak q})\bigr| \\ +\cdots+(-1)^k\bigl|(\mathcal U_1\cap\cdots\cap\,\mathcal U_{k})({F_\mathfrak q})\bigr|. \end{multline*} It remains to apply Theorem~\ref{weil1} to every intersection $\mathcal U_{i_1}\cap\cdots\cap\,\mathcal U_{i_s}$. \ifhmode\unskip\nobreak\fi\quad $\square$\end{pf} \begin{thm} \label{m-zero} Let $\mathcal X$ be a regular integral $S$-scheme and $\mathcal Z\subset \mathcal X$ a closed reduced subscheme of codimension $\ge1$. Then the subset $\mathcal Z(R) \subset \mathcal X(R)$ has zero measure with respect to the canonical $p$-adic measure $\dd\mu$ on $\mathcal X(R)$. \end{thm} \begin{pf} Using a covering of $\mathcal X$ by Zariski open affine subsets $\mathcal U_1, \dots, \mathcal U_k$, we can always reduce to the case when $\mathcal X$ is an affine regular integral $S$-scheme and $\mathcal Z\subset\mathcal X$ an irreducible principal divisor defined by an equation $f=0$, where $f$ is a prime element of $A=\Ga(\mathcal X, \mathcal O_\mathcal X)$. Consider the special case $\mathcal X=\A^n_S=\operatorname{Spec} R[X_1,\dots,X_n]$ and $\mathcal Z=\A^{n-1}_{S} =\operatorname{Spec} R[X_2,\dots,X_n]$, that is, $f=X_1$. For every positive integer $m$, we denote by $\mathcal Z_m(R)$ the subset in $\A^n(R)$ consisting of all points $x=(x_1,\dots,x_d)\in R^n $ such that $x_1\in\mathfrak q^m$. One computes the $p$-adic integral in the straightforward way: \[ \int\limits_{\mathcal Z_m(R)} \bdx=\int\limits_{\A^1(\mathfrak q^m)} \kern-2mm {\dd x_1}\kern2mm \prod_{i=2}^n \left(\int_{\A^1(R)}{\dd x_i}\right) =\frac{1}{q^m}. \] On the other hand, we have \[ \mathcal Z(R)=\bigcap_{m =1}^{\infty} \mathcal Z_m(R). \] Hence \[ \int_{\mathcal Z(R)} \bdx=\lim_{m \to\infty }\int_{\mathcal Z_m(R)} \bdx=0, \] and in this case the statement is proved. Using the Noether normalization theorem reduces the more general case to the above special one. \ifhmode\unskip\nobreak\fi\quad $\square$\end{pf} \section{The Betti numbers} \begin{prop} \label{main-th2} Let $X$ and $Y$ be birational smooth projective $n$-dimensional algebraic varieties over $\C$ having trivial canonical line bundles. Then there exist Zariski open dense subsets $U\subset X$ and $V \subset Y$ such that $U$ is isomorphic to $V$ and $\operatorname{codim}_X(X\setminus U),\operatorname{codim}_Y(Y\setminus V)\ge2$. \end{prop} \begin{pf} Consider a birational rational map $\fie\colon X\dasharrow Y$. Since $X$ is smooth and $Y$ is projective, $\fie$ is regular at the general point of any prime divisor of $X$, so that there exists a maximal Zariski open dense subset $U\subset X$ with $\operatorname{codim}_X(X\setminus U)\ge2$ such that $\fie$ extends to a regular morphism $\fie_0\colon U\to Y$. Since $\fie^*\om_Y$ is proportional to $\om_X$, the morphism $\fie_0$ is \'etale, that is, $\fie_0$ is an open embedding of $U$ into the maximal open subset $V \subset Y$ where $\fie^{-1}$ is defined. Similarly $\fie^{-1}$ induces an open embedding of $V$ into $U$, so we conclude that $\fie_0$ is an isomorphism of $U$ onto $V$. \ifhmode\unskip\nobreak\fi\quad $\square$\end{pf} \bigskip \begin{pfof}{Theorem~\ref{main-th}} Let $X$ and $Y$ be smooth projective birational varieties of dimension $n$ over $\C$ with trivial canonical bundles. By Proposition~\ref{main-th2}, there exist Zariski open dense subsets $U\subset X$ and $V\subset Y$ with $\operatorname{codim}_X(X\setminus U)\ge2$ and $\operatorname{codim}_Y(Y\setminus V)\ge2$ and an isomorphism $\fie\colon U \to V$. By standard arguments, one can choose a finitely generated $\Z$-subalgebra $\mathcal R\subset\C$ such that the projective varieties $X$ and $Y$ and the Zariski open subsets $U\subset X$ and $V\subset Y$ are obtained by base change $*\times_\mathcal S\operatorname{Spec}\C$ from regular projective schemes $\mathcal X$ and $\mathcal Y$ over $\mathcal S:=\operatorname{Spec}\mathcal R$ together with Zariski open subschemes $\mathcal U\subset\mathcal X$ and $\mathcal V\subset\mathcal Y$ over $\mathcal S$. Moreover, one can choose $\mathcal R$ in such a way that both relative canonical line bundles $\Om^n_{\mathcal X/\mathcal S}$ and $\Om^n_{\mathcal Y/\mathcal S}$ are trivial, both codimensions $\operatorname{codim}_\mathcal X(\mathcal X\setminus\mathcal U)$ and $\operatorname{codim}_\mathcal Y(\mathcal Y\setminus\mathcal V)$ are $\ge2$, and the isomorphism $\fie\colon U\to V$ is obtained by base change from an isomorphism $\Phi\colon\mathcal U\to\mathcal V$ over $\mathcal S$. For almost all prime numbers $p\in\N$, there exist a regular $R$-integral point $\pi\in\mathcal S \times_{\operatorname{Spec}\Z}{\operatorname{Spec}\Z_p}$, where $R$ is the maximal compact subring in a local $p$-adic field $F$; let $\mathfrak q$ be the maximal ideal of $R$. By an appropriate choice of $\pi\in\mathcal S\times_{\operatorname{Spec}\Z}{\operatorname{Spec}\Z_p}$, we can ensure that both $\mathcal X$ and $\mathcal Y$ have good reduction modulo $\mathfrak q$. Moreover, we can assume that the maximal ideal $I(\overline{\pi})$ of the unique closed point in \[ S: =\operatorname{Spec} R \stackrel{\pi}{\hookrightarrow} \mathcal S\times_{\operatorname{Spec}\Z}{\operatorname{Spec}\Z_p} \] is obtained by base change from some maximal ideal $J(\overline{\pi})\subset\mathcal R$ lying over the prime ideal $(p)\subset\Z$. Let $\om_\mathcal X$ and $\om_\mathcal Y$ be gauge forms on $\mathcal X$ and $\mathcal Y$ respectively and $\om_\mathcal U$ and $\om_\mathcal V$ their restriction to $\mathcal U$ (respectively $\mathcal V$). Since $\Phi^*$ is an isomorphism over $\mathcal S$, $\Phi^*\om_\mathcal Y$ is another gauge form on $\mathcal U$. Hence there exists a nowhere vanishing regular function $h\in\Ga(\mathcal U,\mathcal O^*_\mathcal X)$ such that \[ \Phi^* \om_\mathcal V=h \om_\mathcal U. \] The property $\operatorname{codim}_\mathcal X(\mathcal X\setminus \mathcal U)\ge2$ implies that $h$ is an element of $\Ga(\mathcal X, \mathcal O^*_\mathcal X)=\mathcal R^*$. Hence, one has $\| h(x) \| =1$ for all $x\in\mathcal X(F)$, that is, the Weil $p$-adic measures on $\mathcal U(F)$ associated with $\Phi^* \om_\mathcal V$ and $\om_\mathcal U$ are the same. The latter implies the following equality of the $p$-adic integrals \[ \int_{\mathcal U(F)} \dd\mu_\mathcal X=\int_{\mathcal V(F)} \dd\mu_\mathcal Y. \] By Theorem~\ref{m-zero} and Remark~\ref{point}, (ii), we obtain \[ \int_{\mathcal U(F)} \dd\mu_\mathcal X=\int_{\mathcal X(F)} \dd\mu_\mathcal X =\int_{\mathcal X(R)} \dd\mu_\mathcal X \] and \[\int_{\mathcal V(\mathcal F)} \dd\mu_\mathcal Y=\int_{\mathcal Y(\mathcal F)} \dd\mu_\mathcal Y=\int_{\mathcal Y(R)} \dd\mu_\mathcal Y. \] Now, applying the formula in Theorem~\ref{integ2}, we come to the equality \[ \frac{|\mathcal X({F_\mathfrak q})|}{q^n}=\frac{|\mathcal Y({F_\mathfrak q})|} {q^n}. \] This shows that the numbers of $F_\mathfrak q$-rational points in $\mathcal X$ and $\mathcal Y$ modulo the ideal $J(\overline{\pi})\subset\mathcal R$ are the same. We now repeat the same argument, replacing $R$ by its cyclotomic extension $\mathcal R^{(r)}\subset\C$ obtained by adjoining all complex $(q^r-1)$th roots of unity; we deduce that the projective schemes $\mathcal X$ and $\mathcal Y$ have the same number of rational points over $F_\mathfrak q^{(r)}$, where $F_\mathfrak q^{(r)}$ is the extension of the finite field $F_\mathfrak q$ of degree $r$. We deduce in particular that the Weil zeta functions \[ Z(\mathcal X,p,t)=\exp \left(\sum_{r =1}^{\infty} |\mathcal X({F_\mathfrak q^{(r)}})| \frac{t^r}{r} \right) \] and \[ Z(\mathcal Y, p, t) =\exp \left(\sum_{r =1}^{\infty} |\mathcal Y({F_\mathfrak q^{(r)}})| \frac{t^r}{r} \right) \] are the same. Using the Weil conjectures proved by Deligne \cite{Deligne} and the comparison theorem between the \'etale and singular cohomology, we obtain \begin{equation} \label{eq_zeta} Z(\mathcal X,p,t)=\frac {P_1(t)P_3(t)\cdots P_{2n-1}(t)} {P_0(t)P_2(t)\cdots P_{2n}(t)} \end{equation} and \[ Z(\mathcal Y,p, t)=\frac{ Q_1(t) Q_3(t) \cdots Q_{2n-1}(t)}{ Q_0(t) Q_2(t) \cdots Q_{2n}(t) }, \] where $P_i(t)$ and $Q_i(t)$ are polynomials with integer coefficients having the properties \begin{equation} \label{eq_betti} \deg P_i(t)=\dim H^i(X, \C), \quad \deg Q_i(t)=\dim H^i(Y, \C) \quad \text{for all $i\ge0$.} \end{equation} Since the standard archimedean absolute value of each root of polynomials $P_i(t)$ and $Q_i(t)$ must be $q^{-i/2}$ and $P_i(0)=Q_i(0)=1$ for all $i\ge0$, the equality $Z(\mathcal X,p,t)=Z(\mathcal Y,p,t)$ implies $P_i(t)=Q_i(t)$ for all $i\ge0$. Therefore, we have $\dim H^i(X,\C)=\dim H^i(Y,\C)$ for all $i\ge0$. \ifhmode\unskip\nobreak\fi\quad $\square$\end{pfof} \section{Further results} \begin{dfn} Let $\fie\colon X\dasharrow Y$ be a birational map between smooth algebraic varieties $X$ and $Y$. We say that $\fie$ {\em does not change the canonical class}, if for some Hironaka resolution $\al\colon Z\to X$ of the indeterminacies of $\fie$ the composite $\al\circ\fie$ extends to a morphism $\be\colon Z\to Y$ such that $\be^*\Om^n_Y\cong\al^*\Om^n_X$. \end{dfn} The statement of Theorem~\ref{main-th} can be generalized to the case of birational smooth projective algebraic varieties which do not necessary have trivial canonical classes as follows: \begin{thm} \label{main-th3} Let $X$ and $Y$ be irreducible birational smooth $n$-dimensional projective algebraic varieties over $\C$. Assume that the exists a birational rational map $\fie\colon X\dasharrow Y$ which does not change the canonical class. Then $X$ and $Y$ have the same Betti numbers. \end{thm} \begin{pf} We repeat the same arguments as in the proof of Theorem~\ref{main-th} with the only difference that instead of the Weil $p$-adic measures associated with gauge forms we consider the canonical $p$-adic measures (see Definition~\ref{can-m}). Using the birational morphisms $\al\colon\mathcal Z\to\mathcal X$ and $\be\colon\mathcal Z\to\mathcal Y$ having the property \[ \be^*\Om^n_{\mathcal Y/S} \cong \al^*\Om^n_{\mathcal X/S}, \] we conclude that for some prime $p\in\N$, the integrals of the canonical $p$-adic measures $\mu_\mathcal X$ and $\mu_\mathcal Y$ over $\mathcal X(R)$ and $\mathcal Y(R)$ are equal, since there exists a dense Zariski open subset $\mathcal U\subset\mathcal Z$ on which we have $\al^*\mu_\mathcal X=\be^*\mu_\mathcal Y$. By Theorem~\ref{integ2}, the zeta functions of $\mathcal X$ and $\mathcal Y$ must be the same. \ifhmode\unskip\nobreak\fi\quad $\square$\end{pf} Another immediate application of our method is related to the McKay correspondence \cite{R}. \begin{thm} Let $G \subset\operatorname{SL}(n,\C)$ be a finite subgroup. Assume that there exist two different resolutions of singularities on $W:=\C^n/G$: \[ f\colon X \to W, \quad g\colon Y \to W \] such that both canonical line bundles $\Om^n_X$ and $\Om^n_Y$ are trivial. Then the Euler numbers of $X$ and $Y$ are the same. \end{thm} \begin{pf} We extend the varieties $X$ and $Y$ to regular schemes over a scheme $\mathcal S$ of finite type over $\operatorname{Spec}\Z$. Moreover, one can choose $\mathcal S$ in such a way that the birational morphisms $f$ and $g$ extend to birational $\mathcal S$-morphisms \[ F\colon \mathcal X \to\mathcal W, \quad G\colon \mathcal Y \to\mathcal W, \] where $\mathcal W$ is a scheme over $\mathcal S$ extending $W$. Using the same arguments as in the proof of Theorem~\ref{main-th}, one obtains that there exists a prime $p\in\N$ such that $Z(\mathcal X,p,t)=Z(\mathcal Y,p,t)$. On the other hand, in view of (\ref{eq_betti}), the Euler number is determined by the Weil zeta function (\ref{eq_zeta}) as the degree of the numerator minus the degree of the denominator. Hence $e(X)=e(Y)$. \ifhmode\unskip\nobreak\fi\quad $\square$\end{pf} With a little bit more work one can prove even more precise statement: \begin{thm} Let $G\subset\operatorname{SL}(n,\C)$ be a finite subgroup and $W:=\C^n/G$. Assume that there exists a resolution \[ f\colon X \to W \] with trivial canonical line bundle $\Om^n_X$. Then the Euler number of $X$ equals the number of conjugacy classes in $G$. \end{thm} \begin{rem} As we saw in the proof of Theorem~\ref{main-th2}, the Weil zeta functions of $Z(\mathcal X,p,t)$ and $Z(\mathcal Y,p,t)$ are equal for almost all primes $p\in\operatorname{Spec}\Z$. This fact being expressed in terms of the associated $L$-functions indicates that the isomorphism $H^i(X,\C)\cong H^i(Y,\C)$ for all $i\ge0$ we have established must have some more deep motivic nature. Recently Kontsevich suggested an idea of a motivic integration \cite{K}, developed by Denef and Loeser \cite{DL}. In particular, this technique allows to prove that not only the Betti numbers, but also the Hodge numbers of $X$ and $Y$ in \ref{main-th} must be the same. \end{rem}

9710

alg-geom/9710001

en

https://arxiv.org/abs/alg-geom/9710001

alg-geom/9710001

Jonathan Fine

Jonathan Fine

Convex polytopes and linear algebra

LaTeX2e. 14 pages

This paper defines, for each convex polytope $\Delta$, a family $H_w\Delta$ of vector spaces. The definition uses a combination of linear algebra and combinatorics. When what is called exact calculation holds, the dimension $h_w\Delta$ of $H_w\Delta$ is a linear function of the flag vector $f\Delta$. It is expected that the $H_w\Delta$ are examples, for toric varieties, of the new topological invariants introduced by the author in "Local-global intersection homolog" (preprint alg-geom/9709011).

alg-geom

\section{Introduction} The goal, towards which this paper is directed, is as follows. Suppose $\Delta$ is a convex polytope. One wishes to construct from $\Delta$ vector spaces whose dimension is a combinatorial invariant of $\Delta$. The smaller the dimension of these spaces, the better. The convex polytope $\Delta$ has both a linear structure, due to the ambient affine linear space, and a combinatorial structure, due to the incidence relations among the faces. The construction in the paper uses both structures to produce many `vector-weighted inclusion-exclusion formulae', to each one of which corresponds a complex of vector spaces. When such a complex exactly computes its homology (a concept to be explained later) the result is a vector space of the type that is sought. The proof of exact calculation, which is not attempted in this paper, is expected to be difficult. In a special case, part of this problem has already been solved. If $\Delta$ has rational vertices then from $\Delta$ a projective algebraic variety $\PDelta$ can be constructed, and the middle perversity intersection homology (mpih) Betti numbers $h\Delta$ are combinatorial invariants of $\Delta$, by virtue of the Bernstein-Khovanskii-MacPherson formula \cite{bib.JD-FL.IHNP,bib.KF.IHTV,bib.RS.GHV}, that express the $h_i$ as linear functions of the flag vector $f\Delta$ of $\Delta$. In this case Braden and MacPherson (personal communication) have proved an exact calculation result. Their proof relies on deep results in algebraic geometry, and in particular on Deligne's proof \cite{bib.PD.WC1,bib.PD.WC2} of the Weil conjectures. Elsewhere \cite{bib.JF.LGIH}, the author has defined local-global intersection homology groups. The construction in this paper corresponds to the extension and unwinding (\cite{bib.JF.LGIH}, formulae (8)--(10)) of the extended $h$-vector defined in that paper. This correspondence, which is an exercise in combinatorics, is left to the reader. It might also be presented elsewhere. This paper has been written to be independent of \cite{bib.JF.LGIH}. Central to this paper is the study of flags. They too have linear and combinatorial strucutre. A \emph{flag} $\delta$ on a convex polytope $\Delta$ is a sequence \[ \delta = ( \delta_1 \subset \delta_2 \subset \dots \subset \delta_r \subset \Delta ) \] of faces $\delta_i$ of $\Delta$, each strictly contained in the next. The \emph{dimension} $d = \dim \delta$ is the sequence \[ d = ( d_1 < d_2 < \dots < d_r < n = \dim \Delta ) \] of the dimensions $d_i$ of the \emph{terms} $\delta_i$ of $\delta$. Altogether there are $2^n$ possible flag dimensions. The number $r$ is the \emph{order} $\ord \delta$ of the flag $\delta$ (and of the dimension vector $d$). If $\delta$ is a face of $\Delta$, the order one flag whose only term is $\delta$, namely $(\delta\subset\Delta)$, will also be denoted by $\delta$. The empty flag will be denoted by $\Delta$. Similarly, if $V$ is a vector space then a flag $U$ on $V$ is a sequence \[ U = ( U_1 \subset U_2 \subset \dots \subset U_r \subset V ) \] of subspaces $U_i$ of $V$, each stricly contained in the next. The \emph{dimension} $d=\dim V$ is the sequence \[ d = ( d_1 < d_2 < \dots < d_r < n = \dim V ) \] where now the $d_i$ are the dimensions of the $U_i$. Throughout $V$ will be the vector space $\langle\Delta\rangle$ spanned by the vectors that lie on $\Delta$. Each face $\delta_i$ similarly determines a subspace $U_i=\langle\delta_i\rangle$ of $V$. Thus, each flag $\delta$ of faces on $\Delta$ determines a flag $\langle\delta\rangle$ of subspaces of $V=\langle\Delta\rangle$. The construction of this paper is, in general terms, as follows. Suppose $U$ is a flag on $V$. From $U$ many vector spaces can be constructed. This paper constructs for each $U$, and for each $w$ lying in an as yet unspecified index set, a vector space $U_w$. Now let $U'$ be obtained from $U$ by deleting from $U$ one of its terms $U_i$. It so happens that this deletion operator induces a natural map \[ U_w \to U'_w \] between the associated $w$-spaces. (In the simplest case, which corresponds to mpih, all the $U_w$ are subspaces of a single space $V_w$, and the maps are inclusions. In general there is no such a global space $V_w$.) Now suppose $U'$ is obtained from $U$ by deleting two or more terms. By choosing an order for the deletion of these terms, a map $U_w \to U'_w$ can be obtained. The single-deletion map is natural (or geometric) in that the induced multiple deletion maps are independent of the choice of deletion order. Suppose now that such spaces $U_w$ have been defined for every flag on $V$, and that $\Delta$ is a convex polytope whose vector space $\langle\Delta\rangle$ is $V$. Each flag $\delta$ on $\Delta$ thus determines a flag $\langle\delta\rangle$ on $V$, and thus a vector space $\langle\delta\rangle_w$, or $\delta_w$ for short. These vector spaces will be assembled into a complex, according to the order $r$ of $\delta$. Define the space $\Delta(w,r)$ of \emph{$w$-weighted $r$-flags} to be the direct sum of the $\delta_w$, where $\delta$ has order $r$. Each vector in $\Delta(w,r)$ can be thought of as a formal sum $\sum v_\delta [\delta]$, where $[\delta]$ is a formal object representing an $r$-term flag $\delta$, and where the coefficient $v_\delta$ is drawn from the vector space $\delta_w$.) By the assumptions of the previous paragraph, the deletion operator on flags determines a differential \[ \bound: \Delta(w,r) \to \Delta(w,r-1) \] and so induces a complex \[ 0 \to \Delta(w,r) \to \Delta(w,r-1) \to \dots \Delta(w,1) \to \Delta(w,0) \to 0 \] of vector spaces. (As is usual, $\bound=\sum (-1)^{i+1}\partial_i$, where $\partial_i$ is the operator induced by deletion of the $i$-th term. Because the maps $U_w\to U'_w$ are natural, $\bound^2$ is zero, and thus one indeed has a complex.) \emph{Exact calculation} is when this complex is exact except at one point, say $\Delta(w,j)$. In that case the homology $H_w\Delta$ at that point is a suitable alternating sum of the dimensions of the $\Delta(w,i)$, and thus of the $\delta_w$. By construction the dimension of $\delta_w$ will depend only on the dimension vector $d$ of $\delta$, and thus (provided exact calculation holds) one has that the dimension $h_w\Delta$ of $H_w\Delta$ is a linear function of the flag vector $f\Delta$ of $\Delta$, and so is a combinatorial invariant. (This is because the $f_d\Delta$ component of $f\Delta$ counts how many $d$-flags there are on $\Delta$. Each $f_d\Delta$ contributes, up to an alternating sign, the quantity $\lambda_d=\dim\delta_w$ to $h_\Delta$, where $\delta_w$ is a coefficient space due to any flag $\delta$ of dimension $d$.) Convex polytopes are not the only combinatorial objects for which the concept of a flag can be defined. In \cite{bib.JF.QTHGFV,bib.JF.SFV}, the author defines flag vectors for $i$-graphs, or more generally any object that is a union of cells (or edges), and which can be shelled. There seems to be no reason why the general form of the construction described here cannot also be applied in this new context. This is not to say the the proper choice of the vector spaces $\delta_w$ to be associated to the flags $\delta$ is not expected to be a deep question. Both for convex polytopes and for $i$-graphs there are subtle and presently unknown combinatorial inequalities on the flag vectors. An exact homology theory, using vector spaces such as the $\delta_w$, is the only method the author can envision, that will lead to the proof such inequalities. The exposition is organised as follows. The next section (\S2) give the definition of exact homology. The deletion operator applied to flags yields the flag complex~(\S3). Next comes a complex~(\S4) that corresponds to middle perversity intersection homology. The local-global variant is more complicated, and is the substance of the paper. First the coefficient spaces to be used are defined~(\S5), then the maps between them~(\S6), and finally the local-global homology~(\S7). The purpose of \S\S3--7 is to present the definition as the result of a study of the geometric resources and constraints. Conversely, in \S8 decorated bar diagrams are used to reformulate the definition, and allow the properties to be demonstrated, in a concise manner. Finally, \S9 provides a wider discussion of what has been done, and what remains to be done. The reader may at first find \S\S5--7 somewhat abstruse. They contain a study of the linear algebra of a segmented flag of vector spaces. Once the problem is understood, the key definitions come out in a fairly natural way. In \S8, the same definitions are presented, but this time via coordinates. Here, the definitions are clear, but may appear somewhat arbitrary. Each point of view informs the other. This paper attempts to show how the definitions arise naturally out of the logic of the situation, and thus places \S\S5--7 before \S8. The reader may wish to reverse this order. The basic devices are the combinatorics of flags, and linear algebra. The exterior algebra on a vector space is widely used. The fibre of the moment map from a projective toric variety to its defining polytope is always a product of circles, and so the homology of the fibre is isomorphic to an exterior algebra. This fact is the beginning of the connection between the linear algebra of flags and the existence of cycles on the toric variety. The reader does not need to know this. Throughout this paper $\Delta$ will be a convex polytope of some fixed dimension~$n$, and $V$ will be the vector space spanned by the vectors lying on $\Delta$. It will do no harm to think of $\Delta$ as lying in $V$. \section{Exact Homology} This section explains the concept of exact homology. Suppose that a sequence \[ A = ( 0 \to A_n \to A_{n-1} \to \dots \to A_1 \to A_0 \to 0 ) \] of vector spaces is given, together with a map \[ \bound:A_i \to A_{i-1} \] between successive terms. If, for each $i$, the composite map \[ \bound \circ \bound = \bound^2 : A_i \to A_{i-2} \] is the zero map, then $A$ is called a \emph{complex}, and $d$ its \emph{boundary map}. If $A$ is a complex then the statement \[ \im (\bound:A_{i+1} \to A_i ) \> \subseteq \> \ker (\bound:A_i \to A_{i-1}) \] restates the condition $\bound^2=0$, and the quotient of the above kernel by the image is called the $i$-th homology $H_iA$ of the complex $A$. This formalism originated in the definition, via chains and cycles, of the homology groups $H_iX$ of a topological space $X$. There, most of the homology groups were expected to be non-zero. The present use will be different. Suppose $A$ is a complex. If all the homology groups $H_iA$ are zero (i.e.~at each $A_i$ the kernel and image are equal) then $A$ is called a (long) \emph{exact sequence}, or \emph{exact} for short. If $A$ is exact then the alternating sum \begin{equation} \label{eqn:chi} \sum \nolimits _{i=0}^{n}\> (-1)^i \dim A_i \end{equation} of the dimensions of the $A_i$ (assumed finite) will be zero. To prove this, introduce in each $A_i$ a subspace $B_i$ that is a complement to $\bound A_{i+1}$ in $A_i$. The dimension of $A_i$ is the sum of the dimensions of $\bound A_{i+1}$ and of $B_i$. The exactness assumption implies that the restricted form \[ \bound:B_i \to \bound A_i \subseteq A_{i-1} \] of the boundary map is an isomorphism. Thus, the contributions of $B_i$ and $\bound A_i$ to the alternating sum are equal but opposite, and so the result follows. Suppose $A$ is a complex. If $H_iA$ is zero then $A$ is said to be \emph{exact at $A_i$}. Now suppose that $A$ is known to be exact at all its $A_i$ except perhaps one, say $A_r$. In this case the alternating sum (\ref{eqn:chi}) of the dimensions gives not zero, but the dimension of the only non-zero homology $H_rA$ of the complex $A$, multiplied by $(-1)^r$. If a complex is exact at all but one location, we shall say that it \emph{exactly computes} the homology at that location. Now suppose that the $A_i$ are constructed from the convex polytope $\Delta$, and that their dimensions depend only on the combinatorial structure of $\Delta$. The same will then be true for $H_rA$, provided that for each $\Delta$ the complex exactly computes its homology at $A_r$. If this holds we shall say that $H_rA$ is an \emph{exact homology} group of $\Delta$. It is of course one thing to define a complex $A$, as is done in this paper, and quite another to prove its exactness. This is expected to require new concepts and methods. \section{The flag complex} The convex polytope $\Delta$ has both a combinatorial structure (incidence relations among faces) and a linear structure (the vectors lying on a face $\delta$ span a subspace $\langle\delta\rangle$ of $V$). In general both structures will be used to define the complex $A$. This section defines a complex that uses the combinatorial structure alone. The general case will arise by allowing vectors to be used instead of numbers in the construction that follows. Recall the definition, in \S1, of a flag $\delta$ on $\Delta$, its dimension vector $d$, and its order $r$. Now suppose $\delta$ is a flag on $\Delta$, of order $r$. By removing one or more of the terms $\delta_i$ from $\delta$, new flags can be obtained, of lower order. Let $\partial_j$ be the \emph{deletion operator} that removes from a flag $\delta$ the $j$-th term. The operators $\partial_j$ do not commute. Removing say the $2$nd term from a sequence, and then say the $4$th, gives the same final result as does first removing the $5$th and then the $2$nd. This is because removing the $2$nd term will cause the subsequent items to move down one place in the sequence. The equation \[ \partial_j \partial_k = \partial_{k+1} \partial_j \qquad \mbox{for $j<k$} \] is an example of the \emph{commutation law} for these deletion operators. Now let $A_i$ consist of all formal weighted sums $x$ of the form \[ x = \sum \nolimits _ { \ord \delta = i } x_\delta [\delta] \] where the coefficients $x_\delta$ are numbers and, as indicated, the sum is over all flags $\delta$ on $\Delta$, which have $i$ terms. Here $[\delta]$ denotes $\delta$ considered as a formal object. Where confusion will not then result, $\delta$ will be written in its place. Thus, a vector $x$ in $A_i$ is a formal sum of $i$-term flags, with numeric coefficients $x_\delta$. The deletion operation $\partial_j$ induces a map $A_i \to A_{i-1}$. (It is zero if $j$ is larger than $i$.) Now use the formula \[ \bound = \partial_1 - \partial_2 + \partial_3 + \dots + (-1)^{i-1}\partial_i + \ldots \] to define a map $\bound:A_i \to A_{i-1}$. It follows immediately from the commutation laws, that $\bound^2=0$, and so $A$ is a complex. It will be called the \emph{flag complex} of $\Delta$. It depends only on the combinatorial structure of $\Delta$. \section{Global coefficient spaces} Instead of using numeric coefficients for the formal sums that constitute the space $A_i$, one could instead write \begin{equation} \label{eqn.sum.vdelta.delta} v = \sum \nolimits _ { \ord \delta = i } v_\delta \delta \end{equation} where now $v_\delta$ is to lie in a vector space $\Lambda(\delta)$ that is in some way associated to $\delta$. Thus, $A_i$ is again to be formal sums of order $i$ flags, where now the coefficients are to be vectors rather than numbers. This section will describe the simplest way of constructing such \emph{coefficient spaces} $\Lambda(\delta)$. Suppose $\delta'$ is obtained from $\delta$ by the deletion of one or more terms. For such a definition to produce a complex, there must also be a natural map \[ \Lambda(\delta) \to \Lambda(\delta') \] between the corresponding coefficient spaces. One way to do this, which will be used in this section, is to have this map be an inclusion. Thus, all the coefficients will lie in the global coefficient space $\Lambda(\Delta)$ associated to the empty flag. Each flag $\delta$ will then define a subspace $\Lambda(\delta)$ of $\Lambda(\Delta)$. For this to work, one must have that $\Lambda(\delta)$ is a subspace of $\Lambda(\delta')$. One way to do this is to have each individual face $\delta_j$ in $\delta$ define a condition (or set of conditions) on $\Lambda(\Delta)$. Now define $\Lambda(\delta)$ to be those vectors in $\Lambda(\Delta)$ that satisfy the condition(s) due to the faces $\delta_j$ in $\delta$. Provided the conditions due to $\delta_j$ depend only on the face $\delta_j$ (and not on its location in $\delta$, or whatever), it will follow automatically that $\delta'$ will provide fewer conditions, and so it will be certain that $\Lambda(\delta')$ will contain $\Lambda(\delta)$ as a subspace. To summarise this section so far, suppose that a vector space $\Lambda(\Delta)$ is given, and for each face $\delta$ of $\Delta$ a subspace $\Lambda(\delta)$ is given (here $\delta$ stands for the flag which has $\delta$ as its only term). From this a complex $A$ can be constructed. First define $\Lambda(\delta)$ to be the intersection of the spaces $\Lambda(\delta_j)$ associated to the terms $\delta_j$ of $\delta$. Next define $A_i$ to be all formal sums (\ref{eqn.sum.vdelta.delta}, where the coefficients $v_\delta$ are to lie in $\Lambda(\delta)$. Finally, the boundary map \[ \bound (v_\delta \delta ) = v_\delta [\partial_1\delta] - v_\delta [\partial_2\delta] + \cdots \] is defined just as before. To complete such a definition, one must provide a vector space $\Lambda(\Delta)$, and derive from each face $\delta\subset\Delta$ a subspace $\Lambda(\delta)$ of $\Lambda(\Delta)$. Recall that $V$ stands for the span of the vectors lying on $\Delta$. Let $V^*$ be the dual space of linear functions. That such a linear function $\alpha$ is constant on $\delta$ (i.e.~zero on the vectors lying on $\delta$) describes a subspace $\delta^\perp$ of $V^*$. Exterior algebra will now be used. Fix a degree $r$, and let $\Lambda(\Delta)$ be the $r$-fold exterior product $\Lambda^r= \bigwedge^rV^*$ of arbitrary linear functions on $V$. The face $\delta$ defines a filtration of $\Lambda(\Delta)$ in the following way. For each decomposition $r=s+t$ one can take the span of expressions of the form \[ \alpha_1 \wedge \alpha_2 \wedge \dots \wedge \alpha_{s} \quad \wedge \quad \beta_1 \wedge \beta_2 \wedge \dots \wedge \beta_{t} \] where the $\alpha_i$ are to vanish on $\delta$. No conditions are placed on the $\beta_i$. The result is of course a subspace of $\Lambda(\Delta)=\bigwedge^rV^*$. To conclude this definition it is enough, for each face $\delta$ of $\Delta$, to choose one of these subspaces of $\Lambda(\Delta)$. One would like the resulting complex to produce an exact homology group, a matter which is presently not well understood, and which involves concepts that lie outside the scope of this paper. The condition \[ s > \codim \delta - s \] is satisfied by some smallest value of $s$. (The \emph{codimension} $\codim\delta$ is defined as usual to be $\dim\Delta-\dim\delta$.) Use this value to define for each face $\delta$ the subspace $\Lambda(\delta)$. If the resulting $s$ is greater than $r$, then $\Lambda(\delta)$ is taken to be zero. This choice of spaces corresponds to middle perversity intersection homology (for the associated toric variety $\PDelta$, if it exists). There is little doubt that this gives the correct choice of subspaces, for reasons that will be discussed in the final section. \section{Local coefficient spaces} To produce a complex $A$ one requires a coefficient space $\Lambda(\delta)$ for each flag $\delta$ on $\Delta$, and natural maps $\Lambda(\delta) \to \Lambda(\delta')$ whenever $\delta'$ is obtained from $\delta$ by the deletion of one or more terms. The previous section assumed the maps were inclusions (and so all the $\Lambda(\delta)$ were subspaces of $\Lambda(\Delta)$). This section will relax this assumption, to obtain the coefficient spaces that will later be used to define further complexes. In the next section, the boundary map will be defined. Suppose $V_1$ is a subspace of $V$. A basic construction of the previous section was to use $V_1$ to define subspaces (in fact a filtration) of the $r$-fold exterior produce $\bigwedge^rV^*$. Such a construction is in fact forced upon us, provided we assume that the deletion operator on flags induces inclusion, and also that $\Lambda(\Delta)$ is $\bigwedge^rV^*$. Relaxing this assumption allows the following. Given $V_1\subset V$ one can form the vector spaces $V_1$ and $V/V_1$ and then form the tensor product \[ \bigwedge \nolimits ^{r_1} V_1^* \otimes \bigwedge \nolimits ^{r_2} V_1^\perp \] of the corresponding exterior products. Here, $V_1^\perp$ consists of the $\alpha$ in $V^*$ that vanish on $V_1$. It is of course naturally isomorphic to $(V/V_1)^*$. In this way $V$ can be broken into two or more segments, within each of which the construction of the previous section can be applied. For example, each subspace $U$ of $V_1$ will determine a filtration of the first factor $\bigwedge ^{r_1} V_1^*$ above. Similarly, if $U$ lies beween $V_1$ and $V$, a filtration of the second factor $\bigwedge ^{r_2} V_1^\perp$ will arise. All the coefficient spaces $\Lambda(\delta)$ will be obtained in this way. Given a flag $\delta$ use some (perhaps all or none) of its terms $\delta_i$ to segment $V$. This gives a tensor product of exterior algebras. For each factor choose a component, i.e.~a degree. The resulting space is used in the same way as $\Lambda(\Delta)$ was, in the previous section. Note that this space depends on the flag, or more exactly the subflag used to segment $V$. There is no longer one global space, in which all the coefficient vectors lie. Now choose a term $\delta_j$ of $\delta$. This can be used to filter $\bigwedge^{r_{k+1}} (V_{k+1}/V_k)^*$, where $\delta_k$ is the largest segmenting face contained in $\delta_j$. (If there is none such, set $k=0$ and use $\bigwedge^{r_1}V_1^*$ instead.) Now, as before, use the condition \[ s_j > \codim \delta_j - s_j \] to select a term in the filtration. Here, however, \[ \codim \delta_j = \dim \delta_{k+1} - \dim \delta_j \] is to be the codimension of $\delta_j$ not within $\Delta$ but within its segment. The above filtration is to be applied for all the terms in the flag, including those used to segment. Such terms can produce of course only a trivial filtration. However, when \[ r_k > ( \dim \delta_{k} - \dim \delta_{k-1} ) - r_k \] holds, the whole of $\bigwedge^{r_{k}} (V_{k}/V_{k-1})$ is to be used, as the coefficient space for $[\delta]$. If the condition fails, zero is the only coefficient to be used with $[\delta]$. This concludes the definition of the coefficient spaces $\Lambda(\delta)$. Note that to specify such a space the following is required, in addition to $\delta$. One must select some (or all or none) of the faces of $\delta$, to be used for segmentation. One must also specify a degree $r_k$ for each segment. A more explicit notation might be \[ \Lambda ( \delta , -r , s ) \] where $r$ is the sequence $(r_1, r_2, \ldots )$ of degrees, and $s$ gives the subflag $\delta_s$ of $\delta$ that is used to segment $\delta$. The minus sign in $-r$ is to distinguish this notation from $\Lambda(\delta,r,s)$, which will be introduced later. As already noted, each degree must be greater than half the length of the corresponding segment, for the coefficient space to be non-zero. \section{The boundary map} Suppose that $\delta$ is a flag of $\Delta$, and that some segmentation $s$ of $\delta$ is chosen, which breaks $\delta$ (and $V$) into $l$ segments. Suppose also that a multi-degree $r=(r_1,\dots, r_l)$ is given. The construction of the previous section yields a coefficient space $\Lambda(\delta,-r,s)$. Now suppose that $\delta'$ is obtained from $\delta$ be removing one of the terms $\delta_j$ from $\delta$. Provided $\delta_j$ was not used to segment $\delta$, the space $\Lambda(\delta',-r,s)$ will contain $\Lambda(\delta,-r,s)$. One could obtain a complex by not allowing deletion to occur only at the faces $\delta_j$ used to segment $\delta$, or rather setting the result of deleting such a face to be zero. The resulting complex and its homology will not however properly speaking be an invariant of the polytope $\Delta$. Rather, for each choice of a segmenting flag $\delta_s$ one will have a complex, and any invariant of $\Delta$ so defined will simply be the direct sum of these flag contributions. The purpose of this section is to define a map from $\Lambda(\delta)$ to $\Lambda(\delta')$, where $\delta'$ is obtained from $\delta$ by the removal of a segmenting face $\delta_i$. This will have the effect of producing a single global complex, that will in general be indecomposable. It can be thought of as a gluing together of the local complexes of the previous paragraph. Here is an example. Suppose one has subspaces $V_1 \subset V_2 \subset V$, where $V_i$ is the span $\langle\delta_i\rangle$ of vectors lying on the face $\delta_i$. The segmentation of $V$ due to $V_1$ will be compared to that due to $V_2$. In both cases, the raw materials are firstly linear functions $\alpha$ defined on $V_i$, and secondly linear functions $\beta$ vanishing on $V_i$ (and defined on the whole of $V$). Now suppose that $\alpha_2$ is defined on $V_2$. Because $V_1\subset V_2$, the linear function $\alpha_2$ can be restricted to give $\alpha_1$ defined on $V_1$. This is straightforward. Now suppose that $\beta_2$ vanishes on $V_2$ (and is defined on $V$). Again because $V_1 \subset V_2$, the linear function $\beta_2$ also vanishes on $V_1$. Thus there is a linear map (restriction of range $\otimes$ relaxation of condition) \begin{equation} \label{eqn:rrmap} \bigwedge \nolimits ^r V_2^* \otimes \bigwedge \nolimits ^s V_2^\perp \to \bigwedge \nolimits ^r V_1^* \otimes \bigwedge \nolimits ^s V_1^\perp \end{equation} between the basic spaces associated to the two segmentations. This map has an interesting relation to the conditions used to define the coefficient spaces $\Lambda(\delta)$. When the condition \[ s > \codim V_i - s \] holds, the coefficient space due to the flag $(\delta_i)$ will be one of the above tensor products. When the condition fails, the coefficient space is zero. Because $V_1$ is a subspace of $V_2$, the condition for $(\delta_1)$ is more onerous than that for $(\delta_2)$, and so the map (\ref{eqn:rrmap}) is going in the wrong direction, to map the one coefficient space to the other. Regarding conditions however, the situation is different. The trick is to think of the conditions that define the coefficient spaces $\Lambda(\delta_i)$ to be themselves subspaces of an exterior algebra (or more exactly a tensor product of such). If $U$ is a vector space of dimension $l+m$, then each subspace of $\bigwedge^l U$ determines a subspace of $\bigwedge^m U$, and vice versa. This is via the nondegenerate pairing \[ \bigwedge \nolimits ^l U \otimes \bigwedge \nolimits ^m U \to \bigwedge \nolimits ^{l+m} U \] provided by the exterior algebra. By a slight abuse of language, this will be called duality. (The value space $\bigwedge^{l+m}U$ is has dimension one, but is not naturally isomorphic to $\bfR$.) It is now necessary to formulate the conditions using this new point of view. Suppose $U\subset V$ is a subspace, and $W\subseteq \bigwedge^r V^*$ is spanned by \[ \alpha_1 \wedge \dots \wedge \alpha_ l \quad \wedge \quad \beta_1 \wedge \dots \wedge \beta_ m \] where the $\alpha_i$ are to vanish on $U$. Consider the space $W'\subseteq\bigwedge^{r'}V^*$ spanned by \[ \alpha_1 \wedge \dots \wedge \alpha_{l'} \quad \wedge \quad \beta_1 \wedge \dots \wedge \beta_{m'} \] where $r+r'=\dim V$, $l+l'= \codim U + 1 $, and as before the $\alpha_i$ vanish on $U$; this is the subspace of the complementary component of the exterior algebra, determined mutually by $W$. The $\beta_i$ are arbitrary, and so by a further application of duality, they can be dropped from the definition of the condition space. Thus, take the span of \[ \alpha_1 \wedge \dots \wedge \alpha_{l'} \] for $l+l'=\codim U + 1$, $\alpha_i$ vanishing on $U$, as the condition space for $W$. We now return to the change of segmentation map (\ref{eqn:rrmap}). As already noted, the conditions for $V_2$ are less onerous that those for $V_1$. The condition space for $V_2$ is either zero (no conditions) or $\bigwedge^{\codim V_2} V_2^\perp$ (zero is the only solution to the conditions), and similarly for $V_1$. The map (\ref{eqn:rrmap}) will in either case respect these conditions. (To make sense of this statement, one should think of the conditions as being an ideal in the exterior algebra, and then the image under (\ref{eqn:rrmap}) of the one ideal is contained in the other.) The goal of this section is now in sight. It is the conditions that are respected by the natural map that is due to change of segmentation, not the coefficient spaces defined by the conditions. Interpret each coefficient vector $v_\delta$ as a linear function, taking values in a tensor product of top-degree exterior products, that vanishes on the condition space. Recall that the example $V_1\subset V_2\subset V$ gives rise to a natural map \[ \bigwedge \nolimits ^r V_2^* \otimes \bigwedge \nolimits ^s V_2^\perp \to \bigwedge \nolimits ^r V_1^* \otimes \bigwedge \nolimits ^s V_1^\perp \] which respects conditions. Now let $v_\delta$ be a coefficient vector on $V_1$, interpreted as a linear function on the range of the above map. Using the map, this linear function on the range can be pulled back to give a linear function on the domain. Because $v_\delta$ vanishes on the $V_1$ condition space, the pull-back vanishes on the $V_2$ condition space. By duality, the pull-back linear function is associated to a unique coefficient vector $v'_\delta$ for the $V_2$ segmentation. This is an example of a \emph{change of segmentation component} of the boundary map. Note that it takes a $V_1$-segmentation coefficient to a $V_2$ such. In other words, under boundary the segmenting term(s) may move rightwards, or in other words, increase in dimension. To finish, there are some details to be taken care of. First, although the one-dimensional value spaces for $V_1$ and $V_2$ are not the same, they are naturally isomorphic. This is good enough. Secondly, because the map (\ref{eqn:rrmap}) is natural, the argument that shows $d^2=0$ works just as before. Thirdly, the above argument has been applied only to the conditions due to the segmenting faces. The reader may wish to show that it works also for the conditions, as imposed in the previous section. (An alternative way of defining the map and checking the statements made will be outlined, as part of the discussion of bar diagrams.) The final matter concerns the degree of the coefficient $v_\delta$. The map (\ref{eqn:rrmap}) preserves the degree of (tensor products of) exterior powers. It induces the map on coefficients via duality and pull-back, and so as (tensor products of) exterior powers the coefficients $v_\delta$ on $V_1$ and $v_\delta'$ on $V_2$ will have different degrees. However, they will by definition have the same \emph{co-degree}, by which is meant the amount by which they fall short of being of top degree. For this reason, in the rest of the paper the coefficient spaces will be indexed by co-degree. For this the notation $\Lambda(\delta,r,s)$ will be used. Here, $\delta$ stands for a flag that has been broken into $l$ segments by segmentating data $s$, and $r=(r_1,r_1,\ldots,r_l)$ provides a \emph{co-degree} for each segment. The same flag can perhaps be segmented in many ways, evn if $l$ is fixed. The boundary map preserves co-degree. \section{Local-global homology} The main definition of this paper can now be given. Recall that if $\delta$ is a flag on the convex polytope $\Delta$, and $s$ breaks $\delta$ (and $V$) into $l$ segments, and if a multi-component co-degree $r=(r_1,\dots,r_l)$ is given, then from all this a coefficient space $\Lambda(\delta,r,s)$ has been defined. Recall also that if $\delta'$ is obtained from $\delta$ by removal of the $i$-th term from $\delta$, then there is a natural map \[ \partial_i : \Lambda(\delta,r,s) \to \Lambda(\delta',r,s') \] such that the usual definition of $\bound$ will produce a boundary map on \[ A_r = (0 \to A_{r,n} \to A_{r,n-1} \to \dots \to A_{r,1} \to A_{r,0} \to 0 ) \] where $A_{r,i}$ is the direct sum of the $\Lambda(\delta_r)$ for $\ord \delta=i$. (The following detail is important. When $\delta_i$ is removed from $\delta$ to obtain $\delta'$, one might as a result have to change the segmentation $s$. This happens when $\delta_i$ is used to segment $\delta$. In this case the next term $\delta_{i+1}$ is used to segment $\delta'$. If $\delta_{i+1}$ is already used by $s$ to segment $\delta$, or does not exist because $\delta_i$ is the last term in $\delta$, then this component of the boundary map is treated as zero. In this way, one obtains either a satisfactory $s'$, or the zero map.) Thus, for each co-degree $r=(r_1,\ldots,r_l)$, a complex $A_r$ has been defined. The boundary map $\bound$ of $A_r$ may cause the terms of the segmenting flag $\delta_s$ to move rightwards. This observation leads to the following. One can define subcomplexes of $A_r$ by placing conditions of the segmentation subflags that are to be used. These conditions must of course allow the movement to the right of the segmentation terms. One way to formulate this is to introduce a multi-dimension $s=(s_1 < \dots < s_{l-1})$ and allow only those segmentations $\delta$ to be used, for which the dimension $d=(d_1<\dots<d_l)$ is term by term at least as large as $s$. In this way one obtains for each $r$ and $s$ a complex $A_{r,s}$. Of course, if $s$ is too great compared to $r$, then the complex will be zero. The complex $A_{r,s}$ is the $A_w$ mentioned in \S1.) We now define the $(r,s)$ \emph{homology space} $H_{r,s}\Delta$ of $\Delta$ to be the homology of $A_{r,s}$ at the level where $\ord\delta=i$ is equal to the total degree of $\Lambda(\delta,r,s)$. In other words, it is the homology at $A_{r,s;i}$, where $ i + r_1 + \dots + r_l = n$, for $r$ gives the co-degree. If $A_{r,s}$ exactly computes $H_{r,s}\Delta$ then its dimension $h_{r,s}\Delta$ is of course a linear function of the flag vector. The next section clarifies this. As mentioned in the introduction, these spaces correspond to the local-global intersection homology spaces of $\PDelta$, introduced in \cite{bib.JF.LGIH}. \section{Decorated bar diagrams} The previous discussion made no use of coordinates, and does not give the dimension of the various $\Lambda(\delta, r, s)$ spaces involved in the construction of local-global homology. This section provides another approach. The contribution made by a flag $\delta$ to the homology $H_{r,s}\Delta$, and the maps between these contributions, can be described using coordinates, via the use of decorated bar diagrams. (Part of the theory of bar diagrams was first published in the survey paper~\cite{bib.MB.TASI}.) Suppose, for example, that the dimension $n$ of $\Delta$ is eleven, and that $\delta$ is a flag of dimension $d=(3 < 5 < 9 < 11)$. The (undecorated) \emph{bar diagram} \[ \bardiagram{...|..|....|..} \] expresses this situation. There are eleven dots, and a bar `\bardiagram{|}' is placed after the $d_i$-th dots. Each bar represents a term $\delta_i$ of $\delta$. Now choose a basis $e_1, \dots , e_n$ of $V$ such that for each $i$, the initial sequence $e_1, \dots , e_j$ (with $j=d_i$) is a basis for the span $\langle\delta_i\rangle$ of the vectors lying on the $i$-th face. Finally, let each dot represent not $e_i$ but the corresponding linear function $\alpha_i$, that vanishes on each $e_j$ except $e_i$. Call this a \emph{system of coordinates} for $V$, that is \emph{subordinate} to $\delta$. By construction, each dot represent a linear function that vanishes on the faces $\delta_i$ of $\delta$ represented by the bars that lie to its left. So that we can speak more concisely, we shall think of the dots and bars as actually being the linear functions and terms of the flag respectively. Each subset of the dots represents an element of the exterior algebra generated by the linear functions on $V$. To represent the selection of a subset, promote the chosen dots `\bardiagram{.}' into circles `\bardiagram{o}'. Thus \begin{equation} \label{eqn:dbd} \bardiagram{..o|o.|o.o.|oo} \end{equation} represents (or more concisely is) a degree six element of the exterior algebra. It is an example of a \emph{decorated bar diagram}. (Each circle represents an element in the homology of the torus, that is the generic or central fibre of the moment map. As the fibre is moved towards a face that lies to the left of the circle, so the circle shrinks to a point. Thus, circles represent $1$-cycles with specified vanishing properties. Similarly, the promotion of say $6$ dots to circles produces a diagram that represent a $6$-cycle in the generic fibre, with specific shrinking properties, as the fibre is moved to the faces of the flag. This will be important, in the topological interpretation of exact homology.) The exterior form (\ref{eqn:dbd}) has certain vanishing properties, with respect to the faces of $\delta$. The condition of \S4 is equivalent to the following: That between each bar and the right hand end of the diagram, there should be strictly more `\bardiagram{o}'s than `\bardiagram{.}'s. Our example satisfies this condition, and so is an \emph{admissable} decorated bar diagram. Now fix an unsegmented degree $r=(r_1)$ and define the coefficient space $\Lambda(\delta,r)$ to be the span of the admissable $r$-circle bar diagrams (considered as exterior forms). (Because there is no segmentation, $s$ is trivial, and will be omitted.) For the $\Lambda(\delta,r)$ to come together to produce a complex, the following must hold. First, $\Lambda(\delta,r)$ as a subspace of the degree~$r$ forms should not depend on the choice of a basis subordinate to $\delta$. Second, the boundary map should not depend on the basis (or in other words should be covariant for such change). Third, when a bar is removed from an admissable diagram, the result should also be admissable. The last two conditions are immediately seen to be true. There are two ways to see that the first requirement (that $\Lambda(\delta,r)$ not move when the subordinate basis is changed) is true. One method is to show that the span $\Lambda(\delta,s)$ of the admissable diagrams is the solution set to a problem that can be formulated without recourse to use of a basis. This is the approach taken in \S\S3--7. The other method is to show directly that $\Lambda(\delta,r)$ does not move, under change of subordinate basis. Any change of subordinate basis can be obtained as a result of applying the following moves. First, one can multiply basis elements by non-zero scalars. Second, one can permute the basis elements (dots and circles), provided so doing does not cause a dot or circle to pass over a bar. Thirdly, one can increase a basis element by some multiple of another basis element, that lies to its right. It is clear that applying either of the first two moves to the basis will not change $\Lambda(\delta,r)$. Regarding the third move, if a diagram such as (\ref{eqn:dbd}) is admissable, then the result of moving one or more `\bardiagram{o}'s to the right, perhaps over bars, will also be admissable. This is obvious, from the nature of the conditions defining admissability. The third type of move makes such changes. Thus, the span $\Lambda(\delta,r)$ of the admissable decorated diagrams does not move. It has now been shown that the admissable bar diagrams, such as (\ref{eqn:dbd}), define coefficient spaces $\Lambda(\delta,r)$ that can be assembled to produce a complex. It is left to the reader, to check that it is exactly the same complex, as was defined in \S4. The remainder of this section is devoted to the description of the segmented form of the above construction. As before, let \[ \bardiagram{...|..|....|..} \] denote a flag of dimension $d=(3 < 5 < 9 < 11)$, and now choose some of the bars, say just the second, to segment the diagram. The result is \[ \bardiagram{...|..} \qquad \bardiagram{|....|..} \] or more concisely \[ \bardiagram{...|..!....|..} \] where the promotion of a `\bardiagram{|}' to a `\bardiagram{!}' indicates that is is being used for segmentation. Note that each segmenting face is also used as the first face in the following segment. As before, one can promote some of the `\bardiagram{.}'s to `\bardiagram{o}'s to obtain a \emph{decorated bar diagram}, which will be \emph{admissable} if between any bar (either `\bardiagram{|}' or `\bardiagram{!}') and the end of the segment, there are more `\bardiagram{o}'s than `\bardiagram{.}'s. To be able to assemble the span of the admissable diagrams into a complex, certain requirements must be met. They have already been formulated. The first is that the span should not depend on the choice of a subordinate basis. Scalar and permutation moves on the basis clearly leave the span unchanged, as in the single segment case. Adding to one basis vector another, lying to its right, has no effect on the span, provided the second lies in the same segment as the first. Now consider the situation, where one of the basis linear forms is changed, by adding to it another basis linear form, as before lying to the right, but this time in a different segment. The way to have this have no effect on the span is to have the dots and circles in a segmented diagram represent not linear functions on $V$, but rather linear functions on the (span of vectors lying on) the face at the right end of the segment (which is $V$ for the last segment). Thus, by having each decorated diagram (possibly segmented) represent an element in a tensor product of exterior algebras, the span of all the admissable diagrams becomes a space that does not move, when the subordinate basis is changed. This agrees with \S5. For these spaces to be assembled into a complex, the boundary map must be well defined. As in the single segment situation, all is well when a non-segmenting term (a `\bardiagram{|}' rather than a `\bardiagram{!}') is removed from a decorated bar diagram. The removal of a `\bardiagram{!}' terms has a more subtle effect, for the segmentation will have to change. Whatever rule is used, it must respect admissability of decorated diagrams, and it must respect change of subordinate basis. The rule, which we state without prior justification, is this: whenever something such as \[ \bardiagram{!ooooo|} \] occurs in a decorated diagram, one can obtain a component of the boundary by replacing it with \[ \bardiagram{ooooo!} \] while the result of removing any other `\bardiagram{!}' terms is zero. (The number of `\bardiagram{o}'s is not relevant, that one has `\bardiagram{!}' followed by some circles, and then a `\bardiagram{|}' is.) It is clear that this rule will, from admissable diagrams, generate only admissable diagrams. This is because it can but only cause some extra `\bardiagram{o}'s to appear, whenever the `\bardiagram{.}' and `\bardiagram{o}' counts are to be compared. The same is of course not true, for the replacement of `\bardiagram{!.o.|}' by `\bardiagram{.o.!}' and similar situations. The next task is to show that this part of the boundary respects change of subordinate basis. Given a fragment such as `\bardiagram{!.o.|}' that contributes zero to this part of the boundary, whatever change of basis is made, the contribution will still be zero. Consider now a fragment such as `\bardiagram{!ooo|}'. Scalar and permutation moves will have no effect on the boundary. This is obvious. The only way that adding something that lies to the right can have any effect is if that something lies to the right of the whole fragment `\bardiagram{!ooo|}'. This is because the exterior algebra is antisymmetric. Consider now the boundary contribution `\bardiagram{ooo!}'. As already described, the `\bardiagram{!}' induces a restriction of linear functions, and so the just considered change of basis will after all have no effect on the boundary map. This concludes the presentation of the complex $A$ via decorated bar diagrams. The details of co-degree (number of `\bardiagram{.}'s in each segment) and so forth are as in the earlier exposition~(\S5--7). It is left to the reader to verify that the two approaches lead to exactly the same family $A_{r,s}$ of complexes. \section{Summary and conclusions} The flag vector of a convex polytope satisfies subtle linear inequalities, and also non-linear inequalities, that are at present not known. Exact homology seems to be the only general method available to us, to prove such results. The difficulty with purely combinatorial means is firstly that there are no natural maps, other than the deletion operators, between the flags on a polytope, and secondly that the convexity of the polytope has to be allowed to enter into the discussion in a significant way. This said, very few results relating to exact homology are known at present. For the usual middle perversity intersection homology theory, the three approachs mentioned in the introduction are known to have significant areas of agreement. The recursive formula in \cite{bib.RS.GHV} for $h\PDelta$ can be unwound to express each Betti number as a sum of contributions due to flags. These contributions are exactly the same as those implicit in the definitions of \S4. The method of decorated bar diagrams provides a reformulation of the recursive formula for $h\PDelta$, where now one simply counts the number of valid diagrams. As was indicated in \S8, once the valid diagrams are known, it is not then hard to determine what the corresponding space of coefficients should be. Thus, the derivation from \cite{bib.RS.GHV} of \S4 is without difficulty. Finally, via the identification of the exterior algebra $\bigwedge^\bullet V^*$ with the homology of the generic fibre of the moment map, one can interpret the complex in \S4 in terms of the construction of intersection homology cycles on $\PDelta$, and relations between them. This last will be presented elsewhere. The important property of intersection homology (for middle perversity only) is that it seems to produce exact homology groups. That there are formula for such Betti numbers, and moreover as somewhat geometric alternating sums, supports this view. The local-global construction defined in \S\S5--7 can, via the moment map, be translated into a class of cycles on $\PDelta$, and relations between them. These cycles and relations have special properties with respect to the strata of $\PDelta$. In this way one can translate the definition of local-global homology in this paper into a topological one. Indeed, if intersection homology were not already known, it could have been discovered via its similarity to the $H\Delta$ theory presented here. There is a subtlety connected to the concept of a linear function of the flag vector. The flag vectors of convex polytopes span a proper subspace of the space of all possible flag vectors. Now, a linear function on a subspace is not the same as a linear function on the whole space. The latter contains more information. The constructions of this paper provide linear functions of arbitrary (not necessarily polytope) flag vectors. As noted, they agree with the formulae (8)--(10) of~\cite{bib.JF.LGIH}. The construction presented here of $H\Delta$ is probably not the only one. In particular, it ought to be possible to define a theory, with the same expected Betti numbers, but where $H\Delta$ is built out of the $H\delta$, for all proper faces $\delta\subset\Delta$, together with perhaps a little gluing information. This can certainly be done in the simple case, where it corresponds to the natural formula in that context for $h\Delta$ in terms of the face vector. For general polytopes, such a theory would correspond to a linear function on arbitrary flag vectors, which agrees on polytope flag vectors with the $h$-vector presented in this paper. Such a theory may be part of a geometric proof of exactness of homology. One cannot, of course, prove that which is not true; and exact homology cannot hold for a complex if the expected Betti number turns out to be negative for some special polytope. Bayer (personal communication) has an example of a 5~dimensional polytope (the bipyramid on the cylinder on a 3-simplex) where this in fact happens. This is discussed further in \cite{bib.JF.LGIH}. All this indicates that there are as yet unrealised subtleties in the concept and proof of exact calculation. The central definitions of this paper, namely of the coefficient spaces $\Lambda(\delta,r,s)$ and the maps between them, use only linear algebra and the deletion operator on flags. Put another way, the largest part of this paper has been the study of the flag \[ V_1 \subset V_2 \subset \dots \subset V_m \subset V \] of vector spaces associated to a flag $\delta$ of faces on $\Delta$, and the vector spaces that can be defined from it. Given the requirements of exact homology, there is little else available that one could study. However, these definitions are very closely linked to the usual intersection homology theory and also its local-global variant. These connections indicate that there may be unremarked subtleties in the linear algebra of a flag, and undiscovered simplicity in intersection homology. A better understanding of the intersection homology of Schubert varieties would be very useful. Finally, as mentioned in the Introduction, one could wish for a similar theory that applies to $i$-graphs, and similar combinatorial objects. This will probably require a different sort of linear algebra.

9710

alg-geom/9710014

en

https://arxiv.org/abs/alg-geom/9710014

alg-geom/9710014

Kai Behrend

Kai Behrend

The product formula for Gromov-Witten invariants

LaTeX

We prove that the system of Gromov-Witten invariants of the product of two varieties is equal to the tensor product of the systems of Gromov-Witten invariants of the two factors.

alg-geom

\subsection{Introduction} \newcommand{\mbox{$\tilde{\GG}_s(V)$}}{\mbox{$\tilde{\GG}_s(V)$}} \newcommand{\mbox{$\tilde{\GG}_s(W)$}}{\mbox{$\tilde{\GG}_s(W)$}} \newcommand{\mbox{$\tilde{\GG}_s(0)$}}{\mbox{$\tilde{\GG}_s(0)$}} \newcommand{\mbox{$\tilde{\GG}_s(V\times W)$}}{\mbox{$\tilde{\GG}_s(V\times W)$}} Let $V$ and $W$ be smooth and projective varieties over the field $k$. In this article we treat the question how to express the Gromov-Witten invariants of $V\times W$ in terms of the Gromov-Witten invariants of $V$ and $W$. On an intuitive level, the answer is quite obvious. For example, assume $V=W={\Bbb P}^1$ and let us ask the question how many curves in ${\Bbb P}^1\times{\Bbb P}^1$ of genus $g$ and bidegree $(d_1,d_2)$ pass through $n=2(d_1+d_2)+g-1$ given points $P_1,\ldots,P_n$ of ${\Bbb P}^1\times {\Bbb P}^1$ in general position. The answer is given by the Gromov-Witten invariant $$I^{{\Bbb P}^1\times{\Bbb P}^1}_{g,n}(d_1,d_2)(\gamma^{\otimes n}),$$ where $\gamma\in H^4({\Bbb P}^1\times{\Bbb P}^1,{\Bbb Q})$ is the cohomology class Poincar\'e dual to a point. We rephrase the question by asking how many triples $(C,x_1,\ldots,x_n,f)$, where $C$ is a curve of genus $g$, $x_1,\ldots,x_n$ are marked points on $C$ and $f:C\to{\Bbb P}^1\times{\Bbb P}^1$ is a morphism of bidegree $(d_1,d_2)$ exist (up to isomorphism) which satisfy $f(x_i)=P_i$, for all $i=1,\ldots,n$. Now a morphism $f:C\to{\Bbb P}^1\times{\Bbb P}^1$ of bidegree $(d_1,d_2)$ is given by two morphisms $f_1:C\to{\Bbb P}^1$ and $f_2:C\to{\Bbb P}^1$ of degrees $d_1$ and $d_2$, respectively. The requirement that $f(x_i)=P_i$ translates into $f_1(x_i)=Q_i$ and $f_2(x_i)=R_i$, if we write the components of $P_i$ as $P_i=(Q_i,R_i)$. The family of all marked curves $(C,x_1,\ldots,x_n)$ admitting such an $f_1$ is some cycle, say $\Gamma_1$, in $\overline{M}_{g,n}$. Of course, the family of all curves $(C,x_1,\ldots,x_2)$ admitting an $f_2$ as above is another cycle $\Gamma_2$ in $\overline{M}_{g,n}$ and the family of all $(C,x_1,\ldots,x_n)$ admitting an $f_1$ and an $f_2$ is the intersection $\Gamma_1\cdot\Gamma_2$. So the Gromov-Witten number we are interested in is \[I^{{\Bbb P}^1\times{\Bbb P}^1}_{g,n}(d_1,d_2)(\gamma^{\otimes n}) = \Gamma_1\cdot\Gamma_2.\] In fact, the dual cohomology classes of $\Gamma_1$ and $\Gamma_2$ are Gromov-Witten invariants themselves, namely $I^{{\Bbb P}^1}_{g,n}(d_1)(\tilde\gamma^{\otimes n})$ and $I^{{\Bbb P}^1}_{g,n}(d_2)(\tilde\gamma^{\otimes n})$, where $\tilde\gamma\in H^2({\Bbb P}^1,{\Bbb Q})$ is the cohomology class dual to a point. Thus we have \[I^{{\Bbb P}^1\times{\Bbb P}^1}_{g,n}(d_1,d_2)(\gamma^{\otimes n}) = I^{{\Bbb P}^1}_{g,n}(d_1)(\tilde\gamma^{\otimes n}) \cup I^{{\Bbb P}^1}_{g,n}(d_2)(\tilde\gamma^{\otimes n}) \] in $H^{\ast}(\overline{M}_{g,n}),{\Bbb Q})$. This is the simplest instance of the product formula, which we shall prove in this article. (Note that we have identified, as usual, top degree cohomology classes on $\overline M_{g,n}$ with their integrals over the fundamental cycle $[\overline M_{g,n}]$.) We get a more general statement by letting $V$ and $W$ be arbitrary smooth projective varieties over $k$. We fix cohomology classes $\gamma_1,\ldots,\gamma_n\in H^{\ast}(V)$ and $\epsilon_1,\ldots,\epsilon_n\in H^{\ast}(W)$, which we assume to be homogeneous, for simplicity. Then the product formula says that \begin{eqnarray}\label{bpf} \lefteqn{ I^{V\times W}_{g,n}(\beta) (\gamma_1\otimes\epsilon_1\otimes\ldots \otimes\gamma_n\otimes\epsilon_n)}\nonumber\\ & = & (-1)^s I^{V}_{g,n}(\beta_V)(\gamma_1\otimes\ldots \otimes\gamma_n) \cup I^{W}_{g,n}(\beta_W)(\epsilon_1\otimes\ldots \otimes\epsilon_n) \end{eqnarray} in $H^{\ast}(\overline{M}_{g,n},{\Bbb Q})$. Here $\beta\in H_2(V\times W)^+$ and $\beta_V={p_V}_{\ast}\beta$, $\beta_W={p_W}_{\ast}\beta$, where $p_V$ and $p_W$ are the projections onto the factors of $V\times W$. The sign is given by \[s=\sum_{i>j}\deg\gamma_i\deg\epsilon_j.\] This formula is already stated in \cite{KM} as a property expected of Gromov-Witten invariants. In the case of $g=0$ and $V$ and $W$ (and hence $V\times W$) convex, it is not difficult to prove, once the properties of stacks of stable maps are established, as they are, for example, in \cite{BM}. Essentially, the above intuitive argument can then be translated into a rigorous proof. In the general case, the enumerative meaning of Gromov-Witten invariants is much less clear, since one has to use `virtual' fundamental classes to define them. (This is done in \cite{BF} and \cite{gwi} or \cite{litian}.) So the theorem follows from properties of virtual fundamental classes. This is what we prove in the present paper. Formula~(\ref{bpf}) has been used by various authors to understand the quantum cohomology of a product. (See \cite{KMK}, \cite{KMZ} and \cite{Kauf}.) By Formula~(\ref{bpf}), the codimension zero Gromov-Witten invariants (ie.\ those that are numbers, like $I^{{\Bbb P}^1\times{\Bbb P}^1}_{g,n}(d_1,d_2)(\gamma^{\otimes n})$, above) of a product are determined by the Gromov-Witten invariants of higher codimension of the factors and by the intersection theory of $\overline{M}_{g,n}$. To explain the treatment in this article, let us reformulate~(\ref{bpf}) by saying that \[\begin{array}{ccc} h(V\times W)^{\otimes n} & \stackrel{I^{V\times W}_{g,n}(\beta)} {\longrightarrow} & h(\overline M_{g,n}) \\ \parallel & & \rdiagup{\Delta^{\ast}} \\ h(V)^{\otimes n}\otimes h(W)^{\otimes n} & \stackrel{I^{V}_{g,n}(\beta_V)\otimes I^{W}_{g,n}(\beta_W)}{\longrightarrow} & h(\overline M_{g,n})\otimes h(\overline M_{g,n}), \end{array}\] where $\Delta:\overline{M}_{g,n}\to \overline{M}_{g,n}\times\overline{M}_{g,n}$ is the diagonal, commutes. Here we have passed to the motivic Gromov-Witten invariants. These are homomorphisms between DMC-motives. (These are like Chow motives, except that they are made from smooth and proper Deligne-Mumford stacks, instead of varieties. For details see \cite{BM}, Section~8.) To summarize all of their functorial properties, Gromov-Witten invariants where defined in \cite{BM} as natural transformations between the functors $h(V)^{\otimes S}$ and $h(\overline M)$, which are functor from a certain graph category $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ to the category of graded DMC-motives. To explain, let us start by reviewing some graph theory. The category $\tilde{\GG}_s=\mbox{$\tilde{\GG}_s(0)$}$ is the category of stable modular graphs (graphs whose vertices are labeled with genuses; see \cite{BM}, Definition~1.5) with so called extended isogenies as morphisms. An {\em extended isogeny }is either a morphism gluing two tails to an edge, or it is a proper isogeny (or a composition of the two). An {\em isogeny }is a morphism which contracts various edges or tails or both. (The name isogeny comes from the fact that such morphisms do not affect the genus of the components of the graphs involved.) For the definition of composition of extended isogenies, see \cite{BM}, Page~36. The category $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ is called the {\em cartesian extended isogeny category }over $V$. The most important objects of $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ are pairs $(\tau,(\beta_i)_{i\in I})$, where $\tau$ is a stable modular graph and $(\beta_i)_{i\in I}$ is a family of $H_2(V)^+$ markings on $\tau$. This means that each $\beta_i$ is a function $\beta_i:V\t\to H_2(V)^+$, where $V\t$ is the set of vertices of $\tau$. (The indexing set $I$ is finite.) The fundamental property of $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ is that it is {\em fibered } over $\tilde{\GG}_s$. This means that there is a functor $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}\to\tilde{\GG}_s$ (projection onto the first component) and that given an object $(\tau,(\beta_i)_{i\in I})$ of $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ and a morphism $\phi:\sigma\to\tau$ there exists, up to isomorphism, a unique object $(\sigma,(\gamma_j)_{j\in J})$ of $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ together with a morphism $\Phi:(\sigma,(\gamma_j)_{j\in J}):\to(\tau,(\beta_i)_{i\in I})$ covering $\phi:\sigma\to\tau$. When constructing $\Phi$, the basic non-obvious case is that where $\phi$ contracts a non-looping edge of $\sigma$ and $I$ has only one element. Then we have the graph $\tau$ with an $H_2(V)^+$-marking $\beta$ and there are two vertices $v_1$, $v_2$ of $\sigma$ corresponding to one vertex $w$ of $\tau$. Then $(\sigma,(\gamma_j)_{j\in J})$ is defined such that $J$ counts the ways to write $\beta(w)=\beta_1+\beta_2$ in $H_2(V)^+$ and $\gamma_j$ assigns $\beta_1$ to $v_1$ and $\beta_2$ to $v_2$, and otherwise does not differ from $\beta$. Things get more complicated, if one also considers the less important objects of $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$. These are of the form $(\tau,(\tau_i)_{i\in I})$, where, as above, $\tau$ is a stable modular graph, but now each $\tau_i$ is a stable $H_2(V)^+$-marked graph (as opposed to an $H_2(V)^+$-marked stable graph), together with a {\em stabilizing morphism }$\tau_i\to\tau$. For the complete picture, see \cite{BM}, Definition~5.9. On objects, the morphisms $h(V)^{\otimes S}$ and $h(\overline M)$ from $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ to $(\mbox{graded DMC-motives})$ are defined as follows: For an object $(\tau,(\beta_i)_{i\in I})$ of $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ we have \[h(V)^{\otimes S}(\tau,(\beta_i)_{i\in I})=h(V)^{\otimes S\t},\] where $S\t$ is the set of tails of $\tau$ and \[h(\overline M)(\tau,(\beta_i)_{i\in I})=h(\overline M(\tau)),\] where $$\overline M(\tau)=\prod_{v\in V\t}\overline{M}_{g(v),F\t(v)}$$ and $F\t(v)$ is the set of flags meeting the vertex $v$ of $\tau$. So both of these functors only depend on the first component $\tau$ of $(\tau,(\beta_i)_{i\in I})$. For the definition of these functors on morphisms, see \cite{BM}, Section~9. Note that $h(V)^{\otimes S}$ actually comes with a twist (ie.\ a degree shift) $\chi\dim V$. This we shall ignore here, to shorten notation and since nothing interesting happens to it, anyway. The {\em Gromov-Witten transformation } of $V$ is now defined as a natural transformation \[I^V:h(V)^{\otimes S}\longrightarrow h(\overline M)\] of functors from $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ to $(\mbox{graded DMC-motives})$. In this paper (Theorem~\ref{st}), we shall prove that \[I^{V\times W}=I^V\cup I^W,\] where $I^V\cup I^W$ is defined as $\Delta^{\ast}(I^V\otimes I^W)$. Since Gromov-Witten invariants are defined in terms of virtual fundamental classes on moduli stacks of stable maps, this theorem follows from a certain compatibility between virtual fundamental classes. This is our main result (Theorem~\ref{pt}) and takes up most of this paper. \subsection{Virtual Fundamental Classes} Fix a ground field $k$. For a smooth projective $k$-variety $V$ let $\mbox{$\tilde{\GG}_s(V)$}$ be the category of extended isogenies of stable $H_2(V)^+$-graphs bounded by the characteristic of $k$ (see \cite{BM}, Definition~5.6 and Example~II following Definition~5.11). Let $J(V,\tau)\in A_{\dim(V,\tau)}(\overline{M}(V,\tau))$, for $\tau\in\mathop{\rm ob}\mbox{$\tilde{\GG}_s(V)$}$, be the `virtual fundamental class', or orientation (\cite{BM}, Definition~7.1) of $\overline{M}$ over $\mbox{$\tilde{\GG}_s(V)$}$ constructed in \cite{gwi}, Theorem~6, using the techniques from \cite{BF}. Now let us consider two smooth projective $k$-varieties $V$ and $W$; denote the two projections by $p_V:V\times W\to V$ and $p_W:V\times W\to W$. If $\tau$ is a stable $H_2(V\times W)^+$-graph, we denote by ${p_V}_{\ast}(\tau)$ and ${p_W}_{\ast}(\tau)$ the stabilizations of $\tau$ with respect to ${p_V}_{\ast}:H_2(V\times W)^+\to H_2(V)^+$ and ${p_W}_{\ast}:H_2(V\times W)^+\to H_2(W)^+$ (see \cite{BM}, Remark~1.15), by $\tau^s$ the absolute stabilization of $\tau$. Applying the functor $\overline{M}$ to the commutative diagram \[\comdia{(V\times W,\tau)}{}{(W,{p_W}_{\ast}(\tau))}{}{}{}{(V,{p_V}_{\ast}(\tau))}{}{(\mathop{\rm Spec}\nolimits k,\tau^s)} \] in ${\frak V}\GG_s$ (see \cite{BM}, Remark~3.1 and the remark following Theorem~3.14) we get a commutative diagram of proper Deligne-Mumford stacks \[\comdia{\overline{M}(V\times W,\tau)}{}{\overline{M}(W,{p_W}_{\ast}(\tau))} {}{}{} {\overline{M}(V,{p_V}_{\ast}(\tau))}{}{\overline{M}(\tau^s).}\] In general, this diagram is not cartesian; let $P$ be the cartesian product \[\comdia{P}{}{\overline{M}(W,{p_W}_{\ast}(\tau))} {}{}{} {\overline{M}(V,{p_V}_{\ast}(\tau))}{}{\overline{M}(\tau^s).}\] Rewrite these diagrams as follows: \[\begin{array}{ccccc} \overline{M}(V\times W,\tau) & \stackrel{h}{\longrightarrow} & P & \longrightarrow & \overline{M}(V,{p_V}_{\ast}\tau)\times\overline{M}(W,{p_W}_{\ast}\tau) \\ & \searrow & \ldiag{} & & \rdiag{} \\ & & \overline{M}(\tau^s) & \stackrel{\Delta}{\longrightarrow} & \overline{M}(\tau^s)\times\overline{M}(\tau^s).\end{array}\] To shorten notation, write $J(V\times W)=J(V\times W,\tau)$, $J(V)=J(V,{p_V}_{\ast}\tau)$ and $J(W)=J(W,{p_W}_{\ast}\tau)$. \begin{them} \label{pt} We have \[\Delta^{!}(J(V)\times J(W))=h_{\ast}(J(V\times W)).\] \end{them} For a stable $A$-graph $\tau$ ($A=H_2(V\times W)^+$, $H_2(V)^+$ etc.) we denote by ${\frak M}(\tau)$ the algebraic $k$-stack of $\tau$-marked prestable curves, forgetting the $A$-structure, and thinking of $\tau$ simply as a (possibly not stable) modular graph. We consider the diagram \begin{equation} \label{bigd} \begin{array}{cccccc} & \overline{M}(V\times W,\tau) & \stackrel{h}{\longrightarrow} & P & \longrightarrow & \overline{M}(V,{p_V}_{\ast}\tau)\times\overline{M}(W,{p_W}_{\ast}\tau) \\ \phantom{M}/ & \rdiag{c} & & \rdiag{} & & \rdiag{a} \\ b\mid & {\frak D}(\tau) & \stackrel{l}{\longrightarrow} & \PP & \stackrel{\phi}{\longrightarrow} & {\frak M}({p_V}_{\ast}\tau)\times{\frak M}({p_W}_{\ast}\tau) \\ \phantom{nM}\searrow& \rdiag{e} & \searrow & \rdiag{} & & \rdiag{s\times s} \\ & {\frak M}(\tau) & & \overline{M}(\tau^s) & \stackrel{\Delta}{\longrightarrow} & \overline{M}(\tau^s)\times\overline{M}(\tau^s). \end{array}\end{equation} Here $s\times s$ is given by stabilizations and $\PP$ is defined as the fibered product of $\Delta$ and $s\times s$. The morphisms $a$ and $b$ are given by forgetting maps, retaining only marked curves. The algebraic stack ${\frak D}(\tau)$ is defined as follows. For a $k$-scheme $T$ the groupoid ${\frak D}(\tau)(T)$ has as objects diagrams \begin{equation}\label{ddd} \begin{array}{ccc} (C,x) & \longrightarrow & (C'',x'') \\ \ldiag{} & & \\ (C',x') & & \end{array}\end{equation} where $(C,x)$ is a $\tau$-marked prestable curve over $T$, $(C',x')$ a ${p_V}_{\ast}(\tau)$-marked prestable curve over $T$ and $(C'',x'')$ a ${p_W}_{\ast}(\tau)$-marked prestable curve over $T$. The arrow $(C,x)\to(C',x')$ is a morphism of marked prestable curves covering the morphism $\tau\to{p_V}_{\ast}(\tau)$ of modular graphs. Similarly, $(C,x)\to(C'',x'')$ is a morphism of marked prestable curves covering $\tau\to{p_W}_{\ast}(\tau)$. This concept has not been defined in \cite{BM}; the definition (in this special case) is as follows. Let us explain it for the case of $W$ instead of $V$, since this will lead to less confusion of notation with the set of vertices of a graph. The morphism $\tau\to{p_W}_{\ast}(\tau)$ is given by a combinatorial morphism of $0$-marked graphs $a:{p_W}_{\ast}(\tau)\to\tau$ (see \cite{BM}, Definition~1.7). So there are maps $a:V_{{p_W}_{\ast}(\tau)}\to V_{\tau}$ and $a:F_{{p_W}_{\ast}(\tau)}\to F_{\tau}$. The morphism $(C,x)\to(C'',x'')$ is given by a family $p=(p_v)_{v\in V_{{p_W}_{\ast}(\tau)}}$ of morphisms of prestable curves (\cite{BM}, Definition~2.1) $p_v:C_{a(v)}\to C''_v$ such that for every $i\in F_{{p_W}_{\ast}(\tau)}$ we have $p_{\partial(i)}(x_{a(i)})=x''_i$. \newcommand{{p_V}}{{p_V}} \newcommand{{p_W}}{{p_W}} There are morphisms of stacks $e:{\frak D}(\tau)\to{\frak M}(\tau)$, ${\frak D}(\tau)\to{\frak M}({p_V}_{\ast}\tau)$ and ${\frak D}(\tau)\to{\frak M}({p_W}_{\ast}\tau)$, given, respectively, by mapping Diagram~(\ref{ddd}) to $(C,x)$, $(C',x')$ and $(C'',x'')$. Let us denote the product of the latter two by \[\tilde{\Delta}:{\frak D}(\tau)\longrightarrow {\frak M}({p_V}_{\ast}\tau)\times{\frak M}({p_W}_{\ast}\tau). \] \begin{lem} \label{lem2} In Diagram~(\ref{ddd}) both morphisms induce isomorphisms on stabilizations. \end{lem} \begin{pf} This follows from the fact that any morphism of stable marked curves (with identical dual graphs) is an isomorphism. This fact is proved in \cite{BM}, at the very end of the proof of Theorem~3.6, which immediately precedes Definition~3.13. \end{pf} By this lemma there is a commutative diagram \[\begin{array}{ccc} {\frak D}(\tau) & \stackrel{\tilde{\Delta}}{\longrightarrow} & {\frak M}({p_V}_{\ast}\tau)\times {\frak M}({p_W}_{\ast}\tau) \\ \ldiag{e} & & \\ {\frak M}(\tau) & & \rdiag{s\times s} \\ \ldiag{s} & & \\ \overline{M}(\tau^s) & \stackrel{\Delta}{\longrightarrow} & \overline{M}(\tau^s)\times\overline{M}(\tau^s), \end{array}\] which gives rise to the morphism $l:{\frak D}(\tau)\to\PP$ of Diagram~(\ref{bigd}). \begin{prop} \label{podd} The morphisms $\Delta$ and $\tilde{\Delta}$ are proper regular local immersions. Their natural orientations satisfy $$l_{\ast}[\tilde{\Delta}]=(s\times s)^{\ast}[\Delta].$$ \end{prop} \begin{pf} Let $S_1$ and $S_2$ be finite sets, set $S=S_1\amalg S_2$. Let the modular graph ${p_V}_{\ast}(\tau)'$ be obtained from ${p_V}_{\ast}(\tau)$ by adding (in any fashion) $S_1$ to the set of tails of ${p_V}_{\ast}(\tau)$. Similarly, let ${p_W}_{\ast}(\tau)'$ be obtained form ${p_W}_{\ast}(\tau)$ by adding $S_2$ to the set of tails, arbitrarily. Now let $\tau'$ be obtained from $\tau$ by adding the set $S$ to the tails of $\tau$ in the unique way such that $\tau\to{p_V}_{\ast}(\tau)$ induces a morphism $\tau'\to{p_V}_{\ast}(\tau)'$, which gives the inclusion $S_1\subset S$ on tails, and $\tau\to{p_W}_{\ast}(\tau)$ induces a morphism $\tau'\to{p_W}_{\ast}(\tau)'$, which gives the inclusion $S_2\subset S$ in tails. With these choices we have a cartesian diagram of $k$-stacks \begin{equation}\label{poddi} \comdia{\overline{M}(\tau')}{\delta} {\overline{M}({p_V}_{\ast}(\tau)') \times \overline{M}({p_W}_{\ast}(\tau)')} {}{}{\chi} {{\frak D}(\tau)} {\tilde{\Delta}} {{\frak M}({p_V}_{\ast}(\tau)) \times {\frak M}({p_W}_{\ast}(\tau)).} \end{equation} The proof that this is the case is similar to the proof of Proposition~\ref{prop5}, below. The morphism $\chi$ in Diagram~(\ref{poddi}) is a local presentation of ${\frak M}({p_V}_{\ast}(\tau)) \times {\frak M}({p_W}_{\ast}(\tau))$, (see \cite{gwi}, remarks following Lemma~1). Moreover, by choosing $S$ and the primed graphs correctly, any point of ${\frak M}({p_V}_{\ast}(\tau)) \times {\frak M}({p_W}_{\ast}(\tau))$ can be assumed to be in the image of $\chi$. So to prove that $\tilde{\Delta}$ is a proper regular local immersion, it suffices to prove that $\delta$ is a proper regular local immersion. Properness is clear; the stacks $\overline{M}(\tau')$, $\overline{M}({p_V}_{\ast}(\tau)')$ and $\overline{M}({p_W}_{\ast}(\tau)')$ are proper. The regular local immersion property follows from injectivity on tangent spaces which can be proved by a deformation theory argument. The proof for $\Delta$ is comparatively trivial. To prove the fact about the orientations, first note that $s\times s$ is flat (see \cite{gwi}, Proposition~3) and so $\phi$ is a regular local immersion and $(s\times s)^{\ast}[\Delta]=[\phi]$. To prove that $l_{\ast}[\tilde{\Delta}]=[\phi]$, it suffices to identify dense open substacks ${\frak D}(\tau)'\subset{\frak D}(\tau)$ and $\PP'\subset\PP$ such that $l$ induces an isomorphism ${\frak D}(\tau)'\to\PP'$. We define ${\frak D}(\tau)'$ to be the open substack of ${\frak D}(\tau)$ over which $C_v\to C_v'$ is an isomorphism for all $v\in V_{{p_V}_{\ast}(\tau)}$ and $C_v\to C_v''$ is an isomorphism for all $v\in V_{{p_W}_{\ast}(\tau)}$. We define $\PP'$ to be the pullback via $\Delta$ of ${\frak M}({p_V}_{\ast}(\tau))' \times {\frak M}({p_W}_{\ast}(\tau))'$, where ${\frak M}({p_V}_{\ast}(\tau))'$ is the open substack over which $(C_v,(x_i)_{i\in F(v)})$ is stable, for all $v\in V_{\tau^s}$, similarly for ${\frak M}({p_W}_{\ast}(\tau))'$. Note the slight abuse of notation; we have denoted vertices of different graphs by the same letter. \end{pf} \begin{lem} The morphism $e:{\frak D}(\tau)\to{\frak M}(\tau)$ is \'etale. \end{lem} \begin{pf} Similar to the proof of \cite{gwi}, Lemma~7. \end{pf} To complete Diagram~(\ref{bigd}), define a morphism $\overline{M}(V\times W,\tau)\to{\frak D}(\tau)$ by mapping a stable $(V\times W,\tau)$-map $(C,x,f)$ first to the diagram \[\begin{array}{ccc} (C,x,f) & \longrightarrow & (C,x,{p_W}\mathbin{{\scriptstyle\circ}} f)^{\mbox{\tiny stab}} \\ \ldiag{} & & \\ (C,x,{p_V}\mathbin{{\scriptstyle\circ}} f)^{\mbox{\tiny stab}} & & \end{array}\] and then passing to the underlying prestable curves. \begin{prop}\label{prop5} The diagram \[\comdia{\overline{M}(V\times W,\tau)} {} {\overline{M}(V,{p_V}_{\ast}\tau)\times\overline{M}(W,{p_W}_{\ast}\tau)} {c}{}{a}{{\frak D}(\tau)} {\tilde{\Delta}} {{\frak M}({p_V}_{\ast}\tau)\times{\frak M}({p_W}_{\ast}\tau)}\] is cartesian. \end{prop} \begin{pf} We have to construct a morphism from the fibered product of $\tilde{\Delta}$ and $a$ to $\overline{M}(V\times W,\tau)$. So let there be given a Diagram~(\ref{ddd}), representing an object of ${\frak D}(\tau)(T)$, for a $k$-scheme $T$. Moreover, let there be given families of maps $(f')_{v\in V_{{p_V}_{\ast}\tau}}$, $f_v':C_v'\to V$ and $(f'')_{v\in V_{{p_W}_{\ast}\tau}}$, $f_v'':C_v''\to W$, making $(C',x',f')$ and $(C'',x'',f'')$ stable maps. We need to construct a stable map from $(C,x)$ to $V\times W$. So let $v\in V_{\tau}$ be a vertex of $\tau$. Let us construct a map $h_v:C_v\to W$. In case $v$ is in the image of $V_{{p_W}_{\ast}\tau}\to V_{\tau}$, and $w\mapsto v$ under this map, we take $h_v$ to be the composition $$C_v\stackrel{{p_W}}{\longrightarrow} C_w''\stackrel{f_w''}{\longrightarrow} W.$$ In case $v$ is not in the image of $V_{{p_W}_{\ast}\tau}\to V_{\tau}$, then $v$ partakes in a long edge or a long tail associated to and edge $\{i,\overline{i}\}$ or a tail $i$ of ${p_W}_{\ast}\tau$ (see the discussion of {\em stabilizing morphisms}, Definition~5.7, in \cite{BM} for this terminology). Then we define $f_v:C_v\to W$ to be the composition $$C_v\longrightarrow T\stackrel{x_i''}{\longrightarrow} C_{\partial(i)}''\stackrel{f_{\partial(i)}''}{\longrightarrow} W.$$ In the same manner, construct a map $g_v:C_v\to V$. Finally, let $f_v:C_v\to V\times W$ be the product $g_v\times h_v$. Then the family $(f_v)_{v\in V_{\tau}}$ makes $(C,x,f)$ a stable map over $T$ to $V\times W$. One checks that $(C,x,{p_V}\mathbin{{\scriptstyle\circ}} f)^{\mbox{\tiny stab}}=(C',x',f')$ and $(C,x,{p_W}\mathbin{{\scriptstyle\circ}} f)^{\mbox{\tiny stab}}=(C'',x'',f'')$, using the universal mapping property of stabilization and the fact already alluded to in the proof of Lemma~\ref{lem2}. \end{pf} Let $E^{\scriptscriptstyle\bullet}(V)=E^{\scriptscriptstyle\bullet}(V,{p_V}_{\ast}\tau)$ and $E^{\scriptscriptstyle\bullet}(W)=E^{\scriptscriptstyle\bullet}(W,{p_W}_{\ast}\tau)$ denote the relative obstruction theories for $\overline{M}(V,{p_V}_{\ast}\tau)\to{\frak M}({p_V}_{\ast}\tau)$ and $\overline{M}(W,{p_W}_{\ast}\tau)\to{\frak M}({p_W}_{\ast}\tau)$, respectively, which were defined in \cite{gwi}. As in \cite{BF} Proposition~7.4 there is an induced obstruction theory $E^{\scriptscriptstyle\bullet}(V)\boxplus E^{\scriptscriptstyle\bullet}(W)$ for the morphism $a$. Pulling back via $\tilde{\Delta}$ (as in \cite{BF} Proposition~7.1) we get an induced obstruction theory $\tilde{\Delta}^{\ast}(E^{\scriptscriptstyle\bullet}(V)\boxplus E^{\scriptscriptstyle\bullet}(W))$ for the morphism $c$. On the other hand, we have the relative obstruction theory $E^{\scriptscriptstyle\bullet}(V\times W)=E^{\scriptscriptstyle\bullet}(V\times W,\tau)$ for the morphism $b$. Since $e:{\frak D}(\tau)\to{\frak M}(\tau)$ is \'etale, we may think of $E^{\scriptscriptstyle\bullet}(V\times W)$ as a relative obstruction theory for $c$. \begin{prop} The two relative obstruction theories $\tilde{\Delta}^{\ast}(E^{\scriptscriptstyle\bullet}(V)\boxplus E^{\scriptscriptstyle\bullet}(W))$ and $E^{\scriptscriptstyle\bullet}(V\times W)$ for the morphism $c$ are naturally isomorphic. \end{prop} \begin{pf} Let ${\cal C}(V\times W,\tau)\to\overline{M}(V\times W,\tau)$, ${\cal C}(V,{p_V}_{\ast}\tau)\to\overline{M}(V,{p_V}_{\ast}\tau)$ and ${\cal C}(W,{p_W}_{\ast}\tau)\to\overline{M}(W,{p_W}_{\ast}\tau)$ be the universal curves. Recall from \cite{gwi} that they are constructed by gluing the curves associated to the vertices of a graph according the the edges of that graph. Let us denote the pullbacks of the latter two universal curves to $\overline{M}(V\times W,\tau)$ by ${\cal C}(V)$ and ${\cal C}(W)$, respectively. We have maps $f_{V\times W}$, $f_V$ and $f_W$, constructed from the universal stable maps, which fit into the following commutative diagram: \[\begin{array}{ccccc} V & \stackrel{{p_V}}{\longleftarrow} & V\times W & \stackrel{{p_W}}{\longrightarrow} & W \\ \ldiagup{f_V} & & \rdiagup{f_{V\times W}} & & \rdiagup{f_W} \\ {\cal C}(V) & \stackrel{q_{V}}{\longleftarrow} & {\cal C}(V\times W,\tau) & \stackrel{q_W}{\longrightarrow} & {\cal C}(W) \\ & \sediag{\pi_V} & \rdiag{\pi_{V\times W}} & \swdiag{\pi_W} & \\ && \overline{M}(V\times W,\tau) & & \end{array}\] By base change it is clear that $$\tilde{\Delta}^{\ast}(\dual{E^{\scriptscriptstyle\bullet}(V)}\boxplus\dual{E^{\scriptscriptstyle\bullet}(W)}) = R{\pi_V}_{\ast} f_V^{\ast} T_V\oplus R{\pi_W}_{\ast} f_W^{\ast} T_W.$$ For any vector bundle $F$ on ${\cal C}(W)$ the canonical homomorphism $F\to R{q_W}_{\ast} q_W^{\ast} F$ is an isomorphism. Of course, the same property is enjoyed by $q_V$. Hence we have a canonical isomorphism \begin{eqnarray*} \lefteqn{R{\pi_V}_{\ast} f_V^{\ast} T_V\oplus R{\pi_W}_{\ast} f_W^{\ast} T_W \stackrel{\textstyle\sim}{\longrightarrow} }\\ &&R{\pi_V}_{\ast} R{q_V}_{\ast} q_V^{\ast} f_V^{\ast} T_V\oplus R{\pi_W}_{\ast} R{q_W}_{\ast} q_W^{\ast} f_W^{\ast} T_W \\ & = & R{\pi_{V\times W}}_{\ast} f_{V\times W}^{\ast}(T_V\boxplus T_W) \\ & = & \dual{E^{\scriptscriptstyle\bullet}(V\times W)}. \end{eqnarray*} To conclude, we have a canonical isomorphism $$\tilde{\Delta}^{\ast}(\dual{E^{\scriptscriptstyle\bullet}(V)}\boxplus\dual{E^{\scriptscriptstyle\bullet}(W)}) \stackrel{\textstyle\sim}{\longrightarrow} \dual{E^{\scriptscriptstyle\bullet}(V\times W)}$$ and by dualizing $$E^{\scriptscriptstyle\bullet}(V\times W) \stackrel{\textstyle\sim}{\longrightarrow} \tilde{\Delta}^{\ast}({E^{\scriptscriptstyle\bullet}(V)}\boxplus{E^{\scriptscriptstyle\bullet}(W)}).$$ \end{pf} By this proposition, we have \begin{eqnarray*} J(V\times W) & = & [\overline{M}(V\times W,\tau), E^{\scriptscriptstyle\bullet}(V\times W)] \\ & = & [\overline{M}(V\times W,\tau), \tilde{\Delta}^{\ast}(E^{\scriptscriptstyle\bullet}(V)\boxplus E^{\scriptscriptstyle\bullet}(W))] \\ & = & \tilde{\Delta}^{!} [ \overline{M}(V,{p_V}_{\ast}\tau) \times \overline{M}(W,{p_W}_{\ast}\tau) , E^{\scriptscriptstyle\bullet}(V)\boxplus E^{\scriptscriptstyle\bullet}(W)] \\ & & \quad\quad\mbox{(by \cite{BF} Proposition~7.2)} \\ & = & \tilde{\Delta}^{!} ( [\overline{M}(V,{p_V}_{\ast}\tau),E^{\scriptscriptstyle\bullet}(V)] \times [\overline{M}(W,{p_W}_{\ast}\tau),E^{\scriptscriptstyle\bullet}(W)]) \\ & & \quad\quad\mbox{(by \cite{BF} Proposition~7.4)} \\ & = & \tilde{\Delta}^{!} (J(V)\times J(W)). \end{eqnarray*} So we may now calculate as follows: \begin{eqnarray*} \tilde{\Delta}^{!} (J(V)\times J(W)) & = & a^{\ast}(s\times s)^{\ast} [\Delta] (J(V)\times J(W)) \\ & = & a^{\ast} l_{\ast}[\tilde{\Delta}] (J(V)\times J(W)) \\ & & \quad\quad\mbox{(by Proposition~\ref{podd})} \\ & = & h_{\ast} \tilde{\Delta}^{!}(J(V)\times J(W)) \\ & = & h_{\ast} J(V\times W), \end{eqnarray*} which is the product property. This finishes the proof of Theorem~\ref{pt}. \subsection{Gromov-Witten Transformations} Theorem~\ref{pt} easily implies that the system of Gromov-Witten invariants for $V\times W$ is equal to the tensor product (see \cite{KM}, 2.5) of the systems of Gromov-Witten invariants for $V$ and $W$, respectively. To get the full `operadic' picture, we need a few graph theoretic preparations. \begin{prop} \label{psi} There is a natural functor of categories fibered over $\mbox{$\tilde{\GG}_s(0)$}$ \[\Psi:\mbox{$\tilde{\GG}_s(V\times W)$}_{{\mbox{\tiny cart}}} \longrightarrow \mbox{$\tilde{\GG}_s(V)$}_{{\mbox{\tiny cart}}}\times_{\mbox{$\tilde{\GG}_s(0)$}}\mbox{$\tilde{\GG}_s(W)$}_{{\mbox{\tiny cart}}}.\] This functor is cartesian. \end{prop} \begin{pf} Let $p:V\to W$ be a morphism of smooth projective varieties over $k$. We shall construct a natural functor of fibered categories over $\mbox{$\tilde{\GG}_s(0)$}$ $$\Psi_p:\mbox{$\tilde{\GG}_s(V)$}_{{\mbox{\tiny cart}}}\longrightarrow\mbox{$\tilde{\GG}_s(W)$}_{{\mbox{\tiny cart}}}.$$ This functor will be cartesian. For an object $(\tau,(\overline{a}_i,\tau_i)_{i\in I})$ of $\mbox{$\tilde{\GG}_s(V)$}_{{\mbox{\tiny cart}}}$ let the image under $\Psi_p$ be $(\tau,(\overline{b}_i,p_{\ast}(\tau_i))_{i\in I})$. Here $p_{\ast}(\tau_i)$ is the stabilization of $\tau_i$ covering $p_{\ast}:H_2(V)^+\to H_2(W)^+$. It comes with a natural morphism $\overline{b}_i:p_{\ast}(\tau_i)\to \tau$. To make $\overline{b}_i$ a stabilizing morphism, we have to endow it with an orbit map (see \cite{BM}, Definition~5.7). Let $\overline{a}_i^m:E\t\cup S\t\to E_{\tau_i}\cup S_{\tau_i}$ be the orbit map of $\overline{a}_i:\tau_i\to \tau$. Let $\{f,\overline{f}\}$ be an edge of $\tau$. Then there exists a unique factor $\{f',\overline{f}'\}$ of the long edge of $p_{\ast}(\tau_i)$ associated to $\{f,\overline{f}\}$, such that $\overline{a}_i^m(\{f,\overline{f}\})$ is a factor of the long edge of $\tau_i$ associated to $\{f',\overline{f}'\}$. We set $\overline{b}_i^m(\{f,\overline{f}\})=\{f',\overline{f}'\}$. This defines the orbit map $\overline{b}_i^m$ of $\overline{b}_i$ on edges. On tails it is defined entirely analogously. This achieves the definition of $\Psi_p$ on objects. We leave it to the reader to explicate the action of $\Psi_p$ on morphisms; it boils down to checking that the {\em pullback }of \cite{BM}, Definition~5.8 is compatible with applying $p_{\ast}$. Now we define $\Psi$ by \begin{eqnarray*} \Psi:\mbox{$\tilde{\GG}_s(V\times W)$}_{{\mbox{\tiny cart}}}& \longrightarrow & \mbox{$\tilde{\GG}_s(V)$}_{{\mbox{\tiny cart}}}\times_{\mbox{$\tilde{\GG}_s(0)$}}\mbox{$\tilde{\GG}_s(W)$}_{{\mbox{\tiny cart}}} \\ (\tau,(\tau_i)) & \longmapsto & (\Psi_{{p_V}}(\tau,(\tau_i)),\Psi_{{p_W}}(\tau,(\tau_i))). \end{eqnarray*} \end{pf} Let us denote, for any $V$, the Gromov-Witten transformation for $V$ (see \cite{BM}, Theorem~9.2), by \[I^V:h(V)^{\otimes S}(\chi\dim V)\longrightarrow h(\overline{M}).\] Recall that $I^V$ is a natural transformation between functors \[\mbox{$\tilde{\GG}_s(V)$}_{{\mbox{\tiny cart}}}\longrightarrow(\mbox{graded DMC-motives}),\] where the two functors $h(V)^{\otimes S}(\chi\dim V)$ and $h(\overline{M})$ are induced from functors (with the same names) $\mbox{$\tilde{\GG}_s(0)$}\to(\mbox{DMC-motives})$, constructed in \cite{BM}, Section~9. \begin{numrmk} \label{lem8} By our various definitions we have \[h(V)^{\otimes S}(\chi\dim V)\otimes h(W)^{\otimes S}(\chi\dim W) = h(V\times W)^{\otimes S}(\chi\dim V\times W)\] as functors $\mbox{$\tilde{\GG}_s(0)$}\to(\mbox{\rm DMC-motives})$. \end{numrmk} The transformations $I^V$ and $I^W$ induce a transformation \[I^V\otimes I^W: h(V)^{\otimes S}(\chi\dim V) \otimes h(W)^{\otimes S} (\chi\dim W) \longrightarrow h(\overline{M})\otimes h(\overline{M})\] between functors \[\mbox{$\tilde{\GG}_s(V)$}_{{\mbox{\tiny cart}}}\times_{\mbox{$\tilde{\GG}_s(0)$}}\mbox{$\tilde{\GG}_s(W)$}_{{\mbox{\tiny cart}}}\longrightarrow (\mbox{graded DMC-motives}).\] It is defined as follows. Let $((\tau,(\tau_i)_{i\in I}),(\tau,(\sigma_j)_{j\in J}))$ be an object of $\mbox{$\tilde{\GG}_s(V)$}_{{\mbox{\tiny cart}}}\times_{\mbox{$\tilde{\GG}_s(0)$}}\mbox{$\tilde{\GG}_s(W)$}_{{\mbox{\tiny cart}}}$. The value of $I^V\otimes I^W$ on this object is the morphism \[I^V(\tau,(\tau_i))\otimes I^W(\tau,(\sigma_j)): \]\[ h(V)^{\otimes S\t}(\chi(\tau)\dim V)\otimes h(W)^{\otimes S\t}(\chi(\tau)\dim W) \longrightarrow h(\overline{M}(\tau))\otimes h(\overline{M}(\tau)).\] Composing with $\Delta^{\ast}:h(\overline{M})\otimes h(\overline{M})\to h(\overline{M})$ we get the transformation $$\Delta^{\ast}(I^V\otimes I^W):h(V)^{\otimes S}(\chi\dim V) \otimes h(W)^{\otimes S} (\chi\dim W) \longrightarrow h(\overline{M}),$$ which we shall also denote by $I^V\cup I^W=\Delta^{\ast}(I^V\otimes I^W)$. Pulling back via the functor $\Psi$ of Proposition~\ref{psi} and using Remark~\ref{lem8}, we may think of $I^V\cup I^W$ as a natural transformation \[I^V\cup I^W:h(V\times W)^{\otimes S}(\chi\dim V\times W) \longrightarrow h(\overline{M})\] between functors $\mbox{$\tilde{\GG}_s(V\times W)$}_{{\mbox{\tiny cart}}}\to(\mbox{graded DMC-motives})$. \begin{them} \label{st} We have \[I^V\cup I^W=I^{V\times W}.\] \end{them} \begin{pf} This follows from Theorem~\ref{pt} and the identity principle for DMC-motives, Proposition~8.2 of \cite{BM}. \end{pf}

9710

alg-geom/9710029

en

https://arxiv.org/abs/alg-geom/9710029

alg-geom/9710029

Tomas L. Gomez

Tomas L. Gomez

Irreducibility of the moduli space of vector bundles on surfaces and Brill-Noether theory on singular curves

Revised PhD thesis (Princeton, 1997), 64 pages, 1 figure, LaTeX2e, Xy-pic

We prove the irreducibility of the moduli space of rank 2 semistable torsion free sheaves (with a generic polarization and any value of c_2) on a K3 or a del Pezzo surface. In the case of a K3 surface, we need to prove a result on the connectivity of the Brill-Noether locus for singular curves on the surface. In the case of a del Pezzo surface, we reduce the problem to the case of P^2 by first relating the moduli spaces of the plane and the blown-up plane, and then studying how the moduli space changes when we change the polarization.

alg-geom

\chapter*{Acknowledgments} I would like to thank my advisor, Robert Friedman, for suggesting me this problem, for many discussions and for his encouragement. I am very grateful to him for introducing me to the study of moduli spaces of vector bundles. I consider myself very fortunate for having found his generous help when I was looking for a thesis problem. I would also like to thank Ignacio Sols Lucia, who introduced me to algebraic geometry. With the time that he generously dedicated to teach me, he made possible the transition from my physics undergraduate background to algebraic geometry. Thanks for his patience and his friendship. I have also benefited from discussions with R. Lazarsfeld, R. MacPherson and many other people as well as from many seminars and courses at Princeton. I want to thank the ``Banco de Espa{\~n}a'' for the fellowship ``Beca para la ampliaci{\'o}n de estudios en el extranjero'' that supported my graduate studies. Without their generous support, this thesis couldn't have been made. \chapter{Introduction} In this chapter we will explain the main results of the thesis using as little mathematical background as possible. We will always work over the complex numbers, i.e all manifolds will be complex manifolds. Also we assume that manifolds are projective, i.e. there is an embedding in $\mathbb{P}^n_\mathbb{C}$ (in particular they are K{\"a}hler). $X$ will be a variety (or a manifold with singularities). We will consider holomorphic vector bundles over $X$ (the transition functions are assumed to be holomorphic). To distinguish non-isomorphic vector bundles of fixed rank we can define certain invariants called Chern classes. These are cohomology classes $c_i(V)\in H^{2i}(X,\mathbb{Z})$, $1\leq i\leq \dim_\mathbb{C} X$. We have $c_i(V)=0$ if $i>\operatorname{rank}(V)$. Even after fixing these discrete invariants, we can have continuous families of non-isomorphic vector bundles. I.e., to specify the isomorphism class of a vector bundle it is not enough to fix some discrete invariants, but we have to fix also some continuous parameters. For fixed rank and Chern classes we would like to define a variety called the moduli space of vector bundles. Each point of this variety will correspond to a different vector bundle. In this thesis we will study the irreducibility of this space for rank 2 vector bundles over certain surfaces. Unfortunately, in order to construct this moduli space we have to restrict our attention to \textit{stable} vector bundles (this is not a very strong restriction, since it can be proved that in some sense all vector bundles can be constructed starting from stable ones). There are different notions of stability (see chapter \ref{Preliminaries}). Here we will only discuss \textit{Mumford stability} (also called slope stability). Let $X$ be a projective variety and $H$ an ample divisor. For a vector bundle $V$ we define the slope of $V$ with respect to $H$ as: $$ \mu_H (V) = \frac { c_1(V) H^{n-1}}{\operatorname{rank}(V)} $$ (the product is the intersection product or cup product in cohomology) $V$ is called $H$-stable if for every subbundle $W$ of $V$ we have $$ \mu_H (W) < \mu_H (V). $$ It can be proved that there is a space $\mathfrak{M}^0_H(r,c_i)$, called the moduli space of $H$-stable vector bundles of rank $r$ and Chern classes $c_i$ (if the rank is clear from the context we will drop it from the notation). In many situations it will be a variety (maybe with singularities) but in general it can have a very singular behavior. In general $\mathfrak{M}^0_H(r,c_i)$ is not compact. To get a compact moduli space we need to consider a larger class of objects. Instead of vector bundles we consider torsion free sheaves (they can be thought as ``singular'' vector bundles that fail to be locally a product on a subvariety of X). Also we have to relax the stability condition, and we will consider \textit{Gieseker semistable} sheaves (see chapter \ref{Preliminaries} for the definition). The moduli space of Gieseker semistable torsion free sheaves is compact, and we denote it by $\mathfrak{M}_H(r,c_i)$. There is an important relationship between the theory of holomorphic vector bundles and gauge theory: there is a bijection between the set of $H$-stable vector bundles and differentiable bundles with a Hermite-Einstein connection. If $c_1=0$ and $\dim_\mathbb{C}=2$ then a Hermite-Einstein connection is the same thing as an anti-selfdual (ASD) connection. Note that the metric of the manifold appears in the ASD equation. This is reflected in the fact that the stability condition depends on the polarization. This relationship has been used, for instance, to calculate Donaldson polynomials for the study of the topology of four-manifolds. Now we are going to consider some particular cases. Let $C$ be a curve (i.e., a Riemann surface) of genus $g$. Then the moduli space of rank $r$ vector bundles on $C$ is a smooth variety of dimension $(g-1)r^2+1$. For line bundles (rank=1) over a variety $X$ ($\dim_\mathbb{C} X=n$) the moduli space is called the Jacobian and we have a very explicit description of it. First we note that all line bundles are stable (because they don't have subbundle). Recall that $q=h^1({\mathcal{O}}_X)=b_1/2$, where $b_1$ is the first Betti number (if $X$ is simply connected then $b_1=0$. If $X$ is a curve then $b_1=2g$). The Jacobian $J$ is the moduli space of line bundles with $c_1=0$ and it is of the form $\mathbb{C}^q/\mathbb{Z}^{2q}$, where $\mathbb{Z}^{2q}$ is a lattice in $\mathbb{C}^q$. If $c_1 \neq 0$ then the moduli space $J^{c_1}$ is isomorphic to the Jacobian, but the isomorphism is not canonical. If $X$ is a curve $c_1$ is called the degree $d$. We define some subsets of $J^d$ as follows $$ W^a_d=\{L \in J^d: \dim(H^0(L))=a+1\} $$ The study of the properties of these subsets is called Brill-Noether theory. If the curve is generic, then $W^a_d$ is a subvariety of the expected dimension $\rho(a,d)=g-(a+1)(g-d+a)$. If $\rho(a,d)>0$ then (for any curve) $W^a_d$ is connected. If the curve is singular, one has to consider also torsion free sheaves in order to get a compact moduli space (in this case the moduli space will be singular). This moduli space has been constructed, but little is known about its Brill-Noether theory. In this thesis, to prove the irreducibility of the moduli space of rank 2 vector bundles on a $K3$ surface, we will need the connectivity of $W^a_d$ for certain singular curves that lie in the surface. In chapter \ref{bn}, theorem I, we will prove that if $\rho(a,d)>0$, $W^a_d$ is still connected for singular curves satisfying certain conditions. If $X$ is a complex surface ($\dim_\mathbb{C} X=2$), then in general the moduli space of vector bundles of rank $r$ is very singular, but in many situations it will be a variety (maybe with singularities) of the expected dimension $$ \dim \mathfrak{M}^0_H(r,c_1,c_2)=2rc_2-(r-1)c_1^2-(r^2-1)\chi({\mathcal{O}}_X)+h^1({\mathcal{O}}_X). $$ By a slight abuse of language we have denoted by $c_2$ the integral of the second Chern class on the variety $\int_X c_2(V)$, and by $c_1^2$ the integral $\int_X c_1(V) \wedge c_1(V)$. Recall that $\chi({\mathcal{O}}_X)=1-h^1({\mathcal{O}}_X)+h^2({\mathcal{O}}_X)$. In this thesis we will consider the case rank=2. For rank 2, $\mathfrak{M}^0_H(c_1,c_2)$ is known to be irreducible and of the expected dimension if $c_2>N$, where $N$ is a constant that depends on $X$, $c_1$ and $H$. In chapter \ref{k3} we present a new proof of the theorem of O'Grady \smallskip\noindent \textbf{Theorem II.} \textit{Let $X$ be a projective $K3$ surface (a $K3$ surface is a simply connected complex surface with $c_1(T_X)=0$, where $T_X$ is the tangent bundle). Let $H$ be a generic polarization (see chapter \ref{Preliminaries} for a definition). Assume that $c_1$ is a nonzero primitive element of $H^2(X,\mathbb{Z})$ (under this condition $\mathfrak{M}^0_H(c_1,c_2)$ is smooth of the expected dimension).} \textit{Then $\mathfrak{M}^0_H(c_1,c_2)$ is irreducible.} \smallskip We should note that in chapter \ref{k3} we work with the moduli space of Gieseker semistable torsion free sheaves $\mathfrak{M}_H(c_1,c_2)$. But it can be shown that under the conditions of the theorem the points of $\mathfrak{M}_H(c_1,c_2)$ that are not in $\mathfrak{M}^0_H(c_1,c_2)$ are in a subvariety of positive codimension, then the irreducibility of one of them is equivalent to the irreducibility of the other. Using the relationship of holomorphic vector bundles with gauge theory this theorem can be stated as follows: \begin{theorem} Let $X$ be a projective $K3$ surface with a generic K{\"a}hler metric $g$. Then the moduli space of $SO(3)$ anti-selfdual connections (with fixed instanton number $k$ and second Stiefel-Whitney class $w_2$) is smooth and connected. \end{theorem} In chapter \ref{dp} we study moduli spaces of vector bundles on \textit{del Pezzo} surfaces. A del Pezzo surface is a surface whose anticanonical bundle is ample. It can be shown that these are all the del Pezzo surfaces: $\mathbb{P}^2$, $\mathbb{P}^1\times\mathbb{P}^1$, or the projective plane $\mathbb{P}^2$ blown up at most at 8 generic points. For rank 2 and fixed Chern classes, the moduli space of stable vector bundles is known to be empty or irreducible for $X=\mathbb{P}^2$ or $\mathbb{P}^1\times\mathbb{P}^1$. In chapter \ref{dp} we prove this result for a projective plane $\mathbb{P}^2$ blown up at most at 8 generic points, and then we get the general result: \smallskip\noindent \textbf{Theorem III.} \textit{Let $X$ be a del Pezzo surface with a generic polarization $H$. Fix some Chern classes $c_1$ and $c_2$. Then $\mathfrak{M}^0_H(c_1,c_2)$ is irreducible (or empty).} \smallskip As in the $K3$ case, in this case irreducibility is equivalent to connectedness, and we have the same result for $\mathfrak{M}_H(c_1,c_2)$. We can also translate this theorem into gauge theory language: \begin{theorem} Let $X$ be a del Pezzo surface with a generic K{\"a}hler metric $g$. Then the moduli space of $SO(3)$ anti-selfdual connections (with fixed Stiefel-Whitney class $w_2$ and instanton number $k$) is connected (or empty). The same is true for $SU(2)$ anti-selfdual connections with fixed instanton number $k$. \end{theorem} \chapter{Preliminaries} \label{Preliminaries} \section{Brill-Noether theory} Let $C$ be a smooth curve of genus $g$ (we will always assume that the base field is $\mathbb{C}$), $J(C)$ its Jacobian, and $W^r_d(C)$ the Brill-Noether locus corresponding to line bundles $L$ of degree $d$ and $h^0(L) \geq r+1$ (see \cite{ACGH}). The expected dimension of this subvariety is $\rho(r,d)= g - (r+1)(g-d+r)$. Fulton and Lazarsfeld \cite{F-L} proved that $W^r_d(C)$ is connected when $\rho > 0$. We are going to generalize this result for certain singular curves, but before stating our result (theorem I), we need to recall some concepts. Let $C$ be an integral curve (not necessarily smooth). We still have a generalized Jacobian $J(C)$, defined as the variety parametrizing line bundles, but it will not be complete in general. Define the degree of a rank one torsion-free sheaf on $C$ to be $$ \deg(A)=\chi(A)+p_a-1, $$ where $p_a$ is the arithmetic genus of $C$. One can define a scheme $\overline J^d(C)$ parametrizing rank one torsion-free sheaves on $C$ of degree $d$ (see \cite{AIK}, \cite {D}, \cite {R}). If $C$ lies on a surface, then $\overline J^d(C)$ is integral, and furthermore the generalized Jacobian $J(C)$ is an open set in $\overline J^d(C)$, and then $\overline J^d(C)$ is a natural compactification of $J(C)$. We will need to consider families of sheaves parametrized by a scheme $T$, and furthermore the curve will vary as we vary the parameter $t\in T$. All this can be done using a relative version of $\overline J^d(C)$, but we will proceed in a different way. We will use the fact that all these curves are going to lie on a fixed surface $S$. Then we will think of the coherent sheaves on $C$ as torsion sheaves on $S$ (all sheaves in this paper will be coherent). To define precisely which sheaves we will consider we need some notation. For any sheaf $F$ on $S$, let $d(F)$ be the dimension of its support. We say that $F$ has pure dimension $n$ if for any subsheaf $E$ of $F$ we have $d(E)=d(F)=n$. Note that if the support is irreducible, then having pure dimension $n$ is equivalent to being torsion-free when considered as a sheaf on its support. The following theorem follows from \cite[theorem 1.21]{S}. \smallskip \begin{theorem}[Simpson] Let $C$ be an integral curve on a surface $S$. Let $\overline {\mathcal{J}}^d_{|C|}$ be the functor which associates to any scheme $T$ the set of equivalence classes of sheaves ${\mathcal{A}}$ on $S\times T$ with (a) ${\mathcal{A}}$ is flat over $T$. (b) The induced sheaf $A_t$ on each fiber $S\times \{t\}$ has pure dimension 1, and its support is an integral curve in the linear system $|C|$. (c) If we consider $A_t$ as a sheaf on its support, it is torsion-free and has rank one and degree $d$. Sheaves ${\mathcal{A}}$ and ${\mathcal{B}}$ are equivalent if there exists a line bundle $L$ on $T$ such that ${\mathcal{A}} \cong {\mathcal{B}} \otimes p_T^*L$, where $p_T:S \times T \to T$ is the projection on the second factor. Then there is a coarse moduli space that we also denote by $\overline {\mathcal{J}}^d_{|C|}$. I.e., the points of $\overline {\mathcal{J}}^d_{|C|}$ correspond to isomorphism classes of sheaves, and for any family ${\mathcal{A}}$ of such sheaves parametrized by $T$, there is a morphism $$ \phi :T \to \overline {\mathcal{J}}^d_{|C|} $$ such that $\phi(t)$ corresponds to the isomorphism class of $A_t$. \end{theorem} Note that $\overline{\mathcal{J}}^d_{|C|}$ parametrizes pairs $(C',A)$ with $C'$ an integral curve linearly equivalent to $C$ and $A$ a torsion-free rank one sheaf on $C$. We denote by $\pi :\overline{\mathcal{J}}^d_{|C|} \to U \subset |C|$ the obvious projection giving the support of each sheaf, where $U$ is the open subset of $|C|$ corresponding to integral curves. A family of curves on a surface $S$ parametrized by a curve $T$ is a subvariety ${\mathcal{C}} \subset S \times T$, flat over $T$, such that the fiber ${\mathcal{C}}|_t=C_t$ over each $t\in T$ is a curve on $S$. Analogously, a family of sheaves on a surface $S$ parametrized by a curve $T$ is a sheaf ${\mathcal{A}}$ on $S\times T$, flat over $T$. For each $t\in T$ we will denote the corresponding member of the family by $A_t={\mathcal{A}}|_t.$ Altman, Iarrobino and Kleiman \cite{AIK} proved the following theorem \begin{theorem}[Altman--Iarrobino--Kleiman] With the same notation as before, $\overline {\mathcal{J}}^d_{|C|}$ is flat over $U$ and its geometric fibers are integral. The subset of $\overline {\mathcal{J}}^d_{|C|}$ corresponding to line bundles (i.e., the relative generalized Jacobian) is open and dense in $\overline {\mathcal{J}}^d_{|C|}$. \end{theorem} We also consider the family of generalized Brill-Nother loci $\overline {\mathcal{W}}^r_{d,|C|} \subset \overline {\mathcal{J}}^d_{|C|}$, and the projection $q:\overline{\mathcal{W}}^r_{d,|C|} \to U$. We can define the generalized Brill-Noether locus $\overline W^r_d(C)$ as the set of points in $\overline J^d(C)$ corresponding to sheaves $A$ with $h^0(A) \geq r+1$ (note that it is complete because of the upper semicontinuity of $h^0(\cdot)$). There is also a determinantal description that gives a scheme structure. This description is a straightforward generalization of the description for smooth curves (see \cite {ACGH}), but we are only interested in the connectivity of $\overline W^r_d(C)$, so we can give it the reduced scheme structure. We will consider curves that lie on a surface $S$ with the following property: \bigskip $h^1(\SO_S)=0$ \textit{, and }$-K_S$ \textit{ is generated by global sections.}\hfill (*) \bigskip We will need this condition to prove proposition \ref{bn1.5}. For instance, $S$ can be a K3 surface or a del Pezzo surface with $K_S^2\neq 1$. Now we can state the theorem that we are going to prove in chapter \ref{bn}. \smallskip \noindent\textbf{Theorem I.} \textit{ Let $C$ be a reduced irreducible curve of arithmetic genus $p_a$ that lies in a surface $S$ satisfying \textup{(*)}. Let $\overline J^d(C)$, $d>0$, be the compactification of the generalized Jacobian. Then for any $r \geq 0$ such that $\rho(r,d)= p_a - (r+1)(p_a-d+r)>0$, the generalized Brill-Noether subvariety $\overline W^r_d(C)$ is nonempty and connected.} \smallskip \section{Moduli space of torsion free sheaves} Let $X$ be a smooth projective variety over $\mathbb{C}$ of dimension $n$. We are interested in constructing a moduli space that will parametrize torsion free sheaves on $X$, with some fixed rank $r$ and Chern classes $c_i$. To do so, we have to introduce some notion of stability. \begin{definition}\textup{\textbf{(Mumford-Takemoto stability)}} Fix a polarization $H$. For any nonzero torsion free sheaf $F$ we define the slope of $F$ with respect to $H$ as $$ \mu_H(F)={\frac {c_1(F)\cdot H^{n-1}} {\operatorname{rk} (F)}} $$ We will say that a torsion free sheaf $V$ on $X$ is Mumford H-stable (resp. semistable) if for every nonzero subsheaf $W$ of $V$ we have $$ \mu_H(W) < \mu_H(V) \ \ \ (\text{resp. }\leq ). $$ \end{definition} Using this definition one can construct the coarse moduli space of Mumford stable vector bundles. This notion of stability turns out to have an analog in differential geometry: Mumford stable holomorphic vector bundles are in one to one correspondence with differentiable vector bundles with a Hermite-Einstein connection (the choice of a polarization is replaced by the choice of a Riemannian metric). This correspondence was proved by Narasimhan and Seshadri \cite{N-S} if $X$ is a curve, by Donaldson \cite{Do1,D-K} for a surface, and it was then generalized for any dimension by \cite{U-Y}. This correspondence has been useful to calculate differentiable invariants of 4-manifolds (the so called Donaldson invariants). The moduli space of Mumford stable sheaves is in general not compact. To define a compactification we have to introduce a refined notion of stability. \begin{definition}\textup{\textbf{(Gieseker stability).}} Fix a polarization $H$. For any nonzero torsion free sheaf $F$ define the Hilbert polynomial of $F$ with respect to $H$ as $$ p_H(F)(n) =\frac {\chi(F \otimes {\mathcal{O}}_X(nH))}{rank(F)}. $$ Given two polynomials $f$ and $g$, we will write $f\prec g$ (resp. $\preceq$) if $f(n)<g(n)\ ($resp. $\leq)$ for $n\gg0$. We will say that $V$ is Gieseker stable (resp. semistable) if for every nonzero torsion free subsheaf $W$ we have $$ p_H(W) \preceq p_H(V) \ \ \ (\text{resp. }\preceq ). $$ \end{definition} Using Hirzebruch-Riemann-Roch theorem we can see that Gieseker semistability implies Mumford semistability, and Mumford stability implies Gieseker stability. In order to have a separated moduli space it is not enough to consider isomorphism classes of sheaves. We will introduce the notion of S-equivalence. For any Gieseker semistable sheaf $V$ there is a filtration (\cite{G,Ma}) $$ 0 =V_0 \subseteq V_1\subseteq \cdots \subseteq V_t=V $$ such that $V_i/V_{i-1}$ is stable and $p_H(V_i)=p_H(V)$. We define $\operatorname{gr} (V)=\oplus (V_i/V_{i-1})$. It can be proved that $\operatorname{gr}(V)$ doesn't depend on the filtration chosen. We will say that $V$ is S-equivalent to $V'$ if $\operatorname{gr}(V)=\operatorname{gr}(V')$. There is another characterization of S-equivalence that is more illuminating from the point of view of moduli problems. Assume that we have a family of Gieseker semistable sheaves parametrized by a curve $T$. I.e., we have a sheaf ${\mathcal{V}}$ on $X \times T$, flat over $T$ inducing torsion free Gieseker stable sheaves ${\mathcal{V}}|_t$ on the slices $X \times \{t\}$. Assume that for one point $0\in T$ we have ${\mathcal{V}}|_0\cong V$ and for the rest of the points ${\mathcal{V}}|_t$ is isomorphic to some other fixed $V'$. We will say that $V$ and $V'$ are equivalent. The equivalence relation generated by this definition is S-equivalence. It can be proved that there is a coarse moduli space for S-equivalence classes of Gieseker semistable torsion free sheaves (with fixed rank and Chern classes). This moduli space is projective. In general the moduli space can be very singular, but if $X$ is a surface and for fixed rank $r$, $c_1$ and polarization, it is known that for $c_2$ large enough the singular locus is a proper subset of positive codimension of the moduli space \cite{Do2,F,Z,G-L2}. The moduli space has the expected dimension $$ 2rc_2-(r-1)c_1^2-(r^2-1)\chi({\mathcal{O}}_X), $$ and is irreducible \cite{G-L1,G-L2,O1,O2}. In the rank two case it is also known, again for $c_2$ large enough, that the moduli space is normal and has local complete intersection singularities at points corresponding to stable sheaves, and if the surface $X$ is of general type (with some technical conditions), then also the moduli space is of general type \cite{L2}. It is natural to ask what is the effect of changing the choice of polarization. From now on we will assume that $X$ is a surface $S$ and that the rank is 2 unless otherwise stated. We will denote the moduli space of rank 2 torsion free sheaves that are Gieseker semistable with respect to the polarization $H$ by $\mathfrak{M}_H(c_1,c_2)$. \begin{definition} Fix $S$, the first and second Chern classes $c_1$, $c_2$. Let $\zeta$ be some class in $H^2(S,\mathbb{Z})$ with $$ \zeta\equiv c_1\ (\text{mod}\ 2),\ \ c_1^2-4c_2\leq \zeta^2 <0. $$ The wall of type $(c_1,c_2)$ associated to $\zeta$ is a hyperplane of $H^2(S,\mathbb{Q})$ with nonempty intersection with the ample cone $\Omega_S$ $$ W^\zeta=\{x\in \Omega_S |\ x \cdot \zeta =0\}. $$ The connected components of the complement of the walls in the ample cone are called chambers. \smallskip A polarization $H$ is called $(c_1,c_2)$-generic if it doesn't lie on a wall (i.e., it lies in a chamber). \end{definition} The walls of type $(c_1,c_2)$ are known to be locally finite on the ample cone \cite{F-M}. If a polarization is $(c_1,c_2)$-generic it is easy to see that Mumford and Gieseker stability coincide, and furthermore there are no strictly semistable sheaves. Stable sheaves are \textit{simple} ($\operatorname{Hom}(V,V)=\mathbb{C}$). If $-K_S$ is effective then this fact and the Kuranishi local model for the moduli space proves that the moduli space is smooth of the expected dimension (if not empty) \cite{F}. If $H_1$ and $H_2$ are two generic polarizations in the same chamber, then every sheaf that is $H_1$-stable is also $H_2$-stable, and we can identify the corresponding moduli spaces \cite{F,Q1,Q2}. If we restrict our attention to some particular class of algebraic surfaces we can obtain more properties of the moduli space (without the condition on $c_2$). The moduli space of sheaves on a $K3$ surface with a generic polarization has been studied by Mukai \cite{M} when the expected dimension is 0 or 2. In particular he proved that the moduli space is irreducible. O'Grady \cite{O3} has proved irreducibility for any $c_2$ (and any rank), as well as having obtained results about the Hodge structure. In chapter \ref{k3} (theorem II), we give a new proof of the irreducibility for any $c_2$ (and rank 2) based on our results about Brill-Noether theory on singular curves. Now let's consider the case $S=\mathbb{P}^2$ and rank 2. Tensoring with a line bundle we can assume that $c_1$ is either $0$ or $1$. If $c_1=0$, then the moduli space is empty for $c_2<2$ and is irreducible of the expected dimension for $c_2\geq2$. If $c_1=1$ then the moduli space is empty for $c_2<1$ and irreducible of the expected dimension for $c_2\geq1$. In the case $S=\mathbb{P}^1\times\mathbb{P}^1$ (and also rank 2), if we take a $(c_1,c_2)$-generic polarization, the moduli space is also known to be either empty or irreducible. We will generalize this result for any del Pezzo surface (i.e., a surface with $-K_S$ ample) in chapter \ref{dp} (theorem III). \chapter{Connectivity of Brill-Noether loci for singular curves} \label{bn} \setcounter{proposition}{0} Recall (see chapter \ref{Preliminaries}) that we are going to study the Brill-Noether locus of singular irreducible curves that lie on a smooth surface $S$ satisfying \bigskip $h^1(\SO_S)=0$ \textit{, and }$-K_S$ \textit{ is generated by global sections.}\hfill (*) \bigskip Now we state the theorem that we are going to prove: \smallskip \noindent\textbf{Theorem I.} \textit{ Let $C$ be a reduced irreducible curve of arithmetic genus $p_a$ that lies in a surface $S$ satisfying \textup{(*)}. Let $\overline J^d(C)$, $d>0$, be the compactification of the generalized Jacobian. Then for any $r \geq 0$ such that $\rho(r,d)= p_a - (r+1)(p_a-d+r)>0$, the generalized Brill-Noether subvariety $\overline W^r_d(C)$ is nonempty and connected.} \smallskip \begin{remark} \label{remark} \textup{If $r \leq d-p_a$, by Riemann-Roch inequality we have $\overline W^r_d(C) = \overline J^d(C)$, and this is connected. Then, in order to prove theorem I we can assume $r>d-p_a$. Note that if $A$ corresponds to a point in $\overline W^r_d(C)$ with $r>d-p_a$, then by Riemann-Roch theorem $h^1(A) > 0$.} \end{remark} \bigskip \textbf{Outline of the proof of theorem I} \bigskip Note that $\overline W^r_d(C)$ is the fiber of $q$ over the point $u_0 \in |C|$ corresponding to the curve $C$. Let $U$ be the open subset of $|C|$ corresponding to integral curves, and $V$ the subset of smooth curves. Define $(\overline {\mathcal{W}}^r_d)_V$ to be the Brill-Noether locus of sheaves with smooth support, i.e. $(\overline{\mathcal{W}}^r_d)_V=q^{-1}(V)$. By \cite{F-L}, the restriction $q^{}_V: (\overline {\mathcal{W}}^r_d)_V \to V$ has connected fibers. We want to use this fact to show that $\overline W^r_d(C)$ is connected. Let $A$ be a rank one torsion-free sheaf on $C$ corresponding to a point in $\overline W^r_d(C)$, and assume that it is generated by global sections. We think of $A$ as a torsion sheaf on $S$. Then we have a short exact sequence on $S$ $$ 0 \to E \stackrel{f_0} \to H^0(A) \otimes \SO_S \to A \to 0, $$ where the map on the right is evaluation. This sequence has already appeared in the literature (see \cite{La}, \cite{Ye}). Our idea is to deform $f_0$ to a family $f_t$. The cokernel of $f_t$ will define a family of sheaves $A_t$ with $h^0(A_t) \geq h^0(A)$ (because $h^0(E)=0$), and then for each $t$ the point in $\overline {\mathcal{J}}^d_{|C|}$ corresponding to $A_t$ lies in $\overline {\mathcal{W}}^r_{d,|C|}$. Assume that there are 'enough' homomorphisms from $E$ to $H^0 \otimes \SO_S$ and the family $f_t$ can be chosen general enough, so that for a general $t$, the support of $A_t$ is smooth (the details of this construction are in section \ref{bnParticular case}). The family $A_t$ shows that the point in $\overline W^r_d(C)$ corresponding to $A$ is in the closure of $(\overline {\mathcal{W}}^r_d)_V$ in $\overline {\mathcal{J}}^d_{|C|}$. It can be shown that this closure has connected fibers. Let $X$ be the fiber over $u_0$ of this closure. Then all sheaves for which this construction works are in the connected component $X$ of $\overline W^r_d(C)$. If this could be done for all sheaves in $\overline W^r_d(C)$ this would finish the proof, but there are sheaves for which this construction doesn't work. For these sheaves we show in section \ref{bnGeneral case} that they can be deformed (keeping the support $C$ unchanged) to a sheaf for which a refinement of this construction works. This shows that all points in $\overline W^r_d(C)$ are in the connected component $X$. \section{The main lemma} \label{bnMain lemma} The precise statement that we will use to prove theorem I is the following lemma. \begin{lemma} \label{bn0.2} Let $C$ be an integral complete curve in a surface $S$. Assume that for each rank one torsion-free sheaf $A$ on $C$ with $h^0(A)=r+1>0$ and $\deg(A)=d>0$ such that $\rho(r,d)>0$ we have the following data: \smallskip (a) A family of curves ${\mathcal{C}}$ in $S$ parametrized by an irreducible curve $T$ (not necessarily complete). (b) A connected curve $T'$ (not necessarily irreducible nor complete) with a map $\psi:T' \to T$. (c) A rank one torsion-free sheaf ${\mathcal{A}}$ on ${\mathcal{C}}' = {\mathcal{C}} \times _T T'$, flat over $T'$, inducing rank one torsion-free sheaves on the fibers of ${\mathcal{C}}' \to T'$. \smallskip Assume that the following is satisfied: \smallskip (i) ${\mathcal{C}}|_{t^{}_0} \cong C$ for some $t^{}_0 \in T$, ${\mathcal{C}}|_t$ is linearly equivalent to $C$ for all $t\in T$, and ${\mathcal{C}}|_t$ is smooth for $t \neq t^{}_0$. (ii) One irreducible component of $T'$ is a finite cover of $T$, and the rest of the components of $T'$ are mapped to $t^{}_0 \in T$. (iii) ${\mathcal{A}}|_{t'_0} \cong A$ for some $t'_0 \in T'$ mapping to $t^{}_0 \in T$. (iv) $h^0({\mathcal{A}}|_{t'}) \geq r+1$ for all $t' \in T'$. \smallskip Then the generalized Brill-Noether subvariety $\overline W^r_d(C)$ of the compactified generalized Jacobian $\overline J^d(C)$ is connected. \end{lemma} \begin{proof} We will use the notation introduced in the previous section. The map $q:\overline {\mathcal{W}}^r_{d,|C|} \to U$ is a projective morphism. Recall that $\overline W^r_d(C)$ is the fiber of $q$ over $u_0$, where $u_0$ is the point corresponding to $C$. By \cite{F-L} the morphism $q$ has connected fibers over $V$, thus a general fiber of $q$ is connected, and we want to prove that the fiber over $u_0 \in U$ is also connected. $$ \begin{array}{ccc} \overline W^r_d(C) & \hookrightarrow & \overline {\mathcal{W}}^r_{d,|C|} \\ \fcndown{} & {} & \fcndown{q} \\ u_0 & \hookrightarrow & U \end{array} $$ Let $\overline {\mathcal{W}}^r_{d,|C|} \stackrel{q'} \to U' \stackrel{g} \to U$ be the Stein factorization of $q$ (see \cite[III Corollary 11.5]{H}), i.e. $q'$ has connected fibers and $g$ is a finite morphism. A general fiber of $q$ is connected, and then $U'$ has one irreducible component $Z$ that maps to $U$ birationally. The subset $U$ is open in $|C|$ and hence normal, the restriction $g|_Z:Z \to U$ is finite and birational, $Z$ and $U$ are integral, thus by Zariski's main theorem (see \cite[III Corollary 11.4]{H}) each fiber of $g|_Z$ consists of just one point. Let $z_0$ be the point of $Z$ in the fiber $g^{-1}(u_0)$. \textbf{\textit{Claim.}} Let $y_0$ be a point in the fiber $q^{-1}(u_0)= \overline W^r_d(C)$. Then $y_0$ is mapped by $q'$ to $z_0$. This claim implies that that $\overline W^r_d(C)$ is connected. Now we will prove the claim. Let $A$ be the sheaf on $S$ corresponding to the point $y_0$. Let $T'$, $T$, $t'_0\in T'$, $t^{}_0\in T$, $\psi:T' \to T$ be the curves points and morphism given by the hypothesis of the lemma. Let $\phi:T' \to \overline{\mathcal{J}}^d_{|C|}$ be the morphism given by the universal property of the moduli space $\overline{\mathcal{J}}^d_{|C|}$. Item (iv) imply that the image of $\phi$ is in $\overline {\mathcal{W}}^r_{d,|C|}$. \centerline{ \xymatrix{ {}\save[]+<0.7cm,0cm>*{t'_0\in} \restore & T' \ar[d]_\psi \ar[r]^\phi & \overline{\mathcal{W}}^r_{d,|C|} \ar[d]^{q'} \ar@(dl,ul)[dd]_q & {}\save[]+<-0.7cm,0cm>*{\ni y_0} \restore\\ {}\save[]+<0.7cm,0cm>*{t^{}_0\in} \restore& T & U' \ar[d]^{g}& Z \ar@{_{(}->}[l] \ar[dl]^{g|_Z} & {}\save[]-<0.6cm,0cm>*{\ni z^{}_0} \restore\\ & & U & {}\save[]+<-0.8cm,0cm>*{\ni u^{}_0} \restore} } \begin{figure}[ht] \centerline{\epsfig{file=figure.eps,height=3in,width=4.5in, bbllx=0in,bblly=9.187in,bburx=4.5in,bbury=12.187in,clip=}} \end{figure} The restriction of $q' \circ \phi$ to $T' \setminus \psi^{-1}(t^{}_0)$ maps to $Z$, because for $t' \in T' \setminus \psi^{-1}(t^{}_0)$ the sheaf ${\mathcal{A}}|_{t'}$ has smooth support by item (i). Items (c) and (i) imply that $g\circ q' \circ \phi (\psi^{-1}(t^{}_0))=u_0$ . Thus $q' \circ \phi (\psi^{-1}(t^{}_0))$ is a finite number of points (because it is in the fiber of $g$ over $u_0$). The facts that $q' \circ \phi(T' \setminus \psi^{-1}(t^{}_0))$ is in $Z$ and that $q' \circ \phi (\psi^{-1}(t^{}_0))$ is a finite number of points imply that $q' \circ \phi (\psi^{-1}(t^{}_0))$ is also in $Z$ (because by item (b) the curve $T'$ is connected and thus also its image under $q'\circ \phi$), and in fact $q' \circ \phi (\psi^{-1}(t^{}_0))=z_0$ because $q' \circ \phi (\psi^{-1}(t^{}_0))$ is in the fiber of $g$ over $u_0$. By item (ii), $t'_0 \in \psi^{-1}(t^{}_0)$. Then $q' \circ \phi (t'_0)=z_0$, and by item (iii) we have $y_0=\phi(t'_0)$, then $q'(y_0)= q'(\phi(t'_0))=z_0$ and the claim is proved. \end{proof} In section \ref{bnParticular case} we will construct this family under some assumptions on $A$ (proposition \ref{bn1.5}), and in section \ref{bnGeneral case} we will show how to use that to construct a family for any $A$. Note that because of remark \ref{remark} we can assume $h^1(A)>0$. \section{A particular case} \label{bnParticular case} Given a rank one torsion-free sheaf $A$ on an integral curve lying on a surface $S$, we define another sheaf $A^*$ that is going to be some sort of dual. Let $j$ be the inclusion of the curve $C$ in the surface $S$. We define $A^*$ as follows: $$ A^*= Ext^1(j_*A,\omega_S). $$ The operation $A \to A^*$ is a contravariant functor. Note that the support of $A^*$ is $C$. It will be clear from the context when we are referring to $A^*$ as a torsion sheaf on $S$ or as a sheaf on $C$. In the case in which $A$ is a line bundle, then $A^*=A^\vee \otimes \omega_C$. Now we prove some properties of this ``dual''. \begin{lemma} \label{bn1.1} Let $A$ be a rank one torsion-free sheaf on an integral curve lying on a surface. Then $A^{**}=A$ \end{lemma} \begin{proof} First observe that if $L$ is a line bundle on $C$, then $(A \otimes L)^* \cong A^* \otimes L^\vee$. To see this, take an injective resolution of $\omega_S$ $$ 0 \to \omega_S \to {\mathcal{I}}_0 \to {\mathcal{I}}_1 \to \cdots $$ Now we use this resolution to calculate the $Ext$ sheaf. $$ Ext^1(A\otimes L,\omega_S) = h^1(Hom(A \otimes L,{\mathcal{I}}_\bullet))= h^1(L^\vee\otimes Hom(A,{\mathcal{I}}_\bullet)) = $$ $$ =L^\vee\otimes h^1 (Hom(A,{\mathcal{I}}_\bullet))=L^\vee\otimes Ext^1(A,\omega_S) $$ The third equality follows from the fact that $Hom(A,{\mathcal{I}}_\bullet)$ is supported on the curve and $L^\vee$ is locally free It follows that $(L\otimes A)^{**}\cong L\otimes A^{**}$, and then proving the lemma for $A$ is equivalent to proving it for $L\otimes A$. Multiplying with an appropriate very ample line bundle, we can assume that $A$ is generated by global sections. Then we have an exact sequence \begin{equation} 0 \to E \to V \otimes \SO_S \to A \to 0, \label{eqbn1.1} \end{equation} where $V=H^0(A)$. The following lemma proves that $E$ is locally free. \begin{lemma} \label{bn1.2} Let $M$ be a torsion-free sheaf on an integral curve $C$ that lies on a smooth surface $S$. Let $j:C \to S$ be the inclusion. Let $F$ be a locally free sheaf on the surface. Let $f:F \to j_* M$ be a surjection. Then the elementary transformation $F'$ of $F$, defined as the kernel of $f$ \begin{equation} \label{eqbn1.1bis} 0 \to F' \to F \stackrel{f}{\to} j_* M \to 0, \end{equation} is a locally free sheaf. \end{lemma} \begin{proof} $M$ is torsion-free sheaf on $C$, and then $j_* M$ has depth at least one, and because $S$ is smooth of dimension 2, this implies that the projective dimension of $j_* M$ is at most one ($Ext^i(j_* M,{\mathcal{O}}_S)=0$ for $i \geq 2$). Now $Ext^i(F,{\mathcal{O}}_S)=0$ for $i \geq 1$ because $F$ is locally free, and then from the exact sequence \ref{eqbn1.1bis}, we get $$ 0 \to Ext^i(F',\SO_S) \to Ext^{i+1}(j_* M,\SO_S) \to 0, \quad i\geq 1, $$ and then $Ext^i(F',{\mathcal{O}}_S)=0$ for $i \geq 1$, and this implies that $F'$ is locally free. \end{proof} In particular, $E^{\vee\vee}=E$. Applying the functor $Hom(\cdot,\omega_S)$ twice to the sequence \ref{eqbn1.1}, we get $$ 0 \to E \to V \otimes \SO_S \to A^{**} \to 0. $$ Comparing with \ref{eqbn1.1} we get the result (because the map on the left is the same for both sequences). \end{proof} \begin{lemma} \label{bn1.3} $\operatorname{Ext}^1(A,\omega_S) \cong H^0(A^*)$, and this is dual to $H^1(A)$. \end{lemma} \begin{proof} The local to global spectral sequence for Ext gives the following exact sequence $$ 0 \to H^1(Hom(A,\omega_S)) \to \operatorname{Ext}^1(A,\omega_S) \to H^0(A^*) \to H^2(Hom(A,\omega_S)) $$ But $Hom(A,\omega_S)=0$ because $A$ is supported in $C$ and then the first and last terms in the sequence are zero and we have the desired isomorphism. \end{proof} Now we will prove a lemma that we will need. The proof can also be found in \cite{O}, but for convenience we reproduce it here. \begin{lemma} \label{bn1.4} Let $E$ and $F$ be two vector bundles of rank $e$ and $f$ over a smooth variety $X$. Assume that $E^\vee \otimes F$ is generated by global sections. If $\phi: E \to F$ is a sheaf morphism, we define $D_k(\phi)$ to be the subset of X where $\operatorname{rk}(\phi_x) \leq k$ (there is an obvious determinantal description of $D_k(\phi)$ that gives a scheme structure). Let $d_k$ be the expected dimension of $D_k(\phi)$ $$ d_k=\dim (X) - (e-k)(f-k). $$ Then there is a Zariski dense set $U$ of $\operatorname{Hom}(E,F)$ such that if $\phi\in U$, then we have that $D_k(\phi) \setminus D_{k-1}(\phi)$ is smooth of the expected dimension (if $d_k < 0$ then it will be empty). \end{lemma} \begin{proof} Let $M_k$ be the set of matrices of dimension $e \times f$ and of rank at most $k$ (there is an obvious determinantal description that gives a scheme structure to this subvariety). It is well known that the codimension of $M_k$ in the space of all matrices is $(e-k)(f-k)$, and that the singular locus of $M_k$ is $M_{k-1}$. Now, because $E^\vee \otimes F$ is generated by global sections, we have a surjective morphism $$ H^0(E^\vee \otimes F) \otimes {\mathcal{O}}_X \to E^\vee \otimes F $$ that gives a morphism of maximal rank between the varieties defined as the total space of the previous vector bundles $$ p: X \times H^0(E^\vee \otimes F) \to \mathbb{V} (E^\vee \otimes F). $$ Define $\Sigma_k \subset \operatorname{V}(E^\vee \otimes F)$ as the set such that $\operatorname{rk}(\phi_x) \leq k$. The fiber of $\Sigma_k$ over any point in $X$ is obviously $M_k$. Define $Z_k$ to be $p^{-1}(\Sigma_k)$. The fact that $p$ has maximal rank implies that $Z_k$ has codimension $(e-k)(f-k)$ in $X \times H^0(E^\vee \otimes F)$ and that the singular locus of $Z_k$ is $Z_{k-1}$. Now observe that the restriction of the projection $$ q|_{Z_k \setminus Z_{k-1}} :Z_k \setminus Z_{k-1} \to H^0(E^\vee \otimes F) $$ has fiber ${q|_{Z_k \setminus Z_{k-1}}}^{-1}(\phi) \cong D_k(\phi) \setminus D_{k-1}(\phi)$. Finally, by generic smoothness, for a general $\phi \in H^0(E^\vee \otimes F)$ this is smooth of the expected dimension (or empty). \end{proof} Now we will construct the deformation of $A$ that we described in the section \ref{bnMain lemma} in the particular case in which both $A$ and $A^*$ are generated by global sections. \begin{proposition} \label{bn1.5} Let $A$ be a rank one torsion-free sheaf on an integral curve $C$ lying on a surface $S$ with $h^1(\SO_S)=0$ and $-K_S$ generated by global sections. Denote $j:C \hookrightarrow S$. If $A$ and $A^*$ are both generated by global sections, then there exists a (not necessarily complete) smooth irreducible curve $T$ and a sheaf ${\mathcal{A}}$ on $S \times T$ flat over $T$, such that \smallskip (a) the sheaf induced on the fiber of $S \times T \to T$ over some $t^{}_0 \in T$ is $j_* A$ (b) the sheaf $A_t$ induced on the fiber over any $t\in T$ with $t \neq t^{}_0$ is supported on a smooth curve $C_t$ and it is a rank one torsion-free sheaf when considered as a sheaf on $C_t$ (c) $h^0(A_t) \geq h^0(A)$ for every $t\in T$. \end{proposition} Note that these are the hypothesis of lemma \ref{bn0.2} for the particular case in which both $A$ and $A^*$ are generated by global sections. We will lift this condition in the next section. \begin{proof} The fact that $A$ is generated by global sections implies that there is an exact sequence $$ 0 \to E \stackrel{f_0}{\to} V \otimes \SO_S \to A \to 0 \; \; \; \; \; V=H^0(A), $$ with $E$ locally free (by proposition \ref{bn1.2}). Taking global sections in this sequence we see that $H^0(E)=0$, because $$ 0 \to H^0(E) \to V \stackrel{\cong}{\to} H^0(A). $$ Consider a curve $T$ mapping to $\operatorname{Hom}(E,V \otimes \SO_S)$ with $t^{}_0 \in T$ mapping to $f_0$ (so that item (a) is satisfied). Denote by $f_t$ the morphism given for $t \in T$ by this map. After shrinking $T$ we can assume that $f_t$ is still injective. Let $\pi_1$ be the projection of $S \times T$ onto the first factor and let ${\mathcal{E}}=\pi_1^*E$. Using the universal sheaf and morphism on $\operatorname{Hom}(E,V \otimes \SO_S)$ we can construct (by pulling back to $S \times T$) an exact sequence on $S \times T$ $$ 0 \to {\mathcal{E}} \stackrel{f}{\to} V \otimes {\mathcal{O}}_{S \times T} \to {\mathcal{A}} \to 0 $$ that restricts for each $t$ to an exact sequence \begin{equation} \label{eqbn1.2} 0 \to E \stackrel{f_t}{\to} V \otimes \SO_S \to A_t \to 0, \end{equation} where $A_t$ is a sheaf supported in the degeneracy locus of $f_t$. It is clear that $\deg(A)=\deg(A_t)$. Now we are going to prove that if the curve $T$ and the mapping to $\operatorname{Hom}(E, V \otimes \SO_S)$ are chosen generically, the quotient of the map gives the desired deformation. The flatness of ${\mathcal{A}}$ over $T$ follows from the fact that it has a short resolution and from the local criterion of flatness (we can apply \cite[III Lemma 10.3.A]{H}). The condition on $h^0(A_t)$ follows because $H^0(E)=0$ and we have a sequence $$ 0 \to H^0(E)=0 \to V \to H^0(A_t), $$ and then $h^0(A) \leq h^0(A_t)$. This proves item (c). Using the long exact sequence obtained by applying $Hom(\cdot,\SO_S)$ to \ref{eqbn1.2}, and the fact that $E$ is locally free, we obtain that $Ext^i(A_t,\SO_S)$ vanishes for $i \geq 2$, and so the projective dimension of $A_t$ is 1, and this implies that $A_t$, when considered as a sheaf on its support $C_t$, is torsion-free. We have to prove that we can choose the curve $T$ and the map to $\operatorname{Hom}(E,V \otimes \SO_S)$ such that $C_t$ is smooth for $t \neq t^{}_0$ (here we will use that $A^*$ is generated by global sections). First note that $Ext^1(A,\SO_S)$ is generated by global sections, because $Ext^1(A,\SO_S)=A^*\otimes \omega_S^{-1}$, and both $A^*$ and $\omega_S^{-1}$ are generated by global sections. Now we see that $E^\vee$ is generated by global sections, because we have $$ 0 \to V^\vee \otimes \SO_S \to E^\vee \to Ext^1(A,\SO_S) \to 0, $$ $Ext^1(A,\SO_S)$ is generated by global sections and $H^1(V^\vee \otimes \SO_S)=0$. Then $E^\vee \otimes (V \otimes \SO_S)$ is generated by global sections. Now apply lemma \ref{bn1.4} with $F=V \otimes \SO_S$. Then $n=m=r+1$, $k=r$ and the expected dimension is 1. And the lemma gives that for $\phi$ in a Zariski open subset of $\operatorname{Hom}(E, V\otimes \SO_S)$, the degeneracy locus $D_r(\phi)$ of $\phi$ is smooth away from the locus $D_{r-1}(\phi)$ where $\phi$ has rank $r-1$, but again by lemma \ref{bn1.4} the locus $D_{r-1}(\phi)$ is empty. This proves item (b). \end{proof} \section{General case} \label{bnGeneral case} Now we don't assume that $A$ satisfies the properties of the particular case (i.e., $A$ and $A^*$ now might not be generated by global sections). We will find a new sheaf that satisfies those conditions. We know how to deform this new sheaf, and we will show how we can use this deformation to construct a deformation of the original $A$. We start with a rank one torsion-free sheaf $A$ with $h^0(A)$, $h^1(A)>0$ on an integral curve $C$ lying on a surface. First we define $A'$ as the base point free part of $A$, i.e. $A'$ is the image of the evaluation map $$ H^0(A)\otimes {\mathcal{O}}_C \to A. $$ We have assumed that $h^0(A)>0$, and then $A'$ is a (nonzero) rank one torsion-free sheaf. Obviously, $H^0(A)=H^0(A')$. We have a short exact sequence $$ 0 \to A' \to A \to Q \to 0, $$ where $Q$ has support of dimension 0. Now consider ${A'}^*$, and define $B$ to be its base point free part. We have $h^0({A'}^*)=h^1(A')=h^1(A)+h^0(Q) \geq h^1(A)>0$. The first equality by lemma \ref{bn1.3}, and the last inequality by assumption. Then $B$ is a (nonzero) rank one torsion-free sheaf. Finally define $A''$ to be equal to $B^*$. \begin{lemma} \label{bn2.1} Both $A''$ and ${A''}^*$ are generated by global sections. \end{lemma} \begin{proof} Since $B$ is the base point free part of ${A'}^*$, we have a sequence $$ 0 \to B \to {A'}^* \to R \to 0 $$ where $R$ has support of dimension zero. Applying $Hom(\cdot,\omega_S)$ we get $$ 0 \to A' \to B^*=A'' \to \widetilde R \to 0 \; \; \; \; \; \widetilde R=Ext^2(R,\omega_S), $$ whose associated cohomology long exact sequence gives $$ 0 \to H^0(A') \to H^0(B^*) \to H^0(\widetilde R) \to H^1(A') \to H^1(B^*) \to 0. $$ To see that $A''$ is generated by global sections, it is enough to prove that the last map is an isomorphism, because then the first three terms make a short exact sequence, and the fact that $A'$ and $\widetilde R$ are generated by global sections (the first by definition, the second because its support has dimension zero) will imply that $B^*$ (that is equal to $A''$ by definition) is generated by global sections. To prove that the last map is an isomorphism, we only need to show that $h^1(A')=h^1(B^*)$, and this is true because $$ h^1(A')=h^0({A'}^*)=h^0(B)=h^0(B^{**})=h^1(B^*). $$ The first equality is by lemma \ref{bn1.3}, the second because $B$ is the base point free part of ${A'}^*$, the third by lemma \ref{bn1.1}, and the last again by lemma \ref{bn1.3}. To see that ${A''}^*$ is generated by global sections, note that by definition ${A''}^*=B^{**}=B$, and this is generated by global sections. \end{proof} We started with a rank one torsion-free sheaf $A$ with $h^0(A)$ and $h^1(A)>0$, and we have constructed new sheaves $A'$ and $A''$ with (nontrivial) maps $A' \to A$ and $A' \to A''$. They give rise to exact sequences \begin{eqnarray} 0 \to A' \to A \to Q \to 0 \nonumber \\ \label{eqbn2.1} 0 \to A' \to A'' \to \widetilde Q \to 0 \end{eqnarray} \begin{lemma} \label{equalit} With the previous definitions we have $h^0(A')=h^0(A)$ and $h^1(A'')=h^1(A')$. \end{lemma} \begin{proof} By construction $h^0(A')=h^0(A)$ and $h^0({A''}^*)=h^0({A'}^*)$. By lemma \ref{bn1.3} this last equality is equivalent to $h^1(A'')=h^1(A')$. \end{proof} As $A''$ and ${A''}^*$ are generated by global sections, then by proposition \ref{bn1.5} the sheaf $A''$ can be deformed in a family $A''_t$ in such a way that the support of a general member of the deformation is smooth. The idea now is to find (flat) deformations of $A'$ and $A$, so that for every $t$ we still have maps like \ref{eqbn2.1}. {}From the existence of these maps we will be able to obtain the condition that $h^0(A_t) \geq h^0(A)$, then we will be able to apply lemma \ref{bn0.2} and then theorem I will be proved. The details are in section \ref{bnProof of the main theorem}. We will start by showing how the condition on $h^0(A_t)$ is obtained, and then how we can find the deformations of $A'$ and $A$. \begin{proposition} \label{bn2.2} Let $A$, $A'$, $A''$ be rank one torsion-free sheaves on an integral curve $C$. Assume that they fit into exact sequences like \ref{eqbn2.1} and that $h^0(A')=h^0(A)$ and $h^1(A')=h^1(A'')$. Let $P$ be a curve (not necessarily complete), and let ${\mathcal{A}}$, ${\mathcal{A}}'$, and ${\mathcal{A}}''$ be sheaves on $S \times P$, flat over $P$, inducing for each $p \in P$ rank one torsion-free sheaves $A^{}_p$, $A'_p$, $A''_p$, supported on a curve $C_p$ of $S$, where $A^{}_{p^{}_0}=A$, $A'_{p^{}_0}=A'$, and $A''_{p^{}_0}=A''$ for some $p^{}_0 \in P$. Assume that $h^0(A''_p) \geq h^0(A''_{p^{}_0})$ for all $p\in P$ and that we have short exact sequences $$ 0 \to {\mathcal{A}}' \to {\mathcal{A}} \to {\mathcal{Q}} \to 0 $$ $$ 0 \to {\mathcal{A}}' \to {\mathcal{A}}'' \to \widetilde {\mathcal{Q}} \to 0 $$ with ${\mathcal{Q}}$ and $\widetilde {\mathcal{Q}}$ flat over $P$ (i.e., the induced sheaves $Q_p$, $\widetilde Q_p$ have constant length, equal to $l(Q)$ and $l(\widetilde Q)$ respectively). Then we have $h^0(A^{}_p) \geq h^0(A^{}_{p^{}_0})$ for all $p\in P$. \end{proposition} \begin{proof} For each $p \in P$ we have sequences $$ 0 \to A'_p \to A^{}_p \to Q^{}_p \to 0 $$ $$ 0 \to A'_p \to A''_p \to \widetilde Q^{}_p \to 0. $$ The maps on the left are injective because they are nonzero and the sheaves have rank one and are torsion-free. Using the associated long exact sequences and the hypothesis we have $$ h^0(A^{}_p) \geq h^0(A'_p) \geq h^0(A''_p)-l(\widetilde Q^{}_p) \geq h^0(A'')- l(\widetilde Q)=h^0(A')=h^0(A). $$ \end{proof} It only remains to prove that those sheaves can be ``deformed along'', and that those deformations are flat, i.e. that given $A$, $A'$ and $A''$ we can construct ${\mathcal{A}}'$ and ${\mathcal{A}}''$. This is proved in the following propositions. \begin{proposition} \label{bn2.3} Let $L$ and $M$ be rank one torsion-free sheaves on an integral curve $C$ that lies on a surface $S$. Assume we have a short exact sequence \begin{equation} \label{eqbn2.2} 0 \to L \to M \to Q \to 0. \end{equation} Assume furthermore that we are given a sheaf ${\mathcal{M}}$ on $S \times P$ (where $P$ is a connected but not necessarily irreducible curve) that is a deformation of $M$, flat over $P$. I.e., ${\mathcal{M}}|_{p^{}_0} \cong M$ for some $p^{}_0 \in P$, and for all $p\in P$ we have that $M_p={\mathcal{M}}|_p$ are torsion-free sheaves on $C_p$, where $C_p$ is a curve on $S$. Then, there is a connected curve $P'$ with a map $f:P' \to P$ and a sheaf ${\mathcal{L}}'$ over $S \times P'$ with the following properties: One irreducible component of $P'$ is a finite cover of $P$ and the rest of the components map to $p^{}_0 \in P$. The sheaf ${\mathcal{L}}'$ is a deformation of $L$, in the sense that ${\mathcal{L}}'|_{p'_0} \cong L$ for some $p'_0 \in P'$ mapping to $p^{}_0 \in P$, the sheaf ${\mathcal{L}}'$ is flat over $P'$ and induces rank one torsion-free sheaves on the fibers over $P'$. And if we define ${\mathcal{M}}'$ to be the pullback of ${\mathcal{M}}$ to $S \times P'$, there exists an exact sequence $$ 0 \to {\mathcal{L}}' \to {\mathcal{M}}' \to {\mathcal{Q}}' \to 0, $$ inducing short exact sequences $$ 0 \to L'_{p'} \to M'_{p'} \to Q'_{p'} \to 0 $$ for every $p'\in P'$. \end{proposition} \begin{proof} If the support of $Q$ were in the smooth part of the curve, we would have $M \cong L \otimes {\mathcal{O}}_C(D)$, with $D$ an effective divisor of degree $l(Q)$. Then, if we are given a deformation $M_p$ of $M$, we only need to find a deformation $D_p$ of the effective divisor $D$, with the only condition that $D_p$ is an effective divisor on $C_p$, with degree $l(Q)$. This can easily be done if we are in the analytic category. In general we might need to do a base change of the parametrizing curve $P$ and we will obtain a finite cover $P'$ of $P$ (What we are doing is moving a dimension zero and length $l(Q)$ subscheme of $S$, with the only restriction that for each $p$ the corresponding scheme is in $C_p$). Then we only need to define $L_{p'}=M_{p'} \otimes {\mathcal{O}}_{C_{p'}}(-D_{p'})$ and the proposition would be proved (with $P'$ a finite cover of $P$). To be able to apply this, we will have to make first a deformation of $L$, keeping $M$ fixed, until we get $Q$ to be supported in the smooth part of $C$ (the curve $C$ also remains fixed in this deformation). This is the reason for the need of the curve $P'$ with some irreducible components mapping to $p^{}_0$. We will prove this by induction on the length of the intersection of the support of $Q$ and the singular part of $C$. \begin{lemma} \label{bn2.4} Let $L$ and $M$ be rank one torsion-free sheaves on an integral curve $C$ that lies on a surface $S$. Assume we have a short exact sequence $$ 0 \to L \to M \to Q \to 0. $$ Assume that $Q=R \oplus Q'$ where $Q'$ has length $l(Q)-1$ and it is supported in the smooth part of $C$, and $R$ has length one as it is supported in a singular point of $C$ (``the length of the intersection of the support of $Q$ and the singular part of $C$ is one''). Then there is a flat deformation $L_y$ of $L$ parametrized by a connected curve $Y$ (it might not be irreducible) such that $L_{y_0}=L$ for some $y_0\in Y$ and for every $y\in Y$ there is an exact sequence $$ 0 \to L_y \to M \to Q_y \to 0 $$ and there is some $y_1\in Y$ such that the support of $Q_{y_1}$ is in the smooth part of $C$. \end{lemma} \begin{proof} In this situation, the exact sequence \ref{eqbn2.2} gives rise to another exact sequence $$ 0 \to L \otimes I_Z^\vee \to M \to R \to 0 $$ where the map on the right is the composition of $M \to Q$ and the projection $Q \to R$, and we denote by $I_Z$ the ideal sheaf of the support $Z$ of $Q'$. Because $Z$ is in the smooth part of $C$, $I_Z$ is an invertible sheaf. Note that $Q'$ is the quotient of ${\mathcal{O}}_C$ by this ideal sheaf. Define $\widehat L$ to be $L \otimes I_Z^\vee$. If we know how to make a flat deformation $\widehat L_y$ of $\widehat L$ so that the quotient $R_y$ is supported in the smooth part of $C$ for some $y_1\in Y$, then we can construct a deformation $L_y$ of $L$ defined as $$ L_y = \widehat L_y \otimes I_Z. $$ Note that this deformation is also flat. The cokernel $Q_y$ of $L_y \to M$ is supported in the smooth part of $C$ for the points $y\in Y$ for which $R_y$ is supported in the smooth part of $C$. This shows that to prove the lemma we can assume that $Q$ has length one and its support is a singular point of $C$, i.e. $Q={\mathcal{O}}_x$, where $x$ is a singular point of $C$. Consider the scheme $\operatorname{Quot}^1(M)$ representing the functor of quotients of $M$ of length 1. If the support $x$ of the quotient $Q$ is in the smooth part of $C$, then there is only one surjective map (up to scalar) because $\dim \operatorname{Hom} (M,Q)=1$, whose kernel is $M\otimes {\mathcal{O}}_C(-x)$. If $x$ is in the singular part, then in general $\dim\operatorname{Hom}(M,Q)>1$, and the quotients are parametrized by $\mathbb{P}\operatorname{Hom}(M,Q)$ (the universal bundle is flat over $\mathbb{P}\operatorname{Hom}(M,Q)$). We want to show that $\operatorname{Quot}^1(M)$ is connected by constructing a flat family of quotients $M \to \widetilde Q_{\tilde c}$ (the family $\widetilde Q_{\tilde c}$ will be parametrized by an open set of the normalization $\widetilde C$ of $C$) such that for a general ${\tilde c}$ the support of $\widetilde Q_{\tilde c}$ is in the smooth part of $C$, and for some point $\tilde c_0$ the support of $\widetilde Q_{\tilde c_0}$ is a singular point of $C$. Consider the normalization $\widetilde C$ of $C$, and let $F$ be an open set of $\widetilde C$ \begin{equation} \label{eqbn2.3} \CD F @>j>> C \times F \\ @. @V{\pi_1}VV \\ @. C \\ \endCD \end{equation} Where $\pi_1$ is the projection to the first factor and $j=(\nu,i)$, the morphism $\nu:F\hookrightarrow \widetilde C\to C $ being the restriction to $F$ of the normalization map and $i$ the identity map. Note that $j$ is a closed immersion, and its image is just $C \times_C F \cong F$. Let $\tilde c_0$ be a point of $\widetilde C$ in $\nu^{-1}(x)$ (the family is going to be parametrized by an open neighborhood $F$ of $\tilde c_0$). We have to construct a surjection of $\widetilde {\mathcal{M}}=\pi_1^* M$ onto $\widetilde {\mathcal{Q}} = j_* {\mathcal{O}}_{F}$. Note that $\widetilde{\mathcal{Q}}|_{C \times \tilde c }=\widetilde Q_{\tilde c} \cong {\mathcal{O}}_{\nu (\tilde c)}$ and that $\widetilde{\mathcal{Q}}$ is flat over $F$. Now, to define that quotient, it is enough to define it in the restriction to the image of $j$ (because this is exactly the support of $\widetilde {\mathcal{Q}}$). So the map we have to define is $$ j^*\widetilde{\mathcal{M}} \to {\mathcal{O}}_{F} \; . $$ But $j^*\widetilde{\mathcal{M}}=\nu^*M$ is a rank one sheaf on the smooth curve $F$, so it is the direct sum of a line bundle and a torsion part $T$. Shrinking $F$ if necessary, the line bundle part is isomorphic to ${\mathcal{O}}_F$, and we have $$ j^*\widetilde{\mathcal{M}} \cong T \oplus {\mathcal{O}}_F \; , $$ and then to define the quotient we just take an isomorphism in the torsion-free part. This finishes the proof of the lemma. \end{proof} Now we go to the general case: the intersection of the support of $Q$ with the singular part of $C$ has length $n$. We are going to see how this can be reduced to the case $n=1$. Take a surjection from $Q$ to a sheaf $Q'$ of length $n-1$, such that $Q$ is isomorphic to $Q'$ at the smooth points. The kernel $R$ of this surjection will have length 1, and will be supported in a singular point of $C$. It is isomorphic to ${\mathcal{O}}_x$, for some singular point $x$. We have a diagram $$ \CD @. @. 0 @. 0 @. \\ @. @. @VVV @VVV @. \\ 0 @>>> L @>>> L'@>>> R @>>> 0 \\ @. @| @VVV @VVV @. \\ 0 @>>> L @>>> M @>>> Q @>>> 0 \\ @. @. @VVV @VVV @. \\ @. @. Q' @= Q' @. \\ @. @. @VVV @VVV @. \\ @. @. 0 @. 0 @. \\ \endCD $$ Observe that $L$, $L'$ and $R$ satisfy the hypothesis of lemma \ref{bn2.4}, so we can find deformations $L_y$, $R_y$ (parametrized by some curve $Y$ and with $L_{y_0}=L$ and $R_{y_0}=R$ for some $y_0 \in Y$) such that for some $y_1 \in Y$ we have that the support of the corresponding sheaf $R_{y_1}$ is a smooth point of $C$. All the maps of the previous diagram can be deformed along. To do this, we change $L$ by $L_y$, $R$ will be deformed to $R_y$ and $L'$ is kept constant. Then $Q$ is deformed to a family $Q_y$ defined as $M/L_y$. The cokernel of $R_y \to Q_y$ will be $Q_y/R_y=M/L'=Q'$, and hence we keep it constant. Then for each $y$ we still have a commutative diagram, and furthermore it is easy to see that all deformations are flat (note that $R_y$ is a flat deformation and $Q'$ is kept constant, and then $Q_y$ is a flat deformation). An important point is that $M$ remains fixed, and the injection $L \to M$ is deformed to $L_y \to M$. $$ \CD @. @. 0 @. 0 @. \\ @. @. @VVV @VVV @. \\ 0 @>>> L_y @>>> L'@>>> R_y @>>> 0 \\ @. @| @VVV @VVV @. \\ 0 @>>> L_y @>>> M @>>> Q_y @>>> 0 \\ @. @. @VVV @VVV @. \\ @. @. Q' @= Q' @. \\ @. @. @VVV @VVV @. \\ @. @. 0 @. 0 @. \\ \endCD $$ For $y_1$ we have that the length of the intersection of the support of $Q_{y_1}$ with the singular part of $C$ is $n-1$. We repeat the process (starting now with $L_{y_1}$, $M$ and $Q_{y_1}$), until all the points of the support of $Q$ are moved to the smooth part of $C$. This finishes the proof of the proposition. \end{proof} The following proposition is similar to proposition \ref{bn2.3}, but now the roles of $L$ and $M$ are changed: we are given a deformation of $L$ and we have to deform $M$ along. \begin{proposition} \label{bn2.5} Let $L$ and $M$ be rank one torsion-free sheaves on an integral curve $C$ that lies on a surface $S$. Assume we have a short exact sequence \begin{equation} \label{eqbn2.4} 0 \to L \to M \to Q \to 0. \end{equation} Assume furthermore that we are given a sheaf ${\mathcal{L}}$ on $S \times P$ (where $P$ is a connected but not necessarily irreducible curve) that is a deformation of $L$, flat over $P$ , i.e., ${\mathcal{L}}|_{p^{}_0} \cong L$ for some $p^{}_0\in P$, and for all $p\in P$, we have that $L_p={\mathcal{L}}|_p$ are torsion-free sheaves on $C_p$, where $C_p$ is a curve on $S$. Then, there is a connected curve $P'$ with a map $f:P' \to P$ and a sheaf ${\mathcal{M}}'$ over $S \times P'$ with the following properties: One irreducible component of $P'$ is a finite cover of $P$ and the rest of the components map to $p^{}_0 \in P$. The sheaf ${\mathcal{M}}'$ is a deformation of $M$, in the sense that ${\mathcal{M}}'|_{p'_0} \cong M$ for some $p'_0 \in P'$ mapping to $p^{}_0 \in P$, the sheaf ${\mathcal{M}}'$ is flat over $P'$ and induces rank one torsion-free sheaves on the fibers over $P'$. And if we define ${\mathcal{L}}'$ to be the pullback of ${\mathcal{L}}$ to $S \times P'$, there exists an exact sequence $$ 0 \to {\mathcal{L}}' \to {\mathcal{M}}' \to {\mathcal{Q}}' \to 0, $$ inducing short exact sequences $$ 0 \to L'_{p'} \to M'_{p'} \to Q'_{p'} \to 0 $$ for every $p'\in P'$. \end{proposition} \begin{proof} The proof is very similar to the proof of proposition \ref{bn2.3}. Again we start by observing that if the support of $Q$ were in the smooth part of the curve, we would have $M \cong L \otimes {\mathcal{O}}_C (D)$, with $D$ an effective divisor. Then if we are given a flat deformation $L_p$ of $L$, we find a deformation $D_p$ of $D$ as in the first part, and the proposition would be proved. So again we need a lemma that deforms $Q$ so that its support is in the smooth part of $C$. \begin{lemma} \label{bn2.6} Let $L$ and $M$ be rank one torsion-free sheaves on an integral curve $C$ that lies on a surface $S$. Assume we have a short exact sequence $$ 0 \to L \to M \to Q \to 0. $$ Assume that the part of $Q$ with support in the smooth part of $C$ has length $l(Q)-1$, i.e. $Q=R \oplus Q'$, where $R$ has length one and is supported in a singular point of $C$ and $Q'$ has length $l(Q)-1$ and is supported in the smooth part of $C$. Then there is a flat deformation $M_y$ of $M$ parametrized by a curve $Y$, such that for every $y\in Y$ there is an exact sequence $$ 0 \to L \to M_y \to Q_y \to 0 $$ with $M_y$ a torsion-free sheaf, and there is some $y_1 \in Y$ such that the support of $Q_{y_1}$ is in the smooth part of $C$. \end{lemma} \begin{proof} Arguing as in the proof of lemma \ref{bn2.4}, we see that it is enough to prove the case $l(Q)=1$, and $Q={\mathcal{O}}_x$ for $x$ a singular point of $C$, then we can assume that the extension of the hypothesis of the lemma is \begin{eqnarray} 0 \to L \to M \to {\mathcal{O}}_x \to 0. \label{assumeext} \end{eqnarray} Now we will consider all extensions of ${\mathcal{O}}_x$ (for $x$ any point in $C$) by $L$. If $x$ is a smooth point, then there is only one extension that is not trivial (up to equivalence) $$ 0 \to L \to M \cong L\otimes {\mathcal{O}}_C(x) \to {\mathcal{O}}_x \to 0. $$ All these extensions are then parametrized by the smooth part of $C$. But if $x$ is a singular point, we could have more extensions, because in general $s=\dim \operatorname{Ext}^1({\mathcal{O}}_x,L)>1$. They will be parametrized by a projective space $\mathbb{P} ^{s-1}$. We call this space $E_x$. Note that there is a universal extension on $C \times E_x$ that is flat over $E_x$. We denote by $e_1$ the point in $E_x$ corresponding to the extension \ref{assumeext}. Assume that $\widetilde Q_{\tilde c}$ is a family of torsion sheaves on $C$ with length 1, parametrized by a curve $F$ such that for a general point $\tilde c\in F$ of the parametrizing curve the support of $\widetilde Q_{\tilde c}$ in $C$ is a smooth point, and for a special point $\tilde c_0\in F$ the support of $\widetilde Q_{\tilde c_0}$ is a singular point. Now assume that we can construct a flat family (parametrized by $F$) of nontrivial extensions of $\widetilde Q_{\tilde c}$ by $L$. The extension corresponding to $\tilde c_0$ gives a point $e_2$ in $E_x$. The space $E_x$ is a projective space, thus connected, and then there is a curve containing $e_1$ and $e_2$. Using this curve (together with the universal extension for $E_x$) and the curve $F$ (together with the family of extensions that it parametrizes) we construct the curve $Y$ that proves the lemma. Now we need to construct $F$. As in the proof of lemma \ref{bn2.4}, the parametrizing curve $F$ will be an affine neighborhood of $\tilde c_0$ in the normalization $\widetilde C$ of $C$, where $\tilde c_0$ is a point that maps to the singular point $x$ of $C$. Consider again the diagram \ref{eqbn2.3} of the proof of lemma \ref{bn2.4}. The family will be given by an extension of $\widetilde{\mathcal{Q}} = j_*{\mathcal{O}}_F$ by $\widetilde{\mathcal{L}}=\pi^*_1 L$ on $C \times F$. These extensions are parametrized by the group $\operatorname{Ext}^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})$. The following lemma gives information about this group and relates this extension with the extensions that we get after restriction for each slice $C \times \tilde c$. We will call $\widetilde Q_{\tilde c}$ and $\widetilde L_{\tilde c}$ the restrictions of $\widetilde{\mathcal{Q}}$ and $\widetilde{\mathcal{L}}$ to the slice $C \times {\tilde c}$. Note that the restriction $\widetilde L_{\tilde c}$ is isomorphic to $L$. \begin{lemma} \label{bn2.7} With the previous notation, we have 1) $\operatorname{Ext}^1(\widetilde{{\mathcal{Q}}},\widetilde{\mathcal{L}}) \cong H^0(Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}}))$ 2) $Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})$ has rank zero outside of the support of $\widetilde{\mathcal{Q}}$, and rank 1 on the smooth points of the support of $\widetilde{\mathcal{Q}}$ 3) Let $I$ be the ideal sheaf corresponding to a slice $C \times {\tilde c}$. Then the natural map $$ Ext^1_{{\mathcal{O}}_{C \times F}} (\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}}) \otimes {\mathcal{O}}_{C \times F}/I \to Ext^1_{{\mathcal{O}}_{C \times {\tilde c}}}(\widetilde Q_{\tilde c},\widetilde L_{\tilde c}) $$ is injective. \end{lemma} \begin{proof} Item 1 follows from the fact that $Hom(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})=0$ and the exact sequence $$ 0 \to H^1(Hom(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})) \to \operatorname{Ext}^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}}) \to H^0(Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})) \to H^2(Hom(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})). $$ To prove item 2 note that the stalk of $Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})$ at a point $p$ is isomorphic to $\operatorname{Ext}^1(R/I,R)$, where $R$ is the local ring at the point $p$, and $I$ is the ideal defining the support or $\widetilde{\mathcal{Q}}$. The ideal $I$ is principal if the point $p$ is smooth, then $R/I$ has a free resolution $$ 0 \to I \to R \to R/I \to 0 $$ and it follows that $\operatorname{Ext}^1(R/I,R)\cong R/I$. For item 3, consider the exact sequence $$ 0 \to \widetilde{\mathcal{Q}} \stackrel{\cdot f}{\to} \widetilde{\mathcal{Q}} \to \widetilde Q_{\tilde c} \to 0 $$ where the first map is multiplication by the local equation $f$ of the slice $C \times {\tilde c}$. Applying $Hom(\cdot,\widetilde L_{\tilde c})$ we get $$ Hom(\widetilde{\mathcal{Q}},\widetilde L_{\tilde c})=0 \to Ext^1(\widetilde Q_{\tilde c},\widetilde L_{\tilde c}) \to Ext^1(\widetilde{\mathcal{Q}},\widetilde L_{\tilde c}) \to Ext^1(\widetilde{\mathcal{Q}},\widetilde L_{\tilde c}), $$ but the last map is zero. To see this, take a locally free resolution of $\widetilde{\mathcal{Q}}$. The map induced on the resolution by the multiplication with the equation $f$ is just multiplication by the same $f$ on each term $$ \CD {\mathcal{F}}^\bullet @>>> \widetilde{\mathcal{Q}} @>>> 0 \\ @V{\cdot f}VV @V{\cdot f}VV @. \\ {\mathcal{F}}^\bullet @>>> \widetilde{\mathcal{Q}} @>>> 0 \\ \endCD $$ A local section of the sheaf $Ext^i(\widetilde{\mathcal{Q}},\widetilde L_{\tilde c})$ is represented by some local section $\varphi(\cdot)$ of $Hom({\mathcal{F}}^i,\widetilde L_{\tilde c})$, and the endomorphism induced by multiplication by $f$ on $Ext^i(\widetilde {\mathcal{Q}},\widetilde L_{\tilde c})$ is given by precomposition with multiplication $\varphi(f\cdot)$, but $\varphi$ is a morphism of sheaves of modules and then this is equal to $f \varphi(\cdot)$, and this is equal to zero because $f \widetilde L_{\tilde c}=0$. Then we have that \begin{equation} \label{iso} Ext^1(\widetilde Q_{\tilde c},\widetilde L_{\tilde c}) \cong Ext^1(\widetilde{\mathcal{Q}},\widetilde L_{\tilde c}). \end{equation} Taking the exact sequence $$ 0 \to \widetilde{\mathcal{L}} \stackrel{\cdot f}{\to} \widetilde{\mathcal{L}} \to \widetilde L_{\tilde c} \to 0 $$ and applying $Hom(\widetilde{\mathcal{Q}},\cdot)$ we get $$ Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}}) \stackrel{\cdot f}{\to} Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}}) \to Ext^1(\widetilde{\mathcal{Q}},\widetilde L_{\tilde c}) $$ and using this and the isomorphism \ref{iso} we have an injection $$ Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}}) \otimes {\mathcal{O}}_{C \times F}/I \cong Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})/(f\cdot Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})) \hookrightarrow Ext^1(\widetilde Q_{\tilde c},\widetilde L_{\tilde c}). $$ \end{proof} Now we are going to construct the family of extensions. By item 2 of the lemma the sheaf ${\mathcal{E}}=Ext^1(\widetilde {\mathcal{Q}},\widetilde {\mathcal{L}})$ is isomorphic to ${\mathcal{O}}_X \oplus T({\mathcal{E}})$ (shrinking $F$ if necessary) where $X$ is the support of $\widetilde{\mathcal{Q}}$ and $T({\mathcal{E}})$ is the torsion part. Take a nonvanishing section of the torsion-free part, and by item 1 this gives a nonzero element $\psi$ of $\operatorname{Ext}^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})$. This element gives a nontrivial extension $$ 0 \to \widetilde{\mathcal{L}} \to \widetilde{\mathcal{M}} \to \widetilde{\mathcal{Q}} \to 0. $$ Observe that $\widetilde{\mathcal{M}}$ is flat over the base, because both $\widetilde{\mathcal{L}}$ and $\widetilde{\mathcal{Q}}$ are flat. By items 3 and 1 we have that the image of $\psi$ under the restriction map $$ \operatorname{Ext}^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}}) \to \operatorname{Ext}^1(\widetilde Q_{\tilde c},L) $$ is nonzero for any ${\tilde c}$ (recall that $\widetilde L_{\tilde c}=L$ for all ${\tilde c}$), and this means that the extensions that we obtain after restriction to the corresponding slices \begin{equation} \label{eqbn2.5} 0 \to L \to \widetilde M_{\tilde c} \to \widetilde Q_{\tilde c} \to 0 \end{equation} are non trivial. Furthermore $\widetilde M_{\tilde c}$ is torsion-free. To prove this claim, let $T(\widetilde M_{\tilde c})$ be the torsion part of $\widetilde M_{\tilde c}$. The map $L \to T(\widetilde M_{\tilde c})$ coming from \ref{eqbn2.5} is zero, because $L$ is torsion-free, i.e. $T(\widetilde M_{\tilde c})$ injects in $\widetilde Q_{\tilde c}$. Then we have $$ \widetilde Q_{\tilde c} \cong {\frac {\widetilde M_{\tilde c}} L} \cong {\frac{\widetilde M_{\tilde c}/T(\widetilde M_{\tilde c}) \oplus T(\widetilde M_{\tilde c})} L} \cong {\frac{\widetilde M_{\tilde c}/T(\widetilde M_{\tilde c})} L} \oplus T(\widetilde M_{\tilde c}). $$ $\widetilde Q_{\tilde c}$ doesn't decompose as the direct sum of two sheaves, and then one of these summands must be zero. The first summand cannot be zero, because this would imply that $L \cong \widetilde M_{\tilde c}/T(\widetilde M_{\tilde c})$ and then $\widetilde M_{\tilde c} \cong L \oplus \widetilde Q_{\tilde c}$, contradicting the hypothesis that the extension is not trivial. Then we must have $T(\widetilde M_{\tilde c})=0$, and the claim is proved. \end{proof} Now we are going to consider the general case, in which the part of $Q$ supported in singular points has length $n$. We are going to see that this can be reduced to the case $n=1$, in a similar way to proposition \ref{bn2.3}. Let $R={\mathcal{O}}_x$, where $x$ is a singular point in the support of $Q$, and take a surjection from $Q$ to $R$. We have a diagram $$ \CD @. @. @. 0 @. \\ @. @. @. @VVV @. \\ @. @. @. L'/L @. \\ @. @. @. @VVV @. \\ 0 @>>> L @>>> M @>>> Q @>>> 0 \\ @. @VVV @| @VVV @. \\ 0 @>>> L' @>>> M @>>> R @>>> 0 \\ @. @. @. @VVV @. \\ @. @. @. 0 @. \\ \endCD $$ Note that $L'$, $M$ and $R$ satisfy the hypothesis of lemma \ref{bn2.6}, then we can find (flat) deformations $M_y$ and $R_y$ parametrized by a curve $Y$ such that for some $y_1\in Y$ we have that the support of the corresponding sheaf $R_{y_1}$ is a smooth point of $C$. All sheaves and maps can be deformed along. To do this we define $Q_y=M_y/L$ (we have $L \hookrightarrow L' \hookrightarrow M_y$, thus this quotient is well defined). The kernel of $Q_y \to R_y$ is $L'/L$. Then $Q_y$ is a flat deformation (being the extension of a flat deformation $R_y$ by a constant and hence flat deformation $L'/L$). Then for each $y$ we have a commutative diagram $$ \CD @. @. @. 0 @. \\ @. @. @. @VVV @. \\ @. @. @. L'/L @. \\ @. @. @. @VVV @. \\ 0 @>>> L @>>> M_y @>>> Q_y @>>> 0 \\ @. @VVV @| @VVV @. \\ 0 @>>> L' @>>> M_y @>>> R_y @>>> 0 \\ @. @. @. @VVV @. \\ @. @. @. 0 @. \\ \endCD $$ Observe that the length of the part of $Q_{y_1}$ supported in singular points is $n-1$, so repeating this process we can deform $Q$ until its support lies in the smooth part of $C$. This finishes the proof of the proposition. \end{proof} \section{Proof of theorem I} \label{bnProof of the main theorem} In this section we will prove theorem I: \begin{proof} Nonemptyness follows from the fact that the Brill-Nother loci for smooth curves is nonempty, and by upper semicontinuity of $h^0(\cdot)$. By remark \ref{remark} we can assume $r>d-p_a$. We will prove theorem I by applying lemma \ref{bn0.2}. We start with a rank one torsion-free sheaf $A$ corresponding to a point in $\overline W^r_d$, $d>0$, $r\geq 0$, with $\rho(r,d)>0$ (recall that we are assuming $r>d-p_a$). We have $h^0(A)$, $h^1(A)>0$. As we explained at the beginning of the section \ref{bnGeneral case}, we call $A'$ its base point free part. Then we take $B$ to be the base point free part of ${A'}^*$, and finally define $A''$ to be $B^*$. By lemma \ref{bn2.1}, $A''$ and ${A''}^*$ are rank one locally free sheaves on $C$ generated by global sections. Then by proposition \ref{bn1.5} we find a deformation ${\mathcal{A}}''$ of $A''$ parametrized by a some smooth irreducible curve $T$. The support of ${\mathcal{A}}''$ defines a family of curves ${\mathcal{C}}$ parametrized by the irreducible curve $T$. note that ${\mathcal{C}}|_t$ is smooth for $t\neq0$. By the definition of $A'$ and $A''$ we have exact sequences \begin{equation} 0 \to A' \to A \to Q \to 0 \label{short1} \end{equation} \begin{equation} 0 \to A' \to A'' \to \widetilde Q \to 0 \label{short2} \end{equation} with $h^0(A')=h^0(A)$ and $h^1(A'')=h^1(A')$ (lemma \ref{equalit}). If we look at \ref{short2} we see that we are in the situation of proposition \ref{bn2.3}, with $L=A'$, $M=A''$, ${\mathcal{M}}={\mathcal{A}}''$, $P=T$. Then we get a family ${\mathcal{A}}'$ (parametrized by some connected but in general not irreducible curve). Now we use this family ${\mathcal{A}}'$ and the sequence \ref{short1} to apply \ref{bn2.5} with $L=A'$, $M=A$ and ${\mathcal{L}}={\mathcal{A}}'$. We get a new family ${\mathcal{A}}$. We denote by $T'$ the curve parametrizing the family ${\mathcal{A}}$. This family satisfies all the hypothesis of lemma \ref{bn0.2} (item (iv) is given by proposition \ref{bn2.2}), and then theorem I is proved. \end{proof} \chapter{Irreducibility of the moduli space for $K3$ surfaces} \label{k3} In this chapter we will prove the following theorem: \smallskip \noindent\textbf{Theorem II.} \textit{ With the notation of chapter \ref{Preliminaries}, if $L$ is a primitive nonzero element of $\operatorname{Pic}(S)$, and $H$ is an $(L,c_2)$-generic polarization, then $\mathfrak{M}_H(L,c_2)$ is irreducible.} \smallskip Due to the fact that the moduli space is smooth, irreducibility is equivalent to connectedness. \bigskip \textbf{Outline of the proof of theorem II} \bigskip First we will prove the theorem for the case in which Pic$(S)=\mathbb{Z}$ For $H$ to be $(L,c_2)$-generic we need $L$ to be an odd multiple of a generator of Pic$(S)$, and tensoring the vector bundles with a line bundle we can assume that $H=L$ is a generator of Pic$(S)$. After proving the theorem for this case, in section \ref{General K3 surface} we show, by considering families of surfaces, that if the result is true for Pic$(S)=\mathbb{Z}$, then it is also true under the conditions of the theorem (this part is very similar to an argument in \cite{G-H}). From now on we will assume that Pic$(S)=\mathbb{Z}$ and that $H=L$ is the ample generator. The proof is divided into two parts. In section \ref{Small second Chern class} we handle the case in which $c_2 \leq \frac{1}{2} L^2 + 3$. First we see (proposition \ref{2.1}) that the sheaves satisfying this inequality are exactly those which are nonsplit extensions of the form $$ 0 \to \SO_S \to V \to L \otimes I_Z \to 0,$$ with $l(Z)=c_2$. Then we study the set $X \subset \Hilb ^{c_2} (S)$ for which there exist nonsplit extensions like these above, and we see, using theorem I, that it is connected (proposition \ref{2.2}). Finally we use this to prove (proposition \ref{2.4}) the connectedness of $\FM (L,c_2)$ for $\dim \FM (L,c_2)>0$ (if the dimension is zero the result is known \cite{M}). Note that for $c_2=\frac{1}{2} L^2 + 3$ we have $\dim \FM (L,c_2) = L^2+6>0$, and then we can continue the proof by induction on $c_2$. Let $C(n)$ be the set of irreducible components of $\mathfrak{M}(L,n)$. We construct a map $$\Phi _n :C(n) \to C(n+1).$$ To define this map, take a sheaf $E$ in a component $A$ of $\mathfrak{M}(L,n)$. Take a point $p \in S$ and a surjection $E \to {\mathcal{O}}_p$. Let $F$ be the kernel $$0 \to F \to E \to {\mathcal{O}}_p \to 0.$$ $F$ is clearly stable, and $c_2(F)=c_2(E)+1$. Now we define $\Phi _n (A)$ to be the component in which $F$ lies. It is easy to see that this is independent of all the choices made, so that $\Phi _n$ is well defined. Now we assume that $\FM (L,c_2)$ is irreducible for $c_2 < n$. We are going to see that if every connected component of $\mathfrak{M}(L,n)$ has a non-locally free sheaf $F$, $\Phi _{n-1}$ is surjective, and then by induction $\mathfrak{M}(L,n)$ will be irreducible. Let $B$ be a component of $\mathfrak{M}(L,n)$ with non-locally free sheaves. By lemma \ref{3.3}, it has a non-locally free sheaf $F$ such that $F^{\vee\vee} \in \mathfrak{M}(L,n-1)$. By smoothness of the moduli space, $F^{\vee\vee}$ is in only one irreducible component. Call this component $A$. By construction $\Phi_{n-1}(A)=B$. In other words, we have seen that if $\mathfrak{M}(L,n-1)$ is irreducible, then there is only one component $\mathfrak{M}_0$ of $\mathfrak{M}(L,n)$ that has sheaves that are not locally free, and then to prove that the later has only one component, it will be enough to check that every component has a non-locally free sheaf. We divide the possible values of $c_2$ in regions labeled by $n \geq 1$, with $c_2$ satisfying $$ ((n-1)^2 + (n-1) + \frac{1}{2}) L^2 + 3 < c_2 \leq (n^2 + n + \frac{1}{2}) L^2 + 3. $$ If $V$ is locally free, we prove that then $V$ fits in a short exact sequence $$ 0 \to L^{\otimes -m} \to V \to L^{\otimes m+1} \otimes I_{Z_m} \to 0 $$ with $0 \leq m \leq n$ (proposition \ref{3.1}). We call it an extension of type $m$. We will also say that $V$ is of type $m$. Next (proposition \ref{3.3}) we show that the set of sheaves that are not locally free has positive codimension, and then we prove (proposition \ref{3.4}) that the generic sheaf is a vector bundle of type $n$. But this is not enough, and we need more information about the generic vector bundle. Let $C$ be the set of vector bundles $V$ such that for any exact sequence \begin{equation} 0 \to L^{\otimes -n} \to V \to L^{\otimes n+1} \otimes I_{Z_n} \to 0, \label{eq0.1} \end{equation} $L^{\otimes n+1} \otimes I_{Z_n}$ has no sections whose zero locus is an irreducible reduced curve. In proposition \ref{3.7} we prove that this set has positive codimension. The reason to look at this set is because it is precisely because of these sheaves that we cannot apply the generalization of Fulton-Lazarsfeld's theorem to proof that the set of type $n$ vector bundles is connected. But now we know that we can ignore $C$, because it has positive codimension, and then conclude that the generic vector bundle $V$ sits in an extension like \ref{eq0.1} such that $L^{\otimes 2n+1} \otimes I_{Z_n}$ has a section whose zero locus is an irreducible reduced curve. In proposition \ref{3.6} we prove that those vector bundles make a connected set. We will need the induction hypothesis to prove this proposition. \section{Preliminaries} \label{secPreliminaries} In this section we will prove some propositions that will be useful later. \begin{lemma} \label{1.1} Let $S$ be a smooth surface and $C$ a smooth (not necessarily complete) curve. Let $p$ be a point in the curve and $j:S \hookrightarrow S \times C$ the corresponding injection. Let $L$ be a line bundle on $S$ and $I_W$ an ideal sheaf on $S$ corresponding to a subscheme of dimension zero. Let ${\mathcal{V}}$ be a family of rank two sheaves on $S$, i.e. a sheaf on $S \times C$ flat over $C$. If we have the following elementary transformation: $$ 0 \to {\mathcal{W}} \to {\mathcal{V}} \to j_*(L \otimes I_W) \to 0 $$ then ${\mathcal{W}}$ is a flat family of rank two sheaves on $S$, and furthermore $$ c_i({\mathcal{W}}_{p'}) = c_i({\mathcal{V}}_{p'})$$ for $i=1,2$ and $p'$ any point of $C$. \end{lemma} \begin{proof} We calculate the Chern classes of $j_*(L \otimes I_W)$ by the Grothendieck-Riemann-Roch theorem, and then the classes of ${\mathcal{W}}$ by Whitney's formula. The fact that ${\mathcal{W}}$ is flat is proved in \cite{F}. \end{proof} Now we will apply this lemma to take limits of stable extensions. Let $S$ be a smooth surface with $\operatorname{Pic} (S)=\mathbb{Z}$. Consider a family of extensions parametrized by a curve $T$ $$ 0 \to L^{\otimes -n} \to V_t \to L^{\otimes n+1} \otimes I_Z \to 0,$$ where L is a generator of $\operatorname{Pic} (S)$, $t \in T$, and $Z$ is a subscheme of dimension zero. Assume that $V_t$ is stable for $t \not= 0$, where $0$ is some fixed point of T, and unstable for $t = 0$. This defines a map $\varphi : T-\{0\} \to \mathfrak{M}$ to the moduli space of stable sheaves. By properness of $\mathfrak{M}$, this can be extended to a map $\varphi : T \to \mathfrak{M}$, i.e. we can take the limit of the family as $t$ goes to $0$ and we obtain a stable sheaf corresponding to $\varphi (0)$. \begin{proposition} \label{1.2} The stable sheaf $V'$ corresponding to $\varphi (0)$ is not locally free or can be written as an extension \begin{equation} 0 \to L^{\otimes -m} \to V' \to L^{\otimes m+1} \otimes I_{Z'} \to 0 \label{eq1.1} \end{equation} with $m < n$. \end{proposition} \begin{proof} $V_0$ is unstable, so we have $$ 0 \to L^{\otimes a} \otimes I_W \to V_0 \to L^{\otimes 1-a} \otimes I_{W'} \to 0$$ with $0 < a \leq n$. Consider the elementary transformation $$ 0 \to {\mathcal{W}} \to {\mathcal{V}} \to j_*[L^{\otimes 1-a} \otimes I_{W'}] \to 0.$$ By lemma \ref{1.1} we have a new family ${\mathcal{W}}$. By standard arguments the member $W_0$ of the new family corresponding to $t=0$ can be written as an extension \begin{equation} 0 \to L^{\otimes 1-a} \otimes I_{W'} \to W_0 \to L^{\otimes a} \otimes I_W \to 0. \label{eq1.2} \end{equation} Note that $0 \geq 1-a > -n$. If $W_0$ is not stable, repeat the process: unstability gives an injective map $L^{\otimes a'}\otimes I_{W''} \to W_0$, and by \ref{eq1.2} we have $a' < a$. Then in each step $1-a$ grows. We are going to see that eventually we are going to get a stable sheaf. Assume we reach $1-a=0$ and $W_0$ is still unstable. The destabilizing sheaf has to be $L \otimes I_{Z_d}$ with $l(Z_d)>l(W)$ and gives a short exact sequence $$ 0 \to L \otimes I_{Z_d} \to W_0 \to I_{Z'_d} \to 0.$$ Note that $l(Z'_d)<l(W')$. Performing the corresponding elementary transformation we get a new family $\overline {\mathcal{W}}$ and the sheaf corresponding to $0$ sits in an exact sequence \begin{equation} 0 \to I_{Z'_d} \to \overline W_0 \to L\otimes I_{Z_d} \to 0. \label{eq1.3} \end{equation} This sequence is like \ref{eq1.2} but with $0\leq l(Z'_d)<l(W')$. If we still don't get a stable sheaf, repeat this. In each step $l(Z'_d)$ decreases, but this must stop because if $l(Z'_d)=0$, the sheaf given by \ref{eq1.3} is stable, as the following lemma shows. Now, once we have obtained a stable sheaf, if it is not locally free, we are done. If it is locally free, then necessarily the subscheme $W'$ is empty, and we get an extension like \ref{eq1.1} as desired. \end{proof} \begin{lemma} \label{1.3} Let $V$ be a torsion free sheaf on a surface $S$ with $\operatorname{Pic} (S)=\mathbb{Z}$, given by an extension $$0 \to \SO_S \to V \to L \otimes I_Z \to 0,$$ where $L$ is the effective generator of $\operatorname{Pic} (S)$. Then $V$ is stable. \end{lemma} \begin{proof} A destabilizing subsheaf should be of the form $L^{\otimes m} \otimes I_W$, with $m>0$. By standard arguments, it is enough to check stability with subsheaves whose quotients are torsion free, so we can assume this. The composition $L^{\otimes m} \otimes I_W \to V \to L \otimes I_Z$ is nonzero, because otherwise it would factor through $\SO_S$, but this is impossible because $m>0$. Then $m=1$ and we have $I_W \hookrightarrow I_Z$. Furthermore, $l(W) > l(Z)$ because if $W=Z$, the sequence would split. Then we have a sequence $$ 0 \to L \otimes I_W \to V \to I_{W'} \to 0,$$ but we reach a contradiction because $c_2 = l(W) + l(W') > l(Z) + l(W') = c_2 + l(W')$. Then there is no destabilizing subsheaf, and $V$ is stable. \end{proof} \begin{proposition} \label{1.4} Let $S$ be a smooth $K3$ surface with Picard group $\operatorname{Pic} (S)=\mathbb{Z}$. If $\dim \operatorname{Ext} ^1(L' \otimes I_Z,L) \geq 2$, then there is a nonsplit extension \begin{equation} 0 \to L \to V \to L' \otimes I_Z \to 0 \label{eq1.4} \end{equation} such that V is not locally free. \end{proposition} \begin{proof} We have an exact sequence $$ 0 \to H^1(L\otimes(L')^{-1}) \to \operatorname{Ext} ^1(L' \otimes I_Z, L) \to H^0({\mathcal{O}}_Z).$$ If $L=L'$, then $H^1(L\otimes(L')^{-1})=0$ because $S$ is a K3 surface. If $L \neq L'$, then due to the fact that $\operatorname{Pic}(S)=\mathbb{Z}$, applying Kodaira's vanishing theorem we also have $H^1(L\otimes(L')^{-1})=0$. We have then an injection $$ 0 \to \operatorname{Ext} ^1(L' \otimes I_Z,L) \stackrel{f}{\rightarrow} H^0({\mathcal{O}}_Z).$$ An extension corresponding to $\xi$ is locally free iff the section $f(\xi)$ generates the sheaf ${\mathcal{O}}_Z$, i.e., iff $$ f(\xi) \notin W=\{ s \in H^0({\mathcal{O}}_Z) : 0=s \otimes k(p) \in H^0({\mathcal{O}}_p) \text { for some } p \in \operatorname{Supp}(Z) \}.$$ $W$ is a union of codimension 1 linear subspaces, hence if $\dim \operatorname{Ext} ^1(L' \otimes I_Z, L) \geq 2$, then $\dim \text{im}(f) \cap W > 0$, and we have a nonzero $\xi$ corresponding to an extension \ref{eq1.4} with $V$ not locally free. \end{proof} Usually we will apply the following corollary \begin{corollary} \label{1.5} Let $S$ be a smooth $K3$ surface with $\operatorname{Pic} (S)=\mathbb{Z}$. If $$ \dim \operatorname{Ext} ^1 (L^{\otimes n+1}\otimes I_Z,L^{\otimes -n}) \geq 2,$$ and there is a stable extension $$0 \to L^{\otimes -n} \to V \to L^{\otimes n+1} \otimes I_Z \to 0,$$ then there is a sheaf $V'$, in the same irreducible component of $\FM (L,c_2)$ as $V$, that is not locally free or sits in an extension $$0 \to L^{\otimes -m} \to V \to L^{\otimes m+1} \otimes I_Z \to 0$$ for some $m<n$. \end{corollary} \begin{proof} There is an open set in $\mathbb{P}(\operatorname{Ext} ^1 (L^{\otimes n+1} \otimes I_Z,L^{\otimes -n}))$ whose points correspond to stable extensions, due to the openness of the stability condition. All these points get mapped to the same irreducible component of $\FM (L,c_2)$. By proposition \ref{1.4}, there is an extension $V$ that is not locally free. If it is not stable, we can take a curve as in proposition \ref{1.2}, and applying the proposition we get a family of stable sheaves. All get mapped to the same component of $\FM (L,c_2)$, and the sheaf corresponding to $t=0$ has the required properties. \end{proof} \section{Small second Chern class} \label{Small second Chern class} In this section we will consider the case in which $c_2 \leq \frac{1}{2} L^2 +3$. Recall that we are assuming that $S$ is a K3 surface with $\operatorname{Pic} (S)=\mathbb{Z}$. In this case we have the following characterization of the stable torsion free sheaves. \begin{proposition} \label{2.1} Let $V$ be a torsion free stable rank two sheaf with $c_1=L$, $c_2 \leq \frac{1}{2} L^2 + 3$. Then $V$ fits in an exact sequence \begin{equation} 0 \to \SO_S \to V \to L \otimes I_Z \to 0. \label{eq2.1} \end{equation} Conversely, every nonsplit extension of $L \otimes I_Z$ by $\SO_S$ is a torsion free stable sheaf. \end{proposition} \begin{proof} Take $V$ stable. Using the Riemann-Roch theorem, $$ h^0(V) + h^2(V) \geq \frac{L^2}{2} - c_2 + 4 \geq 1.$$ If $h^2(V)$ were different from zero, by Serre duality we would have $\text {Hom} (V,{\mathcal{O}}) \not= 0$, contradicting stability because this would give a map $V \to \SO_S$ with image $L^{\otimes -n} \otimes I_Z$ ($n \geq 0$) and kernel $L^{\otimes n+1} \otimes I_{Z'}$. Then $h^0(V) \not= 0$. Take a section of $V$. By stability, the quotient of the section is torsion free, and we have an extension like \ref{eq2.1}. The extension is not split because $V$ is stable. The converse is lemma \ref{1.3}. \end{proof} Now that we know that all sheaves can be written as extensions of $L \otimes I_Z$ by $\SO_S$, the obvious strategy is to construct families of extensions $\mathbb{P}(\Ext ^1 (L \otimes I_Z, \OOS))$ for each $Z$ such that $\dim \Ext ^1 (L \otimes I_Z, \OOS) \geq 1$. Ideally we would like to put all these families together in a bigger family parametrized by a variety $M$. This $M$ would map to $\FM (L,c_2)$ surjectively, so it would be enough to prove the connectedness of $M$, and because $M$ maps to $\Hilb ^{c_2} (S)$ with connected fibers, it would be enough to prove that the set $X=\{ Z \in \Hilb ^{c_2} (S) : \dim \Ext ^1 (L \otimes I_Z, \OOS) \geq 1\}$ (i.e., the image of the map $M \to \Hilb ^{c_2} (S)$) is connected. Unfortunately we cannot construct $M$ because $\dim \Ext ^1 (L \otimes I_Z, \OOS)$ is not constant. We will use a somewhat more elaborate argument to bypass this difficulty, but we will still use the connectivity of $X$, that we prove in the following proposition. \begin{proposition} \label{2.2} The set $X=\{ Z \in \Hilb ^{c_2} (S) :\dim \Ext ^1 (L \otimes I_Z, \OOS) \geq 1\}$ is connected. \end{proposition} \begin{proof} By Serre duality and looking at the sequence $$ 0 \to H^0(L \otimes I_Z) \to H^0(L) \to H^0({\mathcal{O}}_Z) \to H^1(L \otimes I_Z) \to 0,$$ we have $\dim \Ext ^1 (L \otimes I_Z, \OOS) \geq 1\ \iff h^0(L \otimes I_Z)\geq \frac{1}{2} L^2 +3-c_2$. Now consider the following commutative diagram $$ \CD @. @. 0 @. 0 @.\\ @. @. @VVV @VVV @.\\ 0 @>>> \SO_S @>>> L \otimes I_Z @>>> j_*(\omega _C \otimes I_Z) @>>> 0\\ @. @| @VVV @VVV @.\\ 0 @>>> \SO_S @>>> L @>>> L| _C=j_*\omega _C @>>> 0\\ @. @. @VVV @VVV @.\\ @. @. {\mathcal{O}}_Z @= {\mathcal{O}}_Z @.\\ @. @. @VVV @VVV @.\\ @. @. 0 @. 0 @.\\ \endCD $$ where $C \in \mathbb{P}(H^0(L\otimes I_Z))$ (maybe $C$ is singular, but we know it is irreducible and reduced because $\operatorname{Pic} (S)=\mathbb{Z}$ and $L$ is a generator of the group), $j:C \hookrightarrow S$ is the inclusion, and $\omega_C=L|_C$ is the dualizing sheaf on $C$. Using the top row we get $h^0(L \otimes I_Z)\geq \frac{1}{2} L^2 +3-c_2 \iff h^0(\omega _C \otimes I_Z) \geq \frac{1}{2} L^2 +2-c_2$. This condition can be restated in terms of Brill-Noether sets $W^r_d$: $$ \omega _C \otimes I_Z \in W^r_d $$ where $r=\frac{1}{2}L^2 +1-c_2$, and $d=L^2 -c_2$. By a theorem of Fulton and Lazarsfeld \cite {F-L}, the Brill-Noether set $W^r_d$ of a smooth curve is nonempty and connected if the expected dimension $\rho (r,d) = g-(r+1)(g-d+r)$ is greater than zero. In the case of an irreducible reduced curve lying on a $K3$ the generalized Jacobian can be compactified, and the connectedness result is still true (theorem I). In our case we have $$ \rho (r,d)=2c_2-\frac{L^2}{2}-3= \frac{{\dim \FM (L,c_2)}}{2} > 0,$$ (recall that for $\dim \FM (L,c_2)=0$ the irreducibility of the moduli space is known by the work of Mukai \cite{M}) and we can apply the theorem. Now consider the variety $$N = \{ (Z,C): Z \subset C, \dim \Ext ^1 (L \otimes I_Z, \OOS) \geq 1\} \subset \Hilb ^{c_2} (S) \times \mathbb{P}(H^0(L))$$ and the projections $$ \CD N @>p_2>> \mathbb{P}(H^0(L)) \\ @Vp_1VV @. \\ \Hilb ^{c_2} (S) @. \\ \endCD $$ By theorem I, $p_2$ is surjective with connected fibers. Then $N$ is connected, and also the image of $p_1$, that is equal to $X$. \end{proof} For the following proposition we will need this lemma: \begin{lemma} \label{2.3} Let $T$ be a smooth curve, $p$ a point in $T$ and $S$ a variety. Consider the diagram $$ \CD S @>v>> S\times T \\ @VgVV @VfVV \\ p @>u>> T \\ \endCD $$ Then for every coherent torsion free sheaf ${\mathcal{F}}$ on $S\times T$, there exist a natural map $$ (f_* {\mathcal{F}} )(p) \to H^0({\mathcal{F}} _{S\times \{p\} }), $$ where ${\mathcal{F}}(p)=v^* {\mathcal{F}}$. Furthermore, this map is injective. \end{lemma} \begin{proof} The question is local in $T$, so we can assume that $T$ is affine, $T=\operatorname{Spec} A$, and there is an element $x\in A$ such that $p$ is the zero locus of $x$. We have \begin{equation} 0 \to {\mathcal{F}} \stackrel{\cdot x}{\to} {\mathcal{F}} \to {\mathcal{F}} / x\cdot {\mathcal{F}} \to 0, \label{eq2.2} \end{equation} where the map on the left is multiplication by $f^*x$. On the other hand we have the sequence $$ 0 \to {\mathcal{O}}_{S \times T}(-f^*p) \to {\mathcal{O}}_{S\times T} \to {\mathcal{O}}_{S\times \{p\} } \to 0. $$ Tensoring with ${\mathcal{F}}$ is right exact, so we get an exact sequence $$ {\mathcal{F}} \otimes {\mathcal{O}}_{S \times T}(-f^*p) \to {\mathcal{F}} \to {\mathcal{F}}_p \to 0. $$ Note that the image of the left map is $x\cdot {\mathcal{F}}$, and then we conclude that ${\mathcal{F}} / x\cdot {\mathcal{F}}$ is isomorphic to ${\mathcal{F}}_p$. Taking cohomology in the sequence \ref{eq2.2} we get $$ {H^0({\mathcal{F}})}/(x\cdot H^0({\mathcal{F}})) \hookrightarrow H^0({\mathcal{F}}_p), $$ but the first group is exactly $(f_* {\mathcal{F}})|_p$. \end{proof} Finally we can prove: \begin{proposition} \label{2.4} The moduli space $\FM (L,c_2)$ of torsion free, rank two sheaves with $c_2 \leq \frac{1}{2}L^2 + 3$ over a K3 surface with $\operatorname{Pic} (S)=\mathbb{Z}$ is connected (hence irreducible, because we know it is smooth). \end{proposition} \begin{proof} We have a stratification of $X$ $$ X=\bigcup _{r \geq 1} H_r, H_r=\{ Z \in \Hilb ^{c_2} (S) : \dim \Ext ^1 (L \otimes I_Z, \OOS) =r\}. $$ On each stratum $H_r$ we can construct a projective bundle $M_r \to H_r$ with fiber $\mathbb{P} (\Ext ^1 (L \otimes I_Z, \OOS))$, because the dimension of the group is constant. Each point of $M_r$ corresponds to an extension (up to weak isomorphism of extensions). We have then morphisms $M_r \to \mathfrak{M} (L,c_2)$ with fiber $\mathbb{P}(H^0(V))$ over $V$ (see proposition \ref{3.8}). We have $h^0(V)=\frac{1}{2}L^2 +3-c_2 + h^1(L \otimes I_Z)$, and a corresponding stratification of $\FM (L,c_2)$ $$ \FM (L,c_2)=\bigcup _{r \geq 1} \mathfrak{M} _r ,\ \ \ \mathfrak{M} _r = \{V \in \mathfrak{M} (L,c_2) : h^0(V) = \frac{1}{2}L^2 +3-c_2 + r \} $$ (the reason for this dependence on $r$ in the definition is that $h^0(V) = \frac{1}{2}L^2 +3-c_2+h^1(L\otimes I_Z)$. The condition is equivalent to requiring that if $V \in \FM (L,c_2)$ and $V$ is an extension of $L\otimes I_Z$ by $\SO_S$, then $h^1(L\otimes I_Z)=r$. The previous formula proves that $r$ only depends on $V$). Note that $M_r$ can be thought also as a projective bundle over $\mathfrak{M} _r$ with fiber $\mathbb{P}(H^0(V))$. To prove that $\FM (L,c_2)$ is connected, we will show that for any two sheaves $V_a \in \mathfrak{M} _a$, $V_b \in \mathfrak{M} _b$, we can construct a family ${\mathcal{V}}$ of stable sheaves on a connected parameter space, with ${\mathcal{V}} |_0 = V_a$, ${\mathcal{V}} |_1 = V_b$ Due to the fact that $X$ is connected, it is enough to prove this for $V_a$, $V_b$ given by extensions $$ \CD 0 \to \SO_S @>s_a>> V_a \to L \otimes I_{Z_a} \to 0 \\ 0 \to \SO_S @>s_b>> V_b \to L \otimes I_{Z_b} \to 0 \endCD $$ such that $Z_a \in H_a$, $Z_b \in H_b$ and $Z_a \in \overline H_b$, the closure of $H_b$. Take a curve $f:T \to \Hilb ^{c_2} (S)$ with $f(0)=Z_a$, $f(1)=Z_b$ and $\operatorname{im} (T-\{0\}) \in H_b$. $T$ doesn't need to be complete. By shrinking $T$ to a smaller open set, we can assume that there is a lift $f$ to a map $f:T\setminus\{0\} \to M_b$. This gives a family of sheaves $\widetilde {\mathcal{V}}$ and sections $s_t \in H^0({\mathcal{V}}|_t)$ parametrized by $T\setminus\{0\}$. We want to extend this to a family ${\mathcal{V}}$ of sheaves and sections parametrized by $T$, in such a way that the cokernel of $s_0:\SO_S \to {\mathcal{V}} |_0$ is $L \otimes I_{Z_a}$. Maybe ${\mathcal{V}} |_0$ won't be isomorphic to $V_0$, but at least both are extensions of the same sheaf $L \otimes I_{Z_a}$ by $\SO_S$, and then they are in the same connected component, and this is enough. This family gives a morphism $T-\{0\}\to \FM (L,c_2)$ that extends to a unique $T \to \FM (L,c_2)$ by properness (see section \ref{secPreliminaries}, just before proposition \ref{1.2}). With this we have already extended the family $\widetilde {\mathcal{V}}$ to a family ${\mathcal{V}}$ parametrized by $T$, and we only need to extend the sections $s_t$. Shrinking $T$ to a smaller neighborhood of $\{0\}$ if necessary, we can assume that $\pi _2{}_* {\mathcal{V}}$ is trivial ($\pi _1$ and $\pi _2$ are the first and second projections of $S \times T$). Then the sections $s_t$ fit together to give ${\mathcal{O}}_{S \times (T-\{0\})} \to \widetilde {\mathcal{V}}$, i.e. an element $\tilde s \in ({\pi_2} _* {\mathcal{V}})(T-\{0\})$. This can be extended to some $s\in ({\pi_2} _* {\mathcal{V}})(T)$ that is nonzero on the fiber of $t=0$. Using the previous lemma, we have an injection $({\pi_2} _* {\mathcal{V}})|_{t=0} \to H^0({\mathcal{V}} |_0)$ that gives a nonzero section $s_0$ of ${\mathcal{V}} |_0$. Now we only have to check that the cokernel of this section is $L \otimes I_{Z_a}$. We have a short exact sequence over $S \times T$ $$ 0 \to {\mathcal{O}}_{S \times T} \stackrel{s}{\to} {\mathcal{V}} \to {\mathcal{Q}} \to 0. $$ Then ${\mathcal{Q}}$ is torsion free, flat over $T$, and then it is of the form $${\mathcal{Q}} = {\pi _1}^* (L) \otimes I_{\mathcal{Z}} \otimes {\pi _2}^* (L')$$ for some line bundle $L'$ over $T$ and some subscheme ${\mathcal{Z}}$ of $S \times T$ flat over $T$. This subscheme gives a morphism $g:T \to \Hilb ^{c_2} (S)$. By construction $f(t)=g(t)$ for $t \not= 0$, and by properness the equality also holds for $t=0$, then $Z_0 = Z$ as desired. \end{proof} \section{Large second Chern class} \label{Large second Chern class} In this section we will handle the case in which $c_2$ is large. \begin{proposition} \label{3.1} Assume that $c_2$ satisfies $$ ((n-1)^2 + (n-1) + \frac{1}{2}) L^2 + 3 < c_2 \leq (n^2 + n + \frac{1}{2}) L^2 + 3. $$ If $V$ is locally free, then $V$ fits in a short exact sequence $$ 0 \to L^{\otimes -m} \to V \to L^{\otimes m+1} \otimes I_{Z_m} \to 0, l(Z_m) = c_2 + (m+1) m L^2 , $$ with $0 \leq m \leq n$. We will call such an exact sequence an extension of type $m$. \end{proposition} \begin{proof} For any sheaf $V$, $h^2(V\otimes L^{\otimes n}) = \dim \operatorname{Hom}(V, L^{\otimes -n}) = 0$ by stability, and then using the Riemann-Roch theorem we have \begin{equation} h^0(V\otimes L^{\otimes n}) \geq \frac{L^2}{2} - c_2 + 4 +(n + n^2) L^2 \geq 1, \label{eq3.1} \end{equation} so that we have an inclusion $L^{\otimes -n} \hookrightarrow V$. If $V$ is locally free, this will give an exact sequence on type $m$, with $m \leq n$. \end{proof} \begin{proposition} \label{3.2} If a component $\mathfrak{M}'$ has sheaves of type $m$, with $0 \leq m \leq n-1$, then it has sheaves that are not locally free. \end{proposition} \begin{proof} Choose $m$ such that there is no $V$ of type $m'$ for $m' < m$. Let $V$ be of type $m$. By Serre duality $$ \dim \operatorname{Ext} ^1(L^{\otimes m+1} \otimes I_{Z_m}, L^{\otimes -m}) = h^1(L^{\otimes 2m+1} \otimes I_{Z_m}), $$ and this is greater than 2. By proposition \ref{1.5}, $\mathfrak{M}'$ has a sheaf that is not locally free or is of type $m'<m$, but the later cannot happen because of the choice of m. \end{proof} \begin{lemma} \label{3.3} Let $X$ be an irreducible component of $\FM (L,c_2)$. Let $X_0$ be the subset corresponding to non-locally free sheaves. If $X_0$ is not empty, then it has codimension one. Furthermore, there is a dense subset $Y$ of $X_0$ such that for any $F \in Y$, we have $$ c_2(F^{\vee\vee})=c_2(F)+1.$$ \end{lemma} \begin{proof} By \cite {O1}, prop. 7.1.3, we know that \begin{equation} \operatorname{codim}(X_0,X)\leq 1. \label{bound} \end{equation} On the other hand, let $F\in X_0$. It fits in an exact sequence $$ 0 \to F \to F^{\vee\vee} \to {\mathcal{Q}} \to 0 $$ where ${\mathcal{Q}}$ is an Artinian sheaf with length $l=H^0({\mathcal{Q}})=c_2(F^{\vee\vee})-c_2(F)$. We use this to bound the dimension of $X_0$ by a parameter count. First we choose a locally free sheaf $E\in \mathfrak{M}(L,c_2-l)$. These requires $4(c_2-l)-L^2-6$ parameters. Now we have to choose a quotient to a sheaf of length $l$ concentrated on a subset of dimension zero. These quotients are parametrized by the Grothendieck Quot scheme $\operatorname{Quot} (E,l)$, whose dimension is $3l$ (this follows from \cite {L1} Appendix, where it is proved that $\operatorname{Quot} ^0(E,l)$, the Quot scheme corresponding to quotients supported in $l$ distinct points, is dense in $\operatorname{Quot}(E,l)$). Define the following stratification on $X_0$: $$ X_0=\bigcup_{l\geq 1} X_0^l \text{, }X_0^l=\{F: c_2(F^{\vee\vee})-c_2(F)=l\} $$ Then we have $$ \dim (X_0^l) \leq 4(c_2-l)-L^2-6 + 3l $$ and together with the previous bound \ref{bound} we obtain that $Y=X_0^1$ is dense and $\operatorname{codim}(X_0,X)=1$. \end{proof} \begin{proposition} \label{3.4} If $c_2$ satisfies the inequalities of the hypothesis of proposition \ref{3.1}, then there is an open dense set on $\FM (L,c_2)$ that corresponds to sheaves of type n. \end{proposition} \begin{proof} We will prove this by showing that the codimension of sheaves of type $m \leq n-1$ is greater than zero. We will divide the proof into two cases: \textbf{Case 1.} \textit{Extensions of type $m$ with $Z_m$ such that $h^0(L^{\otimes 2m+1} \otimes I_{Z_m}) = 0$.} We have then $h^1(L^{\otimes 2m+1} \otimes I_{Z_m}) = c_2 -(m^2 +m+\frac{1}{2}) L^2 -2$. The dimension of the family $M$ of extensions of this kind is bounded (via Serre duality) by $$ \dim M \leq 2 l(Z_m) + h^1(L^{\otimes 2m+1} \otimes I_{Z_m}) -1 = 3 c_2 +(m^2 +m-\frac{1}{2}) L^2 -3. $$ There is a map $\pi:M \to \FM (L,c_2)$ with fiber over each $V$ equal to $\mathbb{P}(H^0(V\otimes L^{\otimes n}))$, and we can give a bound to its dimension (see proof of proposition \ref{3.1}). $$ \dim \mathbb{P}(H^0(V\otimes L^{\otimes n})) \geq (n^2 + n + \frac{1}{2}) L^2 - c_2 + 3 $$ Then the dimension of the image of $\pi$ is bounded by \begin{eqnarray*} \dim (\operatorname{im} \pi) & \leq & \dim (M) - \min \dim \mathbb{P}(H^0(V\otimes L^{\otimes n})) \\ & \leq & 4 c_2 - L^2 -6 - 2nL^2 \end{eqnarray*} And then $$ \operatorname{codim} (\operatorname{im} \pi) \geq 2nL^2 >0 $$ \textbf{Case 2.} \textit{Extensions of type $m$ with $Z_m$ such that $h^0(L^{\otimes 2m+1} \otimes I_{Z_m}) \geq 1$.} Let $M$ be the family of stable extensions with $Z_m$ satisfying this inequality. Let $l = l(Z_m) = c_2 + (m+1)mL^2$. The subscheme $Z_m$ is in a curve $C$ defined as the zeroes of a section of $L^{\otimes 2m+1}$, i.e. $Z_m \in \operatorname{Hilb} ^l (C)$. Although this curve will be reducible and not reduced in general, the fact that $C$ is in a smooth surface allows us to prove: \begin{lemma} \label{3.5} $\dim \operatorname{Hilb} ^d (C) = d$ \end{lemma} \begin{proof} We have a natural stratification of $\operatorname{Hilb} ^d (C)$ given by the number of points in the support of a subscheme $$ \operatorname{Hilb} ^d (C) = \bigcup _{1 \leq r \leq d} \operatorname{Hilb} ^d _r (C), $$ where $\operatorname{Hilb} ^d _r (C) = \{ Z : \# \operatorname{Supp} Z = r\}$. We have natural maps giving the support of a subscheme: $$ z^C _r : \operatorname{Hilb} ^d _r(C) \to \operatorname{Sym} ^r(C). $$ In the same way we define maps $z^S _r$, when we consider subschemes of the surface $S$. Clearly, if $x \in \operatorname{Sym} ^r (C) \subset \operatorname{Sym} ^r (S)$, then $(z^C _r)^{-1} (x) \subset (z^S _r)^{-1} (x)$. Taking this into account, and using a result of Iarrobino about zero dimensional subschemes of a smooth surface \cite {I}: \begin{eqnarray*} \dim \operatorname{Hilb} ^d _r (C) &\leq & \dim \operatorname{Sym} ^r (C) + \operatorname{max.dim} (z^C _r)^{-1} (x) \\ &\leq & \dim \operatorname{Sym} ^r (C) + \operatorname{max.dim} (z^S _r)^{-1} (x)\\ &= & r + d - r = d \end{eqnarray*} This gives $\dim \operatorname{Hilb} ^d (C) \leq d$, and the opposite direction is trivial. This finishes the proof the lemma. \end{proof} Now we are going to bound the dimension of the set $$ H' = \{ Z \in \operatorname{Hilb}^l (S): h^0(L^{\otimes 2m+1} \otimes I_Z) \geq 1\}. $$ Consider the diagram $$ \CD Y = \{ (Z,C) \in \operatorname{Hilb}^l \times \mathbb{P}(H^0(L^{\otimes 2m+1}) : Z \in C\} @>p_2>> \mathbb{P}(H^0(L^{\otimes 2m+1})) \\ @Vp_1VV @. \\ \operatorname{Hilb} ^l (S) @. \\ \endCD $$ We have $H' = \operatorname{im} (p_1)$ and the fiber of $p_2$ is $\operatorname{Hilb} ^l (C)$. $p_2$ is clearly surjective and the fiber of $p_1$ is $\mathbb{P} (H^0(L^{\otimes 2m+1} \otimes I_Z))$. Then $$ \dim H' = \dim \mathbb{P} (H^0(L^{\otimes 2m+1})) + \dim \operatorname{Hilb} ^l (C) - \dim \mathbb{P} (H^0(L^{\otimes 2m+1} \otimes I_Z)) = 2l-1.$$ Again we have a map $\pi:M \to \FM (L,c_2)$ with fiber $\mathbb{P} (H^0(V\otimes L^{\otimes m}))$, and then $$ \operatorname{codim} (\operatorname{im} \pi) \geq (2m+1) L^2 +1 > 0. $$ \end{proof} As a corollary to this proposition we learn that to prove connectedness of $\FM (L,c_2)$ it is enough to prove that all type $n$ sheaves are in one component. \begin{proposition} \label{3.6} All stable extensions of $L^{\otimes n+1} \otimes I_Z$ by $L^{\otimes -n}$ such that $L^{\otimes 2n+1} \otimes I_Z$ has a section corresponding to an integral curve, are in one component. \end{proposition} \begin{proof} Define the sets \begin{eqnarray*} \widetilde X_r & = & \{Z \in \Hilb ^{c_2 + n(n+1)L^2} (S) : \dim \Ext ^1 (L^{\otimes n+1} \otimes I_Z, L^{\otimes -n}) = r \text { and} \\ & &L^{\otimes 2n+1} \otimes I_Z \text { has a section corresponding to an integral curve} \} \\ & & \\ X_r &= &\{ Z \in \widetilde X_r : \text { there is a stable extension of } L^{\otimes n+1} \otimes I_Z \text { by } L^{\otimes -n} \} \\ \widetilde M_r &=& \{ \text {extensions of } L^{\otimes n+1} \otimes I_Z \text {by } L^{\otimes -n} \text { with } Z \in X_r \}\\ & & \\ M_r &=& \{ m \in M_r : m \text { corresponds to a stable extension} \} \\ & & \\ N_r &=& \{(Z,C) \in \Hilb ^{c_2 + n(n+1)L^2} (S) \times \mathbb{P}(H^0(L^{\otimes 2n+1})) : Z \subset C \\ & & \text { and } \dim \Ext ^1 (L^{\otimes n+1} \otimes I_Z, L^{\otimes -n}) = r \} \\ & & \\ U &=& \{ C \in \mathbb{P}(H^0(L^{\otimes 2m+1})) : C \text { is irreducible and reduced} \} \end{eqnarray*} We construct $\widetilde M_r$ as parameter spaces of universal families of extensions by standard techniques. These techniques require that the dimension of the $\operatorname{Ext} ^1$ group is constant on the whole family. This why we have to introduce the subscript $r$ and break everything into pieces according to the dimension of the group $\operatorname{Ext} ^1$. We also consider the unions $$ \widetilde M = \bigcup _{r \geq 1} \widetilde M_r \text {, } M = \bigcup _{r \geq 1} M_r \text {, } X = \bigcup _{r \geq 1} X_r \text {, } \ldots $$ Note that $X$, being a subset of $\Hilb ^{c_2 + n(n+1)L^2} (S)$, has a natural scheme structure. This is also true for $N \subset C \times \mathbb{P}(H^0(L^{\otimes 2n+1}))\ $. On the other hand, for $M_r$ there is no natural way of ``putting them together'', so we take just the disjoint union. We have the following maps $$ \CD \widetilde M @<<< M @>>> \FM (L,c_2) \\ @VVV @VVV @. \\ \widetilde X @<<< X @. \\ @A{p_1}AA @. @. \\ N @>{p_2}>> \mathbb{P}(H^0(L^{\otimes 2n+1})) @. \\ \endCD $$ By construction $\widetilde X = p_1 {p_2}^{-1}(U)$. Now we are going to prove that the fibers of $p_2$ over $U$ are nonempty and connected. For each point in $N$ we have a commutative diagram $$ \CD @. @. 0 @. 0 @.\\ @. @. @VVV @VVV @.\\ 0 @>>> \SO_S @>>> L^{\otimes 2n+1} \otimes I_Z @>>> j_*(\omega _C \otimes I_Z) @>>> 0\\ @. @| @VVV @VVV @.\\ 0 @>>> \SO_S @>>> L^{\otimes 2n+1} @>>> L^{\otimes 2n+1}|_C=j_*\omega _C @>>> 0\\ @. @. @VVV @VVV @.\\ @. @. {\mathcal{O}}_Z @= {\mathcal{O}}_Z @.\\ @. @. @VVV @VVV @.\\ @. @. 0 @. 0 @.\\ \endCD $$ We argue in the same way as in proposition \ref{2.2}. Here we have $$ r=(n^2+n+\frac{1}{2})L^2 +1 -c_2 \text{, } d=(3n^2+3n+1)L^2-c_2 $$ $$ \rho (r,d)=2c_2-\frac{L^2}{2}-3=\frac{{\dim \FM (L,c_2)}}{2} > 0.$$ But now we don't know if $p_2$ is surjective with connected fibers, because theorem I only applies for irreducible reduced curves. This is the reason why we introduce the open set $U$. For the fibers on $U$ we can apply the theorem, and we conclude that $p_2$ is surjective over $U$ with connected fibers. Then $\widetilde X=p_1 {p_2}^{-1}(U)$ is connected. \textbf{Case 1.} \textit{$p_1 {p_2}^{-1} (U) = X$.} If $X_1 = X$, then we can construct a (connected, because $X$ is connected) family $M_1$ parametrizing all sheaves with the required properties, and we are done. Now, if $X_1 \not= X$, then there are extensions with $r \geq 2$. By corollary \ref{1.5}, $M_r$ with $r \geq 2$ is mapped to $\mathfrak{M} _0$, the irreducible component that has sheaves that are not locally free. There is only one irreducible component with this property, because by induction hypothesis the moduli space when the second Chern class is smaller than $c_2$ is irreducible (see the outline of the proof). Now we have to show that all the connected components of $M_1$ also go to this component $\mathfrak{M} _0$. The connectivity of $X=\widetilde X$ and the fact that $\dim \Ext ^1 (L^{\otimes n+1} \otimes I_Z, L^{\otimes -n})$ is upper semicontinuous allows us to take a curve $f:T \to X$ with $f(T-\{0\})$ in any given connected component of $X_1$ and $f(0) \in X_r$, with $r \geq 2$. Lift $f$ to a map $f:T-\{0\} \to M_1$. Note that $M_1$ won't be in general a projective bundle because we have removed the points corresponding to unstable extensions, but these make a closed set, and (maybe after restricting $T$ to a smaller open set) we can construct the lift without hitting this set. $M_1$ maps to $\FM (L,c_2)$, and then we have a map $\phi : T-\{0\} \to \FM (L,c_2)$. As in the proof of proposition \ref{2.4}, this gives us a family of stable sheaves and sections parametrized by $T-\{0\}$ that we can extend to a family parametrized by $T$. Now there are two possibilities: If $\phi(0)$ is of type $n$, then we have a family of extensions $$ 0 \to L^{\otimes -n} \to V_t \to \SO_S(L^{\otimes n+1} \otimes I_{Z_t}) \to 0 $$ and a corresponding map $\psi : T \to \Hilb ^{c_2 + n(n+1)L^2} (S)$, $t \mapsto Z_t$. By construction $\psi(t) = f(t)$ for $t \not= 0$, and by properness also for $t=0$. The extension corresponding to $t=0$ has to be in $M_r$ with $r \geq 2$, and then $M_1$ is also mapped to $\mathfrak{M} _0$. On the other hand, if $\phi(0)$ is not a vector bundle of type $n$, then it is either of type $m$ for $m<n$ or it is not locally free. In either case, we conclude that $M_1$ is also mapped to $\mathfrak{M} _0$ \textbf{Case 2.} \textit{$p_1 {p_2}^{-1} (U) \not= X$.} Again, $M_r$, $r \geq 2$, gets mapped to $\mathfrak{M} _0$. No connected component of $X_1$ can be closed, because by connectedness of $\widetilde X$ and upper semicontinuity of $\dim \Ext ^1 (L^{\otimes n+1} \otimes I_Z, L^{\otimes -n})$, we would have $\widetilde X = X_1$, and then $X = \widetilde X$, contrary to the hypothesis. Now we can prove that every connected component of $M_1$ gets mapped to $\mathfrak{M} _0$. Take the corresponding connected component of $X_1$. Take a curve $f:T \to \widetilde X$, with $f(T-\{0\})$ in the given connected component of $X_1$, and $f(0) \notin X_1$. As in the previous case, lift $f$ to a map $f:T-\{0\} \to M_1$, and now the proof finishes like the end of case 1. \end{proof} \begin{proposition} \label{3.7} The set of sheaves $V$ of type $n$ such that for any extension $$ 0 \to L^{\otimes -n} \to V \to L^{\otimes n+1} \otimes I_Z \to 0,$$ $L^{\otimes 2n+1} \otimes I_Z$ has no section whose zero locus is an integral curve, has positive codimension. \end{proposition} \begin{proof} Define $\widetilde P=\{ Z \in \Hilb ^{c_2 + n(n+1)L^2} (S) : L^{\otimes 2n+1} \otimes I_Z$ has no sections whose zero locus is an irreducible reduced curve $\}$. For each point of $\widetilde P$ we have a family of extension of type $n$ given by the projectivization of the corresponding $\operatorname{Ext} ^1$ group. Writing $\widetilde P = \cup \widetilde P_r$, with $r$ equal to the dimension of the group, we can construct a family of extensions $\widetilde M_r^P$ for each $r$. As is the previous proposition, let $P_r \subset \widetilde P_r$ be the subset that has stable extensions. We have a natural map $\pi_1:M_r^P \to \FM (L,c_2)$, where $M_r^P$ is the subset of $\widetilde M_r^P$ corresponding to stable sheaves. \begin{lemma} \label{3.8} The fiber of $\pi_1$ over $V \in \FM (L,c_2)$ is $\mathbb{P} (H^0(V\otimes L^{\otimes n}))$. \end{lemma} \begin{proof} The fiber consists of all extensions giving the same $V$. Now, given a point in $\mathbb{P} (H^0(V\otimes L^{\otimes n}))$, we have an injection $f: L^{\otimes -n} \hookrightarrow V$ (up to scalar). $V$ is locally free and of type $n$, then the quotient is torsion free and we get an element $Z_f$ of $\Hilb ^{c_2 + n(n+1)L^2} (S)$: $$0 \to L^{\otimes -n} \to V \to L^{\otimes n+1} \otimes I_{Z_f} \to 0.$$ This defines a map from $\mathbb{P} (H^0(V\otimes L^{\otimes n}))$ to the fiber of $\pi$. It is clearly surjective. Now we will check that it is also injective. If $Z_f = Z_{f'}$, then $f$ and $f'$ have to differ at most by scalar multiplication, because all nonsplit extensions of $L^{\otimes n+1} \otimes I_{Z_f} = L^{\otimes n+1} \otimes I_{Z_{f'}}$ by $L^{\otimes -n}$ that give the same $V$ are weak isomorphic, so we get a diagram: $$ \CD 0 @>>> L^{\otimes -n} @>f>> V @>>> L^{\otimes n+1} \otimes I_{Z_f} @>>> 0 \\ @. @V{\alpha}VV @V{\cong}VV @V{\beta}VV @. \\ 0 @>>> L^{\otimes -n} @>f'>> V @>>> L^{\otimes n+1} \otimes I_{Z_{f'}} @>>> 0 \\ \endCD $$ where $\alpha$ is multiplication by scalar. \end{proof} In $\mathbb{P} (H^0(L^{\otimes 2n+1}))$ we have a subvariety $Y$ corresponding to reducible curves. This subvariety is the image of the natural map $$\bigcup _{0<a<2n+1} \mathbb{P}(H^0(L^{\otimes a})) \times \mathbb{P}(H^0(L^{\otimes 2n+1-a})) \to \mathbb{P}(H^0(L^{\otimes 2n+1}))$$ We define the set \begin{eqnarray*} \widetilde N_r^P & = &\{ (Z,C) \in X_r \times \mathbb{P}(H^0(L^{\otimes 2n+1})): \\ & &Z \subset C, \dim (p_1)^{-1}(Z)=(n^2+n+ \frac{1}{2})L^2-c_2 +1+r \} \end{eqnarray*} and the maps $$ \CD \widetilde N_r^P @>p_2>> \mathbb{P}(H^0(L^{\otimes 2n+1})) \\ @Vp_1VV @. \\ \Hilb ^{c_2 + n(n+1)L^2} (S) @. \\ @A{\pi_2}AA @. \\ M_r^P @>{\pi_1}>> \FM (L,c_2) \\ \endCD $$ By construction we have $P_r \subset \widetilde P_r \subset p_2 p_1^{-1} (Y)$. Then $$ \dim P_r \leq \dim p_2((p_1)^{-1} (Y)) = \dim Y + \dim\operatorname{fiber} (p_2) - \dim\operatorname{fiber} (p_1), $$ where $\dim Y$ is the maximum of the dimensions of its irreducible components. Finally $$\operatorname{codim} (\operatorname{im} \pi_1) = \dim \FM (L,c_2) - \dim P_r -\dim\operatorname{fiber} \pi_2 + \dim\operatorname{fiber} \pi_1,$$ and putting everything together we have $\operatorname{codim} (\operatorname{im} \pi_1) > (2n-a)(a-1)L^2$ for every $0<a<n+1$, and then $\operatorname{codim} (\operatorname{im} \pi_1) > 0$. \end{proof} \section{General K3 surface (proof of theorem II)} \label{General K3 surface} In this section we finally prove theorem II by showing that if the result is true for a surface $S$ with $\operatorname{Pic}(S)=\mathbb{Z}$, then it also holds under the hypothesis of theorem II. The idea is to deform the given surface to a generic surface with $\operatorname{Pic}(S)=\mathbb{Z}$. We also deform the moduli space, and then the irreducibility of the moduli space for the deformed surface will imply the irreducibility for the surface we started with. This is very similar to an argument in \cite{G-H}. Because we are going to vary the surface, in this section we will denote the moduli space of semistable sheaves with $\mathfrak{M}_H(S,L,c_2)$, where $S$ is the surface on which the sheaves are defined. \smallskip \begin{proof2} \textit{of theorem II} Recall that we have a surface $S$ with a $(L,c_2)-$generic polarization $H$. By 2.1.1 in \cite{G-H}, there is a connected family of surfaces ${\mathcal{S}}$ parametrized by a curve $T$ and a line bundle ${\mathcal{L}}$ on ${\mathcal{S}}$ such that $({\mathcal{S}}_0,{\mathcal{L}}_0) = (S,L)$ and $\operatorname{Pic}({\mathcal{S}}_t)={\mathcal{L}}_t \cdot \mathbb{Z}$ for $t\neq 0$. By proposition 2.3 in \cite{G-H}, there is a connected smooth proper family ${\mathcal{Z}} \to T$ such that ${\mathcal{Z}}_0 \cong \mathfrak{M}_H(S,{\mathcal{L}}_0,c_2)$ (note that the polarization is $H$ and not ${\mathcal{L}}_0$) and ${\mathcal{Z}}_t \cong \mathfrak{M}_{{\mathcal{L}}_t}({\mathcal{S}}_t,{\mathcal{L}}_t,c_2)$ for $t \neq 0$. By propositions \ref{2.4} and \ref{3.4} we know that ${\mathcal{Z}}_t$ is irreducible for $t \neq 0$, and then by an argument parallel to lemma \ref{bn0.2}, we obtain that ${\mathcal{Z}}_0$ is connected, but ${\mathcal{Z}}_0$ is smooth (because $H$ is generic), and then this implies that ${\mathcal{Z}}_0$ is irreducible. \end{proof2} \chapter{Irreducibility of the moduli space for del Pezzo surfaces} \label{dp} In this chapter we will consider the case in which $S$ is a del Pezzo surface. We will prove the following theorem (see chapter \ref{Preliminaries} for the notation). \smallskip \noindent\textbf{Theorem III.} \textit{ Let $S$ be a del Pezzo surface. Let $\mathfrak{M}_L(S,c_1,c_2)$ be the moduli space of rank 2, Gieseker L-semistable torsion free sheaves with Chern classes $c_1$, $c_2$, with $L$ a $(c_1,c_2)$-generic polarization. Then $\mathfrak{M}_L(S,c_1,c_2)$ is either empty or irreducible.} \smallskip As we explained in chapter \ref{Preliminaries}, this is already known for $\mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1$, so we will assume from now on that $S$ is a surface isomorphic to $\mathbb{P}^2$ blown up at most at 8 points in general position. We denote the blow up map $\pi:S \to \mathbb{P}^2$. We will denote by $H$ the effective generator of $\operatorname{Pic}(\mathbb{P}^2)$. We also denote by $H$ the pullback $\pi^*(H)$. $\mathfrak{M}_L(X,c_1,c_2)$ will denote the moduli space of $L$-semistable torsion free rank two sheaves on $X$ with Chern classes $c_1$, $c_2$. In the case $X=\mathbb{P}^2$ we will drop $L$ from the notation, because there is only one possible polarization. This is a particular case of the conjecture of Friedman and Qin \cite{F-Q} that states that the moduli space of stable vector bundles on a rational surface with effective anticanonical line bundle is either empty or irreducible (for any choice of polarization and Chern classes). \begin{proposition} \label{dp1} $\mathfrak{M}_{L_0}(S,c_1,c_2)$ is either irreducible or empty, where $L_0$ is a polarization of $S$ that lies in a chamber whose closure contains $H$. \end{proposition} \begin{proof} By \cite{B}, $\mathfrak{M}_{L_0}(S,bH+a_1E_1+\cdots+a_nE_n,c_2)$ is birational to a $(\mathbb{P}^1)^m$ bundle over $\mathfrak{M}(\mathbb{P}^2,bH,c_2)$ (where $m$ is the number of $a_i$'s that are odd), but it is well known that this moduli space is either irreducible or empty. \end{proof} To prove this statement for any generic polarization, we will need some lemmas about the following system of equations on integer numbers: $$ \left. \begin{array}{rl} a_1^2+ \cdots +a_8^2 &= x+b^2\\ -a_1- \cdots -a_8&=x-2+3b \end{array} \right \} \ \ \ \begin{array}{c} (\dagger) \end{array} $$ where $x$ is some given (integer) number, and $b$,$a_1,\ldots,a_8$ are the unknowns. \begin{lemma} \label{dp2} If $x\geq3$ then any integer solution of $(\dagger)$ with $b>0$ has $b\leq2$. \end{lemma} \begin{proof} This is obtained by an elementary argument. We can interpret geometrically these equations as the intersection of a one-sheeted hyperboloid and a plane. We will look first at real solutions. Rewrite $(\dagger)$ as $$ \left. \begin{array}{rl} b &= \frac {1}{3}(-a_1 - \cdots -a_8 +2-x) \\ a_1^2+ \cdots +a_8^2 &= x+\frac {1}{9}(-a_1 - \cdots -a_8 +2-x)^2 \end{array} \right \} $$ The first equation defines a function, and the second equation is a constrain. Using the method of Lagrange multipliers we obtain that the maximum and minimum values of $b$ are at points of the form $a_i=t$ for some $t$. Then $(\dagger)$ become $$ \left . \begin{array}{rl} 8t^2 & =b^2+x \\ -8t-3b & =-2+x \end{array} \right \} $$ Looking at the real solutions of these equations, we find \begin{equation} \label{bpm} b^\pm = \frac {-6(x-2)\pm \sqrt{36(x-2)^2-4((x-2)^2-8x)}}{2}. \end{equation} For $x\geq 3$, $b^-$ is always negative. If we want to have solutions with $b>0$ we need $b^+>0$. Using \ref{bpm}, this is equivalent to $(x-2)^2<8x$, and this implies $x<(12+\sqrt{128})/2<12$. This bound, together with the hypothesis $3\leq x$ and \ref{bpm} implies $b^+<3$, but if we are only interested in integer solutions this gives $b\leq 2$. \end{proof} \begin{lemma} \label{dp3} If $1\leq b \leq 2$, then the integer solutions of $(\dagger)$ (up to permutation of $a_i$) are: \begin{eqnarray*} b=1,\ a_1=\cdots=a_{x+1}=-1,\ a_{x+2}=\cdots=a_8=0 & \\ b=2,\ a_1=\cdots=a_{x+4}=-1,\ a_{x+5}=\cdots=a_8=0 & \! . \end{eqnarray*} \end{lemma} \begin{proof} In both cases ($b=1$ or $b=2$), the right hand sides of the two equations are equal, and then, substracting the equations we have $$ \sum a_i^2+a_i=0. $$ But for $a_i$ integer we have $a_i^2+a_i\geq0$, and then $a_i$ must be equal to -1 or 0. Now, looking at the equations we see that the number of nonzero $a_i$'s is given by $x+b^2$, and we obtain the result. \end{proof} \begin{theorem} \label{dp4} $\mathfrak{M}_L(S,c_1,c_2)$, for any generic polarization $L$, is either empty or irreducible. \end{theorem} \begin{proof} We will denote by $L_0$ a polarization lying in a chamber whose closure contains $H$. If $\mathfrak{M}_L(S,c_1,c_2)$ has more than one irreducible component, then there must be a wall between $L$ and $L_0$ that created the extra component. Recall that a wall $W^\zeta$ is a hyperplane in the ample cone perpendicular to a class $\zeta$ with $\zeta \equiv c_1$ (mod 2), and $c_1^2-4c_2\leq\zeta^2<0$. By \cite{F-Q}, if $L_1\cdot\zeta>0 >L_2\cdot\zeta$, then the sheaves that are $L_1$-unstable and $L_2$-stable make an irreducible family of dimension $N_\zeta+2l_\zeta$, where $N_\zeta=h^1(\zeta)+l_\zeta-1$ and $l_\zeta=(4c_2-c_1^2+\zeta^2)/4$. In the case of a rational surface we have $$ h^1(\zeta)=\frac{\zeta\cdot K_S}{2}-\frac{\zeta^2}{2}-1. $$ The wall creates a new component if $N_\zeta+2l_\zeta$ is equal to the dimension of the moduli space, in our case $4c_2-c_1^2-3$. For a rational surface we have $N_\zeta+N_{-\zeta}+2l_\zeta=4c_2-c_1^2-4$, and then this condition is equivalent to $N_{-\zeta}=-1$. Denoting $\zeta=bH+a_1E_1+\cdots+a_nE_n$ and $x=4c_2-c_1^2$ we get the system of equations ($\dagger$) ($n\leq8$ so without loss of generality we can study the equations with $n=8$). Furthermore, this wall will create a new component if $L_0\cdot\zeta<0<L\cdot\zeta$. By the definition of chamber, the last equality is equivalent to $0<H\cdot\zeta$, and this translates to $b>0$. We will prove the proposition by showing that for given $(S,c_1,c_2)$ there is at most one such wall in the ample cone and that in this case $\mathfrak{M}_{L_0}(S,c_1,c_2)$ is empty, so that $\mathfrak{M}_L(S,c_1,c_2)$ is always either empty or only has one irreducible component. If the moduli space is not empty, its dimension should be nonnegative, and this translates to $x\geq3$. Then by lemmas \ref{dp2} and \ref{dp3} we know all the solutions of ($\dagger$), i.e. all the walls creating components. By tensoring with a line bundle (and relabeling the exceptional curves), we can assume that $c_1$ is either $H+E_1+\cdots+E_m$ or $E_1+\cdots+E_m$ for some $m\leq8$. Now we will use the condition $c_1 \equiv \zeta$ (mod 2). In the first case this implies that the only possible solution for ($\dagger$) is $\zeta=H-E_1-\cdots-E_m$, and $m=x+1$. Then $4c_2-c_1^2=x$ implies $c_2=0$. By \cite{B}, $\mathfrak{M}_{L_0}(S,H+E_1+\cdots+E_m,0)$ is birational to a $(\mathbb{P}^1)^m$ bundle over $\mathfrak{M}(\mathbb{P}^2,H,0)$, but it is well known that this moduli space is empty, then the same is true for $\mathfrak{M}_{L_0}(S,H+E_1+\cdots+E_m,0)$. In the second case ($c_1=E_1+\cdots+E_m$), the condition $c_1 \equiv \zeta$ (mod 2) implies that the only possible solution is $\zeta=2H-E_1-\cdots-E_m$ with $m=x+4$. Then $c_2=-1$ and then $\mathfrak{M}_{L_0}(S,c_1,c_2)$ is empty (because its expected dimension is negative). \end{proof} This techniques can also be used to study the irreducibility of the moduli space of stable torsion free sheaves on $\mathbb{P}^2$ with more than 8 blown up points. We obtain equations similar to $(\dagger)$, but with more variables. Unfortunately now we don't have a bound on $b$ like the one given by lemma \ref{dp2}, and then it becomes more difficult to classify the solutions. This is still work in progress and it will appear elsewhere.

9710

alg-geom/9710004

en

https://arxiv.org/abs/alg-geom/9710004

alg-geom/9710004

Uli Walther

Uli Walther (University of Minnesota)

Algorithmic Computation of Local Cohomology Modules and the Cohomological Dimension of Algebraic Varieties

20 pages, amsart, uses amstex, amssymb, xypic.tex, corrected some typos

In this paper we present algorithms that compute certain local cohomology modules associated to a ring of polynomials containing the rational numbers. In particular we are able to compute the local cohomological dimension of algebraic varieties in characteristic zero. Our approach is based on the theory of D-modules.

alg-geom

\section{Introduction} \subsection{} Let $R$ be a commutative Noetherian ring, $I$ an ideal in $R$ and $M$ an $R$-module. The $i$-th {\em local cohomology functor} with respect to $I$ is the $i$-th right derived functor of the functor $H^0_I(-)$ which sends $M$ to the $I$-torsion $\bigcup_{k=1}^\infty (0:_MI^k)$ of $M$ and is denoted by $H^i_I(-)$. Local cohomology was introduced by Grothendieck as an algebraic analog of (classical) relative cohomology. A brief introduction to local cohomology may be found in appendix 4 of \cite{E}. The {\em cohomological dimension} of $I$ in $R$, denoted by $\operatorname{cd}(R,I)$, is the smallest integer $c$ such that the local cohomology modules $H^q_I(M)=0$ for all $q>c$ and all $R$-modules $M$. If $R$ is the coordinate ring of an affine variety $X$ and $I\subseteq R$ is the defining ideal of the Zariski closed subset $V\subseteq X$ then the {\em local cohomological dimension} of $V$ in $X$ is defined as $\operatorname{cd}(R,I)$. It is not hard to show that if $X$ is smooth, then the integer $\dim(X)-\operatorname{cd}(R,I)$ depends only on $V$ but neither on $X$ nor on the embedding $V\hookrightarrow X$. \subsection{} Knowledge of local cohomology modules provides interesting information, illustrated by the following three situations. Let $I\subseteq R$ and $c=\operatorname{cd}(R,I)$. Then $I$ cannot be generated by fewer than $c$ elements. In fact, no ideal $J$ with the same radical as $I$ will be generated by fewer than $c$ elements. Let $H^i_{dR}$ stand for the $i$-th de Rham cohomology group. A second application is a family of results commonly known as Barth theorems which are a generalization of the classical Lefschetz theorem that states that if $Y\subseteq {\Bbb P}^n_{\Bbb C}$ is a hypersurface then $H^i_{dR}({\Bbb P}^n_{\Bbb C})\to H^i_{dR}(Y)$ is an isomorphism for $i<\dim (Y)-1$ and injective for $i=\dim (Y)$. For example, let $Y\subseteq {\Bbb P}_{\Bbb C}^n$ be a closed subset and $I\subseteq R={\Bbb C}[x_0,\ldots,x_n]$ the defining ideal of $Y$. Then $H^i_{dR}({\Bbb P}^n_{\Bbb C})\to H^i_{dR}(Y)$ is an isomorphism for $i\le \operatorname{depth}_{{\cal O}_{{\Bbb P}^n_{\Bbb C}}}({\cal O}_Y)-\operatorname{cd}(R,I)$ (compare \cite{Og}, 4.7 and \cite{DRCAV}, the theorem after III.7.6). Finally, it is also a consequence of the work of Ogus and Hartshorne (\cite{Og}, 2.2, 2.3 and \cite{DRCAV}, IV.3.1) that if $I\subseteq R={\Bbb C}[x_0,\ldots,x_n]$ is the defining ideal of a complex smooth variety $V\subseteq {\Bbb P}^n_{\Bbb C}$ then, for $i<n-\operatorname{codim} (V)$, $\dim_{\Bbb C}\operatorname{soc}_R (H^0_{\frak m}(H^{n-i} (R)))$ equals $\dim_{\Bbb C} H^i_x(\tilde V,{\Bbb C})$ where $H^i_x(\tilde V,{\Bbb C})$ stands for the $i$-th singular cohomology group of the affine cone $\tilde V$ over $V$ with support in the vertex $x$ of $\tilde V$ and with coefficients in ${\Bbb C}$ ($\operatorname{soc}_R(M)$ denotes the socle $(0:_M{\frak m})\subseteq M$ for any $R$-module $M$). \subsection{} The cohomological dimension has been studied by many authors, for example R.~Hartshorne (\cite{CDAV}), A.~Ogus (\cite{Og}), R.~Hartshorne and R.~Speiser (\cite{H-Sp}), C.~Peskine and L.~Szpiro (\cite{P-S}), G.~Faltings (\cite{F}), C.~Huneke and G.~Lyubeznik (\cite{Hu-L}). Yet despite this extensive effort, the problem of finding an algorithm for the computation of cohomological dimension remained open. For the determination of $\operatorname{cd}(R,I)$ it is in fact enough to find an algorithm to decide whether or not the local cohomology module $H^i_I(R)=0$ for given $i, R, I$. This is because $H^q_I(R)=0$ for all $q>c$ implies $\operatorname{cd}(R,I)\le c$ (see \cite{CDAV}, section 1). In \cite{L-Fmod} G.~Lyubeznik gave an algorithm for deciding whether or not $H^i_I(R)=0$ for all $I\subseteq R=K[x_1,\ldots,x_n]$ where $K$ is a field of positive characteristic. One of the main purposes of this work is to produce such an algorithm in the case where $K$ is a field containing the rational numbers and $R=K[x_1,\ldots,x_n]$. Since in such a situation the local cohomology modules $H^i_I(R)$ have a natural structure of finitely generated left $D(R,K)$-modules (\cite{L-Dmod}), $D(R,K)$ being the ring of $K$-linear differential operators of $R$, explicit computations may be performed. Using this finiteness we employ the theory of Gr\"obner bases to develop algorithms that give a representation of $H^i_I(R)$ and $H^i_{\frak m} (H^j_I(R))$ for all triples $i,j\in {\Bbb N},I\subseteq R$ in terms of generators and relations over $D(R,K)$ (where ${\frak m}=(x_1,\ldots,x_n)$). This also leads to an algorithm for the computation of the invariants $\lambda_{i,j}(R/I)=\dim_K\operatorname{soc}_R(H^i_{\frak m}(H^{n-j}_I(R)))$ introduced in \cite{L-Dmod}. We remark that if $R$ is an arbitrary finitely generated $K$-algebra and $I$ is an ideal in $R$ then, if $R$ is regular, our algorithms can be used to determine $\operatorname{cd}(R,I)$ for all ideals $I$ of $R$, but if $R$ is not regular, then the problem of algorithmic determination of $\operatorname{cd}(R,I)$ remains open (see also the comments in subsection \ref{singular_spaces}). \subsection{} The outline of the paper is as follows. The next section is devoted to a short survey of results on local cohomology and $D$-modules as they apply to our work, as well as their interrelationship. In section \ref{sec-gb} we review the theory of Gr\"obner bases as it applies to $A_n$ and modules over the Weyl algebra. Most of that section should be standard and readers interested in proofs and more details are encouraged to look at the book by D.~Eisenbud (\cite{E}, chapter 15 for the commutative case) or the fundamental article \cite{KR-W} (for the more general situation. In section \ref{sec-mal-kash} we generalize some results due to B.~Malgrange and M.~Kashiwara on $D$-modules and their localizations. The purpose of sections \ref{sec-mal-kash} and \ref{sec-oaku} is to find a representation of $R_f\otimes N$ as a cyclic $A_n$-module if $N$ is a given holonomic $D$-module (for a definition and some properties of holonomic modules, see subsection \ref{D-modules} below). Many of the essential ideas in section \ref{sec-oaku} come from T.~Oaku's work \cite{Oa}. In section \ref{sec-lc} we describe our main results, namely algorithms that for arbitrary $i,j,k,I$ determine the structure of $H^k_I(R), H^i_{\frak m} (H^j_I(R))$ and find $\lambda_{i,j}(R/I)$. Some of these algorithms have been implemented in the programming language C and the theory is illustrated with examples. The final section is devoted to comments on implementations, effectivity and examples. It is a pleasure to thank my advisor Gennady Lyubeznik for suggesting the problem of algorithmic computation of cohomological dimension to me and pointing out that the theory of $D$-modules might be useful for its solution. \section{Preliminaries} \label{sec-prelim} \subsection{Notation} Throughout we shall use the following notation: $K$ will denote a field of characteristic zero, $R=K[x_1,\ldots,x_n]$ the ring of polynomials over $K$ in $n$ variables, $A_n=K\langle x_1,\partial_1,\ldots,x_n,\partial_n\rangle$ the Weyl algebra over $K$ in $n$ variables, or, equivalently, the ring of $K$-linear differential operators on $R$, ${\frak m}$ will stand for the maximal ideal $(x_1,\ldots,x_n)$ of $R$, $\Delta$ will denote the maximal left ideal $(\partial_1,\ldots,\partial_n)$ of $A_n$ and $I$ will stand for the ideal $(f_1,\ldots,f_r)$ in $R$. All tensor products in this work will be over $R$ and all $A_n$-modules (resp.~ideals) will be left modules (resp.~left ideals). \subsection{Local cohomology} It turns out that $H^k_I(M)$ may be computed as follows. Let $C^\bullet(f_i)$ be the complex $0\to R\stackrel{1\to\frac{1}{1}}{\longrightarrow} R_{f_i}\to 0$. Then $H^k_I(M)$ is the $k$-th cohomology group of the {\em \v Cech complex} defined by $C^\bullet(M;f_1,\ldots,f_r) =\bigotimes_1^r C^\bullet(f_i)\otimes M$. Unfortunately, explicit calculations are complicated by the fact that $H^k_I(M)$ is rarely finitely generated as $R$-module. This difficulty disappears for $H^k_I(R)$ if we enlarge the ring to $A_n$, in essence because $R_f$ is finitely generated over $A_n$ for all $f\in R$. \subsection{$D$-modules} \mylabel{D-modules} A good introduction to $D$-modules is the book by Bj\"ork, \cite{B}. Let $f\in R$. Then the $R$-module $R_f$ has a structure as left $A_n$-module: $x_i(\frac{g}{f^k})=\frac{x_ig}{f^k}, \partial_i(\frac{g}{f^k})=\frac{\partial_i(g)f-k\partial_i(f)g}{f^{k+1}}$. This may be thought of as a special case of localizing an $A_n$-module: if $M$ is an $A_n$-module and $f\in R$ then $R_f\otimes_R M$ becomes an $A_n$-module via $\partial_i(\frac{g}{f^k}\otimes m)=\partial_i(\frac{g}{f^k})\otimes m+\frac{g}{f^k}\otimes \partial_i m$. Localization of $A_n$-modules lies at the heart of our arguments. Of particular interest are the {\em holonomic} modules which are those finitely generated $A_n$-modules $N$ for which $\operatorname{Ext}^j_{A_n}(N,A_n)$ vanishes unless $j=n$. Holonomic modules are always cyclic and of finite length over $A_n$. Besides that, if $N=A_n/L$, $f\in R$, $s$ is an indeterminate and $g$ is some fixed generator of $N$, then there is a nonzero polynomial $b(s)$ in $K[s]$ and an operator $P(s)\in A_n[s]$ such that $P(s)(f\cdot f^s\otimes g)=b(s)\cdot f^s\otimes g$. The unique monic polynomial that divides all other polynomials satisfying an identity of this type is called the {\em Bernstein polynomial} of $L$ and $f$ and denoted by $b_f^L(s)$. Any operator $P(s)$ that satisfies $P(s)f^{s+1}\otimes g=b_f^L(s)f^s\otimes g$ we shall call a {\em Bernstein operator} and refer to the roots of $b_f^L(s)$ as {\em Bernstein roots} of $f$ on $A_n/L$. Localizations of holonomic modules at a single element (and hence at any finite number of elements) of $R$ are holonomic (see \cite{B}, section 5.9) and in particular cyclic over $A_n$, generated by $f^{-a}g$ for sufficiently large $a\in {\Bbb N}$ (see also our proposition \ref{kashiwara}). So the complex $C^\bullet(N;f_1,\ldots,f_r)$ consists of holonomic $A_n$-modules whenever $N$ is holonomic. This facilitates the use of Gr\"obner bases as computational tool for maps between holonomic modules and their localizations. As a special case we note that localizations of $R$ are holonomic, and hence finite, over $A_n$ (since $R=A_n/\Delta$ is holonomic). \subsection{The \v Cech complex} In \cite{L-Dmod} it is shown that local cohomology modules are not only $D$-modules but in fact holonomic: we know already that the modules in the \v Cech complex are holonomic, it suffices to show that the maps are $A_n$-linear. All maps in the \v Cech complex are direct sums of localization maps. Suppose $R_f$ is generated by $f^s$ and $R_{fg}$ by $(fg)^t$. We may replace $s,t$ by their minimum $u$ and then we see that the inclusion $R_f\to R_{fg}$ is nothing but the map $A_n/\operatorname{ann}(f^u)\to A_n/\operatorname{ann}((fg)^u)$ sending the coset of the operator $P$ to the coset of the operator $P\cdot g^u$. So $C^i(N;f_1,\ldots,f_r)\to C^{i+1}(N;f_1,\ldots,f_r)$ is an $A_n$-linear map between holonomic modules for every holonomic $N$. One can prove that kernels and cokernels of $A_n$-linear maps between holonomic modules are holonomic. Holonomicity of $H^k_I(R)$ follows. In the same way it can be seen that $H^i_{\frak m} (H^j_I(R))$ is holonomic for $i,j\in{\Bbb N}$ (since $H^j_I(R)$ is holonomic). \section{Gr\"obner bases of modules over the Weyl algebra} \mylabel{sec-gb} In this section we review some of the concepts and results related to the Buchberger algorithm in modules over Weyl algebras. It turns out that with a little care many of the important constructions from the theory of commutative Gr\"obner bases carry over to our case. For an introduction into non-commutative monomial orders and related topics, \cite{KR-W} is a good source. Let us agree that every time we write an element in $A_n$, we write it as a sum of terms $c_{\alpha\beta}x^\alpha \partial^\beta$ in multi-index notation. That is, $\alpha,\beta\in {\Bbb N}^n$, $c_{\alpha\beta}$ are scalars, $x^\alpha=x_1^{\alpha_1}\cdot \ldots\cdot x_n^{\alpha_n}, \partial^\beta=\partial_1^{\beta_1}\cdot\ldots\cdot \partial_n^{\beta_n}$ and in every monomial we write first the powers of $x$ and then the powers of the differentials. Further, if $m=c_{\alpha\beta}x^\alpha\partial^\beta, c_{\alpha\beta}\in K$, we will say that $m$ has degree $\deg m=|\alpha+\beta|$ and an operator $P\in A_n$ has degree equal to the largest degree of any monomial occuring in $P$. Recall that a {\em monomial order} $<$ in $A_n$ is a total order on the monomials of $A_n$, subject to $m<m'\Rightarrow mm''<m'm''$ for all nonzero monomials $m,m',m''$. Since the product of two monomials in our notation is not likely to be a monomial (as $\partial_i x_i=x_i\partial_i+1$) it is not obvious that such orderings exist at all. However, the commutator of any two monomials $m_1,m_2$ will be a polynomial of degree at most $\deg m_1+\deg m_2-2$. That means that the degree of an operator and its component of maximal degree is independent of the way it is represented. Thus we may for example introduce a monomial order on $A_n$ by taking any monomial order on $\tilde A_n=K[x_1,\ldots,x_n,\partial_1,\ldots,\partial_n]$ (the polynomial ring in $2n$ variables) that refines the partial order given by total degree, and saying that $m_1>m_2$ in $A_n$ if and only if $m_1>m_2$ in $\tilde A_n$. Let $<$ be a monomial order on $A_n$. Let $G=\bigoplus_1^dA_n\cdot \gamma_i$ be the free $A_n$-module on the symbols $\gamma_1,\ldots,\gamma_d$. We define a monomial order on $G$ by $m_i\gamma_i>m_j\gamma_j$ if either $m_i>m_j$ in the order on $A_n$, or $m_i=m_j$ and $i>j$. The largest monomial $m\gamma$ in an element $g\in G$ will be denoted by $\operatorname{in}(g)$. Of fundamental importance is \begin{alg}[Remainder] \mylabel{remainder} Let $h$ and $\_ g=\{g_i\}_1^s$ be elements of $G$. Set $h_0=h, \sigma_0=0, j=0$ and let $\varepsilon_i=\varepsilon(g_i)$ be symbols. Then \[ \begin{array}{llll} &{\tt Repeat}&&\\ &&{\tt If} \operatorname{in}(g_i)|\operatorname{in}(h_j) {\tt \,set}&\\ &&&\{h_{j+1}:=h_j-\frac{\operatorname{in}(h_j)}{\operatorname{in}(g_i)}g_i,\\ &&&\sigma_{j+1}:=\sigma_j+\frac{\operatorname{in}(h_j)}{\operatorname{in}(g_i)}\varepsilon_i,\\ &&&j:=j+1\}\\ &{\tt Until}&{\tt No} \operatorname{in}(g_i)|\operatorname{in}(h_j). \end{array} \] The result is $h_a$, called a {\em remainder $\Re(h,\_ g)$ of $h$ under division by $\_ g$}, and an expression $\sigma_a=\sum_{i=1}^s a_i\varepsilon_i$ with $a_i\in A_n$. By Dickson's lemma (\cite{KR-W}, 1.1) the algorithm terminates. It is worth mentioning that $\Re(h,\_ g)$ is not uniquely determined, it depends on which $g_i$ we pick amongst those whose initial term divides the initial term of $h_j$. Note that if $h_a$ is zero, $\sigma_a$ tells us how to write $h$ in terms of $\_ g$. Such a $\sigma_a$ is called a \em{standard expression for $h$} with respect to $\{g_1,\ldots,g_s\}$. \end{alg} \begin{df}\mylabel{schreyer} If $\operatorname{in}(g_i)$ and $\operatorname{in}(g_j)$ involve the same basis element of $G$, then we set $s_{ij}=m_{ji}g_i-m_{ij}g_j$ and $\sigma_{ij}=m_{ji}\varepsilon_i-m_{ij}\varepsilon_j$ where $m_{ij}=\frac{\operatorname{lcm}(\operatorname{in}(g_j),\operatorname{in}(g_i))}{\operatorname{in}(g_j)}$. Otherwise, $\sigma_{ij}$ and $s_{ij}$ are defined to be zero. $s_{ij}$ is the {\em Schreyer-polynomial} to $g_i$ and $g_j$. Suppose $\Re(s_{ij},\_ g)$ is zero for all $i,j$. Then we call $\_ g$ a {\em Gr\"obner basis} for the module $A_n\cdot(g_1,\ldots,g_s)$. \end{df} The following proposition (\cite{KR-W}, Lemma 3.8) indicates the usefulness of Gr\"obner bases. \begin{prop} \mylabel{gb-char} Let $\_ g$ be a finite set of elements of $G$. Then $\_ g$ is a Gr\"obner basis if and only if $h\in A_n\_ g$ implies $\exists i: \operatorname{in}(g_i)|\operatorname{in}(h)$.\hfill$\Box$ \end{prop} Computation of Gr\"obner bases over the Weyl algebra works just as over polynomial rings: \begin{alg}[Buchberger] \mylabel{buchberger} Input: $\_ g=\{g_1,\ldots,g_s\}\subseteq G$. Output: a Gr\"obner basis for $A_n\cdot(g_1,\ldots,g_s)$. Begin. \[ \begin{array}{llll} &{\tt Repeat}&&\\ &&{\tt If\,} h=\Re(s_{ij},\underline g)\not =0&\\ &&&{\tt add\,} h {\tt \,to \,} \_ g\\ &{\tt Until}&{\tt \, all\,} \Re(s_{ij},\_ g)=0.\\ &{\tt Return } \,\,\_ g. \end{array} \] \indent End. \end{alg} \subsection{} \mylabel{free-kernel} Now we shall describe the construction of kernels of $A_n$-linear maps using Gr\"obner bases. Again, this is similar to the commutative case and we first consider the case of a map between free $A_n$-modules. Let $E=\bigoplus_1^sA_n\varepsilon_i, G=\bigoplus_1^rA_n\gamma_j$ and $\phi:E\to G$ be a $A_n$-linear map. Assume $\phi(\varepsilon_i)=g_i$. Suppose that in order to make $\_ g$ a Gr\"obner basis we have to add $g'_1,\ldots,g'_{s'}$ to $\underline g$ which satisfy $g'_i=\sum_{k=1}^s a_{ik}g_k$. We get an induced map $\diagram \bigoplus_1^{s+s'}A_n\varepsilon_i\dto_\pi\drto_{\tilde\phi}\\ \bigoplus_1^sA_n\varepsilon_i\rto_\phi &\bigoplus_1^rA_n\gamma_j \enddiagram $ where $\pi$ is the identity on $\varepsilon_i$ for $i\le s$ and sends $\varepsilon_{i+s}$ into $\sum_{k=1}^s a_{ik}\varepsilon_k$. Of course, $\tilde\phi=\phi\pi$. The kernel of $\phi$ is just the image of the kernel of $\tilde\phi$ under $\pi$. So in order to find kernels of maps between free modules one may assume that the generators of the source are mapped to a Gr\"obner basis and replace $\phi$ by $\tilde\phi$. Recall from definition \ref{schreyer} that $\sigma_{ij}=m_{ji}\varepsilon_i-m_{ij}\varepsilon_j$ or zero, depending on the leading terms of $g_i$ and $g_j$. \begin{prop} \mylabel{syz} Assume that $\{g_1,\ldots,g_s\}$ is a Gr\"obner basis. Let $s_{ij}=\sum d_{ijk}g_k$ be standard expressions for the Schreyer polynomials. Then $\{\sigma_{ij}-\sum_k d_{ijk}\varepsilon_k\}_{1\le i<j\le s}$ generate the kernel of $\phi:\bigoplus_1^sA_n\varepsilon_i\to \bigoplus_1^rA_n\gamma_j$, sending $\varepsilon_i$ to $g_i$. \end{prop} The proof proceeds exactly as in the commutative case, see for example \cite{E}, section 15.10.8. \subsection{} \mylabel{kernel} We explain now how to find a set of generators for the kernel of an arbitrary $A_n$-linear map. Let $E, G$ be as in subsection \ref{free-kernel} and suppose $A_n(p_1,\ldots,p_a)=P\subseteq E, A_n(q_1,\ldots,q_b)=Q\subseteq G$ and $\phi:\bigoplus_1^s A_n\varepsilon_i/P\to \bigoplus_1^r A_n\gamma_j/Q$. It will be sufficient to consider the case $P=0$ since we may lift $\phi$ to the free module $E$ surjecting onto $E/P$. Let as before $\phi(\varepsilon_i)=g_i$. A kernel element in $E$ is a sum $\sum_ia_i\varepsilon_i, a_i\in A_n$, which if $\varepsilon_i$ is replaced by $g_i$ can be written in terms of the generators $q_j$ of $Q$. Let $\_ \beta=\{\beta_1,\ldots,\beta_c\}$ be such that $\_ g\cup \_ q\cup \_\beta$ is a Gr\"obner basis for $A_n(\_ g,\_ q)$. We may assume that the $\beta_i$ are the results of applying algorithm \ref{buchberger} to $\_ g\cup\_ q$. Then from algorithm $\ref{remainder}$ we have expressions \begin{equation} \label{star} \beta_i=\sum_j c_{ij}g_j+\sum_k c'_{ik}q_k, \end{equation} with $c_{ij}, c'_{ik}\in A_n$. Furthermore, by proposition \ref{syz}, algorithm \ref{buchberger} returns a generating set $\_\sigma$ for the syzygies on $\_g\cup\_ q\cup\_\beta$. Write \begin{equation} \sigma_i=\sum_j a_{ij}\varepsilon_{g_j}+\sum_k a'_{ik}\varepsilon_{q_k}+\sum_l a''_{il}\varepsilon_{\beta_l} \end{equation} and eliminate the last sum using the relations (\ref{star}) to obtain syzygies \begin{equation} \tilde \sigma_i=\sum_j a_{ij}\varepsilon_{g_j}+\sum_k a'_{ik}\varepsilon_{q_k}+\sum_l a''_{il}\left(\sum_v c_{lv}\varepsilon_{g_v}+\sum_w c'_{lw}\varepsilon_{q_w}\right). \end{equation} These will then form a generating set for the syzygies on $\_g\cup \_q$. Cutting off the $q$-part of these syzygies we get a set of forms \[ \left\{\sum_j a_{ij}\varepsilon_{g_j}+\sum_l a''_{il}\left(\sum_v c_{lv}\varepsilon_{g_v}\right)\right\} \] which generate the kernel of the map $E\to G/Q$. \subsection{} The comments in this subsection will find their application in algorithm \ref{lclc-alg} which computes the structure of $H^i_{\frak m}(H^j_I(R))$ as $A_n$-module. Let \mylabel{double-kernel} \[ \diagram M_3'\rto^\alpha& M_3\rto^{\alpha '}& M_3''\\ M_2'\rto^\beta\uto_{\phi'}& M_2\rto^{\beta'}\uto_{\psi'}& M_2''\uto_{\rho'}\\ M_1'\rto^\gamma\uto_{\phi}& M_1\rto^{\gamma'}\uto_\psi& M_1''\uto_{\rho} \enddiagram \] be a commutative diagram of $A_n$-modules. Note that the row cohomology of the column co\-ho\-mo\-lo\-gy at $N$ is given by \[ \left[\ker(\psi ')\cap {\beta '}^{-1}(\operatorname{im}\rho)+\operatorname{im}(\psi)\right]\,\, /\,\,\left[\beta(\ker(\phi '))+\operatorname{im}(\psi )\right]. \] In order to compute this we need to be able to find: \begin{itemize} \item preimages of submodules, \item kernels of maps, \item intersections of submodules. \end{itemize} It is apparent that the second and third calculation is a special case of the first: kernels are preimages of zero, intersections are images of preimages (if ${A_n}^r\stackrel{\phi}{\rightarrow} {A_n}^s/M \stackrel{\psi}{\leftarrow} {A_n}^t$ is given, then $\operatorname{im}(\phi)\cap \operatorname{im}(\psi)=\psi(\psi^{-1}(\operatorname{im}(\phi)))$ ). So suppose in the situation $\phi:{A_n}^r/M\to {A_n}^s/N$, $\psi:{A_n}^t/P\to {A_n}^s/N$ we want to find the preimage under $\psi$ of the image of $\phi$. We may reduce to the case where $M$ and $P$ are zero and then lift $\phi,\psi$ to maps into ${A_n}^s$. The elements $x$ in $\psi^{-1}(\operatorname{im} \phi)\subseteq {A_n}^t$ are exactly the elements in $\ker({A_n}^t\stackrel{\psi}{\to} {A_n}^s/N\to {A_n}^s/(N+\operatorname{im}\phi))$ and this kernel can be found according to the comments in \ref{kernel}. \section{$D$-modules after Kashiwara and Malgrange} \mylabel{sec-mal-kash} The purpose of this and the following section is as follows. Given $f\in R$ and an ideal $L\subseteq A_n$ such that $A_n/L$ is holonomic and $L$ is $f$-saturated (i.e.~$f\cdot P\in L$ only if $P\in L$), we want to compute the structure of the module $R_f\otimes A_n/L$. It turns out that it is useful to know the ideal $J^L(f^s)$ which consists of the operators $P(s)\in A_n[s]$ that annihilate $f^s\otimes \bar 1\in M:=R_f[s]f^s\otimes A_n/L$ where the bar denotes cosets in $A_n/L$. In order to find $J^L(f^s)$, we will consider the module $M$ over the ring $A_{n+1}=A_n\langle t,\partial_t\rangle$. It will turn out in \ref{malgrange} that one can easily compute the ideal $J^L_{n+1}(f^s)\subseteq A_{n+1}$ consisting of all operators that kill $f^s\otimes\bar 1$. In section \ref{sec-oaku} we will then show how to compute $J^L(f^s)$ from $J_{n+1}^L(f^s)$. The second basic fact in this section (proposition \ref{kashiwara}) shows how to compute the structure of $R_f\otimes A_n/L$ as $A_n$-module once $J^L(f^s)$ is known. \subsection{} Consider $A_{n+1}=A_n\langle t,\partial_t\rangle$, the Weyl algebra in $x_1,\ldots,x_n$ and the new variable $t$. B.~Malgrange has defined an action of $t$ and $\partial_t$ on $M=R_f[s]\cdot f^s\otimes_R A_n/L$ by $t(g(x,s)\cdot f^s\otimes \bar P)=g(x,s+1)f\cdot f^s\otimes \bar P$ and $\partial_t(g(x,s)\cdot f^s\otimes \bar P)=\frac{-s}{f}g(x,s-1)\cdot f^s\otimes \bar P$ for $\bar P\in A_n/L$. $A_n$ acts on $M$ as expected, the variables by multiplication on the left, the $\partial_i$ by the product rule. One checks that this actually defines an structure of $M$ as a left $A_{n+1}$-module and that $-\partial_tt$ acts as multiplication by $s$. We denote by $J^L_{n+1}(f^s)$ the ideal in $A_{n+1}$ that annihilates the element $f^s\otimes \bar 1$ in $M$. Then we have an induced morphism of $A_{n+1}$-modules $A/J^L_{n+1}(f^s)\to M$ sending $P+J^L_{n+1}(f^s)$ to $P(f^s\otimes\bar 1)$. The operators $t$ and $\partial_t$ were introduced in \cite{M}. The following lemma generalizes lemma 4.1 in \cite{M} (as well as part of the proof given there) where the special case $L=(\partial_1,\ldots,\partial_n), A_n/L=R$ is considered. Note that $J^L_{n+1}(f^s)$ makes perfect sense even if $L$ is not holonomic. \begin{lem} \mylabel{malgrange} Suppose that $L=A_n\cdot (P_1,\ldots,P_r)$ is $f$-saturated (i.e., if $f\cdot P\in L$, then $P\in L$). With the above definitions, $J^L_{n+1}(f^s)$ is the ideal generated by $f-t$ together with the images of the $P_j$ under the automorphism $\phi$ of $A_{n+1}$ induced by $x\to x, t\to t-f$. \pf The automorphism sends $\partial_i$ to $\partial_i+f_i\partial_t$ and $\partial_t$ to $\partial_t$. So if we write $P_j=P_j(\partial_1,\ldots,\partial_n)$, then $\phi P_j=P_j(\partial_1+f_1\partial_t,\ldots,\partial_n+f_n\partial_t)$. One checks that $(\partial_i+f_i\partial_t)(f^s\otimes \bar Q)=f^s\otimes \bar{\partial_i Q}$ for all differential operators $Q$ so that $\phi(P_j(\partial_1,\ldots,\partial_n))(f^s\otimes \bar 1)=f^s\otimes \bar{P_j(\partial_1,\ldots,\partial_n)}=0$. By definition, $f\cdot (f^s\otimes \bar 1)=t\cdot (f^s\otimes \bar 1)$. So $t-f\in J^L_{n+1}(f^s)$ and $\phi(P_j)\in J^L_{n+1}(f^s)$ for $i=1,\ldots,r$. Conversely let $P(f^s\otimes \bar 1)=0$. We may assume, that $P$ does not contain any $t$ since we can eliminate $t$ using $f-t$. Now rewrite $P$ in terms of $\partial_t$ and the $\partial_i+f_i\partial_t$. Say, $P=\sum c_{\alpha\beta}\partial_t^\alpha x^\beta Q_{\alpha\beta}(\partial_1+f_1\partial_t,\ldots,\partial_n+f_n\partial_t)$, where the $Q_{\alpha\beta}$ are polynomials in $n$ variables and $c_{\alpha\beta}\in K$. Application to $f^s\otimes \bar 1$ results in $\sum \partial_t^\alpha(f^s\otimes c_{\alpha\beta}x^\beta \bar{Q_{\alpha\beta}(\partial_1,\ldots,\partial_n)})$. Let $\bar\alpha$ be the largest $\alpha\in{\Bbb N}$ for which there is a nonzero $c_{\alpha\beta}$ occuring in $P=\sum c_{\alpha\beta}\partial_t^\alpha x^\beta Q_{\alpha\beta}(\partial_1+f_1\partial_t,\ldots,\partial_n+f_n\partial_t)$. We show that the sum of terms that contain $\partial_t^{\bar\alpha}$ is in $A_{n+1}\cdot \phi(L)$ as follows. In order for $P(f^s\otimes\bar 1)$ to vanish, the sum of terms with the highest $s$-power, namely $s^{\bar\alpha}$, must vanish, and so $\sum_\beta c_{\bar\alpha\beta}(-1/f)^{\bar\alpha}f^s\otimes x^\beta Q_{\bar\alpha\beta}(\partial_1,\ldots,\partial_n)\in R_ff^s\otimes L$ as $R_f$ is $R$-flat. It follows, that $\sum_\beta c_{\bar\alpha\beta}x^\beta Q_{\bar\alpha\beta}(\partial_1,\ldots,\partial_n)\in L$ ($L$ is $f$-saturated!) and hence $\sum_\beta \partial_t^{\bar\alpha}c_{\bar\alpha\beta} x^\beta Q_{\bar\alpha\beta}(\partial_1+f_1\partial_t,\ldots,\partial_n+f_n\partial_t)\in A_{n+1}\cdot \phi(L)$. So by the first part, $P-\sum_\beta c_{\bar\alpha\beta}\partial_t^{\bar\alpha} x^\beta Q_{\bar\alpha\beta}(\partial_1+f_1\partial_t,\ldots,\partial_n+f_n\partial_t)$ kills $f^s\otimes \bar 1$, but is of smaller degree in $\partial_t$ than $P$ was. The claim follows.\hfill$\Box$ \end{lem} \subsection{} \mylabel{jt-js} Let $J^L(f^s)$ stand for the ideal in $A_n[s]\cong A_n[-\partial_tt]$ that kills $f^s\otimes\bar 1\in R_f[s]f^s\otimes_R A_n/L$. Note that $J^L(f^s)=J^L_{n+1}(f^s)\cap A_n[-\partial_tt]$. Again, we may talk about $J^L(f^s)$ independently of the holonomicity of $L$. We will in the next section show how the lemma can be used to determine $J^L(f^s)$. Now we show why $J^L(f^s)$ is useful, generalizing \cite{K}, proposition 6.2. Recall that the Bernstein polynomial $b^L_f(s)$ is defined to be the monic generator of the ideal of polynomials $b(s) \in K[s]$ for which there exists an operator $P(s)\in A_n[s]$ such that $P(s)(f^{s+1}\otimes \bar 1)=b(s)f^s\otimes \bar 1$ (\cite{B}, chapter 1), and that $b_f^L(s)$ will exist for example if $L$ is holonomic. \begin{prop} \mylabel{kashiwara} If $L$ is holonomic and $a\in \Bbb Z$ is such that no integer root of $b_f^L(s)$ is smaller than $a$, then we have isomorphisms \begin{equation} R_f\otimes A_n/L\cong A_n[s]/J^L(f^s)|_{s=a}\cong A_n\cdot f^a\otimes \bar 1. \end{equation} \pf We mimick the proof given by Kashiwara, who proved the proposition for the case $L=(\partial_1,\ldots,\partial_n), A_n/L=R$ (\cite{K}, proposition 6.2). Let us first prove the second equality. Certainly $J^L(f^s)|_{s=a}$ kills $f^a\otimes\bar 1$. So we have to show that if $P(f^a\otimes\bar 1)=0$ then $P\in J^L(f^s)+A_n[s]\cdot(s-a)$. To that end note that $st$ acts as $t(s-1)$ which means that $t\cdot (A_n[s]/J^L(f^s))$ is a left $A_n[s]$-module. Identify $A_n[s]/J^L(f^s)$ with ${\cal N}^L_f:=A_n[s]\cdot (f^s\otimes\bar 1)$. By definition, $b^L_f(s)$ is the minimal polynomial for which there is $P(s)$ with $b^L_f(s)(f^s\otimes \bar 1)=P(s)f^{s+1}=t\cdot P(s-1)(f^s\otimes\bar 1)$. So $b^L_f(s)$ multiplies $A_n[s]\cdot(f^s\otimes\bar 1)$ into $t\cdot A_n[s](f^s\otimes\bar 1)$ and whenever the polynomial $b(s)\in K[s]$ is relatively prime to $b_f^L(s)$ its action on ${\cal N}_f^L/t\cdot {\cal N}_f^L$ is injective. Since by hypothesis $s-a+j$ is not a divisor of $b^L_f(s)$ for $0<j\in \Bbb N$, \begin{equation} \label{kash-eqn} (s-a+j){\cal N}^L_f\cap t\cdot {\cal N}^L_f\subseteq (s-a+j)t\cdot {\cal N}^L_f. \end{equation} So $(s-a+m){\cal N}^L_f\cap t^m{\cal N}^L_f\subseteq (s-a+m)t{\cal N}^L_f\cap t^m{\cal N}^L_f=t[(s-a+m-1) {\cal N}^L_f\cap t^{m-1}{\cal N}^L_f]$ whenever $m\geq 1$. We show now by induction on $m$ that $(s-a+m){\cal N}^L_f\cap t^m{\cal N}^L_f\subseteq (s-a+m)t^m{\cal N}^L_f$ for $m\geq 1$. The claim is clear for $m=1$ from equation (\ref{kash-eqn}). So let $m>1$. The inductive hypothesis states that $(s-a+m-1){\cal N}^L_f\cap t^{m-1}{\cal N}^L_f\subseteq (s-a+m-1)t^{m-1}{\cal N}^L_f$. The previous paragraph shows that $(s-a+m){\cal N}^L_f\cap t^m{\cal N}^L_f\subseteq t\left[(s-a+m-1){\cal N}^L_f\cap t^{m-1}{\cal N}^L_f\right]$. Combining these two facts we get \begin{eqnarray*} (s-a+m){\cal N}^L_f\cap t^m{\cal N}^L_f&\subseteq & t(s-a+m-1)t^{m-1}{\cal N}^L_f\\ &=&(s-a+m)t^m{\cal N}^L_f. \end{eqnarray*} Now if $P(s)\in A_n[s]$ is of degree $m$ in the $\partial_i$ and $P(a)(f^a\otimes \bar 1)=0$, then $P(s+m)\cdot f^m+J^L(f^s)\in (s-a+m)\cdot {\cal N}^L_f$ because we can interprete $P(s+m)(f^{s+m}\otimes \bar 1)$ as a polynomial in $s+m$ with root $a$. But then $P(s+m) (f^{s+m}\otimes\bar 1)=P(s+m)(f^{m}f^s\otimes\bar 1)$ is in \[ (s-a+m){\cal N}^L_f\cap t^m{\cal N}^L_f\subseteq (s-a+m)t^m{\cal N}^L_f, \] implying $P(s+m)(f^{s+m}\otimes \bar 1)=(s-a+m)Q(s)(f^{s+m}\otimes\bar 1)$ for some $Q(s)\in A_n[s]$ (note that $J^L(f^s)$ kills $f^s\otimes\bar 1$). In other words, $P(s)-(s-a)Q(s-m)\in J^L(f^s)$. For the first isomorphism we have to show that $A_n\cdot(f^a\otimes\bar 1)=R_f\otimes A_n/L$. It suffices to show that every term of the form $f^mf^a\otimes\bar Q$ is in the module generated by $(f^a\otimes\bar 1)$ for all $m\in \Bbb Z$. Furthermore, we may assume that $Q$ is a monomial in $\partial_1,\ldots,\partial_n$. Existence and definition of $b^L_f(s)$ provides an operator $P(s)$ with $[b^L_f(s-1)]^{-1}P(s-1)(f^s\otimes \bar 1)=f^{-1}f^s\otimes \bar 1$. As $b_f^L(a-m)\not=0$ for all $0<m\in \Bbb N$ we have $f^{m}f^a\otimes\bar 1\in A_n\cdot (f^a\otimes \bar 1)$ for all $m$. Now let $Q$ be a monomial in $\partial_1,\ldots,\partial_n$ of $\partial$-degree $j>0$ and assume that $f^mf^a\otimes \bar{Q'}\in A_n\cdot (f^a\otimes \bar 1)$ for all $m$ and all operators $Q'$ of $\partial$-degree lower than $j$. Then $Q=\partial_i Q'$ for some $1\le i\le n$. Fix $m\in{\Bbb Z}$. By assumption on $j$, for some $P'$ we have $P'(f^a\otimes \bar 1)=f^mf^a\otimes\bar{Q'}$. So \begin{equation} f^mf^a\otimes\bar Q= \partial_iP'(f^a\otimes\bar 1)-f_i\cdot(a+m)f^{m-1}f^a\otimes \bar{Q'} \in A_n\cdot (f^a\otimes \bar 1). \end{equation} The claim follows by induction. This completes the proof of the proposition.\hfill$\Box$ \end{prop} We remark that if any $a\in{\Bbb Z} $ satisfies the conditions of the proposition, then so does every integer smaller than $a$. \section{An algorithm of Oaku} \mylabel{sec-oaku} The purpose of this section is to review and generalize an algorithm due to Oaku. In \cite{Oa} (algorithm 5.4.), Oaku showed how to construct a generating set for $J^L(f^s)$ in the case where $L=(\partial_1,\ldots,\partial_n)$. According to \ref{jt-js}, $J^L(f^s)$ is the intersection of $J^L_{n+1}(f^s)$ with $A_n[-\partial_tt]$. We shall explain how one may calculate $J\cap A_n[-\partial_tt]$ whenever $J\subseteq A_{n+1}$ is any given ideal and as a corollary develop an algorithm that for $f$-saturated $A_n/L$ computes $J^L(f^s)$. The proof follows closely Oaku's argument. On $A_{n+1}[y_1,y_2]$ define weights $w(t)=w(y_1)=1, w(\partial_t)=w(y_2)=-1, w(x_i)=w(\partial_i)=0$. If $P=\sum_i P_i\in A_{n+1}[y_1,y_2]$ and all $P_i$ are monomials, then we will write $(P)^h$ for the operator $\sum_i P_i\cdot y_1^{d_i}$ where $d_i=\max_j(w(P_j))-w(P_i)$ and call it the {\em $y_1$-homogenization} of $P$. Note that the Buchberger algorithm preserves homogeneity in the following sense: if a set of generators for an ideal is given and these generators are homogeneous with respect to the weights above, then any new generator for the ideal constructed with the classical Buchberger algorithm will also be homogeneous. (This is a consequence of the facts that the $y_i$ commute with all other variables and that $\partial_t t=t\partial_t+1$ is homogeneous of weight zero.) \begin{prop} \mylabel{oaku} Let $J=A_{n+1}\cdot(Q_1,\ldots,Q_r)$ and let $y_1,y_2$ be two new variables. Let $I$ be the left ideal in $A_{n+1}[y_1]$ generated by the $y_1$-homogenizations $(Q_i)^h$ of the $Q_i$, relative to the weight $w$ above, and let $\tilde I=A_{n+1}[y_1,y_2]\cdot (I,1-y_1y_2)$. Let $G$ be a Gr\"obner basis for $\tilde I$ under a monomial order that eliminates $y_1,y_2$. For each $P\in G$ set $P'=t^{-w(P)}P$ if $w(P)<0$ and $P'=\partial_t^{w(P)}P$ if $w(P)>0$ and let $G'=\{ P': P\in G\}$. Then $G_0=G'\cap A_n[-\partial_tt]$ generates $J\cap A_n[-\partial_tt]$. \pf Note first that $G$ consists of $w$-homogeneous operators and so $w(P)$ is well defined for $P\in G$. Suppose $P\in G_0$. Hence $P\in\tilde I$. So $P=Q_{-1}\cdot (1-y_1y_2)+\sum a_i\cdot (Q_i)^h$ where the $a_i$ are in $A_{n+1}[y_1,y_2]$. Since $P\in A_n[-\partial_tt]$, the substitution $y_i\to 1$ shows that $P=\sum a_i(1,1)\cdot (Q_i)^h(1,1)=\sum a_i(1,1)\cdot Q_i\in J$. Therefore $G_0\subseteq J\cap A_n[-\partial_tt]$. Now assume that $P\in J\cap A_n[-\partial_tt]$. So $P$ is $w$-homogeneous of weight 0. Also, $P\in J$ and $J$ is contained in $I(1)$, the ideal of operators $Q(1)\subseteq A_{n+1}$ for which $Q(y_1)\in I$. By lemma \ref{oaku-lemma} below (taken from \cite{Oa}), $y_1^a P\in I$ for some $a\in \Bbb N$. Therefore $P=(1-(y_1y_2)^a)P+(y_1y_2)^aP\in \tilde I$. Let $G=\{P_1,\ldots,P_b,P_{b+1},\ldots,P_c\}$ and assume that $P_i\in A_{n+1} $ if and only if $i\le b$. Buchberger algorithm gives a standard expression $P=\sum a_iP_i$ with all $\operatorname{in}(a_iP_i)\le \operatorname{in} (P)$. That implies that $a_{b+i}$ is zero for positive $i$ and $a_i$ does not contain $y_1,y_2$ for any $i$. Since $P, P_i$ are $w$-homogeneous, the same is true for all $a_i$, from Buchberger algorithm. In fact, $w(P)=w(a_i)+w(P_i)$ for all $i$. As $w(P)=0$ (and $t, \partial_t$ are the only variables with nonzero weight that may appear in $a_i$) we find $a'_i\in A_n$ with $a_i=a_i'\cdot t^{-w(P_i)}$ or $a_i=a_i'\cdot \partial_t^{w(P_i)}$, depending on whether $w(P_i)$ is negative or positive. It follows that $P=\sum_1^b a_iP_i=\sum_1^b a_i'P_i'\in A_n[-\partial_tt]\cdot G_0$. \hfill$\Box$ \end{prop} \begin{lem} \mylabel{oaku-lemma} Let $I$ be a $w$-homogeneous ideal in $A_{n+1}[y_1]$ with respect to the weights introduced before the proposition and $I(1)$ defined as in the proof of the proposition. Assume $P\in A_{n+1}$ is a $w$-homogeneous operator. Then $P\in I(1)$ implies $y_1^aP\in I$ for some $a$. \pf Note first that $y_1\to 1$ will not lead to cancellation of terms in any homogeneous operator as $w(y_1)\not =0$. If $P\in I(1)$, $P=\sum Q_i(1)$, with all $Q_i$ $w$-homogeneous in $I$. Then the $y_1$-homogenization of $Q_i(1)$ will be a divisor of $Q_i$ and the quotient will be some power of $y_1$, say $y_1^{\eta_i}$. Homogenization of the equation $P=\sum Q_i(1)$ results in $y_1^{\eta}P=\sum Q_i(1)^h$ (since $P$ is homogeneous) so that \[\parbox{12.65cm} {\hfill$y_{1}^{\eta+\max(\eta_i)}P=\sum y_1^{\max(\eta_i)-\eta_i}Q_i\in I.$\hfill$ \Box$} \] \end{lem} So we have \begin{alg} \mylabel{ann-fs} Input: $f\in R, L\subseteq A_n$ such that $L$ is $f$-satuarated. Output: Generators for $J^L(f^s)$. Begin \begin{enumerate} \item For each generator $Q_i$ of $L$ compute the image $\phi(Q_i)$ under $x_i\to x_i, t\to t-f, \partial_i\to \partial_i+f_i\partial_t,\partial_t\to\partial_t$. Add $t-f$ to the list. \item Homogenize all $\phi(Q_i)$ with respect to the new variable $y_1$ relative to the weight $w$ introduced before proposition \ref{oaku}. \item Compute a Gr\"obner basis for the ideal generated by $(\phi(Q_1))^h$, $\ldots$, $(\phi(Q_r))^h$, $1-y_1y_2$, $t-y_1f$ in $A_{n+1}[y_1,y_2]$ using an order that eliminates $y_1,y_2$. \item Select the operators $\{ P_j\}_1^b$ in this basis which do not contain $y_1, y_2$. \item For each $P_j$, $1\le j\le b$, if $w(P_j)>0$ replace $P_j$ by $P_j'=\partial_t^{w(P_j)}P_j$. Otherwise replace $P_j$ by $P_j'=t^{-w(P_j)}P_j$. \item Return the new operators $\{P_j'\}_1^b$. \end{enumerate} End. \end{alg} In order to guarantee existence of the Bernstein polynomial $b^L_f(s)$ we assume for our next result that $L$ is holonomic. \begin{cor} \mylabel{b-poly} Suppose $L$ is a holonomic ideal. If $J^L(f^s)$ is known or it is known that $L$ is $f$-saturated, then the Bernstein polynomial $b_f^L(s)$ of $R_f\otimes_R A_n/L$ can be found from $(b^L_f(s))=A_n[s]\cdot(J^L(f^s),f)\cap K[s]$. Moreover, if $K\subseteq {\Bbb C}$, suppose $b^L_f(s)=s^d+b_{d-1}s^{d-1}+\ldots+b_0$ and define $B=\max_{i}\{|b_i|^{1/(d-i)}\}$. In order to find the smallest integer root of $b^L_f(s)$, one only needs to check the integers between $-2B$ and $2B$. If in particular $L=(\partial_1,\ldots,\partial_n)$, it suffices to check the integers between $-b_{d-1}$ and -1. \pf If $L$ is $f$-saturated, propositions \ref{malgrange} and \ref{oaku} enable us to find $J^L(f^s)$. The first part follows then easily from the definition of $b^L_f(s)$: as $(b_f^L(s)-P_f^L\cdot f)(f^s\otimes\bar 1)=0$ it is clear that $b_f^L(s)$ is in $K[s]$ and in $A_n[s](J^L(f^s),f)$. Using an elimination order on $A_n[s]$, $b^L_f(s)$ will be (up to a scalar factor) the unique element in the reduced Gr\"obner basis for $J^L(f^s)+(f)$ that contains no $x_i$ nor $\partial_i$. Now suppose $K\subseteq {\Bbb C}$, $|s|=2B\rho$ where $B$ is as defined above and $\rho>1$. Assume also that $s$ is a root of $b_f^L(s)$. We find \begin{eqnarray} (2B\rho)^d=|s|^d&=&|-\sum_0^{d-1}b_is^i| \le\sum_0^{d-1}B^{d-i}|s|^i\\ &=&B^d\sum_0^{d-1}(2\rho)^i \le B^d((2\rho)^d-1), \end{eqnarray} using $\rho\geq 1$. By contradiction, $s$ is not a root. The final claim is a consequence of Kashiwara's work \cite{K} where it is proved that if $L=(\partial_1,\ldots,\partial_n)$ then all roots of $b_f^L(s)$ are negative and hence $-b_{n-1}$ is a lower bound for each single root. \hfill$\Box$ \end{cor} For purposes of reference we also list algorithms that compute the Bernstein polynomial to a holonomic module and the localization of a holonomic module. \begin{alg} \mylabel{b-poly-L} Input: $f\in R, L\subseteq A_n$ such that $A_n/L$ is holonomic and $f$-torsionfree. Output: The Bernstein polynomial $b^L_f(s)$. Begin \begin{enumerate} \item Determine $J^L(f^s)$ following algorithm \ref{ann-fs}. \item Find a reduced Gr\"obner basis for the ideal $J^L(f^s)+A_n[s]\cdot f$ using an elimination order for $x$ and $\partial$. \item Pick the unique element in that basis contained in $K[s]$ and return it. \end{enumerate} End. \end{alg} \begin{alg} \mylabel{D/L-loc-f} Input: $f\in R, L\subseteq A_n$ such that $A_n/L$ is holonomic and $f$-torsionfree. Output: Generators for an ideal $J$ such that $R_f\otimes A_n/L\cong A_n/J$. Begin \begin{enumerate} \item Determine $J^L(f^s)$ following algorithm \ref{ann-fs}. \item Find the Bernstein polynomial $b_f^L(s)$ using algorithm \ref{b-poly-L}. \item Find the smallest integer root $a$ of $b_f^L(s)$ (using corollary \ref{b-poly}, if $K\subseteq {\Bbb C}$). \item Replace $s$ by $a$ in all generators for $J^L(f^s)$ and return these generators. \end{enumerate} End. \end{alg} The algorithms \ref{ann-fs} and \ref{b-poly-L} appear already in \cite{Oa} in the special case $L=(\partial_1,\ldots,\partial_n), A_n/L=R$. \section{Local cohomology as $A_n$-module} \mylabel{sec-lc} In this section we will combine the results from the previous sections to obtain algorithms that compute for given $i,j,k\in {\Bbb N}, I\subseteq R$ the local cohomology modules $H^k_I(R), H^i_{\frak m}(H^j_I(R))$ and the invariants $\lambda_{i,j}(R/I)$ associated to $I$. \subsection{Computation of $H^k_I(R)$} \mylabel{subsec-lc} Here we will describe an algorithm that takes in a finite set of polynomials $\underline f=\{f_1,\ldots,f_r\}\subset R$ and returns a presentation of $H^k_I(R)$ where $I=(f_1,\ldots,f_r)$. In particular, if $H^k_I(R)$ is zero, then the algorithm will return the zero presentation. Consider the \v Cech complex associated to $f_1,\ldots,f_r$ in $R$, \begin{equation} \label{cechcomplex} 0\to R\to \bigoplus_1^r R_{f_i}\to \bigoplus_{1\le i<j\le r}R_{f_if_j} \to\cdots\to R_{f_1\cdot\ldots\cdot f_r}\to 0. \end{equation} Its $k$-th cohomology group is the local cohomology module $H^k_I(R)$. The map \begin{equation} \label{cechmap} C^k=\bigoplus\limits_{1\le i_1<\cdots<i_k\le r}R_{f_{i_1}\cdot\ldots\cdot f_{i_k}}\to \bigoplus\limits_{1\le j_1<\cdots<j_{k+1}\le r}R_{f_{j_1}\cdot\ldots\cdot f_{j_{k+1}}}=C^{k+1} \end{equation} is the sum of maps \begin{equation} \label{cechmap-parts} R_{f_{i_1}\cdot\ldots\cdot f_{i_k}}\to R_{f_{j_1}\cdot\ldots\cdot f_{j_{k+1}}} \end{equation} which are either zero (if $\{i_1,\ldots,i_k\}\not\subseteq \{j_1,\ldots,j_{k+1}\}$) or send $\frac{1}{1}$ to $\frac{1}{1}$, up to sign. Recall that $A_n/\Delta= A_n/A_n\cdot(\partial_1,\ldots,\partial_n)\cong R$ and identify $R_{f_{i_1}\cdot\ldots\cdot f_{i_k}}$ with $A_n/J^\Delta((f_{i_1}\cdot\ldots\cdot f_{i_k})^s)|_{s=a}$ and $R_{f_{j_1}\cdot\ldots\cdot f_{j_{k+1}}}$ with $A_n/J^\Delta((f_{j_1}\cdot\ldots\cdot f_{j_{k+1}})^s)|_{s=b}$ where $a,b$ are sufficiently small integers. By proposition \ref{kashiwara} we may assume that $a=b\le 0$. Then the map (\ref{cechmap}) is in the nonzero case multiplication from the right by $(f_l)^{-a}$ where $l=\{j_1,\ldots,j_{k+1}\}\setminus \{i_1,\ldots,i_k\}$, again up to sign. It follows that the matrix representing the map $C^k\to C^{k+1}$ in terms of $A_n$-modules is very easy to write down once the annihilator ideals and Bernstein polynomials for all $k$- and $(k+1)$-fold products of the $f_i$ are known: the entries are 0 or $\pm f_l^{-a}$ where $f_l$ is the new factor. Let $\Theta^r_k$ be the set of $k$-element subsets of $1,\ldots,r$ and for $\theta\in \Theta^r_k$ write $F_\theta$ for the product $\prod_{i\in \Theta^r_k}f_{i}$. We have demonstrated the correctness and finiteness of the following algorithm. \begin{alg} \mylabel{lc-alg} Input: $f_1,\ldots,f_r\in R; k\in {\Bbb N}$. Output: $H_I^k(R)$ in terms of generators and relations as finitely generated $A_n$-module. Begin \begin{enumerate} \item Compute the annihilator ideal $J^\Delta((F_\theta)^s)$ and the Bernstein polynomial $b^\Delta_{F_\theta}(s)$ for all $(k-1)$-, $k$- and $(k+1)$-fold products of ${f_1}^s,\ldots,{f_r}^s$ as in \ref{ann-fs} and \ref{b-poly-L} (so $\theta$ runs through $\Theta^r_{k-1}\cup \Theta^r_k\cup \Theta^r_{k+1}$). \item Compute the smallest integer root $a_\theta$ for each $b^\Delta_{F_\theta}(s)$, let $a$ be the minimum and replace $s$ by $a$ in all the annihilator ideals. \item Compute the two matrices $M_{k-1},M_k$ representing the $A_n$-linear maps $C^{k-1}\to C^k$ and $C^k\to C^{k+1}$ as explained in subsection \ref{subsec-lc}. \item Compute a Gr\"obner basis $G$ for the kernel of the map \[ \bigoplus_{\theta\in\Theta_k^r}A_n\to \bigoplus_{\theta\in\Theta^r_k} A_n/J^\Delta((F_\theta)^s)|_{s=a}\stackrel{M_k}{\longrightarrow} \bigoplus_{\theta\in \Theta^r_{k+1}}A_n/J^\Delta((F_\theta)^s)|_{s=a} \] as in \ref{kernel}. \item Compute a Gr\"obner basis $G_0$ for the module \[ \operatorname{im}(M_{k-1})+\bigoplus_{\theta\in \Theta^r_k} J^\Delta((F_\theta)^s)|_{s=a}\subseteq \bigoplus_{\theta\in\Theta^r_k} A_n/J^\Delta((F_\theta)^s)|_{s=a}. \] \item Compute the remainders of all elements of $G$ with respect to lifts of $G_0$ to $ \bigoplus_{\theta\in\Theta_k^r}A_n$. \item Return these remainders and $G_0$. \end{enumerate} End. \end{alg} The nonzero elements of $G$ generate the quotient $G/G_0\cong H^k_I(R)$ so that $H^k_I(R)=0$ if and only if all returned remainders are zero. \subsection{Computation of $H^i_{\frak m}( H^j_I(R))$} As a second application of Gr\"ob\-ner basis computations over the Weyl algebra we show now how to compute $H^i_{\frak m} (H^j_I(R))$. Note that we cannot apply lemma \ref{malgrange} to $A_n/L=H^j_I(R)$ since $H^j_I(R)$ may well contain some torsion. As in the previous sections, $C^j(R;f_1,\ldots,f_r)$ denotes the $j$-th module in the \v Cech complex to $R$ and $\{f_1,\ldots,f_r\}$. Let $C^{\bullet\bullet}$ be the double complex with $C^{i,j}=C^i(R;x_1,\ldots,x_n)\otimes_R C^j(R;f_1,\ldots,f_r)$, the vertical maps $\phi^{\bullet\bullet}$ induced by the identity on the first factor and the usual \v Cech maps on the second, whereas the horizontal maps $\xi^{\bullet\bullet} $ are induced by the \v Cech maps on the first factor and the identity on the second. Since $C^i(R;x_1,\ldots,x_n)$ is $R$-projective, the column co\-ho\-mo\-lo\-gy of $C^{\bullet\bullet}$ at $(i,j)$ is $C^i(R;x_1,\ldots,x_n)\otimes_RH^j_I(R)$ and the induced horizontal maps in the $j$-th row are simply the maps in the \v Cech complex $C^\bullet(H^j_I(R);x_1,\ldots,x_n)$. It follows that the row cohomology of the column cohomology at $(i_0,j_0)$ is $H^{i_0}_{\frak m}(H^{j_0}_I(R))$. Now note that $C^{i,j}$ is a direct sum of modules $R_g$ where $g=x_{\alpha_1}\cdot\ldots\cdot x_{\alpha_i}\cdot f_{\beta_1}\cdot\ldots\cdot f_{\beta_j}$. So the whole double complex can be rewritten in terms of $A_n$-modules and $A_n$-linear maps using \ref{D/L-loc-f}: \[ \diagram {\,C^{i-1,j+1}\,}{\rto^{\,\,\xi^{i-1,j+1}}}& C^{i,j+1}\rto^{\xi^{i,j+1}}& C^{i+1,j+1}\\ C^{i-1,j}\rto^{\xi^{i-1,j}}\uto_{\phi^{i-1,j}}& C^{i,j}\rto^{\xi^{i,j}}\uto_{\phi^{i,j}}& C^{i+1,j}\uto_{\phi^{i+1,j}}\\ C^{i-1,j-1}\rto^{\xi^{i-1,j-1}}\uto_{\phi^{i-1,j-1}}& C^{i,j-1}\rto^{\xi^{i,j-1}}\uto_{\phi^{i,j-1}}& C^{i+1,j-1}\uto_{\phi^{i+1,j-1}} \enddiagram \] Using the comments in subsection \ref{double-kernel}, we may now compute the modules $H^i_{\frak m} (H^j_I(R))$. More concisely, we have the following \begin{alg} \mylabel{lclc-alg} Input: $f_1,\ldots,f_r\in R; i_0,j_0\in \Bbb N$. Output: $H^{i_0}_{\frak m} (H^{j_0}_I(R))$ in terms of generators and relations as finitely generated $A_n$-module. Begin. \begin{enumerate} \item For $i=i_0-1, i_0, i_0+1$ and $j=j_0-1,j_0,j_0+1$ compute the annihilators $J^\Delta((F_\theta\cdot X_{\theta'})^s)$ and Bernstein polynomials $b^\Delta_{F_\theta\cdot X_{\theta'}}(s)$ of $F_\theta\cdot X_{\theta'}$ where $\theta \in \Theta^r_j, \theta'\in \Theta^n_i$ and $X_{\theta'}$ denotes in analogy to $F_\theta$ the product $\prod_{\alpha\in \theta'}x_\alpha$. \item Let $a$ be the minimum integer root of the product of all these Bernstein polynomials and replace $s$ by $a$ in all the annihilators computed in the previous step. \item Compute the matrices to the $A_n$-linear maps $\phi^{i,j}:C^{i,j}\to C^{i,j+1}$ and $\xi^{i,j}:C^{i,j}\to C^{i+1,j}$, again for $i=i_0-1,i_0,i_0+1$ and $j=j_0-1,j_0,j_0+1$. \item Compute Gr\"obner bases for the modules \[ G=\ker(\phi^{i_0,j_0})\cap \left[ (\xi^{i_0,j_0})^{-1}(\operatorname{im}(\phi^{i_0+1,j_0-1}))\right]+\operatorname{im}(\phi^{i_0,j_0-1}) \] and $G_0=\xi^{i_0-1,j_0}(\ker(\phi^{i_0-1,j_0}))+\operatorname{im}(\phi^{i_0,j_0-1})$. \item Compute the remainders of all elements of $G$ with respect to $G_0$ and return these remainders together with $G_0$. \end{enumerate} End. \end{alg} The elements of $G$ will be generators for $H^{i_0}_{\frak m} (H^{j_0}_I(R))$ and the elements of $G_0$ generate the relations that are not syzygies. \subsection{Computation of $\lambda_{i,n-j}(R/I)$} In \cite{L-Dmod} it has been shown that $H^i_{\frak m} (H^j_I(R))$ is an injective ${\frak m}$-torsion $R$-module of finite socle dimension $\lambda_{i,n-j}$ (which depends only on $i,j$ and $R/I$) and so isomorphic to $(E_R(K))^{\lambda_{i,n-j}}$ where $E_R(K)$ is the injective hull of $K$ over $R$. We are now in a position that allows computation of these invariants of $R/I$. For, let $H^i_{\frak m} (H^j_I(R))$ be generated by $g_1,\ldots,g_l\in {A_n}^d$ modulo the relations $h_1,\ldots,h_e\in {A_n}^d$. Let $H$ be the module generated by the $h_i$. We know that $(A_n\cdot g_1+H)/H$ is ${\frak m}$-torsion and so it is possible (with trial and error) to find a multiple of $g_1$, say $mg_1$ with $m$ a monomial in $R$, such that $(A_n\cdot mg_1+H)/H$ is nonzero but $x_img_1\in H$ for all $1\le i\le n$. Then the element $mg_1+H/H$ has annihilator equal to ${\frak m}$ and hence generates an $A_n$-module isomorphic to $A_n/A_n\cdot {\frak m}\cong E_R(K)$. The injection $A_n\cdot mg_1+H/H\hookrightarrow A_n\cdot(g_1,\ldots,g_l)+H/H$ splits as map of $R$-modules because of injectivity and so the cokernel $A_n(g_1,\ldots,g_l)/A_n(mg_1,h_1,\ldots,h_e)$ is isomorphic to $(E_R(K))^{\lambda_{i,n-j}-1}$. Reduction of the $g_i$ with respect to a Gr\"obner basis of the new relation module and repetition of the previous will lead to the determination of $\lambda_{i,n-j}$. \subsection{Local cohomology in ambient spaces different from ${\Bbb A}^n_K$} \mylabel{singular_spaces} If $A$ equals $K[x_1,\ldots,x_n]$, $I\subseteq A$, $X=\operatorname{Spec} (A)$ and $V=\operatorname{Spec}(A/I)$, knowledge of $H^i_I(A)$ for all $i\in {\Bbb N}$ answers of course the question about the local cohomological dimension of $V$ in $X$. It is worth mentioning, that if $W\subseteq X$ is a smooth variety containing $V$ then our algorithm \ref{lc-alg} for the computation of $H^i_I(A)$ also leads to a determination of the local cohomological dimension of $V$ in $W$. Namely, if $J$ stands for the defining ideal of $W$ in $X$ so that $R=A/J$ is the affine coordinate ring of $W$ and if we set $c=\operatorname{ht}(J)$, then it can be shown that $H^{i-c}_{I}(R)=\operatorname{Hom}_A(R,H^i_I(A))$ for all $i\in{\Bbb N}$. As $H^i_I(A)$ is $I$-torsion (and hence $J$-torsion), $\operatorname{Hom}_A(R,H^i_I(A))$ is zero if and only if $H^i_I(A)=0$. It follows that the local cohomological dimension of $V$ in $W$ equals $\operatorname{cd}(A,I)-c$ and $\{q\in {\Bbb N}:H^q_I(A)\not =0\}=\{q\in {\Bbb N}:H^{q-c}_I(R)\not =0\}$. If however $W$ is not smooth, no algorithms for the computation of either $H^i_I(R)$ or $\operatorname{cd}(R,I)$ are known, irrespective of the characteristic of the base field. \section{Implementation and examples} Some of the algorithms described above have been implemented as C-scripts and tested on some examples. \subsection{} The algorithm \ref{ann-fs} with $L=\Delta$ has been implemented by Oaku using the package Kan (see \cite{T}) which is a postscript language for computations in the Weyl algebra and in polynomial rings. An implementation for general $L$ is written by the current author and part of a program that deals exclusively with computations around local cohomology (\cite{W}). \cite{W} is theoretically able to compute $H^i_I(R)$ for arbitrary $i, R={\Bbb Q}[x_1,\ldots,x_n], I\subseteq R$ in the above described terms of generators and relations, using algorithm \ref{lc-alg}. It is expected that in the near future \cite{W} will work for $R=K[x_1,\ldots,x_n]$ where $K$ is an arbitrary field of characteristic zero and also algorithms for computation of $H^i_{\frak m} (H^j_I(R))$ and $\lambda_{i,j}(R)$ will be implemented, but see the comments in \ref{efficiency} below. \begin{ex} \label{example} Let $I$ be the ideal in $R=K[x_1,\ldots,x_6]$ that is generated by the $2\times 2$ minors $f,g,h$ of the matrix $\left(\begin{array}{ccc}x_1&x_2&x_3\\x_4&x_5&x_6\end{array}\right)$. Then $H_I^i(R)=0$ for $i<2$ and $H^2_I(R)\ne 0$ because $I$ is a height 2 prime and $H^i_I(R)=0$ for $i>3$ because $I$ is 3-generated, so the only remaining case is $H^3_I(R)$. This module in fact does not vanish, but until the discovery of our algorithm, its non-vanishing was a rather non-trivial fact. Our algorithm provides the first known proof of this fact by direct calculation. Previously, Hochster pointed out that $H^3_I(R)$ is nonzero, using the fact that the map $K[f,g,h]\to R$ splits (compare \cite{Hu-L}, Remark 3.13) and Bruns and Schw\"anzl (\cite{Br-S}, the corollary to Lemma 2) provided a topological proof of the nonvanishing of $H^3_I(R)$ via \'etale cohomology. Both of these proofs are quite ingenious and work only in very special situations. Using the program \cite{W}, one finds that $H^3_I(R)$ is isomorphic to a cyclic $A_6$-module generated by $1\in A_6$ subject to relations $x_1=\ldots =x_6=0$. This is a straightforward computational proof of the non-vanishing of $H^3_I(R)$. Of course this proof gives more than simply the non-vanishing. Since the quotient of $A_6$ by the left ideal generated by $x_1,\dots,x_6$ is known to be isomorphic as an $R$-module to $E_R(R/(x_1,\dots,x_6))$, the injective hull of $R/(x_1,\dots,x_6)=K$ in the category of $R$-modules, our proof implies that $H^3_I(R)\cong E_R(K)$. \end{ex} \subsection{} \mylabel{efficiency} Computation of Gr\"obner bases in many variables is in general a time- and space consuming enterprise. Already in (commutative) polynomial rings the worst case performance for the number of elements in reduced Gr\"obner bases is doubly exponential in the number of variables and the degrees of the generators. In the (relatively small) example above $R$ is of dimension 6, so that the intermediate ring $A_{n+1}[y_1,y_2]$ contains 16 variables. In view of these facts the following idea has proved useful. The general context in which lemma \ref{malgrange} and proposition \ref{kashiwara} were stated allows successive localization of $R_{fg}$ in the following way. First one computes $R_f$ according to algorithm \ref{D/L-loc-f} as quotient of $A_n$ by a certain holonomic ideal $L=J^\Delta(f^s)|_{s=a}, a\ll 0$. Then $R_{fg}$ may be computed using \ref{D/L-loc-f} again since $R_{fg}\cong R_g\otimes A_n/L$. (Note that all localizations of $R$ are automatically $f$-torsion free for $f\in R$ as $R$ is a domain.) This process may be iterated for products with any finite number of factors. Note also that the exponents for the various factors might be different. This requires some care as the following situations illustrate. Assume first that $-1$ is the smallest integer root of the Bernstein polynomials of $f$ and $g$ (both in $R$) with respect to the holonomic module $R$. Assume further that $R_{fg}\cong A_n\cdot f^{-2}g^{-1}\supsetneq A_n\cdot (fg)^{-1}$. Then $R_f\to R_{fg}$ can be written as $A_n/\operatorname{ann}(f^{-1})\to A_n/\operatorname{ann}(f^{-2}\cdot g^{-1})$ sending $P\in A_n$ to $P\cdot f\cdot g$. Suppose on the other hand that we are interested in $H^2_I(R)$ where $I=(f,g,h)$ and we know that $R_f=A_n\cdot f^{-2}\supsetneq A_n\cdot f^{-1}, R_g=A_n\cdot g^{-2}$ and $R_{fg}=A_n\cdot f^{-1}g^{-2}$. (In fact, the degree 2 part of the \v Cech complex of example \ref{example} consists of such localizations.) It is tempting to write the embedding $R_f\to R_{fg}$ with the use of a Bernstein operator (if $P_f(s) f^{s+1}=b^\Delta_f(s)f^s$ then take $s=-2$) but as $f^{-1}$ is not a generator for $R_f$, $b^\Delta_f(-2)$ will be zero. In other words, we must write $R_{fg}$ as $A_n/\operatorname{ann}((fg)^{-2})$ and then send $P\in \operatorname{ann}(f^{-2})$ to $P\cdot g^2$. The two examples indicate how to write the \v Cech complex in terms of generators and relations over $A_n$ while making sure that the maps $C^k\to C^{k+1}$ can be made explicit using the $f_i$: the exponents used in $C^i$ have to be at least as big as those in $C^{i-1}$ (for the same $f_i$). \begin{rem} \label{remark} We suspect that for all holonomic $fg$-torsionfree modules $M=A_n/L$ we have (with $R_g\otimes M\cong A_n/L'$): \[\min\{s\in{\Bbb Z}:b_f^L(s)=0\}\le \min\{s\in{\Bbb Z}:b_f^{L'}(s)=0\}.\] This would have two nice consequences. First of all, it would guarantee, that successive localization at the factors of a product does not lead to matrices in the \v Cech complex with entries of higher degree than localization at the product at once. Secondly, if \ref{remark} were known to be true, we could proceed as follows for the computation of $C^i(R;f_1,\ldots,f_r)$. First compute $J^\Delta((f_i)^s)$ for all $i$, find all minimal integer Bernstein roots $\beta_i$ of $f_i$ on $R$ and substitute them into the appropriate annihilator ideals. If from now on we want to use algorithm \ref{D/L-loc-f} in order to compute $R_{f_{i_1}\cdot\ldots\cdot f_{i_k}\cdot f_{i_{k+1}}}$ from $R_{f_{i_1}\cdot\ldots\cdot f_{i_k}}$ then we can skip steps 2 and 3 of \ref{D/L-loc-f} as the remark gives us a lower bound for the minimal integer Bernstein root of $f_{i_{k+1}}$ on $R_{f_{i_1}\cdot\ldots\cdot f_{i_k}}$. (From the comments before \ref{remark} it is also clear that we cannot hope to use a larger value.) \end{rem} The advantage of localizing $R_{fg}$ as $(R_f)_g$ is twofold. For one, it allows the exponents of the various factors to be distinct which is useful for the subsequent cohomology computation: it helps to keep the degrees of the maps small. (So for example $R_{x\cdot (x^2+y^2)}$ can be written as $A_n\cdot x^{-1} (x^2+y^2)^{-2}$ instead of $A_n\cdot (x^{-2}\cdot (x^2+y^2)^{-2})$. On the other hand, since the computation of Gr\"obner bases is doubly exponential it seems to be advantageous to break a big problem (localization at a product) into many ``easy'' problems (successive localization). An extreme case of this behaviour is our example \ref{example}: if we compute $R_{fgh}$ as $((R_f)_g)_h$, the calculation uses approximately 2.5 kB and lasts 32 seconds on a Sun workstation using \cite{W}. If one tries to localize $R$ at the product of the three generators at once, \cite{W} crashes after about 30 hours having used up the entire available memory (1.2 GB).

9710

alg-geom/9710022

en

https://arxiv.org/abs/alg-geom/9710022

alg-geom/9710022

Victor V. Batyrev, Ionunt Ciocan-Fontanine, Bumsig Kim, and Duco van Straten

Conifold Transitions and Mirror Symmetry for Calabi-Yau Complete Intersections in Grassmannians

36 pages, LaTeX 2.09

10.1016/S0550-3213(98)00020-0

In this paper we show that conifold transitions between Calabi-Yau 3-folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection Calabi-Yau 3-folds in Grassmannians. Using a natural degeneration of Grassmannians $G(k,n)$ to some Gorenstein toric Fano varieties $P(k,n)$ with conifolds singularities which was recently described by Sturmfels, we suggest an explicit mirror construction for Calabi-Yau complete intersections $X \subset G(k,n)$ of arbitrary dimension. Our mirror construction is consistent with the formula for the Lax operator conjectured by Eguchi, Hori and Xiong for gravitational quantum cohomology of Grassmannians.

alg-geom

\section{Introduction} One of the simplest ways to connect moduli spaces of two Calabi-Yau $3$-folds $X$ and $Y$ is a so called {\em conifold transition} that attracted interest of physicists several years ago in connection with {\em black hole condensation} \cite{S,GMS,CGGK}. The idea of the conifold transition goes back to Miles Reid \cite{R}, who proposed to connect the moduli spaces of two Calabi-Yau $3$-folds $X$ and $Y$ by choosing a point $x_0$ on the moduli space of complex structures on $X$ corresponding to a Calabi-Yau $3$-fold $X_0$ whose singularities consist of finitely many nodes. If $Y$ is a small resolution of singularities on $X_0$ which replaces the nodes by a union of ${\bf P}^1$'s with normal bundle ${\cal O}(-1) \oplus {\cal O}(-1)$, one obtains another smooth Calabi-Yau $3$-fold $Y$. Let $p$ be the number of nodes on $X_0$, and let $\alpha$ be the number of relations between the homology classes of the $p$ vanishing $3$-cycles on $X$ shrinking to nodes in $X_0$. Then the Hodge numbers of $X$ and $Y$ are related by the following equations \cite{C}: \[ h^{1,1}(Y) = h^{1,1}(X) + \alpha, \] \[ h^{2,1}(Y) = h^{2,1}(X) -p + \alpha. \] The Hodge numbers of mirrors $X^*$ and $Y^*$ of $X$ and $Y$ must satisfy the equations \[ h^{1,1}(X) = h^{2,1}(X^*), \; h^{1,1}(X^*) = h^{2,1}(X) \] and \[ h^{1,1}(Y) = h^{2,1}(Y^*), \; h^{1,1}(Y^*) = h^{2,1}(Y). \] It is natural to expect that the moduli spaces of mirrors $X^*$ and $Y^*$ are again connected in the same simplest way: i.e., that $X^*$ can be obtained by a small resolution of some Calabi-Yau $3$-fold $Y^*_0$ with $p^*$ nodes and $\alpha^*$ relations, corresponding to a point $y_0^*$ on the moduli space of complex structures on $Y^*$. Hence, as suggested in \cite{CGGK,GMS,LS} and \cite{DM1}, the conifold transition can be used to find mirrors of $X$, provided one knows mirrors $Y^*$ of $Y$. For this to work, one then needs $$p^*=\alpha+\alpha^*=p, $$ i.e., $X_0$ and $Y_0^*$ have the same number of nodes and complementary number of relations between them. We remark that even for the simplest family of Calabi-Yau $3$-folds, quintic hypersurfaces in ${\bf P}^4$, it is an open problem to determine all possible values of $p$ \cite{Straten}. One of the problems solved in this paper is an explicit geometric construction of mirrors $X^*$ for Calabi-Yau complete intersections $3$-folds $X$ in Grassmannians $G(k,n)$ (this was only known for quartics in $G(2,4)$, as a particular example of complete intersections in projective space \cite{LT}). Our method is based on connecting $X$ via a conifold transition to complete intersections $Y$ in a toric manifold. This manifold is a small crepant desingularization $\widehat{P(k,n)}$ of a Gorenstein toric Fano variety $P(k,n)$, which in turn is a flat degeneration of $G(k,n)$ in its Pl\"ucker embedding, constructed by Sturmfels (see \cite{St}, Ch. 11). Since one knows how to construct mirrors for Calabi-Yau complete intersections in $\widehat{P}(k,n)$ \cite{BS,LB}, it remains to find an appropriate specialization of the toric mirrors $Y^*$ for $Y$ to conifolds $Y_0^*$ whose small resolutions provide mirrors $X^*$ of $X$. The choice of the $1$-parameter subfamily of $Y_0^*$ among toric mirrors $Y^*$ is determined by the monomial-divisor mirror correspondence and the embedding $${\bf Z} \cong Pic(P(k,n)) \hookrightarrow Pic(\widehat{P}(k,n)) \cong {\bf Z}^{ 1 + (k-1)(n-k-1)}.$$ We expect that this method of mirror constructions can be applied to all Calabi-Yau $3$-folds whose moduli spaces are connected by conifolds transitions to the web of Calabi-Yau complete intersections in Gorenstein toric Fano varieties. This web has been studied in \cite{ACJM,AKMS,BLS} as a generalization of the earlier results on Calabi-Yau complete intersection in products of projective spaces and in weighted projective spaces \cite{CDLS,CGH}. In order to obtain the instanton numbers of rational curves on Calabi-Yau complete intersections in Grassmannians, we compute a generalized hypergeometric series $\Phi_X(z)$, describing the monodromy invariant period of $X^*$, by specializing a $(1 + (k-1)(n-k-1))$-dimensional generalized (Gelfand-Kapranov-Zelevinski) $GKZ$-hypergeometric series for the main period of toric mirrors $Y^*$ to a single monomial parameter $z$. Since $h^{1,1}(X) =1$, the corresponding Picard-Fuchs differential system for periods of $X^*$ reduces to an ordinary differential equation ${\cal D} \Phi =0$ of order $4$ for $\Phi_X(z)$. The Picard-Fuchs differential operator $P$ can be computed from the recurrent relation satisfied by the coefficients of the series $\Phi_X(z)$. Applying the same computational algorithm as in \cite{BS}, one computes the instanton numbers of rational curves on all possible Calabi-Yau complete intersection $3$-folds $X \subset G(k,n)$. The numbers of lines and conics on these Calabi-Yau $3$-folds have been verified by S.-A. Str{\o}mme using classical methods and the Schubert package for MapleV. Another new ingredient of the present paper is the so called {\em Trick with the Factorials}. This is a naive form of a {\em Lefschetz hyperplane section theorem} in quantum cohomology, which goes back to Givental's idea \cite{G2} about the relation between solutions of quantum ${\cal D}$-module for Fano manifolds $V$ and complete intersections $X \subset V$. The validity of this procedure has been established recently for all homogeneous spaces by B. Kim in \cite{K3}. If the Trick with the Factorials works for a Fano manifold $V$, one is able to compute the instanton numbers of rational curves on Calabi-Yau complete intersections $X \subset V$ without knowing a mirror $X^*$ for $X$, provided one knows a special regular solution $A_V$ to the quantum ${\cal D}$-module for $V$. In the case of Grassmannians we conjecture in \ref{flagmirror} that this special solution $A_{G(k,n)}(q)$ to the quantum ${\cal D}$-module determined by the small quantum cohomology of $G(k,n)$) can be obtained from a natural specialization of the $GKZ$-hypergeometric series associated with the Gorenstein toric degeneration $P(k,n)$ of $G(k,n)$. Conjecture \ref{flagmirror} has been checked by direct computation for all Grassmanians containing Calabi-Yau $3$-folds $X$ as complete intersections. In fact, there is no essential difficulty in checking the conjecture in each particular case at hand, because such a check involves only calculations in the small quantum cohomology ring of $G(k,n)$, whose structure is well-known \cite{Ber}. This last result implies that the instanton numbers for rational curves on $3$-dimensional Calabi-Yau complete intersections in Grassmannians are correct in all computed cases. We remark that our conjecture \ref{flagmirror} on the coincidence of $A_{G(k,n)}(q)$ with the specialization of the multidimensional generalized $GKZ$-hypergeometric series corresponding to the Gorenstein toric Fano variety $P(k,n)$ strongly supports the idea that Gromov-Witten invariants of $G(k,n)$ and complete intersections $X \subset G(k,n)$ behave well under flat deformation and conifold transitions. Using the degeneration of $G(k,n)$ to $P(k,n)$, we propose in arbitrary dimension an explicit construction for mirrors of Calabi-Yau complete intersections $X \subset G(k,n)$ whose monodromy invariant period coincide with the power series $\Phi_X(z)$ obtained by applying the Trick with the Factorials to $A_{G(k,n)}(q)$. We observe that our mirror construction is consistent with the formula for the Lax operator of Grassmannians conjectured by Eguchi, Hori and Xiong in \cite{EHX}. Many results formulated in this paper have been generalized and proved in \cite{BCKS} for toric degenerations of partial flag manifolds which have been introduced and investigated by N. Gonciulea and V. Lakshmibai in \cite{GL0,GL1,GL2}. These results are most easily interpreted in terms of certain {\em diagrams} associated to a partial flag manifold, generalizing the one used in \cite{G2} for the case of the complete flag manifold. \section{Simplest Examples} \subsection{Quartics in $G(2,4)$} First we illustrate our method by analyzing a simple case, for which the mirror construction is already known: the case of quartics in $G(2,4)$, the Grassmannian of $2$-planes in ${\bf C} ^4$ \cite{BS,LT}. The Pl\"{u}cker embedding realizes the Grassmannian $G(2,4)$ as a nonsingular quadric in ${\bf P}^5$, defined by the homogeneous equation: \[ z_{12} z_{34} - z_{13}z_{24} + z_{14}z_{23} = 0, \] where $z_{ij}$ $(1\leq i<j \leq 4)$ are homogeneous coordinates on ${\bf P}^5$. Let $P(2,4) \subset {\bf P}^5$ be the $4$-dimensional Gorenstein toric Fano variety defined by the quadratic equation \[ z_{13}z_{24} = z_{14}z_{23}. \] Denote by $X$ the intersection of $G(2,4)$ with a generic hypersurface $H$ of degree $4$ in ${\bf P}^5$, so that $X$ is a nonsingular Calabi-Yau hypersurface in $G(2,4)$. Its topological invariants are $h^{1,1}(X) = 1$, $h^{2,1}(X) = 89$, and $\chi(X)= -176$. Denote by $X_0$ the intersection of $P(2,4)$ with a generic hypersurface $H$ of degree $4$ in ${\bf P}^5$. Then $X_0$ is a Calabi-Yau $3$-fold with $4$ nodes which are the intersection points of $H$ with the line $l \subset P(2,4)$ of conifold singularities. Considering $X_0$ as a deformation of $X$, it follows from general formulas proved in \ref{formul-2} that the homology classes of the vanishing $3$-cycles on $X$ shrinking to $4$ nodes in $X_0$ satisfy a single relation. Denote by $Y$ a simultaneous small resolution of all $4$ nodes. One obtains this resolution by restriction of a small toric resolution of singularities in $P(2,4)$: $\rho\,: \; \widehat{P}(2,4) \rightarrow P(2,4)$. The smooth toric variety $\widehat{P}(2,4)$ is a toric ${\bf P}^3$-bundle over ${\bf P}^1$: \[ \widehat{P}(2,4) = {\bf P}_{{\bf P}^1}({\cal O} \oplus {\cal O} \oplus {\cal O}(1) \oplus {\cal O}(1)) \] and the morphism $\rho$ contracts a $1$-parameter family of sections of this ${\bf P}^3$-bundle with the normal bundle ${\cal O} \oplus {\cal O}(-1) \oplus {\cal O}(-1)$. A smooth Calabi-Yau hypersurface $Y \subset \widehat{P}(2,4)$ has a natural $K3$-fibration over ${\bf P}^1$ and the following topological invariants: $\chi(Y) = -168$, $h^{1,1}(Y) = 2$, and $h^{2,1}(Y)=86$. The Gorenstein toric Fano variety $P(2,4)$ can be described by a $4$-dimensional fan $\Sigma(2,4) \subset {\bf R}^4$ consisting of cones over the faces of a $4$-dimensional reflexive polyhedron $\Delta(2,4)$ with $6$ vertices: \[ u_{1,0} := f_{1,1}, \; u_{2,0} = f_{2,1} - f_{1,1}, \; u_{2,1}: = f_{2,2} - f_{1,2}, \] \[ v_{2,2} := - f_{2,2}, \; v_{2,1}: = f_{2,2} - f_{2,1}, \; v_{1,1} = f_{1,2} - f_{1,1}, \] where $\{ f_{1,1}, f_{1,2}, f_{2,1}, f_{2,2} \}$ is a basis of the lattice ${\bf Z}^4 \subset {\bf R}^4$. The regular fan $\widehat{\Sigma}(2,4)$ defining the smooth projective toric variety $\widehat{P}(2,4)$ is obtained by a subdivision of $\Sigma(2,4)$. The combinatorial structure of $\widehat{\Sigma}(2,4)$ is defined by the following primitive collections (see notations in \cite{Ba1}): \[ {\cal R} = \{ u_{1,0}, v_{1,1}, u_{2,1}, v_{2,2} \} , \; {\cal C}_{1,1} = \{ v_{2,1}, u_{2,0} \}. \] The fan $\widehat{\Sigma}(2,4)$ contains $8$ cones of dimension $4$, obtained by deleting one vector from each primitive collection. The primitive relations corresponding to ${\cal R}_0$ and ${\cal C}_{1,1}$ are \[ u_{1,0} + v_{1,1} + u_{2,1} + v_{2,2} = 0 \] and \[ v_{2,1} + u_{2,0} = v_{1,1} + u_{2,1}. \] Let ${\bf P}_{\Delta(2,4)}$ be the Gorenstein toric Fano variety associated with the reflexive polyhedron $\Delta(2,4)$. By the toric method of \cite{Ba2}, the mirror $Y^*$ of $Y$ can be obtained as a crepant desingularization of the closure in ${\bf P}_{\Delta(2,4)}$ of an affine hypersurface $Z_f$ with the equation \[ f(X) = -1 + a_1X_1 + a_2X_2 + a_3 X_3 + a_4 X_4 + a_5(X_1X_2X_3)^{-1} + a_6(X_4^{-1}X_1X_2), \] where $a_1, \ldots, a_6$ are some complex numbers (the Newton polyhedron of $f$ is isomorphic to $\Delta(2,4)$). We choose a subfamily of Laurent polynomials $f_0$ with coefficients $\{a_i \}$ satisfying an additional monomial equation \[ a_1 a_2 = a_4 a_6. \] The affine Calabi-Yau hypersurfaces $Z_{f_0}$ of this subfamily are not $\Delta(2,4)$-regular anymore, because the closures $\overline{Z}_{f_0}$ in ${\bf P}_{\Delta(2,4)}$ have a singular intersection with the stratum $T_{\Theta} \subset {\bf P}_{\Delta(2,4)}$ corresponding to the face \[ \Theta = {\rm Conv}\left\{ (1,0,0,0),\, (0,1,0,0), \, (0,0,0,1),\,(1,1,0,-1) \right\}. \] Without loss of generality, we can assume that $a_1=a_2=a_3=a_4 =1$ (this condition can be satisfied using the action of $({\bf C}^*)^4$ on $X_1, \ldots, X_4$). Thus we obtain a $2$-parameter family of Laurent polynomials defining $Z_f$: \[ f(X) = -1 + X_1 + X_2 + X_3 + X_4 + a_5(X_1X_2X_3)^{-1} + a_6(X_4^{-1}X_1X_2), \] and a $1$-parameter subfamily of Laurent polynomials \[ f_0(X) = -1 + X_1 + X_2 + X_3 + X_4 + a_5(X_1X_2X_3)^{-1} + (X_4^{-1}X_1X_2) \] defining $Z_{f_0}$. The monodromy invariant period $\Phi$ of the toric hypersurface $Z_f$ can be computed by the residue theorem: \[ \Phi_X(a_5,a_6) = \frac{1}{(2\pi i )^4} \int_{\gamma} \frac{1}{(-f)} \frac{dX_1}{X_1} \wedge \cdots \wedge \frac{dX_4}{X_4}. \] By this method, we obtain the generalized hypergeometric series corresponding to $f(X)$: \[ \Phi_X(a_5,a_6) = \sum_{k, l \geq 0} \frac{(4k + 4l)!}{(k!)^2 (l!)^2 ((k+l)!)^2} a_5^{k+l} a_6^l. \] By the substitution $a_6 =1$ $(a_1a_2 = a_4a_6)$ and $a_5 = z$, we obtain the series corresponding to the $1$-parameter family of Laurent polynomials $f_0$: \[ \Phi_X(z) = \sum_{m \geq 0} \frac{(4m)!}{(m!)^2} \left(\sum_{k+l = m} \frac{1}{(k!)^2 (l!)^2} \right)z^m. \] Using the identity \[ \sum_{k+l =m} \frac{(m!)^2}{(k!)^2 (l!)^2} = { 2m \choose m}, \] we transform $\Phi_X(z)$ to the form \[ \Phi_X(z) = \sum_{m \geq 0} \frac{(4m)!(2m)!}{(m!)^6} z^m. \] This is a well-known series, satisfying a Picard-Fuchs differential equation \[ \left( D^4 - 16z(2 D +1)^2(4 D + 1)(4 D + 3)\right) \Phi_X(z) = 0,\;\; D = z\frac{\partial}{\partial z}, \] predicting the instanton numbers of rational curves on $X$ (cf. \cite{LT}). The correctness of these numbers now follows from the work of Givental, \cite{G1}. \subsection{Complete intersections of type $(1,1,3)$ in $G(2,5)$} \vskip 10pt Let $z_{ij}$ $(1 \leq i < j \leq 5$ ) be homogeneous coordinates on the projective space ${\bf P}^9$. The Grassmannian $G(2,5)$ of $2$-planes in ${\bf C}^5$ can be identified with the subvariety in ${\bf P}^9$ defined by the quadratic equations: \[ z_{23}z_{45} - z_{24}z_{35} + z_{25}z_{34}= 0, \] \[ z_{13}z_{45} - z_{14}z_{35} + z_{15}z_{34}= 0, \] \[ z_{12}z_{45} - z_{14}z_{35} + z_{15}z_{34}= 0, \] \[ z_{12}z_{35} - z_{13}z_{25} + z_{15}z_{23}= 0, \] \[ z_{12}z_{34} - z_{13}z_{24} + z_{14}z_{23}= 0. \] We associate with $G(2,5)$ a $6$-dimensional Gorenstein toric Fano variety $P(2,5) \subset {\bf P}^9$ defined by the equations \[ z_{24}z_{35} = z_{25}z_{34}, \; z_{14}z_{35} = z_{15}z_{34},\; z_{14}z_{35} = z_{15}z_{34}, \] \[ z_{13}z_{25} = z_{15}z_{23}, \; z_{13}z_{24} = z_{14}z_{23}. \] The following statement is due to Sturmfels (see \cite{St}, Example 11.9 and Proposition 11.10): \begin{prop} The Gorenstein toric Fano variety $P(2,5)$ is a degeneration of the Grassmannian $G(2,5)$, i.e., $P(2,5)$ is the special fibre of a flat family whose generic fibre is $G(2,5)$. \hspace*{\fill}\hbox{$\Box$} \end{prop} The toric variety $P(2,5)$ can be described by a fan $\Sigma(2,5) \subset {\bf R}^6$ consisting of cones over the faces of a $6$-dimensional reflexive polyhedron $\Delta(2,5)$ with $9$ vertices \[ u_{1,0} := f_{1,1}\; u_{2,i}: = f_{2,i+1} - f_{1,i+1}, \,\; i = 0, 1, 2,\] \[ v_{2,3} := - f_{2,3}, \; v_{i,j}: = f_{i,j+1} - f_{i,j}, \, \; i = 1, 2,\; j =1,2, \] where $\{ f_{1,1}, f_{1,2}, f_{1,3}, f_{2,1}, f_{2,2}, f_{2,3} \}$ is a basis of the lattice ${\bf Z}^6 \subset {\bf R}^6$. There exists a subdivison of the fan $\Sigma(2,5)$ into a regular fan $\widehat{\Sigma}(2,5)$ defined by the primitive collections: \[ {\cal R} = \{ u_{1,0}, v_{1,1}, v_{1,2}, u_{2,2}, v_{2,3} \}, \] \[ {\cal C}_{1,1} = \{ u_{2,0}, v_{2,1} \}, \; {\cal C}_{1,2} = \{ u_{2,1}, v_{2,2} \}, \] i.e., $\widehat{\Sigma}(2,5)$ contains exactly $20$ cones of dimension $6$ generated by the $6$-element sets obtained by taking all but one of the vectors from each primitive collection. The primitive relations corresponding to ${\cal R}$, ${\cal C}_{1,1}$ and ${\cal C}_{1,2}$ are \[ u_{1,0} + v_{1,1} + v_{1,2} + u_{2,2} + v_{2,3} = 0, \] \[ u_{2,0} + v_{2,1} = v_{1,1} + u_{2,1}, \; \; u_{2,1} + v_{2,2} = v_{1,2} + u_{2,2}. \] Denote by $\widehat{P}(2,5)$ the smooth toric variety associated with the fan $\widehat{\Sigma}(2,5)$. It is easy to check that $P(2,5)$ is a Gorenstein toric Fano variety and $\widehat{P}(2,5)$ is a small crepant resolution of singularities of $P(2,5)$. The toric manifold $\widehat{P}(2,5)$ has nonnegative first Chern class and it can be identified with a toric bundle over ${\bf P}^1$ with the $5$-dimensional fiber \[ F: = {\bf P}_{{\bf P}^1} ({\cal O} \oplus {\cal O} \oplus {\cal O} \oplus {\cal O}(1) \oplus {\cal O}(1) ) \] There is another description of $P(2,5)$. We remark that variables $z_{12}$ and $z_{45}$ do not appear in the equations for $P(2,5)$. Thus $P(2,5)$ is a cone over a Gorenstein $4$-dimensional toric Fano variety \[ P'(2,5) : = P(2,5) \cap \{ z_{12} = z_{45} = 0\} \subset {\bf P}^7. \] We can describe $P'(2,5)$ by a $4$-dimensional fan $\Sigma'(2,5)$ consisting of cones over a $4$-dimensional reflexive polyhedron $\Delta'(2,5)$ with $7$ vertices \[ e_1 = (1,0, 0,0), \; e_2= (0,1,0,0),\; e_3 = (-1,-1, 0,0), \] \[ e_4 = (0,0, 1,0), \; e_5= (0,0,0,1),\; e_6 = (0,0,-1,-1), \] \[ e_7 = (1,1,1,1). \] The only singularities of $P'(2,5)$ are nodes along two lines $l_1, l_2 \in P'(2,5) \subset {\bf P}^7$ corresponding to the $3$-dimensional cones $$ \sigma_1 = {\bf R}_{\geq 0} < e_1, e_2, e_6, e_7 >\;\; \mbox{\rm and } \;\;\sigma_2 = {\bf R}_{\geq 0} < e_4, e_5, e_3, e_7 >$$ in $\Sigma'(2,5)$. Subdividing each of these cones into the union of $2$ simplicial ones, we obtain a small crepant resolution $\widehat{P'}(2,5)$ of singularities of $P'(2,5)$. The smooth toric $4$-fold $\widehat{P'}(2,5)$ can be identified with the blow up of a point on ${\bf P}^2 \times {\bf P}^2$. Let $X = X_{1,1,3} \subset G(2,5)$ be a smooth $3$-dimensional Calabi-Yau complete intersection of $3$ hypersurfaces of degrees $1$, $1$ and $3$ in ${\bf P}^9$ with $G(2,5)$. One can compute $h^{1,1}(X) = 1$, $h^{2,1}(X) = 76$, and $\chi(X) = -150$. Now let $X_0$ be the intersection of $P'(2,5)$ with a generic hypersurface $H \subset {\bf P}^7$ of degree $3$. Then $X_0$ is a deformation of $X$, having $6$ nodes obtained from the intersections $H \cap l_1$ and $H \cap l_2$. The $3$ nodes on each intersection $H \cap l_i$ $(i =1,2)$ are described by $3$ vanishing $3$-cycles on $X$, satisfying a single linear relation. Resolving singularities of $X_0$, we obtain another smooth Calabi-Yau $3$-fold $Y$ with \[ h^{1,1}(Y) = h^{1,1}(X) + 2 = 3, \;\; h^{2,1}(Y) = h^{2,1}(X) + 2 - 6 = 72. \] The mirror $Y^*$ of the Calabi-Yau $3$-fold $Y$ can be obtained by the toric construction \cite{Ba2}. The Calabi-Yau $3$-fold $Y^*$ is a toric desingularization $\widehat{Z}_f$ of a $\Delta'(2,5)$-compactification of a generic hypersurface $Z_f$ in $({\bf C}^*)^4$ defined by a Laurent polynomial $f(X)$ with the Newton polyhedron $\Delta'(2,5)$: \[ f(X) = -1 + a_1 X_1 + a_2 X_2 + a_3 (X_1X_2)^{-1} + a_4 X_3 + a_5X_4+ \] \[ + a_6(X_3X_4)^{-1} + a_7X_1X_2X_3X_4. \] As it was shown in \cite{Ba2}, one has $h^{1,1}(\widehat{Z}_f) = h^{2,1}(Y) = 72$ and $h^{2,1}(\widehat{Z}_f) = h^{1,1}(Y) = 3$. We identify the mirror $X^*$ of $X$ with a desingularization $\widehat{Z}_{f_0}$ of a $\Delta'(2,5)$-compactification $\overline{Z}_{f_0}$ of a generic hypersurface $Z_{f_0}$ in $({\bf C}^*)^4$ defined by Laurent polynomials $f_0$ whose coefficients $\{ a_i\} $ satisfy two additional monomial equations $$ a_1a_2 = a_6a_7\ \ \mbox{\rm and} \ \ a_4 a_5 = a_3 a_7. $$ Without loss of generality, we can put $a_1 = a_2 = a_4 = a_7$. So one obtains \[ f(X) = -1 + X_1 + X_2 + a_3 (X_1X_2)^{-1} + X_3 + a_5X_4 \] \[ + a_6(X_3X_4)^{-1} + X_1X_2X_3X_4 \] and \[ f_0(X) = -1 + X_1 + X_2 + a_3 (X_1X_2)^{-1} + X_3 + a_3X_4 \] \[ + (X_3X_4)^{-1} + X_1X_2X_3X_4. \] It is easy to see that the Laurent polynomial $f_0$ is not $\Delta'(2,5)$-regular (this regularity fails exactly for two $2$-dimensional faces $\Theta_1 := {\rm Conv}(e_1,e_2,e_6,e_7)$ and $\Theta_2:= {\rm Conv}(e_4,e_5,e_3,e_7)$ of $\Delta'(2,5)$ (see definition of $\Delta$-regularity in \cite{Ba2}). The $4$-dimensional Gorenstein toric Fano variety ${\bf P}_{\Delta'(2,5)}$ associated with the reflexive polyhedron $\Delta'(2,5)$-closure has singularities of type $A_2$ along of the $2$-dimensional strata $T_{\Theta_1}$ and $T_{\Theta_2}$. The projective hypersurfaces $\overline{Z}_{f_0} \subset {\bf P}_{\Delta'(2,5)}$ defined by the equation $f_0 =0$ have non-transversal intersections with $T_{\Theta_1}$ and $T_{\Theta_2}$ (each intersection is a union of two rational curves with a single normal crossing point). After toric resolution of $A_2$-singularities along $T_{\Theta_i}$ on ${\bf P}_{\Delta'(2,5)}$, we obtain $3$ new $2$-dimensional strata over each $T_{\Theta_i}$. This shows that we cannot resolve all singularities of $\overline{Z}_{f_0}$ by a toric resolution of singularities on the ambient toric variety ${\bf P}_{\Delta'(2,5)}$. Let $Y_0^* := \widehat{Z}_{f_0}$ be the pullback of $\overline{Z}_{f_0}$ under a $MPCP$-desingularization $$\rho\, : \, \widehat{{\bf P}}_{\Delta'(2,5)} \rightarrow {\bf P}_{\Delta'(2,5)}.$$ Then $Y_0^*$ is a Calabi-Yau 3-fold with $3 + 3 = 6$ nodes obtained as singular points of intersections of $Y_0^*$ with the $6$ strata of dimension $2$ in $\widehat{{\bf P}}_{\Delta'(2,5)}$ over $T_{\Theta_1},\; T_{\Theta_2} \subset \widehat{{\bf P}}_{\Delta'(2,5)}$. One can show that the vanishing $3$-cycles associated with the $3$ nodes over each $T_{\Theta_i}$ $(i=1,2)$ satisfy $2$ linear relations (see \ref{formul-2}). If $X^*$ denotes a small resolution of these $6$ nodes on $Y_0^*$, then \[ h^{1,1}(X^*) = h^{1,1}(\widehat{Z}_{f}) + 4 = 76 \] and \[ h^{2,1}(X^*) = h^{2,1}(\widehat{Z}_f) + 4 - 6 = 1. \] Thus the Hodge numbers of $X^*$ and $X$ satisfy the mirror duality. Finally, we explain the computation of the instanton numbers of rational curves of degree $m$ in the case of Calabi-Yau complete intersections $X$ of type $(1,1,3)$ in $G(2,5)$. As shown in \cite{BS}, one obtains the following monodromy invariant period for $Z_f$: $$\Phi(a_3,a_5,a_6) = \sum_{k,l,n \geq 0} \frac{(3k+3l+3n)!}{(k!)^2(n!)^2l!(k+l)!(l+n)!}a_3^{k+l}a_5^n a_6^{n+l}.$$ By the substitution $a_3 = a_5 =z$ and $a_6 = 1$, we obtain the monodromy invariant period for $Z_{f_0}$: \[ \Phi_X(z) = \left( \sum_{k + l +n = m} \frac{(3m)!}{(k!)^2(n!)^2l!(k+l)!(l+n)!} \right) z^m. $$ It remains to apply to the series $\Phi_X(z)$ the general algorithm from \cite{BS} (see 6.2 and 7.1 for details, and the instanton numbers). \vskip 10pt \section{Toric Degenerations of Grassmannians} In this section we review without proof some results, which we prove for arbitrary partial flag manifolds in \cite{BCKS}. \subsection{The toric variety $P(k,n)$ and its singular locus} Let $G(k,n)$ be the Grassmannian of $k$-dimensional ${\bf C}$-vector subspaces in a $n$-dimensional complex vector space ($k < n$). Denote by $$X_{i,j} \;\; i = 1, \ldots, k, \; j = 1, \ldots, n-k$$ $k(n-k)$ independent variables. We denote by $T(k,n)$ the algebraic torus $ {\rm Spec}\, {\bf C}[X_{i,j},X_{i,j}^{-1} ]\cong ({\bf C}^*)^{k(n-k)}$ of dimension $k(n-k)$. We put $N(k,n): = {\bf Z}^{k(n-k)} $ to be a free abelian group of rank $k(n-k)$ with a fixed ${\bf Z}$-basis $f_{i,j}$ $(i = 1, \ldots, k, \; j = 1, \ldots, n - k)$. Define the set of $2(k-1)(n-k-1) + n$ elements in $N(k,n)$ as follows: \[ u_{1,0} := f_{1,1},\; u_{i,j}: = f_{i,j+1} - f_{i-1,j+1}, \, \; i = 2, \ldots, k,\; j =0, \ldots, n-k-1\; \] \[ v_{k,n-k}: = - f_{k,n-k}, \; v_{i,j}: = f_{i,j+1} - f_{i,j}, \,\; i = 1, \ldots, k,\; j =1, \ldots, n-k-1. \] We set $N(k,n)_{\bf R} = N(k,n) \otimes {\bf R}$. \begin{definition} {\rm Define a convex polyhedron $\Delta(k,n) \subset N(k,n)_{\bf R}$ as the convex hull of all lattice points $\{ u_{i,j}, v_{i',j'} \}$. We set $\Sigma(k,n) \subset N(k,n)_{\bf R}$ to be the fan over all proper faces of the polyhedron $\Delta(k,n)$.} \label{polyh} \end{definition} \begin{definition} {\rm Define $P(k,n)$ to be the toric variety associated with the fan $\Sigma(k,n)$. } \end{definition} \begin{theorem} The polyhedron $\Delta(k,n)$ is reflexive. In particular, $P(k,n)$ is a Gorenstein toric Fano variety. \end{theorem} \begin{definition} {\rm Let $\widehat{\Sigma}(k,n)$ be a complete regular fan whose $1$-dimensional cones are generated by the lattice vectors $\{ u_{i,j}, v_{l,m}\} $ and whose combinatorics is defined by the following $1 + (k-1)(n-k-1)$ primitive collections: \[ {\cal R}_0: = \{ u_{1,0}, v_{1,1}, v_{1,2}, \ldots, v_{1,{n-k-1}}, u_{2, n-k-1}, u_{3, n-k -1}, \ldots, u_{k,n-k-1}, v_{k,n-k} \}, \] \[ {\cal C}_{i,j} = \{ u_{k+1-i, j-1}, v_{k+1-i, j},\; \; i=1, \ldots, k-1,\; j =1, \ldots, n-k-1 \}.\] In particular, the fan $\widehat{\Sigma}(k,n)$ consists of $n2^{(k-1)(n-k-1)}$ cones of dimension $k(n-k)$. } \end{definition} \begin{remark} {\rm We notice that the lattice vectors $u_{i,j}$ and $v_{l,m}$ satisfy the following $1+ (k-1)(n-k-1)$ independent primitive relations: \[ u_{1,0} + v_{1,1} + \cdots + v_{1,{n-k-1}} + u_{2, n-k-1} + \cdots + u_{k,n-k-1} + v_{k,n-k} = 0, \] \[ u_{k+1-i, j-1} + v_{k+1-i, j} = u_{k+1-i, j} + v_{k-i,j}, \] \[ i=1, \ldots, k-1,\; j =1, \ldots, n-k-1 .\] According to Theorem 4.3 in \cite{Ba1}, the toric variety $\widehat{\Sigma}(k,n)$ can be obtained as $ (k-1)(n-k-1)$-times iterated toric bundle over ${\bf P}^1$'s: we start with ${\bf P}^{n-1}$ and construct on each step a toric bundle over ${\bf P}^1$ whose fiber is the toric variety constructed in the previous step. At each stage of this process, we obtain a smooth projective toric variety with the nonnegative first Chern class which is divisible by $n$. In particular we obtain that the smooth projective toric variety $\widehat{P}(k,n)$ defined by the fan $\widehat{\Sigma}(k,n)$ has Picard number $1 + (k-1)(n-k-1)$. Moreover, the first Chern class $\widehat{c}_1(k,n)$ of $\widehat{P}(k,n)$ is nonnegative and it is divisible by $n$ in ${\rm Pic}(\widehat{P}(k,n))$. } \end{remark} \begin{definition} We denote by $\widehat{P}(k,n)$ $( 1 \leq i \leq k-1, \; 1 \leq j \leq n- k-1)$ $(k-1)(n-k-1)$ codimension-$2$ subvarieties of $\widehat{P}(k,n)$ corresponding to the $2$-dimensional cones $ \sigma_{ij} \in \widehat{\Sigma}(k,n)$: \[ \sigma_{ij} = {\bf R}_{\geq 0} <u_{k+1-i, j-1}, v_{k+1-i, j} >. \] \end{definition} \begin{theorem} The small contraction $\rho\, : \, \widehat{P}(k,n) \rightarrow {P}(k,n)$ defined by the semi-ample anticanonical divisor on $\widehat{P}(k,n)$ contracts smooth toric varieties $\widehat{W}_{i,j}$ to codimension-$3$ toric subvarieties $W_{i,j} \subset {P}(k,n)$ whose open strata consist of conifold singularities, i.e., singularities whose $3$-dimensional cross-sections are isolated nondegenerate quadratic singularities (nodes, ordinary double points). \label{sing-l} \end{theorem} The proof of a generalized version of \ref{sing-l} for arbitrary partial flag manifolds is contained in \cite{BCKS}(Th. 3.1.4). \subsection{The flat degeneration of $G(k,n)$ to $P(k,n)$} \begin{definition} {\rm Denote by $A(k,n)$ the set of all sequences of integers \[ a= (a_1, a_2, \ldots, a_k) \in {\bf Z}^k \] satisfying the condition \[ 1 \leq a_1 < a_2 < \cdots < a_k \leq n. \] We consider $A(k,n)$ as a partially ordered set with the following natural partial order: \[ a = (a_1, \ldots, a_k) \prec a'= (a_1', \ldots, a_k') \] if and only if $a_i \leq a_i'$ for all $i =1, \ldots, k$. We set \[ \min{(a,a')} : = (\min{(a_1,a_1')}, \ldots, \min{(a_k,a_k')}) \] and \[ \max{(a,a')} : = (\max{(a_1,a_1')}, \ldots, \max{(a_k,a_k')}). \] } \end{definition} \begin{theorem} There exists a natural one-to-one correspondence between faces of codimension $1$ of the polyhedron $\Delta(k,n)$ and elements of $A(k,n)$. \end{theorem} \noindent {\em Proof.} See \cite{BCKS} (Th. 2.2.3). \begin{theorem} The first Chern class of the Gorenstein toric Fano variety ${P}(k,n)$ is equal to $n[H]$, where $[H]$ is the class of the ample generator of ${\rm Pic}({P}(k,n)) \cong {\bf Z}$. Moreover, there exists a natural one-to-one correspondence between the elements of the monomials basis of \[ H^0({P}(k,n), {\cal O}(H)) \] and elements of $A(k,n)$. In particular, \[ {\rm dim} \, H^0({P}(k,n), {\cal O}(H)) = { n \choose k }. \] \label{gl-sections} \end{theorem} \noindent {\em Proof.} See \cite{BCKS} (Prop. 3.2.5). \begin{theorem} The ample line bundle ${\cal O}(H)$ on $P(k,n)$ defines a projective embedding into the projective space ${\bf P}^{{ n \choose k}-1}$ whose homogeneous coordinates $z_a$ are naturally indexed by elements $a \in A(k,n)$. Moreover, the image of $P(k,n)$ in ${\bf P}^{{ n \choose k}-1}$ is defined by the quadratic homogeneous binomial equations \[ z_a z_{a'} - z_{min(a,a')}z_{max(a,a')} \] for all pairs $(a,a')$ of non-comparable elements $a, a' \in A(k,n)$. \end{theorem} \noindent {\em Proof.} See \cite{BCKS} (Th. 3.2.13). \begin{example} {\rm The following ${ n \choose 4 }$ quadratic equations in homogeneous coordinates $\{ z_{i,j} \}$ $( 1\leq i < j \leq n)$ are defining equations for the toric variety $P(2,n)$ in ${\bf P}^{{ n \choose 2}-1}$: \[ z_{i_1,i_4}z_{i_2,i_3} - z_{i_1,i_3}z_{i_2,i_4} = 0, \;\; ( 1 \leq i_1 < i_2 < i_3 < i_4 \leq n). \] } \end{example} The following theorem is due to B. Sturmfels (\cite{St}, Prop. 11.10.) \begin{theorem} There exists a natural flat deformation of the Pl\"ucker-embedded Grassmannian $$G(k,n) \subset {\bf P}^{{ n \choose k}-1}$$ whose special fiber is isomorphic to the subvariety defined quadratic homogeneous binomial equations \[ z_a z_{a'} - z_{min(a,a')}z_{max(a,a')} \] for all pairs $(a,a')$ of noncomparable elements $a, a' \in A(k,n)$. \end{theorem} \begin{corollary} The toric variety $P(k,n) \subset {\bf P}^{{ n \choose k}-1}$ is isomorphic to a flat degeneration of the Pl\"ucker embedding of the Grassmannian $G(k,n)$. \label{def-gr} \end{corollary} \section{Equations for Mirror Manifolds} \subsection{The mirror construction} Recall the definition of nef-partions for Gorenstein toric Fano varieties and the mirror construction for Calabi-Yau complete intersections associated with nef-partitions \cite{LB} (we will follow the notations in \cite{BB2}). \begin{definition} {\rm Let $\Delta \subset M_{\bf R}$ be a reflexive polyhedron, $\Delta^* \subset N_{\bf R}$ its dual, and $\{ e_1, \ldots ,e_l\}$ the set of vertices of $\Delta^*$ corresponding to torus invariant divisors $D_1, \ldots, D_l$ on the Gorenstein toric Fano variety ${\bf P}_{\Delta}$. We set $I := \{ 1, \ldots, l \}$. A partition $I= J_1 \cup \cdots \cup J_r$ of $I$ into a disjoint union of subsets $J_i \subset I$ is called a {\bf nef-partition}, if \[ \sum_{j \in J_i} D_j \] is a semi-ample Cartier divisor on ${\bf P}_{\Delta}$ for all $i =1, \ldots, r$.} \end{definition} \begin{definition} {\rm Let $I= J_1 \cup \cdots \cup J_r$ be a nef-partition. We define the polyhedron $\nabla_i$ $(i =1, \ldots, r)$ as the convex hull of $0 \in \Delta$ and all vertices $e_j$ with $j \in J_i$. By $\Delta_i \subset M_{\bf R}$ $(i =1, \ldots, r)$ we denote the supporting polyhedron for global sections of the corresponding semi-ample invertible sheaf ${\cal O}( \sum_{j \in J_i} D_j)$ on ${\bf P}_{\Delta}$. For each $i =1, \ldots, r$, we denote by $g_i$ (resp. by $h_i$) a generic Laurent polynomial with the Newton polyhedron $\Delta_i$ (resp. $\nabla_i$). } \end{definition} The mirror construction in \cite{LB} says that the mirror of a compactified generic Calabi-Yau complete intersection $g_1 = \cdots = g_r = 0$ is a compactified generic Calabi-Yau complete intersection defined by the equations $h_1 = \cdots = h_r = 0$. Now we specialize the above mirror construction for the case $\Delta = \Delta^*(k,n)$, $\Delta^* = \Delta(k,n)$, and ${\bf P}_{\Delta} = P(k,n)$, where $\Delta(k,n)$ is a reflexive polyhedron defined in \ref{polyh}, $\Delta^*(k,n)$ its polar-dual reflexive polyhedron and $P(k,n)$ the Gorenstein toric Fano degeneration of the Grassmannian $G(k,n)$. \begin{definition} {\rm Define the following $n$ subsets $E_1, \ldots, E_n$ of the set of vertices $\{ u_{i,j}, v_{i',j'} \}$ of the polyhedron $\Delta(k,n)$: \[ E_1 : =\{u_{1,0}\}, \; E_i = \{ u_{i,0}, u_{i,1}, \ldots, u_{i,n-k-1} \}, \; i =2, \ldots,k, \] \[ E_{k+j}: = \{ v_{1,j}, v_{2,j}, \ldots, v_{k,j} \}, \; j =1, \ldots, n-k-1, \; E_n: = \{ v_{k,n-k}\}. \]} \end{definition} \begin{proposition} Let $D(E_i) \subset P(k,n)$ $(i =1, \ldots, n)$ be the torus invariant divisor whose irreducible components have multiplicity $1$ and correspond to vertices of $\Delta(k,n)$ from the subset $E_i$. Then the class of $D(E_i)$ is an ample generator of $Pic(P(k,n))$. \label{gener} \end{proposition} \noindent {\em Proof.} By a direct computation, one obtains that for all $i, j \in \{ 1, \ldots, n \}$ the difference $D(E_i) - D(E_j)$ is a principal divisor, i.e, all divisors $D(E_1), \ldots, D(E_n)$ are linearly equivalent. On the other hand, \[ D(E_1) + \cdots + D(E_n) \] is the ample anticanonical divisor on $P(k,n)$. By \ref{gl-sections}, the anticanonical divisor on $P(k,n)$ is linearly equivalent to $nH$, where $H$ is an ample generator of $Pic(P(k,n))$. Hence, each divisor $D(E_i)$ is linearly equivalent to $H$. \hspace*{\fill}\hbox{$\Box$} \begin{definition} {\rm Let $1 \leq d_1 \leq \cdots \leq d_r$ be positive integers satisfying the equation $$d_1 + \cdots + d_r = n$$ and $I:= \{ 1, \ldots, n \}$. We denote by $X:= X_{d_1, \ldots, d_r} \subset G(k,n)$ a Calabi-Yau complete intersection of hypersurfaces of degrees $d_1, \ldots, d_r$ with $G(k,n) \subset {\bf P}^{ { n \choose k } -1}$. Consider a partition $I= J_1 \cup \cdots \cup J_r$ of $I$ into a disjoint union of subsets $J_i \subset I$ with $|J_i| = d_i$. } \label{J's} \end{definition} \begin{definition} {\rm Let $\nabla_{J_i}$ $(i =1, \ldots, r)$ be the convex hull of $0 \in N(k,n)_{\bf R}$ and all vertices of $\Delta(k,n)$ contained in the union \[ \bigcup_{j \in J_i } E_j. \] We denote by $h_{J_i}(X)$ a generic Laurent polynomial in variables $X_{i',j'} := X^{f_{i',j'}}$ $( 1 \leq i' \leq k, \; 1 \leq j' \leq n-k)$ having $\nabla_{J_i}$ as a Newton polyhedron. } \label{nablas} \end{definition} By \ref{gener}, one immediately obtains the following: \begin{corollary} Let $Y:= Y_{d_1, \ldots, d_r} \subset P(k,n)$ a Calabi-Yau complete intersection of hypersurfaces of degrees $d_1, \ldots, d_r$ with the Gorenstein toric Fano variety $P(k,n) \subset {\bf P}^{ { n \choose k } -1}$. Then the mirror $Y^*$ of $Y$ $($according to \cite{BS} and \cite{LB}$)$ is a compactified generic Calabi-Yau complete intersection defined by the equations \[ h_{J_1}(X) = \cdots = h_{J_r}(X) = 0. \] \end{corollary} \begin{definition} {\rm Define $n$ Laurent polynomials in $k \times (n-k)$ variables $X_{i,j} := X^{f_{i,j}}$ as follows: \[ p_1(X) = a_{1,0} X^{u_{1,0}}, \; p_i(X) = \sum_{j =0}^{n-k-1} a_{i,j}X^{u_{i,j}}, \; i =2, \ldots,k, \] \[ p_{k+j}(X) = \sum_{i =1}^{k} b_{i,j}X^{v_{i,j}}, \; j =1, \ldots, n-k-1, \; p_n(X) = b_{k,n-k}X^{v_{k,n-k}}, \] where $a_{i,j}$ and $b_{l,m}$ are generically choosen complex numbers. In particular, the Newton polyhedron of $p_i(X)$ is the convex hull of $E_i$. } \end{definition} \begin{conjecture} Let $I = \{1, \ldots, n\} = J_1 \cup \cdots \cup J_r$ be a partition of $I$ into a disjoint union of subsets $J_i \subset I$ with $|J_i| = d_i$ as in $($\ref{J's}$)$ and $Y^*_0$ be a Calabi-Yau compactification of a general complete intersection in $({\bf C}^*)^{k(n-k)}$ defined by the equations \[ 1 - \sum_{j \in J_i} p_j(X) = 0 \;\; ( i =1, \ldots, n), \] where the coefficients $a_{i,j}$ and $b_{l,m}$ satisfy the following $(k-1)(n-k-1)$ conditions \[ a_{k+1 -i, j-1} b_{k+1 -i,j} = a_{k+1 -i,j}b_{k-i,j}. \] Then a minimal desingularization $X^*$ of $Y_0^*$ is a mirror of a generic Calabi-Yau complete intersection $X:= X_{d_1, \ldots, d_r} \subset G(k,n)$. \label{mirror-c} \end{conjecture} \begin{example} {\rm If $X: =X_{1,1,3} \subset G(2,5)$, we take $J_1 = \{1\}$, $J_2 = \{5\}$ and $J_3 = \{2,3,4 \}$. Then the mirror construction for $X$ proposed by \ref{mirror-c} coincides with the one considered in 2.2. } \end{example} \subsection{Lax operators of Grassmannians} In the paper \cite{EHX} Eguchi, Hori, and Xiong have computed the Lax operator $L$ for various Fano manifolds V: projective spaces, Del Pezzo surfaces and Grassmannians. In particular for $V = {\bf P}^n$ the corresponding Lax operator $L$ is given by the formula: \[ L = X_1 + X_2 + \cdots + X_n + qX_1^{-1} X_2^{-1} \cdots X_n^{-1}, \] where $\log q $ is an element of $H_2({\bf P}^n)$. On the other hand, if $Z$ is an affine hypersurface defined by the equation $ L(X_1, \ldots, X_n) = 1$ in the algebraic torus $ T \cong ({\bf C}^*)^n = {\rm Spec}\, {\bf C} [ X_1^{\pm 1}, \ldots, X_n^{\pm 1}]$, then, according to \cite{Ba2}, a suitable compactification of $Z$ is a Calabi-Yau variety which is mirror dual to Calabi-Yau hypersurfaces of degree $n+1$ in ${\bf P}^n$. \begin{remark} {\rm It is natural to suggest that the last observation can be used as a guiding principle for the construction of mirror manifolds of Calabi-Yau hypersurfaces $X$ in Fano manifolds $V$.} \end{remark} Let $V$ be a Fano manifold of dimension $n$. Denote by $P$ (resp. by $[V]$) the class of unity (resp. the class of the normalized by unity volume form on $V$) in the cohomology ring $H^*(V)$. Let \[ \omega = \frac{dX_1}{X_1} \wedge \cdots \wedge \frac{dX_n}{X_n} \] be the invariant differential $n$-form on the $n$-dimensional algebraic torus $T \cong ({\bf C}^*)^n$. According to \cite{EHX}, the Lax operator $L(X)$ of the Fano manifold $V$ is a Laurent polynomial in $X_1, \ldots, X_n$ with coefficients in the group algebra ${\bf Q}[ H_2(V,{\bf Z})]$ satisfying for all $m \geq 0$ the equation \[ \langle \sigma_m([V])P) \rangle = \frac{1}{m+1} \int_{\gamma} L^{m+1}(X) \omega. \] where $\sigma_m([V])$ is the $m$-gravitational descendent of $[V]$ on the moduli spaces of stable maps of curves of genus $g =0$ to $V$, $ \langle \sigma_m([V])P) \rangle$ is the corresponding two point correlator function, and $\gamma$ is the standard generator of $H_n(T, {\bf Z})$. For the case $V= G(r,s)$ $(n = r(s-r))$ the following was conjectured in \cite{EHX}: \begin{conjecture} The Lax operator of the Grassmannian $G(r,s)$ has the following form \[ L(X) = X_{[1,1]} + \sum_{\begin{array}{c} {\scriptstyle 1 \leq a \leq s-r } \\ {\scriptstyle 1 \leq b \leq r } \end{array} } X_{[a,b]}^{-1}(X_{[a+1,b]} + X_{[a,b+1]}) + q X_{[s-r,r]}^{-1}, \] where $\log q \in H_2(G(r,s))$ and $X_{a,b} = 0$ if $a > s-r$ or $b > r$. \label{lax} \end{conjecture} \begin{proposition} Let $P(r,s)$ be the toric degeneration of the Grassmannian $G(r,s)$. Then the equation $L(X) = 1$ defines a $1$-parameter subfamily in the family of toric mirrors of Calabi-Yau hypersurfaces in $P(r,s)$ $($see {\rm \cite{Ba2}}$)$. \end{proposition} {\it Proof:} According to \cite{Ba2}, we have to identify the Newton polyhedron of the Laurent polynomial $L(X)$ in Conjecture \ref{lax} with the reflexive polyhedron $\Delta(r,s)$. The latter follows immediately from the explicit description of $\Delta(r,s)$ in \ref{polyh} and from the $1$-to-$1$-correspondence $f_{i,j} \leftrightarrow X_{[j,i]}$.\hspace*{\fill}\hbox{$\Box$} \vskip 10pt \begin{proposition} The equations for the mirrors to Calabi-Yau hypersurfaces conjectured in \ref{mirror-c} in $G(r,s)$ coincide with the equations $L(X) =1$ where $L(X)$ is the Lax operator conjectured for $G(r,s)$ in \cite{EHX}. \end{proposition} \noindent {\it Proof:} It is easy to see that the coefficients of the polynomial $L(X)$ satisfy all $r(s-r)$ monomial relations which reduce to the equality $1 \cdot 1 = 1 \cdot 1$. On the other hand, using the action of the $r(s-r)$-dimensional torus on the coefficients of the Laurent polynomial \[ 1 - (p_1(X) + \cdots + p_s(X)) \] defining the mirror in \ref{mirror-c}, one can reduce to only one independent parameter, for instance, the unique coefficient $b_{r,s-r}$ of $p_s(X) = b_{r,s-r}X_{r,s-r}^{-1}$. By setting $q : = b_{r,s-r}$ and $X_{[i,j]}: = X_{j,i}$, we can identify the variety $Y_0^*$ in \ref{mirror-c} with a toric compactification of the affine hypersurface $L(X) =1$. \hspace*{\fill}\hbox{$\Box$} \vskip 10pt Using the explicit description of the multiplicative structure of the small quantum cohomology of $G(k,n)$, it is not difficult to check Conjecture \ref{lax} for each given $r$ and $s$: \begin{example} {\rm The Lax operator of the Grassmannian $G(2,4)$ is \[ X_{[1,1]} + X_{[1,1]}^{-1}(X_{[2,1]} + X_{[1,2]}) + X_{[2,1]}^{-1}X_{[2,2]} + X_{[1,2]}^{-1}X_{[2,2]} + q X_{[2,2]}^{-1}. \] Its Newton polyhedron is isomorphic to $\Delta(2,4)$ from 2.1.} \end{example} \begin{example} {\rm The Lax operator of the Grassmannian $G(2,5)$ is \[ X_{[1,1]} + X_{[1,1]}^{-1}(X_{[2,1]} + X_{[1,2]}) + X_{[2,1]}^{-1}( X_{[3,1]} + X_{[2,2]} ) + X_{[1,2]}^{-1} X_{[2,2]} + X_{[2,2]}^{-1} X_{[3,2]} + q X_{[3,2]}^{-1}. \] Its Newton polyhedron is isomorphic to $\Delta(2,5)$ from 2.2.} \end{example} \section{Hypergeometric series} \subsection{The Trick with the Factorials}\label{trick} If $X$ is a Calabi-Yau the complete intersection of hypersurfaces of degree $l_1, l_2, \ldots, l_r$ in ${\bf P}^n$, then the generalised hypergeometric series $$ \Phi_X(q)=\sum_{m=0}^{\infty} \frac{(l_1m)!(l_2m)!\ldots(l_rm)!}{(m!)^{n+1}}q^m$$ is main period of its mirror $X^*$. As is well-known, one can obtain the instanton numbers for $X$ by a formal manipulation with this series, see e.g. \cite{BS} and \ref{instanton}. More precisely, one transforms the Picard-Fuchs differential operator $P$ annihilating the series $\Phi_X$ to the form $D^2 \frac{1}{K(q)}D^2$ (where $D=q\partial/\partial q$) and reads off the the $n_d$ from the power series expansion of the function $K$:\\ $$K(q)=l_1 l_2 \ldots l_r +\sum_{d=1}^{\infty} n_d d^3 \frac{q^d}{1-q^d}.$$ It is important to observe that the power series $\Phi_X$ can be obtained from a power series $$A_V(q)=\sum_{m=0}^{\infty} \frac{1}{(m!)^{n+1}}q^m$$ by the multiplication of its $m$-th coefficient by the product of factorials $(l_1m)!(l_2m)!\ldots(l_rm)!$. On the other hand, the power series $A_V$ can be characterized as the unique series $A_V=1+\ldots$ solving the differential equation $((q\frac{\partial}{\partial q})^{n+1}-q)A_V=0$ associated with the small quantum cohomology of ${\bf P}^n$. This differential equation arizes as the reduction of the first order differential {\em system} $$q\frac{\partial}{\partial q} \vec{S} = p \circ \vec{S}$$ for a $H^*({\bf P}^n)$-valued function $\vec{S}=S_0+S_1p+\ldots+S_np^n,$ where $p \in H^2({\bf P}^{n+1})$ is an ample generator, $\{1, p, p^2, \ldots, p^n\}$ is a basis for $H^*({\bf P}^n)$, and $p \circ$ is the operation of {\em quantum multiplication} with $p$ in the small quantum cohomology of ${\bf P}^{n}$. Since it is well-known that the small quantum cohomology ring of ${\bf P}^n$ is defined by the relation $(p \circ)^{n+1} -q=0$, one finds immediately comes to the differential equation. In particular, we see that the function $A_V$ is uniquely determined by the small quantum cohomology ring of $V={\bf P}^n$. It is natural to try to use these ideas to obtain $\Phi_X$ from $A_V$ for varieties other than ${\bf P}^n$, for example for Grassmannians or other Fano varieties. If it works, this method allows one to find instanton numbers without knowing an explicit mirror manifold. We will formulate this trick in some generality below. Let $V$ be a smooth projective variety, which for reasons of simplicity of exposition is assumed to have only even cohomology and and that $H^2(V,{\bf Z} ) \cong H_2(V,{\bf Z}) \cong {\bf Z}$. Let $p$ be the ample generator of $H^2(V, {\bf Z})$, $\gamma$ a positive generator for $H_2(V, {\bf Z})$. We denote by $1_V \in H^0(V)$ the fundamental class of $V$ and by $<-,->$ the Poincar\'e pairing. The small quantum cohomology ring $QH^*(V)$ of $V$ is the free ${\bf Q}[[q]]$-module $H^*(V,{\bf Q}[[q]])$ with a new multiplication $\circ$ determined by $<A \circ B,C>=<A,B,C>=\sum_{m=0}^{\infty}<A,B,C>_mq^m>$ where $$<A,B,C>_m=I^V_{0,3,m\gamma}=\int_{[\overline{M}_{0,3}]}e^*_1(A) \cup e_2^*(B) \cup e^*_3(C)$$ are the {\em $3$-point, genus 0, Gromov-Witten invariants}, see \cite{FP}. The operator of quantum multiplication with the ample generator $p \in H^2(V, {\bf Z})$ defines the {\em Quantum Differential System}, see e.g. \cite{G1}: $$\frac{\partial}{\partial t}\vec{S}=p \circ \vec{S}$$ where $\vec{S}$ is an series in the variable $t=log $ with coefficients from $H^*(V,{\bf Q})$. The {\em Quantum Cohomology ${\cal D}$-module}is the ${\cal D}$-module generated by the top components $<\vec{S}, 1_V>$ of all solutions $\vec{S}$ to the above differential system. In the case under consideration, it will be of the form ${\cal D}/{\cal D} P$, for a certain differential operator $P$. \begin{definition}{\em The {\em $A$-series of $V$} is the unique solution of the Quantum Cohomology ${\cal D}$-module of the form $A_V=\sum_{m=1}^{\infty} a_m q^m$ with $a_0=1$. } \end{definition} Let $X$ be the intersection of hypersurfaces of degree $l_1, l_2, \ldots, l_r$ in $V$. In other words, $X$ is the zero-set of a generic section of the decomposable bundle ${\cal E}:={\cal O}(l_1p)\oplus {\cal O}(l_2p)\oplus\ldots\oplus{\cal O}(l_rp)$. \begin{definition} {\em Let $A_V=\sum_{m=1}^{\infty} a_m q^m$ be the $A$-series of a Fano manifolds $V$. Define the {\em ${\cal E}$-modification of $A_V$} as follows: $$\Phi_{{\cal E}}(q): =\sum_{m=0}^{\infty} a_m \prod_{i=1}^r (m l_i)! q^m.$$ } \end{definition} \begin{definition} {\em Assume that $X \subset V$ has trivial canonical class, i.e, $X$ is a Calabi-Yau variety. We say that the {\em Trick with the Factorials works}, if the function $\Phi_{{\cal E}}$ is equal to the monodromy invariant period $\Phi_X$ of the mirror family $X^*$ in some algebraic parametrization.} \end{definition} If the Trick with the Factorials works, the usual formal manipulation (see \cite{BS}, \ref{instanton}) with the series $\Phi_{{\cal E}}$ produces the instanton numbers for $X$!\\ \begin{remark} {\rm (i) It is possible to formulate the Trick with the Factorials in much greater generality \cite{K3}, \cite{BCKS}. (ii) The $A$-series $A_V$ very well can be identically $1$, but if $V$ is Fano, it will contain interesting information and it is in such cases that the Trick with the Factorials has a chance to work. (iii) A better formulation uses instead of $A_V$ a certain cohomology-valued series $S_V$, whose components make up a complete solution set to the quantum ${\cal D}$-module. Instead of the factorially modified series $\Phi_{{\cal E}}$ one has a factorially modified cohomological function $F_{{\cal E}}$. We say that Trick with the Factorials works, if $S_V$ and $F_{{\cal E}}$ differ by a coordinate change \cite{K3}, \cite{BCKS}. Such a theorem is a form of the Lefschets hyperplane section theorem in quantum cohomology. (iv) Givental's mirror theorem for toric varieties, \cite{G3}, implies that the Trick with the Factorials works for complete intersections in toric varieties. (v) More generally, it follows from a recent theorem of Kim, \cite{K3}, that the Trick with the Factorials works for arbitrary homogeneous spaces. (vi) E. Tj{\o}tta has applied the Trick with the Factorials succesfully in a non-homogeneous case, \cite{tjotta}.} \end{remark} \subsection{Hypergeometric solutions for Grassmannians} In this paragraph we apply the above ideas to the case of Grassmannians. In \cite{BCKS}, we describe a simple rule for writing down the $GKZ$-hypergeometric series $A_{P(k,n)}$ assiciated with the Gorenstein toric Fano variety $P(k,n)$ in terms of the combinatorics of a certain graph. Here we give a formula for $A_{P(k,n)}$ without going into the details: $$A_{P(k,n)}(q,\tilde{q})= \sum_{s_{i,j}\ge 0}\frac{1}{(m!)^{n}} \prod_{i=1}^{k-1}\prod_{j=1}^{n-k-1}{s_{i+1,j} \choose s_{i,j}}{s_{i,j+1}\choose s_{i,j}}q^m \tilde{q}_{i,j}^{s_{i,j}}$$ where we put $s_{i,j}=m$ if $i > k-1$ or $j > n-k-1$. \begin{example} {\rm $G(2,5)$: $$ A_{P(2,5)}(q, \tilde{q})= \sum_{m,r,s \ge 0}\frac{1}{(m!)^5}{m \choose r}{s \choose r} {m \choose s}^2q^m \tilde{q}_1^r\tilde{q}_2^s.$$ } \end{example} \vskip 10pt \begin{example} {\rm $G(3,6)$ : $$ A_{P(3,6)}(q, \tilde{q})= \sum_{m,r,s,t,u}\frac{1}{(m!)^6}{s \choose r}{t \choose r} {m \choose s}{u \choose s}{u \choose t}{m \choose t} {m \choose u}^2 q^m \tilde{q}_1^r \tilde{q}_2^s \tilde{q}_3^u \tilde{q}_4^v.$$ } \end{example} \vskip 10pt We conjecture an explicit general formula for the series $A_{G(k,n)}(q)$ of an arbitrary Grassmannian: \begin{conjecture} Let $A_{P(k,n)}(q, \tilde{q})$ be the A-hypergeometric series of the toric variety $\widehat{P(k,n)}$ as above. Then \[A_{G(k,n)}(q) = A_{P(k,n)}(q, {\bf 1}). \] \label{flagmirror} \end{conjecture} Using the explicit formulas for multiplication in the quantum cohomology of Grassmannians \cite{Ber}, one can write down the Quantum Differential System for $G(k,n)$ and reduce this first order system to a higher order differential equation satisfied by its components. In particular, one can write down the differential operator $P$ annihilating the component $<\vec{S},1>$ of any solution $\vec{S}$. Below we record some of the (computer aided) calculations of the operator $P$ we did ($D$ denotes the operator $\partial/\partial t=q \partial/\partial q$): $$ \begin{array}{|lcl|} \hline & & \\ G(2,4)&:&D^5-2q(2D+1)\\ & & \\ G(2,5)&:&D^7(D-1)^3-qD^3(11D^2+11D+3)-q^2 \\ & & \\ G(2,6)&:&D^9(D-1)^5-qD^5(2D+1)(13D^2+13D+4)\\ & & \\ & &-3q^2(3D+4)(3D+2)\\ & & \\ G(3,6)&:&D^{10}(D-1)^4-qD^4(65D^4+130D^3+105D^2+40D+6)\\ & & \\ & &+4q^2(4D+3)(4D+5)\\ & & \\ \hline \end{array} $$ The operator for $G(2,7)$ is: $$ \begin{array}{c} D^{11}(D-1)^7(D-2)^7(D-3)^7(D-4)^3\\ \\ -\frac{1}{3}qD^7(D-1)^7(D-2)^7(D-3)^3(173D^4+340D^3+272D^2+102D+15)\\ \\ -\frac{2}{9}q^2D^7(D-1)^7(D-2)^3(1129D^4+5032D^3+7597D^2+4773D+1083)\\ \\ +\frac{2}{9}q^3D^7(D-1)^3(843D^4+2628D^3+2353D^2+675D+6)\\ \\ -\frac{1}{9}q^4D^3(295D^4+608D^3+478D^2+174D+26)+\frac{1}{9}q^5,\\ \end{array} $$ while the one for $G(2,8)$ takes about two pages. Clearly, since both the structure of the quantum cohomology ring and the hypergeometric series are very explicit, one should seek a better way to prove Conjecture \ref{flagmirror}. Nevertheless, using the above operators one obtains by direct computation the following: \begin{theorem} The conjecture \ref{flagmirror} is true for $G(2,4)$, $G(2,5)$, $G(2,6)$, $G(2,7)$, $G(3,6)$. \hspace*{\fill}\hbox{$\Box$} \label{grasscheck} \end{theorem} \section{Complete Intersection Calabi-Yau $3$-folds} \subsection{Conifold transitions and mirrors} Now we turn our attention to the main point of the paper, namely the construction, via conifold transitions, of mirrors for Calabi-Yau $3$-folds $X$ which are complete intersections in Grassmannians $G(k,n)$. By \ref{sing-l}, the singular locus of a generic $3$-dimensional complete intersection $X_0$ of $P(k,n)$ with $r$ hypersurfaces $H_1, \ldots, H_r$ of degrees $d_1, \ldots, d_r$ ($[H_i] = d_i[H]$, $i =1, \ldots, r$) consists of \[ p = d_1 d_2 \cdots d_r \left( \sum_{i =1,\, j =1}^{k-1,\,n-k-1} d(W_{i,j}) \right) \] nodes, where $d(W_{i,j})$ is the degree of $W_{i,j}$ with respect to the generator $H$ of the Picard group of $P(k,n)$. On the other hand, by \ref{def-gr}, $X_0$ is a flat degeneration of the smooth Calabi-Yau $3$-fold $X \subset G(k,n)$. The small crepant resolution $\widehat{P}(k,n) \longrightarrow P(k,n)$ of the ambient toric variety induces a small crepant resolution $Y\longrightarrow X_0$. Hence $Y$ is a {\em smooth} Calabi-Yau complete intersection in the toric variety $\widehat{P}(k,n)$, which is obtained from $X$ by a {\em conifold transition}. \begin{theorem} Let $p$ be the number of nodes of $X_0$, and let $\alpha = (k-1)(n-k-1)$. Then the Hodge numbers of $X$ and $Y$ are related by $$h^{1,1}(Y)=h^{1,1}(X)+\alpha$$ and $$h^{2,1}(Y)=h^{2,1}(X)+\alpha -p.$$ \label{formul-2} \end{theorem} \noindent {\it Proof:} By construction, $Y$ is a complete intersection of general sections of big {\it semiample} line bundles on $\widehat{P}(k,n)$ (i.e., line bundles which are generated by global sections and big). Using the explicit formula for $h^{1,1}(Y)$ from (\cite{BB}, Corollary 8.3) and the fact that the only boundary lattice points of $\Delta(k,n)$ are its vertices, we obtain the isomorphism ${\rm Pic}(Y)\cong{\rm Pic}(\widehat{P})$, which gives the first relation. On the other hand, the $p$ vanishing 3-cycles on X that shrink to nodes in the degeneration must satisfy $\alpha$ linearly independent relations by \cite{C}, and the second relation follows. \hspace*{\fill}\hbox{$\Box$} \vskip 10pt The mirror construction for complete intersection Calabi-Yau manifolds in toric varieties given in \cite{Ba2, BB} provides us with the mirror family of Calabi-Yau manifolds $Y^*$. The generic member of this family is nonsingular (it is obtained by a MPCP-resolution of the ambient toric variety). There is a natural isomorphism of the Hodge groups $H^{1,1}(Y)\longrightarrow H^{2,1}(Y^*)$ (see \cite{Ba2, BB}). During the conifold transition from $X$ to $Y$, we have increased the "K\"ahler moduli", that is, the rank of $H^{1,1}$. This says that we should really look at the one-parameter subfamily of mirrors given by the subspace of $H^{2,1}(Y^*)$ corresponding via the isomorphism above to the divisors on $Y$ which come from $X$. For this reason, the generalized hypergeometric series $\Phi_X$ of $X^*$ is a specialization of the monodromy invariant period integral of the mirror family $Y^*$ to the subfamily $Y_0^*$ defined in \ref{mirror-c}. Let $\nabla_{J_1}, \ldots, \nabla_{J_r}$ be convex polyhedra as in \ref{nablas}. Denote by $\nabla(k,n)$ the Minkowski sum of $\nabla_{J_1}, \ldots, \nabla_{J_r}$. Then $\nabla(k,n)$ is a reflexive polyhedron and ${\bf P}_{\nabla(k,n)}$ is a Gorenstein toric Fano variety defined by a nef-partition corresponding to the equation \[ d_1 + \cdots + d_r = n. \] \begin{conjecture} After a MPCP-desingularization of the ambient toric variety ${\bf P}_{\nabla(k,n)}$, the general member $Y_0^*$ of the special $1$-parameter subfamily is a Calabi-Yau variety with the same number $p$ of nodes as $X_0$, satisfying $\alpha -p$ relations. A small resolution $X^*$ of $Y^*_0$ is a mirror of $X$. \label{toricmirror} \end{conjecture} \begin{remark} {\rm The statement \ref{toricmirror} has been easily checked for the two simplest cases of Section 2, where the toric mirror construction reduces to a hypersurface case. However, singularities of $Y_0^*$ are more difficult control for $4$ remaining cases which can not be reduced to Calabi-Yau hypersurfaces in $4$-dimensional Gorenstein toric Fano varieties.} \end{remark} \subsection{The computation of instanton numbers}\label{instanton} We denote by $X_{d_1,\ldots, d_r}\subset G(k,n)$ a Calabi-Yau complete intersection of $r$ hypersurfaces of degrees $d_1, \ldots, d_r$ with the Grassmannian $G(k,n) \subset {\bf P}^{{n \choose k} -1}$. We denote by $Y$ the toric Calabi-Yau complete intersection in $\widehat{P}(k,n)$ obtained by a conifold transition via resolution of $p$ nodes on the degeneration $X_0$ of $X$ $(h^{1,1}(X) = 1$), and by $\alpha$ the number of relations satisfied by the homology classes of the corresponding $p$ vanishing $3$-cycles on $X$. Now we list all cases of Calabi-Yau complete intersection $3$-folds $X$ in Grassmannians and collect the information about topological invariants of $X$ and their conifold modifications $Y$. \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline $X$ & $h^{2,1}(X)$ & $\chi(X)$ & $h^{1,1}(Y)$ & $h^{2,1}(Y)$ & $\chi(Y)$ & $\alpha$ & $p$ \\ \hline $X_4 \subset G(2,4)$ & $89$ & $- 176$ & $2$ & $86$ & $-168$ & $1$ & $4$ \\ \hline $X_{1,1,3} \subset G(2,5)$ & $76$ & $-150$ & $3$ & $72$ & $-138$ & $2$ & $6$ \\ \hline $X_{1,2,2} \subset G(2,5)$ & $61$ & $-120$ & $3$ & $55$ & $ -104$ & $2$ & $8$ \\ \hline $X_{1,1,1,1,2} \subset G(2,6)$ & $59$ & $-116$ & $4$ & $52$ & $-96$ & $3$ & $10$ \\ \hline $X_{1, \ldots, 1} \subset G(2,7)$ & $50$ & $ -98$ & $5$ & $40$ & $-70$ & $4$ & $14$ \\ \hline $X_{1, \ldots, 1} \subset G(3,6)$ & $49$ & $-96$ & $5$ & $37$ & $-64$ & $4$ & $16$ \\ \hline \end{tabular} \end{center} Recall the (standard) formal procedure used to compute the instanton numbers. (More details can be found e.g in \cite{BS}.) We set $$\Phi_X(z) := \sum_{m \geq 0} b_mz^m $$ to be the generalized hypergeometric series (with variable $z$) corresponding to the monodromy invariant period of the mirror $X^*$. As was explained in \ref{trick}, one can start with the the $A$-series for the grassmannian, and apply the Trick with the Factorials to find the coefficients $b_m$. Then $\Phi_X(z)$ satisfies a Picard-Fuchs differential equation \[ P \Phi_X(z) = 0, \] where $P$ is a differential operator of order $4$ having a maximal unipotent monodromy at $z= 0$. We compute $P$ by finding an explicit recursion relation among coefficients $b_m$ of the generalized hypergeometric series $\Phi_X(z)$. To bring $P$ into the form $D^2 \frac{1}{K}D^2$, one has to change the coordinate $z$ to $q = {\rm exp}(\Phi_1(z)/\Phi_X(z))$, where $\Phi_1$ is the logarithmic solution of $P$. To obtain $K$, it is convenient to use the Yukawa coupling. In the coordinate $z$ it has form \[ K_{zzz} = \frac{K_z^{(3)}}{\Phi_X^2(z)} \left( \frac{dz}{z} \right)^{\otimes 3}, \] where $K_z^{(3)}$ is some rational function of $z$ that can be determined directly from $P$. The Yukawa coupling in coordinate $q$ then is of the form \[ K_{qqq} = K_q^{(3)} \left( \frac{dq}{q} \right)^{\otimes 3}, \] where \[ K_q^{(3)} = n_0 + \sum_{m =1}^{\infty} n_m \frac{m^3 q^m}{1 -q^m} \] and $n_m$ are the instanton numbers for rational curves of degree $m$ on $X$. From proposition \ref{grasscheck} and Kim's Quantum Hyperplane Theorem (\cite{K3}), we have the following \begin{theorem} The virtual numbers of rational curves on a general complete intersection Calabi-Yau three-fold in a Grassmannian are the ones listed in the tables of the next section. \end{theorem} \section{Picard-Fuchs Operators and Yukawa Couplings} {\tiny \noindent \subsection{$X_{1,1,3} \subset G(2,5)$} \begin{center} \begin{tabular}{|c|c|} \hline & \\ $ b_m $ & ${\displaystyle \frac{(m!)(m!)(3m)!}{(m!)^5}\sum_{r,s}{m\choose r}{s \choose r} {m\choose s}^2} $ \\ & \\ \hline & \\ ${P}$ & $ D^4 - 3z (3 D +2)( 3 D + 1) (11 D^2 + 11 D + 3) $\\ & $ - 9z^2 (3 D + 5)(3 D + 2) (3 D + 4)(3 D + 1)$ \\ & \\ \hline & \\ $ K_z^{(3)}$ & ${\displaystyle \frac{15}{1-11\cdot3^3z - 3^9z^2} }$ \\ & \\ \hline & \\ $ n_m$ & $n_1 =540 ,\; n_2 =12555 , \; n_3 =621315 ,\; n_4 =44892765 ,\; n_5 = 3995437590 $ \\ & \\ \hline \end{tabular} \end{center} } \noindent {\tiny \subsection{$X_{1,2,2} \subset G(2,5)$} \begin{center} \begin{tabular}{|c|c|} \hline & \\ $ b_m $ & ${\displaystyle \frac{ (m!)(2 m)!)^2}{(m!)^5} \sum_{r,s } {m\choose r}{s \choose r}{m\choose s}^2}$ \\ & \\ \hline & \\ ${P}$ & $ D^4 - 4z (11 D^2 + 11 D + 3) (1 + 2 D)^2$\\ & $ - 16z^2 (2 D + 3)^2 (1 + 2 D)^2$ \\ & \\ \hline & \\ $ K_z^{(3)}$ & ${\displaystyle \frac{20}{1 - 11 \cdot 2^4z - 2^8z^2} }$ \\ & \\ \hline & \\ $ n_m$ & $n_1 =400 ,\; n_2 =5540 , \; n_3 = 164400,\; n_4 =7059880 ,\; n_5 = 373030720 $ \\ & \\ \hline \end{tabular} \end{center} } {\small The locus of conifold singularities in the toric variety $P(2,5)$ consists of $2$ codimesion-$3$ toric strata of degree $1$. This gives $6$ nodes on the generic complete intersection of type $(1,1,3)$ in $P(2,5) \subset {\bf P}^{10}$ and $8$ nodes on the generic complete intersection of type $(1,2,2)$ in $P(2,5) \subset {\bf P}^{10}$ } \noindent \subsection{$X_{1,1,1,1,2} \subset G(2,6)$} {\tiny \begin{center} \begin{tabular}{|c|c|} \hline & \\ $ b_m $ & ${\displaystyle \frac{ (m!)^4(2 m)!)}{(m!)^6} \sum_{r,s,t} {m\choose r}{s\choose r}{m\choose s}{t\choose s}{m\choose t}^2}$\\ & \\ \hline & \\ ${P}$ & $ D^4 - 2z (4 + 13 D + 13 D^2)(1 + 2 D)^2 $\\ & $ -12 z^2 (3 D +2)( 2 D + 3)(1 + 2 D)( 3 D + 4)$ \\ & \\ \hline & \\ $ K_z^{(3)}$ & ${\displaystyle \frac{28}{1-26\cdot2^2z - 27\cdot 2^4z^2} }$ \\ & \\ \hline & \\ $ n_m$ & $n_1 =280 ,\; n_2 =2674 , \; n_3 =48272 ,\; n_4 = 1279040,\; n_5 = 41389992 $ \\ & \\ \hline \end{tabular} \end{center} } {\small The locus of conifold singularities in the toric variety $P(2,6)$ consists of $2$ codimesion-$3$ toric strata of degree $2$ and $1$ codimension-$3$ toric stratum of degree $1$. This gives $10$ nodes on the generic complete intersection of type $(1,1,1,1,2)$ in $P(2,6) \subset {\bf P}^{14}$. } \noindent \subsection{$X_{1,1,1,1,1,1,1} \subset G(2,7)$} {\tiny \begin{center} \begin{tabular}{|c|c|} \hline & \\ $ b_m $ & ${\displaystyle \frac{ (m!)^7}{(m!)^7} \sum_{r,s,t,u } {m\choose r}{s\choose r}{m\choose s}{t\choose s} {m\choose t}{u\choose t}{m\choose u}^2}$\\ & \\ \hline & \\ ${P}$ & $ 9 D^4 - 3z (15 + 102 D + 272 D^2 + 340 D^3 + 173 D^4) $\\ & $ - 2z^2 (1083 + 4773 D + 7597 D^2 + 5032 D^3 + 1129 D^4)$ \\ & $ + 2z^3 (6 + 675 D + 2353 D^2 + 2628 D^3 + 843 D^4)$ \\ & $ - z^4(26 + 174 D + 478 D^2 + 608 D^3 + 295 D^4) + z^5 ( D + 1)^4 $ \\ & \\ \hline & \\ $ K_z^{(3)}$ & ${\displaystyle \frac{42-14z}{1-57z -289z^2 + z^3} }$ \\ & \\ \hline & \\ $ n_m$ & $n_1 = 196,\; n_2 =1225 , \; n_3 =12740 ,\; n_4 =198058 ,\; n_5 = 3716944 $ \\ & \\ \hline \end{tabular} \end{center} } {\small The locus of conifold singularities in the toric variety $P(2,7)$ consists of $2$ codimesion-$3$ toric strata of degree $2$ and $2$ codimension-$3$ toric stratum of degree $5$. This gives $14$ nodes on the generic complete intersection of type $(1,1,1,1,1,1)$ in $P(2,7) \subset {\bf P}^{20}$. } \noindent \subsection{$X_{1,1,1,1,1,1} \subset G(3,6)$} {\tiny \begin{center} \begin{tabular}{|c|c|} \hline & \\ $ b_m $ & ${\displaystyle \frac{ (m!)^6}{(m!)^6} \sum_{r,s,t,u } {s \choose r}{t \choose r} {m \choose s}{u \choose s}{u \choose t}{m \choose t} {m \choose u}^2}$\\ & \\ \hline & \\ ${P}$ & $ D^4 - z (6 + 40 D + 105 D^2 + 130 D^3 + 65 D^4) $\\ & $ +4z^2 (4 D + 5)(4 D +3)( D + 1)^2$ \\ & \\ \hline & \\ $ K_z^{(3)}$ & ${\displaystyle \frac{42}{1 - 65 z - 64z^2} }$ \\ & \\ \hline & \\ $ n_m$ & $n_1 =210 ,\; n_2 =1176 , \; n_3 =13104 ,\; n_4 = 201936 ,\; n_5 =3824016 $ \\ & \\ \hline \end{tabular} \end{center} } {\small The locus of conifold singularities in the toric variety $P(3,6)$ consists of $2$ codimesion-$3$ toric strata of degree $2$ and $2$ codimension-$3$ toric strata of degree $6$. This gives $16$ nodes on the generic complete intersection of type $(1,1,1,1,1,1)$ in $P(3,6) \subset {\bf P}^{19}$. } \vskip .2truein \noindent \section{Acknowledgement} \noindent We would like to thank S. Katz, S.-A. Str{\o}mme, E. R{\o}dland and E. Tj{\o}tta for helpful discussions and the Mittag-Leffler Institute for hospitality. The second and third named authors have been supported by Mittag-Leffler Institute postdoctoral fellowships. \newpage \vskip .2truein

9710

alg-geom/9710007

en

https://arxiv.org/abs/alg-geom/9710007

alg-geom/9710007

Shu Kawaguchi and Atsushi Moriwaki

Inequalities for semistable families of arithmetic varieties

Version 3.0 (75 pages), the new version of the paper titled "Relative Bogomolov's inequality in the arithmetic case"

In this paper, we will consider a generalization of Bogomolov's inequality and Cornalba-Harris-Bost's inequality to semistable families of arithmetic varieties under the idea that geometric semistability implies a certain kind of arithmetic positivity. The first one is an arithmetic analogue of the relative Bogomolov's inequality proved by the second author. We also establish the arithmetic Riemann-Roch formulae for stable curves over regular arithmetic varieties and generically finite morphisms of arithmetic varieties.

alg-geom

\section*{Introduction} \renewcommand{\theTheorem}{\Alph{Theorem}} In this paper, we will consider a generalization of Bogomolov's inequality and Cornalba-Harris-Bost's inequality to the case of semistable families of arithmetic varieties. An underlying idea of these inequalities as in \cite{Bo}, \cite{BGS}, \cite{Ga}, \cite{MiBi}, \cite{MoBG}, \cite{MoABG}, \cite{MoBU}, \cite{MorFh}, \cite{SoVan}, and \cite{Zh} is that geometric semistability implies a certain kind of arithmetic positivity. The first one is related to the semistability of vector bundles, and the second one involves the Chow (or Hilbert) semistability of cycles. \medskip First of all, let us consider Bogomolov's inequality. Let $X$ and $Y$ be smooth algebraic varieties over an algebraically closed field of characteristic zero, and $f : X \to Y$ a semi-stable curve. Let $E$ be a vector bundle of rank $r$ on $X$, and $y$ a point of $Y$. In \cite{MoRB}, the second author proved that if $f$ is smooth over $y$ and $\rest{E}{X_{\bar{y}}}$ is semistable, then $\operatorname{dis}_{X/Y}(E) = f_* \left( 2r c_2(E) - (r-1)c_1^2(E) \right)$ is weakly positive at $y$. In the first half of this paper, we would like to consider an arithmetic analogue of the above result. Let us fix regular arithmetic varieties $X$ and $Y$, and a semistable curve $f : X \to Y$. Since we have a good dictionary for translation from a geometric case to an arithmetic case, it looks like routine works. There are, however, two technical difficulties to work over the standard dictionary. The first one is how to define a push-forward of arithmetic cycles in our situation. If $f_{{\mathbb{Q}}} : X_{{\mathbb{Q}}} \to Y_{{\mathbb{Q}}}$ is smooth, then, according to Gillet-Soul\'{e}'s arithmetic intersection theory \cite{GSArInt}, we can get the push-forward $f_* : \widehat{\operatorname{CH}}^{p+1}(X) \to \widehat{\operatorname{CH}}^p(Y)$. We would not like to restrict ourselves to the case where $f_{{\mathbb{Q}}}$ is smooth because in the geometric case, the weak positivity of $\operatorname{dis}_{X/Y}(E)$ gives wonderful applications to analyses of the boundary of the moduli space of stable curves. Thus the usual push-forward for arithmetic cycles is insufficient for our purpose. A difficulty in defining the push-forward arises from a fact: if $f_{{\mathbb{C}}} : X({\mathbb{C}}) \to Y({\mathbb{C}})$ is not smooth, then $(f_{{\mathbb{C}}})_*(\eta)$ is not necessarily $C^{\infty}$ even for a $C^{\infty}$ form $\eta$. This suggests us that we need to extend the usual arithmetic Chow groups defined by Gillet-Soul\'{e} \cite{GSArInt}. For this purpose, we will introduce an arithmetic $L^1$-cycle of codimension $p$, namely, a pair $(Z, g)$ such that $Z$ is a cycle of codimension $p$, $g$ is a current of type $(p-1,p-1)$, and $g$ and $dd^c(g) + \delta_{Z({\mathbb{C}})}$ are represented by locally integrable forms. Thus, dividing by the usual arithmetical rational equivalence, an arithmetic Chow group, denoted by $\widehat{\operatorname{CH}}_{L^1}^p$, consisting of arithmetic $L^1$-cycles of codimension $p$ will be defined (cf. \S\ref{subsec:var:arith:chow}). In this way, we have the natural push-forward \[ f_* : \widehat{\operatorname{CH}}_{L^1}^{p+1}(X) \to \widehat{\operatorname{CH}}_{L^1}^p(Y) \] as desired (cf. Proposition~\ref{prop:push:forward:arith:cycle}). The second difficulty is the existence of a suitable Riemann-Roch formula in our situation. As before, if $f_{{\mathbb{Q}}} : X_{{\mathbb{Q}}} \to Y_{{\mathbb{Q}}}$ is smooth, we have the arithmetic Riemann-Roch theorem due to Gillet-Soul\'{e} \cite{GSRR}. If we ignore Noether's formula, then, under the assumption that $f_{{\mathbb{Q}}} : X_{{\mathbb{Q}}} \to Y_{{\mathbb{Q}}}$ is smooth, their Riemann-Roch formula can be written in the following form: \begin{multline*} \widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) - \operatorname{rk} (E) \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X} \right) \\ = f_* \left( \frac{1}{2} \left( \widehat{{c}}_1 (\overline{E})^2 - \widehat{{c}}_1 (\overline{E}) \cdot \widehat{{c}}_1 (\overline{\omega}_{X/Y}) \right) - \widehat{{c}}_2 (\overline{E}) \right) \end{multline*} where $\overline{E} = (E, h)$ is a Hermitian vector bundle on $X$ and $\overline{\omega}_{X/Y}$ is the dualizing sheaf of $f : X \to Y$ with a Hermitian metric. If we consider a general case where $f_{{\mathbb{Q}}} : X_{{\mathbb{Q}}} \to Y_{{\mathbb{Q}}}$ is not necessarily smooth, the right hand side in the above equation is well defined and sits in $\widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}}$. On the other hand, the left hand side is rather complicated. If we admit singular fibers of $f_{{\mathbb{C}}} : X({\mathbb{C}}) \to Y({\mathbb{C}})$, then the Quillen metric $h_Q^{\overline{E}}$ is no longer $C^{\infty}$. According to \cite{BBQm}, it extends to a generalized metric. Thus, we may define $\widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right)$ (cf. \S\ref{subsec:arith:div:gen:metric}). In general, this cycle is not an $L^1$-cycle. However, using Bismut-Bost's formula \cite{BBQm}, we can see that \[ \widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) - \operatorname{rk} (E) \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X} \right) \] is an element of $\widehat{\operatorname{CH}}_{L^1}^1(Y)$. Thus, we have a way to establish a Riemann-Roch formula in the arithmetic Chow group $\widehat{\operatorname{CH}}_{L^1}^1(Y)_{{\mathbb{Q}}}$. Actually, we will prove the above formula in our situation (cf. Theorem~\ref{thm:arith:Riemann:Roch:stable:curves}). The idea of comparing two sides in $\widehat{\operatorname{CH}}_{L^1}^1(Y)_{{\mathbb{Q}}}$ is the tricky Lemma~\ref{lem:criterion:linear:equiv:B:cycle}. Let us go back to our problem. First of all, we need to define an arithmetic analogue of weak positivity. Let $\alpha$ be an element of $\widehat{\operatorname{CH}}_{L^1}^1(Y)_{{\mathbb{Q}}}$, $S$ a subset of $Y({\mathbb{C}})$, and $y$ a closed point of $Y_{{\mathbb{Q}}}$. We say $\alpha$ is semi-ample at $y$ with respect to $S$ if there are an arithmetic $L^1$-cycle $(E, f)$ and a positive integer $n$ such that (1) $dd^c(f) + \delta_{E({\mathbb{C}})}$ is $C^{\infty}$ around each $z \in S$, (2) $E$ is effective, (3) $y \not\in \operatorname{Supp}(E)$, (4) $f(z) \geq 0$ for all $z \in S$, and (5) $n \alpha$ coincides with the class of $(E, f)$ in $\widehat{\operatorname{CH}}_{L^1}^1(Y)_{{\mathbb{Q}}}$. Moreover, $\alpha$ is said to be weakly positive at $y$ with respect to $S$ if it is the limit of semi-ample cycles at $y$ with respect to $S$ (for details, see \S\ref{subsec:wp:div}). For example, if $Y = \operatorname{Spec}(O_K)$, $y$ is the generic point, and $S = Y({\mathbb{C}})$, then, $\alpha$ is weakly positive at $y$ with respect to $S$ if and only if $\widehat{\operatorname{deg}}(\alpha) \geq 0$, where $K$ is a number field and $O_K$ is the ring of integers in $K$ (cf. Proposition~\ref{prop:wp:for:curve}). Let $(E, h)$ be a Hermitian vector bundle of rank $r$ on $X$, and $\widehat{\operatorname{dis}}_{X/Y}(E, h)$ the arithmetic discriminant divisor of $(E, h)$ with respect to $f : X \to Y$, that is, the element of $\widehat{\operatorname{CH}}_{L^1}^1(Y)$ given by $f_* \left( 2r \widehat{{c}}_2(E, h) - (r-1)\widehat{{c}}_1(E, h)^2 \right)$. We assume that $f$ is smooth over $y$ and $\rest{E}{X_{\bar{y}}}$ is poly-stable. In the case where $\dim X = 2$ and $Y = \operatorname{Spec}(O_K)$, Miyaoka \cite{MiBi}, Moriwaki \cite{MoBG,MoABG,MoBU}, and Soul\'{e} \cite{SoVan} proved that $\widehat{\operatorname{deg}} \left( \widehat{\operatorname{dis}}_{X/Y}(E, h) \right) \geq 0$, consequently, $\widehat{\operatorname{dis}}_{X/Y}(E, h)$ is weakly positive at $y$ with respect to $Y({\mathbb{C}})$. One of the main theorems of this paper is the following generalization. \begin{Theorem}[cf. Theorem~\ref{thm:relative:Bogomolov:inequality:arithmetic:case}] \label{thm:A:relative:Bogomolov:inequality} Under the above assumptions, $\widehat{\operatorname{dis}}_{X/Y}(E, h)$ is weakly positive at $y$ with respect to any subsets $S$ of $Y({\mathbb{C}})$ with the following properties: \textup{(1)} $S$ is finite, and \textup{(2)} $f_{{\mathbb{C}}}^{-1}(z)$ is smooth and $\rest{E_{{\mathbb{C}}}}{f_{{\mathbb{C}}}^{-1}(z)}$ is poly-stable for all $z \in S$. In particular, if the residue field of $x$ is $K$, and the canonical morphism $\operatorname{Spec}(K) \to X$ induced by $x$ extends to $\tilde{x} : \operatorname{Spec}(O_K) \to X$, then $\widehat{\operatorname{deg}}\left( \tilde{x}^*\left(\widehat{\operatorname{dis}}_{X/Y}(E, h)\right)\right) \geq 0$. \end{Theorem} \medskip Next, let us consider Cornalba-Harris-Bost's inequality. Motivated by the work of Cornalba and Harris \cite{CoHa} in the geometric case, Bost \cite[Theorem~I]{Bo} proved that, roughly speaking, if $X(\overline{{\mathbb{Q}}}) \subset {\mathbb{P}}^{r-1}(\overline{{\mathbb{Q}}})$ has the $\operatorname{\mathbf{SL}}_r(\overline{{\mathbb{Q}}})$ semi-stable Chow point, then the height of $X$ has a certain kind of positivity. We call this result Cornalba-Harris-Bost's inequality. Zhang \cite{Zh} then gave precision to it and also showed the converse of Bost's result. Further, Gasbarri \cite{Ga} considered a wide range of actions instead of $\operatorname{\mathbf{SL}}_r(\overline{{\mathbb{Q}}})$-action. In the second half of this paper, we would like to consider a relative version of Cornalba-Harris-Bost's inequality. First, let us fix a terminology. Let $V$ be a set, $\phi$ a non-negative function on $V$, and $S$ a finite subset of $V$. We define the geometric mean $\operatorname{g.\!m.}(\phi; S)$ of $\phi$ over $S$ to be \[ \operatorname{g.\!m.}(\phi; S) = \left( \prod_{s \in S} \phi(s) \right)^{1/\#(S)}. \] Then, the following is our solution. \begin{Theorem}[cf. Theorem~\ref{thm:semistability:imply:average:semi-ampleness}] \label{thm:intro:B} Let $Y$ be a regular projective arithmetic variety, and $\overline{E} = (E,h)$ a Hermitian vector bundle of rank $r$. Let $\pi : {\mathbb{P}}(E) = \operatorname{Proj} (\bigoplus_{n \geq 0} \operatorname{Sym}^n(E^{\lor})) \to Y$ be the projection and $\overline{{\mathcal{O}}_{E}(1)}$ the tautological line bundle with the quotient metric induced from $f^*(h)$. Let $X$ be an effective cycle in ${\mathbb{P}} (E)$ such that $X$ is flat over $Y$ with the relative dimension $d$ and degree $\delta$ on the generic fiber. For each irreducible component $X_i$ of $X_{red}$, let $\tilde{X}_i \to X_i$ be a proper birational morphism such that $(\tilde{X}_i)_{{\mathbb{Q}}}$ is smooth over ${\mathbb{Q}}$. Let $Y_0$ be the maximal open set of $Y$ such that the induced morphism $\tilde{X}_i \to Y$ is smooth over $Y_0$ for every $i$. Let $(B, h_B)$ be a line bundle equipped with a generalized metric on $Y$ given by the equality: \[ \widehat{{c}}_1(B, h_B) = r \pi_* \left( \widehat{{c}}_1(\overline{{\mathcal{O}}_{E}(1)})^{d+1} \cdot (X,g_X) \right) + \delta (d+1) \widehat{{c}}_1(\overline{E}). \] \textup{(}Here we postpone the definition of $g_X$, i.e., a suitable compactification of $X$ in the arithmetic sense.\textup{)} Then, $h_B$ is $C^{\infty}$ over $Y_0$. Moreover, there are a positive integer $e=e(r,d,\delta)$, a positive integer $l=l(r,d,\delta)$, a positive constant $C=C(r,d,\delta)$, and sections $s_1, \ldots, s_l \in H^0(Y, B^{\otimes e})$ with the following properties. \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item $e$, $l$, and $C$ depend only on $r$, $d$, and $\delta$. \item For a closed point $y$ of $Y_{{\mathbb{Q}}}$, if $X_y$ is Chow semistable, then $s_i(y) \not= 0$ for some $i$. \item For all $i$ and all closed points $y$ of $(Y_0)_{{\mathbb{Q}}}$, \[ \operatorname{g.\!m.}\left( \left( h_B^{\otimes e} \right)(s_i, s_i);\ O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)\right) \leq C, \] where $O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)$ is the orbit of $y$ by the Galois action in $Y_0(\overline{{\mathbb{Q}}})$. \end{enumerate} \end{Theorem} Compared with the geometric analogue (cf. Remark~\ref{rem:geom:analog:Cornalba-Harris-Bost}), the difficult part of this theorem is the estimate of the geometric mean of the norm over the Galois orbits of closed points. We will do this by reducing to the absolute case. For this purpose, we have to associate $X$ with a `nice' Green current $g_X$. How do we do? One way is to fix a K\"{a}hler metric $\mu \in A^{1,1}({\mathbb{P}}(E)_{{\mathbb{R}}})$ and to attach a $\mu$-normalized Green current for $X$, namely, a Green current $g$ such that $dd^c g + \delta_X = H(\delta_Y)$ and $H(g_Y) = 0$, where $H : D^{p,p}({\mathbb{P}}(E)_{{\mathbb{R}}}) \to H^{p,p}({\mathbb{P}}(E)_{{\mathbb{R}}})$ is the harmonic projection (cf. \cite[2.3.2]{BGS}). This construction however is not suitable for our purpose because it does not behave well when restricted on fibers. Thus we are led to define an $\Omega$-normalized Green form which is given, roughly speaking, by attaching a Green form fiberwisely (Here $\Omega = {c}_1(\overline{{\mathcal{O}}_{E}(1)})$). Precisely, an $\Omega$-normalized Green form $g_X$ for $X$ is characterized by the following three conditions; (i) $g_X$ is an $L^1$-form on ${\mathbb{P}}(E)$, (ii) $dd^c([g_X]) + \delta_X = \sum_{i=0}^{d} \left[ \pi^*(\gamma_i) \wedge \Omega^i \right]$. where $\gamma_i$ is a $d$-closed $L^1$-form of type $(d-i,d-i)$ on $Y$ ($i=0, \ldots, d$). (iii) $\pi_*(g_X \wedge \Omega^{r - d}) = 0$ (cf. Proposition~\ref{prop:normalized:Green:form}). Then we can show that it has a desired property when restricted on fibers (cf. Remark~\ref{rem:norm:Green:general:fiber}). Suppose now $X$ is regular. Let $i : X \to {\mathbb{P}}(E)$ be the inclusion map and $f : X \to Y$ the restriction of $\pi$. If we set $\overline{L} = i^*(\overline{{\mathcal{O}}_{E}(1)})$, then $\pi_* ( \widehat{{c}}_1(\overline{{\mathcal{O}}_{E}(1)})^{d+1} \cdot (X,g_X) ) = f_*(\widehat{{c}}_1(\overline{L})^{d+1})$ (cf. Proposition~\ref{prop:when:Bost:divisor:smooth}). Since $f_*(\widehat{{c}}_1(\overline{L})^{d+1})$ is in general only an element of $\widehat{\operatorname{CH}}^1_{L^1}(Y)$, the above equality explains why we need to consider $(X,g_X)$ in the enlarged arithmetic Chow group $\widehat{\operatorname{CH}}^{r-d-1}_{L^1}({\mathbb{P}}(E))$. Moreover, a similar equality when $X$ is not necessarily regular shows that $\pi_* ( \widehat{{c}}_1(\overline{{\mathcal{O}}_{E}(1)}) \cdot (X,g_X) )$ is independent of the choice of an $\Omega$-normalized Green form $g_X$ for $X$ (cf. Proposition~\ref{prop:when:Bost:divisor:smooth}). Suppose now $Y=\operatorname{Spec}(O_K)$, $y$ is the generic point, and $X_y$ is Chow semistable, where $K$ is a number field. In this case, there exists a generic resolution of $X$ smooth over $y$. Then Theorem~\ref{thm:intro:B} tells us that \[ r \widehat{\operatorname{deg}} ( \widehat{{c}}_1(\overline{L})^{d+1} ) + \delta (d+1) \widehat{\operatorname{deg}} (\overline{E}) + [K:{\mathbb{Q}}] \alpha(r,d,\delta) \geq 0 \] for some constant $\alpha(r,d,\delta)$ depending only on $r$, $d$ and $\delta$, which is nothing but Theorem~I of Bost \cite{Bo}. We can also think a wide range of actions like \cite{Ga}. Namely, let $\rho : \operatorname{GL}_r \to \operatorname{GL}_R$ be a morphism of group schemes such that there is an integer $k$ with $\rho(t I_r) = t^k I_R$ for any $t$, and that $\rho$ commutes with the transposed morphism. For a Hermitian vector bundle $\overline{E}$, we then get the associated Hermitian vector bundle $\overline{E}^{\rho}$ (cf. \S\ref{subsec:associated:herm:vb}). If $X$ is a flat cycle on ${\mathbb{P}}(E^{\rho})$ and $y$ is a closed point of $Y_{{\mathbb{Q}}}$, then $\operatorname{\mathbf{SL}}_r(\overline{{\mathbb{Q}}})$ acts a Chow form ${\Phi_X}_y$. The stability of ${\Phi_X}_y$ under this action yields a similar inequality (cf. Theorem~\ref{thm:semistability:imply:average:semi-ampleness}). Finally, in \S\ref{section:Bogomolov:to:Bost} we make a comparison between the relative Bogomolov's inequality (Theorem~\ref{thm:relative:Bogomolov:inequality:arithmetic:case}) and the relative Cornalba-Harris-Bost's inequality (Theorem~\ref{thm:semistability:imply:average:semi-ampleness}). \renewcommand{\theTheorem}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}} \section{Locally integrable forms and their push-forward} \subsection{Locally integrable forms} \setcounter{Theorem}{0} Let $M$ be an $n$-dimensional orientable differential manifold. We assume that $M$ has a countable basis of open sets. Let $\omega$ be a $C^{\infty}$ volume element of $M$, and $C_c^0(M)$ the set of all complex valued continuous functions on $M$ with compact supports. Then, there is a unique Radon measure $\mu_{\omega}$ defined on the topological $\sigma$-algebra of $M$ such that \[ L\!\!\!\int_M f d\mu_{\omega} = \int_M f \omega \] for all $f \in C_c^0(M)$, where ${\displaystyle L\!\!\!\int_M f d\mu_{\omega}}$ is the Lebesgue integral arising from the measure $\mu_{\omega}$. Let $f$ be a complex valued function on $M$. We say $f$ is {\em locally integrable}, denoted by $f \in L^1_{\operatorname{loc}}(M)$, if $f$ is measurable and, for any compact sets $K$, \[ L\!\!\!\int_K |f| d\mu_{\omega} < \infty. \] Let $\omega'$ be another $C^{\infty}$ volume form on $M$. Then, there is a positive $C^{\infty}$ function $a$ on $M$ with $\omega' = a \omega$. Thus, \[ L\!\!\!\int_K |f| d\mu_{\omega'} = L\!\!\!\int_K |f| a d\mu_{\omega}, \] which shows us that local integrability does not depend on the choice of the volume form $\omega$. Moreover, it is easy to see that, for a measurable complex valued function $f$ on $M$, the following are equivalent. \begin{enumerate} \renewcommand{\labelenumi}{(\alph{enumi})} \item $f$ is locally integrable. \item For each point $x \in M$, there is an open neighborhood $U$ of $x$ such that the closure of $U$ is compact and ${\displaystyle L\!\!\!\int_U |f| d\mu_{\omega} < \infty}$. \end{enumerate} Let $\Omega_M^p$ be a $C^{\infty}$ vector bundle consisting of $C^{\infty}$ complex valued $p$-forms. Let $\pi_p : \Omega_M^p \to M$ be the canonical map. We denote $C^{\infty}(M, \Omega_M^p)$ (resp. $C^{\infty}_c(M, \Omega_{M}^p)$) by $A^p(M)$ (resp. $A_c^p(M)$). Let $\alpha$ be a section of $\pi_p : \Omega_M^p \to M$. We say $\alpha$ is {\em locally integrable}, or simply an {\em $L^1$-form} if, at any point of $M$, all coefficients of $\alpha$ in terms of local coordinates are locally integrable functions. The set of all locally integrable $p$-forms is denoted by $L^1_{\operatorname{loc}}(M, \Omega^p_M)$. For an maximal form $\alpha$ on $M$, there is a unique function $g$ on $M$ with $\alpha = g \omega$. We denote this function $g$ by $c_{\omega}(\alpha)$. Let us define the Lebesgue integral of locally integrable $n$-forms with compact support. Let $\alpha$ be an element of $L^1_{\operatorname{loc}}(M, \Omega^n_M)$ such that the support of $\alpha$ is compact. Then $c_{\omega}(\alpha) \in L^1_{\operatorname{loc}}(M)$ and $\operatorname{supp}(c_{\omega}(\alpha))$ is compact. Thus, ${\displaystyle L\!\!\!\int_{M} c_{\omega}(\alpha) d\mu_{\omega}}$ exists. Let $\omega'$ be another $C^{\infty}$ volume element of $M$. Then, there is a positive $C^{\infty}$ function $a$ on $M$ with $\omega' = a \omega$. Here $a c_{\omega'}(\alpha) = c_{\omega}(\alpha)$. Thus, \[ L\!\!\!\int_{M} c_{\omega'}(\alpha) d\mu_{\omega'} = L\!\!\!\int_{M} c_{\omega'}(\alpha) a d\mu_{\omega} = L\!\!\!\int_{M} c_{\omega}(\alpha) a d\mu_{\omega}. \] Hence, ${\displaystyle L\!\!\!\int_{M} c_{\omega}(\alpha) d\mu_{\omega}}$ does not depend on the choice of the volume form $\omega$. Thus, the Lebesgue integral of $\alpha$ is defined by \[ L\!\!\!\int_M \alpha = L\!\!\!\int_{M} c_{\omega}(\alpha) d\mu_{\omega}. \] Moreover, we denote by $D^p(M)$ the space of currents of type $p$ on $M$. Then, there is the natural homomorphism \[ [\ ] : L^1_{\operatorname{loc}}(M, \Omega^p_M) \to D^p(M) \] given by ${\displaystyle [\alpha](\phi) = L\!\!\!\int_M \alpha \wedge \phi}$ for $\phi \in A_c^{n-p}(M)$. It is well known that the kernel of $[ \ ]$ is $\{ \alpha \in L^1_{\operatorname{loc}}(M, \Omega^p_M) \mid \alpha = 0 \ (\operatorname{a.e.}) \}$. A topology on $D^p(M)$ is defined in the following way. For an sequence $\{ T_n \}_{n=1}^{\infty}$ in $D^p(M)$, $T_n \to T$ as $n \to \infty$ if and only if $T_n(\phi) \to T(\phi)$ as $n \to \infty$ for each $\phi \in A_c^{n-p}(M)$. For an element $T \in D^n(M)$, by abuse of notation, we denote by $c_{\omega}(T)$ a unique distribution $g$ on $M$ given by $T = g \omega$. \begin{Proposition} \label{prop:criterion:loc:int} Let $T$ be a current of type $p$ on $M$. Then, the following are equivalent. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item $T$ is represented by a $L^1$-form. \item For any $\phi \in A^{n-p}(M)$, $c_{\omega}(T \wedge \phi)$ is represented by a locally integrable function. \end{enumerate} \end{Proposition} {\sl Proof.}\quad (1) $\Longrightarrow$ (2): Let $\phi \in A^{n-p}(M)$. Then, by our assumption, for any point $x \in M$, there are an open neighborhood $U$ of $x$, $C^{\infty}$ functions $a_1, \ldots, a_r$ on $U$, and locally integrable functions $b_1, \ldots, b_r$ on $U$ such that \[ \rest{c_{\omega}(T \wedge \phi)}{U} = \sum_{i=1}^r [a_i b_i]. \] Thus, if $K$ is a compact set in $U$, then \[ L\!\!\!\int_K \left|\sum_{i=1}^r a_i b_i \right| d\mu_{\omega} \leq L\!\!\!\int_K \sum_{i=1}^r |a_i| |b_i| d\mu_{\omega} \leq \max_i \sup_{x \in K} \{ |a_i(x)| \} \sum_{i=1}^r L\!\!\!\int_K |b_i| d\mu_{\omega} < \infty. \] Thus, we get (2). \medskip (2) $\Longrightarrow$ (1): Before starting the proof, we would like to claim the following fact. Let $\{ U_{\alpha} \}_{\alpha \in A}$ be an open covering of $M$ such that $A$ is at most a countable set. Let $\lambda_{\alpha}$ be a locally integrable form $U_{\alpha}$ with $\lambda_{\alpha} = \lambda_{\beta} \ (\operatorname{a.e.})$ on $U_{\alpha} \cap U_{\beta}$ for all $\alpha, \beta \in A$. Then, there is a locally integrable form $\lambda$ on $M$ such that $\lambda = \lambda_{\alpha} \ (\operatorname{a.e.})$ on $U_{\alpha}$ for all $\alpha \in A$. Indeed, let us fix a map $a : M \to A$ with $x \in U_{a(x)}$ and define a form $\lambda$ by $\lambda(x) = \lambda_{a(x)}(x)$. Then, $\lambda$ is our desired form because for each $\alpha \in A$, \[ \{ x \in U_{\alpha} \mid \lambda(x) \not= \lambda_{\alpha}(x) \} \subseteq \bigcup_{\beta \in A \setminus \{ \alpha \}} \{ x \in U_{\alpha} \cap U_{\beta} \mid \lambda_{\beta}(x) \not= \lambda_{\alpha}(x) \} \] and the right hand side has measure zero. \medskip Let $U$ be an open neighborhood of a point $x \in M$ and $(x_1, \ldots, x_n)$ a local coordinate of $U$ such that $dx_1 \wedge \cdots \wedge dx_n$ coincides with the orientation by $\omega$. Then, there is a positive $C^{\infty}$ function $a$ on $U$ with $\omega = a dx_1 \wedge \cdots \wedge dx_n$ over $U$. We set \[ T = \sum_{i_1 < \cdots < i_p } T_{i_1 \cdots i_p} dx_{i_1} \wedge \cdots \wedge dx_{i_p} \] for some distributions $T_{i_1 \cdots i_p}$. We need to show that $T_{i_1 \cdots i_p}$ is represented by a locally integrable function. Since $M$ has a countable basis of open sets, by the above claim, it is sufficient to check that $T_{i_1 \cdots i_p}$ is represented by an integral function on every compact set $K$ in $U$. Let $f$ be a non-negative $C^{\infty}$ function on $M$ such that $f = 1$ on $K$ and $\operatorname{supp}(f) \subset U$. Choose $i_{p+1}, \ldots, i_{n}$ such that $\{ i_1, \ldots, i_{n} \} = \{ 1, \ldots, n \}$. Here we set $\phi = f a dx_{i_{p+1}} \wedge \cdots \wedge dx_{i_n}$. Then, $\phi \in A^{n-p}(M)$ and \[ T \wedge \phi = \epsilon T_{i_1 \cdots i_p} f a dx_{1} \wedge \cdots \wedge dx_{n} = \epsilon T_{i_1 \cdots i_p} f \omega, \] where $\epsilon = 1$ or $-1$ depending on the orientation of $\{ x_{i_1}, \ldots, x_{i_n} \}$. By our assumption, there is a locally integrable function $h$ on $M$ with $c_{\omega}(T \wedge \phi) = [h]$. Thus, $[\epsilon h] = T_{i_1 \cdots i_p} f$. Therefore, $T_{i_1 \cdots i_p}$ is represented by $\epsilon h$ on $K$ because $f = 1$ on $K$. Thus, we get (2). \QED \subsection{Push-forward of $L^1$-forms as current} \label{subsec:push:forward:L1:current} \setcounter{Theorem}{0} First of all, we recall the push-forward of currents. Let $f : M \to N$ be a proper morphism of orientable manifolds with the relative dimension $d = \dim M - \dim N$. Then, \[ f_* : D^p(M) \to D^{p-d}(N) \] is defined by $(f_*(T))(\phi) = T(f^*(\phi))$ for $\phi \in A_c^{\dim N - p + d}(N)$. It is easy to see that $f_*$ is a continuous homomorphism. Let us begin with the following lemma. \begin{Lemma} \label{lem:push:forward:product} Let $F$ be an orientable compact differential manifold and $Y$ an orientable differential manifold. Let $\omega_F$ \textup{(}resp. $\omega_Y$\textup{)} be a $C^{\infty}$ volume element of $F$ \textup{(}resp. $Y$\textup{)}. Let $p : F \times Y \to Y$ be the projection to the second factor. Then, we have the following. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item If $g$ is a continuous function on $F \times Y$, then $\int_F g \omega_F$ is a continuous function on $Y$. \item If $\alpha$ is a continuous maximal form on $F \times Y$, then $p_*([\alpha])$ is represented by a unique continuous from. This continuous form is denoted by $\int_p \alpha$. \item For a continuous function $g$ on $F \times Y$, \[ \left| c_{\omega_Y}\left( \int_{p} g \omega_F \wedge \omega_Y \right) \right| \leq c_{\omega_Y} \left( \int_{p} |g| \omega_F \wedge \omega_Y \right). \] \end{enumerate} \end{Lemma} {\sl Proof.}\quad (1) This is standard. \medskip (2) Since $\omega_F \wedge \omega_Y$ is a volume form on $F \times Y$, there is a continuous function $g$ on $F \times Y$ with $\alpha = g \omega_F \wedge \omega_Y$. Thus, it is sufficient to show that \[ p_*([\alpha]) = \left[ \left(\int_F g \omega_F \right) \omega_Y \right]. \] Indeed, by Fubini's theorem, for $\phi \in A_c^0(Y)$, \[ p_*([\alpha])(\phi) = \int_{F \times Y} \phi \alpha = \int_Y \left( \int_F g \omega_F \right) \phi \omega_Y = \left[ \left(\int_F g \omega_F \right) \omega_Y \right](\phi). \] \medskip (3) This is obvious because \[ \left| \int_F g \omega_F \right| \leq \int_F |g| \omega_F. \] \QED \begin{Corollary} \label{cor:cont:ineq:integral:fiber} Let $f : X \to Y$ be a proper, surjective and smooth morphism of connected complex manifolds. Let $\omega_X$ and $\omega_Y$ be volume elements of $X$ and $Y$ respectively. Then, \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item For a continuous maximal form $\alpha$ on $X$, $f_*([\alpha])$ is represented by a unique continuous form. We denote this continuous form by $\int_{f} \alpha$. \item For any continuous functions $g$ on $X$, \[ \left| c_{\omega_Y}\left( \int_{f} g \omega_X \right) \right| \leq c_{\omega_Y} \left( \int_{f} |g| \omega_X \right). \] \end{enumerate} \end{Corollary} {\sl Proof.}\quad (1) This is a local question on $Y$. Thus, we may assume that there are a compact complex manifold $F$ and a differomorphism $h : X \to F \times Y$ such that the following diagram is commutative: \[ \begin{CD} X @>{\sim}>{h}> F \times Y \\ @V{f}VV @VV{p}V \\ Y @= Y, \end{CD} \] where $p : F \times Y \to Y$ is the natural projection. Hence, (1) is a consequence of (2) of Lemma~\ref{lem:push:forward:product}. \medskip (2) First, we claim that if the above inequality holds for some special volume elements $\omega_X$ and $\omega_Y$, then the same inequality holds for any volume elements. Let $\omega'_{X}$ and $\omega'_{Y}$ be another volume elements of $X$ and $Y$ respectively. We set $\omega'_X = a \omega_X$ and $\omega'_Y = b \omega_Y$. Then, $a$ and $b$ are positive $C^{\infty}$ functions. Let $g$ be any continuous function on $X$. Then, by our assumption, \[ \left| c_{\omega_Y}\left( \int_{f} g \omega'_X \right) \right| = \left| c_{\omega_Y}\left( \int_{f} g a \omega_X \right) \right| \leq c_{\omega_Y} \left( \int_{f} |g| a \omega_X \right) = c_{\omega_Y} \left( \int_{f} |g| \omega'_X \right). \] On the other hand, for any maximal forms $\alpha$ on $Y$, \[ c_{\omega_Y}\left( \alpha \right) = b c_{\omega'_Y}\left( \alpha \right). \] Thus, we get our claim. Hence, as in the proof of (1), using the differomorphism $h$ and (3) of Lemma~\ref{lem:push:forward:product}, we can see (2). \QED \begin{Remark} In the situation of Corollary~\ref{cor:cont:ineq:integral:fiber}, if $\alpha$ is a $C^{\infty}$-form on $X$, then $f_*([\alpha])$ is represented by a unique $C^{\infty}$-form. \end{Remark} \begin{Proposition} \label{prop:loc:int:integral:fiber} Let $f : X \to Y$ be a proper and surjective morphism of connected complex manifolds. Let $U$ be a non-empty Zariski open set of $Y$ such that $f$ is smooth over $U$. Let $\alpha$ be a compactly supported continuous maximal form on $X$. If we set \[ \lambda = \begin{cases} {\displaystyle \int_{f^{-1}(U) \to U} \alpha} & \text{on $U$}, \\ {} & {} \\ 0 & \text{on $Y \setminus U$,} \end{cases} \] then $\lambda$ is integrable. Moreover, $f_*([\alpha]) = [\lambda]$. \end{Proposition} {\sl Proof.}\quad Let $\omega_X$ and $\omega_Y$ be volume forms of $X$ and $Y$ respectively. Let $h$ be a function on $Y$ with $\lambda = h \omega_Y$. Then, $h$ is continuous on $U$ by Corollary~\ref{cor:cont:ineq:integral:fiber}. Moreover, let $g$ be a continuous function on $X$ with $\alpha = g \omega_X$. We need to show that $h$ is an integrable function. First note that $\int_X |g| \omega_X < \infty$ because $g$ is a compactly supported continuous function. Let $\{ U_n \}_{n=1}^{\infty}$ be a sequence of open sets such that $\overline{U_n} \subset U$, $\overline{U_n}$ is compact, $U_1 \subseteq U_2 \subseteq \cdots \subseteq U_n \subseteq \cdots$, and $\bigcup_{n=1}^{\infty} U_n = U$. Here we set \[ h_n(y) = \begin{cases} |h(y)| & \text{if $y \in U_n$} \\ 0 & \text{otherwise}. \end{cases} \] Then, $0 \leq h_1 \leq h_2 \leq \cdots \leq h_n \leq \cdots$ and ${\displaystyle \lim_{n \to \infty} h_n(y) = |h(y)|}$. By Corollary~\ref{cor:cont:ineq:integral:fiber}, \[ \left| \rest{h}{U} \right| \leq c_{\omega_{Y}}\left( \int_{f^{-1}(U) \to U} |g| \omega_X \right). \] Thus, \begin{align*} \int_{U_n} |h| \omega_Y & \leq \int_{U_n} c_{\omega_{Y}}\left( \int_{f^{-1}(U_n) \to U_n} |g| \omega_X \right) \omega_Y = \int_{U_n} \int_{f^{-1}(U_n) \to U_n} |g| \omega_X \\ & = \int_{f^{-1}(U_n)} |g| \omega_X \leq \int_X |g| \omega_X. \end{align*} Therefore, \[ L\!\!\!\int_{Y} h_n d\mu_{\omega_Y} = \int_{U_n} |h| \omega_Y \leq \int_{X} |g| \omega_X < \infty. \] Thus, by Fatou's theorem, \[ L\!\!\!\int_{Y} |h| d\mu_{\omega_Y} = \lim_{n \to \infty} L\!\!\!\int_{Y} h_n d\mu_{\omega_Y} \leq L\!\!\!\int_{X} |g| \omega_X < \infty. \] Hence, $h$ is integral. Let $\phi$ be any element of $A^{0}_c(Y)$. Then, since ${\displaystyle \lim_{n \to \infty} \mu_{\omega_Y}(Y \setminus U_n) = 0}$ and $h \phi$ is integrable, by the absolute continuity of Lebesgue integral, \[ \lim_{n \to \infty} L\!\!\!\int_{Y \setminus U_n} h \phi d\mu_{\omega_Y} = 0. \] Thus, \begin{align*} L\!\!\!\int_Y \lambda \phi &= \lim_{n \to \infty} \left( L\!\!\!\int_{U_n} h \phi d\mu_{\omega_{Y}} + L\!\!\!\int_{Y \setminus U_n} h \phi d\mu_{\omega_{Y}} \right) \\ & = \lim_{n \to \infty} L\!\!\!\int_{U_n} h \phi d\mu_{\omega_{Y}} = \lim_{n \to \infty} \int_{U_n} h \phi \omega_Y = \lim_{n \to \infty} \int_{U_n} \lambda \phi. \end{align*} In the same way, \[ \int_X \alpha f^*(\phi) = \lim_{n \to \infty} \int_{f^{-1}(U_n)} \alpha f^*(\phi). \] On the other hand, we have \[ \int_{U_n} \lambda \phi = \int_{f^{-1}(U_n)} \alpha f^*(\phi). \] Hence \begin{align*} f_*([\alpha])(\phi) & = [\alpha](f^*(\phi)) = \int_X \alpha \wedge f^*(\phi) = \lim_{n \to \infty} \int_{f^{-1}(U_n)} \alpha f^*(\phi) \\ & = \lim_{n \to \infty} \int_{U_n} \lambda \phi = L\!\!\!\int_{Y} \lambda \phi = [\lambda](\phi) \end{align*} Therefore, $f_*([\alpha]) = [\lambda]$. \QED Let $X$ be an equi-dimensional complex manifold, i.e., every connected component has the same dimension. We denote by $A^{p,q}(X)$ the space of $C^{\infty}$ complex valued $(p,q)$-forms on $X$. Let $A^{p,q}_c(X)$ be the subspace of compactly supported forms. Let $D^{p,q}(X)$ be the space of currents on $X$ of type $(p,q)$. As before, there is a natural homomorphism \[ [\ ] : L^1_{\operatorname{loc}}(\Omega_{X}^{p,q}) \to D^{p,q}(X). \] Then, as a corollary of Proposition~\ref{prop:loc:int:integral:fiber}, we have the following main result of this section. \begin{Proposition} \label{prop:push:forward:B:pq} Let $f : X \to Y$ be a proper morphism of equi-dimensional complex manifolds. We assume that every connected component of $X$ maps surjectively to a connected component of $Y$. Let $\alpha$ be an $L^1$-form of type $(p+d,q+d)$ on $X$, where $d = \dim X - \dim Y$. Then there is a $\lambda \in L^1_{\operatorname{loc}}(\Omega_Y^{p,q})$ with $f_*([\alpha]) = [\lambda]$. \end{Proposition} {\sl Proof.}\quad Clearly we may assume that $Y$ is connected. Since $f$ is proper, there are finitely many connected components of $X$, say, $X_1, \ldots, X_e$. If we set $\alpha_i = \rest{\alpha}{X_i}$ and $f_i = \rest{f}{X_i}$ for each $i$, then $f_*([\alpha]) = (f_1)_*([\alpha_1]) + \cdots + (f_e)_*([\alpha_e])$. Thus, we may assume that $X$ is connected. Further, since $f_*([ \alpha \wedge f^*(\phi) ]) = f_*([\alpha]) \wedge \phi$ for all $\phi \in A^{\dim Y - p, \dim Y - q}(Y)$, we may assume that $\alpha$ is a maximal form by Proposition~\ref{prop:criterion:loc:int}. Let $g$ be a locally integrable function on $X$ with $\alpha = g \omega_X$. Since the question is local with respect to $Y$, we may assume that $g$ is integrable. Thus, since $C_c^0(Y)$ is dense on $L^1(Y)$ (cf. \cite[Theorem~3.14]{Ru}), there is a sequence $\{ g_n \}_{n=1}^{\infty}$ of compactly supported continuous functions on $X$ such that \[ \lim_{n \to \infty} L\!\!\!\int_X |g_n - g| d\mu_{\omega_X} = 0. \] By Proposition~\ref{prop:loc:int:integral:fiber}, for each $n$, there is an integrable function $h_n$ on $Y$ such that $f_*([g_n \omega_X]) = [h_n \omega_Y]$. Moreover, by (2) of Corollary~\ref{cor:cont:ineq:integral:fiber}, \[ |h_n - h_m| \leq c_{\omega_Y} \left( \int_{f^{-1}(U) \to U} |g_n - g_m | \omega_X \right) \] over $U$. Thus, we can see \[ L\!\!\!\int_Y |h_n - h_m| d\mu_{\omega_Y} \leq L\!\!\!\int_X |g_n - g_m| d\mu_{\omega_X} \] for all $n, m$. Hence, $\{ h_n \}_{n=1}^{\infty}$ is a Cauchy sequence in $L^1(Y)$. Therefore, by the completeness of $L^1(Y)$, there is an integrable function $h$ on $Y$ with $h = \lim_{n \to \infty} h_n$ in $L^1(Y)$. Then, for any $\phi \in A_c^{0,0}(Y)$, \[ \lim_{n \to \infty} L\!\!\!\int_Y h_n \phi \omega_Y = L\!\!\!\int_Y h \phi \omega_Y \quad\text{and}\quad \lim_{n \to \infty} L\!\!\!\int_X g_n f^*(\phi) \omega_X = L\!\!\!\int_X g f^*(\phi) \omega_X. \] Thus, \begin{align*} f_*([\alpha])(\phi) & = L\!\!\!\int_X g f^*(\phi)\omega_X = \lim_{n \to \infty} L\!\!\!\int_X g_n f^*(\phi) \omega_X \\ & = \lim_{n \to \infty} L\!\!\!\int_Y h_n \phi \omega_Y = L\!\!\!\int_Y h \phi \omega_Y = [h\omega_Y](\phi). \end{align*} Therefore, $f_*([\alpha]) = [h\omega_Y]$. \QED \section{Variants of arithmetic Chow groups} \subsection{Notation for arithmetic varieties} \label{subsec:notation:arith:variety} \setcounter{Theorem}{0} An {\em arithmetic variety $X$} is an integral scheme which is flat and quasi-projective over $\operatorname{Spec}({\mathbb{Z}})$, and has the smooth generic fiber $X_{{\mathbb{Q}}}$. Let us consider the ${\mathbb{C}}$-scheme $X \otimes_{{\mathbb{Z}}} {\mathbb{C}}$. We denote the underlying analytic space of $X \otimes_{{\mathbb{Z}}} {\mathbb{C}}$ by $X({\mathbb{C}})$. We may view $X({\mathbb{C}})$ as the set of all ${\mathbb{C}}$-valued points of $X$. Let $F_{\infty} : X({\mathbb{C}}) \to X({\mathbb{C}})$ be the anti-holomorphic involution given by the complex conjugation. For an arithmetic variety $X$, every $(p,p)$-form $\alpha$ on $X({\mathbb{C}})$ is always assumed to be compatible with $F_{\infty}$, i.e., $F_{\infty}^*(\alpha) = (-1)^p \alpha$. Let $E$ be a locally free sheaf on $X$ of finite rank, and $\pi : \pmb{E} \to X$ the vector bundle associated with $E$, i.e., $\pmb{E} = \operatorname{Spec}\left( \bigoplus_{n=0}^{\infty} \operatorname{Sym}^n(E) \right)$. As before, we have the analytic space $\pmb{E}({\mathbb{C}})$ and the anti-holomorphic involution $F_{\infty} : \pmb{E}({\mathbb{C}}) \to \pmb{E}({\mathbb{C}})$. Then, $\pi_{{\mathbb{C}}} : \pmb{E}({\mathbb{C}}) \to X({\mathbb{C}})$ is a holomorphic vector bundle on $X({\mathbb{C}})$, and the following diagram is commutative: \[ \begin{CD} \pmb{E}({\mathbb{C}}) @>{F_{\infty}}>> \pmb{E}({\mathbb{C}}) \\ @V{\pi_{{\mathbb{C}}}}VV @VV{\pi_{{\mathbb{C}}}}V \\ X({\mathbb{C}}) @>>{F_{\infty}}> X({\mathbb{C}}) \end{CD} \] Here note that $F_{\infty} : \pmb{E}({\mathbb{C}}) \to \pmb{E}({\mathbb{C}})$ is anti-complex linear at each fiber. Let $h$ be a $C^{\infty}$ Hermitian metric of $\pmb{E}({\mathbb{C}})$. We can think $h$ as a $C^{\infty}$ function on $\pmb{E}({\mathbb{C}}) \times_{X({\mathbb{C}})} \pmb{E}({\mathbb{C}})$. For simplicity, we denote by $F_{\infty}^*(h)$ the $C^{\infty}$ function $\left( F_{\infty} \times_{X({\mathbb{C}})} F_{\infty} \right)^*(h)$ on $\pmb{E}({\mathbb{C}}) \times_{X({\mathbb{C}})} \pmb{E}({\mathbb{C}})$. Then, $\overline{F_{\infty}^*(h)}$ is a $C^{\infty}$ Hermitian metric of $\pmb{E}({\mathbb{C}})$. We say {\em $h$ is invariant under $F_{\infty}$} if $F_{\infty}^*(h) = \overline{h}$. Moreover, the pair $(E, h)$ is called {\em a Hermitian vector bundle on $X$} if $h$ is invariant under $F_{\infty}$. Note that even if $h$ is not invariant under $F_{\infty}$, $h + \overline{F_{\infty}^*(h)}$ is an invariant metric. \subsection{Variants of arithmetic cycles} \setcounter{Theorem}{0} \label{subsec:var:arith:chow} Let $X$ be an arithmetic variety. We would like to define three types of arithmetic cycles, namely, arithmetic $A$-cycles, arithmetic $L^1$-cycles, and arithmetic $D$-cycles. In the following definition, $g$ is compatible with $F_{\infty}$ as mentioned in \S\ref{subsec:notation:arith:variety}. \begin{enumerate} \renewcommand{\labelenumi}{(\alph{enumi})} \item (arithmetic $A$-cycle on $X$ of codimension $p$) : a pair $(Z, g)$ such that $Z$ is a cycle on $X$ of codimension $p$ and $g$ is represented by a Green form $\phi$ of $Z({\mathbb{C}})$, namely, $\phi$ is a $C^{\infty}$ form on $X({\mathbb{C}}) \setminus \operatorname{Supp}(Z({\mathbb{C}}))$ of logarithmic type along $\operatorname{Supp}(Z({\mathbb{C}}))$ with $dd^c([\phi]) + \delta_{Z({\mathbb{C}})} \in A^{p,p}(X({\mathbb{C}}))$. \item (arithmetic $L^1$-cycle on $X$ of codimension $p$) : a pair $(Z, g)$ such that $Z$ is a cycle on $X$ of codimension $p$ and, there are $\phi \in L^1_{\operatorname{loc}}(\Omega_{X({\mathbb{C}})}^{p-1,p-1})$ and $\omega \in L^1_{\operatorname{loc}}(\Omega_{X({\mathbb{C}})}^{p,p})$ with $g = [\phi]$ and $dd^c(g) + \delta_{Z({\mathbb{C}})} = [\omega]$. \item (arithmetic $D$-cycle on $X$ of codimension $p$) : a pair $(Z, g)$ such that $Z$ is a cycle on $X$ of codimension $p$ and $g \in D^{p-1,p-1}(X({\mathbb{C}}))$. \end{enumerate} The set of all arithmetic $A$-cycles (resp. $L^1$-cycles, $D$-cycles) of codimension $p$ is denoted by $\widehat{Z}_A^p(X)$ (resp. $\widehat{Z}_{L^1}^p(X)$, $\widehat{Z}_D^p(X)$). Let $\widehat{R}^p(X)$ be the subgroup of $\widehat{Z}^p(X)$ generated by the following elements: \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item $((f), - [\log |f|^2])$, where $f$ is a rational function on some subvariety $Y$ of codimension $p-1$ and $[\log |f|^2]$ is the current defined by \[ [\log |f|^2](\gamma) = L\!\!\!\int_{Y({\mathbb{C}})} (\log |f|^2)\gamma. \] \item $(0, \partial(\alpha) + \bar{\partial}(\beta))$, where $\alpha \in D^{p-2, p-1}(X({\mathbb{C}}))$, $\beta \in D^{p-1, p-2}(X({\mathbb{C}}))$. \end{enumerate} Here we define \[ \begin{cases} \widehat{\operatorname{CH}}_A^p(X) = \widehat{Z}_A^{p}(X)/\widehat{R}^p(X) \cap \widehat{Z}_A^{p}(X), \\ \widehat{\operatorname{CH}}_{L^1}^p(X) = \widehat{Z}_{L^1}^{p}(X)/\widehat{R}^p(X) \cap \widehat{Z}_{L^1}^{p}(X), \\ \widehat{\operatorname{CH}}_D^p(X) = \widehat{Z}_D^{p}(X)/\widehat{R}^p(X). \end{cases} \] \begin{Proposition} \label{prop:AChow:equal:Chow} The natural homomorphism $\widehat{\operatorname{CH}}_A^p(X) \to \widehat{\operatorname{CH}}^p(X)$ is an isomorphism. \end{Proposition} {\sl Proof.}\quad Let $(Z, g) \in \widehat{Z}^p(X)$. By \cite[Theorem~1.3.5]{GSArInt}, there is a Green form $g_Z$ of $Z({\mathbb{C}})$. Then, $dd^c(g - [g_Z]) \in A^{p,p}(X({\mathbb{C}}))$. Hence, by \cite[Theorem~1.2.2]{GSArInt}, there are $a \in A^{d,d}(X({\mathbb{C}}))$ and $v \in \operatorname{Image}(\partial) + \operatorname{Image}(\bar{\partial})$ with $g - [g_Z] = [a] + v$. Since $g - [g_Z]$ is compatible with $F_{\infty}$, replacing $a$ and $v$ by $(1/2)(a + (-1)^pF_{\infty}^*(a))$ and $(1/2)(v + (-1)^p F_{\infty}^*(v))$ respectively, we may assume that $a$ and $v$ are compatible with $F_{\infty}$. Here, $g_Z + a$ is a Green form of $Z$. Thus, $(Z, [g_Z + a]) \in \widehat{Z}_A^p(X)$. Moreover, since $(Z, g) - (Z, [g_Z + a]) \in \widehat{R}^p(X)$, our proposition follows. \QED Let $f : X \to Y$ be a proper morphism of arithmetic varieties with $d = \dim X - \dim Y$. Then, we have a homomorphism \[ f_* : \widehat{Z}_D^{p+d}(X) \to \widehat{Z}_D^{p}(Y) \] defined by $f_*(Z, g) = (f_*(Z), f_*(g))$. In the same way as in the proof of \cite[Theorem~3.6.1]{GSArInt}, we can see $f_*(\widehat{R}^{p+d}(X)) \subseteq \widehat{R}^p(Y)$. Thus, the above homomorphism induces \[ f_* : \widehat{\operatorname{CH}}_D^{p+d}(X) \to \widehat{\operatorname{CH}}_D^{p}(Y). \] Then we have the following. \begin{Proposition} \label{prop:push:forward:arith:cycle} If $f$ is surjective, then $f_* : \widehat{\operatorname{CH}}_D^{p+d}(X) \to \widehat{\operatorname{CH}}_D^{p}(Y)$ gives rise to \[ f_* : \widehat{\operatorname{CH}}_{L^1}^{p+d}(X) \to \widehat{\operatorname{CH}}_{L^1}^{p}(Y). \] In particular, we have the homomorphism $f_* : \widehat{\operatorname{CH}}^{p+d}(X) \to \widehat{\operatorname{CH}}_{L^1}^{p}(Y)$. \end{Proposition} {\sl Proof.}\quad Clearly we may assume that $p \geq 1$. It is sufficient to show that if $(Z, g) \in \widehat{Z}_{L^1}^{p+d}(X)$, then $(f_*(Z), f_*(g)) \in \widehat{Z}_{L^1}^{p}(Y)$. By the definition of $L^1$-arithmetic cycles, $g$ and $dd^c(g) + \delta_{Z({\mathbb{C}})}$ are represented by $L^1$-forms. Thus, by Proposition~\ref{prop:push:forward:B:pq}, there is an $\omega \in L^1_{\operatorname{loc}}(\Omega_{Y({\mathbb{C}})}^{p,p})$ with \[ f_*\left( dd^c(g) + \delta_{Z({\mathbb{C}})} \right) = [\omega]. \] On the other hand, \[ f_*\left( dd^c(g) + \delta_{Z({\mathbb{C}})} \right) = dd^c(f_*(g)) + \delta_{f_*(Z({\mathbb{C}}))}. \] Moreover, by Proposition~\ref{prop:push:forward:B:pq}, $f_*(g)$ is represented by an $L^1$-form on $Y({\mathbb{C}})$. Thus, $(f_*(Z), f_*(g))$ is an element of $\widehat{Z}_{L^1}^p(Y)$. \QED \subsection{Scalar product for arithmetic $L^1$-cycles and arithmetic $D$-cycles} \renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}} \setcounter{Theorem}{0} Let $X$ be a regular arithmetic variety. The purpose of this subsection is to give a scalar product on $\widehat{\operatorname{CH}}_D^{*}(X)_{{\mathbb{Q}}} = \bigoplus_{p \geq 0} \widehat{\operatorname{CH}}_D^p(X)_{{\mathbb{Q}}}$ by the arithmetic Chow ring $\widehat{\operatorname{CH}}^{*}(X)_{{\mathbb{Q}}} = \bigoplus_{p \geq 0} \widehat{\operatorname{CH}}^p(X)_{{\mathbb{Q}}}$. Roughly speaking, the scalar product is defined by \[ (Y, f) \cdot (Z, g) = ( Y \cap Z, f \wedge \delta_Z + \omega((Y,f)) \wedge g) \] for $(Y, f) \in \widehat{Z}^p(X)$ and $(Z, g) \in \widehat{Z}_D^q(X)$. This definition, however, works only under the assumption that $Y$ and $Z$ intersect properly. Usually, by using Chow's moving lemma, we can avoid the above assumption. This is rather complicated, so that in this paper we try to use the standard arithmetic intersection theory to define the scalar product. Let $x \in \widehat{\operatorname{CH}}^p(X)$, $(Z, g) \in \widehat{Z}_D^q(X)$, and $g_Z$ a Green current for $Z$. First we shall check that \[ x \cdot [(Z, g_Z)] + [(0, \omega(x) \wedge (g-g_Z))] \] in $\widehat{\operatorname{CH}}_D^{p+q}(X)_{{\mathbb{Q}}}$ does not depend on the choice of $g_Z$. For, let $g'_Z$ be another Green current for $Z$. Then, there are $\eta \in A^{p-1,p-1}(X({\mathbb{C}}))$, and $v \in \operatorname{Image}(\partial) + \operatorname{Image}(\bar{\partial})$ with $g'_Z = g_Z + [\eta] + v$. Then, since $[(0, [\eta] + v)] \in \widehat{\operatorname{CH}}^p(X)$, \begin{align*} x \cdot [(Z, g'_Z)] + [(0, \omega(x) \wedge (g-g'_Z))] & = x \cdot [(Z, g_Z)] + x \cdot [(0, [\eta] + v)] \\ & \qquad\qquad + [(0, \omega(x) \wedge (g-g_Z-[\eta]-v))] \\ & = x \cdot [(Z, g_Z)] + [(0, \omega(x) \wedge ([\eta] + v))] \\ & \qquad\qquad + [(0, \omega(x) \wedge (g-g_Z-[\eta]-v))] \\ & = x \cdot [(Z, g_Z)] + [(0, \omega(x) \wedge (g-g_Z))]. \end{align*} Thus, we have the bilinear homomorphism \[ \widehat{\operatorname{CH}}^p(X) \times \widehat{Z}_D^q(X) \to \widehat{\operatorname{CH}}_D^{p+q}(X)_{{\mathbb{Q}}} \] given by $x \cdot (Z, g) = x \cdot [(Z, g_Z)] + [(0, \omega(x) \wedge (g-g_Z))]$. Moreover, if $(Z, g) \in \widehat{R}^q(X)$, then, by \cite[Theorem~4.2.3]{GSArInt}, $x \cdot (Z, g) = 0$ in $\widehat{\operatorname{CH}}^{p+q}(X)_{{\mathbb{Q}}}$. Thus, the above induces \addtocounter{Theorem}{1} \begin{equation} \label{eqn:scalar:product:1} \widehat{\operatorname{CH}}^p(X) \otimes \widehat{\operatorname{CH}}_D^q(X) \to \widehat{\operatorname{CH}}_D^{p+q}(X)_{{\mathbb{Q}}}, \end{equation} which may give rises to a natural scalar product of $\widehat{\operatorname{CH}}_D^{*}(X)_{{\mathbb{Q}}}$ over the arithmetic Chow ring $\widehat{\operatorname{CH}}^{*}(X)_{{\mathbb{Q}}}$. To see that this is actually a scalar product, we need to check that \[ (x \cdot y) \cdot z = x \cdot (y \cdot z) \] for all $x \in \widehat{\operatorname{CH}}^p(X)$, $y \in \widehat{\operatorname{CH}}^q(X)$ and $z \in \widehat{\operatorname{CH}}_D^r(X)$. If $z \in \widehat{\operatorname{CH}}^r(X)$, then this is nothing more than the associativity of the product of the arithmetic Chow ring (cf. \cite[Theorem~4.2.3]{GSArInt}). Thus, we may assume that $z = [(0, g)]$ for some $g \in D^{r-1,r-1}(X({\mathbb{C}}))$. Then, since \[ (x \cdot y) \cdot z = [(0, \omega(x \cdot y) \wedge g)] = [(0, \omega(x) \wedge \omega(y) \wedge g)] \] and \[ x \cdot (y \cdot z) = x \cdot [(0, \omega(y) \wedge g)] = [(0, \omega(x) \wedge \omega(y) \wedge g)], \] we can see $(x \cdot y) \cdot z = x \cdot (y \cdot z)$. Therefore, we get the natural scalar product. Moreover, (\ref{eqn:scalar:product:1}) induces \addtocounter{Theorem}{1} \begin{equation} \label{eqn:scalar:product:2} \widehat{\operatorname{CH}}^p(X) \otimes \widehat{\operatorname{CH}}_{L^1}^q(X) \to \widehat{\operatorname{CH}}_{L^1}^{p+q}(X)_{{\mathbb{Q}}}. \end{equation} Indeed, if $(Z, g) \in \widehat{Z}_{L^1}^{q}(X)$ and $g_Z$ is a Green form of $Z$, then, \[ x \cdot [(Z, g)] = x \cdot [(Z, g_Z)] + [(0, \omega(x) \wedge (g - g_Z))]. \] Thus, in order to see that $x \cdot [(Z, g)] \in \widehat{\operatorname{CH}}_{L^1}^{p+q}(X)_{{\mathbb{Q}}}$, it is sufficient to check that \[ \begin{cases} \omega(x) \wedge (g - g_Z) \in L^1_{\operatorname{loc}}(\Omega_{X({\mathbb{C}})}^{p+q-1,p+q-1}), \\ dd^c \left( \omega(x) \wedge (g - g_Z) \right) \in L^1_{\operatorname{loc}}(\Omega_{X({\mathbb{C}})}^{p+q,p+q}). \end{cases} \] The first assertion is obvious because $g$ and $g_Z$ are $L^1$-forms. Further, we can easily see the second assertion because \[ dd^c \left( \omega(x) \wedge (g - g_Z) \right) = \pm \omega(x) \wedge dd^c(g - g_Z) = \pm \omega(x) \wedge (\omega(g) - \omega(g_Z)). \] Gathering all observations, we can conclude the following proposition, which is a generalization of \cite[Theorem~4.2.3]{GSArInt}. \begin{Proposition} \label{prop:module:structure:arith:D:cycle} $\widehat{\operatorname{CH}}_{L^1}^{*}(X)_{{\mathbb{Q}}}$ and $\widehat{\operatorname{CH}}_D^{*}(X)_{{\mathbb{Q}}}$ has a natural module structure over the arithmetic Chow ring $\widehat{\operatorname{CH}}^{*}(X)_{{\mathbb{Q}}}$. \end{Proposition} Moreover, we have the following projection formula. \begin{Proposition} \label{prop:projection:formula:regular:smooth} Let $f : X \to Y$ be a proper morphism of regular arithmetic varieties such that $f_{{\mathbb{Q}}} : X_{{\mathbb{Q}}} \to Y_{{\mathbb{Q}}}$ is smooth. Then, for any $\alpha \in \widehat{\operatorname{CH}}^p(Y)$ and $\beta \in \widehat{\operatorname{CH}}_{L^1}^q(X)$, \[ f_* (f^*(\alpha) \cdot \beta) = \alpha \cdot f_* (\beta) \] in $\widehat{\operatorname{CH}}_{L^1}^{p + q -(\dim X - \dim Y)}(Y)_{{\mathbb{Q}}}$. \end{Proposition} {\sl Proof.}\quad If $\alpha \in \widehat{\operatorname{CH}}^p(Y)$ and $\beta \in \widehat{\operatorname{CH}}^q(X)$, then this is well known (cf. \cite{GSArInt}). Thus, we may assume that $\beta = (0,[\phi]) \in \widehat{Z}_{L^1}^q(Y)$. Then \begin{align*} f_* (f^*(\alpha) \cdot \beta) & = f_* ((0,\omega (f^*(\alpha)) \wedge [\phi]) \\ & = (0, [f_*\left( \omega (f^*(\alpha) \wedge \phi) \right)]). \end{align*} On the other hand, \begin{equation*} \alpha \cdot f_* (\beta) = \alpha \cdot (0,[f_*(\phi)]) = (0, \omega(\alpha) \wedge [f_*(\phi)]). \end{equation*} Since $f_* (\omega (f^*(\alpha))) = \omega(\alpha)$, we have proven the projection formula. \QED \renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}} \subsection{Scalar product, revisited (singular case)} \setcounter{Theorem}{0} Let $X$ be an arithmetic variety. Here $X$ is not necessarily regular. Let $\operatorname{Rat}_X$ be the sheaf of rational functions on $X$. We denote $H^0(X, \operatorname{Rat}_X^{\times}/{\mathcal{O}}_X^{\times})$ by $\operatorname{Div}(X)$. An element of $\operatorname{Div}(X)$ is called {\em a Cartier divisor on $X$}. For a Cartier divisor $D$ on $X$, we can assign a divisor $[D] \in Z^1(X)$ in a natural way. This gives rise to the homomorphism \[ c_X : \operatorname{Div}(X) \to Z^1(X). \] Note that $c_X$ is neither injective nor surjective in general. An exact sequence \[ 1 \to {\mathcal{O}}_X^{\times} \to \operatorname{Rat}_X^{\times} \to \operatorname{Rat}_X^{\times}/{\mathcal{O}}_X^{\times} \to 1 \] induces to the homomorphism $\operatorname{Div}(X) \to H^1(X, {\mathcal{O}}_X^{\times})$. For a Cartier divisor $D$ on $X$, the image of $D$ by the above homomorphism induces the line bundle on $X$. We denote this line bundle by ${\mathcal{O}}_X(D)$. Here we set \[ \widehat{\operatorname{Div}}(X) = \{ (D, g) \mid \text{$D \in \operatorname{Div}(X)$ and $g$ is a Green function for $D({\mathbb{C}})$ on $X({\mathbb{C}})$} \}. \] Similarly, we can define $\widehat{\operatorname{Div}}_{L^1}(X)$ and $\widehat{\operatorname{Div}}_D(X)$. The homomorphism $c_X : \operatorname{Div}(X) \to Z^1(X)$ gives rise to the homomorphism $\hat{c}_X : \widehat{\operatorname{Div}}(X) \to \widehat{Z}^1(X)$. Then, we define $\widehat{\operatorname{Pic}}(X)$, $\widehat{\operatorname{Pic}}_{L^1}(X)$, and $\widehat{\operatorname{Pic}}_D(X)$ as follows. \[ \begin{cases} \widehat{\operatorname{Pic}}(X) = \widehat{\operatorname{Div}}(X)/\hat{c}_X^{-1}(\widehat{R}^1(X)), \\ \widehat{\operatorname{Pic}}_{L^1}(X) = \widehat{\operatorname{Div}}_{L^1}(X)/\hat{c}_X^{-1}(\widehat{R}^1(X)), \\ \widehat{\operatorname{Pic}}_D(X) = \widehat{\operatorname{Div}}_D(X)/\hat{c}_X^{-1}(\widehat{R}^1(X)). \end{cases} \] Note that if $X$ is regular, then \[ \widehat{\operatorname{Pic}}(X) = \widehat{\operatorname{CH}}^1(X), \quad \widehat{\operatorname{Pic}}_{L^1}(X) = \widehat{\operatorname{CH}}_{L^1}^1(X) \quad\text{and}\quad \widehat{\operatorname{Pic}}_D(X) = \widehat{\operatorname{CH}}_D^1(X). \] Let $(E, h)$ be a Hermitian vector bundle on $X$. Then, by virtue of \cite[Theorem~4]{GSRR}, we have a cap product of $\widehat{\operatorname{ch}}(E, h)$ on $\widehat{\operatorname{CH}}^{*}(X)_{{\mathbb{Q}}}$, i.e., a homomorphism $\widehat{\operatorname{CH}}^{*}(X)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}^{*}(X)_{{\mathbb{Q}}}$ given by $x \mapsto \widehat{\operatorname{ch}}(E, h) \cap x$ for $x \in \widehat{\operatorname{CH}}^{*}(X)_{{\mathbb{Q}}}$. In the same way as before, we can see that the above homomorphism extends to \[ \widehat{\operatorname{CH}}_D^{*}(X)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}_D^{*}(X)_{{\mathbb{Q}}} \quad\text{and}\quad \widehat{\operatorname{CH}}_{L^1}^{*}(X)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}_{L^1}^{*}(X)_{{\mathbb{Q}}} \] as follows. If $(Z, g) \in \widehat{Z}_D^p(X)$ and $g_Z$ is a Green current of $Z$, then \[ \widehat{\operatorname{ch}}(E, h) \cap (Z, g) = \widehat{\operatorname{ch}}(E, h) \cap (Z, g_Z) + a(\operatorname{ch}(E, h) \wedge (g - g_Z)). \] In particular, we have intersection pairings \[ \widehat{\operatorname{Pic}}(X)_{{\mathbb{Q}}} \otimes \widehat{\operatorname{CH}}_D^{p}(X)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}_D^{p+1}(X)_{{\mathbb{Q}}} \quad\text{and}\quad \widehat{\operatorname{Pic}}(X)_{{\mathbb{Q}}} \otimes \widehat{\operatorname{CH}}_{L^1}^{p}(X)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}_{L^1}^{p+1}(X)_{{\mathbb{Q}}}. \] For simplicity, the cap product ``$\cap$'' is denoted by the dot ``$\cdot$''. Note that \[ \widehat{\operatorname{Pic}}(X)_{{\mathbb{Q}}} \otimes \widehat{\operatorname{CH}}_D^{p}(X)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}_D^{p+1}(X)_{{\mathbb{Q}}} \] is actually defined by \[ (D, g) \cdot (Z, f) = (D \cdot Z, g \wedge \delta_Z + \omega(g) \wedge f) \] if $D$ and $Z$ intersect properly. Then, we have the following projection formula. \begin{Proposition} \label{prop:projection:formula:line:bundle} Let $f : X \to Y$ be a proper morphism of arithmetic varieties. Let $(L, h)$ be a Hermitian line bundle on $Y$, and $z \in \widehat{\operatorname{CH}}_D^p(X)$. Then \[ f_*(\widehat{{c}}_1 (f^*L, f^*h) \cdot z) = \widehat{{c}}_1(L, h) \cdot f_*(z). \] \end{Proposition} {\sl Proof.}\quad Let $(Z, g)$ be a representative of $z$. Clearly, we may assume that $Z$ is reduced and irreducible. We set $T = f(Z)$ and $\pi = \rest{f}{Z} : Z \to T$. Let $s$ be a rational section of $\rest{L}{T}$. Then, $\pi^*(s)$ gives rise to a rational section of $\rest{f^*(L)}{Z} = \pi^* \left( \rest{L}{T} \right)$. Thus, $\widehat{{c}}_1 (f^*L, f^*h) \cdot z$ can be represented by \[ \left( \operatorname{div}(\pi^*(s)), \left[ -\log \pi^*\left( \rest{h}{T} \right)(\pi^*(s), \pi^*(s)) \right] + c_1(f^*L, f^*h) \wedge g \right), \] where $\left[ -\log \pi^*\left( \rest{h}{T} \right)(\pi^*(s), \pi^*(s)) \right]$ is the current given by \[ \left[ -\log \pi^*\left( \rest{h}{T} \right)(\pi^*(s), \pi^*(s)) \right](\phi) = \int_{Z({\mathbb{C}})} \left( - \log \pi^*\left( \rest{h}{T} \right)(\pi^*(s), \pi^*(s)) \right) \phi. \] If we set \[ \deg(\pi) = \begin{cases} 0 & \text{if $\dim T < \dim Z$} \\ \deg(Z \to T) & \text{if $\dim T = \dim Z$,} \end{cases} \] then \begin{align*} \int_{Z({\mathbb{C}})} \left( - \log \pi^*\left( \rest{h}{T} \right)(\pi^*(s), \pi^*(s)) \right) f^*(\psi) & = \int_{Z({\mathbb{C}})} \pi^* \left( \left( -\log \left( \rest{h}{T} \right) (s, s) \right) \psi \right) \\ & = \deg(\pi) \int_{T({\mathbb{C}})} \left( -\log \left( \rest{h}{T} \right) (s, s) \right) \psi \end{align*} for a $C^{\infty}$-form $\psi$ on $Y({\mathbb{C}})$. Thus, we have \[ f_* \left[ -\log \pi^*\left( \rest{h}{T} \right)(\pi^*(s), \pi^*(s)) \right] = \deg(\pi) \left[ -\log \left( \rest{h}{T} \right) (s, s) \right]. \] Therefore, \begin{align*} f_*(\widehat{{c}}_1 (f^*L, f^*h) \cdot z) & = \left( \deg(\pi) \operatorname{div}(s), \deg(\pi) \left[ -\log \left( \rest{h}{T} \right) (s, s) \right] + c_1(L, h) \wedge f_*(g) \right) \\ & = \widehat{{c}}_1 (L, h) \cdot (\deg(\pi) T, f_*(g)) = \widehat{{c}}_1 (L, h) \cdot f_*(z). \end{align*} Hence, we get our proposition. \QED Let $Z$ be a quasi-projective integral scheme over ${\mathbb{Z}}$. Then, by virtue of Hironaka's resolution of singularities \cite{Hiro}, there is a proper birational morphism $\mu : Z' \to Z$ of quasi-projective integral schemes over ${\mathbb{Z}}$ such that $Z'_{{\mathbb{Q}}}$ is non-singular. The above $\mu : Z' \to Z$ is called a {\em generic resolution of singularities of $Z$}. As a corollary of the above projection formula, we have the following proposition. \begin{Proposition} \label{prop:formula:restriction:intersection} Let $X$ be a arithmetic variety, and $\overline{L}_1 = (L_1, h_1), \ldots, \overline{L}_{n} = (L_{n}, h_{n})$ be Hermitian line bundles on $X$. Let $(Z, g)$ be an arithmetic $D$-cycle on $X$, and $Z = a_1 Z_1 + \cdots + a_r Z_r$ the irreducible decomposition as cycles. For each $i$, let $\tau_i : Z'_i \to Z_i$ be a proper birational morphism of quasi-projective integral schemes. We assume that if $Z_i$ is horizontal with respect to $X \to \operatorname{Spec}({\mathbb{Z}})$, then $\tau_i$ is a generic resolution of singularities of $Z_i$. Then, we have \[ \widehat{{c}}_1(\overline{L}_1) \cdots \widehat{{c}}_1(\overline{L}_n) \cdot (Z, g) = \sum_{i=1}^r a_i {\mu_i}_* \left( \widehat{{c}}_1(\mu_i^* \overline{L}_1) \cdots \widehat{{c}}_1(\mu_i^* \overline{L}_n) \right) + a(c_1(\overline{L}_1) \wedge \cdots \wedge c_1(\overline{L}_n) \wedge g) \] in $\widehat{\operatorname{CH}}_D^*(X)_{{\mathbb{Q}}}$, where $\mu_i$ is the composition of $Z'_i \overset{\tau_i}{\longrightarrow} Z_i \hookrightarrow X$ for each $i$. \end{Proposition} {\sl Proof.}\quad We prove this proposition by induction on $n$. First, let us consider the case $n = 1$. Clearly we may assume that $Z$ is integral, i.e., $Z = Z_1$. Let $h_1$ be the Hermitian metric of $\overline{L}_1$, and $s$ a rational section of $\rest{L_1}{Z}$. Then, \[ \left(\operatorname{div}(s), - \log (\rest{h_1}{Z})(s, s) + c_1(\overline{L}_1) \wedge g \right) = \left(\operatorname{div}(s), - \log (\rest{h_1}{Z})(s, s) \right) + a(c_1(\overline{L}_1) \wedge g) \] is a representative of $\widehat{{c}}_1(\overline{L}_1) \cdot (Z, g)$. Moreover, \[ \left(\operatorname{div}(\tau_1^*(s)), - \log \tau_1^*(\rest{h_1}{Z})(\tau_1^*(s), \tau_1^*(s)) \right) \] is a representative of $\widehat{{c}}_1(\mu_1^* \overline{L}_1)$. Hence, we have our assertion in the case $n=1$ because \[ \left({\mu_1}_* (\operatorname{div}(\tau_1^*(s)), - \log \tau_1^*(\rest{h_1}{Z})(\tau_1^*(s), \tau_1^*(s)) \right) = (\operatorname{div}(s), - \log (\rest{h_1}{Z})(s, s)). \] Thus, we may assume that $n > 1$. Therefore, using Proposition~\ref{prop:projection:formula:line:bundle} and hypothesis of induction, \begin{align*} \widehat{{c}}_1(\overline{L}_1) \cdots \widehat{{c}}_1(\overline{L}_n) \cdot (Z, g) & = \widehat{{c}}_1(\overline{L}_1) \cdot \left( \widehat{{c}}_1(\overline{L}_2) \cdots \widehat{{c}}_1(\overline{L}_n) \cdot (Z, g) \right) \\ & = \sum_{i=1}^r a_i \widehat{{c}}_1(\overline{L}_1) {\mu_i}_* \left( \widehat{{c}}_1(\mu_i^* \overline{L}_2) \cdots \widehat{{c}}_1(\mu_i^* \overline{L}_n) \right) + \\ & \qquad\qquad\qquad\qquad \widehat{{c}}_1(\overline{L}_1) a(c_1(\overline{L}_2) \wedge \cdots \wedge c_1(\overline{L}_n) \wedge g) \\ & = \sum_{i=1}^r a_i {\mu_i}_* \left( \widehat{{c}}_1(\mu_i^* \overline{L}_1) \cdots \widehat{{c}}_1(\mu_i^* \overline{L}_n) \right) + a(c_1(\overline{L}_1) \wedge \cdots \wedge c_1(\overline{L}_n) \wedge g). \end{align*} \QED \subsection{Injectivity of $i^*$} \setcounter{Theorem}{0} Let $X$ be an arithmetic variety, $U$ a non-empty Zariski open set of $X$, and $i : U \to X$ the inclusion map. Then, there is a natural homomorphism \[ i^* : \widehat{Z}_{L^1}^1(X) \to \widehat{Z}_{L^1}^1(U) \] given by $i^*(D, g) = (\rest{D}{U}, \rest{g}{U({\mathbb{C}})})$. Since $i^* \left( \widehat{(f)} \right) = \widehat{(\rest{f}{U})}$ for any non-zero rational functions $f$ on $X$, the above induces the homomorphism \[ i^* : \widehat{\operatorname{CH}}_{L^1}^1(X) \to \widehat{\operatorname{CH}}_{L^1}^1(U). \] Then, we have the following useful lemma. \begin{Lemma} \label{lem:criterion:linear:equiv:B:cycle} If $X \setminus U$ does not contain any irreducible components of fibers of $X \to \operatorname{Spec}({\mathbb{Z}})$, then \[ i^* : \widehat{\operatorname{CH}}_{L^1}^1(X) \to \widehat{\operatorname{CH}}_{L^1}^1(U). \] is injective. In particular, $i^* : \widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}_{L^1}^1(U)_{{\mathbb{Q}}}$ is injective. \end{Lemma} {\sl Proof.}\quad Suppose that $i^*(\alpha) = 0$ for some $\alpha \in \widehat{\operatorname{CH}}_{L^1}^1(X)$. Let $(D,g) \in \widehat{Z}_{L^1}^1(X)$ be a representative of $\alpha$. Since $i^*(\alpha) = 0$, there is a non-zero rational function $f$ on $X$ with \[ (\rest{D}{U}, \rest{g}{U({\mathbb{C}})}) = (\rest{(f)}{U}, \rest{-[\log |f|^2]}{U({\mathbb{C}})}). \] Pick up $\phi \in L^1_{\operatorname{loc}}(X({\mathbb{C}}))$ with $g = [\phi]$. Then, the above implies that $\rest{[\phi]}{U({\mathbb{C}})} = \rest{-[\log |f|^2]}{U({\mathbb{C}})}$. Thus, $\phi = - \log |f|^2 \ (\operatorname{a.e.})$. Therefore, we have \addtocounter{Claim}{1} \begin{equation} \label{eqn:1:lem:criterion:linear:equiv:B:cycle} g = [\phi] = - [\log |f|^2]. \end{equation} Here, $dd^c(g) + \delta_{D({\mathbb{C}})} = [h]$ for some $h \in L^1_{\operatorname{loc}}(\Omega_{X({\mathbb{C}})}^{1,1})$ and $dd^c(-[\log |f|^2]) + \delta_{(f)({\mathbb{C}})} = 0$. Thus, by (\ref{eqn:1:lem:criterion:linear:equiv:B:cycle}), $\delta_{D({\mathbb{C}})} - \delta_{(f)({\mathbb{C}})} =[h]$. This shows us that $h = 0 \ (\operatorname{a.e.})$ over $X({\mathbb{C}}) \setminus \left( \operatorname{Supp}(D({\mathbb{C}})) \cup \operatorname{Supp}((f)({\mathbb{C}})) \right)$. Hence $h = 0 \ (\operatorname{a.e.})$ on $X({\mathbb{C}})$. Therefore, we have $D({\mathbb{C}}) = (f)({\mathbb{C}})$, which implies $D = (f)$ on $X_{{\mathbb{Q}}}$. Thus, $D - (f)$ is a linear combination of irreducible divisors lying on finite fibers. On the other hand, $D = (f)$ on $U$ and $X \setminus U$ does not contain any irreducible components of fibers. Therefore, $D = (f)$. Hence $\alpha = 0$ because $(D, g) = \widehat{(f)}$. \QED \section{Weakly positive arithmetic divisors} \subsection{Generalized metrics} \setcounter{Theorem}{0} \label{subsec:gen:metric} Let $X$ be a smooth algebraic scheme over ${\mathbb{C}}$ and $L$ a line bundle on $X$. We say $h$ is {\em a generalized metric on $L$} if there is a $C^{\infty}$ Hermitian metric $h_0$ of $L$ over $X$ and $\varphi \in L^1_{\operatorname{loc}}(X)$ with $h = e^{\varphi}h_0$. To see when a metric of a line bundle defined over a dense Zariski open set extends to a generalized metric, the following lemma is useful. \begin{Lemma} \label{lem:criterion:gen:metric} Let $X$ be a smooth algebraic variety over ${\mathbb{C}}$ and $L$ a line bundle on $X$. Let $U$ be a non-empty Zariski open set of $X$ and $h$ a $C^{\infty}$ Hermitian metric of $L$ over $U$. We fix a non-zero rational section $s$ of $L$. Then, $h$ extends to a generalized metric of $L$ on $X$ if and only if $\log h(s, s) \in L^1_{\operatorname{loc}}(X)$. \end{Lemma} {\sl Proof.}\quad If $h$ extends to a generalized metric of $L$ on $X$, then $\log h(s, s) \in L^1_{\operatorname{loc}}(X)$ by the definition of generalized metrics. Conversely, we assume that $\log h(s, s) \in L^1_{\operatorname{loc}}(X)$. Let $h_0$ be a $C^{\infty}$ Hermitian metric of $L$ over $X$. Here we consider the function $\phi$ given by \[ \phi = \frac{h(s,s)}{h_0(s, s)}. \] Let $y \in U$ and $\omega$ be a local frame of $L$ around $y$. If we set $s = f \omega$ for some meromorphic function $f$ around $y$, then \[ \phi = \frac{h(s,s)}{h_0(s, s)} = \frac{|f|^2 h(\omega, \omega)}{|f|^2 h_0(\omega, \omega)} = \frac{h(\omega, \omega)}{h_0(\omega, \omega)}. \] This shows us that $\phi$ is a positive $C^{\infty}$ function on $U$ and $h = \phi h_0$ over $U$. On the other hand, \[ \log \phi = \log h(s, s) - \log h_0(s, s). \] Here since $\log h(s, s), \log h_0(s, s) \in L^1_{\operatorname{loc}}(X)$, we have $\log \phi \in L^1_{\operatorname{loc}}(X)$. Thus, if we set $\varphi = \log \phi$, then $\varphi \in L^1_{\operatorname{loc}}(X)$ and $h = e^{\varphi} h_0$. \QED \subsection{Arithmetic $D$-divisors and generalized metrics} \label{subsec:arith:div:gen:metric} \setcounter{Theorem}{0} Let $X$ be an arithmetic variety, $L$ a line bundle on $X$, and $h$ a generalized metric of $L$ on $X({\mathbb{C}})$ with $F_{\infty}^*(h) = \overline{h} \ (\operatorname{a.e.})$. We would like to define $\widehat{{c}}_1(L, h)$ as an element of $\widehat{\operatorname{CH}}_D^1(X)$. Let $s, s'$ be two non-zero rational sections of $L$, and $f$ a non-zero rational function on $X$ with $s' = fs$. Then, it is easy to see that \[ (\operatorname{div}(s'), [-\log h(s', s')]) = (\operatorname{div}(s), [-\log h(s,s)]) + \widehat{(f)} \] in $\widehat{Z}_D^1(X)$. Thus, we may define $\widehat{{c}}_1(L, h)$ as the class of $(\operatorname{div}(s), [-\log h(s,s)])$ in $\widehat{\operatorname{CH}}_D^1(X)$. Let us consider the homomorphism \[ \omega : \widehat{Z}_D^p(X) \to D^{p,p}(X({\mathbb{C}})) \] given by $\omega(Z, g) = dd^c(g) + \delta_{Z({\mathbb{C}})}$. Since $\omega\left( \widehat{R}^{p}(X) \right) = 0$, the above homomorphism induces the homomorphism $\widehat{\operatorname{CH}}_D^p(X) \to D^{p,p}(X({\mathbb{C}}))$. Hence, we get the homomorphism $\widehat{\operatorname{CH}}_D^p(X)_{{\mathbb{Q}}} \to D^{p,p}(X({\mathbb{C}}))$ because $D^{p,p}(X({\mathbb{C}}))$ has no torsion. By abuse of notation, we denote this homomorphism by $\omega$. \begin{Proposition} \label{prop:B:cycle:produce:hermitian:line:bundle} Let $X$ be an arithmetic variety, $(Z, [\phi]) \in \widehat{\operatorname{Div}}_D(X)$ with $\phi \in L^1_{\operatorname{loc}}(X({\mathbb{C}}))$, and $1$ a rational section of ${\mathcal{O}}_X(Z)$ with $\operatorname{div}(1) = Z$. Then, there is a unique generalized metric $h$ of ${\mathcal{O}}_X(Z)$ such that $F_{\infty}^*(h) = \overline{h} \ (\operatorname{a.e.})$ and $[-\log h(1, 1)] = [\phi]$. \textup{(}Here uniqueness of $h$ means that if $h'$ is another generalized metric with the same property, then $h = h' \ (\operatorname{a.e.})$.\textup{)} Moreover, $\omega(Z, [\phi])$ is $C^{\infty}$ around $x \in X({\mathbb{C}})$ if and only if $h$ is $C^{\infty}$ around $x$. We denote this line bundle $({\mathcal{O}}_X(Z), h)$ with the generalized metric $h$ by ${\mathcal{O}}_Z((Z, [\phi]))$. With this notation, for $(Z_1, [\phi_1]), (Z_2, [\phi_2]) \in \widehat{\operatorname{Div}}_D(X)$ with $\phi_1, \phi_2 \in L^1_{\operatorname{loc}}(X({\mathbb{C}}))$, if $(Z_1, [\phi_1]) \sim (Z_2, [\phi_2])$, then ${\mathcal{O}}_X((Z_1, [\phi_1]))$ is isometric to ${\mathcal{O}}_X((Z_2, [\phi_2]))$ at every point around which $\omega(Z_1, [\phi_1]) = \omega(Z_2, [\phi_2])$ is a $C^{\infty}$ form. \end{Proposition} {\sl Proof.}\quad First, let us see uniqueness. Let $h$ and $h'$ be generalized metrics of ${\mathcal{O}}_X(Z)$ with $[-\log h(1, 1)] = [-\log h'(1, 1)] = [\phi]$. Take $a \in L^1_{\operatorname{loc}}(X({\mathbb{C}}))$ with $h' = e^a h$. Then, by our assumption, $a = 0 \ (\operatorname{a.e.})$. Thus, $h = h' \ (\operatorname{a.e.})$. Pick up an arbitrary point $x \in X({\mathbb{C}})$. Let $s$ be a local basis of ${\mathcal{O}}_X(Z)$ around $x$. Then, there is a non-zero rational rational function $f$ on $X({\mathbb{C}})$ with $1 = f s$. Let us consider \[ \exp(-\phi - \log |f|^2) \] around $x$. Let $s'$ be a another local basis of ${\mathcal{O}}_X(Z)$ around $x$. We set $s' = us$ and $1 = f' s'$. Then, \[ \exp(-\phi - \log |f'|^2) = \exp(-\phi - \log |f/u|^2) = |u|^2 \exp(-\phi - \log |f|^2), \] which means that if we define the generalized metric $h$ by \[ h(s, s) = \exp(-\phi - \log |f|^2), \] then $h$ is patched globally, and $h$ is a generalized metric by Lemma~\ref{lem:criterion:gen:metric}. Moreover, \[ -\log h(1,1) = -\log h(fs, fs) = -\log \left( |f|^2 h(s,s) \right) = \phi. \] Here, since $F_{\infty}^*(\phi) = \phi \ (\operatorname{a.e.})$, we can see $F_{\infty}^*(h) = \overline{h} \ (\operatorname{a.e.})$. Thus, we can construct our desired metric. We set $b = \omega(Z, [\phi]) \in D^{1,1}(X({\mathbb{C}}))$. Then, since $1 = fs$ around $x$, we have $Z = (f)$ around $x$. Thus, since $dd^c([\phi]) + \delta_{Z({\mathbb{C}})} = b$ and $dd^c(-[\log |f|^2]) + \delta_{(f)} = 0$, \[ dd^c(-[\phi + \log |f|^2]) = \delta_{Z({\mathbb{C}})} - b - \delta_{(f)} = - b \] around $x$. Therefore, \begin{align*} \text{$h$ is $C^{\infty}$ around $x$} & \Longleftrightarrow \text{$-\phi - \log |f|^2$ is $C^{\infty}$ around $x$} \\ & \Longleftrightarrow \text{$dd^c(-[\phi + \log |f|^2])$ is $C^{\infty}$ around $x$} \qquad \text{($\because$ \cite[Theorem~1.2.2]{GSArInt})}\\ & \Longleftrightarrow \text{$b$ is $C^{\infty}$ around $x$} \end{align*} Finally, let us consider the last assertion. By our assumption, there is a rational function $z$ on $X$ such that \[ (Z_1, [\phi_1]) = (Z_2, [\phi_2]) + \widehat{(z)}. \] Then, $Z_1 = Z_2 + (z)$ and $\phi_1 = \phi_2 - \log |z|^2$. Let us consider the homomorphism $\alpha : {\mathcal{O}}_X(Z_1) \to {\mathcal{O}}_X(Z_2)$ defined by $\alpha(s) = zs$. Then, $\alpha$ is an isomorphism. Let $1$ be the unit in the rational function field of $X$. Then, $1$ gives rise to canonical rational sections of ${\mathcal{O}}_X(Z_1)$ and ${\mathcal{O}}_X(Z_2)$. Let $x$ be a point of $X({\mathbb{C}})$ such that $\omega(Z_1, [\phi_1])$ is $C^{\infty}$ around $x$, and $s$ a local basis of ${\mathcal{O}}_X(Z_1)$ around $x$. Then, $\alpha(s) = zs$ is a local basis of ${\mathcal{O}}_X(Z_2)$ around $x$. Choose a rational function $f$ with $1 = fs$. Then, $1 = z^{-1}f\alpha(s)$. Thus, if $h_1$ and $h_2$ are metrics of ${\mathcal{O}}_X((Z_1, [\phi_1]))$ and ${\mathcal{O}}_X((Z_2, [\phi_2]))$ respectively, then \[ h_1(s, s) = \exp(-\phi_1 -\log |f|^2) = \exp( -\phi_2 - \log |z^{-1}f|^2) = h_2(\alpha(s), \alpha(s)) \] Hence, $\alpha$ is an isometry. \QED \subsection{Semi-ampleness and small sections} \setcounter{Theorem}{0} Let $X$ be an arithmetic variety and $S$ a subset of $X({\mathbb{C}})$. We set \[ \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}} = \{ \alpha \in \widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}} \mid \text{$\omega(\alpha)$ is $C^{\infty}$ around $z$ for all $z \in S$} \}. \] In the same way, we can define $\widehat{\operatorname{CH}}_{L^1}^1(X;S)$, $\widehat{Z}_{L^1}^1(X;S)$, $\widehat{Z}_{L^1}^1(X;S)_{{\mathbb{Q}}}$, $\widehat{\operatorname{Div}}_{L^1}(X;S)$, $\widehat{\operatorname{Div}}_{L^1}(X;S)_{{\mathbb{Q}}}$, $\widehat{\operatorname{Pic}}_{L^1}(X;S)$ and $\widehat{\operatorname{Pic}}_{L^1}(X;S)_{{\mathbb{Q}}}$. Let $x$ be a closed point of $X_{{\mathbb{Q}}}$. An element $\alpha$ of $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ is said to be {\em semi-ample at $x$ with respect to $S$} if there are a positive integer $n$ and $(E, g) \in \widehat{Z}_{L^1}^1(X;S)$ with the following properties: \begin{enumerate} \renewcommand{\labelenumi}{(\alph{enumi})} \item $E$ is effective and $x \not\in \operatorname{Supp}(E)$. \item $g(z) \geq 0$ for each $z \in S$. (Note that $g(z)$ might be $\infty$.) \item $n \alpha$ coincides with $(E,g)$ in $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$. \end{enumerate} We remark that $\alpha \in \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ by the condition (c). Moreover, it is easy to see that if $\alpha_1$ and $\alpha_2$ are semi-ample at $x$ with respect to $S$, so is $\lambda \alpha_1 + \mu \alpha_2$ for all non-negative rational numbers $\lambda$ and $\mu$. In terms of the natural action of $\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ on $X(\overline{{\mathbb{Q}}})$, we can consider the orbit $O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(x)$ of $x$. If we fix an embedding $\overline{{\mathbb{Q}}} \to {\mathbb{C}}$, we have an injection $X(\overline{{\mathbb{Q}}}) \to X({\mathbb{C}})$. It is easy to see that the image of $O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(x)$ does not depend on the choice of the embedding $\overline{{\mathbb{Q}}} \to {\mathbb{C}}$. By abuse of notation, we denote this image by $O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(x)$. Then, $O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(x)$ is one of the examples of $S$. \medskip Let $U$ be a Zariski open set of $X$, and $F$ a coherent ${\mathcal{O}}_X$-module such that $F$ is locally free over $U$. Let $h_F$ be a $C^{\infty}$ Hermitian metric of $F$ over $U({\mathbb{C}})$. We assume that $S \subseteq U({\mathbb{C}})$. For a closed point $x$ of $U_{{\mathbb{Q}}}$, we say $(F, h_F)$ is {\em generated by small sections at $x$ with respect to $S$} if there are sections $s_1, \ldots, s_n \in H^0(X, F)$ such that $F_x$ is generated by $s_1, \ldots, s_n$, and that $h_F(s_i, s_i)(z) \leq 1$ for all $1 \leq i \leq n$ and $z \in S$. \begin{Proposition} \label{prop:comparion:semiample:gen:small:sec} We assume that $S \subseteq U({\mathbb{C}})$. For an element $(Z, g)$ of $\widehat{\operatorname{Div}}_{L^1}(X;S)$, $(Z,g)$ is semi-ample at $x$ with respect to $S$ if and only if there is a positive integer $n$ such that ${\mathcal{O}}_X(n(Z,g))$ is generated by small sections at $x$ with respect to $S$. \end{Proposition} {\sl Proof.}\quad First, we assume that $(Z,g)$ is semi-ample at $x$ with respect to $S$. Then, there is $(E, f) \in \widehat{Z}_{L^1}^1(X;S)$ and a positive integer $n$ such that $n(Z, g) \sim (E, f)$, $E$ is effective, $x \not\in \operatorname{Supp}(E)$, and $f(z) \geq 0$ for each $z \in S$. Note that $E$ is a Cartier divisor. Then, by Proposition~\ref{prop:B:cycle:produce:hermitian:line:bundle}, ${\mathcal{O}}_X(n(Z,g)) \simeq {\mathcal{O}}_X((E, f))$. Moreover, if $h$ is the metric of ${\mathcal{O}}_X((E, f))$ and $1$ is the canonical section of ${\mathcal{O}}_X(E)$ with $\operatorname{div}(1) = E$, then $-\log(h(1,1)) = f$. Here $f(z) \geq 0$ for each $z \in S$. Thus, $h(1,1)(z) \leq 1$ for each $z \in S$. Therefore, ${\mathcal{O}}_X((E, f))$ is generated by small sections at $x$ with respect to $S$. Next we assume that ${\mathcal{O}}_X(n(Z,g))$ is generated by small sections at $x$ with respect to $S$ for some positive integer $n$. Then, there is a section $s$ of ${\mathcal{O}}_X(nZ)$ such that $h(s,s)(z) \leq 1$ for each $z \in S$. Thus, if we set $E = \operatorname{div}(s)$ and $f = -\log h(s,s)$, then we can see $(Z,g)$ is semi-ample at $x$ with respect to $S$. \QED \begin{Proposition} \label{prop:finite:sup:imply:globalsup} Let $U$ be a Zariski open set of $X$, and $L$ a line bundle on $X$. Let $h$ be a $C^{\infty}$ Hermitian metric of $L$ over $U({\mathbb{C}})$. Fix a closed point $x$ of $U_{{\mathbb{Q}}}$. If $X$ is projective over ${\mathbb{Z}}$, then the followings are equivalent. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item $(L, h)$ is generated by small sections at $x$ with respect to $U({\mathbb{C}})$. \item $(L, h)$ is generated by small sections at $x$ with respect to any finite subsets $S$ of $U({\mathbb{C}})$. \end{enumerate} \end{Proposition} {\sl Proof.}\quad Clearly, (1) implies (2). So we assume (2). First of all, we can easily find $z_1, \ldots, z_n \in U({\mathbb{C}})$ such that, for any $s \in H^0(X({\mathbb{C}}), L_{{\mathbb{C}}})$, if $s(z_1) = \cdots = s(z_n) = 0$, then $s = 0$. Thus, if we set \[ \Vert s \Vert = \sqrt{h(s, s)(z_1)} + \cdots + \sqrt{h(s, s)(z_n)} \] for each $s \in H^0(X({\mathbb{C}}), L_{{\mathbb{C}}})$, then $\Vert \ \Vert$ gives rise to a norm on $H^0(X({\mathbb{C}}), L_{{\mathbb{C}}})$. Here we set \[ B_z = \{ s \in H^0(X,L) \mid \text{$h(s,s)(z) \leq 1$} \} \] for each $z \in U({\mathbb{C}})$. Then, since $H^0(X, L)$ is a discrete subgroup of $H^0(X({\mathbb{C}}), L_{{\mathbb{C}}})$, $\bigcap_{i=1}^{n} B_{z_i}$ is a finite set. Thus, adding finite points $z_{n+1}, \ldots, z_{N} \in U({\mathbb{C}})$ to $z_1, \ldots, z_n$ if necessary, we have \[ \bigcap_{z \in U({\mathbb{C}})} B_z = \bigcap_{i=1}^N B_{z_i}. \] By our assumption, there is a section $s \in H^0(X, L)$ such that $s(x) \not= 0$ and $h(s, s)(z_i) \leq 1$ for all $i=1, \ldots, N$. Then, $s \in \bigcap_{i=1}^N B_{z_i} = \bigcap_{z \in U({\mathbb{C}})} B_z$. Thus, we get (2). \QED \begin{comment} \begin{Example} We note that even if a Hermitian line bundle $\overline{L}=(L,h_L)$ on an arithmetic variety $X$ is generated by locally small sections at every $x \in X({\mathbb{C}})$, it does not necessarily have a non-zero global section whose sup-norm is less than or equal to $1$. For example, let $X = {\mathbb{P}}^1_{{\mathbb{Z}}}$ and $L = {\mathcal{O}}(1)$. We shall give a metric on $L$ as follows. First let $U$ be a neighborhood of a point $(4:3) \in {\mathbb{P}}^1({\mathbb{C}})$ such that \[ 0 < \frac{\vert x_0-x_1 \vert}{\sqrt{\vert x_0 \vert^2 +\vert x_1 \vert^2 }} \leq \frac{1}{4} \] for every $(x_0:x_1) \in U$. Let $f$ be a smooth positive function on ${\mathbb{P}}^1({\mathbb{C}})$ such that $1 \leq f \leq 4$, $f((4:3)) = 4$ and $f = 1$ on the compliment of $U$. We define a metric $\Vert\cdot\Vert$ on $L$ by \[ \Vert s \Vert (x) = f(x) \frac{\vert ax_0 +bx_1 \vert}{\sqrt{\vert x_0 \vert^2 +\vert x_1 \vert^2 }} \] for $s = aX +bY \; (a,b \in {\mathbb{Z}})$ and $x=(x_0:x_1) \in {\mathbb{P}}^1({\mathbb{C}})$. Now we shall show that, for this metric, $L$ is generated by locally small sections for every $x \in {\mathbb{P}}^1({\mathbb{C}})$. If $x \in U$, then we take the global section $x_0-x_1$. In this case, by the definition of $U$ and $f$, $0 < \Vert x_0-x_1 \Vert (x) \leq 1$. If $x$ is in the compliment of $U$, then we take the global section $x_0$ or $x_1$. One of them does not vanish at $x$, and it is easy to see that the value is less than or equal to $1$ since $f(x) = 1$. Next, let us see that \[ \sup_{x \in {\mathbb{P}}^1({\mathbb{C}})} \Vert s \Vert(x) >1 \] for any non-zero global sections $s = aX_0 +bX_1 \; (a,b \in {\mathbb{Z}})$. If $(a,b) = (\pm 1 , 0)$, then \[ \sup_{x \in {\mathbb{P}}^1({\mathbb{C}})} \Vert s \Vert(x) \geq \Vert s \Vert ((4:3)) = 4 \cdot \frac {4}{5} > 1. \] If $(a,b) = (0, \pm 1)$, then \[ \sup_{x \in {\mathbb{P}}^1({\mathbb{C}})} \Vert s \Vert(x) \geq \Vert s \Vert ((4:3)) = 4 \cdot \frac {3}{5} > 1. \] If $(a,b) \not= (\pm 1 , 0)$ or $(0, \pm 1)$, then \[ \sup_{x \in {\mathbb{P}}^1({\mathbb{C}})} \Vert s \Vert(x) \geq \Vert s \Vert ((a : b)) \geq \sqrt{a^2 + b^2} >1. \] \end{Example} \end{comment} \subsection{Restriction to arithmetic curves} \setcounter{Theorem}{0} Let $X$ be an arithmetic variety, $S$ a subset of $X({\mathbb{C}})$, $x$ a closed point of $X_{{\mathbb{Q}}}$, $K$ the residue field of $x$, and $O_K$ the ring of integers in $K$. We assume that the orbit of $x$ by $\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ is contained in $S$, namely, $O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(x) \subseteq S$, and that the canonical morphism $\operatorname{Spec}(K) \to X$ induced by $x$ extends to $\tilde{x} : \operatorname{Spec}(O_K) \to X$. \begin{Proposition} \label{prop:homo:ChowB:to:Chow:on:OK} There is a natural homomorphism \[ \tilde{x}^* : \widehat{\operatorname{Pic}}_{L^1}(X;S)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}^1(\operatorname{Spec}(O_K))_{{\mathbb{Q}}} \] such that the restriction of $\tilde{x}^*$ to $\widehat{\operatorname{Pic}}(X)_{{\mathbb{Q}}}$ coincides with the usual pull-back homomorphism. \end{Proposition} {\sl Proof.}\quad Let $\alpha \in \widehat{\operatorname{Pic}}_{L^1}(X;S)_{{\mathbb{Q}}}$. Choose $(Z, g) \in \widehat{\operatorname{Div}}_{L^1}(X;S)$ and a positive integer $e$ such that the class of $(1/e)(Z, g)$ in $\widehat{\operatorname{Pic}}_{L^1}(X;S)_{{\mathbb{Q}}}$ coincides with $\alpha$. We would like to define $\tilde{x}^*(\alpha)$ by \[ (1/e) \widehat{{c}}_1\left( \tilde{x}^*({\mathcal{O}}_X((Z, g))) \right). \] For this purpose, we need to check that the above does not depend on the choice $(Z, g)$ and $e$. Let $(Z', g')$ and $e'$ be another $L^1$-cycle of codimension $1$ and positive integer such that the class of $(1/e')(Z', g')$ is $\alpha$. Then, there is a positive integer $d$ such that $de'(Z, g) \sim de(Z', g')$. Thus, by Proposition~\ref{prop:B:cycle:produce:hermitian:line:bundle}, ${\mathcal{O}}_Z(de'(Z, g))$ is isometric to ${\mathcal{O}}_Z(de(Z', g'))$. Hence, \begin{align*} de' \widehat{{c}}_1\left( \tilde{x}^*({\mathcal{O}}_X((Z, g))) \right) & = \widehat{{c}}_1\left( \tilde{x}^*({\mathcal{O}}_X(de'(Z, g))) \right) \\ & = \widehat{{c}}_1\left( \tilde{x}^*({\mathcal{O}}_X(de(Z', g'))) \right) \\ & = de \widehat{{c}}_1\left( \tilde{x}^*({\mathcal{O}}_X((Z', g'))) \right). \end{align*} Therefore, \[ (1/e) \widehat{{c}}_1\left( \tilde{x}^*({\mathcal{O}}_X((Z, g))) \right) = (1/e') \widehat{{c}}_1\left( \tilde{x}^*({\mathcal{O}}_X((Z', g'))) \right). \] Thus, we can define $\tilde{x}^*$. \QED \subsection{Weak positivity of arithmetic $L^1$-divisors} \label{subsec:wp:div} \setcounter{Theorem}{0} Let $X$ be an arithmetic variety, $S$ a subset of $X({\mathbb{C}})$, and $x$ a closed point of $X_{{\mathbb{Q}}}$. Let $\alpha \in \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ and $\{ \alpha_n \}_{n=1}^{\infty}$ a sequence of elements of $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$. We say $\alpha$ is the limit of $\{ \alpha_n \}_{n=1}^{\infty}$ as $n \to \infty$, denoted by ${\displaystyle \alpha = \lim_{n \to \infty} \alpha_n}$, if there are (1) $Z_1, \ldots, Z_{l_1} \in \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$, (2) $g_1, \ldots, g_{l_2} \in L^1_{\operatorname{loc}}(X({\mathbb{C}}))$ with $a(g_j) \in \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ for all $j$, (3) sequences $\{ a_n^{1} \}_{n=1}^{\infty}, \ldots, \{ a_n^{l_1} \}_{n=1}^{\infty}$ of rational numbers, and (4) sequences $\{ b_n^{1} \}_{n=1}^{\infty}, \ldots, \{ b_n^{l_2} \}_{n=1}^{\infty}$ of real numbers with the following properties: \begin{enumerate} \renewcommand{\labelenumi}{(\alph{enumi})} \item $l_1$ and $l_2$ does not depend on $n$. \item ${\displaystyle \lim_{n \to \infty} a_n^{i} = \lim_{n \to \infty} b_n^{j} = 0}$ for all $1 \leq i \leq l_1$ and $1 \leq j \leq l_2$. \item ${\displaystyle \alpha = \alpha_n + \sum_{i=1}^{l_1} a_n^{i}Z_i + \sum_{j=1}^{l_2} a(b_n^{j} g_j)}$ in $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ for all $n$. \end{enumerate} It is easy to see that if ${\displaystyle \alpha = \lim_{n \to \infty} \alpha_n}$ and ${\displaystyle \beta = \lim_{n \to \infty} \beta_n}$ in $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$, then ${\displaystyle \alpha + \beta = \lim_{n \to \infty} (\alpha_n + \beta_n)}$. An element $\alpha$ of $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ is said to be {\em weakly positive at $x$ with respect to $S$} if it is the limit of semi-ample ${\mathbb{Q}}$-cycles at $x$ with respect to $S$, i.e., there is a sequence $\{ \alpha_n \}_{n=1}^{\infty}$ of elements of $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ such that $\alpha_n$'s are semi-ample at $x$ with respect to $S$ and ${\displaystyle \alpha = \lim_{n \to \infty} \alpha_n}$. Let $K$ be the residue field of $x$ and $O_K$ the ring of integers in $K$. We assume that $O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(x) \subseteq S$, and the canonical morphism $\operatorname{Spec}(K) \to X$ induced by $x$ extends to $\tilde{x} : \operatorname{Spec}(O_K) \to X$. Then, we have the following proposition. \begin{Proposition} \label{prop:non:negative:wp:div:via:beta} If $X$ is regular and an element $\alpha$ of $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ is weakly positive at $x$ with respect to $S$, then $\widehat{\operatorname{deg}}(\tilde{x}^*(\alpha)) \geq 0$. \end{Proposition} {\sl Proof.}\quad Take $Z_1, \ldots, Z_{l_1}$, $g_1, \ldots, g_{l_2}$, $\{ a_n^{1} \}_{n=1}^{\infty}, \ldots, \{ a_n^{l_1} \}_{n=1}^{\infty}$, $\{ b_n^{1} \}_{n=1}^{\infty}, \ldots, \{ b_n^{l_2} \}_{n=1}^{\infty}$, and $\{ \alpha_n \}_{n=1}^{\infty}$ as in the previous definition of weak positive arithmetic divisors. Then, \[ \widehat{\operatorname{deg}} ( \tilde{x}^*(\alpha) ) = \widehat{\operatorname{deg}} ( \tilde{x}^*(\alpha_n) ) + \sum_{i=1}^{l_1} a_n^{i} \widehat{\operatorname{deg}} (\tilde{x}^*(Z_i)) + \sum_{j=1}^{l_2} b_n^{j} \widehat{\operatorname{deg}}(\tilde{x}^*a(g_j)). \] Thus, since ${\displaystyle \lim_{n \to \infty} a_n^{i} = \lim_{n \to \infty} b_n^{j} = 0}$ for all $1 \leq i \leq l_1$ and $1 \leq j \leq l_2$ and $\widehat{\operatorname{deg}} ( \tilde{x}^*(\alpha_n) ) \geq 0$ for all $n$, we have $\widehat{\operatorname{deg}} ( \tilde{x}^*(\alpha) ) \geq 0$. \QED \subsection{Characterization of weak positivity} \setcounter{Theorem}{0} Let $X$ be a regular arithmetic variety, $S$ a subset of $X({\mathbb{C}})$, and $x$ a closed point of $X_{{\mathbb{Q}}}$. For an element $\alpha \in \widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}}$, we say $\alpha$ is {\em ample at $x$ with respect to $S$} if there are $(A, f) \in \widehat{Z}_{L^1}^1(X;S)$ and a positive integer $n$ such that $A$ is an effective and ample Cartier divisor on $X$, $x \not\in \operatorname{Supp}(A)$, $f(z) > 0$ for all $z \in S$, and $n \alpha$ is equal to $(A, f)$ in $\widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}}$. First, let us consider the case where $X = \operatorname{Spec}(O_K)$. \begin{Proposition} \label{prop:wp:for:curve} We assume that $X = \operatorname{Spec}(O_K)$, $x$ is the generic of $X$, and $S = X({\mathbb{C}})$. For an element $\alpha \in \widehat{\operatorname{CH}}^1(X;S)_{{\mathbb{Q}}}$, we have the following. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item $\alpha$ is ample at $x$ with respect to $S$ if and only if $\widehat{\operatorname{deg}}(\alpha) > 0$. \item $\alpha$ is weakly positive at $x$ with respect to $S$ if and only if $\widehat{\operatorname{deg}}(\alpha) \geq 0$. \end{enumerate} \end{Proposition} {\sl Proof.}\quad (1) Clearly, if $\alpha$ is ample at $x$ with respect to $S$, then $\widehat{\operatorname{deg}}(\alpha) > 0$. Conversely, we assume that $\widehat{\operatorname{deg}}(\alpha) > 0$. We take a positive integer $e$ and a Hermitian line bundle $(L, h)$ on $X$ such that $\widehat{{c}}_1(L, h) = e \alpha$. Then, $\widehat{\operatorname{deg}}(L, h) > 0$. Thus, by virtue of Riemann-Roch formula and Minkowski's theorem, there are a positive integer $n$ and a non-zero element $s$ of $L^{\otimes n}$ with $(h^{\otimes n})(s, s)(z) < 1$ for all $z \in S$. Thus, $\alpha$ is ample at $x$ with respect to $S$. \medskip (2) First, we assume that $\alpha$ is weakly positive at $x$ with respect to $S$. Then, by Proposition~\ref{prop:non:negative:wp:div:via:beta}, $\widehat{\operatorname{deg}}(\alpha) \geq 0$. Next, we assume that $\widehat{\operatorname{deg}}(\alpha) \geq 0$. Let $\beta$ be an element of $\widehat{\operatorname{CH}}^1(X;S)_{{\mathbb{Q}}}$ such that $\beta$ is ample at $x$ with respect to $S$. Then, for any positive integer $n$, $\widehat{\operatorname{deg}}(\alpha + (1/n) \beta) > 0$. Thus, $\alpha + (1/n)\beta$ is ample at $x$ with respect to $S$ by (1). Hence, $\alpha$ is weakly positive at $x$ with respect to $S$. \QED Before starting a general case, let us consider the following lemma. \begin{Lemma} \label{lem:make:semiample:by:ample} We assume that $S$ is compact. Let $\alpha$ be an element of $\widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}}$ such that $\alpha$ is ample at $x$ with respect to $S$. Then, we have the following. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item Let $\beta$ be an element of $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$. Then, there is a positive number $\epsilon_0$ such that $\alpha + \epsilon \beta$ is semi-ample at $x$ with respect to $S$ for all rational numbers $\epsilon$ with $|\epsilon| \leq \epsilon_0$. \item Let $g$ be a locally integrable function on $X({\mathbb{C}})$ with $a(g) \in \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$. Then, there is a positive number $\epsilon_0$ such that $\alpha + a(\epsilon g)$ is semi-ample at $x$ with respect to $S$ for all real numbers $\epsilon$ with $|\epsilon| \leq \epsilon_0$. \end{enumerate} \end{Lemma} {\sl Proof.}\quad (1) First, we claim that there is a positive number $t_0$ such that $t \alpha + \beta$ is semi-ample at $x$ with respect to $S$ for all rational numbers $t \geq t_0$. Let us choose $(A, f) \in \widehat{Z}_{L^1}^1(X;S)$ and a positive integer $n_0$ such that $A$ is an effective and ample Cartier divisor on $X$, $x \not\in \operatorname{Supp}(A)$, $f(z) > 0$ for all $z \in S$, and $n_0 \alpha$ is equal to $(A, f)$ in $\widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}}$. Moreover, we choose $(D, g) \in \widehat{Z}_{L^1}^1(X;S)$ and a positive integer $e$ such that $e \beta$ is equal to $(D, g)$ in $\widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}}$. Since $A$ is ample, there is a positive integer $n_1$ such that ${\mathcal{O}}_X(n_1 A) \otimes {\mathcal{O}}_X(D)$ is generated by global sections at $x$. Thus, there are $(Z, h) \in \widehat{Z}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ such that $Z$ is effective, $x \not\in \operatorname{Supp}(Z)$ and $(Z, h) \sim n_1(A, f) + (D, g)$. We would like to find a positive integer $n_2$ with $n_2 f(z) + h(z) \geq 0$ for all $z \in S$. Let $U$ be an open set of $X({\mathbb{C}})$ such that $S \subseteq U$, and $\omega(A, f)$ and $\omega(Z,h)$ are $C^{\infty}$ over $U$. We set $\phi = \exp(-f)$ and $\psi = \exp(-h)$. Then, $\phi$ and $\psi$ are continuous on $U$, and $0 \leq \phi < 1$ on $S$. Since $n_2 f + h = -\log(\phi^{n_2} \psi)$, it is sufficient to find a positive integer $n_2$ with $\phi^{n_2} \psi \leq 1 $ on $S$. If we set $a = \sup_{z \in S} \phi(z)$ and $b = \sup_{z \in S} \psi(z)$, then $0 \leq a < 1$ and $0 \leq b$ because $S$ is compact. Thus, there is a positive integer $n_2$ with $a^{n_2} b \leq 1$. Therefore, $\phi^{n_1} \psi \leq 1$ on $S$. Here we set $t_0 = (n_1+n_2)n_0e^{-1}$. In order to see that $t\alpha + \beta$ is semi-ample at $x$ with respect to $S$ for $t \geq t_0$, it is sufficient to show that $(n_1+n_2)n_0 \alpha + e \beta$ is semi-ample at $x$ with respect to $S$ because $e t \geq (n_1+n_2)n_0$. Here \begin{align*} (n_1 + n_2)n_0 \alpha + e \beta & \sim n_2(A, f) + \left( n_1(A, f) + (D, g) \right) \\ & \sim n_2(A, f) + (Z, h) \\ & = (n_2 A + Z, n_2 f + h), \end{align*} $x \not\in \operatorname{Supp}(n_2 A + Z)$, and $(n_2 f + h)(z) \geq 0$ for all $z \in S$. Thus, $(n_1 + n_2)n_0 \alpha + e \beta$ is semi-ample at $x$ with respect to $S$. Hence, we get our claim. In the same way, we can find a positive number $t_1$ such that $t \alpha - \beta$ is semi-ample with respect to $S$ for all $t \geq t_1$. Thus, if we set $\epsilon_0 = \min \{ 1/t_0, 1/t_1 \}$, then we have (1). \medskip (2) In the same way as in the proof of (1), we can find a positive number $\epsilon_0$ such that $(f + \epsilon n_0 g)(z) \geq 0$ for all $z \in S$ and all real number $\epsilon$ with $|\epsilon| \leq \epsilon_0$. Thus we have (2) because $n_0( \alpha + a(\epsilon g)) \sim (A, f + \epsilon n_0 g)$. \QED \begin{Proposition} \label{prop:characterization:wp:div} We assume that $S$ is compact. Let $\beta$ be an element of $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$. Then the following are equivalent. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item $\beta$ is weakly positive at $x$ with respect to $S$. \item $\beta + \alpha$ is semi-ample at $x$ with respect to $S$ for any ample $\alpha \in \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ at $x$ with respect to $S$. \end{enumerate} \end{Proposition} {\sl Proof.}\quad (1) $\Longrightarrow$ (2): Since $\beta$ is weakly positive at $x$ with respect to $S$, there is a sequence of $\{ \beta_n \}$ such that $\beta_n \in \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$, $\beta_n$'s are semi-ample at $x$ with respect to $S$, and $\lim_{n \to \infty} \beta_n = \beta$. Take $Z_1, \ldots, Z_{l_1}$, $g_1, \ldots, g_{l_2}$, $\{ a_n^{1} \}_{n=1}^{\infty}, \ldots, \{ a_n^{l_1} \}_{n=1}^{\infty}$, and $\{ b_n^{1} \}_{n=1}^{\infty}, \ldots, \{ b_n^{l_2} \}_{n=1}^{\infty}$ as in the definition of the limit in $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$. Then, by Lemma~\ref{lem:make:semiample:by:ample}, there is a positive number $\epsilon_0$ such that $\alpha + \epsilon Z_i$'s are semi-ample at $x$ with respect to $S$ for all rational numbers $\epsilon$ with $|\epsilon| \leq \epsilon_0$, and $\alpha + a(\epsilon g_j)$'s are semi-ample at $x$ with respect to $S$ for all real numbers $\epsilon$ with $|\epsilon| \leq \epsilon_0$. We choose $n$ such that $(l_1 + l_2)|a_n^{i}| \leq \epsilon_0$ and $(l_1 + l_2)|b_n^{j}| \leq \epsilon_0$ for all $i$ and $j$. Then, \[ \beta + \alpha = \beta_n + \sum_{i=1}^{l_1} \frac{\alpha + (l_1 + l_2)a_n^{i} Z_i}{l_1 + l_2} + \sum_{j=1}^{l_2} \frac{\alpha + a((l_1+ l_2)b_n^{j} g_j)}{l_1 + l_2}. \] Here, $\alpha + (l_1 + l_2)a_n^{i} Z_i$ and $\alpha + a((l_1+ l_2)b_n^{j} g_j)$ are semi-ample $x$ with respect to $S$. Thus, we get the direction (1) $\Longrightarrow$ (2). \medskip (2) $\Longrightarrow$ (1): Let $\alpha$ be an element of $\widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}}$ such that $\alpha$ is ample at $x$ with respect to $S$. We set $\beta_n = \beta + (1/n)\alpha$. Then, by our assumption, $\beta_n$ is semi-ample at $x$ with respect to $S$. Further, $\beta = \lim_{n \to \infty} \beta_n$. \QED \subsection{Small sections via generically finite morphisms} \setcounter{Theorem}{0} Let $g : V \to U$ be a proper and \'{e}tale morphism of complex manifolds. Let $(E, h)$ be a Hermitian vector bundle on $V$. Then, a Hermitian metric $g_*(h)$ of $g_*(E)$ is defined by \[ g_*(h)(s, t)(y) = \sum_{x \in g^{-1}(y)} h(s, t)(x) \] for any $y \in U$ and $s, t \in g_*(E)_y$. \begin{Proposition} \label{prop:find:small:section} Let $X$ be a scheme such that every connected component of $X$ is a arithmetic variety. Let $Y$ be a regular arithmetic variety, and $g : X \to Y$ a proper and generically finite morphism such that every connected component of $X$ maps surjectively to $Y$. Let $U$ be a Zariski open set of $Y$ such that $g$ is \'{e}tale over $U$. Let $S$ be a subset of $U({\mathbb{C}})$ and $y$ a closed point of $U_{{\mathbb{Q}}}$. Then, we have the following. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item Let $\phi : E \to Q$ be a homomorphism of coherent ${\mathcal{O}}_X$-modules such that $\phi$ is surjective over $g^{-1}(U)$, and $E$ and $Q$ are locally free over $g^{-1}(U)$. Let $h_E$ be a $C^{\infty}$ Hermitian metric of $E$ over $g^{-1}(U)({\mathbb{C}})$, and $h_Q$ the quotient metric of $Q$ induced by $h_E$. If $(g_*(E), g_*(h_E))$ is generated by small sections at $y$ with respect to $S$, then $(g_*(Q), g_*(h_Q))$ is generated by small sections at $y$ with respect to $S$. \item Let $E_1$ and $E_2$ be coherent ${\mathcal{O}}_X$-modules such that $E_1$ and $E_2$ are locally free over $g^{-1}(U)$. Let $h_1$ and $h_2$ be $C^{\infty}$ Hermitian metrics of $E_1$ and $E_2$ over $g^{-1}(U)({\mathbb{C}})$. If $(g_*(E_1), g_*(h_1))$ and $(g_*(E_2), g_*(h_2))$ are generated by small sections at $y$ with respect to $S$, then so is $(g_*(E_1 \otimes E_2), g_*(h_1 \otimes h_2))$. \item Let $E$ be a coherent ${\mathcal{O}}_X$-module such that $E$ is locally free over $g^{-1}(U)$. Let $h_E$ be a $C^{\infty}$ Hermitian metric of $E$ over $g^{-1}(U)({\mathbb{C}})$. If $(g_*(E), g_*(h_E))$ is generated by small sections at $y$ with respect to $S$, then $(g_*(\operatorname{Sym}^n(E)), g_*(\operatorname{Sym}^n(h_E)))$ is generated by small sections at $y$ with respect to $S$. \textup{(}For the definition of $\operatorname{Sym}^n(h_E)$, see \S\textup{\ref{subsec:formula:chern:sym:power}}.\textup{)} \item Let $F$ be a coherent ${\mathcal{O}}_Y$-module such that $F$ is locally free over $U$. Let $h_F$ be a $C^{\infty}$ Hermitian metric of $F$ over $U({\mathbb{C}})$. Since $\rest{\det(F)}{U}$ is canonically isomorphic to $\det(\rest{F}{U})$, $\det(h_F)$ gives rise to a $C^{\infty}$ Hermitian metric of $\det(F)$ over $U({\mathbb{C}})$. If $(F, h_F)$ is generated by small sections at $y$ with respect to $S$, then so is $(\det(F), \det(h_F))$. \end{enumerate} \end{Proposition} {\sl Proof.}\quad (1) By our assumption, $g_*(\phi) : g_*(E) \to g_*(Q)$ is surjective over $U$. Let $s_1, \ldots, s_l \in H^0(Y, g_*(E)) = H^0(X, E)$ such that $g_*(E)_{y}$ is generated by $s_1, \ldots, s_l$, and that $g_*(h_E)(s_i, s_i)(z) \leq 1$ for all $i$ and $z \in S$. Then, $g_*(Q)_y$ is generated by $g_*(\phi)(s_1), \ldots, g_*(\phi)(s_l)$. Moreover, by the definition of the quotient metric $h_Q$, \[ g_*(h_Q)(g_*(\phi)(s_i), g_*(\phi)(s_i))(z) = \sum_{x \in g^{-1}(z)} h_Q(\phi(s_i), \phi(s_i))(x) \leq \sum_{x \in g^{-1}(z)} h_E(s_i, s_i)(x) \leq 1 \] for all $z \in S$. Hence, $g_*(Q)$ is generated by small sections at $y$ with respect to $S$. \medskip (2) Since $g$ is \'{e}tale over $U$, $\alpha : g_*(E_1) \otimes g_*(E_2) \to g_*(E_1 \otimes E_2)$ is surjective over $U$. By our assumption, there are $s_1, \ldots, s_l \in H^0(Y, g_*(E_1))$ and $t_1, \ldots, t_m \in H^0(Y, g_*(E_2))$ such that $g_*(E_1)_y$ (resp. $g_*(E_2)_y$) is generated by $s_1, \ldots, s_l$ (resp. $t_1, \ldots, t_m$), and that $g_*(h_1)(s_i, s_i)(z) \leq 1$ and $g_*(h_2)(t_j, t_j)(z) \leq 1$ for all $i$, $j$ and $z \in S$. Then, $g_*(E_1 \otimes E_2)_y$ is generated by $\{ \alpha(s_i \otimes t_j) \}_{i,j}$. Moreover, {\allowdisplaybreaks \begin{align*} g_*(h_1 \otimes h_2)(\alpha(s_i \otimes t_j), \alpha(s_i \otimes t_j))(z) & = \sum_{x \in g^{-1}(z)} (h_1 \otimes h_2)(s_i \otimes t_j, s_i \otimes t_j)(x) \\ & = \sum_{x \in g^{-1}(z)} h_1(s_i, s_i)(x) h_2(t_j,t_j)(x) \\ & \leq \left( \sum_{x \in g^{-1}(z)} h_1(s_i, s_i)(x) \right) \left( \sum_{x \in g^{-1}(z)} h_2(t_j, t_j)(x) \right) \\ & \leq 1 \end{align*}} for all $z \in S$. Thus, we get (2). \medskip (3) This is a consequence of (1) and (2). \medskip (4) Let $r$ be the rank of $F$. Since $F$ is generated by small sections at $y$ with respect to $S$, there are $s_1, \ldots, s_r \in H^0(Y, F)$ such that $F \otimes \kappa(y)$ is generated by $s_1, \ldots, s_r$ and $h(s_i, s_i)(z) \leq 1$ for all $i$ and $z \in S$. Let us consider an exact sequence: \[ 0 \to F_{tor} \to F \to F/F_{tor} \to 0. \] Then, $\det(F) = \det(F/F_{tor}) \otimes \det (F_{tor})$. Noting that $F_{tor} = 0$ on $U$, let $g$ be a Hermitian metric of $\det(F/F_{tor})$ over $U({\mathbb{C}})$ given by $\det(h_F)$. Then, there is a Hermitian metric $k$ of $\det(F_{tor})$ over $U({\mathbb{C}})$ such that $(\det(F), \det(h_F)) = (\det(F/F_{tor}), g) \otimes (\det (F_{tor}), k)$ over $U({\mathbb{C}})$. If we identify $\det (F_{tor})$ with ${\mathcal{O}}_{Y}$ over $U$, $k$ is nothing more than the canonical metric of ${\mathcal{O}}_{Y}$ over $U({\mathbb{C}})$. Let us fix a locally free sheaf $P$ on $Y$ and a surjective homomorphism $P \to F_{tor}$. Let $P'$ be the kernel of $P \to F_{tor}$. Here $\left( \bigwedge^{\operatorname{rk} P'} P' \right)^{*}$ is an invertible sheaf on $Y$ because $Y$ is regular. Thus we may identify $\det(F_{tor})$ with \[ \bigwedge^{\operatorname{rk} P} P \otimes \left( \bigwedge^{\operatorname{rk} P'} P' \right)^{*}. \] Further, a homomorphism $\bigwedge^{\operatorname{rk} P'} P' \to \bigwedge^{\operatorname{rk} P} P$ induced by $P' \hookrightarrow P$ gives rise to a non-zero section $t$ of $\det(F_{tor})$ because \[ \operatorname{\mathcal{H}\textsl{om}}\left(\bigwedge^{\operatorname{rk} P'} P', \bigwedge^{\operatorname{rk} P} P\right) = \operatorname{\mathcal{H}\textsl{om}}\left(\bigwedge^{\operatorname{rk} P'} P', {\mathcal{O}}_Y\right) \otimes \bigwedge^{\operatorname{rk} P} P. \] Here $F_{tor} = 0$ on $U$. Thus, $\det(F_{tor})$ is canonically isomorphic to ${\mathcal{O}}_{Y}$ over $U$. Since $P' = P$ over $U$, under the above isomorphism, $t$ goes to the determinant of $P' \overset{\operatorname{id}}{\longrightarrow} P$, namely $1 \in {\mathcal{O}}_{Y}$ over $U$. Thus, $k(t,t)(z) = 1$ for each $z \in S$. Let $\overline{s}_i$ be the image of $s_i$ in $F/F_{tor}$. Then, $\overline{s}_1 \wedge \cdots \wedge \overline{s}_r$ gives rise to a section $s$ of $\det(F/F_{tor})$. Thus, $s \otimes t$ is a section of $\det(F)$. By our construction, $(s \otimes t)(y) \not= 0$. Moreover, using Hadamard's inequality, \begin{align*} \det(h_F)(s \otimes t, s \otimes t)(z) & = g(s, s)(z) \cdot k(t,t)(z) = \det \left( h( s_i, s_j)(z) \right) \\ & \leq h(s_1, s_1)(z) \cdots h(s_r, s_r)(z) \leq 1 \end{align*} for each $z \in S$. Thus, we get (4). \QED \section{Arithmetic Riemann-Roch for generically finite morphisms} \subsection{Quillen metric for generically finite morphisms} \setcounter{Theorem}{0} Before starting Proposition~\ref{prop:Quillen:metric:generalised:gen:finite:morph}, we recall the tensor product of two matrices, which we will use in the proof. For an $r\times r$ matrix $A=(a_{ij})$ and an $n\times n$ matrix $B=(b_{kl})$, consider the following $rn\times rn$ matrix \begin{equation*} \begin{pmatrix} a_{11}B & a_{12}B & \cdots & a_{1r}B \\ a_{21}B & a_{22}B & \cdots & a_{2r}B \\ \vdots & \vdots & \ddots & \vdots \\ a_{r1}B & a_{r2}B & \cdots & a_{rr}B \\ \end{pmatrix}. \end{equation*} This matrix, denoted by $A\otimes B$, is called {\em the tensor product of $A$ and $B$}. Then for $r \times r$ matrices $A,A'$ and $n \times n$ matrices $B,B'$, we immediately see \begin{equation*} (A \otimes B) (A' \otimes B') = AA' \otimes BB', \cr \det (A \otimes B) = (\det A)^n (\det B)^r. \end{equation*} Let $X$ be a smooth algebraic scheme over ${\mathbb{C}}$, $Y$ a smooth algebraic variety over ${\mathbb{C}}$, and $f : X \to Y$ a proper and generically finite morphism. We assume that every connected component of $X$ maps surjectively to $Y$. Let $W$ be the maximal open set of $Y$ such that $f$ is \'{e}tale over there. Let $(E, h)$ be a Hermitian vector bundle on $X$ such that on every connected component of $X$, $E$ has the same rank $r$. \begin{Proposition} \label{prop:Quillen:metric:generalised:gen:finite:morph} With notation and assumptions being as above, the Quillen metric $h_Q^{\overline E}$ on $\det Rf_*(E)$ over $W$ extends to a generalized metric on $\det Rf_*(E)$ over $Y$. \end{Proposition} {\sl Proof.}\quad Let $n$ be the degree of $f$. Since $f$ is \'{e}tale over $W$, $f_*(E)$ is a locally free sheaf of rank $rn$ and $R^if_*(E) = 0$ for $i \ge 1$ over there. Thus \[ \det Rf_*(E) \vert {}_W = \bigwedge^{rn} f_*(E) \vert {}_W. \] If $y \in W$ is a complex point and $X_y=\{x_1,x_2,\cdots,x_n\}$ the fiber of $f$ over $y$, then we have \[ \det Rf_*(E)_y = \det H^0(X_y,E). \] The Quillen metric on $\det Rf_*(E)$ over $W$ is defined as follows. On $H{}^0(X_y,E)$ the $L^2$-metric is defined by the formula: \[ h_{L^2}(s,t) = \sum_{\alpha=1}^n h(s,t)(x_\alpha), \] where $s,t \in H{}^0(X_y,E)$. This metric naturally induces the $L^2$-metric on $\det H{}^0(X_y,E)$. Since $X_y$ is zero-dimensional, there is no need for zeta function regularization to obtain the Quillen metric. Thus the Quillen metric $h_Q^{\overline E}$ on $\det Rf_*(E) \vert {}_W$ is defined by the family of Hermitian line bundles $\{\det H^0(X_y,E)\}_{y \in W}$ with the induced $L^2$-metrics pointwisely. To see that the Quillen metric over $W$ extends to a generalized metric over $Y$, let $s_1,s_2,\cdots,s_r$ be rational sections of $E$ such that at the generic point of every connected component of $X$, they form a basis of $E$. Also let $\omega_1,\omega_2,\cdots,\omega_n$ be rational sections of $f_*({\mathcal{O}}_X)$ such that at the generic point they form a basis of $f_*({\mathcal{O}}_X)$. Since \addtocounter{Claim}{1} \begin{equation} \label{eqn:2:prop:Quillen:metric:generalised:gen:finite:morph} \det Rf_*(E) = \left(\bigwedge^{rn} (f_*(E)) \right)^{**}. \end{equation} over $Y$, we can regard $\bigwedge_{ik}s_i\omega_k=s_1\omega_1\operatornamewithlimits{\wedge} s_1\omega_2\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge} s_1\omega_n\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge} s_r\omega_n$ as a non-zero rational section of $\det Rf_*(E)$. Shrinking $W$, we can find a non-empty Zariski open set $W_0$ of $W$ such that $s_i$'s and $\omega_j$'s has no poles or zeros over $f^{-1}(W_0)$. To proceed with our argument, we need the following lemma. \begin{Lemma} \label{lem:formula:for:Quillen:metric} Let $L$ be the total quotient field of $X$, and $K$ the function field of $Y$. Then, \[ \log h_Q^{\overline E} \left( \bigwedge_{ik}s_i\omega_k, \bigwedge_{ik}s_i\omega_k \right) = r \log \left| \det (\operatorname{Tr}_{L/K}(\omega_i \cdot \omega_j)) \right| + f_* \log \det (h(s_i, s_j)) \] over $W_0$. \end{Lemma} {\sl Proof.}\quad Let $y \in W_0$ be a complex point, and $\{x_1,x_2,\ldots,x_n\}$ the fiber of $f^{-1}(y)$ over $y$. Then, {\allowdisplaybreaks \newcommand{\smash{\hbox{\Large 0}}}{\smash{\hbox{\Large 0}}} \newcommand{\smash{\lower1.7ex\hbox{\Large 0}}}{\smash{\lower1.7ex\hbox{\Large 0}}} \begin{align*} \log h_Q^{\overline E} \left( \bigwedge_{ik}s_i\omega_k,\bigwedge_{ik}s_i\omega_k \right)(y) & = \log \det \left( \sum_{\alpha =1}^n h(s_i\omega_k,s_j\omega_l)(x_{\alpha}) \right)_{ij,kl} \\ & = \log \det \left( \sum_{\alpha =1}^n \omega_k(x_\alpha) h(s_i,s_j)(x_{\alpha}) \overline{\omega_l(x_\alpha)} \right)_{ij,kl} \\ & = \log \det \left( (I_r \otimes \Omega) \begin{pmatrix} H(x_1) & & \smash{\lower1.7ex\hbox{\Large 0}} \\ & \ddots & \\ \smash{\hbox{\Large 0}} & & H(x_n) \\ \end{pmatrix} \overline{{}^t(I_r \otimes \Omega)} \right) \\ & = \log \det \left\{ \vert \det (\Omega) \vert ^{2r} \prod_{\alpha=1}^n \det \left(h(s_i,s_j)(x_{\alpha})\right)_{ij} \right\} \\ & = r \log \det \vert \det (\Omega) \vert ^{2} + \sum_{\alpha=1}^n \log \det \left(h(s_i,s_j)\right)(x_{\alpha}), \end{align*}} where $\Omega = (\omega_k(x_\alpha))_{k \alpha}$ and $H(x_\alpha) = (h(s_i,s_j)(x_{\alpha}))_{ij}$. On the other hand, we have \[ \sum_{\alpha=1}^n \log \det \left(h(s_i,s_j)\right) (x_{\alpha}) = \left(f_* \log \det \left(h(s_i,s_j)\right)\right)(y). \] Moreover, using the following Lemma~\ref{lemma:trace:global:local}, we have \[ \vert \det (\Omega) \vert ^{2} = \vert \det (\Omega {}^t \Omega) \vert = \left\vert \det \left(\sum_{\alpha=1}^n \omega_k(x_\alpha) \omega_l(x_\alpha) \right)_{kl} \right\vert = \vert \det \left(\operatorname{Tr}_{L/K}(\omega_k\cdot \omega_l)\right)_{kl} \vert. \] Thus we get the lemma. \QED \begin{Lemma} \label{lemma:trace:global:local} Let $f : \operatorname{Spec}(B) \to \operatorname{Spec}(A)$ be a finite \'{e}tale morphism of regular affine schemes. Let ${\mathfrak m}$ be the maximal ideal of $A$ and ${\mathfrak n}_1,{\mathfrak n}_2,\cdots,{\mathfrak n}_n$ the prime ideals lying over ${\mathfrak m}$. Assume that $\kappa ({\mathfrak m})$ is algebraically closed and hence $\kappa ({\mathfrak n}_i)$ is \textup{(}naturally\textup{)} isomorphic to $\kappa ({\mathfrak m})$ for each $1 \leq i \leq n$. Let $b$ be an element of $B$ and $b({\mathfrak n}_i)$ the value of $b$ in $\kappa ({\mathfrak n}_i) \cong \kappa ({\mathfrak m})$. Then \[ \operatorname{Tr}_{B/A}(b)({\mathfrak m}) = \sum_{i=1}^n b({\mathfrak n}_i) \] in $\kappa ({\mathfrak m})$, where $\operatorname{Tr}_{B/A}(b)({\mathfrak m})$ is the value of $\operatorname{Tr}_{B/A}(b)$ in $\kappa ({\mathfrak m})$. \end{Lemma} {\sl Proof.}\quad It is easy to see that every ${\mathfrak n}_i$ is the maximal ideal and that ${\mathfrak m}B = {\mathfrak n}_1{\mathfrak n}_2 \cdots {\mathfrak n}_n$. Let $\widehat{A}$ be the completion of $A$ with respect to ${\mathfrak m}$, $\widehat{B}$ the completion of $B$ with respect to ${\mathfrak m}B$, and $\widehat{B_i}$ the completion of $B$ with respect to ${\mathfrak n}_i$ for each $1 \leq i \leq n$. Then by Chinese remainder theorem, $\widehat{B} = \prod_{i=1}^n \widehat{B_i}$ as an $\widehat{A}$-algebra. Note that $\widehat{A} /{\mathfrak m} \widehat{A} = \kappa ({\mathfrak m})$ and $\widehat{B_i} /{\mathfrak n}_i \widehat{B_i} = \kappa ({\mathfrak n}_i)$. Since $\widehat{A} \to \widehat{B_i}$ is \'{e}tale and $\kappa ({\mathfrak m}) \cong \kappa ({\mathfrak n}_i)$, we have $\widehat{A} \cong \widehat{B_i}$. Let $e_1=(1,0,\cdots,0),e_2=(0,1,\cdots,0),\cdots,e_n=(0,0,\cdots,1) \in \prod_{i=1}^n \widehat{B_i} =\widehat{B}$ be a free basis of $\widehat{B}$ over $\widehat{A}$. We put $b e_i = b_i e_i$ with $b_i \in \widehat{B_i} \cong \widehat{A}$ for each $1 \leq i \leq n$. Then $b_i \equiv b({\mathfrak n}_i) \pmod{{\mathfrak n}_i}$. Now the lemma follows from \[ \operatorname{Tr}_{B/A}(b) = \operatorname{Tr}_{\widehat{B} / \widehat{A}}(b) = \sum_{i=1}^n b_i \] in $\widehat{A}$. \QED Let us go back to the proof of Proposition~\ref{prop:Quillen:metric:generalised:gen:finite:morph}. Since $\rest{\det (\operatorname{Tr}_{L/K}(\omega_i \cdot \omega_j))}{W_0}$ extends to a rational function $\det (\operatorname{Tr}_{L/K}(\omega_i \cdot \omega_j))$ on $Y$, \[ \log \left | \det (\operatorname{Tr}_{L/K}(\omega_i \cdot \omega_j)) \right| \in L^1_{\operatorname{loc}}(Y). \] Moreover, by Proposition~\ref{prop:push:forward:B:pq}, $f_* \log \det (h(s_i, s_j)) \in L^1_{\operatorname{loc}}(Y)$. Thus, by Lemma~\ref{lem:formula:for:Quillen:metric}, \[ \rest{\log h_Q^{\overline E} \left( \bigwedge_{ik}s_i\omega_k, \bigwedge_{ik}s_i\omega_k \right)}{W_0} \] extends to a locally integrable function on $Y$. Hence by Lemma~\ref{lem:criterion:gen:metric} the Quillen metric over $W$ extends to a generalized metric over $Y$. \QED \begin{Remark} \label{rmk:Quillen:metric:continuous:finite:morphism} In the above situation, Let $W'$ be a open set of $Y$ such that $f$ is flat and finite over there. Then the Quillen metric extends to a continuous function over $W'$ by the same formula as in (\ref{lem:formula:for:Quillen:metric}) \end{Remark} \subsection{Riemann-Roch for generically finite morphisms} \setcounter{Theorem}{0} In this subsection, we formulate the arithmetic Riemann-Roch theorem for generically finite morphisms. \begin{Theorem} \label{thm:arith:Riemann:Roch:gen:finite:morphism} Let $X$ be a scheme such that every connected component of $X$ is an arithmetic variety. Let $Y$ be a regular arithmetic variety, and $f : X \to Y$ a proper and generically finite morphism such that every connected component of $X$ maps surjectively to $Y$. Let $(E, h)$ a Hermitian vector bundle on $X$ such that on each connected component of $X$, $E$ has the same rank $r$. Then, \[ \widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) - r \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X} \right) \in \widehat{\operatorname{CH}}_{L^1}^1(Y) \] and \[ \widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) - r \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X} \right) = f_* \left( \widehat{{c}}_1 (E, h) \right) \] in $\widehat{\operatorname{CH}}_{L^1}^1(Y)_{{\mathbb{Q}}}$, where $h_Q^{\overline{E}}$ and $h_Q^{\overline{{\mathcal{O}}}_X}$ are the Quillen metric of $\det Rf_*(E)$ and $\det Rf_*({\mathcal{O}}_X)$ respectively. \end{Theorem} {\sl Proof.}\quad Let $X = \coprod_{\alpha \in A} X_{\alpha}$ be the decomposition into connected components of $X$. Since $f$ is proper, $A$ is a finite set. We set $f_{\alpha} = \rest{f}{X_{\alpha}}$ and $(E_{\alpha}, h_{\alpha}) = \rest{(E, h)}{X_{\alpha}}$. Then \[ Rf_*(E) = \bigoplus_{\alpha \in A} R (f_{\alpha})_*(E_{\alpha}),\quad Rf_*({\mathcal{O}}_X) = \bigoplus_{\alpha \in A} R (f_{\alpha})_*({\mathcal{O}}_{X_{\alpha}}),\quad \widehat{{c}}_1 (E, h) = \sum_{\alpha \in A} \widehat{{c}}_1 (E_{\alpha}, h_{\alpha}). \] Hence we have the following: \[ \begin{cases} {\displaystyle \widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) = \sum_{\alpha \in A} \widehat{{c}}_1 \left( \det R(f_{\alpha})_*(E_{\alpha}), h_Q^{\overline{E}_{\alpha}} \right)}, \\ {\displaystyle \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X} \right) = \sum_{\alpha \in A} \widehat{{c}}_1 \left( \det R(f_{\alpha})_*({\mathcal{O}}_{X_{\alpha}}), h_Q^{\overline{{\mathcal{O}}}_{X_{\alpha}}} \right)}, \\ {\displaystyle f_* \left( \widehat{{c}}_1 (E, h) \right) = \sum_{\alpha \in A} f_* \left( \widehat{{c}}_1 (E_{\alpha}, h_{\alpha}) \right)}. \end{cases} \] Thus, we may assume that $X$ is connected, i.e., $X$ is an arithmetic variety. \medskip Let $K=K(Y)$ and $L=K(X)$ be the function fields of $Y$ and $X$ respectively. Let $n$ be the degree of $f$ and $\omega_1,\omega_2,\cdots,\omega_n$ rational functions on $X$ such that at the generic point they form a basis of $K$-vector space $L$. Further, let $s_1,s_2,\ldots,s_r$ be rational sections of $E$ such that at the generic point they form a basis of $L$-vector space $E_L$. Then $s_1\omega_1\operatornamewithlimits{\wedge} s_1\omega_2\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge} s_1\omega_n\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge} s_r\omega_n$, $s_1\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge} s_r$ and $\omega_1\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge}\omega_n$ are non-zero rational sections of $\det f_*(E)$, $\det (E)$ and $\det f_*({\mathcal{O}}_X)$ respectively. Here we shall prove the following equality in $\widehat{Z}_D^1(Y)$: \addtocounter{Claim}{1} \begin{multline} \label{eqn:desired:thm:arith:Riemann:Roch:gen:finite:morphism} \left(\operatorname{div} \left(\bigwedge_{ik}s_i\omega_k\right), \left[-\log h_Q^{\overline E} \left(\bigwedge_{ik}s_i\omega_k,\bigwedge_{ik}s_i\omega_k\right)\right] \right) \\ - r \left(\operatorname{div}\left(\bigwedge_k\omega_k\right), \left[-\log h_Q^{\overline {\mathcal{O}}_X} \left(\bigwedge_{k}\omega_k,\bigwedge_{k}\omega_k\right)\right] \right) \\ = f_*\left(\operatorname{div}\left(\bigwedge_i s_i\right), \left[-\log \det h\left(\bigwedge_i s_i,\bigwedge_i s_i\right)\right] \right), \end{multline} where $\bigwedge_{ik}s_i\omega_k=s_1\omega_1\operatornamewithlimits{\wedge} s_1\omega_2\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge} s_1\omega_n\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge} s_r\omega_n$, $\bigwedge_k\omega_k =\omega_1\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge}\omega_n$ and $\bigwedge_i s_i=s_1\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge} s_r$. First we shall show the equality of divisors. Let $Y_0$ be the maximal Zariski open set of $X$ such that $f$ is flat over $Y_0$. Then, $\operatorname{codim}_Y(Y \setminus Y_0)\ge 2$ by \cite[III,Proposition~9.7]{Hartshorne}. Since $f$ is generically finite, $f$ is in fact finite over $Y_0$. Then $Z^1(Y)=Z^1(Y_0)$ and thus it suffices to prove the equality of divisors over $Y_0$. Since it suffices to prove it locally, let $U=\operatorname{Spec}(A)$ be an affine open set of $Y_0$ and $f^{-1}(U)=\operatorname{Spec}(B)$ the open set of $X_0=f^{-1}(Y_0)$. Shrinking $U$ if necessary, we may assume that $B$ is a free $A$-module of rank $n$ and that $E$ is a free $B$-module of rank $r$. Let $d_1,d_2,\cdots,d_n$ be a basis of $B$ over $A$, and $e_1,e_2,\cdots,e_r$ be a basis of $E$ over $B$. Note that $K$ and $L$ are the quotient fields of $A$ and $B$ respectively. In the following we freely identify a rational function (or section) by the corresponding element at the generic point. In this sense, we set \begin{align*} \omega_k = \sum_{l=1}^{n}a^{kl}d_l \quad (k=1,2,\cdots,n) \\ s_i = \sum_{j=1}^{r}\sigma_{ij}e_j \quad (i=1,2,\cdots,r), \end{align*} where $a^{kl} \in K\,(1\le k,l\le n)$ and $\sigma_{ij}\in L\,(1\le i,j\le r)$. For each $\sigma_{ij}(1\le i,j\le r)$, let $T_{\sigma_{ij}} : L \to L$ be multiplication by $\sigma_{ij}$. With respect to a basis $\omega_1,\omega_2,\cdots,\omega_n$ of $L$ over $K$, $T_{\sigma_{ij}}$ gives rise to the matrix $(c_{ij}^{kl})_{1\le k,l\le n}\in M_n(K)$ defined by \[ \sigma_{ij}\omega_k =\sum_{l=1}^n c_{ij}^{kl}\omega_l \quad (k=1,2,\cdots,n). \] We also denote this matrix by $T_{\sigma_{ij}}$. Then, {\allowdisplaybreaks \begin{align*} \bigwedge_{ik}s_i\omega_k & = \bigwedge_{ik}\left(\sum_{j=1}^{r} \sigma_{ij}e_j\right)\omega_k = \bigwedge_{ik}\left(\sum_{j=1}^{r}\sum_{l=1}^{n} c_{ij}^{kl}\right)e_j\omega_l \\ & = \det (c_{ij}^{kl})_{ik,jl} \bigwedge_{ik}e_i\omega_k = \det (c_{ij}^{kl})_{ik,jl} \bigwedge_{ik}e_i\left(\sum_{l=1}^{n}a^{kl}d_l\right) \\ & = \det (c_{ij}^{kl})_{ik,jl} \bigwedge_{ik}\left(\sum_{j=1}^{r} \delta_{ij}a^{kl}\right)e_j \omega_l = \det (c_{ij}^{kl})_{ik,jl} \det (\delta_{ij}a^{kl})_{ik,jl} \bigwedge_{ik}e_i d_l. \end{align*}} On the other hand, since the matrices $T_{\sigma_{ij}}$ and $T_{\sigma_{i'j'}}$ commute with each other, we have {\allowdisplaybreaks \begin{align*} \det (c_{ij}^{kl})_{ik,jl} & = \det \begin{pmatrix} T_{\sigma_{11}} & T_{\sigma_{12}} & \cdots & T_{\sigma_{1r}} \\ T_{\sigma_{21}} & T_{\sigma_{22}} & \cdots & T_{\sigma_{2r}} \\ \vdots & \vdots & \ddots & \vdots \\ T_{\sigma_{r1}} & T_{\sigma_{r2}} & \cdots & T_{\sigma_{rr}} \\ \end{pmatrix}\\ & = \det \left( \sum_{\tau\in {\mathfrak{S}}_r} \operatorname{sign}(\tau) T_{\sigma_{1\tau(1)}}\cdot\cdots\cdot T_{\sigma_{r\tau(r)}} \right) \\ & = \det(T_{\det(\sigma_{ij})_{ij}}) \\ & = \operatorname{Norm}_{L/K}(\det(\sigma_{ij})_{ij}). \end{align*}} Moreover, we have \begin{align*} \det (\delta_{ij}a^{kl})_{ik,jl} & = \det (I_r \otimes (a^{kl})_{kl}) \\ & = (\det (a^{kl})_{kl})^r. \end{align*} >From the above three equalities, $\operatorname{div}\left(\bigwedge_{ik}s_i\omega_k\right)$ is given by the rational function \[ \operatorname{Norm}_{L/K}(\det(\sigma_{ij})_{ij}) (\det (a^{kl})_{kl})^r. \] Further \[ \bigwedge_{i}s_i = (\det (\sigma_{ij})_{ij}) \bigwedge_{i} e_k \quad\text{and}\quad \bigwedge_{k}\omega_k = (\det (a^{kl})_{kl}) \bigwedge_{k} d_k. \] Hence we have \[ \operatorname{div}\left(\bigwedge_{ik}s_i\omega_k\right) -r\left(\operatorname{div}\left(\bigwedge_k\omega_k\right)\right) = f_*\left(\operatorname{div}\left(\bigwedge_i s_i\right)\right). \] Next we shall show the equality of currents. Since all the currents in the equality come from locally integrable functions by Proposition~\ref{prop:push:forward:B:pq} and Proposition~\ref{prop:Quillen:metric:generalised:gen:finite:morph}, it suffices to show the equality over a non-empty Zariski open set of every connected component of $Y({\mathbb{C}})$. So let $W_0$ be a non-empty Zariski open set of a connected component of $Y({\mathbb{C}})$ such that $f_{{\mathbb{C}}}$ is \'{e}tale and that $s_i\, (1 \le i \le r)$ or $\omega_k \,(1 \le k \le n)$ have no poles or zeroes over there. Then over $W_0$ all these currents are defined by $C^{\infty}$ functions. Let $y \in Y({\mathbb{C}})$ be a complex point and $x_1,x_2,\cdots,x_n$ be the fiber $f_{{\mathbb{C}}}^{-1}(y)$ over $y$. >From the proof of Lemma~\ref{lem:formula:for:Quillen:metric}, as $C^{\infty}$ functions around $y$, {\allowdisplaybreaks \begin{align*} -\log h_Q^{\overline E} \left(\bigwedge_{ik}s_i\omega_k,\bigwedge_{ik}s_i\omega_k\right)(y) & = -\log \det \left\{ \vert \det (\Omega) \vert ^{2r} \prod_{\alpha=1}^n \det \left(h(s_i,s_j)(x_{\alpha})\right)_{ij} \right\}, \end{align*}} where $\Omega = (\omega_k(x_\alpha))_{k \alpha}$ and $H(x_\alpha) = (h(s_i,s_j)(x_{\alpha}))_{ij}$. Also, {\allowdisplaybreaks \begin{align*} -\log h_Q^{\overline {\mathcal{O}}_X} \left(\bigwedge_{k}\omega_k,\bigwedge_{k}\omega_k\right)(y) & = -\log \det \vert \det (\Omega) \vert ^{2}. \end{align*}} On the other hand, by the definition of the push-forward $f_*$, \begin{align*} f_* \left( -\log \det h\left(\bigwedge_i s_i,\bigwedge_i s_i\right) \right)(y) & = \sum_{\alpha =1}^n -\log \det h\left(\bigwedge_i s_i,\bigwedge_i s_i\right)(x_\alpha) \\ & = \sum_{\alpha =1}^n -\log \det \left( h(s_i,s_j)(x_\alpha) \right)_{ij}. \end{align*} Hence we have the desired equality of currents by the above three equalities. Thus we have showed the equality (\ref{eqn:desired:thm:arith:Riemann:Roch:gen:finite:morphism}). Since the right hand side belongs in fact to $\widehat{Z}_{L^1}^1(Y)$, the left hand side must also belong to $\widehat{Z}_{L^1}^1(Y)$, and thus we have the equality in $\widehat{Z}_{L^1}^1(Y)$. \QED \section{Arithmetic Riemann-Roch for stable curves} \subsection{Bismut-Bost formula} \setcounter{Theorem}{0} Let $X$ be a smooth algebraic variety over ${\mathbb{C}}$, $L$ a line bundle on $X$, and $h$ a generalized metric of $L$ over $X$. Let $s$ be a rational section of $L$. Then, by the definition of the generalized metric $h$, $-\log h(s, s)$ gives rise to a current $-[\log h(s,s)]$. Moreover, it is easy to see that a current \[ dd^c(-[\log h(s,s)]) + \delta_{\operatorname{div}(s)} \] does not depend on the choice of $s$. Thus, we define $c_1(L, h)$ to be \[ c_1(L, h) = dd^c(-[\log h(s,s)]) + \delta_{\operatorname{div}(s)}. \] Let $f : X \to Y$ be a proper morphism of smooth algebraic varieties ${\mathbb{C}}$ such that every fiber of $f$ is a reduced and connected curve with only ordinary double singularities. We set $\Sigma = \{ x \in X \mid \text{$f$ is not smooth at $x$.} \}$ and $\Delta = f_*(\Sigma)$. Let $|\Delta|$ be the support of $\Delta$. We fix a Hermitian metric of $\omega_{X/Y}$. Then, in \cite{BBQm}, Bismut and Bost proved the following. \begin{Theorem} \label{thm:Quillen:metric:stable:curve} Let $\overline{E} = (E, h)$ be a Hermitian vector bundle on $X$. Then, the Quillen metric $h_Q^{\overline{E}}$ of $\det Rf_*(E)$ on $Y \setminus |\Delta|$ gives rise to a generalized metric of $\det Rf_*(E)$ on $Y$. Moreover, \[ c_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) = - f_* \left[ \operatorname{td}({\overline{\omega}_{X/Y}}^{-1}) \operatorname{ch}(\overline{E}) \right]^{(2,2)} - \frac{\operatorname{rk} E}{12} \delta_{\Delta}. \] \end{Theorem} \subsection{Riemann-Roch for stable curves} \setcounter{Theorem}{0} In this subsection, we prove the arithmetic Riemann-Roch theorem for stable curves. \begin{Theorem} \label{thm:arith:Riemann:Roch:stable:curves} Let $f : X \to Y$ be a projective morphism of regular arithmetic varieties such that every fiber of $f_{{\mathbb{C}}} : X({\mathbb{C}}) \to Y({\mathbb{C}})$ is a reduced and connected curve with only ordinary double singularities. We fix a Hermitian metric of the dualizing sheaf $\omega_{X/Y}$. Let $\overline{E} = (E, h)$ be a Hermitian vector bundle on $X$. Then, \[ \widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) - \operatorname{rk} (E) \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X} \right) \in \widehat{\operatorname{CH}}_{L^1}^1(Y) \] and \begin{multline*} \widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) - \operatorname{rk} (E) \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X} \right) \\ = f_* \left( \frac{1}{2} \left( \widehat{{c}}_1 (\overline{E})^2 - \widehat{{c}}_1 (\overline{E}) \cdot \widehat{{c}}_1 (\overline{\omega}_{X/Y}) \right) - \widehat{{c}}_2 (\overline{E}) \right) \end{multline*} in $\widehat{\operatorname{CH}}_{L^1}^1(Y)_{{\mathbb{Q}}}$, where $h_Q^{\overline{E}}$ and $h_Q^{\overline{{\mathcal{O}}}_X}$ are the Quillen metric of $\det Rf_*(E)$ and $\det Rf_*({\mathcal{O}}_X)$ respectively. \end{Theorem} {\sl Proof.}\quad We prove the theorem in two steps. {\bf Step 1.}\quad First, we assume that $f_{{\mathbb{Q}}} : X_{{\mathbb{Q}}} \to Y_{{\mathbb{Q}}}$ is smooth. In this case, by \cite{GSRR}, \[ \widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) = f_* \left( \widehat{\operatorname{ch}}(E, h)\widehat{\operatorname{td}}(Tf, h_f) - a(\operatorname{ch}(E_{{\mathbb{C}}})\operatorname{td}(Tf_{{\mathbb{C}}})R(Tf_{{\mathbb{C}}})) \right)^{(1)}. \] in $\widehat{\operatorname{CH}}^1(Y)_{{\mathbb{Q}}}$. Since \[ \widehat{\operatorname{ch}}(\overline{E}) = \operatorname{rk}(E) + \widehat{{c}}_1(\overline{E}) + \left( \frac{1}{2} \widehat{{c}}_1(\overline{E})^2 - \widehat{{c}}_2(\overline{E}) \right) + \text{(higher terms)} \] and \[ \widehat{\operatorname{td}}(Tf, h_f) = 1 - \frac{1}{2} \widehat{{c}}_1(\overline{\omega}_{X/Y}) + \widehat{\operatorname{td}}_2(Tf, h_f) + \text{(higher terms)}, \] we have \[ \left( \widehat{\operatorname{ch}}(E, h)\widehat{\operatorname{td}}(Tf, h_f) \right)^{(2)} = \frac{1}{2} \left( \widehat{{c}}_1 (\overline{E})^2 - \widehat{{c}}_1 (\overline{E}) \cdot \widehat{{c}}_1 (\overline{\omega}_{X/Y}) \right) - \widehat{{c}}_2 (\overline{E}) + \operatorname{rk}(E) \widehat{\operatorname{td}}_2(Tf, h_f). \] On the other hand, since the power series $R(x)$ has no constant term, the $(1,1)$ part of \[ \operatorname{ch}(E_{{\mathbb{C}}})\operatorname{td}(Tf_{{\mathbb{C}}})R(Tf_{{\mathbb{C}}}) \] is $\operatorname{rk}(E) R_1(Tf_{{\mathbb{C}}})$, where $R_1(Tf_{{\mathbb{C}}})$ is the $(1,1)$ part of $R(Tf_{{\mathbb{C}}})$. Therefore, we obtain \addtocounter{Claim}{1} \begin{multline} \label{eqn:1:thm:arith:Riemann:Roch:stable:curves} \widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) = f_* \left( \frac{1}{2} \left( \widehat{{c}}_1 (\overline{E})^2 - \widehat{{c}}_1 (\overline{E}) \cdot \widehat{{c}}_1 (\overline{\omega}_{X/Y}) \right) - \widehat{{c}}_2 (\overline{E}) \right) \\ + \operatorname{rk}(E) f_* \left( \widehat{\operatorname{td}}_2(Tf, h_f) - a(R_1(Tf_{{\mathbb{C}}})) \right). \end{multline} Applying (\ref{eqn:1:thm:arith:Riemann:Roch:stable:curves}) to the case $(E, h) = ({\mathcal{O}}_X, h_{can})$, we have \addtocounter{Claim}{1} \begin{equation} \label{eqn:2:thm:arith:Riemann:Roch:stable:curves} \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X} \right) = f_* \left( \widehat{\operatorname{td}}_2(Tf, h_f) - a(R_1(Tf_{{\mathbb{C}}})) \right). \end{equation} Thus, combining (\ref{eqn:1:thm:arith:Riemann:Roch:stable:curves}) and (\ref{eqn:2:thm:arith:Riemann:Roch:stable:curves}), we have our formula in the case where $f_{{\mathbb{Q}}} : X_{{\mathbb{Q}}} \to Y_{{\mathbb{Q}}}$ is smooth. {\bf Step 2.}\quad Next, we consider the general case. The first assertion is a consequence of Theorem~\ref{thm:Quillen:metric:stable:curve} because using Theorem~\ref{thm:Quillen:metric:stable:curve}, \begin{multline*} {c}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) - \operatorname{rk} (E) {c}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X} \right) \\ = - f_* \left[ \operatorname{td}({\overline{\omega}_{X/Y}}^{-1}) \operatorname{ch}(\overline{E}) \right]^{(2,2)} + \operatorname{rk} (E) f_* \left[ \operatorname{td}({\overline{\omega}_{X/Y}}^{-1}) \operatorname{ch}(\overline{{\mathcal{O}}}_X) \right]^{(2,2)} \end{multline*} belongs to $L^1_{\operatorname{loc}}(\Omega_{Y({\mathbb{C}})}^{1,1})$ by Proposition~\ref{prop:push:forward:B:pq}. The second assertion is a consequence of the useful Lemma~\ref{lem:criterion:linear:equiv:B:cycle}. In fact, both sides of the second assertion are arithmetic $L^1$-cycles on Y by the first assertion and the Proposition~\ref{prop:push:forward:arith:cycle}: If we take $\Delta=\{y \in Y_{{\mathbb{Q}}} \mid \text{$f_{{\mathbb{Q}}}$ is not smooth over $y$} \}$ and define $\overline {\Delta}$ to be the closure of $\Delta$ in $Y$, then the compliment $U=Y \setminus \overline {\Delta}$ contains no irreducible components of fibers of $Y \to \operatorname{Spec}({\mathbb{Z}})$ and $f_{{\mathbb{C}}}$ is smooth over $U({\mathbb{C}})$: The arithmetical linear equivalence of both sides restricted to $U$ is a consequence of Step~1. Thus by Lemma~\ref{lem:criterion:linear:equiv:B:cycle}, we also have our formula in the general case. \QED \section{Asymptotic behavior of analytic torsion} \renewcommand{\theTheorem}{\arabic{section}.\arabic{Theorem}} \renewcommand{\theequation}{\arabic{section}.\arabic{Theorem}} Let $M$ be a compact K\"{a}hler manifold of dimension $d$, $\overline{E} = (E, h_E)$ a flat vector bundle of rank $r$ on $M$ with a flat metric $h_E$, and $\overline{A} = (A, h_A)$ a Hermitian vector bundle on $M$. For $0 \leq q \leq d$, let $\Delta_{q,n}$ be the Laplacian on $A^{0,q}\left( \operatorname{Sym}^n(\overline{E}) \otimes \overline{A} \right)$ and $\Delta_{q,n}'$ the restriction of $\Delta_{q,n}$ to $\operatorname{Image} \partial \oplus \operatorname{Image} \overline{\partial}$. Let $\sigma(\Delta_{q,n}') = \{ 0<\lambda_1 \leq \lambda_2 \leq \cdots \}$ be the sequence of eigenvalues of $\Delta_{q,n}'$. Here we count each eigenvalue up to its multiplicity. Then, the associated zeta function $\zeta_{q,n}(s)$ is given by \[ \zeta_{q,n}(s) = \operatorname{Tr}\left[ (\Delta_{q,n}')^{-s} \right] = \sum_{i=1}^{\infty} \lambda_i^{-s}. \] It is well known that $\zeta_{q,n}(s)$ converges absolutely for $\Re(s)>d$ and that it has a meromorphic continuation to the whole complex plane without pole at $s=0$. The analytic torsion $T \left( \operatorname{Sym}^n(\overline{E}) \otimes \overline{A} \right)$ is defined by \[ T \left( \operatorname{Sym}^n(\overline{E}) \otimes \overline{A} \right) = \sum_{q=0}^{d} (-1)^q q \zeta_{q,n}'(0). \] In the following we closely follow \cite[\S 2]{Vojta}. The Theta function associated with $\sigma(\Delta_{q,n}')$ is defined by \[ \Theta_{q,n}(t) = \operatorname{Tr} \left[ \exp (-t \Delta_{q,n}') \right] = \sum_{i=1}^{\infty} e^{- \lambda_i t}. \] By taking Mellin transformation, we have, for $\Re(s)>d$, \[ \zeta_{q,n}(s) = \frac{1}{\Gamma(s)} \int_0^{\infty} \Theta_{q,n}(t) t^s \frac{dt}{t}. \] We put \[ \tilde \zeta_{q,n}(s) = \frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \frac{1}{\Gamma(s)} \int_0^{\infty} \Theta_{q,n}\left(\frac{t}{n}\right) t^s \frac{dt}{t}. \] Then we have \[ \frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \zeta_{q,n}(s) = n^{-s} \tilde \zeta_{q,n}(s) \] and thus \addtocounter{Theorem}{1} \begin{equation} \label{eqn:1:relation:zeta:zero} \frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \zeta_{q,n}'(0) = -(\log n) \tilde \zeta_{q,n}(0) + \zeta_{q,n}'(0) \end{equation} Bismut and Vasserot~\cite[(14),(19)]{BVAT} showed that $\Theta_{q,n}(t)$ has the following properties (note that these parts of \cite{BVAT} do not depend on the assumption of positivity of a line bundle, as indicated in Vojta \cite[Proposition~2.7.3]{Vojta}): \begin{enumerate} \renewcommand{\labelenumi}{(\alph{enumi})} \item For every $k \in {\mathbb{N}}$, $0 \leq q \leq d$ and $n \in {\mathbb{N}}$, there are real numbers $a_{q,n}^j\;(-d \leq j \leq k)$ such that \[ \frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \Theta_{q,n}\left(\frac{t}{n}\right) = \sum_{j=-d}^{k} a_{q,n}^j t^j + o(t^k) \] as $t \downarrow 0$, with $o(t^k)$ uniform with respect to $n \in {\mathbb{N}}$. \item For every $0 \leq q \leq d$ and $j \geq -d$, there are real numbers $a_{q}^j$ such that \[ a_{q,n}^j = a_{q}^j + O\left(\frac{1}{\sqrt n}\right) \] as $n \to \infty$. \end{enumerate} Also by (b), we can replace the $o(t^k)$ in (a) by $O(t^{k+1})$ and still have the uniformity statement. Thus we can write, for every $k \in {\mathbb{N}}$, \[ \frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \Theta_{q,n}\left(\frac{t}{n}\right) = \sum_{j=-d}^{k} a_{q,n}^k t^j + \rho_{q,n}^k(t) \] with $\rho_{q,n}^k(t) = o(t^{k+1})$. Then {\allowdisplaybreaks \begin{align*} \tilde \zeta_{q,n}(s) & = \frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \frac{1}{\Gamma(s)} \int_1^{\infty} \Theta_{q,n}\left(\frac{t}{n}\right) t^s \frac{dt}{t} \\ & \qquad + \frac{a_{q,n}^j}{\Gamma(s)} \int_0^1 t^{j+s-1} dt + \sum_{j=-d}^{k} \frac{1}{\Gamma(s)} \int_0^1 \rho_{q,n}^k(t) dt \\ & = \frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \frac{1}{\Gamma(s)} \int_1^{\infty} \Theta_{q,n}\left(\frac{t}{n}\right) t^s \frac{dt}{t} \\ & \qquad + \sum_{j=-d}^{k} \frac{a_{q,n}^j}{\Gamma(s) (j+s)} + \frac{1}{\Gamma(s)} \int_0^1 \rho_{q,n}^k(t)t^s \frac{dt}{t}. \end{align*} } In the last expression, the first integral is holomorphic for all $s \in {\mathbb{C}}$, while the second integral is holomorphic for $\Re (s) > -k-1$; the middle term is a meromorphic function in the whole complex plane. Putting $k=0$ and $s=0$ in the above equation, we have \addtocounter{Theorem}{1} \begin{equation} \label{eqn:3:zeta:zero} \tilde \zeta_{q,n}(0) = a_{q,n}^0. \end{equation} Moreover, by differentiating the above equation when $k=0$, we have \addtocounter{Theorem}{1} \begin{multline} \label{eqn:4:diff:zeta:zero} \tilde \zeta_{q,n}'(0) = \frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \int_1^{\infty} \Theta_{q,n}\left(\frac{t}{n}\right) \frac{dt}{t} \\ + \sum_{j=-d}^{-1} \frac{a_{q,n}^j}{j} - a_{q,n}^0 \Gamma'(1) + \frac{1}{\Gamma(s)} \int_0^1 \rho_{q,n}^0(t) \frac{dt}{t}. \end{multline} We have now the following Proposition. \begin{Proposition} \label{prop:lower:bound:zeta:function} There exists a constant c such that for all $n \in {\mathbb{N}}$, \[ \zeta_{q,n}'(0) \geq -c n^{d+r-1} \log n \] \end{Proposition} {\sl Proof.}\quad By (\ref{eqn:1:relation:zeta:zero}), (\ref{eqn:3:zeta:zero}) and (\ref{eqn:4:diff:zeta:zero}), we have \begin{multline*} \zeta_{q,n}'(0) = - \operatorname{rk} (\operatorname{Sym}^n(E)) n^d (\log n) a_{q,n}^0 \\ + \operatorname{rk} (\operatorname{Sym}^n(E)) n^d \left( \frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \int_1^{\infty} \Theta_{q,n}\left(\frac{t}{n}\right) \frac{dt}{t} \right. \\ \left. + \sum_{j=-d}^{-1} \frac{a_{q,n}^j}{j} - a_{q,n}^0 \Gamma'(1) + \frac{1}{\Gamma(s)} \int_0^1 \rho_{q,n}^0(t) \frac{dt}{t} \right) \end{multline*} In the first term of the right hand side, $a_{q,n}^0$ is bounded with respect to $n$ by (b). In the second term of the right hand side, the first integral is non-negative; the sum of $a_{q,n}^j$'s is bounded with respect to $n$ by (b); the term $- a_{q,n}^0 \Gamma'(1)$ is also bounded with respect to $n$ by (b); the second integral is also bounded with respect to $n$, for $\rho_{q,n}^0(t)=O(t)$ uniformly with respect to $n$. Moreover, \[ \operatorname{rk} (\operatorname{Sym}^n(E)) = \binom{n+r-1}{r-1}=O(n^{r-1}) \] as $n \to \infty$. Thus, there is a constant $c$ such that for all $n \in {\mathbb{N}}$, \[ \zeta_{q,n}'(0) \geq -c n^{d+r-1} \log n. \] \QED \medskip In the following sections, we only need the case of $d=1$, namely where $M$ is a compact Riemann surface. In this case, the above Proposition~\ref{prop:lower:bound:zeta:function} gives an asymptotic upper bound of analytic torsion. \begin{Corollary} \label{cor:asymp:analytic:torsion} Let $C$ be a compact Riemann surface, $\overline{E} = (E, h_E)$ a flat vector bundle of rank $r$ on $C$ with a flat metric $h$, and $\overline{A} = (A, h_A)$ a Hermitian vector bundle on $C$. Then, there is a constant $c$ such that for all $n \in {\mathbb{N}}$, \[ T \left( \operatorname{Sym}^n(\overline{E}) \otimes \overline{A} \right) \leq c n^r \log n. \] \end{Corollary} {\sl Proof.}\quad Since $\dim C = 1$ \[ T \left( \operatorname{Sym}^n(\overline{E}) \otimes \overline{A} \right) = -\zeta_{1,n}'(0). \] Now the corollary follows from Proposition~\ref{prop:lower:bound:zeta:function}. \QED \renewcommand{\theTheorem}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}} \renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}} \section{Formulae for arithmetic Chern classes} \subsection{Arithmetic Chern classes of symmetric powers} \label{subsec:formula:chern:sym:power} \setcounter{Theorem}{0} Let $M$ be a complex manifold and $(E, h)$ a Hermitian vector bundle on $M$. Since $E^{\otimes n}$ has the natural Hermitian metric $h^{\otimes n}$, we can define a Hermitian metric $\operatorname{Sym}^n(h)$ of $\operatorname{Sym}^n(E)$ to be the quotient metric of $E^{\otimes n}$ in terms of the natural surjective homomorphism $E^{\otimes n} \to \operatorname{Sym}^n(E)$. We denote $(\operatorname{Sym}^n(E), \operatorname{Sym}^n(h))$ by $\operatorname{Sym}^n(E, h)$. If $x \in M$ and $\{ e_1, \ldots, e_r \}$ is an orthonormal basis of $E_x$ with respect to $h_x$, then it is easy to see that \[ (\operatorname{Sym}^n(h))_x \left( e_1^{\alpha_1} \cdots e_r^{\alpha_r}, e_1^{\beta_1} \cdots e_r^{\beta_r} \right) = \begin{cases} {\displaystyle \frac{\alpha_1 ! \cdots \alpha_r !}{n !}} & \text{if $(\alpha_1, \ldots, \alpha_r) = (\beta_1, \ldots, \beta_r)$}, \\ 0 & \text{otherwise}. \end{cases} \] Then we have the following proposition. \begin{Proposition} \label{prop:chern:class:sym:power} Let $X$ be an arithmetic variety and $\overline{E} = (E, h)$ a Hermitian vector bundle of rank $r$ on $X$. Then, we have the following. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item ${\displaystyle \widehat{{c}}_1 \left( \operatorname{Sym}^n(\overline{E}) \right) = \frac{n}{r} \binom{n+r-1}{r-1} \widehat{{c}}_1(\overline{E}) + a \left( \sum_{\substack{\alpha_1 + \cdots + \alpha_r = n, \\ \alpha_1 \geq 0, \ldots, \alpha_r \geq 0}} \log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_r !} \right) \right)}$. \item If $X$ is regular, then \begin{multline*} \widehat{\operatorname{ch}}_2 \left( \operatorname{Sym}^n(\overline{E}) \right) = \binom{n+r}{r+1} \widehat{\operatorname{ch}}_2(\overline{E}) + \frac{1}{2} \binom{n+r-1}{r+1} \widehat{{c}}_1(\overline{E})^2 \\ + a \left( \frac{n}{r} \sum_{\substack{\alpha_1 + \cdots + \alpha_r = n, \\ \alpha_1 \geq 0, \ldots, \alpha_r \geq 0}} \log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_r !} \right) c_1(\overline{E}) \right). \end{multline*} \end{enumerate} \end{Proposition} {\sl Proof.}\quad In \cite{SoVan}, C. Soul\'{e} gives similar formulae in implicit forms. We follow his idea to calculate them. \medskip (1) First of all, we fix notation. We set \[ S_{r, n} = \{ (\alpha_1, \ldots, \alpha_r) \in ({\mathbb{Z}}_{+})^r \mid \alpha_1 + \cdots + \alpha_r = n \}, \] where ${\mathbb{Z}}_{+} = \{ x \in {\mathbb{Z}} | x \geq 0 \}$. For $I = (\alpha_1, \ldots, \alpha_r) \in S_{r, n}$ and rational sections $s_1, \ldots, s_r$ of $E$, we denote $s_1^{\alpha_1} \cdots s_r^{\alpha_r}$ by $s^I$ and $\alpha_1 ! \cdots \alpha_r !$ by $I !$. Let $s_1, \ldots, s_r$ be independent rational sections of $E$. Then, $\{ s^I \}_{I \in S_{r,n}}$ forms independent rational sections of $\operatorname{Sym}^n(E)$. First, let us see that \addtocounter{Claim}{1} \begin{equation} \label{eqn:1:prop:chern:class:sym:power} \operatorname{div} \left( \bigwedge_{I \in S_{r,n}} s^I \right) = \frac{n}{r} \binom{n+r-1}{r-1} \operatorname{div} ( s_1 \wedge \cdots \wedge s_r ). \end{equation} This is a local question. So let $x \in X$ and $\{ \omega_1, \ldots, \omega_r \}$ be a local basis of $E$ around $x$. We set $s_i = \sum_{j=1}^r a_{ij}\omega_j$. Then, $s_1 \wedge \cdots \wedge s_r = \det(a_{ij}) \omega_1 \wedge \cdots \wedge \omega_r$. Let $K$ be a rational function field of $X$. Since the characteristic of $K$ is zero, any $1$-dimensional representation of $\operatorname{GL}_r(K)$ is a power of the determinant. Thus, there is an integer $N$ with \[ \bigwedge_{I \in S_{r,n}} s^I = \det(a_{ij})^N \bigwedge_{I \in S_{r,n}} \omega^I. \] Here, by an easy calculation, we can see that \[ N = \frac{n}{r} \binom{n+r-1}{r-1}. \] Thus, we get (\ref{eqn:1:prop:chern:class:sym:power}). Next, let us see that \addtocounter{Claim}{1} \begin{multline} \label{eqn:2:prop:chern:class:sym:power} - \log \det \left( \operatorname{Sym}^n(h)(s^I, s^J) \right)_{I,J \in S_{r,n}} = \\ - \frac{n}{r} \binom{n+r-1}{r-1} \log \det (h(s_i, s_j))_{i,j} + \sum_{I \in S_{r,n}} \log \left( \frac{n !}{I !} \right). \end{multline} Let $x \in X({\mathbb{C}})$ and $\{ e_1, \ldots, e_r \}$ an orthonormal basis of $E \otimes \kappa(x)$. We set $s_i = \sum_{i=1}^r b_{ij}e_j$. Moreover, we set $s^I = \sum_{J \in S_{r,n}} b_{IJ}e^J$. Then, in the same way as before, $\det (b_{IJ}) = \det (b_{ij})^N$. Further, since \[ \operatorname{Sym}^n(h)(s^I, s^J) = \sum_{I', J' \in S_{r,n}} b_{II'}\operatorname{Sym}^n(h)(e^{I'}, e^{J'}) \overline{b_{J'J}}, \] we have \begin{align*} \det \left( \operatorname{Sym}^n(h)(s^I, s^J) \right)_{I,J \in S_{r,n}} & = |\det(b_{IJ})|^2 \det \left( \operatorname{Sym}^n(h)(e^I, e^J) \right)_{I,J \in S_{r,n}} \\ & = |\det(b_{ij})|^{2N} \prod_{I \in S_{r,n}} \frac{I !}{n !}. \end{align*} Thus, we get (\ref{eqn:2:prop:chern:class:sym:power}). Therefore, combining (\ref{eqn:1:prop:chern:class:sym:power}) and (\ref{eqn:2:prop:chern:class:sym:power}), we obtain (1). \medskip (2) First, we recall an elementary fact. Let $\Phi \in {\mathbb{R}}[X_1, \ldots, X_r]$ be a symmetric homogeneous polynomial, and $M_r({\mathbb{C}})$ the algebra of complex $r \times r$ matrices. Then, there is a unique polynomial map $\underline{\Phi} : M_r({\mathbb{C}}) \to {\mathbb{C}}$ such that $\underline{\Phi}$ is invariant under conjugation by $\operatorname{GL}_r({\mathbb{C}})$ and its value on a diagonal matrix $\operatorname{diag}(\lambda_1, \ldots, \lambda_r)$ is equal to $\Phi(\lambda_1, \ldots, \lambda_r)$. Let us consider the natural homomorphism \[ \rho_{r,n} : \operatorname{Aut}_{{\mathbb{C}}}({\mathbb{C}}^r) \to \operatorname{Aut}_{{\mathbb{C}}}(\operatorname{Sym}^n({\mathbb{C}}^r)) \] as complex Lie groups, which induces a homomorphism \[ \gamma_{r, n} = d(\rho_{r,n})_{\operatorname{id}} : \operatorname{End}_{{\mathbb{C}}}({\mathbb{C}}^r) \to \operatorname{End}_{{\mathbb{C}}}(\operatorname{Sym}^n({\mathbb{C}}^r)) \] as complex Lie algebras. Let $\{ e_1, \ldots, e_r \}$ be the standard basis of ${\mathbb{C}}^r$. Then, $\{ e_I \}_{I \in S_{r,n}}$ forms a basis of $\operatorname{Sym}^n({\mathbb{C}}^r)$, where $e_I = e_{1}^{\alpha_1} \cdots e_r^{\alpha_r}$ for $I = (\alpha_1, \ldots, \alpha_r)$. Let us consider the symmetric polynomial \[ \operatorname{ch}_2^{r,n} = \frac{1}{2} \sum_{I \in S_{r,n}} X_I^2 \] in ${\mathbb{R}}[ X_I ]_{I \in S_{r,n}}$. Then, by the previous remark, using the basis $\{ e_I \}_{I \in S_{r,n}}$, we have a polynomial map \[ \underline{\operatorname{ch}_2^{r,n}} : \operatorname{End}_{{\mathbb{C}}}(\operatorname{Sym}^n({\mathbb{C}}^r)) \to {\mathbb{C}} \] such that $\underline{\operatorname{ch}_2^{r,n}}$ is invariant under conjugation by $\operatorname{Aut}_{{\mathbb{C}}}(\operatorname{Sym}^n({\mathbb{C}}^r))$ and \[ \underline{\operatorname{ch}_2^{r,n}} \left( \operatorname{diag}(\lambda_I)_{I \in S_{r, n}} \right) = \operatorname{ch}_2^{r,n}(\ldots, \lambda_I, \ldots). \] Here we consider a polynomial map given by \[ \begin{CD} \theta_{r, n} : \operatorname{End}_{{\mathbb{C}}}({\mathbb{C}}^r) @>{\gamma_{r,n}}>> \operatorname{End}_{{\mathbb{C}}}(\operatorname{Sym}^n({\mathbb{C}}^r)) @>{\underline{\operatorname{ch}_2^{r,n}}}>> {\mathbb{C}}. \end{CD} \] Since $\gamma_{r,n}(P A P^{-1}) = \rho_{r,n}(P)\gamma_{r,n}(A)\rho_{r,n}(P)^{-1}$ for all $A \in \operatorname{End}_{{\mathbb{C}}}({\mathbb{C}}^r)$ and $P \in \operatorname{Aut}_{{\mathbb{C}}}({\mathbb{C}}^r)$, $\theta_{r,n}$ is invariant under conjugation by $\operatorname{Aut}_{{\mathbb{C}}}({\mathbb{C}}^r)$. Let us calculate \[ \theta_{r,n}(\operatorname{diag}(\lambda_1, \ldots, \lambda_r)). \] First of all, \[ \gamma_{r,n}(\operatorname{diag}(\lambda_1, \ldots, \lambda_r)) = \operatorname{diag}\left( \ldots, \left( \alpha_1 \lambda_1 + \cdots + \alpha_r \lambda_r \right), \ldots \right)_{(\alpha_1, \ldots, \alpha_r) \in S_{r,n}}. \] Thus, \[ \theta_{r,n}(\operatorname{diag}(\lambda_1, \ldots, \lambda_r)) = \frac{1}{2} \sum_{(\alpha_1, \ldots, \alpha_r) \in S_{r,n}} \left( \alpha_1 \lambda_1 + \cdots + \alpha_r \lambda_r \right)^2. \] On the other hand, by easy calculations, we can see that \[ \sum_{(\alpha_1, \ldots, \alpha_r) \in S_{r,n}} \left( \alpha_1 \lambda_1 + \cdots + \alpha_r \lambda_r \right)^2 = \binom{n+r}{r+1}\left( \lambda_1^2 + \cdots + \lambda_r^2 \right) + \binom{n+r-1}{r+1}\left( \lambda_1 + \cdots + \lambda_r \right)^2. \] Therefore, we get \[ \theta_{r,n}(\operatorname{diag}(\lambda_1, \ldots, \lambda_r)) = \frac{1}{2} \binom{n+r}{r+1}\left( \lambda_1^2 + \cdots + \lambda_r^2 \right) + \frac{1}{2} \binom{n+r-1}{r+1}\left( \lambda_1 + \cdots + \lambda_r \right)^2. \] Hence, \addtocounter{Claim}{1} \begin{equation} \label{eqn:1:(2):prop:chern:class:sym:power} \theta_{r,n} = \binom{n+r}{r+1} \underline{\operatorname{ch}_2} + \frac{1}{2} \binom{n+r-1}{r+1} \underline{(c_1)^2}, \end{equation} where ${\displaystyle \operatorname{ch}_2(X_1, \ldots, X_r) = \frac{1}{2}(X_1^2 + \cdots + X_r^2)}$ and $c_1(X_1, \ldots, X_r) = X_1 + \cdots + X_r$. Let $M$ be a complex manifold and $\overline{F} = (F, h_F)$ a Hermitian vector bundle of rank $r$ on $M$. Let $K_{\overline{F}}$ be the curvature form of $\overline{F}$, and $K_{\operatorname{Sym}^n(\overline{F})}$ the curvature form of $\operatorname{Sym}^n(\overline{F})$. Then, \[ K_{\operatorname{Sym}^n(\overline{F})} = \left( \gamma_{r, n} \otimes \operatorname{id}_{A^{1,1}(M)} \right) (K_{\overline{F}}). \] Thus, by (\ref{eqn:1:(2):prop:chern:class:sym:power}), \addtocounter{Claim}{1} \begin{equation} \label{eqn:2:(2):prop:chern:class:sym:power} \operatorname{ch}_2 \left( \operatorname{Sym}^n(F,h_F) \right) = \binom{n+r}{r+1} \operatorname{ch}_2(F,h_F) + \frac{1}{2} \binom{n+r-1}{r+1} {c}_1(F,h_F)^2. \end{equation} Now let $\overline{E} = (E, h)$ be a Hermitian vector bundle on a regular arithmetic variety $X$. Let $h'$ be another Hermitian metric of $E$. Then, using the definition of Bott-Chern secondary characteristic classes and (\ref{eqn:2:(2):prop:chern:class:sym:power}), \begin{multline*} \widehat{\operatorname{ch}}_2 \left( \operatorname{Sym}^n(E, h) \right) - \widehat{\operatorname{ch}}_2 \left( \operatorname{Sym}^n(E, h') \right) = \\ a \left( \binom{n+r}{r+1} \widetilde{\operatorname{ch}_2}(E, h, h') + \frac{1}{2} \binom{n+r-1}{r+1} \widetilde{{c}_1^2}(E, h, h') \right). \end{multline*} Thus, \[ \widehat{\operatorname{ch}}_2 \left( \operatorname{Sym}^n(E, h) \right) - \binom{n+r}{r+1} \widehat{\operatorname{ch}}_2(E, h) - \frac{1}{2} \binom{n+r-1}{r+1} \widehat{{c}}_1(E, h)^2 \] does not depend on the choice of the metric $h$. Therefore, in order to show (2), by using splitting principle \cite[3.3.2]{GSCh}, we may assume that \[ (E, h) = \overline{L}_1 \oplus \cdots \oplus \overline{L}_r, \] where $\overline{L}_i = (L_i, h_i)$'s are Hermitian line bundles. Then, \[ \operatorname{Sym}^n(\overline{E}) = \bigoplus_{\substack{\alpha_1 + \cdots + \alpha_r = n, \\ \alpha_1 \geq 0, \ldots, \alpha_r \geq 0}} \overline{L}_1^{\otimes \alpha_1} \otimes \cdots \otimes \overline{L}_r^{\otimes \alpha_r} \otimes \left( {\mathcal{O}}_X, \frac{\alpha_1 ! \cdots \alpha_r !}{n!} h_{can} \right). \] Therefore, $\widehat{\operatorname{ch}}_2 \left( \operatorname{Sym}^n(\overline{E}) \right)$ is equal to \[ \sum_{\substack{\alpha_1 + \cdots + \alpha_r = n, \\ \alpha_1 \geq 0, \ldots, \alpha_r \geq 0}} \left\{ \widehat{\operatorname{ch}}_2 \left( \overline{L}_1^{\otimes \alpha_1} \otimes \cdots \otimes \overline{L}_r^{\otimes \alpha_r} \right) -\log \left( \frac{\alpha_1 ! \cdots \alpha_r !}{n!} \right) a\left( {c}_1 \left( \overline{L}_1^{\otimes \alpha_1} \otimes \cdots \otimes \overline{L}_r^{\otimes \alpha_r} \right) \right) \right\}. \] On the other hand, since \begin{multline*} \sum_{(\alpha_1, \cdots, \alpha_r) \in S_{r,n}} \log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_r !} \right) \left( \alpha_1 X_1 + \cdots + \alpha_r X_r \right) \\ = \left( \frac{n}{r} \sum_{(\alpha_1, \cdots, \alpha_r) \in S_{r,n}} \log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_r !} \right) \right)(X_1 + \cdots + X_r), \end{multline*} we have {\allowdisplaybreaks \begin{align*} \widehat{\operatorname{ch}}_2 \left( \operatorname{Sym}^n(\overline{E}) \right) = & \binom{n+r}{r+1} \widehat{\operatorname{ch}}_2(\overline{E}) + \frac{1}{2} \binom{n+r-1}{r+1} \widehat{{c}}_1(\overline{E})^2 \\ & + \sum_{(\alpha_1, \cdots, \alpha_r) \in S_{r,n}} \log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_r !} \right) a \left( \alpha_1 {c}_1(\overline{L_1}) + \cdots + \alpha_r{c}_1(\overline{L_r}) \right) \\ = & \binom{n+r}{r+1} \widehat{\operatorname{ch}}_2(\overline{E}) + \frac{1}{2} \binom{n+r-1}{r+1} \widehat{{c}}_1(\overline{E})^2 \\ & \qquad + \left( \frac{n}{r} \sum_{(\alpha_1, \cdots, \alpha_r) \in S_{r,n}} \log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_r !} \right) \right) a ( {c}_1(\overline{E})). \end{align*} } Thus, we get (2). \QED \subsection{Arithmetic Chern classes of $\overline{E} \otimes \overline{E}^{\vee}$} \setcounter{Theorem}{0} Here, let us consider arithmetic Chern classes of $\overline{E} \otimes \overline{E}^{\vee}$. \begin{Proposition} \label{prop:ch2:end} Let $X$ be a regular arithmetic variety and $(E, h)$ a Hermitian vector bundle of rank $r$ on $X$. Then, \[ \widehat{\operatorname{ch}}_2(E \otimes E^{\vee}, h \otimes h^{\vee}) = 2r \widehat{\operatorname{ch}}_2(E, h) - \widehat{{c}}_1(E, h)^2 = (r-1) \widehat{{c}}_1(E, h)^2 - 2r \widehat{{c}}_2(E, h). \] \end{Proposition} {\sl Proof.}\quad Since $\widehat{\operatorname{ch}}_i(E^{\vee}, h^{\vee}) = (-1)^i \widehat{\operatorname{ch}}_i(E, h)$ and $\widehat{\operatorname{ch}}(E \otimes E^{\vee}, h \otimes h^{\vee}) = \widehat{\operatorname{ch}}(E, h) \cdot \widehat{\operatorname{ch}}(E^{\vee}, h^{\vee})$, we have \begin{align*} \widehat{\operatorname{ch}}_2(E \otimes E^{\vee}, h \otimes h^{\vee}) & = r \widehat{\operatorname{ch}}_2(E, h) + \widehat{{c}}_1(E, h) \cdot \widehat{{c}}_1(E^{\vee}, h^{\vee}) + r \widehat{\operatorname{ch}}_2(E^{\vee}, h^{\vee}) \\ & = 2r \widehat{\operatorname{ch}}_2(E, h) - \widehat{{c}}_1(E, h)^2. \end{align*} The last assertion is derived from the fact \[ \widehat{\operatorname{ch}}_2(E, h) = \frac{1}{2} \widehat{{c}}_1(E,h)^2 - \widehat{{c}}_2(E, h). \] \QED \section{The proof of the relative Bogomolov's inequality in the arithmetic case} \renewcommand{\theTheorem}{\arabic{section}.\arabic{subsection}} The purpose of this section is to give the proof of the following theorem. \addtocounter{subsection}{1} \begin{Theorem}[Relative Bogomolov's inequality in the arithmetic case] \label{thm:relative:Bogomolov:inequality:arithmetic:case} Let $f : X \to Y$ be a projective morphism of regular arithmetic varieties such that every fiber of $f_{{\mathbb{C}}} : X({\mathbb{C}}) \to Y({\mathbb{C}})$ is a reduced and connected curve with only ordinary double singularities. Let $(E, h)$ be a Hermitian vector bundle of rank $r$ on $X$, and $y$ a closed point of $Y_{{\mathbb{Q}}}$. If $f$ is smooth over $y$ and $\rest{E}{X_{\bar{y}}}$ is semi-stable, then \[ \widehat{\operatorname{dis}}_{X/Y}(E, h) = f_* \left( 2r \widehat{{c}}_2(E, h) - (r-1)\widehat{{c}}_1(E, h)^2 \right) \] is weakly positive at $y$ with respect to any subsets $S$ of $Y({\mathbb{C}})$ with the following properties: \textup{(1)} $S$ is finite, and \textup{(2)} $f_{{\mathbb{C}}}^{-1}(z)$ is smooth and $\rest{E_{{\mathbb{C}}}}{f_{{\mathbb{C}}}^{-1}(z)}$ is poly-stable for all $z \in S$. \end{Theorem} \renewcommand{\theTheorem}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}} \subsection{Sketch of the proof of the relative Bogomolov's inequality} The proof of the relative Bogomolov's inequality is very long, so that for reader's convenience, we would like to give a rough sketch of the proof of it. \bigskip {\bf Step 1.}\quad Using the Donaldson's Lagrangian, we reduce to the case where the Hermitian metric $h$ of $E$ along $f_{{\mathbb{C}}}^{-1}(z)$ is Einstein-Hermitian for each $z \in S$. \medskip {\bf Step 2.}\quad We set \[ \overline{F}_n = \operatorname{Sym}^n \left( \operatorname{\mathcal{E}\textsl{nd}}(\overline{E}) \otimes f^*(\overline{H}) \right) \otimes \overline{A} \otimes f^*(\overline{H}), \] where $\overline{A}$ is a Hermitian line bundle on $X$ and $\overline{H}$ is a Hermitian line bundle on $Y$. Later we will specify these $\overline{A}$ and $\overline{H}$. By virtue of the arithmetic Riemann-Roch for stable curves (cf. Theorem~\ref{thm:arith:Riemann:Roch:stable:curves}) and formulae of arithmetic Chern classes for symmetric powers (cf. \S\ref{subsec:formula:chern:sym:power}), we can see that \[ \frac{1}{(r^2 + 1)!} \widehat{\operatorname{dis}}_{X/Y}(\overline{E}) = - \lim_{n \to \infty} \frac{\widehat{{c}}_1(\det Rf_*(F_n), h_n)}{n^{r^2+1}}, \] where $h_n$ is a generalized metric of $\det Rf_*(F_n)$ such that $\widehat{{c}}_1(\det Rf_*(F_n), h_n) \in \widehat{\operatorname{CH}}_{L^1}^1(Y)$ and $h_n$ coincides with the Quillen metric $h_Q^{\overline{F}_n}$ at each $z \in S$. \medskip {\bf Step 3.}\quad We assume that $A$ is very ample and $A \otimes \omega_{X/Y}^{-1}$ is ample. We choose an arithmetic variety $B \subset X$ such that $B \in |A^{\otimes 2}|$, $B \to Y$ is \'{e}tale over $y$, and $B({\mathbb{C}}) \to Y({\mathbb{C}})$ is \'{e}tale over each $z \in S$. (Exactly speaking, $B$ is not realized as an element of $|A^{\otimes 2}|$. For simplicity, we assume it.) We set $\overline{G}_n = \rest{\overline{F}_n}{B}$ and $g = \rest{f}{B}$. Here we suppose that $g_*(\rest{\operatorname{\mathcal{E}\textsl{nd}}(\overline{E})}{B}) \otimes \overline{H}$ and $g_*(\rest{\overline{A}}{B}) \otimes \overline{H}$ are generated by small sections at $y$ with respect to $S$. Applying the Riemann-Roch formula for generically finite morphisms (cf. Theorem~\ref{thm:arith:Riemann:Roch:gen:finite:morphism}), we can find a generalized metric $g_n$ of $\det g_*(G_n)$ such that $g_n$ is equal to the Quillen metric of $\overline{G}_n$ at each $z \in S$, $\widehat{{c}}_1(\det g_*(G_n), g_n) \in \widehat{\operatorname{CH}}_{L^1}^1(Y)$, and \[ \lim_{n \to \infty} \frac{\widehat{{c}}_1(\det g_*(G_n), g_n)}{n^{r^2+1}} = 0. \] Let us consider the exact sequence: \[ 0 \to f_*(F_n) \to g_*(G_n) \to R^1f_*(F_n \otimes A^{\otimes -2}) \] induced by $0 \to F_n \otimes A^{\otimes -2} \to F_n \to G_n \to 0$. Let $Q_n$ be the image of \[ g_*(G_n) \to R^1f_*(F_n \otimes A^{\otimes -2}). \] The natural $L^2$-metric of $g_*(G_n)$ around $z$ induces the quotient metric $\tilde{q}_n$ of $Q_n$ around $z$ for each $z \in S$. Thus, we can find a $C^{\infty}$ metric $q_n$ of $\det Q_n$ such that $q_n$ is equal to $\det \tilde{q}_n$ at each $z \in S$. Since \[ \det Rf_*(F_n) = \det g_*(G_n) \otimes (\det Q_n)^{\otimes -1} \otimes \left( \det R^1f_*(F_n) \right)^{\otimes -1}, \] we have the generalized metric $t_n$ of $\det R^1f_*(F_n)$ such that \[ (\det Rf_*(F_n), h_n) = (\det g_*(G_n), g_n) \otimes (\det Q_n, q_n)^{\otimes -1} \otimes (\det R^1f_*(F_n), t_n)^{\otimes -1}. \] {\bf Step 4.}\quad We set $ a_n = \max_{z \in S} \{ \log t_n(s_n,s_n)(z) \}$, where $s_n$ is the canonical section of $\det R^1f_*(F_n)$. In this step, we will show that $\widehat{{c}}_1(\det Q_n, q_n)$ is semi-ample at $y$ with respect to $S$ and $a_n \leq O(n^{r^2}\log(n))$. The semi-ampleness of $\widehat{{c}}_1(\det Q_n, q_n)$ at $y$ is derived from Proposition~\ref{prop:find:small:section} and the fact that $g_*(\rest{\operatorname{\mathcal{E}\textsl{nd}}(\overline{E})}{B}) \otimes \overline{H}$ and $g_*(\rest{\overline{A}}{B}) \otimes \overline{H}$ are generated by small sections at $y$ with respect to $S$. The estimation of $a_n$ involves asymptotic behavior of analytic torsion (cf. Corollary~\ref{cor:asymp:analytic:torsion}) and a comparison of sup-norm with $L^2$-norm (cf. Lemma~\ref{lem:comparison:sup:L2}). {\bf Step 5.}\quad Thus, using the last equation in Step 3, we can get a decomposition \[ -\frac{\widehat{{c}}_1(\det Rf_*(F_n), h_n)}{n^{r^2+1}} = \alpha_n + \beta_n \] such that $\alpha_n$ is semi-ample at $y$ with respect to $S$ and $\lim_{n \to \infty} \beta_n = 0$. \subsection{Preliminaries} \setcounter{Theorem}{0} First of all, we will prepare three lemmas for the proof of the relative Bogomolov's inequality. \begin{Lemma} \label{lem:comparison:sup:L2} Let $M$ be a $d$-dimensional compact K\"{a}hler manifold, $\overline{E} = (E, h)$ a flat Hermitian vector bundle of rank $r$ on $M$, and $\overline{V} = (V, k)$ a Hermitian line bundle. Then, there is a constant $c$ such that, for any $n > 0$ and any $s \in H^0(M, \operatorname{Sym}^n(E) \otimes V)$, \[ \Vert s \Vert_{\sup} \leq c n^{d+r-1} \Vert s \Vert_{L^2}. \] \end{Lemma} {\sl Proof.}\quad Let $f : P = \operatorname{Proj} \left(\bigoplus_{i \geq 0} \operatorname{Sym}^i(E) \right) \to M$ be the projective bundle of $E$, and $L = {\mathcal{O}}_P(1)$ the tautological line bundle of $E$ on $P$. Let $h_L$ be the quotient metric of $L$ induced by the surjective homomorphism $f^*(E) \to L$ and the Hermitian metric $f^*(h)$ of $f^*(E)$. Let $\Omega_M$ be a K\"{a}hler form of $M$. Since $\overline{E}$ is flat, $c_1(L, h_L)$ is positive semi-definite of rank $r-1$. Thus, $f^*(\Omega_M) + c_1(L, h_L)$ gives rise to a fundamental $2$-form $\Omega_P$ on $P$. Moreover, by virtue of the flatness of $\overline{E}$, we have $c_1(L, h_L)^r = 0$. Thus, \[ \Omega_P^{d+r-1} = \binom{d+r-1}{d} f^*(\Omega_M^d) \wedge c_1(L, h_L)^{r-1}. \] By \cite[Lemma~30]{GSRR}, there is a constant $c$ such that \[ \Vert s' \Vert_{\sup} \leq c n^{d+r-1} \Vert s' \Vert_{L^2} \] for any $n > 0$ and any $s' \in H^0(P, L^{\otimes n} \otimes f^*(V))$, where ${\displaystyle \Vert s' \Vert_{L^2} = \int_{P} |s'|^2 \Omega_P^{d+r-1}}$. We denote a homomorphism \[ f^*(\operatorname{Sym}^n(E)) \otimes f^*(V) \to L^{\otimes n} \otimes f^*(V) \] by $\alpha_n$. As in the proof of \cite[(44)]{GSRR}, we can see that, for any $s \in H^0(M, \operatorname{Sym}^n(E) \otimes V)$, \[ |s|^2 = \binom{n+r-1}{r-1} \int_{P \to M} | \alpha_n(s) |^2 c_1(L, h_L)^{r-1}. \] Thus, \[ |s|^2 \leq \binom{n+r-1}{r-1} \int_{P \to M} \Vert \alpha(s) \Vert_{\sup}^2 c_1(L, h_L)^{r-1} = \binom{n+r-1}{r-1} \Vert \alpha(s) \Vert_{\sup}^2. \] Therefore, we get \[ \Vert s \Vert_{\sup}^2 \leq \binom{n+r-1}{r-1} \Vert \alpha_n(s) \Vert_{\sup}^2 \] for all $s \in H^0(M, \operatorname{Sym}^n(E) \otimes V)$. On the other hand, {\allowdisplaybreaks \begin{align*} \Vert \alpha_n(s) \Vert_{L^2}^2 & = \int_{P} |\alpha_n(s)|^2 \Omega_P^{r} \\ & = \binom{d+r-1}{d} \int_M \int_{P \to M} |\alpha_n(s)|^2 f^*(\Omega_M^d) \wedge c_1(L, h_L)^{r-1} \\ & = \binom{d+r-1}{d} \int_M \Omega_M^d \int_{P \to M} |\alpha_n(s)|^2 c_1(L, h_L)^{r-1} \\ & = \binom{d+r-1}{d} \binom{n+r-1}{r-1}^{-1} \int_M |s|^2 \Omega_M^d \\ & = \binom{d+r-1}{d} \binom{n+r-1}{r-1}^{-1} \Vert s \Vert_{L^2}^2. \end{align*} } Therefore, \begin{align*} \Vert s \Vert_{\sup}^2 & \leq \binom{n+r-1}{r-1} \Vert \alpha_n(s) \Vert_{\sup}^2 \\ & \leq \binom{n+r-1}{r-1} c^2 n^{2(d+r-1)} \Vert \alpha_n(s) \Vert_{L^2}^2 \\ & = \binom{d+r-1}{d} c^2 n^{2(d+r-1)} \Vert s \Vert_{L^2}^2. \end{align*} Thus, we get our lemma. \QED Here we recall Einstein-Hermitian metrics of vector bundles. Let $M$ be a $d$-dimensional compact K\"{a}hler manifold with a K\"{a}hler form $\Omega_M$, and $E$ a vector bundle on $M$. We say $E$ is {\em stable} (resp. {\em semistable}) {\em with respect to $\Omega_M$} if, for any subsheaves $F$ of $E$ with $0 \subsetneq F \subsetneq E$, \[ \frac{1}{\operatorname{rk} F} \int_M c_1(F) \wedge \Omega_M^{d-1} < \frac{1}{\operatorname{rk} E} \int_M c_1(E) \wedge \Omega_M^{d-1}. \] \[ \left( \text{resp.}\quad \frac{1}{\operatorname{rk} F} \int_M c_1(F) \wedge \Omega_M^{d-1} \leq \frac{1}{\operatorname{rk} E} \int_M c_1(E) \wedge \Omega_M^{d-1}. \right) \] Moreover, $E$ is said to be {\em poly-stable with respect to $\Omega_M$} if $E$ is semistable with respect to $\Omega_M$ and $E$ has a decomposition $E = E_1 \oplus \cdots \oplus E_s$ of vector bundles such that each $E_i$ is stable with respect to $\Omega_M$. Let $h$ be a Hermitian metric of $E$. We say $h$ is {\em Einstein-Hermitian with respect to $\Omega_M$} if there is a constant $\rho$ such that $K(E, h) \wedge \Omega_M^{d-1} = \rho \Omega_M^d \otimes \operatorname{id}_E$, where $K(E, h)$ is the curvature form given by $(E, h)$ and $\operatorname{id}_E$ is the identity map in $\operatorname{\mathcal{H}\textsl{om}}(E, E)$. The Kobayashi-Hitchin correspondence tells us that $E$ has an Einstein-Hermitian metric with respect to $\Omega_M$ if and only if $E$ is poly-stable with respect to $\Omega_M$. \begin{Lemma} \label{lem:sum:EH:metric} Let $M$ be a compact K\"{a}hler manifold with a K\"{a}hler form $\Omega_M$, and $E$ a poly-stable vector bundle with respect to $\Omega_M$ on $M$. If $h$ and $h'$ are Einstein-Hermitian metrics of $E$ with respect to $\Omega_M$, then so is $h+h'$. \end{Lemma} {\sl Proof.}\quad Let $E = E_1 \oplus \cdots \oplus E_s$ be a decomposition into stable vector bundles. If we set $h_i = \rest{h}{E_i}$ and $h'_i = \rest{h'}{E_i}$ for each $i$, then $h_i$ and $h'_i$ are Einstein-Hermitian metrics of $E_i$ and we have the following orthogonal decompositions: \[ (E, h) = \bigoplus_{i=1}^s (E_i, h_i) \quad\text{and}\quad (E, h') = \bigoplus_{i=1}^s (E_i, h'_i) \] (cf. \cite[Chater~IV, \S~3]{Ko}). Thus, we may assume that $E$ is stable. In this case, by virtue of the uniqueness of Einstein-Hermitian metric, there is a positive constant $c$ with $h' = ch$. Thus, $h + h' = (1+c)h$. Hence $h+h'$ is Einstein-Hermitian. \QED \begin{Lemma} \label{lem:polystable:complex:conjugation} Let $C$ be a compact Riemann surface. Considering $C$ as a projective variety over ${\mathbb{C}}$, let $\overline{C} = C \otimes_{{\mathbb{C}}} {\mathbb{C}}$ be the tensor product via the complex conjugation. Let $E$ be a vector bundle on $C$, and $\overline{E} = E \otimes_{{\mathbb{C}}} {\mathbb{C}}$ on $\overline{C}$. Then, $E$ is poly-stable on $C$ if and only if $\overline{E}$ is poly-stable on $\overline{C}$. \end{Lemma} {\sl Proof.}\quad This is an easy consequence of the fact that if $F$ is a vector bundle on $C$, then $\deg(F) = \deg(\overline{F})$. \QED \subsection{Complete proof of the relative Bogomolov's inequality} \setcounter{Theorem}{0} \renewcommand{\theClaim}{\arabic{section}.\arabic{subsection}.\arabic{Claim}} \renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Claim}} Let us start the complete proof of the relative Bogomolov's inequality. Considering $S \cup F_{\infty}(S)$ instead of $S$, we may assume that $F_{\infty}(S) = S$ by virtue of Lemma~\ref{lem:polystable:complex:conjugation}. For each $z \in S$, let $\Omega_z$ be the K\"{a}hler form induced by the metric of $\overline{\omega}_{X/Y}$ along $f_{{\mathbb{C}}}^{-1}(z)$. Since $\rest{E_{{\mathbb{C}}}}{f_{{\mathbb{C}}}^{-1}(z)}$ is poly-stable for all $z \in S$, there is a $C^{\infty}$ Hermitian metric $h'$ of $E_{{\mathbb{C}}}$ such that $\rest{h'}{f_{{\mathbb{C}}}^{-1}(z)}$ is Einstein-Hermitian with respect to $\Omega_z$ for all $z \in S$. It is easy to see that $\rest{\overline{F_{\infty}^*(h')}}{f_{{\mathbb{C}}}^{-1}(z)}$ is Einstein-Hermitian with respect to $\Omega_z$ for all $z \in S$. Thus, if $h'$ is not invariant under $F_{\infty}$, then, considering $h' + \overline{F_{\infty}^*(h')}$, we may assume that $h'$ is invariant under $F_{\infty}$. For, by Lemma~\ref{lem:sum:EH:metric}, $h' + \overline{F_{\infty}^*(h')}$ is Einstein-Hermitian with respect to $\Omega_z$ on $f_{{\mathbb{C}}}^{-1}(z)$ for each $z \in S$. Here we claim: \begin{Claim} \label{claim:assume:Einstein:Hermitian} There is a $\gamma \in L^1_{\operatorname{loc}}(Y({\mathbb{C}}))$ such that $a(\gamma) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)$ and $\gamma(z) \geq 0$ for each $z \in S$, and \[ \widehat{\operatorname{dis}}_{X/Y}(E, h) = \widehat{\operatorname{dis}}_{X/Y}(E, h') + a(\gamma). \] \end{Claim} {\sl Proof.}\quad We set ${\displaystyle \phi = \sqrt[r]{\det(h')/\det(h)}}$. Then, it is easy to see that $\widehat{\operatorname{dis}}_{X/Y}(E, \phi h) = \widehat{\operatorname{dis}}_{X/Y}(E, h)$. Thus, we may assume that $\det(h) = \det(h')$. Then, we have \[ \widehat{\operatorname{dis}}_{X/Y}(E, h) - \widehat{\operatorname{dis}}_{X/Y}(E, h') = a \left( - f_* (2r \widetilde{\operatorname{ch}_2}(E, h, h')) \right). \] Hence if we set $\gamma = - f_* (2r \widetilde{\operatorname{ch}_2}(E, h, h'))$, then $a(\gamma) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)$. On the other hand, by \cite[(ii) of Corollary~1.30]{BGSAT}, $- f_*(\widetilde{\operatorname{ch}_2}(E, h, h'))(z)$ is nothing more than Donaldson's Lagrangian (for details, see \cite[\S6]{MoBG}). Thus, we get $\gamma(z) \geq 0$ for each $z \in S$. \QED \medskip By the above claim, we may assume that $\rest{h}{f_{{\mathbb{C}}}^{-1}(z)}$ is Einstein-Hermitian for each $z \in S$. Let $\overline{A} = (A, h_A)$ be a Hermitian line bundle on $X$ such that $A$ is very ample, and $A \otimes \omega_{X/Y}^{\otimes -1}$ is ample. If we take a general member $M'$ of $|A_{{\mathbb{Q}}}^{\otimes 2}|$, then, by Bertini's theorem (cf. \cite[Theorem~6.10]{JB}), $M'$ is smooth over ${\mathbb{Q}}$, and $M' \to Y_{{\mathbb{Q}}}$ is \'{e}tale over $y$. Note that if $Z$ is an algebraic set of ${\mathbb{P}}^N_{{\mathbb{C}}}$, $U$ is a non-empty Zariski open set of ${\mathbb{P}}^N_{{\mathbb{Q}}}$, and $U({\mathbb{Q}}) \subseteq Z({\mathbb{C}})$, then $Z = {\mathbb{P}}^N_{{\mathbb{C}}}$. Hence, we may assume that $M'({\mathbb{C}}) \to Y({\mathbb{C}})$ is \'{e}tale over $z$ for all $z \in S$. Let $M' = M'_1 + \cdots + M'_{l_1} + M'_{l_1 + 1} + \cdots + M'_{l_2}$ be the decomposition of $M'$ into irreducible components (actually, the decomposition into connected components because $M'$ is smooth over ${\mathbb{Q}}$) such that $f_{{\mathbb{Q}}}(M'_i) = Y_{{\mathbb{Q}}}$ for $1 \leq i \leq l_1$ and $f_{{\mathbb{Q}}}(M'_j) \subsetneq Y_{{\mathbb{Q}}}$ for $l_1 + 1 \leq j \leq l_2$. Let $M_i$ ($i=1, \ldots, l_1$) be the closure of $M'_i$ in $X$. We set $M = M_1 + \cdots + M_{l_1}$ and $B = M_1 \coprod \cdots \coprod M_{l_1}$ (disjoint union). Then, there is a line bundle $L$ on $X$ with $M \in | A^{\otimes 2} \otimes L |$. Note that $\rest{L}{X_y} \simeq {\mathcal{O}}_{X_y}$ and $ \rest{L_{{\mathbb{C}}}}{f_{{\mathbb{C}}}^{-1}(z)} \simeq {\mathcal{O}}_{f_{{\mathbb{C}}}^{-1}(z)}$ for all $z \in S$ because $y \not\in \bigcup_{j=l_1+1}^{l_2} f_{{\mathbb{Q}}}(M'_j)$ and $z \not\in \bigcup_{j=l_1+1}^{l_2} f_{{\mathbb{C}}}(M'_j({\mathbb{C}}))$. We denote the morphism $B \to M \to X$ by $\iota$, and the morphism $B \overset{\iota}{\longrightarrow} X \overset{f}{\longrightarrow} Y$ by $g$. We remark that the morphism $B \to M$ is an isomorphism over ${\mathbb{Q}}$. Further, we set \[ \overline{F} = \operatorname{\mathcal{E}\textsl{nd}}(E, h) = (E \otimes E^{\vee}, h \otimes h^{\vee}). \] Then, $h \otimes h^{\vee}$ is a flat metric along $f_{{\mathbb{C}}}^{-1}(z)$ for each $z \in S$ because $h \otimes h^{\vee}$ is Einstein-Hermitian and $\deg \left( E \otimes E^{\vee} \right) = 0$ along $f_{{\mathbb{C}}}^{-1}(z)$. We choose a Hermitian line bundle $\overline{H} = (H, h_H)$ on $Y$ such that $g_*(\iota^*(A)) \otimes H$ and $g_*(\iota^*(F)) \otimes H$ are generated by small sections at $y$ with respect to $S$. Moreover, we set \[ \overline{F}_n = \operatorname{Sym}^n \left( \overline{F} \otimes f^*(\overline{H}) \right) \otimes \overline{A} \otimes f^*(\overline{H}) = \left( \operatorname{Sym}^n \left( F \otimes f^*(H) \right) \otimes A \otimes f^*(H), k_n \right). \] \begin{Claim} \label{claim:terms:right:R:R:formla} There are $Z_0, \ldots, Z_{r^2} \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}$ and $\beta \in L^1_{\operatorname{loc}}(Y({\mathbb{C}}))$ such that $a(\beta) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)$, and \[ f_* \left( \widehat{\operatorname{ch}}_2(\overline{F}_n) - \frac{1}{2} \widehat{{c}}_1(\overline{F}_n) \cdot \widehat{{c}}_1(\overline{\omega}_{X/Y}) \right) = \frac{n^{r^2 + 1}}{(r^2 + 1) !} f_* (\widehat{\operatorname{ch}}_2(\overline{F})) + \sum_{i=0}^{r^2} Z_i n^i + a(b_n \beta), \] where ${\displaystyle b_n = \sum_{\substack{\alpha_1 + \cdots + \alpha_{r^2} = n, \\ \alpha_1 \geq 0, \ldots, \alpha_{r^2} \geq 0}} \log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_{r^2} !} \right)}$. \end{Claim} {\sl Proof.}\quad Since $\operatorname{Sym}^n(\overline{F} \otimes f^*(\overline{H})) \otimes \overline{A} \otimes f^*(\overline{H})$ is isometric to $\operatorname{Sym}^n(\overline{F}) \otimes f^*(\overline{H})^{\otimes (n+1)} \otimes \overline{A}$, \begin{multline*} \widehat{\operatorname{ch}}_2(\overline{F}_n) = \widehat{\operatorname{ch}}_2(\operatorname{Sym}^n(\overline{F})) + \widehat{{c}}_1(\operatorname{Sym}^n(\overline{F})) \cdot \widehat{{c}}_1(f^*(\overline{H})^{\otimes (n+1)} \otimes \overline{A}) \\ +\binom{n+r^2-1}{r^2-1} \widehat{\operatorname{ch}}_2(f^*(\overline{H})^{\otimes (n+1)} \otimes \overline{A}). \end{multline*} Here since $\det(\overline{F}) = \overline{{\mathcal{O}}}_X$, by Proposition~\ref{prop:chern:class:sym:power}, \[ \widehat{{c}}_1(\operatorname{Sym}^n(\overline{F})) = a(b_n) \quad\text{and}\quad \widehat{\operatorname{ch}}_2(\operatorname{Sym}^n(\overline{F})) = \binom{n+r^2}{r^2+1} \widehat{\operatorname{ch}}_2(\overline{F}). \] Thus, by Proposition~\ref{prop:projection:formula:line:bundle}, \begin{align*} f_* \left( \widehat{{c}}_1(\operatorname{Sym}^n(\overline{F})) \cdot \widehat{{c}}_1(f^*(\overline{H})^{\otimes (n+1)} \otimes \overline{A}) \right) & = f_* \left( b_n a \left((n+1) f^*(c_1(\overline{H})) + c_1(\overline{A}) \right) \right) \\ & = a \left( b_n f_*(c_1(\overline{A})) \right). \end{align*} On the other hand, using the projection formula (cf. Proposition~\ref{prop:projection:formula:line:bundle}), \begin{align*} f_* \left( \widehat{\operatorname{ch}}_2(f^*(\overline{H})^{\otimes (n+1)} \otimes \overline{A}) \right) & = \frac{1}{2} f_* \left[ \left( (n+1) \widehat{{c}}_1(f^*(\overline{H})) + \widehat{{c}}_1(\overline{A}) \right)^2 \right] \\ & = \frac{1}{2} f_* \left[ (n+1)^2 \widehat{{c}}_1(f^*(\overline{H}))^2 + 2(n+1) \widehat{{c}}_1(f^*(\overline{H})) \cdot \widehat{{c}}_1(\overline{A}) + \widehat{{c}}_1(\overline{A})^2 \right] \\ & = (n+1) \deg_f(A) \widehat{{c}}_1(\overline{H}) + \frac{1}{2} f_* \left( \widehat{{c}}_1(\overline{A})^2 \right), \end{align*} where $\deg_f(A)$ is the degree of $A$ on the generic fiber of $f$. Therefore, we have \begin{multline*} f_* \widehat{\operatorname{ch}}_2(\overline{F}_n) = \binom{n+r^2}{r^2+1} f_* \widehat{\operatorname{ch}}_2(\overline{F}) + \\ \binom{n+r^2-1}{r^2-1} \left( (n+1) \deg_f(A) \widehat{{c}}_1(\overline{H}) + \frac{1}{2} f_* \left( \widehat{{c}}_1(\overline{A})^2 \right) \right) + a \left( b_n f_*(c_1(\overline{A})) \right). \end{multline*} Thus, there are $Z'_0, \ldots, Z'_{r^2} \in \widehat{\operatorname{CH}}^1(Y;S)_{{\mathbb{Q}}}$ such that \addtocounter{Claim}{1} \begin{equation} \label{eqn:ch2:Fn} f_* \widehat{\operatorname{ch}}_2(\overline{F}_n) = \frac{n^{r^2+1}}{(r^2+1)!} f_* \widehat{\operatorname{ch}}_2(\overline{F}) + \sum_{i=0}^{r^2} Z'_i n^i + a \left( b_n f_*(c_1(\overline{A})) \right). \end{equation} Further, since $\widehat{{c}}_1(\overline{F}_n) \cdot \widehat{{c}}_1(\overline{\omega}_{X/Y})$ is equal to \[ \left( \widehat{{c}}_1(\operatorname{Sym}^n(\overline{F})) + \binom{n+r^2-1}{r^2-1} ((n+1) \widehat{{c}}_1(f^*(\overline{H})) + \widehat{{c}}_1(\overline{A}) ) \right) \cdot \widehat{{c}}_1(\overline{\omega}_{X/Y}), \] we have \begin{align*} f_* \left( \widehat{{c}}_1(\overline{F}_n) \cdot \widehat{{c}}_1(\overline{\omega}_{X/Y}) \right) & = a \left( b_n f_*(c_1(\overline{\omega}_{X/Y})) \right) + \\ & \qquad \binom{n+r^2-1}{r^2-1} \left( (n+1) (2g-2) \widehat{{c}}_1(\overline{H}) + f_* \left( \widehat{{c}}_1(\overline{A}) \cdot \widehat{{c}}_1(\overline{\omega}_{X/Y}) \right) \right). \end{align*} Thus, there are $Z''_0, \ldots, Z''_{r^2} \in \widehat{\operatorname{CH}}^1(Y;S)_{{\mathbb{Q}}}$ such that \addtocounter{Claim}{1} \begin{equation} \label{eqn:c1:Fn:c1:w} f_* \left( \widehat{{c}}_1(\overline{F}_n) \cdot \widehat{{c}}_1(\overline{\omega}_{X/Y}) \right) = \sum_{i=0}^{r^2} Z''_i n^i + a \left( b_n f_*(c_1(\overline{\omega}_{X/Y})) \right). \end{equation} Thus, combining (\ref{eqn:ch2:Fn}) and (\ref{eqn:c1:Fn:c1:w}), we get our claim. \QED Let $h_{X/Y}$ be a $C^{\infty}$ Hermitian metric of $\det Rf_* {\mathcal{O}}_X$ over $Y({\mathbb{C}})$ such that $h_{X/Y}$ is invariant under $F_{\infty}$. Then, since the Quillen metric $h^{\overline{{\mathcal{O}}_X}}_Q$ of $\det Rf_* {\mathcal{O}}_X$ is a generalized metric, there is a real valued $\phi \in L^1_{\operatorname{loc}}(Y({\mathbb{C}}))$ such that $h^{\overline{{\mathcal{O}}_X}}_Q = e^{\phi} h_{X/Y}$ and $F_{\infty}^*(\phi) = \phi \ (\operatorname{a.e.})$. Adding a suitable real valued $C^{\infty}$ function $\phi'$ with $F_{\infty}^*(\phi') = \phi'$ to $\phi$ (replace $h_{X/Y}$ by $e^{-\phi'}h_{X/Y}$ accordingly), we may assume that $\phi(z) = 0$ for all $z \in S$. Here, we set ${\displaystyle h_n = \exp \left( - \binom{n+r^2-1}{r^2-1} \phi \right) h^{\overline{F}_n}_Q}$. Then, $h_n$ is a generalized metric of $\det Rf_* F_n$ with $F_{\infty}^*(h_n) = \overline{h}_n \ (\operatorname{a.e.})$. Moreover, \begin{multline*} \widehat{{c}}_1 \left( \det Rf_* F_n, h_n \right) - \binom{n+r^2-1}{r^2-1} \widehat{{c}}_1 \left( \det Rf_* {\mathcal{O}}_X, h_{X/Y} \right) \\ = \widehat{{c}}_1 \left( \det Rf_* F_n, h^{\overline{F}_n}_Q \right) - \binom{n+r^2-1}{r^2-1} \widehat{{c}}_1 \left( \det Rf_* {\mathcal{O}}_X, h^{\overline{{\mathcal{O}}_X}}_Q \right). \end{multline*} Here, since \[ \widehat{{c}}_1 \left( \det Rf_* F_n, h^{\overline{F}_n}_Q \right) - \binom{n+r^2-1}{r^2-1} \widehat{{c}}_1 \left( \det Rf_* {\mathcal{O}}_X, h^{\overline{{\mathcal{O}}_X}}_Q \right) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}} \] by Theorem~\ref{thm:arith:Riemann:Roch:stable:curves} and $\widehat{{c}}_1(\det Rf_* {\mathcal{O}}_X, h_{X/Y}) \in \widehat{\operatorname{CH}}^1(Y;S)$, we have \[ \widehat{{c}}_1 \left( \det Rf_* F_n, h_n \right) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}. \] Further, by the arithmetic Riemann-Roch theorem for stable curves (cf. Theorem~\ref{thm:arith:Riemann:Roch:stable:curves}), \begin{multline*} \widehat{{c}}_1 \left( \det Rf_*(F_n), h_n \right) - \binom{n+r^2-1}{r^2-1} \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_{X/Y} \right) \\ = f_* \left( \widehat{\operatorname{ch}}_2(\overline{F}_n) - \frac{1}{2} \widehat{{c}}_1 (\overline{F}_n) \cdot \widehat{{c}}_1 (\overline{\omega}_{X/Y}) \right). \end{multline*} Therefore, by Claim~\ref{claim:terms:right:R:R:formla}, there are $W_0, \ldots, W_{r^2} \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}$ and $\beta \in L^1_{\operatorname{loc}}(Y({\mathbb{C}}))$ such that $a(\beta) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)$, and \addtocounter{Claim}{1} \begin{equation} \label{eqn:1:proof:arith:BG:inq} \widehat{{c}}_1 \left( \det Rf_*(F_n), h_n \right) = \frac{n^{r^2 + 1}}{(r^2 + 1) !} f_* (\widehat{\operatorname{ch}}_2(\overline{F})) + \sum_{i=0}^{r^2} W_i n^i + a(b_n \beta). \end{equation} \begin{Claim} \label{claim:dis:lim:c1:Fn} ${\displaystyle \frac{1}{(r^2+1)!} \widehat{\operatorname{dis}}_{X/Y}(\overline{E}) = -\lim_{n \to \infty} \frac{\widehat{{c}}_1 \left( \det Rf_*(F_n), h_n \right)}{n^{r^2+1}} }$ in $\widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}$. \end{Claim} {\sl Proof.}\quad By virtue of Proposition~\ref{prop:ch2:end}, $f_*(\widehat{\operatorname{ch}}_2(\overline{F})) = -\widehat{\operatorname{dis}}_{X/Y}(\overline{E})$. Thus, by (\ref{eqn:1:proof:arith:BG:inq}), it is sufficient to show that $0 \leq b_n \leq O(n^{r^2})$. It is well known that \[ \frac{\log(\theta_1) + \cdots + \log(\theta_N)}{N} \leq \log \left( \frac{\theta_1 + \cdots + \theta_N}{N} \right) \] for positive numbers $\theta_1, \ldots, \theta_N$. Thus, noting ${\displaystyle \sum_{\substack{\alpha_1 + \cdots + \alpha_{r^2} = n, \\ \alpha_1 \geq 0, \ldots, \alpha_{r^2} \geq 0}} \frac{n!}{\alpha_1 ! \cdots \alpha_{r^2} !} = (r^2)^n}$, we have \[ 0 \leq \sum_{\substack{\alpha_1 + \cdots + \alpha_{r^2} = n, \\ \alpha_1 \geq 0, \ldots, \alpha_{r^2} \geq 0}} \log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_{r^2} !} \right) \leq \binom{n+r^2-1}{r^2-1} \log \left( \frac{ (r^2)^n }{\binom{n+r^2-1}{r^2-1}} \right) \leq O(n^{r^2}). \] \QED We set $\overline{G}_n = \iota^*(\overline{F}_n)$. Then, by Theorem~\ref{thm:arith:Riemann:Roch:gen:finite:morphism}, \[ \widehat{{c}}_1 \left( \det Rg_*(G_n), h_Q^{\overline{G}_n} \right) - \binom{n+r^2-1}{r^2 -1} \widehat{{c}}_1 \left( \det Rg_*({\mathcal{O}}_{B}), h_Q^{\overline{{\mathcal{O}}}_{B}} \right) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S) \] and \[ \widehat{{c}}_1 \left( \det Rg_*(G_n), h_Q^{\overline{G}_n} \right) - \binom{n+r^2-1}{r^2 -1} \widehat{{c}}_1 \left( \det Rg_*({\mathcal{O}}_{B}), h_Q^{\overline{{\mathcal{O}}}_{B}} \right) = g_* \left( \widehat{{c}}_1 (\overline{G}_n) \right). \] As before, we can take a $C^{\infty}$ Hermitian metric $h_{B/Y}$ of $\det Rg_*({\mathcal{O}}_{B})$ over $Y({\mathbb{C}})$ and a real valued $\varphi \in L^1_{\operatorname{loc}}(Y({\mathbb{C}}))$ such that $h_Q^{\overline{{\mathcal{O}}}_{B}} = e^{\varphi} h_{B/Y}$, $F_{\infty}^*(h_{B/Y}) = \overline{h}_{B/Y}$, $F_{\infty}^*(\varphi) = \varphi \ (\operatorname{a.e.})$, and $\varphi(z) = 0$ for all $z \in S$. We set \[ g_n = \exp \left( - \binom{n+r^2-1}{r^2-1} \varphi \right) h^{\overline{G}_n}_Q. \] Then, \[ \widehat{{c}}_1 \left( \det Rg_*(G_n), g_n \right) - \binom{n+r^2-1}{r^2 -1} \widehat{{c}}_1 \left( \det Rg_*({\mathcal{O}}_{B}), h_{B/Y} \right) = g_* \left( \widehat{{c}}_1 (\overline{G}_n) \right) \] and $\widehat{{c}}_1 \left( \det Rg_*(G_n), g_n \right) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)$. Moreover, in the same as in Claim~\ref{claim:terms:right:R:R:formla}, we can see that \[ g_* \left( \widehat{{c}}_1 (\overline{G}_n) \right) = a(\deg(g)b_n) + \binom{n+r^2-1}{r^2-1} \left( (n+1) g_* \widehat{{c}}_1(g^*(\overline{H})) + g_* \widehat{{c}}_1(\iota^*(\overline{A})) \right). \] Thus, there are $W'_0, \ldots, W'_{r^2} \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}$ such that \[ \widehat{{c}}_1 \left( \det Rg_*(G_n), g_n \right) = \sum_{i=0}^{r^2} W'_i n^i + a(b_n \deg(g)). \] Therefore, we have \addtocounter{Claim}{1} \begin{equation} \label{eqn:2:proof:arith:BG:inq} \lim_{n \to \infty} \frac{\widehat{{c}}_1 \left( \det Rg_*(G_n), g_n \right)}{n^{r^2+1}} = 0 \end{equation} in $\widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}$. Let us consider an exact sequence: \[ 0 \to F_n \otimes A^{\otimes -2} \otimes L^{\otimes -1} \to F_n \to \rest{F_n}{M} \to 0. \] Since $F$ is semi-stable and of degree $0$ along $X_y$ and $\rest{L}{X_y} = {\mathcal{O}}_{X_y}$, we have \[ f_*(F_n \otimes A^{\otimes -2}\otimes L^{\otimes -1}) = 0 \] on $Y$. Thus, the above exact sequence gives rise to \[ 0 \to f_*(F_n) \to (\rest{f}{M})_*(\rest{F_n}{M}) \to R^1f_*(F_n \otimes A^{\otimes -2}\otimes L^{\otimes -1}) \to R^1 f_*(F_n). \] Let $Q_n$ be the cokernel of \[ f_*(F_n) \to (\rest{f}{M})_*(\rest{F_n}{M}) \to g_*(G_n). \] Let $U$ be the maximal Zariski open set of $Y$ such that $f$ is smooth over $U$ and $g$ is \'{e}tale over $U$. Moreover, let $U_n$ be the maximal Zariski open set of $Y$ such that \[ \begin{cases} \text{(a) $U_n \subset U$, } \\ \text{(b) $(\rest{f}{M})_*(\rest{F_n}{M})$ coincides with $g_*(G_n)$ over $U_n$,} \\ \text{(c) $R^1f_*(F_n) = 0$ over $U_n$, and} \\ \text{(d) $f_*(F_n)$, $g_*(G_n)$ and $Q_n$ are locally free over $U_n$.} \end{cases} \] Then, $y \in (U_n)_{{\mathbb{Q}}}$ and $S \subseteq U_n({\mathbb{C}})$. For, since $A \otimes \omega_{X/Y}^{-1}$ is ample on $X_y$ and $E$ is semi-stable on $X_y$, we can see that $R^1 f_*(F_n) = 0$ around $y$, which implies that $f_*(F_n)$ is locally free around $y$. Further, since $f_*(F_n)$ and $(\rest{f}{M})_*(\rest{F_n}{M})$ are free at $y$, $R^1 f_*(F_n) = 0$ around $y$, and $(\rest{f}{M})_*(\rest{F_n}{M})$ coincides with $g_*(G_n)$ around $y$, we can easily check that $Q_n$ is free at $y$. Thus, $y \in (U_n)_{{\mathbb{Q}}}$. In the same way, we can see that $S \subseteq U_n({\mathbb{C}})$. Next let us consider a metric of $\det Q_n$. $g_*(G_n)$ has the Hermitian metric $(\rest{f}{M})_*\left( \rest{k_n}{M} \right)$ over $U_n({\mathbb{C}})$, where $k_n$ is the Hermitian metric of $\overline{F}_n$. Let $\tilde{q}_n$ be the quotient metric of $Q_n$ over $U_n({\mathbb{C}})$ induced by $(\rest{f}{M})_*\left( \rest{k_n}{M} \right)$. Let $q_n$ be a $C^{\infty}$ Hermitian metrics of $\det Q_n$ over $Y({\mathbb{C}})$ such that $F_{\infty}^*(q_n) = q_n$ and $q_n(z) = \det \tilde{q}_n(z)$ for all $z \in S$. (If $q_n$ is not invariant under $F_{\infty}$, then consider $(1/2)\left(q_n + \overline{F_{\infty}^*(q_n)}\right)$.) Here since $\det Rf_*(F_n) \simeq \det f_*(F_n) \otimes \left(\det R^1 f_*(F_n)\right)^{-1}$ and $\det f_*(F_n) \simeq \det g_*(G_n) \otimes (\det Q_n)^{-1}$, we have \[ \det Rf_*(F_n) \simeq \det g_*(G_n) \otimes (\det Q_n)^{-1} \otimes \left(\det R^1 f_*(F_n)\right)^{-1}. \] Further, we have generalized metrics $h_n$, $g_n$ and $q_n$ of $\det Rf_*(F_n)$, $\det g_*(G_n)$ and $\det Q_n$. Thus, there is a generalized metric $t_n$ of $\det R^1 f_*(F_n)$ such that the above is an isometry. As in the proof of Proposition~\ref{prop:find:small:section}, let us construct a section of $\det R^1f_*(F_n)$. First, we fix a locally free sheaf $P_n$ on $Y$ and a surjective homomorphism $P_n \to R^1 f_*(F_n)$. Let $P'_n$ be the kernel of $P_n \to R^1 f_*(F_n)$. Then, $P'_n$ is a torsion free sheaf and has the same rank as $P_n$ because $R^1 f_*(F_n)$ is a torsion sheaf. Noting that $\left( \bigwedge^{\operatorname{rk} P'_n} P'_n \right)^{*}$ is an invertible sheaf on $Y$, we can identify $\det R^1 f_*(F_n)$ with \[ \bigwedge^{\operatorname{rk} P_n} P_n \otimes \left( \bigwedge^{\operatorname{rk} P'_n} P'_n \right)^{*}. \] Moreover, the homomorphism $\bigwedge^{\operatorname{rk} P'_n} P'_n \to \bigwedge^{\operatorname{rk} P_n} P_n$ induced by $P'_n \hookrightarrow P_n$ gives rise to a non-zero section $s_n$ of $\det R^1 f_*(F_n)$. Note that $s_n(y) \not= 0$ and $s_n(z) \not= 0$ for all $z \in S$ because $R^1 f_*(F_n) = 0$ at $y$ and $z$. Here we set \[ a_n = \max_{z \in S} \{ \log t_n(s_n,s_n)(z) \}. \] By our construction, we have \[ \widehat{{c}}_1(\det R^1 f_*(F_n), e^{-a_n}t_n) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S). \] and an isometry \addtocounter{Claim}{1} \begin{multline} \label{eqn:isometry:det:bundle} (\det Rf_*(F_n), h_n) \simeq \\ (\det g_*(G_n), g_n) \otimes (\det Q_n, q_n)^{-1} \otimes (\det R^1 f_*(F_n), e^{-a_n}t_n)^{-1} \otimes ({\mathcal{O}}_Y, e^{-a_n}h_{can}). \end{multline} Here we claim: \begin{Claim} \label{claim:gen:small:sec:Q:n} $(\det Q_n, q_n)$ is generated by small sections at $y$ with respect to $S$. \end{Claim} {\sl Proof.}\quad First of all, \[ g_*\left( \iota^*(F) \otimes g^*(H) \right) = g_*(\iota^*(F)) \otimes H \quad\text{and}\quad g_*\left( \iota^*(A) \otimes g^*(H) \right) = g_*(\iota^*(A)) \otimes H \] are generated by small section at $y$ with respect to $S$. Thus, by (2) and (3) of Proposition~\ref{prop:find:small:section}, \[ g_*(G_n) = g_* \left( \operatorname{Sym}^n (\iota^*(F) \otimes g^*(H)) \otimes \iota^*(A) \otimes g^*(H) \right) \] is generated by small sections at $y$ with respect to $S$. Thus, by (1) of Proposition~\ref{prop:find:small:section}, $(Q_n, \tilde{q}_n)$ is generated by small sections at $y$ with respect to $S$. Hence, by (4) of Proposition~\ref{prop:find:small:section}, $(\det Q_n, q_n)$ is generated by small sections at $y$ with respect to $S$ because $q_n(z) = \det \tilde{q}_n (z)$ for all $z \in S$. \QED Next we claim: \begin{Claim} \label{claim:estimate:sequences:an:bn} $a_n \leq O(n^{r^2} \log(n))$. \end{Claim} {\sl Proof.}\quad It is sufficient to show that $\log t_n(s_n,s_n)(z) \leq O(n^{r^2}\log(n))$ for each $z \in S$. Let $\{ e_1, \ldots, e_{l_n} \}$ be an orthonormal basis of $g_*(G_n) \otimes \kappa(z)$ with respect to $g_*(\rest{k_n}{B})(z)$ such that $\{ e_1, \ldots, e_{m_n} \}$ forms a basis of $f_*(F_n) \otimes \kappa(z)$. Then, $e_1 \wedge \cdots \wedge e_{m_n}$, $e_1 \wedge \cdots \wedge e_{l_n}$ and $\bar{e}_{m_n + 1} \wedge \cdots \wedge \bar{e}_{l_n}$ form bases of $\det (f_*(F_n)) \otimes \kappa(z)$, $\det (g_*(G_n)) \otimes \kappa(z)$, and $\det (Q_n) \otimes \kappa(z)$ respectively, and $(e_1 \wedge \cdots \wedge e_{m_n}) \otimes (\bar{e}_{m_n + 1} \wedge \cdots \wedge \bar{e}_{l_n}) = e_1 \wedge \cdots \wedge e_{l_n}$, where $\bar{e}_{m_n + 1}, \ldots, \bar{e}_{l_n}$ are images of $e_{m_n + 1}, \ldots, e_{l_n}$ in $Q_n \otimes \kappa(z)$. Then, \[ \left| (e_1 \wedge \cdots \wedge e_{m_n}) \otimes s_n^{\otimes -1} \right|_{h_n}^2(z) = \frac{|e_1 \wedge \cdots \wedge e_{l_n}|_{g_n}^2(z)}{ |\bar{e}_{m_n + 1} \wedge \cdots \wedge \bar{e}_{l_n}|_{q_n}^2(z) |s_n|_{t_n}^2(z)} = |s_n|_{t_n}^{-2}(z), \] where $|a|_{\lambda} = \sqrt{\lambda(a,a)}$ for $\lambda = h_n, g_n, q_n, t_n$. Moreover, let $\Omega_{z}$ be the K\"{a}hler form induced by the metric of $\overline{\omega}_{X/Y}$ along $f_{{\mathbb{C}}}^{-1}(z)$. Then, there is a Hermitian metric $v_n$ of $H^0(f_{{\mathbb{C}}}^{-1}(z), F_n)$ defined by \[ v_n(s, s') = \int_{f_{{\mathbb{C}}}^{-1}(z)} k_n(s, s') \Omega_{z}. \] Here $R^1 f_*(F_n) = 0$ at $z$. Thus, $(\det R^1 f_*(F_n))_z$ is canonically isomorphic to ${\mathcal{O}}_{Y({\mathbb{C}}), z}$. Since $(P'_n)_z = (P_n)_z$, under the above isomorphism, $s_n$ goes to the determinant of $(P_n)_z \overset{\operatorname{id}}{\longrightarrow} (P_n)_z$, namely $1 \in {\mathcal{O}}_{Y({\mathbb{C}}), z}$. Hence, by the definition of Quillen metric, \[ \left| (e_1 \wedge \cdots \wedge e_{m_n}) \otimes s_n^{\otimes -1} \right|_{h_n}^2(z) = \det( v_n(e_i, e_j) ) \exp \left( -T \left( \rest{\overline{F}_n}{f_{{\mathbb{C}}}^{-1}(z)} \right) \right). \] Therefore, \[ \log |s_n|_{t_n}^2(z) = T \left( \rest{\overline{F}_n}{f_{{\mathbb{C}}}^{-1}(z)} \right) - \log \det( v_n(e_i, e_j) ). \] By Corollary~\ref{cor:asymp:analytic:torsion}, \[ T \left( \rest{\overline{F}_n}{f_{{\mathbb{C}}}^{-1}(z)} \right) \leq O(n^{r^2} \log(n)). \] Thus, in order to get our claim, it is sufficient to show that \[ - \log \det( v_n(e_i, e_j) ) \leq O(n^{r^2-1}\log(n)). \] Let $s$ be an arbitrary section of $H^0(f_{{\mathbb{C}}}^{-1}(z), F_n)$. Then, by Lemma~\ref{lem:comparison:sup:L2}, \[ g_*\left( \rest{k_n}{B} \right)(s, s) = \sum_{x \in g_{{\mathbb{C}}}^{-1}(z)} |s|_{k_n}^2(x) \leq \deg(g) \sup_{x \in f_{{\mathbb{C}}}^{-1}(z)} \{ |s|_{k_n}^2(x) \} \leq \deg(g) c^2 n^{2r^2} \Vert s \Vert^2_{L^2} \] for some constant $c$ independent of $n$. Thus, by \cite[Lemma~3.4]{MoBG} and our choice of $e_i$'s, \[ 1 = \det \left( g_*\left(\rest{k_n}{B}\right)(e_i, e_j) \right) \leq \left( \deg(g) c^2 n^{2r^2} \right)^{\dim_{{\mathbb{C}}} H^0(f_{{\mathbb{C}}}^{-1}(z), F_n)} \det \left( v_n(e_i, e_j) \right). \] Using Riemann-Roch theorem, we can easily see that \[ \dim_{{\mathbb{C}}} H^0(f_{{\mathbb{C}}}^{-1}(z), F_n) \leq O(n^{r^2-1}). \] Thus, we have \[ -\log \det( v_n(e_i, e_j) ) \leq O(n^{r^2-1}\log(n)). \] Hence, we obtain our claim. \QED Let us go back to the proof of our theorem. By the isometry (\ref{eqn:isometry:det:bundle}), we get \begin{align*} -\widehat{{c}}_1(\det Rf_*(F_n), h_n) & = -\widehat{{c}}_1( \det g_*(G_n), g_n) + \widehat{{c}}_1(\det Q_n, q_n) + \widehat{{c}}_1(\det R^1 f_*(F_n), e^{-a_n}t_n) - a(a_n) \\ & = \left[ \widehat{{c}}_1(\det Q_n, q_n) + \widehat{{c}}_1(\det R^1 f_*(F_n), e^{-a_n}t_n) + a\left( \max \{ -a_n, 0 \} \right) \right] \\ & \qquad\qquad + \left[ -\widehat{{c}}_1( \det g_*(G_n), g_n) + a (\min \{ -a_n, 0 \}) \right]. \end{align*} Here we set \[ \begin{cases} {\displaystyle \alpha_n = \frac{(r^2+1)!}{n^{r^2+1}} \left[ \widehat{{c}}_1(\det Q_n, q_n) + \widehat{{c}}_1(\det R^1 f_*(F_n), e^{-a_n}t_n) + a\left( \max \{ -a_n, 0 \} \right) \right],} \\ {} \\ {\displaystyle \beta_n = \frac{(r^2+1)!}{n^{r^2+1}} \left[ -\widehat{{c}}_1( \det g_*(G_n), g_n) + a (\min \{ -a_n, 0 \}) \right].} \end{cases} \] Then, \[ \frac{-(r^2+1)!\widehat{{c}}_1(\det Rf_*(F_n), h_n)}{n^{r^2+1}} = \alpha_n + \beta_n. \] By (\ref{eqn:2:proof:arith:BG:inq}) and Claim~\ref{claim:estimate:sequences:an:bn}, ${\displaystyle \lim_{n \to \infty} \beta_n = 0}$ in $\widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}$. Therefore, by Claim~\ref{claim:dis:lim:c1:Fn}, \[ \widehat{\operatorname{dis}}_{X/Y}(\overline{E}) = \lim_{n \to \infty} \frac{-(r^2+1)! \widehat{{c}}_1 \left( \det Rf_*(F_n), h_n \right)} {n^{r^2+1}} = \lim_{n\to\infty} (\alpha_n + \beta_n) = \lim_{n \to \infty} \alpha_n \] in $\widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}$. On the other hand, it is obvious that \[ \widehat{{c}}_1(\det R^1 f_*(F_n), e^{-a_n}t_n) \quad\text{and}\quad a\left( \max \{ -a_n, 0 \} \right) \] is semi-ample at $y$ with respect to $S$. By Claim~\ref{claim:gen:small:sec:Q:n}, $\widehat{{c}}_1(\det Q_n, q_n)$ is semi-ample at $y$ with respect to $S$. Thus, $\alpha_n$ is semi-ample at $y$ with respect to $S$. Hence we get our theorem. \QED \renewcommand{\theClaim}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}} \renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}} \section{Preliminaries for Cornalba-Harris-Bost's inequality} \label{section:Bost:type:inequality:preparation} This section is a preparatory one for the next section, where we will prove the relative Cornalba-Harris-Bost's inequality (cf. Theorem~\ref{thm:semistability:imply:average:semi-ampleness}). Moreover, in the next section, we will see how the relative Bogomolov's inequality (Theorem~\ref{thm:relative:Bogomolov:inequality:arithmetic:case}) and the relative Cornalba-Harris-Bost's inequality (Theorem~\ref{thm:semistability:imply:average:semi-ampleness}) are related (cf. Proposition~\ref{prop:Bogomolov:to:Bost}). \subsection{Normalized Green forms} \label{subsec:normalized:Green:form} \setcounter{Theorem}{0} Let $Y$ be a smooth quasi-projective variety over ${\mathbb{C}}$, $\overline{E}=(E,h)$ a Hermitian vector bundle of rank $r$ on $Y$. Let $\pi : {\mathbb{P}}(E) \to Y$ be the canonical morphism, where ${\mathbb{P}}(E) = \operatorname{Proj} (\bigoplus_{i \ge 0} \operatorname{Sym}^{i}(E^{\lor}))$. We equip the canonical quotient bundle ${\mathcal{O}}_{E}(1)$ on ${\mathbb{P}}(E)$ with the quotient metric via $\pi ^* (E^{\vee}) \to {\mathcal{O}}_{E}(1)$. We will denote this Hermitian line bundle by $\overline{{\mathcal{O}}_{E}(1)}$. Furthermore, let $\Omega = {c}_1(\overline{{\mathcal{O}}_{E}(1)})$ be the first Chern form. The purpose of this subsection is that, for every cycle $X \subset {\mathbb{P}}(E)$ whose all irreducible components map surjectively to $Y$, we give a Green form $g_X$ such that on a general fiber, it is an $\Omega$-normalized Green current in the sense of \cite[2.3.2]{BGS}. Let $X$ be a cycle of codimension $p$ on ${\mathbb{P}}(E)$ such that every irreducible component of $X$ maps surjectively to $Y$. An $L^1$-form $g_X$ on ${\mathbb{P}}(E)$ satisfying the following conditions is called an {\em $\Omega$-normalized Green form}, (or simply a {\em normalized Green form} when no confusion is likely). \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item There are $d$-closed $L^1$-forms $\gamma_i$ of type $(p-i, p-i)$ on $Y$ ($i=0, \ldots, p$) with \[ dd^c([g_X]) + \delta_X = \sum_{i=0}^{p} \left[ \pi^*(\gamma_i) \wedge \Omega^i \right]. \] \item $\pi_*(g_X \wedge \Omega^{r-p}) = 0$. \end{enumerate} Note that $\gamma_p$ is the degree of $X$ along a general fiber of $\pi$. Let $X = \sum_i a_i X_i$ be the irreducible decomposition of $X$ as cycles. Let $\tilde{X}_i \to X_i$ be a desingularization of $X_i$, and $\tilde{f}_i : \tilde{X}_i \to Y$ the induced morphism. The main result of this subsection is the following. \begin{Proposition} \label{prop:normalized:Green:form} With notation as above, there exists an $\Omega$-normalized Green form $g_X$ on ${\mathbb{P}}(E)$ satisfying the following property. If $y \in Y$ and $\tilde{f}_i$ is smooth over $y$ for every $i$, then there is an open set $U$ containing $y$ such that $\gamma_0, \ldots, \gamma_p$ are $C^{\infty}$ on $U$ and that $\rest{g_X}{\pi^{-1}(U)}$ is a Green form of logarithmic type for $X_U$, where $\gamma_0, \ldots, \gamma_p$ are $L^1$-forms in the definition of $\Omega$-normalized Green form. \end{Proposition} To prove the above proposition, let us begin with the following two lemmas. \begin{Lemma} \label{lemma:auxiliary:green:form} There exist a Green form $g$ of logarithmic type along $X$, and $d$-closed $C^{\infty}$ forms $\beta_i$ of type $(p-i, p-i)$ on $Y$ \textup{(}$i=0, \ldots, p$\textup{)} such that \[ dd^c([g]) + \delta_X = \sum_{i=0}^{p} \left[ \pi^*(\beta_i) \wedge \Omega^i \right]. \] \end{Lemma} {\sl Proof.}\quad We divide the proof into three steps. {\bf Step 1.} : The case where $Y$ is projective. Let $g_1$ be a Green form of logarithmic type along $X$ such that \[ dd^c([g_1]) + \delta_X = [\omega] \] where $\omega$ is a smooth form on ${\mathbb{P}}(E)$. Then, we can find a smooth form $\eta$ on ${\mathbb{P}}(E)$ of the form \[ \eta = \sum_{i=0}^{p} \pi^*(\beta_i) \wedge \Omega^i \] which represents the same cohomology class as $\omega$, where $\beta_i$ is a $d$-closed $C^{\infty}$-form of type $(p-i, p-i)$ on $Y$. Since $\omega - \eta$ is $d$-exact $(p,p)$-form, by the $dd^c$-lemma, there is a smooth $(p-1,p-1)$-form $\phi$ with $\omega - \eta = dd^c(\phi)$. Thus, if we set $g = g_1 - \phi$, then $g$ is of logarithmic type along $X$ and \[ dd^c([g]) + \delta_X = dd^c([g_1]) - dd^c(\phi) + \delta_X = [\eta]. \] \medskip {\bf Step 2.} : Let $h'$ be another Hermitian metric of $E$, and $\Omega'$ the Chern form of ${\mathcal{O}}_{E}(1)$ arising from $h'$. In this step, we will prove that if the lemma holds for $h'$, then so does it for $h$. By our assumption, there exist a Green form $g'$ of logarithmic type along $X$, and $d$-closed $C^{\infty}$ forms $\beta'_i$ ($i=0, \ldots, p$) of type $(p-i, p-i)$ on $Y$ such that \[ dd^c([g']) + \delta_X = \sum_{i=0}^{p} \left[ \pi^*(\beta'_i) \wedge {\Omega'}^i \right]. \] On the other hand, there is a real $C^{\infty}$-function $a$ on ${\mathbb{P}}(E)$ with $\Omega' - \Omega = dd^c(a)$. Here note that if $v$ is a $\partial$ and $\overline{\partial}$-closed form on ${\mathbb{P}}(E)$, then $dd^c(v \wedge a) = v \wedge dd^c(a)$. Thus, it is easy to see that there is a $C^{\infty}$ form $\theta$ on ${\mathbb{P}}(E)$ such that \[ \sum_{i=1}^p \pi^*(\beta'_i) \wedge {\Omega'}^i = dd^c(\theta) + \sum_{i=1}^p \pi^*(\beta'_i) \wedge {\Omega}^i. \] Therefore, if we set $g = g' - \theta$ and $\beta_i = \beta'_i$, then we have our assertion for $h$. \medskip {\bf Step 3.} : General case. Using Hironaka's resolution \cite{Hiro}, there is a smooth projective variety $Y'$ over ${\mathbb{C}}$ such that $Y$ is an open set of $Y'$. Moreover, using \cite[Exercise~5.15 in Chapter~II]{Hartshorne}, there is a coherent sheaf $E'$ on $Y'$ with $\rest{E'}{Y} = E$. Further, taking a birational modification along $Y' \setminus Y$ if necessary, we may assume that $E'$ is locally free. Let $h'$ be a Hermitian metric of $E'$ over $Y'$. Since ${\mathbb{P}}(E)$ is an Zariski open set of ${\mathbb{P}}(E')$, let $X'$ be the closure of $X$ in ${\mathbb{P}}(E')$. Then, by Step~1, our assertion holds for $(E', h')$ and $X'$. Thus, so does it for $(E, \rest{h'}{Y})$ and $X$. Therefore, by Step~2, we can conclude our lemma. \QED \begin{Lemma} \label{lemma:push:g:is:L1} Let $g$ be a Green form of logarithmic type along $X$ and $\omega$ a $C^{\infty}$-form with $dd^c([g]) + \delta_X = [\omega]$. If we set $\varsigma = \pi_* (g \wedge \Omega^{r-p})$, then $\varsigma \in L^1_{loc}(Y)$ and $dd^c([\varsigma]) \in L^1_{loc}(\Omega_Y^{1,1})$. Moreover, if $y \in Y$ and $\tilde{f}_i$ is smooth over $y$ for every $i$, then $\varsigma$ is $C^{\infty}$ around $y$. \end{Lemma} {\sl Proof.}\quad By Proposition~\ref{prop:push:forward:B:pq}, $\varsigma$ is an $L^1$-function on $Y$ and \begin{align*} dd^c([\varsigma]) & = dd^c( \pi_* ([g \wedge \Omega^{r-p}])) = \pi_* dd^c([g \wedge \Omega^{r-p}]) \\ & = \pi_* dd^c([g]) \wedge \Omega^{r-p} = \pi_* ([\omega] \wedge \Omega^{r-p}) - \pi_* (\delta_X \wedge \Omega^{r-p}) \\ & = \pi_* [\omega \wedge \Omega^{r-p}] - \sum_i a_i \pi_* (\delta_{X_i} \wedge \Omega^{r-p}) \\ & = \pi_* [\omega \wedge \Omega^{r-p}] - \sum_i a_i (\tilde{f}_i)_* [\tilde{f}_i^* (\Omega^{r-p})]. \end{align*} Thus, $dd^c([\varsigma]) \in L^1_{loc}(\Omega_Y^{1,1})$. Moreover, if $y \in Y$ and $\tilde{f}_i$ is smooth over $y$ for every $i$, then, by the above formula, $dd^c([\varsigma])$ is $C^{\infty}$ around $y$. Thus, by virtue of \cite[(i) of Theorem~1.2.2]{GSArInt}, $\varsigma$ is $C^{\infty}$ around $y$. \QED Let us start the proof of Proposition~\ref{prop:normalized:Green:form}. Let $g$ be a Green form constructed in Lemma~\ref{lemma:auxiliary:green:form}. Then, there are $d$-closed $\beta_i$'s with $\beta_i \in A^{p-i,p-i} (Y)$ and \[ dd^c([g]) + \delta_X = \sum_{i=0}^{p} \left[ \pi^*(\beta_i) \wedge \Omega^i \right]. \] If we set $\varsigma = \pi_* (g \wedge \Omega^{r-p})$, then by Lemma~\ref{lemma:push:g:is:L1}, $\varsigma$ is locally an $L^1$-form. We put \[ g_X = g - \pi^*(\varsigma) \Omega^{p-1}, \] which is clearly locally an $L^1$-form on ${\mathbb{P}}(E)$. We will show that $g_X$ satisfies the conditions (i) and (ii). Using $\int_{{\mathbb{P}}(E) \to Y} \Omega^{r-1} = 1$, (ii) can be readily checked. Moreover, \begin{align*} dd^c([g_X]) + \delta_X & = \sum_{i=0}^{p} \left[ \pi^*(\beta_i) \wedge \Omega^i \right] - dd^c [\pi^*(\varsigma) \Omega^{p-1}] \\ & = \beta_p \Omega^p + \pi^*([\beta_{p-1}] - dd^c([\varsigma])) \wedge \Omega^{p-1} + \sum_{i=0}^{p-2} \left[ \pi^*(\beta_i) \wedge \Omega^i \right]. \end{align*} The remaining assertion is easily derived from Lemma~\ref{lemma:push:g:is:L1}. \QED \begin{Remark} \label{rem:norm:Green:general:fiber} Let $y$ be a point of $Y$ such that $\tilde{f}_i$ is smooth over $y$ for every $i$. Then, by Proposition~\ref{prop:normalized:Green:form}, on the fiber $\pi^{-1}(y)$, $\rest{g_X}{\pi^{-1}(y)}$ is a Green form of logarithmic type along $X_{y}$. Moreover, \[ dd^c ([\rest{g_X}{\pi^{-1}(y)}]) + \delta_{X_y} = \deg (X_y) [ \rest{\Omega^p}{\pi^{-1}(y)} ] \] and \[ \int_{\pi^{-1}(y)} \left( \rest{g_X}{\pi^{-1}(y)} \right) \left( \rest{\Omega^{r-p}}{\pi^{-1}(y)} \right) = 0. \] Thus, $\rest{g_X}{\pi^{-1}(y)}$ is a $\Omega$-normalized Green form on $\pi^{-1}(y)$, and it is also a $\Omega$-normalized Green current in the sense of \cite[2.3.2]{BGS}. \end{Remark} \subsection{Associated Hermitian vector bundles} \label{subsec:associated:herm:vb} \renewcommand{\theClaim}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}} \renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}} \setcounter{Theorem}{0} Let $\operatorname{\mathbf{GL}}_r = \operatorname{Spec} {\mathbb{Z}} [X_{11},X_{12},\cdots,X_{rr}]_{\det(X_{ij})}$ be the general linear group of rank $r$ and $\operatorname{\mathbf{SL}}_r = \operatorname{Spec} {\mathbb{Z}} [X_{11},X_{12},\cdots,X_{rr}]/(\det(X_{ij})-1)$ be the special linear group of rank $r$. Let $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ be a morphism of group schemes. First, we note that \[ \rho({\mathbb{C}}) (\overline{A}) = \overline{\rho({\mathbb{C}}) (A)}, \] where $\rho({\mathbb{C}}) : \operatorname{\mathbf{GL}}_r({\mathbb{C}}) \to \operatorname{\mathbf{GL}}_R({\mathbb{C}})$ is the induced morphism and $A \in \operatorname{\mathbf{GL}}_r({\mathbb{C}})$. Indeed, the above equality is nothing but the associativity of the map \[ \operatorname{Spec} {\mathbb{C}} \overset{-}{\longrightarrow} \operatorname{Spec} {\mathbb{C}} \overset{A}{\longrightarrow} \operatorname{\mathbf{GL}}_r \overset{\rho}{\longrightarrow} \operatorname{\mathbf{GL}}_R. \] Next, we consider the following condition for $\rho$; \addtocounter{Theorem}{1} \begin{equation} \label{eqn:condition} \rho({}^t A) = {}^t \rho (A) \qquad \text{for any $A \in \operatorname{\mathbf{GL}}_r$}. \end{equation} In the group scheme language, this condition means $\rho$ commutes with the transposed morphism. Let $\operatorname{U}_r({\mathbb{C}}) = \{A \in \operatorname{\mathbf{GL}}_r({\mathbb{C}}) \mid {}^t A \cdot \overline{A} = I_r \}$ be the unitary group of rank $r$. If a group morphism $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ commutes with the transposed morphism, then \begin{align*} I_R & = \rho({\mathbb{C}}) (I_r) = \rho({\mathbb{C}}) ({}^t A \cdot \overline{A}) \\ & = \rho({\mathbb{C}})({}^t A) \cdot \rho({\mathbb{C}})(\overline{A}) = {}^t \rho({\mathbb{C}})(A) \cdot \overline{\rho({\mathbb{C}})(A)}, \end{align*} namely, $\rho({\mathbb{C}})$ maps $\operatorname{U}_r({\mathbb{C}})$ into $\operatorname{U}_R({\mathbb{C}})$. Let $k$ be an integer. A morphism $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ of group schemes is said to be {\em of degree $k$} if \[ \rho(t I_r) = t^k I_R \qquad \text{for any $t$}. \] In the group scheme language, this means that the diagram \begin{equation*} \begin{CD} \operatorname{\mathbf{GL}}_1 @>{\lambda_r}>> \operatorname{\mathbf{GL}}_r \\ @V{\alpha}VV @VV{\rho}V \\ \operatorname{\mathbf{GL}}_1 @>{\lambda_R}>> \operatorname{\mathbf{GL}}_R \end{CD} \end{equation*} commutes, where $\lambda_r$ and $\lambda_R$ are given by $t \mapsto \operatorname{diag}(t,t,\cdots,t)$ and $\alpha$ is given by $t \mapsto t^k$. \medskip Let $Y$ be an arithmetic variety, $\overline{E} = (E,h)$ a Hermitian vector bundle of rank $r$ on $Y$ and $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ be a morphism of group schemes satisfying commutativity with the transposed morphism. In the following, we will show that we can naturally construct a Hermitian vector bundle $\overline{E}^{\rho} = (E^{\rho},h^{\rho})$, which we will call the {\em associated Hermitian vector bundle} with respect to $\overline{E}$ and $\rho$. First, we construct $E^{\rho}$. Let $\{ Y_{\alpha} \}$ be an affine open covering such that $\phi_{\alpha}: E \vert_{Y_{\alpha}} \overset{\sim}{\longrightarrow} {\mathcal{O}}_{Y_{\alpha}}^{\oplus r}$ gives a local trivialization. On $Y_{\alpha} \cap Y_{\beta}$, we set the transition function $g_{\alpha\beta} = \phi_{\alpha} \cdot \phi_{\beta}^{-1}$, which can be seen as an element of $\operatorname{\mathbf{GL}}_r(\Gamma({\mathcal{O}}_{Y_{\alpha} \cap Y_{\beta}}))$. Then we define the associated vector bundle $E^{\rho}$ as the vector bundle of rank $R$ on $Y$ with the transition functions $\rho(\Gamma({\mathcal{O}}_{Y_{\alpha} \cap Y_{\beta}})) (g_{\alpha\beta})$; \[ E^{\rho} = \coprod_{\alpha} {\mathcal{O}}_{Y_{\alpha}}^{\oplus R} / \sim. \] Next, we define metric on $E^{\rho}$. Let $h^{\alpha}$ be the Hermitian metric on ${\mathcal{O}}_{Y_{\alpha}}^{\oplus r}$ over $Y_{\alpha}$ such that $\phi_{\alpha}: E \vert_{Y_{\alpha}} \overset{\sim}{\longrightarrow} {\mathcal{O}}_{Y_{\alpha}}^{\oplus r}$ becomes isometry over $Y_{\alpha}({\mathbb{C}})$. Let \begin{gather*} e_1^{\alpha} = {}^t (1,0,\cdots,0), \cdots, e_r^{\alpha} = {}^t (0,\cdots,0,1) \in \Gamma({\mathcal{O}}_{Y_{\alpha}}^{\oplus r}), \\ f_1^{\alpha} = {}^t (1,0,\cdots,0), \cdots, f_R^{\alpha} = {}^t (0,\cdots,0,1) \in \Gamma({\mathcal{O}}_{Y_{\alpha}}^{\oplus R}) \end{gather*} be the standard local frames of ${\mathcal{O}}_{Y_{\alpha}}^{\oplus r}$ and ${\mathcal{O}}_{Y_{\alpha}}^{\oplus R}$. We set \[ H_{\alpha} = (h^{\alpha}(e_{i}^{\alpha},e_{j}^{\alpha}))_{1 \leq i,j \leq r}. \] Then $H_{\alpha}$ is a $C^{\infty}$-map over $Y_{\alpha}({\mathbb{C}})$ and, for each point $y$ in $Y_{\alpha}({\mathbb{C}})$, $H_{\alpha}(y)$ is a positive definite Hermitian matrix. Let $\rho(C^{\infty}(Y_{\alpha}({\mathbb{C}}))) : \operatorname{\mathbf{GL}}_r(C^{\infty}(Y_{\alpha}({\mathbb{C}}))) \to \operatorname{\mathbf{GL}}_R(C^{\infty}(Y_{\alpha}({\mathbb{C}})))$ be the induced map. \addtocounter{Theorem}{1} \begin{Claim} $\rho(C^{\infty}(Y_{\alpha}({\mathbb{C}}))) (H_{\alpha})$ is a $C^{\infty}$-map over $Y_{\alpha}({\mathbb{C}})$ and, for each point $y$ in $Y_{\alpha}({\mathbb{C}})$, $\rho(C^{\infty}(Y_{\alpha}({\mathbb{C}})))(H_{\alpha})(y)$ is a positive definite Hermitian matrix. \end{Claim} {\sl Proof.}\quad The first assertion is obvious. For the second one, we note that there is a matrix $A \in \operatorname{\mathbf{GL}}_r({\mathbb{C}})$ such that ${}^t A \cdot \overline{A} = H_{\alpha}(y)$. Then it is easy to see that $\rho(C^{\infty}(Y_{\alpha}({\mathbb{C}})))(H_{\alpha})(y)$ is a positive definite Hermitian matrix by using \eqref{eqn:condition}. \QED Now we define a metric $h^{\rho_{\alpha}}$ on ${\mathcal{O}}_{Y_{\alpha}}^{\oplus R}$ over $Y_{\alpha}$ by \[ h^{\rho_{\alpha}}(f_{k}^{\alpha},f_{l}^{\alpha}) = \rho(C^{\infty}(Y_{\alpha}({\mathbb{C}}))) (H_{\alpha})_{kl} \] for $1 \leq k,l \leq R$. \addtocounter{Theorem}{1} \begin{Claim} $\{h^{\rho_{\alpha}}\}_{\alpha}$ glue together to form a Hermitian metric on $E^{\rho}$. \end{Claim} {\sl Proof.}\quad Let $s_{\alpha} = {}^t (s_1^{\alpha},\cdots,s_R^{\alpha}) \in \Gamma({\mathcal{O}}_{Y_{\alpha}}^{\oplus R} \vert_{Y_{\alpha} \cap Y_{\beta}})$ and $s_{\beta} = {}^t (s_1^{\beta},\cdots,s_R^{\beta}) \in \Gamma({\mathcal{O}}_{Y_{\beta}}^{\oplus R} \vert_{Y_{\alpha} \cap Y_{\beta}})$. Then they give the same section of $E^{\rho} \vert_{Y_{\alpha} \cap Y_{\beta}}$ if ${}^t (s_1^{\alpha},\cdots,s_R^{\alpha}) = \rho(g_{\alpha\beta}) {}^t (s_1^{\beta},\cdots,s_R^{\beta})$. In this case, we write $s_{\alpha} \sim s_{\beta}$. Now we take $s_{\alpha} \sim s_{\beta}$ and $t_{\alpha} \sim t_{\beta}$. Then by a straightforward calculation using \eqref{eqn:condition} and $H_{\beta} = {}^t g_{\alpha\beta} H_{\alpha} \overline{g_{\alpha\beta}}$, we get $h^{\rho_{\alpha}}(s_{\alpha},t_{\alpha}) = h^{\rho_{\beta}}(s_{\beta},t_{\beta})$ on $Y_{\alpha} \cap Y_{\beta}$. \QED \begin{Remark} Let $\operatorname{id}_r : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_r$ be the identity morphism, $\rho_1 = (\operatorname{id}_r)^{\otimes k}$, $\rho_2 = \operatorname{Sym}^k(\operatorname{id}_r)$, and $\rho_3 = \bigwedge^k(\operatorname{id}_r)$. Further, let $\rho_4 : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_r$ be the group homomorphism given by $A \mapsto {}^t A^{-1}$. Then $\rho_1$, $\rho_2$, $\rho_3$ and $\rho_4$ are of degree $k$, $k$, $k$ and $-1$, respectively. Let $(E,h)$ be a Hermitian vector bundle of rank $r$. Then the associated vector bundles are $(E^{\otimes k}, h^{\otimes k})$, $(\operatorname{Sym}^k(E), h^{\rho_2})$, $(\bigwedge^k(E), h^{\rho_3})$ and $(E^{\lor}, h^{\lor})$. Note, for example, that $h^{\rho_2}$ is not the quotient metric $h_{quot}$ given by $E^{\otimes k} \to \operatorname{Sym}^k(E)$; Indeed, for a locally orthogonal basis $e_1,\cdots,e_r$ of $\overline{E}$ and $\alpha_1,\cdots,\alpha_r \in {\mathbb{Z}}$, $h^{\rho_2}(e_1^{\alpha_1}\cdots e_r^{\alpha_r}, e_1^{\alpha_1}\cdots e_r^{\alpha_r}) = 1$, while $h_{quot}(e_1^{\alpha_1}\cdots e_r^{\alpha_r}, e_1^{\alpha_1}\cdots e_r^{\alpha_r}) = {\alpha_1 !\cdots\alpha_r !}/ r !$. \end{Remark} \renewcommand{\theClaim}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}} \renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}} \subsection{Chow forms and their metrics} \label{subsec:Chow:forms:and:their:metrics} \renewcommand{\theClaim}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}} \renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}} \setcounter{Theorem}{0} Let $Y$ be a regular arithmetic variety, and $\overline{E}=(E,h)$ a Hermitian vector bundle of rank $r$ on $Y$. Let $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ be a group scheme morphism of degree $k$ commuting with the transposed morphism and $\overline{E}^{\rho}=(E^{\rho},h^{\rho})$ the associated Hermitian bundle of rank $R$. We give the quotient metric on ${\mathcal{O}}_{E^{\rho}}(1)$ via $\pi^*({E^{\rho}}^{\vee}) \to {\mathcal{O}}_{E^{\rho}}(1)$. We denote this Hermitian line bundle by $\overline{{\mathcal{O}}_{E^{\rho}}(1)}$. Further, let $\Omega_{\rho} = {c}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})$ be the first Chern form. Let $X$ be an effective cycle in ${\mathbb{P}} (E^{\rho})$ such that $X$ is flat over $Y$ with the relative dimension $d$ and degree $\delta$ on the generic fiber. Let $g_X$ be a $\Omega_{\rho}$-normalized Green form for $X$ and we set $\widehat{X}= (X, g_X)$. Then $\widehat{X} \in \widehat{Z}^{R-1-d}_{L^1}({\mathbb{P}}(E^{\rho}))$. Thus $\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}\cdot\widehat{X}$ belongs to $\widehat{\operatorname{CH}}^{R}_{L^1}({\mathbb{P}} (E^{\rho}))_{{\mathbb{Q}}}$. Hence, \[ \pi_*\left( \widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}\cdot\widehat{X} \right) \in \widehat{\operatorname{CH}}^{1}_{L^1}(Y)_{{\mathbb{Q}}}. \] Let us consider elementary properties of $\pi_*\left( \widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}\cdot\widehat{X} \right)$. \begin{Proposition} \label{prop:when:Bost:divisor:smooth} Let $X = \sum_{k=1}^l a_k X_k$ be the irreducible decomposition of $X$ as cycles. Let $\phi_k : \tilde{X}_k \to X_k$ be a generic resolution of singularities of $X_k$ for each $k$, i.e., $\phi_k$ is a proper birational morphism such that $(\tilde{X}_k)_{{\mathbb{Q}}}$ is smooth over ${\mathbb{Q}}$. Let $i_k : X_k \hookrightarrow {\mathbb{P}}(E^{\rho})$ be the inclusion map and $j_k : \tilde{X}_k \to {\mathbb{P}}(E^{\rho})$ the composition map $i_k \cdot \phi_k$. Also we let $f_k : X_k \to Y$ be the composition map $\pi \cdot i_k$ and $\tilde{f}_k : \tilde{X}_k \to Y$ the composition map $\pi \cdot j_k$. Let $Y_0$ be the maximal open set of $Y$ such that $\tilde{f}_k$ is smooth over there for every $k$. Then, we have the following. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item ${\displaystyle \pi_* \left(\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X} \right) = \sum_{k=1}^l a_k \tilde{f}_k{}_* (\widehat{{c}}_1(j_k^* \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}). }$ \\ In particular, $\pi_*\left( \widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}\cdot\widehat{X} \right)$ is independent of the choice of normalized Green forms $g_X$ for $X$, and $\pi_* \left(\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X} \right) \in \widehat{\operatorname{CH}}_{L^1}^1(Y; Y_0({\mathbb{C}}))$. \item Let $y$ be a closed point of $(Y_0)_{{\mathbb{Q}}}$, and $\Gamma'$ the closure of $\{ y \}$ in $Y$. Here we choose $g_X$ as in Proposition~\textup{\ref{prop:normalized:Green:form}}. Then, there is a representative $(Z, g_Z)$ of $\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X}$ such that $\pi^{-1}(\Gamma')$ and $Z$ intersect properly, and $\rest{g_Z}{\pi^{-1}(z)}$ is locally integrable for each $z \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)$. \end{enumerate} \end{Proposition} {\sl Proof.}\quad We may assume that $X$ is reduced and irreducible, so that we will omit index $k$ in the following. (1) Let $g_X$ be a $\Omega_{\rho}$-normalized Green form for $X$. Then, by virtue of Proposition~\ref{prop:formula:restriction:intersection}, \[ \widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X} = j_* \left( \widehat{{c}}_1(j^* \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \right) + a(\Omega_{\rho}^{d+1} \wedge [g_X]). \] Therefore, since $\pi_* (g_X \wedge \Omega_{\rho}^{d+1}) = 0$ by the definition of $g_X$, we get \[ \pi_* \left(\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X} \right) = \pi_* j_* \left( \widehat{{c}}_1(j^* \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \right) = \tilde{f}_* \left( \widehat{{c}}_1(j^* \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \right). \] \medskip (2) First of all, we need the following lemma. \begin{Lemma} \label{lem:intersect:proper} Let $T$ be a quasi-projective integral scheme over ${\mathbb{Z}}$, $L_1, \ldots, L_n$ line bundles on $T$, and $\Gamma$ a cycle on $T$. Then, there is a cycle $Z$ on $T$ such that $Z$ is rationally equivalent to $c_1(L_1) \cdots c_1(L_n)$, and that $Z$ and $\Gamma$ intersect properly. \end{Lemma} {\sl Proof.}\quad We prove this lemma by induction on $n$. First, let us consider the case $n=1$. Let $\Gamma = \sum_{i=1}^r a_i \Gamma_i$ be the irreducible decomposition as cycles. Let $\gamma_i$ be a closed point of $\Gamma_i \setminus \bigcup_{j \not= i} \Gamma_j$, and $m_i$ the maximal ideal at $\gamma_i$. Let $H$ be an ample line bundle on $X$. Choose a sufficiently large integer $N$ such that \[ H^1(T, H^{\otimes N} \otimes m_1 \otimes \cdots \otimes m_r) = H^1(T, H^{\otimes N} \otimes L_1 \otimes m_1 \otimes \cdots \otimes m_r) = 0. \] Then, the natural homomorphisms \[ H^0(T, H^{\otimes N}) \to \bigoplus_{i=1}^r H^{\otimes N} \otimes \kappa(\gamma_i) \quad\text{and}\quad H^0(T, H^{\otimes N} \otimes L_1) \to \bigoplus_{i=1}^r H^{\otimes N} \otimes L_1 \otimes \kappa(\gamma_i) \] are surjective. Thus, there are sections $s_1 \in H^0(T, H^{\otimes N})$ and $s_2 \in H^0(T, H^{\otimes N} \otimes L_1)$ such that $s_1(\gamma_i) \not= 0$ and $s_2(\gamma_i) \not= 0$ for all $i$. Then, $\operatorname{div}(s_2) - \operatorname{div}(s_1) \sim c_1(L_1)$, and $\operatorname{div}(s_2) - \operatorname{div}(s_1)$ and $\Gamma$ intersect properly. Next we assume $n > 1$. Then, by hypothesis of induction, there is a cycle $Z'$ such that $Z' \sim c_1(L_1) \cdots c_1(L_{n-1})$, and $Z'$ and $\Gamma$ intersect properly. Let $Z' = \sum_{j} b_j T_j$ be the decomposition as cycles. We set $\Gamma_j = (T_j \cap \operatorname{Supp}(\Gamma))_{red}$. Then, using the case $n=1$, there is a cycle $Z_j$ such that $Z_j \sim c_1(\rest{L_n}{T_j})$, and $Z_j$ and $\Gamma_j$ intersect properly. Thus, if we set $Z = \sum_j b_j Z_j$, then $Z \sim c_1(L_1) \cdots c_1(L_n)$, and $Z$ and $\Gamma$ intersect properly. \QED Let us go back to the proof of (2) of Proposition~\ref{prop:when:Bost:divisor:smooth}. By virtue of Lemma~\ref{lem:intersect:proper}, there is a cycle $Z$ on $X$ such that $Z \sim c_1\left(i^* {\mathcal{O}}_{E^{\rho}}(1) \right)^{d+1}$, and that $Z$ and $f^{-1}(\Gamma')$ intersect properly. Then, $Z \sim c_1({\mathcal{O}}_{E^{\rho}}(1))^{d+1} \cdot X$, and $Z$ and $\pi^{-1}(\Gamma')$ intersect properly. Let $\phi_X$ be a Green form of logarithmic type for $X$. Then, since \[ \widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot (X, \phi_X) \in \widehat{\operatorname{CH}}^{R}({\mathbb{P}}(E^{\rho})), \] there is a Green form $\phi_Z$ of logarithmic type for $Z$ such that $(Z, \phi_Z)$ is a representative of $\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot (X, \phi_X)$. Thus, if we set \[ g_Z = \phi_Z + c_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \wedge (g_X - \phi_X), \] then $(Z, g_Z)$ is a representative of $\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X}$. Since $Z$ and $\pi^{-1}(\Gamma')$ intersect properly and $g_X$ has the property in Proposition~\ref{prop:normalized:Green:form}, we can easily see that $g_Z$ is locally integrable along $\pi^{-1}(z)$ for each $z \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)$. \QED Here we recall some elementary results of Chow forms. Details can be found in \cite{Bo}. Consider the incidence subscheme $\Gamma$ in the product \[ {\mathbb{P}}(E^{\rho}) \times_{Y} {\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1} = {\mathbb{P}}(E^{\rho}) \times_{Y} {\mathbb{P}}(E^{\rho}{}^{\lor}) \times_{Y} \cdots \times_{Y} {\mathbb{P}}(E^{\rho}{}^{\lor}). \] Let $\imath : \Gamma \to {\mathbb{P}}(E^{\rho})$ and $\jmath : \Gamma \to {\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1}$ be projection maps. The Chow divisor $\operatorname{Ch}(X)$ of $X$ is defined by \[ \operatorname{Ch}(X) = \jmath_* \imath^* (X). \] The following facts are well-known: \begin{enumerate} \item $\operatorname{Ch}(X)$ is an effective cycle of codimension $1$ in ${\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1}$; \item $\operatorname{Ch}(X)$ is flat over $Y$; \item For any $y \in Y$, $\operatorname{Ch}(X)_y$ is a divisor of type $(\delta,\delta,\cdots,\delta)$ in ${\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1}_{y}$. \end{enumerate} Let $p : {\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1} \to Y$ be the canonical morphism, and $p_i : {\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1} \to {\mathbb{P}}(E^{\rho}{}^{\lor})$ the projection to the $i$-th factor. Then, by the above properties, there is a line bundle $L$ on $Y$ and a section $\Phi_{X}$ of \[ H^0 \left({\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1}, p^*(L)\otimes\bigotimes_{i=1}^{d+1}p_i^* {\mathcal{O}}_{E^{\rho}{}^{\lor}}(\delta) \right) \] such that $\operatorname{div}(\Phi_{X}) = \operatorname{Ch}(X)$. Since \[ p_* \left( p^* (L) \otimes \bigotimes_{i=1}^{d+1} p_i^* {\mathcal{O}}_{E^{\rho}{}^{\lor}}(\delta) \right) = L \otimes (\operatorname{Sym}^{\delta}(E^{\rho}))^{{}\otimes {d+1}}, \] we may view $\Phi_{X}$ as an element of $H^0(Y,L \otimes (\operatorname{Sym}^{\delta}(E^{\rho}))^{{}\otimes {d+1}})$. We say $\Phi_{X}$ is a {\em Chow form} of $X$. Clearly $\Phi_{X}$ is uniquely determined up to $H^0(Y,{\mathcal{O}}_{Y}^{\times})$. As in \cite[Proposition 1.2 and its remark]{Bo}, we have \[ {c}_1(L) = \pi_* \left({c}_{1}({\mathcal{O}}_{E^{\rho}}(1))^{d+1}\cdot {X}\right). \] We give a generalized metric $h_{L}$ on $L$ so that $\overline{L} = (L,h_L)$ satisfies the equality \addtocounter{Theorem}{1} \begin{equation} \label{eqn:metric:L} \widehat{{c}}_1(\overline{L}) = \pi_* \left(\widehat{{c}}_{1}(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^ {d+1}\cdot\widehat{X}\right) \end{equation} in $\widehat{\operatorname{CH}}^1_{L^1}(Y)$. Note that we can also give a metric $L$ by the equation \[ {\mathcal{O}}_{{\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1}}(\operatorname{Ch}(X)) = p^*(L)\otimes\bigotimes_{i=1}^{d+1}p_i^* {\mathcal{O}}_{E^{\lor}}(\delta) \] and by suitably metrizing other terms, as is implicitly done in \cite[1.5]{Zh}. We do not however pursue this here. \renewcommand{\theClaim}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}} \renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}} \subsection{Restriction of Chow forms on fibers} \label{subsec:pullback} \setcounter{Theorem}{0} In this section we will consider the restriction of Chow forms on fibers. Let $Y,\overline{E},\rho,X$ be as in \S \ref{subsec:Chow:forms:and:their:metrics}. Let $y$ be a closed point of $Y_{{\mathbb{Q}}}$. Let $\Gamma'$ be the closure of $\{ y \}$ in $Y$, and $\Gamma$ the normalization of $\Gamma'$. Let $f : \Gamma \to Y$ be the natural morphism. We set $E_{\Gamma} = f^*(E)$ and $\overline{E_{\Gamma}} = (E_{\Gamma},f^*(h))$. Also we put $(E^{\rho})_{\Gamma} = f^*(E^{\rho})$ and $\overline{(E^{\rho})_{\Gamma}} = ((E^{\rho})_{\Gamma}, f^*(h^{\rho}))$. Then $(\overline{E_{\Gamma}})^{\rho}$ is equal to $\overline{(E^{\rho})_{\Gamma}}$, so that we denote $(E^{\rho})_{\Gamma}$ by $E^{\rho}_{\Gamma}$, and $\overline{(E^{\rho})_{\Gamma}}$ by $\overline{E^{\rho}_{\Gamma}}$. Considering the following fiber product \begin{equation*} \begin{CD} {\mathbb{P}}(E^{\rho}_{\Gamma}) @>{f '}>> {\mathbb{P}}(E^{\rho}) \\ @VV{\pi'}V @VV{\pi}V \\ \Gamma @>{f}>> Y \end{CD} \end{equation*} we set $X_{\Gamma} = {f'}^*(X)$. Then $X_{\Gamma}$ is flat over $\Gamma$ with the relative dimension $d$ and the degree $\delta$ on the generic fiber. For this quadruplet $(\Gamma,\overline{E_{\Gamma}},\rho,X_{\Gamma})$ in place of the quadruplet $(Y,\overline{E},\rho,X)$, we can define in the same way the Hermitian line bundle $\overline{{\mathcal{O}}_{E^{\rho}_{\Gamma}}(1)}$ on ${\mathbb{P}}(E^{\rho}_{\Gamma})$, an arithmetic $L^1$-divisor $\widehat{X_{\Gamma}} = (X_{\Gamma}, g_{X_{\Gamma}})$ on ${\mathbb{P}}(E^{\rho}_{\Gamma})$ and the arithmetic $L^1$-divisor $\pi '_* \left(\widehat{{c}}_{1}( \overline{{\mathcal{O}}_{E^{\rho}_{\Gamma}}(1)})^{d+1}\cdot \widehat{X_{\Gamma}}\right)$ on $\Gamma$. Further, we have $\overline{L_{\Gamma}} = (L_{\Gamma}, h_{\Gamma})$ satisfying \[ \widehat{{c}}_1(\overline{L_{\Gamma}}) = \pi '_* \left(\widehat{{c}}_{1}(\overline{{\mathcal{O}}_{E^{\rho}_{\Gamma}}(1)})^ {d+1}\cdot\widehat{X_{\Gamma}}\right) \] in $\widehat{\operatorname{CH}}_{L^1}^1(\Gamma)$. We also have $\operatorname{Ch}(X_{\Gamma})$. Moreover, letting $p_i ' : {\mathbb{P}}((E^{\rho}_{\Gamma})^{\lor})^{d+1} \to {\mathbb{P}}((E^{\rho}_{\Gamma})^{\lor})$ be the $i$-th projection, there is a section $\Phi_{X_{\Gamma}}$ of \[ H^0 \left( {\mathbb{P}}((E^{\rho}_{\Gamma})^{\lor})^{d+1}, {p '}^*(L_{\Gamma})\otimes \bigotimes_{i=1}^{d+1} {p_i'}^* {\mathcal{O}}_{(E^{\rho}_{\Gamma})^{\lor}}(\delta) \right) = H^0 \left(\Gamma, L_{\Gamma} \otimes (\operatorname{Sym}^{\delta} ((E^{\rho}_{\Gamma})^{\lor}))^{\otimes d+1} \right), \] such that $\operatorname{div}(\Phi_{X_{\Gamma}}) = \operatorname{Ch}(X_{\Gamma})$. Let us consider the following fiber product, \begin{equation*} \begin{CD} {\mathbb{P}}((E^{\rho}_{\Gamma})^{\lor})^{d+1} @>{g '} >> {\mathbb{P}}((E^{\rho})^{\lor})^{d+1} \\ @VV{p '}V @VV{p}V \\ \Gamma @>{f}>> Y \end{CD} \end{equation*} Then, we have the following proposition. \begin{Proposition} \label{proposition:pullback} \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item ${g'}^* \operatorname{Ch}(X) = \operatorname{Ch}(X_{\Gamma})$. Moreover, we can choose $\Phi_{X_{\Gamma}}$ to be $f ^* \Phi_X$. \item Let $X_1,\cdots,X_l$ be the irreducible components of $X_{red}$. Assume that, for every $1 \leq i \leq l$, there is a generic resolution of singularities $\phi_i : \tilde{X_i} \to X_i$ such that the induced map $\tilde{X_i} \to Y$ is smooth over $y$ for every $i$. Then the equality \begin{align*} f^* \pi_* \left(\widehat{{c}}_{1}( \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X} \right) & = \pi '_* f'{}^* \left(\widehat{{c}}_{1}( \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X} \right) \\ & = \pi '_* \left(\widehat{{c}}_{1}( \overline{{\mathcal{O}}_{E^{\rho}_{\Gamma}}(1)})^{d+1} \cdot \widehat{X_{\Gamma}} \right). \end{align*} holds in $\widehat{\operatorname{CH}}_{L^1}^1(\Gamma)$. In other words, $f^* (L,h_L) = (L_{\Gamma}, h_{L_{\Gamma}})$. \end{enumerate} \end{Proposition} {\sl Proof.}\quad (i) If $f$ is flat, then this follows from the base change theorem. In the case $f$ is not flat, we refer to the remark \cite[4.3.2(i)]{BGS}, or we can easily see this using Appendix~A. (ii) We take $g_X$ as in Proposition~\ref{prop:normalized:Green:form}. Let $\alpha = \widehat{{c}}_{1}( \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X} \in \widehat{\operatorname{CH}}_{L^1}^1(Y)$. By Proposition~\ref{prop:when:Bost:divisor:smooth}, we can take a representative $(Z,g_Z)$ of $\alpha$ such that $Z$ and $\pi^{-1}(\Gamma')$ intersect properly, and $g_Z$ is locally integrable along $\pi^{-1}(w)$ for all $w \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)$. Now we have \begin{align*} f^* \pi_*(\alpha) & = f^* (\pi_*Z, [\pi_* g_Z]) \\ & = \left( f^* \pi_*Z, \sum_{w \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)} \left( \int_{\pi^{-1} (w)} g_Z \right) \cdot w \right). \end{align*} On the other hand, we have \begin{equation*} \pi '_* f'{}^*(\alpha) = \left( \pi '_* f'{}^* Z, \sum_{w \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)} \left( \int_{\pi^{-1} (w)} g_Z \right) \cdot w \right). \end{equation*} Moreover, by Appendix A, $f^* \pi_*Z$ is equal to $\pi '_* f'{}^* Z$ as a cycle. Thus we have proven the first equality. Now we will prove the second equality. Let $\phi_X$ be a Green form of logarithmic type for $X$. Since \[ {f'}^* : \bigoplus_{i \geq 0} \widehat{\operatorname{CH}}^i({\mathbb{P}}(E^{\rho})) \to \bigoplus_{i \geq 0} \widehat{\operatorname{CH}}^i({\mathbb{P}}(E^{\rho}_{\Gamma})) \] is a homomorphism of rings (cf. \cite[5) of Theorem in 4.4.3]{GSArInt}). Thus, \[ {f'}^* \left(\widehat{{c}}_{1}( \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot (X, \phi_X) \right) = {f'}^* \widehat{{c}}_{1}( \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot {f'}^*(X, \phi_X). \] Therefore, since we take $g_X$ as in Proposition~\ref{prop:normalized:Green:form}, we can see \begin{align*} {f'}^* \left(\widehat{{c}}_{1}( \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X} \right) & = {f'}^* \left(\widehat{{c}}_{1}( \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot ((X, \phi_X) + (0, g_X - \phi_X)) \right) \\ & = {f'}^* \left(\widehat{{c}}_{1}( \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot (X, \phi_X) \right) + a \left( {f'}^*(c_{1}(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \wedge (g_X - \phi_X))\right) \\ & = {f'}^* \widehat{{c}}_{1}( \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot {f'}^*(X, \phi_X) + a \left( {f'}^* (c_{1}(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \wedge (g_X - \phi_X)) \right)\\ & = {f'}^* \widehat{{c}}_{1}( \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot {f'}^*(X, g_X) \end{align*} Moreover, as pointed out in Remark~\ref{rem:norm:Green:general:fiber}, ${f'}^* g_X$ is a normalized Green form for ${f'}^* X$. Thus we have got the second equality. \QED \renewcommand{\theClaim}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}} \renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}} \subsection{Chow stability and field extensions} \label{subsec:Chow:stability:field:ext} \setcounter{Theorem}{0} Let $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ be a group scheme morphism of degree $k$ commuting with the transposed morphism. Let $S$ be a ring (commutative, with the multiplicative identity). For a positive integer $\delta$ and $d$, we consider $\operatorname{Sym}^{\delta} (S^R)^{\otimes d+1}$. Then through the induced group homomorphism $\rho(S) : \operatorname{\mathbf{GL}}_r(S) \to \operatorname{\mathbf{GL}}_R(S)$, $\operatorname{\mathbf{GL}}_r(S)$ (or $\operatorname{\mathbf{SL}}_r(S)$) acts on $\operatorname{Sym}^{\delta} (S^R)^{\otimes d+1}$. \begin{Proposition} \label{prop:stability:and:det} Let $K$ be an infinite field and $L$ an extension field of K. Let $P$ be a homogeneous polynomial of degree $e$ on $\operatorname{Sym}^{\delta} (L^R)^{\otimes d+1}$, i.e., $P \in \operatorname{Sym}^e(\operatorname{Sym}^{\delta} (L^R)^{\otimes d+1}{}^{\lor})$. Then we have the following. \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item $P$ is $\operatorname{\mathbf{SL}}_r(K)$-invariant if and only if $P$ is $\operatorname{\mathbf{SL}}_r(L)$-invariant. \item If P is $\operatorname{\mathbf{SL}}_r(K)$-invariant, then \[ P(v^{\sigma}) ^r = (\det \sigma) ^{ek(d+1)\delta} P(v)^r \] in $L$ for all $ v \in \operatorname{Sym}^{\delta} (L^R)^{\otimes d+1}$ and $\sigma \in \operatorname{\mathbf{GL}}_r(L)$. \end{enumerate} \end{Proposition} \proof (i) We only need to prove the `only if' part. Let $S_L(P) = \{ \sigma \in \operatorname{\mathbf{SL}}_r(L) \mid P^{\sigma} = P \}$ be the stabilizer of $P$ in $\operatorname{\mathbf{SL}}_r(L)$. $S_L(P)$ is a closed algebraic set of $\operatorname{\mathbf{SL}}_r(L)$ and contains $\operatorname{\mathbf{SL}}_r(K)$. Since $\operatorname{\mathbf{SL}}_r(K)$ is Zariski dense in $\operatorname{\mathbf{SL}}_r(L)$, $S_L(P)$ must coincide with $\operatorname{\mathbf{SL}}_r(L)$. (ii) Let $M$ be an extension field of $L$ such that it has an $r$-th root $\xi$ of $\det \sigma$. If $\sigma '$ is defined by $\sigma = \xi\sigma '$, then $\sigma ' \in \operatorname{\mathbf{SL}}_r(M)$. Since $P$ is $\operatorname{\mathbf{SL}}_r(M)$-invariant by (i), we find \begin{align*} P(v^{\sigma})^r & = P\left( (\rho({\mathbb{C}})(\xi\sigma ')) \cdot v\right) ^r = P\left( (\xi^k \rho({\mathbb{C}})(\sigma ')) \cdot v \right) ^r \\ & = \xi^{rek(d+1)\delta} P(v)^r = (\det \sigma) ^{ek(d+1)\delta} P(v)^r. \end{align*} in $M$. Hence the equality holds in $L$ because both sides belong to $L$. \QED \begin{Remark} More strongly, we can show that, for any integral domain $S$ of characteristic zero, if $P \in \operatorname{Sym}^e(\operatorname{Sym}^{\delta} (S^R)^{\otimes d+1}{}^{\lor})$ is $\operatorname{\mathbf{SL}}_r({\mathbb{Z}})$-invariant, then $P$ is $\operatorname{\mathbf{SL}}_r(S)$-invariant. \end{Remark} Now let $Y$, $\overline{E}$, $\rho$ and $X$ be as in \S \ref{subsec:Chow:forms:and:their:metrics}. Recall that for a closed point $y$ of $Y_{{\mathbb{Q}}}$, $\operatorname{Ch}(X)_y$ is a divisor of type $(\delta,\delta,\cdots,\delta)$ in ${\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1}_{y}$. Hence the Chow form restricted on $y$, i.e., $\Phi_X \vert_y = \Phi_{X_y}$ is an element of $\operatorname{Sym}^{\delta}(K^R)^{d+1}$. We say that $X_y$ is {\em Chow semi-stable} if $\Phi_{X_y} \in \operatorname{Sym}^{\delta}(K^R)^{d+1}$ is semi-stable under the action of $\operatorname{\mathbf{SL}}_r(K)$, where $K$ is the residue field of $y$. \begin{Lemma} \label{lemma:stability:generators:over:Z} There are a positive integer $e$ and $\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$-invariant homogeneous polynomials $P_1,\cdots,P_l \in \operatorname{Sym}^{e}(\operatorname{Sym}^{\delta}({\mathbb{Z}}^R)^{d+1}{}^{\lor})$, which depend only on $\rho$, $d$ and $\delta$, with the following property. For any closed points $y \in Y_{{\mathbb{Q}}}$, if $X_y$ is Chow semistable, then there is a $P_i$ such that $P_i (\Phi_{X_y}) \neq 0$. \end{Lemma} \proof $\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$ acts linearly on $\operatorname{Sym}^{\delta}({\mathbb{Q}}^R)^{d+1}$. Since $\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$ is a reductive group, we can take $\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$-invariant homogeneous polynomials $Q_1,\cdots,Q_l$ such that they form generators of the algebra of $\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$ invariant polynomials on $\operatorname{Sym}^{\delta}({\mathbb{Q}}^R)^{d+1}$. By clearing the denominators, we may assume that $Q_1,\cdots,Q_l$ is defined over ${\mathbb{Z}}$. Let $e_i$ be the degree of $Q_i$ for $i = 1,\cdots,l$. We take a positive integer $e$ such that $e_i \vert e$ for $i = 1,\cdots,l$. We set $P_i = Q_i^{e/e_i}$. Let us check that $P_i$'s have the desired property. Since $X_y$ is Chow semistable, there is a $\operatorname{\mathbf{SL}}_r(K)$-invariant homogeneous polynomial $F$ on $\operatorname{Sym}^{\delta}(K^R)^{d+1}$ with $F(\Phi_{X_y}) \not= 0$, where $K$ is the residue field of $y$. Let us choose $\alpha_1, \ldots, \alpha_n \in K$ and homogeneous polynomials $F_1, \ldots, F_n$ over ${\mathbb{Q}}$ such that $F = \alpha_1 F_1 + \cdots + \alpha_n F_n$ and that $\alpha_1, \ldots, \alpha_n$ are linearly independent over ${\mathbb{Q}}$. Here, for $\sigma \in \operatorname{\mathbf{SL}}_r({\mathbb{Q}})$, \[ F^{\sigma} = \alpha_1 F_1^{\sigma} + \cdots + \alpha_n F_{n}^{\sigma} \] and $F_{i}^{\sigma}$'s are homogeneous polynomials over ${\mathbb{Q}}$. Thus, we can see that $F_i$'s are $\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$-invariant. Moreover, since \[ F(\Phi_{X_y}) = \alpha_1 F_1(\Phi_{X_y}) + \cdots + \alpha_n F_{n}(\Phi_{X_y}), \] there is $F_i$ with $F_i(\Phi_{X_y}) \not= 0$. On the other hand, $F_i$ is an element of ${\mathbb{Q}}[Q_1, \ldots, Q_l]$. Thus, we can find $Q_j$ with $Q_j(\Phi_{X_y}) \not= 0$, namely $P_j(\Phi_{X_y}) \not= 0$. \QED \medskip \section{Semi-stability and positiveness in a relative case} \label{section:semistability:positiveness:relative} \subsection{Cornalba-Harris-Bost's inequality in a relative case} \label{subsec:CHB:inequality} \setcounter{Theorem}{0} Let $Y$ be an arithmetic variety and $\overline{E} = (E,h)$ a Hermitian vector bundle of rank $r$ on $Y$. Let $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ be a group scheme morphism of degree $k$ commuting with the transposed morphism. Before we prove the relative Cornalba-Harris-Bost's inequality, we need three lemmas. \begin{Lemma} \label{lemma:P:to:sheafhom} Let $L$ be a line bundle on $Y$. Let $P \in \operatorname{Sym}^e(\operatorname{Sym}^{\delta}({\mathbb{Z}}^R)^{d+1}{}^{\lor}) \backslash \{0\}$ and suppose that $P$ is $\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$-invariant. Then there is a polynomial map of sheaves \[ \varphi_P : L \otimes \operatorname{Sym}^{\delta} (E^{\rho})^{\otimes d+1} \to L^{\otimes er} \otimes (\det E)^{\otimes ek(d+1)\delta} \] given by $P^r$, namely, $\varphi_P$ is locally defined by the evaluation in terms of $P^r$. \end{Lemma} {\sl Proof.}\quad Let $U$ be a Zariski open set, and $\phi : \rest{E}{U} \overset{\sim}{\longrightarrow} {\mathcal{O}}_{U}^{\oplus n}$ and $\psi : \rest{L}{U} \overset{\sim}{\longrightarrow} {\mathcal{O}}_{U}$ local trivializations of $E$ and $L$ respectively. Then, by the construction of $E^{\rho}$, we have \[ \phi_{\rho,\delta,d}: \rest{\left( \operatorname{Sym}^{\delta} \left(E^{\rho}\right)^{\otimes d+1} \right)}{U} \overset{\sim}{\longrightarrow} \operatorname{Sym}^{\delta} \left({\mathcal{O}}_{U}^{\oplus R}\right)^{\otimes d+1}. \] Thus we get \[ \psi \otimes \phi_{\rho,\delta,d}: \rest{\left( L \otimes \operatorname{Sym}^{\delta} \left(E^{\rho}\right)^{\otimes d+1} \right)}{U} \overset{\sim}{\longrightarrow} \operatorname{Sym}^{\delta} \left({\mathcal{O}}_{U}^{\oplus R}\right)^{\otimes d+1}. \] Here, we define \[ \rest{\varphi_P}{U} : \rest{\left( L \otimes \operatorname{Sym}^{\delta} (E^{\rho})^{\otimes d+1} \right)}{U} \to \rest{\left( L^{\otimes er} \otimes (\det E)^{\otimes ek(d+1)\delta} \right)}{U} \] such that the following diagram is commutative. \[ \begin{CD} \rest{\left( L \otimes \operatorname{Sym}^{\delta} \left(E^{\rho}\right)^{\otimes d+1} \right)}{U} @>{\psi \otimes \phi_{\rho,\delta,d}}>> \operatorname{Sym}^{\delta} \left({\mathcal{O}}_{U}^{\oplus R}\right)^{\otimes d+1} \\ @V{\rest{\varphi_P}{U}}VV @VV{P^r}V \\ \rest{\left( L^{\otimes er} \otimes (\det E)^{\otimes ek(d+1)\delta} \right)}{U} @>{\psi^{er} \otimes \det(\phi)^{ek(d+1)\delta}}>> {\mathcal{O}}_U, \end{CD} \] where $P^r$ is the map given by the evaluation in terms of the polynomial $P^r$. In order to see that the local $\rest{\varphi_P}{U}$ glues together on $Y$, it is sufficient to show that $\rest{\varphi_P}{U}$ does not depend on the choice of local trivializations $\phi$ and $\psi$. Let $\phi' : \rest{E}{U} \overset{\sim}{\longrightarrow} {\mathcal{O}}_{U}^{\oplus n}$ and $\psi' : \rest{L}{U} \overset{\sim}{\longrightarrow} {\mathcal{O}}_{U}$ be another local trivializations. In the same way, we have the following commutative diagram. \[ \begin{CD} \rest{\left( L \otimes \operatorname{Sym}^{\delta} \left(E^{\rho}\right)^{\otimes d+1} \right)}{U} @>{\psi' \otimes \phi'_{\rho,\delta,d}}>> \operatorname{Sym}^{\delta} \left({\mathcal{O}}_{U}^{\oplus R}\right)^{\otimes d+1} \\ @V{\rest{\varphi'_P}{U}}VV @VV{P^r}V \\ \rest{\left( L^{\otimes er} \otimes (\det E)^{\otimes ek(d+1)\delta} \right)}{U} @>{{\psi'}^{er} \otimes \det(\phi')^{ek(d+1)\delta}}>> {\mathcal{O}}_U \end{CD} \] We set the transition functions $g = \phi \cdot(\phi')^{-1}$ and $h = \psi \cdot(\psi')^{-1}$. Then by a straightforward calculation using (ii) of Proposition~\ref{prop:stability:and:det}, we get, on $U$, \[ P^r \cdot (\psi \otimes \phi_{\rho,\delta,d}) = h^{re}\det(g)^{ek (d+1)\delta} P^r \cdot (\psi' \otimes \phi'_{\rho,\delta,d}), \] which implies \[ \left({\psi}^{er} \otimes \det(\phi)^{ek(d+1)\delta}\right) \cdot \left(\rest{\varphi_P}{U}\right) = h^{re}\det(g)^{ek (d+1)\delta} \left({\psi'}^{er} \otimes \det(\phi')^{ek(d+1)\delta}\right) \cdot \left(\rest{\varphi'_P}{U}\right). \] Here note that \[ h^{re}\det(g)^{ek (d+1)\delta} = \left({\psi}^{er} \otimes \det(\phi)^{ek(d+1)\delta}\right) \cdot \left({\psi'}^{er} \otimes \det(\phi')^{ek(d+1)\delta}\right)^{-1}. \] Thus, we obtain $\rest{\varphi_P}{U} = \rest{\varphi'_P}{U}$. \QED Suppose now $L$ is given a generalized metric $h_L$. Since both sides of \[ \varphi_P : L \otimes \operatorname{Sym}^{\delta} (E^{\rho})^{\otimes d+1} \to L^{\otimes er} \otimes (\det E)^{\otimes ek(d+1)\delta} \] in the lemma above are then equipped with metrics, we can consider the norm of $\varphi_P$. Before evaluating the norm of $\varphi_P$, we define the norm of $P$ as follows; We first define the metric $\Vert \cdot \Vert_{can}$ on $\operatorname{Sym}^{\delta}({\mathbb{C}}^{n})^{\otimes d+1}$ induced from the usual Hermitian metric on ${\mathbb{C}}$; We then define $|\!|\!| P |\!|\!|$ by \[ |\!|\!| P |\!|\!| = \sup_{v \in \operatorname{Sym}^{\delta}({\mathbb{C}}^{n})^{\otimes d+1} \setminus \{0\}} \frac{\vert P(v) \vert}{\Vert v \Vert_{can}^e}, \] where $P$ is regarded as an element of $\operatorname{Sym}^e((\operatorname{Sym}^{\delta}({\mathbb{C}}^{m}))^{\otimes d+1})^{\lor})$. \begin{Lemma} \label{lemma:norm:of:morphism:P} For any section $s \in H^0(Y,L \otimes (\operatorname{Sym}^{\delta}(E^{\rho}))^{\otimes d+1})$ and any complex point $y \in Y({\mathbb{C}})$ around which $h_L \otimes (\operatorname{Sym}^{\delta}(h^{\rho}))^{\otimes d+1}$ is $C^{\infty}$, we have \[ \Vert \varphi_P(s) \Vert (y) \le |\!|\!| P |\!|\!|^r \, \Vert s \Vert^{er} (y). \] \end{Lemma} \proof By choosing bases, $\overline{E}(y)$ and in $\overline{L}(y)$ are isometric to ${\mathbb{C}}^n$ and ${\mathbb{C}}$ with the canonical metrics, respectively. Then, with respect to these bases, $\overline{E}^{\rho}$ is by its construction isomorphic to ${\mathbb{C}}^R$ with the canonical metric. Recalling that $\varphi_P$ is given by the evaluation by $P^r$ once we fix local trivializations of $E$ and $L$, the desired inequality follows from the definition of $|\!|\!| P |\!|\!|$. \QED Now let $X$ be an effective cycle in ${\mathbb{P}}(E^{\rho})$ such that $X$ is flat over $Y$ with the relative dimension $d$ and the degree $\delta$ on the generic fiber. In \S\ref{subsec:Chow:forms:and:their:metrics} we constructed a Chow form $\Phi_X$ of $X$, which is an element of $H^0(Y, L \otimes (\operatorname{Sym}^{\delta}(E^{\rho}))^{{}\otimes {d+1}})$. Recall that $L$ is given a generalized metric by \eqref{eqn:metric:L}. For each irreducible component $X_i$ of $X_{red}$, let $\tilde{X_i} \to X_i$ be a generic resolution of singularities of $X_i$. Moreover, let $Y_0$ be the maximal open set of $Y$ such that the induced morphism $\tilde{X}_i \to Y$ is smooth over $Y_0$ for every $i$. Further, we fix terminologies. Let $T$ be a quasi-projective scheme over ${\mathbb{Z}}$, $t$ a closed point of $T_{{\mathbb{Q}}}$, and $K$ the residue field of $t$. By abuse use of notation, let $t : \operatorname{Spec}(K) \to T$ be the induced morphism by $t$. We say $t$ is {\em extensible in $T$} if $t : \operatorname{Spec}(K) \to T$ extends to $\tilde{t} : \operatorname{Spec}(O_K) \to T$, where $O_K$ is the ring of integers in $K$. Note that if $T$ is projective over ${\mathbb{Z}}$, then every closed point of $T_{{\mathbb{Q}}}$ is extensible in $T$. Let $V$ be a set, $\phi$ a non-negative function on $V$, and $S$ a finite subset of $V$. We define the geometric mean $\operatorname{g.\!m.}(\phi; S)$ of $\phi$ over $S$ to be \[ \operatorname{g.\!m.}(\phi; S) = \left( \prod_{s \in S} \phi(s) \right)^{1/\#(S)}. \] We will evaluate the norm of $\Phi_X$. \begin{Lemma} \label{lemma:evaluation:of:norm:of:Phi} There is a constant $c_1 (R,d,\delta)$ depending only on $R,d$ and $\delta$ with the following property. For any closed points $y$ of $(Y_0)_{{\mathbb{Q}}}$ with $y$ extensible in $Y$, \[ \operatorname{g.\!m.}\left( \Vert \Phi_{X} \Vert_{ \overline{L}\otimes (\operatorname{Sym}^{\delta}(\overline{E}^{\rho}))^{{}\otimes {d+1}}} ;\ O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{{\mathbb{Q}}})}(y) \right) \leq c_1 (R,d,\delta). \] \end{Lemma} {\sl Proof.}\quad Let $K$ be the residue field of $y$. Let $\Gamma$ be the normalization of the closure of $\{ y \}$ in $Y$. Then, since $y$ is extensible in $Y$, $\Gamma = \operatorname{Spec}(O_K)$. Thus, by virtue of Proposition~\ref{proposition:pullback}, we may assume $Y = \operatorname{Spec}(O_K)$. In this case, the estimate of the Chow form was already given in \cite[Proposition~1.3]{Bo} and \cite[4.3]{BGS}. Indeed if we let $k_L$ be the metric on $L$ such that \[ \Vert \Phi_{X} \Vert_ {(L,k_L) \otimes (\operatorname{Sym}^{\delta}(\overline{E}^{\rho}))^{{}\otimes {d+1}}} (w) = 1 \] for every $w \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{{\mathbb{Q}}})}(y)$, then $\widehat\deg (L,h_L) = h_{\overline{{\mathcal{O}}_E(1)}}(X)$ and $\widehat\deg (L,k_L) = h_{Herm}(\operatorname{Ch}(X))$, in the notation of \cite{BGS}. \QED Now we will state a relative case of Cornalba-Harris-Bost's inequality. \begin{Theorem} \label{thm:semistability:imply:average:semi-ampleness} Let $Y$ be a regular arithmetic variety, $\overline{E} = (E,h)$ a Hermitian vector bundle of rank $r$ on $Y$, $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ a group scheme morphism of degree $k$ commuting with the transposed morphism. Let $X$ be an effective cycle in ${\mathbb{P}} (E^{\rho})$ such that $X$ is flat over $Y$ with the relative dimension $d$ and degree $\delta$ on the generic fiber. Let $X_1,\ldots,X_l$ be the irreducible components of $X_{red}$, and $\tilde{X}_i \to X_i$ a generic resolution of singularities of $X_i$. Let $Y_0$ be the maximal open set of $Y$ such that the induced morphism $\tilde{X}_i \to Y$ is smooth over $Y_0$ for every $i$. Let $(B, h_B)$ be a line bundle equipped with a generalized metric on $Y$ given by the equality: \[ \widehat{{c}}_1(B, h_B) = r \pi_* \left(\widehat{{c}}_{1}(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot\widehat{X} \right) + k \delta (d+1) \widehat{{c}}_{1}(\overline{E}). \] Then, $h_B$ is $C^{\infty}$ over $Y_0$. Moreover, there are a positive integer $e=e(\rho,d,\delta)$, a positive integer $l=l(\rho,d,\delta)$, a positive constant $C=C(\rho,d,\delta)$, and sections $s_1, \ldots, s_l \in H^0(Y, B^{\otimes e})$ with the following properties. \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item $e$, $l$, and $C$ depend only on $\rho$, $d$, and $\delta$. \item For a closed point $y$ of $Y_{{\mathbb{Q}}}$, if $X_y$ is Chow semistable, then $s_i(y) \not= 0$ for some $i$. \item For all $i$ and all closed points $y$ of $(Y_0)_{{\mathbb{Q}}}$ with $y$ extensible in $Y$, \[ \operatorname{g.\!m.}\left( \left( h_B^{\otimes e} \right)(s_i, s_i);\ O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)\right) \leq C. \] \end{enumerate} In particular, if we set \[ \beta = e \left( r \pi_* \left(\widehat{{c}}_{1}(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot\widehat{X} \right) + k \delta (d+1) \widehat{{c}}_{1}(\overline{E})\right) + a (\log C), \] then, for any closed point $y \in (Y_0)_{{\mathbb{Q}}}$ with $X_y$ Chow semistable, there is a representative $(D, g)$ of $\beta$ such that $D$ is effective, $y \not\in D$, and that \[ \sum_{w \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(z)} g(w) \geq 0 \] for all $z \in (Y_0)_{{\mathbb{Q}}}$ with $z$ extensible in $Y$. \end{Theorem} Note that if $\rho$ is the identity morphism, then, by the proof below, $C(\rho,d,\delta)$ is depending only on $r,d,\delta$. \\ {\sl Proof.}\quad First of all, by Proposition~\ref{prop:when:Bost:divisor:smooth}, \[ r \pi_* \left(\widehat{{c}}_{1}(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot\widehat{X} \right) + k \delta (d+1) \widehat{{c}}_{1}(\overline{E}) \in \widehat{\operatorname{CH}}_{L^1}^1(Y; Y_0({\mathbb{C}})). \] Thus, $h_B$ is $C^{\infty}$ over $Y_0({\mathbb{C}})$. By Lemma~\ref{lemma:stability:generators:over:Z}, there are a positive integer $e$ and $\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$-invariant homogeneous polynomials $P_1,\cdots,P_l \in \operatorname{Sym}^{e}(\operatorname{Sym}^{\delta}({\mathbb{Z}}^R)^{d+1}{}^{\lor})$ depending only on $\rho$, $d$ and $\delta$ such that if $X_y$ is Chow semistable for a closed point $y$ of $Y_{{\mathbb{Q}}}$, then $P_i(\Phi_{X_{y}}) \ne 0$ for some $P_i$. For later use, we put $c_2 (\rho,d,\delta) = \max \{ |\!|\!| P_1 |\!|\!|, \cdots, |\!|\!| P_l |\!|\!| \}$, which is a constant depending only on $\rho$, $d$ and $\delta$. Recall that the Chow form $\Phi_X$ is an element of $H^0(Y,L \otimes (\operatorname{Sym}^{\delta}(E^{\rho}))^{{}\otimes {d+1}})$ and by Lemma~\ref{lemma:P:to:sheafhom} $P_i$ induces a polynomial map of sheaves \[ \varphi_{P_i} : L \otimes \operatorname{Sym}^{\delta} (E^{\rho})^{\otimes d+1} \to L^{\otimes er} \otimes (\det E)^{ek(d+1)\delta}. \] Hence we have \[ \varphi_{P_i} (\Phi_{X}) \in H^0 \left(Y,L^{\otimes e r} \otimes (\det E)^{e k(d+1)\delta} \right) = H^0 (Y, B^{\otimes e}) \] by \eqref{eqn:metric:L}. Here we set $s_i = \varphi_{P_i} (\Phi_{X})$. Then, the property (ii) is obvious by the construction of $\varphi_{P_i}$ and (i) of Proposition~\ref{proposition:pullback}. Now we will evaluate $\Vert s_i \Vert$. Let $y$ be a closed point of $(Y_0)_{{\mathbb{Q}}}$ with $y$ extensible in $Y$. Combining Lemma~\ref{lemma:norm:of:morphism:P} and Lemma~\ref{lemma:evaluation:of:norm:of:Phi}, we have \begin{align*} \operatorname{g.\!m.}\left( \Vert s_i \Vert ;\ O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{{\mathbb{Q}}})}(y) \right) & \leq \operatorname{g.\!m.}\left( |\!|\!| P_i |\!|\!|^r \, \Vert \Phi_{X} \Vert^{e r};\ O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{{\mathbb{Q}}})}(y) \right) \\ & \leq c_2(\rho,d,\delta) ^{r} c_1 (R,d,\delta) ^{e r}. \end{align*} Now we put \[ C(\rho,d,\delta) = c_1(R,d,\delta)^{2r} c_2(\rho,d,\delta)^{2e r}, \] which is a positive constant depending only on $\rho$, $d$ and $\delta$. Thus, we get (iii). \QED \begin{Remark} \label{rem:geom:analog:Cornalba-Harris-Bost} Here let us consider the geometric analogue of Theorem~\ref{thm:semistability:imply:average:semi-ampleness}. Let $Y$ be an algebraic variety over an algebraically closed field $k$, $E$ a vector bundle of rank $r$, $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ a group scheme morphism of degree $l$ commuting with the transposed morphism. Let $X$ be an effective cycle in ${\mathbb{P}} (E^{\rho})$ such that $X$ is flat over $Y$ with the relative dimension $d$ and degree $\delta$ on the generic fiber. Here we set \[ b_{X/Y}(E, \rho) = r \pi_* \left(c_{1}({\mathcal{O}}_{E^{\rho}}(1))^{d+1} \cdot X \right) + l \delta (d+1) c_{1}(E), \] which is a divisor on $Y$. In the same way as in the proof of Theorem~\ref{thm:semistability:imply:average:semi-ampleness} (actually, this case is much easier than the arithmetic case), we can show the following. \begin{quote} There is a positive integer $e$ depending only on $\rho$, $d$, and $\delta$ such that, if $X_y$ is Chow semi-stable for some $y \in Y$, then \[ H^0(Y, {\mathcal{O}}_Y(e b_{X/Y}(E, \rho))) \otimes {\mathcal{O}}_Y \to {\mathcal{O}}_Y(e b_{X/Y}(E, \rho)) \] is surjective at $y$. \end{quote} This gives a refinement of \cite[Theorem~3.2]{Bo}. \end{Remark} \subsection{Relationship of two theorems} \label{section:Bogomolov:to:Bost} \setcounter{Theorem}{0} In this subsection we will see some relationship between the relative Bogomolov's inequality (Theorem~\ref{thm:relative:Bogomolov:inequality:arithmetic:case}) and the relative Cornalba-Harris-Bost's inequality (Theorem~\ref{thm:semistability:imply:average:semi-ampleness}). For this purpose, we will first show a more intrinsic version of Theorem~\ref{thm:semistability:imply:average:semi-ampleness}. \begin{Proposition} \label{prop:intrinsic:relbost} Let $f : X \to Y$ be a flat morphism of regular projective arithmetic varieties with $\dim f = d$. Let $L$ be a relatively very ample line bundle such that $E = f_* (L)$ is a vector bundle of rank $r$ on $Y$. Let $\eta$ be the generic point of $X$ and $\delta = \deg (L_{\eta}^d)$. Moreover, let $i : X \to {\mathbb{P}}(E^{\lor})$ be the embedding over $Y$. Assume that $E$ is equipped with an Hermitian structure $h$ so that $L$ is also endowed with the Hermitian structure by $i^* {\mathcal{O}}_{E^{\lor}}(1) \simeq L$. Let $Y_0$ be the maximal open set of $Y$ such that $f$ is smooth over $Y_0$. Then, there is a positive integer $e(r,d,\delta)$ and a positive constant $C(r,d,\delta)$ depending only on $r,d,\delta$ with the following properties. If we set \[ \beta = e(r,d,\delta) \left( r f_* (\widehat{{c}}_1 (\overline{L})^{d+1}) - \delta (d+1) \widehat{{c}}_{1}(\overline{E})\right) + a (\log C(r,d,\delta)), \] then, for any closed point $y \in (Y_0)_{{\mathbb{Q}}}$ with $X_y$ Chow semistable, there is a representative $(D, g)$ of $\beta$ such that $D$ is effective, $y \not\in D$, and \[ \sum_{w \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(z)} g(w) \geq 0 \] for all $z \in (Y_0)_{{\mathbb{Q}}}$. \end{Proposition} {\sl Proof.}\quad We identify $X$ with its image by $i$. Let $\pi : {\mathbb{P}}(E) \to Y$ be the projection. Then, by Proposition~\ref{prop:when:Bost:divisor:smooth}, we get \[ \pi_* \left(\widehat{{c}}_{1}(\overline{{\mathcal{O}}_{E^{\lor}}(1)})^{d+1} \cdot\widehat{X} \right) = f_* (\widehat{{c}}_1 (\overline{L})^{d+1}) \] Thus, applying Theorem~\ref{thm:semistability:imply:average:semi-ampleness} for $(Y,E^{\lor},\operatorname{id},X)$, we get our assertion. \QED The following proposition will be derived from Theorem~\ref{thm:relative:Bogomolov:inequality:arithmetic:case}. \begin{Proposition} \label{prop:Bogomolov:to:Bost} Let $f : X \to Y$ be a projective morphism of regular arithmetic varieties such that every fiber of $f_{{\mathbb{C}}} : X({\mathbb{C}}) \to Y({\mathbb{C}})$ is a reduced and connected curve with only ordinary double singularities. We assume that the genus $g$ of the generic fiber of $f$ is greater than or equal to $1$. Let $L$ be a line bundle on $X$ such that \textup{(1)}the degree $\delta$ of $L$ on the generic fiber is greater than or equal to $2g+1$, \textup{(2)} $E = f_* (L)$ is a vector bundle of rank $r$ on $Y$ \textup{(}actually $r = \delta + 1 - g$\textup{)}, and that \textup{(3)} $f^*(E) \to L$ is surjective. Assume that $E$ is equipped with an Hermitian structure $h$ so that $L$ is also endowed with the quotient metric by $f^* (E) \to L$. Let $Y_0$ be the maximal open set of $Y$ such that $f$ is smooth over $Y_0$. Then, for any closed points $y$ of $(Y_0)_{{\mathbb{Q}}}$, \[ r f_* (\widehat{{c}}_1 (\overline{L})^{2}) - 2 \delta \widehat{{c}}_1 (\overline{E}) \] is weakly positive at $y$ with respect to any finite subsets of $Y_0({\mathbb{C}})$. \end{Proposition} Note that if the base space is $\operatorname{Spec}(O_K)$, then the second author showed in \cite[Theorem~1.1]{MorFh} the above inequality (under weaker assumptions) using \cite[Corollary~8.9]{MoBG}. Since we can prove Proposition~\ref{prop:Bogomolov:to:Bost} in the same way as \cite[Theorem~1.1]{MorFh}, we will only sketch the proof. {\sl Proof.}\quad Let $S = \operatorname{Ker} (f^* (E) \to L)$ and $h_S$ the submetric of $S$ induced by $h$. Then, by \cite{EL}, $S_{z}$ is stable for all $z \in Y_0({\mathbb{C}})$. Applying Theorem~\ref{thm:relative:Bogomolov:inequality:arithmetic:case} for $\overline{S} = (S,h_S)$, we obtain that if $y$ is a closed point of $(Y_0)_{{\mathbb{Q}}}$, then \[ f_* ( 2 (r-1) \widehat{{c}}_2(\overline{S}) - (r-2) \widehat{{c}}_1(\overline{S})^2) \] is weakly positive at $y$ with respect to any finite subsets of $Y_0({\mathbb{C}})$. If we set $\rho = \widehat{{c}}_2(f^*\overline{E}) - \widehat{{c}}_2(\overline{S}\oplus \overline{L})$, then there is $g \in L^1_{loc}(Y({\mathbb{C}}))$ such that $f_*(\rho) = a(g)$, $g$ is $C^{\infty}$ over $Y_0({\mathbb{C}})$, and $g > 0$ on $Y_0({\mathbb{C}})$. Now by a straightforward calculation, we have \begin{multline*} f_* ( 2 (r-1) \widehat{{c}}_2(\overline{S}) - (r-2) \widehat{{c}}_1(\overline{S})^2) + 2 (r-1) f_*(\rho) \\ = f_* \left( 2 (r-1) \widehat{{c}}_2(f^* \overline{E}) - (r-2) \widehat{{c}}_1(f^* \overline{E})^2 \right) + f_* \left( r \widehat{{c}}_1(\overline{L})^2 - 2 \widehat{{c}}_1(f^* \overline{E}) \cdot \widehat{{c}}_1(\overline{L}) \right) \\ = r f_* (\widehat{{c}}_1(\overline{L})^2) - 2 \delta \widehat{{c}}_1(\overline{E}). \end{multline*} \QED Let us compare Proposition~\ref{prop:intrinsic:relbost} with Proposition~\ref{prop:Bogomolov:to:Bost}. Both of them give some arithmetic positivity of the same divisor (although $d=1$ in Proposition~\ref{prop:Bogomolov:to:Bost}), under the assumption of some semi-stability (of Chow or of vector bundles). The former has advantage since it treats varieties of arbitrary relative dimension. On the other hand, the latter has advantage since it shows that the anonymous constant in the former is zero (see also \cite{Zh}). Moreover, in the complex case, the counterpart of the relative Bogomolov's inequality of Theorem~\ref{thm:relative:Bogomolov:inequality:arithmetic:case} has a wonderful application to the moduli of stable curves (\cite{MoRB}). \renewcommand{\thesection}{Appendix \Alph{section}} \renewcommand{\theTheorem}{\Alph{section}.\arabic{Theorem}} \renewcommand{\theClaim}{\Alph{section}.\arabic{Theorem}.\arabic{Claim}} \renewcommand{\theequation}{\Alph{section}.\arabic{Theorem}.\arabic{Claim}} \setcounter{section}{0} \section{Commutativity of push-forward and pull-back} \label{sec::comm:push:pull} Let $f : X \to Y$ be a smooth proper morphism of regular noetherian schemes, and $u : Y' \to Y$ a morphism of regular noetherian schemes. Let $X' = X \times_{Y} Y'$ and \[ \begin{CD} X @<{u'}<< X' \\ @V{f}VV @VV{f'}V \\ Y @<{u}<< Y' \end{CD} \] the induced diagram. Let $Z$ be a cycle of codimension $p$ and $|Z|$ the support of $Z$. We assume that $\operatorname{codim}_{X'}({u'}^{-1}(|Z|)) \geq p$. Then, it is easy to see that $\operatorname{codim}_{Y'}(u^{-1}(|f_*(Z)|)) \geq p - d$, where $d = \dim X - \dim Y$. Thus, we can define $f'_*({u'}^*(Z))$ and $u^*(f_*(Z))$ as elements of $Z^{p-d}(Y')$. It is well known, we believe, that $f'_*({u'}^*(Z)) = u^*(f_*(Z))$ in $Z^{p-d}(Y')$. We could not however find any suitable references for the above fact, so that in this section, we would like to give the proof of it. \bigskip Let $X$ be a regular noetherian scheme, and $T$ a closed subscheme of $X$. We denote by $K'_T(X)$ the Grothendieck group generated by coherent sheaves $F$ with $\operatorname{Supp}(F) \subseteq T_{red}$ modulo the following relation: $[F] = [F'] + [F'']$ if there is an exact sequence $0 \to F' \to F \to F'' \to 0$. Let $p$ be a non-negative integer, and $X^{(p)}$ the set of all points $x$ of $X$ with $\operatorname{codim}_X \overline{\{ x \}} = p$. We define $Z^p_T(X)$ to be \[ Z^p_T(X) = \bigoplus_{ x \in X^{(p)} \cap T} {\mathbb{Z}} \cdot \overline{\{ x \}}. \] We assume that $\operatorname{codim}_X T \geq p$. Then, we can define the natural homomorphism \[ z^p : K'_T(X) \to Z^p_T(X) \] to be \[ z^p([F]) = \sum_{x \in X^{(p)} \cap T} l_{{\mathcal{O}}_{X, x}}(F_x) \cdot \overline{\{ x \}}, \] where $l_{{\mathcal{O}}_{X, x}}(F_x)$ is the length of $F_x$ as ${\mathcal{O}}_{X,x}$-modules. Note that if $\operatorname{codim}_X T > p$, then $z^p = 0$. \medskip Let $f : X \to Y$ be a proper morphism of regular noetherian schemes, and $T$ a closed subscheme of $X$. Then, we define the homomorphism $f_* : K'_T(X) \to K'_{f(T)}(Y)$ to be \[ f_*([F]) = \sum_{i \geq 0} [R^i f_*(F)]. \] Here we set $d = \dim X - \dim Y$. Let $p$ be a non-negative integer with $\operatorname{codim}_X T \geq p$ and $p \geq d$. Then, $\operatorname{codim}_Y f(T) \geq p - d$. First, let us consider the following proposition. \begin{Proposition} \label{prop:comm:push:forward} With notation as above, the diagram \[ \begin{CD} K'_T(X) @>{z^p}>> Z^p_T(X) \\ @V{f_*}VV @VV{f_*}V \\ K'_{f(T)}(Y) @>{z^{p-d}}>> Z^{p-d}_{f(T)}(Y) \end{CD} \] is commutative. \end{Proposition} {\sl Proof.}\quad For a coherent sheaf $F$ on $X$ with $\operatorname{Supp}(F) \subseteq T_{red}$, there is a filtration $0 = F_0 \subseteq F_1 \subseteq \cdots \subseteq F_n = F$ with $F_i/F_{i-1} \simeq {\mathcal{O}}_X/P_i$ for some prime ideal sheaves $P_i$ on $X$. Then, \[ \begin{cases} f_*(z^p([F])) = \sum_{i=1}^n f_*(z^p([{\mathcal{O}}_X/P_i])) \\ z^{p-d}(f_*([F])) = \sum_{i=1}^n z^{p-d}(f_*([{\mathcal{O}}_X/P_i])). \end{cases} \] Thus, we may assume that $F = {\mathcal{O}}_X/P$ for some prime ideal sheaf $P$ with $\operatorname{Supp}({\mathcal{O}}_X/P) \subseteq T_{red}$ and $\operatorname{codim}_X(\operatorname{Spec}({\mathcal{O}}_X/P)) = p$. We set $Z = \operatorname{Spec}({\mathcal{O}}_X/P)$. Then, $z^p([{\mathcal{O}}_X/P]) = Z$. First, let us consider the case where $\dim f(Z) < \dim Z$. In this case, $f_*(Z) = 0$. On the other hand, since $\operatorname{Supp}(R^i f_*({\mathcal{O}}_X/P))\subseteq f(Z)$, we can see that $z^{p-d}([R^i f_*({\mathcal{O}}_X/P)]) = 0$ for all $i \geq 0$. Thus, $z^{p-d}(f_*([{\mathcal{O}}/P])) = 0$. Next, we assume that $\dim f(Z) = \dim Z$. Then, $Z \to f(Z)$ is generically finite. Thus, $\operatorname{Supp}(R^i f_*({\mathcal{O}}_X/P))$ is a proper closed subset of $f(Z)$ for each $i \geq 1$. Therefore, we have \[ z^{p-d}(f_*([{\mathcal{O}}_X/P])) = z^{p-d}([f_*({\mathcal{O}}_X/P)]) = f_*(Z). \] \QED Let $g : Z \to X$ be a morphism of regular noetherian schemes, and $T$ a closed subscheme of $X$. Then, we define the homomorphism $g^* : K'_T(X) \to K'_{f^{-1}(T)}(Z)$ to be \[ g^*([F]) = \sum_{i \geq 0} (-1)^i [L_if^*(F)]. \] Let $p$ be a non-negative integer with $\operatorname{codim}_X T \geq p$ and $\operatorname{codim}_Z (g^{-1}(T)) \geq p$. Here let us consider the following proposition. \begin{Proposition} \label{prop:well:def:pull:back} Let $F$ and $G$ be coherent sheaves on $X$ with $\operatorname{Supp}(F), \operatorname{Supp}(G) \subseteq T_{red}$. If $z^p([F]) = z^p([G])$, then $z^{p}(g^*([F])) = z^{p}(g^*([G]))$. \end{Proposition} {\sl Proof.}\quad Let $0 = F_0 \subseteq F_1 \subseteq \cdots \subseteq F_n = F$ and $0 = G_0 \subseteq G_1 \subseteq \cdots \subseteq G_m = F$ be filtrations of $F$ and $G$ respectively such that $F_i/F_{i-1} \simeq {\mathcal{O}}_X/P_i$ and $G_j/G_{j-1} \simeq {\mathcal{O}}_X/Q_j$ for some prime ideal sheaves $P_i$ and $Q_j$ on $X$. Then, \[ \begin{cases} z^{p}(g^*([F])) = \sum_{i=1}^n z^{p}(g^*([{\mathcal{O}}_X/P_i])) \\ z^{p}(g^*([G])) = \sum_{j=1}^m z^{p}(g^*([{\mathcal{O}}_X/Q_j])) \end{cases} \] Thus, it is sufficient to show that $z^{p}(g^*([{\mathcal{O}}_X/P])) = 0$ for all prime ideals $P$ with \[ \text{ $\operatorname{Supp}({\mathcal{O}}_X/P) \subseteq T_{red}$, $\operatorname{codim}_X (\operatorname{Supp}({\mathcal{O}}_X/P)) > p$ and $\operatorname{codim}_Z (g^{-1}(\operatorname{Supp}({\mathcal{O}}_X/P))) = p$. } \] This is a consequence of the following lemma. \QED \begin{Lemma} \label{lem:vanshing:tor:alt:sum} Let $(A, m)$ and $(B, n)$ be regular local rings, $\phi : A \to B$ a homomorphism of local rings, and $M$ an $A$-module of finite type. If $\operatorname{Supp}(M \otimes_A B) = \{ n \}$ and \[ \operatorname{codim}_{\operatorname{Spec}(B)} (\operatorname{Supp}(M \otimes_A B)) < \operatorname{codim}_{\operatorname{Spec}(A)} (\operatorname{Supp}(M)), \] then \[ \sum_{i \geq 0} (-1)^i l_B(\operatorname{Tor}_i^A(M, B)) = 0. \] \end{Lemma} {\sl Proof.}\quad We freely use notations in \cite[Chapter~I]{SoAr}. Let $f : \operatorname{Spec}(B) \to \operatorname{Spec}(A)$ be a morphism induced by $\phi : A \to B$. We set $Y = \operatorname{Supp}(M)$ and $q = \operatorname{codim}_{\operatorname{Spec}(A)} (\operatorname{Supp}(M))$. Let $P_{\cdot} \to M$ be a free resolution of $M$. Then, $[P_{\cdot}] \in F^q K_0^Y(\operatorname{Spec}(A))$. Thus, by \cite[(iii) of Theorem~3 in I.3]{SoAr}, \[ [f^*(P_{\cdot})] = [ P_{\cdot} \otimes_A B ] \in F^q K_0^{\{ n \}}(\operatorname{Spec}(B))_{{\mathbb{Q}}} \] because $f^{-1}(Y) = \operatorname{Supp}(M \otimes_A B) = \{ n \}$. On the other hand, since \[ q > \operatorname{codim}_{\operatorname{Spec}(B)} (\operatorname{Supp}(M \otimes_A B)) = \dim B, \] we have $F^q K_0^{\{ n \}}(\operatorname{Spec}(B))_{{\mathbb{Q}}} = \{ 0 \}$. Thus, $[ P_{\cdot} \otimes_A B ] = 0$ in $K_0^{\{ n \}}(\operatorname{Spec}(B))$ because \[ K_0^{\{ n \}}(\operatorname{Spec}(B)) \simeq {\mathbb{Z}} \] has no torsion. This shows us our assertion. \QED As a corollary of Proposition~\ref{prop:well:def:pull:back}, we have the following. \begin{Corollary} \label{cor:comm:pull:back} With notation as in Proposition~\textup{\ref{prop:well:def:pull:back}}, \[ \begin{CD} K'_T(X) @>{z^p}>> Z^p_T(X) \\ @V{g^*}VV @VV{g^*}V \\ K'_{f^{-1}(T)}(Z) @>{z^{p}}>> Z^{p}_{f^{-1}(T)}(Z) \end{CD} \] is commutative. Note that $g^* : Z^p_T(X) \to Z^{p}_{f^{-1}(T)}(Z)$ is defined by $g^*(Z) = z^{p}(g^*([{\mathcal{O}}_Z]))$ for each integral cycle $Z$ in $Z^p_T(X)$. \end{Corollary} Let $f : X \to Y$ be a flat proper morphism of regular noetherian schemes, and $u : Y' \to Y$ a morphism of regular noetherian schemes. Let $X' = X \times_{Y} Y'$ and \[ \begin{CD} X @<{u'}<< X' \\ @V{f}VV @VV{f'}V \\ Y @<{u}<< Y' \end{CD} \] the induced diagram. We assume that $X'$ is regular. Note that if $f$ is smooth, then $X'$ is regular. We set $d = \dim X - \dim Y = \dim X' - \dim Y'$. Let $T$ be a closed subscheme of $X$, and $p$ a non-negative integer with $\operatorname{codim}_X T \geq p$, $\operatorname{codim}_{X'} ({u'}^{-1}(T)) \geq p$ and $p \geq d$. Note that $\operatorname{codim}_Y f(T) \geq p-d$ and $\operatorname{codim}_{Y'} (u^{-1}(f(T))) \geq p - d$ because $u^{-1}(f(T)) = f'({u'}^{-1}(T))$. Then, we have the following proposition. \begin{Proposition} \label{prop:comm:cycle:push:pull} The diagram \[ \begin{CD} Z^p_T(X) @>{{u'}^*}>> Z^{p}_{{u'}^{-1}(T)}(X') \\ @V{f_*}VV @VV{f'_*}V \\ Z^{p-d}_{f(T)}(Y) @>{u^*}>> Z^{p-d}_{u^{-1}(f(T))}(Y') \end{CD} \] is commutative. \end{Proposition} {\sl Proof.}\quad Since $f$ is flat, by \cite[Proposition 3.1.0 in IV]{SGA6}, for any coherent sheaves $F$ on $X$, \[ L_{\cdot}u^* \left( R^{\cdot} f_*(F) \right) \overset{\sim}{\longrightarrow} R^{\cdot}f'_* \left( L_{\cdot}{u'}^*(F)\right), \] which shows that the diagram \[ \begin{CD} K'_T(X) @>{{u'}^*}>> K'_{{u'}^{-1}(T)}(X') \\ @V{f_*}VV @VV{f'_*}V \\ K'_{f(T)}(Y) @>{u^*}>> K'_{u^{-1}(f(T))}(Y') \end{CD} \] is commutative. Thus, by virtue of Proposition~\ref{prop:comm:push:forward} and Corollary~\ref{cor:comm:pull:back}, we can see our proposition. \QED \bigskip

9710

alg-geom/9710023

en

https://arxiv.org/abs/alg-geom/9710023

alg-geom/9710023

Norbert A'Campo

Norbert A'Campo

Real deformations and complex topology of plane curve singularities

16 pages TeX with 11 fig-xx.eps

Ann. Fac. Sci. Toulouse Math. (6) 8 (1999), no. 1, 5-23

This is the paper as published. The topology of a complex plane curve singularity with real branches is deduced from any real deformation having delta crossings. An example of the computation of the global geometric monodromy of a polynomial mapping is added.

alg-geom

\section{1}{Introduction} The geometric monodromy $T$ of a curve singularity in the complex plane is a diffeomorphism of a compact surface with boundary $(F,\partial F)$ inducing the identity on the boundary, which is well defined up to isotopy relative to the boundary. The geometric monodromy of a curve singularity in the complex plane determines the local topology of the singularity. As element of the mapping class group of the surface $(F,\partial F)$, the diffeomorphism $T$ can be written as a composition of Dehn twists. In section 3 of this paper the geometric monodromy of an isolated plane curve singularity is written explicitly as a composition of right Dehn twists. In fact, a global graphical algorithm for the construction of the surface $(F,\partial F)$ with a system of simply closed curves on it is given in section 4, such that the curves of this system are the vanishing cycles of a real morsification of the singularity. In section 5, as an illustration, the global geometric monodromy of the polynomial $y^4-2y^2x^3+x^6-x^7-4yx^5,$ which has two critical fibers, is computed. \br The germ of a curve singularity in $\Bbb C^2$ is a finite union of parametrized local branches $b_i:\Bbb C \to \Bbb C^2, 1 \leq i \leq r.$ First observe, that without loss of generality for the local topology, we can assume that the branches have a real polynomial parameterization. The combinatorial data used to describe the geometric monodromy of a curve singularity come from generic real polynomial deformations of the parameterizations of the local branches $b_{i,t}:\Bbb C \to \Bbb C^2, 1 \leq i \leq r, t \in [0,1]$, such that:\br (i) $b_{i,0}=b_i, 1 \leq i \leq r,$\br (ii) for some $\rho > 0$ the intersection of the union of the branches with the $\rho-$ball $B$ at the singular point of curve in $\Bbb C^2$ is a representative of the germ of the curve and $B$ is a Milnor ball for the germ,\br (iii) the images of $b_{i,t},1 \leq i \leq r, t \in [0,1],$ intersect the boundary of the ball $B$ transversally,\br (iv) the union of the images $b_{i,t}(\Bbb R), 1 \leq i \leq r,$ has for every $t \in (0,1]$ the maximal possible number of double points in the interior of $B$.\br Such deformations correspond to real morsifications of the defining equation of the singularity and were used to study the local monodromy in [AC2],[AC3],[G-Z]. Real deformations of singu\-larities of plane algebraic curves with the maximal possible number of double points in the real plane were discovered by Charlotte Angas Scott [S1,S2]. I thank Egbert Brieskorn for having drawn my attention on the references [S1],[S2]. In section 6 we start with a connected divide, which defines as explained in section 3 a classical link. We will construct a map from the complement of the link of a connected divide to the circle and prove that this map is a fibration. This fibration is for a divide of a plane curve singularity a model for the Milnor fibration of the singularity. The link of most connected divides are hyperbolic. In a forth coming paper we will study the geometry of a link of a divide. We used MAPLE for the drawings of parametrized curves and for the computation of suitable deformations of the polynomial equations. Of great help for the investigation of topological changes in families of polynomial equations is the mathematical software SURF which has been developed by Stefan Endrass. I thank Stefan Endrass warmly for permitting me to use SURF. Part of this work was done in Toulouse and I thank the members of the Laboratory \'Emile Picard for their hospitality. \par\noindent \section{2}{\fam\bffam\tenbf Real deformations of plane curve singularities} Let $f:\Bbb C^2 \to \Bbb C$ be the germ at $0 \in \Bbb C^2$ of an holomorphic map with $f(0)=0$ and having an isolated singularity $S$ at $0$. We are mainly interested in the study of topological properties of singularities, therefore we can assume without loss of generality that the germ $f$ is a product of locally irreducible real polynomials. Having chosen a Milnor ball $B(0,\rho)$ for $f$, there exists a real polynomial deformation family $f_t,t \in [0,1],$ of $f$ such that for all $t$ the $0$-level of $f_t$ is transversal to the boundary of the ball $B(0,\rho)$ and such that for all $t \in ]0,1]$ the $0$-level of $f_t$ has $\delta$ transversal double points in the interior of the disk $D(0,\rho):=B(0,\rho) \cap \Bbb R^2$, where the Milnor number $\mu$ and the number $r$ of local branches of $f$ satisfy $\mu=2\delta-r+1.$ In particular, the $0$-level of $f_t, t \in ]0,1],$ has in $D(0,\rho)$ no self tangencies or triple intersections. It is possible to choose for $f_t,t \in [0,1],$ a family of defining equations for the union of the images of $b_{i,t}, 1 \leq i \leq r.$ The deformation $f_t,t \in [0,1]$ is called a real morsification with respect to the Milnor ball $B(0,\rho)$ of $f.$ So, the $0$-level of the restriction of $f_t,t \in ]0,1],$ to $D(0,\rho)$ is an immersion without self-tangencies and having only transversal self-intersections of $r$ copies of an interval (see [AC2],[AC3],[G-Z]). The $0$-level of the restriction of $f_t,t \in ]0,1],$ to $D(0,\rho)$ is up to a diffeomorphism independent of $t,$ it is called a divide ("partage" in [AC2]) and it is shown that for instance the divide determines the homological monodromy group of the versal deformation of the singularity. Figure 1 represents a divide for the singularity at $0 \in \Bbb C^2$ of the curve $(y^5-x^3)(x^5-y^3)=0$. \midinsert \cline{\epsffile{fig-1.eps}} \medskip \centerline{Figure 1: A divide for $(y^5-x^3)(x^5-y^3)=0.$} \endinsert \remark{\fam\bffam\tenbf Remark} The transversal isotopy class of the divide of a singularity with real branches is not a topological invariant of the singularity. The singularities of $y^4-2y^2x^3+x^6+x^7$ and $y^4-2y^2x^3+x^6-x^7$ have congruent but not transversal isotopic divides. The singularities $(x^2-y^2)(x^2-y^3)(y^2-x^3)$ and $(x^2-3xy+2y^2)(x^2-y^3)(y^2-x^3)$ are topologically equivalent but can not have congruent divides. The singularity $y^3-x^5$ admits two divides, which give a model for the smallest possible transition, according to the mod 4 congruence of V. Arnold [A] and its celebrated strengthening to a mod 8 congruence of V. A. Rohlin [R1,R2], of odd ovals to even ovals for projective real M-curves of even degree. I owe this remark to Oleg Viro [V]. More precisely, there exist a polynomial family $f_s(x,y), s \in \Bbb R,$ of polynomials of degree $6,$ having the central symmetry $f_s(x,y)=-f_{-s}(-x,-y)$ such that the levels $f_s(x,y)=0,\ s \not=0,$ are divides for the singularity $f_0(x,y)=y^3-x^5.$ Moreover, for $s \in \Bbb R,\ s\not=0,$ the divide $f_s(x,y)=0$ has for regions on which the function has the sign of the parameter $s.$ Therefor, at $s=0$ four regions of $f_s(x,y)=0$ collapse and hence, four ovals of $f_s(x,y)=(s/2)^E$ collapse and change parity at $s=0$ if the exponent $E$ is big and odd. \midinsert \cline{\epsffile{divide_links.eps} \ \epsffile{divide_rechts.eps}} \medskip \centerline{Figure 2a: The divides $f_{\pm 1}(x,y)=0.$} \endinsert In Figure $2b$ are drawn the smoothings of the divides $f_{-1}(x,y)=0$ and $f_{+1}(x,y)=0$ of the singularity of $f_0(x,y)=y^3-x^5$ at $0$. Each of the smoothings $\{f_{\pm1}(x,y)=\pm \epsilon \}\cap D$ consists of four ovals and a chord, such that the ovals lie on the positive, respectively on the negative side, of the chord. Such a family $f_s(x,y)$ is for instance given by: \midinsert \cline{\epsffile{links.eps} \ \epsffile{rechts.eps}} \medskip \centerline{Figure 2b: Four ovals change parity.} \endinsert $$ y^3-x^5 -{{125}\over{8}}\,{s}^{3}{x}^{6} +({{375}\over{64}}\,{s}^{6} +{{245}\over{16}}\,{s}^{4} -{{25}\over{4}}\,{s}^{2}){x}^{5} +{{75}\over{4}}\,{s}^{2}{x}^{4}y $$ $$ +({{2695}\over{128}}\,{s}^{7} +{{21625}\over{256}}\,{s}^{9} -{{35}\over{8}}\,{s}^{3} -{{4847}\over{160}}\,{s}^{5}){x}^{4} -(5\,s +{{159}\over{4}}\,{s}^{3} -{{75}\over{4}}\,{s}^{5}){x}^{3}y $$ $$ -{{15}\over{2}}\,s{x}^{2}{y}^{2} -({{2703}\over{500}}\,{s}^{6} +{{1281}\over{32}}\,{s}^{8} -{{29625}\over{512}}\,{s}^{12} +{{2345}\over{128}}\,{s}^{10})x^3 $$ $$ -({{17625}\over{256}}\,{s}^{8} +{{3583}\over{32}}\,{s}^{6} +{{5793}\over{400}}\,{s}^{4}){x}^{2}y -({{95}\over{2}}\,{s}^{4} +{{53}\over{10}}\,{s}^{2})x{y}^{2} $$ $$ -({{997}\over{4000}}\,{s}^{9} +{{42875}\over{2048}}\,{s}^{15} +{{4575}\over{256}}\,{s}^{13} +{{857}\over{320}}\,{s}^{11}){x}^{2} $$ $$ -({{177325}\over{2048}}\,{s}^{11} +{{1803}\over{200}}\,{s}^{7} +{{35441}\over{512}}\,{s}^{9})xy -({{6395}\over{128}}\,{s}^{7} +{{317}\over{40}}\,{s}^{5}){y}^{2} $$ $$ +({{19871}\over{1280}}\,{s}^{14} +{{10165}\over{1024}}\,{s}^{16} -{{59125}\over{4096}}\,{s}^{18} +{{4171}\over{2000}}\,{s}^{12})x $$ $$ +({{51025}\over{4096}}\,{s}^{12} +{{54223}\over{25600}}\,{s}^{10} -{{153725}\over{16384}}\,{s}^{14})y $$ \endremark \remark{\fam\bffam\tenbf Problem} Classify up to transversal isotopy, i.e. isotopy through immersions with only transversal double and triple point crossings, the divides for an isolated real plane curve singularity. \endremark \goodbreak \section{3}{Complex topology of plane curve singularities} In this section we wish to explain how one can read off from the divide of a plane curve singularity $S$ the local link $L$, the Milnor fiber and the geometric monodromy group of the singularity. In particular, we will give the geometric monodromy of the singularity explicitly as a product of Dehn twists. Let $P \subset D(0,\rho)$ be the divide of the singularity $f$. For a tangent vector $v \in TD(0,\rho)=D(0,\rho) \times\Bbb R^2$ of $D$ at the point $p\in D(0,\rho)$ let $J(v) \in \Bbb C^2$ be the point $p+iv.$ The Milnor ball $B$ can be viewed as $$ B(0,\rho)=\{J(v) \mid v \in T(D(0,\rho)) \ \text{ and }\ \|J(v)\| \leq \rho\}. $$ Observe that $$L(P):=\{J(v) \mid v \in T(P) \ \text{ and }\ \|J(v)\|=\rho \}$$ is a closed submanifold of dimension one in the boundary of the Milnor ball $B(0,\rho)$. We call $L(P)$ the link of the divide $P$. Note further that $$R(P):=\{J(v) \mid v \in T(P) \ \text{ and }\ \|J(v)\| \leq \rho \}$$ is an immersed surface in $B(0,\rho)$ with boundary $L(P)$ having only transversal double point singularities. Let $F(P)$ be the surface obtained from $R(P)$ by replacing the local links of its singularities by cylinders. The differential model of those replacements is as follows: let $\chi: \Bbb C^2 \to \Bbb R$ be a smooth bump function at $0 \in \Bbb C^2$; replace the immersed surface $\{(x,y) \in \Bbb C^2 \mid xy=0\}$ by the smooth surface $\{(x,y) \in \Bbb C^2 \mid xy=\tau^2\chi(x/\tau,y/\tau)\}$, where $\tau$ is a sufficiently small positive real number. We call $R(P)$ the singular and $F(P)$ the regular ribbon surface of the divide $P.$ The connected, compact surface $F(P)$ has genus $g:=\delta-r+1$ and $r$ boundary components. Note, that $g$ is the number of regions of the divide $P$. A region of $P$ is a connected component of the complement of $P$ in $D(0,\rho)$, which lies in the interior of $D(0,\rho)$. For the example drawn in Figure $1,$ we have $r=2,\delta=17,g=16.$ The ribbon surface $R(P)$ carries a natural orientation, since parametrized by an open subset of the tangent space $T(\Bbb R).$ Hence the surface $F(P)$ and the link $L(P)$ are also naturally oriented. We orient $B$ as a submanifold of $-T\Bbb R^2,$ which is the orientation of $B$ as a submanifold in $\Bbb C^2.$ \proclaim{Theorem 1} Let $P$ be the divide for an isolated plane curve singularity $S.$ The submanifold $(F(P),L(P))$ is up to isotopy a model for the Milnor fiber of the singularity $S$. \endgroup\bigbreak \demo{ \fam\bffam\tenbf Proof} Choose $0 < \rho_{-} < \rho$ such that $P \cap D(0,\rho_{-})$ is still a divide for the singularity $S.$ Along the divide the singular level $F_{t,0}:=\{(x,y) \in B \mid f_t(x,y)=0 \}$ is up to order $1$ tangent to the immersed surface $R(P)$. Hence, for $B_{-}':=\{u+iv \in B(0,\rho_{-}) \mid u,v \in \Bbb R^2, ||v|| \leq \rho'\}$ with $0 < \rho' << \rho,$ the intersections $R'(P):=\partial B_{-}' \cap R(P)$ and $F'_{t,0}:=\partial B_{-}' \cap F_{t,0}$ are transversal and are regular collar neighbourhoods of the divide in $R(P)$ and in $F_{t,0}.$ Therefore the nonsingular level $F_{t,\eta}:=\{(x,y) \in B(0,\rho) \mid f_t(x,y)=\eta \}$, where $\eta \in \Bbb R$ is sufficiently small, contains in its interior $F'_{t,\eta}:=B_{-}' \cap F_{t,\eta},$ which is a diffeomorphic copy of the surface with boundary $F(P)$. Since $F'_{t,\eta}$ and $F_{t,\eta}$ are connected surfaces both with $r$ boundary components and the intersection forms on the first homology are isomorphic, the difference $F_{t,\eta}\setminus F'_{t,\eta}$ is a union of open collar tubular neighbourhoods of the boundary components of the surface $F_{t,\eta}$. So, the surfaces $F_{t,\eta},F'_{t,\eta}$ and $F(P)$ are diffeomorphic. We conclude by observing that the nonsingular levels $F_{t,\eta}$ and the Milnor fiber are connected in the local unfolding through nonsingular levels. \penalty-100\null\hfill\qed\bigbreak From this proof it follows also that the local link $L(S)$ of the singularity $S$ in $\partial B$ is cobordant to the sub\-manifold $\partial F'_{t,0}$ in $\partial B_{-}'.$ The cobordism is given by the pair $(B \setminus \text{int}(B_{-}'),F_{t,\eta} \setminus \text{int}(F'_{t,\eta})).$ It is clear, that the pairs $(\partial B,L(P)),$ $(\partial B_{-}',\partial F'_{t,0})$ and $(\partial B_{-}',\partial F'_{t,\eta})$ are diffeomorphic. One can prove even more: \proclaim{Theorem 2} Let $P$ be the divide for an isolated plane curve singularity $S.$ The pairs $(\partial B,L(S))$, where $L(S)$ is the local link of the singularity $S,$ and $(\partial B,L(P))$ are diffeomorphic. \endgroup\bigbreak The proof is given in section 6. \remark{\fam\bffam\tenbf Remark} The signed planar Dynkin diagram of the divide determines up to isotopy the divide of the singularity. It follows from Theorem 2, that the signed planar Dynkin diagram determines geometrically the topology of the singularity. Using the theorem of Burau and Zariski stating that the topological type of a plane curve singularity is determined by the mutual intersection numbers of the branches and the Alexander polynomial of each branch, the authors L. Balke and R. Kaenders [B-K] have proved that the signed Dynkin diagram, without its planar embedding, determines the topology of the singularity. \endremark \midinsert \cline{\epsffile{fig-3.eps}} \medskip \centerline{Figure $3:$ The link $P_v.$} \endinsert We need a combinatorial description of the surface $F(P).$ For a divide $P$ we define: A vertex of $P$ is double point of $P$, and an edge of $P$ is the closure of a connected component of the complement of the vertices in $P$. Now we choose an orientation of $\Bbb R^2$, and a small deformation $\bar f$ of the polynomial $f$ such that the $0$-level of $\bar f$ is the divide $P.$ We call a region of the divide positive or negative according to the sign of $\bar f$. We orient the boundaries of the positive regions such that the outer normal and the oriented tangents of the boundary agree in this order with the chosen orientation of $\Bbb R^2$. We choose a midpoint on each edge, which connects two vertices. The link $P_v$ of a vertex $v$ is the closure of the connected component of the complement of the midpoints in $P$ containing the given vertex $v$. \midinsert \cline{\epsffile{fig-4.eps}} \medskip \centerline{Figure $4:$ A piece of surface $F_v.$} \endinsert For each vertex $v$ of $P$ we will construct a piece of surface $F_v$, such that those pieces glue together and build $F(P)$. Let $P_v$ be the link of the vertex $v$. Call $c_v,c_v'$ the endpoints of the branches of $P_v,$ which are oriented towards $v,$ and $d_v,d_v'$ the endpoints of the branches of $P_v,$ which are oriented away from $v$. Thus, $c_v,c_v',d_v,d_v'$ are midpoints or endpoints of the divide $P$ (see Figure $3.$) Using an orientation of the divide $P$ we label $c_v,c_v'$ such that $c_v'$ comes after $c_v$, and we label $d_v,d_v'$ such that the sector $c_v,d_v$ is in a positive region. Then $F_v$ is the surface with boundary and corners drawn in Figure $4.$ There are 8 corners and there are 8 boundary components in between the corners, 4 of them will get a marking by $c_v,c_v',d_v,d_v'$, which will determine the gluing with the piece of the next vertex and 4 do not have a marking. The gluing of the pieces $F_v$ along the marked boundary components according to the gluing scheme given by the divide $P$ yields the surface $F(P)$. On the pieces $F_v$ we have drawn oriented curves colored red, white, and blue. The white curves are simple closed pairwise disjoint curves. The surface $F(P)$ will be oriented such that the curves taken in the order red-white-blue have nonnegative intersections. The remaining red curves glue together and build a red graph on $F(P)$. The remaining blue curves build a blue graph. After deleting each contractible component of the red or blue graph, each of the remaining components contains a simple closed red or blue curve. All together, we have constructed on $F(P)$ a system of $\mu$ simple closed curves $\delta_1,\delta_2, ... ,\delta_{\mu-1},\delta_\mu$, which we list by first taking red, then white and finally blue. We denote by $n_{+}$ the number of red curves which is also the number of positive regions, by $n_{\cdot}$ the number of crossing points and by $n_{-}$ the number of blue which equals the number of negative regions of the divide $P$. Let $D_i$ be the right Dehn twist along the curve $\delta_i.$ A model for the right Dehn twist is the linear action $(x,y) \mapsto (x+y,y)$ on the cylinder $\{(x,y) \in \Bbb R/\Bbb Z \times \Bbb R \mid 0 \leq y \leq 1\}$ with as orientation the product of the natural orientations of the factors. A right Dehn twist around a simply closed curve $\delta$ on an oriented surface is obtained by embedding the model as an oriented bicollar neighbourhood of $\delta$ such that $\delta$ and $\Bbb R/\Bbb Z \times \{1/2\}$ of the model match. The local geometric monodromy of the singularity of $xy=0$ is as diffeomorphism a right Dehn twist (voir le Th\'eor\`eme Fondamental, page 23 de [L], et page 95 de [P-S]). Using as in [AC2] a local version of a Theorem of Lefschetz, one obtains: \proclaim{Theorem 3} Let $P$ be the divide for an isolated plane curve singularity $S.$ The Dehn twists $D_i$ are generators for the geometric monodromy group of the unfolding of the singularity $S$. The product $T:=D_{\mu}D_{\mu-1} ... D_2D_1$ is the local geometric monodromy of the singularity $S$. \endgroup\bigbreak \section{4}{The singularity $D_5$ and a graphical algorithm in general} We will work out the picture for the singularity $D_5$ with the equation $x(x^3-y^2)$ and the divide given by the deformation $(x-s)(x^3+5sx^2-y^2), s \in [0,1],$ which is shown for $s=1$ in Figure $5.$ There are one positive triangular region, one negative region and three crossings. By gluing three pieces together, one gets the Milnor fiber with a system of vanishing cycles as depicted in Figure $6.$ \midinsert \cline{\epsffile{fig-5.eps}} \medskip \centerline{Figure $5:$ A divide for the singularity $D_5.$} \endinsert An easy and fast graphical algorithm of visualizing the Milnor fiber with a system of vanishing cycles directly from the divide is as follows: think the divide as a road network which has $\delta$ junctions, and replace every junction by a roundabout, which leads you to a new road network with $4\delta$ T-junctions. Realize now every road section in between two T-junctions by a strip with a half twist. Do the same for every road section in between a T-junction and the boundary of the divide. Altogether you will need $6\delta+r$ strips. The core line of the four strips of a roundabout is a white vanishing cycle, the strips corresponding to boundary edges and corners of a positive or negative region have as core line a red or blue vanishing cycle. \midinsert \cline{\epsffile{fig-6.eps}} \medskip \centerline{Figure $6:$ Milnor fiber with vanishing cycles for $D_5.$} \endinsert In Figure $7$ is worked out the singularity with two Puiseux pairs and $\mu=16$, where we used the divide from Figure $9.$ \midinsert \cline{\epsffile{fig-7.eps}} \medskip \centerline{Figure $7:$ Milnor fiber with vanishing cycles for $y^4-2y^2x^3+x^6-x^7-4yx^5.$} \endinsert We have drawn for convenience in Figure $7$ only one red, white, or blue cycle. We have also indicated the position of the arc $\alpha,$ which will play a role in the next section. \goodbreak \section{5}{An example of global geometric monodromy} Let $b:\Bbb C \to \Bbb C^2, b(t):=(t^6+t^7,t^4)$ be the parametrized curve $C$ having at $b(0)=(0,0)$ the singularity with two essential Puiseux pairs and with local link the compound cable knot $(2,3)(2,3).$ The polynomial $f(x,y):=y^4-2y^2x^3+x^6-x^7-4yx^5$ is the equation of $C.$ The function $f:\Bbb C^2 \to \Bbb C$ has besides $0$ the only other critical value $c:=14130940973168155968/558545864083284007.$ The fiber of $0$ has besides its singularity at $(0,0)$ a nodal singularity at $(-8,-4)$, which corresponds to the node $b(-1+i)=b(-1-i)=(-8,-4).$ The geometric monodromy of the singularity at $(0,0),$ which is up to isotopy piecewise of finite order, is described in [AC1]. The fiber of $c$ has a nodal singularity at $(1014/343,16807/79092).$ The singularity at infinity of the curve $C$ is at the point $(0:1:0)$ and its local equation is $z^3-2z^2x^4+zx^6-x^7-4zx^5$, whose singularity is topologically equivalent to the singularity $u^3-v^7$ with Milnor number $12.$ The function $f$ has no critical values coming from infinity. We aim at a description of the global geometric monodromy of the function $f.$ Working with the distance on $\Bbb C^2$ given by $||(x,y)||^2:= |x|^2+ 4|y|^2,$ we have that the parametrized curve $b$ is transversal to the spheres $S_r:=\{(x,y) \in \Bbb C^2 \mid |x|^2+ 4|y|^2=r^2\}$ with center $0 \in \Bbb C^2$ and radius $r > 0.$ So for $0<r< 8\sqrt{2},$ the intersection $K_r:=C \cap S_r$ is the local knot in $S_r$ of the singularity at $0 \in \Bbb C^2$ (see Figure $8$), at $r=8\sqrt{2}$ the knot $K_r$ is singular with one transversal crossing, and for $8\sqrt{2} < r$ the knot $K_r$ is the so called knot at infinity of the curve $C.$ The crossing at the bottom of Fig. $8$ flips for $r=8\sqrt{2}$ and the knot $K_r, 8\sqrt{2} < r,$ becomes the $(4,7)$ torus knot. By making one extra total twist in a braid presentation of the knot $K_r$ one gets the local knot of the singularity at infinity of the projective completion of the curve $C.$ \midinsert \cline{\epsffile{fig-8.eps}} \medskip \centerline{Figure $8$: The torus cable knot $(2,3)(2,3).$} \endinsert From the above we get the following partial description of the global geometric monodromy. The typical regular fiber $F:=f^{-1}(c/2)$ is the interior of the oriented surface obtained as the union of two pieces $A$ and $B,$ where $A$ is a surface of genus $8$ with one boundary component and $B$ is a cylinder. The pieces are glued together in the following way: in each boundary component of $B$ there is an arc, which is glued to an arc in the boundary of $A.$ The interior of $A$ or $B$ can be thought of as a Milnor fiber of the singularity at $0$ or $(-8,-4).$ So, the geometric monodromy around $0$ is a diffeomorphism with support in the interior of $A$ and $B,$ given for instance by a construction as in Paragraph $2.$ The piece $A$ can be constructed from the divide in Figure $9.$ \midinsert \leftline{\epsffile{fig-9.eps}} \medskip \centerline{Figure $9$: The curve $(x_s(t),y_s(t)),\, s:=1,$ as divide for the singularity of $C.$} \endinsert Clearly, the monodromy in $B$ is a positive Dehn twist around the simple essential closed curve $\delta_{17}$ in $B,$ whereas the monodromy in $A$ is a product of positive Dehn twists around a system $(\delta_1, \dots ,\delta_{16})$ of $16$ red, white or blue curves. The monodromy around the critical point $c$ is a positive Dehn twist around a simple curve $\delta_{18},$ in $F,$ which is the union of two simple arcs $\alpha \subset A$ and $\beta \subset B.$ The arcs $\alpha$ and $\beta$ have their endpoints $p,q \in A \cap B$ in common, and moreover the points $p$ and $q$ lie in different components of $A \cap B.$ The arc $\beta$ cuts the curve $\delta_{17}$ transversally in one point. The arc $\alpha$ intersects the curves $(\delta_1, \dots ,\delta_{16})$ transversally in some way. For the position of the system $(\delta_{17},\, \beta)$ in B there is up to a diffeomorphism of the pair $(B,A \cap B)$ only one possibility. To obtain a complete description of the global monodromy it remains to describe the position of the system $(\delta_1, \dots ,\delta_{16},\, \alpha)$ in $(A,A \cap B).$ We consider the family with parameter $s$ of parametrized curves with para\-meter $t$: $$x_s(t)=T(4,t)/8={t}^{4}-{t}^{2}+1/8,\,\, y_s(t)=sT(6,t)/32+T(7,t)/64=$$ $$s{t}^{6}+{t}^{7}-3/2\,s{t}^{4}-7/4\,{t}^{5}+{{9}/{16}}\,s{t}^{2}+ {{7}/{8}}\,{t}^{3}-1/32\,s-{{7}/{64}}\,t,$$ where $T(d,t)$ is the Chebychev polynomial of degree $d.$ Let $$ f_s(x,y):={s}^{4}{x}^{6}-{{3}/{128}}\,{s}^{4}{x}^{4}+{{1}/{1024}}\,{s} ^{4}{x}^{3}-2\,{s}^{2}{y}^{2}{x}^{3}-4\,sy{x}^{5}- $$ $${x}^{7}+{{9}/{65536}}\,{s}^{4}{x}^{2}-{{3}/{262144}}\,{s}^{4}x+{{3}/{128}} \,{s}^{2}{y}^{2}x-{{1}/{4096}}\,{s}^{2}{x}^{3}+ $$ $$ {{5}/{64}}\,s y{x}^{3}+{{7}/{256}}\,{x}^{5}+{{1}/{4194304}}\,{s}^{4}-{{1}/{1024}}\,{s}^{2}{y}^{2}-{{1}/{1024}}\,sy{x}^{2}+ $$ $$ {y}^{4}+ {{3}/{1048576}}\,{s}^{2}x- {{5}/{16384}}\,syx-{{7}/{ 32768}}\,{x}^{3}-{{1}/{8388608}}\,{s}^{2}+ $$ $$ {{1}/{131072}}\,sy-{{1}/{4096}}\,{y}^{2}+{{7}/{16777216}}\,x+{{1}/{134217728}} $$ be the equation, monic in $y,$ for the curve $(x_s(t),y_s(t),$ whose real image is for $s=1$ a divide (see Figure $9$) for the singularity of $C$ at $(0,0).$ The $0-$level of $f_s$ for $s=1$ consists of this divide and an isolated minimum not in one of its regions, which corresponds to the minimum of the restriction of $f$ to $\Bbb R^2$ at $(-8,-4).$ For $a$ small, we call $\delta_{17,a,s} \subset \{f_s=a\}$ the vanishing cycle of the local minimum of $f_s,s=1,$ which does not belong to a region. The curve $(x_s(t),y_s(t)), t \in \Bbb R^2,s=7/24\,\sqrt{2},$ has $8$ nodes, a cusp at $t=-1/2\,\sqrt{2},$(see Figure $10$) and at infinity a singularity with Milnor number 12. We now vary the parameter $s \in [7/24\,\sqrt{2}-\sigma,1]$ from $1$ to $7/24\,\sqrt{2}-\sigma$ for a very small $\sigma >0.$ \midinsert \cline{\epsffile{fig-10.eps}} \medskip \centerline{Figure $10$: The curves $(x_s(t),y_s(t))$ for $ s:=7/24\,\sqrt{2}.$} \endinsert The value of the local minimum, which does not belong to a region, becomes smaller and by adjusting the parameter $a$ we can keep the cycle $\delta_{17,a,s}$ in the new region which emerges from the cusp at $s=7/24\,\sqrt{2}.$ Since the total Milnor number of $f_s$ is $(\hbox{\fam0 \tenrm degree}(f_s)-1)(\hbox{\fam0 \tenrm degree}(f_s)-2)-12=18,$ it follows that all its singularities have Milnor number $1$ for $s=7/24\,\sqrt{2}-\sigma.$ The vanishing cycle of the node, which appears when deforming the cusp singularity, will be called $\delta_{18,s}$ and the vanishing cycle in the region of the divide of Figure $9$, in whose boundary the cusp appeared, will be called $\delta_{16,s}.$ \midinsert \cline{\epsffile{fig-11.eps}} \medskip \centerline{Figure $11$: A divide, which does not come from a singularity.} \endinsert We label the vanishing cycles on the regular ribbon surface $F_{-}$ of the divide of Figure $11$ by $\delta_{1}, ... ,\delta_{15}.$ The cycles $\delta_{17,a,s},\delta_{18,s},\delta_{16,s}$ deform without changing their intersection pattern and $\delta_{16,s}$ becomes the cycle $\delta_{16}$ of the regular ribbon surface $F_{+}$ of the divide of Figure $9.$ Observe that the regular ribbon surface $F_{-}$ of the divide of Figure $11$ is naturally a subset of the regular ribbon surface $F_{+}$ of the divide of Figure $9.$ The description of the position of the system $(\delta_1, \dots ,\delta_{16},\, \alpha)$ in $(A,A \cap B),$ for which we are looking, is the system $(\delta_1, \dots ,\delta_{16})$ on $F_{+},$ where the relative cycle $\alpha$ is a simple arc on $F_{+}-F_{-}$ with endpoints on the boundary of $F_{+}$ and cutting the cycle $\delta_{16}$ transversally in one point. Observe that $F_{+}-F_{-}$ is a strip with core $\delta_{16}$ (see Figure $7$). \goodbreak \section{6}{Connected divides and fibered knots. Proof of Theorem 2} In this section we assume, without loss of generality, that a divide is linear and orthogonal near its crossing points. For a connected divide $P \subset D(0,\rho),$ let $f_P:D(0,\rho) \to \Bbb R$ be a generic $C^{\infty}$ function, such that $P$ is its $0$-level and that each region has exactly one local maximum or minimum. Such a function exist for a connected divide and is well defined up to sign and isotopy. In particular, there are no critical points of saddle type other then the crossing points of the divide. We assume moreover that the function $f_P$ is quadratic and euclidean in a neighborhood of its critical points, i.e. for euclidean coordinates (X,Y) with center at a critical point $c$ of $f_P$ we have in a neighborhood of $c$ the expression $f_P(X,Y)=f_P(c)+XY$ or $f_P(X,Y)=f_P(c)+X^2+Y^2.$ Let $\chi:D(0,\rho) \to [0,1]$ be a $C^{\infty},$ positive function, which evaluates to zero outside of the neighborhoods where $f_P$ is quadratic and to $1$ in some smaller neighborhood of the critical points of $f_P.$ Let $\theta_P:\partial B(0,\rho) \to \Bbb C$ be given by: $\theta_{P,\eta}(J(v)):=f_P(x)+ i \eta df_P(x)(u)-{1\over 2}\eta^2\chi(x)H_{f_P}(v)$ for $J(v)=(x,u) \in TD(0,\rho)=D \times \Bbb R^2$ and $\eta \in \Bbb R, \eta > 0.$ Observe that the Hessian $H_{f_P}$ is locally constant in a the neighborhood of the critical points of $f_P.$ The function $\theta_{P,\eta}$ is $C^{\infty}$. Let $\pi_{P,\eta}:\partial B(0,\rho) \setminus L(P) \to S^1$ be defined by:$\pi_{P,\eta}(J(v)):=\theta_{P,\eta}(J(v))/ |\theta_{P,\eta}(J(v))|.$ \proclaim{Theorem 4} Let $P \subset D(0,\rho)$ be a divide, such that the system of immersed curves is connected. The link $L(P)$ in $\partial B(0,\rho)=\{J(v) \mid v \in T(D(0,\rho)) \ \text{ and }\ \|J(v)\| =\rho\}$ is a fibered link. The map $\pi_P:=\pi_{P,\eta}$ is for $\eta$ sufficiently small, a fibration of the complement of $L(P)$ over $S^1.$ Moreover the fiber of the fibration $\pi_P$ is $F(P)$ and the geometric monodromy is the product of Dehn twist as in Theorem 3. \endgroup\bigbreak The map $\pi_P$ is compatible with a regular product tubular neighborhood of $L(P)$ in $\partial B(0,\rho).$ The map $\pi_P$ is a submersion, so, since already a fibration near $L(P),$ it is a fibration by a theorem of Ehresmann. The graphical algorithm, see Figure $7$, produces in fact, up to a small isotopy of the image, the projection of the fiber $\pi_P^{-1}(i)$ on $D(0,\rho).$ This projection is except above the twist of the strips a submersion. The proof of Theorem $4$ is given in the forthcoming paper [AC4] on generic immersions of curves and knots. \demo{ \fam\bffam\tenbf Proof of Theorem $2$} The oriented fibered links $L(S)$ and $L(P)$ have the same geometric monodromies according to the Theorem 3 and 4. So, the links $L(S)$ and $L(P)$ are diffeomorphic. \penalty-100\null\hfill\qed\bigbreak \remark{\fam\bffam\tenbf Remark} Let $f(x,y)=0$ be a singularity $S,$ such that written in the canonical coordinates of the charts of the embedded resolution the branches of the strict transform have equations of the form $u=a, a \in \Bbb R.$ Let $f_t(x,y), t \in [0,1]$ be a morsification, with its divide $P$ in $D(0,\rho),$ obtained by blowing down generic real linear translates of the strict transforms, as in [AC2]. We strongly believe that with the use of [B-C1,B-C2] the following transversallity property can be obtained, and which we state as a problem:\br Their exists $\rho'_0>0,$ such that for all $t \in [0,1]$ and for all $\rho' \in (0,\rho'_0]$ the $0$-levels of $f_t(x,y)$ in $\Bbb C^2$ meet transversally the boundary of $$ B(0,\rho,\rho'):=\{(x+iu,y+iv) \in \Bbb C^2 \mid x^2+y^2+u^2+v^2 \leq \rho^2, u^2+v^2 \leq {\rho'}^2\}. $$ It is easy to deduce from this transversallity statement an isotopy between the links $L(S)$ and $L(P).$ \endremark \remark{\fam\bffam\tenbf Remark} Bernard Perron has given a proof for the triviallity of the cobordism from $L(S)$ to $L(P),$ which uses the holomorphic convexity of the balls $B(0,\rho,\rho')$ of the previous remark [P]. \endremark \par \noindent \Refs \parskip=0pt \par \key{AC1} Norbert A'Campo, \it Sur la monodromie des singularit\'es isol\'ees d'hypersurfaces complexes, \rm Invent. Math. \bf 20 \rm (147--170), 1973. \par \key{AC2} Norbert A'Campo, \it Le Groupe de Monodromie du D\'eploiement des Singularit\'es Isol\'ees de Courbes Planes I, \rm Math. Ann. \bf 213 \rm (1--32), 1975. \par \key{AC3} Norbert A'Campo, \it Le Groupe de Monodromie du D\'eploiement des Singularit\'es Isol\'ees de Courbes Planes II, \rm Actes du Congr\`es International des Math\'e\-ma\-ti\-ciens, Vancouver \rm (395--404), 1974. \par \key{AC4} Norbert A'Campo, \it Generic immersions of curves, knots, monodromy and \"Uber\-schneidungszahl, \rm Publ. Math. IHES, to appear \br http://xxx.lanl.gov/abs/math/9803081. \par \key{A} V. Arnold, \it On the arrangement of the ovals of real plane curves, involutions of 4-dimensional smooth manifolds, and the arithmetic of integral quadratic forms, \rm Funct. Anal. Appl. \bf 5 \rm (1--9), 1971. \par \key{B-K} Ludwig Balke and Rainer Kaenders, \it On certain type of Coxeter-Dynkin diagrams of plane curve singularities, \rm Topology \bf 35 \rm (39--54), 1995. \par \key{B-C1} F. Bruhat, H. Cartan, \it Sur la structure des sous-ensembles analytiques r\'eels, \rm C. R. Acad. Sci. Paris \bf 244 \rm (988--990), 1957. \par \key{B-C2} F. Bruhat, H. Cartan, \it Sur les composantes irr\'eductibles d'un sous-ensemble analytique r\'eel, \rm C. R. Acad. Sci. Paris \bf 244 \rm (1123--1126), 1957. \par \key{G-Z} S. M. Gusein-Zade, \it Matrices d'intersections pour certaines singularit\'es de fonctions de 2 variables, \rm Funkcional. Anal. i Prilozen \bf 8 \rm (11--15), 1974. \par \key{L} S. Lefschetz, \it L'Analysis Situs et la G\'eom\'etrie Alg\'ebrique, \rm Collection de Monographies sur la Th\'eorie des Fonctions, Gauthier- Villars et $C^{ie}$, \rm Paris, 1924. \par \key{M} J. Milnor, \it Singular Points on Complex Hypersurfaces, \rm Ann. of Math. Studies \bf 61 \fam0 \tenrm Princeton University Press, \rm Princeton, 1968. \par \key{P} B. Perron, \it Preuve d'un Th\'eor\`eme de N. A'Campo sur les d\'eformations r\'eelles des singularit\'es alg\'ebriques complexes planes, \rm Preprint, Universit\'e de Bourgogne, \rm Dijon, 1998. \par \key{P-S} \'Emile Picard et Georges Simart, \it Th\'eorie des Fonctions Alg\'ebriques de deux variables ind\'ependantes, Tome I, Gauthier- Villars et Fils, \rm Paris, 1897. \par \key{R1} V. A. Rohlin, \it Congruence modulo 16 in Hilbert's sixteenth problem I, \rm Funct. Anal. Appl. \bf 6 \rm (301--306), 1972. \par \key{R2} V. A. Rohlin, \it Congruence modulo 16 in Hilbert's sixteenth problem II, \rm Funct. Anal. Appl. \bf 7 \rm (163--164), 1973. \par \key{S1} Charlotte Angas Scott, \it On the Higher Singularities of Plane Curves, \rm Amer. J. Math. \bf 14 \rm (301--325), 1892. \par \key{S2} Charlotte Angas Scott, \it The Nature and Effect of Singularities of Plane Algebraic Curves, \rm Amer. J. Math. \bf 15 \rm (221--243) 1893. \par \key{V} Oleg Viro, \it Private communication, \rm \bf \rm \rm Sapporo, 1990. \par\endgroup \bigskip \address{Universit\"at Basel \br Rheinsprung 21 \br CH-4051 Basel} \null\firstpagetrue\vskip\bigskipamount \title{Erratum: \br Real deformations and complex topology \br of plane curve singularities} \shorttitle{Real deformations and complex topology.} \vskip2\bigskipamount In Section $5$ the parametrized curve $C$ should be $b(t):=(t^4,t^6+t^7)$ instead of $b(t):=(t^6+t^7,t^4)$ and accordingly $(-8,-4)$ has to be $(-4,-8)$. We intersect $C$ with the family of spheres $S_r:=\{(x,y) \in \Bbb C^2 \mid 4|x|^2+ |y|^2=r^2\}$. For $0<r< 8\sqrt{2},$ the intersection $K_r:=C \cap S_r$ is the local knot in $S_r$ of the singularity at $0 \in \Bbb C^2$, at $r=8\sqrt{2}$ the knot $K_r$ is singular with one transversal crossing at $(-8,-4)$, and for $8\sqrt{2} < r$ the knot $K_r$ is the so called knot at infinity of the curve $C.$ Fig. $8$ of the text is a knot projection of $K_r$ for small $r$. It is not possible to obtain from this projection with only one crossing flip the type of the knot $K_r$ for $r > 8\sqrt{2}$. The figure here below is the stereographic knot projection of $K_r$ for $r = 8\sqrt{2}-1$, which is not a minimal knot projection. For $r=8\sqrt{2}$ the crossing at the bottom flips and the knot $K_r, 8\sqrt{2} < r,$ becomes the $(4,7)$ torus knot. The knot projection is a braid projection, where the axis is in the central pentagonal region. The braid word is $acabcaAabacabacabacab$ and flips at $r=8\sqrt{2}$ to $acabcaaabacabacabacab$. \midinsert \cline{\epsffile{vor4_7.ps}} \medskip \endinsert \vskip 6.6cm This picture was made with KNOTSCAPE. \bye

9710

alg-geom/9710025

en

https://arxiv.org/abs/alg-geom/9710025

alg-geom/9710025

Stephan Endrass

Stephan Endrass

Minimal even sets of nodes

LaTeX 2e, 17 pages

We extend some results on even sets of nodes which have been proved for surfaces up to degree 6 to surfaces up to degree 10. In particular, we give a formula for the minimal cardinality of a nonempty even set of nodes.

alg-geom

\section{Setup}\label{sect:setup} Let $S\subset\Pthree\left(\CC\right)$ be a hypersurface of degree $s$ with $\mu$ ordinary double points (nodes) as its only singularities. Such a surface will be called a {\em nodal surface} in the sequel. Denote by $N=\left\{P_1,\ldots,P_\mu\right\}\subset S$ the set of nodes of $S$. The maximum number of nodes of a nodal surface of degree $d$ is denoted classically by $\mu\left(d\right)$. There is a lot of (old) literature on nodal surfaces and estimates for $\mu\left(d\right)$ (see \cite{endrass}). For $d=1,2,\ldots,6$ the numbers $\mu\left(d\right)$ are $0,1,4,16,31,65$ and for every $k\in\left\{0,1,\ldots,\mu\left(d\right)\right\}$ there exists at least one nodal surface of degree $d$ with exactly $k$ nodes. In the case of cubic nodal surfaces ($d=3$), this follows from Cayley's and Schl\"afli's classification of singular cubic surfaces \cite{cayley}, \cite{schlaefli}. For quartic nodal surfaces ($d=4$) the fact that $\mu\left(4\right)=16$ is due to Kummer \cite{kummer}, whereas the construction of arbitrary nodal quartics goes back to Rohn \cite{rohn}. The first quintic nodal surface ($d=5$) with 31 nodes has been constructed by Togliatti in 1940 \cite{togliatti}. In 1971, Beauville \cite{beauville} showed that this is in fact the maximal number. The construction of sextic nodal surfaces ($d=6$) with $1,\ldots,64$ nodes has been given by Catanese and Ceresa \cite{cataneseceresa}. In 1994, Barth \cite{barth} constructed a sextic nodal surface with 65 nodes. Shortly afterwards, Jaffe and Ruberman \cite{jafferuberman} proved that 65 is the maximal number. Both Beauville and Jaffe/Ruberman use the code of a nodal surface in their proofs. This code is a $\mathbb{F}_2$ vector space which carries the information of the low degree contact surfaces of the nodal surface. If a nodal surface has ``nearly'' $\mu\left(d\right)$ nodes, its code often becomes accessible. Let $v\in\mathbb{N}$ and denote $\delta\left(v\right)=2\left(v/2-\left\lfloor v/2\right\rfloor\right)$. This number is 0 if v is even and 1 if v is odd. We want to study surfaces $V\subset\mathbb{P}_3$ of degree $v$ with $S.V=2D$ for a (not necessarily smooth or reduced) curve $D$. In other words, surfaces $V$ which have contact to $S$ along a curve. Let $\pi\colon\tilde{\mathbb{P}}_3\rightarrow\mathbb{P}_3$ be the embedded resolution of all nodes of $S$. Given such a surface $V$, the proper transforms of $S$ and $V$ are calculated as \begin{equation*} % \tilde{S}=\pi^\ast\negthinspace S-2\sum_{i=1}^\mu E_i \quad\text{and}\quad \tilde{V}=\pi^\ast\negthinspace V-\sum_{i=1}^\mu \nu_i E_i, % \end{equation*} where $E_i=\pi^{-1}\left(P_i\right)$ is the exceptional divisor corresponding to $P_i$ and $\nu_i=\operatorname{mult}\left(V,P_i\right)$ for every node $P_i\in N$. On the smooth surface $\tilde{S}$ we have $\tilde{V}\sim_{lin}2\tilde{D}+\sum_{i=1}^\mu\theta_iE_i$, where $\tilde{D}$ is the proper transform of $D$ and the $\theta_i$'s are nonnegative integers. Let $H\in\operatorname{Div}\left(\mathbb{P}_3\right)$ be a hyperplane section, then \begin{equation*} % 2\tilde{D}\sim_{lin}v\pi^\ast\negthinspace H- \sum_{i=1}^\mu\left(\nu_i+\theta_i\right)E_i, % \end{equation*} where $\tilde{D} .E_i=\nu_i+\theta_i=\operatorname{mult}\left(D,P_i\right)=\eta_i$. This shows that in $\operatorname{Pic}\left(\smash{\tilde{S}}\right)$ the divisor class $\left[\delta\left(v\right)\pi^\ast\negthinspace H+\sum_{\text{$\eta_i$ odd}}E_i\right]$ is divisible by $2$. This is a remarkable fact, since every $E_i$ is on $\tilde{S}$ a smooth, rational curve with self intersection $-2$. In particular $E_i\not\sim_{lin}E_j$ for $i\neq j$. For any set of nodes $M\subseteq N$, let $E_M=\sum_{P_i\in M}E_i$ be the sum of exceptional curves corresponding to the nodes in $M$. \begin{definition}\label{definition:even} % A set $M\subseteq N$ of nodes of $M$ is called {\em strictly even}, if the cocycle class $\operatorname{cl}\left[E_M\right]\in\cohom{2}{\smash{\tilde{S}},\mathbb{Z}}$ is divisible by $2$. $M$ is called {\em weakly even}, if the cocycle class $\operatorname{cl}\left[\pi^\ast\negthinspace H+E_M\right]\in\cohom{2}{\smash{\tilde{S}},\mathbb{Z}}$ is divisible by $2$. $M$ is called {\em even} if $M$ is strictly or weakly even. % \end{definition} So the set of nodes $M=\left\{P_i\in N\mid \text{$\operatorname{mult}\left(D,P_i\right)$ is odd}\right\}$ through which $D$ passes with odd multiplicity is strictly even if $v$ is even and weakly even if $v$ is odd. \begin{definition}\label{definition:cut} % Let $M\subseteq N$ be an even set of nodes of $S$. If $V\subset\mathbb{P}_3$ is a surface with $S.V=2D$ and $M$ is the set of nodes of $S$ through which $D$ passes with odd multiplicity, we say that {\em $M$ is cut out by $V$ via $D$}. % \end{definition} Conversely, if $M\subseteq N$ is even, consider the linear system $\mylinsys{v}{-}{M}$ for $v\in\mathbb{N}$ even if $M$ is strictly even and odd if $M$ is weakly even. For $v\gg 0$ this linear system is nonempty by R.R.~and Serre duality. Then for every $v$ such that $\mylinsys{v}{-}{M}\neq\emptyset$ and for every divisor $\overline{D}\in\mylinsys{v}{-}{M}$ we can find a surface $V\subset\mathbb{P}_3$ of degree $v$ which cuts out $M$. The construction is as follows: $\overline{D}$ is effective, so it admits a decomposition $\overline{D}=\tilde{D}+\sum_{i=1}^\mu \tau_iE_i$ such that $\tilde{D}$ is effective and contains no exceptional component and all the numbers $\tau_i$ are nonnegative. In particular we have for all $j\in\left\{1,\ldots,\mu\right\}$ that \begin{equation*} % \tilde{D} .E_j=\left(\frac{1}{2}\left(v\pi^\ast\negthinspace H-E_M\right) -\sum_{i=1}^\mu\tau_i E_i\right).E_j= \left\{\begin{array}{l@{\quad}l} 2\tau_j & \text{is even if $P_j\not\in M$,}\\ 2\tau_j +1 & \text{is odd if $P_j\in M$.} \end{array}\right. % \end{equation*} But $2\overline{D}\in\left|v\pi^\ast\negthinspace H-E_M\right|$ on the surface $\tilde{S}$, so $2\overline{D}$ is cut out by a surface $\overline{V}\in\left|v\pi^\ast\negthinspace H-E_M\right|$ in ${\tilde{\mathbb{P}}}_3$. Let $V=\pi_\ast\left(\overline{V}\right)$ and $D=\pi_\ast\left(\overline{D}\right)=\pi_\ast\left(\smash{\tilde{D}}\right)$, then by construction $S.V=2D$ and $\operatorname{mult}\left(D,P_i\right)=\tilde{D} .E_i$ for all $i$. So $M$ is exactly the set of nodes of $S$ through which $D$ passes with odd multiplicity. This shows that $M$ is cut out by $V$ via $D$. Furthermore we see that only nodal surfaces of even degree do admit weakly even sets of nodes. If the surface $V$ cuts out an even set of nodes $M$ on $S$ via $D$, then in general $D$ is not unique with respect to $M$. The set of these contact curves is parameterized by the linear system $L_M=\mylinsys{v}{-}{M}$ which is a projective space of dimension $\mycohomd{0}{v}{-}{M}-1$. In particular, if $\mycohomd{0}{v}{-}{M}\geq 1$ then there exists a surface of degree $v$ which cuts out $M$. It is funny to compute these dimensions, though often not possible. The canonical divisor of $\tilde{S}$ is $\smash{K_{\tilde{S}}}\sim_{lin}\left(s-4\right)\pi^\ast\negthinspace H$. Define $\binom{n}{k}=0$ for $n<k$, then Riemann Roch for the bundle $\mybundle{v}{-}{w}$ reads as \begin{equation*} % \mychi{v}{-}{w}=\frac{sv}{8}\left(v-2s+8\right) +\binom{s-1}{3}+1-\frac{\left|w\right|}{4}. % \end{equation*} The symmetric difference of two strictly even sets of nodes is strictly even again, so the set $C_S=\left\{M\subseteq N\mid\text{$M$ is strictly even}\right\}$ carries the natural structure of a $\mathbb{F}_2$ vector space sitting inside $\mathbb{F}_2^\mu$. Hence $C_S$ is a binary linear code, which is called {\em the code of $S$}. The symmetric difference of two weakly even sets of nodes is strictly even and the symmetric difference of a strictly even set and a weakly even set is weakly even. Thus the set $\overline{C}_S=\left\{M\subseteq N\mid\text{$M$ is even}\right\}$ is a binary code of dimension $\dim_{\mathbb{F}_2}\left(C_S\right)\leq \dim_{\mathbb{F}_2}\left(\overline{C}_S\right)\leq \dim_{\mathbb{F}_2}\left(C_S\right)+1$ sitting also inside $\mathbb{F}_2^\mu$. The elements of $\overline{C}_S$ are called {\em words}, and for every word $w\in \overline{C}_S$ its weight $\left|w\right|$ is its number of nodes. Let $e_1,\ldots,e_\mu,h$ be the canonical basis of $\mathbb{F}_2^\mu\oplus\mathbb{F}_2$ and consider \begin{equation*} % \begin{array}{ccccc} % \mathbb{F}_2^\mu & \overset{j}{\longrightarrow} & \mathbb{F}_2^\mu\oplus\mathbb{F}_2 & \overset{\lambda}{\longrightarrow} & \cohom{2}{\smash{\tilde{S}},\mathbb{F}_2} \\ & & e_i & \longmapsto & \operatorname{cl}\left[E_i\right] \bmod 2 \\ & & h & \longmapsto & \operatorname{cl}\left[\pi^\ast\negthinspace H\right]\bmod 2 % \end{array} % \end{equation*} The projection of $\ker\left(\lambda\right)$ onto the first factor is nothing but $\overline{C}_S$, and $\ker\left(\lambda\circ j\right) = C_S$. If $s$ is even (resp.~odd), then $\operatorname{im}\left(\lambda\right)$ (resp.~$\operatorname{im}\left(\lambda\circ j\right)$) is a total isotropic subspace of $\cohom{2}{\smash{\tilde{S}},\mathbb{F}_2}$ with respect to the intersection product. This shows \cite{beauville} that \begin{align*} % \dim_{\mathbb{F}_2}\left(C_S\right) & \geq\mu -\frac{1}{2}b_2\left(\smash{\tilde{S}}\right),\\ \dim_{\mathbb{F}_2}\left(\overline{C}_S\right) & \geq\mu +1 -\frac{1}{2}b_2\left(\smash{\tilde{S}}\right) \quad\text{($s$ even)}. % \end{align*} The weight of every word $w\in C_S$ is divisible by 4. If $s=\deg S$ is even, then the weight of every word is divisible by 8 \cite{catanese}. \subsection{Coding theory} We recall some definitions and facts from coding theory \cite{lint}, \cite{wall}. Let $C\subseteq\mathbb{F}_2^n$ be a linear code and let $e_1,\ldots,e_n$ be the canonical basis of $\mathbb{F}_2^n$. $C$ is called {\em even} if $2\mid\left|w\right|$ for every $w\in C$ and {\em doubly even} if $4\mid\left|w\right|$ for every $w\in C$. The dual code of $C$ is defined as \begin{equation*} % C^\perp = \left\{v\in\mathbb{F}_2^n\mid\left< v,w\right>_{\mathbb{F}_2}=0\ \forall w\in C \right\}. % \end{equation*} If $C$ is doubly even, then $C\subseteq C^\perp$. Since $n=\dim_{\mathbb{F}_2}\left(C\right)+\dim_{\mathbb{F}_2}\left(C^\perp\right)$ we also get $2\dim\left(C\right)\leq n$ with equality iff $C$ is self dual. For $w\in C$ the support of $w$ is the linear subspace of $\mathbb{F}_2^n$ which is spanned by the ones of $w$, i.e.~ \begin{equation*} % \operatorname{supp}\left(w\right) = \operatorname{span}_{\mathbb{F}_2}\left\{e_i\mid \left<e_i,w\right>=1\right\}. % \end{equation*} The image of the projection $p_w\colon C\rightarrow\operatorname{supp}\left(w\right)$ is called projection of $C$ onto the support of $w$ and denoted by $C_w$. Assume that $2d\mid\left|v\right|$ for all $v\in C$ for some $d\in\mathbb{N}$. Since $\left|v+w\right|+2\left|v\cap w\right| = \left|v\right|+\left|w\right|$ and $p_w\left(v\right)=v\cap w$ we see that $d\mid v'$ for all $v'\in C_w$. Now the code $C_S$ of the nodal surface $S$ is always doubly even. If $s=\deg\left(S\right)$ is even, then $\left(C_S\right)_w$ is doubly even for all $w\in C_S$. A $\left[n,k,d\right]$-code is a $k$-dimensional linear code $C\subseteq \mathbb{F}_2^n$ with $\left|w\right|\geq d$ for all $w\in C\setminus\left\{0\right\}$. Many methods have been found to give bounds on $k$ for fixed $n$ and $d$. One of the simplest to apply is the \begin{theorem}\label{theorem:griesmer} % (Griesmer bound) For a $\left[n,k,d\right]$ code always $n\geq \sum_{i=0}^{k-1} \left\lceil d/2^i\right\rceil$. % \end{theorem} \subsection{Examples} The following examples exhibit the trivial and some of the the well known cases of even sets of nodes \cite{beauville}. \begin{example}\label{example:cone} % Let $S$ be a quadratic cone and let $P_1$ be its node. Every line $L\subset S$ runs through $P_1$ and there exists exactly one plane $H$ with $S.H=2L$. So $\overline{C}_S$ is spanned by $w=\left\{P_1\right\}$ and $\mycohomd{0}{}{-}{w}=2$. % \end{example} \begin{example}\label{example:cubic} % Let $S$ be a cubic nodal surface, then $C_S$ can only be non trivial if $S$ has exactly $\mu\left(3\right)=4$ nodes $P_1,\ldots,P_4$. But $b_2\left(\smash{\tilde{S}}\right)=7$, so $\dim_{\mathbb{F}_2}\left(C_S\right)\geq 1$. It follows that $\dim_{\mathbb{F}_2}\left(C_S\right)=1$ and $C_S$ is spanned by $w=\left\{P_1,\ldots,P_4\right\}$. But $w$ is cut out by a quadric: Riemann-Roch on $\tilde{S}$ gives $\mychi{2}{-}{w}=3$. From Serre duality we get $\mycohomd{2}{2}{-}{w}=\mycohomd{0}{-4}{+}{w}=0$. One easily checks that $\mybundle{4}{-}{w}$ is ample, so by Kodaira vanishing also $\mycohomd{1}{2}{-}{w}=\mycohomd{1}{-4}{+}{w}=0$. This implies that $\mycohomd{0}{2}{-}{w}=3$, so there exists a two parameter family of quadric surfaces which cut out $w$. % \end{example} \begin{example}\label{example:quartic} % A quartic nodal surface $S$ with $\mu\left(4\right)=16$ nodes is a Kummer surface. Since $b_2\left(\smash{\tilde{S}}\right)=22$, we have $\dim_{\mathbb{F}_2}\left(C_S\right)\geq 5$. On the other hand all nonzero words of $C_S$ must have weight $8$ or $16$. So $C_S$ is a $\left[16,k,8\right]$ code for some $k\geq 5$. The Griesmer bound implies $k\leq 5$, so $C_S$ is a $\left[16,5,8\right]$ code. Every such code has exactly one word of weight 16 and 30 words of weight 8. Moreover $C_S$ is (up to permutation of columns) spanned by the rows of the following table. % \begin{equation*} % \newcommand{\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}}{\smash{\hspace*{0.06cm}\blacksquare\hspace*{0.06cm}}} \newcommand{\\\hline}{\\\hline} % \begin{array}{*{16}{|@{}c@{}}|}\hline % \hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & & & & & \\\hline \hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & \\\hline \hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & \\\hline \hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& \\\hline \hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}\\\hline % \end{array} % \end{equation*} % \end{example} \begin{example}\label{example:quintic} % A quintic nodal surface $S$ with $\mu\left(5\right)=31$ nodes is called Togliatti surface. One computes $b_2\left(\smash{\tilde{S}}\right)=53$, so again $\dim_{\mathbb{F}_2}\left(C_S\right)\geq 5$. By \cite{beauville}, all even sets of nodes on $S$ have weight $16$ or $20$. So $C_S$ is a $\left[31,k,16\right]$ code for some $k\geq 5$. The Griesmer bound gives $31\geq 16+8+4+2+1+\left(k-5\right)$, so $k\leq 5$. This shows that $C_S$ is a $\left[31,5,16\right]$ code. Every such code has exactly 31 words of weight 16 and no word of weight 20. Moreover, $C_S$ is (up to a permutation of columns) spanned by the rows of the following table. % \begin{equation*} % \newcommand{\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}}{\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}} % \begin{array}{*{31}{|@{}c@{}}|}\hline % \hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & & & & & & & & & & & & \\\hline \hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & & & & & & \hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & & & & \\\hline \hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & & \hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & \\\hline \hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & \hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& \\\hline \hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & \hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}\\\hline % \end{array} % \end{equation*} % \end{example} \begin{example}\label{example:sextic} % Let $S$ be a nodal sextic surface with $\mu\left(6\right)=65$ nodes. Every nonzero word $w\in C_S$ must have weight $24$, $32$, $40$ or $56$ \cite{jafferuberman}. We have $b_2\left(\smash{\tilde{S}}\right)=106$, so $\dim_{\mathbb{F}_2}\left(C_S\right)\geq 12$. If $C_S$ contains no word of weight $56$, then $\dim_{\mathbb{F}_2}\left(C_S\right)=12$ \cite{jafferuberman}, \cite{wall}. A short argument runs as follows: By the Griesmer bound $C_S$ contains a word $w$ of weight $24$. Clearly $p_w\colon C_S\rightarrow\left(C_S\right)_w$ has trivial kernel, so $\left(C_S\right)_w$ is a doubly even $\left[24,\dim_{\mathbb{F}_2}\left(C_S\right),4\right]$ code. Hence $\dim_{\mathbb{F}_2}\left(C_S\right)\leq 12$. It is not clear if $C_S$ is unique up to permutation. It is also not known if any nodal sextic surface can have even sets of $56$ or $64$ nodes. % \end{example} \subsection{The theorem} For nodal surfaces of degree $6$, Jaffe and Ruberman proved that the smallest possible nonzero strictly even sets of nodes are the ones cut out by quadrics. This seems to be true for nodal surfaces of arbitrary degree, though we only can prove a few cases. For weakly even sets of nodes, the corresponding statement is proved. \begin{definition}\label{definition:eminmax} % For $s\in\mathbb{N}$ the \emph{minimal cardinality of an even set of nodes on a nodal surface of degree $s$} is defined as % \begin{align*} % e_{min}\left(s\right) &= \min\left\{ \left|w\right|\mid\text{$w\in C_S$, $S$ nodal of degree $s$} \right\}, \\ \overline{e}_{min}\left(s\right) &= \min\left\{ \left|w\right|\mid\text{$w\in \overline{C}_S$, $S$ nodal of degree $s$} \right\}. % \end{align*} % \end{definition} Our main result is the following \begin{theorem}\label{theorem:main} % \begin{itemize} % \item[\romannum{1}] (Strictly even sets of nodes) Let $s\in\left\{3,4,5,6,7,8,10\right\}$. Then % \begin{equation*} % e_{min}\left(s\right) =\left\{\begin{array}{cl} s\left(s-2\right) & \text{if $s$ is even,}\\ \left(s-1\right)^2 & \text{if $s$ is odd.} \end{array}\right. % \end{equation*} % Moreover $\left|w\right|=e_{min}\left(s\right)$ if and only if $w$ is cut out by a quadric surface. % \item[\romannum{2}] (Weakly even sets of nodes) Let $s\in\left\{2,4,6,8\right\}$. Then % \begin{equation*} % \overline{e}_{min}\left(s\right)=\frac{s\left(s-1\right)}{2}. % \end{equation*} % Moreover $\left|w\right|=\overline{e}_{min}\left(s\right)$ if and only if $w$ is cut out by a plane. % \end{itemize} % \end{theorem} A close examination of the proof of theorem \ref{theorem:main} exhibits that certain weights strictly greater than $e_{min}\left(s\right)$ and $\overline{e}_{min}\left(s\right)$ cannot appear. \begin{corollary}\label{corollary:main} % For any nodal surface $S$ of degree $s$, there exist no even sets of nodes with the following weights. % \begin{equation*} % \begin{array}{|c||c|c|c|c|}\hline s & 6 & 7 & 8 & 10 \\\hline\hline \text{weakly even} & 19,23 & & 32,36,\ldots,56 & \\\hline \text{strictly even} & & 40 & 56 & 88,96,104,112 \\\hline \end{array} % \end{equation*} % \end{corollary} If $w\in\overline{C}_S$ is cut out by a smooth cubic surface, then $\left|w\right|=3s\left(s-3\right)/2$ \cite{catanese}. The corollary states that all weights in the open interval $\left] \overline{e}_{min}\left(s\right),3s\left(s-3\right)/2\right[$ do not appear for weakly even set of nodes. In the case of strictly even sets of nodes, the gap is the interval $\left] e_{min}\left(s\right),2s\left(s-4\right)\right[$. Note that if $w\in C_S$ is cut out by a smooth quartic surface, then $\left|w\right|=2s\left(s-4\right)$. \begin{remark} % It follows from example \ref{example:cone} and example \ref{example:cubic} that the theorem is true for $s=2,3$. % \end{remark} \subsubsection*{Acknowledgments} % I would like to thank D.~van Straten for valuable discussions. % \section{The formula of Gallarati} The contact of hypersurfaces in $\mathbb{P}_r$ along a $r-2$ dimensional variety has been (to our knowledge) studied first by D.~Gallarati \cite{gallarati}. He stated the following \begin{theorem}\label{theorem:gallarati} % Let $F_m$, $G_n\subset\mathbb{P}_r$ be hypersurfaces of degree $m$ and $n$ with $F_m.G_n=qC$ for some $r-2$ dimensional variety $C$. Assume that $F_m$ and $G_n$ have at most double points on $C$. If the singular locus of $F_m$ on $C$ (resp.~$G_n$ on $C$) is a $r-3$ dimensional variety of degree $t$ (resp.~$s$), then % \begin{equation*} % q\left(t-s\right) = mn\left(m-n\right). % \end{equation*} % \end{theorem} If one allows the surfaces $F_m$ and $G_n$ to have points of higher multiplicity on $C$, then simple examples show that this number is dependent on the local geometry. But the philosophy of Gallarati's theorem is that in the situation of contact of hypersurfaces the hypersurface of higher degree must have more or harder singularities on the contact variety than the hypersurface of lower degree. We will prove a variant of the above theorem which gives a lower bound for the size of an even set of nodes. If $S\subset\mathbb{P}_3$ is a nodal surface recall that for every even set of nodes $w$ on $S$ there exists a surface $V\subset\mathbb{P}_3$ such that $S.V=2D$ and $w$ is just the set of nodes of $S$ through which $D$ passes with odd multiplicity. We estimate the number of nodes through which $D$ passes with multiplicity one. For a slightly more general setup, let $M$ be a smooth projective threefold and let $S\subset M$ be a nodal surface. Assume that a surface $V\subset M$ intersects $S$ as $S.V=rD+D'$, $r\geq 2$, for an irreducible curve $D$ which is not contained in the support of $D'$. \begin{definition}\label{definition:smooth} % A node $P$ of $S$ is called $D$-smooth if $P\in D$ and $P$ is a smooth point of $V$. % \end{definition} This definition is justified by the following \begin{lemma}\label{lemma:smooth} % Let $P$ be a node of $S$. If $P$ is $D$-smooth, then $P$ is a smooth point of $D$. Moreover $r=2$ and $P\not\in\operatorname{supp}\left(D'\right)$. % \end{lemma} {\noindent\bf Proof:\ } There exists a neighborhood $U$ of $P$ in $M$ which is biholomorphic to some open neighborhood of the origin $\mathbf{0}\in\mathbb{C}^3$, so it suffices to prove the lemma for two affine hypersurfaces $S$, $V\subset\mathbb{C}^3$. We study the intersection with a general plane through $\mathbf{0}$. Let $L\cong\mathbb{P}_2$ be the set of all planes $H\subset\mathbb{C}^3$ through $\mathbf{0}$ and let $T=T_{\mathbf{0}}V\in L$ be the tangent plane to $V$ in $\mathbf{0}$. Then for all $H\in L\setminus\left\{T\right\}$, the curve $C_H=V.H$ is smooth in $\mathbf{0}$. The set of all planes $H\in L$ which have contact to the tangent cone $C_{\mathbf{0}}S$ of $S$ in $\mathbf{0}$ is parameterized by a smooth conic $Q\subset L$. For all $H\in L\setminus Q$, the curve $F_H=S.H$ has an ordinary double point in $\mathbf{0}$. While varying $H$ in $L\setminus\left(Q\cup\left\{T\right\}\right)$, the tangent lines $T_{\mathbf{0}}C_H$ sweep out $T_{\mathbf{0}}V$, while the tangent lines to both branches of $F_H$ in $\mathbf{0}$ sweep out $C_{\mathbf{0}}S$. So there exists a plane $\tilde{H}\in L\setminus\left(Q\cup\left\{T\right\}\right)$ such that $T_{\mathbf{0}}C_{\tilde{H}}$ is not contained in $C_{\mathbf{0}}S$. Therefore $C_{\tilde{H}}$ and $F_{\tilde{H}}$ meet transversal in $\mathbf{0}$, hence on $\tilde{H}$ we have local intersection multiplicity $\left(F_{\tilde{H}}.C_{\tilde{H}}\right)_{\mathbf{0}}=2$. Then of course \begin{align*} % 2 &= \left(F_{\tilde{H}}.C_{\tilde{H}}\right)_{\mathbf{0}} = \left(\left.S\right|_{\tilde{H}}.\left.V\right|_{\tilde{H}}\right)_{\mathbf{0}} = \left(S.V.\smash{\tilde{H}}\right)_{\mathbf{0}} \\ &= \left(\left(rD+D'\right).H\right)_{\mathbf{0}} = r\left(D.\smash{\tilde{H}}\right)_{\mathbf{0}}+ \left(D'.\smash{\tilde{H}}\right)_{\mathbf{0}}. % \end{align*} Now $\mathbf{0}\in D$ implies $\left(D.\smash{\tilde{H}}\right)_{\mathbf{0}}\geq 1$. Since $r\geq 2$ we get $\left(D.\smash{\tilde{H}}\right)_{\mathbf{0}}=1$, $r=2$ and $\left(D'.\smash{\tilde{H}}\right)_{\mathbf{0}}=0$. This proves the lemma.$\square$\medskip\par Now we give the lower bound for the number of $D$-smooth nodes of $S$. \begin{proposition}\label{proposition:irreducible} % Assume that $D\not\subseteq\operatorname{sing}\left(V\right)$ and let $\beta$ be the number of singular points of $V$ on $D$ which are smooth points of $S$. Then $S$ has at least $D.\left(S-V\right)+\beta$ nodes which are $D$-smooth. % \end{proposition} {\noindent\bf Proof:\ } To prove the theorem we would like to have everything smooth. There exists a sequence of blowups (embedded resolution of the singular locus of $S$, $V$ and $D$) \begin{equation*} % \tilde{M} =M_n\overset{\pi_n }{\longrightarrow} M_{n-1} \overset{\pi_{n-1}}{\longrightarrow} \ldots \overset{\pi_2 }{\longrightarrow} M_1 \overset{\pi_1 }{\longrightarrow} M_0=M. % \end{equation*} Let $S_i$, $V_i$ and $D_i$ denote the proper transforms of $S$, $V$ and $D$ with respect to $\pi_i\circ\pi_{i-1}\circ\ldots\circ\pi_1$. We can define divisors $D_i'$ by $S_i.V_i=rD_i+D_i'$ with $D_i\not\subset\operatorname{supp}\left(D_i'\right)$, $1\leq i\leq n$. Moreover we can arrange the maps $\pi_i$ in such a way that the following conditions hold. \begin{itemize} % \item[\romannum{1}] $\pi_1\colon M_1\rightarrow M$ is the blowup of $M$ in all points which are singular for both $S$ and $V$. \item[\romannum{2}] $\pi=\pi_{n-1}\circ\ldots\circ\pi_2\colon M_{n-1}\rightarrow M_1$ is the embedded resolution of the singular locus of $V_1$, i.e.~every map $\pi_{i+1}$ is a blowup of $M_i$ centered in a smooth variety $Z_i\subset M_i$ such that $Z_i$ is either a point or a smooth curve, $1\leq i\leq n-2$. \item[\romannum{3}] $\pi_n\colon\tilde{M}\rightarrow M_{n-1}$ is the embedded resolution of the singularities of $S_{n-1}$ and $D_{n-1}$. % \end{itemize} Now one has to keep track of the intersection numbers $D_i.\left(S_i-V_i\right)$ as $i$ increases. We study each of the three maps separately. {\bf\romannum{1}} Let $Z=\left\{P_1,\ldots,P_\alpha\right\}= \operatorname{sing}\left(S\right)\cap\operatorname{sing}\left(V\right)$, then $M_1=\operatorname{Blow}_Z M$. Denote by $E_j=\pi_0^{-1}\left(P_j\right)$ the exceptional divisor corresponding to $P_j$. The proper transforms are calculated as $S_1=\pi_1^\ast S-2\sum_{j=1}^\alpha E_j$ and $V_1=\pi_1^\ast V-\sum_{j=1}^\alpha m_jE_j$, where $m_j=\operatorname{mult}\left(V,P_j\right)\geq 2$, $1\leq j\leq\alpha$. Then the intersection number can be estimated as \begin{equation*} % D_1.\left(S_1-V_1\right) = D_1.\pi_1^\ast\left(S-V\right) +\sum_{j=1}^\alpha\left(m_j-2\right)D_1.E_j \geq D.\left(S-V\right). % \end{equation*} This is just the information we need, so let us consider the second case. {\bf\romannum{2}} Every blowup $\pi_{i+1}$ gives rise to an exceptional divisor $F_{i+1}=\pi_{i+1}^{-1}\left(Z_i\right)$. In the $\left(i+1\right)$-st step always $S_i$ is smooth in all points of $S_i\cap Z_i$, whereas $V_i$ is singular in all points of $Z_i$. So the proper transforms are \begin{equation*} % S_{i+1}=\pi_{i+1}^\ast S_i-n_iF_{i+1} \quad\text{and}\quad V_{i+1}=\pi_{i+1}^\ast V_i-p_iF_{i+1} % \end{equation*} where $n_i=\operatorname{mult}\left(S_i,Z_i\right)\in\left\{0,1\right\}$ and $p_i=\operatorname{mult}\left(S_i,Z_i\right)\geq 2$. So this time the intersection number in question is just \begin{align*} % D_{i+1}.\left(S_{i+1}\right. &-\left.V_{i+1}\right) = D_{i+1}.\pi_{i+1}^\ast\left(S_i-V_i\right)+ \left(p_i-n_i\right)D_{i+1}.F_{i+1} \\ &= D_i.\left(S_i-V_i\right)+ \left\{\begin{array}{l@{\quad}l} \left(p_i-1\right)\operatorname{mult}\left(D_i,Z_i\right) & \text{if $Z_i$ is a point,}\\ p_i\sum_{P\in Z_i\cap D_i}\operatorname{mult}\left(D_i,P\right) & \text{if $Z_i$ is a curve,} \end{array}\right. \\ &\geq D_i.\left(S_i-V_i\right)+ \#\left(Z_i\cap D_i\right). % \end{align*} But every singularity of $V$ on $D$ outside the singular locus of $S$ counts at least once. So by induction \begin{equation*} % D_{n-1}.\left(S_{n-1}-V_{n-1}\right)\geq D_1.\left(S_1-V_1\right)+\beta\geq D.\left(S-V\right)+\beta. % \end{equation*} where $\beta=\#\left(\left(\operatorname{sing}\left(V\right)\cap D\right)\setminus \operatorname{sing}\left(S\right)\right)$. {\bf\romannum{3}} As for the third case we note that $V_{n-1}$ is smooth and $S_{n-1}$ is nodal with $S_{n-1}.V_{n-1}=rD_{n-1}+D_{n-1}'$. Either $D_{n-1}\cap\operatorname{sing}\left(S_{n-1}\right)=\emptyset$ or $D_{n-1}$ contains at least one node of $S$. But then $r=2$ by lemma \ref{lemma:smooth} and $D_{n-1}$ is smooth in all nodes of $S_{n-1}$. In both cases, $S_{n-1}$ and $D_{n-1}$ do not have common singularities. Let $P_{\alpha+1},\ldots,P_{\alpha+\eta}$ be the nodes of $S_{n-1}$ on $D_{n-1}$ and let $P_{\alpha+\eta+1},\ldots,P_{\alpha+\eta+\tau}$ be the remaining nodes of $S_{n-1}$. Moreover let $E_j=\pi_n^{-1}\left(P_j\right)$, $\alpha+1\leq j\leq \alpha+\eta+\tau$. But the embedded resolution of the singularities of $D_{n-1}$ on $V_{n-1}$ is the same as on $S_{n-1}$, so the proper transforms are \begin{align*} % \tilde{S} = S_n &= \pi_n^\ast S_{n-1}-E_D-2\sum_{j=\alpha+1}^{\alpha+\eta+\tau} E_j, \\ \tilde{V} = V_n &= \pi_n^\ast V_{n-1}-E_D-\sum_{j=\alpha+1}^{\alpha+\eta}E_j- \sum_{k=\alpha+\eta+1}^{\alpha+\eta+\tau} q_kE_k, % \end{align*} where $q_k=\operatorname{mult}\left(V_k,P_k\right)\in\left\{0,1\right\}$ and $E_D$ is a sum of exceptional divisors corresponding to the singularities of $D_{n-1}$. Set $\tilde{D}=D_n$ and calculate \begin{align*} % \tilde{D}.\left(\smash{\tilde{S}-\tilde{V}}\right) &= \tilde{D}.\pi_n^\ast\left(S_{n-1}-V_{n-1}\right) - \sum_{j=\alpha+1}^{\alpha+\eta}\tilde{D}.E_j \\ &= D_{n-1}.\left(S_{n-1}-V_{n-1}\right)-\eta \\ &\geq D.\left(S-V\right)-\eta+\beta. % \end{align*} On the other hand the smooth surfaces $\tilde{S}$ and $\tilde{V}$ have contact of order $r-1\geq 1$ along the smooth curve $\tilde{D}$. So the tangent bundles $T_{\tilde{S}}$ and $T_{\tilde{V}}$ agree along $\tilde{D}$. This implies that the normal bundles $N_{\tilde{D}\mid\tilde{S}}$ and $N_{\tilde{D}\mid\tilde{V}}$ coincide, thus \begin{equation*} % \left(\smash{{\tilde{D}}^2}\right)_{\tilde{S}}= \deg\left(\smash{N_{\tilde{D}\mid\tilde{S}}}\right)= \deg\left(\smash{N_{\tilde{D}\mid\tilde{V}}}\right)= \left(\smash{{\tilde{D}}^2}\right)_{\tilde{V}}. % \end{equation*} Now by adjunction formula $\tilde{D}.K_{\tilde{V}}=\tilde{D}.K_{\tilde{S}}$. Using the adjunction formula again we see that \begin{equation*} % 0 = \tilde{D}.K_{\tilde{S}}-\tilde{D}.K_{\tilde{V}} = \tilde{D}.\left.\left(\smash{K_{\tilde{M}}+\tilde{S}}\right)\right|_{\tilde{S}}- \tilde{D}.\left.\left(\smash{K_{\tilde{M}}+\tilde{V}}\right)\right|_{\tilde{V}} = \tilde{D}.\left(\smash{\tilde{S}-\tilde{V}}\right). % \end{equation*} This gives the desired formula $\eta\geq D.\left(S-V\right)+\beta$. If $V$ is also a nodal surface one can see easily that we have equality.$\square$\medskip\par The application to surfaces in $M=\mathbb{P}_3$ gives the following \begin{corollary}\label{corollary:irreducible} % Let a nodal surface $S\subset\mathbb{P}_3$ of degree $s$ and an irreducible surface $V\subset\mathbb{P}_3$ of degree $v$ intersect as $S.V=2D$ for curve $D$ on $S$. Assume that $V$ is not singular along a curve contained in $S$ and let $\beta$ be the number of singular points of $V$ which are smooth for $S$. If $s>v$, then $D$ is reduced. Moreover $V$ cuts out an even set of at least $sv\left(s-v\right)/2+\beta$ nodes on $S$ with equality if $V$ is also nodal. % \end{corollary} {\noindent\bf Proof:\ } Just run proposition \ref{proposition:irreducible} on every irreducible component of $D$.$\square$\medskip\par It is possible to extend proposition \ref{proposition:irreducible} to the case when the surface $V$ is not irreducible, but reduced. The proof however works different. \begin{proposition}\label{proposition:reduced} % Let $S\subset\mathbb{P}_3$ be a nodal surface, $n\in\mathbb{N}$ and let $V_1,\ldots,V_n\subset\mathbb{P}_3$ be different irreducible surfaces of degrees $v_1,\ldots,v_n$ satisfying the following conditions: % \begin{itemize} % \item[\romannum{1}] $V_i$ is not singular along a curve contained in $S$, $1\leq i\leq n$, \item[\romannum{2}] $v_i=\deg\left(V_i\right)<s$, $1\leq i\leq n$ and \item[\romannum{3}] $S.\left(V_1+\ldots +V_n\right)=2D$ for a (not necessarily reduced) divisor $D$ on $S$. % \end{itemize} % Then the reduced surface $V=V_1+\ldots +V_n$ of degree $v=v_1+\ldots +v_n$ cuts out an even set of nodes $w\in\overline{C}_S$ of weight $\left|w\right|\geq sv\left(s-v\right)\negthinspace /2$. % \end{proposition} {\noindent\bf Proof:\ } Since $v_i<s$ and $V_i$ is not singular along a curve contained in $S$ there exist reduced divisors $D_i$ and $R_i$ on $S$ which do not have a common component such that $S.V_i=2D_i+R_i$, $1\leq i\leq n$. But \begin{equation*} % S.V=S.\left(V_1+\ldots +V_n\right)= 2\left(D_1+\ldots +D_n\right)+R_1+\ldots +R_n. % \end{equation*} This implies that $R_i\subset\bigcup_{j\neq i}V_j$ and thus $R_i$ has a decomposition $R_i=\sum_{j\neq i}R_{i,j}$ such that $R_{i,j}\subset V_i\cap V_j$. Now we count the nodes of $S$ through which $D$ passes with multiplicity 1. Denote $d_i=\deg\left(D_i\right)$, $r_i=\deg\left(R_i\right)$ and $r_{i,j}=\deg\left(R_{i,j}\right)$. By corollary \ref{corollary:irreducible}, $V_i$ contains at least $d_i\left(s-v_i\right)$ nodes of $S$ through which $D_i$ passes with multiplicity 1. All these nodes lie outside $R_i$. We cannot simply add these numbers: some nodes might be counted more than once. But every node $P\in w$ on $V_i$ which is counted more than once is contained also in some $V_j$ for a $j\neq i$, hence in $F_{i,j}=V_i.V_j$. Let $f_{i,j}=\deg\left(F_{i,j}\right)$. Let $C$ be an irreducible component of $F_{i,j}$ and let $c=\deg\left(C\right)$. We have the following possibilities: \begin{itemize} % \item $C\not\subset S$. In this case $C$ contains at most $cs/2$ nodes of $S$. \item $C\subset S$ is a component of $R_i$. Here $C$ does not contain any node that we counted. \item $C\subset S$ is a component of $D_i$ and $D_j$. Here $V_i$ and $V_j$ have contact to $S$ along $C$ and thus $C$ appears in $F_{i,j}$ with multiplicity $\geq 2$. Clearly $C$ contains at most $c\left(s-1\right)$ nodes of $S$. \item $C\subset S$ is a component of $D_i$, but not of $D_j$. Then $V_i$ and $V_j$ meet transversal along $C$, so $S.\left(V_i+V_j\right)=3C+other\ curves$. So there exist a $k\not\in\left\{i,j\right\}$ such that $C\in F_{i,k}$. So $C$ appears with multiplicity $\geq 2$ in $\sum_{j\neq i}F_{i,j}$. Again $C$ contains at most $c\left(s-1\right)$ nodes of $S$. % \end{itemize} Since every component of $R_{i,j}$ is contained in $F_{i,j}$, this shows that $F_{i,j}$ contains at most $\left(f_{i,j}-r_{i,j}\right) s/2$ nodes that we counted. So $V_i$ contains at least \begin{equation*} % d_i\left(s-v_i\right)-\sum_{j\neq i}\left(f_{i,j}-r_{i,j}\right) % \end{equation*} nodes through which $D$ passes with multiplicity 1. This implies \begin{align*} % \left|w\right| &\geq \sum_{i=1}^n d_i\left(s-v_i\right) -\frac{s}{2}\sum_{j\neq i}\left(f_{i,j}-r_{i,j}\right)\\ &=\frac{1}{2}\left(\sum_{i=1}^n\left(sv_i-r_i\right)\left(s-v_i\right) +sr_i -s\sum_{j\neq i}v_iv_j\right)\\ &=\frac{1}{2}\left(s^2\left(v_1+\ldots v_n\right) -s\left(v_1^2+\ldots +v_n^2+2\sum_{i<j}v_iv_j\right) +r_1v_1+\ldots +r_nv_n\right)\\ &\geq \frac{sv}{2}\left(s-v\right) % \end{align*} This completes the proof.$\square$\medskip\par \section{Contact surfaces and quadratic systems} In this section we apply the previous results to our initial situation. So let again $S\subset\mathbb{P}_3$ be a nodal surface of degree $s$ and $V\subset\mathbb{P}_3$ a reduced surface of degree $v$ such that $S.V=2D$ for some curve $D$. We give a complete analysis of the situation when $V$ is a plane or a quadric. Using the notation of the first paragraph, $V$ cuts out an even set of nodes $w\in\overline{C}_S$. Recall that the linear system $L_w=\mylinsys{v}{-}{w}$ parameterizes all contact curves of the form $D'=(1/2)S.V'$ where $V'$ is a surface of degree $v$ which cuts out $w$. In some cases, $V$ will be the unique surface of degree $v$ which cuts out $w$. Now $2D\in\mathbb{P}\left(\cohom{0}{\obundletS{vH}}\right)$ is the restriction of $V\in\mathbb{P}\left(\cohom{0}{\obundle{\mathbb{P}_3}{vH}}\right)$ to $S$. Consider the exact sequence \begin{equation*} % 0\longrightarrow \obundle{\mathbb{P}_3}{\left(v-s\right)H}\longrightarrow \obundle{\mathbb{P}_3}{vH}\longrightarrow \obundle{S}{vH}\longrightarrow 0. % \end{equation*} Since $\cohom{i}{\obundle{\mathbb{P}_3}{\left(v-s\right)H}}=0$ for $s>v$, $i=0,1$, the induced map $\cohom{0}{\obundle{\mathbb{P}_3}{vH}}\rightarrow \cohom{0}{\obundle{S}{vH}}$ is an isomorphism. So if $v<s$, then $V$ is the unique surface of degree $v$ cutting out $w$ via $D$ and $L_w=\mylinsys{v}{-}{w}$ in fact parameterizes the space of all surfaces of degree $v$ which cut out $w$. This space is not a linear system, but the quadratic system \begin{equation*} % Q_w=\left\{V'\mid S.V'=2D'\text{\ with\ }D'\in L_w\right\}. % \end{equation*} It is constructed as follows: If $\mycohomd{0}{v}{-}{w}=n+1\geq 2$ we can find $n+1$ linearly independent sections $s_0,\ldots,s_n\in\mycohom{0}{v}{-}{w}$. Clearly all products $s_is_j\in\cohom{0}{\obundletS{v\pi^\ast\negthinspace H-E_w}}$ for $0\leq i\leq j\leq n$. So there exist sections $g_{i,j}\in\cohom{0}{\obundle{{\tilde{\mathbb{P}}}_3}{v\pi^\ast\negthinspace H-E_w}}$ over ${\tilde{\mathbb{P}}}_3$ which restrict to $s_is_j$ under the identification $\obundle{{\tilde{\mathbb{P}}}_3}{v\pi^\ast\negthinspace H-E_w}\otimes \obundles{\tilde{S}}\cong \obundletS{v\pi^\ast\negthinspace H-E_w}$. Outside the exceptional locus we can view the $g_{i,j}$ as sections of $\obundle{\mathbb{P}_3}{vH}$. Since $w$ has codimension $\geq 2$ in $\mathbb{P}_3$, these sections extend also to $w$. This implies that \begin{align*} % Q_w &= Q\left(g_{i,j}\mid 0\leq i\leq j\leq n\right) \\ &= \left\{\sum_{i=0}^n\lambda_i^2g_{i,i}+ 2\sum_{0\leq i<j\leq n}\lambda_i\lambda_j g_{i,j}=0\mid \left(\lambda_0:\ldots:\lambda_n\right)\in\mathbb{P}_n\right\}. % \end{align*} Therefore the quadratic system $Q_w$ is the image of an embedding of Veronese type of $\mathbb{P}_n$ into the space $\mathbb{P}_{\binom{v+3}{3}}$ parameterizing all surfaces of degree $v$. In general, $Q_w$ will not contain any linear subspace. The quadratic system $Q_w$ admits a decomposition $Q_w=B_w+F_w$ where $B_w$ is a reduced surface of degree $b\leq v$ and the base locus of $F_w$ (if any) consists only of curves and points. If $F_w$ has no basepoints then $B_w$ cuts out $w$ and so $\mycohomd{0}{b}{-}{w}=1$. \begin{definition}\label{definition:stable} % An even set of nodes $w\in\overline{C}_S$ is called % \begin{equation*} % \left.\begin{array}{c} \text{semi stable}\\\text{stable}\\\text{unstable} \end{array}\right\} \quad\text{in degree $v$ if}\quad \left\{\begin{array}{c} \text{$F_w$ is basepointfree,}\\ \text{$F_w=\emptyset$,}\\ \text{$F_w$ has basepoints.} \end{array}\right. % \end{equation*} % \end{definition} The base locus of $Q_w$ is $B\left(Q_w\right)=\left\{g_{i,j}=0\mid 0\leq i\leq j\leq n\right\}$. It is contained in the discriminant locus $Z\left(Q_w\right)=\left\{g_{i,i}g_{j,j}=g_{i,j}^2 \mid 0\leq i<j\leq n\right\}$. There is also a Bertini type theorem for quadratic systems. \begin{lemma}\label{lemma:bertini} % (Bertini for quadratic systems) The general element of $Q_w$ is smooth outside $Z\left(Q_w\right)$. % \end{lemma} {\noindent\bf Proof:\ } The proof runs like the proof of the Bertini theorem in \cite{griffithsharris}.$\square$\medskip\par Next we give a different characterization of stability. \begin{proposition}\label{proposition:stable} % Let $w\in\overline{C}_S$. The surface $B_w$ is always reduced and % \begin{itemize} \item[\romannum{1}] $w$ is stable in degree $v$ if and only if $\mycohomd{0}{v}{-}{w}=1$. \item[\romannum{2}] $w$ is semi stable in degree $v$ if and only if $F_w$ contains a square. Then $B_w$ cuts out $w$ and either every surface in $F_w$ is a square or the general surface in $F_w$ is reduced. \item[\romannum{3}] $w$ is unstable in degree $v$ if and only if $F_w$ contains no square. Then $B_w$ does not cut out $w$ and the general surface in $Q_w$ is reduced. % \end{itemize} % \end{proposition} {\noindent\bf Proof:\ } \romannum{1} follows from the definition. So let $w$ be not stable in degree $v$. We use induction on $n$. $n=1$: By construction $\gcd\left(g_{0,0},g_{0,1},g_{1,1}\right)=g$ is reduced. Let $\overline{g}_{i,j}=g_{i,j}/g$ and let $\overline{Q}_w=Q\left(\overline{g}_{0,0},\overline{g}_{0,1},\overline{g}_{1,1}\right)$. Now we have two cases. a) If $\overline{g}_{0,0}\overline{g}_{1,1}=\overline{g}_{0,1}^2$, then $\overline{g}_{0,0}$ and $\overline{g}_{1,1}$ must be squares. So $\overline{g}_{0,0}=a_0^2$, $\overline{g}_{1,1}=a_1^2$ and thus $\overline{g}_{0,1}=a_0a_1$. Hence $B_{w}=\left\{g=0\right\}$ cuts out $w$. So by construction the quadratic system $F_w=\left\{\smash{\left(\lambda_0a_0+\lambda_1a_1\right)^2} \mid\left(\lambda_0:\lambda_1\right)\in\mathbb{P}_1\right\}$ contains only squares. Then $F_w$ is free and $w$ is semi stable in degree $v$. b) $Z\left(\overline{Q}_w\right)= \left\{\overline{g}_{0,0}\overline{g}_{1,1}=\overline{g}_{0,1}^2\right\}$ is a surface. If all surfaces of $Q_w$ are not reduced, then by lemma \ref{lemma:bertini} all surfaces of $\overline{Q}_w$ contain a component of $Z\left(Q_w\right)$. So this component is constant for all surfaces in $\overline{Q}_w$, which contradicts $\gcd\left(\overline{g}_{0,0},\overline{g}_{0,1},\overline{g}_{1,1}\right)=1$. So the general surface in $Q_w$ is reduced. Now assume $B_w$ cuts out $w$. Then by construction $F_w$ contains squares, so $F_w$ is free and $w$ is semi stable in degree $v$. Otherwise $B_w$ does not cut out $w$, so $F_w$ must have basepoints in $w$. Then $w$ is unstable in degree $v$. $n-1\Rightarrow n$: Again $\gcd\left(g_{i,j}\mid 0\leq i\leq j\leq n\right)=g$ is reduced. Consider the quad\-ra\-tic system $Q=Q\left(g_{i,j}\mid 0\leq i\leq j\leq n-1\right)$. Either $\gcd\left(g_{i,j}\mid 0\leq i\leq j\leq n-1\right)$ is reduced and we're done or it's not reduced. For $\lambda=\left(\lambda_0:\ldots:\lambda_{n-1}\right)\in\mathbb{P}_{n-1}$ let \begin{equation*} % g_\lambda =\sum_{i=0}^{n-1}\lambda_i^2g_{i,i} +2\sum_{0\leq i<j\leq n-1}\lambda_i\lambda_j g_{i,j} \quad\text{and}\quad h_\lambda=\sum_{i=0}^{n-1}\lambda_ig_{i,n}. % \end{equation*} Now consider the quadratic system \begin{equation*} % R_\lambda =\left\{ t^2 g_\lambda+2t\lambda_nh\lambda+\lambda_n^2g_{n,n}=0 \mid\left(t:\lambda_n\right)\in\mathbb{P}_1\right\}. % \end{equation*} While varying $\lambda\in\mathbb{P}_{n-1}$, $\gcd\left(g_\lambda,h_\lambda,g_{n,n}\right)$ is constant on an open dense subset, since it contains only factors of $g_{n,n}$. So for general $\lambda$ $\gcd\left(g_\lambda,h_\lambda,g_{n,n}\right)= \gcd\left(g_{i,j}\mid 0\leq i\leq j\leq n\right)=g$ is reduced. By the first part either $g_\lambda g_{n,n}=h_\lambda^2$ for all $\lambda$, so $g_\lambda$ and $g_{n,n}$ are always squares modulo $g$. Then $w$ is semi stable in degree $v$. Or the general surface in $R_\lambda$ and hence in $Q_w$ is reduced. Again either $B_w$ cuts out $w$ and we're in the semi stable case or $B_w$ does not cut out $w$. Then $w$ is unstable in degree $v$.$\square$\medskip\par \begin{corollary}\label{lemma:unique} % If $2v<s$ then $w$ is semi stable in degree $v$. % \end{corollary} {\noindent\bf Proof:\ } On $S$ we have $g_{0,0}g_{1,1}-g_{0,1}^2=s_0^2s_1^2-\left(s_0s_1\right)^2=0$. Let $S=\left\{f=0\right\}$, then either $g_{0,0}g_{1,1}=g_{0,1}^2$ or $f\mid g_{0,0}g_{1,1}-g_{0,1}^2$. But the second case implies $2v\geq s$, so we are in the first case. Then we always run into case a) in the proof of proposition \ref{proposition:stable}.$\square$\medskip\par \begin{corollary} % If $w$ is semi stable in degree $v$ and $2\deg\left(F_w\right)<s$, then $F_w$ contains only squares. % \end{corollary} {\noindent\bf Proof:\ } $F_w$ contains a square $W_0=\left\{g_{0,0}=g_0^2=0\right\}$. Now take any other $W_1=\left\{g_{1,1}=0\right\}\in F_w$ and consider the quadratic system generated by $g_{0,0}$ and $g_{1,1}$: $g_{0,0}$, $g_{1,1}$ give rise to sections $s_0^2$, $s_1^2$ over $\tilde{S}$. Then $s_0s_1$ is the restriction of a section $g_{0,1}$ to $\tilde{S}$. The quadratic system in question is just $Q=Q\left(g_{0,0},g_{0,1},g_{1,1}\right)$. But $g_{0,0}g_{1,1}-g_{0,1}^2$ vanishes on $S$. Since $\deg\left(g_{0,0}g_{1,1}-g_{0,1}^2\right)=2\deg\left(F_w\right)<s$, we have $g_{0,0}g_{1,1}=g_0^2g_{1,1}=g_{0,1}^2$. This implies that also $g_{1,1}$ is a square.$\square$\medskip\par \begin{proposition}\label{proposition:unstable} % If $w$ is unstable in degree $v$, then there exists a surface $W$ of degree $2v-s$ such that $w$ is cut out by a reduced surface $V$ of degree $v$ satisfying: % \begin{itemize} % \item[\romannum{1}] $V$ is not singular on $S$ outside $W$. \item[\romannum{2}] If $V$ is singular along a curve $C\subset S$, then $C$ is a curve of triple points of $W$. % \end{itemize} % \end{proposition} {\noindent\bf Proof:\ } $w$ is cut out by a reduced surface, so we can assume that $g_{0,0}$ is square free. Again $g_{0,0}g_{1,1}-g_{0,1}^2$ vanishes on $S$ and $g_{0,0}$, $g_{1,1}$ are linearly independent. So there exists a polynomial $\alpha$ of degree $2v-s$ such that $\alpha f=g_{0,0}g_{1,1}-g_{0,1}^2$. Let $W=\left\{\alpha =0\right\}$ and let $V_\lambda=\left\{ \lambda_0^2 g_{0,0}+2\lambda_0\lambda_1 g_{0,1}+\lambda_1^2 g_{1,1}=0 \mid\lambda=\left(\lambda_0:\lambda_1\right)\in\mathbb{P}_1\right\}$. For every point $P\in\mathbb{P}_3$ we can choose affine coordinates $\left(z_1,z_2,z_3\right)$ on an affine neighborhood $U$ of $P$. For any function $h$ on $U$, we identify the total derivative $Dh$ with the gradient $\nabla h$ and $D^2h$ with the Hesse matrix $H\left(h\right)$. We find that \begin{align*} % D\left(\alpha f\right) &= \alpha\nabla f+f\nabla\alpha = g_{0,0}\nabla g_{1,1}+g_{1,1}\nabla g_{0,0}- 2g_{0,1}\nabla g_{0,1},\\ D^2\left(\alpha f\right) &= \alpha H\left(f\right)+f H\left(\alpha\right)+ \nabla\alpha{\nabla f}^t+\nabla f{\nabla\alpha}^t\\ &= g_{0,0}H\left(g_{1,1}\right)+ g_{1,1}H\left(g_{0,0}\right)- 2g_{0,1}H\left(g_{0,1}\right)\\ &\hspace*{4ex}+ \nabla g_{0,0}{\nabla g_{1,1}}^t+ \nabla g_{1,1}{\nabla g_{0,0}}^t- 2\nabla g_{0,1}{\nabla g_{0,1}}^t. % \end{align*} Now let $P\in S$. We have to consider two different cases: a) $P\in\operatorname{sing}\left(S\right)\setminus W$, so $f\left(P\right)=0$, $\nabla f\left(P\right)=0$ and $\operatorname{rk}\left(H\left(f\right)\left(P\right)\right)=3$. If $P$ is a basepoint of $Q$ then \begin{align*} % H\left(\alpha f\right)\left(P\right) &= \alpha\left(P\right)H\left(f\right)\left(P\right) \\ &= \left(\nabla g_{0,0}{\nabla g_{1,1}}^t+ \nabla g_{1,1}{\nabla g_{0,0}}^t- 2\nabla g_{0,1}{\nabla g_{0,1}}^t\right)\left(P\right). % \end{align*} But $P\not\in W$, so $\alpha\left(P\right))\neq 0$ and $\operatorname{rk}\left(H\left(\alpha f\right)\left(P\right)\right)=3$. This is only possible if $\nabla g_{0,0}$, $\nabla g_{1,1}$ and $\nabla g_{0,1}$ are linearly independent in $P$. So every surface $V_\lambda$ is smooth in $P$. If $P$ is not a basepoint of $Q$ then the general surface $V_\lambda$ will not contain $P$. b) Let $P\in\operatorname{smooth}\left(S\right)\setminus W$. Here $f\left(P\right)=0$, $\nabla f\left(P\right)\neq 0$ and $\alpha\left(P\right)\neq 0$. Then $\nabla\left(\alpha f\right)\left(P\right)= \alpha\left(P\right)\nabla f\left(P\right)\neq 0$, so $P$ is not a basepoint of $Q$. Assume now we have chosen $\lambda$ such that $P\in V_\lambda$. After a permutation of indices we can assume $\lambda_0=1$, so $V_\lambda=\left\{ g_{0,0}+2\lambda_1 g_{0,1}+\lambda_1^2 g_{1,1}=0\right\}$. Since $P$ is not a basepoint we have $g_{1,1}\left(P\right)\neq 0$. Together with $\left(g_{0,0}g_{1,1}-g_{0,1}^2\right)\left(P\right)=0$ we get $\lambda_1=-\left(g_{0,1}/g_{1,1}\right)\left(P\right)$. Then \begin{align*} % \nabla &\left(g_{0,0}+2\lambda_1 g_{0,1}+\lambda_1^2 g_{1,1}\right)\left(P\right)=\\ &\hspace*{5ex}=\frac{1}{g_{1,1}\left(P\right)} \left(g_{1,1}\nabla g_{0,0}-2 g_{0,1}\nabla g_{0,1} +g_{0,0}\nabla g_{1,1}\right)\left(P\right)\\ &\hspace*{5ex}=\frac{1}{g_{1,1}\left(P\right)} \alpha\left(P\right)\nabla f\left(P\right). % \end{align*} We see that $P$ is a smooth point of $V_\lambda$, so together with a) we have proved \romannum{1}. c) Assume that $V_\lambda$ is singular along a curve $C_\lambda\subset S$ and let $m_\lambda=\operatorname{mult}\left(V_\lambda,C_\lambda\right)$. Then $C_\lambda$ is a continuous family of curves and $m=\min\left\{m_\lambda\mid\lambda\in\mathbb{P}_1\right\}$ is equal to $m_\lambda$ on an open dense subset of $\mathbb{P}_1$. Now \romannum{1} says that $C_\lambda\subseteq S\cap W$ for all $\lambda\in\mathbb{P}_1$. But $S\cap W$ is itself a curve, so this family is in fact constant. So let $C=C_{(0:1)}$. Now $g_{0,0}g_{1,1}-g_{0,1}^2=\alpha f$ vanishes to the $2m$-th order along $C$ and $\operatorname{mult}\left(S,C\right)=1$, so $\alpha$ vanishes to the $\left(2m-1\right)$-st order along $C$.$\square$\medskip\par \begin{corollary}\label{corollary:unstable} % Let $w\in\overline{C}_S$. % \begin{itemize} % \item[\romannum{1}] If $w$ is unstable in degree $s/2$, then $\left|w\right|=s^3/8$. \item[\romannum{2}] If $w$ is unstable in degree $\left(s+1\right)\!/2$ (resp.~$\left(s+2\right)/2$), then $\left|w\right|\geq s\left(s-1\right)^2/8$ (resp.~$s\left(s-2\right)^2/8$). % \end{itemize} % \end{corollary} {\noindent\bf Proof:\ } In the first case $W=\emptyset$. So the general surface in $Q_w$ is not singular on $S$, hence irreducible. Now apply corollary \ref{corollary:irreducible}. In the second case $\deg\left(W\right)\leq 2$, so $W$ has no triple curve. Now apply proposition \ref{proposition:reduced}.$\square$\medskip\par Now here comes our analysis what happens if $V$ is a plane or a quadric. \begin{proposition}\label{proposition:planequadric} % Let $w\in\overline{C}_S$. % \begin{itemize} % \item[\romannum{1}] If $w$ is cut out by a plane $H$, then $\left|w\right|=s\left(s-1\right)\!/2$. Moreover $w$ is stable in degree $1$ if $s>2$ and unstable in degree $1$ otherwise. \item[\romannum{2}] If $w$ is cut out by a reduced quadric $Q$, then % \begin{equation*} % \left|w\right| = \left\{\begin{array}{c@{\quad}l} s\left(s-2\right) & \text{if $s$ is even,}\\ \left(s-1\right)^2 & \text{if $s$ is odd.} \end{array}\right. % \end{equation*} % Moreover $w$ is stable in degree $2$ if $s>4$ and unstable in degree $2$ otherwise. % \end{itemize} % \end{proposition} {\noindent\bf Proof:\ } \romannum{1} $H$ is smooth, so $\left|w\right|=s\left(s-1\right)/2$ by corollary \ref{corollary:irreducible}. If $s>2$ then $2\deg\left(H\right)=2<s$, so $w$ is semi stable in degree $1$ by lemma \ref{lemma:unique}. But then $w$ is stable in degree $1$. In the case $s=2$ example \ref{example:cone} shows that $w$ is unstable in degree $1$. \romannum{2} Assume first $Q$ is nodal. Then $\left|w\right|\in\left\{s\left(s-2\right),s\left(s-2\right)+1\right\}$ by corollary \ref{corollary:irreducible}. But $s\left(s-2\right)+1=\left(s-1\right)^2$ and $4\mid\left|w\right|$ imply the above formula for $\left|w\right|$. Now let $Q=H_1+H_2$ where $H_1\neq H_2$ are planes and set $L=H_1\cap H_2$. If $L\subset S$, then $S.H_i=2D_i+L$ and each $D_i$ contains exactly $\left(s-1\right)^2\!/2$ nodes which are $D_i$-smooth by proposition \ref{proposition:irreducible}. Clearly $L$ cannot contain any node of $w$. So $\left|w\right|=\left(s-1\right)^2$. If $L\not\subset S$, then $S.H_i=2D_i$. Every $D_i$ is reduced and contains exactly $s(s-1)/2$ $D_i$-smooth nodes. In every point $P\in L\cap S$ both $H_1$ and $H_2$ are tangent to $S$, so $P$ is a node of $S$. Both $H_1$ and $H_2$ have contact to the tangent cone $C_PS$ of $S$ at $P$. This implies $L\not\subset C_PS$, hence $\operatorname{mult}\left(S,L;P\right)=2$. Therefore $L$ contains exactly $s/2$ such nodes and $\left|w\right|=s\left(s-2\right)$. If $s>4$, then $w$ is stable in degree $2$. Now let $s\leq 4$. In any case $F_w$ cannot contain a square. So $w$ is unstable if $\mycohomd{0}{2}{-}{w}>1$. For $s=3$ this follows from example \ref{example:cubic}. If $s=4$ then we find using Serre duality that $\mycohomd{2}{2}{-}{w}=\mycohomd{0}{-2}{+}{w}=0$. Therefore it follows that $\mycohomd{0}{2}{-}{w}\geq\mychi{2}{-}{w}=2$.$\square$\medskip\par \section{The proof of theorem \ref{theorem:main}} This section is devoted entirely to the proof of theorem \ref{theorem:main}. Let $S$ and $V$ with $S.V=2D$ as in the first section. \begin{lemma}\label{lemma:equal} % \cite{catanese} Let $w\in \overline{C}_S$ be an even set of nodes. Let $n\in\mathbb{N}$ be even if $w$ is strictly even and odd if $w$ is weakly even. Then for all $i\geq 0$ % \begin{equation*} % \cohomd{i}{\obundletS{\left(n\pi^\ast\negthinspace H +E_w\right)\negthinspace /2}} = \cohomd{i}{\obundletS{\left(n\pi^\ast\negthinspace H -E_w\right)\negthinspace /2}}. % \end{equation*} % \end{lemma} \proofwith{ of theorem \ref{theorem:main}} \romannum{1} By proposition \ref{proposition:planequadric} the even sets $w\in\overline{C}_S$ cut out by planes satisfy $\left|w\right|=s\left(s-1\right)\!/2$. We show that no smaller even sets can occur. The proof also explains the ``gaps'' of corollary \ref{corollary:main}. So let $w\in\overline{C}_S\setminus\left\{0\right\}$ be weakly even. $s=4$: Since $\mychi{}{-}{w}=\left(10-\left|w\right|\right)\!/4$ is an integer we must have $\left|w\right|\in\left\{2,6,10,14\right\}$. By Serre duality and lemma \ref{lemma:equal} $\mycohomd{2}{}{-}{w}=\mycohomd{0}{-}{-}{w}=0$. Now let $\left|w\right|\leq 6$, then $\mycohomd{0}{}{-}{w}\geq 1$. So $w$ is cut out by a plane, hence $\left|w\right|=6$. $s=6$: Here $\mychi{}{-}{w}=\mychi{3}{-}{w}=\left(35-\left|w\right|\right)\!/4$, so $\left|w\right|\in\left\{3,7,11,\ldots\right\}$. Let $\left|w\right|<27$, so $\left|w\right|\leq 23$ and $\mychi{}{-}{w}\geq 3$. Following proposition \ref{proposition:stable}, we see that $w$ is either stable in degree $1$ or no plane cuts out $w$, hence $\mycohomd{2}{3}{-}{w}=\mycohomd{0}{}{-}{w}\in\left\{0,1\right\}$. This implies $\mycohomd{0}{3}{-}{w}\geq 2$. Now corollary \ref{corollary:unstable} tells us that $w$ is semi stable in degree $3$. But $w$ cannot be stable in degree $3$, so $w$ is stable in degree $1$. Hence $w$ is cut out by a plane. It follows that $\left|w\right|=15$ and that there are no weakly even sets of $19$ and $23$ nodes on a nodal sextic surface. $s=8$: Now $\mychi{3}{-}{w}=\mychi{5}{-}{w}=21-\left|w\right| /4$, so $\left|w\right|\in\left\{4,8,12,\ldots\right\}$. Let $\left|w\right|<60$, then $\left|w\right|\leq 56$ and $\mychi{3}{-}{w}\geq 7$. Assume that $\mycohomd{0}{3}{-}{w}=\mycohomd{2}{5}{-}{w}\leq 1$. It follows that $\mycohomd{0}{5}{-}{w}\geq 6$, hence $w$ is unstable in degree $5$. So $\left|w\right|\geq 60$ by corollary \ref{corollary:unstable}, contradiction. This implies that $w$ is semi stable in degree $3$ and stable in degree $1$. Again $w$ is cut out by a plane, hence $\left|w\right|=28$. Moreover, a nodal octic surface cannot have weakly even sets of $32, 36, 40,\ldots,56$ nodes. \romannum{2} This is essentially a copy of the methods of \romannum{1}. Let $w\in C_S\setminus\left\{0\right\}$ be strictly even. $s=4$: We have $\left|w\right|\in\left\{8,16\right\}$. If $\left|w\right|=9$ then $\mychi{2}{-}{w}=2$. But $\mycohomd{2}{2}{-}{w}=\mycohomd{0}{-2}{-}{w}=0$, so by Serre duality and lemma \ref{lemma:equal} $\mycohomd{0}{2}{-}{w}\geq 2$ and $w$ is cut out by a quadric. $s=5$: Now $\left|w\right|\in\left\{4,8,12,\ldots\right\}$ and $\mychi{2}{-}{w}=5-\left|w\right|/4$. As usual we find that $\mycohomd{2}{2}{-}{w}=\cohomd{0}{\obundletS{-E_w/2}}=0$. So if $\left|w\right|\leq 16$ then $\mycohomd{0}{2}{-}{w}\geq 1$ and $w$ is cut out by a quadric. Then $\left|w\right|=16$ by proposition \ref{proposition:planequadric}. $s=6$: Here $\left|w\right|\in\left\{8,16,24,\ldots\right\}$ and $\mychi{2}{-}{w}=8-\left|w\right| /4$. This time we find that $\mycohomd{2}{2}{-}{w}=\mycohomd{0}{2}{-}{w}$. So if $\left|w\right|\leq 24$, then $\mycohomd{0}{2}{-}{w}\geq 1$ and $w$ is cut out by a quadric surface. But then $\left|w\right|=24$. $s=7$: We modify the proof as follows. One calculates $\mychi{2}{-}{w}=\mychi{4}{-}{w}=14-\left|w\right|/4$. Let $\left|w\right|<44$, so $\left|w\right|\leq40$. By proposition \ref{proposition:planequadric} $\mycohomd{0}{2}{-}{w}\in\left\{0,1\right\}$, so $\mycohomd{0}{4}{-}{w}\geq 3$. If $w$ is unstable in degree $4$ then $\left|w\right|\geq42$, contradiction. So $w$ is semi stable in degree $4$ and stable in degree $2$. Now $w$ is cut out by a quadric and thus $\left|w\right|=36$. In particular, there is no even set of $40$ nodes on a nodal septic surface. $s=8$: Using the same argument as for $s=7$, we get $\mychi{4}{-}{w}=20-\left|w\right|/4$ and $\mycohomd{2}{4}{-}{w}=\mycohomd{0}{4}{-}{w}$. Let $\left|w\right|<64$, then $\left|w\right|\leq56$ and $\mycohomd{0}{4}{-}{w}\geq 3$. Again by corollary \ref{corollary:unstable} $w$ cannot be unstable in degree $4$. Hence $w$ is semi stable in degree $4$ and stable in degree $2$. In particular $\left|w\right|=48$ and there is no even set of $56$ nodes on a nodal octic surface. $s=10$: Finally we calculate $\mychi{6}{-}{w}=40-\left|w\right|/4$ and as before $\mycohomd{2}{6}{-}{w}=\mycohomd{0}{6}{-}{w}$. Let $\left|w\right|<120$, so $\left|w\right|\leq 112$ and $\mycohomd{0}{6}{-}{w}\geq 6$. As before $w$ is semi stable in degree $6$. If $w$ was stable in degree $4$ then $\mycohomd{0}{6}{-}{w}=4$, contradiction. So $w$ is semi stable in degree $w$ and stable in degree $2$. Again $\left|w\right|=80$, hence there are no strictly even sets of $88$, $96$, $104$ and $112$ nodes on a nodal surface of degree $10$.$\square$\medskip\par \section{Examples revisited} We want to go a little more into the example of quartics. Many of the facts stated in this example can be found in \cite{gallarati}. \begin{example} % Let $S$ be a nodal quartic surface and let $w\in C_S$ with $\left|w\right|=8$. We have seen that $\mycohomd{0}{2}{-}{w}\geq 2$ and that $w$ is unstable in degree $2$. Let $Q_w$ be the quadratic system of quadrics which cut out $w$. The base locus of $Q_w$ is contained in the surface $W$ of proposition \ref{proposition:unstable}. But here $W=\emptyset$, so the only basepoints of $Q_w$ are the nodes of $S$. It follows from lemma \ref{lemma:smooth} and Bertini that the general element in $L_w=\mylinsys{2}{-}{w}$ is a smooth elliptic curve. In fact $Q_w$ defines an elliptic fibration of $\tilde{S}$. If $\mycohomd{0}{2}{-}{w}>2$, then we will find two smooth elliptic curves on $\tilde{S}$ intersecting in at least one point, contradiction. This shows that $\mycohomd{0}{2}{-}{w}=2$. Taking into account that $\mydivisor{4}{-}{w}$ is nef and big on $\tilde{S}$, we can calculate the numbers of the next table. If $\left|w\right|=16$ one computes $\mycohomd{i}{4}{-}{w}$ in the same fashion. % \begin{equation*} % \begin{array}{|c||c|c|c|}\hline % \left|w\right|=8 & h^0 & h^1 & h^2 \\\hline\hline \mydivisor{2}{-}{w} & 2 & 0 & 0 \\\hline \mydivisor{4}{-}{w} & 8 & 0 & 0 \\\hline % \end{array} \quad\quad \begin{array}{|c||c|c|c|}\hline % \left|w\right|=16 & h^0 & h^1 & h^2 \\\hline\hline \mydivisor{2}{-}{w} & 0 & 0 & 0 \\\hline \mydivisor{4}{-}{w} & 6 & 0 & 0 \\\hline % \end{array} % \end{equation*} % Now let $w\in\overline{C}_S$ be weakly even. Here $\mydivisor{3}{-}{w}$ is big and nef, so we find the following table. % \begin{equation*} % \begin{array}{|c||c|c|c|}\hline % \left|w\right|=6 & h^0 & h^1 & h^2 \\\hline\hline \mydivisor{ }{-}{w} & 1 & 0 & 0 \\\hline \mydivisor{3}{-}{w} & 5 & 0 & 0 \\\hline % \end{array} \quad\quad \begin{array}{|c||c|c|c|}\hline % \left|w\right|=10 & h^0 & h^1 & h^2 \\\hline\hline \mydivisor{ }{-}{w} & 0 & 0 & 0 \\\hline \mydivisor{3}{-}{w} & 4 & 0 & 0 \\\hline % \end{array} % \end{equation*} % \end{example} \section{Concluding remarks} It is very likely that theorem \ref{theorem:main} is true for surfaces of arbitrary degree. Unfortunately I cannot prove this. The main obstruction is to exclude the possibility that an irreducible contact surface is singular along a curve which is contained in the nodal surface. In \cite{barth2} Barth gave a construction of nodal surfaces admitting even sets of nodes. The surfaces are constructed as degeneracy locus of a generic quadratic form on a globally generated vector bundle on $\mathbb{P}_3$. For convenience we give a list of the strictly even sets of nodes which have been obtained so far. Note that Barth's construction gives exactly the gap of corollary \ref{corollary:main}. \begin{center} % \begin{tabular}{l|l} % degree & even sets \\\hline 3 & 4 \\ 4 & 8,16 \\ 5 & 16,20 \\ 6 & 24,32,40 \\ 8 & 48,64,72,80,\ldots,128 \\ 10 & 80,120,128,136,\ldots,208 % \end{tabular} % \end{center}

9710

alg-geom/9710011

en

https://arxiv.org/abs/alg-geom/9710011

alg-geom/9710011

Andrew Kresch

Andrew Kresch

Canonical rational equivalence of intersections of divisors

LaTeX2e, 14 pages; expanded intro; new first section fixes some errors

We consider the operation of intersecting with a locally principal Cartier divisor (i.e., a Cartier divisor which is principal on some neighborhood of its support). We describe this operation explicitly on the level of cycles and rational equivalences and as a corollary obtain a formula for rational equivalence between intersections of two locally principal Cartier divisors. Such canonical rational equivalence applies quite naturally to the setting of algebraic stacks. We present two applications: (i) a simplification of the development of Fulton-MacPherson-style intersection theory on Deligne-Mumford stacks, and (ii) invariance of a key rational equivalence under a certain group action (which is used in developing the theory of virtual fundamental classes via intrinsic normal cones).

alg-geom

\section{Introduction} One way to define an operation in intersection theory is to define a map on the group of algebraic cycles together with a map on the group of rational equivalences which commutes with the boundary operation. Assuming the maps commute with smooth pullback, the extension of the operation to the setting of algebraic stacks is automatic. The goal of the first section of this paper is to present the operation of intersecting with a principal Cartier divisor in this light. We then show how this operation lets us obtain a rational equivalence which is fundamental to intersection theory. A one-dimensional family of cycles on an algebraic variety always admits a unique limiting cycle, but a family of cycles over the punctured affine plane may yield different limiting cycles if one approaches the origin from different directions. An important step in the historical development of intersection theory was realizing how to prove that any two such limiting cycles are rationally equivalent. The results of the first section yield, as a corollary, a new, explicit formula for this rational equivalence. Another important rational equivalence in intersection theory is the one that is used to demonstrate commutativity of Gysin maps associated to regularly embedded subschemes. In section 2, we exhibit a two-dimensional family of cycles such that the cycles we obtain from specializing in two different ways are precisely the ones we need to show to be rationally equivalent to obtain the commutativity result. Our explicit rational equivalence respects smooth pullback, and hence the generalization to stacks is automatic. This simplifies intersection theory on Deligne-Mumford stacks as in \cite{v}, where construction of such a rational equivalence fills the most difficult section of that important paper. Since our rational equivalence arises by considering families of cycles on a larger total space, we are able to deduce (section 3) that the rational equivalence is invariant under a certain naturally arising group action. The key observation is that we can manipulate the situation so that the group action extends to the total space. This equivariance result is used, but appears with mistaken proof, in \cite{bf}, where an important new tool of modern intersection theory --- the theory of virtual fundamental classes --- is developed. The author would like to thank S. Bloch, W. Fulton, T. Graber, and R. Pandharipande for helpful advice and the organizers and staff of the Mittag-Leffler Institute for hospitality during the 1996--97 program in algebraic geometry. \section{Intersection with divisors} In this section we work exclusively on schemes of finite type over a fixed base field. The term variety denotes integral scheme, and by a subvariety we mean an integral closed subscheme. We denote by $Z_*X$, $W_*X$, and $A_*X$, respectively, the group of algebraic cycles, group of rational equivalences, and Chow group of a scheme $X$. The boundary map $W_*X\to Z_*X$ is denoted $\partial$. We refer to \cite{f} for basic definitions and properties from intersection theory. Given a Cartier divisor $D$ we denote by $[D]$ the associated Weil divisor (it is important to note that the notion of Weil divisor makes sense on arbitrary varieties, \cite{f} \S 1.2). If $X$ is a variety then we denote by $X^1$ the set of subvarieties of codimension 1. \begin{defn} Let $X$ be a variety and let $D$ be a Cartier divisor. Let $\pi\colon \widehat X\to X$ be the normalization map. The {\em support} of $D$, denoted $|D|$, is defined to be $\pi(\bigcup_{\substack{W\in\widehat X^1\\ \mathop{\rm ord}\nolimits_W \pi^*D\ne 0}} W).$ \end{defn} \begin{rem} This agrees with the na\"\i{}ve notion of support (the union of all subvarieties appearing with nonzero coefficient in $[D]$) when $X$ is normal or when $D$ is effective. \end{rem} \begin{rem} There is yet another notion of support which appears in \cite{f}. There, the support of a divisor is a piece of data that must be specified along with the divisor. Given a Cartier divisor $D$ on a variety $X$, let $Z$ be any closed subscheme such that away from $Z$ the canonical section of ${\mathcal O}(D)$ is well-defined and nonvanishing. Then, \cite{f} defines an intersection operation $A_k(X)\to A_{k-1}(Z)$. Unfortunately, the support $|D|$ which we have defined is not generally a support in this sense. Hence in the definition below we require that our divisors be specified by defining functions which are regular away from their supports. \end{rem} We shall denote by $|D|^0$ the set of irreducible components of $|D|$. \begin{defn} \label{pdivisor} Let $X$ be a variety. A {\em $P$-divisor} on $X$ is a tuple $(U,U',x)$ such that \begin{itemize} \vspace{-12pt} \item[(i)] $U$ and $U'$ are nonempty open subschemes of $X$ such that $U\cup U'=X$; \item[(ii)] $x\in k(U)^*$; \item[(iii)] $x|_{U\cap U'}\in {\mathcal O}^*(U\cap U')$; and \item[(iv)] the data $(x\in k(U)^*, 1\in k(U')^*)$ specifies a Cartier divisor $D$ such that $|D|=X\setminus U'$. \end{itemize} \end{defn} By abuse of terminology, we call $D$ a $P$-divisor if $D$ is the Cartier divisor associated to a $P$-divisor as in (iv). Given a $P$-divisor as above, we call $x$ the {\em local defining function}. A $P$-divisor may be pulled back via a morphism of varieties provided that the image of the morphism is not contained in the support of the underlying Cartier divisor. \begin{exas} \begin{itemize} \item[] \hspace{-30pt} (i)\hspace{5pt}Let $X$ be a normal variety. Let $x\in k(X)^*$ specify a principal Cartier divisor $D$. Then $(X, X\setminus |D|, x)$ is a $P$-divisor. \item[(ii)] Let $X$ be a variety. Every effective principal Cartier divisor is a $P$-divisor. \item[(iii)] Let $X$ be a variety, and let $\pi\colon X\to \mathbb P^1$ be a dominant morphism. Then the fiber of $\pi$ over $\{0\}$ is a $P$-divisor. \end{itemize} \end{exas} The operation of intersecting with a Cartier divisor is generally defined only on the level of rational equivalence classes of cycles. When $V\subset |D|$, we have $D\cdot[V]=c_1({\mathcal O}(D)|_V)\mathbin{\raise.4pt\hbox{$\scriptstyle\cap$}}[V]$, and there is generally no way to pick canonically a cycle representing this first Chern class. The exception is when ${\mathcal O}(D)|_{|D|}$ is trivial, or in our terminology, $D$ is a $P$-divisor. Then, we may define a cycle-level intersection operation (see \cite{f}, Remark 2.3). \begin{defn} Let $X$ be a variety, and let $D$ be a $P$-divisor on $X$. The cycle-level intersection operation $$D\cdot {-}\colon Z_k(X)\to Z_{k-1}(|D|)$$ is given by $$D\cdot [V] = \begin{cases} {}[D|_V] & \text{if $V\not\subset |D|$}; \\ 0& \text{if $V\subset |D|$}. \end{cases}$$ \end{defn} The claim that this map passes to rational equivalence and hence gives an intersection operation $D\cdot{}\colon A_k(X)\to A_{k-1}(|D|)$ is proved in \cite{f}, but not in a way that makes it easy to see how $D\cdot\alpha$ is to be rationally equivalent to zero if $\alpha$ is a cycle that is rationally equivalent to zero. Following the program set out in the introduction, we would like to demonstrate this fact by giving an explicit map on rational equivalences which commutes with the boundary operation. \begin{defn} Let $X$ be a variety, and let $D$ be a $P$-divisor on $X$ with local defining function $x$. Say $V$ is a subvariety of $X$ with normalization $\pi\colon \widehat V\to V$, and suppose $y\in k(V)^*$. We define the intersection operation on the level of rational equivalences $$D\cdot{-}\colon W_k(X)\to W_{k-1}(|D|)$$ by \begin{equation} \label{maponrat} D\cdot y = \begin{cases} {}\pi_*\bigl(\bigoplus_{W\in |\pi^*D|^0} (y^{\mathop{\rm ord}\nolimits_W x} / x^{\mathop{\rm ord}\nolimits_W y}) |_W \bigr) & \text{if $V\not\subset |D|$}; \\ 0& \text{if $V\subset |D|$}. \end{cases} \end{equation} Here, $\pi_*\colon W_*\widehat V\to W_*V$ is pushforward of rational equivalence. \end{defn} \begin{rem} This definition explains why we a require the definition of a $P$-divisor to include more data than just that of the underlying Cartier divisor. The map (\ref{maponrat}) actually depends on the choice of defining function. \end{rem} \begin{pr} Let $X$ be a variety and let $D$ be a $P$-divisor on $X$. Then the diagram $$ \xymatrix{ W_k(X) \ar[r]^(.43){D\cdot{}}\ar[d]_\partial & W_{k-1}(|D|) \ar[d]_\partial \\ Z_k(X) \ar[r]^(.43){D\cdot{}} & Z_{k-1}(|D|) } $$ commutes. \end{pr} This follows easily from \begin{pr} Let $X$ be a normal variety and let $x$ and $y$ be rational functions with associated principal Cartier divisors $D$ and $E$. For $V\in X^1$ set $a_V=\mathop{\rm ord}\nolimits_V x$ and $b_V=\mathop{\rm ord}\nolimits_V y$. Then \begin{align} \sum_{V\in X^1} \partial(y^{a_V}/x^{b_V}|_V)&=0; \label{eqone} \\ \partial(D\cdot y) &= D\cdot(\partial\, y); \label{eqtwo} \\ D\cdot[E] - E\cdot[D] &= \sum_{V\in |D|^0\cap |E|^0} \partial(y^{a_V}/ x^{b_V}|_V). \label{eqthree} \end{align} \end{pr} \begin{proof} If we split the sum in (\ref{eqone}) into a sum over $V\in |D|^0$ and a sum over $V\not\in |D|^0$ we obtain (\ref{eqtwo}). Similarly if we split away the terms with $V\in |D|^0\cap |E|^0$ we obtain (\ref{eqthree}) from (\ref{eqone}). So, for a fixed variety $X$ and fixed divisors $D$ and $E$, the three assertions are equivalent. Now, we get (\ref{eqone}) as a consequence of the tame symbol in $K$-theory, cf.\ \cite{q} \S7, or by the following elementary geometric argument. We quickly reduce to the case where $D$ and $E$ are effective. Then, when $D$ and $E$ meet properly, (\ref{eqthree}) follows from \cite{f}, Theorem 2.4, case 1. An induction on {\em excess of intersection} $$\varepsilon(D,E)=\max_{V\in X^1} a_V\cdot b_V$$ completes the proof: if we denote the normalized blow-up along the ideal $(x,y)$ by $\sigma\colon X'\to X$ and denote the exceptional divisor by $Z$ then we may write $\sigma^*D = Z + D'$ and $\sigma^*E = Z + E'$, and now $|D'|\cap |E'|=\emptyset$ and $\max(\varepsilon(D',Z), \varepsilon(E',Z))< \varepsilon(D,E)$ (assuming $D$ and $E$ do not meet properly), cf.\ \cite{f}, Lemma 2.4. The result pushes forward. \end{proof} \begin{cor} \label{canrat} Let $D$ and $E$ be $P$-divisors on a variety $X$, with respective local defining functions $x$ and $y$. Let $\pi\colon\widehat X\to X$ be the normalization map. Then $$D\cdot[E] - E\cdot[D] = \partial\, \omega$$ where $\omega\in W_*(|D|\cap |E|)$ is given by $$\omega = \sum_{V\in |\pi^*D|^0\cap |\pi^*E|^0} \pi_*(y^{\mathop{\rm ord}\nolimits_Vx}/ x^{\mathop{\rm ord}\nolimits_Vy}|_V).$$ \end{cor} \section{Application to intersection theory on stacks} All stacks (and schemes) in this section are algebraic stacks of Artin type, \cite{a}, \cite{l}, which are locally of finite type over the base field. The notion of $P$-divsor on a stack makes sense (it is as in Definition \ref{pdivisor} with ``open subscheme'' replaced by ``open substack,'' where by ``Cartier divisor'' in part (iv) of the definition we mean a global section of the sheaf ${\mathcal K}^*/{\mathcal O}^*$ for the Zariski topology, and where normalization, order along a substack of codimension 1, and support of a Cartier divisor are well defined on stacks because they all respect smooth pullback and hence can be defined locally). Since an Artin stack possesses a smooth cover by a scheme, the operation of intersecting with a $P$-divisor on a stack comes for free once we know that this operation on schemes commutes with smooth pullback. Also for free we get Corollary \ref{canrat} in the setting of stacks: the formation of $\omega$ from $X$, $D$, and $E$ commutes with smooth pullback. \begin{pr} Let $X$ be a variety, let $Y$ be a scheme, and let $f\colon Y\to X$ be a smooth morphism. Let $D$ be a $P$-divisor on $X$. Then $f^*\smallcirc D\cdot{} = (f^*D)\cdot{}\smallcirc f^*$, both as maps on cycles and as maps on rational equivalences. \end{pr} We now turn to an application of Corollary \ref{canrat} to intersection theory on Deligne-Mumford stacks (where a reasonable intersection theory exists, cf.\ \cite{g}, \cite{v}). Central to intersection theory on schemes is the Gysin map corresponding to a regularly embedded subscheme, since the diagonal of a smooth scheme is a regular embedding and this way we obtain an intersection product on smooth schemes. The diagonal morphism for a smooth Deligne-Mumford stack is not generally an embedding, but it is representable and unramified. \begin{lm} \label{unram} Let $f\colon F\to G$ be a representable morphism of Artin stacks. Then $f$ is unramified if and only if there exists a commutative diagram $$ \xymatrix{ U\ar[r]^g\ar[d] & V \ar[d] \\ F\ar[r]^f & G } $$ such that the vertical maps are smooth surjective, $g$ is a closed immersion of schemes, and the induced morphism $U\to F\times_GV$ is \'etale. \end{lm} \begin{proof} This is \cite{v}, Lemma 1.19. Because this is such a basic fact about properties of morphisms in algebraic geometry, we present an elementary proof in the Appendix. \end{proof} To describe a representable morphism, we use the terminology {\em local immersion} as a synonym for {\em unramified} and call $f$ above a {\em regular local immersion} if moreover $g$ is a regular embedding of schemes. Since formation of normal cone is of a local nature, an obvious patching construction produces the normal cone $C_XY$ to a local immersion $X\to Y$; the cone is a bundle in case $X\to Y$ is a regular local immersion. To get Fulton-MacPherson-style intersection theory on Deligne-Mumford stacks we clearly need to have Gysin maps for regular local immersions. In \cite{v}, the author supplies this needed Gysin map by giving a (long, difficult) proof of the stack analogue of \cite{f}, Theorem 6.4, namely \begin{pr} \label{bigrat} Let $X\to Y$ and $Y'\to Y$ be local immersions of Artin stacks. Then $[C_{X\times_YC_{Y'}Y}C_{Y'}Y]=[C_{C_XY\times_YY'}C_XY]$ in $A_*(C_XY\times_YC_{Y'}Y)$. \end{pr} \begin{rem} Though our focus is on applications to intersection theory on Deligne-Mumford stacks, we continue to make use of constructions which behave well locally with respect to smooth pullback, and hence our results are valid in the more general setting of Artin stacks. \end{rem} \begin{rem} Given a stack $X$ which is only locally of finite type over a base field, we must take $Z_*X$ to be the group of {\em locally finite} formal linear combinations of integral closed substacks. More intrinsically, $Z_*X$ is the group of global sections of the sheaf for the smooth topology $\mathcal Z_*$ which associates to a stack of finite type the free abelian group on integral closed substacks. Similarly, $W_*X$ is the group of global sections of sheaf $\mathcal W_*$. As always, $A_*X$ is defined to be $Z_*X/\partial W_*X$. \end{rem} The methods of the last section allow us to supply a new, simpler proof of this proposition. \begin{proof} Recall that given a closed immersion $X\to Y$ there are associated spaces \begin{align*} M_XY &= \mathop{{\rm B}\ell}\nolimits_{X\times\{0\}}Y\times\mathbb P^1, \\ M^\circ_XY &= M_XY \setminus \mathop{{\rm B}\ell}\nolimits_{X\times\{0\}}Y\times\{0\}, \end{align*} cf.\ \cite{f} \S 5.1. Given a locally closed immersion, say with $U$ is an open subscheme of $Y$ and $X$ a closed subscheme of $U$, then $M^\circ_XY:=M^\circ_XU\amalg_{U\times\mathbb A^1} Y\times\mathbb A^1$ makes sense and is independent of the choice of $U$. This lets us define $M^\circ_FG$ when $F\to G$ is a local immersion of stacks, as follows. Assume we have a diagram as in the statement of Lemma \ref{unram}, and set $R=U\times_FU$ and $S=V\times_GV$. There are projections $q_1, q_2\colon S\to G$. Define $s_i\colon M^\circ_RS\to M^\circ_UV$ ($i=1,2$) to be the composite $M^\circ_RS\to M^\circ_{U\times_GV}S\to M^\circ_UV$, where the first map is induced by the open immersion $R\to U\times_GV$ and the second, by pullback via $q_i$. Then $[M^\circ_RS\rightrightarrows M^\circ_UV]$ is the smooth groupoid presentation of a stack which we denote $M^\circ_FG$. We have, by descent, a morphism $M^\circ_FG\to\mathbb P^1$, which is flat and has as general fiber a copy of $G$ and as special fiber the normal cone $C_FG$. In the situation at hand, this construction gives $$(s\times t)\colon M^\circ_XY\times_Y M^\circ_{Y'}Y\to \mathbb P^1\times\mathbb P^1,$$ and hence a pair of $P$-divisors, $D$ (corresponding to $s$) and $E$ (corresponding to $t$). We note that $(s\times t)^{-1}(\{0\}\times\{0\})=C_XY\times_YC_{Y'}Y$. Since the restriction of $s\times t$ to $\mathbb P^1\times\mathbb P^1\setminus \{0\}\times\{0\}$ is flat, we have \begin{align*} {}[D] &= [C_XY\times_Y M^\circ_{Y'}Y] \mod Z_*(C_XY\times_YC_{Y'}Y), \\ {}[E] &= [M^\circ_XY\times_Y C_{Y'}Y] \mod Z_*(C_XY\times_YC_{Y'}Y). \end{align*} We examine the fiber of $s\times t$ over $\mathbb P^1\times\{0\}$ more closely. The fiber square $$\xymatrix{ i^*C_{Y'}Y \ar[r] \ar[d] & C_{Y'}Y \ar[d] \\ X \ar[r]^i & Y }$$ gives rise to a closed immersion $f$ making $$\xymatrix@C=2pt{ M^\circ_{i^*C_{Y'}Y}C_{Y'}Y \ar[rr]^(.48)f\ar[dr]_h && M^\circ_XY \times_Y C_{Y'}Y \ar[dl]^g \\ & {\mathbb P^1} }$$ commute (where $g$ is first projection followed by $s$). Since $f$ is an isomorphism away from the fiber over 0, we see in fact that $$[E] = [M^\circ_{i^*C_{Y'}Y}C_{Y'}Y] \mod Z_*(C_XY\times_YC_{Y'}Y),$$ and since $h$ is flat we find $$D\cdot [E] = [C_{i^*C_{Y'}Y}C_{Y'}Y].$$ Similarly, if $j$ denotes the map $Y'\to Y$ then $$E\cdot [D] = [C_{j^*C_XY}C_XY]$$ and so the rational equivalence $\omega\in W_*(C_XY\times_YC_{Y'}Y)$ of Corollary \ref{canrat} satisfies $$\partial\,\omega = [C_{X\times_YC_{Y'}Y}C_{Y'}Y] - [C_{C_XY\times_YY'}C_XY]. \qed $$ \renewcommand{\qed}{}\end{proof} \begin{rem} The map $M^\circ_FG\to G$ associated to a local immersion of stacks is not generally separated, though this should cause the reader no concern, since intersection theory is valid even on non-separated schemes and stacks. In fact, even those operations of \cite{v} which require a so-called finite parametrization may be carried out on arbitrary Deligne-Mumford stacks which are of finite type over a field (no such operations show up in this paper). This is so thanks to the proof, \cite{l} (10.1), that every Deligne-Mumford stack of finite type over a field possesses a finite parametrization, i.e., admits a finite surjective map from a scheme. \end{rem} \begin{rem} The reader who wishes greater generality may see easily that all results in this section are valid in the setting of Artin stacks which are locally of finite type over an excellent Dedekind domain. \end{rem} \section{Equivariance for tangent bundle action} We continue to work with stacks which are locally of finite type over some base field. A special case of Proposition \ref{bigrat} is when $i\colon X\to Y$ is a local immersion of smooth Deligne-Mumford stacks. Suppose $j\colon Y'\to Y$ is a local immersion, with $Y'$ an arbitrary Deligne-Mumford stack. Recall that the local immersion $j$ gives rise to a natural group action of $j^*T_Y$ on $C_{Y'}Y$. In short, the action is given locally (say $Y$ is an affine scheme and $Y'$ is the closed subscheme given by the ideal $I$) by considering the action of $T_Y|_{Y'}$ on $\mathop{\rm Spec}\nolimits \mathop{\rm Sym}\nolimits (I/I^2)$ induced by the map $I/I^2\to \Omega^1_Y$ and proving (\cite{bf}, Lemma 3.2) that the normal cone $\mathop{\rm Spec}\nolimits \bigoplus I^k/I^{k+1}$ is invariant under the group action. If we let $N_XY$ be the normal bundle to $X$ in $Y$ and denote simply by $N$ its pullback to $X':=X\times_YY'$, then $C_XY\times_YC_{Y'}Y$ is identified with $N\times_{X'} i^*C_{Y'}Y$. Viewing $T_{Y'}$ as a subbundle of $j^*T_Y$, we have the natural action of $T_{Y'}|_{X'}$ on $i^*C_{Y'}Y$. This plus the trivial action on $N$ gives an action of $T_{Y'}|_{X'}$ on $N\times_{X'}i^*C_{Y'}Y$. \begin{thm} The rational equivalence between $[C_{i^*C_{Y'}Y}C_{Y'}Y]$ and $[N\times_{X'}C_{X'}Y']$ produced in the proof of Proposition \ref{bigrat} is invariant under the action of $T_{Y'}|_{X'}$ on $N\times_{X'}C_{X'}Y'$ described above. \end{thm} As a consequence, the rational equivalence descends to a rational equivalence on the stack quotient $[N\times_{X'}i^*C_{Y'}Y\,/\,T_{Y'}|_{X'}]$. This fact is exploited in \cite{bf} (Lemma 5.9, where the authors invoke the incorrect stronger claim appearing in Proposition 3.5 that the rational equivalence is equivariant for the bigger group $T_Y|_{X'}$). \begin{proof} The question is local, so we may assume $Y$ is an irreducible scheme, smooth and of finite type over the base field, $X$ is an smooth irreducible closed subscheme of $Y$, and $Y'$ is a closed subscheme of $Y$. If $X\subset Y'$ then the group action is trivial and there is nothing to prove, so we assume the contrary. \begin{lm} Let $Y$ be a smooth irreducible scheme of finite type over a field $k$, of dimension $n$, let $X$ be a smooth irreducible closed subscheme of $Y$ of codimension $d$, and let $Y'$ be a closed subscheme of $Y$ such that $X\not\subset Y'$. Let $x$ be a closed point of $Y'\cap X$. Then, after suitable base change by a finite separable extension of the base field, and after shrinking $Y$ to a neighborhood of $x$ in $Y$, there exists an \'etale map $f\colon Y\to \mathbb A^n$ such that $X$ maps into a linear subspace of $\mathbb A^n$ of codimension $d$ and such that $Y'\to f(Y')$ is \'etale. \end{lm} \begin{proof} We may assume $x$ is a $k$-valued point, and moreover that $Y$ sits in $\mathbb A^l$ with $X=\mathbb A^{l-d}\cap Y$ (for suitable $l$). We may take $x$ to be the origin of $\mathbb A^l$. We consider as candidates for $f$ all linear functions mapping the flag $\mathbb A^{l-d}\subset \mathbb A^l$ into the flag $\mathbb A^{n-d}\subset \mathbb A^n$. Those $f$ with $f_*\colon T_{x,Y}\to T_{f(x),\mathbb A^n}$ surjective form an open subscheme $U$ of $\mathbb A^{nl-dl+d^2}$. Define locally closed subschemes $V_1$ and $V_2$ of $Y\times U$ by $$V_1=\{(y,f)\in (Y'\cap X\setminus\{x\})\times U\,|\, f(y)=0\}$$ and $$V_2=\{(y,f)\in (Y'\setminus X)\times U\,|\,f(y)=0\},$$ and let $pr_2\colon Y\times U\to U$ be projection. A dimension count using the fact that $X\not\subset Y'$ gives $\dim(V_1) < \dim(U)$ and $\dim(V_2) < \dim(U)$, and hence $U\setminus \bigl( \overline{pr_2(V_1)}\cup \overline{pr_2(V_2)} \bigr)$ is nonempty. \end{proof} Since the rational equivalence of the proof of Proposition \ref{bigrat} commutes with \'etale base change, we are reduced by the Lemma to the case where $Y=\mathbb A^n$ and $X=\mathbb A^m$ (as a linear subspace of $\mathbb A^n$). Now we need the \begin{keyobs} Assume $Y=\mathbb A^n$ and $Y'$ is a closed subscheme of $Y$. Identify $T_Y$, as a group scheme over $Y$, with the additive group $\mathbb A^n$. Then there is a group action of $\mathbb A^n$ on $\widetilde M^\circ_{Y'}Y$ (which we define to be the fiber of $M^\circ_{Y'}Y\to\mathbb P^1$ over $\mathbb A^1$) which restricts to the natural action of $T_Y$ on $C_{Y'}Y$. \end{keyobs} Indeed, we let $\mathbb A^n$ act on $Y\times\mathbb A^1$ by $$(a_1,\ldots,a_n)\cdot(x_1,\ldots,x_n,t)=(x_1+ta_1,\ldots,x_n+ta_n).$$ By the universal property of blowing up, this extends uniquely to an action of $\mathbb A^n$ on $\widetilde M^\circ_{Y'}Y$. If $Y'$ is given by the ideal $(f_1,\ldots,f_k)$, and if we view $\widetilde M^\circ_{Y'}Y$ as the closure of the graph of $(f_1/t,\ldots,f_k/t)\colon Y\times(\mathbb A^1\setminus\{0\})\to \mathbb A^k=\mathop{\rm Spec}\nolimits k[z_1,\ldots,z_k]$, then the action is given coordinatewise by $${\mathbf a}=(a_1,\ldots,a_n)\colon z_i\mapsto z_i+ (f_i({\mathbf x} + t{\mathbf a}) - f_i({\mathbf x}))/t,$$ so at $t=0$ we recover $z_i\mapsto z_i+D_{\mathbf a}f_i({\mathbf x})$. This is the natural action of $T_Y$ on $C_{Y'}Y$. Concluding the proof of equivariance, we observe that $\widetilde M^\circ_{\mathbb A^m}\mathbb A^n$ fits into the fiber diagram $$ \xymatrix{ {\widetilde M^\circ_{\mathbb A^m}\mathbb A^n\times_{\mathbb A^n}\widetilde M^\circ_{Y'}\mathbb A^n} \ar[r] \ar[d] & {\widetilde M^\circ_{Y'}\mathbb A^n} \ar[d] \\ {\widetilde M^\circ_{\mathbb A^m}\mathbb A^n} \ar[r] \ar[d] & {\mathbb A^n} \ar[d] \\ {\widetilde M^\circ_{\{0\}}\mathbb A^{n-m}} \ar[r] & {\mathbb A^{n-m}} }$$ and now the action from the Key Observation of $\mathbb A^m\subset \mathbb A^n$ on $\widetilde M^\circ_{Y'}Y$, plus the trivial action of $\mathbb A^m$ on $M^\circ_{\{0\}}\mathbb A^{n-m}$, combine to give a group action of $\mathbb A^m$ on $\widetilde M^\circ_XY\times_Y\widetilde M^\circ_{Y'}Y$. The function $\widetilde M^\circ_XY\times_Y\widetilde M^\circ_{Y'}Y\to \mathbb A^1\times\mathbb A^1$ which is used in Corollary \ref{canrat} is invariant for this $\mathbb A^m$-action. Since the rational equivalence of the proof of Proposition \ref{bigrat} is compatible with smooth pullback, we get the desired equivariance result. \end{proof} \section{Appendix: unramified morphisms} We give an elementary algebraic proof of the following fact. \begin{lm} Let $S\to T$ be an unramified morphism of affine schemes which are of finite type over a base field $k$. Then there exists a commutative diagram of affine schemes $$\xymatrix{ U \ar[r]^g \ar[d] & V \ar[d] \\ S \ar[r]^f & T }$$ such that the vertical maps are \'etale surjective and such that $g$ is a closed immersion. \end{lm} This fact plus the local nature of the property of being unramified gives us Lemma \ref{unram}. \begin{proof} Say $S=\mathop{\rm Spec}\nolimits A$, $T=\mathop{\rm Spec}\nolimits B$, and $f$ is given algebraically by $f^*\colon B\to A$. Recall that for $f$ to be unramified means that for every maximal ideal ${\mathfrak p}$ of $A$ with ${\mathfrak q}=f({\mathfrak p})$, we have $f^*({\mathfrak q})\cdot A_{\mathfrak p}={\mathfrak p}A_{\mathfrak p}$, and the induced field extension $B/{\mathfrak q}\to A/{\mathfrak p}$ is separable. {\em Case 1:} The induced field extension $B/{\mathfrak q}\to A/{\mathfrak p}$ is an isomorphism. Then, if $x_1,\ldots,x_n$ are generators of $A$ as a $k$-algebra, we may write $$x_i=f^*(t_i)+w_i$$ with $t_i\in B$ and $w_i\in {\mathfrak p}$, for each $i$. Since $f$ is unramified, we have $$w_i=\sum_{j=1}^{m_i} \frac{f^*(y_{ij})p_{ij}}{q_i}$$ for some $y_{ij}\in{\mathfrak q}$, $p_{ij}\in A$, and $q_i\in A\setminus{\mathfrak p}$. Choose representative polynomials $P_{ij}$ and $Q_i$ in $k[X_1,\ldots,X_n]$ such that $P_{ij}(x_1,\ldots,x_n)=p_{ij}$ and $Q_i(x_1,\ldots,x_n)=q_i$. Let \begin{eqnarray*} \lefteqn{ V=\mathop{\rm Spec}\nolimits B[X_1,\ldots, X_n]\bigm/\big(\,X_1Q_1-t_1Q_1-\sum_{j=1}^{m_1} y_{1j}P_{1j}\,,\ \ldots,}\hspace{130pt} \\ & & X_nQ_n-t_nQ_n-\sum_{j=1}^{m_n}y_{nj}P_{nj}\,\big), \end{eqnarray*} and define $g\colon S\to V$ by $B\stackrel{f^*}\rightarrow A$ and $X_i\mapsto x_i$, and let $\varphi\colon V\to T$ be given by inclusion of $B$. Then $g$ is a closed immersion, and by the Jacobian criterion $\varphi$ is \'etale in some neighborhood of $g({\mathfrak p})$. {\em Case 2:} The field extension $B/{\mathfrak q}\to A/{\mathfrak p}$ is separable. Let $k'$ be the maximal subfield of $A/{\mathfrak p}$ which is separable over $k$, and make the \'etale base change $\mathop{\rm Spec}\nolimits k'\to \mathop{\rm Spec}\nolimits k$ to get $f'\colon S'\to T'$. Now $S'$ has an $A/{\mathfrak p}$-valued point which maps to ${\mathfrak p}\in S$, and since $k'$ together with $B/{\mathfrak q}$ generates all of $A/{\mathfrak p}$ we are now in the situation of Case 1. \end{proof}

9710

alg-geom/9710032

en

https://arxiv.org/abs/alg-geom/9710032

alg-geom/9710032

Sergey Barannikov

Sergey Barannikov, Maxim Kontsevich

Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields

12 pages, AMS-TeX; typos and a sign corrected, appendix added. Submitted to IMRN

International Mathematics Research Notices, Volume 1998, Issue 4, Pages 201-215

10.1155/S1073792898000166

We construct a generalization of the variations of Hodge structures on Calabi-Yau manifolds. It gives a Mirror partner for the theory of genus=0 Gromov-Witten invariants

alg-geom

\section{1. Frobenius manifolds} Remind the definition of formal Frobenius (super) manifold as given in [D], [M], [KM]. Let $\bold H$ be a finite-dimensional $\bold Z_2$-graded vector space over $\C$.\footnote{One can use an arbitrary field of characteristic zero instead of $\C$ everywhere} It is convenient to choose some set of coordinates $x_{\bold H}=\{x^a\}$ which defines the basis $\{\p_a:=\p/\p x^a\}$ of vector fields. One of the given coordinates is distinguished and is denoted by $x_0$. Let $A^c_{ab}\in \C [[x_{\bold H}]]$ be a formal power series representing 3-tensor field, $g_{ab}$ be a nondegenerate symmetric pairing on $\bold H$. To simplify notations in superscripts we replace $\text{deg}\, (x^a)$ by $\bar a$. One can use the $A_{ab}^c$ in order to define a structure of $\C [[x_{\bold H}]]$-algebra on $\bold H\otimes\C [[x_{\bold H}]]$, the (super)space of all continuous derivations of $\C[[x_{\bold H}]]$, with multiplication denoted by $\circ$: $$ \p_a\circ\p_b:=\sum_c A_{ab}^c\p_c$$ One can use $g_{ab}$ to define the symmetric $\C[[x_{\bold H}]]$-pairing on $\bold H\otimes\C[[x_{\bold H}]]$: $$\langle \p_a,\p_b\rangle:=g_{ab}$$ These data define the structure of formal Frobenius manifold on $\bold H$ iff the following equations hold: \item{(1)} (Commutativity/Associativity) $$\forall a,b,c \,\,\,\,\,\, A^c_{ba}=(-1)^{\bar a\bar b}A_{ab}^c\eqno (1a)$$ $$\forall a,b,c,d\,\,\,\,\,\,\sum_{e}A^e_{ab} A_{ec}^d =(-1)^{\bar a(\bar b+\bar c)}\sum_{e}A_{bc}^e A_{ea}^d\eqno (1b)$$ equivalently, $A^c_{ab}$ are structure constants of a supercommutative associative \break $\C[[x_{\bold H}]]$-algebra \item {(2)}(Invariance) Put $A_{abc}=\sum_e A^e_{ab}g_{ec}$ $$\forall a,b,c \,\,\,\,\,\,A_{abc}= (-1)^{\bar a(\bar b+\bar c)}A_{bca}\,\,\,\, ,$$ equivalently, the pairing $g_{ab}$ is invariant with respect to the multiplication $\circ$ defined by $A_{ab}^c$. \item {(3)} (Identity) $$\forall a,b \,\,\,\,\,\,A_{0a}^b=\delta^b_a$$ equivalently $\p_0$ is an identity of the algebra $\bold H\otimes\C[[x_{\bold H}]]$ \item {(4)} (Potential) $$\forall a,b,c,d\,\,\,\,\,\, \p_d A^c_{ab}=(-1)^{\bar a\bar d}\p_a A^c_{db}\,\,\,\, ,$$ which implies, assuming (1a) and (2), that the series $A_{abc}$ are the third derivatives of a single power series $\Phi\in \bold H\otimes \C[[x_{\bold H}]]$ $$A_{abc}=\p_a\p_b\p_c\Phi$$ \section{2. Moduli space via deformation functor} The material presented in this section is standard (see [K2] and references therein). Let us remind the definition of the functor ${\Def_\gtg}$ associated with a differential graded Lie algebra $\goth g$ . It acts from the category of Artin algebras to the category of sets. Let $\goth A$ be an Artin algebra with the maximal ideal denoted by $\gt m$. Define the set $${\Def_\gtg}(\gt A):=\{d\g+{[\g,\g]\over 2}=0 |\g \in (\gt g\otimes\gt m)^1\}/\Gamma_{\gt A}^0$$ where the quotient is taken modulo action of the group $\Gamma_{\gt A}^0$ corresponding to the nilpotent Lie algebra $(\gt g\otimes\gt m)^0$. The action of the group can be described via the infinitesimal action of its Lie algebra: $$\alpha\in \gt g\otimes\gt m \to \dot \g=d\alpha +[\g,\alpha]$$ Sometimes functor $\Def_\gtg$ is represented by some topological algebra $\Cal O_{\Cal M_\gtg}$ (projective limit of Artin algebras) in the sense that the functor $\Def_\gtg$ is equivalent to the functor $\text{Hom}_{continuous}(\Cal O_{\Cal M_\gtg},\,\cdot\,)$. For example, $H^0(\gtg)=0$ is a sufficient condition for this. If $\Def_\gtg$ is representable then one can associate formal moduli space to $\gtg$ by defining the ``algebra of functions'' on the formal moduli space to be the \break algebra $\Cal O_{\Cal M_\gtg}$. We will need the $\Z$-graded extension of the functor $\Def_\gtg$. The definition of $\Def^{\,\Z}_\gtg$ is obtained from the definition of $\Def_\gtg$ via inserting $\Z-$graded Artin algebras instead of the usual ones everywhere. A sufficient and probably necessary condition for the functor $\Def^{\,\Z}_\gtg$ to be representable is that $\gtg$ must be quasi-isomorphic to an abelian graded Lie algebra. We will see in \S 2.1 that this is the case for $\gtg=\bt$. Hence one can associate formal (graded) moduli space \footnote{We will omit the superscript $\Z$ where it does not seem to lead to a confusion.} $\Cal M_\bt$ to the Lie algebra $\gtg$. \subhead {2.1 Extended moduli space of complex structure} \endsubhead Let $M$ be a connected compact complex manifold of dimension $n$, with vanishing $1$-st Chern class $c_1(T_M)=0\in \text{Pic}(M)$. We assume that there exists a K\"ahler metric on $M$, although we will not fix it. By Yau's theorem there exists a Calabi-Yau metric on $M$. It follows from the condition $c_1(T_M)=0$ there exists an everywhere nonvanishing holomorphic volume form $\Omega\in \Gamma(X,\Lambda^n T_M^*)$. It is defined up to a multiplication by a constant. Let us fix a choice of $\om$. It induces isomorphism of cohomology groups: $H^q(M,\Lambda^pT_M)\simeq H^q(M,\Omega^{n-p})$; $\g\mapsto \gamma\vdash\om$. Define differential $\dv$ of degree $-1$ on $\bt$ by the formula : $$(\dv\g)\vdash\om=\p(\g\vdash\om)$$ The operator $\dv$ satisfies the following identity (Tian-Todorov lemma) : $$ [\g_1,\g_2]=(-1)^{\text{deg}\g_1+1}(\dv(\g_1\wedge\g_2) - (\dv \g_1)\wedge\g_2 - (-1)^{\text{deg}\g_1+1}\g_1\wedge\dv\g_2)\eqno (2.1)$$ where $\text{deg}\g=p+q-1$ for $\g\in\Gamma(M,\Lambda^q\bar T^*_M\otimes\Lambda^q T_M)$. Denote by $\bold H$ the total homology space of $\dv$ acting on $\bt[1]$. Let $\{\dl_a\}$ denote a graded basis in the vector space $\oplus_{p,q}H^q(M,\Lambda^pT_M)$, $\dl_0=1\in H^0(M,\Lambda^0T_M)$ . Let us redefine the degree of $\dl_a$ as follows $$|\dl_a|:=p+q-2\,\,\,\,\,\text{for}\, \,\,\,\,\,\dl_a\in H^q(M,\Lambda^pT_M)$$ Then $\{\dl_a\}$ form a graded basis in $\bold H$. Denote by $\{t^a\}, t^a\in \bold H^*,\,\text{deg}\,t^a=-|\dl_a|$ the basis dual to $\{\dl_a\}$. Denote by $\C[[t_{\bold H}]]$ the algebra of formal power series on $\Z$-graded vector space $\bold H$. \proclaim{Lemma 2.1} The functor $\Def^{\,\Z}_\bt$ associated with $\bt$ is canonically equivalent to the functor represented by the algebra $\C[[t_{\bold H}]]$. \endproclaim \proof It follows from (2.1) that the maps $$(\bt,\db)\gets(\Ker\dv,\db)\to (\bold H[-1],d=0)\eqno (2.2)$$ are morphisms of differential graded Lie algebras. Then the $\p\db$-lemma, which says that $$\Ker\db\cap \Ker\dv\cap(\im\dv\oplus\im\db)=\im\dv\circ\db,\eqno (2.3) $$ shows that these morphisms are quasi-isomorphisms (this argument is standard in the theory of minimal models, see [DGMS]). Hence (see e.g. theorem in \S 4.4 of [K2]) the deformation functors associated with the three differential graded Lie algebras are canonically equivalent. The deformation functor associated with trivial algebra $(\bold H[-1],d=0)$ is represented by the algebra $\C[[t_{\bold H}]]$. \endproof \corollary 2.2 The moduli space $\Cal M_\bt$ associated to $\bt$ is smooth. The dimension of $\Cal M_\bt$ is equal to $\sum_{p,q}\text{dim }H^q(M,\Lambda^pT_M)$ of the dimension of the space of first order deformations associated with $\bt$.\endproclaim \Remark The Formality theorem proven in [K2] implies that the differential graded Lie algebra controlling the $A_\infty$-deformations of $\Cal D^bCoh(M)$ is quasi-isomorphic to $\bt$. Here we have proved that $\bt$ is quasi-isomorphic to an abelian graded Lie algebra. Therefore, the two differential graded Lie algebras are formal, i.e. quasi-isomorphic to their cohomology Lie algebras endowed with zero differential.\endremark \corollary 2.3 There exists a solution to the Maurer-Cartan equation $$\db{\hat\g}(t)+{[{\hat\g}(t),{\hat\g}(t)]\over 2}=0\eqno (2.4)$$ in formal power series with values in $\bt$ $${\hat\g}(t)=\sum_a\hat\g_at^a+{1\over {2!}}\sum_{a_1,a_2}\hat\g_{a_1a_2}t^{a_1}t^{a_2} +\ldots\in (\bt\otimes\bold C[[t_{\bold H}]])^1$$ such that the cohomology classes $[\hat\g_a]$ form a basis of cohomology of the complex $(\bt,\db)$\endproclaim \Remark The deformations of the complex structure are controlled by the differential graded Lie algebra $$\bt_{(0)}:=\bigoplus_k \bt_{(0)}^k, \,\,\bt_{(0)}^k= \Gamma(M,\Lambda^{k}\overline{T}^*_M\otimes T_M)$$ The meaning of this is that the completion of the algebra of functions on the moduli space of complex structures on $M$ represents $\Def_{\bt_{(0)}}$ (or $\Def_{\bt_{(0)}}^{\,\Z}$ restricted to the category of Artin algebras concentrated in degree $0$). The natural embeddings $\bt_{(0)}\hookrightarrow \bt$ induces embedding of the corresponding moduli spaces. In terms of the solutions to Maurer-Cartan equation the deformations of complex structure are singled out by the condition $\g(t)\in \Gamma(M,\Lambda^1\overline{T}^*_M\otimes\Lambda^1T_M)$.\endremark \Remark Similar thickening of the moduli space of complex structures were considered by Z\.~Ran in [R].\endremark \section{3. Algebra structure on the tangent sheaf of the moduli space } Let $R$ denotes a $\Z$-graded Artin algebra over $\C$, $\g^R \in (\bt\otimes R)^1$ denotes a solution to the Maurer-Cartan equation (2.4). The linear extension of the wedge product gives a structure of graded commutative algebra on $\bt\otimes R[-1]$. Let $\g^R$ be a solution to the Maurer-Cartan equation in $(\bt\otimes R)^1$. It defines a differential $\db_{\g^R}=\db+\{\g^R,\cdot\}$ on $\bt\otimes R[1]$. Denote the cohomology of $\db_{\g^R}$ by $T_{\g^R}$. The space of first order variations of $\g^R$ modulo gauge equivalence is identified with $T_{\g^R}$. Geometrically one can think of $\g^R$ as a morphism from the formal variety corresponding to algebra $R$ to the formal moduli space. An element of $T_{\g^R}$ corresponds to a section of the preimage of the tangent sheaf. Note that $\db_{\g^R}$ acts as a differentiation of the (super)commutative $R-$algebra $t\otimes R[-1]$. Therefore $T_{\g^R}[-2]$ inherits the structure of (super)commutative associative algebra over $R$. This structure is functorial with respect to the morphisms $\phi_*: T_{\g^{R_1}}\to T_{\g^{R_2}}$ induced by homomorphisms $\phi : R_1\to R_2$. Let ${\hat\g}(t)\in (\bt\widehat\otimes\C[[t_{\bold H}]])^1$ be a solution to the Maurer-Cartan satisfying the condition of corollary 2.3. It follows from this condition that the $\C[[t_{\bold H}]]$-module $T_{{\hat\g}(t)}$ is freely generated by the classes of partial derivatives $[\p_a{\hat\g}(t)]$. Therefore we have \proclaim{Proposition 3.1} There exists formal power series $A^c_{ab}(t)\in \bt\widehat\otimes\C[[t_{\bold H}]]$ satisfying $$ \p_a{\hat\g}\wedge\p_b{\hat\g}=\sum_c A^c_{ab}\p_c{\hat\g}\,\,\text{mod} \,\db_{{\hat\g}(t)}$$ The series $A_{ab}^c(t)$ are the structure constants of the commutative associative \break $\C[[t_{\bold H}]]$@-algebra structure on $\bold H\otimes\C[[t_{\bold H}]][-2]$. \endproclaim \qed \Remark Note that on the tangent space at zero this algebra structure is given by the obvious multiplication on $\oplus_{p,q}H^q(M,\Lambda^pT_M)$. This is "Mirror dual" to the ordinary multiplication on $\oplus_{p,q}H^q(M,\om^p_{\widetilde M})$. \endremark \section{4. Integral} Introduce linear functional on $\bt$ $$\int \g:=\int_M (\g\vdash\om)\wedge \om$$ \claim 4.1 It satisfies the following identyties: $$\aligned\int\db\g_1\wedge\g_2=(-1)^{\text{deg}\g_1}\int\g_1\wedge\db\g_2\\ \int\dv\g_1\wedge\g_2=(-1)^{\text{deg}\g_1+1}\int\g_1\wedge\dv\g_2 \endaligned\eqno (4.1)$$ for $\g_i\in \Gamma(M,\Lambda^{q_i}\overline T^*_M\otimes\Lambda^{p_i}T_M), \,\, i=1,2$ where $\text{deg}\,\g_i=p_i+q_i-1$ .\endproclaim\qed \section{5. Metric on $\Cal{T_M}$} There exists a natural metric (i.e. a nondegenerate (super)symmetric $\Cal O_{M_\bt}$-linear pairing) on the sheaf $\Cal T_{\Cal M_\bt}$. In terms of a solution to the Maurer-Cartan equation $\g^R \in (\bt\otimes R)^1$ it means that there exists an $R$-linear graded symmetric pairing on $T_{\g^R}$, which is functorial with respect to $R$. Here $T_{\g^R}$ denotes the cohomology of $\db_{\g^R}$ defined in \S 3. The pairing is defined by the formula $$\langle h_1,h_2\rangle:=\int h_1\wedge h_2\,\,\,\text{for}\,\,h_1,h_2\in T_{\g^R} $$ where we assumed for simplicity that $\g^R\in \Ker\dv\otimes R$. It follows from (2.2) (see also lemma 6.1) that such a choice of $\g^R$ in the given class of gauge equivalence is always possible. \claim 5.1 The pairing is compatible with the algebra structure.\endproclaim \noindent{\qed} \section{6. Flat coordinates on moduli space.} Another ingredient in the definition of Frobenius structure is the choice of affine structure on the moduli space associated with $\bt$. The lemma 2.1 identifies $\Cal M_{\bt}$ with the moduli space associated with trivial algebra $(\bold H[-1],d=0)$. The latter moduli space is the affine space $\bold H$. The affine coordinates $\{t_a\}$ on $\bold H$ give coordinates on $\Cal M_{\bt}$. This choice of coordinates on the moduli space corresponds to a specific choice of a universal solution to the Maurer-Cartan equation over $\C[[t_{\bold H}]]$. \proclaim{Lemma 6.1} There exists a solution to the Maurer-Cartan equation in formal power series with values in $\bt$ $$\db{\hat\g}(t)+{[{\hat\g}(t),{\hat\g}(t)]\over 2}=0,\,{\hat\g}(t)=\sum_a\hat\g_at^a+{1\over {2!}}\sum_{a_1,a_2}\hat\g_{a_1a_2}t^{a_1}t^{a_2} +\ldots\in (\bt\widehat\otimes\C[[t_{\bold H}]])^1$$ such that \roster \item (Universality) the cohomology classes $[\hat\g_a]$ form a basis of cohomology of the complex $(\bt,\db)$ \item (Flat coordinates) $\hat\g_a\in\Ker\dv,\,\,\hat\g_{a_1\ldots a_k}\in \im\dv\,\,\, \text{for}\,\,\, k\geq 2$ \item (Flat identity) $\p_0\hat\g(t)=\bold 1$, where $\p_0$ is the coordinate vector field corresponding to $[\bold 1]\in \bold H[-1]$ \endroster \endproclaim \proof The theorem of \S 4.4 in [K2] shows that there exists $L_\infty$ morphism $f$ homotopy inverse to the natural morphism $(\Ker\dv,\db)\to \bold H[-1]$ (for the definition of $L_\infty$-morphism see \S 4.3 in [K2]). Put $\Delta(t)=\sum_a(\Delta_a[-1]) t^a$ , where $\Delta_a[-1]$ denotes the element $\dl_a$ having degree shifted by one. Then $${\g}(t)=\sum_n{1\over{n!}}f_n(\Delta(t)\wedge\dots\wedge\Delta(t))$$ satisfies the conditions $(1)-(2)$. To fulfill the condition $(3)$ $\g(t)$ must be improved slightly. Define the differential graded Lie algebra $\widetilde{\Ker}$ as follows \roster \item $\widetilde{\Ker}_i=\Ker \dv\subset \bt_i\,\text{for}\,i\geq 0$ \item $\widetilde{\Ker}_{-1}=\im \dv\subset \bt_{-1}$ \endroster Note that the algebra $\Ker \dv$ is the sum of the algebra $\widetilde{\Ker}$ and trivial algebra of constants $\C\otimes\bold 1[-1]$. Let $\tilde f$ be a homotopy inverse to the natural quasi-isomorphism $\widetilde{\Ker}\to \bold H[-1]_{\geq 0}$. Put $\tilde\Delta(t)=\sum_{a\neq 0}(\Delta_a[-1]) t^a$. Then $$\hat{\g}(t)=\bold 1t_0 +\sum_n{1\over{n!}}\tilde f_n(\tilde\Delta(t)\wedge\dots\wedge\tilde\Delta(t))$$ satisfies all the conditions. \endproof \Remark Any two formal power series satisfying conditions of lemma 6.1 are equivalent under the natural action of the group associated with the \break Lie algebra $(\widetilde\Ker \dv\widehat\otimes\C[[t_{\bold H}]])^0$. \endremark \Remark It is possible to write down an explicit formula for the components of the morphism $f$ in terms of Green functions of the Laplace operator acting on differential forms on $M$. \endremark \Remark After the identification of the moduli space $\Cal M_\bt$ with $\bold H$, provided by lemma 2.1, the complex structure moduli space corresponds to the \break subspace $\bold H^1(M,\Lambda^1T_M)$. In the case of classical moduli space of the complex structures on $M$ the analogous lemma was proved in [T]. The coordinates arising on the classical moduli space of complex structures coincide with so called "special" coordinates of [BCOV].\endremark \remark{Notation} Denote $${\hat\g}(t)=\sum_a\hat\g_at^a+{1\over {2!}}\sum_{a_1,a_2}\hat\g_{a_1a_2}t^{a_1}t^{a_2} +\ldots\,\in (\bt\otimes\bold C[[t_{\bold H}]])^1$$ a solution to the Maurer-Cartan equation satisfying conditions of lemma 6.1. \endremark The parameters of a miniversal solution to the Maurer-Cartan equation over $\C[[t_{\bold H}]]$ serve as coordinates on the moduli space. The specific choice of coordinates corresponding to the solution to the Maurer-Cartan equation satisfying conditions (1)-(2) of lemma 6.1 corresponds to choice of coordinates on moduli space that are flat with respect to the natural (holomorphic) metric $g_{ab}$. \claim 6.2 The power series $\langle \p_a{\hat\g}(t),\p_b{\hat\g}(t)\rangle\in\C[[t_{\bold H}]]$ has only constant term in the power series expansion at $t=0$.\endproclaim \proof $\langle x,y\rangle=0$ for $x\in \Ker\dv, y\in \im\dv$.\endproof \remark{Notation} Denote $g_{ab}:=\langle \p_a{\hat\g}(t),\p_b{\hat\g}(t)\rangle$. \endremark Thus we have constructed all the ingredients of the Frobenius structure on $\Cal M_{\bt}$: the tensors $A^c_{ab}(t)$, $g_{ab}$ and the coordinates $\{t_a\}$ that are flat with respect to $g_{ab}$. \Remark The 3-tensor $A_{ab}^c(t)$ on $\Cal M_\bt$ does not depend on the choice of $\om$. The 2-tensor $g_{ab}$ is multiplied by $\lambda^2$ when $\om$ is replaced by $\lambda\om$\endremark \claim 6.3 The structure constants satisfy $A_{0a}^b=\delta^b_a$.\endproclaim \proof It follows from the condition (3) imposed on $\hat\g(t)$\endproof We have checked that the tensors $A^c_{ab}(t),\,g_{ab}$ have the properties (1)-(3) from the definition of the Frobenius structure. It remains to us to check the \break property (4). \section{ 7. Flat connection and periods} Let ${\hat\g}(t)\in (\bt\otimes\bold C[[t_{\bold H}]])^1$ be a solution to the Maurer-Cartan equation satisfying conditions (1)-(2) of lemma 6.1. Then the formula $$\om(t):=e^{\hat\g(t)}\vdash\om $$ defines a closed form of mixed degree depending formally on $t\in \bold H$. For $t\in \bold H^{-1,1}$ $\hat\g(t)\in \Gamma(M,\overline T\otimes T^*)$ represents a deformation of complex structure. Then $\om(t)$ is a holomorphic $n$-form in the complex structure corresponding to $t\in \bold H^{-1,1}$, where $n=\text{dim}_\C\, M$. Let $\{p^a\}$ denote the set of sections of $\Cal T^*_{\bold H}$ that form a framing dual to $\{\p_a\}$. Define a (formal) connection on $\Cal T^*_{\bold H}$ by the covariant derivatives: $$\nabla_{\p_a}(p^c):=\sum_b A_{ab}^c p^b \eqno (7.1)$$ Strictly speaking this covariant derivatives are formal power series sections of $\Cal T^*_{\bold H}$. Let us put $$\Pi_{ai}={\p\over\p t^a}\int_{\Gamma_i} \om(t) \eqno (7.2)$$ where $\{\Gamma_i\}$ form a basis in $\bold H_*(M,\C)$. In particular $$\Pi_{0i}=\int_{\Gamma_i} \om(t)$$ if $\hat\g(t)$ satisfies the condition (3) of lemma 6.1. \lemma 7.1 The periods $\Pi_{i}=\sum_{a}\Pi_{ai} p^a$ are flat sections of $\nabla$\endproclaim \noindent{\it Proof.\,}Let $\p_\ta=\sum_a \ta^a \p_a$ be an even constant vector field, i.e. $\ta^a$ are even constants for even $\p_a$ and odd for odd $\p_a$. It is enough to prove that $$\p_\ta\p_\ta\int_{\Gamma_i}\om(t)=\sum_c A^c_{\ta\ta}\p_c \int_{\Gamma_i} \om(t)$$ where $A^c_{\ta\ta}$ are the algebra structure constants defined via $\sum_a\ta^a\p_a \circ\sum_a\ta^a\p_a=\sum_c A^c_{\ta\ta}\p_c$ (see \S 1). Note that the operators $\dv$ and $\db_{\hat\g(t)}$ acting on $\bt\widehat\otimes\C[[t_{\bold H}]]$ satisfy a version of $\p\db$-lemma : $$\Ker \dv\cap\Ker\db_{\hat\g(t)}\cap(\im\dv\oplus\im\db_{\hat\g(t)})= \im\dv\circ\db_{\hat\g(t)}\eqno(7.3)$$ Equivalently, there exists decomposition of $\bt\widehat\otimes\C[[t_{\bold H}]]=X_0\oplus X_1\oplus X_2\oplus X_3\oplus Y$ into direct sum of graded vector spaces, such that the only nonzero components of $\db_{\hat\g(t)}$, $\dv$ are isomorphisms $\db_{\hat\g(t)}:X_0\mapsto X_1,\,\,\,X_2\mapsto X_3;\,\,\,\,\, \dv:X_0\mapsto X_2,\,\,\,X_1\mapsto X_3$. Differentiating twice the Maurer-Cartan equation (2.4) with respect to $\p_\ta$ and using (2.1) one obtaines $$\dv (\p_{\ta}\hat\g(t)\wedge\p_{\ta}\hat\g(t)-\sum_c A_{\ta\ta}^c\p_c\hat\g(t))=-\db_{\hat\g(t)}\p_{\ta}\p_{\ta}\hat\g(t) \eqno (7.4)$$ It follows from $\p\db$-lemma for $\dv$, $\db_{\hat\g(t)}$ and the equation (7.4) that there exist formal power series $\alpha_{\ta}(t)\in \bt\widehat\otimes \C[[t_{\bold H}]]$ such that $$\eqalign{&\p_{\ta}\hat\g(t)\wedge\p_{\ta}\hat\g(t)-\sum_c A_{\ta\ta}^c\p_c\hat\g(t)=\db_{\hat\g(t)}\alpha_{\ta}(t),\cr & \p_{\ta}\p_{\ta}\hat\g(t)=\dv(\alpha_{\ta}(t)) \cr}$$ Therefore $$\p_{\ta}\p_{\ta}\om(t)=\sum_c A_{\ta\ta}^c{\p_c}\om(t) + d(\alpha_{\ta}e^{\hat\g(t)}\vdash\om)$$\qed It follows from the condition (1) imposed on $\hat\g(t)$ that $\Pi_i$ form a (formal) framing of $\Cal T^*_{\bold H}$. \corollary 7.2 The connection $\nabla$ is flat\endproclaim \claim 7.3 The structure constants $A^c_{ab}$ satisfy the potentiality condition (4) in flat coordinates.\endproclaim \proof If one puts symbolically $\nabla=\nabla_0 + A$ then the flatness of $\nabla$ implies that $$\nabla_0 A+{1\over 2}[A,A]=0$$ Notice that associativity and commutativity of the algebra defined by $A^c_{ab}$ imply that $$[A,A]=0 .$$ Therefore $$\nabla_0(A)=0. $$ \endproof We have completed the proof of the fact that $A^c_{ab}(t)$ and $g_{ab}$ define the Frobenius structure on $\Cal M_\bt$ in the flat coordinates $\{t_a\}$. \Remark In fact one can write an explicit formula for the potential of the Frobenius structure. Let us put ${\hat\g}(t)=\sum_a\hat\g_a t^a+\dv\alpha(t), \alpha(t)\in (\bt\widehat\otimes t_{\bold H}^2\C[[t_{\bold H}]])^0$. Put $$\Phi=\int -{1\over 2}\db\alpha\wedge\dv\alpha+{1\over 6}{\hat\g}\wedge{\hat\g}\wedge{\hat\g}$$ Then one checks easily that $A_{abc}=\p_a\p_b\p_c\Phi$ (see Appendix). In the case $\text{dim}_\C M=3,\, \hat \g \in \bt^{-1,1}=\Gamma(M,\overline{T}^*_M\otimes T_M)$ this formula gives the critical value of so called Kodaira-Spencer Lagrangian of [BCOV]. \endremark \Remark Define differential Batalin-Vilkovisky algebra as $\Z_2$-graded commutative associative algebra $A$ equipped with odd differentiation $\db,\,\, \db^2=0$ and odd differential operator $\dv$ of order $\leq 2$ such that $\dv^2=0,\,\,\,\dv\db+\db\dv=0,\,\,\,\dv(1)=0$. One can use the formula (2.1) to define the structure of $\Z_2$-graded Lie algebra on $\Pi A$. Assume that the operators $\db,\dv$ satisfy $\p\db$-lemma (2.3). Assume in addition that $A$ is equipped with a linear functional $\int:A\to \C$ satisfying (4.1) such that the metric defined as in \S 5 is nodegenerate. Then the same construction as above produces the Frobenius structure on the $\Z_2$-graded moduli space $\Cal M_{\Pi A}$. One can define the tensor product of two such Batalin-Vilkovisky algebras. Operator $\dv$ on $A_1\otimes A_2$ is given by $\dv_1\otimes 1+1\otimes\dv_2$. Also, $\db$ on $A_1\otimes A_2$ is $\db_1\otimes 1+1\otimes\db_2$. It is naturally to expect that the Frobenius manifold corresponding to $A_1\otimes A_2$ is equal to the tensor product of Frobenius manifolds corresponding to $A_1,A_2$, defined in [KM] in terms of the corresponding algebras over operad $\{H_*(\overline M_{0,n+1})\}$. \endremark \section{8. Scaling transformations} The vector field $E=\sum_{a}-{1\over 2}|\dl_a|t^a\p_a$ generates the scaling symmetry on $\bold H$. \proclaim{Proposition 8.1} ${\Cal Lie}_EA_{abc}=({1\over 2}(|\dl_a|+|\dl_b|+|\dl_c|)+3-\text{dim}_\C M)A_{abc}$ \endproclaim \proof $A_{abc}=\int_M \p_{a}{\hat\g}\wedge\p_{b}{\hat\g}\wedge\p_{c}{\hat\g}$. Note that $\int \g\neq 0$ implies that $\g\in \bt_{2n-1}$. The proposition follows from the grading condition on $\hat\g(t)$. \endproof \corollary 8.2 ${\Cal Lie}_EA_{ab}^c=({1\over 2}(|\dl_a|+|\dl_b|-|\dl_c|)+1)A_{ab}^c$ \endproclaim Note that the vector field $E$ is conformal with respect to the metric $g_{ab}$. Therefore the proposition 8.~1 shows that $E$ is the Euler vector field of the Frobenius structure on $\bold H$ (see [M]). Such a vector field is defined uniquely up to a multiplication by a constant. The simplest invariant of Frobenius manifolds is the spectrum of the operator $[E,\cdot]$ acting on infinitesimal generators of translations and the weight of the tensor $A_{ab}^c$ under the ${\Cal Lie}_E$-action. Usually the normalization of $E$ is chosen so that $[E,\p_0]=1$. In our case the spectrum of $[E,\cdot]$ is equal to $$\bigcup_d \{1-d/2\} \,\,\text{\eightrm with multiplicity}\sum_{q-p=d-n}\text{dim}H^q(M,\Omega^p)$$ Note that this spectrum and the weight of $A_{ab}^c$ coincide identically with the corresponding quantities of the Frobenius structure arising from the Gromov-Witten invariants of the dual Calabi-Yau manifold $\widetilde M$. \section {9.Further developments.} Conjecturally the constructed Frobenius manifold is related to the Gromov-Witten invariants of $\widetilde M$ in the following way. One can rephrase the present construction in purely algebraic terms using \v Cech instead of Dolbeault realization of the simplicial graded Lie algebra $\Lambda^*\Cal T_M$. The only additional ``antiholomorphic'' ingredient that is used is the choice of a filtration on $H^*(M,\C)$ complementary to the Hodge filtration. The Frobenius structure, arising from the limiting weight filtration corresponding to a point with maximal unipotent monodromy on moduli space of complex structures on M, coincides conjecturally with Frobenius structure on $H^*(\widetilde M,\C)$, obtained from the Gromov-Witten invariants. Our construction of Frobenius manifold is a particular case of a more general construction. Other cases of this construction include the Frobenius manifold structure on the moduli space of singularities of analytic functions found by K.\,Saito, the Frobenius manifold structure on the moduli space of ``exponents of algebraic functions''. The latter case is a Mirror Symmetry partner to the structure arising from Gromov-Witten invariants on Fano varieties. All these cases seem to be related with yet undiscovered generalization of theory of Hodge structures. We hope to return to this in the next paper. \specialhead Appendix \endspecialhead Let ${\hat\g}(t)=\sum_a\hat\g_a t^a+\dv\alpha(t), \alpha(t)\in ({\bold t}\otimes t_{\bold H}^2{\bold C}[[t_{\bold H}]])^1$ be a solution to Maurer-Cartan equation satisfying conditions (1)-(2) of lemma 6\.1. Put $$\Phi=\int -{1\over 2}\db\alpha\wedge\dv\alpha+{1\over 6}{\hat\g}\wedge{\hat\g}\wedge{\hat\g}$$ \proclaim{Proposition } $A_{abc}=\p_a\p_b\p_c\Phi$ \endproclaim \proof. Let $\p_{\ta}=\sum_a \ta_a\p_a$ be an even constant vector field in $\bold H$. It is enough to prove that $\int (\p_{\ta}{\hat\g}\wedge\p_{\ta}{\hat\g}\wedge{\p_{\ta}\hat\g})= \p^3_{\ta\ta\ta}\Phi$. Let us differentiate the terms in $\Phi$ $$\eqalign{&\p^3_{\ta\ta\ta}({\hat\g}\wedge\hat\g\wedge\hat\g)= 18\p^2_{\ta\ta}{\hat\g} \wedge\p_{\ta}{\hat\g}\wedge{\hat\g}+ 3\p^3_{\ta\ta\ta}{\hat\g}\wedge{\hat\g}\wedge\hat\g+ 6\p_{\ta}{\hat\g}\wedge\p_{\ta}{\hat\g} \wedge\p_{\ta}{\hat\g} \cr}$$ $$\eqalign{&\p^3_{\ta\ta\ta}(\db\alpha\wedge\dv\alpha)= \db\alpha\wedge(\p^3_{\ta\ta\ta}\dv\alpha)+ (\p^3_{\ta\ta\ta}\db\alpha)\wedge\dv\alpha +3(\p_{\ta}\db\alpha)\wedge(\p^2_{\ta\ta}\dv\alpha)+\cr &+3(\p^2_{\ta\ta}\db\alpha)\wedge(\p_{\ta}\dv\alpha)\cr}\eqno(*)$$ Notice that $$\eqalign{&\int(\p^3_{\ta\ta\ta}\db\alpha)\wedge\dv\alpha= (-1)^{\text{deg}\db\alpha+1}\int(\p^3_{\ta\ta\ta}\dv\db\alpha)\wedge\alpha= \int(\p^3_{\ta\ta\ta}\dv\db\alpha)\wedge\alpha=\cr &=-\int(\p^3_{\ta\ta\ta}\db\dv\alpha)\wedge\alpha= -(-1)^{\text{deg}\dv\alpha}\int(\p^3_{\ta\ta\ta}\dv\alpha)\wedge\db\alpha= \int(\p^3_{\ta\ta\ta}\dv\alpha)\wedge\db\alpha=\cr &=(-1)^{(\text{deg}\dv\alpha+1)(\text{deg}\db\alpha+1)}\int \db\alpha\wedge(\p^3_{\ta\ta\ta}\dv\alpha)= \int\db\alpha\wedge(\p^3_{\ta\ta\ta}\dv\alpha)\cr}$$ \noindent Hence, the first two terms in (*) give the same contribution. \noindent We have $$\eqalign{&\int\db\alpha\wedge(\p^3_{\ta\ta\ta}\dv\alpha)= (-1)^{\text{deg}\db\alpha+1}\int\dv\db\alpha\wedge(\p^3_{\ta\ta\ta}\alpha) =\int\dv\db\alpha\wedge(\p^3_{\ta\ta\ta}\alpha)=\cr &=-\int\db\dv\alpha\wedge(\p^3_{\ta\ta\ta}\alpha)= -\int\db\hat\g\wedge(\p^3_{\ta\ta\ta}\alpha)= {1\over 2}\int[\hat\g,\hat\g]\wedge(\p^3_{\ta\ta\ta}\alpha)=\cr &={1\over 2}\int\dv(\hat\g\wedge\hat\g)\wedge(\p^3_{\ta\ta\ta}\alpha)= (-1)^{(\text{deg}(\hat\g\wedge\hat\g)+1)}{1\over 2}\int(\hat\g\wedge\hat\g)\wedge(\p^3_{\ta\ta\ta}\dv\alpha)=\cr &={1\over 2}\int(\hat\g\wedge\hat\g)\wedge\p^3_{\ta\ta\ta}\hat\g\cr}$$ \noindent Similarly,$$\eqalign{ \int(\p_{\ta}\db\alpha)\wedge(\p^2_{\ta\ta}\dv\alpha)&={1\over 2}\int \p_{\ta}(\hat\g\wedge\hat\g)\wedge\p^2_{\ta\ta}\hat\g \cr \int(\p^2_{\ta\ta}\db\alpha)\wedge(\p_{\ta}\dv\alpha)&={1\over 2}\int (\p^2_{\ta\ta}\hat\g)\wedge\p_{\ta}(\hat\g\wedge\hat\g) \cr }$$ \endproof \Refs \widestnumber\key{DGMS} \ref\key BCOV\by M.Bershadsky, S.Cecotti, H.Ooguri, C.Vafa \paper Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes \jour Comm.Math.Phys.\vol 164\yr 1994\pages 311--428\endref \ref\key DGMS \by P.Deligne, Ph.Griffiths, J.Morgan, D.Sullivan\paper Real homotopy theory of K\"ahler \break manifolds\jour Inventiones Math.\vol 29\yr 1975\pages 245--274 \endref \ref\key D\by B.Dubrovin \paper\nofrills Geometry of 2d topological field theories;\inbook LNM 1620\publ Springer\yr 1996\break\pages 120@-348\endref \ref\key K1 \by M.Kontsevich \paper Homological algebra of Mirror Symmetry \jour Proccedings of the International Congress of Mathematicians\vol I \pages 120-139\yr 1994 \publ Birkh\"auser \publaddr Z\"urich\endref \ref\key K2\bysame\paper Deformation quantization of Poisson manifolds, \rm I \jour q-alg/9709040\endref \ref\key KM \by M.Kontsevich, Yu.I.Manin \paper Gromov-Witten classes, quantum cohomology, and enumerative geometry\jour Comm.Math.Phys.\vol 164 \yr 1994 \pages 525--562\endref \ref\key M \by Yu.I.Manin \paper Frobenius manifolds, quantum cohomology and moduli spaces {\rm I,II,III} \break \paperinfo Preprint MPI 96-113\publ Max-Planck-Institut f\"ur Mathematik \endref \ref\key T \by A.Todorov\paper The Weil-Petersson geometry of the moduli space of $su(n\geq 3)$ (Calabi-Yau) manifolds \rm I \jour Comm.Math.Phys.\vol 126 \yr 1989\endref \ref\key R \by Z.Ran\paper\nofrills Thickening Calabi-Yau moduli spaces;\inbook {\sl in} Mirror Symmetry \rm II\eds B.R.Greene, S.Yau\publ AMS/IP International Press \bookinfo Studies in advanced mathematics\yr 1997 \pages 393--400 \endref \endRefs \enddocument \end

9710

alg-geom/9710008

en

https://arxiv.org/abs/alg-geom/9710008

alg-geom/9710008

Wolfgang Ebeling

Wolfgang Ebeling and Sabir M. Gusein-Zade

On the index of a vector field at an isolated singularity

AMS-LaTeX, 11 p. with 1 fig.; remarks, definition, and references added to Section 1

Dept. of Math., University of Hannover, Preprint No. 285

We consider manifolds with isolated singularities, i.e., topological spaces which are manifolds (say, $C^\infty$--) outside discrete subsets (sets of singular points). For (germs of) manifolds with, so called, cone--like singularities, a notion of the index of an isolated singular point of a vector field is introduced. There is given a formula for the index of a gradient vector field on a (real) isolated complete intersection singularity. The formula is in terms of signatures of certain quadratic forms on the corresponding spaces of thimbles.

alg-geom

\section*{Introduction}\label{sec0} An isolated singular point of a vector field on ${\Bbb{R}}^n$ or on an $n$-dimensional smooth manifold has a natural integer invariant~--- the index. The formula of Eisenbud, Levine and Khimshiashvili (\cite{EL}, \cite{Kh}) expresses the index of an (algebraically) isolated singular point of a vector field as the signature of a quadratic form on a local algebra associated with the singular point. For a singular point of a gradient vector field there is a formula in terms of signatures of certain quadratic forms defined by the action of the complex conjugation on the corresponding Milnor lattice (\cite{GZ}, \cite{V}). We define a generalisation of the notion of the index of an isolated singular point of a vector field on a manifold with isolated cone-like singularities in such a way that the Poincar\'e--Hopf theorem (the sum of indices of singular points of a vector field on a closed manifold is equal to its Euler characteristic) holds. It seems that this notion (though very natural) cannot be found in the literature in an explicit form. In particular, the index is defined for vector fields on a germ of a real algebraic variety with an isolated singularity. We give a generalisation of a formula from \cite{GZ} for the gradient vector field on an isolated complete intersection singularity. For that we define (in a somewhat formal way) the notion of the variation operator for a complete intersection singularity. We show that this operator is an invariant of the singularity. \section{Basic definitions}\label{sec1} A {\em manifold with isolated singularities} is a topological space $M$ which has the structure of a smooth (say, $C^\infty$--) manifold outside of a discrete set $S$ (the {\em set of singular points} of $M$). A {\em diffeomorphism} between two such manifolds is a homeomorphism which sends the set of singular points onto the set of singular points and is a diffeomorphism outside of them. We say that $M$ has a {\em cone-like singularity} at a (singular) point $P\in S$ if there exists a neighbourhood of the point $P$ diffeomorphic to the cone over a smooth manifold $W_P$ ($W_P$ is called the {\em link} of the point $P$). In what follows we assume all manifolds to have only cone-like singularities. A (smooth or continuous) {\em vector field} on a manifold $M$ with isolated singularities is a (smooth or continuous) vector field on the set $M\setminus S$ of regular points of $M$. The {\em set of singular points} $S_X$ of a vector field $X$ on a (singular) manifold $M$ is the union of the set of usual singular points of $X$ on $M\setminus S$ (i.e., points at which $X$ tends to zero) and of the set $S$ of singular points of $M$ itself. For an isolated {\em usual} singular point $P$ of a vector field $X$ there is defined its index $\mbox{ind}_PX$ (the degree of the map $X/\Vert X\Vert:\partial B\to S^{n-1}$ of the boundary of a small ball $B$ centred at the point $P$ in a coordinate neighbourhood of $P$; $n=\mbox{dim}\,M$). If the manifold $M$ is closed and has no singularities ($S=\emptyset$) and the vector field $X$ on $M$ has only isolated singularities, then \begin{equation}\label{eq1} \sum_{P\in S_X}\mbox{ind}_PX=\chi(M) \end{equation} ($\chi(M)$ is the Euler characteristic of $M$). Let $(M, P)$ be a cone-like singularity (i.e., a germ of a manifold with such a singular point) and let $X$ be a vector field defined on an open neighbourhood $U$ of the point $P$. Suppose that $X$ has no singular points on $U\setminus\{P\}$. Let $V$ be a closed cone--like neighbourhood of $P$ in $U$ ($V\cong CW_P$, $V\subset U$). On the cone $CW_P=(I\times W_P)/(\{0\}\times W_P)$ ($I=[0, 1]$) there is defined a natural vector field $\partial/\partial t$ ($t$ is the coordinate on $I$). Let $X_{rad}$ be the corresponding vector field on $V$. Let $\widetilde X$ be a smooth vector field on $U$ which coincides with $X$ near the boundary $\partial U$ of the neighbourhood $U$ and with $X_{rad}$ on $V$ and has only isolated singular points. \begin{definition} The {\em index} $\mbox{ind}_PX$ of the vector field $X$ at the point $P$ is equal to $$1+\sum_{Q\in S_{\widetilde X}\setminus\{P\}}\mbox{ind}_Q\widetilde X$$ (the sum is over all singular points $Q$ of $\widetilde X$ except $P$ itself). \end{definition} For a cone-like singularity at a point $P\in S$, the link $W_P$ and thus the cone structure of a neighbourhood are, generally speaking, not well-defined (cones over different manifolds may be {\em locally} diffeomorphic). However it is not difficult to show that the index $\mbox{ind}_PX$ does not depend on the choice of a cone structure on a neighbourhood and on the choice of the vector field $\widetilde X$. \begin{example} The index of the ``radial" vector field $X_{rad}$ is equal to $1$. The index of the vector field $(-X_{rad})$ is equal to $1-\chi(W_P)$ where $W_P$ is the link of the singular point $P$. \end{example} \begin{proposition}\label{prop1} For a vector field $X$ with isolated singular points on a closed manifold $M$ with isolated singularities, the relation {\rm(\ref{eq1})} holds. \end{proposition} \begin{definition} One says that a singular point $P$ of a manifold $M$ (locally diffeomorphic to the cone $CW_P$ over a manifold $W_P$) is {\em smoothable} if $W_P$ is the boundary of a smooth compact manifold. \end{definition} The class of smoothable singularities includes, in particular, the class of (real) isolated complete intersection singularities. For such a singularity, there is a distinguished cone--like structure on its neighbourhood. Let $(M, P)$ be a smoothable singularity (i.e., a germ of a manifold with such a singular point) and let $X$ be a vector field on $(M, P)$ with an isolated singular point at $P$. Let $V=CW_P$ be a closed cone--like neighbourhood of the point $P$; $X$ is supposed to have no singular points on $V\setminus\{P\}$. Let the link $W_P$ of the point $P$ be the boundary of a compact manifold $\widetilde V_P$. Using a smoothing one can consider the union $\widetilde V_P\cup_{W_P}(W_P\times[1/2, 1])$ of $\widetilde V_P$ and $W_P\times[1/2, 1]\subset CW_P$ with the natural identification of $\partial\widetilde V_P=W_P$ with $W_P\times\{1/2\}$ as a smooth manifold (with the boundary $W_P\times\{1\}$). The restriction of the vector field $X$ to $W_P\times[1/2, 1]\subset CW_P$ can be extended to a smooth vector field $\widetilde X$ on $\widetilde V_P\cup_{W_P}(W_P\times[1/2, 1])$ with isolated singular points. \begin{proposition}\label{prop2} The index $\mbox{ind}_PX$ of the vector field $X$ at the point $P$ is equal to $$\sum_{Q\in S_{\widetilde X}}\mbox{ind}_Q\widetilde X - \chi(\widetilde V_P)+1$$ (the sum is over all singular points of $\widetilde X$ on $\widetilde V_P$). \end{proposition} \begin{remark} In \cite{S}, \cite{GSV} there was defined a notion of the index of a vector field at an isolated singular point of a complex variety (satisfying some conditions). That definition does not coincide with the one given here. These definitions differ by the Euler characteristic of the smoothing of the singularity of the variety. One can say that the index of \cite{S}, \cite{GSV} depends on the Euler characteristic of a smoothing and thus is well-defined only for a singularity with well-defined topological type of a smoothing (at least with well-defined Euler characteristic of it). It is valid, e.g., for {\em complex} isolated complete intersection singularities. A closely related notion has been discussed in \cite{BG}. That notion can be considered as a relative version of the index defined here. After a previous version of this paper had been submitted and put on the Duke preprint server as alg-geom/9710008, the authors' attention was drawn to the preprint \cite{ASV}, where a somewhat more general notion is defined, which coincides with the index considered here for real analytic varieties with isolated singularities. \end{remark} A generic (smooth or continuous) vector field on a (singular) analytic variety has zeroes only at isolated points. Thus it is desirable to have a definition of the index of such a point. One can use the following definition. Let $(V,0)\subset ({\Bbb{R}}^N, 0)$ be a germ of a real algebraic variety and let $X$ be a continuous vector field on $(V,0)$ (i.e., the restriction of a continuous vector field on $({\Bbb{R}}^N, 0)$ tangent to $V$ at each point) which has an isolated zero at the origin (in $V$). Let ${\EuScript S}=\{\Xi\}$ be a semianalytic Whitney stratification of $V$ such that its only zero-dimensional stratum $\Xi^0$ consists of the origin. Let $\Xi$ be a stratum of the stratification ${\EuScript S}$ and let $Q$ be a point of $\Xi$. A neighbourhood of the point $Q$ in $V$ is diffeomorphic to the direct product of a linear space ${\Bbb{R}}^k$ (the dimension $k$ of which is equal to the dimension of the stratum $\Xi$) and the cone $CW_Q$ over a compact singular analytic variety $W_Q$. (A diffeomorphism between two stratified spaces is a homeomorphism which is a diffeomorphism on each stratum.) In particular a neighbourhood $U(0)$ of the origin is diffeomorphic to the cone $CW_0$ over a singular variety $W_0$. It is not difficult to show that there exists a (continuous) vector field $\widetilde X$ on $(V, 0)$ such that: \begin{enumerate} \item the vector field $\widetilde X$ is defined on the neighbourhood $U(0)\cong CW_0$ of the origin; \item $\widetilde X$ coincides with the vector field $X$ in a neighbourhood of the base $\{1\}\times W_0$ of the cone $CW_0$; \item the vector field $\widetilde X$ has only a finite number of zeroes; \item each point $Q\in U(0)$ with $\widetilde X(Q)=0$ has a neighbourhood diffeomorphic to $({\Bbb{R}}^k,0)\times CW_Q$ in which $\widetilde X(y,z)$ ($y\in {\Bbb{R}}^k$, $z\in CW_q$) is of the form $Y(y)+Z_{rad}(z)$, where $Y$ is a germ of a vector field on $({\Bbb{R}}^k,0)$ with an isolated singular point at the origin, $Z_{rad}$ is the radial vector field on the cone $CW_Q$. \end{enumerate} Let $S_{\widetilde X}$ be the set of zeroes of the vector field $\widetilde X$ (including the origin). For a point $Q\in S_{\widetilde X}$, let $\widetilde{ind}(Q):=ind_0 Y$, where $Y$ is the vector field on $({\Bbb{R}}^k, 0)$ described above. We define $ind(0)$ to be equal to $1$ (in this case $k=0$). \begin{definition} $ind_{(V,0)}X=\sum\limits_{Q\in S_{\widetilde X}} \widetilde{ind}(Q)$. \end{definition} \section{On the topology of isolated complete intersection singularities}\label{sec2} Let $(V,0) \subset ({\Bbb{C}}^{n+p},0)$ be an $(n-1)$-dimensional isolated complete intersection singularity (abbreviated {\em icis} in the sequel) defined by a germ of an analytic mapping $$f=(f_1, \ldots , f_{p+1}): ({\Bbb{C}}^{n+p},0) \to ({\Bbb{C}}^{p+1},0).$$ (We use somewhat strange notations for the dimension and the number of equations in order to be consistent with the notations in Section~\ref{sec3}.) For $\delta > 0$, let $B_\delta$ be the ball of radius $\delta$ around the origin in ${\Bbb{C}}^{n+p}$. For $\delta > 0$ small enough and for a generic $t \in {\Bbb{C}}^{p+1}$ with $0< \| t \| << \delta$, the set $$V_t = f^{-1}(t) \cap B_\delta$$ is a manifold with boundary and is called a {\em Milnor fibre} of the {\em icis} $(V,0)$ (or of the germ $f$). The diffeomorphism type of $V_t$ does not depend on $t$. The manifold $V_t$ is homotopy equivalent to the bouquet of $\mu$ spheres of dimension $(n-1)$, where $\mu$ is the Milnor number of the {\em icis} $(V,0)$. For $t \in {\Bbb{C}}^{p+1}$, $0 \leq i \leq p$ we define \begin{eqnarray*} (V^{(i)},0) & := & (\{ x \in B_\delta: f_1(x)= \ldots = f_{p-i+1}(x) = 0 \},0), \\ V^{(i)}_t & := & \{ x \in B_\delta: f_j(x)=t_j, 1 \leq j \leq p-i+1 \} \end{eqnarray*} and we set $(V^{(p+1)}, 0) := ({\Bbb{C}}^{n+p}, 0)$. We assume that $(f_1, \ldots , f_{p+1})$ is a system of functions such that for $0\leq i \leq p $ the germ $(V^{(i)},0)$ is an $(n+i-1)$-dimensional {\em icis}. For any $t \in {\Bbb{C}}^{p+1}$ with $0 < | t_1 | << |t_2| << \ldots << |t_{p+1} | << \delta$, the set $V^{(i)}_t$ is the Milnor fibre of the {\em icis} $(V^{(i)},0)$. Here the condition $0 < | t_1 | << |t_2| << \ldots << |t_{p+1} |$ means that $t_1, \ldots ,t_{p+1}$ have to be chosen in such a way that for each $i$, $1 \leq i \leq p+1$, all the critical values of the function $f_i$ on $V^{(p-i+2)}_t$ are contained in the disc of radius $|t_i|$ around $0$. We put \begin{eqnarray*} \hat{H}^{(i)} & := & H_{n+i}(V^{(i+1)}_t,V^{(i)}_t) \quad \mbox{for} \ 0 \leq i \leq p-1 , \\ \hat{H}^{(p)} & := & H_{n+p}(B_\delta, V^{(p)}_t). \end{eqnarray*} We have short exact sequences (cf.\ \cite{Eb}, \cite{AGV}): $$ \begin{array}{ccccccccc} 0 & \rightarrow & H_n(V'_t) & \rightarrow & \hat{H} & \rightarrow & H_{n-1}(V_t) & \rightarrow & 0 \\ 0 & \rightarrow & H_{n+1}(V^{(2)}_t) & \rightarrow & \hat{H}' & \rightarrow & H_n(V'_t) & \rightarrow & 0 \\ & & \vdots & & \vdots & & \vdots & & \\ 0 & \rightarrow & H_{n+p-1}(V^{(p)}_t) & \rightarrow & \hat{H}^{(p-1)} & \rightarrow & H_{n+p-2}(V^{(p-1)}_t) & \rightarrow & 0 \\ & & 0 & \rightarrow & \hat{H}^{(p)} & \rightarrow & H_{n+p-1}(V^{(p)}_t) & \rightarrow & 0 \end{array} $$ They give rise to a long exact sequence (cf.\ \cite[p.~163]{AGV}) $$ 0 \rightarrow \hat{H}^{(p)} \rightarrow \hat{H}^{(p-1)} \rightarrow \ldots \rightarrow \hat{H}' \rightarrow \hat{H} \rightarrow H_{n-1}(V_t) \rightarrow 0 .$$ Each of the modules in this sequence is a free ${\Bbb{Z}}$-module of finite rank. Let $\nu_i:= {\rm rank}\, \hat{H}^{(i)}$. Then $$\mu = {\rm rank}\, H_{n-1}(V_t) = \sum_{i=0}^{p} (-1)^i \nu_i.$$ On each of the modules we have an intersection form $\langle\cdot,\,\cdot\rangle$ defined as in \cite{Eb}. On the module $H_{n+i-1}(V^{(i)}_t)$ ($0\le i\le p$) it is the usual intersection form; on the module $\hat{H}^{(i)}$ it is the pullback of the intersection form by the natural (boundary) homomorphism $\hat{H}^{(i)}\to H_{n+i-1}(V^{(i)}_t)$. The form on $H_{n-1}(V_t)$ is symmetric if $n$ is odd and skew-symmetric if $n$ is even. The form on $\hat{H}^{(i)}$ is symmetric if $n+i$ is odd and skew-symmetric if $n+i$ is even. Denote by $\hat{H}^\ast$ the dual module of $\hat{H}=H_n(V'_t,V_t)$ and let $(\cdot,\,\cdot) : \hat{H}^\ast \times \hat{H} \to {\Bbb{Z}}$ be the Kronecker pairing. We want to define a variation operator ${\rm Var}: \hat{H}^\ast \to \hat{H}$ or rather its inverse ${\rm Var}^{-1} : \hat{H} \to \hat{H}^\ast$. For this purpose we need the notion of a distinguished basis of thimbles. Let $\tilde{f}_{p+1}:V'_t\to {\Bbb{C}}$ be a generic perturbation of the restriction of the function $f_{p+1}$ to $V'_t$ which has only non-degenerate critical points with different critical values $z_1, \ldots , z_\nu$ ($\nu=\nu_0$). Let $z_0$ be a non-critical value of $\tilde{f}_{p+1}$ with $\| z_0 \| > \| z_j\|$ for $j=1, \ldots , \nu$. The level set $\{ x\in V'_t : \tilde{f}_{p+1}(x) = z_0 \}$ is diffeomorphic to the Milnor fibre $V_t$ of the {\em icis} $(V,0)$. Let $u_j$, $j=1, \ldots , \nu$, be non-self-intersecting paths connecting the critical values $z_j$ with the non-critical value $z_0$ in such a way that they lie inside the disc $D_{\| z_0 \|} = \{ z \in {\Bbb{C}} : \| z \| \leq \| z_0 \| \}$ and every two of them intersect each other only at the point $z_0$. We suppose that the paths $u_j$ (and correspondingly the critical values $z_j$) are numbered clockwise according to the order in which they arrive at $z_0$ starting from the boundary of the disc $D_{\| z_0 \|}$. Each path $u_j$ defines up to orientation a thimble $\hat{\delta}_j$ in the relative homology group $\hat{H}$. The system $\{ \hat{\delta}_1 , \ldots , \hat{\delta}_\nu \}$ is a basis of $\hat{H}$. A basis obtained in this way is called {\em distinguished}. The self-intersection number of a thimble $\hat{\delta}$ is equal to $$\langle \hat{\delta}, \hat{\delta} \rangle = (-1)^{n(n-1)/2}(1+(-1)^{n-1}).$$ The {\em Picard-Lefschetz transformation} $h_{\hat{\delta}}:\hat{H} \to \hat{H}$ corresponding to the thimble $\hat{\delta}$ is given by (cf.\ \cite{Eb}) $$h_{\hat{\delta}} (y) = y + (-1)^{n(n+1)/2}\langle y,\hat{\delta} \rangle \hat{\delta} \quad {\rm for} \ y \in \hat{H}.$$ Going once around the disc $D_{\| z_0 \|}$ in the positive direction (counterclockwise) along the boundary induces an automorphism of $\hat{H}$, the {\em (classical) monodromy operator} $h_\ast$. If $\{ \hat{\delta}_1, \ldots, \hat{\delta}_\nu\}$ is a distinguished basis of $\hat{H}$, then the monodromy operator is given by $$h_\ast = h_{\hat{\delta}_1} \circ h_{\hat{\delta}_2} \circ \cdots \circ h_{\hat{\delta}_\nu}.$$ \begin{definition} Let $\{ \hat{\delta}_1, \ldots , \hat{\delta}_\nu \}$ be a distinguished basis of thimbles of $\hat{H}$ and let $\{ \nabla_1, \ldots , \nabla_\nu \}$ be the corresponding dual basis of $\hat{H}^\ast$. The linear operator ${\rm Var}^{-1} : \hat{H} \to \hat{H}^\ast$ (inverse of the {\em variation operator}) is defined by $$ {\rm Var}^{-1}(\hat{\delta}_i) = (-1)^{n(n+1)/2} \nabla_i - \sum_{j < i} \langle \hat{\delta}_i, \hat{\delta}_j \rangle \nabla_j.$$ \end{definition} \begin{proposition} The definition of the operator ${\rm Var}^{-1}$ does not depend on the choice of the distinguished basis. \end{proposition} \begin{proof} Any two distinguished bases of thimbles can be transformed into each other by the braid group transformations $\alpha_j$, $j=1, \ldots , \nu-1$, and by changes of orientations (see, e.g., \cite{AGV}, \cite{Eb}). Here the operation $\alpha_j$ is defined as follows: $$ \alpha_j (\hat{\delta}_1, \ldots , \hat{\delta}_\nu) = (\hat{\delta}'_1, \ldots , \hat{\delta}'_\nu)$$ where $\hat{\delta}'_j = h_{\hat{\delta}_j}(\hat{\delta}_{j+1})= \hat{\delta}_{j+1} + (-1)^{n(n+1)/2}\langle \hat{\delta}_{j+1},\hat{\delta}_{j} \rangle \hat{\delta}_j$, $\hat{\delta}'_{j+1} = \hat{\delta}_j$, and $\hat{\delta}'_i = \hat{\delta}_i$ for $i \neq j, j+1$. It is easily seen that the definition of ${\rm Var}^{-1}$ is invariant under a change of orientation. Therefore it suffices to show that the definition of ${\rm Var}^{-1}$ is invariant under the transformation $\alpha_j$. One easily computes: \begin{eqnarray*} \langle \hat{\delta}'_r ,\hat{\delta}'_s \rangle & = & \langle \hat{\delta}_r, \hat{\delta}_s \rangle \quad \mbox{for} \ 1 \leq r,s \leq \nu, \ r,s \neq j,j+1, \\ \langle \hat{\delta}'_j ,\hat{\delta}'_{j+1} \rangle & = & -\langle \hat{\delta}_j, \hat{\delta}_{j+1} \rangle , \\ \langle \hat{\delta}'_r ,\hat{\delta}'_j \rangle & = & \langle \hat{\delta}_r, \hat{\delta}_{j+1} \rangle +(-1)^{n(n+1)/2}\langle \hat{\delta}_{j+1}, \hat{\delta}_j \rangle \langle \hat{\delta}_r, \hat{\delta}_j \rangle \ \mbox{ for} \ r \neq j,j+1, \\ \langle \hat{\delta}'_r ,\hat{\delta}'_{j+1} \rangle & = & \langle \hat{\delta}_r, \hat{\delta}_j \rangle \quad \mbox{for} \ r \neq j,j+1. \end{eqnarray*} Let $(\nabla'_1, \ldots , \nabla'_\nu)$ be the dual basis corresponding to $(\hat{\delta}'_1, \ldots ,\hat{\delta}'_\nu)$. Then \begin{eqnarray*} \nabla_{j+1} & = & \nabla'_j \\ \nabla_{j} & = & \nabla'_{j+1} + (-1)^{n(n+1)/2} \langle \hat{\delta}_{j+1} , \hat{\delta}_j \rangle \nabla'_{j}. \end{eqnarray*} One has \begin{eqnarray*} {\rm Var}^{-1}(\hat{\delta}'_j) & = & {\rm Var}^{-1}(\hat{\delta}_{j+1}) + (-1)^{n(n+1)/2} \langle \hat{\delta}_{j+1} , \hat{\delta}_j \rangle {\rm Var}^{-1}(\hat{\delta}_j) \\ & = & (-1)^{n(n+1)/2} \nabla_{j+1} - \sum_{k<j+1} \langle \hat{\delta}_{j+1} , \hat{\delta}_k \rangle \nabla_k \\ & & + (-1)^{n(n+1)/2}\langle \hat{\delta}_{j+1} , \hat{\delta}_j \rangle ((-1)^{n(n+1)/2}\nabla_j - \sum_{k<j} \langle \hat{\delta}_j , \hat{\delta}_k \rangle \nabla_k ) \\ & = & (-1)^{n(n+1)/2}\nabla_{j+1} \\ & & - \sum_{k<j} (\langle \hat{\delta}_{j+1} , \hat{\delta}_k \rangle + (-1)^{n(n+1)/2} \langle \hat{\delta}_{j+1} , \hat{\delta}_j \rangle \langle \hat{\delta}_j , \hat{\delta}_k \rangle ) \nabla_k \\ & = & (-1)^{n(n+1)/2}\nabla'_j - \sum_{k<j} \langle \hat{\delta}'_j ,\hat{\delta}'_k \rangle \nabla'_k, \end{eqnarray*} \begin{eqnarray*} {\rm Var}^{-1}(\hat{\delta}'_{j+1}) & = & (-1)^{n(n+1)/2}\nabla_j - \sum_{k<j} \langle \hat{\delta}_j , \hat{\delta}_k \rangle \nabla_k \\ & = & (-1)^{n(n+1)/2}(\nabla_j - (-1)^{n(n+1)/2}\langle \hat{\delta}_{j+1} , \hat{\delta}_j \rangle \nabla_{j+1} ) \\ & & + \langle \hat{\delta}_{j+1} , \hat{\delta}_j \rangle \nabla_{j+1} - \sum_{k<j} \langle \hat{\delta}_j , \hat{\delta}_k \rangle \nabla_k \\ & = & (-1)^{n(n+1)/2}\nabla'_{j+1} - \sum_{k<j+1} \langle \hat{\delta}'_{j+1} , \hat{\delta}'_k \rangle \nabla'_k . \end{eqnarray*} For $i>j+1$ we have \begin{eqnarray*} {\rm Var}^{-1}(\hat{\delta}'_i) & = & (-1)^{n(n+1)/2}\nabla_i - \sum_{k<i} \langle \hat{\delta}_i , \hat{\delta}_k \rangle \nabla_k \\ & = & (-1)^{n(n+1)/2}\nabla'_i - \hspace{-1.5mm} \sum\limits_{\substack{k<i \\ k \neq j,j+1}} \langle \hat{\delta}'_i , \hat{\delta}'_k \rangle \nabla'_k -\langle \hat{\delta}_i , \hat{\delta}_{j+1} \rangle \nabla_{j+1} - \langle \hat{\delta}_i, \hat{\delta}_j \rangle \nabla_j \\ & = & (-1)^{n(n+1)/2}\nabla'_i - \hspace{-1.5mm} \sum\limits_{\substack{k<i \\ k \neq j,j+1}} \! \! \langle \hat{\delta}'_i , \hat{\delta}'_k \rangle \nabla'_k \\ & & - \langle \hat{\delta}'_i, \hat{\delta}'_j \rangle \nabla'_j + (-1)^{n(n+1)/2}\langle \hat{\delta}_{j+1} , \hat{\delta}_j \rangle \langle \hat{\delta}_i , \hat{\delta}_j \rangle \nabla'_j \\ & & - \langle \hat{\delta}'_i , \hat{\delta}'_{j+1} \rangle \nabla'_{j+1} - (-1)^{n(n+1)/2}\langle \hat{\delta}_{j+1} , \hat{\delta}_j \rangle \langle \hat{\delta}'_i , \hat{\delta}'_{j+1} \rangle \nabla'_j \\ & = & (-1)^{n(n+1)/2}\nabla'_i - \sum_{k<i} \langle \hat{\delta}'_i , \hat{\delta}'_k \rangle \nabla'_k \end{eqnarray*} The corresponding formula for $i<j$ is obvious. \end{proof} \begin{remark} There is an interesting problem to give an invariant (topological) definition of the variation operator. \end{remark} Let $S: \hat{H} \to \hat{H}^\ast$ be the mapping defined by the intersection form on $\hat{H}$: $(Sx, y) =\langle x , y \rangle$, $x,\,y\in \hat{H}$. The mapping ${\rm Var}^{-1}$ is defined in such a way that one has the equality $S = -{\rm Var}^{-1} + (-1)^n ({\rm Var}^{-1})^T$ where $({\rm Var}^{-1})^T$ means the transpose operator $({\rm Var}^{-1})^T : \hat{H}^{\ast\ast}= \hat{H} \to \hat{H}^\ast$. \begin{remark} We emphasize that the intersection number $\langle \hat{\delta}_i , \hat{\delta}_j \rangle$ is the entry of the matrix of the operator $S$ with {\em column} index $i$ and {\em row} index $j$: see the remark in \cite[p.~45]{AGV}. \end{remark} \begin{remark} The operator $V: \hat{H}\to \hat{H}^\ast$ defined in \cite[p.~18]{Eb} differs from ${\rm Var}^{-1}$ by sign. \end{remark} \begin{proposition} The monodromy operator $h_\ast$ and the operator ${\rm Var}^{-1}$ are related by the following formula: $$h_\ast = (-1)^n {\rm Var} ({\rm Var}^{-1})^T. $$ \end{proposition} \begin{proof} This can be computed directly; see also \cite[Chap.~V, \S 6, Exercice 3]{B}. \end{proof} In the same way, for each $i$ with $1 \leq i \leq p$ an operator ${\rm Var}^{-1}_i : \hat{H}^{(i)} \to (\hat{H}^{(i)})^\ast$ is defined. \section{The index of the gradient vector field on an isolated complete intersection singularity}\label{sec3} Let $(V', 0)=\{f_1=f_2=\ldots=f_p=0\}\subset({\Bbb{C}}^{n+p}, 0)$ be a real $n$-dimensional {\em icis} (it means that the function germs $f_i:({\Bbb{C}}^{n+p}, 0)\to({\Bbb{C}}, 0)$ are real). We assume that its real part $V'\cap{\Bbb{R}}^{n+p}$ does not coincide with the origin (and thus is $n$-dimensional). Let $g=f_{p+1}:({\Bbb{C}}^{n+p}, 0)\to({\Bbb{C}}, 0)$ be a germ of a real analytic function such that its restriction to $V'\setminus\{0\}$ has no critical points. A Riemannian metric on ${\Bbb{R}}^{n+p}$ determines the gradient vector field $X=\mbox{grad}\,g$ of the restriction of the function $g$ to $(V'\cap{\Bbb{R}}^{n+p})\setminus\{0\}$. This vector field has no singular points on a punctured neighbourhood of the origin in $V'\cap{\Bbb{R}}^{n+p}$. Since the space of Riemannian metrics is connected, the index \,$\mbox{ind}_0\,X$ of the gradient vector field doesn't depend on the choice of a metric. In the case $p=0$ (and thus $V'={\Bbb{C}}^{n}$) the index of the gradient vector field of a function germ $g$ can be expressed in terms of the action of the complex conjugation on the Milnor lattice of the singularity $g$ (\cite{GZ}, \cite{V}). We give a generalisation of such a formula for {\em icis}. Let $0<\varepsilon_1\ll\varepsilon_2\ll\ldots\ll\varepsilon_{p+1}$ be real and small enough, let $s=(s_1,\,\ldots,\, s_{p+1})$ with $s_i=\pm 1$. For $0\le i\le p$, let $\hat{H}^{(i)}=\hat{H}^{(i)}_{s\varepsilon}$ be the corresponding space of thimbles: $\hat{H}^{(i)}=H_{n+i}(V^{(i+1)}_{s\varepsilon}, V^{(i)}_{s\varepsilon})$ for $0\le i\le p-1$ ($s\varepsilon=(s_1\varepsilon_1, s_2\varepsilon_2, \ldots, s_{p+1}\varepsilon_{p+1}))$; see Section~\ref{sec2} for $i=p$. Let $\sigma^{(i)}_s$ be the action of the complex conjugation on the space $\hat{H}^{(i)}$, let ${\rm Var}^{-1}_i:\hat{H}^{(i)}\to (\hat{H}^{(i)})^\ast$ be the inverse of the corresponding variation operator. The operator ${\rm Var}^{-1}_i\sigma^{(i)}_s$ acts from the space $\hat{H}^{(i)}$ to its dual $(\hat{H}^{(i)})^\ast$ and thus defines a bilinear form on $\hat{H}^{(i)}$. \begin{theorem}\label{theo1} The bilinear forms ${\rm Var}^{-1}_i\sigma^{(i)}_s$ are symmetric and non-degener\-ate, and we have \begin{eqnarray}\label{eq2} {\rm ind}_0\,{\rm grad}\,g & = & s_{p+1}^{n}(-1)^{\frac{n(n+1)}{2}} {\rm sgn}\,{\rm Var}^{-1}_{}\sigma^{}_s \nonumber \\ & & +\sum\limits_{i=1}^p(-1)^{\frac{(n+i)(n+i+1)}{2}}{\rm sgn}\,{\rm Var}^{-1}_{i}\sigma^{(i)}_s. \end{eqnarray} \end{theorem} \begin{corollary} The right-hand side of the equation {\rm (\ref{eq2})} does not depend on $s=(s_1,\,\ldots,\, s_{p+1})$. \end{corollary} \begin{proof} Let us consider the restriction of the function $f_i$ to the manifold $V^{(p-i+2)}_{s\varepsilon}$. It may have degenerate critical points. Let $\widetilde f_i:V^{(p-i+2)}_{s\varepsilon}\to{\Bbb{C}}$ be its real morsification (i.e., a perturbation of $f_i$ which is a Morse function on $V^{(p-i+2)}_{s\varepsilon}$ and maps its real part $V^{(p-i+2)}_{s\varepsilon}\cap{\Bbb{R}}^{n+p}$ to ${\Bbb{R}}$). For $c\in{\Bbb{R}}$, let $M_c^{(i)}=\{x\in V^{(p-i+2)}_{s\varepsilon}\cap{\Bbb{R}}^{n+p}:\widetilde f_i\le c\}$. The topological space $M_c^{(i)}$ is homotopy equivalent to $V^{(p-i+2)}_{s\varepsilon}\cap{\Bbb{R}}^{n+p}$ or to $V^{(p-i+1)}_{(s_1\varepsilon_1,\ldots, s_{i-1}\varepsilon_{i-1}, -\varepsilon_i)}\cap{\Bbb{R}}^{n+p}$ for $c$ greater than or less than all the critical values of $\widetilde f_i$ respectively. The standard arguments of Morse theory give $$ \chi(V^{(p-i+2)}_{s\varepsilon}\cap{\Bbb{R}}^{n+p})=\chi(V^{(p-i+1)}_{(s_1\varepsilon_1,\ldots, s_{i-1}\varepsilon_{i-1}, -\varepsilon_i)}\cap{\Bbb{R}}^{n+p}) + \sum\limits_{Q\in S_{{\rm grad}\,\widetilde f_i}}{\rm ind}_Q\,{\rm grad}\,\widetilde f_i. $$ Applying the same reasonings to the function $-\widetilde f_i$ one has \begin{eqnarray*} \chi(V^{(p-i+2)}_{s\varepsilon}\cap{\Bbb{R}}^{n+p}) & = & \chi(V^{(p-i+1)}_{(s_1\varepsilon_1,\ldots, s_{i-1}\varepsilon_{i-1}, \varepsilon_i)}\cap{\Bbb{R}}^{n+p}) \\ & & + (-1)^{n+p-i+1}\sum\limits_{Q\in S_{{\rm grad}\,\widetilde f_i}}{\rm ind}_Q\,{\rm grad}\,\widetilde f_i. \end{eqnarray*} Thus $$ \chi(V^{(p-i+2)}_{s\varepsilon}\cap{\Bbb{R}}^{n+p})=\chi(V^{(p-i+1)}_{s\varepsilon}\cap{\Bbb{R}}^{n+p}) + (-s_i)^{n+p-i+1} \sum\limits_{Q\in S_{{\rm grad}\,\widetilde f_i}}{\rm ind}_Q\,{\rm grad}\,\widetilde f_i. $$ (cf.\ \cite[Lemma in \S 2]{A}). From Proposition 1 one has \begin{eqnarray*} \lefteqn{{\rm ind}_{0}{\rm grad}\, g} \\ & & =\hspace{-3mm}\sum\limits_{Q\in S_{{\rm grad}\,\widetilde f_{p+1}}}\hspace{-3mm}{\rm ind}_Q{\rm grad}\,\widetilde f_{p+1} - \chi(V'_{s\varepsilon}\cap{\Bbb{R}}^{n+p}) + 1 \\ & & =\hspace{-3mm}\sum\limits_{ Q\in S_{{\rm grad}\,\widetilde f_{p+1}}}\hspace{-4.5mm}{\rm ind}_Q{\rm grad}\,\widetilde f_{p+1} + (-s_p)^{(n+1)}\hspace{-3mm}\sum\limits_{Q\in S_{{\rm grad}\,\widetilde f_{p}}}\hspace{-3.5mm}{\rm ind}_Q{\rm grad}\,\widetilde f_{p} - \chi(V''_{s\varepsilon}\cap{\Bbb{R}}^{n+p}) + 1 \\ & & = \ldots =\\ & & = \hspace{-3mm}\sum\limits_{Q\in S_{{\rm grad}\,\widetilde f_{p+1}}} \hspace{-3mm}{\rm ind}_Q{\rm grad}\,\widetilde f_{p+1} + \sum\limits_{i=1}^p(-s_{p-i+1})^{(n+i)}\hspace{-3mm}\sum\limits_{Q\in S_{{\rm grad}\,\widetilde f_{p-i+1}}}\hspace{-3mm}{\rm ind}_Q{\rm grad}\,\widetilde f_{p-i+1}. \end{eqnarray*} Now Theorem~\ref{theo1} follows from the following statement. \end{proof} \begin{theorem}\label{theo2} $$ \sum\limits_{Q\in S_{{\rm grad}\,\widetilde f_{p-i+1}}} {\rm ind}_Q{\rm grad}\,\widetilde f_{p-i+1} =(s_{p-i+1})^{n+i}(-1)^{\frac{(n+i)(n+i+1)}{2}}{\rm sgn}\,{\rm Var}^{-1}_i\sigma^{(i)}_s. $$ \end{theorem} \begin{proof} Let us suppose that $s_{p-i+1} =1$. The case $s_{p-i+1}=-1$ can be reduced to this by multiplying the function $f_{p-i+1}$ (and the function $\widetilde{f}_{p-i+1}$) by $(-1)$. Let $\widetilde{s} = (s_1, \ldots , s_{p-i}, -1, s_{p-i+2}, \ldots, s_{p+1})$. Without loss of generality we can suppose that all critical values of the function $\widetilde{f}_{p-i+1}$ lie inside the circle $\{z: \| z \|\le\frac{\varepsilon_{p-i+1}}{2} \}$ and have different real parts (except, of course, values at complex conjugate points). Let us identify the space $\hat{H}^{(i)}_{\widetilde{s}}$ with the space $\hat{H}^{(i)}_s$ using a path which connects $-\varepsilon_{p-i+1}$ with $+\varepsilon_{p-i+1}$ in the upper half plane outside the circle $\{z: \| z\| < \frac{\varepsilon_{p-i+1}}{2} \}$ (e.g., the half circle $\{z: \| z\| = \varepsilon_{p-i+1}\}$). This identification permits to consider $\sigma_s^{(i)}$ and $\sigma_{\widetilde{s}}^{(i)}$ as operators on the space $\hat{H}^{(i)} =\hat{H}^{(i)}_{s\varepsilon}$. Just as in \cite{GZ} the classical monodromy operator $h_{\ast}^{(i)}: \hat{H}^{(i)} \to \hat{H}^{(i)}$ can be represented in the form \begin{equation}\label{eq3} h_{\ast}^{(i)} = \sigma^{(i)}_s \sigma^{(i)}_{\widetilde{s}}. \end{equation} A distinguished basis of the space $\hat{H}^{(i)}$ is defined by a system of paths connecting the critical values of the function $\widetilde{f}_{p-i+1}$ with the non-critical value $\varepsilon_{p-i+1}$. Let us choose the following system of paths (cf.\ Fig.~\ref{fig1}). \begin{figure} \vspace{2cm} \centering \unitlength1cm \begin{picture}(9,5) \put(0,0){\includegraphics{Fig1.eps}} \put(0.7,2.7){$-\varepsilon_{p-i+1}$} \put(8.4,2.7){$\varepsilon_{p-i+1}$} \end{picture} \caption{The choice of paths}\label{fig1} \end{figure} The paths from real critical values go vertically upwards up to the boundary of the circle $\{ z: \| z \| \leq \frac{\varepsilon_{p-i+1}}{2} \}$. The paths from complex conjugate critical values go vertically (upwards or downwards to the real axis and then go vertically upwards to the boundary of the circle $\{ z: \| z \| \leq \frac{\varepsilon_{p-i+1}}{2} \}$ avoiding from the right side a neighbourhood of the critical value with positive imaginary part. From the boundary of the circle $\{ z: \| z \| \leq \frac{\varepsilon_{p-i+1}}{2} \}$ all the paths go to the non-critical value $\varepsilon_{p-i+1}$ in the upper half plane (see Fig.~\ref{fig1}). The cycles are ordered in the usual way which in this case means that they follow each other in the order of decreasing real parts of the corresponding critical values; the vanishing cycle corresponding to the critical value with negative imaginary part precedes that with the positive one. In the sequel we shall consider the matrices of the operators $\sigma^{(i)}_s$, $\sigma^{(i)}_{\widetilde{s}}$, ${\rm Var}^{-1}_i$, etc.\ as block matrices with blocks of size $1 \times 1$, $1 \times 2$, $2 \times 1$, and $2 \times 2$ corresponding to real critical values and to pairs of complex conjugate critical values of the function $\widetilde{f}_{p-i+1}$. The matrix of the operator $\sigma_s^{(i)}$ is an upper triangular block matrix. Its diagonal entry corresponding to a real critical value is equal to $(-1)^m$ where $m$ is the Morse index of the critical point. A diagonal block of size $2 \times 2$ corresponding to a pair of complex conjugate critical values is equal to $\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$. The matrix of the operator $\sigma_{\widetilde{s}}^{(i)}$ is a lower triangular block matrix (we do not need a precise description of its diagonal blocks). The matrix of the operator ${\rm Var}^{-1}_i$ is upper triangular with diagonal entries equal to $(-1)^{(n+i)(n+i+1)/2}$ (the dual of the space $\hat{H}^{(i)}$ is endowed with the basis dual to the one of $\hat{H}^{(i)}$). One has $h_{\ast}^{(i)} = \sigma_s^{(i)} \sigma_{\widetilde{s}}^{(i)} = (-1)^{n+i} {\rm Var}_i ({\rm Var}^{-1}_i)^T$. Thus ${\rm Var}^{-1}_i \sigma_s^{(i)} = (-1)^{n+i} ({\rm Var}^{-1}_i)^T \sigma_{\widetilde{s}}^{(i)}$. The matrices ${\rm Var}^{-1}_i\sigma_s^{(i)}$ and $({\rm Var}^{-1}_i)^T \sigma_{\widetilde{s}}^{(i)}$ are upper triangular and lower triangular respectively. Thus the matrix ${\rm Var}^{-1}_i \sigma_s^{(i)}$ is in fact block diagonal with the diagonal entry $(-1)^{((n+i)(n+i+1)/2) +m}$ corresponding to a real critical point of the function $\widetilde{f}_{p-i+1}$ ($m$ is the Morse index) and with the diagonal block of the form $$(-1)^{\frac{(n+i)(n+i+1)}{2}} \left( \begin{array}{cc} a & 1 \\ 1 & 0 \end{array} \right) $$ corresponding to a pair of complex conjugate critical points (up to a sign $a$ is the intersection number of the corresponding cycles). This description implies Theorem~\ref{theo2}. \end{proof} The formula~(\ref{eq2}) expresses the index of a gradient vector field in terms of bilinear forms on the spaces of thimbles. Actually each second summand of it can be expressed in terms of bilinear forms on the corresponding spaces of vanishing cycles. Let $\Sigma_s^{(i)}$ be the quadratic form on the space $H_{n+i-1}(V_{s\varepsilon}^{(i)})$ of vanishing cycles defined by $\Sigma_s^{(i)}(x,y)=\langle\sigma_s^{(i)}x,y\rangle$. As above, let $\widetilde{s} = (s_1,\linebreak[0] \ldots,\linebreak[0] s_{p-i},\linebreak[0] -1,\linebreak[0] s_{p-i+2}, \ldots, s_{p+1})$. \begin{theorem}\label{theo3} For $n+i$ odd $$ \sum\limits_{Q\in S_{{\rm grad}\,\widetilde f_{p-i+1}}} {\rm ind}_Q{\rm grad}\,\widetilde f_{p-i+1} =s_{p-i+1}(-1)^{\frac{n+i+1}{2}}({\rm sgn}\,\Sigma_{\widetilde s}^{(i)}-{\rm sgn}\,\Sigma_s^{(i)})/2. $$ \end{theorem} \begin{proof} It is essentially the same as in \cite{GZ}. One has to notice that the kernel of the natural (boundary) homomorphism $\hat{H}^{(i)}_{s\varepsilon}\to H_{n+i-1}(V_{s\varepsilon}^{(i)})$ is contained in the kernel of the quadratic form $\Sigma_s^{(i)}$ and thus \ ${\rm sgn}\,\Sigma_{\widetilde s}^{(i)}$ coincides with the signature of the form $\langle\sigma_s^{(i)}\cdot,\cdot\rangle$ on the space $\hat{H}^{(i)}_{s\varepsilon}$. \end{proof} \begin{remark} For $n+i$ even, it is not possible to express the number $$\sum\limits_{Q\in S_{{\rm grad}\,\widetilde f_{p-i+1}}} {\rm ind}_Q{\rm grad}\,\widetilde f_{p-i+1}$$ in terms of invariants defined by the space of vanishing cycles. It can be understood from the following example. Let $n=2$, $p=1$, $f_1=x_1^2+x_2^2-x_3^2$, $f_2=x_3$, $i=0$. The discussed sum is different for $s_1=1$ and for $s_1=-1$ (i.e., for $t_1=s_1\varepsilon_1$ positive or negative). However the line $\ell=\{t_1=0\}$ is not in the bifurcation set for vanishing cycles: it doesn't lie in the discriminant of the map $(f_1, f_2):({\Bbb{C}}^3, 0)\to({\Bbb{C}}^2, 0)$. On the other hand the discriminant of the map $f_1:({\Bbb{C}}^3, 0)\to({\Bbb{C}}, 0)$ coincides with the origin $0\in{\Bbb{C}}$ and thus the line $\ell$ is in the bifurcation set for thimbles. \end{remark} \begin{remark} It seems that a relation between complex conjugation and the monodromy operator (similar to (\ref{eq3})) was first used in \cite{A'C}. In \cite{GZ} it was written in an explicit way. In \cite{MP} a similar relation was used to find some properties of the Euler characteristics of links of complete intersection varieties. This paper partially inspired our work. \end{remark}

9710

alg-geom/9710009

en

https://arxiv.org/abs/alg-geom/9710009

alg-geom/9710009

Sandra DiRocco

Gian Mario Besana, Sandra Di Rocco

On The Projective Normality of Smooth Surfaces of degree nine

22 pages, AmsLatex, see home pages http://www.emunix.emich.edu/~gbesana/ and http://www.math.kth.se/~sandra/Welcome

We investigate the projective normality of smooth, linearly normal surfaces of degree 9. All non projectively normal surfaces which are not scrolls over a curve are classified. Results on the projective normality of surface scrolls are also given. One of the reasons that brought us to look at this question is our desire to find examples for a long standing problem in adjunction theory. Andreatta followed by a generalization by Ein and Lazarsfeld posed the problem of classifying smooth n-dimensional varieties (X,L) polarized with a very ample line bundle L, such that the adjoint linear system |H| = |K + (n-1)L| gives an embedding which is not projectively normal. After a detailed check of the non projectively normal surfaces found in this work no examples were found except possibly a blow up of an elliptic P^1-bundle whose existence is uncertain.

alg-geom

\section{introduction} Smooth projective varieties of small degree have been classified over the years and thoroughly studied, e.g. \cite{Io1}, \cite{Io2}, \cite{ok2}, \cite{ok8}, \cite {Ale}, \cite{ADS}. A variety $X~\subset~\Pin{n}$ is {\it projectively normal} if the maps $H^0(\Pin{n},\oofp{n}{k}) \to H^0(X, \oof{X}{k})$ are surjective for all $k \ge 1$ or in other words if hypersurfaces of degree $k$ cut complete linear systems on $X$ for every $k\ge 1.$ In \cite{Alibaba} the projective normality of varieties of degree $d \le 8$ of any dimension was investigated. This work is concerned with the projective normality of smooth projective surfaces, embedded by the complete linear system associated with a very ample line bundle $L$ of degree $d=9.$ Such surfaces are either embedded in $\Pin{4}$ or have sectional genus $g \le 7.$ Therefore they are completely classified in \cite{au-ra} and \cite{LiAq}. One of the reasons that brought us to look at this question is our desire to find examples for a long standing problem in adjunction theory. Andreatta \cite{ce}, followed by a generalization by Ein and Lazarsfeld \cite{EL}, posed the problem of classifying smooth $n$-dimensional varieties $(X, \cal{L})$ polarized with a very ample line bundle $\cal{L}$, such that the adjoint linear system $|L| = |K + (n-1) \cal{L}|$ gives an embedding which is not projectively normal. Andreatta and Ballico \cite{an-ba1} gave examples of surfaces $(S, \cal{L})$ with the above behavior, where $d=\deg{S} = 10$ under the adjoint embedding. Alzati, Bertolini and the first author in \cite{Alibaba} found no example with $d\le 8.$ After a detailed check of the non projectively normal surfaces found in this work no examples were found except possibly a blow up of an elliptic $\Pin{1}$-bundle whose existence is uncertain. See section \ref{K+L} for details. Our findings concerning the projective normality of surfaces of degree nine are collected in the the following theorem (see \brref{notation} for notation): \begin{theo} \label{thetheorem} Let $S$ be a smooth surface embedded by the complete linear system associated with a very ample line bundle $L$ as a surface of degree $9$ and sectional genus $g$ in $\Pin{N}. $ Assume $(S, L)$ is not a scroll over a curve. Then $(S, L)$ fails to be projectively normal if and only if it belongs to the following list: \begin{center} \begin{tabular}{|c|c|c|c|} \hline $\Pin{N} $ & $g$& $S$& $L$\\ \hline\hline $\Pin{5}$ & $4$ & $Bl_3X$ where $X$ is a $\Pin{1}$-bundle & \\ & & over an elliptic curve, $e=0$& $2\frak{C_0}+3\frak{f}-\sum_iE_i$ \\ \hline $\Pin{5}$ & $5$ & Rational conic bundle $S=Bl_{15} \bold{F_e}$ , $0\le e \le 5$ & $2 \frak{C_0} + (6+e)\frak{f} - \sum_iE_i $ \\ \hline $\Pin{4}$ & $6$ &$Bl_{10}(\Pin{2})$&$3p^*(\oofp{2}{1})-\sum_i4E_i$ \\ \hline $\Pin{4}$ & $7$ &$Bl_{15}(\Pin{2})$&$9p^*(\oofp{2}{1})-\sum_1^63E_i-$\\ & & &$\sum_7^92E_j-\sum_{10}^{15}E_k $\\ \hline $\Pin{4}$ & $6$ & Projection of an Enriques surface & \\ & & of degree 10 in ${\bf P}^5$& cf. \cite{au-ra} \\ \hline $\Pin{4}$ & $7$ & Minimal elliptic surface & cf. \cite{au-ra} \\ \hline $\Pin{4}$ & $8$ & Minimal surface of general type& cf. \cite{au-ra}\\ \hline \end{tabular} \end{center} \end{theo} The projective normality of surfaces which are scrolls over a curve of genus $g,$ not included in the above theorem, was also investigated. Results are collected in Proposition \ref{scrollprop}. We were not able to prove or disprove the projective normality of scrolls over trigonal curves of genus $3,4,5.$ The authors would like to thank Ciro Ciliberto, Antonio Lanteri and Andrew J. Sommese for many helpful conversations and W. Chach\'{o}lski for his insight and patience. \section{Background material} \subsection{NOTATION} \label{notation} Throughout this article $S$ denotes a smooth connected projective surface defined over the complex field {\bf C}. Its structure sheaf is denoted by ${\cal O_S}$ and the canonical sheaf of holomorphic $2$-forms on $S$ is denoted by $K_S$. For any coherent sheaf $\Im$ on $S$, $h^i(\Im )$ is the complex dimension of $H^i(S,\Im)$ and $\chi=\chi(\cal{O}_S)=\chi(S)=\sum_i(-1)^ih^i(\cal{O}_S).$ Let $L$ be a line bundle on $S.$ If $L$ is ample the pair $(S, L)$ is called a {\it polarized surface}. The following notation is used:\\ $|L|$, the complete linear system associated with L;\\ $d = L^2,$ the degree of $L$;\\ $g=g(S, L)$, the {\it sectional genus} of $(S, L)$, defined by $2g-2=L\cdot (K_S+L).$ If $C\in |L|$ is an irreducible and reduced element then $g=g(C)$ is the arithmetic genus of $C$;\\ $\Delta (S, L) = \Delta = 2+L^2-h^0(L)$ the Delta genus of $(S, L)$;\\ $\bold{F_e}$, the Hirzebruch surface of invariant $e$ ;\\ $E^*$ the dual of a vector bundle $E$.\\ Cartier divisors, their associated line bundles and the invertible sheaves of their holomorphic sections are used with no distinction. Given two divisors $L$ and $M$ we denote linear equivalence by $L \sim M$ and numerical equivalence by $L \equiv M.$ The blow up of a surface $X$ at $n$ points is denoted by $p: S=Bl_nX \to X.$ When $X$ is a $\Pin{1}$-bundle over a curve with fundamental section $C_0$ and generic fibre $f$ it is $Num(X) ={\Bbb Z}[C_0] \oplus {\Bbb Z}[f]$ and the following shorthand is used: $\frak{C_0}=p^*(C_0)$ and $\frak{f}= p^*(f).$ A polarized surface $(S, L)$ is a {\em scroll} or a {\em conic bundle} over a curve $C$ if there exists a surjective morphism $p: S \to C$ with connected fibers and an ample line bundle $H$ on $C$ such that, respectively, $ K_S + 2L = p^*(H)$ or $ K_S + L = p^*(H).$ If $(S, L)$ is a scroll then $S$ is a $\Pin{1}$-bundle over $C$ and $L \cdot f = 1$ for every fibre $f.$ In section \ref{genere6} the notion of {\it reduction} of a smooth polarized surface is shortly used. The best reference is \cite{BESO}. \subsection{\small CASTELNUOVO BOUND} Let $C\subset \Pin{N}$, then by Castelnuovo's lemma \begin{equation} \label{cast} g(C)\leq\left[ \frac{d-2}{N-2}\right ](d-N+1- (\left[ \frac{d-2}{N-2}\right ]-1)\frac{N-2}{2}) \end{equation} where$[x]$ denotes the greatest integer $\leq x$. \subsection{\small PROJECTIVE NORMALITY} Let $S$ be a surface embedded in $\Pin{N}.$ $S$ is said to be {\it k-normal} if the map $$H^0({\cal O}_{\Pin{N}}(k))\longrightarrow H^0({\cal O}_S(k))$$ is surjective . $S$ is said to be {\it projectively normal} if it is $ k$-normal for every $k\geq 1$. An ample line bundle $L$ on $S$ is {\it normally generated} if $S^kH^0(L)\to H^0(L^k)$ is surjective for every $k\geq 1$. If $L$ is normally generated then it is very ample and $S$, embedded in $\Pin{N}$ via $|L|$ is projectively normal. A polarized surface $(S, L)$ is said to be projectively normal if $L$ is very ample and $S$ is projectively normal under the embedding given by $L.$ A polarized surface $(S, L)$ has a {\it ladder} if there exists an irreducible and reduced element $C \in |L|.$ The ladder is said to be {\it regular} if $H^0(S, L)\to H^0(C, \restrict{L}{C})$ is onto. If $L$ is very ample $(S, L)$ clearly has a ladder. We recall the following general result due to Fujita: \begin{theo}[\cite{fu}] \label{fujitatheo} Let $(S, L)$ be a polarized surface with a ladder. Assume $g(L)\geq\Delta$. Then \\ i) The ladder is regular if $d\geq 2\Delta -1$;\\ ii) L is normally generated, $g=\Delta$ and $H^1(S, tL)= 0$ for any t, if $d\geq 2\Delta +1$. \end{theo} \subsection{\small $k$ - REGULARITY} \label{kreg} A good reference is \cite{mu1}. A coherent sheaf ${\cal F}$ over $\Pin{n}$ is {\em k-regular} if $h^i({\cal F}(k-i))=0$ for all $i >0.$ If ${\cal F}$ is $k$-regular then it is $k+1$-regular. If $X\subset \Pin{n} $ is an irreducible variety such that ${\cal I}(X)$ is $k$-regular then the homogeneous ideal $I_X=\oplus H^0(\iof{X}{t})$ is generated in degree $\le k.$ This fact implies that if ${\cal I}_X$ is $k$-regular then $X$ cannot be embedded with a $t\ge( k+1)$- secant line. \subsection{\small CLIFFORD INDEX} Good references are \cite{mart}, \cite{GL}. Let $C$ be a projective curve and $H$ be any line bundle on $C$. The Clifford index of $H$ is defined as follows: $$cl(H)=d-2(h^0(H)-1).$$ The Clifford index of the curve is $cl(C)=\text {min}\{cl(H) | h^0(H)\geq 2 \text{ and }h^1(H)\geq 2 \}$. $ H$ {\it contributes }to the Clifford index of $C$ if $h^0(H)\geq 2 \text{ and }h^1(H)\geq 2$ and $H$ {\it computes} the Clifford index of $C$ if $cl(C)=cl(L)$. For a general curve $C$ it is $cl(C)=\left [\frac{g-1}{2}\right ]$ and in any case $cl(C)\leq\left [\frac{g-1}{2}\right ]$. By Clifford's theorem a special line bundle $L$ on $C$ has $cl(L)\geq 0$ and the equality holds if and only if $C$ is hyperelliptic and $L$ is a multiple of the unique $g^1_2$.\\ If $cl(C)=1$ then $C$ is either a plane quintic curve or a trigonal curve. The following results dealing with the projective normality of curves and relating it to the Clifford index are listed for the convenience of the reader. \begin{theo}[\cite{GL}] \label{glcliff} Let L be a very ample line bundle on a smooth irreducible complex projective curve C with: $$deg(L)\geq 2g+1-2h^1(L)-cl(C)$$ then $L$ is normally generated. \end{theo} In the case of hyperelliptic curves, because there are no special very-ample line bundles, the following is true. \begin{prop}[\cite{la-ma}] \label{hyper} A hyperelliptic curve of genus $g$ has no normally generated line bundles of degree $\leq 2g$. \end{prop} \begin{lemma}[ \cite{la-ma}] \label{2norm} Let $L$ be a base point free line bundle of degree $d\geq g+1$ on a curve of genus $g$. Then $L$ is normally generated if and only if the natural map $H^0(L)\otimes H^0(L)\to H^0(2L)$ is onto. \end{lemma} It follows that a polarized curve $(C, L)$ with $d\geq g+1$ and $L$ very ample is projectively normal if and only if it is 2-normal. The projective normality of a polarized surface $(S, L)$ will be often established by investigating the property for a general hyperplane section. The main tools used are the following results. \begin{theo}[\cite{fu}] \label{fujitatheo2} Let $(S, L)\supset (C,\restrict{L}{C})$ be a polarized surface with a ladder. If the ladder is regular and $\restrict{L}{C}$ is normally generated then $L$ is normally generated. \end{theo} \begin{lemma}[\cite{Alibaba}] \label{besanaignorans} Let $(S, L)\supset (C,\restrict{L}{C})$ be a polarized surface with a regular ladder. Assume $h^1(L)=0$ and $\Delta=g.$ Then $L$ is normally generated if and only if $\restrict{L}{C}$ is normally generated. \end{lemma} \subsection{\small SURFACES EMBEDDED IN QUADRIC CONES} \label{qcones} As Lemma \ref{2norm} suggests, the hyperplane section technique will often reduce the projective normality of a surface to its $2$-normality. It is useful then to recall the detailed investigation of surfaces in $\Pin{5}$ contained in singular quadrics, done in \cite{Alibabaquad}. Let $\Gamma$ be a four dimensional quadric cone in $\Pin{5}$ and let $\sigma:\Gamma^*\longrightarrow\Gamma$ be the blow up of $\Gamma$ along the vertex, with exceptional divisor $T$. Suppose $S\subset\Gamma$ is a smooth surface and let $S'$ be the strict transform of $S$ in $\Gamma^*$ under $\sigma$. The Chern classes of $S'$ and $\Gamma^*$ satisfy the following standard relation: \begin{equation} \label{DPF} \restrict{c_2(\Gamma^*)}{S'}=S'S'+\restrict{c_1(\Gamma^*)}{{S'}}c_1(S')- K_{S'}^2+c_2(S') \end{equation} If $rank(\Gamma)=5$, \ $\Gamma$ is a cone with vertex a point $P$ over a smooth quadric $Q' \subset \Pin{4}.$ Following \cite{Alibabaquad} let \begin{xalignat}{2} C(W):&=& \text{ the cone over the cycle $W \subset Q'$ with vertex $V$} & \notag \\ \sigma:& \Gamma^* \to \Gamma &\text{ the blow up map}& \notag \\ H_{Q'}: &=& \text {the hyperplane section of $Q'$}& \notag \\ l_{Q'}:&=& \text {the generator of $A_1(Q')$}& \notag \\ p_{Q'}:&=& \text {the generator of $A_0(Q')$ }&\notag \end{xalignat} According to the above notation it is \begin{gather} Pic(Q') = <H_{Q'}> \notag \\ H^2_{Q'} = 2 l_{Q'} \notag\\ A_0(Q') =<p_{Q'}> \notag \end{gather} Further, let \begin{gather} Z = \sigma^{-1}(C(H_{Q'})) \notag \\ S = \sigma^{-1}(C( l_{Q'})) \notag \\ F = \sigma^{-1}(C(p_{Q'})), \notag \end{gather} and denote by $\overline{H}$ the cycle $H_{Q'}$ in $\Gamma^*$ and by $\overline{l}$ the cycle $l_{Q'}$ in $\Gamma^* .$ The Chow Rings of $\Gamma^*$ are then given by: \begin{gather} Pic(\Gamma^*) = <Z, \tau>\\ A_2(\Gamma^*) = <\overline{H}, S>\\ A_1(\Gamma^*) = < \overline{l}, F>. \end{gather} and it is \begin{xalignat}{3} c_1(\Gamma^*) &= 2(\tau + Z) & c_2(\Gamma^*) & = Z^2 + 6\tau Z &T&=\tau-Z \end{xalignat} \begin{xalignat}{2} T&=\tau-Z& S'&=\alpha \overline{H} +\beta X \end{xalignat} The intersection table is then the following: \vskip .5 cm \setlength{\unitlength}{ .4cm} \begin{center} \begin{picture}(13,13) \put(1,0){\line(0,1){13}} \put(5,0){\line(0,1){13}} \put(9,0){\line(0,1){13}} \put(13,0){\line(0,1){13}} \put(0,0){\line(1,0){13}} \put(0,4){\line(1,0){13}} \put(0,8){\line(1,0){13}} \put(0,12){\line(1,0){13}} \put(0,.5){$F$} \put(0,2.5){$\overline{l}$} \put(0,4.5){$X$} \put(0,6.5){$\overline{H}$} \put(0,8.5){$Z$} \put(0,10.5){$\tau$} \put(12,12.5){$F$} \put(9.5,12.5){$\overline{l}$} \put(8,12.5){$X$} \put(6,12.5){$\overline{H}$} \put(3.5,12.5){$Z$} \put(1.5,12.5){$\tau$} \put(1.5,.5){$1$} \put(1.5,2.5){$1$} \put(1.5,4.5){$\overline{l}$} \put(1.5,6.5){$2 \overline{l}$} \put(1.5,8.5){$\overline{H}$} \put(1.5,10.5){$\overline{H}$} \put(3.5,.5){$0$} \put(3.5,2.5){$1$} \put(3.5,4.5){$F$} \put(3.5,6.5){$2 \overline{l}$} \put(3.5,8.5){$2X$} \put(3.5,10.5){$\overline{H}$} \put(5.5,4.5){$1$} \put(5.5,6.5){$2$} \put(5.5,8.5){$2\overline{l}$} \put(5.5,10.5){$2\overline{l}$} \put(7.5,4.5){$0$} \put(7.5,6.5){$1$} \put(7.5,8.5){$F$} \put(7.5,10.5){$\overline{l}$} \put(6.5,2){0} \put(9.5,10.5){$1$} \put(9.5,8.5){$1$} \put(11.5,10.5){$1$} \put(11.5,8.5){$0$} \put(10.5,5.5){$0$} \put(10.5,2){$0$} \end{picture} \end{center} If $rank(\Gamma)=4$, \ $\Gamma$ is a cone with vertex a line $r$ over a smooth quadric surface $Q\subset \Pin{3}.$ Following \cite{Alibabaquad} let $\tau$ be the tautological divisor on $\Gamma^*$ and Let $C(W)$ denote the cone with vertex $r$ over the cycle $W\subset \overline{Q}.$ Let \begin{xalignat}{3} Pic \ \overline{Q} : &= <\ell_1, \ell_2> & A_0(\overline{Q}) : &= <p> &Q : &= \sigma^{-1}(\overline{Q}) \notag \\ P_1 :&= \sigma^{-1}(C( \ell_1)) & p_1: &=\tau \cdot P_1 & \overline{\ell}_1:&= \tau \cdot p_1 \notag \\ P_2 :&= \sigma^{-1}(C( \ell_2)) & p_2: &=\tau \cdot P_2 & \overline{\ell}_2:&= \tau \cdot p2 \notag \\ F :&= \sigma^{-1}(C(p))& \ell:&= \tau \cdot F & & \notag \end{xalignat} With the above notation it is: \begin{xalignat}{3} Pic (\Gamma^*) &= <\tau, P_1, P_2> & A_2(\Gamma^*) &= < Q,p_1, p_2, F> & A_1(\Gamma^*) &= < \overline{\ell}_1, \overline{\ell}_2, \ell> \notag \end{xalignat} It is also $T=\tau-P_1-P_2$, $c_1(\Gamma^*)=4\tau-T$, $c_2(\Gamma^*)=3Q+4p_1+4p_2.$ Because $S'$ is an effective cycle it is $S'=\alpha Q+\beta p_1+\gamma p_2+\delta F$ with $\alpha\geq 0$, $\alpha +\beta\geq 0$, $\alpha+ \gamma \ge 0$ and $deg(S)=\tau\cdot \tau\cdot S'=2\alpha+\beta+\gamma+\delta.$ With the above notation we have the following intersection table: \vskip .5cm \setlength{\unitlength}{ .7cm} \begin{center} \begin{picture}(11,11) \multiput(0,0)(0,3){2}{\line(1,0){11}} \multiput(0,7)(0,3){2}{\line(1,0){11}} \multiput(1,0)(3,0){2}{\line(0,1){11}} \multiput(8,0)(3,0){2}{\line(0,1){11}} \multiput(1.2,0.2)(1,1){10}{$1$} \multiput(2.2,0.2)(1,1){9}{$0$} \multiput(1.2,1.2)(0,1){2}{$1$} \multiput(2.2,2.2)(0,1){2}{$0$} \multiput(3.2,3.2)(0,1){2}{$0$} \multiput(7.2,7.2)(0,1){2}{$0$} \multiput(6.2,6.2)(1,0){2}{$1$} \multiput(4.2,4.2)(0,1){2}{$1$} \multiput(.2,9.2)(1,1){2}{$\tau$} \multiput(.2,8.2)(2,2){2}{$P_1$} \multiput(.2,7.2)(3,3){2}{$P_2$} \multiput(.2,6.2)(4,4){2}{$Q$} \multiput(.2,5.2)(5,5){2}{$p_1$} \multiput(.2,4.2)(6,6){2}{$p_2$} \multiput(.2,3.2)(7,7){2}{$F$} \multiput(.2,2.2)(8,8){2}{$\overline{\ell_1}$} \multiput(.2,1.2)(9,9){2}{$\overline{\ell_2}$} \multiput(.2,.2)(10,10){2}{$\ell$} \multiput(1.2,8.2)(1,1){2}{$p_1$} \multiput(1.2,7.2)(2,2){2}{$p_2$} \multiput(1.2,6.2)(3,3){2}{$\star$} \multiput(1.2,5.2)(4,4){2}{$\overline{\ell_1}$} \multiput(1.2,4.2)(5,5){2}{$\overline{\ell_2}$} \multiput(1.2,3.2)(1,1){3}{$\ell$} \multiput(5.2,7.2)(1,1){3}{$\ell$} \multiput(3.2,6.2)(1,1){2}{$\overline{\ell_1}$} \multiput(2.2,5.2)(3,0){2}{$0$} \multiput(2.2,6.2)(2,2){2}{$\overline{\ell_1}$} \multiput(2.2,7.2)(1,1){2}{$F$} \multiput(2.2,8.2)(3,0){2}{$0$} \multiput(3.2,7.2)(3,0){2}{$0$} \multiput(8.2,9.2)(1,0){2}{$1$} \put(1.2,9.2){$Q$} \put(4.2,6.2){$2$} \put(5.2,6.2){$1$} \put(8.2,8.2){$0$} \put(.2,10.2){$\cdot$} \end{picture} \end{center} where $\star = \overline{\ell_1} + \overline{\ell_2}$ and the empty spaces are intended to be $0.$ \section{Surfaces in $\Pin{4}$} Throughout this section $S$ will be a surface embedded in $\Pin{4}$ by a very ample line bundle $L,$ non degenerate of degree $9,$ with sectional genus $g.$ Surfaces of degree $9$ in $\Pin{4}$ have been completely classified by Aure and Ranestad in \cite{au-ra}. The investigation of the projective normality of such surfaces is essentially contained in their work. For completeness we present the global picture in this section. \begin{theo}[ \cite{au-ra}]\label{class} Let $(S, L)$ be as above and $\chi=\chi(\cal {O}_S)$ then $S$ is a regular surface with $K^2=6\chi-5g+23$, where: \begin{itemize} \item[1)] $g=6$, $\chi=1$ and $S$ is rational or the projection of an Enriques surface of degree $10$ in $\Pin{5}$ with center of projection on the surface; \item[2)] $g=7$ and $\chi =1$ and $S$ is rational, or $\chi =2$ and $S$ is a minimal elliptic surface; \item[3)] $g=8$ and $\chi =2$ and $S$ is a K3-surface with $5$ $(-1)$-lines, or $\chi =3$ and $S$ is a minimal surface of general type; \item[4)] $g=9$, $\chi=4$ and $S$ is linked $(3,4)$ to a cubic scroll; \item[5)] $g=10$, $\chi=6$ and $S$ is a complete intersection $(3,3)$; \item[6)] $g=12$, $\chi = 9$ and $S$ is linked to a plane. \end{itemize} Moreover if $g\geq 7$ then $S$ is contained in at least two quartic surfaces. \end{theo} \begin{prop}$(S, L)$ as above is projectively normal if and only if: \begin{itemize} \item [a)] $g=8$, $\chi =2$ and $S$ is a K3-surface with $5$ $(-1)$-lines. \item[b)] $g=9$, $\chi=4$ and $S$ is linked $(3,4)$ to a cubic scroll \item[c)] $g=10$, $\chi=5$ and S is a complete intersection (3.3); \item[d)] $g=12$, $\chi = 9$ and $S$ is linked to a plane. \end{itemize} \end{prop} \begin{pf} Let us examine the surfaces in Theorem \ref{class};\\ Let $C\in|L|$ be a generic smooth element. Since all the surfaces are regular we always have $h^0(\restrict{L}{C})=4$. Note that for $g\leq 9$ $d(2\restrict{L}{C})<2g-2$, then $h^1(2\restrict{L}{C})=0$. If $g=6$ then $h^1(\restrict{L}{C})=0$ and thus $h^1(L)=0$. Because $h^0(2\restrict{L}{C})=h^1(2\restrict{L}{C})+19-6=13$ the following exact sequence \begin{equation} \label{duelle} 0\longrightarrow L\longrightarrow 2L\longrightarrow 2\restrict{L}{C}\longrightarrow 0 \end{equation} gives $h^0(2L)=h^0(L)+h^0(2\restrict{L}{C})=18.$ Thus the map $H^0({\cal O}_{{\bf P}^4}(2))\longrightarrow H^0({\cal O_S}(2))$ cannot be surjective, being $h^0({\cal O} _{{\bf P}^4}(2))=15.$ This means that $S$ is not $2$-normal and therefore it is not projectively normal. \\ If $g=7$ then $h^0(2\restrict{L}{C})=12$, $h^1(\restrict{L}{C})=1$ and $h^1(2L)=0$ by \cite[2.10]{au-ra}. Therefore from \brref{duelle} and the regularity of S it follows that $16\leq h^0(\restrict{L}{C})\leq 17$, which implies that $S$ is not projectively normal, as above. If $g=8$ then $h^0(2\restrict{L}{C}) =11$, $h^1(L)\leq 1$ and $h^1(2L)=0$ by \cite[2.11]{au-ra}. For degree reasons $S$ cannot be contained in any quadric hypersurface being contained in at least a quartic. Therefore $S$ is $2$-normal if and only if $H^0(L)=15$. >From \brref{duelle} we get $ h^0(2L)\leq 15,16$ respectively if $\chi=2,3$. If $\chi=3$ $S$ is not projectively normal as above. If $\chi=2$ $S$ is $2$-normal and therefore projectively normal by Lemma \ref{2norm}. If $g=9$ $(S, L)$ is projectively normal by linkage, see \cite[2.13]{au-ra}. If $g=10$ $S$ is a complete intersection and thus projectively normal. If $g=12$ $S$ is linked to a plane and therefore it is arithmetically Cohen-Macaulay by linkage, which implies $S$ projectively normal. \end{pf} \section{Surfaces embedded in $\Pin{N} ,$ $N\ge 5.$} Let $ S$ be a smooth surface, let $L$ be a very ample line bundle on $S$ and let $${\cal S}_g =\{(S, L) \text{ as above} \ | L^2 = 9, h^0(L)\ge 6, g(S, L)=g \text{ and} (S, L) \text{ is not a scroll}\}.$$ If $(S, L)\in{\cal S}_g$, by Castelnuovo's Lemma $g\leq 7.$ Let $\cal{S} = \bigcup_{g=0}^7 \cal{S}_g.$ In the following lemmata a few preliminary results are collected. \begin{lemma} Let $(S, L)\in{\cal S}$ and let $C\in|L|$ be a smooth generic element. Then \begin{itemize} \item[a)] if $h^1(\restrict{L}{C})=0$ then $g(L)\leq5$; \item[b)] if $h^1(\restrict{L}{C})=1$ then $g(L)=7$ or $6$; \item[c)] if $h^1(\restrict{L}{C})\geq 2$ then $g(L)=7$; \end{itemize}\label{h1LC} \end{lemma} \begin{pf} a) For $g\geq 1$ we have $h^0(\restrict{L}{C})\geq 3$. If $h^0(\restrict{L}{C})=3, 4$ then $h^0(L)\le 4, 5$ respectively, which is a contradiction. Thus $h^0(\restrict{L}{C}) \ge 5$ i.e. $5\leq h^0(\restrict{L}{C})=10-g$, i.e. $g\leq 5.$\\ b) Since $h^0(K_C-\restrict{L}{C})=1$ then $d\leq 2g-2$. Moreover being $d$ odd it is $d\leq 2g-3$ c) If $h^1(\restrict{L}{C})\geq 2$ then $K_S|_C$ is a special divisor on $C$ and it contributes to $cl(C),$ thus $cl(C)\geq 0$, i.e \\ $0\leq d(K_C-\restrict{L}{C})-2h^0(K_C-\restrict{L}{C})+2\leq 2g-13$ , from which we get $g\geq 7$. \end{pf} \begin{lemma}\label{nonP41} Let $(S, L) \in \cal S$ . Then $(S, L)$ is projectively normal or \begin{itemize} \item[a)] $(S, L) \subset \Pin{5}$, $g=4$, $S$ is $\Pin{1}$-bundle over an elliptic curve, $e=-1$, $L=3C_0.$ \item[b)] $(S, L) \subset \Pin{5}$ is a conic bundle over an elliptic curve, $g=4$. \item[c)] $(S, L) \subset \Pin{5}$ $g = 5,6.$ \end{itemize} \end{lemma} \begin{pf} In our hypothesis it is $\Delta(S, L)=11-h^0(L)$. If codim$ (S) =1$ then $S$ is projectively normal. Because $S$ is not embedded in $\Pin{4}$, we can assume codim$ (S) \ge 3$ i.e. $h^0(L) \geq 6$, i.e $\Delta(S)\leq 5$. First assume $\Delta < 5.$ It is $9=d\geq 2\Delta + 1$, then if $g\geq\Delta$ $(S, L)$ is projectively normal by Theorem \ref{fujitatheo}. Because $g=0$ implies $\Delta = 0$ it follows that $(S, L)$ is projectively normal if $\Delta=0,1.$ Moreover if $g=1$ then $(S, L)$ is an elliptic scroll which is impossible. Therefore $(S, L)$ is projectively normal if $\Delta = 2.$ Let now $\Delta=3.$ By \cite{Io1} it is $g=3$ and therefore $(S, L)$ is projectively normal Let now $\Delta= 4.$ By \cite{Io4} Theorem 3, $(S, L)$ is projectively normal unless, possibly, if it is a scroll over a curve of genus $g=2,$ which is impossible. Let now $\Delta(S)=5$, i.e $h^0(L)=6$. If $g(L)=7$ then $S$ is a Castelnuovo Surface, see \cite{Har1}, and thus projectively normal. Let $g=2,3.$ Simple cohomological computations, using the classification given in \cite{Io1} show that there are no such surfaces in $\Pin{5}.$ Let $g=4$ and let $C \in |L|$ be a generic hyperplane section. By Lemma \ref{h1LC} it is $h^1(\restrict{L}{C})=0$ and therefore $h^0(\restrict{L}{C})=6$ by Riemann Roch. This shows that $q(S) \ne 0.$ By \cite{LiAq} and \cite{Io4} the only possible cases are b) and c) in the statement . \end{pf} \subsection{SECTIONAL GENUS $g=4$ } \label{genusfour} In this subsection the projective normality of pairs $(S, L)\in{ \cal S}_4$ is studied. By Lemma \ref{nonP41} and \cite{LiAq} we have to investigate the following cases : \begin{itemize} \item[Case 1.] $(S, L) \subset \Pin{5}$ is a $\Pin{1}$-bundle over an elliptic curve, $e=-1$, $L\simeq3C_0.$ \item[Case 2.] $(S, L) \subset \Pin{5}$ is the blow up $p: S \to X$ of a $\Pin{1}$-bundle over an elliptic curve at $3$ points, $e=-1$, $L\equiv 2\frak{C_0}+2\frak{f}-\sum_1^3E_i$ where $\frak{C_0} = p^*(C_0)$ and $\frak{f}=p^*{f}.$ \item[Case 3.]$(S, L) \subset \Pin{5}$ is the blow up $p: S \to X$ of a $\Pin{1}$-bundle over an elliptic curve at $3$ points, $e=0$, $L\equiv 2\frak{C_0}+3\frak{f}-\sum_1^3E_i$ where $\frak{C_0} = p^*(C_0)$ and $\frak{f}=p^*{f}.$ \end{itemize} \begin{lemma} Let $(S, L) \in \cal{S}_4$ be as in \brref{genusfour} Cases 1,2,3 above. The following are equivalent: \begin{enumerate} \item[i)]$(S, L)$ is $2$- normal; \item[ii)]$\cal{I}_S$ is $3$-regular; \item[iii)] $(S, L)$ is projectively normal; \item[iv)] $h^0(\cal{I}_S(2)) = 0,$ i.e. $S$ is not contained in any quadric hypersurface. \end{enumerate} \label{g4noquad} \end{lemma} \begin{pf} Assume $(S, L)$ is $2$-normal, i.e. $h^1(\cal{I}_S(2)) = 0.$ Since $p_g(S) = 0$ and $h^1(L)=0$ in all the cases under consideration, it is not hard to check that $\cal{I}_S$ is $3$-regular. By \cite[pg. 99]{mu1} it follows that $h^1(\cal{I}_S(k)) = 0$ for all $k \ge 2$ and thus $(S, L)$ is projectively normal. Therefore i), ii) and iii) are equivalent. Since $h^0(\oofp{5}{2}) = 21$ and $h^0(\oof{S}{2}) = 21,$ $S$ is $2$- normal if and only if $h^0(\cal{I}_S(2)) = 0.$ \end{pf} \begin{rem} $(S, L)$ as in \brref{genusfour} Case 3 is a congruence of lines of $\Pin{3}$ of bi-degree $(3,6),$ cf. \cite{gr} and thus not projectively normal by Lemma \ref{g4noquad}. \end{rem} \begin{rem} $(S, L)$ as in \brref{genusfour} Case 1 was shown to be projectively normal by Homma \cite{Ho2}. Following an idea due to Sommese we offer below a different proof. \end{rem} \begin{prop} \label{propdiandrew} Let $(S, L)$ be as in \brref{genusfour} Case 1, then $(S, L)$ is projectively normal. \end{prop} \begin{pf} Lemma \ref{g4noquad} shows that it is enough to show that $(S, L)$ is $2$-normal. Let $E$ be an elliptic curve with fixed origin $O.$ It was shown in \cite{be-so2} that $S$ can be viewed as the quotient of $X=E\times E$ under the involution $ \iota: X\to X$ given by $ \iota(x,y) = (y,x).$ Let $q: X \to S$ be the quotient map and $p_i: X \to E,$ $i=1,2$ be the projections onto the factors. One can see that $ q^*(C_0) = p_1^*(\oof{E}{P}) \otimes p_2^*(\oof{E}{P}),$ where $P$ is a point on $E.$ Let $L_i = p^*_i(\oof{E}{P}).$ It is $$ H^0(X, q^*(L)) = H^0(X, q^*(3C_0)) = H^0(3L_1) \otimes H^0(3L_2),$$ and therefore $h^0(q^*(L)) = 9.$ Let $H^0(X, q^*L)^{\iota}=\{ \sigma \in H^0(X,q^*L) \text{ such that } \sigma \iota = \sigma \}$, i.e. the subspace of global holomorphic sections of $q^*(L)$ which are invariant under $\iota.$ Notice that there is a natural isomorphism $H^0(S, L) \simeq H^0(X,q^*(L))^{\iota}.$ Because $deg (3L_i) \ge 2g(E) + 1$ the map $H^0(3L_i) \otimes H^0(3L_i) \to H^0(2(3L_i))$ is surjective. It follows that $$H^0(q^*(L)) \otimes H^0(q^*(L)) \to H^0(q^*(2L))$$ is surjective. To conclude it is enough to show that \begin{equation} \label{2normiota} H^0(q^*(L))^{\iota} \otimes H^0(q^*(L))^{\iota} \to H^0(q^*(2L))^{\iota} \end{equation} is surjective. Let $\alpha_1, \alpha _2, \alpha_3$ be a base for $H^0(E, \oof{E}{3P})$ and let $a_i=p_1^*(\alpha_i)$ and $b_i=p_2^*(\alpha_i)$ be bases for $H^0(3L_1)$ and $H^0(3L_2)$ respectively. With the above notation $\{ a_i \otimes b_j\}$ form a base for $H^0(q^*(L)).$ Notice that an element $a_i \otimes b_j$ with $i \ne j$ cannot be $\iota$-invariant. To see this assume $a_i \otimes b_j$ were $\iota$-invariant and let $z \in E$ be a point such that $\alpha_i(z) = 0$ and $\alpha_j(z) \ne 0.$ Then $(a_i \otimes b_j)(z,y) = (a_i \otimes b_j)(y,z)$ implies $\alpha_i(y)\alpha_j(z) = 0$ for every $y,$ which is impossible. Simple direct checks show that $$ a_1 \otimes b_1, \ a_2 \otimes b_2, \ a_3 \otimes b_3, a_1 \otimes b_2 + \ a_2 \otimes b_1, a_1 \otimes b_3 + \ a_3 \otimes b_1, a_2 \otimes b_3 + \ a_3 \otimes b_2 $$ form a basis for $H^0(q^*L)^{\iota}$ and that $$t_1= a_1 \otimes b_2,t_2= a_1 \otimes b_3,t_3= a_2 \otimes b_3$$ span a $3$-dimensional subspace $V$ such that $ H^0(q^*(L)) = H^0(q^*(L))^{\iota} \oplus V.$ To conclude the proof of the surjectivity of (\ref{2normiota}) it is enough to show that elements of the form $v \otimes s$ or $s \otimes v$ where $v \in V$ and $s \in H^0(q^*L)^{\iota}$ , and elements of the form $v_1 \otimes v_2$ where $v_i \in V$, cannot be $\iota$-invariant. Let $v= p_1^*(\gamma_1) \otimes p_2^*(\gamma_2)$ and let $s= p_1^*(\sigma_1) \otimes p_2^*(\sigma_2).$ For every $(x,y) \in X$ it is \begin{gather} \label{onesidevs} (v \otimes s)(x,y) = \gamma_1(x) \gamma_2(y)\sigma_1(x) \sigma_2(y) = \gamma_1(x) \gamma_2(y) \sigma_1(y) \sigma_2(x)\\ \label{othersidevs} (v \otimes s)(y,x) = \gamma_1(y) \gamma_2(x) \sigma_1(y) \sigma_2(x). \end{gather} Equating \brref{onesidevs} and \brref{othersidevs} shows that $v$ is invariant, a contradiction. The same argument takes care of the case $s \otimes v.$ Let now $v_1 = \sum c_i t_i$ and $v_2 = \sum d_i t_i,$ where the $t_i's$ are as above and assume that $v_1 \otimes v_2$ is invariant. Since $\oof{E}{3P}$ is generated by global sections there is a point $z \in E$ such that $\alpha_1(z) \ne 0,$ while $\alpha_2(z) = \alpha_3(z) = 0.$ The fact that $(v_1 \otimes v_2)(z,y) = (v_1 \otimes v_2)(y, z)$ for every $y$ implies$$(c_1 + d_1)\alpha_1(z) \alpha_2(y) = 0$$ for every $y,$ which gives $c_1 = -d_1.$ Repeating the argument permuting the indices it follows that $v_2 = - v_1.$ It is then enough to show that $v_1 \otimes (-v_1)$ is not $\iota$-invariant. If it were it would follow that $(v_1(x, y) - v _1(y, x))(v_1(x, y) + v_1(y, x)) = 0$ for all $x,y \in E.$ Because $v_1$ is not invariant and it is not zero everywhere, this is a contradiction. \end{pf} \begin{lemma}\label{tipidicurve} Let $(S, L)$ be as in \brref{genusfour} Case 2 . Then \begin{enumerate} \item[i)] If $r$ is a line contained in $S,$ then $r=E_i$ or $r=\frak{f}-E_i$, $i=1,2,3.$ \item[ii)] If $C\subset S$ is a reduced irreducible cubic with $C^2 \ge 0$ then $C$ is a curve whose numerical class is $C \equiv \frak{C_0} + \frak{f} - \sum_1^3 E_i$ \item[iii)] If $C\subset S$ is a reduced irreducible quartic with $C^2 \ge 0$ then $C$ is a curve whose numerical class is one of the following: \begin{enumerate} \item[a)] $C\equiv \frak{C_0}$ \item[b)] $C \equiv \frak{C_0} +\frak{f} - E_i -E_j $ \end{enumerate} \end{enumerate} \end{lemma} \begin{pf} Let $r=a\frak{C_0}+b\frak{f}-\sum_1^3a_iE_i$ be a line in $S$. Then $L \cdot r=4a+2b-\sum_1^3a_i=1$ and $0=2g(r)=2+a(a-1)+2b(a-1)-\sum_1^3a_i(a_1-1)$. It follows that \begin{gather} \sum_1^3a_i^2=a^2+3a+2ab+1\geq\frac{1}{3}(4a+2b-1)^2 \notag \\ \text{i.e.} \ \ \ \ \ 4b(b-1)+13a(a-1)+10ab-4a-2\leq 0 \label{diseqretta} \end{gather} Since $r$ is an irreducible smooth curve either $r=E_i$ or $r$ is the strict transform of an irreducible curve on the $\Pin{1}$-bundle and the following cases can occur: \begin{itemize} \item $a=0$ and $b=1$, that gives us the fibers through the points blown up, i.e. $r=\frak{f}-E_i$ for $i=1,2,3$. \item $a=1$ and $b\geq 0$ , for which \brref{diseqretta} would imply $b=0$, $\sum_1^3a_i =3$ and $\sum a_i^2=5$, i.e. $r=\frak{C_0}-2E_i-E_j$ for $i,j=1,2,3$. But this would imply the existence of an irreducible curve in $|C_0|$ passing through a point $P_i$ with multiplicity $2,$ that would imply $C_0 \cdot f=2$, where $f$ is the fiber through $P_i$, which is a contradiction. \item $a\geq 2$ and $b\geq -\frac{a}{2}$. Let $a=2+h$ with $h\geq 0.$ >From \brref{diseqretta} it follows that $4b^2+16+8h^2+27h+5h(h+2b)+8(2b+h)\leq 0$ and therefore $ 4b^2+8h^2+17h\leq 0$, which is impossible unless $b=0$, $a=2$ which contradicts \brref{diseqretta}. \end{itemize} Cases ii) and iii) follow from similar computations and Castelnuovo's bound on the arithmetic genus of curves. \end{pf} Techniques found in \cite{Alibabaquad} and a detailed analysis of the geometry of hyperplane sections will be used to deal with Case 2. \begin{prop} \label{selincones} Let $(S, L) \in \cal{S}_4$ be as in Case 2. Then $(S, L)$ is projectively normal. \end{prop} \begin{pf} By contradiction assume $(S, L)$ is not projectively normal. Lemma \ref{g4noquad} then implies that $(S, L)$ must be contained in a quadric hypersurface $\Gamma.$ From \cite{ar-so2} it follows that $\Gamma$ must be singular. \begin{case} $rk(\Gamma)=5.$ \end{case} Let $P$ be the vertex of the quadric cone $\Gamma.$ Following the notation of subsection \ref{qcones} let $S'=\alpha \overline{H} +X.$ If $P$ is not contained in $S$, then $S=S',$ $deg(S)=2\alpha$, which is impossible. We can assume $P\in S$. Then $S'=4\overline{H}+X$ because $deg(S')=\tau^2 S'$ and $E^2=(\restrict{T}{S'})^2=-1.$ Moreover $c_1(\Gamma^*)|_{S'} \cdot c_1(S')=2(2L-E)(-K_S'-E)$, since $\tau|_ {S'}=\sigma^*(L)$. Plugging the above obtained values into \brref{DPF} a contradiction is reached. \begin{case} $rk(\Gamma)=4$ \end{case} Let $r$ be the line vertex of the cone. >From $deg(S')=9$ we have \begin{gather} 9=2\alpha+\beta+\gamma+\delta \end{gather} If $r\subset S$ then by Lemma \ref{tipidicurve} $r=E_i$ or $r=\frak{f}-E_i$. Notice that in this case $S'= S.$ Let $T|_{S'}=\lambda r,$ since $T|_{S'}\cdot\tau|_{S'}= \delta$ and $(T|_{S'})^2=-\delta^2$ we have $\lambda=\delta$ and $\beta+\gamma=\delta- \delta^2.$ Moreover $(S')^2=2(\alpha +\beta)(\alpha +\gamma)+2\alpha \gamma$, $c_2(\Gamma^*)|_{S'}=14\alpha+7(\beta + \gamma)+3\delta$, $c_1(\Gamma^*)|_{S'}=4\sigma^*(L)-\delta r$. But from $9=2\alpha+\beta+\gamma+ \delta$ the only possible values are $(\alpha,\delta,\beta+\gamma)=(3,1,2)$ which give a contradiction in \brref{DPF}. If $S\cap r= \emptyset$ then $9=2\alpha$, since $T|_{S'}=0,$ which is a contradiction. If $S\cap r=\{P_1,...,P_k\}$, let $\mu_j$ be the multiplicity of intersection at $P_j$ and let $s=\sum\mu_j$. Then $(T|_{S'})^2=-\sum \mu_j=-s.$ If any of the $\mu_j's$ is strictly greater then $1,$ $S'$ acquires a singularity of type $A_{\mu_j - 1}$ at a point of $\overline{E_j},$ where $\overline{E_j}$ are the exceptional divisors of $\restrict{\sigma}{S'}.$ Notice that $(\tau T)S'=0$ gives $S'=\alpha Q+\beta p_1+\gamma p_2$, with $\beta+\gamma=s.$ Moreover it is $\alpha\geq 2$ because $\alpha$ can be viewed as the degree of the generically finite rational map $\psi: S\longrightarrow Q$ induced by the projection from the vertex of $\Gamma$, where $S$ is not birational to $Q$. Thus the only possible values are $(\alpha, s)=(4,1),(3,3),(2,5)$. Let us assume at first that $\mu_j = 1$ for all $j,$ so that $S'$ is smooth. It is $(S')^2=2\alpha(\alpha+h) +2\beta\gamma$, $c_2(\Gamma^*)|_{S'}=63$, $c_1(\Gamma^*)|_{S'}=4\sigma^*(L)-\sum\overline{E_j}.$ Using the admissible values for $\alpha$ and $s$ we get a contradiction in \brref{DPF}. Therefore for at least one $j$ it is $\mu_j \ge 2$ and $(\alpha,s) = (4,1)$ does not occur. Let $\Pi\subset \Pin{3}$ be a general 2-plane tangent to $Q.$ Then $\Pi \cap Q = \ell_1 \cup \ell_2$ where $\ell_i$ is a line in $\Pi.$ Cutting $S$ with the hyperplane spanned by $\Pi$ and $r$ we get a degree nine divisor $D\in |L|$ which must be reducible as $D=D_1 \cup D_2$ where $\psi(D_i) =\ell_i.$ Moving $\Pi$ along $\ell_i$ we can see that $D_j$ moves at least in a pencil. Therefore $h^0(D_i)\ge 2$ for $i=1,2.$ Moreover the above argument shows that $D_i$ is spanned away from $S\cap r.$ In particular $D_i$ cannot have a fixed component, therefore $D_i^2 \ge 0.$ Let $d_i = L \cdot D_i.$ Then $(d_1,d_2) = (1,8), (2,7), (3,6),(4,5).$ Lemma \ref{tipidicurve} shows that $S$ contains only a finite number of lines, therefore the first case cannot happen. In the second case, moving $\Pi$ along $\ell_2$, $S$ could be given a conic bundle structure over $\Pin{1}$ which is not possible. When $(d_1,d_2) = (3,6)$ notice that $D_1$ must be reduced and irreducible because $S$ contains only a finite number of lines. Therefore $D_1\equiv \frak{C_0} + \frak{f} - \sum_i E_i$ as in Lemma \ref{tipidicurve} ii). When $s=3$, $\psi $ is a generically $3:1$ map while when $s=5$ $\psi $ is a generically $2:1$ map. Therefore there is always at least a point $P\in S\cap r$ such that $P \not \in D_1.$ pertanto meno con Because $h^0(X, C_0 + f)=3$ and $C_0 + f$ is spanned, $h^0(D_1) \le 2.$ Since $|D_1|$ must be at least a pencil, it is $h^0(D_1) = 2.$ This shows that the complete linear system $|D_1|$ is obtained by moving $\Pi$ along $\ell_2.$ A member of $|D_1|$ passing through $P$ can then be found, contradiction. Let now $(d_1, d_2) = (4,5).$ Assume $D_1$ reduced and irreducible. Then $D_1$ must be as in Lemma \ref{tipidicurve} iii). Because $h^0(\frak{C_0}) =1$ it is $D_1\equiv \frak{C_0} + \frak{f} - E_i -E_j.$ We claim that $D_1$ is then a smooth elliptic quartic embedded in $\Pin{3}.$ To see this notice that every element of $|C_0 + f|$ on $X$ is smooth with the only exception of one curve, reducible as the union $C_0 \cup f.$ Moreover notice that the same argument used above shows that $h^0(D_1) = 2$ and $|D_1|$ is obtained by moving $\Pi$ along $\ell_2.$ Because $\psi(D_1) =\ell_1$, for degree reasons $D_1$ must go through at least one point in $S\cap r.$ Because $\mu_j\ge 2$ for at least one $j$ and $h^0(D_1)=2$ we can always assume that $D_1$ has a $(k\ge 3)$-secant line. It is known (see \cite{Io2}) that the ideal of such quartics in $\Pin{3}$ is generated by quadrics and therefore they cannot have $(k \ge 3)$-secant lines. Let $D_1$ now be reducible or non reduced. $D_1$ cannot be reducible with lines as components since $S$ contains only a finite number of lines. A simple numerical check shows that the only smooth conics on $S$ have numerical class $\frak{f}$. Therefore we can assume $D_1 \equiv 2\frak{f.}$ As it was pointed out above $D_1$ must pass through at least a point $P \in S \cap r$ but this contradicts $\frak{f}^2 = 0.$ \begin{case} $rk(\Gamma)=3$ \end{case} Assume $S \subset \Gamma$ where $\Gamma$ is a quadric cone with $rk \Gamma = 3$ and vertex $ V \simeq \Pin{2}$ over a smooth conic $\gamma\subset\Sigma \simeq \Pin{2}.$ Let $\psi: S -->\gamma$ be the rational linear projection from $V.$ Let $\ell_1, \ell_2$ and $\ell_3$ be distinct lines in $\Sigma$ such that $\gamma \cap \ell_1 =\{P_1, P_2\}$,\ \ $\gamma \cap \ell_2 =\{P_2, P_3\}$, and $\gamma \cap \ell_3 =\{P_1, P_3\}$, where $P_i \neq P_j.$ Let $D_i$ be the hyperplane sections of $S$ given by the hyperplanes spanned by $V$ and $\ell_i.$ Notice that $D_i$ must be reducible. Assume that $D_i$ has no components contained in $V$ for at least one $i$, say $i=1.$ Then $D_1\sim A+B$ where $\psi(A)=P_1$ and $\psi(B)=P_2.$ Let $L\cdot A = a$ and $L \cdot B=b.$ It follows that $D_2 \sim A'+B$ and $D_3 \sim A'+A$. Notice that $A \sim A'$ and so $L\cdot A'=a.$ This leads to the contradiction $2a=L\cdot D_3=9.$ It follows that for any hyperplane section $D$ obtained with the hyperplane spanned by a line $\ell \subset \Sigma$ and $V$ it must be $D \sim 2C + F$ where $F\subset V$ and no component of $C$ is contained in $V.$ Because $|C|$ is at least a pencil and it cannot clearly have fixed components, it must be $C^2\ge 0.$ Since $S$ contains only a finite number of lines and it is not a rational conic bundle it is $L\cdot C = 3,4.$ If $L\cdot C =3$ $C$ must be irreducible and therefore as in Lemma \ref{tipidicurve}, i.e. $C\equiv \frak{C_0}+\frak{f} - \sum_i E_i.$ It follows that $F \equiv E_1+E_2+E_3$ which is impossible because these three lines are disjoint. If $L\cdot C=4$ and $C$ is reduced and irreducible then $C \equiv \frak{C_0} + \frak{f} - E_i-E_j$ as in Lemma \ref{tipidicurve} iii) (the case $C\equiv \frak{C_0}$ cannot happen since $h^0(\frak{C_0})=1$). Then $F \equiv E_i+E_j - E_k$ which is impossible. If $C$ is reducible or non reduced then $C \equiv 2\frak{f}$ and $F\equiv 2\frak{C_0}-2\frak{f} -\sum_iE_i $ which is not effective because $C_0$ is ample on $X$ and $C_0\cdot(2C_0-2f) =0.$ \end{pf} \subsection{SECTIONAL GENUS $g =5$} In this section we will study the projective normality of pairs $(S, L)\in{ \cal S}_5.$ From Lemma \ref{nonP41} it follows that either $(S, L)$ is known to be projectively normal or $S\subset \Pin{5}$ and hence $\Delta(S, L)=5.$ Since $g=\Delta$ and $d\geq 2\Delta-1$ the ladder is regular and therefore $(S, L)$ is projectively normal if $(C,\restrict{L}{C})$ is projectively normal. \begin{lemma} \label{equivalent} Let $(S, L)\in{\cal S}_5$ then if $h^1(L)=0$ the following statements are equivalent: \begin{itemize} \item[1)] $(S, L)$ is projectively normal \item[2)] $(C,\restrict{L}{C})$ is projectively normal \item[3)] $S$ is contained in exactly one quadric hypersurface in $\Pin{5}$. \end{itemize} \end{lemma} \begin{pf} Since $h^1(L)=0,$ $(S, L)$ is projectively normal if and only if $(C, \restrict{L}{C})$ is projectively normal by Lemma \ref{besanaignorans}. Moreover $(C, \restrict{L}{C})$ is projectively normal if and only if it is 2-normal, by Lemma \ref{2norm}. Consider the exact sequence: $$0\longrightarrow{\cal I}_S(1)\longrightarrow{\cal I}_S(2)\longrightarrow{\cal I}_C(2)\longrightarrow 0.$$ It is $h^1({\cal I}_S(1))=0$ since $L$ is linearly normal and $h^0({\cal I}_S(1))=0$ since $S$ is non degenerate. It follows that $h^0({\cal I}_S(2))=h^0({\cal I}_C(2)).$ Because $h^0({\cal O}_{\Pin{4}}(2))=15$ and $h^0(2\restrict{L}{C})=14$ we have that $(C, \restrict{L}{C})$ is projectively normal if and only if $h^0({\cal I}_C(2))=1$ and therefore $(S, L)$ is projectively normal if and only if $S$ is contained in exactly one quadric. \end{pf} \begin{lemma} \label{trighyper} Let $(S, L) \in \cal{S}_5.$ Then either $(S, L)$ is projectively normal or it has a hyperelliptic or trigonal section $C \in |L|.$ \end{lemma} \begin{pf} Since $g(C)=5$ then $cl(C)\le 2.$ If $cl(C)=2$ then by Theorem \ref{glcliff} $(C, \restrict{L}{C})$ is projectively normal and thus $(S, L)$ is projectively normal by regularity of the ladder. \end{pf} \begin{lemma} \label{cbhyper} Let $(S, L)\in{\cal S_5}$ with a hyperelliptic section $C\in |L|.$ Then $S$ is a rational conic bundle, not projectively normal. \end{lemma} \begin{pf} Surfaces with hyperelliptic sections are classified in \cite{so-v}. By degree considerations the only possible case is a rational conic bundle. Since $h^1(L)=0$ Proposition \ref{hyper} and Lemma \ref{equivalent} imply that $(S, L)$ is not projectively normal. \end{pf} \begin{lemma} \label{cbtrig} There are no conic bundles $(S, L)$ with a trigonal section $C\in |L|$ in ${\cal S}_5$ . \end{lemma} \begin{pf} If $S$ is a conic bundle with trigonal section then \cite{fa} Lemma 1.1 gives $g=2q+2$ which is impossible because \cite{LiAq} gives $q\leq 1.$ \end{pf} \begin{theo} \label{g5theo} Let $(S, L)\in{\cal S}_5.$ Then $(S, L)$ fails to be projectively normal if and only if it is \begin{itemize} \item[1)] A rational conic bundle; \item[2)] $(S, L)=(Bl_{12}\bold{ F_1}, 3\frak{C_0}-5\frak{f}-12p)$ with trigonal section $C \in |L|$ and \\$\restrict{L}{C}=K_C-g^1_3+D$. \end{itemize} \end{theo} \begin{pf} >From \cite{LiAq}, Lemma \ref{trighyper}, Lemma \ref{cbhyper} and Lemma \ref{cbtrig}, the following cases are left to investigate: \begin{center} \begin{tabular}{|l|l|l|l|r|} \hline Case &$S$ & $L$& existence\\ \hline\hline 1 & $Bl_{10}\Pin{2}$ & $7p^*(\cal O_{\Pin{2}}(1))-10\sum_1^{10} 2E_i$ & Yes \\ \hline 2 & $Bl_{12}\Pin{2}$ & $6p^*(\cal O_{\Pin{2}}(1))-\sum_1^5 2E_i-\sum_6^{12} E_j$ &Yes\\ \hline 3 & $Bl_{10}\bold {F_e}$, $e=0,1,2$ & $4\frak{C_0}+(2e+5)\frak{f}-\sum_1^7 2E_i-\sum_8^{ 10} E_j$ & Yes\\ \hline 4 & $Bl_{12} \bold{ F_1}$ & $3\frak{C_0}+5\frak{f}-\sum_1^{12}E_i$ & ? \\ \hline \end{tabular} \end{center} where the hyperplane section is trigonal. In case 2 $(S, L)$ admits a first reduction $(S',L')$ with $d'=16$. By \cite{fa} $(S, L)$ cannot have trigonal section and therefore it is projectively normal. In case 3 it is $K_S^2=-2$ so that $K_S(K_S+L)=-3$. Then by \cite{bri-la} Theorem 2.1 $(S, L)$ cannot have trigonal section thus it is projectively normal. Case 1 is a congruence of lines of $\Pin{3}$ of bi-degree $(3,6)$ studied in detail in \cite{ar-so2}. In particular if ${\cal I}_S^*$ is the ideal of $S$ in the grassmanian $G(1,3)$ of lines of $\Pin{3},$ it is $h^0({\cal I}_S^*(2))=0$. From: $$ 0\longrightarrow {\cal I}_G\longrightarrow{\cal I}_S\longrightarrow{\cal I}_S^*\longrightarrow 0$$ recalling that $G\in|{\cal O}_{\Pin{5}}(2)|$ we get $h^0({\cal I}_S(2))=1$ and therefore $(S, L)$ is projectively normal by Lemma \ref{equivalent}. By \cite{GL} Corollary 1.6 a trigonal curve of genus 5 and degree 9 in $\Pin{4}$ fails to be projectively normal if and only if it is embedded via a line bundle $\restrict{L}{C}=K_C-g^1_3+D$ where $D$ is an effective divisor of degree 4. Notice that this means that $ C$ is embedded in $\Pin{4}$ with a foursecant line. \end{pf} \begin{rem} (ADDED IN PROOF) \label{bali} After this work was completed the first author and A. Alzati proved in in \cite{bali} that there exist no surfaces as in Theorem \ref{g5theo} Case 2). Therefore this case does not appear in the table of Theorem 1.1. \end{rem} \subsection{SECTIONAL GENUS $g= 6$} \label{genere6} In this subsection we will study the projective normality of pairs $(S, L)\in{ \cal S}_6$. Notice that by Lemma \ref{nonP41} $L$ embeds $S$ in $\Pin{5}$ and the ladder is regular. \begin{lemma} \label{2normg6}Let $C$ be a curve of genus $6$ embedded in $\Pin{4}$ by the complete linear system associated with a very ample line bundle $\restrict{L}{C}$ of degree $9$. Then $C$ is $2$-normal. \end{lemma} \begin{pf} Consider the exact sequence: $$0\longrightarrow {\cal I}_C\otimes{\cal O}_{ \Pin{4}}(2)\to{\cal O}_{\Pin{4}}(2)\to{\cal O}_C(2)\longrightarrow 0$$ Since $h^0({\cal O}_{\Pin{4}}(2))=15$ and $h^0({\cal O}_C(2))=h^0(2\restrict{L}{C})=18+1-6=13$ it is $h^0(\iof{C}{2}) \ge 2$ with the map $H^0({\cal O}_{\Pin{4}}(2))\longrightarrow H^0({\cal O}_C(2))$ surjective if and only if $h^0( {\cal I}_C(2))=2.$ Assume $h^0(\iof{C}{2})\ge 3,$ and let $Q_i$ for $i =1,2,3$ be three linearly independent quadric hypersurfaces containing $C.$ Because deg$C =9$ and $C$ is non degenerate it must be $dim(\cap_iQ_i) = 2.$ Let $\frak{S} = \cap_iQ_i$ then $h^0(\frak{S}) \ge 3.$ But $\frak{S}$ is a complete intersection $(2,2)$ in $\Pin{4}$ and it is easy to see that $h^0(\frak{S}) = 2$, contradiction. \end{pf} \begin{cor}\label{gen6} Let $(S, L) \in \cal{S}_6.$ Then $(S,L)$ is projectively normal. \end{cor} \begin{pf} Let $C\in|L|$ be a generic section. From Lemma \ref{2normg6} and Lemma \ref{2norm} it follows that $(C,\restrict{L}{C})$ is projectively normal. Since the ladder is regular this implies that $(S, L)$ is projectively normal. \end{pf} \section{Results on Scrolls} A $n$-dimensional polarized variety $(X, L)$ is said to be a scroll over a smooth curve $C$ of genus $g$ if there is a vector bundle \map{\pi}{E}{C} of rank $r = rk\,E = n+1$ such that $(X, L) \simeq ({\Bbb P}(E), \oof{{\Bbb P}(E)}{1}).$ Recall that given a vector bundle $E$ over a curve $C$, $\mu (E)$ and $\mu^- (E)$ of $E$ are defined as ( see \cite{bu} for details) $$\mu(E) = \frac{deg E}{rk E} = \frac{d}{r}.$$ $$\mu^-(E)=min\{\mu(Q)|E\rightarrow Q\rightarrow 0\}$$ $E$ will be called {\em very ample} to signify that the tautological line bundle $\taut{E}$ is a very ample line bundle on ${\Bbb P}(E).$ From \cite[Th 5.1.A]{bu} and general properties of projectivized bundles it follows that: \begin{prop}[\cite{bu}]\label{buscroll} Let $(X, L)$ be a scroll. If $\mu^-(E)>2g$ then $(X, L)$ is projectively normal. \end{prop} The following Lemma is essentially due to Ionescu \cite{fa-li9} : \begin{lemma}[Ionescu]\label{scrolls} Let $(X,L)$ be an n-dimensional scroll over a hyperelliptic curve $C$ of genus $g$ with $L$ very ample. Then $\Delta=ng.$ \end{lemma} \begin{pf} a) Let $X=\Proj{E}$, $L=\taut{E}$ and $\pi:X\longrightarrow C$.\\ By the Riemann-Roch theorem and the fact that $\pi_*({\cal O}_X(1))=E$ it follows that: $$h^0(L)=h^0(C,E)=h^1(C,E)+d-n(g-1)$$ Thus it is enough to show that $h^1(C,E)=0$.\\ By Serre duality $h^1(C,E)=h^0(C,K_C\bigotimes E^{*})$. Assume $h^1(C,E)\neq 0.$ A non trivial section $\sigma\in h^0(C, K_C\bigotimes E^*)$ gives the following surjection: \begin{equation} \label{sigmasur} (K_C\bigotimes E^*)^*\longrightarrow{\cal O}_C(-D)\longrightarrow 0 \end{equation} where $D$ is the divisor on $C$ associated to $\sigma$. Tensoring \brref{sigmasur} with $K_C$ we obtain $$E\longrightarrow K_C-D\longrightarrow 0.$$ Because $\taut{E}$ is very ample, $K_C-D$ is very ample on C. Moreover $K_C-D$ is a special line bundle on $C$ because $h^1(K_C-D)=h^0(D)>0$. This is impossible because $C$ is hyperelliptic. \end{pf} \begin{lemma} \label{scrollg2} Let $(S,L)$ be a two-dimensional scroll over a curve of genus 2 and degree 9 in ${\Bbb P}^{6}$ with $L$ very ample. Then $X={\Bbb P}(E)$ with $E$ stable and $(X, L)$ is projectively normal. \end{lemma} \begin{pf} Let $S={\Bbb P}(E)$ where $E$ is a rank $2$ vector bundle of degree $9$ over a smooth curve of genus $2$. If $E$ is stable then $\mu^-(E)=\mu(E)=\frac{9}{2}>4$ and so by Proposition \ref{buscroll} $(S,L)=({\Bbb P}(E),{\cal O}_{{\Bbb P}(E)})$ is projectively normal. Assume now $E$ non stable. Then there exists a line bundle $Q$ with deg($Q)\leq 4$ such that $E\to Q\to 0$. This contradicts the very ampleness of $Q$ as a quotient of a very ample $E.$ \end{pf} \begin{prop} \label{scrollprop} Let $(S, L)$ be a scroll of degree $d=9$ over a smooth curve $C$ of genus $g.$ Then $(S, L)$ is projectively normal unless possibly if $C$ is trigonal, $3\le g \le 5$ and $S\subset \Pin{5}.$ \end{prop} \begin{pf} Following the proof of Lemma \ref{nonP41} if $\Delta \ge 2$ and $g=1$ then $(S, L)$ is an elliptic scroll (see \cite{fu}) and therefore projectively normal by \cite{Alibaba} or \cite{Ho1},\cite{Ho2}. If $\Delta =4$ and $g=2$ then $(S, L) $ is projectively normal by Lemma \ref{scrollg2}. Let $\Delta=5.$ If $g=6$ then $(S, L)$ is projectively normal by \ref{gen6}. If $g=5$ by Theorem \ref{fujitatheo}, \ref{fujitatheo2}, \ref{glcliff} $(S, L)$ is projectively normal unless $cl(C)\le 1.$ If $g=3,4$ then it is always $cl(C)\le 1.$ By Lemma \ref{scrolls} $C$ must be trigonal. \end{pf} \section{An Adjunction Theoretic Problem} \label{K+L} The question of finding examples for the problem posed by Andreatta, Ein and Lazarsfeld (see introduction) is addressed below. \begin{cor} Let $(S, L)$ be a surface polarized with a very ample line bundle of degree $d=9$ such that the embedding given by $|L|$ is not projectively normal. Then there does not exist a very ample line bundle $\cal{L}$ such that $L= K_S + \cal{L}$ unless $(S, L)$ is the blow up of an elliptic $\Pin{1}$-bundle as in the first case of Theorem \ref{thetheorem}. \end{cor} \begin{pf} Let $(S, L)$ be as in the Table of Theorem \ref{thetheorem}, not as in the first case. Assume $L=K + \cal{L}$ with $\cal{L}$ very ample. Computing $\cal{L}^2$ and $g(\cal{L})$ and using \cite{LiAq} lead to a contradiction in every case. Similarly a contradiction is reached if $(S, L)$ is a scroll over a curve of genus $3,4,5.$ \end{pf} \begin{rem} The existence of an example of a surface as in case 1 of Theorem \ref{thetheorem} where $L=K+\cal{L}$ with $\cal{L}$ very ample is a very delicate question. Let $E$ be an indecomposable rank $2$ vector bundle over an elliptic curve $\cal{C}$ with $c_1(E) =0$ and let $X =\Proj{E}.$ Let $C_0$ be the fundamental section, let $M$ be any line bundle whose numerical class is $2C_0 + f$ and let $p: S=Bl_3X \to X$ be the blow up of $X$ at three points $P_i$ $i=1,..,3.$ Using the same notation for the blow up introduced in subsection \ref{notation} consider a line bundle $L\equiv 2\frak{C_0} + 3\frak{f} - \sum_iE_i.$ Notice that $L\equiv K_S + \cal{L}$ where $\cal {L}\equiv 4\frak{C_0} + 3\frak{f} - \sum_i2E_i.$ Moreover $\cal{L} \equiv K_S + H,$ $H^2 = 9,$ $H \equiv 3T$ where $T= p^*(M) - \sum_1^3 E_i.$ Recent results of Yokoyama and Fujita \cite{fuyo}, \cite{Yoko}) show that the $P_i's $ can be chosen generally enough to have $T$ ample but not effective. Reider's theorem then shows that $\cal{L}$ is very ample if it is possible to choose the $P_i's$ such that for \underline{every} line bundle $M$ whose numerical class is $2C_0 + f$ it is $|p^*(M)-\sum_i E_i|= \emptyset.$ \end{rem}

9710

alg-geom/9710013

en

https://arxiv.org/abs/alg-geom/9710013

alg-geom/9710013

Ilya Zakharevich

Ilya Zakharevich

Quasi-algebraic geometry of curves I. Riemann-Roch theorem and Jacobian

99 pages, AmsLaTeX

We discuss an analogue of Riemann-Roch theorem for curves with an infinite number of handles. We represent such a curve X by its Shottki model, which is an open subset U of CP^{1} with infinite union of circles as a boundary. An appropriate bundle on X is \omega^{1/2} \otimes L, L being a bundle with (say) constants as gluing conditions on the circles. An admissible section of an appropriate bundle on X is a holomorphic half-form on U with given gluing conditions and H^{1/2}-smoothness condition. We study the restrictions on the mutual position of the circles and the gluing constants which guarantee the finite dimension of the space of appropriate sections of admissible bundles, and make the Riemann-Roch theorem hold. The resulting Jacobian variety is described as an infinite-dimension analogue of a torus.

alg-geom

\section{Introduction } The need to carve out a set of curves of infinite genus for which ``most'' theorems of algebraic geometry are true comes out from the following observations: \begin{enumerate} \item The existence of algebro-geometric description of solutions of infinite-dimensional integrable systems; \item The ability to describe the series of perturbation theory for string amplitudes as integrals over moduli spaces; \item The hope that the union of compactifications of moduli spaces may have a simpler geometry than the moduli spaces themselves. \end{enumerate} Different approaches which would result in different sets of curves are possible (as in papers of Feldman, Kn\"orrer and Trubowitz cf.~\cite{FelKnoTru96Inf}), thus we first motivate our choice of tools (Shottki model, $ H^{1/2} $-topology, capacities and half-forms) as (probably) a best one to fulfill the expectations of the above origins of the theory. Until Section~\ref{s0.10} we discuss how the above topics motivate the choice of the Shottki model as a way to describe a curve of infinite genus. After this (up to section~\ref{s0.40}) we describe motivations for the choice of half-forms, $ H^{1/2} $-topology, and generalized-Sobolev-spaces (or capacities) to describe admissible sections on a given curve. In the remaining part of the introduction we do a walk-through the methods and results one can find in this paper, as well as some historic remarks. \subsection{Integrable systems and algebro-geometric solutions } A great break-through in the first topic came with the work \cite{McKTru76Hil}. A hyperelliptic curve of genus $ g $ with real branching points and a divisor of degree $ g $ on this curve allows one to construct a so-called {\em algebro-geometric\/} solution of KdV equation \begin{equation} u_{xx x}+12uu_{x}-4u_{t}=0. \notag\end{equation} Such solutions are called $ g $-{\em gap potentials}. McKean and Trubowitz studied what substitutes algebraic geometry for solutions of KdV which are {\em not\/} finite gap potentials. To such a solution they associated a curve (i.e., a complex manifold of dimension 1) which was hyperelliptic of {\em infinite genus}, i.e., had infinitely many branching points, and some substitution for the notion of a divisor of degree $ g $. It was shown that for the curves which are related to KdV equation one can construct a well-behaved analogue of $ 1 $-dimensional algebraic geometry. One should consider this analogue as a generalization of algebraic geometry to hyperelliptic curves of infinite genus (in fact only to some special curves of this type). Note that other integrable systems lead to different classes of curves which appear in algebro-geometric solutions for the systems. Thus one may expect that infinite-dimensional integrable systems may lead to generalizations of algebraic geometry to different classes of curves of infinite genus. For this approach one of the richest systems is so called KP system \begin{equation} \left(u_{x xx}+12uu_{x}-4u_{t}\right)_{x} +3u_{yy}=0. \notag\end{equation} To describe an algebro-geometric solution of a KP system one starts with an arbitrary algebraic curve, and an arbitrary linear bundle on this curve. Thus to generalize the approach of \cite{McKTru76Hil} to the KP equation, one needs to study curves of infinite genus of {\em generic\/} form (as opposed to a hyperelliptic curve) and bundles on them. Suppose for a minute that we have such a generalization, i.e., a collection of curves and bundles on them. Call the members of these families {\em admissible}. We assume that to any admissible curve and a bundle on it we can associate some solution of KP equation. Let us investigate what can we deduce about this collection from the known properties of KP system (the properties below are applicable at least to algebro-geometric solutions). The dynamics of KP leaves the curve the same, but changes the bundle. Thus the collection of admissible bundles should be reach enough to include all the bundles obtained by time-flow of KP. In fact KP can be generalized to include an infinite collection of commuting flows (with different time variables), and they (taken together) can transform any bundle to any other one (at least in finite-genus case). Thus we should expect that we need our collection to include {\em all\/} the possible bundles, thus the whole Jacobian. The dynamics of KP is described in terms of locally affine structure on the Jacobian, hence one should be able to describe the Jacobian as a quotient of a vector space by a lattice. Algebraic geometry identifies the Jacobian with a quotient of the space of global holomorphic forms by the forms with integer periods. Thus one needs something similar to this description. There is an alternative description of a solution of KP equation in terms of so called $ \tau $-{\em functions}. The relation of algebro-geometric description with the description in terms of $ \tau $-functions needs a construction of a Laurent series of a meromorphic global section of an admissible bundle. This ceases to be trivial in infinite-genus case, since already the description in \cite{McKTru76Hil} shows that one may need to apply this operation at the {\em infinity\/} of the curve, i.e., at the points added to the curve to compactify it. These points {\em are not\/} smooth points of the curve, the curve is not even a topological manifold near these points. Finally, to get somewhat {\em explicit\/} description of solutions of KP as functions of several variables, one needs a way to explicitly describe global sections of linear bundles. Moreover, the bundles we need to consider should have a finite number (preferably one!) of independent global sections. A tool to construct such bundles in the case of finite genus is the Riemann--Roch theorem, which gives an estimate on the dimension of sections of the given bundles, and this estimate is precise in the case of bundles in generic position. These sections are described in terms of $ \theta $-functions, thus we will also need to describe $ \theta $-functions. Collecting all this together, we see that we need to describe curves $ X $ which can be equipped with a linear bundle $ {\cal L} $ such that the space of global sections is finite-dimensional. One should be able to define what does it mean that two bundles $ {\cal L} $ and $ {\cal L}' $ are equivalent, describe the equivalence classes of bundles in terms of global holomorphic $ 1 $-forms, give a local description of sections of the bundles, and give a global description of sections of the bundles in terms of $ \theta $-functions. In this paper we do not complete this program. However, we describe all the ingredients but the last one. \subsection{Universal Grassmannian }\label{s0.4}\myLabel{s0.4}\relax The other two origins, the string theory and geometry of moduli spaces, come into play if we consider the algebro-geometric methods of solving integrable systems in the other direction, as a way to find information of algebro-geometric type from solutions of integrable systems (similar to solution of Shottki problem in \cite{Shio86Char}). The key idea is that in the investigated cases the set of algebro-geometric solutions is {\em dense\/} in the set of all solutions, thus the set of the solutions is a {\em completion\/} of the set of algebro-geometric solutions. Since the the set of algebro-geometric solutions is a moduli space of appropriate structures, and the set of solutions is a linear space (due to the possibility to solve the Cauchy problem), we see that the moduli space has a completion which has a topology of a linear space. While the moduli spaces carry remarkable measures \cite{BeiMan86Mum} used in the integrals of the string theory, this linear space has a symplectic structure, so one may expect that this simplectic structure may have a relationship to the measure on the subsets corresponding to algebro-geometric solutions. To carry out this program one needs to investigate how an arbitrary solution may be approximated by algebro-geometric solutions. For KdV equation this question was answered by \cite{McKTru76Hil} (for a fixed spectrum), ameliorated to describe the inclusion of the subset of finite gap potentials up to a diffeomorphism in \cite{ZakhFinGap2,ZakhFinGap3}, up to a symplectomorphism in \cite{BatBloGuil95Sym}. The answer is that the above inclusion is isomorphic to inclusion of trigonometric polynomials into the space of all functions (for a very wide range of spaces of functions). Since trigonometric polynomials fill finite-dimensional coordinate subspaces in spaces of functions (using Fourier coefficients as coordinates), we see that this picture is very similar to one of a divisor with normal crossings. Much simpler problem is how an algebro-geometric solution may be approximated by (simpler) algebro-geometric solutions. A remarkable fact is that such problems for different classes of integrable systems may be solved via a uniform approach: compactification of moduli spaces via {\em Universal Grassmannian}. The reason is that the standard way to associate a solution of an integrable system to an algebro-geometric data comes from consideration of the mapping to {\em Universal Grassmannian}. The Universal Grassmannian gives a convenient way to say that the space of sections of a linear bundle on one curve is close to the space of sections on another linear bundle on another curve. Consider an algebraic curve $ X $ with a fixed point $ P\in X $ and a local coordinate system $ x $ in a neighborhood of $ P $ which maps $ P $ to $ 0\in{\Bbb C} $. Let $ {\cal L} $ be a linear bundle on $ X $, fix a trivialization of this bundle in a neighborhood of $ P $. One can associate a Laurent series $ {\frak l}\left(\varphi\right) $ to any section $ \varphi $ of $ {\cal L} $ in a punctured neighborhood of $ P $: using a coordinate system on $ X $ and the trivialization of $ {\cal L} $, one can write this section as a function of $ z\approx0 $, $ z\not=0 $. Let $ V $ be the space of meromorphic sections of $ {\cal L} $ which are holomorphic outside of $ P $. (Abusing divisor notations, one can write $ V $ as $ \Gamma\left(X,{\cal L}\left(\infty\cdot P\right)\right)=\bigcup_{k}\Gamma\left(X,{\cal L}\left(k\cdot P\right)\right) $.) Then $ {\frak l}\left(V\right) $ is a subspace of the space $ {\frak L} $ of Laurent series. Let the universal Grassmannian $ \operatorname{Gr}\left({\frak L}\right) $ be the Grassmannian of subspaces of $ {\frak L} $. Then $ {\frak l}\left(V\right)\in\operatorname{Gr}\left(V\right) $ depends on $ X $, $ {\cal L} $, the coordinate system near $ P $, and on the trivialization of $ {\cal L} $ near $ P $. On $ {\frak L} $ there are natural actions of the group of formal diffeomorphisms\footnote{I.e., invertible $ \infty $-jets of mappings $ {\Bbb C} \to {\Bbb C} $ with $ 0\in{\Bbb C} $ being a fixed point.} of $ \left({\Bbb C},0\right) $ and the group of multiplication by invertible Taylor series. Let $ G $ be the semidirect product of these groups. The action of $ G $ on $ {\frak L} $ corresponds (via $ {\frak l} $) to changes of coordinate system on $ X $, and a change of the trivialization of $ {\cal L} $. We see that to a triple $ \left(X,P,{\cal L}\right) $ we can naturally associate a point $ {\frak l}\left(X,P,{\cal L}\right)\in\operatorname{Gr}\left({\frak L}\right)/G $. Note that while only the mapping to $ \operatorname{Gr}\left({\frak L}\right)/G $ is invariantly defined, one can get a canonical lifting to $ \operatorname{Gr}\left({\frak L}\right) $ as far as $ X\not={\Bbb C}P^{1} $. The description below does not behave well w.r.t. deformations of the curve/bundle, but can be easily modified to do so. Given a bundle $ {\cal L} $, we can find $ k\in{\Bbb Z} $ such that $ {\cal L}\left(k\cdot P\right) $ has only one independent section, and $ {\cal L}\left(\left(k+1\right)\cdot P\right) $ has another one. The ratio $ z $ of these two sections identifies a neighborhood of $ P $ with a neighborhood of $ \infty\in{\Bbb C}P^{1} $, and this identification is defined up to an affine transformation. Now the only section of $ {\cal L}\left(k\cdot P\right) $ gives a local identification of $ {\cal L} $ with $ {\cal O}\left(-k\cdot P\right) $. Together with a coordinate system in a neighborhood of $ P $ it gives a local section of $ {\cal L} $ defined up to multiplication by a constant. What remains is to pick up a coordinate system from those which differ by an affine transformation. If $ X\not={\Bbb C}P^{1} $, then for some $ l>k+1 $ there is a section of $ {\cal L}\left(l\cdot P\right) $ of the form $ z^{l}+p_{l-1}\left(z\right)+az^{-m}+bz^{-m-1}+O\left(z^{-m-2}\right) $ with $ a\not=0 $, $ p_{l-1} $ being a polynomial of degree $ l-1 $, and $ m>0 $. Taking the minimal possible $ l $, we see that the condition that $ a=1 $, $ b=0 $ picks up a coordinate system out of the above class, unique up to multiplication by a root of degree $ l+m $ of 1. (If $ X={\Bbb C}P^{1} $, then of course there is no canonically defined coordinate system, since $ \left(X,P,{\cal L}\right) $ has automorphisms.) Now using this coordinate system and the corresponding trivialization of $ {\cal L} $ one gets a canonically defined image of $ \left(X,P,{\cal L}\right) $ in $ \operatorname{Gr}\left({\frak L}\right) $. Let the {\em Teichmuller\/}--{\em Jacoby\/} space $ {\cal N}_{g,1,d} $ be the moduli space (i.e., the ``set'' of equivalence classes) of triples $ \left(X,P,{\cal L}\right) $ with $ g\left(X\right)=g $, and $ \deg \left({\cal L}\right)=d $. What is of primary importance to us is the fact that $ {\frak l}|_{{\cal N}_{g,1,d}} $ is an injection (even in the case when $ X $ may have double points). Indeed, \begin{enumerate} \item Meromorphic functions on $ X $ may be described as ratios of elements in $ {\frak l}\left(X,P,{\cal L}\right) $, thus normalization $ \operatorname{Norm}\left(X\right) $ of $ X $ may be reconstructed basing on $ {\frak l}\left(X,P,{\cal L}\right) $; \item The point $ P $ can be reconstructed since we know all the meromorphic functions on $ X $, and know the order of pole at $ P $; \item To reconstruct the lifting $ \bar{{\cal L}} $ of $ {\cal L} $ to $ \operatorname{Norm}\left(X\right) $ note that if $ \varphi\in V $, then the divisor (of zeros) of $ \varphi $ can be reconstructed as poles of functions in $ {\frak l}\left(\varphi\right)^{-1}{\frak l}\left(V\right) $. \item Finally, to describe the gluings one needs to perform to get $ X $ from $ \operatorname{Norm}\left(X\right) $ it is sufficient to consider a complement to $ \Gamma\left({\cal L}\left(\infty\cdot P\right)\right) $ in $ \Gamma\left(\bar{{\cal L}}\left(\infty\cdot P\right)\right) $. \end{enumerate} As a corollary, the mapping $ {\frak l} $ defines an inclusion of $ {\cal N}_{g,1,d} $ into $ \operatorname{Gr}\left({\frak L}\right)/G $. Since the image has a natural structure of a topological space, this inclusion defines some natural {\em compactification\/} $ \bar{{\cal N}}_{g,1,d} $ of the Teichm\"uller--Jacoby space. Indeed, consider the closure of $ {\frak l}\left({\cal N}_{g,1,d}\right) $ in $ \operatorname{Gr}\left({\frak L}\right)/G $. As we will see it shortly, this closure is an image of a compact smooth manifold\footnote{In fact we will see that already the {\em normalization\/} of the image is smooth.}. The description of this compactification is very similar to the description of Deligne--Mumford compactification, it is carried out by adding to $ {\cal N}_{g,1,d} $ so called {\em semistable\/} objects. We will see that these object are non-smooth curves and sheaves on them. The mapping to Universal Grassmannian is a way to collect all the moduli spaces for different genera into ``one big heap''. It is relevant to curves of infinite genus since on a back yard of this heap one expects to find moduli spaces of curves of infinite genus. \subsection{Semistable elliptic curves }\label{s0.5}\myLabel{s0.5}\relax In this section we consider the case $ g=1 $, where one can calculate the mapping to the Universal Grassmannian explicitly. We will see that in this case the closure of the image is smooth\footnote{As we will see it later, $ g=1 $ is last case when one does not need any normalization of the image.}. Fix an elliptic curve $ E $. Then the set of classes of equivalence of linear bundles of degree 0 (i.e., the Jacobian variety) is isomorphic to the curve itself (after we fix a point $ P $ on the curve). Given any point $ Q\in E $, the divisor $ Q-P $ determines a linear bundle $ {\cal O}\left(Q-P\right) $ on $ E $, and any bundle of degree 0 is isomorphic to exactly one bundle of this form. In the case of $ g=1 $ it is not instructive to consider the Teichm\"uller--Jacoby manifold {\em literally}, since it is tainted by the fact that an involution $ \sigma $ of an elliptic curve $ E $ sends any bundle $ {\cal L} $ of degree 0 to $ {\cal L}^{-1} $, thus the moduli space of bundles of degree 0 on an elliptic curve fixed {\em up to an isomorphism\/} is $ E/\sigma $, i.e., a rational curve. To fix the problem one can add some harmless discrete parameter which would prohibit $ \sigma $ to be an automorphism, say, consider collections $ \left(X,P,{\cal L},\alpha\right) $, $ \alpha $ being a homology class modulo $ {\Bbb Z}/3Z $. However, knowing that we can fix this problem, we are going to ignore it whatsoever, since it can also be avoided by considering curves close to the given one, what we are going to do anyway. For every elliptic curve $ E $ there is a unique number $ j\in{\Bbb C} $ such that $ E $ is isomorphic to the curve \begin{equation} y^{2}=\left(500j-1\right)x^{3}-15jx-j. \notag\end{equation} The number $ j $ is called the $ j $-{\em invariant\/} of the curve. The moduli (Teichm\"uller) space of curves of genus 1 may be identified with $ {\Bbb C} $ via $ j $-invariant. The Teichm\"uller--Jacoby space is fibered over the Teichm\"uller space with the fiber being the Jacobian, i.e., the elliptic curve itself (i.e., in the standard notations this space coincides with $ {\cal M}_{1,2} $). To compactify the Teichm\"uller space one adds a point with $ j=\infty $, obtaining $ {\Bbb C}P^{1} $ as the compactified Teichm\"uller space. To visualize the compactification of the Teichm\"uller space it is more convenient to consider the family of elliptic curves $ X_{\varepsilon} $ given by $ y^{2}=x^{2}-x^{3}-\varepsilon $. When $ \varepsilon \to 0 $ (so $ j=\frac{1}{15^{3}\varepsilon\left(4-27\varepsilon\right)} \to \infty $) one gets a rational curve with a double point. Below we discuss how the corresponding compactification of Teichm\"uller--Jacoby space looks like. We will see that instead of adding one point, we need to add a rational curve with a selfintersection. The compactified Teichm\"uller--Jacoby space maps to the compactified Teichm\"uller space, with the fiber over $ \infty $ being the above singular curve. Consider an arbitrary elliptic curve $ E $ with large a $ j $-invariant, and a bundle of degree 0 on $ E $. One can identify $ E $ with $ X_{\varepsilon} $ for an appropriate small $ \varepsilon $. Fix a point $ P_{\varepsilon} $ on $ X_{\varepsilon} $, then for any bundle $ {\cal L} $ of degree 0 on $ X_{\varepsilon} $ one can identify $ {\cal L} $ with $ {\cal O}\left(Q_{\varepsilon}-P_{\varepsilon}\right) $, $ Q_{\varepsilon} $ being an appropriate point on $ X_{\varepsilon} $. Fix $ P_{\varepsilon} $ to be the point of $ X $ which has $ y $-coordinate being 0, and is close to $ \left(1,0\right)=P_{0} $. Suppose that $ {\cal L}\not={\cal O} $, thus $ Q_{\varepsilon}\not=P_{\varepsilon} $. Let us describe meromorphic sections of $ {\cal L}={\cal O}\left(Q_{\varepsilon}-P_{\varepsilon}\right) $ with the only pole being at $ P_{\varepsilon} $. It is the same as to describe sections of $ {\cal O}\left(Q_{\varepsilon}+kP_{\varepsilon}\right) $, $ k\gg0 $, i.e., meromorphic functions on $ X_{\varepsilon} $ with a (possible) simple pole at $ Q_{\varepsilon} $, and any pole at $ P_{\varepsilon} $. These sections are uniquely determined by their singular part at $ P_{\varepsilon} $ (since there is no meromorphic function on an elliptic curve $ X_{\varepsilon} $ which has only a simple pole at $ Q_{\varepsilon} $). Thus to describe the space of these sections, it is sufficient to describe functions $ f_{k} $ on $ X_{\varepsilon} $ which have a pole of prescribed order $ k $ at $ P_{\varepsilon} $ and possibly an additional pole at $ Q_{\varepsilon} $. (Note that $ f_{0}\equiv 1 $ should be considered as a section of $ {\cal O}\left(Q_{\varepsilon}-P_{\varepsilon}\right) $ with a pole at $ P_{\varepsilon} $, since holomorphic sections of $ {\cal O}\left(Q_{\varepsilon}-P_{\varepsilon}\right) $ are identified with functions having a zero at $ P_{\varepsilon} $.) Note that $ y $ is a meromorphic function on $ X_{\varepsilon} $ with a triple pole at infinity of $ X_{\varepsilon} $, $ x $ has a double pole at infinity of $ X_{\varepsilon} $. Let $ P_{\varepsilon} $ have coordinates $ \left(a_{\varepsilon},0\right) $, the line through $ P_{\varepsilon} $ and $ Q_{\varepsilon} $ have the equation $ c_{\varepsilon}x+y=d_{\varepsilon} $, the third point $ R_{\varepsilon} $ of intersection of this line and $ X_{\varepsilon} $ has $ x $-coordinate $ b_{\varepsilon} $. If $ k=2k_{1} $ is even, one can take $ f_{k}=\frac{1}{\left(x-a_{\varepsilon}\right)^{k_{1}}} $. If $ k=2k_{1}-1 $, and $ k_{1}>1 $, then $ yf_{k+1} $ has no pole at infinity, thus can be considered as $ f_{k} $. What remains is to describe $ f_{1} $. Note that $ \frac{1}{c_{\varepsilon}x+y-d_{\varepsilon}} $ has a pole at $ P_{\varepsilon} $, $ Q_{\varepsilon} $, and at $ R_{\varepsilon} $. Moreover, it has a zero of third order at infinity. Multiplying this function by $ x-b_{\varepsilon} $, we kill the pole at $ R_{\varepsilon} $, thus get $ f_{1}=\frac{x-b_{\varepsilon}}{c_{\varepsilon}x+y-d_{\varepsilon}} $. Let us investigate how the space spanned by $ f_{i} $, $ i\geq0 $, depends on $ X_{\varepsilon} $ and $ Q_{\varepsilon} $. First of all, $ \varepsilon $ depends smoothly on $ Q_{\varepsilon} $ (if coordinates of $ Q_{\varepsilon} $ are $ q_{1},q_{2} $, then $ \varepsilon=q_{2}^{2}-q_{1}^{2}-q_{1}^{3} $). From this moment on, we may denote $ Q_{\varepsilon} $ by just $ Q $, since $ \varepsilon $ is a function of $ Q $. Second, $ x $ depends smoothly on $ y $ and $ \varepsilon $ in a neighborhood of $ P_{\varepsilon} $ on $ X_{\varepsilon} $, {\em including\/} the case $ \varepsilon=0 $. Thus $ P_{\varepsilon} $ depends smoothly on $ Q $, {\em including\/} cases $ \varepsilon=0 $ and $ Q=\left(0,0\right)\in{\Bbb C}^{2} $. Hence $ c_{\varepsilon} $, $ d_{\varepsilon} $ depend smoothly on $ Q $. Substituting $ y=-c_{\varepsilon}x+d_{\varepsilon} $ into $ y^{2}=x^{2}-x^{3}-\varepsilon $, we get a cubic equation in $ x $, which has $ q_{1} $ as a root, and two well-separated roots at $ a_{\varepsilon} $ and $ b_{\varepsilon} $. In particular, $ b_{\varepsilon} $ depends smoothly on $ Q $. Note that one can consider $ y $ as a local coordinate near $ P_{\varepsilon} $, thus the functions $ f_{k} $ can be written as function of $ y $ if $ y $ is small. We see that all the functions $ f_{k}\left(y\right) $ depend smoothly on $ \left(q_{1},q_{2}\right) $ (at least as far as $ q_{1} $, $ q_{2} $ are small, $ y $ is small). If $ q_{2}^{2}-q_{1}^{2}-q_{1}^{3}\not=0 $, the described above space $ V_{q_{1},q_{2}}=\left<f_{i}\left(y\right)\right>_{i\geq0}\subset{\frak L} $ is the image of $ \left(X_{\varepsilon},P_{\varepsilon},{\cal L}\right) $ in the Universal Grassmannian, i.e., it is in $ {\frak l}\left({\cal N}_{1,1,0}\right) $. We see that the closure of $ {\frak l}\left({\cal N}_{1,1,0}\right) $ contains at least the subspaces $ V_{q_{1},q_{2}} $ with $ q_{2}^{2}-q_{1}^{2}-q_{1}^{3}=0 $. What remains is to give a description of these subspaces as spaces of sections of sheaves on non-smooth curves. We will see that if $ \left(q_{1},q_{2}\right)\not=\left(0,0\right) $, it is sufficient to consider {\em linear bundles\/} on non-smooth curves, but if $ \left(q_{1},q_{2}\right)=\left(0,0\right) $, a consideration of a sheaf is unavoidable. It is obvious {\em which\/} non-smooth curve we need to consider: the limit $ X_{0} $ of $ X_{\varepsilon} $ when $ \varepsilon \to $ 0, with the equation $ y^{2}=x^{2}-x^{3} $. There is a parameterization mapping $ {\Bbb C}P^{1} \to X_{0} $, $ \lambda \mapsto \left(x,y\right) $ if $ y=\lambda x $. It is a bijection outside of $ \lambda=\pm1 $, these two points are both mapped to (0,0). Thus $ X_{0} $ is a rational curve with two points glued together. To describe a linear bundle $ {\cal L} $ on $ X_{0} $ one needs to describe its lifting to $ {\Bbb C}P^{1} $, and the identification of two fibers over $ \lambda=\pm1 $. If the lifting has degree 0, it is a trivial bundle, thus one can describe the gluing by a number $ \theta\in{\Bbb C}^{*} $. Denote this bundle $ {\cal L}_{\theta} $, it has a global holomorphic section iff $ \theta=1 $. Note that if $ P_{0}=\left(1,0\right)\in X_{0} $, $ Q\in X_{0} $, $ Q\not=\left(0,0\right) $, then the mapping $ f\left(\lambda\right) \to \frac{\lambda-\lambda_{0}}{\lambda}f\left(\lambda\right) $ gives an isomorphism of $ {\cal O}\left(Q-P_{0}\right) $ to $ {\cal L}_{\theta} $ if $ \theta=\frac{1-\lambda_{0}}{1+\lambda_{0}} $, $ \lambda_{0} $ being the coordinate of $ Q $. The defined above functions $ f_{k}\left(y\right) $ form a basis of meromorphic sections of $ {\cal O}\left(Q-P_{0}\right) $, thus we see that for $ Q\in X_{0} $ the defined above subspace $ V_{q_{1},q_{2}}\subset{\frak L} $ may be interpreted as $ {\frak l}\left(X_{0},P_{0},{\cal L}_{\theta}\right) $. Now investigate what happens if $ Q \to \left(0,0\right) $ along $ X_{0} $, i.e., $ \theta \to 0 $ or $ \theta \to \infty $. Instead of $ y $, consider the coordinate $ \lambda $ in a neighborhood of $ P_{0} $ on $ X_{0} $. In this coordinate the space $ V\left(X_{0},P_{0},{\cal L}_{\theta}\right) $ is the set of polynomials $ p\left(\mu\right) $ in $ \mu=\lambda^{-1} $ which satisfy $ p\left(1\right)=\theta p\left(-1\right) $. Denote this subspace of $ {\Bbb C}\left[\mu\right] $ by $ V_{\theta} $. Obviously, the subspace $ V_{\theta} $ depends smoothly on $ \theta\in{\Bbb C}P^{1} $, $ V_{0} $ consists of polynomials with a zero at 1, $ V_{\infty} $ consists of polynomials with a zero at $ -1 $. Moreover, images of $ V_{0} $ and $ V_{\infty} $ in $ \operatorname{Gr}\left({\frak L}\right)/G $ coincide, since $ V_{\infty}=\frac{\mu+1}{\mu-1}V_{0} $, and $ \frac{\mu+1}{\mu-1} $ is smooth near $ \mu=\infty $, therefore $ f\left(\mu\right) \to \frac{\mu+1}{\mu-1}f\left(\mu\right) $ corresponds to an element of $ G $. Returning back to the representation of $ V_{q_{1},q_{2}} $, $ q_{1}=q_{2}=0 $, in terms of $ f_{k} $, we see that $ f_{1}=x/y=\lambda^{-1} $, $ f_{2}=\left(x-1\right)^{-1}=-\lambda^{-2} $, thus $ f_{2k}=\pm\lambda^{-2k} $; $ y=\lambda\left(1-\lambda^{2}\right) $, thus $ f_{2k+1}=\frac{1-\lambda^{2}}{\lambda^{2k-1}}=\lambda^{-2k-1}-\lambda^{-2k+1} $. We see that this space is $ {\Bbb C}\left[\lambda^{-1}\right] $. Hence the limit of spaces of sections of $ {\cal O}\left(Q-P\right) $ when $ Q \to \left(0,0\right) $ is isomorphic to the space of meromorphic functions on the normalization $ {\Bbb C}P^{1} $ of $ X_{0} $ with the only pole at $ \lambda=0 $, i.e., with $ {\frak l}\left({\Bbb C}P^{1},0,{\cal O}\right) $. If we want to describe this space in the same terms as we described the mapping $ {\frak l} $, we should consider sections of $ {\cal O} $ on $ {\Bbb C}P^{1} $ as sections of the direct image $ \pi_{*}{\cal O} $ on $ X_{0} $, $ \pi $ being the projection of $ {\Bbb C}P^{1} $ to $ X_{0} $. Since $ \pi_{*}{\cal O} $ is no longer a sheaf of sections of a linear bundle, we conclude that we {\em need\/} to consider sheaves instead of bundles. The only thing which remains to prove is that the completion of the image $ {\frak l}\left({\cal N}_{1,1,0}\right) $ is smooth. We had already shown the part of the completion we consider here is an image of a neighborhood of 0 in $ {\Bbb C}^{2} $. What remains to prove is that the mapping $ {\Bbb C}^{2} \xrightarrow[]{V_{\bullet}} \operatorname{Gr}\left({\frak L}\right)/G $ has a non-degenerate derivative at (0,0). It is sufficient to pick up a section of the projection $ \operatorname{Gr}\left({\frak L}\right) \to \operatorname{Gr}\left({\frak L}\right)/G $, and show that a lifting of $ V_{\bullet} $ to this section has a non-degenerate derivative at (0,0). Again, it is sufficient to restrict this mapping to $ X_{0} $ and show that the derivatives along two branches of $ X $ at (0,0) are independent. One branch of $ X_{0} $ gives a family of subspaces $ V_{\theta} $, $ \theta\approx0 $, $ V_{\theta}\subset{\Bbb C}\left[\lambda^{-1}\right] $. It is a simple but tedious calculation to show that the lifting defined in Section~\ref{s0.4} results in a smooth lifting of $ V_{\bullet} $, and that the smooth liftings resulting from two branches are non-degenerate, and have different derivatives indeed. We see that to describe the elements in the completion of $ {\frak l}\left({\cal N}_{1,1,0}\right) $ in a way similar to the description of $ {\frak l} $ one needs to consider both degenerated curves and degenerated bundles (i.e., sheaves) on these curves. Using the fact that the space of sections of $ \pi_{*}{\cal O} $ is close to the set of sections of $ {\cal O}\left(Q\right) $ if $ Q $ is close to the double point (0,0) of $ X_{0} $, it is reasonable to abuse notations and write $ \pi_{*}{\cal O} $ as $ {\cal O}\left(O\right) $, here $ O=\left(0,0\right) $. \subsection{Compactified moduli spaces } We start with a description of elements of the would-be compactified moduli spaces. As we have seen it in Section ~\ref{s0.5}, we need to allow the curves to have double points, and allow a generalization of a notion of bundle. We call such bundles {\em non-smooth\/} bundles, in fact they are sheaves which are (locally) direct images of bundles on the normalization\footnote{Recall that in the case $ \dim =1 $ {\em normalization\/} coincides with ungluing double points and straightening out cusps.} of the curve. \begin{definition} Let $ Y $ be a connected curve with only singularities $ \operatorname{Sing}\left(Y\right) $ being double points, let $ \widetilde{Y} $ be the normalization of $ Y $. Define $ g\left(Y\right)=g\left(\widetilde{Y}\right)+\operatorname{card}\left(\operatorname{Sing}\left(Y\right)\right) $, for a subset $ S_{0}\subset\operatorname{Sing}\left(Y\right) $ let $ Y_{S_{0}} $ be the curve obtained by ungluing points in $ S_{0} $. A {\em smooth linear bundle\/} $ {\cal L} $ on $ Y $ is a linear bundle $ \widetilde{{\cal L}} $ on $ \widetilde{Y} $ with a fixed identification of fibers at points of $ \widetilde{Y} $ over the same point on $ Y $. Define {\em degree\/} of a smooth linear bundle as the $ \deg \left(\widetilde{{\cal L}}\right) $. A {\em section\/} of $ {\cal L} $ is a section of $ \widetilde{{\cal L}} $ compatible with identifications of fibers. A (not necessarily smooth) {\em linear bundle\/} $ {\cal L} $ on $ Y $ is a collection $ \left(S_{0},\bar{{\cal L}}\right) $ of $ S_{0}\subset\operatorname{Sing}\left(Y\right) $ and a smooth linear bundle $ \bar{{\cal L}} $ on $ Y_{S_{0}} $. A {\em section\/} of $ {\cal L} $ is a section of $ \bar{{\cal L}} $. Let $ \deg \left({\cal L}\right)=\deg \left(\bar{{\cal L}}\right)+\operatorname{card}\left(S_{0}\right) $, call $ S_{0} $ the set of {\em double points\/} of $ {\cal L} $. Let $ P\subset\operatorname{Smooth}\left(Y\right) $ be a finite collection of points. Call the collection $ \left(Y,P,{\cal L}\right) $ {\em semistable\/} if $ \left(Y,P,{\cal L}\right) $ has no infinitesimal automorphisms, i.e., if any rational connected component of $ \widetilde{Y} $ has at least three points in $ P $ or over double points of $ Y $, and if $ Y $ is a (smooth) elliptic curve, then either $ P\not=\varnothing $, or $ \deg {\cal L}\not=0 $. \end{definition} In other words, for a ``non-smooth linear bundle'' $ {\cal L} $ the double points of the curve are broken into two subsets: for one (smooth points of $ {\cal L} $) we fix identifications of fibers of the lifting to normalizations, for another one (as we have seen in Section 0.5, one should consider them as {\em poles\/} of $ {\cal L} $) local sections of $ {\cal L} $ may have ``different'' values at two branches of $ Y $ which meat at a given point. Note that the smooth structures on the Teichm\"uller space and on the Jacobian of a smooth curve provide an atlas on the set of equivalence classes of smooth curves/bundles. The explicit deformation which we are going to describe now will provide an atlas in a neighborhood of a class of a semistable curve. Consider a curve $ Y $ as in the definition, let $ Q\in Y $ be a double point, and $ y_{1} $, $ y_{2} $ be coordinates on the branches of $ Y $ near $ Q $, so $ Y $ is locally isomorphic to $ y_{1}y_{2}=0 $. Let $ {\cal L} $ be a linear bundle over $ Y $ with a fixed trivialization near $ Q $. Suppose that $ Q $ is a double point of $ {\cal L} $, thus local sections of $ {\cal L} $ can be written as (unrelated) functions $ f_{1}\left(y_{1}\right) $, $ f_{2}\left(y_{2}\right) $. Note that trivialization of $ {\cal L} $ induces identification of fibers of $ \widetilde{{\cal L}} $ over $ Q $. Let $ \overset{\,\,{}_\circ}{{\cal L}} $ be the bundle on $ Y $ (smooth near $ Q $) obtained by addition of this identification to $ {\cal L} $. Extend the bundle $ \overset{\,\,{}_\circ}{{\cal L}} $ from $ Y $ to $ Y\cup\left\{\left(y_{1},y_{2}\right) | |y_{1}|,|y_{2}|\ll 1\right\} $ trivially using the trivialization of $ {\cal L} $ near $ Q $, denote this extension as $ \bar{{\cal L}} $. Define a deformation $ \left(Y_{\varepsilon_{1},\varepsilon_{2}},{\cal L}_{\varepsilon_{1},\varepsilon_{2}}\right) $ of $ \left(Y,{\cal L}\right) $ (here $ \varepsilon_{1},\varepsilon_{2}\in{\Bbb C} $ are fixed small numbers) by the following recipe: \begin{enumerate} \item if $ \varepsilon_{1}\varepsilon_{2}=0 $, then $ Y_{\varepsilon_{1},\varepsilon_{2}}=Y $; \item if $ \varepsilon_{1}=\varepsilon_{2}=0 $, then $ {\cal L}_{\varepsilon_{1},\varepsilon_{2}}={\cal L} $; \item if $ \varepsilon_{1}\not=0 $, but $ \varepsilon_{2}=0 $, then $ {\cal L}_{\varepsilon_{1},\varepsilon_{2}} $ is the bundle $ \overset{\,\,{}_\circ}{{\cal L}}\left(P_{\varepsilon_{1}}\right) $, here $ P_{\varepsilon_{1}} $ is the point on $ Y $ with $ y_{1}=\varepsilon_{1} $; similarly for $ \varepsilon_{1}=0 $, but $ \varepsilon_{2}\not=0 $; \item If $ \varepsilon_{1}\not=0 $, $ \varepsilon_{2}\not=0 $, then $ Y_{\varepsilon_{1},\varepsilon_{2}} $ is obtained by gluing the hyperbola $ y_{1}y_{2}=\varepsilon_{1}\varepsilon_{2} $ to $ Y $ (with a small neighborhood of $ Q $ removed) via coordinate projections, and $ {\cal L}_{\varepsilon_{1},\varepsilon_{2}}=\overset{\,\,{}_\circ}{{\cal L}}\left(P_{\varepsilon_{1},\varepsilon_{2}}\right) $, here $ \overset{\,\,{}_\circ}{{\cal L}}_{\varepsilon_{1},\varepsilon_{2}} $ is the restriction of $ \bar{{\cal L}} $ to $ Y_{\varepsilon_{1},\varepsilon_{2}} $, the point $ P_{\varepsilon_{1},\varepsilon_{2}} $ is the point on hyperbola with coordinates $ \left(\varepsilon_{1},\varepsilon_{2}\right) $. \end{enumerate} If $ Y $ had some marked points $ \left\{P_{i}\right\}\subset\operatorname{Smooth}\left(Y\right) $, then $ Y_{\varepsilon_{1},\varepsilon_{2}} $ has the same marked points (correctly defined since $ Y_{\varepsilon_{1},\varepsilon_{2}} $ is identified with $ Y $ outside a small neighborhood of $ Q $). Note that if $ P\in\operatorname{Smooth}\left(Y\right) $, then $ {\frak l}\left(Y,P,{\cal L}\right) $ is well-defined in $ \operatorname{Gr}\left({\frak L}\right)/G $. \begin{lemma} $ g\left(Y_{\varepsilon_{1},\varepsilon_{2}}\right)=g\left(Y\right) $, $ \deg {\cal L}_{\varepsilon_{1},\varepsilon_{2}}=\deg {\cal L} $. If $ \left(Y,P,{\cal L}\right) $ is semistable, then $ \left(Y_{\varepsilon_{1},\varepsilon_{2}},P,{\cal L}_{\varepsilon_{1},\varepsilon_{2}}\right) $ is semistable too. If $ P\in\operatorname{Smooth}\left(Y\right) $, then $ {\frak l}\left(Y_{\varepsilon_{1},\varepsilon_{2}},P,{\cal L}_{\varepsilon_{1},\varepsilon_{2}}\right)\in\operatorname{Gr}\left({\frak L}\right)/G $ depends smoothly on $ \varepsilon_{1},\varepsilon_{2} $. \end{lemma} Similarly, if $ Q $ is a double point of $ Y $, but not a double point of $ {\cal L} $, define a one-parametric deformation $ \left(Y_{\varepsilon},{\cal L}_{\varepsilon}\right) $ by gluing to $ Y $ a hyperbola $ y_{1}y_{2}=\varepsilon $ without changing $ {\cal L} $ (far from $ Q $) and the trivialization of $ {\cal L} $. A statement similar to the above lemma continues to be true. Let $ \bar{{\cal N}}_{g,1,d} $ be the set of equivalence classes of semistable collections $ \left(Y,P,{\cal L}\right) $, let $ \bar{{\cal N}}_{g,1,d}^{\left(m,k\right)}\subset\bar{{\cal N}}_{g,1,d} $ be the subset of $ \bar{{\cal N}}_{g,1,d} $ consisting of curves with exactly $ m $ double points and bundles with exactly $ k $ double points $ \left(k\leq m\right) $. Note that $ \bar{{\cal N}}_{g,1,d}^{\left(0,0\right)}={\cal N}_{g,1,d} $, and the complement to $ {\cal N}_{g,1,d} $ in $ \bar{{\cal N}}_{g,1,d} $ is a disjoint union of $ \bar{{\cal N}}_{g,1,d}^{\left(m,k\right)} $. Call these subsets {\em strata\/} of $ \bar{{\cal N}}_{g,1,d} $. Moreover, note that each $ \bar{{\cal N}}_{g,1,d}^{\left(m,k\right)} $ carries a natural smooth structure (inherited from the smooth structures on Teichm\"uller spaces and Jacobians). (In fact the above smooth structures can be refined to structures of orbifolds, but we ignore this refinement here.) Let $ \left(Y,y_{0},{\cal L}\right)\in\bar{{\cal N}}_{g,1,d}^{\left(m,k\right)} $, $ \varepsilon\in{\Bbb C}^{m+k} $ is small. Taking coordinate systems in neighborhoods of double points, and trivializations of $ {\cal L} $ in these neighborhoods, we obtain a deformation $ \left(Y_{\varepsilon},P,{\cal L}_{\varepsilon}\right)\in\bar{{\cal N}}_{g,1,d} $. Since $ \bar{{\cal N}}_{g,1,d}^{\left(m,k\right)} $ has a natural smooth structure, we see that {\em a piece\/} of $ \bar{{\cal N}}_{g,1,d} $ is fibered over $ \bar{{\cal N}}_{g,1,d}^{\left(m,k\right)} $ with fibers being small balls in $ {\Bbb C}^{m+k} $. Define a structure of an manifold on $ \bar{{\cal N}}_{g,1,d} $ using the above fibration as an atlas in a neighborhood of a point in $ \bar{{\cal N}}_{g,1,d}^{\left(m,k\right)} $. \begin{proposition} $ \bar{{\cal N}}_{g,1,d} $ is a smooth compact manifold (orbifold). $ {\frak l}\left(\bar{{\cal N}}_{g,1,d}\right) $ coincides with the closure of $ {\frak l}\left({\cal N}_{g,1,d}\right) $. \end{proposition} \begin{proposition} \label{prop0.33}\myLabel{prop0.33}\relax $ {\frak l}\left(X,P,{\cal L}\right)={\frak l}\left(X',P',{\cal L}'\right) $ iff $ \left(X,P,{\cal L}\right) $ becomes isomorphic to $ \left(X',P',{\cal L}'\right) $ after ungluing of double points of $ {\cal L} $ and $ {\cal L}' $. \end{proposition} This means that $ \bar{{\cal N}}_{g,1,d} $ is a natural smooth compactification of $ {\cal N}_{g,1,d} $. Moreover, the explicit coordinates in a neighborhood of a stratum show that the {\em boundary\/} $ \bar{{\cal N}}_{g,1,d}\smallsetminus{\cal N}_{g,1,d} $ is a divisor with normal intersections. As it follows from Proposition~\ref{prop0.33}, the image $ {\frak l}\left(\bar{{\cal N}}_{g,1,d}\right) $ can be obtained from $ \bar{{\cal N}}_{g,1,d} $ by contracting smooth submanifolds into points, thus the normalization of the image coincides with $ \bar{{\cal N}}_{g,1,d} $. \subsection{Adjacency of moduli spaces and Shottki model }\label{s0.8}\myLabel{s0.8}\relax We have seen that the consideration of the mapping to the universal Grassmannian leads to a remarkable compactification of Teichm\"uller--Jacoby space. Moreover, one can use this compactification to define inclusions of compactified Teichm\"uller--Jacoby spaces for different $ g $, $ d $ one into another. These inclusions are going to be compatible with the mapping $ {\frak l} $ to Universal Grassmannian, and may be constructed studying the properties of the mapping $ {\frak l} $. Consider a curve $ X $ with a bundle $ {\cal L} $ (we allow $ X $ and $ {\cal L} $ to have double points). Given two smooth points $ Q_{1} $, $ Q_{2} $ on $ X $ one can construct a curve $ X_{Q_{1}Q_{2}} $ obtained by gluing $ Q_{1} $ with $ Q_{2} $. This curve has one more double point, and genus $ g\left(X\right)+1 $. One can consider the bundle $ {\cal L}_{Q_{1}Q_{2}} $ on $ X_{Q_{1}Q_{2}} $ which has an extra double point at $ Q_{1}\sim Q_{2} $, otherwise coincides with $ {\cal L} $. Note that sections of $ {\cal L} $ can be considered as sections of $ {\cal L}_{Q_{1}Q_{2}} $ and visa versa, hence $ {\frak l}\left(X,P,{\cal L}\right) = {\frak l}\left(X_{Q_{1}Q_{2}},P,{\cal L}_{Q_{1}Q_{2}}\right) $ for any $ P $. Using this remark one can easily include the compactification of one moduli space into the boundary of another one. Indeed, one can generalize the construction of $ \bar{{\cal N}}_{g,1,d} $ to the case of $ n $ marked points $ \left\{P_{i}\right\} $ (instead of one $ P $), the only change being that one needs to consider $ \operatorname{Gr}\left({\frak L}^{n}\right) $ instead of $ \operatorname{Gr}\left({\frak L}\right) $. Now associate an element of $ \bar{{\cal N}}_{g+l,n-2l,d+l} $ to an element $ \left(Y,\left\{P_{i}\right\},{\cal L}\right)\in\bar{{\cal N}}_{g,n,d} $ as follows: take first $ 2l $ points out of $ \left\{P_{i}\right\} $, and glue them pairwise. Since we do not glue fibers of $ {\cal L} $ at these points, the resulting double points on the resulting curve $ Y_{1} $ are double points of a bundle $ {\cal L}_{1} $. Clearly, $ \left(Y_{1},\left\{P_{2l+i}\right\},{\cal L}_{1}\right)\in\bar{{\cal N}}_{g+l,n-2l,d+l} $. Thus we obtain a mapping $ \bar{{\cal N}}_{g,n,d} \to \bar{{\cal N}}_{g+l,n-2l,d+l} $. Note that this correspondence does not change the space of global sections of $ {\cal L} $, thus the image under the map $ {\frak l} $. Since we are especially interested in $ \bar{{\cal N}}_{g,1,d} $, let us modify the above construction to get a mapping $ \bar{{\cal N}}_{g,1,d} \to \bar{{\cal N}}_{g+l,1,d+l} $. To do this it is sufficient to define a mapping $ \bar{{\cal N}}_{g,1,d} \to \bar{{\cal N}}_{g,1+2l,d} $. We will define such a mapping for every {\em most degenerate\/} curve, which is a semistable curve with the class on a stratum of $ \dim =0 $ on the moduli space. The mapping will be given by {\em gluing\/} this curve to the given curve. Since there are finitely many most degenerate curves (they are enumerated by appropriately colored trees), we will obtain a finite number of mappings $ \bar{{\cal N}}_{g,1,d} \to \bar{{\cal N}}_{g,1+2l,d} $. A most degenerate curve $ \left(Z,\left\{z_{j}\right\}\right) $ of genus 0 with $ 2l+2 $ marked points $ z_{j} $, $ 0\leq j\leq2l+1 $ has $ 2l-1 $ double points, and its normalization has $ 2l $ connected components. Any bundle of degree 0 over $ Z $ is trivial. Given a curve $ \left(Y,P_{0},{\cal L}\right) $, glue $ Z $ to $ Y $ by identifying $ P_{0} $ and $ z_{0} $ and identify $ {\cal L}|_{P_{0}} $ with $ {\cal O}_{Z}|_{z_{0}} $ arbitrarily, denote the resulting bundle $ {\cal L}' $. The resulting collection $ \left(Y\cup Z,\left\{z_{j}\right\},{\cal L}'\right) $ is obviously an element of $ \bar{{\cal N}}_{g,1+2l,d} $, and gluing together $ z_{j} $, $ 1\leq j\leq2l $, we obtain an element of $ \bar{{\cal N}}_{g+l,1,d+l} $. We see that $ {\frak l}\left(X,P,{\cal L}\right) $ can be approximated by $ {\frak l}\left(X',P',{\cal L}'\right) $ (here $ X $ and $ X' $ are smooth curves) with $ g\left(X'\right)<g\left(X\right) $ if $ \left(X,P,{\cal L}\right) $ is close to a subset of codimension 2 which consists of semistable curves/bundles such that a bundle has a double point. To visualize better the above surgery we improve the description in two ways. First, let us revisit the process of $ \varepsilon $-deformation. We glue a hyperbola $ y_{1}y_{2}=\varepsilon $ to the coordinate lines $ y_{1}=0 $, $ y_{2}=0 $ via the coordinate projections. One can momentarily see that this is equivalent to gluing together two regions $ \left\{|y_{1}|\geq\sqrt{|\varepsilon|}\right\} $, $ \left\{|y_{2}|\geq\sqrt{|\varepsilon|}\right\} $ along the boundary via $ y_{2}=\varepsilon/y_{1} $. Given a bundle $ {\cal L} $ with trivializations near $ Q_{1} $ and $ Q_{2} $ one obtains a bundle $ \overset{\,\,{}_\circ}{{\cal L}} $ on the resulting curves (its sections are sections of $ {\cal L} $ outside of the disks with compatible restrictions to the disk boundaries). The deformed bundle is $ \overset{\,\,{}_\circ}{{\cal L}}\left(Q\right) $, here $ Q $ is close to the circle $ |y_{1}|=|y_{2}|=\sqrt{|\varepsilon|} $ on the deformed curve. There is an alternative description of the deformed curve. Above we were removing two disks of radii $ \varepsilon_{1} $ and $ \varepsilon_{2} $ near points $ y_{1} $, $ y_{2} $ and gluing the boundaries. The radii of disks were variable, so it was hard to visualize the picture. Remove instead bigger disks of (fixed) radii $ \delta_{1} $, $ \delta_{2} $. What remains is to glue two annuli $ \left\{\varepsilon_{i}<|z|<\delta_{i}\right\} $, $ i=1,2 $, to the resulting manifold with a boundary, and glue inner boundaries of annuli together. Each annulus is conformally equivalent to a cylinder with ratio length/radius being $ \log \frac{\delta_{i}}{\varepsilon_{i}} $. Thus we need to glue in a cylinder $ S^{1}\times\left(0,L\right) $ of {\em conformal length\/} $ L=\log \frac{\delta_{1}\delta_{2}}{\varepsilon_{1}\varepsilon_{2}} $. If $ \varepsilon_{1,2} $ decrease, it is equivalent to gluing longer and longer {\em handles\/} between circles of radii $ \delta_{1,2} $. Note also that we modify $ {\cal L} $ by adding a pole inside this handle. Second, note that the above gluing mapping $ y_{2}=\varepsilon/y_{1} $ is defined not only on the boundary of the above regions, but in the regions themselves. Thus continuing the identification of boundaries, one can identify the region $ \left\{|y_{2}|\geq\sqrt{|\varepsilon|}\right\} $ with $ \left\{|y_{1}|\leq\sqrt{|\varepsilon|}\right\} $. If a part of curve corresponds to a subset of $ \left\{|y_{2}|\geq\sqrt{|\varepsilon|}\right\} $, then it is identified with a subset of $ \left\{|y_{1}|\leq\sqrt{|\varepsilon|}\right\} $. Suppose that the curve we deform has rational smooth components. If one component $ Y_{i} $ contains $ l_{i} $ double points, then to perform the above deformation we remove $ l_{i} $ disks of radii $ \varepsilon_{ik} $, $ k=1,2 $, around these points. What remains is a sphere without $ l_{i} $ disks, and we need to glue several such spheres together identifying boundaries by fraction-linear mappings. Say, if $ i\not=j $ and $ Y_{j} $ is glued to $ Y_{i} $ via gluing $ \varphi_{i} $, then $ \varphi\left(Y_{j}\right) $ is a subset of a removed disk for $ Y_{i} $. Consider now the union $ Y_{i}\cup\varphi_{i}\left(Y_{j}\right) $. It is again a sphere with several disks removed, and the boundaries of this disks are identified with the boundaries of other components (or different circles on $ Y_{i}\cup\varphi_{i}\left(Y_{j}\right) $) via fraction-linear mappings. Continuing this process as long as we can, the result is {\em one\/} sphere with several disks removed, and boundaries of these disks are identified pairwise via fraction-linear mappings. This is so called {\em Shottki model\/} of the curve, and we see that the analogue of Deligne--Mumford compactification we consider here leads naturally to this model of deformed curves. One can do a similar thing in the case when the initial curve contains non-rational components as well. In this case one should restrict attention to the ``connected part'' of the curve which consists of rational components. We conclude that in the simplest case the above process of deformation may be described as this (for one particular choice of degenerate rational curve with 3 marked points): Take two points $ Q_{1} $, $ Q_{2} $ which are close to the marked point $ P $ and much closer to each other than to $ P $. Now remove two non-intersecting disks around $ Q_{1} $ and $ Q_{2} $, and glue a very long handle between boundaries of these disks. Here a disk which contains $ P $, $ Q_{1} $ and $ Q_{2} $ plays the r\^ole of the ``old'' marked point, we assume that $ z_{0} $ and $ z_{4} $ (using the notations from the beginning of this section) are on the same smooth component of the degenerate rational curve. \subsection{Second compactification } The previous section shows that compactified moduli spaces are included one into another, thus one can consider the direct limit, i.e., the {\em union\/} of these spaces. The mapping $ {\frak l} $ identifies this union $ {\frak N}_{d-g}=\bigcup_{l}\bar{{\cal N}}_{g+l,1,d+l} $, with a subset of the Universal Grassmannian, and (at last!) we have the ingredients necessary for the discussion what the closure of this subset may look like. Consider an element $ \left(Y,P,{\cal L}\right) $ of $ \bar{{\cal N}}_{g+l,1,d+l} $ for a very big $ l $. Suppose that it is close to an element $ \left(Y',P',{\cal L}'\right) $ of $ \bar{{\cal N}}_{g,1,d} $. To get $ Y $ we glue a most degenerate rational curve to $ Y' $, the result is a curve with $ 2l $ components and $ 2l+2 $ marked points. Next we glue marked points pairwise, and $ \varepsilon $-deform the resulting curve at the double points. However, the double points are naturally broken into two categories: double points on the attached rational curve, and doubled marked points. Let us change the order of gluing and deformations: glue in the degenerate curve, $ \varepsilon $-deform the double points on this curve, then glue marked points together and $ \varepsilon $-deforming them. Note that the former double points are not double points of the linear bundle, but the latter ones are double points for the bundle. Deformation of double points on a degenerate rational curve leads to a Shottki model for some curve, i.e., a sphere with several disks removed. However, since the genus is 0, there is no removed disks at all, thus we get a sphere with $ 2l+2 $ marked points. The condition that the parameters of deformation are small is translated into the fact that double ratios of marked points on the sphere are very big, i.e., that {\em conformally\/} the centers are {\em well separated}. Deform now the double point where the degenerated curve is glued to $ Y' $. This leads to the part of the above sphere being identified (by a fraction-linear mapping) with a disk around $ P' $, and the marked points go to points of this disk. We conclude that this part of deformation corresponds to picking up a collection of points which are close to $ P' $ and conformally well separated. The second part of the deformation is the removing of small disks around the marked points, and gluing together the boundaries of these disks (or, what is the same, gluing long handles to boundaries of bigger fixed disks). In the most important case $ g=0 $, thus the description of a neighborhood of a point on $ {\frak N}_{k} $ is related to studying curves of genus $ g $ obtained by gluing together $ 2g $ small conformally well separated disks on $ {\Bbb C}P^{1} $. Note that the notion of being {\em well separated\/} depends on the combinatorics of degenerated curves we glue in to get the limit point of $ {\frak N}_{k} $. Since we expect that $ {\frak l}\left({\frak N}_{k}\right) $ is dense in the space of solutions of an integrable system, one can approximate a point in this space by a sequence of curves and bundles. Assuming the best case scenario, we can obtain a term of this sequence by a small deformation of the previous term (here a small deformation is taken in the sense of algebraic geometry, i.e., one takes a couple of points on the curve, glues them together and deforms the double point slightly). We see that such a point of the space can be described as a {\em curve of potentially infinite\/} genus, i.e., a sequence of the curves where the next one is obtained by gluing long handles to the previous one. In this paper we show that one can consider a {\em curve of actually infinite\/} genus instead. Such a curve is obtained in the same manner as the above sequence of finite-genus curves: one takes an infinite collection of ``well-separated'' points on $ {\Bbb C}P^{1} $, removes a disk of a small radius around each point, and glues the boundaries pairwise.\footnote{The exact meaning of the above terms ``well-separated collection'' and circles being ``of small radii'' is the main topic of this paper. The answers vary a little bit depending on what problem one considers, for the list of results see Section~\ref{s0.120}.} We show that under suitable conditions the standard theorems of algebraic geometry hold for the resulting curves as well. This shows that the {\em completed\/} moduli space $ \bar{{\frak N}}_{-1} $ of such curves$ + $bundles may be important in algebraic geometry. {\em If\/} this moduli space coincides with the whole space of solutions (i.e., the best case scenario has place indeed), one can see that the completed moduli space has a very simple topology. Indeed, by Cauchy--Kovalevskaya theorem the space of solutions is identified with the space of initial data for the solutions, which is a topological linear space. One can see that under assumption that the moduli space described in this paper coincides with the set of solutions of integrable systems, the study of compactified moduli spaces simplifies a lot by a transition to the case of infinite genus. This simplification may suggest additional approaches to the problem of studying the moduli spaces in finite genus as well. Unfortunately, at this moment it is unclear whether the best case scenario is applicable to the moduli space as a whole. As we noted above, the results of \cite{McKTru76Hil} show that this is true for the real part of the hyperelliptic subset of the moduli space. \subsection{Growth conditions and divisors }\label{s0.10}\myLabel{s0.10}\relax The main target of this paper is to describe how to quantify the conditions on the collection of circles to be ``conformally well separated'' and ``of small radii'' so that the main theorems of algebraic geometry are still valid for the resulting curves of infinite genus. However, before we discuss the conditions on the nonlinear data (curves and bundles), we should discuss the simpler conditions on the linear data (sections of above bundles). The basic theorem of $ 1 $-dimensional algebraic geometry states that any linear bundle on a compact analytic curve has a finite-dimensional space of global sections. The moment we try to drop the compactness condition this theorem breaks, since any non-compact analytic curve is a Stein manifold, thus any sheaf on it has a giant space of global sections. One way to fix this situation is to consider growth conditions. Let $ \bar{X} $ be a compact curve with a linear bundle $ \bar{{\cal L}} $. Suppose that $ \bar{X} $ and $ \bar{{\cal L}} $ are provided with Hermitian metrics. Pick up a point $ \infty\in\bar{X} $, let $ X=\bar{X}\smallsetminus\left\{\infty\right\} $, $ {\cal L}=\bar{{\cal L}}|_{X} $. Then bounded sections of $ {\cal L} $ can be uniquely extended to analytic sections of $ \bar{{\cal L}} $, thus $ {\cal L} $ has a finite-dimensional space of bounded sections. Similarly, if we consider sections of $ {\cal L} $ with magnitude going to 0 when $ x \to \infty $, it is the same as to consider sections of $ \bar{{\cal L}}\left(-1\cdot\infty\right) $ on $ \bar{X} $, here $ -1\cdot\infty $ is a divisor on $ \bar{X} $. Going in a different direction, consider $ L_{2} $-sections of $ {\cal L} $. It is easy to see that this is also equivalent to consideration of sections of $ \bar{{\cal L}} $. If we consider a different metric on $ X $, say, $ \frac{dx^{2}}{\operatorname{dist}\left(x,\infty\right)^{2\alpha}} $, then to consider holomorphic on $ X L_{2} $-sections is the same as to consider $ \bar{{\cal L}}\left(\left[\alpha\right]\cdot\infty\right) $. Similarly, different choice of metric on $ {\cal L} $ will lead to different shift of $ \bar{{\cal L}} $ by a divisor at $ \infty $. We see that different growth conditions applied to sections of $ {\cal L} $ lead to different ``effective'' continuations of $ {\cal L} $ to $ \bar{X} $. Obviously, the situation becomes more complicated when $ \bar{X}\smallsetminus X $ consists of more than one point. If $ \bar{X}\smallsetminus X $ is discrete, different possible choices of growth conditions lead to a lattice which is isomorphic to the lattice of divisors on $ \bar{X} $ concentrated on $ \bar{X}\smallsetminus X $. Situation goes out of control if $ \bar{X}\smallsetminus X $ is a ``massive'' set, or $ \bar{X} $ is not a smooth manifold at all. In such cases different choices of metrics on $ X $ and $ {\cal L} $ (or some other data controlling the growth of sections) form a very complicated lattice. However, there is a way to select ``good'' elements of this lattice. \subsection{Riemann--Roch and growth control }\label{s0.20}\myLabel{s0.20}\relax The Riemann--Roch theorem says that on a compact analytic curve there is a relationship between a {\em degree of\/} $ {\cal L} $, which is an easily calculable geometric characteristics of $ {\cal L} $, and the dimensions of spaces of global sections of $ {\cal L} $ and $ \omega\otimes{\cal L}^{-1} $, $ \omega $ being the linear bundle of holomorphic forms: \begin{equation} \dim \Gamma\left({\cal L}\right)-\dim \Gamma\left(\omega\otimes{\cal L}^{-1}\right) = \deg {\cal L}-g+1. \notag\end{equation} Let us show how this theorem might be applied to picking up ``correct'' growth conditions on non-compact analytic curves. Consider some fixed growth conditions for $ X $. They select a new sense for the functor $ \Gamma\left(X,\bullet\right) $. Suppose again that $ \bar{X}\smallsetminus X $ is discrete, $ {\cal L}=\bar{{\cal L}}|_{X} $. We have seen above that consideration of these growth conditions is equivalent to consideration of $ \bar{{\cal L}}\left(D\right) $, here $ D $ is some divisor at infinity, i.e., $ \Gamma\left(X,{\cal L}\right)=\Gamma\left(\bar{X},\bar{{\cal L}}\left(D\right)\right) $. One should expect that $ D $ does not depend on $ {\cal L} $ (at least when $ {\cal L} $ does not change a lot). Suppose that there is a way to determine $ \deg \bar{{\cal L}} $ basing on $ {\cal L} $, say, we describe bundles by divisors on $ X $. If the dimensions of sections of linear bundles on $ X $ with given growth conditions satisfy the Riemann--Roch theorem, then an easy calculation shows that $ 2D\sim0 $. In particular, suppose that $ {\cal L} $ was a restriction of a bundle $ \bar{{\cal L}} $ of half-forms on $ \bar{X} $, i.e., $ \bar{{\cal L}}\otimes\bar{{\cal L}}\simeq\omega $. Then we see that the bundle on $ \bar{X} $ which corresponds to $ \bar{{\cal L}}|_{X} $ with given growth conditions is $ \bar{{\cal L}}\left(D\right) $. Note that $ \bar{{\cal L}}\left(D\right)\otimes\bar{{\cal L}}\left(D\right)=\omega $. In particular, the growth conditions which satisfy Riemann--Roch theorem ``preserve'' the set of bundles of half-forms (or $ \theta $-{\em characteristics\/}): the sections on $ X $ of one $ \theta $-characteristic $ \bar{{\cal L}} $ which satisfy the growth conditions can be naturally identified with sections on $ \bar{X} $ of some other $ \theta $-characteristic $ \bar{{\cal L}}\left(D\right) $. Suppose now that the growth conditions on sections of $ {\cal L} $ are picked up in some invariant way---whatever it means. Because of that we expect that $ D $ is a linear combination of points on $ \bar{X}\smallsetminus X $ with the same coefficients. Since $ \deg D=0 $, thus $ D $ is 0. We see that Riemann--Roch provides a selection criterion for picking up growth conditions which ``do not add'' points at infinity to a given divisor. By analogy, one can apply the same criterion in cases when $ \bar{X}\smallsetminus X $ is massive or $ \bar{X} $ does not exist: If there is an invariant way to describe the growth conditions, and the Riemann--Roch theorem holds for a class of bundles, then one may describe the part at infinity of the divisor of a section of such a bundle: a section has no ``poles'' at infinity if it satisfies the growth conditions. \subsection{Handles, Sobolev spaces, and representations of $ \protect \operatorname{SL}_{2}\left({\Bbb C}\right) $ }\label{s0.30}\myLabel{s0.30}\relax Let us apply heuristics from the previous section to the case of a curve $ X $ of infinite genus, i.e., a Riemannian sphere with an infinite number of disks removed, and infinite number of handles glued in along the cut lines (or just any surface glued in from ``pants'' in such a way that the graph of gluing is of infinite genus). This is not a compact Riemannian surface, so one needs to consider ``infinities'' of this surface, and fix some growth conditions near these infinities. Moreover, if the genus is infinite, one needs to be especially careful with Riemann--Roch theorem, since the right-hand side contains genus $ g\left(X\right) $ (i.e., number of handles) of the curve, which is not a number any more, but infinity. To save the theorem, let us consider the quantity $ \deg {\cal L}-g+1 $ as a unity. One way to do it is to consider some fixed bundle $ {\cal M} $ on $ X $ which is ``naturally constructed'' and satisfies the condition $ \deg {\cal M}=g-1 $ for finite-genus $ X $, and consider sections of $ {\cal L}\otimes{\cal M} $ instead of sections $ {\cal L} $. Then the Riemann--Roch theorem may be rewritten as \begin{equation} \dim \Gamma\left({\cal L}\otimes{\cal M}\right)-\dim \Gamma\left(\omega\otimes{\cal M}^{-1}\otimes{\cal L}^{-1}\right) = \deg {\cal L}. \notag\end{equation} Note that $ \deg \omega\otimes{\cal M}^{-1}=\deg {\cal M} $, so $ {\cal N}=\omega\otimes{\cal M}^{\otimes-2} $ is a ``naturally constructed'' bundle of degree 0. There are $ 2^{g} $ different square roots of a given bundle of degree 0, and fixing one solution $ {\cal R} $ to $ {\cal R}^{2}=\omega\otimes{\cal M}^{\otimes-2} $ we may change $ {\cal M} $ to $ {\cal M}\otimes{\cal R}^{-1} $. After such a change the formula simplifies to \begin{equation} \dim \Gamma\left({\cal L}\otimes{\cal M}\right)-\dim \Gamma\left({\cal L}\otimes{\cal M}^{-1}\right) = \deg {\cal L}, \notag\end{equation} which has an additional advantage of being symmetric w.r.t. $ {\cal L} \mapsto {\cal L}^{-1} $. Moreover, $ {\cal M} $ is now a solution of $ {\cal M}^{2}=\omega $. We come to the following heuristic: to consider Riemann--Roch theorem for a curve $ X $ with infinitely many handles one should fix a bundle of half-forms $ {\cal M}=\omega^{1/2} $ on $ X $, consider a linear bundle $ {\cal L} $ of finite degree, and apply appropriate growth conditions to sections of $ {\cal L}\otimes{\cal M} $. We expect that the choice of growth conditions is very restricted, since any possible change is equivalent to consideration of different square root of $ \omega $ (see Section~\ref{s0.20}). One should expect that thus obtained growth conditions are in some way ``invariant'' (since so rigid), thus geometrically defined. It is still meaningful to apply these conditions in the case $ {\cal L}={\cal O} $, when $ {\cal L}\otimes{\cal M} $ is the sheaf of half-forms. We come to the following conclusions: Conditions of boundness (i.e., $ L_{\infty} $-topology) cannot be applied (since there is no geometrically-defined norm on the fibers of the bundle of half-forms), as well as $ L_{2} $-type restrictions (since a square of a holomorphic half-form is a half-form on $ X_{{\Bbb R}} $, thus cannot be invariantly integrated). One can easily see that the fourth degree of a half form $ \alpha $ (more precise, $ \alpha^{2}\otimes\bar{\alpha}^{2} $) is a top-degree-form on $ X_{{\Bbb R}} $, thus a good candidate is the condition of integrability of fourth degree, i.e., $ L_{4} $-topology. It turns out that generic Banach spaces (like $ L_{4} $) are not convenient for cut-and-glue operations we are going to perform, but there is a way to get a Hilbert structure instead of the Banach one. Note that $ s $-th derivative of $ \alpha $-form changes as a $ \left(\alpha+s\right) $-form under coordinate transformations (in the main term), thus its square is a top-degree form on $ X_{{\Bbb R}} $ (in the main term) if $ \alpha+s=1 $. We conclude that the notion that $ 1/2 ${\em -th derivative of a half-form is square-integrable\/} has a chance to be geometrizable. This notion leads us to consideration of $ H^{1/2} $-Sobolev spaces with values in (holomorphic) half-forms. (We discuss the basics of the theory of Sobolev spaces in Section~\ref{h2}) In fact taking different $ s $ one gets an entire hierarchy of Banach spaces $ W_{\frac{2}{s+1/2}}^{s} $ which interpolate between $ L_{4} $ and $ H^{1/2} $. However, only one of these spaces is a Hilbert one, and having a Hilbert space is very convenient for our method of divide and conquer. As we will see it in Section~\ref{s2.70}, the spaces of functions we obtain in such a way are closest possible analogues of Hardy spaces. Note also that they are very small modifications of a particular representation of $ \operatorname{SL}_{2}\left({\Bbb C}\right) $ from a supplementary series (see Section ~\ref{s2.60}). Indeed, the supplementary series is realized in $ s $-forms on $ \left({\Bbb C}P^{1}\right)_{{\Bbb R}} $, and the Hilbert structure in these spaces is equivalent to $ H^{1-2s} $-structure. Taking $ s=1/4 $, we see that the $ H^{1/2} $-structure on sections of $ \omega^{1/4}\otimes\bar{\omega}^{1/4} $ may be defined\footnote{Recall that $ H^{s} $-structure on sections of bundles on manifolds is defined only up to equivalence of topologies.} to be invariant w.r.t. fraction-linear transformations. One should compare this result with the above heuristic about $ H^{1/2} $-structure on sections of $ \omega^{1/2} $ being ``invariant in the main term'' w.r.t. diffeomorphisms (we use an additional heuristic that $ \omega^{1/4}\otimes\bar{\omega}^{1/4} $ is ``close'' to $ \omega^{1/2} $). \subsection{Thick infinity }\label{s0.40}\myLabel{s0.40}\relax We conclude that a working definition of {\em admissible\/} sections on a sphere with some disks removed is that the section is a half-form on the domain which can be extended as a $ H^{1/2} $-section of $ \omega^{1/2} $ to the whole sphere (though a usual heuristic about $ H^{1/2} $ is that it is a smoothness class, it restricts the growth as well). The usual definition of $ H^{s} $ on a manifold with a boundary (see Section~\ref{s2.20}) allows one to work with function within the interior of the manifold only. In our case the smooth part of the boundary (union of circles) is not closed (if the number of circles is infinite), so one loses the info about what happens on the {\em dust}, i.e., in the accumulation points of the disks. This has some very inconvenient consequences. Let $ S $ be the sphere with interiors of disks removed, $ V $ be the closure of $ {\Bbb C}P^{1}\smallsetminus S $. Then the sections we consider are elements of $ H^{1/2}\left({\Bbb C}P^{1}\smallsetminus V\right) $. The condition of such a $ \varphi $ being holomorphic can be written as $ \bar{\partial}\varphi=0\in H^{-1/2}\left({\Bbb C}P^{1}\smallsetminus V\right) $ (since taking derivative moves one notch down on Sobolev scale). However, if the dust $ S\cap V $ is massive enough, there are functions of smoothness $ H^{-1/2} $ with support in $ S\cap V $, thus there functions $ \varphi $ of smoothness $ H^{1/2} $ such that $ \bar{\partial}\varphi $ has support on the dust only. Since $ \bar{\partial}\varphi=0\in H^{-1/2}\left({\Bbb C}P^{1}\smallsetminus V\right) $, such functions look like holomorphic ones, but in fact the dust is the part of infinity on our curve of infinite genus, thus one should consider the divisor of these functions as having components at infinity (since $ \bar{\partial}\varphi $ is not zero there). This ruins our preparations to count the divisor at infinity correctly. One possible way out of this deadlock is to prohibit collections of circles with a massive dust, say, restrict the Hausdorff dimension of the dust to be smaller than 1. We pick up a different approach: we massage the definition of the Sobolev space for a subset of a manifold in such a way that it now takes into account the behaviour on the boundary even if the boundary is massive. These are so called {\em generalized\/} Sobolev spaces. As a corollary, we can consider pretty monstrous curves of infinite genus. For example, one can take a Serpinsky carpet\footnote{Which is a two-dimensional analogue of the Cantor set, obtained by repeated removing of a smaller rectangle inside a bigger one (or removing a triangle with vertices on the sides of a bigger triangle).} on a complex plane, and take one disk inside each thrown away triangle/rectangle. Gluing the boundaries of disk pairwise gives a curve we can deal with (if radii of disks decrease quick enough). In particular, the dust can have a positive measure (in fact in our theory the only condition on the dust is that it is nowhere dense, so can be a set of accumulation points). One can see that smooth points on the resulting curve have infinitely many connected components, and it is the dust which keeps these components together. The components are tubes, and the boundaries of these tubes are glued to the dust, which is connected. We see that points of the dust need to be considered as legitimate points on the curve of infinite genus, and the curve is {\em not a topological manifold\/} in neighborhoods of these points. The striking fact is that it is possible to strengthen the restriction on the radii of the circles in such a way that any global holomorphic function has an {\em asymptotic\/} Taylor expansion near any point of the dust (see Section~\ref{s4.95})! Thus the dust would consist of points which one has a full right to call {\em smooth points}. \subsection{Fight with $ H^{1/2} $ } While consideration of half-forms of smoothness $ H^{1/2} $ {\em enormously\/} simplifies the work with infinities on the curve, it is one of the worst choices when we consider gluing conditions on sections on the glued together circles on the complex sphere. The reason is that Sobolev spaces with half-integer indices do not satisfy a lot of properties of other Sobolev spaces when restrictions to hypersurfaces is considered (this is the case when one usually considers Besov spaces instead of Sobolev ones). To fight with this, we define a notion of {\em mollified restriction\/} to a hypersurface. Applying this restriction to the case of one removed disk $ \left\{|z|>1\right\} $ one can momentarily see that the space $ H^{1/2} $ we consider coincides with the Hardy space $ {\cal H}^{2} $ for the circle $ \left\{|z|=1\right\}! $ We see that the space of ``global holomorphic functions'' we consider is a generalization of the Hardy space to the case when the boundary consists of many circles. What is more, the Hardy space $ {\cal H}^{\infty} $ (of multiplicators in $ {\cal H}^{2} $) also plays an important r\^ole when we consider equivalence of bundles in Section~\ref{s8.7}. \subsection{Main results }\label{s0.120}\myLabel{s0.120}\relax Since technical details take a lot of place in the course of discussion, we make a concise list of main results (without discussing what the different conditions on the curve/bundle mean). In Section~\ref{s35.40} we prove (an analogue of) Riemann--Roch theorem for $ \bar{\partial}\colon {\cal O} \to \bar{\omega} $ in assumption that the matrix $ {\bold M}_{2}=\left(e^{-l_{ij}}-\delta_{ij}\right) $ (formed basing on conformal distances $ l_{ij} $ between disks) gives a compact mapping $ l_{2} \to l_{2} $. In Section~\ref{s7.90} we show that the mapping $ \bar{\partial}\colon {\cal O} \to \bar{\omega} $ has a maximal rank compatible with its index. In Section~\ref{s9.20} we show an analogues result for $ \bar{\partial}\colon \omega \to \omega\otimes\bar{\omega} $ under an additional assumption that disks have a {\em thickening\/} (in fact we show that the mapping of taking $ A $-periods of global homomorphic forms is an isomorphisms). In Section~\ref{s9.40} we describe the image of the $ \left(A,B\right) $-period mapping. In all these cases the mappings in question have an infinite index, so we massage these mappings to get a mapping of index 0 (by extending the domain or target spaces) which is in fact an isomorphism. In Section~\ref{s7.30} we show that the space of global holomorphic sections of $ \omega^{1/2}\otimes{\cal L} $ is finite-dimensional provided the matrix $ {\bold M}_{1}=\left(a_{i}e^{-l_{ij}/2}-a_{i}\delta_{ij}\right) $: give a compact mapping $ l_{2} \to l_{2} $. Here $ \left(a_{i}\right) $ is some sequence associated to the bundle $ {\cal L} $ (it consists of norms of gluing cocycle for $ {\cal L} $). In the section~\ref{s7.40} we show that the duality \begin{equation} \operatorname{Coker}\left(\bar{\partial}\colon \omega^{1/2}\otimes{\cal L} \to \omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) \simeq \operatorname{Ker}\left(\bar{\partial}\colon \omega^{1/2}\otimes{\cal L}^{-1} \to \omega^{1/2}\otimes{\cal L}^{-1}\otimes\bar{\omega}\right) \notag\end{equation} holds as far as $ {\bold M}_{1} $ gives a bounded operator $ l_{2} \to l_{2} $. In Section~\ref{s5.60} we show that the above mappings satisfy the Riemann--Roch theorem as far as both $ \left(a_{i}e^{-l_{ij}/2}-a_{i}\delta_{ij}\right) $ and $ \left(a'_{i}e^{-l_{ij}/2}-a'_{i}\delta_{ij}\right) $ give compact operators $ l_{2} \to l_{2} $ (here $ a'_{i} $ is the sequence associated to $ {\cal L}^{-1} $). In the same section we show that these mappings are Fredholm under an additional condition that $ \left(a_{i}\right) $, $ \left(a_{i}'\right) $ are bounded. In Section~\ref{s4.50} we show that the above conditions give no restrictions on the {\em dust\/} of the corresponding M\"obius group. In Section ~\ref{s7.70} we discuss heuristics which show that our version of the Riemann--Roch theorem is close to the strongest possible. In Section~\ref{s8.40} we show that an existence of so called {\em Hilbert\/}--{\em Schmidt\/} bundle on a curve shows that $ \omega^{1/2} $ is also Hilbert--Schmidt. In Section~\ref{s8.50} we show that the set of Hilbert--Schmidt bundles is convex (in the sense of multiplication of bundles), thus it is possible to define a {\em group\/} of {\em strong Hilbert\/}--{\em Schmidt\/} bundles. In Section~\ref{s8.7} we define the notion of two bundles being {\em equivalent}, and define the Jacobian. In Section~\ref{s8.80} we define divisors, and in Section~\ref{s8.90} the mapping to the Universal Grassmannian (using asymptotic series at points of the dust defined in Section~\ref{s4.95}). In Section~\ref{s7.90} we show that if $ \sum_{i\not=j}e^{-l_{ij}}<\infty $, then any admissible bundle of degree 0 can be described by locally-constant cocycles (and some amplifications of this result). In Section~\ref{s9.60} we show that the period matrix is symmetric, and the imaginary part is positive. In Section~\ref{s9.70} we show that the Jacobian coincides with the quotient of the space of possible periods by the lattice of {\em integer\/} periods, as far as $ \sum_{i\not=j}e^{-l_{ij}}<\infty $. Finally, in Section~\ref{s9.80} we show how to modify the notion of Hodge structure to describe the curves we study in this paper. \subsection{Main tools } We already noted that to deal with massive infinities we need the notion of generalized Sobolev space (introduced in Section ~\ref{s2.40}), and to deal with gluing conditions along the boundaries of disks we need mollified restriction/extension mappings (defined in Section~\ref{s5.31}). We discuss the relationship of the ``invariant'' growth conditions (we need them to have a good enumeration of divisor at infinity, see Section ~\ref{s0.20}) with the supplementary series of representations of $ \operatorname{SL}_{2}\left({\Bbb C}\right) $ in Section~\ref{s2.60}. In Section~\ref{s2.70} we demonstrate that the classical Hardy space $ {\cal H}^{2} $ is a particular case of spaces we consider here (for ``curves of genus 1/2'', when the fundamental domain is a disk). In Section~\ref{s8.7} we show that the space of multiplicators (needed for defining equivalence of bundles) gives $ {\cal H}^{\infty} $ in the particular case of a disk as a fundamental domain. To handle divide-and-conquer strategy of dealing with curves of infinite genus we consider {\em almost perpendicular\/} families of subspaces (studied in Section~\ref{s5.61}). To study duality we employ spaces of {\em strong sections\/} of bundles (see Section~\ref{s6.50}), to study $ \bar{\partial} $ as a Fredholm operator we employ spaces of {\em weak sections\/} (see Section ~\ref{s5.61}). (In most important cases these spaces coincide.) To define $ B $-periods on a curve we apply a method of averaging (Section~\ref{s9.41}) of paths connecting two sides of an $ A $-cycle. \subsection{Historic remarks } In the spring of 1981 Yu.~I.~Manin was overwhelmed by unusual amount of sophomore students who asked him to be their undergraduate advisor. Contrary to his habits of the time, Yuri Ivanovich presented a list of topics he thought would be interesting and instructive for us to work on. One of these topics was a generalization of algebro-geometric description of solutions of KP to a curve of infinite genus represented by its Shottki model. (It was R.~Ismailov who started to work on this topic.) Five years later, when integrable systems reached the peak of their popularity, we obtained first results on the Riemann--Roch theorem which were similar to the results of this paper. The curves were represented in the way that was in fashion that time: as collections of pants glued along boundaries. The restrictions on the curves/bundles were jaw-breaking and very $ \operatorname{ad} $-hockish. Since Manin's suggestions were completely forgotten already, there was nothing similar to the Shottki model approach. However, half-forms and $ H^{1/2} $-topology were already present at that time. When in the end of the eighties the first variant of this paper was taking shape, it became clear that the conditions on curves/sheaves simplify {\em enormously\/} if one restricts attention to gluings of boundary circles which are fraction-linear, and gluing conditions for the bundles are given by locally constant cocycles (relative to half-forms). To our great surprise, with these restrictions the pants would glue together into the Shottki model of the curve (see Section~\ref{s0.8}), and sheaves glue into the bundle of half-forms on $ {\Bbb C}P^{1} $. This finished a complete circle returning the settings back into context of the question of Manin's of 1981! Note that though the importance of Shottki model, of half-forms and of $ H^{1/2} $-growth conditions was clear long time ago, it was the formalization of the mollified restriction mapping which emphasized the capacity of infinity and the parallelism with the usual study of Hardy spaces $ {\cal H}^{2} $ and $ {\cal H}^{\infty} $. This formalization (together with the rest of this paper) would not appear if not the fruitful discussions with A.~Tyurina in spring of 1996. During the last several years Feldman, Kn\"orrer and Trubowitz made a major breakthrough using an unrelated approach (cf.~\cite{FelKnoTru96Inf}). The author is most grateful to I.~M.~Gelfand, A.~Givental, A.~Goncharov, D.~Kazhdan, M.~Kontsevich, Yu.~I.~Manin, H.~McKean, V.~Serganova, A.~Tyurina and members of A.~Morozov's seminar for discussions which directed many approaches applied here. It was the patient work of V.~Serganova which improved readability of the most obscure pieces of this paper. This work was partially supported by NSF grant and Sloan foundation fellowship. \section{Geometry of half-forms } \subsection{Half-forms and holomorphic half-forms } Consider an oriented real manifold $ M $. For $ \alpha\in{\Bbb C} $ consider a (complex) linear bundle $ \Omega_{M}^{\alpha} $ with transition functions $ \left(\det \frac{\partial Y}{\partial X}\right)^{\alpha} $ if $ Y=Y\left(X\right) $. We will write a section of this bundle as $ \varphi\left(X\right)dX^{\alpha} $, so $ \varphi\left(X\right)dX^{\alpha}=\psi\left(Y\right)dY^{\alpha} $ if $ Y=Y\left(X\right) $ and $ \varphi\left(X\right)\left(\det \frac{\partial Y}{\partial X}\right)^{\alpha}=\psi\left(Y\right) $. When $ M $ is clear from context, we will denote $ \Omega_{M}^{\alpha} $ by $ \Omega^{\alpha} $. In what follows we will be most interested in $ \Omega^{1/2} $. Note that $ \Omega^{1/2}\otimes\Omega^{1/2} $ is canonically isomorphic to the bundle of top de Rham forms. For any two global sections $ \varphi $, $ \psi $ of $ \Omega^{1/2} $ the product $ \varphi\bar{\psi} $ is a top de Rham form, thus the pairing \begin{equation} \left(\varphi,\psi\right) \mapsto \int\varphi\bar{\psi} \notag\end{equation} gives a structure of pre-Hilbert space on $ \Gamma\left(M,\Omega^{1/2}\right) $. In particular, a notion of $ L_{2} $-section of $ \Omega^{1/2} $ is canonically defined. If $ M $ is a complex-analytic manifold, the power $ \left(\det \frac{\partial Y}{\partial X}\right)^{\alpha} $ is not uniquely defined (unless $ \alpha\in{\Bbb Z} $), thus $ \Omega^{\alpha} $ is not canonically defined. We use an $ \operatorname{ad} $ hoc definition of an analogue of $ \Omega^{1/2} $ as follows: Denote by $ \omega $ the top holomorphic de Rham bundle for $ M $. Consider a linear bundle $ l $ with an isomorphism $ i\colon l\otimes l\simeq\omega $. We call sections of such a bundle {\em holomorphic half-forms}, and will denote $ l = \omega^{1/2} $. In what follows we will consider at most one bundle of half-forms on a given manifold. Note that any local diffeomorphism $ M \to N $ induces a bundle of half-forms on $ M $ given a bundle of half-forms on $ N $. \subsection{Half-forms on complex curves }\label{s1.20}\myLabel{s1.20}\relax In what follows we are most interested in complex curves. On a compact complex curve of genus $ g $ there are $ 2^{g} $ different bundles of half-forms. If $ g=0 $, then there is a uniquely defined bundle of half-forms, isomorphic to $ {\cal O}\left(-1\right) $. To pick up an isomorphism of $ {\cal O}\left(-1\right)^{\otimes2}={\cal O}\left(-2\right) $ with $ \omega $, one must fix an element $ v\in\Lambda^{2}V^{*} $, here $ V $ is $ 2 $-dimensional space, projectivization of which is $ {\Bbb C}P^{1} $. Indeed, let $ {\bold e} $ be the Euler vector field on $ V $, $ {\bold e} = z_{1}\frac{\partial}{\partial z_{1}} + z_{2}\frac{\partial}{\partial z_{2}} $. Consider $ v $ as a constant $ 2 $-form on $ V $, then $ {\bold e}\lrcorner v $ is a $ 1 $-form on $ V $ which induces a global section of $ \omega\otimes{\cal O}\left(2\right) $ on $ PV. $ Multiplication on this section gives an isomorphism of $ {\cal O}\left(-2\right) $ and $ \omega $. This shows that there should be an action of $ \operatorname{SL}\left(2,{\Bbb C}\right) $ on the bundle of half-forms, compatible with the action of $ \operatorname{SL}\left(2,{\Bbb C}\right) $ on $ {\Bbb C}P^{1} $. Indeed, in affine coordinate $ z $ on $ {\Bbb C}P^{1}\smallsetminus\left\{\infty\right\} $, a half-form may be written as $ \varphi\left(z\right)dz^{1/2} $. Here if domain of $ \varphi $ contains $ \infty $, then $ \varphi\left(\infty\right)=0 $. To write an action of $ m=\left( \begin{matrix} a & b \\ c & d \end{matrix} \right)\in\operatorname{SL}\left(2,{\Bbb C}\right) $, note that the derivative of the corresponding action $ z'=\displaystyle\frac{az+b}{cz+d} $ on $ {\Bbb C}P^{1} $ has a canonical square root $ \frac{1}{cz+d} $. Thus $ \varphi\left(z'\right)dz'{}^{1/2}\sim\varphi\left(z'\right)\frac{1}{cz+d}dz^{1/2} $. Note that it is the restriction of this action on $ \operatorname{SL}\left(2,{\Bbb R}\right) $ what is used to define automorphic forms on $ \left\{\operatorname{Im} z>0\right\} $. Note also that this action cannot be pushed down to an action of $ \operatorname{PGL}\left(2,{\Bbb C}\right) $. \subsection{Restriction onto $ S^{1} $ } Consider now a annulus $ B=\left\{r<|z|<R\right\} $ on a complex plane. Obviously, there are only two different bundles of half-forms on $ B $. The inclusion of $ B $ into $ {\Bbb C}P^{1} $ induces a bundle $ \omega^{1/2} $ of half-forms on $ B $. Another bundle of half-forms is $ \omega^{1/2}\otimes\mu $, here $ \mu $ is a locally constant sheaf with monodromy $ -1 $ on $ B $, obviously $ \mu^{2} $ is the constant sheaf $ \underline{{\Bbb C}}_{ } $. Note that the holomorphic form $ \frac{dz}{iz} $ on $ B $ has no square root in $ \omega^{1/2} $, but has two in $ \omega^{1/2}\otimes\mu $, one may write these square roots symbolically as $ \pm\frac{\left(dz\right)^{1/2}}{i^{1/2}z^{1/2}} $. Consider now a restriction of $ \omega^{1/2}\otimes\mu $ onto $ S^{1}=\left\{|z|=1\right\}\subset B $ (we assume that $ r<1<R $). Consider a coordinate $ t = \operatorname{Im} \ln z $ on $ S^{1} $, and identify $ \frac{dz^{1/2}}{i^{1/2}z^{1/2}}\in\Gamma\left(S^{1},\omega^{1/2}\otimes\mu\right) $ with $ dt^{1/2}\in\Gamma\left(S^{1},\Omega_{S^{1}}^{1/2}\right) $. A simple calculation shows that this identification gives an isomorphism of $ \omega^{1/2}|_{S^{1}} $ with $ \Omega_{S^{1}}^{1/2}\otimes\mu $, which is preserved by real-analytic diffeomorphisms of $ S^{1} $ with a given lift to $ \mu $. In general, an immersion of $ S^{1} $ in a complex curve $ X $ with a bundle of half-forms $ \omega $ induces an isomorphism of $ \omega^{1/2}|_{S^{1}} $ either with $ \Omega_{S^{1}}^{1/2}\otimes\mu $, or with $ \Omega_{S^{1}}^{1/2} $. One may call immersions of the first type $ A $-cycles, of the second one $ B $-cycles. Until Section~\ref{h9} the immersions we consider are going to be $ A $-cycles, thus we will always have a factor $ \mu $. Note that the above isomorphism is defined up to multiplication by $ \pm1 $, and both $ \Omega_{S^{1}}^{1/2}\otimes\mu $ and $ \Omega_{S^{1}}^{1/2} $ have natural pre-Hilbert structures, thus $ \omega^{1/2}|_{S^{1}} $ has a natural pre-Hilbert structure as well. \section{Sobolev spaces }\label{h2}\myLabel{h2}\relax In what follows $ {\cal S} $ denotes the space of rapidly decreasing smooth functions, $ {\cal D} $ denotes the space of smooth functions with compact support, and $ {\cal S}' $ and $ {\cal D}' $ are the dual spaces. $ l_{2} $ denotes the Hilbert space of square-integrable sequences, and $ L_{2} $ denotes the space of square-integrable functions. The symbol $ \bigoplus V_{i} $ denotes the space of sequences with finite number of non-zero terms. If $ V_{i} $ are Hilbert spaces, $ \bigoplus_{l_{2}}V_{i} $ denotes the space of square-integrable sequences $ \left(v_{i}\right) $ with $ v_{i}\in V_{i} $. The material of this section is mostly standard, however, note that we also discuss some topics which are not parts of standard curriculum of Sobolev spaces: in Section~\ref{s2.40} we introduce {\em generalized\/} Sobolev spaces, in Sections~\ref{s2.60} and~\ref{s2.70} we give descriptions of supplementary series of representations of $ \operatorname{SL}\left(2,{\Bbb C}\right) $ and of Hardy space in terms of Sobolev spaces. In addition to the results which are covered in this section, in Section~\ref{s2.10} we introduce {\em mollifications\/} of mappings of restriction to a submanifold (in the case $ s=1/2 $), and of mapping of extension from submanifold (in the case $ s=0 $), both in the cases when the non-mollified mappings are not continuous. \subsection{Euclidean case }\label{s2.02}\myLabel{s2.02}\relax Consider a vector space $ {\Bbb R}^{n} $ with the standard Euclidean structure. Consider $ s\in{\Bbb R} $ and a norm $ \|\bullet\|_{s} $ on the set $ {\cal S} $ of rapidly decreasing $ C^{\infty} $-functions: \begin{equation} \|f\left(x\right)\|_{s} = \int\left(1+|\xi|^{2}\right)^{s}|\widehat{f}\left(\xi\right)|^{2}d\xi, \notag\end{equation} here $ \widehat{f}\left(\xi\right) $ is the Fourier transform of $ f $. By definition, {\em the Sobolev space\/} $ H^{s}\left({\Bbb R}^{n}\right) $ is the completion of $ {\cal S} $ w.r.t. this norm. It is naturally isomorphic to the set of $ L_{2} $-functions $ g\left(\xi\right) $ w.r.t. the measure $ \left(1+|\xi|^{2}\right)^{s/2}d\xi $. Since the inverse Fourier transform of $ g\left(\xi\right) $ is a well-defined generalized function on $ {\Bbb R}^{n} $, there is a natural inclusion $ H^{s}\left({\Bbb R}^{n}\right)\hookrightarrow{\cal S}' $, compatible with the inclusion $ {\cal S}\hookrightarrow H^{s}\left({\Bbb R}^{n}\right) $. The pairing \begin{equation} \left(f,g\right) \mapsto \int f\bar{g}\,dx \notag\end{equation} extends to a pairing $ H^{s}\left({\Bbb R}^{n}\right)\otimes H^{-s}\left({\Bbb R}^{n}\right) \to {\Bbb C} $ which is a pairing of Hilbert spaces. This pairing is compatible with the pairing between $ {\cal S} $ and $ {\cal S}' $. Let $ D $ be a closed subset of $ {\Bbb R}^{n} $, $ U $ be an open subset of $ {\Bbb R}^{n} $. Define a Hilbert subspace \begin{equation} \overset{\,\,{}_\circ}{H}^{s}\left(D\right)=H^{s}\left({\Bbb R}^{n}\right) \cap \left\{f\in{\cal S}' \mid \operatorname{Supp} f\subset D\right\}, \notag\end{equation} and a quotient Hilbert space \begin{equation} H^{s}\left(U\right)=H^{s}\left({\Bbb R}^{n}\right)/\overset{\,\,{}_\circ}{H}\left({\Bbb R}^{n}\smallsetminus U\right). \notag\end{equation} The following properties of Sobolev spaces are most important in analysis: \begin{enumerate} \item If $ D $ is compact, $ V $, $ U $ are open, $ V\subset D\subset U\subset{\Bbb R}^{n} $, and $ \varphi\colon U \to \varphi\left(U\right) $ is a diffeomorphism, then $ \varphi^{*} $ induces invertible bounded operators $ \overset{\,\,{}_\circ}{H}^{s}\left(\varphi\left(D\right)\right) \to \overset{\,\,{}_\circ}{H}^{s}\left(D\right) $, $ H^{s}\left(V\right) \to H^{s}\left(V\right) $. \item If $ D $ is compact, then any differential operator $ A $ of degree $ d $ with smooth coefficients gives a bounded operators $ \overset{\,\,{}_\circ}{H}^{s}\left(D\right) \to \overset{\,\,{}_\circ}{H}^{s-d}\left(D\right) $, $ H^{s}\left(V\right) \to H^{s-d}\left(V\right) $. \item In the same way a pseudodifferential operator of degree $ d $ gives a bounded operators $ \overset{\,\,{}_\circ}{H}^{s}\left(D\right) \to H^{s-d}\left(U\right) $. \item The mapping of restriction of smooth functions onto $ {\Bbb R}^{n-k}\subset{\Bbb R}^{n} $ extends to a bounded operator \begin{equation} r\colon H^{s}\left({\Bbb R}^{n}\right) \to H^{s-\frac{k}{2}}\left({\Bbb R}^{n-k}\right)\text{ if }s>\frac{k}{2}. \notag\end{equation} \item Dually, extension-by-$ \delta $-function of generalized functions on $ {\Bbb R}^{n-k} $ to generalized functions on $ {\Bbb R}^{n} $ gives a bounded operator $ e\colon H^{s}\left({\Bbb R}^{n-k}\right) \to H^{s-\frac{k}{2}}\left({\Bbb R}^{n}\right) $ if $ s<0 $. \item If $ D $ is compact and has a smooth boundary, then $ \overset{\,\,{}_\circ}{H}^{s}\left(D\right)^{\perp}\subset H^{-s}\left({\Bbb R}^{n}\right) $ coincides with $ \overset{\,\,{}_\circ}{H}^{-s}\left(\overline{{\Bbb R}^{n}\smallsetminus D}\right) $. \end{enumerate} \subsection{Sobolev spaces on manifolds }\label{s2.20}\myLabel{s2.20}\relax If $ M $ is a paracompact $ C^{\infty} $-manifold, then $ H^{s}\left(M\right) $ can be defined as a subspace of generalized functions on $ M $ which consists of functions $ f $ such that for any $ U\subset M $ and $ \varphi\colon \widetilde{U} \overset{\sim}\to \widetilde{V}\subset{\Bbb R}^{n} $ the restriction $ f|_{\widetilde{U}} $ satisfies $ \left(\varphi^{-1}\right)^{*}f\in H^{s}\left(V\right) $ (here $ \widetilde{U} $ is a neighborhood of $ \bar{U} $, $ V=\varphi\left(U\right) $). An alternative definition is that for an appropriate partition of unity $ \left(U_{\alpha},\sigma_{\alpha}\right) $, $ \operatorname{Supp} \sigma_{\alpha}\subset\subset U_{\alpha} $, $ \sum_{\alpha}\sigma_{\alpha}=1 $, the products $ \sigma_{\alpha}f $ satisfy $ \left(\varphi_{\alpha}^{-1}\right)^{*}\left(\sigma_{\alpha}f\right) \in \overset{\,\,{}_\circ}{H}\left(V_{\alpha}\right) $, here $ \varphi_{a}\colon U_{\alpha} \overset{\sim}\to V_{\alpha}\subset{\Bbb R}^{n} $. In the same way one may define $ H^{s} $-sections of $ C^{\infty} $-vector bundles over $ M $. We denote the space of $ H^{s} $-section of a bundle $ E $ by $ H^{s}\left(M,E\right) $. Obviously, if $ M $ is compact, one may define a structure of Hilbert space on $ H^{s}\left(M\right) $, but this structure is not uniquely defined. However, the corresponding topology on $ H^{s}\left(M\right) $ is well-defined. For a general manifold $ M $ one defines a topology on $ H^{s}\left(M\right) $ as an inverse limit w.r.t. topologies on $ H^{s}\left(U\right) $, $ U $ being open subsets of $ M $ diffeomorphic to bounded open subsets of $ {\Bbb R}^{n} $. If $ D $ is a closed subset of $ M $, and $ U $ is an open subset, define $ \overset{\,\,{}_\circ}{H}^{s}\left(D\right)\subset H^{s}\left(M\right) $ as subspace of function with support in $ D $, and $ H^{s}\left(U\right) $ as $ H^{s}\left(M\right)/\overset{\,\,{}_\circ}{H}^{s}\left(M\smallsetminus U\right) $. Note that if $ M={\Bbb R}^{n} $, the above definition produces a different space $ H_{\text{loc}}^{s}\left({\Bbb R}^{n}\right) $ of functions on $ {\Bbb R}^{n} $ than the space $ H^{s}\left({\Bbb R}^{n}\right) $ defined in Section ~\ref{s2.02}. A generalized function $ f $ on $ {\Bbb R}^{n} $ is in $ H_{\text{loc}}^{s}\left({\Bbb R}^{n}\right) $ if in any bounded domain it is equal to a function from $ H^{s}\left({\Bbb R}^{n}\right) $. The properties of Sobolev spaces $ H^{s}\left({\Bbb R}^{n}\right) $ have direct analogues for $ H^{s}\left(M\right) $, thus diffeomorphisms of manifolds and differential operators act on $ H^{s}\left(M\right) $, and one can restrict/extend Sobolev sections to/from submanifolds. In particular, the existence of a parametrix for an elliptic differential operator shows that on a compact manifold $ M $ an elliptic operator $ A $ of degree $ d $ gives a Fredholm operator $ H^{s}\left(M\right) \to H^{s-d}\left(M\right) $ (with suitable changes if $ A $ acts in vector bundles). Note that on compact manifolds one can define the Sobolev spaces using arbitrary elliptic (pseudo)differential operators: \begin{proposition} Consider a compact manifold $ M $ with a metric and a positive self-adjoint elliptic operator $ A $ of degree $ d $. Then the $ s $-Sobolev norm is equivalent to the norm \begin{equation} \|f\| = \int_{M}|\left(1+A\right)^{s/d}f|^{2}d\mu. \notag\end{equation} \end{proposition} We will also use the following statement: \begin{proposition} Consider two linear bundles $ {\cal L}_{1} $ and $ {\cal L}_{2} $ on a compact manifold $ M $ such that $ {\cal L}_{1}\otimes{\cal L}_{2}\simeq\Omega^{\text{top}}\left(M\right) $. Then the pairing $ \int\alpha\beta $ between smooth sections of $ {\cal L}_{1} $ and $ {\cal L}_{2} $ can be extended to non-degenerate pairing between Hilbert spaces $ H^{s}\left(M,{\cal L}_{1}\right) $ and $ H^{-s}\left(M,{\cal L}_{2}\right) $ for an arbitrary $ s\in{\Bbb R} $. \end{proposition} \subsection{Capacity } Consider a manifold $ M $. We say that $ s $-{\em capacity\/} of a closed subset $ S\subset M $ is non-zero, if there is a non-zero function $ f\in H^{s}\left(M\right) $ such that $ \operatorname{Supp} f\subset S $. We say that $ S $ has a {\em capacity dimension\/} $ \geq d $, if $ s $-capacity of $ S $ is non-zero for $ s=-\frac{\dim M - d}{2} $. Since $ \delta $-function of a point belongs to $ H^{s}\left(M\right) $ with $ s<-\frac{\dim M}{2} $, a capacity dimension of a point is $ \geq d $ if $ d<0 $. It is clear that this estimate cannot be improved. In the same way the capacity dimension of a submanifold $ S $ is $ \geq d $ iff $ d<\dim S $. For a general subset $ S $ the same is true if one considers Hausdorff dimension \cite{Hor83Dis}. \subsection{Generalized Sobolev spaces }\label{s2.40}\myLabel{s2.40}\relax In Section~\ref{s5.10} we consider the space of $ H^{1/2} $-half-forms which are holomorphic outside of a closure $ \bar{U} $ of a union $ U $ of disks in $ {\Bbb C}P^{1} $. We will see that this space is too big for our purposes if $ \bar{U} $ is ``much bigger'' than $ U $. We will need to consider half-forms which ``are holomorphic'' in $ \bar{U}\smallsetminus U $ as well as in $ {\Bbb C}P^{1}\smallsetminus\bar{U} $. To have this we need the half-forms to be {\em defined\/} in $ {\Bbb C}P^{1}\smallsetminus U $ instead of $ {\Bbb C}P^{1}\smallsetminus\bar{U} $. The problem with this is that $ U $ is not closed, so the usual definition of $ H^{1/2} $-half-forms as of a quotient-space does not work. This shows the need for the following \begin{definition} Consider a subset $ U $ of the manifold $ M $. Let $ \overset{\,\,{}_\circ} H^{s}\left(U\right) $ denotes the closure of the subspace $ L\subset H^{s}\left(M\right) $ \begin{equation} L=\left\{f\in H^{s}\left(M\right) \mid \operatorname{Supp} f\subset U\right\}. \notag\end{equation} Consider $ V\subset M $. Let $ H^{s}\left(V\right)=H^{s}\left(M\right)/\overset{\,\,{}_\circ} H^{s}\left(M\smallsetminus V\right) $. \end{definition} Note that we do not require that the subset $ U $ is closed, and $ V $ is open. If they are, then we get the standard definitions of $ \overset{\,\,{}_\circ}{H}^{s} $ and $ H^{s} $. Obviously, $ \overset{\,\,{}_\circ} H^{s}\left(U\right)\subset\overset{\,\,{}_\circ} H^{s}\left(\bar{U}\right) $, but this inclusion may be proper, as the following construction shows\footnote{The construction can be simplified, but in the current form it is an example of domains we are going to deal with.}. Consider a disjoint family of closed subsets $ V_{i}\subset M $, Let $ {\cal V}=\overline{\bigcup V_{i}}\smallsetminus \bigcup V_{i} $. Suppose that \begin{enumerate} \item The natural mapping $ \bigoplus_{l_{2}}\overset{\,\,{}_\circ} H^{s}\left(V_{i}\right) \xrightarrow[]{\iota} H^{s}\left(M\right) $ is a (continuous) injection\footnote{I.e., the image is closed, and the mapping is an isomorphism on the image.}. \item The $ s $-capacity of $ {\cal V} $ is positive. \end{enumerate} The first condition insures that the space $ \overset{\,\,{}_\circ} H^{s}\left(\bigcup V_{i}\right) $ is the image of the mapping $ \iota $. Hence any non-zero function $ f\in\overset{\,\,{}_\circ} H^{s}\left(\bigcup V_{i}\right) $ satisfies the condition $ \operatorname{Supp} f \cap \bigcup V_{i}\not=\varnothing $. On the other hand, the second condition shows that there is a non-zero function $ f\in H^{s}\left(M\right) $ such that $ \operatorname{Supp} f\subset{\cal V} $. Obviously, $ f\in\overset{\,\,{}_\circ} H^{s}\left(\overline{\bigcup V_{i}}\right) $, but $ f\notin\overset{\,\,{}_\circ} H^{s}\left(\bigcup V_{i}\right) $. In Section~\ref{s4.50} we show how to construct a family of disks $ V_{i} $ which satisfy the first condition. The centers of these disks may be an arbitrary prescribed locally discrete set. Moreover, one can easily find an appropriate set of centers such that the corresponding set $ {\cal V} $ does not depend on radii and coincides with an arbitrary given closed set $ {\cal V}_{0} $ with empty interior. In particular, if there exist a closed set $ {\cal V} $ with an empty interior and non-zero $ s $-capacity, then one can construct a subset $ U $ of $ {\Bbb C}P^{1} $ such that the inclusion $ \overset{\,\,{}_\circ} H^{s}\left(U\right)\subset\overset{\,\,{}_\circ} H^{s}\left(\bar{U}\right) $ is proper. Taking appropriate Cantor sets, one gets that this is possible with any $ s<0 $. On the other hand, if $ U $ has smooth boundary, then $ \overset{\,\,{}_\circ} H^{s}\left(U\right)=\overset{\,\,{}_\circ} H^{s}\left(\bar{U}\right) $. \subsection{Rescaling on $ {\Bbb R}^{n} $ and $ {\Bbb C}^{n} $ }\label{s2.50}\myLabel{s2.50}\relax Consider an $ H^{s} $-function $ f\left(x\right) $ on $ {\Bbb R}^{n} $ with compact support. The Fourier transform $ \widehat{f}\left(\xi\right) $ is real-analytic, thus the integral \begin{equation} \int|\xi|^{2s}|\widehat{f}\left(\xi\right)|^{2}d\xi, \label{equ3.7}\end{equation}\myLabel{equ3.7,}\relax can be defined in the sense of generalized functions (i.e., analytic continuation in $ s $) near $ \xi=0 $. Since $ f\in H^{s} $, the integral converges near $ \infty $. Moreover, if $ s>-\frac{n}{2} $, the integral converges near $ \xi=0 $, thus its value is a limit of Riemann sums, hence is non-negative. In any case the integral defines a quadratic form on $ \overset{\,\,{}_\circ}{H}^{s}\left(D\right) $ for any compact $ D $. \begin{nwthrmi} If $ s>-\frac{n}{2} $, the integral defines a norm on $ \overset{\,\,{}_\circ}{H}^{s}\left(D\right) $, and this form is equivalent to the Hilbert norm. \end{nwthrmi} The advantage of the norm~\eqref{equ3.7} is the fact that it is covariant with respect to dilatations. Define a mapping $ {\frak D}_{a}^{s}\colon {\cal S} \to {\cal S} $ by \begin{equation} \left({\frak D}_{a}^{s}f\right)\left(x\right) = a^{s}f\left(ax\right),\qquad a,s\in{\Bbb R},\quad a\not=0. \notag\end{equation} Then $ {\frak D}_{a}^{-s+\frac{n}{2}} $ is an isometry w.r.t. the norm~\eqref{equ3.7}. Note also that that $ {\frak D}_{a}^{tn} $ is a natural action of a dilatation on sections of the linear bundle $ \Omega^{t} $. Combining this with the previous statement, we conclude that any similarity transform\footnote{I.e., a composition of translations, rotations and dilatations.} $ {\bold T} $ acts as a uniformly bounded operator in the space of $ H^{s} $-sections of $ \Omega^{\alpha} $ \begin{equation} {\bold T}^{*}\colon \overset{\,\,{}_\circ}{H}^{s}\left({\bold T}D,\Omega^{\alpha}\right) \to \overset{\,\,{}_\circ}{H}^{s}\left(D,\Omega^{\alpha}\right),\qquad \alpha = \frac{1}{2}-\frac{s}{n}, \notag\end{equation} as far as both $ D $ and $ {\bold T}D $ remain in the same disk $ \left\{|x|<R\right\} $, and $ R $ is fixed. If we consider $ {\Bbb C}^{n}={\Bbb R}^{2n} $ and the linear bundle $ \omega^{1/2} $ over $ {\Bbb C}^{n} $, then the action of dilatations\footnote{In fact one needs to consider $ 2 $-sheeted covering of the group of similarity transforms.} on this bundle is the same\footnote{Strictly speaking, differs on a multiplication by a constant of magnitude 1.} as on $ \Omega_{{\Bbb R}^{2n}}^{1/4} $. We conclude that similarity transforms act as uniformly bounded transformations \begin{equation} \overset{\,\,{}_\circ}{H}^{n/2}\left({\bold T}D,\omega^{1/2}\right) \to \overset{\,\,{}_\circ}{H}^{n/2}\left(D,\omega^{1/2}\right) \notag\end{equation} as far as both $ D $ and $ {\bold T}D $ remain in the same disk $ \left\{|z|<R\right\} $, and $ R $ is fixed. \subsection{$ \protect \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant realization of $ H^{1/2}\left({\Bbb C}P^{1}\right) $. }\label{s2.60}\myLabel{s2.60}\relax Consider double ratio \begin{equation} \left(a:b:c:d\right) = \frac{a-b}{b-d}:\frac{a-c}{c-d},\qquad a,b,c,d\in{\Bbb C}P^{1}. \notag\end{equation} Double ratio $ \left(z_{1}:z_{1}+\delta z_{1}:z_{2}:z_{2}+\delta z_{2}\right) $ gives a section $ \rho $ of the bundle $ T^{*}{\Bbb C}P^{1}\boxtimes T^{*}{\Bbb C}P^{1} $ over $ {\Bbb C}P^{1}\times{\Bbb C}P^{1} $. In affine coordinates it can be written as $ \frac{1}{\left(z_{1}-z_{2}\right)^{2}}dz_{1}\,dz_{2} $. By construction this section is invariant with respect to the action of $ \operatorname{PGL}\left(2,{\Bbb C}\right) $. Consider $ K_{s}=\frac{\rho^{s} \bar{\rho}^{s}}{\Gamma\left(-2s+1\right)} $. It is a section of $ \Omega^{s}\left({\Bbb C}P_{{\Bbb R}}^{1}\right) \boxtimes \Omega^{s}\left({\Bbb C}P_{{\Bbb R}}^{1}\right) $, which in local coordinates looks like $ K_{s}\left(z_{1},z_{2}\right)=\frac{1}{|z_{1}-z_{2}|^{4s}\Gamma\left(-2s+1\right)}dZ_{1}^{s}\,dZ_{2}^{s} $ and has no singularity outside of $ z_{1}=z_{2} $. Here $ dZ=dx\wedge dy $ if $ z=x+iy. $ The operator with kernel $ K_{s} $ defines a pairing \begin{equation} \left(\alpha,\beta\right)_{s}=\int_{{\Bbb C}P^{1}\times{\Bbb C}P^{1}}K_{s}\left(z_{1},z_{2}\right)\alpha\left(z_{1}\right)\bar{\beta}\left(z_{1}\right) \notag\end{equation} on the space $ \Gamma\left(\Omega^{1-s}\left({\Bbb C}P_{{\Bbb R}}^{1}\right)\right) $ if $ s<\frac{1}{2} $, this pairing depends on $ s $ analytically \cite{GelShil58Gen}, and may be continued to an arbitrary $ s $. \begin{proposition} $ \left(,\right)_{s} $ is a positive-definite Hermitian form on $ \Gamma\left(\Omega^{1-s}\left({\Bbb C}P_{{\Bbb R}}^{1}\right)\right) $, if $ \left|s-\frac{1}{2}\right|<\frac{1}{2} $. It is equivalent to the $ H^{-1+2s} $-norm on this space. \end{proposition} \begin{proof} For $ s=\frac{1}{2} $ we get a standard pairing on $ \Omega^{1/2} $ (since $ K_{1/2} $ is a $ \delta $-function), so it is sufficient to consider $ s\not=\frac{1}{2} $. Take an affine coordinate system $ z $ on $ {\Bbb C}P^{1}\smallsetminus\left\{\infty\right\} $. A smooth section of $ \Omega^{t} $ near $ \infty $ is represented by a smooth $ t $-form $ f\left(z\right)dZ^{t} $ with an asymptotic \begin{equation} |z|^{-4t}g\left(1/z\right)dZ^{t}, \notag\end{equation} near $ \infty $. Here $ g\left(w\right) $ is a smooth function of $ x $ and $ y $, $ w=x+yi. $ Thus the Fourier transform $ \widehat{f}\left(\zeta\right) $ of $ f $ is rapidly decreasing, smooth outside of $ \zeta=0 $, and has an asymptotic expansion \begin{equation} h\left(\zeta\right)+|\zeta|^{2-4t}h_{1}\left(\zeta\right)+O\left(|\zeta|^{N}\right) \notag\end{equation} near 0, here $ N>0 $ is an arbitrary integer, $ h $ and $ h_{1} $ are smooth functions, and $ t\not=\frac{1}{2} $. The pairing $ \left(,\right)_{s} $ written in terms of $ \widehat{\alpha} $, $ \widehat{\beta} $ is the pairing \begin{equation} \operatorname{const}\cdot\int\widehat{\alpha}\left(\zeta\right)\bar{\widehat{\beta}}\left(\zeta\right)|\zeta|^{4s-2}d\zeta d\bar{\zeta} \label{equ3.74}\end{equation}\myLabel{equ3.74,}\relax (in the sense of generalized functions) and we see that for $ t=1-s $ and $ \left|s-\frac{1}{2}\right|<\frac{1}{2} $ the integral converges, thus the pairing is positive. On the other hand, the local equivalence of the norm~\eqref{equ3.74} with $ H^{2s-1} $ is already proved in the previous section (cf. Equation~\eqref{equ3.7}). \end{proof} \begin{remark} Note that what we got is a complementary series of unitary representations of $ \operatorname{SL}\left(2,{\Bbb C}\right) $. \end{remark} \begin{remark} \label{rem2.60}\myLabel{rem2.60}\relax The defined pairings are compatible with duality $ \int\alpha\beta $ between $ \Omega^{s} $ and $ \Omega^{1-s} $. Note also that in the case $ s=0 $ the pairing has $ \operatorname{rk}=1 $, dually, in the case $ s=1 $ the pairing becomes $ -\left(\Delta\alpha,\beta\right) $, which has constants in the null-space. \end{remark} Put $ s=\frac{1}{4} $. We get a $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant implementation\footnote{I.e., a Hilbert structure which is equivalent to the Sobolev Hilbert structure which is in turn defined up to equivalence (compare with Section~\ref{s2.20}).} of the space $ H^{-1/2}\left(\Omega^{3/4}\right) $. Note now that $ \Omega^{3/4}=\omega^{3/4}\otimes\bar{\omega}^{3/4} $ differs not so much from the space $ \omega^{1/2}\otimes\bar{\omega} $ we are most interested in. The difference $ \omega^{1/4}\otimes\bar{\omega}^{-1/4} $ is a bundle with transitions functions $ \left(\frac{D}{\bar{D}}\right)^{1/4} $ of magnitude 1 (here $ D=dz/dw $). In particular, a choice of affine coordinate $ z $ on $ {\Bbb C}P^{1} $ gives an identification of $ \omega^{1/2}\otimes\bar{\omega} $ with $ \Omega^{3/4} $ (outside of infinity). This identification commutes with natural actions of $ \operatorname{Aff}\left({\Bbb C}\right) $ on $ \Omega^{3/4} $ and $ \omega^{1/2}\otimes\bar{\omega} $ (up to multiplication by a constant of magnitude 1). Here $ \operatorname{Aff}\left({\Bbb C}\right) $ is the group of affine transformations on $ {\Bbb C} $ (in fact to get an action on $ \omega^{1/2}\otimes\bar{\omega} $ one needs to consider $ 2 $-covering of $ \operatorname{Aff}\left({\Bbb C}\right) $). This explains the almost-invariance of $ H^{-1/2}\left({\Bbb C},\omega^{1/2}\otimes\bar{\omega}\right) $ with respect to affine transformations, discussed in Section~\ref{s2.50}. Indeed, consider a disk $ K\subset{\Bbb C}P^{1} $. On $ {\Bbb C}P^{1}\smallsetminus K $ multiplication by $ dz^{1/4}d\bar{z}^{-1/4} $ gives an isomorphism of the sheaves $ \omega^{1/2}\otimes\bar{\omega} $ and $ \Omega^{3/4} $. Taking $ \infty $ as a center of $ K $, we obtain the results of Section~\ref{s2.50} in the case $ n=1 $. In what follows the following result is sufficient for us to show conformal invariance of the objects we introduce: \begin{amplification} \label{amp2.65}\myLabel{amp2.65}\relax Fix a metric on $ {\Bbb C}P^{1} $ and $ \varepsilon>0 $. Consider $ \varphi\in\operatorname{SL}\left(2,{\Bbb C}\right) $, let $ A\subset{\Bbb C}P^{1} $, $ B=\varphi\left(A\right) $. If both $ {\Bbb C}P^{1}\smallsetminus A $ and $ {\Bbb C}P^{1}\smallsetminus B $ contain disks of radius $ \varepsilon $ with centers at $ c_{A} $, $ c_{B} $, then the norm of $ \varphi^{*}\colon \overset{\,\,{}_\circ}{H}^{-1/2}\left(B,\omega^{1/2}\otimes\bar{\omega}\right) \to \overset{\,\,{}_\circ}{H}^{-1/2}\left(A,\omega^{1/2}\otimes\bar{\omega}\right) $ is bounded by \begin{equation} C\left(\varepsilon\right)\left(\frac{\operatorname{diam}\left(A\right)}{\operatorname{dist}\left(\varphi^{-1}\left(c_{B}\right),A\right)}\right)^{1/2}. \notag\end{equation} By duality, the same bound is valid for action of $ \varphi^{*} $ in $ H^{1/2}\left(\bullet,\omega^{1/2}\right) $. \end{amplification} \begin{proof} Indeed, $ SU\left(2\right) $ is compact, thus acts on $ H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right) $ by uniformly bounded operators. Thus it is sufficient to consider the case when a disk $ K $ of radius $ \varepsilon $ is fixed (say, if $ c_{A}=c_{B}=\infty $, thus $ |z| > 1/\varepsilon $), and $ A\cap K=B\cap K=\varnothing $. Let $ \alpha $, $ \beta $ be sections of $ \omega^{1/4}\otimes\bar{\omega}^{-1/4} $ and $ \omega^{-1/4}\otimes\bar{\omega}^{1/4} $ which coincide with $ \bar{d}z^{1/4}d\bar{z}^{-1/4} $ and $ dz^{-1/4}d\bar{z}^{1/4} $ inside $ {\Bbb C}P^{1}\smallsetminus K $ correspondingly, and have a compact support. Then $ \varphi^{*}|_{A}f $, $ f\in\omega^{1/2}\otimes\bar{\omega} $, is equal to \begin{equation} \mu_{\varphi}\beta\varphi_{3/4}^{*}|_{A}\left(\alpha f\right),\qquad \mu_{\varphi}=\left(\frac{\varphi'}{|\varphi'|}\right)^{1/2} \notag\end{equation} (here $ \varphi_{3/4}^{*} $ acts on $ \Omega^{3/4} $). On the other hand, multiplications by $ \alpha $ and $ \beta $ are bounded operators, and $ \varphi_{3/4}^{*} $ is unitary, so what remains to prove is a bound on operator of multiplication by $ \mu_{\varphi} $ in $ H^{-1/2} $ or in $ H^{1/2} $. It is sufficient to consider $ H^{1/2} $. Let us apply interpolation theorem now: the norm of multiplication by $ \mu_{\varphi} $ in $ H^{1/2} $ is no more than geometric mean of norms in $ H^{0}=L_{2} $ and in $ H^{1} $. Since $ |\mu_{\varphi}|=1 $, it is unitary in $ H^{0} $, so it is sufficient to estimate the norm in $ H^{1} $, which is bounded by $ \operatorname{const}\cdot\max |\mu_{\varphi}'| $, i.e., by $ \operatorname{const}\cdot\operatorname{dist}\left(\varphi^{-1}\left(\infty\right),A\right)^{-1} $. To get the estimate in the theorem, note that we have a freedom of rescaling $ A $, thus may assume that $ \operatorname{diam}\left(A\right)=1 $. \end{proof} \subsection{Hardy space }\label{s2.70}\myLabel{s2.70}\relax Let $ K = \left\{z \mid |z|\leq1\right\} $. The Hardy space $ {\cal H} $ is the subspace of $ L_{2}\left(\partial K\right) $ consisting of functions with Fourier coefficients $ \left(a_{n}\right) $ which vanish for $ n<0 $. For a function $ f\in{\cal H} $, $ f=\sum a_{k}z^{k} $, $ |z|=1 $, let $ c_{f}=\sum a_{k}z^{k} $ be defined for $ |z|<1 $. The latter series converges, and defines a holomorphic function inside $ K $. \begin{lemma} The mapping $ f \mapsto c_{f} $ is an injection $ L_{2}\left(\partial K\right) \to H^{1/2}\left(K\right) $. The image of this injection coincides with \begin{equation} \operatorname{Ker}\left(H^{1/2}\left(K\right) \xrightarrow[]{\bar{\partial}} H^{-1/2}\left(K\right)\right). \notag\end{equation} \end{lemma} \begin{proof} It is sufficient to prove that $ \|z^{k}\|_{H^{1/2}\left(K\right)} $ is bounded from above and from below when $ k\in{\Bbb N} $. Consider a concentric disk $ K_{1} $ of radius $ R<1 $. Since $ \|z^{k}\|_{H^{1/2}\left(K_{1}\right)}\leq\|z^{k}\|_{H^{1}\left(K_{1}\right)}=O\left(kR^{k}\right) $, it is $ o\left(1\right) $, thus it is sufficient to consider the norm of $ z^{k} $ in a narrow annulus with external boundary $ \partial K $. In turn, taking coordinate $ \log z=a+ib $, $ b\in{\Bbb R}/2\pi{\Bbb Z} $, it is sufficient to consider $ H^{1/2} $-norm of $ e^{ka} $ on the half-line $ a\leq0 $, more precise, it is sufficient to consider $ L_{2} $-norm of $ \left(\frac{d}{da}+k\right)^{1/2}e^{ka} $, $ a\leq0 $. Indeed, we need to show that there is a continuation of $ e^{k\left(a+ib\right)} $ outside of $ a\leq0 $ such that the $ H^{1/2} $-norm of this continuation is bounded, and that the $ H^{1/2} $-norm of any such continuation is bounded from below. One can suppose that the continuation is $ e^{ik b}\varphi\left(a\right) $, so one needs to estimate $ \int\left(1+k^{2}+\alpha^{2}\right)|\widehat{\varphi}\left(\alpha\right)|^{2}d\alpha $. The operator $ \left(\frac{d}{da}+k\right)^{1/2} $ maps this norm to a norm equivalent to $ L_{2} $-norm, moreover, it sends functions with support in $ a\geq0 $ into itself (since $ k\geq0 $). Thus it defines an invertible operator $ H^{1/2}\left({\Bbb R}_{\leq0}\right) \to H^{0}\left({\Bbb R}_{\leq0}\right)=L_{2}\left({\Bbb R}_{\leq0}\right) $. This means that $ \left(\frac{d}{da}+k\right)^{1/2}e^{ka}|_{a\leq0}=\left(\frac{d}{da}+k\right)^{1/2}\varphi\left(a\right)|_{a\leq0} $, thus we should not care about the choice of continuation $ \varphi $. Moreover, since $ \left(\frac{d}{da}+k\right)^{1/2} $ does not move support to the left, $ \left(\frac{d}{da}+k\right)^{1/2}e^{ka}=C\left(k\right)e^{ka} $, $ a\leq0 $. Since $ \left(\frac{d}{da}+k\right)^{1/2} $ is an operator of convolution with $ \frac{a_{+}^{-3/2}e^{-ka}}{\Gamma\left(-1/2\right)} $ (in the sense of generalized functions, $ x_{+}^{s} $ is 0 if $ x<0 $, $ x^{s} $ if $ x>0 $ and $ s>0 $, and is defined by analytic continuation from the region $ s>0 $, where the above description defines it completely \cite{GelShil58Gen}), to estimate $ C\left(k\right) $ one needs to calculate \begin{align} -\int_{0}^{\infty}x_{+}^{-3/2}e^{-kx}e^{-kx}dx & \buildrel{\text{def}}\over{=} -\int_{0}^{\infty}x^{-3/2}\left(e^{-2kx}-1\right)dx \notag\\ & = k^{1/2}\int_{0}^{\infty}\left(kx\right)^{-1/2}\frac{1-e^{-2kx}}{kx}d\left(kx\right) = k^{1/2}C. \notag\end{align} Thus we need to show that $ k^{1/2}\|e^{ka}|_{a\leq0}\|_{L_{2}} $ is bounded from above and from below, which is obvious. \end{proof} \begin{remark} It is clear that if $ f\in{\cal H}\cap C^{\infty} $, then $ f=c_{f}|_{\partial K} $. Moreover, if $ f\cdot\delta_{\partial K} $ is the continuation of $ f $ to $ {\Bbb C} $ by $ \delta $-function, then $ c_{f}=\operatorname{const}\cdot\bar{\partial}^{-1}\left(f\cdot\delta_{\partial K}\right) $ (see Section~\ref{s3.30} for description of $ \bar{\partial}^{-1} $). Several following sections are dedicated to defining something similar in the case $ f\in{\cal H} $. \end{remark} \begin{remark} \label{rem2.10}\myLabel{rem2.10}\relax In Section~\ref{h4} we will construct a generalization of the following modification of this result: instead of $ K $ we consider an isomorphic domain $ {\Bbb C}P^{1}\smallsetminus K $. Instead of functions on $ {\Bbb C}P^{1} $ and $ \partial K $ we consider sections of $ \omega^{1/2} $ and $ \Omega^{1/2}\otimes\mu $ correspondingly. \end{remark} \section{Cauchy kernel for $ \omega^{1/2} $ } \subsection{$ \bar{\partial} $ on piecewise-analytic functions }\label{s3.05}\myLabel{s3.05}\relax Let $ M $ be a complex curve with a linear bundle $ {\cal L} $. Consider a domain $ D\subset M $ with a smooth boundary $ \partial D=\gamma $. Let $ f\left(z\right) $, $ z\in\bar{D} $, be an analytic section of $ {\cal L} $ in $ D $ which continues as a smooth section to $ \bar{D} $. Extend $ f $ outside of $ D $ as 0. Consider $ \bar{\partial}f $. It is a section of $ {\cal L}\otimes\bar{\omega} $, which vanishes on $ M\smallsetminus\gamma $. Since $ f $ has a jump of the first kind along $ \gamma $, and $ \bar{\partial} $ is a differential operator of the first order, $ \bar{\partial}f $ has at most a $ \delta $-function singularity along $ \gamma $. One may locally write \begin{equation} \bar{\partial}f = g\left(z\right)\cdot\delta_{h\left(z\right)=0}. \notag\end{equation} Here $ g\left(z\right) $ is a smooth section of $ {\cal L}\otimes\bar{\omega} $, $ h $ is a local equation of $ \gamma $. Let us calculate $ g|_{\gamma} $ in terms of $ f $. First of all, if one changes the equation $ h=0 $ by a different equation $ hh_{0}=0 $ (here $ h_{0} $ has no zeros close to the point in question), the coefficient at $ g\left(z\right) $ changes to \begin{equation} \delta_{hh_{0}=0}=\frac{1}{h_{0}}\delta_{h=0}, \notag\end{equation} since $ \delta $-function has homogeneity degree $ -1 $. Thus $ g\left(z\right)|_{\gamma} $ should be multiplied by $ h_{0}|_{\gamma} $ to preserve the same value of the product. This shows that to get an invariant description of $ g\left(z\right)|_{\gamma} $ one needs to write \begin{equation} g\left(z\right)|_{\gamma} = G\left(z\right)\otimes dh|_{\gamma}. \notag\end{equation} Here $ dh|_{\gamma} $ is considered as a section of the conormal bundle $ N^{*}\gamma $ to $ \gamma $, thus $ G\left(z\right) $ should be a section of $ \left({\cal L}\otimes\bar{\omega}\right)|_{\gamma}\otimes\left(N^{*}\gamma\right)^{*} = \left({\cal L}\otimes\bar{\omega}\right)|_{\gamma}\otimes N\gamma $. Now $ G\left(z\right) $ does not depend on the parameterization of a neighborhood of $ \gamma $, so it should be directly expressible in terms of $ f\left(z\right) $. \begin{lemma} $ \bar{\omega}|_{\gamma} $ is canonically isomorphic to $ \left(N^{*}\gamma\right)\otimes{\Bbb C} $. \end{lemma} \begin{proof} Indeed, $ \bar{\omega} $ is defined as a quotient of $ \left(T^{*}M_{{\Bbb R}}\right)\otimes{\Bbb C} $ by holomorphic forms, moreover, $ \bar{\omega}_{{\Bbb R}} $ is an isomorphic image of $ T^{*}M_{{\Bbb R}} $. Since $ N^{*}\gamma\subset T^{*}M_{{\Bbb R}} $, there is a natural non-zero mapping $ N^{*}\gamma \to \bar{\omega}_{{\Bbb R}} $, which gives the required isomorphism after complexification. \end{proof} As a corollary, $ \left({\cal L}\otimes\bar{\omega}\right)|_{\gamma}\otimes N\gamma \simeq {\cal L}|_{\gamma} $. Now the following fact is obvious from calculations in local coordinates: \begin{proposition} $ G\left(z\right) = -f\left(z\right)|_{\gamma} $. \end{proposition} In particular, we see that one can write \begin{equation} \bar{\partial}f=-e\circ r\left(f\right) \label{equ3.02}\end{equation}\myLabel{equ3.02,}\relax in terms of operators $ r $ of restriction and $ e $ of extension-by-$ \delta $-function. Thus we can consider $ -e\circ r $ as an ``approximation'' to $ \bar{\partial} $ which is good on functions which are analytic far from $ \gamma $. This approximation is exact on functions with a jump of the first kind. Yet another way to treat this identity is to write it as \begin{equation} -\bar{\partial}^{-1}\circ e\circ r\left(F\right) = \vartheta_{D}F. \notag\end{equation} Here $ F $ is a function which is holomorphic in a neighbourhood of $ \bar{D} $, $ \vartheta_{D} $ is a function which is 1 on $ D $ and 0 outside of $ D $. We give a mollified version of this statement in Section~\ref{s4.35}. \subsection{Self-duality of $ \bar{\partial} $ } Consider a mapping $ \bar{\partial}\colon \omega^{1/2} \to \omega^{1/2}\otimes\bar{\omega} $. Note that the spaces of sections of these bundles are dual w.r.t. the pairing \begin{equation} \left(\varphi,\psi\right) = \int_{M}\varphi\psi. \label{equ3.20}\end{equation}\myLabel{equ3.20,}\relax \begin{lemma} $ \bar{\partial} $ is skew-symmetric w.r.t. the above pairing, \begin{equation} \left(\varphi_{1},\bar{\partial}\varphi_{2}\right)+\left(\varphi_{2},\bar{\partial}\varphi_{1}\right)=0. \notag\end{equation} \end{lemma} \begin{proof} It is sufficient to show that $ \left(\varphi_{1},\bar{\partial}\varphi_{1}\right)=0 $, i.e., to study \begin{equation} \int_{M}\varphi\bar{\partial}\varphi = \frac{1}{2}\int_{M}\bar{\partial}\varphi^{2} = \frac{1}{2}\int_{M}d\varphi^{2} \notag\end{equation} which obviously vanishes. \end{proof} Note that this supports the heuristic that the bundle $ \omega^{1/2} $ is ``the best one'' of the powers of $ \omega $. Used literally, the above considerations were applicable to the space of smooth sections of the bundles in question. On the other hand, the pairing~\eqref{equ3.20} extends to a continuous pairing between $ H^{1/2}\left(M,\omega^{1/2}\right) $ and $ H^{-1/2}\left(M,\omega^{1/2}\otimes\bar{\omega}\right) $, and the operator $ \bar{\partial} $ maps one of these spaces to another. Using the facts from Section~\ref{s2.20} We get a \begin{proposition} The operator $ \bar{\partial} $ gives a canonically defined Fredholm symplectic form on $ H^{1/2}\left(M,\omega^{1/2}\right) $. This form is non-degenerate if $ M={\Bbb C}P^{1} $. \end{proposition} Here we call a bilinear form $ \alpha $ on a Hilbert space $ {\cal H} $ a {\em Fredholm form}, if the corresponding operator $ \alpha\colon H \to H^{*}=H $ is Fredholm. Recall that in the case $ M={\Bbb C}P^{1} $ the space $ H^{1/2}\left(M,\omega^{1/2}\right) $ is the ``best'' of Sobolev spaces for $ \omega^{1/2} $, since it allows the action of $ \operatorname{SL}\left(2,{\Bbb C}\right) $ (at least ``in small'') by bounded operators. Here we see another ``nice'' property of this particular Sobolev space: there is a canonically defined invertible pairing on this space (invertible in the sense that the corresponding operator is bounded). \begin{remark} If we could construct a self-adjoint operator which enjoys the above properties of the operator $ \bar{\partial} $, i.e., is $ \operatorname{SL}\left(2,{\Bbb C}\right) $-invariant and Fredholm, one would be (almost) able to define an invariant Sobolev space structure on $ H^{1/2}\left(M,\omega^{1/2}\right) $. The only thing missing would be positive definiteness. \end{remark} \subsection{The kernel of $ \bar{\partial}^{-1} $ }\label{s3.30}\myLabel{s3.30}\relax Suppose $ M={\Bbb C}P^{1} $. In this case the operator $ \bar{\partial}: {\cal D}'\left(\omega^{1/2}\right) \to {\cal D}'\left(\omega^{1/2}\otimes\bar{\omega}\right) $ is invertible, thus the operator $ \bar{\partial}^{-1} $ is canonically defined. Consider a kernel $ K\left(x,y\right) $ of this operator, \begin{equation} \left(\bar{\partial}^{-1}f\right)\left(x\right) = \int K\left(x,y\right)f\left(y\right). \notag\end{equation} Obviously, $ K\left(x,y\right) $ is a section of $ \omega^{1/2} \boxtimes \omega^{1/2} $. To calculate $ K\left(x,y\right) $ consider a $ \delta $-section $ \delta_{z_{0}} $ of $ \omega^{1/2} $ at $ z_{0} $ (defined up to a scalar multiple). Then $ K\left(\bullet,z_{0}\right)=\operatorname{const}\cdot\bar{\partial}^{-1}\delta_{z_{0}} $ is a holomorphic section of $ \omega^{1/2} $ on $ {\Bbb C}P^{1}\smallsetminus\left\{z_{0}\right\} $. It is easy to see that it has a simple pole at $ z=z_{0} $, thus has no zeros (since $ \omega^{1/2}\simeq{\cal O}\left(-1\right) $). We conclude that $ K\left(x,y\right) $ has a simple pole at $ x=y $. There is only one such a section of $ \omega^{1/2} \boxtimes \omega^{1/2} $. Now one may easily recognize the kernel for $ \bar{\partial}^{-1} $ in the Cauchy formula \begin{equation} \bar{\partial}^{-1}\delta_{z=z_{0}}dz^{1/2}d\bar{z} = \frac{1}{2\pi i} \frac{1}{z-z_{0}}dz^{1/2} \notag\end{equation} The right-hand side $ f\left(z,z_{0}\right) $ has no pole at $ \infty $, and is uniquely determined by this condition and the equation \begin{equation} \bar{\partial}f=\delta_{z=z_{0}}dz^{1/2}d\bar{z} \notag\end{equation} in $ {\Bbb C}P^{1}\smallsetminus\left\{\infty\right\} $. We conclude that \begin{equation} K\left(x,y\right) = \frac{1}{2\pi i} \frac{1}{x-y}dx^{1/2}dy^{1/2}. \notag\end{equation} \subsection{Cauchy kernel and $ L_{2} $ } Consider a smooth embedded curve $ \gamma\hookrightarrow{\Bbb C}P^{1} $ (we do not suppose that $ \gamma $ is compact). Any smooth function $ f $ on $ \gamma $ with a compact support may be extended as a generalized $ \delta $-function $ e\left(f\right) $ on $ {\Bbb C}P^{1} $ with a support on $ \gamma $, same for half-forms on $ \gamma $. Applying $ \bar{\partial}^{-1} $ to the result, we obtain a form on $ {\Bbb C}P^{1} $ which is holomorphic outside of $ \gamma $. Here we discuss when this mapping $ \bar{\partial}^{-1}\circ e $ allows $ f\in L_{2}\left(\gamma\right) $ instead of $ f\in{\cal D}\left(\gamma\right) $. \begin{remark} In what follows the main example of $ \gamma $ is a disjoint union of infinite number of circles. \end{remark} Fix a point $ z\in{\Bbb C}P^{1} $ and suppose that some neighborhood of $ z $ does not intersect $ \gamma $. Then for a half-form $ f $ \begin{align} \frac{\bar{\partial}^{-1}\circ e\left(f\right)}{dz^{1/2}d\bar{z}}|_{z_{0}} & \buildrel{\text{def}}\over{=} \left< \bar{\partial}^{-1}\circ e\left(f\right), \delta_{z_{0}}dz^{1/2}\right> \notag\\ & = -\left< e\left(f\right), \bar{\partial}^{-1}\left(\delta_{z_{0}}dz^{1/2}\right)\right> \notag\\ & =-\left< e\left(f\right), K\left(\bullet,z_{0}\right)dz^{-1/2} \right> = \int_{C}f\left(x\right)K\left(x,z_{0}\right)dz^{-1/2}. \notag\end{align} We conclude that the linear functional of calculating $ \bar{\partial}^{-1}\circ e\left(f\right) $ at $ z_{0} $ is given by the kernel $ K\left(\bullet,z_{0}\right)|_{\bullet\in C} $. However, since $ K\left(\bullet,z_{0}\right) $ is bounded outside of a neighborhood of $ z_{0} $, we see that \begin{proposition} \label{prop4.12}\myLabel{prop4.12}\relax The mapping $ \bar{\partial}^{-1}\circ e $ extends to a mapping from $ L_{2}\left(\gamma\right) $ to the space $ \operatorname{Hol}\left({\Bbb C}P^{1}\smallsetminus\bar{\gamma}\right) $ of forms holomorphic outside of $ \gamma $ iff the length of $ \gamma $ is finite. \end{proposition} \section{Toy theory }\label{h35}\myLabel{h35}\relax In this section we study an example which shows the main framework of our approach without any of the complications related to the necessity to consider $ H^{1/2} $-spaces. This is an example related to dealing with {\em partial period mapping\/} $ \Gamma\left(M,\omega\right) \to {\Bbb C}^{g} $ of taking periods of global holomorphic forms along $ A $-cycles on the Riemann surface. In fact, we start from studying a related space of global holomorphic functions which are allowed to have constant jumps along $ A $-cycles. \subsection{Toy global space } Consider a complex curve $ M $ and the mapping $ \partial\bar{\partial}\colon \Omega^{0} \to \Omega^{1} $ (let us recall that $ \Omega^{\text{top}}=\Omega^{1} $ in our notations for fractional forms). The integral $ -\int_{M}if\cdot\partial\bar{\partial} \bar{f} $ defines a sesquilinear form on functions with compact support, it is Hermitian, and the only functions which are in the null-space of this form are constants, since the integral is equal to $ \int_{M}i\partial f\cdot\bar{\partial} \bar{f}\geq0 $. This integral defines a pre-Hilbert structure on global sections of $ \Omega^{0} $ modulo constants, this structure is compatible with the Sobolev $ H^{1} $-topology. Moreover, if $ M={\Bbb C}P^{1} $, this structure is $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant, and coincides with the structure of supplementary series of representations of $ \operatorname{SL}\left(2,{\Bbb C}\right) $ (in notations of Section~\ref{s2.60} it is the case $ s=1 $, compare with Remark ~\ref{rem2.60}). As a result, we obtain a $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant realization of Hilbert space $ H^{1}\left({\Bbb C}P^{1}\right)/\operatorname{const} $. The dual space is the subspace $ H_{\int=0}^{-1}\left({\Bbb C}P^{1},\Omega^{1}\right) $ of $ H^{-1}\left({\Bbb C}P^{1},\Omega^{1}\right) $ consisting of forms with integral 0, thus the latter space also carries an $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant Hilbert structure. To describe it, note that Remark ~\ref{rem2.60} is still applicable, thus in the limit $ s \to 0 $ the pre-Hilbert structure of Section~\ref{s2.60} becomes a form of $ \operatorname{rk}=1 $ with $ H_{\int=0}^{-1}\left({\Bbb C}P^{1},\Omega^{1}\right) $ in the null-space. As a corollary, $ \frac{d}{ds}|_{s=0}\left(\alpha,\beta\right)_{s} $ gives a correctly defined positive form on $ H_{\int=0}^{-1}\left({\Bbb C}P^{1},\Omega^{1}\right) $ which should be dual to the pairing on $ H^{1}\left({\Bbb C}P^{1}\right)/\operatorname{const} $. Taking the $ s $-derivative of the kernel $ K_{s} $ from Section~\ref{s2.60}, we get $ \log |z_{1}-z_{2}| $ (when restricted on $ {\Bbb C}\subset{\Bbb C}P^{1} $), which is indeed scaling-invariant up to addition of a constant. (Note that the constant is irrelevant, since the forms $ \alpha $ and $ \beta $ we are going to pair $ K $ with have vanishing integral.) What the above argument shows is that the continuation of this kernel to $ {\Bbb C}P^{1} $ is invariant w.r.t. fraction-linear mappings. \subsection{Almost-perpendicularity } The description of the pairing on $ H_{\int=0}^{-1}\left({\Bbb C}P^{1},\Omega^{1}\right) $ immediately implies \begin{proposition} Consider two disjoint disks $ K_{1} $, $ K_{2} $ on $ {\Bbb C}P^{1} $ of conformal distance $ l $. Let $ H_{1,2} = \overset{\,\,{}_\circ}{H}_{\int=0}^{-1}\left(K_{1,2},\Omega^{1}\right)\subset H_{\int=0}^{-1}\left({\Bbb C}P^{1},\Omega^{1}\right) $ . Let $ P_{l} $ be the orthogonal projector from one subspace to another. Then \begin{equation} \|P_{l}\| \sim e^{-l}. \notag\end{equation} The equivalence means that the quotient of two sides remains bounded and separated from 0 when $ l $ varies. \end{proposition} \begin{proof} Since the natural norm on $ H_{\int=0}^{-1}\left(\Omega^{1}\right) $ is invariant with respect to $ \operatorname{PGL} $, the angle between the subspaces $ H_{1,2} $ depends on $ l $ only. Note that it is sufficient to prove the statement in the case $ l>\varepsilon $ for some fixed $ \varepsilon>0 $. Now proceed as in the proof of Proposition~\ref{prop3.170}. The only change is that the kernel is now $ \log |L+z_{1}-z_{2}| $, and on forms with integral 0 it is equivalent to $ \left(\log |L+z_{1}-z_{2}|-\log L-\operatorname{Re}\frac{z_{1}-z_{2}}{L}\right) $, which is $ L^{-2} $ times an operator of rank 1, plus much smaller operator. \end{proof} \begin{remark} Note that norm of this projector is much smaller than the norm of the projector from Proposition~\ref{prop3.170} (if $ l $ is big enough). \end{remark} \begin{corollary} \label{cor35.20}\myLabel{cor35.20}\relax Consider a family of disjoint disks $ \left\{K_{i}\right\} $ in $ {\Bbb C}P^{1} $ with pairwise conformal distances $ l_{ij} $, $ i\not=j $. Let $ l_{i i}=0 $. If the matrix $ \left(e^{-l_{ij}}\right) $ gives a bounded operator $ l_{2} \to l_{2} $, then the natural extension-by-0 mapping \begin{equation} \bigoplus_{l_{2}}\overset{\,\,{}_\circ}{H}_{\int=0}^{-1}\left(K_{i},\Omega^{1}\right) \to H_{\int=0}^{-1}\left({\Bbb C}P^{1},\Omega^{1}\right) \notag\end{equation} is a continuous injection, and, dually, the natural restriction mapping \begin{equation} H^{1}\left({\Bbb C}P^{1}\right)/\operatorname{const} \to \bigoplus_{l_{2}}H^{1}\left(K_{i}\right)/\operatorname{const} \notag\end{equation} is a continuous surjection. \end{corollary} \subsection{Topology on the boundary }\label{s35.20}\myLabel{s35.20}\relax Consider a disk $ K\subset{\Bbb C}P^{1} $. Given a function $ f $ on $ \partial K $, one can consider its decomposition into a sum $ f_{+}+f_{-} $ of functions which can be analytically extended into/outside of $ K $. Such a decomposition exists if $ f $ is in $ L_{2} $, and the summands are uniquely defined up to addition of a constant. If $ f\in H^{1/2}\left(\partial K\right) $, then $ f_{+} $ and $ f_{-} $ are $ H^{1} $-functions on $ K $ and on $ {\Bbb C}P^{1}\smallsetminus K $ (compare with Section~\ref{s2.70}), if $ f\in H^{1/2}\left(\partial K\right)/\operatorname{const} $, then $ f_{+}\in H^{1}\left(K\right)/\operatorname{const} $, $ f_{-}\in H^{1}\left({\Bbb C}P^{1}\smallsetminus K\right)/\operatorname{const} $, and $ f_{\pm} $ are uniquely defined by $ f $. Thus $ H^{1} $-norms of $ f_{\pm} $ are correctly defined. In the same way as in Section~\ref{s2.70} one can prove \begin{lemma} \label{lm35.30}\myLabel{lm35.30}\relax The described above extension mapping \begin{equation} H^{1/2}\left(\partial K\right) \to H^{1}\left(K\right)/\operatorname{const} \oplus H^{1}\left({\Bbb C}P^{1}\smallsetminus K\right)/\operatorname{const} \notag\end{equation} is a continuous injection. \end{lemma} Since the Hilbert structure on the right-hand side can be made $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant, the extension mapping defines the $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant realization of the Hilbert structure on $ H^{1/2}\left(\partial K\right) $. The property of invariance can be described in the following way: let $ \varphi $ be a fraction-linear mapping, $ \varphi\left(K\right)=K' $. Then $ \varphi^{*} $ defines a unitary operator \begin{equation} \varphi^{*}\colon H^{1/2}\left(\partial K'\right) \to H^{1/2}\left(\partial K\right). \notag\end{equation} \begin{remark} \label{rem35.35}\myLabel{rem35.35}\relax In the same way as we defined a $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant pairing on sections of $ \Omega^{1-s} $ on $ {\Bbb C}P^{1} $ (in Section~\ref{s2.60}), one can define a $ \operatorname{PGL}\left(2,{\Bbb R}\right) $-invariant pairing on section of $ \Omega_{{\Bbb R}P^{1}}^{1-s} $ on $ {\Bbb R}P^{1}=S^{1} $. In this way one gets so called supplementary series of representations of $ \operatorname{SL}\left(2,{\Bbb R}\right) $. By expressing the pairing in appropriate coordinate systems on $ {\Bbb R}P^{1}=S^{1} $ one can easily see that the described above $ \operatorname{SL}\left(2,{\Bbb R}\right) $-invariant Hilbert structure coincides with one of these structures. \end{remark} \begin{definition} The representation $ f=f_{+}+f_{-} $ decomposes $ H^{1/2}\left(\partial K\right)/\operatorname{const} $ into a direct sum of two subspaces which we denote $ H_{+}^{1/2}\left(\partial K\right)/\operatorname{const} $ and $ H_{-}^{1/2}\left(\partial K\right)/\operatorname{const} $. \end{definition} Combining the above description of $ H^{1/2}\left(\partial K\right) $ with results of the previous section, one obtains \begin{corollary} In the conditions of Corollary~\ref{cor35.20} the natural restriction mapping \begin{equation} H^{1}\left({\Bbb C}P^{1}\right)/\operatorname{const} \to \bigoplus_{l_{2}}H^{1/2}\left(K_{i}\right)/\operatorname{const} \notag\end{equation} is a continuous surjection. \end{corollary} \subsection{Space of holomorphic functions }\label{s35.30}\myLabel{s35.30}\relax Consider a family $ \left\{K_{i}\right\} $ of disjoint disks in $ {\Bbb C}P^{1} $. \begin{definition} We say that a generalized function $ f $ on $ {\Bbb C}P^{1} $ is $ H^{1} $-{\em holomorphic in\/} $ {\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $ if $ f\in H^{1}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i}\right) $, and $ \bar{\partial}f=0\in H^{0}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\bar{\omega}\right)= L_{2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\bar{\omega}\right) $. Denote the the space of $ H^{1} $-holomorphic functions in $ {\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $ by $ {\cal H}^{\left(1\right)} $. \end{definition} Note that the Sobolev spaces in this definition are generalized ones. Note also that the Hilbert norm on $ L_{2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\bar{\omega}\right) $ is canonically defined by $ \|\alpha\|^{2}=-i\int\bar{\alpha}\alpha $. \begin{theorem} \label{th35.15}\myLabel{th35.15}\relax Suppose that the conformal distance between $ \partial K_{i} $ and $ \partial K_{j} $ is $ l_{ij} $, and the matrix $ \left(e^{-l_{ij}}\right) $ gives a bounded operator $ l_{2} \to l_{2} $. Consider the mapping of taking the boundary value: \begin{equation} b\colon {\cal H}^{\left(1\right)}/\operatorname{const} \to \bigoplus_{l_{2}}H^{1/2}\left(\partial K_{i}\right)/\operatorname{const}. \notag\end{equation} Let $ b_{-} $ be the component of this mapping going into $ \bigoplus_{l_{2}}H_{-}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $. Then $ b_{-} $ is a bounded invertible mapping. \end{theorem} \begin{proof} We already know that $ b_{-} $ is bounded. To show that one can reproduce a function by the component $ b_{-} $ of its restriction to a boundary consider a function $ f\in{\cal H}^{\left(1\right)} $. By definition, it is a restriction of some function $ g\in H^{1}\left({\Bbb C}P^{1}\right) $, and this function $ g $ is defined up to a function with support in $ \bigcup K_{i} $. By the condition of $ f $ being holomorphic, $ \bar{\partial}g $ is an element of $ L_{2}\left(\bigcup K_{i},\bar{\omega}\right)=\bigoplus_{l_{2}}L_{2}\left(K_{i},\bar{\omega}\right) $, and $ \bar{\partial}g $ is defined up to addition of $ \bar{\partial}\overset{\,\,{}_\circ}{H}^{1}\left(\bigcup K_{i}\right) $. On the other hand, the topology on $ H^{1}\left(X\right) $ (here $ X $ is a manifold with coordinates $ x_{k} $) can be defined by the norm $ \|f\|_{L_{2}}^{2}+\sum\|\partial_{x_{k}}f\|_{L_{2}}^{2} $, thus $ \overset{\,\,{}_\circ}{H}^{1}\left(\bigcup K_{i}\right) = \bigoplus_{l_{2}}\overset{\,\,{}_\circ}{H}^{1}\left(K_{i}\right) $. We conclude that $ \bar{\partial}g $ is a canonically defined element of \begin{equation} \bigoplus_{l_{2}}L_{2}\left(K_{i},\bar{\omega}\right)/\bar{\partial}\overset{\,\,{}_\circ}{H}^{1}\left(K_{i}\right). \notag\end{equation} \begin{lemma} Consider a disk $ K\subset{\Bbb C}P^{1} $. Then there exists a canonical isomorphism \begin{equation} L_{2}\left(K,\bar{\omega}\right)/\bar{\partial}\overset{\,\,{}_\circ}{H}^{1}\left(K\right) \simeq H_{-}^{1/2}\left(\partial K\right)/\operatorname{const}. \notag\end{equation} \end{lemma} \begin{proof} Let us start with a left-to-right mapping. Let $ \alpha\in L_{2}\left(K,\bar{\omega}\right)/\bar{\partial}\overset{\,\,{}_\circ}{H}^{1}\left(K\right) $. Since $ \bar{\partial} $ is an elliptic operator without cokernel and with $ 1 $-dimensional null-space, $ \bar{\partial}^{-1}\alpha $ is an $ H^{1} $-function defined up to addition of a constant and addition of an element of $ \overset{\,\,{}_\circ}{H}^{1}\left(K\right) $. Thus the restriction of $ f $ to $ \partial K $ is defined up to a constant, so it is an element of $ H^{1/2}\left(\partial K\right)/\operatorname{const} $. Moreover, it is in $ H_{-}^{1/2}\left(\partial K\right)/\operatorname{const} $ since $ \bar{\partial}^{-1}\alpha $ is holomorphic outside $ K $. To get right-to-left mapping note that any element of $ H_{-}^{1/2}\left(\partial K\right)/\operatorname{const} $ can be (by definition) continued to a holomorphic outside of $ K $ function $ f $, and this function is of class $ H^{1} $. Thus to this continuation one can apply the arguments which precede the lemma. \end{proof} To finish the proof of the theorem note that the knowledge of $ r_{-}\left(f\right) $ allows one to construct $ \bar{\partial}f $ (by lemma), so the application of $ \bar{\partial}^{-1} $ reconstructs $ f $.\end{proof} \subsection{Toy Riemann--Roch theorem }\label{s35.40}\myLabel{s35.40}\relax Consider a family of disjoint disks $ K_{i} $, $ i\in I $, and the corresponding space of $ H^{1} $-holomorphic functions. Suppose that $ I $ has an involution ' which interchanges two halves of $ I=I_{+}\amalg I_{+}' $. Fix automorphisms $ \varphi_{i} $ of $ {\Bbb C}P^{1} $, $ i\in I $, such that $ \varphi_{i'}=\varphi_{i}^{-1} $, and $ \varphi_{i}\left(\partial K_{i}\right) $ is $ \partial K_{i'} $ with inverted orientation. Note that if the set $ I $ is finite, then after gluing $ \partial K_{i} $ via $ \varphi_{i} $ one gets a curve of genus $ \operatorname{card}\left(I_{+}\right)=\operatorname{card}\left(I\right)/2 $. Associate to a function $ f\in{\cal H}^{\left(1\right)} $ the jump of its boundary value after such a gluing: \begin{equation} {\cal J}\colon f \mapsto \left(f|_{\partial K_{j}}-\varphi_{j}^{*}\left(f|_{\partial K_{j'}}\right)\right),\quad j\in I_{+}. \label{equ4.101}\end{equation}\myLabel{equ4.101,}\relax \begin{theorem} \label{th35.45}\myLabel{th35.45}\relax Suppose that the conformal distance between $ \partial K_{i} $ and $ \partial K_{j} $ is $ l_{ij} $, and the matrix $ \left(e^{-l_{ij}}-\delta_{ij}\right) $ gives a compact operator $ l_{2} \to l_{2} $. Then the mapping \begin{equation} {\cal J}\colon {\cal H}^{\left(1\right)}/\operatorname{const} \to \bigoplus\Sb l_{2} \\ j\in I_{+}\endSb H^{1/2}\left(\partial K_{j}\right)/\operatorname{const} \notag\end{equation} is a Fredholm operator of index 0. \end{theorem} \begin{proof} Consider the composition $ \widetilde{{\cal J}}={\cal J}\circ b_{-}^{-1} $. Let us show that $ \widetilde{{\cal J}} $ is Fredholm of index 0, this would immediately imply the statement of the theorem. The mapping $ \varphi_{j}^{*} $ interchanges $ \pm $-components of $ H^{1/2}\left(\partial K_{j}\right)/\operatorname{const} $ and $ H^{1/2}\left(\partial K_{j'}\right)/\operatorname{const} $, thus one can identify \begin{equation} H^{1/2}\left(\partial K_{j}\right)/\operatorname{const} =H_{-}^{1/2}\left(\partial K_{j}\right)/\operatorname{const} \oplus H_{+}^{1/2}\left(\partial K_{j}\right)/\operatorname{const} \notag\end{equation} with $ H_{-}^{1/2}\left(\partial K_{j}\right)/\operatorname{const} \oplus H_{-}^{1/2}\left(\partial K_{j'}\right)/\operatorname{const} $. Denote the composition of $ \widetilde{{\cal J}} $ and this identification by $ {\cal K}\colon \bigoplus_{l_{2}}H_{-}^{1/2}\left(\partial K_{j}\right)/\operatorname{const} \to \bigoplus_{l_{2}}H_{-}^{1/2}\left(\partial K_{j}\right)/\operatorname{const} $. For $ f\in{\cal H}^{\left(1\right)} $ denote by $ f_{j\pm} $ the $ \pm $-components of $ f|_{\partial K_{j}} $. One can easily see that $ {\cal K}\left(\left(f_{j-}\right)_{j\in I}\right)=\left(f_{j-}-\varphi_{j}^{*}\left(f_{j'+}\right)\right)_{j\in I} $, in other words, $ {\cal K}=\operatorname{id}-\varphi^{*}\circ b_{+}\circ b_{-}^{-1} $, here $ b_{+}=b-b_{-} $. Since $ \varphi^{*}=\bigoplus_{l_{2}}\varphi_{j}^{*} $ is an isometry, to prove the theorem it is enough to show that $ b_{+}\circ b_{-}^{-1} $ is compact. Now we investigate how to reconstruct $ H_{+}^{1/2} $-components of restriction to a boundary via $ H_{-}^{1/2} $-components. Consider the operator with the Cauchy kernel $ \left(y-x\right)^{-1}dy $, $ x,y\in{\Bbb C}P^{1} $. Restrict this kernel to $ \bigcup\partial K_{i} $. One gets an operator which sends functions on $ \bigcup\partial K_{i} $ to functions on $ \bigcup\partial K_{i} $. Put zeros instead of the diagonal terms (which send functions on $ \partial K_{i} $ to functions on $ \partial K_{i} $), and call the resulting operator $ {\bold K} $. Obviously, the functions in the image can be holomorphically extended to $ K_{i} $. Moreover, if a function was non-zero on $ \partial K_{i} $ only, and could be analytically continued inside $ K_{i} $, then this function is in the zero-space of $ {\bold K} $. Thus the operator $ {\bold K} $ maps $ \bigoplus H_{-}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $ to $ \bigoplus H_{+}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $. Moreover, as the proof of Theorem~\ref{th35.15} shows, $ b_{+}\circ b_{-}^{-1}= \frac{{\bold K}}{2\pi i} $, since the pseudodifferential operator $ \bar{\partial}^{-1} $ has $ \frac{dy}{2\pi i\left(x-y\right)} $ as the null-space. We conclude that the image $ V_{1}=b\left({\cal H}^{\left(1\right)}/\operatorname{const}\right) $ in $ \bigoplus H^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $ is a graph of operator $ \frac{{\bold K}}{2\pi i}\colon \bigoplus H_{-}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} \to \bigoplus H_{+}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $. On the other hand, consider the subspace $ V_{2} $ of $ \bigoplus H^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $ consisting of functions which are in the null-space of $ {\cal J} $, i.e., the subspace given by the condition \begin{equation} f|_{\partial K_{j}}=\varphi_{j}^{*}\left(f|_{\partial K_{j'}}\right),\qquad j\in I_{+}. \notag\end{equation} Since $ \varphi_{j}^{*} $ interchanges $ H_{-}^{1/2}\left(\partial K_{j}\right)/\operatorname{const} $ and $ H_{+}^{1/2}\left(\partial K_{j'}\right)/\operatorname{const} $ and is unitary, we conclude that the subspace $ V_{2} $ is a graph of a unitary mapping $ \bigoplus\varphi_{i}^{*}|_{H_{+}^{1/2}\left(\partial K_{i}\right)} $. Thus we are in conditions of the abstract Riemann--Roch theorem (Theorem~\ref{th6.50}), which finishes the proof. \end{proof} \begin{remark} This theorem is an infinite-genus variant of Riemann--Roch theorem for the case of the bundle $ {\cal O} $ on an algebraic curve $ M $. Indeed, the latter theorem says that the mapping $ \bar{\partial}\colon H^{s}\left(M,{\cal O}\right) \to H^{s-1}\left(M,\bar{\omega}\right) $ between Sobolev spaces is a Fredholm mapping of index $ 1-g\left(M\right) $. The theory of elliptic operators says that in a case of smooth compact $ M $ the value of $ s $ is irrelevant, but in our context we are forced to use the value $ s=1 $. The relation of the mapping $ \bar{\partial} $ to the mapping $ {\cal J} $ is the standard (in mathematical physics) trick of {\em reduction to boundary}. We do cuts in the curve $ M $ to obtain a region $ S={\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $. Instead of $ H^{s}\left(M,{\cal O}\right) $ we consider subspace $ V $ of $ H^{s}\left(S,{\cal O}\right) $ which consists of functions which satisfy gluing conditions on the boundary. These gluing conditions are similar to $ {\cal J}f=0 $, but are applied to functions (instead of functions modulo constants), thus there is an extra condition per cut, total $ g $ extra conditions. Let \begin{equation} {\cal J}_{H^{1}}\colon H^{1}\left(S,{\cal O}\right)/\operatorname{const} \to \bigoplus_{l_{2}}H^{1/2}\left(\partial K_{j}\right)/\operatorname{const} \notag\end{equation} be defined by the formula~\eqref{equ4.101}. Thus $ V/\operatorname{const} $ is a subspace of $ \operatorname{Ker}{\cal J}_{H^{1}} $ of codimension $ g-1 $. Now the translation of the classical Riemann--Roch theorem is that the mapping $ \bar{\partial}\colon \operatorname{Ker}{\cal J}_{H^{1}} \to H^{s-1}\left(S,\bar{\omega}\right) $ is of index 0. (Note that the choice $ s=1 $ insures that there should be no gluing conditions for elements of $ H^{s-1}\left(M,\bar{\omega}\right) $.) Now to solve the equation $ \bar{\partial}f=h $, $ {\cal J}_{H^{1}}f=0 $ we reduce it to a boundary value problem: we use the fact that $ \bar{\partial} $ is a surjection on $ {\Bbb C}P^{1} $, thus one can immediately find a function $ f_{0} $ such that $ \bar{\partial}f_{0}=h $. Thus $ f_{1}=f-f_{0} $ should satisfy $ \bar{\partial}f_{1}=0 $, $ {\cal J}_{H^{1}}f_{1}=-{\cal J}_{H^{1}}f_{0} $. This shows that the index of the mapping $ {\cal J} $ is 0 in the algebraic case $ g<\infty $. We see that Theorem~\ref{th35.45} is indeed an infinite-dimensional analogue of Riemann--Roch theorem for the bundle $ {\cal O} $. \end{remark} \begin{remark} It is possible to modify Theorem~\ref{th35.45} to make it applicable to deformations of the bundle $ {\cal O} $ as well (see Section~\ref{s5.30} for details). Since to describe Jacobian we need the special case $ {\cal O} $ only, we do not pursue this venue here. \end{remark} \begin{remark} For the particular case of the bundle $ {\cal O} $ it is possible to prove a much stronger result than Riemann--Roch theorem: that $ {\cal J} $ is an isomorphism. However, since this result is not true out of context of toy theory, we postpone its proof until Section~\ref{s7.90}, when it is needed for description of Jacobian. \end{remark} \section{$ H^{1/2} $-theory } As we have seen in Section~\ref{h35}, Riemann--Roch theorem can be easily proven given appropriate ingredients, such as {\em almost perpendicularity\/} of functions with far-separated supports, ability to {\em restrict\/} a global section to the boundary, ability to {\em invert\/} $ \bar{\partial} $ basing on boundary values, and an ability to glue a function provided we know the values inside and outside the given curve with appropriate compatibility conditions along the curve. However, when one tries to apply the same technique to the topology we are most interested in, i.e., $ H^{1/2} $-topology, significant difficulties arise. \subsection{In an ideal world } In what follows we use the (complex) case $ n=1 $ of Section~\ref{s2.50}. We have seen that the space $ H^{1/2}\left(M,\omega^{1/2}\right) $ should have a special significance in studying the holomorphic half-forms. Moreover, the results of Section~\ref{s2.70} suggest that one would be able to describe holomorphic elements of $ H^{1/2}\left(D,\omega^{1/2}\right) $, $ D\subset M $, by their restrictions on $ \partial D $. If our world were the ideal world, then the following properties would be satisfied: \begin{enumerate} \item The action of $ \operatorname{SL}\left(2,{\Bbb C}\right) $ on $ H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) $ would be an action by uniformly bounded operators; \item For an embedded curve $ \gamma \to {\Bbb C}P^{1} $ the restriction mappings $ r\colon H^{s}\left({\Bbb C}P^{1},\omega^{1/2}\right) \to H^{s-1/2}\left(\gamma,\Omega^{1/2}\otimes\mu\right) $ (defined for $ s>\frac{1}{2} $) would be defined for $ s=\frac{1}{2} $ as well; \item For the same curve the mapping $ e\colon H^{s}\left(\gamma,\Omega^{1/2}\otimes\mu\right) \to H^{s-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right) $ of continuation by $ \delta $-function (defined for $ s<0 $) would be defined for $ s=0 $ as well. \item If a curve $ \gamma $ divides a domain $ S $ into two parts $ S_{1} $, $ S_{2} $, then the natural restriction mapping $ H^{1/2}\left(S\right) \to H^{1/2}\left(S_{1}\right)\oplus H^{1/2}\left(S_{2}\right) $ would be an isomorphism. \end{enumerate} To make a long story short, in the ideal world the main technical tools of this paper would behave in a civilized manner, which would simplify the exposition a lot. Identification of the Hardy space with the subspace of holomorphic functions would be provided by the restriction $ r $. The composition \begin{equation} H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) \xrightarrow[]{r} H^{0}\left(\gamma,\Omega^{1/2}\otimes\mu\right) \xrightarrow[]{e} H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right) \notag\end{equation} would be a ``model'' of $ \bar{\partial} $-operator\footnote{In the sense of Section~\ref{s3.05}, i.e., it would coincide with $ -\bar{\partial} $ on piecewise holomorphic functions.} \begin{equation} H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) \xrightarrow[]{\bar{\partial}} H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right), \notag\end{equation} so that the composition (Cauchy formula) \begin{equation} H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) \xrightarrow[]{r} H^{0}\left(\gamma,\Omega^{1/2}\otimes\mu\right) \xrightarrow[]{e} H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right)\xrightarrow[]{\bar{\partial}^{-1}} H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) \notag\end{equation} would be continuous. By construction the functions in the image of this operator are holomorphic outside of $ \gamma $, and the operator $ r $ gives an injection of this subspace into $ H^{0}\left(\gamma,\Omega^{1/2}\otimes\mu\right) $. Now one may consider this as a Cauchy formula, since the composition is an identity operator on the subspace of holomorphic functions in the domain $ D $ bounded by $ \gamma $. All these operators would be canonically defined by $ D $, and compatible with projective mappings $ {\Bbb C}P^{1} \to {\Bbb C}P^{1} $, $ D \to D_{1} $. Thus one would be able to consider the image of $ r $ on subspace of holomorphic forms in $ H^{1/2}\left(D,\omega^{1/2}\right) $ as a ``model'' of this space. The last property in the list would allow us to glue together the forms which are provided by different means on pieces $ S_{1} $ and $ S_{2} $. Since we are confined to the current world, the above program will not work, thus we need some workarounds against above three non-facts. We will consider a Riemannian structure on $ {\Bbb C}P^{1} $, and will need to control the size of domains in question, in order for results of Section~\ref{s2.50} to be applicable. We will also need mollifications of operators $ r $ and $ e $, and will need some restrictions on what we can glue together. \subsection{Mollification }\label{s2.10}\myLabel{s2.10}\relax Here we introduce a mapping \begin{equation} H^{0}\left(S^{1},\Omega^{1/2}\otimes\mu\right) \xrightarrow[]{\widetilde{e}} H^{-1/2}\left(S^{1}\times\left(-\varepsilon,\varepsilon\right),\omega^{1/2}\right) \notag\end{equation} which a closest existing analogue for the (non-existing) mapping $ e $ from the previous section. It will be an invertible mapping onto its image, and it will depend on additional parameters $ a_{n} $, $ n\in{\Bbb Z} $. (Later we will need some particular choice of parameters $ \left(a_{n}\right) $, appropriate for the elliptic operator $ \bar{\partial} $ we study.) Fix $ \varepsilon>0 $ and a sequence $ \left(a_{n}\right)_{n\in{\Bbb Z}} $ such that $ 0<a<a_{n}<A $ for fixed constants $ a $ and $ A $. Fix a smooth function $ \sigma\left(y\right) $, $ y\in\left(-\varepsilon,\varepsilon\right) $, with a compact support and integral 1. Now map a half-form $ e^{2\pi i kx}dx^{1/2} $, $ x\in S^{1}={\Bbb R}/{\Bbb Z} $, $ k\in{\Bbb Z}+\frac{1}{2} $, into \begin{equation} a_{k}e^{2\pi i kx}\frac{\sigma\left(|k|y\right)}{|k|}dz^{1/2},\qquad x\in S^{1}\text{, }y\in\left(-\varepsilon,\varepsilon\right),\quad z=x+iy, \notag\end{equation} and continue this mapping linearly to $ L_{2}\left(S^{1},\Omega^{1/2}\otimes\mu\right)=H^{0}\left(S^{1},\Omega^{1/2}\otimes\mu\right) $. \begin{lemma} This mapping is an injection, i.e., an invertible mapping onto its image. \end{lemma} The dual mapping $ H^{1/2}\left(S^{1}\times\left(-\varepsilon,\varepsilon\right),\omega^{1/2}\right) \xrightarrow[]{\widetilde{r}} H^{0}\left(S^{1},\Omega^{1/2}\otimes\mu\right) $ given by \begin{align} f\left(x,y\right)dz^{1/2} \mapsto g\left(x\right) & = \sum_{k}g_{k}e^{2\pi kx}dx^{1/2}, \notag\\ g_{k} & = \int f\left(x,y\right)e^{-2\pi kx}\frac{\sigma\left(|k|y\right)}{|k|}dx\,dy \notag\end{align} has a similar property: it is a surjection, i.e., an invertible mapping from the quotient by its null-space. It is a close analogue of the (non-existing) mapping $ r $ from the previous section. Suppose $ a_{k}=1 $ for any $ k $. If $ \varepsilon \to $ 0, then in weak topology for mappings $ {\cal D} \to {\cal D}' $ the constructed mappings converge to the mappings of extension as $ \delta $-function and restriction, but on Sobolev spaces the norms of these mappings go to $ \infty $, \begin{remark} In what follows we will use this mapping with following modifications: we suppose that $ \operatorname{Supp}\sigma\subset\left[0,\varepsilon\right] $, thus we map a half-form on $ S^{1} $ to a half-form concentrated on a small collar to the {\em right\/} of $ S^{1} $ in $ S^{1}\times\left[-\varepsilon,\varepsilon\right] $. In fact $ S^{1}\times\left[-\varepsilon,\varepsilon\right] $ will be identified with a annulus $ \left\{R\,e^{-2\pi\varepsilon}<|z-z_{0}|<R\,e^{2\pi\varepsilon}\right\} $ in $ {\Bbb C} $, and the mapping would send half-forms on $ \left\{|z|=R\right\} $ to half-forms concentrated on the {\em outside\/} collar. Similarly, the dual mapping \begin{equation} H^{1/2}\left(R\,e^{-2\pi\varepsilon}<|z-z_{0}|<R\,e^{2\pi\varepsilon}\right) \xrightarrow[]{\widetilde{r}} H^{0}\left(\left\{|z|=R\right\}\right) \notag\end{equation} will depend only on value of $ f\left(z\right) $ on the outside collar. \end{remark} \subsection{Mollification suitable for $ \bar{\partial} $ }\label{s5.31}\myLabel{s5.31}\relax Here we are going to fix the values for the coefficients $ \left(a_{n}\right) $ from Section~\ref{s2.10} which are most suitable for the operator $ \bar{\partial} $. \begin{proposition} \label{prop4.15}\myLabel{prop4.15}\relax Fix $ \varepsilon>0 $. Let $ \gamma= \left\{z \mid |z|=1\right\} $, $ \widetilde{K} = \left\{z \mid |z|<e^{2\pi\varepsilon}\right\} $. For appropriate $ A $ and $ a $ there exists a sequence $ \left(a_{n}\right) $, $ 0<a<a_{n}<A $, such that the corresponding restriction mapping $ \widetilde{r}\colon H^{1/2}\left(\widetilde{K},\omega^{1/2}\right) \to L_{2}\left(\gamma,\Omega^{1/2}\otimes\mu\right) $ coincides with the usual restriction on half-forms which are holomorphic between $ \gamma $ and $ \widetilde{K} $. The numbers $ a_{n} $, thus the operator $ \widetilde{r} $, are uniquely determined. \end{proposition} Dually, \begin{proposition} \label{prop4.16}\myLabel{prop4.16}\relax Let $ U=K\cup\left({\Bbb C}P^{1}\smallsetminus\widetilde{K}\right) $. Consider the Cauchy kernel restricted to $ \widetilde{K} $. It gives two operators: an operator $ \overset{\,\,{}_\circ} H^{-1/2}\left(\widetilde{K}\right) \xrightarrow[]{\bar{\partial}^{-1}} H^{1/2}\left(U\right) $, and an operator $ C^{\infty}\left(\gamma\right) \xrightarrow[]{\bar{\partial}^{-1}} {\cal D}'\left(U\right) $. Consider the mollification $ \widetilde{e} $ of the extension mapping $ e $ corresponding to a sequence $ \left(a_{n}\right) $. With appropriate choice of the sequence $ \left(a_{n}\right) $ the composition \begin{equation} C^{\infty}\left(\gamma\right)\hookrightarrow L_{2}\left(\gamma\right) \xrightarrow[]{\widetilde{e}} \overset{\,\,{}_\circ} H^{-1/2}\left(\widetilde{K}\right) \xrightarrow[]{\bar{\partial}^{-1}} H^{1/2}\left(U\right) \hookrightarrow {\cal D}'\left(U\right) \notag\end{equation} coincides with the mapping $ C^{\infty}\left(\gamma\right) \xrightarrow[]{\bar{\partial}^{-1}} {\cal D}'\left(U\right) $. The numbers $ a_{n} $, thus the operator $ \widetilde{e} $, are uniquely determined by the above condition. \end{proposition} \begin{remark} Note that the mapping $ \widetilde{r} $ is a left inverse to the mapping $ f \mapsto c_{f} $ from Section~\ref{s2.70}. \end{remark} \subsection{$ \protect \widetilde{e}\circ\protect \widetilde{r} $ as an approximation to $ \bar{\partial} $ }\label{s2.25}\myLabel{s2.25}\relax Consider a circle $ \gamma $ in $ {\Bbb C} $. In what follows we consider particular mollifications of the $ \delta $-inclusion $ C^{\infty}\left(\gamma\right) \xrightarrow[]{e} {\cal D}'\left({\Bbb C}\right) $ and restriction $ {\cal D}\left({\Bbb C}\right) \xrightarrow[]{r} C^{\infty}\left(S^{1}\right) $, which correspond to the only sequences $ \left(a_{n}\right) $ which satisfy the conditions of Propositions~\ref{prop4.15}, ~\ref{prop4.16}. The notations $ \widetilde{e}_{\gamma} $ and $ \widetilde{r}_{\gamma} $ are reserved for these two mappings, we may denote them $ \widetilde{e} $ and $ \widetilde{r} $ if the circle to apply them for is clear from context. A central tool in the following discussion is the mollification of identity~\eqref{equ3.02}: \begin{proposition} Consider a circle $ \gamma $ or radius $ \rho $ which bounds a disk $ K $. Let $ \widetilde{K} $ be a concentric disk of radius $ \rho e^{2\pi\varepsilon} $. Let $ F $ be a holomorphic half-form in $ \widetilde{K}\smallsetminus K $. Let \begin{equation} G=\bar{\partial}^{-1}\circ\widetilde{e}_{\gamma}\circ\widetilde{r}_{\gamma}\left(F\right). \notag\end{equation} If $ F $ is a holomorphic half-form in $ {\Bbb C}P^{1}\smallsetminus K $, then $ G $ coincides with $ F $ outside of $ \widetilde{K} $, and is 0 in $ U $. If $ F $ is a holomorphic half-form in $ \widetilde{K} $, then $ G $ coincides with $ -F $ inside of $ K $, and is 0 outside of $ \widetilde{K} $. \end{proposition} \begin{amplification} \label{amp4.21}\myLabel{amp4.21}\relax Fix $ \varepsilon,R>0 $. Consider disks $ K_{\rho}=\left\{|z|<\rho\right\} $, $ K'_{\rho}=\left\{|z|<\rho e^{2\pi\varepsilon}\right\} $, $ \widetilde{K}_{\rho}=\left\{|z|<\rho e^{4\pi\varepsilon}\right\} $ in $ {\Bbb C} $. Let $ \gamma=\partial K_{\rho} $. There exists a mapping \begin{equation} \lambda_{\rho}\colon H^{1/2}\left(\widetilde{K}_{\rho},\omega^{1/2}\right) \to H^{1/2}\left(\widetilde{K}_{\rho},\omega^{1/2}\right) \notag\end{equation} such that \begin{enumerate} \item For $ 0<\rho<R $ the mapping $ \lambda_{\rho} $ is continuous with the norm uniformly bounded by a constant depending on $ \varepsilon $ and $ R $ only; \item $ \lambda_{\rho}\left(f\right)|_{K_{\rho}}=0 $ for any $ f\in H^{1/2}\left(\widetilde{K}_{\rho}\right) $; \item $ \left(f-\lambda_{\rho}\left(f\right)\right)|_{\widetilde{K}_{\rho}\smallsetminus K'_{\rho}}=0 $ for any $ f\in H^{1/2}\left(\widetilde{K}_{\rho}\right) $; \item if $ f $ is holomorphic in $ \widetilde{K}_{\rho}\smallsetminus K_{\rho} $, then $ \bar{\partial}\lambda_{\rho}\left(f\right) = \left(\widetilde{e}_{\gamma}\circ\widetilde{r}_{\gamma}\right)\left(f\right) $. \end{enumerate} \end{amplification} \begin{proof} During the proof we abuse notations and do not mention the bundle $ \omega^{1/2} $ in notations for Sobolev spaces. Note that for a half-form $ f $ which is holomorphic outside of $ K $ all the statements of the amplification are true if we take $ \lambda_{\rho}\left(f\right)=\left(\bar{\partial}^{-1}\circ\widetilde{e}\circ\widetilde{r}\right)\left(f\right) $. If $ f $ is holomorphic inside of $ \widetilde{K} $, then $ \lambda_{\rho}\left(f\right)=f-\left(\bar{\partial}^{-1}\circ\widetilde{e}\circ\widetilde{r}\right)\left(f\right) $ works. This uniquely defines $ \lambda_{\rho}\left(f\right) $ if $ f $ is holomorphic inside of $ \widetilde{K}_{\rho}\smallsetminus K_{\rho} $. What we need is to adjust this formula to the case of non-holomorphic half-forms. Since $ R $ is fixed, we can consider the norm from Section~\ref{s2.50} instead of the equivalent Sobolev $ H^{1/2} $-norm. Since for the former norm the Sobolev spaces $ H^{1/2}\left(\widetilde{K}_{\rho}\right) $ with different $ \rho $ are naturally isomorphic, it is enough to consider $ \rho=1 $. Note that for a holomorphic $ f $ the image $ \lambda_{\rho}\left(f\right) $ depends on $ f|_{\widetilde{K}_{\rho}\smallsetminus K_{\rho}} $ only. We are going to define $ \lambda_{\rho} $ in general case such that it satisfies the same condition. Thus $ \lambda_{\rho} $ is a mapping $ H^{1/2}\left(\widetilde{K}_{\rho}\smallsetminus K_{\rho}\right) \to H^{1/2}\left(\widetilde{K}_{\rho}\right) $ such that a half-form in the image is 0 inside $ K_{\rho} $. We may substitute conformally equivalent domain $ S^{1}\times\left(-2\varepsilon,2\varepsilon\right) $ instead of $ \widetilde{K}_{\rho}\smallsetminus K_{\rho} $. Now $ H^{1/2}\left(S^{1}\times\left(-2\varepsilon,2\varepsilon\right)\right) $ is an orthogonal sum of subspaces $ L_{n} $, $ n\in{\Bbb Z}+\frac{1}{2} $, spanned by half-forms of the form $ \varphi\left(y\right)e^{2\pi i\quad nx}dz^{1/2} $, $ \left(x,y\right)\in S^{1}\times\left(-2\varepsilon,2\varepsilon\right) $, $ z=x+iy $, thus it is enough to construct uniformly bounded mappings in these subspaces. For a given $ \lambda $ let $ \lambda^{\left(n\right)} $ be defined as \begin{equation} \lambda\left(\varphi\left(y\right)e^{2\pi i nx}dz^{1/2}\right)=\lambda^{\left(n\right)}\left(\varphi\left(y\right)\right)e^{2\pi i nx}dz^{1/2}. \notag\end{equation} Note that the conditions on $ \lambda^{\left(n\right)} $ are: $ \lambda^{\left(n\right)}\left(\varphi\left(y\right)\right) $ vanishes if $ y<0 $, $ \lambda^{\left(n\right)}\left(\varphi\left(y\right)\right)=\varphi\left(y\right) $ if $ y>\varepsilon $, and $ \lambda^{\left(n\right)}\left(e^{ny}\right) $ has a prescribed value (obtained basing on the last condition of the amplification). Define $ \widetilde{\sigma}_{n}\left(y\right) $ via \begin{equation} \lambda^{\left(n\right)}\left(e^{ny}\right) = \widetilde{\sigma}_{n}\left(y\right)e^{ny}. \notag\end{equation} Obviously, $ \widetilde{\sigma}_{n}\left(y\right)=0 $ if $ y<0 $, $ \widetilde{\sigma}_{n}\left(y\right)=1 $ if $ y>\varepsilon $. The function $ \widetilde{\sigma}_{n}\left(y\right) $ is uniquely determined by the last condition of the theorem. After all these remarks {\em define\/} $ \lambda|_{L_{n}} $ by \begin{equation} \varphi\left(y\right)e^{2\pi i nx}dz^{1/2} \to \sigma_{n}\left(y\right)\varphi\left(y\right)e^{2\pi i nx}dz^{1/2}. \notag\end{equation} This mapping satisfies all the conditions of the amplification, with a possible exception of uniform boundness of these mappings for different $ n $. To prove the boundness, note that \begin{equation} \widetilde{\sigma}_{n}\left(y\right)=\widetilde{\sigma}_{1}\left(ny\right), \notag\end{equation} and the mappings \begin{equation} L_{1} \xrightarrow[]{m_{n}} L_{n}\colon \varphi\left(y\right)e^{2\pi i x} \mapsto n^{-1/2}\varphi\left(ny\right)e^{2\pi i nx} \notag\end{equation} are uniformly bounded together with their inverse mappings. \end{proof} \subsection{Mollification for $ L_{2}\left(\Omega^{1/2}\otimes\mu\right) $ }\label{s4.35}\myLabel{s4.35}\relax Consider a family of circles $ \gamma_{i} $ on $ {\Bbb C}P^{1} $. Fix a projective isomorphism of $ \gamma_{i} $ with $ \left\{|z|=1\right\} $ for every $ i $. This isomorphism identifies an annulus $ \left\{e^{-2\pi\varepsilon}<|z|<e^{2\pi\varepsilon}\right\} $ with some neighborhood $ U_{i} $ of $ \gamma_{i} $. The $ \bar{\partial} $-adjusted mollification of $ e $ gives a mapping \begin{equation} \widetilde{e}_{i}\colon L_{2}\left(\gamma_{i},\Omega^{1/2}\otimes\mu\right) \to H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right). \notag\end{equation} This mapping is a injection of topological vector spaces. In the following sections we discuss under which conditions $ \sum_{i}\widetilde{e}_{i} $ is an injection. Here $ \sum_{i}\widetilde{e}_{i} $ is considered as a mapping from $ L_{2}\left(\coprod\gamma_{i},\Omega^{1/2}\otimes\mu\right) $ to $ H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right) $. This injection is going to be a central tool in Section~\ref{s5.10}, which allows one to identify the space of ``admissible'' holomorphic half-forms with the space of their boundary values. To provide the criterion for this mapping to be an injection, we need to introduce a notion of ``almost perpendicular'' subspaces, to study relative position of Sobolev subspaces corresponding to different domains, and to investigate the relative position of domains inside $ {\Bbb C}P^{1} $. First, we suppose that $ \varepsilon $ is small enough and the subsets $ U_{i} $ do not intersect. Then it is clear that $ \sum_{i}\widetilde{e}_{i} $ has no null-vectors. Obviously, $ \sum_{i}\widetilde{e}_{i} $ were an injection if the images of $ \widetilde{e}_{i} $ were perpendicular for different $ i $. We are going to investigate when these images are {\em almost perpendicular\/} for big $ i $. In other words, when the sum of images is direct in the sense of Hilbert topology (i.e., it is an orthogonal sum after an appropriate change of the Hilbert norm to an equivalent one). \subsection{Almost perpendicular subspaces }\label{s5.61}\myLabel{s5.61}\relax \begin{lemma} \label{lm4.41}\myLabel{lm4.41}\relax Consider a Hilbert space $ H $ and a family of subspaces $ H_{i} $, $ i\in{\Bbb N} $. Denote by $ a_{ij} $ the orthogonal projector $ H_{i} \to H_{j} $. Let $ A=\left(a_{ij}\right) $ be the matrix of an operator $ {\bold A}\colon \bigoplus H_{i} \to \coprod H_{i} $, $ \coprod H_{i} $ being the space of arbitrary sequences $ \left(h_{i}\right) $, $ h_{i}\in H_{i} $. Then the natural mapping $ \bigoplus H_{i} \to H $ extends to a Fredholm mapping \begin{equation} \bigoplus_{l_{2}}H_{i} \xrightarrow[]{i} \operatorname{Im} i\subset H,\qquad \left(h_{i}\right) \mapsto \sum h_{i}, \notag\end{equation} onto its image iff {\bf A }induces a Fredholm mapping $ \bigoplus_{l_{2}}H_{i} \to \bigoplus_{l_{2}}H_{i} $. \end{lemma} \begin{proof} Consider $ {\bold A} $ as $ i^{*}\circ i $. \end{proof} \begin{corollary} Suppose that the conditions of the previous lemma the matrix $ \alpha = \left(\|a_{ij}\| - \delta_{ij}\right) $ corresponds to a compact mapping $ l_{2} \to l_{2} $. Then the mapping $ \oplus_{l_{2}}H_{i} \xrightarrow[]{i} H $ is a Fredholm operator onto its image. In particular, this is true if \begin{equation} \sum_{i\not=j}\|a_{ij}\|^{2}<\infty. \notag\end{equation} Moreover, if there is a number $ C $ and a family of unit vectors $ v_{i}\in V_{i} $ such that \begin{equation} \|a_{ij}\| < C|\left(v_{i},v_{j}\right)|, \notag\end{equation} then the mapping $ i $ is continuous iff $ \alpha $ provides a bounded mapping $ l_{2} \to l_{2} $. \end{corollary} \subsection{Conformal distance } In the applications we consider the curve $ \gamma $ is a union of countably many connected components $ \gamma=\coprod\gamma_{i} $, and all the components $ \gamma_{i} $ but a finite number are circles. Let $ \widetilde{e}_{i}=\widetilde{e}_{\gamma_{i}} $ be the mollified extension mapping, and \begin{equation} H_{i}=\operatorname{Im} \widetilde{e}_{i}. \notag\end{equation} If $ H_{i} $ satisfy the conditions of Lemma~\ref{lm4.41}, then the mapping $ \sum_{i}\widetilde{e}_{i} $ from Section~\ref{s4.35} is a Fredholm mapping onto its image. So the next thing to study is how to calculate the norm of the projection \begin{equation} \operatorname{Im} \widetilde{e}_{i} \to \operatorname{Im} \widetilde{e}_{j}. \label{equ4.33}\end{equation}\myLabel{equ4.33,}\relax The subspace $ \operatorname{Im} \widetilde{e}_{i} $ lies inside a subspace $ \overset{\,\,{}_\circ}{H}\left(\widetilde{K}_{i}\right) $, $ \widetilde{K}_{i} $ being any disk which contains an appropriate neighborhood of $ \gamma_{i} $. Thus to majorate the norm of the projection~\eqref{equ4.33} we start with the case of two disjoint disks $ \widetilde{K}_{i} $, $ \widetilde{K}_{j} $ on $ {\Bbb C}P^{1} $, and consider the orthogonal projection from $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(\widetilde{K}_{i}\right) $ to $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(\widetilde{K}_{j}\right) $ inside $ H^{-1/2}\left({\Bbb C}P^{1}\right) $. As we will see, the norm of this projection can be majorated by a number which depends only on some kind of {\em distance\/} between $ \widetilde{K}_{i} $ and $ \widetilde{K}_{j} $. Two disjoint simple curves $ \gamma_{1} $, $ \gamma_{2} $ on $ {\Bbb C}P^{1} $ bound a tube $ U $, which is conformally equivalent to exactly one tube $ S^{1}\times\left[0,l\right] $, $ l>0 $. Here $ S^{1} $ is a circle of circumference $ 2\pi $. \begin{definition} We call the number $ l $ the {\em conformal distance\/} between $ \gamma_{1} $ and $ \gamma_{2} $. \end{definition} \begin{proposition} Consider three simple curves $ \gamma_{1} $, $ \gamma_{2} $, $ \gamma_{3} $ such that $ \gamma_{2} $ separates $ \gamma_{1} $ and $ \gamma_{3} $. Then the conformal distance $ l\left(\gamma_{1},\gamma_{3}\right) \geq l\left(\gamma_{1},\gamma_{2}\right) +l\left(\gamma_{2},\gamma_{3}\right) $. \end{proposition} \begin{proof} Conformal distance between $ \gamma $ and $ \gamma' $ is $ \geq l $ iff there exists a function $ \varphi $ defined between $ \gamma $ and $ \gamma' $ such that $ \varphi|_{\gamma}=0 $, $ \varphi|_{\gamma'}=l $, and the ``energy'' $ \int\partial\varphi\bar{\partial}\varphi \leq l $. Combining two such functions, one defined between $ \gamma_{1} $ and $ \gamma_{2} $, another between $ \gamma_{2} $ and $ \gamma_{3} $, we obtain the statement. \end{proof} \begin{lemma} The only $ \operatorname{PGL}_{2}\left({\Bbb C}\right) $-invariant of a couple of disjoint disks on $ {\Bbb C}P^{1} $ is the conformal distance. \end{lemma} \subsection{Subspaces of $ \Omega^{3/4} $ }\label{s3.8}\myLabel{s3.8}\relax Consider two disjoint disks $ K_{1} $, $ K_{2} $ on $ {\Bbb C}P^{1} $ of conformal distance $ l $. Since the natural norm on $ H^{-1/2}\left(\Omega^{3/4}\right) $ (see Section~\ref{s2.60}) is invariant with respect to $ \operatorname{PGL} $, the angle between the subspaces $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(K_{1},\Omega^{3/4}\right) $ and $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(K_{2},\Omega^{3/4}\right) $ depends on $ l $ only. Let $ P_{l} $ be the orthogonal projector from one subspace to another. \begin{proposition} \label{prop3.170}\myLabel{prop3.170}\relax Fix $ \varepsilon>0 $. If $ l\geq\varepsilon $, then \begin{equation} \|P_{l}\| \sim e^{-l/2}. \notag\end{equation} The equivalence means that the quotient of two sides remains bounded and separated from 0. \end{proposition} \begin{proof} The norm $ \|P_{l}\| $ is a smooth function of $ l $, thus one needs to prove only the asymptotic when $ l \to \infty $. One may represent two disks of conformal distance $ \approx l\gg 1 $ as $ |z|\leq1 $ and $ |z-e^{l/2}|\leq1 $. Let $ L=e^{l/2} $. Consider the coordinate $ z_{1}=z $ in the first disk, and $ z_{2}=z-L $ in the second one. What we need to prove is that the kernel $ K_{1}\left(z_{1},z_{2}\right)=\frac{1}{|z_{1}-z_{2}+L|} $ in $ \left\{\left(z_{1},z_{2}\right) | |z_{1}|,|z_{2}|\leq1\right\} $ gives an operator of the norm $ \sim1/L $. Since the radii are fixed now, we can consider functions instead of $ 3/4 $-forms. Since the norm of operator with the kernel $ K_{0}=\frac{1}{L} $ is $ O\left(\frac{1}{L}\right) $, it is enough to prove that the kernel $ K_{2}\left(z_{1},z_{2}\right)=\frac{1}{|z_{1}-z_{2}+L|} - \frac{1}{L} $ corresponds to an operator of norm $ o\left(1/L\right) $. Let us estimate Hilbert--Schmidt norm of this operator. It is equal to the $ H^{1/2}\otimes_{l_{2}}H^{1/2} $-norm of $ K_{2} $. On the other hand, the last norm is bounded by $ H^{1} $-norm, which is obviously $ O\left(L^{-2}\right) $.\end{proof} \begin{remark} \label{rem3.75}\myLabel{rem3.75}\relax Note that if $ 0<s<1 $, then a similar statement is true in $ H^{1-2s}\left(\Omega^{s}\right) $. We will need only the following statement: \begin{equation} \|P_{l}\| = O\left(e^{-l/2}\right)\text{ if }0<s<\frac{3}{4}, \notag\end{equation} and we will use it in the case $ s=\frac{1}{4} $ only. \end{remark} \begin{corollary} \label{cor3.80}\myLabel{cor3.80}\relax Consider a family of disjoint closed disks $ K_{i}\subset{\Bbb C}P^{1} $, $ i\in{\Bbb N} $, with conformal distance between $ K_{i} $ and $ K_{j} $ being $ l_{ij} $. Put $ l_{i i}=0 $. Let $ {\cal L} $ be a linear bundle on $ {\Bbb C}P^{1} $. Suppose that the closure of $ \bigcup K_{i} $ does not coincide with $ {\Bbb C}P^{1} $. Suppose that $ \inf _{i\not=j}l_{ij}>0 $, and that the matrix $ \left(e^{-l_{ij}/2}\right) $ corresponds to a bounded operator $ l_{2} \to l_{2} $. Then the inclusions $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(K_{i},{\cal L}\right)\hookrightarrow H^{-1/2}\left({\Bbb C}P^{1},{\cal L}\right) $ extend to a continuous injection \begin{equation} \bigoplus_{l_{2}}\overset{\,\,{}_\circ}{H}^{-1/2}\left(K_{i},{\cal L}\right)\hookrightarrow H^{-1/2}\left({\Bbb C}P^{1},{\cal L}\right). \notag\end{equation} Dually, the restrictions $ H^{1/2}\left({\Bbb C}P^{1},{\cal L}\right) \to H^{1/2}\left(K_{i},{\cal L}\right) $ extend to a continuous surjection \begin{equation} H^{1/2}\left({\Bbb C}P^{1},{\cal L}\right) \to \bigoplus_{l_{2}}H^{1/2}\left(K_{i},{\cal L}\right). \notag\end{equation} \end{corollary} \begin{proof} If $ {\cal L}=\Omega^{3/4} $ in the first part of the theorem, or $ {\cal L}=\Omega^{1/4} $ in the second one, then everything is proved. Otherwise let $ U $ be an open subset of $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $. An isomorphism of $ {\cal L} $ and $ \Omega^{3/4} $ (or $ \Omega^{1/4} $) on $ {\Bbb C}P^{1}\smallsetminus U $ proves the rest. \end{proof} Using the results of Remark~\ref{rem3.75} we obtain the following statement: \begin{proposition} In the conditions of Corollary~\ref{cor3.80} \begin{equation} \bigoplus_{l_{2}}\overset{\,\,{}_\circ}{H}^{1/2}\left(K_{i},{\cal L}\right) \to H^{1/2}\left({\Bbb C}P^{1},{\cal L}\right). \notag\end{equation} (components being inclusions) is an isomorphism to its image. The image of this mapping is $ \overset{\,\,{}_\circ}{H}^{1/2}\left(\bigcup_{i}K_{i},{\cal L}\right) $. \end{proposition} \section{Generalized Hardy space }\label{h4}\myLabel{h4}\relax \subsection{Hilbert space of holomorphic half-forms }\label{s5.10}\myLabel{s5.10}\relax In this section we are going to construct a Hilbert space (or at least a space with Hilbert topology) which models global holomorphic half-forms on a curve which is not necessarily compact. Consider a family of disjoint closed disks $ K_{i}\subset{\Bbb C}P^{1} $, $ i\in I $. Let $ \varepsilon>0 $, $ \widetilde{K}_{i} $ be the concentric disk to $ K_{i} $ of radius $ e^{2\varepsilon}\cdot\operatorname{radius}\left(K_{i}\right) $. Suppose that the disks $ \widetilde{K}_{i} $ do not intersect, and denote the conformal distance between $ \widetilde{K}_{i} $ and $ \widetilde{K}_{j} $ by $ l_{ij} $. Put $ l_{i i}=0 $. Suppose that \begin{nwthrmii} Say that the collection of disks is {\em well-separated\/} if \begin{enumerate} \item $ \inf _{i\not=j}l_{ij}>0 $; \item $ \overline{\bigcup\widetilde{K}_{i}}\not={\Bbb C}P^{1} $; and \item the matrix $ \left(e^{-l_{ij}/2}\right) $ gives a bounded operator $ l_{2} \to l_{2} $; \end{enumerate} \end{nwthrmii} Note that the interior of the closure $ \overline{{\Bbb C}P^{1}\smallsetminus\bigcup K_{i}} $ is non-empty by the above conditions, and \begin{proposition} \label{prop5.16}\myLabel{prop5.16}\relax If disks $ K_{i} $ are well-separated, then $ \sum_{i}\operatorname{radius}\left(K_{i}\right)<\infty $, thus Cauchy kernel defines a mapping from $ L_{2}\left(\bigcup\partial K_{i},\Omega^{1/2}\otimes\mu\right) $ to the space of analytic functions in $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $. \end{proposition} \begin{proof} Indeed, since rows of $ \left(e^{-l_{ij}/2}\right) $ are in $ l_{2} $, $ \sum_{j} e^{-l_{ij}} <\infty $. On the other hand, $ e^{-l_{ij}} $ is approximately proportional to radius of $ K_{j} $ when $ i $ is fixed. \end{proof} Moreover, if we slightly decrease the value of $ \varepsilon $, then the first two conditions on the family $ \left\{K_{i}\right\} $ will be automatically satisfied, so only the last condition is the new one. \begin{definition} The {\em generalized Hardy space\/} $ {\cal H}\left({\Bbb C}P^{1}, \left\{K_{i}\right\}\right) $ is the space \begin{equation} \left\{f\left(z\right)\in H^{1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K_{i}, \omega^{1/2}\right) \mid \bar{\partial}f=0\in H^{-1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K_{i},\omega^{1/2}\otimes\bar{\omega}\right)\right\}. \notag\end{equation} \end{definition} Note that elements of $ {\cal H}\left({\Bbb C}P^{1}, \left\{K_{i}\right\}\right) $ are holomorphic half-forms in $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup\widetilde{K}_{i}} $. Note also that the Sobolev space in the definition is a generalized Sobolev space, since $ {\Bbb C}P^{1}\smallsetminus\bigcup_{i}K_{i} $ is not open if the set $ I $ is infinite. \begin{remark} Note that if $ I $ consists of one element, then $ {\cal H} $ is the usual Hardy space from Section~\ref{s2.70} with modifications outlined in Remark ~\ref{rem2.10}. \end{remark} In Section~\ref{s4.90} we introduce a slightly weaker condition on domains $ K_{i} $, and will use it instead of Condition A. Note that we are going to mention the space $ {\cal H} $ only in cases when the Condition A is satisfied. \subsection{Hilbert operator }\label{s5.20}\myLabel{s5.20}\relax A section of $ \Omega^{1/2}\otimes\mu $ on a circle can be decomposed into the Fourier series \begin{equation} f\left(t\right)dt^{1/2}=\sum_{n\in{\Bbb Z}+1/2}a_{n}e^{2\pi i nt}dt^{1/2} \notag\end{equation} (half-integers appear because of the factor $ \mu $), hence it can be written as a sum of a component $ f_{+} $ which can be holomorphically extended to a section of $ \omega^{1/2} $ inside the circle, and a component $ f_{-} $ which may be extended outside of the circle. Moreover, norms of $ f_{\pm} $ are bounded by the norm of $ f $, and the decomposition is unique. We denote this decomposition $ L_{2}^{+}\left(\gamma,\Omega^{1/2}\otimes\mu\right) \oplus L_{2}^{-}\left(\gamma,\Omega^{1/2}\otimes\mu\right) $. \begin{proposition} Consider the operator $ {\bold K}_{+} $ with Cauchy kernel acting in the space $ \bigoplus_{l_{2}}L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $. Let $ {\bold K} $ be the operator $ \widetilde{{\bold K}} $ with diagonal blocks $ L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) \to L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $ removed. If the disks $ K_{i} $ are well-separated, then $ {\bold K} $ is bounded. If we decompose \begin{equation} L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) = L_{2}^{+}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) \oplus L_{2}^{-}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right), \notag\end{equation} then the only nonzero blocks of $ {\bold K} $ act from $ L_{2}^{-}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $ to $ L_{2}^{+}\left(\partial K_{j},\Omega^{1/2}\otimes\mu\right) $, $ i\not=j $. \end{proposition} This statement is an immediate corollary of the first part of \begin{lemma} \label{lm5.05}\myLabel{lm5.05}\relax Consider a Hilbert space $ H=\bigoplus_{l_{2}}H_{i} $ and a linear operator $ A\colon H \to H $ with bounded blocks $ A_{ij}\colon H_{j} \to H_{i} $. Let $ a_{ij}=\|A_{ij}\| $. Suppose that the matrix $ \left(a_{ij}\right) $ gives a bounded operator $ \alpha\colon l_{2} \to l_{2} $. Then \begin{enumerate} \item $ A $ is bounded, $ \|A\|\leq\|\alpha\| $; \item $ A $ is compact if $ \alpha $ is compact and each of operators $ A_{ij} $ is compact. \end{enumerate} \end{lemma} \begin{proof} To bound the operator $ A $ it is enough to bound $ |\left(x, Ay\right)| $ for $ |x|=|y|=1 $, $ x,y\in H $. On the other hand, decomposition of $ H $ gives $ x=\left(x_{i}\right) $, $ y=\left(y_{i}\right) $, and $ |x|^{2}=\sum|x_{i}|^{2} $, $ |y|^{2}=\sum|y_{i}|^{2} $. This gives \begin{equation} |\left(x,Ay\right)| = \left|\sum_{ij}\left(x_{i},A_{ij}y_{j}\right)\right| \leq \sum_{ij}|\left(x_{i},A_{ij}y_{j}\right)| \leq \sum_{ij}\|A_{ij}\||x_{i}||y_{j}| \notag\end{equation} and the latter sum is just $ \left(\xi,\alpha\eta\right) $ with $ \xi=\left(|x_{i}|\right)\in l_{2} $, similarly for $ \eta $. To prove the second part it is enough to show that we can approximate $ A $ (in norm) by compact operators. Since $ \alpha $ is compact, we can approximate it (in norm) by a matrix with finite number of non-zero elements. The relation of norm of $ A $ and norm of $ \alpha $ shows that $ A $ can be approximated by an operator $ A' $ which has the same blocks as $ A $ in finite number of places, all the rest is 0. Since blocks of $ A $ are compact, $ A' $ is compact as well. We conclude that $ A $ can be approximated with arbitrary precision by a compact operator, thus $ A $ is compact.\end{proof} \begin{corollary} \label{cor4.35}\myLabel{cor4.35}\relax Any block-row and block-column of $ {\bold K} $ gives a compact operator. \end{corollary} \subsection{Boundary map } Here we consider which of the facts from Section~\ref{s2.70} have sense for the generalized Hardy space as well. Fix $ i\in I $, let $ S_{i}=\partial K_{i} $. Consider the mollified restriction mapping \begin{equation} H^{1/2}\left(\widetilde{K}_{i}\smallsetminus K_{i},\omega^{1/2}\right) \xrightarrow[]{\widetilde{r}_{i}} L_{2}\left(S_{i},\Omega^{1/2}\otimes\mu\right). \notag\end{equation} Since any element $ f $ of the generalized Hardy space is $ H^{1/2} $ in $ \widetilde{K}_{i}\smallsetminus K_{i} $ (and holomorphic inside this annulus), the restriction $ \widetilde{r}_{i}\left(f\right) =f|_{S_{i}} $ of this half-form on $ S_{i} $ is an $ L_{2} $-section of $ \Omega^{1/2}\left(S_{i}\right)\otimes\mu $. In the rest of this section we are going to abuse notations and denote $ \widetilde{r}_{i}\left(f\right) $ as $ f|_{S_{i}} $. \begin{theorem} \label{th4.40}\myLabel{th4.40}\relax Suppose that the disks $ K_{i} $ are well-separated. Then the mappings $ \widetilde{r}_{i} $ taken together provide a mapping \begin{equation} {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) \xrightarrow[]{\widetilde{r}} \displaystyle\coprod L_{2}\left(S_{i},\Omega^{1/2}\otimes\mu\right), \notag\end{equation} which is fact is a continuous mapping \begin{equation} {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) \xrightarrow[]{\widetilde{r}} \bigoplus_{l_{2}}L_{2}\left(S_{i},\Omega^{1/2}\otimes\mu\right). \notag\end{equation} The image of $ \widetilde{r} $ is closed. Moreover, the mapping $ \widetilde{r} $ is invertible onto its image. \end{theorem} \begin{proof} In fact, all the main ingredients for the proof of this theorem are already here. An element $ f\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ is by definition a restriction of some $ H^{1/2} $-section $ g $ of $ \omega^{1/2} $ on the whole sphere $ {\Bbb C}P^{1} $. The restrictions of $ g $ to $ \widetilde{K}_{i} $ give an element of $ \bigoplus_{l_{2}}H^{1/2}\left(\widetilde{K}_{i},\omega^{1/2}\right) $ by Corollary~\ref{cor3.80}. On the other hand, given $ f $, the restriction of $ g $ on $ \widetilde{K}_{i} $ is defined up to a section with support on $ K_{i} $. Hence the mollified restriction on $ S_{i} $ is correctly defined, and it has a norm majorated by some multiple of the norm of the restriction on $ \widetilde{K}_{i} $. Hence the restriction mapping $ \widetilde{r} $ is bounded indeed. To show that $ \operatorname{Im} \widetilde{r} $ is closed, let us construct a left inverse $ l $ this operator. Then $ \widetilde{r}l $ is going to be a projection on $ \operatorname{Im} \widetilde{r} $, which will prove the closeness. Consider the mapping $ \lambda_{i} $ from Section~\ref{s2.25}, associated to the disks $ K_{i} $, $ \widetilde{K}_{i} $. Then $ g - \lambda_{i}\left(g\right) $ \begin{enumerate} \item vanishes outside of $ \widetilde{K}_{i} $; \item coincides with $ g $ inside of $ K_{i} $; \item has a norm bounded by $ C\cdot\|g|_{\widetilde{K}_{i}}\| $; \item depends on values of $ g $ outside of $ K $ only. \end{enumerate} Combining all this together, we get \begin{equation} F = g+\sum_{i}\left(\lambda_{i}\left(g\right)-g\right) \notag\end{equation} which is a half-form of the norm bounded by \begin{equation} \|g\|+C'\cdot\left(\sum\|g|_{\widetilde{K}_{i}}\|^{2}\right)^{1/2}= O\left(\|g\|\right). \notag\end{equation} The half-form $ F $ is equal to $ g $ (thus to $ f $) outside of $ \bigcup\widetilde{K}_{i} $, and is equal to 0 inside all $ K_{i} $. Moreover, since $ f $ is holomorphic, one can calculate $ \bar{\partial}F|_{\widetilde{K}_{i}} $ as $ \widetilde{e}_{i}\left(f|_{\partial K_{i}}\right) $. \begin{proposition} Let $ f $ be a half-form from the generalized Hardy space. There exists a half-form $ F\in H^{1/2}\left({\Bbb C}P^{1}\right) $ such that: \begin{enumerate} \item $ \|F\| < C\cdot\|f\| $; \item $ f=F $ outside of $ \overline{\bigcup\widetilde{K}_{i}} $; \item $ F=0 $ inside of $ K_{i} $ for any $ i $; \item $ \bar{\partial}F = \sum\widetilde{e}_{i}\left(f|_{\partial K_{i}}\right) $. \end{enumerate} \end{proposition} \begin{proof} The only part which needs proof is the last one. First of all, the sum in the right-hand side obviously converges in $ H^{-1/2} $, since the disks $ \widetilde{K}_{i} $ are separated far enough, and norms of the restrictions of $ f $ onto $ \partial K_{i} $ form a sequence in $ l_{2} $. The same arguments show that the right-hand side is contained in the generalized Sobolev subspace $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(\bigcup\widetilde{K}_{i}\right) $. Moreover, $ \bar{\partial}\left(F-g\right) $ is contained in the same space, and $ \bar{\partial}f $ is there by the definition of the generalized Hardy space. Since $ g $ coincides with $ f $ outside of $ \bigcup K_{i} $, $ \bar{\partial}F\in\overset{\,\,{}_\circ}{H}^{-1/2}\left(\bigcup\widetilde{K}_{i}\right) $. Thus we know that the difference of the right-hand side and left-hand side is contained in $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(\bigcup\widetilde{K}_{i}\right) $ and is 0 inside any $ \widetilde{K}_{i} $. On the other hand, since $ \widetilde{K}_{i} $ are well-separated, $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(\bigcup\widetilde{K}_{i}\right) = \bigoplus_{l_{2}}\overset{\,\,{}_\circ}{H}^{-1/2}\left(\widetilde{K}_{i}\right) $, what finishes the proof.\end{proof} \begin{remark} It is obvious that $ f \mapsto F $ is a continuous linear mapping $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) \to H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) $. Moreover, it is an injection. To show this, one needs only to prove that $ f|_{\bigcup\widetilde{K}_{i}} $ is determined by $ F $ (and bounded by the norm of $ F $). Since the disks $ \widetilde{K}_{i} $ are well-separated, it is enough to show this for one particular disk $ \widetilde{K}_{i} $. Since $ \widetilde{e}_{i} $ is an injection, $ \bar{\partial}F $ determines $ \widetilde{r}_{i}\left(f\right) $, thus $ \lambda_{i}\left(g\right) $ (by construction of $ \lambda $), thus $ \left(F-f\right)|_{\widetilde{K}_{i}} $. We obtained \end{remark} \begin{corollary} The relation $ f \mapsto F $ is an injection $ {\cal H}\left({\Bbb C}P^{1}, \left\{K_{i}\right\}\right) \to H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) $. Since $ \bar{\partial}\colon H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) \to H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right) $ is an isomorphism, the relation $ f \mapsto \bar{\partial}F $ gives an injection $ {\cal H}\left({\Bbb C}P^{1}, \left\{K_{i}\right\}\right) \to H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right) $. Moreover, the last mapping may be pushed through $ {\cal H}\left({\Bbb C}P^{1}, \left\{K_{i}\right\}\right) \to \bigoplus_{l_{2}}H^{-1/2}\left(\widetilde{K}_{i},\omega^{1/2}\otimes\bar{\omega}\right) $. \end{corollary} Since the operator $ \bar{\partial} $ has no null-space on $ H^{1/2}\left({\Bbb C}P^{1}, \omega^{1/2}\right) $, we obtain \begin{corollary} $ \bar{\partial}^{-1}\left(\sum\widetilde{e}_{i}\left(f|_{\partial K_{i}}\right)\right) $ equals $ f $ outside of $ \bigcup\widetilde{K}_{i} $, i.e., modulo $ \overset{\,\,{}_\circ}{H}^{1/2}\left(\bigcup\widetilde{K}_{i}\right) $. \end{corollary} Since $ \bar{\partial}F $ is determined by $ \widetilde{r}\left(f\right) $, we found a left inverse to the mapping $ \widetilde{r} $, thus it is an injection and the image is closed. This finishes the proof of Theorem~\ref{th4.40}. {}\end{proof} The next step is to describe the image of the operator in question. Let $ \gamma_{i}=\partial K_{i} $, $ \gamma=\bigcup\gamma_{i} $. We claim that an element of the image of the mapping $ \widetilde{r} $ is uniquely determined by the minus-components, and any collection of minus-components of bounded norm is possible. \begin{proposition} \label{prop5.28}\myLabel{prop5.28}\relax Consider a decomposition of the space $ L_{2}\left(\gamma_{i}, \Omega^{1/2}\right) = L_{2}^{+}\left(\gamma_{i}\right) \oplus L_{2}^{-}\left(\gamma_{i}\right) $ into subspaces of half-forms which may be holomorphically extended inside the circle and outside the circle. The image of $ \widetilde{r} $ consists of sequences $ \left(f_{i}^{\pm}\right) $ such that \begin{equation} f_{i}^{+}=\sum_{j\not=i}{\bold K}f_{j}^{-}. \notag\end{equation} Here $ {\bold K} $ is the Hilbert operator, i.e., the operator with Cauchy kernel. \end{proposition} \begin{proof} First of all, note that the proof of Theorem~\ref{th4.40} together with Propositions~\ref{prop4.12} and~\ref{prop5.16} shows that knowing $ \widetilde{r}\left(f\right) $ one can write $ f $ by an explicit formula outside of $ \overline{\bigcup\widetilde{K}_{i}} $. On the other hand, it is easy to see that if we throw away the restriction that we want our operators to be continuous in $ H^{s} $-topology, one can reconstruct $ f $ outside of $ \bigcup\bar{K}_{i} $. Indeed, since the curve $ \gamma $ has a finite length, restriction of any smooth section $ \alpha $ of $ \omega^{1/2} $ to $ \gamma $ is in $ L_{2}\left(\gamma\right) $, thus $ \int\alpha|_{S}\widetilde{r}\left(f\right) $ is correctly defined. By duality, this means that extension $ e\left(\widetilde{r}\left(f\right)\right) $ of $ \widetilde{r}\left(f\right) $ to $ {\Bbb C}P^{1} $ by $ \delta $-function is a correctly defined generalized section of $ \omega^{1/2}\otimes\bar{\omega} $. (Note that we consider $ e $, not $ \widetilde{e}! $) \begin{lemma} Half-form $ f $ coincides with $ \bar{\partial}^{-1}e\left(\widetilde{r}\left(f\right)\right) $ outside of $ \bigcup\bar{K}_{i} $. \end{lemma} \begin{proof} Indeed, by the construction of $ \widetilde{e}_{i} $, $ \bar{\partial}^{-1} \widetilde{e}_{i}\left(\widetilde{r}_{i}\left(f\right)\right) $ coincides with $ \bar{\partial}^{-1}e\left(\widetilde{r}_{i}\left(f\right)\right) $ outside of $ \widetilde{K}_{i}\smallsetminus K_{i} $, thus $ \bar{\partial}^{-1}e\left(\widetilde{r}\left(f\right)\right) $ coincides with $ f $ outside of $ \bigcup\widetilde{K}_{i} $. On the other hand, $ \bar{\partial}^{-1}e\left(\widetilde{r}\left(f\right)\right) $ is holomorphic outside of $ \bar{\gamma} $, thus this equality can be extended up to $ \gamma $. \end{proof} Consider now $ f=\bar{\partial}^{-1}e\left(\widetilde{r}\left(f\right)\right) $ near $ \gamma_{i} $. Breaking $ \widetilde{r}\left(f\right) $ into two components, $ \left(\widetilde{r}\left(f\right)-\widetilde{r}_{i}\left(f\right)\right) $ and $ \widetilde{r}_{i}\left(f\right) $, we conclude that \begin{equation} f=\bar{\partial}^{-1}e\left(\widetilde{r}\left(f\right)-\widetilde{r}_{i}\left(f\right)\right) + \bar{\partial}^{-1}e\left(\widetilde{r}_{i}\left(f\right)\right). \notag\end{equation} The first summand is holomorphic inside $ \widetilde{K}_{i} $ and coincides with $ \sum_{j\not=i}{\bold K}f_{j}^{-} $, since $ {\bold K} $ kills $ f_{j}^{+} $. The second summand is holomorphic outside of $ \partial K_{i} $, so its $ + $-part vanishes on $ \partial K_{i} $, which finishes the proof of Proposition ~\ref{prop5.28}. {}\end{proof} \subsection{Gluing conditions }\label{s5.30}\myLabel{s5.30}\relax We continue using notations of Section ~\ref{s5.10}. Suppose that the set $ I $ has an involution $ '\colon I\to I $ which interchanges two subsets $ I_{+} $ and $ I_{+}' $, $ I=I_{+}\amalg I_{+}' $. Thus all the disks $ K_{i} $ are divided into pairs $ \left(K_{i},K_{i'}\right) $, $ i\in I_{+} $. Fix a fraction-linear identification $ \varphi_{i} $ of $ \partial K_{i'} $ and $ \partial K_{i} $ which reverses the orientation of the circles, $ \varphi_{i'}=\varphi_{i}^{-1} $. Let $ \varphi_{i} $ identifies the boundary of the disk $ \widetilde{K}_{i'} $ with the boundary of $ \overset{\,\,{}_\circ}{K}_{i} $, and $ \partial\widetilde{K}_{i} $ with $ \partial\overset{\,\,{}_\circ}{K}_{i'} $, thus $ \overset{\,\,{}_\circ}{K}_{i}\subset K_{i}\subset\widetilde{K}_{i} $. Let $ R_{i} $ be the annulus between $ \widetilde{K}_{i} $ and $ \overset{\,\,{}_\circ}{K}_{i} $. The mapping $ \varphi_{i} $ identifies $ R_{i} $ with $ R_{i'} $. Let $ S $ be the part of $ {\Bbb C}P^{1} $ which lies outside of all the disks $ \overset{\,\,{}_\circ}{K}_{i} $. Glue the annuli $ R_{i}\subset S $ with $ R_{i'}\subset S $ using $ \varphi_{i} $. \begin{definition} A {\em model space\/} $ \bar{M} $ is the set obtained from $ S $ by identifying the annuli $ R_{i} $ and $ R_{i'} $ using $ \varphi_{i} $. \end{definition} Note that $ \bar{M} $ consists of two parts: a smooth manifold $ M $ which is the image of $ S\smallsetminus\overline{\bigcup\overset{\,\,{}_\circ}{K}_{\bullet}} $, and the rest, which one should consider as ``infinity'' $ M_{\infty} $ of the manifold $ M $ (compare with Section~\ref{s0.10}). Unfortunately, the topology on $ \bar{M} $ in neighborhood of infinity is not suitable for studying the Riemann--Roch theorem, so we will not consider it in this paper. Note that Section~\ref{s4.95} suggests a different topology on $ \bar{M} $. We will use some features of this topology when we discuss a mapping into universal Grassmannian. \subsection{Strong sections and duality }\label{s6.50}\myLabel{s6.50}\relax In the notations of the previous section consider now holomorphic functions $ \psi_{i} $ defined in $ R_{i} $. Suppose that $ \psi_{i} $ are nowhere 0, and $ \psi_{i'}\circ\varphi_{i}=\psi_{i}^{-1} $. Define $ {\cal L} $ to be a sheaf on $ \bar{M} $ associated with gluing conditions $ \psi_{i} $, i.e., for $ U\subset\bar{M} $ the section of $ {\cal L} $ on $ U $ is a function $ f $ on $ \widetilde{U}\subset{\Bbb C}P^{1} $ such that $ f\left(\varphi_{i}\left(x\right)\right)=\psi_{i}\left(x\right)f\left(x\right) $ whenever both sides have sense (here $ \widetilde{U} $ is an appropriate covering subset of $ {\Bbb C}P^{1} $). Let $ {\cal L}^{-1} $ be the sheaf associated with gluing conditions $ \psi_{i}^{-1} $. Similarly define the tensor product of two sheaves defined via gluing conditions. The identifications $ \varphi_{i} $ fixes an identification $ \varphi_{i}^{*} $ of half-forms on $ R_{i} $ and on $ R_{i'} $ up to a sign. Choose this sign for all $ i\in I_{+}. $\footnote{This corresponds to picking a representative of $ \varphi_{i} $ in $ 2 $-sheet cover $ \operatorname{SL}\left(2,{\Bbb C}\right) \to \operatorname{PGL}\left(2,{\Bbb C}\right) $.} Let $ \omega^{1/2}\otimes{\cal L} $ be the sheaf on $ \bar{M} $ consisting of half-forms on $ S $ such that \begin{equation} \varphi_{i}^{*}\left(\alpha|_{R_{i'}}\right) = \psi_{i}\cdot\alpha|_{R_{i}},\qquad i\in I. \label{equ5.31}\end{equation}\myLabel{equ5.31,}\relax Similarly define $ \omega^{1/2}\otimes{\cal L}\otimes\bar{\omega} $. To write an analogue of~\eqref{equ5.31} for sections in Sobolev spaces, consider a subannulus $ \overset{\,\,{}_\circ}{R}_{i}\subset R_{i} $ such that $ \overset{\,\,{}_\circ}{R}_{i'}=\varphi_{i}\left(\overset{\,\,{}_\circ}{R}_{i}\right) $. We suppose that one can find numbers $ C,D>0 $ and a sequence $ \left(s_{i}\right) $, $ i\in I $, $ |s_{i}|+C+D<1 $, such that the annulus $ \overset{\,\,{}_\circ}{R}_{i} $ is described as $ \left(\left(s_{i}-C\right)\varepsilon,\left(s_{i}+C\right)\varepsilon\right)\times S^{1} $ in the conformal coordinate system such that $ R_{i} $ is $ \left(-\varepsilon,\varepsilon\right)\times S^{1} $. Let $ K'_{i} $ be the disk bounded by the inner boundary of $ \overset{\,\,{}_\circ}{R}_{i} $. Consider a subset $ \overset{\,\,{}_\circ}{S}\subset S $, $ \overset{\,\,{}_\circ}{S}={\Bbb C}P^{1}\smallsetminus\bigcup_{i}K'_{i} $. One obtains the same manifold $ \bar{M} $ by gluing $ \overset{\,\,{}_\circ}{R}_{i}\subset\overset{\,\,{}_\circ}{S} $ as by gluing $ R\subset S $, but in what follows it will be more convenient to have a choice of annuli $ \overset{\,\,{}_\circ}{R}_{i} $, and have them separated from boundary of $ R_{i} $. \begin{definition} Let $ H^{s}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ be the subspace of $ H^{s}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K'_{i},\omega^{1/2}\right) $ consisting of sections which satisfy the gluing conditions~\eqref{equ5.31}, similarly define $ H^{s}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) $. \end{definition} Dually, \begin{definition} Let $ \overset{\,\,{}_\circ}{H}^{s}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ be the quotient of $ \overset{\,\,{}_\circ}{H}^{s}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K'_{i}, \omega^{1/2}\right) $ by the subspace of $ \overset{\,\,{}_\circ}{H}^{s}\left(\bigcup_{i}\overset{\,\,{}_\circ}{R}_{i}, \omega^{1/2}\right) $ consisting of sections which satisfy the gluing conditions \begin{equation} \varphi_{i}^{*}\left(\alpha|_{\overset{\,\,{}_\circ}{R}_{i'}}\right) = -\psi_{i}\cdot\alpha|_{\overset{\,\,{}_\circ}{R}_{i}},\qquad i\in I, \label{equ5.32}\end{equation}\myLabel{equ5.32,}\relax similarly define $ \overset{\,\,{}_\circ}{H}^{s}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) $. \end{definition} Indeed, these definitions are dual due to \begin{lemma} \label{lm5.22}\myLabel{lm5.22}\relax If the disks $ K_{i} $ are well-separated, the spaces $ H^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ and $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}^{-1}\otimes\bar{\omega}\right) $ are mutually dual w.r.t. the pairing $ \int\alpha\beta $. \end{lemma} \begin{proof} The spaces $ H^{s}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K'_{i},\omega^{1/2}\otimes{\cal L}\right) $ and $ \overset{\,\,{}_\circ}{H}^{-s}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K'_{i},\omega^{1/2}\otimes{\cal L}^{-1}\otimes\bar{\omega}\right) $ are mutually dual w.r.t. this pairing by definition. What remains to prove is that the orthogonal complement to the subspace of $ H^{s}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K'_{i},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) $ given by~\eqref{equ5.31} is given by \begin{equation} \varphi_{i}^{*}\left(\alpha|_{R_{i'}}\right) = -\psi_{i}^{-1}\cdot\alpha|_{R_{i}},\qquad i\in I, \label{equ5.33}\end{equation}\myLabel{equ5.33,}\relax in $ \overset{\,\,{}_\circ}{H}^{-s}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K'_{i},\omega^{1/2}\otimes{\cal L}^{-1}\otimes\bar{\omega}\right) $, provided $ s=1/2 $. This statement if obvious for any fixed $ i\in I $. On the other hand, solutions to~\eqref{equ5.33} form a direct sum over $ i $, and by the second part of Corollary~\ref{cor3.80}, solutions to~\eqref{equ5.31} form a ``direct'' intersection (i.e., an intersection of subspaces with almost orthogonal complements).\end{proof} \begin{lemma} \label{lm5.25}\myLabel{lm5.25}\relax Suppose that the disks $ K_{i} $ are well-separated. The natural mapping \begin{equation} \overset{\,\,{}_\circ}{H}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) \to H^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) \label{equ5.36}\end{equation}\myLabel{equ5.36,}\relax is an isomorphism. Dually, \begin{equation} \overset{\,\,{}_\circ}{H}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \to H^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \notag\end{equation} is an isomorphism. \end{lemma} \begin{proof} We may suppose that all the rings $ \overset{\,\,{}_\circ}{R}_{i} $ have conformal distance between boundaries greater than $ 2C\varepsilon $, $ C>0 $. Consider a function $ \sigma\left(x\right) $, $ x\in\left(-C\varepsilon,C\varepsilon\right) $, such that $ \sigma\left(x\right)=0 $ near the left end, $ \sigma\left(x\right)+\sigma\left(1-x\right)=1 $. This gives a cut-off function in all the rings $ \overset{\,\,{}_\circ}{R}_{i} $, and we may extend it by 0 into all $ K_{i}' $, and by 1 into $ S $. We obtain a function on $ {\Bbb C}P^{1} $. By the results of Section~\ref{s3.8} and Amplification~\ref{amp2.65} the multiplication by this function is a bounded operator in $ H^{1/2} $. Clearly, this operator provides an inverse to~\eqref{equ5.36}. \end{proof} The operator $ \bar{\partial} $ gives mappings \begin{align} \overset{\,\,{}_\circ}{H}^{s}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) & \to \overset{\,\,{}_\circ}{H}^{s-1}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right), \notag\\ H^{s}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) & \to H^{s-1}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right), \notag\end{align} which by the previous lemma induce mappings \begin{align} \overset{\,\,{}_\circ}{H}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) & \xrightarrow[]{\bar{\partial}} H^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right), \notag\\ H^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) & \xrightarrow[]{\bar{\partial}} \overset{\,\,{}_\circ}{H}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) \notag\end{align} (obviously, it does not matter which of isomorphisms of Lemma~\ref{lm5.25} we use to obtain these mappings). \begin{definition} Define the space $ \Gamma_{\text{strong}}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ of {\em strong global holomorphic sections\/} of $ \omega^{1/2}\otimes{\cal L} $ as \begin{equation} \Gamma_{\text{strong}}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) = \operatorname{Ker} \left(H^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \xrightarrow[]{\bar{\partial}} H^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right)\right). \notag\end{equation} \end{definition} Note that the conditions on strong global sections $ \alpha $ are: they are holomorphic sections of $ \omega^{1/2}\otimes{\cal L}|_{M} $ (since $ \bar{\partial}\alpha $ vanishes on $ M $), they do not grow very quick near $ M_{\infty} $ (since they belong to $ H^{1/2} $), and they have no residue on $ M_{\infty} $ (since $ \bar{\partial}\alpha $ vanishes on $ M_{\infty} $). \subsection{Weak sections }\label{s6.60}\myLabel{s6.60}\relax The space $ \Gamma_{\text{strong}} $ from the last section is good for studying the duality conditions, but it is not suitable for for description of global sections via boundary conditions. \begin{definition} Define the space $ \Gamma_{\text{weak}}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ of {\em weak global holomorphic sections\/} of $ \omega^{1/2}\otimes{\cal L} $ as forms from the generalized Hardy space $ \alpha\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ such that the (mollified) restrictions on the circles $ \partial K_{i} $ satisfy the gluing conditions. In other words, \begin{equation} \Gamma_{\text{weak}}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) = \left\{\alpha\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) \mid \psi_{i}\widetilde{r}_{i}\left(\alpha\right)=\varphi_{i}^{*}\left(\widetilde{r}_{i'}\left(\alpha\right)\right) \right\}. \notag\end{equation} \end{definition} There is a natural mapping \begin{equation} \Gamma_{\text{strong}} \to \Gamma_{\text{weak}}. \notag\end{equation} \begin{theorem} \label{th5.31}\myLabel{th5.31}\relax Suppose that the disks $ K_{i} $ are well-separated, and for some $ A>1 $ either $ |\psi_{i}\left(z\right)|<A $, or $ |\psi_{i}\left(z\right)|>1/A $ for any $ i $ and $ z\in R_{i} $. Then the above mapping is an isomorphism for an appropriate choice of annuli $ \overset{\,\,{}_\circ}{R}_{i} $. \end{theorem} \begin{proof} Indeed, any weak section $ \alpha $ is a holomorphic form inside $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup_{i}K_{i}} $. Consider $ \psi_{i}^{-1}\cdot\varphi_{i}^{*}\left(\alpha\right) $, it is a holomorphic form inside $ K_{i}\smallsetminus\overset{\,\,{}_\circ}{K}_{i} $. Since $ \alpha $ satisfies gluing conditions, $ \alpha|_{\widetilde{K}_{i}\smallsetminus K_{i}} $ has the same Laurent coefficients as $ \psi_{i}^{-1}\cdot\varphi_{i}^{*}\left(\alpha\right) $ (we used compatibility of $ \widetilde{r} $ with $ r $ on holomorphic forms), thus these two forms are restrictions of the same holomorphic form defined inside $ \widetilde{K}_{i}\smallsetminus\overset{\,\,{}_\circ}{K}_{i} $. We see that $ \alpha $ can be extended (as a holomorphic form) into $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup_{i}\overset{\,\,{}_\circ}{K}_{i}} $, and this holomorphic form satisfies the gluing conditions~\eqref{equ5.31}. What remains to prove is that we can bound the norm of this extension. On the other hand, this is a local statement, since one can represent $ \alpha=\alpha_{1}+\alpha_{2} $, and $ \alpha=0 $ inside an appropriate circle concentric with $ \widetilde{K}_{i} $, $ \alpha_{2}=0 $ outside of $ \bigcup\widetilde{K}_{i} $. Thus the only thing we need to prove is that inside $ R_{i} $ the form $ \alpha $ can be extended across $ \partial K_{i} $ without increasing its norm too much. Take $ C=\frac{1}{4} $, let $ \varepsilon_{i}=C $ if $ |\psi_{i}\left(z\right)|<A $, $ \varepsilon_{i}=-C $ otherwise. With this choice of $ \overset{\,\,{}_\circ}{R}_{i} $ we know that the $ H^{1/2} $-norms of $ \alpha|_{\widetilde{K}_{i}\smallsetminus K_{i}} $ and $ \alpha|_{K_{i}\smallsetminus K'_{i}} $ are bounded by the norms of $ \alpha|_{\widetilde{K}_{i}\smallsetminus K_{i}} $ and $ \alpha|_{\widetilde{K}_{i'}\smallsetminus K_{i'}} $. Now the theorem becomes a corollary of the following \begin{lemma} Fix two numbers $ A>a>0 $. Consider an annulus $ R $ with concentric boundaries and conformal distance between boundaries between $ A $ and $ a $. Let $ V_{R} $ be the space of holomorphic half-forms in $ R $ which belong to $ H^{1/2}\left(R\right) $. Then the mapping of taking boundary value \begin{equation} b\colon H^{1/2}\left(R,\omega^{1/2}\right) \to L_{2}\left(\partial R,\Omega^{1/2}\right)\colon \alpha \mapsto \alpha|_{\partial R} \notag\end{equation} is an invertible mapping to its (closed) image, and the norms of this mapping and its inverse are bounded by numbers depending on $ A $ and $ a $ only. \end{lemma} This lemma is a variation of what we did in Section~\ref{s2.70}, with a disk substituted with an annulus. It can be proven in the same way as the case of a disk. This finishes the proof of the theorem. \end{proof} \begin{remark} To complement the notion of weak holomorphic sections, let us define the spaces of ``weak'' $ H^{\pm1/2} $-section in such a way that \begin{equation} \Gamma_{\text{weak}}\left(\bar{M}, \omega^{1/2}\otimes{\cal L}\right) = \operatorname{Ker}\left(H_{\text{weak}}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \xrightarrow[]{\bar{\partial}} H_{\text{weak}}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) \right): \notag\end{equation} \end{remark} \begin{definition} Let $ H_{\text{weak}}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ be the subspace of $ H^{1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K_{i},\omega^{1/2}\right) $ consisting of sections $ \alpha $ which satisfy the gluing conditions: \begin{equation} \psi_{i}\widetilde{r}_{i}\left(\alpha\right)=\varphi_{i}^{*}\left(\widetilde{r}_{i'}\left(\alpha\right)\right),\qquad i\in I. \notag\end{equation} Let $ H_{\text{weak}}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) = H^{-1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K_{i},\omega^{1/2}\otimes\bar{\omega}\right) $. \end{definition} \subsection{Finite-degree bundles }\label{s4.90}\myLabel{s4.90}\relax Let $ d_{i} $ be the {\em index\/} of $ \psi_{i} $, i.e., the degree of the mapping $ \arg \psi_{i}\colon K_{i} \to S^{1} $. We say that the collection of gluing data $ \left\{\psi_{i}\right\} $ is of {\em finite degree\/} if $ d_{i}=0 $ for all but the finite number of indices $ i\in I $. The collection $ \left\{\psi_{i}\right\} $ is {\em semibounded\/} if for some fixed number $ C $ and any $ i $ either $ |\psi_{i}\left(z\right)| $ or $ |\psi\left(z\right)|_{i}^{-1} $ is bounded by $ C $ if $ z\in R_{i} $. The {\em degree\/} of the finite-degree collection $ \left\{\psi_{i}\right\} $ is the sum \begin{equation} \sum_{i\in I_{+}}d_{i} = \frac{1}{2}\sum_{i\in I}d_{i}. \notag\end{equation} \begin{definition} The {\em degree\/} of a bundle $ {\cal L} $ defined by gluing conditions $ \psi_{i} $ is the degree of the collection $ \left\{\psi_{i}\right\} $. \end{definition} \begin{amplification} In what follows we are going to use the following generalization of these constructions: we allow a substitution of a finite number of simply-connected domains with smooth boundaries instead of disks $ K_{i} $. (For such an $ i $ one should substitute any bigger domain instead of $ \widetilde{K}_{i} $.) However, we still require that the identifications $ \varphi_{i} $ are fraction-linear. \end{amplification} \subsection{Stratification of infinity }\label{s4.95}\myLabel{s4.95}\relax Consider a smooth curve $ \gamma $ on $ {\Bbb C}P^{1} $ of finite length. Suppose that a metric on $ {\Bbb C}P^{1} $ is fixed, and $ z\notin\gamma $. Let $ \rho_{k}\left(z,\gamma\right)=\|\operatorname{dist}\left(z,y\right)^{-k}\|_{L_{2}\left(\gamma\right)} $, here $ y\in\gamma $. \begin{lemma} $ \rho_{k}\left(x,\gamma\right) $ is a semicontinuous function of $ x $, thus \begin{equation} D_{k,R}=\left\{z\in{\Bbb C}P^{1} \mid \rho_{k}\left(x,\gamma\right)\leq R \right\} \notag\end{equation} is a compact subset of $ {\Bbb C}P^{1}\smallsetminus\gamma $. \end{lemma} Let $ D_{k}=\bigcup_{R\in{\Bbb R}}D_{k,R} $. It is a subset of $ {\Bbb C}P^{1}\smallsetminus\gamma $, moreover, $ {\Bbb C}P^{1}\smallsetminus\bar{\gamma}\subset D_{k} $. Let $ \overset{\,\,{}_\circ}{D}_{k,R}=D_{k,R}\cap\left({\Bbb C}P^{1}\smallsetminus\bar{\gamma}\right) $, $ \overset{\,\,{}_\circ}{D}_{k}=\bigcup_{R\in{\Bbb R}}\overline{\overset{\,\,{}_\circ}{D}_{k,R}} $. Since length of $ \gamma $ is finite, $ \overset{\,\,{}_\circ}{D}_{0}={\Bbb C}P^{1} $. \begin{definition} Define a filtration of $ {\cal K}=\bar{\gamma}\smallsetminus\gamma $ by $ {\cal K}^{\left(k\right)}={\cal K}\cap\overset{\,\,{}_\circ}{D}_{k+1} $, $ k\geq-1 $. Let $ {\cal K}^{\left(\infty\right)}=\bigcap{\cal K}^{\left(k\right)} $. \end{definition} Suppose that $ \gamma $ is the boundary of a well-separated family of circles, $ \gamma=\bigcup\partial K_{i} $. \begin{theorem} Let $ z_{0}\in{\cal K}^{\left(k\right)} $, $ z $ be a coordinate system near $ z_{0} $, $ f\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $. Then $ f $ has an asymptotic decomposition \begin{equation} f\left(z\right)dz^{-1/2} = f_{0}+f_{1}\left(z-z_{0}\right)+f_{2}\left(z-z_{0}\right)^{2}+\dots +f_{k}\left(z-z_{0}\right)^{k}+o\left(z^{k}\right) \notag\end{equation} when $ z \to z_{0} $ along $ \overset{\,\,{}_\circ}{D}_{k,R} $ for an appropriate $ R\gg0 $. \end{theorem} \begin{proof} The half-form $ f $ is holomorphic inside $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $. The Cauchy formula show that inside $ \overset{\,\,{}_\circ}{D}_{k,R} $ the derivatives $ f^{\left(l\right)} $, $ l\leq k $, are bounded. Moreover, these derivatives are given by some integrals along $ \gamma $, and these integrals remain well-defined in $ D_{k,R} $ as well. Let $ f_{l} $ be the values of these integrals in $ z_{0}\in D_{k,R_{1}} $ (here $ R_{1}\gg0 $). Since $ z_{0}\in\overset{\,\,{}_\circ}{D}_{k+1} $, it is in the closure of $ \overset{\,\,{}_\circ}{D}_{k+1,R} $ for an appropriate $ R $. It is easy to see that $ f^{\left(k\right)}\left(z\right) $ has a limit $ f_{k} $ when $ z \to z_{0} $ along $ \overset{\,\,{}_\circ}{D}_{k+1,R} $. Same is true for $ f^{\left(l\right)}\left(z\right) $, $ l\leq k $. Consider the integral for \begin{equation} \frac{f\left(z\right)dz^{-1/2} - \left(f_{0}+f_{1}\left(z-z_{0}\right)+f_{2}\left(z-z_{0}\right)^{2}+\dots +f_{k}\left(z-z_{0}\right)^{k}\right)}{\left(z-z_{0}\right)^{k}}. \notag\end{equation} It is \begin{equation} \int_{\gamma}K_{k}\left(z,z_{0},\zeta\right)f\left(\zeta\right), \notag\end{equation} here \begin{align} K\left(z,\zeta\right) & =\frac{d\zeta^{1/2}}{\zeta-z}, \notag\\ K_{k}\left(z,z_{0},\zeta\right) & =\frac{K\left(z,\zeta\right)-\sum_{l=0}^{k}\frac{d^{k}K}{dz^{k}}|_{z=z_{0}}\frac{\left(z-z_{0}\right)^{k}}{k!}}{\left(z-z_{0}\right)^{k}}. \notag\\ & = \frac{z-z_{0}}{\left(\zeta-z_{0}\right)^{k+1}}K\left(z,\zeta\right) = \frac{1}{\left(\zeta-z_{0}\right)^{k}}\left(K\left(z,\zeta\right)-K\left(z_{0},\zeta\right)\right). \notag\end{align} We need to show that this integral goes to 0 when $ z \to z_{0} $ along $ \overset{\,\,{}_\circ}{D}_{k+1,R} $. Consider a function $ \rho\left(r\right) $ such that $ \lim _{r\to0}\rho\left(r\right)=0 $, $ \lim _{r\to0}\rho\left(r\right)/r=\infty $. Here $ \gamma_{r} $ is the intersection of $ \gamma $ with the disk of radius $ r $ about $ z_{0} $, instead of $ \widetilde{r}_{\gamma}\left(f\right)\in L_{2}\left(\gamma\right) $ we write just $ f $. Break the integral into three parts: two (which we do not want to separate yet) along $ \gamma_{\rho\left(|z-z_{0}|\right)} $, the other along $ \gamma\smallsetminus\gamma_{\rho\left(|z-z_{0}|\right)} $. Since \begin{equation} K_{k}= \frac{z-z_{0}}{\zeta-z_{0}}\frac{K\left(z,\zeta\right)}{\left(\zeta-z_{0}\right)^{k}} = o\left(\frac{K\left(z_{0},\zeta\right)}{|\zeta-z_{0}|^{k}}\right) \notag\end{equation} along the second part of $ \gamma $ if $ |z-z_{0}| \to $ 0, and since \begin{equation} \frac{K\left(z_{0},\zeta\right)}{|z_{0}-\zeta|^{k}}\in L_{2}\left(\gamma\right) \notag\end{equation} as a function of $ \zeta $, it is enough to show that the first two part of the integral go to 0. Since $ \lim _{r\to0} \int_{\gamma_{\rho\left(r\right)}}\left|f\left(\zeta\right)\right|^{2}=0 $, we need to show only that the $ L_{2} $-norm of $ K_{k}\left(z,z_{0},\zeta\right) $ is bounded when $ \zeta\in\gamma_{\rho\left(|z-z_{0}|\right)} $. Subdivide $ \gamma_{\rho\left(|z-z_{0}|\right)} $ once more: into $ \gamma_{\varepsilon|z-z_{0}|} $ and $ \gamma_{\rho\left(|z-z_{0}|\right)}\smallsetminus\gamma_{\varepsilon|z-z_{0}|} $. Here $ \varepsilon\ll1 $. Since on the second part $ K\left(z_{0},\zeta\right)=O\left(K\left(z,\zeta\right)\right) $, we see that $ K_{k}\left(z,z_{0},\zeta\right) = O\left(\frac{K\left(z,\zeta\right)}{\left(\zeta-z\right)^{k}}\right) $, thus $ K_{k} $ has a bounded $ L_{2} $-norm. We conclude that the integral along the second part goes to 0 when $ z \to z_{0} $. On the first part $ K\left(z,\zeta\right)=O\left(K\left(z_{0},\zeta\right)\right) $, thus \begin{equation} K_{k}\left(z,z_{0},\zeta\right) = \frac{1}{\left(\zeta-z_{0}\right)^{k}}\left(K\left(z,\zeta\right)-K\left(z_{0},\zeta\right)\right) = O\left(\frac{K\left(z_{0},\zeta\right)}{\left(\zeta-z_{0}\right)^{k}}\right)=O\left(\frac{1}{\left(z-z_{0}\right)^{k+1}}\right), \notag\end{equation} so it has a bounded $ L_{2} $-norm as well. \end{proof} \begin{definition} Consider a model $ \bar{M} $ of a curve. Say that a point $ z\in M_{\infty} $ is of smoothness $ C^{k} $, if $ z\in{\cal K}^{\left(k\right)} $, here $ {\cal K}=M_{\infty} $. \end{definition} \begin{remark} Note that the stratification we used is related to the following inclusion of $ M $ into $ L_{2}\left(\gamma,\Omega^{1/2}\otimes\mu\right) $: \begin{equation} z \mapsto \widetilde{r}_{\gamma}\left(K_{z}\right), \notag\end{equation} here $ K_{z}\left(\zeta\right)=\frac{d\zeta^{1/2}}{\zeta-z} $ is the Cauchy kernel. In other words, \begin{equation} z \mapsto \frac{d\zeta^{1/2}}{\zeta-z}|_{\gamma} \notag\end{equation} after a choice of coordinate $ \zeta $ on $ {\Bbb C}P^{1} $. The points of smoothness $ C^{0} $ correspond to limit points of this inclusion, the points of smoothness $ C^{k} $ correspond to limit points of $ k $-jets continuation of this mapping. \end{remark} \section{Riemann--Roch theorems } \subsection{Abstract Riemann--Roch theorem } We say that two vector subspaces $ V_{1} $, $ V_{2} $ of a topological vector space $ H $ {\em satisfy the Riemann\/}--{\em Roch theorem\/} if $ \dim V_{1}\cap V_{2}<\infty $, and $ \operatorname{codim}\left(\overline{V_{1}+V_{2}}\right)<\infty $. We call the number \begin{equation} \dim V_{1}\cap V_{2} - \operatorname{codim}\left(\overline{V_{1}+V_{2}}\right) \notag\end{equation} the {\em index\/} of two subspaces. If $ V_{1}+V_{2}=\overline{V_{1}+V_{2}} $, we say that $ V_{1},V_{2} $ satisfy the {\em strong form\/} of the theorem. \begin{remark} Note that if $ V_{1} $, $ V_{2} $ satisfy the strong form of the theorem, then the natural mapping $ V_{1} \to V/V_{2} $ is a Fredholm mapping with the index being the index of $ V_{1} $, $ V_{2} $. If $ V_{1} $, $ V_{2} $ satisfy the weak form of the theorem, then this mapping is a continuous mapping $ p $ with $ \dim \operatorname{Ker} p - \dim \operatorname{Coker} p $ being the index of $ V_{1},V_{2} $. Here $ \operatorname{Coker} $ is the quotient by the closure of the image. \end{remark} Consider a direct sum of two Hilbert spaces $ H=H_{1}\oplus H_{2} $. Consider two closed vector subspaces $ V_{1},V_{2}\subset H $ such that the projection of $ V_{i} $ on $ H_{i} $ has no null-space and a dense image. This means that one can consider $ V_{1} $ as a graph of a mapping $ A_{1}\colon H_{1} \to H_{2} $, similarly $ V_{2} $ is a graph of $ A_{2}\colon H_{2} \to H_{1} $. Mappings $ A_{1,2} $ are closed, but not necessarily bounded. \begin{lemma}[abstract finiteness] If $ A_{2} $ is bounded, and $ A_{1}\circ A_{2} $ is compact, then $ V_{1}\cap V_{2} $ is finite dimensional. \end{lemma} \begin{proof} The projection of $ V_{1}\cap V_{2} $ to $ H_{2} $ is a subspace of $ \operatorname{Ker}\left(A_{1}\circ A_{2}-\boldsymbol1\right) $, thus is finite-dimensional. \end{proof} \begin{proposition}[strong form] If $ A_{1} $, $ A_{2} $ are bounded, and $ A_{1}\circ A_{2} $ is compact, then $ V_{1} $ and $ V_{2} $ satisfy the strong form of Riemann--Roch theorem with index 0. \end{proposition} \begin{proof} The projection of $ V_{1}\cap V_{2} $ to $ H_{2} $ is $ \operatorname{Ker}\left(A_{1}\circ A_{2}-\boldsymbol1\right) $, which implies the statement about $ \dim V_{1}\cap V_{2} $. The statement about $ \operatorname{codim}\overline{V_{1}+V_{2}} $ follows from the fact that the orthogonal complements to $ V_{1} $ and $ V_{2} $ satisfy the same conditions as $ V_{2} $ and $ V_{1} $ with linear mappings being $ -A_{1}^{*} $, $ -A_{2}^{*} $. To show that $ \dim V_{1}\cap V_{2} = \operatorname{codim}\left(\overline{V_{1}+V_{2}}\right) $ note that $ \operatorname{Ker}\left(A_{1}\circ A_{2}-\boldsymbol1\right) $ is dual to $ \operatorname{Coker}\left(A_{2}^{*}\circ A_{1}^{*}-\boldsymbol1\right) $. \end{proof} \begin{proposition}[weak form] If $ A_{2} $ is bounded, and both $ A_{1}\circ A_{2} $ and $ A_{1}^{*}\circ A_{2}^{*} $ are compact, then $ V_{1} $ and $ V_{2} $ satisfy the Riemann--Roch theorem with index 0. \end{proposition} \begin{proof} We already know that $ \dim V_{1}\cap V_{2} $ and $ \operatorname{codim} \overline{V_{1}+V_{2}} $ are finite. The only thing to prove is that \begin{equation} \dim \operatorname{Ker}\left(A_{1}\circ A_{2}-\boldsymbol1\right) = \dim \operatorname{Coker}\left(A_{1}^{*}\circ A_{2}^{*}-\boldsymbol1\right). \notag\end{equation} Obviously, $ \left(A_{1}^{*}\circ A_{2}^{*}\right)^{*} $ is the closure of $ A_{2}\circ A_{1} $, thus \begin{equation} A_{2}\left(\operatorname{Ker}\left(A_{1}\circ A_{2}-\boldsymbol1\right)\right) \subset \operatorname{Ker}\left(\left(A_{1}^{*}\circ A_{2}^{*}\right)^{*}-\boldsymbol1\right), \notag\end{equation} hence \begin{equation} \dim \operatorname{Ker}\left(A_{1}\circ A_{2}-\boldsymbol1\right) \leq \dim \operatorname{Ker}\left(\left(A_{1}^{*}\circ A_{2}^{*}\right)^{*}-\boldsymbol1\right) = \dim \operatorname{Coker}\left(A_{1}^{*}\circ A_{2}^{*}-\boldsymbol1\right). \notag\end{equation} Application of the same argument to the dual operators shows the opposite unequality. \end{proof} We say that two vector subspaces $ V $, $ V' $ of a vector space $ H $ are {\em comparable}, if $ V\cap V' $ is of finite codimension in both $ V $ and $ V' $. The {\em relative dimension\/} $ \operatorname{reldim}\left(V,V'\right) $ is $ \operatorname{codim}\left(V\cap V'\subset V\right)-\operatorname{codim}\left(V\cap V'\subset V'\right) $. The following theorem is a direct corollary of the above statement: \begin{theorem}[Riemann--Roch theorem] \label{th6.50}\myLabel{th6.50}\relax Consider two vector subspaces $ V_{1,2}\subset H $ of a Hilbert space $ H=H_{1}\oplus H_{2} $. Suppose that $ V_{1} $ is comparable with the graph of a closed mapping $ A_{1}\colon H_{1} \to H_{2} $ and the relative dimension of $ V_{1} $ and this graph is $ d_{1} $. Suppose $ V_{2} $ is comparable with the graph of a closed mapping $ A_{2}\colon H_{2} \to H_{1} $ and the relative dimension of $ V_{2} $ and this graph is $ d_{2} $. \begin{enumerate} \item {\bf(weak form) }If $ A_{2} $ is bounded, and both $ A_{1}\circ A_{2} $ and $ A_{1}^{*}\circ A_{2}^{*} $ are compact, then $ V_{1} $ and $ V_{2} $ satisfy the Riemann--Roch theorem with the index being $ d_{1}+d_{2} $. \item {\bf(strong form) }If both $ A_{1} $ and $ A_{2} $ are bounded, and $ A_{1}\circ A_{2} $ is compact, then $ V_{1} $ and $ V_{2} $ satisfy the strong form of Riemann--Roch theorem with the index being $ d_{1}+d_{2} $. \end{enumerate} \end{theorem} \subsection{Riemann problem }\label{s5.5}\myLabel{s5.5}\relax Consider a holomorphic function $ \psi\left(z\right) $ defined in a annulus $ U=\left\{z \mid 1-\varepsilon\leq|z|\leq1+\varepsilon\right\} $, let $ S^{1}=\left\{z \mid |z|=1\right\} $. Suppose that $ \psi\left(z\right) $ is nowhere 0, and consider the subspace $ V_{\psi} $ in $ L_{2}\left(S^{1},\omega^{1/2}\right)\oplus L_{2}\left(S^{1},\omega^{1/2}\right) $ consisting of pairs of the form $ \left(f\left(z\right),\psi\left(z\right)f\left(z\right)\right) $. The Hilbert space $ L_{2}\left(S^{1},\omega^{1/2}\right) $ is a direct sum of subspaces $ L_{2}^{\pm}\left(S^{1},\omega^{1/2}\right) $ consisting of forms which can be holomorphically continued into two regions $ S^{1} $ divides $ {\Bbb C}P^{1} $ into. Let $ p $ be the projection of $ V_{\psi} $ to \begin{equation} L_{2}^{+}\left(S^{1},\omega^{1/2}\right)\oplus L_{2}^{-}\left(S^{1},\omega^{1/2}\right)\subset L_{2}\left(S^{1},\omega^{1/2}\right)\oplus L_{2}\left(S^{1},\omega^{1/2}\right). \notag\end{equation} \begin{lemma} If $ \operatorname{ind}\psi=0 $, then $ p $ is invertible. \end{lemma} \begin{proof} Consider a linear bundle $ {\cal L} $ over $ {\Bbb C}P^{1} $ with isomorphisms to $ \omega^{1/2} $ over $ U^{+}=\left\{|z|<1+\varepsilon\right\} $ and $ U^{-}=\left\{|z|>1-\varepsilon\right\} $, and the gluing data being $ l^{+}=\psi l^{-} $. Since $ \deg {\cal L}=\deg \omega^{1/2}+\operatorname{ind}\psi $, and a linear bundle over $ {\Bbb C}P^{1} $ is determined by its degree up to an isomorphism, we see that $ {\cal L}\simeq\omega^{1/2} $ if $ \operatorname{ind}\psi=0 $. Since $ \operatorname{Ker} p $ consists of global sections of $ {\cal L} $, $ \operatorname{Ker} p=\left\{0\right\} $. Similarly, consideration of orthogonal complement to $ V_{\psi} $ shows that $ \operatorname{Coker} p=\left\{0\right\} $. This finishes the proof, since the operator is obviously Fredholm. \end{proof} Consider now another linear bundle $ {\cal L}' $ with trivializations over $ U^{+} $ and over $ U^{-} $ with gluing data $ l^{+}=\psi l^{-} $. The same arguments as above show that $ {\cal L}' $ is trivial, thus it has a (unique up to multiplication by a constant) global section. This means that there are functions $ l^{\pm} $ defined on $ U^{\pm} $ such that $ l^{+}=\psi l^{-} $. Since this global section has no zeros, $ l^{\pm} $ have no zeros inside the domain of definition, thus $ \psi=\left(l^{+}\right)^{-1}l^{-} $. We see that any function $ \psi $ such that $ \operatorname{ind}\psi=0 $ can be represented as a product $ \psi=\psi_{+}\psi_{-} $ of a parts $ \psi_{+} $, $ \psi_{-} $ which can be holomorphically extended inside/outside a circle without zeros. Let as write the mapping $ p^{-1} $ in terms of $ \psi_{\pm} $. For any $ L_{2} $-section $ \omega $ of $ \Omega^{1/2}\otimes\mu $ on $ \left\{|z|=1\right\} $ let $ \omega=\omega_{+}+\omega_{-} $ be the (unique) decomposition of $ \omega $ into a sum of forms which can holomorphically extended inside/outside of the unit circle. Given $ \omega_{+} $ and $ \omega_{-}' $ we want to find $ \omega_{-} $ and $ \omega'_{+} $ from the equality \begin{equation} \psi\left(\omega_{+}+\omega_{-}\right) = \omega'_{+}+\omega'_{-},\qquad \text{or\qquad }\psi_{-}\left(\omega_{+}+\omega_{-}\right) = \psi_{+}^{-1}\left(\omega'_{+}+\omega'_{-}\right). \notag\end{equation} Taking the $ + $-part we see that $ \left(\psi_{-}\omega_{+}\right)_{+} = \psi_{+}^{-1}\omega'_{+}+\left(\psi_{+}^{-1}\omega'_{-}\right)_{+} $, thus \begin{equation} \omega'_{+} = \psi_{+}\left(\psi_{-}\omega_{+}\right)_{+} - \psi_{+}\left(\psi_{+}^{-1}\omega'_{-}\right)_{+} = \psi\omega_{+} - \psi_{+}\left(\psi_{-}\omega_{+}\right)_{-} - \psi_{+}\left(\psi_{+}^{-1}\omega'_{-}\right)_{+}, \notag\end{equation} similarly $ \left(\psi_{-}\omega_{+}\right)_{-}+\psi_{-}\omega_{-} = \left(\psi_{+}^{-1}\omega'_{-}\right)_{-} $, thus \begin{equation} \omega_{-} = \psi_{-}^{-1}\left(\psi_{+}^{-1}\omega'_{-}\right)_{-}-\psi_{-}^{-1}\left(\psi_{-}\omega_{+}\right)_{-} = \psi^{-1}\omega'_{-} - \psi_{-}^{-1}\left(\psi_{+}^{-1}\omega'_{-}\right)_{+} - \psi_{-}^{-1}\left(\psi_{-}\omega_{+}\right)_{-}. \notag\end{equation} These two formulae express $ \omega_{-} $ and $ \omega'_{+} $ in terms of $ \omega'_{-} $ and $ \omega_{+} $, thus give an inverse mapping to $ p $. Let $ |\psi|_{++}=\frac{\max |\psi_{+}|}{\min |\psi_{+}|} $, $ |\psi|_{--}=\frac{\max |\psi_{-}|}{\min |\psi_{-}|} $, $ |\psi|_{+-}=\max |\psi_{+}| \max |\psi_{-}| $, $ |\psi|_{-+}=\max |\psi_{+}|^{-1} \max |\psi_{-}|^{-1} $, $ |\psi|_{0}=\max \left(|\psi|_{++},|\psi|_{--},|\psi|_{+-},|\psi|_{-+}\right) $. Then the norm of $ p^{-1} $ is bounded by $ C\cdot|\psi|_{0} $. \begin{definition} \label{def5.155}\myLabel{def5.155}\relax The {\em Riemann norm\/} $ \|\psi\|_{{\bold R}} $ of $ \psi $ is the norm of the operator $ p^{-1} $. \end{definition} The following lemma is a corollary of the fact that one can find factorization $ \psi=\psi_{+}\psi_{-} $ using integral operators applied to $ \log \psi $: \begin{lemma} \label{lm5.160}\myLabel{lm5.160}\relax Let $ \log \psi\left(z\right) $ is defined using any branch of logarithm. Then \begin{equation} \|\psi\|_{{\bold R}}< C \exp C \max _{z\in U} |\log \psi\left(z\right)| \notag\end{equation} for an appropriate $ C $ (which depends on $ \varepsilon $ only). \end{lemma} Let $ \Pi_{\psi} $ be the composition of $ p^{-1} $ with the projection of $ L_{2}\left(S^{1},\omega^{1/2}\right)\oplus L_{2}\left(S^{1},\omega^{1/2}\right) $ to $ L_{2}^{-}\left(S^{1},\omega^{1/2}\right)\oplus L_{2}^{+}\left(S^{1},\omega^{1/2}\right) $, \begin{equation} \Pi_{\psi}\colon L_{2}^{+}\left(S^{1},\omega^{1/2}\right)\oplus L_{2}^{-}\left(S^{1},\omega^{1/2}\right) \to L_{2}^{-}\left(S^{1},\omega^{1/2}\right)\oplus L_{2}^{+}\left(S^{1},\omega^{1/2}\right). \notag\end{equation} In other words, $ \left(\left(f_{+},g_{+}\right),\left(f_{-},g_{-}\right)\right) $ lies on the graph of $ \Pi_{\psi} $ if $ g_{+}+g_{-}=\psi\left(f_{+}+f_{-}\right) $. We see that the norm of $ \Pi_{\psi} $ is bounded by $ C\cdot|\psi|_{0} $. Moreover, $ \Pi_{\psi} $ can be written as a sum \begin{equation} \Pi_{\psi} = \left( \begin{matrix} 0 & \psi^{-1} \\ \psi & 0 \end{matrix} \right) + k. \label{equ5.52}\end{equation}\myLabel{equ5.52,}\relax Obviously, $ k $ is compact (as any Hankel operator). Indeed, components of $ k $ look like $ f_{+} \mapsto \left(mf_{+}\right)_{-} $. The Hilbert operator $ f=f_{+}+f_{-} \buildrel{H}\over{\mapsto} f_{+}-f_{-} $ is a pseudodifferential operator of degree 0, thus components of $ K $ may be written as $ \left[m,H\right] $, thus are pseudodifferential operators of degree $ -1 $, thus compact. \begin{lemma} \label{lm5.60}\myLabel{lm5.60}\relax Consider an invertible function $ \psi $ defined in a neighborhood of $ |z|=1 $ and having $ \operatorname{ind}=k $. Let $ V_{\psi}\subset H=L_{2}\left(S^{1}\right)\oplus L_{2}\left(S^{1}\right) $ be \begin{equation} V_{\psi}=\left\{\left(f_{1},f_{2}\right) \mid f_{2}=\psi f_{1}\right\}. \notag\end{equation} Define $ H^{\pm}=L_{2}\left(S^{1}\right)^{\pm}\oplus L_{2}\left(S^{1}\right)^{\mp} $. Then for an appropriate bounded operator $ \pi\colon H^{+} \to H^{-} $ the $ \operatorname{graph}\left(\pi\right) $ is compatible with $ V_{\psi} $, and $ \operatorname{reldim}\left(H_{2},\operatorname{graph}\left(\pi\right)\right)=k $. \end{lemma} \begin{proof} Let $ \psi\left(z\right)=z^{k}\psi_{0}\left(z\right) $. The function $ \psi_{0} $ has $ \operatorname{ind}=0 $, thus the corresponding subspace $ V_{\psi_{0}} $ is the graph of $ \Pi_{\psi_{0}} $. Let $ H_{0}^{\pm}=L_{2}\left(S^{1}\right)^{\pm}\oplus z^{k}L_{2}\left(S^{1}\right)^{\mp} $. We see that $ V_{\psi} $ is a graph of a bounded mapping $ H_{0}^{+} \to H_{0}^{-} $. Since $ H_{0}^{+} $ is compatible with $ H_{0} $ of relative dimension $ k $, we momentarily obtain the required statement about $ V_{\psi} $. \end{proof} \begin{remark} Note that $ \|\pi\| $ may be bounded in the same way as in Lemma ~\ref{lm5.160}, but neither this result, nor~\eqref{equ5.52} are going to be needed in what follows. \end{remark} \subsection{Finiteness theorem }\label{s7.30}\myLabel{s7.30}\relax Consider a family of disks and gluing conditions $ K_{\bullet} $, $ \varphi_{\bullet} $, $ \psi_{\bullet} $ from Section~\ref{s5.30}. Let $ {\cal H}^{\pm}=\bigoplus_{l_{2}}L_{2}^{\pm}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $, $ i\in I $. Then the operator with Cauchy kernel defines a Hilbert mapping $ {\bold K}\colon {\cal H}^{-} \to {\cal H}^{+} $ (see Section~\ref{s5.20}). This mapping depends on the circles $ \partial K_{i} $ only, not on the gluing conditions $ \varphi_{\bullet} $, $ \psi_{\bullet} $. Since the Hilbert structure on $ L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $ is invariant w.r.t. fraction-linear mappings (as is decomposition into $ \pm $-parts), the identification $ \varphi_{i}\colon \partial K_{i'} \to \partial K_{i} $ gives an isomorphism of Hilbert spaces $ L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $ and $ L_{2}\left(\partial K_{i'},\Omega^{1/2}\otimes\mu\right) $ which interchanges $ + $-part and $ - $-part. Suppose that $ \operatorname{ind}\psi_{i}=0 $. Then the operator $ \Pi_{\psi_{i}} $ from Section~\ref{s5.5} together with an identification given by $ \varphi_{i} $ gives a mapping $ \Pi_{\varphi_{i},\psi_{i}} $ \begin{equation} L_{2}^{+}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right)\oplus L_{2}^{+}\left(\partial K_{i'},\Omega^{1/2}\otimes\mu\right) \to L_{2}^{-}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right)\oplus L_{2}^{-}\left(\partial K_{i'},\Omega^{1/2}\otimes\mu\right), \notag\end{equation} the graph of this mapping consists of half-forms on $ \partial K_{i} $ and $ \partial K_{i'} $ which differ by multiplication by $ \psi_{i} $. In other words, $ \left(\left(f_{+},g_{+}\right),\left(f_{-},g_{-}\right)\right) $ lies on this graph if \begin{equation} \left(g_{+}+g_{-}\right)\left(\varphi_{i}^{-1}\left(z\right)\right)=\psi_{i}\left(z\right)\left(f_{+}+f_{-}\right)\left(z\right),\qquad z\in\partial K_{i}. \notag\end{equation} If $ \operatorname{ind}\psi_{i}\not=0 $, instead of $ \Pi_{\varphi_{i},\psi_{i}} $ consider an arbitrary operator between the same spaces such that the graph of $ P $ is comparable with the set of half-forms which differ by multiplication by $ \psi_{i} $ (see Lemma ~\ref{lm5.60}). The only fact important in what follows is that this is a bounded operator (see Corollary~\ref{cor4.35}). Let $ \Pi=\Pi_{\left\{\psi\right\}}=\bigoplus_{i\in I_{+}}\Pi_{\varphi_{i},\psi_{i}}\colon {\cal H}^{+} \to {\cal H}^{-} $. Similarly, let $ \Pi_{\left\{\psi^{-1}\right\}} $ corresponds to gluing data $ \varphi_{i} $, $ \psi_{i}^{-1} $. Since $ \Pi $ consists of bounded diagonal blocks, it is a closed operator. Let $ \bar{M} $ be a curve determined by gluing conditions $ \varphi_{\bullet} $, $ {\cal L} $ be a bundle on $ \bar{M} $ determined by gluing conditions $ \psi_{\bullet} $. \begin{theorem} Suppose that the disks $ K_{i} $ are well separated, $ {\cal L} $ is a finite-degree bundle, and $ \Pi_{\left\{\psi\right\}}\circ{\bold K} $ is compact. Then both $ \Gamma_{\text{strong}}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ and $ \Gamma_{\text{weak}}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ are finite-dimensional. \end{theorem} \begin{proof} We give only a sketch of a proof, since the details are the same as in the case of Riemann--Roch theorem (see Section~\ref{s5.60}). Since $ \Gamma_{\text{strong}} $ is identified with a subspace of $ \Gamma_{\text{weak}} $, it is enough to show that $ \dim \Gamma_{\text{weak}}<\infty $. On the other hand, $ \Gamma_{\text{weak}} $ is defined as an intersection of the generalized Hardy space $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ with the subspace of forms which satisfy the gluing conditions. We are going to reduce the statement of the theorem to the abstract finiteness theorem. To do this, note that Proposition~\ref{prop5.28} identifies $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ with the graph of $ {\bold K}\colon {\cal H}^{-} \to {\cal H}^{+} $, thus the only thing to note is that fact that the subspace of forms which satisfy the gluing conditions is compatible with the graph of $ \Pi_{\left\{\psi\right\}} $. \end{proof} \subsection{Duality }\label{s7.40}\myLabel{s7.40}\relax Consider the curve $ \bar{M} $ and a sheaf $ {\cal L} $ from the previous section and the dual sheaf $ {\cal L}^{-1} $ with inverse gluing conditions $ \left\{\psi_{i}^{-1}\right\} $. \begin{theorem} \label{th5.70}\myLabel{th5.70}\relax Suppose that the disks $ K_{i} $ are well-separated, and $ {\cal L} $ is of finite degree. The mappings \begin{equation} H^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \xrightarrow[]{\bar{\partial}} H^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) \notag\end{equation} and \begin{equation} H^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}^{-1}\right) \xrightarrow[]{-\bar{\partial}} H^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}^{-1}\otimes\bar{\omega}\right) \notag\end{equation} are mutually dual, thus dimension of null-space of one mapping is equal to the dimension of cokernel of another one. \end{theorem} \begin{proof} This is a direct corollary of Lemmas~\ref{lm5.22} and~\ref{lm5.25}. \end{proof} Using Theorems~\ref{th5.70} and~\ref{th5.31} together with selfduality of $ {\bold K} $ and the fact that $ \Pi_{\psi}^{t}=\Pi_{\psi^{-1}} $, we obtain \begin{corollary} Suppose that the disks $ K_{i} $ are well separated, and $ {\cal L} $ is finite-degree and semibounded, and both $ \Pi_{\left\{\psi\right\}}\circ{\bold K} $ and $ \Pi_{\left\{\psi^{-1}\right\}}\circ{\bold K} $ are compact. Then the bounded operator \begin{equation} H_{\text{weak}}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \xrightarrow[]{\bar{\partial}} H_{\text{weak}}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) \notag\end{equation} has finite-dimensional null-space and cokernel. \end{corollary} \subsection{Riemann--Roch for curves }\label{s5.60}\myLabel{s5.60}\relax Now we are ready to state \begin{theorem}[Riemann--Roch] \label{th5.15}\myLabel{th5.15}\relax Suppose that the disks $ K_{i} $ are well-separated, and $ \psi_{i} $ is a finite-degree family. \begin{enumerate} \item If both $ \Pi_{\left\{\psi\right\}}\circ{\bold K} $ and $ \Pi_{\left\{\psi^{-1}\right\}}\circ{\bold K} $ are compact operators then the mapping \begin{equation} H_{\text{weak}}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \xrightarrow[]{\bar{\partial}} H_{\text{weak}}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) \notag\end{equation} has finite-dimensional null-space and cokernel, and \begin{equation} \dim \operatorname{Ker} \bar{\partial} - \dim \operatorname{Coker} \bar{\partial} = \sum_{i\in I_{+}} \operatorname{ind} \psi_{i}. \notag\end{equation} \item If $ \Pi_{\left\{\psi\right\}} $ is bounded, and $ {\bold K} $ is compact, then the mapping is Fredholm of index $ \sum_{i\in I_{+}} \operatorname{ind} \psi_{i} $. \end{enumerate} \end{theorem} \begin{proof} The only thing we need to do is to describe the null-space and the image of $ \bar{\partial} $. We are going to reduce this description to the abstract Riemann--Roch theorem. First of all, the null-space consists of half-forms in the generalized Hardy space, so $ \widetilde{r} $ maps it injectively to a subspace of $ \bigoplus_{l_{2}}L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $. On the other hand, the image of $ \operatorname{Ker}\bar{\partial} $ in $ \bigoplus_{l_{2}}L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $ is described as intersection of the image of the generalized Hardy space and a subspace in $ \bigoplus_{l_{2}}L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $ consisting of half-forms which satisfy the gluing conditions. The first condition can be written as \begin{equation} f_{i}^{+}=\sum_{j\not=i}{\bold K}_{ij}f_{j}^{-}, \notag\end{equation} here $ f_{i}^{\pm} $ are $ \pm $-parts of the component of $ f $ in $ L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $. The second condition is \begin{equation} f'_{i'}=\psi_{i}f_{i}. \notag\end{equation} Here $ f\left(t\right)'=f\left(-t\right) $, and we suppose that we use compatible parameterizations of $ \partial K_{i} $ and of $ \partial K_{i'} $, i.e., such that $ \varphi_{i} $ send parameter $ t $ on $ \partial K_{i} $ to parameter $ -t $ on $ \partial K_{i'} $. Consider the decomposition $ {\cal H}={\cal H}^{+}\oplus{\cal H}^{-} $. Vectors which satisfy the first condition are in the graph of operator $ {\bold K}\colon {\cal H}^{-} \to {\cal H}^{+} $. Consider the vector space of vectors $ {\cal H}_{2} $ which satisfy the second conditions. It is a direct sum over $ i\in I_{+} $ of subspaces of \begin{equation} L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right)\oplus L_{2}\left(\partial K_{i'},\Omega^{1/2}\otimes\mu\right), \notag\end{equation} and all the components but a finite number have $ \psi $ with $ \operatorname{ind}=0 $, thus are described as graphs of $ \Pi_{\varphi_{i},\psi_{i}} $. As a corollary, we conclude that this subspace is compatible with the graph of the mapping $ \Pi $. To finish the description of the null-space the only thing which remains to prove is to show that the relative dimension of $ {\cal H}_{2} $ and $ \operatorname{graph}\left(\Pi\right) $ is $ \sum_{i\in I_{+}}\operatorname{ind}\psi_{i} $. However, because of decomposition of $ {\cal H}_{2} $ into a direct sum it follows from the corresponding fact for each component, i.e., from Lemma~\ref{lm5.60}. To describe the image, consider \begin{equation} \alpha\in H_{\text{weak}}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) = H^{-1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\omega^{1/2}\otimes\bar{\omega}\right). \notag\end{equation} Since $ \bar{\partial} $ is an isomorphism \begin{equation} H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) \to H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right), \notag\end{equation} the element $ \bar{\partial}^{-1}\alpha\in H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) $ is defined up to addition of an element of $ \bar{\partial}^{-1} \overset{\,\,{}_\circ}{H}^{-1/2}\left(\bigcup K_{i},\omega^{1/2}\otimes\bar{\omega}\right) $. On the other hand, the latter space is $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $, i.e., $ \bar{\partial}^{-1} $ gives a correctly defined isomorphism \begin{equation} H^{-1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\omega^{1/2}\otimes\bar{\omega}\right) \to H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right)/{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right), \notag\end{equation} hence $ \bar{\partial}^{-1}\alpha $ is an element of the latter space. Thus \begin{equation} \alpha\in\operatorname{Im}\left(H_{\text{weak}}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \xrightarrow[]{\bar{\partial}} H_{\text{weak}}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right)\right) \notag\end{equation} is equivalent to \begin{equation} \partial^{-1}\alpha\in {\cal H}_{2}/{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) \buildrel{\text{def}}\over{=}\left({\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right)+{\cal H}_{2}\right)/{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right)\text{ .} \notag\end{equation} In particular, if $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right)+{\cal H}_{2} $ is closed, then $ \operatorname{Im} \bar{\partial} $ is closed, and in any case the codimension of the closure of $ \operatorname{Im}\bar{\partial} $ is equal to codimension of the closure of $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right)\oplus{\cal H}_{2} $. \end{proof} From Lemma~\ref{lm5.160} we momentarily obtain \begin{corollary} Suppose that $ {\bold K} $ is compact, and $ \psi_{i}=\exp \Phi_{i} $ for all $ i $ but a finite number. If for an appropriate $ C $ and any $ i $ \begin{equation} |\Phi_{i}| < C, \notag\end{equation} then the operator \begin{equation} H_{\text{weak}}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \xrightarrow[]{\bar{\partial}} H_{\text{weak}}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) \notag\end{equation} is Fredholm. \end{corollary} \begin{definition} Call a pair $ \left(\bar{M},{\cal L}\right) $ {\em admissible\/} if $ \bar{M} $ is given by a family of well-separated disks, $ {\cal L} $ is of finite degree, and both $ \Pi_{\left\{\psi\right\}}\circ{\bold K} $ and $ \Pi_{\left\{\psi^{-1}\right\}}\circ{\bold K} $ are compact. Call $ \bar{M} $ {\em admissible\/} if $ \left(\bar{M},\boldsymbol1\right) $ is admissible. Here {\bf1 }is a bundle over $ \bar{M} $ with $ \psi_{i}\equiv 1 $. \end{definition} \subsection{Criterion of admissibility of a curve }\label{s4.50}\myLabel{s4.50}\relax Recall that in Section ~\ref{s5.10} we considered a matrix $ \left(e^{-l_{ij}/2}\right) $ which was supposed to give a bounded operator in $ l_{2} $. \begin{proposition} If the matrix $ \left(e^{-l_{ij}/2}-\delta_{ij}\right) $ gives a compact operator $ l_{2} \to l_{2} $, then $ \bar{M} $ is admissible. \end{proposition} \begin{proof} Indeed, for the bundle $ {\cal L}=\boldsymbol1 $ the operator $ \Pi $ is an isometry, thus we need to show that $ {\bold K} $ is compact, which is a corollary of Lemma~\ref{lm5.05}. \end{proof} \begin{corollary} If $ \sum_{i\not=j}e^{-l_{ij}}<\infty $, then $ \bar{M} $ is admissible. \end{corollary} \begin{corollary} Fix a locally discrete subset $ I\subset{\Bbb C}P^{1} $ (i.e., for any point $ i $ of $ I $ there is a punctured neighborhood of $ i $ which does not intersect $ I $). Then there exists a family of disks $ K_{i} $, $ i\in I $ with $ \operatorname{center}\left(K_{i}\right)=i $ such that for any involution ' and any gluing data $ \varphi_{i} $ the corresponding curve $ \bar{M} $ is admissible. Moreover, one can chose $ K_{i} $ in such a way that $ M_{\infty} $ consists of points of smoothness $ C^{\infty} $ (see Section~\ref{s4.95}). \end{corollary} \begin{corollary} For an arbitrary nowhere dense subset $ N $ of $ {\Bbb C}P^{1} $ there exists an admissible curve $ \bar{M} $ such that $ M_{\infty}=N $. Moreover, it is possible to make every point of $ N $ to be of smoothness $ C^{\infty} $. \end{corollary} \begin{remark} While the above statements are obvious, note that construction of examples and counterexamples may be simplified a lot by an additional restriction: \begin{equation} \operatorname{dist}\left(i,j\right) \geq \varepsilon\cdot\operatorname{dist}\left(i,N\right)\qquad \text{for any }i,j\in I\text{, }i\not=j. \notag\end{equation} Here $ \varepsilon\ll1 $. To construct such a family $ I $ for a given nowhere dense set $ N $, let $ N_{k}=\left\{z \mid 2^{-k-1}\leq\operatorname{dist}\left(z,N\right)\leq2^{-k}\right\} $. Fix $ k $, and let $ \delta=2^{-k} $. Consider a $ \delta/8 $-net for $ N_{k} $. By removing some points from this net one can obtain $ \delta/4 $-net such that it does not have two points closer than $ \delta/8 $. This net is necessarily finite. Now consider the union of these finite sets over $ k\in{\Bbb N} $, and again remove net points from $ N_{k+1} $ which are closer than $ \delta/8 $ to net points in $ N_{k} $. One obviously obtains a set $ I $ with required properties. Now to chose the radius of $ K_{i} $ denote by $ n_{k} $ the number of points in $ I\cap N_{k} $. Let $ \operatorname{radius}\left(K_{i}\right)=\frac{f\left(k\right)}{n_{k}} $ if $ i\in N_{k} $, here $ f\left(k\right) $ is rapidly decreasing function. Picking appropriate $ f\left(k\right) $, one obtains disks which satisfy the given above requirements. \end{remark} The following property of admissibility is obvious: \begin{nwthrmiii} If we change a finite number of contours $ \partial K_{i} $ and/or a finite numbers of identifications $ \varphi_{i} $, this does not change the admissibility of the resulting curve. \end{nwthrmiii} \subsection{Filling the gap }\label{s7.70}\myLabel{s7.70}\relax This section contains heuristics only, so anyone interested exceptionally in exact results should proceed directly to Section~\ref{h8}. Consider once more the criterion of admissibility of a curve. It says that if the operator $ {\bold K} $ is compact, the curve is admissible (together with any bundle which is defined by gluing functions $ \psi_{i} $ with bounded Riemann norm $ |\psi|_{{\bold R}} $). On the other hand, to obtain this result we study the generalized Hardy space, which is correctly defined if $ {\bold K} $ is bounded. Thus the tools we use leave a gap between the objects for which the analysis is applicable (i.e., $ {\bold K} $ is bounded), and objects for which we get the admissibility (i.e., $ {\bold K} $ is compact). How can we use the existence of this gap? We propose to consider this gap as a confirmation that the Riemann--Roch theorem we obtained is {\em almost unimprovable}. Indeed, we know that the finiteness condition can be written as $ \dim \operatorname{Ker}\left(\Pi\circ{\bold K}-1\right)<\infty $. On the other hand, the invertible operator $ \Pi $ depends on the family $ \left(\psi_{i}\right) $ which has very high degree of freedom (even if we consider the strong form of Riemann--Roch theorem, so $ \Pi $ is required to be bounded), and one should expect that the condition \begin{equation} \dim \operatorname{Ker}\left(\Pi\circ{\bold K}-1\right)<\infty\text{ for every choice of }\psi_{i}\text{ with }|\psi_{\bullet}|_{{\bold R}}<\infty \notag\end{equation} should be very close to the condition \begin{equation} {\bold K}\text{ has discrete spectrum near }|\lambda|=1. \label{equ4.88}\end{equation}\myLabel{equ4.88,}\relax In other words, the pairs $ \left(\Pi,{\bold K}\right) $ with an invertible $ \Pi $, bounded $ {\bold K} $, and {\em infinite-dimensional\/} null-space of $ \Pi{\bold K}-1 $ are plentiful (at least if we drop ``geometric'' conditions on $ \Pi $ and $ {\bold K} $, and consider abstract operators), and they form a ``natural boundary'' of the set of curves for which Riemann--Roch theorem has a chance to be true. This natural boundary is quite close to the boundary of the set of compact operators, which is another confirmation of our thesis. Note that~\eqref{equ4.88} may lead to a different description of possible $ {\bold K} $, like $ \frac{{\bold K}}{1+{\bold K}^{2}} $ being compact. It is unclear, however, whether the above boundary separates the set of compact operators as a connected component of the set of operators $ {\bold K} $ with compact $ \frac{{\bold K}}{1+{\bold K}^{2}} $. \subsection{Moduli space } The description of a complex curve by disks $ \left\{K_{i}\right\} $, involution ' and gluings $ \varphi_{i} $ leaves a feeling of being incomplete, since in the case of finite genus it is enough to provide just gluings $ \varphi_{i} $ which generate a subgroup of $ \operatorname{SL}\left(2,{\Bbb C}\right) $. To describe the quotient by this subgroup one can take any fundamental domain for the subgroup, and different choices of the fundamental domain result in the same geometric data. To get a similar description in the case of infinite genus, note that in Section~\ref{s5.10} instead of the restriction that $ \widetilde{K}_{i} $ {\em is\/} a concentric with $ K_{i} $ disk of radius $ e^{2\varepsilon}\cdot\operatorname{radius}\left(K_{i}\right) $ one can require that $ \widetilde{K}_{i} $ {\em contains\/} such a circle. This leads to the construction of strong sections of $ \omega^{1/2}\otimes{\cal L} $ in the same way as in Section~\ref{s6.50}, and Theorem~\ref{th5.31} can be refined as \begin{amplification} Fix a metric on $ {\Bbb C}P^{1} $ and $ \varepsilon>0 $. Consider a family of elements $ \varphi_{i} $ of $ \operatorname{SL}\left(2,{\Bbb C}\right) $. Suppose that there exists a family of disjoint domains $ \widetilde{K}_{i}\subset{\Bbb C}P^{1} $, $ i\in I $, which satisfy the following properties: \begin{enumerate} \item All $ \widetilde{K}_{i} $ but a finite number are disks; \item Let $ K'_{i} $ be a concentric with $ \widetilde{K}_{i} $ disk of radius $ \left(1-\varepsilon\right)\operatorname{radius}\left(\widetilde{K}_{i}\right) $ (or any domain in $ \widetilde{K}_{i} $ if $ \widetilde{K}_{i} $ is not a disk). Suppose that for an involution ': $ I\to $I one has $ \varphi_{i'}=\varphi_{i}^{-1} $, $ \varphi_{i}\left(K'_{i}\right)\cup K'_{i'}={\Bbb C}P^{1} $; \item Let $ \psi_{i} $ be holomorphic functions defined inside $ \widetilde{K}_{i}\cap\varphi_{i}^{-1}\left(\widetilde{K}_{i'}\right) $ which satisfy $ \psi_{i'}\circ\varphi_{i}=\psi_{i}^{-1} $, and $ \operatorname{ind}\psi_{i}=0 $ for all but a finite number of $ i\in I $. \item Suppose that the pairwise conformal distances $ l_{ij} $ between $ \widetilde{K}_{i} $, $ \widetilde{K}_{j} $ satisfy the condition that the matrix $ \left(e^{-l_{ij}/2}-\delta_{ij}\right) $ gives a compact mapping $ l_{2} \to l_{2} $, and all the functions $ \psi_{i} $ are bounded taken together. \end{enumerate} Let $ \bar{M} $ be a curve given by gluing $ S=\bigcap\varphi_{i}^{-1}\left(K'_{i}\right) $ together via $ \varphi_{i} $, $ {\cal L} $ be a bundle on $ \bar{M} $ given by cocycle $ \psi_{i} $. Define strong sections of $ \omega^{1/2}\otimes{\cal L} $ associated with family $ \left\{\widetilde{K}_{i}\right\} $ as forms $ \alpha $ in $ H^{1/2}\left(\bigcap\varphi_{i}^{-1}\left(K'_{i}\right),\omega^{1/2}\right) $ which satisfy $ \varphi_{i}^{*}\left(\alpha\right)=\psi_{i}\cdot\alpha $ whenever both sides have sense. Let $ I=I_{+}\amalg I_{+}' $. For any choice of circles $ \gamma_{i} $ in $ K_{i}'\cap\varphi_{i}^{-1}\left(K'_{i'}\right) $, $ i\in I_{+} $, let $ K_{i} $ be the disk bounded by $ \gamma_{i} $ inside $ K'_{i} $ (with appropriate modifications if $ K_{i} $ is not a disk). Let $ K_{i'}={\Bbb C}P^{1}\smallsetminus\varphi_{i}\left(K_{i}\right) $, $ i\in I_{+} $. Then the space of strong sections of $ \omega^{1/2}\otimes{\cal L} $ coincides with the set of weak sections associated to the family $ \left\{K_{i}\right\} $. In particular, the Riemann--Roch theorem (in the strong form) is valid for strong sections associated to the family $ \left\{\widetilde{K}_{i}\right\} $. \end{amplification} The proof of this statement is a corollary of the proof of Theorem~\ref{th5.31}. We see that at least for bundles described by bounded cocycles one does not need to specify {\em precisely\/} the circles which cut the curve, one can vary them in wide ranges (which depend on the Kleinian group) without any change to the geometric data. One possible objection to usability of the above theorem is that for different choices $ \left\{\gamma_{i}^{\left(1\right)}\right\} $, $ \left\{\gamma_{i}^{\left(2\right)}\right\} $ of the circles the cocycles $ \psi_{i} $ which describe the bundle need to be defined everywhere between the circles for the theorem to be applicable. However, in Section~\ref{s7.90} we will see that any bounded bundle of degree 0 may be described by constant cocycles, thus this objection becomes void. (If the degree is not 0, one can take all the cocycles to be constant except one pair described by rational functions.) Call a Kleinian group {\em admissible\/} if it has generators $ \varphi_{i} $, $ i\in I_{+} $, which satisfy the conditions of the amplification. The following conjecture would show that the curve (with a fixed family of $ A $-cycles) is completely described by the corresponding Kleinian group, and the restriction on distances between $ \widetilde{K}_{i} $ is in fact the restriction on the Kleinian group. Consider an admissible Kleinian group. A {\em fundamental family\/} is a family of domains $ \widetilde{K}_{\bullet} $ which satisfies the conditions of amplification. If $ \widetilde{K}_{i}^{\left(1\right)}\subset\widetilde{K}_{i}^{\left(2\right)} $, $ i\in I $, then call these families {\em equivalent}, and continue this relation by transitivity. The amplification shows that equivalent families lead to the same spaces of sections of bundles. \begin{conjecture} Any two fundamental families for an admissible Kleinian group are equivalent. \end{conjecture} Let the {\em fine moduli space\/} be the set of admissible Kleinian groups up to conjugation in $ \operatorname{PGL}\left(2,{\Bbb C}\right) $. \begin{conjecture} Consider a complex curve $ M $. Let $ \gamma_{i}^{\left(1\right)} $, $ i\in I^{\left(1\right)} $, be a disjoint family of smooth embedded cycles in $ M $ such that $ M\smallsetminus\bigcup\gamma_{i} $ is conformally equivalent to a fundamental domain of an admissible Kleinian group. Let $ \gamma_{i}^{\left(2\right)} $ be a different family which satisfies same conditions. Then all the cycles $ \gamma_{i}^{\left(1\right)} $ except a finite number are homotopic to cycles in $ \gamma_{i}^{\left(2\right)} $. \end{conjecture} \begin{conjecture} Two families of cuts from the previous conjecture lead to the same set of $ C^{n} $-points at infinity (see Section~\ref{s4.95}) for any $ n\geq0 $. \end{conjecture} \begin{definition} Suppose that families $ \left\{K_{i}^{\left(1\right)},',\varphi_{i}^{\left(1\right)}\right\} $ and $ \left\{K_{i}^{\left(2\right)},',\varphi_{i}^{\left(2\right)}\right\} $ have all the accumulation points of class $ C^{0} $. Say that these families {\em describe the same curve\/} if the set of finite points of the corresponding curves are equivalent as complex manifolds, and the equivalence continues by continuity to the points at infinity. Let the {\em moduli space\/} be the set of equivalence classes of such families up to relationship that they describe the same curve. \end{definition} Note that it is not reasonable to drop the consideration of points at infinity. \begin{example} Indeed, consider a curve $ \bar{M} $ with the Serpinsky carpet as the set $ M_{\infty} $ of accumulation points of disks. The Serpinsky carpet breaks $ {\Bbb C}P^{1} $ into a union of triangles. Suppose that each triangle contains exactly one disk $ K_{i} $. Then the smooth points on $ M $ form a disjoint union of tubes, so the only invariant is the conformal lengths of these tubes. This is one parameter per handle, much smaller than three parameters per handle as one would expect to have from finite-genus theory. The remaining parameters must be contained in the data for gluing the boundary of each smooth component to the set of points at infinity. \end{example} \section{Set of admissible bundles }\label{h8}\myLabel{h8}\relax Here we investigate the structure of the set $ {\frak L} $ of admissible bundles $ {\cal L} $ over the given curve (described by the model space $ \bar{M} $, as in Section ~\ref{s5.30}). The mapping $ {\cal L} \mapsto \left(\Pi_{\left\{\psi\right\}}\circ{\bold K},\Pi_{\left\{\psi^{-1}\right\}}\circ{\bold K} \right) $ into a pair of compact operators allows one to consider the topology on $ {\frak L} $ induced by the operator topology on the space of compact operators. Thus $ {\frak L} $ is a topological space. We will describe some remarkable subsets of $ {\frak L} $ and algebraic structures on these subsets. \subsection{Exceptional indices } Since operators $ \Pi_{\varphi_{i},\psi_{i}} $ are bounded, and row-blocks and column-blocks of the operator $ {\bold K} $ are compact, we can conclude that instead of compactness of $ \Pi{\bold K} $ one can require compactness of restriction of $ \Pi{\bold K} $ to the direct sum of all but a finite number of $ L_{2}\left(\partial K_{i},\Omega^{1}\otimes\mu\right) $. We call the indices of excluded contours {\em exceptional indices}. In particular, we can include all the non-circular contours and all contours $ \partial K_{i} $ such that $ \operatorname{ind}\psi_{i}\not=0 $ in the set of exceptional indices. Thus whenever we discuss admissibility conditions we can suppose that $ \operatorname{ind} \psi_{i}=0 $ and all the $ K_{i} $ are disks. The following facts are obvious: \begin{proposition} If we change a finite number of functions $ \psi_{i} $, this does not change the admissibility of the resulting bundle. If we multiply functions $ \psi_{i} $ by constants $ c_{i} $ with $ |c_{i}| $, $ |c_{i}^{-1}| $ being bounded, this does not change the admissibility of the bundle. \end{proposition} \subsection{Hilbert--Schmidt bundles } In practice the condition of admissibility is very hard to use directly, since there is no practically useful criterion of compactness which is necessary and sufficient. The closest simple-to-check approximation is the Hilbert--Schmidt condition. \begin{definition} We say that a curve with a bundle $ \left(\bar{M},{\cal L}\right) $ is {\em Hilbert\/}--{\em Schmidt\/} if both operators $ \Pi_{\left\{\psi\right\}}{\bold K} $ and $ \Pi_{\left\{\psi^{-1}\right\}}{\bold K} $ are Hilbert--Schmidt operators. \end{definition} \begin{remark} Note that since the Hilbert structure on the sections of the bundle of half-forms on a curve is canonically defined, so is the notion of Hilbert--Schmidt operator. \end{remark} \subsection{Involution } Consider an admissible bundle $ {\cal L} $ and the dual bundle $ {\cal L}^{-1} $ (defined by inverse gluing conditions). The following statement is a direct corollary of definitions: \begin{proposition} If $ {\cal L} $ is admissible, so is $ {\cal L}^{-1} $. If $ {\cal L} $ is Hilbert--Schmidt, so is $ {\cal L}^{-1} $. \end{proposition} \subsection{Hilbert--Schmidt criterion }\label{s8.40}\myLabel{s8.40}\relax From the description of a solution of the Riemann problem one can easily get the following criterion: \begin{theorem} Consider a family of disks $ K_{i} $, $ i\in I $, with an involution ' on I and gluing data $ \varphi_{i} $, $ \psi_{i} $. \begin{enumerate} \item If \begin{equation} \sum_{i\not=j}\left(\left|\psi_{i}\right|_{{\bold R}}^{2}+\left|\psi_{i}^{-1}\right|_{{\bold R}}^{2}\right)e^{-l_{ij}} < \infty \notag\end{equation} then $ {\cal L} $ is Hilbert--Schmidt (here $ ||_{{\bold R}} $ is the Riemann norm, see Definition ~\ref{def5.155}). \item Suppose that all but a finite number of functions $ \psi_{i} $ are constant. Then the corresponding curve $ M $ with a bundle $ {\cal L} $ is Hilbert--Schmidt iff \begin{equation} \sum_{i\not=j}\left(\left|\psi_{i}\right|^{2}+\left|\psi_{i}\right|^{-2}\right)e^{-l_{ij}} < \infty \notag\end{equation} (here the indices $ i $ for which $ \psi_{i} $ is not constant are excluded from summation). \end{enumerate} Here $ l_{ij} $ is the conformal distance between $ \partial K_{i} $ and $ \partial K_{j} $. \end{theorem} This theorem follows immediately from the following \begin{lemma} \label{lm6.30}\myLabel{lm6.30}\relax Let $ k_{ij} $, $ i,j\in I $ be the Hilbert--Schmidt norm of the block of $ {\bold K} $ which maps $ L_{2}^{-}\left(\partial K_{i},\omega^{1/2}\otimes\mu\right) \to L_{2}^{+}\left(\partial K_{j},\omega^{1/2}\otimes\mu\right) $. Then \begin{equation} k_{ij} = \sum\Sb s>0 \\ s\in{\Bbb Z}+1/2\endSb e^{-sl_{ij}} = O\left(e^{-l_{ij}/2}\right). \notag\end{equation} \end{lemma} \begin{proof} It is enough to prove that the characteristic numbers of the block of $ {\bold K} $ which is the mapping \begin{equation} L_{2}^{-}\left(\partial K_{i},\omega^{1/2}\otimes\mu\right) \to L_{2}^{+}\left(\partial K_{j},\omega^{1/2}\otimes\mu\right) \notag\end{equation} are $ e^{-sl_{ij}} $ for $ s>0 $, $ s\in{\Bbb Z}+\frac{1}{2} $. Since $ {\bold K} $ is invariant with respect to fraction-linear transformations, we can assume that $ \partial K_{i} $ and $ \partial K_{j} $ bound a tube $ S^{1}\times\left(0,l_{ij}\right) $ of circumference $ 2\pi $ and length $ l_{ij} $. Then characteristic vectors of $ {\bold K} $ correspond to holomorphic $ 1/2 $-forms \begin{equation} e^{i sx}e^{-sy}dz^{1/2},\qquad z=x+iy,\quad \left(x,y\right)\in S^{1}\times\left(0,l_{ij}\right), \notag\end{equation} the restrictions on $ y=0 $ being the characteristic vectors themselves, the restrictions on $ y=l_{ij} $ being their images, which are $ e^{-sl_{ij}} $ times smaller. Now the condition that the restriction on $ y=0 $ is a section of $ \Omega\otimes\mu $ gives the condition that $ s\in{\Bbb Z}+\frac{1}{2} $, the condition that this restriction is in $ - $-part of $ L_{2} $ gives $ s>0 $.\end{proof} \begin{corollary} \begin{enumerate} \item A curve $ \bar{M} $ is Hilbert--Schmidt iff \begin{equation} \sum_{i\not=j}e^{-l_{ij}} < \infty; \notag\end{equation} \item If a curve $ \bar{M} $ allows a Hilbert--Schmidt bundle, it is Hilbert--Schmidt itself. \end{enumerate} \end{corollary} \begin{proof} Indeed, only the second part requires proof, and it follows from \begin{equation} \sum_{ij}k_{ij}^{2} \leq \frac{1}{2}\sum_{ij}\left(|\psi_{i}|^{2}+|\psi_{i}|^{-2}\right)k_{ij}^{2}. \notag\end{equation} \end{proof} \subsection{$ \protect \log $-convexity }\label{s8.50}\myLabel{s8.50}\relax We were not able to prove the following \begin{conjecture} Let $ {\cal L} $ and $ {\cal M} $ are two bundles on $ \bar{M} $ given by gluing conditions. Suppose that both $ {\cal M}\otimes{\cal L} $ and $ {\cal M}\otimes{\cal L}^{-1} $ are admissible. Then $ {\cal M} $ is admissible as well. \end{conjecture} However, the following statement is true: \begin{proposition} \label{prop6.70}\myLabel{prop6.70}\relax Let $ {\cal L} $ and $ {\cal M} $ are two bundles on $ \bar{M} $ given by gluing conditions with functions $ \psi_{i} $ and $ \gamma_{i} $, and all but a finite number of these functions are constant. Suppose that both $ {\cal M}\otimes{\cal L} $ and $ {\cal M}\otimes{\cal L}^{-1} $ are Hilbert--Schmidt. Then $ {\cal M} $ is Hilbert--Schmidt as well. \end{proposition} \begin{proof} We may suppose that all $ \psi_{i} $ and $ \gamma_{i} $ are constant. Then in notations of Lemma~\ref{lm6.30} we know that \begin{gather} \sum_{ij}\left(\left|\psi_{i}\gamma_{i}\right|^{2}+\left|\psi_{i}\gamma_{i}\right|^{-2}\right)k_{ij}^{2} < \infty, \notag\\ \sum_{ij}\left(\left|\psi_{i}\gamma_{i}^{-1}\right|^{2}+\left|\psi_{i}\gamma_{i}^{-1}\right|^{-2}\right)k_{ij}^{2} < \infty, \notag\end{gather} and want to prove that \begin{equation} \sum_{ij}\left(\left|\psi_{i}\right|^{2}+\left|\psi_{i}\right|^{-2}\right)k_{ij}^{2} < \infty. \notag\end{equation} However, this is an obvious corollary of relation between geometric mean and arithmetic mean. \end{proof} \subsection{Types of admissible bundles } In what follows we are going to study Hilbert--Schmidt bundles, thus we may assume that the curve is Hilbert--Schmidt itself. \begin{definition} \begin{enumerate} \item Call the bundle $ \omega\otimes{\cal L} $ on a curve $ \bar{M} $ {\em real\/} if all the gluing functions $ \psi_{i} $ for this bundle are constants of magnitude 1. \item Call the bundle $ \omega\otimes{\cal L} $ on a curve $ \bar{M} $ {\em bounded\/} if all the gluing functions $ \psi_{i} $ for this bundle taken together are bounded in $ ||_{{\bold R}} $-norm. \item Call the bundle $ \omega\otimes{\cal L} $ on a curve $ \bar{M} $ {\em strongly Hilbert\/}--{\em Schmidt\/} if the bundle $ \omega\otimes{\cal L}^{n} $ is Hilbert--Schmidt for any $ n\in{\Bbb Z} $. \end{enumerate} \end{definition} \begin{lemma} If the bundles $ \omega\otimes{\cal L} $ and $ \omega\otimes{\cal M} $ on a curve $ \bar{M} $ are strongly Hilbert--Schmidt, then $ \omega\otimes{\cal L}\otimes{\cal M} $ is also strongly Hilbert--Schmidt. \end{lemma} \begin{proof} Since $ \omega\otimes{\cal L}^{2n} $ and $ \omega\otimes{\cal M}^{2n} $ are Hilbert--Schmidt, such is $ \omega\otimes{\cal L}^{n}\otimes{\cal M}^{n} $ by the $ \log $-convexity. \end{proof} \begin{lemma} Any real bundle is bounded. Any bounded bundle is strongly Hilbert--Schmidt. \end{lemma} \begin{proposition} Consider an admissible curve $ \bar{M} $ and a bundle $ {\cal L} $ defined by gluing conditions $ \psi_{i} $. Suppose that for any $ N>0 $ \begin{equation} \sum_{i\not=j}\left(\max _{R_{i}} \left|\psi_{i}\right|^{N}+\max _{R_{i}}\left|\psi_{i}\right|^{-N}\right)e^{-l_{ij}} < \infty. \notag\end{equation} Then $ {\cal L} $ is strongly Hilbert--Schmidt. \end{proposition} \begin{proof} Indeed, this is a direct corollary of Lemma~\ref{lm5.160}. \end{proof} Topology on $ {\frak L} $ defines a topology on the set of bounded bundles. Note that the latter topology is very easy to describe. Fix a number $ M>1 $, and consider the subset of bundles with $ |\psi_{i}|_{{\bold R}}<M $, $ i\in I $. Then the topology on this subset is the topology of direct product. This topology is important for the description of Jacobian in Section~\ref{s9.70}. \subsection{Multiplicators, equivalence and Jacobians }\label{s8.7}\myLabel{s8.7}\relax Consider what can play a r\^ole of a mapping $ {\frak m}\colon {\cal L}_{1} \to {\cal L}_{2} $ between two bundles on $ \bar{M} $ defined by gluing conditions. Inside the ``smooth'' part of the curve such mapping should be a multiplication by a section $ a $ of a holomorphic bundle, thus $ {\frak m} $ is determined by a function $ a $ which is holomorphic inside $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $. We start with discussing heuristics for the properties of the function $ a $. On one hand, $ {\frak m} $ should send holomorphic sections to holomorphic one. If $ \bar{\partial}a $ is defined in a neighborhood of the infinity $ M_{\infty} $, and is not 0, then by Leibniz rule $ {\frak m} $ will not send holomorphic sections of $ {\cal L}_{1} $ to holomorphic section of $ {\cal L}_{2} $. On the other hand, $ {\frak m} $ should send $ H^{1/2} $-sections to $ H^{1/2} $-sections. Since $ 1\in H^{1/2} $, $ a\in H^{1/2} $. Moreover, if $ {\frak m} $ is bounded, multiplication by $ a $ should send $ L_{2} $-sections of $ \Omega^{1/2}\left(\bigcup_{i}\partial K_{i}\right)\otimes\mu $ to $ L_{2} $-sections. Since the only multiplicators in $ L_{2} $ are essentially bounded functions, we conclude that the restriction that $ a $ is bounded on the domain of definition and is in $ H^{1/2} $ looks like a particularly good candidate. \begin{proposition} \label{prop8.140}\myLabel{prop8.140}\relax Consider a function $ a $ which is holomorphic inside $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $. Identify $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ with a subspace of the space of analytic functions on $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $. Suppose that the disks $ K_{i} $ are well separated. Multiplication by $ a $ preserves $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ if and only if $ a $ is a restriction of an element $ \widetilde{a}\in H^{1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i}\right) $ such that $ \bar{\partial}\widetilde{a}=0\in H^{-1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i}\right) $ and the restriction $ \widetilde{r}\left(\widetilde{a}\right)\in L_{2}\left(\bigcup\partial K_{i}\right) $ of $ \widetilde{a} $ to the boundary is essentially bounded. \end{proposition} \begin{proof} Indeed, we may suppose that $ \infty\in K_{i} $ for some $ i\in I $. Let us show the ``only if'' part first. Considering $ a\cdot dz^{1/2} $ we see that if $ a $ is a multiplicator in $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $, then $ \widetilde{a} $ satisfying the first two conditions of the proposition exists. Since the operator of multiplication by $ a $ is automatically bounded in $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $, we see that the boundary value of $ \widetilde{a} $ is essentially bounded by the norm $ M $ of this operator. Indeed, if $ \varepsilon>0 $ and the essential supremum of $ \widetilde{r}\left(\widetilde{a}\right) $ is bigger than $ M/\left(1-\varepsilon\right) $, then there is an arc in $ \bigcup\partial K_{i} $ such that $ |\widetilde{r}\left(\widetilde{a}\right)|>M/\left(1-\varepsilon\right) $ on a subset of this arc of relative measure greater than $ 1-\varepsilon $. Taking $ \|a\cdot\frac{dz^{1/2}}{z-z_{0}}\|_{L_{2}\left(\bigcup\partial K_{i}\right)} $ with $ z_{0} $ close to this arc and inside $ \bigcup K_{i} $, we obtain a contradiction. The proof of the ``if'' part consists of three parts. First, let us show that if $ \alpha\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $, then $ a\alpha $ is a generalized half-form correctly defined up to addition of a half-form with support in $ \bigcup\partial K_{i} $ (here we say that the support of a generalized function $ \beta $ is in a set $ U $---not necessarily closed---if it is a weak limit of generalized functions $ \beta_{n} $ such that $ \operatorname{Supp}\beta_{n}\subset U $). Indeed, for any manifold $ M $ the formula $ \left<\alpha\beta,\varphi\right>\buildrel{\text{def}}\over{=}\left<\alpha,\varphi\beta\right> $, $ \varphi\in{\cal D}\left(M\right) $, $ \alpha\in H^{s}\left(M\right) $, $ \beta\in H^{-s}\left(M,\Omega^{\text{top}}\right) $, shows that there is a natural pairing $ \left(\alpha,\beta\right) \mapsto \alpha\beta $ of $ H^{s}\left(M\right) $ with $ H^{-s}\left(M,\Omega^{\text{top}}\right) $ with values in $ {\cal D}'\left(M\right) $. This pairing is weakly bicontinuous, moreover, for any smooth vector field $ v\in\operatorname{Vect}\left(M\right) $ the Leibniz identity \begin{equation} v\left(\alpha\beta\right)=\left(v\alpha\right)\beta+\alpha\left(v\beta\right) \notag\end{equation} holds if $ \alpha\in H^{s}\left(M\right) $, $ \beta\in H^{1-s}\left(M,\Omega^{\text{top}}\right) $. Since both $ a $ and $ \alpha $ are of smoothness $ H^{1/2} $, we see that $ a\alpha $ is indeed a generalized function defined with the described above ambiguity (take $ s=1/2 $, and make appropriate changes to adjust the above discussion to half-forms). Moreover, $ \bar{\partial}\left(a\alpha\right) =\left(\bar{\partial}a\right)\alpha + a\left(\bar{\partial}\alpha\right) $, thus $ \bar{\partial}\left(a\alpha\right)=0 $ as a generalized function defined up to addition of a function with support in $ \bigcup\partial K_{i} $. What remains to prove is that $ \bar{\partial}\left(a\alpha\right)\in H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right) $ after an appropriate choice of continuation of $ a\alpha $ to $ {\Bbb C}P^{1} $. As a second step, fix $ i\in I $ and show that $ a\alpha $ has an appropriate extension into $ K_{i} $. Consider restriction of $ a\alpha $ to a small collar outside $ \partial K_{i} $. \begin{lemma} Consider two concentric circles $ K\subset\widetilde{K}\subset{\Bbb C} $, and a holomorphic half-form $ \beta $ in $ \widetilde{K}\smallsetminus K $. Consider a $ L_{2} $-half-form $ B $ on $ \partial K $, and suppose that Laurent coefficients of $ \beta $ coincide with Fourier coefficients of $ B $. Then $ \beta $ has an $ H^{1/2} $-continuation into $ K $ with the $ H^{1/2} $-norm being $ O\left(\|B\|_{L_{2}}\right) $. \end{lemma} \begin{proof} Indeed, we know that positive Fourier coefficients of $ B $ are $ O\left(e^{-\varepsilon k}\right) $, and negative are in $ l_{2} $. Consider positive and negative parts of $ \beta $ separately. The positive part automatically continues into $ K $, and the bound is a corollary of results of Section~\ref{s2.70}. Consider now the negative part. We can suppose that $ K=\left\{z \mid |z|<1 \right\} $, and $ \beta $ is $ \sum B_{k}z^{-k}dz^{1/2} $, $ \left(B_{k}\right)\in l_{2} $. Extend $ \beta $ into $ K $ as $ \sum B_{k}\bar{z}^{k}dz^{1/2} $. Let us show that this extension satisfies the conditions of the lemma. It is sufficient to show that the $ H^{-1/2} $-norm of $ \bar{\partial}\beta=\sum kB_{k}\bar{z}^{k-1}\vartheta_{K_{1}}dz^{1/2}d\bar{z} $ is bounded (here $ \vartheta_{K_{1}} $ is the characteristic function of $ K_{1} $). Since different components of this form are perpendicular in $ H^{-1/2} $, it is sufficient to bound the norm of one component. To do this it is sufficient to apply the methods of Section~\ref{s2.70}. {}\end{proof} Application of this lemma to $ a\alpha $ shows that $ a\alpha $ may be extended into every disk $ K_{i} $, moreover, that after these extensions the norms $ \|\bar{\partial}\left(a\alpha\right)|_{K_{i}}\|_{H^{-1/2}} $ form a sequence in $ l_{2} $. Indeed, $ \widetilde{r}_{i}\left(a\right) $ is a bounded function on $ \partial K_{i} $, thus $ \widetilde{r}_{i}\left(a\right)\widetilde{r}_{i}\left(\alpha\right) $ has its $ L_{2} $-norm bounded by $ L_{2} $-norm of $ \widetilde{r}_{i}\left(\alpha\right) $, and the latter norms (for different $ i $) form a sequence in $ l_{2} $. Third, we need to show that the above extensions can be glued together to an extension to $ {\Bbb C}P^{1} $. Consider an arbitrary generalized-function-extension of $ a\alpha $ to $ {\Bbb C}P^{1} $ and a generalized function $ \bar{\partial}\left(a\alpha\right) $. It is a generalized function with support in the disjoint union $ \bigcup K_{i} $, thus the components $ \bar{\partial}\left(a\alpha\right)|_{K_{i}} $ are well defined generalized forms $ b_{i} $, $ \operatorname{Supp} b_{i}\subset K_{i} $. On the other hand, above we constructed an $ H^{1/2} $-extension of $ a\alpha $ to $ K_{i} $, let $ \overset{\,\,{}_\circ}{b}_{i} $ be $ \bar{\partial}\left(a\alpha\right) $ obtained from this extension. \begin{lemma} $ b_{i}-\overset{\,\,{}_\circ}{b}_{i}=\bar{\partial}\beta_{i} $, $ \beta_{i} $ being a generalized function with support in $ K_{i} $. \end{lemma} This lemma is an obvious corollary of the fact that restriction of $ \bar{\partial}^{-1}\left(b_{i}-\overset{\,\,{}_\circ}{b}_{i}\right) $ to a collar around $ \partial K_{i} $ can be holomorphically extended inside of $ K_{i} $. Since both series $ \sum b_{i} $ and $ \sum\overset{\,\,{}_\circ}{b}_{i} $ converge (one in $ {\cal D}' $, another in $ H^{-1/2} $), we conclude that $ \sum\left(b_{i}-\overset{\,\,{}_\circ}{b}_{i}\right) $ converges in $ {\cal D}' $, thus $ \sum\beta_{i} $ converges in $ {\cal D}' $. We conclude that $ \beta=a\alpha-\sum\beta_{i} $ is a generalized function such that \begin{enumerate} \item $ \bar{\partial}\beta $ has support in $ \bigcup K_{i} $; \item $ \beta $ coincides with $ a\alpha $ inside $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $; \item $ \bar{\partial}\beta $ is in $ H^{-1/2} $. \end{enumerate} Last condition implies $ \beta\in H^{1/2} $, which shows that $ a\alpha $ can be extended to an element of $ H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) $, thus $ a\alpha\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $. This finishes the proof of Proposition~\ref{prop8.140}. {}\end{proof} \begin{definition} Call a bounded operator $ {\cal M}\colon {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) \to {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ a {\em multiplicator}, if for some point $ z_{0}\in{\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $ there is a formal power series $ \nu $ at $ z_{0} $ such that for any $ f\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ formal power series $ {\cal M}f $ and $ \nu f $ coincide. \end{definition} (This is just a formal way to say that $ {\cal M} $ is a bounded operator of multiplication by a holomorphic function.) \begin{definition} Let $ {\cal H}^{\infty}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right)\subset H^{1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i}\right) $ consists of functions $ f $ such that the restriction $ \widetilde{r}\left(f\right) $ to $ \bigcup\partial K_{i} $ (which is automatically in $ L_{2}\left(\bigcup\partial K_{i}\right) $) is essentially bounded. Define a norm on this space by taking the essential maximum of $ |\widetilde{r}\left(f\right)| $. \end{definition} \begin{amplification} Multiplicators form an algebra. The set of multiplicators coincides with $ {\cal H}^{\infty}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $. One can choose a norm in $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ in such a way that the operator norm of any multiplicator coincides with the $ {\cal H}^{\infty} $-norm. \end{amplification} \begin{proof} The only statement which needs a proof is the last one, and the norm in question is the norm induced by the inclusion into $ \bigoplus_{l_{2}}L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $. \end{proof} \begin{corollary} Let $ z_{0}\in{\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $, $ f\in{\cal H}^{\infty}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $. Then $ |f\left(z_{0}\right)|\leq \|f\|_{{\cal H}^{\infty}} $. \end{corollary} \begin{proof} One can suppose that $ \infty\in K_{i} $ for some $ i\in I $, so that $ dz^{1/2}\in{\cal H} $. Since $ L_{2} $-norm on $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $ is majorated by $ H^{1/2} $-norm, we see that $ \|f^{n}dz^{1/2}\|_{L_{2}}\leq C\cdot\|f\|_{{\cal H}^{\infty}}^{n} $, $ n\in{\Bbb N} $. Hence $ |f\left(z_{0}\right)|^{n}\leq C\cdot\|f\|_{{\cal H}^{\infty}}^{n} $, thus $ |f\left(z_{0}\right)|\leq\|f\|_{{\cal H}^{\infty}} $. \end{proof} Last conditions on $ {\frak m} $ is that it should preserve the gluing conditions. In particular, if $ {\cal L}_{1} $ is defined by gluing conditions $ \psi_{i} $, and $ {\cal L}_{2} $ by $ \xi_{i} $, then \begin{equation} \xi_{i}=\frac{a\circ\varphi_{i}}{a}\psi_{i}. \label{equ7.65}\end{equation}\myLabel{equ7.65,}\relax \begin{definition} Say that linear bundles $ {\cal L}_{1} $, $ {\cal L}_{2} $ defined by gluing conditions $ \left\{\psi_{i}\right\} $ and $ \left\{\xi_{i}\right\} $ are {\em bounded-equivalent\/} if there exists a function $ a\in{\cal H}^{\infty}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ such that $ \xi_{i}=\frac{a\circ\varphi_{i}}{a}\psi_{i} $ and $ a^{-1}\in{\cal H}^{\infty}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $. \end{definition} \begin{remark} It is obvious that a bundle which is bounded-equivalent to a bounded bundle is bounded itself. Moreover, if a bundle is bounded-equivalent to a (strongly) Hilbert--Schmidt bundle, it is (strongly) Hilbert--Schmidt itself. \end{remark} \begin{definition} The {\em Jacobian\/} is the set of equivalence classes of admissible bundles of degree 0. The {\em bounded Jacobian\/} is the subset of Jacobian which consists of classes of bounded bundles, similarly for {\em(strongly) Hilbert\/}--{\em Schmidt Jacobian}, and {\em real Jacobian}. {\em Constant Jacobian\/} is formed from classes of bundles defined by constant gluing functions, similarly one can define different flavors of constant Jacobians. \end{definition} Multiplication by an appropriate rational function with zeros and poles inside $ \bigcup\overset{\,\,{}_\circ}{K}_{i} $ shows that \begin{proposition} Any admissible bundle of degree 0 is bounded-equivalent to a bundle with all the gluing functions $ \psi_{i} $ of index 0. \end{proposition} \subsection{Divisors }\label{s8.80}\myLabel{s8.80}\relax Consider a model $ \left({\Bbb C}P^{1}, \left\{K_{\bullet}\right\}, \left\{\varphi_{\bullet}\right\}\right) $ of a curve $ \bar{M} $, and a rational function $ a $ on $ {\Bbb C}P^{1} $ with a divisor $ D $. Suppose that the part of $ D $ inside $ \bigcup R_{i} $ is invariant w.r.t. $ \varphi_{\bullet} $, and $ D $ does not intersect with the infinity $ M_{\infty}\subset{\Bbb C}P^{1} $ of the curve $ M $. If the bundle $ {\cal L} $ with gluing data $ \left\{\psi_{i}\right\} $ is admissible, so is the bundle $ L\left(D\right) $ given by the gluing data \begin{equation} \xi_{i}=\frac{a\circ\varphi_{i}}{a}\psi_{i}. \notag\end{equation} Obviously, $ \deg {\cal L}\left(D\right)=\deg {\cal L}+\deg ' D $, if $ \deg ' D $ is the degree of the part of $ D $ inside $ M\smallsetminus\bigcup R_{i} $ plus half the degree of the part of $ D $ inside $ \bigcup R_{i} $. Moreover, if we change the part of $ D $ inside $ \bigcup\overset{\,\,{}_\circ}{K}_{i} $, the bundle $ {\cal L}\left(D\right) $ will change to a bounded-equivalent one. In particular, to any finite subset $ D_{0} $ of $ M $ (with integer multiplicities fixed for any point of $ D_{0} $) we associate a transformation $ {\cal L} \mapsto {\cal L}\left(D_{0}\right) $ where the right-hand side is defined up to equivalence. Note that if we fix a point $ Z\in\bigcup\overset{\,\,{}_\circ}{K}_{i} $, then we can complete any divisor $ D_{0} $ on $ M $ to a divisor on $ {\Bbb C}P^{1} $ of degree 0 by adding some multiple of $ Z $, thus one can define $ {\cal L}\left(D_{0}\right) $ uniquely. Moreover, if $ D_{0} $ is of degree 0, then $ {\cal L}\left(D_{0}\right) $ does not depend on the choice of $ Z $. Consider now two sequences of points $ \left(x_{k}\right),\left(y_{k}\right)\subset{\Bbb C}P^{1} $. Let $ a_{k}\left(z\right)=\frac{z-x_{k}}{z-y_{k}} $, and $ a\left(z\right)=\prod_{k}a_{k}\left(z\right) $. If $ x_{k} $ is sufficiently close to $ y_{k} $, and both these points are in appropriate neighborhood of $ K_{k} $, then the infinite product converges and defines a bounded function on $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup\overset{\,\,{}_\circ}{K}_{i}} $. We conclude that it is possible to consider also some ``infinite'' divisors (of finite degree) on $ M $. Consider now a finite divisor $ D\subset{\Bbb C}P^{1} $ such that $ D\cap M_{\infty} $ consists of one point $ z_{0}\in M_{\infty} $ (with some multiplicity). Changing $ D $ to $ D-\left(\deg D\right)\cdot Z $, we can consider $ D $ as a divisor of a rational function $ a $, thus $ {\cal L}\left(D\right) $ is correctly defined by gluing conditions conditions~\eqref{equ7.65}. For $ {\cal L}\left(D\right) $ to be admissible for any such $ D $ and any admissible $ {\cal L} $ it is sufficient that for some constant $ C $ \begin{equation} \operatorname{dist}\left(z_{0},\operatorname{center}\left(K_{i'}\right)\right) \leq C\cdot\operatorname{dist}\left(z_{0},\operatorname{center}\left(K_{i}\right)\right),\qquad i\in I \notag\end{equation} (since the disks $ K_{i} $ are well-separated). On the other hand, suppose that all the bundles with gluing functions \begin{equation} \Psi_{i}=\left(\frac{\operatorname{dist}\left(z_{0},\operatorname{center}\left(K_{i'}\right)\right)}{\operatorname{dist}\left(z_{0},\operatorname{center}\left(K_{i}\right)\right)}\right)^{N},\qquad n\in{\Bbb Z}, \notag\end{equation} are Hilbert--Schmidt. Then by $ \log $-convexity the bundle $ L\left(D\right) $ is strongly Hilbert--Schmidt if $ {\cal L} $ is strongly Hilbert--Schmidt. \begin{definition} Say that the point $ z_{0}\in M_{\infty} $ is {\em bounded\/} if for any $ N\in{\Bbb Z} $ the sheaf with constant gluing functions \begin{equation} \Psi_{i}=\left(\frac{\operatorname{dist}\left(z_{0},\operatorname{center}\left(K_{i'}\right)\right)}{\operatorname{dist}\left(z_{0},\operatorname{center}\left(K_{i}\right)\right)}\right)^{N},\qquad n\in{\Bbb Z}\text{, }i\in I, \notag\end{equation} is Hilbert--Schmidt. \end{definition} We see that if the point $ z_{0}\in M_{\infty} $ is bounded, then the function $ a_{N}\left(z\right)=\left(\frac{z-z_{0}}{z-Z}\right)^{N} $ defines a strongly Hilbert--Schmidt sheaf of degree 1. In particular, one can consider finite divisors on $ \bar{M} $ which consist of points of $ M $ and bounded points on $ M_{\infty} $. Similarly, one can also consider some infinite divisors on $ \bar{M} $. \subsection{Universal Grassmannian and bundles }\label{s8.90}\myLabel{s8.90}\relax \begin{definition} Let $ V $ is the vector space of sequences $ \left(a_{k}\right) $, $ k\in{\Bbb Z} $, such that $ a_{k}=0 $ for $ k\ll0 $. The vector space $ V $ carries a natural topology of inductive limit of projective limits. Let \begin{equation} V_{+}=\left\{\left(a_{k}\right) \mid a_{k}=0\text{ if }k<0\right\},\qquad V_{-}=\left\{\left(a_{k}\right)\in V \mid a_{k}=0\text{ if }k\geq0\right\}. \notag\end{equation} Say that the vector subspace $ W\subset V $ is {\em admissible\/} if $ W\cap V_{-} $ is finite-dimensional, and $ W+V_{-} $ is closed and has a finite codimension in $ V $. Let {\em universal Grassmannian\/} $ {\cal G} $ be the set of admissible vector subspaces with natural topology. \end{definition} Consider an admissible pair $ \left(\bar{M},{\cal L}\right) $ and a smooth point $ z_{0}\in M $. Pick up a coordinate system $ z $ in neighborhood of $ z_{0} $, and a half-form $ f $ defined in the same neighborhood. Now any global section of $ {\cal L} $ may be written as $ g\left(z\right)f\left(z\right) $ with $ g\left(z\right) $ being a holomorphic function which is correctly defined in the neighborhood of $ z_{0} $. Call $ f $ a {\em local section\/} of $ {\cal L} $ near $ z_{0} $. If $ {\cal L}' $ is an equivalent to $ {\cal L} $ bundle, and the equivalence is given by multiplication by $ a $, we obtain a local section $ af $ of $ {\cal L}' $. Say that local sections $ f $ and $ af $ are {\em equivalent}. Consider now a bundle $ {\cal L}\left(k\cdot z_{0}\right) $, defined using a point $ Z\in\overset{\,\,{}_\circ}{K}_{i} $ (as in the previous section). A section $ h\left(\zeta\right) $, $ \zeta\in{\Bbb C}P^{1} $, of this bundle can be identified (via multiplication by $ \left(\frac{\zeta-Z}{\zeta-z_{0}}\right)^{k} $) with a ``meromorphic section'' of $ {\cal L} $, i.e., one can write it as $ g\left(z\right)f\left(z\right) $, $ g\left(z\right) $ being a meromorphic function correctly defined in the neighborhood of $ z_{0} $. Moreover, $ g\left(z\right) $ has poles only at $ z_{0} $. One can momentarily see that $ g\left(z\right) $ does not depend on the choice of the point $ Z $, more precise, for a different choice of $ Z $ there is a different choice of $ h $ which gives the same $ g\left(z\right) $. Associate to any such function $ g\left(z\right) $ the sequence of its Laurent coefficients. Consider this sequence as an element of $ V $. Let $ W $ be the vector subspace of $ V $ spanned by all possible functions $ g\left(z\right) $ for bundles $ {\cal L}\left(k\cdot z_{0}\right) $, $ k\in{\Bbb Z} $. The Riemann--Roch theorem momentarily implies that $ W $ is admissible. Indeed, the condition that $ W\cap V_{-} $ is finite-dimensional means that $ {\cal L} $ has a finite-dimensional space of global sections. The condition on $ W+V_{-} $ is implied by the following fact: \begin{lemma} Consider a admissible semibounded bundle $ {\cal L} $ on a curve $ \bar{M} $, and a point $ z_{0}\in M $. Then \begin{enumerate} \item for big enough $ k $ the bundle $ {\cal L}\left(-k\cdot z_{0}\right) $ has no sections. \item for big enough $ k $ the bundle $ {\cal L}\left(\left(k+1\right)\cdot z_{0}\right) $ has one more section than $ {\cal L}\left(k\cdot z_{0}\right) $. \end{enumerate} \end{lemma} \begin{proof} The second statement is a corollary of the first one, of Riemann--Roch theorem and duality. The first one is obvious, since the global sections of $ {\cal L}\left(-k\cdot z_{0}\right) $ are naturally identified with global sections of $ {\cal L} $ which have a zero of $ k $-th order at $ z_{0} $. \end{proof} We obtained \begin{proposition} To each admissible pair $ \left(\bar{M},{\cal L}\right) $ with a fixed smooth point $ z_{0}\in M $, a coordinate system near $ z_{0} $, and a local section $ f $ of $ {\cal L} $ one can associate a point $ W\in{\cal G} $. If we change $ {\cal L} $ and $ f $ to an equivalent bundle with a local section, $ W $ does not change. \end{proposition} One can generalize this proposition to some points at infinity. If $ z_{0}\in M_{\infty} $, define a {\em local section\/} of $ {\cal L} $ near $ z_{0} $ in the same way as above, i.e., as a non-zero section of $ \omega^{1/2} $ near $ z_{0} $. In what follows we use only the $ \infty $-jet of this section. \begin{amplification} Consider an admissible pair $ \left(\bar{M},{\cal L}\right) $ with a fixed point at infinity $ z_{0}\in M_{\infty} $, a coordinate system near $ z_{0} $, and a local section $ f $ of $ {\cal L} $. Suppose that $ {\cal L} $ is Hilbert--Schmidt, and $ z_{0} $ is bounded and of class $ C^{\infty} $. To this data we can associate a point $ W\in{\cal G} $. If we change $ {\cal L} $ and $ f $ to an equivalent bundle with a local section, $ W $ does not change. \end{amplification} \begin{proof} Since $ z_{0} $ is bounded, and $ {\cal L} $ is Hilbert--Schmidt, $ {\cal L}\left(k\cdot z_{0}\right) $ is Hilbert--Schmidt too, thus the Riemann--Roch theorem is applicable. Since $ z_{0} $ is of class $ C^{\infty} $, any section of $ {\cal L}\left(k\cdot z_{0}\right) $ has a (formal) Taylor series near $ z_{0} $, thus the corresponding section of $ {\cal L} $ has a (formal) Laurent series. What remains to be proved is the fact that equivalence between bundles can be pushed to infinity points of class $ C^{\infty} $. Note that one can associate an element of the generalized Hardy space to any bounded function $ a $. Indeed, suppose that $ \zeta=\infty\in K_{i} $, then $ a\cdot d\zeta^{1/2} $ is an element of $ {\cal H} $. In particular, $ a $ has an asymptotic expansion near $ z_{0} $, thus multiplication by $ a $ maps (formal) Laurent series at $ z_{0} $ to themselves. \end{proof} \section{Structure of Jacobian }\label{h9}\myLabel{h9}\relax \subsection{Constant Jacobian }\label{s7.90}\myLabel{s7.90}\relax Consider the involution $ '\colon I\to I $. It defines a transposition matrix $ t=\left(t_{ij}\right) $, $ i,j\in I $, $ t_{ij}=\delta_{ij'} $. We use the results of toy theory (see Section~\ref{h35}) to obtain the following \begin{theorem} \label{th7.25}\myLabel{th7.25}\relax Let $ l_{ij} $ be pairwise conformal distances between disks $ K_{i} $. Suppose that the matrix $ {\cal R}=\left(e^{-l_{ij}}-\delta_{ij}\right) $ defines a compact mapping $ l_{2} \to l_{2} $ and a bounded mapping $ l_{\infty} \to l_{\infty} $, and that for some $ N>0 $ the matrix $ \left(t{\cal R}\right)^{N} $ defines a mapping $ l_{\infty} \to l_{2} $. Then for any admissible curve obtained by gluing circles $ \partial K_{i} $ the constant bounded Jacobian coincides with the bounded Jacobian. \end{theorem} \begin{proof} Indeed, to show this we need to show that for any cycle $ \left\{\psi_{i}\right\}_{i\in I} $ with $ \deg \psi_{i}=0 $ and $ |\psi_{i}|<C $ one can find a bounded collection of constants $ c_{i} $ such that $ c_{i}\psi_{i} =\frac{a\circ\varphi_{i}}{a} $ for some $ a $ such that $ a,a^{-1}\in{\cal H}^{\infty}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $. Taking logarithms, we see that it is sufficient to show that the mapping \begin{equation} {\cal J}_{b}\colon f+C \mapsto \left(f|_{\partial K_{j}}-\varphi^{*}\left(f|_{\partial K_{j'}}\right)+C_{j}\right)_{j\in I_{+}} \notag\end{equation} from bounded analytic functions (modulo constants) to functions on boundary (modulo constants) with bounded $ \pm $-parts is surjective. In Section~\ref{s35.40} we have seen that (given the first condition of the theorem) a similar mapping \begin{equation} {\cal J}\colon {\cal H}^{\left(1\right)}/\operatorname{const} \to \bigoplus\Sb l_{2} \\ j\in I_{+}\endSb H^{1/2}\left(\partial K_{j}\right)/\operatorname{const} \notag\end{equation} is a bounded mapping of index 0. We are going to prove the surjectivity by using a combination of following lemmas: \begin{lemma} \label{lm8.31}\myLabel{lm8.31}\relax If $ \left(e^{-l_{ij}}-\delta_{ij}\right) $ gives a compact operator $ l_{2} \to l_{2} $, then the mapping $ {\cal J} $ is a bijection. \end{lemma} \begin{proof} Since we know the index of $ {\cal J} $, it is sufficient to show that $ \operatorname{Ker}{\cal J}=0 $. Let $ f\in{\cal H}^{\left(1\right)}/\operatorname{const} $ and $ {\cal J}f=0 $. We are going to show that $ \|f\|_{H^{1}}=0 $. Since $ \partial $ is elliptic, and $ \bar{\partial}f=0 $, it is sufficient to show that $ \|\partial f\|_{L_{2}}=0 $, i.e., that $ \int\partial f\wedge\bar{\partial}\bar{f}=0 $, the integral is taken over $ {\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $. Denote the domain of integration by $ S $. Note that the above integral is well defined, since $ \partial f\in H^{0}\left(S\right)=L_{2}\left(S\right) $. First of all, since $ \wedge $-product of any two sections of $ \omega $ is 0, one can add $ \partial\bar{f} $ to $ \bar{\partial}\bar{f} $ without changing the integral, thus it is sufficient to show that $ \int_{S}\partial f\wedge d\bar{f} = $ 0. Additionally, since $ \bar{\partial}f=0 $ on $ S $, one can change $ \partial f $ to $ df $ without changing the integral, thus it is sufficient to show that $ \int_{S}df\wedge d\bar{f}=0 $. Second, use the duality identity $ \int_{S}d\alpha=\int_{\partial S}\alpha $. Applying it (at first formally), we get \begin{align} \int_{S}df\wedge d\bar{f} & = \int_{S}d\left(f\,d\bar{f}\right) = \int_{\partial S}f\,d\bar{f} = -\sum_{i\in I}\int_{\partial K_{i}}f\,d\bar{f} \notag\\ & = -\sum_{i\in I_{+}}\left(\int_{\partial K_{i}}f\,d\bar{f} + \int_{\partial K_{i'}}f\,d\bar{f}\right). \notag\end{align} We want to show that $ \int_{\partial K_{i}}f\,d\bar{f} + \int_{\partial K_{i'}}f\,d\bar{f}=0 $. Indeed, $ \int_{\partial K_{i'}}f\,d\bar{f}=-\int_{\partial K_{i}}\varphi_{i'}^{*}\left(f\,d\bar{f}\right) $, the sign appears since $ \varphi_{i} $ changes the orientations of $ \partial K_{\bullet} $. Now the above equality becomes obvious, since $ f\circ\varphi_{i}=f+C_{i} $ when both sides are defined, thus the sum of the integrals is $ -C_{i}\int_{\partial K_{i}}d\bar{f}=0 $. What remains to be proved is that one can indeed apply the formula $ \int_{S}d\alpha=\int_{\partial S}\alpha $ in our situation, when $ \alpha $ is not smooth, and $ S $ has non-smooth boundary. Consider an $ H^{1} $-extension $ g $ of $ f $ to $ {\Bbb C}P^{1} $. Since $ dg\wedge d\bar{g}\in L_{1}\left(\Omega^{\text{top}}\right) $, the integral $ \int_{S}df\wedge d\bar{f} $ can be represented as \begin{equation} \int_{{\Bbb C}P^{1}}dg\wedge d\bar{g} - \int_{\bigcup K_{i}}dg\wedge d\bar{g} = \int_{{\Bbb C}P^{1}}dg\wedge d\bar{g} - \sum\int_{K_{i}}dg\wedge d\bar{g}. \notag\end{equation} It remains to prove that $ \int_{{\Bbb C}P^{1}}dg\wedge d\bar{g} = $ 0, and $ \int_{K_{i}}dg\wedge d\bar{g} = \int_{\partial K_{i}}g\,d\bar{g}. $ Note that in both identities the boundary is already smooth (it is empty in the first one!). Let us prove that if $ M $ is a two-dimension manifold, $ S $ is a compact subset of $ M $ with a smooth boundary, and $ g_{1},g_{2}\in H^{1}\left(M\right) $, then \begin{equation} \int_{S}dg_{1}\wedge dg_{2}=\int_{\partial S}g_{1}\,dg_{2}. \notag\end{equation} Here we understand the right-hand side as a natural pairing between $ g_{1}|_{\partial S}\in H^{1/2}\left(\partial S\right) $ and $ d\left(g_{2}|_{\partial S}\right)\in H^{-1/2}\left(\partial S,\Omega_{\partial S}^{\text{top}}\right) $. However, both sides define bounded bilinear functionals on $ H^{1}\left(M\right) $, thus it is sufficient to check them on a dense subset $ C^{\infty}\left(M\right) $, where they are true due to de Rham theory. \end{proof} To continue the proof of the theorem, note that the operator $ {\cal J} $ is related to the operator $ {\bold K} $ in the following way: identify $ {\cal H}^{\left(1\right)}/\operatorname{const} $ with $ \bigoplus\Sb l_{2} \\ i\in I\endSb H_{-}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $. Let $ \Pi_{i} $ be the identification of $ H_{+}^{1/2}\left(\partial K_{i}\right) $ with $ H_{-}^{1/2}\left(\partial K_{i'}\right) $ via $ \varphi^{*} $, let $ \Pi^{\left(1\right)} $ be the direct sum of such identifications. As we have seen it in Section~\ref{s35.40}, the image of $ {\cal J} $ coincides with the image of $ \Pi^{\left(1\right)}\circ{\bold K}-1 $ (which is a mapping \begin{equation} \bigoplus_{l_{2}}H_{-}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} \to \bigoplus_{l_{2}}H_{-}^{1/2}\left(\partial K_{i}\right)/\operatorname{const}, \notag\end{equation} but we identify $ \bigoplus_{l_{2}}H_{-}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $ with $ \bigoplus\Sb l_{2} \\ j\in I_{+}\endSb H^{1/2}\left(\partial K_{j}\right)/\operatorname{const} $ via $ \varphi_{\bullet}^{*} $). \begin{lemma} \label{lm8.33}\myLabel{lm8.33}\relax \begin{enumerate} \item Consider two disjoint disks $ K_{1} $, $ K_{2} $ on $ {\Bbb C}P^{1} $ of conformal distance $ l $. The operator $ {\bold K} $ with smooth kernel $ \frac{dy}{y-x} $ defines bounded operators $ H^{1/2}\left(\partial K_{1}\right) \to L_{\infty}\left(\partial K_{2}\right) $, $ L_{\infty}\left(\partial K_{1}\right) \to L_{\infty}\left(\partial K_{2}\right) $, $ H^{1/2}\left(\partial K_{1}\right) \to H^{1/2}\left(\partial K_{2}\right) $, $ L_{\infty}\left(\partial K_{1}\right) \to H^{1/2}\left(\partial K_{2}\right) $ with the norms being $ O\left(e^{-l}\right) $. \item Consider a family of disjoint disks $ \left\{K_{i}\right\} $ on $ {\Bbb C}P^{1} $ with conformal distances $ l_{ij} $, and suppose that the matrix $ {\cal R}=\left(e^{-l_{ij}}-\delta_{ij}\right) $ defines bounded mappings $ l_{2} \to l_{2} $ and $ l_{\infty} \to l_{\infty} $, and for some $ N>0 $ the matrix $ \left(t{\cal R}\right)^{N} $ defines a mapping $ l_{\infty} \to l_{2} $. Then the operator $ {\bold K} $ gives bounded mappings $ \bigoplus_{l_{2}}H^{1/2}\left(\partial K_{i}\right) \to \bigoplus_{l_{2}}H^{1/2}\left(\partial K_{i}\right) $ and $ L_{\infty}\left(\bigcup\partial K_{i}\right) \to L_{\infty}\left(\bigcup\partial K_{i}\right) $, moreover, the operator $ \left(\Pi^{\left(1\right)}{\bold K}\right)^{N} $ gives a bounded mapping $ L_{\infty}\left(\bigcup\partial K_{i}\right) \to \bigoplus_{l_{2}}H^{1/2}\left(\partial K_{i}\right) $. \item In the conditions of the previous part of the lemma let $ f_{-}\in L_{\infty}\left(\bigcup\partial K_{i}\right) $. Suppose that $ + $-parts of $ f_{-} $ on all the $ \partial K_{i} $ vanish. Then $ {\bold K}f_{-} $ is a bounded analytic function. \end{enumerate} \end{lemma} Now we are able to invert the mapping $ {\cal J}_{b} $. Consider a fixed function $ f $ on $ \bigcup\partial K_{i} $ with bounded $ + $-part and bounded $ - $-part. If $ f-f\circ\varphi_{i} $ is a constant for any $ i\in I $, then $ + $-part of $ f|_{\partial K_{i}} $ can be reconstructed basing on -part of $ f|_{\partial K_{i'}} $, thus one can identify $ f $ (modulo constants) with the collection of $ - $-parts of $ f $ on $ \bigcup_{i\in I}\partial K_{i} $ (modulo constants). The restrictions on $ l_{ij} $ show that the radii of disks $ K_{i} $ for a sequence from $ l_{1} $. Hence the Cauchy kernel allows one to construct an analytic function $ \widetilde{f} $ on $ S $ such that the $ - $-parts of $ \widetilde{f}|_{\partial K_{i}} $ and $ f|_{\partial K_{i}} $ coincide (modulo constants). Moreover, $ {\cal J}_{b}\widetilde{f} = \left(\Pi^{\left(1\right)}\circ{\bold K}-1\right)\left(f\right) $. Since $ f $ is bounded, it is in $ L_{2} $, thus $ \widetilde{f}\in H^{1/2} $. Since $ \widetilde{f}|_{\bigcup\partial K_{i}} $ is bounded, $ \widetilde{f} $ is a multiplicator. We see that $ \Pi^{\left(1\right)}\circ{\bold K}\left(f\right) $ is a well-defined bounded function on $ \bigcup_{j\in I_{+}}\partial K_{j} $, moreover, $ \left(\Pi^{\left(1\right)}\circ{\bold K}\right)^{N}\left(f\right)\in\bigoplus_{l_{2}}H^{1/2}\left(\partial K_{j}\right) $. In particular, $ \left(\Pi^{\left(1\right)}\circ{\bold K}\right)^{N}\left(f\right)={\cal J}F $ for some function $ F\in{\cal H}^{\left(1\right)} $. To show that $ f $ is in the image of $ \Pi^{\left(1\right)}\circ{\bold K}-1 $, it remains to prove is that $ f-\left(\Pi^{\left(1\right)}\circ{\bold K}\right)^{N}\left(f\right) $ is in the image of $ \Pi^{\left(1\right)}\circ{\bold K}-1 $, what is obvious. This finishes the proof of Theorem~\ref{th7.25}. {}\end{proof} To demonstrate a particular case in which the assumptions of Theorem ~\ref{th7.25} hold, consider \begin{lemma} Suppose that the matrix $ \left(a_{ij}\right) $ gives a Hilbert--Schmidt operator $ l_{2} \to l_{2} $, i.e., $ \sum|a_{ij}|^{2}<\infty $. Then the matrix $ \left(a_{ij}^{2}\right) $ gives a bounded operator $ l_{\infty} \to l_{2} $. If $ a_{ij}>0 $ for any $ i,j\in I $, and the matrix $ \left(a_{ij}\right) $ gives a compact mapping $ l_{2} \to l_{2} $, then $ \left(a_{ij}^{2}\right) $ also gives a compact mapping $ l_{2} \to l_{2} $. \end{lemma} \begin{corollary} If $ \bar{M} $ is a Hilbert--Schmidt curve, then the bounded Jacobian of $ \bar{M} $ coincides with the constant bounded Jacobian. \end{corollary} In Section~\ref{s9.70} we show that under suitable restrictions on $ \bar{M} $ the above Jacobians are in $ 1 $-to-1 correspondence with quotients of topological vector spaces by $ {\Bbb Z} $-lattices. On the other hand, this statement is obvious when applied to real Jacobian, since it is a direct product of circles $ |\psi_{i}|=1 $. To construct an isomorphism of this product with a quotient by a lattice, one can take the Hilbert space $ l_{2} $ with the standard basis $ e_{i} $, and the lattice spanned by $ \alpha_{i}e_{i} $, with $ \alpha_{i} \to $ 0. The description of the real part of Jacobian was first obtained in \cite{McKTru76Hil} in the case of a real hyperelliptic curve of a special form. \subsection{The partial period mapping }\label{s9.20}\myLabel{s9.20}\relax Consider a theory similar to one discussed in Section~\ref{s35.30}, but related to holomorphic $ 1 $-forms instead of holomorphic functions. Consider a family $ \left\{K_{i}\right\} $ of disjoint disks in $ {\Bbb C}P^{1} $. \begin{definition} We say that a generalized-function section $ \alpha $ of the linear bundle $ \omega $ on $ {\Bbb C}P^{1} $ is $ H^{0} $-{\em holomorphic in\/} $ {\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $ if $ \alpha\in H^{0}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\omega\right) = L_{2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\omega\right) $, and $ \bar{\partial}\alpha=0\in H^{-1}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\omega\otimes\bar{\omega}\right) $. Denote the the space of $ H^{0} $-holomorphic forms in $ {\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $ by $ {\cal H}^{\left(0\right)} $. \end{definition} Since one can extend a $ H^{0} $-holomorphic form $ \alpha $ into any disk $ K_{i} $ by 0 without increasing $ H^{0} $-norm, $ \alpha $ has a canonical extension to $ {\Bbb C}P^{1} $, and $ \bar{\partial}\alpha $ is concentrated on $ \bigcup\partial K_{i} $. Moreover, if $ K_{i} $ has a neighborhood which does not intersect with other disks, one can define a {\em boundary value\/} $ \alpha|_{\partial K_{i}} $ of $ \alpha $ on $ \partial K_{i} $, which is a element of $ H^{-1/2}\left(\partial K_{i},\Omega_{\partial K_{i}}^{1}\right) $ (note that if $ \gamma $ is a curve in $ {\Bbb C}P^{1} $, then $ \omega|_{\gamma}\simeq\Omega_{\gamma}^{1} $; one can get the $ H^{-1/2} $-restriction on the smoothness of the boundary value in the same way as in Section~\ref{s2.70}). Now $ \bar{\partial}\alpha $ can be described as the extension of $ -\alpha|_{\partial K_{i}} $ from $ \partial K_{i} $ to $ {\Bbb C}P^{1} $ by $ \delta $-function (see ~\eqref{equ3.02}). (Such an extension is a correctly defined continuous mapping $ H^{-1/2}\left(\partial K_{i},\Omega_{\partial K_{i}}^{1}\right) \to H^{-1}\left({\Bbb C}P^{1},\omega\otimes\bar{\omega}\right) $.) Since the norm in $ H^{1}\left({\Bbb C}P^{1}\right) $ can be described by a local (non-invariant) formula $ \|\alpha\|_{l_{2}}^{2}+\|\alpha_{,x}\|_{l_{2}}^{2}+\|\alpha_{,y}\|_{l_{2}}^{2} $ (here $ x+iy $ is the local coordinate on $ {\Bbb C}P^{1} $), functions with non-intersecting support are orthogonal in $ H^{1} $. Dually, if $ U_{i} $ are pairwise non-intersecting, then the natural mapping \begin{equation} H^{-1}\left({\Bbb C}P^{1},\omega\otimes\bar{\omega}\right) \to \bigoplus_{l_{2}}H^{-1}\left(U_{i},\omega\otimes\bar{\omega}\right) \notag\end{equation} is a continuous epimorphism. \begin{definition} Fix $ \varepsilon>0 $. Say that disks $ \left\{K_{i}\right\} $ have a {\em thickening\/} $ \left\{U_{i}\right\} $ if $ K_{i} $ have non-intersecting neighborhoods $ U_{i} $ such that a pair $ K_{i}\subset U_{i} $ is conformally equivalent to $ \left\{|z|<1\right\}\subset\left\{|z|<1+\varepsilon\right\} $. Say that $ \left\{K_{i}\right\} $ have a {\em uniform thickening\/} if the neighborhood $ U_{i} $ can be picked up to be concentric to $ K_{i} $ circles (assume that a metric on $ {\Bbb C}P^{1} $ is fixed). \end{definition} (Note that the existence of uniform thickening does not depend on the metric on $ {\Bbb C}P^{1} $.) In particular, if $ \left\{K_{i}\right\} $ have a thickening $ \left\{U_{i}\right\} $, then the above arguments show that $ \left(\|\alpha|_{\partial K_{i}}\|_{H^{-1/2}}\right)_{i\in I}\in l_{2} $. Note also that $ H^{-1/2}\left(\partial K_{i},\Omega_{\partial K_{i}}^{1}\right) $ is dual to $ H^{1/2}\left(\partial K_{i}\right) $, thus the subspace $ H_{\int=0}^{-1/2}\left(\partial K_{i},\Omega_{\partial K_{i}}^{1}\right) $ of forms with vanishing integral has an $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant Hilbert structure (in the same sense as in Section~\ref{s35.20}). \begin{definition} Call the mapping $ \alpha \mapsto \left(\int_{\partial K_{i}}\alpha\right)_{i\in I} $, $ H^{0}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\omega\right) \to l_{2} $ the {\em integration mapping}. Fix an involution $ '\colon I\to I $ which interchanges two parts of $ I $, $ I=I_{+}\amalg I_{+}' $, and fraction-linear identifications $ \varphi_{i}\colon \partial K_{i} \to \partial K_{i} $ (with the same conditions as in Section~\ref{s35.40}), then a {\em global holomorphic form\/} $ \alpha $ is an element of $ {\cal H}^{\left(0\right)} $ such that $ \varphi_{i}^{*}\left(\alpha|_{\partial K_{i'}}\right)=\alpha|_{\partial K_{i}} $. Define the {\em partial period mapping\/} $ P $ by restriction of the integration mapping to global holomorphic forms, and taking integrals only for $ i\in I_{+} $. Let $ \bar{M} $ be the curve obtained by gluing $ \partial K_{i} $ together via $ \varphi_{i} $. Denote by $ \Gamma\left(\bar{M},\omega\right) $ the space of global holomorphic forms on $ \bar{M} $, i.e., the space of global holomorphic forms compatible with $ \varphi_{\bullet} $. \end{definition} The significant difference of this case and the case of Section ~\ref{s35.30} is that the mapping $ \bar{\partial}\colon H^{0}\left({\Bbb C}P^{1},\omega\right) \to H^{-1}\left({\Bbb C}P^{1},\omega\otimes\bar{\omega}\right) $ has no null-space, but has $ 1 $-dimensional cokernel, thus it is not easy to reconstruct $ \alpha $ basing on $ \bar{\partial}\alpha $ by local formulae. However, it is easy to prove \begin{lemma} \label{lm9.90}\myLabel{lm9.90}\relax Suppose that $ \left\{K_{i}\right\} $ has a thickening. Then the partial period mapping $ P $ satisfies the identity \begin{equation} \dim \operatorname{Ker} P = \dim \operatorname{Ker} {\cal J}\colon {\cal H}^{\left(1\right)}/\operatorname{const} \to \bigoplus_{l_{2}}H^{1/2}\left(\partial K_{j}\right)/\operatorname{const}. \notag\end{equation} \end{lemma} \begin{proof} We claim that $ \partial $ gives a mapping from one null-space to another one. Let $ \alpha\in\operatorname{Ker} P $. Then $ \alpha|_{\partial K_{i}}\in H^{-1/2}\left(\partial K_{i},\Omega_{\partial K_{i}}^{1}\right) $ has an antiderivative $ f_{i}\in H^{1/2}\left(\partial K_{i}\right) $, and the harmonic extension of $ f_{i} $ into $ K_{i} $ has a bounded $ H^{1} $-norm. Since $ \alpha $ is closed in $ U_{i}\smallsetminus K_{i} $, it has an antiderivative. Since $ \partial $ is elliptic, the antiderivative has smoothness $ H^{1} $. It is possible to pick up the constant in such a way that two antiderivatives coincide on $ \partial K_{i} $. Taking de Rham differential of resulting $ H^{1} $-function, we see that one can extend $ \alpha $ into $ K_{i} $ preserving the closeness, and the related increase of the norm of $ \alpha $ is bounded by $ \|\alpha|_{\partial K_{i}}\|_{H^{-1/2}} $. Repeating this operation for all the $ K_{i} $, we obtain a closed extension of $ \alpha $ to $ {\Bbb C}P^{1} $ with the $ H^{0} $-norm bounded by $ \|\alpha\|_{H^{0}} $ (here we use locality of $ L_{2} $-norm). Thus $ \alpha $ has a $ H^{1} $-antiderivative $ f $ on $ {\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $, thus $ \alpha=df $, hence $ \alpha=\partial f $. Obviously, $ f\in\operatorname{Ker} {\cal J} $. On the other hand, if $ f\in\operatorname{Ker} {\cal J} $, then $ \partial f\in\operatorname{Ker} P $.\end{proof} \begin{remark} As we have seen in Section~\ref{s7.90}, in assumptions of Theorem ~\ref{th35.45} the dimension in the lemma is 0, thus $ P $ defines an inclusion of global holomorphic forms into $ l_{2} $. Note that we do not claim that this inclusion is a monomorphism---estimates in Section~\ref{s9.12} show that it is not if $ \operatorname{card}\left(I\right)=\infty $. \end{remark} \subsection{$ A $-Periods }\label{s9.12}\myLabel{s9.12}\relax To describe the image of the integration mapping, consider the mapping $ \bar{\partial}^{-1}\colon H_{\int=0}^{-1}\left({\Bbb C}P^{1},\omega\otimes\bar{\omega}\right) \to H^{0}\left({\Bbb C}P^{1},\omega\right) $. It is a continuous elliptic operator. To make the formulae localizable, fix a top form $ \beta_{0} $ with integral 1, and extend $ \bar{\partial}^{-1} $ to any form on $ {\Bbb C}P^{1} $ by $ \bar{\partial}^{-1}\beta\buildrel{\text{def}}\over{=} \bar{\partial}^{-1}\left(\beta-\beta_{0}\int_{{\Bbb C}P^{1}}\beta\right) $. This operator is still elliptic, but is not an isomorphism any more. Let $ \alpha\in{\cal H}^{\left(0\right)} $, consider $ \alpha|_{\partial K_{i}} $. In contrast with the cases of Section ~\ref{s35.20} and Proposition~\ref{prop5.28} the $ - $-components of $ \left(\alpha|_{\partial K_{i}}\right)_{i\in I} $ are not arbitrary. To describe possible values of $ \left(\alpha|_{\partial K_{i}}\right)_{i\in I} $, break $ \alpha|_{\partial K_{i}} $ into two parts, one $ \alpha_{i}^{\left(0\right)} $ with integral 0, another one $ \alpha_{i}^{\left(1\right)} $ proportional to some fixed $ 1 $-form $ \mu_{i} $ on $ \partial K_{i} $. Use the same letters for $ \delta $-function-extensions of these forms to $ {\Bbb C}P^{1} $. Suppose that $ 1 $-forms $ \mu_{i} $ (one per disk boundary) are normalized to have integral 1 and have uniformly bounded $ H^{-1/2} $-norms when $ i $ varies, and that disks $ K_{i} $ have a thickening. Then $ \sum\alpha_{i}^{\left(1\right)} $ converges in $ \bigoplus_{l_{2}}H^{-1/2}\left(\partial K_{i}\right) $, same for $ \sum\alpha_{i}^{\left(0\right)} $. Note that if conditions of Theorem~\ref{th35.45} hold, then $ \alpha^{\left(0\right)}=\bar{\partial}^{-1}\left(\sum\alpha_{i}^{\left(0\right)}\right)\in H^{0}\left({\Bbb C}P^{1},\omega\right) $. Since $ \alpha=\bar{\partial}^{-1}\sum_{i}\alpha|_{\partial K_{i}} $, $ \alpha^{\left(1\right)}=\bar{\partial}^{-1}\left(\sum\alpha_{i}^{\left(1 \right)}\right)=\alpha-\bar{\partial}^{-1}\left(\sum\alpha_{i}^{\left(0\right)}\right) $ is also in $ H^{0}\left({\Bbb C}P^{1},\omega\right) $. We see that any form $ \alpha\in{\cal H}^{\left(0\right)} $ can be represented as a sum $ \alpha^{\left(0\right)}+\alpha^{\left(1\right)} $, $ \alpha^{\left(0\right)},\alpha^{\left(1\right)}\in{\cal H}^{\left(0\right)} $ such that integrals of $ \alpha^{\left(0\right)} $ around each disk $ K_{i} $ vanishes, and $ \alpha^{\left(1\right)} $ is a linear combination of $ 1 $-forms $ \bar{\partial}^{-1}\mu_{i} $, $ i\in I $. Consider the subspace $ {\cal H}_{0}^{\left(0\right)}\subset{\cal H}^{\left(0\right)} $ consisting of $ 1 $-forms which satisfy $ \int_{\partial K_{i}}\alpha=0 $ for any $ i\in I $, and the subspace $ {\cal H}_{1}^{\left(0\right)} $ of forms satisfying $ \int_{\partial K_{i}}\alpha=\int_{\partial K_{i}'}\alpha $ for any $ i\in I $. Let $ {\frak c}={\cal H}^{\left(0\right)}/{\cal H}_{0}^{\left(0\right)} $, $ {\frak c}''={\cal H}_{1}^{\left(0\right)}/{\cal H}_{0}^{\left(0\right)} $. The decompositon $ \alpha=\alpha^{\left(0\right)}+\alpha^{\left(1\right)} $ gives a splitting of $ {\cal H}^{\left(0\right)} $ into a direct sum of $ {\cal H}_{0}^{\left(0\right)} $ and the span of $ \bar{\partial}^{-1}\mu_{i} $, $ i\in I $. As a corollary, the images of forms $ \bar{\partial}^{-1}\mu_{i} $ form a basis in $ {\frak c} $, and the Gram matrix of the pairing in $ {\frak c} $ can be calculated as $ \left(\bar{\partial}^{-1}\mu_{i},\bar{\partial}^{-1}\mu_{j}\right)_{L_{2}\left({\Bbb C}P^{1}\right)} $. Similarly, projections of $ \bar{\partial}^{-1}\mu_{i}-\bar{\partial}^{-1}\mu_{i'} $, $ i\in I_{+} $, form a basis in $ {\frak c}'' $, and one can easily calculate the Gram matrix of this basis. Let us estimate elements of these Gram matrices. We may assume that $ \infty\in K_{i_{0}} $ for some $ i_{0}\in I $, and that $ \operatorname{Supp}\beta_{0}\subset K_{i_{0}} $. Then the operator $ \bar{\partial}^{-1} $ restricted to $ {\Bbb C}P^{1}\smallsetminus K_{i_{0}} $ has $ \frac{dx}{y-x} $ as a kernel, and we may suppose that $ \mu_{i} $ is $ d\vartheta_{i}/2\pi $, $ \vartheta_{i} $ being the natural angle coordinate on $ \partial K_{i}\subset{\Bbb C} $. Let us calculate the Gram matrix $ \left(G_{ij}\right) $ for $ \bar{\partial}^{-1}d\vartheta_{i}\in{\cal H}^{\left(0\right)} $. If the radius of $ K_{i} $ is $ r_{i} $, and the distance between centers of $ K_{i} $ and $ K_{j} $ is $ d_{ij} $, then $ G_{ij} $ can be estimated as \begin{equation} \int\Sb r_{i}<|z|<C \\ |z-d_{ij}|>r_{j}\endSb\frac{1}{z\overline{\left(z-d_{ij}\right)}}dx\,dy \notag\end{equation} here $ C\gg0 $ (and depends on $ \beta_{0} $), $ z=x+iy. $ Note that the integral over $ |z|>C/2 $ is greater than a constant which does not depends on $ d_{ij} $ and $ r_{i},r_{j} $, thus this integral is bounded from below. If $ i=j $, then it behaves as $ \log \frac{1}{r} $. If $ i\not=j $, then the part outside the disk of radius $ 2d_{ij} $ is $ \sim\log \frac{1}{d_{ij}} $, and the part inside this disk is scaling-invariant. Thus to estimate the second part one may assume $ d_{ij}=1 $. However, the integral converges absolutely inside the whole disk $ |z|<2 $, thus this part is bounded. Thus we have estimates for the elements of the Gram matrix\footnote{Note that these estimates may be not sufficient to describe the Hilbert structure on $ {\frak c} $ up to equivalence.}. Knowledge of this Gram matrix gives a complete description of the subspace $ {\cal B} $ of $ H^{-1}\left({\Bbb C}P^{1},\omega\otimes\bar{\omega}\right) $ spanned by extensions-by-$ \delta $-function of forms on $ \partial K_{i} $, $ i\in I $. Similarly one can estimate the elements of the Gram matrix for $ {\frak c}'' $. Since elements of $ {\frak c}'' $ correspond to $ 2 $-forms with integral 0, this Gram matrix does not change when we apply a conformal transformation to disks $ K_{i} $. The diagonal elements are $ \frac{l_{i i'}}{\pi} $, and the off-diagonal ones are $ C \frac{1}{\pi}\log |\lambda\left(c_{i},c_{i'},c_{j},c_{j'}\right)|+O\left(\sum\varepsilon_{k}r_{k}\right) $, here $ \lambda $ is the double ratio, $ c_{\bullet} $ and $ r_{\bullet} $ are the center and radius of $ K_{\bullet} $. The constants $ \varepsilon_{\bullet} $ can be calculated as $ c_{i}=\frac{c_{j}-c_{j'}}{\left(c_{i}-c_{j}\right)\left(c_{i}-c_{j'}\right)} $. \begin{remark} Note that we have estimates for the elements of this Gram matrix. It is easy to check that if the matrix $ \left(r_{i}/d_{ij}\right)_{i,j\in I} $ gives a compact operator $ l_{2} \to l_{2} $, these estimates allow one to reconstruct the Hilbert norm up to equivalence, thus to describe the space $ {\frak c}'' $ completely. \end{remark} Note that the diagonal entries of the Gram matrix for $ {\frak c} $ are not bounded. Since any element in the image of the integration mapping has a finite norm w.r.t. this matrix, this shows, in particular, that the image of the integration mapping does not coincide with $ l_{2} $. Similarly, the Gram matrix for $ {\frak c}'' $ contains arbitrarily big elements, thus $ {\frak c}'' $ also differs from $ l_{2} $, thus image of partial period mapping differs from $ l_{2} $. As we will see it in Section~\ref{s9.40}, this image coincides with the space $ {\frak c}' $ defined below. So far the spaces we consider depended on the relative position of disks $ K_{i} $ only. Now suppose that a fraction-linear orientation-changing identification $ \varphi_{i} $ of $ \partial K_{i} $ and $ \partial K_{i'} $ is fixed, and consider the subspace $ {\cal B}' $ of $ {\cal B} $ which is spanned by extensions-by-$ \delta $-function of forms on $ \partial K_{i} $, $ i\in I $, which are preserved by the identifications $ \varphi_{i} $. Let $ {\cal B}_{0} $ be the subspace of $ {\cal B} $ spanned by extensions-by-$ \delta $-function of forms on $ \partial K_{i} $, $ i\in I $, with integral 0 along each $ \partial K_{i} $, let $ {\cal B}_{0}'={\cal B}'\cap{\cal B}_{0} $. It is clear that $ {\cal B}/{\cal B}_{0}\simeq{\frak c} $, Let us describe $ {\cal B}'/{\cal B}_{0}' $. Obviously, the image of $ {\cal B}'/{\cal B}_{0}' $ in $ {\frak c} $ is a subspace of $ {\frak c}'' $, however, if the identifications $ \varphi_{i} $ ``squeeze'' the circles $ \partial K_{i} $ too much, this image may be a proper subspace of $ {\frak c}'' $. Indeed, the dual statement to Remark~\ref{rem35.35} shows that one can define a $ \operatorname{PGL}\left(2,{\Bbb R}\right) $-invariant norm on $ H_{\int=0}^{-1/2}\left(S^{1},\Omega^{\text{top}}\right) $ which is equivalent to Hilbert norm. However, there is no $ \operatorname{PGL}\left(2,{\Bbb R}\right) $-invariant norm on the whole space $ H^{-1/2}\left(S^{1},\Omega^{\text{top}}\right) $. For $ g\in\operatorname{PGL}\left(2,{\Bbb R}\right) $ consider how $ g $ changes the Sobolev norm on $ H^{-1/2}\left(S^{1},\Omega^{\text{top}}\right) $. It is enough to estimate $ N\left(g\right)=\|g^{*}\left(\beta\right)\| $, $ \beta $ being an arbitrary form with non-zero integral. It is clear that different choices of Sobolev norm on $ H^{-1/2}\left(S^{1},\Omega^{\text{top}}\right) $\footnote{Recall that it is defined up to equivalence only.} and different choices of $ \beta $ would change $ N\left(g\right) $ to something of the form $ CN\left(g\right)+O\left(1\right) $ only. Since $ \operatorname{PGL}\left(2,{\Bbb R}\right) $ has a compact subgroup $ U\left(1\right) $ of dimension 1, it is enough to estimate $ N\left(g\right) $ on double classes $ U\left(1\right)\backslash\operatorname{PGL}\left(2,{\Bbb R}\right)/U\left(1\right) $ only, thus one can assume $ g=\operatorname{diag}\left(\lambda,1\right) $, $ \lambda\geq1 $. In turn, it is sufficient to estimate $ H^{-1/2} $-norm of $ \frac{\lambda dx}{1+\lambda^{2}x^{2}} $ on $ {\Bbb R} $, and one can do it explicitly since the Fourier transfrom can be easily calculated, it is proportional to $ e^{-\lambda|\xi|} $. Thus $ N\left(g\right)=C \log \lambda + O\left(1\right) $. Using this, one can easily obtain \begin{proposition} Consider the image $ {\frak c}' $ of $ {\cal B}'/{\cal B}_{0}' $ in $ {\frak c}={\cal B}/{\cal B}_{0} $. Consider $ \varphi_{i} $ as elements of $ \operatorname{SL}\left(2,{\Bbb R}\right)\subset{\Bbb R}^{4} $. The space $ {\frak c}' $ consists of sequences $ \left(p_{i}\right)\in{\frak c}'' $ which satisfy $ \left(p_{i}\log \|\varphi_{i}\|\right)\in l_{2} $. \end{proposition} We see that $ {\frak c}' $ can be described by Gram matrix $ G'_{ij}=\delta_{ij}\log \|\varphi_{i}\|+G_{ij}'' $, $ G'' $ being the Gram matrix for $ {\frak c}'' $. \subsection{$ B $-periods }\label{s9.41}\myLabel{s9.41}\relax What we defined in Section~\ref{s9.20} was integration of global holomorphic forms along $ A $-cycles. Finite-genus theory shows that it is important to study additional integrals along $ B $-cycles. They should depend on the choice of cuts on the Riemann surface, and we are not in the conditions when one can easily proceed with such cuts. Since we have only $ H^{0} $-smoothness of global holomorphic forms, one cannot invent a priori bounds on integrals of these forms along arbitrary curves. Indeed, the ``infinity'' $ M_{\infty}\subset{\Bbb C}P^{1} $ can break the complex sphere into infinitely many connected components (see the example in Section~\ref{s0.40}), thus one cannot assume that the cuts do not intersect $ M_{\infty} $. Moreover, even if there is only one connected component, it is not clear how to make infinitely many cuts in $ {\Bbb C}P^{1} $ which would not intersect each other. In fact it {\em is\/} possible to make such cuts, but in general the lengths of these cuts form a quickly increasing sequence. However, if one has a strip $ \left(-\varepsilon,\varepsilon\right)\times\left(a,b\right) $ embedded into the complex curve, then the value of $ \int_{a}^{b} $ averaged along $ \left(-\varepsilon,\varepsilon\right) $ is well defined and may be bounded as $ O\left(\varepsilon^{-1/2}\right) $. Indeed, this average is $ L_{2} $-pairing \begin{equation} \frac{1}{2\varepsilon}\int\alpha\wedge dx \notag\end{equation} with $ dx/\varepsilon $, here $ \left(x,y\right) $ are coordinates on the strip $ -\varepsilon<x<\varepsilon $. Thus we are not going to define the $ B $-periods as integrals over curves, but as some averaged integrals. On the level of would-be homology of the surface we will pair the form with cycles on the surface which are not ``geometric'' cycles, but linear combinations of them. Consider the involution ' and gluings $ \varphi_{i} $. Suppose that $ 0\in K_{i_{0}} $, $ \infty\in K_{i_{0}'} $, and that the disks $ \left\{K_{i}\right\} $ have a uniform thickening. Parameterize the set of rays going from 0 to $ \infty $ by the angle $ \vartheta $, and deform each ray slightly in such a way that it avoids all the circles $ K_{i} $. We require that deformations of two rays which differ by the angle $ \Delta\vartheta $ do not become closer than $ \varepsilon\cdot\Delta\vartheta $ (outside of $ K_{i_{0}} $). Say, let $ \widetilde{K}_{i} $ be the concentric with $ K_{i} $ disk of radius $ \left(1+2\varepsilon\right)\operatorname{radius}\left(K_{i}\right) $, and deform the ray $ {\cal R} $ inside $ \widetilde{K}_{i} $ so that it moves along an appropriate arc going between $ K_{i} $ and $ \widetilde{K}_{i} $. If the $ \operatorname{dist}\left({\cal R},\operatorname{center}\left(K_{i}\right)\right)/\operatorname{radius}\left(\widetilde{K}_{i}\right)=\rho\leq1 $, one can take the radius of the arc to be $ \left(1+\varepsilon+\varepsilon\rho\right)\operatorname{radius}\left(K_{i}\right) $. In fact we need to choose whether the arcs are going to leave the disk $ K_{i} $ on the right or on the left. We use the following algorithm: let $ O_{i} $ be the fixed point of $ \varphi_{i} $ inside $ K_{i} $. If the ray $ {\cal R} $ leaves $ O_{i} $ on the right, let the deformation leave $ K_{i} $ on the right, otherwise leave it on the left. (This choice is going to be important in the proof of Proposition ~\ref{prop8.33}. Note that the choice of the direction of the turnout does not depend on the metric on $ {\Bbb C}P^{1} $.) Now we assume that 0 and $ \infty $ are fixed points of $ \varphi_{i_{0}} $. As a last correction, in a neighborhood of $ K_{i_{0}'} $ change a ray $ {\cal R} $ to a logarithmic spiral so that the intersection of $ {\cal R} $ with $ \partial K_{i_{0}} $ and intersection of $ {\cal R} $ with $ \partial K_{i_{0}'} $ are glued together by $ \varphi_{i_{0}} $. Say, if $ \partial K_{i_{0}'} = \left\{z \mid |z|=R\right\} $, take the part of a spiral $ d\vartheta = C\,dr/\varepsilon r $ inside $ \left\{z \mid e^{-\varepsilon}R <|z|<R\right\} $, here $ C=\operatorname{Arg} \varphi_{i_{0}} $ is defined up to addition of a multiple of $ 2\pi $. Now each deformed ray represents a closed curve after the gluing by $ \varphi_{i_{0}} $ is performed. Under these conditions the averaged over $ \vartheta\in\left[0,2\pi\right] $ integral over the deformed rays (between $ \partial K_{i_{0}} $ and $ \partial K_{i_{0}'} $) is correctly defined. It represents a combination of cycles, thus is a cycle itself. Denote this linear functional on $ {\cal H}^{\left(0\right)}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ by $ {\cal Q}_{i_{0}} $. Restricting this linear functional to global holomorphic forms, we call the integral along this cycle $ B $-{\em period\/} of the global holomorphic form {\em from\/} $ \partial K_{i_{0}} $ {\em to\/} $ \partial K_{i_{0}'} $. The above description shows that \begin{lemma} $ B $-period of $ \alpha $ from $ \partial K_{i} $ to $ \partial K_{i'} $ is bounded by $ C\cdot l_{i i'}\|\alpha\|_{L_{2}} $, here $ l_{i i'} $ is the conformal distance between $ \partial K_{i_{0}} $ and $ \partial K_{i_{0}'} $. The constant $ C $ depends on $ \varepsilon $ only. \end{lemma} \begin{definition} Associate to $ \alpha\in\Gamma\left(\bar{M},\omega\right) $ the sequence $ \left(q_{j}\right)_{j\in I_{+}} $, $ q_{j} $ being the $ B $-period of $ \alpha $ from $ \partial K_{j} $ to $ \partial K_{j'} $. Denote this mapping by $ Q $. \end{definition} \begin{remark} Note that in the case of finite genus one chooses the $ B $-cycles to be non-intersecting. This assures that the matrix of periods is symmetric. The above construction takes average of different cycles connecting given points, thus one can get an impression that the resulting matrix of periods will have much worse properties than in the standard settings. However, the properties turn out to be exactly the same, due to our choice of direction of turnouts, and the following surprising result: \end{remark} \begin{proposition} Consider 4 different points $ x_{0},x_{1},y_{0},y_{1}\in{\Bbb C}P^{1} $. Parameterize the set of (circular) arcs connecting $ x_{0} $ with $ x_{1} $ by the angle at $ x_{0} $, and do the same with arcs connecting $ y_{0} $ with $ y_{1} $. This provides a measure on the set of arcs connecting $ x_{0} $ with $ x_{1} $, same for $ y_{0} $ and $ y_{1} $. Then the average index of intersection of an arc $ x_{0}x_{1} $ with an arc $ y_{0}y_{1} $ is 0. \end{proposition} \subsection{Space of periods }\label{s9.40}\myLabel{s9.40}\relax In Section~\ref{s9.12} we have shown that under mild assumption the space $ {\cal H}^{\left(0\right)} $ (which, loosely speaking, consists of holomorphic forms with jumps along the cuts on the curve) is a sum of two components: forms with integrals 0 along $ A $-cycles (which can be described by locale data on each cut), and some explicitely defined Hilbert subspace $ {\frak c} $ of ``small'' dimension (two basis vectors per each cut on the curve). Only the elements of $ {\cal H}^{\left(0\right)} $ which have equal integrals along two sides of the cut have a chance to correspond to a global holomorphic form on $ \bar{M} $, thus only subspace $ {\frak c}'\subset{\frak c} $ is interesting for us. \begin{theorem} Suppose that the matrix $ \left(e^{-l_{ij}}-\delta_{ij}\right) $ defines a compact operator $ l_{2} \to l_{2} $, and the disks $ K_{i} $ have a thickening. Then the partial period mapping $ P $ sends global holomorphic forms to elements of $ {\frak c}' $. Moreover, it is an isomorphism onto $ {\frak c}' $. \end{theorem} \begin{proof} The first statement is an immediate corollary of the description of the elements of $ {\cal H}^{\left(0\right)} $ via the space $ {\cal B}' $ in Section~\ref{s9.12}. We start the proof of the second one by showing that $ P $ is a component of a Fredholm operator of index 0. In the notations of Section~\ref{s9.12}, consider the mapping $ {\cal H}_{1}^{\left(0\right)} \xrightarrow[]{\pi} {\frak c}'' $. Let $ {\cal H}_{2}^{\left(0\right)}=\pi^{-1}\left({\frak c}'\right) $. Then the mapping \begin{equation} {\cal J}_{\omega}\colon {\cal H}_{2}^{\left(0\right)} \to \bigoplus\Sb l_{2} \\ i\in I_{+}\endSb H_{\int=0}^{-1/2}\left(\partial K_{i},\Omega_{\partial K_{i}}^{1}\right)\colon \alpha \mapsto \left(\alpha|_{\partial K_{i}}-\varphi^{*}\left(\alpha|_{\partial K_{i'}}\right)\right). \notag\end{equation} is continuous. The same arguments as in the proof of Theorem~\ref{th35.45} show that $ {\cal J}_{\omega}|_{{\cal H}_{0}^{\left(0\right)}} $ is a Fredholm operator of index 0. Combining this mapping with the projection $ {\cal H}_{1}^{\left(0\right)} \xrightarrow[]{\pi} {\cal H}_{2}^{\left(0\right)}/{\cal H}_{0}^{\left(0\right)}={\frak c}' $, we see that $ \alpha \mapsto \left({\cal J}_{\omega}\left(\alpha\right),\pi\left(\alpha\right)\right) $ is a Fredholm mapping of index 0. If $ \alpha $ is in the null-space of this mapping, then $ \alpha $ is a non-trivial global holomorphic form with vanishing $ A $-periods, thus $ \alpha\in\operatorname{Ker} P $, and $ \alpha=0 $. Hence this mapping is an isomorphism. Since $ P=\pi|_{\operatorname{Ker}{\cal J}_{\omega}} $, it is an isomorphism as well. \end{proof} Let $ \alpha_{i}=\bar{\partial}^{-1}\beta_{i} $. The theorem says that for any element $ \left(c_{i}\right)\in{\frak c}' $ one can find an element $ \alpha\in{\cal H}^{\left(0\right)} $ such that integrals of $ \alpha $ along $ \partial K_{i} $ are 0 for any $ i\in I $, and $ \sum c_{i}\alpha_{i}+\alpha $ is a global holomorphic form (which automatically has $ A $-periods $ c_{i} $). Moreover, $ \|\alpha\|_{{\cal H}^{\left(0\right)}}\leq C\cdot\|\left(c_{i}\right)\|_{{\frak c}'} $. \begin{proposition} \label{prop9.42}\myLabel{prop9.42}\relax Let $ \beta\in{\cal H}^{\left(0\right)} $, and integrals of $ \beta $ along $ \partial K_{i} $ are 0 for any $ i\in I $, let $ \left(c_{i}\right)\in{\frak c}' $. Then $ \left|\sum_{j\in I_{+}}c_{j}{\cal Q}_{j}\left(\beta\right)\right| \leq C\|\beta\|_{{\cal H}^{\left(0\right)}}\cdot\|\left(c_{i}\right)\|_{{\frak c}'} $ for an appropriate $ C $ which does not depend on $ \beta $ and $ \left(c_{i}\right) $. \end{proposition} \begin{proof} Suppose that $ \beta $ can be holomorphically continued into all disks $ K_{i} $ except $ K_{i_{0}} $. Then $ \beta=\partial f $, and $ f $ is holomorphic outside $ K_{i_{0}} $. Let $ j,j'\not=i_{0} $. The construction of $ {\cal Q}_{j} $ shows that one can calculate $ {\cal Q}_{j}\left(\beta\right) $ as $ f\left(y_{1}\right)-f\left(y_{0}\right) $, $ y_{0} $, $ y_{1} $ being two fixed points of $ \varphi_{j} $. Let $ \Psi_{j} $ be the fraction-linear function with a zero and a pole at $ y_{0} $ and $ y_{1} $. One can momentarily see that $ {\cal Q}_{j}\left(\beta\right)=\int_{\partial K_{i_{0}}}\beta\log \Psi_{j} $. Similarly, $ {\cal Q}_{i_{0}}\left(\beta\right)=0 $. Similar statements are true for forms $ \beta $ such that they can be holomorphically continued into all the disks $ K_{i} $ except a finite number. Since any form $ \beta $ which satisfies conditions of the proposition can be approximated by such forms, we see that it is sufficient to show that $ \sum_{j\not=i,i'}c_{j}\log \Psi_{j} $ converges in $ \bigoplus_{l_{2}}H^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $, or that $ \sum_{j\not=i,i'}c_{j}d\Psi_{j}/\Psi_{j} $ converges in $ \bigoplus H^{-1/2}\left(\partial K_{i},\Omega^{1}\right) $. In turn, it is sufficient to show convergence of $ \sum_{j\not=i,i'}c_{j}d\Psi_{j}/\Psi_{j} $ in $ \bigoplus_{l_{2}}H^{0}\left(K_{i},\omega\right)=L_{2}\left(\bigcup K_{i},\omega\right) $. On the other hand, if $ \sigma_{j} $ is 1 in $ K_{j} $ and $ K_{j'} $, and 0 otherwise, then $ \bar{\partial}\sum_{j\not=i,i'}c_{j}\sigma_{j}d\Psi_{j}/\Psi_{j} =\sum_{j\not=i,i'}c_{j}\beta_{j}\in H^{-1} $, hence $ \sum c_{j}\sigma_{j}d\Psi_{j}/\Psi_{j} $ converges in $ L_{2} $. \end{proof} \begin{proposition} \label{prop9.25}\myLabel{prop9.25}\relax Let $ \left(d_{i}\right),\left(c_{i}\right)\in{\frak c}' $, and $ \alpha\in{\cal H}^{\left(0\right)} $ such that integrals of $ \alpha $ along $ \partial K_{i} $ are 0 for any $ i\in I $, and $ \alpha_{c}=\sum c_{i}\alpha_{i}+\alpha $ is a global holomorphic form on $ \bar{M} $ (automatically with $ A $-periods $ c_{i} $). Then $ \sum d_{j}{\cal Q}_{j}\left(\alpha_{c}\right) \leq C\|\left(c_{i}\right)\|_{{\frak c}'}\cdot\|\left(d_{i}\right)\|_{{\frak c}'} $. \end{proposition} \begin{proof} It is sufficient to show that $ \sum d_{j}{\cal Q}_{j}\left(\sum c_{i}\alpha_{i}\right)\leq C_{1}\|\left(c_{i}\right)\|_{{\frak c}'}\cdot\|\left(d_{i}\right)\|_{{\frak c}'} $. In the notations of the previous proposition $ \alpha_{i}-\alpha_{i'}=d\Psi_{i}/\Psi_{i} $ outside of $ K_{i} $ and $ K_{i'} $. One can assume that only a finite number of $ c_{i} $ and $ d_{i} $ is non-zero. Then $ \sum_{i\in I}c_{i}\alpha_{i} $ is $ \sum_{i\in I_{+}}c_{i}d\Psi_{i}/\Psi_{i} $. Now it should be obvious that $ {\cal Q}_{j}\left(\alpha_{i}\right)=\int\sigma_{i}\sigma_{j}\alpha_{i}\wedge\alpha_{j} $, which finishes the proof. \end{proof} \subsection{Period matrix }\label{s9.60}\myLabel{s9.60}\relax Since $ P $ is an isomorphism $ \Gamma\left(\bar{M},\omega\right) \to {\frak c}' $, one can consider the mapping $ \Omega=Q\circ P^{-1} $. Write this mapping using coordinate ``basis'' in $ {\frak c}' $: \begin{definition} Let $ \widetilde{\alpha}_{j} $, $ j\in I_{+} $, be the global holomorphic form on $ \bar{M} $ such that $ P\left(\widetilde{\alpha}_{i}\right) $ has 1 on $ j $-th position, $ -1 $ at $ j' $-position, 0 at the other positions. Let $ \Omega_{ij} $ be the $ B $-period of $ \widetilde{\alpha}_{j} $ from $ \partial K_{i} $ to $ \partial K_{i'} $, $ i\in I_{+} $. \end{definition} \begin{proposition} \label{prop8.33}\myLabel{prop8.33}\relax Let $ \alpha\in\Gamma\left(\bar{M},\omega\right) $ and the sequence $ \left(p_{j}\right)=P\left(\alpha\right) $ has only a finite number of non-zero elements. Let $ Q\left(\alpha\right)=\left(q_{j}\right) $. Then $ \|\alpha\|_{L_{2}}^{2}= i\sum_{I_{+}}\left(\bar{p}_{j}q_{j}-p_{j}\bar{q}_{j}\right) $. \end{proposition} \begin{proof} Consider a representative of $ \alpha $ in $ {\cal H}^{\left(0\right)}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $. Since all the $ A $-periods of $ \alpha $ but a finite number are 0, $ \alpha $ can be extended (without changing the norm too much) into all the disks but a finite number preserving the closeness. Consider the remaining disks. Since the integral of $ \alpha $ around $ K_{i} $ is opposite to the integral around $ K_{i'} $, we see that if we connect $ K_{i} $ and $ K_{i'} $ by a cut, then the integral of $ \alpha $ around the resulting hole is 0. Make smooth cuts which connect the remaining disks pairwise (according to $ ' $) and do not intersect. One can suppose that two ends of the cut---one on $ \partial K_{i} $, another on $ \partial K_{i'} $---are identified by $ \varphi_{i} $. After the cuts are performed, on the resulting domain $ 1 $-form $ \alpha $ is closed, and the integral along any component of (piecewise-smooth) boundary is 0. Thus one can write $ \alpha = df $, $ f $ being a function of smoothness $ H^{1} $. The restriction of $ f $ to any smooth curve is well-defined, and is of smoothness $ H^{1/2} $. When one goes from one side of the cut to another one along $ \partial K_{i} $, $ f $ grows by $ p_{i} $. Let $ \overset{\,\,{}_\circ}{q}_{i} $ be the change of $ f $ when one goes from $ \partial K_{i} $ to $ \partial K_{i'} $ along the cut (choosing the side of the cut so that the direction is counterclockwise). Note that $ f $ is holomorphic near $ \partial K_{i} $ and $ \partial K_{i'} $ thus the {\em value\/} of $ f $ at points is well-defined, thus the change of $ f $ along the cut is well-defined. (In generic point $ z $ of the cut $ f\left(z\right) $ is not correctly defined, since $ f $ is only of smoothness $ H^{1/2} $.) Let $ \gamma_{i} $ be the part of the boundary of the domain consisting of the circles $ \partial K_{i} $, $ \partial K_{i'} $ and both sides of the cut which connects them. Now take into account that $ \|\alpha\|_{L_{2}}^{2}=i\int\alpha\wedge\bar{\alpha}=i\int\partial f\wedge\bar{\partial}\bar{f} $. Proceeding as in Section~\ref{s7.90}, we see that the only change to the arguments is that instead of taking integrals along $ \partial K_{i}\cup\partial K_{i'} $, one needs to take some integrals along $ \gamma_{i} $. As there, the integral along $ \partial K_{i}\cup\partial K_{i'} $ vanishes, so what remains is \begin{equation} \|\alpha\|_{L_{2}}^{2}=-i\sum_{i}\int_{\gamma_{i}}f\,d\bar{f} \notag\end{equation} summation being over $ i $ such that $ p_{i}\not=0 $. The cycle $ \gamma_{i} $ consists of 4 parts: two going around $ \partial K_{i} $ and $ \partial K_{i'} $, another two going along sides of the cut. On the first two parts $ d\bar{f} $ are identified via $ \varphi_{i} $ (with opposite signs), and $ f $ differs by $ \overset{\,\,{}_\circ}{q}_{i} $, thus the total integral is $ -\bar{p}_{i}\overset{\,\,{}_\circ}{q}_{i} $. On the second two $ d\bar{f} $ coincide (but the orientation is opposite), and $ f $ differs by $ p_{i} $, thus the total integral is $ p_{i}\overline{\overset{\,\,{}_\circ}{q}_{i}} $. What remains to prove is that we may substitute $ q_{i} $ instead of $ \overset{\,\,{}_\circ}{q}_{i} $. To do this one needs to investigate the relationship between $ q_{i} $ and $ \overset{\,\,{}_\circ}{q} $. First, one can describe $ \overset{\,\,{}_\circ}{q}_{i} $ as an integral of $ df $ along one side of the cut. Indeed, though $ d\left(f|_{\gamma_{i}}\right) $ is of smoothness $ H^{-1/2} $ on each part of $ \gamma_{i} $, it is actually analytic near circles $ \partial K_{i} $, $ \partial K_{i'} $, thus the pairing with the fundamental cycle of the interval (which has jumps at the ends of the interval!) is well-defined. Second, one can suppose that the cuts are in fact piecewise-smooth (as far as non-smooth points are in $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $), and consist of arcs of circles. Let us recall that $ q_{i} $ is the average integral of $ \alpha $ along ``rays'' which connect 0 and $ \infty $ (after an appropriate choice of coordinate system). Here ``rays'' are curves with differ from rays inside circles $ \widetilde{K}_{j} $ only, and consist of arcs of circles. What is more, one does not need to deform a ray into a ``ray'' inside the circle $ K_{j} $ as far as $ p_{j}=0 $, since a closed continuation of $ \alpha $ inside $ K_{j} $ is already fixed, thus the deformation does not change the value of the integral along the ``ray''. Note that each ray intersects cuts along a finite number of points (except a finite number of rays which may contain whole pieces of cuts), thus the same is true for ``rays''. We see that the integral of $ \alpha $ along each ``ray'' is now well defined (recall that in general setting only the average was well-defined), moreover, it is easy to calculate this integral using representation $ \alpha=df. $ The integral along a ``ray'' $ {\cal R} $ is equal to the change of $ f $ on the ends minus the jumps of $ f $ at the finite number of points where ``ray'' intersects cuts, thus it is \begin{equation} f\left(\operatorname{end}\left({\cal R}\right)\right)-f\left(\operatorname{start}\left({\cal R}\right)\right)-\sum_{j}n_{ij}p_{j},\qquad n_{ij}\in{\Bbb Z}\text{, }i,j\in I. \notag\end{equation} The change of $ f $ is equal to $ \overset{\,\,{}_\circ}{q}_{i} $, thus the average value of the integral is $ \overset{\,\,{}_\circ}{q}_{i} $ minus sum of some real multiples of $ p_{j} $: \begin{equation} q_{i}=\overset{\,\,{}_\circ}{q}_{i}-\sum_{j}\nu_{ij}p_{j},\qquad \nu_{ij}\in{\Bbb R}\text{, }i,j\in I, \notag\end{equation} and $ \nu_{ij} $ are averaged values of $ n_{ij} $. Finally, use our choice of direction of turnout around the disk $ K_{i} $. It insures that the following fact is true: \begin{lemma} $ \nu_{ij}=\nu_{ji} $ if $ i\not=i $. \end{lemma} \begin{proof} Extend the cut between $ \partial K_{j} $ and $ \partial K_{j'} $ to fixed points of $ \varphi_{j} $ (one inside each of $ K_{j} $ and $ K_{j'} $) along straight intervals. Call the resulting curve $ \gamma'_{j} $. Then $ 2\pi\nu_{ij} $ is the change of $ \operatorname{Arg} z $ along the curve $ \gamma_{j}' $ (we again suppose that 0 and $ \infty $ are fixed points of $ \varphi_{i} $). Similarly, $ \pi\nu_{ji} $ is the change of $ \operatorname{Arg}\Psi_{j}\left(z\right) $ along $ \gamma'_{i} $, here $ \Psi_{j}\left(z\right) $ is a fraction-linear function with a pole and a zero at fixed points of $ \varphi_{j} $. We need to show that $ \operatorname{Im} \int_{\gamma_{j}'}\frac{dz}{z}+\int_{\gamma_{i}'}\frac{d\Psi_{j}\left(z\right)}{\Psi_{j}\left(z\right)}=0 $. However, \begin{equation} \int_{\gamma_{j}'}\frac{dz}{z}=\frac{1}{2\pi i} \int_{\gamma_{j}'}\frac{dz}{z}\operatorname{Jump}\left(\log \Psi_{j}\left(z\right)\right) =\frac{1}{2\pi i}\int_{\gamma_{j}''}\log \Psi_{j}\left(z\right) d \log z \notag\end{equation} here we take an arbitrary branch of $ \log \Psi\left(z\right) $ defined outside of $ \gamma_{j}' $, and $ \gamma_{j}'' $ is a loop around $ \gamma_{j}' $. Now the identity is obvious, since $ \gamma_{j}'' $ is homotopic to $ -\gamma_{i}'' $. \end{proof} The lemma implies that plugging in $ \overset{\,\,{}_\circ}{q}_{i} $ instead of $ q_{i} $ into the formula of the proposition gives the same value, which finishes the proof of Proposition~\ref{prop8.33}. {} \end{proof} \begin{proposition} \label{prop8.35}\myLabel{prop8.35}\relax Let $ \alpha,\beta\in\Gamma\left(\bar{M},\omega\right) $ and the sequences $ \left(p_{j}\right)=P\left(\alpha\right) $, $ \left(p_{j}'\right)=P\left(\beta\right) $ have only a finite number of non-zero elements. Let $ Q\left(\alpha\right)=\left(q_{j}\right) $, $ Q\left(\beta\right)=\left(q_{j}'\right) $. Then $ \sum_{I_{+}}\left(p_{j}q'_{j}-p'_{j}q_{j}\right) = $ 0. \end{proposition} \begin{proof} The proof of Proposition~\ref{prop8.33} with minor changes is applicable, the integral to consider is $ 0=\int\alpha\beta $. \end{proof} The following statement is an immediate corollary of Proposition ~\ref{prop9.25}: \begin{proposition} In the conditions of the previous proposition $ \sum_{I_{+}}p_{j}q'_{j}\leq C\|\alpha\|\cdot\|\beta\| $. \end{proposition} \begin{corollary} The period matrix $ \Omega_{ij} $ is symmetric, $ \operatorname{Im}\Omega_{ij} $ is a positive real symmetric matrix, and the matrix $ \Omega_{ij} $ defines a bounded symmetric form on the Hilbert space with the pairing given by the Gram matrix $ \operatorname{Im}\Omega_{ij} $. \end{corollary} \subsection{Bounded Jacobian as a torus }\label{s9.70}\myLabel{s9.70}\relax In Section~\ref{s7.90} we have seen that (under mild assumptions) the Jacobian coincides with the constant Jacobian. Given a topological space $ S $ and a set $ I $, let $ S_{l_{\infty}}^{I}\buildrel{\text{def}}\over{=}\bigcup_{K}K^{I} $, here $ K $ runs over compact subsets of $ S $. Obviously, there is a surjection from $ \left({\Bbb C}^{*}\right)_{l_{\infty}}^{I_{+}} $ to the constant bounded Jacobian, which sends a sequence $ \left(\psi_{i}\right)_{i\in I} $ such that $ \psi_{i'}=\psi_{i}^{-1} $ into a corresponding line bundle. Here we are going to show that the kernel of this surjection is a lattice in $ \left({\Bbb C}^{*}\right)_{l_{\infty}}^{I_{+}} $, as in finite-genus case. To avoid defining a lattice in $ \left({\Bbb C}^{*}\right)_{l_{\infty}}^{I_{+}} $, consider $ \left({\Bbb C}^{*}\right)_{l_{\infty}}^{I_{+}} $ as the set of coordinate-wise exponents of $ {\Bbb C}_{l_{\infty}}^{I_{+}} $. We obtain a mapping from $ {\Bbb C}_{l_{\infty}}^{I_{+}} $ to the bounded Jacobian which sends $ \left(2\pi i{\Bbb Z}\right)_{l_{\infty}}^{I_{+}} $ to the origin in the Jacobian. We are going to prove that there is a lattice $ L\subset{\Bbb C}^{I_{+}} $ which goes to the origin. First, construct generators of this lattice: \begin{proposition} \label{prop9.72}\myLabel{prop9.72}\relax Suppose that the matrix $ \left(e^{-l_{ij}}-\delta_{ij}\right) $ gives a compact operator $ l_{2} \to l_{2} $ and a compact operator $ l_{1} \to l_{1} $. Let $ j\in I_{+} $. There exists a global holomorphic form $ \alpha^{\left(j\right)} $ on $ \bar{M} $ such that the $ A $-periods of $ \alpha^{\left(j\right)} $ vanish except for the $ j $-th one, which is equal to $ 2\pi i $. Then $ a_{j}=\exp \int\alpha^{\left(j\right)} $ is an element of $ {\cal H}^{\infty} $. The corresponding cocycle $ \psi_{kj}=a_{j}\circ\varphi_{k}/a_{j} $ is (locally) constant, and coincides with $ \exp 2\pi i\Omega_{kj} $. \end{proposition} \begin{proof} Existence of $ \alpha^{\left(j\right)} $ is a corollary of results of Section~\ref{s9.40}. The only statement we need to prove is that $ a_{j}\in{\cal H}^{\infty} $. In turn, it is sufficient to prove that $ \operatorname{Im}\Omega_{kj} $ is bounded (for a fixed $ j $). On the other hand, $ \operatorname{Im}\Omega_{kj} $ consists of two parts which correspond to decomposition $ \alpha^{\left(j\right)}=2\pi i\alpha_{j}+o^{\left(j\right)} $, here $ \alpha_{j} $ is defined as in Section~\ref{s9.40}, and $ o^{\left(j\right)}\in{\cal H}^{\left(0\right)} $ and has $ A $-periods 0. The description of the cycle for $ B $-period shows that first part of $ \operatorname{Im}\Omega_{kj} $ is bounded by $ \log |l_{jj'}| $, thus it is sufficient to estimate the second part $ \operatorname{Im}{\cal Q}_{k}\left(o^{\left(j\right)}\right) $. This estimate follows from the following lemma: \begin{lemma} Suppose that the matrix $ \left(e^{-l_{ij}}-\delta_{ij}\right) $ gives a compact operator $ l_{2} \to l_{2} $ and a compact operator $ l_{1} \to l_{1} $. \begin{enumerate} \item Let $ \alpha\in H^{-1/2}\left(\partial K_{i},\Omega^{\text{top}}\right) $. Denote by the same letter extension-by-$ \delta $-function of $ \alpha $ to $ {\Bbb C}P^{1} $. Then $ \bar{\partial}^{-1}\alpha\in{\cal H}^{\left(0\right)} $, and $ {\cal Q}_{k}\left(\bar{\partial}^{-1}\alpha\right)=O\left(\|\alpha\|_{H^{-1/2}}\right) $ uniformly in $ k\not=i $. \item The sequence $ \left(\|o^{\left(j\right)}|_{\partial K_{k}}\|_{H^{-1/2}}\right)_{k\in I}\in l_{1} $. \end{enumerate} \end{lemma} \begin{proof} The first part follows from the explicit construction of cycles for $ B $-periods. Prove the second part. Since sum of radii of disks $ K_{i} $ is finite, one can estimate that the sequence $ \left(\|\alpha_{j}|_{\partial K_{k}}\|_{H^{-1/2}}\right)_{k\in I}\in l_{1} $. Denote the space of such $ 1 $-forms on $ \bigcup_{i}\partial K_{i} $ by $ {\cal H}_{1} $. On the other hand, $ o^{\left(j\right)} $ is uniquely determined by the conditions $ {\cal J}_{\omega}o^{\left(j\right)}=-2\pi i{\cal J}_{\omega}\alpha_{j} $ and $ o^{\left(j\right)}\in{\cal H}_{0}^{\left(0\right)} $, here $ {\cal J}_{\omega} $ is the operator from Section~\ref{s9.40}. What remains to prove is that $ {\cal J}_{\omega} $ sends the subspace $ {\cal H}_{0}^{\left(0\right)}\cap{\cal H}_{1} $ onto itself. We know that $ {\cal J}_{\omega} $ sends $ {\cal H}_{0}^{\left(0\right)} $ onto itself. However, from the restrictions on $ l_{ij} $ one immediately obtains that $ {\cal J}_{\omega} $ gives a Fredholm operatorar $ {\cal H}_{1} \to {\cal H}_{1} $ of index 0. Since $ {\cal J}_{\omega} $ has no kernel, this operator is an isomorphism. \end{proof} This finishes the proof of Proposition~\ref{prop9.72}. {}\end{proof} \begin{definition} Call a multiplicator $ a\in{\cal H}^{\infty} $ {\em lattice-like}, if $ a^{-1}\in{\cal H}^{\infty} $ and $ \psi_{k}=a^{-1}\cdot\left(a\circ\varphi_{k}\right) $ is a constant function on $ \partial K_{k} $ for every $ k\in I $. \end{definition} Now to each $ j\in I_{+} $ we associated a lattice-like multiplicator $ a_{j} $, which induces a cocycle $ \left(\psi_{k}\right)_{k\in I_{+}} $, $ \psi_{k}=\exp 2\pi i\Omega_{kj} $. The bundle given by this cocycle is isomorphic to a trivial one (via $ a_{j} $). Since $ \operatorname{Im}\Omega_{kj} $ gives a positive Hermitian form (defined at least on real sequences of finite length), columns of $ \operatorname{Im}\Omega_{kj} $ generate the subspace which is dense in the set of real sequences (with topology of direct product), which shows in conditions of Proposition~\ref{prop9.72} \begin{proposition} Consider the space $ {\Bbb C}_{l_{\infty}}^{I_{+}} $ of bounded sequences $ \left(\Psi_{k}\right) $ with topology induced from the direct product $ {\Bbb C}^{I_{+}} $. Consider the subgroup $ L $ generated by rows $ \left(\Omega_{kj}\right) $, $ j\in I_{+} $, and $ {\Bbb Z}_{l_{\infty}}^{I_{+}} $. This subgroup is a lattice, i.e., its $ {\Bbb R} $-span is dense in $ {\Bbb C}_{l_{\infty}}^{I_{+}} $. \end{proposition} To finish the description of bounded Jacobian, it remains to prove that any lattice-like multiplicator $ a $ is a product of powers of $ a_{j} $, $ j\in I_{+} $. Recall that the index of function $ a $ is the change of $ \frac{\operatorname{Arg} a}{2\pi i} $ along a contour. Note that the {\em index\/} of $ a_{j} $ around $ \partial K_{k} $ is $ \pm1 $ if $ k=j,j' $, and is 0 otherwise. Thus one reconstruct the needed powers of $ a_{j} $ by taking indices of $ a $ around $ \partial K_{j} $, $ j\in I_{+} $. The only things we need to prove is that a lattice-like multiplicator $ a $ with all indices around $ \partial K_{j} $ being 0 is constant, and only a finite number of indices of a lattice-like multiplicator is non-zero. The first statement is a direct corollary of Lemma~\ref{lm8.31} and the following \begin{lemma} \label{lm9.91}\myLabel{lm9.91}\relax Suppose that the conformal distances $ l_{ij} $ satisfy the condition that $ \left(e^{-l_{ij}}-\delta_{ij}\right) $ gives a bounded operator $ l_{\infty} \to l_{2} $. Then any lattice-like multiplicator is of class $ {\cal H}^{\left(1\right)} $. \end{lemma} Recall that the space $ {\cal H}^{\left(1\right)} $ was defined in Section~\ref{s35.30}. \begin{proof} Since $ a\,dz^{1/2}\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{j}\right\}\right) $, Theorem~\ref{th4.40} shows that $ a $ can be reconstructed from its restriction to $ \bigcup_{j\in I}\partial K_{j} $ using Cauchy formula. In particular, $ + $-part of $ \partial a $ on $ \partial K_{j} $ is given by an integral along $ \bigcup_{k\not=j}\partial K_{k} $. The restrictions on $ l_{ij} $ guarantie that the $ + $-parts of $ \partial a $ on the circles $ \partial K_{j} $ is in $ \oplus_{l_{2}}H^{-1/2}\left(\partial K_{j}\right) $ modulo constants. Since $ a^{-1}\cdot\left(a\circ\varphi_{k}\right) $ is constant and bounded, $ - $-parts of $ \partial a $ are also in this space. Since the integral of $ \partial a $ along each circle vanishes, arguments similar to ones in Section ~\ref{s35.30} show that $ \partial a\in L_{2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\omega\right) $, thus $ a\in{\cal H}^{\left(1\right)} $. \end{proof} Iteration of the procedure used in the proof of the lemma shows that \begin{amplification} \label{amp9.94}\myLabel{amp9.94}\relax Conclusions of Lemma~\ref{lm9.91} remain true if $ \left(e^{-l_{ij}}-\delta_{ij}\right) $ gives a bounded operator $ l_{\infty} \to l_{\infty} $, and some power of this operator sends $ l_{\infty} $ to $ l_{2} $. \end{amplification} To prove that only a finite number of indices of a lattice-like multiplicator $ a $ is non-zero, it is enough to show that $ a^{-1}\partial a\in L_{2}\left(U,\omega\right) $, $ U $ being homotopic to $ {\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $. Indeed, indices of $ a $ are proportional to periods of $ a^{-1}\partial a $, and periods of $ L_{2} $-form form a sequence in $ l_{2} $. In turn, since $ a^{-1} $ is a multiplicator, thus is bounded, this follows from the fact that $ \partial a\in L_{2}\left(U,\omega\right) $. We obtain \begin{theorem} Consider a family of non-intersecting disks $ K_{i}\subset{\Bbb C}P^{1} $ with pairwise conformal distances. Let $ {\cal R} $ be the matrix $ \left(e^{-l_{ij}}-\delta_{ij}\right) $. Suppose that $ {\cal R} $ gives a compact mapping $ l_{2} \to l_{2} $ and a bounded mapping $ l_{\infty} \to l_{\infty} $, and that some power of $ {\cal R} $ gives a bounded mapping $ l_{\infty} \to l_{2} $. If disks $ K_{i} $ have a uniform thickening, then the bounded Jacobian coincides with the quotient of $ {\Bbb C}_{l_{\infty}}^{I_{+}} $ by the lattice generated by $ {\Bbb Z}_{l_{\infty}}^{I_{+}} $ and rows of the period matrix $ \left(\Omega_{ij}\right) $. \end{theorem} \begin{remark} Formally speaking, we defined the Jacobians in the case when $ \left(e^{-l_{ij}/2}-\delta_{ij}\right) $ gives a compact operator $ l_{2} \to l_{2} $, thus the statement of theorem is abuse of notations. However, it is easy to define all the ingredients needed for the definition of the bounded Jacobian as far as the conditions of Amplification~\ref{amp9.94} are satisfied. \end{remark} Let us recall that the conditions of the theorem are automatically satisfied for any Hilbert--Schmidt curve, thus we get a complete description of the bounded Jacobian in this case. It is similar to the finite-genus case, where Jacobian is a quotient of a finite-dimensional complex vector space by a lattice. Moreover, note that the topology on the constant bounded Jacobian is inherited from the topology of direct product on $ {\Bbb C}^{I_{+}} $, thus the above description allows one to reconstruct the topology on the bounded Jacobian as well. \subsection{Rigged Hodge structure }\label{s9.80}\myLabel{s9.80}\relax Let us wrap the results of the previous section into the familiar form of Hodge structures. In fact the resulted structure will be a hybrid of a Hodge structure and a structure of a rigged topological vector space. Let $ I_{+} $ be an arbitrary set. Let $ H_{{\Bbb Z}}^{A} $, $ H_{{\Bbb R}}^{A} $, $ H_{{\Bbb C}}^{A} $ be the spaces of sequences $ \left(p_{i}\right)_{i\in I_{+}} $ with only a finite number of non-zero terms (with integer/real/complex terms), $ H_{{\Bbb Z}}^{B} $, $ H_{{\Bbb R}}^{B} $, $ H_{{\Bbb C}}^{B} $ be the spaces of sequences $ \left(q_{i}\right)_{i\in I_{+}} $ without any restriction on growth, and $ H_{{\Bbb Z}}= H_{{\Bbb Z}}^{A}\oplus H_{{\Bbb Z}}^{B} $, $ H_{{\Bbb R}}=H_{{\Bbb R}}^{A}\oplus H_{{\Bbb R}}^{B} $, $ H_{{\Bbb C}}=H_{{\Bbb C}}^{A}\oplus H_{{\Bbb C}}^{B} $. One can define an operation of complex conjugation on the spaces $ H_{{\Bbb C}}^{\bullet} $, this operation leaves $ H_{{\Bbb R}}^{\bullet} $ fixed. The spaces $ H_{{\Bbb Z}} $, $ H_{{\Bbb R}} $, $ H_{{\Bbb C}} $ have a natural symplectic structure \begin{equation} \left[\left(\left(p_{i}\right),\left(q_{i}\right)\right),\left(\left(p'_{i}\right),\left(q_{i}'\right)\right)\right] = \sum_{i}\left(p_{i}q_{i}'-p'_{i}q_{i}\right) \notag\end{equation} such that the components $ H_{\bullet}^{A,B} $ are Lagrangian and mutually dual. Fix an arbitrary mapping $ \Omega\colon H_{{\Bbb C}}^{A} \to H_{{\Bbb C}}^{B} $. Let $ \Omega_{1}\colon H_{{\Bbb C}}^{A} \to H_{{\Bbb C}}^{A}\oplus H_{{\Bbb C}}^{B}=H_{{\Bbb C}} $ be $ \operatorname{id}\oplus\Omega $. Denote $ \operatorname{Im}\Omega_{1}\subset H_{{\Bbb C}} $ by $ H^{1,0} $. Suppose that (compare Proposition~\ref{prop8.35}) $ H^{1,0} $ is Lagrangian. Moreover, suppose (compare Proposition~\ref{prop8.33}) that if $ \alpha\in H^{1,0} $, then $ \operatorname{Im} \left[\alpha,\bar{\alpha}\right]\gg0 $ in the following sense: \begin{enumerate} \item $ \operatorname{Im} \left[\alpha,\bar{\alpha}\right]\geq0 $ (here $ \alpha \mapsto \bar{\alpha} $ is the complex conjugation on $ H_{{\Bbb C}} $); \item the equality is achieved only if $ \alpha=0 $; \item the completion $ {\frak h}^{1,0} $ of $ H^{1,0} $ w.r.t. the norm $ \|\alpha\|^{2}=\operatorname{Im} \left[\alpha,\bar{\alpha}\right] $ has no vectors of length 0. \end{enumerate} Let $ H^{0,1} = \left\{\alpha\in H_{{\Bbb C}} \mid \bar{\alpha}\in H^{1,0}\right\} $. The mapping $ \Omega_{1} $ gives an inclusion $ i $ of $ H_{{\Bbb C}}^{A} $ into $ {\frak h}^{1,0} $. \begin{definition} A {\em rigged Hodge structure\/} is a subspace $ H^{1,0}\subset H_{{\Bbb C}} $ which satisfies the above conditions, and such that the projection $ p\colon H^{1,0} \to H_{{\Bbb C}}^{B} $ can be continuously extended to a mapping $ p\colon {\frak h}^{1,0} \to H_{{\Bbb C}}^{B} $. \end{definition} Note the mappings $ H_{{\Bbb C}}^{A} \xrightarrow[]{i} {\frak h}^{1,0} \xrightarrow[]{p} H_{{\Bbb C}}^{B} $ equip $ {\frak h}^{1,0} $ with a structure of a rigged topological vector space \cite{GelVil64Gen}. It is clear that $ H^{1,0} $ is a rigged Hodge structure iff $ \Omega $ is symmetric, $ \operatorname{Im}\Omega>0 $, $ -C\cdot\operatorname{Im}\Omega<\operatorname{Re}\Omega<C\cdot\operatorname{Im}\Omega $. From now on suppose that $ H^{1,0} $ is a rigged Hodge structure. Let $ {\frak h}^{0,1} $ be the complexly conjugated space to $ {\frak h}^{1,0} $, $ {\frak h}_{{\Bbb C}}={\frak h}^{1,0}\oplus{\frak h}^{0,1} $. The vector space $ {\frak h}_{{\Bbb C}} $ has a natural operation of complex conjugation $ \left(\alpha,\alpha'\right) \mapsto \left(\bar{\alpha}',\bar{\alpha}\right) $, let $ {\frak h}_{{\Bbb R}} $ be the subspace $ \left(\alpha,\bar{\alpha}\right) $ of fixed points of this complex conjugation. There is a natural extension of the projection $ {\frak h}^{1,0} \to H_{{\Bbb C}}^{B} $ to $ {\frak h}_{{\Bbb C}} \to H_{{\Bbb C}}^{B} $. This mapping is compatible with the complex conjugation, thus induces a mapping $ {\frak h}_{{\Bbb R}} \to H_{{\Bbb R}}^{B} $. Let $ {\frak h}_{{\Bbb C}}^{A} $ be the kernel of the mapping $ {\frak h}_{{\Bbb C}} \to H_{{\Bbb C}}^{B} $, similarly for $ {\frak h}_{{\Bbb R}}^{A} $. Let $ \bar{\Omega}_{1}\colon H_{{\Bbb C}}^{A} \to {\frak h}^{0,1} $ be the complex conjugate to the mapping $ \Omega_{1}\colon H_{{\Bbb C}}^{A} \to {\frak h}^{1,0} $, $ \bar{\Omega}_{1}v=\overline{\Omega_{1}\bar{v}} $. Define a pre-Hilbert structure on $ H_{{\Bbb C}}^{A} $ via $ \|v\|_{A}^{2}=\|\Omega_{1}v\|_{{\frak h}^{1,0}} $. Then $ \Omega $ extends to a mapping from Hilbert completion $ \left(H_{{\Bbb C}}^{A}\right)^{\text{compl}} $ of $ H_{{\Bbb C}}^{A} $ to $ {\frak h}^{1,0} $, same for $ \bar{\Omega}_{1} $ and $ {\frak h}^{0,1} $. Let $ {\frak h}_{{\Bbb C}}^{B} $ be the image of this completion w.r.t. $ \Omega_{1}-\bar{\Omega}_{1} $. Identify $ {\frak h}_{{\Bbb C}}^{B} $ with a subspace of $ H_{{\Bbb C}}^{B} $ via the projection $ {\frak h}_{{\Bbb C}} \to H_{{\Bbb C}}^{B} $. The restriction on $ \Omega $ immediately imply \begin{proposition} $ {\frak h}_{{\Bbb C}} $ carries a natural symplectic structure, the corresponding mapping $ {\frak h}_{{\Bbb C}} \to {\frak h}_{{\Bbb C}}^{*} $ is invertible, and $ {\frak h}_{{\Bbb C}} $ is a direct sum of Lagrangiann subspaces $ {\frak h}_{{\Bbb C}}^{A} $ and $ {\frak h}_{{\Bbb C}}^{B} $. \end{proposition} Since the subspace $ {\frak h}_{{\Bbb C}}^{B} $ is stable w.r.t. complex conjugation, one can define $ {\frak h}_{{\Bbb R}}^{B} $. Similarly, define $ {\frak h}_{{\Bbb Z}}^{A} $ as the image of $ H_{{\Bbb Z}}^{A} $ in $ {\frak h}_{{\Bbb C}}^{A} $. Note that $ {\frak h}_{{\Bbb C}}^{A} $ and $ {\frak h}_{{\Bbb C}}^{B} $ are mutually dual, so $ {\frak h}_{{\Bbb Z}}^{A} $ is identified with a lattice in $ {\frak h}_{{\Bbb C}}^{B} $. Say that $ v\in{\frak h}_{{\Bbb Z}}^{B}\subset{\frak h}_{{\Bbb C}}^{B} $ if $ \left(v,w\right)=0 $ for any $ w\in{\frak h}_{{\Bbb Z}}^{A} $. \begin{proposition} The subset $ {\frak h}_{{\Bbb Z}}^{B} $ generates $ {\frak h}_{{\Bbb C}}^{B} $. \end{proposition} \begin{proof} Let $ {\bold e}_{j} $, $ j\in I_{+} $, be the natural basis of $ H_{{\Bbb Z}}^{A} $. Let $ V_{j} $ be a subspace of $ {\frak h}_{{\Bbb C}}^{A} $ generated by $ {\bold e}_{k} $, $ k\not=j $. If $ V_{j}={\frak h}_{{\Bbb C}}^{A} $, then $ {\bold e}_{j} $ is a linear combination of $ {\bold e}_{k} $, $ k\not=j $, which contradicts $ \operatorname{Im}\Omega \gg $ 0. Appropriate multiple of a normal vector to $ V_{i} $ is in $ {\frak h}_{{\Bbb Z}}^{B} $. It is obvious that these elements generate $ {\frak h}_{{\Bbb C}}^{B} $. \end{proof} \begin{definition} Say that the rigged Hopf structure is {\em integer\/} if $ {\frak h}_{{\Bbb Z}}^{A} $ is closed in $ {\frak h}_{{\Bbb C}}^{A} $. \end{definition} The condition of being integer is a restriction from below on $ \operatorname{Im}\Omega $, say, it prohibits $ \operatorname{Im}\Omega=\operatorname{diag}\left(\lambda_{i}\right) $, $ \lambda_{i}\in l_{2} $. We obtained 4 subspaces $ {\frak h}_{{\Bbb C}}^{A} $, $ {\frak h}_{{\Bbb C}}^{B} $, $ {\frak h}^{1,0} $ and $ {\frak h}^{0,1} $ of $ {\frak h}_{{\Bbb C}} $. Note that $ {\frak h}_{{\Bbb R}}^{A} $, $ {\frak h}_{{\Bbb R}}^{B} $ are generated by $ {\Bbb Z} $-lattices $ {\frak h}_{{\Bbb Z}}^{A} $, $ {\frak h}_{{\Bbb Z}}^{B} $ in them. If we consider $ {\frak h}_{{\Bbb C}} $ and $ {\frak h}^{0,1} $ as real vector spaces, then $ {\frak h}_{{\Bbb C}} ={\frak h}_{{\Bbb R}}\oplus{\frak h}^{0,1} $. Note that $ {\frak h}^{1,0} $ is naturally identified with $ {\frak h}_{{\Bbb C}}/{\frak h}^{0,1} $, thus $ {\frak h}^{1,0}\simeq{\frak h}_{{\Bbb R}} $. Let $ {\frak L} $ be the image of $ {\frak h}_{{\Bbb Z}}^{A}\oplus{\frak h}_{{\Bbb Z}}^{B}\subset{\frak h}_{{\Bbb R}}\subset{\frak h}_{{\Bbb C}} $ in $ {\frak h}_{{\Bbb C}}/{\frak h}^{0,1} $, identify $ {\frak L} $ with a subgroup of $ {\frak h}^{1,0} $. It is clear that $ {\frak L} $ is a lattice in $ {\frak h}^{1,0}\simeq{\frak h}_{{\Bbb R}} $. Start from these 4 subspaces of $ {\frak h}_{{\Bbb C}} $, the complex conjugation and symplectic structure on $ {\frak h}_{{\Bbb C}} $, and lattices in the real parts of the first two spaces. Try to reconstruct the initial rigged Hodge structure. Suppose that $ {\frak h}_{{\Bbb C}}^{A} $, $ {\frak h}_{{\Bbb C}}^{B} $ are stable w.r.t. the complex conjugation, and that $ {\frak h}^{1,0} $, $ {\frak h}^{0,1} $ are interchanged by complex conjugation. Suppose that the last two subspaces project isomorphically on any subspace of the first two (along the other one). Thus $ {\frak h}^{1,0} $ and $ {\frak h}^{0,1} $ are graphs of invertible mappings $ \Omega_{1},\bar{\Omega}_{1}\colon {\frak h}_{{\Bbb C}}^{A} \to {\frak h}_{{\Bbb C}}^{B} $, these mappings are mutually complex conjugate. Now the symplectic structure on $ {\frak h}_{{\Bbb C}} $ identifies $ {\frak h}_{{\Bbb Z}}^{B} $ with a subset of the dual lattice to $ {\frak h}_{{\Bbb Z}}^{A} $. This reconstructs the groups $ H_{{\Bbb Z}}^{A} $, $ H_{{\Bbb Z}}^{B} $, together with $ \Omega_{1} $ they allow to reconstruct the mapping $ \Omega $ from $ H_{{\Bbb C}}^{A}=H_{{\Bbb Z}}^{A}\otimes{\Bbb C} $ to $ H_{{\Bbb C}}^{B}=H_{{\Bbb Z}}^{B}\otimes{\Bbb C} $. The positivity of $ \operatorname{Im}\Omega $ is translated into positivity of $ \left(\alpha,\bar{\alpha}\right) $, $ \alpha\in{\frak h}^{1,0} $, and the condition $ {\frak h}^{1,0}\oplus{\frak h}^{0,1}={\frak h} $. Now the results of this chapter imply \begin{proposition} Consider a family of disks $ K_{i}\subset{\Bbb C}P^{1} $, $ i\in I $, with pairwise conformal distances $ l_{ij} $. Let $ {\cal R}=\left(e^{-l_{ij}}-\delta_{ij}\right) $. Suppose that $ {\cal R} $ gives a compact mapping $ l_{2} \to l_{2} $ and a bounded mapping $ l_{\infty} \to l_{\infty} $, and that some power of $ {\cal R} $ gives a bounded mapping $ l_{\infty} \to l_{2} $. Consider an involution $ ': I\to $I, $ I=I_{+}\coprod I'_{+} $, glue boundaries of disks pairwise using fraction-linear mappings $ \varphi_{i}=\varphi_{i'}^{-1} $, let $ \left(\Omega_{jk}\right) $, $ j,k\in I_{+} $, be the matrix of periods of the resulting curve $ \bar{M} $. Then $ \Omega_{jk} $ gives rise to a rigged Hodge structure. \end{proposition} Note that it is natural to call the space $ {\frak h}_{{\Bbb C}} $ of this rigged Hodge structure {\em the first cohomology space\/} of the curve $ \bar{M} $. The space $ {\frak h}^{1,0} $ can be identified with $ \Gamma\left(\bar{M},\omega\right) $. As in finite-genus case, the quotient $ {\frak h}^{1,0}/{\frak L} $ is identified with the (bounded) Jacobian of the curve.

9710

alg-geom/9710019

en

https://arxiv.org/abs/alg-geom/9710019

alg-geom/9710019

Barbara Russo

B. Russo and M. Teixidor i Bigas

On a conjecture of Lange

13 pages, amslatex, deleted the result of irreducibility in theorems 0.2 and 0.3

Let C be a projective smooth curve of genus g> 1. Let E be a vector bundle of rank r on C. For each integer r'<r, associate to E the invariant s_{r'}(E)=r'deg(E)-rdeg(E') where E'is a subbundle of E of rank r' and maximal degree. For every r', one can stratify the moduli space of stable vector bundles according to the value of the invariant. Lange's conjecture says that this strata are non-empty and of the right dimension if s_{r'}>0. The conjecture has recently been solved thanks to work of Lange- Narasimhan, Lange-Brambila-Paz, Ballico and the authors. The purpose of this paper is to give a simpler proof of the result valid without further assumptions. The method of proof provides additional information on the geometry of the strata. We can prove that each strata (which is irreducible) is contained in the closure of the following one. We also show the unicity of the maximal subbundle when s\le r'(r-r')(g-1). Our methods can be used to study twisted Brill-Noether loci and to give a new proof of Hirschowitz Theorem about the non-speciality of the tensor product of generic vector bundles.

alg-geom

\section*{Introduction} Let $C$ be a projective non-singular curve of genus $g\ge 2$ . Let $E$ be a vector bundle of rank $r$ and degree $d$. Fix a positive integer $r'<r$. Define $$s_{r'}(E)=r'd-r\max _{E'}\{ degE'|rk E'=r', E'\subset E\} $$ Notice that $E$ is stable if and only if $s_{r'}(E)>0$ for every $r'<r$. On the other hand, for a generic stable $E$ $$r'(r-r')(g-1)\le s_{r'}(E)<r'(r-r')(g-1)+r$$ (cf [L] Satz 2.2 p.452 and [Hi] Th.4.4). One can then stratify the moduli space $U(r,d)$ of vector bundles of rank $r$ and degree $d$ according to the value of $s$. Define $$U_{r',s}(r,d)=\{ E\in U(r,d)|s_{r'}(E)=s\} $$ We want to study this stratification. A vector bundle $E\in U_{r',s}(r,d)$ can be written in an exact sequence $$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0$$ with $E',E''$ vector bundles of ranks $r',r''$ and degrees $d',d''$ satisfying $r=r'+r'', d=d'+d'', r'd-rd'=r'd''-r''d'=s$. Note that the condition $s>0$ is equivalent to the inequality of slopes $\mu (E')<\mu (E'')$. One expects that when this condition is satisfied, a generic such extension will yield a stable $E$. We call this statement Lange's conjecture (cf. [L]). The conjecture is now solved and a great deal is known about the geometry of the strata : the rank two case is treated in [L,N], the case $s\leq \min (r',r'')(g-1)$ in [B,B,R]. In [T1], the result is proved for the generic curve and for every curve if $E$ is assumed to be only semistable. This apparently implies the result also for $E$ stable (cf.[B]). In [B,L], a proof is provided for $g\ge (r+1)/2$. The purpose of this paper is to give a simpler proof of the result valid without further assumptions. The method is somehow the converse of the one used by Brambila-Paz and Lange in [B,L]. They start with the most general $E$ in $U(r,d)$ and then show that a suitable transformation of $E$ gives a new bundle with smaller $s$. They need to check then that such an $E$ is in fact stable. Here, we start with an $E$ with the smallest possible $s$ and produce an $E'$ with larger $s$. Stability then comes for free because $E'$ is more general than $E$. The drawback is that one needs to prove existence of stable vector bundles with small $s$ but this is surprisingly easy. Our method of proof provides additional information on the geometry of the strata. We can prove that $U_{r',s}(r,d)$ is contained in the closure of $U_{r',s+r}(r,d)$ as well as the unicity of the subbundle (see also [T2]) Our results can be stated in the following \begin{Thm} \label{Theorem} Assume that $0<s\le r'(r-r')(g-1) ,s\equiv r'd(r)$. Write $d'={r'd-s\over r}$. If $g\ge 2$, then $U_{r's}(r,d)$ is non-empty, irreducible of dimension $$dim U_{r's}(r,d)= r^2(g-1)+1+s-r'(r-r')(g-1)$$ Moreover, a generic $E\in U_{r',s}(r,d)$ can be written in an exact sequence $$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0$$ with both $E',E''$ stable and $E'$ is the unique subbundle of $E$ of rank $r'$ and degree $d'$. \end{Thm} \begin{Thm} \label{Theorem2} If $s\ge r'(r-r')(g-1)$, every stable vector bundle has subbundles of rank $r'$ and degree $d'$. Denote by $$A_{r',d'}(E)= \{ E'|rk E'=r',deg E'=d', E'\subset E ,E' {\rm saturated}\} .$$ Then, for generic $E$, $A_{r',d'}(E)$ has dimension $$dimA_{r',d'}(E) =r'(r-r')(g-1)-s.$$ \end{Thm} These results and our methods of proof can be used to study twisted Brill-Noether loci. We can show the following \begin{Thm} \label{BrillNoether} (twisted Brill-Noether for one section). Let $E$ be a generic vector bundle of rank $r_E$ and degree $d_E$. Consider the twisted Brill-Noether loci $W^0_{r_F,d_F}(E)$. This is defined as the subset of the moduli space $U(r_F,d_F)$ consisting of those $F$ such that $h^0(F^*\otimes E)\ge 1$. Then the dimension of $W^0_{r_F,d_F}(E)$ is the expected dimension given by the Brill-Noether number $$ \rho ^0_{r_F,d_F}(E)=r_F(r_F-r_E)(g-1)+r_Fd_E-r_Ed_F$$ if this number is positive and is empty otherwise. Moreover, when non-empty its generic elements considered as maps $F\rightarrow E$ have maximal rank. \end{Thm} We also include a proof of Hischowitz's Theorem that states that the tensor product of two generic vector bundles is non-special. \bigskip Acnowledgments: The first author was partially supported by MURST GNSAGA of CNR (Italy), Max-Planck Institut of Bonn and the University of Trento. The second author is visiting the Mathematics Department of the University of Cambridge. This collaboration started during the Europroj meeting ``Vector Bundles and Equations'' in Madrid. Both authors are members of the Europroj group VBAC and received support from Europroj and AGE to attend this conference. \section{Existence and dimensionality} In this section we prove the existence of extensions with central term stable and we compute the dimension of the set of vector bundles that fit in such exact sequences. We need several preliminary results. \begin{Lem} \label{h^0=0} Let $E$ be a stable vector bundle. Assume that we have an exact sequence $$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0.$$ Then $h^0(E^{''*}\otimes E')=0$. \end{Lem} \begin{pf} A non-zero map $E''\rightarrow E'$ induces an endomorphism of $E$ that is not an homothethy. This is impossible if $E$ is stable. \end{pf} \begin{Thm}[Hirschowitz] \label{Hirschowitz} The tensor product of two generic vector bundles is not special. \end{Thm} This result was stated and proved in [Hi], 4.6. As this is , unfortunately, still unpublished, we provide an alternative proof below. \begin{pf} We shall denote by $r_G,d_G$ the rank and degree of a given sheaf say $G$. By Serre duality, it is enough to show that if $E,F$ are generic vector bundles, then $h^0(F^*\otimes E)>0$ implies $\chi (F^*\otimes E)>0$. Assume $h^0(F^*\otimes E)\not= 0$. Then, there is a non-zero map $F\rightarrow E$. Denote by $F'$ its kernel, $I$ its image, $E''$ its cokernel. Let $T$ be the torsion subsheaf of $E''$ and $\bar E=E''/T$. We then have the following exact sequences of sheaves $$0\rightarrow F'\rightarrow F\rightarrow I\rightarrow 0$$ $$ \begin{array}{ccccccccc} & & & & & & 0& & \\ & & & & & & \downarrow & & \\ & &0 & & & & T & & \\ & &\downarrow & & & & \downarrow & & \\ 0&\rightarrow & I&\rightarrow &E&\rightarrow &E''&\rightarrow &0\\ & &\downarrow & &\downarrow & &\downarrow & & \\ 0&\rightarrow &\bar I&\rightarrow &E&\rightarrow &\bar E&\rightarrow &0\\ & &\downarrow & & & &\downarrow & & \\ & &T & & & &0 & & \\ & &\downarrow & & & & & & \\ & & 0& & & & & & \\ \end{array}$$ As $T$ is a torsion sheaf, $I$ is determined by $\bar I$, the support of $T$ and for every point in the support a map from the fiber of $\bar I$ at the point to the basefield. Hence, $$dim\{ \bar I \}\le dim \{ I\} +r_I degT$$ As any vector bundle can be deformed to a stable vector bundle, (cf.[N,R]Prop.2.6), $F',\bar I, \bar E$ depend at most on $r_{F'}^2(g-1)+1, r_{\bar E}^2(g-1)+1, r_{\bar I}^2(g-1)+1$ moduli respectively. From \ref{h^0=0} and the stability of $E,F$, $h^0(I^*\otimes F)=0, h^0(\bar E^*\otimes \bar I)=0$. Notice that $F$ is determined by $F',I$ and an extension class in $H^1(I^*\otimes F')$ up to homotethy. Similarly, $E$ is determined by $\bar I,\bar E$ and an extension class in $H^1(\bar E^*\otimes \bar I)$ up to homotethy. From the genericity of the pair $E,F$, we find $$r_F^2(g-1)+1+r_E^2(g-1)+1=dimU(r_F,d_F)+dimU(r_E,d_E)\le$$ $$\le dimU(r_{F'},d_{F'})+dim U(r_I,d_I)+dim U(r_{\bar E},d_{\bar E}) +r_IdegT+$$ $$+h^1(I^*\otimes F')-1+h^1(\bar E^*\otimes \bar I)-1$$ $$\le (r_{F'}^2+r_I^2+r_{\bar E^2}+r_{F'}r_I+r_Ir_{\bar E})(g-1) +1+r_{F'}d_I-r_Id_{F'}+r_Id_{\bar E}-r_{\bar E}d_I+r_IdegT$$ This condition can be written as $$(*)r_{F'}d_I-r_Id_{F'}+r_Id_{\bar E}-r_{\bar E}d_I+r_IdegT- r_I(r_I+r_{\bar E}+r_{F'})(g-1)-1\ge 0$$ From the genericity of $E,F$ and [L] Satz 2.2, we obtain $$\mu (I)-\mu(F')\ge g-1, \mu(\bar E)-\mu (\bar I)\ge g-1$$ Hence $$\mu (\bar E)-\mu (F')\ge 2(g-1)+degT/r_I$$ Equivalently $$r_{F'}d_{\bar E}-r_{\bar E}d_{F'}\ge 2r_{F'}r_{\bar E}(g-1)+ {r_{F'}r_{\bar E}\over r_I}deg T$$ Adding (*) and this last inequality, we find $$\chi (F^*\otimes E)\ge 1+r_{F'}r_{\bar E}(g-1)+{r_{F'}r_{\bar E} \over r_I}deg T+r_{F'}degT\ge 1$$ \end{pf} \begin{Lem} \label{cotasubf} Denote by $V_{r',s}(r,d)$ the set of stable $E$ that can be written in an exact sequence of vector bundles $$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0$$ with $rk E'=r', deg E'=d'$. Assume that $V_{r's}(r,d)$ is non-empty. If $s\le r'(r-r')(g-1)$, then the generic such $E$ has only a finite number of subbundles of rank $r'$ and degree $d'$ such that $ r'd-rd'=s$. If $s\ge r'(r-r')(g-1)$, then the dimension of the space of subbundles of rank $r'$ and degree $d'$ of the generic $E$ is at most $s-r'(r-r')(g-1)$ \end{Lem} \begin{pf} Several proofs of this fact appear in the literature . We sketch a proof here for the convenience of the reader. The set of subbundles of rank $r'$ and degree $d'$ of $E$ is parametrised by the quotient scheme of $E$ of the corresponding rank and degree. The tangent space to this quotient scheme at the point corresponding to a bundle $E$ with subbundle $E'$ and quotient $E''$ is $H^0(E^{'*}\otimes E'')$. As $E$ is generic, we can assume $E',E''$ generic. Then, from \ref{Hirschowitz}, $E^{'*}\otimes E''$ is non-special. Hence, if $s\le r'(r-r')(g-1) ,h^0(E^{'*}\otimes E'') =0$ while if $s\ge r'(r-r')(g-1)$, then $ h^1(E^{'*}\otimes E'')=0$ and so $h^0(E^{'*}\otimes E'')=s-r'(r-r')(g-1)$. \end{pf} \begin{Prop} \label{Irr} With the notations of \ref{cotasubf}, if $V_{r',s}(r,d)$ is non-empty, then, it is irreducible and the generic $E\in V_{r',s}(r,d)$ can be written in an exact sequence as above with $E',E''$ stable. Moreover, $dimV_{r',s}(r,d)= \min [r^2(g-1)+1, r^2(g-1)+1+s-r'(r-r')(g-1)]$ \end{Prop} \begin{pf} This proof appears in [T]. We give a sketch here for the convenience of the reader . Consider an extension $$0\rightarrow E'_0\rightarrow E_0 \rightarrow E''_0\rightarrow 0$$ with $E_0$ stable. From [N,R] Prop.2.6, there are irreducible families of vector bundles ${\cal M}', {\cal M}''$ containing $E'_0, E''_0$ respectively and whose generic member is stable. Consider the universal family of extensions ${\bf P}$ of an $E''\in {\cal M''}$ by an $E'\in {\cal M'}$. Consider the open subset $U\subset {\cal M'}\times {\cal M''}$ consisting of those pairs $(E',E'')$ such that $h^0(E^{''*}\otimes E')=0$. As $\mu (E')<\mu (E'')$, $U$ contains all pairs in which both $E',E''$ are stable. From \ref{h^0=0}, $(E'_0,E''_0)\in U$. As $h^1(E^{''*}\otimes E')$ is constant on $U$, the inverse image ${\bf P}(U)$ of $U$ in ${\bf P}$ is irreducible. This proves that the given extension can be deformed to an extension with both $E',E''$ stable. By the stability of $E_0$, the generic central term in an extension in ${\bf P}(U)$ is stable. Consider the canonical rational map $\pi :{\bf P}(U)\rightarrow U(r,d)$. By definition $V_{r',d'}(r,d)$ is the image of this map. Hence, it is irreducible. The dimension of ${\bf P}$ can be computed as $$dim{\bf P}=dim{\cal M}+dim{\cal M}'+h^1(E^{''*}\otimes E')-1= (r^2-r'r'')(g-1)+1+s$$ From \ref{cotasubf}, the fibers of $\pi $ have dimension $\max [0,s-r'(r-r')(g-1)]$. Hence, the result follows. \end{pf} \begin{Prop} \label{Irrsubf} Assume $s>r'(r-r')(g-1)$ and $E$ is a generic stable vector bundle. If $A_{r'd'}(E)$ is non-empty, then it has dimension $s-r'(r-r')(g-1)$. \end{Prop} \begin{pf} With the notations in the proof of \ref{Irr}, $A_{r',d'}(r,d)$ are the fibers of $\pi$. Its dimension has been computed already. \end{pf} \begin{Def} Let $E$ be a vector bundle. A vector bundle $\tilde E$ is called an elementary transformation of $E$ if there is an exact sequence $$0\rightarrow \tilde E \rightarrow E \rightarrow {\bf C}_P \rightarrow 0$$ Here ${\bf C}_P$ denotes the skyscraper sheaf isomorphic to the base field with support on the point $P$. A vector bundle $\bar E$ is called a dual elementary transformation of $E$ if $E$ is an elementary transformation of $\bar E$. Equivalently, the dual of $\bar E$ is an elementary transformation of the dual of $E$ or equivalently there is an exact sequence $$\bar E \rightarrow E(Q)\rightarrow {\bf C}_Q^{r-1}\rightarrow 0.$$ \end{Def} \begin{Lem} \label{TE} Let $$0\rightarrow E' \rightarrow E\rightarrow E''\rightarrow 0$$ be an exact sequence of vector bundles. Then, for a generic elementary transformation $\tilde E$ of $E$, we have an exact sequence of vector bundles $$0\rightarrow \tilde E' \rightarrow \tilde E\rightarrow E''\rightarrow 0$$ where $\tilde E'$ is a generic elementary transformation of $E$. \end{Lem} \begin{pf} There is an injective map $0\rightarrow E'_P \rightarrow E_P$. The elementary transformation depends on the choice of a map $E_P\rightarrow {\bf C}_P\rightarrow 0$. If this map is generic, it induces a non-zero map $E'_P\rightarrow {\bf C}_P$. Hence, we have a diagram $$\begin{array}{ccccccccc} & &0 & &0 & & & & \\ & &\downarrow & &\downarrow & & & & \\ 0&\rightarrow& \tilde E'&\rightarrow &\tilde E& \rightarrow &\tilde E''&\rightarrow &0\\ & &\downarrow & &\downarrow & & \downarrow & & \\ 0&\rightarrow& E'&\rightarrow & E& \rightarrow & E''&\rightarrow &0\\ & &\downarrow & &\downarrow & & & & \\ & &{\bf C}_P&\rightarrow & {\bf C}_P& & & & \\ & &\downarrow & &\downarrow & & & & \\ & & 0 & & 0 & & & & \\ \end{array}$$ This proves the statement. \end{pf} \begin{Lem} \label{TE*} Let $$0\rightarrow E' \rightarrow E\rightarrow E''\rightarrow 0$$ be an exact sequence of vector bundles. Then, for a generic dual elementary transformation $\bar E$ of $E$, we have an exact sequence of vector bundles $$0\rightarrow E' \rightarrow \bar E\rightarrow \bar E''\rightarrow 0$$ where $\bar E''$ is a generic dual elementary transformation of $E''$. \end{Lem} \begin{pf}: Dualise the proof above \end{pf} \begin{Prop} \label{exist} Let $$0\rightarrow E' \rightarrow E\rightarrow E''\rightarrow 0$$ be an exact sequence of vector bundles. Assume that $E$ is stable. Then, there exists an exact sequence of vector bundles $$0\rightarrow \hat E' \rightarrow \hat E \rightarrow \hat E''\rightarrow 0$$ satisfying \begin{description} \item[i)] $deg \hat E'=deg E'-1, deg \hat E=deg E$\medskip \item[ii)] $\hat E$ is stable. \end{description} \end{Prop} \begin{pf} Take first an elementary transformation of the exact sequence based at a point $P$. Take next a dual elementary transformation based at a point $Q$. From the two Lemmas above, $deg \hat E'=deg E+1$. We now construct a family of these transformations which contains $E$ as one of its members: let the point $Q$ vary until it coincides with $P$. Then, with a suitable choice of the dual transformation, one can go back to $E$. The existence of this family of vector bundles together with the stability of $E$, implies the stability of the generic $\hat E$. \end{pf} \begin{Cor} \label{inclusio} If $U_{r's}(r,d)$ is non empty, then it is contained in the closure of $U_{r', s+r}(r,d)$. \end{Cor} \begin{pf} Take $E\in U_{r',s}(r,d)$ and consider an exact sequence $$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0$$ with $E'$ of rank $r'$ and maximal degree $d'$. In the proof above, we construct a family with special member $E$ and generic member $\tilde E$ that fits in an exact sequence $$0\rightarrow \tilde E'\rightarrow \tilde E\rightarrow \tilde E'' \rightarrow 0$$ with $deg(\tilde E')=d'+1$. Hence, this $\tilde E\in V_{r',s+r}(r,d)$. From \ref{Irr}, $V_{r',s+r}(r,d)$ is irreducible and from the dimensionality statement in \ref{Irr}, $V_{r',s+r}(r,d)\not\subseteq V_{r',s-kr}(r,d), k\ge 0$. Hence the generic element in $V_{r',s+r}(r,d)$ is in $U_{r',s+r}(r,s)$. \end{pf} \begin{Prop} \label{spetita} Let $C$ be a projective non-singular curve of genus $g\ge 2$. Consider an exact sequence of vector bundles on $C$ $$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0.$$ Denote by $r',r'',r,d',d'',d$ the ranks and degrees of $E',E'',E$. Assume that $E',E''$ are generic stable vector bundles of their ranks and degrees. If $0<r'd-rd'\le r$, then, the generic such $E$ is stable. \end{Prop} \begin{pf} Assume that $E$ is not stable. Let $F$ be a subbundle of $E$ such that $\mu (F)\ge \mu (E)$. Up to replacing $F$ by a subbundle of smaller rank or by its saturation, we can assume $F$ stable and $E/F$ without torsion. As $E'$ is stable and $\mu (E')<\mu (E)$, $F$ gives rise to a non-zero map $\phi :F\rightarrow E''$. Denote by $F'$ its kernel, $F''$ its image. Denote by $r_{F'}, r_{F}, r_{F''}, d_{F'}, d_{F}, d_{F''}$ the ranks and degrees of the bundles $F',F,F''$. {\it Claim 1.} $ r_{F''}=r''$. Proof of Claim 1: Assume $ r_{F''}<r''$. By the genericity of $E''$ this implies $r_{F''}d''-r''d_{F''}\ge r_{F''}(r''-r_{F''})(g-1)$ (cf.[L] Satz 2.2). Equivalently $$\mu (F'')\le \mu (E'')-(1-(r_{F''}/r''))(g-1)$$ By the initial assumption $r'd-rd'\le r$, $$(*)\mu (E'')\le \mu (E) +1/r''.$$ As $F$ is a destabilizing subbundle, $$\mu (E)\le \mu (F)$$ and from the stability of $F$ $$\mu (F) \le \mu (F'')$$ with equality if and only if $F=F''$. Notice that $$1/r''-(1-(r_{F''}/r''))(g-1)\le 1/r''-(1-(r_{F''}/r''))\le 0$$ With equalities if and only if $g=2, r_{F''}=r''-1$. Puting together all of the above inequalities, we find that they are all equalities. This proves Claim 1 except in the case when all of the following properties are satisfied: \begin{description} \item[i)] $g=2, r_{F''}=r''-1$ \item[ii)]$\mu(E)=\mu(F)$, $F'=0$ and $F''=F$ is a subsheaf of $E''$ \item[iii)] $ (r''-1)d''-r''d_F=r''-1$ \end{description} We shall see at the end of the proof that this situation does not correspond to a generic $E$. This will finish the proof of Claim 1. {\it Claim 2.} $F'=0$ Proof of Claim 2. Note that $E^*$ satisfies the hypothesis in \ref{spetita} . If $F$ is a maximal destabilising subbundle of $E$ and we write $G=E/F$, then $G^*$ is a maximal destabilising subbundle of $E^*$. Then, Claim 2 follows from Claim 1. From now on, we assume that $F$ is a subbundle of $E''$ and $r_F=r''$ As $F$ is a destabilising subbundle, $\mu (F)\ge \mu (E)$. As $F$ is a subbundle of $E''$, $\mu (F)\le \mu (E'')$. Using (*) and $r_F=r''$, we obtain either $d_F=d''$ or $d_{F}= d''-1$ and $\mu (F)=\mu (E)=\mu (E'')-1/r''$. If $F= E''$, the sequence splits and the extension is not generic. If $d_F=d''-1$, we have an exact sequence $$0\rightarrow F\rightarrow E''\rightarrow T\rightarrow 0$$ where $T$ is a torsion sheaf of degree one supported at one point, say $P$. Then, $F$ is determined by the choice of $P$ and a map from $E_P$ to the base field defined up to homothety. Therefore the number of moduli for such $F$ is at most $r''$. Consider the pull-back diagram $$\begin{array}{ccccccccc} 0& \rightarrow &E'&\rightarrow &E&\rightarrow &E''& \rightarrow &0\cr & &\uparrow & &\uparrow & &\uparrow & & \cr 0& \rightarrow &E'&\rightarrow &E\times _{E''}F&\rightarrow &F& \rightarrow &0\cr \end{array} $$ As $F$ is a subsheaf of $E$, the bottom row splits. Hence the top row corresponds to an element in the kernel of the map $$ H^1(E^{''*}\otimes E')\rightarrow H^1(F^*\otimes E')\rightarrow 0.$$ This kernel has dimension at most $h^0(T\otimes E')=deg(T)\times rk(E')=r'$. Therefore the dimension of the subspace of $ H^1(E^{''*}\otimes E')$ that may correspond to unstable extensions is at most $$dim\{ F \} +r'\le r''+r'=r.$$ On the other hand, $$ h^1(E^{''*}\otimes E')=r'd-rd'+r'(r-r')(g-1)= r'(r-r')(g-1)+r$$ where the last equality comes from the condition $ \mu (E)=\mu (E'')-1/r''$ . It is then clear that the generic extension is stable. We now prove that the special situation at the end of Claim 1 does not occur: Notice that from condition iii) and the stability of $E''$, $E''$ does not have a subbundle of rank $r''-1$ and degree higher than $d_F$. Hence, we have a pull-back diagram as above but $E/F=L$ is a line bundle. From condition iii) and \ref{cotasubf}, there is only a finite number of possible $F$ for a given $E''$. From the genericity of $E''$, both $F$ and $L$ are generic (and depend only on $E''$ and not on $E'$). Hence, it is enough to show that the canonical map $$ H^1(E^{''*}\otimes E')\rightarrow H^1(F^*\otimes E')$$ is non-zero. As this map is surjective, this is equivalent to $ H^1(F^*\otimes E')\not= 0$. From Riemann-Roch $ h^1(F^*\otimes E')\ge r'(r''-1)+r''(r''-1)(\mu (F)-\mu (E'))>0$ where the last inequality comes from ii) $\mu (E')<\mu (E)$ and $r_F=r''-1>0$. \end{pf} \begin{pf}. We now prove \ref{Theorem} except for the unicity of the subbundle that we postpone to next section. From \ref{cotasubf} , \ref{Irr} and \ref{inclusio} , it is enough to prove the non-emptiness of $U_{r's}(r,d)$. Take now any positive $s$. Write $s=ar+\bar s, \ 0<\bar s\le r$. Then, $U_{r',\bar s}(r,d)$ is non empty by \ref{spetita}. Applying \ref{exist} $a$-times, we obtain the non-emptiness of $V_{r',s}(r,d)$. From the definitions of $V_{r',s}(r,d),U_{r',s}(r,d)$, a generic element of $V_{r',s}(r,d)$ belongs to an $U_{r',\tilde s}(r,d)$ for some $\tilde s\le s$. In order to prove the non-emptiness of $U_{r',s}(r,d)$, it is enough to see that $V_{r',s}(r,d)\not\subset V_{r',\tilde s}(r,d), \tilde s<s$. From the dimensionality statement in Lemma 5, this is true. \end{pf} \begin{pf} The proof of \ref{Theorem2} is similar to the proof of \ref{Theorem}: use \ref{spetita}, \ref{exist} and the dimensionality statement in \ref{Irr}. \end{pf} \section{Brill-Noether for twisted bundles and unicity of subbundles} In this section we prove the result that we stated in the introduction about twisted Brill-Noether Theory. The corresponding result for the untwisted case (i.e. $E={\cal O}$) is well-known (cf[S] Th IV 2.1). We then use this result to show the unicity of the Lange subbundle. \begin{pf} (of \ref{BrillNoether}) Assume that $ W^0_{r_F,d_F}(E)$ is non-empty. Consider an element $F$ in this set. This gives rise to a non-zero map $F\rightarrow E$. Denote by $F'$ its kernel, $F''$ its image. Then $F''$ is a subbundle of $E$. Moreover, we have an exact sequence $$0\rightarrow F'\rightarrow F\rightarrow F''\rightarrow 0.$$ Assume first $r_{F''}<r_E$. From \ref{Theorem}, the set of saturated subbundles of $E$ of rank $r_{F''}$ and degree $d_{F''}$ has dimension $$dim\{ F''\} = r_{F''}(r_{F''}-r_{E})(g-1)+r_{F''}d_{E}-r_{E}d_{F''}$$ if this number is positive and is empty otherwise. The set of non-saturated subbundles has dimension smaller than this number. Consider then the case in which $r_{F''}=r_E$. Then, the quotient $E/F''$ is torsion. The choice of $F''$ depends on the choice of the support of this quotient and for each point $P$ on the support the choice of a map (up to homothety) to the base field $E_P\rightarrow {\bf C}$ . Hence, $dim\{ F''\} \le r_E(d_E-d_{F''})$. This coincides with the bound above for the case $r_{F''}=r_E.$ Any family of vector bundles can be embedded in a family with generic member stable(cf [NR] Prop.2.6). Hence, $F'$ varies in a parameter space of dimension at most $$dim \{ F' \} =r_{F'}^2(g-1)+1.$$ From \ref{h^0=0}, $h^0(F*{''*}\otimes F')=0$. Using Riemann-Roch, $$h^1(F^{''*}\otimes F')= r_{F'}r_{F''}(g-1)+r_{F'}d_{F''}-r_{F''}d_{F'}.$$ The choice of $F$ depends on the choice of the pair $F',F''$ and the class of the extension up to scalar. Therefore, the dimension of all possible $F$ is bounded by $$dim \{ F\} \le [r_{F'}^2+r_{F''}^2-r_{F''}r_E+r_{F'}r_{F''}](g-1)+ r_{F'}d_{F''}-r_{F''}d_{F'}+r_{F''}d_E-r_Ed_{F''}=$$ $$=\rho ^0_{r_F,d_F}(E)-[r_{F'}d_{E''}-d_{F'}r_{E''}- r_{F'}r_{E''}(g-1)]$$ where $E''$ denotes the quotient of $E$ by $F''$ Let us check that $$(*)[r_{F'}d_{E''}-d_{F'}r_{E''}-r_{F'}r_{E''}(g-1)]\ge 0.$$ By the genericity of $E$, if $F''$ exists, then $\mu (E'')-\mu (F'')\ge g-1$ (cf Prop. 2.4) By the stability of $F$, $\mu (F')<\mu (F'')$. Hence, $\mu (E'')-\mu (F')\ge (g-1)$. This is equivalent to the inequality (*) and proves the upper bound for the dimension. Notice also that (*) vanishes if and only if either $F'=0$ or $F''=E$. In both these cases, the map has maximal rank. It only remains to prove existence in case the Brill-Noether number is positive. If $r_F\not= r_E$, this is equivalent to the existence of a stable subbundle or quotient of $E$ and is contained in \ref{Theorem2}. If $r_F=r_E$, one needs to check that the generic elementary transformation of a stable vector bundle is again stable. This is well known. \end{pf} \begin{Prop} Let $E$ be a stable vector bundle obtained as a generic extension $$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0.$$ Assume that $0<s=r'd-rd'\le r'(r-r')(g-1)$. Then the only subbundle of $E$ of rank $r'$ and degree $d'$ is $E'$. \end{Prop} \begin{pf} Assume that there were another subbundle $ F'$ of rank $r'$ and degree $d'$. Denote by $F''$ the quotient sheaf $E/F'$. Claim: If $E$ is general, both $F',F''$ are generic vector bundles of the given ranks and degrees (i.e. as $E$ varies, $F',F''$ vary in an open dense subset of the corresponding moduli spaces). Proof of the claim: If $F''$ had torsion, then $E$ would have a subbundle of higher degree and from \ref{Theorem} it could not be general. If $F'$ or $F''$ were not general or were not stable, then they would move in varieties of dimension strictly smaller than those parametrising $E',E''$. From \ref{h^0=0}, $h^1(E^{''*}\otimes E')= h^1(F^{''*}\otimes F')$. From \ref{cotasubf}, every $E$ appears in at most a finite number of extensions of an $E''$ by an $E'$. Hence, $E$ could not be general. This proves the claim. We obtain non-zero maps $ F'\rightarrow E''$ and $E'\rightarrow F''$. From the genericity of $E', E''$ and \ref{BrillNoether} , the dimension of the sets of these $ F',F''$ is at most $$dim\{ F'\} =r'(r'-r'')(g-1)+r'd''-r''d'=r'(r'-r'')(g-1)+s$$ $$dim \{ F''\} =r''(r''-r')(g-1)+r'd''-r''d'=r''(r''-r')(g-1)+s$$ and both these numbers are positive. From \ref{h^0=0} and the stability of $E$, $h^0(F^{''*}\otimes F')=0$. Hence, from Riemmann-Roch $$h^1(F^{''*}\otimes F')=r'r''(g-1)-[r''d'-r'd'']= r'r''(g-1)+s$$ We obtain then a bound for the dimension of the set of $E$ for which there is an exact sequence $$0\rightarrow F'\rightarrow E\rightarrow F''\rightarrow 0$$ given by $$dim \{ E\} \le (r^{'2}+r^{''2}-r'r'')(g-1)+3s-1$$ On the other hand, from \ref{Irr}, $$dim \{ E\} =(r^{'2}+r^{''2}+r'r'')(g-1)+s+1$$ It follows then that $2r'r''(g-1)\le 2s-2$. This contradicts our assumption on $s$. \end{pf}

9710

alg-geom/9710033

en

https://arxiv.org/abs/alg-geom/9710033

alg-geom/9710033

Gian Mario Besana

Alberto Alzati and Gian Mario Besana

On the k-normality of some projective manifolds

AMS-LaTeX, 20 pages, to appear in Collect. Math. special volume in memory of F. Serrano

A long standing conjecture, known to us as the Eisenbud Goto conjecture, states that an n-dimensional variety embedded with degree $d$ in the $N$- dimensional projective space is $(d-(N-n)+1)$-regular in the sense of Castelnuovo-Mumford. In this work the conjecture is proved for all smooth varieties $X$ embedded by the complete linear system associated with a very ample line bundle $L$ such that $\Delta (X,L) \le 5$ where $\Delta (X,L) = \dim{X} + \deg{X} -h^0(L).$ As a by-product of the proof of the above result the projective normality of a class of surfaces of degree nine in $\Pin{5}$ which was left as an open question in a previous work of the second author and S. Di Rocco alg-geom/9710009 is established. The projective normality of scrolls $X =\Proj{E}$ over a curve of genus 2 embedded by the complete linear system associated with the tautological line bundle assumed to be very ample is investigated. Building on the work of Homma and Purnaprajna and Gallego alg-geom/9511013, criteria for the projective normality of three-dimensional quadric bundles over elliptic curves are given, improving some results due to D. Butler.

alg-geom

\section{Introduction} A complex projective variety $X\subset \Pin{N}$ is $k$-regular in the sense of Castelnuovo-Mumford if $h^i(\iof{X}{k-i}) = 0$ for all $i\ge 1$ where $\iofo{X}$ is the ideal sheaf of $X.$ If $X$ is $k$-regular then the minimal generators of its homogeneous ideal have degree less than or equal to $k.$ A long standing conjecture, known to us as the Eisenbud Goto conjecture, states that an $n$-dimensional variety $X \subset \Pin{N}$ of degree $\deg X = d$ is $(d-(N-n)+1)$-regular. Gruson Lazarsfeld and Peskine \cite{GLP} established the conjecture for curves, Lazarsfeld \cite{Laz1} for smooth surfaces and Ran \cite{Ran1} for threefolds with high enough codimension. A nice historical account of the conjecture and further results can be found in \cite{Kw}. In section \ref{deltacongettura} the conjecture is proved for all smooth varieties $X$ embedded by the complete linear system associated with a very ample line bundle $L$ such that $\Delta (X, L) \le 5$ where $\Delta (X, L) = \dim{X} + \deg{X} -h^0(L).$ Notice also that in recent times computer algebra systems like Macaulay have made possible the explicit construction and study of examples of algebraic varieties starting from minimal generators of the homogeneous ideal of the variety. A priori information on the $k$-regularity of a variety is therefore useful for these constructions. Strictly related to the notion of $k$-regularity is the notion of $k$-normality of a projective variety. A variety $X \subset \Pin{N}$ is $k$-normal if hypersurfaces of degree $k$ cut a complete linear system on $X$ or, equivalently, if $h^1(\iof{X}{k})=0.$ If $X$ is $k$-regular it is clearly $(k-1)$-normal. $X$ is said to be projectively normal if it is $k$-normal for all $k\ge 1.$ As a by-product of the proof of the above result the projective normality of a class of surfaces of degree nine in $\Pin{5}$ which was left as an open question in \cite{gisa} is established in Lemma \ref{blowupF1}. The non existence of a class of scrolls of degree $10$, left as an open problem in \cite{fa-li10}, is also established in Remark \ref{scr10}. In section \ref{g2scrolls} we deal with the projective normality of scrolls $X =\Proj{E}$ over a curve of genus $2$ embedded by the complete linear system associated with the tautological line bundle $\taut{E},$ assumed to be very ample. Two-dimensional such scrolls are shown to be always projectively normal except for a class $S$ of non $2$-normal surfaces of degree eight in $\Pin{5}$ studied in detail in \cite{alibaba}. Three-dimensional scrolls $X=\Proj{E}$ of degree $\deg X \ge 13$ are then shown to be projectively normal if and only if $E$ does not admit a quotient $E \to \cal{E} \to 0$ where $P(\cal{E})$ belongs to the class $S$ of non quadratically normal surfaces mentioned above. In section \ref{ellipticpkbundles}, building on the work of Homma \cite{Ho1},\cite{Ho2} and Purnaprajna and Gallego \cite{pu-ga}, criteria for the projective normality of three-dimensional quadric bundles over elliptic curves are given, improving some results contained in \cite{bu}. \section{General Results and Preliminaries} \label{prelimsec} \subsection{Notation} \label{notation} The notation used in this work is mostly standard from Algebraic Geometry. Good references are \cite{H} and \cite{gh}. The ground field is always the field $\Bbb{ C}$ of complex numbers. Unless otherwise stated all varieties are supposed to be projective. $\Bbb{P}^{N}$ denotes the N-dimensional complex projective space. Given a projective n-dimensional variety $X$, ${\cal O}_X$ denotes its structure sheaf and $Pic(X)$ denotes the group of line bundles over $X.$ Line bundles, vector bundles and Cartier divisors are denoted by capital letters as $L, M,\cal{M} \dots.$ Locally free sheaves of rank one, line bundles and Cartier divisors are used interchangeably as customary. Let $L, M \in Pic(X)$, let $E$ be a vector bundle of rank $r$ on $X$, let $\cal{F}$ be a coherent sheaf on $X$ and let $Y\subset X$ be a subvariety of $X.$ Then the following notation is used: \begin{enumerate} \item[ ] $LM$ the intersection of divisors $L$ and $M$ \item[ ] $L^{n}$ the degree of $L,$ \item[ ] $|L|$ the complete linear system of effective divisors associated with $L$, \item[ ]$L_Y$ or $\restrict{L}{Y}$ the restriction of $L$ to $Y,$ \item[ ] $L \sim M$ linear equivalence of divisors \item[ ] $L \equiv M$ numerical equivalence of divisors \item[ ] Num$(X)$ the group of line bundles on $X$ modulo numerical equivalence \item[ ] $\Bbb{P}(E)$ the projectivized bundle of $E,$ see \cite{H} \item[ ] $H^i(X, \cal{F})$ the $i^{th}$ cohomology vector space with coefficients in ${\cal F},$ \item[ ] $h^i(X,\cal{F})$ the dimension of $H^i(X, \cal{F}),$ here and immediately above $X$ is sometimes omitted when no confusion arises. \end{enumerate} If $C$ denotes a smooth projective curve of genus $ g$, and $E$ a vector bundle over $C$ of deg $E= c_1(E)= d$ and rk$E=r$, we need the following standard definitions: \begin{enumerate} \item[ ] $E$ is $\it normalized$ if $h^0(E)\ne 0$ and $h^0(E \otimes \cal{L})=0$ for any invertible sheaf $\cal{L}$ over $C$ with deg$\cal{L}<0$. \item[ ] $E$ has slope $\mu(E) = \frac{d}{r}$. \item[ ] $E$ is $\it semistable$ if and only if for every proper subbundle $S$, $\mu(S) \leq \mu(E)$. It is $\it stable$ if and only if the inequality is strict. \item[ ] The Harder-Narasimhan filtration of $E$ is the unique filtration: $$0=E_0\subset E_1\subset ....\subset E_s=E$$ such that $\frac{E_i}{E_{i-1}}$ is semistable for all $i$, and $\mu_i(E)=\mu (\frac{E_i}{E_{i-1}})$ is a strictly decreasing function of $\it i$. \end{enumerate} A few definitions from \cite{bu} needed in the sequel are recalled. Let $0=E_0 \subset E_1 \subset ....\subset E_s=E$ be the Harder-Narasimhan filtration of a vector bundle $E$ over $C$. Then \begin{enumerate} \item[]$\mu^-(E)=\mu_s(E)=\mu (\frac{E_s}{E_{s-1}})$ \item[]$\mu^+(E)=\mu_1(E)=\mu (E_1)$ \item[]or alternatively \item[]$\mu^+(E)= \text{ max }\{\mu(S) |0 \to S \to E \}$ \item[]$\mu^-(E)= \text{ min }\{\mu(Q) |E \to Q \to 0 \}$. \end{enumerate} It is also $\mu^+(E) \geq \mu(E) \geq \mu^-(E)$ with equality if and only if $E$ is semistable. In particular if $C$ is an elliptic curve, an indecomposable vector bundle $E$ on $C$ is semistable and hence $\mu(E) = \mu^-(E)=\mu^+(E)$. The following definitions are standard in the theory of polarized varieties. A good reference is \cite{fu}. A {\em polarized variety } is a pair $(X, L)$ where $X$ is a smooth projective n-dimensional variety and $L$ is an ample line bundle on $X$. Its {\em sectional genus}, denoted $g(X, L)$, is defined by $2g(X, L) - 2 = (K_X + (n-1)L) L^{ n-1}$. Given any $n$-dimensional polarized variety $(X, L)$ its $\Delta${\em - genus } is defined by $\Delta (X, L) = dim (X) + L^{ n} - h^0(X, L).$ A polarized variety $(X, L)$ has a {\em ladder} if there exists a sequence of reduced and irreducible subvarieties $X = X_n \supset X_{n-1} \dots \supset X_1$ of $X$ where $X_j\in|L_{j+1}| = |\restrict{L}{X_{j+1}}|.$ Each $(X_j, L_j)$ is called a {\em rung} of the ladder. If $L$ is generated by global sections $(X, L)$ has a ladder. A rung $(X_j, L_j)$ is {\em regular} if $H^0(X_{j+1},\restrict{L}{X_{j+1}}) \to H^0(X_j,\restrict{L}{X_j})$ is onto. The ladder is regular if all the rungs are regular. If the ladder is regular $\Delta (X_j,L_j) = \Delta(X,L)$ for all $1\le j \le n.$ A variety $X \subset \Pin{N}$ is {\it k-normal} for some $k \in \Bbb{Z}$ if $H^0(\Pin{N}, \oofp{N}{k}) \to H^0(\oof{X}{k})$ is onto. Equivalently, if $\iofo{X}$ is the ideal sheaf of $X,$ $X$ is $k$-normal if $h^1(\iof{X}{k}) = 0.$ $X$ is {\it projectively normal} if it is $k$-normal for all $k \ge 1.$ A polarized pair $(X, L)$ with $L$ very ample is called $k$-normal or projectively normal if $X$ is $k$-normal or p.n. in the embedding given by $|L|.$ A polarized variety $(X, L)$ with $L$ very ample is always $1$-normal (linearly normal). A line bundle $L$ on $X$ is {\em normally (or simply) generated} if the graded algebra $G(X, L) = \bigoplus_{t\ge 0}H^0(X,tL)$ is generated by $H^0(X,L).$ $L$ is very ample and normally generated if and only if $(X, L)$ is p.n. A variety $X\subset \Pin{N}$ is $k$-regular, in the sense of Castelnuovo-Mumford, if for all $i\ge 1$ it is $h^i(\iof{X}{k-i})=0.$ A polarized pair $(X, L)$ with $L$ very ample is $k$-regular if $X$ is $k$-regular in the embedding given by $|L|.$ If $X$ is $k$-regular then it is $(k+1)$-regular. \medskip \subsection{General Results} \medskip Let $C$ be a smooth projective curve of genus $g$, $E$ a vector bundle of rank $n$, with $n \ge 2$, over $C$ and $\pi : X =\Bbb{P}(E) \to C $ the projectivized bundle associated to $E$ with the natural projection $\pi$. Denote with $\cal{T} = \taut{E} $ the tautological sheaf and with $\frak{F}_P= \pi^*\cal {O}_{C}(P) $ the line bundle associated with the fiber over $P\in C.$ Let $T$ and $F$ denote the numerical classes respectively of $\cal T$ and $\frak{F}_P$. In this work we refer to a polarized variety $(X,\cal{T})$ as a {\it scroll} over a curve $C$ if there is a vector bundle $E$ over $C$ such that $(X,\cal{T})= \scroll{E}$ and $\cal{T}$ is very ample. \begin{rem} \label{leray} Let $\pi : \scroll{E} \to C$ be a $n$-dimensional projectivized bundle over a curve $C.$ From Leray's Spectral sequence and standard facts about higher direct image sheaves (see for example \cite{H} pg. 253) it follows that \begin{gather} H^1(\tautof{E}{t})=H^1(C, S^tE) \text{ for } t\ge 0 \notag\\ H^i(\tautof{E}{t}) = 0 \text { for } i \ge 2 \text { and } t>-n. \notag \end{gather} \end{rem} Let $D\sim a\cal{T} + \pi^*B$, with $ a\in \Bbb{Z}$, $B\in Pic(C)$ and $\deg B = b$, then $ D \equiv aT+bF .$ Moreover $\pi_*(\oof{\Proj{E}}{D}) = S^{a}(E) \otimes \cal {O}_{C}(B) $ and hence $\mu^-(\pi_*(\oof{\Proj{E}}{D})=a\mu^-(E) +b$ (see \cite{bu}). Regarding the ampleness, the global generation, and the normal generation of $D$, a few known criteria useful in the sequel are listed here: \begin{theo}[Miyaoka \cite{Miyao3}] \label{miyaoteo} Let $E$ be a vector bundle over a smooth projective curve $C$ of genus $g$, and $X =\Bbb{P}(E)$ . If $D\equiv aT+bF$ is a line bundle over $X$, then $D$ is ample if and only if $a>0$ and $b+a \mu^-(E) >0$. \end {theo} \begin{lemma} \label{buongg} (see e.g. \cite{bu}, Lemma 1.12) Let $E$ be a vector bundle over $C$ of genus $g$. \begin{itemize} \item[i)]if $\mu^-(E) > 2g-2$ then $h^1(C, E)=0$ \item[ii)]if $\mu^-(E) > 2g-1$ then $E$ is generated by global sections. \end{itemize} \end{lemma} \begin{lemma}[Butler \cite{bu} Theorem 5.1A] \label{criteriodelbutler} Let $E$ be a vector bundle on a smooth projective curve of genus $g$ and let $D\equiv aT + bF$ be a divisor on $X =\Bbb{P}(E).$ If \begin{equation} \label{condizionedelbutler} b+a\mu^-(E) > 2g. \end{equation} then $D$ is normally generated. \end{lemma} A few basic facts on the {\it Clifford Index} of a curve are recalled. Good references are \cite{mart} and \cite{GL}. Let $C$ be a projective curve and $L$ be any line bundle on $C$. The Clifford index of $L$ is defined as follows: $$cl(L)=deg(L) -2(h^0(L)-1).$$ The Clifford index of the curve is $cl(C)=\text {min}\{cl(L) | h^0(L)\geq 2 \text{ and }h^1(L)\geq 2 \}$. For a general curve $C$ it is $cl(C)=\left [\frac{g-1}{2}\right ]$ and in any case $cl(C)\leq\left [\frac{g-1}{2}\right ]$. By Clifford's theorem a special line bundle $L$ on $C$ has $cl(L)\geq 0$ and the equality holds if and only if $C$ is hyperelliptic and $L$ is a multiple of the unique $g^1_2$.\\ If $cl(C)=1$ then $C$ is either a plane quintic curve or a trigonal curve. \begin{theo}[\cite{GL}] \label{glcliff} Let L be a very ample line bundle on a smooth irreducible complex projective curve $C.$ If $$\deg(L)\geq 2g+1-2h^1(L)-cl(C)$$ then $(C, L)$ is projectively normal. \end{theo} \section{The Eisenbud Goto conjecture for low values of $\Delta.$} \label{deltacongettura} Let $X \subset \Pin{N}$ be an $n$ dimensional projective variety of degree $d$. A long standing conjecture, known to us as the {\it Eisenbud Goto} conjecture, states that $X$ should be $(d-(N-n) +1)$-regular, i.e. $(degree-codimension+1)$-regular. Many authors worked on the conjecture for low values of the dimension and codimension of X. A nice historic account is found in \cite{Kw}. Some of their results are collected in the following Theorem. \begin{theo}[\cite{GLP}, \cite{Laz1}] \label{lazran} If $X\subset \Pin{N}$ is any smooth curve or any smooth surface then $X$ is $(d-c+1)$-regular where $d =\deg{(X)}$ and $c = \text{codimension} (X)$. \end{theo} In this section we would like to offer a proof of the conjecture for linearly normal smooth varieties with low $\Delta$-genus. Let $(X, L)$ be a polarized variety with $L$ very ample. The above conjecture can be restated for the embedding given by $|L|$ in terms of $\Delta$-genus as follows: {\bf Conjecture } {\it Let $(X, L)$ be a polarized variety with $L$ very ample. Then $(X, L)$ is $(\Delta + 2)$-regular.} \begin{rem} \label{ipers} It is straightforward to check that hypersurfaces of degree $d$ are always $d$-regular and not $(d-1)$-regular. This shows that the conjecture is indeed sharp. On the other hand there are varieties $X\subset \Pin{N}$ which are $k$-regular for $k < d-c+1.$ This motivates Definition \ref{extremal}. \end{rem} \begin{rem} It is a classical adjunction theoretic results that given $(X, L)$ with $L$ very ample, $K_X + tL$ is globally generated, and in particular $h^0(K_X + tL) \ne 0,$ for $ t \ge n$ unless $t=n$ and $(X, L) = (\Pin{n}, \oofp{n}{1}).$ This fact, Remark \ref{ipers} and the sequence $0 \to \iofo{X}\to {\cal O}_{\Pin{N}} \to {\cal O}_X \to 0$ suitably twisted show that no linearly normal non degenerate $n$-dimensional variety $X \subset \Pin{N}$ can be $k$-regular for $k \le 1.$ Therefore in what follows we will always assume $k \ge 2$ when dealing with $k$-regularity. \end{rem} \begin{dfntn} \label{extremal} Let $X \subset \Pin{N}$ be a $n$-dimensional variety of degree $d.$Let $$r(X)= Min \{k \in \Bbb{Z} | X \text{ is } k-\text{regular}\}.$$ A variety $X$ is {\bf extremal} if $r (X) = d-(N-n)-1.$ A polarized variety $(X, L)$ with $L$ very ample is {\bf extremal} if it is extremal in the embedding given by $|L|,$ i.e. if $r(X,L) = Min \{k \in \Bbb{Z} | (X, L) \text{ is } k-\text{regular}\} = \Delta + 2.$ \end{dfntn} In what follows we will prove the above conjecture for all linearly normal manifolds with $\Delta \le 5$ obtaining along the way the value of $r (X, L)$ for most of the same manifolds. \begin{lemma} \label{hyperplanesec} Let $X \subset \Pin{N}$ be a smooth n-dimensional variety and let $Y \subset \Pin{N-1}$ be a generic hyperplane section. \begin{itemize} \item[i)] If $X$ is $k$-regular then $Y$ is $k$-regular \item[ii)] If $Y$ is $k$-regular and $X$ is $(k-1)$-normal then $X$ is $k$-regular. \item[iii)] If $X$ is $(r(Y)-1)$-normal then $r(X)=r(Y)$ \end{itemize} \end{lemma} \begin{pf} The exact sequence \begin{equation} \label{ideali} 0 \to \iof{X}{k-i} \to \iof{X}{k-i+1} \to\iof{Y}{k-i+1} \to 0. \end{equation} immediately gives i). To see ii) consider again sequence \brref{ideali}. The $k$-regularity of $Y$ gives $h^{i-1}(\iof{Y}{k-i+1})=0$ for all $i\ge 2.$ Since $k$ regularity implies $k+1$-regularity it is $h^i(\iof{Y}{k-i+1})=0$ for all $i \ge 1.$ Therefore $h^i(\iof{X}{k-i})=h^i(\iof{X}{k-i+1})$ for all $i\ge 2$ from \brref{ideali} and iteratively $h^i(\iof{X}{k-i})=h^i(\iof{X}{k-i+t}$ for all $i\ge 2$ and for all $t\ge 1.$ Letting $t$ grow, Serre's vanishing theorem gives $h^i(\iof{X}{k-i+t}=0$ for all $i\ge 2$ and all $t\ge 1$ and thus $h^i(\iof{X}{k-i})=0$ for all $i\ge 2.$ Because $X$ is assumed $(k-1)$-normal it is $h^1(\iof{X}{k-1})=0$ which concludes the proof of $ii).$ Now $iii)$ follows immediately from $i)$ and $ii).$ \end{pf} \begin{lemma} \label{hyperplanesecpol} Let $(X, L)$ be a polarized variety with $L$ very ample. Let $Y \in |L|$ be a generic element and assume $H^0(X, L)\to H^0(Y, \restrict{L}{Y})$ is onto. Then $i), ii),iii)$ as in Lemma \ref{hyperplanesec} hold if we replace $X$ by $(X, L)$ and $Y$ by $(Y, \restrict{L}{Y}).$ \end{lemma} \begin{pf} Let $h^0(L)=N+1.$ The surjectivity condition on the restriction map between global sections of $L$ and $\restrict{L}{Y}$ guarantees that $|\restrict{L}{Y}|$ embedds $Y$ as a linearly normal manifold in $\Pin{N-1}$ , therefore the same proof as in Lemma \ref{hyperplanesec} applies. \end{pf} \begin{rem} \label{liftingpn} Let $(X, L)$ be a polarized variety with $L$ very ample. Let $Y \in |L|$ be a generic element and assume $H^0(X, L)\to H^0(Y, \restrict{L}{Y})$ is onto. Then \cite{fu} Corollary 2.5 shows that if $(Y, \restrict{L}{Y})$ is projectively normal, so is $(X, L).$ Therefore when the ladder is regular and $Y$ is p.n. Lemma \ref{hyperplanesecpol} gives $r (X, L) = r (Y, \restrict{L}{Y}).$ \end{rem} \begin{lemma} \label{rofpncurves} Let $(C, L)$ be a projectively normal curve with $g\ge 1.$ Then $r (C, L) = Min \{ t \ge 3 | h^1((t-2) L) =0\}.$ \end{lemma} \begin{pf} Let $h^0(L)=N+1$ so that $C\subset \Pin{N}.$ It is $h^1(\iof{C}{k-1})=0$ for all $k\ge 2$ because of the projective normality assumption. The sequence $$0 \to \iof{C}{k-i} \to \oofp{N}{k-i} \to (k-i) L \to 0$$ easily gives $h^i(\iof{C}{k-i})=0$ for all $i \ge 3$ and $k \ge 2.$ The same sequence gives $h^2(\iof{C}{k-2})=h^1((k-2) L)$ and since $h^1({\cal O}_C) = g \ge 1$ it is $r (C, L) = Min \{ t \ge 3 | h^1((t-2) L) =0\}.$ \end{pf} In order to apply the above lemmata in one occasion the projective normality of a particular class of surfaces of degree nine needs to be established. The following Lemma also improves \cite{gisa}. Here $\Bbb{F}_1$ denotes the Hirzebruch rational ruled surface of invariant $e=1,$ $\pi : Bl_{t}S \to S$ denotes the blow up of a surface $S$ at $t$ points, $E_i$ are the exceptional divisors of the blow up, $\frak{C}_0=\pi^*(C_0)$ denotes the pull back of the line bundle associated with the fundamental section of $\Bbb{F}_1$ and $\frak{f}=\pi^*(f)$ the pull back of the one associated with any fibre $f$ of the natural projection $p: \Bbb{F}_1 \to \Pin{1}.$ \begin{lemma} \label{blowupF1} Let $(S, L)=(Bl_{12} {\Bbb F}_1, 3{\frak C_0} + 5{\frak f} - \sum_i E_i)$. Then $(S, L)$ is projectively normal. \end{lemma} \begin{pf} The projective normality of linearly normal degree nine surfaces was studied in \cite{gisa}. Let $(S, L)$ be a surface of degree $9$ and sectional genus $5,$ embedded in $\Pin{5}.$ The surface under consideration was established to be projectively normal unless its generic curve section $C$ is trigonal and $\restrict{L}{C}=K_C-M+D$ where $M$ is a divisor in the $g^1_3$ and $D$ is a divisor of degree $4$ giving a foursecant line for $C.$ Therefore if $S$ were not p.n. it would admit an infinite number of $k\ge 4$-secant lines. On the other hand a careful study of the embedding shows that $S$ contains only a finite number of lines and that the only lines with self intersection $\ge -1$ are the $12$ exceptional divisors $E_i.$ Thus the formulas contained in \cite{LeBz1} can be used. A straightforward calculation using \cite{LeBz1} shows that $S$ cannot have a infinite number of $k\ge 4$-secants, contradiction. \end{pf} \begin{theo} \label{alafujita} Let $(X, L)$ be a $n$ dimensional polarized pair, $n\ge 2,$ with a ladder. Assume $g=g (X, L) \ge \Delta (X, L) = \Delta$ and $d=L^n \ge 2 \Delta + 1.$ Then : \begin{itemize} \item[i)] The curve section $(C,\restrict{L}{C})$ is $k$-regular if and only if $(X, L)$ is $k$-regular and $r (X, L)= r (C,\restrict{L}{C}).$ \item[ii)] Either $\Delta=0,1$ and $(X, L)$ is extremal or $\Delta\ge 2$ and $r (X, L)=3.$ \end{itemize} \end{theo} \begin{pf} From \cite{fu} Theorem (3.5) and from the fact that a normally genereated ample line bundle is automatically very ample it follows that $L$ is very ample, $g=\Delta$, the ladder is regular and every rung of the ladder is projectively normal. Therefore Lemma \ref{hyperplanesecpol} immediately gives $i).$ Then $(X, L)$ is extremal if and only if the curve section $(C,\restrict{L}{C})$ is such. Extremal linearly normal curves were classified in \cite{GLP} and they are either rational or elliptic normal curves. Therefore $(X, L)$ is extremal if and only if $\Delta=g=0,1.$ Now assume $\Delta \ge 2$ and thus $(X, L)$ not extremal. The curve section $(C,\restrict{L}{C})$ is embedded in $\Pin{M}$where $M= d - \Delta.$ Since $h^M(\oofp{M}{k-M})=h^0(\oofp{M}{-1-k})=0$ for all $k \ge 0$ the sequence $0 \to \iof{C}{k-i} \to \oofp{M}{k-i} \to \oof{C}{k-i} \to 0$ shows that $h^i(\iof{C}{k-i}) = h^{i-1}(\oof{C}{k-i})$ for all $i \ge 2$ and all $k\ge 0.$ Therefore, because $h^1({\cal O}_C) = g \ge 2,$ it must be $r (X, L) \ge 3.$ If $i\ge 3$ then clearly $h^{i-1}(\oof{C}{3-i}) =0$ and thus $h^i(\iof{C}{3-i})=0.$ It is also $h^2(\iof{C}{1}) = h^1(\oof{C}{1}) = h^1(\restrict{L}{C})=0$ because $g=\Delta$ and $d\ge 2 \Delta+1 > 2g-2.$ Since every rung of the ladder is projectively normal, in particular $h^1(\iof{C}{2}) =0$ and thus $(C,\restrict{L}{C})$ is $3$-regular. We can conclude that $r (X, L)=r (C,\restrict{L}{C}) = 3.$ \end{pf} \begin{prop} \label{elscr} Let $(X,\cal{T}) = (\Proj{E}, \taut{E}) $ be a scroll over an elliptic curve. Then $r (X,\cal{T}) = 3.$ \end{prop} \begin{pf} Because $h^2(\iofo{X})=h^1({\cal O}_X)=1$ it is $r (X, L) \ge 3.$ We need to show that $h^i(\iof{X}{3-i})=0$ for all $i\ge 1.$ Notice that $|\cal{T}|$ embeds $X$ into $\Pin{N}$ as a variety of degree $d$ where $N=d-1.$ Let $i=1.$ It is known, cf. \cite{bu} and \cite{alibaba}, that elliptic scrolls are projectively normal, so $ h^1(\iof{X}{2}) = 0.$ Let $i=2.$ From Remark \ref{leray} it is $h^2(\iof{X}{1}) = h^1(\oof{X}{1}) = h^1(C, E).$ Because $E$ is very ample it is $\mu^-(E) >0$ which, by Lemma \ref{buongg} implies $h^1(C, E) = 0.$ For $i=N$ it is $h^N( \oofp{N}{3-N}) = h^0(\oofp{N}{-4})=0.$ Therefore it follows that $h^i(\iof{X}{3-i}) = h^{i-1}(\oof{X}{3-i})$ for all $i\ge 3.$ Remark \ref{leray} gives $h^{i-1}(\oof{X}{3-i}) = 0$ for $3\le i \le n+1$ while clearly $h^{i-1}(\oof{X}{3-i})=0$ for $i>n + 1$ since $n=\dim X.$ Therefore $h^i(\iof{X}{3-i}) = 0$ for all $i\ge 3.$ \end{pf} \begin{lemma} \label{diseqonscr} Let $(\Proj{E},\taut{E})$ be a n-dimensional scroll over a curve of genus $g \ge 2.$ If $\deg (E) >2g-2$ then $\Delta \ge 2n+g-3.$ \end{lemma} \begin{pf} Because $d=\deg(det E)=\deg(E) > 2g-2$, it is $h^0(det E) = 1 + d - g$ by Riemann Roch. Combining this with the inequality $h^0(det E) \ge h^0(E) + r-2$ found in \cite{Io-To}, it follows that $h^0(E) \le d - n + 3-g$ and therefore $\Delta \ge 2n + g -3.$ \end{pf} \begin{rem} \label{scr10} Notice that the above Lemma \ref{diseqonscr} rules out the existence of scrolls of degree $10$ over a curve of genus $g=3$ left as an open possibility in \cite{fa-li10}. \end{rem} We can now prove the main theorems of this section. For $\Delta \le 3$ we establish the conjecture and give the value of $r (X, L)$ for all pairs. For $\Delta=4,5$ we establish the conjecture and collect in a remark the known values of $r (X, L).$ \begin{theo} \label{Delta+2thm} Let $(X, L)$ be a n-dimensional polarized pair with $X$ smooth, $L$ very ample and $\Delta \le 5.$ Then $(X, L)$ is $\Delta + 2$-regular. \end{theo} \begin{pf} Because of Theorem \ref{lazran} and Remark \ref{ipers}, the blanket hypothesis $n\ge 3$ and $codim X \ge 2$ will be in place throughout this proof. \begin{case} $\Delta \le 1$ \end{case} If $\Delta = 0$ then $(X, L)$ is extremal by Theorem \ref{alafujita}. Assume $\Delta=1$, because $g=0$ implies $\Delta=0,$ see \cite{fu} Prop. (3.4), it is $g \ge1.$ Because $(X, L)$ is not a hypersurface it is $d\ge 3$ and again Theorem \ref{alafujita} gives $(X, L)$ extremal. \begin{case} $\Delta=2$ \end{case} If $g\le 1$ then $(X, L)$ must be a two dimensional elliptic scroll, see \cite{fu} Theorem (10.2). Proposition \ref{elscr} gives $r=3.$ Let $g\ge 2.$ Because $X$ is not a hypersurface it is $h^0(L) \ge n+3.$ This implies $\Delta\le d-3$ i.e. $d\ge 5$ and then Theorem \ref{alafujita} gives $r=3.$ \begin{case} $\Delta=3$ \end{case} From \cite{BEL2} it follows that complete intersections of type $(2,3)$ have $r=4.$ Following \cite{Io1} Theorem 4.8 and section 7, it follows from Theorem \ref{alafujita} and Proposition \ref{elscr} that the only varieties left to investigate are Bordiga threefolds scrolls in $\Pin{5}.$ They have the following resolution with $N=5.$ \begin{equation} \label{bordigares} 0 \to \oofp{N}{-4}^{\oplus 3} \to \oofp{N}{-3}^{\oplus 4} \to \iofo{X} \to 0. \end{equation} Equalities $h^{N-1}(\iof{X}{3-N}) = h^0({\cal O}_{\Pin{N}})=1$ and $h^i(\iof{X}{3-i})=0$ for all $i \ge 1$ are straightforward to see, therefore $r=3.$ \begin{case} $\Delta=4$ \end{case} Varieties with $\Delta=4$ are classified. Let us follow the list of varieties given in \cite{Io4} Theorem 3. Threefolds in $\Pin{5}$ with $d=7$, $g=5$ or $g=6$ have respective resolutions as in \brref{D4g5res} and \brref{D4g6res} with $N=5.$ \begin{equation} \label{D4g5res} 0 \to \oofp{N}{-5} \oplus \oofp{N}{-4} \to \oofp{N}{-3}^{\oplus 3} \to \iofo{X} \to 0. \end{equation} \begin{equation} \label{D4g6res} 0 \to \oofp{N}{-5}^{ \oplus 2} \to \oofp{N}{-2} \oplus \oofp{N}{-4}^{\oplus 2} \to \iofo{X} \to 0. \end{equation} The resolutions \brref{D4g5res} and \brref{D4g6res} quickly show that $r=4.$ Complete intersections of type $(2,2,2)$ have $r=4$ by \cite{BEL2}. Scrolls over a genus $2$ curve must be two-dimensional while elliptic scrolls are taken care of by Proposition \ref{elscr}. Let now $q=0$ and $g=4.$ If $d\ge 9$ Theorem \ref{alafujita} gives $r=3.$ On the other hand since $\Delta=4$ and the codimension must be at least two, it follows that $d\ge 7.$ Let us now compare the varieties under consideration with the lists of manifolds of degree $7$ and $8$ given in \cite{Io1} and \cite{Io2}. If $d=8$ then $X\subset \Pin{6}$ is a threefold scroll over the quadric surface. Since $q=0$ the ladder is regular. Consider the curve section $(C,\restrict{L}{C}).$ Such a $(C, \restrict{L}{C})$ is known to be non hyperelliptic (see \cite{Io2}) and thus Theorem \ref{glcliff} gives $(C, \restrict{L}{C})$ p.n. Since $d =8 > 2g-2 = 6$ it is $h^1(\restrict{L}{C})=0$ and thus $r(C, \restrict{L}{C})=3$ by lemma \ref{rofpncurves}. Because the ladder is regular $r (X, L)=3$ by Remark \ref{liftingpn}. If $d=7$ then $X \subset \Pin{5}$ is Palatini's scroll over the cubic surface. A resolution for $\iofo{X}$ is found in \cite{BSS3}: $$0 \to \oofp{5}{-4}^{\oplus 4} \to \Omega^1(-2) \to \iofo{X} \to 0.$$ A simple cohomological calculation gives $r=4.$ \begin{case} $\Delta=5$ \end{case} Theorem \ref{alafujita} takes care of cases with $g\ge 5$ and $d\ge 11.$ Manifolds with degree $d\le 10$ were classified by various authors and we will examine them later in the proof. Let us now assume $d \ge 11$ and $g\le 4.$ Because $\Delta=5$ and elliptic scrolls are dealt with in Proposition \ref{elscr}, it must be $g \ge 2.$ Varieties of low sectional genus were classified in \cite{Io1}. Let us follow the lists given there. If $g=2$ scrolls over a curve are the only manifolds to be considered. On the other hand such scrolls of genus $2$ have $\Delta=2n$ (cf. \cite{gisa}) so there are no manifolds to examine. If $g=3$ scrolls over curves are ruled out by Lemma \ref{diseqonscr} and scroll over $\Pin{2},$ having $q=0,$ are ruled out by \cite{Io1} Theorem 4.8 iv). If $g=4$ scrolls over curves are again ruled out by Lemma \ref{diseqonscr}. Using standard numerical relations (see for example \cite{fa-li9} ( 0.14)) one sees that there are no hyperquadric fibrations of dimension $n\ge 3,$ $g=4,$ $\Delta=5$ over $\Pin{1}$ or over an elliptic curve. Let now $(X, L)$ be a threefold which is a scroll over a surface $(Y, \cal{L})$ with $q(Y)=0,$ $g (X, L)=4.$ Because $h^1({\cal O}_X)=q(Y)=0,$ recalling that a general hyperplane section of $X$ is birational to $Y$ and thus regular, the ladder is regular and then $\Delta (X, L) = \Delta(C, \restrict{L}{C}) = 4$ by Riemann Roch. Let us now consider the cases with $d\le 10$ by looking at the classification found in \cite{Io2}, \cite{fa-li9}, \cite{fa-li10}. The first non trivial case occurs with $d=8.$ $(X, L)$ is a threefold in $\Pin{5},$ admitting a fibration over $\Pin{1}$ with generic fibers complete intersections of type $(2,2)$ in $\Pin{4}.$ A resolution of the ideal of this variety can be found in \cite{BSS3}. A standard cohomological calculation shows that $r (X, L) = 4.$ Let now $d=9.$ From \cite{fa-li9} all varieties to be considered are threefolds in $\Pin{6}$ with $g=5,6,7 \ge \Delta$ and $d=9\ge 2\Delta -1.$ Thus the ladder is regular, see \cite{fu} Theorem 3.5. Let $(S,\restrict{L}{S})$ be the surface section and let $(C,\restrict{L}{C})$ be the curve section. The projective normality of linearly normal surfaces of degree nine was studied in \cite{gisa}. Comparing the list given there with \cite{fa-li9} and using Lemma \ref{blowupF1} $(S,\restrict{L}{S})$ is seen to be projectively normal. Remark \ref{liftingpn} then gives $r (X, L)= r (S,\restrict{L}{S}) = r (C,\restrict{L}{C}).$ Let now $g=5.$ Then $h^1(tL_C)=0$ for all $t\ge 1$ and from the structural sequence of $C$ in $\Pin{4}$ it is easy to see that $ r (C,\restrict{L}{C}) = 3$ if and only if $(C,\restrict{L}{C})$ is $2$-normal. On the other hand \cite{fa-li9} shows that in this case $h^1({\cal O}_S)=0$ and since $h^1(L_C)=0$ it must be $h^1(L_S)= 0$ and thus $ h^2(\iof{S}{1}) = 0.$ Now the $2$-normality of $(S,\restrict{L}{S})$ implies the $2$-normality of $C$ as can be seen from $0 \to \iof{S}{1} \to \iof{S}{2} \to \iof{C}{2} \to 0$ and therefore $r (X, L) = 3.$ Let now $g=6.$ First notice that since $h^0(L_C)=5$ it is $h^1(L_C)=1$ and thus $0 \to \iof{C}{1} \to \oofp{4}{1} \to L_C \to 0$ shows that $h^2(\iof{C}{1})=h^1(L_C) = 1$ i.e. $(C,\restrict{L}{C})$ cannot be $3$-regular. Consider the sequence \begin{equation} \label{powersofLS} 0 \to tL_S \to (t+1)\restrict{L}{S} \to (t+1) \restrict{L}{C} \to 0 \end{equation} for all $t \ge 1.$ Because $\deg{(t+1)\restrict{L}{C}} = 9(t+1) > 2g-2$ it is $h^1((t+1)\restrict{L}{C}) =0$ for all $t \ge 1.$ Therefore the above sequence gives $h^2(t\restrict{L}{S}) =h^2((t+1)\restrict{L}{S})=0$ for all $t\ge 1$ and thus $h^2(t\restrict{L}{S}) =0$ for all $t \ge 1$ by Serre's Theorem. From \cite{fa-li9} we know that $q(S)=0$ and $p_g(S) = 1.$ Thus the sequence $0 \to {\cal O}_S \to \restrict{L}{S} \to\restrict{L}{C}\to 0$ gives $h^1(\restrict{L}{S}) = h^2(\restrict{L}{S}) = 0.$ Then the sequence \brref{powersofLS} for $t=1$ gives $h^1(2\restrict{L}{S}) =0.$ The sequence $0 \to \iof{S}{2} \to \oofp{5}{2} \to 2\restrict{L}{S} \to 0$ gives $h^2(\iof{s}{2})= h^1(2L_S) =0.$ Then the sequence $0 \to \iof{S}{2} \to \iof{S}{3}\to \iof{C}{3} \to 0$, recalling that $(S,\restrict{L}{S})$ is projectively normal, gives $h^1(\iof{C}{3})=h^2(\iof{S}{2}) =0$, i.e. $(C,\restrict{L}{C})$ is $3$-normal. The structure sequence for $C$ in $\Pin{4}$ then easily shows that $(C,\restrict{L}{C})$ is $4$-regular and thus $r (X, L) = 4.$ Let now $g=7.$ Noticing that $h^1(\restrict{L}{C})=2$ and recalling from \cite{fa-li9} that in this case $q(S) =0$ and $p_g(S) = 2$, the same argument as above shows that $(C,\restrict{L}{C})$ is not $3$-regular but it is $4$-regular thus $r (X, L) = 4.$ Let now $d=10.$ From \cite{fa-li10} we see that $h^1({\cal O}_X)=0$ and therefore the ladder of these manifolds is regular. Following the list given in \cite{fa-li10} let $X$ be a sectional genus $6$, codimension $4$ Mukai manifold of dimension $3$ or $4.$ The curve section $(C,\restrict{L}{C})$ is then a canonical curve in $\Pin{5}$ and as such it is projectively normal. Because $h^1(K_C) =1,$ $h^1(2K_C) = 0$ and the ladder is regular, it follows from Lemma \ref{rofpncurves} and Remark \ref{liftingpn} that $r (X, L) = r (C,\restrict{L}{C})=4.$ Let now $X$ be any of the remaining threefolds of degree $10$ in $\Pin{7},$ all of which have $g=5,$ according to \cite{fa-li10}. Let $(C,\restrict{L}{C})$ be a generic curve section. From the classification of manifolds with hyperelliptic section (see \cite{BESO}) it follows that either $X$ is a hyperquadric fibration over $\Pin{1}$ or $C$ is not hyperelliptic. In the latter case it is $cl(C)\ge 1$ and therefore Theorem \ref{glcliff} gives the projective normality of $C.$ Because $g=5$ it is $h^1(\restrict{L}{C})=0$ and then Lemma \ref{rofpncurves}, the regularity of the ladder and Remark \ref {liftingpn} give $r (X, L) = r (C,\restrict{L}{C})= 3.$ Let $(X, L) \stackrel{\pi}{\to}\Pin{1}$ now be a hyperquadric fibration. Consider $ W= \Proj{\oofp{1}{1,1,1,1}}$ and let $\cal{T}={\cal O}_W(1).$ From \cite{fa-li10} it follows that $X \in|2\cal{T} + \pi^*(\oofp{1}{2})|$ and $L= \restrict{\cal{T}}{X}.$ The higher vanishings $h^i(\iof{X}{k-i})=0$ for $i\ge 2$ required for the $k$-regularity of $X$ are easily obtained for all $k \ge 3$ from the sequences \begin{gather} 0\to \iof{X}{k-i}\to \oofp{7}{k-i} \to \oof{X}{k-i} \to 0\notag \\ 0\to (k-2-i)\cal{T} +\pi^*(\oofp{1}{-2}) \to (k-i)\cal{T} \to \oof{X}{k-i} \to 0\notag \end{gather} recalling Remark \ref{leray}. Notice that $|\cal{T}|$ embeds $W$ in $\Pin{7}$ and the embedding is projectively normal, i.e. $H^0(\oofp{7}{k}) \to H^0(W, \oof{W}{k})$ is onto for all $k\ge 1.$ Therefore $X$ is $k$-normal in the embedding given by $\restrict{\cal{T}}{X},$ for some $k,$ if and only if $H^0(W,\oof{W}{k}) \to H^0(X,\oof{X}{k})$ is surjective and this happens if and only if $H^1(W, (k-2)\cal{T} +\pi^*(\oofp{1}{-2})=0.$ It is $H^1(W, (k-2)\cal{T} +\pi^*(\oofp{1}{-2})) = H^1(\Pin{1}, \oofp{1}{-2} \otimes S^{k-2}\oofp{1}{1,1,1,1}).$ \\ Combining Lemma \ref{buongg} and the fact that $\mu^-(\oofp{1}{-2} \otimes S^{k-2}\oofp{1}{1,1,1,1}) = k-4$ it is $H^1(\Pin{1}, \oofp{1}{-2} \otimes S^{k-2}\oofp{1}{1,1,1,1})=0$ for all $k \ge 3.$ On the other hand $H^1(\Pin{1},\oofp{1}{-2}) =1$ so $r (X, L) =3.$ \end{pf} \begin{cor} Let $(X, L)$ be a $n$-dimensional polarized pair with $X$ smooth, $L$ very ample and $\Delta \le 3.$ Then \begin{itemize} \item[i)] $(X, L)$ is extremal if and only if it is either a hypersurface or $\Delta=0,1.$ \item[ii)] If $\Delta=2$ then $r (X, L) = 3.$ \item[iii)] If $\Delta=3$ then $r (X, L) = 3$ unless $(X, L)$ is a complete intersection of type $(2,3)$ or a curve of genus $3$ embedded in $\Pin{3}$ as a curve of type $(2,4)$ on a smooth quadric hypersurface. In both these cases $r (X, L) = 4. $ \end{itemize} \end{cor} \begin{pf} From the proof of Theorem \ref{Delta+2thm} there are only curves and surfaces with $\Delta=3$ to consider. If $X$ is a curve, since $c\ge 2$, it must be $g\ge 3$ and $d \ge 6$. If $d\ge 7$ Theorem \ref{alafujita} gives $r=3.$ If $d=6$ then $X \subset \Pin{3}$ and \cite{Io1} section 7 gives three possible types for $X.$ $X$ is linked to a twisted cubic by two cubic hypersurfaces, $X$ is of type $(2,4)$ on a smooth quadric or $X$ is a complete intersection of type $(2,3).$ In the first case $\iofo{X}$ has a resolution as in \brref{bordigares} for $N=3$ and therefore $r=3.$ In the second case $X$ is not $2$-normal and therefore $r\ge 4.$ By Theorem \ref{lazran} $r \le 5.$ By \cite{GLP} $X$ cannot be extremal, therefore $r=4.$ From \cite{BEL2} it follows that complete intersections of type $(2,3)$ have $r=4.$ Assume $n=2.$ As above complete intersections of type $(2,3)$ have $r=4.$ Following \cite{Io1} Theorem 4.8 and section 7, it follows from Theorem \ref{alafujita} and Proposition \ref{elscr} that the only varieties left to investigate are Bordiga surfaces in $\Pin{4}.$ They have resolutions as in \brref{bordigares} with $N=4.$ It is straightforward to check $r=3.$ \end{pf} \begin{cor} Let $(X, L)$ be a $n$-dimensional polarized pair with $n \ge 3,$ $X$ smooth, $L$ very ample, $\Delta=4$ and $(X, L)$ not a hypersurface. Then $r (X, L) =3$ unless $(X, L)$ is a complete intersection of type $(2,2,2)$ or any threefold in $\Pin{5}$ of degree $7$ in which cases $r(X, L) =4.$ \end{cor} \begin{pf} Immediate from the proof of Theorem \ref{Delta+2thm}. \end{pf} \begin{cor} Let $(X, L)$ be a $n$-dimensional polarized pair with $n \ge 3,$ $X$ smooth, $L$ very ample, $\Delta=5$ and $(X, L)$ not a hypersurface. Then $r (X, L) =3$ unless $(X, L)$ is in the following list, in which cases $r(X, L) =4.$ \begin{itemize} \item[i)] $(X, L) \subset \Pin{5}$ is a threefold of degree $8$ fibered over $\Pin{1}$ with generic fibres complete intersections of type $(2,2)$ (see \cite{BSS3}). \item[ii)] $(X, L) \subset \Pin{6}$ is a threefold of degree $9$, $g=6$, obtained by blowing up a point on a Fano manifold in $\Pin{7}$, (see \cite{fa-li9}). \item[iii)] $(X, L) \subset \Pin{6}$ is a threefold of degree $9$, $g=7$, obtained by a cubic section of a cone over the Segre embedding of $\Pin{1} \times \Pin{2} \subset \Pin{5}$, (see \cite{fa-li9}). \item[iv)] $(X, L) \subset \Pin{n+4}$ is a Mukai manifold of degree $10,$ $n=3,4$, $g=6$, (see \cite{fa-li10}). \end{itemize} \end{cor} \begin{pf} Immediate from the proof of Theorem \ref{Delta+2thm}. \end{pf} \section{Scrolls over curves of genus two} \label{g2scrolls} Let $(X,\cal{T})= (\Proj{E}, \taut{E})$ be an $n$-dimensional scroll over a curve $C$ of genus $2.$ From \cite{gisa} (Lemma 5.2) it follows that $\Delta(X,\cal{T}) = 2n$ and $h^1(\cal{T})=0.$ These facts will be used without further mention throughout this section. The same conclusions can also be drawn from Lemma \ref{buongg} and the following lemma: \begin{lemma} \label{muscrollg2} Let $E$ be a rank $r$ very ample vector bundle over a genus $2$ curve. Then $\mu^-(E) > 3$ and $h^1(C, S^t(E))=0$ for $t\ge 1.$ \end{lemma} \begin{pf} By induction on $r.$ If $r=1$ then $E$ is semistable and very ample, therefore $\mu^-(E) = \mu(E) \ge 5.$ Let now $r\ge 2$ and assume $\mu^-(E) >3$ for every very ample vector bundle of rank up to $r-1.$ From \cite{Io-To} it is $c_1(E) \ge 3r+1.$ If $E$ is semistable then $\mu^-(E) = \mu(E) =\frac{d}{r}\ge 3 + \frac{1}{r}>3.$ Let now $E$ be non semistable. Then there is a quotient bundle $E \to Q\to 0$ such that $rk(Q) < rk(E)$ and $\mu(Q) = \mu^-(E).$ Being a quotient of a very ample bundle on a curve, $Q$ is also very ample and by induction $\mu^-(E) = \mu(Q) \ge \mu^-(Q) >3.$ Because $\mu^-(S^t(E))=t \mu^-(E) > 3t \ge 3$ it is $h^1(S^t(E))=0$ from Lemma \ref{buongg}. \end{pf} \begin{prop} \label{pnscrg2surf} Let $(X,\cal{T})$ be a surface scroll over a curve of genus $g=2$ with degree $T^2=d.$ Then $(X,\cal{T})$ is projectively normal unless $d=8.$ In this case $X$ is as in \cite{Io2} (4.2). \end{prop} \begin{pf} The projective normality of such scrolls up to degree $8$ was studied in \cite{alibaba} where the non projectively normal surfaces in the statement can be found. Let us assume $d \ge 9.$ If $E$ is semistable then $\mu^-(E) = \mu(E) = d/2 >4$ and therefore $(X,\cal{T})$ is p.n. by Lemma \ref{criteriodelbutler}. Let now $E$ be non semistable. Then $E$ admits a Harder Narasimhan filtration of the form $0\to D\to E$ where $D$ is a line bundle. Let now $Q$ be the quotient line bundle $0\to D \to E\to Q \to 0.$ From the definition of $\mu^-$ it is $\mu^-(E)=\mu(Q) = \deg Q.$ Because $E$ is very ample so must be $Q.$ A line bundle $Q$ on a curve of genus $2$ is very ample if and only if $\deg Q \ge 5.$ Thus $\mu^-(E) >4$ and $(X,\cal{T})$ is p.n. by Lemma \ref{criteriodelbutler}. \end{pf} The above non projectively normal surface scrolls are such because they are not $2$-normal, (see \cite{alibaba}). Indeed the next Proposition and Lemma \ref{muscrollg2} show that $2$-normality is equivalent to projective normality for scrolls of genus $2,$ extending a result found in \cite{pu-ga}. \begin{prop} \label{pn2n} Let $(X,\cal{T})=(\Proj{E},\taut{E})$ be a scroll over a smooth curve $C$ of genus $g$ such that $\mu^-(E)>2g-2.$ Then $(X,\cal{T})$ is projectively normal if and only if it is $2$-normal. \end{prop} \begin{pf} If $(X,\cal{T})$ is p.n. it is obviously $2$-normal. \\ To see the converse let $n=dim X=rk E$ and let $\pi:X\to C$ be the natural projection. Reasoning as in \cite{pu-ga} Lemma 1.4, projective normality of $(X,\cal{T})$ follows from the surjectivity of the maps \begin{equation} \label{ontomaps} H^0((j-1)\cal{T}) \otimes H^0(\cal{T}) \to H^0(j\cal{T}) \text{ \ \ \ for all \ } j \ge 2. \end{equation} This in turns follows, according to \cite{mu} Theorem 2, from the vanishing of $$H^i(X,(j-1-i)\cal{T})=0 \text{\ \ for all } n\ge i\ge 1 \text{ \ \ and for all } j \ge 2.$$ Because $i\le n$ and $j\ge 2$ it is $j-1-i>-n$ and therefore Remark \ref{leray} shows that $(X,\cal{T})$ is p.n. if $H^1(X,(j-2)\cal{T})=H^1(C, S^{j-2}E)=0$ for all $j\ge 2.$ The hypothesis $\mu^-(E)>2g-2$ implies $\mu^-(S^{j-2}E) = (j-2)\mu^-(E)>2g-2$ for all $j\ge 3.$ From Lemma \ref{buongg} it follows that $H^1(X,(j-2)\cal{T})=0$ for all $j\ge 3.$ This gives all necessary surjectivity in \brref{ontomaps} but for $j=2.$ Thus $(X,\cal{T})$ is p.n. if $H^0(\cal{T}) \otimes H^0(\cal{T}) \to H^0(2\cal{T})$ is onto, i.e. if $(X,\cal{T})$ is $2$-normal. \end{pf} \begin{cor} A scroll $(X,\cal{T})$ over a curve of genus $2$ is p.n. if and only if it is $2$ normal. \end{cor} \begin{pf} Immediate from Proposition \ref{pn2n} and Lemma \ref{muscrollg2} \end{pf} Results on threefold scrolls are collected in the following proposition. \begin{prop} Let $(X,\cal{T})=(\Proj{E}, \taut{E})$ be a threefold scroll over a curve of genus $2.$ Let $d \ne 12.$ Then $(X,\cal{T})$ fails to be projectively normal if and only if one of the following cases occur \begin{enumerate} \item[i)] $d=11.$ \item[ii)] $d\ge 13$, $E$ is not semistable and it admits a quotient $E \to \cal{E}\to 0$ of rank two and degree eight. \end{enumerate} \end{prop} \begin{pf} It is known, see \cite{Io-To} or \cite{Io2} and \cite{fa-li9}, \cite{fa-li10}, that there do not exist threefold scrolls of genus two and $d\le 10.$ So assume $d\ge 11.$ Because $h^1(\cal{T})=0$ it is $h^0(\cal{T}) = d-3.$ A simple computation shows that $h^0(X,\oof{X}{2}) = 4d-6 > h^0(\oofp{d-4}{2}) = \frac{(d-2)(d-3)}{2}$ if $d \le 11,$ so that degree $11$ scrolls cannot be $2$-normal. Assume $d\ge 13.$ If $E$ is semistable then $\mu^-(E) = \mu(E) = d/3 > 4$ and thus $(X,\cal{T})$ is p.n. by Lemma \ref{criteriodelbutler}. Let now $E$ be not semistable. Assume $E$ does not admit a degree $8$ and rank $2$ quotient. All quotients $E \to Q \to 0$ must be very ample and thus it must be $rank Q = 1$ and $\deg Q \ge 5$ or $rank Q =2$ and $\deg Q \ge 9.$ Therefore for all $Q$ it is $\mu(Q) >4$ and thus $\mu^-(E) >4$ and then $(X,\cal{T})$ is p.n. by Lemma \ref{criteriodelbutler}. Let now $E$ be not semistable with a quotient $E\to \cal{E} \to 0$ of degree $8$ and rank $2.$ Notice that $(\Projcal{E},\tautcal{E})$ is one of the non $2$-normal surfaces of degree eight embedded in $\Pin{5}$ studied in \cite{alibaba}. Let $D$ be the line bundle of degree $d -8$ such that \begin{equation} \label{quotient} 0\to D \to E \to \cal{E} \to 0. \end{equation} Since $d - 8 > 2$ it is $h^1(D) =0$ and thus $H^0(E) = H^0(\cal{E}) \oplus H^0(D).$ Therefore \begin{equation} \label{essedueE} S^2(H^0(E))=S^2(H^0(\cal{E})) \bigoplus H^0(\cal{E}) \otimes H^0(D)\bigoplus S^2(H^0(D)) \end{equation} Consider the sequence obtained by tensoring \brref{quotient} with $D:$ \begin{equation} \label{tensorD} 0 \to D \otimes D \to E \otimes D \to \cal{E} \otimes D \to 0. \end{equation} Because $\deg(D\otimes D) =2(d-8) > 2$ and $\mu^-(\cal{E} \otimes D)=\mu^-(\cal{E}) + \mu^-(D) = d-4 > 2$ it follows that $h^1(D\otimes D) = h^1( \cal{E} \otimes D)=0$ and thus $H^0(E \otimes D) = H^0(D\otimes D) \oplus H^0(\cal{E} \otimes D)$ and $h^1(E \otimes D) = 0.$ Considering now the exact sequence \begin{equation} \label{simm} 0 \to D \otimes E \to S^2(E) \to S^2(\cal{E}) \to 0 \end{equation} it follows that \begin{equation} \label{hzero} H^0(S^2(E)) = H^0(S^2(\cal{E})) \bigoplus H^0(\cal{E} \otimes D)\bigoplus H^0(D \otimes D). \end{equation} Putting together \brref{hzero} and \brref{essedueE} it follows that the map $\phi: S^2(H^0(E)) \to H^0(S^2(E))$ decomposes as \begin{gather}[S^2(H^0(\cal{E})) \stackrel{\alpha}{\rightarrow}H^0(S^2(\cal{E}))] \notag \\ \bigoplus \notag \\ [H^0(\cal{E}) \otimes H^0(D) \stackrel{\beta}{\rightarrow} H^0(\cal{E}\otimes D)] \bigoplus [S^2(H^0(D)) \stackrel{\gamma}{\rightarrow} H^0(D \otimes D)]. \notag \end{gather} It was proven in \cite{alibaba} that $\alpha$ is not surjective, therefore $\phi$ cannot be surjective, i.e. $(X,\cal{T})$ cannot be $2$-normal. \end{pf} \begin{rem} The existence of degree $11$ and $12$ threefold scrolls over curves of genus $2$ is an open problem. If a degree $12$ such scroll $(X,\cal{T})=(\Proj{E},\taut{E})$ exists then it is not difficult to see that $E$ must be semistable. If it were not semistable then there would be a destabilizing subbundle\ $\cal{F}$ either of rank $1$ such that $ \deg \cal{F} > 4$ or of rank $2$ such that $ \deg \cal{F} > 8.$ In both cases the resulting quotient $0 \to \cal{F} \to E \to Q \to 0$ could not be very ample for degree reasons, which is a contradiction. \end{rem} \begin{rem} Let $(X,\cal{T})=(\Proj{E},\taut{E})$ again be a $3$ dimensional scroll over a curve of genus $2.$ If $(X, L)$ is projectively normal, recalling that $h^1(E) =0,$ the same argument used in Proposition \ref{elscr} gives $r (X, L) = 3.$ If $(X, L)$ is not p.n. , notice that if $d\ge 13$ it follows that $h^0(\cal{T}) \ge 10$ and thus $(X, \cal{T})$ is $(\Delta + 2)$-regular, i.e. $8$-regular from \cite{Ran1}. When $d=11,12$ it is easy to check that $h^i(\iof{X}{8-i}) = 0$ for all $i\ge 2$ while we were not able to establish the $7$-normality of these manifolds. \end{rem} \section{ $\Pin{r}$ bundles over an elliptic curve } \label{ellipticpkbundles} Throughout this section let $E$ be a vector bundle of rank $r$ and degree $d$ over an elliptic curve $C$. Let $(X,\cal{T}) = \scroll{E}$ and let $D$ be a divisor on $X$ numerically equivalent to $aT + bF.$ Assume $D$ is very ample. The projective normality of the embedding given by $D$ was studied by Homma \cite{Ho1}, \cite{Ho2} when $r=2,$ and in a more general setting by Butler \cite{bu} (See also \cite{alibaba}). In this section the case of $a=2$ and $r=3$ is addressed and Butler's results are improved in some cases. \begin{lemma} \label{mupiudec} Let $E = \bigoplus_i E_i$ be a decomposable vector bundle over an elliptic curve. Then $$\mu^+(E) = max_i \{{\mu(E_i)}\}.$$ \end{lemma} \begin{pf} This is essentially \cite{alibaba2} Lemma 2.8, reinterpreted from the point of view of $\mu^+$ instead of $\mu^-.$ \end {pf} \begin{lemma} \label{mupiu} Let $(X,\cal{T})$ be as above and let $M_{s}$ be a divisor on $X$ whose numerical class is $M_{s}\equiv T+sF. $ Let $m=min\{ t \in \Bbb{Z}| h^0(M_t)> 0.\}$ Then $m=-[\mu^+(E)]$ and there exists a smooth $S \equiv T+mF.$ \end{lemma} \begin{pf} From \cite{bu} it follows that for any vector bundle $\cal{G}$ over an elliptic curve $\mu^+(\cal{G}) <0$ implies $h^0(\cal{G}) = 0.$ For simplicity of notation let $m^*=-[\mu^+(E)].$ We need to show that $m=m^*.$ Let $\cal{L}_t$ be a line bundle on $C$ with degree $t.$ If $t < m^*,$ then $t=m^*- x$ for some integer $x\ge 1.$ Then $\mu^+(E \otimes\cal{L}_t )= \mu^+(E) + t = \mu^+(E) +m^*- x < 1-x \le 0$ Therefore $ h^0(M_t) = 0$ if $t< m^*$ and thus \begin{equation} \label{mandm1} m^*\le m.\end{equation} Let now $E$ be indecomposable and thus semistable. Because $\mu(E) =\mu^+(E)$ and $-m^* \le \mu^+(E)$ it is $d+rm^* \ge 0.$ If $d +rm^*>0$ then $ h^0(M_{m^*}) >0.$ If $d + rm^*= 0$ then, as in \cite{At}, a line bundle $\cal{L}_{m^*}$ of degree $m^*$ can be found by a suitable twist of degree zero, such that $h^0(E \otimes \cal{L}_{m^*}) =1.$ Let now $E = \bigoplus_i E_i$ be decomposable. Then $h^0(E) = \bigoplus_ih^0(E_i).$ Let $E_{\hat{i}}$ be one of the components such that $\mu^+(E) = \mu(E_{\hat{i}}).$ As $E_{\hat{i}}$ is indecomposable it follows from above that there exists a line bundle $\cal{L}_{m^*}$ of degree $m^*$ such that $h^0(E_{\hat{i}} \otimes \cal{L}_{m^*})>0.$ Therefore $h^0(M_{m^*}) > 0$ and thus $m\le m^*$ which combined with \brref{mandm1} gives $m=m^*.$ If $S\equiv T+m^*F$ is singular it must be reducible as $S' \cup (m^*-t) F$ where $S'\equiv T+tF$ with $t<m^*$ which is not possible because of the minimality of $m^*=m.$ Therefore there must be a smooth $S\equiv T +m^*F.$ \end{pf} \begin{lemma} \label{D2npn} Let $(X,\cal{T})$ and $D$ be as above with $r=3,$ $a \ge 2,$ and $D$ very ample. If \begin{itemize} \item[i)] there exists an ample smooth surface $S \equiv T+x F$ for some $x \in \Bbb Z;$ \item[ii)] $(a-1)\mu^-(E) + b - x>1$ \end{itemize} then the embedding of $X$ given by $D$ is projectively normal if and only if it is $2$-normal. \end{lemma} \begin{pf} If the embedding is p.n. it is obviously $2$-normal. As in Lemma \ref{pn2n} the projective normality follows from the surjectivity of the maps \begin{equation} \label{ontomaps2} H^0((j-1)D) \otimes H^0(D) \to H^0(jD) \text{ \ \ \ for all \ } j \ge 2. \end{equation} Assume $j\ge 4.$ Surjectivity in \brref{ontomaps2} follows, according to \cite{mu}, from the vanishing of $$H^i(X,(j-1-i)D)=0 \text{\ \ for all } 3 \ge i\ge 1 \text { \ \ and for all } j \ge 4.$$ Notice that $R^q\pi_*((j-1-i)D)=R^q\pi_*(a(j-1-i) \cal{T}) \otimes \cal{M}_{i,j}$ where $\cal{M}_{i,j}$ is a line bundle on $C$ of degree $(j-1-i)b.$ Notice also, e.g. \cite{H}, that $R^q\pi_*(a(j-1-i)\cal{T})=0$ unless $q=0$ and $j-1-i\ge 0,$ or $q=2$ and $a(j-1-i)\le -3.$ Since $a\ge 2,$ $i\le 3$ and $j\ge 4,$ the last inequality is never satisfied . For $j \ge 4$ Leray's spectral sequence shows that it is enough to show $H^1(X,(j-2)D)=H^1(C, S^{a(j-2)}E\otimes \cal{M}_{1,j})=0$ which is guaranteed by $D$ being ample. This gives all necessary surjectivity in \brref{ontomaps2} but for $j=2,3.$ If $j=2$ the map $H^0(D) \otimes H^0(D) \to H^0(2D)$ is onto by assumption , being the embedding $2$-normal. Assume now $j=3.$ Let $S \equiv T+x F$ be the smooth element whose existence is given by assumption $i).$ Ampleness of $D$ gives $a \mu^-(E) + b > 0.$ Combining this with condition $ii)$ it follows from Lemma \ref{buongg} that $H^1(tD-S) =0$ for $t=1,2,3.$ In particular, following Homma, the commutative diagram below is obtained : \begin{alignat}{5} 0 \to H^0(D-S)&\otimes H^0(2D)& &\to& H^0(D)&\otimes H^0(2D) &&\to& H^0(S,\restrict{D}{S}) &\otimes H^0(2D)\to 0 \notag \\ &\downarrow \alpha& & & &\downarrow \beta & & & &\downarrow \gamma\\ 0 \longrightarrow H^0(3&D-S)& &\longrightarrow& H^0(3&D)& &\longrightarrow& H^0(S,&\restrict{3D}{S}) \to 0 \notag \end{alignat} The surjectivity of $\beta$ will follow from the surjectivity of $\alpha$ and $\gamma.$ From \cite{Ho1} and \cite{Ho2} it follows that $\restrict{D}{S}$ is normally generated. Because $H^0(2D) \to H^0(\restrict{2D}{S})$ is surjective from above, it follows that $\gamma$ is onto. Lemma \ref{buongg} and condition $ii)$ give $D-S\equiv (a-1)T + (b-x)F$ generated by global sections. Using this fact and noticing that $H^1(D+S)=0$ being $D$ very ample and $S$ ample, it is straightforward to check that $H^i(2D - i (D-S)) = 0$ for all $i \ge 1.$ Therefore by \cite{mu} $\alpha$ is onto. \end{pf} \begin{prop} \label{homma} Let $(X,\cal{T})$ and $D \equiv 2T + bF$ be as above. If \begin{itemize} \item[i)] there exists an ample smooth divisor $Y \equiv T+x F$ for some $x \in \Bbb Z;$ \item[ii)] $\mu^-(E) + b - x>1$ \end{itemize} then $|D|$ gives a $2$-normal embedding of $X$ if $|\restrict {D}{Y}|$ gives a $2$-normal embedding of $Y$. \end{prop} \begin{pf} The proof proceeds along the same lines of the case $j=3$ in the proof of Lemma \ref{D2npn}. Let $Y \equiv T+x F$ be the smooth element whose existence is given by assumption $i).$ Ampleness of $D$ gives $2 \mu^-(E) + b > 0.$ Combining this with condition $ii)$ it follows that $H^1(tD-Y) =0$ for $t=1,2.$ In particular the following commutative diagram is obtained : \begin{alignat}{5} 0 \to H^0(D-Y)&\otimes H^0(D)& &\to& H^0(D)&\otimes H^0(D) &&\to& H^0(Y,\restrict{D}{Y}) &\otimes H^0(D)\to 0 \notag \\ &\downarrow \alpha& & & &\downarrow \beta & & & &\downarrow \gamma\\ 0 \longrightarrow H^0(2&D-Y)& &\longrightarrow& H^0(2&D)& &\longrightarrow& H^0(Y,&\restrict{2D}{Y}) \to 0. \notag \end{alignat} The surjectivity of $\beta$ will follow from the surjectivity of $\alpha$ and $\gamma.$ Because $H^0(D) \to H^0(\restrict{D}{Y})$ is onto from above and $H^0(\restrict{D}{Y}) \otimes H^0(\restrict{D}{Y}) \to H^0(\restrict{2D}{Y})$ is onto by assumption it follows that $\gamma$ is onto. Condition $ii)$ is equivalent to $D-S\equiv T + (b-x)F$ being generated by global sections (see Lemma \ref{buongg} and \cite{alibaba2} Lemma 2.9). Using this fact and noticing that $H^1(Y)=0$ being $Y$ ample, it is straightforward to check that $H^i(D - i(D-Y)) = 0$ for all $i \ge 1.$ Therefore by \cite{mu} $\alpha$ is onto. \end{pf} \begin{cor} \label{hommacor} Let $(X,\cal{T})$ and $D$ be as above with $r=3$ and $a=2.$ If \begin{itemize} \item[i)]$\mu^-(E) > [\mu^+(E)]$ \item[ii)] $\mu^-(E) + b >1-[\mu^+(E)]$ \end{itemize} then $|D|$ gives a projectively normal embedding. \end{cor} \begin{pf} Let $x=-[\mu^+(E)]$, notice that condition i) and Theorem \ref{miyaoteo} give ampleness of $T+xF$. Now combine Lemma \ref{mupiu}, Lemma \ref{D2npn}, Proposition \ref{homma} and the fact that a very ample line bundle on a $2$ dimensional scroll over an elliptic curve is always normally generated by \cite{Ho1} and \cite{Ho2}. \end{pf} \begin{rem} Let $E$ be an indecomposable vector bundle of rank $r=3$ and degree $d\equiv 1 (3).$ For simplicity let us assume that $E$ has been normalized, so $d=1.$ Since $E$ is indecomposable it is semistable and $\mu^-(E) = \mu ^+(E) = \mu(E) = 1/3.$ The hypothesis of Corollary \ref{hommacor} are satisfied for $D\equiv 2T + F.$ and such a $D$ is very ample from \cite{alibaba2} Theorem 4.5. Therefore $|D|$ gives a projectively normal embedding. Notice that Butler's results \cite{bu}, see Lemma \ref{criteriodelbutler}, were not able to establish the normal generation of such a $D.$ \end{rem} \begin{rem} It is straightforward to check that for a divisor $D$ as in Corollary \ref{hommacor} it is always $h^0(D) \ge 10$ and therefore the embedding given by $|D|$ satisfies the Eisenbud Goto conjecture . \end{rem}

9710

alg-geom/9710031

en

https://arxiv.org/abs/alg-geom/9710031

alg-geom/9710031

Gregor Masbaum

Jorgen Ellegaard Andersen and Gregor Masbaum

Involutions on Moduli Spaces and Refinements of the Verlinde Formula

33 pages, Latex, minor modifications, to appear in Mathematische Annalen

MSRI 1997-102

The moduli space $M$ of semi-stable rank 2 bundles with trivial determinant over a complex curve carries involutions naturally associated to 2-torsion points on the Jacobian of the curve. For every lift of a 2-torsion point to a 4-torsion point, we define a lift of the involution to the determinant line bundle $\L$. We obtain an explicit presentation of the group generated by these lifts in terms of the order 4 Weil pairing. This is related to the triple intersections of the components of the fixed point sets in $M$, which we also determine completely using the order 4 Weil pairing. The lifted involutions act on the spaces of holomorphic sections of powers of $\L$, whose dimensions are given by the Verlinde formula. We compute the characters of these vector spaces as representations of the group generated by our lifts, and we obtain an explicit isomorphism (as group representations) with the combinatorial-topological TQFT-vector spaces of [BHMV]. As an application, we describe a `brick decomposition', with explicit dimension formulas, of the Verlinde vector spaces. We also obtain similar results in the twisted (i.e., degree one) case.

alg-geom

\section*{Introduction and Motivation.} The celebrated Verlinde formula \cite{Ve} gives the dimension of certain vector spaces of so-called `conformal blocks' appearing in conformal field theory. In this paper, we will take the point of view of algebraic geometry and think of the conformal blocks as holomorphic sections of powers of the determinant line bundle over moduli spaces of semi-stable bundles with fixed determinant over a complete non-singular complex curve $\Sigma$. According to Atiyah \cite{At} and Witten \cite{Wi}, the spaces of holomorphic sections should also fit into $2+1$-dimensional `Topological Quantum Field Theories' (TQFT). This geometric construction has been studied quite a lot (see {\em e.g.} \cite{ADW,MS,RSW,Hi,CLM,Th,Th2,Sz,BSz,BL,KNR,Fa}). In \cite{BHMV}, a combinatorial-topological construction of TQFT-functors was described, based on a particularly simple construction of the Witten-Reshetikhin-Turaev $3$-manifold invariant \cite{RT} in the $SU(2)$-case through the Kauffman bracket. In that paper one also constructed certain involutions on the TQFT-vector spaces which were then used to decompose the vector spaces into direct summands. These involutions are associated to simple closed curves on the underlying smooth surface of $\Sigma$, and they generate a kind of Heisenberg group presented in terms of the $mod $ $4$ intersection form. These ideas were developed further in \cite{BM} to construct spin-refined TQFT's. The starting point for the present paper was the idea that the involutions of \cite{BHMV} should correspond on the algebraic-geometric side to the involutions on moduli space naturally associated to $2$-torsion points on the Jacobian of $\Sigma$. More precisely, the $2$-torsion points should correspond to the $mod $ $2$ homology classes of the simple closed curves. Note that the involutions on the spaces of holomorphic sections require a choice of lift to the determinant line bundle ${\mathcal{L}}$. It is easy to see that these lifts generate a central extension of the group of $2$-torsion points whose alternating form is given by the order $2$ Weil pairing. It follows that this extension is indeed abstractly isomorphic to the one that appeared in \cite{BHMV}. The ambiguity in the choice of lift is reflected on the topological side by the choice of a simple closed curve within its $mod $ $2$ homology class. One of our motivations in this paper is to establish a more precise correspondance between the two theories. The key idea is to define, for every lift of a $2$-torsion point ${\alpha}$ to a $4$-torsion point $a$, an involution $\rho_a$ on ${\mathcal{L}}$ which covers the action of ${\alpha}$ on the moduli space $M$. The sign of this lift $\rho_a$ is fixed by requiring it to act as the identity over a certain component of the fixed point set of the involution acting on $M$; this component is simply the one containing the class of the semi-stable bundle $L_a\oplus L_a^{-1}$ (see section \ref{prel} for more details). We will see in section \ref{taugamma} that the action of this lift $\rho_a$ on holomorphic sections corresponds precisely to the involution $\tau_{\gamma}$ which is associated in the \cite{BHMV}-theory to a simple closed curve ${\gamma}$ whose homology class is Poincar\'e dual to $a$. In this way, we obtain an explicit isomorphism (as group representations) with the TQFT-vector spaces of \cite{BHMV}. We believe this constitutes a nice confirmation that there should be a natural correspondence between the two theories. The main work in this paper is, however, algebraic-geometric in nature and consists of a detailed study of our lifts $\rho_a$ and their action on the Verlinde vector spaces. Our main results are as follows (see section \ref{smr} for more complete statements). We show in theorem \ref{1.1} that our lifts satisfy $$\rho_a \,\rho_b\,=\,\lambda_4(a,b) \,\rho_{a+b}, $$ where $\lambda_4$ is the order $4$ Weil pairing (which is the algebraic-geometric analogue of the {\em mod} $4$ intersection form). Along the way, we also determine in theorem \ref{evencasei} the triple intersections of the components of the fixed point sets of the action on the various moduli spaces. The {\em r\^ole} played by the order $4$ Weil pairing in this context doesn't seem to have been observed before. We then compute in theorem \ref{1.2} the trace of the induced involutions $\rho_a^{\otimes k}$ on the spaces of holomorphic sections of the $k$-th tensor power of ${\mathcal{L}}$. This determines the characters of these vector spaces as representations of the group generated by our lifts. We also obtain similar results in the twisted ({\em i.e.}, degree one) case, where the situation is somewhat simpler, as only the order $2$ Weil pairing is needed. A corollary of our results is a `brick decomposition' of the spaces of holomorphic sections, the structure of which depends on the value of the level $k$ modulo $4$. This is, of course, analogous to the decomposition in \cite{BHMV}, but we will derive it, as well as explicit dimension formulas, directly from the character of the representation. At low levels, similar decompositions have appeared previously in the work of Beauville \cite{Be2} (see also Laszlo \cite{L}) for $k=2$, and of van Geemen and Previato \cite{vGP1,vGP2}, Oxbury and Pauly \cite{OP}, and Pauly \cite{P} for $k=4$ (in our notation). Their approach is based on the relationship with abelian theta-functions and seems quite different from ours. This paper is organized as follows. After giving the basic definitions in section \ref{prel}, we state our main results in section \ref{smr}. The relationship with the \cite{BHMV}-theory is discussed in more detail in section \ref{taugamma}, and the `brick decompositions' are described in section \ref{3}. The remainder of the paper is devoted to the proofs. (See remark \ref{logstruct} for the logical structure of the proofs.) The reader interested only in the algebraic geometry may skip section \ref{taugamma}, and no familiarity with \cite{BHMV} is necessary to understand the results and their proofs. {\small \vskip 8pt\noindent{\bf Acknowledgment.} We would like to thank Ch. Sorger for discussions on this project.} \section{Basic definitions and notation.}\label{prel} Let $\Sigma$ be a complete, non-singular curve over the complex numbers of genus $g\geq 2$. Let $M_d$ be the moduli space of semi-stable bundles of rank $2$ and degree $d\in {\mathbb Z}$ on $\Sigma$. There is a natural algebraic action of the degree zero Picard group $\mathop{\fam0 Pic}\nolimits_0(\Sigma)$ on $M_d$ gotten by tensoring. In this paper, we will use the standard identification of $\mathop{\fam0 Pic}\nolimits_0(\Sigma)$ with the Jacobian $J(\Sigma)$ and speak of an action of $J(\Sigma)$ on $M_d$. Fix a point $p\in \Sigma$ and let $[p]$ be the associated line bundle. We have the determinant morphism $\det : M_d \rightarrow \mathop{\fam0 Pic}\nolimits_d(\Sigma)$. We put $M=\det^{-1}({\mathcal{O}}_\Sigma)$ and $M'=\det^{-1}([p])$. As it is explained in \cite{DN} there is a natural construction of a determinant line bundle ${\mathcal{L}}_d$ over $M_d$. We will denote the restriction of ${\mathcal{L}}_0$ to $M$ by ${\mathcal{L}}$ and the restriction of ${\mathcal{L}}_1$ to $M'$ by ${\mathcal{L}}'$. These two determinant bundles are generators respectively of the Picard group of $M$ and of $M'$ (see \cite{DN}). Let $k\geq 1$ be an integer called the {\em level}. Put $$Z_k(\Sigma) = H^0(M,{\mathcal{L}}^k)$$ $$Z'_k(\Sigma) = \left\{\begin{array}{ccl} H^0(M',{{\mathcal{L}}'}^{k/2}) & \ {\rm if} \ k\equiv 0 \ {\rm mod} \ 2\\ 0& \ {\rm if} \ k\equiv 1 \ {\rm mod} \ 2\end{array}\right.$$ It is by now well-known that the dimensions of these vector spaces are given by the celebrated Verlinde formulas (to be recalled in section \ref{3}). Following Thaddeus \cite{Th}, we will refer to $Z'_k(\Sigma)$ as the {\em twisted case}, and to $Z_k(\Sigma)$ as the {\em untwisted case.} Let $J^{(r)}$ be the subgroup of order $r$ points on $J(\Sigma)=\mathop{\fam0 Pic}\nolimits_0(\Sigma)$. We denote by $L_{\alpha}$ (resp. $L_a$) the line bundle on $\Sigma$ corresponding to ${\alpha}\in J^{(2)}$ (resp. $a\in J^{(4)}$). Note that the group $J^{(r)}$ is identified with $H^1(\Sigma;\mu_r)$, where $\mu_r\subset {\mathbb C}$ is the group of $r$-th roots of unity. The action of ${\alpha}\in J^{(2)}$ on $M_d$ is given by tensoring with $L_{\alpha}$. Since $L_{\alpha}^{\otimes 2}\cong {\mathcal{O}}_\Sigma$, this preserves $M$ and $M'$. By abuse of notation, the automorphisms of $M$ and $M'$ induced by ${\alpha}\inJ^{(2)}$ will again be denoted by ${\alpha}$. \vskip 8pt\noindent{\bf The lifts $\rho_a$ and $\rho'_{\alpha}$.} By a lift of ${\alpha}$ to ${\mathcal{L}}$ we mean an invertible bundle map from ${\mathcal{L}}$ to itself covering ${\alpha}$. For a lift to exist it suffices that ${\alpha}^*{\mathcal{L}}\cong {\mathcal{L}}$. This is the case for every ${\alpha}\in J^{(2)}$, since the Picard group of $M$ is isomorphic to ${\mathbb Z}$, and ${\mathcal{L}}$ is ample, so that ${\alpha}$ must act trivially on the Picard group. Therefore the action of any ${\alpha}\in J^{(2)}$ can be lifted to ${\mathcal{L}}$, and also to ${\mathcal{L}}'$, for the same reason. Since the only algebraic functions on $M$ and $M'$ are the constant ones, it follows that we can actually choose involutive lifts of each element of $J^{(2)}$. Any two involutive lifts of ${\alpha}$ agree up to sign. To fix the signs of the lifts of ${\alpha}$ to ${\mathcal{L}}$ and ${\mathcal{L}}'$, we use the fact that the sign can be read off in the fiber over a fixed point of ${\alpha}$. Given $0\neq {\alpha}\in J^{(2)}$, we use the notation $|X|_\alpha$ for the fixed point variety of the automorphism induced by ${\alpha}$ on the various moduli spaces $X$. It is well-known that $|M'|_{\alpha}$ is isomorphic to the {\em Prym variety $P_{\alpha}$} associated to ${\alpha}$, and that $|M|_{\alpha}$ is isomorphic to the disjoint union of two copies of the {\em Kummer variety} $P_{\alpha}/{\langle} \pm 1{\rangle}$. This result is due to Narasimhan and Ramanan \cite{NR}. In particular, $|M'|_{\alpha}$ is connected and non-empty, and $|M|_{\alpha}$ has two components. For any $a\in J^{(4)}$ such that $2a={\alpha}$, we denote by $|M|_a^+$ the component of $|M|_{\alpha}$ containing the S-equivalence class of the semistable bundle $L_a\oplus L^{-1}_a$. (Note that this is indeed fixed under tensoring with $L_{\alpha}$, since $L_a\otimes L_{\alpha}\cong L_a^{-1}$.) The other component of $|M|_\alpha$ is denoted by $|M|_a^-$. \begin{definition}\label{def11} For $0\neq {\alpha}\in J^{(2)}$, we define $\rho'_\alpha$ to be the involutive lift to ${\mathcal{L}}'$ of $\alpha$ acting on $M'$ such that $\rho'_\alpha$ acts by {\em minus} the identity on the restriction of ${\mathcal{L}}'$ to $|M'|_\alpha$. \end{definition} \begin{definition}\label{def12} For $a\in J^{(4)}$ such that $2a={\alpha}\neq 0$, we define $\rho_a$ to be the involutive lift to ${\mathcal{L}}$ of $\alpha$ acting on $M$ such that $\rho_a$ acts by the identity on the restriction of ${\mathcal{L}}$ to the component $|M|^+_a$ of $|M|_\alpha$ specified by $a$. \end{definition} It will be convenient to extend this definition by letting $\rho'_\alpha$ (resp. $\rho_a$) be the identity if ${\alpha}=0$ (resp. $2a=0$). \vskip 8pt\noindent{\bf Note.} According to Theorem F in \cite{DN}, we have that ${{\mathcal{L}}'}^{-2} \cong K$, where $K$ is the canonical bundle of $M'$. The natural action of ${\alpha}$ on $K$ coincides with the one induced by both possible involutive lifts $\pm \rho'_\alpha$. This is because ${\alpha}$ acts as the identity on the fiber of $K$ over the fixed point set $|M'|_\alpha$, since $|M'|_\alpha$ has even codimension (see section \ref{fpv}). Similar comments apply in the untwisted case. \vskip 8pt \noindent{\bf The Weil pairing.} (See {\em e.g.} \cite{Ho}.\footnote{We thank A. Beauville for pointing out this reference.}) Let ${\mathcal M}(\Sigma)$ be the field of meromorphic functions on $\Sigma$. The divisor of $f\in {\mathcal M}(\Sigma)$ is denoted by $(f)$. As already mentioned, we consider $J^{(r)}$ to be the $r$-torsion points on the group $\mathop{\fam0 Pic}\nolimits_0(\Sigma)$ which is naturally identified with $\mathop{\fam0 Div}\nolimits_0(\Sigma)/\mathop{\fam0 Div}\nolimits^{pr}(\Sigma)$. (Here, $\mathop{\fam0 Div}\nolimits_d(\Sigma)$ is the group of degree $d$ divisors, and $\mathop{\fam0 Div}\nolimits^{pr}(\Sigma)$ are the principal divisors.) If $D=\sum n_j x_j$ is a divisor and $f\in {\mathcal M}(\Sigma)$ a meromorphic function, we put $f(D)=\prod f(x_j)^{n_j}$. The Weil pairing $$\lambda_r : J^{(r)}\times J^{(r)} \rightarrow \mu_r$$ is defined as follows. Given $a,b\in J^{(r)}$, represent them by divisors $D_a,D_b$ with disjoint support, and pick $f,g\in {\mathcal M}(\Sigma)$ such that $(f)=r D_a$ and $(g)=r D_b$. Then \begin{equation} \lambda_r(a,b)=\frac{g(D_a)}{f(D_b)}.\nonumber \end{equation} The Weil pairing is antisymmetric and non-degenerate. The fact that it takes values in $\mu_r$ follows from Weil reciprocity (see \cite{GH}, p. 242). \section{Statement of the main results.}\label{smr} Let ${\mathcal G}(J^{(2)},{\mathcal{L}})$ be the group of automorphisms of the determinant line bundle ${\mathcal{L}}$ covering the action of $J^{(2)}$ on $M$. Since $\alpha^*({\mathcal{L}})\approx {\mathcal{L}}$ for every $\alpha\in J^{(2)}$, the group ${\mathcal G}(J^{(2)},{\mathcal{L}})$ is a central extension \begin{equation} {\mathbb C}^* \ \longrightarrow \ {\mathcal G}(J^{(2)},{\mathcal{L}})\ \longrightarrow \ J^{(2)} \label{tildE}. \end{equation} The same holds for the group ${\mathcal G}(J^{(2)},{\mathcal{L}}')$ of automorphisms of ${\mathcal{L}}'$ covering the action of $J^{(2)}$ on $M'$. \vskip 8pt\noindent{\bf Notation.} Let ${\mathcal{E}}\subset{\mathcal G}(J^{(2)},{\mathcal{L}})$ be the subgroup generated by the involutions $\rho_a$ ($a \in J^{(4)}$), and let ${\mathcal{E}}'\subset{\mathcal G}(J^{(2)},{\mathcal{L}}')$ be the subgroup generated by the involutions $\rho'_\alpha$ (${\alpha}\in J^{(2)}$). (See definitions \ref{def12} and \ref{def11}.) \vskip 8pt Our first main result gives a presentation of the groups ${\mathcal{E}}$ and ${\mathcal{E}}'$, as follows. \begin{theorem}\label{1.1} The involutions $\rho_a$ and $\rho'_\alpha$ satisfy the following relations: \begin{equation} \rho_a \,\rho_b\,=\, \lambda_4(a,b)\, \rho_{a+b}\label{rhoa} \end{equation} \begin{equation} \rho'_\alpha\, \rho'_\beta\,=\, \lambda_2({\alpha},{\beta})\, \rho'_{\alpha+\beta}\label{rhoprima} \end{equation} \end{theorem} It follows that the group ${\mathcal{E}}$ is a central extension \begin{equation} \mu_4 \ \longrightarrow \ {\mathcal{E}} \ \longrightarrow \ J^{(2)}.\label{tildE2} \end{equation} This extension is non-trivial, since the associated alternating form on $J^{(2)}$ is the order $2$ Weil pairing $\lambda_2$. Indeed, this form is given by the commutator $$ c({\alpha},{\beta})=\rho_a \rho_b\rho_a^{-1} \rho_b^{-1}=(\rho_a \rho_b)^2=\lambda_4(a,b)^2=\lambda_2({\alpha},{\beta}).$$ The group ${\mathcal{E}}'$ is a trivial extension of $J^{(2)}$, {\em i.e.}, it is isomorphic to $\mu_2 \times J^{(2)}$. \begin{remarks} {\em (i) The group ${\mathcal G}(J^{(2)},{\mathcal{L}})$ is known as the {\em Heisenberg group}. The fact that its alternating form is the order $2$ Weil pairing $\lambda_2$ is well-known. For example, it follows already from Beauville's isomorphism of $Z_1(\Sigma)$ with the space of abelian theta-functions \cite{Be0}. This fact is also very easy to see from our point of view (see remark \ref{coc}). Note that the alternating form determines the extension (\ref{tildE}) (but not the extension (\ref{tildE2})) up to isomorphism. (ii) If the alternating form is known, one knows {\em a priori} that our lifts $\rho_a$ satisfy $\rho_a\rho_b=\pm \rho_{a+b}$ if $\lambda_2({\alpha},{\beta})=1$, and $\rho_a\rho_b=\pm i \rho_{a+b}$ if $\lambda_2({\alpha},{\beta})=-1$, where $2a={\alpha}$ and $2b={\beta}$. (This is because the lifts $\rho_a,\rho_b,\rho_{a+b}$ are involutions.) But the sign of the prefactors $\pm 1$ and $\pm i$ in these relations is, of course, not determined by the alternating form. The contribution of theorem \ref{1.1} is to show that with our `geometric' choice of lifts in terms of components of the fixed point sets, the prefactors $\pm 1$ and $\pm i$ are given by the order $4$ Weil pairing. Similar remarks apply in the twisted case. }\end{remarks} \noindent{\bf Note.} In the literature, the Heisenberg group ${\mathcal G}(J^{(2)},{\mathcal{L}})$ is often described explicitly in terms of a `theta-structure' (see {\em e.g.} \cite{Be2}), that is, ${\mathcal G}(J^{(2)},{\mathcal{L}})$ is described as a certain group structure on the set ${\mathbb C}^*\times ({\mathbb Z}/2)^g\times ({\mathbb Z}/2)^g$. A theta-structure allows one to write the extensions (\ref{tildE}) and (\ref{tildE2}) as push-outs of an extension of $J^{(2)}$ by $\mu_2$. But this `reduction' to an extension by $\mu_2$ is not canonical, as it depends on the choice of theta-structure (which comes down, essentially, to the choice of a symplectic basis of $J^{(2)}\approx H^1(\Sigma,{\mathbb Z}/2)$).\footnote{The {\em existence} of such a `reduction' follows already from the fact that the alternating form $c({\alpha},{\beta})=\lambda_2({\alpha},{\beta})$ takes values in $\mu_2$. But there are many choices for this `reduction'.} {}From our point of view, such a choice is neither necessary nor useful, as it would break the symmetry of our description of the group ${\mathcal{E}}$, which is defined completely intrinsically in terms of the involutions $\rho_a$. Therefore we won't use theta-structures in this paper. \vskip 8pt Theorem \ref{1.1} is related to the triple intersections of the components of the fixed point set on $M$. In fact, a key step in the proof is the following result which does not seem to have been observed before. \begin{theorem}\label{evencasei} Assume that ${\alpha}$ and ${\beta}$ are distinct non-zero elements of $J^{(2)}$ such that $\lambda_2({\alpha},{\beta})=1\in\mu_2$. Let $a,b\in J^{(4)}$ such that $2a={\alpha}$ and $2b={\beta}$. Then $$ |M|^+_a \cap |M|^+_b \cap |M|^+_{a + b} \neq \emptyset \ \Leftrightarrow\ \lambda_4(a,b)=1\in\mu_4$$ $$ |M|^+_a \cap |M|^+_b \cap |M|^-_{a + b} \neq \emptyset \ \Leftrightarrow\ \lambda_4(a,b)=-1\in\mu_4$$ \end{theorem} \noindent{\bf Note.} Given theorem \ref{evencasei}, theorem \ref{1.1} in the case $\lambda_4(a,b)=1$ follows immediately. Indeed, if the triple intersection $|M|_a^+\cap|M|_b^+\cap |M|_{a+b}^+$ is non-empty, it follows from the definition of the lifts that $\rho_{a+b}=\rho_a \rho_b$ (since $\rho_a \rho_b \rho_{a+b}$ must be a constant, and this constant can be computed in the fiber of ${\mathcal{L}}$ over a triple intersection point.) The proof in the general case, however, requires some further arguments. The most interesting case is when $\lambda_4(a,b)=\pm i$. In this situation, the fixed point sets $|M|_{\alpha}$, $|M|_{\beta}$, $|M|_{{\alpha}+{\beta}}$ don't intersect, but there is a ${\mathbb P}^1$ intersecting each one of the six components $|M|_a^+$, $|M|_a^-$, $|M|_b^+$, $|M|_b^-$, $|M|_{a+b}^+$, $|M|_{a+b}^-$, in a point. See section \ref{Geoinvest}. \vskip 8pt Our next result describes $Z_k(\Sigma)$ (resp. $Z'_k(\Sigma)$) as representations of the group ${\mathcal{E}}$ (resp. ${\mathcal{E}}'$). (Here, ${\mathcal{E}}$ acts on $Z_k(\Sigma)$ {\em via} the natural action of $\rho_a^{\otimes k}$ on ${\mathcal{L}}^{\otimes k}$, and similarly for ${\mathcal{E}}'$.) This is based on the following result obtained by applying the Lefschetz-Riemann-Roch formula. We assume $\alpha\in J^{(2)}$ is non-zero, and $2a={\alpha}$. \begin{theorem}\label{1.2} One has $$ \mathop{\fam0 Tr}\nolimits(\rho_a^{\otimes k}) = \frac{1+(-1)^k}{2}\left(\frac{k+2}{2}\right)^{g-1}$$ and (for even $k$) $$\mathop{\fam0 Tr}\nolimits({\rho'_\alpha}^{\otimes k/2}) = (-1)^{k/2}\left( \frac{k+2}{2}\right)^{g-1}$$ \end{theorem} \noindent{\bf Note.} In the twisted case and for levels divisible by $4$, this result is due to Pantev \cite{Pa}. In the untwisted case and for levels divisible by $4$, it is also contained in Beauville's recent paper \cite{Be3}. Our computation was done independently of his. \vskip 8pt The characters of $Z_k(\Sigma)$ and $Z'_k(\Sigma)$ (as representations of the groups ${\mathcal{E}}$ and ${\mathcal{E}}'$, respectively) are determined by the formulas in theorem \ref{1.2} together with the trace of the identity, given by the Verlinde formulas. Therefore the above result determines these representations up to isomorphism. A remarkable consequence of this is the following theorem, which was actually the main motivation for the present paper. \begin{theorem}\label{2.4} The spaces $Z_k(\Sigma)$ and $Z'_k(\Sigma)$ are isomorphic, as representations of the groups ${\mathcal{E}}$ and ${\mathcal{E}}'$, respectively, to the TQFT-vector spaces constructed in \cite{BHMV}. \end{theorem} This will be discussed in more detail in section \ref{taugamma}. \begin{corollary} \label{2.5} If the level $k$ is even, the spaces $Z_k(\Sigma)$ and $Z'_k(\Sigma)$ are decomposed as direct sums of isotypic components (called `bricks' in what follows) for the action of ${\mathcal{E}}$ and ${\mathcal{E}}'$. If $k\equiv 0$ mod $4$, the bricks are indexed by characters of $J^{(2)}$. If $k\equiv 2$ mod $4$, the bricks are indexed by $\theta$-characteristics on the curve $\Sigma$ (or, equivalently, by spin structures on $\Sigma$). If the level is odd, $Z_k(\Sigma)$ is isomorphic, as representation of the group ${\mathcal{E}}$, to a direct sum of copies of $Z_1(\Sigma)$ or to a direct sum of copies of the conjugate representation, $\overline{Z_1(\Sigma)}$, according to the parity of $(k-1)/2$. \end{corollary} \begin{remarks}{\em (i) Note that formula (\ref{rhoa}) implies \begin{equation} \rho_a^{\otimes k} \,\rho_b^{\otimes k}\,= \,\lambda_4(a,b)^k\,\rho_{a+b}^{\otimes k}. \label{rhoak} \end{equation} Hence the action of ${\mathcal{E}}$ on $Z_k(\Sigma)$ factors through an action of ${\mathcal{E}}'$ if $k$ is even, and furthermore through an action of $J^{(2)}$ if $k\equiv 0$ mod $4$. This explains why the bricks are indexed by characters of $J^{(2)}$ if $k\equiv 0$ mod $4$. If $k\equiv 2$ mod $4$, the index set are the characters of ${\mathcal{E}}'$ which do not factor through $J^{(2)}$; such characters can be identified with $\theta$-characteristics. (ii) The dimensions of the bricks are the same for all non-trivial characters of $J^{(2)}$ in the case $k\equiv 0$ mod $4$, and depend only on the parity of the $\theta$-characteristic in the case $k\equiv 2$ mod $4$. In section \ref{3}, we will give explicit formulas for their dimensions.} \end{remarks} \noindent{\bf Note.} Our brick decomposition can be viewed as a generalisation of an old result of Beauville's \cite{Be2} (see also Laszlo \cite{L}) at level $2$. Beauville constructed bases of $Z_2(\Sigma)$ (resp. $Z'_2(\Sigma)$) whose basis elements are indexed by even (resp. odd) $\theta$-characteristics. This corresponds from our point of view to the fact that the bricks in level $2$ are zero, if the $\theta$-characteristic has the `wrong' parity, and one-dimensional otherwise. Note, however, that the situation in level $2$ is very special; in general, the bricks are non-zero for both parities. We would like to mention also the work of van Geemen and Previato \cite{vGP1,vGP2}, Oxbury and Pauly \cite{OP}, and Pauly \cite{P}, whose work contains a description of the bricks in level $4$. \begin{remark}\label{logstruct}{\em The proofs of our results are organized as follows. After a description of the fixed point varieties in section \ref{fpv}, theorem \ref{evencasei} concerning their triple intersections is proved in section \ref{triple}. As already observed, theorem \ref{evencasei} implies part of theorem \ref{1.1}; the remainder of the proof of \ref{1.1} is given in section \ref{Geoinvest}, using some results about the Hecke correspondence proved in section \ref{Heckecorr}. Finally, theorem \ref{1.2} is proved in section \ref{tracecomp}. Both section \ref{taugamma} (where theorem \ref{2.4} is explained) and section \ref{3} (where the brick decompositions in corollary \ref{2.5} are derived) require only the statements of theorem \ref{1.1} and theorem \ref{1.2}, but not their proofs. }\end{remark} \section{Relationship with the \cite{BHMV}-theory.}\label{taugamma} In \cite{BHMV}, the Kauffman bracket (at a $2p$-th root of unity called $A$) was used to give a combinatorial-topological construction of TQFT-functors $V_p$ on a certain $2+1$-dimensional cobordism category. It is expected that these functors for $p=2k+4$ correspond (in some natural sense) to Witten's ones for $SU(2)$ and level $k$. For example, the dimensions of the vector space $V_p(\Sigma)\otimes {\mathbb C}$ \footnote{By definition, $V_p(\Sigma)$ is a module over a certain abstract cyclotomic ring $k_{p}$. By $V_p(\Sigma)\otimes {\mathbb C}$, we mean the vector space obtained by extending coefficients from $k_p$ to ${\mathbb C}$; this depends on a choice of a $2p$-th root of unity $A$ in ${\mathbb C}$.} associated to a closed oriented surface $\Sigma$ is also given by the Verlinde formula (\cite{BHMV}, cor. 1.16). The aim of the present section is to explain the following more precise statement of theorem \ref{2.4}. (As already mentioned in the introduction, the reader interested only in the algebraic geometry may proceed directly to section \ref{3}.) \begin{theorem}\label{2.1} Let $p=2k+4$. For the `right' choice of $2p$-th root of unity $A$, the vector spaces $V_p(\Sigma)\otimes {\mathbb C}$ and $Z_k(\Sigma)$ are isomorphic as representations of ${\mathcal{E}}$ (this group is denoted by $\Gamma(\Sigma)$ in \cite{BHMV}). This isomorphism sends the involution $\rho_a^{\otimes k}$ acting on $Z_k(\Sigma)$ to the involution $\tau_\gamma$ acting on $V_p(\Sigma)\otimes {\mathbb C}$, where ${\gamma}$ is a simple closed curve on $\Sigma$ representing the Poincar\'e dual of $a$. A similar statement holds in the twisted case. \end{theorem} \noindent{\bf Note.} For simplicity of exposition, a few technical details related to framing issues will be suppressed from the discussion here. The reader is also referred to the survey article \cite{MV}. \vskip 8pt By definition, elements of $V_p(\Sigma)$ are represented by linear combinations of compact oriented $3$-manifolds $M$ with boundary equal to $\Sigma$. (No complex structure on $\Sigma$ is needed here.) The $3$-manifolds may also contain colored links or, more generally, colored trivalent graphs. In the following, we assume $p$ is even and put $p=2k+4$. Then a color is just an element of $\{0,1,2,\ldots,k\}$. In the \cite{BHMV}-theory, one has involutions $\tau_\gamma$ of $V_p(\Sigma)$ associated to unoriented simple closed curves $\gamma$ on $\Sigma$. They are defined as follows. Consider a vector represented by some $(M,L)$, where $M$ is a $3$-manifold with boundary $\Sigma$, and $L$ stands for some colored link inside $M$. Then the action of $\tau_\gamma$ consists of adding to the link $L$ already present in $M$, an additional component consisting of the curve $\gamma$ pushed slightly inside $M$ in a neighborhood of $\Sigma=\partial M$, where $\gamma$ is colored by $k$ (the last color). It was shown in section 7 of \cite{BHMV} that (because of the relations which hold in $V_p(\Sigma)$) this endomorphism $\tau_\gamma$ is an involution. Following \cite{BHMV}, p. 917, the groups generated by these involutions acting on $V_p(\Sigma)$ can be described as follows. Let $\Gamma(\Sigma)$ be the group with one generator $[a]$ for each $a\in H_1(\Sigma;{\mathbb Z}/4)$ plus one additional generator $u$, and the following relations: the element $u$ is central, $u^4=1$, $[a]^2=1$ for all $a\in H_1(\Sigma;{\mathbb Z}/4)$, and \begin{equation} [a]\,[b]\, = \, u^{a\cdot b} \,[a+b]\label{uu} \end{equation} for all $a,b\in H_1(\Sigma;{\mathbb Z}/4)$. Here, $a\cdot b\in {\mathbb Z}/4$ denotes the {\em mod} $4$ intersection form determined by the orientation of $\Sigma$. If a {\em mod} $2$ class $\alpha\in H_1(\Sigma;{\mathbb Z}/2)$ is given, every lift of $\alpha$ to a {\em mod} $4$ class $a\in H_1(\Sigma;{\mathbb Z}/4)$ determines an element $[a]\in \Gamma(\Sigma)$. However, up to multiplication by $u^2$, this element $[a]$ depends only on $\alpha$. (Indeed, if $a_1,a_2$ are two such lifts, then $[a_1]=u^{2 a_2\cdot x}[a_2]$ where $a_1-a_2=2x$.) From this it follows easily that the group $\Gamma(\Sigma)$ is a central extension of $H_1(\Sigma;{\mathbb Z}/2)$ by ${\mathbb Z}/4$. (It can be viewed as some kind of `reduced' Heisenberg group associated to twice the intersection form.) Here is the relationship of the group $\Gamma(\Sigma)$ with the involutions $\tau_\gamma$. Given $\alpha \in H_1(\Sigma;{\mathbb Z}/2)$, a lift of $\alpha$ to an element of $\Gamma(\Sigma)$ can also be specified by an unoriented simple closed curve $\gamma$ representing $\alpha$, as follows. The curve $\gamma$ determines an element $a \in H_1(\Sigma;{\mathbb Z}/4)$ up to sign, and the induced element $[a]\in \Gamma(\Sigma)$ is well-defined, since $a\cdot a=0$ and hence $[-a]=[a]$. It is clear from the geometric description of the $\tau_\gamma$'s given above that the commutation properties of these involutions are related to the intersection properties of the associated curves; for example two such involutions obviously commute if the corresponding simple closed curves on $\Sigma$ don't intersect. The fact that the $\tau_\gamma$'s satisfy precisely the relations in $\Gamma(\Sigma)$ is shown in prop. 7.5 of \cite{BHMV}. Moreover, it is shown there that the central element $u$ acts on $V_{2k+4}(\Sigma)$ as multiplication by $(-1)^{k+1}A^{(k+2)^2}$. Now let us identify ${\mathbb Z}/4=\mu_4$ such that $1\in {\mathbb Z}/4$ corresponds to $i\in\mu_4$. Also, let us use Poincar\'e duality to identify $ J^{(4)}(\Sigma)=H^1(\Sigma;\mu_4)$ with $H_1(\Sigma;{\mathbb Z}/4)$. By a well-known folk theorem, there exists a sign $\varepsilon =\pm 1$ such that the Weil pairing is related to the intersection form as follows: \footnote{The value of $\varepsilon$ should be known, but we have been unable to locate it in the literature.} $$ \lambda_4(a,b)=(\varepsilon i)^{a\cdot b}.$$ We now choose the primitive root of unity $A$ of order $2p=4k+8$ such that $(-1)^{k+1}A^{(k+2)^2}=(\varepsilon i)^k$. For instance, we may choose $A=-\varepsilon e^{2i\pi/(4k+8)}$. Comparing (\ref{rhoa}) and (\ref{uu}), we see that the assignment $$\tau_\gamma \mapsto \rho_a^{\otimes k}$$ (where the curve $\gamma$ represents the Poincar\'e dual of $a$) defines an isomorphism from the image of $\Gamma(\Sigma)$ in the general linear group of $V_{2k+4}(\Sigma)\otimes {\mathbb C}$ to the image of the group ${\mathcal{E}}$ in the general linear group of $Z_k(\Sigma)$. (In particular, for $k=1$ we have an isomorphism $\Gamma(\Sigma)\cong {\mathcal{E}}$.) Using the dimension formulas given in section 7 of \cite{BHMV}, one can check that the traces of the involutions $\tau_\gamma$ acting on $V_p(\Sigma)$ coincide precisely with the traces computed in our theorem \ref{1.2}. This verification will be omitted. Since representations of finite groups are determined by their characters, this proves theorem \ref{2.1} in the untwisted case. The twisted case, where $Z_k(\Sigma)$ is replaced with $Z'_k(\Sigma)$, is similar. Here we must replace $V_p(\Sigma)$ with $V_p(\Sigma')$, where $\Sigma'$ is $\Sigma$ with one puncture colored by $k$ (the last color). (We maintain the convention $p=2k+4$.) Then we have again an isomorphism of representations (the group corresponding to ${\mathcal{E}}'$ is called $\Gamma'(\Sigma)$ in \cite{BHMV}). The dimension of $V_p(\Sigma')$ is given by the `twisted Verlinde formula', see \cite{BHMV}, Remark 5.11. This space is zero if $k$ is odd, and this is why we define $Z'_k(\Sigma)$ to be zero for odd $k$. (See also Thaddeus \cite{Th}.) (N.b., the vector spaces $V_p$ are also defined for surfaces with colored punctures; they are called `surfaces with colored structure' in \cite{BHMV}. The theory of the involutions $\tau_\gamma$ works the same for these, with the {\em caveat} that the curves $\gamma$ must avoid the punctures. We expect analogues of our results to hold in the case of general colored punctures, using moduli spaces of parabolic bundles.) \vskip 8pt \noindent{\bf Remark.} Although it has been known for some time that $V_p(\Sigma)$ and $Z_k(\Sigma)$ have the same dimensions, it seems, to the best of our knowledge, that a {\em natural} isomorphism between the two theories is still missing. Of course, the fact that these two spaces are isomorphic also as representations (of canonically isomorphic groups) gives further evidence that there should be a natural isomorphism between the two theories. \section{Brick decomposition and dimension formulas.}\label{3} In this section, we describe $Z_k(\Sigma)$ and $Z'_k(\Sigma)$ as representations of the groups ${\mathcal{E}}$ and ${\mathcal{E}}'$, respectively. We also give explicit dimension formulas. The results of this section are immediate consequences of the isomorphism of theorem \ref{2.1} and the computations in \cite{BHMV} and \cite{BM}. In order to make this paper self-contained, we will, however, derive them directly from theorems \ref{1.1} and \ref{1.2}. We first recall the Verlinde formula and its twisted counterpart. Put $d_g(k)=dim\ Z_k(\Sigma_g)$ and $d'_g(k)=dim\ Z'_k(\Sigma_g)$, where the subscript $g$ indicates the genus of the curve $\Sigma_g$. \footnote{These numbers are denoted by $d_g(p)$ and $\widehat d_g(p)$ in \cite{BHMV,MV,BM}, where $p=2k+4$.} Then one has $$d_g(k)=\left(\frac{k+2}{2}\right)^{g-1}\ \sum_{j=1}^{k+1} \ \left( \sin \frac{ \pi j}{k+2}\right)^{2-2g}$$ $$d'_g(k)=\left(\frac{k+2}{2}\right)^{g-1}\ \sum_{j=1}^{k+1} \ (-1)^{j+1}\left( \sin \frac{ \pi j}{ k+2}\right)^{2-2g}$$ \noindent{\bf The case $k \equiv 0$ mod $4$.} In view of formula (\ref{rhoak}) in section \ref{smr}, the action of the group ${\mathcal{E}}$ on $Z_k(\Sigma_g)$ factors through an action of $J^{(2)}(\Sigma_g)=H^1(\Sigma_g;\mu_2)$. We have a direct sum decomposition $$Z_k(\Sigma_g)=\bigoplus_{h} Z_k(\Sigma_g;h)$$ where $h$ runs through the characters of $J^{(2)}$, and $Z_k(\Sigma_g;h)$ is the subspace of $Z_k(\Sigma_g)$ where ${\alpha}$ acts as multiplication by $h({\alpha})\in \mu_2={\pm 1}$, for all $\alpha\in J^{(2)}$. We will refer to $Z_k(\Sigma_g;h)$ as the {\em brick} associated to $h$. By theorem \ref{1.2}, the character of the representation $Z_k(\Sigma_g)$ takes the same value on all non-trivial elements of $J^{(2)}$, and is therefore invariant under the automorphism group of $J^{(2)}$. It follows that the dimension of the brick $Z_k(\Sigma_g;h)$ is the same for all non-trivial characters $h$ of the group, since the automorphism group acts transitively on the set of non-trivial characters. We denote this dimension by $d_g^{(1)}(k)$, and we put $d_g^{(0)}(k)=dim \ Z_k(\Sigma_g;0)$, where $0$ denotes the trivial character. (Thus, the space $Z_k(\Sigma_g;0)$ is the fixed point set of the action of the group $J^{(2)}$ on $Z_k(\Sigma_g)$.) These numbers can be computed from the following two formulas: $$d_g(k)=d_g^{(0)}(k) +(2^{2g}-1)\, d_g^{(1)}(k)$$ $$d_g^{(0)}(k)-d_g^{(1)}(k)=tr(\alpha)= \left(\frac{k+2}{2}\right)^{g-1}, \ \ \alpha\neq 0$$ For example, one has $$d_g^{(0)}(k)=\frac{1}{2^{2g}} \left(d_g(k) + (2^{2g}-1) \left(\frac{k+2}{2}\right)^{g-1}\right)$$ Similarly, $Z'_k(\Sigma_g)$ is the direct sum of bricks $Z'_k(\Sigma_g;h)$, and the dimensions ${d'_g}^{(0)}(k)=dim \ Z'_k(\Sigma_g;0)$ and ${d'_g}^{(1)}(k)=dim \ Z'_k(\Sigma_g;h)$ for $h\neq 0$, can be computed as before (just replace $d_g(k)$ with $d'_g(k)$ and $d_g^{(\varepsilon)}(k)$ with ${d'_g}^{(\varepsilon)}(k)$ in the above). \vskip 8pt\noindent{\bf Example: The case $k=4$.} The numbers $d_g^{(1)}(4)$ and ${d'_g}^{(1)}(4)$ are equal to $(3^{g-1}+1)/2$ and $(3^{g-1}-1)/2$, respectively, and the numbers $d_g^{(0)}(4)$ and ${d'_g}^{(0)}(4)$ are obtained by adding $3^{g-1}$. These numbers have appeared in Oxbury and Pauly \cite{OP} and Pauly \cite{P}. \vskip 8pt \noindent{\bf The case $k\equiv 2$ mod $4$.}\nopagebreak In this case, the action of ${\mathcal{E}}$ factors through an action of the group ${\mathcal{E}}'$ but not through an action of $J^{(2)}$. Indeed, the involutions $\rho_a^{\otimes 2}$ acting on ${\mathcal{L}}^2$ depend only on $\alpha$, and satisfy the same relations as the $\rho'_\alpha$'s (see formulas (\ref{rhoprima}) and (\ref{rhoak}) in section \ref{smr}, and note that $\lambda_4(a,b)^2=\lambda_2({\alpha},{\beta})$). Therefore one has a direct sum decomposition $$Z_k(\Sigma_g)=\bigoplus_{q} Z_k(\Sigma_g;q)$$ where $q$ runs through the characters of the group ${\mathcal{E}}'$ which do not factor through $J^{(2)}$, {\em i.e.}, such that $q$ takes the value $-1$ on the central element $-1\in\mu_2\subset {\mathcal{E}}'$. Such characters are in $1$-to-$1$ correspondence with functions $q\colon J^{(2)}\rightarrow \mu_2$ such that $$q(\alpha+\beta)\,=\,q(\alpha)\,q(\beta)\, \lambda_2(\alpha,\beta).$$ In other words, $q$ runs through the set of quadratic forms on $J^{(2)}\cong H^1(\Sigma_g;{\mathbb Z}/2)$ inducing the Weil pairing, {\em i.e.}, the {\em mod} $2$ intersection form. It is well-known that such quadratic forms correspond to spin structures, or equivalently, $\theta$-characteristics, on $\Sigma$. (See Atiyah \cite{At2}, Johnson \cite{Jo}.) Let ${{\mathcal G}}\cong Sp(2g;{\mathbb Z}/2)$ be the group of automorphisms of $J^{(2)}$ preserving the order $2$ Weil pairing $\lambda_2$. The group ${\mathcal G}$ acts on ${\mathcal{E}}'$. As in the case $k\equiv 0$ mod $4$, it follows from theorem \ref{1.2} that the character of the representation $Z_k(\Sigma_g)$ is invariant under the action of ${\mathcal G}$. It is well-known that the induced action of ${\mathcal G}$ on quadratic forms has two orbits which are characterized by the {\em Arf invariant}, {\em i.e.}, the action is transitive on the set of forms $q$ with the same Arf invariant $\mathop{\fam0 Arf}\nolimits(q)\in {\mathbb Z}/2$. (The Arf invariant of the quadratic form corresponds to the parity of the $\theta$-characteristic.) This shows that the dimension of the brick $Z_k(\Sigma_g;q)$ depends only on $\mathop{\fam0 Arf}\nolimits(q)$. Put $d_g^{(\varepsilon)}(k)=dim \ Z_k(\Sigma_g;q_\varepsilon)$, where $q_\varepsilon$ has Arf invariant $\varepsilon\in {\mathbb Z}/2$. These dimensions can be computed from the following two formulas: $$d_g(k)=2^{g-1}(2^g+1)\,d_g^{(0)}(k) +2^{g-1}(2^g-1)\, d_g^{(1)}(k)$$ $$2^{g-1}(d_g^{(0)}(k)-d_g^{(1)}(k))=tr(\rho_a^{\otimes k})= \left(\frac{k+2}{2}\right)^{g-1}, \ \ \alpha\neq 0$$ The first formula follows from the fact that the number of quadratic forms with zero Arf invariant is equal to $2^{g-1}(2^g+1)$. The second formula follows from the fact\footnote{A nice way to think about this is to observe that there is a natural bijection between quadratic forms $q$ with fixed Arf invariant and such that $q({\alpha})=-1$, and quadratic forms with arbitrary Arf invariant on the $2g-2$-dimensional space ${\langle} {\alpha}{\rangle} ^\bot/{\langle} {\alpha}{\rangle}$.} that for $\alpha\neq 0$, one has $$\sharp \{q\,|\,q(\alpha)=-1,\ \mathop{\fam0 Arf}\nolimits(q)=0\}=2^{2g-2}=\sharp \{q\,|\, q(\alpha)=-1,\ \mathop{\fam0 Arf}\nolimits(q)=1\}$$ Similarly, $Z'_k(\Sigma_g)$ is the direct sum of bricks $Z'_k(\Sigma_g;q)$, and the dimensions ${d'_g}^{(\varepsilon)}(k)=dim \ Z'_k(\Sigma_g;q_\varepsilon)$ can be computed as before (just replace $d_g(k)$ with $d'_g(k)$ and $d_g^{(\varepsilon)}(k)$ with ${d'_g}^{(\varepsilon)}(k)$ in the above, but notice that $tr({\rho'_{\alpha}}^{\otimes k/2})$ is now equal to $-\left((k+2)/2\right)^{g-1}$.) Here are explicit formulas for the dimensions. They are equivalent to the formulas on p. 264 of \cite{BM} \footnote{Warning: Note that $k$ does not denote the level in \cite{BM}; one has $p=8k$ in \cite{BM} while in the present paper $p=2k+4$.}. \begin{eqnarray} d_g^{(\varepsilon)}(k)&=&{\frac{1}{2^{2g}}} \left(d_g(k)+\left(\frac{k+2}{2}\right)^{g-1} \left((-1)^{\varepsilon} 2^g-1\right)\right)\nonumber\\ {d'_g}^{(\varepsilon)}(k)&=& {\frac{1}{2^{2g}}}\left(d'_g(k) +\left(\frac{k+2}{2}\right)^{g-1} \left(1-(-1)^{\varepsilon} 2^g\right)\right)\nonumber \end{eqnarray} \vskip 8pt\noindent{\bf Example: The case $k=2$.} The numbers $d_g^{(0)}(2)$ and ${d'_g}^{(1)}(2)$ are equal to $1$, and the numbers $d_g^{(1)}(2)$ and ${d'_g}^{(0)}(2)$ are zero. Therefore $d_g(2)=2^{g-1}(2^g+1)$ and $d'_g(2)=2^{g-1}(2^g-1)$ (see Beauville \cite{Be2}). \vskip 8pt\noindent{\bf Note.} In \cite{BM}, a ${\mathbb Z}/2$-graded TQFT-functor is constructed on a cobordism category of surfaces equipped with spin structures (and other things). Given a connected surface $\Sigma$ with a spin structure $\sigma$, the even (resp. odd) part of this functor is isomorphic to $Z_k(\Sigma_g;q_\sigma)$ (resp. $Z'_k(\Sigma_g;q_\sigma)$), where $q_\sigma$ is the quadratic form corresponding to $\sigma$. \begin{remark}{\em The action of the symplectic group ${\mathcal G}\cong Sp(2g;{\mathbb Z}/2)$ permuting the bricks has the following geometric interpretation. Recall that all elements of ${\mathcal G}$ can be represented by diffeomorphisms of $\Sigma$. In the \cite{BHMV}-theory, one has, more or less by definition, a natural action of a certain extended mapping class group on $V_{2k+4}(\Sigma)$. On the geometric side, there is also a (projective-linear) action of the mapping class group of $\Sigma$ on $Z_k(\Sigma)$; this action is constructed using Hitchin's projectively-flat connection \cite{Hi}. It is, of course, expected that $V_{2k+4}(\Sigma)\otimes {\mathbb C}$ and $Z_k(\Sigma)$ are isomorphic as representations of the extended mapping class group. In any case, it is easy to see in both theories that the action of a diffeomorphism $f$ takes the brick associated to a character $h$ (resp. a quadratic form $q$) to the brick associated to $f^*(h)$ (resp. $f^*(q)$). On the geometric side, the main reason for this is that Hitchin's connection is (projectively) invariant under the actions of both the mapping class group and the group ${\mathcal{E}}$. }\end{remark} \noindent{\bf The odd-level case.} \begin{proposition} If $k\equiv 1$ mod $2$, $Z_k(\Sigma)$ is isomorphic, as representation of the group ${\mathcal{E}}$, to a direct sum of copies of $Z_1(\Sigma)$, if $k\equiv 1$ mod $4$, and to a direct sum of copies of the conjugate representation, $\overline{Z_1(\Sigma)}$, if $k\equiv 3$ mod $4$. \end{proposition} \noindent {\bf Proof.} Note that the character of the representation $Z_1(\Sigma)$ is the function $\chi\colon {\mathcal{E}}\rightarrow {\mathbb C}$ which is zero on all group elements not in the central subgroup $\mu_4\subset {\mathcal{E}}$, while the trace of a central element $\lambda\in \mu_4$ is $\chi(\lambda)=2^g\,\lambda$. The character of the conjugate representation, $\overline{Z_1(\Sigma)}$, is of course the conjugate character $\overline{\chi}$. Now it follows from theorems \ref{1.1} and \ref{1.2} and formula (\ref{rhoak}) that the character of $Z_k(\Sigma)$ is a multiple of $\chi$ if $k\equiv 1$ mod $4$, and a multiple of $\overline{\chi}$, if $k\equiv 3$ mod $4$. It is however well-known that $\chi$ is an irreducible character, and this proves the proposition. \vskip 8pt \noindent{\bf Note.} This result corresponds, {\em via} the isomorphism $Z_k(\Sigma)\cong V_{2k+4}(\Sigma) \otimes {\mathbb C}$, to theorems 1.5 and 1.6 of \cite{BHMV}. It is shown there that (for odd $k$) $$V_{2k+4}(\Sigma)\cong V_2'(\Sigma)\otimes V_{k+2}(\Sigma)$$ as representations of the group ${\mathcal{E}}$, where $V_2'(\Sigma)$ and $V_{k+2}(\Sigma)$ are defined in \cite{BHMV}. Moreover, the group ${\mathcal{E}}$ acts trivially on $V_{k+2}(\Sigma)$, and $V_2'(\Sigma_g)$ is isomorphic to $V_6(\Sigma)$, and hence to $Z_1(\Sigma)$ or to $\overline{Z_1(\Sigma)}$, after a change of coefficients. It would be interesting to have an algebro-geometric interpretation of the space $V_{k+2}(\Sigma)$, which in the \cite{BHMV}-theory can be interpreted as a $SO(3)$-TQFT vector space. Here, the name $SO(3)$-TQFT just means that the allowed colors in this TQFT are even, or in other words, are $SU(2)$-representations which lift to $SO(3)$. \section{The fixed point varieties.}\label{fpv} We need to analyze the action of $J^{(2)}$ on $M$ and $M'$ in order to obtain information about our lifts. In this section, we describe the various fixed point varieties, mainly following Narasimhan and Ramanan \cite{NR}. We also discuss when two lifts of ${\alpha}\in J^{(2)}$ to $J^{(4)}$ determine the same component of the fixed point set $|M|_{\alpha}$. \vskip 8pt\noindent{\bf Notation.} Throughout this paper, we denote by $L_{\alpha}$ (resp. $L_a$) the line bundle on $\Sigma$ corresponding to ${\alpha}\in J^{(2)}$ (resp. $a\in J^{(4)}$). \vskip 8pt Let ${\alpha}\inJ^{(2)}$ be nonzero. Let $\pi_{\alpha} : \Sigma^\alpha \rightarrow \Sigma$ be the 2-sheeted unramified covering of $\Sigma$ corresponding to $\alpha$, and let $\phi_{\alpha}$ be the covering transformation of $\pi_{\alpha}$. Using the line bundle $L_{\alpha}$, we can explicitly construct $\Sigma^{\alpha}$ as $$\Sigma^{\alpha} = \left\{ \xi \in L_{\alpha} | \xi\otimes\xi = 1\right\}$$ using an isomorphism $L^2_{\alpha}\cong {\mathcal{O}}_\Sigma$. The involution $\phi_{\alpha}$ is then induced by multiplication by $-1$ on $L_{\alpha}$. Given a line bundle $L$ over $\Sigma^{\alpha}$, the push-down $E={\pi_{\a*}}(L)$ can be obtained by descending $L\oplus \phi_{\alpha}^*(L)$ (which is naturally an equivariant bundle) to $\Sigma$: \begin{equation} E=(L\oplus \phi_{\alpha}^*(L)){\big/}\langle \phi_{\alpha}\rangle.\nonumber \end{equation} Here, the natural involution of $L\oplus \phi_{\alpha}^*(L)$ covering $\phi_{\alpha}$ is again denoted by $\phi_{\alpha}$. The fundamental observation is that $E\otimes L_{\alpha}$ is isomorphic to $E$. This follows formally from the pull-push formula ${\pi_{\a*}}(L)\otimes L'\cong {\pi_{\a*}}(L\otimes \pi_{\alpha}^*L') $ and the fact that $L_{\alpha}$ pulls back to the trivial bundle on $\Sigma^{\alpha}$. \begin{remark} \label{fundobs} {\em We will later need the following explicit isomorphism from $E={\pi_{\a*}}(L)$ to $E\otimes L_{\alpha}$. Note that $\pi_{\alpha}^*L_{\alpha}$, as an equivariant bundle, is isomorphic to ${\mathcal{O}}_{\Sigma^{\alpha}}^-$, that is, the trivial line bundle $\Sigma^{\alpha}\times {\mathbb C}$, but with non-trivial action, given by $(x,z)\mapsto (\phi_{\alpha}(x), -z)$. Therefore $$E\otimes L_{\alpha}=(L\oplus \phi_{\alpha}^*(L)){\big/}\langle -\phi_{\alpha}\rangle.$$ It follows that the diagonal automorphism $1\oplus(-1)$ of $L\oplus \phi_{\alpha}^*(L)$ descends to an isomorphism from $E$ to $E\otimes L_{\alpha}$.}\end{remark} Let $\mathop{\fam0 Nm}\nolimits_{\alpha}\colon \mathop{\fam0 Pic}\nolimits(\Sigma^\alpha) \rightarrow \mathop{\fam0 Pic}\nolimits(\Sigma)$ be the classical albanese or norm map induced by the following map on divisors. If $D= \sum n_j x_j\in \mathop{\fam0 Div}\nolimits(\Sigma^{\alpha})$ then $\mathop{\fam0 Nm}\nolimits_{\alpha}(D) = \sum n_j \pi_{\alpha}(x_j)\in \mathop{\fam0 Div}\nolimits(\Sigma)$. The {\em Prym variety} $P_\alpha$ associated to $\alpha$ is by definition the connected component of $\mathop{\fam0 Nm}\nolimits_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$ containing ${\mathcal{O}}_{\Sigma^\alpha}$. It is a principally polarized abelian variety of dimension $g-1$. The quotient variety $P_\alpha/\langle\pm 1\rangle$ is called the {\em Kummer variety.} We define $\theta_{\alpha} : \mathop{\fam0 Pic}\nolimits(\Sigma^\alpha) \rightarrow \mathop{\fam0 Pic}\nolimits(\Sigma)$ by $$\theta_{{\alpha}}(L) = \det({\pi_{\a*}}(L)) =\mathop{\fam0 Nm}\nolimits_{\alpha}(L)\otimes L_\alpha$$ (see \cite{NR} for the second equality). Note that $\phi_{\alpha}$ acts on $\mathop{\fam0 Pic}\nolimits(\Sigma^\alpha)$ by sending $L$ to $\phi_{\alpha}^*(L)$. \begin{proposition}[Narasimhan and Ramanan \cite{NR}]\label{6.1} (i) The map $L\mapsto (\pi_{\alpha})_*(L)$ induces isomorphisms \begin{equation}\label{thetu} \theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)/{\langle}\phi_{\alpha} {\rangle} \,\mapright\sim \,|M|_{\alpha} \end{equation} \begin{equation}\label{thett}\theta_{\alpha}^{-1}([p])/{\langle}\phi_{\alpha}{\rangle} \,\mapright\sim \,|M'|_{\alpha} \end{equation} (ii) Moreover, $\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)/{\langle}\phi_{\alpha}{\rangle}$ is isomorphic to two copies of $P_\alpha/{\langle}\pm 1{\rangle}$, while $\theta_{\alpha}^{-1}([p])/{\langle}\phi_{\alpha}{\rangle}$ is isomorphic to $P_{\alpha}$. \end{proposition} \begin{remark}{\em In the twisted case, ${\pi_{\a*}}\colon \theta_{\alpha}^{-1}([p]) \rightarrow |M'|_{\alpha}$ is a double covering, with covering transformation $\phi_{\alpha}$. In the untwisted case, the same holds on the open subvariety of $\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$ where $\phi_{\alpha}$ acts freely, while the fixed points of $\phi_{\alpha}$ are sent by the map ${\pi_{\a*}}$ bijectively to the points in $|M|_\alpha$ represented by semi-stable but not stable bundles. (See remark \ref{check} below.) }\end{remark} For later use, we need the following more explicit description of $\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$ and $\theta_{\alpha}^{-1}([p])$. For $d=0,1$, define $ \Phi^d_{\alpha} : \mathop{\fam0 Pic}\nolimits_d(\Sigma^{\alpha}) \rightarrow \mathop{\fam0 Pic}\nolimits_0(\Sigma^{\alpha})=J(\Sigma^{\alpha})$ by \begin{equation} \Phi^d_{\alpha}(L) = L\otimes\phi^*_{\alpha} L^{-1}.\nonumber \end{equation} The following two properties (\ref{imPhi1}) and (\ref{imPhi2}) of the maps $ \Phi^d_{\alpha}$ are elementary facts from the classical theory of line bundles on curves, see {\em e.g.} Appendix B in \cite{ACGH}. First, one has the disjoint union \begin{equation} \mathop{\fam0 Nm}\nolimits_{\alpha}^{-1}({\mathcal{O}}_\Sigma)=\mathop{\fam0 Im}\nolimits\Phi^0_{\alpha} \cup \mathop{\fam0 Im}\nolimits \Phi^1_{\alpha} \label{imPhi1} \end{equation} and the Prym variety $P_{\alpha}$ is equal to the component $\mathop{\fam0 Im}\nolimits\Phi^0_{\alpha}$. Second, note that $\mathop{\fam0 Nm}\nolimits_{\alpha}(\pi_{\alpha}^*(L_{\beta}))=L_{\beta}^{\otimes 2}={\mathcal{O}}_\Sigma$ for all $\beta\in J^{(2)}$. Moreover, one has \begin{equation} \pi_{\alpha}^*L_{\beta} \in \mathop{\fam0 Im}\nolimits \Phi^d_{\alpha} \ \Leftrightarrow\ \lambda_2({\alpha},{\beta})=(-1)^d. \label{imPhi2} \end{equation} Here, $\lambda_2 : J^{(2)}\times J^{(2)} \rightarrow \mu_2$ is the order $2$ Weil pairing. Now pick $a\in J^{(4)}$ such that $2a={\alpha}$, and ${\beta}\inJ^{(2)}$ such that $\lambda_2({\alpha},{\beta})=-1$. Note that $a'=a+{\beta}$ is another element of $ J^{(4)}$ such that $2a'={\alpha}$. Also, pick a point ${p_\a}\in\pi_{\alpha}^{-1}(p)\subset \Sigma^{\alpha}$. Note that $\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$ and $\theta_{\alpha}^{-1}([p])$ are both isomorphic to $\mathop{\fam0 Nm}\nolimits_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$. {}From (\ref{imPhi1}) and (\ref{imPhi2}), we have the following description of $\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$ and $\theta_{\alpha}^{-1}([p])$ as disjoint unions: \begin{eqnarray} \theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)&=&\pi_{\alpha}^*L_{a}\otimes \mathop{\fam0 Im}\nolimits\Phi^0_{\alpha} \,\cup\, \pi_{\alpha}^*L_{a'} \otimes \mathop{\fam0 Im}\nolimits \Phi^0_{\alpha} \label{untw1}\\ \theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)&=&\pi_{\alpha}^*L_{a}\otimes \mathop{\fam0 Im}\nolimits\Phi^0_{\alpha} \,\cup\, \pi_{\alpha}^*L_{a} \otimes \mathop{\fam0 Im}\nolimits \Phi^1_{\alpha} \label{untw2}\\ \theta_{\alpha}^{-1}([p])&=&\pi_{\alpha}^*L_{a}\otimes[p_{\alpha}]\otimes \mathop{\fam0 Im}\nolimits\Phi^0_{\alpha} \,\cup\, \pi_{\alpha}^*L_{a} \otimes [\phi_{\alpha}(p_{\alpha})]\otimes\mathop{\fam0 Im}\nolimits \Phi^0_{\alpha} \label{tw} \end{eqnarray} \begin{remark} {\em It follows from (\ref{untw2}) and (\ref{tw}) that the action of $\phi_{\alpha}$ preserves the two components of $\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$, while it exchanges the two components of $\theta_{\alpha}^{-1}([p])$. By (\ref{imPhi1}), the action of $\phi_{\alpha}$ on $\mathop{\fam0 Pic}\nolimits(\Sigma^\alpha)$ restricts to multiplication by $-1$ (that is, the map $L\mapsto L^{-1}$) on $\mathop{\fam0 Nm}\nolimits_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$. This proves \ref{6.1}(ii). }\end{remark} We can now describe when two lifts of ${\alpha}\in J^{(2)}$ to $J^{(4)}$ determine the same component of the fixed point set $|M|_{\alpha}$. Recall that $|M|_a^+$ was defined to be the component of $|M|_{\alpha}$ containing the S-equivalence class of the semi-stable bundle $L_a\oplus L^{-1}_a$. \begin{proposition}\label{4.6} (i) Let $a_1,a_2 \in J^{(4)}$ such that $2a_1=2a_2={\alpha}$. Then $$|M|_{a_1}^+ = |M|_{a_2}^+ \ \Leftrightarrow\ \lambda_2(a_1-a_2,{\alpha})=1\in\mu_2.$$ (ii) For ${\beta}\in J^{(2)}$, the action of ${\beta}$ on $|M|_{\alpha}$ interchanges the two components of $|M|_{\alpha}$ if and only if $\lambda_2({\alpha},{\beta})=-1$. \end{proposition} \noindent {\bf Proof.} Note that ${\pi_{\a*}}(\pi_{\alpha}^* (L_{a_i}))\cong L_{a_i}\oplus L_{a_i}\otimes L_{\alpha}\cong L_{a_i}\oplus L_{a_i}^{-1}$. Therefore $ |M|^+_{a_i}$ is the component $\pi_{{\alpha}*}(\pi_{\alpha}^*L_{{a_i}}\otimes \mathop{\fam0 Im}\nolimits\Phi^0_{\alpha})$, and (i) follows from formula (\ref{untw1}). Now part (ii) follows from (\ref{imPhi2}), since the action of ${\beta}$ on $|M|_{\alpha}$ lifts to tensoring with $\pi_{\alpha}^*L_{\beta}$ on $ \theta_{\alpha}^{-1} ({\mathcal{O}}_\Sigma)$. This completes the proof. \vskip 8pt\noindent{\bf Note.} Translating the `multiplicative' notation of the Weil pairing into the `additive' notation in section \ref{taugamma}, the condition $\lambda_2(a_1-a_2,{\alpha})=1\in\mu_2$ becomes the condition $((a_1-a_2)/2)\cdot {\alpha}=0\in {\mathbb Z}/2$. Thus, prop. \ref{4.6}(i) implies that $|M|_{a_1}^+ = |M|_{a_2}^+$ if and only if $[a_1]=[a_2]$, where $[a_i]$ is the lift of ${\alpha}$ to the group $\Gamma(\Sigma)$ defined in section \ref{taugamma}. \begin{remark}\label{check}{\em Let $M^{sing}$ denote the set of points of $M$ represented by semistable, but not stable, bundles. A semi-stable bundle $E$ with $\mathop{\fam0 Gr}\nolimits(E) \cong L_1\oplus L_2$ represents a point in $M^{sing}$ if and only if $L_2\cong L_1^{-1}$, and this point lies in $|M|_{\alpha}$ if and only if $L_1\otimes L_{\alpha}\cong L_2$. This shows that $M^{sing}\cap |M|_{\alpha}$ is precisely the set of points represented by bundles of the form $L_a\oplus L_a^{-1}$ with $a\in J^{(4)}$ and $2a={\alpha}$. Since $L_a\oplus L_a^{-1}\cong{\pi_{\a*}}(\pi_{\alpha}^*(L_a))$, and $\pi_{\alpha}^*(\{L\in \mathop{\fam0 Pic}\nolimits_0(\Sigma) | L^2 \cong L_{\alpha}\})$ is precisely the fixed point set of $\phi_{\alpha}$ on $\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$, we see that $\pi_{{\alpha}*}$ induces a bijection from that fixed point set to $M^{sing}\cap |M|_{\alpha}$, as asserted above. }\end{remark} \section{Intersections and triple intersections.} \label{triple} \vskip 8pt In this section, we describe how the various fixed point varieties intersect. The triple intersections will be used in the proof of theorem \ref{1.1}. \vskip 8pt\noindent{\bf Note.} The intersection properties of the fixed point sets in relation to the order $2$ Weil pairing are well-known; they are used for example in van Geemen and Previato \cite{vGP1}. On the other hand, the relationship of the triple intersections of their individual {\em components} (in the untwisted case) with the order $4$ Weil pairing seems to be new. \vskip 8pt \begin{proposition}\label{6.5} For $\alpha,\beta\in J^{(2)}$ non-zero distinct elements, we have that $$|M|_\alpha\cap |M|_\beta \neq \emptyset\ \Leftrightarrow\ \lambda_2(\alpha,\beta) = 1\in\mu_2$$ $$|M'|_\alpha\cap |M'|_\beta \neq \emptyset\ \Leftrightarrow\ \lambda_2(\alpha,\beta) = -1\in\mu_2$$ \end{proposition} \noindent {\bf Proof.} The morphism $\pi_{{\alpha}*}\colon \mathop{\fam0 Pic}\nolimits_d(\Sigma^\alpha)\rightarrow |M_d|_\alpha$ is surjective and $J^{(2)}$-equivariant. (Here, ${\beta}\in J^{(2)}$ acts on $\mathop{\fam0 Pic}\nolimits_d(\Sigma^\alpha)$ by $L\mapsto L\otimes \pi_{\alpha}^*L_{\beta}$.) Hence we get the following description of the intersection $$|M_d|_\alpha \cap |M_d|_\beta = {\pi_{\a*}}( \{L\in \mathop{\fam0 Pic}\nolimits_d(\Sigma^\alpha) | L\otimes \pi_{\alpha}^*L_\beta \cong \phi_{\alpha}^* L\mbox{ or } L\}).$$ Hence we see that $$|M_d|_\alpha \cap |M_d|_\beta \neq \emptyset$$ if and only if $$\pi_{\alpha}^*L_{\beta} \in \mathop{\fam0 Im}\nolimits \Phi^d_{\alpha}.$$ {}From (\ref{imPhi2}), this is the case if and only if $\lambda_2({\alpha},{\beta})=(-1)^d $. Using the action of $J(\Sigma)$ on $|M_d|_\alpha \cap |M_d|_\beta$, the results for $|M|_\alpha\cap |M|_\beta$ and $|M'|_\alpha\cap |M'|_\beta$ follow from this. \begin{proposition}\label{trans} If $\lambda_2({\alpha},{\beta})=1$, the quotient group $J^{(2)}/{\langle}{\alpha},{\beta}{\rangle}$ acts simply transitively on $|M|_{\alpha}\cap |M|_{\beta}$. In particular, this intersection has $2^{2g-2}$ elements. If $\lambda_2({\alpha},{\beta})=-1$, the same holds for $|M'|_{\alpha}\cap |M'|_{\beta}$. \end{proposition} \noindent {\bf Proof.} Put $I_{{\alpha},{\beta}}=\{L\in \theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)\,|\, \pi_{\alpha}^*(L_{\beta})\cong L\otimes \phi_{\alpha} ^*L^{-1}\}$. Then ${\pi_{\a*}}\colon I_{{\alpha},{\beta}}\rightarrow |M|_{\alpha}\cap |M|_{\beta}$ is a double covering. Note that $J^{(2)}/{\langle} {\alpha}{\rangle}$ acts simply transitively on $I_{{\alpha},{\beta}}$. The involution $\phi_{\alpha}$ is on $ I_{{\alpha},{\beta}}$ the same as tensoring with $\pi_{\alpha}^*L_{\beta}$, in other words, the action of ${\beta}$. This proves the result in the untwisted case. The twisted case is proved similarly. \vskip 8pt \noindent{\bf Note.} In view of remark \ref{check}, this description shows that $|M|_{\alpha}\cap |M|_{\beta}$ is contained in the stable part of $M$. \vskip 8pt \begin{proposition}\label{count} $|M|_{\alpha}\cap |M|_{\beta}$ is the disjoint union of the sets $|M|_a^\varepsilon\cap |M|_b^\mu $, where $\varepsilon=\pm$ and $\mu=\pm$, each of which sets has $2^{2g-4}$ elements. \end{proposition} \noindent {\bf Proof.} Recall from proposition \ref{4.6}(ii) that the action of $\gamma\in J^{(2)}$ exchanges the components of $|M|_{\alpha}$ if and only if $\lambda_2({\alpha},{\gamma})=-1$. Thus, the result follows by exploiting the fact that for every choice of signs $\varepsilon=\pm 1$ and $\mu=\pm 1$, there exists ${\gamma}$ such that $\lambda_2({\alpha},{\gamma}) = \varepsilon$ and $\lambda_2({\beta},{\gamma})= \mu$. \vskip 8pt We now turn to the triple intersections. The first observation is the following easy lemma. \begin{lemma} Let ${\alpha},{\beta},{\gamma}$ be distinct non-zero elements of $J^{(2)}$ such that the triple intersection $|M|_\alpha \cap |M|_\beta \cap |M|_{\gamma}$ is non-empty. Then ${\gamma}={\alpha}+{\beta}$. The same holds for the triple intersections in the twisted case. \end{lemma} \noindent {\bf Proof.} Indeed, it follows from the description in prop. \ref{trans} that the triple intersection can only be non-empty if $\pi_{\alpha}^*(L_{\beta})=\pi_{\alpha}^*(L_{\gamma})$ which implies ${\gamma}={\alpha}+{\beta}$. \vskip 8pt\noindent{\bf Note.} Since the group $J^{(2)}$ is commutative, we have $$|M|_\alpha \cap |M|_\beta = |M|_\alpha \cap |M|_\beta\cap |M|_{\alpha + \beta},$$ and similarly in the twisted case. \vskip 8pt In the untwisted case, the fixed point sets have two components each, and we may ask about the triple intersections of the individual components. Our answer was already stated in theorem \ref{evencasei}. We will prove it in the following equivalent form. \begin{theorem}\label{6.8} Assume that ${\alpha}$ and ${\beta}$ are distinct non-zero elements of $J^{(2)}$ such that $\lambda_2({\alpha},{\beta})=1\in\mu_2$. Let $a,b\in J^{(4)}$ such that $2a={\alpha}$ and $2b={\beta}$. Let $\varepsilon,\mu,\nu=\pm 1$ be three signs. Then \begin{equation} |M|^\varepsilon_a \cap |M|^\mu_b \cap |M|^\nu_{a + b} \neq \emptyset \ \Leftrightarrow\ \lambda_4(a,b)=\varepsilon\mu\nu\nonumber \end{equation}\end{theorem} \noindent{\bf Note.} It follows that if $|M|^\varepsilon_a \cap |M|^\mu_b \cap |M|^\nu_{a + b}$ is non-empty, then it is equal to all of $|M|^\varepsilon_a \cap |M|^\mu_b$. This fact can of course be seen directly using prop. \ref{count}. The important information in the theorem is that it tells us when $|M|^\varepsilon_a \cap |M|^\mu_b$ intersects $|M|^+_{a + b}$ and when $|M|^-_{a + b}$. \vskip 8pt \noindent {\bf Proof.} To simplify notation, we put ${\gamma}={\alpha}+{\beta}$ and $c=a+b$. Let $E$ represent a point in $|M|_\alpha \cap |M|_\beta \cap |M|_{\gamma}$. The description of $|M|_\alpha$ in section \ref{fpv} tells us that there exists ${\mathcal{L}}_a\in \pi_{\alpha}^*(L_a)\otimes \mathop{\fam0 Im}\nolimits \Phi_{\alpha}^d \subset \mathop{\fam0 Pic}\nolimits_0(\Sigma^{\alpha})$ such that $E\cong{\pi_{\a*}}{\mathcal{L}}_a$; moreover $E$ lies in $|M|_a^+$ if and only $d$ is even. (See formula (\ref{untw2}).) Similarly we have $E\cong{\pi_{\b*}}{\mathcal{L}}_b\cong{\pi_{\g*}}{\mathcal{L}}_c$, where ${\mathcal{L}}_b\in \pi_{\beta}^*(L_b)\otimes \mathop{\fam0 Im}\nolimits \Phi_{\beta}^{d'}\subset \mathop{\fam0 Pic}\nolimits_0(\Sigma^{\beta})$ and ${\mathcal{L}}_c\in \pi_{\gamma}^*(L_c)\otimes \mathop{\fam0 Im}\nolimits \Phi_{\gamma}^{d''}\subset \mathop{\fam0 Pic}\nolimits_0(\Sigma^{\gamma})$. Thus, the theorem is equivalent to the following lemma. \begin{lemma}\label{6.9} One has $\lambda_4(a,b)=(-1)^{d+d'+d''}$. \end{lemma} The proof of this lemma will occupy the remainder of this section. There is a curve $\widetilde{\S}$ naturally double covering $\Sigma^{\alpha}$, $\Sigma^{\beta}$ and $\Sigma^{{\gamma}}$: $$\widetilde{\S} = \left\{ (\xi,\eta)\in L_{\alpha}\oplus L_{\beta}\,|\, \xi^2=1=\eta^2\right\}.$$ The projections onto the two factors induce projections $\pi^{\alpha} \colon \widetilde{\S} \rightarrow \Sigma^{\alpha}$ and $\pi^{\beta} \colon \widetilde{\S} \rightarrow \Sigma^{\beta}.$ The bilinear map $L_{\alpha}\oplus L_{\beta} \rightarrow L_{\alpha}\otimes L_{\beta}$ induces the projection $\pi^{{\gamma}} \colon \widetilde{\S} \rightarrow \Sigma^{{\gamma}}.$ $$\begin{array}{lll} &\widetilde{\S}&\\ {}^{\textstyle\pi^{\alpha}}\hskip -5pt\swarrow &\downarrow\pi^{\beta}&\searrow^{\textstyle \pi^{\gamma}}\\ \hskip -10pt\Sigma^{\alpha} & \Sigma^{\beta} & \ \ \ \ \Sigma^{\gamma}\\ {}_{\textstyle\pi_{\alpha}}\hskip -5pt\searrow&\downarrow\pi_{\beta}&\swarrow_{\textstyle \pi_{\gamma}}\\ &\Sigma& \end{array}$$ The deck-transformations of the coverings $\pi^{\alpha}, \pi^{\beta}, \pi^{{\gamma}}$ will be denoted respectively by $\phi^{\alpha}, \phi^{\beta},\phi^{\gamma}$. Note that $\phi^{\alpha}$ (resp. $ \phi^{\beta}$) is induced by multiplication by $-1$ in the fibers of $L_{\beta}$ (resp. $L_{\alpha}$), and that $\phi^{\gamma}=\phi^{\alpha}\circ \phi^{\beta}$. We denote the projection $\widetilde{\S}\rightarrow \Sigma$ by $\tilde \pi$, so that $$\tilde \pi=\pi_{\alpha}\circ \pi^{\alpha} =\pi_{\beta}\circ \pi^{\beta}=\pi_{{\gamma}}\circ \pi^{{\gamma}}.$$ Notice also that the involution $\phi_{\alpha}$ of $\Sigma^{\alpha}$ is covered by both $\phi^{\beta}$ and $\phi^{\gamma}$ (but of course not by $\phi^{\alpha}$, since $\Sigma^{\alpha}=\widetilde{\S}/{\langle}\phi^{\alpha}{\rangle}$). Similar comments apply to $\phi_{\beta}$ and $\phi_{\gamma}$. \begin{lemma}\label{6.10} One has $\pi^{{\alpha}*}({\mathcal{L}}_a)\cong \pi^{{\beta}*}({\mathcal{L}}_b)\cong\pi^{{\gamma}*}({\mathcal{L}}_c)$. \end{lemma} \noindent {\bf Proof.} Since $E\cong \pi_{{\alpha}*}({\mathcal{L}}_a)$ lies in $|M|_{\alpha}\cap|M|_{\beta}$, we have $\phi_{\alpha}^*({\mathcal{L}}_a)\cong {\mathcal{L}}_a\otimes \pi_{\alpha}^*(L_{\beta})$ (see the proof of prop. \ref{trans}). Since $\pi^{{\alpha}*}\pi_{\alpha}^*(L_{\beta})=\tilde\pi^*(L_{\beta})$ is trivial, it follows that $$\tilde \pi^*(E)\cong\tilde \pi^*{\pi_{\a*}}({\mathcal{L}}_a)\cong\pi^{{\alpha}*}({\mathcal{L}}_a\oplus \phi_{\alpha}^*({\mathcal{L}}_a))\cong\pi^{{\alpha}*}({\mathcal{L}}_a)\oplus\pi^{{\alpha}*}({\mathcal{L}}_a).$$ Similarly $\tilde \pi^*(E)\cong \pi^{{\beta}*}({\mathcal{L}}_b)\oplus\pi^{{\beta}*}({\mathcal{L}}_b)\cong \pi^{{\gamma}*}({\mathcal{L}}_c)\oplus\pi^{{\gamma}*}({\mathcal{L}}_c)$. Since line bundles are simple, the lemma follows. \vskip 8pt We now turn to the computation of the Weil pairing $\lambda_4(a,b)$. Represent $a,b\in J^{(4)}$ by divisors $D_a,D_b\in \mathop{\fam0 Div}\nolimits_0(\Sigma)$ with disjoint support, and put $D_c=D_a+D_b$. Pick $D\in Div_d(\Sigma^{\alpha})$ (resp. $D'\in Div_{d'}(\Sigma^{\beta})$, resp. $D''\in Div_{d''}(\Sigma^{\gamma})$) such that $\pi_{\alpha}^*(D_a) +(1-\phi_{\alpha}^*)(D)$ (resp. $\pi_{\beta}^*(D_b) +(1-\phi_{\beta}^*)(D')$, resp. $\pi_{\gamma}^*(D_c) +(1-\phi_{\gamma}^*)(D'')$) represents ${\mathcal{L}}_a$ (resp. ${\mathcal{L}}_b$, resp. ${\mathcal{L}}_c$). Pulling everything up to $\widetilde{\S}$, we get divisors $$F_a=\tilde\pi^{*}(D_a)+\pi^{{\alpha}*}(1-\phi_{\alpha}^*)(D)$$ $$F_b=\tilde\pi^{*}(D_b)+\pi^{{\beta}*}(1-\phi_{\beta}^*)(D')$$ $$F_c=\tilde\pi^{*}(D_c)+\pi^{{\gamma}*}(1-\phi_{\gamma}^*)(D'')$$ such that $F_a$ represents $\pi^{{\alpha}*}({\mathcal{L}}_a)$, $F_b$ represents $\pi^{{\beta}*}({\mathcal{L}}_b)$, and $F_c$ represents $\pi^{{\gamma}*}({\mathcal{L}}_c)$. Since these three bundles are isomorphic by lemma \ref{6.10}, there exist meromorphic functions $h_1,h_2\in {\mathcal M}(\widetilde{\S})$ such that $$(h_2)+F_a\,=\,F_c\,=\, (h_1)+F_b.$$ Let $\mathop{\fam0 Nm}\nolimits^{\alpha}\colon {\mathcal M}(\widetilde{\S})\rightarrow {\mathcal M}(\Sigma^{\alpha})$ be the norm map on meromorphic functions associated to the covering $\pi^{\alpha}$. The norm maps associated to the various other coverings will similarly be denoted by $\mathop{\fam0 Nm}\nolimits^{\beta},\mathop{\fam0 Nm}\nolimits^{\gamma},\mathop{\fam0 Nm}\nolimits_{\alpha},\mathop{\fam0 Nm}\nolimits_{\beta},\mathop{\fam0 Nm}\nolimits_{\gamma},$ and $\widetilde \mathop{\fam0 Nm}\nolimits$. \begin{lemma}\label{6.12} (i) Define $f,g\in {\mathcal M}(\Sigma)$ by $f=\widetilde \mathop{\fam0 Nm}\nolimits(h_1)$, $g=\widetilde \mathop{\fam0 Nm}\nolimits(h_2)$. Then $$(f)=4 \,D_a, \ \ (g)=4\, D_b.$$ (ii) Define $f_{\alpha}=\mathop{\fam0 Nm}\nolimits^{\alpha}(h_1)\in {\mathcal M}(\Sigma^{\alpha})$, $f_{\beta}=\mathop{\fam0 Nm}\nolimits^{\beta}(h_2)\in {\mathcal M}(\Sigma^{\beta})$, $f_{\gamma}=\mathop{\fam0 Nm}\nolimits^{\gamma}(h_1/h_2)\in {\mathcal M}(\Sigma^{\gamma})$. Then $$f_{\alpha}\circ \phi_{\alpha}=-f_{\alpha}, \ f_{\beta}\circ \phi_{\beta}=-f_{\beta}, \ \ f_{\gamma}\circ \phi_{\gamma}=-f_{\gamma}.$$ \end{lemma} \noindent {\bf Proof.} Using that $\phi^{\alpha}$ covers $\phi_{\beta}$ and $\phi_{\gamma}$, one computes that $$\pi^{{\alpha}*}((f_{\alpha}))=(h_1)+\phi^{{\alpha}*}(h_1)=2\,\tilde\pi^*(D_a).$$ It follows that $(f_{\alpha})=2\pi_{\alpha}^*(D_a)$ and hence $(f)=(\mathop{\fam0 Nm}\nolimits_{\alpha}(f_{\alpha}))=4D_a$, as asserted. This also shows that the divisor $(f_{\alpha})$ is $\phi_{\alpha}$-invariant. Therefore one has $f_{\alpha}\circ \phi_{\alpha}=\pm f_{\alpha}$. But $f_{\alpha}$ itself cannot be $\phi_{\alpha}$-invariant, because if it were, it would descend to a function $h\in{\mathcal M}(\Sigma)$ such that $(h)=2D_a$, which is impossible since $a$ has order $4$. Therefore $f_{\alpha}\circ \phi_{\alpha}=-f_{\alpha}$. The other assertions of the lemma are proved similarly. \vskip 8pt By lemma \ref{6.12}(i), we can compute the Weil pairing $\lambda_4(a,b)$ using the functions $f$ and $g$ (see the definition in section \ref{prel}). Note that by lemma \ref{6.12}(ii), we have that $$h_1(\pi^{{\alpha}*}(1-\phi_{\alpha}^*)(D))=f_{\alpha}((1-\phi_{\alpha}^*)(D))=\frac{f_{\alpha}(D)}{(f_{\alpha}\circ \phi_{\alpha} )(D)}=(-1)^{\deg D}.$$ Thus \begin{eqnarray*}\label{calc} \lambda_4(a,b)&=&\frac{g(D_a)}{f(D_b)} =\frac{f(-D_b)}{g(-D_a)}=\frac{h_1(-\tilde\pi^*(D_b))}{h_2(-\tilde\pi^*(D_a))} \\ &=&\frac{h_1(-\tilde\pi^*(D_b) +(h_2))}{h_2(-\tilde\pi^*(D_a) +(h_1))} \\ &=&\frac{h_1(\pi^{{\gamma}*}(1-\phi_{\gamma}^*)(D'') - \pi^{{\alpha}*}(1-\phi_{\alpha}^*)(D))} {h_2(\pi^{{\gamma}*}(1-\phi_{\gamma}^*)(D'') - \pi^{{\beta}*}(1-\phi_{\beta}^*)(D'))}\\ &=&f_{\gamma}((1-\phi_{\gamma}^*)(D'')) \frac {f_{\beta}((1-\phi_{\beta}^*)(D'))}{f_{\alpha}((1-\phi_{\alpha}^*)(D))}\\ &=&(-1)^{\deg(D) +\deg (D')+\deg(D'')}=(-1)^{d+d'+d''} \end{eqnarray*} where we have used Weil reciprocity in the fourth equality. This proves lemma \ref{6.9} and hence theorem \ref{6.8}. \section{The action of $J^{(2)}$ on the Hecke correspondence.}\label{Heckecorr} We will make use of the Hecke correspondence in our analysis of the involutions in section \ref{Geoinvest}. This is a pair of morphisms \begin{center} \begin{picture}(120,80) \put(55,60){${\mathcal P}$} \put(0,10){$M$} \put(100,10){$M'$} \put(50,55){\vector(-1,-1){30}} \put(65,55){\vector(1,-1){30}} \put(20,45){$q$} \put(90,45){$q'$} \end{picture} \end{center} which allows one to `transfer' information from $M$ to $M'$. In this section, we describe the fixed point varieties $|{\mathcal P}|_{\alpha}$ of the action of the various ${\alpha}\inJ^{(2)}$ on ${\mathcal P}$. \vskip 8pt \noindent{\bf Notation.} Given a bundle $E$ over $\Sigma$, we denote by $E_x$ the fiber of $E$ at the point $x\in\Sigma$. Also, for a bundle $E$ representing a point in the moduli spaces $M$ or $M'$, we use the notation $[E]$ for that point. \vskip 8pt We briefly review the construction of ${\mathcal P}$. (See {\em e.g.} Bertram and Szenes \cite{BSz}.) Let ${\mathcal{U}}$ be a Poincar\'{e} bundle over $\Sigma\times M'$. Thus, if $[E']\in M'$, the restriction of ${\mathcal{U}}$ to $\Sigma\times \{[E']\}$ is isomorphic to $E'$. We can uniquely fix ${\mathcal{U}}$ by requiring that $\det({\mathcal{U}}|_{\{p\}\times M'})$ is an ample generator of $\mathop{\fam0 Pic}\nolimits(M')$. We put ${\mathcal P} = {\mathbb P}({\mathcal{U}}|_{\{p\}\times M'})$ and let $ q' : {\mathcal P} \rightarrow M'$ be the projection. Note that $q'$ is a ${\mathbb P}^1$-fibration and for $[E']\in M'$, the fiber $(q')^{-1}([E'])$ is isomorphic to the projective space ${\mathbb P}(E'_p)$. In fact, ${\mathcal P}$ can be viewed as the moduli space of pairs $(E', {\mathcal F})$ where $E'$ is a stable rank $2$ bundle with $\det(E')=[p]$, and ${\mathcal F}\subset E'_p$ is a one-dimensional subspace, {\em i.e.}, ${\mathcal P}$ is a moduli space of semi-stable parabolic bundles. We will refer to ${\mathcal F}$ as a {\em line} in $E'_p$. Points in ${\mathcal P}$ will be denoted as $[(E',{\mathcal F})]$, and we have $q'([(E',{\mathcal F})])=[E']$. The map $q$ is obtained by the operation of elementary modification at $p$. This means that we have $q([(E',{\mathcal F})])=[E]$ if and only if there is a short exact sequence (of sheaves) $$0\rightarrow E\rightarrow E'\mapright{\lambda}{\mathbb C}_p\rightarrow 0$$ such that $\ker_p(\lambda)={\mathcal F}\subset E'_p$. Here, ${\mathbb C}_p$ is the skyscraper sheaf at $p$. The group $J^{(2)}$ acts naturally on ${\mathcal P}$. The action of ${\alpha}\in J^{(2)}$ on ${\mathcal P}$ sends $[(E',{\mathcal F})]$ to $[(E'\otimes L_{\alpha},{\mathcal F}\otimes L_{\alpha})]$. The morphisms $q$ and $q'$ are $J^{(2)}$-equivariant. Let ${\alpha}\in J^{(2)}$ be non-zero. We now describe the fixed point variety $|{\mathcal P}|_{\alpha}$. Recall that ${\pi_{\a*}}\colon \theta_{\alpha}^{-1}([p]) \rightarrow |M'|_{\alpha}$ is a double covering, where $\theta_{\alpha}^{-1}([p])\subset \mathop{\fam0 Pic}\nolimits_1(\Sigma^{\alpha})$ consists of two `translates' of the Prym variety $P_{\alpha}$. Let $p_\alpha\in\Sigma^\alpha$ be such that $\pi_{\alpha}(p_\alpha) = p$. If $L\in \mathop{\fam0 Pic}\nolimits_1(\Sigma^\alpha)$, the projection gives a canonical isomorphism $$(L\oplus \phi_{\alpha}^*(L))_{p_\a} \mapright \sim ({\pi_{\a*}}(L))_p.$$ \begin{proposition} \label{fixPa} We have an isomorphism $$j_{{p_\a}}\colon \theta_{\alpha}^{-1}([p]) \,\mapright\sim\, |{\mathcal P}|_{\alpha}$$ defined by $j_{{p_\a}}(L)=[({\pi_{\a*}}(L),(L\oplus 0)_{p_a})]$. \end{proposition} \noindent {\bf Proof.} Put $E'={\pi_{\a*}}(L)$. It is clear that $|{\mathcal P}|_{\alpha}\subset (q')^{-1} (|M'|_{\alpha})$. Therefore the only question is which lines in $E'_p$ correspond to fixed points of ${\alpha}$ acting on ${\mathbb P}(E'_p)\cong (q')^{-1}([E'])$. Let $\psi\colon E'\mapright\sim E'\otimes L_{\alpha}$ be the isomorphism described in remark \ref{fundobs}. It is covered by the diagonal automorphism $\tilde\psi=1\oplus(-1)$ of $L\oplus \phi_{\alpha}^*(L)$. Since $E'$ is stable, it is simple, hence any other isomorphism is a non-zero multiple of $\psi$. Therefore $(E',{\mathcal F})$ represents a point in $|{\mathcal P}|_{\alpha}$ if and only if $\psi({\mathcal F})={\mathcal F}\otimes L_{\alpha}$. Letting $\widetilde {\mathcal F}$ denote the line in $(L\oplus \phi_{\alpha}^*(L))_{p_\a}$ projecting down to ${\mathcal F}\subset E'_p$, this condition is equivalent to $\tilde\psi(\widetilde {\mathcal F})=\widetilde {\mathcal F}$. The only lines in $(L\oplus \phi_{\alpha}^*(L))_{p_\a}$ that $\tilde\psi$ preserves are $(L\oplus 0)_{p_\a}$ and $(0\oplus \phi_{\alpha}^*(L))_{p_\a}$. The first line defines the point $j_{p_\a}(L)$ in ${\mathcal P}$, and the second line defines the point $j_{p_\a}(\phi_{\alpha}^*(L))$. This shows that $j_{p_\a}$ is bijective. It is clearly an algebraic morphism, and since its domain is smooth, this shows $j_{p_\a}$ is an isomorphism. \begin{remark}\label{qeq} {\em One has $q'\circ j_{p_\a}={\pi_{\a*}}$. In other words, $j_{p_\a}$ is an isomorphism of coverings over the identity of $|M'|_{\alpha}$. Note also that $j_{\phi_{\alpha}({p_\a})}=j_{p_\a} \circ \phi_{\alpha}^*$.}\end{remark} \noindent{\bf Notation.} Given $a\inJ^{(4)}$ such that $2a={\alpha}$, we denote by $|{\mathcal P}|_a^+$ the component of $|{\mathcal P}|_{\alpha}$ containing the point $j_{p_\a}(\pi_{\alpha}^*(L_a)\otimes [p_{\alpha}])$. (See formula (\ref{tw}).) Note that this point, and hence the definition of the component $|{\mathcal P}|_a^+$, depends only on $a$, not on the choice of $p_{\alpha}$. \begin{proposition}\label{q-+} One has $q(|{\mathcal P}|_a^+)=|M|_a^-$. \end{proposition} \noindent {\bf Proof.} Put $L=\pi_{\alpha}^*(L_a)$ and $L'=L\otimes [p_{\alpha}]$. Then $E={\pi_{\a*}}(L)$ represents a point in $|M|_a^+$, and $E'={\pi_{\a*}}(L')$ represents a point in $|M'|_{\alpha}$. The short exact sequence of sheaves $$0\rightarrow L\rightarrow L'\rightarrow {\mathbb C}_{p_\a}\rightarrow 0$$ induces the short exact sequence \begin{equation} 0\rightarrow E\rightarrow E'\,\mapright{\lambda}\,{\mathbb C}_p\rightarrow 0. \label{ses} \end{equation} We need to determine the line ${\mathcal F}=\ker_p (\lambda)\subset E'_p$. Pulling (\ref{ses}) back to $\Sigma^{\alpha}$ and restricting to the fiber at $p_{\alpha}$, the map $\lambda$ becomes $$(L'\oplus \phi_{\alpha}^*(L'))_{p_\a}\rightarrow ({\mathbb C}_{p_\a} \oplus \phi_{\alpha}^*({\mathbb C}_{p_\a}))_{p_\a} ={\mathbb C}_{p_\a} \oplus 0.$$ This shows that ${\mathcal F}=\ker_p (\lambda)$ is the projection of the line $(0\oplus \phi_{\alpha}^*(L'))_{p_\a}$. Hence $$[(E', {\mathcal F})]=j_{p_\a}(\phi_{\alpha}^*(L'))=j_{\phi_{\alpha}(p_{\alpha})}(L').$$ Since $j_{p_\a}(L')\in |{\mathcal P}|_a^+$, this shows that $[(E', {\mathcal F})]$ lies in $|{\mathcal P}|_a^-$ (see remark \ref{qeq}). Recalling that $q([(E', {\mathcal F})]=[E]$, it follows that $q(|{\mathcal P}|_a^-)=|M|_a^+ $, and also that $q(|{\mathcal P}|_a^+)=|M|_a^-$. \vskip 8pt The following observation will be used in section \ref{tracecomp}. \begin{proposition}\label{qtriv} Let $\nu$ be the relative cotangent sheaf of $q'\colon {\mathcal P}\rightarrow M'$. Then the restriction of $\nu$ to $|{\mathcal P}|_{\alpha}$ is numerically trivial. \end{proposition} \noindent {\bf Proof.} Note that $\nu|_{|{\mathcal P}|_{\alpha}}$ is the dual of the normal bundle, $N$, say, of the inclusion $|{\mathcal P}|_{\alpha}\subset (q')^{-1}(|M'|_{\alpha})$. By prop. \ref{fixPa}, it suffices to show that $j_{p_\a}^*(N) $ is numerically trivial. Let $\Lambda$ be a Poincar\'e bundle over $\mathop{\fam0 Pic}\nolimits_1(\Sigma^{\alpha})\times \Sigma^{\alpha}$. For $x\in \Sigma^{\alpha}$, let $\Lambda_{x}$ denote its restriction to $\theta_{\alpha}^{-1}([p])\times \{x\}$. We have a commutative diagram $$\begin{array}{cccc} {\mathbb P}( \Lambda_{p_\alpha} \oplus \Lambda_{\phi_{\alpha}({p_\a})} ) &\mapright{\Pi_{{\alpha}*}} & (q')^{-1}(|M'|_{\alpha})&\ \subset \ {\mathcal P} \\ \downarrow & & \downarrow \, q' &\\ \theta_{\alpha}^{-1}([p]) & \mapright{{\pi_{\a*}}} & |M'|_{\alpha}& \end{array}$$ Here, $\Pi_{{\alpha}*}$ is the obvious map covering ${\pi_{\a*}}$. (A point in ${\mathbb P}( \Lambda_{p_\alpha} \oplus \Lambda_{\phi_{\alpha}({p_\a})} )$ is a point $[L]\in \theta_{\alpha}^{-1}([p])$ together with a line in $ L_{{p_\a}}\oplus L_{\phi_{\alpha}({p_\a})}=(L\oplus \phi_{\alpha}^*(L))_{p_\a}$. This is sent by $\Pi_{{\alpha}*}$ to the point represented by ${\pi_{\a*}}(L)$ and the induced line in $({\pi_{\a*}}(L))_p$.) Let $s_{{p_\a}}$ be the section of the fibration on the left defined by $s_{p_\a}(L)=L_{{p_\a}}\oplus 0$, for $[L]\in \theta_{\alpha}^{-1}([p])$. Then $j_{p_\a}=\Pi_{{\alpha}*}\circ s_{{p_\a}}$, and hence the inclusion $|{\mathcal P}|_{\alpha}\subset (q')^{-1}(|M'|_{\alpha})$ corresponds to the inclusion of the image of $s_{p_\a}$ in ${\mathbb P}( \Lambda_{p_\alpha} \oplus \Lambda_{\phi_{\alpha}({p_\a})} )$. This shows $$j_{p_\a}^*(N) \cong \Lambda_{p_\a}^*\otimes \Lambda_{\phi_{\alpha}({p_\a})}.$$ Tensoring $\Lambda$ by the pull-back of a bundle over $\mathop{\fam0 Pic}\nolimits_1(\Sigma^{\alpha})$ if necessary, we may assume $\Lambda_{p_\a}$ is trivial. Hence $j_{p_\a}^*(N)$ is numerically trivial, proving the proposition. \section{Investigation of the involutions $\rho_a$ and $\rho'_{\alpha}$.} \label{Geoinvest} Let $a\inJ^{(4)}$ such that $2a={\alpha}\neq0$. Recall that the involution $\rho_a$ is the lift of ${\alpha}$ to ${\mathcal{L}}$ which acts as the identity over the fixed point component $|M|_a^+$. \begin{proposition}\label{rhominus} The involution $\rho_a$ acts as minus the identity over the fixed point component $|M|_a^-$. \end{proposition} \noindent {\bf Proof.} Let $E'$ represent a point $[E']$ in $|M'|_\alpha$. Now $\alpha$ acts on the fiber of $q'$ over $[E']$ and from our description of $|{\mathcal P}|_\alpha$, we have that $\alpha$ has exactly two fixed points on $(q')^{-1}([E'])$. Consider now $q^*{\mathcal{L}}|_{(q')^{-1}([E'])}$ with its lift of $\alpha$ induced by $\rho_a$. Let $s_1,s_2$ be the signs by which $\rho_a$ acts over the two fixed points. By lemma 2.1 in \cite{BSz} we have that $q^*{\mathcal{L}}|_{(q')^{-1}([E'])}$ is isomorphic to ${\mathcal{O}}(1)$ over $(q')^{-1}([E'])\cong {\mathbb P}(E'_p)$. From this we conclude that $s_1s_2 = -1$, proving the proposition. \begin{remark}\label{coc} {\em At this point, it follows easily that the alternating form of the extension ${\mathcal{E}}$ generated by the involutions $\rho_a$ is equal to the order $2$ Weil pairing $\lambda_2$. Indeed, recall that the alternating form is defined by the commutator pairing $c({\alpha},{\beta})=\rho_a\rho_b\rho_a^{-1}\rho_b^{-1}$ where $a$ is a lift of $\alpha$ and $b$ is a lift of $\beta$. Assume first that $\lambda_2(\alpha, \beta)=1$. Let us then evaluate $\rho_a\rho_b\rho_a^{-1}\rho_b^{-1}$ in a point in $|M|^+_a$. Recalling from \ref{4.6}(ii) that $\beta$ preserves this component of $|M|_\alpha$, we get that $$c(\alpha,\beta) = 1 \,\rho_b \,(1)^{-1}\, \rho_b^{-1} = 1.$$ If however $\lambda_2(\alpha, \beta)=-1$, then $\beta$ exchanges the two components, and $$c(\alpha,\beta) = 1 \,\rho_b \,(-1)^{-1}\, \rho_b^{-1} = -1.$$ Thus $c=\lambda_2$, as asserted.} \end{remark} \begin{theorem}\label{8.4} We have $\rho'_\alpha \rho'_\beta=\lambda_2({\alpha},{\beta})\rho'_{\alpha+\beta}$. \end{theorem} \noindent {\bf Proof.} We may assume none of the classes ${\alpha}$, ${\beta}$, ${\alpha}+{\beta}$, is zero, the result being obvious otherwise. Consider first the case $\lambda_2({\alpha},{\beta})=-1$. Then by prop. \ref{6.5} we have that the triple intersection $ |M'|_\alpha \cap |M'|_\beta \cap |M'|_{\alpha + \beta}$ is non-empty. Since by definition $\rho'_\alpha$ acts as minus the identity on the fiber over $|M'|_\alpha$, it follows that $$\rho'_\alpha \rho'_\beta=-\rho'_{\alpha + \beta}, $$ proving the result in this case. Now consider the case $\lambda_2({\alpha},{\beta})=1$. Pick $a,b\in J^{(4)}$ such that $2a={\alpha}$ and $2b={\beta}$, and consider the involutions $\rho_a^{\otimes 2}$, $\rho_b^{\otimes 2}$, and $\rho_{a+b}^{\otimes 2}$, acting on ${\mathcal{L}}^2$. In fact, those involutions depend only on ${\alpha}$ and ${\beta}$, and not on the choice of $a$ and $b$. By prop. \ref{6.5} we have that the triple intersection $ |M|_\alpha \cap |M|_\beta \cap |M|_{\alpha + \beta}$ is non-empty. Note that $\rho_a^{\otimes 2}$ acts as the identity over both components of $|M|_{\alpha}$. Hence \begin{equation}\label{rhoH} \rho_a^{\otimes 2} \rho_b^{\otimes 2}=\rho_{a+b}^{\otimes 2}. \end{equation} Now consider the Hecke correspondence. From Corollary 2.2 in \cite{BSz} (see also Lemma 10.3 in \cite{BLS}) we have that the canonical bundle $K_{{\mathcal P}}$ of ${\mathcal P}$ satisfies \begin{equation}\label{Hecke} K_{{\mathcal P}} \cong (q')^*({{\mathcal{L}}'}^{-1})\otimes q^*({\mathcal{L}}^{-2}). \end{equation} From Proposition \ref{fixPa} we see that $|{\mathcal P}|_\alpha$ has odd codimension, hence $\alpha$ acts by $-1$ on the restriction of $K_{\mathcal P}$ to $|{\mathcal P}|_\alpha$. Our lifts $\rho_a^{\otimes 2}$ and $\rho'_\alpha$ thus make the isomorphism (\ref{Hecke}) a $J^{(2)}$-equivariant isomorphism. The action of $J^{(2)}$ on $K_{{\mathcal P}}$ is obviously a group action. This enables us to compare the lift $\rho'_\alpha$ acting on ${{\mathcal{L}}'}$ and the lift $\rho_a^{\otimes 2}$ acting on ${\mathcal{L}}^{2}$. Thus (\ref{rhoH}) implies $$\rho'_\alpha \rho'_\beta=\rho'_{\alpha + \beta},$$ proving the result in the case $\lambda_2({\alpha},{\beta})=1$. This completes the proof. \vskip 8pt\noindent{\bf Note.} The equivariance of the isomorphism (\ref{Hecke}) is the reason why we defined $\rho'_\alpha$ to be the lift which acts as {\em minus} the identity over the fixed point set. \begin{theorem}\label{8.5} We have $ \rho_a \rho_b=\lambda_4(a,b)\rho_{a+b}$. \end{theorem} \noindent {\bf Proof.} We first deal with the case where $\lambda_4(a,b)=\pm 1$, or, equivalently, $\lambda_2({\alpha},{\beta})=1$. If ${\alpha}={\beta}=0$, there is nothing to show. If ${\alpha}={\beta}\neq 0$, then $\rho_a$ and $\rho_b$ are lifts of the same class, hence $\rho_a=\pm \rho_b$. By prop. \ref{rhominus}, we have $\rho_a=\rho_b$ if and only if $a$ and $b$ define the same component of $|M|_{\alpha}$, which in turn is equivalent, by prop. \ref{4.6}(i), to $\lambda_2(b-a,{\alpha})=1$. But if ${\alpha}={\beta}$ then $\lambda_2(b-a,{\alpha})=\lambda_4(a,b)$ and $\rho_{a+b}$ is the identity (by definition). This proves the result in the case ${\alpha}={\beta}$. Finally, if ${\alpha},{\beta},$ and ${\alpha}+{\beta}$ are all three non-zero, the triple intersection $ |M|_\alpha \cap |M|_\beta \cap |M|_{\alpha + \beta}$ is non-empty, and we can compute $\rho_a \rho_b \rho_{a+b}$ in the fiber over an intersection point. By theorem \ref{evencasei} and prop. \ref{rhominus}, it follows that $\rho_a \rho_b \rho_{a+b}=\lambda_4(a,b)$, completing the proof in the case $\lambda_4(a,b)=\pm 1$. The remainder of this section is devoted to the proof in the case where $\lambda_4(a,b)=\pm i.$ We will again use the notations ${\gamma}={\alpha}+{\beta}$ and $c=a+b$. Let the bundle $E'$ represent a point $[E']$ in $|M'|_\alpha\cap |M'|_{\beta}\cap |M'|_{\gamma}$. The three involutions ${\alpha},{\beta},{\gamma}$ induce involutions on $(q')^{-1}([E'])\subset {\mathcal P}$; recall that $(q')^{-1}([E'])$ is identified with the projective space ${\mathbb P}(E'_p)$. Each of these involutions has two fixed points on ${\mathbb P}(E'_p)$; these are precisely the intersection points of ${\mathbb P}(E'_p)$ with the fixed point varieties $|{\mathcal P}|_{\alpha},|{\mathcal P}|_{\beta}$, and $|{\mathcal P}|_{\gamma}$. Note that $\rho_a$ acts as $\mp 1$ on the fiber of $q^*{\mathcal{L}}$ at the intersection point of ${\mathbb P}(E'_p)$ with the component $|{\mathcal P}|_a^{\pm}$, since $q(|{\mathcal P}|_a^+)=|M|_a^-$ by prop. \ref{q-+}. Let ${\mathcal F}_a^{\pm}$, ${\mathcal F}_b^{\pm}$, ${\mathcal F}_c^{\pm}$ be the lines in $E'_p$ corresponding to the intersection points of ${\mathbb P}(E'_p)$ with the components $|{\mathcal P}|_a^{\pm},|{\mathcal P}|_b^{\pm}$, and $|{\mathcal P}|_c^{\pm}$. As already used in the proof of prop. \ref{rhominus}, the restriction of $q^*{\mathcal{L}}$ to ${\mathbb P}(E'_p)$ is the bundle ${\mathcal{O}}(1)$. It will be convenient to transfer the calculation to the tautological bundle ${\mathcal{O}}(-1)$, whose fiber over a point represented by a line ${\mathcal F}$ is that line. Of course, ${\mathcal{O}}(-1)$ is the restriction of $q^*{\mathcal{L}}^{-1}$ to ${\mathbb P}(E'_p)$. For $a\inJ^{(4)}$, let us denote by $\hat\rho_a$ the involution $\rho_a^{\otimes(-1)}$ acting on ${\mathcal{L}}^{-1}$. Then $\hat\rho_a$ acts as $\mp 1$ on the line ${\mathcal F}_a^{\pm}$, and similarly for $\hat\rho_b$ and $\hat\rho_c$ on the lines ${\mathcal F}_b^{\pm}$ and ${\mathcal F}_c^{\pm}$. It follows easily from this description (or from the computation of the alternating form in remark \ref{coc}) that one has $\hat\rho_a\hat\rho_b=\varepsilon \hat\rho_c$ where $\varepsilon\in\{\pm i\}$. (In fact, the involutions $\hat\rho_a, \hat\rho_b$ generate a quaternion subgroup $Q_8\subset Sl_2({\mathbb C})$, covering the commutative subgroup ${\mathbb Z}/2\times {\mathbb Z}/2 \subset PSl_2({\mathbb C})$ generated by ${\alpha},{\beta}$.) Of course, the sign of $\varepsilon$ is determined by the relative position of the six lines. The following lemma computes this relative position in terms of the Weil pairing $\lambda_4(a,b)$. \begin{lemma}\label{ml} Let $\lambda=\lambda_4(b,a)\in\{\pm i\}$. There is an isomorphism of $E'_p$ with ${\mathbb C}^2$ sending the six lines ${\mathcal F}_a^{+}$, ${\mathcal F}_a^{-}$, ${\mathcal F}_b^{+}$, ${\mathcal F}_b^{-}$, ${\mathcal F}_c^{+}$, ${\mathcal F}_c^{-}$, to the lines generated by the vectors $$\left(\begin{array}{c} 1\\0 \end{array}\right), \ \left(\begin{array}{c} 0\\1 \end{array}\right), \ \left(\begin{array}{c} 1\\1 \end{array}\right), \ \left(\begin{array}{c} 1\\-1\end{array}\right), \ \left(\begin{array}{c} 1\\-\lambda \end{array}\right), \ \left(\begin{array}{c} 1\\ \lambda \end{array}\right) \ . $$ \end{lemma} The proof will be given later. Assuming lemma \ref{ml} for the moment, we see that $\hat\rho_a, \hat\rho_b$ and $\hat\rho_c$ correspond to the matrices $$T_a=\left( \begin{array}{cc} -1& 0 \\ 0 & 1 \end{array}\right), \ \ T_b=\left( \begin{array}{cc} 0& -1 \\ -1 & 0 \end{array}\right), \ \ T_c=\left( \begin{array}{cc} 0& -\lambda \\ \lambda & 0 \end{array}\right).$$ Note that $T_aT_b=\lambda T_c$, and hence $\hat\rho_a\hat\rho_b=\lambda \hat \rho_c$ and $\rho_a \rho_b=\lambda^{-1}\rho_c$. Thus the remaining case of theorem \ref{8.5} follows directly from lemma \ref{ml}. \vskip 8pt Now let us prove lemma \ref{ml}. The proof uses again the coverings $\Sigma^{\alpha}$, $\Sigma^{\beta}$, $\Sigma^{\gamma}$, and their common covering $\widetilde{\S}$ (see section \ref{triple}). Choose a point $\tilde p\in\tilde\pi^{-1}(p)\subset \widetilde{\S}$ and put \begin{equation} p_{\alpha}=\pi^{\alpha}(\tilde p), \ p_{\beta}=\pi^{\beta}(\tilde p), \ p_{\gamma}=\pi^{\gamma}(\tilde p). \label{palpha} \end{equation} Since $[E']\in |M'|_\alpha\cap |M'|_{\beta}\cap |M'|_{\gamma}$, there exist line bundles ${\mathcal{L}}_a$ over $\Sigma^{\alpha}$, ${\mathcal{L}}_b$ over $\Sigma^{\beta}$, and ${\mathcal{L}}_c$ over $\Sigma^{\gamma}$, such that $ E'\cong {\pi_{\a*}}({\mathcal{L}}_a)\cong {\pi_{\b*}}({\mathcal{L}}_b)\cong {\pi_{\g*}}({\mathcal{L}}_c)$. We can fix ${\mathcal{L}}_a$ (resp. ${\mathcal{L}}_b$, resp. ${\mathcal{L}}_c$) uniquely up to isomorphism by requiring that ${\mathcal{L}}_a\in \pi_{\alpha}^*(L_a)\otimes[p_{\alpha}]\otimes \mathop{\fam0 Im}\nolimits\Phi_{\alpha}^0$ (resp. ${\mathcal{L}}_b\in \pi_{\beta}^*(L_b)\otimes[p_{\beta}]\otimes \mathop{\fam0 Im}\nolimits\Phi_{\beta}^0$, resp. ${\mathcal{L}}_c\in \pi_{\gamma}^*(L_c)\otimes[p_{\gamma}]\otimes \mathop{\fam0 Im}\nolimits\Phi_{\gamma}^0$) (see formula (\ref{tw})). Let us denote the bundle $\pi^{{\gamma}*}({\mathcal{L}}_c)$ on $\widetilde{\S}$ by $L$. Being a pull-back bundle, $L$ has a canonical involution, $C$, say, covering the involution $\phi^{\gamma}$ on $\widetilde{\S}$, and such that $L/{\langle} C{\rangle}$ is the bundle ${\mathcal{L}}_c$ on $\Sigma^{\gamma}=\widetilde{\S}/{\langle}\phi^{\gamma}{\rangle}$. Proceeding as in lemma \ref{6.10}, it is easy to check that the bundles $\pi^{{\alpha}*}({\mathcal{L}}_a)$ and $\pi^{{\beta}*}({\mathcal{L}}_b)$ are isomorphic to $L=\pi^{{\gamma}*}({\mathcal{L}}_c)$. Therefore $L$ also has canonical involutions $A$ and $B$, covering $\phi^{\alpha}$ and $\phi^{\beta}$, respectively, such that $L/{\langle} A{\rangle}\cong {\mathcal{L}}_a$ and $L/{\langle} B{\rangle}\cong {\mathcal{L}}_b$. \begin{lemma}\label{ABC} One has $AB=\lambda_4(b,a)C$. \end{lemma} \noindent {\bf Proof.} As in the proof of lemma \ref{6.9}, represent $\pi^{{\alpha}*}({\mathcal{L}}_a)$, $\pi^{{\beta}*}({\mathcal{L}}_b)$, and $L=\pi^{{\gamma}*}({\mathcal{L}}_c)$, by divisors $F_a$, $F_b$, $F_c$, respectively, such that $$F_a=\tilde\pi^{*}(D_a)+\pi^{{\alpha}*}(p_{\alpha}+(1-\phi_{\alpha}^*)(D))$$ $$F_b=\tilde\pi^{*}(D_b)+\pi^{{\beta}*}(p_{\beta}+(1-\phi_{\beta}^*)(D'))$$ $$F_c=\tilde\pi^{*}(D_c)+\pi^{{\gamma}*}(p_{\gamma}+(1-\phi_{\gamma}^*)(D''))$$ where $D_a,D_b\in \mathop{\fam0 Div}\nolimits_0(\Sigma)$ represent $a,b\in J^{(4)}$, $D_c=D_a+D_b$, $D\in Div_0(\Sigma^{\alpha})$, $D'\in Div_{0}(\Sigma^{\beta})$, and $D''\in Div_{0}(\Sigma^{\gamma})$. As before, since the three bundles are isomorphic, there exist meromorphic functions $h_1,h_2\in {\mathcal M}(\widetilde{\S})$ such that $$(h_2)+F_a\,=\,F_c\,=\, (h_1)+F_b.$$ The action of our involutions $A,B$ and $C$ on $L$ can be described on local sections as follows. Since $L={\mathcal{O}}(F_c)$, a local section over some open set $U\subset \widetilde{\S}$ is just a meromorphic function $s$ on $U$ such that $(s)+F_c|_U\geq 0$. The action of $C$ is simply given by $$ C\ : s \mapsto s\circ \phi^{\gamma},$$ since the divisor $F_c$ was pulled back from $\Sigma^{\gamma}$, and $C$ is the canonical involution of the pull-back bundle. The involution $A$ is nothing but the canonical involution of the pull-back bundle $\pi^{{\alpha}*}({\mathcal{L}}_a)={\mathcal{O}}(F_a)$, conjugated by an isomorphism with $L={\mathcal{O}}(F_c)$. Since $(h_2)=F_c-F_a$, multiplication by $h_2$ gives such an isomorphism ${\mathcal{O}}(F_c)\,\mapright\sim \,{\mathcal{O}}(F_a)$. Therefore the action of $A$ on local sections of $L={\mathcal{O}}(F_c)$ is $$ A\ : s \mapsto ((s h_2)\circ \phi^{\alpha})h_2^{-1}=(s\circ \phi^{\alpha})\,k_A= (s\ k_A^{-1})\circ \phi^{\alpha}$$ where we have put $k_A=(h_2\circ \phi^{\alpha})/h_2$. (N.b., one may think about this as follows: $s\circ \phi^{\alpha}$ is a local section of $\phi^{{\alpha}*}L={\mathcal{O}}(\phi^{{\alpha}*}F_c)$, and multiplication by $k_A$ describes an isomorphism ${\mathcal{O}}(\phi^{{\alpha}*}F_c)\,\mapright\sim \,{\mathcal{O}}(F_c)$.) Similarly, $B$ acts on local sections of ${\mathcal{O}}(F_c)$ as $$ B\ : s \mapsto ((s h_1)\circ \phi^{\beta})h_1^{-1}=(s\circ \phi^{\beta})\,k_B= (s\ k_B^{-1})\circ \phi^{\beta}$$ where $k_B=(h_1\circ \phi^{\beta})/h_1$. Put \begin{equation} \lambda= \frac{k_B}{k_A} = \frac{h_2}{h_1} \,\frac {h_1\circ \phi^{\beta}}{h_2\circ \phi^{\alpha}}\label{la1} \end{equation} Note that $\lambda$ is a constant, since $(k_B)=(k_A)$. Since $AB$ acts on local sections by $$AB\ : s\mapsto (((s\circ \phi^{\beta})\,k_B) \circ \phi^{\alpha})\, k_A=((s\circ \phi^{\beta})\,k_B\,k_A^{-1})\circ \phi^{\alpha} =\lambda \, s\circ \phi^{\gamma},$$ we have $AB=\lambda C$. Now let us show that $\lambda=\lambda_4(b,a)$. As in section \ref{triple}, we use the functions $f,g\in {\mathcal M}(\Sigma)$ defined by $f=\widetilde \mathop{\fam0 Nm}\nolimits(h_1)$ and $g=\widetilde \mathop{\fam0 Nm}\nolimits(h_2)$. A computation shows that the statements of lemma \ref{6.12} hold word for word. We can therefore compute $\lambda_4(a,b)=g(D_a)/f(D_b)$ exactly as before. Note that the divisors $D,D',D''$ have degree zero, so that the terms involving the functions $f_{\alpha},f_{\beta}, f_{\gamma}$ are now equal to $1$. We thus obtain \begin{eqnarray*} \lambda_4(a,b)&=&\frac{h_1(\pi^{{\gamma}*}(p_{\gamma})-\pi^{{\alpha}*}(p_{\alpha}))} {h_2(\pi^{{\gamma}*}(p_{\gamma})-\pi^{{\beta}*}(p_{\beta}))} =\frac{h_1(\phi^{\gamma}(\tilde p)-\phi^{\alpha}(\tilde p))}{h_2(\phi^{\gamma}(\tilde p)-\phi^{\beta}(\tilde p))} \\ &=& \frac{h_1}{h_2} \,\frac {h_2\circ \phi^{\alpha}}{h_1\circ \phi^{\beta}} \,(\phi^{\gamma}(\tilde p)) \end{eqnarray*} where we have used (\ref{palpha}) in the last but one step. Comparing this with formula (\ref{la1}), we have $\lambda=\lambda_4(a,b)^{-1}=\lambda_4(b,a)$, proving lemma \ref{ABC}. We return to the proof of lemma \ref{ml}. Recall that \begin{equation} E'\cong {\pi_{\a*}}({\mathcal{L}}_a)=({\mathcal{L}}_a\oplus\phi_{\alpha}^*{\mathcal{L}}_a)/{\langle}\phi_{\alpha}{\rangle}. \label{uniqueflag} \end{equation} Since ${\mathcal{L}}_a\in \pi_{\alpha}^*(L_a)\otimes[p_{\alpha}]\otimes \mathop{\fam0 Im}\nolimits\Phi_{\alpha}^0$, we have $$j_{p_\a}({\mathcal{L}}_a)\in |{\mathcal P}|_a^+,$$ where $j_{p_\a}$ is the isomorphism of prop. \ref{fixPa}. Thus, the line ${\mathcal F}_a^+\subset E'_p$ (representing the unique intersection point in ${\mathbb P}(E'_p)\cap |{\mathcal P}|_a^+$) is the image of the line $({\mathcal{L}}_a\oplus 0)_{{p_\a}}\subset ({\mathcal{L}}_a\oplus\phi_{\alpha}^*{\mathcal{L}}_a)_{{p_\a}}$ under the natural projection. Also, the line ${\mathcal F}_a^-$ is the image of $(0\oplus \phi_{\alpha}^*({\mathcal{L}}_a))_{{p_\a}}$. Note that since $E'$ is stable, the isomorphism (\ref{uniqueflag}) is unique up to scalar multiples, hence the lines ${\mathcal F}_a^{\pm}$ are well-determined by this description. Similarly, using the isomorphism of $E'$ with ${\pi_{\b*}}({\mathcal{L}}_b)$ and with ${\pi_{\g*}}({\mathcal{L}}_c)$, the lines ${\mathcal F}_b^+$ and ${\mathcal F}_b^-$ correspond to the projections of the lines $({\mathcal{L}}_b\oplus 0)_{{p_\b}}$ and $(0\oplus \phi_{\beta}^*({\mathcal{L}}_b))_{{p_\b}}$, and the lines ${\mathcal F}_c^+$ and ${\mathcal F}_c^-$ correspond to the projections of the lines $({\mathcal{L}}_c\oplus 0)_{{p_\g}}$ and $(0\oplus \phi_{\gamma}^*({\mathcal{L}}_c))_{{p_\g}}$. Let us now understand the relative position of the six lines in $E'_p$. Note that since $AB=-BA$, the involution $B$ of $L$ covering the involution $\phi^{\beta}$ on $\widetilde{\S}$ induces a map $$L/{\langle} -A{\rangle}\,\mapright{}\,L/{\langle} A{\rangle}$$ covering the involution $\phi_{\alpha}$ on $\Sigma^{\alpha}$. We may choose isomorphisms ${\mathcal{L}}_a\cong L/{\langle} A{\rangle}$ and $\phi_{\alpha}^*{\mathcal{L}}_a\cong L/{\langle} -A{\rangle}$ such that this map becomes the canonical map $\phi_{\alpha}^*{\mathcal{L}}_a \rightarrow {\mathcal{L}}_a$ covering $\phi_{\alpha}$. Therefore ${\pi_{\a*}}({\mathcal{L}}_a)$ is isomorphic to the bundle $E_a$ defined by $$E_a\ = \ L\oplus L {\Big/} \bigl< \left( \begin{array}{cc} A& 0 \\ 0 & -A \end{array}\right), \left( \begin{array}{cc} 0& B \\ B & 0 \end{array}\right)\bigr>.$$ (Notice that the two matrices commute, so that $E_a$ is indeed a well-defined bundle on $\Sigma=\widetilde{\S}/\langle\phi^{\alpha},\phi^{\beta}{\rangle}$.) Moreover, the lines ${\mathcal F}_a^+$ and ${\mathcal F}_a^-$ in $E'_p$ correspond, {\em via} an isomorphism $E'\cong E_a$ (which is unique up to scalar multiples), to the images of the lines $(L\oplus 0)_{\tilde p}$ and $(0\oplus L)_{\tilde p}$ under the projection from $L\oplus L$ to $E_a$. Here, we have used that $\pi^{\alpha}(\tilde p)={p_\a}$ by our choice of $p_{\alpha}$ in (\ref{palpha}). Similarly, ${\pi_{\b*}}({\mathcal{L}}_b)$ is isomorphic to the bundle $E_b$ defined by $$E_b\ = \ L\oplus L {\Big/} \bigl< \left( \begin{array}{cc} B& 0 \\ 0 & -B \end{array}\right), \left( \begin{array}{cc} 0& A \\ A & 0 \end{array}\right)\bigr>,$$ and since $\pi^{\beta}(\tilde p)={p_\b}$, the lines ${\mathcal F}_b^+$ and ${\mathcal F}_b^-$ in $E'_p$ correspond to the images of the same lines $(L\oplus 0)_{\tilde p}$ and $(0\oplus L)_{\tilde p}$, but now projected from $L\oplus L$ to $E_b$. Lastly, ${\pi_{\g*}}({\mathcal{L}}_c)$ is isomorphic to the bundle $E_c$ defined by $$E_c\ = \ L\oplus L {\Big/} \bigl< \left( \begin{array}{cc} C& 0 \\ 0 & -C \end{array}\right), \left( \begin{array}{cc} 0& A \\ A & 0 \end{array}\right)\bigr>, $$ and as before, since $\pi^{\gamma}(\tilde p)={p_\g}$, the lines ${\mathcal F}_c^+$ and ${\mathcal F}_c^-$ in $E'_p$ correspond to the images of the lines $(L\oplus 0)_{\tilde p}$ and $(0\oplus L)_{\tilde p}$ under the projection from $L\oplus L$ to $E_c$. The three bundles $E_a$,$E_b$, $E_c$ are all isomorphic to $E'$. In fact, we have isomorphisms $\psi_{X}\colon E_b\mapright\sim E_a$ and $\psi_{Y}\colon E_c\mapright\sim E_a$ induced by the endomorphisms of $L\oplus L$ defined by the matrices $$X=\left( \begin{array}{cc} 1& 1 \\ 1 & -1 \end{array}\right), \ \ \ Y=\left( \begin{array}{cc} 1& 1 \\ -\lambda & \lambda \end{array}\right).$$ The verification, which uses that $AB=\lambda C$ by lemma \ref{ABC}, is left to the reader. Let us identify the fiber $(L\oplus L)_{\tilde p}$ in the obvious way with ${\mathbb C}\oplus {\mathbb C}$ and consider the isomorphism $$E'_p\,\mapright\sim\, (E_a)_p \stackrel{\sim}\longleftarrow (L\oplus L)_{\tilde p} = {\mathbb C}\oplus {\mathbb C},$$ where the first map is induced by an isomorphism $E'\cong E_a$ and the second map is the projection. Then the six lines ${\mathcal F}_a^{+}$, ${\mathcal F}_a^{-}$, ${\mathcal F}_b^{+}$, ${\mathcal F}_b^{-}$, ${\mathcal F}_c^{+}$, ${\mathcal F}_c^{-}$, correspond to the lines in ${\mathbb C}\oplus {\mathbb C}$ generated by the vectors $$\left(\begin{array}{c} 1\\0 \end{array}\right), \ \left(\begin{array}{c} 0\\1 \end{array}\right), \ X\left(\begin{array}{c} 1\\0 \end{array}\right) , \ X\left(\begin{array}{c} 0\\1 \end{array}\right), \ Y\left(\begin{array}{c} 1\\0 \end{array}\right) , \ Y\left(\begin{array}{c} 0\\1 \end{array}\right) \ . $$ These vectors are precisely the ones in the statement of lemma \ref{ml}. This proves lemma \ref{ml}, and completes the proof of theorem \ref{8.5}. \section{The trace computation.}\label{tracecomp} In this section, we prove theorem \ref{1.2} by computing the trace of ${\rho'_\alpha}^{\otimes k/2}$ and $\rho_a^{\otimes k}$ using the Lefschetz-Riemann-Roch fixed point formula. In the twisted case, the computation is rather straightforward, since $M'$ is smooth, and the relevant cohomology classes on the fixed point set are given in \cite{NR}. This computation has been done in a different context by Pantev \cite{Pa}. We repeat the calculation here and in the process we correct a misprint in his formula. In the untwisted case, the moduli space and the fixed point sets are not smooth. We circumvent this problem by transferring the computation to ${\mathcal P}$, using some results of \cite{BSz}. \vskip 8pt\noindent{\bf Note.} Beauville \cite{Be3} has recently computed the traces in the untwisted case in a different way by transferring the calculation to $M'$. \footnote{Our computation was done independently of his.} Beauville considers, more generally, rank $r$ bundles, and his formula agrees with ours in the case $r=2$. He does not, however, choose lifts to the line bundle ${\mathcal{L}}$, and his result (for $r=2$) concerns only the case $k\equiv 0$ mod $4$, where one has a group action of $J^{(2)}$. \begin{proposition} The trace of the involution ${\rho'_\alpha}^{\otimes k/2}$ is given by $$\mathop{\fam0 Tr}\nolimits( {\rho'_\alpha}^{\otimes k/2})= (-1)^{k/2}\left( \frac{k+2}{2}\right)^{g-1}.$$ \end{proposition} \noindent {\bf Proof.} The Lefschetz-Riemann-Roch fixed point formula \cite{AS} states that \[ \mathop{\fam0 Tr}\nolimits({\rho'_\alpha}^{\otimes k/2}) = {\widetilde {Ch}}({{\mathcal{L}}'}|_{|M'|_\alpha})^{k/2}\, {\widetilde {Ch}}(\lambda_{-1}N_\alpha)^{-1}\,\mathop{\fam0 Td}\nolimits(|M'|_{\alpha})\cap [|M'|_\alpha].\] Here $\mathop{\fam0 Td}\nolimits(|M'|_{\alpha})$ is the Todd class, $N_\alpha$ is the conormal bundle of $|M'|_\alpha$, $\lambda_{t}$ is the operation defined by $\lambda_t E= \sum t^i \Lambda^i E$, and $${\widetilde {Ch}}(E)=Ch(E_+ -E_-),$$ where $Ch$ is the Chern character, and, for any ${\mathbb Z}/2$-equivariant bundle $E$ over the fixed point set $|M'|_\alpha$, $E_+$ and $E_-$ are the $\pm 1$-eigenbundles. Since the fixed point set $|M'|_\alpha$ is isomorphic to the Prym variety $P_\alpha$, its Todd class is $1$. The cohomology class ${\widetilde {Ch}}(\lambda_{-1}N_\alpha)$ was computed in Proposition 4.2 in \cite{NR}. Let $\Theta$ be the restriction to $P_\alpha$ of the principal polarization of $J_0(\Sigma^\alpha)$. The result of Narasimhan and Ramanan then states that \begin{equation} {\widetilde {Ch}}(\lambda_{-1}N_\alpha) = 2^{2(g-1)}e^{-2\Theta}. \label{Thet2} \end{equation} By the construction of ${{\mathcal{L}}'}$ in \cite{DN}, we have the following relation between $\Theta$ and $Ch({{\mathcal{L}}'}|_{|M'|_\alpha})$. \begin{equation} Ch({{\mathcal{L}}'}|_{|M'|_\alpha}) = e^{2\Theta}. \label{Thet1} \end{equation} Note that ${\widetilde {Ch}}({{\mathcal{L}}'}|_{|M'|_\alpha})=- Ch({{\mathcal{L}}'}|_{|M'|_\alpha})$, by our definition of $\rho'_{\alpha}$. Hence we get that \[ \mathop{\fam0 Tr}\nolimits({\rho'_\alpha}^{\otimes k/2})= (-1)^{k/2}2^{-2(g-1)} e^{(k+2)\Theta}\cap [P_\alpha].\] Using Corollary 4.16 in \cite{NR}, which states that \begin{equation} \Theta^{g-1}\cap [P_\alpha] = (g-1)!2^{g-1}, \label{Thet3} \end{equation} the result follows. \begin{proposition} The trace of the involution $\rho_a^{\otimes k}$ is given by $$\mathop{\fam0 Tr}\nolimits(\rho_a^{\otimes k}) = \frac{1+(-1)^k}{2}\left( \frac{k+2}{2}\right)^{g-1}.$$ \end{proposition} \noindent {\bf Proof.} The morphism $q$ induces a $J^{(2)}$-equivariant morphism $$q^* : H^0(M,{\mathcal{L}}^k) \rightarrow H^0({\mathcal P},q^*{\mathcal{L}}^k).$$ According to \cite{BSz} we have that this morphism is an isomorphism and that $$H^i({\mathcal P},q^*{\mathcal{L}}^k) = 0,$$ for $i>0$. Hence we just need to apply the Lefschetz-Riemann-Roch fixed point theorem to $({\mathcal P},q^*{\mathcal{L}}^k)$. The fixed point set $|{\mathcal P}|_{\alpha}$ has two components, $|{\mathcal P}|_a^+$ and $|{\mathcal P}|_a^-$, each of which is isomorphic to the Prym variety $P_{\alpha}$, and hence has trivial Todd class. In order to understand $q^*{\mathcal{L}}|_{|{\mathcal P}|_{\alpha}}$ and the conormal bundle $N(|{\mathcal P}|_\alpha)$, consider the following exact sequence $$ 0 \rightarrow (q')^*T^*_{M'} \rightarrow T^*_{\mathcal P} \rightarrow \nu \rightarrow 0,$$ where $\nu$ is the relative cotangent sheaf of $q'$. We conclude that $$K_{\mathcal P} \cong \nu \otimes (q')^*K_{M'}\cong \nu \otimes (q')^* ({\mathcal{L}}')^{-2},$$ since $K_{M'} \cong ({\mathcal{L}}')^{-2}$ \cite{DN}. By cor. 2.2 in \cite{BSz} (see equation (\ref{Hecke}) in the proof of theorem \ref{8.4}), we get that $$q^*{\mathcal{L}}^2 \cong (q')^*{{\mathcal{L}}'} \otimes \nu^{-1}.$$ Now recall from prop. \ref{qtriv} that the line bundle $\nu|_{|{\mathcal P}|_{\alpha}}$ is numerically trivial. It follows that $$Ch(q^*{\mathcal{L}}|_{|{\mathcal P}|_a^{\pm}})=\bigl(Ch((q')^*{{\mathcal{L}}'}|_{|{\mathcal P}|_a^{\pm}})\bigr)^{1/2}=e^\Theta$$ where in the last equality we have used formula (\ref{Thet1}), after identifying both $|{\mathcal P}|_a^+$ and $|{\mathcal P}|_a^-$ with the Prym variety $P_{\alpha}$. Next, observe that $\alpha$ acts as $-1$ on $\nu|_{|{\mathcal P}|_{\alpha}}$, since one has an exact sequence (of bundles over $|{\mathcal P}|_{\alpha}$) \begin{equation} 0\rightarrow (q')^*N_{\alpha} \rightarrow N(|{\mathcal P}|_\alpha)\rightarrow \nu|_{|{\mathcal P}|_\alpha}\rightarrow 0.\label{exse} \end{equation} where $N_{\alpha}=N(|M'|_\alpha)$ as before. Therefore $${\widetilde {Ch}}(\lambda_{-1}\nu|_{|{\mathcal P}|_a^{\pm}})={\widetilde {Ch}}(1-\nu|_{|{\mathcal P}|_a^{\pm}})=Ch(1+1)=2,$$ and the exact sequence (\ref{exse}) gives us that $${\widetilde {Ch}}(\lambda_{-1}(N(|{\mathcal P}|_a^{\pm})) )= 2\, {\widetilde {Ch}}(\lambda_{-1}(N_{\alpha}))=2\cdot 2^{2(g-1)}e^{-2\Theta}, $$ where we have used formula (\ref{Thet2}) in the last equality, after again identifying $|{\mathcal P}|_a^+$ and $|{\mathcal P}|_a^-$ with $P_{\alpha}$. Now recall that $\rho_a$ acts with opposite signs on the restriction of $q^*{\mathcal{L}}$ to the two components $|{\mathcal P}|_a^+$ and $|{\mathcal P}|_a^-$. In fact, it acts as $\mp 1$ over $|{\mathcal P}|_a^{\pm }$, by prop. \ref{q-+}. Therefore $${\widetilde {Ch}}(q^*{\mathcal{L}}|_{|{\mathcal P}|_a^{\pm }})=\mp Ch(q^*{\mathcal{L}}|_{|{\mathcal P}|_a^{\pm }})=\mp e^\Theta.$$ Putting everything together, the fixed point formula gives $$\mathop{\fam0 Tr}\nolimits(\rho_a^{\otimes k}) = (1 + (-1)^k) e^{k\Theta} \, 2^{-1} 2^{-2(g-1)} e^{2\Theta} \cap [P_\alpha].$$ The proposition now follows as in the twisted case from formula (\ref{Thet3}).

9710

alg-geom/9710018

en

https://arxiv.org/abs/alg-geom/9710018

alg-geom/9710018

Sandra DiRocco

Sandra Di Rocco

Generation of $k$-jets on Toric Varieties

14M25, 14J60, 14C20(14C25, 14E25), 17 pages, AmsLatex, see home page http://www.math.kth.se/~sandra/Welcome

In this notes we study $k$-jet ample line bundles $L$ on a non singular toric variety $X$, i.e. line bundles with global sections having arbitrarily prescribed $k$-jets at a finite number of points. We introduce the notion of an associated $k$-convex $\D$-support function, $\psi_L$, requiring that the polyhedra $P_L$ has edges of length at least $k$. This translates to the property that the intersection of $L$ with the invariant curves, associated to every edge, is $\geq k$. We also state an equivalent criterion in terms of a bound of the Seshadri constant $\e(L,x)$. More precisely we prove the equivalence of the following: (1) $L$ is $k$-jet ample; (2) $L\cdot C\geq k$, for any invariant curve $C$; (3) $\psi_L$ is $k$-convex; (4) the Seshadri constant $\e(L,x)\geq k$ for each $x\in X$.

alg-geom

\section*{Introduction} The notion of $k$-jet ampleness has been introduced by Demailly to describe line bundles $L$ whose global sections can have arbitrarily prescribed $k$-jets at every single point $x\in X$, see \cite{Dem}. Beltrametti and Sommese generalized it by considering $k$-jets supported on a finite number of points. \vskip6pt {\em A line bundle $L$ is said to be $k$-jet ample on $X$ if for any collection of $r$ points, $(x_1,\cdots,x_r)$, and any $r$-ple of positive integers $(k_1,\cdots,k_r)$, with $\sum k_i=k+1$ the natural map $$H^0(X,L)\to H^0(L\otimes{\cal O}_X/{\frak m}_{x_1}^{k_1}\otimes \cdots \otimes {\frak m}_{x_r}^{k_r})$$ is surjective}. \vskip6pt Notice that $0$-jet ampleness is equivalent to being spanned by global sections and $1$-jet ampleness is equivalent to being very ample. During the last years many results on $k$-jet ample line bundles on surfaces have been established, \cite{BeSok, EiLa, BaSz, BaDRSz}. Up to our knowledge the problem is still quite open for higher dimensional varieties, besides few cases like $\pn{n}$ and Fano varieties \cite{BeSok, BeDRSo}. In \cite{Cox1} D. Cox has introduced ``homogeneous coordinates" on a toric variety $X(\Delta)$. For the points invariant under the torus action the situation looks similar to the projective space case. Using this system of local coordinates we give a description of the fibers of the $k$-jet bundle $J_k(L)$ on fixed points, see section \ref{kjet}. According to Oda and Demazure a line bundle $L$ on $X(\Delta)$ is generated by global sections (respectively very ample) if the $\Delta$-support function $\psi_L$ is {\em convex (respectively strictly convex)}. This suggests to use a ``higher convexity" property for $\psi_L$ in the cases $k\geq 2$.\\ In section \ref{kconvex} we introduce the notion of a {\em $k$-convex $\Delta$-support function}, which for $k=0,1$ agrees with being convex or strongly convex.\\ The $\Delta$-support function $\psi_L$ is $k$-convex if the polyhedra $P_L$, associated to $L$, has edges of length at least $k$. This translates to the property that the intersection of $L$ with the invariant curves, associated to every edge, is $\geq k$, which is a generalization of the toric Nakai criterion for ample line bundle. A key step in the proof is the reduction to the case where the considered points are invariant under the torus action. We are grateful to T. Ekedahl for suggesting to use the Borel's fixed point theorem, and for pointing out the sufficiency of the reduction argument. Our result states that in order to check the $k$-jet ampleness of a line bundle $L$ it is enough to have a bound on the intersection $L\cdot C$, for all the invariant curves $C$. This can be applied to the study of ``local positivity". In section \ref{applications} we report a series of results on blow-ups and higher adjoint bundles, which in the toric case can be shown by means of a direct checking on intersections. We also state an equivalent criterion for $k$-jet ampleness in terms of a bound of the Seshadri constant $\epsilon(L,x)$. This can be thought as a toric version of the Seshadri criterion for ample line bundles, generalized to $k$-jet ampleness. In this paper we prove:\\ {\em Let $L$ be a line bundle on a non singular toric variety $X(\Delta)$, then the following statements are equivalent: \begin{itemize} \item $L$ is $k$-jet ample; \item $L\cdot C\geq k$, for any $T$-invariant curve $C$, [Proposition \ref{inter}]; \item $\psi_L$ is $k$-convex, [Theorem \ref{main}]; \item the Seshadri constant $\epsilon(L,x)\geq k$ for each $x\in X$, [Proposition \ref{sesh}]. \end{itemize}} \vskip6pt It is a pleasure to acknowledge valuable discussions with T. Ekedahl, M. Boij, D. Laksov and D. Cox. This research was initiated during the author's stay at Mittag-Leffler Institute and KTH and reached a final stage at the Max Planck Institute. To all those institutions we owe thanks for their support. \section*{ Notation} We will use standard notation in Algebraic Geometry. The groundfield will always be the field of complex numbers.\\ By abuse of terminology the words line bundle and Cartier divisor will be used with no distinction, as well as the multiplicative and additive structure.\\ For basic notions on toric varieties we refer to \cite{Fu, Oda, Ew} and for a nice survey on the resent progress on toric geometry we refer to \cite{Cox2}.\\ When not stated $X$ will always denote a smooth $n$-dimensional toric variety and $L$ a line bundle on it.\\ \section{Toric Varieties} Let $N$ be an $n$-dimensional lattice and $\Delta=\cup \sigma_i$ be a complete and regular fan, meaning: \begin{itemize} \item $supp(\Delta)=N_{\Bbb R}=N\otimes {\Bbb R}$ and \item for every $r$-dimensional cone $\sigma\in\Delta$, there exists a $\Bbb Z$-basis of $N_{\Bbb R}$, $\{\rho_1,\cdots,\rho_n\}$, such that the subset $\{\rho_1,\cdots,\rho_r\}$ spans $\sigma$. \end{itemize} We will denote by $X=X(\Delta)$ the associated non singular $n$-dimensional toric variety and by $\Delta(t)$ the set of $t$-dimensional cones in $\Delta$.\\ Let $M=Hom_{\Bbb Z}(N,{\Bbb Z})$ be the dual lattice so that $X$ is obtained by gluing together the affine toric varieties $X_{\sigma}=Spec({\Bbb C}[\check{\sigma}\cap M])$, where $\check{\sigma}=\{v\in M_{\Bbb R}:\langle v,\sigma\rangle\geq 0\}$, and $\sigma\in\Delta$.\\ Each $m\in M$ can be viewed as a rational function $\chi^m:T=N\otimes{\Bbb C}^*\to \Bbb C^*.$\\ There is a $1-1$ correspondence between $r$-dimensional cones $\tau\in\Delta(r)$ and $T$-invariant codimension $r$ subvarieties of $X$, which will be denoted by $V(\tau)$.\\ Let $D_i=V(\rho_i)$ be the $T$-invariant divisors corresponding to the one dimensional cones $\rho_i\in\Delta(1)$. The set $\{D_i\}_{\rho_i\in\Delta(1)}$ form a set of generators for the Picard group of $X$ and thus every line bundle $L$ can be written in terms of those principal divisors: $$L=\sum_{\rho_i\in\Delta(1)} a_i D_i$$ We will denote by $P_L$ the associated convex polyhedra: $$P_L=\{m\in M_{\Bbb R}:\langle m, \rho_i\rangle\geq -a_i\}$$ This gives a nice way of expressing the global sections of $L$: $$H^0(X,L)=\bigoplus_{m\in P_L\cap M}{\Bbb C}{\cal X}^m$$ Recently David Cox \cite{Cox1}, has introduced the notion of homogeneous coordinates on a toric variety. There is a $1-1$ correspondence between $T$-invariant principal divisors $D_i$ and linear monomials ${\cal X}_i$ on $X$. The polynomial ring is then defined as: $$S={\Bbb C}[{\cal X}_i:\rho_i\in\Delta(1)]$$ and the grading is given by the group of divisors modulo rational equivalence, $Pic(X)$, i.e. two rationally equivalent divisors $D$ and $E$ are associated to monomials ${\cal X}_D$ and ${\cal X}_E$ of the same degree.\\ Considering the exact sequence: $$0\to M\to \oplus_{\rho_i\in\Delta(1)}{\Bbb Z}\cdot D_i\to Pic(X)\to 0$$ we associate to each $m\in M$ a divisor $\sum \langle m,\rho_i\rangle D_i=div(\chi^m)$.\\ The global sections of $L$ are generated by the monomials of the form $$(\Pi_i {\cal X}_i^{\langle m,\rho_i\rangle+a_i})_{m\in P_D\cap M}$$ The notion of $\Delta$-support function will be use constantly throughout this paper: \begin{definition}\cite[2.1]{Oda} A real valued function $f:\cup_i\sigma_i\to\bf R$ is a $\Delta$-linear support function if it is $\Bbb Z$-valued on $N\cap(\cup_i\sigma_i)$ and it is linear on each $\sigma_i$. \end{definition} This means that for each $\sigma$ there exists $m_\sigma\in M$ such that $f(n)=\langle m_\sigma,n\rangle$ for $n\in\sigma$ and $\langle m_\sigma,n\rangle=\langle m_\tau,n\rangle$ when $\tau$ is a face of $\sigma$. To each divisor $L$ we associate a $\Delta$-support function $\psi_L$ defined by: $$\psi_L(\rho_i):=-a_i$$ \section{$k$-jet Bundles}\label{kjet} Let $\Delta$ be the diagonal in $X\times X$ and $p:X\times X\to X$ the projection onto the first factor. The $k$-th jet-bundle associated to $L$ is the vector bundle associated to the sheaf: $$p^*L/p^*L\otimes {\cal I}_{\Delta}^{k+1}$$ Where ${\cal I}_{\Delta}$ is the ideal sheaf of $\Delta$. It is a vector bundle of rank ${k+n}\choose n$ whose fiber is $$J_k(L)_x=L_x\otimes{\cal O}_X/m_x^{k+1}$$ For details on jet bundles we refer the reader to \cite[Ch.I]{KuSp}. There are natural maps (defined on the sheaf level): $$i_k:L\to J_k(L)$$ sending the germ of a section $s$ at a point $x\in X$ to its $k$-th jet. More specifically for $s\in H^0(X,L)$ $i_k(s(x))\in \bigoplus_1^{{k+n}\choose n}\Bbb C$ is the ${k+n}\choose n$-ple determined by the coefficients of the terms of degree up to $k$, in the Taylor expansion of $s$ around $x$.\\ So if $(x_1,\cdots,x_n)$ are local coordinates around $x_0=(0,0,\cdots,0)$ and $s=\sum c_{i_1,\cdots,i_r}\prod x_i^{i_j}$ then $$i_k(s(x_0))=(\cdots,\frac{\partial s}{\pr{x_1}{t_1}\cdots\pr{x_r}{t_r}},\cdots)|_{x=x_0}=(\cdots,(\text{{\Tiny constant}})\cdot c_{t_1,\cdots,t_r},\cdots) $$ where $t_1+\cdots+t_r\leq k$. For example $i_1(s(x))$ consists in the constant and linear term. The following definition formalizes the property for a linear series $|L|$ on $X$ to generate $k$-jets on one or more points of $X$. When more points are considered $L$ is said to generate ``simultaneous jets" at those points. \begin{definition} Let ${\cal Z}=\{x_1,\cdots,x_r\}$ be a finite collection of distinct points on $X$. $L$ is said to be $k$-jet ample on ${\cal Z}$ (or equivalently the series $|L|$ is said to generate all $k$-jets on ${\cal Z}$) if for any $r$-ple of positive integers $(k_1,\cdots,k_r)$, such that $\sum_1^r k_i=k+1$ the map: $$H^0(X,L)\to H^0(L\otimes {\cal O}_X/{\frak m}_{x_1}^{k_1}\otimes\cdots\otimes {\frak m}_{x_r}^{k_r})$$ is surjective, where ${\frak m}_{x}^k$ is $k$-th tensor power of the maximal ideal sheaf ${\frak m}_x$. $L$ is $k$-jet ample on $X$ if it is $k$-jet ample on each such ${\cal Z}$ in $X$. \end{definition} Clearly from the definition: \begin{itemize} \item We can rewrite the map above as: $$\psi_{{\cal Z}}^{k_1,\cdots,k_r}:H^0(X,L)\to \bigoplus_1^r (J_{k_i-1}(L))_{x_i}$$ defined by $\psi_{{\cal Z}}^{k_1,\cdots,k_r}(s)=(i_{k_1-1}(s(x_1)),\cdots,i_{k_r-1}(s(x_1)))$. We say then that $L$ is $k$-jet ample on $X$ if the map $\psi_{{\cal Z}}^{k_1,\cdots,k_r}$ is surjective for any ${\cal Z}$ and any $(k_1,\cdots,k_r)\in\Bbb Z^r_+$, such that $\sum k_i=k+1$. \item If $L$ is $0$-jet ample then $L$ is generated by its global sections; \item if $L$ is $1$-jet ample then using the sections in $H^0(X,L)$ we can define an embedding $i:X\to \pn{N}$ and thus $L$ is very ample. \end{itemize} Using the homogeneous coordinates introduced in the previous section the $k$-jets at the $T$-invariant points $x(\sigma)=V(\sigma)$ can be better described in terms of the polyhedra associated to $L$.\\ Let $\sigma=\langle \rho_1,\cdots,\rho_n\rangle$, where $\sigma\in\Delta(n)$ and $\rho_i\in\Delta(1)$ are the one dimensional cones generating $\sigma$. The point $x(\sigma)$, lies on the intersection of the divisors $D_i$, $i=1,\cdots,n$: $x(\sigma)\in\cap_1^n({\cal X}_i=0)$. Then the maximal ideal is generated by the linear monomials in ${\cal X}_i$: $${\frak m}_{x(\sigma)}=\langle {\cal X}_1,\cdots,{\cal X}_n\rangle$$ and thus $${\frak m}_{x(\sigma)}^{k+1}=\langle \Pi_{\rho_i\subset\sigma}{\cal X}_i^{t_i}|t_1+\cdots+t_n=k+1\rangle$$ i.e. the generators are the monomials of ``degree$=k+1$" in the variables ${\cal X}_1,\cdots,{\cal X}_n$ ( here by degree we mean the sum of the powers of the variables, i.e. the usual one). \\ Each ${{\cal X}}^m$, generator of $H^0(X,L)$, can be written in the local coordinates $({\cal X}_1,\cdots,{\cal X}_n)$ as follows. Fix $\{\rho_1,\cdots,\rho_n\}$ as basis of $N$ and let $\{m_1,\cdots,m_n\}$ the dual basis. In this coordinate system $m=\sum \langle m,\rho_i\rangle m_i$ and the germ of ${{\cal X}}^m$ at $x(\sigma)$ is: $${{\cal X}}^m|_{x(\sigma)}=\prod_{i=1}^n {\cal X}_i^{\langle m,\rho_i\rangle+a_i}$$ Taking its $k$-th jet means ``killing" all the monomials of degree $\geq k+1$ in the variables ${\cal X}_1,\cdots{\cal X}_n$: $$ i_k({{\cal X}}^m(x(\sigma)))=(\cdots, \frac{\partial{{\cal X}}^m}{\pr{x_1}{t_1}\cdots\pr{x_r}{t_r}},\cdots)|_{x=x(\sigma)}$$ \ex\label{delpezzo} Let $N={\Bbb Z}^2$ and $\Delta$ be the $2$-dimensional fan composed by the following $6$ cones, and their edges:\\ $\sigma_1=\langle (0,1),(1,1)\rangle,\,\sigma_2=\langle (1,1),(1,0)\rangle,\,\sigma_3=\langle (1,0),(0,-1)\rangle\\ \sigma_4=\langle (0,-1),(-1,-1)\rangle,\,\sigma_5=\langle (-1,-1),(-1,0)\rangle,\,\sigma_6=\langle (-1,0),(0,-1)\rangle\\$ $X(\Delta)$ is the equivariant blow up of $\pn{2}$ in the $3$ fixed points, i.e. a Del Pezzo surface of degree $6$.\\ Let $L=D_1+D_2+D_3+D_4+D_5+D_6=-K_{X(\Delta)}$, where the $D_i's$ are associated to the edges in the order given above. Let $\sigma=\langle (0,1),(1,1)\rangle$ and let $\{m_1,m_2\}$ be the basis dual to $\{(0,1),(1,1)\}$.\\ In this basis $P_L$ is the convex hull of the points $$\{(0,1),(1,1),(1,0),(-1,0),(-1,-1),(-1,0)\}$$ and thus the generators of $H^0(X,L)$ are $$\{1,{\cal X}_1,{\cal X}_2,{\cal X}_1{\cal X}_2,{\cal X}_1^2{\cal X}_2,{\cal X}_1{\cal X}_2^2,{\cal X}_1^2{\cal X}_2^2\}$$ Moreover ${\frak m}_{x(\sigma)}^2=\langle {\cal X}_1{\cal X}_2,{\cal X}_1^2,{\cal X}_2^2\rangle$ and then $$J_1(L)_{x(\sigma)}={\Bbb C}\oplus {\Bbb C}{\cal X}_1\oplus {\cal X}_2$$ \ex\label{pn1} Let $X=\pn{n}$, then $\Delta$ is the fan composed by $(n+1)$ $n$-dimensional cones spanned by the $(n+1)$ edges \begin{itemize} \item $\rho_i=(0\cdots,0,\underbrace{1}_{i-th},0,\cdots,0)$ for $i=1,\cdots n$ \item $\rho_{n+1}=(-1,\cdots,-1)=-rho_1-\cdots -\rho_n$ \end{itemize} Let $D_1,\cdots,D_{n+1}$ be the associated $T$-invariant principal divisors and let $L=D_1+\cdots+D_k={\cal O}_{\pn{n}}(k)$.\\ recall that the Picard group is generated by one principal divisor $D_i$ and that $D_i\equiv D_j$ for $i\neq j$. So we can think of $L$ as $$L=t_1D_1+\cdots+t_nD_n;\,\,t_1+\cdots+t_n=k$$ Let $\sigma=\langle\rho_1,\cdots\rho_n\rangle$, and fix the basis $\{\rho_1,\cdots,\rho_n\}$ with dual $\{m_1,\cdots,m_n\}$. In this basis the polyhedra $P_L$ is the convex hall of the $(n+1)$ points $$\{ (-1,\cdots,-1),(k,-1,\cdots,-1),\cdots,(-1,\cdots,-1,k,-1,\cdots,-1),\cdots,(-1,\cdots,k)\}$$ Then for any decomposition $t_1,\cdots,t_n$ of positive integers such that $\sum_1^n(t_1+1)=k$ the lattice point $m=\sum _1^n t_1m_1\in P_L$ and any lattice point $m\in P_L$ can be written in this form. The situation stays the same if we consider another $\sigma\in\Delta$. It follows that $$J_k(L)_{x(\sigma)}=H^0(X,L)=\bigoplus_{t_1+\cdots+t_r=k}{\Bbb C}\prod_1^r{\cal X}_1^{t_i}$$ In particular if $L={\cal O}(1)$ then $J_1(L)$ is trivial. This is in fact a characterization of the projective space, cf.\cite{So}. \section{ $k$-convex functions}\label{kconvex} In order to study positivity properties of line bundles Demazure and Oda introduced the definition of convex and strictly convex $\Delta$-support function. \begin{theorem}\cite[Th. 2.13]{Oda} A line bundle $L$ on $X$ is globally generated ( i.e. $0$-jet ample) if and only if $\psi_L$ is convex and it is very ample (i.e. $1$-jet ample) if and only if $\psi_L$ is strictly convex. \end{theorem} A natural way of generalizing such a criterion to higher jets is to introduce a definition of ``higher convexity". \begin{definition} Let $\psi$ be a $\Delta$-linear support function with $\psi(v)=\langle m_\sigma, v\rangle$ for each $v\subset\sigma\in\Delta$. we will say that $\psi$ is {\em $k$-convex} if for any $\sigma\in\Delta$ and $v\not\subset\sigma$ $$\langle m_\sigma,v\rangle\geq \psi(v)+k $$ \end{definition} Confronting the notion of convex and strictly convex function, see \cite{Oda}, it is clear that \begin{itemize} \item $\psi$ is $0$-convex if and only if it is convex; \item $\psi$ is $1$-convex if and only if it is strongly convex. \end{itemize} \rem\label{additive}It is clear from the definition that: \begin{itemize} \item If $\psi$ is $k$ convex then it is $t$-convex for any $t\leq k$; \item If $\psi_1$ is $t_1$-convex and $\psi_2$ is $t_2$-convex, then $(\psi_1+\psi_2)$ is $(t_1+t_2)$-convex. \end{itemize} \vskip10pt The meaning of convexity and strong convexity of a $\Delta$-support function associated to a line bundle $L$ is quite clear at least at the fixed points $x(\sigma)\in\Delta$. If $\psi_L$ is convex then: $${{\cal X}}^{m_{\sigma}}(x(\sigma))=\prod_1^n {\cal X}_i^{a_1-a_i}\neq 0$$ If $\sigma=\langle \rho_1,\cdots,\rho_n\rangle$ and $\sigma'=\langle \rho_0,\rho_2,\cdots,\rho_n\rangle$ then $${{\cal X}}^{m_{\sigma'}}(x(\sigma))=\prod_2^n {\cal X}_i^{a_1-a_i}{\cal X}_1^{\langle m_{\sigma'},\rho_1\rangle+a_1}=0$$ in the case $\langle m_{\sigma'},\rho_1\rangle+a_1>0$, i.e. $\psi_L$ strictly convex. In other words if $\psi_L$ is convex then for each invariant point there is a non vanishing section, and $\psi_L$ strictly convex implies that different invariant points can be separated.\\ The notion of $k$-convexity generalizes the above property to more points with possible multiplicities. More geometrically a $\Delta$-support function $\psi$ is $k$-convex if for each $\sigma\in\Delta$ the graph of the defining linear function $\langle m_{\sigma},\,\,\rangle$ is ``very" high compared to the graph of $\psi$.\\ Recall that if $\psi_L$ is convex then the polyhedra $P_L$ is the convex hull of the points $m_{\sigma}$ in $M_{\Bbb R}$. If $\psi_L$ is strictly convex then there is a correspondence between the faces in $\Delta$ and the set of non empty faces of $P_L$ (cf. \cite[2.12]{Oda}). Any face $F\subset P_L$ corresponds to $$F^*=\{n\in N_{\Bbb R}|<m,n>=\psi_L(n),\text{ for any }m\in F\}\in\Delta$$ and any cone $\sigma\in\Delta$ corresponds to $$\sigma^*=\{m\in M_{\Bbb R}|<m,n>=\psi_L(n),\text{ for any }n\in \sigma\}\subset P_L$$ Then $\psi_L$ being $k$-convex means that the length of the edges of the polyhedra, corresponding to $\tau=\sigma_i\cap\sigma_j$, are bigger or equal to $k$. This implies that $P_L$ is ``big enough" to choose points in it, corresponding to sections with an arbitrary prescribed jet. \begin{lemma}\label{lenght} Let $P_L$ be the polyhedra associated to $L$ and assume $\psi_L$ is $k$-convex, then \begin{enumerate} \item For each $\tau=\sigma_i\cap\sigma_j\in\Delta(n-1)$ the length of the associated edge $\tau^*$ is\\ $l(\tau^*)=|m_{\sigma_i}-m_{\sigma_j}|=l_{i,j}\geq k$; \item Let $\sigma_1,\cdots,\sigma_r$ be the $n$-dimensional cones in $\Delta$. For each partition \\ $(t_1^1,\cdots,t_n^1,t_1^2,\cdots,t_n^2,\cdots,t_n^r)$ where $t_i^j\geq 0$, $\sum_{j=1}^nt_j^i=k_i-1$ and $\sum_1^rk_i=k+1$, and for any $\sigma_i\in\Delta(n)$ we can find $m\in P_L$ such that \begin{itemize} \item $\langle m, \rho_l\rangle=-a_l+t_l^i$ for all $\rho_j\subset\sigma_i$ \item $\langle m, \rho_l\rangle\geq-a_l+ t_l^j$ for all $\rho_j\not\subset\sigma_i$ with equality only if $t_l=0$ \end{itemize} \end{enumerate} \end{lemma} \begin{pf} of (1). Let $\sigma_i=\langle \rho_1,\cdots,\rho_n\rangle$ and $\sigma_j=\langle \rho_2,\cdots,\rho_{n+1}\rangle$. Then using the basis $\{m_1,\cdots,m_n\}$ dual to $\{\rho_1,\cdots,\rho_n\}$ $m_{\sigma_i}=(-a_1,\cdots,-a_n)$ and $m_{\sigma_j}=(-a_1+l_{i,j},\cdots,-a_n)$, where $l_{i,j}\geq k$ since $\psi_L$ is $k$-convex.\end{pf} \begin{pf} of (2). Fix $\sigma=\langle \rho_1,\cdots,\rho_n\rangle$ for simplicity of notation and let \\$\sigma_i=\langle \rho_1,\cdots,\check{\rho_i},\cdots,\rho_n,\overline{\rho_i}\rangle$ the $n$-cone so that $\sigma\cap\sigma_i=\tau_i=\langle \rho_1,\cdots,\check{\rho_i},\cdots,\rho_n\rangle$. By (1) the edge $\tau_i^*$ has lenght at least $k$. Choose the $t_i$-th lattice point next to $m_{\sigma}$ traveling on $\tau^*$ towards $m_{\sigma_i}$, i.e $$\overline{m_i}=(-a_1,\cdots,-a_i+t_i,\cdots,-a_n)=m_{\sigma}+(\frac {t_i}{l_i})(m_{\sigma_i}-m_{\sigma_i})$$ in the basis dual to $\{\rho_1,\cdots,\rho_n\}$, where $\langle m_{\sigma_i},\rho_i\rangle=-a_i+l_i\geq -a_i+k$ by hypothesis. Traveling on the $n$ edges next to $m_{\sigma}$ we get $$m=m_{\sigma}+\sum_1^n(\frac {t_i}{l_i})(m_{\sigma_i}-m_{\sigma})$$ Rewriting it in the form $$m=(1-\sum_1^n(\frac {t_i}{l_i}))m_{\sigma}+\sum_1^n(\frac {t_i}{l_i})m_{\sigma_i}$$ It is clear that $m$ is a convex combination of $\{m_{\sigma},m_{\sigma_1},\cdots,m_{\sigma_n}\}$, since \\$0\leq \sum_1^n(\frac {t_i}{l_i})\leq \frac{1}{k}\sum_1^n t_i\leq 1$ and therefore $m\in P_L=Conv(m_{\sigma})_{\sigma\in\Delta(n)}$. Moreover \begin{itemize} \item $\langle m,\rho_i\rangle=-a_i+t_i$ for $i=1,\cdots,n$ \item if $\rho_l\not\subset\sigma$ then \begin{align*} \langle m,\rho_l\rangle&=(1-\sum (\frac{t_i}{l_i}))\langle m_{\sigma},\rho_l\rangle+\sum [(\frac{t_i}{l_i})\langle m_{\sigma_i},\rho_l\rangle]\\ &> (1-\sum (\frac{t_i}{l_i}))(-a_l+k)+\sum [(\frac{t_i}{l_i})(-a_l)]\\ &=-a_l-k(\sum \frac{t_i}{l_i})+k\geq -a_l+(k-\sum t_i) \end{align*} If $k_j=0$ for all $j\neq i$ then $\sum t_i=k_i-1=k$ and thus $\langle m,\rho_l\rangle\geq -a_l.$ Otherwise $k-\sum t_i<t_l$ and thus $\langle m,\rho_l\rangle>-a_l+t_l.$ \end{itemize} \end{pf} \vskip10pt \section{The Main Result} Let us first formulate an equivalent criterion for $\psi_L$ to be $k$-convex in terms of the intersections of the divisor $L$ with the $T$-invariant rational curves associated to each $\tau\in\Delta(n-1)$.\\ In fact the polyhedra $P_L$ having edges of length at least $k$ translates to the restriction of $L$ to each curve, corresponding to such edges, being at least $k$.\\ This is in a way a generalization of the ``toric Nakai criterion", cfr. \cite[Th. 2.18]{Oda}. \begin{proposition}\label{inter} Let $L$ be a line bundle on a smooth $n$-dimensional toric variety $X$. Then $\psi_L$ is $k$-convex if and only if the restriction $L|_{V(\tau)}$ has degree $\geq k$, for every $\tau\in\Delta(n-1)$.\\ Equivalently if and only if $L_{V(\tau)}={\cal O}_{\pn{1}}(a)$ with $a\geq k$ for every $\tau\in\Delta(n-1)$. \end{proposition} \proof Let $\tau=\sigma_0\cap\sigma_1$ and assume $\sigma_i=\langle \tau, n_i\rangle$ for $i=0,1$. Then, since we are assuming $X$ to be non singular, there exists a $\Bbb Z$-basis $\langle n_i, n_2,\cdots,n_n\rangle$ and $(n-1)$ integers $(s_2,\cdots,s_n)$ such that: $$n_0+n_1-\sum_2^{n-2}s_in_i=0$$ Write $L=-\sum_i \psi_L(n_i)D_i$ where $D_1, D_0$ are the principal divisors associated to the edges $n_0, n_2$ and $D_i$ are the ones associated to $n_i$, $i=2,\cdots,n$. then \begin{itemize} \item $D_1\cdot V(\tau)=D_0\cdot V(\tau)=1$ \item $D_i\cdot V(\tau)=-s_i$ for $i=2,\cdots,n$ \item $D_j\cdot V(\tau)=0$ otherwise \end{itemize} \[\begin{array}{ll} L\cdot V(\tau)&=\sum (-\psi_L(n_i))D_i\cdot V(\tau)=\\ &=-\psi_L(n_0)-\psi_L(n_1)+\sum_2^n \psi_L(n_i)s_i\\ &=\langle m_{\sigma_1}, n_0\rangle -\psi_L(n_0) \end{array}\] It follows that $L\cdot V(\tau)\geq k$ for all $\tau\in\Delta(n-1)$ if and only if for any $\sigma\in\Delta$ and $\langle\rho_j, \sigma\cap\sigma'\rangle=\sigma'$ the inequality $$\langle m_{\sigma}, \rho_j\rangle -\psi_L(\rho_j)\geq k$$ In other words $L\cdot V(\tau)\geq k$ if and only if the support function $\psi_L$ is $k$-convex.\qed The relation between $k$-convex $\Delta$-support functions and $k$-very ampleness is given by the following: \begin{theorem}\label{main} A line bundle $L$ generates $k$-jets on $X$ if and only if the $\Delta$-support function $\psi_L$ is $k$-convex. \end{theorem} We need the following reduction step:\\ \noindent{\rm C\sc laim} {\em If $L$ is $k$-jet ample on any $r$-ple of fixed points $\{x(\sigma_1),\cdots,x(\sigma_r)\}$, $\sigma_i\neq \sigma_j$, then it is $k$-jet ample on $X$.} \begin{pf}{\rm \sc of the claim.} To every $(x_1,\cdots,x_{k+1})\in X^{k+1}$ we can associate ${\cal Z}=(x_1,\cdots,x_r)$, a collection of $r\leq k+1$ distinct points of $X$, and $(k_1,\cdots,k_r)$, an $r$-ple of positive integers such that $\sum k_i=k+1$, simply counting the multeplicities of each $x_i$: $$(\underbrace{x_1,\cdots,x_1}_{k_1},\underbrace{x_2,\cdots,x_2}_{k_2},\cdots,\underbrace{x_r,\cdots,x_r}_{k_r})=[{\cal Z}=(x_1,\cdots,x_r),(k_1,\cdots,k_r)]$$ Then to each $\underline{x}\in X^{k+1}$ we can associate the map: $$\psi_{\underline{x}}=\psi_{{\cal Z}}^{k_1,\cdots,k_r}:H^0(X,L)\to \bigoplus_1^r (J_{k_{i-1}}(L))_{x_i} $$ Let $C=\{\underline{x}\in X^{k+1}\text{ such that }coker(\psi_{\underline{x}})\neq 0\}$. Since $\psi_{\underline{x}}$ is an equivariant map, $C$ inherits the torus action from $X$, i.e. it is an invariant closed subvariety of $X^{k+1}$, and hence proper. If $L$ is not $k$-jet ample on $X$, then $\psi_{\underline{x}}$ is not surjective for some $\underline{x}=[{\cal Z},(k_1,\cdots,k_r)]$, which means $C\neq\emptyset$. But this implies $C^{T}\neq\emptyset$, where $C^{T}$ is the set of the fixed points in $C$. To see this one can apply Borel's fixed point theorem ( see \cite[21.1]{Hum}) or more directly observe that $C$ is a lower dimensional toric variety and thus it must contain fixed points. It follows that there exists $\underline{x}\in X^{k+1}$, fixed by the torus action, for which $\psi_{\underline{x}}$ is not surjective. Such $\underline{x}$ must have all the components fixed so it is of the form $(x(\sigma_1),\cdots,x(\sigma_r))$, which is a contradiction.\end{pf} \begin{pf}{\sc of the theorem.} If $L$ is a $k$-jet ample line bundle then the restriction $L|_{V(\tau)}$ to every $\tau\in\Delta(n-1)$ is $k$-jet ample, i.e. $L_{V(\tau)}={\cal O}_{\pn{1}}(a)$ with $a\geq k$ for every $\tau\in\Delta(n-1)$. Proposition \ref{inter} then implies that $\psi_L$ is $k$-convex.\\ Assume now that $\psi_L$ is $k$-convex. By the reduction step it suffices to prove that the map $\psi_{{\cal Z}}^{(k_1,\cdots,k_r)}$ is surjective for each ${\cal Z}=\{x(\sigma_1),\cdots,x(\sigma_r)\}$, with $k_1+\cdots+k_r=k+1$. This follows immediately from Lemma \ref{lenght}. For each $k_i$ and for each partition $t_1^i+\cdots+t_n^i=k_{i}-1$ we can choose $m\in P_L$ such that \begin{itemize} \item ${\cal X}^{m}=\prod_{\rho_j\subset\sigma_i}{\cal X}_j^{t_j^i}$ around $x(\sigma_i)$ and \item ${\cal X}^{m}=\prod_{\rho_j\subset\sigma_l}{\cal X}_j^{t_j^l+c_j^l}$ around $x(\sigma_l)\neq x(\sigma_i)$, with $c_j^l>o$ for some $j$ and for any partition $\sum t_j^l\leq k_l$ \end{itemize} This means that $$(i_{k_i}( {\cal X}^{m}(x(\sigma_1)),\cdots,i_{k_r}( {\cal X}^{m}(x(\sigma_r)))=(0,\cdots,1,0,\cdots,0)$$ the non zero term corresponding to $\frac{\partial {\cal X}^{m}}{\pr{x_1}{t_1}\cdots\pr{x_r}{t_r}}|_{x=x(\sigma_i)}$. Enough to prove the surjectivity.\end{pf} \rem\label{add}From \ref{additive} it follows immediately that: \begin{itemize} \item if $L$ is $k$-jet ample then it is $t$-jet ample for any $t\leq k$; \item If $L$ is $k$-jet ample and $E$ is $t$-jet ample then the line bundle $E\otimes L$ is $(k+t)$-jet ample. In fact $\psi_{E\otimes L}=\psi_L+\psi_E$. \end{itemize} It is worth observing that in the toric case the notion of $k$-jet ampleness is equivalent to the notion of $k$-very ampleness (which is weaker in general). For basic properties of $k$-very ample line bundles we refer to \cite{BeSoB}. \begin{definition} $L$ is said to be $k$-very ample if for every zeroscheme $({\cal Z},{\cal O}_{{\cal Z}})$ of length $h^0({\cal O}_{{\cal Z}})=k+1$ the map $H^0(X,L)\to H^0({\cal Z},L\otimes{\cal O}_{{\cal Z}})$ is onto. \end{definition} \begin{proposition} A line bundle $L$ on a smooth toric variety $X$ is $k$-very ample if and only if it is $k$-jet ample. \end{proposition} \proof If $L$ is $k$-jet ample then it is $k$-very ample ( see \cite[Prop. 2.2]{BeSok}).\\ If $L$ is $k$-very ample then the degree of $L$ restricted to any irreducible curve must be $\geq k$, i.e. $L\cdot V(\tau)\geq k$ for all $\tau\in\Delta(n-1)$ and thus it is $k$-jet ample by \ref{inter}. \qed \section{Examples} In this section we work out few examples, for which the $k$-jet ampleness has been studied otherwise, to convince the reader that this is in fact the right way to formulate the result. \ex\label{pn}{\sc The projective space $\pn{n}$.} Notation as in \ref{pn1}. Let $L=t_1D_1+\cdots+t_{n+1}D_{n+1}\cong (t_1+\cdots+t_{n+1})D_1$. By \ref{inter} $L$ is $k$-jet ample if and only if $V({\sigma_i})\cdot L\geq k$ where $\sigma_i$ is the n-dimensional cone $\langle \rho_1,\cdots\check{\rho_i},\cdots,\rho_{n+1}\rangle$ and $D_i$ is the divisor associated to the edge $\rho_i$. Since $\rho_{n+1}+\rho_i+\sum_{j\neq i}\rho_j=0$ $$V({\sigma_i})\cdot L=t_1+\cdot+t_{n+1}\geq k$$ In other words ${\cal O}_{\pn{N}}(a)$ is $k$-jet ample if and only if it is $k$-very ample if and only if $a\geq k$, as proven in \cite{BeSok}. \vskip10pt \ex\label{fn}{\sc The Hirzebruch surface ${\Bbb F}_n$.} Let $\{e_1,e_2\}$ be the standard basis for ${\Bbb R}^2$. The Hirzebruch surface ${\Bbb F}_n$ is the toric surface associated to the fan $\Delta$ spanned by the following $2$-cones: $$\sigma_1=\langle e_1,e_2\rangle,\,\sigma_2=\langle e_2,-e_1\rangle,\,\sigma_3=\langle -e_2,-e_1+ne_2\rangle,\,\sigma_4=\langle -e_1+ne_2,e_2\rangle$$ Let $D_1,\cdots,D_4$ be the divisors associated respectively to $e_2,e_1,-e_2,-e_1+ne_2$. Then we have the following intersection matrix: \[\begin{pmatrix}D_1^2&D_1\cdot D_2&D_1\cdot D_3&D_1\cdot D_4\cr D_2\cdot D_1&D_2^2&D_2\cdot D_3&D_2\cdot D_4\cr D_3\cdot D_1&D_2\cdot D_3&D_3^2&D_3\cdot D_4\cr D_4\cdot D_1&D_2\cdot D_4&D_4\cdot D_3&D_4^2\cr \end{pmatrix}= \begin{pmatrix}-n&1&0&1\cr 1&0&1&0\cr 0&1&n&1\cr 1&0&1&0\cr \end{pmatrix}\] Recall that $D_3\equiv D_1+nD_2$ and $D_2\equiv D_4$. Let $L=a_1D_1+\cdots+a_2D_4=(a_1+a_3)D_1+(a_4+a_2+na_3)D_2$. Then by \ref{inter} $L$ is $k$-jet ample if and only if it is $k$-very ample if and only if \begin{itemize} \item $L\cdot D_1=a_4-na_1+a_2\geq k$ \item $L\cdot D_2=a_1+a_3\geq k$ \item $L\cdot D_3=a_2+na_3+a_4\geq k$ \item $L\cdot D_4=a_1+a_2\geq k$ \end{itemize} which of course is equivalent to saying that $L=aE_o+bf$, where $E_0$ is the section of minimal selfintersection $-n$ ( i.e. $D_1$) and $f$ the general fiber of the projection onto $\pn{1}$ (i.e. $D_2$), is $k$-jet ample if and only if \begin{itemize} \item $a=a_1+a_3\geq k$ \item $-an+b=-na_1-na_3+a_4+na_3+a_2=a_4-na_1+a_2\geq k$ \end{itemize} This conditions have been given by Beltrametti-Sommese in \cite{BeSok}, using a decomposition argument. \vskip10pt \ex{\sc Del Pezzo surfaces.} The toric Del Pezzo surfaces are $\pn{2}$, ${\Bbb F}_1$ and the equivariant blow up of $\pn{2}$ in $2$ or $3$ points. The most interesting one is the last one. Let $S$ be the equivariant blow up of $\pn{2}$ in the $3$ invariant points as described in \ref{delpezzo}. The principal divisors $D_1,\cdots,D_6$ are the $6$ $(-1)$-curves on the surfaces, i.e. the three exceptional divisors and the pull back of the three lines passing through two of the $3$ points blown up.\\ Proposition \ref{inter} says that $L$ is $k$-jet ample if and only if it is $k$-very ample if and only if the intersection with all the $(-1)$-curves on the surface is $\geq k$.\\ This criterion has been given for $k$-very ampleness in \cite{DR} using a generalization of Reider's theorem.\\ Note that the equivalence between $k$-jet ampleness and $k$-very ampleness is not always true for Del Pezzo surfaces. In fact it is not hard to see that if $S$ is the blow up of $\pn{2}$ in $7$ points in general position ( so it is not toric), then the line bundle $L=-2K_S$ is $2$-very ample but it is not $2$-jet ample. \section{ Local positivity applications}\label{applications} In this section we report some nice applications of $k$-jet ampleness to the study of ``local positivity" of line bundles. Most of it is a survey on well known results. We think it is interesting to show how these results can be established rather easily in the case of toric varieties. \vskip10pt \noindent{\sc Blow ups}. A $k$-jet ample line bundle carries its positivity along blow ups at a finite number of points. The following property has been proved by Beltrametti and Sommese in \cite{BeSo96} and it has a very ``visible" proof in the toric case. We refer to \cite{Oda} for notation and definition of equivariant blow ups. \begin{proposition}\label{blowup} Let $p:X(\Delta')\to X(\Delta)$ be the equivariant blow up of $X(\Delta)$ at $r$ points, $x_1,\cdots,x_r$, and let $L$ be a $k$-jet ample line bundle on $X(\Delta)$. Then $p^*(L)-\sum\epsilon_iE_i$ is min$(k-\sum\epsilon_i,\epsilon_1,\cdots,\epsilon_r)$-jet ample on $X(\Delta)$, where $E_i$'s are the exceptional divisors. \end{proposition} \begin{pf}Use induction on $r$. Assume the number of edges in $\Delta$ is $N$. \\ If $r=1$, let $x=x(\sigma)$, $\sigma=\langle \rho_1,\cdots,\rho_n\rangle$ and $E_1=E$ be the divisor associated to the edge $\overline{\rho}=\rho_1+\cdots+\rho_n$. In the new fan $\Delta'$ there are $n$ new $n$-cones $\sigma_i=\langle \overline{\rho},\rho_1,\cdots,\check{\rho_i},\cdots,\rho_n\rangle$. Moreover let $\overline{D_i}$ be the divisors in $Pic(X(\Delta'))$ corresponding to the edges $\rho_i$, then \begin{itemize} \item $\overline{D_i}=p^*(D_i)-E$ for $i=1,...,n$; \item $\overline{D_i}=D_i$ for $i=n+1,...,N$. \end{itemize} If $L=\sum a_i D_i$ then $${\cal H}=p^*(L)-\epsilon E=\sum_1^N a_ip^*(D_i)-\epsilon E=\sum_1^N a_i\overline{D_i}-(\epsilon+\sum_i^n a_i)E$$ Let $\tau\in\Delta'(n-1)$. If $\tau\in\Delta(n-1)$ then clearly ${\cal H}\cdot V(\tau)=L\cdot V(\tau)\geq k$. If $\tau\in\Delta'-\Delta$ then it is one of the following: \begin{itemize} \item[(a)] $\sigma_i\cap\sigma_j=\langle \overline{\rho},\rho_1,\cdots,\check{\rho_i},\check{\rho_j},\cdots,\rho_n\rangle$ \item[(b)] $\sigma_i\cap\langle\rho_{n+1},\rho_1,\cdots,\check{\rho_i},\cdots,\rho_n\rangle$ \end{itemize} Following the lines of \ref{inter}:\\ In case (a), since $\rho_i+\rho_j-\overline{\rho}+\sum_{l\neq i,j}\rho_l=0$ $${\cal H}\cdot V(\tau)=-a_1-a_2+\epsilon+\sum_1^n a_l+ \sum_{l\neq i,j}a_l=\epsilon$$ In case (b), assume $\rho_{n+1}+\rho_i-\sum_{j\neq i}s_j\rho_j=0$, then $\rho_n+1+\overline{\rho}-\sum_{j\neq i}(s_j-1)\rho_j=0$ and $${\cal H}\cdot V(\tau)=-a_{n+1}-\epsilon-\sum_1^n a_j+\sum_{j\neq i}s_j+\sum_{j\neq i}a_i=L\cdot V(\tau')-\epsilon$$ where $\tau'=\langle \rho_1,\cdots,\check{\rho_i},\cdots,\rho_n,\rho_{n+1}\rangle\cap\sigma\in\Delta(n-1)$\\ If $r>1$, iterating this process, after $r$ blow-ups ${\cal H}=p_{r}^*(L')-\epsilon_rE$, where \\$p^*_r:X(\Delta')\to X(\Delta_{r-1})$ is the $r$-th blow up map. By induction $L'$ is min$(k-\sum_1^{r-1}\epsilon_1,\epsilon_r,\cdots,\epsilon_{r-1})$-jet ample on $X(\Delta_{r-1})$. Clearely from what done before $${\cal H}\cdot V(\tau)\geq min(L'\cdot V(\tau')-\epsilon_r,\epsilon_r)\geq min(k-\sum_1^n\epsilon_i,\epsilon_1,\cdots,\epsilon_r)$$ for any $\tau\in \Delta'(n-1)$.\end{pf} \vskip10pt \noindent{\sc Toric Seshadri criterion}. An ample line bundle on a smooth projective variety, $X$, is characterized by the positive value of its Seshadri constant at each point.\\ Let $L$ be a nef line bundle on $X$. For every irreducible curve $C\subset X$, $m_x(C)$ denotes the multiplicity of $C$ at the point $x\in C$ and $$m(C)=\sup_{x\in C}\{m_x(C)\}$$. \begin{theorem}(Seshadri \cite[7.1]{Ha}) A line bundle $L$ on $X$ is ample if and only if there exists $\epsilon>0$ such that $L\cdot C\geq \epsilon\cdot m(C)$ for every irreducible curve $C\subset X$. \end{theorem} As for the Nakai criterion we can generalize the Seshadri criterion to $k$-jet ampleness on toric varieties.\\ Let us first reformulate the Seshadri's theorem in the ``modern languige" of Seshadri constants.\\ For a nef line bundle $L$ the Seshadri constant of $L$ at a point $x\in X$ is the real number $$\epsilon(L,x)=\inf_{x\in C}\frac{L\cdot C}{m_x(C)}=\sup_{\epsilon}\{\epsilon\in {\Bbb R}|p^*(L)-\epsilon L\text{ is nef }\}$$ where the inf is taken over all the irreducible curves containing $x$ and $p$ is the blow-up map of $X$ at $x$. Then one can immediately see that The Seshadri criterion says that $L$ is ample if and only if $\epsilon(L,x)>0$ for every $x\in X$.\\ Demailly showed that the Seshadri constant is a measure of the highest degree jets that can be generated by the global sections of $L$: \begin{proposition}\label{dem}\cite{Dem} Let $s(L,x)$ be the largest integer such that $|L|$ generates $s$-jets at $x$. Then $$\epsilon(L,x)= \limsup_{n\to\infty}\frac{s(nL,x)}{n}$$ \end{proposition} Since every ample line bundle on a toric variety is very ample the toric Seshadri criterion says that $L$ is a ample if and only if $\epsilon(L,x)\geq 1$ for every $x\in X(\Delta)$. More generally: \begin{proposition}\label{ses} A line bundle $L$ on a non singular toric variety $X(\Delta)$ is $k$-jet ample if and only if there exists an $\epsilon\geq k$ such that $$L\cdot V(\tau)\geq \epsilon \cdot m(V(\tau))$$ for every invariant curve $V(\tau)$.\end{proposition} \begin{pf} Assume $L$ is $k$-jet ample, where $k$ is the biggest integer such that the property is true. Then by \ref{dem} $\epsilon(L,x)=k$, since $nL$ is $(nk)$-jet ample by \ref{add}. It follows that $\frac{L\cdot V(\tau)}{m(V(\tau))}\geq k$ for every $\tau\in\Delta$.\\ Assume now that $L\cdot V(\tau)\geq k\cdot m(V(\tau))$ for any $\tau\in\Delta$. Let $\tau=\sigma_i\cap\sigma_j$, we have to prove that $L\cdot V(\tau)\geq k$. Consider the invariant point $x=V(\sigma_i)\in V(\tau)$, clearely $m_{x}(V(\tau))=1$. Then $L\cdot V(\tau)\geq k\cdot m(V(\tau))\geq k\cdot m_x(V(\tau))\geq k$. \end{pf} \begin{corollary}\label{sesh} A line bundle $L$ on a non singular toric variety is $k$-jet ample if and only if $\epsilon(L,x)\geq k$ for every $x\in X$.\end{corollary} \begin{pf} If $L$ is $k$-jet ample then $\epsilon(L,x)=k$ by \ref{dem}.\\ If $\epsilon(L,x)\geq k$ for every $x\in X$ then in particular $L\cdot V(\tau)\geq k \cdot m(V(\tau))$ for every invariant curve $V(\tau)$ and thus $L$ is $k$-jet ample by \ref{ses}.\end{pf} \vskip10pt \noindent{\sc Higher adjoint series}. By use of bounds on the Seshadry constant of $L$ at a sufficiently general point Ein and Lazarsfeld showed that: \begin{proposition}\cite[5.14]{Laz} Let $L$ be an ample line bundle on a smooth surface $S$. Then the adjoint series $|K_S+(k+3)L|$ generates $k$-jets at a sufficiently general point $x\in S$.\end{proposition} The fact that for a line bundle on a toric variety being ample is equivalent to being very ample implies a simple generalization. \begin{proposition} Let $S$ be a non singular toric surface and let $L$ an ample line bundle on it such that $L^2>1$. Then \begin{itemize} \item[(a)] $|K_S+(k+2)L|$ generates $k$-jets at every point $x\in S$; \item[(b)] $|K_S+(2k+2)|$ generates $k$-jets on $S$. \end{itemize} \end{proposition} \begin{pf}(a) By the Nakai toric criterium $L$ is in fact very ample and by \ref{add} $(k+2)L$ is $(k+2)$-jet ample. Let $x\in S$, using the long exact sequence: $$\to H^0(K_S+(k+2)L)\to H^0((K_S+(k+2)L)/{\frak m}_x^{k+1}))\to H^1((K_S+(k+2)L)\otimes {\frak m}_x^{k+1})\to$$ the vanishing of $H^1((K_S+(k+2)L)\otimes {\frak m}_x^{k+1})$ would imply the result. Let $p:\overline{S}\to S$ be the blow up of $S$ at $x$ with $E=p^{-1}(x)$, then by Leray spectral sequence and Serre duality $$H^1((K_S+(k+2)L)\otimes {\frak m}_x^{k+1})=H^1(K_{\overline{S}}+[p^*((k+2)L)-(k+2)E])$$ Kawamata vanishing theorem applies since $p^*((k+2)L)-(k+2)E$ is nef and big and thus $K_S+(k+2)L$ is $k$-jet ample at any point $x\in S$.\\ (b) Consider now simoultaneus jets supported on $\{x_1,\cdots x_r\}\in S$ and $(k_1,\cdots,k_r)\in{\Bbb Z}_+^r$ such that $\sum k_1=k+1$. Let $p:\overline{S}\to S$ the blow up of $S$ at $x_1,\cdots, x_r$ with $E_i=p^{-1}(x_i)$. Using the same exact sequence as above it suffices to prove that: $$H^1((K_S+(2k+2)L)\otimes({\frak m}_{x_1}^{k_1}\otimes\cdots\otimes {\frak m}_{x_r}^{k_r}))=H^1(K_{\overline{S}}+[p^*((2k+2)L)-\sum (k_i+1)E_i])=0$$ Since $(2k+2)L$ is $(2k+2)$-jet ample and $\sum(k_i+1)\leq 2k+2$, $p^*((2k+2)L)-\sum (k_i+1)E_i$ is spanned by \ref{blowup}. Moreover $$(p^*((2k+2)L)-\sum (k_i+1)E_i)^2>(2k+2-\sum(k_1+1))(2k+2+\sum(k_1+1))\geq 0$$ Then Kawamata vanishing theorem applies to gine the nedeed vanishing.\end{pf} \vskip10pt \noindent{\sc The $k$-reduction}. In the case of surfaces we can make some further remarks about the ``$k$-reduction" process (see \cite{BeSo}). Let $S$ be a non singular toric surface and $L$ a $k$-very ample line bundle on it. Since $k$-jet very ampleness and $k$-very ampleness are equivalent we will use freely the property of being $k$-very ample according to the criterion given in \ref{main}. Assume the adjoint bundle is not $k$-very ample i.e., the surface contains $(-1)$-curves whose intersection with $L$ is exactly $k$. If the $k$-adjoint bundle $kK_S+ L$ is nef we can contract down those curves and get the $k$-reduction $(S',L')$. Notice that if $L=-kK_{S}$ then $-K_{S}$ is ample and hence $S$ is a toric Del Pezzo surface. We can then compute directly the nefness of the $k$-adjoint bundle obtaining the same result as in \cite{BeSo}. Notation as in \ref{pn},\ref{fn}. \begin{proposition} Let $L\neq -kK_{S}$ be a $k$-very ample line bundle on $S$. Then $kK_{S}+L$ is nef unless: \begin{itemize} \item $S=\pn{2}$ and $L=aD_1$ with $a<3k$ ; \item $S={\Bbb F}_r$ and $L=aD_1+bD_2$ with $a< 2k$. \end{itemize} \end{proposition} \begin{pf}Let $L=\sum a_iD_i$ and $D_i^2=-s_i$, then $$(kK_S+L)\cdot D_i=L\cdot D_i+k(s_i-2)\geq 0\text{ if }s_i\geq 1$$ Recall that $S$ is isomorphic to $\pn{2}$, ${\Bbb F}_n$ or their equivariant blow up in a finite number of points. If $S$ is minimal then intersecting $L$ with the basic generators of $Pic(S)$ and imposing at least one intersection to be less then $k$ gives the cases in the statement. Assume now $S$ not minimal. If $S=Bl_r(\pn{2})$ (i.e. the blow up of $\pn{2}$ in $r$ points) let $D_1,D_j,D_l$ be the divisors associated to the edges $(0,1),(1,0),(-1,-1)$ respectively. If $r\geq 2$, for each $D_i$ generator of $Pic(S)$, the corresponding weight $D_i^2=-s_i\leq -1$ unless possibly only one among $(s_1,s_j,s_l)$, say $s_1=-1$. But in this case $D_1\equiv D_2+\sum_1^rD_i$, the $D_i$'s being the exceptional divisors, $L\cdot D_1\geq (r+1)k$ and $$(kK_S+L)D_1\geq (r+1)k-3k\geq 0$$ If $r=1$ then $S\cong {\Bbb F}_1$.\\ If $S=Bl_r({\Bbb F}_n)$, let $D_l,D_j,D_3,D_h$ be the divisors corresponding to the edges $(0,1),\\ (1,0),(0,-1),(-1,n)$ respectively. We can assume the weights $s_i\geq 1 $, unless possibly $s_3=-n+s<1$, since $Bl_1{\Bbb F}_0\cong Bl_2(\pn{2})$ and $D_j\equiv -D_h$. But in this case $D_3\equiv D_l+nD_j+\sum_1^{r-s}D_i$, $L\cdot D_3\geq (n+1+r-s)k$ and $$(kK_S+L)D_3\geq (n+r+1-s)k-(n-s+2)k=(r-1)k\geq 0$$\end{pf} \small

9710

alg-geom/9710015

en

https://arxiv.org/abs/alg-geom/9710015

alg-geom/9710015

Zvezdelina E. Stankova-Frenkel

Moduli of Trigonal Curves

69 pages, 34 figures, Latex2e

We study the moduli of trigonal curves. We establish the exact upper bound of ${36(g+1)}/(5g+1)$ for the slope of trigonal fibrations. Here, the slope of any fibration $X\to B$ of stable curves with smooth general member is the ratio $\delta_B/\lambda_B$ of the restrictions of the boundary class $\delta$ and the Hodge class $\lambda$ on the moduli space $\bar{\mathfrak{M}}_g$ to the base $B$. We associate to a trigonal family $X$ a canonical rank two vector bundle $V$, and show that for Bogomolov-semistable $V$ the slope satisfies the stronger inequality ${\delta_B}/{\lambda_B}\leq 7+{6}/{g}$. We further describe the rational Picard group of the {trigonal} locus $\bar{\mathfrak T}_g$ in the moduli space $\bar{\mathfrak{M}}_g$ of genus $g$ curves. In the even genus case, we interpret the above Bogomolov semistability condition in terms of the so-called Maroni divisor in $\bar{\mathfrak T}_g$.

alg-geom

\section*{\hspace*{1.9mm}1. Introduction} \setcounter{section}{1} \setcounter{subsection}{0} \setcounter{lem}{0} \setcounter{thm}{0} \setcounter{prop}{0} \setcounter{defn}{0} \setcounter{cor}{0} \setcounter{conj}{0} \setcounter{claim}{0} \setcounter{remark}{0} \setcounter{equation}{0} \label{introduction} In this paper $\overline{\mathfrak M}_g$ denotes the Deligne-Mumford compactification of the moduli space of smooth curves over $\mathbb{C}$ of genus $g\geq 2$. The boundary locus $\Delta$ of $\overline{\mathfrak M}_g$ consists of nodal curves with finite automorphism groups, which together with the smooth curves are referred to as {\it stable} curves. The locus of hyperelliptic curves will be denoted by $\overline{\mathfrak{I}}_g$, and the closure of the locus of trigonal curves will be denoted by $\overline{\mathfrak{T}}_g$. \medskip The main objects of our study will be families of genus $g$ stable curves, whose general members are smooth. Associated to any such {\it flat and proper} family $f\!:\!X\!\rightarrow\! B$ are three basic invariants $\lambda|_B$, $\delta|_B$ and $\kappa|_B$. We define these in Section~\ref{definition} as divisors on $B$, but for most purposes one can think of them as integers by considering their respective degrees. The invariant $\delta|_B$ counts, with appropriate multiplicities, the number of singular fibers of $X$. The self-intersection of the relative dualizing sheaf $\omega_f$ on $X$ defines $\kappa|_B$, and its pushforward to $B$ is a rank $g$ locally free sheaf, whose determinant is $\lambda|_B$. \smallskip The basic relation $12\lambda|_B=\delta|_B+\kappa|_B$ and the positivity of the three invariants for non-isotrivial families force the {\it slope} $\displaystyle{\frac{\delta|_B}{\lambda|_B}}$ of $X/\!_{\displaystyle{B}}$ to fall into the interval $[0,12)$ (cf.~Sect.~\ref{slope-non-isotrivial}). In fact, Cornalba-Harris and Xiao establish for this slope an exact upper bound of $8+4/g$, which is achieved only for certain hyperelliptic families (cf.~Theorem~\ref{CHX}). However, if the base curve $B$ passes through a {\it general} point of $\overline{\mathfrak{M}}_g$, Mumford-Harris-Eisenbud give the better bound of $6+\on{o}(1/g)$ (cf.~Theorem~\ref{generalbound1}). The families violating this inequality are entirely contained in the closure $\overline{\mathcal{D}}_k$ of the locus of $k$-sheeted covers of ${\mathbf P}^1$, for a suitably chosen $k$. In particular, for $k=2$ we recover the {hyperelliptic} locus $\overline{\mathfrak{I}}_g$, for $k=3$ - the {trigonal} locus $\overline{\mathfrak{T}}_g$, etc. Therefore, higher than the above ``generic'' ratio can be obtained only for families with special linear series, such as $g^1_2$, $g^1_3$, etc. These observations clearly raise the following \medskip \noindent{\bf Question.} {\it According to the possession of special linear series, is there a stratification of $\overline{\mathfrak M}_g$ which would give successively smaller slopes $\delta/\lambda$? What would be the successive upper bounds with respect to such a stratification?} \smallskip The following result, whose proof will be given in the paper, answers this question for an exact upper bound for families with linear series $g^1_3$. \medskip \noindent{\bf Theorem I.} {\it If $f\!:\!X{\rightarrow} B$ is a trigonal nonisotrivial family with smooth general member, then the slope of $X/\!_{\displaystyle{B}}$ satisfies: \begin{equation*} \frac{\delta|_B}{\lambda|_B}\leq \frac{36(g+1)}{5g+1}\cdot \end{equation*} Equality is achieved if and only if all fibers are irreducible, $X$ is a triple cover of a ruled surface $Y$ over $B$, and a certain divisor class $\eta$ on $X$ is numerically zero.} \smallskip To understand the importance of this result and the above question, consider Mumford's alternative description of the basic invariants (cf.~ Sect.~\ref{linebundles}): $\lambda|_B$, $\delta|_B$ and $\kappa|_B$ are restrictions of certain rational divisor classes $\lambda, \delta, \kappa\in \on{Pic}_{\mathbb{Q}}\overline{\mathfrak{M}}_g$. Specifically, $\delta=\delta_0+\cdots+\delta_{[g/2]}$, where $\delta_i$ the class of the boundary divisor $\Delta_i$ of $\overline{\mathfrak{M}}_g$, and $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{M}}_g$ is freely generated by the Hodge class $\lambda$ and the boundary classes $\delta_i$ for $g\geq 3$ (cf.~\cite{Ha2}). Thus, our question about a stratification of $\overline{\mathfrak{M}}_g$ translates into a question about the relations among the fundamental classes of various subvarieties defined by geometric conditions in $\overline{\mathfrak{M}}_g$ . Moreover, such a stratification would provide a link between the {\it global} invariant $\lambda$ (the degree of the Hodge bundle on $\overline{\mathfrak M}_g$) and the {\it locally defined} invariant $\delta$ of the singularities of our families. In the process of estimating the ratio $\delta / \lambda$ we hope to understand the geometry of interesting loci in $\overline{\mathfrak M}_g$, and describe their rational Picard groups. \smallskip Such a program for the hyperelliptic locus $\overline{\mathfrak{I}}_g$ is completed by Cornalba-Harris (cf.~Theorem~\ref{theoremCHPic}), who exhibit generators and relations for $\on{Pic}_{\mathbb{Q}}{\overline{\mathfrak{I}}_g}$. The typical examples of families with maximal ratio of $8+{4}/{g}$ are constructed as blow-ups of pencils of hyperelliptic curves, embedded in the same ruled surface. \smallskip Similar examples for {trigonal families} yield the slope $7+{6}/{g}$, but as Theorem I shows, this ratio is {\it not} an upper bound. This happens because of an ``extra'' {\it Maroni} locus in $\overline{\mathfrak{T}}_g$ (cf.~ Sect.~12). While a general trigonal curve embeds in ${\mathbf F}_0={\mathbf P}^1\times {\mathbf P}^1$ or in the blow-up ${\mathbf F}_1$ of ${{\mathbf P}^2}$ at a point, the remaining trigonal curves embed in other rational ruled surfaces and comprise a closed subset in $\overline{\mathfrak{T}}_g$, called the Maroni locus. The proof of Theorem II, stated below, implies that all trigonal families achieving the maximal bound lie entirely in the Maroni locus, and moreover, their members are embedded in ruled surfaces ``as far as possible from the generic'' ruled surfaces ${\mathbf F}_0$ and ${\mathbf F}_1$. \smallskip The ratio $7+{6}/{g}$, though not the ``correct'' maximum, plays a significant role in understanding the geometry of the trigonal locus, and in describing its rational Picard group. In particular, in a linear relation is established between the Hodge class, the boundary classes on $\overline{\mathfrak{T}}_g$, and a canonically defined vector bundle $V$ of rank 2 on a ruled surface $\widehat{Y}$ (cf.~Sect.~9): \medskip \noindent{\bf Theorem II.} {\it Let $\delta_0$ denote the boundary class in $\overline {\mathfrak{T}}_g$ corresponding to irreducible singular curves, and let $\delta_{k,i}$ be the remaining boundary classes. For any trigonal non-isotrivial family with general smooth member, we have \begin{equation*} (7g+6)\lambda|_B=g\delta_0|_B+\sum_{k,i} \widetilde{c}_{k,i}\delta_{k,i}|_B +\frac{g-3}{2}(4c_2(V)-c_1^2(V)), \end{equation*} where $\widetilde{c}_{k,i}$ is a quadratic polynomial in $i$ with linear coefficients in $g$, and it is determined by the geometry of $\delta_{k,i}$.} \smallskip For example, $\widetilde{c}_{1,i}=3(i+2)(g-i)/2$ corresponds to the boundary divisor $\Delta{\mathfrak{T}}_{1,i}$, whose general member is the join in three points of two trigonal curves of genera $i$ and $g-i-2$, respectively (cf.~Fig.~\ref{Delta-k,i}). \medskip Recall that the vector bundle $V$ is called {\it Bogomolov semistable} if its Chern classes satisfy $4c_2(V)\geq c_1^2(V)$ (cf.~\cite{Bo,Re}). We show in Section~9 the following \medskip \noindent{\bf Theorem III.} {\it For any trigonal nonisotrivial family $X\rightarrow B$ with general smooth member, if $V$ is Bogomolov semistable, then \begin{equation*} \frac{\delta|_B}{\lambda|_B}\leq 7+\frac{6}{g}\cdot \end{equation*}} \medskip In the even genus case, the Maroni locus is in fact a divisor on $\overline{\mathfrak{T}}_g$, whose class we denote by $\mu$. We further recognize the ``Bogomolov quantity'' $4c_2(V)-c_1^2(V)$ as counting roughly four times the number of Maroni fibers in $X$, and deduce \medskip \noindent{\bf Theorem IV.} {\it For even $g$, $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$ is freely generated by all boundary divisors $\delta_0$ and $\delta_{k,i}$, and the Maroni divisor $\mu$. The class of the Hodge bundle on $\overline{\mathfrak{T}}_g$ is expressed in terms of these generators as the following linear combination: \begin{equation*} (7g+6)\lambda|_{\overline{\mathfrak{T}}_g}=g\delta_0+ \sum_{k,i}\widehat{c}_{k,j}\delta_{k,i}+2(g-3){\mu}. \end{equation*}} Consequently, the condition $\eta\equiv 0$ in Theorem I can be interpreted as a relation among the number of irreducible singular curves and the ``Maroni'' fibers in our family: $(g+2)\delta_0|_B=-72(g+1)\mu|_B$, and hence maximal slope families are entirely contained in the Maroni locus of $\overline{\mathfrak{T}}_g$ (cf.~Theorem~\ref{maximalmaroni}). The stated theorems complete the program for the trigonal locus $\overline{\mathfrak{T}}_g$, which was outlined earlier in this section. \smallskip An important interpretation of these results can be traced back to \cite{MHE}, where it is shown that the moduli space $\mathfrak{M}_g$ is of {\it general type}. The $k$-gonal locus $\overline{\mathcal{D}}_k$ is realized in terms of the generating classes as: $[\overline{\mathcal{D}}_k]=a\lambda-b\delta-\mathcal{E}$ for some $a,b>0$, and an effective boundary combination $\mathcal{E}$. Restricting to a general curve $B\subset \overline{\mathfrak{M}}_g$, we have $\overline{\mathcal{D}}_k|_B> 0$, and hence $a\lambda|_B-b\delta|_B>0$. Because of Seshadri's criterion for ampleness of line bundles, in effect, we are asking for all positive numbers $a$ and $b$ such that the linear combination $a\lambda-b\delta$ is ample on $\overline{\mathfrak{M}}_g$. The smaller the ratio $a/b$ is, the stronger result we obtain. In other words, we are aiming at a maximal bound of $\delta /\lambda$, when we think of these classes as restricted to any curve $B\subset \overline{\mathfrak{M}}_g$. In view of this, part of this paper can be described as looking for all {\it ample} divisors on $\overline{\mathfrak{T}}_g$ of the form $a\lambda-b\delta$ with $a,b>0$. Theorem I then gives the necessary condition $(5g+1)a\geq 36(g+1)b$ (compare with \cite{M2,MHE,CH}). Some of the results can be applied to the study of the discriminant loci of a certain type of triple covers of surfaces. \smallskip The methods and ideas for the trigonal case are in principal extendable to more general families of $k$-gonal curves. We refer the reader to Sect.~13 for a general maximal bound for tetragonal curves (for $g$ odd), and conjectures for the maximal and general bounds for any $d$-gonal and other families of stable curves. \bigskip \begin{center}{\sc Acknowledgments}\end{center} \medskip This paper is based on my Ph.D. thesis at Harvard University. Joe Harris, my advisor, introduced me to the problem of finding a stratification of the moduli space with respect to a descending sequence of slopes of one--parameter families. I am very grateful to him for his advice and support throughout my work on the present thesis. I would like to thank Fedor Bogomolov, David Eisenbud, Benedict Gross, Brendan Hassett, David Mumford, Tony Pantev and Emma Previato for the helpful discussions that I have had with them at different stages of the project, as well as Kazuhiro Konno for providing me with his recent results on trigonal families. A source of inspiration and endless moral support has been my husband, Edward Frenkel, to whom goes my gratitude and love. \bigskip \section*{\hspace*{1.9mm}2. Preliminaries} \setcounter{section}{2} \setcounter{subsection}{0} \setcounter{lem}{0} \setcounter{thm}{0} \setcounter{prop}{0} \setcounter{defn}{0} \setcounter{cor}{0} \setcounter{conj}{0} \setcounter{claim}{0} \setcounter{remark}{0} \setcounter{equation}{0} \label{preliminaries} \subsection{Definition of $\lambda|_B,\,\,\delta|_B$ and $\kappa|_B$ in Pic$B$} \label{definition} Let $f:X\rightarrow B$ be a flat proper one-parameter family of stable curves of genus $g$, where $B$ is a smooth projective curve. Assume in addition that the general member of $X$ is {\it smooth} (cf.~Fig.~\ref{family}). \medskip \begin{figure}[h] $$\psdraw{family}{1.5in}{1.5in}$$ \hspace*{3mm} \caption{Trigonal family $f\!:\!X\!\rightarrow\! B$} \label{family} \end{figure} Let $\omega_f=\omega_X\otimes f^*\omega_B^{-1}$ be the relative dualizing sheaf of $f$. Its pushforward $f_*(\omega_f)$ is a locally free sheaf on $B$ of rank $g$, and we set \[\lambda|_B=\lambda_X:=\wedge^gf_*(\omega_f)\in \on{Pic}B\] to be its determinant. The sheaf $f_*(\omega_f)$ is known as the ``Hodge bundle'' on $B$, and $\lambda|_B$ - as the ``Hodge class'' of $B$. In a similar way, we set $\kappa|_B$ to be the self-intersection of $\omega_f$: \[\kappa|_B=\kappa_X:=f_*(c_1^2(\omega_f))\in\on{Pic}B.\] \bigskip The definition of $\delta|_B$, on the other hand, is local and requires some notation. Let $q$ be any singular point of a fiber $X_b$, $b\in B$. Since the general fiber of $X$ is smooth, the total space of $X$ near $q$ is given locally analytically by $xy=t^{m_q}$, where $x$ and $y$ are local parameters on $X_b$, $t$ is a local parameter on $B$ near $b$, and $m_q\geq 1$. (This follows from the one-dimensional versal deformation space of the nodal singularity at $q$.) For any other point $q$ of $X$ we set $m_q=0$. Now we can define \[\delta|_B=\delta_X:=f_*(\sum_{q\in X}m_qq)\in \on{Pic}B.\] By abuse of notation, we shall use the same letters for the line bundles $\lambda|_B, \,\,\kappa|_B$ and $\delta|_B$ and for their respective degrees, e.g. $\lambda|_B=\on{deg}\lambda|_B$. \medskip\noindent{\bf Remark 2.1.} It is possible to define the three basic invariants for a wider variety of families. In particular, dropping the assumption of smoothness of the general fiber roughly means that the base curve $B$ is contained entirely in the boundary locus of $\overline{\mathfrak M}_g$. Since such families are not discussed in our paper, we shall not give here these definitions. The existence, however, of such invariants for any one-parameter family of stable curves will follow from the description of $\lambda,\,\,\delta$ and $\kappa$ as ``global'' classes in $\on{Pic}_{\mathbb Q}\overline{\mathfrak M}_g$ (cf. Sect.~\ref{linebundles}). \medskip\noindent{\bf Remark 2.2.} It is also possible to consider families whose special members are not stable, e.g. cuspidal, tacnodal and other types of singular curves. One reduces to the above cases by applying {\it semistable reduction} (cf.~\cite{FM}). \subsection{The line bundles $\lambda,\,\,\delta$ and $\kappa$ in Pic$_{\mathbb Q}\overline{\mathfrak M}_g$} \label{linebundles} Another way to interpret the classes $\lambda|_B,\,\,\delta|_B$ and $\kappa|_B$ is to think of them as rational divisor classes on $\overline{\mathfrak{M}}_g$. In fact, Mumford shows that such invariants, defined for {\it any} proper flat family $X\rightarrow S$ and natural under base change, induce line bundles in Pic$_{\mathbb Q}\overline{\mathfrak M}_g$. Here follows a rough sketch of the argument (cf.~\cite{M2}). \smallskip Consider $\on{Hilb}^{p(x)}_r$, the Hilbert scheme parametrizing all closed subschemes of ${\mathbf P}^r$ with Hilbert polynomial $p(x)=dx-g+1$ for some $d=2n(g-1)\gg 0$ and $r=d-g$. Let $\mathcal H\subset \on{Hilb}^{p(x)}_r$ be the locally closed smooth subscheme of $n$-canonical stable curves of genus $g$. Then $\overline{\mathfrak M}_g$ is the GIT-quotient of $\mathcal H$ by ${\on{PGL}}_r$. Let \[\rho:\mathcal H\rightarrow \overline{\mathfrak M}_g=\mathcal H/{\on{PGL}}_r\] be the natural surjection, and let $(\on{Pic}\mathcal H)^{{\on{PGL}}_r}$ be the subgroup of isomorphism classes of line bundles on $\mathcal H$ invariant under the action of ${\on{PGL}}_r$. Consider also $\on {Pic}_{\on{fun}}\overline{\mathfrak M}_g$, the group of line bundles on the {\it moduli functor}. An element $L$ of $\on {Pic}_{\on{fun}}\overline{\mathfrak M}_g$ consists of the following data: for any proper flat family $f:X\rightarrow S$ of stable curves a line bundle $L_S$ on $S$ natural under base change. Two such elements are declared isomorphic under the obvious compatibility conditions. Naturally, a line bundle on $\overline{\mathfrak M}_g$ gives rise by pull-back to a line bundle on the moduli functor. In fact, this map is an inclusion with a torsion cokernel, and $\on{Pic}_{\on{fun}}\overline{\mathfrak M}_g$ is torsion free and isomorphic to $(\on{Pic}\mathcal H)^{{\on{PGL}}_r}$: \[\on{Pic}\overline{\mathfrak M}_g\stackrel{\rho^*}{\hookrightarrow} \on{Pic}_{\on{fun}}\overline{\mathfrak M}_g\cong (\on{Pic}\mathcal H)^{{\on{PGL}}_r}.\] Hence we may regard all these groups as sublattices of $\on{Pic}_{\mathbb Q}\overline{\mathfrak M}_g$. In particular, \[\on{Pic}_{\on{fun}}\overline{\mathfrak M}_g\otimes {\mathbb Q}\cong \on{Pic}_{\mathbb Q}\overline{\mathfrak M}_g,\] and any line bundle on the moduli functor can be thought of as a rational class on $\overline{\mathfrak M}_g$. These identifications allow us to make the following \medskip\noindent{\bf Definition 2.1.} In $\on{Pic}_{\mathbb Q} \overline{\mathfrak M}_g$ we define the line bundles $\lambda,\,\,\kappa$ and $\delta$ by \[\lambda=\det\pi_*(\omega_{\mathcal {C}/\mathcal H}),\,\, \kappa=\pi_*c_1(\omega _{\mathcal {C}/\mathcal H})^2,\,\, \delta=\mathcal{O}_{\mathcal H}(\Delta\mathcal H),\] where ${\mathcal{C}}\subset \mathcal H\times {\mathbb P}^r$ is the universal curve over $\mathcal H$, $\pi:\mathcal C \rightarrow \mathcal H$ is the projection map, $\omega_{\mathcal{C}/\mathcal H}$ is the relative dualizing sheaf of $\pi$, and $\Delta\mathcal H\subset \mathcal H$ is the divisor of singular curves on $\mathcal H$. \medskip As defined, $\lambda,\kappa$ and $\delta$ lie in $\on{Pic}_{\mathbb Q}\overline{\mathfrak M}_g$, and as such they are only {\it rational} Cartier divisors on $\overline{\mathfrak M}_g$. In \cite{MHE} one can find examples where $\lambda$ does {\it not} descend to a line bundle on $\overline{\mathfrak M}_g$. On the other hand, it is obvious from which divisor on $\overline{\mathfrak M}_g$ our $\delta$ comes: $\delta=\mathcal O_{\overline{\mathfrak M}_g}(\Delta)$, where $\Delta$ denotes the divisor on $\overline{\mathfrak M}_g$ comprised of all singular stable curves. Again, due to singularities of the total space of $\overline{\mathfrak M}_g$, $\Delta$ is only a {\it rational} Cartier divisor. In fact, the only locus of $\overline{\mathfrak{M}}_g$ on which $\lambda,\,\,\delta$ and $\kappa$ are necessarily {\it integer} divisor classes is $(\overline{\mathfrak{M}}_g)_0$ - the locus of automorphism-free curves. \smallskip We can further define the {\it boundary} classes $\delta_0, \delta_1,...,\delta_{[\frac{g}{2}]}$ in $\on{Pic} _{\mathbb Q}\overline {\mathfrak M}_g$. Let $\Delta_i$ be the $\mathbb Q$--Cartier divisor on $\overline {\mathfrak M}_g$ whose general member is an irreducible nodal curve with a single node (if $i=0$), or the join of two irreducible smooth curves of genera $i$ and $g-i$ intersecting transversally in one point (if $i>0$). Setting $\delta_i={\mathcal O}_{\overline {\mathfrak M}_g}(\Delta_i)$, we have $\Delta=\sum_i\Delta_i$ and $\delta=\sum_i\delta_i$. \smallskip As the following result of Harer \cite{Ha1,Ha2} suggests that, in order to describe the geometry of the moduli space $\overline{\mathfrak{M}}_g$, it will be useful to study the divisor classes defined above, and to understand the relations between them. \begin{thm}[Harer] The Hodge class $\lambda$ and the boundary classes $\delta_0,\delta_1,...,\delta_{[\frac{g}{2}]}$ generate $\on{Pic}_{\mathbb Q} \overline{\mathfrak M}_g$, and for $g\geq 3$ they are linearly independent. \end{thm} It is easy to recognize the restrictions of $\lambda,\,\,\delta$ and $\kappa$ to a curve $B$ in $\overline{\mathfrak{M}}_g$ as the previously defined $\lambda|_B,\,\,\delta|_B$ and $\kappa|_B$. For example, the restriction of $\delta$ to the base curve $B$ counts, with appropriate multiplicities, the number of intersections of $B$ with the boundary components $\Delta_i$ of $\overline{\mathfrak{M}}_g$. As a final remark, applying Grothendieck Riemann-Roch Theorem (GRR) to the map $\pi:\mathcal C \rightarrow \mathcal H$ and the sheaf $\omega_{\mathcal{C}/\mathcal H}$, implies the basic relation: \begin{equation} 12\lambda=\kappa+\delta. \label{GRR} \end{equation} \subsection{Slope of non-isotrivial families} \label{slope-non-isotrivial} Let $f:X\rightarrow B$ be our family of stable curves with a smooth general member. By definition, $\delta_B\geq 0$. Further, all locally free quotients of the Hodge bundle $f_*(\omega_f)$ have non-negative degrees \cite{key13}. If $X$ is a non-isotrivial family, then $\lambda|_B>0$, and since the relative canonical divisor $K_{X/B}$ is nef, $\kappa|_B>0$ \cite{key32}. In particular, we can divide by $\lambda|_B$. \medskip \noindent{\bf Definition 2.2.} The {\it slope} of a non-isotrivial family $f:X\rightarrow B$ of stable curves with a smooth general member is the ratio \[\on{slope}(X/\!_{\displaystyle{B}}):=\frac{\delta|_B}{\lambda|_B} \cdot\] Suppose we make a base change $B_1\rightarrow B$ of degree $d$, and set $X_1=X\times_{B}B_1$ to be the pull-back of our family over the new base $B_1$ (cf.~Fig.~\ref{basechange}). Then the three invariants on $B$ will pull-back to the corresponding invariants on $B_1$, and their degrees will be multipied by $d$, e.g. $\lambda|_{B_1}=d\lambda|_B$, etc. In particular, the slope of $X/_{\displaystyle{B}}$ will be preserved. \setlength{\unitlength}{10mm} \begin{figure}[t] \begin{picture}(1.8,1.8)(-0.2,3.9) \put(0,4){$B_1\stackrel{d}{\longrightarrow} B$} \put(0,5.1){$X_1\,\,{\longrightarrow}\,X$} \multiput(0.2,5)(1.25,0){2}{\vector(0,-1){0.6}} \end{picture} \hspace*{3mm} \vspace*{-1mm} \caption{Base change} \label{basechange} \end{figure} \smallskip In view of (\ref{GRR}), restricting to the base curve $B$ yields \begin{equation} 12\lambda|_B=\kappa|_B+\delta|_B. \end{equation} From the positivity conditions above, we deduce that $0\leq \on{slope}(X/\!_{\displaystyle{B}})<12.$ \subsection{Statement of the problem and what is known} \label{statement} It is natural to ask whether we can find a better estimate for the slope of $X$. The first fundamental result in this direction is the following \begin{thm}[Cornalba-Harris, Xiao] Let $f:X\rightarrow B$ be a nonisotrivial family with smooth general member. Then the slope of the family satisfies: \begin{equation} \frac{\delta|_B}{\lambda|_B}\leq 8+\frac{4}{g}\cdot \label{8+4/g} \end{equation} Equality holds if and only if the general fiber of $f$ is hyperelliptic, and all singular fibers are irreducible. \label{CHX} \end{thm} Note that the upper bound is achieved only for hyperelliptic families. Such families are of very special type since the hyperelliptic locus $\overline{\mathfrak I}_g$ has codimension $g-2$ in $\overline{\mathfrak M}_g$. On the other hand, if the base curve $B$ is general enough, a much better estimate can be shown (cf.~\cite{MHE}): \begin{thm}[Mumford-Harris-Eisenbud] If $B$ passes through a general point $[C]\in \overline{\mathfrak{M}}_g$, then \vspace*{-4mm} \begin{equation} \frac{\delta|_B}{\lambda|_B}\leq 6+\on{o}(\frac{1}{g})\cdot \label{6+o(1/g)} \end{equation} \label{generalbound1} \end{thm} For example, when $g$ is odd, we can set $k=(g+1)/2$ and define the divisor $\overline{\mathcal{D}}_k$ in ${\overline{\mathfrak M}}_g$ as the closure of the $k$-sheeted covers of ${\mathbf P}^1$: \[\overline{\mathcal{D}}_k=\overline{\{C\in{\mathfrak{M}}_g\,\,|\,\,C\,\, \on{has}\,\,{g}^1_k\}}.\] \begin{figure}[h] $$\psdraw{general}{1.5in}{1.3in}$$ \caption{General curve $B\not \subset \overline{\mathcal{D}}_k$} \label{generalcurve} \end{figure} If our family is not entirely contained in $\overline{\mathcal{D}}_k$, or equivalently, if $B$ passes through a point $[C]\not\in\overline{\mathcal{D}}_k$ (cf.~Fig.~\ref{generalcurve}), \begin{equation} \frac{\delta|_B}{\lambda|_B}\leq 6+\frac{12}{g+1}\cdot \label{6+12/(g+1)} \end{equation} \smallskip\noindent Higher than the ``generic'' ratio can be obtained, therefore, only for a very special type of families: those entirely contained in $\overline{\mathcal{D}}_k$, and hence possessing ${g}^1_2$, ${g}^1_3$, etc. \subsubsection{The rational Picard group of the hyperelliptic locus $\overline{\mathfrak{I}}_g$} \label{rationalhyper} In proving the maximal bound $8+4/g$, Cornalba-Harris also describe $\on{Pic}_{\mathbb Q}\overline{\mathfrak I}_g$ by exhibiting generators and relations (cf.~\cite{CH}). Here we briefly discuss their result. \smallskip Recall the irreducible divisors $\Delta_i$ on $\overline{\mathfrak M}_g$. For $i=1,...,[g/2]$, $\Delta_i$ cuts out an irreducible divisor on $\overline{\mathfrak I}_g$, while the intersection $\Delta_0\cap\overline{\mathfrak I}_g$ breaks up into several components: \[\Delta_0\cap\overline{\mathfrak I}_g=\Xi_0\cap\Xi_1\cap\cdots \cap\Xi_{[\frac{g-1}{2}]}.\] Set $\xi_i=\mathcal O_{\overline{\mathfrak I}_g}(\Xi_i)$ for the class of $\Xi_i$ in $\overline{\mathfrak I}_g$, and retain the symbols $\lambda$ and $\delta_i$ for their corresponding restrictions to $\on{Pic}_{\mathbb Q}\overline{\mathfrak I}_g$. Thus, $\delta_i:=\mathcal O_{\overline{\mathfrak I}_g}(\Delta_i\cap\overline{\mathfrak I}_g)$ for all $i$. Finally note that the class $\delta_0$ is realised in $\on{Pic}_{\mathbb Q}\overline{\mathfrak I}_g$ as the sum \[\delta_0=\xi_0+2\xi_1+\cdots+2\xi_{[\frac{g-1}{2}]}.\] The coefficient $2$ roughly means that $\Delta_0$ is {\it double} along $\Xi_i$, for $i>0$. \begin{thm}[Cornalba-Harris] The classes $\xi_0,\cdots,\xi_{[\frac{g-1}{2}]}$ and $\delta_1,\cdots,\delta_{[\frac{g} {2}]}$ freely generate $\on{Pic}_{\mathbb Q}{\overline{\mathfrak I}_g}$. The Hodge class $\lambda\in \on{Pic}_{\mathbb Q}{\overline{\mathfrak I}_g}$ is expressed in terms of these generators as the following linear combination: \begin{equation} (8g+4)\lambda=g\xi_0+\sum_{i=1}^{[(g-1)/2]}2(i+1)(g-i)\xi_i +\sum_{j=1}^{[g/2]}4j(g-j)\delta_j. \label{CHPic} \end{equation} \label{theoremCHPic} \end{thm} For a specific family $f:X\rightarrow B$ of hyperelliptic stable curves this relation reads: \[(8g+4)\lambda|_B=g\xi_{0}|_B+\sum_{i=1}^{[(g-1)/2]}2(i+1)(g-i)\xi_{i}|_B +\sum_{j=1}^{[g/2]}4j(g-j)\delta_{j}|_B\] \[\Rightarrow\,\, (8+4/g)\lambda|_B\geq \xi_{0}|_B+\sum_i2\xi_{i}|_B+ \sum_j2\delta_{j}|_B=\delta|_B.\] This yields the desired $8+4/g$ inequality for the slope of a hyperelliptic family, and shows that the maximum can be obtained exactly when all $\xi_1,\cdots,\xi_{[\frac{g-1}{2}]},\delta_1,\cdots,\delta_{[\frac{g}{2}]}$ vanish on $B$. In other words, the singular fibers of $X$ belong only to the boundary divisor $\Xi_0$, and hence are irreducible. In Appendix we review the description of the divisors $\Xi_i$ via admissible covers, and give an alternative proof of Theorem~\ref{theoremCHPic}. \subsubsection{Example of a hyperelliptic family with maximal slope} \label{example} We present here a typical example in which the upper bound $8+4/g$ is achieved, and show how to calculate explicitly the basic invariants $\lambda|_B$ and $\delta|_B$ for this family. \medskip \noindent{\bf Example 2.1.} Consider a pencil $\mathcal{P}$ of hyperelliptic curves of genus $g$ on ${\mathbf P}^1\!\times \!{\mathbf P}^1$. Because of genus considerations, its members must be of type $(2,g+1)$. Our family \newline $f\!:\!X\!\rightarrow \!{\mathbf P}^1$ will be obtained by blowing-up ${\mathbf P}^1\!\times\! {\mathbf P}^1$ at the $4(g+1)$ base points of the pencil in order to separate its members (cf.~Fig.~\ref{ratio8+4/g}). Hence, $\chi(X)=\chi({\mathbf P}^1\!\times\! {\mathbf P}^1)+4(g+1)$ for the corresponding topological Euler characteristics. Riemann-Hurwitz formula for the map $f$ gives a second relation: $\chi(X)=\chi({\mathbf P}^1)\chi(X_b)+\delta|_B$, where ${\mathbf P}^1$ is the base $B$ and $X_b$ is the general fiber of $X$. Putting together, $\delta|_B=8g+4.$ \begin{figure}[t] \begin{picture}(1,3)(0,3.1) \put(-0.7,5.5){$X\,\,\hookrightarrow\,\,{\mathbf P}^1\! \!\times\! {\mathbf P}^1\!\!\times\!{\mathbf P}^1$} \put(-0.5,5.4){\vector(1,-1){1}} \put(1.7,5.4){\vector(-1,-1){1}} \put(0,3.9){${\mathbf P}^1\!\!\times\! {\mathbf P}^1$} \put(0.4,2.7){${\mathbf P}^1$} \put(0.6,3.8){\vector(0,-1){0.6}} \end{picture} \caption{Ratio $8+4/g$} \label{ratio8+4/g} \end{figure} The total space of $X$ is a divisor on ${\mathbf P}^1\!\!\times\! {\mathbf P}^1\!\!\times\! {\mathbf P}^1$ of type $(2,g+1,1)$, and the map $f\!:\!X\!\rightarrow \!{\mathbf P}^1$ is the restriction to $X$ of the third projection $\pi_3\!:\!{\mathbf P}^1\!\!\times\! {\mathbf P}^1\!\!\times\!{\mathbf P^1}\!\rightarrow \!{\mathbf P}^1$. Using standard methods, we compute $h^0( (f_*(\omega_f))(-2))=0$. From the positivity of all free quotients of the Hodge bundle on ${\mathbf P}^1$, $f_*(\omega_f)$ splits as a direct sum $\bigoplus_{i=1}^g {\mathcal O}_{{\mathbf P}^1}(a_i)$ for some $a_i>0$. Then, for $f_*(\omega_f)(-2) =\bigoplus_i\mathcal{O}_{\mathbf P^1}(a_i-2)$ to have no sections, all $a_i$'s must be at most $1$. Finally, \begin{equation*} f_*(\omega_f)=\bigoplus_{i=1}^g{\mathcal O}_{{\mathbf P}^1}(+1),\,\,\lambda|_B=g, \,\,\on{and}\,\,\frac{\delta|_B}{\lambda|_B}=8+\frac{4}{g} \cdot \end{equation*} \subsubsection{The Trigonal Locus $\overline{\mathfrak{T}}_g$} \label{trigonallocus} In a similar vein as in the above example, we consider pencils of trigonal curves on ruled surfaces, and obtain the slope $7+6/g$. It is somewhat reasonable to expect that this is the maximal ratio. Recall that a bundle $\mathcal{E}$ on a curve $B$ is {\it semistable} if for any proper subbundle $\mathcal{F}$, we have $q(\mathcal{F})\leq q(\mathcal{E})$, where $q$ is the quotient of the degree and the rank of the corresponding bundle. Following Xiao's approach in the proof of Theorem~\ref{CHX}, Konno shows that for non-hyperelliptic fibrations of genus $g$ with semistable Hodge bundle $f_*\omega_{f}$ (cf.~\cite{HN,key5}): \begin{equation} \frac{\delta|_B}{\lambda|_B}\leq 7+\frac{6}{g}\cdot \label{7+6/g} \end{equation} As for any trigonal families, he establishes the inequality (cf.~\cite{key6}): \begin{equation} \frac{\delta|_B}{\lambda|_B}\leq \frac{22g+26}{3g+1}\sim 7\frac{1}{3}+\on{o} (\frac{1}{g})\cdot \end{equation} Examples of trigonal families achieving this ratio were not found, which suggested that this bound might be too big. On the other hand, in trying to disprove the smaller bound $7+6/g$, we naturally arrived at counterexamples pointing to a third intermediate ratio (cf.~Theorem~\ref{maximal bound2}): \begin{equation} \frac{36(g+1)}{5g+1}\sim 7\frac{1}{5}+\on{o} (\frac{1}{g})\cdot \end{equation} The difference between the last two estimates may seem negligible, but this would not be so when both $\lambda|_B$ and $\delta|_B$ become large and we attempt to bound $\lambda|_B$ from below by $\delta|_B$. What is more important, the second ratio is in fact {\it exact}, and we give equivalent conditions for it to be achieved (cf.~Sect.~\ref{whenmaximal}, \ref{Maroni-maximal}). This maximal bound confirms Chen's result for genus $g=4$ in \cite{Chen}. \smallskip The reader may ask why the ``generic'' examples for the maximum in the hyperelliptic case fail to provide also the maximum in the trigonal case. As we noted in the Introduction, the answer is closely related to the so-called {\it Maroni} locus in $\overline{\mathfrak{T}}_g$. More precisely, if ${\mathbf F}_k={\mathbf P}({\mathcal O}_{{\mathbf P}^1}\oplus {\mathcal O}_{{\mathbf P}^1}(k))$ denotes the corresponding rational ruled surface, a general curve $C$ embeds in ${\mathbf F}_0$ is $g$ is even, and in ${\mathbf F}_1$ if $g$ is odd. The Maroni locus consists of those curves that embed in ${\mathbf F}_k$ with $k\geq 2$. The number $k/2$ is referred to as the {\it Maroni invariant} of $C$. In these terms, the examples of pencils of trigonal curves on ${\mathbf F}_0$ and ${\mathbf F}_1$ have the lowest possible constant Maroni invariant, and we shall see that the maximum bound can be obtained only for families entirely contained in the Maroni locus, and having very high Maroni invariants. \medskip The ``semistable'' bound $7+6/g$ appears in Theorem~\ref{7+6/g Bogomolov2}, where we give instead a sufficient {\it Bogomolov-semistability} condition $4c_2(V)-c_1^2(V)\geq 0$ for a canonically associated to $X$ vector bundle $V$ of rank 2. The rational Picard group of $\overline{\mathfrak{T}}_g$ is described in terms of generators and relations in Section~\ref{generators}, providing thus in the trigonal case an analog of Theorem~\ref{theoremCHPic}. Note the apparent similarity of the coefficients $\widetilde{c}_{k,i}$ of the trigonal boundary classes and the coefficients of the hyperelliptic boundary classes. This is not coincidental. In fact, the $\widetilde{c}_{k,i}$'s are in a sense the ``smallest'' coefficients that could have been associated to the corresponding classes $\delta_{k,i}$ (cf.~Fig.~\ref{Delta-k,i}): they are symmetric with respect to the two genera of the components in the general member of $\delta_{k,i}$. A crucial role in the proof of Theorem~\ref{Pic trigonal} is played by the interpretation of the above Bogomolov semistability condition in terms of the Maroni locus it $\overline{\mathfrak{T}}_g$ (cf.~Sect.~\ref {interpretation}). \medskip \subsection{The idea of the proof} \label{idea} Let $f:X\rightarrow B$ be a family of stable curves, whose general member $X_b$ is a smooth trigonal curve. By definition, $X_b$ is a triple cover of ${\mathbf P}^1$. We would like to study how this triple cover varies as $X_b$ moves in the family $X$. Thus, it would be desirable to represent $X$, by analogy with $X_b$, as a triple cover of a ruled surface $Y$, comprised by the image lines ${\mathbf P}^1$. Unfortunately, due to existence of hyperelliptic and other special singular fibers, this is not always possible. \setlength{\unitlength}{10mm} \begin{figure}[h] \begin{picture}(3,4.9)(-1,2) \put(0,4){$\widetilde{X}\,\stackrel{\widetilde{\phi}} {\longrightarrow}\, \widetilde{Y}$} \put(0,5.1){$\widehat{X}\,\stackrel{\widehat{\phi}}{\longrightarrow}\, \widehat{Y}$} \multiput(0.2,3.85)(-1.3,-0.7){2}{\vector(1,-1){0.5}} \put(1.4,3.85){\vector(-1,-1){0.5}} \multiput(0.5,3)(-0.5,0.85){2}{\vector(-2,-1){0.8}} \put(0.6,2.9){$\widetilde{B}$} \put(-1.3,3.3){$X$} \put(-0.7,2.3){$B$} \multiput(0.1,5)(1.5,0){2}{\vector(0,-1){0.6}} \put(1.6,6.2){\vector(0,-1){0.6}} \put(1.35,6.3){${\mathbf P}V$} \put(0.2,5.6){\vector(2,1){1.2}} \end{picture} \vspace*{-3mm} \caption{Basic construction} \label{Basic construction idea} \end{figure} \subsubsection{The basic construction} The ``closest'' model of such a triple cover can be obtained after a finite number of birational transformations on $X$, and a possible base change over the base $B$. This way we construct a {\it quasi-admissible} cover $\widetilde{\phi}:{\widetilde{X}} \rightarrow {\widetilde{Y}}$ over a new base ${\widetilde{B}}$ (cf.~Prop.~\ref{propquasi}). Here ${\widetilde{Y}}$ is a {\it birationally} ruled surface over $\widetilde{B}$ with reduced, but non necessarily irreducible, special fibers: $\widetilde{Y}$ allows for {\it pointed stable} rational fibers, i.e. trees of ${\mathbf P}^1$'s with points marked in a certain (stable) way. The map $\widetilde{\phi}$ expresses any fiber $\widetilde{X}_ b$ as a triple quasi-admissible cover of the corresponding {\it pointed stable} rational curve $\widetilde{Y}_b$. To calculate effectively our invariants $\lambda,\delta$ and $\kappa$, we need that $\widetilde{\phi}$ be {\it flat}, which could force a few additional blow-ups on $\widetilde{X}$ and $\widetilde{Y}$. We end up with a flat proper triple cover $\widehat{\phi}: \widehat{X}\rightarrow \widehat{Y}$, where certain fibers of $\widehat{X}$ and $\widehat{Y}$ are allowed to be {\it non-reduced}: these are the scheme-theoretic preimages under the blow-ups on $\widetilde{X}$ and $\widetilde{Y}$. We call such covers $\widehat{\phi}$ {\it effective}. \smallskip\quad We observe next that any smooth trigonal curve $C$ can be naturally embedded in a ruled surface ${\mathbf F}_k$ over $B$. If $\alpha:C\rightarrow {{\mathbf P}^1}$ is the corresponding triple cover, there is an exact sequence of locally free sheaves on ${{\mathbf P}^1}$: \begin{equation*} 0\rightarrow {V}\rightarrow {\alpha}_*{\mathcal O}_{C}\stackrel {\on{tr}}{\rightarrow}{\mathcal O}_{{\mathbf P}^1}\rightarrow 0. \end{equation*} The projectivization $\mathbf P V$ of the rank 2 vector bundle $V$ is the ruled surface ${\mathbf F}_k$. \smallskip This construction can be extended as $C$ moves in the effective cover $\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$. The flatness of $\widehat{\phi}$ forces the pushforward ${\phi}_*{\mathcal O}_{\widehat{X}}$ to be a locally free sheaf of rank 3 on $\widehat{Y}$, and the finiteness of $\widehat{\phi}$ ensures the existence of a {\it trace map} $\on{tr}:{\phi}_*{\mathcal O}_{\widehat{X}}\rightarrow {\mathcal O}_{\widehat{Y}}$. Again, the kernel $V$ of $\on{tr}$ is the desired rank 2 vector bundle on $\widehat{Y}$, in whose projectivization, ${\mathbf P}V$, we embed $\widehat{X}$ (cf.~Fig.~\ref{Basic construction idea}). \subsubsection{Chow Rings Calculations} We can now use the relations in the Chow rings of ${\mathbb{A}}({\mathbf P}V)$, $\mathbb{A}\widehat{Y}$ and $\mathbb{A}\widehat{X}$ to calculate the invariants $\lambda_{\widehat{X}}$ and $\delta_{\widehat{X}}$, appropriately defined for the new family $\widehat{X}\rightarrow {\widetilde{B}}$ of semistable and occasionally non-reduced fibers. Then, of course, we translate $\lambda_{\widehat{X}}$ and $\delta_{\widehat{X}}$ into $\lambda_{{X}}$ and $\delta_{{X}}$ with the necessary adjustments from the birational transformations on $X$ and the base change on $B$. We compare the resulting expressions to obtain a relation among $\lambda_X$ and $\delta_X$. \subsubsection{Boundary of the Trigonal Locus} As we vary the base curve $B$ inside $\overline{\mathfrak{T}}_g$, we actually obtain a relation among the restrictions of $\lambda$ and $\delta$ in $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$, rather than just among $\lambda|_B=\!\lambda_X$ and $\delta|_B=\!\delta_X$ in $\on{Pic}B$. \smallskip {\it In terms of what} have we thus represented and linked $\lambda|_{\overline{\mathfrak{T}}_g}$ and $\delta|_{\overline {\mathfrak{T}}_g}$? To answer this question, we need first to understand the boundary divisors of the trigonal locus $\overline{\mathfrak{T}}_g$. As we shall see, there are seven types of such divisors, denoted by $\Delta{\mathfrak{T}}_{0}$ and $\Delta{\mathfrak{T}}_{k,i}$ for $k=1,...,6$. Each type is determined by the specific geometry of its general member. For example, $\Delta{\mathfrak{T}}_0$ is the closure of all irreducible trigonal curves with one node, while $\Delta{\mathfrak{T}}_{2,i}$ corresponds to joins in two points of a trigonal and a hyperelliptic curve with genera $i$ and $g-1-i$, respectively (cf.~Fig.~\ref{Delta-k,i}). Naturally, we derive an expression for the restriction of the divisor class $\delta\in\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{M}}_g$ to $\overline{\mathfrak{T}}_g$: \begin{equation*} \delta|_{\displaystyle{\overline{\mathfrak{T}}_g}}=\delta_0+ \sum_{i=1}^{\scriptscriptstyle{[(g-2)/2]}}3\delta_{1,i} +\sum_{i=1}^{\scriptscriptstyle{g-2}}2\delta_{2,i} +\sum_{i=1}^{\scriptscriptstyle{[g/2]}}\delta_{3,i} +\sum_{i=1}^{\scriptscriptstyle{[(g-1)/2]}}3\delta_{4,i}+ \sum_{i=1}^{\scriptscriptstyle{g-1}}\delta_{5,i} +\sum_{i=1}^{\scriptscriptstyle{[g/2]}}\delta_{6,i}. \label{delta} \end{equation*} Here $\delta_0$ and $\delta_{k,i}$ are the divisor classes of $\Delta{\mathfrak{T}}_0$ and $\Delta{\mathfrak{T}}_{k,i}$ in $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$. \subsubsection{Relations among $\lambda$ and $\delta$} For a fixed family $X\rightarrow B$ with a smooth trigonal general member, we establish a relation among the Hodge class $\lambda|_B$, the boundary classes $\delta_{k,i}|_B$, and the Bogomolov quantity $4c_2(V)-c_1^2(V)$ for the associated vector bundle $V$: \begin{equation} (7g+6)\lambda|_B=g\delta_0|_B+\sum_{k,i}\widetilde{c}_ {k,i}\delta_{k,i}|_B+\frac{g-3}{2}(4c_2(V)-c_1^2(V)). \label{tobelifted} \end{equation} The polynomial coefficients $\widetilde{c}_{k,i}$ are comparatively larger than the corresponding coefficients of the boundary divisors in the expression for $\delta|_{\displaystyle{\overline{\mathfrak{T}}_g}}$. As a result, we rewrite (\ref{tobelifted}) as \begin{equation} (7g+6)\lambda|_B=g\delta|_B+\mathcal{E}|_B+\frac{g-3}{2}(4c_2(V)-c_1^2(V)), \label{E-argument} \end{equation} where $\mathcal{E}$ is an effective combination of the boundary classes on $\overline{\mathfrak{T}}_g$. In particular, if $V$ is Bogomolov semistable, the slope satisfies (cf.~Theorem~\ref{7+6/g Bogomolov2}): \begin{equation} \on{slope}(X/_{\displaystyle{B}})\leq 7+\frac{6}{g}. \label{idea7+6/g} \end{equation} Further, we describe $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$ as generated freely by the restriction $\lambda|_{\overline{\mathfrak{T}}_g}$ and the boundary classes of $\overline{\mathfrak{T}}_g$. In the even genus $g$ case, we can replace $\lambda|_{\overline{\mathfrak{T}}_g}$ by a geometrically defined class $\mu$, corresponding to the so-called Maroni divisor in $\overline{\mathfrak{T}}_g$. This, of course, means that the Hodge class $\lambda|_{\overline{\mathfrak{T}}_g}$ must be some linear combination of the boundary classes and $\mu$. The Bogomolov quantity is interpreted as \[4c_2(V)-c_1^2(V)=4\mu|_B+0\cdot \delta_0|_B+\sum_{k,i}\alpha_{k,i} \delta_{k,i}|_B,\] which in turn ``lifts'' (\ref{tobelifted}) to the wanted relation in $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$: \begin{equation*} (7g+6)\lambda|_{\overline{\mathfrak{T}}_g}=g\delta_0+ \sum_{k,i}\widehat{c}_{k,i}\delta_{k,i}+2(g-3){\mu}. \end{equation*} We have not yet computed explicitly all coefficients $\widehat{c}_{k,i}$. In the cases which we have completed ($\Delta_{0}\mathfrak{T}_g$ and $\Delta_{1,i}\mathfrak{T}_g$), these coefficients turn out again sufficiently large so that we can repeat the argument in (\ref{E-argument}). Thus, if $X$ has at least one non-Maroni fiber, and its singular fibers belong to $\Delta_{0}\mathfrak{T}_g\cup\Delta_{1,i}\mathfrak{T}_g$, then $\mu|_B\geq 0$, and hence the stronger bound of (\ref{idea7+6/g}) holds (cf.~Prop.~\ref{Maroni inequality} and Conj.~\ref{Maroni-conj}). \subsubsection{Maximal Bound} Since the Bogomolov semistability condition $4c_1^2(V)-c_2(V)\geq 0$ is not always satisfied, the above discussion shows that $7+6/g$ is {\it not} the maximal bound for the slope of trigonal families, Therefore, we need another, more subtle, estimate. The expressions for $\lambda|_B$ and $\delta|_B$ suggest that any maximal bound would be equivalent to an inequality involving $c_1^2(V)$, $c_2(V)$, and possibly some other invariants. We construct a specific divisor class $\eta$ on $\widetilde{X}$, for which the {\it Hodge Index} theorem implies $\eta^2\leq 0$, and we translate this into $9c_2(V)-2c_1^2(V)\geq 0$ (cf.~Prop.~\ref{genindex}). We notice that the only reasonable way to replace Bogomolov's condition $4c_1^2(V)-c_2(V)\geq 0$ by the newly found inequality is by subtracting the following quantities: \begin{equation*} 36(g+1)\lambda|_B-(5g+1)\delta|_B= \mathcal{E}^{\prime}|_B+(g-3)\big(9c_2(V)- 2c_1^2(V)\big), \label{maximum1} \end{equation*} so that the ``left-over'' linear combination of boundary divisors $\mathcal{E}^{\prime}$ is again effective (cf.~Theorem~\ref{maximal relation2}). Hence, we conclude that for {\it all} trigonal families: \[\on{slope}(X/_{\displaystyle{B}})\leq \frac{36(g+1)}{5g+1} \cdot\] \subsection{The organization of the paper} \label{organization} The presentation of the {\it Basic Construction} is done in several stages. Fig.~\ref{stages} shows schematically the connection between the three types of covers, admissible, quasi-admissible and effective, in relation to the original family $X\rightarrow B$ of stable curves. \vspace*{4mm} \begin{figure}[h] $$\psdraw{stages}{1.6in}{1.35in}$$ \caption{Types of covers} \label{stages} \end{figure} We start in Section~\ref{hurwitz} by introducing a compactification $\overline{\mathcal{H}}_{d,g}$ of the Hurwitz scheme, parametrizing {\it admissible} $d$-uple covers of stable pointed rational curves. Using its coarse moduli properties, we show in Section ~\ref{admissible} the existence of admissible covers of surfaces $X^a\rightarrow Y^a$ associated to the original family $f\!:\!X\!\rightarrow \!B$. Next we modify these covers to {\it quasi-admissible} covers $\widetilde{\phi}:\widetilde{X} \rightarrow \widetilde{Y}$ (cf.~Prop.~\ref{propquasi}), and further to {\it effective} covers $\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$ in order to resolve the technical difficulties arising from the non-flatness of $\widetilde{\phi}$ (cf.~Sect.~\ref{effectivecovers}). \bigskip We devote Section~4 to the study of the boundary components of the trigonal locus $\overline{\mathfrak{T}}_g$ inside the moduli space $\overline{\mathfrak{M}}_g$, and express the restriction $\Delta|_{\overline{\mathfrak{T}}_g}$ as a linear combination of the boundary divisors (cf.~Prop.~\ref{divisorrel}). In Section~6 we complete the Basic Construction by embedding the effective cover $\widehat{X}$ in a rank 1 projective bundle ${\mathbf P}V$ over $\widehat{Y}$. \medskip For convenience of the reader, the proofs of the maximal $36(g+1)/(5g+1)$ and the semistable $7+6/g$ bounds are presented first in the special, but fundamental case when the original family $f\!:\!X\!\rightarrow \!B$ is already an effective triple cover of a ruled surface $Y$ (cf.~Sect.~7). The discussion results in finding the coefficients of $\delta_0$ in two different expressions of $\lambda|_{\overline{\mathfrak{T}}_g}$, but, as it turns out, the knowledge of these coefficients is enough to determine the desired two bounds. We refer to this as the {\it global} calculation. The Hodge Index Theorem and Nakai-Moishezon criterion on $X$ complete the global calculation in Sect.~\ref{indextheorem}. A discussion of maximal bound examples can be found in Section~\ref{whenmaximal}. \medskip The {\it local} calculations in Sections~8-10 compute the contributions of the other boundary \vspace*{-1mm}classes $\delta_{k,i}$, and express $\lambda|_{\overline{\mathfrak{T}}_g}$ in terms of these contributions and the Chern classes of the rank 2 vector bundle $V$ on $\widehat{Y}$. For clearer exposition, the proofs of the two bounds are shown first for a {\it general} base curve $B$ (i.e. $B$ intersects transversally the boundary components in general points), and then in Section~11 the results are extended to {\it any} base curve $B$. We develop the necessary notation and techniques for the local calculations in Section ~\ref{conventions}. \medskip Section~12 discusses the relation between the Bogomolov semistability condition and the Maroni locus, and describes the structure of $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$. In Section~\ref{Maroni-maximal} we give another interpretation of the conditions for the maximal bound. \medskip We present further results and conjectures for $d$-gonal families in Section~13. In the Appendix, we give another proof of the $8+4/g$ bound in the hyperelliptic case and show an application of the maximal trigonal bound to the study of the discriminant locus of certain triple covers. \bigskip \section*{\hspace*{1.9mm}3. Quasi-Admissible Covers of Surfaces} \setcounter{section}{3} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{lem}{0} \setcounter{thm}{0} \setcounter{prop}{0} \setcounter{defn}{0} \setcounter{cor}{0} \setcounter{conj}{0} \setcounter{claim}{0} \setcounter{remark}{0} \setcounter{equation}{0} \label{quasi-admissible} We first review briefly the theory of admissible covers. For more details, we refer the reader to \cite{MHE,HM}. \subsection{The Hurwitz scheme $\overline{\mathcal H}_{d,g}$} \label{hurwitz} Let ${\mathcal H}_{d,g}$ be the {\it small Hurwitz scheme} parametrizing the pairs $(C,\phi)$, where $C$ is a smooth curve of genus $g$ and $\phi:C\rightarrow {\mathbf P}^1$ is a cover of degree $d$, simply branched over $b=2d+2g-2$ distinct points. Since $C\in {\mathfrak M}_g$, there is a natural map ${\mathcal H}_{d,g}\rightarrow {\mathfrak M}_g$, whose image contains an open dense subset of ${\mathfrak M}_g$. The theory of admissible covers provides the commutative diagram in Fig.~\ref{Hurwitz figure}. \begin{figure}[h] \begin{picture}(5,3.5)(-0.8,2.2) \put(0,4){${\mathcal H}_{d,g}\hookrightarrow \overline{\mathcal H}_{d,g}$} \put(0.4,3.85){\vector(1,-1){0.9}} \put(1.9,3.85){\vector(1,-1){0.9}} \put(0.4,4.2){\vector(1,1){0.9}} \put(1.9,4.2){\vector(1,1){0.9}} \put(2.4,4.5){${pr}_1$} \put(2.4,3.5){${pr}_2$} \put(1.1,2.5){${\mathfrak P}_{0,b}\hookrightarrow \overline{\mathfrak P}_{0,b}$} \put(1.1,5.2){${\mathfrak M}_g\hookrightarrow \overline{\mathfrak M}_g$} \end{picture} \caption{Hurwitz scheme} \label{Hurwitz figure} \end{figure} There ${\mathfrak P}_{0,b}$ (resp. $\overline{\mathfrak P}_{0,b}$) is the moduli space of $m$-pointed ${\mathbf P}^1$'s (resp. of stable $m$-pointed rational curves), and $\overline{\mathcal H}_{d,g}$ is a compactification of the Hurwirz scheme. The points of $\overline{\mathcal H}_{d,g}$ correspond to triples $(C,(P;p_1,...,p_m),\phi)$, where $C$ is a connected reduced nodal curve of genus $g$, $(P;p_1,...,p_m)$ is a stable $m$-pointed rational curve, and $\phi:C\rightarrow P$ is a so-called {\it admissible cover}. \medskip \noindent{\bf Definition 3.1.} Given the curves $C$ and $P$ as above, an {\it admissible cover} $\phi:C\rightarrow P$ is a regular map satisfying the following conditions: \smallskip (A1) $\phi^{-1}(P_{\on{sm}})=C_{\on{sm}}$ and $\phi:C_{\on{sm}} \rightarrow P_{\on{sm}}$ is simply branched over the distinct points $p_1,...,p_b\in P_{\on{sm}}$; (A2) for every $q\in C_{\on{sing}}$ lying over a node $p\in P$, the two branches through $q$ map with the same ramification index to the two branches through $p$. \begin{figure}[h] $$\psdraw{nonstable}{1in}{1in}$$ \caption{Admissible model} \label{nonstable} \end{figure} \smallskip Note that $C$ is not necessarily a stable curve, but contracting its destabilizing rational chains yields the corresponding stable curve $pr_1(C)\in \overline{\mathfrak M}_g$. In such a case, we say that $C$ is the ``admissible model'' for $pr_1(C)$ (cf.~Fig.~\ref{nonstable}). Harris-Mumford have shown that the compactification $\overline{\mathcal H}_{d,g}$ is in fact a {\it coarse moduli space} for the admissible covers $\phi:C\rightarrow P$. \begin{figure}[h] $$\psdraw{admfamilies}{3.4in}{1.4in}$$ \hspace*{4.5mm}\vspace*{-5mm} \caption{Admissible family} \label{admfamilies} \end{figure} \subsection{Local properties of admissible covers} \label{localproperties} When we vary the admissible covers of curves in families, the local structure of the corresponding total spaces becomes apparent. Let $\phi:\mathcal C\rightarrow \mathcal P$ be a proper flat family (over a scheme $\mathcal B$) of admissible covers of curves (cf.~Fig.~\ref{admfamilies}). Assume that $\phi$ is \'{e}tale everywhere except over the nodes of the fibers of $\mathcal P/_ {\textstyle{\mathcal B}}$, and except over some sections $\sigma_i:\mathcal B\rightarrow \mathcal C$ and their images $\omega_i:\mathcal B\rightarrow \mathcal P$: there $\phi$ is simply branched along $\sigma_i$ over $\omega_i$ for all $i$. If $q\in {\mathcal C}_b$ is a point lying above a node $p\in {\mathcal P}_b$ for some $b\in \mathcal B$, then $\mathcal C_b$ has a node at $q$, and locally analytically we can describe $\mathcal C,\mathcal P$ and $\phi$ near $q$ and $p$ by: \[\left\{\begin{array}{lll} \mathcal C: & xy=a, &x,y\,\,\on{generate}\,\,\widehat{\mathfrak m}_{q,\mathcal C_b},\,\, a\in \widehat{\mathcal O}_{b,\mathcal B},\\ \mathcal P: & uv=a^n, &u,v\,\,\on{generate}\,\,\widehat{\mathfrak m}_{p,\mathcal P_b},\\ \phi: & u=x^n,v=u^n. \end{array}\right.\] \smallskip One can see that $n$ is the index of ramification of $\phi$ at $q$, and that fiberwise $\mathcal C_b\rightarrow \mathcal P_b$ is an admissible cover (of curves). From now on, by {\it admissible covers} we mean, more generally, families $\mathcal C\rightarrow \mathcal P$ over $\mathcal B$ with the above description. \smallskip The local properties of the admissible cover $\phi:\mathcal C\rightarrow \mathcal P$ over the nodes in $\mathcal P_b$ forces singularities on the total spaces of $\mathcal C$ and $\mathcal P$. Since we will be interested only in the cases when the base $\mathcal B$ is a smooth projective curve $B$ and the general fiber of $\mathcal C$ is smooth, we can always pick a generator $t$ for $\widehat{\mathcal O}_{b,B}$, and express $a=t^l$ for some $l\in {\mathbb N}$. \begin{figure}[h] $$\psdraw{singular}{1.2in}{1.2in}$$ \caption{Singularity of $\mathcal C$} \label{singular} \end{figure} \noindent{\bf Example 3.1.} Let the triple admissible cover $\phi\!:\!\mathcal C\!\rightarrow\!\mathcal P$ contain the fiber $\mathcal C_b$ as in Fig.~\ref{singular}. At $q$, ${\mathcal C}$ is given by $\,\,xy=t^{l}$, and at $p$, $\mathcal P$ is given by $uv=t^{2l}$, where $u=x^2,\,\,v=y^2$. This forces at $r$ the local equation $xy=t^{2l}$ ($u=x,\,\,v=y$). Even if $\mathcal C$ is smooth at $q$ ($l=1$), $\mathcal C$ and $\mathcal P$ will be singular at $r$ and $p$, respectively ($xy=t^2,\,\,uv=t^2$). Compare this with the non-flat cover of ramification index 1 in Fig.~\ref{mult4}. \smallskip Recall that a rational double point $s$ on a surface $S$ is of type $A_{l-1}$ if locally analytically $S$ is given at $s$ by the equation $xy=t^l$. Thus, $r$ and $p$ above are rational double points on $\mathcal C$ and $\mathcal P$, respectively, of type $A_{l-1}$. \medskip \noindent{\bf Remark 3.1.} In the sequel, we use the fact that the projection $pr_1\!:\!\overline{\mathcal H}_{d,g}\!\rightarrow \!\overline{\mathfrak P}_{0,b}$ is a {\it finite} map. From the weak valuative criterion for properness, this means that given a family of admissible covers $\phi:{\mathcal C}^*\rightarrow {\mathcal P}^*$ over the punctured disc ${\on {Spec}}\,{\mathbb C}((t))$, there is some $n\in {\mathbb N}$ for which $\phi$ extends to a family $\phi_n:{\mathcal C}_n\rightarrow {\mathcal P}_n$ of admissible covers over ${\on {Spec}}\,{\mathbb C}[[t^{1/n}]]$. In particular, if the base for the admissible cover $\phi:X^*\rightarrow Y^*$ is an open set $B^*$ of a smooth projective curve $B$, modulo a finite base change, we can extend this to a family of admissible covers $X^a\rightarrow Y^a$ over the whole curve $B$. \subsection{Admissible covers of surfaces} \label{admissible} Consider a family $f:X\rightarrow B$ of stable curves of genus $g$, whose general member is smooth and $d$-gonal. Let $\psi:B\rightarrow {\overline{\mathfrak M}_g}$ be the canonical map, and let $\overline B$ denote the fiber product $B\times_{\overline{\mathfrak M}_g} \overline{\mathcal H}_{d,g}$. \begin{figure} \begin{picture}(5,4)(-0.7,1.9) \put(-2,4){$\overline{B}_0\subset \overline B\stackrel{\eta}{\longrightarrow} \overline{\mathcal H}_{d,g}$} \multiput(-0.8,3.85)(1.4,0){2}{\vector(0,-1){0.9}} \put(-1,2.5){$B\stackrel{\psi}{\longrightarrow} \,\overline{\mathfrak M}_{g}$} \put(-2.5,3.3){$X$} \put(0.7,3.35){$\scriptstyle{pr_1}$} \put(-3.5,4.8){$\overline{X}$} \multiput(-3.2,4.7)(1.5,-0.8){2}{\vector(2,-3){0.7}} \multiput(-2.1,3.4)(-1,1.5){2}{\vector(3,-2){1.1}} \put(3,3.7){$B^*\subset B\stackrel{\eta}{\longrightarrow} \overline{\mathcal H}_{d,g}$} \multiput(4.2,5.05)(1.4,-1.5){2}{\vector(0,-1){0.9}} \put(3.2,5.05){\vector(0,-1){0.9}} \put(5.4,2.2){$\overline{\mathfrak M}_{g}$} \put(5.7,3.1){$\scriptstyle{pr_1}$} \put(3,5.2){$X^*\subset X$} \put(4.3,3.6){\vector(1,-1){1}} \put(4.8,3.1){$\scriptstyle{\psi}$} \end{picture} \caption{$\eta:\overline{B}\rightarrow\overline{\mathcal {H}}_{d,g}$ \hspace*{10mm}{\sc Figure 12.} Simply branched $C$\hspace*{-20mm}} \label{map eta} \end{figure} \addtocounter{figure}{1} If the general member of $X$ has infinitely many ${g}_d^1$'s, the variety $\overline B$ will have dimension $\geq 2$. We can resolve this by considering an intersection of the appropriate number of hyperplane sections of $\overline B$, and picking a one-dimensional component $\overline B_0$ dominating $B$. The curve $\overline B_0$ might be singular, but by normalizing it and pulling $X$ over it, we get another family of stable curves (cf.~Fig.~\ref{map eta}): \[\overline X=X\times_B(\overline B_0)^{\on{norm}} \rightarrow (\overline B_0)^{\on{norm}}.\] Since the two families have the same basic invariants, we can replace the original with the new one, and assume the existence of a map $\eta:B\rightarrow \overline{\mathcal H}_{d,g}$ compatible with $\psi: B\rightarrow \overline{\mathfrak M}_g$. In other words, $\eta$ associates to every fiber $C$ of $X$ a specific $g^1_d$ on $C$ or, possibly, a $g^1_d$ on an admissible model $C^a$ of $C$. \smallskip Let $B^*$ be the open subset of $B$ over which {\it all} fibers are smooth and $d$-gonal. For simplicity, assume for now that all the fibers over $B^*$ can be represented as admissible covers of ${\mathbf P}^1$ via the chosen $g^1_d$'s, i.e. they are {\it simply branched} covers of ${\mathbf P}^1$ over $m$ distinct points of ${\mathbf P}^1$. Denote by $X^*$ the restriction of $X$ over $B^*$ (cf.~Fig.12). \smallskip The map $\eta:B^*\rightarrow {\mathcal H}_{d,g}$ induces a section \[\sigma:B^*\rightarrow {\on{Pic}}^d(X^*/B^*),\] where ${\on{Pic}}^d(X^*/B^*)$ is the {\it relative degree $d$ Picard variety} of $X^*$ over $B^*$. ${\on{Pic}}^d(X^*/B^*)$ parametrizes the line bundles on $X^*$ of relative degree $d$. The image $\sigma(B^*)\subset {\on{Pic}}^d(X^*/B^*)$ is a class of line bundles on $X^*$ whose fiberwise restrictions are the chosen $g^1_d$'s. Let $\mathcal L$ be a representative of this class, and let $Y^*$ be the ruled surface ${\mathbf P}((f_*{\mathcal L})^{\widehat {\phantom{n}}})$ over $B^*$. The map $\phi:X^*\rightarrow Y^*$ induced by $\mathcal L$ defines an admissible cover over $B^*$, as shown in Fig.~\ref{construction Y*}. \setlength{\unitlength}{10mm} \begin{figure}[h] \begin{picture}(5,2.2)(-0.3,2.6) \put(0,4){$X^*\stackrel{\phi}{\longrightarrow} Y^*={\mathbf P}((f_*{\mathcal L})^ {\widehat{\phantom{n}}})$} \put(0.2,3.85){\vector(1,-1){0.5}} \put(1.4,3.85){\vector(-1,-1){0.5}} \put(0.6,2.9){$B^*$} \put(0.05,3.4){$f$} \put(1.3,3.4){$h$} \end{picture} \caption{ Construction of $Y^*$} \label{construction Y*} \end{figure} \bigskip From Remark 3.1, $\phi$ extends to a family of admissible covers ${\phi}^a:{X}^a\rightarrow {Y}^a$ over the whole base $B$. Since ${X}^a$ and $X$ are isomorphic over $B^*$, they are birational to each other. In other words, the fibers $C$ of $X$, over which $\mathcal L$ does not extend to the base-point free linear series $g^1_d=\sigma_1(b)$, are modified by blow-ups and blow-downs so as to arrive at their admissible models in ${X}^a$. We have thus proved the following \begin{lem} Let $f:X\rightarrow B$ be a family of stable curves, whose general member over an open subset $B^*\subset B$ is a smooth $d$-uple admissible cover of ${\mathbf P}^1$. Then, modulo a finite base change, there exists an admissible cover of surfaces ${X}^a\rightarrow {Y}^a$ over $B$ such that ${X}^a$ is obtained from $X$ by a finite number of birational transformations performed on the fibers over $B-B^*$. \label{quasicov} \end{lem} \subsection{Quasi-admissible covers} \label{quasi-covers} In case the general member of $X$ is {\it not} an admissible cover of ${\mathbf P}^1$, e.g. it is trigonal with a total point of ramification, we have to modify the above construction. To start with, we cannot expect to obtain an {\it admissible} cover $X^*\rightarrow Y^*$, even modulo a finite base change. This leads us to consider a different kind of covers, which we call {\it quasi-admissible}. \medskip \noindent{\bf Definition 3.2.} A {\it quasi-admissible cover} $\widetilde{\phi}: C\rightarrow P$ of a nodal curve $C$ over a semistable pointed rational curve $P$ is a regular map which behaves like an admissible cover over the singular locus of $P$, i.e. for any $q\in C$ lying over a node $p\in P$ the two branches through $q$ map with the same ramification index to the two branches through $p$. \smallskip \begin{figure}[h] $$\vspace*{5mm}\psdraw{quasi}{4.5in}{0.5in}$$ \vspace*{-6mm} \caption{Quasi-admissible covers over $\mathbf P^1$} \label{quasicovers} \end{figure} Quasi-admissible covers differ from admissible covers in allowing more diverse behavior of $C$ over $P_{\on{sm}}$, e.g. having singularities, higher ramification points and multiple simple ramification points. Fig.~\ref{quasicovers} displays several degree 3 quasi-admissible covers over ${\mathbf P}^1$: However, any quasi-admissible cover can be obtained from an admissible cover $\phi^a\!:\!C^a\!\rightarrow\! P^a$ by simultaneous contractions of components in $P^a$ and their (rational) inverses on $C^a$. \medskip \noindent{\bf Definition 3.3.} A {\it minimal} quasi-admissible cover $\widetilde{\phi}:C\rightarrow P$ is minimal with respect to the number of components of $P$. In other words, one cannot apply more simultaneous contractions on $C\rightarrow P$ and end with another quasi-admissible cover. \smallskip \noindent{\bf Example 3.2.} A smooth trigonal curve $C$ with a total point of ramification $q$ is a minimal quasi-admissible cover of $P={\mathbf P}^1$. Blowing up $q$ on $C$ and $p=\widetilde{\phi}(q)\in P$, gives an admissible cover $C^a=C\cup C_1\rightarrow P\cup P_1$, where $C_1\cong {\mathbf P}^1$ maps three-to-one onto $P_1\cong {\mathbf P}^1$ with a total point of ramification $q=C_1\cap C$ (cf.~Fig.~\ref{quasi/adm}). \bigskip \begin{figure}[h] $$\psdraw{quasiadm}{2.4in}{0.8in}$$ \caption{Quasi virsus admissible covers} \label{quasi/adm} \end{figure} \medskip The motivation for using {\it minimal} quasi-admissible covers, instead of just admissible or quasi-admissible covers, is that the former are the closest covers to the original families $X\rightarrow B$ of stable curves, and calculations on them will yield the best possible estimate for the ratio $\delta_X/\lambda_X $ (cf.~Fig.~\ref{stages}). \subsubsection{Quasi-admissible covers for families with higher ramification sections} Now let us consider the remaining case of a family $X\rightarrow B$, whose general member over $B^*$ is smooth and $d$-gonal, but {\it not} an admissible cover of ${\mathbf P}^1$. After a possible base change, we still have the map (cf.~Fig.~{12}) \[\eta:B\longrightarrow {\overline {\mathcal H}}_{d,g}.\] It associates to every fiber $C$ a $g^1_d$ on its admissible model $C^a$. Let $C^a\rightarrow P^a$ be the corresponding admissible cover. Since $C$ itself is $d$-gonal, and by assumption it does not possess a $g^1_e$ with $e<d$, $C$ must be a $d$-uple cover of some component of $P^a$. In particular, the $g^1_d$ on $C^a$ restricts to a $g^1_d$ on $C$. Thus, in effect, $\eta$ gives again a section $\sigma:B^*\rightarrow {\on{Pic}}^d(X^*/B^*)$. As before, we obtain a degree $d$ finite map $\phi:X^*\rightarrow Y^*$ to the ruled surface ${\mathbf P}((f_*{\mathcal L})^{\widehat{\phantom{n}}})$ over $B^*$. Note that this is a family of {\it minimal quasi-admissible} covers. \medskip We extend $\phi$ over the curve $B$ as follows. For simplicity, assume that $d=3$. Let $R$ be the ramification divisor of $\phi$ in $X^*$. By hypothesis, there is a component $R_0$ of $R$ which passes through total ramification points and dominates $B^*$. Letting $\overline{R}_0$ be the closure of $R_0$ in $X$, we can normalize it and pull the family $X$ over it. So we may assume that $\overline{R}_0$ is a section of $X\rightarrow B$. If there are some other components $R_1,R_2,...,R_l$ of the ramification divisor $R$ passing through higher ramification points, we repeat the same procedure for them, until we ``straighten out'' all $\overline{R}_i$'s into sections of $X\rightarrow B$. Let $E_i=\phi(R_i)$ be the corresponding sections of $Y^*$ over $B^*$. We can shrink $B^*$ in order to exclude any fibers with isolated higher ramification points. \smallskip Consider a fiber $C$ in $X^*$. Let $\{r_i=C\cap R_i\}$ be its total ramification points, and let $\{p_i=\phi(r_i)\}$ be their images on $P=\phi(C)$ in $Y^*$. It is clear that blowing-up all $r_i$'s and $p_i$'s will give an admissible triple cover $C^a= \on {Bl}_{\{r_i\}}(C)\rightarrow P^a=\on {Bl}_{\{p_i\}}(P)$. The $g^1_d$, giving this cover, is the original one assigned by $\eta:B^*\rightarrow {\overline{\mathcal H}}_{d,g}$. We globalize this construction by blowing-up the sections ${R}_i$ on $X^*$ and $E_i$ on $Y^*$. Similarly as above, we obtain a triple admissible cover of surfaces $\phi^*:\on{Bl}_{\cup R_i}(X^*)\rightarrow \on{Bl}_{\cup E_i} (Y^*)$ over $B^*$. The properness of $pr_1: \overline{\mathcal H}_{d,g}\rightarrow \overline{\mathfrak M}_g$ allows us to extend this to an admissible cover ${\phi}^a: \overline{\on{Bl}_{\cup R_i}(X^*)} \rightarrow\overline{\on{Bl}_{\cup E_i}(Y^*)}$ over $B$ (cf.~ Fig.~\ref{blowing up}). \begin{figure}[h]\hspace*{-30mm} \begin{picture}(3,5)(4.3,-0.5) \put(2,3.7){$\overline{{\mathcal R}_i}\subset \overline{\on{Bl}_{\cup R_i}(X^*)}\stackrel{{\phi}^a} {\longrightarrow} \overline{\on{Bl}_{\cup E_i}(Y^*)}\supset \overline{{\mathcal E}_i}$} \put(2,2.2){${{\mathcal R}_i}\subset {\on{Bl}_{\cup R_i}(X^*)} \stackrel{\phi^*}{\longrightarrow} {\on{Bl}_{\cup E_i}(Y^*)}\supset{{\mathcal E}_i}$} \put(2,0.7){$R_i\hspace{0.5mm}\subset \hspace{6.6mm} X^*\hspace{6mm}\stackrel{\phi}{\longrightarrow}\hspace{7.3mm}Y^* \hspace*{5.6mm}\supset E_i$} \multiput(2.2,3.5)(6.1,0){2}{\vector(0,-1){0.9}} \multiput(3.9,3.5)(2.8,0){2}{\vector(0,-1){0.9}} \multiput(2.2,2)(6.1,0){2}{\vector(0,-1){0.9}} \multiput(3.9,2)(2.8,0){2}{\vector(0,-1){0.9}} \end{picture} \vspace*{-10mm} \caption{Blowing up ${R}_i$ and ${E}_i$ \hspace*{15mm} {\sc Figure 17.} Over $B^*$\hspace*{-10mm}} \label{blowing up} \hspace*{80mm}\begin{picture}(3,0)(6.7,-2.3) \put(7.8,2.2){${X}^q\stackrel{\phi^q}{\longrightarrow} {Y}^q$} \multiput(8,2)(1.4,0){2}{\vector(0,-1){0.9}} \multiput(8.1,1.5)(1.4,0){2}{$\wr$} \put(7.8,0.7){$X^*\longrightarrow Y^*$} \put(8,0.5){\vector(1,-1){0.5}} \put(9.4,0.5){\vector(-1,-1){0.5}} \put(8.5,-0.4){$B^*$} \end{picture} \label{over B*} \end{figure} \addtocounter{figure}{1} \smallskip Denote by ${\mathcal R}_i$ the component of $\on {Bl}_{\cup R_i}(X^*)$, obtained by blowing up ${R}_i\subset X^*$, and let $\overline{{\mathcal R}_i}$ be its closure in $\overline{\on{Bl}_{\cup R_i}(X^*)}$. Define similarly ${\mathcal E}_i\subset \on{Bl}_{\cup E_i}(Y^*)$ and $\overline{\mathcal E}_i \subset \overline{\on{Bl}_{\cup E_i}(Y^*)}$. The admissible cover $\phi^a$ maps $\overline{\mathcal R}_i$ to $\overline{\mathcal E}_i$, so that after removing all the $\overline{\mathcal R}_i$'s and $\overline{\mathcal E}_i$'s we still have a triple cover \[{\phi}^q:{X}^q=\overline{\on{Bl}_{\cup R _i} (X^*)}-\cup\overline{\mathcal R_i}\longrightarrow {Y}^q=\overline{\on{Bl}_{\{{ E_i}\}} (Y^*)}-\cup\overline{\mathcal E_i}.\] Note that ${X}^q\cong X$ and ${Y}^q\cong Y$ over the open set $B^*$, and that ${Y}^q$ is a birationally ruled surface over $B$ (cf.~Fig.~17). Finally, note that from the quasi-admissible cover ${\phi}^q:{X}^q\rightarrow {Y}^q$ we obtain a family $\widetilde{\phi}:\widetilde{X}\rightarrow \widetilde{Y}$ of {\it minimal} quasi-admissible covers: simply contract the unnecessary rational components in the fibers of ${X}^q$ and ${Y}^q$, and observe that the triple map ${\phi}^q$ restricts to the corresponding triple map $\widetilde{\phi}$. \smallskip This completes the construction of minimal quasi-admissible covers for any family $X\rightarrow B$ with general smooth trigonal member. The cases $d>3$ are only notationally more difficult. One has to keep track of the possibly different higher multiplicities in $C$ and multiple double points in $C$ over the same $p\in P$. The construction of an admissible cover ${X}^a\rightarrow {Y}^a$ goes through with minimal modifications. We combine the results of this section in the following \begin{prop} Let $f:X\rightarrow B$ be a family of stable curves, whose general member over an open subset $B^*\subset B$ is smooth and $d$-gonal. Then, modulo a finite base change, there exists a minimal quasi-admissible cover of surfaces $\widetilde{X}\rightarrow \widetilde{Y}$ over $B$ such that $\widetilde{X}$ is obtained from $X$ by a finite number of birational transformations performed on the fibers over $B-B^*$. \label{propquasi} \end{prop} \medskip \section*{\hspace*{1.9mm}4. The Boundary $\Delta{\mathfrak{T}}_g$ of the Trigonal Locus $\overline{\mathfrak{T}}_g$} \setcounter{section}{4} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{lem}{0} \setcounter{thm}{0} \setcounter{prop}{0} \setcounter{defn}{0} \setcounter{cor}{0} \setcounter{conj}{0} \setcounter{claim}{0} \setcounter{remark}{0} \setcounter{equation}{0} \label{boundarycomponents} \subsection{Description and notation for the boundary of $\overline{\mathfrak{T}}_g$} \label{description} In this section we shall see that there are {\it seven types} of boundary divisors of $\overline{\mathfrak{T}}_g$, each denoted by $\Delta{\mathfrak{T}}_{k,i}$ for $k=0,1,...,6$. The second index $i$ is determined in the following way. Let $C=C_1\cup C_2$ be the general member of $\Delta{\mathfrak{T}}_{k,i}$, where $C_1$ and $C_2$ are smooth curves. If $C_1$ and $C_2$ are both trigonal or both hyperelliptic, then we set $i$ to be the smaller of the two genera $p(C_1)$ or $p(C_2)$. If, say, $C_1$ is a trigonal, but $C_2$ is hyperelliptic, then we set $i$ to be genus of the trigonal component $C_1$. The only exception to this rule occurs when $C$ is irreducible (and hence of genus $g$ with exactly one node). We denote this boundary component by $\Delta{\mathfrak{T}}_0$. \smallskip When we view a general member $C$ roughly as a triple cover of ${\mathbf P}^1$'s in the Hurwitz scheme (consider the pull-back $pr_1[C]\in\overline{\mathcal{H}}_{3,g}$), then it may or may not be ramified. If there is no ramification, then $C$ lies in one of the first four types of trigonal boundary divisors $\Delta{\mathfrak{T}}_{k,i}$, $k=0,1,2,3$. Ramification index 1 characterizes the general members of $\Delta{\mathfrak{T}}_{4,i}$ and $\Delta{\mathfrak{T}}_{5,i}$, and in case of $\Delta{\mathfrak{T}}_{6,i}$ the ramification index is 2 (cf.~Fig.~\ref{Delta-k,i}). \smallskip There is an alternative description of the boundary components $\Delta{\mathfrak{T}}_{k,i}$'s of $\overline{\mathfrak{T}}_g$. \vspace*{-1mm}If one such $\Delta{\mathfrak{T}}_{k,i}$ lies in the restriction $\Delta_0\big|_{\displaystyle{\overline{\mathfrak{T}}_g}}$ of the divisor $\Delta_0$ in $\overline{\mathfrak{M}}_g$, \vspace*{-1mm}then $\Delta{\mathfrak{T}}_{k,i}$ is one of $\Delta{\mathfrak{T}}_0,\,\, \Delta{\mathfrak{T}}_{1,i},\,\,\Delta{\mathfrak{T}}_{2,i},$ or $\Delta{\mathfrak{T}}_{4,i}$. The partial normalization of their general members $C$ is still connected, i.e. $C$ is either irreducible, or the join of two smooth curves meeting in at least two points. Correspondingly, for the general member $C$ of the remaining three types of boundary components, $\Delta{\mathfrak{T}}_{3,i},\,\,\Delta{\mathfrak{T}}_{5,i}$ and $\Delta{\mathfrak{T}}_{6,i}$, the irreducible components of $C$ intersect transversely in exactly one point, so that the normalization of $C$ is disconnected. \bigskip \begin{figure}[h] $$\psdraw{boundary}{4.5in}{1in}$$ \begin{picture}(6,1)(2.7,-1.4) \put(-0.7,2.45){$\Delta{\mathfrak{T}}_{0} \hspace{19mm}\Delta{\mathfrak{T}}_{1,i} \hspace{25mm}\Delta{\mathfrak{T}}_{2,i} \hspace{23mm}\Delta{\mathfrak{T}}_{3,i}$} \put(2.8,1.2){$\scriptstyle{i=1,2,...,}\left[\frac{g-2}{2}\right]\hspace{17mm} \scriptstyle{i=1,2,..., g-2}\hspace{17mm}\scriptstyle{i=1,2, ...,} \left[\frac{g}{2}\right]$} \put(-0.2,0.5){$\Delta{\mathfrak{T}}_{4,i}\hspace{31mm}\Delta{\mathfrak{T}}_{5,i} \hspace{31mm}\Delta{\mathfrak{T}}_{6,i}$} \put(01.1,-0.6){$\scriptstyle{i=1,2,...,\left[\frac{g-1}{2}\right]\hspace{24mm} i=1,2,...,g-1 \hspace{22mm}i=1,2,...,\left[\frac{g}{2}\right]}$} \end{picture} \vspace*{-10mm} \caption{Boundary Components $\Delta\mathfrak{T}_{k,i}$ of $\overline{\mathfrak{T}}_g$} \label{Delta-k,i} \end{figure} \begin{prop} The boundary divisors of $\overline{\mathfrak{T}}_g$ can be grouped in seven types: $\Delta{\mathfrak{T}}_{0}$ and $\Delta{\mathfrak{T}}_{k,i}$ for $k=1,...,6$. Their general members and range of index $i$ are shown in Fig.~\ref{Delta-k,i}. The boundary of $\overline{\mathfrak{T}}_g$ consists of $\Delta{\mathfrak{T}}_{0}$, $\Delta{\mathfrak{T}}_{k,i}$, and the codimension 2 component $\overline{\mathfrak{I}}_g$ of hyperelliptic curves. \label{boundary} \end{prop} \medskip Consider the projection map $pr_1:\overline{\mathcal{H}}_{3,g}\rightarrow \overline{\mathfrak{M}}_g$, whose image is the trigonal locus $\overline{\mathfrak{T}}_g$. Thus, the inverse image of each boundary divisor $\Delta{\mathfrak{T}}_{k,i}$ will be a boundary divisor $\Delta{\mathcal{H}}_{k,i}$ in $\overline{\mathcal{H}}_{3,g}$. The converse, however, is not always true, i.e. certain boundary divisors of $\overline{\mathcal{H}}_{3,g}$ contract under $pr_1$ to smaller subschemes of $\overline{\mathfrak{T}}_g$, e.g. the hyperelliptic locus $\overline{\mathfrak{I}}_g$. With the description of the Hurwitz scheme $\overline{\mathcal{H}}_{3,g}$, given in Section~3, it is easier to determine first $\overline{\mathcal{H}}_{3,g}$'s boundary divisors. Thus, we postpone the proof of Proposition~\ref{boundary} until the end of the next subsection. \subsubsection{The Boundary of $\overline{\mathcal{H}}_{3,g}$.} \label{admissibleaboundary} \begin{prop} The boundary divisors of $\overline{\mathcal{H}}_{3,g}$ can be grouped in six types: $\Delta{\mathcal{H}}_{k,i}$ for $k=1,...,6$. Their general members and range of index $i$ are shown in Fig.~\ref{admissible-k,i}. \begin{figure}[h] \bigskip $$\psdraw{admissible}{4.5in}{1in}$$ \begin{picture}(5,1)(3,-2) \put(-0.9,1.85){$\Delta{\mathcal{H}}_{1,i} \hspace{33mm}\Delta{\mathcal{H}}_{2,i} \hspace{32mm}\Delta{\mathcal{H}}_{3,i}$} \put(0.3,0.7){$\scriptstyle{i=1,2,...,}\left[\frac{g-2}{2}\right]\hspace{29mm} \scriptstyle{i=1,..., g-1}\hspace{24mm}\scriptstyle{i=0,1, ...,} \left[\frac{g}{2}\right]$} \put(-0.9,0.15){$\Delta{\mathcal{H}}_{4,i}\hspace{33mm}\Delta{\mathcal{H}}_{5,i} \hspace{32mm}\Delta{\mathcal{H}}_{6,i}$} \put(0.3,-1.2){$\scriptstyle{i=1,2,...,\left[\frac{g-1}{2}\right]\hspace{29mm} i=1,2,...,g-1 \hspace{23mm}i=1,2,...,\left[\frac{g}{2}\right]}$} \end{picture} \vspace*{-5mm} \caption{Boundary Components of $\overline{\mathcal{H}}_{3,g}$} \label{admissible-k,i} \end{figure} \label{boundary2} \end{prop} \begin{proof} A general member $A$ of the boundary $\Delta{\mathcal{H}}$ is a triple admissible cover of a chain of {\it two} ${\mathbf P}^1$. (From the dimension calculations that follow it will become clear that an admissible cover of a chain of three or more $\mathbf P^1$'s will generate a subscheme in $\overline{\mathcal{H}}_{3,g}$ of codimension $\geq 2$.) Note that {\it three} connected components of $A$ over one ${\mathbf P}^1$ means that they are all smooth ${\mathbf P}^1$'s themselves, and hence they can all be contracted simultaneously, leaving us with a smooth trigonal curve, or with a hyperelliptic curve with an attached ${{\mathbf P}}^1$, neither of which cases by dimension count corresponds to a {\it general} member of a boundary component $\Delta{\mathcal{H}}_{k,i}$. Considering all combinations of one or two connected components of $A$ over each ${\mathbf P}^1$, we generate a list of the possible general members of the boundary divisors $\Delta{\mathcal{H}}_{k,i}$. To see which of these are indeed of codimension 1 in $\overline{\mathcal{H}}_{3,g}$, we do the following calculation. First we note that, for a fixed set of $2i+4$ ramification points in ${\mathbf P}^1$, there are finitely many covers of degree $3$ and genus $i$, that is, \[\on{dim}\overline{\mathfrak{T}}_i=2i+4-3=2i+1.\] Substracting $3$ takes into account the projectively equivalent triples of points on ${\mathbf P}^1$. In particular, $\on{dim}\overline{\mathfrak{T}}_g=2g+1$. A similar agrument (with $2i+2$ ramification points) shows that for the hyperelliptic locus: \[\on{dim}\overline{\mathfrak{I}}_i=2i+2-3=2i-1.\] These computations are valid for $i>0$, whereas $0= \on{dim}\overline{\mathfrak{T}}_i=\on{dim}\overline{\mathfrak{I}}_i$. \smallskip Thus, to compute the dimensions of the six types of subschemes of $\overline{\mathcal{H}}_{3,g}$, one adds the corresponding dimensions of $\overline{\mathfrak{T}}_i$ and $\overline{\mathfrak{I}}_j$, making the necessary adjustments for the choice of intersection points on the components of each curve $A$. For example, when $i>0$ the dimension of the subscheme with general member $A$, shown in Fig.~\ref{admissible-k,i}, is \[\on{dim}\overline{\mathfrak{T}}_i+\on{dim}\overline{\mathfrak{T}}_{g-i-2}+1+1=2g.\] The final 1's account for the choice of triples of points in the $g^1_3$'s on each component. We conclude that for $i=1,2,...,[(g-2)/2]$ the join at three points of two trigonal curves, one of genus $i$ and the other of genus $g-i-2$, is the general member of a boundary component of $\overline{\mathcal{H}}_{3,g}$. We denote it by $\Delta{\mathcal{H}}_{1,i}$. The range of $i$ stops at $[(g-2)/2]$ for symmetry considerations. When $i=0$, the corresponding subscheme has a smaller dimension of $2g-2$ and hence no boundary divisor is generated by such curves. \smallskip As another example, consider the fifth sketch in Fig.~\ref{admissible-k,i}. It corresponds to the join at one point of a trigonal curve $C_1$ of genus $i$, a hyperelliptic curve $C_2$ of genus $g-i$, and an attached ${\mathbf P}^1$ to $C_2$ to make the whole curve a triple cover. Note that $C_1$ and $C_2$ intersect transversally at a point $q$, but when presented as covers of ${\mathbf P}^1$ they both have ramifications at $q$ of index 1. On all such curves $C_1$ and $C_2$ the total number of ramification points over ${\mathbf P}^1$ is finite, and hence their choice does not affect the dimension of our subscheme. Thus, \[\on{dim}\overline{\mathfrak{T}}_i+\on{dim}\overline{\mathfrak{I}}_{g-i}=2i+1+2(g-i)= 2g.\] Therefore, this subscheme is in fact a divisor in $\overline{\mathcal{H}}_{3,g}$, which we denote by $\Delta {\mathcal{H}}_{5,i}$. The cases of $i=0$ or $i=g$ lead to contractions of unstable rational components ($C_1$ or $C_2$), and do not yield the necessary dimension of $2g$. Hence, $i=1,2,...,g-1$. \smallskip In the case of $\Delta{\mathcal{H}}_{6,i}$, the two components $C_1$ and $C_2$ meet transversally in one point $q$, but both have ramification of index $2$ at $q$ as triple covers of ${\mathbf P}^1$. Smooth trigonal curves of genus $i$ with such high ramification form a codimension 1 subscheme of the trigonal locus ${\mathfrak{T}}_i$, hence the dimension of $\Delta{\mathcal{H}}_{6,i}$ is \[\on{\dim}\overline{\mathfrak{T}}_i-1+\on{dim}\overline{\mathfrak{T}}_{g-i}-1= (2i+1)-1+(2(g-i)+1)-1=2g.\] Thus, $\Delta{\mathcal{H}}_{6,i}$ is a boundary divisor in $\overline{\mathcal{H}}_{3,g}$ for $i=1,2,...,[g/2]$. The case of $i=0$ yields dimension $2g-1$, and hence we disregard it. \smallskip The remaining cases are treated similarly. We conclude that $\overline{\mathcal{H}}_{3,g}$ has six types of boundary divisors, $\Delta{\mathcal{H}}_{k,i}$, whose general members and range of indices are indicated in Fig.~\ref{admissible-k,i}. \end{proof} \subsubsection{Boundary of $\overline{\mathfrak{T}}_g$. Proof of Proposition~\ref{boundary}} \label{trigonalboundary} Having described the boundary of $\overline{\mathcal{H}}_{3,g}$, it remains to check which of the divisors $\Delta\mathcal{H}_{k,i}$ preserve their dimension under the map $pr_1$ and hence map into divisors of $\overline {\mathfrak{T}}_g$. The only ``surprises'' can be expected where $pr_1$ contracts unstable ${\mathbf P}^1$, such as in $\Delta{\mathcal{H}}_{2,i}$, $\Delta{\mathcal{H}}_{3,i}$, and $\Delta{\mathcal{H}}_{5,i}$. In fact, only $\Delta{\mathcal{H}}_{2,g-1}$ and $\Delta{\mathcal{H}}_{3,0}$ diverge from the common pattern; in all other cases, we set $\Delta{\mathfrak{T}}_{k,i}:= pr_1\left(\Delta{\mathcal{H}}_{k,i}\right)$ to be the corresponding boundary divisor in $\overline{\mathfrak{T}}_g$. \smallskip The map $pr_1$ contracts the three rational components of the general member of $\Delta{\mathcal{H}}_{3,0}$, leaving only a smooth hyperelliptic curve of genus $g$. Thus, the image $pr_1\left(\Delta{\mathcal{H}}_{3,0}\right)$ is the hyperelliptic locus $\overline{\mathfrak{I}}_g$, which is of dimension $2g-1$. Hence $\Delta{\mathcal{H}}_{3,0}$ does not yield a divisor in $\overline{\mathfrak{T}}_g$, but a boundary component of codimension 2. \smallskip Finally we consider $\Delta{\mathcal{H}}_{2,g-1}$. After we contract its two rational components, we arrive at an {\it irreducible nodal} trigonal curve with exactly one node. The dimension of the subscheme of such curves is \[\on{dim}\overline{\mathfrak{T}}_{g-1}+1=2(g-1)+1+1=2g,\] where the final $1$ indicates the choice of a triple of points on a smooth trigonal curve (belonging to the $g^1_3$), two of which will be identified as a node. Correspondingly, we obtain another divisor in $\overline{\mathfrak{T}}_g$, which we denote by $\Delta{\mathfrak{T}}_0.$ \qed \subsection{Multiplicities of the boundary divisors $\Delta{\mathfrak{T}}_{k,i}$ in the restriction $\delta|_{\overline{\mathfrak{T}}_g}$} \label{multiplicities} By abuse of notation, we will denote by $\delta_0$ and $\delta_{k,i}$ the classes in $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$ of $\Delta{\mathfrak{T}}_0$ and $\Delta{\mathfrak{T}}_{k,i}$, respectively. \begin{prop} The divisor class $\delta\in\on{Pic}_{\mathbb{Q}} \overline{\mathfrak{M}}_g$ restricts to $\overline{\mathfrak{T}}_g$ as the following linear combination of the boundary classes in $\overline{\mathfrak{T}}_g$: \begin{equation} \delta|_{\displaystyle{\overline{\mathfrak{T}}_g}}=\delta_0+ \sum_{i=1}^{\scriptscriptstyle{[(g-2)/2]}}3\delta_{1,i} +\sum_{i=1}^{\scriptscriptstyle{g-2}}2\delta_{2,i} +\sum_{i=1}^{\scriptscriptstyle{[g/2]}}\delta_{3,i} +\sum_{i=1}^{\scriptscriptstyle{[(g-1)/2]}}3\delta_{4,i}+ \sum_{i=1}^{\scriptscriptstyle{g-1}}\delta_{5,i} +\sum_{i=1}^{\scriptscriptstyle{[g/2]}}\delta_{6,i}. \label{divisorrel} \end{equation} \end{prop} \noindent{\it Proof.} Let us rewrite equation (\ref{divisorrel}) in the form \[\delta|_{\displaystyle{\overline{\mathfrak{T}}_g}}=(\on{mult}_{\delta} \delta_0)\delta_0+\sum_{k,i}(\on{mult}_{\delta}\delta_{k,i})\delta_{k,i},\] and call $\on{mult}_{\delta}\delta_{k,i}$ the {\it multiplicity} of $\delta_{k,i}$ in $\delta|_{\overline{\mathfrak{T}}_g}$. This linear relation simply counts the contribution of each singular curve of a specific boundary type in $\Delta\mathfrak{T}_g$ to the degree of $\delta$. Recall that for any trigonal family $f:X\rightarrow B$: \[\on{deg}\delta|_B=\sum_{q\in X}m_q.\] Here $m_q$ denotes the local analytic multiplicity of the total space of $X$ nearby $q$ measured by the equation $xy=t^{m_q}$, where $x$ and $y$ are local parameters on the singular fiber $X_b$, and $t$ is a local parameter on $B$ near $b=f(q)$. \smallskip For each boundary class $\Delta{\mathfrak{T}}_{k,i}$ of $\overline{\mathfrak{T}}_g$, we consider its general member $C\!=\!C_1\cup C_2$, and a base curve $B$ in $\overline{\mathfrak{T}}_g$ which intersects transversally $\Delta{\mathfrak{T}}_{k,i}$ in $[C]$. In the corresponding one-parameter trigonal family $f:X\rightarrow B$, we must find the sum of the multiplicities $m_q$ of the singularities of $C$. Thus, \[\on{mult}_{\delta}\delta_{k,i}=\sum_{\,\,q\in C_{\on{sing}}}\!\!m_q.\] For most of the divisors classes, this sum is actually quite straight forward. For example, the general member $[C]\in \Delta{\mathfrak{T}}_{3,i}$ is the join of two smooth hyperelliptic curves $C_1$ and $C_2$, which intersect transversally in one point $q$. The family $X$, constructed as above, will be given locally analytically nearby $q$ by $xy=t$, and hence $\on{mult}_{\delta}\delta_{k,i}=m_q=1$. A similar situation occurs in the cases of $\Delta\mathfrak{T}_0, \Delta\mathfrak{T}_{5,i}$ and $\Delta\mathfrak{T}_{6,i}$: there is one point of transversal intersection (or one node) forcing \[\on{mult}_{\delta}\delta_0= \on{mult}_{\delta}\delta_{k,i}=1\,\,\on{for}\,\, k=3,5,6.\] \smallskip In the cases of $\Delta\mathfrak{T}_{2,i}$ and $\Delta\mathfrak{T}_{1,i}$ there are correspondingly two or three points of transversal intersection, forcing \[\on{mult}_{\delta}\delta_{2,i}=2\,\,\on{and}\,\, \on{mult}_{\delta}\delta_{1,i}=3.\] This can be also interpreted by the fact that $\Delta\mathfrak{T}_{2,i}$ and $\Delta\mathfrak{T}_{1,i}$ lie entirely in the divisor $\Delta_0$ in $\overline{\mathfrak{M}}_g$ with, $\Delta_0$ being {\it double} along $\Delta\mathfrak{T}_{2,i}$ and {\it triple} along $\Delta\mathfrak{T}_{1,i}$. \medskip A slightly more complex situation occurs in the case of $\Delta\mathfrak{T}_{4,i}$. The general member $C$ consists of two curves $C_1$ and $C_2$, meeting transversally in two points $q$ and $r$ (see Fig.~\ref{mult4}). But, as in an admissible triple cover of two ${\mathbf P}^1$'s, the points $q$ and $r$ behave differently: at one of them, say $r$, the triple cover is {\it not} ramified, while at $q$ there is ramification of index $1$. In the local analytic rings of $p,q$ and $r$ the generators of $\widehat{\mathcal{O}}_{Y,p}$ map into the squares of the generators of $\widehat{\mathcal{O}}_{X,q}$: $u\mapsto x^2, v\mapsto y^2$, and of course, $t\mapsto t$, so that the local equation of $Y$ near $p$ is $uv=t^2$, and that of $X$ near $q$ is $xy=t$. But since the triple cover is a local isomorphism of $\widehat{\mathcal{O}}_{Y,p}$ into $\widehat{\mathcal{O}} _{X,r}$, the total space of $X$ near $r$ is given locally analytically by $zw=t^2$ ($u\mapsto z, v\mapsto w, t\mapsto t$). Therefore, $m_q=1$, but $m_r=2$, and \[\on{mult}_{\delta_0}\delta_{4,i}=m_q+m_r=3.\,\,\,\qed\] \begin{figure} $$\psdraw{mult4}{2.8in}{1.3in}$$ \caption{The multiplicity mult$_{\delta_0}\delta_{4,i}$} \label{mult4} \end{figure} \subsection{The hyperelliptic locus $\overline{\mathfrak{I}}_g$ inside $\overline{\mathfrak{T}}_g$} \label{hyperelliptic locus} Although the relations proved in this paper will be valid on the Picard group $\on{Pic}_{\mathbb Q}\overline{\mathfrak{T}}_g$, it will be interesting to check what happens with the hyperelliptic curves inside the trigonal locus $\overline{\mathfrak{T}}_g$. We noted that $\overline{\mathfrak{I}}_g$ is the only boundary component of $\overline{\mathfrak{T}}_g$ of codimension 2. It is obtained as the image $pr_1(\Delta{\mathcal H}_{3,0})$. By blowing up a point on a smooth hyperelliptic curve $C$, we add a $\mathbf P^1$--component to $C$ to make it a triple cover $C^{\prime}$ (cf.~Fig.~\ref{smoothhyper}). It terms of the quasi-admissible covers, such $C^{\prime}$ behaves exactly as an irreducible singular trigonal curve in $\Delta{\mathfrak{T}}_0$. However, $C$ does not contribute to the invariant $\delta|_B$, as defined in Section~\ref{definition}. In fact, in a certain sense, it even decreases $\delta|_B$. To simplify the exposition, we shall postpone the discussion of families with hyperelliptic fibers until Section~11, where we will explain the behavior of trigonal families with finitely many hyperelliptic fibers in terms of the exceptional divisor $\Delta{\mathcal H}_{3,0}$ of the projection $pr_1$. A similar phenomenon occurs with the boundary component $\Delta\mathfrak{T}_{1,0}=pr_1(\Delta\mathcal{H}_{1,0})$, but it does not make sense to exclude its members from our discussion, since they behave exactly as members of the boundary divisor $\Delta\mathfrak{T}_{1,i}$ for $i\geq 1$. \subsection{The invariants $\mu(C)$} \label{The invariants} In the transition from the original family $X\rightarrow B$ to the minimal quasi-admissible family $\widetilde{X}\rightarrow \widetilde{Y}$ over $\widetilde{B}$, certain changes occur in the calculation of the basic invariants. To start with, it is easy to redefine $\lambda_{\widetilde{X}},\kappa_{\widetilde{X}}$ and $\delta_{\widetilde{X}}$ for $\widetilde{X}\rightarrow \widetilde{B}$: simply use the corresponding definitions from Section~\ref{definition}. Since we are interested in the slope of the family, which is invariant under base change, we may assume that $\widetilde{B}:=B$ and that $X$ is the pull-back over the new base $\widetilde{B}$. Now the difference between $X$ and $\widetilde{X}$ is reduced to several ``quasi-admissible'' blow-ups on $X$. \smallskip Blowing up smooth or rational double points on a surface does not affect its structure sheaf. Therefore, the degrees of the Hodge bundles on the two surfaces $X$ and $\widetilde{X}$ will be the same: $\lambda_{\widetilde{X}}=\lambda_X$. On the other hand, blowing up a smooth point on a surface decreases the square of its dualizing sheaf by 1, while there is no effect when blowing up a rational double point. Each type of singular fibers $C$ in $X$ requires apriori different quasi-admissible modifications (or no modifications at all), and thus decreases $\kappa_X$ by some nonnegative integer, denoted by $\mu(C)$: \begin{equation} \kappa_X=\kappa_{\widetilde{X}}+\sum_{C}\mu(C). \end{equation} Thus, $\mu(C)$ counts the number of ``smooth blow-ups'' on $C$, which are needed to obtain the minimal quasi-admissible cover $\widetilde{C}\rightarrow C$ within the surface quasi-admissible cover $\widetilde{\phi}: \widetilde{X}\rightarrow \widetilde{Y}$. \smallskip In the following Lemma, we compute the invariants $\mu(C)$ only for the general members of the boundary $\Delta{\mathfrak{T}}_g$ (cf.~Fig.~\ref{Delta-k,i}). The remaining, more special, singular curves in $\Delta{\mathfrak{T}}_g$ will be linear combinations of these $\mu(C)$'s (cf.~Sect.~11). \begin{lem} If $\mu_{k,i}$ denotes the invariant $\mu(C)$ for a general curve $C\in\Delta\mathfrak{T}_{k,i}$, then \begin{eqnarray*} \on{(a)}&&\mu_0=\mu_{1,i}=\mu_{4,i}=\mu_{6,i}=0;\\ \on{(b)}&&\mu_{2,i}=1;\\ \on{(c)}&&\mu_{3,i}=\mu_{5,i}=2. \end{eqnarray*} \label{mu(C)}\vspace*{-5mm} \end{lem} \noindent{\it Proof.} The general members of the boundary $\Delta{\mathcal H}$ are in fact the minimal quasi-admissible covers associated to the general members of the boundary $\Delta{\mathfrak{T}}$, except for $\Delta_0$ which has $\mu_0=0$. Thus, we trace the blow-ups necessary to transform the curves in Fig.~\ref{Delta-k,i} to the curves in Fig.~\ref{admissible-k,i}. For example, no blow-ups are needed in the case of $\Delta_{1,i}$, so that $\mu_{1,i}=0$, while we need 2 blow-ups in the case of $\Delta_{3,i}$, and hence $\mu_{3,i}=2$. \smallskip The only interesting situation occurs for $\Delta_{5,i}$. Apparently, there is only {\it one} added component $\mathbf P ^1$ to the original $C\in\Delta_{3,i}$, but the lemma states that $\mu_{3,i}=2$. The difference comes from the fact that near the intersection $r=C\cup\mathbf P^1$ the surface $\widetilde{X}$ has equation $xy=t^2$, i.e. $r$ is a rational double point on $\widetilde{X}$ of type $A_1$ (a similar situation occurred in Fig.~\ref{mult4}). To obtain such a point $r$ in place of a smooth point $r_1$ on $X$, we first blow up $r_1$, and then on the obtained exceptional divisor we blow up another point $r_2$, so as to end with a {\it chain of two} $\mathbf P^1$'s (cf.~Fig.~\ref{mu-5,i}). Finally, we blow down the first $\mathbf P^1$, and develop the required rational double point $r$. As a result, we have two ``smooth'' and one ``singular'' blow-ups, which implies $\mu_{5,i}=2.$ \qed \bigskip \begin{figure}[h] $$\psdraw{mu}{3in}{1in}$$ \caption{Quasi-admissible blow-ups on $\Delta_{5,i}$} \label{mu-5,i} \end{figure} \section*{\hspace*{1.9mm}5. Effective Covers} \setcounter{section}{5} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{lem}{0} \setcounter{thm}{0} \setcounter{prop}{0} \setcounter{defn}{0} \setcounter{cor}{0} \setcounter{conj}{0} \setcounter{claim}{0} \setcounter{remark}{0} \setcounter{equation}{0} \label{effectivecovers} In this section we construct the final type of triple covers in the Basic Construction. These will not be necessary for the global calculation in Section~7, so the reader may wish to skip this more technical part on a first reading, and assume in Section~6 that all covers are flat. \subsection{Construction of effective covers $\widehat{X}\rightarrow\widehat{Y}$} \label{constructioneffective} Consider a quasi-admissible cover $\widetilde{\phi}:\widetilde{X}\rightarrow \widetilde{Y}$, as given in Prop.~\ref{propquasi}. In order to use the fact that the pushforward $\widetilde{\phi}_*{\mathcal{O}_{\widetilde{X}}}$ is locally free on $\widetilde{Y}$, we need to assure that the map $\widetilde{\phi}$ is {\it flat}. Unfortunately, there are certain points on $\widetilde{X}$ where this fails to be true: exactly where the fibers of $\widetilde{X}$ are ramified as triple covers of the corresponding fibers of $\widetilde{Y}$. The situation can be resolved by several further blow-ups. \smallskip Namely, we work locally analytically near the points in $\widehat{X}$ of ramifications index 1 or 2, and consider correspondingly two cases. \subsubsection{Case of ramification index 1} \label{caseram1} This case involves members of the boundary divisors $\Delta{\mathfrak{T}}_{4,i}$ and $\Delta{\mathfrak{T}}_{5,i}$. Let $q$ be the point of ramification in the fiber of $\widetilde{X}$ over the point $p$ in the fiber of $\widetilde{Y}$ (cf.~Fig.~\ref{ram}). We use the pull-back of the map $\widetilde{\phi}$ to study the embedding of the completion of the local ring of $p$ into that of $q$: \[\widehat{\mathcal{O}}_{\widetilde{Y}\!,p}=\mathbb{C}[[u,v,t]] \big/_{\displaystyle{(uv-t^2)}} \stackrel{\widetilde{\phi}^{\#}} {\hookrightarrow}\widehat{\mathcal{O}}_{\widetilde{X}\!,q}= \mathbb{C}[[x,y,t]]\big/_{\displaystyle{(xy-t)}}.\] \begin{figure}[t] $$\psdraw{ram}{2.7in}{1.2in}$$ \caption{Non-flat cover of ramification index 1} \label{ram} \end{figure} \noindent Therefore, as an $\widehat{\mathcal{O}}_{\widetilde{Y}\!,p}$-module, \[\widehat{\mathcal{O}}_{\widetilde{X}\!,q}= \widehat{\mathcal{O}}_{\widetilde{Y}\!,p}+ \widehat{\mathcal{O}}_{\widetilde{Y}\!,p}x+ \widehat{\mathcal{O}}_{\widetilde{Y}\!,p}y.\] However, this is not a locally-free $\widehat{\mathcal{O}}_{\widetilde{Y},p}$-module: for instance, one relation among the generators is $(v-t)x+(u-t)y=0$. \smallskip Alternatively, the fiber of $\phi$ over $p$ is supported at $q$, but it is of degree 3 rather than 2, which would have been necessary for the flatness of a degree $2$ map. Indeed, as $\mathbb{C}-$vector spaces: \[\widehat{\mathcal{O}}_{\widetilde{X}\!,q}\otimes_{\widehat{\mathcal{O}}_ {\widetilde{Y}\!,p}} \on{Spec}k(p)\cong \widehat{\mathcal{O}}_{\widetilde{X}\!,q}\big/_{\displaystyle {\widehat{\mathfrak{m}}_{Y\!,p}\widehat{\mathcal{O}}_{\widetilde{X}\!,q}}}\cong \mathbb{C}[[x,y]]\big/_{\displaystyle{(x^2,y^2,xy)}}=\mathbb{C}\oplus \mathbb{C}x\oplus\mathbb{C}y.\] In Fig.~\ref{ram} one can visually observe the two distinct tangent directions at $q$ making it a {\it fat} point of degree $3$. \medskip We conclude that $\widetilde{\phi}$ is indeed non-flat at $q$. To resolve this, we blow-up $\widetilde{Y}$ at $p$ and $\widetilde{X}$ at $q$, denoting the new surfaces by $\widehat{Y}$ and $\widehat{X}$. It is easy to see that they fit into the following coming diagram: \smallskip \begin{figure}[h] $$\psdraw{blow1}{3.5in}{1.4in}$$ \caption{Resolving the case of ram. index 1} \label{resolve1} \end{figure} \smallskip In order to keep the map $\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$ of degree 3, we need to blow-up one further point on $\widetilde{X}$: if the inverse image of $p$ is $\{q,r\}$ we blow-up $r$, and thus we add the necessary component to $\widehat{X}$ to make it a triple cover of $\widehat{Y}$ (cf.~Fig~\ref{coef2.fig}). \medskip \subsubsection{Case of ramification index 2} \label{caseram2} The only boundary component, where ramification index 2 occurs, is $\Delta{\mathfrak{T}}_{6,i}$. Similarly as above, $\widetilde{\phi}: \widetilde{X}\rightarrow \widetilde{Y}$ is non-flat at $q$. Indeed, $\widehat{\mathcal{O}}_{\widetilde{X}\!,q}$ is generated as an $\widehat{\mathcal{O}}_{\widetilde{Y}\!,p}$-module by $1,x,y,x^2,y^2$, but not-freely due to the relation $u\cdot x+v\cdot y-t\cdot x^2-t\cdot y^2=0$. To resolve the apparent non-flatness of $\widetilde{\phi}$, we can blow-up once $\widetilde{X}$ and $\widetilde{Y}$ at $q$ and $p$, but this would not be sufficient. In fact, we must make further blows-ups on each surface, as Fig.~\ref{resolve2} suggests: two more on $\widetilde{X}$ and one more on $\widetilde{Y}$. \medskip \begin{figure}[h] $$\psdraw{blow2}{5.3in}{1.4in}$$ \caption{Resolving the case of ram. index 2} \label{resolve2} \end{figure} \medskip In both cases of ramification index 1 or 2, the new map $\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$ is obtained from $\widetilde{\phi}$ by a base change, and hence $\widehat{\phi}$ is {\it proper} and {\it finite}, and by construction, also a {\it flat} morphism. We call such covers {\it effective}. \medskip The above considerations combined with Prop.~\ref{quasicov} imply the existence of effective covers for our families of trigonal curves: \begin{prop} Let $X/\!_{\displaystyle{B}}$ be a family of trigonal curves with smooth general member. After several blow-ups (and possibly modulo a base change) we can associate to it an effective cover $\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$. \label{effexist} \end{prop} Here $\widehat{Y}$ is a birationally ruled surface over $B$. If the base curve $B$ is {\it not} tangent to the boundary divisors $\Delta {\mathfrak{T}}_{k,i}$, then the resulting surfaces $\widehat{X}$ and $\widehat{Y}$ will have smooth total spaces. If, moreover, $B$ intersects the $\Delta_{k,i}$'s only in their general points (as given in Fig.~\ref{Delta-k,i}), then the special fibers of $\widehat{Y}$ and $\widehat{X}$ are easy to describe (cf.~Fig.~\ref{coef1.fig}-\ref{coef3.fig}). For example, $\widehat{Y}$'s special fibers are either chains of two or three reduced projective lines, or chains of five smooth rational curves with non-reduced middle component of multiplicity two. The special fibers of $\widehat{X}$ can also contain nonreduced components (of multiplicity 2 or 3), and this occurs only in the ramification cases discussed above ($\Delta{\mathfrak{T}}_{k,i}$ for $k=4,5,6$). \subsection{Change of $\lambda_X,\kappa_X$ and $\delta_X$ in the effective covers} \label{change} This is an analog to the discussion in Section~\ref{The invariants}. After the necessary base changes we again identify, without loss of generality, the new base curve $\widetilde{B}$ with $B$, and the pull-back of $X$ over $\widetilde{B}$ with $X$, and we redefine the basic invariants $\lambda_{\widehat{X}}$ and $\kappa_{\widehat{X}}$ for the effective family $\widehat{X}$ over $\widetilde{B}$. (It doesn't make sense to define directly $\delta_{\widehat{X}}$, because of the nonreduced fiber components in $\widehat{X}$. We could, of course, set $\delta_{\widehat{X}}=12\lambda_{\widehat{X}}-\kappa_{\widehat{X}}$, but we will not need this in the sequel.) Now the original $X$ and the effective $\widehat{X}$ differ by ``quasi-admissible'' and ``effective'' blow-ups. The connections between the invariants of $X$, $\widetilde{X}$ and $\widehat{X}$ are expresssed by the following \begin{lem} With the above notation, \begin{eqnarray*} \on{(a)}&\!\!\!\!&\displaystyle{\lambda_X}=\lambda_{\widetilde{X}}= \lambda_{\widehat{X}};\\ \on{(b)}&\!\!\!\! &\kappa_X=\kappa_{\widetilde{X}}+\sum_{C}\mu(C);\\ \on{(c)}&\!\!\!\! &\displaystyle{\kappa_{\widetilde{X}}= \kappa_{\widehat{X}}+\sum_{\on{ram}1}1+\sum_{\on{ram}2}3}. \end{eqnarray*} \label{changeinv}\vspace*{-10mm} \end{lem} \begin{proof} In view of Lemma~\ref{mu(C)}, the first and the second statements are obvious. Obtaining a flat cover $\widehat{X}\rightarrow \widehat{Y}$ requires blowing up on $\widetilde{X}$ one smooth point for each ramification index 1, and three smooth points for each ramification index 2. Hence the relation between $\kappa_{\widehat{X}}$ and $\kappa_{\widetilde{X}}.$ \end{proof} \section*{\hspace*{1.9mm}6. Embedding $\widehat{X}$ in a Projective Bundle over $\widehat{Y}$} \setcounter{section}{6} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{lem}{0} \setcounter{thm}{0} \setcounter{prop}{0} \setcounter{defn}{0} \setcounter{cor}{0} \setcounter{conj}{0} \setcounter{claim}{0} \setcounter{remark}{0} \setcounter{equation}{0} \label{embedding} Given the effective degree $3$ map $\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$, our next step is to embed $\widehat{X}$ into a projective bundle $\mathbf P V$ of rank $1$ over the birationally ruled surface $\widehat{Y}$. We shall consider a degree $3$ morphism $\widehat{\phi}$, but the same discussion is valid for any degree $d$. \subsection{Trace map} \label{tracemap} Since $\widehat{\phi}$ is flat and finite, the pushforward $\widehat{\phi}_*(\mathcal O_{\widehat{X}})$ is a locally free sheaf on $\widehat{Y}$ of rank $3$. Define the {\it trace} map \[\on{tr}:\widehat{\phi} _*(\mathcal O_{\widehat{X}})\rightarrow \mathcal O_{\widehat{Y}}\] as follows. The finite field extension $K(\widehat{X})$ of $K(\widehat{Y})$ induces the {\it algebraic} trace map $\on{tr^\#}:K(\widehat{X})\rightarrow K(\widehat{Y})$, defined by $\on{tr^\#}(a)=\textstyle{\frac{1}{3}}(a_1+a_2+a_3)$. Here the $a_i$'s are the conjugates of $a$ over $K(\widehat{Y})$ in an algebraic closure of $K(\widehat{X})$. The restriction $\on{tr^\#}{|}_{K(\widehat{Y})}=\on{id}_{K(\widehat{Y})}$. Over an affine open $U={\on {Spec}}\,A\subset \widehat{Y}$ and its affine inverse $\widehat{\phi}^{-1}(U)= {\on {Spec}}\,B\subset \widehat{X}$, $B$ is the integral closure of $A$ in its field of fractions $K(\widehat{X})$. Therefore, the trace map restricts to the $A$-module homomorphism $\on{tr^\#}:B\rightarrow A$. We have a commutative diagram: \medskip \begin{figure}[h] \vspace*{2mm} \begin{picture}(4,2.5)(4,0) \put(7.8,2.2){$B\hspace{1mm}\hookrightarrow \hspace{1mm}K(\widehat{X})$} \multiput(8,1.2)(1.6,0){2}{\vector(0,1){0.9}} \put(7.8,0.7){$A\hspace{1mm}\hookrightarrow \hspace{1mm}K(\widehat{Y})$} \put(7.8,1.65){\oval(0.35,0.9)[l]} \put(7.8,1.2){\vector(1,0){0.1}} \put(7.8,2.1){\line(1,0){0.1}} \put(9.8,1.65){\oval(0.35,0.9)[r]} \put(9.8,1.2){\vector(-1,0){0.1}} \put(9.8,2.1){\line(-1,0){0.1}} \multiput(7,1.5)(3.1,0){2}{$\on{tr}^{\#}$} \put(3.2,0.7){$U=\on{Spec}\,A$} \put(2.3,2.2){$\widehat{\phi}^{-1}(U)=\on{Spec}\,B$} \put(3.7,1.5){$\big\downarrow$} \put(3.7,1.4){$\big\downarrow$} \put(3.3,1.5){$\widehat{\phi}$} \end{picture} \vspace*{-6mm} \caption{The trace map} \end{figure} The so-defined local maps $\on{tr}:\widehat{\phi}_*\mathcal O_{\on {Spec}\,B} \twoheadrightarrow \mathcal O_{\on {Spec}\,A}$ patch up to give a global trace map $\on{tr}:\widehat{\phi}_*\mathcal O_{\widehat{X}} \twoheadrightarrow \mathcal O_{\widehat{Y}}.$ Let $\mathcal V$ be the kernel of $\on{tr}$: \begin{equation} 0\rightarrow {\mathcal V}\rightarrow {\widehat{\phi}} _*{\mathcal O}_{\widehat X}\stackrel {\on{tr}}{\rightarrow}{\mathcal O}_{\widehat Y}\rightarrow 0. \label{splitting} \end{equation} Note that $\mathcal V$ is locally free of rank $2$. The natural inclusion $\mathcal O_{\widehat{Y}}\hookrightarrow \widehat{\phi} _*\mathcal O_{\widehat{X}}$, composed with $\on{tr}$, is the identity on $\mathcal O_{\widehat{Y}}$, hence the exact sequence splits: \begin{equation} {\widehat{\phi}}_*{\mathcal O}_{\widehat X}={\mathcal O}_{\widehat Y}\oplus {\mathcal V}. \label{directsum} \end{equation} \smallskip \subsubsection{Geometric interpretation of the trace map} It is useful to interpret the trace map geometrically in terms of the corresponding vector bundles $\widehat{\phi}_*{O_{\widehat X}}$, $O_{\widehat Y}$ and $V$ associated to the sheaves $\widehat{\phi}_*{\mathcal O}_{\widehat X}$, ${\mathcal O}_{\widehat Y}$ and $\mathcal V$. We again localize over affine opens, and if necessary, we shrink $U=\on{Spec}\,A$ so that $\widehat{\phi}_* \mathcal O_{\widehat X}$ becomes a {\it free} $\mathcal O_{\widehat Y}$-module. \smallskip Let $p$ be a closed point in $\on{Spec}\,A$ with maximal ideal $\mathfrak{p} \subset A$, having three distinct preimages $q,r,s\in\on{Spec}\,B$ with maximal ideals $\mathfrak{q,r,s}\subset B$. Since $B$ is a free $A$-module, the quotient $B/{\mathfrak{p}}B$ is a 3-dim'l algebra over the ground field $\on{k}(p)=A/{\mathfrak{p}}$, i.e. a 3-dim'l vector space over ${\mathbb C}$. On the other hand, from $\mathfrak{qrs}=\mathfrak{q}\cap \mathfrak{r}\cap \mathfrak{s}$ and the Chinese Remainder Theorem, it is clear that $B/{\mathfrak{p}}B\cong B/{\mathfrak{q}} \oplus B/{\mathfrak{r}} \oplus B/{\mathfrak{s}}\cong {\mathbb C}\overline{f}_q\oplus{\mathbb C}\overline{f}_r\oplus{\mathbb C}\overline{f}_s.$ The generators $\overline{f}_q,\overline{f}_r,\overline{f}_s$ are chosen as usual: $\overline{f}_q$, for instance, is the residue in $\on{k}(q)$ of a function $f_q\in B$ such that $f_q\equiv 1(\on{mod}\mathfrak{q}),\,\, f_q\equiv 0(\on{mod}\mathfrak{r,s})$. \smallskip In the Groethendieck style, the vector bundle over $\widehat Y$ associated to $\widehat{\phi}_*\mathcal O_{\widehat X}$ is $\on{ Spec}\on{S}(B_A)$, where $\on{S}(B_A)$ is the symmetric algebra of $B$ over $A$. Its fiber over $p$ is the pull-back $\on{ Spec}\on{S}(B_A)\times _{\on{Spec\,A}}\on{Spec\,k}(p) = \on{ Spec}(\on{S}(B_A)\times_A A/\mathfrak {p})=\on{Spec}\on{S(B}/{\mathfrak{p}}B).$ We prefer to work with the dual $\widehat{\phi}_*{O_{\widehat X}}$ of this bundle, and the same goes for projectivizations: we projectivize the 1-dim'l subspaces of $\widehat{\phi}_*{O_{\widehat X}}$ rather than its 1-dim'l quotients. In view of this convention, the fiber of the bundle $\widehat{\phi}_*{O_{\widehat X}}$ is canonically identified as \[(\widehat{\phi}_*{O_{\widehat X}})_p=B/{\mathfrak p}B\cong {\mathbb C}\overline{f}_q \oplus{\mathbb C}\overline{f}_r\oplus{\mathbb C}\overline{f}_s \cong {\bf A}^3_{\mathbb C}.\] The generators $\overline{f}_q,\overline{f}_r,\overline{f}_s$ span three lines in ${\bf A}^3_{\mathbb C}$, which can be canonically described: the line $\Lambda_q= {\mathbb C}\overline{f}_q$, for example, corresponds to all functions regular at $q,r$ and $s$, and vanishing at $r$ and $s$. \smallskip Similarly, the vector bundle $O_{\widehat Y}$ associated to $\mathcal O_{\widehat Y}$ has fiber $(O_{\widehat Y})_p=A/{\mathfrak {p}} \cong {\mathbb C}\overline{h}_p$, where $h_p$ is a function near $p$ having residue $h_p(p)=1$ in $\on{k}(p)$. The trace map $\on{tr}:\widehat{\phi}_*\mathcal O_{\widehat{X}} \twoheadrightarrow \mathcal O_{\widehat{Y}}$ translates fiberwise in terms of the vector bundles $\widehat{\phi}_*{O_{\widehat X}}$ and $O_{\widehat Y}$ as: \[\on{tr}_p:{\mathbb C}\overline{f}_q\oplus{\mathbb C}\overline{f}_r\oplus{\mathbb C}\overline{f}_s \rightarrow {\mathbb C}\overline{h}_p,\,\,\overline{f}\mapsto \frac{1}{3} (f(q)+f(r)+f(s)).\] Finally, the locally free subsheaf ${\mathcal V}=\on{Ker(tr)}\subset \widehat{\phi}_*{\mathcal O}_{\widehat X}$ is associated to a vector bundle $V$ with fiber $V_p=\{\overline{f}\,\,|\,\,f(q)+f(r)+f(s)=0\} \subset (\widehat{\phi}_*{O_{\widehat X}})_q.$ Equivalently, from the direct sum (\ref{directsum}), $V_p=(\widehat{\phi}_*{O_{\widehat X}})_p \big{/}_{\textstyle{\Lambda}}$, where the line $\Lambda=\{\overline{f}\,\,|\,\,f(q)=f(r)=f(s)\}$ corresponds to pull-backs of functions regular at $p$. \begin{figure}[h] $$\psdraw{embedding}{3in}{1.88in}$$ \caption{Geometric interpretation of $tr$} \label{geometric} \end{figure} Since the four lines $\Lambda_q,\Lambda_r,\Lambda_s$ and $\Lambda$ are in general position in the fiber $(\widehat{\phi} _*{O_{\widehat X}})_p$, modding out by $\Lambda$ yields three distinct lines in the fiber $V_p$ (cf.~Fig.~\ref{geometric}). Therefore, projectivizing $V_p$ naturally induces three distinct points $Q,R,S$ in the fiber ${\mathbf P}^1$ of ${\mathbf P}V$. Going the other way around the diagram, we first projectivize $(\widehat{\phi}_*{O_{\widehat X}})_q\cong{\bf A}^3$, and then we project from the point $[\Lambda]$ onto the fiber of ${\mathbf P}V$. In other words, $\pi_{[\Lambda]} :{\mathbf P}^2\dashrightarrow {\mathbf P}^1$ is well-defined at $[\Lambda_q],[\Lambda_r]$ and $[\Lambda_s]$. \medskip This completes the interpretation of the trace map in the case of three distinct preimages $q,r,s$ in $\widehat{X}$. In case of only two or one preimage of $p$ in $\widehat{X}$, one modifies correspondingly the above interpretation. \subsection{$\widehat{X}$ embeds naturally in $\mathbf P V$ over $\widehat{Y}$} \label{naturally} We construct the map $i:\widehat{X}\hookrightarrow \mathbf P V$ via the use of an invertible relative dualizing sheaf $\omega_{\widehat X/ \widehat Y}$. Its existence imposes a mild technical condition on the schemes $\widehat X$ and $\widehat Y$: we assume that they are Gorenstein, i.e. Cohen-Macaulay with invertible dualizing sheaves $\omega_{\widehat X/{\mathbb C}}$ and $\omega_{\widehat Y/{\mathbb C}}$. In our situation this will be sufficient. As we noted in Section~\ref{constructioneffective}, when the base curve $B$ is {\it not} tangent to the boundary divisors $\Delta{\mathfrak{T}}_{k,i}$, then $\widehat{X}$ and $\widehat{Y}$ are smooth surfaces. The remaining cases are ``local'' base changes of these, and the construction carries over. \begin{prop} Let $\widehat{\phi}:\widehat X\rightarrow \widehat Y$ be a flat and finite degree $d$ morphism of Gorenstein schemes, with $\widehat Y$ integral. Then $\widehat{\phi}$ factors through a natural embedding of $\widehat X$ into the projective bundle $\mathbf P V$, followed by the projection $\pi: \mathbf P V\rightarrow \widehat Y$ (cf.~Fig.~\ref{basicconstruction}). \label{propembedding} \end{prop} For easier referencing in the sequel, we have kept the notation $\widehat X$ and $\widehat Y$, but the statement is true for {\it any} schemes satisfying the Gorenstein condition. For another proof of Prop.~\ref{propembedding}, see \cite{embedding}. \medskip \noindent{\it Proof of Prop.~\ref{propembedding}}. Here we construct the map $i:\widehat X\rightarrow \mathbf P V$, give the proof of its regularity, and point out how to show its injectivity. \begin{figure}[h] \begin{picture}(6,2.5)(-2.3,2.5) \put(-1.7,4){$\mathbf P({\mathcal O}_{\widehat Y})\stackrel {\on{tr}^{\#}}{\hookrightarrow}\mathbf P({\widehat{\phi}}_*{\mathcal O}_{\widehat X}) \stackrel{\rho}{\dashrightarrow} \mathbf P V$} \put(0.95,3.05){\vector(0,1){0.8}} \put(0.75,2.6){$\widehat X$} \put(0.6,3.35){$\psi$} \put(1.1,2.95){\vector(2,1){1.85}} \put(1.7,3.35){$i$} \end{picture} \caption{Embedding $i:\widehat X\hookrightarrow \mathbf P V$} \label{construction of i} \end{figure} \subsubsection{Construction of the embedding map} According to Prop. II.7.12 \cite{Hartshorne}, to give a morphism $\psi:\widehat X\rightarrow {\mathbf P}(\widehat{\phi} _*(\mathcal O_{\widehat X}))$ over $\widehat Y$ is equivalent to give an invertible sheaf $\mathcal L$ on $\widehat Y$ and a surjective map of sheaves $\widehat{\phi}^*(\widehat{\phi}_*(\mathcal O_{\widehat X}){\textstyle {\widehat{\phantom{n}}}})\twoheadrightarrow \mathcal L$. Recall from {\it relative Serre-duality} that $(\widehat{\phi}_*\mathcal O_{\widehat X}){\textstyle {\widehat{\phantom{n}}}}\cong \widehat{\phi} _*\omega_{\widehat X/\widehat Y}$, and let $\mathcal L=\omega_{\widehat X/\widehat Y}$. The natural morphism \[\sigma:\widehat{\phi}^*\widehat{\phi} _*\omega_{\widehat X/\widehat Y}\rightarrow\omega_{\widehat X/\widehat Y}\] is {\it surjective}. This is in fact true for any quasicoherent sheaf $\mathcal F$ on $\widehat X$. Indeed, restricting to the affine open sets $\widehat{\phi}:\on{Spec}\,B\rightarrow \on{Spec}\,A$, we have $\mathcal F=M^{\sim}$ for some finitely generated $B$-module $M$, and $\widehat{\phi}^*\widehat{\phi} _*{\mathcal F}=\widehat{\phi}^*(M_A)^{\sim}=(M_A\otimes_A B)^{\sim}.$ The surjective $B$-module homomorphism $M_A\otimes_A B \twoheadrightarrow M$, given by $m\otimes b \mapsto b\circ m$, induces $\widehat{\phi}^*\widehat{\phi}_*{\mathcal F}\twoheadrightarrow \mathcal F$. Thus, the above map $\sigma$ naturally defines a morphism $\psi:\widehat X\rightarrow {\mathbf P}(\widehat{\phi}_*(\mathcal O_ {\widehat X}))$ over $\widehat Y$. Projectivizing $0\rightarrow {\mathcal O}_{\widehat Y} \rightarrow {\widehat{\phi}} _*{\mathcal O}_{\widehat X}\rightarrow {\mathcal V} \rightarrow 0$ gives a sequence of projective bundles, as in Fig.~\ref{construction of i}. The map $\rho$ is undefined exactly on the image of $\on{tr} ^{\#}$. Composing $\rho$ with the map $\psi$ yields the map $i:\widehat{X}\dasharrow \mathbf P V$, which we claim is a regular map. \subsubsection{Regularity and injectivity of $i$.} To see regularity, we show that the restriction of $\sigma|_{\widehat{\phi}^*(\mathcal V{{\widehat{\phantom{n}}}})} :{\widehat{\phi}^*(\mathcal V{{\widehat{\phantom{n}}}})} \rightarrow \omega_{\widehat X/\widehat Y}$ is also surjective. Indeed, we again work locally, and let $B=A\oplus C$ be the decomposition of $B$ via the trace map as a free $A$-module, where $C=A\cdot b_1\oplus A\cdot b_2$ with $\on{tr}b_1=\on{tr}b_2=0$. Let $\omega_{\widehat X/\widehat Y}=M^{\sim}$ for some finitely generated $B$-module $M$. Recall that $\widehat{\phi}_*\omega_{\widehat X/\widehat Y}\cong (\widehat{\phi}_*\mathcal O_{\widehat X}){\textstyle{\widehat{\phantom{n}}}}$, so that as $A$-modules: $M\cong (B_A){{\widehat{\phantom{n}}}}=\on{Hom}_A (B,A)$, and $B$ acts on $M$ by \[(b\circ f)(x)=f(bx)\,\,\,\on{for}\,\,\,f\in \on{Hom}_A(B,A),\,x\in B.\] Naturally, the sheaf $\mathcal V=C^{\sim}$, and $\widehat{\phi}^* (\mathcal V{{\widehat{\phantom{n}}}})=(\on{Hom}_A(C,A)\otimes_A B)^{\sim}$, where we think of $f\in \on{Hom}_A(C,A)$ as an element of $\on{Hom}_A(B,A)$ by extending it via $f(1)=0$. Our statement is equivalent to showing that the $B$-module homomorphism \[\sigma:\on{Hom}_A(C,A)\otimes_A B \rightarrow \big{(}\on{Hom}_A(B,A) \big{)_B},\,\,f\otimes b \mapsto b\circ f,\] is surjective. In fact, it suffices to show that the trace map is in the image of $\sigma$, i.e. to find $c_1,c_2\in B$ such that \begin{equation} \on{tr}\equiv c_1\circ {\pi_1}+c_2\circ {\pi_2}. \label{trace equation} \end{equation} Here $\pi_j:B\rightarrow A$ gives the $b_j$-coordinate of $b\in B$, $j=1,2$. Set $c_1=b_1-\pi_1(b_1^2)$ and $c_2=-\pi_1(b_1b_2)$. Evaluating both sides of (\ref{trace equation}) at $1,b_1$ and $b_2$ yields the same result, hence the identity is established, and $\sigma|_{\widehat{\phi}^*(\mathcal V{{\widehat{\phantom{n}}}})} :{\widehat{\phi}^*(\mathcal V{{\widehat{\phantom{n}}}})} \rightarrow \omega_{\widehat X/\widehat Y}$ is surjective. \smallskip We have shown that the composition $\rho\circ \psi= i:\widehat{X}\dasharrow \mathbf P V$ is a regular map on $\widehat{X}$. Alternatively, one could employ the geometric interpretation of the trace map. A {\it general} point $p\in {\widehat Y}$ has three preimages $q,r,s$ in $\widehat X$, each of which defines canonically a distinct point $[\Lambda_q],[\Lambda_r]$ or $[\Lambda_s]$ in the fiber of ${\mathbf P}(\widehat{\phi}_*{\mathcal O}_{\widehat X})$. As we pointed above, the projection $\pi_{[\Lambda]} :{\mathbf P}^2\dashrightarrow {\mathbf P}^1$ is well-defined at $[\Lambda_q],[\Lambda_r]$ and $[\Lambda_s]$. But $\pi_{[\Lambda]}$ is precisely the fiberwise restriction of $\mathbf P({\widehat{\phi}}_* {\mathcal O}_{\widehat X}) \stackrel{\rho}{\dashrightarrow} \mathbf P V$, which shows again that the composition $i=\rho\circ\psi:\widehat{X}\dasharrow \mathbf P V$ is regular on an open set of $\widehat X$. One makes the necessary modifications in the cases of fewer preimages of $p$ in $\widehat{X}$. Finally, one can show, using similar methods (either algebraically or geometrically), that the map $i$ is also an embedding. \qed \medskip \noindent{\bf Remark 6.1.} Since the general fiber $C$ of $\widehat{X}$ is a smooth trigonal curve, the restriction $i|_{\displaystyle{C}}$ embeds $C$ in a ruled surface ${\mathbf F}_k$ over the corresponding fiber $F_{\widehat{Y}}= \mathbf P^1$ of $\widehat{Y}$, where ${\mathbf F}_k=\mathbf P(V|_{F_{\widehat{Y}}})$. In Section~\ref{Maroniinvariant} we will describe how the ruled surface ${\mathbf F}_k$ varies as the fiber $C$ varies in $\widehat{X}$. \bigskip \section*{\hspace*{1.9mm}7. Global Calculation on a Triple Cover $X\rightarrow Y$} \setcounter{section}{7} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{lem}{0} \setcounter{thm}{0} \setcounter{prop}{0} \setcounter{defn}{0} \setcounter{cor}{0} \setcounter{conj}{0} \setcounter{claim}{0} \setcounter{remark}{0} \setcounter{equation}{0} \label{triplecover} In this section we consider the simplest case of effective covers, namely, when the original family $X$ is itself a triple cover of a {\it ruled surface} $Y$ over the base curve $B$. This happens exactly when all fibers of $X$ are irreducible, and the linear system of $g^1_3$'s on the smooth fibers extends over the singular fibers to base-point free line bundles of degree 3 with two linearly independent sections. As we saw in Section~\ref{quasi-admissible}, we can patch together all these $g^1_3$'s in a line bundle ${\mathcal L}$ on the total space of $X$. Thus, $X$ will map to ${\mathbf P}(H^0(X,{\mathcal L})^{\widehat{\phantom{n}}})$ via $\phi_{\mathcal L}$, and this map will factor through our ruled surface $Y$ over $B$: \begin{figure}[h] $$\psdraw{triple}{2.7in}{1.31in}$$ \caption{Basic triple cover case.} \label{fig.triplecover} \end{figure} Equivalently, we can describe such a family $X\rightarrow B$ via the classification of the boundary components of the trigonal locus in Section~4: in $\overline{\mathfrak T}_g$ the base curve $B$ meets only the boundary component $\Delta{\mathfrak{T}}_0$ of irreducible curves ($\delta_0|_B>0$), and there are no hyperelliptic fibers in $X$ ($B\cap \overline{\mathfrak I}_g=\emptyset$). \subsection{Global versus local calculation} \label{global} As it will turn out, the calculation of the slope $\delta_X /\lambda_X$ in this basic case yields the actual maximal bound $\frac{36(g+1)}{5g+1}$: any addition of singular fibers belonging to other boundary components of $\overline{\mathfrak T}_g$ will only decrease the ratio. Henceforth, we distinguish among two types of calculation: {\it global} and {\it local}. The {\it global} calculation refers to finding the coefficients of $\delta_0$ and the Hodge bundle $\lambda|_{\overline{\mathfrak T}_g}$ in a relation in $\on {Pic}_{\mathbb Q}\overline{\mathfrak T}_g$ involving {\it all} boundary classes. The {\it local} calculation provides the remaining coefficients by considering {\it local invariants} of each individual boundary class (cf.~Sect.~8). \subsection{The basic construction} \label{basic} For the remainder of this section, we consider a family $X\rightarrow B$ such that, as above, $X$ is a triple cover of the corresponding ruled surface $Y$, and we carry out the proposed global calculation. \unitlength 0.11in \begin{figure}[h] \begin{picture}(10,9)(25,21) \put(26.5,24){$X$} \put(28.4,24.2){\vector(1,0){3.1}} \put(32,24){$Y$} \put(29,20){$B$} \put(27.3,23.5){\vector(1,-1){2}} \put(32.3,23.5){\vector(-1,-1){2}} \put(26.6,22){$f$} \put(31.75,22){$h$} \put(29.45,24.7){$\phi$} \put(27.3,25.4){\vector(1,1){2}} \put(30.3,27.4){\vector(1,-1){2}} \put(29,27.8){${\mathbf P}V$} \put(32,26.4){$\pi$} \put(27.4,26.4){$i$} \end{picture} \caption{Triple Case} \label{basicconstruction} \end{figure} Recall that the pushforward of the structure sheaf ${\mathcal O}_X$ to $Y$ is a locally free sheaf of rank $3$. In the exact sequence: \[0\rightarrow E\rightarrow {\phi}_*{\mathcal O}_X\stackrel {\on{tr}}{\rightarrow}{\mathcal O}_Y\rightarrow 0,\] the kernel of the trace map $\on{tr}$ is a vector bundle $E$ on $Y$ of rank $2$, and $X$ naturally embeds in the rank 1 projective bundle ${\mathbf P}V$ over $Y$, where $V=E\,\widehat{\phantom{n}}$. Any rank 2 vector bundle $E$ has the same projectivizations as its dual bundle $V$ since $E\cong \bigwedge^2E\otimes V$, where $\bigwedge^2E$ is a line bundle. For easier notation, in the trigonal case we use the dual $V$ instead of $E$ from Section~\ref{embedding}. A basis for $\on{Pic}Y$ can be chosen by letting $F_Y$ be the fiber of $Y$, and $B^{\prime}$ be any section of $Y\rightarrow B$. Hence $\on{Pic}Y={\mathbb Z}B^{\prime}\oplus {\mathbb Z}F_Y$. We normalize by replacing $B^{\prime}$ with the ${\mathbb Q}$-linear combination $B_0=B^{\prime}-\displaystyle{\frac{(B^{\prime})^2}{2}}{F_Y}$, and provide a $\mathbb{Q}$-basis for $\on{Pic}_{\mathbb Q}Y$: \begin{equation} \on{Pic}_{\mathbb Q}Y={\mathbb Q}B_0\oplus {\mathbb Q}F_Y\,\,\,\on{with}\,\,\, B_0^2=F_Y^2=0\,\,\,\on{and}\,\,\,B_0\cdot F_Y=1. \label{normalize} \end{equation} Let $\zeta$ denote the class of the hyperplane line bundle ${\mathcal O}_{{\mathbf P}V}(+1)$ on ${\mathbf P}V$, and let $c_1(V)$ and $c_2(V)$ be the Chern classes of $V$ on $Y$. The Chow ring $A({\mathbf P}V)$ is generated as a $\pi^*(A(Y))$-module by $\zeta$ with the only relation: \begin{equation} \zeta^2+\pi^*c_1(V)\zeta+\pi^*c_2(V)=0. \label{zeta-relation} \end{equation} In particular, for the Picard groups: \begin{equation} \on {Pic}_{\mathbb Q}({\mathbf P}V)=\pi^*(\on{Pic} _{\mathbb Q}Y)\oplus {\mathbb Q}\zeta. \label{Q-basis} \end{equation} \smallskip \noindent As a divisor on ${\mathbf P}V$, the surface $X$ meets the fiber $F_{\pi}$ of $\pi$ generically in three points ($X$ maps three-to-one onto $Y$). Thus in the Chow ring $A({\mathbf P}V)$ we have $[X]\cdot [F_{\pi}]=3$, which simply means that $X$ can be expressed as \[X\sim 3\zeta + \pi^*D\] for some divisor $D$ on $Y$ (see (\ref{Q-basis})). With respect to the chosen basis for $\on{Pic}_{\mathbb Q}Y$: \begin{equation} D\sim aB_0+bF_Y\,\,\on{and}\,\, c_1(V)\sim cB_0+dF_Y \label{define D,c1(V)} \end{equation} for some $a,b,c,d\in {\mathbb Z}$. Note that $\deg(D|_{B_0})=b$ and $\deg(c_1(V)|_{B_0})=d$. \subsection{Relation among the divisor classes $D$ and $c_1(V)$} \label{relation} It is evident that the divisors $D$ and $c_1(V)$ cannot be independent on ruled surface $Y$ since both are canonically determined by the surface $X$. The relation is in fact quite straightforward. \begin{lem} With the above notation for the triple cover $\phi:X\rightarrow Y$, we have $D=2c_1(V)$ in $\on{Pic}Y$. \label{D=2c_1(V)} \end{lem} \begin{proof} We start with the standard exact sequence for the divisor $X$ on ${\mathbf P}V$: \begin{equation} 0\rightarrow \mathcal{O}_{{\mathbf P}V}(-X)\rightarrow \mathcal{O}_{{\mathbf P}V} \rightarrow \mathcal{O}_X\rightarrow 0. \label{X-divisorsequence} \end{equation} Pushing to $Y$ yields: \begin{equation} 0 \!\rightarrow \!\pi_*\mathcal{O}_{{\mathbf P}V}(-X) \!\rightarrow\! \pi_*\mathcal{O}_{{\mathbf P}V}\!\rightarrow\! \pi_*\mathcal{O}_X \!\rightarrow \! R^1\pi_*\mathcal{O}_{{\mathbf P}V}(-X) \!\rightarrow \!R^1 \pi_*\mathcal{O}_{{\mathbf P}V} \!\rightarrow \cdots \label{pushforward} \end{equation} \noindent It is easy to show that $R^1\pi_*\mathcal{O}_{{\mathbf P}V}=0$ and $\pi_*\mathcal{O}_{{\mathbf P}V}(-X)=0$. This follows from Grauert's theorem \cite{Hartshorne}: $h^1(\mathcal{O}_{{\mathbf P}V}|_{F_{\pi}})=h^1(\mathcal O_{{\mathbf P}^1})=0,$ and \[h^0(\mathcal{O}_{{\mathbf P}V}(-X)|_{F_{\pi}})=h^0(\mathcal{O}_{{\mathbf P}V} (-3\zeta-\pi^*D)|_{F_{\pi}})=h^0(\mathcal{O}_{{\mathbf P}^1}(-3))=0.\] Furthermore, $\pi_*\mathcal{O}_{{\mathbf P}V}= \mathcal{O}_Y$ and $\pi_*\mathcal{O}_X=\phi_*\mathcal{O}_X$, so that (\ref{pushforward}) becomes \begin{equation} 0\rightarrow \mathcal{O}_Y\rightarrow \phi_*\mathcal{O}_X\rightarrow R^1\pi_*\mathcal{O}_{{\mathbf P}V}(-X)\rightarrow 0. \label{remainingnonzero} \end{equation} From relative Serre-duality, and using the first Chern class of the relative dualizing sheaf, $c_1(\omega_{\pi})$ (cf.~(\ref{omega-pi})): \[R^1\pi_*\mathcal{O}_{{\mathbf P}V}(-X) \cong \big(\pi_*(\omega_{\pi}\otimes \mathcal {O}_{{\mathbf P}V}(X))\big)\widehat{\phantom{t}}= \big(\pi_*\mathcal{O}_{{\mathbf P}V}(\zeta+\pi^*D-\pi^*c_1(V))\big) \widehat{\phantom{t}}.\] \noindent Since $\pi_*\mathcal{O}_{{\mathbf P}V}(\zeta)=V\widehat{\phantom{t}}$ (cf.~ ~\cite{BPV}), \begin{equation} R^1\pi_*\mathcal{O}_{{\mathbf P}V}(-X) \cong \big[V\widehat{\phantom{t}} \otimes \mathcal{O}_Y(D-c_1(V))\big]\widehat{\phantom{t}}. \label{tranformedsequence} \end{equation} Finally, combining (\ref{tranformedsequence}) with (\ref{remainingnonzero}) and $\phi_*\mathcal{O}_X/\mathcal{O}_Y=V\widehat{\phantom{t}}$, we arrive at \[V\cong V\widehat{\phantom{t}}\otimes \mathcal{O}_Y(D-c_1(V))\,\,\Rightarrow\,\, \mathcal{O}_Y(D-c_1(V))\cong \bigwedge ^2 V\cong\mathcal{O}_Y(c_1(V)),\] and hence $D=2c_1(V)$ in $\on{Pic}Y. \,$ \end{proof} \subsection{Global calculation of $\lambda_X$ and $\kappa_X$.} \label{globalcalculation} In the following proposition we express $\lambda_X$ and $\kappa_X$ in terms of $\deg(c_1(V)|_{B_0})=d$ and the Chern class polynomial $c_1^2(V)-4c_2(V)$, both of which are independent of the choice of the vector bundle $V$. Indeed, if we twist $V$ by a line bundle $M$ on $Y$ and set $V^{\prime}=V\otimes M$, then \[c_1(V^{\prime})=c_1(V)+2c_1(M),\,\, c_2(V^{\prime})=c_2(V)+c_1(V)c_1(M)+c_1(M)^2,\] \[\Rightarrow\,\,c_1(V^{\prime})^2-4c_2(V^{\prime})=c_1(V)^2-4c_2(V).\] Recall the notation of (\ref{define D,c1(V)}). In order to make $d$ also invariant, we use $b=2d$ from Lemma~\ref{D=2c_1(V)} and write $d=2b-3d$. Now, if we replace ${\mathbf P}V$ with its isomorphic ${\mathbf P}V^{\prime}$ (cf.~Fig.~\ref{invariance}), and set $\zeta^{\prime}=i^* \zeta\otimes(\pi^{\prime})^*M^{-1}$ to be the new hyperplane bundle, then in $\on{Pic}({\mathbf P}V)$: $X\sim 3\xi^{\prime}+(\pi^{\prime})^*D^{\prime}$ with $D^{\prime}\sim D+3c_1(M)$. Hence \[2D^{\prime}-3c_1(V^{\prime})=2D+6c_1(M)-3c_1(V)-6c_1(M)=2D-3c_1(V),\] and equating their degrees on $B_0$, we obtain $2b^{\prime}-3d^{\prime}=2b-3d$. \unitlength 0.11in \begin{figure}[h] \begin{picture}(10,7)(25,20.5) \put(26,24){${\mathbf P}V^{\prime}$} \put(28.4,24.2){\vector(1,0){3.1}} \put(32,24){${\mathbf P}V$} \put(29,20){$Y$} \put(27.3,23.5){\vector(1,-1){2}} \put(32.3,23.5){\vector(-1,-1){2}} \put(26.5,22){$\pi^{\prime}$} \put(31.75,22){$\pi$} \put(29.45,24.7){$i$} \end{picture} \caption{$V^{\prime}=V\otimes M$} \label{invariance} \end{figure} In other words, the following formulas for $\lambda_X$ and $\kappa_X$ would be valid for any vector bundle $V^{\prime}$ in place of the canonically defined $V$, as long as the diagram of the basic construction (cf.~Fig.~\ref{basicconstruction}) is satisfied, and as long as we adjust the degree $d=\deg(c_1(V)|_{B_0})$ by its invariant form $2b-3d= 2\deg(D|_{B_0})-3\deg(c_1(V)|_{B_0})$. \begin{prop} Let $\phi:X\rightarrow Y$ be a degree 3 map from the original family $X$ of trigonal curves to the ruled surface $Y$ over $B$. The invariants $\lambda_X$ and $\kappa_X$ are given by the formulas: \begin{eqnarray*} \lambda_X&=&\displaystyle{\frac{g}{2}\deg\big(c_1(V)|_{B_0}\big)+\frac{1}{4} \big(c_1(V)^2-4c_2(V)\big)},\\ \kappa_X&=&\displaystyle{\frac{5g-6}{2}\deg\big(c_1(V)|_{B_0}\big)+\frac{3}{4} \big(c_1(V)^2-4c_2(V)\big)}. \end{eqnarray*} \label{lambda_X,kappa_X}\vspace*{-6mm} \end{prop} \noindent We defer the proof of Prop.~\ref{lambda_X,kappa_X} to Subsections 7.4.2-3. \subsubsection{Notation and Basic Tools.} \label{notation} The proof of Prop.~\ref{lambda_X,kappa_X} consists of two calculations in the Chow ring of ${\mathbf P}V$; one uses versions of Riemann-Roch theorem on $X$ and ${\mathbf P}V$, and the other uses the adjunction formula on ${\mathbf P}V$ for the divisor $X$. Here we discuss these statements and set up the necessary notation. \medskip In order to work in ${\mathbb A}({{\mathbf P}}V)$, we express the Chern classes of ${\mathbf P}V$ in terms of the hyperplane class $\zeta$ and the Chern classes of $Y$. We first recall that $\pi_*\mathcal{O}_{{\mathbf P}V}(+1)\cong V\widehat{\phantom{t}}$. In the Euler sequence on ${{\mathbf P}V}$: \begin{equation} 0\rightarrow \mathcal{O}_{{\mathbf P}V} \rightarrow \mathcal{O}_{{\mathbf P}V}(+1)\otimes \pi^*V\rightarrow \mathcal{T}_{\pi} \rightarrow 0, \label{Eulersequence} \end{equation} we compare the Chern polynomials $c_t(\mathcal{O}_{{\mathbf P}V}(+1)\otimes \pi^*V)=c_t(\mathcal{T}_{\pi})$, and obtain: \begin{eqnarray} K_{{\mathbf P}V}&\!\!\!=&\!\!\!\!-2\zeta-\pi^*c_1(V)+\pi^*K_Y, \label{K_PV}\\ c_1(\omega_{\pi})&\!\!\!=&\!\!\!\!-2\zeta-\pi^*c_1(V), \label{omega-pi}\label{omega_pi}\\ c_2({\mathbf P}V)&\!\!\!=&\!\!\!\!-2\zeta\pi^*K_Y+\pi^*\big(c_1(V)K_Y+c_2(Y)\big). \label{c_2(PV)} \end{eqnarray} Here $\mathcal{T}_{\pi}$ and $\omega_{\pi}$ are correspondingly the relative tangent and the relative dualizing sheaves of $\pi$, while $K_{{\mathbf P}V}$ is the class of the canonical sheaf on ${\mathbf P}V$. On the ruled surface $Y$ over the curve $B$ of genus $g_{\scriptscriptstyle B}$ we similarly have \begin{eqnarray} \hspace{9mm}K_Y&\!\!\!=&\!\!\!\!-2B_0+h^*(K_B) \equiv -2B_0+(2g_{\scriptscriptstyle B}-2)F_Y \label{K_Yglobal},\\ \hspace{9mm}c_2(Y)&\!\!\!=&\!\!4(1-g_{\scriptscriptstyle B}). \label{c_2(Y)global} \end{eqnarray} \medskip Now let $C$ be the general fiber of $X$, i.e. a smooth trigonal curve of genus $g$. Assuming the Basic construction for the triple cover $X\rightarrow Y$ (cf.~Fig.~\ref{basicconstruction}), we have the following lemmas. \begin{lem}If $\chi(\mathcal{E})$ denotes the holomorphic Euler characteristic of any sheaf $\mathcal{E}$, then the invariant $\lambda_X$ is expressible as $\lambda_X=\chi({\mathcal O}_X)-\chi({\mathcal O}_B)\cdot \chi({\mathcal O}_C).$ \label{Euler} \end{lem} \noindent{\it Proof.} From Grothendieck-Riemann-Roch theorem for the map $f:X\rightarrow B$, \[\on{ch}(f_{!}\mathcal{O}_X).\on{td}\mathcal{T}_B= f_*(\on{ch}\mathcal{O}_X.\on{td}\mathcal {T}_X),\] where $\mathcal{T}_X$ and $\mathcal{T}_B$ are the corresponding tangent sheaves. Since the fibers of $f$ are one-dimensional, $f_{!}\mathcal{O}_X= f_*\mathcal{O}_X-R^1\!f_*\mathcal{O}_X=\mathcal{O}_B-(f_*\omega_f)\widehat{\phantom{t}}$. Substituting: \[\big(1-g+c_1(f_*\omega_f)\big).\big(1-\frac{1}{2}K_B\big)= f_*\big(1-\frac{1}{2}K_X+\frac{1}{12}(K^2_X+c_2(X))\big),\] \[\Rightarrow\,\, c_1(f_*\omega_f)=\frac{1}{12}f_*(K^2_X+c_2(X))-\frac{g-1}{2}K_B,\] \[\,\,\,\,\,\,\,\,\,\,\Rightarrow\,\, \lambda_X=\on{deg}(f_*\omega_f)=\chi(\mathcal{O}_X)-\chi(\mathcal{O}_B)\cdot \chi(\mathcal{O}_C).\,\,\,\qed\] \medskip Note the similarity between this formula and the formula for $\delta_B$ in Example 2.1. Both quantities are expressed as differences of the Euler characteristic (holomorphic or topological) on the total space of $X$ and the product of the corresponding Euler characteristics on the base $B$ and the general fiber $C$. Lemma~\ref{Euler} suggests that in order to calculate $\lambda_X$, we must have control over $\chi(\mathcal{O}_X)$. \begin{lem} In the Chow ring of ${\mathbf P}V$: \[\chi({\mathcal O}_X)=\frac{1}{12}X\big[\big(X+K_{{\mathbf P}V}\big) \big(2X+K_{{\mathbf P}V}\big)+ c_2({\mathbf P}V)\big]\] \label{holomorphicEuler}\vspace*{-7mm} \end{lem} \begin{proof} From the standard exact sequence (\ref{X-divisorsequence}) for the divisor $X$ on ${\mathbf P}V$ we have $\chi({\mathcal O}_X)=\chi({\mathcal O}_{{\mathbf P}V})-\chi({\mathcal O}_{\mathbf {P}V}(-X))$. On the other hand, Hirzebruch-Riemann-Roch claims that for any sheaf $\mathcal E$ on ${\mathbf P}V$: $\chi(\mathcal{E})=\on{deg}\big(\on{ch}(\mathcal{E}).\on{td}\mathcal{T}_{{\mathbf P}V} \big)_3$. Applying this to the line bundles $\mathcal{O}_{{\mathbf P}V}$ and ${\mathcal O}_{{\mathbf P}V}(-X)$, and subtracting the results completes the proof of the lemma. \end{proof} \medskip The reader may have noticed that all quantities discussed in the above lemmas are elements of the third graded piece ${\mathbb A}^3({\mathbf P}V)\otimes \mathbb{Q}$ of the Chow ring ${\mathbb A}({\mathbf P}V)\otimes \mathbb{Q}$. Hence they are cubic polynomials in the class $\zeta$, whose coefficients are appropriate products of pull-backs from ${\mathbb A}(Y)\otimes \mathbb{Q}$. The higher degrees $\zeta^3$ and $\zeta^2$ can be decreased using the basic relation (\ref{zeta-relation}), while $\zeta$ itself can be altogether eliminated by noticing that for any $\vartheta\in {\mathbb A}^2(Y)$: \begin{equation} \zeta.\pi^*(\on{point})=\zeta.F_{\pi}=1\,\, \Rightarrow\,\,\zeta . \pi^*\vartheta=\on{deg}\vartheta. \label{trivial} \end{equation} It is also useful to remember the trivial fact that for any divisors $D_i$ on $Y$, $\on{dim}Y=2$ implies $D_1.D_2.D_3=0=D_i.c_2(V)$. \smallskip \begin{figure}[h] $$\psdraw{intersection}{1.8in}{1.8in}$$ \caption{Intersection of $X$ and $\pi^*F_Y$} \label{intersection} \end{figure} \begin{lem}[Adjunction formula] The canonical bundle $\omega_Z$ of a smooth divisor $Z$ on the smooth variety $T$ can be expressed as $\omega_Z\cong \omega_T\otimes \mathcal{O}_T(Z)\otimes \mathcal{O}_Z$. Consequently, \[K_X^2=\big(K_{{\mathbf P}V}+X\big)^2X\,\,\,{and}\,\,\, g+2=\on{deg}c_1(V)|_{F_Y}.\] \label{adjunction}\vspace*{-4mm} \end{lem} \noindent{\it Proof.} For the general statement of the adjunction formula see \cite{Hartshorne}. The expression for $K_X^2$ is a straightforward application to the divisor $X$ on ${\mathbf P}V$: $K_X=\big(K_{{\mathbf P}V}+X\big)\big|_X$ is being squared in ${\mathbb A}({\mathbf P}V)$. As for the genus $g$ of the general member $C$ of our family, we consider a general fiber $F_Y$ of $Y$ (cf.~Fig.~\ref{intersection}). Its pullback $\pi^*F_Y$ is a rational ruled surface $\mathbf F$ over $F_Y$, embedded in the 3-fold ${\mathbf P}V$. The intersection of $\mathbf F$ with the surface $X$ is the trigonal fiber $C=X\cdot\pi^*F_Y= (3\zeta+2\pi^*c_1(V))|_{\pi^*F_Y}$. From the adjunction formulas for $C\subset \pi^*F_Y$ and $\pi^*F_Y\subset {\mathbf P}V$: \begin{eqnarray*} 2g-2&\!\!=&\!\!(K_{\pi^*F_Y}+C)\cdot C=\big((K_{{\mathbf P}V}+\pi^*F_Y)\big|_{\pi^*F_Y} +X\big|_{\pi^*F_Y}\big)\cdot X\big|_{\pi^*F_Y}\\ &\!\!=&\!\!\big(\zeta+\pi^*c_1(V)+\pi^*K_Y+\pi^*F_Y\big)\big( 3\zeta+2\pi^*c_1(V)\big)\cdot \pi^*F_Y\\ &\!\!=&\!\!(2c_1(V)+3K_Y)\cdot F_Y=2\on{deg}c_1(V)\big|_{F_Y}-6. \,\,\,\qed \end{eqnarray*} \subsubsection{Global Calculation of $\lambda_X$.} \label{globallambda} We substitute in Lemma~\ref{holomorphicEuler} the expressions (\ref{K_PV}--\ref{c_2(PV)}) for $X,\,\, K_{{\mathbf P}V}$ and $c_2(V)$, as well as the identity $D=2c_1(V)$: \[\chi(\mathcal{O}_X)=\frac{3\xi+2\pi^*c_1(V)}{12}\big[ \big(\xi+\pi^*c_1(V)+\pi^*K_Y\big) \big(4\xi+3\pi^*c_1(V)+\pi^*K_Y\big)\] \[-2\xi\pi^*K_Y+\pi^* c_1(V)\pi^*K_Y+\pi^*c_2(Y)\big].\] Applying the necessary reductions, we arrive at: \[\chi({\mathcal O}_X)=\frac{1}{2}\big(c_1^2(V)-2c_2(V)\big)+\frac{1}{2}c_1(V)K_Y+ \frac{1}{4}\big(K_Y^2+c_2(Y)\big).\] We expect our formula for $\lambda_X$ to be independent of the base curve $B$. The contribution of $g_{\scriptscriptstyle B}$ in $\chi(\mathcal{O}_X)$ can be written as: $(g_{\scriptscriptstyle B}-1)\on{deg}c_1(V)\big|_{F_Y}+\chi(\mathcal{O}_Y)= (g_{\scriptscriptstyle B}-1)(g-1)$, but this is precisely the adjustment $\chi(\mathcal{O}_B)\chi(\mathcal{O}_C)$ given in Lemma~\ref{Euler}. Thus, \[\lambda_X=\frac{1}{2}\big(c_1^2(V)-2c_2(V)\big)-\on{deg}c_1(V)\big|_{B_0}.\] It remains to notice that $c_1^2(V)=2\on{deg}c_1(V)|_{F_Y} \on{deg}c_1(V)|_{B_0}=2(g+2)\on{deg}c_1(V)|_{B_0}$ and rewrite $\lambda_X$ in the form \[\lambda_X=\frac{1}{4}\big (c_1^2(V)-4c_2(V)\big)+\frac{g}{2}\on{deg}c_1(V)\big|_{B_0}.\,\,\,\qed\] \subsubsection{Global Calculation of $\kappa_X$.} \label{globalkappa} Since $\omega_f= \omega_X\otimes \omega_B^{-1}$, \begin{equation} \kappa_X=(K_X-\pi^*K_B)^2=K_X^2-8(g_{\scriptscriptstyle B}-1)(g-1). \label{kappa} \end{equation} \noindent From Lemma~\ref{adjunction} we calculate \begin{eqnarray*} K_X^2&\!\!=\!\!\!&(K_{{\mathbf P}V}+X)^2X=\big(\xi+\pi^*c_1(V)+\pi^*K_Y\big)^2 (3\xi+2\pi^*c_1(V))\\ &\!\!=\!\!\!\!&2c_1^2(V)-3c_2(V)+4c_1(V)K_Y+3K_Y^2. \end{eqnarray*} \noindent We calculate the contribution of $g_{\scriptscriptstyle B}$ in $K_X^2$: $8(g_{\scriptscriptstyle B}-1)\on{deg}c_1(V)|_{F_Y}+ 24(1-g_{\scriptscriptstyle B})=8(g_{\scriptscriptstyle B}-1)(g-1)$, which is exactly the necessary adjustment for $\kappa_X$ in (\ref{kappa}). Therefore, \begin{eqnarray*} \kappa_X&\!\!\!=\!\!\!&2c_1^2(V)-3c_2(V)-8\on{deg}c_1(V)\big|_{B_0}\\ &\!\!\!=\!\!\!&\frac{3}{4}\big(c_1^2(V)-4c_2(V)\big)+\frac{5}{2} \on{deg}c_1(V)\big|_{B_0}\on{deg}c_1(V)\big|_{F_Y}-8\on{deg}c_1(V)\big|_{B_0}\\ &\!\!\!=\!\!\!&\frac{3}{4}(c_1(V)^2-4c_2(V))+\frac{5g-6}{2} \on{deg}c_1(V)\big|_{B_0}.\,\,\,\qed \end{eqnarray*} \subsection{Index theorem on the surface $X$.} \label{indextheorem} Now that we have completed the proof of Prop.~\ref{lambda_X,kappa_X}, we notice that any bound on the ratio $\delta_X/\lambda_X$ would be equivalent to some inequality involving the genus $g$ and the two invariants discussed earlier: $\on{deg}c_1(V)|_{B_0}$ and the quantity $c_1(V)^2\!-\!4c_2(V)$. This inequality should be a fairly general one, since the only relevant information in our situation is that $X$ is a triple cover of a ruled surface $Y$. One way of obtaining such general inequalities in ${\mathbb A}^2(X)\otimes \mathbb{Q}$ is via \begin{thm}[Hodge Index] Let $H$ be an ample divisor on the smooth surface $X$, and let $\eta$ be a divisor on $X$, numerically not equivalent to 0. If $\eta \cdot H=0$, then $\eta^2<0$. \label{Index} \end{thm} The question here, of course, is how to find suitable divisors $H$ and $\eta$ that would yield our result for the maximal slope bound. For that, we make use of the triple cover $\phi:X\rightarrow Y$. If $H$ is any {\it ample} divisor on $Y$, then its pullback $\phi^*H$ to $X$ is also ample. This follows from \begin{thm}[Nakai-Moishezon Criterion] A divisor $A$ on the smooth surface $X$ is ample if and only if $A^2>0$ and $A\cdot C>0$ for all irreducible curves $C$ in $X$. \label{Nakai} \end{thm} \noindent Since $H$ is ample itself, $(\phi^*H)^2=3H^2>0$ and $(\phi^*H)\!\cdot\! C=H\!\cdot\!\phi_*(C)>0$ for any curve $C$ on $X$, so that $\phi^*H$ is also ample on $X$. Now, if we find a divisor $\eta$ on $X$ such that $\eta\cdot\phi^*\on{Pic} Y=0$, we will have assured that $\eta\cdot\phi^*H=0$, and then the Index theorem will assert $\eta^2\leq 0$. As $X$ is a divisor itself on ${\mathbf P}V$, its Picard group naturally contains the restriction of $\on{Pic}{\mathbf P}V$ to $X$. We look for $\eta$ inside this subgroup, and for our purposes we may write it in the form $\eta=\big(\zeta+\pi^*C_1\big)\big|_X$ for some $C_1\in \on{Pic}_{\mathbb Q}Y$. Let $C$ be any divisor class $\on{Pic}_{\mathbb {Q}}Y$. We compute \[\eta\cdot \phi^*C=\big(\zeta+\pi^*C_1\big)\big(3\zeta+2\pi^*c_1(V)\big) \pi^*C=C(3C_1-c_1(V)).\] We want this to be zero for all $C$, so we naturally take $C_1={\displaystyle\frac{1}{3}}c_1(V)\in\on{Pic}_{\mathbb {Q}}Y$. We summarize the above discussion in \begin{lem}[Index Theorem on $X$] The divisor class $\eta=\big(\zeta+\frac{1}{3}\pi^*c_1(V)\big)\big|_X$ on $X$ satisfies $\eta\cdot \phi^*\on{Pic}(Y)=0$. In particular, for an ample divisor $H$ on $Y$, the pullback $\phi^*H$ is also ample on $X$ and $\eta \cdot \phi^*H=0$. Consequently, $\eta^2\leq 0$ with equality if and only if $\eta$ is numerically equivalent to $0$ on $X$. \label{Eta} \end{lem} We have shown that \begin{equation} 0\geq3\eta^2=3\big(\zeta+\frac{1}{3}\pi^*c_1(V)\big)^2 \big(3\zeta+2\pi^*c_1(V)\big)=2c_1^2(V)-9c_2(V), \label{eta} \end{equation} or equivalently, \begin{equation} 2(g+2)\on{deg}c_1(V)\big|_{B_0}-9\big(c_1^2(V)-4c_2(V)\big)\geq 0. \label{indexinequality} \end{equation} We are now ready to find a maximal bound for the slope of $X$. Recall the formulas for $\lambda_X$ and $\kappa_X$ (cf.~Prop.~\ref{lambda_X,kappa_X}), and write \[\delta_X=12\lambda_X-\kappa_X=\displaystyle{ \frac{7g+6}{2}\on{deg}c_1(V)\big|_{B_0} +\frac{9}{4}\big(c_1^2(V)-4c_2(V)\big)}.\] \medskip In view of the type of bound for the ratio $\delta_X/\lambda_X$, which we aim to achieve, we have to eliminate any extra terms and use inequality (\ref{eta}). Our only choice is to subtract \begin{eqnarray*} 36(g+1)\lambda_X-(5g+1)\delta_X&\!\!\!=\!\!\!& \frac{1}{2}\big(36(g+1)g-(5g+1)(7g+6)\big) \on{deg}c_1(V)\big|_{B_0}+\\ && +\frac{1}{4}\big(9(g+1)-9(5g+1)\big)\big(c_1^2(V)-4c_2(V)\big)\\ &\!\!\!=\!\!\!& \frac{1}{2}(g^2-g-6)\on{deg}c_1(V)\big|_{B_0}-\frac{9}{4}(g-3) \big(c_1^2(V)-4c_2(V)\big)\\ &\!\!\!=\!\!\!&\frac{g-3}{4}\big[2(g+2)\on{deg}c_1(V)\big|_{B_0}- 9\big(c_1^2(V)-4c_2(V)\big)\big]\\ &\!\!\!=\!\!\!&(g-3)(9c_2(V)-2c_1^2(V))\geq 0 \end{eqnarray*} \smallskip As a result, we establish an exact maximal bound for the slopes of our triple covers: \begin{thm}[Main Theorem in Triple Cover Case] Given a triple cover \newline$\phi:X\!\rightarrow \!Y$ satisfying in the Basic construction, the slope of $X$ satisfies \[\frac{\delta_X}{\lambda_X}\leq \frac{36(g+1)}{5g+1}\cdot\] Equality is achieved if and only if $g=3$, or $g>3$ and $\eta\equiv 0$ on $X$. \label{maintheorem} \end{thm} \subsection{When is the maximal bound achieved?} \label{whenmaximal} \subsubsection{The branch divisor of $\phi$} From GRR, applied to $\phi:X\rightarrow Y$ and the sheaf $\mathcal{O}_X$, we obtain a description of $c_1(V)$: \[\on{ch}(\phi_{!}\mathcal{O}_X).\on{td}\mathcal{T}_Y= \phi_*(\on{ch}\mathcal{O}_X.\on{td}\mathcal {T}_X),\] \[\on{ch}(\phi_*\mathcal{O}_X)\big(1-\frac{1}{2}K_Y+\frac{1}{12}(K_Y^2+c_2(Y))\ \big)=\phi_*\big(1-\frac{1}{2}K_X+\frac{1}{12}(K_X^2+c_2(X)\big)\] \[\Rightarrow c_1(\phi_*\mathcal{O}_X)=-\frac{1}{2}\big(\phi_*K_X-3K_Y).\] For the {\it ramification} divisor $R$ on $X$ we know $K_X=\phi^*K_Y+R$, so that $\phi_*K_X=3K_Y+\phi_*R$. Hence $c_1(V)=-c_1(\phi_*\mathcal{O}_X)= \frac{1}{2}\phi_* R$. In other words, from Lemma~\ref{D=2c_1(V)} we conclude that $c_1(V)$ is half of the {\it branch} divisor $D$ on $Y$. On the other hand, we can rewrite the condition $\eta\equiv 0$ in the following way: \[0 \equiv 3\eta=\big(3\zeta+\pi^*c_1(V)\big)\big|_X= \big(X-\pi^*c_1(V)\big)\big|_X= c_1\big(\mathcal{O}_{{\mathbf P}V}(X)\big|_X\big)-\pi^*c_1(V)\big|_X\] \[\Leftrightarrow\,\,c_1\big(\mathcal{O}_{{\mathbf P}V}(X)\big|_X\big)\equiv \frac{1}{2}\phi^*D.\] The self-intersection of $X$ on ${\mathbf P}V$ satisfies (cf.~ \cite{self-intersection}) \[i^*i_*(1_X)=c_1(\mathcal{N}_{X/{\mathbf P}V})\,\,\Rightarrow\,\, X\cdot X=i_*c_1(\mathcal{N}_{X/{\mathbf P}V}).\] In particular, our condition $\eta\equiv 0$ can be expressed as $\displaystyle {c_1(\mathcal{N}_{X/{\mathbf P}V})\equiv \displaystyle{\frac{1}{2}}\phi^*D}$. \subsubsection{Examples of the maximal bound} Constructing examples of families achieving the maximal bound is not so easy, considering that the condition $\eta\equiv 0$ is not useful in practice. Instead, we start from the Basic construction and attempt to find a ruled surface $Y$ and a rank 2 vector bundle $V$ on it satisfying the equality in (\ref{indexinequality}), as well as the ``genus condition'' given in Lemma~\ref{adjunction}. The former will ensure the maximal ratio $\delta/\lambda = 36(g+1)/(5g+1)$, while the latter will imply that the fibers of our family are indeed of genus $g$. The remaining question is what linear series $3\zeta+\phi^*D$ has an irreducible member with at most rational double points as singularities, which would serve as the total space of our family $X$. \medskip It is hard to work with the canonically defined bundle $V= \phi_*(\mathcal{O}_X)/\mathcal{O}_Y$, since not every vector bundle $W$ of rank $2$ on $Y$ is of this form for some surface $X$. But any $W$ is congruent to some $V$ after a twist by an appropriate line bundle $M$: $V=W\otimes M$, and ${\mathbf P}V\cong {\mathbf P}W$. So, it seems reasonable to start with $W$ rather than $V$, and use the invariant forms of our required equalities (cf.~Sect.~\ref{globalcalculation}). This means replacing the degrees of $c_1(V)$ on $B_0$ and $F_Y$ by the corresponding invariant degrees of $2D-3c_1(V)$. Thus, we need for some divisor $\widehat{D}$ on $Y$: \begin{equation} 2(g+2)(2\on{deg}\widehat{D}\big|_{B_0}-3\on{deg}c_1(W)\big|_{B_0}) =9\big(c_1^2(W)-4c_2(W)\big), \label{condition1} \end{equation} \begin{equation} g+2=2\on{deg}\widehat{D}\big|_{F_Y}-3\on{deg}c_1(W)\big|_{F_Y}. \label{condition2} \end{equation} For a general fiber $F_Y$ of $Y$ consider the rational ruled surface (cf.~Fig.~\ref{intersection}): \[{\mathbf F}_e=\pi^*F_Y={\mathbf P}(W|_{F_Y})= {\mathbf P}(\mathcal{O}_{{\mathbf P}^1}\oplus \mathcal{O}_{{\mathbf P}^1}(e)),\,\,\,\on{with}\,\, e\geq 0,\] Let $S^{\prime}$ be the section in ${\mathbf F}_e$ with self-intersection $(S^{\prime})^2=-e$, and let $F_{\pi}$ be the fiber of ${\mathbf F}_e$ (in terms of the map $\pi:{\mathbf P}V\rightarrow Y$, $F_{\pi}=\pi^*(\on{pt})$). Since a general fiber $C$ of our family is embedded in ${\mathbf F}_e$, the linear system \[|C|=\big|3S^{\prime}+\frac{g+2+3e}{2}F_{\pi}\big|\] has an irreducible nonsingular member. Equivalently, $C\cdot S^{\prime}\geq 0$, i.e. \begin{equation}e\leq (g+2)/3\,\,\,\on{and}\,\,\, e\equiv g(\on{mod}2) \label{e-conditions} \end{equation} (compare with Lemma~\ref{gentrig}). This forces three types of extremal examples according to the remainders $g(\on{mod}3)$. \bigskip \noindent{\bf Example 7.1 ($g\equiv 0(\on{mod}3)$).} Let $g=3e$ for some $e\in \mathbb{N}$. Set the base curve $B={\mathbf P}^1$, and the ruled surface \[Y={\mathbf P}(\mathcal{O}_B\oplus \mathcal{O}_B(6))={\mathbf F}_6.\] Let $B^{\prime}$ be the section in $Y$ with smallest self-intersection: $(B^{\prime})^2=-6$, thus $B_0=B^{\prime}+3F_Y$ with $B_0^2=0$. Let $Q=B^{\prime}+6F_Y$, and we choose two divisors $\widehat{D}$ and $E$ on $Y$ as follows: \[\widehat{D}=(g+1)Q\,\,\,\on{and}\,\,\,E=eB^{\prime}+2(g+1)F_Y.\] For the vector bundle $W$ on $Y$ we set $W=\mathcal{O}_Y\oplus \mathcal{O}_Y(E)$ so that $c_1(W)=E$ and $c_2(W)=0$. We claim that the linear system $L=|3\zeta+\pi^*\widehat{D}|$ on the 3-fold ${\mathbf P}W$ contains an irreducible smooth member, which we set to be our surface $X$ with maximal ratio $\delta/\lambda$. \begin{figure}[h] $$\psdraw{maximal1}{2.6in}{1.2in}$$ \caption{Example with $g\equiv 0$(mod$3$)} \label{maximal1.fig} \end{figure} Indeed, it is trivial to check conditions (\ref{condition1}--20). Further, for {\it any} fiber $F_Y$ of $Y$: \[\pi^*F_Y={\mathbf P}\big(\mathcal{O}_{{\mathbf P}^1}\oplus \mathcal{O}_{{\mathbf P}^1} (E\cdot F_Y)\big)={\mathbf P}\big(\mathcal{O}_{{\mathbf P}^1}\oplus \mathcal{O}_{{\mathbf P}^1} (e)\big)={\mathbf F}_e,\] so that $e=g/3$ satisfies the required conditions (\ref{e-conditions}). \medskip The only nontrivial fact is the existence of the wanted member $X$ in the linear system $L$ on ${\mathbf P}W$. Consider two sections $\Sigma_0$ and $\Sigma_1$ of ${\mathbf P}W$ corresponding to the subbundles $\mathcal{O}_Y$ and $\mathcal{O}_Y(E)$ of $W$, respectively: $\Sigma_0\in |\zeta|,\,\,\,\Sigma_1\in |\zeta+\pi^*(E)|,$ so that $\Sigma_1\sim \Sigma_0+E$ (cf.~Fig.~\ref{maximal1.fig}). Note that $\Sigma_0\cdot \Sigma_1=0$ and $\Sigma_0\cdot L=\Sigma_0\cdot \pi^*B^{\prime}$. In other words, if $G=\pi^*B^{\prime}$ is the ruled surface over $B^{\prime}$, then $\Sigma_0$ intersects every irreducible member of $L$ in the curve $R=\Sigma_0\cap G$. On the other hand, if a member of $L$ meets $\Sigma_0$ in a point outside $R$, then this member contains entirely $\Sigma_0$. Thus, $L$ does not distinguish the points on $\Sigma_0$, and $R$ is in the base locus of $L$. Similarly, the restriction $L|_G=|3\Sigma_0|_G|=|3R|$ has exactly one section on $G$, namely, $3R$. Again it follows that $L$ does not distinguish the points on $G$. \smallskip Away from the closed subset $Z=\Sigma_0\cup G$, the linear system $L$ is in fact very ample. This can be checked by showing directly that $L$ separates points and tangent vectors on ${\mathbf P}W-Z$. Therefore, $L$ defines a rational map \[\phi_L:{\mathbf P}W \rightarrow {\mathbf P}(H^0(L)^{\widehat{ \phantom{o}}})={\mathbf P}^N.\] The map $\phi_L$ is regular on ${\mathbf P}W-R$, embeds ${\mathbf P}W-Z$, and contracts $\Sigma_0-R$ and $G-R$ to two distinct points $p$ and $q$ in ${\mathbf P}^N$. By Bertini's theorem (cf.~\cite{Hartshorne}), the {\it general} member of $L$ is {\it smooth} away from the base locus $R$. Let $H$ be a general hyperplane in ${\mathbf P}^N$ not passing through $p$ and $q$. Pulling $H$ back to ${\mathbf P}W$ yields a member $X$ of $L$ not containing $\Sigma_0$ or $G$, and hence irreducible. \medskip It remains to show that the total space of $X$ has at most finitely many double point singularities along the curve $R$. Since the member $3\Sigma_1+G\in|L|$ is smooth along $R$, then the general member of $|L|$ must be smooth along $R$. Hence our surface $X$ has, in fact, a smooth total space. This concludes the construction of our maximal bound family of trigonal curves. \bigskip \noindent{\bf Example 7.2 ($g\equiv 1(\on{mod}3)$).} Set $g=3e-2$ for $e\in{\mathbb N}$. Then $e$ satisfies the requirements of our construction: $e=(g+2)/3$ and $e\equiv g (\on{mod}2)$. For the ruled surface $Y$ we choose $Y=\mathbf P^1\times\mathbf P^1$. Let $E$ and $\widehat{D}$ be the following divisors on $Y$: $E=eB_0+fF_Y\,\,\on{and}\,\,\widehat{D}=3E,$ where $f\in{\mathbb N}$. The vector bundle $W$ on $Y$ is then defined by $W=\mathcal{O}_Y\oplus\mathcal{O}_Y(E)$. Finally, we indentify the total space of the surface $X$ with an irreducible smooth member of the linear system $L=|3\zeta+\pi^*\widehat{D}|$ on the 3-fold $\mathbf P W$. \smallskip The verification of this construction is similar to the previous example. Here $L$ is very ample everywhere on $\mathbf P W$ except on the section $\Sigma_0$, which is contracted to a point under the map $\phi_{L}$. This example, in somewhat different context, is shown in \cite{small}. \medskip \noindent{\bf Remark 7.1} The case of $g\equiv 2(\on{mod}3)$ is complicated by the fact that we cannot take $e=(g+1)/3$, for then $e\not \equiv g(\on{mod}2)$. For example, if $g=5$, then the only possibility is $e=1$. In the notation of Section~12, all trigonal curves have lowest Maroni invariant of $1$, and there is no Maroni locus. For now, in this case we have not been able to construct a trigonal family with singular general member, whose ratio is $36(g+1)/(5g+1)$. \bigskip \section*{\hspace*{1.9mm}8. Local Calculation of $\lambda,\delta$ and $\kappa$ in the General Case} \setcounter{section}{8} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{lem}{0} \setcounter{thm}{0} \setcounter{prop}{0} \setcounter{defn}{0} \setcounter{cor}{0} \setcounter{conj}{0} \setcounter{claim}{0} \setcounter{remark}{0} \setcounter{equation}{0} \label{generalcase} \subsection{Notation and conventions} \label{Conventions} In this section we consider the general case of a trigonal family $X\rightarrow B$. For convenience of notation, we shall assume that the base curve $B$ intersects transversally and in general points the boundary divisors of $\overline{\mathfrak{T}}_g$ (cf.~Fig.~\ref{Delta-k,i}). We will call such a base curve {\it general}, and use this definition throughout Section~8-10. Since we work in the rational Picard group of $\overline{\mathfrak{M}}_g$, all arguments and statements in the remaining cases are shown similarly in Sect.~11. From Prop.~6.1, we may assume that modulo a base change, our family $X\!\rightarrow\! B$ fits in the following commutative diagram: \setlength{\unitlength}{10mm} \begin{figure}[h] \begin{picture}(3,4.3)(-1,2) \put(0,3.9){$\widetilde{X}\,\stackrel{\phi}{\longrightarrow} \widetilde{Y}$} \put(0,5.1){$\widehat{X}\,\stackrel{\widehat{\phi}}{\longrightarrow} \widehat{Y}$} \multiput(0.2,5)(0,-1.2){3}{\vector(0,-1){0.6}} \put(1.45,3.8){\vector(-1,-2){1}} \put(0,2.8){$X$} \put(0,1.6){$B$} \put(1.55,5){\vector(0,-1){0.6}} \put(1.55,6.2){\vector(0,-1){0.6}} \put(1.65,5.8){$\pi$} \put(1.3,6.3){${\mathbf P}V$} \put(0.2,5.6){\vector(2,1){1.2}} \end{picture} \caption{General base $B$} \label{general B}\vspace*{-4mm} \end{figure} \subsubsection{Relations in ${\rm Pic}_{\mathbb{Q}}\widehat{Y}$ and ${\rm Pic}_{\mathbb{Q}}{\mathbf P}V$} \label{relations} The special fibers of of $\widehat{X}$ and of the birationally ruled surface $\widehat{Y}$ over $B$ are described in Fig.~\ref{coef1.fig}--\ref{coef3.fig}. Since each such fiber in $\widehat{Y}$ is a {\it chain} $T$ of rational components, we can fix one of the end components to be the {\it root} $R$. We keep the notation $E^-$ ($E^+$, respectively) for the ancestor (descendants, respectively) of a component $E$ in $T$. We also fix a general fiber $F_{\widehat{Y}}\cong {{\mathbf P}}^1$ of $\widehat{Y}$, and a section $B_{\widehat{Y}}$, which is the pullback of the corresponding section $B_0$ in $\widetilde{Y}$ (cf.~(\ref{normalize})). The rational Picard group of $\widehat{Y}$ is generated by $F_{\widehat{Y}}$, $B_{\widehat{Y}}$ and all non-root components $E$ of the special fibers of $\widehat{Y}$: \[\on{Pic}_{\mathbb{Q}}\widehat{Y}=\mathbb{Q} B_{\widehat{Y}}\bigoplus \mathbb{Q}F_{\widehat{Y}}\!\!\!\bigoplus _{E-\on{not}\,\on{root}}\!\!\!\mathbb{Q}E.\] The intersection numbers of these generators are as follows: $B_{\widehat{Y}}^2=0=F_{\widehat{Y}}^2,\,\,\, B_{\widehat{Y}}\cdot F_{\widehat{Y}}=1,\,\,\on{and}\,\, E\cdot B_{\widehat{Y}}=E\cdot F_{\widehat{Y}}=0.$ \vspace*{-3mm} \setlength{\unitlength}{7mm} \begin{figure}[h] \begin{picture}(3,2.3)(12.6,4.5) \put(6,5.3){\line(1,0){2}} \multiput(5,4.6)(0,-1.7){2}{\line(2,1){1.6}} \multiput(6,3.6)(0,-0.1){2}{\line(1,0){2}} \put(9.1,4.2){\line(-2,-1){1.6}} \multiput(4.9,5)(0,-1.8){2}{$E^-$} \multiput(6.8,5.45)(0,-1.7){2}{$E$} \put(8.7,4.3){$E^+$} \end{picture} \vspace*{9mm} \caption{$m_{E}$\hspace*{100mm}} \label{m E} \begin{picture}(7,4)(3,-2.4) \put(4.45,2.9){\line(2,1){1.6}} \put(5.5,3.6){\line(1,0){3}} \put(6.5,3.4){\line(-1,1){1.2}} \put(7.5,3.4){\line(1,1){1.2}} \put(4.5,3.2){$E^-$} \put(6.8,2.9){$E$} \put(5.1,4.75){$E^+$} \put(5.9,4.85){$E^+$} \put(7.6,4.85){$E^+$} \put(8.4,4.75){$E^+$} \put(6.75,3.4){\line(-1,2){0.65}} \put(7.25,3.4){\line(1,2){0.65}} \end{picture}\vspace*{-38mm} \caption{$E^2$} \label{E^2} \begin{picture}(3,4)(-1,-1.9) \multiput(6,5.3)(0,-0.1){2}{\line(1,0){2}} \multiput(5,4.6)(0,-1.65){2}{\line(2,1){1.6}} \put(6,3.6){\line(1,0){2}} \multiput(9.1,4.2)(0,1.8){2}{\line(-2,-1){1.6}} \put(9.1,4.3){\line(-2,-1){1.6}} \multiput(4.8,5)(0,-1.7){2}{$R$} \multiput(6.8,5.45)(0,-1.7){2}{$E^-$} \multiput(8.7,4.3)(0,1.8){2}{$E$} \end{picture} \vspace*{-37.5mm} \caption{$\theta_E$\hspace*{-90mm}} \label{theta_E} \end{figure} We also set $m_{\!\stackrel{\phantom{.}}{E}}=E\cdot E^-$ (cf.~Fig.~\ref{m E}): \[m_{\!\stackrel{\phantom{.}}{E}}=\left\{\begin{array}{l} 0\,\,\on{if}\,\,E=R\,\,\on{root},\\ 1\,\,\on{if}\,\,E\,\,\on{and}\,\,E^-\,\,\on{reduced},\\ 2\,\,\on{if}\,\,E\,\,\on{or}\,\,E^-\,\,\on{nonreduced}. \end{array}\right.\] In this notation, due to the fact that $E\cdot T=E\cdot F_{\widehat{Y}}=0$, the self-intersection of any $E$ is computed by (cf.~Fig.~\ref{E^2}): \[E^2=-\sum_{\!\stackrel{\scriptstyle{E^{\prime}\not= E}} {E^{\prime}\cap E\not= \emptyset}} E\cdot E^{\prime}=-\sum_{\!\stackrel{\scriptstyle{E^{\prime}=E^+}} {\on{or}\,\,E^{\prime}=E}}m(E^{\prime})\] \smallskip In order to express the dualizing sheaf $K_{\widehat{Y}}$ in terms of the above generators of $\on{Pic}_{\mathbb{Q}}\widehat{Y}$, for each component $E$ in $\widehat{Y}$ we denote by $\theta_E$ the length of the path $\stackrel{\longrightarrow} {RE}$, omitting any nonreduced components except for $E$ itself. For example, in the two cases in Fig.~\ref{theta_E} we have $\theta_E=1$ and $\theta_E=2$. Note that $\theta_R=0$. Considering the ``effective'' blow-ups on $\widetilde{Y}$, necessary to construct $\widehat{Y}$, we immediately obtain the following identities (compare with (\ref{K_PV}) and (\ref{K_Yglobal})). \begin{lem} In $\on{Pic}_{\mathbb{Q}}\widehat{Y}$ and $\on{Pic}_{\mathbb{Q}}{\mathbf P}V$: \begin{eqnarray*} \on{(a)}&\!\!\!\!&\!\! \displaystyle{K_{\widehat{Y}} \equiv -2B_{\widehat{Y}}+(2g_B-2)F_{\widehat{Y}}+\sum_E \theta_EE},\\ \vspace*{-2mm} \on{(b)}&\!\!\!\!&\!\!K_{{\mathbf P}V}\equiv -2\zeta -\pi^* c_1(V)+\pi^*K_{\widehat{Y}},\\ \on{(c)}&\!\!\!\!&\!\!K_{{\mathbf P}V/\!_{\scriptstyle{B}}} \equiv -2\zeta -\pi^*c_1(V)-2\pi^*B_{\widehat{Y}}+\sum_E \theta_E\pi^*E. \end{eqnarray*} \label{Kdivisors}\vspace*{-8mm} \end{lem} The hyperplane section $\zeta$ of ${\mathbf P}V$ and the rank 2 vector bundle $V$ on $\widehat{Y}$ are defined similarly as in Section~7. Thus, in $\on{Pic}_{\mathbb{Q}}{\mathbf P}V$ we have $\widehat{X}\sim 3\zeta +\pi^*D$ for a certain divisor $D$ on $\widehat{Y}$. By analogy with Lemma~\ref{D=2c_1(V)}, one shows that $D\equiv 2c_1(V)$ in $\on{Pic}_{\mathbb{Q}}Y$, so that \begin{equation} \widehat{X}\equiv 3\zeta+ 2\pi^*c_1(V). \label{genX} \end{equation} Using the above notation for $\on{Pic}_{\mathbb{Q}}\widehat{Y}$ we can write for some half-integers $c,d,\gamma_{\!\stackrel{\phantom{.}}{E}}$: \begin{equation} \displaystyle{c_1(V)\equiv cB_{\widehat{Y}}+dF_{\widehat{Y}}+ \sum_{E}\gamma_{\!\stackrel{\phantom{.}}{E}}E}. \label{genc_1(V)} \end{equation} Here we can assume that $\gamma_{\!\stackrel{\phantom{.}}{R}}=0$ by replacing $R$ with a linear combination of the remaining components $E$ in its chain $T$ (compare with (\ref{define D,c1(V)})). Finally, we need the top Chern classes of $\widehat{Y}$ and ${\mathbf P}V$ in terms of intersections of known divisors and other known invariants of the two surfaces (compare with (\ref{c_2(PV)}) and (\ref{c_2(Y)global})). \begin{lem} In the Chow rings $\mathbb{A}(\widehat{Y})$ and $\mathbb{A}({\mathbf P}V)$ the following equalities are true: \begin{eqnarray*} \on{(a)}&\!\!&\!\!\!\!c_2(\widehat{Y})=c_2(Y)+\sum_{E\not =R} 1=4(1-g_B)+\sum_ {E\not = R} 1, \\ \on{(b)}&\!\!&\!\!\!\!c_2({\mathbf P}V)= c_2(\widehat{Y})-\pi^*K_{\widehat{Y}}(2\zeta+\pi^*c_1(V)),\\ \on{(c)}&\!\!&\!\!\!\!\displaystyle{ c_2({\mathbf P}V/\!_{\displaystyle{B}})= -\pi^*K_{\widehat{Y}/B}(2\zeta+\pi^*c_1(V))+\sum_{E\not =R} 1}. \end{eqnarray*} \end{lem} \label{conventions} \subsubsection{A technical lemma} \label{technicallemma} In the sequel, we will work with several functions defined on the set of components $\{E\}$ in $\widehat{Y}$. For easier calculations, to any such function $f$ we associate the {\it difference function} $F$ by setting $F_{\!\stackrel{\phantom{.}}{E}}:= f_{\!\stackrel{\phantom{.}}{E}}-f_{\!\stackrel{\phantom{.}} {E^-}}$ for all $E$. Since $R^-$ does not exist, we define $f_{R^-}=0$ for all roots $R$ in $\widehat{Y}$. \begin{lem}For any functions $f$ and $h$ defined on the set of components $\{E\}$ in $\widehat{Y}$, the following identity holds true: \[\sum_E f_{\!\stackrel{\phantom{.}}{E}}E \cdot \sum_E h_{\!\stackrel{\phantom{.}}{E}}E=-\sum_E (m\!\cdot\! F\!\cdot\! H)_{\!\stackrel{\phantom{.}}{E}}.\] \label{technical}\vspace*{-7mm} \end{lem} \noindent{\it Proof.} We rewrite the lefthand side as $\displaystyle{\sum_{E_1\not= E_2}f_{\!\stackrel{\phantom{.}}{E_1}} h_{\!\stackrel{\phantom{.}}{E_2}}E_1E_2+\sum_E f_{\!\stackrel{\phantom{.}}{E}}h_{\!\stackrel{\phantom{.}}{E}}E^2=}$ \[=\sum_{E}\left(f_{\!\stackrel{\phantom{.}}{E^-}} h_{\!\stackrel{\phantom{.}}{E}}+f_{\!\stackrel{\phantom{.}}{E}} h_{\!\stackrel{\phantom{.}}{E^-}}\right)m_{\!\stackrel{\phantom{.}}{E}}- \sum_E \left(f_{\!\stackrel{\phantom{.}}{E}}h_{\!\stackrel{\phantom{.}}{E}}+ f_{\!\stackrel{\phantom{.}}{E^-}} h_{\!\stackrel{\phantom{.}}{E^-}}\right)m_{\!\stackrel{\phantom{.}}{E}}=\] \[=\sum_E\left(f_{\!\stackrel{\phantom{.}}{E^-}}- f_{\!\stackrel{\phantom{.}}{E}}\right)\left(h_{\!\stackrel{\phantom{.}}{E}} -h_{\!\stackrel{\phantom{.}}{E^-}}\right)m_{\!\stackrel{\phantom{.}}{E}} =\sum_E(m\!\cdot\! F\!\cdot\! H) _{\!\stackrel{\phantom{.}}{E}}.\,\,\,\qed\] We have noted that all three functions $m,\theta$ and $\gamma$ are zero on the roots $R$ in $\widehat{Y}$. Since we shall be working specifically with these three functions, it makes sense to restrict from now on all sums $\sum_E$ only to the non-roots $E$ in $\widehat{Y}$. With this in mind, in every application of Lemma~\ref{technical} one must check that the corresponding functions $f$ and $h$ have the same property: $f_R=0=h_R$, so that we can restrict the sums in Lemma~\ref{technical} also to all {\it non-roots} $E$ in $\widehat{Y}$. In fact, in all cases this verification will be obvious as $f$ and $h$ will be, for the most part, linear combinations of $\theta$ and $\gamma$. \smallskip \noindent{\bf Example 8.1.} From expression (\ref{genc_1(V)}) for $c_1(V)$ as a divisor on $\widehat{Y}$, and Lemma~\ref{technical}: \begin{equation} c_1^2(V)=2cd+ \sum_{E}\gamma_{\!\stackrel{\phantom{.}}{E}}E\cdot \sum_{E}\gamma_{\!\stackrel{\phantom{.}}{E}}E= 2cd-\sum_{E}m_{\!\stackrel{\phantom{.}}{E}} \Gamma^2_{\!\stackrel{\phantom{.}}{E}}. \label{c^2_1(V)} \end{equation} \subsection{Computation of the invariants $\lambda_{\widehat{X}}, \kappa_{\widehat{X}}$ and $\delta$} \label{computation} The following proposition~\ref{hatlambda} is a generalization of the corresponding statement in Section~7 (cf.~ Prop.~\ref{lambda_X,kappa_X}). We set $\Gamma_{ \!\stackrel{\phantom{.}}{E}}=\gamma_{\!\stackrel {\phantom{.}}{E}}-\gamma_{\!\stackrel{\phantom{.}}{E^-}}$ and $\Theta_{\!\stackrel{\phantom{.}}{E}} =\theta_{\!\stackrel{\phantom{.}}{E}} -\theta_{\!\stackrel{\phantom{.}}{E^-}}$ to be the difference functions of $\gamma$ and $\theta$. \begin{prop} The degrees of the invariants $\lambda_{\widehat{X}}$ and $\kappa_{\widehat{X}}$ on $\widehat{X}$ are given by \[\lambda_{\widehat{X}}=d(g+1)-c_2(V)-\frac{1}{4}\sum_E \left\{m_{\!\stackrel{\phantom{.}}{E}}\cdot\left(2\Gamma^2+2\Gamma\cdot \Theta +\Theta^2\right)_{\!\stackrel{\phantom{.}}{E}}-1\right\},\] \vspace*{-6mm}\[\,\,\,\,\kappa_{\widehat{X}}=4dg-3c_2(V)-\sum_Em_{\!\stackrel{\phantom{.}}{E}}\left (2\Gamma^2+4\Gamma\Theta+3\Theta^2\right)_{\!\stackrel{\phantom{.}}{E}}.\] \label{hatlambda}\vspace*{-6mm} \end{prop} \begin{proof} One starts with the Euler characteristic formula $\lambda_{\widehat{X}}=\chi(\mathcal{O}_{\widehat{X}})- \chi(\mathcal{O}_C)\cdot \chi(\mathcal{O}_B),$ or the adjunction formula $\kappa_{\widehat{X}}=\big(\widehat{X}+K_{{\mathbf P}V/\!_{\scriptstyle{B}}}) ^2\widehat{X}.$ The rest of the proof is a straight forward calculation, which uses the equalities given in~\ref{conventions}, and is substantially simplified by Lemma~\ref{technical}. \end{proof} \begin{cor} The degree $\delta$ on the original family $X$ is given by \[\delta=4d(2g+3)-9c_2(V)-\!\sum_T\mu(T)-\!\sum_{\on{ram}1} 1-\! \sum_{\on{ram}2} 3-\!\sum_E\left\{m_{\!\stackrel{\phantom{.}}{E}} \left(4\Gamma^2+2\Gamma\Theta\right)_{\!\stackrel{\phantom{.}}{E}}-3\right\}\] \label{hatdelta}\vspace*{-4mm} \end{cor} Here $\mu(T)$ stands for the quasi-admissible contribution to $\kappa_{\widehat{X}}$ of the preimage $C=\widehat{\phi}^*T$ in $\widehat{X}$, as defined in Lemma~\ref{mu(C)}. \smallskip \begin{proof} Since $\lambda=\lambda_{\widehat{X}}$, $\kappa=\kappa_{\widehat{X}}+\sum_T \mu(T)+ \sum_{\on{ram}1} 1+\sum_{\on{ram}2}3$, and $\delta=12\lambda-\kappa$, the statement immediately follows from Prop.~\ref{hatlambda}.\end{proof} \subsection{The arithmetic genus $p_{{E}}$, and the invariants $\Gamma_{{E^{\prime}}}$ and $\Theta_{{E^{\prime}}}$} \label{arithmetic}For a component $E$ in a special fiber $T$ of $\widehat{Y}$, we define $T(E)$ to be the subtree of $T$ generated by the component $E$. In other words, $T(E)$ is the union of all components $E^{\prime}\in T$ such that $E^{\prime}\geq E$ (cf. Fig.~\ref{subtree}). For simplicity, we set $p_{\!\stackrel{\phantom{.}}{E}}:=p_a\big(\phi^*(T(E))\big)$ to be the arithmetic genus of the inverse image $\phi^*(T(E))$ in $\widehat{X}$. It can be easily computed via the following analog of Lemma~\ref{adjunction}, where $T$ consisted of a single component $E=R$. \begin{figure}[h] $$\psdraw{ex5}{1.2in}{1.2in}$$ \caption{Subtree $T(E)$} \label{subtree} \end{figure} \begin{lem} For a general base curve $B$ and for any non-root component $E\in T$: \begin{equation} \displaystyle{p_{\!\stackrel{\phantom{.}}{E}}=-m_{\!\stackrel {\phantom{.}}{E}}\left(\Gamma_E+\frac{3(\Theta_E+1)}{2}\right)+1}. \label{arithmetic equation} \end{equation} \label{arithgenus}\vspace*{-6mm} \end{lem} \noindent{\it Proof.} From the adjunction formula for the divisor $\phi^*(T(E))$ in $\widehat{X}$: \[2p_{\!\stackrel{\phantom{.}}{E}}-2=\left(K_{\widehat{X}}+\phi^*(T(E))\right)\phi^*(T(E))= \left((K_{{\mathbf P}V}+\widehat{X})|_{\widehat{X}}+\sum_{E^{\prime}} \delta_{E^{\prime}}\widehat{\phi}^*E^{\prime}\right)\sum_{E^{\prime}} \delta_{E^{\prime}}\widehat{\phi}^*E^{\prime}.\] Here $\delta_{E^{\prime}}=0$ if $E^{\prime}<E$, and $\delta_{E^{\prime}}=1$ otherwise. Thus, the sums above are effectively taken over all $E^{\prime}\geq E$. Substituting the expressions for $K_{{\mathbf P}V}$ and $\widehat{X}$ as divisors in ${{\mathbf P}V}$ from Lemma~\ref{Kdivisors} and (\ref{genX}), we arrive at \[2p_{\!\stackrel{\phantom{.}}{E}}-2=\sum_{E^{\prime}}\left(2 \gamma_{\!\stackrel{\phantom{.}}{E^{\prime}}}+ 3\theta_{\!\stackrel{\phantom{.}}{E^{\prime}}}+ 3\delta_{\!\stackrel{\phantom{.}}{E^{\prime}}} \right)E^{\prime} \sum_{E^{\prime}} \delta_{\!\stackrel{\phantom{.}}{E^{\prime}}}E^{\prime}.\] Set $\Delta_{\!\stackrel{\phantom{.}}{E}} =\delta_{\!\stackrel{\phantom{.}}{E}} -\delta_{\!\stackrel{\phantom{.}}{E^-}}$, i.e. $\Delta_{\!\stackrel{\phantom{.}}{E^{\prime}}}=1$ only if $E^{\prime}=E$; otherwise, $\Delta_{\!\stackrel{\phantom{.}}{E^{\prime}}}=0$. By Lemma~\ref{technical}, \[2p_{\!\stackrel{\phantom{.}}{E}}-2=-\sum_{E^{\prime}} m_ {\!\stackrel{\phantom{.}}{E^{\prime}}} \left(2\Gamma_{\!\stackrel{\phantom{.}}{E^{\prime}}}+ 3\Theta_{\!\stackrel{\phantom{.}}{E^{\prime}}}+3 \Delta_{\!\stackrel{\phantom{.}}{E^{\prime}}}\right)\Delta_{\!\stackrel{\phantom{.}}{E^{\prime}}}\,\, \Rightarrow\,\,2p_{\!\stackrel{\phantom{.}}{E}}-2= -m_{\!\stackrel{\phantom{.}}{E}}\left(2\Gamma_{\!\stackrel{\phantom{.}}{E}} +3\Theta_{\!\stackrel{\phantom{.}}{E}}+3\right).\,\,\,\qed\] \smallskip Now we can easily compute the invariants $m_{\!\stackrel{\phantom{.}}{E}}$, $\Theta_{\!\stackrel{\phantom{.}}{E^{\prime}}}$ and $\Gamma_{\!\stackrel{\phantom{.}}{E^{\prime}}}$, appearing in the formulas for $\lambda_{X}$ and $\kappa_{X}$. \begin{cor} There are three possibilities for the triple $(m_{\!\stackrel{\phantom{.}}{E}}, \Theta_{\!\stackrel{\phantom{.}}{E^{\prime}}}, \Gamma_{\!\stackrel{\phantom{.}}{E^{\prime}}})$, depending on whether the components $E$ and $E^-$ of $T$ are reduced: \begin{eqnarray*} \on{(a)}\,\on{if}\,E,E^-\,\on{reduced},\,\on{then}&\!\!\!&\!\!\!\! m_{\!\stackrel{\phantom{.}}{E}}=1,\,\,\Theta_{\!\stackrel{\phantom{.}}{E}} =1,\,\,\Gamma_{\!\stackrel{\phantom{.}}{E}}= -(p_{\!\stackrel{\phantom{.}}{E}}+2).\\ \on{(b)}\, \on{if}\,E\,\,\on{nonreduced}, \,\on{then}&\!\!\!&\!\!\!\! m_{\!\stackrel{\phantom{.}}{E}}=2,\,\,\Theta_{\!\stackrel{\phantom{.}}{E}} =1,\,\,\Gamma_{\!\stackrel{\phantom{.}}{E}}= -({p_{\!\stackrel{\phantom{.}}{E}}+5})/{2}.\\ \on{(c)}\,\on{if}\,E^-\!\on{nonreduced}, \,\on{then}&\!\!\!&\!\!\!\! m_{\!\stackrel{\phantom{.}}{E}}=2,\,\,\Theta_{\!\stackrel{\phantom{.}}{E}} =0,\,\,\Gamma_{\!\stackrel{\phantom{.}}{E}}= -({p_{\!\stackrel{\phantom{.}}{E}}+2})/{2}. \end{eqnarray*} \label{constants}\vspace*{-5mm} \end{cor} \begin{proof} Note that for the list all possible special fibers $T$ of $\widehat{Y}$, each component $E$ fits in exactly one of the three cases above (cf.~Fig.~\ref{coef1.fig}--\ref{coef3.fig}). The proof of the statement is immediate from the definitions of $m_{\!\stackrel{\phantom{.}}{E}}$ and $\Theta_{\!\stackrel{\phantom{.}}{E^{\prime}}}$, and from Lemma~\ref{arithgenus}. \end{proof} \bigskip\section*{\hspace*{1.9mm}9. The Bogomolov Condition $4c_2-c_1^2$ and the $7+6/g$ Bound in $\overline{\mathfrak{T}}_g$} \setcounter{section}{9} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{lem}{0} \setcounter{thm}{0} \setcounter{prop}{0} \setcounter{defn}{0} \setcounter{cor}{0} \setcounter{conj}{0} \setcounter{claim}{0} \setcounter{remark}{0} \setcounter{equation}{0} \label{Bogomolov1} With the conventions of Section~8, we state the main proposition of the section. \begin{prop} There exists an effective $\mathbb Q$-linear combination $\mathcal{E}$ of boundary divisors $\Delta{\mathfrak{T}}_{k,i}$, not containing $\Delta{\mathfrak{T}}_0$, such that for a general base curve $B$ in $\overline{\mathfrak{T}}_g$: \[(7g+6)\lambda|_B=g\delta|_B+\mathcal{E}|_B+\frac{g-3}{2}\left(4c_2(V)-c_1^2(V) \right).\] \label{bogomolov1}\vspace*{-10mm} \end{prop} For a shorthand notation, we denote by $\mathfrak{S}$ the difference \[\mathfrak{S}:=(7g+6)\lambda|_B-g\delta|_B -\frac{g-3}{2}\left( 4c_2(V)-c_1^2(V)\right).\] Using the results of the previous section, we can write: \begin{eqnarray*} \mathfrak{S}&=& -\frac{1}{4}\sum_E\left\{m_{\!\stackrel{\phantom{.}}{E}}\left(6\Gamma^2+ 6(g+2)\Gamma\Theta+(7g+6)\Theta^2\right)_{\!\stackrel{\phantom{.}}{E}}+5g-6 \right\}\\ &&+\sum_T g\mu(T)+\sum_{\on{ram}1} g +\sum_{\on{ram}3} 3g. \end{eqnarray*} We defer the proof of Prop.~\ref{bogomolov1} until the end of this section, when all of the terms in this sum will be computed. \subsection{Grouping the contributions of each $\Delta{\mathfrak{T}}_{k,i}$ in $\mathfrak{S}$} \label{Grouping} Substituting the results of Corollary~\ref{constants} in the expression for $\mathfrak{S}$, we eliminate $m_{\!\stackrel{\phantom{.}}{E}}$, $\Theta_{\!\stackrel{\phantom{.}}{E^{\prime}}}$ and $\Gamma_{\!\stackrel{\phantom{.}}{E^{\prime}}}$: \begin{eqnarray*} \mathfrak{S}&\!\!=\!\!&\sum_T g\mu(T)+\sum_{\on{ram}1} g+\sum_{\on{ram}2} 3g+ \frac{1}{4}\sum_{E,E^-\on{red}}\!\!\left(6(2+p_{\!\stackrel{\phantom{.}}{E}}) (g-p_{\!\stackrel{\phantom{.}}{E}})-12g\right)\\ &\!\!-\!\! &\frac{1}{4}\sum_{E^-\on{nonred}}\!\!\!\!\!\left(3(p_{\!\stackrel{\phantom{.}}{E}}+2)^2+5g-6\right)+\frac{1}{4}\sum_{E\on{nonred}}\!\!\left(3(p_{\!\stackrel{\phantom{.}}{E}}+5)(2g-p_{\!\stackrel{\phantom{.}}{E}}-1)-19g+6\right). \end{eqnarray*} For each chain $T$ in $\widehat{Y}$, the inverse image $\widehat{\phi}^*(T)$ in $\widehat{X}$ is a member (or a blow-up of a member) of exactly one boundary divisor $\Delta{\mathfrak{T}}_{k,i}$. Consequently, to find the contribution to $\mathfrak{S}$ of a specific $\Delta{\mathfrak{T}}_{k,i}$, we calculate the sum in $\mathfrak{S}$ corresponding to all types of special fibers $\widehat{\phi}^*(T)$. \begin{figure}[h] $$\hspace*{-7mm}\psdraw{coef1}{4.5in}{1.3in}$$ \caption{Coefficients with no ramification} \label{coef1.fig} \end{figure} \subsubsection{Contributions of $\Delta{\mathfrak{T}}_{1,i}, \Delta{\mathfrak{T}}_{2,i}$ and $\Delta{\mathfrak{T}}_{3,i}$} \label{contribution1} Fig.~\ref{coef1.fig} presents the special fibers corresponding to the boundary divisors $\Delta{\mathfrak{T}}_{1,i},\,\,\Delta{\mathfrak{T}}_{2,i},\,\,\Delta{\mathfrak{T}}_{3,i}$. In each of these cases, there is only one component $E$ in $T$ besides the root $R=E^-$. Thus, the subchain $T(E)$ in $T$ is trivial -- it consists only of $E$. Its inverse image $\widehat{\phi}^*(E)$ is connected for $\Delta{\mathfrak{T}}_{1,i}$, and consists of two connected curves for $\Delta{\mathfrak{T}}_{2,i}$ and $\Delta{\mathfrak{T}}_{3,i}$. Setting the genus of the inverse image of $R$ to be $i$, it is easy to see that the genus $p_{\!\stackrel{\phantom{.}}{E}}$ of $\phi^*(E)$ is $g-i-2$ in the first two cases, and $g-i-1$ in the third case.(The total genus of the original fiber of $X$, drawn in full lines, must be $g$.) Finally, counting the number of ``quasi-admissible'' blow-ups (drawn by dashed lines), we see that $\mu(T)=0$ for $\Delta{\mathfrak{T}}_{1,i}$, $\mu(T)=1$ for $\Delta{\mathfrak{T}}_{2,i}$, and $\mu(T)=2$ for $\Delta{\mathfrak{T}}_{3,i}$ (cf.~Lemma~\ref{mu(C)}). Note that there are no ramification modifications. The contribution of each such fiber $\widehat{\phi}^*T$ to the sum $\mathfrak{S}$ is only one summand of the first type ($E,E^-$reduced), plus the quasi-admissible adjustment $g\mu(T)$. If $\widehat{\phi}^*T$ corresponds to the boundary divisor $\Delta{\mathfrak{T}}_{k,i}$, we denote this contribution by $c_{k,i}$. In conclusion, \[c_{k,i}= \frac{1}{4}\big(6(2+p_{\!\stackrel{\phantom{.}}{E}})(g-p_{\!\stackrel {\phantom{.}}{E}})-12g\big)+g\mu(T)\,\,\Rightarrow\,\, c_{k,i}=\frac{3}{2}(i+2)(g-i)-(4-k)g,\,\,k=1,2,3.\] \subsubsection{Contributions of $\Delta{\mathfrak{T}}_{4,i}$ and $\Delta{\mathfrak{T}}_{5,i}$: ramification index 1} \label{contribution2} In each of these cases, the fiber $T$ of $\widehat{Y}$ consists of two rational curves $E_1$ and $E_2$, and the root $R=E_1^-$ (cf.~Fig.~\ref{coef2.fig}). There are no nonreduced components in $T$, so the contribution to $\mathfrak{S}$ consists of two summands of the first type ($E,E^-$ nonreduced), plus a quasi-admissible adjustment of $\mu(T)=1$ for $\Delta{\mathfrak{T}}_{5,i}$, and a ramification adjustment of $g$ in both cases: \[c_{k,i}=\frac{1}{4}\sum_{j=1,2} \big(6(2+p_{\!\stackrel{\phantom{.}}{E_j}})(g-p_ {\!\stackrel{\phantom{.}}{E_j}}) -12g\big)+g\mu(T)+g\,\,\on{for}\,\,k=4,5.\] \begin{figure}[h] $$\psdraw{coef2}{3.25in}{2in}$$ \caption{Coefficients for ramification index 1} \label{coef2.fig} \end{figure} The arithmetic genus of the nonreduced component of $\widehat{X}$ is $-2$, and its intersection number with each of the neighboring components is 2. Setting $p_a(\widehat{\phi}^*R)=i$ forces $p_a(\widehat{\phi}^*E_2)= g-i-1$. Hence, $p_{\!\stackrel{\phantom{.}}{E_1}}=g-i-1$ and $p_{\!\stackrel{\phantom{.}}{E_2}}=g-i-2$. Substituting: \[c_{k,i}=3(g-i)(i+1)-\frac{7g-3}{2}+g\mu(T),\] \[\vspace*{-5mm} c_{4,i}=3(i+1)(g-i)-\frac{7g-3}{2},\,c_{5,i}=3(i+1)(g-i)-\frac{7g-3}{2}+2g.\] \subsubsection{Contribution of $\Delta{\mathfrak{T}}_{6,i}$: ramification index 2} \label{contribution3} It remains to consider the case of ramification index 2. Here there are four components $E$ besides the root $R$ in the special fiber $T\subset \widehat{Y}$. Consequently, there are four summands in $\mathfrak{S}$ corresponding to the $E_i$'s: $E_1$ and $E_4$ yield summands of the first type ($E,E^-$ reduced), $E_2$ yields a summand of the second type ($E$ nonreduced), and $E_3$ yields a summand of the third type ($E^-$ nonreduced). \begin{figure}[h] $$\psdraw{coef3}{3.5in}{1.5in}$$ \caption{Coefficients for ramification index 2} \label{coef3.fig} \end{figure} Since $\mu(T)=0$, and the ramification adjustment is $3g$, we obtain for the contribution of $\Delta{\mathfrak{T}}_{6,i}$ to $\mathfrak{S}$ the following expression: \begin{eqnarray*} c_{6,i}\!\!&\!\!=\!\!&\!\!\frac{1}{4}\big(6(2+ p_{\!\stackrel{\phantom{.}}{E_1}})(g- p_{\!\stackrel{\phantom{.}}{E_1}})-12g\big)+ \frac{1}{4}\big(6(2+p_{\!\stackrel{\phantom{.}}{E_4}})(g- p_{\!\stackrel{\phantom{.}}{E_4}})-12g\big)+\\ \!\!&\!\!+\!\!&\!\!\frac{1}{4}\left(3( p_{\!\stackrel{\phantom{.}}{E_2}}+5)(2g- p_{\!\stackrel{\phantom{.}}{E_2}}-1)-19g+6\right) -\frac{1}{4}\left(3(p_{\!\stackrel{\phantom{.}}{E_3}}+2)^2+5g-6\right)+3g. \end{eqnarray*} The arithmetic genera of the components in $\widehat{X}$ are denoted in the Fig.~\ref{coef3.fig}. It is easy to see that $p_{\!\stackrel{\phantom{.}}{E_4}} =i$, $p_{\!\stackrel{\phantom{.}}{E_3}}=i-3$, $p_{\!\stackrel{\phantom{.}}{E_2}}=i-2$, $p_{\!\stackrel{\phantom{.}}{E_1}}=i-2$. Finally, \[c_{6,i}=\frac{9}{2}i(g-i)-\frac{3}{2}(g-1).\] \subsection{Proof of Proposition~\ref{bogomolov1}} \label{Proof} In the above discussion we calculated the contributions of the boundary divisors $\Delta{\mathfrak{T}}_{k,i}$ to the sum $\mathfrak{S}$, so that $\mathfrak{S}=\sum_{k,i}c_{k,i}$ with $k=1,...,6$, and the corresponding limits for the index $i$ (cf.~Prop.~\ref{boundary}). It is now clear what the divisor $\mathcal{E}$ should be. We set $\mathcal{E}:=\sum_{k,i}c_{k,i}\Delta{\mathfrak{T}}_{k,i}$, and thus, $\mathfrak{S}=\mathcal{E}|_B$, \[\Rightarrow\,\,\, (7g+6)\lambda|_B=g\delta|_B+\mathcal{E}|_B+\frac{g-3}{2}(4c_2(V)-c_1^2(V)).\] Using the restrictions on the index $i$ for each type of boundary divisor $\Delta{\mathfrak{T}}_{k,i}$, one can easily deduce that all coefficients $c_{k,i}> 0$. For instance, when $i=1,...,[g/2]$: \[c_{6,i}=\frac{9}{2}i(g-i)-\frac{3}{2}(g-1)>\frac{9}{2}1\cdot (g-1)-\frac{3}{2}(g-1)=3(g-1)>0.\] In other words, $\mathcal{E}$ is an effective rational linear combination of boundary divisors in $\overline{\mathfrak{T}}_g$, which by construction does not contain $\Delta{\mathfrak{T}}_0.\,\,\,\qed$ \subsection{The slope bound $7+6/g$ and a relation restricted to the base curve $B$} \label{slopebound} Recall that a vector bundle $V$ of rank 2 is {\it Bogomolov semistable} if $4c_2(V)\geq c^2_1(V)$. \begin{prop}[$7+6/g$ bound] For a general base curve $B$, if the canonically associated vector bundle $V$ is Bogomolov semistable, then the slope of $X/\!_{\displaystyle{B}}$ is bounded by \[\frac{\delta|_B}{\lambda|_B}\leq 7+\frac{6}{g}\cdot\] \label{7+6/g Bogomolov} \end{prop} \vspace*{-5mm}\begin{proof} The statement follows directly from Prop.~\ref{bogomolov1}. Indeed, since $\mathcal{E}$ is effective, then $\mathcal{E}|_B\geq 0$. By hypothesis, $4c_2(V)-c^2_1(V)\geq 0$, and $g\geq 3$. Hence, $(7g+6)\lambda|_B\geq g\delta|_B.$ \end{proof} \begin{cor} For a general base curve $B$ the following relation holds true: \begin{eqnarray*} \!(7g+6)\lambda|_B&\!\!\!=\!\!\!&g\delta_0|_B+ \frac{g-3}{2}\left(4c_2(V)-c_1^2(V)\right)\\ &\!\!\!+\!\!\!\!\!\!& \sum_{i=1}^{[(g-2)/2]}\frac{3}{2}(i+2)(g-i)\delta_{1,i}|_B+ \sum_{i=1}^{g-2}\frac{3}{2}(i+2)(g-i)\delta_{2,i}|_B\\ &\!\!\!+&\sum_{i=1}^{[g/2]}\frac{3}{2}(i+1)(g-i+1)\delta _{3,i}|_B+\sum_{i=1}^{[(g-1)/2]} \big(3(i+1)(g-i)-\frac{g-3}{2}\big)\delta_{4,i}|_B\\ &\!\!\!+&\sum_{i=1}^{g-1}\big(3(i+1)(g-i)-\frac{g-3}{2}\big) \delta_{5,i}|_B+\sum_{i=1}^{[g/2]} \big(\frac{9}{2}i(g-i)-{\frac{g-3}{2}}\big)\delta_{6,i}|_B. \end{eqnarray*} \label{analog1}\vspace*{-7mm} \end{cor} \noindent{\it Proof.} This is an immediate consequence of the established relation in Prop.~\ref{bogomolov1}. We replace $\delta$ by the linear combination (\ref{divisorrel}) of the boundary classes of $\overline{\mathfrak{T}}_g$, and write \[(7g+6)\lambda=g\delta_0|_B+\sum_{k,i}\widetilde{c}_{k,i}\delta_{k,i}|_B +\frac{g-3}{2}(4c_2(V)-c_1^2(V)),\] for some new coefficients $\widetilde{c}_{k,i}$. Recall that $\on{mult}_{\delta}(\delta_{k,i})$ denotes the {\it multiplicity} of $\delta_{k,i}$ in $\delta$, so that $\widetilde{c}_{k,i}=c_{k,i}+\on{mult}_{\delta}(\delta_{k,i})g$. For example, the coefficient of $\delta_{1,i}$ is \[\widetilde{c}_{1,i}=\left\{\frac{3}{2}(i+2)(g-i)-3g\right\}+3g= \frac{3}{2}(i+2)(g-i),\] or the coefficient of $\delta_{5,i}$ is \[\widetilde{c}_{5,i}=\left\{3(i+2)(g-i)-\frac{7g-3}{2}+2g\right\}+g= 3(i+1)(g-i)-\frac{g-3}{2}.\,\,\,\qed\] \medskip\section*{10. Generalized Index Theorem and Upper Bound} \setcounter{section}{10} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{lem}{0} \setcounter{thm}{0} \setcounter{prop}{0} \setcounter{defn}{0} \setcounter{cor}{0} \setcounter{conj}{0} \setcounter{claim}{0} \setcounter{remark}{0} \setcounter{equation}{0} \label{Index1} \begin{prop}[Index Theorem on $\widehat{X}$] For a general base curve $B$ and for the rank 2 vector bundle $V$ on $\widehat{Y}$, we have $9c_2(V)-2c_1^2(V)\geq 0.$ \label{genindex} \end{prop} \begin{proof} The proof is identical to that of Theorem~\ref{indextheorem}. One considers the divisor $\eta$ on $\widehat{X}$ defined by \[\eta:=\left(\zeta+\frac{1}{3}\pi^*c_1(V)\right)\big|_ {\widehat{X}},\] and shows that $\eta$ kills the pullback of any divisor on $\widehat{Y}$. In particular, $\eta$ kills an ample divisor on $\widehat{X}$. By the index theorem on $\widehat{X}$, $\eta^2 \leq 0$. From expression (\ref{genX}), this can be also written as $9c_2(V)-2c_1^2(V)\geq 0.$ \end{proof} \medskip As in Section~7, the index theorem on $\widehat{X}$ suggests to replace the Bogomolov difference $4c_2(V)-c^2_1(V)$ by another linear combination of $c_2(V)$ and $c^2_1(V)$, which would behave in a more ``predictable'' way, namely, by $9c_2(V)-2c_1^2(V)$. In the process of doing so, the only way to eliminate the unnecessary global terms $d$ and $c$ from a relation among $\lambda|_B$ and $\delta|_B$ is to subtract: $36(g+1)\lambda|_B-(5g+1)\delta|_B.$ \begin{prop} For a general base curve $B$ and an effective rational combination $\mathcal{E}^{\prime}$ of the boundary divisors $\Delta{\mathfrak{T}}_{k,i}$, not containing $\Delta{\mathfrak{T}}_0$, we have: \[36(g+1)\lambda|_B=(5g+1)\delta|_B+\mathcal{E}^{\prime}|_B+(g-3) \big(9c_2(V)-2c_1^2(V)\big).\] \label{indexrelation}\vspace*{-5mm} \end{prop} Note the apparent similarity between this relation and Prop.~\ref{bogomolov1}. One may use the latter to prove the former, but the calculations are not simpler than if one starts from scratch. We will show a sketch of this proof, leaving the details to the reader, and referring to the proof of Prop.~\ref{bogomolov1} for comparison. \begin{proof} We denote by $\mathfrak{S}^{\prime}$ the difference \[\mathfrak{S}^{\prime}:= 36(g+1)\lambda|_B-(5g+1)\delta|_B -(g-3)\left(9c_2(V)-2c_1^2(V)\right).\] Substituting for $\delta|_B,\lambda|_B$ and $c_1^2(V)$ the corresponding identities from Prop.~\ref{hatlambda} and Example 8.1, and recalling that $c=g+2$ (cf.~Lemma~\ref{adjunction}), we write $\mathfrak{S}^{\prime}$ as \begin{eqnarray*} \mathfrak{S}^{\prime}&= &-\sum_E\left\{m_{\!\stackrel{\phantom{.}}{E}}\left(8\Gamma^2+ 8(g+2)\Gamma\Theta+9(g+1)\Theta^2\right)_{\!\stackrel{\phantom{.}}{E}}+6(g-1) \right\}\\ &&+\,(5g+1)\left(\sum_T \mu(T)+\sum_{\on{ram}1}1 +\sum_{\on{ram}3} 3\right). \end{eqnarray*} As in Lemma~\ref{bogomolov1}, we group the above summands for every special fiber in $\widehat{X}$, and correspondingly, for every chain $T$ in $\widehat{Y}$. Recall Corollary~\ref{constants}, and the computations of the arithmetic genera $p_{\!\stackrel{\phantom{.}}{E}}$ in the previous section: \begin{eqnarray*} \mathfrak{S}^{\prime} &\!\!=\!\!&(5g+1)\big(\sum_T \mu(T)+\sum_{\on{ram}1} 1+\sum_{\on{ram}2} 3\big)+\!\!\sum_{E,E^-\on{red}}\!\!\left(8(p_{\!\stackrel{\phantom{.}}{E}}+2) (g-p_{\!\stackrel{\phantom{.}}{E}})-3(5g+1)\right)\\ &\!\!-\!\!\!\!&\sum_{E^-\on{nonred}}\!\!\!\!\!\left(4 (p_{\!\stackrel{\phantom{.}}{E}}+2)^2+6(g-1)\right)\,+ \sum_{E\on{nonred}}\!\!\left(4(p_{\!\stackrel{\phantom{.}}{E}}+5) (2g-1-p_{\!\stackrel{\phantom{.}}{E}})-12(g-1)\right). \end{eqnarray*} With this at hand, it is not hard to calculate the contributions $d_{k,i}$ of each boundary component $\Delta{\mathfrak{T}}_{k,i}$ to the sum $\mathfrak{S}^{\prime}$: \[\begin{array}{|l|l|}\hline \!d_{1,i}\stackrel{\phantom{l}}{=}8(i+2)(g-i)\phantom{+1}\!-3(5g+1)\!& \!d_{4,i}=16(i+1)(g-i)-2(g-3)-3(5g+1)\\ \!d_{2,i}\stackrel{\phantom{l}}{=}8(i+2)(g-i)\phantom{+1}-\!2(5g+1)\!& \!d_{5,i}\stackrel{\phantom{l}}{=}16(i+1)(g-i)-2(g-3)-\phantom{3}(5g+1)\\ \!d_{3,i}\stackrel{\phantom{l}}{=}8(i+1)(g-i+1)-(5g+1)\!& \!d_{6,i}\stackrel{\phantom{l}}{=}24i(g-i)-(5g+1).\phantom{\big)}\\\hline \end{array}\] \label{d_{k,i}table} Let $\mathcal{E}^{\prime}=\sum_{k,i}d_{k,i}\Delta{\mathfrak{T}}_{k,i}$. Then $\mathfrak{S}^{\prime}=\mathcal{E}^{\prime}|_B$, and the desired relation would be established if $\mathcal{E}^{\prime}$ is effective. Given the restrictions on the indices $i$ of the coefficients $d_{k,i}$ in Prop.~\ref{boundary}, one easily shows that all $d_{k,i}>0.$ \end{proof} \begin{prop}[Maximal Bound] For a general base curve $B$, the slope satisfies: \[\frac{\delta}{\lambda}\leq \frac{36(g+1)}{5g+1},\] with equality if and only all fibers of $X$ are irreducible curves, and either $g=3$ or the divisor $\eta$ on the total space of ${X}$ is numerically zero. \label{genmaximal} \end{prop} \begin{proof} From the Index Theorem on $\widehat{X}$, it follows that $9c_2(V)-2c_1^2(V)\geq 0$. Since $\mathcal{E}^{\prime}$ is effective, $\mathcal{E}^{\prime}|_B\geq 0$. Then Prop.~\ref{indexrelation} implies $36(g+1)\lambda|_B\geq(5g+1)\delta|_B$, with equality exactly when $9c_2(V)-2c_1^2(V)=0$ and $\mathcal{E}^{\prime}|_B=0$. The latter means that $B\cap \Delta{\mathfrak{T}}_{k,i}=\emptyset$ because all coefficients $d_{k,i}$ of $\mathcal{E}^{\prime}$ are strictly positive. In other words, the family $\widehat{X}$ has only {\it irreducible} fibers ($B\cap\Delta{\mathfrak{T}}_{0}\not = \emptyset$). This takes us back to Section~7, where we presented the global calculation on the triple cover $X\rightarrow Y$. There we concluded that the {\it index condition} $9c_2(V)-2c_1^2(V)=0$ was equivalent to $\eta\equiv 0$ on $X$($=\widehat{X}$), or the genus $g=3$. \end{proof} \begin{cor} For a general base curve $B$, \begin{eqnarray*}\vspace*{-1mm} \!36(g+1)\lambda|_B&\!\!\!\!=\!\!\!\!&(5g+1)\delta_0|_B+ (g-3)\left(9c_2(V)-2c_1^2(V)\right)\\ &\!\!\!\!+\!\!\!\!\!&\sum_{i=0}^{[(g-2)/2]}8(i+2)(g-i)\delta_{1,i}|_B+ \sum_{i=1}^{g-2}8(i+2)(g-i)\delta_{2,i}|_B\\ &\!\!\!\!+\!\!\!\!\!&\sum_{i=1}^{[g/2]}8(i+1)(g-i+1)\delta_{3,i}|_B+ \!\!\!\sum_{i=1}^{[(g-1)/2]}\!\!\!\!\! \big(16(i+1)(g-i)-2(g-3)\big)\delta_{4,i}|_B\\ &\!\!\!\!+\! \!\!\!&\sum_{i=1}^{g-1}\big(16(i+1)(g-i)-2(g-3)\big)\delta_{5,i}|_B+ \sum_{i=1}^{[g/2]}24i(g-i)\delta_{6,i}|_B. \end{eqnarray*} \label{analog2} \end{cor} \label{page analog2} \noindent{\it Proof.} We only need to substitute the known expressions for the divisors $\mathcal{E}^{\prime}$ and $\delta$ into Prop.~\ref{indexrelation}: \[36(g+1)\lambda|_B=(5g+1)\delta_0|_B+\sum_{k,i}\big((5g+1)\on{mult}_{\delta} (\delta_{k,i})+d_{k,i}\big)+(g-3)\big(9c_2(V)-2c_1^2(V)\big).\] The rest is a simple calculation. For example, the total coefficient $\widetilde{d}_{3,i}$ of $\delta_{3,i}$ equals \begin{eqnarray*} d_{3,i}+(5g+1)\on{mult}_{\delta}(\delta_{3,i})&=&\{8(i+1)(g-i+1)-(5g+1)\}+ (5g+1)\cdot 1\\ &=&8(i+1)(g-i+1).\,\,\,\qed \end{eqnarray*} \medskip \section*{11. Extension to an Arbitrary Base $B$} \setcounter{section}{11} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{lem}{0} \setcounter{thm}{0} \setcounter{prop}{0} \setcounter{defn}{0} \setcounter{cor}{0} \setcounter{conj}{0} \setcounter{claim}{0} \setcounter{remark}{0} \setcounter{equation}{0} \label{arbitrary} We extend now the results of Sect.~8-10 to arbitrary nonisotrivial families $X\!\rightarrow \!B$ with smooth trigonal general member. The essential case is when $B$ is {\it not} tangent to the boundary $\Delta{\mathfrak{T}}$, from which the remaining cases easily follows. \subsection{The base curve $B$ not tangent to $\Delta\mathfrak{T}$} \label{nontangentB} We now drop the hypothesis of the base curve $B$ intersecting the boundary divisors in general points. Instead, for now we only assume that the base curve $B$ is not tangent the boundary $\Delta{\mathfrak{T}}$. This means that all special fibers of $X$ locally look like the general ones (cf.~Fig.~\ref{coef1.fig}--\ref{coef3.fig}). Therefore, from the quasiadmissible cover $\widetilde{X}\rightarrow \widetilde{Y}$ we can construct an effective cover $\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$ of {\it smooth} surfaces $\widehat{X}$ and $\widehat{Y}$. (The {\it smoothness} indicates that $B$ is {\it not} tangent to any $\Delta{\mathfrak{T}}_{k,i}$. Otherwise, there would be a higher local multiplicity $xy=t^n$ near a node of a special fiber $C_{X}$, $n> 1$. Hence $\widehat{X}$ would be obtained locally via a base change from a smooth surface, but $\widehat{X}$ would have a singular total space.) Now the special fibers of $\widehat{Y}$ are {\it trees} $T$ (rather than just chains) of reduced smooth rational curves with occasional nonreduced rational components of multiplicity 2. The latter occur again exactly for each singular point in $C_{\widetilde{X}}$ of ramification index 2 under the quasiadmissible cover $\widetilde{\phi}: \widetilde{X}\rightarrow\widetilde{Y}$ (cf.~Fig.~\ref{coef3.fig}). \smallskip The notation and conventions from Sections \ref{conventions} are also valid here. In particular, for any tree $T$, we fix one of its end (nonreduced) components to be its root $R$, and we define as before the functions $m,\theta,\gamma$ on the components $E$ of $T$. Moreover, since Lemma \ref{technical} can be applied also for any tree $T$, the calculations of $\lambda_{\widehat{X}},\kappa_{\widehat{X}}$ and $\delta$ in Prop.~\ref{hatlambda} and Cor.~\ref{hatdelta} go through without any modifications. \smallskip Finally, we wish to extend all results of Sections~8-10 over the new base $B$. The only difference arises in the final calculation of the coefficients $c_{k,i}$ and $d_{k,i}$. The fiber $C_X$ in $X$, corresponding to a tree $T$, may now lie in the intersection of {\it several} boundary divisors $\Delta{\mathfrak{T}}_{k,i}$. Such a trigonal curve $C_X$ is called a {\it special boundary} curve. Accordingly, its contribution $c_{\!\stackrel{\phantom{.}}{T}}$ to $\mathfrak{S}$ (or $d_{\!\stackrel{\phantom{.}}{T}}$ to $\mathfrak{S}^{\prime}$) will be distributed among these divisors $\Delta{\mathfrak{T}}_{k,i}$'s, rather than just yielding a single coefficient $c_{k,i}$ (or $d_{k,i}$) as before. \begin{figure}[h] $$\psdraw{arbitrary}{1.6in}{1.5in}$$ \caption{Moving $B$} \label{arbitrary1} \end{figure} This can be easily resolved. The idea is to replace any special singular fiber in $\widehat{X}$ by a suitable combination of {\it general} fibers, without changing the sums $\mathfrak{S}$ and $\mathfrak{S}^{\prime}$. We can imagine this as ``moving'' the base curve $B$ in $\overline{\mathfrak{T}}_g$ {\it away from} the special singular locus of $\overline{\mathfrak{T}}_g$, and replacing it with a {\it general} base curve $B^{\prime}$, as defined in Section~8. For example, in Fig.~\ref{arbitrary1} the base $B$ passes through a point $p$ in the intersection of two boundary divisors $\Delta\mathfrak{T}_{k,i}$. Two new general points $p_1$ and $p_2$, each lying on a single $\Delta\mathfrak{T}_{k,i}$, replace the special point $p$, and thus $B$ moves to a {\it general} curve $B^{\prime}$. \begin{lem} Let $C_X$ be a special boundary curve in $\overline{\mathfrak{T}}_g$. Denote by $\alpha_{k,i}$ the degree of the point $[C_X]$ in the intersection $\Delta{\mathfrak{T}}_{k,i}\cdot B$. Then the contributions of $T=\widehat{\phi}(C_{\widehat{X}})$ to $\mathfrak{S}$ and to $\mathfrak{S}^{\prime}$ are $c_{\!\stackrel{\phantom{.}}{T}}=\sum_{k,i}\alpha_{k,i}c_{k,i}$ and $d_{\!\stackrel{\phantom{.}}{T}}=\sum_{k,i}\alpha_{k,i}d_{k,i}$, respectively. \label{contributions d_Tc_T} \end{lem} \noindent{\it Proof:} Rewrite $\mathfrak{S}$ and $\mathfrak{S}^{\prime}$ as sums over the non-root components $E$ of the special trees $T$: \begin{eqnarray*} \mathfrak{S}&\!\!=\!\!& \sum_{E,E^-\on{red}}F_1(p_{\!\stackrel{\phantom{.}}{E}})+ \sum_{E^-\on{nonred}}\!\!\!\!\!F_2(p_{\!\stackrel{\phantom{.}}{E}}) +\sum_{E\on{nonred}}\!\!F_3(p_{\!\stackrel{\phantom{.}}{E}}) +gH,\\ \mathfrak{S}^{\prime}&\!\!=\!\!& \sum_{E,E^-\on{red}}G_1(p_{\!\stackrel{\phantom{.}}{E}})+ \sum_{E^-\on{nonred}}\!\!\!\!\!G_2(p_{\!\stackrel{\phantom{.}}{E}}) +\sum_{E\on{nonred}}\!\!G_3(p_{\!\stackrel{\phantom{.}}{E}})+(5g+1)H, \end{eqnarray*} where $H=\sum_T\mu(T)+\sum_{\on{ram}1}1+\sum_{\on{ram}2}3$ is the quasi-admissible and effective adjustment, and the functions $F_i$ and $G_j$ are quadratic polynomials in $p_{\!\stackrel{\phantom{.}}{E}}$ with linear coefficients in $g$. Recall that in these sums each non-root component $E$ appears exactly once, and $p_{\!\stackrel{\phantom{.}}{E}}$ is the arithmetic genus of the inverse image $\widehat{\phi}^*(T(E))$ of the subtree $T(E)$ generated by $E$. \smallskip There is a simple way to recognize the boundary divisors $\Delta{\mathfrak{T}}_{k,i}$ in which a special trigonal fiber $C_X$ lies. Consider the corresponding ``effective'' fiber $C_{\widehat{X}}=\widehat{\phi}^*T$ in $\widehat{X}$. For any non-root component $E$ in $T$ there are two possibilities: either $\widehat{\phi}^*E$ and $\widehat{\phi}^*{E^-}$ are both reduced, or $E$ is part of a chain of length 3 or 5, constructed to resolve ramifications in the quasi-admissible fiber $C_{\widetilde{X}}$. \smallskip \begin{figure}[h] $$\psdraw{notchain}{3.8in}{1.8in}$$ \vspace*{-2mm} \caption{$E\not\subset$ chain $\rightarrow\,\, \alpha_{1,i},\alpha_{2,i},\alpha_{3,i}$} \label{notchain} \end{figure} \subsubsection{Contributions to the degrees $\alpha_{1,i},\alpha_{2,i},\alpha_{3,i}$} Consider the first situation, and denote by $C^{\prime}$ the preimage $\widehat{\phi}^*E\cup \widehat{\phi}^*{E^-}$ in $\widehat{X}$. Thus, $C^{\prime}$ corresponds to a general member of $\Delta{\mathfrak{T}}_{1,i},\Delta{\mathfrak{T}}_{2,i}, \Delta{\mathfrak{T}}_{3,i}$, possibly of lower genus (cf.~Fig.~\ref{notchain}). As part of the fiber $C_{\widehat{X}}$, the curve $C^{\prime}$ is represented for simplicity by the triple intersection of two {\it smooth} trigonal curves (in $\Delta{\mathfrak{T}}_{1,i}$), but it could have corresponded to any general member of $\Delta{\mathfrak{T}}_{2,i}$ or $\Delta{\mathfrak{T}}_{3,i}$. The solid box encompasses the preimage $\widehat{\phi}^*T(E)$, and the dashed box encompasses the preimage of the rest, $\widehat{\phi}^*\big(T-T(E)\big)$. Each of these boxes represents a limit of a quasi-admissible curve, $C_1$ or $C_2$, which is naturally a triple cover of ${\mathbf P}^1$. Thus, we can {\it ``smoothen''} each box to such a curve $C_i$. As a result we obtain a quasiadmissible curve $C_1\cup C_2$ of total genus $g$, which corresponds to a general member of $\Delta{\mathfrak{T}}_{1,i},\Delta{\mathfrak{T}}_{2,i}$ or $\Delta{\mathfrak{T}}_{3,i}$. Depending on which divisor $\Delta{\mathfrak{T}}_{k,i}$ is evoked, there is a corresponding contribution of $1$ to the coefficient $\alpha_{k,i}$: $[C_X]\in\Delta{\mathfrak{T}}_{k,i}$. \smallskip Note that the arithmetic genus of $C_2$ is the previously defined $p_{\!\stackrel{\phantom{.}}{E}}$. The contribution of $E$ to $\mathfrak{S}$ is $F_1(p_{\!\stackrel{\phantom{.}}{E}})$ plus the possible quasi-admissible adjustment in $\mu(T)$ needed to obtain $\widetilde {\phi}^*(E\cup E^-)$. In view of the above ``smoothening'', this can be thought of as the contribution of $C_2$ in the effective curve $C_1\cup C_2$, and by Prop.~\ref{bogomolov1} this is exactly the coefficient $c_{k,i}$. The same argument works in the case of $\mathfrak{S}^{\prime}$ from Prop.~\ref{indexrelation}. We conclude that $\alpha_{k,i}$ (for $k=1,2,3$) equals the number of $c_{k,i}$'s and $d_{k,i}$'s in $\mathfrak{S}$ and $\mathfrak{S}^{\prime}$, respectively. \subsubsection{Contributions to the degrees $\alpha_{4,i},\alpha_{5,i},\alpha_{6,i}$} We treat analogously the remaining case when the component $E$ is part of a chain of length 3 or 5. Here, however, one must consider {\it simultaneously all} the components $E$ of $T$ participating in such a chain, and take a quasi-admissible limit only {\it over the reduced} curves in $C_{\widehat{X}}$. In Fig.~\ref{chain} one can see all three ramification cases, or equivalently, the boundary divisors $\Delta{\mathfrak{T}}_{4,i},\Delta{\mathfrak{T}}_{5,i}$ and $\Delta{\mathfrak{T}}_{6,i}$. For simplicity, we have again depicted the reduced components in $\widehat{X}$ by smooth trigonal curves, which may not always be true for every tree $T$: they could, for instance, be singular or reducible, but they will keep the ramification index 1 or 2 at the appropriate points. \begin{figure}[h] $$\psdraw{chain}{5.5in}{2in}$$ \caption{$E\subset$ chain $\rightarrow\,\, \alpha_{4,i},\alpha_{5,i},\alpha_{6,i}$} \label{chain} \end{figure} \newline To see how $c_{k,i}$ and $d_{k,i}$ are obtained, let us calculate, for example, the contributions of $E_1,E_2,E_3$ and $E_4$ in the case of $\Delta{\mathfrak{T}}_{6,i}$. The inverse images in $\widehat{X}$ of $T-T(E_1)$ and $T(E_4)$ are marked by dashed and solid boxes, respectively. We {\it ``smoothen''} each box by a smooth trigonal curve, $C_1$ or $C_2$, and keep the inverse images of $E_1$,$E_2$ and $E_3$. Thus, we obtain a general member ${C}^{\prime\prime}$ of $\Delta{\mathfrak{T}}_{6,i}$. The arithmetic genera, necessary to calculate the contribution of ${C}^{\prime\prime}$ to $\mathfrak{S}$, are given from right to left by: \[p_a(C_2)=p_{\!\stackrel{\phantom{.}}{E_4}},\,\, p_{\!\stackrel{\phantom{.}}{E_3}}=p_{\!\stackrel{\phantom{.}}{E_4}}\!\!-3,\,\, p_{\!\stackrel{\phantom{.}}{E_2}}=p_{\!\stackrel{\phantom{.}}{E_4}}\!\!-2,\,\, p_{\!\stackrel{\phantom{.}}{E_1}}=p_{\!\stackrel{\phantom{.}}{E_4}}\!\!-2.\] As in the proof of Prop.~\ref{bogomolov1}, we substitute these in the sum $\mathfrak{S}$, and for $i=p_{\!\stackrel{\phantom{.}}{E_4}}$ we obtain \[F_1(E_1)+F_1(E_4)+F_2(E_2)+F_3(E_3)+3g=\frac{9}{2} p_{\!\stackrel{\phantom{.}}{E_4}}(g-p_{\!\stackrel{\phantom{.}}{E_4}})- \frac{3}{2}(g-1)=c_{6,i}.\] Combining all of the above results, we conclude that the contributions of any tree $T$ to the sums $\mathfrak{S}$ and $\mathfrak{S}^{\prime}$ are $c_{\!\stackrel{\phantom{.}}{T}}=\sum_{k,i}\alpha_{k,i}c_{k,i}\,\,\on{and}\,\, d_{\!\stackrel{\phantom{.}}{T}}=\sum_{k,i}\alpha_{k,i}d_{k,i}.$\qed \smallskip This allows us to extend all results of Sect.~\ref{Bogomolov1}--\ref{Index1} to the case of a base curve $B$ meeting transversally the boundary $\Delta{\mathfrak{T}}_g$. \subsection{Extension to an arbitrary base $B$, not contained in $\Delta{\mathfrak{T}}_g$} \label{extension} If the base curve $B$ happens to be {\it tangent} to a boundary divisor $\Delta\mathfrak{T}_{k,i}$ at a point $[C_{X}]$, then over some node $p$ of the corresponding tree $T=\widehat{\phi}(C_{\widehat{X}})$ {\it all} local analytic multiplicities $m_q$ (cf.~Sect.~\ref{definition}) will be multiplied by the degree of tangency of $B$ and $\Delta\mathfrak{T}_{k,i}$. Fig.~\ref{Local multiplicities} presents a few examples of possible fibers in $\widetilde{X}$: \begin{figure}[h] $$\psdraw{tangent}{5.4in}{1.2in}$$ \caption{Local multiplicities} \label{Local multiplicities} \end{figure} \smallskip In the nonramification cases of $\Delta\mathfrak{T}_{1,i},\Delta\mathfrak{T}_{2,i}$ and $\Delta\mathfrak{T}_{3,i}$, this would force rational double points as singularities on the total spaces of $\widehat{X}$ and $\widehat{Y}$, whereas in the ramification cases of $\Delta\mathfrak{T}_{4,i},\Delta\mathfrak{T}_{5,i}$ and $\Delta\mathfrak{T}_{6,i}$, one may arrive at surfaces $\widehat{X}$ and $\widehat{Y}$, nonnormal over some nonreduced fibers. But in both cases, one can roughly view the corresponding fibers as being obtained by a base change from the general or special fibers of Sect.~8 and Sect. ~11.1. Alternatively, one can go through the arguments of the paper for the new surfaces $\widehat{X}$ and $\widehat{Y}$ (normalizing, if necessary), and notice that all formulas (e.g. Euler characteristic formula for $\lambda$, Index theorem on $\widehat{X}$, adjunction formula in $\mathbf P V$, etc.) hold for surfaces with double point singularities. \smallskip Thus, in effect, one may replace a given singular fiber $C_X$ by a bunch of general boundary curves $C$, following the procedure described in Section~\ref{nontangentB}. Furthermore, if some of these general curves $C$ are ``multiple'' (i.e. $B$ is tangent to $\Delta\mathfrak{T}_{k,i}$ at $[C]$), one may in turn replace each $C$ by several ``transversal'' general boundary curves, and refer to the statements in Sections~8.3 and 9.3. The only notational difference in this approach will appear in the definition of the invariants $m,\theta$ and $\gamma$ from Sect.~8: now we have to allow for them to be {\it rational}, rather than integral, due to possible rational intersections $E\cdot E^-$. This will be ``compensated'' in the final calculations, which will take into account the multiplicity of the corresponding fibers, and roughly speaking, will ``multiply back'' our invariants $\delta,\lambda$ and $\kappa$ by what they were divided by in the beginning of the calculations. \bigskip This concludes the proof of our results for all families of stable curves $X\rightarrow B$ with general smooth trigonal members. \subsection{Statements of the results for any family $X\rightarrow B$} \label{results} In the following list of results, Theorems~\ref{7+6/g relation2} and \ref{maximal relation2} can be viewed as local trigonal analogs of Cornalba-Harris's relation in the Picard group of the hyperelliptic locus $\overline{\mathfrak{I}}_g$ (cf.~Theorem~\ref{CHPic}). Similarly, Theorem~\ref{maximal bound2} is the analog of the $8+4/g$ maximal bound in the hyperelliptic case (cf.~Theorem~\ref{CHX}). \begin{thm}[$7+6/g$ relation] For any family $X\rightarrow B$ of stable curves with smooth trigonal general member, if $V$ is the canonically associated to $X$ vector bundle of rank 2, then the following relation holds true \[(7g+6)\lambda|_B=g\delta|_B+\mathcal{E}|_B+\frac{g-3}{2}\left(4c_2(V)-c_1^2(V) \right),\] where $\mathcal{E}$ is an effective rational linear combination of boundary components of $\overline{\mathfrak{T}}_g$, not containing $\Delta{\mathfrak{T}}_0$. In particular, \[(7g+6)\lambda|_B=g\delta_0|_B+\sum_{k,i}\widetilde{c}_{k,i} \delta_{k,i}|_B+\frac{g-3}{2}\left(4c_2(V)-c_1^2(V)\right),\] where $\widetilde{c}_{k,i}$ is quadratic polynomial in $i$ with linear coefficients in $g$, and it is determined by the geometry of $\delta_{k,i}$ (cf.~p.~\pageref{analog1}). \label{7+6/g relation2} \end{thm} \begin{thm}[$7+6/g$ bound] For any nonisotrivial family $X\rightarrow B$ of stable curves with smooth trigonal general member, if the canonically associated to $X$ vector bundle $V$ is Bogomolov semistable, then the slope of $X/\!_{\displaystyle{B}}$ is bounded from above by \[\frac{\delta}{\lambda}\leq 7+\frac{6}{g}\cdot\vspace*{-5mm}\] \label{7+6/g Bogomolov2} \end{thm} \begin{thm}[Index relation] For any family $X\rightarrow B$ of stable curves with smooth trigonal general member, if $V$ is the canonically associated to $X$ vector bundle of rank 2, then the following relation holds true \[36(g+1)\lambda|_B=(5g+1)\delta|_B+\mathcal{E}^{\prime}|_B+(g-3) \big(9c_2(V)-2c_1^2(V)\big),\] where $\mathcal{E}^{\prime}$ is an effective rational combination of the boundary divisors $\Delta{\mathfrak{T}}_{k,i}$, not containing $\Delta{\mathfrak{T}}_0$. In particular, \[36(g+1)\lambda|_B=(5g+1)\delta_0|_B+\sum_{k,i}\widetilde{d}_{k,i} \delta_{k,i}|_B+(g-3)\left(9c_2(V)-2c_1^2(V)\right),\] where $\widetilde{d}_{k,i}$ is quadratic polynomial in $i$ with linear coefficients in $g$, and it is determined by the geometry of $\delta_{k,i}$ (cf.~p.~\pageref{page analog2}). \label{maximal relation2} \end{thm} \begin{thm}[Maximal bound] For any nonisotrivial family $X\rightarrow B$ of stable curves with smooth trigonal general member, the slope of $X/\!_{\displaystyle{B}}$ satisfies: \[\frac{\delta}{\lambda}\leq \frac{36(g+1)}{5g+1},\] with equality if and only all fibers of $X$ are irreducible curves, and either $g=3$ or the divisor $\eta$ on the total space of ${X}$ is numerically zero. \label{maximal bound2} \end{thm} \label{list of theorems} \subsection{What happens with the hyperelliptic locus $\overline{\mathfrak{I}}_g$} As we promised in Section~\ref{hyperelliptic locus}, we consider the contribution of the hyperelliptic locus to the above theorems. For any hyperelliptic curve $C$, we need to blow up a point on $C$ before it starts ``behaving'' like a trigonal curve in the quasi-admissible and effective covers. Below we have shown what happens to a smooth or general singular hyperelliptic curve (cf.~Fig.~\ref{hyperboundary} for the admissible classification of the boundary locus $\Delta\mathfrak{I}_g$). \begin{figure}[h] $$\psdraw{smoothhyper}{1in}{1in}$$ \caption{$\overline{\mathfrak{I}}_g\cap\Delta\mathfrak{T}_0$} \label{smoothhyper} \end{figure} \subsubsection{Smooth hyperelliptic curves} We blow up $C$ at a point, and thus add a smooth rational component $\mathbf P^1$ to make it a triple cover $C^{\prime}$ (cf.~Fig.~\ref{smoothhyper}). The quasi-admissible adjustment of $C$ is $\mu(C^{\prime})=1$. From here on, $C$ will behave essentially like a smooth trigonal curve. Therefore, in all relations $C$ is going to contribute $g$ or $(5g+1)$, depending on what $\delta$ is multiplied by. \subsubsection{Singular hyperelliptic curves in $\Delta\mathfrak{T}_{2,i}$ and $\Delta\mathfrak{T}_{5,i}$} The necessary effective and quasi-admissible modifications are shown in Fig.~\ref{singularhyper}--47. In the first case, there are two hyperelliptic components intersecting transversally in two points. For the quasi-admissible cover, we need two ``smooth'' blow-ups, which makes $\mu=2$. From now on, this curve will behave like a element of $\Delta\mathfrak{T}_{2,i}$, where $\mu_{2,i}=1$. Thus, the coefficient in, say, the maximal bound relation will be: $\widetilde{d}_{2,i}+(5g+1)$, due to the extra blow-up in $\mu$. \begin{figure}[h] $$\psdraw{singularhyper}{5in}{1.1in}$$ \caption{$\overline{\mathfrak{I}}_g\cap\Delta\mathfrak{T}_{2,i}$\hspace{22mm} {\sc Figure 47.} $\overline{\mathfrak{I}}_g\cap\Delta\mathfrak{T}_{5,i}$ \hspace*{25mm}} \label{singularhyper} \end{figure} \addtocounter{figure}{1} In the second case, two hyperelliptic components meet transversally in one point, but have a ramification index 1 at this point when viewed as double covers. Fig.46 presents first the quasi-admissible modification: as in the case of $\Delta\mathfrak{T}_{5,i}$, the local analytic multiplicity between the two rational components is $2$, which means that we must have made three ``smooth'' blow-ups and one ``singular'' blow-down. As a result, $\mu=3$. From here on, this curve behaves exactly as a general member of $\Delta\mathfrak{T}_{5,i}$. Recall that $\mu_{5,i}=2$, and the extra $1$ in the hyperelliptic case accounts for the one extra blow-up. Therefore, the coefficient of this fiber $C$, say, in the maximal bound relation, will be $\widetilde{d}_{5,i}+(5g+1)$. \smallskip We conclude that a base curve $B$, passing through the hyperelliptic locus, will contribute in the results listed in Section~\ref{results} roughly $g$, or $(5g+1)$, times the number of elements in $B\cap\overline{\mathfrak{I}}_g$. We cannot write the latter in the form of a scheme-theoretic intersection, since $\overline{\mathfrak{I}}_g$ is of a larger codimension in $\overline{\mathfrak{T}}_g$. \label{hypercalculations} \smallskip One can explain these extra summands in the expressions for $\lambda$ in the following way. Recall the projection map $pr_1:\overline{\mathcal{H}}_{3,g} \rightarrow \overline{\mathfrak{T}}_g$. The exceptional locus of $pr_1$ is the admissible boundary divisor $\Delta{\mathcal{H}}_{3,0}$, which is blown down to the codimension 2 hyperelliptic locus $\overline{\mathfrak{I}}_g$ inside $\overline{\mathfrak{T}}_g$. For calculation purposes, it will be easier to work instead with the space of minimal quasi-admissible covers $\overline{\mathcal Q}_{3,g}$, which replaces $\overline{\mathcal{H}}_{3,g}$. The same situation of a blow-down occurs, where the exceptional divisor in $\overline{\mathcal Q}_{3,g}$ consists of reducible curves $C^{\prime}$, as shown in Fig.~\ref{smoothhyper}. \begin{figure}[h] $$\psdraw{exceptional}{1.3in}{1.8in}$$ \caption{$B\cap \overline{\mathfrak{I}}_g$} \label{exceptional.fig} \end{figure} Let $D$ be the linear combination of divisors in $\overline{\mathfrak{T}}_g$ given by the restriction $\Delta|_{\overline{\mathfrak{T}}_g}$, and consider a curve $B\subset \overline{\mathfrak{T}}_g$, intersecting the hyperelliptic locus in finitely many points. By abuse of notation, we denote by $pr_1$ the projection from $\overline{\mathcal Q}_{3,g}$ to $\overline{\mathfrak{T}}_g$. Then for the intersection $D\cdot B$ we have: \[D\cdot B=pr_1^*(D)\cdot pr_1^*(B)=pr_1^*(D)\cdot(\overline{B}+\sum E_j),\] where $\overline{B}$ is the proper transform of $B$, and the $E_j$'s are the corresponding exceptional curves above $B$. Note that each $E_j$ is in fact a line $\mathbf P^1$ representing all possible quasi-admissible covers, arising from a hyperelliptic curve $[C]\in B\cap \overline{\mathfrak{I}}_g$. From Fig.~\ref{smoothhyper}, these are the blow-ups of $C$ at a point, one for each involution pair $\{p_1,p_2\}\in g^1_2$, and that is \vspace*{3mm}why $E_j\cong\mathbf P^1$. The extra summands on p.~\pageref{list of theorems}, induced by the base curve $B$, are result of the extra intersections $pr_1^*(D)\cdot E_j$ from above. Indeed, the relations, as they stand, compute only $pr_1^*(D)\cdot\overline{B}$, the component corresponding to families with general smooth members. From the calculations on p.~\pageref{hypercalculations}, we expect that each $pr_1^*(D)\cdot E_j=1$, and this will account for the extra $1$ apprearing in all $\mu$'s. \smallskip To verify this, we only needs to show $\delta|_{E_j}=1$. Since we cannot pick out canonically one point $p_i$ from each hyperelliptic pair $\{p_1,p_2\}$ on $C$, and thus construct a family of blow-ups at $p_i$ of $C$ over $E_j\cong \mathbf P^1$, we make a base change of degree two $C\rightarrow E_j$. \begin{figure}[h] $$\psdraw{deltahyper}{1.1in}{1.3in}$$ \caption{$\delta|_{C}=2$} \label{deltahyper.fig} \end{figure} We construct a family over $C$, corresponding to {\it all} blow-ups of $C$ at point $p\in C$. This is simply the products $C\times C$ and $\mathbf P^1\times C$, identified at two sections $S_i$: $S_1$ is the diagonal on $C\times C$, and $S_2$ is a trivial section of $\mathbf P^1\times C$ over $C$ (cf.~Fig.~\ref{deltahyper.fig}). From \cite{CH}, for the base curve $C$ of this family, the degree $\delta|_C$ is computed as \[\delta|_C=\delta_{C\times C}+\delta_{\mathbf P^1\times C}+S_1^2+S_2^2= 0+0+2+0=2.\] Taking into account the base \vspace*{2mm}change $C\rightarrow E_j$, $\delta|_{E_j}=1$. \smallskip Finally, if we allow for our families to have finitely many hyperelliptic fibers, we adjust the relation in ~\ref{7+6/g relation2} by $g\Delta{\mathcal H}_{3,0}\cdot B$, and the relation in ~\ref{maximal relation2} by $(5g+1)\Delta{\mathcal H}_{3,0}\cdot B$. The two bounds in Theorems~\ref{7+6/g Bogomolov2}-\ref{maximal bound2} are unaffected by the above discussion. \setcounter{section}{12} \bigskip\section*{12. Interpretation of the Bogomolov Index $4c_2-c_1^2$ via the Maroni Divisor} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{lem}{0} \setcounter{thm}{0} \setcounter{prop}{0} \setcounter{defn}{0} \setcounter{cor}{0} \setcounter{conj}{0} \setcounter{claim}{0} \setcounter{remark}{0} \setcounter{equation}{0} \label{Bog-Maroni} \subsection{The Maroni invariant of trigonal curves} \label{Maroniinvariant} For any smooth trigonal curve $C$, consider the triple cover $f:C\rightarrow {{\mathbf P}^1}$. The pushforward $f_*(\mathcal{O}_{C})$, as we noted before, is a locally free sheaf of rank 3 on ${{\mathbf P}^1}$, and hence decomposes into a direct sum of three invertible sheaves on ${{\mathbf P}}^1$: \[f_*(\mathcal{O}_{C})=\mathcal{O}_{{\mathbf P}^1}\oplus \mathcal{O}_{{\mathbf P}^1}(a)\oplus \mathcal{O}_{{\mathbf P}^1}(b).\] The first summand is trivial due to the split exact sequence \[0\rightarrow {V}\rightarrow {\alpha}_*{\mathcal O}_{C}\stackrel {\on{tr}}{\rightarrow}{\mathcal O}_{{\mathbf P}^1}\rightarrow 0,\] where $V=\mathcal{O}_{{\mathbf P}^1}(a)\oplus\mathcal{O}_{{\mathbf P}^1}(b)$. From GRR, $a+b=g+2$. We have observed in Section~6 that $C$ embeds in the rational ruled surface ${\mathbf P}V={\mathbf F}_k$, for $k=|b-a|$. \medskip \noindent{\bf Definition 12.1.} The {\it Maroni invariant} of an irreducible trigonal curve $C$ is the difference $|b-a|$. The {\it Maroni locus} in $\overline{\mathfrak{T}}_g$ is the closure of the set of curves with Maroni invariants $\geq 2$ (cf.~[Ma]). \begin{lem} For a general trigonal curve $C$ the vector bundle $V$ is {\it balanced}, i.e. the integers $a$ and $b$ are equal or 1 apart according to $g(\on{mod}2)$. \label{gentrig} \end{lem} \begin{proof} Let $a\leq b$. The statement follows from a dimension count of the linear system $L=|3B_0+\frac{g+2}{2}F|$ on the ruled surface ${\mathbf F}_{b-a}={\mathbf F}_k$. Indeed, all trigonal curves with Maroni invariant $(b-a)/2$ are elements of $L$. If $p:{\mathbf F}_k\rightarrow {\mathbf P}^1$ is the projection map, the projective dimension of $L$ equals \[r(L)=h^0\big(p_*\mathcal{O}_{{\mathbf F}_k}(3B_0+\textstyle{\frac{g+2}{2}} F)\big)-1.\] Denoting by $\widetilde{B}=B_0-\frac{k}{2}F$ the section of ${\mathbf F}_k$ with smallest self-intersection of $-k$, we have $p_*\mathcal{O}_{{\mathbf F}_k}(\widetilde{B})\cong \mathcal{O}_{{\mathbf P}^1}\oplus \mathcal{O}_{{\mathbf P}^1}(-k)$. The necessary pushforward from above is: \begin{equation*} p_*\mathcal{O}_{{\mathbf F}_k}(3\widetilde{B}+\textstyle{\frac{g+2+3k}{2}}F)= \on{Sym}^3(\mathcal{O}_{{\mathbf P}^1}\!\!\oplus\mathcal{O}_{{\mathbf P}^1}(-k)) \otimes \mathcal{O}_{{\mathbf P}^1}({\textstyle{\frac{g+2+3k}{2}}})= \!\!\!\!\displaystyle{\bigoplus_{j=\pm 1,\pm 3}} \!\!\!\!\mathcal{O}_{{\mathbf P}^1}({\textstyle{\frac{g+2+jk}{2}}}). \end{equation*} Since an irreducible trigonal curve $C$ lies in $L$, we have $C\cdot \widetilde{B}\geq 0$, hence $g+2-3k\geq 0$ and $g\equiv k (\on{mod}2)$. Evaluating the sections of this sum of sheaves, we obtain $r(L)=2g+7.$ The ruled surface ${\mathbf F}_k$ has automorphisms, inducing automorphisms of the linear system $L$. We need to mod out these in order to obtain the dimension of the space of trigonal curves embedded in ${\mathbf F}_k$. The group $\on{Aut}{\mathbf F}_k$ is a product (not necessarily direct) of the base automorphisms $\on{Aut}{\mathbf P}^1=\on{PGL}_2$, and the projective automorphisms of the vector bundle $V$. The latter is an open set (up to projectivity) of the homomorphisms of $V$ into $V$, and hence has the same dimension as: \[\on{Hom}(V,V)\cong H^0(V\otimes V\,\,\widehat{\phantom{n}})= H^0\big(\mathcal{O}_{{\mathbf P}^1}(-k)\oplus\mathcal{O}_{{\mathbf P}^1}\oplus \mathcal{O}_{{\mathbf P}^1}\oplus\mathcal{O}_{{\mathbf P}^1}(k)\big).\] For $k>0$, $\on{dim}\on{Aut}V=k+3$, while for $k=0$, $\on{dim}\on{Aut}V=4$. We conclude that the dimension of the set of trigonal curves with Maroni invariant $k/2$ is \[r(L)-\on{dim}\on{Aut}{\mathbf F}_k= \left\{\begin{array}{l} 2g+1\,\,\on{if}\,\,k=0,\\ 2g+2-k\,\,\on{if}\,\,k>0. \end{array}\right.\] \medskip When $k=0$ or $k=1$, this space corresponds to an open dense set of $\overline{\mathfrak{T}}_g$. For an even $g$ a general trigonal curve has Maroni invariant $0$ and therefore embeds in ${\mathbf F}_0= {\mathbf P}^1\times{\mathbf P}^1$, while for an odd $g$ a general trigonal curve has Maroni invariant $1$ and embeds in ${\mathbf F}_1=\on{Bl}_{\on{pt}}(\mathbf P^2)$. In both cases, the vector bundle $V$ is balanced. \end{proof} \begin{cor} For $g$ even, the Maroni locus is a divisor in $\overline{\mathfrak{T}}_g$ whose general member embeds in ${\mathbf F}_2$. For $g$ odd, the Maroni locus has codimension 2 in $\overline{\mathfrak{T}}_g$ and its general member embeds in ${\mathbf F}_3$. \label{maronilocus} \end{cor} \noindent{\bf Remark 12.1.} It will be useful to identify precisely the group of authomorphisms of the linear system $L$ for $k=0,1$. We have $\on{Aut}(\mathbf P^1\!\times \mathbf P^1)\cong PGL_2\times PGL_2\times {\mathbb Z}/2{\mathbb Z}$. The last factor comes from the exchange of the fiber and the base of $\mathbf P^1\times \mathbf P^1$ and it is relevant only for $g=4$: then $L=|3B_0+3F|$. Otherwise, for any even $g>4$: \[\on{Aut}L\cong PGL_2\times PGL_2.\] When $g$ is odd, the ruled surface ${\mathbf F}_1$ can be thought of as the blow-up of $\mathbf P^2$ at the point $q=[0,0,1]$. Any automorphism of $\on{Bl}_q{\mathbf P^2}$ carries the exceptional divisor $E_q$ of $\mathbf F_1$ to itself, and hence is induced by an automorphism of the plane preserving the point $q$. The group of such automorphisms of $\mathbf P^2$ is the subgroup of $PGL_3$ corresponding to matrices: \[\left(\begin{array}{ccc} a_{11} & a_{12} & 0\\ a_{21} & a_{22} & 0\\ a_{31} & a_{32} & a_{33} \end{array}\right).\] Taking into account the discriminant of these matrices, we easily identify for odd $g$: \[\on{Aut}L\cong \mathbf A^2\times GL_2.\] Note that all of the above groups $\on{Aut}L$ have dimension $6$, which was claimed already in Lemma~\ref{gentrig}. \subsection{Generators of Pic$_{\mathbb{Q}}\overline{\mathfrak{T}}_g$} \label{generators} \begin{prop} The rational Picard group of $\overline{\mathfrak{T}}_g$, $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$, is freely generated by the boundary classes $\delta_0$, $\delta_{k,i}$, and one additional class, which for even genus $g$ coincides with the Maroni class $\mu$. \label{genPic} \end{prop} \begin{proof} Since a general trigonal curve $C$ embeds in the ruled surface ${\mathbf F}_k$ ($k=0,1$), $C$ is a member of the linear system $L=|3B_0+ \frac{g+2}{2}F|$ on ${\mathbf F}_k$. Let $U$ be the open set inside ${\mathbf P}L\cong \mathbf P^{2g+7}$ corresponding to the {\it smooth trigonal} members of $L$. The surjection \[{\mathbb Z}=\on{Pic}\mathbf P^{2g+7}\twoheadrightarrow \on{Pic}U \] has a nontrivial kernel, because the set of singular trigonal curves in ${\mathbf F}_k$ is a divisor in ${\mathbf P}L$. Hence $\on{Pic}U={\mathbb Z}/n{\mathbb Z}$ for some integer $n\!>\!0$, and $\on{Pic}_{\mathbb{Q}}U\!=\!0$. \medskip The image of the natural projection map $p:U\rightarrow \overline{{\mathfrak{T}}}_g$ is the open dense set $W$ of smooth trigonal curves with lowest Marone invariant of $0$ or $1$. Let $F$ denote the fiber of $p$. From Remark 12.1, \[F\cong\left\{\begin{array}{l} PGL_2\times PGL_2\,\,\on{if}\,\,g-\on{even},g>4;\\ PGL_2\times PGL_2\times {\mathbb Z}/2{\mathbb Z}\,\,\on{if}\,\,g=4;\\ \on{Aut}L\cong {\mathbf A}^2\times GL_2\,\,\on{if}\,\,g-\on{odd}. \end{array}\right.\] Leray spectral sequence or other methods (cf.~~\cite{Gr-Ha,Milne}) yield: \[H^1(W,f_*{\mathcal O}^*_U)\hookrightarrow H^1(U,{\mathcal O}^*_U).\] Pushing the exponential sequence on $U$ to $W$: \[0\rightarrow {\mathbb Z}\rightarrow {\mathcal O}_U \rightarrow {\mathcal O}^*_U \rightarrow 0\,\, \Rightarrow \,\,0\rightarrow {\mathbb Z}\rightarrow {\mathcal O}_{W} \rightarrow f_*{\mathcal O}^*_U\rightarrow R^1\!\!f_*{\mathbb Z}.\] Combining with the exponential sequence on $W$: \[0\rightarrow {\mathcal O}^*_{W} \rightarrow f_*{\mathcal O}_U^* \rightarrow R^1\!\!f_*{\mathbb Z}\,\, \Rightarrow\,\,H^1(W,{\mathcal O}^*_{W}) \stackrel{p^*}{\rightarrow}H^1(U,{\mathcal O}_X^*),\] with $\on{ker}p^*\subset H^0(W,R^1\!\!f_*{\mathbb Z})\subset H^1(F,{\mathbb Z})$. For even $g$, $H^1(F,{\mathbb Z})$ is torsion (a direct sum of copies of ${\mathbb Z}/2{\mathbb Z}$), but for odd $g$ it is isomorphic to ${\mathbb Z}$. \smallskip Hence, for even $g$ we have the natural embedding $p^*:\on{Pic}_{\mathbb Q}W\hookrightarrow \on{Pic}_{\mathbb Q}U$, and in view of $\on{Pic}_{\mathbb{Q}}U=0$, it follows that $\on{Pic}_{\mathbb{Q}} W=0$. The complement of $W$ in $\overline{\mathfrak{T}}_g$ is the union of the boundary of $\overline{\mathfrak{T}}_g$ and the Maroni divisor. Therefore, $\delta_0$, $\delta_{k,i}$ and $\mu$ generate $\on{Pic}_{\mathbb Q}\overline{\mathfrak{T}}_g$. Since the class of the Hodge bundle $\lambda$ is {\it not} a linear combination of the boundary classes (cf.~p.~\pageref{list of theorems}), the boundary divisors are {\it not} sufficient to generate the rational Picard group of $\overline{\mathfrak{T}}_g$, and $\mu$ must be linearly independent of them. We conclude that $\delta_0$, $\delta_{k,i}$, and $\mu$ generate freely $\on{Pic}_{\mathbb Q}\overline{\mathfrak{T}}_g$ for even genus $g$. \smallskip For $g$-odd, $p^*:\on{Pic}_{\mathbb{Q}}W\rightarrow \on{Pic}_{\mathbb{Q}}U$ is either an inclusion, or has a kernel with one generator. Since the Maroni locus for $g$-odd is not a divisor, an inclusion would imply as above that $\lambda$ is a linear combination of the boundary classes, which is not true. Hence, $\on{ker}p^*={\mathbb Q}$ and $\on{Pic}_{\mathbb{Q}}W$ is generated freely by the boundary classes $\delta_0$ and $\delta_{k,i}$, and one additional class. \end{proof} \subsection{The Bogomolov condition and the Maroni divisor} \label{interpretation} \begin{prop} For even genus $g$ and a base curve $B$, not contained in $\Delta{\mathfrak{T}}_g$: \[(7g+6)\lambda=g\delta_0+ \sum_{k,i}\widehat{c}_{k,i}\delta_{k,i}+2(g-3)\mu,\] where $\widehat{c}_{k,i}$ are certain polynomial coefficients computed similarly as $\widetilde{c}_{k,i}$. (cf.~p.~\pageref{analog1}) \label{Maroni2} \end{prop} \begin{proof} We set $g=2(m-1)$. Let us consider for now only families with irreducible trigonal fibers, i.e. the base curve $B$ intersects only the boundary component $\Delta{\mathfrak{T}}_0$. \smallskip \noindent{\it Case 1.} If $B$ does not intersect the Maroni divisor $\mu$, then the Maroni invariant of the fibers in $X$ is constant, and equal to $0$. The fibers $C$ of $X$ embed in the projectivization ${\mathbf P}(V|_{F_Y})\cong {\mathbf P}^1\times {\mathbf P}^1$. Since $\on{deg}V|_{F_Y} =g+2$ and $V$ is balanced, the restriction of $V$ to the fiber $F_Y$ on the ruled surface $Y$ is \[V|_{F_Y}=\mathcal{O}_{{\mathbf P}^1}(m)\oplus \mathcal{O}_{{\mathbf P}^1}(m).\] Moreover, $V|_{F_Y}$ does not jump as $F_Y$ moves, so that $V$ can be written as: \[V\cong h^*M\otimes\mathcal{O}_{Y}\left(mB_0\right)\] for some vector bundle $M$ of rank 2 on $B$. But the Bogomolov index $4c_2(V)-c_1^2(V)$ is independent of twisting $V$ by line bundles, in particular, by $\mathcal{O}_{Y}\left(mB_0\right)$, so that \[4c_2(V)-c_1^2(V)=4c_2(h^*M)-c_1^2(h^*M)= 4c_2(M)-c_1^2(M)=0.\] The last equality follows from $c_2(M)=0=c_1^2(M)$ for any bundle on the curve $B$. We conclude that $4c_2(V)-c_1^2(V)=4\mu|_B=0$. \medskip \begin{figure}[h] $$\psdraw{marone1}{1.5in}{1.3in}$$ \caption{$B\cap \mu$ in $\overline{\mathfrak{T}}_g$} \label{intersectBandmu} \end{figure} {\it Case 2.} Now let $B$ intersect the Maroni divisor $\mu$ in {\it finitely} many points. Assume also that these points are {\it general} in $\mu$, i.e. they correspond to trigonal curves $C$ embeddable in the ruled surface ${\mathbf F}_2$. We twist $V$ by a line bundle $M= \mathcal{O}_{Y}\left(mB_0\right)$, and set $\widetilde{V}=V\otimes M$, so that $\on{deg}\widetilde{V}|_{F_Y}=0$ and \[\,\,\,\,\left\{\begin{array}{l} \widetilde{V}|_{F_Y}= \mathcal{O}_{{\mathbf P}^1}\oplus \mathcal{O}_{{\mathbf P}^1}\,\,\, \phantom{(-1)(1)}\on{when}\,\,F_Y\,\,\on{is\,\,generic,}\\ \widetilde{V}|_{F_Y}=\mathcal{O}_{{\mathbf P}^1}(-1)\oplus \mathcal{O}_{{\mathbf P}^1}(1)\,\,\,\on{when}\,\,F_Y\,\,\on{is\,\,special}. \end{array}\right.\] Then $\widetilde{V}$ is the middle term of a short exact sequence on $Y$ \[0\rightarrow \widetilde{V}^{\prime} \rightarrow \widetilde{V} \rightarrow \mathcal{I} \rightarrow 0,\] where $\widetilde{V}^{\prime}=h^*(h_*\widetilde{V})$ is a vector bundle of rank 2 on $Y$. In the notation of \cite{Br}, let $W$ be the sum of the special fibers of $Y$, and let $Z$ be the union of certain isolated points on each member of $W$, so that $\mathcal{I}=\mathcal{I}_{Z\subset W}$ is the ideal sheaf of $Z$ inside $W$. Note that the number of the special fibers, which comprise $W$, equals $\on{deg}Z=\mu|_B$. We can now compute the Chern classes of $\widetilde{V}$: \[\left\{\begin{array}{l} c_1(\widetilde{V})=c_1(\widetilde{V}^{\prime})+W=\on{a\,\,sum\,\,of\,\, fibers\,\,of\,\,Y},\\ c_2(\widetilde{V})=c_2(\widetilde{V}^{\prime})+\on{deg}Z=\on{deg}Z. \end{array}\right.\] The last equality follows from the fact that $\widetilde{V}^{\prime}$ is the pull-back of a bundle on the curve $B$, hence of zero higher Chern classes. We conclude that $c_1^2(\widetilde{V})=0$, and \[4c_2({V})-c_1^2({V})= 4c_2(\widetilde{V})-c_1^2(\widetilde{V})=4\on{deg}Z=4\mu|_B.\] Putting the above two cases together, we have for any family with irreducible trigonal members, not entirely contained in the Maroni locus: \begin{equation} 4c_2({V})-c_1^2({V})=4\mu|_B. \end{equation} Prop.~\ref{genPic} then implies that $\lambda$ is a linear combination of the boundary and the Maroni class: \[(7g+6)\lambda|_{\overline{\mathfrak{T}}_g}=g\delta_0+\sum_{k,i} \widehat{c}_{k,i}\delta_{k,i}+2(g-3)\mu,\] where the coefficients $\widehat{c}_{k,i}$ are computed in a similar way, or by direct computation with families of singular trigonal curves (cf.~\cite{CH}). \end{proof} \medskip We can combine the above results in the following \begin{thm} For even $g$, $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$ is freely generated by all boundary classes $\delta_0$ and $\delta_{k,i}$, and the Maroni class $\mu$. The class of the Hodge bundle on $\overline{\mathfrak{T}}_g$ is expressed in terms of these generators as the following linear combination: \begin{equation*} (7g+6)\lambda|_{\overline{\mathfrak{T}}_g}=g\delta_0+ \sum_{k,i}\widehat{c}_{k,i}\delta_{k,i}+2(g-3){\mu}. \end{equation*} \label{Pic trigonal}\vspace*{-5mm} \end{thm} \noindent{\bf Remark 12.2.} Note that the coefficients $\widehat{c}_{k,i}$ depend on the specific decriptions of the Maroni curves that appear in the boundary divisors $\Delta{\mathfrak{T}}_{k,i}$, and they are {\it not} always equal to the corresponding coefficients $\widetilde{c}_{k,i}$ in Theorem \ref{7+6/g relation2}. Indeed, in the above Proposition, we have shown that \begin{equation} 4c_2({V})-c_1^2({V})=4\mu|_B+ \sum_{k,i}\alpha_{k,i}\delta_{k,i}, \label{alpha-coef} \end{equation} for some $\alpha_{k,i}$, which may be non-zero. Hence, $\widehat{c}_{k,i}=\widetilde{c}_{k,i}+\frac{g-3}{2}\alpha_{k,i}$. \smallskip For example, consider the case of $\Delta{\mathfrak{T}}_{1,i}$, and let $C=C_1\cup C_2$ be a general member of it. If $C$ is also Maroni, then there exists a family $X\rightarrow B$, whose general fiber is an irreducible Maroni curve, and one of whose special fibers is our $C$. We can assume, modulo a base change and certain blow-ups not affecting $C$, that this family fits in the basic construction diagram (cf.~ Fig.~\ref{general B}). Let ${\mathbf R}_1$ and ${\mathbf R}_2$ be the two ruled surfaces in which $C_1$ and $C_2$ are embedded, and let $E_1$ and $E_2$ be the projections of $C_1$ and $C_2$ in the birationally ruled surface $\widehat{Y}$. Then $F=E_1+E_2$ is a special fiber of $\widehat{Y}$, with self-intersections $E_1^2=E_2^2=-1$. \smallskip Now, the general member of $X$, being Maroni, is embedded in a ruled surface $\mathbf F_2$ with a section $L$ of self-intersection $-2$. The union of such $L$'s forms a surface in the 3-fold $\mathbf PV$, whose closure we denote by $S$. Evidently, $S\cong \widehat{Y}$, at least outside their special fibers. Let $S$ intersect ${\mathbf R}_1$ and ${\mathbf R}_2$ in curves $L_1$ and $L_2$ (over $E_1$ and $E_2$). We claim that at least one of ${\mathbf R}_1$ and ${\mathbf R}_2$ is {\it not} isomorphic to $\mathbf F_0=\mathbf P^1\times \mathbf P^1$. It will suffice to show that $L_1$ or $L_2$ has negative self-intersection. \smallskip Indeed, suppose to the contrary that $L_m^2\geq 0$ in ${\mathbf R}_m$ ($m=1,2$). Note that $S\cdot \mathbf R_m=L_m$ in $\mathbf PV$, so that \[L_m^2=S|_{\mathbf R_m}\cdot S|_{\mathbf R_m}= S^2\cdot \mathbf R_m\,\, \Rightarrow\,\,S^2(\mathbf R_1+\mathbf R_2)\geq 0.\] On the other hand, $\mathbf R_1+\mathbf R_2$ is the fiber of the projection $\mathbf PV\rightarrow \widehat{Y}$, and as such it is linearly equivalent to the general fiber $\mathbf F_2$. Hence \[0\leq S^2\cdot \mathbf F_2=S|_{\mathbf F_2}\cdot S|_{\mathbf F_2}=L^2=-2,\] a contradiction. We conclude that if $C=C_1\cup C_2$ is a Maroni curve of boundary type $\Delta{\mathfrak{T}}_{1,i}$, then either $C_1$ or $C_2$ (or both) is embedded in a ruled surface $\mathbf F_k$ with $k\geq 1$. This already distinguishes the cases of odd and even genus $i$. \smallskip When $i=g(C_1)$ is even (and hence $j=g(C_2)=g-j-2$ is also even), the general member of $\Delta{\mathfrak{T}}_{1,i}$ is embedded in a join of two $\mathbf F_0$'s (each $C_m\subset \mathbf F_0$), and hence it is {\it not} Maroni. Based on this observation, one can easily find the coefficient $\alpha_{1,i}$ for $i$-even. To do this, consider the birationally ruled surface $Y$ which is the blow-up of $\mathbf F_0$ at one point. Let again the two components of the special fiber of $Y$ be $E_1$ and $E_2$, and projectivize the trivial vector bundle $V=\mathcal{O}_Y\oplus \mathcal{O}_Y$: ${\mathbf P}V=Y\times \mathbf P^1$. By taking an appropriate linear system in ${\mathbf P}V$, one obtains a family of trigonal curves $X$, whose fibers are all irreducible and embedded in $\mathbf F_0$, except for a special reducible curve $C$ over $E_1\cup E_2$ of the specified above type. Hence none of $X$'s members are Maroni, and so $\mu|_B=0$. Further, $4c_2(V)-c_1^2(V)=0$, and $\delta_{1,i}|_B=1$, so that equation~(\ref{alpha-coef}) implies $\alpha_{1,i}=0$, and hence $\widehat{c}_{k,i}=\widetilde{c}_{k,i}$ for $i$-even. \smallskip The situation is quite different when the genus $i$ is odd. Then both components of the general member $C$ of $\Delta{\mathfrak{T}}_{1,i}$ are embedded in $\mathbf F_1$'s, and hence $C$ is potentially Maroni. One can take further the above general argument of intersection theory on ${\mathbf P}V$, and show that the curves $L_1$ and $L_2$ are in fact both sections of negative self-intersection $-1$ in these $\mathbf F_1$'s: consider the product $S\cdot X \cdot {\mathbf F}_2$ and its variation over the special fiber of $\widehat{Y}$. But we know that $L_1$ and $L_2$ intersect, as the fiber of $S$ over $\widehat{Y}$ is connected. Thus, the curve $C$ would be Maroni if and only if the two corresponding ruled surfaces $\mathbf F_1$ are glued along one of their fibers so that their negative sections intersect on that fiber. (This decsription can be alternatively derived by considering the degenerations of the $g^1_3$'s on the irreducible Maroni curves.) To find $\alpha_{1,i}$ in this case, we construct a similar example as above, only changing $V$ to $\mathcal O_Y\oplus \mathcal O_Y(E_1)$. This, while keeping the general fiber embedded in $\mathbf F_0$, has the effect of embedding the special one in a ``Maroni'' gluing of two $\mathbf F_1$'s. We have $4c_2(V)-c_1^2(V)=-E_1^2=1$, $\mu|_B=1$, and $\delta_{1,i}|_B=1$, so that equation~(\ref{alpha-coef}) implies $\alpha_{1,i}=-3$ for $i$-odd, and hence $\widehat{c}_{k,i}=\widetilde{c}_{k,i}-3/2(g-3)$. \bigskip \begin{figure}[t] $$\psdraw{maroni}{1.5in}{1.5in}$$ \caption{Maroni curves in $\Delta_{1,i}\mathfrak{T}_g$, $i$-odd} \label{Maroni-boundary} \end{figure} One can similarly compute the remaining coefficients $\alpha_{k,i}$, by first figuring out which boundary curves in $\Delta_{k,i}\mathfrak{T}_g$ are Maroni, then constructing an appropriate vector bundle $V$, and finally using equation~(\ref{alpha-coef}) to compute $\alpha_{k,i}$, and hence $\widehat{c}_{k,i}$. \qed \begin{prop} For $g$-even, if the base curve $B$ is not entirely contained in the Maroni divisor, and the singular members of $X$ belong only to $\Delta_0\mathfrak{T}_g \cup \Delta_{1,i}\mathfrak{T}_g$, then the slope of the family $X/_B$ satisfies: \[\frac{\delta}{\lambda}\leq 7+\frac{6}{g}.\] \label{Maroni inequality}\vspace*{-5mm} \end{prop} \begin{conj} For $g$-even, if the base curve $B$ is not entirely contained in the Maroni divisor, then the slope of the family $X/_B$ satisfies: \[\frac{\delta}{\lambda}\leq 7+\frac{6}{g}.\] \label{Maroni-conj} \end{conj} \subsection{The Maroni divisor and the maximal bound} \label{Maroni-maximal} Even though for odd genus $g$ the Maroni locus is not large enough to be a divisor in $\overline{\mathfrak{T}}_g$, we can define a {\it generalized Maroni} divisor class by extending the relation from the $g$-even case. \medskip \noindent{\bf Definition 12.2.} For any genus, we define the {\it generalized Maroni} class $\mu$ in $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$ by \[\mu:=\frac{1}{2(g-3)}\big\{(7g+6)\lambda-g\delta_0- \sum_{k,i}\widehat{c}_{k,i}\delta_{3,i}\big\}.\] \begin{thm} The maximal bound $36(g+1)/(5g+1)$ is attained for a trigonal family of curves $X\rightarrow B$ if and only all fibers of $X$ are irreducible and \[\delta_0|_B=-\frac{72(g+1)}{g+2}\mu|_B\] \label{maximalmaroni}\vspace*{-5mm} \end{thm} \begin{proof} The fact that $X$ must have only irreducible fibers in order to attain the maximum bound is already known from Theorem~\ref{genmaximal}. This means $\delta_{k,i}|_B=0$ for all $k,i$. Then, Theorem~\ref{bogomolov1} implies: \begin{equation} (7g+6)\lambda|_B=g\delta_0|_B+\frac{g-3}{2}\mu|_B. \end{equation} Assume that the maximal bound is attained, i.e. $36(g+1)\lambda|_B=(5g+1)\delta_0|_B$. Substituting for $\lambda|_B$ in the above equation, yields the desired equality. The converse follows similarly. \end{proof} \medskip \noindent{\bf Remark 12.3.} In the $g$-even case, this equality has a specific meaning. Since the Maroni class $\mu$ corresponds to an effective divisor on $\overline{\mathfrak{T}}_g$, the equality (and hence the maximal bound) is achieved only for base curves $B$ entirely contained in the Maroni divisor, so that the restriction $\mu|_B$ can be negative. In fact, in all found examples, the base $B$ is contained in a very small subloci of the Maroni loci, defined by the highest possible Maroni invariant. \medskip \noindent{\bf Remark 12.4.} Theorem~\ref{Pic trigonal} and Prop.~\ref{maximalmaroni} do not have analogs in the hyperelliptic case: there is no additional Maroni divisor to generate $\on{Pic}_{\mathbb Q}\overline{\mathfrak{I}}_g$ together with the boundary $\Delta\mathfrak{I}_g$. \medskip \noindent{\bf Remark 12.5.} When $g=3$, there is no Maroni locus in $\overline{\mathfrak{T}}_3$ either. Indeed, since an irreducible trigonal curve of genus $3$ embeds only in ruled surfaces ${\mathbf F}_k$ with $k$-odd and $k\leq (g+2)/3=5/3$, then {\it all} irreducible trigonal curves embed in ${\mathbf F}_1$, and correspondingly they all have the lowest possible Maroni invariant $k=1$. However, $\on{Pic}_{\mathbb Q}\overline{\mathfrak{T}}_3$ is not generated by the boundary classes of $\overline{\mathfrak{T}}_3$: as Prop.~\ref{genPic} asserts, in the odd genus case there is always one additional generating class. \smallskip On the other hand, the results on p.~\pageref{list of theorems} yield apriori {\it two} relations among $\lambda$ and the $\delta_{k,i}$'s. This would have been a contradiction to the {\it freeness} of the generators above, unless these two relations are the same. This is in fact what happens: \[9\lambda=\delta_0+3\delta_{2,1}+3\delta_{3,1}+4\delta_{4,1}+4\delta_{5,1} +3\delta_{5,2}+3\delta_{6,1},\] as restricted to any base curve $B\not\subset\Delta\overline{\mathfrak{T}}_3$. Note the convenient disappearance of the ``extra'' $(g-3)$--summands in the coefficients of $\delta_{4,i},\delta_{5,i}, \delta_{6,i}$). Then the maximal and the semistable ratios both equal $9$, and are attained for families with irreducible trigonal members. \bigskip\section*{13. Further Results and Conjectures} \setcounter{section}{13} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{lem}{0} \setcounter{thm}{0} \setcounter{prop}{0} \setcounter{defn}{0} \setcounter{cor}{0} \setcounter{conj}{0} \setcounter{claim}{0} \setcounter{remark}{0} \setcounter{equation}{0} \label{furtherresults} \subsection{Results and conjectures for $d$-gonal families, $d\geq 4$} I have carried out some preliminary research in the $d$-gonal case, and while the methods and ideas for the trigonal case are in principle extendable, this appears to be a substantially more subtle and complex problem. More precisely, let $\overline{\mathcal{D}}_d$ be the closure in $\overline{\mathfrak{M}}_g$ of the stable curves expressible as $d$-sheeted covers of ${\mathbf P}^1$. One possible goal is to complete the program of describing generators and relations for the rational Picard groups $\on{Pic}_{\mathbb{Q}}\overline{\mathcal{D}}_d$, and to find the exact maximal bounds for the slopes of $d$-gonal families. \smallskip For example, I have obtained the following bound for the slope of a general tetragonal family with smooth general member (for odd genus $g$): \[\frac{\delta}{\lambda}\leq 6\frac{2}{3}+\frac{64}{3(3g+1)}= \frac{4(5g+7)}{3g+1}.\] I have also conjectured formulas for the maximal and general bounds for any $d$-gonal and other families of stable curves. Entering these formulas are the {\it Clifford index} of curves, {\it Bogomolov semistability} conditions for higher rank bundles, and some new geometrically described loci in $\overline{\mathcal{D}}_d$. Generalizing the idea of the Maroni locus in the trigonal case, these loci are characterized, for example, in the tetragonal case by the dimensions of the multiples of the $g^1_4$-series. In particular, there will be another generator of $\on{Pic}_{\mathbb Q}\overline{\mathfrak{T}}_4$ besides the boundary and Maroni divisors. \smallskip In the following I present some of these conjectures on the upper bounds for $\overline{\mathcal{D}}_d$. We start by comparing all known maximal and general bounds functions of the genus $g$: \[\begin{array}{|c|c|c|c|c|c|} \hline\hline \stackrel{\vspace*{1mm}}{\on{locus\,\, in \,\,}\overline{\mathfrak{M}}_g} &\on{bound}& g=1& g=2& g=3& g=5\\ \hline\hline\vspace*{1mm} \on{general}\,\overline{\mathfrak{M}}_g& \stackrel{\vspace*{1mm}}{\displaystyle{ 6+\frac{12}{g+1\vspace*{1mm}}}} & 12 & 10 &9 &8\\ \hline\vspace*{1mm} \on{hyperelliptic}\,\overline{\mathcal{H}}_g=\overline{\mathcal{D}}_2& \stackrel{\vspace*{1mm}}{\displaystyle{8+ \frac{4}{g\vspace*{1mm}}}} & 12 & 10 & - &-\\ \hline\vspace*{1mm} \on{trigonal}\,\overline{\mathcal{T}}_g=\overline{\mathcal{D}}_3 & \stackrel{\vspace*{1mm}}{\displaystyle{ \frac{36(g+1)}{5g+1 \vspace*{1mm}}}} & 12 & - & 9 &- \\ \hline\vspace*{1mm} \on{ gen. tetragonal=\overline{\mathcal{D}}_4} &\stackrel{\vspace*{1mm}}{\displaystyle{ \frac{4(5g+7)}{3g+1\vspace*{1mm}}}}& 12 & - & -&8\\ \hline \end{array}\] \medskip The pattern appearing in this table is clear: the general bound $6+\displaystyle{12/(g+1)}$ coincides with each of the other bounds exactly twice for some special values of the genus $g$. Evidently, $g=1$ is one of these special values, yielding 12 everywhere. (I owe this observation to Benedict Gross.) Let $g_d$ be the other genus $g$ for which the general formula in $\overline{\mathfrak{M}}_g$ and the maximal formula for $\overline{\mathcal{D}}_d$ coincide, i.e. $g_2=2$, $g_3=3$, $g_5=5$. We notice that for these genera $g_d$ the moduli spaces $\overline{\mathfrak{M}}_2,\overline{\mathfrak{M}}_3$ and $\overline{\mathfrak{M}}_5$ consist only of hyperelliptic, trigonal or tetragonal curves, respectively. In general, {\it Brill-Noether} theory (cf.~\cite{ACGH}) asserts that for complete linear series $g^r_d=g^1_d$ the expected dimension of the variety of $g^1_d$'s on a smooth curve of genus $g$ is $\rho=g-(r+1)(g-d+r)=2(d-1)-g,$ and hence the smallest genus $g$ for which $\overline{\mathfrak{M}}_g=\overline{\mathcal{D}}_d\supsetneq \overline{\mathcal{D}}_{d-1}$ is $g=2d-3$. Thus we set $g_d=2d-3$ for $d\geq 3$ and $g_2=2$. Note that this coincides with the previously found $g_3=3$ and $g_5=5$. \begin{conj} If $\mathcal{F}_d(g)$ is an exact upper bound for the slopes of families of stable curves with smooth $d$-gonal general member (locus $\overline{\mathcal{D}}_d$), then \begin{eqnarray*} &&(a)\,\,\mathcal{F}_d(1)=12.\\ &&(b)\,\,\mathcal{F}_d(g_d)=6+\displaystyle{\frac{12}{g_d+1}}\cdot \end{eqnarray*} \label{conj2}\vspace*{-5mm} \end{conj} It is reasonable to expect that the upper bounds for $\overline{\mathcal{D}}_d$ will be ratios of linear functions of the genus $g$: $\mathcal{F}_d(g)=(Ag+B)/(Cg+D)$. Conjecture~\ref{conj2} then estimates the difference between $\mathcal{F}_d(g)$ and the general bound for $\overline{\mathfrak{M}}_g$ up to a factor $f_d=D/C$. \begin{conj} The exact upper bounds $\mathcal{F}_d(g)$ are given by \[\mathcal{F}_d(g)=6+\frac{12}{g+1}+6\frac{(1-f_d)(g-g_d)(g-1)}{(g+f_d)(g_d+1)(g+1)},\]\vspace*{-3mm} or equivalently, \vspace*{-3mm} \[\mathcal{F}_d(g)=6+\frac{6}{g+f_d}\left(1+f_d+\frac{1-f_d}{g_d+1}(g-1)\right).\] \end{conj} I have a conjecture on how to determine the remaining factor $f_d$, which seems to be closely related to the coefficients of the linear expression in [EMH] for the divisor $\overline{\mathcal{D}}_{\frac{g+1}{2}}$ in terms of the Hodge bundle $\lambda$ and the boundary classes $\delta_i$ on $\overline{\mathfrak{M}}_g$. These conjectures are supported by the work of Cornalba-Harris on the { hyperelliptic locus} $\overline{\mathcal{H}}_g=\overline{\mathcal{D}}_2$, by the results of this paper on the { trigonal locus} $\overline{\mathcal{T}}_g=\overline{\mathcal{D}}_3$, and by partial results on the tetragonal locus $\overline{\mathcal{D}}_4$. \smallskip In view of Remark 12.5, the equality between the maximal and semistable trigonal bounds for $g=3$ suggests that a similar situation might occur for other $d$-gonal families. It is reasonable to expect two or more ``semistable'' bounds, depending on the number of extra generators in $\on{Pic}_{\mathbb Q}{\overline{\mathcal D}}_d$. \smallskip One of these ``semistable'' bounds relates to families obtained as blow-ups of pencils of $d$-gonal curves on a ruled surface ${\mathbf F}_k$. Example 2.1 yields the maximal bound $8+4/g$ for hyperelliptic families (no extra generator besides the boundary classes), and a similar example in the trigonal case yields the $7+6/g$ semistable bound (one extra generator, the Maroni locus). We generalize this to any $d$-gonal family of curves embedded in an arbitrary ruled surface ${\mathbf F}_k$. Invariably, the slope of $X/\!_{\displaystyle{B}}$ is: \begin{equation*} \frac{\delta|_B}{\lambda|_B}=\left(6+\frac{2}{d-1}\right)+\frac{2d}{g}\cdot \end{equation*} \medskip \begin{conj} Let $X$ be a family of $d$-gonal curves of genus $g$ whose base $B$ is not contained in a certain codimension 1 closed subset of $\overline{\mathcal D}_d$. Then the slope of $X/\!_{\displaystyle{B}}$ satisfies: \begin{equation*} \frac{\delta|_B}{\lambda|_B}\leq \left(6+\frac{2}{d-1}\right)+\frac{2d}{g}\cdot \end{equation*} \label{clifford} \vspace*{-5mm} \end{conj} Conjectures~\ref{clifford}--4 are modifications of earlier conjectures of Joe Harris. \subsection{A look at families with special $g^r_d$'s, $r\geq 2$} The discussion so far was primarily concerned with the loci $\overline{\mathcal{D}}_d\subset \overline{\mathfrak{M}}_g$ corresponding to linear series $g^1_d$. But all of our problems are well-defined and quite interesting to solve for curves with series $g^r_d$ of dimension $r>1$. Equivalently, we consider the loci $\overline{\mathcal{D}}^r_d$ of curves mapping with degree $d$ to ${\mathbf P}^r$, $r\geq 1$. \medskip \noindent{\bf Definition 13.1.} The {\it Clifford index} $\mathfrak {c}$ of a smooth curve $C$ is defined as \[\mathfrak{c}=\on{min}_L\left\{\on{deg} {L} -2\on{dim}{L}\right\}\] where $L$ runs over all effective special linear series ${L}$ on $C$. \medskip Clifford's theorem implies ${\mathfrak{c}}\geq 0$, with equality if and only if $C$ is hyperelliptic, i.e. ${L}=g^1_2$ (cf.~\cite{ACGH}). On the other hand, ${\mathfrak{c}}=1$ means that there exists a $g^r_d$ on $C$ with $d-2r=1$. From Marten's Theorem, $\on{dim}W^r_d(C)\leq d-2r-1=0$, where $W^r_d$ is the variety parametrizing complete linear series on $C$ of degree $d$ and dimension at least $r$. Therefore, we must have $\on{dim}W^r_d=0$. But then Mumford's theorem asserts that $C$ is either trigonal, or bi-elliptic, or a smooth plane quintic. The bi-elliptic case would mean that $W^r_d$ consists of $g^2_6$'s, which contradicts the dimension of $\on{dim}W^r_d$. In short, $\mathfrak{c}=1$ if and only if $C$ is not hyperelliptic and possesses a $g^1_3$ or a $g^2_5$. \smallskip Thus, according to the Clifford index, the first case with $r\geq 2$ is the space of plane quintics. Consider a general pencil of such, and blow up the plane at its 25 base points. The resulting family $X=\on{Bl}_{25}{\mathbf P^2}\rightarrow\mathbf P^1$ is easily seen to have slope $8=7+6/g$, which corresponds to the bound in Conjecture~\ref{clifford} with $d-2$ replaced by the Clifford index $\mathfrak{c}=1$. Finally, note that for a $d$-gonal curve $C$ of genus $g$, by definition $\mathfrak{c}\leq d-2$, so that when $g\gg d$ we may generalize to: \begin{conj} For a general family $X\rightarrow B$ of genus $g$ stable curves whose general member has Clifford index $\mathfrak{c}$ and whose base $B$ is a general curve in $\overline{\mathcal D}^r_d$, the slope of $X/\!_{\displaystyle{B}}$ satisfies: \begin{equation*} \frac{\delta_X}{\lambda_X}\leq\left(6+\frac{2}{\mathfrak{c}+1}\right)+ \frac{2\mathfrak{c}+4}{g}\,\,\,\on{for}\,\,\mathfrak{c}<\!<g\cdot \label{clifford1} \end{equation*} \end{conj} \noindent{\bf Remark 13.1.} It is worth noting that the stratification of $\overline{\mathfrak{M}}_g$, for which we asked in the Introduction, is not obtained via the Clifford index $\mathfrak{c}$. For example, Xiao constructs families of bi--elliptic curves $C$ with slope $8$ (cf.~\cite{Xiao}), which is between the hyperelliptic and the trigonal maximal bounds. Since $C$ has a $g^1_4$ as bi--elliptic, this already exceeds the conjectured maximal bounds for the tetragonal case. This shows that in some of the above conjectures we have to exclude the subset of bi--elliptic curves from the tetragonal locus $\overline{\mathcal D}_4$, and that similar modifications might be necessary for the other loci $\overline{\mathcal D}_d$. More precisely, it seems plausible that the stratification of $\overline{\mathfrak{M}}_g$ according to successively lower slope bounds is related not just to the existence of a specific linear series $g^r_d$, but also to the number, dimension and description of the irreducible components of corresponding varieties $W^r_d$. \subsection{Other methods via the moduli space $\overline{\mathcal{M}}_{g,n}({\mathbf P}^r,d)$} The approach in the $g^1_d$-cases is based on a modification of the Harris-Mumford's [EHM] {\it Hurwitz scheme of admissible covers}, which parametrized the $d$-uple covers of stable pointed rational curves. However, in the more general situation for linear series with larger dimensions $r>1$, such a compactification via admissible covers does not exist, so we have to look for a different solution. \smallskip Consider moduli spaces of stable maps $\overline{\mathcal{M}}_{g,n}({\mathbf P}^r,d)$. They parametrize {\it stable} maps $(C,p_1,p_2,...,p_n;\mu)$, where $C$ is a projective, connected nodal curve of arithmetic genus $g$, the $p_i$'s are marked distinct nonsingular points on $C$, and the map $\mu:C\rightarrow{\mathbf P}^r$ has image $\mu_*([C])=d[\on{line}]$ and satisfies certain stability conditions (cf.~\cite{K,KM}). The space $\overline{\mathcal{M}}_{g,n}({\mathbf P}^r,d)$ seems to be the right compactification which we need in order to extend our results to families with $g^r_d$-series on the fibers: the moduli space of stable maps is somewhat more ``sensitive'' in describing our loci $\overline{\mathcal{D}}^r_d$ in terms of their geometry. \smallskip Going back to the $g^1_d$-problems, one can also see the combinatorial flavor that stands in the background of these questions. It is probably not coincidental that the spaces $\overline{\mathcal{M}}_{g,n}({\mathbf P}^r,d)$ are also combinatorially defined and give rise to many enumerative problems. It will be useful to understand better the loci $\overline{\mathcal{D}}^r_d$ via their connection with the Kontsevich spaces $\overline{\mathcal{M}}_{g,n}({\mathbf P}^r,d)$, and ultimately to solve the remaining questions on $\on{Pic}_{\mathbb{Q}}\overline{\mathcal{D}}^r_d$ for any $d,r$, as well as related interesting enumerative problems that will inevitably arise from such considerations. \section*{14. Appendix: The Hyperelliptic Locus $\overline{\mathfrak{I}}_g$} \setcounter{section}{14} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{lem}{0} \setcounter{thm}{0} \setcounter{prop}{0} \setcounter{defn}{0} \setcounter{cor}{0} \setcounter{conj}{0} \setcounter{claim}{0} \setcounter{remark}{0} \setcounter{equation}{0} \label{hyperelliptic} In this section we give a proof of Theorems~\ref{theoremCHPic} and~\ref{CHX}, following the same ideas and methods as in the trigonal case. We refer the reader to previous sections for a detailed proof of certain statements. \subsection{Boundary locus of $\overline{\mathfrak{I}}_g$} \label{hyperellipticboundary} Cornalba-Harris describe the boundary of $\overline{\mathfrak{I}}_g$ as consisting of several boundary components, whose general members and indexing are shown in Fig.~\ref{hyperboundary} (cf.~\cite{CH}). The restriction of the divisor class $\delta$ to $\overline{\mathfrak{I}}_g$ is the following linear combination: \begin{equation} \delta\big|_{\overline{\mathfrak{I}}_g}=\delta_0+2\sum_{i=1}^ {[(g-1)/2]}\xi_i+ \sum_{j=1}^{\left[g/2\right]}\delta_j, \label{boundaryrel} \end{equation} where $\xi_i$ and $\delta_i$ are the classes in $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{I}}_g$ of the boundary divisors $\Xi_i$ and $\Delta_j$. \newpage \begin{figure}[h] $$\psdraw{hyper1}{4.5in}{0.9in}$$ \caption{Boundary of the hyperelliptic locus $\overline{\mathfrak{I}}_g$} \label{hyperboundary} \end{figure} \vspace*{-8mm}$$\Xi_0;\,\,\Xi_i,\,{\scriptstyle {i=1,...,[(g-1)/2]}} ;\,\,\Delta_j,\,{\scriptstyle{j=1,...,[g/2]}}$$ \subsection{Effective covers and embedding for hyperelliptic families} \label{embeddinghyperelliptic} In the case of a hyperelliptic family $f:X\rightarrow B$, a minimal quasi-admissible cover coincides with the original family $X$, because no blow-ups are necessary to perform on the fibers of $X$: these are already quasi-admissible double covers. Thus, we have a degree 2 map $\phi=\widetilde{\phi}:X\rightarrow Y$ for some birationally ruled surface $Y$ over $B$. As for an effective cover $\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$, only the boundary divisors $\Delta_i$ require blow-ups (cf.~Fig.~\ref{hyperboundary}). This is analogous to the ``ramification index 1'' discussion in Fig.~\ref{ram}--\ref{resolve1}. Thus, while in $\widehat{X}$ the special fibers may have occasional nonreduced rational components of multiplicity 2, the fibers of $\widehat{Y}$ are always trees of reduced smooth ${\mathbf P}^1$'s. \medskip In the case of a smooth hyperelliptic curve $C$, we consider the natural double sheeted map $f:C\rightarrow {\mathbf P}^1$. The pushforward $f_*{\mathcal{O}_C}$ is a rank 2 vector bundle on ${\mathbf P}^1$, which fits into the short exact sequence \begin{equation*} 0\rightarrow {\mathcal{O}_{{\mathbf P}^1}(g+1)}\rightarrow {f}_*{\mathcal O}_{C}\stackrel {\on{tr}}{\rightarrow}{\mathcal O}_{{\mathbf P}^1}\rightarrow 0. \end{equation*} We can embed $C$ in the rational ruled surface ${\mathbf P}((f_*\mathcal{O}_C)\,\,\hat{})$. We generalize this construction to the effective cover $\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$ by setting $V:=({\phi}_*{\mathcal O}_{X})\,\,\hat{}$. For some line bundle $E$ on $\widehat{Y}$: \begin{equation*} 0\rightarrow {E}\rightarrow {\widehat{\phi}}_*{\mathcal O}_{\widehat{X}}\stackrel {\on{tr}}{\rightarrow}{\mathcal O}_{\widehat{Y}}\rightarrow 0. \end{equation*} Then $\widehat{X}$ naturally embeds in the threefold ${\mathbf P}V$. Let $\pi:{\mathbf P}V\rightarrow \widehat{Y}$ be the corresponding projection map. \subsection{The invariants $\lambda,\delta$ and $\kappa$} \label{Hyperinvariants} As a divisor in ${\mathbf P}V$, $\widehat{X}\equiv 2\zeta+\pi^*D$, for some divisor $D$ on $\widehat{Y}$. From the adjunction formula, $g=\on{deg}c_1(V)|_{F_{\widehat{Y}}}-1=c-1$, where $c_1(V)=cB_0+dF_Y$. The arithmetic genus of the inverse image $\widehat{\phi}^*T(E)$ is given by \[p_{\!\stackrel{\phantom{.}}{E}}=-m_{\!\stackrel{\phantom{.}}{E}} \left(\Gamma_{\!\stackrel{\phantom{.}}{E}}+ \Theta_{\!\stackrel{\phantom{.}}{E}}\right).\] It turns out that these are the only differences between the set-up of the hyperelliptic and the trigonal case. The definitions of the functions $m,\theta$ and $\gamma$, as well as the formulas for $c_1(V), K_{{\mathbf P}V}, c_2({\mathbf P}V)$ and the congruence $D\equiv 2c_1(V)$ are valid without any modifications. \smallskip As in the trigonal case, it will be sufficient to consider only the cases when the base curve $B$ intersects {\it transversally} the boundary divisors of $\overline{\mathfrak{I}}_g$. But then for all non-root components $E$ in $\widehat{Y}$: \[m_{\!\stackrel{\phantom{.}}{E}}=1=\Theta_{\!\stackrel{\phantom{.}}{E}}\,\, \on{and}\,\, \Gamma_{\!\stackrel{\phantom{.}}{E}}=-(p_{\!\stackrel{\phantom{.}}{E}}+1).\] We can now easily calculate the invariants on $X$. \begin{prop} For any family $f:X\rightarrow B$ of hyperelliptic curves with smooth general member and a base curve $B$ intersecting transversally the boundary of $\overline{\mathfrak{I}}_g$:\ \begin{eqnarray*} \lambda_X&\!\!=\!\!&dg+\frac{1}{2}\sum_{E\not =R}\Gamma_ {\!\stackrel{\phantom{.}}{E}} (\Gamma_{\!\stackrel{\phantom{.}}{E}}+1),\\ \kappa_X&\!\!=\!\!&4d(g-1)-2\sum_{E\not =R}(\Gamma_{\!\stackrel{\phantom{.}}{E}}+1)^2+\sum_{\on{ram}1}1,\\ \delta_X&\!\!=\!\!&4d(2g+1)+2\sum_{E\not = R} (\Gamma_{\!\stackrel{\phantom{.}}{E}}+1)(1-2\Gamma_ {\!\stackrel{\phantom{.}}{E}})+\sum_{\on{ram}1}1. \end{eqnarray*} \label{hyperinvariants} \end{prop} With this, we are ready to show the linear relations among $\lambda|_B$ and the boundary restrictions $\delta_i|_B$ and $\xi_i|_B$. It is evident that in order to cancel the ``global'' term $d$, one must subtract $(8g+4)\lambda_X|_B-g\delta|_B$, which is the main idea of the next theorem. \begin{thm} There exists an effective linear combination $\mathcal{E}_h$ of the boundary divisors of $\overline{\mathfrak{I}}_g$, not containing $\Xi_0$, such that for any family $f:X\rightarrow B$ of hyperelliptic curves with smooth general member: \[(8g+4)\lambda_X|_B=g\delta|_B+\mathcal{E}_h|_B\] \label{hyperrelation}\vspace*{-8mm} \end{thm} \begin{proof} We consider the difference \begin{eqnarray*} \mathfrak{S}_h&\!\!=\!\!&(8g+4)\lambda_X|_B-g\delta|_B= 2\sum_{E\not = R}(1+\Gamma_{\!\stackrel{\phantom{.}}{E}})(g+\Gamma_ {\!\stackrel{\phantom{.}}{E}})+\sum_{\on{ram}1}g\\ &\!\!=\!\!&2\sum_{E\not = R}p_{\!\stackrel{\phantom{.}}{E}}(g-1+p_ {\!\stackrel{\phantom{.}}{E}})+\sum_{\on{ram}1}g. \end{eqnarray*} In the hyperelliptic case, as opposed to the trigonal case, there is only {\it one type} of non-root components $E$, namely, such that both $E$ and $E^-$ are reduced. That is why there is just one type of summands in $\mathfrak{S}_h$. \smallskip As in Section~\ref{arbitrary}, it is sufficient to calculate the above sum for general members of $\Xi_{i}$ and $\Delta_i$, as described in Prop.~\ref{Delta-k,i}, i.e. for a {\it general} base curve $B$. \subsubsection{Contribution of the boundary divisors $\Xi_{i}$} \label{hypercontribution1} This case is analogous to the case of $\Delta_{3,i}$ (cf.~ Subsection~\ref{contribution1}). The arithmetic genus $p_{\!\stackrel{\phantom{.}}{E}}=g-i-1$, and the corresponding summand in $\mathfrak{S}_h$ is \[e_i=2p_{\!\stackrel{\phantom{.}}{E}}(g-1+p_ {\!\stackrel{\phantom{.}}{E}}) =2i(g-i-1)>0,\] where $i=1,...,[(g-1)/2]$. \subsubsection{Contribution of the boundary divisors $\Delta_{j}$} \label{hypercontribution2} Compare this with the contribution of $\Delta_{5,j}$ (subsection \ref{contribution2}). There are two non-root components $E_1$ and $E_2$ in the special fiber of $\widehat{Y}$ ($E_1^-=R$), whose invariants are $p_{\!\stackrel{\phantom{.}}{E_1}}=g-j-1$ and $p_{\!\stackrel{\phantom{.}}{E_2}}=g-j$. With the ramification adjustment of $g$, the contribution of $\Delta_j$ to the sum $\mathfrak{S}_h$ is \begin{equation*} f_j= 2p_{\!\stackrel{\phantom{.}}{E_1}}(g-1+p_{\!\stackrel{\phantom{.}}{E}})+ p_{\!\stackrel{\phantom{.}}{E_2}}(g-1+p_{\!\stackrel{\phantom{.}}{E}})+g =4j(g-j)-g>0, \end{equation*} where $j=1,...,[g/2]$. \medskip Finally, for the appropriate indices $i$ and $j$ we set $\displaystyle{\mathcal{E}_h:=\sum_{i>0}e_i\Xi_i+\sum_{j>0}f_j\Delta_j.}$ This is an effective combination of boundary divisors in $\overline{\mathfrak{I}}_g$, not containing $\Delta_0$ by construction, and satisfying $\mathfrak{S}_h=\mathcal{E}_h|_B$. \end{proof} \medskip Theorem~\ref{hyperrelation} implies immediately the following \begin{cor} Let $f:X\rightarrow B$ be a nonisotrivial family with smooth general member. Then the slope of the family satisfies: \begin{equation} \frac{\delta|_B}{\lambda|_B}\leq 8+\frac{4}{g}. \label{second8+4/g} \end{equation} Equality holds if and only if the general fiber of $f$ is hyperelliptic, and all singular fibers are irreducible. \end{cor} It is now straightforward to prove the fundamental relation in $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{I}}_g$, shown first in \cite{CH}. In Theorem~\ref{hyperrelation}, we add to the coefficients $e_i$ and $f_j$ the corresponding multiplicities $\on{mult}_{\delta}\xi_i$ and $\on{mult}_{\delta}\delta_j$: \[\widetilde{e}_i=e_i+2\cdot g=2(i+1)(g-i),\,\,\, \widetilde{f}_j=f_j+1\cdot g=4j(g-j).\] Using the fact that $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{I}}_g$ is generated freely by the boundary classes $\xi_i$ and $\delta_j$ (see \cite{CH}), we obtain \[(8g+4)\lambda=g\delta_0+\sum_{i>0}\widetilde{e_i}\xi_i+ \sum_{j>0}\widetilde{f_j}\delta_j.\] \begin{thm} In the Picard group of the hyperelliptic locus, $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{I}}_g$, the class of the Hodge bundle $\lambda$ is expressible in terms of the boundary divisor classes of $\overline{\mathfrak{I}}_g$ as: \begin{equation*} (8g+4)\lambda=g\xi_0+\sum_{i=1}^{[(g-1)/2]}2(i+1)(g-i)\xi_i +\sum_{j=1}^{[g/2]}4j(g-j)\delta_j. \end{equation*} \label{CHPic2} \end{thm}

9710

alg-geom/9710026

en

https://arxiv.org/abs/alg-geom/9710026

alg-geom/9710026

Dmitry Kaledin

D. Kaledin

Hyperkaehler structures on total spaces of holomorphic cotangent bundles

100 pages, LaTeX2e

Let $M$ be a Kaehler manifold, and consider the total space $T^*M$ of the cotangent bundle to $M$. We show that in the formal neighborhood of the zero section $M \subset T^*M$ the space $T^*M$ admits a canonical hyperkaehler structure, compatible with the complex and holomorphic symplectic structures on $T^*M$. The associated hyperkaehler metric $h$ coincides with the given Kaehler metric on the zero section $M \subset T^*M$. Moreover, $h$ is invariant under the canonical circle action on $T^*M$ by dilatations along the fibers of $T^*M$ over $M$. We show that a hyperkaehler structure with these properties is unique. When the Kaehler metric on $M$ is real-analytic, we show that this formal hyperkaehler structure can be extended to an open neighborhood of the zero section. We also prove a hyperkaehler analog of the Darboux-Weinstein Theorem. To prove these results, we use the machinery of $R$-Hodge structures, following Deligne and Simpson.

alg-geom

\section*{Introduction} A hyperk\"ahler manifold is by definition a Riemannian manifold $M$ equipped with two anti-commuting almost complex structures $I$, $J$ parallel with respect to the Levi-Civita connection. Hyperk\"ahler manifolds were introduced by Calabi in \cite{C}. Since then they have become the topic of much research. We refer the reader to \cite{Bes} and to \cite{HKLR} for excellent overviews of the subject. Let $M$ be a hyperk\"ahler manifold. The almost complex structures $I$ and $J$ generate an action of the quaternion algebra ${\Bbb H}$ in the tangent bundle $\Theta(M)$ to the manifold $M$. This action is parallel with respect to the Levi-Civita connection. Every quaternion $h \in {\Bbb H}$ with $h^2 = -1$, in particular, the product $K = IJ \in {\Bbb H}$, defines by means of the ${\Bbb H}$-action an almost complex structure $M_h$ on $M$. This almost complex structure is also parallel, hence integrable and K\"ahler. Thus every hyperk\"ahler manifold $M$ is canonically K\"ahler, and in many different ways. For the sake of convenience, we will consider $M$ as a K\"ahler manifold by means of the complex structure $M_I$, unless indicated otherwise. One of the basic facts about hyperk\"ahler manifolds is that the K\"ahler manifold $M_I$ underlying a hyperk\"ahler manifold $M$ is canonically holomorphically symplectic. To see this, let $\omega_J$, $\omega_K$ be the K\"ahler forms for the complex structures $M_J$, $M_K$ on the manifold $M$, and consider the $2$-form $\Omega = \omega_J + \sqrt{-1}\omega_K$ on $M$. It is easy to check that the form $\Omega$ is of Hodge type $(2,0)$ for the complex structure $M_I$ on $M$. Since it is obviously non-degenerate and closed, it is holomorphic, and the K\"ahler manifold $M_I$ equipped with the form $\Omega$ is a holomorphically symplectic manifold. It is natural to ask whether every holomorphically symplectic manifold $\langle M,\Omega\rangle$ underlies a hyperk\"ahler structure on $M$, and if so, then how many such hyperk\"ahler structures are there. Note that if such a hyperk\"ahler structure exists, it is completely defined by the K\"ahler metric $h$ on $M$. Indeed, the K\"ahler forms $\omega_J$ and $\omega_K$ are by definition the real and imaginary parts of the form $\Omega$, and the forms $\omega_J$ and $\omega_K$ together with the metric define the complex structures $J$ and $K$ on $M$ and, consequently, the whole ${\Bbb H}$-action in the tangent bundle $\Theta(M)$. For the sake of simplicity, we will call a metric $h$ on a holomorphically symplectic manifold $\langle M,\Omega\rangle$ hyperk\"ahler if the Riemannian manifold $\langle M, h \rangle$ with the quaternionic action associated to the pair $\langle \Omega, h \rangle$ is a hyperk\"ahler manifold. It is known (see, e.g., \cite{Beauv}) that if the holomorphically symplectic manifold $M$ is compact, for example, if $M$ is a $K3$-surface, then every K\"ahler class in $H^{1,1}(M)$ contains a unique hyperk\"ahler metric. This is, in fact, a consequence of the famous Calabi-Yau Theorem, which provides the canonical Ricci-flat metric on $M$ with the given cohomology class. This Ricci-flat metric turns out to be hyperk\"ahler. Thus in the compact case holomorphically symplectic and hyperk\"ahler manifolds are essentially the same. The situation is completely different in the general case. For example, all holomorphically symplectic structures on the formal neighborhood of the origin $0 \in \C^{2n}$ in the $2n$-dimensional complex vector space $\C^{2n}$ are isomorphic by the appropriate version of the Darboux Theorem. On the other hand, hyperk\"ahler structures on this formal neighborhood form an infinite-dimensional family (see, e.g., \cite{HKLR}, where there is a construction of a smaller, but still infinite-dimensional family of hyperk\"ahler metrics defined on the whole $\C^{2n}$). Thus, to obtain meaningful results, it seems necessary to restrict our attention to holomorphically symplectic manifolds belonging to some special class. The simplest class of non-compact holomorphically symplectic manifolds is formed by total spaces $T^*M$ to the cotangent bundle to complex manifolds $M$. In fact, the first examples of hyperk\"ahler manifolds given by Calabi in \cite{C} were of this type, with $M$ being a K\"ahler manifold of constant holomorphic sectional curvature (for example, a complex projective space). It has been conjectured for some time that every total space $T^*M$ of the cotangent bundle to a K\"ahler manifold admits a hyperk\"ahler structure. The goal of this paper is to prove that this is indeed the case, if one agrees to consider only an open neighborhood $U \subset T^*M$ of the zero section $M \subset T^*M$. Our main result is the following. \begin{thm}\label{th.1} Let $M$ be a complex manifold equipped with a K\"ahler metric. The metric on $M$ extends to a hyperk\"ahler metric $h$ defined in the formal neighborhood of the zero section $M \subset T^*M$ in the total space $T^*M$ to the holomorphic cotangent bundle to $M$. The extended metric $h$ is invariant under the action of the group $U(1)$ on $T^*M$ given by dilatations along the fibers of the canonical projection $\rho:T^*M \to M$. Moreover, every other $U(1)$-invariant hyperk\"ahler metric on the holomorphically symplectic manifold $T^*M$ becomes equal to $h$ after a holomorphic symplectic $U(1)$-equivariant automorphism of $T^*M$ over $M$. Finally, if the K\"ahler metric on $M$ is real-analytic, then the formal hyperk\"ahler metric $h$ converges to a real-analytic metric in an open neighborhood $U \subset T^*M$ of the zero section $M \subset T^*M$. \end{thm} Many of the examples of hyperk\"ahler metrics obtained by Theorem~\ref{th.1} are already known. (See, e.g., \cite{K1}, \cite{K2}, \cite{Nak}, \cite{H}, \cite{BG}, \cite{Sw}.) In these examples $M$ is usually a generalized flag manifold or a homogeneous space of some kind. On the other hand, very little is known for manifolds of general type. In particular, it seems that even for curves of genus $g \geq 2$ Theorem~\ref{th.1} is new. We would like to stress the importance of the $U(1)$-invariance condition on the metric in the formulation of Theorem~\ref{th.1}. This condition for a total space $T^*M$ of a cotangent bundle is equivalent to a more general compatibility condition between a $U(1)$-action and a hyperk\"ahler structure on a smooth manifold introduced by Hitchin (see, e.g., \cite{H}). Thus Theorem~\ref{th.1} can be also regarded as answering a question of Hitchin's in \cite{H}, namely, whether every K\"ahler manifold can be embedded as the sub-manifold of $U(1)$-fixed points in a $U(1)$-equivariant hyperk\"ahler manifold. On the other hand, it is this $U(1)$-invariance that guarantees the uniqueness of the metric $h$ claimed in Theorem~\ref{th.1}. We also prove a version of Theorem~\ref{th.1} ``without the metrics''. The K\"ahler metric on $M$ in this theorem is replaced with a holomorphic connection $\nabla$ on the cotangent bundle to $M$ without torsion and $(2,0)$-curvature. We call such connections {\em K\"ah\-le\-ri\-an}. The total space of the cotangent bundle $T^*M$ is replaced with the total space ${\overline{T}M}$ of the complex-conjugate to the tangent bundle to $M$. (Note that {\em a priori} there is no complex structure on ${\overline{T}M}$, but the $U(1)$-action by dilatations on this space is well-defined.) The analog of the notion of a hyperk\"ahler manifold ``without the metric'' is the notion if a hypercomplex manifold (see, e.g., \cite{Bo}). We define a version of Hitchin's condition on the $U(1)$-action for hypercomplex manifolds and prove the following. \begin{thm}\label{th.2} Let $M$ be a complex manifold, and let ${\overline{T}M}$ be the total space of the complex-conjugate to the tangent bundle to $M$ equipped with an action of the group $U(1)$ by dilatation along the fibers of the projection ${\overline{T}M} \to M$. There exists a natural bijection between the set of all K\"ah\-le\-ri\-an connections on the cotangent bundle to $M$ and the set of all isomorphism classes of $U(1)$-equivariant hypercomplex structures on the formal neighborhood of the zero section $M \subset {\overline{T}M}$ in ${\overline{T}M}$ such that the projection $\rho:{\overline{T}M} \to M$ is holomorphic. If the K\"ah\-le\-ri\-an connection on $M$ is real-analytic, the corresponding hypercomplex structure is defined in an open neighborhood $U \subset {\overline{T}M}$ of the zero section. \end{thm} Our main technical tool in this paper is the relation between $U(1)$\--equi\-va\-ri\-ant hyperk\"ahler manifolds and the theory of $\R$-Hodge structures discovered by Deligne and Simpson (see \cite{De2}, \cite{Simpson}). To emphasize this relation, we use the name {\em Hodge manifolds} for the hypercomplex manifolds equipped with a compatible $U(1)$-action. It must be noted that many examples of hyperk\"ahler manifolds equipped with a compatible $U(1)$-action are already known. Such are, for example, many of the manifolds constructred by the so-called hyperk\"ahler redution from flat hyperk\"ahler spaces (see \cite{H} and \cite{HKLR}). An important class of such manifolds is formed by the so-called quiver varieties, studied by Nakajima (\cite{Nak}). On the other hand, the moduli spaces $\M$ of flat connections on a complex manifold $M$, studied by Hitchin (\cite{Hcurves}) when $M$ is a curve and by Simpson (\cite{Smoduli} in the general case, also belong to the class of Hodge manifolds, as Simpspn has shown in \cite{Simpson}. Some parts of our theory, especially the uniqueness statement of Theorem~\ref{th.1}, can be applied to these known examples. \medskip We now give a brief outline of the paper. Sections 1-3 are preliminary and included mostly to fix notation and terminology. Most of the facts in these sections are well-known. \begin{itemize} \item In Section 1 we have collected the necessary facts from linear algebra about quaternionic vector spaces and $\R$-Hodge structures. Everything is standard, with an exception, perhaps, of the notion of {\em weakly Hodge map}, which we introduce in Subsection~\ref{w.H.sub}. \item We begin Section 2 with introducing a technical notion of a {\em Hodge bundle} on a smooth manifold equipped with a $U(1)$-action. This notion will be heavily used throughout the paper. Then we switch to our main subject, namely, various differential-geomteric objects related to a quaternion action in the tangent bundle. The rest of Section 2 deals with almost quaternionic manifolds and the compatibility conditions between an almost quaternionic structure and a $U(1)$-action on a smooth manifold $M$. \item In Section 3 we describe various integrability conditions on an almost quaternionic structure. In particular, we recall the definition of a hypercomplex manifold and introduce $U(1)$-equivariant hypercomplex manifolds under the name of {\em Hodge manifolds}. We then rewrite the definition of a Hodge manifold in the more convenient language of Hodge bundles, to be used throughout the rest of the paper. Finally, in Subsection~\ref{polarization} we discuss metrics on hypercomplex and Hodge manifolds. We recall the definition of a hyperk\"ahler manifold and define the notion of a {\em polarization} of a Hodge manifold. A polarized Hodge manifold is the same as a hyperk\"ahler manifold equipped with a $U(1)$-action compatible with the hyperk\"ahler structure of the sense of Hitchin, \cite{H}. \item The main part of the paper begins in Section 4. We start with arbitrary Hodge manifolds and prove that in a neighborhood of the subset $M^{U(1)}$ of ``regular'' $U(1)$-fixed points every such manifold $M$ is canonically isomorphic to an open neighborhood of the zero section in a total space $\overline{T}M^{U(1)}$ of the tangent bundle to the fixed point set. A fixed point $m \in M$ is ``regular'' if the group $U(1)$ acts on the tangent space $T_mM$ with weights $0$ and $1$. We call this canonical isomorphism {\em the linearization} of the regular Hodge manifold. The linearization construction can be considered as a hyperk\"ahler analog of the Darboux-Weinstein Theorem in the symplectic geometry. Apart from the cotangent bundles, it can be applied to the Hitchin-Simpson moduli space $\M$ of flat connections on a K\"ahler manifold $X$. The regular points in this case correspond to stable flat connections such that the underlying holomorphic bundle is also stable. The linearization construction provides a canonical embedding of the subspace $\M^{reg} \subset \M$ of regular points into the total space $T^*\M_0$ of the cotangent bundle to the space $\M_0$ of stable holomorphic bundles on $X$. The unicity statement of Theorem~\ref{th.1} guarantees that the hyperk\"ahler metric on $\M^{reg}$ provided by the Simpson theory is the same as the canonical metric constructed in Theorem~\ref{th.1}. \item Starting with Section 5, we deal only with total spaces ${\overline{T}M}$ of the complex-conjugate to tangent bundles to complex manifolds $M$. In Section 5 we describe Hodge manifolds structures on ${\overline{T}M}$ in terms of certain connections on the locally trivial fibration ${\overline{T}M} \to M$. ``Connection'' here is understood as a choice of the subbundle of horizontal vectors, regardless of the vector bundle structure on the fibration ${\overline{T}M} \to M$. We establish a correspondence between Hodge manifold structures on ${\overline{T}M}$ and connections on ${\overline{T}M}$ over $M$ of certain type, which we call {\em Hodge connection}. \item In Section 6 we restrict our attention to the formal neighborhood of the zero section $M \subset {\overline{T}M}$. We introduce the appropriate ``formal'' versions of all the definitions and then establish a correspondence between formal Hodge connections on ${\overline{T}M}$ over $M$ and certain differential operators on the manifold $M$ itself, which we call {\em extended connections}. We also introduce a certain canonical algebra bundle $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ on the complex manifold $M$, which we call {\em the Weil algebra} of the manifold $M$. Extended connections give rise to natural derivations of the Weil algebra. \item Before we can proceed with classification of extended connections on the manifold $M$ and therefore of regular Hodge manifolds, we need to derive some linear-algebraic facts on the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. This is the subject of Section 7. We begin with introducing a certain version of the de Rham complex of a smooth complex manifold, which we call {\em the total de Rham complex}. Then we combine it the material of Section 6 to define the so-called {\em total Weil algebra} of the manifold $M$ and establish some of its properties. Section 7 is the most technical part of the whole paper. The reader is advised to skip reading it until needed. \item Section 8 is the main section of the paper. In this section we prove, using the technical results of Section 7, that extended connections on $M$ are in a natural one-to-one correspondence with K\"ah\-le\-ri\-an connections on the cotangent bundle to $M$ (Theorem~\ref{kal=ext}). This proves the formal part of Theorem~\ref{th.2}. \item In Section 9 we deal with polarizations. After some preliminaries, we use Theorem~\ref{th.2} to deduce the formal part of Theorem~\ref{th.1} (see Theorem~\ref{metrics}). \item Finally, in Section~\ref{convergence} we study the convergence of our formal series and prove Theorem~\ref{th.1} and Theorem~\ref{th.2} in the real-analytic case. \item We have also attached a brief Appendix, where we sketch a more conceptual approach to some of the linear algebra used in the paper, in particular, to our notion of a weakly Hodge map. This approach also allows us to describe a simple and conceptual proof of Proposition~\ref{ac}, the most technical of the facts proved in Section 7. The Appendix is mostly based on results of Deligne and Simpson (\cite{De2}, \cite{Simpson}). \end{itemize} \noindent {\bf Acknowledgment.} I would like to thank A. Beilinson, D. Kazhdan, A. Levin, L. Posicelsky, A. Shen and A. Tyurin for stimulating discussions. I am especially grateful to my friends Misha Verbitsky and Tony Pantev for innumerable discussions, constant interest and help, without which this paper most probably would not have been written. I would also like to mention here how much I have benefited from a course on moduli spaces and Hodge theory given by Carlos Simpson at MIT in the Fall of 1993. On a different note, I would like to express my dearest gratitude to Julie Lynch, formerly at International Press in Cambridge, and also to the George Soros's Foundation and to CRDF for providing me with a source of income during the preparation of this paper. \tableofcontents \section{Preliminary facts from linear algebra} \subsection{Quaternionic vector spaces} \refstepcounter{subsubsection Throughout the paper denote by ${\Bbb H}$ the $\R$-algebra of quaternions. \noindent {\bf Definition.\ } A {\em quaternionic vector space} is a finite-dimensional left module over the algebra ${\Bbb H}$. Let $V$ be a quaternionic vector space. Every algebra embedding $I:\C \to {\Bbb H}$ defines by restriction an action of $\C$ on $V$. Denote the corresponding complex vector space by $V_I$. Fix once and for all an algebra embedding $I:\C \to {\Bbb H}$ and call the complex structure $V_I$ {\em the preferred complex structure} on $V$. \refstepcounter{subsubsection \label{u.acts.on.h} Let the group $\C^*$ act on the algebra ${\Bbb H}$ by conjugation by $I(\C^*)$. Since $I(\R^*) \subset I(\C^*)$ lies in the center of the algebra ${\Bbb H}$, this action factors through the map $N:\C^* \to \C^*/\R^* \cong U(1)$ from $\C^*$ to the one-dimensional unitary group defined by $N(a) = a^{-1}\overline{a}$. Call this action {\em the standard action} of $U(1)$ on ${\Bbb H}$. The standard action commutes with the multiplication and leaves invariant the subalgebra $I(\C)$. Therefore it extends to an action of the complex algebraic group $\C^* \supset U(1)$ on the algebra ${\Bbb H}$ considered as a right complex vector space over $I(\C)$. Call this action {\em the standard action} of $\C^*$ on ${\Bbb H}$. \refstepcounter{subsubsection \noindent {\bf Definition.\ } An {\em equivariant} quaternionic vector space is a quaternionic vector space $V$ equipped with an action of the group $U(1)$ such that the action map $$ {\Bbb H} \otimes_\R V \to V $$ is $U(1)$-equivariant. The $U(1)$-action on $V$ extends to an action of $\C^*$ on the complex vector space $V_I$. The action map ${\Bbb H} \otimes_\R V \to V$ factors through a map $$ \Mult:{\Bbb H} \otimes_\C V_I \to V_I $$ of complex vector spaces. This map is $\C^*$-equivariant if and only if $V$ is an equivariant quaternionic vector space. \refstepcounter{subsubsection The category of complex algebraic representations $V$ of the group $\C^*$ is equivalent to the category of graded vector spaces $V = \oplus V^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. We will say that a representation $W$ is {\em of weight $i$} if $W = W^i$, that is, if an element $z \in \C^*$ acts on $W$ by multiplication by $z^k$. For every representation $W$ we will denote by $W(k)$ the representation corresponding to the grading $$ W(k)^i = W^{k+i}. $$ \refstepcounter{subsubsection \label{h=c+c} The algebra ${\Bbb H}$ considered as a complex vector space by means of right multiplication by $I(\C)$ decomposes ${\Bbb H} = I(\C) \oplus \overline{\C}$ with respect to the standard $\C^*$-action. The first summand is of weight $0$, and the second is of weight $1$. This decomposition is compatible with the left $I(\C)$-actions as well and induces for every complex vector space $W$ a decomposition $$ {\Bbb H} \otimes_\C W = W \oplus \overline{\C} \otimes_\C W. $$ If $W$ is equipped with an $\C^*$-action, the second summand is canonically isomorphic to $\overline{W}(1)$, where $\overline{W}$ is the vector space complex-conjugate to $W$. \refstepcounter{subsubsection Let $V$ be an equivariant quaternionic vector space. The action map $$ \Mult:{\Bbb H} \otimes_\C V_I \cong V_I \oplus \overline{\C} \otimes V_I \to V_I $$ decomposes $\Mult = {\sf id} \oplus j$ for a certain map $j:\overline{V_I}(1) \to V_I$. The map $j$ satisfies $j \circ \overline{j} = -{\sf id}$, and we obviously have the following. \begin{lemma}\label{explicit.eqvs} The correspondence $V \longmapsto \langle V_I, j \rangle$ is an equivalence of categories bet\-ween the category of equivariant quaternionic vector spaces and the category of pairs $\langle W, j \rangle$ of a graded complex vector space $W$ and a map $j:\overline{W^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to W^{1-{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ satisfying $j \circ \overline{j} = -{\sf id}$. \end{lemma} \refstepcounter{subsubsection We will also need a particular class of equivariant quaternionic vector spaces which we will call regular. \noindent {\bf Definition.\ } An equivariant quaternionic vector space $V$ is called {\em regular} if every irreducible $\C^*$-subrepresentation of $V_I$ is either trivial or of weight $1$. \begin{lemma}\label{regular.quaternionic} Let $V$ be a regular $\C^*$-equivariant quaternionic vector space and let $V^0_I \subset V_I$ be the subspace of $\C^*$-invariant vectors. Then the action map $$ V^0_I \oplus \overline{V^0_I} \cong {\Bbb H} \otimes_\C V^0_I \to V_I $$ is invertible. \end{lemma} \proof Let $V^1_I \subset V_I$ be the weight $1$ subspace with respect to the $ gm$-action. Since $V$ is regular, $V_I = V^0_I \oplus V^1_I$. On the other hand, $j:\overline{V_I} \to V_I$ interchanges $V^0_I$ and $V^1_I$. Therefore $V^1_I \cong \overline{V^0_I}$ and we are done. \hfill \ensuremath{\square}\par Thus every regular equivariant quaternionic vector space is a sum of several copies of the algebra ${\Bbb H}$ itself. The corresponding Hodge structure has Hodge numbers $h^{1,0} = h^{0,1}$, $h^{p,q} = 0$ otherwise. \subsection{The complementary complex structure}\label{complementary} \refstepcounter{subsubsection Let $J:\C \to {\Bbb H}$ be another algebra embedding. Say that embeddings $I$ and $J$ are {\em complementary} if $$ J(\sqrt{-1}) I(\sqrt{-1}) = - I(\sqrt{-1}) J(\sqrt{-1}). $$ Let $V$ be an equivariant quaternionic vector space. The standard $U(1)$-action on ${\Bbb H}$ induces an action of $U(1)$ on the set of all algebra embeddings $\C \to {\Bbb H}$. On the subset of embeddings complementary to $I$ this action is transitive. Therefore for every two embeddings $J_1,J_2:\C \to {\Bbb H}$ complementary to $I$ the complex vector spaces $V_{J_1}$ and $V_{J_2}$ are canonically isomorphic. We will from now on choose for convenience an algebra embedding $J:\C \to {\Bbb H}$ complementary to $I$ and speak of {\em the complementary complex structure} $V_J$ on $V$; however, nothing depends on this choice. \refstepcounter{subsubsection For every equivariant quaternionic vector space $V$ the complementary embedding $J:\C \to {\Bbb H}$ induces an isomorphism $$ J \otimes {\sf id}:\C \otimes_\R V \to {\Bbb H} \otimes_{I(\C)} V_I $$ of complex vector spaces. Let $\Mult:{\Bbb H} \otimes_{I(\C)} V_I \to V_I$, $\Mult:\C \otimes_\R V \to V_J$ be the action maps. Then there exists a unique isomorphism $H:V_J \to V_I$ of complex vector spaces such that the diagram $$ \begin{CD} \C \otimes_\R V @>{J \otimes {\sf id}}>> {\Bbb H} \otimes_{I(\C)} V_I\\ @V{\Mult}VV @V{\Mult}VV \\ V_J @>{H}>> V_I \end{CD} $$ is commutative. Call the map $H:V_J \to V_I$ {\em the standard isomorphism} between the complementary and the preferred complex structures on the equivariant quaternionic vector space $V$. \refstepcounter{subsubsection Note that both $V_I$ and $V_J$ are canonically isomorphic to $V$ as real vector spaces; therefore the map $H:V_J \to V_I$ is in fact an automorphism of the real vector space $V$. Up to a constant this automorphism is given by the action of the element $I(\sqrt{-1}) + J(\sqrt{-1}) \in {\Bbb H}$ on the ${\Bbb H}$-module $V$. \subsection{$\protect\R$-Hodge structures} \refstepcounter{subsubsection Recall that {\em a pure $\R$-Hodge structure $W$ of weight $i$} is a pair of a graded complex vector space $W = \oplus_{p+q=i} W^{p,q}$ and a {\em real structure} map $\overline{\ }:\overline{W^{p,q}} \to W^{q,p}$ satisfying $\overline{\ } \circ \overline{\ } = {\sf id}$. The bigrading $W^{p,q}$ is called {\em the Hodge type bigrading}. The dimensions $h^{p,q} = \dim_\C W^{p,q}$ are called {\em the Hodge numbers} of the pure $\R$-Hodge structure $W$. Maps between pure Hodge structures are by definition maps of their underlying complex vector spaces compatible with the bigrading and the real structure maps. \refstepcounter{subsubsection Recall also that for every $k$ the {\em Hodge-Tate} pure $\R$-Hodge structure $\R(k)$ of weight $-2k$ is by definition the $1$-dimensional complex vector space with complex conjugation equal to $(-1)^k$ times the usual one, and with Hodge bigrading $$ \R(k)^{p,q} = \begin{cases} \R(k), \quad p=q=-k,\\ 0, \quad \text{otherwise}. \end{cases} $$ For a pure $\R$-Hodge structure $V$ denote, as usual, by $V(k)$ the tensor product $V(k) = V \otimes \R(k)$. \refstepcounter{subsubsection \label{w.1} We will also need special notation for another common $\R$-Hodge structure, which we now introduce. Note that for every complex $V$ be a complex vector space the complex vector space $V \otimes_\R \C$ carries a canonical $\R$-Hodge structure of weight $1$ with Hodge bigrading given by $$ V^{1,0} = V \subset V \otimes_\R \C \qquad\qquad V^{0,1} = \overline{V} \otimes_\R \C. $$ In particular, $\C \otimes_\R \C$ carries a natural $\R$-Hodge structure of weight $1$ with Hodge numbers $h^{1,0} = h^{0,1} = 1$. Denote this Hodge structure by $\W_1$. \refstepcounter{subsubsection Let $\langle W, \overline{\ } \rangle$ be a pure Hodge structure, and denote by $W_\R \subset W$ the $\R$-vector subspace of elements preserved by $\overline{\ }$. Define the {\em Weil operator} $C:W \to W$ by $$ C = (\sqrt{-1})^{p-q}:W^{p,q} \to W^{q,p}. $$ The operator $C:W \to W$ preserves the $\R$-Hodge structure, in particular, the subspace $W_\R \subset W$. On pure $\R$-Hodge structures of weight $0$ the Weil operator $C$ corresponds to the action of $-1 \in U(1) \subset \C^*$ in the corresponding representation. \refstepcounter{subsubsection \label{Weil} For a pure Hodge structure $W$ of weight $i$ let $$ j = C \circ \overline{\ }:\overline{W^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},1-{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}} \to W^{1-{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}. $$ If $i$ is odd, in particular, if $i=1$, then $j \circ \overline{j} = -{\sf id}$. Together with Lemma~\ref{explicit.eqvs} this gives the following. \begin{lemma}\label{eqvs.hodge} The category of equivariant quaternionic vector spaces is equivalent to the category of pure $\R$-Hodge structures of weight $1$. \end{lemma} \refstepcounter{subsubsection Let $V$ be an equivariant quaternionic vector space, and let $\langle W, \overline{\ } \rangle$ be $\R$-Hodge structure of weight $1$ corresponding to $V$ under the equivalence of Lemma~\ref{eqvs.hodge}. By definition the complex vector space $W$ is canonically isomorphic to the complex vector space $V_I$ with the preferred complex structure on $V$. It will be more convenient for us to identify $W$ with the complementary complex vector space $V_J$ by means of the standard isomorphism $H:V_J \to V_I$. The multiplication map $$ \Mult:V_I \otimes_\R \C \cong V \otimes_\C {\Bbb H} \to V_J $$ is then a map of $\R$-Hodge structures. The complex conjugation $\overline{\ }:W \to \overline{W}$ is given by \begin{equation}\label{i.conj} \overline{\ } = C \circ H \circ j \circ H^{-1} = C \circ i:V_J \to \overline{V_J}, \end{equation} where $i:V_J \to \overline{V_J}$ is the action of the element $I(\sqrt{-1}) \subset {\Bbb H}$. \subsection{Weakly Hodge maps}\label{w.H.sub} \refstepcounter{subsubsection Recall that the category of pure $\R$-Hodge structures of weight $i$ is equivalent to the category of pairs $\langle V, F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\rangle$ of a real vector space $V$ and a decreasing filtration $F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ on the complex vector space $V_\C = V \otimes_\R \C$ satisfying $$ V_\C = \bigoplus_{p+q=i}F^pV_\C \cap \overline{F^qV_\C}. $$ The filtration $F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is called {\em the Hodge filtration}. The Hodge type grading and the Hodge filtration are related by $V^{p,q} = F^pV_\C \cap \overline{F^qV_\C}$ and $F^pV_\C = \oplus_{k \geq p}V^{k,i-k}$. \refstepcounter{subsubsection \label{weakly.hodge} Let $\langle V,F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\rangle$ and $\langle W,F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \rangle$ be pure $\R$-Hodge structures of weights $n$ and $m$ respectively. Usually maps of pure Hodge structures are required to preserve the weights, so that $\Hom(V,W)=0$ unless $n=m$. In this paper we will need the following weaker notion of maps between pure $\R$-Hodge structures. \noindent {\bf Definition.\ } An $\R$-linear map $f:V \to W$ is said to be {\em weakly Hodge} if it preserves the Hodge filtrations. Equivalently, the complexified map $f:V_\C \to W_\C$ must decompose \begin{equation}\label{H.type} f = \sum_{0 \leq p \leq m-n}f^{p,m-n-p}, \end{equation} where the map $f^{p,m-n-p}:V_\C \to W_\C$ is of Hodge type $(p,m-n-p)$. Note that this condition is indeed weaker than the usual definition of a map of Hodge structures: a weakly Hodge map $f:V \to W$ can be non-trivial if $m$ is strictly greater than $n$. If $m < n$, then $f$ must be trivial, and if $m=n$, then weakly Hodge maps from $V$ to $W$ are the same as usual maps of $\R$-Hodge structures. \refstepcounter{subsubsection We will denote by ${{\cal W}{\cal H}odge}$ the category of pure $\R$-Hodge structures of arbitrary weight with weakly Hodge maps as morphisms. For every integer $n$ let ${{\cal W}{\cal H}odge}_n$ be the full subcategory in ${{\cal W}{\cal H}odge}$ consisting of pure $\R$-Hodge structures of weight $n$, and let ${{\cal W}{\cal H}odge}_{\geq n}$ be the full subcategory of $\R$-Hodge structures of weight not less than $n$. Since weakly Hodge maps between $\R$-Hodge structures of the same weight are the same as usual maps of $\R$-Hodge structures, the category ${{\cal W}{\cal H}odge}_n$ is the usual category of pure $\R$-Hodge structures of weight $n$. \refstepcounter{subsubsection \label{w.k} Let $\W_1 = \C \otimes_\R \C$ be the pure Hodge structure of weight $1$ with Hodge numbers $h^{1,0}=h^{0,1}=1$, as in \ref{w.1}. The diagonal embedding $\C \to \C \otimes_\R \C$ considered as a map $w_1:\R \to \W_1$ from the trivial pure $\R$-Hodge structure $\R$ of weight $0$ to $\W_1$ is obviously weakly Hodge. It decomposes $w_1 = w_1^{1,0} + w_1^{0,1}$ as in \eqref{H.type}, and the components $w_1^{1,0}:\C \to \W_1^{1,0}$ and $w_1^{0,1}:\C \to \W_1^{0,1}$ are isomorphisms of complex vectors spaces. Moreover, for every pure $\R$-Hodge structure $V$ of weight $n$ the map $w_1$ induces a weakly Hodge map $w_1:V \to \W_1 \otimes V$, and the components $w_1^{1,0}:V_\C \to V_\C \otimes \W_1^{1,0}$ and $w_1^{0,1}:V_\C \to V_\C \otimes \W_1^{0,1}$ are again isomorphisms of complex vector spaces. More generally, for every $k \geq 0$ let $\W_k = S^k\W_1$ be the $k$-th symmetric power of the Hodge structure $\W_1$. The space $\W_k$ is a pure $\R$-Hodge structure of weight $k$, with Hodge numbers $h^{k,0} = h^{k-1,1} =\ldots = h^{0,k} = 1$. Let $w_k:\R \to \W_k$ be the $k$-th symmetric power of the map $w_1:\R \to \W_1$. For every pure $\R$-Hodge structure $V$ of weight $n$ the map $w_k$ induces a weakly Hodge map $w_k:V \to \W_k \otimes V$, and the components $$ w_k^{p,q}:V_\C \to V_\C \otimes \W_k^{p,k-p}, \qquad 0 \leq p \leq k $$ are isomorphisms of complex vector spaces. \refstepcounter{subsubsection \label{w.k.uni} The map $w_k$ is a universal weakly Hodge map from a pure $\R$-Hodge structures $V$ of weight $n$ to a pure $\R$-Hodge structure of weight $n+k$. More precisely, every weakly Hodge map $f:V \to V'$ from $V$ to a pure $\R$-Hodge structure $V'$ of weight $n+k$ factors uniquely through $w_k:V \to \W_k \otimes V$ by means of a map $f':\W_k \otimes V \to V'$ preserving the pure $\R$-Hodge structures. Indeed, $V_\C \otimes \W_k = \bigoplus_{0 \leq p \leq k} V_\C \otimes \W_k^{p,k-p}$, and the maps $w_k^{p,k-p}:V_\C \to V_\C \otimes \W_k^{p,k-p}$ are invertible. Hence to obtain the desired factorization it is necessary and sufficient to set $$ f' = f^{p,k-p} \circ \left(w_k^{p,k-p}\right)^{-1}:V_\C \otimes \W_k^{p,k-p} \to V_\C \to \V'_\C, $$ where $f = \sum_{0 \leq p \leq k}f^{p,k-p}$ is the Hodge type decomposition \eqref{H.type}. \refstepcounter{subsubsection It will be convenient to reformulate the universal properties of the maps $w_k$ as follows. By definition $\W_k = S^k\W_1$, therefore the dual $\R$-Hodge structures equal $\W_k^* = S^k\W_1^*$, and for every $n,k \geq 0$ we have a canonical projection ${\sf can}:\W_n^* \otimes \W_k^* \to \W_{n+k}^*$. For every pure $\R$-Hodge structure $V$ of weight $k \geq 0$ let $\Gamma(V) = V \otimes \W_k^*$. \begin{lemma}\label{g.ex} The correspondence $V \mapsto \Gamma(V)$ extends to a functor $$ \Gamma:{{\cal W}{\cal H}odge}_{\geq 0} \to {{\cal W}{\cal H}odge}_0 $$ adjoint on the right to the canonical embedding ${{\cal W}{\cal H}odge}_0 \hookrightarrow {{\cal W}{\cal H}odge}_{\geq 0}$. \end{lemma} \proof Consider a weakly Hodge map $f:V_n \to V_{n+k}$ from $\R$-Hodge structure $V_n$ of weight $n$ to a pure $\R$-Hodge structure $V_{n+k}$ of weight $n+k$. By the universal property the map $f$ factors through the canonical map $w_k:V_n \to V_n \otimes \W_k$ by means of a map $f_k:V_n \otimes \W_k \to V_{n+k}$. Let $f_k':V_k \to V_{n+k} \otimes \W_k^*$ be the adjoint map, and let \begin{align*} \Gamma(f) = {\sf can} \circ f_k':&\Gamma(V_n) = V_n \otimes \W_k^* \to V_{n+k} \otimes \W_n^* \otimes \W_k^* \to\\ \to &\Gamma(V_{n+k}) = V \otimes \W_{n+k}^*. \end{align*} This defines the desired functor $\Gamma:{{\cal W}{\cal H}odge}_{\geq 0} \to {{\cal W}{\cal H}odge}_0$. The adjointness is obvious. \hfill \ensuremath{\square}\par \noindent {\bf Remark.\ } See Appendix for a more geometric description of the functor $\Gamma:{{\cal W}{\cal H}odge}_{\geq 0} \to {{\cal W}{\cal H}odge}_0$. \refstepcounter{subsubsection \label{l.r} The complex vector space $\W_1 = \C \otimes_\R \C$ is equipped with a canonical skew-symmetric trace pairing $\W_1 \otimes_\C \W_1 \to \C$. Let $\gamma:\C \to \W_1^*$ be the map dual to $w_1:\R \to \W_1$ under this pairing. The map $\gamma$ is not weakly Hodge, but it decomposes $\gamma = \gamma^{-1,0} + \gamma^{0,-1}$ with respect to the Hodge type grading. Denote $\gamma_l = \gamma^{-1,0}$, $\gamma_r = \gamma^{0,-1}$. For every $0 \leq p \leq k$ the symmetric powers of the maps $\gamma_l$, $\gamma_r$ give canonical complex-linear embeddings $$ \gamma_l, \gamma_r:\W_p^* \to \W_k^*. $$ \refstepcounter{subsubsection The map $\gamma_l$ if of Hodge type $(p-k,0)$, while $\gamma_r$ is of Hodge type $(0, p-k)$, and the maps $\gamma_l$, $\gamma_r$ are complex conjugate to each other. Moreover, they are each compatible with the natural maps ${\sf can}:\W_p^* \otimes \W_q^* \to \W_{p+q}^*$ in the sense that ${\sf can} \circ (\gamma_l \otimes \gamma_l) = \gamma_l \circ {\sf can}$ and ${\sf can} \circ (\gamma_r \otimes \gamma_r) = \gamma_r \circ {\sf can}$. For every $p$, $q$, $k$ such that $p+q \geq k$ we have a short exact sequence \begin{equation}\label{p+q} \begin{CD} 0 @>>> \W_{p+q-k} @>{\gamma_r - \gamma_l}>> \W_p^* \oplus \W_q^* @>{\gamma_l + \gamma_l}>> \W_k^* @>>> 0 \end{CD} \end{equation} of complex vector spaces. We will need this exact sequence in \ref{gamma.use}. \refstepcounter{subsubsection \label{gamma.tensor} The functor $\Gamma$ is, in general, not a tensor functor. However, the canonical maps ${\sf can}:\W_n^* \otimes \W_k^* \to \W_{n+k}^*$ define for every two pure $\R$-Hodge structures $V_1$, $V_2$ of non-negative weight a surjective map $$ \Gamma(V_1) \otimes \Gamma(V_2) \to \Gamma(V_1 \otimes V_2). $$ These maps are functorial in $V_1$ and $V_2$ and commute with the associativity and the commutativity morphisms. Moreover, for every algebra $\A$ in the tensor category ${{\cal W}{\cal H}odge}_{\geq 0}$ they turn $\Gamma(\A)$ into an algebra in ${{\cal W}{\cal H}odge}_0$. \subsection{Polarizations} \refstepcounter{subsubsection \label{hyperherm} Consider a quaternionic vector space $V$, and let $h$ be a Euclidean metric on $V$. \noindent {\bf Definition.\ } The metric $h$ is called {\em Qua\-ter\-nionic\--Her\-mi\-tian} if for any algebra embedding $I:\C \to {\Bbb H}$ the metric $h$ is the real part of an Hermitian metric on the complex vector space $V_I$. Equivalently, a metric is Quaternionic-Hermitian if it is invariant under the action of the group $SU(2) \subset {\Bbb H}$ of unitary quaternions. \refstepcounter{subsubsection \label{hermhodge} Assume that the quaternionic vector space $V$ is equivariant. Say that a metric $h$ on $V$ is {\em Hermitian-Hodge} if it is Qua\-ter\-nionic-\-Her\-mi\-tian and, in addition, invariant under the $U(1)$-action on $V$. Let $V_I$ be the vector space $V$ with the preferred complex structure $I$, and let $$ V_I = \bigoplus V^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} $$ and $j:\overline{V^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to V^{1-{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ be as in Lemma~\ref{explicit.eqvs}. The metric $h$ is Hermitian-Hodge if and only if \begin{enumerate} \item it is the real part of an Hermitian metric on $V_I$, \item $h(V^p,V^q) = 0$ whenever $p \neq q$, and \item $h(j(a),b) = - h(a,j(b))$ for every $a,b \in V$. \end{enumerate} \refstepcounter{subsubsection Recall that a {\em polarization} $S$ on a pure $\R$-Hodge structure $W$ of weight $i$ is a bilinear form $S:W \otimes W \to \R(-i)$ which is a morphism of pure Hodge structures and satisfies \begin{align*} S(a,b) &= (-1)^{i}S(b,a)\\ S(a,Ca) &> 0 \end{align*} for every $a,b \in W$. (Here $C:W \to W$ is the Weil operator.) \refstepcounter{subsubsection \label{pol} Let $V$ be an equivariant quaternionic vector space equipped with an Euclidean metric $h$, and let $\langle W, \overline{\ } \rangle$ be the pure $\R$-Hodge structure of weight $1$ associated to $V$ by Lemma~\ref{eqvs.hodge}. Recall that $W = V_J$ as a complex vector space. Assume that $h$ extends to an Hermitian metric $h_J$ on $V_J$, and let $S:W \otimes W \to \R(-1)$ be the bilinear form defined by $$ S(a,b) = h(a,C\overline{b}), \qquad a,b \in W_\R \subset W. $$ The form $S$ is a polarization if and only if the metric $h$ is Hermitian-Hodge. This gives a one-to-one correspondence between the set of Hermitian-Hodge metrics on $V$ and the set of polarizations on the Hodge structure $W$. \refstepcounter{subsubsection Let $W^*$ be the Hodge structure of weight $-1$ dual to $W$. The sets of polarizations on $W$ and on $W^*$ are, of course, in a natural one-to-one correspondence. It will be more convenient for us to identify the set of Hermitian-Hodge metrics on $V$ with the set of polarizations on $W^*$ rather then on $W$. Assume that the metric $h$ on the equivariant quaternionic vector space $V$ is Hermitian-Hodge, and let $S \in \Lambda^2(W) \subset \Lambda^2(V \otimes \C)$ be the corresponding polarization. Extend $h$ to an Hermitian metric $h_I$ on the complex vector space $V$ with the preferred complex structure $V_I$, and let $$ \omega_I \in V_I \otimes \overline{V_I} \subset \Lambda^2(V \otimes_\R \C) $$ be the imaginary part of the corresponding Hermitian metric on the dual space $V_I^*$. Let $i:V_J \to \overline{V_J}$ the action of the element $I(\sqrt{-1}) \in {\Bbb H}$. By \eqref{i.conj} we have $$ \omega_I(a,b) = h(a,i(b)) = h(a,C\overline{b}) = S(a,b) $$ for every $a,b \in V_J \subset V \otimes \C$. Since $\omega_I$ is real, and $V \otimes \C = V_J \oplus \overline{V_J}$, the $2$-forms $\omega_I$ and $S$ are related by \begin{equation}\label{omega.and.Omega} \omega_I = \frac{1}{2}(S + \nu(S)), \end{equation} where $\nu:V \otimes \C \to \overline{V \otimes \C}$ is the usual complex conjugation. \section{Hodge bundles and quaternionic manifolds}\label{hbqm.section} \subsection{Hodge bundles}\label{hb.sub} \refstepcounter{subsubsection Throughout the rest of the paper, our main tool in studying hyperk\"ahler structures on smooth manifolds will be the equivalence between equivariant quaternionic vector spaces and pure $\R$-Hodge structures of weight $1$, established in Lemma~\ref{explicit.eqvs}. In order to use it, we will need to generalize this equivalence to the case of vector bundles over a smooth manifold $M$, rather than just vector spaces. We will also need to consider manifolds equipped with a smooth action of the group $U(1)$, and we would like our generalization to take this $U(1)$-action into account. Such a generalization requires, among other things, an appropriate notion of a vector bundle equipped with a pure $\R$-Hodge structure. We introduce and study one version of such a notion in this section, under the name of a Hodge bundle (see Definition~\ref{hodge.bundles}). \refstepcounter{subsubsection Let $M$ be a smooth manifold equipped with a smooth $U(1)$-action (or {\em a $U(1)$-manifold}, to simplify the terminology), and let $\iota:M \to M$ be the action of the element $-1 \in U(1) \subset \C^*$. \noindent {\bf Definition.\ } \label{hodge.bundles} An {\em Hodge bundle of weight $k$} on $M$ is a pair $\langle\E,\overline{\ }\rangle$ of a $U(1)$-equi\-va\-ri\-ant complex vector bundle $\E$ on $M$ and a $U(1)$-equivariant bundle map $\overline{\ }:\overline{\iota^*\E}(k) \to \E$ satisfying $\overline{\ } \circ \iota^*\overline{\ } = {\sf id}$. Hodge bundles of weight $k$ over $M$ form a tensor $\R$-linear additive category, denoted by ${{\cal W}{\cal H}odge}_k(M)$. \refstepcounter{subsubsection\label{w.hodge} Let $\E$, $\F$ be two Hodge bundles on $M$ of weights $m$ and $n$. Say that a bundle map. or, more generally, a differential operator $f:\E \to \F$ is {\em weakly Hodge} if \begin{enumerate} \item $f = \overline{\iota^*f}$, and \item there exists a decomposition $f = \sum_{0 \leq n-m} f_i$ with $f_i$ being of degree $i$ with respect to he $U(1)$-equivariant structure. (In particular, $f=0$ unless $n \geq m$, and we always have $f_k = \overline{\iota^* f_{n-m-k}}$.) \end{enumerate} Denote by ${{\cal W}{\cal H}odge}(M)$ the category of Hodge bundles of arbitrary weight on $M$, with weakly Hodge bundle maps as morphisms. For every $i$ the category ${{\cal W}{\cal H}odge}_i(M)$ is a full subcategory in ${{\cal W}{\cal H}odge}(M)$. Introduce also the category ${{\cal W}{\cal H}odge}^\D(M)$ with the same objects as ${{\cal W}{\cal H}odge}(M)$ but with weakly Hodge differential operators as morphisms. Both the categories ${{\cal W}{\cal H}odge}(M)$ and ${{\cal W}{\cal H}odge}^\D(M)$ are additive $\R$-linear tensor categories. \refstepcounter{subsubsection\label{H-type} For a weakly Hodge map $f:\E \to \F$ call the canonical decomposition $$ f = \sum_{0 \leq i \leq m-n} f_i $$ {\em the $H$-type decomposition}. \label{gamma.m} For a Hodge bundle $\E$ on $M$ of non-negative weight $k$ let $\Gamma(\E) = \E \otimes \W_k^*$, where $\W_k$ is the canonical pure $\R$-Hodges structure introduced in \ref{w.k}. The universal properties of the Hodge structures $\W_k$ and Lemma~\ref{g.ex} generalize immediately to Hodge bundles. In particular, $\Gamma$ extends to a functor $\Gamma:{{\cal W}{\cal H}odge}_{\geq 0}(M) \to {{\cal W}{\cal H}odge}_0(M)$ adjoint on the right to the canonical embedding. \refstepcounter{subsubsection \label{u.trivial} If the $U(1)$-action on $M$ is trivial, then Hodge bundles of weight $i$ are the same as real vector bundles $\E$ equipped with a Hodge type grading $\E = \oplus_{p+q=i} \E^{p,q}$ on the complexification $\E_\C = \E \otimes_\R \C$. In particular, if $M={\operatorname{pt}}$ is a single point, then ${{\cal W}{\cal H}odge}(M) \cong {{\cal W}{\cal H}odge}^\D(M)$ is the category of pure $\R$-Hodge structures. Weakly Hodge bundle map are then the same as weakly Hodge maps of $\R$-Hodge structures considered in \ref{weakly.hodge}. (Thus the notion of a Hodge bundle is indeed a generalization of the notion of a pure $\R$-Hodge structure.) \refstepcounter{subsubsection The categories of Hodge bundles are functorial in $M$, namely, for every smooth map $f:M_1 \to M_2$ of smooth $U(1)$-manifolds $M_1$, $M_2$ there exist pull-back functors \begin{align*} f^*:{{\cal W}{\cal H}odge}(M_1) &\to {{\cal W}{\cal H}odge}(M_1) \\ f^*:{{\cal W}{\cal H}odge}^\D(M_1) &\to {{\cal W}{\cal H}odge}^\D(M_1). \end{align*} In particular, let $M$ be a smooth $U(1)$-manifold and let $\pi:M \to {\operatorname{pt}}$ be the canonical projection. Then every $\R$-Hodge structure $V$ of weight $i$ defines a constant Hodge bundle $\pi^*V$ on $M$, which we denote for simplicity by the same letter $V$. Thus the trivial bundle $\R = \Lambda^0(M) = \pi^*\R(0)$ has a natural structure of a Hodge bundle of weight $0$. \refstepcounter{subsubsection \label{de.Rham} To give a first example of Hodge bundles and weakly Hodge maps, consider a $U(1)$-manifold $M$ equipped with a $U(1)$-invariant almost complex structure $M_I$. Let $\Lambda^i(M,\C) = \oplus \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},i-{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_I)$ be the usual Hodge type decomposition of the bundles $\Lambda^i(M,\C)$ of complex valued differential forms on $M$. The complex vector bundles $\Lambda^{p,q}(M_I)$ are naturally $U(1)$-equivariant. Let $$ \overline{\ }:\Lambda^{p,q}(M_I) \to \iota^*\overline{\Lambda^{q,p}(M_I)} $$ be the usual complex conjugation, and introduce a $U(1)$-equivariant structure on $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ by setting $$ \Lambda^i(M,\C) = \bigoplus_{0 \leq j \leq i} \Lambda^{j,i-j}(M)(j). $$ The bundle $\Lambda^i(M,\C)$ with these $U(1)$-equivariant structure and complex conjugation is a Hodge bundle of weight $i$ on $M$. The de Rham differential $d_M$ is weakly Hodge, and the $H$-type decomposition for $d_M$ is in this case the usual Hodge type decomposition $d = \partial + \bar\partial$. \refstepcounter{subsubsection \noindent {\bf Remark.\ } Definition~\ref{hodge.bundles} is somewhat technical. It can be heuristically rephrased as follows. For a complex vector bundle $\E$ on $M$ the space of smooth global section $C^\infty(M,\E)$ is a module over the algebra $C^\infty(M,\C)$ of smooth $\C$-valued functions on $M$, and the bundle $\E$ is completely defined by the module $C^\infty(M,\E)$. The $U(1)$-action on $M$ induces a representation of $U(1)$ on the algebra $C^\infty(M,\C)$. Let $\nu:C^\infty(M,\C) \to C^\infty(M,\C)$ be composition of the complex conjugation and the map $\iota^*:C^\infty(M,\C) \to C^\infty(M,\C)$. The map $\nu$ is an anti-complex involution; together with the $U(1)$-action it defines a pure $\R$-Hodge structure of weight $0$ on the algebra $C^\infty(M,\C)$. Giving a weight $i$ Hodge bundle structure on $\E$ is then equivalent to giving a weight $i$ pure $\R$-Hodge structure on the module $C^\infty(M,\E)$ such that the multiplication map $$ C^\infty(M,\C) \otimes C^\infty(M,\E) \to C^\infty(M,\E) $$ is a map of $\R$-Hodge structures. \subsection{Equivariant quaternionic manifolds} \refstepcounter{subsubsection We now turn to our main subject, namely, various dif\-feren\-ti\-al\--ge\-o\-met\-ric structures on smooth manifolds associated to actions of the algebra ${\Bbb H}$ of quaternions. \refstepcounter{subsubsection\noindent {\bf Definition.\ } A smooth manifold $M$ is called {\em quaternionic} if it is equipped with a smooth action of the algebra ${\Bbb H}$ on its cotangent bundle $\Lambda^1(M)$. Let $M$ be a quaternionic manifold. Every algebra embedding $J:\C \to {\Bbb H}$ defines by restriction an almost complex structure on the manifold $M$. Denote this almost complex structure by $M_J$. \refstepcounter{subsubsection Assume that the manifold $M$ is equipped with a smooth action of the group $U(1)$, and consider the standard action of $U(1)$ on the vector space ${\Bbb H}$. Call the quaternionic structure and the $U(1)$-action on $M$ {\em compatible} if the action map $$ {\Bbb H} \otimes_\R \Lambda^1(M) \to \Lambda^1(M) $$ is $U(1)$-equivariant. Equivalently, the quaternionic structure and the $U(1)$-action are compatible if the action preserves the almost complex structure $M_I$, and the action map $$ {\Bbb H} \otimes_\C \Lambda^{1,0}(M_I) \to \Lambda^{1,0}(M_I) $$ is $U(1)$-equivariant. \refstepcounter{subsubsection \noindent {\bf Definition.\ } A quaternionic manifold $M$ equipped with a compatible smooth $U(1)$-action is called {\em an equivariant quaternionic manifold}. For a $U(1)$-equivariant complex vector bundle $\E$ on $M$ denote by $\E(k)$ the bundle $\E$ with $U(1)$-equivariant structure twisted by the $1$-dimensional representation of weight $k$. Lemma~\ref{explicit.eqvs} immediately gives the following. \begin{lemma}\label{explicit.qm} The category of quaternionic manifolds is equivalent to the category of pairs $\langle M_I, j \rangle$ of an almost complex manifold $M_I$ and a $\C$-linear $U(1)$-equivariant smooth map $j:\Lambda^{0,1}(M_I)(1) \to \Lambda^{0,1}(M_I)$ satisfying $j \circ \overline{j} = -{\sf id}$. \end{lemma} \subsection{Quaternionic manifolds and Hodge bundles} \refstepcounter{subsubsection Let $M$ be a smooth $U(1)$-manifold. Recall that we have introduced in Subsection~\ref{hb.sub} a notion of a Hodge bundle on $M$. Hodge bundles arise naturally in the study of quaternionic structures on $M$ in the following way. Define a {\em quaternionic bundle} on $M$ as a real vector bundle $\E$ equipped with a left action of the algebra ${\Bbb H}$, and let $\Bun(M,{\Bbb H})$ be the category of smooth quaternionic vector bundles on the manifold $M$. Let also $\Bun^{U(1)}(M,{\Bbb H})$ be the category of smooth quaternionic bundles $\E$ on $M$ equipped with a $U(1)$-equivariant structure such that the ${\Bbb H}$-action map ${\Bbb H} \to {{\cal E}\!nd\:}(\E)$ is $U(1)$-equivariant. Lemma~\ref{eqvs.hodge} immediately generalizes to give the following. \begin{lemma}\label{eqb.hodge} The category $\Bun^{U(1)}(M,{\Bbb H})$ is equivalent to the category of Hodge bundles of weight $1$ on $M$. \end{lemma} \refstepcounter{subsubsection Note that if the $U(1)$-manifold $M$ is equipped with an almost complex structure, then the decomposition ${\Bbb H} = \overline{\C} \oplus I(\C)$ (see \ref{h=c+c}) induces an isomorphism $$ {\sf can}:{\Bbb H} \otimes_{I(\C)} \Lambda^{0,1} \cong \Lambda^{1,0}(M) \oplus \Lambda^{0,1}(M) \cong \Lambda^1(M,\C). $$ The weight $1$ Hodge bundle structure on $\Lambda^1(M,\C)$ corresponding to the natural quaternionic structure on ${\Bbb H} \otimes_{I(\C)} \Lambda^{0,1}(M)$ is the same as the one considered in \ref{de.Rham}. \refstepcounter{subsubsection Assume now that the smooth $U(1)$-manifold $M$ is equipped with a compatible quaternionic structure, and let $M_I$ be the preferred almost complex structure on $M$. Since $M_I$ is preserved by the $U(1)$-action on $M$, the complex vector bundle $\Lambda^{0,1}(M_I)$ of $(0,1)$-forms on $M_I$ is naturally $U(1)$-equivariant. The quaternionic structure on $\Lambda^1(M)$ induces by Lemma~\ref{eqb.hodge} a weight-$1$ Hodge bundle structure on $\Lambda^{0,1}(M_I)$. The corresponding $U(1)$-action on $\Lambda^{0,1}(M_I)$ is induced by the action on $M_I$, and the real structure map $$ \overline{\ }:\Lambda^{1,0}(M_I)(1) \to \Lambda^{0,1}(M_I) $$ is given by $\overline{\ } = \sqrt{-1} \left( \iota^* \circ j \right)$. (Here $j$ is induced by quaternionic structure, as in Lemma~\ref{explicit.qm}). \refstepcounter{subsubsection \label{lambda_j=lambda_i} Let $M_J$ be the complementary almost complex structure on the equi\-va\-ri\-ant quaternionic manifold $M$. Recall that in \ref{complementary} we have defined for every equivariant quaternionic vector space $V$ the standard isomorphism $H:V_J \to V_I$. This construction can be immediately generalized to give a complex bundle isomorphism $$ H:\Lambda^{0,1}(M_J) \to \Lambda^{0,1}(M_I). $$ Let $P:\Lambda^1(M,\C) \to \Lambda^{0,1}(M_J)$ be the natural projection, and let $\Mult:{\Bbb H} \otimes_{I(\C)} \Lambda^{0,1}(M_I) \to \Lambda^{0,1}(M_I)$ be the action map. By definition the diagram $$ \begin{CD} \Lambda^1(M,\C) @>{{\sf can}}>> {\Bbb H} \otimes_{I(\C)} \Lambda^{0,1}(M_I) \\ @V{P}VV @V{\Mult}VV \\ \Lambda^{0,1}(M_J) @>{H}>> \Lambda^{0,1}(M_I) \end{CD} $$ is commutative. Since the map $\Mult$ is compatible with the Hodge bundle structures, so is the projection $P$. \noindent {\bf Remark.\ } This may seems paradoxical, since the complex conjugation $\overline{\ }$ on $\Lambda^1(M,\C)$ does not preserve $\Ker P = \Lambda^{0,1}(M_J)$. However, under our definition of a Hodge bundle the conjugation on $\Lambda^1(M,\C)$ is $\iota^* \circ \overline{\ }$ rather than $\overline{\ }$. Both $\overline{\ }$ and $\iota^*$ interchange $\Lambda^{1,0}(M_J)$ and $\Lambda^{0,1}(M_J)$. \refstepcounter{subsubsection The standard isomorphism $H:\Lambda^{0,1}(M_J) \to \Lambda^{0,1}(M_I)$ does not commute with the Dolbeault differentials. They are, however, related by means of the Hodge bundle structure on $\Lambda^{0,1}(M_I)$. Namely, we have the following. \begin{lemma}\label{d=d_0+d_0} The Dolbeault differential $D:\Lambda^0(M,\C) \to \Lambda^{0,1}(M_I)$ for the almost complex structure $M_J$ is weakly Hodge. The $U(1)$-invariant component $D_0$ in the $H$-type decomposition $D = D_0 + \overline{D_0}$ of the map $D$ coincides with the Dolbeault differential for the almost complex structure $M_I$. \end{lemma} \proof The differential $D$ is the composition $D = P \circ d_M$ of the de Rham differential $d_M:\Lambda^0(M,\C) \to \Lambda^1(M,\C)$ with the canonical projection $P$. Since both $P$ and $d_M$ are weakly Hodge, so is $D$. The rest follows from the construction of the standard isomorphism $H$. \hfill \ensuremath{\square}\par \subsection{Holonomic derivations} \refstepcounter{subsubsection Let $M$ be a smooth $U(1)$-manifold. In order to give a Hodge-theoretic description of the set of all equivariant quaternionic structures on $M$, it is convenient to work not with various complex structures on $M$, but with associated Dolbeault differentials. To do this, recall the following universal property of the cotangent bundle $\Lambda^1(M)$. \begin{lemma}\label{universal} Let $M$ be a smooth manifold, and let $\E$ be a complex vector bundle on $M$. Every differential operator $\partial:\Lambda^0(M,\C) \to \E$ which is a derivation with respect to the algebra structure on $\Lambda^0(M,\C)$ factors uniquely through the de Rham differential $d_M:\Lambda^0(M,\C) \to \Lambda^1(M,\C)$ by means of a bundle map $P:\Lambda^1(M,\C) \to \E$. \end{lemma} \refstepcounter{subsubsection \label{holonomic} We first use this universal property to describe almost complex structures. Let $M$ be a smooth manifold equipped with a complex vector bundle $\E$. \noindent {\bf Definition.\ } A derivation $D:\Lambda^0(M,\C) \to \E$ is called {\em holonomic} if the associated bundle map $P:\Lambda^1(M,\C) \to \E$ induces an isomorphism of the subbundle $\Lambda^1(M,\R) \subset \Lambda^1(M,\C)$ of real $1$-forms with the real vector bundle underlying $\E$. By Lemma~\ref{universal} the correspondence $$ M_I \mapsto \left\langle \Lambda^{0,1}(M_I), \bar\partial \right\rangle $$ identifies the set of all almost complex structures $M_I$ on $M$ with the set of all pairs $\langle \E, D \rangle$ of a complex vector bundle $\E$ and a holonomic derivation $D:\Lambda^0(M,\C) \to \E$. \refstepcounter{subsubsection Assume now that the smooth manifold $M$ is equipped with smooth action of the group $U(1)$. Then we have the following. \begin{lemma}\label{qm.hodge} Let $\E$ be a weight $1$ Hodge bundle on the smooth $U(1)$-manifold $M$, and let $$ D:\Lambda^0(M,\C) \to \E $$ be a weakly Hodge holonomic derivation. There exists a unique $U(1)$\--equi\-va\-ri\-ant quaternionic structure on $M$ such that $\E \cong \Lambda^{0,1}(M_J)$ and $D$ is the Dolbeault differential for the complementary almost complex structures $M_J$ on $M$. \end{lemma} \proof Since the derivation $M$ is holonomic, it induces an almost complex structure $M_J$ on $M$. To construct an almost complex structure $M_I$ complementary to $M_J$, consider the $H$-type decomposition $D = D_0 + \overline{D_0}$ of the derivation $D:\Lambda^0(M,\C) \to \E$ (defined in \ref{H-type}). The map $D_0$ is also a derivation. Moreover, it is holonomic. Indeed, by dimension count it is enough to prove that the associated bundle map $P:\Lambda^1(M,\R) \to \E$ is injective. Since the bundle $\Lambda^1(M,\R)$ is generated by exact $1$-forms, it is enough to prove that any real valued function $f$ on $M$ with $D_0f = 0$ is constant. However, since $D$ is weakly Hodge, $$ Df = D_0 f + \overline{D_0} f = D_0 f + \overline{ D_0 \overline{f} } = D_0 f + \overline{ D_0 f}, $$ hence $D_0f = 0$ if and only if $Df=0$. Since $D$ is holonomic, $f$ is indeed constant. The derivation $D_0$, being holonomic, is the Dolbeault differential for an almost complex structure $M_I$ on $M$. Since $D_0$ is by definition $U(1)$-equivariant, the almost complex structure $M_I$ is $U(1)$-invariant. Moreover, $\E \cong \Lambda^{0,1}(M_I)$ as $U(1)$-equivariant complex vector bundles. By Lemma~\ref{eqb.hodge} the weight $1$ Hodge bundle structure on $\E$ induces an equivariant quaternionic bundle structure on $\E$ and, in turn, a structure of an equivariant quaternionic manifold on $M$. The almost complex structure $M_I$ coincides by definition with the preferred almost complex structure. It remains to notice that by Lemma~\ref{d=d_0+d_0} the Dolbeault differential $\bar\partial_J$ for the complementary almost complex structure on $M$ indeed equals $D = D_0 + \overline{D_0}$. \hfill \ensuremath{\square}\par Together with Lemma~\ref{d=d_0+d_0}, this shows that the set of equivariant quaternionic structures on the $U(1)$-manifold $M$ is naturally bijective to the set of pairs $\langle \E, D \rangle$ of a weight $1$ Hodge bundle $\E$ on $M$ and a weakly Hodge holonomic derivation $D:\Lambda^0(M,\C) \to \E$. \section{Hodge manifolds} \subsection{Integrability} \refstepcounter{subsubsection There exists a notion of integrability for quaternionic manifolds analogous to that for the almost complex ones. \noindent {\bf Definition.\ } A quaternionic manifold $M$ is called {\em hypercomplex} if for two complementary algebra embeddings $I,J:\C \to {\Bbb H}$ the almost complex structures $M_I,M_J$ are integrable. In fact, if $M$ is hypercomplex, then $M_I$ is integrable for any algebra embedding $I:\C \to {\Bbb H}$. For a proof see, e.g., \cite{K}. \refstepcounter{subsubsection When a quaternionic manifold $M$ is equipped with a compatible $U(1)$-action, there exist a natural choice for a pair of almost complex structures on $M$, namely, the preferred and the complementary one. \noindent {\bf Definition.\ } An equivariant quaternionic manifold $M$ is called a {\em Hodge manifold} if both the preferred and the equivariant almost complex structures $M_I$, $M_J$ are integrable. Hodge manifolds are the main object of study in this paper. \refstepcounter{subsubsection \label{standard} There exists a simple Hodge-theoretic description of Hodge manifolds based on Lemma~\ref{qm.hodge}. To give it (see Proposition~\ref{explicit.hodge}), consider an equivariant quaternionic manifold $M$, and let $M_J$ and $M_I$ be the complementary and the preferred complex structures on $M$. The weight $1$ Hodge bundle structure on $\Lambda^{0,1}(M_J)$ induces a weight $i$ Hodge bundle structure on the bundle $\Lambda^{0,i}(M_J)$ of $(0,i)$-forms on $M_J$. The standard identification $H:\Lambda^{0,1}(M_J) \to \Lambda^{0,1}(M_I)$ given in \ref{lambda_j=lambda_i} extends uniquely to an algebra isomorphism $H:\Lambda^{0,i}(M_J) \to \Lambda^{0,i}(M_I)$. Let $D:\Lambda^{0{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J) \to \Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M_J)$ be the Dolbeault differential for the almost complex manifold $M_J$. \begin{lemma}\label{yet.another.lemma} The equivariant quaternionic manifold $M$ is Hodge if and only if the following holds. \begin{enumerate} \item $M_J$ is integrable, that is, $D \circ D = 0$, and \item the differential $D:\Lambda^{0,i}(M_J) \to \Lambda^{0,i+1}(M_J)$ is weakly Hodge for every $i \geq 0$. \end{enumerate} \end{lemma} \proof Assume first that the conditions \thetag{i}, \thetag{ii} hold. Condition \thetag{i} means that the complementary almost complex structure $M_J$ is integrable. By \thetag{ii} the map $D$ is weakly Hodge. Let $D = D_0 + \overline{D_0}$ be the $H$-type decomposition. The map $D_0$ is an algebra derivation of $\Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_I)$. Moreover, by Lemma~\ref{d=d_0+d_0} the map $D_0:\Lambda^0(M,\C) \to \Lambda^{0,1}(M_J)$ is the Dolbeault differential $\bar\partial_I$ for the preferred almost complex structure $M_I$ on $M$. (Or, more precisely, is identified with $\bar\partial_I$ under the standard isomorphism $H$.) But the Dolbeault differential admits at most one extension to a derivation of the algebra $\Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J)$. Therefore $D_0 = \bar\partial_I$ everywhere. The composition $D_0 \circ D_0$ is the $(2,0)$-component in the $H$-type decomposition of the map $D \circ D$. Since $D \circ D = 0$, $$ D_0 \circ D_0 = \bar\partial_I \circ \bar\partial_I = 0. $$ Therefore the preferred complex structure $M_I$ is also integrable, and the manifold $M$ is indeed Hodge. Assume now that $M$ is Hodge. The canonical projection $P:\Lambda^1(M,\C) \to \Lambda^{0,1}(M_J)$ extends then to a multiplicative projection $$ P:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J) $$ from the de Rham complex of the complex manifold $M_I$ to the Dolbeault complex of the complex manifold $M_J$. The map $P$ is surjective and weakly Hodge, moreover, it commutes with the differentials. Since the de Rham differential preserves the pre-Hodge structures, so does the Dolbeault differential $D$. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection Lemma~\ref{yet.another.lemma} and Lemma~\ref{qm.hodge} together immediately give the following. \begin{prop}\label{explicit.hodge} The category of Hodge manifolds is equivalent to the category of triples $\langle M, \E, D \rangle$ of a smooth $U(1)$-manifold $M$, a weight $1$ Hodge bundle $\E$ on $M$, and a weakly Hodge algebra derivation $$ D = D^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(\E) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(\E) $$ such that $D \circ D = 0$, and the first component $$ D^0:\Lambda^0(M,\C) = \Lambda^0(\E) \to \E = \Lambda^1(\E) $$ is holonomic in the sense of \ref{holonomic}. \end{prop} \subsection{The de Rham complex of a Hodge manifold} \refstepcounter{subsubsection Let $M$ be a Hodge manifold. In this subsection we study in some detail the de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the manifold $M$, in order to obtain information necessary for the discussion of metrics on $M$ given in the Subsection~\ref{polarization}. The reader is advised to skip this subsection until needed. \refstepcounter{subsubsection Let $\Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J)$ be the Dolbeault complex for the complementary complex structure $M_J$ on $M$. By Proposition~\ref{explicit.hodge} the complex vector bundle $\Lambda^{0,i}(M_J)$ is a Hodge bundle of weight $i$ on $M$, and the Dolbeault differential $D:\Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J) \to \Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M_J)$ is weakly Hodge. Therefore $D$ admits an $H$-type decomposition $D = D_0 + \overline{D_0}$. \refstepcounter{subsubsection Consider the de Rham complex $\Lambda^i(M,\C)$ of the smooth manifold $M$. Let $\Lambda^i(M,\C) = \oplus_{p+q} \Lambda^{p,q}(M_J)$ be the Hodge type decomposition for the complementary complex structure $M_J$ on $M$, and let $\nu:\Lambda^{p,q}(M_J) \to \overline{\Lambda^{q,p}(M_J)}$ be the usual complex conjugation. Denote also $$ f^\nu = \nu \circ f \circ \nu^{-1} $$ for any map $f:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. Let $d_M:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ be the de Rham differential, and let $d_M = D + D^\nu$ be the Hodge type decomposition for the complementary complex structure $M_J$ on $M$. Since the Dolbeault differential, in turn, equals $D = D_0 + \overline{D_0}$, we have $$ d_M = D_0 + \overline{D_0} + D_0^\nu + \overline{D_0}^\nu. $$ \refstepcounter{subsubsection \label{6_I} Let $\bar\partial_I:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ be the Dolbeault differential for the preferred complex structure $M_I$ on $M$. As shown in the proof of Lemma~\ref{yet.another.lemma}, the $(0,1)$-component of the differential $\bar\partial_I$ with respect to the complex structure $M_J$ equals $D_0$. Therefore the $(1,0)$-component of the complex-conjugate map $\partial_I = \bar\partial_I^\nu$ equals $D_0^\nu$. Since $d_M = \bar\partial_I + \partial_I$, we have \begin{align*} \bar\partial_I &= D_0 + \overline{D_0}^\nu\\ \partial_I &= \overline{D_0} + D_0^\nu \end{align*} \refstepcounter{subsubsection \label{6_I^H} The standard isomorphism $H:\Lambda^{0,1}(M_J) \to \Lambda^{0,1}(M_I)$ introduced in \ref{standard} extends uniquely to a bigraded algebra isomorphism $H:\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_I)$. By definition of the map $H$, on $\Lambda^0(M,\C)$ we have \begin{align}\label{identities} \begin{split} \bar\partial_I &= H \circ D_0 \circ H^{-1}\\ \partial_I &= H \circ D_0^\nu \circ H^{-1}\\ d_M &= \partial_I + \bar\partial_I = H \circ (D_0 + D_0^\nu) H^{-1}. \end{split} \end{align} The right hand side of the last identity is the algebra derivation of the de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. Therefore, by Lemma~\ref{universal} it holds not only on $\Lambda^0(M,\C)$, but on the whole $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. The Hodge type decomposition for the preferred complex structure $M_I$ then shows that the first two identities also hold on the whole de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. \refstepcounter{subsubsection Let now $\xi = I(\sqrt{-1}):\Lambda^{0,1}(M_J) \to \Lambda^{1,0}(M_J)$ be the operator corresponding to the preferred almost complex structure $M_I$ on $M$. Let also $\xi = 0$ on $\Lambda^0(M,\C)$ and $\Lambda^{1,0}(M_J)$, and extend $\xi$ to a derivation $\xi:\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}-1,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M_J)$ by the Leibnitz rule. We finish this subsection with the following simple fact. \begin{lemma}\label{xi.lemma} On $\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}(M_J) \subset \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ we have \begin{align}\label{xi.eq} \begin{split} \xi \circ D_0 + D_0 \circ \xi &= \overline{D}_0^\nu\\ \xi \circ \overline{D}_0 + \overline{D}_0 \circ \xi &= -D_0^\nu. \end{split} \end{align} \end{lemma} \proof It is easy to check that both identities hold on $\Lambda^0(M,\C)$. But both sides of these identities are algebra derivations of $\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}(M_J)$, and the right hand sides are holonomic in the sense of \ref{holonomic}. Therefore by Lemma~\ref{universal} both identities hold on the whole $\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}(M_J)$. \hfill \ensuremath{\square}\par \subsection{Polarized Hodge manifolds}\label{polarization} \refstepcounter{subsubsection Let $M$ be a quaternionic manifold. A Riemannian metric $h$ on $M$ is called {\em Qua\-ter\-ni\-onic-\-Her\-mi\-ti\-an} if for every point $m \in M$ the induces metric $h_m$ on the tangent bundle $T_mM$ is Qua\-ter\-ni\-onic\--Her\-mi\-ti\-an in the sense of \ref{hyperherm}. \noindent {\bf Definition.\ } A {\em hyperk\"ahler manifold} is a hypercomplex manifold $M$ equipped with a Quaternionic-Hermitian metric $h$ which is K\"ahler for both integrable almost complex structures $M_I$, $M_J$ on $M$. \noindent {\bf Remark.\ } In the usual definition (see, e.g., \cite{Bes}) the integrability of the almost complex structures $M_I$, $M_J$ is omitted, since it is automatically implied by the K\"ahler condition. \refstepcounter{subsubsection Let $M$ be a Hodge manifold equipped with a Riemannian metric $h$. The metric $h$ is called {\em Hermitian-Hodge} if it is Quaternionic-Hermitian and, in addition, invariant under the $U(1)$-action on $M$. \noindent {\bf Definition.\ } Say that the manifold $M$ is {\em polarized} by the Hermitian-Hodge metric $h$ if $h$ is not only Quaternionic-Hermitian, but also hyperk\"ahler. \refstepcounter{subsubsection \label{positive} Let $M$ be a Hodge manifold. By Proposition~\ref{explicit.hodge} the holomorphic cotangent bundle $\Lambda^{1,0}(M_J)$ for the complementary complex structure $M_J$ on $M$ is a Hodge bundle of weight $1$. The holomorphic tangent bundle $\Theta(M_J)$ is therefore a Hodge bundle of weight $-1$. By \ref{pol} the set of all Hermitian-Hodge metrics $h$ on $M$ is in natural one-to-one correspondence with the set of all polarizations on the Hodge bundle $\Theta(M_J)$. Since $\theta(M)$ is of odd weight, its polarizations are skew-symmetric as bilinear forms and correspond therefore to smooth $(2,0)$-forms on the complex manifold $M_J$. A $(2,0)$-form $\Omega$ defines a polarization on $\Theta(M_J)$ if and only if $\Omega \in C^\infty(M,\Lambda^{2,0}(M_J))$ considered as a map $$ \Omega:\R(-1) \to \Lambda^{2,0}(M_J) $$ is a map of weight $2$ Hodge bundles, and for an arbitrary smooth section $\chi \in C^\infty(M,\Theta(M_J))$ we have \begin{equation}\label{P} \Omega(\chi,\overline{\iota^*(\chi)}) > 0. \end{equation} \refstepcounter{subsubsection Assume that the Hodge manifold $M$ is equipped with an Hermitian-Hodge metric $h$. Let $\Omega \in C^\infty(M,\Lambda^{2,0}(M_J))$ be the corresponding polarization, and let $\omega_I \in C^\infty(M,\Lambda^{1,1}(M_I))$ be the $(1,1)$-form on the complex manifold $M_I$ associated to the Hermitian metric $h$. Either one of the forms $\Omega$, $\omega_I$ completely defines the metric $h$, and by \eqref{omega.and.Omega} we have $$ \Omega + \nu(\Omega) = \omega_I, $$ where $\nu:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is the complex conjugation. \begin{lemma}\label{pol.hm} The Hermitian-Hodge metric $h$ polarizes $M$ if and only if the corresponding $(2,0)$-form $\Omega$ on $M_J$ is holomorphic, that is, $$ D\Omega = 0, $$ where $D$ is the Dolbeault differential for complementary complex structure $M_J$. \end{lemma} \proof Let $\omega_I,\omega_J \in \Lambda^2(M,\C)$ be the K\"ahler forms for the metric $h$ and complex structures $M_I$, $M_J$ on $M$. The metric $h$ is hyperk\"ahler, hence polarizes $M$, if and only if $d_M\omega_I = d_M\omega_J = 0$. Let $D = D_0 + \overline{D_0}$ be the $H$-type decomposition and let $H:\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J)$ be the standard algebra identification introduced in \ref{6_I^H}. By definition $H(\omega_I) = \omega_J$. Moreover, by \eqref{identities} $H^{-1} \circ d_M \circ H = D_0 + \overline{D_0}^\nu$, hence $$ H(d_M\omega_J) = D_0 \omega_I + \overline{D_0}^\nu\omega_I, $$ and the metric $h$ is hyperk\"ahler if and only if \begin{equation}\label{K} d_M\omega_I = (D_0+\overline{D_0}^\nu)\omega_I = 0 \end{equation} But $2\omega_I = \Omega + \nu(\Omega)$. Since $\Omega$ is of Hodge type $(2,0)$ with respect to the complementary complex structure $M_J$, \eqref{K} is equivalent to $$ \overline{D_0} \Omega = D_0 \Omega = \overline{D_0}^\nu \Omega = D_0^\nu \Omega = 0. $$ Moreover, let $\xi$ be as in Lemma~\ref{xi.lemma}. Then $\xi(\Omega) = 0$, and by \eqref{xi.eq} $D_0\Omega = \overline{D}_0\Omega = 0$ implies that $D_0^\nu\Omega = \overline{D}_0^\nu\Omega = 0$ as well. It remains to notice that since the metric $h$ is Hermitian-Hodge, $\Omega$ is of $H$-type $(1,1)$ as a section of the weight $2$ Hodge bundle $\Lambda^{2,0}(M_J)$. Therefore $D\Omega = 0$ implies $\overline{D_0}\Omega = D_0\Omega = 0$, and this proves the lemma. \hfill \ensuremath{\square}\par \noindent {\bf Remark.\ } This statement is wrong for general hyperk\"ahler manifolds (eve\-ry\-thing in the given proof carries over, except for the implication $D\Omega=0 \Rightarrow D_0\Omega=\overline{D_0}\Omega=\overline{D_0}^\nu \Omega = D_0^\nu \Omega = 0$, which depends substantially on the $U(1)$-action). To describe general hyperk\"ahler metrics in holomorphic terms, one has to introduce the so-called {\em twistor spaces} (see, e.g., \cite{HKLR}). \section{Regular Hodge manifolds} \subsection{Regular stable points} \refstepcounter{subsubsection Let $M$ be a smooth manifold equipped with a smooth $U(1)$-action with differential $\phi_M$ (thus $\phi_M$ is a smooth vector field on $M$). Since the group $U(1)$ is compact, the subset $M^{U(1)} \subset M$ of points fixed under $U(1)$ is a smooth submanifold. Let $m \in M^{U(1)} \subset M$ be a point fixed under $U(1)$. Consider the representation of $U(1)$ on the tangent space $T_m$ to $M$ at $m$. Call the fixed point $m$ {\em regular} if every irreducible subrepresentation of $T_m$ is either trivial or isomorphic to the representation on $\C$ given by embedding $U(1) \subset \C^*$. (Here $\C$ is considered as a $2$-dimensional real vector space.) Regular fixed points form a union of connected component of the smooth submanifold $M^{U(1)} \subset M$. \refstepcounter{subsubsection Assume that $M$ is equipped with a complex structure preserved by the $U(1)$-action. Call a point $m \in M$ {\em stable} if for any $t \in \R, t \geq 0$ there exists $\exp(\sqrt{-1}t\phi_M)m$, and the limit $$ m_0 \in M, m_0 = \lim_{t \to +\infty} \exp(\sqrt{-1}t\phi_M)m $$ also exists. \enlargethispage{10mm} \refstepcounter{subsubsection For every stable point $m \in M$ the limit $m_0$ is obviously fixed under $U(1)$. Call a point $m \in M$ {\em regular stable} if it is stable and the limit $m_0 \in M^{U(1)}$ is a regular fixed point. Denote by $M^{reg} \subset M$ the subset of all regular stable points. The subset $M^{reg}$ is open in $M$. \noindent {\bf Example.\ } Let $Y$ be a complex manifold with a holomorphic bundle $\E$ and let $E$ be the total space of $\E$. Let $\C^*$ act on $M$ by dilatation along the fibers. Then every point $e \in E$ is regular stable. \refstepcounter{subsubsection Let $M$ be a Hodge manifold. Recall that the $U(1)$-action on $M$ preserves the preferred complex structure $M_I$. \noindent {\bf Definition.\ } A Hodge manifold $M$ is called {\em regular} if $M_I^{reg} = M_I$. \subsection{Linearization of regular Hodge manifolds} \refstepcounter{subsubsection Consider a regular Hodge manifold $M$. Let $\Delta \subset \C$ be the unit disk equipped with the multiplicative semigroup structure. The group $U(1) \subset \Delta$ is embedded into $\Delta$ as the boundary circle. \begin{lemma}\label{regular} The action $a:U(1) \times M \to M$ extends uniquely to a holomorphic action $\tilde{a}:\Delta \times M_I \to M_I$. Moreover, for every $x \in \Delta \setminus \{0\}$ the action map $\wt{a}(x):M_I \to M_I$ is an open embedding. \end{lemma} \proof Since $M$ is regular, the exponential flow $\exp(it\phi_M)$ of the differential $\phi_M$ of the action is defined for all positive $t \in \R$. Therefore $a:U(1) \times M \to M$ extends uniquely to a holomorphic action $$ \tilde{a}:\Delta^* \times M_I \to M_I, $$ where $\Delta^* = \Delta \backslash \{0\}$ is the punctured disk. Moreover, the exponential flow converges as $t \to +\infty$, therefore $\tilde{a}$ extends to $\Delta \times M_I$ continuously. Since this extension is holomorphic on a dense open subset, it is holomorphic everywhere. This proves the first claim. To prove the second claim, consider the subset $\wt{\Delta} \subset \Delta^*$ of points $x \in \Delta$ such that $\wt{a}(x)$ is injective and \'etale. The subset $\wt{\Delta}$ is closed under multiplication and contains the unit circle $U(1) \subset \Delta^*$. Therefore to prove that $\wt{\Delta} = \Delta^*$, it suffices to prove that $\wt{\Delta}$ contains the interval $]0,1] \subset \Delta^*$. By definition we have $\wt{a}(h) = \exp(-\sqrt{-1}\log h \phi_M)$ for every $h \in ]0,1] \subset \Delta^*$. Thus we have to prove that if for some $t \in \R, t \geq 0$ and for two points $m_1,m_2 \in M$ we have $$ \exp(\sqrt{-1}t\phi_M)(m_1) = \exp(\sqrt{-1}t\phi_M)(m_2), $$ then $m_1 = m_2$. Let $m_1$, $m_2$ be such two points and let $$ t = \inf\{t \in \R, t \geq 0, \exp(\sqrt{-1}t\phi_M)(m_1) = \exp(\sqrt{-1}t\phi_M)(m_2)\}. $$ If the point $m_0 = \exp(\sqrt{-1}t\phi_M)(m_1) = \exp(\sqrt{-1}t\phi_M)(m_2) \in M$ is not $U(1)$-invariant, then it is a regular point for the vector field $\sqrt{-1}\phi_M$, and by the theory of ordinary differential equations we have $t = 0$ and $m_1 = m_2 = m_0$. Assume therefore that $m_0 \in M^{U(1)}$ is $U(1)$-invariant. Since the group $U(1)$ is compact, the vector field $\sqrt{-1}\phi_M$ has only a simple zero at $m_0 \subset M^{U(1)} \subset M$. Therefore $m_0 = \exp(\sqrt{-1}t\phi_M)m_1$ implies that the point $m_1 \in M$ also is $U(1)$-invariant, and the same is true for the point $m_2 \in M$. But $\wt{a}(\exp t)$ acts by identity on $M^{U(1)} \subset M$. Therefore in this case we also have $m_1=m_2=m_0$. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection Let $V = M_I^{U(1)} \subset M_I$ be the submanifold of fixed points of the $U(1)$ action. Since the action preserves the complex structure on $M_I$, the submanifold $V$ is complex. \begin{lemma} There exists a unique $U(1)$-invariant holomorphic map $$ \rho_M:M_I \to V $$ such that $\rho_M|_{V} = {\sf id}$. \end{lemma} \proof For every point $m \in M$ we must have $\rho_M(m) = \displaystyle\lim_{t \to +\infty} \exp(i t \phi_M)$, which proves uniqueness. To prove that $\rho_M$ thus defined is indeed holomorphic, notice that the diagram $$ \begin{CD} M_I @>{0 \times {\sf id}}>> \Delta \times M\\ @V{\rho_M}VV @VV{\tilde{a}}V\\ V @>>> M_I \end{CD} $$ is commutative. Since the action $\tilde{a}:\Delta \times M_I \to M_I$ is holomorphic, so is the map $\rho_M$. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection Call the canonical map $\rho_M:M_I \to M_I^{U(1)}$ the {\em canonical projection} of the regular Hodge manifold $M$ onto the submanifold $V \subset M$ of fixed points. \begin{lemma} The canonical projection $\rho_M:M \to M^{U(1)}$ is submersive, that is, for every point $m \in M$ the differential $d\rho_M:T_mM \to T_{\rho(m)}M^{U(1)}$ of the map $\rho_M$ at $m$ is surjective. \end{lemma} \proof Since $\rho_M|_{M^{U(1)}} = {\sf id}$, the differential $d\rho_M$ is surjective at points $m \in V \subset M$. Therefore it is surjective on an open neighborhood $U \supset V$ of $V$ in $M$. For any point $m \in M$ there exists a point $x \in \Delta$ such that $x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} m \in U$. Since $\rho_M$ is $\Delta$-invariant, this implies that $d\rho_M$ is surjective everywhere on $M$. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection Let $\Theta(M/V)$ be the relative tangent bundle of the holomorphic map $\rho:M \to V$. Let $\Theta(M)$ and $\Theta(V)$ be the tangent bundles of $M$ and $V$ and consider the canonical exact sequence of complex bundles $$ 0 \longrightarrow \Theta(M/V) \longrightarrow \Theta(M) \overset{d\rho_M}{\longrightarrow} \rho^*\Theta(V) \longrightarrow 0, $$ where $d\rho_M$ is the differential of the projection $\rho_M:M \to V$. The quaternionic structure on $M$ defines a $\C$-linear map $j:\Theta(M) \to \overline{\Theta}(M)$. Restricting to $\Theta(M/V)$ and composing with $d\rho_M$, we obtain a $\C$-linear map $j:\Theta(M/V) \to \rho^*\overline{\Theta}(V)$. \refstepcounter{subsubsection \label{overline.T} Let ${\overline{T}V}$ be the total space of the bundle $\overline{\Theta}(V)$ complex-conjugate to the tangent bundle $\Theta(V)$, and let $\rho:{\overline{T}V} \to V$ be the projection. Let the group $U(1)$ act on ${\overline{T}V}$ by dilatation along the fibers of the projection $\rho$. Since the canonical projection $\rho_M:M \to V$ is $U(1)$-invariant, the differential $\phi_M$ of the $U(1)$-action defines a section $$ \phi_M \in C^\infty(M,\Theta(V/M)). $$ The section $j(\phi_M) \in C^\infty(M,\rho_M^*\overline{\Theta}(V))$ defines a map $\Lin_M:M \to {\overline{T}V}$ such that $\Lin_M \circ \rho = \rho_M:M \to V$. Call the map $\Lin_M$ {\em the linearization} of the regular Hodge manifold $M$. \begin{prop}\label{linearization} The linearization map $\Lin_M$ is a $U(1)$-equivariant open embedding. \end{prop} \proof The map $j:\Theta(M/V) \to \rho^*\overline{\Theta}(V)$ is of degree $1$ with respect to the $U(1)$-action, while the section $\phi_M \in C^\infty(M,\Theta(M/V))$ is $U(1)$-invariant. Therefore the map $\Lin_M$ is $U(1)$-equivariant. Consider the differential $d\Lin_M:T_m(M) \to T_m({\overline{T}V})$ at a point $m \in V \subset M$. We have $$ d\Lin_M = d\rho_M \oplus d\rho_M \circ j:T_m(M) \to T_m(V) \oplus \overline{T}_m(V) $$ with respect to the decomposition $T_m({\overline{T}V}) = T_m(V) \oplus \overline{T}(V)$. The tangent space $T_m$ is a regular quaternionic vector space. Therefore the map $d\Lin_M$ is bijective at $m$ by Lemma~\ref{regular.quaternionic}. Since $\Lin_M$ is bijective on $V$, this implies that $\Lin_M$ is an open embedding on an open neighborhood $U \subset M$ of the submanifold $V \subset M$. To finish the proof of proposition, it suffices prove that the linearization map $\Lin_M:M_I \to {\overline{T}V}$ is injective and \'etale on the whole $M_I$. To prove injectivity, consider arbitrary two points $m_1,m_2 \in M_I$ such that $\Lin_M(m_1)=\Lin_M(m_2)$. There exists a point $x \in \Delta \setminus \{0\}$ such that $x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} m_1, x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} m_2 \in U$. The map $\Lin_M$ is $U(1)$-equivariant and holomorphic, therefore it is $\Delta$-equivariant, and we have $$ \Lin_M(x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} m_1) = x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} \Lin_M(m_1) = x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} \Lin_M(m_2) = \Lin_M(x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} m_2). $$ Since the map $\Lin_M:U \to {\overline{T}V}$ is injective, this implies that $x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} m_1 = x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} m_2$. By Lemma~\ref{regular} the action map $x:M_I \to M_I$ is injective. Therefore this is possible only if $m_1 = m_2$, which proves injectivity. To prove that the linearization map is \'etale, note that by Lemma~\ref{regular} the action map $x:M_I \to M_I$ is not only injective, but also \'etale. Since $\Lin_M$ is \"etale on $U$, so is the composition $\Lin_M \circ x:M_I \to U \to {\overline{T}V}$ is \'etale. Since $\Lin_M \circ x = x \circ \Lin_M$, the map $\Lin_M:M_I \to {\overline{T}V}$ is \'etale at the point $m_1 \in M_I$. Thus the linearization map is also injective and \'etale on the whole of $M_I$. Hence it is indeed an open embedding, which proves the propostion. \hfill \ensuremath{\square}\par \subsection{Linear Hodge manifold structures} \refstepcounter{subsubsection By Proposition~\ref{linearization} every regular Hodge manifold $M$ admits a canonical open embedding $\Lin_M:M \to {\overline{T}V}$ into the total space ${\overline{T}V}$ of the (complex-conjugate) tangent bundle to its fixed points submanifold $V \subset M$. This embedding induces a Hodge manifold structure on a neighborhood of the zero section $V \subset {\overline{T}V}$. In order to use the linearization construction, we will need a characterization of all Hodge manifold structures on neighborhoods of $V \subset {\overline{T}V}$ obtained in this way (see \ref{lin.def}). It is convenient to begin with an invariant characterization of the linearization map $\Lin_M:M \to {\overline{T}V}$. \refstepcounter{subsubsection \label{tau} Let $V$ be an arbitrary complex manifold, let ${\overline{T}V}$ be the total space of the complex-conjugate to the tangent bundle $\Theta(V)$ to $V$, and let $\rho:{\overline{T}V} \to V$ be the canonical projection. Contraction with the tautological section of the bundle $\rho^*\overline{\Theta(V)}$ defines for every $p$ a bundle map $$ \tau:\rho^*\Lambda^{p+1}(V,\C) \to \rho^*\Lambda^{p}(V,\C), $$ which we call {\em the tautological map}. In particular, the induced map $$ \tau:C^\infty(V,\Lambda^{0,1}(V)) \to C^\infty({\overline{T}V},\C) $$ identifies the space $C^\infty(V,\Lambda^{0,1}(V))$ of smooth $(0,1)$-forms on $V$ with the subspace in $C^\infty({\overline{T}V},\C)$ of function linear along the fibers of the projection ${\overline{T}V} \to V$. \refstepcounter{subsubsection Let now $M$ be a Hodge manifold. Let $V \subset M_I$ be the complex submanifold of $U(1)$-fixed points, and let $\rho_M:M \to V$ be the canonical projection. Assume that $M$ is equipped with a smooth $U(1)$-equivariant map $f:M \to {\overline{T}V}$ such that $\rho_M = \rho \circ f$. Let $\bar\partial_I$ be the Dolbeault differential for the preferred complex structure $M_I$ on $M$, and let $\phi \in \Theta(M/V)$ be the differential of the $U(1)$-action on $M$. Let also $j:\Lambda^{0,1}(M_I) \to \Lambda^{1,0}(M_I)$ be the map induced by the quaternionic structure on $M$. \begin{lemma}\label{lin.char} The map $f:M \to {\overline{T}V}$ coincides with the linearization map if and only if for every $(0,1)$-form $\alpha \in C^\infty(V,\Lambda^{0,1}(V))$ we have \begin{equation}\label{L} f^*\tau(\alpha) = \langle \phi, j(\rho_M^*\alpha) \rangle. \end{equation} Moreover, if $f = \Lin_M$, then we have \begin{equation}\label{LL} f^*\tau(\beta) = \langle \phi, j(f^*\beta) \rangle \end{equation} for every smooth section $\beta \in C^\infty({\overline{T}V},\rho^*\Lambda^1(V,\C))$. \end{lemma} \proof Since functions on ${\overline{T}V}$ linear along the fibers separate points, the correspondence $$ f^* \circ \tau:C^\infty(V,\Lambda^{0,1}(V)) \to C^\infty(M,\C) $$ characterizes the map $f$ uniquely, which proves the ``only if'' part of the first claim. Since by assumption $\rho_M = \rho \circ f$, the equality \eqref{L} is a particular case of \eqref{LL} with $\beta = \rho^*\alpha$. Therefore the ``if'' part of the first claim follows from the second claim, which is a rewriting of the definition of the linearization map $\Lin_M:M \to {\overline{T}V}$ (see \ref{overline.T}). \hfill \ensuremath{\square}\par \refstepcounter{subsubsection Let now $\Lin_M:M \to {\overline{T}V}$ be the linearization map for the regular Hodge manifold $M$. Denote by $U \subset {\overline{T}V}$ the image of $\Lin_M$. The subset $U \subset {\overline{T}V}$ is open and $U(1)$-invariant. In addition, the isomorphism $\Lin_M:M \to U$ induces a regular Hodge manifold structure on $U$. Donote by $\Lin_U$ the linearization map for the regular Hodge manifold $U$. Lemma~\ref{lin.char} implies the following. \begin{corr}\label{lin.lin} We have $\Lin_M \circ \Lin_U = \Lin_M$, thus the linearization map $\Lin_U:U \to {\overline{T}V}$ coincides with the given embedding $U \hookrightarrow {\overline{T}V}$. \end{corr} \proof Let $\alpha \in C^\infty(V,\Lambda^{0,1}(V)$ be a $(0,1)$-form on $V$. By Lemma~\ref{lin.char} we have $\Lin_U^*\tau(\alpha) = \langle \phi_U, j_U(\rho_U^*\alpha)\rangle$, and it suffices to prove that $$ \Lin_M^*(\Lin_U^*(\tau(\alpha))) = \langle \phi_M, j_M(\rho_M^*\alpha) \rangle. $$ By definition we have $\rho_M = \rho_U \circ \Lin_M$. Moreover, the map $\Lin_M$ is $U(1)$-equivariant, therefore it sends $\phi_M$ to $\phi_U$. Finally, by definition it commutes with the quaternionic structure map $j$. Therefore $$ \Lin_M^*(\Lin_U^*(\tau(\alpha))) = \Lin_M^*(\langle \phi_U, j_U(\rho_U^*\alpha) \rangle)= \langle \phi_M, j_M(\rho_M^*\alpha) \rangle, $$ which proves the corollary. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection \noindent {\bf Definition.\ } \label{lin.def} Let $U \subset {\overline{T}V}$ be an open $U(1)$-invariant neighborhood of the zero section $V \subset {\overline{T}V}$. A Hodge manifold structure on ${\overline{T}V}$ is called {\em linear} if the associated linearization map $\Lin_U:U \to {\overline{T}V}$ coincides with the given embedding $U \hookrightarrow {\overline{T}V}$. By Corollary~\ref{lin.lin} every Hodge manifold structure on a subset $U \subset {\overline{T}V}$ obtained by the linearization construction is linear. \refstepcounter{subsubsection We finish this section with the following simple observation, which we will need in the next section. \begin{lemma}\label{aux2} Keep the notations of Lemma~\ref{lin.char}. Moreover, assume given a subspace $\A \subset C^\infty({\overline{T}V},\rho^*\Lambda^1(V,\C))$ such that the image of $\A$ under the restriction map $$ \Res:C^\infty({\overline{T}V},\rho^*\Lambda^1(V,\C)) \to C^\infty(V,\Lambda^1(V,\C)) $$ onto the zero section $V \subset {\overline{T}V}$ is the whole space $C^\infty(V,\Lambda^1(V,\C))$. If \eqref{LL} holds for every section $\beta \in \A$, then it holds for every smooh section $$ \beta \in C^\infty({\overline{T}V},\rho^*\Lambda^1(V,\C)). $$ \end{lemma} \proof By assumptions sections $\beta \in \A$ generate the restriction of the bundle $\rho^*\Lambda^1(V,\C)$ onto the zero section $V \subset {\overline{T}V}$. Therefore there exists an open neighborhood $U \subset {\overline{T}V}$ of the zero section $V \subset {\overline{T}V}$ such that the $C^\infty(U,\C)$-submodule $$ C^\infty(U,\C) {\:\raisebox{3pt}{\text{\circle*{1.5}}}} \A \subset C^\infty(U,\rho^*\Lambda^1(V,\C)) $$ is dense in the space $C^\infty(U,\rho^*\Lambda^1(V,\C))$ of smooth sections of the pull-back bundle $\rho^*\Lambda^1(V,\C)$. Since \eqref{LL} is continuous and linear with respect to multiplication by smooth functions, it holds for all sections $\beta \in C^\infty(U,\rho^*\Lambda^1(V,\C))$. Since it is also compatible with the natural unit disc action on $M$ and ${\overline{T}V}$, it holds for all sections $\beta \in C^\infty({\overline{T}V},\rho^*\Lambda^1(V,\C))$ as well. \hfill \ensuremath{\square}\par \section{Tangent bundles as Hodge manifolds}\label{section.5} \subsection{Hodge connections} \refstepcounter{subsubsection The linearization construction reduces the study of arbitrary regular Hodge manifolds to the study of linear Hodge manifold structures on a neighborhood $U \subset {\overline{T}V}$ of the zero section $V \subset {\overline{T}V}$ of the total space of the complex conjugate to the tangent bundle of a complex manifold $V$. In this section we use the theory of Hodge bundles developed in Subsection~\ref{hb.sub} in order to describe Hodge manifold structures on $U$ in terms of connections on the locally trivial fibration $U \to V$ of a certain type, which we call {\em Hodge connections} (see \ref{hodge.con}). It is this description, given in Proposition~\ref{equiv}, which we will use in the latter part of the paper to classify all such Hodge manifold structures. \refstepcounter{subsubsection \label{conn} We begin with some preliminary facts about connections on locally trivial fibrations. Let $f:X \to Y$ be an arbitrary smooth map of smooth manifolds $X$ and $Y$. Assume that the map $f$ is submersive, that is, the codifferential $\delta_f:f^*\Lambda^1(Y) \to \Lambda^1(X)$ is an injective bundle map. Recall that a {\em connection} on $f$ is by definition a splitting $\Theta:\Lambda^1(X) \to f^*\Lambda^1(Y)$ of the canonical embedding $\delta_f$. Let $d_X$ be the de Rham differential on the smooth manifold $X$. Every connection $\Theta$ on $f:X \to Y$ defines an algebra derivation $$ D = \Theta \circ d_X:\Lambda^0(X) \to f^*\Lambda^1(Y), $$ satisfying \begin{equation}\label{conn.eq} D \rho^* h = \rho^* d_Y h \end{equation} for every smooth function $h \in C^\infty(Y,\R)$. Vice versa, by the universal property of the cotangent bundle (Lemma~\ref{universal}) every algebra derivation $D:\Lambda^0(X) \to \Lambda^1(Y)$ satisfying \eqref{conn.eq} comes from a unique connection $\Theta$ on $f$. \refstepcounter{subsubsection Recall also that a connection $\Theta$ is called {\em flat} if the associated derivation $D$ extends to an algebra derivation $$ D:f^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(Y) \to f^*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(Y) $$ so that $D \circ D = 0$. The splitting $\Theta:\Lambda^1(X) \to f^*\Lambda^1(Y)$ extends in this case to an algebra map $$ \Theta:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(X) \to f^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(Y) $$ compatible with the de Rham differential $d_X:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(X) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(X)$. \refstepcounter{subsubsection We will need a slight generalization of the notion of connection. \noindent {\bf Definition.\ } Let $f:X \to Y$ be a smooth submersive morphism of complex manifolds. A {\em $\C$-valued connection} $\Theta$ on $f$ is a splitting $\Theta:\Lambda^1(Y,\C) \to f^*\Lambda^1(X,\C)$ of the codifferential map $\delta f:f^*\Lambda^1(Y,\C) \to \Lambda^1(X,\C)$ of complex vector bundles. A $\C$-valued connection $\Theta$ is called {\em flat} if the associated algebra derivation $$ D = \Theta \circ d_X:\Lambda^0(X,\C) \to f^*\Lambda^1(Y,\C) $$ extends to an algebra derivation $$ D:f^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(Y,\C) \to f^*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(Y,\C) $$ satisfying $D \circ D = 0$. As in \ref{conn}, every derivation $D:\Lambda^0(X,\C) \to f^*\Lambda^1(Y,\C)$ satisfying \eqref{conn.eq} comes from a unique $\C$-valued connection $\Theta$ on $f:X \to Y$. \noindent {\bf Remark.\ } By definition for every flat connection on $f:X \to Y$ the subbundle of horizontal vectors in the the tangent bundle $\Theta(X)$ is an involutive distribution. By Frobenius Theorem this implies that the connection defines locally a trivialization of the fibration $f$. This is no longer true for flat $\C$-valued connections: the subbundle of horizontal vectors in $\Theta(X) \otimes \C$ is only defined over $\C$, and the Frobenius Theorem does not apply. One can try to correct this by replacing the splitting $\Theta:\Lambda^1(X,\C) \to f^*\Lambda^1(Y,\C)$ with its real part $\Re\Theta:\Lambda^1(X) \to \Lambda^1(Y)$, but this real part is, in general, no longer flat. \refstepcounter{subsubsection For every $\C$-valued connection $\Theta:\Lambda^1(X,\C) \to f^*\Lambda^1(Y,\C)$ on a fibration $f:X \to Y$ the kernel $\Ker\Theta \subset \Lambda^1(X,\C)$ is canonically isomorphic to the quotient $\Lambda^1(X,\C)/\delta_f(f^*\Lambda^1(Y,\C))$, and the composition $$ R = \Theta \circ d_X:\Lambda^1(X,\C)/\delta_f(f^*\Lambda^1(Y,\C) \cong \Ker\Theta \to f^*\Lambda^2(Y,\C) $$ is in fact a bundle map. This map is called {\em the curvature} of the $\C$-valued connection $\Theta$. The connection $\Theta$ is flat if and only if its curvature $R$ vanishes. \refstepcounter{subsubsection \label{conn.setup} Let now $M$ be a complex manifold, and let $U \subset {\overline{T}M}$ be an open neighborhood of the zero section $M \subset {\overline{T}M}$ in the total space ${\overline{T}M}$ of the complex-conjugate to the tangent bundle to $M$. Let $\rho:U \to M$ be the natural projection. Assume that $U$ is invariant with respect to the natural action of the unit disc $\Delta \subset \C$ on ${\overline{T}M}$. \refstepcounter{subsubsection Since $M$ is complex, by \ref{de.Rham} the bundle $\Lambda^1(M,\C)$ is equipped with a Hodge bundle structure of weight $1$. The pullback bundle $\rho^*\Lambda^1(M,\C)$ is then also equipped with a weight $1$ Hodge bundle structure. Our description of the Hodge manifold structures on the subset $U \in {\overline{T}M}$ is based on the following notion. \noindent {\bf Definition.\ } \label{hodge.con} A {\em Hodge connection} on the pair $\langle M, U\rangle$ is a $\C$-valued connection on $\rho:U \to M$ such that the associated derivation $$ D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C) $$ is weakly Hodge in the sense of \ref{w.hodge}. A Hodge connection is called {\em flat} if it extends to a weakly Hodge derivation $$ D:\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \rho^*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C) $$ satisfying $D \circ D = 0$. \refstepcounter{subsubsection Assume given a flat Hodge connection $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ on the pair $\langle U,M \rangle$, and assume in addition that the derivation $D$ is holonomic in the sense of \ref{holonomic}. Then the pair $\langle D, \rho^*\Lambda^1(M,\C)\rangle$ defines by Proposition~\ref{explicit.hodge} a Hodge manifold structure on $U$. It turns out that every Hodge manifold structure on $U$ can be obtained in this way. Namely, we have the following. \begin{prop}\label{equiv} There correspondence $D \mapsto \langle \rho^*\Lambda^1(M,\C), D \rangle$ is a bijection between the set of all flat Hodge connections $D$ on the pair $\langle U, M\rangle$ such that $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is holonomic in the sense of \ref{holonomic}, and the set of all Hodge manifold structures on the $U(1)$-manifold $U$ such that the projection $\rho:U_I \to M$ is holomorphic for the preferred complex structure $U_I$ on $U$. \end{prop} \refstepcounter{subsubsection The crucial part of the proof of Proposition~\ref{equiv} is the following observation. \begin{lemma}\label{ident} Assume given a Hodge manifold structure on the $U(1)$-manifold $U \subset {\overline{T}M}$. Let $\delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)$ be the codifferential of the projection $\rho:U \to M$, and let $P:\Lambda^1(U,\C) \to \Lambda^{0,1}(U_J)$ be the canonical projection. The bundle mao given by the composition $$ P \circ \delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^{0,1}(U_J) $$ is an isomorphism of complex vector bundles. \end{lemma} \proof Since the bundles $\rho^*\Lambda^1(M,\C)$ and $\Lambda^{0,1}(U_J)$ are of the same rank, and the maps $\delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)$ and $P \circ \delta_\rho$ are equivariant with respect to the action of the unit disc on $U$, it suffices to prove the claim on $M \subset U$. Let $m \in M$ be an arbitrary point, and let $V = T^*_m{\overline{T}M}$ be the cotangent bundle at $m$ to the Hodge manifold $U \subset {\overline{T}M}$. Let also $V^0 \subset V$ be the subspace of $U(1)$-invariant vectors in $V$. The space $V$ is an equivariant quaternionic vector space. Moreover, the fibers of the bundles $\rho^*\Lambda^1(M,\C)$ and $\Lambda^{0,1}(U_J)$ at the point $m$ are complex vector spaces, and we have canonical identifications \begin{align*} \rho^*\Lambda^1(M,\C)|_m &\cong V^0_I \oplus \overline{V^0_I},\\ \Lambda^{0,1}(U_J)|_m &\cong V_J. \end{align*} Under these identifications the map $P \circ \delta_\rho$ at the point $m$ coincides with the action map $V^0_I \oplus \overline{V^0_I} \to V_J$, which is invertible by Lemma~\ref{regular.quaternionic}. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection By Proposition~\ref{explicit.hodge} every Hodge manifold structure on $U$ is given by a pair $\langle \E, D\rangle$ of a Hodge bundle $\E$ on $U$ of weight $1$ and a holonomic derivation $D:\Lambda^0(U,\C) \to \E$. Lemma~\ref{ident} gives an isomorphism $\E \cong \rho^*\Lambda^1(M,\C)$, so that $D$ becomes a flat $\C$-valued connection on $U$ over $M$. To prove Proposition~\ref{equiv} it suffices now to prove the following. \begin{lemma} The complex vector bundle isomorphism $$ P \circ \delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^{0,1}(U_J) $$ associated to a Hodge manifold structure on $U$ is compatible with the Hodge bundle structures if and only if the projection $\rho:U_I \to M$ is holomorphic for the preferred complex structure $U_I$ on $U$. \end{lemma} \proof The preferred complex structure $U_I$ induces a Hodge bundle structure of weight $1$ on $\Lambda^1(U,\C)$ by \ref{de.Rham}, and the canonical projection $P:\Lambda^1(U,\C) \to \Lambda^{0,1}(U_J)$ is compatible with the Hodge bundle structures by \ref{lambda_j=lambda_i}. If the projection $\rho:U_I \to M$ is holomorphic, then the codifferential $\delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)$ sends the subbundles $\rho^*\Lambda^{1,0}(M),\rho^*\Lambda^{0,1}(M) \subset \rho^*\Lambda^1(M,\C)$ into, respectively, the subbundles $\Lambda^{1,0}(U_I),\Lambda^{0,1}(U_I) \subset \Lambda^1(U,\C)$. Therefore the map $\delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)$ is compatible with the Hodge bundle structures, which implies the ``if'' part of the lemma. To prove the ``only if'' part, assume that $P \circ \delta_\rho$ is a Hodge bundle isomorphism. Since the complex conjugation $\nu:\overline{\Lambda^{0,1}(U_I)} \to \Lambda^{1,0}(U_J)$ is compatible with the Hodge bundle structures, the projection $\overline{P}:\Lambda^1(U,\C) \to \Lambda^{1,0}(U_J)$ and the composition $\overline{P} \circ \delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^{1,0}(U_J)$ are also compatible with the Hodge bundle structures. Therefore the map $$ P \oplus \overline{P}:\Lambda^1(U,\C) \to \Lambda^{1,0}(U_J) \oplus \Lambda^{0,1}(U_J) $$ is a Hodge bundle isomorphism, and the composition $$ \delta_\rho \circ (P \oplus \overline{P}):\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C) $$ is a Hodge bundle map. Therefore the codifferential $\delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)$ is compatible with the Hodge bundle structures. This means precisely that the projection $\rho:U_I \to M$ is holomorphic, which finishes the proof of the lemma and of Proposition~\ref{equiv}. \hfill \ensuremath{\square}\par \subsection{The relative de Rham complex of $U$ over $M$}\label{relative.de.rham.sub} \refstepcounter{subsubsection Keep the notation of the last subsection. To use Proposition~\ref{equiv} in the study of Hodge manifold structures on the open subset $U \subset {\overline{T}M}$, we will need a way to check whether a given Hodge connection on the pair $\langle U,M \rangle$ is holonomic in the sense of \ref{holonomic}. We will also need to rewrite the linearity condition \ref{lin.def} for a Hodge manifold structure on $U$ in terms of the associated Hodge connection $D$. To do this, we will use the so-called {\em relative de Rham complex} of $U$ over $M$. For the convenience of the reader, and to fix notation, we recall here its definition and main properties. \refstepcounter{subsubsection Since the projection $\rho:U \to M$ is submersive, the codifferential $$ \delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C) $$ is injective. The {\em relative cotangent bundle} $\Lambda^1(U/M,C)$ is by definition the quotient bundle $$ \Lambda^1(U/M,\C) = \Lambda^1(U,\C)/\delta_\rho(\rho^*\Lambda^1(M,\C)). $$ Let $\pi:\Lambda^1(U,\C) \to \Lambda^1(U/M,\C)$ be the natural projection. We have by definition the short exact sequence \begin{equation}\label{ex.seq} \begin{CD} 0 @>>> \rho^*\Lambda^1(M,\C) @>\delta_\rho>> \Lambda^1(U,\C) @>\pi>> \Lambda^1(U/M,\C) @>>> 0 \end{CD} \end{equation} of complex vector bundles on $U$. \refstepcounter{subsubsection The composition $d^r = \pi \circ d_U$ of the de Rham differential $d_U$ with the projection $\pi$ is an algebra derivation $$ d^r:\Lambda^0(U,\C) \to \Lambda^1(U/M,\C), $$ called {\em the relative de Rham differential}. It is a first order differential operator, and $d^rf = 0$ if and only if the smooth function $f:U \to \C$ factors through the projection $\rho:U \to M$. Let $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C)$ be the exterior algebra of the bundle $\Lambda^1(U/M,\C)$. The projection $\pi$ extends to an algebra map $$ \pi:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U,\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C). $$ The differential $d^r$ extends to an algebra derivation $$ d^r:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(U/M,\C) $$ satisfying $d^r \circ d^r = 0$, and we have $\pi \circ d_U = d^r \circ \pi$. The differential graded algebra $\langle \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C),d^r\rangle$ is called {\em the relative de Rham complex} of $U$ over $M$. \refstepcounter{subsubsection Since the relative de Rham differential $d^r$ is linear with respect to multiplication by functions of the form $\rho^*f$ with $f \in C^\infty(M,\C)$, it extends canonically to an operator $$ d^r:\rho^*\Lambda^i(M,\C) \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C) \to \rho^*\Lambda^i(M,\C) \otimes \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(U/M,\C). $$ The two-step filtration $\rho^*\Lambda^1(M,\C) \subset \Lambda^1(U,\C)$ induces a filtration on the de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U,\C)$, and the $i$-th associated graded quotient of this filtration is isomorphic to the complex $\langle \rho^*\Lambda^i(M,\C) \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C), d^r \rangle$. \refstepcounter{subsubsection Since $U \subset {\overline{T}M}$ lies in the total space of the complex-conjugate to the tangent bundle to $M$, we have a canonical algebra isomorphism $$ {\sf can}:\overline{\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)} \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C). $$ Let $\tau:C^\infty(M,\Lambda^1(M,\C)) \to C^\infty(U,\C)$ be the tautological map sending a $1$-form to the corresponding linear function on ${\overline{T}M}$, as in \ref{tau}. Then for every smooth $1$-form $\alpha \in C^\infty(M,\Lambda^1(M,\C))$ we have \begin{equation}\label{can.and.tau} {\sf can}(\rho^*\alpha) = d^r\tau(\alpha). \end{equation} \refstepcounter{subsubsection The complex vector bundle $\Lambda^1(U/M,\C)$ has a natural real structure, and it is naturally $U(1)$-equivariant. Moreover, the decomposition $\Lambda^1(M,\C) = \Lambda^{1,0}(M) \oplus \Lambda^{0,1}(M)$ induces a decomposition $$ \Lambda^1(U/M,\C) = {\sf can}(\Lambda^{1,0}(M)) \oplus {\sf can}(\Lambda^{0,1}(M)). $$ This allows to define, as in \ref{de.Rham}, a canonical Hodge bundle structure of weight $1$ on $\Lambda^1(U/M,\C)$. It gives rise to a Hodge bundle structure on $\Lambda^i(U/M,\C)$ of wieght $i$, and the relative de Rham differential $d^r:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(U/M,\C)$ is weakly Hodge. \refstepcounter{subsubsection The canonical isomorphism ${\sf can}:\overline{\rho^*\Lambda^1(M,\C)} \to \Lambda^1(U/M,\C)$ is not compatible with the Hodge bundle structures. The reason for this is that the real structure on the Hodge bundles $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C)$ is, by definition \ref{de.Rham}, twisted by $\iota^*$, where $\iota:{\overline{T}M} \to {\overline{T}M}$ is the action of $-1 \in U(1) \subset \C$. Therefore, while ${\sf can}$ is $U(1)$-equivariant, it is not real. To correct this, introduce an involution $\zeta:\Lambda^1(M,\C) \to \Lambda^1(M,\C)$ by \begin{equation}\label{zeta} \zeta = \begin{cases} {\sf id} &\text{ on }\Lambda^{1,0}(M) \subset \Lambda^1(M,\C) \\ -{\sf id} &\text{ on }\Lambda^{0,1}(M) \subset \Lambda^1(M,\C) \end{cases} \end{equation} and set \begin{equation}\label{eta} \eta = {\sf can} \circ \rho^*\overline{\zeta}:\rho^*\overline{\Lambda^1(M,\C)} \to \rho^*\overline{\Lambda^1(M,\C)} \to \Lambda^1(U/M,\C) \end{equation} Unlike ${\sf can}$, the map $\eta$ preserves the Hodge bundle structures. It will also be convenient to twist the tautological map $\tau:\rho^*\Lambda^1(M,\C) \to \Lambda^0(U,\C)$ by the involution $\zeta$. Namely, define a map $\sigma:\rho^*\Lambda^1(M,\C) \to \Lambda^0(U,\C)$ by \begin{equation}\label{sigma1} \sigma = \tau \circ \rho^*\overline{\zeta}: \rho^*\overline{\Lambda^1(M,\C)} \to \rho^*\overline{\Lambda^1(M,\C)} \to \Lambda^0(U/M,\C) \end{equation} By \eqref{can.and.tau} the twisted tautological map $\sigma$ and the canonical map $\eta$ satisfy \begin{equation}\label{eta.and.sigma} \eta(\rho^*\alpha) = d^r\sigma(\alpha) \end{equation} for every smooth $1$-form $\alpha \in C^\infty(M,\Lambda^1(M,\C))$. \refstepcounter{subsubsection Let $\phi \in \Theta(U)$ be the differential of the canonical $U(1)$-action on $U \subset {\overline{T}M}$. The vector field $\phi$ is real and tangent to the fibers of the projection $\rho:U \to M$. Therefore the contraction with $\phi$ defines an algebra derivation \begin{align*} \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(U/M,\C) &\to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C)\\ \alpha &\mapsto \langle \phi, \alpha \rangle \end{align*} The following lemma gives a relation between this derivation, the canonical weakly Hodge map $\eta:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U/M,\C)$ given by \eqref{eta}, and the tautological map $\tau:\rho^*\Lambda^1(M,\C) \to \Lambda^0(U,\C)$. \begin{lemma}\label{phi.and.tau} For every smooth section $\alpha \in C^\infty(U,\rho^*\Lambda^1(M,\C))$ we have $$ \sqrt{-1}\tau(\alpha) = \langle \phi, \eta(\alpha) \rangle \in C^\infty(U,\C). $$ \end{lemma} \proof Since the equality that we are to prove is linear with respect to multiplication by smooth functions on $U$, we may assume that the section $\alpha$ is the pull-back of a smooth $1$-form $\alpha \in C^\infty(M,\Lambda^1(M,\C))$. The Lie derivative $\LL_\phi:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U,\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U,\C)$ with respect to the vector field $\phi$ is compatible with the projection $\pi:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U,\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C)$ and defines therefore an algebra derivation $\LL_\phi:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C)$. The Cartan homotopy formula gives \begin{equation}\label{eqqq.1} \LL_\phi \tau(\alpha) = \langle \phi, d^r\tau(\alpha) \rangle. \end{equation} The function $\tau(\alpha)$ on ${\overline{T}M}$ is by definition $\R$-linear along the fibers of the projection $\rho:{\overline{T}M} \to M$. The subspace $\tau(C^\infty(M,\Lambda^1(M,\C))) \subset C^\infty(U,\C)$ of such functions decomposes as $$ \tau(C^\infty(M,\Lambda^1(M,\C))) = \tau(C^\infty(M,\Lambda^{1,0}(M))) \oplus \tau(C^\infty(M,\Lambda^{0,1}(M,\C))), $$ and the group $U(1)$ acts on the components with weight $1$ and $-1$. Therefore the derivative $\LL_\phi$ of the $U(1)$-action acts on the components by multiplication with $\sqrt{-1}$ and $-\sqrt{-1}$. By definition of the involution $\zeta$ (see \eqref{zeta}) this can be written as \begin{equation}\label{eqqq.2} \LL_\phi \tau(\alpha) = \sqrt{-1}\tau(\zeta(\alpha)). \end{equation} On the other hand, by \eqref{can.and.tau} and the definition of the map $\eta$ we have \begin{equation}\label{eqqq.3} d^r\tau(\alpha) = {\sf can}(\alpha) = \eta(\zeta(\alpha)). \end{equation} Substituting \eqref{eqqq.2} and \eqref{eqqq.3} into \eqref{eqqq.1} gives $$ \sqrt{-1}\tau(\zeta(\alpha)) = \langle \phi, \eta(\zeta(\alpha)), $$ which is equivalent to the claim of the lemma. \hfill \ensuremath{\square}\par \subsection{Holonomic Hodge connections} \refstepcounter{subsubsection We will now describe a convenient way to check whether a given Hodge connection $D$ on the pair $\langle U,M \rangle$ is holonomic in the sense of \ref{holonomic}. To do this, we proceed as follows. Consider the restriction $\Lambda^1(U,\C)|_M$ of the bundle $\Lambda^1(U,\C)$ to the zero section $M \subset U \subset {\overline{T}M}$, and let $$ \Res:\Lambda^1(U,\C)|_M \to \Lambda^1(M,\C) $$ be the restriction map. The kernel of the map $\Res$ coincides with the conormal bundle to the zero section $M \subset U$, which we denote by $S^1(M,\C)$. The map $\Res$ splits the restriction of exact sequence \eqref{ex.seq} onto the zero section $M \subset U$, and we have the direct sum decomposition \begin{equation}\label{hor.vert} \Lambda^1(U,\C)|_M = S^1(M,\C) \oplus \Lambda^1(M,\C). \end{equation} \refstepcounter{subsubsection\label{S1} The $U(1)$-action on $U \subset {\overline{T}M}$ leaves the zero section $M \subset U$ invariant and defines therefore a $U(1)$-action on the conormal bundle $S^1(M,\C)$. Together with the usual real structure twisted by the action of the map $\iota:{\overline{T}M} \to {\overline{T}M}$, this defines a Hodge bundle structure of weight $0$ on the bundle $S^1(M,\C)$. Note that the automorphism $\iota:{\overline{T}M} \to {\overline{T}M}$ acts as $-{\sf id}$ on the Hodge bundle $S^1(M,\C)$, so that the real structure on $S^1(M,\C)$ is minus the usual one. Moreover, as a complex vector bundle the conormal bundle $S^1(M,\C)$ to $M \subset {\overline{T}M}$ is canonically isomorphic to the cotangent bundle $\Lambda^1(M,\C)$. The Hodge type grading on $S^1(M,\C)$ is given in terms of this isomorphism by $$ S^1(M,\C) = S^{1,-1}(M) \oplus S^{-1,1}(M) \cong \Lambda^{1,0}(M) \oplus \Lambda^{0,1}(M) = \Lambda^1(M,\C). $$ \refstepcounter{subsubsection Let $$ C_{lin}^\infty(U,\C) = \tau(C^\infty(M,\Lambda^1(M,\C))) \subset C^\infty(U,\C) $$ be the subspace of smooth functions linear along the fibers of the canonical projection $\rho:U \subset {\overline{T}M} \to M$. The relative de Rham differential defines an isomorphism \begin{equation}\label{iso} d^r:C_{lin}^\infty(U,\C) \to C^\infty(M,S^1(M,\C)). \end{equation} This isomorphism is compatible with the canonical Hodge structures of weight $0$ on both spaces, and it is linear with respect to multiplication by smooth functions $f \in C^\infty(M,\C)$. \refstepcounter{subsubsection \label{pr.part} Let now $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ be a Hodge connection on the pair $\langle U,M \rangle$, and let $\Theta:\Lambda^1(U,\C) \to \rho^*\Lambda^(M,\C)$ be the corresponding bundles map. Since $D$ is a $\C$-valued connection, the restriction $\Theta|_M$ onto the zero section $M \subset M$ decomposes as \begin{equation}\label{hor.vert.1} \Theta = D_0 \oplus {\sf id}:S^1(M,\C) \oplus \Lambda^1(M,\C) \to \Lambda^1(M,\C) \end{equation} with respect to the direct sum decomposition \eqref{hor.vert} for a certain bundle map $D_0:S^1(M,\C) \to \Lambda^1(M,\C)$. \noindent {\bf Definition.\ } The bundle map $D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ is called the {\em principal part} of the Hodge connection $D$. \refstepcounter{subsubsection Consider the map $D_0:C^\infty(M,S^1(M,\C)) \to C^\infty(M,\Lambda^1(M,\C))$ on the spaces of smooth sections induced by the principal part $D_0$ of a Hodge connection $D$. Under the isomorphism \eqref{iso} this map coincides with the restriction of the composition $$ \Res \circ D:C^\infty(U,\C) \to C^\infty(U,\rho^*\Lambda^1(M,\C)) \to C^\infty(M,\Lambda^1(M,\C)) $$ onto the subspace $C^\infty_{lin}(U,\C) \subset C^\infty(U,\C)$. Each of the maps $\Res$, $D$ is weakly Hodge, so that this composition also is weakly Hodge. Since the isomorphism \eqref{iso} is compatible with the Hodge bundle structures, this implies that the principal part $D_0$ of the Hodge connection $D$ is a weakly Hodge bundle map. In particular, it is purely imaginary with respect to the usual real structure on the conormal bundle $S^1(M,\C)$. \refstepcounter{subsubsection We can now formulate the main result of this subsection. \begin{lemma}\label{aux1} A Hodge connection $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ on the pair $\langle U,M \rangle$ is holonomic in the sense of \ref{holonomic} on an open neighborhood $U_0 \subset U$ of the zero section $M \subset U$ if and only if its principal part $D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ is a complex vector bundle isomorphism. \end{lemma} \proof By definition the derivation $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is holonomic in the sense of \ref{holonomic} if and only if the corresponding map $$ \Theta:\Lambda^1(U,\R) \to \rho^*\Lambda^1(M,\C) $$ is an isomorphism of real vector bundles. This is an open condition. Therefore the derivation $D$ is holonomic on an open neighborhood $U_0 \supset M$ of the zero section $M \subset U$ if and only if the map $\Theta$ is an isomorphism on the zero section $M \subset U$ itself. According to \eqref{hor.vert.1}, the restriction $\Theta|_M$ decomposes as $\Theta_M = D_0 + {\sf id}$, and the principal part $D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ of the Hodge connection $D$ is purely imaginary with respect to the usual real structure on $\Lambda^1(U,\C)|_M$, while the identity map ${\sf id}:\rho^*\Lambda^1(M,\C) \to \rho^*\Lambda^1(M,\C)$ is, of course, real. Therefore $\Theta_M$ is an isomorphism if and only is $D_0$ is an isomorphism, which proves the lemma. \hfill \ensuremath{\square}\par \subsection{Hodge connections and linearity}\label{hodge.lin.subsec} \refstepcounter{subsubsection \label{aux} Assume now given a Hodge manifold structure on the subset $U \subset {\overline{T}M}$, and let $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ be the associated Hodge connection on the pair $\langle U, M\rangle$ given by Proposition~\ref{equiv}. We now proceed to rewrite the linearity condition \ref{lin.def} in terms of the Hodge connection $D$. Let $j:\Lambda^1(U,\C) \to \overline{\Lambda^1(U,\C)}$ be the canonical map defined by the quaternionic structure on $U$, and let $\iota^*:\Lambda^1(U,\C) \to \iota^*\Lambda^1(U,\C)$ be the action of the canonical involution $\iota:U \to U$. Let also $D^\iota:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ be the operator $\iota^*$-conjugate to the Hodge connection $D$. We begin with the following identity. \begin{lemma}\label{aux.lemma} For every smooth function $f \in C^\infty(U,\C)$ we have $$ d^rf = \frac{\sqrt{-1}}{2}\pi(j(\delta_\rho(D-D^\iota)(f))), $$ where $\pi:\Lambda^1(U,\C) \to \Lambda^1(U/M,\C)$ is the canonical projection, and $$ \delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C) $$ is the codifferential of the projection $\rho:U \to M$. \end{lemma} \proof By definition of the Hodge connection $D$ the Dolbeault derivative $\bar\partial_Jf$ coincides with the $(0,1)$-component of the $1$-form $\delta_\rho(Df) \in \Lambda^1(U,\C)$ with respect to the complementary complex structure $U_J$ on $U$. Therefore $$ \bar\partial_J f = \frac{1}{2} \delta_\rho(Df) + \frac{\sqrt{-1}}{2} j(\delta_\rho(Df)). $$ Applying the complex conjugation $\nu:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U,\C) \to \overline{\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U,\C)}$ to this equation, we get \begin{align*} \begin{split} \partial_J f &= \nu\left(\frac{1}{2} \delta_\rho(D\nu(f)) + \frac{\sqrt{-1}}{2} j(\delta_\rho(D\nu(f))))\right) = \\ &= \frac{1}{2} \nu(\delta_\rho(D\nu(f))) - \frac{\sqrt{-1}}{2} j(\nu(\delta_\rho(D\nu(f)))). \end{split} \end{align*} Since the map $\delta_\rho \circ D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is weakly Hodge, we have $$ \delta_\rho(D(\iota^*\nu(f))) = \iota^*\nu(\delta_\rho(Df)). $$ Therefore $\nu(\delta_\rho(D(f))) = \delta_\rho(D^\iota(\nu(f)))$, and we have $$ \partial_J f = \frac{1}{2} \delta_\rho(D^\iota f) - \frac{\sqrt{-1}}{2} j(\delta_\rho(D^\iota f)). $$ Thus the de Rham derivative $d_Uf$ equals $$ d_Uf = \partial_Jf + \bar\partial_Jf = \frac{1}{2} \delta_\rho((D+D^\iota)f) + \frac{\sqrt{-1}}{2} j(\delta_\rho((D-D^\iota)f)). $$ Now, by definition $\delta_\rho \circ \pi = 0$. Therefore $$ d^rf = \pi(d_Uf) = \frac{\sqrt{-1}}{2}\pi(j(\delta_\rho((D-D^\iota)f))), $$ which is the claim of the lemma. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection We will also need the following fact. It can be derived directly from Lemma~\ref{aux.lemma}, but it is more convenient to use Lemma~\ref{aux1} and the fact that the Hodge connection $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is holonomic. \begin{lemma}\label{aux3} In the notation of Lemma~\ref{aux.lemma}, let $$ \A = \delta_\rho\left((D-D^\iota)\left(C_{lin}^\infty(U,\C)\right)\right) \subset C^\infty(U, \rho^*\Lambda^1(M,\C)) $$ be the subspace of sections $\alpha \in C^\infty(U,\rho^*\Lambda^1(M,\C))$ of the form $\alpha = \delta_\rho((D-D^\iota)f)$, where $f \in C^\infty(U,\C)$ lies in the subspace $C_{lin}^\infty(U,\C) \subset C^\infty(U,\C)$ of smooth functions on $U$ linear along the fibers of the projection $\rho:U \to M$. The restriction $\Res(\A) \subset C^\infty(M,\Lambda^1(M,\C))$ of the subspace $\A$ onto the zero section $M \subset U$ is the whole space $C^\infty(M,\Lambda^1(M,\C))$. \end{lemma} \proof Let $D_0 = \Res \circ D:C^\infty_{lin}(U,\C) \to C^\infty(M,\Lambda^1(M,\C))$ be the principal part of the Hodge connection $D$ in the sense of Definition~\ref{pr.part}. Since the canonical automorphism $\iota:{\overline{T}M} \to {\overline{T}M}$ acts as $-{\sf id}$ on $C_{lin}^\infty(U,\C)$, we have $D_0^\iota = - D_0$. Therefore \begin{multline*} \Res(\A) = \Res \circ (D - D^\iota)\left(C_{lin}^\infty(U,\C)\right) = \\ =(D_0-D^\iota_0)\left(C_{lin}^\infty(U,\C)\right) = D_0\left(C_{lin}^\infty(U,\C)\right). \end{multline*} Since the Hodge connection $D$ is holonomic, this space coincides with the whole $C^\infty(M,\Lambda^1(M,\C))$ by Lemma~\ref{aux1}. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection We now apply Lemma~\ref{aux.lemma} to prove the following criterion for the linearity of the Hodge manifold structure on $U$ defined by the Hodge connection $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$. \begin{lemma}\label{explicit.lin} The Hodge manifold structure on $U \subset {\overline{T}M}$ corresponding to a Hodge connection $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is linear in the sense of \ref{lin.def} if and only if for every smooth function $f \in C^\infty(U,\C)$ linear along the fibers of the projection $\rho:U \subset {\overline{T}M} \to M$ we have \begin{equation}\label{expl.lin} f = \frac{1}{2}\sigma\left((D-D^\iota)f\right), \end{equation} where $\sigma:\rho^*\Lambda^1(M,\C) \to \Lambda^0(U,\C)$ is the twisted tautological map introduced in \eqref{sigma1}, and $D^\iota:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is the operator $\iota^*$-conjugate to $D$, as in \ref{aux}. \end{lemma} \proof By Lemma~\ref{lin.char} the Hodge manifold structure on $U$ is linear if and only if for every $\alpha \in C^\infty(U,\rho^*\Lambda^1(M,\C))$ we have \begin{equation}\label{eee.to.prove} \langle \phi, j(\alpha) \rangle = \tau(\alpha), \end{equation} where $\phi$ is the differential of the $U(1)$-action on $U$, $j:\Lambda^1(U,\C) \to \overline{\Lambda^1(U,\C)}$ is the operator given by the quaternionic structure on $U$, and $\tau:\rho^*\Lambda^1(M,\C) \to \Lambda^0(U,\C)$ is the tautological map sending a $1$-form on $M$ to the corresponding linear function on ${\overline{T}M}$, as in \ref{tau}. Moreover, by Lemma~\ref{aux3} and Lemma~\ref{aux2} the equality \eqref{eee.to.prove} holds for all smooth sections $\alpha \in C^\infty(U,\rho^*\Lambda^1(M,\C))$ if and only if it holds for sections of the form \begin{equation}\label{special} \alpha = \frac{\sqrt{-1}}{2}\delta_\rho((D-D^\iota)f), \end{equation} where $f \in C_{lin}^\infty(U,\C) \subset C^\infty(U,\C)$ is linear along the fibers of $\rho:U \to M$. Let now $f \in C^\infty(U,\C)$ be a smooth function on $U$ linear along the fibers of $\rho:U \to M$, and let $\alpha$ be as in \eqref{special}. Since $\phi$ is a vertical vector field on $U$ over $M$, we have $\langle \phi, j(\alpha) \rangle = \langle \phi, \pi(j(\alpha))\rangle$, where $\pi:\Lambda^1(U,\C) \to \Lambda^1(U/M,\C)$ is the canonical projection. By Lemma~\ref{aux.lemma} \begin{equation}\label{thelema} \langle \phi, j(\alpha) \rangle = \langle \phi, \pi(j(\alpha)) \rangle = \langle \phi, d^rf \rangle. \end{equation} Since the function $f$ is linear along the fibers of $\rho:U \to M$, we can assume that $f = \sigma(\beta)$ for a smooth $1$-form $\beta \in C^\infty(M,\Lambda^1(M,\C)$. Then by \eqref{eta.and.sigma} and by Lemma~\ref{phi.and.tau} the right hand side of \eqref{thelema} is equal to $$ \langle \phi, d^rf \rangle = \langle \phi, d^r(\sigma(\beta)) \rangle = \langle \phi, \eta(\beta) \rangle = \sqrt{-1} \tau(\beta). $$ Therefore, \eqref{eee.to.prove} is equivalent to \begin{equation}\label{horus} \sqrt{-1}\tau(\beta) = \tau\left(\frac{\sqrt{-1}}{2}\delta_\rho((D-D^\iota)\sigma(\beta))\right). \end{equation} But we have $\tau = \sigma \circ \zeta$, where $\zeta:\rho^*\Lambda^1(M,\C) \to \rho^*\Lambda^1(M,\C)$ is the invulution introduced in \eqref{zeta}. In particular, the map $\zeta$ is invertible, so that \eqref{horus} is in turn equivalent to $$ \sigma(\beta) = \frac{1}{2}\sigma(\delta_\rho((D-D^\iota)\sigma(\beta))), $$ or, substituting back $f = \sigma(\beta)$, to $$ f = \frac{1}{2}\sigma(\delta_\rho((D-D^\iota)f)), $$ which is exactly the condition \eqref{expl.lin}. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection \noindent {\bf Definition.\ } \label{hodge.conn.lin} A Hodge connection $D$ on the pair $\langle U, M \rangle$ is called {\em linear} if it satisfies the condition~\ref{expl.lin}. We can now formulate and prove the following more useful version of Proposition~\ref{equiv}. \begin{prop}\label{equiv.bis} Every linear Hodge connection $D\!:\!\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ on the pair $\langle U, M\rangle$ defines a linear Hodge manifold structure on an open neighborhood $V \subset U$ of the zero section $M \subset U$, and the canonical projection $\rho:V_I \to M$ is holomorphic for the preferred complex structure $V_I$ on $V$. Vice versa, every such linear Hodge manifold structure on $U$ comes from a unique linear Hodge connection $D$ on the pair $\langle U,M \rangle$. \end{prop} \proof By Proposition~\ref{equiv} and Lemma~\ref{explicit.lin}, to prove this proposition suffices to prove that if a Hodge connection $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is linear, then it is holonomic in the sense of \ref{holonomic} on a open neighborhood $V \subset U$ of the zero section $M \subset U$. Lemma~\ref{aux1} reduces this to proving that the principal part $D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ of a linear Hodge connection $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is a bundle isomorphism. Let $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ be such connection. By \eqref{expl.lin} we have $$ \frac{1}{2} \sigma \circ (D_0 - D_0^\iota) = {\sf id}:S^1(M,\C) \to \Lambda^1(M,\C) \to S^1(M,\C). $$ Since $\sigma:\Lambda^1(M,\C) \to S^1(M,\C)$ is a bundle isomorphism, so is the bundle map $D_0 - D_0^\iota:S^1(M,\C) \to \Lambda^1(M,\C)$. As in the proof of Lemma~\ref{aux3}, we have $D_0 = - D_0^\iota$. Thus $D_0 = \frac{1}{2}(D_0-D_0^\iota):S^1(M,\C) \to \Lambda^1(M,\C)$ also is a bundle isomorphism, which proves the proposition. \hfill \ensuremath{\square}\par \section{Formal completions}\label{formal.section} \subsection{Formal Hodge manifolds} \refstepcounter{subsubsection Proposition~\ref{equiv.bis} reduces the study of arbitrary regular Hodge manifolds to the study of connections of a certain type on a neighborhood $U \subset {\overline{T}M}$ of the zero section $M \subset {\overline{T}M}$ in the total space ${\overline{T}M}$ of the tangent bundle to a complex manifold $M$. To obtain further information we will now restrict our attention to the {\em formal} neighborhood of this zero section. This section contains the appropriate definitions. We study the convergence of our formal series in Section~\ref{convergence}. \refstepcounter{subsubsection Let $X$ be a smooth manifold and let $\Bun(X)$ be the category of smooth real vector bundles over $X$. Let also $\Diff(X)$ be the category with the same objects as $\Bun(X)$ but with differential operators as morphisms. Consider a closed submanifold $Z \subset X$. For every two real vector bundles $\E$ and $\F$ on $X$ the vector space $\Hom(\E,\F)$ of bundle maps from $\E$ to $\F$ is naturally a module over the ring $C^\infty(X)$ of smooth functions on $X$. Let $\J_Z \subset C^\infty(X)$ be the ideal of functions that vanish on $Z$ and let $\Hom_Z(\E,\F)$ be the $\J_Z$-adic completion of the $C^\infty(X)$-module $\Hom(\E,\F)$. For any three bundles $\E,\F,\G$ the composition map $$ \Mult:\Hom(\E,\F) \otimes \Hom(\F,\G) \to \Hom(\E,\G) $$ is $C^\infty(X)$-linear, hence extends to a map $$ \Mult:\Hom_Z(\E,\F) \otimes \Hom_Z(\F,\G) \to \Hom_Z(\E,\G). $$ Let $\Bun_Z(X)$ be the category with the same objects as $\Bun(X)$ and for every two objects $\E$, $\F \in \Ob\Bun(X)$ with $\Hom_Z(\E,\F)$ as the space of maps between $\F$ anf $\F$. The category $\Bun_Z(X)$, as well as $\Bun(X)$, is an additive tensor category. \refstepcounter{subsubsection The space of differential operators $\Diff(\E,\F)$ is also a $C^\infty(X)$ module, say, by left multiplication. Let $\Diff_Z(\E,\F)$ be its $\J_Z$-completion. The composition maps in $\Diff(X)$ are no longer $C^\infty(X)$-linear. However, they still are compatible with the $\J_Z$-adic topology, hence extend to completions. Let $\Diff_Z(X)$ be the category with the same objects as $\Bun(X)$ and with $\Diff_Z(\E,\F)$ as the space of maps between two objects $\E,\F \in \Ob \Bun(X)$. By construction we have canonical {\em $Z$-adic completion functors} $$ \Bun(X) \to \Bun_Z(X) \text{ and } \Diff(X) \to \Diff_Z(X). $$ Call the categories $\Bun_Z(X)$ and $\Diff_Z(X)$ {\em the $Z$-adic completions} of the categories $\Bun(X)$ and $\Diff(X)$. \refstepcounter{subsubsection When the manifold $X$ is equipped with a smooth action of compact Lie group $G$ fixing the submanifold $Z$, the completion construction extends to the categories of $G$-equivariant bundles on $M$. When $G = U(1)$, the categories ${{\cal W}{\cal H}odge}(X)$ and ${{\cal W}{\cal H}odge}^\D(X)$ defined in \ref{w.hodge} also admit canonical completions, denoted by ${{\cal W}{\cal H}odge}_Z(X)$ and ${{\cal W}{\cal H}odge}_Z^\D(X)$. \refstepcounter{subsubsection Assume now that the manifold $X$ is equipped with a smooth $U(1)$-action fixing the smooth submanifold $Z \subset X$. \noindent {\bf Definition.\ } A {\em formal quaternionic structure} on $X$ along the submanifold $Z \subset X$ is given by an algebra map $$ \Mult: {\Bbb H} \to {{\cal E}\!nd\:}_{\Bun_Z(X)} \left(\Lambda^1(X)\right) $$ from the algebra ${\Bbb H}$ to the algebra ${{\cal E}\!nd\:}_{\Bun_Z(X)} \left(\Lambda^1(X)\right)$ of endomorphisms of the cotangent bundle $\Lambda^1(X)$ in the category $\Bun_Z(X)$. A formal quaternionic structure is called {\em equivariant} if the map $\Mult$ is equivariant with respect to the natural $U(1)$-action on both sides. \refstepcounter{subsubsection Note that Lemma~\ref{universal} still holds in the situation of formal completions. Consequently, everything in Section~\ref{hbqm.section} carries over word-by-word to the case of formal quaternionic structures. In particular, by Lemma~\ref{qm.hodge} giving a formal equivariant quaternionic structure on $X$ along $Z$ is equivalent to giving a pair $\langle \E, D \rangle$ of a Hodge bundle $\E$ on $X$ and a holonomic algebra derivation $D:\Lambda^0(X) \to \E$ in ${{\cal W}{\cal H}odge}_Z^\D(X)$. \refstepcounter{subsubsection The most convenient way to define Hodge manifold structures on $X$ in a formal neighborhood of $Z$ is by means of the Dolbeault complex, as in Proposition~\ref{explicit.hodge}. \noindent {\bf Definition.\ } A {\em formal Hodge manifold structure} on $X$ along $Z$ is a pair of a pre-Hodge bundle $\E \in \Ob {{\cal W}{\cal H}odge}_Z(X)$ of weight $1$ and an algebra derivation $D^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\E \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}\E$ in ${{\cal W}{\cal H}odge}^\D_Z(X)$ such that $D^0:\Lambda^0(\E) \to \E$ is holonomic and $D^0 \circ D^1 = 0$. \refstepcounter{subsubsection Let $U \subset X$ be an open subset containing $Z \subset X$. For every Hodge manifold structure on $U$ the $Z$-adic completion functor defines a formal Hodge manifold structure on $X$ along $Z$. Call it {\em the $Z$-adic completion} of the given structure on $U$. \noindent {\bf Remark.\ } Note that a Hodge manifold structure on $U$ is completely defined by the preferred and the complementary complex structures $U_I$, $U_J$, hence always real-analytic by the Newlander-Nirenberg Theorem. Therefore, if two Hodge manifold structures on $U$ have the same completion, they coincide on every connected component of $U$ intersecting $Z$. \subsection{Formal Hodge manifold structures on tangent bundles} \refstepcounter{subsubsection Let now $M$ be a complex manifold, and let ${\overline{T}M}$ be the total space of the complex-conjugate to the tangent bundle to $M$ equipped with an action of $U(1)$by dilatation along the fibers of the projection $\rho:{\overline{T}M} \to M$. All the discussion above applies to the case $X = {\overline{T}M}$, $Z = M \subset {\overline{T}M}$. Moreover, the linearity condition in the form given in Lemma~\ref{lin.char} generalizes immediately to the formal case. \noindent {\bf Definition.\ } A formal Hodge manifold structure on ${\overline{T}M}$ along $M$ is called {\em linear} if for every smooth $(0,1)$-form $\alpha \in C^\infty(M,\Lambda^{0,1}(M))$ we have $$ \tau(\alpha) = \langle \phi, j(\rho^*) \rangle \in C^\infty_M({\overline{T}M},\C), $$ where $j$ is the map induced by the formal quaternionic structure on ${\overline{T}M}$ and $\phi$ and $\tau$ are as in Lemma~\ref{lin.char}. \refstepcounter{subsubsection As in the non-formal case, linear Hodge manifold structures on ${\overline{T}M}$ along $M \subset {\overline{T}M}$ can be described in terms of differential operators of certain type. \noindent {\bf Definition.\ } \label{formal.hodge.con} A {\em formal Hodge connection} on ${\overline{T}M}$ along $M \subset {\overline{T}M}$ is an algebra derivation $$ D:\Lambda^0({\overline{T}M},\C) \to \rho^*\Lambda^1(M,\C) $$ in ${{\cal W}{\cal H}odge}_M^\D({\overline{T}M})$ such that for every smooth function $f \in C^\infty(M,\C)$ we have $D\rho^*=\rho^*d_Mf$, as in \eqref{conn.eq}. A formal Hodge connection is called {\em flat} if it extends to an algebra derivation $$ D:\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C) $$ in ${{\cal W}{\cal H}odge}_M^\D({\overline{T}M})$ such that $D \circ D = 0$. A formal Hodge connection is called {\em linear} if it satisfies the condition \eqref{expl.lin} of Lemma~\ref{explicit.lin}, that is, for every function $f \in C^\infty_{lin}({\overline{T}M},\C)$ linear along the fibers of the projection $\rho:{\overline{T}M} \to M$ we have $$ f = \frac{1}{2}\sigma\left((D-D^\iota)f\right), $$ where $\sigma:\rho^*\Lambda^1(M,\C) \to \Lambda^0({\overline{T}M},\C)$ is the twisted tautological map introduced in \eqref{sigma1}, the automorphism $\iota:{\overline{T}M} \to {\overline{T}M}$ is the multiplication by $-1 \in \C$ on every fiber of the projection $\rho:{\overline{T}M} \to M$, and $D^\iota:\Lambda^0({\overline{T}M},\C) \to \rho^*\Lambda^1(M,\C)$ is the operator $\iota^*$-conjugate to $D$, as in \ref{aux}. The discussion in Section~\ref{section.5} generalizes immediately to the formal case and gives the following. \begin{lemma}\label{form.hdg} Linear formal Hodge manifold structures on ${\overline{T}M}$ along the zero section $M \subset {\overline{T}M}$ are in a natural one-to-one correspondence with linear flat formal Hodge connections on ${\overline{T}M}$ along $M$. \end{lemma} \subsection{The Weil algebra} \refstepcounter{subsubsection Let, as before, $M$ be a complex manifold and let ${\overline{T}M}$ be the total space of the complex conjugate to its tangent bundle, as in \ref{overline.T}. In the remaining part of this section we give a description of the set of all formal Hodge connections on ${\overline{T}M}$ along $M$ in terms of certain differential operators on $M$ rather than on ${\overline{T}M}$. We call such operators {\em extended connections} on $M$ (see \ref{ext.con} for the definition). Together with a complete classification of extended connections given in the next Section, this description provides a full classification of regular Hodge manifolds ``in the formal neighborhood of the subset of $U(1)$-fixed points''. \refstepcounter{subsubsection Before we define extended connections in Subsection~\ref{ext.con.subsec}), we need to introduce a certain algebra bundle in ${{\cal W}{\cal H}odge}(M)$ which we call {\em the Weil algebra}. We begin with some preliminary facts. Recall (see, e.g., \cite{Del}) that every additive category $\A$ admits a canonical completion $\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}\A$ with respect to filtered projective limits. The category $\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}\A$ is also additive, and it is tensor if $\A$ was tensor. Objects of the canonical completion $\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}\A$ are called {\em pro-objects in $\A$}. \refstepcounter{subsubsection\label{rho_*} Let $\rho:{\overline{T}M} \to M$ be the canonical projection. Extend the pullback functor $\rho^*:\Bun(M) \to \Bun({\overline{T}M})$ to a functor $$ \rho^*:\Bun(M) \to \Bun_M({\overline{T}M}) $$ to the $M$-adic completion $\Bun_M({\overline{T}M})$. The functor $\rho^*$ admits a right adjoint direct image functor $$ \rho_*:\Bun_M({\overline{T}M}) \to \displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}\Bun(M). $$ Moreover, the functor $\rho_*$ extends to a functor $$ \rho_*:\Diff_M({\overline{T}M}) \to \displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}\Diff(M). $$ Denote by $\B^0(M,\C) = \rho_*\Lambda^0({\overline{T}M})$ the direct image under the projection $\rho:{\overline{T}M} \to M$ of the trivial bundle $\Lambda^0({\overline{T}M})$ on ${\overline{T}M}$. The compact Lie group $U(1)$ acts on ${\overline{T}M}$ by dilatation along the fibers, and the functor $\rho_*:\Diff_M({\overline{T}M}) \to \displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}\Diff(M)$ obviously extends to a functor $\rho_*:{{\cal W}{\cal H}odge}^\D_M({\overline{T}M}) \to \displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}{{\cal W}{\cal H}odge}(M)$. The restriction of $\rho_*$ to the subcategory ${{\cal W}{\cal H}odge}_M({\overline{T}M}) \subset {{\cal W}{\cal H}odge}^\D_M({\overline{T}M})$ is adjoint on the right to the pullback functor $\rho^*:{{\cal W}{\cal H}odge}(M) \to {{\cal W}{\cal H}odge}_M({\overline{T}M})$. \refstepcounter{subsubsection The constant bundle $\Lambda^0({\overline{T}M})$ is canonically a Hodge bundle of weight $0$. Therefore $\B^0(M,\C) = \rho_*\Lambda^0(M,\C)$ is also a Hodge bundle of weight $0$. Moreover, it is a commutative algebra bundle in $\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}{{\cal W}{\cal H}odge}_0(M)$. Let $S^1(M,\C)$ be the conormal bundle to the zero section $M \subset {\overline{T}M}$ equipped with a Hodge bundle structure of weight $0$ as in \ref{S1}, and denote by $S^i(M,\C)$ the $i$-th symmetric power of the Hodge bundle $S^1(M,\C)$. Then the algebra bundle $\B^0(M,\C)$ in $\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}{{\cal W}{\cal H}odge}_0(M)$ is canonically isomorphic $$ \B^0(M,\C) \cong \widehat{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) $$ to the completion $\widehat{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the symmetric algebra $S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the Hodge bundle $S^1(M,\C)$ with respect to the augmentation ideal $S^{>0}(M,\C)$. Since the $U(1)$-action on $M$ is trivial, the category ${{\cal W}{\cal H}odge}(M)$ of Hodge bundles on $M$ is equivalent to the category of pairs $\langle \E, \overline{\ } \rangle$ of a complex bundle $\E$ equipped with a Hodge type grading $$ \E = \bigoplus_{p,q} \E^{p,q} $$ and a real structure $\overline{\ }:\E^{p,q} \to \overline{\E^{q,p}}$. The Hodge type grading on $\B^0(M,\C)$ is induced by the Hodge type grading $S^1(M,\C) = S^{1,-1}(M) \oplus S^{-1,1}(M)$ on the generators subbundle $S^1(M,\C) \subset \B^0(M,\C)$, which was described in \ref{S1}. \noindent {\bf Remark.\ } The complex vector bundle $S^1(M,\C)$ is canonically isomorphic to the cotangent bundle $\Lambda^1(M,\C)$. However, the Hodge bundle structures on these two bundles are different (in fact, they have different weights). \refstepcounter{subsubsection \label{Weil.defn} Consider the pro-bundles $$ \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) = \rho_*\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) $$ on $M$. The direct sum $\oplus\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is a graded algebra in $\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}\Bun(M,\C)$. Moreover, since for every $i \geq 0$ the bundle $\Lambda^i(M,\C)$ is a Hodge bundle of weight $i$ (see \ref{de.Rham}), $\B^i(M,\C)$ is also a Hodge bundle of weight $i$. Denote by $$ \B^i(M,\C) = \bigoplus_{p+q=i} \B^{p,q}(M,\C) $$ the Hodge type bigrading on $\B^i(M,\C)$. The Hodge bundle structures on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ are compatible with the multiplication. By the projection formula $$ \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \cong \B^0(M,\C) \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C), $$ and this isomorphism is compatible with the Hodge bundle structures on both sides. \noindent {\bf Definition.\ } Call the algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ in $\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}{{\cal W}{\cal H}odge}(M)$ {\em the Weil algebra} of the complex manifold $M$. \refstepcounter{subsubsection \label{iota.Weil} The canonical involution $\iota:{\overline{T}M} \to {\overline{T}M}$ induces an algebra involution $\iota^*:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. It acts on generators as follows $$ \iota^* = -{\sf id}:S^1(M,\C) \to S^1(M,\C) \qquad \iota^* = {\sf id}:\Lambda^1(M,\C) \to \Lambda^1(M,\C). $$ For every operator $N:\B^p(M,\C) \to \B^q(M,\C)$, $p$ and $q$ arbitrary, we will denote by $$ N^\iota = \iota^* \circ N \circ \iota^*:\B^p(M,\C) \to \B^q(M,\C) $$ the operator $\iota^*$-conjugate to $N$. \refstepcounter{subsubsection \label{sigma} The twisted tautological map $\sigma:\rho^*\Lambda^1(M,\C) \to \Lambda^0({\overline{T}M},\C)$ introduced in \ref{sigma1} induces via the functor $\rho_*$ a map $\sigma:\B^1(M,\C) \to \B^0(M,\C)$. Extend this map to a derivation $$ \sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C) \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) $$ by setting $\sigma = 0$ on $S^1(M,\C) \subset \B^0(M,\C)$. The derivation $\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C) \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is not weakly Hodge. However, it is real with respect to the real structure on the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. \refstepcounter{subsubsection \label{C.Weil} By definition of the twisted tautological map (\ref{sigma1}, \ref{tau}), the derivation $\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C) \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ maps the subbundle $\Lambda^1(M,\C) \subset \B^1(M,\C)$ to the subbundle $S^1(M,\C) \subset \B^0(M,\C)$ and defines a complex vector bundle isomorphism $\sigma:\Lambda^1(M,\C) \to S^1(M,\C)$. To describe this isomorphism explicitly, recall that sections of the bundle $\B^0(M,\C)$ are the same as formal germs along $M \subset {\overline{T}M}$ of smooth functions on the manifold ${\overline{T}M}$. The sections of the subbundle $S^1(M,\C) \subset \B^0(M,\C)$ form the subspace of functions linear along the fibers of the canonical projection $\rho:{\overline{T}M} \to M$. The isomorphism $\sigma:\Lambda^1(M,\C) \to S^1(M,\C)$ induces an isomorphism between the space of smooth $1$-forms on the manifold $M$ and the space of smooth functions on ${\overline{T}M}$ linear long the fibers of $\rho:{\overline{T}M} \to M$. This isomorphism coincides with the tautological one on the subbundle $\Lambda^{1,0} \subset \Lambda^1(M,\C)$, and it is minus the tautological isomorphism on the subbundle $\Lambda^{0,1} \subset \Lambda^1(M,\C)$. Denote by $$ C = \sigma^{-1}:S^1(M,\C) \to \Lambda^1(M,\C) $$ the bundle isomorphism inverse to $\sigma$. Note that the complex vector bundle isomorphism $\sigma:\Lambda^1(M,\C) \to S^1(M,\C)$ is real. Moreover, it sends the subbundle $\Lambda^{1,0}(M) \subset \Lambda^1(M,\C)$ to $S^{1,-1}(M) \subset S^1(M,\C)$, and it sends $\Lambda^{0,1}(M)$ to $S^{-1,1}(M)$. Therefore the inverse isomorphism $C:S^1(M,\C) \to \Lambda^1(M,\C)$ is weakly Hodge. It coincides with the tautological isomorphism on the subbundle $S^{1,-1} \subset S^1(M,\C)$, and it equals minus the tautological isomorphism on the subbundle $S^{-1,1} \subset S^1(M,\C)$. \subsection{Extended connections}\label{ext.con.subsec} \refstepcounter{subsubsection We are now ready to introduce the extended connections. Keep the notation of the last subsection. \noindent {\bf Definition.\ } \label{ext.con} An {\em extended connection} on a complex manifold $M$ is a differential operator $D:S^1(M,\C) \to \B^1(M,\C)$ which is weakly Hodge in the sense of \ref{w.hodge} and satisfies \begin{equation}\label{e.c} D(fa) = fDa + a \otimes df \end{equation} for any smooth function $f$ and a local section $a$ of the pro-bundle $\B^0(M,\C)$. \refstepcounter{subsubsection \label{red} Let $D$ be an extended connection on the manifold $M$. By \ref{Weil.defn} we have canonical bundle isomorphisms $$ \B^1(M,\C) \cong \B^0(M,\C) \otimes \Lambda^1(M,\C) \cong \bigoplus_{i \geq 0} S^i(M,\C) \otimes \Lambda^1(M,\C). $$ Therefore the operator $D:S^1 \to \B^1$ decomposes \begin{equation}\label{aug.con} D = \sum_{p \geq 0}D_p, \quad D_p:S^1(M,\C) \to S^i(M,\C) \otimes \Lambda^1(M,\C). \end{equation} By \eqref{e.c} all the components $D_p$ except for the $D_1$ are weakly Hodge bundle maps on $M$, while $$ D_1:S^1(M,\C) \to S^1(M,\C) \otimes \Lambda^1(M,\C) $$ is a connection in the usual sense on the Hodge bundle $S^1(M,\C)$. \noindent {\bf Definition.\ } The weakly Hodge bundle map $D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ is called {\em the principal part} of the extended connection $D$ on $M$. The connection $D_1$ is called {\em the reduction} of the extended connection $D$. \refstepcounter{subsubsection Extended connection on $M$ are related to formal Hodge connections on the total space ${\overline{T}M}$ of the complex-conjugate to the tangent bundle to $M$ by means of the direct image functor $$ \rho_*:{{\cal W}{\cal H}odge}^\D_M({\overline{T}M}) \to \displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}{{\cal W}{\cal H}odge}^\D(M). $$ Namely, let $D:\Lambda^0(M,\C) \to \rho^*\Lambda^1(M,\C)$ be a formal Hodge connection on ${\overline{T}M}$ along $M$ in the sense of \ref{formal.hodge.con}. The restriction of the operator $$ \rho_*D:\B^0(M,\C) \to \B^1(M,\C) $$ to the generators subbundle $S^1(M,\C) \subset \B^0(M,\C)$ is then an extended connection on $M$ in the sense of \ref{ext.con}. The principal part $D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ of the Hodge connection $D$ in the sense of \ref{pr.part} coincides with the principal part of the extended connection $\rho_*D$. \refstepcounter{subsubsection We now write down the counterparts of the flatness and linearity conditions on a Hodge connection on ${\overline{T}M}$ for the associated extended connection on $M$. We begin with the linearity condition \ref{hodge.conn.lin}. Let $D:S^1(M,\C) \to \B^1(M,\C)$ be an extended connection on $M$, let $\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C) \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ be the algebra derivation introduced in \ref{sigma}, and let $$ D^\iota:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C) $$ be the operator $\iota^*$-conjugate to $D$ as in \ref{iota.Weil}. \noindent {\bf Definition.\ } \label{lin.ext.con} An extended connection $D$ is called {\em linear} if for every local section $f$ of the bundle $S^1(M,\C)$ we have $$ f = \frac{1}{2} \sigma((D-D^\iota)f). $$ This is, of course, the literal rewriting of Definition~\ref{hodge.conn.lin}. In particular, a formal Hodge connection $D$ on ${\overline{T}M}$ is linear if and only if so is the extended connection $\rho_*D$ on $M$. \refstepcounter{subsubsection \label{deriv} Next we rewrite the flatness condition \ref{hodge.con}. Again, let $$ D:S^1(M,\C) \to \B^1(M,\C) $$ be an extended connection on $M$. Since the algebra pro-bundle $\B^0(M,\C)$ is freely generated by the subbundle $S^1(M,\C) \subset \B^1(M,\C)$, by \eqref{e.c} the operator $D:S^1(M,\C) \to \B^1(M,\C)$ extends uniquely to an algebra derivation $$ D:\B^0(M,\C) \to \B^1(M,\C). $$ Moreover, we can extend this derivation even further to a derivation of the Weil algebra $$ D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C) $$ by setting \begin{equation}\label{D=d} D(f \otimes \alpha) = Df \otimes \alpha + f \otimes d\alpha \end{equation} for any smooth section $f \in C^\infty(M,\B^0(M,\C))$ and any smooth form $\alpha \in C^\infty(M,\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C))$. We will call this extension {\em the derivation, associated to the extended connection $D$}. Vice versa, the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is generated by the subbundles $$ S^1(M,\C),\Lambda^1(M,\C) \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C). $$ Moreover, for every algebra derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ the condition \eqref{D=d} completely defines the restriction of $D$ to the generator subbundle $\Lambda^1(M,\C) \subset \B^1(M,\C)$. Therefore an algebra derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ satisfying \eqref{D=d} is completely determined by its restriction to the generators subbundle $S^1(M,\C) \to \B^1(M,\C)$. If the derivation $D$ is weakly Hodge, then this restriction is an extended connection on $M$. \refstepcounter{subsubsection \noindent {\bf Definition.\ } \label{flat.ext.con} The extended connection $D$ is called {\em flat} if the associated derivation satisfies $D \circ D = 0$. If a formal Hodge connection $D$ on ${\overline{T}M}$ is flat in the sense of \ref{hodge.con}, then we have a derivation $D:\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \rho^*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ such that $D \circ D = 0$. The associated derivation $\rho_*D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ satisfies \eqref{D=d}. Therefore the extended connection $\rho_*D:S^1(M,\C) \to \B^1(M,\C)$ is also flat. \refstepcounter{subsubsection It turns out that one can completely recover a Hodge connection $D$ on ${\overline{T}M}$ from the corresponding extended connection $\rho_*D$ on $M$. More precisely, we have the following. \begin{lemma} The correspondence $D \mapsto \rho_*D$ is a bijection between the set of formal Hodge connections on ${\overline{T}M}$ along $M \subset {\overline{T}M}$ and the set of extended connections on $M$. A connection $D$ is flat, resp. linear if and only if $\rho_*D$ is flat, resp. linear. \end{lemma} \proof To prove the first claim of the lemma, it suffices to prove that every extended connection on $M$ comes from a unique formal Hodge connection on the pair $\langle {\overline{T}M},M\rangle$. In general, the functor $\rho_*$ is not fully faithful on the category $\Diff(M)$, in other words, it does not induce an isomorphism on the spaces of differential operators between vector bundles on ${\overline{T}M}$. However, for every complex vector bundle $\E$ on ${\overline{T}M}$ the functor $\rho_*$ does induce an isomorphism $$ \rho_*:\Der_M(\Lambda^0(M,\C),\E) \cong \Der_{\B^0(M,\C)}(\B^0(M,\C),\rho_*\E) $$ between the space of {\em derivations} from $\Lambda^0(M,\C)$ to $\F$ completed along $M \subset {\overline{T}M}$ and the space of derivations from the algebra $\B^0(M,\C) = \rho_*\Lambda^0(M,\C)$ to the $\B^0(M,\C)$-module $\rho_*\E$. Therefore every derivation $$ D':\B^0(M,\C) \to \B^1(M,\C) = \rho_*\rho^*\Lambda^1(M,\C) $$ must be of the form $D'=\rho_*D$ for some derivation $$ D:\Lambda^0({\overline{T}M},\C) \to \rho^*\Lambda^1(M,\C) $$ It is easy to check that $D$ is a Hodge connection if and only if $D'=\rho_*D$ is weakly Hodge and satisfies \eqref{e.c}. By \ref{deriv} the space of all such derivations $D':\B^0(M,\C) \to \B^1(M,\C)$ coincides with the space of all extended connections on $M$, which proves the first claim of the lemma. Analogously, for every extended connection $D' = \rho_*D$ on $M$, the canonical extension of the operator $D'$ to an algebra derivation $D':\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ constructed in \ref{deriv} must be of the form $\rho_*D$ for a certain weakly Hodge differential operator $D:\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$. If the extended connection $D'$ is flat, then $D' \circ D' = 0$. Therefore $D \circ D = 0$, which means that the Hodge connection $D$ is flat. Vice versa, if the Hodge connection $D$ is flat, then it extends to a weakly Hodge derivation $D:\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ so that $D \circ D = 0$. The equality $D \circ D = 0$ implies, in particular, that the operator $\rho_*D$ vanishes on the sections of the form $$ Df = df \in C^\infty(M,\Lambda^1(M,\C)) \subset C^\infty(M,\B^1(M,\C)), $$ where $f \in C^\infty(M,\C)$ is a smooth function on $M$. Therefore $\rho_*D$ coincides with the de Rham differential on the subbundle $\Lambda^1(M,\C) \subset \B^1(M,\C)$. Hence by \ref{deriv} it is equal to the canonical derivation $D':\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$. Since $D \circ D = 0$, we have $D' \circ D' = 0$, which means that the extended connection $D'$ is flat. Finally, the equivalence of the linearity conditions on the Hodge connection $D$ and on the extended connection $D' = \rho_*D$ is trivial and has already been noted in \ref{lin.ext.con}. \hfill \ensuremath{\square}\par This lemma together with Lemma~\ref{form.hdg} reduces the classification of linear formal Hodge manifold structures on ${\overline{T}M}$ along the zero section $M \subset {\overline{T}M}$ to the classification of extended connections on the manifold $M$ itself. \section{Preliminaries on the Weil algebra}\label{Weil.section} \subsection{The total de Rham complex}\label{de.rham.sub} \refstepcounter{subsubsection Before we proceed further in the study of extended connections on a complex manifold $M$, we need to establish some linear-algebraic facts on the structure of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ defined in \ref{Weil.defn}. We also need to introduce an auxiliary Hodge bundle algebra on $M$ which we call {\em the total Weil algebra}. This is the subject of this section. Most of the facts here are of a technical nature, and the reader is advised to skip this section until needed. \refstepcounter{subsubsection We begin with introducing and studying a version of the de Rham complex of a complex manifold $M$ which we call {\em the total de Rham complex}. Let $M$ be a smooth complex $U(1)$-manifold. Recall that by \ref{de.Rham} the de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the complex manifold $M$ is canonically a Hodge bundle algebra on $M$. Let $\Lambda_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M) = \Gamma(\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C))$ be the weight $0$ Hodge bundle obtained by applying the functor $\Gamma$ defined in \ref{gamma.m} to the de Rham algebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. By \ref{gamma.tensor} the bundle $\Lambda_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M)$ carries a canonical algebra structure. By \ref{de.Rham} the de Rham differential $d_M$ is weakly Hodge. Therefore it induces an algebra derivation $d_M:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}(M)$ which is compatible with the Hodge bundle structure and satisfies $d_M \circ d_M = 0$. \noindent {\bf Definition.\ } The weight $0$ Hodge bundle algebra $\Lambda_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M)$ is called {\em the total de Rham complex} of the complex manifold $M$. \refstepcounter{subsubsection By definition $$ \Lambda^i_{tot}(M) = \Gamma(\Lambda^i(M,\C)) = \Lambda^i(M,\C) \otimes \W^*_i, $$ where $\W^*_i = S^i\W^*_1$ is the symmetric power of the $\R$-Hodge structure $\W^*_1$, as in \ref{w.k}. To describe the structure of the algebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$, we will use the following well-known general fact. (For the sake of completeness, we have included a sketch of its proof, see \ref{gen.symm.proof}.) \begin{lemma}\label{symm} Let $A$, $B$ be two objects in an arbitrary $\Q$-linear symmetric tensor category $\A$, and let $\CC^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(A \otimes B)$ be the sum of symmetric powers of the object $A \otimes B$. Note that the object $\CC^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is naturally a commutative algebra in $\A$ in the obvious sense. Let also $\wt{\CC}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \bigoplus_k S^kA \otimes S^kB$ with the obvious commutative algebra structure. The isomorphism $\CC^1 \cong \wt{\CC}^1 \cong A \otimes B$ extends to a surjective algebra map $\CC^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \wt{\CC}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, and its kernel $\J^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \CC^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is the ideal generated by the subobject $\J^2 = \Lambda^2(A) \otimes \Lambda^2(B) \subset S^2(A \otimes B)$. \end{lemma} \refstepcounter{subsubsection The category of complexes of Hodge bundles on $M$ is obviously $\Q$-linear and tensor. Applying Lemma~\ref{symm} to $A = \W^*_1$, $B = \Lambda^1(M,\C) [1]$ immediately gives the following. \begin{lemma}\label{total.rel} The total de Rham complex $\Lambda_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M)$ of the complex manifold $M$ is generated by its first component $\Lambda^1_{tot}(M)$, and the kernel of the canonical surjective algebra map $$ \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(\Lambda_{tot}^1(M)) \to \Lambda_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M) $$ from the exterior algebra of the bundle $\Lambda^1_{tot}(M)$ to $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$ is the ideal generated by the subbundle $$ \Lambda^2\W_1 \otimes S^2(\Lambda^1(M,\C)) \subset S^2(\Lambda^1_{tot}(M)). $$ \end{lemma} \refstepcounter{subsubsection \label{S} We can describe the Hodge bundle $\Lambda^1_{tot}(M)$ more explicitly in the following way. By definition, as a $U(1)$-equivariant complex vector bundle it equals $$ \Lambda^1_{tot}(M) = \Lambda^1(M,\C) \otimes \W_1^* = \left( \Lambda^{1,0}(M)(1) \oplus \Lambda^{0,1}(M)(0)\right) \otimes \left(\C(0) \oplus \C(-1)\right), $$ where $\Lambda^{p,q}(M)(i)$ is the $U(1)$-equivariant bundle $\Lambda^{p,q}(M)$ tensored with the $1$-dimensional representation of weight $i$, and $\C(i)$ is the constant $U(1)$-bundle corresponding to the representation of weight $i$. If we denote \begin{align*} S^1(M,\C) &= \Lambda^{1,0}(M)(1) \oplus \Lambda^{0,1}(M)(-1) \subset \Lambda^1_{tot}(M),\\ \Lambda^1_{ll}(M) &= \Lambda^{1,0}(M) \subset \Lambda^1_{tot}(M),\\ \Lambda^1_{rr}(M) &= \Lambda^{0,1}(M) \subset \Lambda^1_{tot}(M), \end{align*} then we have $$ \Lambda^1_{tot}(M) = S^1(M,\C) \oplus \Lambda^1_{ll}(M) \oplus \Lambda^1_{rr}(M). $$ The complex conjugation $\overline{\ }:\Lambda^1_{tot}(M) \to \iota^*\overline{\Lambda^1_{tot}(M)}$ preserves the subbundle $$ S^1(M,\C) \subset \Lambda^1_{tot}(M,\C) $$ and interchanges $\Lambda^1_{ll}(M)$ and $\Lambda^1_{rr}(M)$. \refstepcounter{subsubsection\label{S.Hodge.type} If the $U(1)$-action on the manifold $M$ is trivial, then Hodge bundles are the same as bigraded complex vector bundles with a real structure. In this case the Hodge bigrading on the Hodge bundle $\Lambda^1_{tot}(M,\C)$ is given by \begin{align*} \left(\Lambda^1_{tot}(M)\right)^{1,-1} &= S^{1,-1}(M,\C) = \Lambda^{1,0}(M)(1),\\ \left(\Lambda^1_{tot}(M)\right)^{-1,1} &= S^{-1,1}(M,\C) = \Lambda^{0,1}(M)(-1),\\ \left(\Lambda^1_{tot}(M)\right)^{0,0} &= \Lambda^1_{ll}(M) \oplus \Lambda^1_{rr}(M) = \Lambda^1(M,\C). \end{align*} Under these identifications, the real structure on $\Lambda^1_{tot}(M,\C)$ is minus the one induced by the usual real structure on the complex vector bundle $\Lambda^1(M,\C)$. \noindent {\bf Remark.\ } The Hodge bundle $S^1(M,\C)$ is canonically isomorphic to the conormal bundle to the zero section $M \subset {\overline{T}M}$, which we have described in \ref{S1}. \refstepcounter{subsubsection \label{gamma.use} Recall now that we have defined in \ref{l.r} canonical embeddings $\gamma_l,\gamma_r:\W_p^* \to \W_k^*$ for every $0 \leq p \leq k$. Since $\W_0^* = \C$, for every $p,q \geq 0$ we have by \eqref{p+q} a short exact sequence \begin{equation}\label{cap.cup} \begin{CD} 0 @>>> \C @>>> \W_p^* \oplus \W_q^* @>{\gamma_l \oplus \gamma_r}>> \W_{p+q}^* @>>> 0 \end{CD} \end{equation} of complex vector spaces. Recall also that the embeddings $\gamma_l$, $\gamma_r$ are compatible with the natural maps ${\sf can}:\W_p^* \otimes \W_q^* \to \W_{p+q}^*$. Therefore the subbundles defined by \begin{align*} \Lambda^k_l(M) &= \bigoplus_{0 \leq p \leq k} \gamma_l(\W_p^*) \otimes \Lambda^{p,k-p}(M) \subset \Lambda^k_{tot}(M) = \bigoplus_{0 \leq p \leq k} \W_k^* \otimes \Lambda^{p,k-p}(M)\\ \Lambda^k_r(M) &= \bigoplus_{0 \leq p \leq k} \gamma_r(\W_p^*) \otimes \Lambda^{k-p,p}(M) \subset \Lambda^k_{tot}(M) = \bigoplus_{0 \leq p \leq k} \W_k^* \otimes \Lambda^{k-p,p}(M) \end{align*} are actually subalgebras in the total de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$. \refstepcounter{subsubsection \label{l.r.rel} To describe the algebras $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l(M)$ and $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r(M)$ explicitly, note that we obviously have $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M) = \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l(M) + \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r(M)$. Moreover, in the notation of \ref{S} we have \begin{align*} \Lambda^1_l(M) &= S^1(M,\C) \oplus \Lambda^1_{ll}(M) \subset \Lambda^1_{tot}(M),\\ \Lambda^1_r(M) &= S^1(M,\C) \oplus \Lambda^1_{rr}(M) \subset \Lambda^1_{tot}(M). \end{align*} By Lemma~\ref{symm}, the algebra $$ \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l(M) = \left(\bigoplus_p \W_p^* \otimes \Lambda^{p,0}(M)\right) \otimes \left( \bigoplus_q \Lambda^{0,q}(M) \right) $$ is the subalgebra in the total de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$ generated by $\Lambda^1_l(M)$, and the ideal of relations is generated by the subbundle $$ S^2(\Lambda^{1,0}(M)) \otimes \Lambda^2(\W_1^*) \subset \Lambda^2(\Lambda^1_l(M)). $$ Analogously, the subalgebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r(M) \subset \Lambda^1_{tot}(M)$ is generated by $\Lambda^1_r(M)$, and the relations are generated by $$ S^2(\Lambda^{0,1}(M)) \otimes \Lambda^2(\W_1^*) \subset \Lambda^2(\Lambda^1_r(M)). $$ \refstepcounter{subsubsection We will also need to consider the ideals in these algebras defined by \begin{align*} \Lambda^k_{ll}(M) &= \bigoplus_{1 \leq p \leq k} \gamma_l(\W_p^*) \otimes \Lambda^{p,k-p}(M) \subset \Lambda^k_l(M)\\ \Lambda^k_{rr}(M) &= \bigoplus_{1 \leq p \leq k} \gamma_r(\W_p^*) \otimes \Lambda^{k-p,p}(M) \subset \Lambda^k_r(M) \end{align*} The ideal $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{ll}(M) \subset \Lambda^1_l(M)$ is generated by the subbundle $\Lambda^1_{ll}(M) \subset \Lambda^1_l(M)$, and the ideal $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{rr}(M) \subset \Lambda^1_r(M)$ is generated by the subbundle $\Lambda^1_{rr}(M) \subset \Lambda^1_r(M)$. \refstepcounter{subsubsection \label{left.right} Denote by $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o(M) = \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l(M) \cap \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r(M) \subset \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$ the intersection of the subalgebras $\Lambda_l^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M)$ and $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r(M)$. Unlike either of these subalgebras, the subalgebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o(M) \subset \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$ is compatible with the weight $0$ Hodge bundle structure on the total de Rham complex. By \eqref{cap.cup} we have a short exact sequence \begin{equation}\label{shrt} \begin{CD} 0 @>>> \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) @>>> \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l(M) \oplus \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r(M) @>>> \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M) @>>> 0 \end{CD} \end{equation} of complex vector bundles on $M$. Therefore the algebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o(M)$ is isomorphic, as a complex bundle algebra, to the usual de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. As a Hodge bundle algebra it is canonically isomorphic to the exterior algebra of the Hodge bundle $S^1(M,\C)$ of weight $0$ on the manifold $M$. Finally, note that the short exact sequence \eqref{shrt} induces a direct sum decomposition $$ \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M) \cong \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{ll}(M) \oplus \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o(M) \oplus \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{rr}(M). $$ \refstepcounter{subsubsection \noindent {\bf Remark.\ } The total de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$ is related to Simpson's theory of Higgs bundles (see \cite{Shiggs}) in the following way. Recall that Simpson has proved that every (sufficiently stable) complex bundle $\E$ on a compact complex manifold $M$ equipped with a flat connection $\nabla$ admits a unique Hermitian metric $h$ such that $\nabla$ and the $1$-form $\theta = \nabla - \nabla^h \in C^\infty(M,\Lambda^1({{\cal E}\!nd\:}\E))$ satisfy the so-called {\em harmonicity condition}. He also has shown that this condition is equivalent to the vanishing of a certain curvature-like tensor $R \in \Lambda^2(M,{{\cal E}\!nd\:}\E)$ which he associated canonically to every pair $\langle \nabla, \theta \rangle$. Recall that flat bundles $\langle \E, \nabla \rangle$ on the manifold $M$ are in one-to-one correspondence with free differential graded modules $\E \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ over the de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. It turns out that complex bundles $\E$ equipped with a flat connection $\nabla$ and a $1$-form $\theta \in C^\infty(M, \Lambda^1({{\cal E}\!nd\:}\E))$ such that Simpson's tensor $R$ vanishes are in natural one-to-one correspondence with free differential graded modules $\E \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$ over the total de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$. Moreover, a pair $\langle\theta,\nabla\rangle$ comes from a variation of pure $\R$-Hodge structure on $\E$ if and only if there exists a Hodge bundle structure on $\E$ such that the product Hodge bundle structure on the free module $\E \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$ is compatible with the differential. \refstepcounter{subsubsection\label{gen.symm.proof} \proof[Proof of Lemma~\ref{symm}] For every $k \geq 0$ let $G = \Sigma_k \times \Sigma_k$ be the product of two copies of the symmetric group $\Sigma_k$ on $k$ letters. Let $\V_k$ be the $\Q$-representation of $G_k$ induced from the trivial representation of the diagonal subgroup $\Sigma_k \subset G_k$. The representation $\V_k$ decomposes as $$ \V_k = \bigoplus_V V \boxtimes V, $$ where the sum is over the set of irreducible representations $V$ of $\Sigma_k$. We obviously have $$ \CC^k = \Hom_{G_k}\left(\V_k, A^{\otimes k} \otimes B^{\otimes k}\right) = \bigoplus_V \Hom_{\Sigma_k}\left(V,A^{\otimes k}\right) \otimes \Hom_{\Sigma_k}\left(V,A^{\otimes k}\right) . $$ Let $\J^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \CC^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the ideal generated by $\Lambda^2A \otimes \Lambda^2B \subset S^2(A \otimes B)$. It is easy to see that $$ \J^k = \sum_{1 \leq l \leq k-1} \bigoplus_V \Hom_{\Sigma_k}\left(V,A^{\otimes k}\right) \otimes \Hom_{\Sigma_k}\left(V,A^{\otimes k}\right) \subset \CC^k, $$ where the first sum is taken over the set of $k-1$ subgroups $\Sigma_2 \subset \Sigma_k$, the $l$-th one transposing the $l$-th and the $l+1$-th letter, while the second sum is taken over all irreducible constituents $V$ of the representation of $\Sigma_k$ induced from the sign representation of the corresponding $\Sigma_2 \subset \Sigma_k$. Now, there is obviously only one irreducible representation of $\Sigma_k$ which is not encountered as an index in this double sum, namely, the trivial one. Hence $\CC^k / \J^k = S^kA \otimes S^kB$, which proves the lemma. \hfill \ensuremath{\square}\par \subsection{The total Weil algebra}\label{t.W.sub} \refstepcounter{subsubsection \label{S.and.Lambda} Assume from now on that the $U(1)$-action on the complex manifold $M$ is trivial. We now turn to studying the Weil algebra of the manifold $M$. Let $S^1(M,\C) = S^{1,-1}(M,\C) \oplus S^{-1,1}(M,\C)$ be the weight $0$ Hodge bundle on $M$ introduced in \ref{S1}. To simplify notation, denote \begin{align*} S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} &= \widehat{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(S^1(M,\C))\\ \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} &= \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C), \end{align*} where $\widehat{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is the completed symmetric power, and let $$ \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) = S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} $$ be the Weil algebra of the complex manifold $M$ introduced in \ref{Weil.defn}. Recall that the algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ carries a natural Hodge bundle structure. In particular, it is equipped with a Hodge type bigrading $\B^i = \sum_{p+q=i} \B^{p,q}$. \refstepcounter{subsubsection \label{aug} We now introduce a different bigrading on the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The commutative algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is freely generated by the subbundles $$ S^1 = S^{1,-1} \oplus S^{-1,1} \subset \B^0 \quad\text{ and }\quad \Lambda^1 = \Lambda^{1,0} \oplus \Lambda^{0,1} \subset \B^1, $$ therefore to define a multiplicative bigrading on the algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ it suffices to assign degrees to these generator subbundles $S^{1,-1},S^{-1,1},\Lambda^{1,0},\Lambda^{0,1} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. \noindent {\bf Definition.\ } The {\em augmentation bigrading} on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is the multiplicative bigrading defined by setting \begin{align*} \deg S^{1,-1} &= \deg \Lambda^{1,0} = ( 1, 0 )\\ \deg S^{-1,1} &= \deg \Lambda^{0,1} = ( 0, 1 ) \end{align*} on generators $S^{1,-1},S^{-1,1},\Lambda^{1,0},\Lambda^{0,1} \subset \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$. We will denote by $\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_{p,q}$ the component of the Weil algebra of augmentation bidegree $(p,q)$. For any linear map $a:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ we will denote by $a = \sum_{p,q}a_{p,q}$ its decomposition with respect to the augmentation bidegree. It will also be useful to consider a coarser {\em augmentation grading} on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, defined by $\deg\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q} = p + q$. We will denote by $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_k = \bigoplus_{p+q=k}\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q}$ the component of $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of augmentation degree $k$. \refstepcounter{subsubsection Note that the Hodge bidegree and the augmentation bidegree are, in general, independent. Moreover, the complex conjugation $\overline{\ }:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \overline{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ sends $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q}$ to $\overline{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{q,p}}$. Therefore the augmentation bidegree components $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ are not Hodge subbundles. However, the coarser augmentation grading is compatible with the Hodge structures, and the augmentation degree $k$-component $\B^i_k \subset \B^i$ carries a natural Hodge bundle structure of weight $i$. Moreover, the sum $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q} + \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{q,p} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is also a Hodge subbundle. \refstepcounter{subsubsection \label{total.Weil} We now introduce an auxiliary weight $0$ Hodge algebra bundle on $M$, called the total Weil algebra. Recall that we have defined in \ref{gamma.m} a functor $\Gamma:{{\cal W}{\cal H}odge}_{\geq 0}(M) \to {{\cal W}{\cal H}odge}_0(M)$ adjoint on the right to the canonical embedding. Consider the Hodge bundle $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \Gamma(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$ of weight $0$ on $M$. By \ref{gamma.tensor} the multiplication on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ induces an algebra structure on $\Gamma(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$. \noindent {\bf Definition.\ } The Hodge algebra bundle $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of weight $0$ is called {\em the total Weil algebra} of the complex manifold $M$. \noindent {\bf Remark.\ } For a more conceptual description of the functor $\Gamma$ and the total Weil algebra, see Appendix. \refstepcounter{subsubsection By definition of the functor $\Gamma$ we have $\B_{tot}^k = \B^k \otimes \W_k^* = S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda^k \otimes \W_k^* = S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda^k_{tot}$, where $\Lambda^k_{tot} = \Lambda^k \otimes \W_{\:\raisebox{3pt}{\text{\circle*{1.5}}}}^* = \Gamma(\Lambda^k)$ is the total de Rham complex introduced in Subsection~\ref{de.rham.sub}. We have also introduced in Subsection~\ref{de.rham.sub} Hodge bundle subalgebras $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o, \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l, \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r \subset \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ in the total de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ and ideals $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{ll} \subset \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l$, $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{rr} \subset \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r$ in the algebras $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l$, $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r$. Let \begin{align*} \B_o^k &= S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda_o^k \subset \B_{tot}^k\\ \B_l^k &= S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda_l^k \subset \B_{tot}^k\\ \B_r^k &= S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda_r^k \subset \B_{tot}^k \end{align*} be the associated subalgebras in the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ and let \begin{align*} \B_{ll}^k &= S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda_{ll}^k \subset \B_l^k\\ \B_{rr}^k &= S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda_{rr}^k \subset \B_r^k \end{align*} be the corresponding ideals in the Hodge bundle algebras $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l$, $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r$. By \ref{left.right} we have bundle isomorphisms $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} = \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l + \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r$ and $\Lambda_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l \cap \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r$, and the direct sum decomposition $\Lambda_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \cong \Lambda_{ll}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \oplus \Lambda_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \oplus \Lambda_{rr}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Therefore we also have \begin{align}\label{drct} \begin{split} \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} &= \B_l^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} + \B_r^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{ll} \oplus \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o \oplus \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{rr}\\ \B_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} &= \B_l^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \bigcap \B_r^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \end{split} \end{align} Moreover, the algebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o$ is isomorphic to the usual de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, therefore the subalgebra $\B_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is isomorphic to the usual Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. These isomorphisms are {\em not} weakly Hodge. \refstepcounter{subsubsection The total Weil algebra carries a canonical weight $0$ Hodge bundle structure, and we will denote the corresponding Hodge type grading by upper indices: $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \oplus_p\left(\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)^{p,-p}$. The augmentation bigrading on the Weil algebra introduced in \ref{aug} extends to a bigrading of the total Weil algebra, which we will denote by lower indices. In general, both these grading and the direct sum decomposition \eqref{drct} are independent, so that, in general, for every $i \geq 0$ we have a decomposition $$ \B_{tot}^i = \bigoplus_{n,p,q} \left(\B_{ll}^i\right)_{p,q}^{n,-n} \oplus \left(\B_o^i\right)_{p,q}^{n,-n} \oplus \left(\B_{rr}^i\right)_{p,q}^{n,-n}. $$ We would like to note, however, that some terms in this decomposition vanish when $i = 0,1$. Namely, we have the following fact. \begin{lemma}\label{total.aug} Let $n,k$ be arbitrary integers such that $k \geq 0$. \begin{enumerate} \item If $n+k$ is odd, then $\left(\B^0_{tot}\right)_k^{n,-n} = 0$. \item If $n+k$ is even, then $\left(\B^1_{ll}\right)_k^{n,-n} = \left(\B^1_{rr}\right)_k^{n,-n} = 0$, while if $n+k$ is odd, then $\left(\B^1_o\right)_k^{n,-n} = 0$. \end{enumerate} \end{lemma} \proof \begin{enumerate} \item The bundle $\B^0_{tot}$ by definition coincides with $\B^0$, and it is generated by the subbundles $S^{1,-1},S^{-1,1} \subset \B^0$. Both these subbundles have augmentation degree $1$ and Hodge degree $\pm 1$, so that the sum $n+k$ of the Hodge degree with the augmentation degree is even. Since both gradings are multiplicative, for all non-zero components $\B^{n,-n}_k \subset \B^0$ the sum $n+k$ must also be even. \item By definition we have $\B^1_{tot} = \B^0 \otimes \Lambda^1_{tot}$. The subbundle $\Lambda^1_{tot} \subset \B^1_{tot}$ has augmentation degree $1$, and it decomposes $$ \Lambda^1_{tot} = \Lambda^1_o \oplus \Lambda^1_{ll} \oplus \Lambda^1_{rr}. $$ By \ref{S} we have $\Lambda^1_o \cong S^1 = S^{1,-1} \oplus S^{-1,1}$ as Hodge bundles, so that the Hodge degrees on $\Lambda^1_o \subset \Lambda^1_{tot}$ are odd. On the other hand, the subbundles $\Lambda^1_{ll},\Lambda^1_{rr} \subset \Lambda^1_{tot}$ are by \ref{S.Hodge.type} of Hodge bidegree $(0,0)$. Therefore the sum $n+k$ of the Hodge and the augmentation degrees is even for $\Lambda^1_o$ and odd for $\Lambda^1_{ll}$ and $\Lambda^1_{rr}$. Together with \thetag{i} this proves the claim. \hfill \ensuremath{\square}\par \end{enumerate} \subsection{Derivations of the Weil algebra} \refstepcounter{subsubsection We will now introduce certain canonical derivations of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ which will play an important part in the rest of the paper. First of all, to simplify notation, for any two linear maps $a,b$ let $$ \{ a,b \} = a \circ b + b \circ a $$ be their anticommutator, and for any linear map $a:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+i}$ let $a = \sum_{p+q=i} a^{p,q}$ be the Hodge type decomposition. The following fact is well-known, but we have included a proof for the sake of completeness. \begin{lemma} For every two odd derivations $P,Q$ of a graded-commutative algebra $\A$, their anticommutator $\{P,Q\}$ is an even derivation of the algebra $\A$. \end{lemma} \proof Indeed, for every $a,b \in \A$ we have \begin{align*} \{P,Q\}(ab) &= P(Q(ab)) + Q(P(ab)) \\ & = P(Q(a)b + (-1)^{\deg a}aQ(b)) + Q(P(a)b + (-1)^{\deg a}aP(b)) \\ & = P(Q(a))b + (-1)^{\deg Q(a)}Q(a)P(b) + (-1)^{\deg a}P(a)Q(b) \\ &\quad + aP(Q(b)) + Q(P(a))b + (-1)^{\deg P(a)}P(a)Q(b) \\ &\quad + (-1)^{\deg a}Q(a)P(b) + aQ(P(b)) \\ & = P(Q(a))b - (-1)^{\deg a}Q(a)P(b) + (-1)^{\deg a}P(a)Q(b) \\ &\quad + aP(Q(b)) + Q(P(a))b - (-1)^{\deg a}P(a)Q(b) \\ &\quad + (-1)^{\deg a}Q(a)P(b) + aQ(P(b)) \\ & = P(Q(a))b + aP(Q(b)) + Q(P(a))b + a Q(P(b)) \\ & = \{P,Q\}(a)b + a\{P,Q\}(b). \end{align*} \hfill \ensuremath{\square}\par \refstepcounter{subsubsection \label{C.and.sigma} Let $C:S^1 \to \Lambda^1$ be the canonical weakly Hodge map introduced in \ref{C.Weil}. Extend $C$ to an algebra derivation $C:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ by setting $C = 0$ on $\Lambda^1 \subset \B^1$. By \ref{C.Weil} the derivation $C$ is weakly Hodge. The composition $$ C \circ C = \frac{1}{2}\{C,C\}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+2} $$ is also an algebra derivation, and it obviously vanishes on generators $S^1,\Lambda^1 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Therefore $C \circ C = 0$ everywhere. Let also $\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the derivation introduced in \ref{sigma}. The derivation $\sigma$ is not weakly Hodge; however, it is real and admits a decomposition $\sigma = \sigma^{-1,0} + \sigma^{0,-1}$ into components of Hodge types $(-1,0)$ and $(0,-1)$. Both these components are algebra derivations of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. We obviously have $\sigma \circ \sigma = \sigma^{-1,0} \circ \sigma^{-1,0} = \sigma^{0,-1} \circ \sigma^{0,-1} = 0$ on generators $S^1,\Lambda^1 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, and, therefore, on the whole Weil algebra. \noindent {\bf Remark.\ } Up to a sign the derivations $C,\sigma$ and their Hodge bidegree components coincide with the so-called {\em Koszul differentials} on the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. \refstepcounter{subsubsection \label{total.C} The derivation $C:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ is by definition weakly Hodge. Applying the functor $\Gamma$ to it, we obtain a derivation $C:\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ of the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ preserving the weight $0$ Hodge bundle structure on $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The canonical identification $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \cong \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o \subset \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is compatible with the derivation $C:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$. Moreover, by \ref{C.and.sigma} this derivation satisfies $C \circ C = 0:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+2}_{tot}$. Therefore the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ equipped with the derivation $C$ is a complex of Hodge bundles of weight $0$. The crucial linear algebraic property of the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ of the manifold $M$ which will allow us to classify flat extended connections on $M$ is the following. \begin{prop}\label{ac} Consider the subbundle \begin{equation}\label{sbcmp} \bigoplus_{p,q \geq 1}\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}\right)_{p,q} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \end{equation} of the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ consisting of the components of augmentation bidegrees $(p,q)$ with $p,q \geq 1$. This subbundle equipped with the differential $C:\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}\right)_{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to \left(\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}\right)_{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is an acyclic complex of Hodge bundles of weight $0$ on $M$. \end{prop} \refstepcounter{subsubsection We sketch a more or less simple and conceptual proof of Proposition~\ref{ac} in the Appendix. However, in order to be able to study in Section~\ref{convergence} the analytic properties of our formal constructions, we will need an explicit contracting homotopy for the complex \eqref{sbcmp}, which we now introduce. The restriction of the derivation $C:\B^0_{tot} \to \B^1_{tot}$ to the subbundle $S^1 \subset \B_0 \cong \B^0_{tot}$ induces a Hodge bundle isomorphism $$ C:S^1 \to \Lambda^1_o \subset \Lambda^1_{tot} \subset \B^1_{tot}. $$ Define a map $\sigma_{tot}:\Lambda^1_{tot} \to S^1$ by \begin{equation}\label{sigma.c.eq} \sigma_{tot} = \begin{cases} 0 \quad &\text{ on }\quad\Lambda^1_{ll},\Lambda^1_{rr} \subset \Lambda^1_{tot},\\ C^{-1} \quad &\text{ on }\quad\Lambda^1_o \subset \Lambda^1_{tot}. \end{cases} \end{equation} The map $\sigma_{tot}:\Lambda^1_{tot} \to S^1$ preserves the Hodge bundle structures of weight $0$ on both sides. Moreover, its restriction to the subbundle $\Lambda^1 \cong \Lambda^1_o \subset \Lambda^1_{tot}$ coincides with the canonical map $\sigma:\Lambda^1 \to S^1$ introduced in \ref{C.and.sigma}. \refstepcounter{subsubsection \label{sigma.l} Unfortunately, unlike $\sigma:\Lambda^1 \to S^1$, the map $\sigma_{tot}:\Lambda^1_{tot} \to S^1$ does {\em not} admit an extension to a derivation $\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot} \to \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$. We will extend it to a {\em bundle map} $\sigma_{tot}:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ in a somewhat roundabout way. To do this, define a map $\sigma_l:\Lambda^1_l \to S^0 \subset \B^0$ by $$ \sigma_l = \begin{cases} 0 \quad &\text{ on } \quad\Lambda_{ll}^1 \subset \Lambda_l^1 \quad \text{ and on }\quad \left(\Lambda^1_o\right)^{-1,1} \subset \Lambda^1_{tot}, \\ C^{-1} \quad &\text{ on } \quad\left(\Lambda^1_o\right)^{1,-1} \subset \Lambda^1_{tot}. \end{cases} $$ and set $\sigma_l = 0$ on $S^1$. By \ref{gamma.use} we have $$ \Lambda^1_l = \Lambda^{1,0} \oplus \left( \Lambda^{0,1} \otimes \W_1^*\right). $$ The map $\sigma_l:\Lambda^1_l \to S^1$ vanishes on the second summand in this direct sum, and it equals $C^{-1}:\Lambda^{1,0} \to S^{1,-1} \subset S^1$ on the first summand. The restriction of the map $\sigma_l$ to the subbundle $$ \Lambda^{1,0} \oplus \Lambda^{0,1} = \Lambda^1 \cong \Lambda^1_o \subset \Lambda^1_l $$ vanishes on $\Lambda^{0,1}$ and equals $C^{-1}$ on $\Lambda^{1,0}$. Thus it is equal to the Hodge type-$(0,-1)$ component $\sigma^{0,-1}:\Lambda^1 \to S^1$ of the canonical map $\sigma:\Lambda^1 \to S^1$. \refstepcounter{subsubsection By \ref{l.r.rel} the algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l$ is generated by the bundles $S^1$ and $\Lambda^1_l$, and the ideal of relations is generated by the subbundle \begin{equation}\label{rell} S^2\left(\Lambda^{0,1}\right) \otimes \Lambda^2\left(\W_1^*\right) \subset \Lambda^2\left(\Lambda^1_l\right). \end{equation} Since the map $\sigma_l:\Lambda^1_l \to S^1$ vanishes on $\Lambda^{0,1} \otimes \W_1^* \subset \Lambda^1_l$, it extends to an algebra derivation $\sigma_l:\B_l^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l$. The restriction of the derivation $\sigma_l$ to the subalgebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \cong \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_0 \subset \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ coincides with the $(0,-1)$-component $\sigma^{0,-1}$ of the derivation $\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Analogously, the $(-1,0)$-component $\sigma^{-1,0}$ of the derivation $\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ extends to an algebra derivation $\sigma_r:\B_r^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B_r^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of the subalgebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r \subset \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. By definition, the derivation $\sigma_l$ preserves the decomposition $\B_l^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \B_{ll} \oplus \B_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, while the derivation $\sigma_r$ preserves the decomposition $\B_r^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \B_{rr}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \oplus \B_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Both these derivations vanish on $\B_{tot}^0$, therefore both are maps of $\B_{tot}^0$-modules. In addition, the compositions $\sigma_l \circ \sigma_l$ and $\sigma_r \circ \sigma_r$ vanish on generator and, therefore, vanish identically. \refstepcounter{subsubsection \label{sigma.c} Extend both $\sigma_l$ and $\sigma_r$ to the whole $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ by setting \begin{equation}\label{nol} \sigma_l = 0 \text{ on }\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r \qquad \sigma_r = 0 \text{ on }\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l, \end{equation} and let $$ \sigma_{tot} = \sigma_l + \sigma_r:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot} \to \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}. $$ On $\Lambda^1_{tot} \subset \B^1_{tot}$ this is the same map as in \eqref{sigma.c.eq}. The bundle map $\sigma_{tot}:\B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ preserves the direct sum decomposition \eqref{drct}, and its restriction to $\B_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ coincides with the derivation $\sigma$. Note that neither of the maps $\sigma_l$, $\sigma_r$, $\sigma_{tot}$ is a derivation of the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. However, all these maps are linear with respect to the $\B_{tot}^0$-module structure on $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ and preserve the decomposition \eqref{drct}. The map $\sigma_{tot}:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is equal to $\sigma_l$ on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{ll} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$, to $\sigma_r$ on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{rr}$ and to $\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \cong \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$. Since $\sigma_l \circ \sigma_l = \sigma_r \circ \sigma_r = \sigma \circ \sigma = 0$, we have $\sigma_{tot} \circ \sigma_{tot} = 0$. \refstepcounter{subsubsection The commutator $$ h = \{C, \sigma_{tot}\}:\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} $$ of the maps $C$ and $\sigma_{tot}$ also preserves the decomposition \eqref{drct}, and we have the following. \begin{lemma}\label{h.acts} The map $h$ acts as multiplication by $p$ on $\left(\B_{ll}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{p,q}$, as multiplication by $q$ on $\left(\B_{rr}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{p,q}$ and as multiplication by $(p+q)$ on $\left(\B_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{p,q}$. \end{lemma} \proof It suffices to prove the claim separately on each term in the decomposition \eqref{drct}. By definition $\sigma_{tot} = \sigma_l + \sigma_r$, and $h = h_l + h_r$, where $h_l = \{\sigma_l,C\}$ and $h_r = \{\sigma_r,C\}$. Moreover, $h_l$ vanishes on $\B_{rr}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ and $h_r$ vanishes on $\B_{ll}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Therefore it suffices to prove that $h_l = p{\sf id}$ on $\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l\right)_{p,q}$ and that $h_r = q{\sf id}$ on $\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r\right)_{p,q}$. The proofs of these two identities are completely symmetrical, and we will only give a proof for $h_l$. The algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l$ is generated by the subbundles $S^1 \subset \B^0_l$ and $\Lambda^1_l \subset \B^1_l$. The augmentation bidegree decomposition of $S^1$ is by definition given by $$ S^1_{1,0} = S^{1,-1} \qquad\qquad S^1_{0,1} = S^{-1,1}, $$ while the augmentation bidegree decomposition of $\Lambda^1_l$ is given by $$ \left(\Lambda^1_l\right)_{1,0} = \Lambda^{1,0} \qquad \left(\Lambda^1_l\right)_{0,1} = \Lambda^{0,1} \otimes \W_1^*. $$ By the definition of the map $\sigma_l:\Lambda^1_l \to S^1$ (see \ref{sigma.l}) we have $h_l = \{C,\sigma_l\} = {\sf id}$ on $\Lambda^{1,0}$ and $S^{1,-1}$, and $h_l = 0$ on $\Lambda^{0,1} \otimes \W_1^*$ and on $S^{-1,1}$. Therefore for every $p,q \geq 0$ we have $h_l=p{\sf id}$ on the generator subbundles $S^1_{p,q}$ and on $\left(\Lambda^1_l\right)_{p,q}$. Since the map $h_l$ is a derivation and the augmentation bidegree is multiplicative, the same holds on the whole algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l = \oplus\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l\right)_{p,q}$. \hfill \ensuremath{\square}\par Lemma~\ref{h.acts} shows that the map $\sigma_{tot}$ is a homotopy, contracting the subcomplex \eqref{sbcmp} in the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$, which immediately implies Proposition~\ref{ac}. \noindent {\bf Remark.\ } In fact, in our classification of flat extended connections given in Section~\ref{main.section} it will be more convenient for us to use Lemma~\ref{h.acts} directly rather than refer to Proposition~\ref{ac}. \refstepcounter{subsubsection We finish this section with the following corollary of Lemma~\ref{total.aug} and Lemma~\ref{h.acts}, which we will need in Section~\ref{convergence}. \begin{lemma}\label{h.on.b1} Let $n=\pm 1$. If the integer $k \geq 1$ is odd, then the map $h:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ acts on $\left(\B_{tot}^1\right)^{n,-n}_k$ by multiplication by $k$. If $k = 2m \geq 1$ is even, then the endomorphism $h:\left(\B_{tot}^1\right)^{n,-n}_k \to \left(\B_{tot}^1\right)^{n,-n}_k$ is diagonalizable, and its only eigenvalues are $m$ and $m-1$. \end{lemma} \proof If $k$ is odd, then $\left(\B_{tot}^1\right)^{n,-n}_k = \left(\B_o^1\right)^{n,-n}_k$ by Lemma~\ref{total.aug}, and Lemma~\ref{h.acts} immediately implies the claim. Assume that the integer $k = 2m$ is even. By Lemma~\ref{total.aug} we have $$ \left(\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)^{n,-n}_k = \left(\B_{ll}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)^{n,-n}_k \oplus \left(\B_{rr}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)^{n,-n}_k = \B^{n,-n}_{k-1} \otimes \left(\Lambda^1_{ll} \oplus \Lambda^1_{rr}\right). $$ The bundle $\B^0$ is generated by subbundles $S^{1,-1}$ and $S^{-1,1}$. The first of these subbundles has augmentation bidegree $(1,0)$, while the second one has augmentation bidegree $(0,1)$. Therefore for every augmentation bidegree component $\B^{n,-n}_{p,q} \subset \B^{n,-n}_{k-1}$ we have $p - q = n$ and $p+q=k-1$. This implies that $\B^{n,-n}_{k-1} = \B^{n,-n}_{p,q}$ with $p=m-(1-n)/2$ and $q=m-(1+n)/2$. By definition the augmentation bidegrees of the bundles $\Lambda^1_{ll}$ and $\Lambda^1_{rr}$ are, respectively, $(0,1)$ and $(1,0)$. Lemma~\ref{h.acts} shows that the only eigenvalue of the map $h$ on $\left(\B^1_{ll}\right)_{k+1}$ is $p=(m-(1-n)/2))$, while its only eigenvalue on $\left(\B^1_{rr}\right)_{k+1}$ is $q=(m-(1+n)/2)$. Since $n = \pm 1$, one of these numbers equals $m$ and the other one equals $m-1$. \hfill \ensuremath{\square}\par \section{Classification of flat extended connections}\label{main.section} \subsection{K\"ah\-le\-ri\-an connections} \refstepcounter{subsubsection Let $M$ be a complex manifold. In Section~\ref{formal.section} we have shown that formal Hodge manifold structures on the tangent bundle ${\overline{T}M}$ are in one-to-one correspondence with linear flat extended connections on the manifold $M$ (see \ref{ext.con}--\ref{flat.ext.con} for the definitions). It turns out that flat linear extended connections on $M$ are, in turn, in natural one-to-one correspondence with differential operators of a much simpler type, namely, connections on the cotangent bundle $\Lambda^{1,0}(M)$ satisfying certain vanishing conditions (Theorem~\ref{kal=ext}). We call such connections {\em K\"ah\-le\-ri\-an}. In this section we use the results of Section~\ref{Weil.section} establish the correspondence between extended connections on $M$ and K\"ah\-le\-ri\-an connections on $\Lambda^{1,0}(M)$. \refstepcounter{subsubsection We first give the definition of K\"ah\-le\-ri\-an connections. Assume that the manifold $M$ is equipped with a connection $$ \nabla:\Lambda^1(M) \to \Lambda^1(M) \otimes \Lambda^1(M) $$ on its cotangent bundle $\Lambda^1(M)$. Let \begin{align*} T &= \Alt \circ \nabla - d_M:\Lambda^1(M) \to \Lambda^2(M) \\ R &= \Alt \nabla \circ \nabla:\Lambda^1(M) \to \Lambda^1(M) \otimes \Lambda^2(M) \end{align*} be its torsion and curvature, and let $R = R^{2,0} + R^{1,1} + R^{0,2}$ be the decomposition of the curvature according to the Hodge type. \noindent {\bf Definition.\ } The connection $\nabla$ is called {\em K\"ah\-le\-ri\-an} if \begin{align*} T &= 0 \tag{i}\\ R^{2,0} &= R^{0,2} = 0 \tag{ii} \end{align*} \noindent {\bf Example.\ } The Levi-Civita connection on a K\"ahler manifold is K\"ah\-le\-ri\-an. \noindent {\bf Remark.\ } The condition $T=0$ implies, in particular, that the component $$ \nabla^{0,1}:\Lambda^{1,0}(M) \to \Lambda^{1,1}(M) $$ of the connection $\nabla$ coincides with the Dolbeault differential. Therefore a K\"ah\-le\-ri\-an connection is always holomorphic. \refstepcounter{subsubsection Recall that in \ref{red} we have associated to any extended connection $D$ on $M$ a connection $\nabla$ on the cotangent bundle $\Lambda^1(M,\C)$ called the reduction of $D$. We can now formulate the main result of this section. \begin{theorem}\label{kal=ext} \begin{enumerate} \item If an extended connection $D$ on $M$ is flat and linear, then its reduction $\nabla$ is K\"ah\-le\-ri\-an. \item Every K\"ah\-le\-ri\-an connection $\nabla$ on $\Lambda^1(M,\C)$ is the reduction of a unique linear flat extended connection $D$ on $M$. \end{enumerate} \end{theorem} The rest of this section is taken up with the proof of Theorem~\ref{kal=ext}. To make it more accessible, we first give an informal outline. The actual proof starts with Subsection~\ref{pf.first.sub}, and it is independent from the rest of this subsection. \refstepcounter{subsubsection Assume given a K\"ah\-le\-ri\-an connection $\nabla$ on the manifold $M$. To prove Theorem~\ref{kal=ext}, we have to construct a flat linear extended connection $D$ on $M$ with reduction $\nabla$. Every extended connection decomposes into a series $D = \sum_{k \geq 0}D_k$ as in \eqref{aug.con}, and, since $\nabla$ is the reduction of $D$, we must have $D_1 = \nabla$. We begin by checking in Lemma~\ref{lin.aug} that if $D$ is linear, then $D_0 = C$, where $C:S^1(M,\C) \to \Lambda^1(M,\C)$ is as in \ref{C.Weil}. The sum $C + \nabla$ is already a linear extended connection on $M$. By \ref{deriv} it extends to a derivation $D_{\leq 1}$ of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the manifold $M$, but this derivation does not necessarily satisfy $D_{\leq 1} \circ D_{\leq 1} = 0$, thus the extended connection $D_{\leq 1}$ is not necessarily flat. We have to show that one can add the ``correction terms'' $D_k, k \geq 2$ to $D_{\leq 1}$ so that $D = \sum_k D_k$ satisfies all the conditions of Theorem~\ref{kal=ext}. To do this, we introduce in \ref{red.Weil} a certain quotient $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$, called the reduced Weil algebra. The reduced Weil algebra ia defined in such a way that for every extended connection $D$ the associated derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ preserves the kernel of the surjection $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, thus inducing a derivation $\wt{D}:\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \wt{\B}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$. Moreover, the algebra $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ has the following two properties: \begin{enumerate} \item \label{first.prop} The derivation $\wt{D}:\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \wt{\B}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ satisfies $\wt{D} \circ \wt{D} = 0$ if and only if the connection $D_1$ is K\"ah\-le\-ri\-an. \item \label{second.prop} Let $\wt{D}$ be the weakly Hodge derivation of the quotient algebra $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ induced by an arbitrary linear extended connection $D_{\leq 1}$ and such that $\wt{D} \circ \wt{D} = 0$. Then the derivation $\wt{D}$ lifts uniquely to a weakly Hodge derivation $D$ of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ such that $D \circ D = 0$, and the derivation also $D$ comes from a linear extended connection on $M$ (see Proposition~\ref{main} for a precise formulation of this statement). \end{enumerate} \refstepcounter{subsubsection The property \ref{first.prop} is relatively easy to check, and we do it in the end of the proof, in Subsection~\ref{pf.last.sub}. The rest is taken up with establishing the property \ref{second.prop}. The actual proof of this statement is contained in Proposition~\ref{main}, and Subsection~\ref{pf.first.sub} contains the necessary preliminaries. Recall that we have introduced in \ref{aug} a new grading on the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$, called the augmentation grading, so that the component $D_k$ in the decomposition $D = \sum_kD_k$ is of augmentation degree $k$. In order to lift $\wt{D}$ to a derivation $D$ so that $D \circ D = 0$, we begin with the given lifting $D_{\leq 1}$ and then add components $D_k, k \geq 2$, one by one, so that on each step for $D_{\leq k} = D_{\leq 1} + \sum_{2 \leq p \leq k}$ the composition $D_{\leq k} \circ D_{\leq k}$ is zero in augmentation degrees from $0$ to $k$. In order to do it, we must find for each $k$ a solution to the equation \begin{equation}\label{tslv} D_0 \circ D_k = - R_k, \end{equation} where $R_k$ is the component of augmentation degree $k$ in the composition $D_{\leq k-1} \circ D_{\leq k-1}$. This solution must be weakly Hodge, and the extended connection $D_{\leq k} = D_{\leq k-1} + D_k$ must be linear. We prove in Lemma~\ref{lin.aug} that since $D_{\leq 0}$ is linear, we may assume that $D_0 = C$. In addition, since $\wt{D} \circ \wt{D} = 0$, we may assume by induction that the image of $R_k$ lies in the kernel $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of the quotient map $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. \refstepcounter{subsubsection In order to analyze weakly Hodge maps from $S^1(M,\C)$ to the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$, we apply the functor $\Gamma:{{\cal W}{\cal H}odge}_{\geq 0}(M) \to {{\cal W}{\cal H}odge}_0(M)$ constructed in \ref{gamma.m} to the bundle $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ to obtain the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) = \Gamma(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C))$ of weight $0$, which we studied in Subsection~\ref{t.W.sub}. The Hodge bundle $S^1(M,\C)$ on the manifold $M$ is of weight $0$, and, by the universal property of the functor $\Gamma$, weakly Hodge maps from $S^1(M,\C)$ to $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ are in one-to-one correspondence with Hodge bundle maps from $S^1(M,\C)$ to the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$. The canonical map $C:S^1(,\C) \to \B^1(M,\C)$ extends to a derivation $C:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C) \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}(M,\C)$. Moreover, the weakly Hodge map $R_k:S^1(M,\C) \to \B^2(M,\C)$ defines a Hodge bundle map $R_k^{tot}:S^1(M,\C) \to \B^2_{tot}(M,\C)$, and solving \eqref{tslv} is equivalent to finding a Hodge bundle map $D_k:S^1(M,\C) \to \B^1_{tot}(M,\C)$ such that \begin{equation}\label{tslvc} C \circ D_k = - R_k. \end{equation} \refstepcounter{subsubsection Recall that by \ref{C.and.sigma} the derivation $$ C:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C) \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}(M,\C) $$ satisfies $C \circ C = 0$, so that the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$ becomes a complex with differential $C$. The crucial part of the proof of Theorem~\ref{kal=ext} consists in noticing that the subcomplex $\I_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) = \Gamma(\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)) \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$ of the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$ corresponding to the kernel $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the quotient map $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is canonically contractible. This statement is analogous to Proposition~\ref{ac}, and we prove it in the same way. Namely, we check that the subcomplex $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is preserved by the bundle map $\sigma_{tot}:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ constructed in \ref{sigma.c}, and that the anticommutator $h = \{\sigma_{tot},C\}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C) \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$ is invertible on the subcomplex $\I_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$ (Corollary~\ref{h.inv} of Lemma~\ref{h.acts}). We also check that $C \circ R_k^{tot} = 0$, which implies that the Hodge bundle map \begin{equation}\label{sltn} D_k = -h^{-1} \circ \sigma_{tot} \circ R_k^{tot}:S^1(M,\C) \to \B^1_{tot}(M,\C) \end{equation} provides a solution to the equation \eqref{tslvc}. \refstepcounter{subsubsection To establish the property \ref{second.prop}, we have to insure additionally that the extended connection $D = D_{\leq k}$ is linear, and and we have to show that the solution $D_k$ of \eqref{tslvc} with this property is unique. This turns out to be pretty straightforward. We show in Lemma~\ref{lin.aug} that $D_{\leq k}$ is linear if and only if \begin{equation}\label{lnr} \sigma_{tot} \circ D_k = 0. \end{equation} Moreover, we show that the homotopy $\sigma_{tot}:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}(M,\C) \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$ satisfies $\sigma_{tot} \circ \sigma_{tot} = 0$. Therefore the solution $D_k$ to \eqref{tslvc} given by \eqref{sltn} satisfies \eqref{lnr} automatically. The uniqueness of such a solution $D_k$ follows from the invertibility of $h = C \circ \sigma_{tot} + \sigma_{tot} \circ C$. Indeed, for every two solutions $D_k,D_k'$ to \eqref{tslv}, both satisfying \eqref{lnr}, their difference $P = D_k - D'_k$ satisfies $C \circ P = 0$. If, in addition, both $D_k$ and $D'_k$ were to satisfy \eqref{lnr}, we would have had $\sigma_{tot} \circ P = 0$. Therefore $h \circ P = 0$, and $P$ has to vanish. \refstepcounter{subsubsection These are the main ideas of the proof of Theorem~\ref{kal=ext}. The proof itself begins in the next subsection, and it is organized as follows. In Subsection~\ref{pf.first.sub} we express the linearity condition on an extended connection $D$ in terms of the associated derivation $D_{tot}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_C \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$ of the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ of the manifold $M$. After that, we introduce in Subsection~\ref{pf.third.sub} the reduced Weil algebra $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ and prove Proposition~\ref{main}, thus reducing Theorem~\ref{kal=ext} to a statement about derivations of the reduced Weil algebra. Finally, in Subsection~\ref{pf.last.sub} we prove this statement. \noindent {\bf Remark.\ } In the Appendix we give, following Deligne and Simpson, a more geomteric description of the functor $\Gamma:{{\cal W}{\cal H}odge}_{\geq 0} \to {{\cal W}{\cal H}odge}$ and of the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$, which allows to give a simpler and more conceptual proof for the key parts of Theorem~\ref{kal=ext}. \subsection{Linearity and the total Weil algebra}\label{pf.first.sub} \refstepcounter{subsubsection Assume given an extended connection $D:S^1 \to \B^1$ on the manifold $M$, and extend it to a derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ of the Weil algebra as in \ref{deriv}. Let $D = \sum_{k\geq 0} D_k$ be the augmentation degree decomposition. The derivation $D$ is weakly Hodge and defines therefore a derivation $D = \sum_{k\geq 0} D_k:\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ of the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Before we begin the proof of Theorem~\ref{kal=ext}, we give the following rewriting of the linearity condition \ref{lin.ext.con} on the extended connection $D$ in terms of the total Weil algebra. \begin{lemma}\label{lin.aug} The extended connection $D$ is linear if and only if $D_0=C$ and $\sigma_{tot} \circ D_k = 0$ on $S^1 \subset \B_{tot}^0$ for every $k \geq 0$. \end{lemma} \proof Indeed, by Lemma~\ref{total.aug} for odd integers $k$ and $n = \pm 1$ the subbundle $\left(\B_o^1\right)^{n,-n}_{k+1} \subset \B_{tot}^1$ vanishes. Therefore the map $D_k:S^1 \to \B_{tot}^1$ factors through $\B_{ll}^1 \oplus \B_{rr}^1$. Since by definition (\ref{sigma.c}) we have $\sigma_{tot}=0$ on both $\B_{ll}^1$ and $\B_{rr}^1$, for odd $k$ we have $\sigma_{tot} \circ D_k = 0$ on $S^1$ regardless of the extended connection $D$. On the other hand, for even $k$ we have $\left(\B_{tot}^1\right)^{n,-n}_{k+1} = \left(\B_o^1\right)^{n,-n}_{k+1}$. Therefore on $S^1$ we have $\sigma_{tot} \circ D_k = \sigma \circ D_k$ (where $\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is as in \ref{C.and.sigma}). Moreover, since $\sigma:\Lambda^1 \to S^1$ is an isomorphism, $D_0 = C$ is equivalent to $\sigma \circ D_0 = {\sf id}:S^1 \to \Lambda^1 \to S^1$. Therefore the condition of the lemma is equivalent to the following \begin{equation}\label{tprv} \sigma \circ D_k = \begin{cases} {\sf id}, \quad &\text{ for } k=0\\ 0, \quad &\text{ for even integers } k > 0. \end{cases} \end{equation} Let now $\iota^*:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the operator given by the action of the canonical involution $\iota:{\overline{T}M} \to {\overline{T}M}$, as in \ref{iota.Weil}, and let $D^\iota = \sum_{k \geq 0} D_k^\iota = \iota^* \circ D \circ(\iota^*)^{-1}$ be the operator $\iota^*$-conjugate to the derivation $D$. The operator $\iota^*$ acts as $-{\sf id}$ on $S^1 \subset \B^0$ and as ${\sf id}$ on $\Lambda^1 \subset \B^1$. Since it is an algebra automorphism, it acts as $(-1)^{i+k}$ on $\B_k^i \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Therefore $D_k^\iota = (-1)^{k+1}D_k$, and \eqref{tprv} is equivalent to $$ \sigma \circ \frac{1}{2}(D - D^\iota) = {\sf id}:S^1 \to \B^1 \to S^1, $$ which is precisely the definition of a linear extended connection. \hfill \ensuremath{\square}\par \subsection{The reduced Weil algebra}\label{pf.third.sub} \refstepcounter{subsubsection We now begin the proof of Theorem~\ref{kal=ext}. Our first step is to reduce the classification of linear flat extended connections $D:S^1 \to \B^1$ on the manifold $M$ to the study of derivations of a certain quotient $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. We introduce this quotient in this subsection under the name of reduced Weil algebra. We then show that every extended connection $D$ on $M$ induces a derivation $\wt{D}:\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \wt{\B}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ of the reduced Weil algebra, and that a linear flat extended connection $D$ on $M$ is completely defined by the derivation $\wt{D}$. \refstepcounter{subsubsection By Lemma~\ref{h.acts} the anticommutator $h = \{C,\sigma_{tot}\}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ of the canonical bundle endomorphisms $C,\sigma_{tot}$ of the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is invertible on every component $\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}\right)_{p,q}$ of augmentation bidegree $(p,q)$ with $p,q \geq 1$. The direct sum $\oplus_{p,q \geq 1}\left(\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{p,q}$ is an ideal in the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$, and it is obtained by applying the functor $\Gamma$ to the ideal $\oplus_{p,q \geq 1}\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q}$ in the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. For technical reasons, it will be more convenient for us to consider the smaller subbundle $$ \I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \bigoplus_{p \geq 2,q \geq 1}\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q} + \bigoplus_{p \geq 1, q \geq 2} \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} $$ of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The subbundle $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is a Hodge subbundle in $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, and it is an ideal with respect to the multiplication in $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. \refstepcounter{subsubsection \noindent {\bf Definition.\ } \label{red.Weil} The reduced Weil algebra $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the manifold $M$ is the quotient $$ \wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} / \I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} $$ of the full Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ by the ideal $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The reduced Weil algebra decomposes as $$ \wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{1,1} \oplus \bigoplus_{p \geq 0}\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,0} \oplus \bigoplus_{q \geq 0}\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{0,q} $$ with respect to the augmentation bigrading on the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The two summands on the right are equal to \begin{align*} \bigoplus_{p \geq 0}\B_{p,0}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} &= \bigoplus_{p geq 0}S^p\left(S^{1,-1}\right) \otimes \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}, \\ \bigoplus_{q \geq 0}\B_{0,q}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} &= \bigoplus_{q geq 0}S^q\left(S^{-1,1}\right) \otimes \Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}, \end{align*} \refstepcounter{subsubsection Since $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is a Hodge subbundle, the reduced Weil algebra carries a canonical Hodge bundle structure compatible with the multiplication. It also obviously inherits the augmentation bigrading, and defines an ideal $\I_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \Gamma(\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}) \subset \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ in the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Lemma~\ref{h.acts} immediately implies the following fact. \begin{corr}\label{h.inv} The map $h=\{C,\sigma_{tot}\}:\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is invertible on $\I_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. \end{corr} \refstepcounter{subsubsection \label{wD} Let now $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the derivation associated to the extended connection $D$ as in \ref{deriv}. The derivation $D$ does not increase the augmentation bidegree, it preserves the ideal $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ and defines therefore a weakly Hodge derivation of the reduced Weil algebra $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, which we denote by $\wt{D}$. If the extended connection $D$ is flat, then the derivation $\wt{D}$ satisfies $\wt{D} \circ \wt{D} = 0$. We now prove that every derivation $\wt{D}:\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \wt{\B}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ of this type comes from a linear flat extended connection $D$, and that the connection $D$ is completely defined by $\wt{D}$. More precisely, we have the following. \begin{prop}\label{main} Let $D:S^1 \to \B^1$ be a linear but not necessarily flat extended connection on $M$, and let $\wt{D}:\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \wt{\B}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ be the associated weakly Hodge derivation of the reduced Weil algebra $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Assume that $\wt{D} \circ \wt{D} = 0$. There exists a unique weakly Hodge bundle map $P:S^1 \to \I^1$ such that the extended connection $D' = D + P:S^1 \to \B^1$ is linear and flat. \end{prop} \refstepcounter{subsubsection \proof Assume given a linear extended connection $D$ satisfying the condition of Proposition~\ref{main}. To prove the proposition, we have to construct a weakly Hodge map $P:S^1 \to \I^1$ such that the extended connection $D+P$ is linear and flat. We do it by induction on the augmentation degree, that is, we construct one-by-one the terms $P_k$ in the augmentation degree decomposition $P = \sum_k P_k$. The identity $\wt{D} \circ \wt{D} = 0$ is the base of the induction, and the induction step is given by applying the following lemma to $D + \sum_{i=2}^k P_k$, for each $k \geq 1$ in turn. \begin{lemma}\label{main.ind} Assume given a linear extended connection $D:S^1 \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ on $M$ and let $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ also denote the associated derivation. Assume also that the composition $D \circ D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+2}$ maps $S^1$ into $\I^2_{>k} = \oplus_{p>k}\I^2_p$. There exists a unique weakly Hodge bundle map $P_k:S^1 \to \I^1_{k+1}$ such that the extended connection $D' = D + P_k:S^1 \to \B^1$ is linear, and for the associated derivation $D':\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ the composition $D' \circ D'$ maps $S^1$ into $\I^2_{>k+1} = \oplus_{p>k+1}\I^2_p$. \end{lemma} \proof Let $D:\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the derivation of the total Weil algebra associated to the extended connection $D$, and let $$ R:\left(\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \left(\B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+2}\right)_{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+k} $$ be the component of augmentation degree $k$ of the composition $D \circ D:S^1 \to \B^2_{tot}$. Note that by \ref{deriv} the map $R$ vanishes on the subbundle $\Lambda^1_{tot} \subset \left(\B^1_{tot}\right)_1$. Moreover, the composition $C \circ R:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+3}$ of the map $R$ with the canonical derivation $C:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$ vanishes on the subbundle $S^1 \subset \B^0_{tot}$. Indeed, since $C$ maps $S^1$ into $\Lambda^1_{tot}$, the composition $C \circ R$ is equal to the commutator $[C,R]:S^1 \to \B^3_{tot}$. Since the extended connection $D$ is by assumption linear, we have $D_0=C$, and \begin{align*} C \circ R = [C,R] &= \sum_{0 \leq p \leq k}[C,D_p \circ D_{k-p}] =\\ &= [C,\{C,D_k\}] + \sum_{1 \leq p \leq k-1}[C,D_p \circ D_{k-p}]. \end{align*} Since $C \circ C = 0$, the first term in the right hand side vanishes. Let $\Theta = \sum_{1 \leq p \leq k-1}D_p:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$. Then the second term is the component of augmentation degree $k$ in the commutator $[C,\Theta \circ \Theta]:S^1 \to \B^3_{tot}$. By assumption $\{D,D\}=0$ in augmentation degrees $<k$. Therefore we have $\{C,\Theta\} = - \{\Theta,\Theta\}$ in augmentation degrees $< k$. Since $\Theta$ increases the augmentation degree, this implies that in augmentation degree $k$ $$ [C,\Theta \circ \Theta] = \{C,\Theta\} \circ \Theta - \Theta \circ \{C,\Theta\} = [\Theta,\{\Theta,\Theta\}], $$ which vanishes tautologically. The set of all weakly Hodge maps $P:S^1 \to \I_k^1$ coincides with the set of all maps $P:S^1 \to \left(\I_{tot}^1\right)_k$ preserving the Hodge bundle structures. Let $P$ be such a map, and let $D':\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ be the derivation associated to the extended connection $D' = D + P$. Since the extended connection $D$ is by assumption linear, by Lemma~\ref{lin.aug} the extended connection $D'$ is linear if and only if $\sigma_{tot} \circ P = 0$. Moreover, since the augmentation degree-$0$ component of the derivation $D$ equals $C$, the augmentation degree-$k$ component $Q:S^1 \to \B^2_{tot}$ in the composition $D' \circ D'$ is equal to $$ Q = R + \{C,(D'-D)\}. $$ By definition $D'-D:\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ equals $P$ on $S^1 \subset \B_{tot}^0$ and vanishes on $\Lambda^1_{tot} \subset \B_{tot}^1$. Since $C$ maps $S^1$ into $\Lambda^1_{tot} \subset \B_{tot}^1$, we have $Q = R + C \circ P$. Thus, a map $P$ satisfies the condition of the lemma if and only if $$ \begin{cases} C \circ P = -R\\ \sigma_{tot} \circ P = 0 \end{cases} $$ To prove that such a map $P$ is unique, note that these equations imply $$ h \circ P = (\sigma_{tot} \circ C + C \circ \sigma_{tot}) \circ P = \sigma_{tot} \circ R, $$ and $h$ is invertible by Corollary~\ref{h.inv}. To prove that such a map $P$ exists, define $P$ by $$ P = -h^{-1} \circ \sigma_{tot} \circ R:S^1 \to \I^1_{k+1}. $$ The map $h = \{C,\sigma_{tot}\}$ and its inverse $h^{-1}$ commute with $C$ and with $\sigma_{tot}$. Since $\sigma_{tot} \circ \sigma_{tot} = C \circ C0$, we have $\sigma_{tot} \circ P = h^{-1} \circ \sigma_{tot} \circ \sigma_{tot} \circ R = 0$. On the other hand, $C \circ R = 0$. Therefore \begin{align*} C \circ P &= - C \circ h^{-1} \circ \sigma_C \circ R = - h^{-1} \circ C \circ \sigma_{tot} \circ R = \\ &= h^{-1} \circ \sigma_{tot} \circ C \circ R - h^{-1} \circ h \circ R = -R. \end{align*} This finishes the proof of the lemma and of Proposition~\ref{main}. \hfill \ensuremath{\square}\par \subsection{Reduction of extended connections}\label{pf.last.sub} \refstepcounter{subsubsection We now complete the proof of Theorem~\ref{kal=ext}. First we will need to identify explicitly the low Hodge bidegree components of the reduced Weil algebra $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The following is easily checked by direct inspection. \begin{lemma} We have \begin{align*} &\wt{\B}^{2,-1} \oplus \wt{\B}^{1,0} \oplus \wt{\B}^{0,1} \oplus \wt{\B}^{-1,2} = \Lambda^1 \oplus \left(S^1 \otimes \Lambda^1\right) \subset \wt{\B}^1\\ &\wt{\B}^{3,-2} \oplus \wt{\B}^{2,-1} \oplus \wt{\B}^{1,1} \oplus \wt{\B}^{-1,2} \oplus \wt{\B}^{-2,3} = \\ &\qquad\qquad\qquad\qquad\qquad\qquad = \left(S^{1,-1} \otimes \Lambda^{2,0}\right) \oplus \left(S^{-1,1} \otimes \Lambda^{0,2}\right) \oplus \Lambda^2 \subset \wt{\B}^2 \end{align*} \end{lemma} \refstepcounter{subsubsection Let now $\nabla:S^1 \to S^1 \otimes \Lambda^1$ be an arbitrary real connection on the bundle $S^1$. The operator $$ D = C + \nabla:S^1 \to \Lambda^1 \oplus \left(\Lambda^1 \otimes S^1\right) \subset \B^1 $$ is then automatically weakly Hodge and defines therefore an extended connection on $M$. This connection is linear by Lemma~\ref{lin.aug}. Extend $D$ to a derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ as in \ref{deriv}, and let $\wt{D}:\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \wt{\B}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ be the associated derivation of the reduced Weil algebra. \begin{lemma}\label{kal=red} The derivation $\wt{D}$ satisfies $\wt{D} \circ \wt{D} = 0$ if and only if the connection $\nabla$ is K\"ah\-le\-ri\-an. \end{lemma} \proof Indeed, the operator $\wt{D} \circ \wt{D}$ is weakly Hodge, hence factors through a bundle map \begin{multline*} \wt{D} \circ \wt{D}: S^1 \to \wt{\B}^{3,-1} \oplus \wt{\B}^{2,0} \oplus \wt{\B}^{1,1} \oplus \wt{\B}^{0,2} \oplus \wt{\B}^{-1,3} = \\ = \left(S^{1,-1} \otimes \Lambda^{2,0}\right) \oplus \left(S^{-1,1} \otimes \Lambda^{0,2}\right) \oplus \Lambda^2 \subset \B^2. \end{multline*} By definition we have $$ \wt{D} \circ \wt{D} = (C + \nabla) \circ (C + \nabla) = \{C, \nabla\} + \{ \nabla, \nabla \}. $$ An easy inspection shows that the sum is direct, and the first summand equals $$ \{C, \nabla\} = T \circ C:S^1 \to \Lambda^2, $$ where $T$ is the torsion of the connection $\nabla$, while the second summand equals $$ \{\nabla, \nabla\} = R^{2,0} \oplus R^{0,2}: S^{1,-1} \oplus S^{-1,1} \to \left(S^{1,-1} \otimes \Lambda^{2,0}\right) \oplus \left(S^{-1,1} \otimes \Lambda^{0,2}\right), $$ where $R^{2,0}$, $R^{0,2}$ are the Hodge type components of the curvature of the connection $\nabla$. Hence $\wt{D} \circ \wt{D} = 0$ if and only if $R^{2,0} = R^{0,2} = T = 0$, which proves the lemma and finishes the proof of Theorem~\ref{kal=ext}. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection We finish this section with the following corollary of Theorem~\ref{kal=ext} which gives an explicit expression for the augmentation degree-$2$ component $D_2$ of a flat linear extended connection $D$ on the manifold $M$. We will need this expression in Section~\ref{metrics.section}. \begin{corr}\label{D.2} Let $D = \sum_{k \geq 0}D_k:S^1 \to \B^1$ be a flat linear extended connection on $M$, so that $D_0 = C$ and $D_1$ is a K\"ah\-le\-ri\-an connection on $M$. We have $$ D_2 = \frac{1}{3}\sigma \circ R, $$ where $\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is the canonical derivation introduced in \ref{sigma}, and $R = D_1 \circ D_1:S^1 \to S^1 \otimes \Lambda^{1,1} \subset \B^2$ is the curvature of the K\"ah\-le\-ri\-an connection $D_1$. \end{corr} \proof Extend the connection $D$ to a derivation $D_C = \sum_{k \geq 0}D^{tot}_k:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$ of the total Weil algebra. By the construction used in the proof of Lemma~\ref{main.ind} we have $D^{tot}_2 = h^{-1} \circ \sigma_{tot} \circ R_{tot}:S^1 \to \left(\B^1_{tot}\right)_3$, where $h:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is as in Lemma~\ref{h.acts}, the map $\sigma_{tot}:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is the canonical map constructed in \ref{sigma.c}, and $R^{tot}:S^1 \to \left(\B^2_{tot}\right)_3$ is the square $R_{tot} = D^{tot}_1 \circ D^{tot}_1$ of the derivation $D^{tot}_1:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$. By Lemma~\ref{h.on.b1} the map $h$ acts on $\left(\B_{tot}^1\right)_3$ by multiplication by $3$. Moreover, it is easy to check that $$ \left(B^2_{tot}\right)_3 = \left(S^1 \otimes \Lambda^2\right) \oplus \left(S^{-1,1} \otimes \Lambda^{2,0}\right)\oplus \left(S^{1,-1} \otimes \Lambda^{0,2}\right), $$ and the map $R^{tot}:S^1 \to \left(B^2_{tot}\right)_3$ sends $S^1$ into the first summand in this decomposition and coincides with the curvature $R:S^1 \to S^1 \otimes \Lambda^2 \subset \left(B^2_{tot}\right)_3$ of the K\"ah\-le\-ri\-an connection $D_1$. Therefore $\sigma_{tot} \circ R^{tot} = \sigma \circ R$, which proves the claim. \hfill \ensuremath{\square}\par \section{Metrics}\label{metrics.section} \subsection{Hyperk\"ahler metrics on Hodge manifolds} \refstepcounter{subsubsection Let $M$ be a complex manifold equipped with a K\"ah\-le\-ri\-an connection $\nabla$, and consider the associated linear formal Hodge manifold structure on the tangent bundle ${\overline{T}M}$. In this section we construct a natural bijection between the set of all polarizations on the Hodge manifold ${\overline{T}M}$ in the sense of Subsection~\ref{polarization} and the set of all K\"ahler metrics on $M$ compatible with the given connection $\nabla$. \refstepcounter{subsubsection \label{restr} Let $h$ be a hyperk\"ahler metric on ${\overline{T}M}$, or, more generally, a formal germ of such a metric in the neighborhood of the zero section $M \subset {\overline{T}M}$. Assume that the metric $h$ is compatible with the given hypercomplex structure and Hermitian-Hodge in the sense of \ref{hermhodge}, and let $\omega_I$ be the K\"ahler form associated to $h$ in the preferred complex structure ${\overline{T}M}_I$ on ${\overline{T}M}$. Let $h_M$ be the restriction of the metric $h$ to the zero section $M \subset {\overline{T}M}$, and let $\omega \in C^\infty(M,\Lambda^{1,1}(M))$ be the associated real $(1,1)$-form on the complex manifold $M$. Since the embedding $M \subset {\overline{T}M}_I$ is holomorphic, the form $\omega$ is the restriction onto $M$ of the form $\omega_I$. In particular, it is closed, and the metric $h_M$ is therefore K\"ahler. \refstepcounter{subsubsection The main result of this section is the following. \begin{theorem}\label{metrics} Restriction onto the zero section $M \subset {\overline{T}M}$ defines a one-to-one correspondence between \begin{enumerate} \item K\"ahler metrics on $M$ compatible with the K\"ah\-le\-ri\-an connection $\nabla$, and \item formal germs in the neighborhood on $M \subset {\overline{T}M}$ of Hermitian-Hodge hyperk\"ahler metrics on ${\overline{T}M}$ compatible with the given formal Hodge manifold structure. \end{enumerate} \end{theorem} \noindent The rest of this section is devoted to the proof of this theorem. \refstepcounter{subsubsection In order to prove Theorem~\ref{metrics}, we reformulate it in terms of polarizations rather than metrics. Recall (see \ref{positive}) that a {\em polarization} of the formal Hodge manifold ${\overline{T}M}$ is by definition a $(2,0)$-form $\Omega \in C^\infty_M({\overline{T}M},\Lambda^{2,0}({\overline{T}M}_J))$ for the complementary complex structure ${\overline{T}M}_J$ which is holomorphic, real and of $H$-type $(1,1)$ with respect to the canonical Hodge bundle structure on $\Lambda^{2,0}({\overline{T}M}_J)$, and satisfies a certain positivity condition \eqref{P}. \refstepcounter{subsubsection By Lemma~\ref{pol.hm} Hermitian-Hodge hyperk\"ahler metrics on ${\overline{T}M}$ are in one-to-one correspondence with polarizations. Let $h$ be a metric on $M$, and let $\omega_I$ and $\omega$ be the K\"ahler forms for $h$ on ${\overline{T}M}_I$ and on $M \subset {\overline{T}M}_I$. The corresponding polarization $\Omega \in C^\infty({\overline{T}M},\Lambda^{2,0}({\overline{T}M}_J))$ satisfies by \eqref{omega.and.Omega} \begin{equation}\label{pol2kal} \omega_I = \frac{1}{2}\left(\Omega + \nu(\Omega)\right) \in \Lambda^2({\overline{T}M},\C), \end{equation} where $\nu:\Lambda^2({\overline{T}M},\C) \to \overline{\Lambda^2({\overline{T}M},\C)}$ is the usual complex conjugation. \refstepcounter{subsubsection Let $\rho:{\overline{T}M} \to M$ be the natural projection, and let $$ \Res:\rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) $$ be the map given by the restriction onto the zero section $M \subset {\overline{T}M}$. Both bundles are naturally Hodge bundles of the same weight on $M$ in the sense of \ref{hodge.bundles}, and the bundle map $\Res$ preserves the Hodge bundle structures. Since $\Omega$ is of $H$-type $(1,1)$, the form $\Res\Omega \in C^\infty(M,\Lambda^2(M,\C))$ is real and of Hodge type $(1,1)$. By \eqref{pol2kal} \begin{multline*} \Res\Omega = \frac{1}{2}\left(\Res\Omega + \overline{\Res\Omega}\right) = \frac{1}{2}(\Omega+\nu(\Omega))|_{M \subset {\overline{T}M}} = \\ = \omega_I|_{M \subset {\overline{T}M}} = \omega \in \Lambda^{1,1}(M). \end{multline*} Therefore to prove Theorem~\ref{metrics}, it suffices to prove the following. \begin{itemize} \item \label{to.prove} For every polarization $\Omega$ of the formal Hodge manifold ${\overline{T}M}$ the restriction $\omega = \Res\Omega \in C^\infty(\Lambda^{1,1}(M))$ is compatible with the connection $\nabla$, that is, $\nabla\omega=0$. Vice versa, every real positive $(1,1)$-form $\omega \in C^\infty(\Lambda^{1,1}(M))$ satisfying $\nabla\omega=0$ extends to a polarization $\Omega$ of ${\overline{T}M}$, and such an extension is unique. \end{itemize} This is what we will actually prove. \subsection{Preliminaries}\label{drm.mod} \refstepcounter{subsubsection\label{descr} We begin with introducing a convenient model for the holomorphic de Rham algebra $\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J)$ of the complex manifold ${\overline{T}M}_J$, which would be independent of the Hodge manifold structure on ${\overline{T}M}$. To construct such a model, consider the relative de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M}/M,\C)$ of ${\overline{T}M}$ over $M$ (see \ref{relative.de.rham.sub} for a reminder of its definition and main properties). Let $\pi:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M},\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M}/M,\C)$ be the canonical projection. Recall that the bundle $\Lambda^i({\overline{T}M}/M,\C)$ of relative $i$-forms on ${\overline{T}M}$ over $M$ carries a natural structure of a Hodge bundle of weight $i$. Moreover, we have introduced in \eqref{eta} a Hodge bundle isomorphism $$ \eta:\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M}/M,\C) $$ between $\Lambda^i({\overline{T}M}/M,\C)$ and the pullback $\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the bundle of $\C$-valued $i$-forms on $M$. \begin{lemma}\label{ident.bis} \begin{enumerate} \item The projection $\pi$ induces an algebra isomorphism $$ \pi:\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M}/M,\C) $$ compatible with the natural Hodge bundle structures. \item Let $\alpha \in C^\infty_M({\overline{T}M},\Lambda^{i,0}({\overline{T}M}_J))$ be a smooth $(i,0)$-form on ${\overline{T}M}_J$, and consider the smooth $i$-form $$ \beta = \eta^{-1}\pi(\alpha) \in C^\infty_M({\overline{T}M},\Lambda^i({\overline{T}M},\C)) $$ on ${\overline{T}M}$. The forms $\alpha$ and $\beta$ have the same restriction to the zero section $M \subset {\overline{T}M}$. \end{enumerate} \end{lemma} \proof Since $\eta$, $\pi$ and the restriction map are compatible with the algebra structure on $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$, it suffices to prove both claims for $\Lambda^1(M,\C)$. For every bundle $\E$ on ${\overline{T}M}$ denote by $\E|_{M \subset {\overline{T}M}}$ the restriction of $\E$ to the zero section $M \subset {\overline{T}M}$. Consider the bundle map $$ \chi = \eta \circ \Res:\Lambda^{1,0}({\overline{T}M}_J)|_{M \subset {\overline{T}M}} \to \Lambda^1(M,\C) \to \Lambda^1({\overline{T}M}/M,\C)|_{M \subset {\overline{T}M}}. $$ The second claim of the lemma is then equivalent to the identity $\chi = \pi$.\! Moreover, note that the contraction with the canonical vector field $\phi$ on ${\overline{T}M}$ defines an injective map $i_\phi:C^\infty(M,\Lambda^1({\overline{T}M}/M,\C)|_M) \to C^\infty({\overline{T}M},\C)$. Therefore it suffices to prove that $i_\phi \circ \chi = i_\phi \circ \pi$. Every smooth section $s$ of the bundle $\Lambda^{1,0}({\overline{T}M}_J)|_{M \subset {\overline{T}M}}$ is of the form $$ s = (\rho^*\alpha + \sqrt{-1}j\rho^*\alpha)|_{M \subset {\overline{T}M}}, $$ where $\alpha \in C^\infty(M,\Lambda^1(M,\C))$ is a smooth $1$-form on $M$, and $j:\Lambda^1({\overline{T}M},\C) \to \overline{\Lambda^1({\overline{T}M},\C)}$ is the map induced by the quaternionic structure on ${\overline{T}M}$. For such a section $s$ we have $\Res(s) = \alpha$, and by \eqref{phi.and.tau} $i_\phi(\chi(s)) = \sqrt{-1}\tau(\alpha)$, where $\tau:C^\infty(M,\Lambda^1(M,\C)) \to C^\infty({\overline{T}M},\C)$ is the tautological map introduced in \ref{tau}. On the other hand, since $\pi \circ \rho^* = 0$, we have $$ i_\phi(\pi(s)) = i_\phi(\pi(\sqrt{-1}j\rho^*\alpha)) = i_\phi(\sqrt{-1}j\rho^*\alpha). $$ Since the Hodge manifold structure on ${\overline{T}M}$ is linear, this equals $$ i_\phi(\pi(s)) = \sqrt{-1}i_\phi(j(\rho^*\alpha)) = \sqrt{-1}\tau(\alpha) = \chi(s), $$ which proves the second claim of the lemma. Moreover, it shows that the restriction of the map $\pi$ to the zero section $M \subset {\overline{T}M}$ is an isomorphism. As in the proof of Lemma~\ref{ident}, this implies that the map $\pi$ is an isomorphism on the whole ${\overline{T}M}$, which proves the first claim and finishes the proof of the lemma. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection Lemma~\ref{ident.bis}~\thetag{i} allows to define the bundle isomorphism $$ \pi^{-1} \circ \eta:\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M}/M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J), $$ between $\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ and $\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J)$, and it induces an isomorphism $$ \rho_*(\eta \circ \pi^{-1}):\rho_*\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \cong \rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J) $$ between the direct images of these bundles under the canonical projection $\rho:{\overline{T}M} \to M$. On the other hand, by adjunction we have the canonical embedding $$ \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \hookrightarrow \rho_*\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C), $$ and by the projection formula it extends to an isomorphism $$ \rho_*\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \cong \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \otimes \B^0, $$ where $\B^0 = \rho_*\Lambda^0({\overline{T}M},\C)$ is the $0$-th component of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of $M$. All these isomorphisms are compatible with the Hodge bundle structures and with the multiplication. \refstepcounter{subsubsection \label{ident.punkt} It will be convenient to denote the image $\rho_*(\eta \circ \pi^{-1})\left(\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)\right) \subset \rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J)$ by $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ or, to simplify the notation, by $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. (The algebra $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is, of course, canonically isomorphic to $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$.) We then have the identification \begin{equation}\label{ident.zero} L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^0 \cong \rho_*\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \cong \rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J). \end{equation} This identification is independent of the Hodge manifold structure on ${\overline{T}M}$. Moreover, by Lemma~\ref{ident.bis}~\thetag{ii} the restriction map $\Res:\rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is identified under \eqref{ident.zero} with the canonical projection $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^0 \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^0_0 \cong L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. By Lemma~\ref{ident} we also have the identification $\rho_*\Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J) \cong \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Therefore \eqref{ident.zero} extends to an algebra isomorphism \begin{equation}\label{ident.formula} \rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J) \cong \rho_*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M}/M,\C) \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \cong L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}. \end{equation} This isomorphism is also compatible with the Hodge bundle structures on both sides. \subsection{The Dolbeault differential on $\protect{\overline{T}M}_J$} \refstepcounter{subsubsection Our next goal is to express the Dolbeault differential $\bar\partial_J$ of the complex manifold ${\overline{T}M}_J$ in terms of the model for the de Rham complex $\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J)$ given by \eqref{ident.formula}. For every $k \geq 0$ denote by $$ D:L^k \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^k \otimes \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}. $$ the differential operator induced by $\bar\partial_J:\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}({\overline{T}M}_J)$ under \eqref{ident.formula}. The operator $D$ is weakly Hodge. It satisfies the Leibnitz rule with respect to the algebra structure on $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, and we have $D \circ D = 0$. By definition for $k=0$ it coincides with the derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ defined by the Hodge manifold structure on ${\overline{T}M}$. For $k > 0$ the complex $\langle L^k \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}, D \rangle$ is a free differential graded module over the Weil algebra $\langle\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}},D\rangle$. \refstepcounter{subsubsection \label{dr.L} The relative de Rham differential $d^r$ (see Subsection~\ref{relative.de.rham.sub}) induces under the isomorphism \eqref{ident.formula} an algebra derivation $$ d^r:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}. $$ The derivation $d^r$ also is weakly Hodge, and we have the following. \begin{lemma}\label{D.dr} The derivations $D$ and $d^r$ commute, that is, $$ \{D,d^r\} = 0:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \otimes \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}. $$ \end{lemma} \proof The operator $\{D,d^r\}$ satisfies the Leibnitz rule, so it suffices to prove that it vanishes on $\B^0$, $\B^1$ and $L^1 \otimes \B^0$. Moreover, the $\B^0$-modules $\B^1$ and $L^1 \otimes \B^0$ are generated, respectively, by local sections of the form $Df$ and $d^rf$, $f \in \B^0$. Since $\{D,d^r\}$ commutes with $D$ and $d^r$, it suffices to prove that it vanishes on $\B^0$. Finally, $\{D,d^r\}$ is continuous in the adic topology on $\B^0$. Since the subspace $$ \{fg|f,g \in \B^0, Df = d^rg = 0\} \subset \B^0 $$ is dense in this topology, it suffices to prove that for a local section $f \in \B^0$ we have $\{D,d^r\}f=0$ if either $d^rf=0$ of $Df=0$. It is easy to see that for every local section $\alpha \in \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ we have $d^r\alpha = 0$ if and only if $\alpha \in \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_0$ is of augmentation degree $0$. By definition the derivation $D$ preserves the component $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_0 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of augmentation degree $0$ in $\B^0$. Therefore $d^rf=0$ implies $d^rDf=0$ and consequently $\{D,d^r\}f=0$. This handles the case $d^rf=0$. To finish the proof, assume given a local section $f \in \B^0$ such that $Df=0$. Such a section by definition comes from a germ at $M \subset {\overline{T}M}$ of a holomorphic function $\wh{f}$ on ${\overline{T}M}_J$. Since $\wh{f}$ is holomorphic, we have $\bar\partial_J\wh{f}=0$ and $d\wh{f} = \partial_J\wh{f}$. Therefore $d^rf = \pi(d\wh{f}) = \pi(\partial_Jf)$, and $$ Dd^rf = \pi(\bar\partial_J\partial_J\wh{f}) = -\pi(\partial_J\bar\partial_J\wh{f}) = 0, $$ which, again, implies $\{D,d^r\}f = 0$. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection Let now $\nabla=D_1:S^1 \to S^1 \otimes \Lambda^1$ be the reduction of the extended connection $D$. It induces a connection on the bundle $L^1 \cong S^1$, and this connection extends by the Leibnitz rule to a connection on the exterior algebra $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of the bundle $L^1$, which we will also denote by $\nabla$. Denote by $R = \nabla \circ \nabla:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda^2$ the curvature of the connection $\nabla$. Since $\nabla \circ \nabla = \frac{1}{2}\{\nabla,\nabla\}$, the operator $R$ also satisfies the Leibnitz rule with respect to the multiplication in $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. \refstepcounter{subsubsection Introduce the augmentation grading on the bundle $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ by setting $\deg L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = 0$. The derivation $D:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ obviously does not increase the augmentation degree, and we have the decomposition $D = \sum_{k \geq 0}D_k$. On the other hand, the derivation $d^r$ preserves the augmentation degree. Therefore Lemma~\ref{D.dr} implies that for every $k \geq 0$ we have $\{D_k,d^r\} = 0$. This in turn implies that $D_0 = 0$ on $L^p$ for $p > 0$, and therefore $D_0 = {\sf id} \otimes C:L^p \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^p \otimes \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$. Moreover, this allows to identify explicitly the components $D_1$ and $D_2$ of the derivation $D:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^1$. Namely, we have the following. \begin{lemma}\label{D.1.2} We have \begin{align*} D_1 &= \nabla:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^1_1 = L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda^1\\ D_2 &= \frac{1}{3}\sigma \circ R:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^1_2, \end{align*} where $\sigma = {\sf id} \otimes \sigma:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is as in \ref{C.and.sigma}. \end{lemma} \proof Since both sides of these identities satisfy the Leibnitz rule with respect to the multiplication in $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, it suffices to prove them for $L^1$. But $d^r:\B^0 \to L^1 \otimes \B^0$ restricted to $S^1 \subset \B^0$ becomes an isomorphism $d^r:S^1 \to L^1$. Since $\{D_1,d^r\} = \{D_2,d^r\} = 0$, it suffices to prove the identities with $L^1$ replaced with $S^1$. The first one then becomes the definition of $\nabla$, and the second one is Corollary~\ref{D.2}. \hfill \ensuremath{\square}\par \subsection{The proof of Theorem~\ref{metrics}} \refstepcounter{subsubsection We can now prove Theorem~\ref{metrics} in the form \ref{to.prove}. We begin with the following corollary of Lemma~\ref{D.1.2}. \begin{corr}\label{red.corr} Let $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the ideal introduced in \ref{red.Weil}. An arbitrary smooth section $\alpha \in C^\infty(M,L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$ satisfies \begin{equation}\label{rrd} D\alpha \in C^\infty(M, L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \I^1) \subset C^\infty(M, L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^1) \end{equation} if and only if $\nabla \alpha = 0$. \end{corr} \proof Again, both the identity \eqref{rrd} and the equality $\nabla\alpha$ are compatible with the Leibnitz rule with respect to the multiplication in $\alpha$. Therefore it suffices to prove that they are equivalent for every $\alpha \in L^1$. By definition of the ideal $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ the equality \eqref{rrd} holds if and only if $D_1\alpha = D_2\alpha = 0$. By Lemma~\ref{D.1.2} this is equivalent to $\nabla\alpha = \sigma \circ R(\alpha) = 0$. But since $R = \nabla \circ \nabla$, $\nabla\alpha = 0$ implies $\sigma \circ R(\alpha) = 0$, which proves the claim. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection Let now $\Omega \in C^\infty(M, \rho_*\Lambda^{2,0}({\overline{T}M}_J) \cong L^2 \otimes \B^0$ be a polarization of the Hodge manifold ${\overline{T}M}_J$, so that $\Omega$ is of Hodge type $(1,1)$ and $D\Omega=0$. Let $\omega = \Res\Omega \in C^\infty(M,\Lambda^{1,1}(M,\C))$ be its restriction, and let $\Omega = \sum_{k \geq 0}\Omega_k$ be its augmentation degree decomposition. As noted in \ref{ident.punkt}, the restriction map $\Res:\rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is identified under the isomorphism \eqref{ident.formula} with the projection $\LL^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^0 \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ onto the component of augmentation degree $0$. Therefore $\omega = \Omega_0$. Since the augmentation degree-$1$ component $\left(L^2 \otimes B^0_{tot}\right)^{1,1}_1=0$ and $D\Omega = 0$, we have $\nabla\omega = D_1\Omega_0 = 0$, which proves the first claim of Theorem~\ref{metrics}. \refstepcounter{subsubsection To prove the second claim of the theorem, let $\omega$ be a K\"ahler form on $M$ compatible with the connection $\nabla$, so that $\nabla\omega = 0$. We have to show that there exists a unique section $\Omega = \sum_{k \geq 0}\Omega_k \in C^\infty(M, L^2 \otimes \B^0)$ which is of Hodge type $(1,1)$ and satisfies $D\Omega=0$ and $\Omega_0 = \omega$. As in the proof of Theorem~\ref{kal=ext}, we will use induction on $k$. Since $\Omega$ is of Hodge type $(1,1)$, we must have $\Omega_1 = 0$, and by Corollary~\ref{red.corr} we have $D(\Omega_0 + \Omega_1) \in C^\infty(M, L^2 \otimes \I^1)$, which gives the base of our induction. The induction step is given by applying the following proposition to $\sum_{0 \geq p \geq k}\Omega_k$ for each $k \geq 1$ in turn. \begin{prop}\label{metrics.ind} Assume given integers $p,q,k; p,q \geq 0, k \geq 1$ and assume given a section $\alpha \in C^\infty(M, L^{p+q} \otimes \B^0)$ of Hodge type $(p,q)$ such that $$ D\alpha \in C^\infty(M, L^{p+q} \otimes \I^1_{\geq k}), $$ where $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{\geq k} = \oplus_{m \geq k}\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_m$. Then there exists a unique section $\beta \in C^\infty(M, L^{p+q} \otimes \B_k)$ of the same Hodge type $(p,q)$ and such that $D(\alpha + \beta) \in C^\infty(M, L^{p+q} \otimes \I^1_{\geq k+1})$. \end{prop} \proof Let $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the total Weil algebra introduced in \ref{total.Weil}, and consider the free module $L^{p+q} \otimes \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ over $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ generated by the Hodge bundle $L^{p+q}$. This module carries a canonical Hodge bundle structure of weight $p+q$. Consider the maps $C:\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$, $\sigma_{tot}:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ introduced in \ref{total.C} and \ref{sigma.c}, and let $C = {\sf id} \otimes C,\sigma_{tot} = {\sf id} \otimes \sigma_{tot}:L^{p+q} \otimes \B_{tot} \to L^{p+q} \otimes \B_{tot}$ be the associated endomorphisms of the free module $L^{p+q} \otimes \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The maps $C$ and $\sigma_{tot}$ preserve the Hodge bundle structure. The commutator $h = \{C, \sigma_{tot}\}: \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is invertible on $\I_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ by Corollary~\ref{h.inv} and acts as $k{\sf id}$ on $\B^0_k \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$. Therefore the endomorphism $$ h = {\sf id} \otimes h = \{C,\sigma_{tot}\}:L^{p+q} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to L^{p+q} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} $$ is invertible on $L^{p+q} \otimes \I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ and acts as $k{\sf id}$ on $L^{p+q} \otimes \B^0_k$. Since the derivation $D:L^{p+q} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{p+q} \otimes \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ is weakly Hodge, it induces a map $D^{tot}:L^{p+q} \otimes \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{p+q} \otimes \B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$, and $D\alpha \in L^{p+q} \otimes \I^1_{\geq k}$ if and only if the same holds for $D^{tot}\alpha$. To prove uniqueness, note that $D_0 = C$ is injective on $L^{p+q} \otimes \B^0_k$. If there are two sections $\beta,\beta'$ satisfying the conditions of the proposition, then $D_0(\beta-\beta')=0$, hence $\beta = \beta'$. To prove existence, let $\gamma = (D^{tot}\alpha)_k$ be the component of the section $D^{tot}\alpha$ of augmentation degree $k$. Since $D^{tot} \circ D^{tot}=0$, we have $C\gamma = D^{tot}_0\gamma = 0$ and $C \circ \sigma_{tot} \gamma = h \gamma$. Let $\beta = - \frac{1}{k} \circ \sigma_{tot}(\gamma)$. The section $\beta$ is of Hodge type $(p,q)$ and of augmentation degree $k$. Moreover, it satisfies $$ D^{tot}_0\beta = C\beta = -Ch^{-1}\sigma_{tot}(\gamma) = -h^{-1} \circ C \circ \sigma_{tot} \gamma = -h^{-1} \circ h\gamma = -\gamma. $$ Therefore $D_{tot}(\alpha+\beta)$ is indeed a section of $L^{p+q} \otimes (I^1_{tot})_{\geq k+1}$, which proves the proposition and finishes the proof of Theorem~\ref{metrics}. \hfill \ensuremath{\square}\par \subsection{The cotangent bundle} \refstepcounter{subsubsection For every K\"ahler manifold $M$ Theorem~\ref{metrics} provides a canonical formal hyperk\"ahler structure on the total space ${\overline{T}M}$ of the complex-conjugate to the tangent bundle to $M$. In particular, we have a closed holomorphic $2$-form $\Omega_I$ for the preferred complex structure ${\overline{T}M}_I$ on $M$. Let $T^*M$ be the total space to the cotangent bundle to $M$ equipped with the canonical holomorphic symplectic form $\Omega$. To obtain a hyperk\"ahler metric of the formal neighborhood of the zero section $M \subset T^*M$, one can apply an appropriate version of the Darboux Theorem, which gives a local symplectic isomorphism $\kappa:{\overline{T}M} \to T^*M$ in a neighborhood of the zero section. However, this theorem is not quite standard in the holomorphic and formal situations. For the sake of completeness, we finish this section with a sketch of a construction of such an isomorphism $\kappa:{\overline{T}M} \to T^*M$ which can be used to obtain a hyperk\"ahler metric on $T^*M$ rather than on ${\overline{T}M}$. \refstepcounter{subsubsection We begin with the following preliminary fact on the holomorphic de Rham complex of the manifold ${\overline{T}M}_I$. \begin{lemma}\label{de.rham.exact} Assume given either a formal Hodge manifold structure on the $U(1)$-manifold ${\overline{T}M}$ along the zero section $M \subset {\overline{T}M}$, or an actual Hodge manifold structure on an open neighborhood $U \subset {\overline{T}M}$ of the zero section. \begin{enumerate} \item For every point $m \in M$ there exists an open neighborhood $U \subset {\overline{T}M}_I$ such that the spaces $\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U)$ of holomorphic forms on the complex manifold ${\overline{T}M}_I$ (formally completed along $M \subset {\overline{T}M}$ if necessary) equipped with the holomorphic de Rham differential $\partial_I:\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U) \to \Omega^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(U)$ form an exact complex. \item If the subset $U \subset {\overline{T}M}$ is invariant under the $U(1)$-action on ${\overline{T}M}$, then the same is true for the subspaces $\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_k(U) \subset \Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U)$ of forms of weight $k$ with respect to the $U(1)$-action. \item Assume further that the canonical projection $\rho:{\overline{T}M}_I \to M$ is holomorphic for the preferred complex structure ${\overline{T}M}_I$ on ${\overline{T}M}$. Then both these claims hold for the spaces $\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M)$ of relative holomorphic forms on $U$ over $M$. \end{enumerate} \end{lemma} \proof The claim \thetag{i} is standard. To prove \thetag{ii}, note that, both in the formal and in the analytic situation, the spaces $\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U)$ are equipped with a natural topology. Both this topology and the $U(1)$-action are preserved by the holomorphic Dolbeault differential $\partial_I$. The subspaces $\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{fin}(U) \subset \Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U)$ of $U(1)$-finite vectors are dense in the natural topology. Therefore the complex $\langle \Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{fin}(U),\partial_I\rangle$ is also exact. Since the group $U(1)$ is compact, we have $$ \Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{fin}(U) = \bigoplus_k\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_k(U), $$ which proves \thetag{ii}. The claim \thetag{iii} is, again, standard. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection We can now formulate and prove the main result of this subsection. \begin{prop} Assume given a formal polarized Hodge manifold structure on the manifold ${\overline{T}M}$ along the zero section $M \subset {\overline{T}M}$ such that the canonical projection $\rho:{\overline{T}M}_I \to M$ is holomorphic for the preferred complex structure ${\overline{T}M}_I$ on ${\overline{T}M}$. Let $\Omega_I$ be the associated formal holomorphic $2$-form on ${\overline{T}M}_I$. Let $T^*M$ be the total space of the cotangent bundle to the manifold $M$ equipped with a canonical holomorphic symplectic form $\Omega$. There exists a unique $U(1)$-equivariant holomorphic map $\kappa:{\overline{T}M}_I \to T^*M$, defined in a formal neighborhood of the zero section, which commutes with the canonical projections onto $M$ and satisfies $\Omega_I = \kappa^*\Omega$. Moreover, if the polarized Hodge manifold structure on ${\overline{T}M}_I$ is defined in an open neighborhood $U \subset {\overline{T}M}$ of the zero section $M \subset {\overline{T}M}$, then the map $\kappa$ is also defined in a (possibly smaller) open neighborhood of the zero section. \end{prop} \proof By virtue of the uniqueness, the claim is local on $M$, so that we can assume that the whole $M$ is contained in a $U(1)$-invariant neighborhood $U \subset {\overline{T}M}_I$ satisfying the conditions of Lemma~\ref{de.rham.exact}. Holomorphic maps $\kappa:U \to T^*M$ which commute with the canonical projections onto $M$ are in a natural one-to-one correspondence with holomorphic sections $\alpha$ of the bundle $\rho^*\Lambda^{1,0}(M)$ on ${\overline{T}M}_I$. Such a map $\kappa$ is $U(1)$-equivariant if and only if the corresponding $1$-form $\alpha \in \Omega^1(U)$ is of weight $1$ with respect to the $U(1)$-action. Moreover, it satisfies $\kappa^*\Omega=\Omega_I$ if and only if $\partial_I\alpha = \Omega_I$. Therefore to prove the formal resp. analytic parts of the proposition it suffices to prove that there exists a unique holomorphic formal resp. analytic section $\alpha \in C^\infty(U,\rho^*\Lambda^{1,0}(M)$ which is of weight $1$ with respect to the $U(1)$-action and satisfies $\partial_I\alpha=\Omega_I$. The proof of this fact is the same in the formal and in the analytic situations. By definition of the polarized Hodge manifold the $2$-form $\Omega_I \in \Omega^2(U)$ is of weight $1$ with respect to the $U(1)$-action. Therefore by Lemma~\ref{de.rham.exact}~\thetag{ii} there exists a holomorphic $1$-form $\alpha \in \Omega^1(U)$ of weight $1$ with respect to the $U(1)$-action and such that $\partial_I\alpha = \Omega_I$. Moreover, the image of the form $\Omega_I$ under the canonical projection $\Omega^2(U) \to \Omega^2(U/M)$ is zero. Therefore by Lemma~\ref{de.rham.exact}~\thetag{iii} we can arrange so that the image of the form $\alpha$ under the projection $\Omega^1(U) \to \Omega^1(U/M)$ is also zero, so that $\alpha$ is in fact a section of the bundle $\rho^*\Lambda^{1,0}(M)$. This proves the existence part. To prove uniqueness, note that every two such $1$-forms must differ by a form of the type $\partial_if$ for a certain holomorphic function $f \in \Omega^0(U)$. Moreover, by Lemma~\ref{de.rham.exact}~\thetag{ii} we can assume that the function $f$ is of weight $1$ with respect to the $U(1)$-action. On the other hand, by Lemma~\ref{de.rham.exact}~\thetag{iii} we can assume that the function $f$ is constant along the fibers of the canonical projection $\rho:{\overline{T}M}_I \to M$. Therefore we have $f=0$ identically on the whole $U$. \hfill \ensuremath{\square}\par \section{Convergence}\label{convergence} \subsection{Preliminaries} \refstepcounter{subsubsection Let $M$ be a complex manifold. By Theorem~\ref{kal=ext} every K\"ah\-le\-ri\-an connection $\nabla:\Lambda^1(M,\C) \to \Lambda^1(M,\C) \otimes \Lambda^1(M,\C)$ on the cotangent bundle $\Lambda^1(M,\C)$ to the manifold $M$ defines a flat linear extended connection $D:S^1(M,\C) \to \B^1(M,\C)$ on $M$ and therefore a formal Hodge connection $D$ on the total space ${\overline{T}M}$ of the complex-conjugate to the tangent bundle to $M$. By Proposition~\ref{equiv.bis}, this formal Hodge connection defines, in turn, a formal Hodge manifold structure on ${\overline{T}M}$ in the formal neighborhood of the zero section $M \subset {\overline{T}M}$. In this section we show that if the K\"ah\-le\-ri\-an connection $\nabla$ is real-analytic, then the corresponding formal Hodge manifold structure on ${\overline{T}M}$ is the completion of an actual Hodge manifold structure on an open neighborhood $U \subset {\overline{T}M}$ of the zero section $M \subset {\overline{T}M}$. We also show that if the connection $\nabla$ comes from a K\"ahler metric $\omega$ on $M$, then the corresponding polarization $\Omega$ of the formal Hodge manifold ${\overline{T}M}$ defined in Theorem~\ref{metrics} converges in a neighborhood $U' \subset U \subset {\overline{T}M}$ of the zero section $M \subset {\overline{T}M}$ to a polarization of the Hodge manifold structure on $U'$. Here is the precise formulation of these results. \begin{theorem}\label{converge} Let $M$ be a complex manifold equipped with a real-analytic K\"ah\-le\-ri\-an connection $\nabla:\Lambda^1(M,\C) \to \Lambda^1(M,\C) \otimes \Lambda^1(M,\C)$ on its cotangent bundle $\Lambda^1(M,\C)$. There exists an open neighborhood $U \subset {\overline{T}M}$ of the zero section $M \subset {\overline{T}M}$ in the total space ${\overline{T}M}$ of the complex-conjugate to the tangent bundle to $M$ and a Hodge manifold structure on $U \subset {\overline{T}M}$ such that its completion along the zero section $M \subset {\overline{T}M}$ defines a linear flat extended connection $D$ on $M$ with reduction $\nabla$. Moreover, assume that $M$ is equipped with a K\"ahler metric $\omega$ such that $\nabla\omega=0$, and let $\Omega \in C^\infty_M({\overline{T}M},\Lambda^2({\overline{T}M},\C))$ be the formal polarization of the Hodge manifold structure on $U \subset {\overline{T}M}$ along $M \subset {\overline{T}M}$. Then there exists an open neighborhood $U' \subset U$ of $M \subset U$ such that $\Omega \in C^\infty(U',\Lambda^2({\overline{T}M},\C)) \subset C^\infty_M({\overline{T}M},\Lambda^2({\overline{T}M},\C))$. \end{theorem} \refstepcounter{subsubsection \label{formal.Weil} We begin with some preliminary observations. First of all, the question is local on $M$, therefore we may assume that $M$ is an open neighborhood of $0$ in the complex vector space $V = \C^n$. Fix once and for all a real structure and an Hermitian metrics on the vector space $V$, so that it is isomorphic to its dual $V \cong V^*$. The subspace $\J \subset C^\infty(M,\C)$ of functions vanishing at $0 \in M$ is an ideal in the algebra $C^\infty(M,\C)$, and $\J$-adic topology on $C^\infty(M,\C)$ extends canonically to the de Rham algebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the manifold $M$ and, further, to the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of $M$ introduced in \ref{Weil.defn}. Instead of working with bundle algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ on $M$, it will be convenient for us to consider the vector space $$ \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = C^\infty_\J(M,B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)), $$ which is by definition the $\J$-adic completion of the space $C^\infty(M,\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C))$ of global sections of the Weil algebra. This vector space is canonically a (pro-)algebra over $\C$. Moreover, the Hodge bundle structure on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ induces an $\R$-Hodge structure on the algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. \refstepcounter{subsubsection The $\J$-adic completion $C^\infty_\J(M,\C)$ of the space of smooth functions on $M$ is canonically isomorphic to the completion $\wh{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V)$ of the symmetric algebra of the vector space $V \cong V^*$. The cotangent bundle $\Lambda^1(M,\C)$ is isomorphic to the trivial bundle $\V$ with fiber $V$ over $M$, and the completed de Rham algebra $C^\infty_\J(M,\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C))$ is isomorphic to the product $$ C^\infty_\J(M,\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)) \cong S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V) \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V). $$ This is a free graded-commutative algebra generated by two copies of the vector space $V$, which we denote by $V_1 = V \subset \Lambda^1(V)$ and by $V_2 = V \subset S^1(V)$. It is convenient to choose the trivialization $\Lambda^1(M,\C) \cong \V$ in such a way that the de Rham differential $d_M:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ induces an identity map $d_M:V_2 \to V_1 \subset C^\infty_\J(M,\Lambda^1(M,\C))$. \refstepcounter{subsubsection The complex vector bundle $S^1(M,\C)$ on $M$ is also isomorphic to the trivial bundle $\V$. Choose a trivialization $S^1(M,\C) \cong \V$ in such a way that the canonical map $C:S^1(M,\C) \to \Lambda^1(M,\C)$ is the identity map. Denote by $$ V_3 = V \subset C^\infty(M,S^1(M,\C)) \subset \B^0 $$ the subset of constant sections in $S^1(M,\C) \cong \V$. Then the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ becomes isomorphic to the product $$ \B^i \cong \wh{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_2 \oplus V_3) \otimes \Lambda^i(V_1) $$ of the completed symmetric algebra $\wh{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_2 \oplus V_3)$ of the sum $V_2 \oplus V_3$ of two copies of the vector space $V$ and the exterior algebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_1)$ of the third copy of the vector space $V$. \refstepcounter{subsubsection Recall that we have introduced in \ref{aug} a grading on the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ which we call {\em the augmentation grading}. It induces a grading on the the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The augmentation grading on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is multiplicative, and it is obtained by assigning degree $1$ to the generator subspaces $V_1,V_3 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ and degree $0$ to the generator subspace $V_2 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. As in \ref{aug}, we will denote the augmentation grading on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ by lower indices. We will now introduce yet another grading on the algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ which we will call {\em the total grading}. It is by definition the multiplicative grading obtained by assigning degree $1$ to {\em all} the generators $V_1,V_2,V_3 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ We will denote by $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{k,n} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ the component of augmentation degree $k$ and total degree $n$. Note that by definition $n,k \geq 0$ and, moreover, $n \geq k$. \noindent {\bf Remark.\ } In \ref{aug} we have also defined a finer {\em augmentation bigrading} on the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ \, and it this bigrading that was denoted by double lower indices throughout Section~\ref{main.section}. We will now longer need the augmentation bigrading, so there is no danger of confusion. \refstepcounter{subsubsection The trivialization of the cotangent bundle to $M$ defines an isomorphism ${\overline{T}M} \cong M \times V$ and a constant Hodge connection on the pair $\langle {\overline{T}M},M \rangle$. The corresponding extended connection $D^{const}:S^1(M,\C) \to \B^1(M,\C)$ is the sum of the trivial connection $$ \nabla^{const}_1:S^1(M,\C) \to S^1(M,\C) \otimes \Lambda^1(M,\C) \subset \B^1(M,\C) $$ on $S^1(M,\C) \cong \V$ and the canonical isomorphism $$ C = {\sf id}:S^1(M,\C) \to \Lambda^1(M,\C) \subset \B^1(M,\C). $$ The derivation $D^{const}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ of the Weil algebra associated to the extended connection $D^{const}$ by \ref{deriv} is equal to \begin{align*} D^{const} &= C = {\sf id}:V_3 \to V_1\\ D^{const} &= d_M = {\sf id}:V_2 \to V_1\\ D^{const} &= d_M = 0 \text{ on }V_1 \end{align*} on the generator spaces $V_1,V_2,V_3 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. In particular, the derivation $D^{const}$ preserves the total degree. \refstepcounter{subsubsection Let now $D:S^1(M,\C) \to \B^1(M,\C)$ be the an arbitrary linear extended connection on the manifold $M$, and let $$ D = \sum_{k \geq 0}D_k:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} $$ be the derivation of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ associated to the extended connection $D$ by \ref{deriv}. The derivation $D$ admits a finer decomposition $$ D = \sum_{k,n \geq 0} D_{k,n}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} $$ according to both the augmentation and the total degree on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The summand $D_{k,n}$ by definition raises the augmentation degree by $k$ and the total degree by $n$. \refstepcounter{subsubsection Since the extended connection $D:S^1(M,\C) \to \B^1(M,\C)$ is linear, its component $D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ of augmentation degree $0$ coincides with the canonical isomorphism $C:S^1(M,\C) \to \Lambda^1(M,\C)$. Therefore the restriction of the derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ to the generator subspace $V_3 \subset S^1(M,\C) \subset \B^0$ satisfies $$ D_0 = C = D^{const}_0 = D^{const}_{0,0}:V_3 \to V_1 \subset \B^1. $$ In particular, all the components $D_{0,n}$ except for $D_{0,0}$ vanish on the subspace $V_3 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The restriction of the derivation $D$ to the subspace $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ by definition coincides with the de Rham differential $d_M:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$. Therefore on the generator subspaces $V_1,V_2 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ we have $D = d_M = D^{const}$. In particular, all the components $D_{k,n}$ except for $D_{1,0}$ vanish on the subspaces $V_1,V_2 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. \refstepcounter{subsubsection The fixed Hermitian metric on the generator spaces $V_1 = V_2 = V_3 = V$ extends uniquely to a metric on the whole Weil algebra such that the multiplication map $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is an isometry. We call this metric {\em the standard metric} on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. We finish our preliminary observations with the following fact which we will use to deduce Theorem~\ref{converge} from estimates on the components $D_{n,k}$ of the derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$. \begin{lemma}\label{est=conv} Let $D = \sum_{n,k}D_{n,k}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ be a derivation associated to an extended connection $D$ on the manifold $M$. Consider the norms $\|D_{k,n}\|$ of the restrictions $D_{k,n}:V_3 \to \B^1_{n+1,k+1}$ of the derivations $D_{k,n}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ to the generator subspace $V_3 \subset \B^0$ taken with respect to the standard metric on the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. If for certain constants $C,\varepsilon > 0$ and for every natural $n \geq k \geq 0$ we have \begin{equation}\label{est} \|D_{k,n}\| < C\varepsilon^n, \end{equation} then the formal Hodge connection on ${\overline{T}M}$ along $M$ associated to $D$ converges to an actual real-analytic Hodge connection on the open ball of radius $\varepsilon$ in ${\overline{T}M}$ with center at $0 \in M \subset {\overline{T}M}$. Conversely, if the extended connection $D$ comes from a real-analytic Hodge connection on an open neighborhood $U \subset {\overline{T}M}$, and if the Taylor series for this Hodge connection converge in the closed ball of radius $\varepsilon$ with center at $0 \in M \subset {\overline{T}M}$, then there exists a constant $C>0$ such that \eqref{est} holds for every $n,k \geq 0$. \end{lemma} \proof The constant Hodge connection $D^{const}$ is obviously defined on the whole ${\overline{T}M}$, and every other formal Hodge connection on ${\overline{T}M}$ is of the form $$ D = D^{const} + d^r \circ \Theta:\Lambda^0({\overline{T}M},\C) \to \rho^*\Lambda^1(M,\C), $$ where $d^r:\Lambda^0({\overline{T}M},\C) \to \Lambda^1({\overline{T}M}/M,\C)$ is the relative de Rham differential, and $\Theta \in C^\infty_M({\overline{T}M},\Lambda^1({\overline{T}M}/M,\C) \otimes \rho^*\Lambda^1(M,\C))$ is a certain relative $1$-form on the formal neighborhood of $M \subset {\overline{T}M}$ with values in the bundle $\rho^*\Lambda^1(M,\C)$. Both bundles $\Lambda^1({\overline{T}M}/M,\C)$ and $\rho^*\Lambda^1(M,\C)$ are canonically isomorphic to the trivial bundle $\V$ with fiber $V$ on ${\overline{T}M}$. Therefore we can treat the $1$-form $\Theta$ as a formal germ of a $\End(V)$-valued function on ${\overline{T}M}$ along $M$. The Hodge connection $D$ converges on a subset $U \subset {\overline{T}M}$ if and only if this formal germ comes from a real-analytic $\End(V)$-valued function $\Theta$ on $U$. The space of all formal Taylor series for $\End(V)$-valued functions on ${\overline{T}M}$ at $0 \in M \subset {\overline{T}M}$ is by definition equal to $\End(V) \otimes \B^0$. Moreover, for every $n \geq 0$ the component $\Theta_n \in \B^0_n \otimes \End(V) = \Hom(V,\B^0_n \otimes V)$ of total degree $n$ of the formal power series for the function $\Theta$ at $0 \in {\overline{T}M}$ is equal to the derivation $$ \sum_{0 \leq k \leq n}D_{k,n}:V=V_3 \to V=V_1 \otimes \bigoplus_{0 \leq k \leq n}\B^0_{k,n}. $$ Every point $x \in {\overline{T}M}$ defines the ``evaluation at $x$'' map $$ \ev_x:C^\infty({\overline{T}M},\End(V)) \to \C, $$ and the formal Taylor series for $\Theta \in \End(V) \otimes \B^0$ converges at the point $x \in {\overline{T}M}$ if and only if the series $$ \Theta(x) = \sum_{n \geq 0}\ev_x(\Theta_n) \in \End(V) $$ converges. But we have $$ \|\ev_x(\Theta_n)\| = \left\|\sum_{0 \leq k \leq n}D_{k,n}\right\||x|^n, $$ where $|x|$ is the distance from the point $x$ to $0 \in {\overline{T}M}$. Now the application of standard criteria of convergence finishes the proof of the lemma. \hfill \ensuremath{\square}\par \subsection{Combinatorics} \refstepcounter{subsubsection We now derive some purely combinatorial facts needed to obtain estimates for the components $D_{k,n}$ of the extended connection $D$. First, let $a_n$ be the Catalan numbers, that is, the numbers defined by the recurrence relation $$ a_n = \sum_{1 \leq k \leq n-1} a_k a_{n-k} $$ and the initial conditions $a_1 = 1$, $a_n = 0$ for $n \leq 0$. As is well-known, the generating function $f(z) = \sum_{k \geq 0}a_kz^k$ for the Catalan numbers satisfies the equation $f(z) = f(z)^2 + z$ and equals therefore $$ f(z) = \frac{1}{2} - \sqrt{\frac{1}{4} - z}. $$ The Taylor series for this function at $z=0$ converges for $4|z| < 1$, which implies that $$ a_k < C(4+\varepsilon)^k $$ for some positive constant $C > 0$ and every $\varepsilon > 0$. \refstepcounter{subsubsection \label{b.n.k} We will need a more complicated sequence of integers, numbered by two natural indices, which we denote by $b_{k,n}$. The sequence $b_{k,n}$ is defined by the recurrence relation $$ b_{k,n} = \sum_{p,q;1 \leq p \leq k-1} \frac{q+1}{k} b_{p,q}b_{k-p,n-q} $$ and the initial conditions $$ \begin{cases} b_{k,n} &= 0 \quad \text{ for } \quad k \leq 0,\\ b_{k,n} &= 0 \quad \text{ for } \quad k = 1, n < 0,\\ b_{k,n} &= 1 \quad \text{ for } \quad k = 1, n \geq 0, \end{cases} $$ which imply, in particular, that if $n < 0$, then $b_{k,n} = 0$ for every $k$. For every $k \geq 1$ let $g_k(z) = \sum_{n \geq 0}b_{k,n}z^n$ be the generating function for the numbers $b_{k,n}$. The recurrence relations on $b_{k,n}$ give \begin{align*} g_k(z) &= \frac{1}{k}\sum_{1 \leq p \leq k-1}g_{k-p}(z)\left(1+z\frac{\partial}{\partial z}\right)(g_p(z)) \\ &=\frac{1}{2k}\sum_{1 \leq p \leq k-1}\left(2+z\frac{\partial}{\partial z}\right)(g_p(z)g_{k-p}(z)), \end{align*} and the initial conditions give $$ g_1(z) = \frac{1}{1-z}. $$ \refstepcounter{subsubsection Say that a formal series $f(z)$ in the variable $z$ is {\em non-negative} if all the terms in the series are non-negative real numbers. The sum and product of two non-negative series and the derivative of a non-negative series is also obviously non-negative. For two formal series $s(z)$, $t(z)$ write $s(z) \ll t(z)$ if the difference $t(z) - s(z)$ is a non-negative power series. Our main estimate for the generating functions $g_k(z)$ is the following. \begin{lemma} For every $k \geq 1$ we have $$ g_k(z) \ll a_k\frac{1}{(1-z)^{2k-1}}, $$ where $a_k$ are the Catalan numbers. \end{lemma} \proof Use induction on $k$. For $k=1$ we have $g_1(z) = \frac{1}{1-z}$ and $a_1 = 1$, which gives the base for induction. Assume that the claim is proved for all $p < k$. Since all the $g_n(z)$ are non-negative power series, this implies that for every $p$, $1 \leq p \leq k-1$ we have $$ g_p(z)g_{k-p}(z) \ll a_pa_{k-p}\frac{1}{(1-z)^{2p-1}}\frac{1}{(1-z)^{2k-2p-1}} = a_pa_{k-p}\frac{1}{(1-z)^{2k-2}}. $$ Therefore \begin{align*} \left(2+z\frac{\partial}{\partial z}\right)(g_p(z)g_{k-p}(z)) &\ll a_pa_{k-p}\left(2+z\frac{\partial}{\partial z}\right)\frac{1}{(1-z)^{2k-2}} \\ &= a_pa_{k-p}\left(\frac{2}{(1-z)^{2k-2}} + \frac{(2k-2)z}{(1-z)^{2k-1}} \right) \\ &= a_pa_{k-p}\left(\frac{2k-2}{(1-z)^{2k-1}} - \frac{2k-4}{(1-z)^{2k-2}}\right) \\ &\ll (2k-2)a_pa_{k-p}\frac{1}{(1-z)^{2k-1}}. \end{align*} Hence \begin{align*} g_k(z) &= \frac{1}{2k}\sum_{1 \leq p \leq k-1}\left(2+z\frac{\partial}{\partial z}\right)(g_p(z)g_{k-p}(z)) \\ &\ll \frac{2k-2}{2k}\sum_{1 \leq p \leq k-1}a_pa_{k-p}\frac{1}{(1-z)^{2k-1}} \\ &\ll \frac{1}{(1-z)^{2k-1}}\sum_{1 \leq p \leq k-1}a_pa_{k-p} = a_k\frac{1}{(1-z)^{2k-1}}, \end{align*} which proves the lemma. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection This estimate yields the following estimate for the numbers $b_{k,n}$. \begin{corr}\label{combin} The power series $$ g(z) = \sum_{k,n} b_{k,n}z^{n+k} = \sum_{k \geq 1}g_k(z)z^k $$ converges for $z < 3-\sqrt{8}$. Consequently, for every $C_2$ such that $(3-\sqrt{8})C_2 > 1$ there exists a positive constant $C > 0$ such that $$ b_{n,k} < CC_2^{n+k} $$ for every $n$ and $k$. (One can take, for example, $C_2 = 6$.) \end{corr} \proof Indeed, we have \begin{equation}\label{g(z)} g(z) \ll \sum_{k \geq 1}a_kz^k\frac{1}{(1-z)^{2k-1}} = (1-z)f\left(\frac{z}{(1-z)^2}\right), \end{equation} where $f(z) = \frac{1}{2} - \sqrt{\frac{1}{4}-z}$ is the generating function for the Catalan numbers. Therefore $$ g(z) \ll (1-z)\left(\frac{1}{2} - \sqrt{\frac{1}{4} - \frac{z}{(1-z)^2}}\right), $$ and the right hand side converges absolutely when $$ \frac{|z|}{(1-z)^2} < \frac{1}{4}. $$ Since $3 - \sqrt{8}$ is the root of the quadratic equation $(1-z)^2 = 4z$, this inequality holds for every $z$ such that $|z| < 3 - \sqrt{8}$. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection \label{b.n.k.m} To study polarizations of Hodge manifold structures on ${\overline{T}M}$, we will need yet another recursive sequence of integers, which we denote by $b_{k,n}^m$. This sequence is defined by the recurrence relation $$ b^m_{k,n} = \sum_{p,q;1 \leq p \leq k-1} \frac{q+m(k-p)}{k} b^m_{p,q}b_{k-p,n-q} $$ and the initial conditions $$ \begin{cases} b^m_{k,n} &= 0 \quad \text{ unless } \quad k,m \leq 0,\\ b^m_{k,n} &= 0 \quad \text{ for } \quad k = 1, n < 0,\\ b^m_{k,n} &= 1 \quad \text{ for } \quad k = 1, n \geq 0. \end{cases} $$ \refstepcounter{subsubsection \label{c.k.n} To estimate the numbers $b^m_{k,n}$, consider the auxiliary sequence $c_{k,n}$ defined by setting $$ c_{k,n} = \sum_{p,q;1 \leq p \leq k-1}c_{p.q}b_{k-p,n-q} \qquad k \geq 2, $$ and $c_{k,n} = b_{k,n}$ for $k \leq 1$. The generating series $c(z) = \sum_{k,n \geq 0}c_{k,n}z^{n+k}$ satisfies $$ c(z) = c(z)g(z) + \frac{z}{1-z}, $$ so that we have $c(z) = \frac{z}{(1-z)(1-g(z))}$, which is non-singular when $|z|, |g(z)| < 1$ and $g(z)$ is non-singular. By \eqref{g(z)} the latter inequality holds if $$ \left|(1-z)\left(\frac{1}{2} - \sqrt{\frac{1}{4} - \frac{z}{(1-z)^2}}\right)\right| < 1, $$ which holds in the whole disc where $g(z)$ converges, that is, for $|z| < 3-\sqrt{8}$. Therefore, as in Corollary~\ref{combin}, we have \begin{equation}\label{cnk} c_{k,n} < C6^{n+k} \end{equation} for some positive constant $C$. \refstepcounter{subsubsection We can now estimate the numbers $b^m_{k,n}$. \begin{lemma}\label{combin.2} For every $m,k,n$ we have \begin{equation}\label{bnkm.indu} b^m_{k,n} \leq (2m)^{k-1}c_{k,n}b_{k,n}, \end{equation} where $c_{k,n}$ are as in \ref{c.k.n} and $b_{k,n}$ are the numbers introduced in \ref{b.n.k}. Consequently, we have $$ b^m_{k,n} < C(72m)^{n+k+m} $$ for some positive constant $C > 0$. \end{lemma} \proof Use induction on $k$. The case $k=1$ follows from the initial conditions. Assume the estimate \eqref{bnkm.indu} proved for all $b^m_{p,q}$ with $p < k$. Note that by the recurrence relations we have $b_{p,q} \leq b_{k,n}$ and $c_{p,q} \leq c_{k,n}$ whenever $p < k$. Therefore \begin{align*} b^m_{k,n} &= \sum_{p,q;1 \leq p \leq k-1} \frac{q+m(k-p)}{k} b^m_{p,q}b_{k-p,n-q} \\ &\leq \sum_{p,q;1 \leq p \leq k-1} \frac{q}{k}b^m_{p,q}b_{k-p,n-q} + \sum_{p,q;1 \leq p \leq k-1} b^m_{p,q}b_{k-p,n-q} \\ &\leq \sum_{p,q;1 \leq p \leq k-1}2^{p-1} c_{p,q} \frac{q}{k} b_{p,q}b_{k-p,n-q} + \sum_{p,q;1 \leq p \leq k-1} 2^{p-1} b_{p,q} c_{p,q} b_{k-p,n-q} \\ &\leq 2^{k-2} c_{k,n}\sum_{p,q;1 \leq p \leq k-1} \frac{q+1}{k} b_{p,q} b_{k-p,n-q} \\ &\quad + 2^{k-2}b_{k,n}\sum_{p,q;1 \leq p \leq k-1} c_{p,q} b_{k-p,n-q} \\ &= 2^{k-2}c_{k,n}b_{k,n} + 2^{k-2}c_{k,n}b_{k,n} = 2^{k-1}c_{k,n}b_{k,n}, \end{align*} which proves \eqref{bnkm.indu} for $b^m_{k,n}$. The second estimate of the lemma now follows from \eqref{cnk} and Corollary~\ref{combin}. \hfill \ensuremath{\square}\par \subsection{The main estimate} \refstepcounter{subsubsection \label{norms} Let now $D = \sum_{k,n}D_{n,k}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ be a derivation of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ associated to a flat linear extended connection on $M$. Consider the restriction $D_{k,n}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}{p+k,q+n}$ of the derivation $D_{k,n}$ to the component $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q}\subset\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of augmentation degree $p$ and total degree $q$. Since both $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q}$ and $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}{p+k,q+n}$ are finite-dimensional vector spaces, the norm of this restriction with respect to the standard metric on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is well-defined. Denote this norm by $\|D_{k,n}\|_{p,q}$. By Lemma~\ref{est=conv} the convergence of the Hodge manifold structure on ${\overline{T}M}$ corresponding to the extended connection $D$ is related to the growth of the norms $\|D_{k,n}\|_{1,1}$. Our main estimate on the norms $\|D_{k,n}\|_{1,1}$ is the following. \begin{prop}\label{estimate} Assume that there exist a positive constant $C_0$ such that for every $n$ the norms $\|D_{1,n}\|_{1,1}$ and $\|D_{1,n}\|_{0,1}$ satisfy $$ \|D_{1,n}\|_{1,1}, \|D_{1,n}\|_{1,0} < C_0^n. $$ Then there exists a positive constant $C_1$ such that for every $n,k$ the norm $\|D_{k,n}\|_{1,1}$ satisfies $$ \|D_{k,n}\|_{1,1} < C_1^n. $$ \end{prop} \refstepcounter{subsubsection \label{metr} In order to prove Proposition~\ref{estimate}, we need some preliminary facts. Recall that we have introduced in \ref{total.Weil} the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the manifold $M$, and let $$ \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} = C^\infty_\J(M,\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)) $$ be the algebra of its smooth sections completed at $0 \subset M$. By definition for every $k \geq 0$ we have $\B_{tot}^k = \B^k \otimes \W_k^*$, where $\W_k$ is the $\R$-Hodge structure of weight $k$ universal for weakly Hodge maps, as in \ref{w.k.uni}. There exists a unique Hermitian metric on $\W_k$ such that all the Hodge components $\W^{p,q} \subset \W_k$ are orthogonal and all the Hodge degree components $w_k^{p,q}$ of the universal weakly Hodge map $w_k:\R(0) \to \W_k$ are isometries. This metric defines a canonical Hermitian metric on $\W_k^*$. \noindent {\bf Definition.\ } The {\em standard metric} on the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is the product of the canonical metric and the standard metric on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. \refstepcounter{subsubsection By Lemma~\ref{total.rel} the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is generated by the subspaces $V_2,V_3 \subset \B^0 = \B^0_{tot}$ and the subspace $V_1 \otimes \W_1^* \subset \B^1_{tot}$, which we denote by $V_1^{tot}$. The ideal of relations for the algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is the ideal in $S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_2 \oplus V_3)\otimes\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_1^{tot})$ generated by $S^2(V_1)\otimes\Lambda^2(\W_1^*) \subset \Lambda^2(V_1^{tot})$. The direct sum decomposition \eqref{drct} induces a direct sum decomposition $$ V^{tot}_1 = V^{ll}_1 \oplus \V^o_1 \oplus V^{rr}_1 $$ of the generator subspace $V^{tot}_1 \subset \B^1_{tot}$. The subspaces $V^o_1 \subset V^{tot}_1$ and $V^{ll}_1 \oplus V^{rr}_1 \subset V^{tot}_1$ are both isomorphic to the vector space $V_1$. More precisely, the universal weakly Hodge map $w_1:\R(0) \to \W_1$ defines a projection $$ P:V^{tot}_1 = V_1 \otimes \W_1^* \to V_1, $$ and the restriction of the projection $P$ to either of the subspaces $V^o_1,V^{ll}_1 \oplus V^{rr}_1 \subset V^{tot}_1$ is an isomorphism. Moreover, either of these restrictions is an isometry with respect to the standard metrics. \refstepcounter{subsubsection \label{P2} The multiplication in $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is not an isometry with respect to this metric. However, for every $b_1,b_2 \subset \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ we have the inequality $$ \|b_1b_2\| \leq \|b_1\|{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\|b_2\|. $$ Moreover, this inequality becomes an equality when $b_1 \subset \B^0_{tot}$. In particular, if we extend the map $P:V^{tot}_1 \to V_1$ to a $\B^0$-module map $$ P:\B^1_{tot} \to \B^1, $$ then the restriction of the map $P$ to either of the subspaces $\B^1_o, \B^1_{ll}\oplus\B^1_{rr} \subset \B^1_{tot}$ is an isometry with respect to the standard metric. Therefore the norm of the projection $P:\B^1_{tot} \to \B^1$ is at most $2$. \refstepcounter{subsubsection The total and augmentation gradings on the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ extend to gradings on the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, also denoted by lower indices. The extended connection $D$ on $M$ induces a derivation $D^{tot} = \sum_{n,k}D_{k,n}^{tot}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$. As in \ref{norms}, denote by $\|D^{tot}_{k,n}\|_{p,q}$ the norm of the map $D^{tot}_{k,n}:\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}\right)_{p,q} \to \left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}\right)_{p+k,q+n}$ with respect to the standard metric on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$. The derivations $D^{tot}_{k,n}$ are related to $D_{k,n}$ by $$ D_{k,n} = P \circ D^{tot}_{k,n}:\B^0 \to \B^1, $$ and we have the following. \begin{lemma}\label{total.ne.total} For every $k$, $n$ and $p=0,1$ we have $$ \|D_{k,n}\|_{p,1} = \|D^{tot}_{k,n}\|_{p,1}. $$ \end{lemma} \proof By definition we have $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{0,1} \oplus \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{1,1} = V_1 \oplus V_2 \oplus V_3$. Moreover, the derivation $D_{k,n}$ vanishes on $V_1$, hence $D^{tot}_{k,n}$ vanishes on $V_1^{tot}$. Therefore it suffices to compare their norms on $V_2 \oplus V_3 \subset \B^0 = \B^0_{tot}$. Since on $S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_2) \subset \B^0$ the derivation $D_{k,n}$ coincides with the de Rham differential, the derivation $D^{tot}_{k,n}$ maps the subspace $V_2$ into $\B^1_{ll}\oplus\B^1_{rr}$. Moreover, by Lemma~\ref{total.aug} the derivation $D_{k,n}^{tot}$ maps $V_3$ either into $\B_o^1$ or into $\B^1_{ll} \oplus \B^1_{rr}$, depending on the parity of the number $k$. Since $D_{k,n} = P \circ D^{tot}_{k,n}$ and the map $P:\B^1_{tot} \to \B^1$ is an isometry on both $\B^1_o \subset \B^1_{tot}$ and $\B^1_{ll}\oplus\B^1_{rr} \subset \B^1_{tot}$, we have $\|D_{k,n}\| = \|D^{tot}_{k,n}\|$ on both $V_2 \subset \B^0$ and $V_3 \subset \B^0$, which proves the lemma. \hfill \ensuremath{\square}\par This lemma allows to replace the derivations $D_{k,n}$ in Proposition~\ref{estimate} with associated derivations $D_{k,n}^{tot}$ of the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. \refstepcounter{subsubsection Since the extended connection $D$ is linear and flat, the construction used in the proof of Lemma~\ref{main.ind} shows that \begin{equation}\label{indu} D^{tot}_k = h^{-1} \circ \sigma_{tot} \circ \sum_{1 \leq p \leq k-1}D_p^{tot} \circ D^{tot}_{k-p}:V_3 \to \left(\B^1_{tot}\right)_{k+1}, \end{equation} where $\sigma_{tot}:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is the canonical map constructed in \ref{sigma.c}, and $h:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is as in Lemma~\ref{h.acts}. Both $\sigma_{tot}$ and $h$ preserve the augmentation degree. In order to obtain estimates on $\|D_{k,n}\|_{1,1} = \|D_{k,n}^{tot}\|_{1,1}$, we have to estimate the norms $\|h^{-1}\|$ and $\|\sigma_{tot}\|$ of the restrictions of maps $h^{-1}$ and $\sigma_{tot}$ on the subspace $\left(\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{k+1} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$. By Lemma~\ref{h.on.b1} the map $h:\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{k+1} \to \left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{k+1}$ is diagonalizable, with eigenvalues $k+1$ if $k$ is even and $(k+1)/2,(k-1)/2$ if $k$ is odd. Since $m \geq 2$, in any case on $\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{k+1}$ we have \begin{equation}\label{h.est} \|h^{-1}\| < \frac{3}{k}. \end{equation} \refstepcounter{subsubsection \label{sigma.est.punkt} To estimate $\sigma_{tot}:\B^2_{tot} \to \B^2_{tot}$, recall that, as noted in \ref{sigma.c}, the map $\sigma_{tot}$ is a map of $\B^0$-modules. The space $\B^2_{tot} = \B^0 \otimes \left(\B^2_{tot}\right)_2$ is a free $\B^0$-module generated by a finite-dimensional vector space $\left(\B^2_{tot}\right)_2$. The map $\sigma_{tot}$ preserves the augmentation degree, hence it maps $\left(\B^2_{tot}\right)_2$ into the finite-dimensional vector space $\left(\B^1_{tot}\right)_2$. Therefore there exists a constant $K$ such that \begin{equation}\label{sigma.est} \|\sigma_{tot}\| \leq K \end{equation} on $\left(\B^2_{tot}\right)_2$. Since we have $\|b_1b_2\|=\|b_1\|{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\|b_2\|$ for every $b_1 \in \B^0$, $b_2 \in \B^2_{tot}$, and the map $\sigma_{tot}$ is $\B^0$-linear, the estimate \eqref{sigma.est} holds on the whole $\B^2_{tot} = \B^0 \otimes \left(\B^2_{tot}\right)_2$. We can assume, in addition, that $K \geq 1$. \noindent {\bf Remark.\ } In fact $K = 2$, but we will not need this. \refstepcounter{subsubsection Let now $b_{k,n}$ be the numbers defined recursively in \ref{b.n.k}. Our estimate for $\|D^{tot}_{k,n}\|_{p,q}$ is the following. \begin{lemma}\label{est.indu} In the assumptions and notations of Proposition~\ref{estimate}, we have $$ \|D^{tot}_{k,n}\|_{p,q} < q(3K)^{k-1} C_0^nb_{k,n} $$ for every $k,n$. \end{lemma} \proof Use induction on $k$. The base of induction is the case $k=1$, when the inequality holds by assumption. Assume that for some $k$ we have proved the inequality for all $\|D^{tot}_{m,n}\|_{p,q}$ with $m < k$, and fix a number $n \geq k$. Consider first the restriction of $D^{tot}_{k,n}$ onto the generator subspace $V_3 \subset \B^0$. Taking into account the total degree, we can rewrite \eqref{indu} as $$ D^{tot}_{k,n} = h^{-1} \circ \sigma_{tot} \circ \sum_{1 \leq p \leq k-1}\sum_q D^{tot}_{k-p,n-q} \circ D^{tot}_{p,q}:V_3 \to \left(\B^1_{tot}\right)_{k+1}. $$ Therefore the norm of the map $D^{tot}_{k,n}:V_3 \to \B_{tot}^1$ satisfies $$ \|D^{tot}_{k,n}\} \leq \|h^{-1}\| {\:\raisebox{3pt}{\text{\circle*{1.5}}}} \|\sigma_{tot}\| {\:\raisebox{3pt}{\text{\circle*{1.5}}}} \sum_{1 \leq p \leq k-1}\sum_q\|D^{tot}_{k-p,n-q}\|_{p+1,q+1} {\:\raisebox{3pt}{\text{\circle*{1.5}}}} \|D^{tot}_{p,q}\|_{1,1}. $$ Substituting into this the estimates \eqref{h.est}, \eqref{sigma.est} and the inductive assumption, we get $$ \|D^{tot}_{k,n}\| < \frac{3}{k}K\sum_{1 \leq p \leq k-1}\sum_q(q+1)(3K)^{k-2}C_0^nb_{k-p,n-q}b_{p,q} = (3K)^{k-1}C_0^nb_{k,n}. $$ Since by definition $D^{tot}_{k,n}$ vanishes on $V_2 \subset \B^0$ and on $V^{tot}_1 \subset \B^1_{tot}$, this proves that $$ \|D^{tot}_{k,n}\|_{p,1} < (3K)^{k-1}C_0^nb_{k,n} $$ when $p=0,1$. Since the map $D^{tot}_{k,n}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$ is a derivation, the Leibnitz rule and the triangle inequality show that for every $p$, $q$ $$ \|D^{tot}_{k,n}\|_{p,q} < q(3K)^{k-1}C_0^nb_{k,n}, $$ which proves the lemma. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection \proof[Proof of Proposition~\ref{estimate}] By Lemma~\ref{total.ne.total} we have $\|D_{k,n}\|_{p,1} = \|D^{tot}_{k,n}\|_{p,1}$, and by Lemma~\ref{est.indu} we have $$ \|D_{k,n}\|_{p,1} = \|D^{tot}_{k,n}\|_{p,1} < (3K)^{k-1}C_0^nb_{n,k}. $$ Since $k \leq n$ and $K \geq 1$, this estimate together with Corollary~\ref{combin} implies that $$ \|D_{k,n}\|_{p,1} < C(3K)^{k-1}C_0^n6^{2n} < C(108KC_0)^n $$ for some positive constant $C > 0$, which proves the proposition. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection Proposition~\ref{estimate} gives estimates for the derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ or, equivalently, for the Dolbeault differential $$ \bar\partial_J:\Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J) \to \Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}({\overline{T}M}_J) $$ for the complementary complex structure ${\overline{T}M}_J$ on ${\overline{T}M}$ associated to the extended connection $D$ on $M$. To prove the second part of Theorem~\ref{converge}, we will need to obtain estimates on the Dolbeault differential $\bar\partial_J:\Lambda^{p,0}({\overline{T}M}_J) \to \Lambda^{p,1}({\overline{T}M}_J)$ with $p > 0$. To do this, we use the model for the de Rham complex $\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J)$ constructed in Subsection~\ref{drm.mod}. Recall that in \ref{ident.punkt} we have identified the direct image $\rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J)$ of the de Rham algebra of the manifold ${\overline{T}M}$ with the free module over the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ generated by a graded algebra bundle $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ on $M$. The Dolbeault differential $\bar\partial_J$ for the complementary complex structure ${\overline{T}M}_J$ induces an algebra derivation $D:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \otimes \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$, so that the free module $\rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J) \cong L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ becomes a differential graded module over the Weil algebra. \refstepcounter{subsubsection The algebra bundle $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is isomorphic to the de Rham algebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. In particular, the bundle $L^1(M,\C)$ is isomorphic to the trivial bundle $\V$ with fiber $V$ over $M$. By \ref{dr.L} the relative de Rham differential $$ d^r:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M}/M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}({\overline{T}M}/M,\C) $$ induces a derivation $$ d^r:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C) \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C), $$ and we can choose the trivialization $L^1(M,\C) \cong \V$ in such a way that $d^r$ identifies the generator subspace $V_3 \subset \B^0$ with the subspace of constant sections of $\V \cong L^1(M,\C) \subset L^1(M,\C) \otimes \B^0(M,\C)$. \refstepcounter{subsubsection \label{LLL} Denote by $$ \LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} = C^\infty_\J(L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \otimes \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)) $$ the $\J$-adic completion of the space of smooth sections of the algebra bundle $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \otimes \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ on $M$. The space $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is a bigraded algebra equipped with the derivations $d^r:\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to \LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$, $D^{tot}:\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to \LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$, which commute by Lemma~\ref{D.dr}. The algebra $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is the free graded-commutative algebra generated by the subspaces $V_1,V_2,V_3 \subset \LL^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} = \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ and the subspace $V = d^r(V_3) \subset \LL^{1,0}$, which we denote by $V_4$. \refstepcounter{subsubsection As in \ref{metr}, the given metric on the generator subspaces $V_1=V_2=V_3=V_4=V$ extends uniquely to a multiplicative metric on the algebra $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$, which we call {\em the standard metric}. For every $k > 0$, introduce the total and augmentation gradings on the free $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$-module $\LL^{k,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} = \Lambda^k(V_4) \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ by setting $\deg \Lambda^k(V_4) = (0,0)$. Let $D = D_{k,n}$ be the decomposition of the derivation $D:\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to \LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ with respect to the total and the augmentation degrees. Denote by $\|D_{k,n}\|^p_q$ the norm with respect to the standard metric of the restriction of the derivation $D_{k,n}:\LL^{p,0} \to \LL^{p,1}$ to the component in $\LL^{p,0}$ of total degree $q$. \refstepcounter{subsubsection Let now $\LL_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} = \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_4) \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ be the product of the exterior algebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_4)$ with the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$. We have the canonical identification $\LL^{p,q}_{tot} = \LL^{p,q} \otimes \W_q^*$, and the canonical projection $P:\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_{tot} \to \LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$, identical on $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}_{tot} = \LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}$. As in \ref{P2}, the norm of the projection $P$ on $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},1}_{tot}$ is at most $2$. The derivation $D:\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0} \to \LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},1}$ induces a derivation $D^{tot}:\LL_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0} \to \LL_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},1}$, related to $D$ by $D = P \circ D^{tot}$. The gradings and the metric on $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ extend to $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_{tot}$, in particular, we have the decomposition $D^{tot} = \sum_{k,n}D^{tot}_{k,n}$ with respect to the total and the augmentation degrees. Denote by $\|D_{k,n}^{tot}\|^p_q$ the norm with respect to the standard metric of the restriction of the derivation $D^{tot}_{k,n}:\LL_{tot}^{p,0} \to \LL_C^{p,1}$ to the component in $\LL^{p,0}$ of total degree $q$. Since $\|P\|\leq 2$ on $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},1}_{tot}$, we have \begin{equation}\label{DcL} \|D_{k,n}^{tot}\|^p_q \leq 2{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\|D_{k,n}\|^p_q. \end{equation} \refstepcounter{subsubsection The estimate on the norms $\|D^{tot}_{k,n}\|^\LL_{p,q}$ that we will need is the following. \begin{lemma}\label{D2est} In the notation of Lemma~\ref{est.indu}, we have $$ \|D^{tot}_{k,n}\|^p_q < 2(q+pk)(3K)^{k-1} C_0^nb_{k,n}, $$ for every $p,q,k,n$. \end{lemma} \proof Since $D_{k,n}$ satisfies the Leibnitz rule, it suffices to prove the estimate for the restriction of the derivation $D^{tot}_{k,n}$ to the generator subspaces $V_2,V_3,V_4 \subset \LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}$. By \eqref{DcL} it suffices to prove that on $V_2,V_3,V_4$ we have $$ \|D_{k,n}\|^p_q < (q+pk)(3K)^{k-1} C_0^nb_{k,n}. $$ On the generator subspaces $V_2,V_3 \subset \B^0$ we have $p=0$, and this equality is the claim of Lemma~\ref{est.indu}. Therefore it suffices to consider the restriction of the derivation $D_{k,n}$ to the subspace $V_4 = d^r(V_3) \subset \LL^{1,0}$. We have $d^r \circ D_{k,n} = D_{k,n} \circ d^r:V_3 \to \LL^{1,1}$. Moreover, $d^r = {\sf id}:V_3 \to V_4$ is an isometry. Since the operator $d^r:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \LL^{1,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ satisfies the Leibnitz rule and vanishes on the generators $V_1,V_2 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, the norm $\|d^r\|_k$ of its restriction to the subspace $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_k$ of augmentation degree $k$ does not exceed $k$. Therefore \begin{align*} \|D_{k,n}\|^1_0 &= \left\|D_{k,n}|_{V_4}\right\| = \left\|D_{k,n}\circ d^r|_{V_3}\right\| = \left\|d^r \circ D_{k,n}|_{V_3}\right\| \\ &\leq \left\|D_{k,n}|_{V_3}\right|{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\left\|d^r_{\B^1_k}\right\| < k (3K)^{k-1} C_0^nb_{k,n}, \end{align*} which proves the lemma. \hfill \ensuremath{\square}\par \subsection{The proof of Theorem~\ref{converge}} \refstepcounter{subsubsection We can now prove Theorem~\ref{converge}. Let $D:S^1(M,\C) \to \B^1(M,\C)$ be a flat linear extended connection on $M$. Assume that its reduction $D_1=\nabla:S^1(M,\C) \to S^1(M,\C) \otimes \Lambda^1(M,\C)$ is a real-analytic connection on the bundle $S^1(M,\C)$. The operator $D_1:S^1(M,\C) \to S^1(M,\C) \otimes \Lambda^1(M,\C) \subset \B^1(M,\C)$ considered as an extended connection on $M$ defines a Hodge connection $D_1:\Lambda^0({\overline{T}M},\C) \to \rho^*\Lambda^1(M,\C)$ on the pair $\langle {\overline{T}M},M \rangle$, and this Hodge connection is also real-analytic. Assume further that the Taylor series at $0 \subset M \subset {\overline{T}M}$ for the Hodge connection $D_1$ converge in the closed ball of radius $\varepsilon>0$. \refstepcounter{subsubsection Let $D = \sum_{n,k}D_{k,n}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ be the derivation of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ associated to the extended connection $D$. Applying Lemma~\ref{est=conv} to the Hodge connection $D_1$ proves that there exists a constant $C>0$ such that for every $n \geq 0$ the norm $\|D_{1,n}\|_{1,1}$ of the restriction of the derivation $D_{1,n}$ to the generator subspace $V_3 = \B^0_{1,1} \subset \B^0$ satisfies $$ \|D_{1,n}\|_{1,1} < CC_0^n, $$ where $C_0 = 1/\varepsilon$. By definition the derivation $D_{1,n}$ vanishes on the generator subspace $V_1 \subset \B^1$. If $n > 0$, then it also vanishes on the generator subspace $V_2 = \B^0_{0,1} \subset \B^0$. If $n=0$, then its restriction to $V_2 = \B^0_{0,1} \subset \B^0$ is the identity isomorphism $D_{1,0}={\sf id}:V_2 \to V_1$. In any case, we have $\|D_{1,n}\|_{0,1} \leq 1$. Increasing if necessary the constant $C_0$, we can assume that for any $n$ and for $p=0,1$ we have $$ \|D_{1,n}\|_{p,1} < C_0^n. $$ \refstepcounter{subsubsection We can now apply our main estimate, Proposition~\ref{estimate}. It shows that there exists a constant $C_1 > 0$ such that for every $k,n$ we have $$ \|D_{k,n}\|_{1,1} < C_1^n. $$ Together with Lemma~\ref{est=conv} this estimate implies that the formal Hodge connection $D$ on ${\overline{T}M}$ along $M \subset {\overline{T}M}$ corresponding to the extended connection $D$ converges to a real-analytic Hodge connection on an open neighborhood $U \subset {\overline{T}M}$ of the zero section $M \subset {\overline{T}M}$. This in turn implies the first claim of Theorem~\ref{converge}. \refstepcounter{subsubsection To prove the second claim of Theorem~\ref{converge}, assume that the manifold $M$ is equipped with a K\"ahler form $\omega$ compatible with the K\"ah\-le\-ri\-an connection $\nabla$, so that $\nabla\omega=0$. The differential operator $\nabla:\Lambda^{1,1}(M) \to \Lambda^{1,1}(M) \otimes \Lambda^1(M,\C)$ is elliptic and real-analytic. Since $\nabla\omega=0$, the K\"ahler form $\omega$ is also real-analytic. For every $p,q \geq 0$ we have introduced in \ref{LLL} the space $\LL^{p,q}$, which coincides with the space of formal germs at $0 \in M \subset {\overline{T}M}$ of smooth forms on ${\overline{T}M}$ of type $(p,q)$ with respect to the complementary complex structure ${\overline{T}M}_J$. The spaces $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ carry the total and the augmentation gradings. Consider $\omega$ as an element of the vector space $\LL^{2,0}_0$, and let $\omega = \sum_n\omega_n$ be the total degree decomposition. The decomposition $\omega = \sum_n \omega_n$ is the Taylor series decomposition for the form $\omega$ at $0 \in M$. Since the form $\omega$ is real-analytic, there exists a constant $C_2$ such that \begin{equation}\label{omega.est} \|\omega_n\| < C_2^n \end{equation} for every $n$. \refstepcounter{subsubsection Let $\Omega = \sum_k \Omega_k = \sum_{k,n} \Omega_{k,n} \subset \LL^{2,0}$ be the formal polarization of the Hodge manifold ${\overline{T}M}$ at $M \subset {\overline{T}M}$ corresponding to the K\"ahler form $\omega$ by Theorem~\ref{metrics}. By definition we have $\omega = \Omega_0$. Moreover, by construction used in the proof of Proposition~\ref{metrics.ind} we have \begin{equation}\label{Omega.rec} \Omega_k = -\frac{1}{k}\sum_{1 \leq p \leq k-1}\sigma_{tot}(D^{tot}_{k-p}\Omega_p), \end{equation} where $D^{tot}:\LL^{2,0} \to \LL^{2,1}$ is the derivation associated to the extended connection $D$ on $M$ and $\sigma_{tot}:\LL^{2,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \LL^{2,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is the extension to $\LL^{2,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} = L^2_0 \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ of the canonical endomorphism of the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ constructed in the proof of Proposition~\ref{metrics.ind}. \refstepcounter{subsubsection The map $\sigma_{tot}:\LL^{2,1} \to \LL^{2,0}$ is a map of $\B^0_{tot}$-modules. Therefore, as in \ref{sigma.est.punkt}, there exists a constant $K_1 > 0$ such that \begin{equation}\label{K1} \|\sigma_{tot}\| < K_1 \end{equation} on $\LL^{2,1}$. We can assume that $K_1 > 3K$, where $K$ is as in \eqref{sigma.est}. Together with the recursive formula \eqref{Omega.rec}, this estimate implies the following estimate on the norms $\|\Omega_{k,n}\|$ of the components $\Omega_{k,n}$ of the formal polarization $\Omega$ taken with respect to the standard metric. \begin{lemma} For every $k,n$ we have $$ \|\Omega_{k,n}\| < (2K_1)^{k-1}C^nb^2_{k,n}, $$ where $C = \max(C_0,C_2)$ is the bigger of the constants $C_0$, $C_2$, and $b^2_{k,n}$ are the numbers defined recursively in \ref{b.n.k.m}. \end{lemma} \proof Use induction on $k$. The base of the induction is the case $k=1$, where the estimate holds by \eqref{omega.est}. Assume the estimate proved for all $\Omega_{p,n}$ with $p < k$, and fix a number $n$. By \eqref{Omega.rec} we have $$ \Omega_{k,n} = -\frac{1}{k}\sum_{1 \leq p \leq k-1}\sum_q\sigma_{tot}(D^{tot}_{k-p,n-q}\Omega_{p,q}). $$ Substituting the estimate \eqref{K1} together with the inductive assumption and the estimate on $\|D^{tot}_{k-p,n-q}\|^2_q$ obtained in Lemma~\ref{D2est}, we get \begin{align*} \|\Omega_{k,n}\| &< \frac{1}{k}\sum_{1 \leq p \leq k-1}\sum_q \|\sigma_{tot}\|{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\|D^{tot}_{k-p,n-q}\|^2_q{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\|\Omega_{p,q}\| \\ &< \frac{1}{k}\sum_{1 \leq p \leq k-1}\sum_qK_1 {\:\raisebox{3pt}{\text{\circle*{1.5}}}} 2(q+2(k-p))(3K)^{k-p-1}C_0^{n-q}b_{k-p,n-q} \\ &\qquad\qquad\qquad\qquad{\:\raisebox{3pt}{\text{\circle*{1.5}}}} (2K_1)^{p-1}C^qb_{p,q} \\ &< \frac{1}{k}(2K_1)^{k-1}C^n\sum_{1 \leq p \leq k-1}\sum_q(q+2(k-p))b_{k-p,n-q}b^2_{p,q} \\ &= (2K_1)^{k-1}C^nb^2_{k,n}, \end{align*} which proves the lemma. \hfill \ensuremath{\square}\par \refstepcounter{subsubsection This estimate immediately implies the last claim of Theorem~\ref{converge}. Indeed, together with Lemma~\ref{combin.2} it implies that for every $k,n$ $$ \|\Omega_{k,n}\| < (C_3)^n $$ for some constant $C_3>0$. But $\Omega = \sum_n\sum_{0 \leq k \leq n}\Omega_{k,n}$ is the Taylor series decomposition for the formal polarization $\Omega$ at $0 \subset M$. The standard convergence criterion shows that this series converges in an open ball of radius $1/C_3 > 0$. Therefore the polarization $\Omega$ is indeed real-analytic in a neighborhood of $0 \subset M \subset {\overline{T}M}$. \hfill \ensuremath{\square}\par \section*{Appendix} \refstepcounter{section} \refstepcounter{subsection} \renewcommand{\thesection}{A} \addcontentsline{toc}{section}{Appendix} \refstepcounter{subsubsection In this appendix we describe a well-known Borel-Weyl type localization construction for quaternionic vector spaces (see, e.g. \cite{HKLR}) which provides a different and somewhat more geometric approach to many facts in the theory of Hodge manifolds. In particular, we establish, following Deligne and Simpson (\cite{De2}, \cite{Simpson}), a relation between Hodge manifolds and the theory of mixed $\R$-Hodge structures. For the sake of simplicity, we consider only Hodge manifold structures on the formal neighborhood of $0 \in \R^{4n}$ instead of actual Hodge manifolds, as in Section~\ref{convergence}. To save the space all proofs are either omitted or only sketched. \refstepcounter{subsubsection Let $\SB$ be the Severi-Brauer variety associated to the algebra ${\Bbb H}$, that is, the real algebraic variety of minimal right ideals in ${\Bbb H}$. The variety $\SB$ is a twisted $\R$-form of the complex projective line ${\C P}^1$. For every algebra map $I:\C \to {\Bbb H}$ let the algebra ${\Bbb H} \otimes_\R \C$ act on the $2$-dimensional complex vector space ${\Bbb H}_I$ by left multiplication, and let $\widehat{I} \subset {\Bbb H} \otimes_\R \C$ be the annihilator of the subspace $I(\C) \subset {\Bbb H}_I$ with respect to this action. The subspace $\widehat{I}$ is a minimal right ideal in ${\Bbb H} \otimes_\R \C$. Therefore it defines a $\C$-valued point $\widehat{I} \subset \SB(\C)$ of the real algebraic variety $\SB$. This establishes a bijection between the set $\SB(\C)$ and the set of algebra maps from $\C$ to ${\Bbb H}$. \refstepcounter{subsubsection Let $\Shv(\SB)$ be the category of flat coherent sheaves on $\SB$. Say that a sheaf $\E \in \Ob\Shv(\SB)$ is {\em of weight $p$} if the sheaf $\E \otimes \C$ on ${\C P}^1 = \SB \otimes \C$ is a sum of several copies of the sheaf ${\cal O}(p)$. Consider a quaternionic vector space $V$. Let $\I \in {\Bbb H} \otimes {\cal O}_\SB$ be the tautological minimal left ideal in the algebra sheaf ${\Bbb H} \otimes {\cal O}_\SB$, and let $\loc{V} \in \Ob\Shv(\SB)$ be the sheaf defined by $$ \loc{V} = V \otimes {\cal O}_\SB / \I {\:\raisebox{3pt}{\text{\circle*{1.5}}}} V \otimes {\cal O}_\SB. $$ The correspondence $V \mapsto \loc{V}$ defines a functor from quaternionic vector spaces to $\Shv(\SB)$. It is easy to check that this functor is a full embedding, and its essential image is the subcategory of sheaves of weight $1$. Call $\loc{V}$ {\em the localization} of the quaternionic vector space $V$. For every algebra map $\I:\C \to {\Bbb H}$ the fiber $\loc{V}|_{\widehat{I}}$ of the localization $\loc{V}$ over the point $\widehat{I} \subset \SB(\C)$ corresponding to the map $i:\C \to {\Bbb H}$ is canonically isomorphic to the real vector space $V$ with the complex structure $V_I$. \refstepcounter{subsubsection The compact Lie group $U(1)$ carries a canonical structure of a real algebraic group. Fix an algebra embedding $I:\C \to {\Bbb H}$ and let the group $U(1)$ act on the algebra ${\Bbb H}$ as in \ref{u.acts.on.h}. This action is algebraic and induces therefore an algebraic action of the group $U(1)$ on the Severi-Brauer variety $\SB$. The point $\widehat{I}:\Spec\C \subset \SB$ is preserved by the $U(1)$-action. The action of the group $U(1)$ on the complement $\SB \setminus \widehat{I}(\Spec\C) \subset \SB$ is free, so that the variety $\SB$ consists of two $U(1)$-orbits. The corresponding orbits of the complexified group $\C^* = U(1) \times \Spec\C$ on the complexification $\SB \times \Spec\C \cong {\C P}$ are the pair of points $0,\infty \subset {\C P}$ and the open complement ${\C P} \setminus \{0,\infty\} \cong \C^* \subset {\C P}$. Let $\Shv^{U(1)}(\SB)$ be the category of $U(1)$-equivariant flat coherent sheaves on the variety $\SB$. The localization construction immediately extends to give the equivalence $V \mapsto \loc{V}$ between the category of equivariant quaternionic vector spaces and the full subcategory in $\Shv^{U(1)}(\SB)$ consisting of sheaves of weight $1$. For an equivariant quaternionic vector space $V$, the fibers of the sheaf $\loc{V}$ over the point $\widehat{I} \subset \SB(\C)$ and over the complement $\SB \setminus \widehat{I}(\Spec\C)$ are isomorphic to the space $V$ equipped, respectively, with the preferred and the complementary complex structures $V_I$ and $V_J$. \refstepcounter{subsubsection The category of $U(1)$-equivariant flat coherent sheaves on the variety $\SB$ admits the following beautiful description, due to Deligne. \begin{lemma}[ (\cite{De2},\cite{Simpson})]\label{DS} \begin{enumerate} \item For every integer $n$ the full subcategory $$ \Shv^{U(1)}_n(\SB) \subset \Shv^{U(1)}(\SB) $$ of sheaves of weight $n$ is equivalent to the category of pure $\R$-Hodge structures of weight $n$. \item The category of pairs $\langle \E, W_{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\rangle$ of a flat $U(1)$-equivariant sheaf $$ \E \in \Ob\Shv^{U(1)}(\SB) $$ and an increasing filtration $W_{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ on $\E$ such that for every integer $n$ \begin{equation}\label{hg} W_n\E/W_{n-1}\E \quad \text{ is a sheaf of weight } \quad n \quad \text{ on } \quad \SB \end{equation} is equivalent to the category of mixed $\R$-Hodge structures. (In particular, it is abelian.) \end{enumerate} \end{lemma} \refstepcounter{subsubsection For every pure $\R$-Hodge structure $V$ call the corresponding $U(1)$-e\-qui\-va\-ri\-ant flat coherent sheaf on the variety $\SB$ {\em the localization} of $V$ and denote it by $\loc{V}$. For the trivial $\R$-Hodge structure $\R(0)$ of weight $0$ the sheaf $\loc{\R(0)}$ coincides with the structure sheaf ${\cal O}$ on $\SB$. If $V$, $W$ are two pure $\R$-Hodge structures, then the space $\Hom(\loc{V},\loc{W})$ of $U(1)$-equivariant maps between the corresponding sheaves coincides with the space of weakly Hodge maps from $V$ to $W$ in the sense of Subsection~\ref{w.H.sub}. For every pure $\R$-Hodge structure $V$ the space $\Gamma(\SB,\loc{V})$ of the global sections of the sheaf $\loc{V}$ is equipped with an action of the group $U(1)$ and carries therefore a canonical $\R$-Hodge structure of weight $0$. This $\R$-Hodge structure is the same as the universal $\R$-Hodge structure $\Gamma(V)$ of weight $0$ constructed in Lemma~\ref{g.ex}. This explains our notation for the functor $\Gamma:{{\cal W}{\cal H}odge}_{\geq 0} \to {{\cal W}{\cal H}odge}_0$. \refstepcounter{subsubsection Assume given a complex vector space $V$ and let $M$ be the formal neighborhood of $0 \in V$. Let $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the Weil algebra of the manifold $M$, as in \ref{formal.Weil}. For every $n \geq 0$ the vector space $\B^n$ is equipped with an $\R$-Hodge structure of weight $n$, so that we can consider the localization $\loc{\B^n}$. The sheaf $\oplus\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is a commutative algebra in the tensor category $\Shv^{U(1)}(\SB)$. We will call {\em the localized Weil algebra}. The augmentation grading on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ defined in \ref{aug} is compatible with the $\R$-Hodge structures. Therefore it defines an augmentation grading on the localized Weil algebra $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$. The finer augmentation bigrading on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ does not define a bigrading on $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$. However, it does define a bigrading on the complexified algebra $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \otimes \C$ of $\C^*$-equivariant sheaves on the manifold $\SB \otimes \C \cong {\C P}$. \refstepcounter{subsubsection Assume now given a flat extended connection $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ on $M$. Since the derivation $D$ is weakly Hodge, it corresponds to a derivation $D:\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to \loc{\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}}$ of the localized Weil algebra $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$. It is easy to check that the complex $\langle \loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}, D \rangle$ is acyclic in all degrees but $0$. Denote by $\HH^0$ the $0$-th cohomology sheaf $H^0(\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}})$. The sheaf $\HH^0$ carries a canonical algebra structure. Moreover, while the derivation $D$ does not preserve the augmentation grading on $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$, it preserves the decreasing {\em augmentation filtration} $\left(\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{\geq {\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$. Therefore we have a canonical decreasing filtration on the algebra $\HH^0$, which we also call the augmentation filtration. \refstepcounter{subsubsection It turns out that the associated graded quotient algebra $\gr \HH^0$ with respect to the augmentation filtration does not depend on the extended connection $D$. To describe it, introduce the $\R$-Hodge structure $W$ of weight $-1$ by setting $W = V$ as a real vector space and \begin{align}\label{W.dfn} \begin{split} W^{-1,0} &= V \subset V \otimes_\R \C,\\ W^{0,-1} &= \overline{V} \subset V \otimes_\R \C. \end{split} \end{align} The $k$-th graded piece $\gr_k \HH^0$ with respect to the augmentation filtration is then isomorphic to the symmetric power $S^k(\loc{W})$ of the localization $\loc{W}$ of the $\R$-Hodge structure $W$. In particular, it is a sheaf of weight $-n$, so that up to a change of numbering the augmentation filtration on $\HH^0$ satisfies the condition \eqref{hg}. The extension data between these graded pieces depend on the extended connection $D$. The whole associated graded algebra $\gr \HH^0$ is isomorphic to the completed symmetric algebra $\widehat{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(\loc{W})$. \refstepcounter{subsubsection Using standard deformation theory, one can show that the algebra map $\HH^0 \to \loc{\B^0}$ is the universal map from the algebra $\HH^0$ to a complete commutative pro-algebra in the tensor category of $U(1)$-equivariant flat coherent sheaves of weight $0$ on $\SB$. Moreover, the localized Weil algebra $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ coincides with the relative de Rham complex of $\loc{\B^0}$ over $\HH^0$. Therefore one can recover, up to an isomorphism, the whole algebra $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ and, consequently, the extended connection $D$, solely from the algebra $\HH^0$ in $\Shv^{U(1)}(\SB)$. Together with Lemma~\ref{DS}~\thetag{ii} this gives the following, due also to Deligne (in a different form). \begin{prop}\label{DS.prop} The correspondence $\langle \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}, D \rangle \mapsto \HH^0$ is a bijection between the set of all isomorphism classes of flat extended connections on $M$ and the set of all algebras $\HH^0$ in the tensor category of mixed $\R$-Hodge structures equipped with an isomorphism $\gr^W_{-1}\HH^0 \cong W$ between the $-1$-th associated graded piece of the weight filtration on $\HH^0$ and the pure $\R$-Hodge structure $W$ defined in \eqref{W.dfn} which induces for every $n \geq 0$ an isomorphism $\gr^W_{-n}\HH^0 \cong S^nW$. \end{prop} \noindent {\bf Remark.\ } The scheme $\Spec\HH^0$ over $\SB$ coincides with the so-called twistor space of the manifold ${\overline{T}M}$ with the hypercomplex structure given by the extended connection $D$ (see \cite{HKLR} for the definition). Deligne's and Simpson's (\cite{De2}, \cite{Simpson}) approach differs from ours in that they use the language of twistor spaces to describe the relation between $U(1)$-equivariant hypercomplex manifolds and mixed $\R$-Hodge structures. Since this requires some additional machinery, we have avoided introducing twistor spaces in this paper. \refstepcounter{subsubsection We will now try to use the localization construction to eludicate some of the complicated linear algebra used in Section~\ref{main.section} to prove our main theorem. As we have already noted, the category ${{\cal W}{\cal H}odge}$ of pure $\R$-Hodge structures with weakly Hodge maps as morphisms is identified by localization with the category $\Shv^{U(1)}(\SB)$ of $U(1)$-equivariant flat coherent sheaves on $\SB$. Moreover, the functor $\Gamma:{{\cal W}{\cal H}odge}_{\geq 0} \to {{\cal W}{\cal H}odge}_0$ introduced in Lemma~\ref{g.ex} is simply the functor of global sections $\Gamma(\SB,{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$. \refstepcounter{subsubsection Consider the localized Weil algebra $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ with the derivation $C:\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to \loc{\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}}$ associated to the canonical weakly Hodge derivation introduced in \ref{C.and.sigma}. The differential graded algebra $\langle \loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}},C \rangle$ is canonically an algebra over the completed symmetric algebra $\widehat{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V)$ generated by the constant sheaf on $\SB$ with the fiber $V$. Moreover, it is a free commutative algebra generated by the complex \begin{equation}\label{cmplx} V \longrightarrow V(1) \end{equation} placed in degrees $0$ and $1$, where $V(1)$ is the $U(1)$-equivariant sheaf of weight $1$ on $\SB$ corresponding to the $\R$-Hodge structure given by $V(1)^{1,0} = V$ and $V(1)^{0,1} = \overline{V}$. \refstepcounter{subsubsection The homology sheaves of the complex \eqref{cmplx} are non-trivial only in degree $1$. This non-trivial homology sheaf is a skyscraper sheaf concentrated in the point $\widehat{I}(\Spec\C) \subset \SB$ with fiber $V$. The associated sheaf on the complexification $\SB \otimes \C \cong {\C P}$ splits into the sum of skyscraper sheaf with fiber $V$ concentrated at $0 \in {\C P}$ and the skyscraper sheaf with fiber $\overline{V}$ concentrated at $\infty \in {\C P}$. This splitting cooresponds to the splitting of the complex \eqref{cmplx} itself into the components of augmentation bidegrees $(1,0)$ and $(0,1)$. \refstepcounter{subsubsection Let now $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{\geq 0,\geq 0}$ be the sum of the components in the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of augmentation bidegree greater or equal than $(1,1)$. The subspace $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is compatible with the $\R$-Hodge structure. The crucial point in the proof of Theorem~\ref{kal=ext} is Proposition~\ref{ac}, which claim the acyclycity of the complex $\langle \Gamma(\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}), C \rangle$. It is this fact that becomes almost obvious from the point of view of the localization construction. To show it, we first prove the following. \begin{lemma} The complex $\langle\loc{\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}},C\rangle$ of $U(1)$-equivariant sheaves on $\SB$ is acyclic. \end{lemma} \proof It suffices to prove that the complex $\loc{I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \otimes \C$ of sheaves on $\SB \otimes \C \cong {\C P}$ is acyclic. To prove it, let $p,q \geq 1$ be arbitrary integer, and consider the component $\left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{p,q}$ of augmentation bidegree $(p,q)$ in the localized Weil algebra $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$. By definition we have $$ \left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{p,q} = \left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{p,0} \otimes \left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{0,q} $$ Since the complex $\left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{p,0} = S^p\left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{1,0}$ has homology concentrated at $0 \in {\C P}$, while the complex $\left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{0,q} = S^q\left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{0,1}$ has homology concentrated at $\infty \in {\C P}$, their product is indeed acyclic. \hfill \ensuremath{\square}\par Now, we have $\Gamma(\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}) = \Gamma(\SB,\loc{\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}})$, and the functor $\Gamma(\SB,{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$ is exact on the full subcategory in $\Shv^{U(1)}(\SB)$ consisting of sheaves of positive weight. Therefore the complex $\langle \Gamma(\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}),C \rangle$ is also acyclic, which gives an alternative proof of Proposition~\ref{ac}. \refstepcounter{subsubsection We would like to finish the paper with the following observation. Proposition~\ref{DS.prop} can be extended to the following claim. \begin{prop} Let $M$ be a complex manifold. There exists a naturla bijection between the set of isomorphism classes of germs of Hodge manifold structures on ${\overline{T}M}$ in the neighborhood of the zero section $M \subset {\overline{T}M}$ and the set of multiplicative filtrations $F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ on the sheaf ${\cal O}_\R(M) \otimes \C$ of $\C$-valued real-analytic functions on $M$ satisfying the following condition: \begin{itemize} \item For every point $m \in M$ let $\widehat{{\cal O}}_m$ be the formal completion of the local ring ${\cal O}_m$ of germs of real-analytic functions on $M$ in a neighborhood of $m$ with respect to the maximal ideal. Consider the filtration $F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ on $\widehat{{\cal O}}_m \otimes \C$ induced by the filtration $f^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ on the sheaf ${\cal O}_\R(M) \otimes \C$, and for every $k \geq 0$ let $W_{-k}\widehat{{\cal O}}_m \subset \widehat{{\cal O}}_m$ be the $k$-th power of the maximal ideal in $\widehat{{\cal O}}_m$. Then the triple $\langle \widehat{{\cal O}}_m, F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}, W^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \rangle$ is a mixed $\R$-Hodge structure. (In particular, $F^k = 0$ for $k > 0$.) \end{itemize} \end{prop} If the Hodge manifold structure on ${\overline{T}M}$ is such that the projection $\rho:{\overline{T}M}_I \to M$ is holomorphic for the preferred complex structure ${\overline{T}M}_I$ on ${\overline{T}M}$, then it is easy to see that the first non-trivial piece $F^0{\cal O}_\R(M) \otimes \C$ of the filtration $F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ on the sheaf ${\cal O}_\R(M) \otimes \C$ coincides with the subsheaf ${\cal O}(M) \subset {\cal O}_\R(M) \otimes \C$ of holomorphic functions on $M$. Moreover, since the filtration $F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is multiplicative, it is completely defined by the subsheaf $F^{-1}{\cal O}_\R(M) \otimes \C \subset {\cal O}_\R(M) \otimes \C$. It would be very interesting to find an explicit description of this subsheaf in terms of the K\"ah\-le\-ri\-an connection $\nabla$ on $M$ which corresponds to the Hodge manifold structure on ${\overline{T}M}$.

9406

alg-geom/9406004

en

https://arxiv.org/abs/alg-geom/9406004

alg-geom/9406004

Fumiharu Kato

Fumiharu Kato

Log Smooth Deformation Theory

29 pages, Latex version 2.09, Kyoto-Math 94-07

This paper gives a foundation of log smooth deformation theory. We study the infinitesimal liftings of log smooth morphisms and show that the log smooth deformation functor has a representable hull. This deformation theory gives, for example, the following two types of deformations: (1) relative deformations of a certain kind of a pair of an algebraic variety and a divisor of it, and (2) global smoothings of normal crossing varieties. The former is a generalization of the relative deformation theory introduced by Makio, and the latter coincides with the logarithmic deformation theory introduced by Kawamata and Namikawa.

alg-geom

\section{Introduction} In this article, we formulate and develop the theory of {\it log smooth deformation}. Here, log smoothness (more precisely, logarithmic smoothness) is a concept in {\it log geometry} which is a generalization of ``usual'' smoothness of morphisms of algebraic varieties. Log geometry is a beautiful geometric theory which succesfully generalizes and unifies the scheme theory and the theory of torus embeddings. This theory was first planned and founded by Fontaine and Illusie, based on their idea of {\it log structures} on schemes, and further developed by Kato \cite {Kat1}. Recently, the importance of log geometry comes to be recognized by many geometers and applied to various fields of algebraic and arithmetic geometry. One of such applications can be seen in the recent work of Steenbrink \cite {Ste1}. In the present paper, we attempt to apply log geometry to extend the usual smooth deformation theory by using the concept of log smoothness. Log smoothness is one of the most important concepts in log geometry, and is a log geometric generalization of usual smoothness. For example, varieties with toric singularities or normal crossing varieties may become log smooth over certain logarithmic points. Kato \cite {Kat1} showed that any log smooth morphism is written \'{e}tale locally by the composition of a usual smooth morphism and a morphism induced by a homomorphism of monoids which essentially determines the log structures (Theorem \ref{lisse}). On the other hand, log smoothness is described by means of {\it log differentials} and {\it log derivations} similarly to usual smoothness by means of differentials and derivations. Hence if we consider the log smooth deformation by analogy with the usual smooth deformation, it is expected that the first order deformation is controled by the sheaf of log derivations. This intuition motivated this work and we shall see later that this is, in fact, the case. In the present paper, we construct log smooth deformation functor by the concept of infinitesimal log smooth lifting. The goal of this paper is to show that this functor has a representable hull in the sense of Schlessinger \cite {Sch1}, under certain conditions on the underlying schemes (Theorem \ref{hull}). At the end of this paper, we give two examples of our log smooth deformation theory, which are summarized as follows: \vspace{3mm} 1. {\sc Deformations with divisors} (\S \ref{exam1}): Let $X$ be a variety over a field $k$. Assume that the variety $X$ is covered by \'{e}tale open sets which are smooth over affine torus embeddings, and there exists a divisor $D$ of $X$ which is the union of the closures of codimension 1 torus orbits. Then, there exists a log structure ${\cal M}$ on $X$ such that the log scheme $(X, {\cal M})$ is log smooth over $k$ with trivial log structure. (The converse is also true in a certain excellent category of log schemes.) In this case, our log smooth deformation is a deformation of the pair $(X,D)$. If $X$ itself is smooth and $D$ is a smooth divisor of $X$, our deformation coincides with the relative deformation studied by Makio \cite {Mak1}. \vspace{3mm} 2. {\sc Smoothings of normal crossing varieties} (\S \ref{exam2}): If a connected scheme of finite type $X$ over a field $k$ is, \'{e}tale locally, isomorphic to an affine normal crossing variety $\mathop{\rm Spec}\nolimits k[z_1,\ldots,z_n]/(z_1\cdots z_d)$, then we call $X$ a normal crossing variety over $k$. If $X$ is $d$--semistable (cf.\ \cite {Fri1}), there exists a log structure ${\cal M}$ on $X$ such that $(X,{\cal M})$ is log smooth over a standard log point $(\mathop{\rm Spec}\nolimits k,{\bf N})$ (Theorem \ref{dss}). Then, our log smooth deformation is nothing but a smoothing of $X$. If the singular locus of $X$ is connected, our deformation theory coincides with the one introduced by Kawamata and Namikawa \cite {K-N1}. \vspace{3mm} The composition of this paper is as follows. We recall some basic notions of log geometry in the next section, and review the definition and basic properties of log smoothness in section 3. In section 4, we study the characterization of log smoothness by means of the theory of torus embeddings according to Illusie \cite {Ill1} and Kato \cite {Kat1}. In section 5, we recall the definitions and basic properties of log derivations and log differentials. In section 6 and section 7, we give proofs of theorems stated in section 4. Section 8 is devoted to the formulation of log smooth deformation theory. This section is the main section of this present paper. We prove the existance of a representable hull of the log smooth deformation functor in section 9. In section 10 and section 11, we give two examples of log smooth deformation. For the reader's convenience, in section 12, we give a proof of the result of Kawamata and Namikawa \cite {K-N1} which is relevant to our log smooth deformation. The author would like to express his thanks to Professors Kazuya Kato and Yoshinori Namikawa for valuable suggestions and advice. The author is also very grateful to Professor Luc Illusie for valuable advice on this paper. \vspace{3mm} {\sc Convention}. We assume that all monoids are commutative and have neutral elements. A homomorphism of monoids is assumed to preserve neutral elements. We write the binary operations of all monoids multiplicatively except in the case of ${\bf N}$ (the monoid of non--negative integers), ${\bf Z}$, etc., which we write additively. All sheaves on schemes are considered with respect to the \'{e}tale topology. \section{Fine saturated log schemes} In this and subsequent sections, we use the terminology of log geometry basically as in \cite {Kat1}. Let $X$ be a scheme. We view the structure sheaf $\O_X$ of $X$ as a sheaf of monoids under multiplication. \begin{dfn}{\rm (cf.\ \cite [\S 1]{Kat1}) A {\it pre--log structure} on $X$ is a homomorphism ${\cal M}\rightarrow\O_X$ of sheaves of monoids where ${\cal M}$ is a sheaf of monoids on $X$. A pre--log structure $\alpha:{\cal M}\rightarrow\O_X$ is said to be a {\it log structure} on $X$ if $\alpha$ induces an isomorphism $$ \alpha:\alpha^{-1}({\cal O}^\times_X)\stackrel{\sim}{\longrightarrow}{\cal O}^\times_X. $$} \end{dfn} \noindent Given a pre--log structure $\alpha:{\cal M}\rightarrow\O_X$, we can construct the {\it associated log structure} $\alpha^{\rm a}:{\cal M}^{\rm a}\rightarrow\O_X$ functorially by \begin{equation}\label{asslog1} {\cal M}^{\rm a}=({\cal M}\oplus{\cal O}^\times_X)/\P \end{equation} and $$ \alpha^{\rm a}(x,u)=u\cdot\alpha(x) $$ for $(x,u)\in{\cal M}^{\rm a}$, where $\P$ is the submonoid defined by $$ \P=\{(x,\alpha(x)^{-1})\: |\: x\in\alpha^{-1}({\cal O}^\times_X)\}. $$ Here, in general, the quotient $M/P$ of a monoid $M$ with respect to a submonoid $P$ is the coset space $M/\sim$ with induced monoid structure, where the equivalence relation $\sim$ is defined by $$ x\sim y\Leftrightarrow xp=yq \,\mbox{ for some $p,q\in P$}. $$ ${\cal M}^{\rm a}$ has a universal mapping property: if $\beta:{\cal N}\rightarrow\O_X$ is a log structure on $X$ and $\varphi:{\cal M}\rightarrow{\cal N}$ is a homomorphism of sheaves of monoids such that $\alpha=\beta\circ\varphi$, then there exists a unique lifting $\varphi^{\rm a}:{\cal M}^{\rm a}\rightarrow{\cal N}$. Note that the monoid ${\cal M}^{\rm a}$ defined by (\ref{asslog1}) is the push--out of the diagram $$ {\cal M}\supset \alpha^{-1}({\cal O}^\times_X)\stackrel{\alpha}{\longrightarrow}{\cal O}^\times_X $$ in the category of monoids, and the homomorphism $\alpha^{\rm a}$ is induced by $\alpha$ and the inclusion ${\cal O}^\times_X\hookrightarrow\O_X$. We sometimes denote the monoid ${\cal M}^{\rm a}$ by ${\cal M}\oplus_{\alpha^{-1}({\cal O}^\times_X)}{\cal O}^\times_X$. Note that we have the natural isomorphism \begin{equation}\label{basic1} {\cal M}/\alpha^{-1}({\cal O}^\times_X)\stackrel{\sim}{\longrightarrow} {\cal M}^{\rm a}/{\cal O}^\times_X. \end{equation} \begin{dfn}{\rm By a {\it log scheme}, we mean a pair $(X,{\cal M})$ with a scheme $X$ and a log structure ${\cal M}$ on $X$. A {\it morphism} of log schemes $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ is a pair $f=(f,\varphi)$ where $f:X\rightarrow Y$ is a morphism of schemes and $\varphi:f^{-1}{\cal N}\rightarrow{\cal M}$ is a homomorphism of sheaves of monoids such that the diagram $$ \begin{array}{ccc} f^{-1}{\cal N}&\stackrel{\varphi}{\longrightarrow}&{\cal M}\\ \vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\ f^{-1}\O_Y&\longrightarrow&\O_X \end{array} $$ commutes. } \end{dfn} \begin{dfn}\label{logequiv}{\rm Let $\alpha:{\cal M}\rightarrow\O_X$ and $\alpha':{\cal M}'\rightarrow\O_X$ be log structures on a scheme $X$. These log structures are said to be {\it equivalent} if there exists an isomorphism $\varphi:{\cal M}\stackrel{\sim}{\rightarrow}{\cal M}'$ such that $\alpha=\alpha'\circ\varphi$, i.e., there exists an isomorphism of log schemes $(X,{\cal M})\stackrel{\sim}{\rightarrow}(X,{\cal M}')$ whose underlying morphism of schemes is the identity ${\rm id}_X$. Let $\beta:{\cal N}\rightarrow\O_Y$ and $\beta':{\cal N}'\rightarrow\O_Y$ be log structures on a scheme $Y$. Let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ and $f':(X,{\cal M}')\rightarrow(Y,{\cal N}')$ be morphisms of log schemes. Then $f$ and $f'$ are said to be {\it equivalent} if there exist isomorphisms $\varphi:{\cal M}\stackrel{\sim}{\rightarrow}{\cal M}'$ and $\psi:{\cal N}\stackrel{\sim}{\rightarrow}{\cal N}'$ such that $\alpha=\alpha'\circ\varphi$, $\beta=\beta'\circ\psi$ and the diagram $$ \begin{array}{ccc} {\cal M}&\stackrel{\varphi}{\longrightarrow}&{\cal M}'\\ \vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\ f^{-1}{\cal N}&\underrel{\longrightarrow}{f^{-1}\psi}&f^{-1}{\cal N}' \end{array} $$ commutes. } \end{dfn} \noindent We denote the category of log schemes by ${\bf LSch}$. For $(S,\L)\in\mathop{\rm Obj}\nolimits({\bf LSch})$, we denote the category of log schemes over $(S,\L)$ by ${\bf LSch}_{(S,\L)}$. The following examples play important roles in the sequel. \begin{exa}\label{trilog}{\rm On any scheme $X$, we can define a log structure by the inclusion ${\cal O}^\times_X\hookrightarrow\O_X$, called the {\it trivial} log structure. Thus, we have an inclusion functor from the category of schemes to that of log schemes. We often denote the log scheme $(X, {\cal O}^\times_X\hookrightarrow\O_X)$ simply by $X$. } \end{exa} \begin{exa}\label{canlog}{\rm Let $A$ be a commutative ring. For a monoid $P$ , we can define a log structure canonically on the scheme $\mathop{\rm Spec}\nolimits A[P]$, where $A[P]$ denotes the monoid ring of $P$ over $A$, as the log structure associated to the natural homomorphism, $$ P\stackrel{\alpha}{\longrightarrow}A[P]. $$ This log structure is called the {\it canonical log structure} on $\mathop{\rm Spec}\nolimits A[P]$. Thus we obtain a log scheme which we denote simply by $(\mathop{\rm Spec}\nolimits A[P], P)$. A monoid homomorphism $P\rightarrow Q$ induces a morphism $(\mathop{\rm Spec}\nolimits A[Q], Q)\rightarrow(\mathop{\rm Spec}\nolimits A[P], P)$ of log schemes. Thus, we have a contravariant functor from the category of monoids to ${\bf LSch}_{\mathop{\rm Spec}\nolimits A}$. } \end {exa} \begin{exa}\label{torlog}{\rm Let $\Sigma$ be a fan on $N_{\bf R}={\bf R}^d$, $N={\bf Z}^d$, and $X_{\Sigma}$ the toric variety determined by the fan $\Sigma$ over a commutative ring $A$. Then, we get an induced log structure on the scheme $X_{\Sigma}$ by gluing the log structures associated to the homomorphisms $$ M\cap\sigma^{\vee}\longrightarrow A[M\cap\sigma^{\vee}], $$ for each cone $\sigma$ in $\Sigma$, where $M=\mathop{\rm Hom}\nolimits_{{\bf Z}}(N, {\bf Z})$. Thus, a toric variety $X_{\Sigma}$ is naturally viewed as a log scheme over $\mathop{\rm Spec}\nolimits A$, which we denote by $(X_{\Sigma}, \Sigma)$. } \end{exa} Next, we define important subcategories of ${\bf LSch}$. These subcategories are closely related with {\it charts} defined as follows. \begin{dfn}{\rm Let $(X,{\cal M})\in\mathop{\rm Obj}\nolimits({\bf LSch})$. A {\it chart} of ${\cal M}$ is a homomorphism $P\rightarrow{\cal M}$ from the constant sheaf of a monoid $P$ which induces an isomorphism from the associated log structure $P^{\rm a}$ to ${\cal M}$. } \end{dfn} \begin{dfn}{\rm Let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ be a morphism in ${\bf LSch}$. A {\it chart} of $f$ is a triple $(P\rightarrow{\cal M},Q\rightarrow{\cal N},Q\rightarrow P)$, where $P\rightarrow{\cal M}$ and $Q\rightarrow{\cal N}$ are charts of ${\cal M}$ and ${\cal N}$, respectively, and $Q\rightarrow P$ is a homomorphism for which the diagram $$ \begin{array}{ccc} Q&\longrightarrow&P\\ \vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\ f^{-1}{\cal N}&\longrightarrow&{\cal M} \end{array} $$ is commutative. } \end{dfn} \begin{dfn}{\rm (cf.\ \cite [\S 2]{Kat1}) A log structure ${\cal M}\rightarrow\O_X$ on a scheme $X$ is said to be {\it fine} if ${\cal M}$ has \'{e}tale locally a chart $P\rightarrow{\cal M}$ with $P$ a finitely generated integral monoid. Here, in general, a monoid $M$ is said to be {\it finitely generated} if there exists a surjective homomorphism ${\bf N}^n\rightarrow M$ for some $n$, and a monoid $M$ is said to be {\it integral} if $M\rightarrow\gp{M}$ is injective, where $\gp{M}$ denotes the Grothendieck group associated with $M$. A log scheme $(X,{\cal M})$ with a fine log structure ${\cal M}\rightarrow\O_X$ is called a {\it fine} log scheme. } \end{dfn} \noindent We denote the category of fine log schemes by ${\bf LSch}^{\rm f}$. Similarly, we denote the category of fine log schemes over $(S,\L)\in\mathop{\rm Obj}\nolimits({\bf LSch}^{\rm f})$ by ${\bf LSch}^{\rm f}_{(S,\L)}$, The category ${\bf LSch}^{\rm f}$ (resp. ${\bf LSch}^{\rm f}_{(S,\L)}$) is a full subcategory of ${\bf LSch}$ (resp. ${\bf LSch}_{(S,\L)}$). Both ${\bf LSch}$ and ${\bf LSch}^{\rm f}$ have fiber products. But the inclusion functor ${\bf LSch}^{\rm f}\hookrightarrow{\bf LSch}$ does not preserve fiber products (cf. Lemma \ref{fpro}). The inclusion functor ${\bf LSch}^{\rm f}\hookrightarrow {\bf LSch}$ has a right adjoint ${\bf LSch}\rightarrow{\bf LSch}^{\rm f}$ \cite[(2.7)]{Kat1}. Then, the fiber product of a diagram $(X,{\cal M})\rightarrow(Z,\P)\leftarrow(Y,{\cal N})$ in ${\bf LSch}^{\rm f}$ is the image of that in ${\bf LSch}$ by this adjoint functor. Note that the underlying scheme of the fiber product of $(X,{\cal M})\rightarrow(Z,\P)\leftarrow(Y,{\cal N})$ in ${\bf LSch}$ is $X\times_{Z}Y$, but this is not always the case in ${\bf LSch}^{\rm f}$. \vspace{3mm} Next, we introduce more excellent subcategory of ${\bf LSch}$. \begin{dfn}\label{satin}{\rm Let $M$ be a monoid and $P$ a submonoid of $M$. The monoid $P$ is said to be {\it saturated} in $M$ if $x\in M$ and $x^n\in P$ for some positive integer $n$ imply $x\in P$. An integral monoid $N$ is said to be {\it saturated} if $N$ is saturated in $\gp{N}$. } \end{dfn} \begin{exa}{\rm Put $M={\bf N}$ and $P=l\cdot M$ for an integer $l>1$. Then $P$ is saturated but not saturated in $M$. } \end{exa} \begin{dfn}{\rm A fine log scheme $(X,{\cal M})\in\mathop{\rm Obj}\nolimits ({\bf LSch}^{\rm f})$ is said to be {\it saturated} if the log structure ${\cal M}$ is a sheaf of saturated monoids. } \end{dfn} \noindent We denote the category of fine saturated log schemes by ${\bf LSch}^{\rm fs}$. Similarly, we denote the category of fine saturated log schemes over $(S,\L)\in\mathop{\rm Obj}\nolimits({\bf LSch}^{\rm fs})$ by ${\bf LSch}^{\rm fs}_{(S,\L)}$. The category ${\bf LSch}^{\rm fs}$ (resp. ${\bf LSch}^{\rm fs}_{(S,\L)}$) is a full subcategory of ${\bf LSch}^{\rm f}$ (resp. ${\bf LSch}^{\rm f}_{(S,\L)}$). The following lemma is an easy consequence of \cite [Lemma (2.10)]{Kat1}. \begin{lem} Let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ be a morphism in ${\bf LSch}^{\rm fs}$, and $Q\rightarrow{\cal N}$ a chart of ${\cal N}$, where $Q$ is a finitely generated integral saturated monoid. Then there exists \'{e}tale locally a chart $(P\rightarrow{\cal M},Q\rightarrow{\cal N},Q\rightarrow P)$ of $f$ extending $Q\rightarrow{\cal N}$ such that the monoid $P$ is also finitely generated, integral and saturated. \end{lem} \begin{lem} The inclusion functor ${\bf LSch}^{\rm fs}\lhook\joinrel\longrightarrow{\bf LSch}^{\rm f}$ has a right adjoint. \end{lem} \noindent{\sc Proof.}\hspace{2mm} Let $M$ be an integral monoid. Define $$ \sat{M}=\{x\in\gp{M}\: | \: x^n\in M \;\mbox{for some positive integer $n$}\}. $$ Then $\sat{M}$ is an integral saturated monoid. For any integral saturated monoid $N$ and homomorphism $M\rightarrow N$, there exists a unique lifting $\sat{M} \rightarrow N$. In this sense, $\sat{M}$ is the universal saturated monoid associated with $M$. Let $(X,{\cal M})$ be a fine log scheme. Then we have \'{e}tale locally a chart $P\rightarrow{\cal M}$. This chart defines a morphism $X\rightarrow\mathop{\rm Spec}\nolimits {\bf Z}[P]$ \'{e}tale locally. Let $X'=X\times_{\mathop{\rm Spec}\nolimits {\bf Z}[P]}\mathop{\rm Spec}\nolimits {\bf Z}[\sat{P}]$. Then $X'\rightarrow \mathop{\rm Spec}\nolimits {\bf Z}[\sat{P}]$ induces a log structure ${\cal M}'$ by the associated log structure of $\sat{P}\rightarrow{\bf Z}[\sat{P}]\rightarrow\O_{X'}$, and $(X',{\cal M}')$ is a fine saturated log scheme. This procedure defines a functor ${\bf LSch}^{\rm f}\rightarrow{\bf LSch}^{\rm fs}$. It is easy to see that this functor is the right adjoint of the inclusion functor ${\bf LSch}^{\rm fs} \hookrightarrow{\bf LSch}^{\rm f}$. $\Box$ \begin{cor} ${\bf LSch}^{\rm fs}$ has fiber products. More precisely, the fiber product of morphisms $(X,{\cal M})\rightarrow(Z,\P)\leftarrow(Y,{\cal N})$ in ${\bf LSch}^{\rm fs}$ is the image of that in ${\bf LSch}^{\rm f}$ by the right adjoint functor of ${\bf LSch}^{\rm fs}\lhook\joinrel\longrightarrow{\bf LSch}^{\rm f}$. \end{cor} \section{Log smooth morphisms} In this section, we review the definition and basic properties of log smoothness (cf. \cite {Kat1}). \begin{dfn}\label{pback}{\rm Let $f:X\rightarrow Y$ be a morphism of schemes, and ${\cal N}$ a log structure on $Y$. Then the {\it pull--back} of ${\cal N}$, denoted by $f^{*}{\cal N}$, is the log structure on $X$ associated with the pre--log structure $f^{-1}{\cal N}\rightarrow f^{-1}\O_Y\rightarrow\O_X$. A morphism of log schemes $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ is said to be {\it strict} if the induced homomorphism $f^{*}{\cal N}\rightarrow{\cal M}$ is an isomorphism. A morphism of log schemes $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ is said to be an {\it exact closed immersion} if it is strict and $f:X\rightarrow Y$ is a closed immersion in the ususal sense. } \end{dfn} \noindent Exact closed immersions are stable under base change in ${\bf LSch}^{\rm f}$ \cite [(4.6)]{Kat1}. \begin{lem}\label{basic2} Let $\alpha:{\cal M}\rightarrow\O_X$ and $\alpha':{\cal M}'\rightarrow\O_X$ be fine log strctures on a scheme $X$ with a homomorphism $\varphi:{\cal M}\rightarrow{\cal M}'$ of monoids such that $\alpha=\alpha'\circ\varphi$. Then, $\varphi$ is an isomorphism if and only if $\varphi\ {\rm mod}\ {\cal O}^\times_X:{\cal M}/{\cal O}^\times_X \rightarrow{\cal M}'/{\cal O}^\times_X$ is an isomorphism. \end{lem} \noindent The proof is straightforward. \begin{lem}\label{basic3} Let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ be a morphism of log schemes. Then, we have the natural isomorphism $$ f^{-1}({\cal N}/{\cal O}^\times_Y)\stackrel{\sim}{\longrightarrow}f^{*}{\cal N}/{\cal O}^\times_X. $$ In particular, $f$ is strict if and only if the induced morphism $$ f^{-1}({\cal N}/{\cal O}^\times_Y)\stackrel{\sim}{\longrightarrow}{\cal M}/{\cal O}^\times_X. $$ \end{lem} \noindent{\sc Proof.}\hspace{2mm} The first part is easy to see. For the second part, apply (\ref {basic1}) and Lemma \ref{basic2}. $\Box$ \begin{lem}{\rm (cf. \cite [(1.7)]{Kaj1})}\label{fpro} Let \begin{equation}\label{fpro1} (X,{\cal M})\rightarrow(Z,\P)\leftarrow(Y,{\cal N}) \end{equation} be morphisms in ${\bf LSch}^{\rm fs}$. If $(Y,{\cal N})\rightarrow(Z,\P)$ is strict, then the fiber product of {\rm (\ref{fpro1})} in ${\bf LSch}^{\rm fs}$ is isomorphic to that in ${\bf LSch}$. In particular, the underlying scheme of the fiber product of {\rm (\ref{fpro1})} in ${\bf LSch}^{\rm fs}$ is isomorphic to $X\times_{Z}Y$. \end{lem} \noindent{\sc Proof.}\hspace{2mm} Let $P\rightarrow\P$ be a chart of $\P$, where $P$ is a finitely generated integral saturated monoid. Since $(Y,{\cal N})\rightarrow(Z,\P)$ is strict, $\rightarrow\P\rightarrow{\cal N}$ is a chart of ${\cal N}$ by Lemma \ref{basic2}, Lemma \ref{basic3}, and (\ref{basic1}). Take a chart $$ \begin{array}{ccc} P&\longrightarrow&M\\ \vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\ \P&\longrightarrow&{\cal M} \end{array} $$ of $(X,{\cal M})\rightarrow(Z,\P)$ extending $P\rightarrow\P$. Set $W=X\times_{Z}Y$. There exists an induced homomorphism $M\rightarrow\O_W$. Define a log structure on $W$ by this homomorphism. Then this log scheme $(W,M\rightarrow\O_W)$ is the fiber product of (\ref{fpro1}) in ${\bf LSch}$. Since the associated log structure of $M\rightarrow\O_W$ is fine and saturated, $(W,M)$ is, indeed, the fiber product of (\ref{fpro1}) in ${\bf LSch}^{\rm fs}$. $\Box$ \begin{dfn}\label{defthick}{\rm The exact closed immersion $t:(T',\L')\rightarrow(T,\L)$ is said to be a {\it thickening of order $\leq n$}, if ${\cal I}=\mathop{\rm Ker}\nolimits(\O_T\rightarrow\O_{T'})$ is a nilpotent ideal such that ${\cal I}^{n+1}=0$. } \end{dfn} \begin{lem}\label{thick}{\rm (cf.\ \cite {Ill1}).} Let $(T,\L)$ and $(T',\L')$ be fine log schemes. If $(t,\theta):(T',\L')\rightarrow(T,\L)$ is a thickening of order 1, there exists a commutative diagram with exact rows: $$ \begin{array}{ccccccccc} 1&\rightarrow&1+{\cal I}&\hookrightarrow&t^{-1}\L& \stackrel{\theta}{\rightarrow}&\L'&\rightarrow&1\\ &&\parallel&&\cap&&\cap\\ 1&\rightarrow&1+{\cal I}&\rightarrow&t^{-1}\gp{\L}& \underrel{\rightarrow}{\gp{\theta}}&\gp{\L'}&\rightarrow&1\rlap{,} \end{array} $$ where ${\cal I}=\mathop{\rm Ker}\nolimits(\O_T\rightarrow\O_{T'})$, such that the right square of this commutative diagram is cartesian. \end{lem} \noindent The proof is straightforward. Note that the multiplicative monoid $1+{\cal I}$ is identified with the additive monoid ${\cal I}$ by $1+x\mapsto x$ since ${\cal I}^2=0$. \begin{dfn}{\rm (cf.\ \cite [(3.3)]{Kat1}) Let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ be a morphism in ${\bf LSch}^{\rm f}$. $f$ is said to be {\it log smooth} if the following conditions are satisfied: \begin{enumerate} \item $f$ is locally of finite presentation, \item for any commutative diagram $$ \begin{array}{ccc} (T',\L')&\stackrel{s'}{\longrightarrow}&(X,{\cal M})\\ \llap{$t$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$f$}\\ (T,\L)&\underrel{\longrightarrow}{s}&(Y,{\cal N}) \end{array} $$ in ${\bf LSch}^{\rm f}$, where $t$ is a thickening of order 1, there exists \'{e}tale locally a morphism $g:(T,\L)\rightarrow(X,{\cal M})$ such that $s'=g\circ t$ and $s=f\circ g$. \end{enumerate}} \end{dfn} \noindent The proofs of the following two propositions are straightforward and are left to the reader. \begin{pro}\label{usulisse} Let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ be a morphism in ${\bf LSch}^{\rm f}$. If $f$ is strict, then $f$ is log smooth if and only if $f$ is smooth in the usual sense. \end{pro} \begin{pro}\label{bextlisse} For $(S,\L)\in\mathop{\rm Obj}\nolimits({\bf LSch}^{\rm f})$ and $(X,{\cal M}),(Y,{\cal N})\in\mathop{\rm Obj}\nolimits({\bf LSch}^{\rm f}_{(S,\L)})$, let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ be a morphism in ${\bf LSch}^{\rm f}_{(S,\L)}$. Assume that $f$ is log smooth. If $(S',\L')$ is a log scheme over $(S,\L)$, then the induced morphism $$ (X,{\cal M})\times_{(S,\L)}(S',\L')\rightarrow(Y,{\cal N})\times_{(S,\L)}(S',\L') $$ is also log smooth. \end{pro} \section{Toroidal characterization of log smoothness} The following theorem is due to Kato \cite{Kat1}, and we prove it in \S \ref{prf1} for the reader's convenience. \begin{thm}\label{lisse}{\rm (\cite [(3.5)]{Kat1})} Let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ be a morphism in ${\bf LSch}^{\rm f}$. and $Q\rightarrow{\cal N}$ a chart of ${\cal N}$, where $Q$ is a finitely generated integral monoid. Then the following conditions are equivalent. \begin{description} \item[{\rm 1.}] $f$ is log smooth. \item[{\rm 2.}] There exists \'{e}tale locally a chart $(P\rightarrow{\cal M},Q\rightarrow{\cal N},Q\rightarrow P)$ of $f$ extending $Q\rightarrow{\cal N}$, where $P$ is a finitely generated integral monoid, such that \begin{description} \item[{\rm (a)}] $\mathop{\rm Ker}\nolimits(\gp{Q}\rightarrow\gp{P})$ and the torsion part of $\mathop{\rm Coker}\nolimits(\gp{Q}\rightarrow\gp{P})$ are finite groups of orders invertible on $X$, \item[{\rm (b)}] $X\rightarrow Y\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ is smooth (in the usual sense). \end{description} \end{description} Moreover, if $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ is a log smooth morphism in ${\bf LSch}^{\rm fs}$ and $Q\rightarrow{\cal N}$ is a chart of ${\cal N}$ such that $Q$ is finitely generated, integral and saturated, then there exists a chart $(P\rightarrow{\cal M},Q\rightarrow{\cal N},Q\rightarrow P)$ of $f$ as above with $P$ also saturated. \end{thm} \begin{rem}\label{lisserem}{\rm The proof of Theorem \ref{lisse} in \S \ref{prf1} shows that we can require in the condition 2. (a) that $\gp{Q}\rightarrow\gp{P}$ is injective without changing the conclusion. } \end{rem} We give some important examples of log smooth morphisms in the following. Let $k$ be a field. \begin{dfn}\label{logpt}{\rm A log structure on $\mathop{\rm Spec}\nolimits k$ is called a log structure of a {\it logarithmic point} if it is equivalent (Definition \ref{logequiv}) to the associated log structure of $\alpha:Q\rightarrow k$, where $Q$ is a monoid having no invertible element other than 1 and $\alpha$ is a homomorphism defined by $$ \alpha(x)=\left\{ \begin{array}{ll} 1&\mbox{if $x=1$,}\\ 0&\mbox{otherwise.} \end{array} \right. $$ Note that this log structure is equivalent to $Q\oplus k^{\times}\rightarrow k$. We denote the log scheme obtained in this way by $(\mathop{\rm Spec}\nolimits k,Q)$. The log scheme $(\mathop{\rm Spec}\nolimits k,Q)$ is called a {\it logarithmic point}. Especially, if $Q={\bf N}$, the logarithmic point $(\mathop{\rm Spec}\nolimits k,{\bf N})$ is said to be the {\it standard log point}. } \end{dfn} \noindent If $k$ is algebraically closed, any log structure on $\mathop{\rm Spec}\nolimits k$ is equivalent to a log structure of logarithmic point (cf.\ \cite {Ill1}). Note that if we set $Q=\{1\}$, then the log structure of the logarithmic point induced by $Q$ is the trivial log structure (Example \ref{trilog}). \begin{exa}\label{toroex}{\rm Let $P$ be a submonoid of a group $M={\bf Z}^d$ such that $\gp{P}=M$ and that $P$ is saturated. Let $Q$ be a submonoid of $P$, which is saturated but not necessarily saturated in $P$. We assume the following: \begin{enumerate} \item the monoid $Q$ has no invertible element other than 1, \item the order of the torsion part of $M/\gp{Q}$ is invertible in $k$. \end{enumerate} Let $R={\bf Z}[1/N]$ where $N$ is the order of the torsion part of $M/\gp{Q}$. The latter assumption implies, by Theorem \ref{lisse}, that $(\mathop{\rm Spec}\nolimits R[P],P)\rightarrow(\mathop{\rm Spec}\nolimits R[Q],Q)$ (see Example \ref{canlog}) is log smooth. Define $\mathop{\rm Spec}\nolimits k\rightarrow\mathop{\rm Spec}\nolimits R[Q]$ by $\alpha:Q\rightarrow k$ as in Definition \ref{logpt}. Let $X$ be a scheme over $k$ which is smooth over $\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits R[Q]}\mathop{\rm Spec}\nolimits R[P]$. Then we have a diagram $$ \begin{array}{ccc} X\\ \vphantom{\bigg|}\Big\downarrow\\ \mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits R[Q]}\mathop{\rm Spec}\nolimits R[P]&\longrightarrow&\mathop{\rm Spec}\nolimits R[P]\\ \vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\ \mathop{\rm Spec}\nolimits k&\longrightarrow&\mathop{\rm Spec}\nolimits R[Q]\rlap{.} \end{array} $$ Define a log structure ${\cal M}$ on $X$ by the pull--back of the canonical log structure on $\mathop{\rm Spec}\nolimits R[P]$. Then we have a morphism $$ f:(X,{\cal M})\longrightarrow(\mathop{\rm Spec}\nolimits k,Q) $$ of fine saturated log schemes. This morphsim $f$ is log smooth by Proposition \ref{usulisse} and Proposition \ref{bextlisse}. We denote this log scheme $(X,{\cal M})$ simply by $(X,P)$. } \end{exa} \begin{exa}\label{toroidal}{\rm (Toric varieties.) In this and the following examples, we use the notation appearing in Example \ref{toroex}. Let $\sigma$ be a cone in $N_{{\bf R}}={\bf R}^d$ and $\sigma^{\vee}$ be its dual cone in $M_{{\bf R}}={\bf R}^d$. Set $P=M\cap\sigma^{\vee}$ and $Q=\{0\}\subset P$. Then, $\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ is $k$--isomorphic to $\mathop{\rm Spec}\nolimits k[P]$ which is nothing but an affine toric variety. Let $X\rightarrow\mathop{\rm Spec}\nolimits k[P]$ be a smooth morphism. Then $(X,P)\rightarrow\mathop{\rm Spec}\nolimits k$ is log smooth. } \end{exa} \begin{exa}\label{ssreduc}{\rm (Variety with normal crossings.) Let $\sigma$ be the cone in $M_{{\bf R}}={\bf R}^d$ generated by $e_1,\ldots,e_d$, where $e_i=(0,\ldots,0,1,0,\ldots 0)\,\mbox{($1$ at the $i$--th entry)}$, $1\leq i\leq d$. Let $\tau$ be the subcone generated by $a_1e_1+\cdots+a_de_d$ with positive integers $a_j$ for $j=1,\ldots,d$. We assume that $\mbox{\rm GCD}(a_1,\ldots,a_d)(=N)$ is invertible in $k$. Set $R={\bf Z}[1/N]$. Then, by setting $P=M\cap\sigma$ and $Q=M\cap\tau$, we see that $\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits R[Q]}\mathop{\rm Spec}\nolimits R[P]$ is $k$--isomorphic to $\mathop{\rm Spec}\nolimits k[z_1,\ldots,z_d]/(z_1^{a_1}\cdots z_d^{a_d})$ and $f$ is induced by $$ \begin{array}{ccl} {\bf N}^d&\longrightarrow&k[z_1,\ldots,z_d]/(z_1^{a_1}\cdots z_d^{a_d})\\ \llap{$\varphi$}\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\ {\bf N}&\longrightarrow&k, \end{array} $$ where the morphism in the first row is defined by $e_i\mapsto z_i,\, (1\leq i\leq d)$, and $\varphi$ is defined by $\varphi(1)=a_1e_1+\cdots+a_de_d$. Let $X\rightarrow \mathop{\rm Spec}\nolimits k[z_1,\ldots,z_d]/(z_1^{a_1}\cdots z_d^{a_d})$ be a smooth morphism. Then, $(X,{\bf N}^d)\rightarrow(\mathop{\rm Spec}\nolimits k,{\bf N})$ is log smooth. } \end{exa} The following theorem is an application of Theorem \ref{lisse}. We prove it in \S \ref{prf2}. \begin{thm}\label{toroch} Let $X$ be an algebraic scheme over a field $k$, and ${\cal M}\rightarrow\O_X$ a fine saturated log structure on $X$. Then, the log scheme $(X,{\cal M})$ is log smooth over $\mathop{\rm Spec}\nolimits k$ with trivial log structure if and only if there exist an open \'{e}tale covering ${\cal U}=\{U_i\}_{i\in I}$ of $X$ and a divisor $D$ of $X$ such that: \begin{description} \item[{\rm 1.}] there exists a smooth morphism $$ h_i:U_i\longrightarrow V_i $$ where $V_i$ is a affine toric variety over $k$ for each $i\in I$, \item[{\rm 2.}] the divisor $U_i\cap D$ of $U_i$ is the pull--back of the union of the closure of codimension 1 torus orbits of $V_i$ by $h_i$ for each $i\in I$, \item[{\rm 3.}] the log structure ${\cal M}\rightarrow\O_X$ is equivalent to the log structure $\O_X\cap j_{\ast}\O^{\times}_{X-D}\hookrightarrow\O_X$ where $j:X-D\hookrightarrow X$ is the inclusion. \end{description} \end{thm} \begin{cor}\label{toroch1} Let $X$ be a smooth algebraic variety over a field $k$, and ${\cal M}\rightarrow \O_X$ a fine saturated log structure on $X$. Then, the log scheme $(X,{\cal M})$ is log smooth over $\mathop{\rm Spec}\nolimits k$ with trivial log structure if and only if there exists a reduced normal crossing divisor $D$ of $X$ such that the log structure ${\cal M}\rightarrow\O_X$ is equivalent to the log structure $\O_X\cap j_{\ast}\O^{\times}_{X-D}\hookrightarrow\O_X$ where $j:X-D\hookrightarrow X$ is the inclusion. \end{cor} \section{Log differentials and log derivations} In this section, we are going to discuss the log differentials and log derivations. These objects are closely related with log smoothness, and play important roles in the sequel. To begin with, we introduce a useful notation which we often use in the sequel. Let $(X,{\cal M})$ be a log scheme. If we like to omit writing the log structure ${\cal M}$, we write this log scheme by $\underline{X}$ to distinguish from the underlying scheme $X$. \begin{dfn}\label{logder} {\rm (cf.\ \cite {Kat1}, in different notation) Let $\underline{X}=(X,{\cal M})$ and $\underline{Y}=(Y,{\cal N})$ be fine log schemes, and $f=(f,\varphi):\underline{X}\rightarrow\underline{Y}$ a morphism, where $\varphi:f^{-1}{\cal N}\rightarrow{\cal M}$ is a homomorphism of sheaves of monoids. \begin{enumerate} \item Let ${\cal E}$ be an $\O_X$--module. The sheaf of {\it log derivations} $\mathop{{\cal D}er}\nolimits_{\underline{Y}}(\underline{X},{\cal E})$ of $\underline{X}$ to ${\cal E}$ over $\underline{Y}$ is the sheaf of germs of couples $(D,D{\rm log} )$, where $D\in\mathop{{\cal D}er}\nolimits_Y(X,{\cal E})$ and $D{\rm log} :{\cal M}\rightarrow{\cal E}$, such that the following conditions are satisfied: \begin{description} \item[{\rm (a)}] $D{\rm log} (ab)=D{\rm log} (a)+D{\rm log} (b),\mbox{ for}\,a,b\in{\cal M}$, \item[{\rm (b)}] $\alpha(a)D{\rm log} (a)=D(\alpha(a)),\mbox{ for}\,a\in{\cal M}$, \item[{\rm (c)}] $D{\rm log} (\varphi(c))=0,\mbox{ for}\,c\in f^{-1}{\cal N}$. \end{description} \item The sheaf of {\it log differentials} of $\underline{X}$ over $\underline{Y}$ is the $\O_X$--module defined by $$ \Omega^1_{\underline{X}/\underline{Y}}= [\Omega^1_{X/Y}\oplus(\O_X\otimes_{{\bf Z}}\gp{{\cal M}})]/{\cal K}, $$ where ${\cal K}$ is the $\O_X$--submodule generated by $$ (d\alpha(a),0)-(0,\alpha(a)\otimes a)\;\mbox{and}\; (0,1\otimes\varphi(b)), $$ for all $a\in{\cal M}$, $b\in f^{-1}{\cal N}$. \end{enumerate}} \end{dfn} \noindent These are coherent $\O_X$--modules if $Y$ is locally noetherian and $X$ locally of finite type over $Y$ (cf.\ \cite {Ill1}). The proofs of the following three propositions are found in \cite [\S 3]{Kat1}. \begin{pro} Let $\underline{X}$, $\underline{Y}$, $f$, and ${\cal E}$ be the same as in Definition \ref{logder}. Then there is a natural isomorphism $$ \mathop{{\cal H}om}\nolimits_{\O_X}(\Omega^1_{\underline{X}/\underline{Y}},{\cal E}) \stackrel{\sim}{\longrightarrow} \mathop{{\cal D}er}\nolimits_{\underline{Y}}(\underline{X},{\cal E}), $$ by $u\mapsto(u\circ d,u\circ d{\rm log} )$, where $d$ and $d{\rm log}$ are defined by $$ d:\O_X\rightarrow\Omega^1_{X/Y}\rightarrow\Omega^1_{\underline{X}/\underline{Y}} $$ and $$ d{\rm log} :{\cal M}\rightarrow\O_X\otimes_{{\bf Z}}\gp{{\cal M}}\rightarrow\Omega^1_{\underline{X}/\underline{Y}}. $$ \end{pro} \begin{pro}\label{genbun} Let $\underline{X}\stackrel{f}{\rightarrow}\underline{Y}\stackrel{g}{\rightarrow}\underline{Z}$ be morphisms of fine log schemes. \begin{enumerate} \item[{\rm 1.}] There exists an exact sequence $$ f^{*}\Omega^1_{\underline{Y}/\underline{Z}}\rightarrow\Omega^1_{\underline{X}/\underline{Z}} \rightarrow\Omega^1_{\underline{X}/\underline{Y}}\rightarrow 0. $$ \item[{\rm 2.}] If $f$ is log smooth, then \begin{equation}\label{diffseq} 0\rightarrow f^{*}\Omega^1_{\underline{Y}/\underline{Z}}\rightarrow\Omega^1_{\underline{X}/\underline{Z}} \rightarrow\Omega^1_{\underline{X}/\underline{Y}}\rightarrow 0 \end{equation} is exact. \item[{\rm 3.}] If $g\circ f$ is log smooth and {\rm (\ref{diffseq})} is exact and splits locally, then $f$ is log smooth. \end{enumerate} \end{pro} \begin{pro}\label{bungen} If $f:\underline{X}\rightarrow\underline{Y}$ is log smooth, then $\Omega^1_{\underline{X}/\underline{Y}}$ is a locally free $\O_X$--module of finite type. \end{pro} \begin{exa}\label{fan1}{\rm (cf.\ \cite [Chap. 3, \S(3.1)]{Oda1}) Let $X_{\Sigma}$ be a toric variety over a field $k$ determined by a fan $\Sigma$ on $N_{{\bf R}}$ with $N={\bf Z}^d$. Consider the log scheme $(X_{\Sigma},\Sigma)$ (Example \ref{torlog}) over $\mathop{\rm Spec}\nolimits k$. Then we have isomorphisms of $O_X$--modules $$ \mathop{{\cal D}er}\nolimits_k(\underline{X},\O_X)\cong\O_X\otimes_{{\bf Z}}N\; \; {\rm and}\; \; \Omega^1_{\underline{X}/k}\cong\O_X\otimes_{{\bf Z}}M, $$ where $M=\mathop{\rm Hom}\nolimits_{{\bf Z}}(N,{\bf Z})$. } \end{exa} \begin{exa}{\rm For $X=\mathop{\rm Spec}\nolimits k[z_1,\ldots,z_{n}]/(z_1\cdots z_l)$, let $f:(X,{\cal M})\rightarrow(\mathop{\rm Spec}\nolimits k,{\bf N}\rightarrow k)$ be the log smooth morphism defined in Example \ref{ssreduc}. Then $\mathop{{\cal D}er}\nolimits_{\underline{k}}(\underline{X},\O_X)$ is a free $\O_X$--module generated by $$ z_1\frac{\partial}{\partial z_1},\ldots,z_l\frac{\partial}{\partial z_l}, \frac{\partial}{\partial z_{l+1}},\ldots,\frac{\partial}{\partial z_n} $$ with a relation $$ z_1\frac{\partial}{\partial z_1}+\cdots+z_l\frac{\partial}{\partial z_l}=0. $$ The sheaf $\Omega^1_{\underline{X}/\underline{k}}$ is a free $\O_X$--module generated by the {\it logarithmic differentials}: $$ \frac{dz_1}{z_1},\ldots,\frac{dz_l}{z_l},dz_{l+1},\ldots,dz_n $$ with a relation $$ \frac{dz_1}{z_1}+\cdots+\frac{dz_l}{z_l}=0. $$ In the complex analytic case, the sheaf $\Omega^1_{\underline{X}/\underline{k}}$ is nothing but the sheaf of {\it relative logarithmic differentials} introduced in, for example, \cite [\S 3]{Fri1}, and \cite [\S 2]{K-N1}. } \end{exa} \section{The proof of Theorem 4.1}\label{prf1} In this section, we give a proof of Theorem \ref{lisse} due to Kato \cite {Kat1}. Before proving the general case, we prove the following proposition. \begin{pro}\label{canlisse} Let $A$ be a commutative ring and $h:Q\rightarrow P$ a homomorphism of finitely generated integral monoids. The homomorphism $h$ induces the morphism of log schemes $$ f:\underline{X}=(\mathop{\rm Spec}\nolimits A[P],P)\longrightarrow\underline{Y}=(\mathop{\rm Spec}\nolimits A[Q],Q). $$ We set $K=\mathop{\rm Ker}\nolimits(\gp{h}:\gp{Q}\rightarrow\gp{P})$ and $C=\mathop{\rm Coker}\nolimits(\gp{h}:\gp{Q}\rightarrow\gp{P})$, and denote the torsion part of $C$ by $\tor{C}$. If both $K$ and $\tor{C}$ are finite groups of order invertible in $A$, then $f$ is log smooth. \end{pro} \noindent{\sc Proof.}\hspace{2mm} Suppose we have a commutative diagram $$ \begin{array}{ccccc} (T',\L')&\stackrel{s'}{\longrightarrow}&\underline{X}&=&(\mathop{\rm Spec}\nolimits A[P],P)\\ \llap{$t$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$f$}\\ (T,\L)&\underrel{\longrightarrow}{s}&\underline{Y}&=&(\mathop{\rm Spec}\nolimits A[Q],Q) \end{array} $$ in ${\bf LSch}^{\rm f}$, where the morphism $t$ is a thickening of order 1. Since we may work \'{e}tale locally, we may assume that $T$ is affine. Set $$ {\cal I}=\mathop{\rm Ker}\nolimits(\O_T\rightarrow \O_{T'}). $$ Since the morphism $t$ is a thickening of order 1, by Lemma \ref{thick}, we have the following commutative diagram with exact rows: $$ \begin{array}{ccccccccc} 1&\longrightarrow&1+{\cal I}&\lhook\joinrel\longrightarrow&\L&\stackrel{t^*} {\longrightarrow}& \L'&\longrightarrow&1\\ &&\parallel&&\cap&&\cap\\ 1&\longrightarrow&1+{\cal I}&\longrightarrow&\gp{\L}& \underrel{\longrightarrow}{\gp{(t^*)}}&\gp{\L'}& \longrightarrow&1\rlap{.} \end{array} $$ \noindent Note that the right square of the above commutative diagram is cartesian. \vspace{3mm} First, consider the following commutative diagram with exact rows: $$ \begin{array}{ccccccccccc} 1&\longrightarrow&K&\longrightarrow&\gp{Q} &\stackrel{\gp{h}}{\longrightarrow}&\gp{P}&\longrightarrow&C &\longrightarrow&1\\ &&\vphantom{\bigg|}\Big\downarrow\rlap{$u$}&&\vphantom{\bigg|}\Big\downarrow\rlap{$v$}&& \vphantom{\bigg|}\Big\downarrow\rlap{$w$}\\ 1&\longrightarrow&1+{\cal I}&\longrightarrow&\gp{\L} &\underrel{\longrightarrow}{\gp{(t^*)}}&\gp{\L'}&\longrightarrow &1\rlap{.} \end{array} $$ The multiplicative monoid $1+{\cal I}$ is isomorphic to the additive monoid ${\cal I}$ by $1+x\mapsto x$ since ${\cal I}^2=0$. If the order of $K$ is invertible in $A$, then we have $u=1$, and hence there exists a morphism $a':R\rightarrow \gp{\L}$ with $R=\mbox{\rm Image}\,(\gp{h}:\gp{Q}\rightarrow\gp{P})$ such that $a'\circ\gp{h}=v$ and $\gp{(t^*)}\circ a'=w$. \vspace{3mm} Next, we consider the following commutative diagram with exact rows: $$ \begin{array}{ccccccccccc} &&1&\longrightarrow&R&\stackrel{i}{\longrightarrow}&\gp{P} &\longrightarrow&C&\longrightarrow&1\\ &&&&\llap{$a'$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$w$}\\ 0&\longrightarrow&{\cal I}&\longrightarrow&\gp{\L} &\underrel{\longrightarrow}{\gp{(t^*)}}&\gp{\L'}&\longrightarrow &1\rlap{.} \end{array} $$ We shall show that there exists a homomorphism $a'':\gp{P}\rightarrow\gp{\L}$ such that $a''\circ t=a'$ and $\gp{(t^*)}\circ a''=w$. The obstruction of existence of $a''$ lies in $\mbox{\rm Ext}^1(C,{\cal I})$. In general, if a positive integer $n$ is invertible in $A$ then we have $\mbox{\rm Ext}^1({\bf Z}/n{\bf Z},{\cal I})=0$. Combining this with $\mbox{\rm Ext}^1({\bf Z},{\cal I})=0$, we have $\mbox{\rm Ext}^1(C,{\cal I})=0$ since the order of the torsion part of $C$ is invertible in $A$. Hence a homomorphism $a''$ exists. Since the diagram $$ \begin{array}{ccc} \L&\stackrel{t^*}{\longrightarrow}&\L'\\ \cap&&\cap\\ \gp{\L}&\underrel{\longrightarrow}{\gp{(t^*)}}&\gp{\L'}\rlap{.} \end{array} $$ is cartesian, we found a homomorphism $$ a:P\longrightarrow\L $$ such that $t^*\circ a=(s')^*$ and $a\circ h=s^*$. Using this $a$, we can construct a morphism of log schemes $$ g:(T,\L)\longrightarrow\underline{X}=(\mathop{\rm Spec}\nolimits A[P],P) $$ such that $g\circ t=s'$ and $s\circ g=f$. $\Box$ \vspace{3mm} Now, let us prove Theorem 4.1. First, we prove the implication $2\Rightarrow 1$. Let $R={\bf Z}[1/(N_1\cdot N_2)]$ where $N_1$ is the order of $\mathop{\rm Ker}\nolimits(\gp{Q}\rightarrow\gp{P})$ and $N_2$ is the order of the torsion part of $\mathop{\rm Coker}\nolimits(\gp{Q}\rightarrow\gp{P})$. By the assumption (a), we have $$ Y\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]\cong Y\times_{\mathop{\rm Spec}\nolimits R[Q]}\mathop{\rm Spec}\nolimits R[P]. $$ Since $X\rightarrow Y\times_{\mathop{\rm Spec}\nolimits R[Q]}\mathop{\rm Spec}\nolimits R[P]$ is smooth by (b), $f$ is log smooth due to Proposition \ref{usulisse}, Proposition \ref{bextlisse} and Proposition \ref{canlisse}. \vspace{3mm} Next, let us prove the converse. Assume the morphism $f$ is log smooth. Then, the sheaf $\Omega^1_{\underline{X}/\underline{Y}}$ is a locally free $\O_X$--module of finite type (Proposition \ref{bungen}). Take any point $x\in X$. We denote by $\bar{x}$ a separable closure of $x$. \begin{ste}{\rm Consider the morphism of $\O_X$--modules $$ 1\otimesd{\rm log}:\O_X\otimes_{{\bf Z}}\gp{{\cal M}}\longrightarrow \Omega^1_{\underline{X}/\underline{Y}}, $$ which is surjective by the definition of $\Omega^1_{\underline{X}/\underline{Y}}$. Then we can take elements $t_1,\ldots,t_r\in{\cal M}_{\bar{x}}$ such that the system $\{d{\rm log} t_i\}_{1\leq i\leq r}$ is a $\O_{X,\bar{x}}$--base of $\Omega^1_{\underline{X}/\underline{Y},\bar{x}}$. Consider the homomorphism $\psi:{\bf N}^r\rightarrow{\cal M}_{\bar{x}}$ defined by $$ {\bf N}^r\ni(n_1,\ldots,n_r)\mapsto t_1^{n_1}\cdots t_r^{n_r}\in{\cal M}_{\bar{x}}. $$ Combining this $\psi$ with the homomorphism $Q\rightarrow f^{-1}({\cal N})_{\bar{x}} \rightarrow{\cal M}_{\bar{x}}$, we have a homomorphism $\varphi:H={\bf N}^r\oplus Q\rightarrow{\cal M}_{\bar{x}}$. } \end{ste} \begin{ste}{\rm Let $k(\bar{x})$ denote the residue field at $\bar{x}$. We have a homomorphism \begin{equation}\label{bunchan} k(\bar{x})\otimes_{{\bf Z}}{\bf Z}^r\longrightarrow k(\bar{x})\otimes_{{\bf Z}} \mathop{\rm Coker}\nolimits(f^{-1}(\gp{{\cal N}}/{\cal O}^\times_Y)_{\bar{x}}\rightarrow\gp{{\cal M}}_{\bar{x}}/ {\cal O}^\times_{X,\bar{x}}) \end{equation} by $k(\bar{x})\otimes_{{\bf Z}}\gp{\psi}:k(\bar{x})\otimes_{{\bf Z}}{\bf Z}^r\rightarrow k(\bar{x})\otimes_{{\bf Z}}\gp{{\cal M}}_{\bar{x}}$ and canonical projectiones $\gp{{\cal M}}_{\bar{x}}\rightarrow\gp{{\cal M}}_{\bar{x}}/{\cal O}^\times_{X,\bar{x}}\rightarrow \mathop{\rm Coker}\nolimits(f^{-1}(\gp{{\cal N}}/{\cal O}^\times_Y)_{\bar{x}}\rightarrow\gp{{\cal M}}_{\bar{x}}/ {\cal O}^\times_{X,\bar{x}})$. We claim that this morphism (\ref{bunchan}) is surjective. In fact, this morphism coincides with the composite morphism $$ k(\bar{x})\otimes_{{\bf Z}}{\bf Z}^r\rightarrow k(\bar{x})\otimes_{\O_{X,\bar{x}}} \Omega^1_{\underline{X}/\underline{Y},\bar{x}}\rightarrow k(\bar{x})\otimes_{{\bf Z}} \mathop{\rm Coker}\nolimits(f^{-1}(\gp{{\cal N}}/{\cal O}^\times_Y)_{\bar{x}}\rightarrow\gp{{\cal M}}_{\bar{x}}/ {\cal O}^\times_{X,\bar{x}}), $$ where the first morphism is induced by $d{\rm log}\circ\psi$ and the second one by the canonical projection, and these morphisms are clearly surjective. Hence the morphism (\ref{bunchan}) is surjective. On the other hand, the homomorphism $$ \gp{Q}\longrightarrow f^{-1}({\cal N}/{\cal O}^\times_Y)_{\bar{x}} $$ is surjective since $Q\rightarrow{\cal N}$ is a chart of ${\cal N}$. Hence, the homomorphism $$ k(\bar{x})\otimes_{{\bf Z}}\gp{Q}\longrightarrow k(\bar{x})\otimes_{{\bf Z}} f^{-1}({\cal N}/{\cal O}^\times_Y)_{\bar{x}} $$ is surjective, and then, the homomorphism $$ 1\otimes_{{\bf Z}}\gp{\varphi}:k(\bar{x})\otimes_{Z}\gp{H}\longrightarrow k(\bar{x})\otimes_{{\bf Z}}({\cal M}_{\bar{x}}/{\cal O}^\times_{X,\bar{x}}) $$ is surjective. This shows that the cokernel $C=\mathop{\rm Coker}\nolimits(\gp{\varphi}:\gp{H}\rightarrow \gp{{\cal M}}_{\bar{x}}/{\cal O}^\times_{X,\bar{x}})$ is annihilated by an integer $N$ invertible in $\O_{X,\bar{x}}$. } \end{ste} \begin{ste}{\rm Take elements $a_1,\ldots,a_d\in\gp{{\cal M}}_{\bar{x}}$ which generates $C$. Then we can write $a_i^n=u_i\varphi(b_i)$ for $u_i\in{\cal O}^\times_{X,\bar{x}}$ and $b_i\in\gp{H}$, for $i=1,\ldots,d$. Since ${\cal O}^\times_{X,\bar{x}}$ is $N$--divisible, we can write $u_i=v_i^N$ for $v_i\in{\cal O}^\times_{X,\bar{x}}$, for $i=1,\ldots,d$, and hence we may suppose $a_i^N=\varphi(b_i)$, replacing $a_i$ by $a_i/v_i$, for $i=1,\ldots,d$. Let $G$ be the push--out of the diagram $$ \gp{H}\longleftarrow{\bf Z}^d\longrightarrow{\bf Z}^d, $$ where ${\bf Z}^d\rightarrow\gp{H}$ is defined by $e_i\mapsto b_i$, and ${\bf Z}^d\rightarrow{\bf Z}^d$ is defined by $e_i\mapsto Ne_i$ for $i=1,\ldots,d$. Then $\gp{\varphi}: \gp{H}\rightarrow\gp{{\cal M}}_{\bar{x}}$ and ${\bf Z}^d\rightarrow\gp{{\cal M}}_{\bar{x}}$, defined by $e_i\mapsto a_i$ for $i=1,\ldots,d$, induce the homomorphism $$ \phi:G\longrightarrow\gp{{\cal M}}_{\bar{x}} $$ which maps $G$ surjectively onto $\gp{{\cal M}}_{\bar{x}}/{\cal O}^\times_{X,\bar{x}}$. Define $P=\phi^{-1}({\cal M}_{\bar{x}})$, then $P$ defines a chart of ${\cal M}$ on some neighborhood of $\bar{x}$ (\cite [Lemma 2.10]{Kat1}). If ${\cal M}$ is saturated, $P$ is also saturated. There exists an induced map $Q\rightarrow P$ which defines a chart of $f$ on some neighborhood of $\bar{x}$. Since $\gp{H}\rightarrow\gp{P}$ is injective, $\gp{Q}\rightarrow\gp{P}$ is injective. The cokernel $\mathop{\rm Coker}\nolimits(\gp{H}\rightarrow\gp{P})$ is annihilated by $N$, hence $\tor{\mathop{\rm Coker}\nolimits(\gp{Q}\rightarrow\gp{P})}$ is finite and annihilated by $N$. } \end{ste} \begin{ste}{\rm Set $X'=Y\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ and $g:X\rightarrow X'$. We need to show that the morphism $g$ is smooth in the usual sense. Since $\underline{X}$ has the log structure induced by $g$ from $\underline{X}'=(X',P)$, it suffice to show that $g$ is log smooth (Proposition \ref{usulisse}). Since $k(\bar{x})\otimes_{{\bf Z}}(\gp{P}/\gp{Q})\cong k(\bar{x})\otimes_{{\bf Z}}{\bf Z}^d\cong k(\bar{x})\otimes_{\O_{X,\bar{x}}}\Omega^1_{\underline{X}/\underline{Y},\bar{x}}$ and $\Omega^1_{\underline{X}/\underline{Y}}$ is locally free, we have $\Omega^1_{\underline{X}/\underline{Y}}\cong \O_X\otimes_{{\bf Z}}(\gp{P}/\gp{Q})$ on some neighborhood of $\bar{x}$. On the other hand, by direct calculations, one sees that $\Omega^1_{\underline{X}'/\underline{Y}}\cong\O_{X'}\otimes_{{\bf Z}[P]}\Omega^1_{{\bf Z}[P]/{\bf Z}[Q]}\cong \O_{X'}\otimes_{{\bf Z}}(\gp{P}/\gp{Q})$. Hence we have $g^*\Omega^1_{\underline{X}'/\underline{Y}} \cong\Omega^1_{\underline{X}/\underline{Y}}$. This implies $g$ is log smooth due to Proposition \ref{genbun} (in fact, $g$ is {\it log \'{e}tale} (cf.\ \cite{Kat1})). } \end{ste} \noindent This completes the proof of the theorem. $\Box$ \section{The proof of Theorem 4.7}\label{prf2} In this section, we give a proof of Theorem \ref{toroch}. If $V=\mathop{\rm Spec}\nolimits k[P]$ is an affine toric variety, then it is easy to see that the log structure associated to $P\rightarrow k[P]$ is equivalent to the log structure $\O_X\cap j_{\ast}\O^{\times}_{V-D}\hookrightarrow \O_X$ where $D$ is the union of the closure of codimention 1 torus orbits of $V$ and $j:V-D\hookrightarrow V$ is the inclusion. Hence, the ``if'' part of Theorem \ref{toroch} is easy to see. Let us prove the converse. Let $(X,{\cal M})$ be as in the assumption of Theorem \ref{toroch} and $f:(X,{\cal M})\rightarrow\mathop{\rm Spec}\nolimits k$ the structure morphism. The key--lemma is the following. \begin{lem}\label{keylem1} We can take \'{e}tale locally a chart $P\rightarrow{\cal M}$ of ${\cal M}$ such that \begin{description} \item[{\rm 1.}] the chart $(P\rightarrow{\cal M},1\rightarrow k^{\times}, 1\rightarrow P)$ of $f$ satisfies the conditions {\rm (a)} and {\rm (b)} in Theorem \ref{lisse}, \item[{\rm 2.}] $P$ is a finitely generated integral saturated monoid, and has no torsion element. \end{description} Here, by a torsion element, we mean an element $x\neq 1$ such that $x^n=1$ for some positive integer $n$. \end{lem} First, we are going to show that the theorem follows from the above lemma. Since the monoid $P$ has no torsion element, $P$ is the saturated submonoid of a finitely generated free abelian group $\gp{P}$. Hence, $X$ is \'{e}tale locally smooth over affine toric varieties, and the log structure ${\cal M}$ on $X$ is \'{e}tale locally equivalent to the pull--back of the log structure induced by the union of the closure of codimension 1 torus orbits. Since, these log structure glue to the log structure ${\cal M}$ on $X$, the pull--back of the union of the closure of codimension 1 torus orbits glue to a divisor on $X$. In fact, this divisor is the compliment of the largest open subset $U$ such that ${\cal M}|_U$ is trivial with the reduced scheme structure. Hence our assertion is proved. \vspace{3mm} Now, we are going to prove Lemma \ref{keylem1}. We may work \'{e}tale locally. Take a chart $(P\rightarrow{\cal M},1\rightarrow k^{\times},1\rightarrow P)$ of $f$ as in Theorem \ref{lisse}. We may assume that $P$ is saturated. Define $$ \tor{P}=\{x\in P\: |\: x^n=1\,(\mbox{for some $n$})\}. $$ $\tor{P}$ is a subgroup in $P$. Take a decomposition $\gp{P}=\fr{G}\oplus\tor{G}$ of the finitely generated abelian group $\gp{P}$, where $\fr{G}$ (resp.\ $\tor{G}$) is a free (resp.\ torsion) subgroup of $\gp{P}$. Then we have the equalities $\tor{P}=P\cap\tor{G}=\tor{G}$ since $P$ is saturated. Define a submonoid $\fr{P}$ by $\fr{P}=P\cap\fr{G}$. \begin{cla} $P=\fr{P}\oplus\tor{P}$. \end{cla} \noindent{\sc Proof.}\hspace{2mm} Take $x\in P$. Decompose $x=yz$ in $\gp{P}$ such that $y\in\fr{G}$ and $z\in\tor{G}=\tor{P}$. Since $y^n=(xz^{-1})^n=x^n\in P$ for a large $n$, we have $y\in P$. Hence $y\in\fr{P}$. $\Box$ \vspace{3mm}\noindent Define $\fr{\alpha}:\fr{P}\rightarrow\O_X$ by $\fr{P}\hookrightarrow P\stackrel{\alpha}{\rightarrow}\O_X$. \begin{cla} The homomorphism $\fr{\alpha}:\fr{P}\rightarrow\O_X$ defines a log structure equivalent to ${\cal M}$. \end{cla} \noindent{\sc Proof.}\hspace{2mm} If $x\in\tor{P}$, then $\alpha(x)\in{\cal O}^\times_X$ since $\alpha(x)^n=1$ for a large $n$. Hence $\alpha(\tor{P})\subset{\cal O}^\times_X$. This implies that the associated log structure of $\fr{P}$ is equivalent to that of $P$. $\Box$ \vspace{3mm}\noindent Hence, the morphism $f$ is equivalent to the morphsim induced by the diagram $$ \begin{array}{ccc} \fr{P}&\stackrel{\fr{\alpha}}{\longrightarrow}&\O_X\\ \llap{$\fr{\varphi}$}\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\ 1&\underrel{\longrightarrow}{\lambda}&k, \end{array} $$ Then we have to check the conditions (a) and (b) in Theorem \ref{lisse}. The condition (a) is easy to verify. Let us check the condition (b). We need to show that the morphism $$ X\longrightarrow\mathop{\rm Spec}\nolimits k[\fr{P}] $$ induced by $X\rightarrow\mathop{\rm Spec}\nolimits{\bf Z}[P]\rightarrow\mathop{\rm Spec}\nolimits{\bf Z}[\fr{P}]$ is smooth. \begin{cla} The morphism \begin{equation}\label{sepext} \mathop{\rm Spec}\nolimits k[P]\longrightarrow \mathop{\rm Spec}\nolimits k[\fr{P}] \end{equation} induced by $\fr{P}\hookrightarrow P$ is \'{e}tale. \end{cla} \noindent{\sc Proof.}\hspace{2mm} Since $P=\fr{P}\oplus\tor{P}$, we have $k[P]=k[\fr{P}]\otimes_{k}k[\tor{P}]$. Since every element in $\tor{P}$ is roots of 1, and the order of $\tor{P}$ is invertible in $k$, the morphism $$ k\lhook\joinrel\longrightarrow k[\tor{P}] $$ is a finite separable extension of the field $k$. This shows that the morphism (\ref{sepext}) is \'{e}tale. $\Box$ \vspace{3mm}\noindent Now we have proved Lemma \ref{keylem1}, and hence, Theorem \ref{toroch}. \section{Formulation of log smooth deformation} {}From now on, we fix the following notation. Let $k$ be a field and $Q$ a finitely generated integral saturated monoid having no invertible element other than 1. Then we have a logarithmic point (Definition \ref{logpt}) $\underline{k}=(\mathop{\rm Spec}\nolimits k, Q)$. Let $f=(f,\varphi):\underline{X}=(X,{\cal M})\rightarrow\underline{k}=(\mathop{\rm Spec}\nolimits k,Q)$ be a log smooth morphism in ${\bf LSch}^{\rm fs}$. \vspace{3mm} Let $\Lambda$ be a complete noetherian local ring with the residue field $k$. For example, $\Lambda=k$ or $\Lambda=\mbox{the Witt vector ring of $k$}$ when $k$ is perfect. We denote, by $\Lambda[[Q]]$, the completion of the monoid ring $\Lambda[Q]$ along the maximal ideal $\mu+\Lambda[Q\setminus\{1\}]$ where $\mu$ denotes the maximal ideal of $\Lambda$. The completion $\Lambda[[Q]]$ is a complete local $\Lambda$--algebra and is noetherian since $Q$ is finitely generated. If the monoid $Q$ is isomorphic to ${\bf N}$ then the ring $\Lambda[[Q]]$ is isomorphic to $\Lambda[[t]]$ as local $\Lambda$--algebras. Let ${\cal C}_{\Lambda[[Q]]}$ be the category of artinian local $\Lambda[[Q]]$--algebras with the residue field $k$, and $\widehat{{\cal C}}_{\Lambda[[Q]]}$ be the category of pro--objects of ${\cal C}_{\Lambda[[Q]]}$ (cf.\ \cite {Sch1}). For $A\in\mathop{\rm Obj}\nolimits(\widehat{{\cal C}}_{\Lambda[[Q]]})$, we define a log structure on the scheme $\mathop{\rm Spec}\nolimits A$ by the associated log structure $$ Q\oplus A^{\times}\longrightarrow A $$ of the homomorphism $Q\rightarrow\Lambda[[Q]]\rightarrow A$. We denote, by $(\mathop{\rm Spec}\nolimits A, Q)$, the log scheme obtained in this way. \begin{dfn}{\rm For $A\in\mathop{\rm Obj}\nolimits({\cal C}_{\Lambda[[Q]]})$, a {\it log smooth lifting} of $f:(X,{\cal M})\rightarrow(\mathop{\rm Spec}\nolimits k,Q)$ on $A$ is a morphism $\widetilde{f}:(\widetilde{X},\widetilde{{\cal M}})\rightarrow(\mathop{\rm Spec}\nolimits A,Q)$ in ${\bf LSch}^{\rm fs}$ together with a cartesian diagram $$ \begin{array}{ccc} (X,{\cal M})&\longrightarrow&(\widetilde{X},\widetilde{\M})\\ \llap{$f$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$\widetilde{f}$}\\ (\mathop{\rm Spec}\nolimits k,Q)&\longrightarrow&(\mathop{\rm Spec}\nolimits A,Q) \end{array} $$ in ${\bf LSch}^{\rm fs}$. Two liftings are said to be {\it isomorphic} if they are isomorphic in ${\bf LSch}^{\rm fs}_{(\mathop{\rm Spec}\nolimits A,Q)}$. } \end{dfn} \noindent Note that $(\mathop{\rm Spec}\nolimits k,Q)\rightarrow(\mathop{\rm Spec}\nolimits A,Q)$ is an exact closed immersion, and hence, the above diagram is cartesian in ${\bf LSch}^{\rm fs}$ if and only if so is in ${\bf LSch}$ (Lemma \ref{fpro}). In particular, the underlying morphisms of log smooth liftings are (not necessarily flat) liftings in the usual sense. Moreover, since exact closed immersions are stable under base changes, $(X,{\cal M})\rightarrow(\widetilde{X},\widetilde{\M})$ is also an exact closed immersion. If either $Q=\{1\}$ or $Q={\bf N}$, the underlying morphisms of any log smooth liftings of $f$ are flat since these morphisms of log schemes are {\it integral} (cf.\ \cite {Kat1}). Hence, in this case, the underlying morphisms of log smooth liftings of $f$ are flat liftings of $f$. \vspace{3mm} Take a local chart $(P\rightarrow {\cal M}, Q\rightarrow Q\oplus k^{\times}, Q\rightarrow P)$ of $f$ extending the given $Q\rightarrow k$ as in Theorem \ref{lisse} such that $\gp{Q}\rightarrow\gp{P}$ is injective (Remark \ref{lisserem}). Then, $f$ factors throught $\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ by the smooth morphism $X\rightarrow\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ and the natural projection \'{e}tale locally. For $A\in\mathop{\rm Obj}\nolimits({\cal C}_{\Lambda[[Q]]})$, a smooth lifting \begin{equation}\label{loclift} \widetilde{X}\longrightarrow\mathop{\rm Spec}\nolimits A\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P] \end{equation} of $X\rightarrow\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$, with the naturally induced log structure, gives a local log smooth lifting of $f$. Note that this local lifting $(\widetilde{X},\widetilde{\M})\rightarrow (\mathop{\rm Spec}\nolimits A,Q)$ is log smooth. Conversely, suppose $\widetilde{f}:(\widetilde{X},\widetilde{\M})\rightarrow (\mathop{\rm Spec}\nolimits A,Q)$ is a local log smooth lifting of $f$ on $A$. \begin{lem}\label{liftlem1} The local chart $(P\rightarrow {\cal M}, Q\rightarrow Q\oplus k^{\times}, Q\rightarrow P)$ of $f$ lifts to the local chart $(P\rightarrow\widetilde{\M}, Q\rightarrow Q\oplus A^{\times}, Q\rightarrow P)$ of $\widetilde{f}$. \end{lem} \noindent{\sc Proof.}\hspace{2mm} The proof is done by the induction with respect to the length of $A$. Take $A'\in\mathop{\rm Obj}\nolimits({\cal C}_{\Lambda[[Q]]})$ with surjective morphism $A\rightarrow A'$ such that $I=\mathop{\rm Ker}\nolimits(A\rightarrow A')\neq 0$ and $I^2=0$. Let $\widetilde{f}':(\widetilde{X}',\widetilde{\M}')\rightarrow (\mathop{\rm Spec}\nolimits A',Q)$ be a pull--back of $\widetilde{f}$. Then, $\widetilde{f}'$ is a log smooth lifting of $f$ to $A'$. By the induction, we have the lifted local chart $(P\rightarrow\widetilde{\M}', Q\rightarrow Q\oplus A'^{\times}, Q\rightarrow P)$ of $\widetilde{f}'$. Since $(\widetilde{X}',\widetilde{\M}')\rightarrow (\widetilde{X},\widetilde{\M})$ is a thickening of order 1, by Lemma \ref{thick}, we have the following commutative diagram with exact rows: $$ \begin{array}{ccccccccc} 0&\rightarrow&{\cal I}&\hookrightarrow&\widetilde{\M}& \rightarrow&\widetilde{\M}'&\rightarrow&1\\ &&\parallel&&\cap&&\cap\\ 0&\rightarrow&{\cal I}&\rightarrow&\gp{\widetilde{\M}}& \rightarrow&\gp{\widetilde{\M}'}&\rightarrow&1\rlap{,} \end{array} $$ where ${\cal I}=\mathop{\rm Ker}\nolimits(\O_{\widetilde{X}}\rightarrow\O_{\widetilde{X}'})$. The right square of this diagram is cartesian. Consider the following commutative diagram of abelian groups with exact rows and columns: $$ \begin{array}{ccccccc} \mathop{\rm Hom}\nolimits(\gp{P},{\cal I})&\rightarrow&\mathop{\rm Hom}\nolimits(\gp{P},\gp{\widetilde{\M}})&\rightarrow& \mathop{\rm Hom}\nolimits(\gp{P},\gp{\widetilde{\M}'})&\rightarrow&\mathop{\rm Ext}\nolimits^1(\gp{P},{\cal I})\\ \vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\ \mathop{\rm Hom}\nolimits(\gp{Q},{\cal I})&\rightarrow&\mathop{\rm Hom}\nolimits(\gp{Q},\gp{\widetilde{\M}})&\rightarrow& \mathop{\rm Hom}\nolimits(\gp{Q},\gp{\widetilde{\M}'})\\ \vphantom{\bigg|}\Big\downarrow\\ \mathop{\rm Ext}\nolimits^1(C,{\cal I})\rlap{,} \end{array} $$ where $C=\mathop{\rm Coker}\nolimits(\gp{Q}\hookrightarrow\gp{P})$. By our assumption, we have $\mathop{\rm Ext}\nolimits^1(\gp{P},{\cal I})=0$ and $\mathop{\rm Ext}\nolimits^1(C,{\cal I})=0$, since the order of the torsion part of each $\gp{P}$ and $C$ is invertible in $A$. Then, by an easy diagram chasing, we can show that the given morphism $\gp{P}\rightarrow \gp{\widetilde{\M}'}$ can be lifted to a morphism $\gp{P}\rightarrow \gp{\widetilde{\M}}$ such that $$ \begin{array}{ccc} \gp{P}&\longrightarrow&\gp{\widetilde{\M}}\\ \vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\ \gp{Q}&\longrightarrow&\gp{Q}\oplus A^{\times} \end{array} $$ is commutative. Then, we have the morphism $P\rightarrow\widetilde{\M}$ and we see that the diagram $$ \begin{array}{ccc} \P&\longrightarrow&\widetilde{\M}\\ \vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\ Q&\longrightarrow&Q\oplus A^{\times} \end{array} $$ is commutative. Since $\widetilde{\M}/{\cal O}^\times_{\widetilde{X}} \stackrel{\sim}{\rightarrow}\widetilde{\M}'/{\cal O}^\times_{\widetilde{X}'}$, we can easily show that the morphism $P\rightarrow\widetilde{\M}$ defines a chart (Lemma \ref {basic3}). $\Box$ \vspace{3mm} \noindent Then, $\widetilde{f}$ is factors throught $\mathop{\rm Spec}\nolimits A\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ by the induced morphism $\widetilde{X}\rightarrow\mathop{\rm Spec}\nolimits A\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]} \mathop{\rm Spec}\nolimits {\bf Z}[P]$ and the natural projection, and we have the following commutative diagram $$ \begin{array}{ccc} X&\longrightarrow&\widetilde{X}\\ \vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\ \mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]&\longrightarrow& \mathop{\rm Spec}\nolimits A\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]\\ \vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\ \mathop{\rm Spec}\nolimits k&\longrightarrow&\mathop{\rm Spec}\nolimits A\rlap{,} \end{array} $$ such that the each square is cartesian. Hence, $\widetilde{X}\rightarrow\mathop{\rm Spec}\nolimits A\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]} \mathop{\rm Spec}\nolimits {\bf Z}[P]$ is smooth since it is a smooth lifting of the smooth morphism $X\rightarrow\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$, and is unique up to isomorphisms by the classical theory. Therefore, we have proved the following proposition. \begin{pro}\label{liftloc}{\rm (cf. \cite [(3.14)]{Kat1})} For $A\in{\cal C}_{\Lambda[[Q]]}$, a log smooth lifting of $f:(X,{\cal M})\rightarrow(\mathop{\rm Spec}\nolimits k,Q)$ on $A$ exists \'{e}tale locally, and is unique up to isomorphisms. In particular, log smooth liftings are log smooth. \end{pro} Let $\widetilde{f}:(\widetilde{X},\widetilde{\M})\rightarrow(\mathop{\rm Spec}\nolimits A,Q)$ be a log smooth lifting of $f$ to $A$, and $u:A'\rightarrow A$ a surjective homomorphism in ${\cal C}_{\Lambda[[Q]]}$ such that $I^2=0$ where $I=\mathop{\rm Ker}\nolimits (u)$. Suppose $\widetilde{f}':(\widetilde{X}',\widetilde{\M}')\rightarrow(\mathop{\rm Spec}\nolimits A',Q)$ is a log smooth lifting of $f$ to $A'$ which is also a lifting of $\widetilde{f}$. Let $(P\rightarrow\widetilde{\M}', Q\rightarrow Q\oplus A'^{\times}, Q\rightarrow P)$ be a local chart of $\widetilde{f}'$ which is a lifting of $(P\rightarrow{\cal M}, Q\rightarrow Q\oplus k^{\times}, Q\rightarrow P)$. Define a local chart $(P\rightarrow\widetilde{\M}, Q\rightarrow Q\oplus A^{\times}, Q\rightarrow P)$ of $\widetilde{f}$ by $P\rightarrow{\cal M}' \rightarrow{\cal M}$ and $Q\rightarrow Q\oplus A'^{\times}\rightarrow Q\oplus A^{\times}$. An automorphism $\Theta:(\widetilde{X}',\widetilde{\M}')\stackrel{\sim}{\rightarrow} (\widetilde{X}',\widetilde{\M}')$ over $(\mathop{\rm Spec}\nolimits A',Q)$ which is identity on $(\widetilde{X},\widetilde{\M})$ induces an automorphism $\theta:\gp{\widetilde{\M}'}\stackrel{\sim}{\rightarrow}\gp{\widetilde{\M}'}$. Consider the diagram $$ \begin{array}{ccccccccc} &&&&\gp{P}&=&\gp{P}\\ &&&&\llap{$\alpha'$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$\alpha$}\\ 1&\rightarrow&1+{\cal I}&\rightarrow&\gp{\widetilde{\M}'}&\rightarrow&\gp{\widetilde{\M}}& \rightarrow&1. \end{array} $$ For $a\in\gp{P}$, the element $\alpha'(a)\cdot[\theta\circ\alpha'(a)]^{-1}$ is in $1+{\cal I}$. Then, we have a morphism $\Delta:\gp{P}\rightarrow{\cal I}=I\cdot\O_{\widetilde{X}'} \cong I\otimes_{A}\O_{\widetilde{X}}$ by $\Delta(a)=\alpha'(a)\cdot[\theta\circ\alpha'(a)]^{-1}-1$. The morphism $\Delta$ lifts to the morphism $\Delta:\gp{P}/\gp{Q}\rightarrow I\otimes_{A}\O_{\widetilde{X}}$ and defines a morphism of $\O_{\widetilde{X}}$--modules $$ \O_{\widetilde{X}}\otimes_{{\bf Z}}(\gp{P}/\gp{Q})\rightarrow I\otimes_{A}\O_{\widetilde{X}}. $$ Since $\Omega^1_{\underline{\widetilde{X}}/\underline{A}}\cong\O_{\widetilde{X}}\otimes_{{\bf Z}}(\gp{P}/\gp{Q})$ \'{e}tale locally, this defines a local section of $$ \mathop{{\cal H}om}\nolimits_{\O_{\widetilde{X}}}(\Omega^1_{\underline{\widetilde{X}}/\underline{A}},I\otimes_{A}\O_{\widetilde{X}})\cong \mathop{{\cal D}er}\nolimits(\underline{\widetilde{X}},\O_{\widetilde{X}})\otimes_{A}I. $$ Conversely, for a local section $(D,D{\rm log})\in\mathop{{\cal D}er}\nolimits(\underline{\widetilde{X}},\O_{\widetilde{X}})\otimes_{A}I$, $D$ induces an automorphism of $O_{\widetilde{X}'}$ and $D{\rm log}$ induces an automorphism of $\widetilde{\M}'$, and then, indues an automorphism of $(\widetilde{X}',\widetilde{\M}')$. By this, applying the arguement in SGA I \cite{Gro1} Expos\'{e} 3, we get the following proposition. \begin{pro}\label{liftauto}{\rm (cf. \cite [(3.14)]{Kat1})} Let $\widetilde{f}:(\widetilde{X},\widetilde{\M})\rightarrow(\mathop{\rm Spec}\nolimits A,Q)$ be a log smooth lifting of $f$ to $A$, and $u:A'\rightarrow A$ a surjective homomorphism in ${\cal C}_{\Lambda[[Q]]}$ such that $I^2=0$ where $I=\mathop{\rm Ker}\nolimits (u)$ (i.e., $(\mathop{\rm Spec}\nolimits A,Q)\rightarrow(\mathop{\rm Spec}\nolimits A',Q)$ is a thickening of order $\leq 1$). \begin{enumerate} \item The sheaf of germs of lifting automorphisms of $\widetilde{X}$ to $A'$ is $$ \mathop{{\cal D}er}\nolimits_{\underline{A}}(\underline{\widetilde{X}},\O_{\widetilde{X}})\otimes_A I. $$ \item The set of isomorphism classes of log smooth liftings of $\widetilde{X}$ to $A'$ is isomorphic to $$ \mbox{\rm H}^1(\widetilde{X},\mathop{{\cal D}er}\nolimits_{\underline{A}}(\underline{\widetilde{X}},\O_{\widetilde{X}}))\otimes_A I. $$ \item The lifting obstructions of $\widetilde{X}$ to $A'$ are in $$ \mbox{\rm H}^2(\widetilde{X},\mathop{{\cal D}er}\nolimits_{\underline{A}}(\underline{\widetilde{X}},\O_{\widetilde{X}}))\otimes_A I. $$ \end{enumerate} \end{pro} Define the {\it log smooth deformation functor} ${\bf LD}={\bf LD}_{\underline{X}/\underline{k}}$ by $$ {\bf LD}_{\underline{X}/\underline{k}}(A)=\{\mbox{isomorphism class of log smooth lifting of $f:\underline{X}\rightarrow\underline{k}$ on $A$}\} $$ for $A\in\mathop{\rm Obj}\nolimits({\cal C}_{\Lambda[[Q]]}$). This is a covariant functor from ${\cal C}_{\Lambda[[Q]]}$ to $\mbox{\bf Ens}$, the category of sets, such that ${\bf LD}_{\underline{X}/\underline{k}}(k)$ consists of one point. We shall prove the following theorem in the next section. \begin{thm}\label{hull} If the underlying scheme $X$ is proper over $k$, then the log deformation functor ${\bf LD}_{\underline{X}/\underline{k}}$ has a representable hull {\rm (cf.\ \cite {Sch1})}. \end{thm} \section{The proof of Theorem 8.5} We are going to prove Theorem \ref{hull} by checking M. Schlessinger's criterion (\cite [Theorem 2.11]{Sch1}) for ${\bf LD}$. Let $u_1:A_1\rightarrow A_0$ and $u_2:A_2\rightarrow A_0$ be morphisms in ${\cal C}_{\Lambda[[Q]]}$. Consider the map \begin{equation}\label{desant} {\bf LD}(A_1\times_{A_0}A_2)\longrightarrow{\bf LD}(A_1)\times_{{\bf LD}(A_0)}{\bf LD}(A_2). \end{equation} Then we shall check the following conditions. \begin{description} \item[(H1)] The map (\ref{desant}) is a surjection whenever $u_2:A_2\rightarrow A_0$ is a surjection. \item[(H2)] The map (\ref{desant}) is a bijection when $A_0=k$ and $A_2=k[\epsilon]$, where $k[\epsilon]=k[E]/(E^2)$. \item[(H3)] $\mbox{dim}_k(t_{{\bf LD}})<\infty$, where $t_{{\bf LD}}={\bf LD}(k[\epsilon])$. \end{description} By Proposition \ref{liftauto}, we have $$ t_{{\bf LD}}\cong\mbox{\rm H}^1(X,\mathop{{\cal D}er}\nolimits_{\underline{k}}(\underline{X},\O_X)). $$ Our assumption implies that $t_{{\bf LD}}$ is finite dimensional since $\mathop{{\cal D}er}\nolimits_{\underline{k}}(\underline{X},\O_X)$ is a coherent $\O_X$--module. Hence we need to check (H1) and (H2). Set $B=A_1\times_{A_0}A_2$. Let $v_i:B\rightarrow A_i$ be the natural map for $i=1,2$. We denote the morphisms of schemes associated to $u_i$ and $v_i$ also by $u_i:\mathop{\rm Spec}\nolimits A_0\rightarrow\mathop{\rm Spec}\nolimits A_i$ and $v_i:\mathop{\rm Spec}\nolimits A_i\rightarrow\mathop{\rm Spec}\nolimits B$ for $i=1,2$, respectively. \vspace{3mm} \noindent{\sc Proof of (H1).}\hspace{2mm} Suppose the homomorphism $u_2:A_2\rightarrow A_0$ is surjective. Take an element $(\eta_1,\eta_2)\in{\bf LD}(A_1)\times_{{\bf LD}(A_0)}{\bf LD}(A_2)$ where $\eta_i$ is an isomorphism class of a log smooth lifting $f_i:(X_i,{\cal M}_i)\rightarrow(\mathop{\rm Spec}\nolimits A_i,Q)$ for each $i=1,2$. The equality ${\bf LD}(u_1)(\eta_1)={\bf LD}(u_2)(\eta_2)(=\eta_0)$ implies that there exists an isomorphism $(u_2)^*(X_2,{\cal M}_2)\stackrel{\sim}{\rightarrow}(u_1)^*(X_1,{\cal M}_1)$ over $(\mathop{\rm Spec}\nolimits A_0,Q)$. Here, $(u_i)^*(X_i,{\cal M}_i)$ is the pull--back of $(X_i,{\cal M}_i)$ by $u_i:\mathop{\rm Spec}\nolimits A_0\rightarrow\mathop{\rm Spec}\nolimits A_i$ for $i=1,2$. Set $(X_0,{\cal M}_0)=(u_1)^*(X_1,{\cal M}_1)$. We denote the induced morphism of log schemes $(X_0, {\cal M}_0)\rightarrow(X_i, {\cal M}_i)$ by $\underline{u}_i$ for $i=1,2$. Then we have the following commutative diagram: $$ \begin{array}{ccccc} (X_1,{\cal M}_1)&\stackrel{\underline{u}_1}{\longleftarrow}&(X_0,{\cal M}_0)& \stackrel{\underline{u}_2}{\longrightarrow}&(X_2,{\cal M}_2)\\ \llap{$f_1$}\vphantom{\bigg|}\Big\downarrow&&\llap{$f_0$}\vphantom{\bigg|}\Big\downarrow&& \vphantom{\bigg|}\Big\downarrow\rlap{$f_2$}\\ (\mathop{\rm Spec}\nolimits A_1,Q)&\stackrel{u_1}{\longleftarrow}&(\mathop{\rm Spec}\nolimits A_0,Q)& \stackrel{u_2}{\longrightarrow}&(\mathop{\rm Spec}\nolimits A_2,Q). \end{array} $$ We have to find an element $\xi\in{\bf LD}(B)$, which represents a lifting of $f$ to $B$, such that ${\bf LD}(v_i)(\xi)=\eta_i$ for $i=1,2$. Consider a scheme $Z=(|X|,\O_{X_1}\times_{\O_{X_0}}\O_{X_2})$ over $\mathop{\rm Spec}\nolimits B$. Define a log structure on $Z$ by the natural homomorphism $$ {\cal N}={\cal M}_1\times_{{\cal M}_0}{\cal M}_2\longrightarrow \O_Z=\O_{X_1}\times_{\O_{X_0}}\O_{X_2}. $$ It is easy to verify that this homomorphism is a log structure. Since the diagram $$ \begin{array}{ccc} {\cal N}&\longrightarrow&\O_Z\\ \vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\ \llap{$Q\cong$\hspace{1mm}}Q\times_{Q}Q&\longrightarrow&B \end{array} $$ is commutative, we have a morphism $g:(Z,{\cal N})\rightarrow(\mathop{\rm Spec}\nolimits B,Q)$ of log schemes. By the construction, we have the morphism $v_i:(X_i,{\cal M}_i)\rightarrow (Z,{\cal N})$ for $i=1,2$ such that the diagram $$ \begin{array}{ccc} (X_1,{\cal M}_1)&\stackrel{\underline{v}_1}{\longrightarrow}&(Z,{\cal N})\\ \llap{$\underline{u}_1$}\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\rlap{$\underline{v}_2$}\\ (X_0,{\cal M}_0)&\underrel{\longrightarrow}{\underline{u}_2}&(X_2,{\cal M}_2) \end{array} $$ is commutative. Since $u_2:A_2\rightarrow A_0$ is surjective, the underlying morphism $X_1\rightarrow Z$ of $\underline{v}_1$ is a closed immersion in the classical sense. We have to show that the morphism $\underline{v}_1$ is an exact closed immersion. Take a local chart $(P\rightarrow{\cal M}, Q\rightarrow Q\oplus k^{\times}, Q\rightarrow P)$ of $f$ as in Theorem \ref{lisse} such that $\gp{Q} \rightarrow\gp{P}$ is injective. By Lemma \ref{liftlem1}, this local chart lifts to the local chart of $f_i$ for each $i=0,1,2$. Since $u_2:A_2\rightarrow A_0$ is surjective, we have the following isomorphism $$ {\cal N}/{\cal O}^\times_Z\stackrel{\sim}{\rightarrow} ({\cal M}_1/{\cal O}^\times_{X_1})\times_{({\cal M}_0/{\cal O}^\times_{X_0})}({\cal M}_2/{\cal O}^\times_{X_2}). $$ By this, one sees that $P\cong P\times_{P}P\rightarrow{\cal N}$ is a local chart of ${\cal N}$. This shows that $(Z,{\cal N})$ is a fine saturated log scheme, and $\underline{v}_1$ is an exact closed immersion. Then, the exact closed immersion $(X,{\cal M})\rightarrow (X_1,{\cal M}_1)\rightarrow (Z,{\cal N})$ gives $g$ a structure of log smooth lifting of $f$ to $(\mathop{\rm Spec}\nolimits B,Q)$. Hence, $g$ represents an element $\xi\in{\bf LD}(B)$. It is easy to verify that ${\bf LD}(v_i)(\xi)=\eta_i$ for $i=1,2$ since the morphism $f_1$,$f_2$ and $g$ have the common local chart. Thus (H1) is now proved. $\Box$ \vspace{3mm} \noindent{\sc Proof of (H2).}\hspace{2mm} We continue to use the same notation as above. First, we prepare the following lemma. \begin{lem}\label{basic4} Let $g':(Z',{\cal N}')\rightarrow (\mathop{\rm Spec}\nolimits B,Q)$ be a log smooth lifting of $f$ with a commutative diagram $$ \begin{array}{ccc} (X_1,{\cal M}_1)&\longrightarrow&(Z',{\cal N}')\\ \llap{$\underline{u}_1$}\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\ (X_0,{\cal M}_0)&\underrel{\longrightarrow}{\underline{u}_2}&(X_2,{\cal M}_2) \end{array} $$ of liftings such that $(v_i)^{*}(Z',{\cal N}')\stackrel{\sim}{\leftarrow} (X_i,{\cal M}_i)$ over $(\mathop{\rm Spec}\nolimits A_i,Q)$ for $i=1,2$. Then, the natural morphism $(Z,{\cal N})\rightarrow(Z',{\cal N}')$ is an isomorphism. \end{lem} \noindent{\sc Proof.}\hspace{2mm} We may work \'{e}tale locally. By Lemma \ref{liftlem1}, the local chart $(P\rightarrow{\cal M}, Q\rightarrow Q\oplus k^{\times}, Q\rightarrow P)$ of $f$ lifts to a local chart of $g'$. Take a local chart $(P\rightarrow{\cal N}, Q\rightarrow Q\oplus B^{\times}, Q\rightarrow P)$ of $g$ by $P\rightarrow{\cal N}'\rightarrow{\cal N}$. Then, the schemes $Z$ and $Z'$ are smooth liftings of $X\rightarrow\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ to $\mathop{\rm Spec}\nolimits B\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$. Hence, we have only to show that the natural morphism $Z\rightarrow Z'$ of underlying schemes is an isomorphism. But this follows from the classical theory \cite [Corollary 3.6]{Sch1} since each $X_i$ is a smooth lifting of $X\rightarrow\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ to $\mathop{\rm Spec}\nolimits A_i\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ for $i=0,1,2$. $\Box$ \vspace{3mm} \noindent Let $g':(Z',{\cal N}')\rightarrow (\mathop{\rm Spec}\nolimits B,Q)$ be a log smooth lifting of $f$ which represents a class $\xi'\in{\bf LD}(B)$. Suppose that the class $\xi'$ is mapped to $(\eta_1,\eta_2)$ by (\ref {desant}). Then, $$ (X_0,{\cal M}_0)\stackrel{\sim}{\rightarrow} (v_1\circ u_1)_{*}(Z',{\cal N}')\cong (v_2\circ u_2)_{*}(Z',{\cal N}') \stackrel{\sim}{\leftarrow}(X_0,{\cal M}_0) $$ defines an automorphism $\theta$ of the lifting $(X_0,{\cal M}_0)$. If this automorphism $\theta$ lifts to an automorphism $\theta'$ of the lifting $(X_1,{\cal M}_1)$ such that $\theta'\circ \underline{u}_1=\underline{u}_1\circ\theta'$, then, replacing $(X_1,{\cal M}_1)\rightarrow(Z',{\cal N}')$ by $(X_1,{\cal M}_1)\stackrel{\theta'}{\rightarrow}(X_1,{\cal M}_1)\rightarrow(Z',{\cal N}')$, we have a commutative diagram as in Lemma \ref{basic4}, and then, we have $\xi=\xi'$. Now if $A=k$ (so that $(X_0,{\cal M}_0)=(X,{\cal M})$, $\theta={\rm id}$), $\theta'$ exists. Thus (H2) is proved. $\Box$ \vspace{3mm}\noindent Thus we have proved Theorem \ref{hull}. \section{Example 1: Log smooth deformation over trivial base} \label{exam1} As we have seen in Theorem \ref{toroch}, any log scheme $(X,{\cal M})$ which is log smooth over $\mathop{\rm Spec}\nolimits k$ with the trivial log structure is smooth over an affine torus embedding \'{e}tale locally. \begin{exa}{\rm (Usual smooth deformations.) Let $X$ be a smooth algebraic variety over a field $k$. Then $X$ with the trivial log structure is log smooth over $\mathop{\rm Spec}\nolimits k$ (this is the case $D=0$ in Corollary \ref{toroch1}), and our log smooth deformation of $X$ is nothing but the usual smooth deformation of $X$. } \end{exa} \begin{exa}{\rm (Generalized relative deformations.) Let $X$ be an algebraic variety over a field $k$. Assume that there exists a fine saturated log structure ${\cal M}$ on $X$ such that $f:(X,{\cal M})\rightarrow\mathop{\rm Spec}\nolimits k$ is log smooth. Then, by Theorem \ref{toroch}, $X$ is covered by \'{e}tale open sets which are smooth over affine toric varieties, and the log structure ${\cal M}\rightarrow\O_X$ is equivalent to the log structure defined by \begin{equation}\label{logdiv} {\cal M}=j_{\ast}{\cal O}^\times_{X-D}\cap\O_X \end{equation} for some divisor $D$ of $X$, where $j$ is the inclusion $X-D\hookrightarrow X$. In this situation, our log smooth deformation of $f$ is the deformation of the pair $(X,D)$. If $X$ is smooth over $k$, then $D$ is a reduced normal crossing divisor (Corollary \ref{toroch1}). Assume $X$ is smooth over $k$, then we have the exact sequence $$ 0\rightarrow\mathop{{\cal D}er}\nolimits_k(\underline{X},\O_X)\rightarrow\mathop{{\cal D}er}\nolimits_k(X,\O_X)(=\Theta_X) \rightarrow{\cal N}\rightarrow 0, $$ where ${\cal N}$ is an $\O_X$--module locally written by $$ ({\cal N}_{D_1|X}\otimes\O_{D_1})\oplus\cdots\oplus ({\cal N}_{D_d|X}\otimes\O_{D_d}), $$ where $D_1,\ldots,D_d$ are local components of $D$, and ${\cal N}_{D_i|X}$ is the normal bundle of $D_i$ in $X$ for $i=1,\ldots,d$. Then, we have an exact sequence $$ \mbox{\rm H}^0(D,{\cal N})\rightarrow t_{{\bf LD}}\rightarrow \mbox{\rm H}^1(X,\Theta_X)\rightarrow \mbox{\rm H}^1(D,{\cal N}). $$ In this sequence, $\mbox{\rm H}^0(D,{\cal N})$ is viewed as the set of isomorphism classes of locally trivial deformations of $D$ in $X$, and $\mbox{\rm H}^1(D,{\cal N})$ is viewed as a set of obstructions of deformations of $D$ in $X$. Hence this sequence explains the relation between the log smooth deformation and the usual smooth deformation. Note that, if $D$ is a smooth divisor on $X$, the log smooth deformation is nothing but the {\it relative deformation} of the pair $(X,D)$ studied by Makio \cite {Mak1}. } \end{exa} \begin{exa}{\rm (Toric varieties.) Let $X_{\Sigma}$ be a complete toric variety over a field $k$ defined by a fan $\Sigma$ in $N_{{\bf R}}$, and consider the log scheme $\underline{X}=(X_{\Sigma},\Sigma)$ (Example \ref{torlog}) over $\mathop{\rm Spec}\nolimits k$. We have seen in Example \ref{fan1} that $$ \mathop{{\cal D}er}\nolimits_k(\underline{X},\O_X)\cong\O_X\otimes_{{\bf Z}}N, $$ and hence is a globally free $\O_X$--module. Since $\mbox{\rm H}^1(X,\mathop{{\cal D}er}\nolimits_k(\underline{X},\O_X))=0$, any toric varieties are infinitesimally rigid with respect to our log smooth deformation. Note that toric varieties without log structures are not necessarily rigid with respect to the usual smooth deformation. } \end{exa} \section{Example 2: Smoothings of normal crossing varieties} \label{exam2} Let $k$ be a field. A {\it normal crossing variety} over $k$ is a seperated, connected, and geometrically reduced scheme $X$ of finite type over $k$ which is covered by an \'{e}tale open covering $\{X_\lambda\}_{\lambda\in\Lambda}$ such that each $X_{\lambda}$ is isomorphic to $\mathop{\rm Spec}\nolimits k[z_1,\ldots,z_n]/(z_1\cdots z_{d_\lambda})$ over $k$ where $n-1={\rm dim}_k X$. Let $(\mathop{\rm Spec}\nolimits k,{\bf N})$ be a standard log point (Definition \ref{logpt}). \begin{dfn}{\rm (cf. \cite[(2.6)]{Kaj1})\ A log structure $\alpha:{\cal M}\rightarrow\O_X$ on a normal crossing variety $X$ is called a log structure of {\it semistable type} over $(\mathop{\rm Spec}\nolimits k,{\bf N})$ if the following conditions are satisfied: \begin{enumerate} \item there exists an \'{e}tale open covering $\{X_\lambda\}_{\lambda\in\Lambda}$ of $X$ such that, for each $\lambda\in\Lambda$, $X_\lambda$ is isomorphic to $\mathop{\rm Spec}\nolimits k[z_1,\ldots,z_n]/(z_1\cdots z_{d_\lambda})$ over $k$ and the log structure ${\cal M}_\lambda={\cal M}|_{X_\lambda}$ has a chart $\beta_{\lambda}:{\bf N}^{d_\lambda} \rightarrow{\cal M}_{\lambda}$ such that $\alpha\circ\beta_{\lambda}(e_i)=z_i$ for $i=1,\ldots, d_{\lambda}$, where $e_i=(0,\ldots,0,1,0,\ldots,0)$ ($1$ at the $i$--th entry), \item there exists a morphism $f:(X,{\cal M})\rightarrow (\mathop{\rm Spec}\nolimits k,{\bf N})$ which has a local chart $(\beta_{\lambda}:{\bf N}^{d_\lambda}\rightarrow{\cal M}, {\bf N}\rightarrow{\bf N}\oplus k^{\times}, \varphi_{\lambda}:{\bf N}\rightarrow{\bf N}^{d_{\lambda}})$ where $\varphi_{\lambda}$ is the diagonal homomorphism, for each $\lambda\in\Lambda$. \end{enumerate} } \end{dfn} \vspace{3mm} \noindent The morphism $f:(X,{\cal M})\rightarrow (\mathop{\rm Spec}\nolimits k,{\bf N})$ defined as above is called a {\it logarithmic semistable reduction}. Note that a logarithmic semistable reduction is log smooth (Example \ref{ssreduc}). Let $f:(X,{\cal M})\rightarrow(\mathop{\rm Spec}\nolimits k,{\bf N})$ be a logarithmic semistable reduction. Then, \'{e}tale locally, $X$ is isomorphic to $k[z_1,\ldots,z_n]/(z_1\cdots z_d)$, and $f$ is induced by the diagram $$ \begin{array}{ccl} {\bf N}^d&\longrightarrow&k[z_1,\ldots,z_n]/(z_1\cdots z_d)\\ \llap{$\varphi$}\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\ {\bf N}&\longrightarrow&k. \end{array} $$ For $A\in\mathop{\rm Obj}\nolimits({\cal C}_{\Lambda[[{\bf N}]]})$, the log structure on $A$ is defined by $\gamma:{\bf N}\rightarrow A$, $\gamma(1)=\pi\in m_A$, where $m_A$ is the maximal ideal of $A$. Then the log smooth lifting of $f$ on $A$ is locally equivalent to the morphism induced by the diagram $$ \begin{array}{ccl} {\bf N}^d&\longrightarrow&A[z_1,\ldots,z_n]/(z_1\cdots z_d-\pi)\\ \llap{$\varphi$}\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\ {\bf N}&\underrel{\longrightarrow}{\gamma}&A. \end{array} $$ Hence, our log smooth deformation carries out the smoothing of the normal crossing variety $X$. \vspace{3mm} Now we assume that the singular locus $X_{\mbox{sing}}=D$ of $X$ is connected. Then our deformation theory coincides with the {\it logarithmic deformation theory of normal crossing varieties} introduced by Kawamata and Namikawa \cite {K-N1} in the complex analytic situation. The following theorem is essentially due to Kawamata and Namikawa \cite {K-N1}, and we prove it in the next section. \begin{thm}\label{dss} Let $X$ be a normal crossing variety over $k$. Then, $X$ has a log structure of semistable type over $(\mathop{\rm Spec}\nolimits k,{\bf N})$ if and only if $X$ is $d$--semistable, i.e., ${\cal E}xt^1(\Omega^1_X,\O_X)\cong\O_D$ {\rm (cf.\ \cite {Fri1})}. \end{thm} \section{The proof of Theorem 11.2} Let $X$ be a normal crossing variety over a field $k$ of dimension $n$. Then the scheme $X$ is covered by an \'{e}tale open covering $\{X_{\lambda}\}_{\lambda\in\Lambda}$ such that each $X_{\lambda}$ is isomorphic to the divisor in the $(n+1)$--affine space over $k$, $V_\lambda=\mathop{\rm Spec}\nolimits k[z^{\ssd{\la}}_0,\ldots, z^{\ssd{\la}}_n]\cong \mbox{\bf A}^{(n+1)}_k$, defined by $z^{\ssd{\la}}_0\cdots z^{\ssd{\la}}_{d_\lambda}=0$ for some $0\leq d_\lambda \leq n$. We call this covering $\U=\{X_\la\hookrightarrow V_\la\}$ a {\it coordinate covering} of $X$. If $\U=\{X_\la\hookrightarrow V_\la\}$ is a coordinate covering, the {\it singular locus} of $X$, denoted by $D$, is the closed subscheme of $X$ defined locally by the equations $z^{\ssd{\la}}_0\cdotsz^{\ssd{\la}}_{i-1}\cdotz^{\ssd{\la}}_{i+1}\cdotsz^{\ssd{\la}}_{d_\lambda}=0$ for $0\leq i \leq d_\lambda$ in $V_\lambda$. We write the defining ideals of $X_\lambda$ and $D$ in $V_\lambda$ as $$ \begin{array}{ccl} {\cal I}_{X_\lambda} &=& (z^{\ssd{\la}}_0\cdots z^{\ssd{\la}}_{d_\lambda}),\vspace{2mm} \\ {\cal I}_{D_\lambda} &=& (z^{\ssd{\la}}_0\cdotsz^{\ssd{\la}}_{i-1}\cdotz^{\ssd{\la}}_{i+1}\cdots z^{\ssd{\la}}_{d_\lambda}\mid 0\leq i \leq d_\lambda). \end{array} $$ These are ideals in $\O_{V_\lambda}$. In the sequel, we fix these notation and conventions, sometimes omitting the index $\lambda$. \vspace{3mm} Let us describe ${\cal T}^1_X={\cal E}xt^1_{\O_X}(\Omega^1_X, \O_X)$ locally. Consider the following exact sequence, $$ 0\rightarrow{\cal I}_X/{\cal I}^2_X\rightarrow\Omega^1_V\otimes\O_X \rightarrow\Omega^1_X\rightarrow 0. $$ Here, we omitted the index $\lambda$. Its dual is $$ 0\rightarrow{\cal T}^0_X\rightarrow\Theta_V\otimes\O_X \rightarrow{\cal N}_{X\mid V}. $$ Then, we have an isomorphism ${\cal T}^1_X\cong \mathop{\rm Coker}\nolimits(\Theta_V\otimes\O_X\rightarrow{\cal N}_{X\mid V})$. \begin{lem}\label{locT1} ${\cal T}^1_X$ is an invertible $\O_D$--module. More precisely, we have an isomorphism of $\O_D$--modules $$ {\cal T}^1_X\cong({\cal I}_X/{\cal I}^2_X)^\vee\otimes\O_D. $$ \end{lem} \noindent{\sc Proof.}\hspace{2mm} First, note that ${\cal I}_X/{\cal I}^2_X=\O_X\cdot d(z_0\cdots z_d)$, and $\Omega^1_V\otimes\O_X=\oplus^n_{i=1}\O_X\cdot dz_i$. Here, we omited the index $\lambda$. Let us denote the inclusion ${\cal I}_X/{\cal I}^2_X\hookrightarrow\Omega^1_V\otimes\O_X$ by $\iota$. For each $f\in\Theta_V\otimes\O_X= {\cal H}om_{\O_X}(\Omega^1_V\otimes\O_X, \O_X)$, set $f_i=f(dz_i)\in\O_X$. Then, $\iota^* f(d(z_0\cdots z_d))= \sum^d_{i=0}f_i z_0\cdots z_{i-1}\cdot z_{i+1}\cdots z_d\in{\cal I}_D\otimes\O_X$. Hence, $\mbox{\rm Image}\,\iota^*\subseteq{\cal H}om_{\O_X}({\cal I}_X/{\cal I}^2_X, {\cal I}_D\otimes\O_X)$. Easy to see the converse. Then, we have $\mbox{\rm Image}\,\iota^*= {\cal H}om_{\O_X}({\cal I}_X/{\cal I}^2_X, {\cal I}_D\otimes\O_X)$. Hence, we get ${\cal T}^1_X\cong\mathop{\rm Coker}\nolimits\iota^*\cong{\cal H}om_{\O_X}({\cal I}_X/{\cal I}^2_X, \O_D) \cong({\cal I}_X/{\cal I}^2_X)^\vee\otimes\O_D$. $\Box$ \vspace{3mm} Let $X$ be a reduced normal crossing variety over a field $k$ and $\U=\{X_\la\hookrightarrow V_\la\}$ a coordinate covering of $X$. A system $\{(\zeta^{\ssd{\la}}_0,\ldots,\zeta^{\ssd{\la}}_n)\}_{\lambda\in\Lambda}$, where $\zeta^{\ssd{\la}}_i\in {\rm H}^0(X_\lambda, \O_X)$ for $\lambda\in\Lambda$ and $i=0,\ldots,n$, is said to be a {\it log system} on $X$ with respect to $\U=\{X_\la\hookrightarrow V_\la\}$ if the following conditions are satisfied: \begin{enumerate} \item for $0\leq i\leq d_\lambda$, we have $ \zeta^{\ssd{\la}}_i=u^{\ssd{\lambda}}_i\cdotz^{\ssd{\la}}_i $ for some $u^{\ssd{\lambda}}_i\in{\rm H}^0(X_\lambda, \O^\times_X)$, \item for $d_\lambda <j\leq n$, $\zeta^{\ssd{\la}}_j$ is invertible on $X_\lambda$. \end{enumerate} Then we have the following lemma. \begin{lem}\label{ltrns} On each $X_{\lambda\mu}=X_\lambda\cap X_\mu\neq\emptyset$, there exists a transition relation such as \begin{equation}\label{trans} \zeta^{\ssd{\la}}_i=u^{\ssd{\la\mu}}_i\zeta^{\ssd{\mu}}_{\sigma^{\ssd{\la\mu}}(i)}\ (0\leq i\leq n), \end{equation} for some invertible function $u^{\ssd{\la\mu}}_i$ on $X_{\lambda\mu}$ and a permutation $\sigma^{\ssd{\la\mu}}$. {\rm (As is seen in the following proof, these $u^{\ssd{\la\mu}}_i$'s and $\sigma^{\ssd{\la\mu}}$'s are {\it not} unique. )} \end{lem} \noindent{\sc Proof.}\hspace{2mm} Set $E^{\ssd{\lambda}}=\{i\mid(\zeta^{\ssd{\la}}_i \mid_{X_{\lambda\mu}})\not\in{\rm H}^0(X_{\lambda\mu}, \O^\times_X)\}$, and set $E^{\ssd{\mu}}$ similarly. There is a one to one correspondence between elements in $E^{\ssd{\lambda}}$ and components of $X_{\lambda\mu}$ by $i\leftrightarrow\{\zeta^{\ssd{\la}}_i=0\}$. This implies that there exists a bijection $\sigma: E^{\ssd{\lambda}}\stackrel{\sim}{\rightarrow}E^{\ssd{\mu}}$ such that $\{\zeta^{\ssd{\la}}_i=0\}=\{\zeta^{\ssd{\mu}}_{\sigma(i)}=0\}$. Hence, for $i\in E^{\ssd{\lambda}}$, we can write $\zeta^{\ssd{\la}}_i =u^{\ssd{\la\mu}}_i\cdot\zeta^{\ssd{\mu}}_{\sigma(i)}$ for some $u^{\ssd{\la\mu}}_i\in{\rm H}^0(X_{\lambda\mu}, \O^\times_X)$. On the other hand, if $i$ is not in $E^{\ssd{\lambda}}$, we can write as (\ref{trans}) for these $i$'s, since $\zeta^{\ssd{\la}}_i$ is invertible on $X_{\lambda\mu}$ (note that the number of these $i$'s coincides with that of those $j$'s which are not in $E^{\ssd{\mu}}$). $\Box$ \begin{pro}\label{cridss}{\rm (cf.\ \cite {K-N1})} The following conditions are equivalent. \begin{enumerate} \item $X$ is $d$--semistable. \item There exist a log system $\{(\zeta^{\ssd{\la}}_0,\ldots,\zeta^{\ssd{\la}}_n)\}$ on $X$ and its transition system $\{(u^{\ssd{\la\mu}}_i, \sigma^{\ssd{\la\mu}})\}$ as in Lemma \ref{ltrns} such that the equality \begin{equation}\label{cdss} u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_n=1 \end{equation} holds on each $X_\lambda\cap X_\mu\neq\emptyset$. \end{enumerate} \end{pro} \noindent{\sc Proof.}\hspace{2mm} ($1\Rightarrow 2$) Assume that the normal crossing variety $X$ is $d$--semistable. Due to Lemma \ref{locT1}, the invertible $\O_X$--module $({\cal I}_X/{\cal I}^2_X)\otimes\O_D\cong{\cal I}_X/{\cal I}_X{\cal I}_D$ is trivial, i.e., $$ {\cal I}_X/{\cal I}_X{\cal I}_D\quad\cong\quad\O_D. $$ Let us denote the natural projection ${\cal I}_X/{\cal I}^2_X\rightarrow{\cal I}_X/{\cal I}_X{\cal I}_D$ by $p$. The sheaf ${\cal I}_X/{\cal I}_X{\cal I}_D$ is a free $\O_D$-module and $p(z_0\cdots z_d)$ is a $\O_D$-free base of it. By the above isomorphism, the global section $1\in\O_D$ correspondes to $p(v\cdot z_0\cdots z_d)$, for some invertible function $v$ on $X$. Set $$ \zeta_i=\left\{ \begin{array}{ll} v\cdot z_0&(i=0), \\ z_i&(1\leq i\leq d), \\ 1&(d<i\leq n). \end{array} \right. $$ Then, the system $\{(\zeta^{\ssd{\la}}_0,\ldots,\zeta^{\ssd{\la}}_n)\}$ is a log system on $X$. Due to Lemma \ref{ltrns}, we can take a transition system $\{(u^{\ssd{\la\mu}}_i, \sigma^{\ssd{\la\mu}})\}$ such that the transition relation (\ref{trans}) holds. On each $X_{\lambda\mu}\neq\emptyset$, we have $p_\lambda(\zeta^{\ssd{\la}}_0\cdots\zeta^{\ssd{\la}}_n)= p_\mu(\zeta^{\ssd{\mu}}_0\cdots\zeta^{\ssd{\mu}}_n)$. Then, we have $u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_n p_\mu(\zeta^{\ssd{\mu}}_0\cdots\zeta^{\ssd{\mu}}_n)= p_\mu(\zeta^{\ssd{\mu}}_0\cdots\zeta^{\ssd{\mu}}_n)$. Since, $p_\mu(v^{\ssd{\mu}}z^{\ssd{\mu}}_0\cdotsz^{\ssd{\mu}}_{d_{\mu}})$ is a $\O_D$--free base of ${\cal I}_X/{\cal I}_X{\cal I}_D$, we have $u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_n=1$ on $D$. Hence, we can write $$ u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_n=1+\sum^n_{j=0}a_j z^{\ssd{\mu}}_0\cdotsz^{\ssd{\mu}}_{j-1}\cdotz^{\ssd{\mu}}_{j+1}\cdotsz^{\ssd{\mu}}_n. $$ Replacing every $u^{\ssd{\la\mu}}_j$ with $$ u^{\ssd{\la\mu}}_j-a_jz^{\ssd{\mu}}_0\cdotsz^{\ssd{\mu}}_{j-1}\cdotz^{\ssd{\mu}}_{j+1}\cdotsz^{\ssd{\mu}}_n (u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_{j-1}\cdotu^{\ssd{\la\mu}}_{j+1}\cdotsu^{\ssd{\la\mu}}_n)^{-1}, $$ we get $u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_n=1$ on $X_{\lambda\mu}$ as desired. \vspace{3mm} ($2\Rightarrow 1$) The local section $\zeta^{\ssd{\la}}_0\otimes\cdots\otimes\zeta^{\ssd{\la}}_n$ is a local generator of $({\cal I}_X/{\cal I}^2_X)\otimes\O_D$. By (\ref{cdss}), these local generators glue to a global section. Hence, the invertible $\O_D$--module $({\cal I}_X/{\cal I}^2_X)\otimes\O_D$ is trivial, and its dual ${\cal T}^1_X$ is also trivial. $\Box$ \vspace{3mm} Let $\U=\{X_\la\hookrightarrow V_\la\}$ be a coordinate covering on $X$, and take a log system $\{(\zeta^{\ssd{\la}}_0,\ldots,\zeta^{\ssd{\la}}_n)\}$ with respect to $\U=\{X_\la\hookrightarrow V_\la\}$. On each $X_\lambda$, consider a pre--log structure $\alpha_\lambda:{\bf N}^{n+1}\longrightarrow\O_{X_\lambda}$, defined by $\alpha_\lambda(e_i)=\zeta^{\ssd{\la}}_i$ for $0\leq i\leq n$. Then, the associate log structure of this pre--log structure is equivalent to the log structure ${\cal M}_\lambda$ of semistable type on the normal crossing variety $X_\lambda$. We write this log structure, according to \S 1, by \begin{equation}\label{loclog} {\cal M}_\lambda={\bf N}^{n+1}\oplus_{\alpha^{-1}_\lambda(\O^\times_{X_\lambda})} \O^\times_{X_\lambda} \stackrel{{\bar{\alpha}}_\lambda}{\longrightarrow}\O_{X_\lambda}, \end{equation} and ${\bar{\alpha}}_\lambda(e_i, u)=u\cdot\zeta^{\ssd{\la}}_i$, for $0\leq i\leq n$. Due to Lemma \ref{ltrns}, on each $X_{\lambda\mu}=X_\lambda\cap X_\mu\neq\emptyset$, there exists a non canonical isomorphism $$ \begin{array}{ccc} {\bf N}^{n+1} \oplus_{\alpha^{-1}_\lambda(\O^\times_{X_{\lambda\mu}})} \O^\times_{X_{\lambda\mu}} & \stackrel{\phi_{\lambda\mu}}{\longrightarrow} & {\bf N}^{n+1} \oplus_{\alpha^{-1}_\mu(\O^\times_{X_{\lambda\mu}})} \O^\times_{X_{\lambda\mu}} \\ \vphantom{\bigg|}\Big\downarrow & & \vphantom{\bigg|}\Big\downarrow \\ \O_{X_{\lambda\mu}} & = & \O_{X_{\lambda\mu}} \end{array} $$ defined by \begin{equation}\label{isom} \phi_{\lambda\mu}(e_i, u)=(e_{\sigma^{\ssd{\la\mu}}(i)}, u\cdotu^{\ssd{\la\mu}}_i), \end{equation} for $0\leq i\leq n$. \begin{lem}\label{patch} Any isomorphism $\phi_{\lambda\mu}$ which makes the above diagram commute is written in the form of (\ref{isom}). \end{lem} \noindent{\sc Proof.}\hspace{2mm} Changing indicies suitably, we may assume that \begin{itemize} \item for $0\leq i\leq d$, $\alpha_\lambda(e_i)$ and $\alpha_\mu(e_i)$ are not invertible on $X_{\lambda\mu}$, \item for $d< i\leq n$, $\alpha_\lambda(e_i)$ and $\alpha_\mu(e_i)$ are invertible $X_{\lambda\mu}$. \end{itemize} Set $\phi_{\lambda\mu}(e_i, 1)=(\sum_j a_i^j e_j, u^{\ssd{\la\mu}}_i)$, for $0\leq i\leq n$. \vspace{3mm} \noindent{\sc Case}\vspace{2mm} $0\leq i\leq d\:$: Since ${\bar{\alpha}}_\mu\circ\phi_{\lambda\mu}={\bar{\alpha}}_\lambda$, the matrix $(a_i^j)_{0\leq i,j\leq d}$ is a permutation matrix (Note that $A,B\in{\rm M}_d({\bf N})$ and $AB=1$ implies that $A$ and $B$ are permutation matrices). Then we may assume $a_i^j=\delta_i^j\quad(0\leq i,j\leq d)$. Hence, we can write $\phi_{\lambda\mu}(e_i, 1)=(e_i, u^{\ssd{\la\mu}}_i)\cdot(b_i, 1)$, where $\alpha_\mu(b_i)\in{\rm H}^0(X_{\lambda\mu}, \O^\times_X)$. Since we have the equality $$ (e_i, u^{\ssd{\la\mu}}_i\cdot\alpha_\mu(b_i))\cdot(b_i, \alpha_\mu(b_i)^{-1}) =(e_i, u^{\ssd{\la\mu}}_i)\cdot(b_i, 1), $$ we get $\phi_{\lambda\mu}(e_i, 1)=(e_i, u^{\ssd{\la\nu}}_i\cdot\alpha_\mu(b_i))$ in the quotient monoid $\mbox{\rm N}^{n+1}\oplus_{\alpha^{-1}_\mu(\O^\times_{X_{\lambda\mu}})} \O^\times_{X_{\lambda\mu}}$. \vspace{3mm} \noindent{\sc Case}\vspace{2mm} $d<i\leq n\:$: Since ${\bar{\alpha}}_\mu\circ\phi_{\lambda\mu}(e_i, 1)$ is invertible, we have $a_i^j=0$, for $d<i\leq n$ and $0\leq j\leq d$. This implies $\phi_{\lambda\mu}(e_i, 1)=(c_i, u^{\ssd{\la\mu}}_i)$, where $\alpha_\mu(c_i)\in{\rm H}^0(X_{\lambda\mu}, \O^\times_X)$. Since $$ (c_i, u^{\ssd{\la\mu}}_i)\cdot(e_i, \alpha_\mu(e_i)^{-1})= (e_i, \alpha_\mu(c_i)u^{\ssd{\la\mu}}_i\alpha_\mu(e_i)^{-1})\cdot (c_i, \alpha_\mu(c_i)^{-1}), $$ we get $\phi_{\lambda\mu}(e_i, 1)= (e_i, \alpha_\mu(c_i)\cdotu^{\ssd{\la\nu}}_i\cdot\alpha_\mu(e_i)^{-1})$ in the quotient monoid $\mbox{\rm N}^{n+1}\oplus_{\alpha^{-1}_\mu(\O^\times_{X_{\lambda\mu}})} \O^\times_{X_{\lambda\mu}}$. \vspace{3mm} Hence, for $0\leq i\leq n$, we have the equality $\phi_{\lambda\mu}(e_i, 1)= (e_i, \widetilde{u^{\ssd{\la\mu}}_i})$, for some invertible function $\widetilde{u^{\ssd{\la\mu}}_i}$ on $X_{\lambda\mu}$, in the quotient monoid $\mbox{\rm N}^{n+1}\oplus_{\alpha^{-1}_\mu(\O^\times_{X_{\lambda\mu}})} \O^\times_{X_{\lambda\mu}}$. Combining this with $\phi_{\lambda\mu}(0, u)=(0, u)$, we have the desired result. $\Box$ \vspace{3mm} \setcounter{ste}{0} Now, let us prove Theorem 11.2. Assume that $X$ is $d$--semistable. Take a coordinate covering $\U=\{X_\la\hookrightarrow V_\la\}$, a log system $\{(\zeta^{\ssd{\la}}_0,\ldots,\zeta^{\ssd{\la}}_n)\}$ and a transition system $\{(u^{\ssd{\la\mu}}_i, \sigma^{\ssd{\la\mu}})\}$ such that (\ref{cdss}) holds. Then, each $X_\lambda$ has a log structure by (\ref{loclog}). Moreover, there exists a system of isomorphisms $\{\phi_{\lambda\mu}\}$ defined by (\ref{isom}). Now we are going to show that $\{{\cal M}_\lambda\}$ glues to a log structure ${\cal M}$ on $X$ of the desired type. \begin{ste}{\rm Let us prove that $\phi_{\mu\nu}\circ\phi_{\lambda\mu}=\phi_{\lambda\nu}$ on each $X_{\lambda\mu\nu}=X_\lambda\cap X_\mu\cap X_\nu\neq\emptyset$, i.e., $\{\phi_{\lambda\mu}\}$ satisfies the 1-cocycle condition. Set \begin{eqnarray*} \phi_{\mu\nu}\circ\phi_{\lambda\mu}(e_i, 1) & = & (e_{\tau\circ\sigma(i)}, v_{\sigma(i)}u_i) \\ \phi_{\lambda\nu}(e_i, 1) & = & (e_{\rho(i)}, w_i), \end{eqnarray*} where we put $\sigma=\sigma^{\ssd{\la\mu}}$, $\tau=\sigma^{\ssd{\mu\nu}}$, $\rho=\sigma^{\ssd{\la\nu}}$, $u=u^{\ssd{\la\mu}}$, $v=u^{\ssd{\mu\nu}}$, and $w=u^{\ssd{\la\nu}}$. Changing indicies suitably, we may assume \begin{itemize} \item for $0\leq i\leq d$, $\zeta^{\ssd{\la}}_i\mid_{X_{\lambda\mu\nu}}$ is not invertible on $X_{\lambda\mu\nu}$, \item for $d<i\leq n$, $\zeta^{\ssd{\la}}_i\mid_{X_{\lambda\mu\nu}}$ is invertible on $X_{\lambda\mu\nu}$. \end{itemize} Since ${\bar{\alpha}}_\nu\circ\phi_{\mu\nu}\circ\phi_{\lambda\mu}= {\bar{\alpha}}_\lambda={\bar{\alpha}}_\nu\circ\phi_{\lambda\nu}$, we have \begin{equation}\label{str1} v_{\sigma(i)}u_i\zeta^{\ssd{\nu}}_{\tau\circ\sigma(i)}=\zeta^{\ssd{\la}}_i= w_i\zeta^{\ssd{\nu}}_{\rho(i)}\quad(0\leq i\leq n). \end{equation} \vspace{3mm} \noindent {\sc Case}\hspace{2mm} $d<i\leq n\:$: Since the equality (\ref{str1}) holds, both $\zeta^{\ssd{\nu}}_{\tau\circ\sigma(i)}$ and $\zeta^{\ssd{\nu}}_{\rho(i)}$ are invertible on $X_{\lambda\mu\nu}$ and we have $$ (e_{\tau\circ\sigma(i)}, v_{\sigma(i)}u_i)\cdot (e_{\rho(i)}, (\zeta^{\ssd{\nu}}_{\rho(i)})^{-1})= (e_{\rho(i)}, w_i)\cdot (e_{\tau\circ\sigma(i)}, (\zeta^{\ssd{\nu}}_{\tau\circ\sigma(i)})^{-1}). $$ Hence, we have $(e_{\tau\circ\sigma(i)}, v_{\sigma(i)}u_i)= (e_{\rho(i)}, w_i)$ in the quotient monoid ${\bf N}^{n+1} \oplus_{\alpha^{-1}_\nu(\O^\times_{X_{\lambda\mu\nu}})} \O^\times_{X_{\lambda\mu\nu}}$. \vspace{3mm} \noindent {\sc Case}\hspace{2mm} $0\leq i\leq d\:$: Since the components $\{\zeta^{\ssd{\nu}}_{\tau\circ\sigma(i)}=0\}$ and $\{\zeta^{\ssd{\nu}}_{\rho(i)}=0\}$ of $X_{\lambda\mu\nu}$ coincides due to (\ref{str1}), we have $\tau\circ\sigma(i)=\rho(i)$. Hence, we have $\zeta^{\ssd{\nu}}_{\rho(i)}=w_i^{-1}v_{\sigma(i)}u_i\zeta^{\ssd{\nu}}_{\rho(i)}$. This imples \begin{equation}\label{str2} w_i^{-1}v_{\sigma(i)}u_i=1+ a_iz^{\ssd{\nu}}_{\rho(0)}\cdotsz^{\ssd{\nu}}_{\rho(i-1)}\cdotz^{\ssd{\nu}}_{\rho(i+1)} \cdotsz^{\ssd{\nu}}_{\rho(d)}, \end{equation} for some $a_i\in\O_{X_{\lambda\mu\nu}}$. Since \begin{eqnarray*} \rho(\{d+1,\ldots,n\})&=&\tau\circ\sigma(\{d+1,\ldots,n\})\\ &=&\{j\mid(\zeta^{\ssd{\nu}}_j\mid_{X_{\lambda\mu\nu}})\in {\rm H}^0(X_{\lambda\mu\nu}, \O^\times_X)\}, \end{eqnarray*} and $\zeta^{\ssd{\nu}}_{\rho(j)}=w_j^{-1}v_{\sigma(j)}u_j \zeta^{\ssd{\nu}}_{\tau\circ\sigma(j)}$ for $d<j\leq n$, we have $$ \prod_{d<j\leq n}w_j^{-1}v_{\sigma(j)}u_j=1. $$ On the other hand, by our assumptions $u_1\cdots u_n=1$, etc., we have $$ \prod_{0\leq i\leq n}w_j^{-1}v_{\sigma(j)}u_j=1. $$ Hence, we get the equality $$ \prod_{0\leq i\leq d}w_j^{-1}v_{\sigma(j)}u_j=1. $$ By this and (\ref{str2}), we have $$ \sum_{0\leq i\leq d}a_iz^{\ssd{\nu}}_{\rho(0)}\cdots z^{\ssd{\nu}}_{\rho(i-1)}\cdotz^{\ssd{\nu}}_{\rho(i+1)}\cdots z^{\ssd{\nu}}_{\rho(d)}=0, $$ and consequently we get $a_i=0$, that is, $w_i^{-1}v_{\sigma(i)}u_i=1$. Hence, also in this case, we have $(e_{\tau\circ\sigma(i)}, v_{\sigma(i)}u_i)=(e_{\rho(i)}, w_i)$. Therefore, we have proved that $\phi_{\mu\nu}\circ\phi_{\lambda\mu}= \phi_{\lambda\nu}$ holds and $\{{\cal M}_\lambda\}$ glues to a log structure ${\cal M}$ on $X$. } \end{ste} \begin{ste}{\rm The system of morphisms $\{f_\lambda\}$, where $f_\lambda:(X_\lambda, {\cal M}_\lambda)\rightarrow (\mathop{\rm Spec}\nolimits k, {\bf N})$ is defined as in Definition \ref{canlog}, glues to the morphism $f$ if and only if the following diagram commutes: $$ \begin{array}{ccc} {\bf N}^{n+1}\oplus_{\alpha^{-1}_\lambda(\O^\times_{X_{\lambda\mu}})} \O^\times_{X_{\lambda\mu}} & \stackrel{\phi_{\lambda\mu}}{\longrightarrow} & {\bf N}^{n+1}\oplus_{\alpha^{-1}_\mu(\O^\times_{X_{\lambda\mu}})} \O^\times_{X_{\lambda\mu}} \\ \vphantom{\bigg|}\Big\uparrow & & \vphantom{\bigg|}\Big\uparrow \\ {\bf N}\oplus k^\times & = & {\bf N}\oplus k^\times, \end{array} $$ where ${\bf N}\oplus k^\times\rightarrow k$ is the associated log structure of ${\bf N}\rightarrow k$, and the homomorphism ${\bf N}\oplus k\rightarrow {\bf N}^{n+1}\oplus_{\alpha^{-1}_\lambda (\O^\times_{X_{\lambda\mu}})}\O^\times_{X_{\lambda\mu}}$ is defined by $(1, 1)\mapsto (e_0+\cdots+e_n, 1)$. The commutativity of this diagram is equivalent to the equality \begin{eqnarray*} (e_0+\cdots+e_n, 1) & = & \phi_{\lambda\mu}(e_0+\cdots+e_n, 1) \\ & = & (e_0+\cdots+e_n, u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_n). \end{eqnarray*} It is easy to see that this is equivalent to $u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_n=1$. Hence, $\{f_\lambda\}$ glues to the morphism $f$ due to Proposition \ref{cridss}. Thus, we have proved that ${\cal M}$ is the log structure on $X$ of the desired type. } \end{ste} \begin{ste}{\rm Let us prove the converse. Assume that $X$ has a log structure of semistavle type. Then, by Lemma \ref{patch}, Step 2 above and Proposition \ref{cridss}, it is easy to show that $X$ is $d$--semistable. } \end{ste} Thus, the proof of Theorem 11.2 is completed.

9406

alg-geom/9406006

en

https://arxiv.org/abs/alg-geom/9406006

alg-geom/9406006

Dr. Yakov Karpishpan

Yakov Karpishpan

Infinite-dimensional Lie algebras and the period map for curves

36 pages, LaTeX. This corrects a reference in the earlier version

We compute higher-order differentials of the period map for curves and show how they factor through the corresponding higher Kodaira-Spencer classes. Our approach is based on the infinitesimal equivariance of the period map, due to Arbarello and De Concini \cite{AD}.

alg-geom

\section{Notation and some preliminaries} \label{sect:notat} The results discussed in this paper owe their explicitness to a very concrete object, the field of Laurent power series $$ \H={\bf C}((z))={\bf C}[[z]][z^{-1}]\ . $$ Most of the time we will regard it merely as an infinite-dimensional vector space. It has several distinguished subspaces: $$ \H_+={\bf C}[[z]]\ \ \ {\rm and} \ \ \ \H_-= \mbox{the span of negative powers of}\ z\ . $$ Thus $\H=\H_+\oplus\H_-$ . Also, $$ \H_+'=z\H_+, \ \ \ {\rm and} \ \ \ \H'=\H'_+\oplus \H_-\ . $$ \begin{Def} $<f,g>={\rm Res}_{z=0}fdg$\ . \end{Def} This is a symplectic form on $\H$, non-degenerate on $\H'$. \begin{Def} $${\bf sp}(\H')=\{\alpha\in \mbox{\rm End}(\H')|<\alpha(x),y>+<\alpha(y),x>=0\ \ \mbox{\rm for all}\ x,y\in\H'\} \ .$$ \end{Def} \noindent{\bf Facts:} (a) $\H$ (and, hence, $\H'$) is a topological vector space with the $z$-adic topology. (b) ${\bf sp}(\H')$ is isomorphic to the completion $\widehat{S^2}(\H')$ of $S^2(\H')$ --- the symmetric square of $\H'$, where $S^2(\H')$ embeds in ${\bf sp}(\H')$ via $$ hk \mapsto \{x\mapsto <h,x>k+<k,x>h\}\ . $$ Finally, $\d=\H\frac{d}{dz}$ will denote the {\em Witt} Lie algebra of formal vector fields on a punctured disc. Its central extension is the more famous Virasoro algebra. We will also use $\d_+=\H_+\frac{d}{dz}$. The Lie algebras $\d$ and ${\bf sp}(\H')$ will appear in the setting of the following \begin{Def} \label{def:inf-hom} A Lie algebra $L$ {\em acts by vector fields} on a manifold $M$ if there is a homomorphism (or anti-homomorphism) of Lie algebras $$ L\longrightarrow \Gamma(M,\Theta_M)\ . $$ If the composed map to the tangent space of $M$ at a point $x$ $$ L\longrightarrow \Gamma(M,\Theta_M)\longrightarrow T_xM $$ is surjective for each $x\in M$, $M$ is called {\em infinitesimally homogeneous}, and one says that $L$ provides an {\em infinitesimal uniformization} for $M$. \end{Def} We will write $\Omega^1_X$ and $\omega_X$ interchangeably when $X$ is a curve. In turn, ``curve" will mean a complex algebraic curve. We will consider the classical topology and the analytic structure on $X$ only when dealing with the cohomology of $X$ with coefficients in ${\bf Z}$ or ${\bf C}$, and when using the exponential sequence in the proof of Lemma \ref{lemma:Lambda}, where we write $X^{an}$. We will also use the following notation: if ${\bf g}$ is a Lie algebra, then ${\cal U}{\bf g}$ is its universal enveloping algebra, and $\overline{\cal U}{\bf g}:={\cal U}{\bf g}/{\bf C}$. ${\cal U}^{(k)}{\bf g}$ (respectively, $\overline{\cal U}^{(k)}{\bf g}$) will denote the elements of order$\leq k$ in the natural filtration of ${\cal U}{\bf g}$ (respectively, $\overline{\cal U}{\bf g}$). Finally, we will use the fact that Lie algebras of endomorphisms of Hodge structures, with or without polarization, carry a Hodge structure of their own, always of weight 0. E.g. if $H=\oplus H^{p,q}$ is a HS and ${\bf g}=\mbox{\rm End}(H)$, then the Hodge decomposition of ${\bf g}$ is $$ {\bf g}=\oplus{\bf g}^{-k,k}\ , $$ where ${\bf g}^{-k,k}=\{f\in\mbox{\rm End}(H)\ |\ f(H^{p,q})\subset H^{p-k,q+k}\ \ \forall p,q\}$. \section{Infinitesimal uniformization of moduli of curves} Let $X$ be a complete curve of genus $g\geq 2$, $p$ --- a point on $X$, and $$ u:\widehat{\cal O}_{X,p}\stackrel{\cong}{\longrightarrow}\H_+ $$ --- a formal local coordinate at $p$; $u$ extends to an isomorphism of fields of fractions, also defining the obvious monomorphisms $$ \Theta_{X,p}\hookrightarrow \H_+\frac{d}{dz}\subset\d\ , \ \ \ \Omega^1_{X,p}\hookrightarrow \H_+dz , \ \ \ \Omega_X^1(*p)_p\hookrightarrow \H dz , \ \ \ {\rm etc.}, $$ all of which will also be denoted by $u$. As any point on a complete curve, $p$ is an ample divisor on $X$. Therefore, $X-p$ is an affine open set in $X$. Choose an affine neighborhood $V$ of $p$ in $X$. Then $\{(X-p), V\}$ is an affine covering of $X$, suitable for computing the \v{C}ech cohomology of $X$ with coefficients in a coherent sheaf. Thus we have an exact sequence \begin{eqnarray} \label{seq:basic} & & \\ & & 0\rightarrow\Gamma(X-p,\Theta_X)\oplus\Gamma(V,\Theta_X) \stackrel{\delta}{\rightarrow}\Gamma(V-p,\Theta_X) \stackrel{\pi}{\rightarrow}H^1(X,\Theta_X)\rightarrow 0\ . \nonumber \end{eqnarray} Exactness on the left is a consequence of $H^0(X,\Theta_X)=0$, which, in turn, follows from the assumption $g\geq 2$. The sheaf $\Theta_X$ is filtered by the subsheaves $\Theta_X(-ip)$ of vector fields vanishing at $p$ to an order $\geq i$ \ ($i\geq 0$). This induces a decreasing filtration $P^i$ on spaces of sections over $V$, and hence over $X-p$ and $V-p$. The \v{C}ech differential $\delta$ is strictly compatible with $P^{\bullet}$, and so is the projection $\pi$, once $H^1(X,\Theta_X)$ receives the induced filtration $P^{\bullet}$ from $\Gamma(V-p, \Theta_X)$. Therefore, the sequence (\ref{seq:basic}) remains exact when reduced modulo $P^i$. \begin{Lemma} The maps $$ u_i:\Gamma(V-p, \Theta_X)/P^i\longrightarrow\d/z^i\d_+ $$ induced via the identification $u$ are isomorphisms for each $i>0$. \end{Lemma} \ \\ \noindent {\bf Proof.\ \ } Suppose the points $q_1,\ldots,q_m$ constitute the complement of $V$ in $X$, and let $Q$ be the effective divisor $q_1+\ldots+q_m$. Since $Q$ is ample, for $N$ sufficiently large $H^1(X,\Theta_X(NQ-ip))$ will vanish. We may also assume $deg{\cal L}\ge q g-1$. With this choice of $N$, set ${\cal L}=\Theta_X(NQ-ip)$. Then the Riemann-Roch Theorem gives $$ H^0(X,{\cal L}(kp))=\deg {\cal L}+k+1-g $$ for each $k\geq 1$, which means that for each $k$ there exists a section of $\Theta_V(-ip)$, regular on $V-p$ and with a pole of order exactly $k$ at $p$. Thisimplies the surjectivity of $u_i$. Now, $$u:\Gamma(V-p,{\cal O}_X)\longrightarrow\H$$ is injective, since any regular function on $V-p$ is completely determined by its Laurent expansion. And $$u^{-1}(z^i\H_+)=\Gamma(V-p,{\cal O}_X(-ip))\ ,$$ implying that the $u_i$'s are injective too. \ $\displaystyle\Box$\\ \ \par \begin{Cor} $\llim\Gamma(V-p,\Theta_X)/P^i=\d$ . \end{Cor} \begin{Lemma} \label{lemma:surj} Passing to the inverse limit in the exact sequence obtained from (\ref{seq:basic}) by reduction $\bmod P^i$ produces an exact sequence \begin{equation} 0\rightarrow u(\Gamma(X,\Theta_X(*p)))\oplus\d_+\longrightarrow\d\longrightarrow H^1(X,\Theta_X)\rightarrow 0 \label{seq:final} \end{equation} \end{Lemma} \ \\ \noindent {\bf Proof.\ \ } First we note that $\Gamma(X-p,\Theta_X)=\Gamma(X,\Theta_X(*p))$ by \cite{Gro}. Also, $P^i\Gamma(X,\Theta_X(*p))=0$ for all $i>0$, since $X$ supports no non-zero global regular vector fields by virtue of the assumption $g\geq 2$. Hence $\Gamma(X,\Theta_X(*p))^{\wedge}=\Gamma(X,\Theta_X(*p))$. Second, $$ \llim H^1(X,\Theta_X)/P^i=H^1(X,\Theta_X)\ , $$ because for all sufficiently large $i$ $$ H^1(X,\Theta_X)/P^i=H^1(X,\Theta_X)\ ; $$ this is simply a consequence of $H^1(X,\Theta_X)$ being finite-dimensional. Finally, inverse limits preserve the exactness of (\ref{seq:basic}) $\bmod P^i$, because the directed system $$ \{A_i=(\Gamma(X-p,\Theta_X)\oplus\Gamma(V,\Theta_X))/P^i\} $$ satisfies the Mittag-Leffler condition (see \cite{L}, III, Prop. (9.3)): {\em For each $n$, the decreasing sequence of images of natural maps $\varphi_{mn}:A_m\rightarrow A_n\ \ (m\geq n)$ stabilizes.} \noindent This is trivially so since all $\varphi_{mn}$ are surjective in our situation. \ $\displaystyle\Box$\\ \ \par Assume now that $X$ moves in a flat family \begin{eqnarray} {\cal X} & \supset & X_t \nonumber\\ \pi\ \downarrow & & \downarrow \\ S & \ni & t\ \ ,\nonumber \label{family:flat} \end{eqnarray} with a section ${\bf p}:S\rightarrow{\cal X}$ and a local coordinate $$ {\bf u}:\widehat{\cal O}_{{\cal X},{\bf p}} \stackrel{\sim}{\longrightarrow}\Gamma(S,{\cal O}_S) \otimes\H_+ $$ on $\cal X$ along $\bf p$, so that the restriction of $\bf u$ to $X_t$ provides a local formal coordinate $u_t$ near $p_t$. For each $t\in S$ one has an analogue of (\ref{seq:final}). In particular, for each $t$ there is a surjection$$ \d\longrightarrow H^1(X_t,\Theta_{X_t}) \ ; $$ these glue together into a map \begin{equation} \d \longrightarrow\Gamma(S,R^1\pi_{*}\Theta_{{\cal X}/S}) \ . \label{map:rel} \end{equation} Assume further that $S$ is a disc centered at $0$ in ${\bf C}^{3g-3}$, and the family \begin{equation} \begin{array}{ccc} {\cal X} & \supset & X \\ \pi\ \downarrow & & \downarrow \\ S & \ni & 0 \end{array} \label{deform:X} \end{equation} is a miniversal deformation of the curve $X$. Then the Kodaira-Spencer map of the family, $$ \kappa:\Theta_S\longrightarrow R^1\pi_{*}\Theta_{{\cal X}/S}\ , $$ isan isomorphism. Composing its inverse with the map in (\ref{map:rel}) yields a linear map \begin{equation} \lambda:\d\longrightarrow\Gamma(S,\Theta_S)\ . \label{map:Lie} \end{equation} \begin{Lemma} \label{lemma:act} The map $\lambda$ in (\ref{map:Lie}) is an anti-homomorphism of Lie algebras. \end{Lemma} \ \\ \noindent {\bf Proof.\ \ } $\cal X$ admits an acyclic Stein covering ${\cal W}=\{W_0,W_1\}$ with $W_0\cong S\times V$ and $W_1={\cal X}-{\bf p}\cong S\times {X-{\bf p}}$. It follows from the proof of the previous lemma that the map $\lambda$ fits in the commutative diagram $$ \begin{array}{ccccc} \Gamma(S,{\cal O}_S)\otimes\d & \hookleftarrow & \d & \stackrel{\lambda}{\longrightarrow} & \Gamma(S,\Theta_S)\\ & & \\ j\uparrow\cong & && & \cong\downarrow\kappa \\ & && & \\ \Gamma(W_0\cap W_1,\Theta_{{\cal X}/S})^{\wedge} & & \stackrel{\textstyle c}{\,\longrightarrow\hspace{-12pt} & & \Gamma(S,R^1\pi_{*}\Theta_{{\cal X}/S}) \end{array}\ , $$ where $ ^{\wedge}$ indicates completion with respect to the filtration by the order of vanishing along $\bf p$, and $j$ is the isomorphism given by taking Laurent expansionsof relative vector fields on $W_0\cap W_1$ along $\bf p$ via $\bf u$. We begin by reviewing the definition of $\kappa$. The Kodaira-Spencer map $\kappa$ is the connecting morphism in the direct-image sequence of the short exact sequence of ${\cal O}_{\cal X}$ modules \begin{equation} 0\rightarrow\Theta_{{\cal X}/S}\longrightarrow\Theta_{\cal X} \longrightarrow\pi^*\Theta_S\rightarrow 0\ \ . \label{seq:KS} \end{equation} This contains an exact subsequence of $\pi^{-1}{\cal O}_S$-modules \begin{equation} 0\rightarrow\Theta_{{\cal X}/S}\longrightarrow \widetilde{\Theta}_{\cal X} \longrightarrow\pi^{-1}\Theta_S\rightarrow 0\ \ . \label{subseq:KS} \end{equation} whose direct-image sequence also has $\kappa$ as a connecting morphism (see \cite{BS} and also \cite{EV}). Furthermore, (\ref{subseq:KS}) is an exact sequence of sheaves of Lie algebras. The ${\bf C}$-linear brackets on $\widetilde{\Theta}_{\cal X}$ and $\pi^{-1}\Theta_S$ are inherited from $\Theta_{\cal X}$ and $\Theta_S$, respectively. The bracket on $\Theta_{{\cal X}/S}$ is even $\pi^{-1}{\cal O}_S$-linear. We are ready to prove the lemma. Take any $\zeta,\xi\in\d$, and let $Z=\lambda(\zeta),\ \Xi=\lambda(\xi)$. We wish to show that $[\zeta,\xi]=-[Z,\Xi]$, where the first bracket istaken in the Witt Lie algebra $\d$, and the second is in $\Gamma(S,\Theta_S)$. The elements $\zeta$ and $\xi$ of $\d$, which we identify with their pre-images under $j$, may be taken as Kodaira-Spencer representatives of $Z$ and $\Xi$. Lift $Z$ to some sections of $\widetilde{\Theta}_{\cal X}$, $\zeta_0\in$ on $W_0$ and $\zeta_1\in$ on $W_2$, and similarly for $\Xi$: $\xi_0\in \Gamma(W_0,\widetilde{\Theta}_{\cal X})$, and $\xi_1\in \Gamma(W_1,\widetilde{\Theta}_{\cal X})$. Then $\zeta_1-\zeta_0$ and $\xi_1-\xi_0$, with all terms restricted to $W_{01}=W_0\cap W_1$, also give KS representatives for $Z$ and $\Xi$. In particular, $$ \zeta=\zeta_1-\zeta_0+\delta\theta\ , $$ and $$ \xi=\xi_1-\xi_0+\delta\eta\ , $$ where $\theta$ and $\eta$ are some elements of $\check{C}^0({\cal W},\Theta_{{\cal X}/S})^{\wedge}$. Then $[Z,\Xi]$ admits as its KS representative the following expression, all terms of which are restricted to $W_{01}$: \begin{eqnarray} \label{brackets} \lefteqn{[\zeta_1,\xi_1]-[\zeta_0,\xi_0]=}\\ & =& [\zeta_1,\xi_1]- [\zeta_1-\zeta+\delta\theta,\xi_1-\xi+\delta\eta] \nonumber\\ & =& -[\zeta,\xi]+[\zeta_1,\xi]+[\zeta,\xi_1]+[\delta\theta,\xi_0]+ [\zeta_0,\delta\eta] \ .\nonumber \end{eqnarray} The Lie bracket of a section of $\widetilde{\Theta}_{\cal X}$ with that of $\Theta_{{\cal X}/S}$ is again a section of $\Theta_{{\cal X}/S}$, which implies that the last two terms in (\ref{brackets}) are in $\delta\check{C}^0({\cal W},\Theta_{{\cal X}/S})$. We may assume that $\bf u$ is induced by an isomorphism $u:\widehat{\cal O}_{X,p} \rightarrow\H_+$ viathe identification $W_{01}\cong S\times\{V-p\}$. The identification allows us to label some vector fields on $W_{01}$ as horizontal or vertical. By construction, $\zeta$ and $\xi$ are vertical and constant in the horizontal direction. The fields $\zeta_1$ and $\xi_1$, on the other hand, may be chosen to be horizontal and constant in the vertical direction. Then $[\zeta_1,\xi]=[\zeta,\xi_1]=0$. Collecting what is left of (\ref{brackets}), we conclude that $-[\zeta,\xi]$ is a Kodaira-Spencer representative for $[Z,\Xi]$, which proves the lemma. \ $\displaystyle\Box$\\ \ \par Recallingdefinition \ref{def:inf-hom}, we may summarize lemmas \ref{lemma:act} and \ref{lemma:surj} in the following theorem, due to\cite{BMS,Ko,BS}, cf. \cite{N,TUY}. \begin{Thm} \label{thm:S:infhom} For any curve $X$ of genus $g\geq 2$ the Witt Lie algebra $\d$ acts by vector fields on the base $S$ of a miniversal deformation of$X$, making $S$ infinitesimally homogeneous. \end{Thm} \refstepcounter{Thm Theaction above clearly depends on the choice of a point $p_t$ on each curve $X_t$, as well as on a formal parameter $u_t$ at $p_t$. For our purposes all these choices are equally good. More canonically, one may consider the moduli space of triples $(X,p,u)$, encompassing all possible choices of $p$ and $u$ on each $X$. The action of $\d$ extends to such ``dressed" moduli spaces $\hat{\cal M}_g$, making them also infinitesimally homogeneous. We will not need these constructions, since the questions we study are local on $\hat{\cal M}_g$. \newpage \section{Infinitesimal uniformization of period domains of weight one} By definition, a {\em Hodge structure of weight one} consists of a lattice $\Lambda\cong{\bf Z}^{2g}$ and a decomposition of its complexification $H=\Lambda\otimes{\bf C}$, $H=H^{1,0}\oplus H^{0,1}$, such that $H^{1,0}=\overline{H^{0,1}}$. The {\em Hodge filtration} $F^{\bullet}$ on $H$ is given by $F^0=H$, $F^1=H^{1,0}$, $F^0=0$. The HS $(\Lambda,H,F^{\bullet})$ is {\em principally polarized} if $\Lambda$ is equipped with a unimodular symplectic form $Q(\ ,\ )$ such that $(u,v)=Q(\bar{u},v)$ is a {\em positive-definite} Hermitian form on $H^{1,0}$ (and on $H^{0,1}$). The data $(\Lambda,H,F^{\bullet},Q)$ defines {\em a principally-polarized abelian variety} $A=H^{0,1}/i(\Lambda)$, where $i$ denotes the composition of the inclusion $\Lambda\rightarrow\Lambda\otimes{\bf C}=H$ with the projection $H=H^{1,0}\oplus H^{0,1}\rightarrow H^{0,1}$. As is well-known, the space ${\bf D}$ of all Hodge structures $(H,F^{\bullet})$ with a given lattice $\Lambda$ and polarization $Q$ (={\em the period domain}) can be identified with the Siegel upper half-space ${\bf H}_g$ of complex symmetric $g\times g$ matrices whose imaginary parts are positive-definite. The moduli space of principally-polarized abelian varieties of dimension $g$, ${\cal A}_g$, is a quotient of ${\bf H}_g$ by the action of $Sp(2g,{\bf Z})$. Note that ${\bf D}$ is a homogeneous space for the group$Sp(2g,{\bf R})$. We wish to present ${\bf D}$ locally as an infinitesimally homogeneous space for ${\bf sp}(\H')$. \begin{Def} An {\em extended Hodge structure} (of weight one) is a triple $(Z,K_0,\Lambda)$, where $Z$ is a maximal isotropic subspace of $\H'$ (with respect to the symplectic form $<\ ,\ >$), $K_0$ is a codimension $g$ subspace of $Z$, and $\Lambda$ is a rank $2g$ lattice in $K^{\perp}_0/K_0$, subject to several conditions. \end{Def} First of all, $Z\cap\H'_+=0$. This implies the splittings $\H'=Z\oplus\H'_+$ and $$ H:=K^{\perp}_0/K_0=H^{1,0}\oplus H^{0,1}\ , $$ where $H^{0,1}=Z/K_0$, and $H^{1,0}=K^{\perp}_0\cap \H'_+$. Let $Q$ be the bilinear form induced on $H$ by $\frac{1}{2\pi i}<\ ,\ >$ on $\H'$. The remaining conditions state that $H=\Lambda\otimes{\bf C}$, defining a real structure on $H$, that $H^{1,0}=\overline{H^{0,1}}$ with respect to this structure, and that $Q$ is unimodular on $\Lambda$. Thus $(\Lambda, H, H^{1,0},H^{0,1},Q)$ is a principally-polarized HS of weight one. Arbarello and De Concini introduced an extended version of the Siegel upper half-space, $\widehat{{\bf H}}_g$, on which $Sp(2g,{\bf Z})$ acts transitively, and the quotient manifold $\widehat{\cal A}_g$ parameterizes extended Hodge structures. The latter may also be regarded as``extended abelian varieties" in view of the following commutative diagram: \begin{equation} \begin{array}{ccc} \widehat{{\bf H}}_g & \longrightarrow & \widehat{\cal A}_g \\ && \\ \downarrow & \swarrow & \downarrow \\ && \\ {\bf H}_g & \longrightarrow & {\cal A}_g \end{array} \label{diag:extended} \end{equation} The horizontal maps are quotients with respect to the $Sp(2g,{\bf Z})$-action. All spaces are manifolds (the top two are infinite dimensional), except ${\cal A}_g$, which is a $V$-manifold. Note that all maps in the upper triangle are smooth. \begin{Prop}[\cite{AD}] $\widehat{\cal A}_g$ is an infinitesimally homogeneous space for ${\bf sp}(\H')$. \label{prop:A:infhom} \end{Prop} Obviously, this also makes ${\bf D}={\bf H}_g$ {\em locally} infinitesimally homogeneous for ${\bf sp}(\H')$. Let us work out the surjection $$ {\bf sp}(\H')\longrightarrow T_H{{\bf D}} $$ explicitly. At any point $H=H^{1,0}\oplus H^{0,1}$ of ${\bf D}$, $$ T_H{\bf D}=\mbox{\rm Hom}^{(s)}(H^{1,0}, H^{0,1})=S^2H^{0,1}\ . $$ Suppose $H$ comes from an extended HS $(Z,K_0,\Lambda)$. Then any $\alpha\in\mbox{\rm End}(\H')$ induces a map \begin{equation} H^{1,0}=K_0^{\perp}\cap \H'_+\longrightarrow\H' \ . \label{sect:of:proj} \end{equation} We use the formulas $\H'=Z\oplus\H'_+$ and $K_0\cap \H'_+=0$ to observe that $$ H^{0,1}\cong Gr_F^{0} H= Z/K_0\cong \H'/K_0+\H'_+\ . $$ Then (\ref{sect:of:proj}), composed with the natural projection $$ \H'\longrightarrow\H'/K_0+\H'_+\ , $$ yields an element $a\in\mbox{\rm Hom}(H^{1,0},H^{0,1})$. When $\alpha\in{\bf sp}(\H')$, $$<\alpha(x),y>=-<x,\alpha(y)>\ ,$$ i.e. $<x,\alpha(y)>=<y,\alpha(x)>$ for all $x,y\in\H'$. Hence $$ Q(x,a(y))=Q(y,a(x)) $$ for all $x,y\in H^{1,0}$, which means $a$ is {\em symmetric}: $$ a\in\mbox{\rm Hom}^{(s)}(H^{1,0},H^{0,1})=S^2H^{0,1}\ . $$ For reasons that will be clear later, we prefer $-a\in S^2H^{0,1}$. Thus $\alpha\mapsto -a$ indeed defines a map \begin{equation} \label{res:rho} \rho:{\bf sp}(\H')\longrightarrow T_H{\bf D}= \mbox{\rm Hom}^{(s)}(H^{1,0},H^{0,1})=S^2H^{0,1}\ . \end{equation} \refstepcounter{Thm \label{rho} For further use we record that the above construction presents the uniformizing map (\ref{res:rho}) as a restriction of a more broadly defined map \begin{equation} \mbox{\rm End}(\H')\longrightarrow \mbox{\rm Hom}(H^{1,0},H^{0,1})\ . \end{equation} Both maps will be denoted $\rho$. \refstepcounter{Thm In view of (\ref{diag:extended}), Proposition \ref{prop:A:infhom} implies that a sufficiently small open set $U$ in ${\bf D}$ is an infinitesimally homogeneous space under the action of ${\bf sp}(\H')$. However, the action is not unique --- it depends on the choice of a lift from $U$ to $\widehat{\cal A}_g$. \section{The extended period map} Let $X$ be a complete smooth curve, $p$ --- a point on $X$, and $u:\widehat{\cal O}_{X,p}\stackrel{\cong}{\longrightarrow}\H_+$ --- a formal local parameter at $p$. In this section we review Arbarello and De Concini's construction associating an extended HS $(Z,K_0,\Lambda)$ to the data $(X,p,u)$. When the triple $(X,p,u)$ varies in a flat family over some base $S$, this construction defines ``an extended period map" $$ \widehat{\Phi}: S\longrightarrow\widehat{\cal A}_g\ , $$ such that the usual period map $\Phi: S\longrightarrow {\bf D}$ naturally factors through $\widehat{\Phi}$. \begin{Def} $K_0:=u(\Gamma(X-p,{\cal O}_X))\cap\H'$\ . \end{Def} \noindent This is the same as putting $K_0=u(\Gamma(X-p,{\cal O}_X))/{\bf C}$. Note that $\Gamma(X-p,{\cal O}_X)=\Gamma(X,{\cal O}_X(*p))$ by a theorem of Grothendieck \cite{Gro}, and that $$ u:\Gamma(X-p,{\cal O}_X)\longrightarrow\H $$ is injective. There are no non-constant regular functions on $X$, and so $K_0\cap\H'_+=0$. \begin{Lemma} $H^1(X,{\cal O}_X)\cong \H/\H_++K_0$. \label{H1:O} \end{Lemma} \ \\ \noindent {\bf Proof.\ \ } This follows from the exact sequence $$ 0\rightarrow\Gamma(V,{\cal O}_X)\oplus\Gamma(X-p,{\cal O}_X)\longrightarrow \Gamma(V-p,{\cal O}_X)\longrightarrow H^1(X,{\cal O}_X)\rightarrow 0 $$ by completion with respect to the order-of-vanishing filtration $P^{\bullet}$ as in (\ref{lemma:surj}). \ $\displaystyle\Box$\\ \ \par Furthermore, $K_0$ is an isotropic subspace of $\H'$, i.e. $K_0$ is contained in $K_0^{\perp}$, the orthogonal complement of$K_0$ in $\H'$ with respect to the symplectic form $<\ ,\ >$. We can be more specific about $K_0^{\perp}$. \begin{Def} $\Omega:=\{f\in\H'\ |\ df\in u(\Gamma(X-p,\Omega^1_X))\}$. \end{Def} We have $\Omega\cong\Gamma(X-p,\Omega^1_X)=\Gamma(X,\Omega^1_X(*p))$. Now, Grothendieck's Algebraic De Rham Theorem \cite{Gro}, coupled wirh the injectivity of the map $d:\H'\rightarrow\H dz$ and of $u$, gives \begin{equation} \Omega/K_0\cong \frac{\Gamma(X,\Omega^1_X(*p))}{d\,\Gamma(X,{\cal O}_X(*p))} \cong H^1(X-p,{\bf C})=H^1(X,{\bf C})\ . \label{Groth:DR} \end{equation} \begin{Lemma} $K_0^{\perp}=\Omega$. \label{K:perp} \end{Lemma} \ \\ \noindent {\bf Proof.\ \ } If $f\in K_0$ and $g\in\Omega$, then $fdg$ is the Laurent expansion of a globally defined one-form on $X$ with poles only at $p$. Then ${\rm Res}_0 fdg=0$, i.e. $<K_0,\Omega>=0$, and so $\Omega\subseteq K_0^{\perp}$. The well-known duality theorem of Serr e \cite{S} implies that the residue pairing induces a duality between $H^0(X,\Omega_X^1)$ and $H^1(X,{\cal O}_X)$. The first of these groups is isomorphic to $\Omega\cap\H'_+$, the second --- to $$ \frac{\H}{\H_++K_0}=\frac{\Omega+\H}{\H_++K_0}\cong\frac{\Omega}{\Omega\cap(\H_++K_0)}= \frac{\Omega}{\Omega\cap\H_++K_0}\ . $$ This implies that the residue pairing on $\Omega/K_0$ is non-degenerate. Coupled with the earlier statements that $\Omega\subseteq K_0^{\perp}$ and $K_0\subset\Omega^{\perp}$, we have $\Omega^{\perp}=K_0$ and $(K_0^{\perp})^{\perp}\subseteq \Omega^{\perp} $, which means that $(K_0^{\perp})^{\perp}=K_0$. This, in turn, says that $<\ ,\ >$ is non-degenerate on $K_0^{\perp}/K_0$. However, the pairing is 0 on $K_0^{\perp}\cap\H_+$ (since it is 0 on all of $\H_+$), and on $$ \frac{K_0^{\perp}}{(K_0^{\perp}\cap\H_+)+K_0}\cong \frac{K_0^{\perp}+\H}{\H_++K_0}=\frac{\H}{\H_++K_0}\cong H^1(X,{\cal O}_X)\ . $$ Then $K_0^{\perp}\cap\H_+$ must be dual to $$ \frac{K_0^{\perp}}{(K_0^{\perp}\cap\H_+)+K_0}\cong H^1(X,{\cal O}_X) $$ under the residue pairing on $K_0^{\perp}/K_0$, which implies $K_0^{\perp}=\Omega$. \ $\displaystyle\Box$\\ \ \par \begin{Cor} $K_0^{\perp}/K_0\cong H^1(X,{\bf C})$. \label{EHS:HS} \end{Cor} At this point we make the observation that the Laurent expansion via $u$ at $p$ can be made well-defined not only for regular functions on a punctured neighborhood of $p$, but also for sections of ${\cal O}_X/{\bf Z}$: \begin{Def} $K:=u(\Gamma(X-p,{\cal O}_X/{\bf Z}))\cap \H'$\ . \end{Def} Of course, by means of the exponential map, $\Gamma(X-p,{\cal O}_X/{\bf Z})$ may be regarded as a subspace of $\Gamma(X-p,{\cal O}^*_{X^{an}})$. In other words, $K$ consists of those $f\in\H'$ for which $e^f$ lies in $u(\Gamma(X-p,{\cal O}^*_{X^{an}}))$. Obviously, $K_0\subset K$. Since the exterior derivative $d$ of a constant function is 0, $d$ is well-defined on ${\cal O}_X/{\bf Z}$, and (\ref{K:perp}) implies that $K\subset K_0^{\perp}$. \Def $\Lambda:=K/K_0$\ . \begin{Lemma} \label{lemma:Lambda} The isomorphism (\ref{EHS:HS}): $K_0^{\perp}/K_0\stackrel{\simeq}{\longrightarrow}H^1(X,{\bf C})$ maps $\Lambda$ onto $H^1(X,{\bf Z})$; in particular, $K_0^{\perp}/K_0\cong\Lambda\otimes{\bf C}$. \end{Lemma} \ \\ \noindent {\bf Proof.\ \ } The starting point in identifying $H^1(X,{\bf Z})$ with $\Lambda$ is the exponential sequence (on $X^{an}$, of course) \begin{equation} \begin{array}{rcccccl} & & & & & {\cal O}^*_{X^{an}} & \\ & & & & & e\,\uparrow\,\cong & \\ 0\longrightarrow & {\bf Z} & \longrightarrow & {\cal O}_{X^{an}} & \longrightarrow& {\cal O}_{X^{an}}/{\bf Z} & \longrightarrow 1\ \end{array} \label{seq:exp} \end{equation} and its cohomology sequence \begin{equation}\textstyle \begin{array}{rcl} H^1(X^{an},{\cal O}^*_{X^{an}})\ & & {\bf Z}\\ e\,\uparrow\,\cong\ \ \ \ \ \ \ \ \ \ & & \ \| \\ H^1(X^{an},{\bf Z}) \hookrightarrow H^1(X^{an},{\cal O}_{X^{an}}) \rightarrow H^1(X^{an}, {\cal O}_{X^{an}}/{\bf Z}) & \makebox[0pt]{$\,\,\longrightarrow\hspace{-12pt$} & H^2(X^{an},{\bf Z}) \end{array} \label{seq:exp:coho} \end{equation} But we also have an algebraic partial analogue of (\ref{seq:exp}) on $X$: $$ 0\longrightarrow{\bf Z}\longrightarrow{\cal O}_X\longrightarrow {\cal O}_X/{\bf Z}\longrightarrow 1\ , $$ with the cohomology sequence $$ \begin{array}{ccccccc} 0 & & & & & & 0 \\ \| & & & & & & \| \\ H^1(X,{\bf Z}) & \longrightarrow & H^1(X,{\cal O}_X) & \longrightarrow & H^1(X,{\cal O}_X/{\bf Z}) & \longrightarrow & H^2(X,{\bf Z}) \end{array} $$ mapping functorially to (\ref{seq:exp:coho}): $$ \begin{array}{rcccl} 0\longrightarrow H^1(X^{an},{\bf Z}) \longrightarrow & H^1(X^{an},{\cal O}_{X^{an}}) & \longrightarrow& {\rm Pic}^0(X) & \longrightarrow 0\\ & \cong\ \uparrow & & \uparrow & \\ & H^1(X,{\cal O}_X) & \stackrel{\cong}{\longrightarrow} & H^1(X,{\cal O}_X/{\bf Z}) & \ . \end{array} $$ The commutativity of the square implies that the right verical arrow is surjective. We also have the commutative ladder with exact columns $$ \begin{array}{ccc} 0 & & \\ \uparrow & & \\ H^1(X,{\cal O}_X) & \stackrel{\cong}{\longrightarrow} & H^1(X,{\cal O}_X/{\bf Z})\\ \uparrow & & \uparrow \\ \Gamma(V-p,{\cal O}_X) & \longrightarrow & \Gamma(V-p,{\cal O}_X/{\bf Z})\\ \uparrow & & \uparrow \\ \Gamma(V,{\cal O}_X)\oplus\Gamma(X-p,{\cal O}_X) & \longrightarrow & \Gamma(V,{\cal O}_X/{\bf Z})\oplus\Gamma(X-p,{\cal O}_X/{\bf Z}) \\ \uparrow & & \uparrow \\ 0 & & 0 \end{array} $$ Again we note that the upper right vertical arrow must be surjective. Splicing the two diagrams, and completing with respect to the order-of-vanishing filtration $P^{\bullet}$ as in (\ref{lemma:surj}), we get \begin{equation} \begin{array}{rcccl} & 0 & & 0 & \\ & \uparrow & & \uparrow & \\ 0\longrightarrow H^1(X^{an},{\bf Z}) \longrightarrow & H^1(X^{an},{\cal O}_{X^{an}}) & \longrightarrow& {\rm Pic}^0(X) & \longrightarrow 0\\ & \uparrow & & \uparrow & \\ & \H & = & \H & \\ & \uparrow & & \uparrow & \\ & \H_++K_0 & \longrightarrow & \H_++K & \\ & \uparrow & & \uparrow & \\ & 0 & & 0 & \end{array} \label{ladder:ABC} \end{equation} The lemma now follows by simple homological algebra. Consider the vertical ladder in the above diagram as a short exact sequence of three complexes $$ 0\longrightarrow A^{\bullet}\longrightarrow B^{\bullet} \longrightarrow C^{\bullet}\longrightarrow 0\ . $$ Then $H^0(C^{\bullet})=H^1(X^{an},{\bf Z})$, $H^1(A^{\bullet})=K/K_0$, and the connecting map in the corresponding cohomology sequence is precisely the sought-after isomorphism \begin{equation} H^1(X^{an},{\bf Z})\stackrel{\cong}{\longrightarrow}\Lambda=K/K_0\ . \label{sought-after} \end{equation} It remains to show that this isomorphism is induced by theone in (\ref{EHS:HS}). The map (\ref{sought-after}) factors through the monomorphism $$ H^1(X^{an},{\bf Z})\longrightarrow H^1(X^{an},{\cal O}_{X^{an},})\ , $$ which, in turn, factors through $H^1(X^{an},{\bf C})$. And the vertical sequences in (\ref{ladder:ABC}) may be amended as in the proof of Lemma \ref{K:perp}. Then we arrive at the following variant of (\ref{ladder:ABC}): $$ \begin{array}{rccl}\textstyle & H^1(X^{an},{\bf C}) & 0 & \ \ \ \ \ 0 \\ & \nearrow\ \ \uparrow\ \ \searrow & \uparrow &\ \ \ \ \ \uparrow \\ H^1(X^{an},{\bf Z}) & \longrightarrow & H^1(X^{an},{\cal O}_{X^{an}}) & \rightarrow {\rm Pic}^0(X)\rightarrow 0\\ & | & \uparrow & \ \ \ \ \ \uparrow \\ & \ \ \ K_0^{\perp}\ = & K_0^{\perp} & = \ K_0^{\perp} \\ & \uparrow & \uparrow & \ \ \ \ \ \uparrow \\ & \ \ \ K_0\ \rightarrow & (K_0^{\perp} \cap \H_+)+K_0 & \rightarrow (K_0^{\perp} \cap \H_+)+K \rightarrow\Lambda\\ & \uparrow & \uparrow & \ \ \ \ \ \uparrow \\ & 0 & 0 & \ \ \ \ \ 0 \end{array} $$ With this diagram it is easy to trace the map $H^1(X^{an},{\bf Z}) \longrightarrow \Lambda$ and see that it fits in the commutative square $$ \begin{array}{ccc} H^1(X^{an},{\bf C}) & \longrightarrow & K_0^{\perp}/K_0\\ \uparrow & & \uparrow\\ H^1(X^{an},{\bf Z}) & \longrightarrow & \Lambda \end{array} $$ with natural inclusions as the vertical arrows. \ $\displaystyle\Box$\\ \ \par \begin{Prop} The isomorphism in (\ref{EHS:HS}) is {\em symplectic}, identifying ${\frac{1}{2\pi i}<\ ,\ >}$ on $\Lambda$ with the intersection form $Q(\ ,\ )$ on $H^1(X,{\bf Z})$. \end{Prop} \ \\ \noindent {\bf Proof.\ \ } The above proposition is established in \cite{AD}, following \cite{SW}, by reasoning similar to that in the proof of the Riemann reciprocity laws. Alternatively, we can identify the residue pairing with the cup product $$ H^0(X,\Omega_X)\otimes H^1(X,{\cal O}_X)\longrightarrow H^1(X,\Omega_X^1)\cong H^2(X,{\bf C})\cong {\bf C}\ , $$ as Serre suggests in \cite{S}, and then relate the cup product to the intersection pairing. \ $\displaystyle\Box$\\ \ \par We now complete the identifications above to include the Hodge structure. First, $U:=K_0^{\perp}\cap\H'_+$ is easily seen to be mapped onto $$ H^0(X,\Omega_X^1)=H^{1,0}(X)=F^1H^1(X,{\bf C}) $$ by the isomorphism (\ref{EHS:HS}). Let $\overline{U}$ be the complex conjugate of $U$ with respect to the real structure which $\Lambda$ defines on $K_0^{\perp}/K_0=\Lambda\otimes{\bf C}$. Then (\ref{EHS:HS}) identifies $\overline{U}$ with $H^{0,1}(X)$. Finally, let $Z\subset K_0^{\perp}$ to be the pre-image of $\overline{U}$ with respect to the projection $$ K_0^{\perp}\,\longrightarrow\hspace{-12pt K_0^{\perp}/K_0\ . $$ It is easy to see that $Z\cap\H'_+=0$, $\H'=Z\oplus\H'_+$, and that $Z$ is a maximal isotropic subspace of $\H'$. To summarize, we have constructed an extended HS $(Z,K_0,\Lambda)$ out of the data $(X,p,u)$. \section{Infinitesimal equivariance of the period map} We will work with a miniversal deformation $\pi:{\cal X}\rightarrow S$ of a complete smooth curve $X$ of genus $g\geq 2$, as in (\ref{deform:X}), with a sufficiently small contractible open Stein manifold $S$ as its base. It was shown in Theorem \ref{thm:S:infhom} that $S$ is an infinitesimally homogeneous space for $\d$. Consider the usual and the extended period maps on $S$: \begin{eqnarray} S & \stackrel{\widehat{\Phi}}{\longrightarrow} & \widehat{\cal A}_g\nonumber\\ & &\nonumber\\ \Phi\downarrow & j\nearrow & \downsurj \\ & &\nonumber\\ U & \subset & {\bf D}={\bf H}_g\ .\nonumber \label{maps:period} \end{eqnarray} Let $U$ be a neighborhood of $\Phi(0)$ in ${\bf D}$ containing the image of $S$; we assume that $U$ is small enough to admit lifts to $\widehat{\cal A}_g$. Choose the lift $j$ making the diagram commutative (i.e. $j\circ\Phi=\widehat{\Phi}$ on $S$). This makes $U$ an infinitesimally homogeneous space for ${\bf sp}(\H')$. \Def $\varphi: \d \longrightarrow {\bf sp}(\H')$ is the Lie-algebra homomorphism given by $$ f\frac{d}{dz}\longmapsto\{g\mapsto fg'\ \ \ \forall g\in \H'\}\ . $$ Using the identification ${\bf sp}(\H')\cong\widehat{S^2}(\H')$ (see Section \ref{sect:notat}), the map $\varphi$ may also be written as \begin{eqnarray*} \varphi:\d & \longrightarrow & \widehat{S^2}(\H')\\ z^{k+1}\frac{d}{dz} & \longmapsto & \frac{1}{2}\sum_{j\in{\bf Z}-\{0\}} z^{-j}z^{j+k}\ . \end{eqnarray*} We note that $\varphi$ is an irreducible representation of the Witt algebra on $\H'$, described in \cite{KR}, (1.2), where it is denoted $V'_{0,0}$. The following is an adaptation of a theorem of Arbarello and De Concini \cite{AD}. \begin{Thm} \label{thm:equi} The period map $\Phi: S\longrightarrow U\subset{\bf D}$ is infinitesimally equivariant, i.e. there exists a commutative diagram \begin{equation} \label{phi:induce:dPhi} \begin{array}{ccc} \d & \stackrel{\varphi}{\longrightarrow} & {\bf sp}(\H') \\ \downarrow & & \downarrow \\ \Gamma(S,\Theta_S) & \stackrel{d\Phi}{\longrightarrow} & \Gamma(U,\Theta_{{\bf D}}) \ . \end{array} \end{equation} The vertical arrows are Lie algebra anti-homomorphisms, while the horizontal ones are Lie algebra homomorphisms. The vertical arrows induce surjections onto $T_tS$ (respectively, $T_{H}{\bf D}$) for any point $t\in S$ (respectively, $H\in U\subset{\bf D}$). \end{Thm} \refstepcounter{Thm The vertical arrows are not unique. \newpage \section{The second differential of the period map} \label{second:diff} We continue with a miniversal deformation(\ref{deform:X}) of $X$. Theorem \ref{thm:equi} allows one to calculate the various differentials of the period map. We begin by specializingdiagram (\ref{phi:induce:dPhi}) to $0\in S$: \begin{equation} \label{diag:equi:at_0} \begin{array}{ccc} \d & \stackrel{\varphi}{\longrightarrow} & {\bf sp}(\H') \\ \downsurj & & \downsurj \\ T_0S & \stackrel{d_0\Phi}{\longrightarrow} & T_{\Phi(0)}{\bf D} \end{array} \end{equation} A well-known theorem of Griffiths \cite{Gri} factors $d_0\Phi$ as \begin{equation} \label{diag:Griffiths} \begin{array}{ccl} T_0S & \stackrel{d_0\Phi}{\longrightarrow} & T_{\Phi(0)}{\bf D} \\ \kappa\downarrow\cong & & \ || \\ H^1(\Theta_X) & \stackrel{\nu}{\longrightarrow} & \mbox{\rm Hom}^{(s)}(H^0(\omega_X),H^1({\cal O}_X))\ , \end{array} \end{equation} where $\kappa$ is the Kodaira-Spencer isomorphism, and $\nu$ is the map defined by the cup-product pairing \begin{equation} \label{cup} H^1(\Theta_X)\otimes H^0(\omega_X)\stackrel{\smile}{\longrightarrow} H^1({\cal O}_X)\ , \end{equation} itself induced by the contraction pairing of sheaves $\Theta_X\otimes\omega_X\stackrel{\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,}{\rightarrow}{\cal O}_X$. Splicing (\ref{diag:equi:at_0}) and (\ref{diag:Griffiths}) yields \begin{equation} \label{diag:post-Griffiths} \begin{array}{ccl} \d & \stackrel{\varphi}{\longrightarrow} & {\bf sp}(\H')\\ \downsurj & & \ \downsurj{\scriptstyle \rho} \\ H^1(\Theta_X) & \stackrel{\nu}{\longrightarrow} & \mbox{\rm Hom}^{(s)}(H^0(\omega_X),H^1({\cal O}_X))\ , \end{array} \end{equation} which we want to work out explicitly. As before, let $V$ be an affine open set in $X$ containing $p$, so that $X-p$ and $V$ form an affine covering of $X$. Let $\xi$ be a vector field on $V-p$ with $u(\xi)=f(z)\frac{d}{dz}$ in $\d$. Let $\omega$ be a global holomorphic 1-form on $X$ with $u(\omega|_{V-p})=dg$ for some $g\in\H'$. Then the cup-product pairing (\ref{cup}) gives $$ [\xi]\smile[\omega]=[-\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,(\omega|_{V-p})]\in H^1({\cal O}_X)\ . $$ We observe that the minus sign is built into $\rho$ (see (\ref{rho})), and that $$ u(\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,(\omega|_{V-p}))=f\frac{d}{dz}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\, dg=fg'=\varphi(f\frac{d}{dz})g\ , $$ which is how (\ref{diag:post-Griffiths}) and, indeed, the theorem of Arbarello and De Concini (\ref{thm:equi}) is proved. We would like to work out an equally explicit realization of the second differential of $\Phi$ (higher-order cases are similar). Our starting point is again Theorem \ref{thm:equi}. We simply pass from Lie algebras to their (reduced) enveloping algebras to obtain a commutative diagram \begin{equation} \label{diag:U:2} \begin{array}{lcl} \overline{\cal U}^{(2)}\d & \stackrel{\varphi^{(2)}}{\longrightarrow} & \overline{\cal U}^{(2)}{\bf sp}(\H') \\ \downarrow & & \downarrow \\ \Gamma(S,\Theta^{(2)}_S) & \stackrel{d^2\Phi}{\longrightarrow} & \Gamma(U,\Theta^{(2)}_{{\bf D}})\ , \end{array} \end{equation} where $\Theta^{(2)}={\cal D}^{(2)}/{\cal O}$ stands for the second-order tangent sheaf, and $\overline{\cal U}^{(2)}$ is the notation introduced in Section \ref{sect:notat}. Again, to be precise, the maps emanating from the upper-left corner reverse the order of products, while the remaining maps are the second-degree parts of filtered ring homomorphisms. Restricting to $0\in S$, we obtain \begin{equation} \label{equi:at_0:two} \begin{array}{ccl} \overline{\cal U}^{(2)}\d & \stackrel{\varphi^{(2)}}{\longrightarrow} & \overline{\cal U}^{(2)}{\bf sp}(\H')\\ \lambda^{(2)}\downsurj\& & \ \downsurj\rho^{(2)} \\ T_0^{(2)}S & \stackrel{d^2_0\Phi}{\longrightarrow} & T_0^{(2)}{\bf D}\ . \end{array} \end{equation} \begin{Prop} \label{split} The second tangent space of the period domain ${\bf D}$ at the point corresponding to a HS $(H,F^{\bullet})$ admits a canonical splitting $$ T_F^{(2)}{\bf D}=T_F{\bf D}\oplus S^2T_F{\bf D}\ . $$ \end{Prop} \ \\ \noindent {\bf Proof.\ \ } Let ${\bf g}=\mbox{\rm End}(H)$ ($={\bf gl}(2g,{\bf C})$), and ${\bf s}={\bf sp}(H)$ (symplectic with respect to the polarization on $H$). Then ${\bf D}$ is infinitesimally homogeneous under the action of $\bf s$, and $$ T_F{\bf D}\cong{\bf s}^{-1,1}\cong\mbox{\rm Hom}^{(s)}(F^1,H/F^1)\ . $$ We also have a natural surjection $\overline{\cal U}^{(2)}{\bf s}\,\longrightarrow\hspace{-12pt T_F^{(2)}{\bf D}$. Its restriction to $\overline{\cal U}^{(2)}{\bf s}^{-1,1}$ is an isomorphism by reason of dimension. But ${\bf s}^{-1,1}$ is an abelian Lie algebra, i.e. $$ \overline{\cal U}^{(2)}{\bf s}^{-1,1}={\bf s}^{-1,1}\oplus S^2{\bf s}^{-1,1}\ .$$ \ $\displaystyle\Box$\\ \ \par In view of Proposition \ref{split}, $d^2_0\Phi:T_0^{(2)}S\rightarrow T_{\Phi(0)}^{(2)}{\bf D}$ breaks up into a direct sum of two components: {\em the symbol map} $$ \begin{array}{ccl} T_0^{(2)}S & \stackrel{\sigma}{\longrightarrow} & S^2{\bf s}^{-1,1}\\ (\Upsilon+\sum_iZ_i\Xi_i)|_0 & \longmapsto & d_0\Phi(Z|_0)\otimes d_0\Phi(\Xi|_0)\ \ \ (\mbox{\rm order does not matter here}), \end{array} $$ where $\Upsilon,Z_i,\Xi_i\in\Gamma(S,\Theta_S)$, and {\em the linear part} $$ \ell: T_0^{(2)}S\longrightarrow{\bf s}^{-1,1}\ . $$ It is the linear part that is really interesting. A typical second-order tangent vector to $S$ at 0, $(\Upsilon+\sum_iZ_i\Xi_i)|_0$, is sent by $\ell$ to $$ d_0\Phi(\Upsilon)+\sum_i\ell((Z_i\Xi_i)|_0)\ . $$ Thus, it suffices to understand $\ell((Z\Xi)|_0)$ for $Z,\Xi\in\Gamma(S,\Theta_S)$. By surjectivity of $\lambda^{(2)}$ in (\ref{equi:at_0:two}), we may assume that the vector fields $Z$ and $\Xi$ on $S$ are the images, respectively, of some $f_1\frac{d}{dz}$ and $f_2\frac{d}{dz}$ in $\d$, under the map $\d\rightarrow\Gamma(S,\Theta_S)$, whose restriction$\lambda$ is. Then (\ref{equi:at_0:two}) implies that \begin{equation} \label{vw} d_0^2\Phi((Z\Xi)|_0)=\rho^{(2)}\circ\varphi^{(2)}(f_1\frac{d}{dz} f_2\frac{d}{dz})=\rho^{(2)}(vw) \, \end{equation} where $v=\varphi(f_1\frac{d}{dz})$ and $w=\varphi(f_2\frac{d}{dz})$. Now, the map$\rho^{(2)}$ in (\ref{equi:at_0:two}) is not induced by $\rho:{\bf sp}(\H')\rightarrow{\bf s}^{-1,1}$, which was a restriction of the map, also denoted $\rho$ in (\ref{rho}), $$ \mbox{\rm End}(\H')\longrightarrow{\bf g}^{-1,1}\ . $$ In fact, the maps $\rho$ are not even Lie algebra morphisms. Nevertheless, there is a way to reduce$\rho^{(2)}$ to $\rho$. This will require a more detailed understanding of the infinitesimal action of ${\bf sp}(\H')$; in fact, we need to work out how the group $Sp(\H')\subset\mbox{\rm Aut}(\H')$ acts on a neighborhood of a point in $\widehat{\cal A}_g$. Since any element in the group $\mbox{\rm Aut}(\H')$ may be written as $I+\alpha$, where $\alpha\in\mbox{\rm End}(\H')$, we have the following map from $\mbox{\rm Aut}(\H')$ to $\mbox{\rm Aut}(H)$: $$ A\longmapsto I-\rho(\alpha), \ {\rm where}\ I+\alpha=A^{-1}\ . $$ This map will be denoted $R$. So \begin{equation} R(A)=I-\rho(A^{-1}-I)\ . \label{R} \end{equation} \noindent {\bf Caution:} $R$ is not a group homomorphism. Let $(H,F^{\bullet}_t)$ be the Hodge structure corresponding to a point in $U$ near $\Phi(0)$. The HS $(H,F^{\bullet}_t)$ comes from an extended HS $(Z_t,K_{0,t},\Lambda_t)$. The assumption that $U$ is small and infinitesimally homogeneous under ${\bf sp}(\H')$ implies that there exists $A_t\in Sp(\H')$ such that $K_{0,t}$ and $Z_t$ are images under $A_t$ of $K_0$ and $Z$, respectively (we refer to the components of the extended HS corresponding to $\Phi(0)$). \begin{Lemma} \label{how:Sp:acts} In this situation $F_t^1=R(A_t)F^1$. \end{Lemma} \ \\ \noindent {\bf Proof.\ \ } We wish to compare the Hodge structures $$ (H=K_0^{\perp}/K_0,F^1=K_0^{\perp}\cap\H'_+) $$ and $$ (H_t=K_{0,t}^{\perp}/K_{0,t},F_t^1=K_{0,t}^{\perp}\cap\H'_+)\ . $$ To do so, we identify $H_t$ with $H$ by $A_t^{-1}$. Then the comparison involves two subspaces of $H$: $F^1=K_0^{\perp}\cap\H'_+$ and $$ A_t^{-1}F_t^1=A_t^{-1}(K_{0,t}^{\perp})\cap A_t^{-1}(\H'_+)= K_0^{\perp}\cap A_t^{-1}(\H'_+)\ . $$ We regard $U$ as a subset of the Grassmannian $$ Grass(F^1,H)=\mbox{\rm Aut}(H)/\{A\,|\,A(F^1)\subseteq F^1\}\ . $$ Any element of $\mbox{\rm Aut}(H)$ may be written as $I+T$ for some $T\in{\bf g}$, and if$I+T\in\{A\,|\,A(F^1)\subseteq F^1\}$, then $T\in{\bf g}^{0,0}$. If some $I+T\in\mbox{\rm Aut}(H)$ moves $F^1$ to $A_t^{-1}F^1_t$, then so does $I+T^{-1,1}$, where the subscript refers to the $(-1,1)$-component of $T$ under the direct sum decomposition ${\bf g}={\bf g}^{-1,1}\oplus{\bf g}^{0,0}\oplus{\bf g}^{1,-1}$. Thus we only need to find the map $$ T^{-1,1}:H^{1,0}=\H'_+\cap K_0^{\perp}\longrightarrow \H'/\H'_++K_0\cong H^{0,1} $$ which measures deviation of $A_t^{-1}F^1_t$ from $F^1$. The above formulas for $F^1$ and $A_t^{-1}F^1_t$ show that $T^{-1,1}$ is induced by $$ A_t^{-1}:\H'_+\longrightarrow \H'/\H'_+\ . $$ But if $A_t^{-1}=I+\alpha$, then $$ \alpha:\H'_+\longrightarrow \H'/\H'_+ $$ induces the same $T^{-1,1}$. Recalling the definition of $\rho$ (\ref{rho}), this says that $T^{-1,1}=-\rho(\alpha)$. It remains to consult the definition of $R$ (\ref{R}) and to abuse notation by putting $F_t^1=A_t^{-1}F_t^1$. \ $\displaystyle\Box$\\ \ \par We are now able to establish the principal formula relating the two infinitesimal uniformizations of $U$ on the second-order level. \begin{Lemma} \label{two:unifs} Let $v,w\in\mbox{\rm End}(\H')$. Then $$ \rho^{(2)}(vw)=\rho(v)\rho(w)-\rho(w\circ v)\ , $$ where $w\circ v$ denotes the composition law in $\mbox{\rm End}(\H')$. \end{Lemma} \ \\ \noindent {\bf Proof.\ \ } Let $V=\rho(v)$, $W=\rho(w)$. $V$ and $W$ are vectors in $T_{\Phi(0)}{\bf D}\cong {\bf s}^{-1,1}$. For any $Z\in{\bf s}^{-1,1}$ we will write $\widetilde{Z}$ to denote the vector field on $U$ correspondingto $Z$ under the Lie algebra homomorphism $$ {\bf s}^{-1,1}\longrightarrow\Gamma(U,\Theta_{{\bf D}})\ . $$ In particular, $\widetilde{V}|_{\Phi(0)}=V$, $\widetilde{W}|_{\Phi(0)}=W$. Let $f$ be any smooth function on $U$. Then \begin{eqnarray*} \lefteqn{\rho^{(2)}(vw)f=} & & \\ & = & \frac{\partial^2}{\partial t\partial s}|_0 f\{R(\exp tv\circ\exp sw)\Phi(0)\}\\ & = & \frac{\partial^2}{\partial t\partial s}|_0 f\{[I-\rho(\exp(-sw)\circ\exp(-tv) -I)]\Phi(0)\}\\ & = & \frac{\partial^2}{\partial t\partial s}|_0 f\{[I-\rho(-sw-tv+tsw\circ v+\ldots)]\Phi(0)\}\\ & = & \frac{\partial^2}{\partial t\partial s}|_0 f\{(I+tV+sW-ts\rho(w\circ v)+\ldots)\Phi(0)\}\\ & = & \frac{d}{dt}|_0\left\{\frac{\partial}{\partial s}|_0 f\{[(I+tV+o(t))+s(W-t\rho(w\circ v)+o(t))+o(s)]\Phi(0)\} \right\}\\ & = & \frac{d}{dt}|_0\left\{ [(\widetilde{W}-t(\rho(w\circ v))^{\sim}+o(t))f] \{(I+tV+o(t))\Phi(0)\}\right\}\\ & = & \frac{d}{dt}|_0\biggl\{ [\widetilde{W}f]\{(I+tV+\ldots)\Phi(0)\}- t[(\rho(w\circ v))^{\sim}f]\{(I+tV+\ldots)\Phi(0)\}+\\ & & +o(t) \biggr\}\\ & = & V\widetilde{W}f-\rho(w\circ v)f\ . \end{eqnarray*}\ $\displaystyle\Box$\\ \ \par Observe that $\rho$ takes its values in ${\bf g}^{-1,1}$, which is an abelian Lie algebra. Thus the lemma gives a splitting of $$ \rho^{(2)}:\overline{\cal U}^{(2)}\mbox{\rm End}(\H')\longrightarrow\overline{\cal U}^{(2)}{\bf g}^{-1,1}={\bf g}^{-1,1} \oplus S^2{\bf g}^{-1,1}\ : $$ $V\widetilde{W}=\rho(v)\rho(w)$ ispurely quadratic (=the symbol part), and $-\rho(w\circ v)$ is the linear part. Going back to (\ref{vw}),this implies that $\ell((Z\Xi)|_0)=-\rho(w\circ v)$, which proves the following \begin{Thm} \label{Thm:main} If $Z,\Xi\in \Gamma(S,\Theta_S)$ lift to $f_1\frac{d}{dz}, f_2\frac{d}{dz}\in\d$, then the linear part of $d^2_0\Phi$, $$ \ell:T_0^{(2)}S\longrightarrow T_{\Phi(0)}{\bf D}={\bf s}^{-1,1}\ , $$ sends $(Z\Xi)|_0$ to the negative of the image under $\rho:\mbox{\rm End}(\H')\rightarrow{\bf g}^{-1,1}$ of the composition in $\mbox{\rm End}(\H')$ of $\varphi(f_1\frac{d}{dz})$ and $\varphi(f_2\frac{d}{dz})$, in reverse order: \begin{equation} g\longmapsto f_2f'_1g'+f_1f_2g''\ . \label{prescription} \end{equation} \end{Thm} \refstepcounter{Thm A composition (in $\mbox{\rm End}(\H')$) of two elements of ${\bf sp}(\H')$ need not be in ${\bf sp}(\H')$. In particular, it is not a priori obvious that the image of $\ell$ is in $$ \mbox{\rm Hom}^{(s)}(H^0(\omega_X),H^1({\cal O}_X))\cong{\bf s}^{-1,1}\ . $$ There is a better-known object which carries part of the information contained in the linear part $\ell$ of the period map's second differential. It is {\em the second fundamental form of the VHS} of \cite{CGGH}, the map $$ {\rm II}: T_0^{(2)}S/T_0S=S^2T_0S\longrightarrow T_{\Phi(0)}{\bf D}/im\,(d_0\Phi) $$ induced by $\ell$. \begin{Thm} \label{Thm:II} The prescription (\ref{prescription}) for computing $\ell$ gives a formula for {\rm II}, which coincides with that in \cite{K1}, $\S 6$: \begin{equation} \label{II} {\rm II}(Z\otimes\Xi)= \{\omega\mapsto\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_\zeta\omega\}\bmod im\,(d_0\Phi) \end{equation} for any$Z,\Xi\in T_0S$ with KS representatives $\zeta,\xi\in\Gamma(V-p,\Theta_X)$, and $\omega\in H^0(X,\omega_X)$. \end{Thm} \ \\ \noindent {\bf Proof.\ \ } Recall that a choice of a point $p$ on the curve $X$ and a local parameter $z$ near $p$ allows one to represent $\omega\in H^0(X,\omega_X)$ by some $g\in\H'$ with $dg=\omega$ near $p$. The vectors $Z$ and $\Xi$ are the images under $\rho$ of some $f_1\frac{d}{dz}$ and $f_2\frac{d}{dz}$ in $\d$, i.e. $f_1\frac{d}{dz}$ and $f_2\frac{d}{dz}$ are the Laurent expansions at $p$ of $\zeta$ and $\xi$, respectively. Working out (\ref{II}) in terms of $z$, using the formula for the Lie derivative $$ \pounds_\zeta\omega=d\zeta\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega+\zeta\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\, d\omega\ , $$ easily yields (\ref{prescription}) ($\bmod\ im\,(d_0\Phi)$),as was already done, in fact, in \cite{K1}, $\S 6$. \ $\displaystyle\Box$\\ \ \par \refstepcounter{Thm Thus the prescription for ${\rm II}$ given in \cite{K1} turns out to be well-defined for second-order differential operators on $S$ and not just their symbols, and the values given by that prescription are not merely equivalence classes modulo $im\,(d_0\Phi)$. \newpage \section{Relation with the second Kodaira-Spencer class} \label{rel:withKS2} As explained at the beginning of the previous section, the first differential of the period map is given bycup product with the (first) Kodaira-Spencer class $\kappa=\kappa_1$ of the deformation. In \cite{K2} we have shown that ${\rm II}$ depends only on the {\em second} Kodaira-Spencer class $\kappa_2$ (more precisely, on $\kappa_2\ \bmod\ im\,(\kappa_1)$) introduced recently in \cite{BG}, \cite{EV} and \cite{R1}. In this section we will explain, in the case of curves, how the full second differential $$ d^2_0\Phi: T_0^{(2)}S\longrightarrow T_{\Phi(0)}^{(2)}{\bf D} $$ factors through the second KS mapping $$ \kappa_2: T_0^{(2)}S\longrightarrow {\bf T}_X^{(2)}\ . $$ Let us recall first the construction of ${\bf T}_X^{(2)}$, the space of second-order deformations of $X$. Our reference is \cite{R1} or \cite{K2}. Let $X_2$ denote the symmetric product of the curve $X$ with itself;write $$ g: X\times X\longrightarrow X_2 $$ for the obvious projection map, and $i:X\hookrightarrow X_2$ for the inclusion of the diagonal. Then ${\bf T}_X^{(2)}={\bf H}^1(X_2,{\cal K}^{\bullet})$, where ${\cal K}^{\bullet}$ is the sheaf complex on $X_2$ $$ \begin{array}{ccc} {\scriptstyle -1} & & {\scriptstyle 0}\\ (g_*(\Theta_X^{\makebox[0pt][l]{$\scriptstyle\times$ 2}))^- & \stackrel{[\ ,\ ]}{\longrightarrow} & i_*\Theta_X\ . \end{array} $$ Here $\makebox[0pt][l]{$\times$$ stands for the exterior tensor product on $X\times X$, $(\ \ )^-$ denotes anti-invariants of the ${\bf Z}/2{\bf Z}$-action, and the differential is the restriction to the diagonal followed by the Lie bracket of vector fields. In practice it seems easier to do the following. Letting $C^{\bullet}$ denote the \v{C}ech cochain complex $\check{C}^{\bullet}({\cal U},\Theta_X)$ of $\Theta_X$ with respect to an affine covering $\cal U$ of $X$, one may compute ${\bf T}_X^{(2)}$ as the cohomology of the simple complex associated to the double complex $$ \begin{array}{cccc} {\scriptstyle 2} & (C^1\otimes C^1)^{(s)} & & \\ & 1\otimes\delta\uparrow -\delta\otimes 1 & & \\ {\scriptstyle 1} & (C^0\otimes C^1 + C^1\otimes C^0)^- & \stackrel{[\ ,\ ]}{\longrightarrow} & C^1 \\ & 1\otimes\delta\uparrow +\delta\otimes 1 & & \uparrow\delta \\ {\scriptstyle 0} & (C^0\otimes C^0)^- & \stackrel{[\ ,\ ]}{\longrightarrow} & C^0 \\ & {\scriptstyle -1} & & {\scriptstyle 0} \end{array} $$ The superscripts $^{(s)}$ and $^-$ denote the invariants and the anti-invariants, respectively, of the ${\bf Z}/2$-action. Working with ${\cal U}=\{X-p,V\}$ and using Laurent expansions at $p$, we may follow the proof of Lemma \ref{lemma:surj} and replace $C^1$ with $\d$ and $C^0$ with the completion of its image in $\d\oplus\d$. The resulting bicomplex still computes ${\bf T}_X^{(2)}$. In particular, ${\bf T}_X^{(2)}$ is a quotient of $\d\oplus(\d\otimes\d)^{(s)}$. \begin{Lemma} \label{ks2:rep} Assume the vector fields $Z$ and $\Xi$ on $S$ are the images of $\zeta,\xi\in\d$ under the infinitesimal uniformization map $\lambda: \d\longrightarrow\Gamma(S,\Theta_S)$. Then $$ \frac{1}{2}([\xi,\zeta]+\zeta\otimes\xi+\xi\otimes\zeta)\in \d\oplus(\d\otimes\d)^{(s)} $$ is a representative for $\kappa_2((Z\Xi)|_0)\in {\bf T}_X^{(2)}$. \end{Lemma} \ \\ \noindent {\bf Proof.\ \ } The Kodaira-Spencer maps are compatible with the symbol map ${\bf T}_X^{(2)}\longrightarrow S^2{\bf T}_X^1$ in the sense that there is a commutative diagram $$ \begin{array}{ccc} T_0^{(2)}S & \stackrel{\kappa_2}{\longrightarrow} & {\bf T}_X^{(2)}\\ \downarrow && \downarrow\\ S^2T_0S & \stackrel{\kappa_1^2}{\longrightarrow} & S^2{\bf T}_X^1\ . \end{array} $$ Thus it is natural to look for a representative of $\kappa_2((Z\Xi)|_0)$ of the form $$ \theta+\frac{1}{2}(\zeta\otimes\xi+\xi\otimes\zeta)\in \d\oplus(\d\otimes\d)^{(s)} $$ for some $\theta\in \d$. The construction of $\kappa_2$ as the connecting morphism in a certain long exact sequence (\cite{EV,R1,R2}), presented more explicitly in \cite{K2}, offers the following way to determine $\theta$. Working with a covering ${\cal W}=\{W_0,W_1\}$ of $\cal X$ as in the proof of Lemma \ref{lemma:act}, and using the subsheaf $\widetilde{\Theta}_{\cal X}$ of $\Theta_{\cal X}$ introduced in (\ref{subseq:KS}), let $\zeta_0,\zeta_1$ be lifts of $Z$ to $\Gamma(W_0, \widetilde{\Theta}_{\cal X})$, $\Gamma(W_1, \widetilde{\Theta}_{\cal X})$. Write $\widetilde{\zeta}$ to denote $\zeta_0+\zeta_1$ viewed as a cochain in $\check{C}^0({\cal W}, \widetilde{\Theta}_{\cal X})$. A slight modification of the proof of Prop. 2 in \cite{K2} shows that $\theta$ should be cohomologous (in $\check{C}^1({\cal W},\Theta_{{\cal X}/S})$) to $$ \frac{1}{2}([\widetilde{\zeta},\xi]+[\xi,\widetilde{\zeta}])= \frac{1}{2}([\zeta_0,\xi]+[\xi,\zeta_1])= \frac{1}{2}[\xi,\zeta_1-\zeta_0]\ . $$ But $\zeta_1-\zeta_0$ is cohomologous to $\zeta$. Hence we can take $\theta=\frac{1}{2}[\xi,\zeta]$. \ $\displaystyle\Box$\\ \ \par In (\ref{split}) we explained how $T_{\Phi(0)}^{(2)}{\bf D}$ splits into $T_{\Phi(0)}{\bf D}\oplus S^2T_{\Phi(0)}{\bf D}$, with the second differential of the period map breaking up accordingly: $$ d_0^2\Phi=\ell\oplus \sigma\ . $$ The symbol part factors through the square of the first KS class: $$ \begin{array}{ccrcl} T_0^{(2)}S && \stackrel{\sigma}{\longrightarrow} & & S^2T_{\Phi(0)}{\bf D} \\ & & & & \\ \downsurj & & {\scriptstyle (d_0\Phi)^2}\nearrow & & \ \ \uparrow \nu_1^2\\ & & & & \\ S^2T_0S & & \stackrel{\kappa_1^2}{\longrightarrow} & & S^2{\bf T}_X=S^2\mbox{\rm Hom}^{(s)}(H^0(\omega_X),H^1({\cal O}_X))\ . \end{array} $$ This diagram may be directly obtained from (\ref{diag:Griffiths}) and carries no additional information. Now to the linear part $\ell$ of $d_0^2\Phi$. ``Recall" the canonical bijection $b$ given by the composition of the obvious maps $$ b: \d\oplus(\d\otimes\d)^{(s)}\hookrightarrow \d\oplus(\d\otimes\d) \,\longrightarrow\hspace{-12pt \overline{\cal U}^{(2)}\d\ . $$ \begin{Lemma} \label{lemma:canon.bij} The canonical bijection $b$ fits in the commutative square $$ \begin{array}{ccc} \overline{\cal U}^{(2)}\d & \stackrel{b}{\longleftarrow} & \d\oplus(\d\otimes\d)^{(s)}\\ \downsurj & & \downsurj \\ T_0^{(2)}S & \stackrel{\kappa_2}{\longrightarrow} & {\bf T}_X^{(2)} \end{array} $$ with bijective horizontal arrows, and surjective vertical ones. \end{Lemma} \ \\ \noindent {\bf Proof.\ \ } It suffices to show that if $\zeta,\xi\in\d$ lift the vector fields $Z$ and $\Xi$ on $S$, then $b^{-1}(\zeta\xi)$ lifts $\kappa_2((\Xi Z)|_0)$ under the projection $$ \d\oplus(\d\otimes\d)^{(s)}\,\longrightarrow\hspace{-12pt {\bf T}_X^{(2)}\ . $$ In other words, we must verify that $b^{-1}(\zeta\xi)$ is a KS representative for $(\Xi Z)|_0\in T_0^{(2)}S$. From the definition of $b$ it easily follows that $$ b^{-1}(\zeta\xi)= \frac{1}{2}([\xi,\zeta]+\zeta\otimes\xi+\xi\otimes\zeta)\ . $$ This, together with Lemma \ref{ks2:rep}, implies our statement. \ $\displaystyle\Box$\\ \ \par \begin{Def} We define $\nu_2:{\bf T}_X^{(2)}\longrightarrow T_{\Phi(0)}{\bf D}= \mbox{\rm Hom}^{(s)}(H^0(\omega_X),H^1({\cal O}_X))$ as the composition $$ {\bf T}_X^{(2)}\stackrel{\kappa_2^{-1}}{\longrightarrow}T_0^{(2)}S \stackrel{\ell}{\longrightarrow}T_{\Phi(0)}{\bf D}\ . $$ \end{Def} Thus we have a commutative triangle \begin{equation} \begin{array}{ccccc} T_0^{(2)}S & & \stackrel{\ell}{\longrightarrow} & & T_{\Phi(0)}{\bf D}\\ & & & & \\ & \searrow\kappa_2 & & \nu_2\nearrow & \\ & & & & \\ & & {\bf T}_X^{(2)} & & \ . \end{array} \end{equation} \begin{Thm} \label{Interpret:coho} $\nu_2:{\bf T}_X^{(2)}={\bf H}^1({\cal K}^{\bullet})\longrightarrow\mbox{\rm Hom}(H^0(\omega_X),H^1({\cal O}_X))$ is induced by the pairing \begin{equation} {\bf H}^1({\cal K}^{\bullet})\otimes H^0(\omega_X)\longrightarrow H^1({\cal O}_X)\ , \label{pairing:coho} \end{equation} defined on the \v{C}ech cochain level by the coupling \begin{equation} \label{pairing:cochain} \begin{array}{cclcl} (\check{C}^1(\Theta_X))^{\otimes 2} \oplus \check{C}^1(\Theta_X) & \times\ & \check{C}^0(\omega_X) & \longrightarrow & \check{C}^1({\cal O}_X) \\ (\zeta\otimes\xi+\upsilon) & \times & \omega & \longmapsto & \xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_{\zeta}\omega-\upsilon\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega\ . \end{array} \end{equation} \end{Thm} \ \\ \noindent {\bf Proof.\ \ } It suffices to study the effect of $\nu_2=\ell\circ\kappa_2^{-1}$ on an element $x$ of ${\bf T}_X^{(2)}$ represented by $$ \frac{1}{2}(\zeta\otimes\xi+\xi\otimes\zeta)+\upsilon\in (\d\otimes\d)^{(s)}\oplus \d\ . $$ According to Lemma \ref{ks2:rep}, $$ \kappa_2^{-1}(x)=(Z\Xi+\Upsilon-\frac{1}{2}[Z,\Xi])|_0\ , $$ where $Z,\Xi$ and $\Upsilon$ are the images of $\zeta,\xi$ and $\upsilon$, respectively, under the uniformization map $\lambda: \d\longrightarrow\Gamma(S,\Theta_S)$. Note that $\lambda([\zeta,\xi])=-[Z,\Xi]$. As explained in the previous section, $\ell((Z\Xi+\Upsilon-\frac{1}{2}[Z,\Xi])|_0)$ is a map $H^0(\omega_X)\longrightarrow H^1({\cal O}_X)$ given by \begin{equation} \omega\longmapsto\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_{\zeta}\omega -(\upsilon-\frac{1}{2}[\zeta,\xi])\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega\ . \label{pre-Cartan} \end{equation} However, Cartan's identity gives $$ \pounds_{\zeta}(\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega)-\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_{\zeta}\omega= [\zeta,\xi]\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega\ , $$ and on a curve $\pounds_{\zeta}(\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega)= \zeta\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_{\xi}\omega$. Hence the right-hand side of (\ref{pre-Cartan}) equals $$ \frac{1}{2}(\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_{\zeta}\omega+ \zeta\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_{\xi}\omega)-\upsilon\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega\ . $$ \ $\displaystyle\Box$\\ \ \par \begin{Cor} \label{II:factors} The second fundamental form of the VHS, {\rm II}, factors through $S^2{\bf T}_X$. \end{Cor} \noindent This fact was already proved in complete generality (for a deformation of any compact K\"{a}hler manifold) in \cite{K2}, using Archimedean cohomology. We conclude with a diagram summarizing the relationships between some of the maps discussed in this section: \begin{equation}\begin{array}{ccccc} & & \ell & & \\ & & & & \\ T_0^{(2)}S & \stackrel{\kappa_2}{\longrightarrow} & {\bf T}^{(2)}_X & \stackrel{\nu_2}{\longrightarrow} & T_{\Phi(0)}{\bf D} \\ \downsurj && \downsurj & & \downsurj\\ S^2T_0S & \stackrel{\kappa_1^2}{\longrightarrow} & S^2{\bf T}_X & \stackrel{\nu_2/\,im\,\nu_1}{\longrightarrow} & T_{\Phi(0)}{\bf D}/\,im\,\nu_1 \\ & & & & \\ & & {\rm II} & & \end{array} \label{relationships} \end{equation} \section{The higher-order case} We have the following analogues of the results in sections \ref{second:diff} and \ref{rel:withKS2}. The proofs, which are notationally cumbersome transcriptions of the $n=2$ case, are omitted. \begin{Prop} The $n^{th}$ tangent space of the period domain ${\bf D}$ at a point corresponding to a HS $(H,F^{\bullet})$ admits a canonical splitting $$ T_F^{(n)}{\bf D}=T_F{\bf D}\oplus S^2T_F{\bf D} \oplus\ldots\oplus S^nT_F{\bf D}\ . $$ \end{Prop} \noindent The $n^{th}$ differential of the period map splits accordingly: $$ d_0^n\Phi=\ell_1^{(n)}+\ldots+\ell_n^{(n)}\ . $$ E.g. what we called $\ell$ and $\sigma$ earlier are $\ell_1^{(2)}$ and $\ell_2^{(2)}$, respectively. Thus it suffices to describe the $k^{th}$ component of $d_0^n\Phi$, $$ \ell_k^{(n)}: T_0^{(n)}{\bf D}\longrightarrow S^k T_{\Phi(0)}{\bf D}\ . $$ \begin{Thm} If $Z_1,\ldots,Z_n\in\Gamma(S,\Theta_S)$ lift to $\zeta_1,\ldots,\zeta_n\in\d$, then \newcounter{bean} \begin{list}{\rm\alph{bean})}{\usecounter{bean}} \item $\ell_1^{(n)}$ sends $(Z_1\ldots Z_n)|_0$ to $(-1)^{n-1}$ times the image under $\rho:\mbox{\rm End}(\H')\longrightarrow{\bf g}^{-1,1}$ of the composition in $\mbox{\rm End}(\H')$ of $\varphi(\zeta_1),\ldots,\varphi(\zeta_n)$ in reverse order; \item $\ell_k^{(n)}$ is the sum, over all partitions of $k$, of the symmetrized tensor products $$ \overline{\bigotimes}_{\sum_i p_i=k}\ell_1^{(p_i)}\ ; $$ \item $d_0^{n}\Phi$, as well as each $\ell_k^{(n)}$, factors through ${\bf T}_X^{(n)}$. \end{list} \label{thm:main:higher} \end{Thm} We may add to (a) that in terms of the covering $\{V,X-p\}$ of $X$ as above, $\ell_1^{(n)}((Z_1\ldots Z_n)|_0)$ can be also described as follows: it is a map $$ H^0(X,\omega_X)\longrightarrow H^1(X,{\cal O}_X) $$ sending the class represented by a form $\omega$ on $V$ to the class represented by the function $$ (-1)^n\zeta_n\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_{\zeta_{n-1}}\ldots \pounds_{\zeta_1} \omega $$ on $V-p$. Here we assume that the lifts $\zeta_i\in \d$ of $Z_i\in\Gamma(S,\Theta_S)$ converge and define regular vector fields on $V-p$. \newpage

9406

alg-geom/9406005

en

https://arxiv.org/abs/alg-geom/9406005

alg-geom/9406005

Charles Walter

Charles H. Walter

Pfaffian Subschemes

26 pages, AMS-LaTeX

A subscheme $X\subset \Bbb P^{n+3}$ of codimension $3$ is {\em Pfaffian} if it is the degeneracy locus of a skew-symmetric map $f:\cal{E}\spcheck(-t) @>>> \cal{E}$ with $\cal{E}$ a locally free sheaf of odd rank on $\Bbb P^{n+3}$. It is shown that a codimension $3$ subscheme $X\subset\Bbb P^{n+3}$ is Pfaffian if and only if it is locally Gorenstein, subcanonical (i.e.\ $\omega_X\cong\cal O_X(l)$ for some integer $l$), and the following parity condition holds: if $n\equiv 0\pmod{4}$ and $l$ is even, then $\chi (\cal O_X (l/2))$ is also even. The paper includes a modern version of the Horrocks correspondence, stated in the language of derived categories. A local analogue of the main theorem is also proved.

alg-geom

\subsection{Outline of the Paper} In the first section we review the proof of the local version of Theorem \ref{main} given by Buchsbaum and Eisenbud (\cite{BE} Theorem 2.1). We show that their proof will work for us if we can replace their minimal projective resolution by a locally free resolution of $\cal O_X$ which satisfies two properties (Proposition \ref{conditions}). The rest of the paper is devoted to finding a locally free resolution of $\cal O_X$ which satisfies these properties. Our main tool for constructing this locally free resolution is the Horrocks correspondence of \cite{H}. In the second section of the paper, we give a modern description of this correspondence using derived categories. This point of view is not identical to Horrocks', so we have felt it prudent to include a full proof of Horrocks' principal result (Theorem \ref{Horrocks}) from this point of view. However, the derived categories viewpoint is useful because it permits us to further develop Horrocks' ideas so as to obtain a method for transfering a portion of the cohomology of the coherent sheaf $\cal O_X$ to a locally free sheaf in a controlled way (Proposition \ref{functorial}). This is critical for our construction. In the third section we apply the Horrocks correspondence to construct a particular locally free resolution of the form (\ref{resol}). The basic idea is to cut in half the cohomology of the subscheme $X$ by using truncations of $\bold R\Gamma_*(\cal I_X)$. Our results on the Horrocks correspondence then permit us to find a vector bundle $\cal F_1$ whose intermediate cohomology is one of the halves of the cohomology of $\cal O_X$. Moreover, there is a natural morphism from this $\cal F_1$ to $\cal I_X$. This more or less gives the right half of the resolution, and the left half comes from the conventional methods of the Serre correspondence. We then show that if the cohomology of $\cal O_X$ was cut in half properly (viz.\ if the subcomplex carries an ``isotropic'' half of the cohomology), then the resolution is self-dual in a very strong way: i.e.\ any chain map from the resolution to its dual which extends the identity on $\cal{O}_X$ is necessarily an isomorphism of complexes. This is one of the properties required of the locally free resolution in order to make the Buchsbaum-Eisenbud proof work. In the fourth section we show that our locally free resolution of $\cal O_X$ can be endowed with a commutative differential graded algebra structure. This is a matter of calculating the obstruction to the lifting of a certain map. This is the second property required of the locally free resolution in order for the Buchsbaum-Eisenbud proof to work. This will complete the proof of Theorem \ref{main}. In the fifth section we consider Theorem \ref{main} in characteristic $2$. Essentially, certain lemmas in the fourth section fail in characteristic $2$ and must be replaced by analogues which are slightly different. In the sixth section we consider the results for regular local rings. Theorem \ref{main} concerning projective spaces has an obvious analogue (Theorem \ref{punc:spec}) for the punctured spectrum of a regular local ring. We show that this analogue is equivalent to Theorem \ref{RLR}. \begin{ack} The author would like to thank R.~M.~Mir\`o-Roig who brought the problem to his attention and with whom he had several discussions concerning it. The paper was written in the context of the Space Curves group of Europroj. \end{ack} \section{The Buchsbaum-Eisenbud Proof} In this section we review Buchsbaum and Eisenbud's proof of the local version of Theorem \ref{main}. In particular, we describe the two conditions that a locally free resolution of $\cal O_X$ must satisfy in order for their proof to show that a subcanonical subscheme $X\subset \Bbb{P}^{n+3}$ is Pfaffian (Proposition \ref{conditions}). \begin{theorem}[\cite{BE} Theorem 2.1] \label{local} Let $R$ be a regular local ring and $I$ an ideal of $R$ of height $3$ such that $R/I$ is a Gorenstein ring. Then $I$ has a minimal projective resolution of the form \[ 0 @>>> R @>{g\spcheck}>> F\spcheck @>f>> F @>g>> R @>>> R/I \] such that $F$ of odd rank $2p+1$, the map $f$ is skew-symmetric, and $g$ is composed of the Pfaffians of order $2p$ of $f$. \end{theorem} \begin{pf*}{Sketch of Buchsbaum and Eisenbud's proof of Theorem \ref{local}} One considers a minimal projective resolution of $R/I$. Since $R/I$ is Gorenstein, it is of the form \[ \bold{P}^*: \qquad\quad 0 @>>> R @>{d_3}>> F_2 @>{d_2}>> F_1 @>{d_1}>> R \] We now seek to find a way of identifying $F_2\cong F_1\spcheck$ so that $d_2$ becomes skew-symmetric. The first step is to endow $\bold{P}^*$ with the structure of a commutative associative differential graded algebra (\cite{BE} pp.~451--453). To define the multiplication, they define $S_2(\bold{P}^*) = (\bold{P}^* \otimes \bold{P}^*)/M^*$ where $M^*$ is the graded submodule of $\bold{P}^*\otimes\bold{P}^*$ generated by \[ \{a\otimes b-(-1)^{(\deg a)(\deg b)}b\otimes a \mid a,b \text{ homogeneous elements of }\bold{P}^*\}. \] Using universal properties of projective modules, they then construct a map of complexes $\Phi : S_2(\bold{P}^*) @>>> \bold{P}^*$ which extends the multiplication $R/I \otimes R/I$ and which is the identity on the subcomplex $R\otimes\bold{P}^* \subset S_2(\bold{P}^*)$. This makes $\bold{P}^*$ into a commutative differential graded algebra. The associativity of this algebra follows from the fact that it of length $3$, i.e. $P_n = 0$ for $n\geq 4$. The next step (p.~455) is to note that the multiplication $F_i \otimes F_{3-i} @>>> F_3 = R$ induces maps $s_i: F_i @>>> F_{3-i}\spcheck$ and a commutative diagram: \begin{equation} \label{dual} \begin{CD} \bold{P}^*: & \qquad & 0 @>>> R @>{d_3}>> F_2 @>{d_2}>> F_1 @>{d_1}>> R \\ &&&& @| @VV{s_2}V @VV{s_1}V @| \\ (\bold{P}^*)\spcheck : && 0 @>>> R @>{d_1\spcheck}>> F_1\spcheck @>{-d_2\spcheck}>> F_2\spcheck @>{d_3\spcheck}>> R \end{CD} \end{equation} This map of complexes is an extension of the Gorenstein duality isomorphism $R/I \cong \omega _{R/I} = \operatorname{Ext} ^3_R(R/I,R)$ to the minimal projective resolutions of $R/I$ and $\omega _{R/I}$. Since any map between minimal projective resolutions which extends an isomorphism in degree $0$ must be an isomorphism, it follows that the $s_i$ are all isomorphisms. We can therefore use the identification $s_2 : F_2\cong F_1\spcheck$. A very simple computation (p.~465) shows that with this identification, the commutativity and associativity of the differential graded algebra structure on $\bold{P}^*$ imply the skew-symmetry of $d_2$. In particular $d_2$ must have even rank (say $2p$), and $F_2$ must have odd rank $2p+1$. The identification of $d_1$ and $d_3$ with the vectors of Pfaffians of order $2p$ of $d_2$ is a lengthy but unproblematic computation (pp.~458--464). \end{pf*} Now let $X$ be a locally Gorenstein subcanonical subscheme of codimension $3$ in $\Bbb{P}^{n+3}$ with $\omega_X\cong\cal O_X (l)$. We wish to repeat the proof we have just sketched only with $\bold{P}^*$ replaced by a locally free resolution of $\cal O_X$: \begin{equation} \label{P} \cal{P}^*: \qquad\quad 0 @>>> \cal{L} @>{d_3}>> \cal{F}_2 @>{d_2}>> \cal{F}_1 @>{d_1}>> \cal O_{\Bbb{P}^{n+3}} \end{equation} where we will write $\cal{L}$ in place of $\omega_{\Bbb{P}^{n+3}}(-l)$ in order to simplify our diagrams. A careful reading yields only two places where the fact that $\bold{P}^*$ is a minimal projective resolution of $R/I$ was used in a way that does not immediately carry over to the locally free resolution $\cal{P}^*$. The first place was in the definition of the map $\Phi : S_2(\bold{P}^*) @>>> \bold{P}^*$ which made $\bold{P}^*$ into a commutative differential graded algebra. Therefore we will need to show directly the existence of a map of complexes \[ \begin{CD} S_2(\cal{P}^*): & \quad & \dotsb & \:\longrightarrow\: & \cal{L} \oplus [\cal{F}_2\otimes \cal{F}_1] & \:\stackrel{\sigma}{\longrightarrow}\: & \cal{F}_2 \oplus \Lambda^2 \cal{F}_1 & \:\longrightarrow\: & \cal{F}_1 & \:\longrightarrow\: & \cal O_{\Bbb{P}^{n+3}} \\ &&&&@VV{\phi _3}V @VV{\phi _2}V @| @| \\ \cal{P}^*: && 0 & \:\longrightarrow\: & \cal{L} & \:\stackrel{d_3}{\longrightarrow}\: & \cal{F}_2 & \:\stackrel{d_2}{\longrightarrow}\: & \cal{F}_1 & \:\stackrel{d_1}{\longrightarrow}\: & \cal O_{\Bbb{P}^{n+3}} \end{CD} \] The critical problem in defining the morphism of complexes is the following. Let $\psi : \Lambda^2 \cal{F}_1 @>>> \ker (d_1)$ be defined by $\psi (a\wedge b)=d_1(a)b-d_1(b)a$. We then must lift \begin{equation} \label{liftdiag} \begin{CD} &&&&&& \Lambda^2 \cal{F}_1 \\ &&&&&& @VV{\psi}V \\ 0 @>>> \cal{L} @>>> \cal{F}_2 @>>> \ker (d_1) @>>> 0 \end{CD} \end{equation} to a $\phi\in\operatorname{Hom} (\Lambda ^2\cal{F}_1,\cal{F}_2)$. Once that is done, the rest of the chain map follows. For one may define $\phi _2 = (1_{\cal{F}_2},\phi )$. Then \[ \phi _2\circ\sigma (\cal{L}\oplus [\cal{F}_2\otimes \cal{F}_1]) \subset \ker (d_2) = \cal{L}. \] So $\phi _2\circ\sigma$ factors through $\cal{L}$, allowing one to define $\phi _3$. Thus one can put a commutative associative differential graded algebra structure on $\cal{P}^*$ provided $\psi$ can be lifted. The obstruction to lifting $\psi$ lies in $\operatorname{Ext} ^1(\Lambda^2 \cal{F}_1,\cal{L}) \cong H^{n+2}(\Lambda^2 \cal{F}_1(l))^*$. Once we have the commutative differential graded algebra structure on $\cal{P}^*$, we may use it to define maps $s_i : \cal{F}_i @>>> \cal{F}_{3-i}\spcheck \otimes \cal{L}$ and a commutative diagram analogous to (\ref{dual}): \begin{equation} \label{dual2} \begin{CD} \cal{P}^*: & \qquad & 0 @>>> \cal{L} @>{d_3}>> \cal{F}_2 @>{d_2}>> \cal{F}_1 @>{d_1}>> \cal O_{\Bbb{P}^{n+3}} \\ &&&& @| @VV{s_2}V @VV{s_1}V @| \\ (\cal{P}^*)\spcheck : &&0 @>>> \cal{L} @>{d_1\spcheck}>> \cal{F}_1\spcheck\otimes\cal{L} @>{-d_2\spcheck}>> \cal{F}_2\spcheck\otimes\cal{L} @>{d_3\spcheck}>> \cal O_{\Bbb{P}^{n+3}} \end{CD} \end{equation} The vertical maps extend the isomorphism $\cal O_X\cong \omega_X(-l)= \cal{E}xt^3(\cal O_X,\cal{L})$. We now run into the second problem with locally free resolutions. Namely, a morphism of locally free resolutions which extends an isomorphism in degree $0$ is not automatically an isomorphism between the resolutions. But we reach the conclusion: \begin{proposition} \label{conditions} Suppose $X$ is a locally Gorenstein subcanonical subscheme of codimension $3$ in $\Bbb{P}^{n+3}$ with $\omega_X\cong\cal O_X (l)$. Then $X$ will be a Pfaffian scheme if $\cal O_X$ has a locally free resolution $\cal{P}^*$ as in \rom{(}\ref{P}\rom{)} satisfying the following two conditions: \rom(a\rom) Any morphism of complexes $\cal{P}^* @>>> (\cal{P}^*)\spcheck$ as in \rom{(}\ref{dual2}\rom{)} which extends the identity of $\cal O_X$ is an isomorphism of complexes, and \rom(b\rom) The morphism $\psi$ of \rom(\ref{liftdiag}\rom) lifts to a map $\phi\in\operatorname{Hom} (\Lambda ^2\cal{F}_1,\cal{F}_2)$. \end{proposition} We will now construct locally free resolutions $\cal{P}^*$ satisfying the conditions of the proposition. Our method involves the Horrocks correspondence. \section{The Horrocks Correspondence} In this section we give a modern description of the Horrocks correspondence of \cite{H} using derived categories. We include a full proof of the principal properties of the correspondence from this point of view (Theorem \ref{Horrocks}). Taking advantage of the greater flexibility of the derived category viewpoint, we develop a technique which allows us to transfer a prescribed portion of the cohomology of $\cal O_X$ to prescribed parts of a locally free resolution (Proposition \ref{functorial}). \subsection{Notation and Generalities} We first recall some generalities about complexes. If $\frak{A}$ is an abelian category, let $C(\frak{A})$ (resp.\ $K(\frak{A})$, $D(\frak{A})$) denote the category (resp.\ homotopy category, derived category) of complexes of objects of $\frak{A}$, and let $C^b(\frak{A})$, $C^-(\frak{A})$, $C^+(\frak{A})$, etc., denote the corresponding complexes of bounded (resp.\ bounded above, bounded below) complexes of objects of $\frak{A}$. When speaking of complexes, we will generally reserve the word ``isomorphism'' for isomorphisms in $C(\frak A)$. Isomorphisms in $K(\frak A)$ (resp.\ $D(\frak A)$) are referred to as homotopy equivalences (resp.\ quasi-isomorphisms). If $r$ is an integer, then any complex $C^*$ of objects of $\frak{A}$ has two {\em canonical truncations} at $r$ and a {\em naive truncation}: \begin{align*} \begin{CD} \tau _{\leq r}(C^*): & \qquad & \cdots & \:\rightarrow\: & C^{r-2} & \:\rightarrow\: & C^{r-1} & \:\rightarrow\: & \ker(\delta^r) & \:\rightarrow\: & 0 & \:\rightarrow\: & 0 & \:\rightarrow\: & \cdots , \\ \tau _{> r}(C^*): & \qquad & \cdots & \:\rightarrow\: & 0 & \:\rightarrow\: & 0 & \:\rightarrow\: & C^r/\ker(\delta^r) & \:\rightarrow\: & C^{r+1} & \:\rightarrow\: & C^{r+2} & \:\rightarrow\: & \cdots . \\ \sigma _{\geq r}(C^*): & \qquad & \cdots & \:\rightarrow\: & 0 & \:\rightarrow\: & 0 & \:\rightarrow\: & C^r & \:\rightarrow\: & C^{r+1} & \:\rightarrow\: & C^{r+2} & \:\rightarrow\: & \cdots . \end{CD} \end{align*} All the truncations are functorial in $C(\frak{A})$. The canonical truncations are functorial in $K(\frak{A})$ and $D(\frak{A})$ as well. We will often find it more convenient to write $\tau_{<r+1}$ instead of $\tau_{\leq r}$. Suppose now that $\frak{A}$ has enough projectives. Every bounded above complex $C^*$ of objects in $\frak{A}$ admits a {\em projective resolution}, i.e.\ a quasi-isomorphism $P^* @>>> C^*$ with $P^*$ a complex of projectives (\cite{Ha} Proposition I.4.6). The projective resolution of a complex is unique up to homotopy equivalence. If $C^*$ and $E^*$ are bounded above complexes of objects in $\frak{A}$, and if $P^* @>>> C^*$ is a projective resolution of $C^*$, then there is a natural isomorphism $\operatorname{Hom}_{D^-(\frak{A})}(C^*,E^*) \cong \operatorname{Hom}_{K^-(\frak{A})}(P^*,E^*)$. In particular if $\frak{P}$ denotes the full subcategory of projective objects of $\frak{A}$, then the natural functor $K^-(\frak{P}) @>>> D^-(\frak{A})$ is an equivalence of categories (\cite{Ha} Proposition I.4.7). This can be refined to the following statement: \begin{lemma} \label{category} Let $\frak{A}$ be an abelian category with enough projectives, and let $\frak{P}$ be the full subcategory of projective objects of $\frak{A}$. Suppose $A\subset D^-(\frak{A})$ and $P\subset K^-(\frak{P})$ are full subcategories such that $\operatorname{ob}(P)\subset \operatorname{ob}(A)$ and every object of $A$ has a projective resolution belonging to $P$. Then the natural functor $P @>>> A$ is an equivalence of categories. \end{lemma} Let $S=k[X_0,\dots,X_N]$ be the homogeneous coordinate ring of $\Bbb{P}^N$, and let $\frak{m}=(X_0,\dots,X_N)$ be its irrelevant ideal. Let $\operatorname{Mod}_{\mit{S},\operatorname{gr}}$ be the category of graded $S$-modules. Then $\operatorname{Mod}_{\mit{S},\operatorname{gr}}$ has enough projectives, namely the free modules. We will call a complex $P^*$ of projectives in $\operatorname{Mod}_{\mit{S},\operatorname{gr}}$ a {\em minimal} if all its objects $P^i$ are free of finite rank and its differential $\delta^*$ satisfies $\delta^i(P^i) \subset \frak m P^{i+1}$ for all $i$. If $C^*$ is a bounded above complex of objects in $\operatorname{Mod}_{\mit{S},\operatorname{gr}}$ whose cohomology modules $H^i(C^*)$ are all finitely generated, then $C^*$ has a {\em minimal projective resolution}, i.e.\ a projective resolution by a minimal complex of projectives. The next lemma, which is a well known consequence of Nakayama's lemma, says that minimal projective resolutions are unique up to isomorphism and not merely up to homotopy equivalence: \begin{lemma} \label{NAK} Let $\phi: P^* @>>> Q^*$ be a homotopy equivalence between {\em minimal} complexes of free graded $S$-modules of finite rank. Then $\phi$ is an isomorphism. \end{lemma} Let $\operatorname{Mod_{\cal O}}$ be the category of sheaves of $\cal{O}_{\Bbb{P}^N}$-modules. For $\cal{E}$ a sheaf of $\cal{O}_{\Bbb{P}^N}$-modules, let $\Gamma _*(\cal{E})=\bigoplus _{t\in\Bbb{Z}} \Gamma(\cal{E}(t))$. Then $\Gamma_*$ defines a left exact functor from $\operatorname{Mod_{\cal O}}$ to $\operatorname{Mod}_{\mit{S},\operatorname{gr}}$. It has a right derived functor $\bold{R}\Gamma_*: D^b(\operatorname{Mod_{\cal O}}) @>>> D^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ whose cohomology functors we denote $H^i_*(\cal{E}) = \bigoplus _{t\in\Bbb{Z}}H^i(\cal{E}(t))$. The functor $\Gamma_*$ has an exact left adjoint $\widetilde{\ \ }$, the functor of associated sheaves. Let $\Gamma_{\frak m}: \operatorname{Mod}_{\mit{S},\operatorname{gr}} @>>> \operatorname{Mod}_{\mit{S},\operatorname{gr}}$ be the functor associating to a graded $S$-module $M$ the maximal submodule $\Gamma_{\frak m}(M) \subset M$ supported at the origin $0$ of $\Bbb A^{N+1}$. This functor is also left exact and has a right derived functor $\bold R\Gamma_{\frak m}: D^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}}) @>>> D^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$. Its cohomology functors are denoted $H^i_{\frak m}$. \begin{lemma} \label{bounds} Let $P^*$ be a bounded complex of free graded $S$-modules of finite rank where $S=k[X_0,\dots,X_N]$. If $P^*$ is minimal, then \begin{align*} \max\{i\mid P^i\neq 0\} = & \max\{i\mid H^i(P^*)\neq 0\},\\ \min\{i\mid P^i\neq 0\} = & \min\{i\mid H^i_{\frak m}(P^*)\neq 0\} -N-1. \end{align*} \end{lemma} \begin{pf} The assertion about maxima is a simple and well-known application of the minimality condition and Nakayama's Lemma. The assertion about minima, which is essentially the Auslander-Buchsbaum theorem, reduces to the assertion about maxima by Serre duality. \end{pf} \subsection{The Horrocks Correspondence} We now begin to describe the components of the Horrocks correspondence. Let $\frak{B}$ be the full subcategory of $\operatorname{Mod_{\cal O}}$ of locally free sheaves of finite rank, and let $\frak{Z}$ denote the full category of $D^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ of complexes $C^*$ such that $H^i(C^*)$ is of finite length for $0<i<N$ and $H^i(C^*)$ vanishes for all other $i$. The Horrocks correspondence consists of a functor $\zeta: \frak{B} @>>> \frak{Z}$ and a map $\cal{H}: \operatorname{ob}(\frak{Z}) @>>> \operatorname{ob}(\frak{B})$ in the opposite direction. The functor $\zeta$ is simply $\tau_{>0}\tau_{< N}\bold{R}\Gamma _*$. For $\cal{E}$ a vector bundle on $\Bbb{P}^N$, the cohomology of $\zeta(\cal{E})$ is of course: \[ H^i(\zeta(\cal{E}))= \begin{cases} H^i_*(\cal{E}) & \text{if }0<i<N,\\ 0 & \text{otherwise.} \end{cases} \] Since $\cal{E}$ is locally free of finite rank, $H^i_*(\cal{E})$ is of finite length for $0<i<N$. So $\zeta(\cal{E})\in\operatorname{ob}(\frak{Z})$. We now define $\cal{H}$. Any $C^*\in\operatorname{ob}(\frak{Z})$ has a minimal projective resolution $P^* @>>> C^*$. We define $\cal{H}(C^*)$ to be the kernel of the differential $\widetilde\delta^0:\widetilde P^0 @>>> \widetilde P^1$. Then $\cal{H}(C^*)$ is a vector bundle because it fits into an exact complex of vector bundles \begin{equation} \label{Hcomp} \dotsb @>>> 0 @>>> \cal{H}(C^*) @>>> \widetilde P^0 @>>> \widetilde P^1 @>>> \dotsb @>>> \widetilde P^{N-1} @>>> 0 @>>> \dotsb . \end{equation} Note that $\cal{H}(C^*)$ is well-defined up to isomorphism because the minimal projective resolution $P^*$ of $C^*$ is unique up to isomorphism because of Lemma \ref{NAK}. However, $\cal{H}$ is not a functor. The principal results of Horrocks' paper \cite{H} can be described in the following way: \begin{theorem}[Horrocks] \label{Horrocks} Let $\frak{B}$ be the category of locally free sheaves of finite rank on $\Bbb{P}^N$, and let $\frak{Z}$ be the full subcategory of $D^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ of complexes $C^*$ such that $H^i(C^*)$ is of finite length if $0<i<N$, and $H^i(C^*)=0$ for all other $i$. Let $\zeta = \tau_{>0}\tau_{< N}\bold{R}\Gamma _*: \frak{B} @>>> \frak{Z}$, and let $\cal{H}:\operatorname{ob}(\frak{Z}) @>>> \operatorname{ob}(\frak{B})$ be the map defined as in \rom(\ref{Hcomp}\rom) above. \rom(a\rom) If $\cal{E}\in\operatorname{ob}(\frak{B})$, then $\cal{E} \cong \cal{H}\zeta(\cal{E}) \oplus \bigoplus _i \cal{O}_{\Bbb{P}^N}(n_i)$ for some integers $n_i$. \rom(b\rom) If $C^*\in\operatorname{ob}(\frak{Z})$, then $\zeta\cal{H}(C^*)\simeq C^*$. \rom(c\rom) If $\cal{E},\cal{F}\in\operatorname{ob}(\frak{B})$, then $\operatorname{Hom}_{\frak{Z}}(\zeta(\cal{E}),\zeta(\cal{F}))\cong \operatorname{Hom}(\cal{E},\cal{F})/\operatorname{Hom}_{\Phi}(\cal{E},\cal{F})$ where $\operatorname{Hom}_{\Phi}(\cal{E},\cal{F})$ is the set of all morphisms which factor through a direct sum of line bundles. \end{theorem} The theorem may be read as saying the following. Call two vector bundles $\cal{E}$ and $\cal{F}$ {\em stably equivalent} if there exist sets of integers $\{n_i\}$ and $\{m_j\}$ such that $\cal{E} \oplus \bigoplus_i\cal{O}_{\Bbb{P}^N}(n_i) \cong \cal{F}\oplus \bigoplus_j \cal{O}_{\Bbb{P}^N}(m_j)$. Then the theorem says that $\zeta$ and $\cal{H}$ induce a one-to-one correspondence between stable equivalence classes of vector bundles on $\Bbb{P}^N$ and quasi-isomorphism classes of complexes in $\frak{Z}$. For Horrocks' proof of the theorem, see \cite{H} Lemma 7.1 and Theorem 7.2 and the discussion between them. However, Horrocks' definition of the category $\frak{Z}$ and the functor $\zeta$ are different from ours, and demonstrating the equivalence of the definitions is somewhat tedious. So instead of referring the reader to Horrocks' paper, we give a new proof. The first step is the following lemma: \begin{lemma} \label{tauresol} \rom(a\rom) Suppose \[ P^*:\qquad \dotsb @>>> 0 @>>> P^0 @>>> P^1 @>>> \dotsb @>>> P^{N-1} @>>> 0 @>>> \dotsb \] is a complex of free graded $S$-modules of finite rank such that $H^i(P^*)$ is a module of finite length for $0<i<N$. Let $\cal{E}= H^0(P^*)\sptilde$. Then $P^*$ is quasi-isomorphic to $\tau_{<N}\bold{R}\Gamma_*(\cal{E})$. \rom(b\rom) Conversely, if $\cal{E}$ is a vector bundle on $\Bbb{P}^N$, then the minimal projective resolution of $\tau_{<N}\bold{R}\Gamma_*(\cal{E})$ is of the above form. \end{lemma} \begin{pf} (a) Note that the complex $\widetilde{P}^*$ of coherent sheaves on $\Bbb{P}^N$ has vanishing cohomology in degrees different from $0$. So it is quasi-isomorphic to $H^0(\widetilde{P}^*) = \cal{E}$. Hence the triangle of functors of \cite{W} Proposition 1.1: \[ \bold{R}\Gamma_{\frak m} @>>> \operatorname{Id} @>>> \bold{R}\Gamma_*\circ\sptilde @>>> \bold{R}\Gamma_{\frak m}[1], \] when applied to $P^*$, yields a triangle \begin{equation} \label{triangle} \bold{R}\Gamma_{\frak m}(P^*) @>>> P^* @>\beta>> \bold{R}\Gamma_*(\cal{E}) @>>> \bold{R}\Gamma_{\frak m}(P^*)[1]. \end{equation} By Lemma \ref{bounds}, we have $H^i_{\frak m}(P^*)=0$ for $i \leq N$. So $H^i(\beta): H^i(P^*) @>>> H^i_*(\cal{E})$ is an isomorphism for $i<N$. Therefore $\beta$ induces a quasi-isomorphism of $P^*$ onto $\tau_{<N}\bold{R}\Gamma_* (\cal{E})$. (b) Conversely, if $\cal{E}$ is a vector bundle on $\Bbb{P}^N$, then $H^i_*(\cal{E})$ is finitely generated for $i<N$. Hence $\tau_{<N}\bold{R}\Gamma_*(\cal{E})$ has a minimal projective resolution $P^*$. For $0<i<N$ the module $H^i(P^*) = H^i_*(\cal E)$ is of finite length because $\cal E$ is locally free. By construction $H^i(P^*) = H^i(\tau_{<N}\bold R\Gamma_*(\cal E)) = 0$ for $i \geq N$. So we have $P^i = 0$ for $i\geq N$ by Lemma \ref{bounds}. Looking again at the triangle (\ref{triangle}), we see by the construction of $P^*$ that $H^i(\beta)$ is an isomorphism for $i<N$ and an injection for $i=N$. So $H^i_{\frak m}(P^*) = 0$ for $i\leq N$. So by Lemma \ref{bounds} we see that $P^i=0$ for $i\leq -1$. Thus $P^*$ has the form asserted by the lemma. \end{pf} We now wish to functorialize the previous lemma. Let $B\subset K^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ be the full subcategory of complexes of the form \begin{equation} \label{rightresol} \dotsb @>>> 0 @>>> P^0 @>>> P^1 @>>> \dotsb @>>> P^{N-1} @>>> 0 @>>> \dotsb \end{equation} such that the $P^i$ are free of finite rank for all $i$, the modules $H^i(P^*)$ are of finite length for $0<i<N$ and the differentials satisfy $\delta^i(P^i)\subset \frak{m}P^{i+1}$ for all $i$. For any vector bundle $\cal E$ on $\Bbb{P}^N$ we now define $P^*(\cal E)$ as the minimal projective resolution of $\tau_{<N}\bold R\Gamma_* (\cal E)$. By Lemma \ref{tauresol}, $P^*(\cal E)$ is always an object of $B$. \begin{lemma} \label{Bequiv} The functor $P^*: \frak{B} @>>> B$ which associates to an $\cal{E} \in\operatorname{ob}(\frak{B})$ the minimal projective resolution of $\tau_{<N} \bold{R}\Gamma_*(\cal{E})$ is an equivalence of categories with inverse given by $C^*\mapsto H^0(C^*)\sptilde$. \end{lemma} \begin{pf} Since the functor $\tau_{<N}\bold R\Gamma_*: \frak B @>>> D^-(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ has a left inverse $H^0(-)\sptilde$, it induces an equivalence between $\frak B$ and the full subcategory $A\subset D^-(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ of complexes quasi-isomorphic to complexes in the image of $\tau_{<N}\bold R\Gamma_*$. But by Lemma \ref{tauresol}, the full subcategory $B\subset K^-(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ has the properties that $\operatorname{ob}(B)\subset\operatorname{ob}(A)$ and that the minimal projective resolution of every object of $A$ belongs to $B$. Hence the natural functor $B @>>> A$ is also an equivalence of categories by Lemma \ref{category}. Since $P^*$ is exactly the composition of the equivalence $\tau_{<N}\bold R\Gamma_*: \frak B @>>> A$ with the inverse of the equivalence $B @>>> A$, it is an equivalence. The inverse of $P^*$ remains the same as that of $\tau_{<N} \bold R\Gamma_*$, namely $H^0(-)\sptilde$. \end{pf} Now the graded module associated to a vector bundle $\cal E$ on $\Bbb{P}^N$ has a minimal projective resolution: \[ 0 @>>> Q^{-(N-1)} @>>> \dotsb @>>> Q^{-1} @>>> Q^0 @>>> \Gamma_*(\cal E) \] For any $\cal E$ we now define the following complexes in addition to the $P^*(\cal E)$ defined above. First we set: \[ Q^*(\cal E):\qquad \dotsb @>>> 0 @>>> Q^{-(N-1)} @>>> \dotsb @>>> Q^{-1} @>>> Q^0 @>>> 0 @>>> \dotsb. \] We then let $R^*(\cal E)$ be the natural concatenation of $Q^*(\cal E)$ with $P^*(\cal E)$ induced by the composition $Q^0 \twoheadrightarrow \Gamma_*(\cal E) \hookrightarrow P^0$: \[ R^*(\cal E):\qquad\dotsb @>>> 0 @>>> Q^{-(N-1)} @>>> \dotsb @>>> Q^0 @>>> P^0 @>>> \dotsb @>>> P^{N-1} @>>> 0 @>>> \dotsb \] Thus $R^i(\cal E) = P^i(\cal E)$ for $i\geq 0$, and $R^i(\cal E) = Q^{i+1}(\cal E)$ for $i<0$. Note that although the projective complexes $P^*(\cal E)$ and $Q^*(\cal E)$ are minimal, $R^*(\cal E)$ may not be minimal, because there may be a direct factor of $Q^0(\cal E)$ which is mapped isomorphically onto a direct factor of $P^0(\cal E)$. However, one may write $R^*(\cal E)$ as the direct sum of a minimal complex of projectives $R^*_{\min}(\cal E)$ \begin{align*} R^*_{\min}(\cal E): \qquad \dotsb @>>> Q^{-2} @>>> Q^{-1} @>>> Q^0_{\min} & @>>> P^0_{\min} @>>> P^1 @>>> P^2 @>>> \dotsb\\ \intertext{and of an exact complex of projectives} \stepcounter{equation}\tag{\theequation}\label{L:module} \dotsb @>>> 0 @>>> L & @>\operatorname{Id}>> L @>>> 0 @>>> \dotsb. \end{align*} The complexes $Q^*(\cal E)$, $R^*(\cal E)$, and $R^*_{\min}(\cal E)$ are all functorial (in the homotopy category) in $\cal E$. Moreover, we may use the identification between the categories $\frak B$ and $B$ to define complexes $Q^*(P^*)$, $R^*(P^*)$, and $R^*_{\min}(P^*)$ for $P^*$ in $B$. Namely, $Q^*(P^*)$ is the minimal projective resolution of $H^0(P^*)$, $R^*(P^*)$ is the concatenation of $Q^*(P^*)$ with $P^*$, etc. We now define a homotopy category of complexes of type $R^*_{\min}$. More formally, let $Z\subset K^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ be the full subcategory of minimal complexes of projective modules of finite rank of the form \begin{equation} \label{R:complex} \dotsb @>>> 0 @>>> R^{-N} @>>> \dotsb @>>> R^{-1} @>>> R^0 @>>> \dotsb @>>> R^{N-1} @>>> 0 @>>> \dotsb \end{equation} such that the cohomology modules $H^i(R^*)$ are of finite length for $0<i<N$ and vanish for all other $i$. We need one more lemma before proving Theorem \ref{Horrocks}. \begin{lemma} \label{Zequiv} The natural functor $Z @>>> \frak{Z}$ is an equivalence of categories. \end{lemma} \begin{pf} Let $R^*$ be the minimal projective resolution of an object $C^*$ of $\frak Z$. Since $H^i(R^*)= H^i(C^*)=0$ for $i\geq N$, we have $R^i=0$ for $i\geq N$ by Lemma \ref{bounds}. Moreover, all the $H^i(C^*)$ are of finite length, so $H^i_{\frak m}(C^*) = H^i(C^*)$ for all $i$. In particular, $H^i_{\frak m}(R^*)=H^i_{\frak m}(C^*) =0$ for $i\leq 0$. So $R^i=0$ for $i\leq -N-1$ by Lemma \ref{bounds}. Thus the minimal projective resolution of any object of $\frak Z$ is in $Z$. The lemma now follows from Lemma \ref{category}. \end{pf} \begin{pf*}{Proof of Theorem \ref{Horrocks}} Lemma \ref{Bequiv} permits us to identify a vector bundle $\cal E$ with the complex $P^*(\cal E)$ of $B$. Since $P^*(\cal E)$ is already quasi-isomorphic to $\tau_{<N}\bold R\Gamma_*(\cal E)$, the complex $\zeta(\cal E) = \tau_{>0}\tau_{<N} \bold R\Gamma_*(\cal E)$ is quasi-isomorphic to the complex \[ \dotsb @>>> 0 @>>> \Gamma_*(\cal E) @>>> P^0 @>>> P^1 @>>> \dotsb @>>> P^{N-1} @>>> 0 @>>> \dotsb \] and hence to the complexes $R^*(\cal E)$ and $R^*_{\min}(\cal E)$. Hence the object $\zeta(\cal E)$ in $\frak Z$ is quasi-isomorphic to the object $R^*_{\min}(\cal E)$ of $Z$. Hence after identifying $\frak B$ and $\frak Z$ with $B$ and $Z$ by Lemmas \ref{Bequiv} and \ref{Zequiv}, the functor $\zeta$ may be identified with the functor from $B$ to $Z$ which associates to any complex $P^*$ in $B$ the corresponding complex $R^*_{\min}$ as described earlier. Similarly, given any object $C^*$ of $\frak Z$ with minimal projective resolution $R^*$, the definitions say that $P^*(\cal H(C^*))= \sigma_{\geq 0}(R^*)$, the naive truncation. Thus the map $\cal H: \operatorname{ob}(\frak Z) @>>> \operatorname{ob}(\frak B)$ may be identified with $\sigma_{\geq 0}: \operatorname{ob}(Z) @>>> \operatorname{ob}(B)$. Note that since all objects of $Z$ and $B$ are minimal complexes of projective modules, homotopy equivalence classes of objects of $Z$ and $B$ coincide with isomorphism classes. Hence the map $\sigma_{\geq 0}: \operatorname{ob}(Z) @>>> \operatorname{ob}(B)$ preserves homotopy equivalence. Since $Z$ and $B$ are subcategories of the homotopy category, this means that $\sigma_{\geq 0}$ is well-defined on objects of $Z$. However, $\sigma_{\geq 0}$ and hence $\cal H$ are not well-defined on morphisms of $Z$. (a) The above identifications now say if $\cal E\in\operatorname{ob}(\frak B)$, then $\cal H\zeta(\cal E)$ is the object of $\frak B$ corresponding to the complex $\sigma_{\geq 0}(R^*_{\min}(\cal E))$: \[ \sigma_{\geq 0}(R^*_{\min}(\cal E)): \qquad \dotsb @>>> 0 @>>> P^0_{\min} @>\mu>> P^1 @>>> \dotsb @>>> P^{N-1} @>>> 0 @>>> \dotsb . \] By Lemma \ref{Bequiv}, the sheaf $\cal H\zeta(\cal E)$ is $\ker(\mu)\sptilde$. So $\cal E = \cal H\zeta(\cal E)\oplus\widetilde{L}$ where $L$ is the projective module of (\ref{L:module}). Since $\widetilde{L}$ is now a direct sum of line bundles, (a) follows. (b) If $C^*$ is an object of $\frak Z$ with minimal projective resolution $R^*$ in $Z$ of the form (\ref{R:complex}), then the above computations identify $\cal H(C^*)$ in $\frak B$ with $P^*(\cal H(C^*)) = \sigma_{\geq 0}(R^*)$ in $B$. Thus $\zeta\cal H(C^*)$ becomes identified with $R^*_{\min}(\cal H(C^*))$ which is just $R^*$ again. Since $R^*$ is quasi-isomorphic to $C^*$, we have $\zeta\cal H(C^*)\simeq C^*$ as desired. (c) After identifying $\frak B$ with $B$ and $\frak Z$ with $Z$, assertion (c) becomes the statement: For any pair of objects $E^*$ and $F^*$ in $B$, the natural map \begin{equation} \label{B:to:Z} \operatorname{Hom}_B(E^*,F^*) @>>> \operatorname{Hom}_Z(R^*_{\min}(E^*),R^*_{\min}(F^*)) \end{equation} is surjective and its kernel is the subspace of morphisms which factor through an object of $B$ of the form \begin{equation} \label{triv:comp} \dotsb @>>> 0 @>>> L @>>> 0 @>>> \dotsb \end{equation} with $L$ a free graded $S$-module of finite rank appearing in degree $0$. We first prove surjectivity. Suppose $\phi\in \operatorname{Hom}_Z (R^*_{\min}(E^*), R^*_{\min}(F^*))$. Since $Z$ is a homotopy category, $\phi$ is actually a homotopy equivalence class of maps in $C(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$. So we may choose a chain map $f$ in the class $\phi$. Then $f$ may be extended to a chain map $\overline f: R^*(E^*) @>>> R^*(F^*)$ by defining it to be $0$ on the exact factor of the type (\ref{L:module}). Then $\sigma_{\geq 0}\overline f$ maps $E^*$ to $F^*$, and its homotopy class in $B$ has image $\phi$ in $Z$. This proves surjectivity. We now compute the kernel of (\ref{B:to:Z}). First if $\alpha\in \operatorname{Hom}_B(E^*,F^*)$ factors through a complex $L^*$ of the form (\ref{triv:comp}), then $R^*_{\min}(\alpha)$ factors through $R^*_{\min}(L^*)=0$ and so vanishes. So the kernel of (\ref{B:to:Z}) contains all morphisms which factor through complexes of the form (\ref{triv:comp}). Conversely, suppose $\alpha$ is in the kernel of (\ref{B:to:Z}). Since $\alpha$ is a morphism in $B$, it is a homotopy class of chain maps from which we may choose a member $\beta$. We may complete $\beta$ to a chain map $\rho: R^*(E^*) @>>> R^*(F^*)$. \[ \begin{CD} R^*(E^*) & \qquad & \dotsb & \:\rightarrow\: & 0 & \:\rightarrow\: & \overline E^{-N} & \:\rightarrow\: & \dotsb & \:\rightarrow\: & \overline E^{-1} & \:\rightarrow\: & E^0 & \:\rightarrow\: & \dotsb & \:\rightarrow\: & E^{N-1} & \:\rightarrow\: & 0 & \:\rightarrow\: & \dotsb \\ @VV{\rho}V && @VVV @VVV && @VVV @VV{\beta}V && @VV{\beta}V @VVV \\ R^*(F^*) & \qquad & \dotsb & \:\rightarrow\: & 0 & \:\rightarrow\: & \overline F^{-N} & \:\rightarrow\: & \dotsb & \:\rightarrow\: & \overline F^{-1} & \:\rightarrow\: & F^0 & \:\rightarrow\: & \dotsb & \:\rightarrow\: & F^{N-1} & \:\rightarrow\: & 0 & \:\rightarrow\: & \dotsb \end{CD} \] The homotopy class of $\rho$ is the image of $\alpha$ under $R^*$ and so must vanish by hypothesis. (Note that $R^*$ and $R^*_{\min}$ are homotopy equivalent.) Thus $\rho$ is homotopic to $0$. Thus if we write $\delta^i$ for the differentials of $R^*(E^*)$, and $\epsilon^i$ for the differentials of $R^*(F^*)$, then there is a chain homotopy $h = (h^i)$ such that $\rho^i = h^{i+1}\delta^i + \epsilon^{i-1}h^i$ for all $i$. Now restrict $h$ to a chain homotopy $\widehat{h} = (\widehat{h}^i)$ with $\widehat{h}^i: E^i @>>> F^{i-1}$ defined by defined by $\widehat{h}^i = h^i$ for all $i\geq 1$, and $\widehat{h}^i = 0$ for all $i\leq 0$. Then $\beta$ is homotopic to a morphism whose components are \[ \beta^i-(\widehat{h}^{i+1}\delta^i + \epsilon^{i-1}\widehat{h}^i)= \begin{cases} \rho^i-(h^{i+1}\delta^i+\epsilon^{i-1}h^i) = 0 & \text{if }i\geq1,\\ \rho^0-h^1\delta^0 = \epsilon^{-1}h^0 & \text{if }i=0,\\ 0 & \text{if }i\leq -1. \end{cases} \] Hence the homotopy class $\alpha$ of $\beta$ factors through the complex \[ \dotsb @>>> 0 @>>> \overline F^{-1} @>>> 0 @>>> \dotsb \] of type (\ref{triv:comp}). So the kernel of (\ref{B:to:Z}) is as asserted. This completes the proof of the theorem. \end{pf*} We will use three further results concerning the Horrocks correspondence. The first will permit us to use the Horrocks correspondence to constuct locally free resolutions of coherent sheaves. \begin{proposition} \label{functorial} Let $\cal{Q}$ be a quasi-coherent sheaf on $\Bbb{P}^N$, let $C^*\in\operatorname{ob}(\frak{Z})$, and let $\beta : C^* @>>> \tau_{>0}\tau_{<N} \bold{R}\Gamma _*(\cal{Q})$ be a morphism in $D^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$. Then there exists a morphism of quasi-coherent sheaves $\widetilde{\beta} : \cal{H}(C^*) @>>> \cal{Q}$ such that $\beta =\tau_{>0}\tau_{<N} \bold{R}\Gamma _*(\widetilde{\beta})$. In particular, the induced morphisms $H^i_*(\cal{H}(C^*)) @>>> H^i_*(\cal{Q})$ are the same as $H^i(\beta)$ for $1\leq i\leq N-1$. \end{proposition} \begin{pf} Let $R^*$ be a minimal projective resolution of $C^*$, and let $\cal{I}^*$ be an injective resolution of $\cal{Q}$. Then $\beta$ may be identified with an actual chain map \[ \begin{CD} \dotsb & \:\rightarrow\: & R^{-2} & \:\rightarrow\: & R^{-1} & \:\rightarrow\: & R^0 & \:\stackrel{\lambda}{\longrightarrow}\: & R^1 & \:\rightarrow\: & \dotsb & \:\rightarrow\: & R^{N-2} & \:\rightarrow\: & R^{N-1} & \:\rightarrow\: & 0 \\ && @VVV @VVV @VVV @VVV && @VVV @VVV \\ \dotsb & \:\rightarrow\: & 0 & \:\rightarrow\: & \Gamma_*(\cal{Q}) & \:\rightarrow\: & \Gamma_*(\cal{I}^0) & \:\stackrel{\mu}{\longrightarrow}\: & \Gamma_*(\cal{I}^1) & \:\rightarrow\: & \dotsb & \:\rightarrow\: & \Gamma_*(\cal{I}^{N-2}) & \:\rightarrow\: & \ker(\delta^{N-1}) & \:\rightarrow\: & 0 \end{CD} \] Thus $\beta$ induces a morphism $\widetilde{\beta}$ from $\cal{H}(C^*) = \ker(\lambda)\sptilde$ to $\cal{Q} = \ker(\mu)\sptilde$. We now need to calculate $\bold{R}\Gamma_*(\widetilde{\beta})$. Consider the complex \[ P^*: \qquad \dotsb @>>> 0 @>>> R^0 @>>> R^1 @>>> \dotsb @>>> R^{N-1} @>>> 0 @>>> \dotsb . \] The previous diagram induces a new commutative diagram \[ \begin{CD} \widetilde{P}^*&: & \qquad\qquad & \dotsb & \:\rightarrow\: & 0 & \:\rightarrow\: & \widetilde R^0 & \:\rightarrow\: & \widetilde R^1 & \:\rightarrow\: & \dotsb & \:\rightarrow\: & \widetilde R^{N-2} & \:\rightarrow\: & \widetilde R^{N-1} & \:\rightarrow\: & 0 & \:\rightarrow\: & 0 & \:\rightarrow\: & \dotsb \\ @VV{\overline \beta}V &&& @VVV @VVV @VVV && @VVV @VVV @VVV @VVV\\ \cal{I}^*&: && \dotsb & \:\rightarrow\: & 0 & \:\rightarrow\: & \cal{I}^0 & \:\rightarrow\: & \cal{I}^1 & \:\rightarrow\: & \dotsb & \:\rightarrow\: & \cal{I}^{N-2} & \:\rightarrow\: & \cal{I}^{N-1} & \:\rightarrow\: & \cal{I}^N & \:\rightarrow\: & 0 & \:\rightarrow\: & \dotsb \end{CD} \] between resolutions of $\cal{H}(C^*)$ and $\cal{Q}$ extending $\widetilde \beta$. Let $\gamma: P^* @>>> J^*$ be an injective resolution of $P^*$. Then $\overline\beta$ factors through $\widetilde\gamma$ as $\widetilde{P}^* @>>> \widetilde{J}^* @>>> \cal{I}^*$. Applying $\Gamma_*$ now gives a factorization \begin{equation} \label{factorize} P^* @>>> \Gamma_*(\widetilde{J}^*) @>>> \Gamma_*(\cal{I}^*). \end{equation} Now $\overline\beta$ is a map between resolutions of $\cal{H}(C^*)$ and $\cal{Q}$, respectively, which extends $\widetilde{\beta}: \cal{H}(C^*) @>>> \cal{Q}$, while $\widetilde\gamma$ is a quasi-isomorphism. So the map $\widetilde{J}^* @>>> \cal{I}^*$ is a map between injective resolutions of $\cal{H}(C^*)$ and $\cal{Q}$ extending $\widetilde\beta$. So by definition, the second arrow of (\ref{factorize}) is $\bold{R}\Gamma_*(\widetilde{\beta}) : \bold{R}\Gamma_*(\cal{H}(C^*)) @>>> \bold{R}\Gamma_*(\cal{Q})$. On the other hand, the proof of Lemma \ref{tauresol}(a) shows that the first arrow of (\ref{factorize}) can be identified with the truncation $\tau_{<N}(\bold{R}\Gamma_*(\cal{H}(C^*))) @>>> \bold{R}\Gamma_*(\cal{H}(C^*))$ because it induces isomorphisms $H^i(P^*) \cong H^i(\Gamma_*(\widetilde{J}^*)) = H^i_*(\cal{H}(C^*))$ for $i < N$. Hence $\Gamma_*(\overline\beta): P^* @>>> \Gamma_*(\cal{I}^*)$ can be identified with the composition of the truncation $\tau_{<N}(\bold{R}\Gamma_*(\cal{H}(C^*))) @>>> \bold{R}\Gamma_*(\cal{H}(C^*))$ with $\bold{R}\Gamma_* (\widetilde{\beta})$. Thus $\tau_{<N}\bold{R}\Gamma_* (\widetilde{\beta})$ may be identified with the diagram \[ \begin{CD} \dotsb & \:\rightarrow\: & 0 & \:\rightarrow\: & R^0 & \:\rightarrow\: & R^1 & \:\rightarrow\: & \dotsb & \:\rightarrow\: & R^{N-2} & \:\rightarrow\: & R^{N-1} & \:\rightarrow\: & 0 & \:\rightarrow\: & \dotsb \\ && @VVV @VVV @VVV && @VVV @VVV @VVV\\ \dotsb & \:\rightarrow\: & 0 & \:\rightarrow\: & \Gamma_*(\cal{I}^0) & \:\rightarrow\: & \Gamma_*(\cal{I}^1) & \:\rightarrow\: & \dotsb & \:\rightarrow\: & \Gamma_*(\cal{I}^{N-2}) & \:\rightarrow\: & \ker(\delta^{N-1}) & \:\rightarrow\: & 0 & \:\rightarrow\: & \dotsb \end{CD} \] induced by $\beta$. Truncating on the left, we reach a diagram equivalent to the first diagram of the proof of the proposition. So $\beta = \tau_{>0}\tau_{<N} \bold{R}\Gamma _*(\widetilde{\beta})$. \end{pf} We now need two homological criteria for maps of vector bundles to be isomorphisms. \begin{lemma} \label{isom} Let $\cal E$ and $\cal F$ be vector bundles on $\Bbb{P}^N$ with neither containing a line bundle as a direct factor. If $\alpha: \cal E @>>> \cal F$ is a map such that $H^i_*(\alpha): H^i_*(\cal E) @>>> H^i_*(\cal F)$ is an isomorphism for $0<i<N$, then $\alpha$ is an isomorphism. \end{lemma} \begin{pf} We use the notation of the proof of Theorem \ref{Horrocks}. Let $E^*=P^*(\cal E)$ and $F^*=P^*(\cal F)$, and let $\overline\alpha: E^* @>>> F^*$ be the map induced by $\alpha$. The hypothesis $\cal E = \cal H\zeta(\cal E)$ implies that $E^*$ is homotopy equivalent to $\sigma_{\geq 0}R^*_{\min}(E^*)$, or equivalently that $R^*(E^*)$ is a minimal complex of projectives. Similarly, $R^*(F^*)$ is a minimal complex of projectives. The hypothesis on $\alpha$ implies that $\zeta(\alpha): \zeta(\cal E) @>>> \zeta(\cal F)$ is a quasi-isomorphism. This in turn translates into $R^*_{\min}(\overline\alpha)$ being a homotopy equivalence. But because of the earlier hypotheses, this means that $R^*(\overline\alpha): R^*(E^*) @>>> R^*(F^*)$ is a homotopy equivalence between the minimal complexes of projectives. Hence by Lemma \ref{NAK} $R^*(\overline\alpha)$ is actually an isomorphism of complexes. So its naive truncation $\sigma_{\geq 0} R^*(\overline\alpha) = \overline\alpha$ is also an isomorphism. Therefore $\alpha$ is an isomorphism. \end{pf} We will also need a slight generalization of the previous lemma. \begin{lemma} \label{second:isom} Let $\cal E = \cal H\zeta(\cal E)\oplus\bigoplus\cal O_{\Bbb{P}^N}(n_i)$ and $\cal F$ be vector bundles on $\Bbb{P}^N$, and let $\cal Q$ be a coherent sheaf on $\Bbb{P}^N$. Suppose that there exist morphisms $\alpha: \cal E @>>> \cal F$ and $\beta: \cal F @>>> \cal Q$ such that \rom(i\rom) $H^i_*(\alpha): H^i_*(\cal E) @>>> H^i_*(\cal F)$ is an isomorphism for $0<i<N$, \rom(ii\rom) $\beta\alpha$ takes the generators of the factors $S(n_i)$ of $\Gamma_*(\cal E)$ onto a minimal set of generators of the module $\overline Q :=\Gamma_*(\cal Q)/\beta\alpha(\Gamma_*(\cal H\zeta(\cal E)))$, \rom(iii\rom) $\cal E$ and $\cal F$ have the same rank. Then $\alpha$ is an isomorphism. \end{lemma} \begin{pf} Write $\cal F = \cal H\zeta(\cal F)\oplus\bigoplus\cal O_{\Bbb{P}^N}(m_j)$. The splittings of $\cal E$ and of $\cal F$ into direct factors are not canonical. But choosing such splittings gives an injection $\cal H\zeta (\cal E) \hookrightarrow \cal E$ and a projection $\cal F \twoheadrightarrow \cal H\zeta (\cal F)$. Then the composition \[ \overline{\alpha}:\quad \cal H\zeta (\cal E) @>>> \cal E @>{\alpha}>> \cal F @>>> \cal H\zeta (\cal F) \] is, like $\alpha$, an isomorphism on $H^i_*$ for $0<i<N$. So $\overline \alpha$ is an isomorphism by Lemma \ref{isom}. Hence by identifying $\cal H\zeta(\cal F)$ with $\alpha (\cal H\zeta(\cal E)) \subset \cal F$, we see that $\alpha$ induces a morphism of diagrams \begin{equation} \label{split} \begin{CD} 0 @>>> \cal H\zeta(\cal E) @>>> \cal E @>>> \bigoplus \cal O_{\Bbb{P}^N}(n_i) @>>> 0\\ && @| @VV{\alpha}V @VV{\alpha_1}V \\ 0 @>>> \cal H\zeta(\cal F) @>>> \cal F @>>> \bigoplus \cal O_{\Bbb{P}^N}(m_j) @>>> 0 \end{CD} \end{equation} The morphisms $\alpha$ and $\beta$ therefore induce maps \[ \bigoplus S(n_i) @>{\Gamma_*(\alpha_1)}>> \bigoplus S(m_j) @>{\overline \beta}>> \overline Q = \Gamma_*(\cal Q)/ \beta\alpha (\Gamma_*(\cal H\zeta(\cal E))). \] The composition is a surjection corresponding to a minimal set of generators of $\overline Q$ by hypothesis (ii). Hence the righthand map $\overline \beta$ must be a surjection corresponding to a set of generators of $\overline Q$. However, the two free modules have the same rank by hypothesis (iii). Hence $\overline \beta$ also corresponds to a minimal set of generators, and $\Gamma_*(\alpha_1)$ must be an isomorphism. So returning to diagram (\ref{split}), $\alpha_1$ and hence $\alpha$ are isomorphisms. \end{pf} \section{The Self-Dual Resolution} Let $X\subset \Bbb{P}^{n+3}$ be a locally Gorenstein subcanonical subscheme of equicodimension $3$ satisfying the parity condition. In this section we use the Horrocks correspondence and especially Proposition \ref{functorial} to construct a locally free resolution of $\cal O_X$. We then use Lemma \ref{second:isom} to show that the resolution satisfies condition (a) of Proposition \ref{conditions}. In the course of the construction we will need a more refined variant of the canonical truncation. Namely, suppose $D^*$ is a complex of objects in an abelian category with differentials $\delta^i: D^i @>>> D^{i+1}$. Suppose $r$ is an integer, and $W\subset H^r(D^*)$ a subobject. Then $W$ may be pulled back to a $\overline W$ satisfying \[ \operatorname{im}(\delta^{r-1})\subset \overline W \subset \ker(\delta^r) \subset D^r \] We then define: \[ \tau_{\leq r,W}(D^*): \qquad \dotsb @>>> D^{r-2} @>>> D^{r-1} @>>> \overline{W} @>>> 0 @>>> 0 @>>> \dotsb \quad . \] The cohomology of this complex is given by \[ H^i(\tau_{\leq r,W}(D^*)) = \begin{cases} H^i(D^*) & \text{if }i<r,\\ W & \text{if }i=r,\\0 & \text{if }i>r. \end{cases}\\ \] We will also use the following conventions. If $\cal{E}$ is a coherent sheaf on $\Bbb{P}^N$ and $\alpha,\beta\in\Bbb{Q}$ are not both integers, then we define $H^{\alpha}(\cal{E}(\beta)) = 0$. Also if $D^*$ is a complex and $\alpha\in\Bbb Q$, we define $\tau_{\leq \alpha}(D^*) = \tau_{\leq [\alpha]}(D^*)$. \subsection{Definition of the Locally Free Resolution} Suppose $X\subset\Bbb{P}^{n+3}$ is a locally Gorenstein subscheme of equidimension $n>0$ such that $\omega_X\cong\cal O_X(l)$ for some integer $l$ and such that $h^{n/2}(\cal O_X(l/2))$ is even. Let $\nu = n/2$ and $l' = l/2$. By hypothesis $H^{\nu}(\cal O_X(l'))$ is an even-dimensional vector space (zero if $n$ or $l$ is odd) equipped with a nondegenerate $(-1)^{\nu}$-symmetric bilinear form \[ H^{\nu}(\cal O_X(l')) \times H^{\nu}(\cal O_X(l')) @>>> H^n(\cal O_X(l)) \cong k . \] Let $U\subset H^{\nu}(\cal O_X(l'))$ be an isotropic subspace of maximal dimension $h^{\nu}(\cal O_X(l'))/2$. Let \begin{equation} \label{w} W = U \oplus \bigoplus_{t>l'} H^{\nu}(\cal O_X(t)) \subset H^{\nu}_*(\cal O_X) \end{equation} We begin the construction of the locally free resolution with the short exact sequence \begin{equation} \label{ideal} 0 @>>> \cal I_X @>>> \cal O_{\Bbb{P}^{n+3}} @>>> \cal O_X @>>> 0. \end{equation} Since $H^i_*(\cal O_X)\cong H^{i+1}_*(\cal I_X)$ for $0<i<n+2$, we have $W\subset H^{\nu+1}_*(\cal I_X)$. Now since $X$ is locally Cohen-Macaulay of equidimension $n$, the modules $H^i_*(\cal I_X)$ are of finite length for $0<i<n+1$. Hence the truncated complex $C^*_X = \tau_{>0} \tau_{\leq\nu+1,W} \bold{R}\Gamma_*(\cal I_X)$ has cohomology modules $H^i(C^*_X)$ of finite length for $0<i\leq \nu+1$, while $H^i(C^*_X)=0$ for all other $i$. Hence $C^*_X$ is in $\frak Z$. The definition of $C^*_X$ as a truncation means that it is endowed with a natural map $\beta: C^*_X @>>> \tau_{>0}\tau_{<n+3} \bold{R}\Gamma_*(\cal I_X)$. By Proposition \ref{functorial} this map induces a morphism $\widetilde{\beta}: \cal{H}(C^*_X) @>>> \cal I_X$. Let $Q$ be the cokernel \[ H^0_*(\cal{H}(C^*_X)) @>{H^0_*(\widetilde{\beta})}>> H^0_*(\cal I_X) @>>> Q @>>> 0. \] Let $d_1,\dots,d_r$ be the degrees of a minimal set of generators of $Q$. These generators lift to $H^0_*(\cal I_X)$, allowing us to define a surjection \begin{equation} \label{gamma} \gamma: \cal{F}_1 := \cal{H}(C^*_X)\oplus\bigoplus\cal O_{\Bbb{P}^{n+3}}(-d_i) \twoheadrightarrow \cal I_X . \end{equation} By construction, $\cal F_1$ is locally free. Let $\cal K = \ker(\gamma)$. We may then attach the short exact sequence $0 @>>> \cal K @>>> \cal F_1 @>>> \cal I_X @>>> 0$ to the short exact sequence (\ref{ideal}) to get an exact sequence \begin{equation} \label{partial} 0 @>>> \cal K @>>> \cal F_1 @>>> \cal O_{\Bbb{P}^{n+3}} @>>> \cal O_X @>>> 0. \end{equation} The construction described above leads immediately to the following conclusions about the cohomology of $\cal F_1$ and about the induced morphisms $H^i_*(\gamma): H^i_*(\cal F_1) @>>> H^i_*(\cal I_X)$ (cf.\ Proposition \ref{functorial}). \begin{itemize} \item $H^i_*(\gamma)$ is surjective (resp.\ an isomorphism) for $i=0$ (resp.\ $0<i<\nu+1$). \item $H^{\nu+1}_*(\gamma): H^{\nu+1}_*(\cal F_1) \cong W \hookrightarrow H^{\nu+1}_*(\cal I_X)$ is injective. \item $H^i_*(\cal F_1)=0$ for $\nu+1<i<n+3$. \end{itemize} One may now draw the following conclusions about the cohomology of $\cal K$. \begin{itemize} \item $H^i_*(\cal K) = 0$ for $0<i<\nu+2$. \item $H^{\nu+2}_*(\cal K)\cong H^{\nu}_*(\cal O_X)/W$. \item $H^i_*(\cal K) \cong H^{i-2}_*(\cal O_X)$ for $\nu+2<i<n+3$. \end{itemize} To finish the definition of the locally free resolution, consider the isomorphisms \[ \operatorname{Ext}^1(\cal K,\omega_{\Bbb{P}^{n+3}}(-l)) \cong H^{n+2}(\cal K(l))^* \cong H^n(\cal O_X(l))^* \cong H^0(\cal O_X). \] The extension class corresponding to $1\in H^0(\cal O_X)$ gives a short exact sequence \begin{equation} \label{extension} 0 @>>> \omega_{\Bbb{P}^{n+3}}(-l) @>>> \cal F_2 @>>> \cal K @>>> 0 \end{equation} which we may attach to (\ref{partial}) to get a complex of the type (\ref{P}) resolving $\cal O_X$ \begin{equation} \label{free:resol} \cal P^*:\qquad 0 @>>> \omega_{\Bbb{P}^{n+3}}(-l) @>>> \cal{F}_2 @>>> \cal{F}_1 @>>> \cal O_{\Bbb{P}^{n+3}} . \end{equation} \begin{lemma} \label{duality} The sheaves $\cal F_1$ and $\cal F_2$ in the resolution \rom{(\ref{free:resol})} satisfy $H^i_*(\cal F_2) \cong \left( H^{n+3-i}_*(\cal F_1) \right)^*(l)$ for $0<i<n+3$. \end{lemma} \begin{pf} If $0<i<\nu+2$, then $H^i_*(\cal F_2) \cong H^i_*(\cal K) = 0$ and $H^{n+3-i}_*(\cal F_1) = 0$. So the lemma holds for these values of $i$. If $i=\nu+2$, then $H^{\nu+2}_*(\cal F_2) \cong H^{\nu+2}_*(\cal K) \cong H^{\nu}_*(\cal O_X)/W$, while $H^{\nu+1}_*(\cal F_1) \cong W$. However, the submodule $W\subset H^{\nu}_*(\cal O_X)$ has been constructed so that it is an isotropic submodule with respect to the perfect pairing of Serre duality \[ H^{\nu}_*(\cal O_X)\times H^{\nu}_*(\cal O_X) @>>> H^n_*(\cal O_X) @>{\operatorname{tr}}>> k(-l). \] Moreover the length of $W$ is half the length of $H^{\nu}_*(\cal O_X)$. Hence $W = W^{\perp}$, and the duality isomorphism $H^{\nu}_*(\cal O_X) \cong \left( H^{\nu}_*(\cal O_X) \right)^*(l)$ carries the submodule $W$ onto $\left(H^{\nu}_*(\cal O_X)/W \right)^*(l)$. If $\nu+2 < i < n+2$, then $H^i_*(\cal F_2) \cong H^i_*(\cal K) \cong H^{i-2}_*(\cal O_X)$, while $H^{n+3-i}_*(\cal F_1) \cong H^{n+3-i}_*(\cal I_X) \cong H^{n+2-i}_*(\cal O_X)$. The asserted duality is then simply the Serre duality pairing \[ H^{i-2}_*(\cal O_X)\times H^{n+2-i}_*(\cal O_X) @>>> H^n_*(\cal O_X) @>{\operatorname{tr}}>> k(-l). \] Finally if $i=n+2$, we have an exact sequence \[ 0 @>>> H^{n+2}_*(\cal F_2) @>>> H^{n+2}_*(\cal K) @>>> H^{n+3}_*(\omega_{\Bbb{P}^{n+3}}(-l)). \] Now $H^{n+2}_*(\cal K) \cong H^{n}_*(\cal O_X)$. Moreover, the fact that the extension class defining $\cal F_2$ corresponded under the Serre duality identifications to $1\in H^0_*(\cal O_X) \cong \left( H^{n+2}_*(\cal K) \right)^*(l)$ implies that the last exact sequence dualizes to \[ H^0_*(\cal O_{\Bbb{P}^{n+3}}) @>1>> H^0_*(\cal O_X) @>>> \left( H^{n+2}_*(\cal F_2) \right)^*(l) @>>> 0. \] Hence $\left( H^{n+2}_*(\cal F_2) \right)^*(l) \cong H^1_*(\cal I_X) \cong H^1_*(\cal F_1)$. Dualizing now gives the last of the asserted isomorphisms. \end{pf} \begin{corollary} The coherent sheaf $\cal F_2$ in the resolution \rom{(\ref{free:resol})} is locally free. \end{corollary} \begin{pf} Since $\cal F_1$ is locally free, $H^i_*(\cal F_1)$ is of finite length for $0<i<n+3$. So by the lemma, $H^i_*(\cal F_2)$ is also of finite length for $0<i<n+3$. But this implies that $\cal F_2$ is locally free. \end{pf} \begin{proposition} \label{cond:a} The locally free resolution \rom{(\ref{free:resol})} satisfies condition \rom(a\rom) of Proposition \ref{conditions}. \end{proposition} \begin{pf} We write $\cal L=\omega_{\Bbb{P}^{n+3}}(-l)$. We have to show that if there is a commutative diagram \begin{equation} \label{self:dual} \begin{CD} 0 @>>> \cal{L} @>{d_3}>> \cal{F}_2 @>{d_2}>> \cal{F}_1 @>{d_1}>> \cal O_{\Bbb{P}^{n+3}}\\ && @| @VV{s_2}V @VV{s_1}V @| \\ 0 @>>> \cal{L} @>{d_1\spcheck}>> \cal{F}_1\spcheck\otimes\cal{L} @>{-d_2\spcheck}>> \cal{F}_2\spcheck\otimes\cal{L} @>{d_3\spcheck}>> \cal O_{\Bbb{P}^{n+3}} \end{CD} \end{equation} such that the vertical maps extend the identity on $\cal O_X$, then $s_1$ and $s_2$ are isomorphisms. By exactness, the image of $d_3\spcheck$ is $\cal I_X$. We will show that $s_1$ is an isomorphism by applying Lemma \ref{second:isom} to the composition \[ \cal F_1 @>{s_1}>> \cal F_2\spcheck\otimes \cal L \twoheadrightarrow \cal I_X. \] Note that this composition is exactly the surjection $\gamma: \cal F_2 \twoheadrightarrow \cal I_X$ of (\ref{gamma}). Hence the composition \[ H^i_*(\cal F_1) @>{H^i_*(s_1)}>> H^i_*(\cal F_2\spcheck\otimes\cal L) @>>> H^i_*(\cal I_X) \] is injective for $0<i<n+3$. A fortiori, $H^i_*(s_1)$ is also injective for $0<i<n+3$. However, by Serre duality $H^i_*(\cal F_2\spcheck\otimes \cal L) \cong \left( H^{n+3-i}_*(\cal F_2) \right)^*(l)$ for all $i$. So by Lemma \ref{duality}, we have $H^i_*(\cal F_2\spcheck\otimes \cal L) \cong H^i_*(\cal F_1)$ for $0<i<n+3$. Hence for each $0<i<n+3$, the morphism $H^i_*(s_1)$ is an injection of modules of the same finite length. Hence $H^i_*(s_1)$ is an isomorphism for $0<i<n+3$. Thus condition (i) of Lemma \ref{second:isom} holds. Condition (ii) of Lemma \ref{second:isom} holds because of the method of construction of $\cal F_1$ and of the surjection $\gamma$ in (\ref{gamma}). Finally exactness in the resolution implies that $\cal F_1$ and $\cal F_2$ have the same rank. Hence $\cal F_1$ and $\cal F_2\spcheck\otimes\cal L$ also have the same rank, which is condition (iii) of Lemma \ref{second:isom}. Hence all three conditions of Lemma \ref{second:isom} hold, and we may conclude that $s_1$ is an isomorphism. The map $s_2$ must now also be an isomorphism by the five-lemma. This completes the proof of the proposition. \end{pf} \section{The Differential Graded Algebra Structure} In this section we finish the proof of Theorem \ref{main} by showing that the locally free resolution (\ref{free:resol}) defined in the previous section satisfies condition (b) of Proposition \ref{conditions}. That is to say, we show that the locally free resolution (\ref{free:resol}) admits a commutative, associative differential graded algebra structure. Throughout this section we assume that the characteristic is not $2$. We recall what needs to be proven. In the previous section we defined a locally free resolution (\ref{free:resol}) of $\cal O_X$ \[ 0 @>>> \cal L @>{d_3}>> \cal F_2 @>{d_2}>> \cal F_1 @>{d_1}>> \cal O_{\Bbb{P}^{n+3}} @>>> \cal O_X. \] Let $\cal K=\ker(d_1)$. We then had a morphism $\psi: \Lambda^2\cal F_1 @>>> \cal K$ defined by $\psi(a\wedge b) = d_1(a)b-d_1(b)a$. We also have a long exact sequence \[ \cdots @>>> \operatorname{Hom}(\Lambda^2\cal F_1,\cal F_2) @>>> \operatorname{Hom}(\Lambda^2\cal F_1,\cal K) @>>> \operatorname{Ext}^1(\Lambda^2\cal F_1,\cal L) @>>> \cdots. \] According to diagram (\ref{liftdiag}), the problem is to lift $\psi\in \operatorname{Hom}(\Lambda^2\cal F_1,\cal K)$ to a $\phi\in\operatorname{Hom}(\Lambda^2\cal F_1,\cal F_2)$. The obstruction to doing this is simply the image of $\psi$ in \[ \operatorname{Ext}^1(\Lambda^2\cal F_1,\cal L) = H^{n+2}(\Lambda^2\cal F_1(l))^*. \] Our first goal will therefore be to compute $H^{n+2}(\Lambda^2\cal F_1(l))$. We begin by considering a complex of locally free sheaves on $\Bbb{P}^N$. \begin{equation} \label{G} \cal G^*: \qquad 0 @>>> \cal G^0 @>>> \cal G^1 @>>> \cdots @>>> \cal G^r @>>> 0. \end{equation} There is an involution \begin{align*} T: \quad \cal G^*\otimes\cal G^* \ & @>>> \ \cal G^*\otimes\cal G^*\\ a\otimes b \ & \mapsto \ (-1)^{(\deg a)(\deg b)}b\otimes a \end{align*} interchanging the factors of $\cal G^*\otimes\cal G^*$. Since the characteristic is not $2$, the complex $\cal G^*\otimes\cal G^*$ splits into a direct sum of subcomplexes on which $T$ acts as multiplication by $\pm 1$, viz.\ $\cal G^*\otimes\cal G^* = S_2(\cal G^*) \oplus \Lambda^2(\cal G^*)$. The complex $\Lambda^2(\cal G^*)$ is of the form \begin{equation} \label{Lambda:length} \Lambda^2(\cal G^*):\qquad 0 @>>> \cal H^0 @>>> \cal H^1 @>>> \cdots @>>> \cal H^{2r} @>>> 0 \end{equation} where (cf.\ \cite{BE} p.\ 452) \begin{equation} \label{Lambda:formula} \cal H^i \cong \bigoplus_{q<i/2} \left( \cal G^q\otimes \cal G^{i-q} \right) \oplus \begin{cases} 0 &\text{if $i$ is odd,}\\ \Lambda^2(\cal G^{i/2}) &\text{if }i\equiv 0\pmod 4,\\ S_2(\cal G^{i/2}) &\text{if }i\equiv 2\pmod 4. \end{cases} \end{equation} \begin{lemma} \label{Lambda} Suppose $\cal G^*$ is a complex of locally free sheaves on $\Bbb{P}^N$ as in \rom{(\ref{G})} which is exact except in degree $0$. Let $\cal E = H^0(\cal G^*)$. Then $\Lambda^2(\cal G^*)$ is an exact sequence of locally free sheaves which is exact except in degree $0$, and $H^0(\Lambda^2(\cal G^*)) = \Lambda^2\cal E$. \end{lemma} \begin{pf} The standard spectral sequences of the double complex $\cal G^*\otimes\cal G^*$ degenerate to show that the simple complex $\cal G^*\otimes \cal G^*$ is exact except in degree $0$, and $H^0(\cal G^*\otimes\cal G^*) = \cal E\otimes\cal E$. Thus the augmented complex $0 @>>> \cal E\otimes\cal E @>>> \cal G^*\otimes\cal G^*$ is exact, and consequently its direct factor $0 @>>> \Lambda^2\cal E @>>> \Lambda^2(\cal G^*)$ is also exact. \end{pf} \begin{lemma} \label{max:cohom} Suppose $\cal E$ is a locally free sheaf on $\Bbb{P}^N$. Let $r<N/2$ be an integer. Suppose that $H^i_*(\cal E)=0$ for $r<i<N$. Then \rom(a\rom) $H^i_*(\Lambda^2\cal E)=0$ for $2r<i<N$, \rom(b\rom) $H^{2r}_*(\Lambda^2\cal E)\cong S_2 (H^r_*(\cal E))$ if $r$ is odd, and $H^{2r}_*(\Lambda^2\cal E)\cong \Lambda^2 (H^r_*(\cal E))$ if $r$ is even. \rom(c\rom) If $H^r(\cal E(t))=0$ for $t<q$ for some integer $q$, then $H^{2r}(\Lambda^2\cal E(t))=0$ for $t<2q$, while $H^{2r}(\Lambda^2\cal E(2q)) \cong S_2 (H^r(\cal E(q)))$ if $r$ is odd, and $H^{2r} (\Lambda^2 \cal E(2q)) \cong \Lambda^2 (H^r(\cal E(q)))$ if $r$ is even. \end{lemma} \begin{pf} By Lemma \ref{tauresol}(b), the minimal projective resolution of $P^*$ of the truncation $\tau_{<N}\bold R\Gamma_*(\cal E)$ is a complex of free graded $S$-modules such that $P^i=0$ unless $0\leq i\leq N-1$. Indeed, since $H^i(\tau_{<N}\bold R\Gamma_*(\cal E))=0$ for all $i>r$, Lemma \ref{bounds} indicates that $P^i=0$ unless $0\leq i\leq r$, i.e.\ $P^*$ is of the form \[ P^*: \qquad 0 @>>> P^0 @>>> \cdots @>>> P^{r-1} @>>> P^r @>>> 0 . \] We now consider the complex of free graded $S$-modules \[ \Lambda^2(P^*):\qquad 0 @>>> \Lambda^2 P^0 @>>> \cdots @>>> P^{r-1} \otimes P^r @>>> T_2 (P^r) @>>> 0. \] where $T_2(P^r) = \Lambda^2(P^r)$ if $r$ is even, and $T_2(P^r) = S_2(P^r)$ if $r$ is odd (cf.\ \eqref{Lambda:length} and \eqref{Lambda:formula}). According to Lemma \ref{tauresol}, the complex of sheaves $\widetilde P^*$ associated to $P^*$ is exact except in degree $0$ where the homology is $\cal E$. So Lemma \ref{Lambda} implies that the complex of sheaves $\Lambda^2(\widetilde P^*)$ is also exact except in degree $0$ where the homology is $\Lambda^2\cal E$. The complex $\Lambda^2(P^*)$ of graded $S$-modules therefore has homology of finite length except in degree $0$. Moreover, the complex $\Lambda^2(P^*)$ vanishes except in degrees between $0$ and $2r<N$, and the coefficients of its differentials lie in $\frak m$ because it those of $P^*$ and therefore $P^*\otimes P^*$ do. It now follows from Lemma \ref{tauresol}(a) that $\Lambda^2(P^*)$ is the minimal projective resolution of $\tau_{<N}\bold R\Gamma_*(\Lambda^2 \cal E)$. Therefore $H^i_*(\Lambda^2\cal E)\cong H^i(\Lambda^2(P^*))$ for all $i<N$. In particular, since $\Lambda^2(P^*)$ is concentrated in degrees between $0$ and $2r$ by \eqref{Lambda:length}, we see that $H^i_*(\Lambda^2\cal E)=0$ for $2r<i<N$. This is part (a) of the lemma. For (b) note that $H^r_*(\cal E)$ and $H^{2r}_*(\Lambda^2\cal E)$ has respective presentations \[ \begin{CD} P^{r-1} & \:\overset{\delta}{\longrightarrow}\: & P^r & \:\rightarrow\: & H^r_*(\cal E) & \:\rightarrow\: & 0 ,\\ P^{r-1}\otimes P^r & \:\overset{\delta_1}{\longrightarrow}\: & T_2(P^r) & \:\rightarrow\: & H^{2r}_*(\Lambda^2\cal E) & \:\rightarrow\: & 0, \end{CD} \] where $\delta_1(e\otimes f) = \delta(e)f\in T_2 (P^r)$. But since the presentation of $T_2(H^r_*(\cal E))$ is of exactly this form, we see that $H^{2r}_*(\Lambda^2\cal E) \cong T_2(H^r_*(\cal E))$, as asserted by the lemma. For (c) write $H=H^r(\cal E(q))$. The hypothesis that $H^r(\cal E(t))=0$ for $t<q$ implies that \( P^r = \left( H\otimes_k S(-q)\right) \oplus F \) with $F=\bigoplus S(-n_i)$ for some $n_i>q$. Then \( T_2(P^r) = \left( T_2 H \otimes_k S(-2q) \right) \oplus G \) with $G = \left( H\otimes_k F(-q) \right) \oplus T_2 F = \bigoplus S(-m_j)$ for some $m_j>2q$. Since the presentation of $H^{2r}_*(\Lambda^2 \cal E)$ given above has the property that no direct factor of $P^{r-1}\otimes P^r$ is mapped surjectively onto a factor of $T_2(P^r)$, it now follows that $H^{2r}(\Lambda^2\cal E(t))=0$ for $t<2q$, and $H^{2r}(\Lambda^2\cal E(2q)) \cong T_2 H$. \end{pf} \begin{corollary} \label{Lambda:cohom} Let $n$, $l$, and $X\subset \Bbb{P}^{n+3}$ be as in Theorem \ref{main}. Suppose that $U\subset H^{n/2}(\cal O_X(l/2))$ is the maximal isotropic subspace defined in \eqref{w}, and that $\cal F_1$ is the locally free sheaf defined in \eqref{gamma}. Then \[ H^{n+2}(\Lambda^2\cal F_1(l)) \cong \begin{cases} 0 &\text{if $n$ or $l$ is odd,}\\ S_2 U &\text{if $l$ is even, and }n\equiv 0\pmod 4,\\ \Lambda^2 U &\text{if $l$ is even, and }n\equiv 2\pmod 4. \end{cases} \] \end{corollary} \begin{pf} If $n$ is odd, then $H^i_*(\cal F_1)=0$ for $(n+1)/2 < i < n+3$. So Lemma \ref{max:cohom}(a) applies with $r=(n+1)/2$. Therefore $H^i_*(\Lambda^2\cal F_1) = 0$ for $n+1<i<n+3$, i.e.\ $H^{n+2}(\Lambda^2\cal F_1(t)) = 0$ for all $t$. If $n$ is even but $l$ is odd, then Lemma \ref{max:cohom}(c) applies with $r=(n+2)/2$ and $q=(l+1)/2$. Then $H^{n+2}(\Lambda^2\cal F_1(t))=0$ for all $t < l+1$. If $l$ and $n$ are even, then Lemma \ref{max:cohom}(c) applies with $r=(n+2)/2$ and $q=l/2$. Since $H^{(n+2)/2}(\cal F_1(l/2))\cong U$, it follows that $H^{n+2}(\Lambda^2\cal F_1(l))\cong \Lambda^2 U$ if $r$ is even, and $H^{n+2}(\Lambda^2\cal F_1(l))\cong S_2 U$ if $r$ is odd. The corollary follows. \end{pf} \begin{lemma} \label{cond:b} If $\cal F_1$ is the locally free sheaf defined in \eqref{gamma}, then the image of the map $\psi$ of \eqref{liftdiag} in $\operatorname{Ext}^1(\Lambda^2\cal F_1,\cal L) \cong H^{n+2}(\Lambda^2\cal F_1(l))^*$ vanishes. \end{lemma} \begin{pf} If $n$ or $l$ is odd, then $H^{n+2}(\Lambda^2\cal F_1(l)) = 0$ according to Corollary \ref{Lambda:cohom}, so the image of $\psi$ is evidently zero. If $n$ and $l$ are even, then we claim that the image of $\psi$ in $H^{n+1}(\Lambda^2\cal F_1(l))^*$ is the map \[ \bigl\{ S_2 U \text{ or } \Lambda^2 U \bigr\} @>>> k \] which is the restriction to $U$ of the pairing $H^{n/2}(\cal O_X(l/2)) \times H^{n/2}(\cal O_X(l/2)) @>>> k$ of \eqref{pairing}. Since $U$ was chosen isotropic, this map vanishes. In order to prove the claim, we consider the diagonal $i: \Bbb{P}^{n+3} = \Delta\subset\Bbb{P}^{n+3}\times\Bbb{P}^{n+3}$. Then there is a natural inclusion $i(X)\subset X\times X$ which corresponds to a restriction map \begin{equation} \label{rest:diag} \cal O_{X\times X} @>>> i_* \cal O_X. \end{equation} This map is essentially the multiplication $\cal O_X\otimes\cal O_X @>>> \cal O_X$. In any case applying $\bold R\Gamma_*$ to \eqref{rest:diag} gives the cup product map \begin{equation} \label{cup:product} \bold R\Gamma_*(\cal O_X)\otimes_k \bold R\Gamma_*(\cal O_X) @>>> \bold R\Gamma_*(\cal O_X). \end{equation} Now consider the ``resolution'' of $\cal O_X$ given in \eqref{partial} \[ \cal K^*:\qquad 0 @>>> \cal K @>>> \cal F_1 @>{d_1}>> \cal O_{\Bbb{P}^{n+3}} @>>> 0. \] The complex $\cal K^*$ is quasi-isomorphic to $\cal O_X$. Hence the restriction to the diagonal map \eqref{rest:diag} corresponds to a morphism in the derived category \[ p_1^*\cal K^*\otimes p_2^*\cal K^* @>>> i_*\cal K^*. \] In fact this morphism in the derived category is represented by an actual map of complexes of sheaves \begin{equation} \label{FFK} \begin{CD} \dotsb & \:\rightarrow\: & p_1^*\cal K \oplus (p_1^*\cal F_1 \otimes p_2^*\cal F_1) \oplus p_2^*\cal K & \:\rightarrow\: & p_1^*\cal F_1 \oplus p_2^*\cal F_1 & \:\rightarrow\: & \cal O_{\Bbb P\times\Bbb P} & \:\rightarrow\: & 0\\ && @VVV @VVV @VVV \\ 0 & \:\rightarrow\: & i_*\cal K & \:\rightarrow\: & i_*\cal F_1 & \:\rightarrow\: & i_*\cal O_{\Bbb P} & \:\rightarrow\: & 0 \end{CD} \end{equation} All the vertical maps are straightforward restrictions to the diagonal except for the component $p_1^*\cal F_1 \otimes p_2^*\cal F_1 @>>> i_*(\cal K)$ which is defined (like $\psi$ of \eqref{liftdiag}) by noting that the composition \begin{equation} \label{FFK2} \begin{CD} p_1^*\cal F_1 \otimes p_2^*\cal F_1 & \:\rightarrow\: p_1^*\cal F_1 \oplus p_2^*\cal F_1 \:\rightarrow\: & i_* \cal F_1 \\ p_1^*(a)\otimes p_2^*(b) & \mapsto & i_*\left(d_1(a)b-d_1(b)a\right) \end{CD} \end{equation} is contained in the kernel of $i_*\cal F_1 @>>> i_*\cal O_{\Bbb P}$. Now since $\cal K^*$ is quasi-isomorphic to $\cal O_X$, if we apply $\bold R\Gamma_*$ to \eqref{FFK} we get a morphism $\bold R\Gamma_*(p_1^*\cal K\otimes p_2^*\cal K) @>>> \bold R\Gamma_*(i_*\cal K^*)$ in $D^b_{\grssmod}$ which is quasi-isomorphic to \eqref{cup:product}. In particular, the maps of hypercohomology are quasi-isomorphic to the cup product \[ H^n_*(p_1^*\cal O_X\otimes p_2^*\cal O_X) \cong \bigoplus_i H^i_*(\cal O_X)\otimes_k H^{n-i}_*(\cal O_X) @>>> H^n_*(\cal O_X). \] The hypercohomology $H^n_*(\cal K^*)\cong H^n_*(\cal O_X)$ is of course the same as the $H^n$ of the total complex of the double complex \[ 0 @>>> \bold R\Gamma_*(\cal K) @>>> \bold R\Gamma_*(\cal F_1) @>>> \bold R\Gamma_*(\cal O_{\Bbb P}) @>>> 0. \] According to the calculations at the beginning of the previous section, this $H^n_*$ is all attributable to $\cal K$, i.e.\ the truncation $\cal K^* @>>> \cal K[2]$ induces an isomorphism $H^n_*(\cal O_X) \cong H^n_*(\cal K^*) \cong H^{n+2}_*(\cal K)$. Similarly, the hypercohomology $H^n_*(p_1^*\cal K^*\otimes p_2^*\cal K^*)$ is the same as the $H^n$ of the total complex of $\bold R\Gamma_*$ of the first row of \eqref{FFK}. The submodule $W\otimes_k W \subset H^n_*(p_1^*\cal O_X\otimes p_2^*\cal O_X)$ is attributable as the $H^{n+2}_*$ of the factor $p_1^*\cal F_1\otimes p_2^*\cal F_1$ in the first row of \eqref{FFK}. Therefore $H^{n+2}_*$ of the vertical map $p_1^*\cal F_1 \otimes p_2^*\cal F_1 @>>> \cal K$ is simply the cup product map $W\otimes_k W @>>> H^n_*(\cal O_X)$. Now the fact that $i_*(\cal K)$ is supported on $\Delta$, plus the symmetry of the product map imply that the vertical map of \eqref{FFK} factors as \begin{equation} \label{factorization} \begin{CD} p_1^*\cal F_1\otimes p_2^*\cal F_1 & \:\rightarrow\: & i_*(\cal F_1\otimes \cal F_1) & \:\rightarrow\: & i_*(\Lambda^2\cal F_1) & \:\overset{i_*(\psi)}{\longrightarrow}\: & i_*(\cal K).\\ p_1^*(a)\otimes p_2^*(b) & \mapsto & i_*(a\otimes b) & \mapsto & i_*(a\wedge b) & \mapsto & d_1(a)b-d_1(b)a \end{CD} \end{equation} We wish to calculate $H^{n+2}_*$ of the above morphisms. Let \[ P^*:\qquad 0 @>>> P^0 @>>> \dotsb @>>> P^{(n+2)/2} @>>> 0 \] be a minimal projective resolution of $\tau_{<n+3}\bold R\Gamma_*(\cal F_1)$ (cf.\ Lemma \ref{tauresol}). Then if one applies $\tau_{<n+3}\bold R\Gamma_*$ to the first two morphisms of \eqref{factorization}, one gets the natural maps \[ P^*\otimes_k P^* @>>> P^*\otimes_S P^* @>>> \Lambda^2(P^*) \] (cf.\ the proof of Lemma \ref{max:cohom}). All three complexes are supported in degrees between $0$ and $n+2$, and applying $H^{n+2}$ gives surjections \[ W\otimes_k W \twoheadrightarrow W\otimes_S W \twoheadrightarrow \bigl\{ S_2 W \text{ or } \Lambda^2 W \bigr\}. \] It therefore follows that $H^{n+2}_*(\psi): H^{n+2}_*(\Lambda^2\cal F_1) @>>> H^{n+2}_*(\cal K)$ is isomorphic to the cup product map $\bigl\{ S_2 U \text{ or } \Lambda^2 U \bigr\} @>>> H^n_*(\cal O_X)$. In particular, in degree $l$ the morphism $H^{n+2}(\cal F_1(l)) @>>> H^{n+2}(\cal K(l))$ is the same as $\bigl\{ S_2 U \text{ or } \Lambda^2 U \bigr\} @>>> H^n(\cal O_X(l))$. We now have to consider the extension of \eqref{extension} \[ 0 @>>> \omega_{\Bbb P}(-l) @>>> \cal F_2 @>>> \cal K @>>> 0. \] (Recall $\cal L = \omega_{\Bbb P}(-l)$.) In the associated long exact sequence of cohomology \[ \dotsb @>>> H^{n+2}(\cal F_2(l)) @>>> H^{n+2}(\cal K(l)) @>\operatorname{tr}>> H^{n+3}(\omega_{\Bbb{P}^{n+3}}) \cong k, \] the differential is the element of $H^{n+2}(\cal K(l))^* \cong \operatorname{Ext}^1(\cal K(l),\omega_{\Bbb{P}^{n+3}})$ corresponding to the extension class. So by construction the differential is the trace map $\operatorname{tr}\in H^{n+2}(\cal K(l))^*\cong H^n(\cal O_X(l))^*$ which corresponds under Serre duality to $1\in H^0(\cal O_X)$. Now the image of $\psi\in \operatorname{Ext}^1(\Lambda^2\cal F_1(l),\omega_{\Bbb P}) \cong H^{n+2}(\Lambda^2\cal F_1(l))^*$ is exactly the composition \[ H^{n+2}(\Lambda^2\cal F_1(l)) @>{H^{n+2}(\psi)}>> H^{n+2}(\cal K(l)) @>\operatorname{tr}>> H^{n+3}(\omega_{\Bbb{P}^{n+3}}) \cong k. \] By our previous calculations, this is the composition of the cup product map $\bigl\{ S_2 U \text{ or } \Lambda^2 U \bigr\} @>>> H^n(\cal O_X(l))$ with the trace map $H^n(\cal O_X(l)) @>>> k$. Therefore this composition is the restriction to $S_2 U$ or $\Lambda^2 U$ of the Serre duality pairing $H^{n/2}(\cal O_X(l/2))\otimes H^{n/2}(\cal O_X(l/2)) @>>> k$. This is what was claimed at the beginning of the proof of the lemma. Since $U$ was chosen isotropic, this composition vanishes, i.e.\ the image of $\psi$ in $\operatorname{Ext}^1(\cal K,\cal L)$ vanishes. \end{pf} \begin{pf*}{Proof of Theorem \ref{main}} According to Proposition \ref{conditions}, in order to prove Theorem \ref{main} it suffices to find a locally free resolution \[ 0 @>>> \cal L @>>> \cal F_2 @>>> \cal F_1 @>>> \cal O_{\Bbb{P}^{n+3}} @>>> \cal O_X \] which satisfies two conditions. But the locally free resolution defined in \eqref{free:resol} was shown to satisfy the first of these conditions was shown in Proposition \ref{cond:a}. Moreover, this resolution was just shown to satisfy the second condition in Lemma \ref{cond:b}. Hence Theorem \ref{main} holds. \end{pf*} \section{Characteristic $2$ Computations} In the introduction, we asserted that Theorem \ref{main} also holds in characteristic $2$ provided the phrase ``$n\equiv 0\pmod 4$'' in the parity condition is replaced by the phrase ``$n$ is even.'' In this section we justify that assertion by proving analogues of Lemmas \ref{Lambda} and \ref{max:cohom} and Corollary \ref{Lambda:cohom} in characteristic $2$. These were the only steps in the proof of Theorem \ref{main} where we used the assumption that the characteristic is not $2$. Throughout this section we assume that the characteristic is $2$. We recall certain simple facts from modular representation theory. Let $R$ be a commutative algebra over a field of characteristic $2$, and let $V$ be a free $R$-module. Let $t\in\operatorname{End}(V\otimes V)$ be the endomorphism $t(a\otimes b) = a\otimes b - b\otimes a$. Set $D_2 V =\ker(t)$, and $\Lambda^2 V =\operatorname{im}(t)$, and $S_2 V = \operatorname{coker}(t)$. Since $t^2=0$ in characteristic $2$, there are inclusions \[ 0 \subset \Lambda^2 V \subset D_2 V \subset V\otimes V \] and corresponding surjection of quotients of $V\otimes V$ \[ V\otimes V \twoheadrightarrow S_2 V \twoheadrightarrow \Lambda^2 V @>>> 0. \] The subquotient $D_2 V /\Lambda^2 V$ is $F(V)$, the Frobenius pullback of $V$. It is a free module of the same rank as $V$. This $F(V)$ is also the kernel of the surjection $S_2 V \twoheadrightarrow \Lambda^2 V$. Note that the natural map from $S_2 V $ to $V\otimes V$ given by $xy \mapsto x\otimes y + y\otimes x$ is not injective because any $x^2 \mapsto 0$. The map is the composition $S_2 V \twoheadrightarrow \Lambda^2 V \hookrightarrow V\otimes V$ with kernel $F(V)$. The operations $D_2$, $\Lambda^2$, $S_2$, and $F$ are all functorial. Therefore we may define $D_2\cal E$, $\Lambda^2\cal E$, $S_2\cal E$, and $F(\cal E)$ for any locally free sheaf $\cal E$ on any scheme $X$ over a field of characteristic $2$. In some ways $F$ has better properties than the others. If $M = (m_{ij}): R^n @>>> R^m$ is a morphism of free $R$-modules, then $F(M) = (m_{ij}^2): R^n @>>> R^m$. One may use this formula together with the Buchsbaum-Eisenbud exactness criterion \cite{BE:exact} to show that $F$ of an exact sequence of locally free sheaves is exact. As a result of this we get the following lemma. \begin{lemma} \label{frob} Let $\cal E$ be a locally free sheaf on $\Bbb{P}^N$. If for some integer $r$ one has $H^i_*(\cal E)=0$ for $r<i<N$, then $H^i_*(F(\cal E))=0$ for $r<i<N$ also. \end{lemma} \begin{pf} Let $P^*$ be the minimal projective resolution of $\tau_{<N}\bold R\Gamma_*(\cal E)$. By Lemmas \ref{tauresol} and \ref{bounds}, $P^*$ has the form \[ P^*:\qquad 0 @>>> P^0 @>>> P^1 @>>> \dotsb @>>> P^r @>>> 0. \] Moreover, $P^*$ is exact except in degree $0$ away from the irrelevant ideal $\frak m\subset S$, and $H^0(\widetilde P^*)=\cal E$. The functoriality and exactness of $F$ now imply that \[ F(P^*):\qquad 0 @>>> F(P^0) @>>> F(P^1) @>>> \dotsb @>>> F(P^r) @>>> 0 \] is exact except in degree in degree $0$ away from the irrelevant ideal, and has $H^0(F(\widetilde P^*))=F(\cal E)$. Applying Lemma \ref{tauresol} again, we conclude that $F(P^*)$ is the minimal projective resolution of $\tau_{<N}\bold R\Gamma_*(F(\cal E))$. So if $r<i<N$, then $H^i_*(F(\cal E)) = H^i(F(P^*))=0$ since $F(P^*)$ vanishes in degrees greater than $r$. \end{pf} \begin{corollary} \label{l:s} Let $\cal E$ be a locally free sheaf on $\Bbb{P}^N$ over a field of characteristic $2$ such that $H^{N-1}_*(\Lambda^2\cal E)=0$. Suppose $r<N$ is an integer such that $H^i_*(\cal E)=0$ for $r<i<N$. Then $H^i_*(S_2\cal E) \cong H^i_*(\Lambda^2\cal E)$ for $r<i<N$. \end{corollary} \begin{pf} We consider the exact sequence $0 @>>> F(\cal E) @>>> S_2\cal E @>>> \Lambda^2\cal E @>>> 0$ and the associated long exact sequence \[ \dotsb @>>> H^i_*(F(\cal E)) @>>> H^i_*(S_2\cal E) @>>> H^i_*(\Lambda^2\cal E) @>>> H^{i+1}_*(F(\cal E)) @>>> \dotsb \] The hypothesis $H^i_*(\cal E)=0$ for $r<i<N$ implies also $H^i_*(F(\cal E))=0$ for $r<i<N$ by Lemma \ref{frob}. Hence the long exact sequence implies that $H^i_*(S_2\cal E)\cong H^i_*(\Lambda^2\cal E)$ for $r<i<N-1$ and that $H^{N-1}_*(S_2\cal E) \hookrightarrow H^{N-1}_*(\Lambda^2\cal E)$ is injective. But by hypothesis $H^{N-1}_*(\Lambda^2\cal E)=0$, so $H^{N-1}_*(S_2\cal E)=0$ as well. \end{pf} We now prove the analogue of Lemma \ref{max:cohom}. \begin{lemma} \label{char:two} Suppose $\cal E$ is a locally free sheaf on $\Bbb{P}^N$ over a field of characteristic $2$. Let $0 < r <N/2$ be an integer such that $H^i_*(\cal E)=0$ for $r<i<N$. Then \rom(a\rom) $H^i_*(\Lambda^2\cal E)=0$ for $2r<i<N$, \rom(b\rom) $H^{2r}_*(\Lambda^2\cal E)\cong S_2 (H^r_*(\cal E))$. \rom(c\rom) If $H^r(\cal E(t))=0$ for $t<q$ for some integer $q$, then $H^{2r}(\Lambda^2\cal E(t))=0$ for $t<2q$, while $H^{2r}(\Lambda^2\cal E(2q)) \cong S_2 (H^r(\cal E(q)))$. \end{lemma} \begin{pf} Let $P^*$ be the minimal projective resolution of $\tau_{<N}\bold R\Gamma_*(\cal E)$ \[ P^*:\qquad 0 @>>> P^0 @>{\delta^0}>> P^1 @>{\delta^1}>> P^2 @>>> \dotsb @>>> P^r @>>> 0 \] Let $\cal P = \widetilde P^0$, and let $\cal F = \widetilde{\ker( \delta^1)}$. Then we have an exact sequence $0 @>>> \cal E @>>> \cal P @>>> \cal F @>>> 0$ such that $\cal P$ is a direct sum of line bundles, $H^i_*(\cal F) = H^{i+1}_*(\cal E)$ for $0<i<N-1$, and $H^{N-1}_*(\cal F)=0$. It is easy to see that there is a natural exact complex \begin{equation} \label{lambda:s} 0 @>>> \Lambda^2 \cal E @>>> \Lambda^2 \cal P @>>> \cal P\otimes \cal F @>>> S_2\cal F @>>> 0. \end{equation} We now prove parts (a) and (b) of the lemma by induction on $r$. If $r=1$, then $\cal F = \widetilde P^1$ is a direct sum of line bundles, and the complex \eqref{lambda:s} is just the augmented complex \[ 0 @>>> \Lambda^2 \cal E @>>> \Lambda^2(\widetilde P^*) \] which is still exact in this case. So we may conclude just as in Lemma \ref{max:cohom} that $H^i_*(\Lambda^2\cal E)=0$ for $2<i<N$, and that $H^2_*(\Lambda^2\cal E) = S_2(H^1_*(\cal E))$. If $r>1$, then $H^i_*(\cal F)=0$ for $r-1<i<N$. So by induction $H^i_*(\Lambda^2\cal F)=0$ for $2r-2<i<N$, and also $H^{2r-2}_* (\Lambda^2\cal F)) \cong S_2(H^{r-1}_*(\cal F)) \cong S_2(H^r_*(\cal E))$. It now follows from Corollary \ref{l:s} that $H^i_*(S_2\cal F)\cong H^i_*(\Lambda^2\cal F)=0$ for $r-1 < i < N$. So in particular $H^i_*(S_2\cal F)=0$ for $2r-2<i<N$ and that $H^{2r-2}_*(S_2\cal F)\cong S_2(H^r_*(\cal E))$. Now since $\cal P$ is a direct sum of line bundles, we have $H^i_*(\Lambda^2\cal P)=0$ for $0<i<N$, and $H^i_*(\cal P\otimes\cal F)=0$ for $r-1<i<N$. So if we break up \eqref{lambda:s} into short exact sequences and take its graded cohomology, we can deduce that $H^i_*(\Lambda^2\cal E)) \cong H^{i-2}_* (S_2\cal F)$ for $r+1<i<N$. Since $r+1<2r$, this gives parts (a) and (b) of the lemma. Part (c) of the lemma follows from part (b) by the same argument as in Lemma \ref{max:cohom}. \end{pf} We have the following corollary in analogy with Corollary \ref{Lambda:cohom}. \begin{corollary} \label{S2U} Let $n$, $l$, and $X\subset \Bbb{P}^{n+3}_k$ be as in Theorem \ref{main} with $k$ a field of characteristic $2$. Suppose that $U\subset H^{n/2}(\cal O_X(l/2))$ is the maximal isotropic subspace defined in \eqref{w}, and that $\cal F_1$ is the locally free sheaf defined in \eqref{gamma}. Then \[ H^{n+2}(\Lambda^2\cal F_1(l)) \cong \begin{cases} 0 &\text{if $n$ or $l$ is odd,}\\ S_2 U &\text{if $n$ and $l$ are even.} \end{cases} \] \end{corollary} The proof of Lemma \ref{cond:b} in characteristic $2$ is essentially the same as in the previous section, only with Lemma \ref{char:two} and Corollary \ref{S2U} replacing their analogues, and with $T_2 = S_2$ always. Hence Theorem \ref{main} also holds in characteristic $2$ as long as one treats all even $n$ the same. \section{The Local Version of the Main Theorem} In this section we consider Theorem \ref{RLR}, the local version of our main result. We state a variant version which is clearly a local analogue of Theorem \ref{main} with an identical proof, and then show that this variant version is equivalent to Theorem \ref{RLR}. Let $(R,\frak m,k)$ be a regular local ring, and let $U = \operatorname{Spec}(R) - \{\frak m\}$ be the punctured spectrum of $R$. We say that a closed subscheme $Y\subset U$ of pure codimension $3$ is {\em Pfaffian} if $O_X$ has a locally free resolution on $U$ \[ 0 @>>> \cal{O}_U @>h>> \cal{E}\spcheck @>f>> \cal{E} @>g>> \cal{O}_U @>>> \cal O_X \] where $\cal E$ is a locally free $\cal O_U$-module of odd rank $2p+1$, $f$ is skew-symmetric, and $g$ and $h=g\spcheck$ are given locally by the Pfaffians of order $2p$ of $f$. The following theorem is the obvious local analogue of Theorem \ref{main}. \begin{theorem} \label{punc:spec} Let $(R,\frak m,k)$ be a regular local ring of dimension $n+4>4$ with residue field not of characteristic $2$, and let $U=\operatorname{Spec}(R)-\{\frak m\}$. Let $X\subset U$ be a closed subscheme of pure codimension $3$. Then $X$ is Pfaffian if and only if the following three conditions hold: \rom(a\rom) $X$ is locally Gorenstein, \rom(b\rom) $\omega_X\cong \cal O_X$, and \rom(c\rom) if $n\equiv 0\pmod 4$, then $H^{n/2}(\cal O_X)$ is of even length. \end{theorem} Theorem \ref{punc:spec} may be proven in exactly the same manner as Theorem \ref{main}. All results concerning the Buchsbaum-Eisenbud proof, the Horrocks correspondence, Serre/local duality, the cohomology of $H^{n+2}(\Lambda^2\cal F_1)$ work identically for graded modules over polynomial rings over $k$ and for modules over regular local $k$-algebras. There is only one point which is in any way more subtle in the local case. Namely, if $n$ is even, then one has a Matlis duality pairing of $R$-modules of even finite length \[ H^{n/2}(\cal O_X) \times H^{n/2}(\cal O_X) @>>> k. \] This pairing is perfect in the sense that for any submodule $M\subset H^{n/2}(\cal O_X)$ one has \[ \operatorname{length}(M)+\operatorname{length}(M^\perp) = \operatorname{length}(H^{n/2}(\cal O_X)). \] In order to be able to define $C^*_X$ and $\cal F_1$ as in \eqref{gamma} one must choose an isotropic submodule $W$ of length equal to half that of $H^{n/2}(\cal O_X)$. But it is not difficult to show that this is possible. We now compare Theorems \ref{RLR} and \ref{punc:spec}. First of all, $E=\Gamma(\cal E)$ gives a bijective correspondence between locally free sheaves $\cal E$ on $U$ and reflexive $R$-modules $E$ such that $E_{\frak p}$ is a free $R_{\frak p}$-module for all prime ideals $\frak p\neq \frak m$. There is also bijective correspondence betweenclosed subschemes $X\subset U$ of pure codimension $3$ and unmixed ideals $I\subset R$ of height $3$ given by $I=\Gamma(\cal I_X)$. Hence an ideal $I$ is Pfaffian in the sense of Theorem \ref{RLR} if and only if the corresponding subscheme $X\subset U$ is Pfaffian in the sense of Theorem \ref{punc:spec}. The three conditions (a), (b), and (c) of the two theorems also correspond. In the case of (a) this is obvious. For (b) note that $\omega_{R/I}\cong \Gamma(\omega_X)$ since for all $\frak p\in U$ one has $\omega_{R/I,\frak p} = \operatorname{Ext}^3_{R_{\frak p}}((R/I)_{\frak p},R_{\frak p}) = \omega_{X,\frak p}$, and $\omega_{R/I}$ is saturated. Similarly $(R/I)^{\operatorname{sat}} \cong \Gamma(\cal O_X)$. This gives the equivalence of the two conditions (b). As for the conditions (c), first note that the dimension $n$ in Theorem \ref{RLR} corresponds to $n+4$ in Theorem \ref{punc:spec}. But if one uses $n$ as in the latter theorem, one has \[ H^{n/2}(U,\cal O_X) \cong H^{(n+2)/2}(U,\cal I_X) \cong H^{(n+4)/2}_{\frak m}(I). \] Hence the two conditions (c) correspond. Therefore the two theorems \ref{RLR} and \ref{punc:spec} are equivalent, as claimed. In equicharacteristic $2$ the computations of Section 5 remain true in the local case. So Theorems \ref{RLR} and \ref{punc:spec} are true in equicharacteristic $2$ provided one changes the phrase ``$n\equiv 0\pmod 4$'' in the parity condition to ``$n$ is even.'' If $R$ is a regular local ring with residue field of characteristic $2$ and quotient field of characteristic $0$, a different set of calculations is needed. These are unfortunately somewhat involved, and we do not reproduce them here.

9305

alg-geom/9305011

en

https://arxiv.org/abs/alg-geom/9305011

alg-geom/9305011

Rita Pardini

Rita Pardini, Francesca Tovena

On the fundamental group of an abelian cover

17 pages Latex Version 2.09

We study the behaviour of the topological fundamental group under totally ramified abelian covers (a special case of abelian Galois covers) of complex projective varieties of dimension at least 2.

alg-geom

\section{Introduction.} \hspace{6 mm}This work generalizes a result of Catanese and the second author, who analyze in \cite{kn:Cato} the fundamental group of a special type of covering ${f:Y\rightarrow X}$, with Galois group $({\bf Z}/m{\bf Z})^{2}$, of a complex smooth projective surface $X$, the so-called "$m$-th root extraction" of a divisor $D$ on $X$. By means of standard topological methods, the fundamental group $\pi_{1}(Y)$ can be described in that case as a central extension of the group $\pi_{1}(X)$, as follows: \begin{equation} \label{ext1} 0\longrightarrow {\bf Z}/r{\bf Z}\longrightarrow \pi_{1}(Y) \longrightarrow \pi_{1}(X) \longrightarrow 1, \end{equation} $r$ being a divisor of $m$ which depends only on the divisibility of $\pi^{*}(D)$ in $H^2(\tilde{X},{\bf Z})$, where ${\pi:\tilde{X}\rightarrow X}$ is the universal covering of $X$. The main result of \cite{kn:Cato} (see Thm.2.16) is that the group cohomology class corresponding to the extension (\ref{ext1}) can be explicitly computed in terms of the first Chern class of $D$. This is an instance of a more general philosophy: in principle, it should be possible to recover all the information about an abelian cover ${f:Y\to X}$ from the "building data" of the cover, i.e., from the Galois group $G$, the components of the branch locus, the inertia subgroups and the eigensheaves of ${f_{*}{\cal O}_{Y}}$ under the natural action of $G$ (see section 2 or \cite{kn:Rita} for more details). Actually, the description of the general abelian cover given in \cite{kn:Rita} enables us to treat (under some mild assumptions on the components $D_{1},\ldots D_{k}$ of the branch locus) the case of any totally ramified abelian covering ${f:Y\rightarrow X}$, with $X$ a complex projective variety of dimension at least $2$ (cf. section 2 for the definition of a totally ramified abelian cover). Using the same methods as in \cite{kn:Cato}, we show that $\pi_{1}(Y)$ is a central extension as before: \begin{equation} \label{ext2} 0\longrightarrow K\longrightarrow \pi_{1}(Y) \longrightarrow \pi_{1}(X) \longrightarrow 1\,, \end{equation} where $K$ is a finite abelian group which is determined by the building data of the cover and the cohomology classes of $\pi^{*}(D_{1}),\ldots \pi^{*}(D_{k})$ (cf. Prop.\ref{top}). Consistently with the above "philosophy", a statement analogous to Thm.2.16 of \cite{kn:Cato} actually holds in the general case: our main result (Thm.\ref{mtgen}, Rem.\ref{dipdaLchi}) can be summarized by saying that the class of the extension (\ref{ext2}) can be recovered from the Chern classes of the $D_j$'s and of the eigensheaves of $f_*{\cal O}_Y$; in some special case, this relation can be described in a particularly simple way (Thm.\ref{mt}, Cor.\ref{mtcor}, Rem.\ref{rem}). Moreover, one can construct examples of not homeomorphic varieties realized as covers of a projective variety $X$ with the same Galois group, branch locus and inertia subgroups (cf. Rem.\ref{dipdaLchi}). The idea of the proof is to exploit a natural representation of $\pi_{1}(Y)$ on a vector bundle on the universal covering $\tilde{X}$ of $X$ and the spectral sequence describing the cohomology of a quotient, in order to relate the group cohomology class of the extension (\ref{ext2}) to the geometry of the covering. These are basically the same ingredients as in the proof of \cite{kn:Cato}, but we think that we have reached here a more conceptual and clearer understanding of the argument. \noindent {\bf Acknowledgements:} we wish to express our heartfelt thanks to Fabrizio Catanese, who suggested that the result of \cite{kn:Cato} was susceptible of generalization and encouraged us to investigate this problem. \section{A brief review of abelian covers.} \hspace{6 mm}In this section we set the notation and, for the reader's convenience, we collect here the definitions and the notions concerning abelian covers that will be needed later. For further details and proofs, we refer to \cite{kn:Rita}, sections 1 and 2. Let $X$, $Y$ be complex algebraic varieties of dimension at least 2, smooth and projective, and let ${f:Y\to X}$ be a finite abelian cover, i.e. a Galois cover with finite abelian Galois group $G$. The bundle $f_{*}({\cal O} _{Y})$ splits as a sum of one dimensional eigensheaves under the action of $G$, so that one has: \begin {equation} \label{splitting} f_{*}({\cal O} _{Y})= \bigoplus_{\chi \in G^{*}}L_{\chi}^{-1} ={\cal O} _{X} \oplus ( \bigoplus _{\chi \in G^{*}\setminus \{1\}} L_{\chi}^{-1}) \end{equation} where $G^{*}$ denotes the group of characters of G and $G$ acts on $L_{\chi}^{-1}$ via the character $\chi$. We warn the reader that the notation here and in the next section is dual to the one adopted in \cite{kn:Rita}; however this does not affect the formulas quoted from there. Under our assumptions, the ramification locus of $f$ is a divisor. Let $D_{1}, \ldots D_{k}$ be the irreducible components of the branch locus $D$ and let $R_{j} = f^{-1}(D_{j})$, ${j = 1,} \ldots k$. For every ${j = 1, \ldots k}$, one defines the {\em inertia subgroup} $G_{j} = \{g\in G | g(y)=y \; {\rm for \;each}\;y\in R_{j}\}$. Given any point $y_{0} \in R_{j}$, one obtains a natural representation of $G_{j}$ on the normal space to $R_{j}$ at $y_{0}$ by taking differentials. The corresponding character, that we denote by $\psi_{j}$, is independent of the choice of the point $y_{0} \in R_{j}$. By standard results, the subgroup $G_{j}$ is cyclic and the character $\psi_{j}$ generates the group $G_{j}^{*}$ of the characters of $G_{j}$. We denote by $m_{j}$ the order of $G_{j}$, by $m$ the least common multiple of the $m_{j}$'s and by $g_{j}$ the generator of $G_{j}$ such that $\psi_{j}(g_{j}) = exp(\frac{2\pi \sqrt{-1}}{m_{j}})$. In what follows we will always assume that the cover ${f:Y\rightarrow X}$ is {\em totally ramified}, i.e., that the subgroups $G_{j}$ generate $G$; then the group of characters $G^{*}$ injects in ${\bigoplus_{j=1}^{k}G_{j}^{*}}$ and every ${\chi \in G^{*}}$ may be written uniquely as: \begin{equation} \label{char0} \chi = \sum_{j=1}^{k} \;a_{\chi,j}\,\psi_{j}, \hspace{10mm}\,0\leq a_{\chi,j}< m_{j} \;{\rm for\,every}\,j\,. \end{equation} In particular, let $\chi_{1}, \ldots \chi_{n} \in G^{*}$ be such that $G^{*}$ is the direct sum of the cyclic subgroups generated by the $\chi $'s, and let $d_{i}$ be the order of $\chi_{i}$, $i = 1, \ldots n$. Write: \begin{equation} \label{char} \chi_{i} = \sum_{j=1}^{k} \;a_{ij}\,\psi_{j}, \;\;\,\;0\leq a_{ij}< m_{j}\,, \;\;\, i= 1, \ldots n\,. \end {equation} Then one has (\cite{kn:Rita}, Prop.2.1): \begin{equation} \label{eqstr} d_{i} L_{\chi_{i}} \equiv \sum_{j=1}^{k}\frac{d_{i}a_{ij}}{m_{j}}\;D_{j} \;\;\;\;i= 1, \ldots n\, \end{equation} the corresponding isomorphism of line bundles being induced by multiplication in the ${\cal O}_{X}$-algebra $f_{*}{\cal O}_{Y}$. More generally, if $\chi=\sum_{i=1}^n b_{\chi,i}\chi_i$, with $0\leq b_{\chi,i}<d_i \;\forall\,i$, one has (\cite{kn:Rita}, Prop.2.1): \begin{equation}\label{Lchigen} L_{\chi}\equiv \sum_{i=1}^n b_{\chi,i}L_{\chi_{i}} - \sum_{j=1}^k q^{\chi}_{j} D_j\,. \end{equation} where $q^{\chi}_{j}$ is the integral part of the rational number ${\sum_{i=1}^n \frac{b_{\chi,i}a_{ij}}{m_j}}$, $j=1,\ldots k$. Equations (\ref{eqstr}) are the characteristic relations of an abelian cover. Actually, since $X$ is complete, for assigned $G$, $D_{j}$, $G_{j}$, $\psi_{j}$, $j = 1, \ldots k$, to each set of line bundles $L_{\chi_{i}}$, $i = 1, \ldots n$, satisfying (\ref{eqstr}) there corresponds a unique, up to isomorphism, $G$-cover of $X$, branched on the $D_{j}$'s and such that $G_{j}$ is the inertia subgroup of $D_{j}$ and $\psi_{j}$ is the corresponding character (\cite{kn:Rita}, Thm.2.1). Moreover, the cover is actually smooth under suitable assumptions on the building data. \section{The fundamental group and the universal covering of $Y$.} \hspace{6 mm}We keep the notation introduced in the previous section. \begin{Definition}{\rm (\cite{kn:MM}, pag.218)} A smooth divisor $\Delta$ on a variety $X$ is called {\rm flexible} if there exists a smooth divisor $\Delta ' \equiv \Delta$ such that $\Delta ' \cap \Delta \neq \emptyset$ and $\Delta$ and $\Delta '$ meet transversely. \end{Definition} We recall that a flexible divisor on a projective surface is connected (see \cite{kn:Ca}, Remark 1.5). Hence, by considering a general linear section, one deduces easily that a flexible divisor on a projective variety of dimension $\geq 2$ is connected. \begin{Proposition} \label{top} Let $X$, $Y$ be smooth projective varieties over {\bf C} of dimension $n$ at least 2. Let ${f:Y\to X}$ be a totally ramified abelian cover branched on irreducible, flexible and ample divisors $\{D_j\}_{j=1,\ldots k}$. Then: \begin{list}% {\alph{alph})}{\usecounter{alph}} \item The natural map ${f_{*}:\pi_{1}(Y)\to \pi_{1}(X)}$ is surjective. \item Let $K = ker(f_{*})$; then $K$ is finite and \begin{equation} \label{ext} 0\longrightarrow K\longrightarrow \pi_{1}(Y) \longrightarrow \pi_{1}(X) \longrightarrow 1. \end{equation} is a central group extension. \item Let ${\pi :\tilde{X}\to X}$ be the universal covering of $X$ and ${\tilde{D} = \pi^{-1}(D)}$; then $\tilde{D}_{j}=\pi^{-1}(D_{j})$ is connected, j = 1, \ldots k. Denote by $H^{i}_{c}$ the cohomology with compact supports and by ${\rho : H^{2n-2}_{c}(\tilde{X}) \to H^{2n-2}_{c}(\tilde{D}) \cong \bigoplus ^{k}_{j=1}\,{\bf Z}\tilde{D}_{j}}$ the restriction map. Finally, let $\sigma$ be the map defined by: \begin{equation} \begin{array}{crclc} \sigma:&H^{2n-2}_{c}(\tilde{D}) \cong \bigoplus ^{k}_{j=1}\,{\bf Z}\tilde{D}_{j} &\to& \bigoplus_{j=1}^{k}G_{j}&\\ &{\tilde{D}_{j}} &\mapsto &g_{j}&. \end{array} \end{equation} Then ${N = \ker(\bigoplus G_{j} \to G)}$ contains ${\rm Im}(\sigma \circ \rho)$ and $K$ is isomorphic to the quotient group $N/{\rm Im}(\sigma \circ \rho)$. \end{list} \end{Proposition} {\sc Proof.} a) and the fact that the extension (\ref{ext}) is central can be proven exactly as in \cite{kn:Ca}, Thm.1.6 and in \cite{kn:Cato}, Lemma 2.1. For the proof of c) (that implies that $K$ is finite), we refer the reader to \cite{kn:Ca}, Prop.1.8 and to \cite{kn:Cato}, proof of Thm.2.16, Step I. One only has to notice that, by Lefschetz theorem (cf. \cite{kn:Bott}, Cor. of Thm.1), $\pi_{1}(D_{j})$ surjects onto $\pi_{1}(X)$, hence $\tilde{D}_{j}=\pi^{-1}(D_{j})$ is connected and smooth for every $j = 1, \ldots k$.\hfill\qed \begin{Remark}{\rm \begin{list}% {\alph{alph})}{\usecounter{alph}} \item {}From Prop.\ref{top}, c), it follows in particular that the kernel $K$ of the surjection ${f_*:\pi_1(Y)\to \pi_1(X)}$ does not depend on the choice of the solution $L_\chi$ of (\ref{eqstr}), once $G$, the $g_j$'s and the class of the $\tilde{D}_j$'s in $H^2(\tilde{X},{\bf Z}/m_j{\bf Z})$, $j=1,\ldots k$, are fixed. \item If $f:Y\to X$ is an abelian cover as in the hypotheses of Prop.\ref{top}, then $H^1(Y,{\cal O}_Y)\cong H_1(X,{\cal O}_X)$ by (\ref{splitting}) and the Kodaira Vanishing Theorem. Moreover, according to Prop.\ref{top}, a) the map ${f_*:H_1(Y,{\bf Z})\to H_1(X,{\bf Z})}$ is surjective; thus the map ${f_*: alb(Y)\to alb(X)}$ between the Albanese varieties is an isomorphism. \end{list} } \end{Remark} \begin{Proposition} \label{tilde} In the same hypotheses as in Prop.\ref{top}, let ${q:\tilde{Y}\to Y}$ be the universal cover of $Y$ and let ${\tilde{f\,}:\tilde{Y}\to \tilde{X}}$ be the map lifting $f:Y\to X$. Then $\tilde{f\,}$ is a totally ramified abelian cover of $\tilde{X}$ with group ${\tilde{G} = (\bigoplus_{j=1}^k G_{j})/{\rm Im}(\sigma \circ \rho)}$, branched on $\tilde{D}$. \end{Proposition} {\sc Proof.} By diagram chasing, it is easy to show that ${\pi_{1}(\tilde{X}\setminus \tilde{D})}$ is isomorphic to the kernel $V$ of the surjection ${\pi_{1}(X\setminus D)\to \pi_{1}(X)}$ induced by the inclusion ${X\setminus D\subset X}$. Since the $D_{i}$'s are flexible, one proves as in (\cite{kn:Cato}, Lemma 2.1) that $V$ is an abelian group. It follows that $\tilde{f\,}$, being branched on $\tilde{D}$, is an abelian cover. Consider now the fiber product $Y'$ of ${f:Y\to X}$ and ${\pi:\tilde{X} \to X}$, with the natural maps ${f':Y'\to\tilde{X}}$ and ${q':Y'\to Y}$; $f'$ is a $G$-cover ramified on $\tilde{D}$ and $q'$ is unramified. According to Prop.\ref{top}, b), the universal covering ${q:\tilde{Y}\to Y}$ of $Y$ factors as ${q=q' \circ q''}$, for a suitable unramified cover ${q'':\tilde{Y} \to Y'}$ with group $K$, giving a commutative diagram as follows: \begin{equation} \label{diagramma} \begin{picture}(60,85)(-4,-40) \put (8,0){\vector(1,0){20}} \put (9,30){\vector(1,-1){20}} \put (0,30){\vector(0,-1){19}} \put (0,-8){\vector(0,-1){19}} \put (42,-8){\vector(0,-1){19}} \put (8,-38){\vector(1,0){20}} \put (-4,-2){$Y'$} \put (36,-2){$\tilde{X}$} \put (-3,36){$\tilde{Y}$} \put (-4,-40){$Y$} \put (36,-40){$X$} \put(23,20){${\scriptstyle \tilde{f\,}}$} \put(13,3){${\scriptstyle f'}$} \put(-10,20){${\scriptstyle q''}$} \put(13,-35){${\scriptstyle f}$} \put(-10,-20){${\scriptstyle q'}$} \put(44,-20){${\scriptstyle \pi}$} \end{picture} \end{equation} In particular, ${K\cong \pi_{1}(Y')}$ and ${\tilde{f\,}=f'\circ q''}$. Hence, the Galois group $\tilde{G}$ of $\tilde{f\,}$ is given as an extension: \begin{equation} \label{extgrouO} 0\longrightarrow K \longrightarrow\tilde{G} \longrightarrow G \longrightarrow 0 \end{equation} Moreover, if one denotes by $\tilde{G}_{j}$ the inertia subgroup of $\tilde{D}_{j}$ with respect to $\tilde{f\,}$, then $\tilde{G}_{j}$ maps isomorphically onto $G_{j}$ for every $j=1, \ldots k$. The isomorphism ${\tilde{G} = (\bigoplus G_{j})/{\rm Im}(\sigma \circ \rho)}$ can be obtained by computing the fundamental group of $\tilde{Y}$ as in \cite{kn:Cato}, proof of Thm.2.16. \hfill\qed \vspace{3mm} The following lemma will be used in the next section. \begin{Lemma} \label{commut} Consider the subgroups $\pi_{1}(Y)$ and $\tilde{G}$ of ${Aut(\tilde{Y})}$; then one has: \begin{equation} \beta g = g\beta\,\hspace{15mm}\forall g\in \tilde{G}, \forall\beta \in \pi_{1}(Y). \end{equation} \end{Lemma} {\sc Proof.} Since the cover $\tilde{f\,}$ is totally ramified, it is enough to show that all the elements of $\pi_{1}(Y)$ commute with $g_{j}$, $j=1, \ldots k$. Let ${\beta \in \pi_{1}(Y)}$ and fix $j=1, \ldots k$. We remark firstly that ${\beta g_{j}\beta^{-1}}$ is actually an element of ${\tilde{G}\subset Aut(\tilde{Y})}$. In fact, consider the classes represented by ${\beta g_{j}}$ and ${g_{j}\beta}$ modulo $K$: they do coincide as automorphisms of ${Y'\subseteq Y\times\tilde{X}}$, since the group ${G\times\pi_{1}(X)}$ acts there via the natural action on the components. So, ${\beta g_{j}\beta^{-1}g_{j}^{-1}\in K}$ and ${\beta g_{j}\beta^{-1}\in g_{j}K\subseteq\tilde{G}}$, as desired. By diagram (\ref{diagramma}), we have $\tilde{R}_{j}=\tilde{f\,}^{-1}(\tilde{D}_{j})=q^{-1}(R_{j})\;\forall$ $j=1, \ldots k$. Since $R_{j}=f^{-1}(D_{j})$ is ample and connected, the same argument as in the proof of Lemma \ref{top}, c) shows that $\tilde{R}_{j}$ is connected. So, ${\beta \tilde{R}_{j}=\tilde{R}_{j}}$ and ${\beta g_{j}\beta^{-1}}$ fixes ${\tilde{R}_{j}}$ pointwise, namely ${\beta g_{j}\beta^{-1}\in \tilde{G}_{j}}$. Finally, recalling the definition of the character ${\psi_{j}\in G_{j}^{*}}$ introduced in section 2, one checks immediately that ${\psi_{j}(\beta g_{j}\beta^{-1})=\psi_{j}(g_{j})}$. The conclusion now follows from the faithfulness of $\psi_{j}$. \hfill\qed \section{Computing the cohomology class of the central extension $0\to K \to\pi_{1}(Y)\to \pi_{1}(X)\to 1$.} \hspace{6 mm}We keep the notation and the assumptions introduced in the previous sections, unless the contrary is explicitly stated. We need two technical Lemmas in order to state the main result of this paper. \begin{Lemma} \label{Lemma} Let ${\cal H}$ be a finite abelian group and $\zeta_1$,\ldots $\zeta_m\,\in\,{\cal H}$ be such that ${\cal H}=\bigoplus_{j=1}^m <\zeta_j>$ is the direct sum of the cyclic subgroups generated by $\zeta_j$, j=1, \ldots m; denote by $h_j$ the order of $\zeta_j$. Let ${p\,\in\,{\bf Z}}$ be a prime and ${\cal H}_{p}$ be the p-torsion subgroup of ${\cal H}$. Let $\chi_{1}, \ldots \chi_{t} \in {\cal H}_{p}$ such that ${<\chi_{1}, \ldots \chi_{t}>=\bigoplus_{i=1}^t<\chi_{i}>}$. Finally, let $d_{i}$ be the order of $\chi_{i}$ and write ${\chi_{i}=\sum_{j=1}^{h} a_{ij}\zeta_{j}}$ with ${0\leq a_{ij}<h_j}$. Then, $\forall$ $x_{1}, \ldots x_{t} \in {\bf Z}$ and $\forall$ $\gamma \geq 1$, the system: \begin{equation} \label{sistlemma} \sum_{j=1}^{m}\frac{d_{i}a_{ij}}{h_{j}}s_{j}\equiv x_{i}\;\;\; {\rm mod}\,p^{\gamma}\;\;\;\;\;\;\;\;\;i=1,\ldots t \end{equation} admits a solution $(s_{1}, \ldots s_{m})\,\in\,{\bf Z}^m$. \end{Lemma} {\sc Proof.} We set $c_{ij}=\frac{d_{i}a_{ij}}{h_{j}}$ and, for $x\,\in\,{\bf Z}$, we denote by $\overline{x}$ the class of $x$ in ${\bf Z}/p{\bf Z}$. We proceed by induction on $\gamma$. Let $\gamma =1$. We show that the matrix $(\overline{c}_{ij})$ has rank $t$. Let $y_{1}, \ldots y_{m} \in {\bf Z}$ and assume that: \begin{equation} \sum_{i}\overline{c}_{ij}\overline{y}_{i}=0\;\;\;\;\;\forall\,j=1, \ldots m\,. \end{equation} This implies that: \begin{equation} \sum_{i=1}^tc_{ij}y_{i}\equiv 0\;\;\;{\rm mod}\,p\;\;\;\;\;\;\;\forall\,j=1, \ldots m \end{equation} so that: \begin{equation} \sum_{i=1}^t\frac{y_{i}d_{i}}{p}\frac{a_{ij}}{h_{j}}\;\in\, {\bf Z}\;\;\;\;\forall\,j=1, \ldots m\,. \end{equation} Recalling that $p$ divides $d_{i}$ $\forall\,i$, we deduce that: \begin{equation} \sum_{i=1}^t\left(\frac{y_{i}d_{i}}{p}\right)\,a_{ij}\equiv 0 \;\;\;{\rm mod}\,h_{j}\;\;\;\;\;\;\;\forall\,j=1, \ldots m\,, \end{equation} so that $\sum_{i}\frac{y_{i}d_{i}}{p}\chi_{i}$ is the zero element in ${\cal H}$. By the hypothesis on the $\chi_{i}$'s, it follows that: \begin{equation} \frac{y_{i}d_{i}}{p}\equiv 0\;\;\;{\rm mod}\,d_{i} \end{equation} and finally: \begin{equation} y_{i}\equiv 0\;\;\;{\rm mod}\,p \end{equation} showing, as desired, that the rows of the matrix $(\overline{c}_{ij})$ are linearly independent over ${\bf Z}/p{\bf Z}$. Let now $\gamma > 1$ and assume by inductive hypothesis that ${(s_{1}, \ldots s_{m})\in {\bf Z}^{m}}$ is a solution of the system (\ref{sistlemma}). We set $s_{j}'=s_{j}+\delta_{j}p^{\gamma}$ and we look for a suitable choice of the integers $\delta_{j}$. We have: \begin{equation} \begin{array}{ccl} \sum_{j}c_{ij}s_{j}'&=&\sum_{j}c_{ij}s_{j}+p^{\gamma} \sum_{j}c_{ij}\delta_{j}\\ &=&x_{i}+p^{\gamma}y_{i}+p^{\gamma} \sum_{j}c_{ij}\delta_{j}\;\;\;\;\;\exists\,y_{i}\,\in\,{\bf Z}, \end{array} \end{equation} so that: \begin{equation} \sum_{j}c_{ij}s_{j}'\equiv x_{i}\;\;{\rm mod}\,p^{\gamma +1} \;\;\;\Longleftrightarrow\;\;\;\sum_{j}c_{ij}\delta_{j} \equiv -y_{i} \;\;\;{\rm mod}\,p \end{equation} and the latter system has a solution, by the case $\gamma = 1$. This conclude the proof.\hfill\qed We come back to the study of the cover $f$: \begin{Lemma} \label{coeffinK} Let $A$ be the subgroup of $Pic(X)$ generated by $D_{1}, \ldots D_{k}$ and $L_{\chi}$, $\chi\in G^{*}$. Then there exist $M_{1}, \ldots M_{q}\in Pic(X)$ such that ${A=\bigoplus_{l=1}^{q}<M_{l}>}$ and \begin{equation} \left(\begin{array}{c} D_{1}\\ \vdots\\ D_{k} \end{array}\right)\;\;\equiv\;\;C\, \left(\begin{array}{c} M_{1}\\ \vdots\\ M_{q} \end{array}\right) \end{equation} where $C=(c_{jl})$ is a matrix with integral coefficients such that each column $(c_{jl})_{j=1,\ldots k}$ represents an element of ${N=\,{\rm ker}\,(\bigoplus_{j=1}^kG_j\to G)}$. \end{Lemma} {\sc Proof.} $A$ is a finitely generated abelian group, so one can write ${A=F\bigoplus T}$, where $T$ is the torsion part of $A$ and $F$ is free. Denote by $\{\xi_{l}\}_{l}$ a set of free generators of $F$ and by $\{\eta_{l}\}_{l}$ a set of generators of $T$ such that ${T=\bigoplus<\eta_{l}>}$ and the order $o(\eta_{l})$ of $\eta_{l}$ is the power of a prime, $\forall l$. Let finally $\chi_{i}$ be generators of $G^{*}$ such that $G^{*}=\bigoplus_{i=1}^n<\chi_{i}>$ and the order $o(\chi_{i})$ of $\chi_{i}$ is the power of a prime, $\forall i$. One can write: \begin{eqnarray} \label{Lchii}L_{\chi_{i}}\equiv \sum_l\lambda_{il}\eta_{l}+\sum_l\lambda_{il}'\xi_{l}\hspace{10mm} \forall\,i=1,\ldots n,\\ \label{Djlemma}D_{j}\equiv\sum_l c_{jl}\eta_{l}+\sum_l c_{jl}'\xi_{l}\hspace{10mm} \forall\,j=1,\ldots k, \end{eqnarray} where the coefficients $\lambda_{il}'$ and $c_{jl}'$ are uniquely determined, whereas $\lambda_{il}$ and $c_{jl}$ are determined only up to a multiple of $o(\eta_{l})$. We can apply the analysis of section 2 to the cover $f$. We write ${\chi_i=\sum_{j=1}^{k}a_{ij}\psi_j}$, with ${0\leq a_{ij}<m_j}$, and we set ${d_{i}=o(\chi_i)}$ as in the previous Lemma; the equations (\ref{eqstr}) become here: \begin{equation} \label{eqstrf} d_{i} L_{\chi_{i}} \equiv \sum_{j=1}^{k}\frac{d_{i}a_{ij}}{m_{j}}\;D_{j}\; \hspace{10mm}i=1,\ldots n, \end{equation} so that we must have: \begin{equation} d_{i}\lambda_{il}'=\sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;c_{jl}' \hspace{10mm}i=1,\ldots n, \end{equation} showing that $(c_{jl}')_{j=1,\ldots k}$ represents an element of $N$, $\forall \,l$: in fact, by duality, ${(t_1,\ldots,t_k)\,\in\, {\bf Z}^k}$ represents an element of $N$ if and only if it satisfies the relations: \begin{equation}\label{relN} \sum_{j=1}^k\frac{a_{ij}}{m_{j}}t_j\;\in\;{\bf Z}\hspace{15mm}\forall\,i=1,\ldots n. \end{equation} For the coefficients of the torsion part, we have: \begin{equation} d_{i}\lambda_{il}\,\eta_{l}=\left( \sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;c_{jl}\right)\;\eta_{l} \end{equation} so that: \begin{equation} \label{cong} d_{i}\lambda_{il}\equiv \sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;c_{jl} \;\;\;{\rm mod}\,o(\eta_{l})\;. \end{equation} We fix an index $l$. Let $p$ be a prime such that $o(\eta_l)=p^{\alpha}$. We want to show that, for a suitable choice of the $c_{jl}$, the following relation holds $\forall\,i=1, \ldots n$: \begin{equation} \label{tesi} d_{i}\lambda_{il}\equiv \sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;c_{jl} \;\;\;{\rm mod}\,d_{i}\,. \end{equation} Let $\chi_{i}$ be a generator such that ${d_{i}\equiv 0}$ mod $p$ and set ${d_{i}=p^{\alpha_{i}}}$. By (\ref{cong}), it is enough to consider the case in which ${\alpha<\alpha_{i}}$. Setting $c_{jl}''= c_{jl}+p^{\alpha}s_{j}$ and recalling (\ref{cong}), one has: \begin{equation} \begin{array}{ccl} \sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;c_{jl}''&=& \sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;c_{jl}+ p^{\alpha}\sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;s_{j}\\ &=&d_{i}\lambda_{il}-p^{\alpha}x_{i}+p^{\alpha} \sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;s_{j} \end{array} \end{equation} for a suitable choice of integers $x_{i}$. One concludes that the relation (\ref{tesi}) holds if and only if: \begin{equation} \sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;s_{j}\equiv x_{i} \;\;\;{\rm mod}\,p^{\alpha_{i}-\alpha}\;. \end{equation} Let $\beta = {\rm max}\,\{\alpha_{i}-\alpha\}_{i}$. The system of congruences: \begin{equation} \sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;s_{j}\equiv x_{i} \;\;\;{\rm mod}\,p^{\beta}\;\hspace{10mm}\forall\,i \mbox{ such that } d_{i}\equiv 0\;\mbox{ mod } p \end{equation} admits a solution by Lemma \ref{Lemma}. So, we can assume that the coefficients $(c_{jl})_{j=1,\ldots k}$ in (\ref{Djlemma}) satisfy (\ref{tesi}) for every $i$ such that $d_{i}\equiv 0$ mod $p$. To complete the proof, let $\gamma$ be an integer $\gg 0$; we can still modify the coefficients as ${c_{jl}''=c_{jl}+ p^{\gamma}t_{j}}$. It is enough to notice that, setting ${d={\rm lcm}\{d_i\,|\,d_{i}\not\equiv 0\;\mbox{ mod } p\}}$, then $d$ and $p$ are coprime and the system of congruences: \begin{equation} c_{jl}+ p^{\gamma}t_{j}\equiv 0\hspace{10mm}{\rm mod}\,d \hspace{15mm}\forall\,j \end{equation} admits a solution. So we can assume that $c_{jl}''\equiv 0$ mod $d$, and the proof is complete. \hfill\qed \vspace{3mm} To any decomposition (\ref{decteor}) as in Lemma \ref{coeffinK}, we associate a cohomology class in $H^2(X,K)$: \begin{Definition}\label{defxi} Given a decomposition (\ref{decteor}) as in Lemma \ref{coeffinK}, consider the map: \begin{equation} \begin{array}{ccl} {\bf Z}^{q}&\to &N\\ (x_1,\ldots x_q) &\mapsto&\sum_{l=1}^q\,x_l\underline{c}_l= (\sum_{l=1}^q\,x_lc_{jl})_{j=1,\ldots k} \end{array} \end{equation} and denote by ${\Theta:{\bf Z}^{q}\to K}$ its composition with the projection ${N\to K}$ (cf. Prop.\ref{top}, c)). Then, set: \begin{equation} \xi=\Theta_*([M_1], \ldots [M_q])\,, \end{equation} where ${\Theta_*:H^2(X,{\bf Z}^q)\cong\bigoplus^q H^2(X,{\bf Z})\to H^2(X,K)}$ is the map induced in cohomology by $\Theta$ and $[M]$ is the Chern class of a divisor $M$ on $X$. \end{Definition} \vspace{3mm} We briefly recall some facts about quotients by a properly discontinuous group action (see for instance \cite{kn:Mu}, Appendix to section 1, \cite{kn:Grot}, ch. 5). Let $\tilde{X}$ be a simply connected variety, let $\Gamma$ be a group acting properly and discontinuously on $\tilde{X}$ and let ${p:\tilde{X}\rightarrow X=\tilde{X}/\Gamma}$ be the projection onto the quotient. Consider the following two functors: $$ \begin{array}{l} M\stackrel{F}{\longrightarrow} M^{\Gamma} ,\,{\rm for}\,M \,{\rm a}\, \Gamma \mbox{-module}\\ {\cal F}\stackrel{H}{\longrightarrow} H^0 (\tilde{X},p^{*}{\cal F}),\,{\rm for}\,{\cal F} \,{\rm a\,locally\,constant\,sheaf \, on}\, X\,. \end{array} $$ The spectral sequence associated to the functor $F\circ H$ yields in this case the exact sequence of cohomology group: \begin{equation} \label{spseq} 0\longrightarrow H^{2}(\Gamma,H^{0}(\tilde{X},p^{*}{\cal F})) \longrightarrow H^{2}(X,{\cal F}) \longrightarrow H^{2}(\tilde{X},p^{*}{\cal F})^{\Gamma} \end{equation} that will be used several times in the following and it is natural with respect to the sheaf maps on $X$. \begin{Theorem}\label{mtgen} Let $X$, $Y$ be smooth projective varieties over {\bf C} of dimension at least 2. Let ${f:Y \rightarrow X}$ be a totally ramified finite abelian cover branched on a divisor with flexible and ample components ${\{D_{j}\}_{j=1, \ldots k}}$. According to Prop.\ref{top}, b), the map $f$ induces a central extension: \begin{equation} \label{extgrouteogen} 0\longrightarrow K \longrightarrow\pi_{1}(Y) \stackrel{f_{*}}{\longrightarrow} \pi_{1}(X) \longrightarrow 1 \end{equation} Denote by ${c(f)\, \in \,H^{2}(\pi_{1}(X), K)\subseteq H^{2}(X, K)}$ the cohomology class classifying the extension (\ref{extgrouteogen}). Let: \begin{equation} \label{decteor} \left(\begin{array}{c} D_{1}\\ \vdots\\ D_{k} \end{array}\right)\;\;\equiv\;\;C\, \left(\begin{array}{c} M_{1}\\ \vdots\\ M_{q} \end{array}\right) \end{equation} be a decomposition as in Lemma \ref{coeffinK} and let $\xi\,\in\, H^{2}(X,K)$ be the class defined in Def.\ref{defxi}. In this notation, one has: \begin{equation} c(f)=\xi; \end{equation} in particular, the class $\xi$ does not depend on the chosen decomposition. \end{Theorem} {\sc Proof.} It is enough to show that $\xi$ and $c(f)$ admit cohomologous representatives. This can be done in three steps. \vspace{3 mm} {\sc Step I}: we compute a cocycle representing $c(f)\,\in\,H^2(X,K)$. We start by choosing suitable trivializations of the line bundles that appear in the computation. Set $\Gamma = \pi_{1}(X)$ and $\tilde{\Gamma} = \pi_{1}(Y)$. Let $\{U_{r}\}$ be a sufficiently fine cover of $X$ such that $\Gamma$ acts transitively on the set of connected components of $\pi^{-1}(U_{r})$, $\forall$ $r$. If we fix a component $V_{r}$ of $\pi^{-1}(U_{r})$, then ${\pi^{-1}(U_{r}) = \cup_{\gamma \in \Gamma}\,\gamma(V_{r})}$; for every ${\gamma \in \Gamma}$ we write: \begin{equation} \gamma(V_{r}) = V_{(\gamma, r)} \end{equation} and, in particular: $V_{(1, r)}=V_{r}$. Such a covering has the following properties: \begin{list}% {\alph{alph})}{\usecounter{alph}} \item For every $(r, s)$ such that ${U_{r}\cap U_{s} \neq \emptyset}$, there exists a unique element ${\beta (r, s) \in \Gamma}$ such that: \begin{equation} V_{(1, r)} \cap V_{(\beta (r, s), s)} \neq \emptyset\;. \end{equation} \item If ${U_{r}\cap U_{s} \neq \emptyset}$, then $V_{(\gamma, r)}$ and $V_{(\gamma \beta (r, s), s)}$ have nonempty intersection. \item Since $\pi$ is a local homeomorphism, if ${U_{r}\cap U_{s}\cap U_{t} \neq \emptyset}$, then: \begin{equation} \emptyset\;\neq\; V_{(\beta (r, s), s)}\cap V_{(\beta (r, t), t)}. \end{equation} Hence the following relation is satisfied for every ${U_{r}\cap U_{s}\cap U_{t} \neq \emptyset}$: \begin{equation} \beta (r, t) = \beta (r, s) \beta (s, t)\;. \end{equation} In particular: $\beta (s, r) = \beta (r, s)^{-1}$. \end{list} \noindent For later use, we set: \begin{equation} V_{(\alpha, r, s)} = \alpha(V_{(1, r)}\cap V_{(\beta (r, s), s)}) = V_{(\alpha, r)}\cap V_{(\alpha\beta (r, s), s)} \end{equation} for every $\alpha \in \Gamma$ and for every $(r, s)$ such that ${U_{r}\cap U_{s} \neq \emptyset\,}$. For every $r$ and for every $j = 1, \ldots k$, we choose a local generator $w^{j}_{r}$ for ${{\cal O}_{X}(-D_{j})}$ on $U_{r}$ (we ask that $w^{j}_{r}$ is a local equation for $D_j$) and for every pair $(r, s)$ such that ${U_{r}\cap U_{s} \neq \emptyset}$ we write: \begin{equation}\label{coc-Dj} w^{j}_{r} = k^{j}_{(r, s)}\,w^{j}_{s}\;\;{\rm on}\,U_{r}\cap U_{s}\,. \end{equation} Now we apply to ${\tilde{f\,}:\tilde{Y}\rightarrow \tilde{X}}$ the analysis of section 2 , most of which can be easily extended to the case of analytic spaces. One has: \begin {equation} \label{split} \tilde{f\,}_{*}({\cal O} _{\tilde{Y}})= \bigoplus_{\tilde{\chi} \in \tilde{G}^{*}}L_{\tilde{\chi}}^{-1} \end{equation} Each element of the group $\tilde{G}$ can be interpreted as an automorphism of the sheaf $\tilde{f\,}_{*}({\cal O} _{\tilde{Y}})$. In particular, by duality, the elements of $K\subseteq \tilde{G}$ are characterized by the property that they induce the identity on the subsheaf $L_{\chi}$ for every ${\chi\,\in\,G^*\subseteq\tilde{G}^*}$. Let $\tilde{\chi}_{1}, \ldots \tilde{\chi}_{h} \in \tilde{G}^{*}$ be such that $\tilde{G}^{*}$ is isomorphic to the direct sum of the cyclic subgroups generated by the $\tilde{\chi}_{i}$'s, and let $\tilde{d_i}$ be the order of $\tilde{\chi}_{i}$, $i = 1, \ldots h$. Let $\tilde{D_{i}}$ be the inverse image of $D_{i}$ via the universal covering map ${\pi :\tilde{X}\to X}$, as before. If ${\tilde{\chi}_i=\sum_{j=1}^{k}\,\tilde{a}_{ij}\psi_j}$, with ${0\leq \tilde{a}_{ij} <m_j}$, the system (\ref{eqstr}) yields in this case: \begin{equation} \label{eqstr1} \tilde{d_i} L_{\tilde{\chi}_{i}} \equiv \sum_{j=1}^{k}\frac{\tilde{d_i}\tilde{a}_{ij}}{m_{j}} \;\tilde{D} _{j}\;\;\;\;i= 1, \ldots h\,. \end{equation} So it is possible to choose local generators $\tilde{z}^{i}_{(\alpha, r)}$ for $L_{\tilde{\chi}_{i}}^{-1}$ on $V_{(\alpha, r)}$ such that for every ${\alpha \in \Gamma}$ and for every pair $(r, s)$ with ${U_{r}\cap U_{s} \neq \emptyset}$ one has: \begin{equation}\label{relforz} \left(\tilde{z}_{(\alpha, r)}^{i}\right)^{\tilde{d_i}}= \prod_{j=1}^k\left(w_r^j\right)^{\frac{\tilde{d_i}\tilde{a}_{ij}}{m_j}} \,. \end{equation} Writing: \begin{equation} \tilde{z}_{(\alpha, r)}^{i} = \tilde{h}^{i}_{(\alpha, r,s)}\, \tilde{z}^{i}_{(\alpha \beta (r, s), s)} \;\;\;\; {\rm on}\,V_{(\alpha, r, s)} \end{equation} we have: \begin{equation} \label{rela} (\tilde{h}^{i}_{(\alpha, r, s)})^{\tilde{d_i}}\;=\;\prod_{j=1}^{k}(k^{j}_{(r, s)}) ^{\frac{\tilde{d_i}\tilde{a}_{ij}}{m_{j}}}\;. \end{equation} and the cocycle condition for $\tilde{h}^{i}_{(\alpha, r, s)}$, that will be often used later on, yields the relation: \begin{equation} \label{coch} 1=\tilde{h}^{i}_{(\alpha, r, s)}\tilde{h}^{i}_{(\alpha\beta(r,s), s, t)} \tilde{h}^{i}_{(\alpha\beta(r,t), t, r)}\; \hspace{15mm}\forall\,\alpha\,\in\,\Gamma,\forall\,i,\forall\, r,s,t\mbox{ with }U_r\cap U_s \cap U_t\neq\emptyset. \end{equation} We observe that the generator $\tilde{z}^{i}_{(\alpha,r)}$ is determined by (\ref{relforz}) only up to a constant of the form $exp2\pi \sqrt{-1} \,(u^i_{(\alpha,r)}/\tilde{d_i})$ with ${u^i_{(\alpha,r)}\in {\bf Z}}$. Moreover, according to (\ref{Lchigen}), every choice of local generators $w^j_r$ for ${\cal O}_X(-D_j)$ and $\tilde{z}_{(\alpha, r)}^{i}$ for $L_{\tilde{\chi}_i}^{-1}$ induces a choice of local generators for $L_{\tilde{\chi}}^{-1}$, ${\forall\,\tilde{\chi}\,\in\,\tilde{G}^*}$, by the rule: \begin{equation}\label{ztildechi} \tilde{z}_{(\alpha, r)}^{\tilde{\chi}}=\prod_{i=1}^n\left( \tilde{z}_{(\alpha, r)}^{i} \right)^{b_{\tilde{\chi},i}} \prod_{j=1}^k\,(w^j_r)^{-q^{\tilde{\chi}}_{j}}\,\hspace{15mm} \mbox{if }\,\tilde{\chi}=\sum_{i=1}^h b_{\tilde{\chi},i}\tilde{\chi}_i,\;\; 0\leq b_{\tilde{\chi},i}<\tilde{d_i}\,. \end{equation} where $q^{\tilde{\chi}}_j$ denotes the integral part of the real number $\sum_{i=1}^h b_{\tilde{\chi},i} \,\tilde{a}_{ij}$. Let now $\chi_1, \ldots \chi_n$ be a set of generators for $G^*$ such that $G^*$ is the direct sum of the cyclic subgroups generated by the $\chi_v$'s and the order $d_v$ of $\chi_v$ is a power of a prime number, $v=1, \ldots n$. We recall that ${G^*\subseteq \tilde{G}^*}$ and, ${\forall\,\chi\,\in\,G^*}$, the corresponding eigensheaf $L_{\chi}$ is a pullback from $X$. We write ${\chi_v=\sum_{i=1}^n b_{vi}\tilde{\chi}_i\,\in\,\tilde{G}^*}$ (${0\leq b_{vi}<\tilde{d_i}}$) and $q^{\chi_v}_j=q^{v}_j$; the corresponding local generator for ${L_{\chi_v}^{-1}}$ chosen in (\ref{ztildechi}) is: \begin{equation}\label{zG*} z_{(\alpha, r)}^{v}=\prod_{i=1}^h\left( \tilde{z}_{(\alpha, r)}^{i} \right)^{b_{vi}}\prod_{j=1}^k\,(w^j_r)^{-q^{v}_j} \;; \end{equation} we show that, for a suitable choice of the $\tilde{z}_{(\alpha, r)}^{i}$, we can assume that the expression in (\ref{zG*}) is independent from $\alpha$. In fact, using the characteristic equations of the cover $f$, one can choose a local base $y^v_r$ of $L_{\chi_v}^{-1}$ on $V_{(\alpha, r)}$ that does not depend on $\alpha$ and satisfies the relation: \begin{equation}\label{eqstrsuX} \left(y^v_r\right)^{d_v}=\prod_{j=1}^k\left(w^j_r \right)^{\frac{a_{vj} d_v}{m_j}}\,. \end{equation} Since $\sum_{i=1}^h b_{vi}\tilde{a}_{ij}=q^v_jm_j+a_{vj}$ $\forall\,j=1,\ldots k$, the two local generators $y^v_r$ and $z_{(\alpha, r)}^{v}$ on $V_{(\alpha,r)}$ differ by a $d_v$-th root of unity, that we denote by $exp( 2 \pi \sqrt{-1}\,(x^v_{(\alpha,r)}/d_v))$. If we multiply $\tilde{z}_{(\alpha, r)}^{i}$ by $exp( 2 \pi \sqrt{-1}(u^i_{(\alpha,r)}/\tilde{d_i}))$, then ${(x^v_{(\alpha,r)}/d_v)}$ becomes ${(x^v_{(\alpha,r)}/d_v)+\sum_{i=1}^h (b_{vi}/\tilde{d_i})\,u^i_{(\alpha, r)}}$. Hence, we only need to solve the linear system of congruences, $\forall\,(\alpha, r)$: \begin{equation} \sum_{i=1}^h\frac{d_v b_{vi}}{\tilde{d_i}}u^i_{(\alpha, r)}\equiv x^v_{(\alpha, r)}\;\;\;\mbox{mod}\,d_v\hspace{20 mm}v=1, \ldots n; \end{equation} since we assume that $d_v$ is a power of a prime number, this system admits a solution according to the Chinese Remainder's Theorem and Lemma \ref{Lemma}. So we can assume that the expression $z_{(\alpha, r)}^{v}$ in (\ref{zG*}) does not depend on $\alpha$ and it is the pullback of a local generator of the corresponding eigensheaf on $X$: we write $z_{r}^{v}=z_{(\alpha, r)}^{v}$. For later use, we define the corresponding cocycle $h^{v}_{(r,s)}$ ($v=1,\ldots n$) by the rule: \begin{equation}\label{cochv} z_{r}^{v}=h^{v}_{(r,s)}z_{s}^{v} \hspace{15 mm}\mbox{on }U_r\cap U_s \end{equation} and we observe that, according to (\ref{eqstrsuX}), the following relation holds: \begin{equation}\label{relhv} (h^{v}_{(r,s)})^{d_v}=\prod_{j=1}^k(k^j_{(r,s)})^{\frac{a_{vj} d_v}{m_j}}\,\hspace{15mm}\mbox{if }\chi_v=\sum_{j=1}^k a_{ij}\psi_j, \;\;0\leq a_{ij}<m_j. \end{equation} \vspace{3 mm} In order to compute the class of the extension (\ref{extgrouteogen}), for every ${\gamma \in \Gamma}$ we choose a lifting ${\tilde{\gamma} \in \tilde{\Gamma}}$. By Lemma \ref{commut}, the induced map ${\tilde{\gamma}_{*}: \tilde{f\,}_{*} {\cal O}_{\tilde {Y}} \to \tilde{f\,}_{*}{\cal O}_{\tilde {Y}}}$ is a ${\cal O}_{\tilde {X}}$-algebra isomorphism lifting ${\gamma : \tilde{X} \to \tilde{X}}$; in terms of the chosen trivializations we may write: \begin{equation} \tilde{z}^{i}_{(\alpha ,r)} \stackrel{\tilde{\gamma}_{*}}{\mapsto} \sigma^{i,\gamma}_{(\alpha,r)}\tilde{z}^{i}_{(\gamma\alpha ,r)}\;\;\;\;\;\; \;\forall\gamma , \alpha \in \Gamma , i=1,\ldots h \end{equation} for a suitable choice of a $\tilde{d}_{i}$-th root of unity $\sigma^{i,\gamma}_{(\alpha,r)}$, $i=1, \ldots h$. For later use, we write down the transition relations for the constants $\sigma^{i,\gamma}_{(\alpha,r)}$. Let $s,t$ be such that ${U_{s}\cap U_{t} \neq \emptyset}$ and let $\alpha,\gamma \in\Gamma$; then, for $i=1,\ldots h$: \begin{equation} \label{rel} \sigma^{i,\gamma}_{(\alpha ,s)}\tilde{h}^{i}_{(\gamma\alpha ,s,t)}= \sigma^{i,\gamma}_{(\alpha\beta (s,t),t)}\left(\tilde{h}^{i}_{(\alpha ,s,t)}\circ \gamma^{-1}\right)\;\;\;\;\;\;\;{\rm on}\;V_{(\gamma\alpha, s,t)}. \end{equation} We now exploit the action of the chosen elements in $\tilde{\Gamma}$ on $\tilde{f\,}_{*} {\cal O}_{\tilde {Y}}$ in order to compute the class ${c(f)\in H^{2}(\Gamma ,K)}$ associated to the extension (\ref{extgrouteogen}), and its image ${c(f)\in H^{2}(X,K)}$. For any given $\delta,\gamma\in\tilde{\Gamma}$, the action of $(\widetilde{\delta\gamma})_{*}^{-1}\tilde{\delta}_{*} \tilde{\gamma}_{*}$ on $L^{-1}_{\tilde{\chi}_{i}}$, $i = 1, \ldots h$, is described with respect to the chosen trivializations by: \begin{equation} \label{comm} \tilde{z}^{i}_{(\alpha ,r)} \mapsto \left(\sigma^{i,(\delta\gamma)}_{(\alpha,r)}\right)^{-1} \sigma^{i,\delta}_{(\gamma\alpha ,r)} \sigma^{i,\gamma}_{(\alpha ,r)} \tilde{z}^{i}_{(\alpha ,r)}\;\;\;\;\;\;\;\;\;\;\forall r,\;\forall\alpha\in\Gamma, \; i=1,\ldots h. \end{equation} Since (\ref{comm}) represents a line bundle automorphism given by a root of the unity, the expression does not depend on $(\alpha ,r)$ by the connectedness of $\tilde{X}$: therefore, we may set $\alpha = 1$. So, the class ${c(f)\in H^{2}(\Gamma ,K)}$ is represented by the cocycle: \begin{equation} c(f)(\delta ,\gamma )=\left( \left(\sigma^{i,(\delta\gamma)}_{(1,r)}\right)^{-1} \sigma^{i,\delta}_{(\gamma,r)} \sigma^{i,\gamma}_{(1,r)}\right)_{i=1,\ldots h}\; \;\;\;\;\;\forall r, \end{equation} where an element of $K\subseteq \tilde{G}$ is represented by its coordinates with respect to the basis dual to $\{\tilde{\chi}_{1}, \ldots \tilde{\chi}_{h}\}$. According to (\cite{kn:Mu}, page 23), the class ${c(f)\in H^{2}(X,K)}$ is represented on $V_{(1,r,s)}\cap V_{(1, r,t)}$ by the cocycle: \begin{equation} \label{uffa} c(f)_{r,s,t}=c(f)(\beta (r,s),\beta (s,t))= \left( \left(\sigma^{i,\beta (r,t)}_{(1,p)}\right)^{-1} \sigma^{i,\beta (r,s)}_{(\beta (s,t),p)} \sigma^{i,\beta (s,t)}_{(1 ,p)} \right)_{i=1,\ldots h}\; \;\;\;\;\;\;\forall p \end{equation} for $r,s,t$ such that ${U_{r}\cap U_{s}\cap U_{t}\neq\emptyset}$. We set $p=t$ and, by the relation (\ref{rel}), we rewrite (\ref{uffa}) as follows: \begin{equation}\label{coccf2} c(f)_{r,s,t}=\left( \left(\sigma^{i,\beta (r,t)}_{(1,p)}\right)^{-1} \sigma^{i,\beta (t,r)}_{(1,r)} \sigma^{i,\beta (s,t)}_{(1 ,t)} \tilde{h}^{i}_{(\beta (r,s),s,t)} \left( \tilde{h}^{i}_{(1,s,t)}\circ \beta(s,r)\right)^{-1} \right)_{i=1, \ldots h}\,; \end{equation} this shows that $c(f)_{r,s,t}$ differs from the following cocycle (that we still denote by $c(f)_{r,s,t}$ by abuse of notation): \begin{equation}\label{coccfnuovo} c(f)_{r,s,t}=\left( \tilde{h}^{i}_{(\beta (r,s),s,t)} \left( \tilde{h}^{i}_{(1,s,t)}\circ \beta(s,r)\right)^{-1}\right)_{i=1, \ldots h}\, \end{equation} by the coboundary of the cochain: \begin{equation} g_{r,t}=\left(\sigma^{i,\beta(r,t)}_{(1,t)}\right)_{i=1, \ldots h}\,. \end{equation} The cochain $g_{r,t}$ actually takes values in $K$: in fact, it is enough to check $g_{r,t}$ acts trivially on the eigensheaves corresponding to the chosen generators $\chi_v$ of $G^*$. This follows easily by the prevous choices since the action of $\beta(r,t)$ on $L_{\chi_v}^{-1}$ is given locally by ${\prod_{i=1}^h\left(\sigma^{i,\beta(r,t)}_{(1,t)}\right)^{b_{vi}}}$. \vspace{3mm} {\sc Step II:} we compute a cocycle representing $\xi\,\in\,H^2(X,K)$. For every $r$ and for every $l = 1, \ldots q$, we choose a local generator $y^{l}_{r}$ for ${\cal O}_X(-M_{l})$ on $U_{r}$; if $M_{l}$ has finite order $e$, then we require: \begin{equation} \label{tors} \left(y^{l}_{r}\right)^{e}=1\,. \end{equation} We set $m={\rm lcm}\,\{m_j\}_{j=1, \ldots k}$. For every pair of indices $(r, s)$ such that ${U_{r}\cap U_{s} \neq \emptyset}$ we write: \begin{equation} y^{l}_{r} = \mu^{l}_{(r, s)}\,y^{l}_{s}\;\;{\rm on}\,U_{r}\cap U_{s} \end{equation} and we choose a $m$-th root $\hat{\mu}^{l}_{(r, s)}$ of $\mu^{l}_{(r, s)}$ in such a way that ${\hat{\mu}^{l}_{(s, r)}=(\hat{\mu}^{l}_{(r, s)})^{-1}}$. Then, as in \cite{kn:Cato}, (2.45), one sees that the image of the class of $-M_{l}$ in $H^2(X,{\bf Z}/m_l{\bf Z})$ is represented on ${U_{r}\cap U_{s}\cap U_{t}}$ by the cocycle ${\left(\hat{\mu}^{l}_{(r, s)} \hat{\mu}^{l}_{(s, t)} (\hat{\mu}^{l}_{(r, t)})^{-1} \right)^{m/m_l}}$, $l=1, \ldots q$. We conclude that the class ${\xi=\Theta_*([M_1], \ldots [M_q])}$ is represented on ${U_{r}\cap U_{s}\cap U_{t}}$ by: \begin{equation}\label{cocc} \xi_{r,s,t}=\left(\prod_{j=1}^k\prod_{l=1}^{q}\,\left(\hat{\mu}^{l}_{(r, s)} \hat{\mu}^{l}_{(s, t)} \hat{\mu}^{l}_{(t, r)}\right)^{-(m/m_j) c_{jl}\tilde{a}_{ij}}\right)_{i=1, \ldots h} \end{equation} \vspace{3mm} {\sc Step III:} we show that ${\xi=c(f)}$. We remark that, according to (\ref{decteor}), the cocycle $k^{j}_{(r, s)}$ in (\ref{coc-Dj}) representing ${{\cal O}_{X}(-D_{j})}$ ($j=1, \ldots k$) and the cocycles $\mu^{l}_{(r,s)}$ representing $-M_l$ ($l=1, \ldots q$) are related as follows: \begin{equation} k^{j}_{(r, s)}=\prod_{l=1}^q\left(\mu^{l}_{(r, s)}\right)^{c_{jl}}\frac{f^j_r}{f^j_s} \end{equation} for suitable nowhere vanishing holomorphic functions $f_r$ on $U_{ r}$. For every $j=1, \ldots k$ and every $r$, we choose a $m$-th root $\hat{f}^j_r$ of $f^j_r$ on $U_{ r}$; then, the expression: \begin{equation}\label{tildekmu} \hat{k}^j_{(r,s)}=\prod_{l=1}^{q}\,\left(\hat{\mu}^{l}_{(r, s)}\right)^{c_{jl}(m/m_j)} \left(\frac{\hat{f\,}^j_r}{\hat{f\,}^j_s}\right)^{(m/m_j)} \end{equation} is a $m_j$-th root of the cocycle $k^j_{(r,s)}$ and, as before, the product $\prod_{l=1}^{q}\,\left(\hat{\mu}^{l}_{(r, s)} \hat{\mu}^{l}_{(s, t)} \hat{\mu}^{l}_{(t, r)}\right)^{ c_{jl}(m/m_j)}$ yields a cocycle representing the image of the class of $-D_j$ in $H^2(X,{\bf Z}/m_j{\bf Z})$. In this notation, by (\ref{tors}), we rewrite as follows the cocycle in (\ref{cocc}) representing the class $\xi$: \begin{equation}\label{cocc1} \xi_{r,s,t}=\left(\prod_{j=1}^k \left( \hat{k}^j_{(r,s)}\hat{k}^j_{(s,t)}\hat{k}^j_{(t,r)} \right)^{-\tilde{a}_{ij}}\right)_{i=1, \ldots, h}\,. \end{equation} Let $\epsilon = exp(\frac{2\pi \sqrt{-1}}{m})$. Then, by (\ref{rela}), one has: \begin{equation}\label{defq} \tilde{h}^{i}_{(\alpha, r, s)}\;=\;\prod_{j=1}^{k}(\hat{k}^{j}_{(r,s)}) ^{\tilde{a}_{ij}}\; \epsilon^{-q^{i}_{(\alpha,r,s)}} \end{equation} where $q^{i}_{(\alpha,r,s)}$ is an integer, multiple of $m/\tilde{d_i}$, and (\ref{coccfnuovo}) may be rewritten as: \begin{equation} c(f)_{r,s,t} = (\epsilon^{q^{i}_{(1,s,t)}-q^{i}_{(\beta(r,s),s,t)}}) _{i=1,\ldots h}\;\;. \end{equation} {}From the cocycle condition (\ref{coch}) for $\tilde{h}^{i}_{(\alpha , r, s)}$, it follows: \begin{equation} \xi_{r,s,t}=(\epsilon^{-q^{i}_{(\alpha ,r,s)}-q^{i}_{(\alpha\beta (r,s),s,t)} -q^{i}_{(\alpha\beta(r,t),t,r)}})_{i=1,\ldots h}\;\;\;\forall\, r,s,t, \forall\,\alpha\in\Gamma\,. \end{equation} In particular, for $\alpha = 1$, one gets: \begin{equation} \xi_{r,s,t}=(\epsilon^{-q^{i}_{(1,r,s)}-q^{i}_{(\beta (r,s),s,t)} -q^{i}_{(\beta (r,t),t,r)}})_{i=1,\ldots h}\;\;. \end{equation} So, one has: \begin{equation} c(f)_{r,s,t} = \xi_{r,s,t} (\epsilon^{q^{i}_{(1,r,s)}+q^{i}_{(1,s,t)} +q^{i}_{(\beta(r,t),t,r)}})_{i=1,\ldots h}\;\;. \end{equation} By the definition of $q^{i}_{(\alpha,r,s)}$, this equality can be rewritten as follows: \begin{equation}\label{cobord} c(f)_{r,s,t} = \xi_{r,s,t} (\epsilon^{q^{i}_{(1,r,s)}-q^{i}_{(1,r,t)} +q^{i}_{(1,s,t)}})_{i=1,\ldots h}\,. \end{equation} To complete the proof of the theorem, we show that we can choose the $m$-th root $\hat{f\,}_j$ of $f_j$ ($j=1, \ldots k$) so that: \begin{equation} \underline{q}_{(1,r,s)}=\left(\epsilon^{q^{i}_{(1,r,s)}} \right)_{i=1,\ldots h} \mbox{ is an element of } K, \;\forall\,(r,s). \end{equation} Let $\chi_v$ one of the chosen generators of $G^*$. According to (\ref{zG*}), (\ref{relhv}) and (\ref{defq}), the action of $\underline{q}_{(1,r,s)}$ on $L_{\chi_v}^{-1}$ is given by a $d_v$-th root of unity, that we denote by $exp({2\pi \sqrt{-1}\,\frac{x_v}{d_v}})$ (for a suitable integer $x_v$). We want to show that we can assume that $x_v\equiv 0$ mod $d_v$, $v=1, \ldots n$. We observe that: \begin{equation}\label{cobinK} exp({2\pi \sqrt{-1}\,\frac{x_v}{d_v}})= \prod_{i=1}^h \epsilon^{q^{i}_{(1,r,s)}b_{vi}}= (h^{v}_{(1,r,s)})^{-1}\prod_{j=1}^k(\hat{k}_{(r,s)}^j)^{a_{vj}} \end{equation} and we compute the right-hand side of (\ref{cobinK}). By (\ref{tildekmu}), one must have: \begin{equation} \begin{array}{cl} h^{v}_{(1,r,s)}&=exp({-2\pi \sqrt{-1}\,\frac{x_v}{d_v}})\prod_{j=1}^k (\hat{k}^j_{(r,s)})^{a_{vj}}\\ &= exp({-2\pi \sqrt{-1}\,\frac{x_v}{d_v}}) \prod_{l=1}^q (\hat{\mu}^l_{(r,s)})^{m\sum_{j=1}^k \frac{c_{jl}a_{vj}}{m_j}} \prod_{j=1}^k\left(\frac{\hat{f\,}^j_r}{\hat{f\,}^j_s} \right)^{a_{vj}(m/m_j)}\\ &= exp({-2\pi \sqrt{-1}\,\frac{x_v}{d_v}}) \prod_{l=1}^q (\hat{\mu}^l_{(r,s)})^{\frac{m}{d_v}\sum_{j=1}^k \frac{c_{jl}d_v a_{vj}}{m_j}} \prod_{j=1}^k\left(\frac{\hat{f\,}^j_r}{\hat{f\,}^j_s} \right)^{a_{vj}(m/m_j)}. \end{array} \end{equation} On the other hand, as in Lemma \ref{coeffinK}, we write $L_{\chi_v}\equiv \sum_{l=1}^q \lambda_{vl}M_l$ and we get the following relation form of cocycles on $V_{(1,r,s)}$: \begin{equation} h^{v}_{(1,r,s)}=\prod_{l=1}^q (\mu^l_{(r,s)})^{\lambda_{vl}} \frac{\varphi^v_{r}}{\varphi^v_{s}} \end{equation} for suitable nowhere vanishing holomorphic functions $\varphi^v_{r}$ on $U_{r}$. According to Lemma \ref{Lemma} and to (\ref{tors}), we can then assume that in the previous equations one has: \begin{equation} \prod_{l=1}^q (\hat{\mu}^l_{(r,s)})^{\left(\frac{m}{d_v}\sum_{j=1}^k \frac{c_{jl}d_v a_{vj}}{m_j}\right)-d_v \lambda_{vl}}=1 \end{equation} so that one gets: \begin{equation} \label{xv} exp({2\pi \sqrt{-1}\,\frac{x_v}{d_v}})= \prod_{j=1}^k\left(\frac{\hat{f\,}^j_r}{\hat{f\,}^j_s} \right)^{a_{vj}(m/m_j)}\frac{\varphi^v_s}{\varphi^v_r}\,. \end{equation} We observe that we may assume that: \begin{equation} \varphi^v_r= exp(2\pi \sqrt{-1}\,\frac{t^v_r}{d_v}) \prod_{j=1}^k(\hat{f\,}^j_r)^{a_{vj}(m/m_j)} \end{equation} for suitable integers $t^v_r$; hence the equation (\ref{xv}) gives: \begin{equation} \frac{x_v}{d_v}- \frac{t^v_s}{d_v}+\frac{t^v_r}{d_v}\;\in\;{\bf Z}. \end{equation} If we replace $\hat{f\,}^j_r$ by $exp(2\pi \sqrt{-1}\,\frac{s^j_r}{m})\hat{f\,}^j_r$, then $\frac{t^u_r}{d_v}$ is replaced by $\frac{t^u_r}{d_v}+\sum_{j=1}^k\frac{a_{vj}s_j}{m_j}$. Therefore, we need to solve the system: \begin{equation} \sum_{j=1}^k\frac{a_{vj}s^j_r}{m_j}\equiv t^v_r \hspace{10mm} \mbox{mod}\,d_v \hspace{15mm}v=1,\ldots n. \end{equation} Since this is possible according to Lemma \ref{Lemma} and the Chinese Remainder's Theorem, the proof is complete. \hfill\qed \begin{Remark}\label{dipdaLchi} The cohomology class of the extension (\ref{extgrouteogen}) of the fundamental groups depends on the choice of the solution $\{L_\chi\}$ of the characteristic relations (\ref{eqstr}) for the covering $f$. Moreover, covers corresponding to different solutions $\{L_\chi\}$ may not be homeomorphic. {\rm This is shown, for instance, by the following class of examples. Denote by $e_i$ the standard generators of the group $({\bf Z}/4{\bf Z})^3$ and let $G$ be the quotient of $({\bf Z}/4{\bf Z})^3$ by the subgroup generated by ${2e_1+2e_2+2e_3}$. Let now $X$ be a smooth projective surface such that ${H^2(\pi_1(X), {\bf Z}/2 {\bf Z})\neq 0}$ and Pic($X$) has a 2-torsion element $\eta$ whose class in ${H^2(X, {\bf Z}/2{\bf Z})}$ is non zero. Fix a very ample divisor $H$ on $X$ and choose suitable divisors $D_i$ ($i=1,2,3$) such that $D_i\equiv 4H$ and the $D_i$'s are in general position. Then there exists a smooth abelian $G$-cover ${f:Y\to X}$ ramified on the $D_i$'s ($i=1,2,3$), with inertia subgroup $G_i=<e_i>={\bf Z}/4{\bf Z}$ and character $\psi_i$ dual to $e_i$, respectively. In fact, taking the characters $\chi_1=\psi_1+3\psi_3$, $\chi_2=\psi_2+3\psi_3$, $\chi_3=2\psi_3$ as generators of $G^*$, the characteristic relations (\ref{eqstr}) of the cover $f$ are: \begin{equation} \label{eqstrex} \left\{ \begin{array}{lll} 4L_1&\equiv&D_1+3D_3\\ 4L_2&\equiv&D_2+3D_3\\ 2L_3&\equiv&D_3 \end{array} \right. \end{equation} and admit, in particular, the solution $L_1= L_2= L_3= 2H$. Under these hypotheses, $L_3$ generates the subgroup $<D_i,L_\chi>$ ($i=1,2,3,\chi\,\in\,G^*$) of Pic($X$) and the decomposition $D_i\equiv 2L_3$ has the properties requested in Prop.\ref{coeffinK}. According to Prop.\ref{top} and Thm.\ref{mtgen}, since the pull back $\tilde{D}_i$ of $D_i$ under the universal cover $\tilde{X}$ of $X$ is 2-divisible, $\forall\,i$, then the map $f$ induces a central extension of the form: \begin{equation} 0\to {\bf Z}/2{\bf Z}\to \pi_1(Y)\to \pi_1(X)\to 1 \end{equation} and the cohomology class of this extension in $H^2(\pi_1(X), {\bf Z}/2{\bf Z})\subseteq H^2(X,{\bf Z}/2{\bf Z})$ is the image $\Psi_*([L_3])$ of the Chern class of $L_3$ under the map induced in cohomology by the standard projection $\Psi:{\bf Z}\to {\bf Z}/2{\bf Z}$: so, this class is trivial. Let now $\overline{Y}$ be the $G$-cover of $X$ corresponding to the solution $\overline{L}_i=2H+\eta$ ($i=1,2,3$) of (\ref{eqstrex}); in this case the cohomology class describing $\pi_1(\overline{Y})$ is given by $\Psi_*([L_3+\eta])$ and, by the hypotheses made, it is not trivial. In particular, when $X$ is a projective variety with $\pi_1(X)={\bf Z}/2{\bf Z}$, the previous construction yields two non homeomorphic $G$-covers $Y$, $\overline{Y}$ of $X$, branched on the same divisor, with the same inertia subgroups and characters, such that: \begin{equation} \pi_1(Y)=({\bf Z}/2{\bf Z})^2\hspace{20mm}\pi_1(\overline{Y})={\bf Z}/4{\bf Z}\,. \end{equation} } \end{Remark} \hfill\qed The following theorem is an attempt to determine to what extent the class $c(f)$ depends on the choice of the $L_\chi$'s, once the branch divisor and the covering structure are fixed. \begin{Theorem} \label{mt} Same hypotheses and notation as in the statement of Thm.\ref{mtgen}. Consider the class $c(f)\,\in\,H^{2}(\pi_1(X), K)$ associated to the central extension (\ref{extgrouteogen}) given by the fundamental groups and denote by ${i(c(f)) \in H^{2}(\pi_1(X),\tilde{G})\subseteq H^{2}(X,\tilde{G})}$ its image via the map induced in cohomology by the inclusion (\ref{extgrouO}) $K\subseteq \tilde{G}$. Denote by $\Phi$ the group homomorphism defined as follows: \begin{equation} \begin{array}{rccl} \Phi :& {\bf Z}^{k}& \rightarrow &\tilde{G}\\ &(x_{1}, \ldots x_{k})&\rightarrow& g_{1}^{x_{1}}\cdots g_{k}^{x_{k}} \end{array} \end{equation} and by ${\Phi_{*}: H^{2}(X,{\bf Z}^{k}) \rightarrow H^{2}(X,\tilde{G})}$ the map induced by $\Phi$ in cohomology. Then: \begin{equation} \label{coc} i(c(f)) = \Phi_{*}([D_{1}], \ldots [D_{k}]) \end{equation} where $[\Delta]$ denotes the class of a divisor $\Delta$ on $X$ in ${H^{2}(X,{\bf Z})}$. \end{Theorem} \begin{Corollary}\label{mtcor} Same hypotheses and notation as in Thm.\ref{mtgen}. Assume moreover that the natural morphism $Hom(\pi_1(X),\tilde{G})\to Hom(\pi_1(X),G)$, induced by the surjection ${\tilde{G}\to G}$, is surjective. Then the map ${i:H^2(\pi_1(X),K)\to H^2(\pi_1(X),\tilde{G})}$ is injective and the class $\Phi_{*}([D_{1}], \ldots [D_{k}])$ in (\ref{coc}) determines uniquely the class $c(f)\,\in\,H^2(\pi_1(X),K)$ of the extension (\ref{extgrouteogen}) of the fundamental groups. This happens, in particular, if $\pi_1(X)$ is torsion free or $Hom(\pi_1(X),G)=0$ (e.g., if $\pi_1(X)$ is finite with order coprime to the order of $G$), or the sequence ${0\to K\to \tilde{G}\to G\to 0}$ splits. \end{Corollary} {\sc Proof of Thm.}\ref{mt}. We keep the notation and the results in Step I of the proof of Thm.\ref{mtgen}, noticing that the cocycle $c(f)_{r,s,t}$ in (\ref{coccfnuovo}) also represents the class $i(c(f))$ in $H^{2}(X,\tilde{G})$. We want to write down a cocycle representing the class ${\Phi_{*}([D_{1}], \ldots [D_{k}])\in H^{2}(X,\tilde{G})}$ and to show that it represents the same cohomology class as the cocycle in (\ref{coccfnuovo}). We consider as before the cocycle $k^{j}_{(r,s)}$ representing ${\cal O}_{X}(-D_{j})$ in the choosen covering ${U_{r}}$ of $X$. For every pair of indices $r$, $s$ with ${U_{r}\cap U_{s}\neq\emptyset}$ and for every $j = 1, \ldots k$, we choose a $m_{j}$-th root $\hat{k}^{j}_{(r,s)}$ of $k^{j}_{(r,s)}$ on ${U_{r}\cap U_{s}}$ in such a way that ${\hat{k}^{j}_{(s,r)}= (\hat{k}^{j}_{(r,s)})^{-1}}$. As before, the image of the class of $-D_{j}$ in $H^{2}(X,{\bf Z}/m_{j}{\bf Z})$ is represented on ${U_{r}\cap U_{s}\cap U_{t}}$ by the cocycle ${\hat{k}^{j}_{(r,s)}\hat{k}^{j}_{(s,t)}\hat{k}^{j}_{(t,r)}}$, ${j = 1,} \ldots k$. Then the class ${-\Phi_{*}([D_{1}], \ldots [D_{k}])\in H^{2}(X,\tilde{G})}$ is represented on ${U_{r}\cap U_{s}\cap U_{t}}$ by: \begin{equation} b_{r,s,t} = (\prod^{k}_{j=1} (\hat{k}^{j}_{(r,s)}\hat{k}^{j}_{(s,t)}\hat{k}^{j}_{(t,r)}) ^{\tilde{a}_{ij}})_{i=1,\ldots h}\;\;. \end{equation} and we have shown in the equality (\ref{cobord}) in the proof of Thm.\ref{mtgen}, Step III, that this cocycle represents the same class then $c(f)_{r,s,t}$ in $H^2(X,\tilde{G})$. \hfill\qed \begin{Remark} \label{rem} {\rm From Thm.\ref{mt} it follows in particular that the class $i(c(f))\in H^{2}(\Gamma,\tilde{G})$ depends only on the class of the $D_{j}$'s in $H^{2}(X,{\bf Z}/m_{j}{\bf Z})$ ($j = 1, \ldots k$), once $G$ and the $g_{j}$'s are fixed. In particular, if $D_j$ is $m_j$-divisible on $X$ ($\forall\; j=1,\ldots k$), then $i(c(f))=0$.} \end{Remark}

9305

alg-geom/9305012

en

https://arxiv.org/abs/alg-geom/9305012

alg-geom/9305012

Claude LeBrun

Claude LeBrun

A Kaehler Structure on the Space of String World-Sheets

13 pages, LaTeX

10.1088/0264-9381/10/9/006

Let (M,g) be an oriented Lorentzian 4-manifold, and consider the space S of oriented, unparameterized time-like 2-surfaces in M (string world-sheets) with fixed boundary conditions. Then the infinite-dimensional manifold S carries a natural complex structure and a compatible (positive-definite) Kaehler metric h on S determined by the Lorentz metric g. Similar results are proved for other dimensions and signatures, thus generalizing results of Brylinski regarding knots in 3-manifolds. Generalizing the framework of Lempert, we also investigate the precise sense in which S is an infinite-dimensional complex manifold.

alg-geom

\section{Introduction} Given a collection of circles in a 4-dimensional oriented Lorentzian space-time, one may consider the space $\cal S$ of unparameterized oriented time-like compact 2-surfaces with the given circles as boundary. The main purpose of the present note is to endow $\cal S$ with the structure of an infinite-dimensional K\"ahler manifold--- i.e. with both a complex structure and a Riemannian metric for which this complex structure is covariantly constant. This was motivated by a construction of Brylinski \cite{bryl}, whereby a K\"ahler structure is given to the space of knots in a Riemannian 3-manifold. In fact, our discussion will be structured so as to apply to codimension 2 submanifolds of a space-time of arbitrary dimension and metrics of arbitrary signature, with the proviso that we only consider those submanifolds for which the normal bundle is orientable and has (positive- or negative-)definite induced metric; thus Brylinski's construction becomes subsumed as a special case. As the reader will therefore see, complex manifold theory thus comes naturally into play when one studies codimension 2 submanifolds of a space-time. On the other hand, complex manifold theory makes a quite different kind of appearance when one attempts to study the intrinsic geometry of 2-dimensional manifolds. If some interesting modification of string theory could be found which invoked both of these observations simultaneously, one might hope to thereby explain the puzzling four-dimensionality of the observed world. Many of the key technical ideas in the present note are straightforward generalizations of arguments due to L\'aszl\'o Lempert \cite{lemp}, whose lucid study of Brylinski's complex structure is based on the theory of twistor CR manifolds \cite{leb}. One of the most striking features of the complex structures in question is that, while they are formally integrable and may even admit legions of local holomorphic functions, they do {\em not} admit enough finite-dimensional complex submanifolds to be locally modeled on any complex topological vector space. This beautifully illustrates the fact, emphasized by Lempert, that the Newlander-Nirenberg Theorem \cite{nn} fails in infinite dimensions. \section{The Space of World-Sheets} Let $(M, g)$ be a smooth oriented pseudo-Riemannian n-manifold. We use the term {\em world-sheet} to refer to a smooth compact oriented codimension-2 submanifold-with-boundary $\Sigma^{n-2}\subset M^n$ for which the inner product induced by $g$ on the conormal bundle $$\nu^{\ast}_{\Sigma}:=\{ \phi \in T^{\ast}M|_{\Sigma}~~~|~~\phi|_{T\Sigma}\equiv 0 \} $$ of $\Sigma$ is definite at each point. If $g$ is Riemannian, this just means an oriented submanifold of codimension 2; on the other hand, if $(M,g)$ is a Lorentzian 4-manifold, a world-sheet is exactly an oriented time-like 2-surface. \begin{defn} Let $(M, g)$ be a smooth oriented pseudo-Riemannian n-manifold, and let $B^{n-3}\subset M^n$ be a smooth codimension-3 submanifold which is compact, without boundary. We will then let ${\cal S}_{M,B}$ denote the space of smooth oriented world-sheets $\Sigma^{n-2}\subset M$ such that $\partial \Sigma =B$. \end{defn} Of course, this space is sometimes empty--- as happens, for example, if $B$ is a single space-like circle in Minkowski 4-space. This said, ${\cal S}_{M,B}$ is automatically a Fr\'echet manifold, and its tangent space at $\Sigma$ is $$T_{\Sigma}{\cal S}_{M,B}=\{ v\in \Gamma (\Sigma, C^{\infty}(\nu_{\Sigma}))~~|~~ v|_{\partial \Sigma}\equiv 0\}~ .$$ Indeed, if we choose a tubular neighborhood of $\Sigma$ which is identified with the normal bundle of an open extension $\Sigma_{\varepsilon}$ of $\Sigma$ beyond its boundary, every section of $\nu_{\Sigma}\to \Sigma$ which vanishes on $\partial \Sigma$ is thereby identified with an imbedded submanifold of $M$, and this submanifold is still a world-sheet provided the $C^1$ norm of the section is sufficiently small. This provides ${\cal S}_{M,B}$ with charts which take values in Fr\'echet spaces, thus giving it the desired manifold structure. Since the normal bundle $\nu_{\Sigma} =(\nu_{\Sigma}^{\ast})^{\ast}=TM/T\Sigma =(T\Sigma)^{\perp}$ of our world-sheet is of rank 2 and comes equipped with an orientation as well as a metric induced by $g$, we may identify $\nu_{\Sigma}$ with a complex line bundle by taking $J: \nu_{\Sigma}\to\nu_{\Sigma}$, $J^2=-1$ to be rotation by $+90^{\circ}$. This then defines an endomorphism ${\cal J}$ of $T{\cal S}$ by $${\cal J} : T_{\Sigma}{\cal S}_{M,B}\to T_{\Sigma}{\cal S}_{M,B}: v \to J\circ v ~ .$$ Clearly ${\cal J}^2=-1$, so that ${\cal J}$ gives ${\cal S}$ the structure of an almost-complex Fr\'echet manifold--- i.e. every tangent space of ${\cal S}$ can now be thought of as a complex Fr\'echet space by defining ${\cal J}$ to be multiplication by $\sqrt{-1}$. In the next sections, we shall investigate the integrability properties of this almost-complex structure. \section{Integrability of the Complex Structure} Let $(M,g)$ denote, as before, an oriented pseudo-Riemannian manifold. Let $Gr_2^{+}(M)$ denote the bundle of oriented 2-planes in $T^{\ast}M$ on which the inner product induced by $g$ is definite. This smooth ($3n-4$)-dimensional manifold then has a natural CR structure \cite{leb,ros} of codimension $n-2$. Let us review how this comes about. Let $\hat{N}\subset [({\Bbb C}\otimes T^{\ast}M)- T^{\ast}M]$ denote the set of non-real null covectors of $g$, and let $N\subset {\Bbb P} ({\Bbb C}\otimes T^{\ast}M)$ be its image in the fiber-wise projectivization of the complexified cotangent bundle. There is then a natural identification of ${N}$ with $Gr_2^{+}(M)$. Namely, using pairs $u,v\in T_x^{\ast}M$ of real covectors satisfying $\langle u, v\rangle =0$ and $\langle u, u\rangle =\langle v, v\rangle$, we define a bijection between these two spaces by $$Gr_2^{+}(M) \ni \mbox{oriented span}(u,v)\leftrightarrow [u+iv]\in {N}\subset {\Bbb P} ({\Bbb C}\otimes T_x^{\ast}M)$$ which is independent of the representatives $u$ and $v$. But, letting $\vartheta=\sum p_jdx^j$ denote the canonical complex-valued 1-form on the total space of ${\Bbb C}\otimes T^{\ast}M\to M$, and letting $\omega$ be the restriction of $d\vartheta$ to $\hat{N}$, the distribution $$ \hat{D}=\ker (\omega:{\Bbb C}\otimes T\hat{N}\to {\Bbb C}\otimes T^{\ast}\hat{N})$$ is involutive by virtue of the fact that $\omega$ is closed; since $ \hat{D}$ also contains no non-zero real vectors as a consequence of the fact that $\hat{N}\cap T^{\ast}M=\emptyset$, $ \hat{D}$ is a CR structure on $\hat{N}$, the codimension of which can be checked to be $n-2$. This CR structure is invariant under the natural action of ${\Bbb C}^{\times}$ on $\hat{N}$ by scalar multiplication, and thus descends to a CR structure $D$ on $N=Gr_2^{+}(M)$, again of codimension $n-2$. Moreover, $\vartheta|_{\hat{N}}$ descends to $N$ as a CR line-bundle-valued 1-form $$\theta\in \Gamma (N, {\cal E}^{1,0} (L))~,~~ \bar{\partial}_b\theta =0~, $$ where, letting $T^{1,0}N:=({\Bbb C}\otimes TN)/D$, $L^{\otimes (n-1)}=\bigwedge^{(2n-3)}T^{1,0}N$, ${\cal E}^{1,0} (L):= C^{\infty} (L\otimes (T^{1,0}N)^{\ast})$, and $\bar{\partial}_b$ is naturally induced by $d|_D$. The CR structure $D$ of $N$ may be expressed in the form $$D=\{ v-iJv~| ~~v\in H\}$$ for a unique rank $2n-2$ sub-bundle $H$ of the real tangent bundle $TN$ and a unique endomorphism $J$ of $H$ satisfying $J^2=-1$. In these terms the geometric meaning of the CR structure of $N$ is fairly easy to describe. Indeed, if $\varpi : Gr_2^{+}(M)\to M$ is the tautological projection, then $H_P = (\varpi_{\ast P})^{-1}(P)$ for every oriented definite 2-plane $P\subset TM$. On vertical vectors, $J$ acts by the standard complex structure on the quadric fibers of $N\to M$; whereas $J$ acts on horizontal vectors by $90^{\circ}$ rotation in the 2-plane $P\subset TM$. This point of view, however, obscures the fact that $D$ is both involutive and unaltered by conformal changes $g\mapsto e^fg$. A compact $(n-2)$-dimensional submanifold-with-boundary $S\subset N$, will be called a {\em transverse sheet} if its tangent space is everywhere tansverse to the CR tangent space of $N$: $$TN|_S= TY\oplus H|_Y~ .$$ As before, let $B^{n-3}\subset M$ denote a compact codimension-3 submanifold, and let $\varpi :N\to M$ be the canonical projection. We will then let $\hat{{\cal S} }_{N,B}$ denote the set of transverse sheets $S\subset N$ such that $\varpi$ maps $\partial S$ diffeomorphically onto $B$. Thus $\hat{{\cal S} }_{N,B}$ is a Fr\'echet manifold whose tangent space at $S$ is given by $$T\hat{{\cal S} }_{N,B}|_S=\{ v\in \Gamma (S, C^{\infty}(H|_S))~ |~\varpi_{\ast}(v|_{\partial S})\equiv 0\} ~ ,$$ and hence $J: H\to H$ induces an almost-complex structure $\hat{\cal J}$ on $ \hat{{\cal S} }_{N,B}$ by $\hat{\cal J} (v):= J\circ v$. \begin{propn} The almost-complex structure $\hat{\cal J}$ on the space $ \hat{{\cal S} }_{N,B}$ of transverse sheets is formally integrable--- i.e. $$\tau (v,w):= \hat{\cal J}[v,w] -[v,\hat{\cal J}w] -[\hat{\cal J}v,w] -\hat{\cal J} [\hat{\cal J}v,\hat{\cal J}w] =0 $$ for all smooth vector fields $v$, $w$ on $\hat{{\cal S} }_{N,B}$. \end{propn} \begin{proof} The Fr\"ohlicher-Nijenhuis torsion $\tau (v,w)$ is tensorial in the sense that its value at $S$ only depends on the values of $v$ and $w$ at $S$. Given $v_S,w_S \in \{ v\in \Gamma (S, C^{\infty}(H|_S))~ |~\varpi_{\ast}(v|_{\partial S})\equiv 0\}$, we will now define preferred extensions of them as vector fields near $S\in \hat{{\cal S} }_{N,B}$ in such a manner as to simplify the computation of $\tau (v,w)= \tau (v_S,w_S)$. To do this, we may first use a partition of unity to extend $v_S$ and $w_S$ as sections $\hat{v}, \hat{w}\in \Gamma (N, C^{\infty}(H))$ defined on all of of $N$ in such a manner that $\hat{v}$ and $\hat{w}$ are tangent to the fibers of $\varpi$ along all of $\varpi^{-1}(B)$. Now let $U\subset N$ be a tubular neighborhood of $S$ which is identified with the normal bundle $H$ of some open extension $S_{\epsilon }$ of $S$, and let $\hat{U}\subset \hat{{\cal S} }_{N,B}$ be the set of transverse sheets $S'\subset U$. We may now define our preferred extensions of $v$ and $w$ of $v_S$ and $w_S$ on the domain $\hat{U}$ by letting the values of $v$ and $w$ at $S'\subset U$ be the restrictions of $\hat{v}$ and $\hat{w}$ to ${S'}$. Notice that $[ v,w ]$ is then precisely the vector field on $\hat{U}$ induced by $[\hat{v}, \hat{w}]$, whereas $\hat{\cal J}v$ is the vector field induced by $J\hat{v}$. Since the integrability condition for $(N,D)$ says that $$J([\hat{v},\hat{w}] -[J\hat{v},J\hat{w}]) =[\hat{v},J\hat{w}] +[J\hat{v},\hat{w}]~ , $$ it therefore follows that $$\hat{\cal J}[v,w] -\hat{\cal J} [\hat{\cal J}v,\hat{\cal J}w] = [v,\hat{\cal J}w] +[\hat{\cal J}v,w] ~ ,$$ so that $\tau (v,w)=0$, as claimed. \end{proof} We now observe that there is a canonical imbedding \begin{eqnarray*} {{\cal S} }_{M,B}&\stackrel{\Psi}{\hookrightarrow} &\hat{{\cal S} }_{N,B}\\ \Sigma &\mapsto & \nu_{\Sigma } \end{eqnarray*} obtained by sending a world-sheet to its normal-bundle, thought of as the image of a section of $Gr_2^+(M)|_{\Sigma }= N|_{\Sigma }$; thought of in this way, it is easy to see that $\nu_{\Sigma}\subset N$ is a transverse submanifold. \begin{thm} The imbedding $\Psi$ realizes $({{\cal S} }_{M,B}, {\cal J})$ as a complex submanifold of $(\hat{{\cal S} }_{N,B}, \hat{\cal J})$. In particular, the almost-complex structure ${\cal J}$ of ${{\cal S} }_{M,B}$ is formally integrable. \label{imb} \end{thm} \begin{proof} The projection $\varpi: N\to M$ induces a map $\hat{\varpi}: \hat{{\cal S} }_{N,B}\to{{\cal S} }_{M,B}$ which is a left inverse of $\Psi$ and satisfies $\hat{\varpi}_{ *}\hat{\cal J} ={\cal J}\hat{\varpi}_{ *}$. It therefore suffices to show that the tangent space of the image of $\Psi$ is $\hat{\cal J}$-invariant. Now the condition for a transverse sheet $S\subset N$ to be the $\Psi$-lifting of the world-sheet $\varpi(S)\subset M$ is exactly that $\theta|_S\equiv 0$. When $S$ satisfies this condition, a connecting field $v\in\Gamma (S, {\cal E}(H))$ then represents a vector $\hat{v}\in T\hat{\cal S}$ which is tangent to the image of $\Psi$ iff \be (v \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\theta)|_{TS}+d(v \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm \theta)|_{TS}\equiv 0~ ;\label{leg} \end{equation} the exterior derivative of $\theta$ may here be calculated in any local trivialization for the line bundle $L$, since the left-hand side rescales properly under changes of trivialization so as define an $L$-valued 1-form on $S$. But since $\theta\in \Gamma (N, {\cal E}^{1,0} (L))$ satisfies $\bar{\partial}_b \theta =0$, it follows that $$(Jv \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\theta)|_{TS}+d(Jv \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm \theta)|_{TS}=i(Jv \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\theta)|_{TS}+ id(Jv \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm \theta)|_{TS} $$ because $\theta$ and $d\theta$ are of types (1,0) and (2,0), respectively. The tangent space of the image of $\Psi$ is therefore $\hat{\cal J}$-invariant, and the claim follows. \end{proof} \begin{defn} Let $({\Frak X}, {\Frak J})$ be an almost-complex Fr\'echet manifold, and let $f: U\to {\Bbb C}$ be a differentiable function defined on an open subset of ${\Frak X}$. We will say that $f$ is ${\Frak J}$-holomorphic if $$({\Frak J}v)f=ivf~~\forall v\in TU~.$$ \end{defn} \begin{defn} An almost-complex Fr\'echet manifold $({\Frak X}, {\Frak J})$ is called {\em weakly integrable} if for each real tangent vector $w\in T{\Frak X}$ there is a ${\Frak J}$-holomorphic function $f$ defined on a neighborhood of the base-point of $w$ such that $wf\neq 0$. \end{defn} \begin{thm} Suppose that $(M,g)$ is real-analytic. Then $(\hat{{\cal S} }_{N,B}, \hat{\cal J})$ is weakly integrable.\label{wint} \end{thm} \begin{proof} If $(M,g)$ is real-analytic, so is the CR manifold $(N,D)$, and we can therefore realize $(N,D)$ as a real submanifold of a complex manifold $(2n-3)$-manifold $\cal N$. This can even be done explicitly by taking $\cal N$ to be a space of complex null geodesics for a suitable complexification of $(M,g)$. Now let $S\subset N\subset {\cal N}$ be any transverse sheet. Then there is a neighborhood $V\subset {\cal N}$ of $S$ which can be holomorphically imbedded in some ${\Bbb C}^{\ell}$. Indeed, let $Y\subset {\cal N}$ be a totally real $(2n-3)$-manifold containing $S$, let $f: Y\to {\Bbb R}^{\ell}$ be a smooth imbedding, and let $Y_0$ be a precompact neighborhood of $S\subset Y$ with smooth boundary. By \cite{wel},the component functions $f^j|_{Y_0}$ are limits in the $C^1$ topology of the restrictions of holomorphic functions. Using such an approximation of $f$, we may therefore imbed $Y_0$ as a totally real submanifold of ${\Bbb C}^{\ell}$ by a map which extends holomorphically to a neighborhood of $Y_0$, and this holomorphic extension then automatically yields a holomorphic imbedding of some open neighborhood $V\supset S$ in ${\Bbb C}^{\ell}$. Now suppose that $v$ is a smooth section of $H$ along $S$. We may express $v$ uniquely as $u+Jw$, where $u$ and $w$ are tangent to the $Y$. By changing $Y$ if necessary, we can furthermore assume that $u\not\equiv 0$. Let $F$ be a smooth function on $Y$ which vanishes on $S$ and such that the derivative $vF$ is non-negative and supported near some interior point of $S$; and let $\varphi$ be a real-valued smooth $(n-2)$-form on $Y$ whose restriction to $S$ is positive on the support of $vF$. Set $\psi =F\varphi$. Using our imbedding of $Y$ in ${\Bbb C}^{\ell}$, we can express $\psi$ as a family of component functions--- e.g. by arbitrarily declaring that all contractions of $\psi$ with elements of the normal bundle $(TY)^{\perp}\subset T{\Bbb C}^{\ell }$ shall vanish. But, again by \cite{wel}, these component functions are $C^1$-limits on $Y_0$ of restrictions of holomorphic functions from a neighborhood of $Y_0\subset {\Bbb C}^{\ell}$. Thus, by perhaps replacing $V$ with a smaller neighborhood, there is a holomorphic $(n-2)$-form $\beta$ on $V$ which approximates $\psi$ well enough that $$\Re e \int_Sv \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\beta > \frac{1}{2} \int_Su \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\psi >0$$ and $$\Re e \int_{\partial S}v \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm \beta > - \frac{1}{2} \int_Su \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\psi~ .$$ Let $\hat{V}:=\{S'\in {\cal S}_{N,B}~|~S'\subset V\}$, and define $f_{\beta}: \hat{V}\to {\Bbb C}$ by $f_{\beta}(S')=\int_{S'}\beta$. Then $f_{\beta}$ is a holomorphic function on the open set $\hat{V}\subset {\cal S}_{N,B}$. Indeed, if $\gamma$ is {\em any} smooth $(n-2)$-form on $V$, and if we set $f_{\gamma}(S')=\int_{S'}\gamma$, then, for $S'\subset V$, the derivative of $f_{\gamma}$ in the direction of $w\in \Gamma (S', C^{\infty}(H))$, $\varpi_*(w)|_{\partial S'}\equiv 0$, is given by $$wf_{\gamma}|_S= \int_{S}w \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\gamma + \int_{\partial S'}w \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm\gamma ~ ;$$ for if $w$ is extended to $V$ as a smooth vector field $\hat{w}$ tangent to the fibers of $\varpi$ and $S_t$ is obtained by pushing $S'$ along the flow of the vector field $\hat{w}$, then \begin{eqnarray*} wf_{\gamma}|_{S'}&=& \left.\frac{d}{dt}\left[\int_{S_t}\gamma\right]\right|_{t=0} \\&=& \int_{S'}\pounds_{\hat{w}} \gamma \\&=& \int_{S'}[\hat{w} \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\gamma +d(\hat{w} \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm\gamma )] \\&=& \int_{S'}w \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\gamma +\int_{\partial S'}w \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm\gamma ~ . \end{eqnarray*} But since $\beta$ is the restriction of a holomorphic $(n-2)$-form from a region of $\cal N$, it therefore follows that \begin{eqnarray*}(\hat{\cal J}w)f|_{S'}&=& \int_{S'}(Jw ) \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\beta +\int_{\partial S'}(Jw) \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm\beta \\&=& i\int_{S'}w \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\beta +i\int_{\partial S'}w \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm\beta \\&=& iwf|_{S'}~ ,\end{eqnarray*} showing that the function $f_{\beta}$ induced by $\beta$ is $\hat{\cal J}$-holomorphic, as claimed. However, we have also carefully chosen $\beta$ so that the real part of the expression $\int_{S}v \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\beta +\int_{\partial S}v \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm\beta = vf_{\beta}$ is positive. For every real tangent vector $v$ on ${\cal S}_{N,B}$, one can thus find a locally-defined $\cal J$-holomorphic function whose derivative is non-trivial in the direction $v$. Hence $\hat{\cal J}$ is weakly integrable, as claimed. \end{proof} \begin{cor} If $(M,g)$ is real-analytic, then $({{\cal S} }_{M,B}, {\cal J})$ is weakly integrable. \end{cor} \begin{proof} By Theorem \ref{imb} and \ref{wint}, $({{\cal S} }_{M,B}, {\cal J})$ can be imbedded in the weakly integrable almost-complex manifold $(\hat{{\cal S} }_{N,B}, \hat{\cal J})$. Since the restriction of a holomorphic function to an almost-complex submanifold is holomorphic, it follows that $({{\cal S} }_{M,B}, {\cal J})$ is weakly integrable. \end{proof} One might instead ask whether $({{\cal S} }_{M,B}, {\cal J})$ is {\em strongly integrable}--- i.e. locally biholomorphic to a ball in some complex vector space. The answer is {\bf no}; in contrast to any strongly integrable almost-complex manifold, $({{\cal S} }_{M,B}, {\cal J})$ contains very few finite-dimensional complex submanifolds: \begin{propn} Suppose that $(M,g)$ is real-analytic. At a generic point $S\in {{\cal S} }_{M,B}$, a generic $(n-1)$-plane is not tangent to any $(n-1)$-dimensional ${\cal J}$-complex submanifold. \end{propn} \begin{proof} Let $S\subset M$ be a world-sheet which is {\sl not} real-analytic near $p\in S$. Let $q\in N=Gr_2^+(M)$ be given by $q=T_pS^{\perp}$, and let $v_1, \ldots , v_{n-1}\in H_q$ be a set of real vectors such that the $v_j+iJv_j$ form a basis for $D_q$. Extend these vectors as sections $\hat{v}_j$ of $H|_{\Psi(S)}$ which satisfy equation (\ref{leg}) along the sheet; this may be done, for example, by first extending each $v_j$ to just a 1-jet at $p$ satisfying (\ref{leg}) at $p$, projecting this to a 1-jet via $\varpi$ to yield a 1-jet of a normal vector field on $S\subset M$, extending this 1-jet as a section of the normal bundle of $S$, and finally lifting this section using $\Psi_{*}$. Let $u_1, \ldots , u_n$ be the elements of $T_S{{\cal S} }_{M,B}$ represented by $\hat{v}_1, \ldots , \hat{v}_{n-1}$, and let $P\subset T^{1,0}_S{{\cal S} }_{M,B}$ be spanned by $u_1-i{\cal J}u_1, \ldots , u_n-i{\cal J} u_n$. Now suppose there were an $\cal J$-holomorphic submanifold $X\subset {{\cal S} }_{M,B}$ through $S$ with (1,0)-tangent plane equal to $P$. Then $X$ represents a family of transverse sheets in $N$ which foliates a neighborhood $U\subset N$ of $q$; moreover, because $X$ represents a {\sl holomorphic} family, the leaf-space projection $\ell : U\to X$ is CR in the sense that $\ell_*(D)\subset T^{0,1}X$. Since we have assumed that $(M,g)$ is real-analytic, we may also assume that $U$ has a real-analytic CR imbedding $U\hookrightarrow {\Bbb C}^{2n-3}$. Moreover the twistor CR manifold $N$ is automatically ``anticlastic,'' by which I mean that the Levi form ${\cal L}: D\to TN/H: v\mapsto [v,\bar{v}] \bmod H$ is {\sl surjective} at each point of $N$. This gives rise to a Bochner-Hartogs extension phenomenon: every CR function on $U$ extends to a holomorphic function on some neighborhood of $U\subset {\Bbb C}^{2n-3}$. In particular, every CR map defined on $U$ must be real-analytic, and this applies in particular to the leaf-space projection $\ell$. Thus $\Psi(S)$ is real-analytic near $q$, and $S$ is therefore real-analytic near $p$. This proves the result by contradiction. \end{proof} \section{K\"ahler Structure} The complex structure $\cal J$ on ${\cal S}$ depends only on the conformal class $[ g]= \{ e^fg\}$ of our metric, but we will now specialize by fixing a specific pseudo-Riemannian metric $g$. Our reason for doing so is that we thereby induce an $L^2$-metric on ${\cal S}$. Indeed, each tangent space $$T_{\Sigma}{\cal S}_{M,B}=\{ v\in \Gamma (\Sigma, C^{\infty}(\nu_{\Sigma}))~~|~~ v|_{\partial \Sigma}\equiv 0\}~$$ may be equipped with a positive-definite inner product by setting $$h( v, w ) := \int_{\Sigma}g(v, w) ~d\mbox{vol}_{g|_{\Sigma }} .$$ We shall now see that this metric has some quite remarkable properties. \begin{thm} The Riemannian metric $h$ on ${\cal S}$ is {\em K\"ahler} with respect to the previously-defined complex structure ${\cal J}$.\end{thm} \begin{proof} Let $\Omega$ denote the volume n-form of $g$, and define a 2-form $\omega$ on ${\cal S}$ by $$\omega ( v, w ) := \int_{\Sigma}(v\wedge w) \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm \Omega ~ .$$ Obviously, $\omega$ is ${\cal J}$-invariant and $$h( v, w )=\omega ({\cal J}v, w).$$ We therefore just need to show that $\omega$ is closed.\footnote{ The reader may ask whether it is actually legitimate to call a Riemannian manifold K\"ahler when the almost-complex structure in question is at best weakly integrable. However, formal integrability and the closure of the K\"ahler form are the only conditions necessary to insure that the almost-complex structure tensor is parallel, even in infinite dimensions.} To check this, let us introduce the universal family \setlength{\unitlength}{1ex} \begin{center}\begin{picture}(20,17)(0,3) \put(9.5,17){\makebox(0,0){${\cal F}$}} \put(1.5,5){\makebox(0,0){${\cal S}$}} \put(17,5){\makebox(0,0){$M$}} \put(14.5,12){\makebox(0,0){$p$}} \put(4.5,12){\makebox(0,0){${\pi }$}} \put(10.5,15){\vector(2,-3){5.5}} \put(8,15){\vector(-2,-3){5.5}} \end{picture}\end{center} where the fiber of $\pi$ over $\Sigma \in {\cal S}$ is defined to be $\Sigma \subset M$. We can then pull $\Omega$ back to ${\cal F}$ to obtain a closed n-form $\alpha=p^{\ast}\Omega$ which vanishes on the boundary $B=\partial \Sigma$ of every fiber of $\pi $. But $\omega$ is just obtained from $\alpha$ by integrating on the fibers of $\pi $: $$\omega=\pi _{\ast}\alpha~ .$$ Since $\pi _{\ast}$ commutes with $d$ on forms which vanish along the fiber-wise boundary (cf. \cite{bt}, Prop. 6.14.1), it follows that $$d\omega=d(\pi _{\ast}\alpha )= \pi _{\ast}d\alpha= \pi _{\ast}d (p^{\ast}\Omega )=\pi _{\ast}p^{\ast}d\Omega= \pi _{\ast}p^{\ast}0=0.$$ Thus $h$ is a K\"ahler metric, with K\"ahler form $\omega$. \end{proof} To conclude this note, we now observe that $({\cal S}, h)$ is formally of Hodge type--- but non-compact, of course! \begin{propn} Modulo a multiplicative constant, the K\"ahler form $\omega$ of $h$ represents an integer class in cohomology. If, moreover, $M$ is non-compact, $\omega$ is actualy an exact form, and its cohomology class thus vanishes. \end{propn} \begin{proof} If $M$ is compact, we may assume that $g$ has total volume 1, so that its volume form $\Omega$ then represents an element of integer cohomology. Since $\omega=\pi _{\ast}p^{\ast}\Omega$, its cohomology class $[\omega ]=\pi _{\ast}p^{\ast}[ \Omega ]$ is therefore integral. If, on the other hand, $M$ is non-compact, $\Omega =d\Upsilon$ for some $(n-1)$-form $\Upsilon$, and hence $\omega = d(\pi _{\ast}p^{\ast}\Upsilon)$. \end{proof} \bigskip \noindent {\bf Acknowledgements.} The author would like to thank L\'aszl\'o Lempert and Edward Witten for their suggestions and encouragement.

9305

alg-geom/9305001

en

https://arxiv.org/abs/alg-geom/9305001

alg-geom/9305001

Kirti

Kirti Joshi

A General Noether-Lefschetz Theorem and applications

30 pages, in LaTeX. replaced to correct earlier e-mail corruption

In this paper we generalize the classical Noether-Lefschetz Theorem to arbitrary smooth projective threefolds. Let $X$ be a smooth projective threefold over complex numbers, $L$ a very ample line bundle on $X$. Then we prove that there is a positive integer $n_0(X,L)$ such that for $n \geq n_0(X,L)$, the Noether-Lefschetz locus of the linear system $H^0(X,L^n)$ is a countable union of proper closed subvarieties of $\P(H^0(X,L^n)^*)$ of codimension at least two. In particular, the {\em general singular member} of the linear system $H^0(X,L^n)$ is not contained in the Noether-Lefschetz locus. As an application of our main theorem we prove the following result: Let $X$ be a smooth projective threefold, $L$ a very ample line bundle. Assume that $n$ is very large. Let $S=\P(H^0(X,L^n)^*)$, let $K$ denote the function field of $S$. Let ${\cal Y}_K$ be the generic hypersurface corresponding to the sections of $H^0(X,L^n)$. Then we show that the natural map on codimension two cycles $$ CH^2(X_{\C}) \to CH^({\cal Y}_K) $$ is injective. This is a weaker version of a conjecture of M. V. Nori, which generalises the Noether-Lefschetz theorem on codimension one cycles on a smooth projective threefolds to arbitrary codimension

alg-geom

\section{Introduction} In this paper we generalize the classical Noether-Lefschetz Theorem (see \cite{Lefschetz}, \cite{GH}) to arbitrary smooth projective threefolds. More specifically, we prove that given any smooth projective threefold $X$ over complex numbers and a very ample line bundle $L$ on $X$, there is an integer $n_0(X,L)$ such that if $n \geq n_0(X,L)$ then the Noether-Lefschetz locus of the linear system $H^0(X,L^n)$ is a countable union of proper closed subvarieties of $\P(H^0(X,L^n)^*)$ of codimension at least two. In particular, the {\em general singular member} of the linear system $H^0(X,L^n)$ is not contained in the Noether-Lefschetz locus. This generalizes the results of \cite{GH}. In \cite{Madhav}, we find a conjecture due to M.~V.~Nori, which generalizes the Noether-Lefschetz theorem for codimension one cycles on smooth projective threefolds to higher codimension cycles on arbitrary smooth projective varieties. As an application of our main theorem we prove a result which can be thought of as a weaker version of Nori's conjecture for {\em codimension two cycles} on smooth projective threefolds. The idea of the proof is borrowed from an elegant paper of M.~Green (\cite{MarkGreen}) and the work of M.~V.~Nori (see \cite{Madhav}). In \cite{MarkGreen} it was shown that the classical Noether-Lefschetz theorem can be reduced to a coherent cohomology vanishing result and the required vanishing was also proved for $\P^3$. Though we have used this idea, we prove the required vanishing by combining the techniques of \cite{Madhav} and of \cite{MarkGreen}. The paper is organized as follows. In the next section (Section \ref{Generalities and notations}) we set up the basic notations and terminology. Here we have also collected a few facts which will be used throughout this paper. In Section~\ref{The Noether-Lefschetz machine} we set up the basic technical machinery. In this section, we reduce the Infinitesimal Noether-Lefschetz theorem to certain coherent cohomology vanishing statements. This is the most crucial part of the paper. Following the unpublished work of N. Mohan Kumar and V. Srinivas (see \cite{Mohan-Srinivas}), in Section~\ref{The Noether-Lefschetz Locus}, we show how the Infinitesimal Noether-Lefschetz theorem can be used to prove the global Noether-Lefschetz theorem. The global Noether-Lefschetz theorem is thus reduced to Infinitesimal Noether-Lefschetz theorems, which in turn are deduced from coherent cohomology vanishing results. In Section~\ref{A General Noether-Lefschetz Theorem}, we prove our main result, Theorem~\ref{main theorem}. By the results of the Sections~\ref{The Noether-Lefschetz machine} and \ref{The Noether-Lefschetz Locus}, to prove our main theorem, we are reduced to proving several cohomology vanishing results. This program is carried out in Section~\ref{A General Noether-Lefschetz Theorem}. The technique of the proofs of this section are based on Green's work (see \cite{MarkGreen}) and a modification of this method due to Paranjape (see \cite{KP}). Finally, in the last section we give an application of our main theorem to codimension two cycles on smooth projective threefolds. This paper could not have been written without all the help that we have received from N. Mohan Kumar and Kapil Paranjape. They have generously shared their ideas and insights on the problems considered in this paper; moreover N. Mohan Kumar suggested the problem and also explained to us his unpublished work (with V. Srinivas) and the work of Green. Kapil Paranjape patiently explained to us the work of Nori and the subsequent simplifications of Nori's work due to him. We would like to thank both N.~Mohan Kumar and Kapil Paranjape for all their help and encouragement (and also their patience) without which this paper might not have been written. We would also like to thank Madhav Nori and V.~Srinivas for numerous conversations and suggestions. \section{Generalities and notations}\label{Generalities and notations} Let $X/\C$ be a smooth projective threefold, $L$ an ample line bundle on $X$, spanned by its global sections. For any linear system $ W \subset H^0(X,L) $ we will write $S=\P(W^*)$, and for any point $s\in S$ we will write $T = \Spec({\cal O}_{\P(W^*)}/m^2_s)$, where $m_s$ is the maximal ideal of the point $s$. The base point free linear system $H^0(X,L)$ gives a universal family of hypersurfaces with the parameter space $\P(H^0(X,L)^*)$. Moreover, if $W\subset H^0(X,L)$ is any sub linear system, then we can restrict the above universal family, to the linear subspace $\P(W^*) \subset \P(H^0(X,L)^*)$, and get a universal family of hypersurfaces ${\cal Y}$ parametrised by $\P(W^*)$. Further, we will also write ${\cal X} = X \times \P(W^*)$. Note that we have suppressed the dependence of ${\cal X},{\cal Y}$ and $T$ on $W$ and $s$. In our discussions, it will be clear which $W,s$ the notation refers to, and there will be no danger of confusion. When the linear system $W$ is base point free, we can also describe the universal family as follows. We denote by $M(L,W)$ the locally free sheaf defined by the exact sequence \begin{equation} 0 \to M(L,W) \to W \tensor {\cal O}_X \to L \to 0. \end{equation} Further as a notational convenience, if $W=H^0(X,L)$, we will write $M(L) = M(L,W)$. Then ${\cal Y} = \P_X(M(L,W)^*)$. The dual of the exact sequence above then gives an embedding ${\cal Y} \into {\cal X}$. For any $S$-scheme $S'$, denote by ${\cal X}_{S'} = {\cal X} \times_{S} S'$. In particular we have the ``universal infinitesimal deformations'' corresponding to a point $s\in S$, given by ${\cal X}_{T}, {\cal Y}_T$. Note that for any $S' \to S$, we have an embedding ${\cal Y}_{S'} \into {\cal X}_{S'}$. We have the projections $p_X : {\cal X} \to X$ and $p_S:{\cal X} \to S$. The restrictions of these morphisms to ${\cal Y}$ will also be denoted by the same symbols. Note that $p_S:{\cal Y} \to S$ is smooth over $s\in S$ if the zero scheme of the section $s\in W$ is a smooth subscheme of $X$. We will denote by $Y_s$ the zero scheme of $s$. We also have on ${\cal X}$ (resp. on ${\cal Y}$) the de Rham complex on ${\cal X}$ (resp. on ${\cal Y}$) denoted by $\Omega_{\cal X}^{\scriptscriptstyle\bullet}$ (resp. $\Omega_{\cal Y}^{\scriptscriptstyle\bullet}$). We define a complex on ${\cal X}$, denoted by $\Omega_{({\cal X},{\cal Y})}^{\scriptscriptstyle\bullet}$ by the exact sequence \begin{equation} 0 \to \Omega_{({\cal X},{\cal Y})}^{\scriptscriptstyle\bullet} \to \Omega_{{\cal X}}^{\scriptscriptstyle\bullet} \to \Omega_{\cal Y}^{\scriptscriptstyle\bullet} \to 0. \end{equation} The first two are complexes of locally free sheaves on ${\cal X}$ and the latter is a complex of locally free sheaves on ${\cal Y}$. Further as the natural projections ${\cal X} \to X$ and ${\cal Y} \to X$ are smooth morphisms, we see that we also have a commutative diagram: \label{main diagram} \begin{equation} \Matrix{ & & 0 & & 0 & & 0 & & \cr & & \da & & \da & & \da & & \cr 0 &\to& p_X^*(\Omega_X^1) \tensor {\cal O}_{\cal X}(-{\cal Y}) &\to& \Omega^1_{({\cal X},{\cal Y})} &\to& \Omega^1_{({\cal X},{\cal Y})/X}&\to&0\cr & & \da & & \da & & \da & & \cr 0 &\to& p_X^*(\Omega^1_X) &\to& \Omega^1_{{\cal X}} &\to& \Omega^1_{{\cal X}/X}& \to & 0\cr & & \da & & \da & & \da & & \cr 0 &\to& p_X^*(\Omega^1_X)\big|_{\cal Y} &\to& \Omega^1_{{\cal Y}} &\to& \Omega^1_{{\cal Y}/X} & \to & 0\cr & & \da & & \da & & \da & & \cr & & 0 & & 0 & & 0 & & \cr } \end{equation} Note that as ${\cal Y}$ is a divisor in ${\cal X}$, the sheaves $\Omega^1_{({\cal X},{\cal Y})}$ and $\Omega^1_{({\cal X},{\cal Y})/X}$ are vector bundles on ${\cal X}$. \begin{defn}\label{VNL} We will say the Vanishing Noether-Lefschetz condition is valid for $(X,L,W,s)$ where $W$ is a linear system contained in $H^0(X,L)$, and $s\in W$ if the following assertion is valid: $$H^2( X , \Omega^1_{({\cal X}_T,{\cal Y}_T)}\big|_{X\times s} ) = 0.$$ We will symbolically denote this hypothesis by $\mathop{\rm VNL}\nolimits(X,L,W,s)$. \end{defn} \begin{defn} If $X,L$ are as above and $W\subset H^0(X,L)$ is any linear system we will say that the Infinitesimal Noether-Lefschetz Theorem is valid at $s\in W$ if in the commutative diagram $$\Matrix{ \Pic(X) & \to & \Pic({\cal Y}_T) \cr \searrow & & \swarrow \cr & \Pic(Y_s) & \cr } $$ the equality $\image(\Pic({\cal Y}_T) \to \Pic(Y_s)) = \image(\Pic(X)\to \Pic(Y_s) ) $ is valid. We will denote this hypothesis by $\mathop{\rm INL}\nolimits(X,L,W,s)$. \end{defn} The key observation which relates these two definitions is the following proposition, which is just a reformulation of Remark 3.10 of \cite{Madhav}. \begin{propose}\label{vnl gives inl} Let $X$ be a smooth projective threefold over complex numbers, $L$ a very ample line bundle on $X$, $W\subset H^0(X,L)$ a base point free linear system. If $s\in W$ cuts out a smooth divisor $Y_s$ on $X$ and if $\mathop{\rm VNL}\nolimits(X,L,W,s)$ is valid then so is $\mathop{\rm INL}\nolimits(X,L,W,s)$. \end{propose} \begin{proof} The morphism $p_S: {\cal X} \to S$ gives rise to an exact sequence $$ 0 \la p_S^*\Omega^1_S \la \Omega^1_{\cal X} \la \Omega^1_{{\cal X}/S} \la 0,$$ and a similar exact sequence for the morphism $p_S:{\cal Y} \to S$. Since $Y_s$ is a smooth divisor on $X$, the morphism $p_S:{\cal Y}\to S$ is smooth over $s\in S$. Hence we have the following commutative diagram: \begin{equation} \Matrix{ & & 0 & & 0 & & 0 & & \cr & & \da & & \da & & \da & & \cr 0 &\to& p_S^*(\Omega_S^1) \tensor {\cal O}_X(-Y_s) &\to& \Omega^1_{({\cal X},{\cal Y})}\big|_{X_s} &\to& \Omega^1_{({\cal X},{\cal Y})/S}\big|_{X_s}&\to&0\cr & & \da & & \da & & \da & & \cr 0 &\to& p_S^*(\Omega^1_S)\big|_{X_s} &\to& \Omega^1_{{\cal X}}\big|_{X_s} &\to& \Omega^1_{{\cal X}/S}\big|_{X_s}& \to & 0\cr & & \da & & \da & & \da & & \cr 0 &\to& p_S^*(\Omega^1_S)\big|_{Y_s} &\to& \Omega^1_{{\cal Y}}\big|_{Y_s} &\to& \Omega^1_{{\cal Y}/S}\big|_{Y_s}& \to & 0\cr & & \da & & \da & & \da & & \cr & & 0 & & 0 & & 0 & & \cr } \end{equation} This diagram gives rise to the following cohomology diagram: \begin{equation}\let\scty=\scriptstyle \Matrix{ & & \scty{H^1( X , \Omega_X^1)} & \scty{ \mapright{}} & \scty{H^2( X , p_S^*(\Omega_S^1) \big|_{X_s})} \cr & & \scty{\mapdown{\beta}} & & \scty{\mapdown{}} \cr \scty{H^1(Y_s,\Omega^1_{{\cal Y}}\big|_{Y_s})} & \scty{\mapright{}} & \scty{H^1(Y_s,\Omega^1_{Y_s})} & \scty{ \mapright{\alpha}} & \scty{H^2(Y_s,p_S^*(\Omega_S^1)\big|_{Y_s})} \cr \scty{\mapdown{}} & & \scty{\mapdown{}} & & \scty{\mapdown{}} \cr \scty{H^2(X,\Omega^1_{({\cal X},{\cal Y})}\big|_{X_s})} & \scty{\mapright{}} & \scty{H^2 ( X, \Omega^1_{({\cal X},{\cal Y})/S}\big|_{X_s})} & \scty{\mapright{}} & \scty{H^3( X , p_S^*(\Omega^1_S)\tensor {\cal O}_X(-Y_s))} \cr } \end{equation} This diagram is obtained from the previous diagram by taking the long exact cohomology sequence and noting that we have the identifications: \begin{eqnarray*} \Omega^1_{{\cal X}/S}\big|_{X_s} & \isom & \Omega^1_{X_s} \\ \Omega^1_{{\cal Y}/S}\big|_{Y_s} & \isom & \Omega^1_{Y_s} \end{eqnarray*} By hypothesis, the first term on the bottom row (from the left) is zero (this vanishing is just $\mathop{\rm VNL}\nolimits(X,L,W,s)$). Then it follows from the injectivity of the bottom row map that $\ker(\alpha) \subset \image(\beta)$. Further note that one has the trivial identifications: \begin{eqnarray*} H^2(X_s,p_S^*(\Omega^1_S)\big|_{X_s}) & \isom & H^2(X,{\cal O}_X)\tensor \Omega^1_S\tensor k(s) \\ H^2(Y_s,p_S^*(\Omega^1_S)\big|_{Y_s}) & \isom & H^2(Y_s,{\cal O}_{Y_s})\tensor \Omega^1_S \tensor k(s) , \end{eqnarray*} where $k(s)$ denotes the residue field of $s$. Using these identifications, one notes that the top two rows of this diagram can be identified with the ``Kodaira-Spencer maps'' for the infinitesimal deformations ${\cal X}_T$ and ${\cal Y}_T$ respectively (see \cite{CGGH}) (strictly speaking this identification above should be carried out on $T$). Since $\ker(\alpha) \subset \image(\beta)$, we see that any any class $c\in H^1(Y_s,\Omega^1_{Y_s})$ which deforms infinitesimally is the image of a class in $H^1(X,\Omega^1_X)$. Further as $Y_s$ is the zero scheme of a section of a very ample line bundle, one has a surjection $\Pic^0(X) \to \Pic^0(Y_s)$ (this follows easily from Kodaira vanishing theorem). Moreover, it is easy to see, using the exponential sequence, that the homomorphism $\Pic(X) \to \Pic(Y_s)$ has torsion free cokernel. So that we see at once that in the commutative diagram $$ \Matrix{ \Pic(X) & \to & \Pic({\cal Y}_T) \cr \searrow & & \swarrow \cr & \Pic(Y_s) & \cr } $$ any line bundle which is in the image of the map $\Pic({\cal Y}_T) \to \Pic(Y_s)$, comes from $\Pic(X)$. Thus we have proved the proposition. \end{proof} This proposition, though a technical assertion, is a crucial result from the point of view of this paper. In the next section we use this result to systematically reduce Infinitesimal Noether-Lefschetz Theorem to a coherent cohomology vanishing which can be checked in practice. \section{The Noether-Lefschetz machine}\label{The Noether-Lefschetz machine} The condition $\mathop{\rm VNL}\nolimits(X,L,W,s)$ is now cast into a more manageable form. The technique which is employed in the proof of the following proposition is going to be applied repeatedly in different contexts throughout this section and hence in the subsequent proofs of this section, we will give the important points and leave the details to the reader. \begin{propose}\label{base free and smooth} Let $(X,L,W,s)$ be such that $W\subset H^0(X,L)$ is a base point free linear system and $s\in W$ cuts out a smooth divisor on $X$. Assume that the following assertions are valid: \begin{description} \item[{\rm(\thepropose.1)}] $H^1( X , \Omega^2_X \tensor L ) = 0$, a nd \item[{\rm(\thepropose.2)}] $H^1( X , M(L,W) \tensor K_X \tensor L ) = 0 $. \end{description} Then $\mathop{\rm VNL}\nolimits(X,L,W,s)$ is valid. \end{propose} \begin{proof} We closely follow the techniques of proof in \cite{Madhav}. Indeed, we make repeated application of this method throughout this section. Since the morphisms $p_X:{\cal X} \to X$ and $p_X\big|_{\cal Y}: {\cal Y} \to X$ are smooth we have the following exact sequence of vector bundles on ${\cal X}$: \begin{equation} 0 \to p_X^*(\Omega^1_X)\tensor {\cal O}_{\cal X}(-{\cal Y}) \to \Omega^1_{({\cal X},{\cal Y})} \to \Omega^1_{({\cal X},{\cal Y})/X} \to 0. \end{equation} This exact sequence is just the top row of the commutative diagram (\ref{main diagram}). Further, note that we can restrict this exact sequence to $X_s$ and get the exact sequence: \begin{equation} 0 \to p_X^*(\Omega^1_X) \tensor {\cal O}_{\cal X}(-{\cal Y}) \big|_{X_s} \to \Omega^1_{({\cal X},{\cal Y})} \big|_{X_s} \to \Omega^1_{({\cal X},{\cal Y})/X}\big|_{X_s} \to 0. \end{equation} Thus to prove that the middle term of the above exact sequence has no $H^2$, it suffices to prove that the extreme terms have no $H^2$. Now as ${\cal O}_{\cal X}(-{\cal Y})\big|_{X_s} = L^{-1}$,we see that \begin{equation} H^2(X , p_X^*(\Omega^1_X)\tensor{\cal O}_{\cal X}(-{\cal Y})\big|_{X_s}) = H^2(X , \Omega^1_X\tensor L^{-1}). \end{equation} And so the vanishing of this is just the hypothesis \specialref{base free and smooth}{1}, by Serre duality. Now we prove that the second term also has no $H^2$. This is done by the following: \begin{claim}\label{from Kapils paper} $$\Omega^1_{({\cal X},{\cal Y})/X}\big|_{X_s} = M(L,W)^* \tensor L^{-1}.$$ \end{claim} \begin{proof} This fact is easily verified, it also follows from an explicit resolution of the sheaves $\Omega^i_{({\cal X},{\cal Y})/X}$ constructed in \cite{KP}. \end{proof} Now the second vanishing is just our hypothesis \specialref{base free and smooth}{2}, after applying Serre duality. This proves Proposition \ref{base free and smooth}. \end{proof} Before we proceed further we need some notations. If $x\in X$ is any point, we denote by ${\tilde X}_x$ the blowup of $X$ along $x$; $\pi_x:{\tilde X}_x \to X$ the blowup morphism and $E_x$ the exceptional divisor. We write $L'_x = \pi_x^*(L)\tensor{\cal O}_{{\tilde X}_x}(-E_x)$. Frequently, when there is no chance of confusion, we will suppress the subscript $x$ from the above notations. Suppose $W\subset H^0(X, m_x \tensor L)=H^0({\tilde X}_x,L'_x)$ is a subspace. We assume that the linear system $W$ is base point free on ${\tilde X}$. By Proposition \ref{base free and smooth}, $\mathop{\rm INL}\nolimits({\tilde X},L'_x,W,s)$ is reduced to a vanishing on ${\tilde X}$. The next proposition reduces the vanishing on ${\tilde X}$ to a coherent cohomology vanishing on $X$. \begin{propose}\label{base point and smooth} Let $(X,L,W,s)$ be as above. Suppose the following assertions are valid: \begin{description} \item[{\rm(\thepropose.1)}] $H^1( X , (\pi_x)_*(\Omega^2_{{\tilde X}_x}(-2E_x))\tensor L) =0$, and \item[{\rm(\thepropose.2)}] \label{b-p-s-2} $H^1( X , (\pi_x)_* (M(L'_x,W))\tensor K_X \tensor L) = 0$. \end{description} Then $\mathop{\rm VNL}\nolimits({\tilde X}_x , L'_x , W, \tilde{s})$ is valid. \end{propose} \begin{proof} The proof is similar to the one given earlier, though it is a bit more involved. For the purpose of the proof let us use the following notations: we write ${\cal X} = {\tilde X}_x \times S$, $S=\P(W^*)$, ${\cal Y}=\P_{{\tilde X}_x}(M(L'_x,W)^*)$ and $D=E_x \times S$. Note that $D$ is a divisor on ${\cal X}$ and that ${\cal O}_{\cal X}(D) = p_{{\tilde X}_x}^*({\cal O}_{{\tilde X}_x}(E_x))$. We have the following exact sequence on ${\cal X}$. $$ 0 \to \Omega^1_{({\cal X},{\cal Y})} \to \Omega^1_{({\cal X},{\cal Y})}(D) \to \Omega^1_{({\cal X},{\cal Y})} (D) \big|_D \to 0.$$ Now we can restrict this exact sequence to the fibre ${\cal X}_s={\tilde X} \times \{\tilde s\}$ of the projection morphism ${\cal X} \to S$ over $\tilde s\in S$. Thus we have an exact sequence $$ 0 \to \Omega^1_{({\cal X},{\cal Y})}\big|_{{\cal X}_s} \to \Omega^1_{({\cal X},{\cal Y})}(D)\big|_{{\cal X}_s} \to \Omega^1_{({\cal X},{\cal Y})} (D) \big|_D\big|_{{\cal X}_s} \to 0.$$ Thus to prove that the first term on the left has no $H^2$ (that is to say $\mathop{\rm VNL}\nolimits({\tilde X}_x , L'_x , W, \tilde{s})$), it suffices to prove the following assertions: \begin{description} \item[{\rm(\thepropose.3)}] $H^2({\cal X}_s, \Omega^1_{({\cal X},{\cal Y})}(D)\big|_{{\cal X}_s} ) = 0$, and \item[{\rm(\thepropose.4)}] $H^1( {\cal X}_s, \Omega^1_{({\cal X},{\cal Y})}(D)\big|_{D}\big|_{{\cal X}_s} )= 0$. \end{description} These are proved in Lemma \ref{proof of 3.2.3} and Lemma \ref{proof of 3.2.4}. \end{proof} \begin{lemma}\label{proof of 3.2.3} The hypotheses of Proposition \ref{base point and smooth} imply \specialref{base point and smooth}{3}. \end{lemma} \begin{proof} First note that $D\big|_{{\cal X}_s} = E_x$ (recall that $D=E_x\times S$, so this equality is obvious). Now we have the exact sequence $$ 0 \to p_{{\tilde X}}^*(\Omega^1_{{\tilde X}}) \tensor {\cal O}_X(-{\cal Y}) \to \Omega^1_{({\cal X}, {\cal Y})} \to \Omega^1_{({\cal X},{\cal Y})/{\tilde X}} \to 0.$$ Twisting this exact sequence by ${\cal O}_{\cal X}(D)$ we get $$\scriptstyle{ 0 \to p_{{\tilde X}_x}^*(\Omega^1_{{\tilde X}_x}) \tensor {\cal O}_X(-{\cal Y}) \tensor {\cal O}_{\cal X}(D) \to \Omega^1_{({\cal X}, {\cal Y})} \tensor {\cal O}_{\cal X}(D) \to \Omega^1_{({\cal X},{\cal Y})/{\tilde X}_x} \tensor {\cal O}_{\cal X}(D) \to 0.}$$ Now restricting to ${\cal X}_s$, we have $$ 0 \to p_{{\tilde X}}^*(\Omega^1_{{\tilde X}}(E_x)) \tensor {\cal O}_{\cal X}(-{\cal Y})\big|_{{\cal X}_s} \to \Omega^1_{({\cal X},{\cal Y})}(D)\big|_{{\cal X}_s} \to \Omega^1_{({\cal X},{\cal Y})/{\tilde X}}(D) \big|_{{\cal X}_s} \to 0,$$ where we have used the fact that ${\cal O}_{\cal X}(D) = p_{{\tilde X}}^*({\cal O}_{{\tilde X}}(E_x))$ in the term on the left. So the middle term has no $H^2$ if the extreme terms have no $H^2$. So we have to check that the hypotheses of the Proposition \ref{base point and smooth} ensure this. Note that ${\cal O}_{\cal X}(-{\cal Y}) \big|_{{\cal X}_s} = L'^{-1}_x$, and as $p^*_{\tilde X}(\Omega^1_{\tilde X} (E_x))\big|_{{\cal X}_s} = \Omega^1_{{\tilde X}}(E_x)$, we have to prove that the hypotheses of the proposition imply that \begin{description} \item[{\rm(\thepropose.5)}] $H^2({\tilde X}, \Omega^1_{\tilde X}(E_x) \tensor L'^{-1}_x) = 0$, and \item[{\rm(\thepropose.6)}] $H^2({\tilde X} ,\Omega^1_{({\cal X},{\cal Y})/{\tilde X}}(D)\big|_{{\cal X}_s}) = 0 $. \end{description} We will use the Leray spectral sequence to prove that the hypothesis \specialref{base point and smooth}{1} will imply \specialref{proof of 3.2.3}{5} and hypothesis \specialref{base point and smooth}{2} will imply \specialref{proof of 3.2.3}{6}. This will prove Lemma \ref{proof of 3.2.3}. \end{proof} \begin{sublem}\label{sublemma1} Hypothesis \specialref{base point and smooth}{1} implies \specialref{proof of 3.2.3}{5}. \end{sublem} \begin{proof} By Serre duality we are reduced to proving: $$H^1( {\tilde X}, \Omega^2_{\tilde X}(-E_x) \tensor L'_x ) = 0.$$ Now by Leray spectral sequence for $\pi_x:{\tilde X} \to X$, we see that the above $H^1$ is vanishes if: \begin{description} \item[{\rm(\thepropose.7)}] $H^1(X , (\pi_x)_*( \Omega^2_{\tilde X}(-E_x) \tensor L'_x)) = 0$, and \item[{\rm(\thepropose.8)}] $R^1(\pi_x)_*(\Omega^2_{\tilde X}(-E_x) \tensor L'_x) = 0$. \end{description} Note that by the projection formula, $$(\pi_x)_*(\Omega^2_{\tilde X}(-E_x) \tensor L'_x) = (\pi_x)_*(\Omega^2_{\tilde X}(-2E_x)) \tensor L.$$ Hence vanishing of \specialref{sublemma1}{7} above is implied by hypothesis \specialref{base point and smooth}{1}. So we have to check that $R^1$ is also zero. By projection formula, it suffices to prove the following: \begin{equation}\label{whats in a name} R^1(\pi_x)_*(\Omega^2_{\tilde X}(-2E_x)) = 0. \end{equation} Since the fibres of $\pi_x$ have dimension $\leq 2$, $R^2$ satisfies base change. Since the $R^2$ is supported on the point $x$, we can compute the $R^2$ by restricting to $E_x$. Providing we show that $R^2$ is 0, we get that $R^1$ satisfies base change. Then $R^1$ is also supported on the point $x$ and can be computed by restricting to $E_x$. On $E_x$ we have the fundamental exact sequence $$ 0 \to {\cal O}_{E_x}(1) \to \Omega^1_{\tilde X}\big|_{E_x} \to \Omega^1_{E_x} \to 0.$$ An easy calculation shows that this exact sequence splits. So that we have $$\Omega^1_{\tilde X}\big|_{E_x} = \Omega^1_{E_x} \oplus {\cal O}_{E_x}(1).$$ So that by taking exteriors, we have $$\Omega^2_{\tilde X}\big|_{E_x} = \Omega^2_{E_x} \oplus \Omega^1_{E_x}(1).$$ Hence we have $$\Omega^2_{\tilde X}(-2E_x)\big|_{E_x} = \Omega^2_{\tilde X}\big|_{E_x}\tensor {\cal O}_{E_x}(2) = \Omega^2_{E_x}(2) \oplus \Omega^1_{E_x}(3).$$ Since $E_x \isom \P^2$, we see easily that $H^1$ and $H^2$ of the latter sheaf vanish. \end{proof} \begin{sublem} Hypothesis \specialref{base point and smooth}{2} implies \specialref{proof of 3.2.3}{6} \end{sublem} \begin{proof} We want to prove that $$H^2( {\tilde X} , \Omega^1_{({\cal X},{\cal Y})/{\tilde X}}(D)\big|_{{\cal X}_s}) = 0. $$ Firstly we recall that $\Omega^1_{({\cal X},{\cal Y})/{\tilde X}} \big|_{{\cal X}_s} = M(L'_x, W)^* \tensor L'^{-1}_x$ (this is just the formula \ref{from Kapils paper}). Further ${\cal O}_{\cal X}(D)\big|_{{\tilde X}} = {\cal O}_{\tilde X}(E_x)$. So that we have to show that $$ H^2( {\tilde X} , M(L'_x,W)^* \tensor {\cal O}_{\tilde X}(E_x) \tensor L'^{-1}_x ) = 0.$$ By Serre duality, we have to prove the following vanishing: $$H^1({\tilde X} , M(L'_x,W) \tensor {\cal O}_{\tilde X}(-E_x) \tensor K_{\tilde X} \tensor L'_x ) = 0.$$ This is done as before by a Leray spectral sequence argument. We observe that $ H^1( X , (\pi_x)_*( M(L'_x,W) \tensor {\cal O}_{\tilde X}(-E_x) \tensor K_{\tilde X} \tensor L'_x)) = 0 $ by the hypothesis \specialref{base point and smooth}{2}. Thus it suffices to prove that $$R^1(\pi_x)_*(M(L'_x,W) \tensor {\cal O}_{\tilde X}(-E_x) \tensor K_{\tilde X} \tensor L'_x) = 0.$$ Using the projection formula and the fact that $K_{\tilde X} = \pi_x^*(K_X) \tensor {\cal O}_{\tilde X}(2E_x)$, we see that it suffices to prove that $$R^1(\pi_x)_*(M(L'_x,W)) = 0.$$ As before we will show vanishing of $H^1$ and $H^2$ after after restricting to $E_x$. This will imply the vanishing of $R^1$ and $R^2$ as before. Since we have the exact sequence $$ 0 \to M(L'_x , W) \to W\tensor {\cal O}_{\tilde X} \to L'_x \to 0, $$ restricting this exact sequence to $E_x$, by the universal property of the tautological bundle ${\cal O}_{E_x}(1)$, we see that we have: $$M(L'_x,W)\big|_{E_x} = \Omega^1_{E_x}(1) \oplus V' \tensor {\cal O}_{E_x},$$ for some subspace $V' \subset W$. Now since $E_x\isom\P^2$ one notes that the $H^1$ and $H^2$ of the latter sheaf vanish. \end{proof} Finally, to complete the proof of Proposition \ref{base point and smooth}, it remains to prove \specialref{base point and smooth}{4}: \begin{lemma}\label{proof of 3.2.4} $$H^1({\cal X}_s, \Omega^1_{({\cal X},{\cal Y})}(D)\big|_{D}\big|_{{\cal X}_s}) =0.$$ \end{lemma} \begin{proof} Observe that as $D=E_x \times S$, we have $D\big|_{{\cal X}_s} =E_x$ and hence $D\big|_{E_x} = {\cal O}_{E_x}(-1)$. We have the divisor $Z = E_x\times S\cap {\cal Y} = D\cap {\cal Y}$ in $D$. Note that $D$ is just a product of $\P^2$ and a projective space. Thus the vanishing to be proved is reduced to a vanishing result on $\P^2$. Further note that by definition $Z = \P_{E_x}(M(L'_x,W)^*\big|_{E_x})$. We have a natural commutative diagram $$\let\scty=\scriptstyle \Matrix{ & & \scty{0} &&\scty{0} & & \scty{0} & & \cr & & \scty{\da} &&\scty{\da} & &\scty{\da} & & \cr \scty{0} & \scty{\to} &\scty{{\cal O}_D(-Z)} &\scty{\to}&\scty{{\cal O}_D} & \scty{\to} &\scty{{\cal O}_Z} &\scty{\to} & \scty{0}\cr & & \scty{\da} && \scty{\da} & & \scty{\da} & & \cr \scty{0} & \scty{\to} & \scty{\Omega^1_{({\cal X},{\cal Y})}(D)\big|_D} &\scty{\to}& \scty{\Omega^1_{\cal X}(D)\big|_D} &\scty{\to} &\scty{\Omega^1_{\cal Y}(D)\big|_D} & \scty{\to} & \scty{0}\cr & & \scty{\da} && \scty{\da} & & \scty{\da} & & \cr \scty{0} & \scty{\to} & \scty{\Omega^1_{(D,Z)}\tensor {\cal O}_{E_x}(-1)} & \scty{\to}& \scty{\Omega^1_D\tensor {\cal O}_{E_x}(-1)} &\scty{\to} &\scty{\Omega^1_Z\tensor {\cal O}_{E_x}(-1)} & \scty{\to} & \scty{0}\cr & & \scty{\da} && \scty{\da} & & \scty{\da} & & \cr & &\scty{0} && \scty{0} & & \scty{0} & & \cr } $$ Further restricting the right column to ${\cal X}_s = {\tilde X}\times \{s\}$ we see that the middle term $H^1({\tilde X} , \Omega^1_{({\cal X},{\cal Y})}(D)\big|_D\big|_{{\cal X}_s}) = 0$ if \begin{description} \item[{\rm(\thepropose.9}] $H^1(E_x , {\cal O}_D(-Z)\big|_{{\cal X}_s}) =0$, and \item[{\rm(\thepropose.10)}] $H^1( E_x , \Omega^1_{(D,Z)}\big|_{E_x} \tensor {\cal O}_{E_x}(-1)) = 0$. \end{description} Note that as $E_x \isom \P^2$, and as ${\cal O}_D(-Z)\big|_{E_x}$ is a line bundle on $E_x$, its $H^1$ is trivially zero. So we have to prove \specialref{proof of 3.2.4}{10}. To do this we proceed as follows: We have the exact sequence (from the definition) $$\Matrix{ 0 &\to& \Omega^1_{(D,Z)}& \to &\Omega^1_D &\to& \Omega^1_Z& \to& 0\cr & & & & \Vert & & & &\cr & & & & \Omega^1_{E_x}\oplus \Omega^1_S & & & &\cr } $$ So restricting to $E_x$, as $\tilde{s}$ cuts out a smooth divisor on ${\tilde X}$, $Z=E_x\cap {\cal Y}$ is a $\P^1$-bundle over $S$ in a Zariski neighbourhood of $\tilde s$. Then we have the exact sequence $$ 0 \to \Omega^1_{(D,Z)}\big|_{E_x} \to \Omega^1_D\big|_{E_x} \to \Omega^1_Z \big|_{E_x} \to 0.$$ Now $\Omega^1_D\big|_{E_x}=\Omega^1_{E_x}\oplus \Omega^1_{S,s}\tensor{\cal O}_{E_x}$, and similarly we have $\Omega^1_Z\big|_{E_x} = \Omega^1_{\P^1} \oplus \Omega^1_{S,s}\tensor{\cal O}_{E_x}$. This gives an exact sequence $$\displaylines{ 0 \to (\Omega^1_{(\P^2,\P^1)} \oplus \Omega^1_{S,s}\tensor{\cal O}_{E_x}) \tensor {\cal O}_{E_x}(-1) \to \Omega^1_{\P^2}(-1) \oplus \Omega^1_{S,s}\tensor {\cal O}_{E_x}(-1) \to \hfill\cr \hfill{}\qquad\to \Omega^1_{\P^1}(-1) \oplus \Omega^1_{S,s} \tensor {\cal O}_{E_x}(-1)\to 0.\hfill\cr } $$ Then taking long exact cohomology sequence we have: $$\displaylines{ H^0(\P^1, \Omega^1_{\P^1}(-1)) \oplus H^0(\P^2,\Omega^1_{S,s} \tensor {\cal O}_{E_x}(-1)) \to \hfill\cr \hfill{}\qquad \to H^1( (\Omega^1_{(\P^2,\P^1)} \oplus \Omega^1_{S,s}\tensor{\cal O}_{E_x})\tensor {\cal O}_{E_x}(-1)) \to \hfill\cr \hfill{} \qquad\qquad \to H^1(\Omega^1_{\P^2}(-1)) \oplus H^1(\Omega^1_{S,s}\tensor{\cal O}_{E_x}(-1) ).\hfill\cr } $$ And the terms on the extreme vanish so that the middle term vanishes. Hence we are done. This finishes the proof of Lemma \ref{proof of 3.2.4} \end{proof} For the next proposition, we need some more notation. Let $$ L''_x = (\pi_x)^*(L)\tensor{\cal O}_{\tilde X}(-2E_x).$$ Observe that image, under $\pi_x$ of smooth sections of $H^0({\tilde X},L''_x)$, are sections in $H^0(X,L)$ which have a single ordinary double point at $x$. We now want to reduce the condition $\mathop{\rm VNL}\nolimits({\tilde X},L''_x,H^0({\tilde X},L''_x),s)$ for smooth $s\in S=\P(H^0({\tilde X},L''_x)^*)$, to a more manageable form. This is done by the following. \begin{propose}\label{base point and ODP} Assume that $L$ is such that $L''_x$ is very ample on ${\tilde X}$. Let $s\in W=H^0({\tilde X}, L''_x)$ be a section which cuts out a smooth divisor on ${\tilde X}$. Assume that the following assertions are valid: \begin{description} \item[{\rm(\thepropose.1)}] $H^1(X, L\tensor m_x^3) = 0$, where $m_x$ is the maximal ideal of $x\in X$. \item[{\rm(\thepropose.2)}] $H^1( X , (\pi_x)_*(\Omega^2_{{\tilde X}_x}(-2E_x)) \tensor L) = 0$, and \item[{\rm(\thepropose.3)}] $H^1( X , (\pi_x)_*( M(L_x'') ) \tensor K_X \tensor L ) = 0$. \end{description} Then $\mathop{\rm VNL}\nolimits({\tilde X},L''_x,H^0({\tilde X},L''_x),s)$ is valid. \end{propose} \begin{proof} Since the linear system $W$ is base point free on ${\tilde X}$, and as the section $s\in W$ cuts out a smooth divisor on ${\tilde X}$, we see that Proposition \ref{base free and smooth} can be applied. By this proposition, it suffices to prove the following assertions: \begin{description} \item[{\rm(\thepropose.4)}] $H^1({\tilde X}, \Omega^2_{\tilde X} \tensor L''_x ) = 0$, and \item[{\rm(\thepropose.5)}] $H^1( {\tilde X} , M(L''_x,W) \tensor K_{\tilde X} \tensor L''_x ) =0$. \end{description} Thus we have to prove that the hypothesis of the proposition ensure the vanishing \specialref{base point and ODP}{4} and \specialref{base point and ODP}{5}. The argument is similar to the one earlier. Firstly, let us observe that the implication \specialref{base point and ODP}{2} implies \specialref{base point and ODP}{4} follows from \ref{sublemma1}. By a Leray spectral sequence argument, we are reduced to proving $$R^1(\pi_x)_*(\Omega^2_{\tilde X}(-2E_x)) = 0.$$ But this has been proved during the proof of Sublemma\ref{sublemma1} as equation (\ref{whats in a name}). Thus we are done in this case. So it remains to prove that \specialref{base point and ODP}{1} and \specialref{base point and ODP}{3} together imply \specialref{base point and ODP}{5}. First we check that $R^1(\pi_x)_*(M(L''_x,W)) = 0$. This can be done by restricting to $E_x$. On $E_x$ we have an exact sequence: $$ 0 \to M(L''_x,W)\big|_{E_x} \to W\tensor {\cal O}_{E_x} \to L''_x \big|_{E_x} \to 0.$$ So on noting that $W= H^0({\tilde X},L''_x)=H^0(X,L\tensor m_x^2)$, the result now follows from \specialref{base point and ODP}{1} and \specialref{base point and ODP}{3}. \end{proof} \section{The Noether-Lefschetz Locus}\label{The Noether-Lefschetz Locus} For any smooth projective threefold $X$, and $L$ be a very ample line bundle on $X$, we say smooth member $Y$ of $H^0(X,L)$ lies in the ``Noether-Lefschetz locus of the linear system $H^0(X,L)$'' if $\Pic(X)\to \Pic(Y)$ is not a surjection. More generally, if $Y$ is normal we say $Y$ lies in the ``Noether-Lefschetz locus'' if $\Pic(X) \to CH^1(Y)$ is not a surjection. Now we deduce the global Noether-Lefschetz theorem from the infinitesimal results. \begin{propose}\label{codim one} Suppose $X$ is a smooth projective threefold over $\C$ and $L$ is a very ample line bundle over $X$. Assume that $\mathop{\rm INL}\nolimits(X,L,H^0(X,L),s)$ is valid for all $s\in H^0(X,L)$ which cut out a smooth divisor on $X$. Then the Noether-Lefschetz locus for the linear system $H^0(X,L)$ has codimension $\geq 1$ in $\P(H^0(X,L)^*)$. \end{propose} \def\bar{K}{\bar{K}} \begin{proof} We argue as in \cite{Mohan-Srinivas}. We can assume that the $X, L$ are defined over a finitely generated field $K/\Q$. Let $\bar{K}$ denote the algebraic closure of $K$. Let $\bar{K}(\eta)$ denote the rational function field in $\dim H^0(X,L)$ variables over $\bar{K}$. Then this defines a point $\eta \into S=\P(H^0(X,L)^*)$, which we will call the $\bar{K}$-generic point of $S$. Let ${\overline\eta} \in S=\P(H^0(X,L)^*)$ be the corresponding $\bar{K}$-geometric generic point of $S$. Then we claim that $\Pic(X_{\overline\eta}) \isom \Pic(Y_{\overline\eta})$. The claim is proved as follows. Suppose that it is not an isomorphism, then in particular the map is not surjective. Let $\alpha\in\Pic(Y_{\overline\eta})$ be any line bundle not in the image of the map. Then this cycle is defined over a finite field extension of the function field $\bar{K}(\eta)$. Then there exists an \'etale open set $U \to S$ such that $\alpha$ ``spreads'' out to a line bundle on ${\cal Y}_U$. By replacing $U$ by a smaller non-empty open subset if necessary, we can further assume that $\alpha$ is a nontrivial element of $\Pic({\cal Y}_U)$. Now we restrict $\alpha$ to the fibres of ${\cal Y}_U \to U$. By shrinking $U$ further we can assume that image of $U$ in $S$ is an open set which does not meet the discriminant locus of the linear system $H^0(X,L)$ in $S$. Thus we can now apply the Infinitesimal Noether-Lefschetz to the fibres over $U$. On the fibres of ${\cal Y}_U \to U$, as $\mathop{\rm INL}\nolimits(X,L,H^0(X,L),s)$ is valid, we see that the restriction of $\alpha$ to the fibres over $u\in U$ is zero in $\Pic(Y_u)$ because infinitesimally there are no extra cycles (note that we have used the fact that we can identify ${\cal O}_U/m_u^2 \isom {\cal O}_S/m_s^2$, where $s$ is the image of $u$ under the map $U\to S$). Then by the semi-continuity theorem we see that the line bundle $\alpha$ on ${\cal Y}_U$ must be trivial. This proves the claim. Now it remains to prove that the Noether-Lefschetz locus has codimension $\geq 1$. We proceed as in \cite{Mohan-Srinivas}. Let $U$ be the subset of $S$ obtained by removing all the divisors of $S$, which are defined over $\bar{K}$. Thus we have removed a countable set of closed subvarieties. If $p\in U$ is a closed point the map $ \Spec(\C(p)) \into U$ factors through the geometric generic point ${\overline\eta}$. Further as ${\cal Y}_{\overline\eta}$ is the zero scheme of a section of an ample line bundle on $X_{\overline\eta}$, we see as $H^1(X_{\overline\eta}, {\cal O}_{X_{\overline\eta}}) \isom H^1(Y_{\overline\eta},{\cal O}_{Y_{\overline\eta}})$. We see that $\Pic(Y_p)(\bar{K})\isom \Pic(Y_{\overline\eta})(\bar{K})$. This proves the claim. \end{proof} An identical argument, with divisors on $S$ which are not contained in the support of the discriminant locus, we can prove the following: \begin{propose}\label{codim two in smooth} If {}$\mathop{\rm INL}\nolimits(X,L,W,s)$ is valid for all $W$ of codimension one in $H^0(X,L)$ and $s\in W$ smooth. Then ``smooth part'' of the Noether-Lefschetz locus has codimension $\geq 2$ in $S$. \end{propose} \begin{proof} Since the proof is almost identical to the one given earlier, we will only indicate the important point. The idea is to work with divisor of $D$ which is not contained in the discriminant locus. For such a divisor, we prove by an argument identical to the one given above that for the $\bar{K}[A$-geometric generic point ${\overline\eta}_D$ of $D$, we have an isomorphism $\Pic(X_{{\overline\eta}_D}) \isom \Pic(Y_{{\overline\eta}_D})$. This is proved as before, except that one has to use $\mathop{\rm INL}\nolimits$ for a codimension one linear system. For which we use the following Lemma. Assuming the Lemma for the moment, we complete the proof as follows. Consider the subset of $D$ obtained by removing all the codimension one subvarieties of $D$ which are defined over $\bar{K}$. Then the one argues as in the proof of Proposition~\ref{codim one}. \end{proof} \begin{lemma} \label{vnl and inl on the blowup} Let $(X,L,W,s)$ be such that $W\subset H^0(X,L)$ is linear system of codimension one with a single base point at $x\in X$. Further assume that $s$ cuts out a smooth divisor $Y_s$ in $X$. Then we have $$ \mathop{\rm INL}\nolimits({\tilde X}_x,L'_x,W,\tilde s) \mbox{$\Rightarrow$} \mathop{\rm INL}\nolimits(X,L,W,s).$$ \end{lemma} \begin{proof} We use the following commutative diagrams: $$ \Matrix{ \Pic({\tilde X}_x)&& \mapright{f_2} && \Pic(\tilde{{\cal Y}_T}) \cr h_2 \searrow && && \swarrow g_2 \cr && \Pic({\tilde Y}_s) && \cr \mapup{\tau_1} && \mapup{\tau_2} && \mapup{\tau_3} \cr && \Pic(Y_s) && \cr h_1 \nearrow && && \nwarrow g_1 \cr \Pic(X) && \mapright{f_1} && \Pic({\cal Y}_T) \cr } $$ And one also has the following commutative diagram: $$ \Matrix{ 0 & \to & \Pic(X) & \mapright{\tau_1} & \Pic({\tilde X}_x) & \mapright{\tau_1'}& \Z & \to & 0 \cr & & \mapdown{h_1} & & \mapdown{h_2}& &\Big\Vert& & \cr 0 & \to & \Pic(Y_s) & \mapright{tau_2}& \Pic({\tilde Y}_s) & \mapright{\tau_2'} & \Z & \to & 0 \cr } $$ As $\mathop{\rm INL}\nolimits({\tilde X}_x,L'_x,W,s)$ is valid, we have $\image{(h_2)}=\image{(g_2)}$. We want to prove that $\image{(h_1)}=\image{(g_1)}$. Since the lower triangle commutes, we have the inclusion $\image{(h_1)} \subset \image{(g_1)}$. So we will prove the reverse inclusion. Let $\alpha\in \Pic({\cal Y}_T)$, as $\tau_2 g_1 = g_2\tau_3$ we have $\tau_2g_1(\alpha) = g_2\tau_3(\alpha)$. This implies that there exists $\alpha'\in \Pic({\tilde X}_x)$ such that $h_2(\alpha') = g_2(\tau_3(\alpha)) = \tau_2(g_1(\alpha))$. Now chasing the image $g_1(\alpha)\in\Pic(Y_s)$ in the second diagram, we see that $\tau_2'(h_2(\alpha')) = \tau_2'(\tau_2(g_1(\alpha))) =0$. Then $\tau_2'(h_2(\alpha')) = \tau_1'(\alpha') = 0$. So that $\alpha' \in \image(\tau_1)$. So there is an $\alpha'' \in \Pic(X)$ such that $\alpha' = \tau_1(\alpha'')$. Now going back to the previous diagram, we see that $$h_2(\alpha')=h_2(\tau_1(\alpha'')) = \tau_2(h_1(\alpha''))= \tau_2(g_1(\alpha)).$$ Hence, $\tau_2(h_1(\alpha'')-g_1(\alpha)) =0$. But as $\tau_2$ is injective, we see that $h_1(\alpha'')=g_1(\alpha)$. This proves Lemma~\ref{vnl and inl on the blowup}. \end{proof} Applying the same technique we can prove: \begin{propose}\label{codim one in disc.} If for all $x\in X$, $\mathop{\rm INL}\nolimits({\tilde X},L''_x,H^0({\tilde X},L''_x),s)$ is valid for all smooth $s\in H^0({\tilde X},L''_x)$, then the intersection of the Noether-Lefschetz locus of $H^0(X,L)$ with the discriminant locus of $H^0(X,L)$ has codimension $\geq 2$ in $S=\P(H^0(X,L)^*)$. \end{propose} \begin{proof} Firstly one notes that the hypothesis of the proposition together with Proposition~\ref{codim one}, implies that the Noether-Lefschetz locus of $H^0({\tilde X},L''_x)$ is a proper of codimension $\geq 1$ in $\P(H^0({\tilde X},L''_x)^*)$. Note that the image of the zero scheme of any smooth $s\in H^0({\tilde X},L''_x)$ in $X$ is an element of the linear system $H^0(X,L)$ with a single ordinary double point at $x$. Then the intersection of the Noether-Lefschetz locus with the discriminant locus is clearly of codimension $\geq 2$. This proves the result. \end{proof} \section{A General Noether-Lefschetz Theorem}\label{A General Noether-Lefschetz Theorem} In this section we are going to apply the general machinery of the previous section to prove the following generalization of the classical Noether-Lefschetz theorem. \begin{thm}\label{main theorem} Let $X/\C$ be a smooth projective threefold, $L$ a very ample line bundle on $X$. Then there exits an integer $n_0(X,L) > 0$ such that for all $n \geq n_0(X,L)$, the Noether-Lefschetz locus of the linear system $H^0(X,L^n)$ has codimension $\geq 2$. \end{thm} \begin{proof} The general machinery for the proof of this result was set up in the previous section. By the Propositions \ref{base free and smooth}, \ref{base point and smooth} and \ref{base point and ODP}, and Propositions \ref{codim one}, \ref{codim two in smooth} and \ref{codim one in disc.} we are reduced to proving the following assertion: There exists and $n_0(X,L)$ such that for all $n\geq n_0$ the following are valid: \begin{enumerate} \item $H^1(X,\Omega^2_X\tensor L^n) =0$, \item for every base point free linear system $W\subset H^0(X,L^n)$ of codimension at most one, we have $$H^1(X,M(L^n,W) \tensor K_X \tensor L^n ) =0.$$ \item For all points $x\in X$, for all $n \geq n_0$ we have $$H^1( X , (\pi_x)_*( \Omega^2_{{\tilde X}_x}(-2E_x)) \tensor L^n ) = 0, $$ \item for any codimension one linear system $W\subset H^0(X,L^n)$ with a single base point at $x\in X$ the following holds for all $n \geq n_0$: $$H^1( X , (\pi_x)_*(M(L'_x,W)) \tensor K_X \tensor L^n) =0.$$ \item For all $x\in X$ and for all $n\geq n_0$, we have $$H^1(X , m^3_x \tensor L^n ) =0,$$ \item $$H^1(X, (\pi_x)_*(\Omega^2_{{\tilde X}_x}(-2E_x)) \tensor L^n ) = 0;$$ \item for $W= H^0({\tilde X}_x,L''_x)$, we have for all $x\in X$ and for all $n\geq n_0$: $$H^1( X , (\pi_x)_*(M(L''_x,W)) \tensor K_X \tensor L^n ) = 0.$$ \end{enumerate} By what has transpired so far, these assertions, by the results of Section~\ref{The Noether-Lefschetz machine}, imply the infinitesimal Noether-Lefschetz results {\it i.e.\/},\ Propositions \ref{base free and smooth}, \ref{base point and smooth} and \ref{base point and ODP}, and finally from these local results by Propositions \ref{codim one}, \ref{codim two in smooth} and \ref{codim one in disc.}, we obtain the above global result. This entire section is devoted to the proof of these seven assertions. Firstly, note that of these seven assertions, the assertions (1), (3), (5) and (6) follow immediately from the Semi-continuity Theorem, Serre's vanishing theorem and from simple Noetherian induction. This will be left to the reader. The remaining assertions are more difficult and require more elaborate arguments. We will now prove the assertions (2), (4) and (7). \end{proof} \begin{propose} There exists a positive integer $n_0$ depending only on $X,L$ such that for any base point free linear system of codimension at most one, we have $$H^1( X , M(L^n,W)\tensor K_X \tensor L^n ) = 0.$$ \end{propose} \begin{proof} The idea is to use a method of Mark Green (see \cite{MarkGreen}) and reduce the result to a regularity computation. Let $V = H^0(X,L)$, by hypothesis, $X$ embeds in $\P:=\P(V)$. Let $j:X \into \P$ be the embedding given by the linear system. Write ${\cal I}_X$ for the ideal sheaf of $X$ in $\P$. Then note that $L = j^*({\cal O}_\P(1))$. We have a surjection: $$H^0( \P, {\cal O}_\P(n)) \to H^0(X,L^n) \to 0.$$ We write $F=H^0(X,L^n), F'=H^0(\P,{\cal O}_\P(n))$. Then the subspace $W\subset F$ gives us a subspace $W'\subset F'$. If we choose $n_0 > \mathop{\rm reg}\nolimits({\cal I}_X)$ then $W'$ is base point free on $\P$. Moreover, one has $\codim_F'(W') = \codim_F(W)$. Then we have corresponding to the triple $(\P,{\cal O}_\P(n),W')$ a vector bundle $M':=M({\cal O}_\P(n),W')$ which is defined by the usual evaluation sequence: $$ 0 \to M' \to F'\tensor {\cal O}_\P \to {\cal O}_\P(n) \to 0.$$ Write $V' = \ker(W' \to W)$. Now we have the following commutative diagram of locally free sheaves on $X$: $$ \Matrix{ & & 0 & & 0 & & & & \cr & & \da & & \downarrow & & & & \cr & &V'\tensor{\cal O}_X& = & V'\tensor{\cal O}_X & & & & \cr & & \da & & \downarrow & & & & \cr 0 & \la & j^*M' & \la & W'\tensor{\cal O}_X & \la & F & \la & 0 \cr & & \downarrow & & \downarrow & & \parallel & & \cr 0 & \la & M & \la & W\tensor{\cal O}_X & \la & F & \la & 0 \cr & & \downarrow & & \downarrow & & & & \cr & & 0 & & 0 & & & & \cr } $$ Now any splitting of the middle column gives a (non-canonical) splitting of the left column. Thus we see that one has a non-canonical splitting: $$ j^*(M') = M(L^n,W) \oplus V'\tensor {\cal O}_X.$$ Now we need the following lemma due to Mark Green, see \cite{MarkGreen}. \begin{lemma} Let $W$ be any base point free linear system $W\subset H^0(\P,{\cal O}_\P(n))$. Then for all $i \geq 1$, and for all $k+i \geq \codim(W)+1$, we have $$H^i( \P , M' \tensor {\cal O}_\P(k)) =0,$$ in other words, $M'$ is $(\codim(W)+1)$-regular. \end{lemma} Now by projection formula we have $j_*j^* M' =j_*M(L^n,W) \oplus V' \tensor j_*({\cal O}_X)$. Thus as $j_*(M(L^n,W))$ is a direct summand of $M' \tensor j_*({\cal O}_X)$, and as the regularity of $M'$ is bounded independent of $n$, and as the regularity of $j_*({\cal O}_X)$ is a fixed constant, it follows that the regularity of $j_*M(L^n,W)$ is bounded independent of $n$. Thus if we assume that $n > (\codim(W)+1+\mathop{\rm reg}\nolimits(K_X))$, then $H^1(X, M(L^n,W)\tensor K_X \tensor L^n ) =0$. This proves the proposition. \end{proof} A slight modification of the same technique gives the following result. \begin{propose} For any codimension one linear system $W\subset H^0(X,L^n)$ with a single base point $x$, there is an $n_0(X,L)$ such that for all $n\geq n_0$, we have: $$H^1( X , (\pi_x)_*(M(L'_x,W)) \tensor K_X \tensor L^n ) =0.$$ \end{propose} \begin{proof} We have to modify the argument we gave for the previous proposition. We need the following variant of the previous lemma. For $x\in \P$ write $\tilde\P$ for the blowup of $\P$ at $x$, and write $\pi_x$ for the blowup morphism. Write $W=H^0(\P,{\cal O}_\P(n)\tensor m_{x,\P})$, where $m_{x,\P}$ is the maximal ideal of $x$ in $\P$. $W$ gives a natural linear system on $\tilde\P$. Further, let $M'$ be the vector bundle defined in Section~\ref{Generalities and notations}, on $\tilde\P$ corresponding to the linear system $W$. Assume that $n \geq 2$ so that this linear system is base point free. We have \begin{lemma} With the above notations, the coherent sheaf $(\pi_x)_*(M')$ on $\P$ is $1$-regular, {\it i.e.\/},\ for all $i\geq 1$ and for all $k+i \geq 1$ we have $$H^i(\P, (\pi_x)_*(M') \tensor {\cal O}_\P(k)) = 0.$$ \end{lemma} \begin{proof} On $\P$ we have the exact sequence $$ 0 \to (\pi_x)_*(M') \to W\tensor {\cal O}_\P \to {\cal O}_\P(n) \tensor m_{x,\P} \to 0.$$ The result now follows from the regularity computation for the sheaf $m_{x,\P}$. \end{proof} \long\def\comment#1\endcomment{} \comment Note that we have assumed here that $n$ is large enough, so that the $W$ spans ${\cal O}_\P(n) \tensor m_{x,\P}$. Now twisting by ${\cal O}_\P(k-i)$ we see that we have $$\to H^{i-1}({\cal O}(n+k-i) \tensor m_x ) \to H^i( (\pi_x)_*(M') \tensor {\cal O}(k-i)) \to H^i({\cal O}(k-i)) \tensor W\to$$ Now for $i=1$, $H^0({\cal O}(k-1)) \tensor W \to H^0({\cal O}(n+k-1)\tensor m_x)$ is a surjection if $k\geq i+1$. And for $1\leq i \leq \dim\P$, we have $H^i(\P,{\cal O}(k-i) =0$ and for $i=\dim\P$, $H^i({\cal O}_\P(k-i)) = 0$ if $k-i > -\dim\P-1$ {\it i.e.\/},\ if $k > -1$. So one has to prove that the first term on the left in the exact sequence above is zero. This is done as follows: we have the exact sequence $$ 0 \to m_x \to {\cal O}_\P \to \C_x \to 0,$$ where $\C_x$ is the skyscraper sheaf supported only at $x$. Then twisting by ${\cal O}(n+k-i)$ we get: $$ 0 \to {\cal O}(n+k-i)\tensor m_x \to {\cal O}_\P(n+k-i) \to \C_x\tensor{\cal O}(n+k-i) \to 0.$$ Note the sheaf on the extreme right is still supported at a point and hence has no cohomology except possibly $H^0$. Hence for $i\geq 1$ we have $H^i(m_x\tensor{\cal O}(n+k-i)) \isom H^i({\cal O}(n+k-i))$. So that for $i=\dim\P$, we have $H^i({\cal O}(n+k-i))=0$ if $n+k-i > -\dim\P-1$ {\it i.e.\/},\ if $n+k > -1 $. Thus finally if $k > -1 $ and $k > -1 -n $ then for $i=\dim\P$, $H^i( (\pi_x)_*(M') \tensor {\cal O}(k) ) = 0 $ if $k+i \geq 1$. Thus we have proved that $(\pi_x)_*(M')$ is $1$-regular. \end{proof} \endcomment Finally, we can now finish the proof of the proposition. We have an exact sequence $$ 0 \to \pi_*M' \to {\cal O}_\P \tensor H^0({\cal O}(n)\tensor m_{x,\P}) \to {\cal O}(n) \tensor m_{x,\P} \to 0.$$ Write $K_x = \ker(L^n \tensor m_{x,\P} \tensor {\cal O}_X \to L^n \tensor m_{x,X})$. Note that $K_x$ is a finite length sheaf, supported at the point $x$. We assume that $n$ is large enough to ensure that there is a surjection $$ \Matrix{ H^0(\P, {\cal O}(n)\tensor m_{x,\P}) & \to & H^0(X, L^n\tensor m_{x,X})& \to& 0\cr \Vert & & \Vert & & \cr F' & & F & & \cr } $$ Write $W'=\ker(F' \to F)$. Then there is a surjection $W'\tensor {\cal O}_X \to K_x$. To see this note that $K_x$ is a sheaf supported at a point, further note that we can assume, by choosing $n$ sufficiently large so that $H^1(X, L^n \tensor I_X ) = 0$, that $F'$ surjects on to $H^0(X, L^n \tensor m_{x,\P} \tensor {\cal O}_X)$. This being done, we have the required surjection. Note that this choice of $n$ depends only on $X,L$. We have then the following commutative diagram on $X$ $$\let\sc=\scriptstyle \Matrix{ & & \sc{0} & & \sc{0} & & \sc{0} & & \cr & & \sc{\da} & & \sc{\da} & & \sc{\da} & & \cr \sc{0} & \sc{\to}& \sc{H} &\sc{\to}&\sc{W'\tensor{\cal O}_X} & \sc{\to} & \sc{K_x} & \sc{\to} & \sc{0}\cr & & \sc{\da} & & \sc{\da} & & \sc{\da} & & \cr \sc{0} & \sc{\to}& \sc{j^*M'/ \mathop{\rm Tor}\nolimits^{{\cal O}_X}_1( \C_x, {\cal O}_X)} &\sc{\to}& \sc{F'\tensor{\cal O}_X} &\sc{\to} & \sc{L^n \tensor m_{x,\P} \tensor {\cal O}_X} & \sc{\to} & \sc{0}\cr & & \sc{\da} & & \sc{\da} & & \sc{\da} & & \cr \sc{0} & \sc{\to}& \sc{M} & \sc{\to}&\sc{F\tensor{\cal O}_X} & \sc{\to} & \sc{m_{x,X} \tensor L^n} & \sc{\to} & \sc{0}\cr & & \sc{\da} & & \sc{\da} & & \sc{\da} & & \cr & & \sc{0} & & \sc{0} & & \sc{0} & & \cr } $$ Note that $\mathop{\rm Tor}\nolimits^{{\cal O}_X}_1(\C_x,{\cal O}_x)$ is supported at $x$ and consequently is of finite length. Further we note that $K_x$ is also a sheaf of finite length. Now we want to compute the regularity of $M$ in terms of $j^*M'$. On one hand we know that the regularity of $j^*M'$ can be computed in terms of regularity of $M'$ on $\P$. By the previous lemma, we know that the regularity of $M'$ on $\P$ is independent of $n$. And so that regularity of $j^*M'$ depends only on regularity of $M'$ and on regularity of $j_*{\cal O}_X$. Thus we can choose $n$ to be larger than the sum of these two numbers. We note that as the $\mathop{\rm Tor}\nolimits$ term is of finite length, the regularity of the sheaf $j^*(M')$ is the same as the regularity of the sheaf $j^*(M')/\mathop{\rm Tor}\nolimits^{{\cal O}_X}_1(\C_x,{\cal O}_X)$ (this is essentially because the $\mathop{\rm Tor}\nolimits$ has no higher cohomologies, being supported on a single point). Similarly, as $K_x$ has finite length, the top row of the diagram says that $F$ and ${\cal O}_X$ have the same regularity. Thus we see that there is an $n_0$ depending only on $X,L$ such that $M$ is $n_0$ regular. Then the vanishing which is required follows. \end{proof} Lastly, the assertion (7) is proved exactly as above. The point to be observed is the following lemma \begin{lemma} Let $W=H^0(\P,{\cal O}(n) \tensor m_x^2 )$, $x\in\P$, define the coherent sheaf $M''$ on $\P$ by the exact sequence: $$ 0 \to M'' \to H^0(\P,{\cal O}(n)\tensor m_x^2) \tensor {\cal O}_\P \to {\cal O}(n) \tensor m_x^2 \to 0.$$ Then $M''$ is $2$-regular on $\P$. \end{lemma} \section{Application to codimension two cycles}\label{Applications to Codimension two cycles} In this section we give an application of our general Noether-Lefschetz Theorem to codimension two cycles. The application which we have, is related to a conjecture of Madhav Nori (see \cite{Madhav}, \cite{KP}). Let $X$ be a smooth projective variety and $L$ a very ample line bundle on $X$. Let $S=\P(H^0(X,L^n)^*)$. Let $K$ be the function field of $S$, and let ${\bar K}$ be its algebraic closure. Let ${\cal X}=X\times S$ and let ${\cal Y}=\P(M(L^n)^*)\into {\cal X}$ be the universal hypersurface corresponding to sections of $L^n$. The natural inclusion $\Spec(K) \into S$ gives rise to a fibre square $$\Matrix{ {\cal Y}_K & \into & {\cal X}_K \cr \downarrow & & \downarrow \cr {\cal Y} & \into & {\cal X} \cr } $$ The inclusion ${\cal Y}_K \into {\cal X}_K$ then gives rise to the restriction map on rational equivalence classes of cycles in codimension $i$: $$ CH^i({\cal X}_K) \to CH^i({\cal Y}_K).$$ Nori's conjecture in this set up is the following: \begin{conj}[M. V. Nori]\label{Madhav's conjecture} If $n$ is sufficiently large then the natural map $$CH^i({\cal X}_{\bar{K}})\tensor\Q \to CH^i({\cal Y}_{{\bar K}})\tensor\Q$$ is an isomorphism for $i < \dim({\cal Y}_K) $ and an inclusion for $i=\dim({\cal Y}_K) $. \end{conj} Observe that the inclusion of fields $\C \into K$ gives rise to a morphism ${\cal X}_K \to X_{\C}$ and hence a pull back morphism $CH^i(X_{\C}) \to CH^i({\cal X}_K)$ on cycle classes. Hence by composition we get an homomorphism $p_X^* : CH^i(X_{\C}) \to CH^i({\cal Y}_K)$. We can now prove the following weaker statement: \begin{thm}\label{codim two cycles} Let $X$ be a smooth projective threefold, $L$ an ample line bundle on $X$. If $n$ is sufficiently large then the natural map (given above): $$CH^2(X_{\C}) \to CH^2({\cal Y}_K)$$ is an inclusion. \end{thm} Before we begin the proof of the theorem, let us note one immediate consequence of Theorem~\ref{codim two cycles}. \begin{cor} Let $X$ be a smooth projective threefold, $L$ a very ample line bundle on $X$. Then we have $$ CH^2(X_{\C})\tensor\Q \into CH^2({\cal Y}_{\bar K})\tensor\Q $$ \end{cor} This is immediate from the fact that the passage from $K$ to ${\bar K}$, annihilates only the torsion in Chow groups. \begin{proof} Let $z\in CH^2(X_{\C})$ be a cycle such that $p_X^*(z) = 0$ in $CH^2({\cal Y}_K)$. Then we note that there exists a divisor $D\into S$ such that $z$ is the image of a cycle in $CH^1({\cal Y}_D)$, where ${\cal Y}_D$ is the pull back of ${\cal Y} \to S$ to $D$ (via the inclusion $D\into S$). We can find a general pencil $\P^1 \into S$, which meets $D$ transversally. Let $F$ denote the function field of the $\P^1$ which parametrises the pencil. We have a commutative diagram in which the rows are complexes: $$\Matrix{ & & CH^2(X_{\C}) && \cr & & \mapdown{\tau} & & \cr CH^1({\cal Y}_D) & \mapright{f} & CH^2({\cal Y}) & \mapright{h} & CH^2({\cal Y}_K) \cr \mapdown{g} & & \mapdown{g'} & & \cr CH^1({\cal Y}_{D\cap\P^1}) & \mapright{f'} & CH^2({\cal Y}_{\P^1}) & \mapright{h'} & CH^2({\cal Y}_F) \cr } $$ Since $p_X^*(z) = h(\tau(z)) = 0$, there is a $\psi\in CH^1({\cal Y}_D)$ such that $f(\psi) = \tau(z)$. Then as the square commutes and as the bottom row is a complex, we see that the image of $z$ in $CH^2({\cal Y}_F)$ is also zero. Thus we are reduced to proving the following stronger assertion. \end{proof} \begin{thm}\label{reduction to pencil} Suppose $n$ is sufficiently large, and let $S=\P(H^0(X,L^n)^*)$, and let ${\cal Y}_P \into X\times \P^1$ be a general Lefschetz pencil in $S$. Let $F$ be the function field of the $\P^1$ corresponding to this pencil. Then the natural morphism $CH^2(X_{\C}) \to CH^2({\cal Y}_F)$ is injective. \end{thm} \begin{proof} Since $n$ is sufficiently large, by Theorem~\ref{main theorem} the Noether-Lefschetz locus is of codimension at least two in $S$. Therefore we can assume that for every closed point $t\in\P^1$, we have a surjection $\Pic(X) \to CH^1({\cal Y}_t)$. Moreover by Bertini's Theorem we can also assume that the base locus of such a pencil is an irreducible, smooth projective curve $C$. Let $E\into {\cal Y}_P$ be the exceptional divisor. Observe that $E=C\times \P^1$. Then it is easy to check that we have an isomorphism $CH^2({\cal Y}_P) \isom CH^2(X_{\C}) \oplus CH^1(C)$ (see \cite{Fulton}). Further, we also have the following exact sequence: $$ \oplus_{t\in\P^1}CH^1({\cal Y}_t) \to CH^2(X)\oplus CH^1(C) \to CH^2({\cal Y}_F) $$ and where the direct sum extends over all the closed points of $\P^1$. Let $z\in CH^2(X_\C)$ be such that $p_X^*(z)=0$ in $CH^2({\cal Y}_F)$. Then $p_X^*(z)$ is in the image of the map $\oplus_{t\in\P^1}CH^1({\cal Y}_t) \to CH^2({\cal Y}_P)$. And hence there are finitely many closed points $t_1,t_2\ldots,t_m\in \P^1$, and cycles $s_i \in CH^1({\cal Y}_{t_i})$ such that $p_X^*(z)=\sum_{i}s_i$. By the choice of our pencil, we see that there are cycles $w_i\in CH^1(X)$ such that $s_i= p_X^*(w_i).{\cal Y}_{t_i}$. But any two fibres of ${\cal Y}_P \to \P^1$ are rationally equivalent. Hence $p_X^*(z) = p_X^*(\sum_{i}w_i).{\cal Y}_t$. Writing $w=\sum_{i}w_i$, we can rewrite the last equality as $p_X^*(z) = p_X^*(w).{\cal Y}_t$. So to prove that $z =0$, by the projection formula, it suffices to prove that $w=0$. Assume, if possible, that $w$ is not zero. Then we have on intersecting with $E$, $p_X^*(z).E = p_X^*(w).{\cal Y}_t.E$. By the projection formula, $(p_X)_*(p_X^*(z).E) = z.(p_X)_*(E)$. Since $E$ is the exceptional divisor, we have $(p_X)_*(E)=0$, an thus $p_X^*(w).{\cal Y}_t.E = p_X^*(z).E =0$. Now the intersection of ${\cal Y}_t$ with $E$ is $C$, so we see that $p_X^*(w).C =0$. To contradict this, we need to prove that $ \Pic(X) \to \Pic(C) $ is injective. This is done as follows. Let $G$ be the Grassmannian of all lines in $S$, and $\tilde{X} \into X \times G$ be the incidence locus. By choosing a large enough $n$, we can ensure that the fibres of $p_G:\tilde{X} \to G$ are irreducible outside a codimension three subset of $G$. This can be done as follows. Let $R \subset \tilde{X}$ be a subset where the morphism $p:\tilde{X} \to G$ is not smooth. Then since $p$ is generically smooth, $R$ is a proper closed subscheme of ${\tilde X}$. \begin{claim} $R$ is an irreducible subscheme of $\tilde{X}$. \end{claim} \begin{proof} For simplicity, we write $S=\P^n$. By definition, $(x,L) \in {\tilde X}$ is in $R$ if and only if the the natural map $T_{\tilde X} \to p^*T_G$ is not surjective. Note that lines in the dual projective space $\tilde{\P^n}$ correspond to codimension two linear spaces in $\P^n$. We will use the same notation to denote a line in $\tilde{\P^n}$ and the corresponding codimension two linear space in $\P^n$. Thus we see that $p$ is not smooth at $(x,L)$ if and only if the tangent spaces $T_{X,x}$ and $T_{L,x}$ do not span the tangent space $T_{\P^n,x}$. One has the following diagram of vector spaces: $$ \Matrix{ 0 & \to & T_{L,x} & \to & T_{\P^n,x} & \to & N_{L/\P^n} & \to & 0 \cr & & & & \parallel & & & & \cr 0 & \to & T_{X,x} & \to & T_{\P^n,x} & \to & N_{X/\P^n} & \to & 0 \cr } $$ Then $R$ is the locus of pairs $(x,L)$ where $\coker(T_{L,x} \to N_{X/\P^n,x}) \neq 0$. So on writing $U=T_{\P^n,x}, E=N_{X/\P^n,x}$, for each $L$, such that $(x,L)\in R$, we have a codimension $n-2$ subspace $U_L$ of $U$. We are interested in those which subspaces $U_L$ whose images under the surjection $U\to E$ of non-zero co-rank. Write $K=\ker(U\to E)$. Clearly the composite map $U_L \to W$ is not surjective if and only if we have $\dim U\cap K \geq \dim U -\dim W +1$. Thus we see that the set of such subspaces $U_L$ is in fact a Schubert subvariety in the Grassmannian of codimension two subspaces of $U$. It is well know that such varieties are irreducible. Now varying $x\in X$ we see that, for every $x\in X$, we have a surjection of vector spaces $T_{\P^n,x} \to N_{X/\P^n,x}$ and the set of pairs $(x,L)\in R$ is a codimension two linear subspace $T_{L,x} \into T_{\P^n,x}$. Furthermore, $(x,L)\in R$ if and only if the composite map $T_{L,x} \to N_{X/\P^n,x}$ is not surjective. Thus $R$ is an irreducible variety. \end{proof} Now the subscheme $R'$ of ${\tilde X}$ where the fibres of the map $p:{\tilde X} \to G$ are of dimension two is in fact a subscheme of $R$. It is in fact a proper subscheme of $R$. This follows from the fact that since $n$ is large enough, the general singular fibre of $p$ (which a complete intersection) is irreducible. This is an easy consequence of Bertini's Theorem. Then the image of $R'$ in $G$ has codimension at least three. Now let $Z$ be the open subset of $G$ where the fibres of $p_G$ are smooth and irreducible. Let $\eta$ be the generic fibre of $Z$. We have a complex $$\oplus_D CH^0(p_G^{-1}(D)) \to \Pic(\tilde{X}) \to \Pic(\tilde{X}_\eta) $$ where the sum extends over all codimension one subvarieties $D$ in $G$. By the choice of $n$ as above, $p_G^{-1}(D) $ is irreducible. Therefore, the image of $CH^0(p_G^{-1}(D)) \to \Pic(\tilde{X})$ is contained in the image of $\Pic(G) \to \Pic(\tilde{X})$. Note that $\Pic(\tilde{X}) =\Pic(X) \oplus \Z$. Thus $\Pic(X) \to \Pic(\tilde{X}_\eta)$ is injective. Now the following proposition, which seems to be well known but for which we have been unable to find a convenient reference, completes the proof. \begin{propose}\label{specialization} Let $\pi: \tilde{X} \to X, f:\tilde{X}\to Z$ be such that $f$ is proper with general fibre irreducible and smooth, and $X,Z$ irreducible, $X$ smooth projective variety over complex numbers. Let $\eta$ be the generic point of $Z$. Suppose $\Pic(X) \to \Pic(\tilde{X}_\eta)$ is injective. Then for $s\in Z$ outside a countable union of proper closed subvarieties of $Z$, the ``specialization map'' $\Pic(X) \to \Pic(\tilde{X}_s)$ is injective. \end{propose} \begin{proof} Recall that the N\'eron-Severi group, $NS(X) =\Pic(X)/\Pic^0(X)$, is a finitely generated group. Let $\tau \in NS(X)$ be a numerical class, then it is well-known (see for instance \cite{Mumford-Curves}) that there exists smooth projective variety $\Pic^\tau(X)$, which parametrises line bundles on $X$ of numerical class $\tau$, and a universal line bundle $P_\tau$ on $X \times \Pic^\tau(X)$ (the ``Poincare bundle'') with the following property: for any $\alpha\in \Pic^\tau(X)$ the restriction $P_\tau\big|_{X\times \{\alpha\}}$ is the line bundle $\alpha$ on $X$. The line bundle $P_\tau$ is unique up to a tensoring with a line bundle pulled back from $\Pic^\tau(X)$. We have then the following diagram of morphisms: \def\mathop{\rm id}\nolimits{\mathop{\rm id}\nolimits} $$\Matrix{ \tilde{X}\times \Pic^\tau(X) & \mapright{f'=f\times\mathop{\rm id}\nolimits} & Z \times \Pic^\tau(X) \cr \mapdown{\pi\times\mathop{\rm id}\nolimits} & & \cr X\times \Pic^\tau(X) & & \cr } $$ Now recall that $\Pic(X) = \coprod_{\tau\in NS(X)} \Pic^\tau(X)$. Let $s\in Z$, then the ``specialization map'' $g:\Pic(X) \to \Pic({\tilde X}_s)$ gives to maps $g_{\tau,s}:\Pic^\tau(X) \to \Pic({\tilde X}_s)$. Note that $0\in g_{0,s}^{-1}(0)$ for any $s\in Z$. Then to prove the proposition, it suffices to prove that for $s$ outside a countable union of proper closed subsets, we have $$ g_{\tau,s}^{-1}(0) = \cases{\{0\}, &if $\tau = 0$;\cr \emptyset, & otherwise.\cr } $$ Let $P'_\tau = (\pi\times\mathop{\rm id}\nolimits)^*(P_\tau)$. Then we are interested in the set of points $Z_{\tau}=\{(s,p)\in Z\times \Pic^\tau(X) \big| P'_\tau\big|_{\tilde{X}_s\times\{p\}}\} = 0$. We claim that $Z_\tau$ is in fact a closed subset of $Z\times\Pic^\tau(X)$. This is accomplished by the following well-known: \begin{schol} Let $X$ be a smooth projective variety, $S$ any irreducible smooth variety. Let $X \to S$ be smooth proper morphism, and let ${\cal E}$ be a coherent sheaf on $X$ which is flat over $S$. Then the set of points $s\in S$ where the $H^0(X_s,{\cal E}_s) \neq 0 $ is a closed subset of $S$. \end{schol} The assertion of the Scholium is clearly local, so we can assume that the base is affine. Now the result is an easy consequence of the existence of a Grothendieck complex for ${\cal E}$ (see \cite{Mumford}), and will be left to the reader. Now we apply the Scholium to the morphism $f':{\tilde X} \times \Pic^\tau(X) \to X\times \Pic^\tau(X)$ and with ${\cal E} = P'_\tau$, and with ${\cal E} = P_\tau^{'-1}$. Now we are done because a line $L$ bundle on an irreducible variety is trivial if and only if $H^0(L) \neq 0$ and $H^0(L^{-1})\neq 0$. So the set $Z_\tau$ is closed. Now the proof breaks up into the above two cases: $\tau=0$ and $\tau\neq 0$. Let us first dispose of the case $\tau \neq 0$. Since by the hypothesis of Proposition~\ref{specialization} $\Pic(X) \to \Pic({\tilde X}_\eta)$ is injective, and since the morphism $f'$ is generically smooth we see that the subset $Z_\tau$ does not meet $\{\eta\}\times \Pic^\tau(X)$. Hence, there is a proper closed subset $D_\tau\subset Z$ such that $Z_\tau \into D_\tau\times \Pic^\tau(X)$. Hence the image of $Z_\tau$ in $Z$ under the first projection is a proper closed subset of $Z$. The argument for $\tau=0$ is almost identical except that at the generic point, the line bundle $P'_0$ is trivial, and hence at the generic point, $Z_0$ is contained in a subset $\eta\times\Pic^0(X) \cup D_0\times\Pic^0(X)$, where $D_0$ is a proper closed subset of $X$. So to sum up this argument, we have shown that for every class $\tau$ in $NS(X)$, there is closed subscheme $Z_\tau$ of $Z\times \Pic^\tau(X)$, such that for $\tau\neq 0$, $Z_\tau\into D_\tau\times \Pic^\tau(X)$, where $D_\tau\subset Z$ is a proper closed subscheme. For $\tau=0$, the subscheme $Z_0$ (at the generic point) is contained in a subscheme of the form $\eta\times\Pic^0(X)\cup D_0\times \Pic^0(X)$, and again $D_0$ is a proper subscheme of $Z$. Since $NS(X)$ is finitely generated, it is countable, so by removing a countable union of closed subsets $D_\tau$ where $\tau\in NS(X)$, we can ensure that the specialization map is injective. \end{proof} \begin{rmk} In a personal communication, Nori has pointed out to us that his conjecture is not valid for Chow groups over $\Z$, in other words one has to work with Chow groups tensor $\Q$. This can been seen as follows. Suppose $X$ is a smooth projective threefold, $L$ an ample line bundle on $X$, assume further that $H^1(X,{\cal O}_X) = 0$. Let $n$ be any positive integer. Then the zero scheme $Y$ of a general section of $H^0(X,L^n)$ has the property that $H^1({\cal Y}_{K},{\cal O}_{{\cal Y}_K}) =0 $. Then by a well-known result of Roitman (see \cite{Roitman}), we know that $CH^2({\cal Y}_{\bar K})$ is torsion free, because the Albanese variety of ${\cal Y}_{\bar K}=0$. So that the restriction $CH^2(X_{K}) \to CH^2({\cal Y}_{\bar K})$ obliterates all the torsion in $CH^2(X_{K})$, consequently if $CH^2(X_{K})$ has non-trivial torsion then the restriction map cannot be injective. It is possible to write down examples of threefolds which have the above properties. A well studied example being that of a smooth general quartic hypersurface in $\P^4$. It is shown in \cite{Bloch} that group of codimension two cycles algebraically equivalent to zero modulo rational equivalence, denoted by $A^2(X)$, is isomorphic as a group to the intermediate Jacobian $J^2(X)$. The latter group is a complex torus (in fact an Abelian variety) and hence has non-trivial torsion. Moreover, it is standard that $A^2(X) \subset CH^2(X)$ (see \cite{Fulton}). A similar example can be found in \cite{Murre}. \end{rmk} \end{proof} \bibliographystyle{plain}

9206

alg-geom/9206005

en

https://arxiv.org/abs/alg-geom/9206005

alg-geom/9206005

Valery Alexeev

Valery Alexeev

Two Two-dimensional Terminations

25 pages, 4 figures, LaTeX 2.09

Varieties with log terminal and log canonical singularities are considered in the Minimal Model Program, see \cite{...} for introduction. In \cite{shokurov:hyp} it was conjectured that many of the interesting sets, associated with these varieties have something in common: they satisfy the ascending chain condition, which means that every increasing chain of elements terminates. Philosophically, this is the reason why two main hypotheses in the Minimal Model Program: existence and termination of flips should be true and are possible to prove. In this paper we prove that the following two sets satisfy the ascending chain condition: 1. The set of minimal log discrepancies for $K_X+B$ where $X$ is a surface with log canonical singularities. 2. The set of groups $(b_1,...b_s)$ such that there is a surface $X$ with log canonical and numerically trivial $K_X+\sum b_jB_j$. The order on such groups is defined in a natural way.

alg-geom

\section{Introduction} Varieties with log terminal and log canonical singularities are considered in the Minimal Model Program, see \cite{kmm} for introduction. In \cite{sh:hyp} it was conjectured that many of the interesting sets, associated with these varieties have something in common: they satisfy the ascending chain condition, which means that every increasing chain of elements terminates (in \cite{sh:hyp} it was called the upper semi-discontinuaty). Philosophically, this is the reason why two main hypotheses in the Minimal Model Program: existence and termination of flips should be true and are possible to prove. As for the latter, one of the main properties of flips is that log discrepancies after doing one do not decrease and some of them actually increase, \cite{sh:old}. Therefore, if one could show that a set of ``the minimal discrepancies'' satisfies the ascending chain condition, that would help to prove the termination of flips. The Shokurov's proof of existence of 3-fold log flips \cite{sh:3f} is another example of applying the same principle. In fact, to complete the induction it uses some 1~-~dimensional statement, 2~-~dimensional analog of which is proved in this paper. For further discussion, see also \cite{ag-kol}. For one of the first examples where the phenomenon is actually proved let us mention the following \begin{utv}[\cite{al:fi},\cite{al:tg}] Let us define the Gorenstein index of an $n$-dimensional Fano variety $X$ with weak log terminal singularities as the maximal rational number $r$ such that the anticanonical divisor $-K_X\equiv rH$ with an ample Cartier divisor $H$. Then a set $$FS_n\cap [n-2,=+\infty]=\{r(X)|X\; is\; a\; Fano\; variety\; and \; r(X)>n-2\}$$ satisfies the ascending chain condition and has only the following limit points: $n-2$ and $n-2+{1\over k}$, $k=1,2,3...$. \end{utv} In this paper we prove that the following two sets satisfy the ascending chain condition: \begin{num} \item (Theorems \ref{utv:local_pasc},\ref{utv:local_asc}) The set of minimal log discrepancies for $K_X+B$ where $X$ is a surface with log canonical singularities. \item (Theorem \ref{global_asc}) The set of groups $(b_1,...b_s)$ such that there is a surface $X$ with log canonical and numerically trivial $K_X+\sum b_jB_j$. The order on such groups is defined in a natural way, see \ref{blessb'}. \end{num} \medskip The proofs heavily use explicit formulae for log discrepancies from \cite{al:lc}. We do not find it possible to prove them here again. (This is quite easy anyway). \begin{askn} Author would like to thank V.V.Shokurov and J.Koll\'ar for asking the questions that this paper gives the answers to and for useful discussions. \end{askn} \section{Definitions and recalling} All varieties in this paper are defined over the algebraically closed field of characteristic zero. $K_X$ or simply $K$ if the variety $X$ is clear from the context always denote the class of the canonical divisor. \subsection{Basics} \begin{opr} A {\bf \bfQ-divisor} on a variety $X$ is a formal combination $D=\sum d_j D_j$ of Weil divisors with rational coefficients. \end{opr} \begin{opr} One says that a \bfQ-divisor $D$ is {\bf \bfQ-Cartier} if some multiple of it is a Weil divisor with integer coefficients that is a Cartier divisor. \end{opr} \begin{opr} \label{opr:discr} Let $f:Y\to X$ be any birational morphism and $F_i$ be exceptional divisors of this morphism. Consider a divisor of the form $K+B$, where $B=\sum b_j B_j$ and $0<b_j\le1$. Coefficients $a_i$ in the following formula $$K_Y+f^{-1}B+\sum F_i = f^*(K+B)+\sum a_i F_i$$ are called {\bf log discrepancies} of $K+B$. \end{opr} \begin{opr} \label{opr:codiscr} Let $f:Y\to X$ be any birational morphism and $F_i$ be exceptional divisors of this morphism. Consider a divisor of the form $K+B$, where $B=\sum b_j B_j$ and $0<b_j\le1$. Coefficients $b_i$ in the following formula $$K_Y+f^{-1}B+\sum b_i F_i = f^*(K+B)$$ are called {\bf codiscrepancies} of $K+B$. \end{opr} \begin{zam} Evidently there is a simple relation between log discrepancy and codiscrepancy: $a_i=1-b_i$. \end{zam} \begin{opr} A \bfQ-divisor of the form $K+B$ is said to be {\bf log canonical (lc)} if \begin{num} \item $K+B$ is \bfQ-Cartier \item there is a resolution of singularities $f:Y\to X$ such that $supp(f^{-1}B)\bigcup F_i$ is a divisor with normal intersections and all the log discrepancies $a_i\ge0$. \end{num} \end{opr} \begin{opr} A \bfQ-divisor of the form $K+B$ is said to be {\bf log terminal (lt)} if \begin{num} \item $K+B$ is \bfQ-Cartier \item there is a resolution of singularities $f:Y\to X$ such that $supp(f^{-1}B)\bigcup F_i$ is a divisor with normal intersections and all the log discrepancies $a_i>0$. \end{num} \end{opr} \subsection{Graphs} With rare exceptions all the varieties in this paper will be two-dimensional. No doubt that the case of surfaces is much easier than that of more-dimensional varieties. One of the reasons for this is that surface has a natural quadratic form defined by intersection of curves. Many statements that we need can be formulated in terms of weighted graphs and become therefore basicly combinatorical problems. So let us start with a system of curves on a surface that are divided into two classes: ``internal'', denoted by $F_i$ and ``external'', denoted by $B_j$. \begin{opr} A weighted graph $\Gam$ is the following data: \begin{num} \item a ``ground graph'': each vertex $v$ of it corresponds to an ``internal'' curve $F$ , two different vertices $v_1$ and $v_2$ are connected by wedge of weight $F_1\cdot F_2$. \item weights: a vertex $v$ has weight $w=-F^2$ \item genera: a vertex $v$ has genus $p_a(F)$ (arithmetical genus of the curve) \item an ``external part'': additional vertices, corresponding to the ``external'' components $B_j$ , connected with vertices $v_i$ if $B_j$ and $F_i$ intersect. \end{num} \end{opr} Vice versa, every weighted graph $\Gam$ corresponds to a system of curves $\{F_i,B_j\}$. \begin{opr} Graph $\Gam$ is said to be {\bf elliptic, parabolic or hyperbolic} if the corresponding quadratic form $F_i\cdot F_k$ is elliptic, parabolic or hyperbolic, that is, has the signature $(0,n)$, $(0,n-1)$ or $(1,n-1)$. \end{opr} The following is the basic case when we shall need such graphs: $X$ is a surface with a divisor $K+B$ and $f:Y\to X$ is a resolution of singularities of $X$. The curves $F_i$ are exceptional divisors of $f$ and the curves $B_j$ are strict transforms of the components of $B$. Note that since a matrix of intersection $(F_i\cdot F_k)$ is negatively defined, the graph is elliptic and all the weights in this case are positive integer numbers. Usually we will examine graphs that correspond to the {\it minimal\/} resolution of singularities. \begin{opr} A graph $\Gam$ is said to be {\bf minimal} if it does not contain internal vertices that have $p_a=0$ and weight 1. \end{opr} \begin{opr} For any graph with a nondegenerate quadratic form $F_i\cdot F_k$ (for example elliptic or hyperbolic) we define {\bf log discrepancies} $a_i$ as the solutions of a system of linear equations $$\sum a_i F_i\cdot F_k=(2p_a(F_k)-2-F^2)+(f^{-1}B+\sum F_i)F_k$$ \end{opr} \begin{opr} For any graph with a nondegenerate quadratic form $F_i\cdot F_k$ we define {\bf codiscrepancies} $b_i$ by the formula $b_i=1-a_i$ \end{opr} Let us explain the meaning of the two previous definitions. The formulae above are equivalent to the following: $$(K+\sum b_jB_j+\sum b_iF_i)F_k=0\quad for \quad any\quad k$$ So if the graph $\Gam$ is an elliptic graph, corresponding to some birational morphism $f:Y\to X$ we get the previous definitions \ref{opr:discr}, \ref{opr:codiscr}. Another situation when we shall use discrepancies and codiscrepancies is the following: $X$ is a surface with numerically trivial $K+B$, $f:Y\to X$ is some resolution. Part of the vertices of $\Gam$ correspond to exceptional curves of $f$ and the other part -- to strict transforms of certain curves on $X$. \begin{opr} A graph $\Gam$ is said to be {\bf log canonical (lc) or log terminal (lt)} with respect to $K+B$ if its log discrepancies $a_i\ge0$ or $a_i>0$ respectively. \end{opr} The main object into consideration in this paper is a surface $X$ with a divisor $K+B$ that is lc. So will be the corresponding graphs. If we ignore the way $B$ meets the ground graph or assume that all the coefficients of $B_j$ equal 1, then all such graphs are classified in \cite{kaw:cb} (see also \cite{al:lc}). They are divided into two classes describing respectively rational and elliptic singularities. In the case of rational singularities all the genera are equal to 0 (and by this reason will be omited), all the edges are simple (of weight 1) the ground graphs are those of types $A_n$, $D_n$ and $E_6,E_7,E_8$. If we fix some number $N$ and consider graphs with weights $\le N$ then the only infinite series of such graphs are the following \begin{picture}(300,150)(-10,0) \multiput(75,75)(150,0){2}{\begin{picture}(0,0) \put(0,0){\oval(100,20)} \put(-7.5,0){\oval(75,10)} \put(40,0){\circle{10}} \put(0,25){\vector(0,-1){15}} \put(-15,-20){\vector(0,1){20}} \put(-5,30){$m$} \put(-20,-35){$q$} \end{picture}} \multiput(265,50)(0,50){2}{ \put(0,0){\circle{10}} \put(10,-5){2} } \multiput(265,55)(0,25){2}{\put(0,0){\line(0,1){15}}} \put(85,30){a)} \put(235,30){b)} \put(130,10){Figure 1} \end{picture} These are typical pictures that we shall use to describe graphs. Long ovals denote chains of vertices. The numbers $q$ and $m$ denote the absolute values of determinants of the submatrices of $F_i\cdot F_k$ that include only rows and columns corresponding to the vertices of the chains. It is very well known (\cite{ri}, comp.\cite{al:lc}) that any chain is uniquely determined by a pair of coprime numbers $(q,m)$ with $1\le q<m$ and vice versa. In the previous example $q$ and $m$ are any such numbers. Generally, graphs shall also have external parts that shall be denoted by crossed vertices. In the case of elliptic singularities one has ``circles'' of vertices with $p_a=0$ and a single vertex with $p_a=1$. $B$ is empty and all the log discrepancies $a_i=0$, codiscrepancies $b_i=1$. \begin{opr} {\bf Du Val graph} is an elliptic graphs with all genera = 0, all weights = 2 and empty external part $B$. It is well known that the ground graph is then one of the graphs $A_n$, $D_n$, $E_6,E_7$ or $E_8$. \end{opr} \begin{opr} We say that a graph $\Gam'$ is a subgraph of $\Gam$ if all the vertices of $\Gam'$ are at the same time vertices of $\Gam$, weights of vertices and edges of $\Gam'$ and $p_a$ of vertices in $\Gam'$ do not exceed the corresponding weights and $p_a$ in $\Gam$ and $F_i'\sum b_j'B_j'\le F_i'\sum b_jB_j $ for the corresponding vertices. \end{opr} The following are easy linear algebra statements. \begin{lem} \label{a_less1} Let $\Gam$ be a minimal elliptic graph. Then all the log discrepancies $a_i\le1$ (codiscrepancies $b_i\ge0$) and if $\Gam$ is not a Du Val graph then $a_i<1$ ($b_i>0$). \end{lem} \par\noindent {\sl Proof:}\enspace Well known. \qed \begin{lem} \label{ag_less_ag'} Let $\Gam'\subset\Gam$, $\Gam'\neq\Gam$ be two minimal elliptic graphs and assume that the weights of the vertices are in both graphs the same. Then for the log discrepancies one has $a_i\le a_i'$ (for codiscrepancies $b_i\ge b_i'$) and if $\Gam$ is not a Du Val graph then $a_i< a_i'$ ($b_i> b_i'$). If the weights of $\Gam'$ and $\Gam$ are different then $a_i\le a_i'$ assuming that $\Gam$ is log canonical. \end{lem} \par\noindent {\sl Proof:}\enspace Compare the corresponding systems of linear equations (see \cite{al:fi}, \cite{al:lc}). \qed \begin{lem} \label{ag_greater_ag'} Let $\Gam'\subset\Gam$, $\Gam'\neq\Gam$ be two graphs such that all the log discrepancies of $\Gam'$ $a_i\le 1$ (codiscrepancies $b_i\ge 0$) and $v_0$ is a fixed vertex of $\Gam'$. Assume that $\Gam$ is hyperbolic and that $\Gam-v_0$ is elliptic. Then for the log discrepancy of $v_0$ one has $a_0\ge a_0' $ ($b_0\le b_0' $) assuming that $\Gam$ is log canonical. \end{lem} \par\noindent {\sl Proof:}\enspace Compare the corresponding systems of linear equations. \qed \begin{sle} \label{max_ell} Let $\Gam$ be a minimal elliptic graph and assume that all the log discrepancies of $\Gam$ $a_i\ge c>0$. Then weights of the vertices are bounded from above by $2/c$. \end{sle} \par\noindent {\sl Proof:}\enspace Consider a graph $\Gam'$ containing a single vertex of weight $n$. Then $a'=2/n$. \qed \begin{sle} \label{max_hyp} Let $\Gam$ be a graph as in \ref{ag_greater_ag'} plus let $v_0$ have weight 1. Assume that the codiscrepancy of $v_0$ $b_0\ge c>0$. Then $$\sum_{i\ne0} F_0F_i\le 2+\frac{2}{c}$$ \end{sle} \par\noindent {\sl Proof:}\enspace Consider a graph $\Gam'$ containing a vertex $v_0$ connected with $n$ vertices of weight 2. Then $b_0'=2/(n+2)$. \qed \subsection{Sequences} \begin{opr} Let $X$ be a variety with a log canonical $K+B$. {\bf A log discrepancy of $\bf K+B$ $ld(K+B)$} is a minimal log discrepancy $a_i$ that appears in \ref{opr:discr} for some birational morphism $f:Y\to X$. \end{opr} It is easy to see that $ld(K+B)$ is well defined and is a nonnegative rational number. \begin{opr} Let $X$ be a surface with a log canonical $K+B$. {\bf A partial log discrepancy of $\bf K+B$ $pld(K+B)$} is a minimal log discrepancy $a_i$ that appears in \ref{opr:discr} for the special birational morphism $h:{\widetilde X}} \def\xtil{{\tilde x}\to X$, where ${\widetilde X}} \def\xtil{{\tilde x}$ is the minimal resolution of singularities. \end{opr} \begin{opr} Let $\xi=\{X^{(n)},K+B^{(n)}| n=1,2...\}$ be a sequence of surfaces. Then we define $\bf ld(\xi)$ and $\bf pld(\xi)$ as the {\bf sequences} of real numbers \{$ld({K+B^{(n)}}{ })$\} and \{$pld({K+B^{(n)}}{ })$\} respectively. \end{opr} \begin{opr} \label{LDPLD} Let $\xi=\{X^{(n)},K+B^{(n)}| n=1,2...\}$ be a sequence of surfaces. Then we define $\bf LD(\xi)$ and $\bf PLD(\xi)$ as the {\bf subsets} of real numbers \{$ld({K+B^{(n)}}{ })$\} and \{$pld({K+B^{(n)}}{ })$\} respectively. \end{opr} \begin{opr} We define $ld,pld,LD,PLD$ for graphs in the same way as we have done it for surfaces. \end{opr} \begin{opr} \label{blessb'} Let $B=(b_1,b_2...b_s)$ an $B'=(b_1',b_2'...b_t')$ be two groups of numbers. One says that $B\le B'$ if \begin{num} \item $s\geq t$ \item for every $j=1...t$ $b_j\le b_j'$ \end{num} If, in addition, one of the inequalities in (i) or in (ii) for some index $j_0$ is strict, one says that $B<B'$. \end{opr} \begin{zam} Because of the part (i) of \ref{blessb'} when considering a nondecreasing sequence {$B^{(n)}$ } we can always assume, passing to a subsequence, that the lengths of {$B^{(n)}$ } are in fact the same. \end{zam} \subsection{Log Del Pezzo surfaces} \begin{opr} A normal surface $X$ is said to be a Del Pezzo surface if $-K$ is an ample \bfQ-divisor. \end{opr} The following is an simple lemma, see \cite{al-nik},\cite{nik} for the proof which is especially easy if $K$ is lt or lc. \begin{lem} \label{Del_Pezzo} Let $X$ be a log Del Pezzo surface an $h:{\widetilde X}} \def\xtil{{\tilde x}\to X$ be the minimal resolution of singularities. Then \begin{num} \item the Kleiman-Mori cone of effective curves $NE({\widetilde X}} \def\xtil{{\tilde x})$ is generated by finitely many extremal rays \item if $X\ne \bfP^2,\bfF_n$ (minimal rational surface) then all the extremal rays are generated by exceptional curves of $f$ and (-1)-curves. \end{num} \end{lem} \begin{lem} \label{log_Del_Pezzo} Let $X$ be a Del Pezzo surface and assume that $K$ is lc. Then $X$ is one of the following: \begin{num} \item a rational surface with rational singularities \item a generalized cone over a smooth elliptic curve \end{num} \end{lem} \par\noindent {\sl Proof:}\enspace Let $h:{\widetilde X}} \def\xtil{{\tilde x}\to X$ be a minimal desingularization. ${\widetilde X}} \def\xtil{{\tilde x}$ is a smooth surface and clearly $h^0(NK_{{\widetilde X}} \def\xtil{{\tilde x}}=0$ for any $N>0$, so ${\widetilde X}} \def\xtil{{\tilde x}$ is ruled. Assume that $X$ has a nonrational singularity. Then by the classification of log canonical singularities ${\widetilde X}} \def\xtil{{\tilde x}$ contains an elliptic curve or a circle of rational curves $F_0$ that is disjoint from other curves, exceptional for $h$. If ${\widetilde X}} \def\xtil{{\tilde x}$ is a locally trivial $\bfP^1$-bundle then $F_0$ should be an exceptional section of this bundle and should be smooth. In this case $X$ is a generalized cone. Otherwise $F_0$ should intersect a curve $E$ with $E^2<0$ that lies in the fiber of a generically $\bfP^1$-bundle giving the structure of a ruled surface and such that $E$ is not exceptional for $h$. By \ref{Del_Pezzo} $E$ is a $(-1)$-curve. The latter is impossible since $-h^*K=-K_{{\widetilde X}} \def\xtil{{\tilde x}}-F_0-...$ and therefore $-h^*K\cdot E\le0$. Now let us assume that $X$ has only rational log canonical singularities. By the classification again one has $-h^*K=-K_{{\widetilde X}} \def\xtil{{\tilde x}}-F_0-\sum b_iF_i$, $0\le b_i<1$ and $F_0$ is a disjoint union of smooth rational curves. Since $-h^*K$ is big, nef, the Kawamata-Fiehweg vanishing gives $$h^1({\widetilde X}} \def\xtil{{\tilde x},-F_0)=h^1({\widetilde X}} \def\xtil{{\tilde x},K_{{\widetilde X}} \def\xtil{{\tilde x}}+\sum b_iF_i+(-h^*K))=0$$ and from the exact sequence $$0\to\cal O_{{\widetilde X}} \def\xtil{{\tilde x}}(-F_0)\to\cal O_{{\widetilde X}} \def\xtil{{\tilde x}}\to\cal O_{F_0}\to 0$$ one gets $h^1(\cal O_{{\widetilde X}} \def\xtil{{\tilde x}})=0$. Therefore ${\widetilde X}} \def\xtil{{\tilde x}$ and $X$ are rational surfaces. \qed \section{Local case: elliptic log canonical graphs} {\bf In this section we consider only local situation. $\bfX$ is a neighbourhood of a surface point $\bfP$ and all the components of $\bfB$ pass through $\bfP$. } \begin{utv}[Local boundness] \label{utv:loc_bound} Let $X,K+B$ be as above a neighbourhood of a surface point $P$ with lc $K+B$ and all of $B_j$ pass through $P$. Then $\sum b_j\leq2$. \end{utv} \par\noindent {\sl Proof:}\enspace Proved in \cite{ag-kol} for the $n$-dimensional case with a bound $n$. \subsection{Minimal resolution} \begin{utv}[Local partial ascending chain condition] \label{utv:local_pasc} Let $\xi=\{X^{(n)},K+B^{(n)}\}$ be a sequence of surfaces such that \begin{num} \item $K+B^{(n)}$ is lc \item $B^{(n)}$ is a nondecreasing sequence (for example, constant) \end{num} Then every increasing subsequence in $pld(\xi)$ terminates. \noindent If, in the addition, one has \begin{num} \setcounter{beam}{2} \item $B^{(n)}$ is an increasing sequence \end{num} then every nondecreasing subsequence in $pld(\xi)$ terminates. \end{utv} We prove \ref{utv:local_pasc} in several steps. \begin{shag} One can assume that {$K+B^{(n)}$ } and moreover $K$ are lt. \end{shag} \par\noindent {\sl Proof:}\enspace Indeed, if in (ii) {$K+B^{(n)}$ } is not lt, then $pld({K+B^{(n)}}{ })=0$ but we are looking for increasing subsequences of $pld(\xi)$. In (iii) if {$K+B^{(n)}$ } is not lt, then there exists a partial resolution $f:Y\to X$ with a single exceptional divisor $F$ such that the corresponding log discrepancy $a=0$. Then $$K_Y+F+f^{-1}B=f^*(K+B)$$ and the log adjunction formula for $F$ (see \cite{sh:3f}) yields $$\sum {k-1+\sum l_j b_j\over k}=2$$ for some positive integers $k,l_j$. It is easy to see that if {$B^{(n)}$ } are increasing then the sequence should terminate. \qed \begin{shag} By the previous step we can assume that there is a constant $\varepsilon$ so that for every $n$ $pld({K+B^{(n)}}{ })>\varepsilon$. Then we prove the following \end{shag} \begin{lem} All the lt elliptic graphs with $pld({K+B^{(n)}}{ })>\varepsilon$ and $b_j>\varepsilon$ can be described as follows: \label{fin_many_graphs} \begin{num} \item finitely many graphs (that includes the way $B_j$ intersect $F_i$) \item the graphs given on the next picture, where there are only finitely many possibilities for the chains of vertices, denoted by ovals and for the ways $B_j$ meet that vertices \end{num} \begin{picture}(320,270)(-10,-30) \multiput(0,0)(0,100){2}{ \put(5,50){$q_1$} \put(15,50){\vector(1,0){25}} \put(65,50){\oval(80,20)} \put(60,50){\oval(50,10)} \put(95,50){\circle{5}} \put(105,50){\line(1,0){12.5}} \put(122.5,50){\line(1,0){15}} \multiput(120,50)(20,0){2}{\circle{5}} \multiput(117.5,55)(20,0){2}{2} \multiput(142.5,50)(27.5,0){2}{\line(1,0){7.5}} \multiput(155,50)(5,0){3}{\circle*{1}} \multiput(35,85)(50,0){2}{\circle{14}} \multiput(30,90)(50,0){2}{\line(1,-1){10}} \multiput(30,80)(50,0){2}{\line(1,1){10}} \multiput(50,80)(10,0){3}{\circle*{1}} \put(35,78){\line(5,-6){15}} \put(85,78){\line(-5,-6){15}} \put(5,20){$m_1$} \put(15,25){\vector(1,1){15}} \put(125,20){\vector(-1,1){25}} \put(125,15){$min$} \put(110,85){\circle{14}} \put(105,90){\line(1,-1){10}} \put(105,80){\line(1,1){10}} \put(95,52.5){\line(3,5){15}} } \multiput(180,30)(0,20){3}{\circle{5}} \multiput(185,27.5)(0,20){3}{2} \multiput(180,32.5)(0,20){2}{\line(0,1){15}} \multiput(180,150)(20,0){2}{\circle{5}} \multiput(177.5,155)(20,0){2}{2} \put(182.5,150){\line(1,0){15}} \put(202.5,150){\line(1,0){12.5}} \put(255,150){\oval(80,20)} \put(260,150){\oval(50,10)} \put(225,150){\circle{5}} \multiput(235,185)(50,0){2}{\circle{14}} \multiput(230,190)(50,0){2}{\line(1,-1){10}} \multiput(230,180)(50,0){2}{\line(1,1){10}} \multiput(250,180)(10,0){3}{\circle*{1}} \put(235,178){\line(5,-6){15}} \put(285,178){\line(-5,-6){15}} \put(210,185){\circle{14}} \put(205,190){\line(1,-1){10}} \put(205,180){\line(1,1){10}} \put(225,152.5){\line(-3,5){15}} \put(195,120){\vector(1,1){25}} \put(175,115){$min$} \put(305,150){\vector(-1,0){25}} \put(310,150){$q_2$} \put(310,120){$m_2$} \put(305,125){\vector(-1,1){15}} \put(140,-15){Figure 2} \end{picture} Moreover, the log discrepancies of any of suchs graphs satisfy the following inequality $$pld(K+B)\ge {1-\sum l_jb_j\over m-q}$$ where $\bar b_j=\lim b_j$ and tend to this number as the chain of 2's gets longer and longer. Here $l_j=\sum( B_j\cdot F_i) r_i$, where $r_i$ is the determinant of the short subchain of the ground graph, ``cut off'' by the vertex $v_i$. \end{lem} \par\noindent {\sl Proof:}\enspace By \ref{max_ell} the weights of vertices in the graph $\Gam$ are bounded by $2/\varepsilon$. Therefore, sacrificing finitely many graphs we can assume that $\Gam$ is one of the graphs on Fig.1. First, assume that we are in the case a) of Fig.1, i.e. $\Gam$ is a chain. Consider the sequence of log discrepancies of vertices in this chain. By \cite{al:lc} $$a_{i-1}-2a_i+a_{i+1}=(w_i-2)a_i+\sum b_jB_jF_j\ge(w_i-2+\sum B_jF_i)\varepsilon,$$ therefore the graph of this function is concave up and, unless $w_i=2$ and all $B_jF_i=0$, it is not a straight line but is ``very concave up''. Now by \ref{a_less1} the discrepancies $a_i\le1$. This implies that all the chains are those on Fig.2 with only finitely many possibilities for the ovals and with the chains of 2's of an arbitrary length $A$. Also, omiting finitely many graphs, we can assume that the minimal log discrepancy is achieved at one of the two vertices, where the arrows point out. Now we use an explicit formula for the log discrepancies of those vertices which follows easily from 3.1.8, 3.1.10 of \cite{al:lc}. Define $\alpha} \def\Alp{\Alpha_1=1-\sum l_j^{(1)}b_j$ for the left part of the chain, the meaning of $l_j$ being explained in the formulation of the statement, $\alpha} \def\Alp{\Alpha_2=1-\sum l_j^{(2)}b_j$ be the corresponding expression for the right par, and let $A$ be the length of the chain of 2's. Then $$a_1={\alpha} \def\Alp{\Alpha_1(A(m_2-q_2)+m_2)+\alpha} \def\Alp{\Alpha_2q_1\over A(m_2-q_2)(m_1-q_1)+m_2(m_1-q_1)+q_1(m_2-q_2)}$$ or $$a_1={{\alpha} \def\Alp{\Alpha_1\over m_1-q_1}(A+{m_2\over m_2-q_2})+ {\alpha} \def\Alp{\Alpha_2\over m_2-q_2}{q_1\over m_1-q_1}\over A+{m_2\over m_2-q_2}+{q_1\over m_1-q_1}}$$ with the symmetric expression for $a_2$. One can note that \begin{enumerate} \item if ${\alpha} \def\Alp{\Alpha_1\over m_1-q_1}\le {\alpha} \def\Alp{\Alpha_2\over m_2-q_2}$ then ${\alpha} \def\Alp{\Alpha_1\over m_1-q_1}\le a_1\le {\alpha} \def\Alp{\Alpha_2\over m_2-q_2}$ \item $\lim_{A\to\infty}a_1={\alpha} \def\Alp{\Alpha_1\over m_1-q_1}$ \end{enumerate} and these two observations complete the proof in the case a) of Fig.1. The case b) of Fig.1 is handled similarly. Let us mention only that in the latter case there is only one possible vertex for the minimal log discrepancy which is given by the formula $$a_1={\alpha} \def\Alp{\Alpha_1\over m_1-q_1},$$ so this case can be treated formally as a subcase of a) with $\alpha} \def\Alp{\Alpha_2=0$ and $m_2=q_2$. \qed \begin{shag} The lemma~\ref{fin_many_graphs} implies \ref{utv:local_pasc}. \end{shag} \par\noindent {\sl Proof:}\enspace For any fixed graph $\Gam$ if the coefficients of the external part $B$ increase, then by \ref{ag_less_ag'} log discrepancy $pld(K+B)$ decreases. Therefore, we can consider only case (ii) of \ref{fin_many_graphs}. Passing to a subsequence we can assume that all the graphs are of the same type and the length of the sequence of 2's increases. But then $$pld({K+B^{(n)}}{ }) \ge{1-\sum l_j\bar b_j\over m-q}$$ and $$\lim pld({K+B^{(n)}}{ }) = {1-\sum l_j\bar b_j\over m-q}$$ where $\bar b_j=\lim b_j$, and we are done. \qed \begin{sle}\label{corol} If $B=\emptyset$, then \ref{fin_many_graphs} says that the set of minimal log discrepancies satisfies the ascending chain condition and the only limit points are 0 and $1/k$, $k=2,3...$ \end{sle} \begin{zam} The statement \ref{corol} is due to V.V.Shokurov (unpublished). \end{zam} \subsection{General case} Later we shall use the local ascending chain condition in the just proved form, i.e. for the minimal resolution of singularities. But the minimal resolution of singularities of $X$ is not necessarily a resolution of singularities for $K+B$, because $supp(f^{-1}B\cup F_i)$ can have nonnormal intersections. Below we prove the statement, corresponding to \ref{utv:local_pasc} but for $ld(\xi)$ instead of $pld(\xi)$. We first consider the case when {$X^{(n)}$ } are nonsingular and then combine our arguments to treat the general situation. \begin{utv} \label{nonsingular} Let $\xi=\{X^{(n)},K+B^{(n)}\}$ be a sequence of {\bf nonsingular} surfaces such that \begin{num} \item $K+B^{(n)}$ is lc \item $B^{(n)}$ is a nondecreasing sequence (for example, constant) \end{num} Then every increasing subsequence in $ld(\xi)$ terminates. \noindent If, in the addition, one has \begin{num} \setcounter{beam}{2} \item $B^{(n)}$ is an increasing sequence \end{num} then every nondecreasing subsequence in $ld(\xi)$ terminates. \end{utv} \par\noindent {\sl Proof:}\enspace As above, we can assume that that {$K+B^{(n)}$ } are in fact lt. Now let us find out what happens with a nonsingular surface $X$ with $K+B$ after a single blow up $f:X\to Y$ at the point $P$, $F$ as usually denotes the exceptional divisor of $f$. The answer is evident: \begin{equation} \label{blowup} f^*(K+\sum b_jB_j)=K_Y+\sum b_jf^{-1}B_j+(-1+\sum mult_PB_jb_j)F \end{equation} and the condition $a>0$ translates to $-1+\sum mult_PB_jb_j<1$. If $-1+\sum mult_PB_jb_j\le0$, then for any further blowups all the log discrepancies $a_i\ge a$, so they are irrelevant in finding the minimal log discrepancy and $K+B$ is lt. However, if this is a positive number, some negative log discrepancies can appear on the following steps. Now let $f:X\to Y$ be a composite of several blow ups. One gets \begin{equation} \label{blowups} f^*(K+\sum b_jB_j)=K_Y+\sum b_jf^{-1}B_j+ \sum(-s_i+\sum t_{ik}b_k)F_i \end{equation} with some nonnegative integers $s_i$, $t_{ik}$ and $s_i\le\rho(Y/X)$. The corresponding log discrepancies are given by $$a_i=1+s_i-\sum t_{ik}b_k$$ and for fixed $s_i$ and nondecreasing/increasing $b_j$ they evidently form a nonincreasing/decreasing sequences. Note that there are only finitely many such sequences with $a_i\ge0$. Therefore \ref{utv:local_pasc} follows from the following lemma. \begin{lem} With the assumptions as above, there is a constant $N(\xi)$ so that for every surface {$X^{(n)}$ } in $\xi$ there exists a birational morphism $g:{Y^{(n)} }\to{X^{(n)}}{ }$ such that \begin{num} \item $\rho({Y^{(n)} }/{X^{(n)}}{ })<N(\xi)$ \item the minimal log discrepancy $ld({K+B^{(n)}}{ })$ is one of the log discrepancies of $g$. \end{num} \end{lem} \par\noindent {\sl Proof:}\enspace Let us remind that we are in the local situation, so {$X^{(n)}$ } is a neighbourhood of a (nonsingular) point $P$. Let $f:{Z^{(n)} }\to{X^{(n)}}{ }$ be a single blow up at $P$. If in the formula~\ref{blowup} the number $C=-1+\sum mult_PB_jb_j$ is positive and on {$Y^{(n)}$ } the strict transforms of $B_j$ intersect at one point and have the same multiplicities as on {$X^{(n)}$ }, then by the formula~\ref{blowup} on the second blowup codiscrepancy of the exceptional divisor equals $2C$, after the third blowup $3C$ and so on (and it should be $\le1$). Since $B^{(n)}$ is nondecreasing, there exists a constant $\varepsilon(\xi)$ so that for any $-1+\sum m_jb_j>0$, one also has $-1+\sum m_jb_j>\varepsilon(\xi)$. The conclusion is that there exists a number $N_1$, depending on $\xi$, so that after $N_1(\xi)$ blowups the configuration of $B_j$ simplifies in some way: either the number of curves, passing through the points, or the multiplicities at those points get smaller; or all the further blowups are irrelevant in finding the minimal discrepancy. Let $X^{(n)} {}'$ be $X^{(n)}$ with blown up points, ${K+B^{(n)}}{ }'=f^*({K+B^{(n)}}{ })$. Note that the coefficients of ${B^{(n)}}{ }'$ are still nonnegative numbers. At the neighbourhood of any point of $X^{(n)}{ }'$\enspace ${B^{(n)}}{ }'$ consists of several curves $B_j+\le 2$ nonsingular curves $F_i$ with coefficients, given by the formula~\ref{blowups} and hence, nondecreasing, and from the finite list of possible combinations. Now we can find the next number $N_2(\xi)$ so that after $N_2(\xi)$ blowups the configuration of ($B_j$ $+$ $\le 2$ nonsingular curves) simplifies even further. By induction we get the desired result. \qed \medskip And finally we prove \begin{utv}[Local ascending chain condition] \label{utv:local_asc} Let $\xi=\{X^{(n)},K+B^{(n)}\}$ be a sequence of surfaces such that \begin{num} \item $K+B^{(n)}$ is lc \item $B^{(n)}$ is a nondecreasing sequence (for example, constant) \end{num} Then every increasing subsequence in $ld(\xi)$ terminates. \noindent If, in the addition, one has \begin{num} \setcounter{beam}{2} \item $B^{(n)}$ is an increasing sequence \end{num} then every nondecreasing subsequence in $ld(\xi)$ terminates. \end{utv} \par\noindent {\sl Proof:}\enspace By \ref{fin_many_graphs} all the singularities with $ld({K+B })\ge\varepsilon$ are divided into finite number of series $+$ finite number of graphs. The latters are taken care by \ref{nonsingular}. So all we have to do is to consider one of the graphs on Fig.2 with the chain of 2's that is getting longer and longer. And a simple calculation shows that for all except finitely many graphs the minimal log discrepancy is in fact one of log discrepancies of $h:{\widetilde X}} \def\xtil{{\tilde x}\to X$. \qed \section{Special hyperbolic log canonical graphs} \noindent {\bf Set-up\enspace } In this section $\Gam$ or $(X,K+B)$ always denote the following: \begin{opr} We say that a graph $\Gam$ is {\bf special hyperbolic} if \begin{num} \item $\Gam$ is hyperbolic and connected \item all the vertices have $p_a=0$, there is a special vertex $v_0$ of weight 1, all other vertices $v_i$ have weights $\ge2$ \item $\Gam-v_0$ is elliptic \item as usually, $\Gam$ may have an external part $B=\sum b_jB_j$ \end{num} \end{opr} Such graphs naturally appear when one considers a minimal resolution of singularities of a Del Pezzo surface $X$ with $\rho(X)=1$ and $B_0$ being a (-1)-curve on the resolution. \begin{utv}[Local-to-global ascending chain condition] \label{hyper} Let $\xi=\{{X^{(n)}}{ },{K+B^{(n)}}{ }\}$ be a sequence of special hyperbolic graphs with a chosen vertex $B_0$ such that \begin{num} \item {$K+B^{(n)}$ } is lc \item \{{$B^{(n)}$ }\} is an increasing sequence, moreover, $\{b_0^{(n)}\}$ is an increasing sequence \item all the log discrepancies $a_i\ge 1-\bar b_0=1-\lim b_0$ \item $K+B^{(n)}$ is numerically trivial \end{num} Then $\xi$ terminates. \end{utv} \par\noindent {\sl Proof:}\enspace {\par\noindent{\sl Case 1:}\enspace} $\bar b_0=\lim b_0=1$. {}From \cite{al:lc} it follows that if $b_0$ is close enough to 1, then all the singularities (that is, the connected components of $\Gam-v_0$) and the ways the components of $B$ meet $F_i$ are exhausted by the following list: \begin{picture}(300,270)(-30,-30) \multiput(0,0)(0,100){2}{ \put(20,45){$B_0$} \put(50,50){\circle{14}} \put(45,55){\line(1,-1){10}} \put(45,45){\line(1,1){10}} \put(130,50){\oval(110,20)} \put(57,50){\line(1,0){18}}} \multiput(170,20)(0,30){3}{\circle{10}} \multiput(170,25)(0,30){2}{\line(0,1){20}} \multiput(180,17)(0,60){2}{2} \put(85,150){\circle{5}} \put(135,150){\oval(80,10)} \put(115,180){\vector(0,-1){25}} \put(105,180){$q$} \put(145,120){\vector(0,1){25}} \put(150,115){$m$} \multiput(135,185)(40,0){2}{\circle{14}} \multiput(130,190)(40,0){2}{\line(1,-1){10}} \multiput(130,180)(40,0){2}{\line(1,1){10}} \multiput(150,180)(5,0){3}{\circle*{1}} \put(135,178){\line(5,-6){15}} \put(175,178){\line(-5,-6){15}} \put(110,-15){Figure 3} \end{picture} The next step is a formula for the coefficient $b_0$ that follows from the explicit calculations of \cite{al:lc}: \begin{equation} \label{horrible} b_0={\sum_{s=1}^{N}{{q_s+\alpha} \def\Alp{\Alpha_s}\over m_s}-(N-1)\over \sum_{s=1}^{N} {q_s\over m_s}-1}=1-{\frac{(N-2)-\sum_{s=1}^{N}{\frac{\alpha} \def\Alp{\Alpha_s}{m_s}}} {\sum_{s=1}^{N}{\frac{q_s}{m_s}}-1}} \end{equation} with denominator $>0$, where $N$ is a number of connected components of $\Gam-v_0$. Here $\alpha} \def\Alp{\Alpha_s=1-\sum l_j^sb_j$ as in \ref{fin_many_graphs}. We consider the second case of the figure~3 formally as a subcase of the first one with $q=m$ and $\alpha} \def\Alp{\Alpha=0$. Note that by \ref{max_hyp} a number of graphs that $v_0^{(n)}$ is connected with in the sequence is bounded. For any fixed $N$ the conditions $\lim b_0=1$ and $b_0<1$ imply $$ \sum_{s=1}^N{\frac{\alpha} \def\Alp{\Alpha_s}{m_s}}<N-2\quad and\quad \lim \sum_{s=1}^N{\frac{\alpha} \def\Alp{\Alpha_s}{m_s}}=N-2.$$ We can assume that some of $m_s$ are fixed and others tend to infinity. For the latters ${\frac{\alpha} \def\Alp{\Alpha_s}{m_s}}\to 0$ and ${\frac{\alpha} \def\Alp{\Alpha_s}{m_s}}>0$. This is so by \ref{fin_many_graphs} (here it is important again that there is a constant $\varepsilon(\xi)$ so that $\sum m_jb_j-1>0$ implies $\sum m_jb_j-1>\varepsilon(\xi)$) and by \ref{utv:loc_bound}. So we can assume that $\sum_{s=1}^M{\frac{\alpha} \def\Alp{\Alpha_s}{m_s}}<N-2$ and $$\lim \sum_{s=1}^M{\frac{\alpha} \def\Alp{\Alpha_s}{m_s}} =\lim\sum_{s=1}^M{\frac{1-\sum l_j^sb_j}{m_s}}=N-2$$ with $m_1...m_M$ being fixed. But this definitely gives a contradiction. Note that $\sum l_j^sb_j\le 2$ by \ref{utv:loc_bound}, so there are only finitely many possibilities for $l_j^s$. Finally, for $N\ge5$ $$b_0\le1-{\frac{(N-2)-\sum{1\over{m_s}}}{N-1}} \le 1-{\frac{{N\over 2}-2}{N-1}}\le {7\over 8}$$ \noindent and we are done. \medskip {\par\noindent{\sl Case 2:}\enspace} $\bar b_0=\lim b_0<1$. Since all the log discrepancies $a_i\ge\varepsilon=1-{\bar b}_0$, the only infinite series of connected components of $\Gam-v_0$ are given by \ref{fin_many_graphs}. Moreover, for the minimal log discrepancies there $$\lim\min a_i\le{\frac{1-\sum(B_0F_i)r_ib_0}{m-k}}$$ \noindent and this should be not less than $1-\bar b_0$. As a conclusion, all the infinite series are given by \begin{picture}(300,250)(-50,-40) \multiput(0,0)(0,100){2}{ \put(-5,45){$B_0$} \put(20,50){\circle{14}} \put(15,55){\line(1,-1){10}} \put(15,45){\line(1,1){10}} \put(27,50){\line(1,0){15.5}} \multiput(45,50)(20,0){2}{\circle{5}} \multiput(42.5,55)(20,0){2}{2} \put(47.5,50){\line(1,0){15}} \multiput(67.5,50)(27.5,0){2}{\line(1,0){7.5}} \multiput(80,50)(5,0){3}{\circle*{1}}} \multiput(105,150)(20,0){2}{\circle{5}} \multiput(102.5,155)(20,0){2}{2} \put(107.5,150){\line(1,0){15}} \put(127.5,150){\line(1,0){17.5}} \put(190,150){\oval(90,20)} \put(155,150){\circle{5}} \put(195,150){\oval(60,10)} \multiput(155,185)(70,0){2}{\circle{14}} \multiput(150,190)(70,0){2}{\line(1,-1){10}} \multiput(150,180)(70,0){2}{\line(1,1){10}} \multiput(180,180)(10,0){3}{\circle*{1}} \put(155,178){\line(5,-6){15}} \put(225,178){\line(-5,-6){15}} \multiput(105,30)(0,20){3}{\circle{5}} \multiput(110,28.5)(0,20){3}{2} \multiput(105,32.5)(0,20){2}{\line(0,1){15}} \put(125,-15){Figure 4} \end{picture} Now we would like to use a variant of the formula~\ref{horrible}. However, $B_0$ can intersect finitely many types of graphs arbitrarily. Still, for any fixed combination, if $b_j$ increase, $b_0$ decreases. The situation is exactly the opposite to the one of elliptic graphs since the signature of the quadratic form is now $(1,n-1)$ instead of $(0,n)$ and the graph $\Gam-v_0$ is still elliptic (cf. \ref{max_hyp}). All the said above implies that for $b_0$ there are only finitely many possible expressions of the form $$b_0=1-{C_1+\sum C_2^jb_j-\sum_{s=1}^{N}{\alpha} \def\Alp{\Alpha_s\over m_s}\over C_3+\sum_{s=1}^{N}{q_s\over m_s}}$$ with fixed $C_1,C_2^j,C_3,m_s-q_s$, $m_s\to+\infty$, $C_2^j\ge0$ and the denominator $>0$. Simplifying, $$1-b_0={C_1+\sum C_2^jb_j-\sum_{s=1}^{N}{\alpha} \def\Alp{\Alpha_s\over m_s}\over C_3'-\sum_{s=1}^{N}{m_s-q_s\over m_s}}$$ Now $\lim b_0=\bar b_0$ implies $(C_1+\sum C_2^j\bar b_j)/C_3'=1-\bar b_0$. And, finally, the inequalities ${\alpha} \def\Alp{\Alpha_s\over m_s-q_s}\ge 1-\bar b_0$ and $C_2^j\ge0$ imply that$1-b_0\le1-\bar b_0$, i.e. $b_0\ge \bar b_0$ that gives a contradiction. \qed \begin{zam} As the proof shows, \ref{hyper} is not true without the assumption (iii). \end{zam} \section{Global case} \begin{utv}[Global boundness] \label{utv:glob_bound} Let $X$ be a surface with a lc divisor $K+B$ and assume that $f:X\to Y$ is a contraction of an extremal ray such that $K+B$ if $f$-nonpositive. Let $B^+=\sum b_j^+B_j^+$ contain all the components in $B$ that are $f$-positive. Then \begin{num} \item if $\dim Y=2$, $\sum b_j^+\le2$ \item if $\dim Y=1$, $\sum b_j^+\le2$ \item if $\dim Y=0$, $\sum b_j^+\le3$ \end{num} \end{utv} \par\noindent {\sl Proof:}\enspace (i) follows from \ref{utv:loc_bound}, because $K_Y+f(B)$ is also lc. (ii) is clear: if $B^+$ is not empty, then $-K$ should be negative on the general fiber, so a general fiber $F$ is isomorphic to $\bfP^1$ and $\sum b_j^+\le B^+F\le -KF=2$. In the case (iii) if $X$ is nonsingular, then $X\simeq {\bfP^2}$ and the statement is evident. If $X$ is singular, consider a partial resolution $g:Z\to X$, dominated by the minimal resolution and such that $\rho(Z)=\rho(X)+1=2$. Then by \ref{Del_Pezzo} there is a second extremal ray and $g^*(K+B)=K_Z+B_Z$ is nonpositive with respect to this extremal ray. Since every curve on $Z$ is positive with respect to at least one of the extremal rays, (iii) with the bound 4 follows immediately. To get the bound 3 is an easy exercise. \qed \begin{zam} \ref{utv:glob_bound}(iii) is also proved in \cite{ag-kol} for arbitrary dimension with a bound $n+1$. \end{zam} \begin{utv}[Global ascending chain condition] \label{global_asc} Let $\xi=\{X^{(n)},K+B^{(n)}\}$ be a sequence of surfaces such that \begin{num} \item $K+B^{(n)}$ is lc \item $B^{(n)}$ is an increasing sequence \item $K+B^{(n)}$ is numerically trivial \end{num} Then $\xi$ terminates. \end{utv} Proof of \ref{global_asc} will be given in several steps. \setcounter{shag}{0} \begin{shag} One can assume that all the surfaces {$X^{(n)}$ } are Del Pezzo surfaces with $\rho({X^{(n)}}{ })=1$. \end{shag} \par\noindent {\sl Proof:}\enspace We can assume that the lengths of the groups {$B^{(n)}$ } in the sequence $\xi$ are constant and that $b_1$ always increases. Now consider a divisor $K+B-\varepsilon B_1$ on $X^{(n)}$. Note here that $B_1$ is $\bfQ$-factorial by the classification of log canonical singgularities. It is lc and is not numerically effective and if $B_1^2\le0$ then $(K+B-\varepsilon B_1)B_1\ge0$. Therefore either $\rho({X^{(n)}}{ })=1$ and then {$X^{(n)}$ } is a Del Pezzo surface with lc $K+B$ or there is an extremal ray that does not contract $B_1$. If the contraction is birational, we make it and repeat the same procedure again. If it is a fibration, the claim follows from the corresponding 1-dimensional statement. \qed \begin{zam} The argument works in the 3-dimensional case as well. \end{zam} \begin{shag} One can assume that there are only finitely many different types of graphs of singularities that the increasing components of {$B^{(n)}$ } are passing through. \end{shag} \par\noindent {\sl Proof:}\enspace As usually, we can assume that the groups {$B^{(n)}$ } have the same length. Now consider the set $PLD(\xi)$. By \ref{utv:local_pasc} this set satisfies the ascending chain condition and has at least one limit point. Let $l$ be the minimal limit point of $PLD(\xi)$. Fix the number $C$ so that all $b_j\ge C$. If the surfaces in $\xi$ contain singularities that correspond to infinitely many elliptic graphs, then by \ref{fin_many_graphs} $l\le1-C$. Passing to a subsequence we can assume that a sequence of minimal log discrepancies, which we shall denote $\{a^{(n)}_s\}$ is a decreasing sequence and $\lim a^{(n)_s}=l$ (the sequence of codiscrepancies is increasing and $\lim b_0^{(n)_s}=1-l\ge C$. Now consider a partial resolution $f:{Y^{(n)} }\to{X^{(n)}}{ }$ which is dominated by the minimal desingularization and which blows up exactly the curve $B^{(n)_s}$. Then $$f^*(K+{B^{(n)}}{ })=K_Y+f^{-1}{B^{(n)}}{ }+b^{(n)_s}B^{(n)_s}$$ The surface {$Y^{(n)}$ } has Picard number 2 and by \ref{Del_Pezzo} there is a second extremal ray, corresponding to a $(-1)$-curve on ${\widetilde Y}} \def\ytil{{\tilde y}={\widetilde X}} \def\xtil{{\tilde x}$. Let $g:{Y^{(n)} }\to{X'^{(n)}}$ be the contraction of this second extremal ray. If $g$ is a fibration then restricting of $K_Y+f^{-1}{B^{(n)}}{ }+b^{(n)_s}B^{(n)_s}$ on the general fibre of $g$ readily gives a contradiction. Hence, we shall assume that $g$ is a birational morphism. A divisor $K+{B'^{(n)} }=g_*f^*(k+{B^{(n)}}{ })$ is lc and numerically trivial, {$B'^{(n)}$ } has either the same number of components as {$B^{(n)}$ } or one more, and, after passing to a subsequence, $B'^{(n)}$ is an increasing sequence. A morphism $g$ can contract one of the components of {$B^{(n)}$ } and we can assume that it is always, say, $B_0$. However, by \ref{hyper} and \ref {utv:local_pasc} the sequence $\{b_0^{(n)}\}$ cannot be an increasing sequence with $\lim b_0^{(n)}\ge1-l$. Therefore, changing the sequence $\xi=\{X^{(n)}\}$ by a new sequence $\xi'=\{X'^{(n)}\}$, we are gaining a new component with increasing coefficient that has the limit $1-l\ge C$. Note that for a new minimal limit point $l'$ of $PLD(\xi')$ one has $l'\ge l$. This is so because a minimal desingularization of {$X'^{(n)}$ } is dominated by the minimal desingularization of {$X^{(n)}$ } and $K+B^{(n)}$, $K+B'^{(n)}$ both are numerically trivial, so $PLD(\xi')$ is a subset in $PLD(\xi)$. Repeating the procedure, we get one more component and so on. After $k$ steps the sum of the coefficients in {$B^{(n)}$ } will be greater than $kC$. This eventually will get into the contradiction with \ref{utv:glob_bound}. \qed \begin{shag} One can assume that all the surfaces {$X^{(n)}$ } are isomorphic to each other. \end{shag} \par\noindent {\sl Proof:}\enspace By \ref{log_Del_Pezzo} a surface ${\widetilde X}} \def\xtil{{\tilde x}^{(n)}$ is either a locally trivial $\bfP^1$-bundle with a section which is a smooth elliptic curve or a rational surface with rational singularities. In the former case the statement follows from the 1-dimensional analog by restricting $B$ to the fiber of the fibration. Now assume we are in the latter case. By the previous step, there exists a constant $N(\xi)$ so that for the increasing component $B_1$ of $B$\enspace $NB_1$ is Cartier. Hence for any curve $D$ on {$X^{(n)}$ } $$-KD=\sum b_jB_jD\ge b_1/N\ge C/N$$ Now theorem~2.~3 of \cite{al:fi} states that for all such surfaces $\rho({\widetilde X}} \def\xtil{{\tilde x})$ is bounded. Therefore one can get $X^{(n)}$ by blowing up finitely many points from the minimal rational surface $\bfF_k$. Threfore there are only finitely many possibilities for the graph of exceptional curves on $X^{(n)}$ except for the fact that one weight $k$ can be arbitrary. Now if $B_1$ does not lie in the fiber for infinitely many $n$ we prove the statement restricting a numerically trivial divisor $K+\sum b_jB_j+\sum b_i F_i$ to the fiber and using \ref{utv:local_pasc}. Otherwise (recall that $\rho(X)=1$) $B_1$ on $X$ should pass through the singularity which graph contains the exceptional curve of $\bfF_k$. By the previous step $k$ is bounded. Hence we can assume that the surfaces $X^{(n)}$ belong to a bounded family and it is enough to consider only finitely many of them. \qed \begin{zam} Theorem 2.3 in \cite{al:fi} is stated for log terminal singularities. But in fact the proof is exactly the same for rational log canonical singularities. \end{zam} \begin{shag} \ref{global_asc} follows. \end{shag} \par\noindent {\sl Proof:}\enspace Indeed, there are only finitely many possibilities for effective Weil divisors $B_j$. \qed \medskip The following example shows that \ref{global_asc} is not true without the assumption~(i). \begin{pri} Consider a sequence of surfaces {$X^{(n)}$ } so that ${\widetilde X}} \def\xtil{{\tilde x}^{(n)}=\bfF_n$ and ${B^{(n)}}{ }=(1-1/n)F_1+3/4(F_2+F_3+F_4)$ where $F_1$ is an image of the infinite section of $\bfF_n$, $F_{2,3,4}$ are fibres. Note that {$K+B^{(n)}$ } is numerically trivial but is not lc. \end{pri}

9206

alg-geom/9206009

en

https://arxiv.org/abs/alg-geom/9206009

alg-geom/9206009

Temporary

G.Mikhalkin

The complex separation and extensions of Rokhlin congruence for curves on surfaces

17 pages, LaTeX

The subject of this paper is the problem of arrangement of real algebraic curves on real algebraic surfaces. In this paper we extend Rokhlin, Kharlamov-Gudkov-Krakhnov and Kharlamov-Marin congruences for curves on surfaces and give some applications of this extension. For some pairs consisting of a surface and a curve on this surface (in particular for M-pairs) we introduce a new structure --- the complex separation that is separation of the complement of curve into two surfaces. In accordance with Rokhlin terminology the complex separation is a complex topological characteristic of real algebraic varieties. The complex separation is similar to complex orientations introduced by O.Ya.Viro (to the absolute complex orientation in the case when a curve is empty and to the relative complex orientation otherwise). In some cases we calculate the complex separation of a surface (for example in the case when surface is the double branched covering of another surface along a curve). With the help of these calculations applications of the extension of Rokhlin congruence gives some new restrictions for complex orientations of curves on a hyperboloid.

alg-geom

\section{Introduction} \subsection{Rokhlin and Kharlamov-Gudkov-Krakhnov congruences for curves on surfaces} If $A$ is a real curve of odd degree on real projective surface $B$ then $A$ divides $B$ into two parts $B_+$ where polynomial determining $A$ is non-negative and $B_-$ where this polynomial is non-positive. V.A.Rokhlin \cite{R} proved the congruence for the Euler characteristic of $B_-$ under such strong hypotheses that they follow that $B$ is an M-surface, $A$ is an M-curve and $B_+$ is contained in a connected component of $B$. V.M.Kharlamov \cite{Kh}, D.A.Gudkov and A.D.Krakhnov \cite{GK} proved a relevant congruence that makes sense sometimes even if $A$ is not M- but (M-1)-curve, but the hypotheses for $B$ and $B_+$ were not weakened. \subsection{Description of the paper} The results of the paper make sense in the case when a pair consisting of a surface and a curve in this surface is of characteristic type (for definition see section \ref{not}). For pairs of characteristic type we introduce a complex separation of the complement of the curve in the surface that is a new complex topological characteristic of pairs consisting of a surface and a curve in this surface. If the curve is empty then the complex separation is a new complex topological characteristic of surfaces relevant to the complex orientation of surfaces introduced by O.Viro \cite{V1}. The main theorem is formulated with the help of the complex separation, this theorem is a generalization of Rokhlin and Kharlamov-Gudkov-Krakhnov congruences for curves in surfaces. The main theorem gives nontrivial restrictions even for curves of odd degree on some surfaces. The paper contains also some applications of the main theorem. We prove a congruence modulo 32 for Euler characteristic of real connected surface of characteristic type. We prove some new congruences for curves on a hyperboloid. We give a direct extension of Rokhlin and Kharlamov-Gudkov-Krakhnov congruences for curves on projective surfaces, there we avoid both of the hypotheses that $B$ is an M-surface and $B_+$ is contained in a connected component of $B$. We apply the main theorem to the classification of curves of low degrees on an ellipsoid. In particular, we get a complete classification of flexible curves on an ellipsoid of bidegree (3,3) (the notion of flexible curve is analogous to one introduced by O.Viro \cite{V} for plane case). One can see that the fact that the theorem can be applied to curves of odd degree follows that this theorem can not be proved in the Rokhlin approach using the double covering of the complexification of the surface branched along the curve (since there is no such a covering for curves of odd degree). We use the Marin approach \cite{Marin}. All results of this paper apply to flexible curves as well as to algebraic. The author is indebted to O.Ya.Viro for his attention to the paper and consultations. \section{Notations and the statement of the main theorem} \label{not} Let ${\bf C} B$ be a smooth oriented 4-manifold such that its first ${\bf Z}_2$-Betti number is zero equipped with an involution $conj$ such that the set ${\bf R} B$ of its fixed points is a surface. Let ${\bf C} A$ be a smooth surface in ${\bf C} B$ invariant under $conj$ and such that the intersection of ${\bf C} A$ and ${\bf R} B$ is a curve. These notations are inspired by algebraic geometry. It is said that $A$ is of even degree if ${\bf C} A$ is ${\bf Z}_2$-homologous to zero in ${\bf C} B$ and that $A$ is of odd degree otherwise. It is said that curve $A$ is of type I if ${\bf R} A$ is ${\bf Z}_2$-homologous to zero in ${\bf C} A$ and that $A$ is of type II otherwise. It is said that surface $B$ is of type I$abs$ if ${\bf R} B$ is ${\bf Z}_2$-homologous to zero in ${\bf C} B$ If $B$ is a real projective plane then it is said that $B$ is of type I$rel$ if ${\bf R} B$ is ${\bf Z}_2$-homologous to a plane section of ${\bf C} B$. We shall say that pair $(B,a)$ is of characteristic type if the sum of ${\bf R} B$ and ${\bf C} A$ is Poincar\'{e} dual to the second Stiefel-Whitney class of ${\bf C} B$. We shall say that surface $B$ is of characteristic type if ${\bf R} B$ is a characteristic surface in ${\bf C} B$. Let $b_*$ denote the total ${\bf Z}_2$-Betti number. It is said that manifold ${\bf C} X$ equipped with involution $conj$ is an (M-$j$)-manifold if $b_*({\bf R} X)+2j=b_*({\bf C} X)$ where ${\bf R} X$ is the fixed point set of $conj$. One can easily see that Smith theory follows that $j$ is a nonnegative integer number. Let $\sigma(M)$ denote the signature of oriented manifold $M$.; $D_M:H^*(M;{\bf Z}_2)\rightarrow H_*(M;{\bf Z}_2)$ denote Poincar\'{e} duality operator; $[N]\in H_*(M;{\bf Z}_2)$ denote ${\bf Z}_2$-homology class of submanifold $N\subset M$. Let $e_A=[{\bf C} A\circ{\bf C} A]_{{\bf C} B}$ denote normal Euler number of ${\bf R} B$ in ${\bf C} B$. If $B_\epsilon$ is a surface contained in ${\bf R} B$ and such that $\partial B_\epsilon={\bf R} A$ that we shall denote by $e_{B_\epsilon}$ the obstruction to extending of line bundle over ${\bf R} A$ and normal to ${\bf R} A$ in ${\bf C} A$ to the line bundle over $B_\epsilon$ normal in ${\bf C} B$ to ${\bf R} B$ evaluated on the twisted fundamental class $[B_\epsilon,\partial B_\epsilon]$ and divided by 2. One can see that if $(B,A)$ is a nonsingular pair consisting of an algebraic surface and algebraic curve then $e_{B_\epsilon}=-\chi(B_\epsilon)$. Let $\beta(q)$ denote the Brown invariant of ${\bf Z}_4$-valued quadratic form $q$. \begin{th} \label{main} If $(B,A)$ is of characteristic type then there is a natural separation of ${\bf R} B-{\bf R} A$ into surfaces $B_1$ and $B_2$ such that $\partial B_1=\partial B_2={\bf R} A$ defined by the condition that ${\bf C} A/conj\cup B_j$ is a characteristic surface in ${\bf C} B/conj (j=1,2)$. There is a congruence for the Guillou-Marin form $q_j$ on $H_1({\bf C} A/conj\cup B_j;{\bf Z}_2)$ \begin{displaymath} e_{B_j}\equiv\frac{e_{{\bf R} B}+\sigma({\bf C} B)}{4}-\frac{e_A}{4}-\beta(q_j)\pmod{8} \end{displaymath} \end{th} \begin{add} Let $q_j|_{H_1({\bf R} A;{\bf Z}_2)}=0$ \begin{itemize} \begin{description} \item[a)] If $A$ is an M-curve then $e_{B_j}\equiv\frac{e_{{\bf R} B}+\sigma({\bf C} B)}{4}-\frac{e_A}{4}-\beta_j\pmod{8}$ \item[b)] If $A$ is an (M-1)-curve then $e_{B_j}\equiv\frac{e_{{\bf R} B}+\sigma({\bf C} B)}{4}-\frac{e_A}{4}-\beta_j\pm1\pmod{8}$ \item[c)] If $A$ is an (M-2)-curve and $e_{B_j}\equiv\frac{e_{{\bf R} B}+\sigma({\bf C} B)}{4}-\frac{e_A}{4}-\beta_j+4\pmod{8}$ then $A$ is of type I \item[d)] If $A$ is of type I then $e_{B_j}\equiv\frac{e_{{\bf R} B}+\sigma({\bf C} B)}{4}-\frac{e_A}{4}\pmod{4}$ \end{description} \end{itemize} where $\beta_j$ is the Brown invariant of the restriction $q_j|_{H_1(B_j;{\bf Z}_2)}$ \end{add} \subsection{Remark} Some of components of ${\bf R} A$ can be disorienting loops in ${\bf C} A$ (it is easy to see that number of such components is even). If $\alpha$ is some 1-dimensional ${\bf Z}_2$-cycle in $B_j$ that is a boundary of some 2-chain $\beta$ in ${\bf R} B$ containing an even number of disorienting ${\bf C} A$ components of ${\bf R} A$ then $q_j(\alpha)=0$; if such a number is odd then $q_j(\alpha)=2$. It follows that if $(B,A)$ is of characteristic type then the number of disorienting ${\bf C} A$ components of ${\bf R} A$ in each component of ${\bf R} B$ is even. \section{The proof of Theorem 1 and Addendum 1} \subsection{Calculation of the characteristic class of ${\bf C} B/conj$} It is not difficult to see the formula for the characteristic classes of double branched covering: if $\pi:Y\rightarrow X$ is a double covering branched along $Z$ then $$w_2(Y)=\pi^*w_2(X)+D^{-1}_Y[Z]$$ Applying this formula to $p:{\bf C} B\rightarrow {\bf C} B/conj$ we get $$tr(Dw_2({\bf C} B/conj))=Dw_2({\bf C} B)+[{\bf R} B]$$ where $tr:H_2({\bf C} B/conj;{\bf Z}_2)\rightarrow H_2({\bf C} B;{\bf Z}_2)$ is transfer (i.e. the inverse Hopf homomorphism to $P$). It is easy to see that transfer can be decomposed as the composition $$H_2({\bf C} B/conj;{\bf Z}_2)\stackrel{k}{\rightarrow}H_2({\bf C} B/conj,{\bf R} B;{\bf Z}_2) \stackrel{h}{\rightarrow}H_2({\bf C} B;{\bf Z}_2)$$ where $k$ is an inclusion homomorphism and $h\circ k=tr$. To prove that $h$ is a monomorphism we use the Smith exact sequence (see e.g. \cite{W}): $$H_3({\bf C} B/conj,{\bf R} B;{\bf Z}_2)\stackrel{\gamma_3}{\rightarrow} H_2({\bf R} B;{\bf Z}_2)\oplus H_2({\bf C} B/conj,{\bf R} B;{\bf Z}_2)\stackrel{\alpha_2}{\rightarrow} H_2({\bf C} B;{\bf Z}_2)$$ In this sequence the first component of $\gamma_3$ is equal to the boundary homomorphism $\partial$ of pair $({\bf C} B/conj,{\bf R} B)$; $\partial$ is a monomorphism since $H_3({\bf C} B/conj;{\bf Z}_2)=0$ (since ${\bf C} B$ and therefore ${\bf C} B/conj$ are simply connected). It means that no element of type $(0,x)\in H_2({\bf R} B;{\bf Z}_2)\oplus H_2({\bf C} B/conj,{\bf R} B;{\bf Z}_2), x\neq 0$ is contained in $Im\gamma_3$ and therefore the restriction of $\alpha_2$ to $H_2({\bf C} B/conj,{\bf R} B;{\bf Z}_2)$ is a monomorphism and thus $H$ is a monomorphism. Now if $[{\bf C} A]=Dw_2({\bf C} B)+[{\bf R} B]$ then $$Dw_2({\bf C} B/conj)=[{\bf C} A/conj]\in H_2({\bf C} B/conj,{\bf R} B;{\bf Z}_2)$$ The exactness of homology sequence of pair $({\bf C} B/conj,{\bf R} B)$ follows since $H$ is a monomorphism that there exists a surface $B_1\subset{\bf R} B$ such that $\partial B_1={\bf R} A$ and $W_1=B_1\cup{\bf C} A/conj$ is dual to $w_2({\bf C} B/conj)$. Let $B_2$ be equal to $Cl({\bf R} B-B_1)$ Surface ${\bf R} B$ is ${\bf Z}_2$-homologous to zero in ${\bf C} B/conj$ since ${\bf C} B$ is a double covering of ${\bf C} B/conj$ branched along ${\bf R} B$. It follows that $W_2=B_2\oplus{\bf C} A/conj$ is also a surface dual to $w_2({\bf C} B/conj)$. Note that the separation of ${\bf R} B$ into $B_1$ and $B_2$ is unique since $$dim(ker(in_*:H_2({\bf R} B;{\bf Z}_2)\rightarrow H_2({\bf C} B/conj)))=1$$ as it follows from exactness of the Smith sequence. Indeed, since $H_3({\bf C} B/conj;{\bf Z}_2)=0$ then this dimension is equal to the dimension of $H_3({\bf C} B/conj,{\bf R} B;{\bf Z}_2)$; the dimension of $H_3({\bf C} B/conj,{\bf R} B;{\bf Z}_2)$ is equal to 1 since $\gamma_4:H_4({\bf C} B/conj,{\bf R} B;{\bf Z}_2)\rightarrow {0}\oplus H_3({\bf C} B/conj,{\bf R} B; {\bf Z}_2)$ is an isomorphism. \subsection{Proof of the congruence for $\chi(B_j)$} \label{prth} Note that since $W_j,j\in \{1,2\}$ is a characteristic surface in ${\bf C} B/conj$ and ${\bf C} B$ is simply connected we can apply the Guillou-Marin congruence to pair $({\bf C} B/conj,W_j)$ $$\sigma({\bf C} B/conj)\equiv W_j\circ W_j+2\beta (q_j)\pmod{16}$$ where $q_j:H_1(W_j;{\bf Z}_2)\rightarrow{\bf Z}_4$ is the quadratic form associated to the embedding of $W_j$ into ${\bf C} B/conj$ (see \cite{GM}). Similar to the calculations in \cite{Marin} we get that $$W_j\circ W_j=\frac{e_A}{2}-2\chi(B_j)$$ The Atiyah-Singer-Hirzebruch formula follows that $$\sigma({\bf C} B/conj)=\frac{\sigma({\bf C} B)-\chi({\bf R} B)}{2}$$ Combining all this we get $$\chi(B_j)\equiv\frac{e_A}{4}+\frac{\chi({\bf R} B)-\sigma({\bf C} B)}{4}+ \beta(q_j)\pmod{8}$$ Now if $q|_{H_1({\bf R} A;{\bf Z}_2)}=0$ then additivity of the Brown invariant (see \cite{KV}) follows that $$\beta(q_j)=\beta(q|_{H_1({\bf C} A/conj;{\bf Z}_2)})+\beta_j$$ It is easy to see that if $A$ is an (M-$j$)-curve then $rkH_1({\bf C} A/conj;{\bf Z}_2)=j$. Points a) and b) of the addendum immediately follow from this. To deduce points c) and d) of the addendum note that $q|_{H_1({\bf C} A/conj;{\bf Z}_2)}$ is even iff ${\bf C} A/conj$ is an orientable surface iff $A$ is of type I (cf. \cite{Marin}, \cite{KV}). \section{Some applications of Theorem 1} \subsection{The case ${\bf C} A$ is empty; congruences for surfaces} \label{pov} \subsubsection{} \label{surface} If $Dw_2({\bf C} B)=[{\bf R} B]$ then there is defined a complex separation of ${\bf R} B$ into two closed surfaces $B_1$ and $B_2$; there is defined a ${\bf Z}_4$-quadratic form $q$ on $H_1({\bf R} B;{\bf Z}_2)= H_1(B_1;{\bf Z}_2)\oplus H_1(B_2;{\bf Z}_2)$ equal to sum of Guillou-Marin forms of $B_1$ and $B_2$ which are characteristic surfaces in ${\bf C} B/conj$ and \begin{displaymath} \chi(B_j)\equiv\frac{\chi({\bf R} B)-\sigma({\bf C} B)}{4}+\beta(q|_{H_1(B_j;{\bf Z}_2)}) \pmod{8} \end{displaymath} \subsubsection{} \label{pusto} If $Dw_2({\bf C} B)=[{\bf R} B]$ and $B_j$ is empty for some $j$ (that is evidently true if ${\bf R} B$ is connected) then $$\chi({\bf R} B)\equiv\sigma({\bf C} B)\pmod{32}$$ \subsubsection{} If $Dw_2({\bf C} B)=[{\bf R} B]$ then $$\chi({\bf R} B)\equiv\sigma({\bf C} B)\pmod{8}$$ {\em\underline{Proof}} It follows from an easy observation that $\chi(B_j) \equiv\beta(q|_{H_1(B_j;{\bf Z}_2)})\pmod{2}$ \subsubsection{Remark} According to O.Viro \cite{V} in some cases one can define some more complex topological characteristics on ${\bf R} B$. Namely, If $B$ is of type I$abs$ then ${\bf R} B$ possesses two special reciprocal orientations (so-called semi-orientation) and a special spin structure. If $Dw_2({\bf C} B)=[{\bf R} B]$ then ${\bf R} B$ possesses the special $Pin_-$-structure corresponding to Guillou-Marin form $q_{{\bf C} B}:H_1({\bf R} B;{\bf Z}_2)\rightarrow{\bf Z}_4$ of surface ${\bf R} B$ in ${\bf C} B$. The complex separation is a new topological characteristic for surfaces and \ref{surface} may be interpreted as a formula for this characteristic. Quadratic form $q$ is not a new complex topological characteristic. \subsubsection{} Form $q$ is equal to $q_{{\bf C} B}$ in the case when these forms are defined (i.e. when $Dw_2({\bf C} B)=[{\bf R} B]$) To prove this one can note that the index of a generic membrane in ${\bf C} B$ bounded by curves in ${\bf R} B$ differs from the index of the image of this membrane in ${\bf C} B/conj$ by number of intersection points of this membrane and ${\bf R} B$. \subsubsection{Remark} If $B$ is a complete intersection in the projective space of hypersurfaces of degrees $m_j,j=1,\ldots,s$ then the condition that $Dw_2({\bf C} B)=[{\bf R} B]$ is equivalent to the condition that $B$ is of type I$abs$ in the case when $\sum^{s}_{j=1}m_j\equiv 0\pmod{2}$ and to the condition that $B$ is of type I$rel$ in the case when $\sum^{s}_{j=1}m_j\equiv 1\pmod{2}$. \subsection{Calculations for double coverings} \label{vych} We see that to apply Theorem 1 one needs to be able to calculate the complex separation and the corresponding Guillou-Marin form. We calculate them in some cases in this subsection. By the semiorientation of manifold $M$ we mean a pair of reciprocal orientations of $M$ (note that this notion is nontrivial only for non-connected manifolds). It is easy to see that two semiorientations determine a separation of $M$; this separation is a difference of two semiorientations, namely, two components of $M$ are of the same class of separation iff the restrictions of the semiorientations on these components are the same. Let ${\bf C} B$ be the double covering of surface ${\bf C} X$ branched along curve ${\bf C} D$ invariant under the complex conjugation $conj_X$ in ${\bf C} X$. Suppose ${\bf C} D$ is of type I. Let ${\bf C} D_+$ be one of two components of ${\bf C} D-{\bf R} D$. Then ${\bf R} D$ possesses a special semiorientation called the complex semiorientation (see \cite{R1}). The invariance of ${\bf C} D$ under $conj_X$ follows that $conj_X$ can be lifted in two different ways into an involution of ${\bf C} B$. Let $conj_B$ be one of these two lifts. Let $X_-=p({\bf R} B))$, $X_+={\bf R} X-int(X_-)$ where $p$ is the covering map. \subsubsection{} \label{calor} Suppose that $Dw_2({\bf C} B)=[{\bf R} B]$. Then for every component $C$ of $X_+$ the complex separation of $\partial C$ induced from complex separation of ${\bf R} B$ via $p$ (namely, two circles of $\partial C$ are of the same class of separation iff the lie in the image under $p$ of the same class of the separation of ${\bf R} B$) is equal to the difference of the semiorientations on $\partial C$ induced by the complex semiorientation of ${\bf R} D$ and the unique (since $C$ is connected) semiorientation of $C$. In particular $C$ is orientable. {\em\underline{Proof}} Let $\alpha$ and $\beta$ be components of $\partial C$. Consider two only possible cases: the first case when the semiorientations induced from ${\bf R} D$ and $C$ are equal on $\alpha$ and $\beta$ and the second case when they are different (see fig.1, arrows indicate one of two orientations induced by the complex semiorientation of ${\bf R} D$). Choose a point $Q_{\alpha}$ in $\alpha$ and $Q_{\beta}$ in $\beta$. Connect these points by a path $\gamma$ inside $C$ and by a path $\delta$ inside ${\bf C} D_+$ (not visible on the picture) without self-intersection points. It is easy to see that there exists a disk $F'\subset{\bf C} X$ bounding loop $\gamma\delta$ and such that the interior of $F'$ does not intersect ${\bf C} D$. Set $F$ to be equal to $F'\cup conj_X F'$. Then $p^{-1}(F)$ gives an element of $H_2({\bf C} B)$ (the construction of this element was suggested by O.Viro \cite{V3}). Since $p^{-1}(F)$ is invariant under $conj_B$ it gives an element in $H_2({\bf C} B/conj_B;{\bf Z}_2)$, say $f\in H_2({\bf C} B/conj_B;{\bf Z}_2)$. Let us calculate the self-intersection number of $f$. It is easy to see that because of symmetry the self-intersection number of $f$ in ${\bf C} B/conj_B$ is equal to the self-intersection number of $\gamma\delta$ in $C\cup{\bf C} D_+$. The definition of complex semiorientation follows that the self-intersection number of $\gamma\delta$ in $C\cup{\bf C} D_+$ (and therefore the self-intersection number of $f$) is equal to zero in the first case and to one in the second case. \subsubsection{Remark} In the case when ${\bf R} X$ is connected \ref{calor} completely determines the complex separation of ${\bf R} B$. \subsubsection{(O.Viro [11])} \label{vychv} If $\lambda$ is a loop in ${\bf R} B$ such that $p(\lambda)=\partial(G)$, where $G\subset{\bf R} X$ then $$q_{{\bf C} B}(\lambda)\equiv 2\chi(G\cap X_+)\pmod{4}$$ \subsubsection{(O.Viro [11])} \label{vychq} Suppose $\gamma$ is a path in $X_-$ connecting points $Q_{\alpha}$ and $Q_{\beta}$ of components $\alpha$ and $\beta$ of ${\bf R} D$ respectively. Then $q_{{\bf C} B}(p^{-1}(\gamma))=0$ if the intersection numbers of $\gamma$ with $\alpha$ and $\beta$ are of opposite sign (case 1 of fig.1) and $q_{{\bf C} B}(p^{-1}(\gamma))=2$ otherwise (case 2 of fig.1) \subsection{New congruences for complex orientations of curves on a hyperboloid} In this subsection we apply the results of \ref{pov} and \ref{vych} to double branched coverings over simplest surfaces of characteristic type, a plane and a hyperboloid. To state congruences it is convenient to use the language of integral calculus based on Euler characteristic developed by O.Viro \cite{V2}. Let ${\bf R} A$ be a curve of type I in the connected surface ${\bf R} X$. We equip ${\bf R} A$ with one of two complex orientations and fix $X_{\infty}$ -- one of the components of ${\bf R} X-{\bf R} A$. If ${\bf R} X-X_{\infty}$ is orientable and ${\bf R} A$ is $R$-homologous to zero for some ring $R$ of coefficient coefficient then there is defined function $ind_R:{\bf R} X-{\bf R} A\rightarrow R$ equal to zero on $X_{\infty}$ and equal to the $R$-linking number with oriented curve ${\bf R} A$ in ${\bf R} X-X_{\infty}$ otherwise. It is easy to see that $ind_R$ is measurable and defined almost everywhere on ${\bf R} X$ with respect to Euler characteristic. Evidently, the function ${ind_R}^2:{\bf R} X-{\bf R} A\rightarrow R\otimes R$ does not depend on the ambiguity in the choice of one of two complex orientation of ${\bf R} A$. \subsubsection{} Consider the case when $X=P^2$, $R={\bf Z}$. Let $A$ be a plane nonsingular real curve of type I given by polynomial $f_A$ of degree $m=2k$. Then ${\bf R} A$ is ${\bf Z}_2$-homologous to zero. Let $X_{\infty}$ be the only nonorientable component of ${\bf R} P^2-{\bf R} A$. Without loss of generality suppose that $f_A|_{X_{\infty}}<0$. Define $X_{\pm}$ to be equal to $\{y\in{\bf R} P^2|\pm f_A(y)\ge 0\}$. Let $p:{\bf C} B\rightarrow{\bf C} P^2$ be the double covering of ${\bf C} P^2$ branched along ${\bf C} A$ (note that such a covering exists and is unique since $m$ is even and ${\bf C} P^2$ is simply connected). Let $conj_B:{\bf C} B\rightarrow{\bf C} B$ be the lift of $conj:{\bf C} P^2\rightarrow{\bf C} P^2$ such that $p({\bf R} B)=X_-$ (as it is usual we denote $Fix(conj_B$ by ${\bf R} B$). Lemmae 6.6 and 6.7 of \cite{W} immediately follow that $Dw_2({\bf C} B)={\bf R} B$. Therefore we can apply \ref{surface} and \ref{calor}. Proposition \ref{calor} follows that $p^{-1}(cl({ind_{{\bf Z}_2}}^{-1}(2+4{\bf Z})))$ is equal to one of two surfaces of the complex separation of ${\bf R} B$. Set $B_1$ to be equal to $p^{-1}(cl({ind_{{\bf Z}}}^{-1}_{{\bf Z}_2}(2+4{\bf Z})))$, note that $B$ is orientable since $ind_{{\bf Z}}|_{X_{\infty}}=0$. \subsubsection{Lemma} $$\beta(q|_{H_1(B_1;{\bf Z}_2)})\equiv 4\chi({ind_{{\bf Z}}}^{-1}(3+8{\bf Z})\cup{ind_{{\bf Z}}}^{-1}(-3+8{\bf Z})) \pmod{8}$$ {\em\underline{Proof}}The boundary of each component $C$ of $p(B_1)$ has one exterior oval and some interior ovals (with respect to $C$). We call an interior oval of $C$ $C$-positive if its complex orientation and the complex orientation of the exterior oval can be extended to some orientation of $C$ and we call it $C$-negative otherwise. Proposition \ref{vychq} follows that $\beta(q|_{H_1(B_1;{\bf Z}_2)})$ is equal to twice the sum of values of $q$ on on all $C$-negative ovals modulo 8. Note that $ind_{{\bf Z}}^{-1}(\pm 3+8{\bf Z})$ is just the part of $X_+$ lying inside odd number of $C$-negative ovals. The lemma follows now from \ref{vychv}. \subsubsection{} The application of \ref{surface} gives that $$\chi(B_1)\equiv\frac{\chi({\bf R} B)-\sigma({\bf C} B)}{4}+\beta(q|_{H_1(B_1;{\bf Z}_2)}) \pmod{8}$$ Note that $p({\bf R} B)={ind_{{\bf Z}}}^{-1}(2{\bf Z})$, $\chi({ind_{{\bf Z}}}^{-1}(2{\bf Z}))= 1-\chi(1+2{\bf Z})$ and $\sigma({\bf C} B)=2-2k^2$. We get $$4\chi({ind_{{\bf Z}}}^{-1}(2+4{\bf Z}))\equiv 1-\chi({ind_{{\bf Z}}}^{-1}(1+2{\bf Z}))-1+k^2+ 8\chi({ind_{{\bf Z}}}^{-1}(\pm 3+8{\bf Z}))\pmod{16}$$ We can reformulate this in integral calculus language $$\int_{{\bf R} P^2}{ind_{{\bf Z}}}^2d\chi\equiv k^2\pmod{16}$$ Thus for projective plane we get nothing new but the reduction modulo 16 of the Rokhlin congruence for complex orientation \cite{R1} $$\int_{{\bf R} P^2}{ind_{{\bf Z}}}^2d\chi=k^2$$ \subsubsection{} Consider now the case when $X=P^1\times P^1$. Let $A$ be a nonsingular real curve of type I in $P^1\times P^1$ of bidegree $(d,r)$, i.e. the bihomogeneous polynomial $f_A$ determining $A$ is of bidegree $(d,r)$ where $d$ and $r$ are even numbers. Let $X_{\infty}$ be a component of ${\bf R} P^2-{\bf R} A$, $X_{\pm}= \{y\in{\bf R} P^1\times{\bf R} P^1|\pm f_A(y)\ge 0\}$. Suppose without loss of generality that $f_A|_{X_{\infty}}<0$. Let $p:{\bf C} B\rightarrow{\bf C} P^2$ be the double covering of ${\bf C} P^2$ branched along ${\bf C} A$ and let $conj_B:{\bf C} B\rightarrow{\bf C} B$ be the lift of $conj:{\bf C} P^1\times{\bf C} P^1\rightarrow{\bf C} P^1\times{\bf C} P^1$ such that $p({\bf R} B)=X_-$. Nonsingularity of $A$ follows that all components of ${\bf R} A$ non-homologous to zero are homologous to each other. Let $e_1,e_2$ form the standard basis of $H_1({\bf R} P^1\times{\bf R} P^1)$ and let $s,t$ be the coordinates in this basis of a non-homologous to zero component of ${\bf R} A$ equipped with such an orientation that $s,t\ge 0$. If all the components of ${\bf R} A$ are homologous to zero then set $s=t=0$. Then ${\bf R} A$ equipped with the complex orientation produces $l'(se_1+te_2)$ in $H_1({\bf R} P^1\times{\bf R} P^1)$. Note that $s$ and $t$ are relatively prime and $l'$ is even since both $d$ and $r$ are even. \subsubsection{Lemma} If $l'\equiv 0\pmod{4}$ then $Dw_2({\bf C} B)=[{\bf R} B]$ The proof follows from Lemma 3.1 of \cite{Ma}. \subsubsection{Lemma} If $l'\equiv 0\pmod{4}$ then $ind_{{\bf Z}_4}$ is defined and if $sd+tr\equiv 0\pmod{4}$ then $$\int_{{\bf R} P^1\times{\bf R} P^1}{ind_{{\bf Z}_4}}^2d\chi\equiv \frac{dr}{2}\equiv 0\pmod{8}$$ {\em\underline{Proof}} Note that the condition that $sd+tr\equiv 0\pmod{4}$ is just equivalent to the orientability of ${\bf R} B$. Therefore $\beta(q|_{H_1(B_1;{\bf Z}_2)})\equiv 0\pmod{4}$. Further arguments are similar to the plane case, we skip them. \subsubsection{Remark} The traditional way of proving of formulae of complex orientations for curve on surfaces (see \cite{Z}) adjusted to the case when the real curve is only ${\bf Z}_4$-homologous to zero gives only congruence $$\int_{{\bf R} P^1\times{\bf R} P^1}{ind_{{\bf Z}_4}}^2d\chi\equiv \frac{dr}{2}\pmod{4}$$ The \ref{l=4} shows the worthiness of modulo 4 in this congruence. If $l'\equiv 0\pmod{8}$ then the traditional way gives that $$\int_{{\bf R} P^1\times{\bf R} P^1}{ind_{{\bf Z}_8}}^2d\chi\equiv \frac{dr}{2}\pmod{8}$$ \subsubsection{} \label{l=4} If $l'\equiv 4\pmod{8}$ and $sd+tr\equiv 2\pmod{4}$ then $$\int_{{\bf R} P^1\times{\bf R} P^1}{ind_{{\bf Z}_4}}^2d\chi\equiv \frac{dr}{2}+4\pmod{8}$$ {\em\underline{Proof}} Form $q|_{H_1(B_1;{\bf Z}_2)}$ is cobordant to the sum of a form on an orientable surface and some forms on Klein bottles. The condition that $l'\equiv 4\pmod{8}$ is equivalent to the condition that the number of forms on Klein bottles non-cobordant to zero is odd. Thus $$\beta(q|_{H_1(B_1;{\bf Z}_2)})\equiv 2\pmod{4}$$ \subsubsection{} If $l'\equiv 0\pmod{8}$ and $sd+tr\equiv 0\pmod{4}$ then $$\int_{{\bf R} P^1\times{\bf R} P^1}{ind}_{{\bf Z}_8}^{2}d\chi\equiv \frac{dr}{2}\pmod{16}$$ The proof is similar to the plane case. \subsubsection{Addendum, new congruences for the Euler characteristic of $B_+$ for curves on a hyperboloid} \label{b10} Let $d\equiv r\equiv 0\pmod{2}$ and $\frac{d}{2}t+\frac{r}{2}s+s+t\equiv 1\pmod{2}$. \begin{itemize} \begin{description} \item[a)] If $A$ is an M-curve then $\chi(B_+)\equiv\frac{dr}{2}\pmod{8}$ \item[b)] If $A$ is an (M-1)-curve then $\chi(B_+)\equiv\frac{dr}{2}\pm 1\pmod{8}$ \item[c)] If $A$ is an (M-2)-curve and $\chi(B_+)\equiv\frac{dr}{2}+4\pmod{8}$ then $A$ is of type I \item[d)] If $A$ is of type I then $\chi(B_+)\equiv 0\pmod{4}$ \end{description} \end{itemize} This theorem follows from Theorem 1 and gives some new restriction on the topology of the arrangement of real nonsingular algebraic curve of even bidegree on a hyperboloid with non-contractible branches. Points a) and b) of \ref{b10} in the case when $\frac{d}{2}t+\frac{r}{2}s\equiv 0\pmod{2}, s+t\equiv 1\pmod{2}$ were proved by S.Matsuoka \cite{Ma1} in another way (using 2-sheeted branched coverings of hyperboloid). Point d) of \ref{b10} is a corollary of the modification of Rokhlin formula of complex orientations for modulo 4 case. \subsection{The case when $A$ is a curve of even degree on projective surface $B$} In this subsection we deduce Rokhlin and Kharlamov-Gudkov-Krakhnov congruences from Theorem 1 and give a direct generalization of these congruences. Let $B$ be the surface in $P^q$ given by the system of equations $P_j(x_0,\ldots,x_q)=0,j=1,\ldots,s-1$, let $A$ be the (M-$k$)-curve given by the system of equations $P_j(x_0,\ldots,x_q)=0,j=1,\ldots,s-1$, where $P_j$ are homogeneous polynomials with real coefficients, $deg P_j=m_j$, $s=q-1$. Suppose $(B,A)$ is a non-singular pair, ${\bf R} A\neq\emptyset$ and $m_s$ is even. Denote $B_+=\{x\in{\bf R} B|P_s(x)\geq 0\},B_-=\{x\in{\bf R} B|P_s(x)\leq 0\}$, $$d=rk(in^{B_+}_*:H_1(B_+;{\bf Z}_2)\rightarrow H_1({\bf R} B;{\bf Z}_2)), e=rk(in^{{\bf R} A}_*:H_1({\bf R} A;{\bf Z}_2)\rightarrow H_1({\bf R} B;{\bf Z}_2))$$ Set $c$ to be equal to the number of non-contractible in ${\bf R} P^q$ components of ${\bf R} B$ not intersecting ${\bf R} A$. Let us reformulate Rokhlin and Kharlamov-Gudkov-Krakhnov congruences for curve on surfaces in the form convenient for generalization and correcting the error in \cite{R}. \subsubsection{(Rokhlin [1], Kharlamov [2], Gudkov-Krakhnov [3])} \label{RKGK} Suppose $B$ is an $M$-surface, $e=0$, $B_+$ is contained in one component of ${\bf R} B$ and in the case when $m_s\equiv 0\pmod{4}$ suppose in addition that $c=0$. \begin{itemize} \begin{description} \item[a)] If $d+k=0$ then $$\chi(B_+)\equiv\frac{m_1\ldots m_{s-1}m_s^2}{4}\pmod{8}$$ \item[b)] If $d+k=1$ then $$\chi(B_+)\equiv\frac{m_1\ldots m_{s-1}m_s^2}{4}\pm 1\pmod{8}$$ \end{description} \end{itemize} Indeed, it is easy to see that the hypothesis of \ref{RKGK} without the condition on $c$ is equivalent to the hypotheses of corresponding theorems in \cite{R}, \cite{Kh} and \cite{GK}, to reformulate them it is enough to apply the Rokhlin congruence for M-surfaces. \subsubsection{Remark (on an error in [1])} Point 2.3 of \cite{R} contains a miscalculation of characteristic class $x$ of the restriction to $B_-$ of the double covering of ${\bf C} B$ branched along ${\bf C} A$. In \cite{R} it is claimed that if $m_s\equiv 2\pmod{4}$ then $x=w_1(B_{-}-A)$. It led to the omission of the condition on $c$ in both Rokhlin and Kharlamov-Gudkov-Krakhnov congruences. \subsubsection{Correction of the error in [1]} If $m_s\equiv 2\pmod{4}$ then $x=in^*\alpha$ where $in$ is the inclusion of $B_-$ into ${\bf R} P^q$ and $\alpha$ is the only non-zero element of $H^1({\bf R} P^q;{\bf Z}_2)$ {\em\underline{Proof}} Consider $E$ -- the auxiliary surface in $P^q$ given by equation $P_s(x_0,\ldots,x_q)=0$. Then the construction of the double covering of ${\bf C} P^q$ branched along ${\bf C} E$ in the weighted-homogeneous projective space by equation $\lambda^2= P_s(x_0,\ldots,x_q)$ follows that the characteristic class of the restriction of the covering to ${\bf R} P^q-{\bf R} E$ is equal to the restriction of $\alpha$ to ${\bf R} P^q-{\bf R} E$. Therefore, the characteristic class of the restriction of the covering to $B_-$ is equal to $in^*\alpha$. Besides, it is claimed in \cite{R} that in the case of curves on surfaces (i.e. $n=1$ in notations of \cite{R}) the endomorphism $\omega:H_*(B_-,A;{\bf Z}_2)\rightarrow H_*(B_-,A;{\bf Z}_2)$ of cap-product with characteristic class $x$ is trivial. This is true only if either $m_s\equiv 0\pmod{4}$ or each non-contractible component of $B_-$ contains at least one component of ${\bf R} A$. The condition on $C$ in \ref{RKGK} allows to correct proofs of congruences of \cite{R}, \cite{Kh} and \cite{GK}. The author though does not know counter-examples to \ref{RKGK} without condition on $c$ (without the condition on $c$ point a) of \ref{RKGK} is equivalent to 3.4 of \cite{R}). \subsubsection{Direct generalization of Rokhlin and Kharlamov-Gudkov-Krakhnov congruences} \label{gen} Suppose that $B$ is of type I$abs$ in the case when $\sum_{j=1}^{s-1}m_j\equiv 0 \pmod{2}$ and that $B$ is of type I$rel$ in the case when $\sum_{j=1}^{s-1}m_j\equiv 1\pmod{2}$. Suppose that $m_s$ is even, $e=0$, $B_+$ is contained in one surface of the complex separation of ${\bf R} B$ and in the case when $m_s\equiv 2\pmod{4}$ suppose in addition that $c=0$. \begin{itemize} \begin{description} \item[a)] If $d+k=0$ then $$\chi(B_+)\equiv\frac{m_1\ldots m_{s-1}m_s^2}{4}\pmod{8}$$ \item[b)] If $d+k=1$ then $$\chi(B_+)\equiv\frac{m_1\ldots m_{s-1}m_s^2}{4}\pm 1\pmod{8}$$ \item[c)] If $d+k=2$ and $$\chi(B_+)\equiv\frac{m_1\ldots m_{s-1}m_s^2}{4}+4\pmod{8}$$ then $A$ is of type I \item[d)] If $A$ is of type I then $$\chi(B_+)\equiv\frac{m_1\ldots m_{s-1}m_s^2}{4}\pmod{4}$$ \end{description} \end{itemize} \subsubsection{Lemma} \label{V+=0} The surface $V_+={\bf C} A/conj\cup B_+$ is ${\bf Z}_2$-homologous to zero in ${\bf C} B/conj$ {\em\underline{Proof}} Consider the diagram \begin{picture}(200,-100)(-50,0) \put(-10,0){${\bf C} Y$}\put(60,0){${\bf C}B$} \put(-20,-40){${\bf C} Y/conj_Y$}\put(60,-40){${\bf C}B/conj .$} \put(10,4){\vector(1,0){46}} \put(3,-3){\vector(0,-1){25}} \put(67,-3){\vector(0,-1){25}} \put(33,8){$p$} \end{picture}\\[42pt] where $p:{\bf C} Y\rightarrow{\bf C} B$ is the double covering over ${\bf C} B$ branched along ${\bf C} A$ and $conj_Y$ is such a lift of $conj:{\bf C} B\rightarrow{\bf C} B$ that ${\bf R} Y=\{y\in{\bf C} Y|conj_Yy=y\}$ is equal to $p^{-1}(B_-)$. It is easy to see that this diagram can be expanded to a commutative one by adding $\phi:{\bf C} Y/conj_Y\rightarrow{\bf C} B/conj$ where $\phi$ is the double covering map branched along $V_+$. It follows that $[V_+]=0\in H_2({\bf C} B/conj)$. \subsubsection{Remark} The construction of \ref{V+=0} allows to define the separation of ${\bf R} B$ into $B_+$ and $B_-$ for any (not necessarily algebraic) curve of even degree and invariant with respect to $conj$ in any (not necessarily projective) complex surface ${\bf C} B$ equipped with almost antiholomorphic involution $conj:{\bf C} B\rightarrow{\bf C} B$ and such that $H_1({\bf C} B;{\bf Z}_2)=0$. In this case there is the unique double covering over ${\bf C} B$ branched along ${\bf C} A$ and two possible lifts of $conj:{\bf C} B\rightarrow{\bf C} B$. The images of the fix point sets of these lifts form the desired separation. Note that this separation is different from the complex separation. Let us give an internal definition of this separation. Let $x,y$ be points in ${\bf R} B-{\bf R} A$. Connect these points by path $\gamma$ inside ${\bf C} B-{\bf C} A$. If the linking number of the loop $\gamma conj\gamma$ and ${\bf C} A$ is equal to zero then $x$ and $y$ are points of the same surface of the complex separation, otherwise $x$ and $y$ are points of two different surfaces of the complex separation. This remark allows to extend \ref{gen} to the case of flexible curves. \subsubsection{The proof of 4.4.4} Note that $Dw_2({\bf C} B)+[{\bf R} B]=0$ since $Dw_2({\bf C} B)\equiv [\infty] \sum_{j=1}^{s-1}m_j$ where $[\infty]\in H_2({\bf C} B;{\bf Z}_2)$ is the class of hyperplane section of ${\bf C} B$ and note that $[{\bf C} A]=0$ since $m_s$ is even. We apply \ref{surface}, let $W$ be the surface of the complex separation not intersecting $B_+$. Then \ref{V+=0} follows that $W\cup B_+$ is the surface of the complex separation of $(B,A)$ since if $W$ is a characteristic surface in ${\bf C} B/conj$ then so is $W_+=W\cup V_+$. Let us prove now that $q_{W_+}|_{H_1(W;{\bf Z}_2)}$ is equal to $q_W$ where $q_{W_+}$ and $q_W$ are Guillou-Marin forms of $W_+$ and $W$ in ${\bf C} B/conj$. Note that $(q_{W_+}-q_W)(x), x\in H_1(W;{\bf Z}_2)$ is equal to the linking number of $x$ and $V_+$ in ${\bf C} B/conj$ that is equal to the linking number of $x$ and ${\bf C} A$ in ${\bf C} B$. It is not difficult to see that the condition on $c$ follows that the linking number of $x$ and ${\bf C} A$ in ${\bf C} B$ is equal to zero. Theorem 1 follows that $$\chi(B_+)+\chi(W)\equiv\frac{e_A}{4}+\frac{\chi({\bf R} B)-\sigma({\bf C} B)}{4}+ \beta(q_W)+\beta(q_{W_+}|_{H_1(W_+;{\bf Z}_2)})\pmod{8}$$ and \ref{surface} follows that $$\chi(W)\equiv\frac{\chi({\bf R} B)-\sigma({\bf C} B)}{4}+\beta(q_W)\pmod{8}$$ Thus, noting that $e_A=m_1\ldots m_{s-1}m_s^2$ we get that $$\chi(B_+)\equiv\frac{m_1\ldots m_{s-1}m_s^2}{4}+ \beta(q_{W_+}|_{H_1(V_+;{\bf Z}_2)})\pmod{8}$$ The rest of the proof is similar to that of \ref{prth}. \subsection{Curves on an ellipsoid} In the congruences of \ref{gen} we was able to avoid the appearance of the complex separation with the help of the separation of ${\bf R} B$ into $B_+$ and $B_-$. For curves of odd degree this does not work since there is no such a separation. Although, if ${\bf R} B$ is connected then the complex separation does not provide the additional information and still can be avoided. Surfaces $B_1$ and $B_2$ of the complex separation are determined by the condition that $B_1\cup B_2={\bf R} B$ and $\partial B_1=\partial B_2={\bf R} A$. The simplest case is the case of curves of odd degree on an ellipsoid. It is well-known that the complex quadric is isomorphic to ${\bf C} P^1\times{\bf C} P^1$, an algebraic curve in quadric is determined by a bihomogeneous polynomial of bidegree $(d,r)$. If the curve is real and the quadric is an ellipsoid then $d=r$, otherwise the curve can not be invariant under the complex conjugation of an ellipsoid since the complex conjugation of an ellipsoid acts on $H_2({\bf C} P^1\times{\bf C} P^1={\bf Z}\times{\bf Z}$ in the following way : $conj_*(a,b)=(-b,-a)$ as it is easy to see considering the behavior of $conj$ on the generating lines of ${\bf C} P^1\times{\bf C} P^1$. Thus a real curve on an ellipsoid is the intersection of the ellipsoid and a surface of degree $d$, this can be regarded as a definition of real curves on an ellipsoid. \subsubsection{Theorem} \label{ell} Let $A$ be a nonsingular real curve of bidegree $(d,d)$ on ellipsoid $B$. Suppose that $d$ is odd. \begin{itemize} \begin{description} \item[a)] If $A$ is an M-curve then $$\chi(B_1)\equiv\chi(B_2)\equiv\frac{d^2+1}{2}\pmod{8}$$ \item[b)] If $A$ is an (M-1)-curve then $$\chi(B_1)\equiv\frac{d^2+1}{2}\pm 1\pmod{8}$$ $$\chi(B_2)\equiv\frac{d^2+1}{2}\mp 1\pmod{8}$$ \item[c)] If $A$ is an (M-2)-curve and $$\chi(B_1)\equiv\frac{d^2+1}{2}+4\pmod{8}$$ then $A$ is of type I \item[d)] If $A$ is of type I then $$\chi(B_1)\equiv\chi(B_2)\equiv 1\pmod{4}$$ \end{description} \end{itemize} {\em\underline{Proof}} $Dw_2({\bf C} B)+[{\bf R} B]+[{\bf C} A]=0$ since an ellipsoid is of type I$rel$. Theorem 1 follows \ref{ell} now since $$\frac{e_A}{4}+\frac{\chi({\bf R} B)-\sigma({\bf C} B)}{4}=\frac{d^2+1}{2}$$ and $\beta_j=0$ because of the triviality of $H_1({\bf R} B;{\bf Z}_2)$. \subsubsection{Low-degree curves on an ellipsoid} Consider the application of \ref{ell} to the low-degree curves on an ellipsoid. Gudkov and Shustin \cite{GSh} classified real schemes of curves of bidegree not greater then (4,4) on an ellipsoid. To prove restrictions for such a classification it was enough to apply the Harnack inequality and Bezout theorem. Using the Bezout theorem avoided the extension of such a classification to the flexible curves. For the curves of bidegree (4,4) one can avoid the using of the Bezout theorem using instead the analogue of the strengthened Arnold inequalities for curve on an ellipsoid. for the classification of the flexible curves of bidegree (3,3) the old restrictions does not give the complete system of restrictions, they do not restrict schemes $2\sqcup 1$$<$$2$$>$, $3\sqcup 1$$<$$1$$>$ and $2\sqcup 1$$<$$1$$>$ (see \cite{V} for the notations). Theorem \ref{ell} restricts these schemes and thus completes the classification of the real schemes of flexible curves of bidegree (3,3) on an ellipsoid. \subsubsection{Theorem} The real scheme of a nonsingular flexible curve of bidegree (3,3) on an ellipsoid is $1$$<1$$<$$1$$>$$>$ or $\alpha\sqcup 1$$<$$\beta$$>, \alpha>\beta,\alpha+\beta\leq 4$. Each of this schemes is the real scheme of some flexible (and even algebraic) curve of bidegree (3,3) on an ellipsoid \subsubsection{Curves of bidegree (5,5) on an ellipsoid} Theorem \ref{ell} gives an essential restriction for the M-curves of bidegree (5,5) on an ellipsoid, \ref{ell} leaves unrestricted 18 possible schemes of flexible M-curves of bidegree (5,5) and 15 of them can be realized as algebraic M-curves given by birational transformations of the appropriate affine curves of degree 6 and another one by Viro technique of small perturbation from the product of five plane sections intersecting in two different points. Thus there are only two schemes of M-curves of bidegree (5,5) unrestricted and unconstructed, namely, $1\sqcup 1$$<$$6$$>$$\sqcup 1$$<$$8$$>$ and $1\sqcup 1$$<$$5$$>$$\sqcup 1$$<$$9$$>$. \subsubsection{The case of bidegree (4,4)} The classification of the real schemes of curves of bidegree (4,4) \cite{GSh} shows that there is no congruence like \ref{ell} for curves of even degree: the Euler characteristic of surfaces $B_1$ and $B_2$ for curves of bidegree (4,4) can be any even number between -8 and 10. Thus the condition that $Dw_2({\bf C} B)+[{\bf R} B]+[{\bf C} A]=0$ is essential. \subsubsection{Addendum, the Fiedler congruence for curves on an ellipsoid} Let $A$ be an M-curve of bidegree $(d,d)$ on ellipsoid $B$. Suppose that $d$ is even, the Euler characteristic of each component of $B_1$ is even and $\chi(B_1)\equiv 2\pmod{4}$ then $$\chi(B_2)\equiv d^2\pmod{16}$$ $$\chi(B_1)\equiv 2-d^2\pmod{16}$$ {\em\underline{Proof}} The formula of complex orientations for curves on an ellipsoid (see \cite{Z1}) follows that the surface $V$ equal to $B_2\cup A_+$ is ${\bf Z}_2$-homologous to zero in ${\bf C} B$ where $A_+$ is one of the components of ${\bf C} A-{\bf R} A$. Thus $V$ is a characteristic surface in ${\bf C} B$. Further arguments are similar to that of \cite{F} \subsection{Curves on cubics} In this subsection we consider the application of Theorem 1 to the curves of degree 2 on cubics, surfaces in $P^3$ given by cubic polynomial (cf.\cite{Mi}) of type I$rel$. In notations of \ref{gen} we have $q=3$, $s=2$, $m_1=3$, $m_2=2$. Rokhlin congruence for curves on surfaces gives the complete system of restrictions for curves of degree 2 on $M$-cubic but for restrictions for curves of degree 2 on another cubic of type I$rel$ -- cubic diffeomorphic to the disjoint sum of ${\bf R} P^2$ and $S^2$ we need some new tools. Theorem 1 suffices for this purpose. To apply Theorem 1 note that the complex separation of the disjoint cubic consists of two surfaces -- ${\bf R} P^2$ and $S^2$. The following is the classification of the real schemes of flexible curves of degree 2 on a hyperboloid obtained in \cite{Mi}, for more details and for the classifications on other real cubics see \cite{Mi}. \subsubsection{Theorem} Each flexible M-curves of degree 2 on non-connected cubic has one of the following real schemes. \begin{itemize} \begin{description} \item[a)] $($$<$$3\sqcup 1$$<$$1$$>$$>$$)_{{\bf R} P^2}\sqcup ($$<$$\emptyset$$>$$)_{S^2}$ \item[b)] $($$<$$1$$<$$4$$>$$>$$)_{{\bf R} P^2}\sqcup ($$<$$\emptyset$$>$$)_{S^2}$ \item[c)] $($$<$$\alpha$$>$$)_{{\bf R} P^2}\sqcup ($$<$$5-\alpha$$>$$)_{S^2}, 0\leq\alpha\leq 5$ \end{description} \end{itemize} Each of these 8 schemes is the real scheme of some flexible curve of degree 2 on an ellipsoid.

9206

alg-geom/9206008

en

https://arxiv.org/abs/alg-geom/9206008

alg-geom/9206008

Ron Donagi

The fibers of the Prym map

71 pages, LATEX, (This is a reformatted version. It should print better than its predecessor.)

In this work we use the bigonal, trigonal and tetragonal constructions to describe the fibers of the Prym map P : R_{g} ---->A_{g-1} inthe cases when it is dominant, i.e. for g < 7. The most interesting cases are g = 5, where the fiber is a double cover of the Fano surface of lines on a cubic threefold, and g=6, where the map is generically finite (of degree 27) with Galois group WE_{6}, so that the general fiber has the structure of the 27 lines on a cubic surface. For g > 6, the map is known to be generically injective. The tetragonal construction gives many counterexamples to injectivity, and we conjecture that all noninjectivity is due to the tetragonal construction.

alg-geom

\section{Pryms.} \subsection{Pryms and parity.} \ \ \ \ Let $$\pi:\widetilde{C}\rightarrow C$$ be an unramified, irreducible double cover of a curve $C\in{\cal M}_g$. The genus of $\widetilde{C}$ is then $2g-1$, and we have the Jacobians $$J:=J(C), \ \ \ \ \ \ \ \widetilde{J}:=J(\widetilde{C})$$ of dimensions $g,\, 2g-1$ respectively, and the norm homomorphism $${\rm Nm}:\widetilde{J}\longrightarrow J.$$ Mumford shows [M2] that $${\rm Ker}({\rm Nm})=P\cup P^-$$ where $P={\cal P}(C, \widetilde{C})$ is an abelian subvariety of $\widetilde{J}$, called the Prym variety, and $P^-$ is its translate by a point of order 2 in $\widetilde{J}$. The principal polarization on $\widetilde{J}$ induces twice a principal polarization on the Prym. This appears most naturally when we consider instead the norm map on line bundles of degree $2g-2$, $${\rm Nm}:{\rm Pic}^{2g-2}(\widetilde{C})\rightarrow {\rm Pic}^{2g-2}(C).$$ Let $\omega_{C}\in {\rm Pic}^{2g-2}(C)$ be the canonical bundle of $C$. \bigskip \noindent {\bf Theorem 1.1} (Mumford [M1], [M2]) \begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}} \item The two components $P_0, \,P_1$ of ${\rm Nm}^{-1}(\omega_C)$ can be distinguished by their parity: $$P_i=\{L\in {\rm Nm}^{-1}(\omega_C) \ \ \ | \ \ \ h^0(L)\equiv i \ \ \ \ \ {\rm mod. \ 2}\}, \ \ \ \ \ \ \ \ \ \ i=0,1.$$ \item Riemann's theta divisor $\widetilde{\Theta}'\subset {\rm Pic}^{2g-2}(\widetilde{C})$ satisfies $$\widetilde{\Theta}'\supset P_1$$ and $$\widetilde{\Theta}'\cap P_0=2\Xi'$$ where $\Xi'\subset P_0$ is a divisor in the principal polarization on $P_0$. \end{list} \subsection{Bilinear and quadratic forms.} \ \ \ \ Let $X\in{\cal A}_g$ be a {\rm PPAV}, and let $Y$ be a torser (=principal homogeneous space) over $X$. By theta divisor in $Y$ we mean an effective divisor whose translates in $X$ are in the principal polarization. $X$ acts by translation on the variety $Y'$ of theta divisors in $Y$, making $Y'$ also into an $X$-torser. In $X'$ there is a distinguished divisor $$\Theta':=\{\Theta\subset X|\Theta\ni 0\}\subset X'$$ which turns out to be a theta divisor, $\Theta'\in X''$. In particular, we have a natural identification $X''\approx X$ sending $\Theta'$ to $0$. Let $X_2$ be the subgroup of points of order $2$ in $X$. Inversion on $X$ induces an involution on $X'$; the invariant subset $X'_2$, consisting of symmetric theta divisors in $X$, is an $X_2$-torser. Let $\langle\, ,\, \rangle$ denote the natural ${\bf F}_2$-valued (Weil) pairing on $X_2$. On $X'_2$ we have an ${\bf F}_2$-valued function $$q=q_X:X'_2\rightarrow{\bf F}_2$$ sending $\Theta\in X'$ to its multiplicity at $0\in X$, taken mod. 2. \bigskip \noindent {\bf Theorem 1.2} [M1] The function $q_X$ is quadratic. Its associated bilinear form, on $X_2$, is $\langle \, ,\, \rangle$. When $(X,\Theta)$ vary in a family, $q_X(\Theta)$ is locally constant. When $X$ is a Jacobian $J=J(C)$, these objects have the following interpretations: \noindent \begin{tabbing} $q(L)$ \= $\approx$ \= $\{ L \in {\rm Pic}^{0}(C) = J \ | \ L^{2} \ \approx {\cal O}_{C} \} \ \approx \ H^{1}(C,{\bf F}_{2})$ \= \ (semi periods) \kill $J'$ \> $\approx$ \> ${\rm Pic}^{g-1}(C) \,$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (use Riemann's theta divisor) \\ $J_{2}$ \> $\approx$ \> $\{ L \in {\rm Pic}^{0}(C) = J \ | \ L^{2} \ \approx {\cal O}_{C} \} \ \approx \ H^{1}(C,{\bf F}_{2})$ \= \ \ \ \ (semi periods) \\ $J'_{2}$ \> $\approx$ \> $\{ L \in {\rm Pic}^{g-1}(C) \ \ \ \ | \ L^{2} \ \approx {\cal \omega}_{C} \ \} \,$ \ \ \ \ \ \ \ \ \ \ \ \ (theta characteristics) \\ $q(L)$ \> $\equiv$ \> $h^{0}(C,L)$ mod. 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (by Riemann-Kempf) \end{tabbing} Explicitly, the theorem says in this case that for $\nu, \sigma\in J_2$ and $L\in J'_2$: \bigskip \noindent ({\bf 1.3}) $\langle \nu,\sigma \rangle \ \ \ \equiv \ \ \ h^0(L)+h^0(L\otimes\nu)+h^0(L\otimes\sigma)+h^0(L\otimes\nu \otimes\sigma)$ \begin{flushright} mod. 2. \end{flushright} We note that non-zero elements $\mu\in J_2$ correspond exactly to irreducible double covers $\pi:\widetilde{C}\rightarrow C$. Let $X$ be the Prym $P={\cal P}(C,\widetilde{C})$, which we also denote $P(C,\mu)$, $P(C,\widetilde{C})$, \ $P(\widetilde{C}/C)$ etc. Now the divisor $\Xi'\subset P_0$ of Theorem 1.1 gives a natural identification $$P'\approx P_0\subset\widetilde{J}'.$$ The pullback $$\pi^*:J\longrightarrow\widetilde{J}$$ sends $J_2$ to $\widetilde{J}_2$. Since ${\rm Nm}\circ\pi^*=2$, we see that $$\pi^*(J_2)\subset P_2\cup P^-_2.$$ Let $(\mu)^\perp$ denote the subgroup of $J_2$ perpendicular to $\mu$ with respect to $\langle \, , \, \rangle$. \bigskip \noindent {\bf Theorem 1.4} [M2]\begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}} \item For $\tau\in J_2$, $\pi^*\tau\in P_2$ iff $\tau\in(\mu)^{\perp}$. \item This gives an exact sequence $$0\rightarrow(\mu)\rightarrow(\mu)^\perp \stackrel{\pi^*}{\rightarrow}P_2\rightarrow 0.$$ \item In (2), $\pi^*$ is symplectic, i.e. $$\langle \nu, \sigma \rangle_J \, = \, \langle \pi^*\nu, \pi^*\sigma\rangle _P, \ \ \ \ \ \ \ \ \ \ \nu, \sigma\in(\mu)^\perp\subset J_2.$$ \end{list} This equality of bilinear forms can be refined to an equality of quadratic functions. The identifications $$J'\approx {\rm Pic}^{g- 1}(C), \ \ \ \ \ \ \ \widetilde{J}'\approx {\rm Pic} ^{2g-2} (\widetilde{C})$$ convert the pullback $$\pi^* :{\rm Pic}^{g-1}(C)\rightarrow{\rm Pic}^{2g-2}(\widetilde{C})$$ into a map of torsers $${\pi^{*}}' : J' \rightarrow \widetilde{J}'$$ over the group homomorphism $$\pi^{*}:J\rightarrow\widetilde{J}.$$ Let $${(\mu)^{\perp }}':=({\pi^{*}}')^{-1}(P'_2).$$ the refinement is: \bigskip \noindent {\bf Theorem 1.5} [D4]\begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}} \item ${(\mu)^{\perp}}'$ is contained in $J'_2$ and is a $(\mu)^\perp$-coset there. \item ${\pi^{*}}':{(\mu)^{\perp}}'\rightarrow P'_2$ is a map of torsers over $\pi^*:(\mu)^\perp\rightarrow P_2$. \item In (2), ${\pi^{*}}'$ is orthogonal, i.e. $$q_J(\nu)=q_P({\pi^{*}}'\nu), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nu\in{(\mu)^{\perp}}'.$$ \end{list} \subsection{The Prym Maps.} \ \ \ \ Let ${\cal R}_g$ be the moduli space of irreducible double covers $\pi:\widetilde{C}\rightarrow C$ of non-singular curves $C\in{\cal M}_g$. Equivalently, ${\cal R}_g$ parametrizes pairs $(C, \mu)$ with $\mu\in J_2(C)\backslash(0)$, a semiperiod on $C$. The assignment of the Prym variety to a double cover gives a morphism $${\cal P}:{\cal R}_g\rightarrow{\cal A}_{g-1}.$$ Let $\iota$ be the involution on $\widetilde{C}$ over $C$. The Abel- Jacobi map $$\varphi:\widetilde{C}\rightarrow J(\widetilde{C})$$ induces the Abel-Prym map $$\psi:\widetilde{C}\rightarrow{\rm Ker}({\rm Nm})$$ $$ \ \ \ \ \ \ \ \ \ \ \ x\longmapsto\varphi(x)-\varphi(\iota x).$$ The image actually lands in the wrong component, $P^-$, but at least $\psi$ is well-defined up to translation (by a point of order 2). In particular, its derivative is well-defined; it factors through $C$, yielding the Prym-canonical map $$\Psi:C\rightarrow{\bf P}^{g-2}$$ given by the complete linear system $|\omega_C\otimes\mu|$. Beauville computed the codifferential of the Prym map: \bigskip \noindent {\bf Theorem 1.6} [B1] The codifferential $$d{\cal P}:T^*_P{\cal A}_{g-1}\rightarrow T^*_{(C,\mu)}{\cal R}_g$$ can be naturally identified with restriction $$\Psi^*:S^2 H^0(\omega_C \otimes \mu) \rightarrow H^0(\omega^2_C).$$ In particular, ${\rm Ker}(d{\cal P})$ is given by quadrics through the Prym-canonical curve $\Psi(C)\subset{\bf P}^{g-2}$. Let $\bar{\cal A}_g$ denote a toroidal compactification of ${\cal A}_g$. Its boundary $\partial\bar{\cal A}_g$ maps to $\bar{\cal A}_{g-1}$, and the fiber over generic $A\in{\cal A}_{g-1}\subset\bar{\cal A}_{g-1}$ is the Kummer variety $K(A):=A/(\pm 1)$. In codimension 1, this picture is independent of the toroidal compactification used. Let ${\cal RA}_g$ denote the level moduli space parametrizing pairs $(A,\mu)$ with $A\in{\cal A}_g \ , \ \mu\in A_2\backslash(0)$, and let $\overline{\cal RA}_g$ be a toroidal compactification. In [D3] we noted that its boundary has 3 irreducible components, distinguished by the relation of the vanishing cycle (mod. 2), $\lambda$, to the semiperiod $\mu$: \noindent{\bf (1.7)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\begin{array}{llll} \partial^{\rm I} \ \ : \!\!\! & \!\!\!\lambda \!\!\! & \!\! = \!\! & \!\!\! \mu \\ \partial^{\rm II} \ : \!\!\! & \!\!\! \lambda \!\!\! & \!\! \neq \!\! & \!\!\! \mu, \ { \ } \!\langle \lambda,\mu\rangle =0\in{\bf F}_2 \\ \partial^{\rm III}:\langle \!\!\! & \!\!\!\lambda \!\!\! & \! , \!\! & \!\!\! \mu\rangle \neq0. \end{array} $ Let $\bar{\cal M}_g$, $\doublebar{\cal R}_g$ denote the Deligne-Mumford stable-curve compactifications of ${\cal M}_g$ and ${\cal R}_g$.At least in codimension one, the Jacobi map extends: $$\bar{\cal M}_g\rightarrow\bar{\cal A}_g \ , \ \doublebar{\cal R}_g\rightarrow \overline{\cal RA}_g.$$ We use $\partial\bar{\cal M}_g$ \ , \ $\partial^i\doublebar{\cal R}_g$ \ \ \ $(i = {\rm I, \, II, \, III})$ to denote the intersections of $\bar{\cal M}_g$, $\doublebar{\cal R}_g$ with the corresponding boundary divisors in $\bar{\cal A}_g$, $\overline{\cal RA}_g$. In [B1], Beauville introduced the notion of an allowable double cover. This leads to the construction ([DS] I, 1.1) of a proper version of the Prym map, $$\bar{\cal P}:\bar{\cal R}_g\rightarrow {\cal A}_{g-1}.$$ Roughly, one extends ${\cal P}$ to $$\doublebar{\cal P}:\doublebar{\cal R}_g\rightarrow\bar{\cal A}_{g-1},$$ then restricts to the open subset $\bar{\cal R}_g\subset\doublebar{\cal R}_g$ of covers which are allowable, in the sense that their Prym is in ${\cal A}_{g-1}$.This condition can be made more explicit: \bigskip \noindent {\bf Theorem 1.8} [B1] A stable curve $\widetilde{C}$ with involution $\iota$, quotient $C$, is allowable if and only if all the fixed points of $\iota$ are nodes of $\widetilde{C}$ where the branches are not exchanged, and the number of nodes exchanged under $\iota$ equals the number of irreducible components exchanged under $\iota$. We illustrate the possibilities in codimension 1: \bigskip \noindent {\bf Examples 1.9} \begin{list}{{\rm(\Roman{butter})}}{\usecounter{butter}} \item $X\in{\cal M}_{g-1}, \ p, q\in X, \ \ p\neq q$; let $X_0, X_1$ be isomorphic copies of $X$. Then $C:=X/(p\sim q)$ is a point of $\partial\bar{\cal M}_g$. The Wirtinger cover $$\widetilde{C}:=(X_0\amalg X_1)/(p_0\sim q_1, p_1\sim q_0)$$ gives a point $$(C,\widetilde{C})\in\partial^{\rm I}\bar{\cal R}_g$$ which is allowable. The Prym is $${\cal P}(C,\widetilde{C})\approx J(X)\in{\cal A}_{g-1}.$$ \item Start with $(\widetilde{X}\rightarrow X)\in{\cal R}_{g-1}$, choose distinct points $p, q\in X$, let $p_i, q_i(i=0, 1)$ be their inverse images in $\widetilde{X}$, and set $$C:=X/(p\sim q), \ \ \ \widetilde{C}:=\widetilde{X}/(p_0\sim q_0, p_1\sim q_1).$$ Then $$(C,\widetilde{C})\in\partial^{\rm II}\doublebar{\cal R}_g$$ is an unallowable cover. Its Prym is a ${\bf C}^*$-extension of ${\cal P}(X,\widetilde{X})$; the extension datum defining this extension is given by $$\psi(p_0)-\psi(q_0)\in{\cal P}(X,\widetilde{X}),$$ which is well defined modulo $\pm 1$ (i.e. in the Kummer), as it should be. \item $X, p, q$ as before, but now $\widetilde{X}\rightarrow X$ is a double cover branched at $p, q$; consider Beauville's cover $$C:=X/(p\sim q), \ \ \ \\widetilde{C}:=\widetilde{X}/(\widetilde{p}\sim\widetilde{q})$$ where $\widetilde{p}, \widetilde{q}$ are the ramification points in $\widetilde{X}$ above $p, q$. Then $(C, \widetilde{C})\in\partial^{\rm III}\bar{\cal R}_g$ is allowable. \end{list} In [M1], Mumford lists all covers $(C, \widetilde{C})\in{\cal R}_g$ whose Pryms are in the Andreotti-Mayer locus (i.e. have theta divisors singular in codimension 4). A major result in [B1] (Theorem (4.10)) is the extension of this list to allowable covers in $\bar{\cal R}_g$. We do not copy Beauville's list here, but we will refer to it when needed. \section{Polygonal constructions} \subsection{The $n$-gonal constructions} \ \ \ \ Let $$f:C\rightarrow K$$ be a map of non singular algebraic curves, of degree $n$, and $$\pi:\widetilde{C}\rightarrow C$$ a branched double cover. These two determine a $2^n$-sheeted branched cover of $K$, $$f_*\widetilde{C}\rightarrow K,$$ whose fiber over a general point $k\in K$ consists of the $2^n$ sections $s$ of $\pi$ over $k$: $$s:f^{- 1}(k)\rightarrow\pi^{-1}f^{-1}(k), \ \ \ \pi\circ s=id.$$ The curve $f_*\widetilde{C}$ can be realized, for instance, as sitting in ${\rm Pic}^n(\widetilde{C})$ or $S^n\widetilde{C}$: \noindent {\bf (2.1)} \ \ \ $f_*\widetilde{C}=\{D\in S^n\widetilde{C} \ \ | \ \ {\rm Nm}(D)=f^{-1}(k), \ {\rm some} \ k\in K\}.$ \noindent (If we think of $\widetilde{C}$ as a local system on an open subset of $C$, this is just the direct image local system on $K$, hence our notation $f_*\widetilde{C}$.) On $f_*\widetilde{C}$ we have two structures: an involution $$\iota:f_*\widetilde{C}\rightarrow f_*\widetilde{C}$$ obtained by changing all $n$ choices in the section $s$ via the involution (also denoted $\iota$) of $\widetilde{C}$, and an equivalence relation $$f_*\widetilde{C}\rightarrow\widetilde{K}\rightarrow K$$ where $\widetilde{K}$ is a branched double cover of $K$: two sections $$s_1, s_2:f^{- 1}(k)\rightarrow\pi^{-1}f^{-1}(k)$$ are equivalent if they differ by an even number of changes. For even $n$, the involution $\iota$ respects equivalence, so we have a sequence of maps \noindent{\bf (2.1.1)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $f_*\widetilde{C}\rightarrow f_*\widetilde{C}/\iota\rightarrow\widetilde{K}\rightarrow K$ \noindent of degrees $2, 2^{n-2}, 2$ respectively. For odd $n$ the equivalence classes are exchanged by $\iota$, so we have instead a Cartesian diagram: \begin{equation} \renewcommand{\theequation}{\bf {\arabic{section}}.{\arabic{subsection}}.{\arabic{equation}}} \setcounter{equation}{2} \begin{diagram}[f ] \node[2]{f_{*}\widetilde{C}} \arrow{sw} \arrow{se} \\ \node{f_{*}\widetilde{C}/\iota} \arrow{se} \node[2]{\widetilde{K}} \arrow{sw} \\ \node[2]{K} \end{diagram} \end{equation} {\bf Remark 2.1.3} In prctice we will often want to allow $C$ to acquire some nodes, over which $\pi$ may be etale (as in (1.9 II)) or ramified (as in \linebreak (1.9 III)). We will always consider this as a limiting case of the non-singular situation, and interpret the $n$-gonal construction in the limit so as to make it depend continuously on the parameters, whenever possible. We will see various examples of this below. \subsection{Orientation} \ \ \ \ We observe that the branched cover $\widetilde{K}\rightarrow K$ depends on $f\circ\pi:\widetilde{C}\rightarrow K$, but not on $f, \pi$ or $C$ directly. More generally, to an $m$-sheeted branched cover $$g:M\rightarrow K$$ we can associate an $m!$-sheeted branched cover (the Galois closure of $M$) $$g!:M!\rightarrow K,$$ with an action of the symmetric group $S_m$; the quotient by the alternating group $A_m$ gives a branched double cover $$O(g):O(M)\rightarrow K$$ which we call the orientation cover of $M$. We say $M$ is orientable (over $K$) if the double cover $O(M)$ is trivial. One verifies easily that the double cover $\widetilde{K}\rightarrow K$ (obtained in \S 2.1 from the maps $\widetilde{C}\stackrel{\pi}{\rightarrow}C\stackrel{f}{\rightarrow}K$ as quotient of $f_*\widetilde{C}$) is the orientation cover $O(f\circ\pi)$ of $\widetilde{C}$. \bigskip \noindent{\bf Corollary 2.2} If $\widetilde{C}$ is orientable over $K$ then $f_*\widetilde{C}=\widetilde{C}_0\cup\widetilde{C}_1$ is reducible: \begin{list}{{\rm(\roman{butter})}}{\usecounter{butter}} \item For $n$ even, the involution $\iota$ acts on each $\widetilde{C}_i$ with quotient $C_i$ of degree $2^{n-2}$ over $K, \ \ \ i=0, 1$. \item For $n$ odd, $\iota$ exchanges $\widetilde{C}_0, \widetilde{C}_1$.Each $\widetilde{C}_i$ has degree $2^{n-1}$ over $K$. \end{list} \bigskip \noindent{\bf Lemma 2.3} Branch $(\widetilde{K}/K)=f_*({\rm Branch} \ (\widetilde{C}/C))$. This means: if one point of $f^{-1}(k)$ is a branch point of $\widetilde{C}\rightarrow C$, then $k$ is a branch point of $\widetilde{K}\rightarrow K$; if two points of $f^{-1}(k)$ are branch points of $\widetilde{C}\rightarrow C$, then $k$ is not a branch point of (the normalization of) $\widetilde{K}\rightarrow K$, but the two sheets of $\widetilde{K}$ there intersect; etc. In particular, the ramification behavior of $f:C\rightarrow K$ does not affect the ramification of $\widetilde{K}$. \bigskip \noindent{\bf Corollary 2.4} Let $f:C\rightarrow{\bf P}^1$ be a branched cover, $\pi:\widetilde{C}\rightarrow C$ an (unramified) double cover. Then $\widetilde{C}$ is orientable over ${\bf P}^1$. (More generally, the conclusion holds whenever $$f_*({\rm Branch} (\pi))=2D$$ for some divisor $D$ on ${\bf P}^1$, since the normalization of $O(\widetilde{C})$ is then an unramified double cover of the simply connected ${\bf P}^1$, by (2.3). In this situation we say that $\pi$ has \underline{cancelling ramification.}) \bigskip \noindent{\bf Remark 2.5} Assume $K={\bf P}^1$ and $\pi$ unramified. The image of $f_*\widetilde{C}$ in ${\rm Pic}(\widetilde{C})$ is: $$\{L\in{\rm Pic}^n(\widetilde{C}) \ \ | \ \ {\rm Nm}(L)=f^*{\cal O}_{{\bf P}^1}(1), \ \ \ h^0(L) > 0\}.$$ This is contained in a translate of $${\rm Nm}^{-1}(\omega_C)=P_0\cup P_1,$$ and the splitting (2.2) of $f_*\widetilde{C}$ is ``explained", in this case, by the splitting (1.1) of ${\rm Ker}({\rm Nm})$, i.e. after translation: $$\widetilde{C}_i\subset P_i, \ \ \ \ \ i=0, 1,$$ cf. [D1, \S 6], [B2]. \bigskip \noindent{\bf Remark 2.6} The splitting of $f_*\widetilde{C}$ can also be explained group theoretically. Let $WC_n$ be the group of signed permutations of $n$ letters, i.e. the subgroup of $S_{2n}$ centralizing a fixed-point-free involution of the $2n$ letters.Let $WD_n$ be its subgroup of index 2 consisting of even signed permutations, i.e. permutations of $n$ letters followed by an even number of sign changes. (These are the Weyl groups of the Dynkin diagrams $C_n,D_n$.) Over an arbitrary space $X$, we have equivalences: \pagebreak[4] $$\{ \ \ \ \ \ n{\rm -sheeted \ cover \ } Y\rightarrow X \ \ \ \ \ \} \ \longleftrightarrow\{ \ \ \ {\rm Representation \ } \pi_1(X)\rightarrow \ \ \ \ \ \ S_n \ \ \}$$ $$ \left\{ \begin{array}{l} n{\rm -sheeted \ cover \ } Y\rightarrow X \\ {\rm with \ a \ double \ cover \ } \widetilde{Y}\rightarrow Y \end{array} \right\} \longleftrightarrow \left\{ \begin{array}{l} {\rm \ Representation \ } \pi_1(X)\rightarrow WC_n \end{array} \right\} $$ $$ \left\{ \begin{array}{l} n{\rm -sheeted \ cover \ } Y\rightarrow X \\ {\rm with \ an \ orientable \ double \ \ } \\ {\rm cover \ } \widetilde{Y}\rightarrow Y \ \end{array} \right\} \longleftrightarrow \left\{ \begin{array}{l} {\rm Representation \ } \pi_1(X)\rightarrow WD_n \end{array} \right\} $$ \medskip The basic construction of $f_*\widetilde{C}$ then corresponds to the standard representation $$\rho:WC_n\hookrightarrow S_{2^n}.$$ The existence of the involution $\iota$ on $f_*\widetilde{C}$ corresponds to the factoring of $\rho$ through $$WC_{2^{n-1}}\subset S_{2^n}.$$ The restriction $\bar{\rho}$ of $\rho$ to $WD_n$ factors through $$S_{2^{n-1}}\times S_{2^{n-1}},$$ explaining the splitting when $\widetilde{C}$ is orientable. \subsection{The bigonal construction} \ \ \ \ The case $n=2$ of our construction (``bigonal") takes a pair of maps of degree 2: $$\widetilde{C}\stackrel{g}{\rightarrow}C\stackrel{f}{\rightarrow}K$$ and produces another such pair $$f_*\widetilde{C}\stackrel{g'}{\rightarrow}\widetilde{K} \stackrel{f'}{\rightarrow} K.$$ Above any given point $k \in K$, the possibilities are: \begin{list}{(\roman{bean})}{\usecounter{bean}} \item If $f$, $g$ are etale then so are $f'$ and $g'$. \item If $f$ is etale and $g$ is branched at one of the two points $f^{-1}(k)$, then $f'$ is branched at $k$, and $g'$ is etale there. \item Vise versa, if $f$ is branched and $g$ is etale then $f'$ is etale and $g'$ is branched at one point of $f'^{-1}(k)$. \item If both $f$ and $g$ are branched over $k$ then so are $f'$, $g'$. \item If $f$ is etale and $g$ is branched at both points $f^{-1}(k)$, then $\widetilde{K}$ will have a node over $k$, and $g' : f_{*}\widetilde{C} \rightarrow \widetilde{K}$ will be a $\partial^{\rm III}$ degeneration, i.e. will look like (1.9 III). \item Vice versa, we can extend the bigonal construction by continuity, as in (2.1.3), to allow $g : \widetilde{C} \rightarrow C$ to degenerate to a $\partial^{\rm III}$-cover. This leads to $f'$ which is etale and $g'$ which is branched at both points of $f'^{-1}(k)$. \end{list} The following general properties are immediately verified: \bigskip \noindent{\bf Lemma 2.7}\begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}} \item The bigonal construction is symmetric, i.e. if it takes $\widetilde{C}\stackrel{g}{\rightarrow}C\stackrel{f}{\rightarrow}K$ to $\widetilde{C}'\stackrel{g'}{\rightarrow}C'\stackrel{f'}{\rightarrow} K$ then it takes $\widetilde{C}'\rightarrow C'\rightarrow K$ to $\widetilde{C}\rightarrow C\rightarrow K$. \item The bigonal construction exchanges branch loci: $${\rm Branch}(g')=f_*({\rm Branch}(g)), \ \ \ \ \ {\rm Branch}(f)=g'_*({\rm Branch}(f')).$$ \end{list} (As in lemma (2.3), this requires the following convention in case (vi) above: the local contribution to Branch($f$) is $2k$, and the contribution to Branch($g$) is 0). The symmetry group of this situation, $WC_2$, is the dihedral group of the square: $$WC_2= \ \ \langle r, f \ \ \ | \ \ \ f^2=r^4=(rf)^2=1\rangle.$$ ($r=90^\circ$ rotation, $f=$flip around $x$-axis, in the 2-dimensional representation.) It has a non-trivial outer automorphism (=conjugation by a $45^\circ$ rotation), which explains why conjugacy classes of representations (of $\pi_1(X)$) in $WC_2$ come in (bigonally related) pairs. We list all conjugacy classes of subgroups of $WC_2$ in the following diagram ($\sim$ denotes conjugate subgroups): \begin{equation} \renewcommand{\theequation}{\bf {\arabic{section}}.{\arabic{equation}}} \setcounter{equation}{8} \begin{diagram}[(fr) \sim] \node[2]{(1)} \arrow{sw,-} \arrow{s,-} \arrow{se,-} \\ \node{(f) \sim (fr^{2})} \arrow{s,-} \node{(r^{2})} \arrow{sw,-} \arrow{s,-} \arrow{se,-} \node{(fr) \sim (fr^{3})} \arrow{s,-} \\ \node{(f,r^{2})} \arrow{se,-} \node{(r)} \arrow{s,-} \node{(fr,r^{2})} \arrow{sw,-} \\ \node[2]{WC_{2}} \end{diagram} \end{equation} Correspondingly, we obtain the diagram of curves and maps of degree 2: \begin{equation} \renewcommand{\theequation}{\bf {\arabic{section}}.{\theau}.{\arabic{equation}}} \setcounter{au}{8} \setcounter{equation}{1} \begin{diagram}[\widetilde{C}] \node[2]{\doubletilde{C}} \arrow{sw} \arrow{s} \arrow{se} \\ \node{\widetilde{C}} \arrow{s} \node{C \times_{K} C'} \arrow{sw} \arrow{s} \arrow{se} \node{\widetilde{C}'} \arrow{s} \\ \node{C} \arrow{se} \node{C''} \arrow{s} \node{C'} \arrow{sw} \\ \node[2]{K} \end{diagram} \end{equation} Here the two sides are bigonally related. Note that $C'$ is $O(\widetilde{C})$; so if $\widetilde{C}$ is orientable (e.g. if $K={\bf P}^1$ and $g$ is unramified) then everything splits: $$\widetilde{C}'=C_0\amalg C_1\rightarrow K\amalg K=C',$$ $\widetilde{C}$ is Galois over $K$ with group $({\bf Z}/2{\bf Z})^2$ and quotients \[ \begin{diagram}[C_{0}] \node[2]{\widetilde{C}} \arrow{sw} \arrow{s} \arrow{se} \\ \node{C_{0}} \arrow{se} \node{C} \arrow{s} \node{C_{1}} \arrow{sw} \\ \node[2]{K} \end{diagram} \] (cf. [M1]), and (2.8.1) simplifies to: \begin{equation} \renewcommand{\theequation}{\bf {\arabic{section}}.{\theau}.{\arabic{equation}}} \setcounter{au}{8} \setcounter{equation}{2} \begin{diagram}[\widetilde{C} {\textstyle \amalg}] \node[2]{\widetilde{C} {\textstyle \amalg} \widetilde{C}} \arrow{sw} \arrow{s} \arrow{se} \\ \node{\widetilde{C}} \arrow{s} \node{C {\textstyle \amalg} C} \arrow{sw} \arrow{s} \arrow{se} \node{C_{0} {\textstyle \amalg} C_{1}} \arrow{s} \\ \node{C} \arrow{se} \node{C} \arrow{s} \node{K {\textstyle \amalg} K} \arrow{sw} \\ \node[2]{K} \end{diagram} \end{equation} Given an arbitrary branched double cover $\widetilde{C}\rightarrow C$, we form its Prym variety $$P(\widetilde{C}/C):={\rm Ker}^0({\rm Nm}:J(\widetilde{C})\rightarrow J(C)).$$ It is an abelian variety (for $C, \widetilde{C}$ non-singular), but in general not a principally polarized one. Nevertheless, there is a simple relationship between the bigonally-related Pryms $P(\widetilde{C}/C)$ and $P(\widetilde{C}'/C'):$ in the case $K={\bf P}^1$, Pantazis [P] showed that these abelian varieties are dual to each other. \subsection{The trigonal construction.} \ \ \ \ The case $n=3$ of our construction was discovered by Recillas [R]. Start with a tower $$\widetilde{C}\stackrel{\pi}{\rightarrow}C\stackrel{f}{\rightarrow}{ \bf P}^1$$ where $f$ has degree 3, and $\widetilde{C}\rightarrow C$ is an unramified double cover. By Corollaries (2.4) and (2.2), $f_*\widetilde{C}$ consists of two copies of a tetragonal curve $g:X\rightarrow{\bf P}^1$. Since $f$ and $g$ have the same branch locus by Lemma (2.3), we find from Hurwitz' formula: $${\rm genus}(X)={\rm genus}(C)-1.$$ All in all, we have constructed a map: $$T: \left\{ \begin{array}{c} {\rm trigonal \ curves \ } C {\rm \ of \ } \\ {\rm genus \ } g {\rm \ with \ a \ double \ cover \ } \widetilde{C} \end{array} \right\} \rightarrow \left\{ \begin{array}{c} {\rm tetragonal \ curves \ } \\ X {\rm \ of \ genus \ } g-1 \end{array} \right\}.$$ We claim that this map is a bijection (except that sometimes a nonsingular object on one side may correspond to a singular one on the other): given $g:X\rightarrow{\bf P}^1$, we recover $\widetilde{C}$ as the relative second symmetric product of $X$ over ${\bf P}^1$, $$\widetilde{C}:=S^2_{{\bf P}^1}X\rightarrow{\bf P}^1,$$ whose fiber over $p\in{\bf P}^1$ consists of all unordered pairs in $g^{-1}(p)$; this has an involution $\iota$ (=complementation of pairs), giving the quotient $C:=\widetilde{C}/\iota$. \begin{center} \begin{tabular}{cccc} \hspace{1in} & \hspace{1in} & \hspace{1in} & \hspace{1in} \\ \begin{picture}(30,30)(2,1) \thicklines \put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(26,1){$\circ$} \put(26,25.8){${\circ}$} \end{picture} & \begin{picture}(30,30)(2,1) \thicklines \put(2,1){$\circ$} \put(7,3.9){\line(1,1){22.4}} \put(2,25.8){$\circ$} \put(7,27.9){\line(1,-1){22}} \put(28,1){$\circ$} \put(28,25.8){${\circ}$} \end{picture} & \begin{picture}(30,30)(2,1) \thicklines \put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(4.5,6){\line(0,1){20}} \put(26,1){$\circ$} \put(28.5,6){\line(0,1){20}} \put(26,25.8){${\circ}$} \end{picture} & \begin{picture}(30,30)(2,1) \thicklines \put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}} \put(7,3.5){\line(1,0){20}} \put(26,1){$\circ$} \put(26,25.8){${\circ}$} \end{picture} \\ & & & \\ $X$ & \multicolumn{3}{c}{$S^{2}_{{\bf P}^{1}}X$ and its involution} \end{tabular} \end{center} In the group-theoretic setup of Remark (2.6), $\bar{\rho}$ induces an isomorphism $$WD_3\stackrel{\sim}{\rightarrow}S_4.$$ (This is the standard isomorphism, reflecting the isomorphism of the Dynkin diagrams $D_3, A_3$.) Recillas' map $T$ then corresponds to composition of a representation with this isomorphism. We list a few of the subgroups of $S_4$: \[ \begin{diagram}[\widetilde{C} {\textstyle \amalg}] \node[2]{\langle (1) \rangle} \arrow{sw,-} \arrow{se,-} \\ \node{\langle (12) \rangle} \arrow[2]{s,-} \arrow{se,-} \node[2]{\langle (12)(34) \rangle} \arrow{sw,-} \arrow{s,-} \\ \node[2]{\langle (12) , (34) \rangle} \arrow{s,-} \node{K} \arrow{s,-} \arrow{sw,-} \\ \node{S_{3}} \arrow{se,-} \node{D} \arrow{s,-} \node{A_{4}} \arrow{sw,-} \\ \node[2]{S_{4}} \end{diagram} \] $D$: The dihedral group $\langle\!(12), (1324)\!\rangle$ \\ $K=D\cap A_4$: The Klein group $\langle\!(12)(34), (13)(24)\!\rangle$. The corresponding curves are: \[ \begin{diagram}[\widetilde{C} {\textstyle \amalg}] \node[2]{X !} \arrow{sw,t}{2} \arrow{se,t}{2} \\ \node{Y} \arrow[2]{s,l}{3} \arrow{se,b}{2} \node[2]{Z} \arrow{sw,b}{2} \arrow{s,r}{2} \\ \node[2]{\widetilde{C}} \arrow{s,r}{2} \node{T !} \arrow{s,r}{3} \arrow{sw,b}{2} \\ \node{X} \arrow{se,b}{4} \node{C} \arrow{s,r}{3} \node{O} \arrow{sw,b}{2} \\ \node[2]{{\bf P}^{1}} \end{diagram} \] \noindent $O\approx O(X)\approx O(C)$: The orientation \\ $Y\approx(X\times_{{\bf P}^1}X)$ $\backslash$ (diagonal) \\ $Z\approx\widetilde{C}\times_{{\bf P}^1}O$. Using either of these constructions, we can easily describe the behavior of $X, C, \widetilde{C}$ around various types of branch points. Keeping $X$ non-singular, there are the following five possible local pictures, cf. [DS, III 1.4]. \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \cline{2-6} \multicolumn{1}{c|}{ } & (i) & (ii) & (iii) & (iv) & (v) \\ \cline{2-6} \hline & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ $X$ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ \hline & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ $\widetilde{C}$ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ \hline & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ $C$ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} \\ \hline \end{tabular} \end{center} \begin{center} \begin{tabular}{cc} \multicolumn{2}{c}{\Large\bf Legend} \\ \begin{tabular}{cl} \hspace{.35in} & \hspace{1.2in} \\ \hspace{.35in} &unramified sheet \\ \hspace{.35in} & \hspace{1.2in} \\ \hspace{.35in} & \hspace{1.2in} \\ \hspace{.35in} & simple ramification \\ \hspace{.35in} & \hspace{1.2in} \\ \hspace{.35in} & \hspace{1.2in} \\ \hspace{.35in} & node (two unramified \\ \hspace{.35in} & sheets glued together) \\ \hspace{.35in} & \hspace{1.2in} \\ \hspace{.35in} & two ramified sheets\\ \hspace{.35in} & glued together \\ \end{tabular} & \begin{tabular}{cl} \hspace{.35in} & \hspace{1.2in} \\ \hspace{.35in} & ramification point \\ \hspace{.35in} & of index 2 \\ \hspace{.35in} & \hspace{1.2in} \\ \hspace{.35in} & \hspace{1.2in} \\ \hspace{.35in} & ramification point \\ \hspace{.35in} & of index 3 \\ \hspace{.35in} & \hspace{1.2in} \\ \hspace{.35in} & \hspace{1.2in} \\ \hspace{.35in} & glueing of two sheets \\ \hspace{.35in} & of different \\ \hspace{.35in} & ramification indices \\ \end{tabular} \end{tabular} \end{center} \begin{list}{{\rm(\roman{butter})}}{\usecounter{butter}} \item $f, \pi, g$ are \'{e}tale.\item $f$ and $g$ have simple ramification points, $\pi$ is \'{e}tale. \item $f$ and $g$ each have a ramification point of index 2, $\pi$ is \'{e}tale. \item $g$ has two simple ramification points, $\pi$ is a Beauville cover: \\ $\bar{f}:N\rightarrow{\bf P}^1$ is trigonal, with a fiber $\{p, q, r\}; \ \bar{\pi}:\widetilde{N}\rightarrow N$ is branched at $p, q: \ \bar{\pi}^{-1}(p)=\widetilde{p}, \ \bar{\pi}^{-1}(q)=\widetilde{q}$; \ and we have $C=N/(p\sim q), \ \ \ \widetilde{C}=\widetilde{N}/(\widetilde{p}\sim\widetilde{q})$, \ \ and $\pi:\widetilde{C}\rightarrow C, \ \ \ f:C\rightarrow{\bf P}^1$ are induced by $\bar{\pi}, \bar{f}$. \item $g$ has a ramification point of index 3, $\pi$ is Beauville, $f$ is ramified at one of the two branches of the node of $C$. \end{list} \bigskip Considering first the first three cases, then all five, we conclude: \noindent{\bf Theorem 2.9} The trigonal construction gives isomorphisms $$T^0:{\cal R}^{{\rm Trig}}_g\stackrel{\sim}{\rightarrow}{\cal M}^{{\rm Tet}, 0}_{g-1}$$ and $$T:\bar{\cal R}^{{\rm Trig}}_g\stackrel{\sim}{\rightarrow}{\cal M}^{{\rm Tet}}_{g-1},$$ where: ${\cal M}^{{\rm Tet}}_{g-1}$ is the moduli space of (non-singular) curves of genus $g-1$ with a tetragonal line bundle. ${\cal M}^{{\rm Tet},0}_{g-1}$ is the open subset of tetragonal curves $X$ with the property that above each point of ${\bf P}^1$ there is at least one etale point of $X$. ${\cal R}^{{\rm Trig}}_g$ is the moduli space of etale double covers of non-singular curves of genus $g$ with a trigonal bundle. $\bar{\cal R}^{{\rm Trig}}_g$ is the partial compactification of ${\cal R}^{{\rm Trig}}_g$ using allowable covers in $\bar{\cal R}_g$ of type $\partial^{\rm III}$ (cf (1.9.III)). \bigskip \noindent{\bf Examples 2.10} \begin{list}{{\rm(\roman{butter})}}{\usecounter{butter}} \item $\widetilde{C}$ is the trivial cover, $\widetilde{C}=C_0\amalg C_1$, iff $X$ is disconnected, \\ $X={\bf P}^1\amalg C$, with $f=g|_C, \ \ \ id_{{\bf P}^1}=g|_{{\bf P}^1}$. \item Wirtinger covers $(C_0\amalg C_1) \ / \ (p_0\sim q_1, \ q_0\sim p_1)\rightarrow C/(p\sim q)$, where $\{p, q, r\}$ form a trigonal fiber in $C$, correspond to reducible $X={\bf P}^1\cup_rC$, the two components meeting at $r\in C$. \item $C$ is reducible: $C=H\cup{\bf P}^1$, with $H$ hyperelliptic, and \\ $\widetilde{C}=\widetilde{H}\cup{\bf P}^1$ with $\widetilde{H}\rightarrow H$ and $\widetilde{{\bf P}^1}\rightarrow{\bf P}^1$ branched over \\ $B:=H\cap{\bf P}^1$. This situation corresponds to $g:X\rightarrow{\bf P}^1$ factoring through a hyperelliptic $H'$. Indeed, such a pair $(C, \widetilde{C})$ is uniquely determined by the tower $\widetilde{H}\rightarrow H\rightarrow{\bf P}^1$. The trigonal construction for $C$ is reduced to the bigonal construction for $H$, which then gives $X=\widetilde{H}'\rightarrow H'\rightarrow{\bf P}^1$. In particular: \item $C=H\amalg{\bf P}^1$ is disconnected iff $X=H_0\amalg H_1$ is disconnected with hyperelliptic pieces, and then $\widetilde{C}=\widetilde{H}\amalg{\bf P}^1\amalg{\bf P}^1$, where $\widetilde{H}$ is the Cartesian cover: $$\widetilde{H}=H_0\times_{{\bf P}^1}H_1.$$ \end{list} So far, we have only used the fact that $\widetilde{C}$ is an orientable double cover of a triple cover. We now use our two assumptions, that $\pi$ is unramified and that the base $K$ equals ${\bf P}^1$, to obtain an identity of abelian varieties. Namely, by Remark 2.5 we have a map, natural up to translation. $$\alpha:X\rightarrow P(\widetilde{C}/C).$$ The result, due to S. Recillas, is: \bigskip \noindent{\bf Theorem 2.11} [R] If $X$ is trigonally related to $(\widetilde{C},C)$, then the above map $\alpha$ induces an isomorphism $$\alpha_*:J(X)\stackrel{\sim}{\rightarrow}P(\widetilde{C}/C).$$ \noindent{\bf Proof.} By naturality of $\alpha$ and irreducibility of ${\cal M}^{{\rm Tet}}_{g-1}$, it suffices to prove this for any one convenient $X$. We take $\widetilde{C}\rightarrow C$ to be a Wirtinger cover as in (2.10)(ii), so $$X={\bf P}^1\cup_rC'.$$ where $p+q+r$ is a trigonal divisor on $C'$, and $C=C'/(p\sim q)$. We have natural identifications: $$J(X)\approx J(C')\approx P(\widetilde{C}/C),$$ in terms of which $\alpha$ becomes the Abel-Jacobi map $\varphi$ on $C'$, and collapses ${\bf P}^1$ to a point.The induced $\alpha_*$ is therefore the identity. \begin{flushright} QED \end{flushright} \noindent{\bf Corollary 2.12} All trigonal Pryms are Jacobians, and all tetragonal Jacobians are Pryms. \subsection{The tetragonal construction} \ \ \ \ Consider now a tower $$\widetilde{C}\rightarrow C\stackrel{f}{\rightarrow}{\bf P}^1$$ where $f$ has degree 4 and $\widetilde{C}$ is a double cover (unramified) of $C$. The general construction yields a sequence of maps of degrees 2, 4, 2: $$f_*\widetilde{C}\rightarrow f_*\widetilde{C}/\iota\rightarrow\widetilde{\bf P}^1\rightarrow{\bf P}^1.$$By (2.2) and (2.4) again, $\widetilde{\bf P}^1$ is unramified, hence we have splittings: \[\begin{array}{rcl} \widetilde{\bf P}^1 & = & {\bf P}_0^1\amalg{\bf P}_1^1 \\ f_*\widetilde{C} & = & \widetilde{C}_0\amalg\widetilde{C}_1 \\ f_*\widetilde{C}/\iota & = & {C}_0\amalg {C}_1. \end{array} \] The tetragonal construction thus associates to a tower $$\widetilde{C}\stackrel{2}{\rightarrow}C\stackrel{4}{\rightarrow}{\b f P}^1$$ two other towers $$\widetilde{C}_i\rightarrow C_i\rightarrow{\bf P}^1, \ \ \ \ \ \ \ \ \ \ \ \ i=0, 1$$ of the same type. \bigskip \noindent{\bf Lemma 2.13} The tetragonal construction is a triality, i.e. starting with \ \ \ $\widetilde{C}_0\rightarrow C_0\rightarrow{\bf P}^1$ \ \ \ it returns \ \ \ $\widetilde{C}\rightarrow C\rightarrow{\bf P}^1$ \ \ \ and \\ $\widetilde{C}_1\rightarrow C_1\rightarrow{\bf P}^1$. \ \ \ \ On the group level, the point is this: Our tower $\widetilde{C}\rightarrow C\rightarrow{\bf P}^1$ corresponds to a representation (of $\pi_1({\bf P}^1\backslash$ (branch locus))) in $WD_4$. Now the Dynkin diagram $D_4$ has an automorphism of order 3: \vspace{3in} \noindent This corresponds to an outer automorphism of $WD_4$, of order 3. Hence representations in $WD_4$ come in packets of three. The various groups involved are described in some detail in the proof of Lemma (5.5), below. \bigskip \pagebreak[4] \noindent{\bf Local pictures 2.14} Given the local behavior of $C$ and $\widetilde{C}$ over a point of ${\bf P}^1$, it is quite straightforward to compute $f_*\widetilde{C}$ and hence $\widetilde{C}_i, C_i \ \ (i=0, 1)$ over the same point. Since these local pictures are needed quite frequently, we record the simplest ones here. \begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}} \item $C, \widetilde{C}$ unramified $\Rightarrow C_i, \widetilde{C}_i$ are also unramified. \item $C$ has one simple ramification point (and two unramified sheets), $\widetilde{C}\rightarrow C$ unramified $\Rightarrow C_i, \widetilde{C}_i$ have the same local picture as $C, \widetilde{C}$ respectively. \item $C$ has two distinct simple ramification points, $\widetilde{C}\rightarrow C$ unramified $\Rightarrow$ One pair, say $C_0, \widetilde{C}_0$, has the same local pictures as $C, \widetilde{C}$, while the other is a Beauville degeneration: $C_1$ is unramified but two of its four sheets are glued, $\widetilde{C}_1\rightarrow C_1$ is ramified over these two sheets (and the ramification points are glued) while the other sheets are unramified. \item $C$ is unramified but two of its sheets are glued, $\widetilde{C}\rightarrow C$ is ramified over these two sheets $\Rightarrow$ $C_i$ has two distinct ramification points, $\widetilde{C}_i\rightarrow C_i$ is unramified $(i=0, 1)$. (This is the same triple as in (3).) \item $C$ has a simple ramification point and the other two sheets are glued, $\widetilde{C}$ is ramified over the glued sheets $\Rightarrow C_i, \widetilde{C}_i$ have the same local pictures as $C, \widetilde{C}$. \item $C$ has a ramification point of index 2 (i.e. 3 of its sheets are permuted by the local monodromy), $\widetilde{C}\rightarrow C$ unramified $\Rightarrow$ same local picture for $\widetilde{C}_i\rightarrow C_i$. \item $C$ has a ramification point of index 3 (all 4 sheets permuted), $\widetilde{C}\rightarrow C$ unramified $\Rightarrow C_0, \widetilde{C}_0$ have the same local picture as $C, \widetilde{C}$, but $C_1$ has a simple ramification point glued to an unramified point, so $\widetilde{C}_1$ must be simply ramified over each. (I.e. $\widetilde{C}_1$ has a point which is simply ramified over ${\bf P}^1$, glued to a point which has ramification index 3 over ${\bf P}^1$!) \end{list} We note that in examples (3) and (7), the tetragonal construction applied to $(\widetilde{C}\rightarrow C)\in{\cal RM}_g$ produces an (allowable) degenerate cover, $(\widetilde{C}_1\rightarrow C_1)\in \partial^{\rm III}({\cal RM}_g)$. \begin{center} \begin{tabular}{ccc} \begin{tabular}{|c|c|c|} \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \end{tabular} & \begin{tabular}{|c|c|c|} \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \end{tabular} & \begin{tabular}{|c|c|c|} \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \end{tabular} \\ (1) & (2) & (3,4) \end{tabular} \end{center} \begin{center} \begin{tabular}{ccc} \begin{tabular}{|c|c|c|} \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \end{tabular} & \begin{tabular}{|c|c|c|} \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \end{tabular} & \begin{tabular}{|c|c|c|} \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \end{tabular} \\ (5) & (6) & (7) \end{tabular} \end{center} \begin{center} \begin{tabular}{c} \begin{tabular}{|c|c|c|} \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in}& \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ $\widetilde{C}$ & $\widetilde{C}_{0}$ & $\widetilde{C}_{1}$ \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ $C$ & $C_{0}$ & $C_{1}$ \\ \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline \hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ ${\bf P}^{1}$ & ${\bf P}^{1}$ & ${\bf P}^{1}$ \\ \hline \end{tabular} \\ (pattern) \end{tabular} \end{center} \pagebreak[4] \noindent{\bf Examples 2.15} \begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}} \item It is perhaps not terribly surprising that the trigonal construction is a degenerate case of the tetragonal construction.Start with \linebreak $\widetilde{C}\rightarrow C$ the split double cover of the curve $C$ with the tetragonal map $f:C\stackrel{4}{\rightarrow}{\bf P}^1$. Then $f_*\widetilde{C}$ splits into 5 components, of degrees 1, 4, 6, 4, 1 respectively over ${\bf P}^1$. The components of degree 4 make up $\widetilde{C}_1\rightarrow C_1$, which is isomorphic to $\widetilde{C}\rightarrow C$. The remaining components give $${\bf P}^1\amalg\widetilde{T}\amalg{\bf P}^1\rightarrow T\amalg{\bf P}^1$$ where $(\widetilde{T}, T)$ is associated to $C$ by the trigonal construction. Vice versa, starting with an (unramified) double cover \\ ${\bf P}^1\amalg\widetilde{T}\amalg{\bf P}^1$ of $T\amalg{\bf P}^1$, the tetragonal construction produces \linebreak $C\amalg C\rightarrow C$, twice. \item Let $p+q+r+s$ be a tetragonal divisor on $C$. Then $C/(p\sim q)$ is still tetragonal. Tacking a node onto the previous example, we see that the Wirtinger cover $$(C'\amalg C'')/(p'\sim q'', q'\sim p'')\rightarrow C/(p\sim q)$$ is taken by the tetragonal construction to : \noindent $\bullet$ Another Wirtinger Cover, $$(C'\amalg C'')/(r'\sim s'', s'\sim r'')\rightarrow C/(r\sim s),$$ and to: \noindent $\bullet$ ${\bf P}^1\cup_{t'}\widetilde{T}\cup_{t''}{\bf P}^1\rightarrow T\cup_t{\bf P}^1$, where $(\widetilde{T}, T)$ is associated by the trigonal construction to $C$. (Each copy of ${\bf P}^1$ meets $\widetilde{T}$ or $T$ in the unique point indicated. $t\in T$ corresponds to the partition $\{\{p, q\}, \{r, s\}\}$.) \item We will see in Lemma (3.5) that if $C\rightarrow{\bf P}^1$ factors through a hyperelliptic curve, so do $C_0, C_1$. An interesting subcase occurs when $C=H^0\cup H^1$ has two hyperelliptic components, cf. Proposition (3.6). \item Let $X$ be a non-singular cubic hypersurface in ${\bf P}^4$, $\ell\subset X$ a line, and $\widetilde{X}$ the blowup of $X$ along $\ell$, with projection from $\ell$: $$\pi:\widetilde{X}\rightarrow{\bf P}^2.$$ This is a conic bundle [CG] whose discriminant is a plane quintic curve $Q\subset{\bf P}^2$. The set of lines $\ell'\subset X$ meeting $\ell$ is a double cover $\widetilde{Q}$ of $Q$. Now choose a plane $A\subset{\bf P}^4$ meeting $X$ in 3 lines $\ell, \ell', \ell''$; we get 3 plane quinties $Q, Q', Q''$, with double covers $\widetilde{Q}, \widetilde{Q}', \widetilde{Q}''$. Note that $\ell', \ell''$ map to a point $p\in Q$, hence determine a tetragonal map $f:Q\rightarrow{\bf P}^1$, given by ${\cal O}_Q(1)(-p)$, and similarly for $Q', Q''$. Our observation is that the 3 objects $$(\widetilde{Q}, Q, f) \ \ ; \ \ (\widetilde{Q}', Q', f') \ \ ; \ \ (\widetilde{Q}'', Q'', f'')$$ are tetragonally related. Indeed, the 3 maps can be realized simultaneously via the pencil of hyperplanes $S\subset{\bf P}^4$ containing $A$. Such an $S$ meets $X$ in a (generally non-singular) cubic surface $Y$. $A$ line in $Y$ (and not in $A$) which meets $\ell'$, also meets 4 of the 8 lines (in $Y$, not in $A$) meeting $\ell$, one in each of 4 coplanar pairs.this gives the desired injection $\widetilde{Q}'\hookrightarrow f_*\widetilde{Q}$. \end{list} Our main interest in the tetragonal construction stems from: \bigskip \noindent{\bf Theorem 2.16} The tetragonal construction commutes with the Prym map, $$P(\widetilde{C}/C)\approx P(\widetilde{C}_0/C_0)\approx P(\widetilde{C}_1/C_1).$$ \bigskip \noindent{\bf Proof} \ \ \ \ As in Remark (2.5), we have a map $$\alpha:\widetilde{C}_i\hookrightarrow f_*\widetilde{C}\rightarrow{\rm Pic}(\widetilde{C}), \ \ \ \ \ \ \ \ \ \ i=0, 1.$$ The image sits in a translate of $P(\widetilde{C}/C)$, so we get induced maps $$\alpha_*:J(\widetilde{C}_i)\rightarrow P(\widetilde{C}/ C)$$ and by restriction $$\beta:P(\widetilde{C}_i/ C_i)\rightarrow P(\widetilde{C}/ C).$$ By Masiewicki's criterion [Ma], $\beta$ will be an isomorphism if we can show: \begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}} \item The image $\alpha(\widetilde{C}_i)$ of $\widetilde{C}_i$ in $P(\widetilde{C}/C)$ is symmetric; \item The fundamental class in $P(\widetilde{C}/C)$ of $\alpha(\widetilde{C}_i)$ is twice the minimal class, $\frac{2}{(g- 1)!}[\Theta]^{g-1}$. \end{list} Now (1) is clear, since the involution on $\widetilde{C}_i$ commutes with $-1$ in $P(\widetilde{C}/C)$. The fundamental class in (2) can be computed directly, as is done in [B2]. Instead, we find it here by a degeneration argument: it varies continuously with $(C, \widetilde{C})\in{\cal RM}^{{\rm Tet}}_g$, which is an irreducible parameter space, so it suffices to do the computation for a single $(C, \widetilde{C})$. We take $$C=T\cup_t{\bf P}^1, \ \ \ \widetilde{C}={\bf P}^1\cup_{t'}\widetilde{T}\cup_{t''}{\bf P}^1,$$ as in Example (2.15)(2). Then $(C_i, \widetilde{C}_i)$ is a Wirtinger cover, $i=0, 1$, and the normalization of $C_i$ is the tetragonal curve $N$ associated to $(T, \widetilde{T})$ by the trigonal construction. We have identifications $$J(N)\approx P(\widetilde{T}/T)\approx P(\widetilde{C}/C)$$ (Theorem (2.11)), in terms of which $\alpha(\widetilde{C}_i)$ consists of the Abel-Jacobi image $\varphi(N)\subset J(N)$ and of its image under the involution.Thus the fundamental class is twice that of $\varphi(N)$, as required. (Note: since this argurment works for any double cover $\widetilde{T}\rightarrow T$, and since any semiperiod on a nearby \mbox{non-singular} $C$ specializes to a semiperiod on $T\cup_t{\bf P}^1$ which is supported on $T$, we need only the irreducibility of ${\cal M}^{{\rm Tet}}_g$, instead of ${\cal RM}^{{\rm Tet}}_g$.) \begin{flushright} QED \end{flushright} \large \section{Bielliptic Pryms.} \ \ \ \ As a first application of the tetragonal construction, we show that some remarkable coincidences occur among the various loci in Beauville's list [B1]. The central role here is played byPryms of bielliptic curves. We see in (3.7), (3.8) that the bielliptic loci can be tetragonally related to various other components in Beauville's list, and therefore give the same Pryms. As suggested in [D1], this leads to a complete, short list of the irreducible components of the Andreotti-Mayer locus in genus $\leq 5$, and of its intersection with the image of the proper Prym map for arbitrary $g$. We do not include here the complete analysis of the Andreotti-Mayer locus itself, since this has already appeared in [Deb1] and [D5] (together with some corrections to the original list in [D1]). Nevertheless, we could not resist describing explicitly the operation of the tetragonal construction on Beauville's list, as it is such a pretty and straightforward application of the results of \S 2. \ \ \ \ We recall Mumford's results on hyperelliptic Pryms. Let $$f_i:C^i\rightarrow K, \ \ \ \ \ \ i= 0, 1$$ be two ramified double covers of a curve $K$. The fiber product $$\widetilde{C}:=C^0\times_KC^1$$ has 3 natural involutions:$\tau_i(i=0, 1)$, with quotient $C^i$, and \\ $\tau:=\tau_0\circ\tau_1$, with a new quotient, $C$. This all fits in a Cartesian diagram: \[ \begin{diagram}[C] \node[2]{\widetilde{C}} \arrow{sw,t}{\pi_{1}} \arrow{s,r}{\pi} \arrow{se,t}{\pi_{0}} \\ \node{C^{0}} \arrow{se,b}{f_{0}} \node{C} \arrow{s,r}{f} \node{C^{1}} \arrow{sw,b}{f_{1}} \\ \node[2]{K} \end{diagram} \] If the branch loci of $f_0, f_1$ are disjoint, then $$\pi:\widetilde{C}\rightarrow C$$ is unramified. We say that a double cover obtained this way is \linebreak \underline{Cartesian}. \bigskip \noindent{\bf Lemma 3.1} Let $f:C\rightarrow K$ be a ramified double cover. A double cover $$\pi:\widetilde{C}\rightarrow C,$$ given by a semiperiod $\eta\in J_2(C)$, is Cartesian if and only if \\ $f_*(\eta)=0\in J_2(K)$. \bigskip \noindent{\bf Proof:} apply the bigonal construction. \begin{flushright} QED \end{flushright} \bigskip \noindent{\bf Proposition 3.2} [M1] \begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}} \item Any double cover $\widetilde{C}$ of a hyperelliptic $C$ is Cartesian. \item Any hyperelliptic Prym is a product of 2 hyperelliptic Jacobians (one of which may vanish): If $\widetilde{C}$ arises as $C^0\times_{{\bf P}^1}C^1$ then $$P(\widetilde{C}/C)\approx J(C^0)\times J(C^1).$$ \end{list} \bigskip \noindent{\bf Proof:} (2) follows from (1), (1) follows from lemma (3.1) with $K = {\bf P}^{1}$. \begin{flushright} QED \end{flushright} \ \ \ \ A \underline{bielliptic} curve (aka elliptic-hyperelliptic, superelliptic, ...) is a branched double cover of an elliptic curve. In this section we apply the tetragonal construction to find various identities between bielliptic Pryms and Pryms of other, usually degenerate, curves. Some of the results extend to \underline{bihyperelliptic} curves, i.e. branched double covers of hyperelliptic curves. To warm up, we consider \underline{Jacobians} of bihyperelliptic curves. Example (2.10)(iii) can be restated: \bigskip {\bf Lemma 3.3} The trigonal construction gives a bijection between: \noindent $\bullet \ \ $ Bihyperelliptic, non singular curves $C$: \[ C \stackrel{f}{\rightarrow} H \stackrel{g}{\rightarrow} {\bf P}^{1}; \] $\bullet \ \ $ Reducible trigonal double covers $\widetilde{X} \rightarrow X$: \begin{center} \begin{tabular}{rlccc} $\begin{diagram}[X] \node{\widetilde{X}} \arrow{s} \\ \node{X} \end{diagram}$ & $\begin{diagram}[X] \node{=} \\ \node{=} \end{diagram}$ & $\begin{diagram}[X] \node{C'} \arrow{s} \\ \node{H'} \end{diagram}$ & $ \begin{diagram}[X] \node{\cup} \\ \node{\cup} \end{diagram} $& $\begin{diagram}[X] \node{H} \arrow{s} \\ \node{{\bf P}^{1}} \end{diagram}$ \end{tabular} \end{center} where $X = H' \cup {\bf P}^{1}$ is reducible $\tau : X \rightarrow {\bf P}^{1}$, the trigonal map, has degree 2 on $H'$ and 1 \linebreak on ${\bf P}^{1}$. $\tau(H' \cap {\bf P}^{1})$ = Branch($g$) $\widetilde{X} \rightarrow X$ is allowable of type $\partial^{\rm III}$ at each point of $H' \cap {\bf P}^{1}$. \bigskip \noindent We note that $C' \rightarrow H' \rightarrow {\bf P}^{1}$ is bigonally related to $C \rightarrow H \rightarrow {\bf P}^{1}$. \bigskip \noindent{\bf Corollary 3.4} The Jacobian of a bihyperelliptic curve $C$, $$C\stackrel{f}{\rightarrow}H\stackrel{g}{\rightarrow}{\bf P}^1,$$ is isogenous to the product $$J(H)\times P(g_*C, \iota)$$ of a hyperelliptic Jacobian and a bihyperelliptic (branched) Prym. \bigskip \ \ \ \ We move to the Pryms of bihyperelliptic curves. First we note that this class is closed under the tetragonal construction: \bigskip \noindent{\bf Lemma 3.5} \ \ Let $(\widetilde{C}_i, C_i)$ be tetragonally related to $(\widetilde{C}, C)$, with $C$ non-singular. If $C\rightarrow{\bf P}^1$ factors through a (possibly reducible) hyperelliptic $H$, so do the $C_i$: $$C_i\stackrel{f_i}{\rightarrow}H_i\stackrel{g_i}{\rightarrow}{\bf P}^1, \ \ \ \ \ \ i=0, 1.$$ \bigskip \noindent{\bf Proof.} The bigonal construction applied to $$\widetilde{C}\stackrel{\pi}{\rightarrow}C\stackrel{f}{\rightarrow}H $$ yields $$f_*\widetilde{C}\rightarrow\widetilde{H}\rightarrow H,$$ and when applied again to $$\widetilde{H}\rightarrow H\stackrel{g}{\rightarrow}{\bf P}^1$$ yields $$g_*\widetilde{H}\rightarrow\widetilde{{\bf P}}^1\rightarrow{\bf P}^1.$$ Since $\pi$ is unramified, so are $\widetilde{H}\rightarrow H$ and $\widetilde{\bf P}^1\rightarrow{\bf P}^1$. Hence $\widetilde{\bf P}^1$ splits: $$\widetilde{\bf P}^1={\bf P}^1_0\amalg{\bf P}^1_1,$$ and this splitting climbs its way up the tower: \[ \begin{array}{rlc} (g\circ f)_{*}\widetilde{C} & = & \widetilde{C}_{0} {\textstyle \amalg} \widetilde{C}_{1} \\ & & \downarrow \\ & & C_{0} {\textstyle \amalg} C_{1} \\ & & \downarrow \\ g_{*}\widetilde{H} & = & {H}_{0} {\textstyle \amalg} {H}_{1} \\ & & \downarrow \\ \widetilde{\bf P}^{1} & = & {\bf P}_{0}^{1} {\textstyle \amalg} {\bf P}_{1}^{1} \\ & & \downarrow \\ & & {\bf P}^{1} \end{array} \] \begin{flushright} QED \end{flushright} \bigskip \noindent {\bf Remark 3.5.1} The rational map $f_{i} : C_{i} \rightarrow H_{i}$ can, in a couple of cases, fail to be a morphism; this is easily remedied by identifying a pair of points in $H_{i}$. Among the local pictures (2.14), the ones that can occur here are (1), (2), (7) and (3) : \begin{itemize} \item In cases (1), (2), the hyperelliptic maps $g$, $g_{0}$, $g_{1}$ are all unramified at the relevant point, and the $f_{i}$ are morphisms. \item In case (7), $g$ and $g_{0}$ are ramified, $g_{1}$ is not, $f$ and $f_{0}$ are (ramified) morphisms, but $f_{1}$ is not, since $C_{1}$ is singular above a point where $H_{1}$, as constructed above, is nonsingular. To make $f_{1}$ into a morphism, we must glue the two points of $g_{1}^{-1}(k)$. \item In case (3) we find two possibilities: \begin{list}{(3\alph{bean})}{\usecounter{bean}} \item $g$ is etale, $f$ is ramified at both points of $g^{-1}(k)$; then $g_{0}$, $g_{1}$ are also etale, $f_{0}$ is ramified at both points of $g_{0}^{-1}(k)$, $C_{1}$ has a node but $f_{1}$ is still a morphism. \item $g$ is ramified, $f$ is etale; then $g_{0}$ is ramified, $f_{0}$ is etale, $g_{1}$ is etale, but the two branches of the node of $C_{1}$ are sent by $f_{1}$ to opposite sheets of $H_{1}$, so $f_{1}$ is again not a morphism. \end{list} \end{itemize} \noindent{\bf Proposition 3.6} Let $\widetilde{C}\rightarrow C$ be a Cartesian double cover of a bihyperelliptic $C$: $$C\stackrel{f}{\rightarrow}H\stackrel{g}{\rightarrow}{\bf P}^1, \; \widetilde{C} = C^{0}\times_{H} C^{1}.$$ The tetragonal construction applied to $\widetilde{C}\rightarrow C\rightarrow{\bf P}^1$ yields: \noindent $\bullet$ A similar Cartesian tower $\widetilde{C}_0\rightarrow C_0\stackrel{f_0}{\longrightarrow}H\stackrel{g_0}{\longrightarrow }{\bf P}^1$, same $H$. \noindent $\bullet$ A tower $\widetilde{C}_1\rightarrow C_1\rightarrow{\bf P}^1$ where: \begin{tabbing} XXXXX \= \kill \>$C_1$ is reducible, $C_1=H^0\cup H^1$, \\ \> $H^0, H^1$ are hyperelliptic, \\ \> $H^0\cap H^1$ maps onto $B:={\rm Branch}(g)\subset{\bf P}^1$, \\ \> $\widetilde{C}_1=\widetilde{H}^0\cup\widetilde{H}^1$ is allowable over $C_1$, \\ \> $C^{i} \rightarrow H \rightarrow {\bf P}^{1}$ is bigonally related to $\widetilde{H}^{i} \rightarrow H^{i} \rightarrow {\bf P}^{1}$, $i$ = 1,2. \end{tabbing} Vice versa, the tetragonal construction takes any tower \linebreak $\widetilde{C}_1\rightarrow C_1\rightarrow{\bf P}^1$ as above to two Cartesian bihyperelliptic towers $$\widetilde{C}\rightarrow C\rightarrow H\rightarrow{\bf P}^1 \ \ \ {\rm and} \ \ \ \widetilde{C}_0\rightarrow C_0\rightarrow H\rightarrow{\bf P}^1.$$ The proof is quite straightforward, and we will simply write down a few of the relationships involved, using the notation of the previous proof: \noindent $\bullet$ $\widetilde{H}$ splits into two copies of $H$, by (3.1).Hence: \noindent $\bullet$ $g_*\widetilde{H}\approx H\cup {\bf P}^1\cup {\bf P}^1$, say $H_0\approx H, \ \ H_1 = R^{0} \cup R^{1}, \; R^{i} \approx {\bf P}^{1}$, $i$ = 0, 1. \begin{tabbing} \noindent $\bullet$ \= Let $H^{i}$, $\widetilde{H}^{i}$ be the inverse image of $R^{i}$ in $C_{1}$, $\widetilde{C}_{1}$ respectively. Then \\ \> $\widetilde{H}^{i} \rightarrow H^{i} \rightarrow {\bf P}^{1}$ is bigonally related to $C^{i} \rightarrow H \rightarrow {\bf P}^{1}$. \end{tabbing} \begin{tabbing} \noindent $\bullet$ \=The intersection properties of the $H^i$ (or $\widetilde{H}^i$) can be read off the \\ \>local pictures (2.14.3). \end{tabbing} \begin{tabbing} \noindent $\bullet$ \=Finally, let $\varepsilon:H\to H$ be the hyperelliptic involution. A cover \\ \>$C^1\to H$ determines a mirror-image $\varepsilon^*C^1$. The remaining tower \\ \>$\widetilde{C}_0\to C_0\to H\to{\bf P}^1$ is given by the Cartesian diagram: \end{tabbing} \[ \begin{diagram}[C] \node[2]{\widetilde{C}_{0}} \arrow{sw} \arrow{s} \arrow{se} \\ \node{C^{0}} \arrow{se} \node{C_{0}} \arrow{s} \node{\varepsilon^{*}C^{1}} \arrow{sw} \\ \node[2]{H} \end{diagram} \] \begin{flushright} QED \end{flushright} {\bf Remarks} \noindent {\bf (3.6.1)} Since the branch points of $C^{i} \rightarrow H$ map to the branch points of $H^{i} \rightarrow {\bf P}^{1}$, we have the relation between the genera: \[ g(H^{i}) = g(C^{i}) - 2\cdot g(H). \] \noindent {\bf (3.6.2)} The possible local pictures are exactly the same as in (3.5.1). (The use of $C_{0}$, $C_{1}$ in (3.6) is consistent with that of (2.14).) \noindent {\bf (3.6.3)} Another way of proving both lemma (3.5) and proposition (3.6) is based on lemma (5.5), which says that the three tetragonal curves $C, C_{0}, C_{1}$ which are tetragonally related are obtained, via the trigonal construction, from one and the same trigonal curve $X$ (with three distinct double covers). Lemma (3.3) characterizes the possible curves $X$, hence proves that the locus of bihyperelliptics is closed under the tetragonal construction, lemma (3.5). To complete the proof of proposition (3.6), one simply needs to characterize the double covers $\widetilde{X}$ which correspond to Cartesian covers of $C$. \ \ \ \ For the rest of this section, we specialize to the case where the hyperelliptic $H$ is an elliptic curve $E$, i.e. $C$ is bielliptic. First, we write out explicitly the content of Proposition (3.6) in this case: \noindent{\bf Corollary 3.7} The Pryms of double covers $\pi :\widetilde{C}\to C$ where \noindent $\bullet$ $C$ is bielliptic, $C\stackrel{f}{\to}E\stackrel{g}{\to}{\bf P}^1$, \noindent $\bullet$ $\widetilde{C}\to C$ is Cartesian, $\widetilde{C}=C^0\times_EC^1, \ \ \ C^0$ is of genus $n$, \noindent are precisely (via the tetragonal construction) the Pryms of the following allowable double covers $\widetilde{X}\to X$: \begin{tabbing} $n=1$: \ \ \=$X$ is obtained from a hyperelliptic curve by identifying \\ \>two pairs of points, $X=H/(p\sim q, \ \ r\sim s)$. \\ \\ $n=2$:\>$X=X_0\cup X_1, \ \ X_0$ rational, $X_1$ hyperelliptic, \\ \>$\#(X_0\cap X_1)\!=\!4$. \\ \\ $n\ge 3$:\>$X=X_0\cup X_1$, \ each $X_i$ hyperelliptic, \ $g(X_0)=n-2$, \\ \>$g(X_1)=g(C)-n-1, \ \ \#(X_0\cap X_1)=4$, and both \\ \>hyperelliptic maps are restrictions of the same tetragonal \\ \>map on $X$ (i.e. they agree on $X_0\cap X_1$). \end{tabbing} \ \ \ \ Everything here follows directly from the proposition, except that for $n=1$ we need to use (twice) the following observation of Beauville. Let $\pi:\widetilde{X}\to X$ be an allowable double cover where $$X=Y\cup R, \ \ \ \ \ \ \ \ R \;\mbox{rational}, \ \ \ \ \ \ \ \ \ Y\cap R=\{a,b\}$$ $$\widetilde{X}=\widetilde{Y}\cup\widetilde{R}, \ \ \widetilde{R}=\pi^{-1}(R)\;\mbox{rational}, \ \ \widetilde{Y}\cap\widetilde{R}=\{\widetilde{a},\widetilde{b}\}$$ and $\pi$ is ramified at $\widetilde{a}, \widetilde{b}$, which map to $a,b$. Construct a new cover $\widetilde{Z}\to Z$ where $$\widetilde{Z}:=\widetilde{Y}/(\widetilde{a}\sim\widetilde{b})$$ $$Z:=Y/(a\sim b).$$ Then this is still allowable, and $$P(\widetilde{Z}/Z)\approx P(\widetilde{X}/X).$$ (Indeed, there are natural isomorphisms of generalized Jacobians $$J(\widetilde{Z})\approx J(\widetilde{X}), \;\;J(Z)\approx J(X)$$ commuting with $\pi_*$ and inducing the desired isomorphisms.) \begin{flushright} QED \end{flushright} \ \ \ \ We are left with the Pryms of non-Cartesian double covers of bielliptic curves. The result here may be somewhat surprising: \bigskip \noindent{\bf Proposition 3.8} Pryms of \underline{non}-Cartesian double covers of bielliptic curves are precisely the Pryms of Cartesian covers (of bielliptic curves) with $n(:=g(C_0))=1.$ (The isomorphism is obtained through a sequence of 2 tetragonal moves.) \ \ \ \ The point is that if $X=H/(p\sim q, \ \ r\sim s)$ with $H$ hyperelliptic, and $\widetilde{X}\to X$ is an allowable double cover, then $P(\widetilde{X}/X)$ is the Prym of a Cartesian cover (with $n=1$) of a bielliptic curve, as we've just seen; but $X$ has another $g^1_4$, and applying the tetragonal construction to it yields a non-Cartesian double cover of a bielliptic curve. \ \ \ \ The $g^1_4$ is obtained as follows: map $H$ to a conic in ${\bf P}^2$ (by the hyperelliptic map), then project the conic to ${\bf P}^1$ from the unique point $x$ in ${\bf P}^2$ (and not on the conic) on the intersection of the lines $\overline{pq}$ and $\overline{rs}$. \vspace{2in} \ \ \ \ We should now check that the tetragonal construction yields a non-Cartesian cover of a bielliptic curve, and that all covers arise this way. We leave the former to the reader, and do the latter. \ \ \ \ Let $\widetilde{C}\to C$ be a non-Cartesian cover of $C$, which is bielliptic: $$ C\stackrel{f}{\to}E\stackrel{g}{\to}{\bf P}^1.$$ \noindent Let $(\widetilde{C}_i,C_i)$, \ \ $i=0,1$, be the tetragonally related covers. By lemma (3.5), $C_i$ is bihyperelliptic:$$C_i\stackrel{f_i}{\to}H_i\stackrel{g_i}{\to}{\bf P}^1.$$ By the local pictures (2.14), $$B :=\mbox{Branch}(g)=B_0\amalg B_1, \ \ \ B_i:=\mbox{Branch}(g_i).$$ (As we saw in Remark (3.5.1), the possible pictures are (1), (2), (7), (3a) and (3b). Of these, (7) and (3b) contribute to $B$, and each contributes also to one of the $B_{i}$.) Since $\#B=4$ \ \ ($E$ is elliptic), and $\# B_i$ is even and $>0$ (non-Cartesian!), we find $$\# B_i=2, \;\;\; i=0,1,$$ hence $H_i$ is rational and $C_i$ is hyperelliptic. Again by the local pictures, $C_i$ will have two nodes, at points lying over $B_{1-i}$. \begin{flushright} QED \end{flushright} \ \ \ \ We observe that the last argument works not only for bielliptics but also for branched double covers of hyperelliptic curves of genus 2, since now $$\#B_0>0, \ \ \#B_1>0, \ \ \#B_0+\#B_1=6, \ \ \#B_i\; \mbox{even}\Rightarrow$$ $$\mbox{either} \ \;\#B_0=2 \ \mbox{or} \ \#B_1=2.$$ However, the resulting hyperelliptic curve with 4 nodes does not carry other $g^1_4$'s and is not necessarily related to any other covers. \ \ \ \ We leave one more corollary of proposition (3.6) to the reader. \bigskip \noindent{\bf Corollary 3.9} Let $K$ be hyperelliptic, $\widetilde{K}\to K$ a double cover with 2 branch points. Then $P(\widetilde{K}/K)$ is a hyperelliptic Jacobian. \noindent(Hint: take both $H$ and $C^0$ in proposition (3.6) to be rational, show $P(\widetilde{C}_1/C_1)\approx J(C^1)$ and $C_1=K\cup_{(2\; \mbox{points})}{\bf P}^1, \ K$ hyperelliptic.) \large \section{Fibers of ${\cal P}:{\cal R}_6\to{\cal A}_5.$} \subsection{The structure} \ \ \ \ We recall the main result of [DS]: \bigskip \noindent{\bf Theorem 4.1 [DS]} ${\cal P}:{\cal R}_6\to{\cal A}_5$ is generically finite, of degree 27. Recall that ${\cal M}_6^{\rm Tet}$ denotes the moduli space of curves of genus 6 with a $g^1_4$. The forgetful map ${\cal M}_6^{\rm Tet}\to{\cal M}_6$ is generically finite, of degree 5 [ACGH]. By base change we get a corresponding object ${\cal R}_6^{\rm Tet}$, with map $${\cal R}_6^{\rm Tet}\to{\cal R}_6$$ of degree 5. The tetragonal construction gives a triality, or (2,2)-corres\-pon\-den\-ce, on ${\cal R}_6^{\rm Tet}$. The image in ${\cal R}_6$ is then a (10,10)-correspondence: \medskip \noindent{\bf (4.1.1)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ${\rm Tet}\subset{\cal R}_6\times{\cal R}_6.$ \bigskip \noindent{\bf Theorem 4.2} The correspondence Tet induced by the tetragonal construction on the fiber ${\cal P}^{-1}(A)$, for generic $A\in{\cal A}_4$, is isomorphic to the incidence correspondence on the 27 lines on a non-singular cubic surface. The monodromy group of ${\cal R}_6$ over ${\cal A}_5$ (i.e. the Galois group of its Galois closure) is the Weyl group $WE_6$, the symmetry group of the incidence of the 27 lines on the cubic surface. This was conjectured in [DS] and announced in [D1]. The proof will be given below. For the symmetry group of the line incidence on a cubic surface, or other del Pezzo surfaces, we refer to [Dem]. \bigskip \noindent{\bf (4.3) The blownup map} Let ${\cal Q}\subset{\cal M}_6$ denote the moduli space of non-singular plane quintic curves, ${\cal R\cal Q}$ its inverse image in ${\cal R}_6$. By Theorem (1.2), it splits: $${\cal R\cal Q}={\cal R\cal Q}^+ \ \ \cup \ \ {\cal R\cal Q}^-$$ with $(Q,\mu)\in{\cal R\cal Q}^+$ (respectively, ${\cal R\cal Q}^-$) iff $h^0(\mu\otimes{\cal O}_Q (1)$) is even (respectively, odd). The point is that ${\cal O}_Q(1)$ gives a uniform choice of theta characteristics over ${\cal Q}$, hence the spaces of theta characteristics and semiperiods over ${\cal Q}$ are identified. Let ${\cal J}$ be the closure in ${\cal A}_5$ of the locus of Jacobians of curves, and let ${\C}$ denote the moduli space of non-singular cubic threefolds. Via the intermediate Jacobian map, we identify ${\C}$ with its image in ${\cal A}_5$. The Prym map sends ${\cal R\cal Q}^+$ to ${\cal J}$ and ${\cal R\cal Q}^-$ to ${\C}$. Since the fiber dimensions can be positive, it is useful to consider the blownup Prym map $$\widetilde{\cal P}:\widetilde{\cal R}_6\to\widetilde{\cal A}_5$$ where ${\cal J},{\C}$ on the right are blown up to divisors $\widetilde{\cal J},\widetilde{\C}$, while on the left we blow up ${\cal R\cal Q}^+,{\cal R\cal Q}^-$, as well as the locus ${\cal R}_6^{Trig}$ of double covers of trigonal curves. The result is a morphism which is generically finite over $\widetilde{\cal J}$ and $\widetilde{\C}$. We recall the geometric description of points of the various loci, and give the map in these geometric terms. This is taken from [CG], [T] and [DS]. \bigskip \noindent{\bf (4.3.1)} \ A point of ${\C}$ is given by a non-singular cubic threefold $X\subset{\bf P}^4$. A point of $\widetilde{\C}$ is given by a pair $(X,H), \ H\in({\bf P}^4)^*$ a hyperplane. \bigskip \noindent{\bf (4.3.2)} \ A point of $\widetilde{{\cal R\cal Q}}$ is given by ($Q, \mu, L$), or $(Q, \widetilde{Q}, L)$, where $Q\subset{\bf P}^2$ is a plane quintic, $L\in({\bf P}^2)^*$ a line, and $\mu$ a semiperiod on $Q$ (or $\widetilde{Q}$ the corresponding double cover). \bigskip \noindent{\bf (4.3.3)} \ The fiber ${\cal P}^{-1}(J(X))\subset{\cal R\cal Q}^-$ over a cubic threefold $X$ can be identified with the Fano surface $F(X)$ of lines $\ell\subset X$.(Projection from $\ell$ puts a conic bundle structure $\pi: X --\!\to{\bf P}^2={\bf P}^4/\ell$ on $X$; the corresponding point of ${\cal R\cal Q}^-$ is $(Q,\widetilde{Q})$, where the plane quintic $Q$ is the discriminant locus of $\pi$, and its double cover $\widetilde{Q}$ parametrizes lines $\ell'\in F(X)$ meeting $\ell$.) \bigskip \noindent{\bf (4.3.4)} \ The fiber $\widetilde{\cal P}^{-1}(X,H)$ corresponds to the lines $\ell$ in the cubic surface $X\cap H$.For general $X,H$, there are 27 of these. The corresponding objects are of the form $(Q,\widetilde{Q},L)$ where $(Q,\widetilde{Q})$ are as above, and $L\subset{\bf P}^2$ is the projection, $L=\pi(H)$. \bigskip \noindent{\bf (4.3.5)} \ A point of ${\cal R}_6^{Trig}$ is given by a curve $T\in{\cal M}_6$ with a trigonal line bundle ${\cal L}\in\mbox{Pic}^3(T),h^0({\cal L})=2$, and a double cover $\widetilde{T}\to T$. The fiber of $\widetilde{\cal R}_6^{Trig}$ above it is given by the linear system $|\omega_T\otimes{\cal L}^{-2}|$, a ${\bf P}^1$. \bigskip \noindent{\bf (4.3.6)} \ A point of ${\cal J}$ is given by the Jacobian of a curve $C\in{\cal M}_5$. The canonical curve $\Phi(C)\subset{\bf P}^4$, for general $C$, is the base locus of a net of quadrics: $$A_p\subset{\bf P}^4, \;\;\;\;\;\;p\in{\bf P}^2={\bf P}^2(C).$$ A point of $\widetilde{\cal J}$ above $C$ is then given by a pair $(C,L)$, where $L$ is a line in ${\bf P}^2(C)$.(Choosing such a line is the same as choosing a quartic del Pezzo surface $$S=S_L=\cap_{p\in L}A_p$$ containing $\Phi(C)$.) \bigskip \noindent{\bf (4.3.7)} \ Consider the map $$\alpha:{\cal M}_5\rightarrow{\cal R\cal Q}^+$$ sending $C\in{\cal M}_5$ to $\alpha(C)=(Q, \widetilde{Q})$, where: $$Q:=\{p\in{\bf P}^2(C) \ | \ A_p {\rm \ is \ singular}\}\subset{\bf P}^2(C),$$ and $\widetilde{Q}$ is the double cover whose fiber over a general $p\in Q$ corresponds to the two rulings on the rank-4 quadric $A_p$. This $\alpha$ is a birational isomorphism; its inverse is the restriction to ${\cal R\cal Q}^+$ of ${\cal P}$. The fiber $\widetilde{\cal P}^{-1}(C, L)$ over generic $(C, L)\in\widetilde{\cal J}$ is given by the following 27 objects: \noindent$\bullet$ The quintic object$(Q,\widetilde{Q},L)\in\widetilde{{\cal R\cal Q}}^+$, where$(Q,\widetilde{Q})=\alpha(C)$ and $L$ is the given line in ${\bf P}^2(C)$. \noindent$\bullet$ Ten trigonals $T^\varepsilon_i, \ 1\le i\le 5, \ \ \varepsilon=0, 1$, each with a double cover $\widetilde{T}^\varepsilon_i$: each of the 5 points $p_i\in Q\cap L$ determines two $g^1_4$'s on $C$, cut out by the rulings $R^\varepsilon_i$ on $A_{p_{i}}$, and the $(T^\varepsilon_i, \widetilde{T}^\varepsilon_i)$ are associated to these by the trigonal construction. \noindent$\bullet$ Sixteen Wirtinger covers $(X_j, \widetilde{X}_j)\in\partial^I{\cal R}_6$: the quartic del Pezzo surface $S_L$ contains 16 lines $\ell_j$ [Dem], each meeting $\Phi(C)$ in two points, say $p_j, q_j$, and then $$X_j=C/(p_j\sim q_j)$$ and $\widetilde{X}_j$ is its unique Wirtinger cover (1.9.I). \bigskip \noindent{\bf (4.3.8)} \ We observe that the generically finite map \ \ ${\cal R}_6^{\mbox{Tet}}\to{\cal R}_6$ \ \ has \\ 1-dimensional fibers over both ${\cal R\cal Q}$ and ${\cal R}^{\mbox{Trig}}_6$. After blowing up and normalizing, we obtain finite fibers generically over the exceptional loci. In the limit: \noindent$\bullet$ Over $(Q,L)$, the 5 $g^1_4$'s correspond to projections of the plane quintic $Q$ from one of the 5 points $p_i\in Q\cap L$. \noindent$\bullet$ Over $(T,D)$, with $T$ a trigonal curve, ${\cal L}$ the trigonal bundle, and \\ $D\in|\omega_T\otimes{\cal L}^{-2}|$, four of the $g^1_4$'s are of the form ${\cal L}(q)$ with $q\in D$ (i.e. they are the trigonal ${\cal L}$ with base point $q$); the fifth $g^1_4$ is $\omega_T\otimes{\cal L}^{-2}$. \noindent$\bullet$ Given $X=C/(p\sim q)\in\partial\bar{\cal M}_5$, there is a pencil $L\subset{\bf P}^2(C)$ of quadrics $A_p, \ p\in L$, which contain both $\Phi(C)$ and its chord $\overline{pq}$.Among these there are 5 quadrics $A_{p_{i}}$ which are singular, generically of rank 4. Each of these has a single ruling $R_i$ containing a plane containing $\overline{pq}$. These $R_i$ cut the 5 $g^1_4$'s on $X$. We conclude that the tetragonal correspondence Tet of (4.1.1) lifts to $$\widetilde{\rm Tet}\subset\widetilde{\cal R}_6 \times \widetilde{\cal R}_6$$ which is generically finite, of type (10,10), over each of our special loci. \bigskip \noindent {\bf Theorem 4.4 \ Structure of the blownup Prym map.} Over each of the following loci, the blownup Prym map $\widetilde{\cal P}$ has the listed monodromy group, and the lifted tetragonal correspondence $\widetilde{\rm Tet}$ induces the listed structure. \begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}} \item $\widetilde{\C}$: The group is $WE_6$, the structure is that of lines on a general non-singular cubic surface. \item $\widetilde{\cal J}$: The group is $WD_5$, the symmetry group of the incidence of lines on a quartic del Pezzo surface, or stabilizer in $WE_6$ of a line. The structure is that of lines on a non-singular cubic surface, one of which is marked. \item ${\cal B}$= the locus of intermediate Jacobians of Clemens' quartic double solids of genus 5 \ [C1]: The group is $WA_5=S_6$, the structure is that of lines on a nodal cubic surface. [Note: ${\cal B}$ is contained in the branch locus of ${\cal P}$ [DS, V.4] and in fact ([D6], and compare also [SV], [I]) equals the branch locus. The monodromy along ${\cal B}$ acting on a nearby, unramified fiber is $({\bf{Z}\rm} \newcommand{\C}{\cal C}/2{\bf{Z}\rm} \newcommand{\C}{\cal C})\times S_6$, or the symmetry group of a double-six, which is a subgroup of $WE_6$. The group $S_6$ thus occurs as a subquotient of $WE_6$.] \item (cf. [I]) $\widetilde{\cal P}$ extends naturally to the boundary $\partial=\partial{\cal A}_5$; the monodromy is $WE_6$ and the structure is that of lines on a general cubic surface. \end{list} We will prove parts (1), (2) and (3) in \S4.2. In the rest of this section we show that theorems (4.1) and (4.2) follow from (4.4). \bigskip \noindent{\bf (4.5) Proofs of Theorem (4.2).} By Theorem (2.16), Tet commutes with ${\cal P}$, therefore $\widetilde{\rm Tet}$ commutes with $\widetilde{\cal P}$. To identify this structure over a generic point, it suffices to do so over any point over which $\widetilde{\cal P}$ and $\widetilde{\rm Tet}$ are etale. These conditions hold, e.g., over a generic $(X,H)\in\widetilde{\C}$, where (4.4.1) identifies the structure.This implies that the monodromy is contained in $WE_6$, but we get all of $WE_6$ already over $\widetilde{\C}$ (by (4.4.1) again), so we are done. We can work instead over $\widetilde{\cal J}$: again, $\widetilde{\cal P}$ and $\widetilde{\rm Tet}$ are etale over generic $(C, L)\in\widetilde{\cal J}$, and $\widetilde{\rm Tet}$ has the right structure there by (4.4.2). This shows $$WD_5\subset \mbox{Monodromy} \subset WE_6.$$ As there are no intermediate groups, the monodromy must equal $WD_5$ or $WE_6$.But if it were the former, $\widetilde{\cal R}_6$ would be reducible (since $WD_5$ is the stabilizer in $WE_6$ of one of the 27 lines), contradiction. \begin{flushright} QED \end{flushright} \bigskip \noindent{\bf Remark 4.5.1} \ Along the same lines, we can also reprove Theorem (4.1). Let $\widetilde{\rm Tet}^i$ denote the $i$-th iterate of the correspondence $\widetilde{\rm Tet}$. On ${\cal R\cal Q}^-$ we have: $$\widetilde{\rm Tet}^2 \mbox{ \ has degree} \ 27, $$ $$\widetilde{\rm Tet}^i=\widetilde{\rm Tet}^2 \;\;\;\mbox{for}\; i\ge 2.$$Since $\widetilde{\rm Tet}$ is etale there, these properties persist generically on $\widetilde{\cal R}_6$. Let $\sim$ be the equivalence relation generated by $\widetilde{\rm Tet}$. We conclude that $\sim$ has degree 27, and that $\widetilde{\cal P}$ factors through a proper quotient: $${\cal P}':\widetilde{\cal R}_6/\sim\longrightarrow\widetilde{\cal A}_5.$$ We still need to verify that $\deg ({\cal P}')=1$. There are several possibilities: \noindent$\bullet$ We can work over $\widetilde{\cal J}$; as we will see in (4.7), the fiber of $\widetilde{\cal P}$ there consists of aunique $\sim$-equivalence class; so we need to check that ${\cal P}'$ is unramified at that equivalence class. This reduces to seeing that $\widetilde{\cal P}$ is unramified at least at one point of the fiber; this is trivial at the plane-quintic point. (This argument avoids some of the detailed computations of the codifferential on the boundary, [DS, Ch., IV], but is still very close in spirit to [DS].) \noindent$\bullet$ We could instead work over any other point of $\widetilde{\cal A}_5$ over which we know the complete fiber, e.g. over Andreotti-Mayer points, coming from bielliptic Pryms, as in \S 3. (This was proposed in [D1], as a way to avoid the boundary computations.) \noindent$\bullet$ Izadi [I] applies a similar argument over boundary points, in $\partial {\cal A}_5$. This lets her reduce the degree computation over ${\cal A}_5$ to her results on ${\cal A}_4$, cf. (4.9). \subsection{Special Fibers} In this section we exhibit the cubic surface of theorem (4.2) explicitly over three special loci in ${\cal A}_{5}$. We do not know how to do this at the generic point of ${\cal A}_{5}$. \noindent{\bf (4.6) Cubic threefolds} {}From (4.3.4) we have an identification of $\widetilde{\cal P}^{-1}(X,H)$, where $X\subset{\bf P}^4$ is a cubic threefold and $H\subset{\bf P}^4$ a hyperplane, with the set of lines $\ell$ on the cubic surface $X\cap H$. For Theorem (4.4.1) we need to check that two of these, say $\ell, \ell'\in F(X)$, intersect each other if and only if the corresponding objects $(Q,\widetilde{Q},L), (Q',\widetilde{Q}',L')$ correspond under $\widetilde{\rm Tet}$. If the lines $\ell,\ell'$ intersect, we are in the situation of (2.15.4), so the corresponding objects $$(Q,\widetilde{Q},f), (Q',\widetilde{Q}',f')$$ (notation of (2.15.4)) are tetragonally related. Since $f,f'$ are both cut out by hyperplanes through the span $A$ of $\ell,\ell'$, we find points $$p\in Q\cap L, \ \ \ \ p'\in Q'\cap L'$$ (namely,the projection of $A$ from $\ell,\ell'$ respectively) such that $f,f'$ are the projections of $Q$ from $p$ and of $Q'$ from $p'$, respectively. The description of $\widetilde{\cal R}_6^{\rm Tet}$ in (4.3.8) then shows that $$((Q,\widetilde{Q},L),(Q',\widetilde{Q}',L'))\in\widetilde{\rm Tet},$$ as required. Since both the line incidence and $\widetilde{\rm Tet}$ are of bidegree (10,10), and we have an inclusion, it must be an equality. This shows that $\widetilde{\rm Tet}$ induces on $\widetilde{\cal P}^{-1}(X,H)$ the structure of line incidence on the cubic surface $X\cap H$. Fix the ambiant ${\bf P}^4$ and the hyperplane $H$, and let the cubic $X$ vary. We clearly get all cubic surfaces in $H$ as intersections $X\cap H$; therefore the monodromy group is the full symmetry group of the line configuration. This completes the proof of (4.4.1), hence also of Theorem (4.2). \bigskip \pagebreak \noindent{\bf (4.7) Jacobians} Start with $(C,L)\in\widetilde{\cal J}$. The fiber $\widetilde{\cal P}^{- 1}(C,L)$ consists of the 27 objects listed in (4.3.7). Each of these comes with the 5 $g^1_4$'s given in (4.3.8). These give the correspondence $\widetilde{\rm Tet}$, which we claim is equivalent to the line incidence on a cubic surface. Let $S=S_L$ be the quartic del Pezzo surface determined by $(C,L)$, as in (4.3.6). Let $S'$ be its blowup at a generic point $r\in S$. Then $S'$ is a cubic surface; its lines correspond to: \noindent$\bullet$ $\ell_Q$, the exceptional divisor over $r$. \noindent$\bullet$ 10 conics through $r$ in $S$; these correspond naturally to the 10 rulings ${\cal R}^\varepsilon_i$ (as in (4.3.7)).[Each ${\cal R}^\varepsilon_i$ contains a unique plane through $r$, which meets $S$ in a conic through $r$.] \noindent$\bullet$ The 16 lines $\ell_j$ in $S$. There is thus a natural bijection between the lines of $S'$ and \linebreak $\widetilde{\cal P}^{-1}(C,L)$. We need to check that this correspondence takes incident lines to covers which are tetragonally related to each other through the $g^1_4$'s of (4.3.8). To that end, we list the effects of the tetragonal constructions on our curves. The details are straightforward, and are omitted. \bigskip \noindent{\bf (4.7.1)} \ The quintic $(Q,\widetilde{Q})$, with the $g_4^1: \ \ {\cal O}_Q(1)(-p_i), \ \ p_i\in Q\cap L$, \ \ is taken to the two trigonals $$(T_i^\varepsilon, \widetilde{T}^\varepsilon_i), \;\;\;\;\;\varepsilon=0,1,$$ each with its unique base-point-free $g_4^1, \ \ \omega_T\otimes{\cal L}^{-2}$. \bigskip \noindent{\bf (4.7.2)} \ The trigonal $(T^\varepsilon_i,\widetilde{T}^\varepsilon_i)$with its base-point-free $g^1_4$ goes to $(Q,\widetilde{Q})$ with ${\cal O}_Q(1)(-p_i)$, and to $(T^{1-\varepsilon}_i, \widetilde{T}^{1- \varepsilon}_i)$ with the base-point-free $g^1_4$. Consider $(T^\varepsilon_i,\widetilde{T}^\varepsilon_i)$ with the $g^1_4 \ \ {\cal L}^\varepsilon_i(p).$ The actual 4-sheeted cover of ${\bf P}^1$ in this case is reducible, consisting of the trigonal $T^\varepsilon_i$ together with a copy of ${\bf P}^1$ glued to it at $p$. We are thus precisely in the situation of Example (2.15.2): both tetragonally related objects are Wirtinger covers $(X_j,\widetilde{X}_j)$. \bigskip \noindent{\bf (4.7.3)} \ A Wirtinger cover $(X_j,\widetilde{X}_j)$ with the $g^1_4$ cut out by the ruling ${\cal R}^\varepsilon_i$ on the singular quadric $A_{p_{i}}$, is taken to the trigonal $(T^\varepsilon_i,\widetilde{T}^\varepsilon_i)$ and to another Wirtinger cover. \bigskip \pagebreak[4] \noindent{\bf (4.8) Quartic double solids and the branch locus of ${\cal P}$}. The fiber of ${\cal P}$ over the Jacobian $J(X)\in{\cal B}$ of a quartic double solid $X$ of genus 5 is described in [DS, V.4], following ideas of Clemens. It consists of 6 objects $(C_i,\widetilde{C}_i), \ \ 0\le i\le 5$, each with multiplicity 2, and 15 objects $(C_{ij},\widetilde{C}_{ij}), \ \ 0\le i<j\le 5$. The monodromy group $S_6$ permutes the six values of $i$: clearly the two sets $\{C_i\}$ and $\{C_{ij}\}$ must be separately permuted, and any permutation of the $C_i$ induces a unique permutation of the $C_{ij}$. The situation is precisely that of lines on a nodal cubic surface: the $C_i$ correspond to lines $\ell_i$ through the node; and the plane through $\ell_i,\ell_j$ meets the cubic residually in a line $\ell_{i,j}$. The best way to see the symmetry is to consider Segre's cubic threefold $Y\subset{\bf P}^4$, image of ${\bf P}^3$ by the linear system of quadrics through 5 points $p_i$, $1\le i\le 5$, in general position in ${\bf P}^3$. (cf. [SR] for the details.) $Y$ contains six irreducible, two-dimensional families of lines, which we call the ``rulings" $R_i, \ \ 0\le i\le 5$: For $1\le i\le 5, \ \ R_i$ consists of proper transforms of lines through $p_i$; while $R_0$ parametrizes twisted cubics through $p_1,\cdots,p_5$.$Y$ also contains 15 planes $\Pi_{ij}, \ \ 0\le i < j\le 5$ \linebreak (= the 5 exceptional divisors and the proper transforms of the 10 planes $\overline{p_ip_jp_k}$); the ruling $R_i$ is characterized as the set of lines in ${\bf P}^4$ meeting the 5 planes $\Pi_{ij}, \ \ j\ne i$. The symmetric group $S_6$ acts linearly on ${\bf P}^4$, preserving $Y$, permuting the $R_i$ and correspondingly the $\Pi_{ij}$. The quartic double solids in question are essentially the double covers $$\zeta: X\to Y$$ branched along the intersection of $Y$ with a quadric $Q\subset{\bf P}^4$. The Prym fiber is obtained as follows: \begin{tabbing} $\bullet$ \=$C_i:= \{$ lines $\ell\in R_i$, tangent to $Q\}$ \\ \> $\widetilde{C}_i:= \{$ irreducible curves $\ell'\subset X$ such that $\zeta(\ell')=\ell\in C_i\}$ \end{tabbing} Thus $(C_i,\widetilde{C}_i)$ is the discriminant of a conic-bundle structure on $X$ given by $\zeta^{-1}(R_i)$. The Prym canonical curve $\Psi(C_i)\subset{\bf P}^4$ is traced by the tangency points of $\ell$ and $Q$; in particular, $\Psi(C_i)\subset Q$, so $(C_i,\widetilde{C}_i)$ is a ramification point of ${\cal P}$, by (1.6). \noindent$\bullet$ $(C_{ij},\widetilde{C}_{ij})$ is similarly obtained as discriminant of a conic bundle structure on $X$ given by projection from $\Pi_{ij}$, cf. [DS, V4.5]. \bigskip \noindent{\bf (4.9) Boundary behavior} In [I], Izadi uses results on the structure of ${\cal P}:{\cal R}_5\to{\cal A}_4$ to find the incidence structure on the fibers of the compactified map $\doublebar{\cal P}:\doublebar{\cal R}_6\to\bar{\cal A}_5$ over boundary points of the toroidal compactification $\bar{\cal A}_5$. The picture is as follows: \[ \begin{array}{cc} \begin{diagram}[AA] \node{\begin{array}{r} \; \doublebar{\cal P} \; \; : \end{array}} \end{diagram} & \begin{diagram}[AA] \node{\doublebar{\cal R}_{6}} \arrow{e} \node{\bar{\cal A}_{5} } \end{diagram} \\ & \begin{diagram}[AA] \node{\cup} \node{\cup} \end{diagram} \\ \begin{diagram}[AA] \node{\begin{array}{r} \partial {\cal P} \; \; : \end{array}} \\ \node{\begin{array}{r} \; {\cal P} \; \; : \end{array} } \end{diagram} & \begin{diagram}[AA] \node{\partial^{\rm II}\doublebar{\cal R}_{6}} \arrow{e} \arrow{s,l}{\alpha} \node{\partial\bar{\cal A}_{5} } \arrow{s,r}{\beta} \\ \node{{\cal R}_{5}} \arrow{e} \node{{\cal A}_{4}} \end{diagram} \end{array} \] Over general $A\in{\cal A}_4$, the fiber $\beta^{-1}(A)$ is isomorphic to the Kummer variety $A/(\pm 1)$. Over $(\widetilde{C},C)\in{\cal R}_5$, the fiber of $\alpha$ is $S^2\widetilde{C}/\iota$, and $\partial{\cal P}$ becomes (cf. [D3, (4.6)]) the map \[ \begin{array}{l} x+y \mapsto \psi (x) + \psi (y) \\ \begin{diagram}[AA] \node{S^{2}\widetilde{C}} \arrow{e} \arrow{s} \node{A} \arrow{s} \\ \node{S^{2}\widetilde{C}/\iota} \arrow{e} \node{A/(\pm 1)} \end{diagram} \end{array} \] \noindent where $\psi$ is the Abel-Prym map $\widetilde{C}\to A$.All in all then, we are considering the map $$\partial{\cal P}:\cup_{(\widetilde{C},C)\in{\cal P}^{-1}(A)}S^2 \widetilde{C}=:E\longrightarrow A.$$ Theorem (4.1) says that its degree is 27, and Theorem (4.2) predicts an incidence structure on its fibers, i.e. a way of associating a cubic surface to each point $a \in A$. In \S5 we associate to $A\in{\cal A}_4$ a cubic threefold $X=\kappa(A)\subset{\bf P}^4$ such that ${\cal P}^{-1}(A)$ is a double cover of the Fano surface $F(X)$ of lines in $X$. For generic $a\in A$, we are looking for a cubic surface; it is reasonable to hope that this should be of the form $H(a)\cap X$, where $H(a)$ is an appropriate hyperplane in ${\bf P}^4$. We thus want a map $$H:A\to({\bf P}^4)^*$$ such that $$pr((\partial{\cal P})^{-1}(a))=\{\mbox{lines in} \ H(a)\cap X\}.$$ \[ \begin{diagram}[A] \node{E} \arrow[2]{e,t}{\partial {\cal P}} \arrow[2]{s,l}{pr} \arrow{se} \node[2]{A} \arrow[2]{s,r}{H} \\ \node[2]{{\cal P}^{-1}(A)} \arrow{sw} \\ \node{F(X)} \node[2]{({\bf P}^{4})^{*}} \end{diagram} \] Izadi's beautiful observation is that such an $H$ is given by the linear system $\Gamma_{00}$ (sections of $|2\Theta|$ vanishing to order $\ge 4$ at 0). The identification of $\Gamma_{00}$ with the ambiant ${\bf P}^4$ of $X$ uses a construction of Clemens relating his double solids to $\Gamma_{00}$, and the interpretaton of (a cover of) $X$ as parametrizing double solids with intermediate Jacobians isomorphic to $A$, cf. [D6] or [I]. \section{Fibers of $P:{\cal R}_5\to A_4.$} \subsection{The general fiber.} \ \ \ \ Our main result in this section is: \bigskip \noindent{\bf Theorem 5.1} For generic $A\in A_4$, the fiber $\overline{\cal P}^{-1}(A)$ is isomorphic to a double cover of the Fano surface $F=F(X)$ of lines on some cubic threefold $X$. Let ${\cal R}{\cal C}$ denote the inverse image in ${\cal R}A_5$ of the locus ${\cal C}$ of (intermediate Jacobians $J(X)$ of) cubic threefolds $X$. We recall from [D4] that it splits into even and odd components: \noindent{\bf (5.1.1)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ${\cal R}{\cal C}={\cal R}{\cal C}^+\amalg {\cal R}{\cal C}^-,$ \noindent distinguished by a parity funciton. This follows from the existence of a natural theta divisor $\Xi\subset J(X)$, characterized (cf. [CG]) by having a triple point at $0:\Xi$ translates the parity function $q$ of (1.2), on theta characteristics, to a parity on semiperiods. More explicitly, pick $(Q,\sigma)\in{\cal P}^{-1}(J(X))\subset{\cal R}{\cal Q}^-$; Mumford's exact sequence (Theorem (1.4)(2)) says that any $\delta\in J_2(X)$ is $\pi^*\nu$ for some $\nu\in(\sigma)^\perp\subset J_2(Q)$. The compatibility result, theorem (1.5), then gives (cf. [D4], Proposition (5.1)): \noindent{\bf (5.1.2)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $q_X(\delta)=q_Q(\nu)=q_Q(\nu \sigma).$ In case $\delta$ is even, we end up with an isotropic subgroup \\ $(\!\nu,\sigma\!) \ \ \subset \ \ J_2(Q)$, with $\sigma$ odd and $\nu, \nu\sigma$ even. The Pryms of the latter are therefore Jacobians of curves: \noindent{\bf (5.1.3)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $P(Q,\nu)\approx J(C), \ \ \ P(Q,\nu\sigma)\approx J(C'),$ \noindent and the image of $\sigma$ gives semiperiods $\mu\in J_2(C),\mu'\in J_2(C')$. Reversing direction, we can construct an involution $$\lambda:{\cal R}_5\longrightarrow {\cal R}_5$$ and a map $$\kappa: {\cal R}_5\longrightarrow{\cal R}{\cal C}^+,$$ as follows: Start with $(C,\mu)\in {\cal R}_5$, pick the unique \ \ $(Q,\nu)$ \ \ in \\ ${\cal P}^{-1}(C)\cap {\cal R}{\cal Q}^+$, and let $\sigma,\nu\sigma\in J_2(Q)$ map to $\mu\in J_2(C).$ Then formula (1.3) reads: \noindent{\bf (5.1.4)} \ \ \ \ \ \ \ \ \ \ \ \ \ $0\equiv3+{\rm \ even}+q(\sigma)+q(\nu\sigma) \ \ \ \ \ {\rm (mod. \ 2)},$ \noindent so after possibly relabeling, we may assume $$(Q,\sigma)\in {\cal RQ}^-, \ \ \ (Q,\nu\sigma)\in{\cal RQ}^+$$ so that there is a well-defined curve $C'\in{\cal M}_5$ and a cubic threefold $X\in{\cal C}$ such that $$P(Q,\sigma)\approx J(X)$$ {\bf (5.1.5)} $$P(Q,\nu\sigma)\approx J(C').$$ \noindent We can thus define $\lambda$ and $\kappa$ by: $$\lambda(C,\mu):=(C',\mu')$$ {\bf (5.1.6)} $$\kappa(C,\mu):=(X,\delta),$$ \noindent where $\mu'\in J_2(C'), \ \ \ \delta\in J_2(X)$ are the images of $\nu\in J_2(Q).$ The precise version of our results is in terms of $\lambda$ and $\kappa$: \bigskip \noindent{\bf Theorem 5.2} \begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}} \item $(C,\mu)$ is related to $\lambda(C,\mu)$ by a sequence of two tetragonal constructions. Hence $\lambda$ commutes with the Prym map:$${\cal P}\circ\lambda={\cal P}, \ \ \ \lambda\circ\lambda=id.$$\item $\kappa$ factors through the Prym map: $$\kappa:{\cal R}_5\stackrel{{\cal P}}{\longrightarrow}A_4\stackrel{\chi}{\longrightarrow}{\cal RC}^+,$$ where $\chi$ is a birational map. \end{list} Recall the Abel-Jacobi map [CG], $$AJ:F(X)\longrightarrow J(X),$$ which is well-defined up to translation in $J(X)$. (It can be identified with the Albanese map of the Fano Surface $F(X)$.)A point $\delta\in J_2(X)$ determines a double cover of $J(X)$, hence of $F(X)$. \bigskip \noindent{\bf Theorem 5.3} For generic $A\in{\cal A}_4$, set $$(X,\delta):=\chi(A)=\kappa({\cal P}^{-1}(A))\in {\cal R C}^+.$$ Let $F(X)$ be the Fano surface of $X$, $\widetilde{F(X)}$ its double cover determined by $\delta$ via the Abel-Jacobi map. \begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}} \item There is a natural isomorphism $$P^{-1}(A)\approx\widetilde{F(X)}.$$ \item The action of $\lambda$ on the left corresponds to the sheet interchange on the right. \item Two objects $(C,\mu), (C',\mu)\in{\cal P}^{-1}(A)$ are tetragonally related if and only of the lines $\ell, \ell'\in F(X)$ which they determine intersect. \end{list} \bigskip \noindent{\bf Remark 5.4} Izadi has recently analyzed the birational map $\chi$, in [I]. In particular, she shows that $\chi$ is an isomorphism on an explicitly described, large open subset of ${\cal A}_4$. \subsection{Isotropic subgroups.} \begin{tabbing} X \= \kill \>By isotropic subgroup of rank $r$ on a curve $C$ we mean an \\$r$-dimensional ${\bf F}_2$-subspace of $J_2(C)$ on which the intersection pairing \\ $\langle \ , \ \rangle$ is identically zero.Choosing an isotropic subgroup of rank 1 is \\ the same as choosing a non-zero semiperiod. \end{tabbing} Start with a trigonal curve $T\in {\cal M}_{g+1}$, with a rank-2 isotropic subgroup $W\subset J_2(T)$ whose non-zero elements we denote $\nu_i, \ \ i=0,1,2.$ The trigonal construction associates to $(T,\nu_i)$ the tetragonal curve $X_i \in {\cal M}_g$.Mumford's sequence (1.4)(2) sends $W$ to an isotropic subgroup of rank $1$ on $X_i$, whose non-zero element we denote $\mu_i$. \bigskip \noindent{\bf Lemma 5.5} The construction above sets up a bijection between the following data: \noindent $\bullet$ A trigonal curve $T\in {\cal M}_{g+1}$ with rank-2 isotropic subgroup. \noindent $\bullet$ A tetragonally related triple $(X_i,\mu_i)\in {\cal R}_g, \ \ \ i=0, 1, 2$. \bigskip \noindent{\bf Proof.} We think of $WD_4$ as the group of signed permutations of the 8 objects $\{x^{\pm}_i\}$, $1\leq i\leq 4$. \ Start with a tetragonal double cover \linebreak $\widetilde{X}_0\longrightarrow X_0\longrightarrow{\bf P}^1.$ It determines a principal $WD_4$-bundle over ${\bf P}^1\backslash$(Branch). The original covers $\widetilde{X}_0, X_0$ are recovered as quotients by the following subgroups of $WD_4$: $$\widetilde{H}_0:={\rm Stab}(x^+_1),$$ $$H_0:={\rm Stab}(x^\pm_1),$$ Consider also the subgroup $$G:={\rm Stab}\{\{x^+_1, x^+_2\}, \{x_1^-,x_2^-\}\}.$$ It has index 12 in $WD_4$. Its normalizer is:$$N(G)={\rm Stab}\{\{x^\pm_1, x^\pm_2\}, \{x^\pm_3, x^\pm_4\}\},$$ of index 3. The quotient is $$N(G)/G\approx({\bf Z}/2{\bf Z})^2,$$ so there are 3 intermediate groups $\widetilde{G}_i,\ \ \ i=0,1,2.$ We single out one of them:$$\widetilde{G}_0:={\rm Stab}\{x^\pm_1, x^\pm_2\}.$$ The three subgroups $\widetilde{G}_i$ are not conjugate to each other, but can be taken to each other by outer automorphisms of $WD_4$. In fact, the action of Out$(WD_4)\approx S_3$ sends $G$, and hence also $N(G)$, to conjugate subgroups; it permutes the $\widetilde{G}_i$ transitively, modulo conjugation; and it also takes $H_0, \widetilde{H}_0$ to non-conjugate subgroups $H_i, \widetilde{H}_i, \ \ \ i=1,2.$ We illustrate each of these subgroups as the stabilizer in $WD_4$ of a corresponding partition of $\left(\begin{array}{llll}x^+_1 & x^+_2 & x^+_3 & x^+_4 \\ x^-_1 & x^-_2 & x^-_3 & x^-_4 \end{array} \right)$: \begin{center} \begin{tabular}{ccc} \hspace{1.5in} & \hspace{1.5in} & \hspace{1.5in} \\ \begin{picture}(82,30)(2,1) \thicklines \put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(4.5,6){\line(0,1){20}} \put(26,1){$\circ$} \put(31,3.5){\line(1,0){20}} \put(28.5,6){\line(0,1){20}} \put(26,25.8){${\circ}$} \put(31,28.3){\line(1,0){20}} \put(50,1){${\circ}$} \put(55,3.5){\line(1,0){20}} \put(52.5,6){\line(0,1){20}} \put(50,25.8){${\circ}$} \put(55,28.3){\line(1,0){20}} \put(74,1){${\circ}$} \put(76.5,6){\line(0,1){20}} \put(74,25.8){${\circ}$} \end{picture} & \begin{picture}(82,30)(2,1) \thicklines \put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}} \put(7,3.5){\line(1,0){20}} \put(26,1){$\circ$} \put(31,3.5){\line(1,0){20}} \put(26,25.8){${\circ}$} \put(31,28.3){\line(1,0){20}} \put(50,1){${\circ}$} \put(55,3.5){\line(1,0){20}} \put(50,25.8){${\circ}$} \put(55,28.3){\line(1,0){20}} \put(74,1){${\circ}$} \put(74,25.8){${\circ}$} \end{picture} & \begin{picture}(82,30)(2,1) \thicklines \put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}} \put(7,3.5){\line(1,0){20}} \put(26,1){$\circ$} \put(31,3.5){\line(1,0){20}} \put(26,25.8){${\circ}$} \put(31,28.3){\line(1,0){20}} \put(50,1){${\circ}$} \put(55,3.9){\line(1,1){22.4}} \put(50,25.8){${\circ}$} \put(55,27.9){\line(1,-1){22}} \put(76,1){${\circ}$} \put(76,25.8){${\circ}$} \end{picture} \\ $H_{0}$ & $H_{1}$ & $H_{2}$ \\ & & \\ & & \\ \begin{picture}(82,30)(2,1) \thicklines \put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,3.5){\line(1,0){20}} \put(26,1){$\circ$} \put(31,3.5){\line(1,0){20}} \put(28.5,6){\line(0,1){20}} \put(26,25.8){${\circ}$} \put(31,28.3){\line(1,0){20}} \put(50,1){${\circ}$} \put(55,3.5){\line(1,0){20}} \put(52.5,6){\line(0,1){20}} \put(50,25.8){${\circ}$} \put(55,28.3){\line(1,0){20}} \put(74,1){${\circ}$} \put(76.5,6){\line(0,1){20}} \put(74,25.8){${\circ}$} \end{picture} & \begin{picture}(82,30)(2,1) \thicklines \put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}} \put(26,1){$\circ$} \put(26,25.8){${\circ}$} \put(31,28.3){\line(1,0){20}} \put(50,1){${\circ}$} \put(50,25.8){${\circ}$} \put(55,28.3){\line(1,0){20}} \put(74,1){${\circ}$} \put(74,25.8){${\circ}$} \end{picture} & \begin{picture}(82,30)(2,1) \thicklines \put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}} \put(26,1){$\circ$} \put(26,25.8){${\circ}$} \put(31,28.3){\line(1,0){20}} \put(50,1){${\circ}$} \put(50,25.8){${\circ}$} \put(55,27.9){\line(1,-1){22}} \put(76,1){${\circ}$} \put(76,25.8){${\circ}$} \end{picture} \\ $\widetilde{H}_{0}$ & $\widetilde{H}_{1}$ & $\widetilde{H}_{2}$ \\ & & \\ & & \\ \begin{picture}(82,30)(2,1) \thicklines \put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}} \put(7,3.5){\line(1,0){20}} \put(4.5,6){\line(0,1){20}} \put(26,1){$\circ$} \put(28.5,6){\line(0,1){20}} \put(26,25.8){${\circ}$} \put(50,1){${\circ}$} \put(55,3.5){\line(1,0){20}} \put(52.5,6){\line(0,1){20}} \put(50,25.8){${\circ}$} \put(55,28.3){\line(1,0){20}} \put(74,1){${\circ}$} \put(76.5,6){\line(0,1){20}} \put(74,25.8){${\circ}$} \end{picture} & & \\ $N(G)$ & & \\ & & \\ & & \\ \begin{picture}(82,30)(2,1) \thicklines \put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}} \put(7,3.5){\line(1,0){20}} \put(4.5,6){\line(0,1){20}} \put(26,1){$\circ$} \put(28.5,6){\line(0,1){20}} \put(26,25.8){${\circ}$} \put(50,1){${\circ}$} \put(50,25.8){${\circ}$} \put(74,1){${\circ}$} \put(74,25.8){${\circ}$} \end{picture} & \begin{picture}(82,30)(2,1) \thicklines \put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}} \put(7,3.5){\line(1,0){20}} \put(26,1){$\circ$} \put(26,25.8){${\circ}$} \put(50,1){${\circ}$} \put(55,3.5){\line(1,0){20}} \put(50,25.8){${\circ}$} \put(55,28.3){\line(1,0){20}} \put(74,1){${\circ}$} \put(74,25.8){${\circ}$} \end{picture} & \begin{picture}(82,30)(2,1) \thicklines \put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}} \put(7,3.5){\line(1,0){20}} \put(26,1){$\circ$} \put(26,25.8){${\circ}$} \put(50,1){${\circ}$} \put(55,3.9){\line(1,1){22.4}} \put(50,25.8){${\circ}$} \put(55,27.9){\line(1,-1){22}} \put(76,1){${\circ}$} \put(76,25.8){${\circ}$} \end{picture} \\ $\widetilde{G}_{0}$ & $\widetilde{G}_{1}$ & $\widetilde{G}_{2}$ \\ & & \\ & & \\ \begin{picture}(82,30)(2,1) \thicklines \put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}} \put(7,3.5){\line(1,0){20}} \put(26,1){$\circ$} \put(26,25.8){${\circ}$} \put(50,1){${\circ}$} \put(50,25.8){${\circ}$} \put(74,1){${\circ}$} \put(74,25.8){${\circ}$} \end{picture} & & \\ $G$& & \\ & & \\ & & \end{tabular} \end{center} Let $X_i, \widetilde{X}_i, T, \stackrel{\approx}{T},\widetilde{T}_i \ \ \ (i=0,1,2)$ be the quotients of the principal $WD_4$-bundle by the subgroups $H_i, \widetilde{H}_i, N(G), G, \widetilde{G}_i$ respectively, compactified to branched covers of ${\bf P}^1$. We see immediately that: \noindent $\bullet$ The trigonal construction takes $X_0\to {\bf P}^1$ to $\widetilde{T}_0\to T\to{\bf P}^1.$ \noindent $\bullet$ The double cover $\widetilde{X}_0\to X_0$ corresponds via (1.4)(2) to the double cover $\stackrel{\approx}{T}\to\widetilde{T}_0$. \noindent $\bullet$ The tetragonal construction acts by outer antomorphisms, hence exchanges the three tetragonal double covers $\widetilde{X}_i\to X_i\to{\bf P}^1.$ Applying the same outer automorphisms, we see that the trigonal construction also takes $X_i\to{\bf P}^1$ to $\widetilde{T}_i\to T\to {\bf P}^1, \ i=1,2.$ To a tetragonally related triple $(\widetilde{X}_i\to X_i \to {\bf P}^1)$ we can thus unambiguously associate the trigonal \ \ $T\to{\bf P}^1$ \ \ together with the rank-2, isotropic subgroup corresponding to the covers $\widetilde{T}_i$. This inverts the construction predecing the lemma. \begin{flushright} QED. \end{flushright} \bigskip \noindent{\bf Note 5.5.1} The basic fact in the above proof is that the 3 tetragonals $X_i$ yield the same trigonal $T$. This can be explained more succinctly: outer automorphisms take the natural surjection $\alpha_0:WD_4\to\!\to S_4$ to homomorphisms $\alpha_1,\alpha_2$ which are not conjugate to it.But the composition $\beta\circ\alpha_i:WD_4\to\!\to S_3$, where $\beta:S_4\to\!\to S_3$ is the Klein map, are conjugate to each other. \bigskip \noindent{\bf Construction 5.6} Now let $T\in{\cal M}_{g+1}$ be a trigonal curve, together with an isotropic subgroup of rank 3, $$V\subset J_2(T).$$ We think of $V$ as a vector space over ${\bf F}_2$; the projective plane ${\bf P}(V)$ is identified with $V\backslash(0)$. For each $i\in {\bf P}(V)$, the trigonal construction gives a tetragonal curve $Y_i\in{\cal M}_g$.Mumford's sequence (1.4)(2) gives an isotropic subgroup of rank 2, $$W_i\subset J_2(Y_i),$$ with a natural identification $W_i\approx V/(i).$ Let $U\subset V$ be a rank-2 subgroup, so ${\bf P}(U)\subset{\bf P}(V)$ is a projective line. Lemma (5.5) shows that the 3 objects $$(Y_i, U/(i))\in{\bf R}_g, \ \ \ \ \ \ \ \ i\in {\bf P}(U)$$ are tetragonally related. In particular, they have a common Prym variety $$P_U\approx{\cal P}(Y_i, U/(i))\in {\cal A}_{g-1}, \ \ \ \ \ \forall i\in {\bf P}(U).$$Applying (1.4) twice, we see that the original rank-3 subgroup $V$ determines a rank-1 subgroup $$V/U\subset(P_{U})_2,$$ so we let $\mu_{U}\in(P_U)_2$ be its non-zero element.Altogether then, we have a map $${\bf P}(V)^*\longrightarrow{\cal R A}_{g-1}$$ $$U\longmapsto(P_{U},\mu_{U}).$$ \bigskip \noindent {\bf (5.6.1)} Assume now that one of the $Y_i$ happens to be trigonal. (This can only happen if $g\le 6.$) Whenever $U\ni i$, we find a tetragonal curve $C_U\in {\cal M}_{g-1}$ such that $P_U\approx J(C_U)$. Lemma (5.5), applied to $(Y_i, W_i)$, shows that the 3 objects $$(C_U,\mu_U)\in{\cal R}_{g-1}, \ \ \ \ \ U\ni i$$ are tetragonally related, so they have a common Prym variety \\ $A=P_V\in{\cal A}_{g-2}.$ \bigskip \noindent {\bf (5.6.2)} Assume instead that $g=6$ and that $P_U$ happens to be a Jacobian $J(C_U)\in{\cal J}_5$, for some $U\in{\bf P}(V)^*.$ Of the three $Y_i, \ \ i\in U$, we claim two are trigonal and the third, a plane quintic. Indeed, by (4.7), the tetragonal triples above $J(C_U)$ consist either of a plane quintic and two trigonals, as claimed, or of a trigonal and two Wirtingers. The latter is excluded since the isomorphism $$J(Y_i)\approx P(T,i)$$ implies that $Y_i$ is non-singular for each $i\in{\bf P}(V)$. Assume from now on that $g=6$. Our data consists of: \noindent $\bullet$ $T\in {\cal M}_7$, trigonal, with $V\subset J_2(T)$ isotropic of rank 3. \noindent $\bullet$ For each $i\in {\bf P}(V)$, a curve $Y_i \in {\cal M}_6$, with a rank-2 isotropic subgroup $W_i\subset J_2(Y_i)$. \noindent $\bullet$ For each $U\in {\bf P}(V)^*$, an object $(P_U,\mu_U)\in {\cal RA}_5$ \noindent $\bullet$ An abelian variety $A=P_V\in {\cal A}_4.$ We display ${\bf P}(V)$ as a graph with 7 vertices $i\in {\bf P}(V)$ and 7 edges \\ $U\in{\bf P}(V)^*$, in (3,3)-correspondence.We write $T$ (or $Q$) on a vertex corresponding to a trigonal (or quintic) curve, and $C$ on an edge corresponding to a Jacobian.We restate our observations: \bigskip \noindent{\bf (5.6.3)}: Edges through a $T$-vertex are $C$-edges. \bigskip \noindent{\bf (5.6.4)}: On a $C$-edge, the vertices are $T,T,Q$. \bigskip It follows that only one configuration is possible: \pagebreak[4] { \ } \vspace{4in} \centerline{ {\bf Figure 5.7}} \noindent Thus four of the $Y_i$ are trigonal, the other three are quintics, and six of the $P_U$, corresponding to the straight lines, are Jacobians of curves. Let $U_0\in{\bf P}(V)^*$ correspond to the circle. For $i\in U_0$, \ \ $Y_i$ is a quintic $Q$. Through $Q$ pass two $C$ edges and $U_0$, and the semiperiods corresponding to the $C$-edges are even; by (1.3), the semiperiod $U_0/(i)$ corresponding to $U_0$ must be \underline{odd}, so there is a cubic threefold $X\in{\cal C}$ such that $${\bf P}_{U_0}\approx J(X).$$ Finally, theorem (1.5), or formula (5.1.2), shows that the semiperiod $\delta:=\mu_{U_0}\in J_2(X)$ is \underline{even}. We observe that the three tetragonally related quintics correspond to 3 lines on the cubic threefold which meet each other and thus form the intersection of $X$ with a (tritangent) plane. We are thus exactly in the situation of (2.15.4). \subsection{Proofs.} \noindent{\bf (5.8)} Theorems (5.1),(5.2) and (5.3) all follow from the following statements: \begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}} \item $(C,\mu)$ is related to $\lambda(C,\mu)$ by a sequence of two tetragonal constructions. \item $\kappa$ is invariant under the tetragonal construction \item For $(X,\delta)\in {\cal R}{\cal C}^+, \ \ \ \kappa^{-1}(X,\delta)\approx\widetilde{F(X)}$, the isomorphism takes $\lambda$ to the involution on $\widetilde{F(X)}$ over $F(X)$, and two objects on the left are tetragonally related iff the corresponding lines intersect. \item Any two objects in ${\cal P}^{-1}(A)$, generic $A\in{\cal A}_4$, are connected by a sequence of (two) tetragonal constructions. \end{list} Indeed, (1) is (5.2)(1); \ (2) and (4) imply the existence of \\ $\chi:{\cal A}_4\longrightarrow{\cal R}{\cal C}^+$ such that $\kappa=\chi\circ{\cal P}$, while (3) shows that any two objects in a $\kappa$-fiber are also connected by a sequence of two tetragonal constructions, so $\chi$ must be birational, giving (5.2)(2). This gives an isomorphism ${\cal P}^{-1}(A)\approx\kappa^{-1}(X,\delta)$, so (5.3) follows. \bigskip \noindent{\bf (5.9)} We let ${\cal R}^2{\cal Q}^+, {\cal R}^2{\cal Q}^-$ denote the moduli spaces of plane quintic curves $Q$ together with: \noindent $\bullet$ A rank-2, isotropic subgroup $W\subset J_2(Q)$, containing one odd and two even semiperiods, and \noindent $\bullet$ a marked even (respectively odd) semiperiod in $W\backslash(0).$ Exchanging the two even semiperiods gives an involution on ${\cal R}^2{\cal Q}^+$, with quotient ${\cal R}^2{\cal Q}^-$. The birational map $$\alpha:{\cal M}_5\widetilde{\longrightarrow}{\cal R}{\cal Q}^+,$$ of (4.3.7), lifts to a birational map \noindent{\bf (5.9.1)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ${\cal R}\alpha:{\cal R}_5\widetilde{\longrightarrow}{\cal R}^2{\cal Q}^+.$ \noindent From the construction of $\lambda$ in (5.1.6) it follows that the involution on the right hand side corresponds to $\lambda$ on the left, so we have a commutative diagram: \begin{equation} \renewcommand{\theequation}{\bf {\arabic{section}}.{\theau}.{\arabic{equation}}} \setcounter{au}{9} \setcounter{equation}{2} \begin{diagram}[AA] \node{{\cal R}_{5}} \arrow{s} \arrow{e,tb}{{\cal R}\alpha}{\sim} \node{{\cal R}^{2}Q^{+}} \arrow{s,lr}{\pi}{2:1} \\ \node{{\cal R}_{5}/\lambda} \arrow{e,b}{\sim} \node{{\cal R}^{2}Q^{-}} \end{diagram} \end{equation} Start with $(C,\mu)\in{\cal R}_5$ and any $g^1_4$ on $C$. The trigonal construction produces a trigonal $Y\in{\cal M}_6$ with rank-2, isotropic subgroup $W_Y$. On $Y$ we have a natural $g^1_4$, namely $w_Y\otimes L^{-2}$, where $L$ is the trigonal bundle; so we bootstrap again, to a trigonal $T\in{\cal M}_7$ with rank-3 isotropic subgroup $V$. Applying construction (4.6) we obtain a diagram like (5.7), including an edge for $(C,\mu)$ and on it a vertex for $(Q,W_Q):=$ \\ $\pi{\cal R}\alpha(C,\mu)$. But then $\lambda(C,\mu)$ and $\kappa(C,\mu)$ also appear in the same diagram, as the two other edges (the line, respectively the circle) through $Q$! Statement (5.8.1) now follows, since any two edges of (5.7) which meet in a trigonal vertex are tetragonally related. (5.8.2) also follows, since any $(C',\mu')$ tetragonally related to $(C,\mu)$ will appear in the same diagram with $(C,\mu)$ (for the obvious initial choice of $g^1_4$ on $C$), so they have the same $\kappa$. {}From the restriction to ${\cal R}{\cal Q}^-$ of the Prym map we obtain, by base change: \begin{equation} \renewcommand{\theequation}{\bf {\arabic{section}}.{\theau}.{\arabic{equation}}} \setcounter{au}{9} \setcounter{equation}{3} \begin{diagram}[AA] \node{{\cal R}^{2}Q^{-}} \arrow{s,l}{\cal RP} \arrow{e} \node{{\cal R}Q^{-}} \arrow{s,r}{\cal P} \\ \node{{\cal RC}^{+}} \arrow{e} \node{{\cal C}} \end{diagram} \end{equation} Combining with (5.8)(1),(2) and (5.9.2), we find that $\kappa$ factors \begin{equation} \renewcommand{\theequation}{\bf {\arabic{section}}.{\theau}.{\arabic{equation}}} \setcounter{au}{9} \setcounter{equation}{4} \begin{diagram}[AA] \node{{\cal R}_{5}} \arrow{s} \arrow{e,tb}{{\cal R}\alpha}{\sim} \node{{\cal R}^{2}Q^{+}} \arrow{s,r}{\pi} \\ \node{{\cal R}_{5}/\lambda} \arrow{e,b}{\sim} \node{{\cal R}^{2}Q^{-}} \arrow{s,r}{\cal RP} \\ \node[2]{{\cal RC}^{+}} \end{diagram} \end{equation} We know ${\cal P}^{-1}(X)$ from (4.6), so by (5.9.3): \noindent{\bf (5.9.5)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ${\cal RP}^{-1}(X,\delta)\approx{\cal P}^{-1}(X)\approx F(X),$ \noindent and $\kappa^{-1}(X,\delta)$ is a double cover, which by the following lemma is identified with $\widetilde{F(X)}$. (The compatibility with $\lambda$ follows from (5.9.4); line incidence in $F(X)$ corresponds by (4.6) to the tetragonal relation among the quintics, which by figure (5.7) corresponds, in turn, to the tetragonal relation in ${\cal R}_5$, so the proof of (5.8)(3) is complete.) \bigskip \noindent{\bf Lemma 5.10} The Albanese double cover $\widetilde{F(X)}$ determined by \\ $\delta\in J_2(X)$ is isomorphic to $\pi^{-1}{\cal R P}^{-1}(X,\delta)$ (notation of (5.9.4)). \bigskip \noindent{\bf Proof.} The second isomorphism in (5.9.5) sends a line $\ell\in F(X)$ to the object $(\widetilde{Q}_{\ell},Q_{\ell})\in{\cal P}^{-1}(X)$, where the curves $\widetilde{Q}_{\ell},Q_{\ell}$ parametrize ordered (respectively, unordered) pairs $\ell',\ell'' \in F(X)$ satisfying: $$\ell + \ell' + \ell''= 0 \ \ \ \ \ \ \ \ \ \ {\rm (sum \ in \ } \ \ \ J(X)).$$ We may of course think of $\widetilde{Q}_{\ell}$ as sitting in $F(X)$, since $\ell'$ uniquely determines $\ell''$: $\widetilde{Q}_{\ell}$ is the closure in $F(X)$ of \noindent{\bf (5.10.1)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\{ \ell'\in F(X) \ \ | \ \ \ell'\cap\ell\neq\phi \, , \, \ell' \neq \ell \}. $ \medskip \noindent The corresponding object of ${\cal R}{\cal P}^{-1}(X,\delta)$ is $(\stackrel{\approx}{Q}_\ell, \widetilde{Q}_\ell, Q_\ell)$, where $\stackrel{\approx}{Q}_\ell$ is the inverse image in $\widetilde{F(X)}$ of $\widetilde{Q}_\ell$ embedded in $F(X)$ via (5.10.1). Now to specify a point in $\pi^{-1}{\cal R}{\cal P}^{-1}(X,\delta)$ we need, additionally, a double cover $\widetilde{Q}_\ell' \to Q_\ell$ satisfying: \noindent{\bf (5.10.2)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\widetilde{Q}_\ell\times_{Q_\ell}\widetilde{Q}_\ell'\approx \stackrel{\approx}{Q}_\ell \ .$ We need to show that a choice of $\widetilde{\ell} \in\widetilde{F(X)}$ over $\ell\in F(X)$ determines such a $\widetilde{Q}_\ell'$.Recall that $\widetilde{F(X)} \to F(X)$ is obtained by base change, via the Albanese map, from the double cover $\widetilde{J(X)} \to J(X)$ determined by $\delta$. \ $\widetilde{Q}_\ell'$ can thus be taken to parametrize unordered pairs $\widetilde{\ell}', \widetilde{\ell}''\in\widetilde{F(X)}$ satisfying: $$\widetilde{\ell} + \widetilde{\ell}' + \widetilde{\ell}'' = 0 \ \ \ \ \ \ \ {\rm (sum \ in} \ \ \ \widetilde{J(X)})\ .$$ The fiber product in (5.10.2) then parametrizes such ordered pairs, so the required isomorphism to $\doubletilde{Q}_\ell$ simply sends $$(\widetilde{\ell}',\widetilde{\ell}'') \mapsto \widetilde{\ell}'.$$ \begin{flushright} Q.E.D. \end{flushright} Finally, we prove (5.8)(4). Let $\overline{{\cal P}}: \overline{{\cal R}}_5 \to {\cal A}_4$ be the proper Prym map. By (5.8)(3) it factors $$\overline{\cal P} = \iota \circ \kappa$$ where $\iota: {\cal R}{\cal C}^+ \to {\cal A}_4$ is a rational map, which we are trying to show is birational. It suffices to find some $A \in{\cal A}_4$ such that: \begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}} \item Any two objects in $\overline{\cal P}^{-1}(A)$ can be related by a sequence of tetragonal constructions. \item The differential $d{\cal P}$ is surjective over $A$. \end{list} In \S5.4 we see that $(1)$ holds for various examples, including generic Jacobians $\in{\cal J}_4$: for generic $C \in {\cal M}_4$, $\overline{\cal P}^{-1}(J(C))$ consists of Wirtinger covers $\widetilde{C} \to C'$ (with normalilzation $C$) and of trigonals $T$, and the two types are exchanged by $\lambda$.It is easier to check surjectivity of $d{\cal P}$ at the Wirtingers: by theorem (1.6), this amounts to showing that the Prym-canonical curve $\Psi(X) \subset {\bf P}^3$ is contained in no quadrics. By [DS] IV, Propo. 3.4.1, $\Psi(X)$ consists of the canonical curve $\Phi(C)$ together with an (arbitrarily chosen) chord. Since $\Phi(C)$ is contained in a unique quadric $Q$, which does not contain the generic chord, we are done. [Another argument: it suffices to show that no one quadric contains $\Psi(T)$ for all trigonal $T$ in ${\cal P}^{-1}(J(C))$.By [DS], III 2.3 we have $$\cup_T \Psi(T) \ \supset \ \Phi(C) ,$$ so the only possible quadric would be $Q$. Consider the $g^1_4$ on $C$ given by $\omega_C$(-$p$-$q$), where $p,q \in C$ are such that the chord $\overline{\Phi(p),\Phi(q)}$ is not in $Q$. Let $T$ be the trigonal curve associated to $(C,\omega_C$(-$p$-$q$)), and choose a plane $A \subset {\bf P}^3$ through $\Phi(p),\Phi(q)$, meeting $Q$ and $\Phi(C)$ transversally, say $$A \cap \Phi(C) = \Phi(p+q+\sum^{4}_{i=1}x_i),$$ then by [DS],III 2.1, \ \ $\Psi(T)$ contains the point $$\overline{\Phi(x_1),\Phi(x_2)} \cap \overline{\Phi(x_3), \Phi(x_4)}$$ which cannot be in $Q$.]\begin{flushright} Q.E.D. \end{flushright} \subsection{Special fibers.} \begin{tabbing} X \= \kill \>We want to illustrate the behavior of the Prym map over some \\ special loci in $\overline{\cal A}_4$. The common feature to all of these examples is that \\ the cubic threefold $X$ given in Theorem (5.1) acquires a node. We thus \\ begin with a review of some results, mostly from [CG], on nodal \\ cubics. \end{tabbing} \bigskip \noindent{\bf (5.11) Nodal cubic threefolds} There in a natural correspondence between nodal cubic threefolds $X\subset{\bf P}^4$ and nonhyperelliptic curves $B$ of genus 4. Either object can be described by a pair of homogeneous polynomials $F_2, \ F_3$, of degrees 2 and 3 respectively, in 4 variables $x_1, ..., x_4:X$ has homogeneous equation $0=F_3+x_0F_2$ \ (in ${\bf P}^4$), and the canonical curve $\Phi(B)$ has equations $F_2=F_3=0$ in ${\bf P}^3$. More geometrically, we express the Fano surface $F(X)$ in terms of $B$. Assume the two $g^1_3$'s on $B$, \ ${\cal L}'$ and ${\cal L}''$, are distinct. They give maps $$\tau', \ \tau'':B\hookrightarrow S^2B$$ sending $r\in B$ to $p+q$ if $p+q+r$ is a trigonal divisor in $|{\cal L}'|$, \ $|{\cal L}''|$ respectively. We then have the identification \noindent{\bf (5.11.1)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $F(X)\approx S^2B/(\tau'(B)\sim\tau''(B))$. Indeed, we have an embedding $$\tau:B\hookrightarrow F(X),$$ identifying $B$ with the family of lines through the node $n = (1,0,0,0,0)$. This gives a map $S^2B\rightarrow F(X)$ sending a pair $\ell_{1}, \ell_{2}$ of lines through $n$ to the residual intersection with $X$ of the plane $(\ell_1, \ell_2)$.this map identifies $\tau'(B)$ with $\tau''(B)$, and induces the isomorphism (5.11.1). \bigskip \noindent{\bf (5.11.2)} A line $\ell\in F(X)$ determines a pair $(Q, \widetilde{Q})\in\overline{\cal RQ}^-$, which must be in $\partial^{\rm II}{\cal RQ}^-$, i.e. for generic $\ell$ we obtain a nodal quintic $Q$ with \'{e}tale double cover $\widetilde{Q}$. We can interpret (5.11.1) in terms of these nodal quintics: Start with a divisor $p+q\in S^2B$. Then $\omega_B(-p-q)$ is a $g^1_4$ on $B$, so the trigonal construction produces a double cover $\widetilde{T}\rightarrow T$, where $T\in{\cal M}_5$ comes with a trigonal bundle ${\cal L}$. The linear system $|\omega_T\otimes{\cal L}^{-1}|$ maps $T$ to a plane quintic $Q$, with a single node given by the divisor $|\omega_T\otimes{\cal L}^{-2}|$ on $T$. \bigskip \noindent{\bf (5.11.3)} In the special case that there exists $r\in B$ such that \\ $p+q=\tau''(r)$, i.e. $p+q+r\in|{\cal L}''|$ is a trigonal divisor, our $g^1_4$ acquires a base point: $$\omega_B(-p-q)\approx{\cal L}'(r).$$ As seen in (2.10.ii), the trigonal construction produces the nodal trigonal curve $$T:=B/(p'\sim q')$$ with its Wirtinger double cover $\widetilde{T}$, where $p', q'\in B$ are determined by: $$p'+q'+r\in|{\cal L}'|,$$ i.e. $p'+q'=\tau'(r)$. In this case, the quintic $Q$ is the projection of $\Phi(B)$ from $\Phi(r)$, with 2 nodes $p\sim q, \ \ p'\sim q'$, and $\widetilde{Q}$ is the reducible double cover with crossings over both nodes. \bigskip \noindent{\bf (5.12) Degenerations in ${\cal RC}^+$.} We fix our notation as in \S 5.1. Thus we have: \begin{tabbing} \=$X\in{\cal C} \ \ \ \ \ \ \ (X, \delta)\in{\cal RC}^+$ \\ \>$(Q,\sigma)\in{\cal RQ}^-, \ \ \ \ \ \ (Q, \nu), \ (Q, \nu\sigma)\in{\cal RQ}^+$ \\ \>$(C, \mu), \ (C', \mu')\in{\cal R}_5$ \\ \>$A\in{\cal A}_4$ \end{tabbing} \noindent and these objects satisfy: \begin{tabbing} \= ${\cal P}(Q, \sigma)$ \= $=$ \= $J(X)$ \= , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= $\nu, \nu\!$ \= $\!\sigma\mapsto\delta$ \\ \>${\cal P}(Q, \nu)$ \>= \>$J(C)$ \>, \>$\sigma, \nu\!$ \>$\!\sigma\mapsto\mu$ \\ \>${\cal P}(Q, \nu\sigma)$ \>= \>$J(C')$ \>, \>$\nu,$ \>$\!\sigma\mapsto\mu'$ \end{tabbing} \bigskip \begin{tabbing} \= ${\cal P}(C, \mu)={\cal P}\!$ \= $\!(C'\!$ \= $, \mu'\!$ \= $\!)=A$ \\ \>$\lambda(C, \mu)=$ \>$\!(C'\!$ \>$, \mu'\!$ \>$\!)$ \\ \>$\kappa(C, \mu)=$ \>$\!(X$ \>$, \delta\!$ \>$\!).$ \end{tabbing} Now let $X$ degenerate, acquiring a node, with $\bar{\varepsilon}\in J_2(X)\backslash(0)$ the vanishing cycle mod. 2. \ From (5.11) we see that $Q$ also degenerates, with a vanishing cycle $\varepsilon$ which maps (via. (1.4)) to $\bar{\varepsilon}$. Lemma (5.9) of [D4] shows that $\varepsilon$, hence also $\bar{\varepsilon}$, must be even. There are 3 types of degenerations of $(X, \delta)$, distinguished as in (1.7) by the relationship of $\delta, \bar{\varepsilon}$. (A fourth type, where $Q$ degenerates but $X$ does not, is explained in (5.13).)The possibilities are summarized below: \bigskip \begin{list}{{\rm(\Roman{butter})}}{\usecounter{butter}} \item If $\bar{\varepsilon}=\delta$ then either $\varepsilon=\nu$ or $\varepsilon=\nu\sigma$, which gives the same picture with $C, C'$ exchanged. In case $\varepsilon = \nu$, $(Q, \nu)$ undergoes a $\partial^{\rm I}$ degeneration, while $(Q, \nu\sigma)$ is $\partial^{\rm II}$. (The notation is that of (1.7).) Thus $A$ is a Jacobian. \ \ \ The double cover $\widetilde{F(X)}$ is itself a $\partial^I$ cover. In terms of the curve $B$ of (5.11), we have $$\widetilde{F(X)}=(S^2B)_0\amalg(S^2B)_1 \ \ / \ \ (\tau'(B)_0\sim\tau''(B)_1, \ \ \ \tau''(B)_0\sim\tau'(B)_1).$$This is clear, either from the definition of $\widetilde{F(X)}$ viathe Albanese map, or by considering the restriction to ${\cal RP}^{-1}(X, \delta)$ of the double cover $${\cal R}^2{\cal Q}^+\stackrel{\pi}{\rightarrow}{\cal R}^2{\cal Q}^-$$ of (5.9). One of the components parametrizes the trigonal objects $(C, \mu)$, the other parametrizes the nodals $(C', \mu')$. \item $\sigma$ is always perpendicular to $\varepsilon, \nu$, and the condition $\langle\bar{\varepsilon}, \delta\rangle=0$ implies $\langle\varepsilon, \nu\rangle=0$ by (1.4.3). Both $(Q, \nu)$ and $(Q, \nu\sigma)$ then give $\partial^{\rm II}$-covers, so $C, C'$ are nodal. Again by (1.4.3), both $(C, \mu)$ and $(C', \mu')$ are $\partial^{\rm II}$, so their common Prym $A$ is in $\partial\bar{\cal A}_4$. \ \ \ \ From the Albanese map we see that $\widetilde{F(X)}$ is an etale cover of $F(X)$. Indeed, $\delta$ comes from a semiperiod $\delta'$ on $B$, giving a double cover $\widetilde{B}$ with involution $\iota$; the normalization of $\widetilde{F(X)}$ is then $S^2\widetilde{B}/\iota$, and $\widetilde{F(X)}$ is obtained by glueing above $\tau(B)$. \item In this case both $(Q, \nu)$ and $(Q, \nu\sigma)$ are $\partial^{\rm III}$, so $C, C'$ are nonsingular. The node of $Q$ represents a quadric of rank 3 through $\Phi(C)$, so ${\cal L}$ is cut out by the unique ruling. By the Schottky-Jung relations [M2], the vanishing theta null on $C$ descends to one on $A$. The double cover $\widetilde{F(X)}$ is again a $\partial^{\rm III}$-cover, in the sense that its normalization is ramified over $\tau'(B), \tau''(B)$, the sheets being glued.Each of the quintics in (5.11.2) gives two points of $\widetilde{F(X)}$, while the two-nodal quintics (5.11.3) land in the branch locus of $\pi$ \ (5.9.4). \end{list} \begin{center} \begin{tabular}{|l|l|l|l|l|} \hline Degeneration & Degeneration & & & \\ type of & type of & & & \\ \multicolumn{1}{|c|}{$(X,\delta)$} & \multicolumn{1}{|c|}{$(Q,\sigma,\nu,\nu\sigma)$} & \multicolumn{1}{|c|}{$(C,\mu)$} & \multicolumn{1}{|c|}{$(C',\mu')$} & \multicolumn{1}{|c|}{$A$} \\ \hline \hline & & & & \\ I : $\bar{\varepsilon} = \delta$ &$\varepsilon = \nu$ & nonsingular & nodal, $\partial^{\rm I}$ & ${\cal J}_{4}$ \\ & & trigonal & & \\ \hline & & & & \\ II : $\bar{\varepsilon} \neq \delta$, & $(\varepsilon, \sigma, \nu)$ & nodal, $\partial^{\rm II}$& nodal, $\partial^{\rm II}$& $\partial \bar{\cal A}_{4}$ \\ \multicolumn{1}{|c|}{$\langle \bar{\varepsilon} , \delta \rangle = 0$} & rank 3 & & & \\ & isotropic & & & \\ & subgroup & & & \\ \hline & & & & \\ III : $\langle \bar{\varepsilon} , \delta \rangle \neq 0$ & $\langle \varepsilon , \sigma \rangle = 0$ & nonsingular, & nonsingular, & $\theta_{\rm null}$ \\ &$\langle \varepsilon , \nu \rangle \neq 0$ & has vanishing & has vanishing & \\ & & thetanull ${\cal L}$, & thetanull ${\cal L}'$, & \\ & & ${\cal L}(\mu)$ even & ${\cal L}'(\mu')$ even & \\ \hline IV : & $\langle \varepsilon , \nu \rangle = 0$ & & nonsingular, & \\ nonsingular & $\langle \varepsilon , \sigma \rangle \neq 0$ & nodal, $\partial^{\rm II}$ & has vanishing & ${\cal A}_{4}$ \\ & & & thetanull ${\cal L}'$, & \\ & & & ${\cal L}'(\mu')$ odd & \\ \hline \end{tabular} \end{center} \bigskip \pagebreak[4] \noindent{\bf (5.13) Degenerations in ${\cal R}^2{\cal Q}^+$.} We have just described the universe as seen by a degenerating cubic threefold. From the point of view of a degenerating plane quintic, there are a few more possibilities though they lead to no new components. We retain the notation: $Q, \nu, \sigma, \varepsilon$ etc. \noindent 0. \ \ $\varepsilon$ cannot equal $\sigma$, since $\varepsilon$ is even, $\sigma$ odd. \noindent I. \ \ $\varepsilon=\nu$ reproduces case I of (5.12), as does: \noindent I$'$. \ $\varepsilon=\nu\sigma$. \begin{tabbing} \noindent II. \ \=Excluding the above, $\nu, \sigma, \varepsilon$ generate a subgroup of rank 3. If \\ \>it is isotropic, we are in case II above. \end{tabbing} \noindent III. If $\langle\varepsilon, \sigma\rangle=0$ but $\langle\varepsilon, \nu\rangle=\langle\varepsilon, \nu\sigma\rangle\neq0$, we're in case III. The only new cases are thus: \noindent IV. \ \ \ \ \ \ \ \ \ \ \ \ \ $\langle\varepsilon, \nu \ \ \rangle=0\neq\langle\varepsilon, \sigma\rangle, \ \ {\rm or}:$ \noindent IV.$'$ \ \ \ \ \ \ \ \ \ \ \ \ \ $\langle\varepsilon, \nu\sigma\rangle=0\neq\langle\varepsilon, \sigma\rangle,$ which is the same as IV after exchanging $C, C'$. In case IV, we find: \noindent $\bullet$ $X$ is non-singular, in fact any $X$ can arise. What is special is the line $\ell\in F(X)$ corresponding to $Q:$ it is contained in a plane which is tangent to $X$ along another line, $\ell'$. \noindent $\bullet$ $(Q, \nu)$ is a $\partial^{\rm II}$ degeneration, so $C$ is nodal, and $(C, \mu)$ is a $\partial^{\rm II}$ degeneration. \noindent $\bullet$ On the other hand, $(Q, \nu\sigma)$ is $\partial^{\rm III}$, so $C'$ is non-singular, and has a vanishing theta null ${\cal L}'$ (corresponding, as before, to the node of $Q$). \noindent $\bullet$ This time though, ${\cal L}'(\mu')$ is odd, so $A\in{\cal A}_4$ does not inherit a vanishing theta null. In fact, any $A\in{\cal A}_4$ arises from a singular quintic with degeneration of type IV. So far, we found three loci in $\bar{\cal A}_4$ which are related to nodal cubics: \[ \begin{array}{lcl} {\cal P}\circ \kappa^{-1}(\partial^{\rm I} {\cal RC}^{+}) & \subset & {\cal J}_{4} \\ \bar{\cal P}\circ \kappa^{-1}(\partial^{\rm II} {\cal RC}^{+}) & \subset & \partial \bar{A}_{4} \\ {\cal P}\circ \kappa^{-1}(\partial^{\rm III} {\cal RC}^{+}) & \subset & \theta_{\rm null} \end{array} \] We are now going to study, one at a time, the fibers of ${\cal P}$ above generic points in these three loci. We note that related results have recently been obtained by Izadi. In a sense, her results are more precise: she knows (cf. Remark 5.4) that $\chi$ is an isomorphism on the open complement ${\cal U}$ of a certain 6-dimensional locus in ${\cal A}_{4}$. In [I] she shows that for $A \in {\cal U}$, $\chi (A)$ is singular if and only if \[ A \in {\cal J}_{4} \cup \theta_{\rm null}. \] Her description of the cubic threefold corresponding to $A \in {\cal J}_{4}$ complements the one we give below. In general her techiques, based on $\Gamma_{00}$, are very different than our degeneration arguments. \bigskip \noindent{\bf (5.14) Jacobians} \bigskip \noindent{\bf Theorem 5.14} Let $B\in{\cal M}_4$ be a general curve of genus 4, and let $(X,\delta)=\chi(J(B)).$ \begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}} \item $X$ is the nodal cubic threefold corresponding to $B$ (5.11). \item $(X,\delta)\in\partial^I$, so $\widetilde{F(X)}$ is reducible, each component is isomorphic to $S^2B.$ \item Let $(Q,\sigma,\nu)$ be the plane quintic with rank-2 isotropic subgroup corresponding to some $\ell\in F(X)$. Then $Q$ is nodal, with trigonal normalization $T, \ \nu$ is the vanishing cycle, and \\ $(Q,\sigma)=(Q,\nu\sigma)\in \partial^{\rm II}$. \item$\bar{\cal P}^{-1}(J(B))$ is isomorphic to $\widetilde{F(X)}$.The component corresponding to $\nu$ (respectively $\nu\sigma$) consists of trigonal curves $T_{p,q}$ (respectively Wirtinger covers of singular curves $S_{p,q}$), \ \ \ $(p,q)\in S^2B.$ \item The tetragonal construction takes both $S_{p,q}$ and $T_{p,q}$ to $S_{r,s}$ and $T_{r,s}$ if and only if $p+q+r+s$ is a special divisor on $B$. The involution $\lambda$ exchanges $S_{p,q}, T_{p,q}$. \item Any two objects in $\bar{\cal P}^{-1}(J(B))$ can be connected by a sequence of two tetragonal moves (generally, in 10 ways). \end{list} \bigskip \noindent{\bf Proof} Since at least some of these results are needed for the proof of (5.8)(4), we do not use Theorem (5.3). For $(p,q)\in S^2B$, we consider: \noindent $\bullet$ $\widetilde{T}_{p,q}\to T_{p,q}$, the trigonal double cover associated by the trigonal construction to $B$ with the $g^1_4$ given by $\omega_B$(-$p$-$q$). \noindent $\bullet$ $\widetilde{S}_{p,q}\to S_{p,q}$, the Wirtinger cover of $S_{p,q}:=B/(p\sim q)$. (When $p=q$, this specializes to $B\cup_pR$, where $R$ is a nodal rational curve in which $p$ is a non singular point.) These objects are clearly in $\bar{\cal P}^{-1}(J(B))$. Beauville's list ([B1], (4.10)) shows that they exhaust the fiber. This proves part (4). Now clearly $\kappa$, as defined in (5.1.6), takes any of these objects to our $(X,\delta)$; so the analysis in (5.12)(I) applies, proving (1)-(3). (Note: this already suffices to complete the proof of (5.8)(4)!) Let $r+s+t+u$ be an arbitrary divisor in $|\omega_B(-p-q)|$.Projection of $\Phi(B)$ from the chord $\overline{\Phi(t),\Phi(u)}$ gives (the general) $g^1_4$ on $S_{p,q}$. The tetragonal construction takes this to the curves $T_{t,u}$ and $S_{r,s}$.(The situation is that of (2.15.2).) On $T_{p,q}$ there are two types of $g^1_4$'s, of the form ${\cal L}(x)$ and $\omega\otimes{\cal L}^{-1}(-x)$, where ${\cal L}$ is the trigonal bundle and $x\in T_{p,q}$. Now $x$ corresponds to a (2,2) partition, say $\{\{r,s\},\{t,u\}\}$, of some divisor in $|\omega_B(-p-q)|$. The tetragonal construction, applied to ${\cal L}(x)$, yields the curves $S_{r,s}$ and $S_{t,u}$; while when applied to $\omega\otimes{\cal L}^{-1}(-x)$, it gives $T_{r,s}$ and $T_{t,u}$. Altogether, this proves (5).We conclude with: \bigskip \noindent{\bf Lemma 5.14.7} Given any $p,q,r,s\in B$, there are points $t,u\in B$ (in general, 5 such pairs) such that both $p+q+t+u, r+s+t+u$ are special. \bigskip \noindent{\bf Proof} Let $\alpha, \beta$ be the maps of degree 4 from $B$ to ${\bf P}^1$ given by \\ $|\omega_B(-p-q)|, |\omega_B(-r-s)|$. Then $$\alpha\times\beta:B\to{\bf P}^1\times{\bf P}^1$$ exhibits $B$ as a curve of type (4,4) on a non-singular quadric surface, hence the image has arithmetic genus $(4-1)^2=9\rangle4=g(B)$, so there must be (in general, 5) singular points; these give the desired pairs $(t,u)$. \begin{flushright} QED \end{flushright} \bigskip \noindent{\bf (5.15) The Boundary.} The results in this case were obtained by Clemens [C2]. A general point $A$ of the boundary $\partial\bar{\cal A}_4$ of a toroidal compactification $\bar{\cal A}_4$ is a ${\bf C}^*$-extension of some $A_0\in{\cal A}_3$. The extension data is given by a point $a$ in the Kummer variety $A_0/(\pm 1)$. Given $a\in A_0$, consider the curve $$\widetilde{B}= \widetilde{B}_a:=\Theta\cap\Theta_a\subset A_0$$ (where $x\in\Theta_a\Leftrightarrow x+a\in\Theta$), and its quotient $B=B_a$ by the involution $x\mapsto -a-x$. We have $$(B,\widetilde{B})\in{\cal R}_4$$ and $${\cal P}(B,\widetilde{B})\approx A_0.$$ The pair $(B,\widetilde{B})$ does not change (up to isomorphism) when $a$ is replaced by $-a$. \bigskip \noindent{\bf Theorem 5.15 ([C2])} Let $A\in\partial\bar{\cal A}_4$ be the ${\bf C}^*$-extension of \\ $A_0\in{\cal A}_3$, a generic $P\!P\!A\!V$, determined by $\pm a\in A_0$. Let $(X, \delta)=\chi(A)$. \begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}} \item $X$ is the nodal cubic threefold corresponding to $B=B_a$. \item $(X,\delta)\in\partial^{\rm II}$, so $\widetilde{F(X)}$ is the etale double cover of $F(X)$ with normalization $S^2\widetilde{B}/\iota$, as in (5.12.II). \item The corresponding quintics $Q$ are nodal; all three of $\sigma,\nu,\nu\sigma$ are of type $\partial^{\rm II}$. \item $\doublebar{\cal P}^{-1}(A)$ is isomorphic to $\widetilde{F(X)}$, and consists of $\partial^{\rm II}$-covers $(C,\widetilde{C})$ whose normalizations (at one point) are of the form $(B_b,\widetilde{B}_b)$ for $b=b_1-b_2, \ \ b_1, b_2\in\psi(\widetilde{B})$. \end{list} \bigskip \noindent{\bf Proof} Clearly $\doublebar{\cal P}^{-1}(A)\subset\partial^{\rm II}\doublebar{\cal R}_5$, so consider a pair $(C,\widetilde{C})\in\partial^{\rm II}$, say $$C=N/(p\sim q),\;\;\;\widetilde{C}=\widetilde{N}/(p'\sim q', p"\sim q")$$ with $(N,\widetilde{N})\in\bar{\cal R}_4$. Then $\doublebar{\cal P}(C,\widetilde{C})$ is a ${\bf C}^*$-extension of $P(N,\widetilde{N})$, with extension data $$\pm(\psi(p')-\psi(q'))\in{\cal P}(N, \widetilde{N})/(\pm 1).$$ We see that $\doublebar{\cal P}(C,\widetilde{C})=A$ if and only if \noindent{\bf (5.15.5)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $(N,\widetilde{N})\in\bar{\cal P}^{-1}(A_0),$ \noindent and: \noindent{\bf (5.15.6)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\psi(p')-\psi(q')=a, \ \ \ p',q'\in\widetilde{N}.$ \medskip Now, (5.15.5) says that $(N,\widetilde{N})$ is taken, by its Abel-Prym map $\psi$, to $(B_b,\widetilde{B}_b)$ for some $b\in A_0$, and then (5.15.6) translates to: $$a=a_1-a_2, \ \ \ \ \ \ \ \ \ \ \ a_1,a_2\in\Theta\cap\Theta_b$$ which is equivalent to$$b=b_1-b_2, \ \ \ \ \ \ \ \ \ b_1,b_2\in\Theta\cap\Theta_a$$ (take $b_1=a_2+b, \ b_2=a_2)$. This proves (4), and everything else follows from what we have already seen. \begin{flushright} QED \end{flushright} \bigskip \noindent{\bf (5.16) Theta nulls} Let $A\in{\cal A}_4$ be a generic $PPAV$ with vanishing thetanull, and $(C,\widetilde{C})$ a generic element of ${\cal P}^{-1}(A)$. By [B1], Proposition (7.3), $C$ has a vanishing thetanull. This implies that the plane quintic $Q$ parametrizing singular quadrics through $\Phi(C)$ has a node, corresponding to the thetanull. The corresponding cubic threefold $X$ is thus also nodal, and we are again in the situation of (5.11.III). I do not see, however, a more direct way of describing the curve $B$ (or the cubic $X$) in terms of $A$. \bigskip \noindent{\bf (5.17) Pentagons and wheels}. In [V], Varley exhibits a two dimensional family of double covers $(C,\widetilde{C})\in{\cal R}_5$ whose Prym is the unique non-hyperelliptic $P\!P\!A\!V$ \linebreak $A\in{\cal A}_4$ with 10 vanishing thetanulls. The curves $C$ involved are Humbert curves, and each of these comes with a distinguished double cover $\widetilde{C}$. As an illustration of our technique, we work out the fiber of $\doublebar{\cal P}$ over $A$ and the tetragonal moves on this fiber. This is, of course, a very special case of (5.12)(III) or (5.16). We recall the construction of Humbert curves and their double covers. Start by marking 5 points $p_1,\cdots,p_5\in{\bf P}^1$.Take 5 copies $L_i$ of ${\bf P}^1$, and let $E_i$ be the double cover of $L_i$ branched at the 4 points $p_j, \ \ j\ne i$. Let \noindent{\bf (5.17.1)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $A:=\coprod^5_{i=1}L_i, \;\;\;\;\; B:=\coprod^5_{i=1}E_i.$ \noindent The \underline{pentagonal} construction applied to \noindent{\bf (5.17.2)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $B\stackrel{g}{\to}A\stackrel{f}{\to}{\bf P}^1$ \noindent($f$ is the forgetful map, of degree 5), yields a 32-sheeted branched cover $f_*B\to{\bf P}^1$ which splits, by (2.1.1), into 2 copies of the Humbert curve $C$, of degree 16 over ${\bf P}^1$. Let ${\beta}_I$, $I\subset S:=\{1,\cdots,5\}$, be the involution of (5.17.2) which fixes $A$ and acts non-trivially on $E_i, \ \ i\in I$. It induces an involution $\alpha_I$ on $f_*B$, hence on its quotient $C$. Let $$G:=\{\alpha_I \ \ | \ \ I\subset S\} \ / \ (\alpha_S).$$ Then $C$ is Galois over ${\bf P}^1$, with group $G\approx({\bf{Z}\rm} \newcommand{\C}{\cal C}/2{\bf{Z}\rm} \newcommand{\C}{\cal C})^4$. Let $G_i, \ \ \ 1\le i\le 5$, be the image in $G$ of $$\{\alpha_I|i \not\in I, \ \ \#(I)=\;\mbox{even}\}.$$Then $$C/\alpha_i\approx C/G_i\approx E_i,$$ and the quotient map $$E_i\approx C/\alpha_i\to C/G_i\approx E_i$$ becomes multiplication by 2 on $E_i$. In particular, the Humbert curve $C$ has 5 bielliptic maps $h_i:C\to E_i$. The branch locus of $h_i$ consists of the 8 points $x\in E_i$ satisfying $g(2x)=p_i$. For ease of notation, set $E:=E_5, \ \ \ p=p_5\in{\bf P}^1$, $$C\stackrel{h}{\to}E\stackrel{g}{\to}{\bf P}^1,$$ and $$\{p^0,p^1\}:=g^{-1}(p)\subset E.$$ Then for $j=0,1, \ \ E$ has a natural double cover $C^{j}$, branched at the four points ${1\over 2}p^j$ and given by the line bundle ${\cal O}_E(2p^j)$.The fiber product \noindent{\bf (5.17.3)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\widetilde{C}:=C^0\times_EC^1$ \noindent gives a Cartesian double cover of $C$. Replacing $E_5$ by another $E_i$, we get an isomorphic double cover $\widetilde{C}$. Here is an invariant description of this cover: Let $p_{i,j}:=L_i\cap f^{-1}(p_j)\in A$, and consider the curve $$Q:=A/(p_{i,j}\sim p_{j,i}, \ \ \ i\ne j).$$ Then $Q$ can be embedded in ${\bf P}^2$ as a pentagon, or completely reducible plane quintic curve: embed ${\bf P}^1$ as a non-singular conic, and take $L_i$ to be the tangent line of the conic at $p_i$. We have two natural branched double covers of $Q$: \noindent{\bf (5.17.4)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\widetilde{Q}_\sigma:=(\coprod^{ \: 5 \ \ \ 1} _{i=1,\varepsilon=0} \ \ L^\epsilon_i)/(p^0_{i,j}\sim p^1_{j,i}, \ \ \ i\ne j)$ \noindent{\bf (5.17.5)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\widetilde{Q}_\nu:=B/(\widetilde{p}_{i,j}\sim\widetilde{p}_{j,i}, \ \ \ i\ne j),$ \noindent where $\widetilde{p}_{i,j}\in E_i$ is the unique (ramification) point above $p_{i,j}\in L_i$. We may think of $\widetilde{Q}_\sigma$ as a "totally $\partial^I$" degeneration, and of $\widetilde{Q}_\nu$ as a "totally $\partial^{\rm III}$" degeneration. We then find: \begin{tabbing} \noindent{\bf (5.17.6)} \=$(Q,\widetilde{Q}_\nu)\in\overline{\cal R\cal Q}^+$ is the quintic double cover corresponding to the \\ \>Humbert curve $C\in{\cal M}_5$. \end{tabbing} \begin{tabbing} \noindent{\bf (5.17.7)} \=The double cover $\widetilde{Q}_\sigma$ of $Q$ corresponds, via (1.4.2), to the double \\ \>cover $\widetilde{C}$ of $C$. \end{tabbing} We note that $\widetilde{Q}_\sigma$ is itself an odd cover, so it corresponds to some (singular) cubic threefold. A moment's reflection shows that this must be Segre's cubic threefold $Y$ which we have already met in (4.8). Indeed, the Fano surface $F(Y)$ consists of the six rulings $R_i, \ \ \ 0\le i\le 5$, plus the 15 dual planes $\Pi_{i,j}^*$ of lines in $\Pi_{i,j}$ (notation of (4.8)). We see that: \begin{tabbing} \noindent{\bf (5.17.8)} \=The discriminant of projection of $Y$ from a line $\ell\in R_i$ is a plane \\ \>pentagon $Q$, with its double cover $\widetilde{Q}_\sigma$ as above. \end{tabbing} The other covers, $\widetilde{Q}_\sigma$, fit together to determine a point $(Y,\delta)\in\overline{\cal R\C}^+$: \noindent{\bf (5.17.9)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $(Y,\delta)=\kappa(C, \widetilde{C}),$ \noindent for any Humbert cover $(C,\widetilde{C})$. The tetragonal construction takes any $(Q,\widetilde{Q}_\sigma)$ to any other (in two steps), so we recover Varley's theorem: \noindent{\bf (5.17.10)} $A:={\cal P}(C, \widetilde{C})\in{\cal A}_4$ is independent of the Humbert cover $(C,\widetilde{C})$. But this is not the complete fiber: we have only used one of the two component types of $F(Y)$. We note: \noindent{\bf (5.17.11)} The discriminant of projection of $Y$ from a line $\ell\subset\Pi_{ij}$ consists of a conic plus three lines meeting at a point; the double cover is split. \vspace{2in} \begin{center} \begin{tabular}{cc} \hspace{2.3in} & \hspace{2.3in} \\ pentagon & wheel \end{tabular} \end{center} Consider a tritangent plane, meeting $Y$ in lines $\ell_i\in R_i, \ \ \ \ell_j\in R_j$, and $\ell_{ij}\in\Pi_{ij}^*$. It corresponds to a tetragonal construction involving two pentagons and a wheel. The other kind of tritangent plane intersects $Y$ in lines $\ell_{ij}\in\Pi_{ij}^*, \ \ell_{kl}\in\Pi_{kl}^*, \\ell_{mn}\in\Pi_{mn}^*$, where $\{i,j,k,l,m,n\}=\{0,1,2,3,4,5\}$; the tetragonal construction then relates three wheels. \noindent{\bf Theorem 5.18} Let $A\in{\cal A}_4$ be the non-hyperelliptic $P\!P\!A\!V$ with 10 vanishing thetanulls. \begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}} \item $\chi(A)$ consists of the Segre cubic threefold $Y$, with its degenerate semi-period $\delta$ (5.17.9). \item The corresponding curve $B\in\bar{\cal M}_4$ (5.11) consists of six ${\bf P}^1$'s: { \ } \vspace{2in} \item The Fano surface $F(Y)$ consists of the 6 rulings $R_i \ \ (0\le i\le 5)$ and the 15 dual planes $\Pi_{i,j}^*$.The plane quintics are pentagons, for $\ell\in R_i$, and wheels, for $\ell\in\Pi_{ij}^*$, \ all with split covers $\sigma$ \ (5.17.4, 5.17.11). (The $\nu$ covers are branched over all the double points.) \item The fiber $\doublebar{\cal P}^{-1}(A)$ is contained in the fixed locus of the involution $\lambda:\doublebar{\cal R}_5\to\doublebar{\cal R}_5$ (5.1.6), so it is a quotient of $F(Y).$ \item $\doublebar{\cal P}^{-1}(A)$ consists of two components: \begin{itemize} \item Humbert double covers $\widetilde{C} \rightarrow C$ (5.17.3). \item Allowable covers $\widetilde{X}_{0} \cup \widetilde{X}_{1} \rightarrow X_{0} \cup X_{1}$, where $X_{0}$, $X_{1}$ are elliptic, meeting at their 4 points of order 2. \end{itemize}\end{list} All of this follows from our previous analysis, except (5). The new, allowable, covers are obtained by applying Corollary (3.7), with $n=3$, to the Cartesian cover $\widetilde{C}\to C$ in (5.17.3). It is also easy to see that the plane quintic parametrizing singular quadrics through the canonical curve $\Phi(X_0\cup X_1)$ is a wheel, and vice versa, that the generalized Prym of any wheel (with its $\partial^{\rm III}$-cover) is the generalized Jacobian $J(X_0\cup X_1)$ of such a curve. Thus every line in $F(Y)$ is accounted for, so we have the complete fiber $\doublebar{\cal P}^{-1}(A)$. \begin{flushright} QED \end{flushright} \large \section{Other genera} \ \ \ \ For $g\le 4$, it is relatively easy to describe the fibers of ${\cal P}:\bar{\cal R}_g\to{\cal A}_{g-1}$. Indeed, every curve in ${\cal M}_g$ is trigonal, and every $A\in{\cal A}_{g-1}$ is a Jacobian (of a possibly reducible curve), so the situation is completely controlled by Recillas' trigonal construction. Similar results can be obtained, for \linebreak $g\le 3$, by using Masiewicki's criterion [Ma]. \bigskip \noindent{\bf (6.1)} \ \underline{$g=1$}. Here $\bar{\cal P}$ sends $\bar{\cal R}_1\approx{\bf P}^1$ to ${\cal A}_0$ (= a point). Thefibers of $\bar{\cal P},{\cal P}$ are then ${\bf P}^1, {\bf C}^*$ respectively. \bigskip \noindent{\bf (6.2)} \ \underline{$g=2$}. All curves of genus 2 are hyperelliptic, and all covers are Cartesian (3.2). An element of ${\cal R}_2$ is thus given by 6 points in ${\bf P}^1$, with 4 of them marked, modulo ${\bf P}GL(2)$; an element $E$ of ${\cal A}_1$ is given by 4 points of ${\bf P}^1$ modulo ${\bf P}GL(2)$; and ${\cal P}$ forgets the 2 unmarked points. The fiber of ${\cal P}$ is thus rational; it can be described as $S/G$ where \[ S := S^{2}\left( {\bf P}^{1} \setminus ({\rm 4 \; points}) \right) \setminus ({\rm diagonal})\] and $G \approx ({\bf Z}/2{\bf Z})^{2}$ is the Klein group, whose action on $S$ is induced from its action on ${\bf P}^{1}$ permuting the 4 marked points. We note that $S$ is ${\bf P}^{2}$ minus a conic $C$ and four lines $L_{i}$ tangent to it. To compactify it we add: $\bullet$ a $\partial^{\rm I}$ cover for each point of $C\backslash\cup L_i$, $\bullet$ a $\partial^{\rm III}$ cover for each point of $L_i\backslash C$, and $\bullet$ an "elliptic tail" cover [DS, IV 1.3] for each point in the exceptional divisor obtained by blowing up one of the points $L_i\cap C$. (The limiting double cover obtained is \[{(E_0\amalg E_1)/\approx} \; \longrightarrow \; {E/\sim}\] where $\sim$ places a cusp at one of the four marked points $p_i$ on $E$ and $\approx$ places a tacnode above it. These curves are unstable, and the family of elliptic-tail covers gives their stable models, each elliptic tail being blown down to the cusp.) The resulting $\overline{S}$ is ${\bf P}^{2}$ with 4 points in general position blown up, and the compactified fiber is $\overline{S}/G$, or ${\bf P}^{2}/G$ with one point blown up. \bigskip \noindent{\bf (6.3)} \ \underline{$g=3$}. Fix $A\in{\cal A}_2$. The Abel-Prym map sends pairs $(C,\tilde{C})\in{\cal P}^{-1}(A)$ to curves $\psi(\tilde{C})$ in the linear system $|2\Theta|$ on $A$, uniquely defined modulo translation by the group $G=A_2\approx({\bf{Z}\rm} \newcommand{\C}{\cal C}/2{\bf{Z}\rm} \newcommand{\C}{\cal C})^4$. The fiber is therefore, birationally, the quotient ${\bf P}^3/G$. Since some curves in $|2\Theta|$ are not stable, some blowing up is required to obtain the biregular model of $\bar{\cal P}^{-1}(A)$. This is carried out in [Ve]. The quotient ${\bf P}^3/G$ is identified with Siegel's modularquartic threefold, or the minimal compactification $\bar{\cal A}_2^{(2)}$ of the moduli space of $P\!P\!A\!V$'s with level-2 structure. To obtain $\bar{\cal P}^{-1}(A)$, Verra shows that we need to blow $\bar{\cal A}_2^{(2)}$ up at a point $A'$, corresponding to a level-2 structure on $A$ itself, and along a rational curve. The 2 exceptional divisors then parametrize hyperelliptic and elliptic-tail covers, respectively. \bigskip \noindent{\bf (6.4)} \ \underline{$g=4$}. As we noted in (5.15), the fiber ${\cal P}^{-1}(A), \ \ A\in{\cal A}_3$, consists of covers $(B_a,\tilde{B}_a), \ \ a\in A/(\pm 1):$ $$\tilde{B}_a=\Theta\cap\Theta_a,\;\;\;B_a=\tilde{B}_a/(x\sim(- a-x)).$$ The fiber is thus (birationally) the Kummer variety $A/(\pm 1)$. \bigskip \noindent{\bf (6.5)} \underline{$g\ge 7$}. In this case, it was proved in [FS], [K], and [W], that ${\cal P}$ is generically injective. The results in \S3 show that it is never injective: on the hyperelliptic loci there are positive-dimensional fibers, and various coincidences occur on the bielliptic loci. In [D1] we conjectured: \bigskip \noindent{\bf Conjecture 6.5.1} Any two objects in a fiber of ${\cal P}$ are connected by a sequence of tetragonal constructions. We state this for ${\cal P}$, rather than $\overline{\cal P}$, since various other phenomena can contribute to non-trivial fibers at the boundary. For example, all fibers of $\overline{\cal P}$ on $\partial^{\rm I}$ are two-dimensional. On the other hand, from the local pictures (2.14) it is clear that the tetragonal construction can take a nonsingular curve to a singular one. In fact proposition (3.8) shows that it is possible for two objects in ${\cal R}_g$ to be tetragonally related through an intermediate object of $\partial{\cal R}_g$, so some care must be taken in clarifying which class of tetragonal covers should be allowed. The conjecture is consistent with our results for $g\le 6$. For $g\ge 13$, Debarre [Deb2] proved it for curves which are neither hyperelliptic, trigonal, or bielliptic. Naranjo [N] extended this to generic bielliptics, $g\ge 10$. 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Mexicana 19 (1974), 9-13. \bib{Ra}D. Radionov, {\it letter}. \bib{SR}J.G. Semple, L. Roth, {\it Introduction to Algebraic Geometry}, Oxford U. Press, 1949. \bib{SV}R. Smith, R. Varley, {\it Components of the locus of singular theta divisors of genus 5}, LNM 1124, Springer-Verlag (1983), 338-416. \bib{T}A. Tjurin, {\it Five lectures on three dimensional varieties}, Russ. Math. Surv. 27 (1972). \bib{V}R. Varley, {\it Weddle's surfaces, Humbert's curves, and a certain 4-dimensional abelian variety}, Amer. J. Math. 108 (1986), 931-952. \bib{Ve}A. Verra, {\it The fibre of the Prym map in genus three}, Math. Ann. 276 (1987), 433-448. \bib{W}G. Welters, {\it Recovering the curve data from a general Prym variety}, Amer. J. of Math 109 (1987), 165-182. \end{document}

9206

alg-geom/9206006

fr

https://arxiv.org/abs/alg-geom/9206006

alg-geom/9206006

Jean Francois Mestre

Jean-Francois Mestre

Corps quadratiques dont le 5-rang du groupe des classes est >=3

4 pages, LaTeX. (A paraitre dans les Comptes-Rendus de l'Acad. des Sciences de Paris.)

We prove the existence of infinitely many real and imaginary fields whose 5-rank of the class group is >=3.

alg-geom

\section*{Bibliographie\markboth {REFERENCES}{REFERENCES}}\list {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth \advance\leftmargin\labelsep \usecounter{enumi}} \def\hskip .11em plus .33em minus -.07em{\hskip .11em plus .33em minus -.07em} \sloppy \sfcode`\.=1000\relax} \let\endthebibliography=\endlist \def\@begintheorem#1#2{\global\advance\@listdepth -1\relax \list{}{\leftmargin 0pt\labelwidth -\labelsep}\item[{\sc #1\ #2}]\it} \def\@endtheorem{\endlist\global\advance\@listdepth 1\relax} \def\section{\@startsection {section}{1}{\z@}{3.5ex plus 1ex minus .2ex}{2.3ex plus .2ex}{\large\bf}} \def\subsection{\@startsection{subsection}{2}{\z@}{3.25ex plus 1ex minus .2ex}{1.5ex plus .2ex}{\normalsize\bf}} \def\subsubsection{\@startsection{subsubsection}{3}{\z@}{3.25ex plus 1ex minus .2ex}{1.5ex plus .2ex}{\normalsize\bf}} \def\section*{Table des mati\`eres{\section*{Table des mati\`eres} \markboth{CONTENTS}{CONTENTS} \@starttoc{toc}} \def\biblio#1#2#3#4#5{{\sc #1}, {\it #2}, #3 {\bf #4} #5} \def\arabic{section}. {\arabic{section}.} \def\thesection\arabic{subsection}. {\arabic{section}.\arabic{subsection}.} \def\thesubsection \arabic{subsubsection}. {\thesection\arabic{subsection}. \arabic{subsubsection}.} \def\rm Cl\;{\rm Cl\;} \documentstyle[A4,11pt]{article} \begin{document} G\'eom\'etrie alg\'ebrique/ {\it Algebraic geometry.} \par\medskip {\large Corps quadratiques dont le $5$-rang du groupe des classes est $\geq 3$.} Jean-Fran\c cois Mestre. {\bf R\'esum\'e.-} Nous prouvons qu'il existe une infinit\'e de corps quadratiques r\'eels (resp. imaginaires) dont le $5$-rang du groupe des classes d'id\'eaux est $\geq 3$. \par\medskip {\large Quadratic fields whose $5$-rank is $\geq 3$.} {\bf Abstract.-} We prove the existence of infinitely many real and imaginary fields whose $5$-rank of the class group is $\geq 3$. \vspace{5ex} Soient $K$ un corps de nombres, $\rm Cl\; K$ le groupe des classes d'id\'eaux de $K$, et $p$ un nombre premier. Par d\'efinition, le $p$-rang du groupe des classes de $K$ est la dimension de $\rm Cl\; K/p\rm Cl\; K$ sur $\hbox{\bf F}_p$. \par\medskip Nous d\'emontrons ici le th\'eor\`eme suivant: \begin{theo} Il existe une infinit\'e de corps quadratiques r\'eels (resp. imaginaires) dont le $5$-rang du groupe des classes est $\geq 3$. \end{theo} L'id\'ee de la d\'emonstration est la suivante: soit $E$ une courbe elliptique d\'efinie sur $\hbox{\bf Q}$, munie d'un point $P$ d'ordre $5$ rationnel sur $\hbox{\bf Q}$; si $F$ est la courbe quotient $E/<P>$, notons $\phi:\;\;E\rightarrow F$ l'isog\'enie canonique de $E$ sur $F$. Le lemme suivant, cons\'equence de \cite{RAYNAUD:schemas} et du lemme $3$ de \cite{RAYNAUD:jacobienne}, m'a \'et\'e indiqu\'e par M. Raynaud: {\sc Lemme.-} {\it Supposons $E$ semi-stable en tout nombre premier $p$; soient $K$ un corps quadratique, et $O_K$ son anneau d'entiers. Si ${\cal E}$ (resp. ${\cal F}$) est le mod\`ele de N\'eron de $E$ (resp. $F$) sur $O_K$, on a une suite exacte (de sch\'emas en groupes sur $O_K$): $$0\rightarrow \hbox{\bf Z}/5\hbox{\bf Z} \rightarrow {\cal E} \rightarrow {\cal F}'\rightarrow 0,$$ o\`u $\cal F'$ est un sous-sch\'ema en groupes ouvert de ${\cal F}$ contenant la composante neutre ${\cal F}^0$ de ${\cal F}$.} \par\medskip Par suite, l'image r\'eciproque par $\phi$ de tout point de ${\cal F}'(O_K)$ engendre une extension ab\'elienne non ramifi\'ee de degr\'e divisant $5$ de $K$. Soit $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ une \'equation minimale de Weierstrass de $F$ sur $\hbox{\bf Z}$. Soit $Q=(x,y)$ un point de $F$ tel que $x\in \hbox{\bf Q}$; $Q$ est alors rationnel sur le corps $K=\hbox{\bf Q}(y)$. Notons $S$ l'ensemble fini des nombres premiers $p$ tels que le nombre de composantes connexes de la fibre en $p$ du mod\`ele de N\'eron de $F$ sur $\hbox{\bf Z}$ est divisible par $5$. Supposons que, pour tout nombre premier $p\in S$, $Q$ ne se r\'eduise pas $\mathop{\;\rm mod}\nolimits p$ en le point singulier de $F_{/\hbox{\bf F}_p}$. Alors le point $Q$ se prolonge en un point de ${\cal F}'(O_K)$. Par suite, si $L=K(\phi^{-1}(Q))$, $L$ est une extension non ramifi\'ee de $K$. Le th\'eor\`eme d'irr\'eductibilit\'e de Hilbert permet de montrer que, pour une infinit\'e de tels $x\in \hbox{\bf Q}$, $L/K$ est de degr\'e $5$ (En fait, comme on le verra plus loin, on peut donner des crit\`eres effectifs de congruence modulo des nombres premiers convenables pour assurer que $L/K$ est de degr\'e $5$.) Nous construisons dans la section suivante une courbe $X$, d\'efinie sur $\hbox{\bf Q}$, poss\'edant les propri\'et\'es suivantes: i) Il existe un rev\^etement $\psi$ de degr\'e $2$ de $X$ sur la droite projective. ii) Pour $1\leq i\leq 3$, il existe une courbe elliptique $E_i$, semi-stable sur $\hbox{\bf Z}$, poss\'edant un point $\hbox{\bf Q}$-rationnel $P_i$, et un rev\^etement ab\'elien $\tau_i$ de groupe de Galois $(\hbox{\bf Z}/2\hbox{\bf Z})^2$ de $X$ sur la courbe $F_i=E_i/<P_i>$. Notons $\phi_i$ le morphisme canonique de $E_i$ sur $F_i$. Nous montrons ensuite que, pour une infinit\'e de nombres rationnels $x$, si $K=\hbox{\bf Q}(\psi^{-1}(x))$, les trois extensions $L_i=K(\phi^{-1}(\tau_i(\psi^{-1}(x))))$ de $K$ sont ab\'eliennes de degr\'e $5$, non ramifi\'ees et ind\'ependantes (i.e. les \'el\'ements de $\mathop{\rm Hom}\nolimits(\rm Cl\; K,\hbox{\bf Z}/5\hbox{\bf Z})$ dont elles proviennent sont ind\'ependants), ce qui prouve le th\'eor\`eme. {\sc Remarque.- } Soit $E$ une courbe elliptique d\'efinie sur $\hbox{\bf Q}$, munie d'un point rationnel $P$ d'ordre $n$, $n$ entier $\geq 1$. Soit $F$ la courbe quotient $E/<P>$, $\phi:\;\;E\rightarrow F$ le morphisme canonique, et $y^2=f(x)$ un mod\`ele de Weierstrass de $F$. Soit $x\in \hbox{\bf Q}$, et $Q$ l'un des deux points de $F$ d'abscisse $x$. Si $K=\hbox{\bf Q}(\sqrt{f(x)})$, le th\'eor\`eme de Chevalley-Weil permet de montrer que, d\`es que la valuation de $x$ est suffisamment n\'egative en chaque nombre premier o\`u $E$ a mauvaise r\'eduction, l'extension $K(\phi^{-1}(Q))/K$ est non ramifi\'ee. L'hypoth\`ese ``{\it $E$ semi-stable}'' n'est donc pas n\'ecessaire. N\'eanmoins, dans le cas o\`u elle n'est pas v\'erifi\'ee, les conditions de congruence sur $x$ sont plus d\'elicates \`a d\'eterminer. \section{Construction de la courbe $X$} \subsection{Construction de $E$ et $F$} La courbe modulaire $X_1(10)$, classifiant les courbes elliptiques munies d'un point d'ordre $10$, est de genre $0$, et a \'et\'e param\'etr\'ee par Kubert \cite{KUBERT:universal}: il construit une courbe elliptique $E$ rationnelle sur $\hbox{\bf Q}(f)$, o\`u $f$ est un param\`etre, poss\'edant un point $P_0$ d'ordre $10$ rationnel sur $\hbox{\bf Q}(f)$. Si l'on pose $f=(u+1)/2$, et apr\`es un changement de variables, on trouve comme \'equation de $E$: $$y^2=(x^2-u(u^2+u-1))(8xu^2+(u^2+1)(u^4-2u^3-6u^2+2u+1)) .$$ Le point $P_0$ d'ordre $10$ est le point d'abscisse $-{\frac {\left (u^{4}-2\,u^{3}-6\,u^{2}+2\,u+1\right )\left (u^{2}+1 \right )}{8\,u^{2}}}.$ Les formules de V\'elu \cite{VELU:isog} permettent alors d'obtenir une \'equation de la courbe $F$ quotient de $E$ par le groupe d'ordre $5$ engendr\'e par $2P_0$; une \'equation de $F$ est donn\'ee par $y^2=g_u(x)$, o\`u $$g_u(x)=(x^2-u(u^2+u-1))h_u(x)\;\;\; {\rm et}\;\;\; h_u(x)=8(u^2+u-1)^2x+(u^2+1)(u^4+22u^3-6u^2-22u+1).$$ De plus, si $u\in \hbox{\bf Q}$, la condition $u\equiv \pm 1 \mathop{\;\rm mod}\nolimits 5$ assure que les courbes $E$ et $F$ sont semi-stables sur $\hbox{\bf Z}$. \subsection{Construction de $X$} Si $$\left\{\begin{array}{l} u_1=(t^2+t-1)/(t^2+t+1),\\ u_2=-(t^2+3t+1)/(t^2+t+1),\\ u_3=-(t^2-t-1)/(t^2+t+1) \end{array}\right.$$ on a $$u_1(u_1^2+u_1-1)=u_2(u_2^2+u_2-1)=u_3(u_3^2+u_3-1).$$ La courbe $X$, d\'efinie sur $\hbox{\bf Q}(t)$, normalis\'ee de la courbe d'\'equations $$\left\{\begin{array}{l} y_1^2=g_{u_1}(x),\\ y_2^2=g_{u_2}(x),\\y_3^2=g_{u_3}(x) \end{array}\right.$$ est de genre $5$. L'application $\phi:\;\;(x,y_1,y_2,y_3)\mapsto (x,v=y_1/y_2,w=y_1/y_3)$ de $X$ sur la courbe $C$ de genre $0$ et d'\'equations $$h_{u_1}(x)=v^2h_{u_2}(x)=w^2h_{u_3}(x)$$ est de degr\'e $2$; les quatre points de $C$ de coordonn\'ees $(x,v,w)=(\infty, \pm u_2/u_1,\pm u_3/u_1)$ sont rationnels sur $\hbox{\bf Q}(t)$ et sont des points de ramification de $\phi$, donc $C$ est $\hbox{\bf Q}(t)$-isomorphe \`a la droite projective, et $X$ est hyperelliptique, rev\^etement double de la droite projective, et poss\`ede quatre points de Weierstrass rationnels sur $\hbox{\bf Q}(t)$. Sp\'ecialisons en $t=4$ les formules de la section pr\'ec\'edente. On trouve $u_1=19/21$, $u_2=-29/21$ et $u_3=-11/21$. Les courbes $E_i$ et $F_i$, pour $1\leq i\leq 3$, sont donc semi-stables sur $\hbox{\bf Z}$. Une param\'etrisation de la courbe $C$ est donn\'ee par $$v=\frac{29}{19}\frac{53719189282 z^2+26766692861}{ 53719189282 z^2 - 283246634396 z- 26766692861},$$ $$w=\frac{11}{19}\frac{53719189282 z^2+26766692861}{ 53719189282 z^2+ 20305766998 z- 26766692861},$$ $$x=\frac{c_4z^4+c_3z^3+c_2z^2+c_1z+c_0}{5167944494559 (4883562662 z+922989409) (11 z -29) z},$$ avec $$\begin{array}{ll} c_0=343898806423252015354080,&c_1=- 411804539876837130626339,\\ c_2=- 642297925780193483509181,&c_3=826467660375890872281118,\\ c_4=1385160622615364964251520.&\end{array}$$ On obtient alors une \'equation hyperelliptique de $X$ en substituant la fraction rationnelle $x(z)$ ci-dessus dans, par exemple, l'\'equation de $E_1$, \`a savoir $$y^2=42(44876601 x-133597561)(9261 x^2-6061).$$ \`A toute valeur rationnelle de $z$ distincte des p\^oles de $x(z)$, on associe ainsi le corps $K=\hbox{\bf Q}(y)$. \subsection{Conditions sur $z$ pour que le $5$-rang de $\rm Cl\; K$ soit $\geq 3$} Si $p$ est un nombre premier, notons $v_p$ la valuation $p$-adique. Le calcul montre qu'un point de $F_1(K)$ (resp. $F_2(K)$, resp. $F_3(K)$) se prolonge \`a ${\cal F_1}'(O_K)$ (resp. ${\cal F_2}'(O_K)$, resp. ${\cal F_3}'(O_K)$) si et seulement si son abscisse $x$ v\'erifie $v_{11}(x)\leq -2$, $v_{29}(x)\leq -2$ et $x\not\equiv 77 \mathop{\;\rm mod}\nolimits 419$ (resp. $v_{11}(x)\leq -2$, $v_{19}(x)\leq -2$, $x\not\equiv 677 \mathop{\;\rm mod}\nolimits 709$, resp. $v_{19}(x)\leq -2$, $v_{29}(x)\leq -2$, $x\not\equiv 36 \mathop{\;\rm mod}\nolimits 151$). Si \begin{equation} z\equiv 0 \mathop{\;\rm mod}\nolimits 11.19.29\;\;\;{\rm et}\;\;\; z\not\equiv \pm 86\mathop{\;\rm mod}\nolimits 419,\end{equation} les conditions de congruence ci-dessus sont remplies, et les points correspondants de $F_i(K)$, $i=1,2,3$, se prolongent en des points de $F'_i(O_K)$. De plus, soit $l_1=163$, $l_2=701$ et $l_3=1277$; supposons \begin{equation} z\equiv 1\mathop{\;\rm mod}\nolimits l_1l_2l_3.\end{equation} Alors: i) Les id\'eaux $(l_i)$, $i=1,2,3$, se d\'ecomposent chacun dans $K$ en deux id\'eaux ${\cal P_i}\overline{\cal P_i}$, ii) ${\cal P_1}$ (resp. ${\cal P_2}$, resp. ${\cal P_3}$) est d\'ecompos\'e (resp. d\'ecompos\'e, resp. inerte) dans $L_1$, iii) ${\cal P_1}$ (resp. ${\cal P_2}$, resp. ${\cal P_3}$) est inerte (resp. d\'ecompos\'e, resp. d\'ecompos\'e) dans $L_2$, iv) ${\cal P_1}$ (resp. ${\cal P_2}$, resp. ${\cal P_3}$) est d\'ecompos\'e (resp. inerte, resp. d\'ecompos\'e) dans $L_3$. Ceci assure que les extensions $L_i$, $i=1,2,3$, sont ind\'ependantes. Par suite, d\`es que $z$ v\'erifie les congruences $(1)$ et $(2)$ ci-dessus, le $5$-rang du groupe des classes de $K$ est $\geq 3$. Le fait que, lorsque $z$ parcourt une infinit\'e de valeurs rationnelles, on obtient une infinit\'e de corps quadratiques $K$ provient par exemple du th\'eor\`eme de Faltings (la courbe $X$, \'etant de genre $>1$, n'a qu'un nombre fini de points rationnels dans un corps $K$ fix\'e.) De plus, comme $X$ a trois points de Weierstrass rationnels, donc r\'eels, on obtient ainsi une infinit\'e de corps quadratiques r\'eels (resp. imaginaires) dont le $5$-rang du groupe des classes est $\geq 3$. Par exemple, si $z$ est $>0$ (resp. $<0$) et suffisamment proche de $0$ (pour la topologie usuelle), $K$ est quadratique imaginaire (resp. r\'eel).

9206

alg-geom/9206007

fr

https://arxiv.org/abs/alg-geom/9206007

alg-geom/9206007

Jean Francois Mestre

Jean-Francois Mestre

Rang de courbes elliptiques d'invariant donne

4 pages. (A paraitre aux Comptes-Rendus de l'Acad. des Sc. de Paris.)

We prove that there exist infinitely many elliptic curves over \Q with given modular invariant, and rank >=2. Furthermore, there exist infinitely many elliptic curves over $\Q$ with invariant equal at 0 (resp. 1728) and rank >=6 (resp. >=4).

alg-geom

\section*{Bibliographie\markboth {REFERENCES}{REFERENCES}}\list {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth \advance\leftmargin\labelsep \usecounter{enumi}} \def\hskip .11em plus .33em minus -.07em{\hskip .11em plus .33em minus -.07em} \sloppy \sfcode`\.=1000\relax} \let\endthebibliography=\endlist \def\@begintheorem#1#2{\global\advance\@listdepth -1\relax \list{}{\leftmargin 0pt\labelwidth -\labelsep}\item[{\sc #1\ #2}]\it} \def\@endtheorem{\endlist\global\advance\@listdepth 1\relax} \def\section{\@startsection {section}{1}{\z@}{3.5ex plus 1ex minus .2ex}{2.3ex plus .2ex}{\large\bf}} \def\subsection{\@startsection{subsection}{2}{\z@}{3.25ex plus 1ex minus .2ex}{1.5ex plus .2ex}{\normalsize\bf}} \def\subsubsection{\@startsection{subsubsection}{3}{\z@}{3.25ex plus 1ex minus .2ex}{1.5ex plus .2ex}{\normalsize\bf}} \def\section*{Table des mati\`eres{\section*{Table des mati\`eres} \markboth{CONTENTS}{CONTENTS} \@starttoc{toc}} \def\biblio#1#2#3#4#5{{\sc #1}, {\it #2}, #3 {\bf #4} #5} \def\arabic{section}. {\arabic{section}.} \def\thesection\arabic{subsection}. {\arabic{section}.\arabic{subsection}.} \def\thesubsection \arabic{subsubsection}. {\thesection\arabic{subsection}. \arabic{subsubsection}.} \documentstyle[A4,11pt]{article} \begin{document} G\'eom\'etrie alg\'ebrique/ {\it Algebraic geometry.} \par\medskip {\large Rang de courbes elliptiques d'invariant donn\'e.} Jean-Fran\c cois Mestre. \par\medskip {\bf R\'esum\'e.-} Nous montrons qu'il existe une infinit\'e de courbes elliptiques d\'efinies sur $\hbox{\bf Q}$, d'invariant modulaire donn\'e, et de rang $\geq 2$. De plus, il existe une infinit\'e de courbes d\'efinies sur $\hbox{\bf Q}$, d'invariant nul (resp. \'egal \`a $1728$), et de rang $\geq 6$ (resp. $\geq 4$). \par\medskip {\large On the rank of elliptic curves with given modular invariant.} {\bf Abstract.-} We prove that there exist infinitely many elliptic curves over $\hbox{\bf Q}$ with given modular invariant, and rank $\geq 2$. Furthermore, there exist infinitely many elliptic curves over $\hbox{\bf Q}$ with invariant equal to $0$ (resp. $1728$), and rank $\geq 6$ (resp. $\geq 4$). \vspace{5ex} Soit $k$ un corps de caract\'eristique nulle, et $t$ une ind\'etermin\'ee. Nous prouvons ici les th\'eor\`emes suivants: \begin{theo} Soit $j$ un \'el\'ement de $k$. Il existe une courbe elliptique d\'efinie sur $k(t)$, d'invariant modulaire $j$, qui n'est pas $k(t)$-isomorphe \`a une courbe elliptique d\'efinie sur $k$, et qui poss\`ede deux points rationnels sur $k(t)$ lin\'eairement ind\'ependants. \end{theo} \begin{theo} Il existe une courbe elliptique d\'efinie sur $k(t)$, dont l'invariant modulaire est \'egal \`a $1728$ (resp. $0$), qui n'est pas $k(t)$-isomorphe \`a une courbe d\'efinie sur $k$, et qui poss\`ede $4$ (resp. $6$) points rationnels sur $k(t)$ lin\'eairement ind\'ependants. \end{theo} On en d\'eduit par sp\'ecialisation les corollaires suivants: \begin{cor} Soit $j$ un \'el\'ement de $\hbox{\bf Q}$. Il existe une infinit\'e de courbes elliptiques d\'efinies sur $\hbox{\bf Q}$, non deux \`a deux $\hbox{\bf Q}$-isomorphes, d'invariant modulaire $j$, dont le rang du groupe de Mordell-Weil est $\geq 2$. \end{cor} \begin{cor} Il existe une infinit\'e de courbes elliptiques d\'efinies sur $\hbox{\bf Q}$, d'invariant modulaire \'egal \`a $1728$ (resp. $0$), non deux \`a deux $\hbox{\bf Q}$-isomorphes, dont le rang du groupe de Mordell-Weil est $\ge 4$ (resp. $\ge 6$). \end{cor} \section{D\'emonstration du th\'eor\`eme $1$} \begin{theo} Soient $k$ un corps de caract\'eristique nulle, et $E$ et $E'$ deux courbes elliptiques d\'efinies sur $k$. On suppose que les invariants modulaires $j(E)$ et $j(E')$ ne sont pas simultan\'ement \'egaux \`a $0$ ou \`a $1728$. Il existe alors une courbe $C$, rev\^etement quadratique de la droite projective, d\'efinie sur $k$, et deux morphismes ind\'ependants $p:\;\;C\rightarrow E$ et $p':\;\;C\rightarrow E'$ d\'efinis sur $k$. \end{theo} (On rappelle que deux morphismes $p:\;\;C\rightarrow E$ et $p':\;\;C\rightarrow E'$ sont dits ind\'ependants si les images r\'eciproques par $p^*$ et $p'^*$ des formes de premi\`ere esp\`ece de $E$ et $E'$ sont lin\'eairement ind\'ependantes.) \par\medskip Soient $y^2=x^3+ax+b$ une \'equation de $E$ et $y^2=x^3+a'x+b'$ une \'equation de $E'$. L'hypoth\`ese sur $j(E)$ et $j(E')$ implique que $a=0\rightarrow a'\neq 0$ et $b=0\rightarrow b'\neq 0$. Posons $f(x)=x^3+ax+b$ et $g(x)=x^3+a'x+b'$. Si $u$ est une ind\'etermin\'ee, l'\'equation (en $x$) $$u^6f(x)=g(u^2x)$$ a pour solution $x=\phi(u)$, avec $\phi(u)=-\fracb{b'-u^6b}{u^2(a'-u^4a)}.$ Soit $C$ la courbe d'\'equation $Y^2=f(\phi(X)).$ Soient $\rho:\;\;\;C\rightarrow E$ et $\rho':\;\;\;C\rightarrow E'$ les morphismes donn\'es par $\rho(X,Y)=(x=\phi(X),y=Y)$ et $\rho'(X,Y)=(x=X^2\phi(X),y=X^3 Y)$. Si $\omega=\rho^*(dx/y)$ et $\omega'=\rho'^*(dx/y)$, on a $$\omega/\omega'= \frac{{ { 3 a { X^{4}} b'} { - 2 { X^{6}} b a'} {- b'a' }}}{{ { X^{3}} {( { { { X^{6}} b a} { - 3 { X^{2}} b a'}+ { 2 a b'}} )}}},$$ fraction rationnelle en $X$ non constante. Par suite, $\omega$ et $\omega'$ sont ind\'ependantes dans l'espace des formes diff\'erentielles de premi\`ere esp\`ece de $C$, d'o\`u le th\'eor\`eme. \par\medskip {\sc Remarque.} Le calcul montre que le genre de $C$ est $\leq 10$. Plus pr\'ecis\'ement, si l'invariant modulaire $j(E)$ de $E$ n'est pas \'egal \`a $j(E')$, et si $j(E)$ et $j(E')$ sont distincts de $0$ et $1728$, le genre de $C$ est \'egal \`a $10$. Si $j(E)=j(E')$, et distinct de $0$ et $1728$, le genre de $C$ est \'egal \`a $6$. Si $j(E)=1728$, et $j(E')\neq 0$, le genre de $C$ vaut $7$. Si $j(E)=0$, et $j(E')\neq 1728$, le genre de $C$ vaut $8$. Enfin, si $j(E)=0$ et $j(E')=1728$, le genre de $C$ vaut $5$. \begin{theo} Soient $k$ un corps de caract\'eristique nulle, et $j$ un \'el\'ement de $k$. Il existe une courbe $C$ d\'efinie sur $k$, rev\^etement quadratique de la droite projective, une courbe elliptique $E$ d\'efinie sur $k$ d'invariant $j$, et deux morphismes ind\'ependants $p$ et $p'$ de $C$ dans $E$ d\'efinis sur $k$. \end{theo} Si $j\in k$, $j\neq 0,1728$, et si $a=b=\fracb{27j}{4(j-1728)}$, la courbe elliptique $E$, d\'efinie sur $k$, d'\'equation $y^2=x^3+ax+b$ a comme invariant modulaire $j$. Le th\'eor\`eme pr\'ec\'edent permet donc de conclure, sauf si $j=0$ ou $j=1728$. Or la jacobienne de la courbe de genre $2$, d\'efinie sur $\hbox{\bf Q}$, d'\'equation $y^2=x^6+1$ est $\hbox{\bf Q}$-isog\`ene au produit de la courbe elliptique $y^2=x^3+1$, d'invariant modulaire \'egal \`a $0$, avec elle-m\^eme. D'o\`u le r\'esultat si $j=0$. De m\^eme, soit $C$ la courbe de genre $2$ d'\'equation $y^2=(t^2+1)(t^2-2)(2t^2-1).$ Les morphismes $(t,y)\mapsto (t^2,y)$ et $(t,y)\mapsto (1/t^2,y/t^3)$ d\'efinissent deux rev\^etements de $C$ sur la courbe elliptique d'\'equation $y^2=(x+1)(x-2)(2x-1)$, dont l'invariant modulaire vaut $1728$. Cela ach\`eve la d\'emonstration du th\'eor\`eme. \par\medskip {\sc Remarques.-} Si $E$ est une courbe elliptique d\'efinie sur $k$, il est parfois possible de trouver une courbe hyperelliptique d\'efinie sur $k$, de genre $<10$, dont la jacobienne est $k$-isog\`ene \`a $E\times E\times A$, o\`u $A$ est une vari\'et\'e ab\'elienne convenable. Par exemple: 1) Soit $E$ une courbe elliptique d\'efinie sur $k$ d'\'equation $y^2=x^3-ax+b$, o\`u $a$ est non nul et de la forme $\alpha^2+3\beta^2$, $\alpha$, $\beta \in k$. La conique $x_1^2+x_1x_2+x_2^2=a$ est alors $k$-isomorphe \`a la droite projective, d'o\`u l'existence de deux fractions rationnelles $x_1(t)$ et $x_2(t)$ telles que la fraction rationnelle $f(t)=x_1^3-ax_1+b$ soit \'egale \`a la fraction rationnelle $x_2^3-ax_2+b$. On en d\'eduit $2$ applications rationnelles $(t,y)\mapsto (x_1(t),y)$ et $(t,y)\mapsto (x_2(t),y)$ de la courbe $C$ d'\'equation $y^2=f(t)$ sur $E$. Les fractions rationnelles $x'_1$ et $x'_2$ n'\'etant pas proportionnelles, et la courbe $C$ \'etant de genre $3$, on en d\'eduit que la jacobienne de la courbe $C$ est $k$-isog\`ene \`a $E\times E\times E_1$, o\`u $E_1$ est une courbe elliptique d\'efinie sur $k$. \par\medskip 2) Soient $E_1$ et $E_2$ deux courbes elliptiques, d\'efinies sur $k$, dont les points d'ordre $2$ appartiennent \`a $k$. Si $y^2=(x-a)(x-b)(x-c)$ (resp. $y^2=(x-a')(x-b')(x-c')$) est une \'equation de $E_1$ (resp. $E_2$), quitte \`a permuter les r\^oles de $a,b,c$, on peut trouver une application affine $x\mapsto h(x)=\alpha x+\beta$ telle que $h(a)=a'$, $h(b)=b'$, et $h(c)\neq c'.$ La jacobienne de la courbe de genre $2$ d'\'equations $$y^2=(x-a)(x-b)(x-c),\;\;\;z^2=\alpha (x-a)(x-b)(x-h^{-1}(c'))$$ est alors isog\`ene \`a $E_1\times E_2$. \par\bigskip Le th\'eor\`eme 1 d\'ecoule ais\'ement du th\'eor\`eme pr\'ec\'edent. En effet, si $j\in k$, d'apr\`es le th\'eor\`eme pr\'ec\'edent, il existe une courbe $C$, d\'efinie sur $k$, rev\^etement quadratique de la droite projective, une courbe elliptique $E$ d\'efinie sur $k$ d'invariant $j$, et deux morphismes ind\'ependants $p_1$ et $p_2$ de $C$ sur $E$. Soit $w$ l'involution hyperelliptique de $C$; les morphismes $p_1\circ w+p_1$ et $p_2\circ w+p_2$ de $C$ dans $E$ sont constants, car $w$ agit sur la jacobienne de $C$ comme $-1$. Par suite, les morphismes $p'_1=p_1\circ w-p_1$ et $p'_2=p_2\circ w-p_2$ sont ind\'ependants; si $y^2=f(t)$ est une \'equation de $C$, et si $E_w$ est la courbe obtenue \`a partir de $E$ par torsion par $\sqrt{f(t)}$, les points $P_1=p'_1(t,\sqrt{f(t)})$ et $P_2=p'_2(t,\sqrt{f(t)})$ sont des points ind\'ependants de $E_w$, rationnels sur $k(t)$. D'o\`u le th\'eor\`eme $1$. \section{D\'emonstration du th\'eor\`eme 2} \subsection{Le cas des courbes d'invariant $j=1728$} Soit $p(x)=x^4+a_2x^2+a_1x+a_0$ un \'el\'ement de $k[x]$, dont les racines $x_i$, $1\leq i\leq 4$, appartiennent \`a $k$, et sont de somme nulle. La courbe $E$ d'\'equation $x^4+a_2y^2+a_1y+a_0=0$ poss\`ede $4$ points $k$-rationnels naturels, \`a savoir les points $P_i=(x_i,x_i)$. Si $a_0=-u^4$, o\`u $u\in k$, $E$ poss\`ede un nouveau point $k$-rationnel, \`a savoir le point $O=(-u,0)$. Si $a_2(a_1^2-4a_0a_2)\neq 0$, la courbe $E$ est de genre $1$, et d'invariant modulaire \'egal \`a $1728$. Or l'\'equation $a_0=-u^4$ s'\'ecrit $x_1x_2x_3(x_1+x_2+x_3)=u^4.$ Comme me l'a indiqu\'e J.-P. Serre, cette \'equation a \'et\'e \'etudi\'ee par Euler ([1], p. $660$), qui a exhib\'e plusieurs courbes unicursales trac\'ees sur $S$, par exemple la courbe $$u=1,\;\;\;x_1=t\fracb{2t^2-1}{2t^2+1},\;\;\; x_2=\fracb{2t^2-1}{2t(2t^2+1)},\;\;\; x_3=\fracb{4t}{2t^2-1}.$$ Soit donc $x_4=-x_1-x_2-x_3$, o\`u les $x_i$ sont donn\'es par les formules ci-dessus, et soit $p=\prod (x-x_i)=x^4+a_2x^2+a_1x+a_0.$ La courbe $E$, d\'efinie sur $k(t)$, d'\'equation $x^4+a_2y^2+a_1y+a_0$ est de genre $1$; elle est $k(t)$-isomorphe \`a la courbe elliptique d'\'equation $y^2=x^3+a_2(a_1^2-4a_0a_2)x$. On v\'erifie que $a_2(a_1^2-4a_0a_2)$ n'est pas une puissance quatri\`eme dans $k(t)$; par suite, $E$ n'est pas $k(t)$-isomorphe \`a une courbe d\'efinie sur $k$. \par\medskip Pour prouver que les $4$ points $P_i$ sont ind\'ependants, le point $O$ \'etant choisi comme origine, et d\'emontrer ainsi l'assertion du th\'eor\`eme $2$ relative aux courbes d'invariant $1728$, il suffit de v\'erifier que, pour une valeur de $t$, les sp\'ecialisations des points $P_i$ sont des points ind\'ependants. Or, pour $t=1$, le calcul, \`a l'aide du logiciel gp, montre que le d\'eterminant de la matrice des hauteurs des sp\'ecialisations des points $P_i$ est \'egal \`a $603.61237\ldots$, et est donc non nul. \subsection{Le cas des courbes d'invariant $0$} Soit $p\in k[X]$ un polyn\^ome unitaire de degr\'e $6$. Il existe alors un unique polyn\^ome unitaire $g\in k[X]$, de degr\'e 2, tel que le polyn\^ome $r=p-g^3$ soit de degr\'e $\leq 3$. Supposons que les racines $x_1,\ldots,x_6$ de $p$ soient dans $k$. La courbe $E$ d'\'equation $r(x)+y^3=0$ contient les $6$ points $k$-rationnels $P_i=(r(x_i),g(x_i))$, $1\leq i\leq 6$. De plus, si le discriminant de $r$ est non nul, la courbe $E$ est de genre $1$ et d'invariant modulaire \'egal \`a $0$. Si le coefficient de degr\'e $3$ de $r$ est le cube d'un \'el\'ement de $k$, l'un des points \`a l'infini de $E$ est $k$-rationnel, et on peut le choisir comme origine $O$ de la courbe elliptique $E$. Nous allons montrer que, si les $x_i$ sont convenablement choisis, les points $P_i$ sont alors ind\'ependants. Sans nuire \`a la g\'en\'eralit\'e du probl\`eme, on peut supposer que la somme des racines $x_i$ de $p$ est nulle. On peut donc \'ecrire $p$ sous la forme $p(x)=x^6+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0.$ On a alors $$g(x)=x^2+a_4/3,\;\;\;r(x)=a_3x^3+(a_2-a_4^2/3)x^2+a_1x-a_4^3/27.$$ Le coefficient $a_3$ du polyn\^ome $p$ est homog\`ene de degr\'e $3$ en les racines $x_i$ de $p$. L'hypersurface cubique (en les variables $u$ et $x_i$, $1\leq i\leq 5$) d'\'equation $u^3=a_3$ poss\`ede des sous-vari\'et\'es lin\'eaires $k$-rationnelles naturelles, par exemple $u=0, x_1=x_2=x_3=-x_4=-x_5$. Par des manipulations classiques, cela permet d'obtenir des courbes unicursales trac\'ees sur cette hypersurface. On trouve par exemple $$\begin{array}{ll} x_1=-126(35t-19)(14t-13)(t+1),& x_2=63(-980t^3+3549 t - 3084 t + 1135),\\ x_3=126(35 t - 19) (14 t - 13) (t + 1),& x_4=63(1127 t^3- 3108 t^2+ 3525 t- 988),\\ x_5=- 113876 t^3+ 265629 t^2- 259980 t + 69103,& x_6=104615 t^3 - 293412 t^2+ 232197 t - 78364. \end{array}$$ On obtient ainsi, par la m\'ethode d\'ecrite ci-dessous, une courbe elliptique $E$, d\'efinie sur $k(t)$, munie de $6$ points $k(t)$-rationnels. Cette courbe est $k(t)$-isomorphe \`a la courbe $y^2=x^3-16D$, o\`u $D$ est le discriminant du polyn\^ome $r$. On v\'erifie que $D$ est un polyn\^ome irr\'eductible sur $k(t)$, et n'est donc pas une puissance sixi\`eme. Par suite, $E$ n'est pas $k(t)$-isomorphe \`a une courbe d\'efinie sur $k$. Pour prouver que les points $P_i$ sont ind\'ependants, le point $O$ \'etant choisi comme origine, il suffit de le montrer pour une valeur convenable de $t$. Or, pour $t=1$, le d\'eterminant de la matrice des hauteurs normalis\'ees des points $P_i$ vaut $38462030713.186929\ldots$, et est donc non nul. \par\medskip {\sc R\'ef\'erence bibliographique} \par\medskip [1] {\sc L. Dickson}, {\it History of the theory of numbers}, vol. 2, Chelsea $1971$. \begin{flushright} UFR de Math\'ematiques, Universit\'e de Paris VII\\ 2 place Jussieu, 75251 Paris Cedex 05.\end{flushright} \end{document}

9602

alg-geom/9602010

en

https://arxiv.org/abs/alg-geom/9602010

alg-geom/9602010

Steven Bradlow

Steven Bradlow and Oscar Garcia-Prada

Non-abelian monopoles and vortices

Revised version, in which some minor clarrifications and a number of unjustly ommitted references have been added. (Apologies to any inadvertantly offended parties.) To appear in the Proceedings of the 1995 Aarhus Conference in Geometry and Physics. 25 pages. AMSLaTeX v 2.09

The Seiberg-Witten equations are defined on certain complex line bundles over smooth oriented four manifolds. When the base manifold is a complex Kahler surface, the Seiberg-Witten equations are essentially the Abelian vortex equations. Using known non-abelian generalizations of the vortex equations as a guide, we explore some non-abelian versions of the Seiberg-Witten equations. We also make some comments about the differences between the vortex equations that have previously appeared in the literature and those that emerge as Kahler versions of Seiberg-witten type equations.

alg-geom

\section{Introduction}\label{introduction} In the short time since their discovery, the Seiberg--Witten equations have already proved to be a powerful tool in the study of smooth four-manifolds. Virtually all the hard-won gains that have been obtained using the heavy machinery of Donaldson invariants, can be recovered with a fraction of the effort if the (SU(2)) anti-self-duality equations are replaced by the ($\mathop{{\fam0 U}}\nolimits(1)$) Seiberg--Witten equations. In addition, the new equations probe deep features of symplectic structures. They have also been used to study geometric questions on K\"ahler surfaces. (See \cite{D3} for a useful survey.) The impressive success of the original equations has naturally led to speculation about possible generalizations and other related sets of equations. The original equations as proposed by Seiberg--Witten are associated with a Hermitian line bundle, and thus with the abelian group $\mathop{{\fam0 U}}\nolimits(1)$. One way to generalize the equations is thus to look for versions based on larger, non-abelian groups. This means replacing the line bundle with a higher rank complex vector bundle. Indeed a number of authors have proposed versions of the equations along these lines. These include, among others, Okonek and Teleman \cite{OT1,OT2}, Pidstrigach and Tyurin \cite{PT}, Labastida and Mari\~no \cite{LM}, as well as the second author \cite{G5}. Some of these (cf. \cite{PT}) play a key role in attempts to prove the conjecture of Witten \cite{W} concerning the equivalence of the old Donaldson and the new Seiberg--Witten\ invariants. (See also \cite{D3}, and \cite{FL} for more recent progress in this direction.) It is striking that no two of the above mentioned authors consider precisely the same set of equations. One conclusion to be drawn from this abundance of equations, is that there is apparently more than one natural way to write down non abelian versions of the Seiberg-Witten equations. This leads to the question: Are some versions more reasonable, or more natural, than others? The material in this paper gives one perspective on this question. The main idea in our point of view is to exploit the special form of the Seiberg-Witten equations in the case where the four manifold is a K\"ahler surface. In this case the original Seiberg--Witten equations are known to reduce essentially to familiar equations in gauge theory known as the abelian vortex equations. Looked at from the opposite direction, the Seiberg--Witten equations serve as a ''Riemannian version'' of the vortex equations. The key point is that there are a number of well motivated, natural generalizations of such vortex equations. All of these are defined on complex vector bundles over K\"ahler manifolds, and thus in particular over K\"ahler surfaces. Our guiding principle is that the generalizations of the Seiberg--Witten equations should provide ''Riemannian versions'' of these vortex-type equations over K\"ahler surfaces. In this paper we explore essentially two such non-abelian generalizations. We also make some remarks concerning a different aspect of the relation between the vortex and the Seiberg-Witten equations. This aspect has to do with the parameters which appear in the vortex equations. In their original form, these were taken to be real numbers, i.e. constant functions on the base manifolds. In the versions that emerge from the Seiberg-Witten equations, the analogous terms turn out to be non-constant functions (related to the scalar curvature). This has prompted a closer look at the affected terms in the vortex equation. We discuss various ways of incorporating --- and interpreting --- this level of generality in the analysis of the vortex equations. In the interests of completeness, we have included a certain amount of standard background material on the Seiberg-Witten and vortex equations. \noindent {\bf Acknowledgements.} The second author wishes to thank the organisers of the Aarhus Conference in Geometry and Physics, and especially J{\o}rgen Andersen, for their kind invitation to participate in the Conference, and to visit the Mathematics Institute in Aarhus, as well as for their warm hospitality. \section{The Seiberg--Witten\ monopole equations}\label{sw} In this section we briefly review the Seiberg--Witten\ equations and the analysis of these equations in the K\"{a}hler\ case. For more details, see the original papers by Witten \cite{W} and Kronheimer and Mrowka \cite{KM}, or any recent survey on the subject (e.g. \cite{D3,G5}). Let $(X,g)$ be a compact, oriented, Riemannian four-manifold. To write the Seiberg--Witten\ equations one needs a $\spin^c$-structure on $X$. This involves the choice of a Hermitian line bundle $L$ on $X$ satisfying that $c_1 (L)\equiv w_2(X)\;\mathop{{\fam0 mod}}\nolimits\;2$. A $\spin^c$-structure is then a lift of the fibre product of the $\mathop{{\fam0 SO}}\nolimits(4)$-bundle of orthonormal frames of $(X,g)$ with the $\mathop{{\fam0 U}}\nolimits(1)$-bundle defined by $L$ to a $\spin^c(4)$-bundle, according to the short exact sequence $$ 0\longrightarrow{\bf Z}_2\longrightarrow\spin^c(4)\longrightarrow\mathop{{\fam0 SO}}\nolimits(4)\times \mathop{{\fam0 U}}\nolimits(1)\longrightarrow 1. $$ Using the two fundamental irreducible 2-dimensional representations of $\spin^c(4)$---the so-called Spin representations---we can construct the associated vector bundles of positive and negative spinors $S_L^\pm$. These are rank 2 Hermitian vector bundles whose determinant is $L$ \cite{H,LaMi}. The set of $\spin^c$-structures on $X$ is thus parametrised, up to the finite group $H^1(X,{\bf Z}_2)$, by $$ \spin^c(X)=\{c\in H^2(X,{\bf Z})\;|\; c\equiv w_2(X)\; \mathop{{\fam0 mod}}\nolimits\;2\}. $$ Let us fix a $\spin^c$-structure $c\in \spin^c(X)$, and let $L=L_c$ be the corresponding Hermitian line bundle, and $S_L^\pm$ the corresponding spinor bundles. The Seiberg--Witten\ {\em monopole equations} are equations for a pair $(A,\Psi)$ consisting of a unitary connection on $L$ and a smooth section of ${S_L^+}$. Using the connection $A$ one has the Dirac operator $$ D_A:\,\Gamma({S_L^+})\longrightarrow\Gamma({S_L^-}). $$ The first condition is that $\Psi$ must be in the kernel of the Dirac operator. The curvature $F_A\in\Omega^2=\Omega^+\oplus\Omega^-$, can be decomposed in the self-dual and anti-selfdual parts $$ F_A=F_A^+ + F_A^-. $$ Using the spinor $\Psi$ we can consider another self-dual 2-form that we may couple to $F_A^+$ to obtain our second equation. Let $\mathop{{\fam0 ad}}\nolimits_0{S_L^+}$ be the subbundle of the adjoint bundle of ${S_L^+}$ consistig of the traceless skew-Hermitian endomorphisms --- its fibres are hence isomorphic to $\gls\glu(2)$. We have a map $$ \Omega^0({S_L^+})\longrightarrow\Omega^0(\mathop{{\fam0 ad}}\nolimits_0{S_L^+}) $$ $$ \Psi\mapsto i(\Psi\otimes\Psi^\ast)_0\ , $$ where $\Psi^\ast$ is the adjoint of $\Psi$, and the 0 subindex means that we are taking the trace-free part. This map is fibrewise modelled on the map ${\bf C}^2\longrightarrow\gls\glu(2)$, given by $v\mapsto i(v\overline{v}^t)_0$. One of the basic ingredients that makes the Seiberg--Witten\ equations possible is the identification between the space of self-dual 2-forms and the skew-Hermitian automorphism of the positive spin representation \cite{AHS}. This is a basic fact in Clifford algebras in dimension four, that takes place at each point of the manifold, and that can be carried out over the whole manifold precisely when one has a $\spin^c$-structure. More specifically, we have the isomorphism \begin{equation} \mathop{{\fam0 ad}}\nolimits_0{S_L^+}\cong\Lambda^+\ . \label{iso} \end{equation} We can now interpret $i(\Psi\otimes\Psi^\ast)_0$ as a section of $\Lambda^+$, i.e. as an element in $\Omega^+$. The monopole equations consist in the system of equations \begin{equation} \left.\begin{array}{l} D_A \Psi=0\\ F_A^+=i(\Psi\otimes\Psi^\ast)_0 \end{array}\right \}. \label{me} \end{equation} In writing the second equation there is an abuse of notation, since we are not especifying what the isomorphism (\ref{iso}) is. Notice also that $F_A^+$ is a purely imaginary self-dual 2-form, and hence we are in fact identifying $\mathop{{\fam0 ad}}\nolimits_0{S_L^+}$ with $i\Lambda^+$. We shall analyse now the monopole equations in the case in which $(X,g)$ is K\"{a}hler. Recall that a K\"{a}hler\ manifold is Spin if and only if there exists a square root of the canonical bundle $K^{1/2}$ \cite{A,H}. Moreover the spinor bundles are $$ S^+=(\Lambda^0\oplus\Lambda^{0,2})\otimes K^{1/2}=K^{1/2}\oplus K^{-1/2} $$ $$ S^-=\Lambda^{0,1}\otimes K^{1/2}. $$ In this situation the spinor bundles for the $\spin^c$-structure $c$ are given by $S_L^\pm=S^\pm\otimes L^{1/2}$ (notice that $L^{1/2}$ exists since $c_1(L)\equiv 0\;\mathop{{\fam0 mod}}\nolimits\;2$). Even if $X$ is not Spin, i.e. even if $K^{1/2}$ and $L^{1/2}$ do not exist, the bundles $S_L^\pm$ do exist. In other words, there exists a square root of $K\otimes L$. Let us denote $$ {\hat{L}}=(K\otimes L)^{1/2}. $$ Then $$ {S_L^+}={\hat{L}}\oplus \Lambda^{0,2}\otimes {\hat{L}},\;\;{S_L^-}=\Lambda^{0,1}\otimes{\hat{L}} $$ and $$ \Gamma({S_L^+})=\Omega^0({\hat{L}})\oplus\Omega^{0,2}({\hat{L}}). $$ We can write $\Psi$ according to this decomposition as a pair $\Psi=(\phi,\beta)$. The Dirac operator can be written in this language as $$ \overline{\partial}_{\hat{A}} +\overline{\partial}_{\hat{A}}^\ast\;\;: \Omega^0({\hat{L}})\oplus\Omega^{0,2}({\hat{L}})\longrightarrow\Omega^{0,1}({\hat{L}}), $$ where $\overline{\partial}_{\hat{A}}$ is the $\overline{\partial}$ operator on ${\hat{L}}$ corresponding to the connection ${\hat{A}}$ on ${\hat{L}}$ defined by the connection $A$ on $L$ and the metric connection on $K$ (cf. \cite{H}). On the other hand recall that $$ \Lambda^+\otimes{\bf C}=\Lambda^0\omega\oplus\Lambda^{2,0} \oplus\Lambda^{0,2}, $$ where $\omega$ is the K\"{a}hler\ form. According to this decomposition, the isomorphism (\ref{iso}) (or rather $\mathop{{\fam0 ad}}\nolimits_0{S_L^+}\cong i\Lambda^+$) is explicitely given by \begin{equation} i(\Psi\otimes\Psi^\ast)_0\mapsto i(|\phi|^2-|\beta|^2)\omega + \beta\overline{\phi}-\phi\overline{\beta}. \end{equation} We may thus write the monopole equations (\ref{me}) as \begin{equation} \begin{array}{l} \overline{\partial}_{\hat{A}}\phi+\overline{\partial}^\ast_{\hat{A}}\beta=0\\ \Lambda F_A=i(|\phi|^2-|\beta|^2) \\ F_A^{2,0}=-\phi \overline{\beta}\\ F_A^{0,2}=\beta\overline{\phi} \end{array}\label{kme} \end{equation} where $\Lambda F_A$ is the contraction of the curvature with the K\"{a}hler\ form. It is not difficult to see (cf. \cite{W}) that the solutions to these equations are such that either $\beta=0$ or $\phi=0$, and moreover it is not possible to have irreducible solutions, i.e. solutions with $\Psi\neq 0$, of both types simultaneously for a fixed $\spin^c$-structure. We thus have one of the following two situations: (i)\ $\beta=0$ and the equations reduce to \begin{equation} \begin{array}{l} F_A^{0,2}=0\\ \overline{\partial}_{\hat{A}}\phi=0\\ \Lambda F_A=i|\phi|^2 \end{array}\label{ve1} \end{equation} (ii)\ $\phi=0$ and then \begin{equation} \begin{array}{l} F_A^{0,2}=0\\ \overline{\partial}_{\hat{A}}^\ast\beta=0\\ \Lambda F_A=-i|\beta|^2\ . \end{array} \label{ve2} \end{equation} \noindent{\em Remark}. We have omitted the equation $F_A^{2,0}=0$, since by unitarity of the connection this is equivalent to $F_A^{0,2}=0$. Clearly if we have solutions to (\ref{ve1}), from the third equation in (\ref{ve1}) we obtain that $\deg L\leq0$, while from (\ref{ve2}) we have $\deg L\geq0$, where $\deg L$ is the degree of $L$ with respect to the K\"{a}hler\ metric defined as in (\ref{degree}). Since we are interested only in irreducible solutions, obviously these two situations cannot occur simultaneously. The Hodge star operator interchanges these two cases, and we can thus concentrate on case (i). Equations (\ref{ve1}) are essentially the equations known as the {\em vortex equations}. These are generalisations of the vortex equations on the Euclidean plane studied by Jaffe and Taubes \cite{T1,T2,JT}, and have been extensively studied (e.g. in \cite{B1,B2,G2,G3} ) for compact K\"{a}hler\ manifolds of arbitrary dimension. In that setting, the equations are the following: Let $(X,\omega)$ be a compact K\"{a}hler\ manifold of arbitrary dimension, and let $(L,h)$ be a Hermitian $C^\infty$ line bundle over $X$. Let $\tau\in{\bf R}$. The $\tau$-{\em vortex equations} \begin{equation} \left. \begin{array}{l} F_A^{0,2}=0\\ \overline{\partial}_A\phi=0 \\ \Lambda F_A =\frac{i}{2}(|\phi|^2 -\tau) \end{array}\right \}, \label{ve} \end{equation} are equations for a pair $(A,\phi)$ consisting of a connection on $(L,h)$ and a smooth section of $L$. The first equation means that $A$ defines a holomorphic structure on $L$, while the second says that $\phi$ must be holomorphic with respect to this holomorphic structure. Coming back to the monopole equations, we first observe that (\ref{ve1}) can be rewritten in terms of ${\hat{A}}$ only, i.e. not involving simultaneously $A$ and ${\hat{A}}$. To do this we recall that ${\hat{A}}=(A\otimes a_K)^{1/2}$, where $a_K$ is the metric connection on $K$. We thus have \begin{equation} F_{{\hat{A}}}=\frac{1}{2}(F_A + F_{a_K}), \label{curvature} \end{equation} and hence $F_A^{0,2}=0$ is equivalent to $F_{{\hat{A}}}^{0,2}=0$ since $F_{a_K}^{0,2}=0$. From (\ref{curvature}), and using that $s=-i\Lambda F_{a_K}$ is the scalar curvature, we obtain that $$ \Lambda F_A=2\Lambda F_{\hat{A}}-is, $$ and hence (\ref{ve1}) is equivalent to \begin{equation} \left. \begin{array}{l} F_{\hat{A}}^{0,2}=0\\ \overline{\partial}_{\hat{A}}\phi=0\\ \Lambda F_{\hat{A}}=\frac{i}{2}(|\phi|^2+ s) \end{array} \right\}.\label{ve1'} \end{equation} These are the vortex equations on ${\hat{L}}$, but with the parameter $\tau$ replaced by minus the scalar curvature. As we will explain in Section \ref{t-vortices} the existence proofs for the vortex equations can be easily modified to give an existence theorem for the equations obtained by replacing the parameter $\tau$ by a function $t\in C^\infty (X,{\bf R})$ in (\ref{ve}). However, to compute the Seiberg--Witten\ invariants one can slightly perturb equations (\ref{me}) in such a way that, when $\beta=0$, equations (\ref{me}) reduce to the constant function vortex equations, i.e. to (\ref{ve}) (see e.g. \cite{G5}). \section{Non-abelian vortex equations}\label{vortices} As we have seen in the previous section, the Seiberg--Witten\ monopole equations can be considered as a four-dimensional Riemannian generalisation of the vortex equations. This suggests that we may find interesting Seiberg--Witten-type equations by considering the corresponding analogues of different equations of vortex-type existing in the literature. With this objective in mind, in this section we shall review three different non-abelian generalisations of the vortex equations described above. The first one consists in studying the vortex equations on a Hermitian vector bundle of arbitrary rank. The other two involve two vector bundles, one of which will actually be a line bundle in most cases. Let $(E,H)$ be a Hermitian vector bundle over a compact K\"{a}hler\ manifold $(X,\omega)$ of complex dimension $n$. Let $\tau\in{\bf R}$. The $\tau$-vortex equations were generalised to this situation in \cite{B2}. As in the line bundle case, one studies equations \begin{equation} \left. \begin{array}{l} F_A^{0,2}=0\\ \overline{\partial}_A\phi=0 \\ \Lambda F_A =i(\phi\otimes\phi^\ast -\tau{\bf I}) \end{array}\right \} \label{nave} \end{equation} for a pair $(A,\phi)$ consisting of a unitary connection on $(E,H)$ and a smooth section of $E$. By $\phi^\ast$ we denote the adjoint of $\phi$ with respect to $H$, and ${\bf I}\in\mathop{{\fam0 End}}\nolimits E$ is the identity. Notice that in the abelian equations (\ref{ve}) there is a $1/2$ in the third equation, while in (\ref{nave}) there is not. This is not essential since by applying a constant complex gauge transformation we can introduce an arbitrary positive constant in front of $i\phi\otimes\phi^\ast $. These vortex equations appear naturally as the equations satisfied by the minima of the Yang--Mills--Higgs\ functional. This is a functional defined on the product of the space ${\cal A}$ of unitary connections on $(E,H)$ and the space of smooth sections $\Omega^0(E)$ by $$ \mathop{{\fam0 YMH}}\nolimits_\tau(A,\phi)=\|F_A\|^2 +2\|d_A\phi\|^2+ \|\phi\otimes\phi^\ast-\tau{\bf I}\|^2, $$ where $\|\;\;\|$ denotes the $L^2$-metric. This is easily seen by rewriting the Yang--Mills--Higgs\ functional --- using the K\"{a}hler\ identities --- as \begin{eqnarray} \mathop{{\fam0 YMH}}\nolimits_\tau(A,\phi) &=& 4\|F_A^{0,2}\|^2+4\|\overline{\partial}_A\phi\|^2+ \|i\Lambda F_A +\phi\otimes\phi^\ast -\tau{\bf I}\|^2 \nonumber \\ & &+4\pi\tau\deg E- \frac{8\pi^2}{(n-2)!} \int_X\mathop{{\fam0 ch}}\nolimits_2(E)\wedge\omega^{n-2},\nonumber \end{eqnarray} where $\deg E$ is the degree of $E$ defined as \begin{equation} \deg E=\int_X c_1(E)\wedge \omega^{n-1}\ , \label{degree} \end{equation} and $\mathop{{\fam0 ch}}\nolimits_2(E)$ is the second Chern character of $E$, which is represented in terms of the curvature by $$ \mathop{{\fam0 ch}}\nolimits_2(E)=-\frac{1}{8\pi^2} \mathop{{\fam0 Tr}}\nolimits(F_A\wedge F_A). $$ Clearly $\mathop{{\fam0 YMH}}\nolimits_\tau$ achieves its minimum value $$ 4\pi\tau\deg E- \frac{8\pi^2}{(n-2)!} \int_X\mathop{{\fam0 ch}}\nolimits_2(E)\wedge\omega^{n-2} $$ if and only if $(A,\phi)$ is a solution to equations (\ref{nave}) (see \cite{B1} for details). As we will explain in Section \ref{t-vortices}, the vortex equations also have a symplectic interpretation as moment map equations. The moment map in question is for a symplectic action of ${\cal G}(E)$, i.e. the unitary gauge group of $E$, on a certain infinite dimensional symplectic space. A natural generalization of these equations is obtained if we regard the section $\phi$ in (\ref{nave}) as a morphism from the trivial line bundle to $E$. One can replace the trivial line bundle by a vector bundle of arbitrary rank and study equations for connections on both bundles and a morphism from one to the other. These are the {\em coupled vortex equations} introduced in \cite{G4}. Let $(E,H)$ and $(F,K)$ be smooth Hermitian vector bundles over $X$. Let $A$ and $B$ be unitary connections on $(E,H)$ and $(F,K)$ resp., and let $\phi\in \Omega^0(\mathop{{\fam0 Hom}}\nolimits(F,E))$. Let $\tau$ and $\tau'$ be real parameters. The coupled vortex equations are \begin{equation} \left. \begin{array}{l} F_A^{0,2}=0\\ F_B^{0,2}=0\\ \overline{\partial}_{A, B}\phi=0\\ i \Lambda F_A+\phi\phi^\ast=\tau {\bf I}_E\\ i \Lambda F_B-\phi^\ast\phi=\tau'{\bf I}_F \end{array}\right \}.\label{gcve} \end{equation} As in the case of the vortex equations described above, equations (\ref{gcve}) correspond to the minima of a certain Yang--Mills--Higgs\ functional and are also moment map equations (cf. \cite{G4}). The appropriate functional in this case is defined as $$ \mathop{{\fam0 YMH}}\nolimits_{\tau,\tau'}(A,B,\phi)=\|F_A\|^2 +\|F_B\|^2 + 2\|d_{A\otimes B}\phi\|^2+ \|\phi\phi^\ast-\tau{\bf I}_E\|^2+\|\phi^\ast\phi-\tau'{\bf I}_F\|^2. $$ The moment map is now for a symplectic action of ${\cal G}(E)\times{\cal G}(F)$, i.e. for the product of the unitary groups of $E$ and $F$. In this paper we will be mostly interested in the case in which $F=L$ is a line bundle. In this situation the equations can be written as \begin{equation} \left. \begin{array}{l} F_A^{0,2}=0\\ F_B^{0,2}=0\\ \overline{\partial}_{A, B}\phi=0\\ i \Lambda F_A+\phi\otimes\phi^\ast=\tau {\bf I}_E\\ i \Lambda F_B-|\phi|^2=\tau' \end{array}\right \}.\label{cve} \end{equation} It is clear, from taking the trace of the last two equations in (\ref{cve}) and integrating, that to solve (\ref{cve}) $\tau$ and $\tau'$ must be related by \begin{equation} \tau\mathop{{\fam0 rank}}\nolimits E +\tau'=\deg E +\deg L,\label{parameters} \end{equation} hence there is only one free parameter. We shall consider next the {\em framed vortex equations}. The situation is very similar to the previous one in that it also involves two vector bundles. In this case, however, the connection on one of the bundles is fixed. More specifically, let $(E,H)$ and $(F,K)$ be two Hermitian vector bundles. Let $B$ a fixed Hermitian connection on $(F,K)$ such that $F_B^{0,2}=0$. The equations are now for a unitary connection $A$ on $(E,H)$, and $\phi\in \Omega^0(\mathop{{\fam0 Hom}}\nolimits(F,E))$. As explained in \cite{BDGW}, the appropriate equations are \begin{equation} \left. \begin{array}{l} F_A^{0,2}=0\\ \overline{\partial}_{A, B}\phi=0\\ i \Lambda F_A+\phi\phi^\ast=\tau {\bf I}_E\\ \end{array}\right \}\label{fve} \end{equation} The relation between these equations and the full coupled vortex equations is perhaps best understood from the symplectic point of view. One sees that the effect of fixing the data on $F$ is to reduce the symmetry group in the problem from ${\cal G}(E)\times{\cal G}(F)$\ to ${\cal G}(E)$. The new equations must thus correspond to the moment map for the subgroup ${\cal G}(E)\subset{\cal G}(E)\times{\cal G}(F)$. But the moment maps for ${\cal G}(E)$ and for ${\cal G}(E)\times{\cal G}(F)$ are related by a projection from the Lie algebra of ${\cal G}(E)\times{\cal G}(F)$ onto the summand corresponding to the Lie algebra of ${\cal G}(E)$. The effect of this projection is to eliminate the last equations in (\ref{cve}) (cf. \cite{BDGW} for more details). The appropriate moduli space problem corresponds to that of studying morphisms from a vector bundle with a fixed holomorphic structure to another vector bundle in which the holomorphic structure is varying. Such moduli spaces have been studied by Huybrecht and Lehn \cite{HL1,HL2}, who refer to these objects as {\em framed modules}. As in the coupled vortex equations, we will be mostly interested in the case in which $F=L$ is a line bundle. All the vortex-type equations that we have considered so far involve one or two real parameters $\tau$ and $\tau'$. As we have seen in Section \ref{sw}, the study of the Seiberg--Witten\ monopole equations leads to abelian vortex equations in which $\tau$ is replaced by a function. The same will happen in the generalizations of the monopole equations that we are about to discuss. This will be analysed in detail in Section \ref{t-vortices}. \section{Non-abelian monopole equations}\label{monopoles} Let us go back to the set-up of Section \ref{sw} and let $(X,g)$ be a compact, oriented, Riemannian, four-dimensional manifold. Let $c\in \spin^c(X)$ be a fixed $\spin^c$-structure, with corresponding Hermitian line bundle $L$ and bundles of spinors ${S_L^\pm}$. Let $(E,H)$ be a Hermitian vector bundle on $X$. Let $\Psi\in \Gamma (S^+_L\otimes E)$. Using the metrics on $S_L^+$ and $E$ one has the antilinear identification $$ S^+_L\otimes E\longrightarrow S^{+*}_L\otimes E^* $$ $$ \Psi \mapsto \Psi^*. $$ and hence $$ \Psi \otimes \Psi^*\in \mathop{{\fam0 End}}\nolimits (S^+_L\otimes E). $$ We shall also need the map $$ \mathop{{\fam0 End}}\nolimits (S^+_L\otimes E)\longrightarrow \mathop{{\fam0 End}}\nolimits_0(S^+_L)\otimes \mathop{{\fam0 End}}\nolimits E $$ $$ \Psi\otimes \Psi^*\mapsto (\Psi\otimes \Psi^*)_0\; , $$ as well as the map $$ \mathop{{\fam0 End}}\nolimits_0(S^+_L)\otimes \mathop{{\fam0 End}}\nolimits E\stackrel{\mathop{{\fam0 Tr}}\nolimits}{\longrightarrow}\mathop{{\fam0 End}}\nolimits_0 (S^+_L) $$ $$ (\Psi\otimes \Psi^*)_0\mapsto \mathop{{\fam0 Tr}}\nolimits(\Psi\otimes \Psi^*)_0 $$ obtained from the trace map $$ \mathop{{\fam0 End}}\nolimits_0\,E\longrightarrow \mathop{{\fam0 End}}\nolimits E \stackrel{\mathop{{\fam0 Tr}}\nolimits}{\longrightarrow}{\bf C}\longrightarrow 0. $$ The endomorphism $(\Psi\otimes \Psi^\ast)_0$ should not be confused with the completely traceless part of $\Psi\otimes \Psi^\ast$ --- here we are only removing the trace corresponding to $S_L^+$. In this paper we shall consider essentially two different non-abelian generalizations of the Seiberg--Witten\ equations. While in the first one we will fix a connection $b$ on $L$ and study equations for a pair $(A,\Psi)$, where $A$ is a unitary connection on $(E,H)$ and $\Psi\in \Gamma(S_L^+\otimes E)$, in the second one we will allow $b$ to vary as well, and hence our system of equations will be one for triples $(A,b,\Psi)$. The first non-abelian version of the Seiberg--Witten\ equations is suggested by the framed vortex equations: Let $b$ be a {\em fixed} unitary connection on $L$ and $A$ be a unitary connection on $E$. Using these two connections and the Levi--Civita connection one can consider the coupled Dirac operator $$ D_{A,b}:\; \Gamma({S_L^+}\otimes E)\longrightarrow \Gamma({S_L^-}\otimes E). $$ and study the equations \begin{equation} \left.\begin{array}{l} D_{A,b}\Psi=0\\ F^+_A=i(\Psi\otimes \Psi^*)_0 \end{array} \right \}\label{name} \end{equation} for the unknowns $A$ and $\Psi\in\Gamma({S_L^+}\otimes E)$. The Bochner--Weitzenb\"ock formula for $D_{A,b}$ is given by \begin{equation} D_{A,b}^\ast D_{A,b}= \nabla_{A,b}^\ast\nabla_{A,b}+\frac{s}{4} +c(F_{A,b}), \ \label{bw} \end{equation} where $$ F_{A,b}=F_A +\frac{1}{2} F_b\otimes I_E $$ and $\nabla_{A,b}$ is the connection on $E\otimes S_L$ determined by the connections $A$ and $b$. The term $c(F_{A,b})$ in (\ref{bw}) means Clifford multiplication by $F_{A,b}$. In fact the action of $F_{A,b}$ on $\Psi\in \Gamma({S_L^+}\otimes E)$ coincides with the Clifford multiplication with $F_{A,b}^+$ (see \cite{LaMi} for example). It is thus natural to perturb the second equation in (\ref{name}) by adding the constant self-dual 2-form $\frac{1}{2} F_b^+$, and consider instead equations \begin{equation} \left.\begin{array}{l} D_{A,b}\Psi=0\\ F^+_{A, b}=i(\Psi\otimes \Psi^*)_0 \end{array} \right \}. \label{name'} \end{equation} These are the equations studied in \cite{OT1}. The abelian Seiberg--Witten\ equations (\ref{me}) for a $\spin^c$-structure with line bundle $\tilde L$ can be recovered from (\ref{name}) or (\ref{name'}) by considering the Hermitian bundle $E=(\tilde L \otimes L^\ast)^{1/2}$. Notice that this square root exists since $c_1(\tilde L)\equiv c_1(L)\;\mathop{{\fam0 mod}}\nolimits\; 2$. Other non-abelian versions of the Seiberg--Witten\ equations that have been considered include replacing the $\mathop{{\fam0 U}}\nolimits(r)$-bundle $(E,H)$ by an ${SU}(r)$-bundle, and study equations (\ref{name}) in which $(\Psi\otimes\Psi^\ast)_0$ is replaced by the completely trace-free part of $\Psi\otimes\Psi^\ast$. These are the equations studied in \cite{LM} (see also \cite{OT2}) In other versions, like the one considered in \cite{PT}, one fixes the connection of $\det E\otimes L$ instead of fixing that of $L$. In all the versions mentioned above one considers spinors coupled to a bundle associated the fundamental representation of $\mathop{{\fam0 U}}\nolimits(r)$ or ${SU}(r)$. Another direction in which the monopole equations can be generalized is by considering any compact Lie group $G$ and/or an arbitrary representation. When the manifold is K\"ahler some of these correspond to the vortex-type equations described in \cite{G1}. These more general equations will be dealt with somewhere else. We shall consider next the case in which the connection on $L$ is also varying. It is clear that we cannot consider the same equations as in the previous situation, since we would not obtain an elliptic complex in the linearization of the equations --- we need an extra equation. As the coupled vortex equations (\ref{cve}) suggest, it is natural to consider the following set of equations for the triple $(A,b,\Psi)$: \begin{equation} \left. \begin{array}{l} D_{A,b} \Psi =0\\ F^+_A=i(\Psi \otimes\Psi^*)_0\\ F^+_b=2i\mathop{{\fam0 Tr}}\nolimits(\Psi\otimes \Psi^*)_0 \end{array} \right\}.\label{cme} \end{equation} \section{The K\"ahler case}\label{kaehler} Let now $(X,\omega)$ be a compact K\"{a}hler\ surface. Let us fix a $\spin^c$-structure $c\in \spin^c(X)$, and let $L=L_c$ be the corresponding Hermitian line bundle. As mentioned in Section \ref{sw}, the corresponding spinor bundles for the $\spin^c$-structure $c$ are given by $$ {S_L^+}={\hat{L}}\oplus \Lambda^{0,2}\otimes {\hat{L}}\;\;\;\mbox{and}\;\;\; {S_L^-}=\Lambda^{0,1}\otimes{\hat{L}}, $$ where $$ {\hat{L}}=(K\otimes L)^{1/2}. $$ Let $(E,H)$ be a Hermitian vector bundle over $X$. \begin{equation} E\otimes{S_L^+}=E\otimes{\hat{L}}\oplus \Lambda^{0,2}\otimes E\otimes{\hat{L}}\label{E-spinors} \end{equation} and hence $$ \Gamma(E\otimes{S_L^+})=\Omega^0(E\otimes{\hat{L}})\oplus\Omega^{0,2}(E\otimes{\hat{L}}). $$ We can write $\Psi$ according to this decomposition as a pair $\Psi=(\phi,\beta)$. Let $b$ and $A$ be unitary connections on $L$ and $E$, respectively. Let $a_K$ be the metric connection on $K$. We shall denote by ${\hat{b}}$ the connection on ${\hat{L}}$ defined by $b$ and $a_K$. The Dirac operator $D_{A,b}$ can be written in this language as $$ \overline{\partial}_{A,{\hat{b}}} +\overline{\partial}_{A,{\hat{b}}}^\ast\;: \Omega^0(E\otimes{\hat{L}})\oplus\Omega^{0,2}(E\otimes{\hat{L}})\longrightarrow \Omega^{0,1}(E\otimes{\hat{L}}), $$ where $\overline{\partial}_{A,{\hat{b}}}$ is the $\overline{\partial}$ operator on $E\otimes{\hat{L}}$ corresponding to the connections $A$ and ${\hat{b}}$. \subsection{Fixed connection on $L$} We shall perturb equations (\ref{name'}) by a self-dual 2-form $\alpha$ and consider \begin{equation} \left.\begin{array}{l} D_{A,b}\Psi=0\\ F^+_{A, b}=i((\Psi\otimes \Psi^*)_0+\alpha {\bf I}) \end{array} \right \}. \label{pname} \end{equation} We will choose the perturbation to be of K\"{a}hler\ type, that is $\alpha= -f\omega$, where $f$ is a smooth real function. Similarly to the abelian case, we can write (\ref{pname}) as \begin{equation} \begin{array}{l} \overline{\partial}_{A,{\hat{b}}}\phi+\overline{\partial}^\ast_{A,{\hat{b}}}\beta=0\\ \Lambda F_{A,b}=i(\phi\otimes \phi^\ast- \Lambda^2 \beta\otimes\beta^\ast-f{\bf I}) \\ F_{A,b}^{2,0}=-\phi\otimes\beta^\ast \\ F_{A,b}^{0,2}=\beta\otimes\phi^\ast\ . \end{array}\label{kname} \end{equation} By $\Lambda^2$ we denote the operation of contracting twice with the K\"{a}hler\ form. As in the abelian case, one can see that the solutions to these equations are such that either $\beta=0$ or $\phi=0$. More precisely \begin{prop}\label{decoupling} Let $\overline{f}=\frac{1}{2\pi}\int_X f$. The only solutions to (\ref{kname}) satisfy either \noindent (i)\ $\beta=0$, \begin{equation} \begin{array}{l} F_{A,b}^{0,2}=0\\ \overline{\partial}_{A,{\hat{b}}}\phi=0\\ \Lambda F_{A,b}=i(\phi\otimes\phi^\ast-f{\bf I})\ , \end{array}\label{nave1} \end{equation} then $\mu(E)-1/2\deg L\leq\overline{f}$, or \noindent (ii)\ $\phi=0$, \begin{equation} \begin{array}{l} F_{A,b}^{0,2}=0\\ \overline{\partial}_{A,{\hat{b}}}^\ast\beta=0\\ \Lambda F_{A,b}=-i(\Lambda^2\beta\otimes\beta^\ast+f{\bf I}) \end{array}\label{nave2} \end{equation} and then $\mu(E)-1/2\deg L\geq\overline{f}$. \end{prop} {\em Proof}. One uses exactly the same method as the one used by Witten \cite{W} in the abelian case. Consider the transformation $(A,\phi,\beta)\mapsto (A,-\phi,\beta)$. Although this is not a symmetry of equations (\ref{kname}), if $(A,\phi,\beta)$ is a solution $(A,\phi,-\beta)$ must also be. This is easily seen by considering the functional $$ \mathop{{\fam0 SW}}\nolimits(A,\Psi)=\|F^+_{A, b}-i((\Psi\otimes \Psi^*)_0+\alpha {\bf I})\|^2 +2\|D_{A,b}\Psi\|^2. $$ Using the Bochner--Weitzenb\"ock formula (\ref{bw}) and the fact that on a K\"{a}hler\ manifold the decomposition (\ref{E-spinors}) is parallel with respect to the connection $\nabla_{A,b}$, we have \begin{eqnarray} \mathop{{\fam0 SW}}\nolimits(A,\Psi)& = &\|F_{A,b}^+\|^2+ 2\|\nabla_{A,b}\phi\|^2+2\|\nabla_{A,b}\beta\|^2+ \|i((\Psi\otimes \Psi^*)_0+\alpha {\bf I})\|^2 \nonumber\\ & & + \int_X\frac{s}{2}(|\phi|^2 +|\beta|^2)- 2\int_X\langle F_{A,b},i\alpha{\bf I}\rangle.\nonumber \end{eqnarray} \noindent{\em Remark}. Notice the analogy between this and the way of rewriting the Yang--Mills--Higgs\ functional in Section \ref{vortices} using the K\"{a}hler\ identities. In fact in the K\"{a}hler\ case both things are essentially equivalent. \hfill$\Box$ Of course the only way in which the two type of solutions can occur simultaneously is if $\mu(E)=1/2\deg L +\overline{f}$, then $\Psi=0$ and the equations reduce essentially to the Hermitian--Einstein\ equations. Since the Hodge operator interchanges the roles of $\phi$ and $\beta$ we may concentrate in the case $\phi\neq 0$. We shall write equations (\ref{nave1}) in a way that we can identify them as the vortex equations discussed in Section \ref{vortices}. To do this denote by $$ {\hat{E}}=E\otimes {\hat{L}}\;\;\;\mbox{and}\;\;\; {\hat{A}}=A\otimes {\hat{b}}, $$ where recall that ${\hat{b}}=b\otimes a_K$. We have that $$ F_{\hat{A}}=F_{A,b}+\frac{1}{2} F_{a_K}, $$ and hence (\ref{nave1}) is equivalent to \begin{equation} \begin{array}{l} F_{\hat{A}}^{0,2}=0\\ \overline{\partial}_{\hat{A}}\phi=0\\ \Lambda F_{\hat{A}}=i(\phi\otimes\phi^\ast+(s/2-f) {\bf I}) \end{array} \label{nave1'} \end{equation} where $s=-i\Lambda F_{a_K}$ is the scalar curvature of $(X,\omega)$, and we have used that $a_K$ is integrable, i.e. $F_{a_K}^{0,2}=0$. These are indeed the vortex equations (\ref{nave}) on the bundle ${\hat{E}}$, with the parameter $\tau$ replaced by the function $t=f-s/2$. Equations (\ref{nave1}) can also be interpreted as the framed vortex equations (\ref{fve}). To see this we choose the fixed connection $b$ to be integrable. We then have that $F_{A,b}^{0,2}=F_A^{0,2}$ and (\ref{nave1}) is equivalent to \begin{equation} \begin{array}{l} F_A^{0,2}=0\\ \overline{\partial}_{A,{\hat{b}}^\ast}\phi=0\\ \Lambda F_A=i(\phi\otimes\phi^\ast- t {\bf I}) \end{array} \label{nave1''} \end{equation} where $t=f+\frac{i}{2}\Lambda F_b$. We thus obtain the framed vortex equations (\ref{fve}) on $(E,{\hat{L}}^\ast)$ with fixed connection ${\hat{b}}^\ast$ on ${\hat{L}}^\ast$, and parameter $\tau$ replaced by the function $t$. These two slightly different points of view in relating (\ref{nave1}) to the vortex equations reflect the close relation between the usual vortex equations and the framed vortex equations, as we will explain in Section \ref{t-vortices}. \subsection{Variable connection on $L$} We come now to equations (\ref{cme}). In the K\"{a}hler\ situation these equations can be written as \begin{equation} \begin{array}{l} \overline{\partial}_{A,{\hat{b}}}\phi+\overline{\partial}^\ast_{A,{\hat{b}}}\beta=0\\ \Lambda F_A=i(\phi\otimes \phi^\ast- \Lambda^2 \beta\otimes\beta^\ast) \\ \Lambda F_b=2i(|\phi|^2-|\beta|^2) \\ F_A^{2,0}=-\phi\otimes\beta^\ast\\ F_A^{0,2}=\beta\otimes\phi^\ast\\ F_b^{2,0}=-2\mathop{{\fam0 Tr}}\nolimits(\phi\otimes\beta^\ast)\\ F_b^{0,2}=2\mathop{{\fam0 Tr}}\nolimits(\beta\otimes\phi^\ast) \ . \label{kcme} \end{array} \end{equation} By taking the third equation, subtracting twice the trace of the second equation in (\ref{kcme}), and integrating we obtain that, in order to have solutions, we need $$ \deg E = \frac{1}{2}\deg L. $$ To avoid this restriction we can perturb, as in the previous case, the coupled monopole equations by adding fixed self-dual forms $\alpha, \gamma$, i.e. by considering \begin{equation} \left. \begin{array}{l} D_{A,b} \Psi =0\\ F^+_A=i(\Psi \otimes\Psi^*)_0+i\alpha {\bf I}_E\\ F^+_b=2i\mathop{{\fam0 Tr}}\nolimits(\Psi\otimes \Psi^*)_0 +i\gamma \label{pcve} \end{array} \right\}. \end{equation} In the K\"{a}hler\ case we shall choose $$ \alpha=-f\omega \;\;\;\;\;\;\;\gamma=2f'\omega, $$ where $f$ and $f'$ are smooth real functions on $X$. With this choice of perturbation the second and third equations in (\ref{kcme}) become respectively $$ i \Lambda F_A + \phi \otimes \phi^\ast - \Lambda^2 \beta \otimes\beta^* = f {\bf I}_E $$ $$ i \Lambda F_b + 2(|\phi|^2 - |\beta|^2) =- 2f'. $$ A necessary condition for existence of solutions is now \begin{equation} \deg E - \frac{1}{2}\deg L = \int_X (rf + f'), \label{ff'} \end{equation} where $r = \mathop{{\fam0 rank}}\nolimits E$. We shall study the more general monopole equations (\ref{pcve}). \begin{prop} Let $f,f'\in C^\infty(X,{\bf R})$ be related by (\ref{ff'}) and denote $$ \overline{f}=\frac{1}{2\pi}\int_X f \;\;\;\;\mbox{and}\;\;\;\; \overline{f'}=\frac{1}{2\pi}\int_X f'. $$ The only solutions to (\ref{kcme}) are such that either \noindent (i)\ $\beta=0$, \begin{equation} \begin{array}{l} F_A^{0,2}=0\\ F_b^{0,2}=0\\ \overline{\partial}_{A,{\hat{b}}}\phi=0\\ i\Lambda F_A+\phi\otimes\phi^\ast=f{\bf I}_E\\ i\Lambda F_b+2|\phi|^2=-2f'\ , \end{array} \label{cve1} \end{equation} then $\mu(E)\leq \overline{f}$, $\deg L\geq 2\overline{f'}$ or \noindent (ii)\ $\phi=0$, \begin{equation} \begin{array}{l} F_A^{0,2}=0\\ F_b^{0,2}=0\\ \overline{\partial}_{A,{\hat{b}}}^\ast\beta=0\\ i\Lambda F_A-\Lambda^2\beta\otimes\beta^\ast=f {\bf I}_E\\ i\Lambda F_b-2|\beta|^2=-2f' \end{array}\label{cve2} \end{equation} and then $\mu(E)\geq \overline{f}$, $\deg L\leq 2\overline{f'}$. \end{prop} {\em Proof}. This is proved similarly to Proposition \ref{decoupling}, by considering the functional $$ \mathop{{\fam0 SW}}\nolimits(A,b,\Psi)= \| F^+_A-i(\Psi \otimes\Psi^*)_0-i\alpha {\bf I}_E\|^2+ 2\|F^+_b-2i\mathop{{\fam0 Tr}}\nolimits(\Psi\otimes \Psi^*)_0 -i\gamma\|^2+ 2\| D_{A,b} \Psi\|^2, $$ and observing that $$ \langle F_A^+,i(\Psi \otimes\Psi^*)_0\rangle +\frac{1}{2} \langle F_b^+,2i\mathop{{\fam0 Tr}}\nolimits(\Psi \otimes\Psi^*)_0\rangle = \langle F_{A,b}^+,i(\Psi \otimes\Psi^*)_0\rangle. $$ \hfill$\Box$ Again, we can focus on case (i), since by means of the Hodge operator we can interchange the roles of $\phi$ and $\beta$. The system of equations (\ref{cve1}) is equivalent to \begin{equation} \begin{array}{l} F_A^{0,2}=0\\ F_{{\hat{b}}^\ast}^{0,2}=0\\ \overline{\partial}_{A,{\hat{b}}^\ast}\phi=0\\ i\Lambda F_A+\phi\otimes\phi^\ast=f{\bf I}_E\\ i\Lambda F_{{\hat{b}}^\ast}-|\phi|^2=(f'-\frac{s}{2}) \end{array} \label{cve1'} \end{equation} where ${\hat{b}} ^\ast$ is the dual connection to ${\hat{b}}$ on ${\hat{L}}^\ast$, which satisfies that $F_{{\hat{b}}^\ast}=-F_{\hat{b}}$. We can now identify (\ref{cve1'}) as the coupled vortex equations (\ref{cve}) on $(E,{\hat{L}}^\ast)$, with the parameters $\tau$ and $\tau'$ replaced by functions. The existence of solutions to these equations as well as the description of the moduli space of all solutions will be dealt with in the next section. \section{Back to the vortex equations}\label{t-vortices} We study now the existence of solutions to the monopole equations in the K\"{a}hler\ case --- equations (\ref{nave1'}), (\ref{nave1''}) and (\ref{cve1'}) --- or equivalently the vortex equations in which the parameters have been replaced by functions. \subsection{The $t$-vortex equations} In this section we will examine what happens if the parameter $\tau$, which appears on the $\tau$-vortex equations (\ref{nave}) is permitted to be a non-constant smooth function, say $t$. We will refer to the resulting equations as the $t$-vortex equations. The main result, namely that existence of solutions is governed entirely by the average value of $t$, has already been observed by Okonek and Teleman. Here we give a somewhat different proof than that in \cite{OT1}. We also discuss some interpretations and implications of the result. Let $E\rightarrow X$\ be a rank $r$, smooth complex bundle over a closed K\"{a}hler\ manifold $(X,\omega)$. It is convenient to look at equations (\ref{nave}) from a different although equivalent point of view. Instead of fixing a Hermitian metric $H$ and solving for $(A,\phi)$ satisfying (\ref{nave}) we fix a $\overline{\partial}$-operator on $E$, $\overline{\partial}_E$ say, and a section $\phi\in H^0(X,{\cal E})$, where ${\cal E}$ is the holomorphic bundle determined by $\overline{\partial}_E$, and solve for a metric $H$ satisfying \begin{equation} i\Lambda F_H +\phi\otimes\phi^{*_H}=\tau{\bf I},\label{mve} \end{equation} where $F_H$\ is the curvature of the metric connection determined by $\overline{\partial}_E$\ and $H$. It will be important to explicitly write $\phi^{*_H}$\ to denote the adjoint of $\phi$ with respect to the metric $H$. Equation (\ref{mve}) can be regarded as the defining condition for a special metric on the holomorphic pair $({\cal E},\phi)$. In \cite{B2} the first author showed that there is a Hitchin--Kobayashi\ correspondence between the existence of such metrics and a stability condition for holomorphic pairs. The appropriate notion of stability is as follows. \begin{definition} Define the degree of any coherent sheaf ${\cal E}'\subset{\cal E}$ to be $$ \deg{\cal E}'=\int_X c_1({\cal E}')\wedge \omega^{n-1}, $$ and define the slope of ${\cal E}'$ by $$ \mu({\cal E}')=\frac{\deg{\cal E}'}{\mathop{{\fam0 rank}}\nolimits {\cal E}'}. $$ Fix $\tau\in{\bf R}$. The holomorphic pair $({\cal E},\phi)$ is called $\tau$-stable\ if (1)\ $\mu({\cal E}')<\tau\;\; \mbox{for every coherent subsheaf}\;\;\;{\cal E}'\subset{\cal E}$, and (2)\ $\mu({\cal E}/{\cal E}'')>\tau\;\; \mbox{for every non-trivial coherent subsheaf} \;{\cal E}''\; \mbox{such that} \; \phi\in H^0(X,{\cal E}'')$. \end{definition} The Hitchin--Kobayashi\ correspondence is expressed by the following two propositions. \begin{prop}[\cite{B2}]\label{easy} Fix $\tau\in {\bf R}$, let $({\cal E},\phi)$\ be a holomorphic pair, and suppose that there exists a metric, H, satisfying the $\tau$-vortex equation (\ref{mve}). Then either (1)\ the holomorphic pair $({\cal E},\phi)$\ is $\tau$-stable, or (2)\ the bundle ${\cal E}$\ splits holomorphically as ${\cal E}={\cal E}_{\phi}\oplus {\cal E}_{ps}$\ with $\phi\in H^0(X,{\cal E}_{\phi})$, and such that the holomorphic pair $({\cal E}_{\phi},\phi)$\ is $\tau$-stable, and ${\cal E}_{ps}$\ is polystable with slope equal to $\tau$. \end{prop} \begin{prop}[\cite{B2}]\label{hard} Fix $\tau\in {\bf R}$, let $({\cal E},\phi)$\ be a $\tau$-stable holomorphic pair. Then there is a unique smooth metric, H, on $E$\ such that the $\tau$-vortex equation (\ref{mve})\ is satisfied. \end{prop} \noindent{\em Remark}. We are assuming that the K\"{a}hler\ metric is normalized so that $\mathop{{\fam0 Vol}}\nolimits(X)=2\pi$. Otherwise we need to introduce the factor $\frac{2\pi}{\mathop{{\fam0 Vol}}\nolimits(X)}$ in the right hand side of (\ref{mve}). Suppose now that we replace $\tau$\ in equation (\ref{mve}) by a smooth real valued function $t\in C^{\infty}(X,{\bf R})$ and study \begin{equation} i\Lambda F_H +\phi\otimes\phi^{*_H}=t{\bf I}.\label{tmve} \end{equation} The question we wish to address is: how does this affect the Hitchin-Kobayashi correspondence? One direction is clear: replacing the constant $\tau$\ by the smooth function $t$\ has absolutely no effect on the proof of Proposition \ref{easy} (Theorem 2.1.6 in \cite{B2}). The same proof thus yields \begin{thm}[cf. also Theorem 3.3, \cite{OT1}] Fix a smooth function $t\in C^{\infty}(X,{\bf R})$. Let $\overline{t}=\frac{1}{2\pi}\int t$. If a holomorphic pair $({\cal E},\phi)$\ supports a solution to the $t$-vortex equation (\ref{tmve}), then either (1)\ the holomorphic pair $({\cal E},\phi)$\ is $\overline t$-stable, or (2)\ the bundle ${\cal E}$\ splits holomorphically as ${\cal E}={\cal E}_{\phi}\oplus{\cal E}_{ps}$\ with $\phi\in H^0(X,{\cal E}_{\phi})$, and such that the holomorphic pair $({\cal E}_{\phi},\phi)$\ is $\overline t$-stable, and ${\cal E}_{ps}$\ is polystable with slope equal to $\overline t$. \end{thm} We now consider the converse result, i.e. the analogue of Proposition \ref{hard} (Theorem 3.1.1 in \cite{B2}). As shown in \cite{OT1}, the $t$-vortex equation can be reformulated as an equation with a constant right hand side. Let $\tau=\overline{t}$. Since $\int_X(\tau-t)=0$, we can find a smooth function $u$\ such that $\Delta(u)=\tau-t$. Thus (\ref{tmve}) is equivalent to \begin{equation} i\Lambda (F_H+\Delta(u){\bf I}) +\phi\otimes\phi^{*_H}=\tau{\bf I}\ .\label{delta-ve} \end{equation} If we define a new metric $K=He^u$, then (\ref{delta-ve}) becomes \begin{equation} i\Lambda F_K +e^{-u}\phi\otimes\phi^{*_K}=\tau{\bf I}\ .\label{uveOT} \end{equation} This is {\it almost} the $\tau$-vortex equation, the only difference being the prefactor $e^{-u}$\ in the second term. In the analysis of this situation by Okonek and Teleman, they indicate how the proof (of Theorem 3.1.1) in [B2] can be modified to accomodate this new wrinkle. (The proof in [B2] employs a modified Donaldson functional on the space of Hermitian metrics on the bundle $\cal E$. In [OT1], the authors generalize the functional so that it accomodates the extra factor of $e^{-u}$, and argue that this has little effect on the proof.) It is interesting to observe that the same result can be achieved without {\it any modification at all} of the proof in [B2], if one enlarges the catgory in which the proof is applied. This can be seen as follows. If we set \begin{equation} \phi_u=e^{-u/2}\phi\ ,\label{phiu} \end{equation} then (\ref{uveOT}) becomes \begin{equation} i\Lambda F_K +\phi_u\otimes\phi_u^{*_K}=\tau{\bf I}\ .\label{uve} \end{equation} We thus see that \begin{lemma} \label{t-tau} Let $u$\ be given by $\Delta(u)=\tau-t$. The pair $({\cal E},\phi)$\ admits a metric $H$ satisfying the $t$-vortex equation if and only if the pair $({\cal E},\phi_u)$\ admits a metric $K$ satisfying the $\tau$-vortex equation. The metrics $H$\ and $K$\ are related by $K=He^u$. \end{lemma} It is important to notice that, unless $u$\ is a constant function, $\phi_u$\ is {\it not} \ a holomorphic section of ${\cal E}$. Indeed $\overline{\partial}_E\phi_u=-\frac{1}{2}\overline{\partial}(u)\phi_u$. Our key observation is that in Theorem 3.1.1 in \cite{B2}, the holomorphicity of $\phi$\ is not required either in the statement or in the proof of the theorem. The proof, and thus the result remains unchanged if the holomorphic section $\phi$\ is replaced by a smooth section $\phi_u$\ related to $\phi$\ by $\phi_u=e^{-u/2}\phi$. The basic reason can be traced back to the following simple fact: \begin{lemma}\label{u-simple} Let $\phi$, $u$, and $\phi_u$\ be as above. (1)\ Let ${\cal E}'\subset{\cal E}$\ be any holomorphic subbundle of ${\cal E}$. Then $\phi$\ is a section of ${\cal E}'$\ if and only if $\phi_u$\ is a section of ${\cal E}'$. (2)\ Let $s$\ be any smooth endomorphism of ${\cal E}$. Then $s\phi=0$\ if and only if $s\phi_u=0$. \end{lemma} Notice, for instance, that if we define $$ \begin{array}{ll} H^0_u(X,{\cal E})&=e^{-u/2}H^0(X,{\cal E})\\ &=\{e^{-u/2}\phi\in\Omega^0(X,E)\ |\ \phi\in H^0(X,{\cal E})\}\ , \end{array} $$ then the definition of $\tau$-stability can be applied to any pair $({\cal E},\phi_u)$\ where $\phi_u\in H^0_u(X,{\cal E})$. \begin{definition} A pair $({\cal E},\phi_u)$, where $\phi$\ is in $H^0_u(X,{\cal E})$, will be called a $u$-{\em holomorphic pair}. \end{definition} In view of Lemma \ref{u-simple}, it follows that the $u$-holomorphic pair $({\cal E},\phi_u)$\ is $\tau$-stable if and only if the holomorphic pair $({\cal E},\phi)$\ is $\tau$-stable (where $\phi$\ and $\phi_u$\ are related by (\ref{phiu})). Furthermore, without any alteration whatsoever, the proof of Theorem 3.1.1 in \cite{B2} can be applied to a $u$-holomorphic pair to prove: \begin{prop}\label{upair-tvor} Fix $u\in C^{\infty}(X,{\bf R})$\ and $\tau\in{\bf R}$. Let $({\cal E},\phi_u)$\ be a $\tau$-stable $u$-holomorphic pair. Then $E$\ admits a unique smooth metric, say $K$, such that the $\tau$-vortex equation is satisfied, i.e. such that $$ i\Lambda F_K +\phi_u\otimes\phi_u^{*_K}=\tau{\bf I}\ . $$ \end{prop} Taken together, Lemma \ref{t-tau} and Proposition \ref{upair-tvor} thus prove \begin{thm} Fix $\tau=\overline t$\ and suppose that $({\cal E},\phi)$\ is a $\tau$-stable pair. Then $E$\ supports a metric satisfying the $t$-vortex equation. \end{thm} The above results describe the sense in which the vortex equation is {\em insensitive} to the precise form of the parameter $t$. This can be made precise by considering the moduli spaces. Let ${\cal C}$\ be the space of holomorphic structures (or, equivalently, integrable $\overline{\partial}_E$-operators) on $E$, and let $$ {\cal H}=\{(\overline{\partial}_E,\phi)\in{\cal C}\times\Omega^0(X,E)\ |\ \overline{\partial}_E\phi=0\}\ $$ be the configuration space of holomorphic pairs on $E$. Let ${\cal V}_t\subset{\cal H}$\ be the set of $t$-{\em vortices}, i.e. $$ {\cal V}_t=\{(\overline{\partial}_E,\phi)\in{\cal H}\ |\ \mbox{there is a metric satisfying the $t$-vortex equation}\}\ . $$ The above results can then be summarized by saying that (1)\ ${\cal V}_t={\cal V}_{\tau} $\ for all functions $t$\ such that $\overline t=\tau$, and (2)\ for generic values of $\tau$, we can identify ${\cal V}_t={\cal H}_{\tau}$\ where ${\cal H}_{\tau}$\ denotes the set of $\tau$-stable\ holomorphic pairs. In fact, as complex spaces, ${\cal V}_t/{\cal G}^c={\cal V}_{\tau}/{\cal G}^c={\cal B}_{\tau}$, where ${\cal B}_{\tau}$\ is the moduli space of $\tau$-stable holomorphic pairs --- which has the structure of a variety (cf. \cite{Be,BD1,BD2,G4,HL1,HL2,Th}). \noindent{\em Remark}. In the case in which $E=L$ is a line bundle the $\tau$-stability condition reduces to\ $\deg L<\tau$\ , and the moduli space of $\tau$-stable\ pairs is nothing else but the space of {\em non-negative divisors} supported by $L$, where by a non-negative divisor we mean either an effective divisor or the zero divisor. Nevertheless, the function $t$\ does carry some information. For example, the metrics which satisfy the $t$-vortex equation (for fixed $\overline{\partial}_E$\ and $\phi$) do depend on $t$. The following observations shed some light on the role played by $t$. As described in \cite{G4}, \cite{BDGW}, the holomorphic pair $({\cal E},\phi)$\ can be identified with the {\em holomorphic triple} $({\cal E},{\cal O},\phi)$, where ${\cal O}$ is the structure sheaf and $\phi$ is a morphism ${\cal O}\rightarrow{\cal E}$. From this point of view, the natural equations to consider are the framed vortex equations (\ref{fve}). Coming back to the set-up of Section \ref{vortices}, we want to study equations (\ref{fve}) for a vector bundle $E$ of arbitrary rank and $F=L_0$, the trivial line bundle. As for the usual vortex equations, we will look at (\ref{fve}) as equations for a metric on $E$. To do this we fix the holomorphic structure $\overline{\partial}_{L_0}$ on $L_0$ to be the trivial one i.e. $(L_0,\overline{\partial}_{L_0})={\cal O}$, and consider a holomorphic structure $\overline{\partial}_E$ on $E$. Then we take $\phi:{\cal O}\rightarrow{\cal E}$ to be a holomorphic morphism, where ${\cal E}=(E,\overline{\partial}_E)$. In contrast with the coupled vortex equations (\ref{cve}) that we will analyse later, here we need to fix a metric $h$ on $L_0$. Then solving (\ref{fve}) is equivalent to solving for a metric $H$ on $E$ satisfying \begin{equation} i\Lambda F_H+\phi\otimes\phi^\ast=\tau{\bf I}\ .\label{mfve} \end{equation} It is important to notice that now $\phi^\ast$ is the adjoint of $\phi$ with respect to both metrics $H$ and $h$. The identification between $({\cal E},\phi)$\ and $({\cal E},{\cal O},\phi)$\ requires a choice of trivializing frame for ${\cal O}$, say $f$. If $h(f,f)=e^u$, then (\ref{mfve}) becomes $$ i\Lambda F_H +e^{-u}\phi\otimes\phi^{*_H}=\tau {\bf I}\ . $$ Thus we recover the usual $\tau$-vortex equation when we select the metric on $L_0$\ for which the holomorphic frame of ${\cal O}$\ is also a unitary frame. For other choices of $h$\ we see that we get essentially equation (\ref{uve}), or equivalently, the $t$-vortex equation. {\em From this point of view, the function $t$\ is determined by the metric on $L_0$.} The impact of non-constant $t$\ can also be understood from the symplectic point of view. If we fix a metric, say $H$, on $E$, the induced inner products on ${\cal C}$\ and on $\Omega^0(X,E)$\ can be combined to give a symplectic structure on the configuration space ${\cal H}$. Denoting the symplectic forms on ${\cal C}$\ and $\Omega^0(X,E)$\ by $\omega_{H,{\cal C}}$\ and $\omega_{H,0}$\ respectively, we take $$ \omega_{H,H}=\omega_{H,{\cal C}}+\omega_{H,0}\ $$ as the symplectic form on ${\cal H}$. Let ${\cal G}_H$\ be the unitary gauge group of $E$\ determined by $H$, and let $\mathop{{\fam0 Lie }}\nolimits {\cal G}_H$\ be its Lie algebra. A moment map $\mu:{\cal H}\rightarrow\mathop{{\fam0 Lie }}\nolimits{\cal G}_H^*$\ for the action of ${\cal G}_H$\ on the symplectic space $({\cal H},\omega_{H,H})$\ is given by $$ \mu_{H,H}(\overline{\partial}_E,\phi)=\Lambda F_{\overline{\partial}_E,H} -i\phi\otimes\phi^{*_H}\ , $$ where we have written $F_{\overline{\partial}_E,H}$ instead of $F_H$ to emphasize that $F_H$ depends also on the holomorphic structure on $E$. The $\tau$-vortex equation is thus equivalent to the condition $\mu_{H,H}(\overline{\partial}_E,\phi)=-i\tau{\bf I}$, and we get an identification of moduli spaces: $$ {\cal B}_{\tau}={\cal V}_{\tau}/{\cal G}^c=\mu_{H,H}^{-1}(-i\tau{\bf I})/{\cal G}_H\ . $$ Replacing $\tau$\ by the non-constant function $t$\ has no effect on the identification ${\cal V}_{t}/{\cal G}^c=\mu_{H,H}^{-1}(-it{\bf I})/{\cal G}_H$: The element $it{\bf I}$\ is still a central element in $\mathop{{\fam0 Lie }}\nolimits{\cal G}_H^*$, so the symplectic quotient at this level is well defined. (The problem comes in proving that ${\cal V}_t/{\cal G}^c={\cal B}_{\tau}$.) An alternative point of view makes use of the equivalence between the equations (\ref{tmve}) and (\ref{uve}). We define $$ \mu_{H,K}(\overline{\partial}_E,\phi)=\Lambda F_{\overline{\partial}_E,H} -i\phi\otimes\phi^{*_K}\ , $$ where $H$\ and $K$\ are metrics on $E$. By the above results, $K=He^u$\ where $\Delta(u)=\tau-t$, then $$ \mu_{H,H}^{-1}(-it{\bf I})= \mu_{H,K}^{-1}(-i\tau{\bf I})\ . $$ The point is that $\mu_{H,K}$\ is also a moment map for the action of ${\cal G}_H$. It arises when the symplectic structure on ${\cal H}$\ is taken to be $$ \omega_{H,K}=\omega_{H,{\cal C}}+\omega_{K,0}\ . $$ {\em From this point of view, the function $t$\ arises from a deformation of the symplectic structure on ${\cal H}$.} \subsection{The coupled vortex equations} Let us consider the set-up in Section \ref{vortices} for the coupled vortex equations\ (\ref{cve}). As in the previous situation, we want to look at (\ref{cve}) as equations for metrics. In order to do this let us fix holomorphic structures $\overline{\partial}_E$ and $\overline{\partial}_L$ on $E$ and $L$ respectively. Denote by ${\cal E}$ and ${\cal L}$ the corresponding holomorphic vector bundles. Let $\phi\in H^0({\cal E}\otimes{\cal L}^\ast)$. Equations (\ref{cve}) are then equivalent to solving \begin{equation} \left. \begin{array}{l} i \Lambda F_H+\phi\otimes\phi^\ast=\tau {\bf I}_E\\ i \Lambda F_K-|\phi|^2=\tau' \end{array}\right \}.\label{mcve} \end{equation} for metrics $H$ and $K$ on ${\cal E}$ and ${\cal L}$ respectively. A Hitchin--Kobayashi\ correspondence was proved in \cite{G4}. The appropriate notion of stability for $({\cal E},{\cal L},\phi)$ can be expressed in terms of the stability of a pair, namely \begin{definition}\label{st} The holomorphic triple $({\cal E},{\cal L},\phi)$ is said to be $\tau$-stable if the holomorphic pair $({\cal E}\otimes{\cal L}^\ast,\phi)$ is $(\tau-\deg L)$-stable. \end{definition} \begin{thm}[\cite{G4}]\label{existence-cve} Let $\tau$ and $\tau'$ be real numbers satisfying (\ref{parameters}). Let $({\cal E},{\cal L},\phi)$ a holomorphic triple. Suppose that there exist metrics $H$ and $K$ satisfying (\ref{mcve}), then either $({\cal E},{\cal L},\phi)$ is $\tau$-stable\ or the bundle ${\cal E}$ splits holomorphically as ${\cal E}_\phi\oplus{\cal E}_{ps}$ with $\phi\in H^0(X,{\cal E}_{\phi}\otimes{\cal L}^\ast)$, and such that $({\cal E}_\phi,{\cal L},\phi)$ is $\tau$-stable\ and ${\cal E}_{ps}$ is polystable with slope equal to $\tau$. Conversely, let $({\cal E},{\cal L},\phi)$ be a $\tau$-stable\ triple then there are unique smooth metrics $H$ and $K$ satisfying the coupled vortex equations (\ref{mcve}). \end{thm} Suppose now that we replace $\tau$ and $\tau'$ in (\ref{mcve}) by smooth functions $t, t'\in C^\infty(X,{\bf R})$, i.e. we consider \begin{equation} \left. \begin{array}{l} i \Lambda F_H+\phi\otimes\phi^\ast=t {\bf I}_E\\ i \Lambda F_K-|\phi|^2=t' \end{array}\right \}.\label{tmcve} \end{equation} The first thing that we observe is that in order to have solutions $t$ and $t'$ must satisfy \begin{equation} \int_X (r t+t')=\deg E +\deg L\label{t-t'} \end{equation} where $r=\mathop{{\fam0 rank}}\nolimits E$. Next, let us recall the main ideas in the proof of Theorem \ref{existence-cve}: The basic fact is that the coupled vortex equations (\ref{mcve}) are a dimensional reduction of the Hermitian--Einstein\ equation for a metric on a certain vector bundle over ${{X\times\bP^1}}$. This bundle ${\cal F}$, canonically associated to the holomorphic triple $({\cal E},{\cal L},\phi)$, is an extension on ${{X\times\bP^1}}$ of the form \begin{equation} 0\longrightarrow p^\ast{\cal E}\longrightarrow {\cal F}\longrightarrow {p^\ast} {\cal L} \otimes q^\ast{\cal O}(2)\longrightarrow 0, \label{bigbun} \end{equation} where $p$ and $q$ are the projections from ${{X\times\bP^1}}$ to $X$ and ${\bf P}^1$ respectively. This is simply because $H^1({{X\times\bP^1}}, p^\ast ({\cal E}\otimes{\cal L}^\ast)\otimes q^\ast{\cal O}(2)) \cong H^0(X,{\cal E}\otimes{\cal L}^\ast)$ Let $SU(2)$ act on $X\times {\bf P}^1$, trivially on $X$, and in the standard way on ${{\bf P}}^1 \cong SU(2)/U(1)$. This action can be lifted to an action on ${\cal F}$, trivial on $p^\ast{\cal E}$ and ${p^\ast}{\cal L}$, and standard on $q^\ast{\cal O}(2)$. The bundle ${\cal F}$ is in this way an $SU(2)$-equivariant holomorphic vector bundle. Let $\tau$ and $\tau'$ be related by (\ref{parameters}) and let \begin{equation} \sigma=\frac{4\pi}{\tau-\tau'} \end{equation} be positive. Consider the $SU(2)$-invariant K\"{a}hler metric on $X\times {\bf P}^1$ whose K\"{a}hler form is $$ \omega_\sigma =p^\ast\omega +\sigma q^\ast\omega_{{\bf P}^1}, $$ where $\omega$ is the K\"{a}hler form on $X$ (normalized such that $\mathop{{\fam0 Vol}}\nolimits(X)=2\pi$), and $\omega_{{\bf P}^1}$ is the {\em Fubini-Study} metric with volume 1. Theorem \ref{existence-cve} is then a consequence of the following two propositions and the Hitchin--Kobayashi\ correspondence proved by Donaldson \cite{D1,D2}, and Uhlenbeck and Yau \cite{UY}). \begin{prop}[\cite{G4}]\label{dr} The triple $({\cal E},{\cal L},\phi)$ admits a solution to (\ref{mcve}) if and only if the vector bundle ${\cal F}$ in (\ref{bigbun}) has a (${SU}(2)$-invariant) Hermitian--Einstein\ metric with respect to $\omega_\sigma$. \end{prop} \begin{prop}[\cite{G4}]\label{s-ts} Suppose that ${\cal E}$ is not isomorphic to ${\cal L}$. Then the triple $({\cal E},{\cal L},\phi)$ is $\tau$-stable\ if and only if ${\cal F}$ is stable with respect to $\omega_\sigma$. If ${\cal E}\cong{\cal L}$, then ${\cal F}\cong {p^\ast} {\cal L}\otimes{q^\ast}{\cal O}(1)\oplus{p^\ast} {\cal L}\otimes{q^\ast}{\cal O}(1)$. \end{prop} Suppose first that the functions $t$ and $t'$, in addition to satisfying (\ref{t-t'}), verify that there is a positive constant $\sigma$ so that \begin{equation} t-t'=\frac{4\pi}{\sigma}.\label{strong-t-t'} \end{equation} The proof of Proposition \ref{dr} then yields \begin{prop} \label{weakdr} The triple $({\cal E},{\cal L},\phi)$ admits a solution to (\ref{tmcve}) if and only if the vector bundle ${\cal F}$ in (\ref{bigbun}) has a (${SU}(2)$-invariant) metric satisfying the weak Hermitian--Einstein\ equation \begin{equation} i\Lambda_\sigma F_{\bf H}= t\ {\bf I} \end{equation} with respect to the K\"{a}hler\ form $\omega_\sigma={p^\ast} \omega_X\oplus \sigma{q^\ast}\omega_{{\bf P}^1}$. \end{prop} But the existence of a weak Hermitian--Einstein\ metric is in fact equivalent to the existence of a Hermitian--Einstein\ metric --- as one can see simply by applying a conformal change to the metric --- and hence equivalent to the stability of the bundle. We can then combine again Propositions \ref{weakdr} and \ref{s-ts} to prove the following. \begin{thm}\label{existence-tcve} Fix smooth functions $t,t'\in C^\infty(X,{\bf R})$ satisfying (\ref{t-t'}) and (\ref{strong-t-t'}). Let $\overline{t}=\frac{1}{2\pi}\int_X t$ and $\overline{t'}=\frac{1}{2\pi}\int_X t'$. Let $({\cal E},{\cal L},\phi)$ a holomorphic triple. Suppose that there exist metrics $H$ and $K$ satisfying (\ref{tmcve}), then either $({\cal E},{\cal L},\phi)$ is $\overline{t}$-stable or the bundle ${\cal E}$ splits holomorphically as ${\cal E}_\phi\oplus{\cal E}_{ps}$ with $\phi\in H^0(X,{\cal E}_{\phi}\otimes{\cal L}^\ast)$, and such that $({\cal E}_\phi,{\cal L},\phi)$ is $\overline{t}$-stable and ${\cal E}_{ps}$ is polystable with slope equal to $\overline{t}$. Conversely, let $({\cal E},{\cal L},\phi)$ be a $\overline{t}$-stable triple then there are unique smooth metrics $H$ and $K$ satisfying equations(\ref{tmcve}). \end{thm} We will show now that the general coupled vortex equations (\ref{tmcve}) with $t$ and $t'$ satisfying simply (\ref{t-t'}) are also a dimensional reduction, but in this case of a metric on ${\cal F}$ satisfying a certain deformation of the Hermitian--Einstein\ condition. We set, as above, $$ \sigma=\frac{4\pi}{\overline{t}-\overline{t'}} $$ where $\overline{t}$ and $\overline{t'}$ denote the average values of $t$ and $t'$ respectively. Again the proof of Proposition \ref{dr} yields \begin{prop} \label{newdr} The triple $({\cal E},{\cal L},\phi)$ admits a solution to (\ref{tmcve}) if and only if the vector bundle ${\cal F}$ in (\ref{bigbun}) has a (${SU}(2)$-invariant) metric satisfying the deformed Hermitian--Einstein\ equation \begin{equation} i\Lambda_\sigma F_{\bf H}= \overline{t}\ {\bf I} + \left(\begin{array}{cc}(t-\overline{t}) {\bf I}_1 & 0\\ 0&(t'-\overline{t'}){\bf I}_2 \end{array}\right), \label{bigbun-eqn} \end{equation} with respect to the K\"{a}hler\ form $$ \omega_\sigma={p^\ast} \omega_X\oplus \sigma{q^\ast}\omega_{{\bf P}^1}. $$ \end{prop} The deformed Hermitian--Einstein\ equation in this Proposition is similar to the kind studied in \cite{BG2}. In \cite{BG2} we considered an extension of holomorphic vector bundles over a compact K\"{a}hler\ manifold \begin{equation} {0\lra \cE_1\lra \cE\lra \cE_2\lra 0}\label{extn} \end{equation} and studied metrics $H$ on ${\cal E}$ satisfying the equation \begin{equation} i\Lambda F_H=\left(\begin{array}{cc}\tau_1 {\bf I}_1 & 0\\ 0&\tau_2 {\bf I}_2 \end{array}\right),\label{ext-eqn} \end{equation} where $\tau_1$ and $\tau_2$ are real numbers, related by $$ \tau_1 r_1 +\tau_2 r_2=\deg {\cal E}. $$ The reason we can write an equation like (\ref{ext-eqn}) is that the metric $H$ gives a $C^\infty$ splitting of (\ref{extn}). We proved an existence theorem for metrics satisfying (\ref{ext-eqn}) in terms of a notion of stability for the extension depending on the parameter $\alpha=\tau_1-\tau_2$. To define this stability condition consider any coherent subsheaf ${\cal E}'\subset{\cal E}$ and write it as a subextension $$ {0\lra \cE_1'\lra\cE'\lra \cE_2'\lra 0}. $$ Define the $\alpha$-slope of ${\cal E}'$ as $$ \mu_\alpha({\cal E}')=\mu({\cal E}')+\alpha\frac{\mathop{{\fam0 rank}}\nolimits{\cal E}_2'}{\mathop{{\fam0 rank}}\nolimits {\cal E}'}. $$ Then we say that (\ref{extn}) is $\alpha$-stable if and only if for every non-trivial subsheaf ${\cal E}'\subset{\cal E}$ $$ \mu_\alpha({\cal E}')<\mu_\alpha({\cal E}). $$ We proved \begin{thm}[\cite{BG2}]Let $\alpha=\tau_1-\tau_2\leq 0$ and suppose that (\ref{extn}) is indecomposable (as an extension), then ${\cal E}$ admits a metric satisfying (\ref{ext-eqn}) if and only (\ref{extn}) is $\alpha$-stable. \end{thm} \noindent{\em Remark}. If $\alpha=0$ (\ref{ext-eqn}) reduces to the Hermitian--Einstein\ equation and the stability condition is the usual stability of the bundle ${\cal E}$. The deformed Hermitian--Einstein\ equation in Proposition \ref{newdr} differs from (\ref{ext-eqn}) only in that the constants $\tau_1$ and $\tau_2$ have been replaced by smooth functions, $t_1$ and $t_2$, satisfying \begin{equation} \int(r_1 t_1 +r_2 t_2)=\deg {\cal E}.\label{t1-t2} \end{equation} By the same methods used in \cite{BG2} one can readily show one direction of the Hitchin-Kobayashi correspondence, namely \begin{thm}\label{alpha-hk}Let\ $t_1$ and $t_2$ be smooth real functions satisfying (\ref{t1-t2}) and such that $\alpha=\int(t_1-t_2)\leq 0$. Then the existence of a metric $H$ on ${\cal E}$ satisfying \begin{equation} i\Lambda F_H=\left(\begin{array}{cc}t_1 {\bf I}_1 & 0\\ 0&t_2 {\bf I}_2 \end{array}\right),\label{var-extn-eqn} \end{equation} implies the $\alpha$-stability of (\ref{extn}). \end{thm} It should likewise be possible to adapt the proof of the other direction of the Hitchin-Kobayashi correspondence. This will then allow one (by taking $\alpha=0$) to establish a more general version of Theorem \ref{existence-tcve}, valid when $t$\ and $t'$\ are smooth functions satisfying just (\ref{t-t'}). We will discuss this in a future publication. To describe the moduli space, let ${\cal C}_E$ and ${\cal C}_L$ the sets of holomorphic structures on $E$ and $L$ respectively. Consider the set \begin{equation} {\cal H}(E,L)= \{(\overline{\partial}_E,\overline{\partial}_L,\phi)\in {\cal C}_E\times{\cal C}_L\times\Omega^0(\mathop{{\fam0 Hom}}\nolimits(L,E)) \;\;|\;\; \phi\in H^0(X,{\cal E}\otimes{\cal L}^\ast)\} \label{ht} \end{equation} of {\em holomorphic triples} on $(E,L)$, where ${\cal E}$ and ${\cal L}$ denote the holomorphic vector bundles defined by $\overline{\partial}_E$ and $\overline{\partial}_L$ respectively. Let ${\cal H}_\tau(E,L)\subset{\cal H}(E,L)$ be the set of $\tau$-stable\ holomorphic triples. This set is invariant under the action of the complex gauge groups of $E$ and $L$, ${\cal G}^c_E$ and ${\cal G}^c_L$, say. The moduli space of $\tau$-stable\ triples is defined as $$ {\cal B}_\tau(E,L)={\cal H}_\tau(E,L)/{\cal G}^c_E\times{\cal G}^c_L. $$ The set ${\cal B}_\tau(E,L)$, which has naturally the structure of a variety (cf. \cite{G4}), is closely related to the moduli space of stable pairs---this is not surprising in view of the definition \ref{st}. More precisely, the map $({\cal E},{\cal L},\phi)\mapsto ({\cal E}\otimes{\cal L}^\ast,\phi)$ exhibits ${\cal B}_\tau(E,L)$ as a $\mathop{{\fam0 Pic}}\nolimits^0$-principal bundle over the moduli space of ($\tau-\deg L$)-stable pairs on $E\otimes L^\ast$. The study of the general equations (\ref{gcve}), i.e. the case in which $F$ is of arbitrary rank, requires the introduction of a new notion of stability. This was carried out in \cite{BG1}. All the results explained above should extend appropriately to the higher rank case when one replaces $\tau$ and $\tau'$ in (\ref{gcve}) by functions $t$ and $t'$.

9602

alg-geom/9602014

en

https://arxiv.org/abs/alg-geom/9602014

alg-geom/9602014

Alice Silverberg

A. Silverberg and Yu. G. Zarhin

Reduction of abelian varieties

We study semistable reduction and torsion points of abelian varieties. In particular, we give necessary and sufficient conditions for an abelian variety to have semistable reduction. We also study N\'eron models of abelian varieties with potentially good reduction and torsion points of small order. We study some invariants that measure the extent to which an abelian variety with potentially good reduction fails to have good reduction.

alg-geom

\section{Introduction} In this paper we study the reduction of abelian varieties. We assume $F$ is a field with a discrete valuation $v$, $X$ is an abelian variety over $F$, and $n$ is an integer not divisible by the residue characteristic. In Part \ref{semistabpart} we give criteria for semistable reduction. Suppose $n \ge 5$. In Theorem \ref{ssredlem} we show that $X$ has semistable reduction if and only if $(\sigma - 1)^2 = 0$ on the $n$-torsion in $X$, for every $\sigma$ in the absolute inertia group. In Theorem \ref{ssredconverse} we show (using Theorem \ref{ssredlem}) that $X$ has semistable reduction if and only if there exists a subgroup of $n$-torsion points such that the absolute inertia group acts trivially on both it and its orthogonal complement with respect to the $e_n$-pairing. We deduce as special cases both Raynaud's criterion (that the abelian variety have full level $n$ structure for $n \ge 3$; see Theorem \ref{raynaud}) and the criterion of \cite{semistab} (that the abelian variety have partial level $n$ structure for $n \ge 5$; see Theorem \ref{ssred}). We also obtain a (near) converse to the criterion of \cite{semistab}. The proofs are based on the fundamental results of Grothendieck on semistable reduction of abelian varieties (see \cite{SGA}). In \S\ref{except} we allow $n<5$. In \S\ref{Gsect} we give a measure of potentially good reduction. We discuss other measures of potentially good reduction in Part \ref{neronpart}. In Part \ref{neronpart} we study N\'eron models of abelian varieties with potentially good reduction and torsion points of small order. Suppose that the valuation ring is henselian and the residue field is algebraically closed. If $X$ has good reduction, then $X_n \subseteq X(F)$ (this is an immediate corollary of the existence of N\'eron models; see Lemma \ref{neronlemma} below). On the other hand, if $X_n \subseteq X(F)$ and $n \ge 3$, then by virtue of Raynaud's criterion for semistable reduction, $X$ has good reduction. Notice that the failure of $X$ to have good reduction is measured by the dimension $u$ of the unipotent radical of the special fiber of the N\'eron model of $X$. In particular, $u = 0$ if and only if $X$ has good reduction. In general, $0 \le u \le \mathrm{dim}(X)$. The equality $u = \mathrm{dim}(X)$ says that $X$ has purely additive reduction. Another measure of the deviation from good reduction is the (finite) group of connected components $\Phi$ of the special fiber of the N\'eron model. If $X$ has good reduction then $\Phi=\{0\}$, but the converse statement is not true in general. The aim of \S\ref{main} is to connect explicitly the invariants $u$ and $\Phi$ with the failure of $X(F)$ to contain all the $n$-torsion points. This failure can be measured by the index $[X_n:X_n(F)]$. We assume that at least ``half'' of the $n$-torsion points are rational over $F$. More precisely, we assume that there exists an $F$-rational polarization $\lambda$ on $X$ and a maximal isotropic (with respect to the pairing $e_{\lambda,n}$ induced from the Weil $e_n$-pairing by $\lambda$) subgroup of $X_n$ consisting of $F$-rational points. If in addition $n \ge 5$, then $X$ has good reduction (see Theorem 7.4 of \cite{semistab}), and therefore $u = 0$, $\Phi=\{0\}$, and $X_n=X_n(F)$. Therefore, we have to investigate only the cases $n = 2$, $3$, and $4$. Let $\Phi'$ denote the prime-to-$p$ part of $\Phi$, where $p$ is the residue characteristic (with $\Phi ' = \Phi$ if $p = 0$). We show that if $n = 2$ then $\Phi'$ is an elementary abelian $2$-group and $[X_2:X_2(F)]\#\Phi' = 4^{u}$, if $n = 3$ then $[X_3:X_3(F)] = 3^{u}$ and $\Phi' \cong ({\mathbf Z}/3{\mathbf Z})^u$, and if $n = 4$ then $X_2 \subseteq X(F)$, $[X_4:X_4(F)] = 4^{u}$, and $\Phi' \cong ({\mathbf Z}/2{\mathbf Z})^{2u}$. If instead of assuming partial level $n$ structure we assume that all the points of order $2$ on $X$ are defined over $F$, then $[X_4:X_4(F)]=4^u$ and $\Phi' \cong ({\mathbf Z}/2{\mathbf Z})^{2u}$. Earlier work on abelian varieties with potentially good reduction and on groups of connected components of N\'eron models has been done by Serre and Tate \cite{Serre-Tate}, Silverman \cite{Silverman}, Lenstra and Oort \cite{LenstraOort}, Lorenzini \cite{Lorenzini}, and Edixhoven \cite{Edixhoven}. Silverberg would like to thank the IHES and the Bunting Institute for their hospitality, and the NSA and the Science Scholars Fellowship Program at the Bunting Institute for financial support. Zarhin would like to thank the NSF for financial support. He also would like to thank the organizers of the NATO/CRM 1998 Summer School on the Arithmetic and Geometry of Algebraic Cycles for twelve wonderful days in Banff. \section{Notation and definitions} If $F$ is a field, let $F^s$ denote a separable closure. Suppose that $X$ is an abelian variety defined over $F$, and $n$ is a positive integer not divisible by the characteristic of $F$. Let $X^\ast$ denote the dual abelian variety of $X$, let $X_n$ denote the kernel of multiplication by $n$ in $X(F^s)$, let $X_n^\ast$ denote the kernel of multiplication by $n$ in $X^\ast(F^s)$, and let $\boldsymbol \mu _n$ denote the $\mathrm{Gal} (F^s/F)$-module of $n$-th roots of unity in $F^s$. The $e_n$-pairing $$e_n : X_n \times X_n^\ast \to {\boldsymbol \mu}_n$$ is a $\mathrm{Gal}(F^s/F)$-equivariant nondegenerate pairing (see \S 74 of \cite{WeilAV}). If $S$ is a subgroup of $X_n$, let $$S^{\perp_n} = \{ y \in X_n^\ast : e_n(x,y) = 1 \text{ for every } x \in S \} \subseteq X_n^\ast.$$ If $\lambda$ is a polarization on $X$, define $$e_{\lambda ,n} : X_n \times X_n \to {\boldsymbol \mu}_n$$ by $e_{\lambda ,n}(x,y) = e_n(x,\lambda(y))$ (see \S 75 of \cite{WeilAV}). Then $$\sigma (e_{\lambda ,n}(x_1,x_2)) = e_{\sigma (\lambda ),n}(\sigma (x_1),\sigma (x_2))$$ for every $\sigma \in \mathrm{Gal} (F^s/F)$ and $x_1$, $x_2 \in X_n$. If $n$ is relatively prime to the degree of the polarization $\lambda$, then the pairing $e_{\lambda ,n}$ is nondegenerate. If $\ell$ is a prime not equal to the characteristic of $F$, and $d = \mathrm{dim}(X)$, let $$\rho_{\ell,X} : \mathrm{Gal}(F^s/F) \to \mathrm{Aut}(T_\ell(X)) \cong \mathrm{M}_{2d}({\mathbf Z}_\ell)$$ denote the $\ell$-adic representation on the Tate module $T_\ell(X)$ of $X$, and let $V_\ell(X) = T_\ell(X) \otimes_{{\mathbf Z}_\ell} {\mathbf Q}_\ell$. Let $I$ denote the identity matrix in $\mathrm{M}_{2d}({\mathbf Z}_\ell)$. If $L$ is a Galois extension of $F$, $v$ is a discrete valuation on $F$, and $w$ is an extension of $v$ to $L$, let ${\mathcal I}(w/v)$ denote the inertia subgroup at $w$ of $\mathrm{Gal}(L/F)$. If $X$ is an abelian variety over $F$, let $X_v$ denote the special fiber of the N\'eron model of $X$ at $v$ and let $X_v^0$ denote its identity connected component. Let $a$, $u$, and $t$ denote, respectively, the abelian, unipotent, and toric ranks of $X_v$. Then $a + u + t = \mathrm{dim}(X)$. If $p$ ( $\ge 0$) is the residue characteristic of $v$, let $\Phi '$ denote the prime-to-$p$ part of the group of connected components of the special fiber of the N\'eron model of $X$ at $v$ (with $\Phi '$ the full group of components if $p = 0$). \begin{defn} If $v$ is a discrete valuation on a field $F$, we say the valuation ring is {\em strictly henselian} if the valuation ring is henselian and the residue field is algebraically closed. \end{defn} \begin{defn} Suppose $L/F$ is an extension of fields, $w$ is a discrete valuation on $L$, and $v$ is the restriction of $w$ to $F$. We say that $w/v$ is {\em unramified} if a uniformizing element of the valuation ring for $v$ induces a uniformizing element of the valuation ring for $w$ and the residue field extension is separable (see Definition 1 on page 78 of \cite{BLR}). \end{defn} \begin{rem}[Remark 5.3 of \cite{semistab}] \label{ramifiedcyclic} Suppose $v$ is a discrete valuation on a field $F$, and $m$ is a positive integer not divisible by the residue characteristic. Then every degree $m$ Galois extension of $F$ totally ramified at $v$ is cyclic. If $F(\zeta_m) = F$, then $F$ has a cyclic extension of degree $m$ which is totally ramified at $v$. In particular, if the residue characteristic is not $2$ then $F$ has a quadratic extension which is (totally and tamely) ramified at $v$. If the valuation ring is henselian and the residue field is separably closed, then $F= F(\zeta_m)$ and therefore $F$ has a cyclic totally ramified extension of degree $m$. (See Remark 5.3 of \cite{semistab}.) Note also that $F$ has no non-trivial unramified extensions if and only if the valuation ring is henselian and the residue field is separably closed. \end{rem} \part{Semistable reduction of abelian varieties} \label{semistabpart} \section{Preliminaries} \begin{defn} If $k$ is a positive integer, define a finite set of prime powers $N(k)$ by $$N(k) = \{\text{prime powers $\ell^m : 0 \le m(\ell - 1) \le k $}\}.$$ \end{defn} For example, $$N(1) = \{1, 2\}, \quad N(2) = \{1, 2, 3, 4\},$$ $$N(3) = \{1, 2, 3, 4, 8\}, \quad N(4) = \{1, 2, 3, 4, 5, 8, 9, 16\}.$$ \begin{thm} \label{quasithm} Suppose $n$ and $k$ are positive integers, ${\mathcal O}$ is an integral domain of characteristic zero such that no rational prime which divides $n$ is a unit in ${\mathcal O}$, $\alpha \in {\mathcal O}$, $\alpha$ has finite multiplicative order, and $(\alpha - 1)^k \in n{\mathcal O}$. If $n \notin N(k)$, then $\alpha = 1$. In particular, if $(\alpha - 1)^2 \in n{\mathcal O}$ and $n \ge 5$, then $\alpha = 1$. \end{thm} \begin{proof} See Corollary 3.3 of \cite{serrelem}. \end{proof} \begin{lem}[Lemma 5.2 of \cite{semistab}] \label{localglobal} Suppose that $d$ and $n$ are positive integers, and for each prime $\ell$ which divides $n$ we have a matrix $A_\ell \in M_{2d}({\mathbf Z}_\ell)$ such that the characteristic polynomials of the $A_\ell$ have integral coefficients independent of $\ell$, and such that $(A_\ell - I)^2 \in nM_{2d}({\mathbf Z}_\ell)$. Then for every eigenvalue $\alpha$ of $A_\ell$, $(\alpha - 1)/\sqrt{n}$ satisfies a monic polynomial with integer coefficients. \end{lem} \begin{lem}[Lemma 4.2 of \cite{degree}] \label{mevals} Suppose $v$ is a discrete valuation on a field $F$ with residue characteristic $p \ge 0$, $m$ is a positive integer, $\ell$ is a prime, $p$ does not divide $m\ell$, $K$ is a degree $m$ extension of $F$ which is totally ramified above $v$, and ${\bar v}$ is an extension of $v$ to a separable closure $K^s$ of $K$. Suppose that $X$ is an abelian variety over $F$, and for every $\sigma \in {\mathcal I}({\bar v}/v)$, all the eigenvalues of $\rho_{\ell,X}(\sigma)$ are $m$-th roots of unity. Then $X$ has semistable reduction at the extension of $v$ to $K$. \end{lem} \section{Criteria for semistable reduction} \begin{thm}[Galois Criterion for Semistable Reduction] \label{galcrit} Suppose $X$ is an abel\-ian variety over a field $F$, $v$ is a discrete valuation on $F$, $\ell$ is a prime not equal to the residue characteristic of $v$, ${\bar v}$ is an extension of $v$ to $F^s$, and ${\mathcal I} = {\mathcal I}({\bar v}/v)$. Then the following are equivalent: \begin{enumerate} \item[(i)] $X$ has semistable reduction at $v$, \item[(ii)] ${\mathcal I}$ acts unipotently on $T_\ell(X)$; i.e., all the eigenvalues of $\rho_{\ell,X}(\sigma)$ are $1$, for every $\sigma \in {\mathcal I}$, \item[(iii)] for every $\sigma \in {\mathcal I}$, $(\rho_{\ell,X}(\sigma) - I)^2 = 0$. \end{enumerate} \end{thm} \begin{proof} See Proposition 3.5 and Corollaire 3.8 of \cite{SGA} and Theorem 6 on p.~184 of \cite{BLR}. \end{proof} \begin{thm}[Raynaud Criterion for Semistable Reduction] \label{raynaud} \hfil Suppose $X$ is an abelian variety over a field $F$ with a discrete valuation $v$, $m$ is a positive integer not divisible by the residue characteristic of $v$, and the points of $X_m$ are defined over an extension of $F$ which is unramified over $v$. If $m \ge 3$, then $X$ has semistable reduction at $v$. \end{thm} \begin{proof} See Proposition 4.7 of \cite{SGA}. \end{proof} \begin{prop} \label{sslem} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$, $n$ is an integer not divisible by the residue characteristic of $v$, ${\bar v}$ is an extension of $v$ to $F^s$, and ${\mathcal I} = {\mathcal I}({\bar v}/v)$. Let $S = X_n^{\mathcal I}$, the elements of $X_n$ on which ${\mathcal I}$ acts as the identity. If $X$ has semistable reduction at $v$, then \begin{enumerate} \item[{(i)}] $(\sigma - 1)^2 X_n = 0$ for every $\sigma \in {\mathcal I}$, and \item[{(ii)}] ${\mathcal I}$ acts as the identity on $S^{\perp_n}$. \end{enumerate} \end{prop} \begin{proof} Suppose $X$ has semistable reduction at $v$. By Theorem \ref{galcrit}, we have (i). It follows that $\sigma^n = 1$ on $X_n$. Since $n$ is not divisible by the residue characteristic, $X_n$ is tamely ramified over $F$. Let ${\mathcal J}$ denote the first ramification group. Then the action of ${\mathcal I}$ on $X_n$ factors through ${\mathcal I}/{\mathcal J}$. Let $\tau$ denote a lift to ${\mathcal I}$ of a topological generator of the pro-cyclic group ${\mathcal I}/{\mathcal J}$. Since $$e_n((\tau - 1)X_n,(X_n^\ast)^{\mathcal I}) = 1,$$ we have $$\#((X_n^\ast)^{\mathcal I})\#((\tau - 1)X_n) \le \#X_n^\ast.$$ The map from $X_n$ to $(\tau - 1)X_n$ defined by $y \mapsto (\tau - 1)y$ defines a short exact sequence $$0 \to S \to X_n \to (\tau - 1)X_n \to 0.$$ Therefore, $$\#S\#((\tau - 1)X_n) = \#X_n = \#S\#S^{\perp_n}.$$ Similarly, $$\#((X_n^\ast)^{\mathcal I})\#((\tau - 1)X_n^\ast) = \#X_n^\ast.$$ Therefore, $$\#S^{\perp_n} = \#((\tau - 1)X_n) \le \#((\tau - 1)X_n^\ast).$$ Since $(\tau - 1)X_n^\ast \subseteq S^{\perp_n}$, we conclude that $$S^{\perp_n} = (\tau - 1)X_n^\ast.$$ By (i), we have $(\tau - 1)^2X_n = 0$. It follows from the natural $\mathrm{Gal}(F^s/F)$-equivariant isomorphism $X_n^\ast \cong \mathrm{Hom}(X_n,\boldsymbol \mu_n)$ that $(\tau - 1)^2X_n^\ast = 0$, and therefore ${\mathcal I}$ acts as the identity on $S^{\perp_n}$. \end{proof} \begin{thm} \label{ssredlem} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$, $n$ is an integer not divisible by the residue characteristic of $v$, $n \ge 5$, ${\bar v}$ is an extension of $v$ to $F^s$, and ${\mathcal I} = {\mathcal I}({\bar v}/v)$. Then $X$ has semistable reduction at $v$ if and only if $(\sigma - 1)^2 X_n = 0$ for every $\sigma \in {\mathcal I}$. \end{thm} \begin{proof} If $X$ has semistable reduction at $v$ then for every $\sigma \in {\mathcal I}$ we have $(\sigma - 1)^2 X_n = 0$, by Proposition \ref{sslem}i. Conversely, suppose $n \ge 5$ and $(\sigma - 1)^2 X_n = 0$ for every $\sigma \in {\mathcal I}$. Let ${\mathcal I}' \subseteq {\mathcal I}$ be the inertia group for the prime below ${\bar v}$ in a finite Galois extension of $F$ over which $X$ has semistable reduction. Take $\sigma \in {\mathcal I}$. Then $\sigma^m \in {\mathcal I}'$ for some $m$. Let $\ell$ be a prime divisor of $n$. Theorem \ref{galcrit} implies that $(\rho_{\ell,X}(\sigma)^m - I)^2 = 0$. Let $\alpha$ be an eigenvalue of $\rho_{\ell,X}(\sigma)$. Then $(\alpha^m - 1)^2 = 0$. Therefore, $\alpha^m = 1$. By our hypothesis, $$(\rho_{\ell,X}(\sigma) - I)^2 \in n\mathrm{M}_{2d}({\mathbf Z}_\ell),$$ where $d = \mathrm{dim}(X)$. By Theorem 4.3 on p.~359 of \cite{SGA}, the characteristic polynomial of $\rho_{\ell,X}(\sigma)$ has integer coefficients which are independent of $\ell$. By Lemma \ref{localglobal}, $(\alpha - 1)^2 \in n{\bar {\mathbf Z}}$, where ${\bar {\mathbf Z}}$ denotes the ring of algebraic integers. Since $n \ge 5$, by Theorem \ref{quasithm} we have $\alpha = 1$ (i.e., ${\mathcal I}$ acts unipotently on $T_\ell(X)$). By Theorem \ref{galcrit}, $X$ has semistable reduction at $v$. \end{proof} \begin{thm} \label{ssredconverse} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$, $n$ is an integer not divisible by the residue characteristic of $v$, $n \ge 5$, ${\bar v}$ is an extension of $v$ to $F^s$, and ${\mathcal I} = {\mathcal I}({\bar v}/v)$. Then $X$ has semistable reduction at $v$ if and only if there exists a subgroup $S$ of $X_n$ such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_n}$. \end{thm} \begin{proof} Suppose there exists a subgroup $S$ as in the statement of the theorem. The map $x \mapsto (y \mapsto e_n(x,y))$ induces a $\mathrm{Gal}(F^s/F)$-equivariant isomorphism from $X_n/S$ onto $\mathrm{Hom}(S^{\perp_n},\boldsymbol \mu_n)$. Suppose $\sigma \in {\mathcal I}$. Then $\sigma = 1$ on $S^{\perp_n}$ and on $\boldsymbol \mu_n$. Therefore, $\sigma = 1$ on $X_n/S$. Thus, $(\sigma - 1)^2X_n \subseteq (\sigma - 1)S = 0$. By Theorem \ref{ssredlem}, $X$ has semistable reduction at $v$. Conversely, suppose $X$ has semistable reduction at $v$. Let $S = X_n^{\mathcal I}$, and apply Proposition \ref{sslem}ii. \end{proof} \begin{thm} \label{ssred} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$, $\lambda$ is a polarization on $X$ defined over an extension of $F$ which is unramified over $v$, $n$ is a positive integer not divisible by the residue characteristic of $v$, and $n \ge 5$. \begin{enumerate} \item[{(i)}] If $\widetilde{X}_n$ is a maximal isotropic subgroup of $X_n$ with respect to $e_{\lambda,n}$, and the points of $\widetilde{X}_n$ are defined over an extension of $F$ which is unramified over $v$, then $X$ has semistable reduction at $v$. \item[{(ii)}] Conversely, if $X$ has semistable reduction at $v$, and the degree of the polarization $\lambda$ is relatively prime to $n$, then there exists a maximal isotropic subgroup of $X_n$ with respect to $e_{\lambda,n}$, all of whose points are defined over an extension of $F$ which is unramified over $v$. \end{enumerate} \end{thm} \begin{proof} Under the hypotheses in (i), let $S = \widetilde{X}_n$. Then $S^{\perp_n} = \lambda(S)$, and $X$ has semistable reduction at $v$ by applying Theorem \ref{ssredconverse}. Conversely, suppose $X$ has semistable reduction at $v$. Let ${\bar v}$ be an extension of $v$ to $F^s$ and let ${\mathcal I} = {\mathcal I}({\bar v}/v)$. Let $S = X_n^{\mathcal I}$. If $G$ is a subgroup of $X_n$, let $$G^{\perp_{\lambda,n}} = \{ y \in X_n : e_{\lambda,n}(x,y) = 1 \text{ for every } x \in G \}.$$ Since the degree of $\lambda$ is relatively prime to $n$, $\lambda$ induces an isomorphism between $S^{\perp_{\lambda,n}}$ and $S^{\perp_n}$. Since $\lambda$ is defined over an unramified extension, ${\mathcal I}$ acts as the identity on $S^{\perp_{\lambda,n}}$ by Proposition \ref{sslem}ii. Therefore, $S^{\perp_{\lambda,n}} \subseteq S = X_n^{\mathcal I}$. The pairing $e_{\lambda,n}$ induces a nondegenerate pairing on $S/S^{\perp_{\lambda,n}}$. Let $H$ be the inverse image in $S$ (under the natural projection) of a maximal isotropic subgroup of $S/S^{\perp_{\lambda,n}}$. It is easy to check that $H$ is a maximal isotropic subgroup of $X_n$ with respect to $e_{\lambda,n}$, proving (ii). \end{proof} \begin{rems} Raynaud's criterion (Theorem \ref{raynaud}) follows from Theorem \ref{ssredconverse} by letting $n = m^2$ and $S = X_m \subset X_{n}$ (since then $S^{\perp_n} = X^\ast_m$, the dual Galois module of $X_m$, and $n \ge 5$ whenever $m \ge 3$). The converse of Raynaud's criterion is clearly false, i.e., semistable reduction does not imply that the $n$-torsion points are unramified (for $n \ge 3$ and $n$ not divisible by the residue characteristic), as can be seen, for example, by comparing Raynaud's criterion with the N\'eron-Ogg-Shafarevich criterion for good reduction, and considering an abelian variety with semistable but not good reduction. Theorem \ref{ssred}i is Theorem 6.2 of \cite{semistab}. Similarly, the other results of \cite{semistab} and of \S 3 of \cite{connected} can readily be generalized to the setting of Theorem \ref{ssredconverse}. Theorem \ref{ssred}ii shows that the sufficient condition for semistability given in Theorem 6.2 of \cite{semistab} comes close to being a necessary condition. Note that Theorem \ref{ssred}ii would be false if the condition on the degree of the polarization were omitted. \end{rems} \begin{defn} Suppose $v$ is a discrete valuation on $F$ of residue characteristic $p$. We say $v$ satisfies (*) if at least one of the following conditions is satisfied: \begin{enumerate} \item[(a)] $p \ne 2$, \item[(b)] the valuation ring is henselian and the residue field is separably closed. \end{enumerate} \end{defn} The techniques of the above proofs can be extended to prove the following result. The proof will appear in \cite{etale}. \begin{thm} \label{highercoh} Suppose $X$ is an abelian variety over a field $F$, and $v$ is a discrete valuation on $F$ of residue characteristic $p \ge 0$. Suppose $k \in {\mathbf Z}$, and $0 < k < 2\mathrm{dim}(X)$. \begin{enumerate} \item[{(i)}] If either $X$ has semistable reduction at $v$, or $k$ is even and $X$ has purely additive reduction at $v$ which becomes semistable over a quadratic extension of $F$, then $$(\sigma - 1)^{k+1}H^k_{\text{\'et}}(X \times_F F^s, {\mathbf Z}_{\ell}) = 0$$ for every $\sigma \in {\mathcal I}$ and every prime $\ell \ne p$, and $$(\sigma - 1)^{k+1}H^k_{\text{\'et}}(X \times_F F^s, {\mathbf Z}/n{\mathbf Z}) = 0$$ for every $\sigma \in {\mathcal I}$ and every positive integer $n$ not divisible by $p$. \item[{(ii)}] Suppose $n$ is a positive integer not divisible by $p$, and $$(\sigma - 1)^{k+1}H^k_{\text{\'et}}(X \times_F F^s, {\mathbf Z}/n{\mathbf Z}) = 0$$ for every $\sigma \in {\mathcal I}$. Suppose $L$ is a degree $R(k+1,n)$ extension of $F$ which is totally ramified above $v$, and let $w$ be the extension of $v$ to $L$. If $k$ is odd, then $X$ has semistable reduction at $w$. If $k$ is even and $v$ satisfies (*), then either $X$ has semistable reduction at $w$, or $X$ has purely additive reduction at $w$ which becomes semistable over a quadratic extension of $L$. \end{enumerate} \end{thm} If we restrict to the case where $n \notin N(k+1)$, we obtain the following result. This result gives necessary and sufficient conditions for semistable reduction, and also necessary and sufficient conditions for $X$ to have either semistable reduction or purely additive reduction which becomes semistable over a quadratic extension. \begin{cor} \label{highercohcor} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$ of residue characteristic $p \ge 0$, $k$ and $n$ are positive integers, $\ell$ is a prime number, $k < 2\mathrm{dim}(X)$, $n$ and $\ell$ are not divisible by $p$, and $n \notin N(k+1)$. \begin{enumerate} \item[{(i)}] Suppose $k$ is odd. Then the following are equivalent: \begin{enumerate} \item[{(a)}] $X$ has semistable reduction at $v$, \item[{(b)}] for every $\sigma \in {\mathcal I}$, $$(\sigma - 1)^{k+1}H^k_{\text{\'et}}(X \times_F F^s, {\mathbf Z}_{\ell}) = 0,$$ \item[{(c)}] for every $\sigma \in {\mathcal I}$, $$(\sigma - 1)^{k+1}H^k_{\text{\'et}}(X \times_F F^s, {\mathbf Z}/n{\mathbf Z}) = 0.$$ \end{enumerate} \item[{(ii)}] Suppose $k$ is even and $v$ satisfies (*). Then the following are equivalent: \begin{enumerate} \item[{(a)}] either $X$ has semistable reduction at $v$, or $X$ has purely additive reduction at $v$ which becomes semistable over a quadratic extension of $F$, \item[{(b)}] for every $\sigma \in {\mathcal I}$, $$(\sigma - 1)^{k+1}H^k_{\text{\'et}}(X \times_F F^s, {\mathbf Z}_{\ell}) = 0,$$ \item[{(c)}] for every $\sigma \in {\mathcal I}$, $$(\sigma - 1)^{k+1}H^k_{\text{\'et}}(X \times_F F^s, {\mathbf Z}/n{\mathbf Z}) = 0.$$ \end{enumerate} \end{enumerate} \end{cor} \section{Exceptional $n$} \label{except} In this section we discuss briefly the ``exceptional'' cases $n=2,3,4$. For the proofs, and for examples which show the results are sharp, we refer the reader to \cite{degree}. First, let us state the following ``one-way" generalization of Theorem 4.5. \begin{thm} \label{oneway} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$, and $n$ is an integer greater than $1$ which is not divisible by the residue characteristic of $v$. Suppose there exists a subgroup $S$ of $X_n$ such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_n}$. Then $X$ has semistable reduction over every degree $R(n)$ extension of $F$ totally ramified above $v$. \end{thm} It turns out that the converse statement is not true. However, the following result gives an ``approximate converse''. \begin{thm} \label{bothways} Suppose $n = 2$, $3$, or $4$, respectively. Suppose $X$ is an abelian variety over a field $F$, and $v$ is a discrete valuation on $F$ whose residue characteristic does not divide $n$. Suppose $L$ is an extension of $F$ of degree $4$, $3$, or $2$, respectively, which is totally ramified above $v$. Then the following are equivalent: \begin{enumerate} \item[(i)] $X$ has semistable reduction over $L$ above $v$, \item[(ii)] there exist an abelian variety $Y$ over a finite extension $K$ of $F$ unramified above $v$, a separable $K$-isogeny $\varphi : X \to Y$, and a subgroup $S$ of $Y_n$ such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_n}$. \end{enumerate} Further, $\varphi$ can be taken so that its kernel is killed by $8$, $9$, or $4$, respectively. If $X$ has potentially good reduction at $v$, then $\varphi$ can be taken so that its kernel is killed by $2$, $3$, or $2$, respectively. \end{thm} In the case of low-dimensional $X$ this result may be improved as follows. \begin{thm} \label{ellcor} In Theorem \ref{bothways}, with $d = \mathrm{dim}(X)$, $\varphi$ can be taken so that its kernel is killed by $4$ if $d = 3$ and $n = 2$, by $3$ if $d =2$ and $n = 3$, and by $2$ if $d = n = 2$. If $d = 1$, then we can take $Y = X$ and $\varphi$ the identity map. \end{thm} In the case of elliptic curves this implies the following statement. \begin{cor} \label{4326cor} Suppose $X$ is an elliptic curve over a field $F$, and $v$ is a discrete valuation on $F$ of residue characteristic $p \ge 0$. \begin{enumerate} \item[(a)] If $p \ne 2$, then $X$ has semistable reduction above $v$ over a totally ramified quartic extension of $F$ if and only if $X$ has an ${\mathcal I}$-invariant point of order $2$. \item[(b)] If $p \ne 3$, then $X$ has semistable reduction above $v$ over a totally ramified cubic extension of $F$ if and only if $X$ has an ${\mathcal I}$-invariant point of order $3$. \item[(c)] If $p \ne 2$, then $X$ has semistable reduction above $v$ over a quadratic extension of $F$ if and only if either $X$ has an ${\mathcal I}$-invariant point of order $4$, or all the points of order $2$ on $X$ are ${\mathcal I}$-invariant. \item[(d)] If $p \ne 2$ and $X$ has bad but potentially good reduction at $v$, then $X$ has good reduction above $v$ over a quadratic extension of $F$ if and only if $X$ has no ${\mathcal I}$-invariant point of order $4$ and all its points of order $2$ are ${\mathcal I}$-invariant. \item[(e)] Suppose $p$ is not $2$ or $3$. Then the following are equivalent: \begin{enumerate} \item[(i)] $X$ has no ${\mathcal I}$-invariant points of order $2$ or $3$, \item[(ii)] there does not exist a finite separable extension $L$ of $F$ of degree less than $6$ such that $X$ has semistable reduction at the restriction of ${\bar v}$ to $L$. \end{enumerate} \item[(f)] Suppose $p$ is not $2$ or $3$. Then the following are equivalent: \begin{enumerate} \item[(i)] $X$ has no ${\mathcal I}$-invariant points of order $4$ or $3$ and not all the points of order $2$ are ${\mathcal I}$-invariant, \item[(ii)] there does not exist a finite separable extension $L$ of $F$ of degree less than $4$ such that $X$ has semistable reduction at the restriction of ${\bar v}$ to $L$. \end{enumerate} \end{enumerate} \end{cor} In the case of potentially good reduction the following statement holds true. \begin{thm} \label{paddcor} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$ of residue characteristic $p \ge 0$, and $X$ has purely additive and potentially good reduction at $v$. \begin{enumerate} \item[(a)] If $p \ne 2$, then $X$ has good reduction above $v$ over a quadratic extension of $F$ if and only if there exists a subgroup $S$ of $X_4$ such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_4}$. \item[(b)] If $p \ne 3$, then $X$ has good reduction above $v$ over a totally ramified cubic extension of $F$ if and only if there exists a subgroup $S$ of $X_3$ such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_3}$. \item[(c)] Suppose $p \ne 2$, and $L/F$ is a degree $4$ extension, totally ramified above $v$, which has a quadratic subextension over which $X$ has purely additive reduction. Then $X$ has good reduction above $v$ over $L$ if and only if there exists a subgroup $S$ of $X_2$ such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_2}$. \end{enumerate} \end{thm} \section{A measure of potentially good reduction} \label{Gsect} Suppose $v$ is a discrete valuation on a field $F$, and $X$ is an abelian variety over $F$ which has potentially good reduction at $v$. Let $F_{v}^{nr}$ denote the maximal unramified extension of the completion of $F$ at $v$, let $L$ denote the smallest extension of $F_{v}^{nr}$ over which $X$ has good reduction, and let $$G_{v,X} = \mathrm{Gal}(L/F_{v}^{nr}).$$ Then $G_{v,X}$ can also be characterized as the inertia group of the extension $F(X_n)/F$, where $n$ is any integer greater than $2$ and not divisible by the residue characteristic of $v$ (see Corollary 2 on p.~497 of \cite{Serre-Tate}). Clearly, $X$ has good reduction at $v$ if and only if $G_{v,X} = 1$. The finite group $G_{v,X}$ is a measure of how far $X$ is from having good reduction at $v$. If $A$ is a matrix, let $P_A$ denote its characteristic polynomial. The following result gives constraints on the group $G_{v,X}$. \begin{thm} \label{Gthm} Suppose $v$ is a discrete valuation on a field $F$, and $X$ is a $d$-dimensional abelian variety over $F$ which has potentially good reduction at $v$. Let $G=G_{v,X}$. Suppose $\ell$ is a prime number not equal to the residue characteristic of $v$. Then the action of $\mathrm{Gal}(F^s/F)$ on the $\ell$-adic Tate module $V_\ell(X)$ induces an embedding $$f : G \hookrightarrow \mathrm{Sp}_{2d}({\mathbf Q}_\ell)$$ which satisfies the following properties. \begin{enumerate} \item[(i)] For every $\sigma \in G$, the coefficients of $P_{f(\sigma)}$ are integers which are independent of $\ell$. If $X$ has an $F$-polarization of degree not divisible by $\ell$, then one may choose $f$ so that its image lies in $\mathrm{Sp}_{2d}({\mathbf Z}_\ell)$. \item[(ii)] If either $(\ell,\#G)=1$ or $\ell > d+1$, then there exists an embedding $$g : G \hookrightarrow \mathrm{Sp}_{2d}({\mathbf Z}_\ell)$$ such that $P_{g(\sigma)} = P_{f(\sigma)}$ for every $\sigma \in G$. \item[(iii)] If $\ell \ge 5$ then there exists an embedding $$h : G \hookrightarrow \mathrm{Sp}_{2d}(\mathbf{F}_\ell)$$ such that $P_{h(\sigma)} \equiv P_{f(\sigma)} \pmod{\ell}$ for every $\sigma \in G$. \end{enumerate} Further, if $\ell \ge 5$ then there exists an embedding $$G \hookrightarrow \mathrm{Sp}_{2d}({\mathbf Z}_\ell)$$ (which does not necessarily ``preserve'' the characteristic polynomials obtained from the embedding $f$). \end{thm} See \cite{Serre-Tate} for (i), and see \cite{inertia} for the case $(\ell,\#G)=1$ of (ii). The remainder of Theorem \ref{Gthm} follows from results whose proofs will appear elsewhere (along with examples which show that the results are sharp). Those results apply more generally to measure how far an abelian variety (not necessarily with potentially good reduction) is from having semistable reduction. In some cases, these results apply to more general finite groups than those obtained as $G_{v,X}$'s. \part{N\'eron models of abelian varieties with potentially good reduction} \label{neronpart} \section{Preliminaries} In \cite{serrelem}, the following result was obtained as a corollary of Theorem \ref{quasithm} above. \begin{prop}[Theorem 6.10a of \cite{serrelem}] \label{randm} Suppose $\ell$ is a prime, $m$ and $r$ are positive integers, ${\mathcal O}$ is an integral domain of characteristic zero with no non-zero infinitely $\ell$-divisible elements, $\ell{\mathcal O}$ is a maximal ideal of ${\mathcal O}$, $M$ is a free ${\mathcal O}$-module of finite rank, and $A$ is an endomorphism of $M$ of finite multiplicative order such that $(A - 1)^{m(\ell - 1)\ell^{r-1}} \in \ell^m\mathrm{End}(M)$. If $r > 1$, then the torsion subgroup of $M/(A - 1)M$ is killed by $\ell^{r-1}$. \end{prop} \begin{prop}[see Proposition 6.1i and Corollary 7.1 of \cite{semistab}] \label{pressred} Suppose $X$ is a $d$-dimensional abelian variety over a field $F$, $v$ is a discrete valuation on $F$ with residue characteristic not equal to $2$, $\lambda$ is a polarization on $X$, $\widetilde{X}_2$ is a maximal isotropic subgroup of $X_2$ with respect to $e_{\lambda,2}$, $\lambda$ and the points of $\widetilde{X}_2$ are defined over an extension of $F$ which is unramified over $v$, ${\bar v}$ is an extension of $v$ to a separable closure of $F$, and $\sigma \in {\mathcal I}({\bar v}/v)$. Then $(\rho_{2,X}(\sigma) - I)^2 \in 2\mathrm{M}_{2d}({\mathbf Z}_2)$, and $X$ has semistable reduction above $v$ over every totally ramified Galois (necessarily cyclic) extension of $F$ of degree $4$. \end{prop} Recall that $u$ denotes the unipotent rank of $X_v$, $a$ denotes the abelian rank, and $\Phi '$ denotes the prime-to-$p$ part of the group of connected components of the special fiber of the N\'eron model of $X$ at $v$, where $p$ is the residue characteristic of the discrete valuation $v$. If $X$ has potentially good reduction, then $\mathrm{dim}(X) = a + u$. \begin{thm}[Theorem 7.5 of \cite{semistab}] \label{neronmod} Suppose $v$ is a discrete valuation on a field $F$ with strictly henselian valuation ring, $X$ is an abelian variety over $F$ which has potentially good reduction at $v$, and either \begin{enumerate} \item[{(a)}] $n = 2$ and the points of $X_2$ are defined over $F$, or \item[{(b)}] $n = 3$ or $4$, $\lambda$ is a polarization on $X$ defined over $F$, and the points of a maximal isotropic subgroup of $X_n$ with respect to $e_{\lambda,n}$ are defined over $F$. \end{enumerate} Suppose the residue characteristic $p$ ( $\ge 0$) of $v$ does not divide $n$. Then $\Phi ' \cong ({\mathbf Z}/2{\mathbf Z})^{2u}$ if $n = 2$ or $4$, and $\Phi ' \cong ({\mathbf Z}/3{\mathbf Z})^u$ if $n = 3$. \end{thm} \begin{lem} \label{neronlemma} Suppose $v$ is a discrete valuation on a field $F$ such that the valuation ring is strictly henselian. Suppose $X$ is an abelian variety over $F$ which has potentially good reduction at $v$, and suppose $n$ is a positive integer not divisible by the residue characteristic of $v$. Let $\Phi_n$ denote the subgroup of $X_v/X_v^0$ of points of order dividing $n$. Then: \begin{enumerate} \item[(i)] $(X_v)_n \cong X_n(F)$, \item[(ii)] $(X_v^0)_n \cong ({\mathbf Z}/n{\mathbf Z})^{2a}$, \item[(iii)] $\Phi_n \cong (X_v)_n/(X_v^0)_n$, and \item[(iv)] if $X_n(F) \cong ({\mathbf Z}/n{\mathbf Z})^{b}$, then $\Phi_n \cong ({\mathbf Z}/n{\mathbf Z})^{b-2a}$. \end{enumerate} \end{lem} \begin{proof} By Lemma 2 of \cite{Serre-Tate}, the reduction map defines an isomorphism of $X_n^{\mathcal I}$ onto $(X_v)_n$, where ${\mathcal I} = {\mathcal I}({\bar v}/v)$ for some extension ${\bar v}$ of $v$ to $F^s$. Under our hypotheses on $v$, we have $X_n^{\mathcal I} \cong X_n(F)$. Therefore, $(X_v)_n \cong X_n(F)$. As shown in the proof of Lemma 1 of \cite{Serre-Tate}, $(X_v^0)_n \cong ({\mathbf Z}/n{\mathbf Z})^{2a+t}$, where $t$ denotes the toric rank of $X_v$. Since $X$ has potentially good reduction at $v$, $t = 0$. Since $X_v^0$ is $n$-divisible, we have $\Phi_n \cong (X_v)_n/(X_v^0)_n$. Part (iv) follows easily from (i), (ii), and (iii). \end{proof} \section{N\'eron models} \label{main} In Theorem \ref{neron} we generalize Theorem \ref{neronmod} to the case of partial level $2$ structure. We can recover Theorem \ref{neronmod}a as a special case. Recall that $u$ denotes the unipotent rank of $X_v$, $a$ denotes the abelian rank, and $\Phi '$ denotes the prime-to-$p$ part of the group of connected components of the special fiber of the N\'eron model of $X$ at $v$, where $p$ is the residue characteristic of $v$ (with $\Phi '$ the full group of components if $p = 0$). \begin{thm} \label{neron} Suppose $v$ is a discrete valuation on a field $F$, suppose the valuation ring is strictly henselian, and suppose the residue field has characteristic $p \ne 2$. Suppose $(X, \lambda)$ is a $d$-dimensional polarized abelian variety over $F$, $X$ has potentially good reduction at $v$, and the points of a maximal isotropic subgroup of $X_2$ with respect to $e_{\lambda,2}$ are defined over $F$. Then: \begin{enumerate} \item[(i)] $\Phi' \cong ({\mathbf Z}/2{\mathbf Z})^{b-2a} = ({\mathbf Z}/2{\mathbf Z})^{b+2u-2d}$, where $b$ is defined by $X_2(F) \cong ({\mathbf Z}/2{\mathbf Z})^{b}$, \item[(ii)] $[X_2 : X_2(F)]\#\Phi' = 2^{2u}$, and \item[(iii)] $X$ has good reduction at $v$ if and only if $\Phi' = \{0\}$ and $X_2 \subseteq X(F)$. \end{enumerate} \end{thm} \begin{proof} Let ${\bar v}$ be an extension of $v$ to a separable closure of $F$, let ${\mathcal I} = {\mathcal I}({\bar v}/v)$, let $k$ be the residue field of $v$, and let ${\mathcal J}$ be the first ramification group (i.e., ${\mathcal J}$ is trivial if $p = 0$ and ${\mathcal J}$ is the pro-$p$-Sylow subgroup of ${\mathcal I}$ if $p > 0$). Suppose $q$ is a prime not equal to $p$, and let $\Phi_q$ denote the $q$-part of the group of connected components of the special fiber of the N\'eron model of $X$. Since $X$ has potentially good reduction at $v$, $\rho_{q,X}(\sigma)$ has finite multiplicative order for every $\sigma \in {\mathcal I}$. Let $\tau$ be a lift to ${\mathcal I}$ of a generator of the pro-cyclic group ${\mathcal I}/{\mathcal J}$. By \S11 of \cite{SGA} (see Lemma 2.1 of \cite{Lorenzini}), $$\Phi_q \text{ is isomorphic to the torsion subgroup of } T_q(X)^{\mathcal J}/(\tau - 1)T_q(X)^{\mathcal J}.$$ By Proposition \ref{pressred} and Remark \ref{ramifiedcyclic}, $X$ has semistable reduction (and therefore good reduction) above $v$ over a totally ramified cyclic Galois extension of $F$ of degree $4$. Therefore ${\mathcal I}$ acts on $T_q(X)$ through a cyclic quotient of order $4$, so $\rho_{q,X}(\sigma)^4 = I$ for every $\sigma \in {\mathcal I}$. Since $p \ne 2$, we have $\rho_{q,X}(\sigma) = I$ for every $\sigma \in {\mathcal J}$. Therefore, $T_q(X)^{\mathcal J} = T_q(X)$. If $q \ne 2$, then $T_q(X)/(\rho_{q,X}(\tau) - I)T_q(X)$ is torsion-free, so $\Phi_q$ is trivial. Further, $$\Phi_2 \text{ is isomorphic to the torsion subgroup of } T_2(X)/(\tau - 1)T_2(X).$$ We have $(\rho_{2,X}(\tau) - I)^2 \in 2\mathrm{M}_{2d}({\mathbf Z}_2)$, by Proposition \ref{pressred}. By Proposition \ref{randm} with $\ell = 2$, $r = 2$, $m = 1$, and ${\mathcal O} = {\mathbf Z}_2$, $\Phi_2$ is annihilated by $2$. Therefore, $\Phi'$ is an elementary abelian $2$-group. By Lemma \ref{neronlemma}, $\Phi' \cong ({\mathbf Z}/2{\mathbf Z})^{b-2a}$. Parts (ii) and (iii) follow immediately. Note that Theorem \ref{neronmod}a is a special case of Theorem \ref{neron}. \end{proof} \begin{thm} \label{neron3} Suppose $v$ is a discrete valuation on a field $F$, suppose the valuation ring is strictly henselian, and suppose the residue field has characteristic $p \ne 3$. Suppose $(X, \lambda)$ is a $d$-dimensional polarized abelian variety over $F$, $X$ has potentially good reduction at $v$, and the points of a maximal isotropic subgroup of $X_3$ with respect to $e_{\lambda,3}$ are defined over $F$. Then: \begin{enumerate} \item[(i)] $X_3(F) \cong ({\mathbf Z}/3{\mathbf Z})^{2d-u}$, \item[(ii)] $X$ has good reduction at $v$ if and only if $X_3(F) = X_3$, and \item[(iii)] $X$ has purely additive reduction at $v$ if and only if $X_3(F) \cong ({\mathbf Z}/3{\mathbf Z})^{d}$. \end{enumerate} \end{thm} \begin{proof} By Theorem \ref{neronmod}, $\Phi' \cong ({\mathbf Z}/3{\mathbf Z})^u$. Write $X_3(F) \cong ({\mathbf Z}/3{\mathbf Z})^b$. By Lemma \ref{neronlemma}, $\Phi' \cong ({\mathbf Z}/3{\mathbf Z})^{b-2d+2u}$. Therefore, $b = 2d - u$, and we obtain the desired result. \end{proof} \begin{thm} \label{neron4} Suppose $v$ is a discrete valuation on a field $F$ with strictly henselian valuation ring, $X$ is an abelian variety over $F$ which has potentially good reduction at $v$, the residue field has characteristic $p \ne 2$, and either \begin{enumerate} \item[{(a)}] the points of $X_2$ are defined over $F$, or \item[{(b)}] $\lambda$ is a polarization on $X$ defined over $F$, and the points of a maximal isotropic subgroup of $X_4$ with respect to $e_{\lambda,4}$ are defined over $F$. \end{enumerate} Then $$X_4(F) \cong ({\mathbf Z}/4{\mathbf Z})^{2a} \times ({\mathbf Z}/2{\mathbf Z})^{2u}.$$ In particular: \begin{enumerate} \item[(i)] $X_2 \subseteq X_4(F) \subseteq X_4$, $[X_4 : X_4(F)] = 2^{2u}$, $[X_4(F) : X_2] = 2^{2a}$, \item[(ii)] $X$ has good reduction at $v$ if and only if $X_4(F) = X_4$, and \item[(iii)] $X$ has purely additive reduction at $v$ if and only if $X_4(F) = X_2$. \end{enumerate} \end{thm} \begin{proof} By Theorem \ref{neronmod}, we have $\Phi' \cong ({\mathbf Z}/2{\mathbf Z})^{2u}$. By Lemma \ref{neronlemma}, we have a short exact sequence $$0 \to ({\mathbf Z}/4{\mathbf Z})^{2a} \to X_4(F) \to ({\mathbf Z}/2{\mathbf Z})^{2u} \to 0.$$ Let $d = \mathrm{dim}(X)$. Since $X_4(F) \subseteq X_4 \cong ({\mathbf Z}/4{\mathbf Z})^{2d}$, we conclude that $X_4(F) \cong ({\mathbf Z}/4{\mathbf Z})^{2a} \times ({\mathbf Z}/2{\mathbf Z})^{2u}$. Note that $X_2 \cong ({\mathbf Z}/2{\mathbf Z})^{2d} = ({\mathbf Z}/2{\mathbf Z})^{2a + 2u}$. The rest of the result follows immediately. \end{proof} As an example, let $X$ be the elliptic curve defined by the equation $y^2 = x^3 - 9x$, and let $F$ be the maximal unramified extension of ${\mathbf Q}_3$. Then $X_2(F) = X_2 = X_4(F)$, $X$ has additive and potentially good reduction, and $\Phi' \cong ({\mathbf Z}/2{\mathbf Z})^{2}$. \begin{rems} If $X$ has a polarization $\lambda$ of odd degree, then $X_2$ is a maximal isotropic subgroup of $X_4$ with respect to $e_{\lambda,4}$. As stated in the Introduction, Theorems \ref{neron3}ii and \ref{neron4}ii are immediate corollaries of Raynaud's criterion for semistable reduction. If $X$ has purely additive reduction, then $X_n(F) \cong \Phi_n$ (see \cite{Lorenzini}). Suppose $v$ is a discrete valuation on a field $F$, $X$ is an abelian variety over $F$ with potentially good reduction at $v$, the valuation ring is strictly henselian, $\ell = 2$ or $3$, and the residue characteristic is not equal to $\ell$. Then Theorem 6.1 of \cite{Edixhoven} implies that if $\Phi'$ is an elementary abelian $\ell$-group, then $\Phi'$ is a subgroup of $({\mathbf Z}/2{\mathbf Z})^{2u}$ if $\ell = 2$ or of $({\mathbf Z}/3{\mathbf Z})^{u}$ if $\ell = 3$. For simplicity of exposition, we do not generalize the results of \S \ref{main} (or the prerequisite results from \cite{semistab}, or related results in \S 3 of \cite{connected}) to the setting of Theorem \ref{ssredconverse}, but leave such generalizations as a straightforward exercise for the reader. \end{rems}

9602

alg-geom/9602005

en

https://arxiv.org/abs/alg-geom/9602005

alg-geom/9602005

Frank Sottile

Frank Sottile

Real enumerative geometry and effective algebraic equivalence

12 pages, LaTeX 2e

J. Pure and Appl. Alg., 117 & 118 (1997) 601--615

We describe an approach to the question of finding real solutions to problems of enumerative geometry, in particular the question of whether a problem of enumerative geometry can have all of its solutions be real. We give some methods to infer one problem can have all of its solutions be real, given that a related problem does. These are used to show many Schubert-type enumerative problems on some flag manifolds can have all of their solutions be real.

alg-geom

\section{Introduction} Determining the common zeroes of a set of polynomials is further complicated over non-algebraically closed fields such as the real numbers. A special case is whether a problem of enumerative geometry can have all its solutions be real. We call such a problem {\em fully real}. Little is known about enumerative geometry from this perspective. A standard proof of B\'ezout's Theorem shows the problem of intersecting hypersurfaces in projective space is fully real. Khovanskii~\cite{Khovanskii_fewnomials} considers intersecting hypersurfaces in a torus defined by few monomials and shows the real zeros are at most a fraction of the complex zeroes. Fulton, and more recently, Ronga, Tognoli and Vust~\cite{Ronga_Tognoli_Vust} have shown the problem of 3264 plane conics tangent to five given conics is fully real. The author~\cite{sottile_real_lines} has shown all problems of enumerating lines incident on linear subspaces of projective space are fully real. There are few methods for studying this phenomenon. We ask: How can the knowledge that one enumerative problem is fully real be used to infer that a related problem is fully real? We give several procedures to accomplish this inference and examples of their application, lengthening the list of enumerative problems known to be fully \smallskip real. We study intersections of any dimension, not just the zero dimensional intersections of enumerative problems. Our technique is to deform general intersection cycles into simpler cycles. This modification of the classical method of degeneration was used by Chiavacci and Escamilla-Castillo~\cite{Chiavacci_Escamilla-Castillo} to investigate these questions. Let $\alpha_1,\ldots,\alpha_a$ be cycle classes spanning the Chow ring of a smooth variety. For cycle classes $\beta_1,\ldots,\beta_b$, there exist integers $c_i$ for $i = 1,\ldots, a$ such that $$ \beta_1\cdots\beta_b \ =\ \sum_{i=1}^a c_i\cdot \alpha_i. $$ When the $c_i\geq 0$, this product formula has a geometric interpretation. Suppose $Y_1,\ldots,Y_b$ are cycles representing the classes $\beta_1,\ldots,\beta_b$ which meet generically transversally in a cycle $Y$. Then $Y$ is algebraically equivalent to $Z:=Z_1\cup\cdots\cup Z_a$, where $Z_i$ has $c_i$ components, each representing the cycle class $\alpha_i$. This algebraic equivalence is effective if there is a family of cycles containing both $Y$ and $Z$ whose general member is a generically transverse intersection of cycles representing the classes $\beta_1,\ldots,\beta_b$. If the cycles $Y_1,\ldots,Y_b$ and each component of $Z$ are defined over ${\Bbb R}$ and both $Y$ and $Z$ are in the same connected component of the real points of that family, then the effective algebraic equivalence is real. Real effective algebraic equivalence can be used to show an enumerative problem is fully real, or more generally, to obtain lower bounds on the maximal number of real solutions. Suppose the cycles $Y_1,\ldots,Y_b, W_1,\ldots, W_c$ give an enumerative problem and the problem obtained by substituting $Z$ for $Y_1,\ldots,Y_b$ has at least $d$ real solutions. Then there exist real cycles $Y'_1,\ldots,Y'_b$ such that the original problem (with $Y'_i$ in place of $Y_i$) has at least $d$ real solutions, since the number of real solutions is preserved by small real deformations. Sections 2 through 5 introduce and develop our basic notions and techniques. Subsequent sections are devoted to elaborations and applications of these ideas. In Section 6, we prove that any enumerative problem on a flag variety involving five Schubert varieties, three of which are special Schubert varieties, is fully real. Given a map $\pi: Y\rightarrow X$ with equidimensional fibres, real effective algebraic equivalence on $X$ and $Y$ is compared in Section 7 and used in Sections 8 and 9 to show that many Schubert-type enumerative problems in two classes of flag varieties are fully real. A proof of B\'ezout's Theorem in Section 10 suggests another method for obtaining fully real enumerative problems. This is applied in Section 11 to show the enumerative problem of $(n-2)$-planes in ${\Bbb P}^n$ meeting $2n-2$ rational normal curves is fully real. The author thanks Bernd Sturmfels for encouraging these investigations. \section{Intersection Problems} \subsection{Conventions} Varieties are reduced, complex, and defined over the real numbers ${\Bbb R}$. Let $X$ and $Y$ denote smooth projective varieties and $U$, $V$, and $W$ smooth quasi-projective varieties. Equip the real points $X({\Bbb R})$ of $X$ with the classical topology. Let $A^*X$ be the Chow ring of cycles modulo algebraic equivalence. Two subvarieties meet {\em generically transversally} if they meet transversally along a dense subset of each component of their intersection. Such an intersection scheme is reduced at the generic point of each component, or generically reduced. A subvariety $\Xi\subset U\times X$ (or $\Xi\rightarrow U$) with generically reduced equidimensional fibres over a smooth base $U$ is a family of {\em multiplicity free cycles on $X$ over $U$.} All fibres of $\Xi$ over $U$ are algebraically equivalent, and we say $\Xi\rightarrow U$ {\em represents} that algebraic equivalence class. \subsection{Chow varieties}\label{sec:Chow} Positive cycles on $X$ of a fixed dimension and degree are parameterized by the Chow variety of $X$. We suppress the dependence on dimension and degree and write $\mbox{\it Chow}\, X$ for any Chow variety of $X$. The open Chow variety ${\mbox{\it Chow}}^\circ X$ is the open subset of $\mbox{\it Chow}\, X$ parameterizing multiplicity free cycles on $X$. There is a tautological family $\Phi \rightarrow {\mbox{\it Chow}}^\circ X$ of cycles on $X$ with the property that $\zeta\in{\mbox{\it Chow}}^\circ X$ represents the the fundamental cycle of the fibre $\Phi_\zeta$. Let $\Xi\rightarrow U$ be a family of multiplicity free cycles on $X$. The association of a point $u$ of $U$ to the fundamental cycle of the fibre $\Xi_u$ defines the {\em fibre function} $\phi$, which is algebraic on a dense open subset $U'$ of $U$. If $U$ is a curve, then $U=U'$. \subsection{Proposition} {\em $\phi(U')$ is dense in the set $\phi(U)$.}\medskip \noindent{\bf Proof:} Let $u\in U$ and $C\subset U$ be a smooth curve with $u\in C$ and $C-\{u\}\subset U'$. Such a curve is not necessarily closed in $U$, but is the smooth points of a closed subvariety. The fibre function $\phi|_C$ of $\Xi|_C \rightarrow C$ is algebraic, hence $\phi(u)\in \overline{\phi(C-\{u\})} \subset \overline{\phi(U')}$. \QED\vspace{10pt} Two families $\Xi\rightarrow U$ and $\Psi\rightarrow V$ of multiplicity free cycles on $X$ are {\em equivalent} if $\overline{\phi(U)} = \overline{\phi(V)}$, that is, if they have essentially the same cycles. Our results remain valid when one family of cycles is replaced by an equivalent family, perhaps with the additional assumption that $\overline{\phi(U({\Bbb R}))} = \overline{\phi(V({\Bbb R}))}$. The varieties $\mbox{\it Chow}\, X$ and ${\mbox{\it Chow}}^\circ X$ as well as $U'$ and the morphism $\phi: U' \rightarrow \mbox{\it Chow}\, X$ are defined over ${\Bbb R}$~(\cite{Samuel}, \S I.9). We use $\phi$ to denote all fibre functions. Any ambiguity may be resolved by context. \subsection{Intersection Problems}\label{sec:intersection_problems} For $1\leq i\leq b$, let $\Xi_i\rightarrow U_i$ be a family of multiplicity free cycles on $X$. Let $U\subset \prod_{i=1}^b U_i$ be the locus where the fibres of the product family $\prod_{i=1}^b \Xi_i$ meet the (small) diagonal $\Delta^b_X$ of $X^b$ generically transversally. Equivalently, $U$ is the locus where fibres of $\Xi_1,\ldots,\Xi_b$ meet generically transversally in $X$. If $U$ is nonempty, then $\Xi_1,\ldots,\Xi_b$ give a {\em (well-posed) intersection problem}. Given an intersection problem as above, let $\delta:X\stackrel{\sim}{\longrightarrow}\Delta^b_X\subset X^b$ and set $\Xi$ to be $$ \Xi \ :=\ (1_U\times \delta)^* \prod_{i=1}^b \Xi_i \ \subset\ U\times X, $$ a family of multiplicity free cycles on $X$ over $U$. We often suppress the dependence on the original families and write $\Xi\rightarrow U$ for this intersection problem. Not all collections of families of cycles give well-posed intersection problems, some transversality is needed to guarantee $U$ is nonempty. When a reductive group acts transitively on $X$, Kleiman's Transversality Theorem~\cite{Kleiman} has the following consequence. \subsection{Proposition}\label{prop:transitive_action} {\em Suppose a reductive group acts transitively on $X$, $\Xi_1$ is a constant family, and for $2\leq i\leq b$, $\Xi_i$ is equivalent to a family of multiplicity free cycles stable under that action. Then $\Xi_1,\ldots,\Xi_b$ give a well-posed intersection problem. }\medskip Grassmannians and flag varieties have such an action. For these, we suppose all families of cycles are stable under that action, and thus give well-posed intersection problems. Suppose a reductive group acts on $X$ with a dense open orbit $X'$. For example, if $X$ is a toric variety, or more generally, a spherical variety~\cite{Brion_spherical_introduction,% Knop_spherical_expository,Luna_Vust_Plongements}. Each family may be stable under that action, but the collection need not give a well-posed intersection problem as Kleiman's Theorem~\cite{Kleiman} only guarantees transversality in $X'$. However, it is often the case that only points of intersection in $X'$ are desired, and suitable blow up of $X$ or a different equivariant compactification of $X'$ exists on which the corresponding intersection problem is well-posed (see~\cite{Fulton_introduction_intersection}, \S 1.4 or~\cite{Fulton_intersection}, \S 9 and \S 10.4). \section{effective algebraic equivalence} \label{sec:effective_algebraic_equivallences} Let $\alpha_1,\ldots,\alpha_a$ be distinct additive generators of $A^*X$, and for $1\leq i\leq a$, suppose $\Psi(\alpha_i)\rightarrow V(\alpha_i)$ is a family of multiplicity free cycles on $X$ representing the cycle class $\alpha_i$. When $X$ is a Grassmannian or flag variety, $\alpha_1,\ldots,\alpha_a$ will be the Schubert classes, and $\Psi(\alpha_i)\rightarrow V(\alpha_i)$ the corresponding families of Schubert varieties. A family of multiplicity free cycles $\Xi\subset U\times X$ has an {\em effective algebraic equivalence} with {\em witness} $Z\in \overline{\phi(U)}\cap {\mbox{\it Chow}}^\circ X$ if each (necessarily multiplicity free) component of $Z$ is a fibre of some family $\Psi(\alpha_i)$. This effective algebraic equivalence is {\em real} if $Z\in \overline{\phi(U({\Bbb R}))}$ and each component of $Z$ is a fibre over a real point of some $V(\alpha_i)$. An intersection problem $\Xi_1,\ldots,\Xi_b$ has {\em (real) effective algebraic equivalences} if its family of intersection cycles $\Xi\rightarrow U$ has (real) effective algebraic equivalences. \subsection{Products in $A^*X$} \label{sec:products} Suppose $\beta_1,\ldots,\beta_b$ are classes from $\{\alpha_1,\ldots,\alpha_a\}$ and the families $\Psi(\beta_1),\ldots,\Psi(\beta_b)$ give an intersection problem $\Psi\rightarrow V$. We say $\Psi\rightarrow V$ is given by $\beta_1,\ldots,\beta_b$. Suppose $\Psi\rightarrow V$ has an effective algebraic equivalence with witness $Z$. Fibres of $\Psi\rightarrow V$ are generically transverse intersections of fibres of $\Psi(\beta_1),\ldots,\Psi(\beta_b)$, and so have cycle class $\beta_1\cdots\beta_b$. As $Z\in \overline{\phi(V)}$, this equals the cycle class of $Z$, which is $\sum_{i=1}^a c_i \alpha_i$, where $c_i$ counts the components of $Z$ lying in the family $\Psi(\alpha_i)$. Thus in $A^*X$, we have $$ \hspace{2.5in} \beta_1\cdots\beta_b \ =\ \sum_{i=1}^a c_i \alpha_i. \hspace{2.2in} (\ref{sec:products}) $$ To compute products in $A^*X$, classical geometers would try to understand a generically transverse intersection of degenerate cycles in special position, as a generic intersection cycle is typically too difficult to describe. Effective algebraic equivalence extends this method of degeneration by also considering limiting positions of such intersection cycles as the subvarieties degenerate further, attaining excess intersection. \subsection{Pieri-type intersection problems}\label{sec:pieri_type} A Schubert subvariety of a flag variety is determined by a complete flag ${F\!_{\DOT}}$ and a coset $w$ of a parabolic subgroup in the symmetric group. Thus Schubert classes $\sigma_w$ are indexed by these cosets and families $\Psi_w$ of Schubert varieties have base ${\Bbb F}\ell$, the variety of complete flags. A {\em special Schubert subvariety} of a Grassmannian is the locus of planes meeting a fixed linear subspace non-trivially, or the image of such a subvariety in the dual Grassmannian. More generally, a special Schubert subvariety of a flag variety is the pullback of a special Schubert subvariety from a Grassmannian projection. If $m$ is the index of a special Schubert class, then the Pieri-type formulas of~\cite{Lascoux_Schutzenberger_polynomes_schubert,Sottile_Pieri_Schubert} show that for any $w$, there exists a subset $I_{m,w}$ of these cosets such that $$ \hspace{2.5in} \sigma_m\cdot \sigma_w\ =\ \sum_{v\in I_{m,w}}\sigma_v. \hspace{2.2in} (\ref{sec:pieri_type}) $$ \subsection{Theorem}\label{thm:pieri_effective_equivalences} {\em The intersection problem $\Xi\rightarrow U$ given by the classes $\sigma_m$ and $\sigma_w$ has real effective algebraic equivalences. }\medskip \noindent{\bf Proof:} The Borel subgroup $B$ of $GL_n{\Bbb C}$ stabilizing a real complete flag ${F\!_{\DOT}}$ acts on the Chow variety with fixed points the $B$-stable cycles, which are sums of Schubert varieties determined by ${F\!_{\DOT}}$. As Hirschowitz~\cite{Hirschowitz} observed, $\overline{\phi(U)}$ is $B$-stable, and must contain a fixed point (\cite{Borel_groups}, III.10.4). In fact, if ${{F\!_{\DOT}}'}$ is a real flag in linear general position with ${F\!_{\DOT}}$, then the $B({\Bbb R})$-orbit of $\Omega_m{F\!_{\DOT}}\bigcap \Omega_w{{F\!_{\DOT}}'}$ is a subset of $\phi(U({\Bbb R}))$. Moreover its closure has a $B({\Bbb R})$-fixed point, as the proof in~\cite{Borel_groups} may be adapted to show that complete $B({\Bbb R})$-stable real analytic sets have fixed points. Since the coefficients of the sum (\ref{sec:pieri_type}) are all 1, $\sum_{v\in I_{m,w}}\Omega_w{F\!_{\DOT}}$ is the only $B({\Bbb R})$-stable cycle in its algebraic equivalence class, and therefore $$ \sum_{v\in I_{b,w}}\Omega_w{F\!_{\DOT}}\ \in\ \overline{\phi(U({\Bbb R}))}. \qquad\qquad\qquad\qquad\QED $$ \section{Fully real enumerative problems} An {\em enumerative problem} of {\em degree} $d$ is an intersection problem $\Xi\rightarrow U$ with zero-dimensional fibres of cardinality $d$. An enumerative problem is {\em fully real} if there exists $u\in U({\Bbb R})$ with all points in the fibre $\Xi_u$ real. In this case, $u = (u_1,\ldots,u_b)$ with $u_i\in U_i({\Bbb R})$ and the cycles $(\Xi_1)_{u_1},\ldots, (\Xi_b)_{u_b}$ meet transversally with all points of intersection real. \subsection{Theorem}\label{thm:real_closure}{\em An enumerative problem $\Xi\rightarrow U$ is fully real if and only if it has real effective algebraic equivalences. That is, if and only if there exists a point $\zeta\in \overline{\phi(U({\Bbb R}))}$ representing distinct real points. }\medskip \noindent{\bf Proof:} The forward implication is a consequence of the definition. For the reverse, let $d$ be the degree of $\Xi\rightarrow U$. Then $\phi:U\rightarrow S^dX$, the Chow variety of effective degree $d$ zero cycles on $X$. The real points $S^dX({\Bbb R})$ of the Chow variety represent degree $d$ zero cycles stable under complex conjugation. Its dense set of multiplicity free cycles have an open subset ${\cal M}$ parameterizing cycles of distinct real points, and $\zeta\in {\cal M}$. Thus $\phi(U({\Bbb R}))\cap {\cal M}\neq \emptyset$, which implies $\Xi\rightarrow U$ is fully real. \QED\vspace{10pt} The set of witnesses to $\Xi\rightarrow U$ being fully real contains an open subset $\phi^{-1}({\cal M})\bigcap U({\Bbb R})$. \section{Curve selection} Subsequent sections use real effective algebraic equivalence for one or more families to infer results about related families. While intuition supports the claim that the functions we define between Chow varieties are algebraic (or at least continuous), we are unaware of general results verifying this intuition. An obvious obstruction is that the Chow variety does not represent a functor. However, weaker claims suffice. Our tool is the Curve Selection Lemma~\cite{Benedetti_Risler} of real semi-algebraic geometry, in the following guise: \subsection{Curve Selection Lemma}\label{lemma:curve_selection} {\em Let $V$ be a real variety and $R\subset V({\Bbb R})$ a semi-algebraic subset. If $\zeta \in \overline{R}$, then there is a real algebraic map $f: C\rightarrow V$ with $C$ a smooth curve, and a point $s$ on a connected arc $S$ of $C({\Bbb R})$ such that $f(S -\{s\}) \subset R$ and $f(s) = \zeta$.\medskip } \noindent{\bf Proof:} By the Curve Selection Lemma (\cite{Benedetti_Risler}, 2.6.20), there exists a semi-algebraic function $g:[0,1]\rightarrow \overline{R}$ with $g(0)=\zeta$, $g(0,1]\subset R$, and $g$ a real analytic homeomorphism onto its image in $\overline{R}$. Let $C^\circ$ be the Zariski closure of $g[0,1]$ in $V$, and $f:C\rightarrow C^\circ$ its normalization. Let $S\subset C({\Bbb R})$ be a connected arc of $f^{-1}(g[0,1])$ whose image contains $g(0)$ and let $s \in S\cap f^{-1}(g(0))$. \QED \section{Pieri-type enumerative problems} \subsection{Theorem}\label{thm:pieri_schubert} {\em Any enumerative problem in any flag variety involving five Schubert varieties, three of which are special, is fully real.} \medskip This generalizes Theorem 5.2 of~\cite{sottile_explicit_pieri}, the analogous result for Grassmannians. It requires an additional transversality result. \subsection{Lemma}\label{lemma:special_transversality} {\em Let $(w_1,w_2)$ and $(v_1,v_2)$ be indices of Schubert subvarieties of a flag variety, with $w_1$ and $w_2$ (respectively $v_1$ and $v_2$) defining defining Schubert varieties of the same dimension. Suppose $m$ is the index of a special Schubert subvariety such that $(w_1,v_1,m)$ gives an enumerative problem. If $(w_1,v_1)\neq (w_2,v_2)$, and ${F\!_{\DOT}}, {{F\!_{\DOT}}'}$ are complete flags in linear general position, then there is an open set $V$ of the variety ${\Bbb F}\ell$ of complete flags consisting of flags ${E_{\DOT}}$ such that $$ \Omega_{w_1}{F\!_{\DOT}} \bigcap \Omega_{v_1}{{F\!_{\DOT}}'} \bigcap \Omega_m {E_{\DOT}} \qquad\mbox{and}\qquad \Omega_{w_2}{F\!_{\DOT}} \bigcap \Omega_{v_2}{{F\!_{\DOT}}'} \bigcap \Omega_m {E_{\DOT}} $$ are transverse intersections which coincide only when empty. }\medskip If the three flags are real, then a nonempty intersection as above is a single real point. \medskip \noindent{\bf Proof:} By Kleiman's Theorem ~\cite{Kleiman}, there is an open subset $U$ of ${\Bbb F}\ell\times{\Bbb F}\ell\times{\Bbb F}\ell$ consisting of triples $({F\!_{\DOT}},{{F\!_{\DOT}}'},{E_{\DOT}})$ such that each intersection is transverse and so is either empty or a single point, by the Pieri-type formulas of~\cite{Lascoux_Schutzenberger_polynomes_schubert,Sottile_Pieri_Schubert}. Suppose neither is empty. Similarly, there is an open subset $V$ of triples for which $$ \left(\Omega_{w_1}{F\!_{\DOT}} \bigcap \Omega_{w_2}{F\!_{\DOT}} \right) \bigcap \left(\Omega_{v_1}{{F\!_{\DOT}}'}\bigcap \Omega_{v_2}{{F\!_{\DOT}}'}\right) \bigcap \Omega_b {E_{\DOT}} $$ is proper. Since $(w_1,v_1)\neq (w_2,v_2)$, it is empty. Thus for triples $({F\!_{\DOT}},{{F\!_{\DOT}}'},{E_{\DOT}})\in U\cap V$, $$ \Omega_{w_1}{F\!_{\DOT}} \bigcap \Omega_{v_1}{{F\!_{\DOT}}'} \bigcap \Omega_b {E_{\DOT}}\ \neq\ \Omega_{w_2}{F\!_{\DOT}} \bigcap \Omega_{v_2}{{F\!_{\DOT}}'} \bigcap \Omega_b {E_{\DOT}}. $$ The lemma follows, as $U\cap V$ is stable under the diagonal action of $GL_n{\Bbb C}$ and the set of pairs $({F\!_{\DOT}},{{F\!_{\DOT}}'})$ in linear general position is the open $GL_n{\Bbb C}$-orbit in ${\Bbb F}\ell\times{\Bbb F}\ell$. \QED \subsection{Proof of Theorem~\ref{thm:pieri_schubert}:} Let $\Xi_1,\Xi_2,\Xi_3, \Gamma_1$, and $\Gamma_2$ be families of Schubert varieties representing the classes $\sigma_{m_1}, \sigma_{m_2}, \sigma_{m_3}, \sigma_{w_1}$, and $\sigma_{w_2}$. Suppose $\sigma_{m_1}, \sigma_{m_2}$, and $\sigma_{m_3}$ are special Schubert classes, and these families give an enumerative problem $\Xi\rightarrow U$. By \S \ref{sec:pieri_type}, for each $i=1,2$, the intersection problem $\Psi_i\rightarrow V_i$ given by the families $\Xi_i$ and $\Gamma_i$ has a real effective algebraic equivalence with witness $\sum_{v_i\in I_{m_i,w_i}} \Omega_{v_i}{F\!_{\DOT}}$, for any real flag ${F\!_{\DOT}}$. Let ${F\!_{\DOT}}$ and ${{F\!_{\DOT}}'}$ be real flags in linear general position and set $$ Z_1 \ := \sum_{v_1\in I_{m_1,w_1}} \Omega_{v_1} {F\!_{\DOT}} \qquad\mbox{and}\qquad Z_2 \ := \sum_{v_2\in I_{m_2,w_2}} \Omega_{v_2} {{F\!_{\DOT}}'}. $$ For $i=1,2$, let $\phi_i$ be the fibre function for $\Psi_i\rightarrow V_i$. Then $Z_i\in \overline{\phi_i(V_i({\Bbb R}))}$ and by Lemma~\ref{lemma:curve_selection}, there is a map $f_i : C_i \rightarrow \overline{\phi_i(V_i)}$ with $C_i$ a smooth curve, and a point $s_i$ on a connected arc $S_i$ of $C_i({\Bbb R})$ such that $f(S_i -\{s_i\})\subset\phi_i(V_i({\Bbb R}))$ and $f_i(s_i) = Z_i$. Then $f_i^*\Phi\rightarrow C_i$ is a family of multiplicity free cycles, where $\Phi\rightarrow {\mbox{\it Chow}}^\circ X$ is the tautological family. Considering pairs of components of $Z_1$ and $Z_2$ separately, Lemma~\ref{lemma:special_transversality} shows there is a real flag ${E_{\DOT}}$ such that $Z_1\bigcap Z_2 \bigcap \Omega_{m_3} {E_{\DOT}}$ is a transverse intersection all of whose points are real. Thus $f_1^*\Phi \rightarrow C_1$, $f_2^*\Phi \rightarrow C_2$, and $\Xi_3\rightarrow {\Bbb F}\ell$ give a well-posed fully real enumerative problem $\Psi \rightarrow V$, as $(s_1,s_2,{E_{\DOT}}) \in V({\Bbb R})$. Let ${\cal M}$ be the open subset of the real points of the Chow variety parameterizing cycles consisting entirely of real points. Then $\phi(s_1,s_2,{E_{\DOT}})\in {\cal M}$ and so $\phi^{-1}({\cal M})$ meets $R:= (S_1-\{s_1\})\times(S_2-\{s_2\})\times \{{E_{\DOT}}\}$. However, fibres of $\Psi$ over points of $R$ are fibres of $\Xi$ over points of $U({\Bbb R})$, showing $\Xi\rightarrow U$ to be fully real. \QED \section{Fibrations} Suppose $\pi: Y\rightarrow X$ has equidimensional fibres. If $\Xi\rightarrow U$ is a family of multiplicity free cycles on $X$ representing the cycle class $\alpha$, its pullback $\pi^*\Xi := (\pi\times 1_U)^{-1}\Xi\rightarrow U$ is a family of multiplicity free cycles on $Y$ representing the cycle class $\pi^*\alpha$. Suppose $\alpha_1\ldots,\alpha_a$ generate $A^*X$ additively and $\Psi(\alpha_1),\ldots,\Psi(\alpha_a)$ are families of cycles representing these generators. The classes $\pi^*\alpha_1,\ldots,\pi^*\alpha_a$ generate the image of $A^*X$ in $A^*Y$ and are represented by the families $\pi^*\Psi(\alpha_1),\ldots,\pi^*\Psi(\alpha_a)$. Effective algebraic equivalence is preserved by pullbacks: \subsection{Theorem}\label{thm:pullbacks} {\em If $\,\Xi\rightarrow U$ is a family of multiplicity free cycles on $X$ having effective algebraic equivalences with witness $Z$, then $\pi^*\Xi\rightarrow U$ is a family of multiplicity free cycles on $Y$ having effective algebraic equivalences with witness $\pi^{-1}Z$. Likewise, if $\,\Xi\rightarrow U$ has real effective algebraic equivalences, then so does $\pi^*\Xi\rightarrow U$. }\medskip Associating a cycle $Z$ on $X$ to $\pi^{-1}Z\subset Y$ defines a function $\pi^*: \mbox{\it Chow}\, X \rightarrow \mbox{\it Chow}\, Y$. If $\phi$ is the fibre function of $\Xi\rightarrow U$, then $\pi^*\circ \phi$ is the fibre function of $\pi^*\Xi\rightarrow U$. Letting $W=\phi(U')$ and $R = \phi(U'({\Bbb R}))$, we see that Theorem~\ref{thm:pullbacks} is a consequence of the following lemma. \subsection{Lemma}\label{lemma:pullback} {\em Let $W\subset {\mbox{\it Chow}}^\circ X$ be constructible and $V := \overline{W}\cap {\mbox{\it Chow}}^\circ X$. Then $\pi^*(V)\subset \overline{\pi^*(W)}$ in $\mbox{\it Chow}\, Y$. Likewise, if $R\subset {\mbox{\it Chow}}^\circ X({\Bbb R})$ is semi-algebraic and $Q:=\overline{R}\cap {\mbox{\it Chow}}^\circ X({\Bbb R})$, then $\pi^*(Q)\subset \overline{\pi^*(R)}$ in $\mbox{\it Chow}\, Y({\Bbb R})$. }\medskip \noindent{\bf Proof:} For the first part, let $\zeta\in V$. We show $\pi^*(\zeta) \in \overline{\pi^*(W)}$. Let $C^\circ\subset {\mbox{\it Chow}}^\circ X$ be an irreducible curve with $\zeta\in C^\circ$ and $C^\circ-\{\zeta\} \subset W$. Let $f: C \rightarrow C^\circ$ be its normalization and let $s\in f^{-1}(\zeta)$. Let $\Phi \subset {\mbox{\it Chow}}^\circ X \times X$ be the tautological family. Then $f^*\Phi$ is a family of multiplicity free cycles on $X$ with fibre function $f$. Similarly, $\pi^*\circ f$ is the fibre function of the family $\pi^*(f^*\Phi)$ of multiplicity free cycles on $Y$ over the smooth curve $C$. As noted in \S \ref{sec:Chow}, this implies $\pi^*\circ f$ is algebraic, and so $\pi^*(\zeta) \in \pi^*(f(C)) \subset \overline{\pi^*(W)}$, since $\pi^*(f(C-\{f^{-1}(\zeta)\}))\subset \pi^*(W)$. For the second part, suppose $R\subset {\mbox{\it Chow}}^\circ X({\Bbb R})$ and $\zeta \in Q = \overline{R}\bigcap{\mbox{\it Chow}}^\circ X$. By Lemma~\ref{lemma:curve_selection}, there is a smooth curve $C$, a connected arc $S\subset C({\Bbb R})$, a point $s\in S$, and an algebraic map $f: C\rightarrow {\mbox{\it Chow}}^\circ X$ such that $f(s) = \zeta$ and $f(S-\{s\}) \subset R$. Arguing as above shows $\pi^*(\zeta) \in \pi^*(f(S)) \subset \overline{\pi^*(R)}$. \QED \section{Schubert-type enumerative problems in ${\Bbb F}\ell_{0,1}{\Bbb P}^n$ are fully real} The variety ${\Bbb F}\ell_{0,1}{\Bbb P}^n$ of partial flags $q\in l \subset {\Bbb P}^n$ with $q$ a point and $l$ a line has projections $$ p\ :\ {\Bbb F}\ell_{0,1}{\Bbb P}^n \longrightarrow {\Bbb P}^n \qquad\mbox{and}\qquad \pi\ :\ {\Bbb F}\ell_{0,1}{\Bbb P}^n\longrightarrow {\Bbb G}_1{\Bbb P}^n, $$ where ${\Bbb G}_1{\Bbb P}^n$ is the Grassmannian of lines in ${\Bbb P}^n$. A Schubert subvariety of ${\Bbb G}_1{\Bbb P}^n$ is determined by a partial flag $F\subset P$ of ${\Bbb P}^n$: $$ \Omega(F,P)\ :=\ \{l \in {\Bbb G}_1{\Bbb P}^n\,|\, l\cap F \neq \emptyset\ \mbox{\ and }\ l\subset P\}. $$ If $F$ is a hyperplane of $P$, then $\Omega(F,P) = {\Bbb G}_1P$, the Grassmannian of lines in $P$. In addition to $\pi^{-1}\Omega(F,P)$, there is another Schubert subvariety of ${\Bbb F}\ell_{0,1}{\Bbb P}^n$ which projects onto $\Omega(F,P)$ in ${\Bbb G}_1{\Bbb P}^n$: $$ \widehat{\Omega}(F,P)\ :=\ \{(q,l) \in {\Bbb F}\ell_{0,1}{\Bbb P}^n\,|\, q\in F \ \mbox{\ and }\ l\subset P\}. $$ Any Schubert subvariety of ${\Bbb F}\ell_{0,1}{\Bbb P}^n$ is one of $\Omega(F,P)$ or $\widehat{\Omega}(F,P)$, for suitable $F\subset P$. The varieties $\widehat{\Omega}(F,P)$ have another description, which is straightforward to verify: \subsection{Lemma}\label{lemma:no_hats} {\em Let $N,P$ be subspaces of $\,{\Bbb P}^n$. Then $$ p^{-1}N \bigcap \pi^{-1}{\Bbb G}_1 P \ =\ \widehat{\Omega}(N\cap P,\,P), $$ and, if $N$ and $P$ meet properly, this intersection is generically transverse. } \subsection{Corollary}\label{cor:reduction} {\em Any Schubert-type enumerative problem on ${\Bbb F}\ell_{0,1}{\Bbb P}^n$ is equivalent to one involving only pullbacks of Schubert subvarieties of $\,{\Bbb P}^n$ and ${\Bbb G}_1{\Bbb P}^n$. }\medskip The next lemma, an exercise in linear algebra, describes Poincar\'e duality for Schubert subvarieties of ${\Bbb F}\ell_{0,1}{\Bbb P}^n$. \subsection{Lemma}\label{lemma:poincare_duality} {\em Suppose a linear subspace $N$ meets a partial flag $F\subset P$ properly in ${\Bbb P}^n$. If $\pi^{-1}\Omega(F,P)$ and $p^{-1}N$ have complimentary dimension in ${\Bbb F}\ell_{0,1}{\Bbb P}^n$, then their intersection is empty unless $F$ and $N\cap P$ are points. In that case, they meet transversally in a single point and $\pi^{-1}\Omega(F,P)\bigcap p^{-1}N = (N\cap P,\ \Span{F, N\cap P})$.} \subsection{Theorem}\label{thm:schubert_fully_real} {\em Any Schubert-type enumerative problem in ${\Bbb F}\ell_{0,1}{\Bbb P}^n$ is fully real. }\medskip \noindent{\bf Proof:} By Corollary~\ref{cor:reduction}, it suffices to consider enumerative problems involving only pullbacks of Schubert subvarieties of ${\Bbb P}^n$ and ${\Bbb G}_1{\Bbb P}^n$. Since the intersection of linear subspaces in ${\Bbb P}^n$ is another linear subspace, we may further suppose the enumerative problem $\Xi\rightarrow U$ is given by families $p^*\Xi_1,\pi^*\Xi_2,\ldots,\pi^*\Xi_b$, where $\Xi_1$ is the family of subspaces of a fixed dimension in ${\Bbb P}^n$ and $\Xi_2,\ldots,\Xi_b$ are families of Schubert subvarieties of ${\Bbb G}_1{\Bbb P}^n$. By Theorem~C$'$ of~\cite{sottile_real_lines}, the intersection problem $\Psi \rightarrow V$ on ${\Bbb G}_1{\Bbb P}^n$ given by $\Xi_2,\ldots,\Xi_b$ has real effective algebraic equivalences. Let $Z$ be a witness. By Theorem~\ref{thm:pullbacks}, $\pi^*\Psi \rightarrow V$ has a real effective algebraic equivalence with witness $\pi^* Z$. By Lemma~\ref{lemma:curve_selection}, there is a real algebraic map $f:C \rightarrow \overline{\phi(W)}\cap {\mbox{\it Chow}}^\circ {\Bbb F}\ell_{0,1}{\Bbb P}^n$ with $C$ a smooth curve, and a point $s$ on a connected arc $S$ of $ C({\Bbb R})$ such that $f(s) = \pi^{-1}Z$ and $f(S-\{s\})\subset \phi(V({\Bbb R}))$. Let $\Phi \rightarrow {\mbox{\it Chow}}^\circ {\Bbb F}\ell_{0,1}{\Bbb P}^n$ be the tautological family and consider the family $f^*\Phi \rightarrow C$. The fibre over $s$ of $f^*\Phi$ is $\pi^{-1}Z$. Let ${\cal L}$ be the lattice of subspaces of ${\Bbb P}^n$ generated by the (necessarily real) subspaces defining components of $Z$, and let $N$ be a real subspace from the family $\Xi_1$ meeting all subspaces of ${\cal L}$ properly. By Lemma~\ref{lemma:poincare_duality}, $p^{-1}N\bigcap \pi^{-1}Z$ is transverse with all points of intersection real. Thus there is a Zariski open subset $C'$ of $C$ such that fibres of $f^*\Phi$ over $C'$ meet $p^{-1} N$ transversally. Then $s\in \overline{(S-\{s\})\bigcap C'({\Bbb R})}$, so there is a point $t\in S-\{s\}$ such that $p^{-1}N\bigcap(f^*\Phi)_t$ is transverse and consists entirely of real points. But $f(t)\in \phi(V({\Bbb R}))$, so $(f^*\Phi)_t=\Phi_{f(t)}$ is a fibre of $\pi^*\Psi$ over $V({\Bbb R})$, and hence a generically transverse intersection of real Schubert varieties from the families $\pi^*\Xi_2,\ldots,\pi^*\Xi_b$. Thus $\Xi\rightarrow U$ is fully real. \QED \subsection{Effective algebraic equivalence for ${\Bbb F}\ell_{0,1}{\Bbb P}^n$} Any Schubert-type intersection problem on ${\Bbb F}\ell_{0,1}{\Bbb P}^n$ has real effective algebraic equivalences. We give an outline, as a complete analysis is lengthy and involves no new ideas beyond~\cite{sottile_real_lines}. By Corollary~\ref{cor:reduction}, it suffices to consider intersection problems $\Xi\rightarrow U$ given by families $p^*\Xi_1,\pi^*\Xi_2,\ldots,\pi^*\Xi_b$, where $\Xi_1$ is a family of subspaces of a fixed dimension in ${\Bbb P}^n$ and $\Xi_2,\ldots,\Xi_b$ are families of Schubert subvarieties of ${\Bbb G}_1{\Bbb P}^n$. In~\cite{sottile_real_lines}, the intersection problem given by $\Xi_2,\ldots,\Xi_b$ is shown to have real effective algebraic equivalences with witness $Z$. Let $\Psi \rightarrow V$ be the intersection problem given by $p^*\Xi_1$ and the constant family $\pi^{-1}Z$. Using Theorem~\ref{thm:pullbacks} and Lemma~\ref{lemma:curve_selection} one may show $$ \phi(V)\subset \overline{\phi(U)} \qquad\mbox{and}\qquad \phi(V({\Bbb R}))\subset \overline{\phi(U({\Bbb R}))}. $$ It suffices to show $\Psi\rightarrow V$ has real effective algebraic equivalences. A proof that $\Psi\rightarrow V$ has real effective algebraic equivalences mimics the proof of Theorem~E of~\cite{sottile_real_lines}, with the following Lemma playing the role of Lemma~2.4 of~\cite{sottile_real_lines}. \subsection{Lemma}\label{lemma:reducible} {\em Let $F,P,N$, and $H$ be linear subspaces of ${\Bbb P}^n$ and suppose that $H$ is a hyperplane containing neither $P$ nor $N$, $F$ is a proper subspace of $P\cap H$, and $N$ meets $F$, and hence $P$ properly. Set $L = N\cap H$. Then $\pi^{-1}\Omega(F,P)$ and $p^{-1}L$ meet generically transversally, $$ \pi^{-1}\Omega(F,P)\bigcap p^{-1}L\ =\ \widehat{\Omega}(N\cap F,\,P) \ +\ \pi^{-1}\Omega(F,P\cap H)\bigcap p^{-1}N, $$ and the second term is itself an irreducible generically transverse intersection.} \medskip The proof of this statement is almost identical to the proof of Lemma~2.4 of~\cite{sottile_real_lines}. \section{Some Schubert-type enumerative problems in ${\Bbb F}\ell_{1,n-2}{\Bbb P}^n$} The variety ${\Bbb F}\ell_{1,n-2}{\Bbb P}^n$ of partial flags $l \subset \Lambda \subset {\Bbb P}^n$, where $l$ is a line and $\Lambda$ an $(n-2)$-plane has projections $$ \pi: {\Bbb F}\ell_{1,n-2}{\Bbb P}^n \rightarrow {\Bbb G}_1{\Bbb P}^n \qquad\mbox{and}\qquad p: {\Bbb F}\ell_{1,n-2}{\Bbb P}^n\rightarrow {\Bbb G}_{n-2}{\Bbb P}^n, $$ where ${\Bbb G}_{n-2}{\Bbb P}^n$ is the Grassmannian of $(n-2)$-planes in ${\Bbb P}^n$. \subsection{Theorem}\label{thm:simple_schubert} {\em Any enumerative problem in ${\Bbb F}\ell_{1,n-2}{\Bbb P}^n$ given by pullbacks of Schubert subvarieties of $\,{\Bbb G}_1{\Bbb P}^n$ and ${\Bbb G}_{n-2}{\Bbb P}^n$ is fully real. }\medskip \noindent{\bf Proof:} Suppose $\pi^*\Xi_1,\ldots,\pi^*\Xi_b, p^*\Gamma_1,\ldots,p^*\Gamma_c$ give an enumerative problem on ${\Bbb F}\ell_{1,n-2}{\Bbb P}^n$ where, for $1\leq i\leq b$, $\Xi_i$ is a family of Schubert subvarieties of ${\Bbb G}_1{\Bbb P}^n$ and for $1\leq j\leq c$, $\Gamma_i$ is a family of Schubert subvarieties of ${\Bbb G}_{n-2}{\Bbb P}^n$. By Theorem~\ref{thm:pullbacks} and~\cite{sottile_real_lines}, $\pi^*\Xi_1,\ldots,\pi^*\Xi_b$ give an intersection problem $\Psi_1\rightarrow V_1$ which has a real algebraic equivalence with witness $Z_1$. Identifying ${\Bbb P}^n$ with its dual projective space gives an isomorphism ${\Bbb G}_{n-2}{\Bbb P}^n \stackrel{\sim}{\rightarrow} {\Bbb G}_1{\Bbb P}^n$, mapping Schubert subvarieties to Schubert subvarieties. It follows that $p^*\Gamma_1,\ldots,p^*\Gamma_c$ give an intersection problem $\Psi_2\rightarrow V_2$ which has a real algebraic equivalence with witness $Z_2$. It suffices to show the enumerative problem $\Psi\rightarrow V$ given by $\Psi_1$ and $\Psi_2$ is fully real. Since $Z_1$ and $Z_2$ may be replaced by any translate by elements of $PGL_{n+1}{\Bbb R}$, we assume $Z_1$ and $Z_2$ intersect transversally. Components of $Z_1$ and $Z_2$ are Schubert varieties defined by real flags. Moreover, each component of $Z_1$ has complementary dimension to each component of $Z_2$. In a flag variety, Schubert varieties of complimentary dimension which meet transversally and are defined by real flags either have empty intersection, or meet in a single real point. Thus $Z_1\bigcap Z_2$ consists entirely of real points. By Lemma~\ref{lemma:curve_selection}, for each $i=1,2$, there is a real algebraic map $f_i : C_i \rightarrow \overline{\phi(V_i)}$ where $C_i$ is a smooth curve, and a point $s_i$ on a connected arc $S_i$ of $C_i({\Bbb R})$ such that $f(S_i -\{s_i\})\subset\phi(V_i({\Bbb R}))$ and $f_i(s_i) = Z_i$. The enumerative problem $\Psi'\rightarrow V'$ given by $f_1^*\Psi_1\rightarrow C_1$ and $f_2^*\Psi_2\rightarrow C_2$ is fully real, as $\Psi'_{(s_1,s_2)} = Z_1\bigcap Z_2$. Since $(s_1,s_2)\in \overline{(S_1-\{s_1\})\times(S_2-\{s_2\})\bigcap V'({\Bbb R})}$, there is a point $(t_1,t_2)\in (S_1-\{s_1\})\times(S_2-\{s_2\})$ such that $\Psi'_{(t_1,t_2)} = (f_1^*\Psi_1)_{t_1}\bigcap(f_2^*\Psi_2)_{t_2}$ is transverse and consists entirely of real points. Since $f_i(t_i)\in \phi(V_i({\Bbb R}))$, we see that $(f_i^*\Psi_i)_{t_i} = (\Psi_i)_{f_i(t_i)}$ is a fibre of $\Psi_i$ over a point of $V_i({\Bbb R})$. This shows $\Psi\rightarrow V$ is fully real. \QED \section{Powers of Enumerative Problems} A method to construct a new fully real enumerative problem out of a given one is illustrated by a proof of B\'ezout's Theorem in the plane. We will formalize this method. \subsection{B\'ezout's Theorem} {\em Let $d_1$ and $d_2$ be positive integers. Then there exist smooth real plane curves $D_1$ and $D_2$ of degrees $d_1$ and $d_2$ meeting transversally in \medskip$d_1\cdot d_2$ real points.} \noindent{\bf Proof:} Two distinct real lines meet in a single real point. Thus if $D_1$ consists of $d_1$ distinct real lines, $D_2$ of $d_2$, and if $D_1$ and $D_2$ meet transversally, then $D_1\bigcap D_2$ is $d_1\cdot d_2$ real points. The family of real reduced degree $d$ plane curves has general member a smooth curve and contains all cycles of $d$ distinct real lines. Thus the enumerative problem of intersecting reduced curves $D_1$ and $D_2$ of respective degrees $d_1$ and $d_2$ is fully real of degree $d_1\cdot d_2$. Moreover, pairs of smooth real curves are dense in the set of pairs of reduced real curves, showing the enumerative problem of intersecting two smooth plane curves of respective degrees $d_1$ and $d_2$ is fully real of degree $d_1\cdot d_2$. \QED \subsection{Powers of intersection problems} Suppose $\Xi\rightarrow U$ is a family of multiplicity free cycles on $X$ and $d$ is a positive integer. If the locus of $d$-tuples $(u_1,\ldots,u_d)$ such that no two of $\Xi_{u_1},\ldots,\Xi_{u_d}$ share a component is dense in $U^d$, then let $U^{(d)}$ be an open subset of that locus. Let $\Xi^{\oplus d}\rightarrow U^{(d)}$ be the family of multiplicity free cycles whose fibre over $(u_1,\ldots,u_d)\in U^{(d)}$ is $\sum_{j=1}^d \Xi_{u_j}$. Suppose $\Xi_1\rightarrow U_1,\ldots,\Xi_b\rightarrow U_b$ are families of multiplicity free cycles on $X$ giving an intersection problem $\Xi\rightarrow U$ and $d_1,\ldots,d_b$ is a sequence of positive integers. Then the families $\Xi_1^{\oplus d_1}\rightarrow U_{1}^{(d_1)},\ldots, \Xi_b^{\oplus d_b}\rightarrow U_{s}^{(d_b)}$ give a well-posed intersection problem if general members of the famililies $\Xi\rightarrow U$ and $\Xi_i\rightarrow U_i$ meet properly, for $1\leq i\leq b$. When a reductive group $G$ acts transitively on $X$ and the families of cycles are $G$-stable, $\Xi_1^{\oplus d_i},\ldots,\Xi_b^{\oplus d_b}$ give an intersection problem. Moreover, if $\Xi\rightarrow U$ is fully real, then so is that intersection problem. We produce a witness with a particular form. \subsection{Lemma}\label{lemma:transverse_technicality} {\em Suppose $\Xi_1\rightarrow U_1,\ldots,\Xi_b\rightarrow U_b$ give a fully real enumerative problem of degree $d$. Let $d_1,\ldots,d_b$ be a sequence of positive integers and suppose that for $1\leq i\leq b$, $V_i$ is $G$-stable subset of $U_{i}^{(d_i)}$ such that $\Delta^{d_i}U_i({\Bbb R})\subset \overline{V_i({\Bbb R})}$, as subsets of $U_i({\Bbb R})^{d_i}$. Then for $1\leq i\leq b$, there exists $v_i\in V_i({\Bbb R})$ such that $(\Xi_1^{\oplus d_1})_{v_1},\ldots,(\Xi_b^{\oplus d_b})_{v_b}$ intersect transversally in $d\cdot d_1\cdots d_b$ real points.} \medskip \noindent{\bf Proof:} The restriction $\Psi_i$ of $\Xi_i^{\oplus d_i}$ to $V_i$ is $G$-stable. Thus $\Psi_1,\ldots,\Psi_b$ give a well-posed enumerative problem $\Psi\rightarrow V$. We show this is fully real and compute its degree. Since $\Xi\rightarrow U$ is fully real, there is an open subset $R$ of points $u \in U({\Bbb R})$ such that $\Xi_u$ is $d$ distinct real points. Since $U({\Bbb R})\subset \prod_{i=1}^b U_i({\Bbb R})$, for $1\leq i\leq b$ there exists an open subset $R_i$ of $U_i({\Bbb R})$ such that $\prod_{i=1}^b R_i \subset R$. Then $V_i({\Bbb R})\bigcap R^{d_i}\neq \emptyset$, as $\Delta^{d_i} R_i\subset \Delta^{d_i}U_i({\Bbb R}) \subset \overline{V_i({\Bbb R})}$. Thus $R' := V({\Bbb R}) \bigcap \prod_{i=1}^b R_i^{d_i}$ is nonempty, as $V({\Bbb R})$ is dense in $ \prod_{i=1}^b V_i({\Bbb R})$. Let $w =(w_{11},\ldots,w_{1d_1},\ldots,w_{b1}\ldots,w_{bd_b})\in R'$. Here, $w_{ij}\in R_i$ and $(w_{i1},\ldots,w_{id_i})\in V_i({\Bbb R})$. If $1\leq j_i\leq d_i$, then $(w_{1j_1},\ldots,w_{bj_b})\in U({\Bbb R})$. Furthermore, $\Psi_w = \bigcap_{i=1}^b (\Psi_i)_{(w_{i1},\ldots,w_{id_i})}$ is a transverse intersection, as $R' \subset V$. Since $(\Psi_i)_{(w_{i1},\ldots,w_{id_i})} = \sum_{j=1}^{d_i} (\Xi_i)_{w_{ij}}$, we have $$ \Psi_w \ =\ \bigcap_{i=1}^b \,\sum_{j=1}^{d_i} (\Xi_i)_{w_{ij}} \ =\ \sum_{\stackrel{\mbox{\scriptsize$j_1,\ldots,j_b$}}{1\leq j_i\leq d_i}} \ \bigcap_{i=1}^b\,(\Xi_i)_{w_{ij_i}} \ =\ \sum_{\stackrel{\mbox{\scriptsize$j_1,\ldots,j_b$}}{1\leq j_i\leq d_i}} \Xi_{(w_{1j_1},\ldots,w_{bj_b})}. $$ Since this intersection is transverse, it consists of $d\cdot d_1\cdots d_b$ real points. \QED \subsection{Real B\'ezout's Theorem} {\em Let $d_1,\ldots,d_b$ be positive integers. Then there exist smooth real hypersurfaces $H_1,\ldots,H_b$ in ${\Bbb P}^b$ of respective degrees $d_1,\ldots,d_b$ which intersect transversally in $d_1\cdots d_b$ real points. }\medskip \noindent{\bf Proof:} Let $\Xi\rightarrow U$ be the family of hyperplanes in ${\Bbb P}^b$. Since $b$ real hyperplanes in general position meet in a real point, either simple checking or Lemma~\ref{lemma:transverse_technicality} with $V:=U^{(d_i)}$ shows that $\Xi^{\oplus d_1},\ldots,\Xi^{\oplus d_b}$ give a fully real enumerative problem of degree $d_1\cdots d_b$. Note that $\Xi^{\oplus d_i}\rightarrow U^{(d_i)}$ is the family of hypersurfaces composed of $d_i$ distinct hyperplanes. Let $W_i\subset {\Bbb P}(\mbox{\em Sym}^{d_i}{\Bbb C}\,^{b+1})$ be the space of forms of degree $d_i$ with no repeated factors and $\Gamma_i\rightarrow W_i$ the family of reduced degree $d_i$ hypersurfaces. Let $W'_i\subset W_i$ be the dense subset of forms determining smooth hypersurfaces. Note that $U^{(d_i)}\subset W_i$ and $\Xi^{\oplus d_i} = \Gamma_i|_{U^{(d_i)}}$. It follows that $\Gamma_1,\ldots,\Gamma_b$ give a fully real enumerative problem of degree $d_1\cdots d_b$. Let $R$ be an open set of witnesses. Since $U^{(d_i)}({\Bbb R})\subset\overline{W'_i({\Bbb R})}$ and $R$ meets $\prod_{i=1}^sU^{(d_i)}({\Bbb R})$, we see that $R$ meets $\prod_{i=1}^sW'_i({\Bbb R})$. That is, there exist smooth real hypersurfaces $H_1,\ldots,H_b$ in ${\Bbb P}^b$ of respective degrees $d_1,\ldots,d_b$ which intersect transversally in $d_1\cdots d_b$ real points. \QED \section{$(n-2)$-planes meeting rational normal curves in ${\Bbb P}^n$} Let ${\Bbb G}_{n-2}{\Bbb P}^n$ be the Grassmannian of $(n-2)$-planes in ${\Bbb P}^n$, a variety of dimension $2n-2$. Those $(n-2)$-planes which meet a curve form a hypersurface in ${\Bbb G}_{n-2}{\Bbb P}^n$. We synthesize ideas of previous sections to prove the following theorem. \subsection{Theorem}\label{thm:rational_normal} {\em The enumerative problem of $\,(n-2)$-planes meeting $2n-2$ general rational normal curves in ${\Bbb P}^n$ is fully real and has degree ${2n-2\choose n-1}n^{2n-3}$. }\medskip \noindent{\bf Proof:} Identifying ${\Bbb P}^n$ with its dual projective space gives an isomorphism ${\Bbb G}_{n-2}{\Bbb P}^n \stackrel{\sim}{\rightarrow} {\Bbb G}_1{\Bbb P}^n$, mapping Schubert subvarieties to Schubert subvarieties. By Theorem~C of~\cite{sottile_real_lines}, any enumerative problem involving Schubert subvarieties of ${\Bbb G}_{n-2}{\Bbb P}^n$ is fully real. In particular, the enumerative problem given by $2n-2$ copies of the family $\Xi \rightarrow U$ is fully real, where $U = {\Bbb G}_1{\Bbb P}^n$ and the fibre of $\Xi$ over $l\in U$ is the Schubert variety $\Omega_l$ of $(n-2)$-planes meeting $l$. We compute its degree, $d$. The image of $\Omega_l$ under the isomorphism ${\Bbb G}_{n-2}{\Bbb P}^n \stackrel{\sim}{\rightarrow} {\Bbb G}_1{\Bbb P}^n$ is the Schubert subvariety of all lines meeting a fixed $(n-2)$-plane. Thus $d$ is the number of lines meeting $2n-2$ general $(n-2)$-planes in ${\Bbb P}^n$. By Corollary 3.3 of~\cite{sottile_real_lines}, this is the number of (standard) Young tableaux of shape $(n-1,n-1)$, which is $\frac{1}{n} {2n-2\choose n-1}$, by the hook length formula of Frame, Robinson, and Thrall~\cite{FRT}. Let $e_0,\ldots,e_n$ be real points spanning ${\Bbb P}^n$. For $1\leq i\leq n$, let $l_i := \Span{e_{i-1},e_i}$. Then $\Omega_{l_1}+ \cdots+\Omega_{l_n}$ is the fibre of $\Xi^{\oplus n}$ over $(l_1,\ldots,l_n)\in U^{(n)}({\Bbb R})$. Let $V = PGL_{n+1}{\Bbb C}\cdot (l_1,\ldots,l_n)\subset U^{(n)}$. For $t\in [0,1]$ and $1\leq i\leq n$, let $$ l_i(t) \ :=\ \Span{te_{i-1} + (1-t)e_{\overline{i -1}},\, t e_i + (1-t) e_{\overline{i}}}, $$ where $\overline{j}\in \{0,1\}$ is congruent to $j$ modulo 2. Let $\gamma(t) := (l_1(t),\ldots,l_n(t))$. If $t\in (0,1]$, then $\gamma(t)\in V({\Bbb R})$. Since $\gamma(0) = (l_1,\ldots,l_1)$ and $\Delta^nU({\Bbb R}) = PGL_{n+1}{\Bbb R}\cdot \gamma(0)$, it follows that $\Delta^nU({\Bbb R})\subset \overline{V{(\Bbb R})}$. Then, by Lemma~\ref{lemma:transverse_technicality}, there exist points $v_1,\ldots,v_{2n-2} \in V({\Bbb R})$ such that $\Xi^{\oplus n}_{v_1},\ldots,\Xi^{\oplus n}_{v_{2n-2}}$ meet transversally in ${2n-2\choose n-1}n^{2n-3}$ points. Let $p(m):= n\cdot m +1$, the Hilbert polynomial of a rational normal curve in ${\Bbb P}^n$. Let ${\cal H}$ be the open subset of the Hilbert scheme parameterizing reduced schemes with Hilbert polynomial $p$. Let $\Psi\subset {\cal H}\times {\Bbb G}_{n-2}{\Bbb P}^n$ be the family of multiplicity free cycles on ${\Bbb G}_{n-2}{\Bbb P}^n$ whose fibre over a curve $C\in {\cal H}$ is the hypersurface of $(n-2)$-planes meeting $C$. Note that $p$ is also the Hilbert polynomial of $l_1\bigcup\cdots\bigcup l_n$. Let $\lambda \in {\cal H}$ be the point representing $l_1\bigcup\cdots\bigcup l_n$. If $V'$ is the $PGL_{n+1}{\Bbb C}$-orbit of $\lambda$ in ${\cal H}$, then $\Psi|_{V'}\rightarrow V'$ is isomorphic to the family $\Xi^{\oplus n}\rightarrow V$, under the obvious isomorphism between $V$ and $V'$. It follows that the enumerative problem given by $2n-2$ copies of $\Psi\rightarrow {\cal H}$ is fully real. Let $W$ be the subset of ${\cal H}$ representing rational normal curves. We claim $V'({\Bbb R})\subset \overline{W({\Bbb R})}$, from which it follows that the enumerative problem of $(n-2)$-planes meeting $2n-2$ rational normal curves in ${\Bbb P}^n$ is fully real and has degree ${2n-2\choose n-1}n^{2n-3}$. Let $[x_0,\ldots,x_n]$ be homogeneous coordinates for ${\Bbb P}^n$ dual to the basis $e_0,\ldots,e_n$. For $t\in {\Bbb C}$, define the ideal ${\cal I}_t$ by $$ {\cal I}_t \ :=\ ( x_ix_j - t x_{i+1}x_{j-1}\, |\, 0\leq i<j\leq n\ \mbox{and}\ j-i\geq 2). $$ For $t\neq 0$, ${\cal I}_t$ is the ideal of a rational normal curve and ${\cal I}_0$ is the ideal of $l_1\bigcup\cdots\bigcup l_n$. This family of ideals is flat. Let $\varphi:{\Bbb C}\rightarrow {\cal H}$ be the map representing this family. Then $\varphi({\Bbb R}-\{0\})\subset W({\Bbb R})$. Noting $\varphi(0)=\lambda$ shows $\lambda \in \overline{W({\Bbb R})}$. Since $W({\Bbb R})$ is $PGL_{n+1}{\Bbb R}$-stable, we conclude that $V'({\Bbb R})\subset \overline{W({\Bbb R})}$. \QED

9602

alg-geom/9602018

en

https://arxiv.org/abs/alg-geom/9602018

alg-geom/9602018

Klaus Altmann

Klaus Altmann

P-Resolutions of Cyclic Quotients from the Toric Viewpoint

10 pages; LaTeX. dvi and ps file available at http://www-irm.mathematik.hu-berlin.de/~altmann/PAPER/paper.html

P-resolutions of two-dimensional, cyclic quotient singularities have been introduced to study deformation theory. Those P-resolutions (as well as the singularities themselves) are toric varieties. In the present paper we give a straight, elementary description of them just by their defining fans.

alg-geom

\section{Introduction}\label{I} \neu{I-1} The break through in deformation theory of (two-dimensional) quotient singularities $Y$ was Koll\'{a}r/Shepherd-Barron's discovery of the one-to-one correspondence between so-called P-resolutions, on the one hand, and components of the versal base space, on the other hand (cf.\ \cite{KS}, Theorem (3.9)). It generalizes the fact that all deformations admitting a simultaneous (RDP-) resolution form one single component, the Artin component.\\ \par According to defintion (3.8) in \cite{KS}, P-resolutions are partial resolutions $\pi:\tilde{Y}\to Y$ such that \begin{itemize} \item the canonical divisor $K_{\tilde{Y}|Y}$ is ample relative to $\pi$ (a minimality condition) and \item $\tilde{Y}$ contains only mild singularities of a certain type (so-called T-singularities). \end{itemize} Despite their definition as those quotient singularities admitting a $I\!\!\!\!Q$-Gorenstein one-parameter smoothing (\cite{KS}, (3.7)), there are at least three further descriptions of the class of T-singularities: An explicit list of their defining group actions on $\,I\!\!\!\!C^2$ (\cite{KS}, (3.10)), an inductive procedure to construct their resolution graphs (\cite{KS}, (3.11)), and a characterization using toric language (\cite{Homog}, (7.3)).\\ The latter one starts with the observation that affine, two-dimensional toric varieties (given by some rational, polyhedral cone $\sigma\subseteqI\!\!R^2$) provide exactly the two-dimensional cyclic quotient singularties. Then, T-singularities come from cones over rational intervals of integer length placed in height one (i.e.\ contained in the affine line $(\bullet,1)\subseteqI\!\!R^2$). \vspace{-2ex} \begin{center} \unitlength=0.4mm \linethickness{0.4pt} \begin{picture}(155.00,68.00) \put(0.00,40.00){\line(1,0){138.00}} \put(10.00,40.00){\circle*{2.00}} \put(40.00,40.00){\circle*{2.00}} \put(70.00,40.00){\circle*{2.00}} \put(100.00,40.00){\circle*{0.00}} \put(100.00,40.00){\circle*{2.00}} \put(10.00,10.00){\line(5,2){145.00}} \put(10.00,10.00){\line(1,2){29.00}} \put(130.00,40.00){\circle*{2.00}} \put(130.00,10.00){\circle*{2.00}} \put(100.00,10.00){\circle*{2.00}} \put(70.00,10.00){\circle*{2.00}} \put(40.00,10.00){\circle*{2.00}} \put(10.00,10.00){\circle*{2.00}} \put(55.00,44.00){\makebox(0,0)[cb]{\footnotesize length 2}} \put(100,60){\makebox(0,0)[cc]{$\sigma$}} \end{picture} \end{center} If the affine interval is of length $\mu+1$, then the corresponding T-singularity will have Milnor number $\mu$ (on the $I\!\!\!\!Q$-Gorenstein one-parameter smoothing).\\ \par \neu{I-2} In \cite{Ch-CQS} and \cite{St-CQS} Christophersen and Stevens gave a combinatorial description of all P-resolutions for two-dimensional, cyclic quotient singularities. Using an inductive construction method (going through different cyclic quotients with step-by-step increasing multiplicity) they have shown that there is a one-to-one correspondence between P-resolutions, on the one hand, and certain integer tuples $(k_2,\dots,k_{e-1})$ yielding zero if expanded as a (negative) continued fraction (cf.\ \zitat{P}{2}), on the other hand.\\ \par The aim of the present paper is to provide an elementary, direct method for constructing the P-resolutions of a cyclic quotient singularity (i.e.\ a two-dimensional toric variety) $Y_{\sigma}$. Given a chain $(k_2,\dots,k_{e-1})$ representing zero, we will give a straight description of the corresponding polyhedral subdivision of $\sigma$. (In particular, the bijection between those 0-chains and P-resolutions will be proved again by a different method.)\\ \par \section{Cyclic Quotient Singularities}\label{CQS} In the following we want to remind the reader of basic notions concerning continued fractions and cyclic quotients. It should be considered a good chance to fix notations. References are \cite{Oda} (\S 1.6) or the first sections in \cite{Ch-CQS} and \cite{St-CQS}, respectively.\\ \par \neu{CQS-1} {\bf Definition:} To integers $c_1,\dots,c_r\inZ\!\!\!Z$ we will assign the continued fraction $[c_1,\dots,c_r]\inI\!\!\!\!Q$, if the following inductive procedure makes sense (i.e.\ if no division by 0 occurs): \begin{itemize} \item $[c_r]:=c_r$ \item $[c_i,\dots,c_r]:= c_i- 1/[c_{i+1},\dots,c_r]$. \end{itemize} If $c_i\geq 2$ for $i=1,\dots,r$, then $[c_1,\dots,c_r]$ is always defined and yields a rational number greater than 1. Moreover, all these numbers may be represented by those continued fractions in a unique way.\\ \par \neu{CQS-2} Let $n\geq 2$ be an integer and $q\in (Z\!\!\!Z/nZ\!\!\!Z)^\ast$ be represented by an integer between $0$ and $n$. These data provide a group action of $Z\!\!\!Z/nZ\!\!\!Z$ on $\,I\!\!\!\!C^2$ via the matrix $\left(\begin{array}{cc}\xi&0\\0&\xi^q\end{array}\right)$ (with $\xi$ a primitive $n$-th root of unity). The quotient is denoted by $Y(n,q)$.\\ \par In toric language, $Y(n,q)$ equals the variety $Y_\sigma$ assigned to the polyhedral cone $\sigma:=\langle (1,0);(-q,n)\rangle\subseteqI\!\!R^2$. ($Y_\sigma$ is defined as $\mbox{Spec}\; \,I\!\!\!\!C[\sigma^{\scriptscriptstyle\vee}\capZ\!\!\!Z^2]$ with \[ \sigma^{\scriptscriptstyle\vee}:=\{r\in(I\!\!R^2)^\ast\,| \;r\geq 0 \mbox{ on } \sigma\}= \langle [0,1];[n,q]\rangle \subseteq (I\!\!R^2)^\ast \cong I\!\!R^2\,.) \] {\bf Notation:} Just to distinguish between $I\!\!R^2$ and its dual $(I\!\!R^2)^\ast\congI\!\!R^2$, we will denote these vector spaces by $N_{I\!\!R}$ and $M_{I\!\!R}$, respectively. (Hence, $\sigma\subseteq N_{I\!\!R}$ and $\sigma^{\scriptscriptstyle\vee}\subseteq M_{I\!\!R}$.) Elements of $N_{I\!\!R}\congI\!\!R^2$ are written in paranthesis; elements of $M_{I\!\!R}\congI\!\!R^2$ are written in brackets. The natural pairing between $N_{I\!\!R}$ and $M_{I\!\!R}$ is denoted by $\langle\,,\,\rangle$ which should not be mixed up with the symbol indicating the generators of a cone. Finally, all these remarks apply for the lattices $N\congZ\!\!\!Z^2$ and $M\congZ\!\!\!Z^2$, too.\\ \par \neu{CQS-3} Let $n,q$ as before. We may write $n/(n-q)$ and $n/q$ (both are greater than 1) as continued fractions \[ n/(n-q) = [a_2,\dots,a_{e-1}] \; \mbox{ and } \; n/q = [b_1,\dots,b_r] \quad (a_i, b_j\geq 2). \] The $a_i$'s and the $b_j$'s are mutually related by Riemenschneider's point diagram (cf.\ \cite{Riem}).\\ \par Take the convex hull of $(\sigma^{\scriptscriptstyle\vee}\cap M)\setminus\{0\}$ and denote by $w^1,w^2,\dots, w^e$ the lattice points on its compact edges. If ordered the right way, we obtain $w^1=[0,1]$ and $w^e=[n,q]$ for the first and the last point, respectively. \begin{center} \unitlength=0.4mm \linethickness{0.4pt} \begin{picture}(141.00,142.00) \put(0,10){\makebox(0,0)[cc]{$0$}} \put(10.00,10.00){\line(2,5){52.67}} \put(10.00,10.00){\line(5,1){131.00}} \put(53.00,118.00){\line(-1,-6){11.67}} \put(41.33,47.00){\line(5,-3){24.67}} \put(66.00,32.33){\line(1,0){54.00}} \put(53.00,117.00){\circle*{3.00}} \put(42.00,47.00){\circle*{3.00}} \put(47.00,82.00){\circle*{3.00}} \put(66.00,32.00){\circle*{3.00}} \put(121.00,32.00){\circle*{3.00}} \put(63.00,117.00){\makebox(0,0)[cc]{$w^1$}} \put(59.00,82.00){\makebox(0,0)[cc]{$w^2$}} \put(95.00,35.00){\makebox(0,0)[cc]{$\dots$}} \put(53.00,52.00){\makebox(0,0)[cc]{$w^3$}} \put(121.00,23.00){\makebox(0,0)[cc]{$w^e$}} \put(106.00,82.00){\makebox(0,0)[cc]{$\sigma^{\scriptscriptstyle\vee}$}} \end{picture} \end{center} Then, $E:=\{w^1,\dots,w^e\}$ is the minimal generating set (the so-called Hilbert basis) of the semigroup $\sigma^{\scriptscriptstyle\vee}\cap M$. These point are related to our first continued fraction by \[ w^{i-1}+w^{i+1}=a_i\,w^i \quad (i=2,\dots,e-1). \] {\bf Remark:} The surjection $I\!\!N^E\longrightarrow\hspace{-1.5em}\longrightarrow \sigma^{\scriptscriptstyle\vee}\cap M$ provides a minimal embedding of $Y_{\sigma}$. In particular, $e$ equals its embedding dimension.\\ \par In a similar manner we can define $v^0,\dots,v^{r+1}\in\sigma\cap N$ in the original cone; now we have $v^0=(1,0)$, $v^{r+1}=(-q,n)$, and the relation to the continued fractions is $v^{j-1}+v^{j+1}=b_j\,v^j$ (for $j=1,\dots,r$). \begin{center} \unitlength=0.4mm \linethickness{0.4pt} \begin{picture}(148.00,146.00) \put(10.00,10.00){\line(1,3){45.33}} \put(10.00,10.00){\line(6,1){138.00}} \put(10.00,10.00){\line(1,2){65.67}} \put(10.00,10.00){\line(5,6){80.00}} \put(10.00,10.00){\line(5,2){116.00}} \put(51.00,133.00){\line(-1,-6){10.00}} \put(41.00,72.00){\line(1,-6){3.33}} \put(44.33,51.00){\line(5,-4){22.67}} \put(67.00,33.00){\line(1,0){81.00}} \put(144.00,23.00){\makebox(0,0)[cc]{$v^0$}} \put(68.00,42.00){\makebox(0,0)[cc]{$v^1$}} \put(40.00,132.00){\makebox(0,0)[cc]{$v^{r+1}$}} \put(0.00,12.00){\makebox(0,0)[cc]{$0$}} \put(123.00,101.00){\makebox(0,0)[cc]{$\sigma$}} \end{picture} \end{center} Drawing rays through the origin and each point $v^j$, respectively, provides a polyhedral subdivision $\Sigma$ of $\sigma$. The corresponding toric variety $Y_\Sigma$ is a resolution of our singularity $Y_\sigma$. The numbers $-b_j$ equal the self intersection numbers of the exceptional divisors; since $b_j\geq 2$, the resolution is the {\em minimal} one.\\ \par \section{The Maximal Resolution}\label{MR} \neu{MR-1} {\bf Definition:} (\cite{KS}, (3.12)) For a resolution $\pi:\tilde{Y}\to Y$ we may write $K_{\tilde{Y}|Y}:=K_{\tilde{Y}}-\pi^\ast K_Y= \sum_{j}(\alpha_j-1)E_j$ ($E_j$ denote the exceptional divisors, $\alpha_j\inI\!\!\!\!Q$). Then, $\pi$ will be called {\em maximal}, if it is maximal with respect to the property $0<\alpha_j<1$.\\ \par The maximal resolution is uniquely determined and dominates all the P-resolutions. Hence, for our purpose, it is more important than the minimal one. It can be constructed from the minimal resolution by sucsessive blowing up of points $E_i\cap E_j$ with $\alpha_i+\alpha_j\geq 0$ (cf.\ Lemma (3.13) and Lemma (3.14) in \cite{KS}).\\ \par \neu{MR-2} {\bf Proposition:} {\em The maximal resolution of $Y_\sigma$ is toric. It can be obtained by drawing rays through $0$ and all interior lattice points (i.e.\ $\in N$) of the triangle $\Delta:= \mbox{\em conv} \,(0,v^0,v^{r+1})$, respectively. }\\ \par {\bf Proof:} We have to keep track of the rational numbers $\alpha_j$. Hence, we will show how they can be ``seen'' in an arbitrary toric resolution of $Y_\sigma$. Let $\Sigma<\sigma$ be a subdivision generated by one-dimensional rays through the points $u^0,\dots,u^{s+1}\in\sigma\cap N$. (In particular, $u^0=v^0=(1,0)$ and $u^{s+1}=v^{r+1}=(-q,n)$; moreover, for the minimal resolution we would have $s=r$ and $u^j=v^j$ ($j=0,\dots,r+1$).) Denote by $c_1,\dots,c_s$ the integers given by the relations \[ u^{j-1}+u^{j+1}=c_j\,u^j \qquad (j=1,\dots,s). \] (In particular, $c_j =b_j$ for the minimal resolution again.)\\ \par As usual, the numbers $-c_j$ equal the self intersection numbers of the exceptional divisors $E_j$ in $Y_\Sigma$: Indeed, $D:=\sum_i u^i\,E_i$ is a principal divisor (if you do not like coefficients $u^i$ from $N$, evaluate them by arbitrary elements of $M$); hence, \[ \begin{array}{rcl} 0 \,=\, E_j\cdot D &=& E_j\cdot (u^{j-1} E_{j-1} + u^j E_j + u^{j+1} E_{j+1})\\ &=& u^{j-1}+ (E_j)^2\,u^j + u^{j+1}\\ &=& ( c_j + (E_j)^2 )\cdot u^j \qquad\qquad (j=1,\dots,s). \end{array} \] \par On the other hand, we can use the projection formula to obtain \[ \begin{array}{rcl} -2 \,= \, 2\,g(E_j)-2 &=& K_{\tilde{Y}|Y}\cdot E_j + (E_j)^2\\ &=& \sum_i \,(\alpha_i-1) \,(E_i\cdot E_j) + (E_j)^2\\ &=& (\alpha_{j-1}-1) + (\alpha_j -1)\,(E_j)^2 + (\alpha_{j+1}-1) + (E_j)^2\,, \vspace{-1ex} \end{array} \] hence \[ \alpha_{j-1} + \alpha_{j+1} = c_j\, \alpha_j \qquad (j=1,\dots,s\,;\; \alpha_0,\,\alpha_{s+1}:=1). \vspace{1ex} \] \par Looking at the definition of the $c_j$'s (via relations among the lattice points $u^j$), there has to be some $R\in M_{I\!\!R}$ such that \[ \alpha_j = \langle u^j,\,R\rangle \qquad\qquad (j=0,\dots,s+1). \] The conditions $\langle u^0,\,R\rangle = \alpha_0 =1$ and $\langle u^{s+1},\,R\rangle = \alpha_{s+1} =1$ determine $R$ uniquely. Now, we can see that $\alpha_j$ measures exactly the quotient between the length of the line segment $\overline{0\,u^j}$, on the one hand, and the length of the $\Delta$-part of the line through $0$ and $u^j$, on the other hand. In particular, $\alpha_j<1$ if and only if $u^j$ sits below the line connecting $u^0$ and $u^{s+1}$. \begin{center} \unitlength=0.4mm \linethickness{0.4pt} \begin{picture}(146.00,140.00) \put(10.00,10.00){\line(2,5){52.00}} \put(10.00,10.00){\line(6,1){136.00}} \put(39.00,127.00){\line(5,-6){93.33}} \put(10.00,10.00){\line(3,4){47.33}} \put(10.00,10.00){\line(5,3){98.00}} \put(51.00,112.00){\circle*{3.00}} \put(121.00,28.00){\circle*{3.00}} \put(57.00,72.00){\circle*{3.00}} \put(108.00,68.00){\circle*{3.00}} \put(1.00,10.00){\makebox(0,0)[cc]{$0$}} \put(117.00,20.00){\makebox(0,0)[cc]{\footnotesize $u^0$}} \put(108.00,78.00){\makebox(0,0)[lc]{\scriptsize $u^i$ with $\alpha_i>1$}} \put(57.00,78.00){\makebox(0,0)[cc]{\scriptsize $\alpha_j<1$}} \put(65.00,115.00){\makebox(0,0)[cc]{\footnotesize $u^{s+1}$}} \put(32.00,135.00){\makebox(0,0)[cc]{\scriptsize line $[R=1]$}} \put(102.00,111.00){\makebox(0,0)[cc]{$\sigma$}} \end{picture} \end{center} This explains how to construct the maximal resolution: Start with the minimal one and continue subdividing each small cone $\langle u^j,u^{j+1}\rangle$ into $\langle u^j, u^j+ u^{j+1}\rangle \,\cup \, \langle u^j + u^{j+1}, u^{j+1}\rangle$ as long as it contains interior lattice points below the line $[R=1]$, i.e.\ belonging to $\mbox{int}\, \Delta$. \hfill$\Box$\\ \par {\bf Corollary:} {\em Every P-resolution is toric.}\\ \par {\bf Proof:} P-resolutions are obtained by blowing down curves in the maximal resolution. \hfill$\Box$\\ \par \neu{MR-3} {\bf Example:} We take the example $Y(19,7)$ from \cite{KS}, (3.15). Since $\sigma=\langle (1,0),\, (-7,19) \rangle$, the interior of $\Delta$ is given by the three inequalities \[ y>0\,,\; 19x+7y>0\,, \mbox{ and}\; 19x+8y<19\;\, (\mbox{corresponding to } R=[1,\,8/19])\,. \] The only primitive (i.e.\ generating rays) lattice points contained in $\mbox{int}\,\Delta$ are \[ u^1\!=(0,1)\,,\; u^2\!=(-1,4)\,,\; u^3\!=(-2,7)\,,\; u^4\!=(-1,3)\,,\; u^5\!=(-5,14)\,,\; u^6\!=(-4,11)\,. \] They provide the maximal resolution. The corresponding $\alpha$'s can be obtained by taking the scalar product with $R=[1,\,8/19]$, i.e.\ they are $8/19$, $13/19$, $18/19$, $5/19$, $17/19$, and $12/19$.\\ The minimal resolution uses only the rays through $\,u^1=(0,1)$, $\,u^4=(-1,3)$, and $\,u^6=(-4,11)$, respectively.\\ \par \section{P-Resolutions}\label{P} \neu{P-1} In this section we will speak about {\em partial} toric resolutions $\pi:Y_\Sigma\to Y_\sigma$. Nevertheless, we use the same notation as we did for the maximal resolution: The fan $\Sigma$ subdividing $\sigma$ is genarated by rays through $u^0,\dots,u^s\in\sigma\cap N$; each ray $u^j$ corresponds to an exceptional divisor $E_j\subseteq Y_\Sigma$. However, since $u^{j-1} + u^{j+1}$ need not to be a multiple of $u^j$, the numbers $c_j$ do not make sense anymore.\\ \par {\bf Lemma:} (\cite{toricMori}, (4.3)) {\em For $K:=K_{Y_\Sigma}$ or $K:=K_{Y_\Sigma|Y_\sigma}$ the intersection number $(E_j\cdot K)$ is positive, zero, or negative, if the line segments $\overline{u^{j-1} u^j}$ and $\overline{u^j u^{j+1}}$ form a strict concave, flat, or strict convex ``roof'' over the two cones, respectively. } \begin{center} \unitlength=0.50mm \linethickness{0.4pt} \begin{picture}(231.00,68.50) \put(18.00,66.00){\makebox(0,0)[cc]{$u^{j+1}$}} \put(46.00,19.00){\makebox(0,0)[cc]{$u^{j-1}$}} \put(27.00,8.00){\makebox(0,0)[cc]{\footnotesize $(E_j\cdot K)>0$}} \put(108.00,66.00){\makebox(0,0)[cc]{$u^{j+1}$}} \put(136.00,19.00){\makebox(0,0)[cc]{$u^{j-1}$}} \put(117.00,8.00){\makebox(0,0)[cc]{\footnotesize $(E_j\cdot K)=0$}} \put(198.00,66.00){\makebox(0,0)[cc]{$u^{j+1}$}} \put(226.00,19.00){\makebox(0,0)[cc]{$u^{j-1}$}} \put(180.00,14.00){\makebox(0,0)[cc]{$0$}} \put(207.00,8.00){\makebox(0,0)[cc]{\footnotesize $(E_j\cdot K)<0$}} \put(231.00,60.00){\makebox(0,0)[cc]{$u^j$}} \put(140.00,47.00){\makebox(0,0)[cc]{$u^j$}} \put(34.00,42.00){\makebox(0,0)[cc]{$u^j$}} \put(0.00,14.00){\makebox(0,0)[cc]{$0$}} \put(90.00,14.00){\makebox(0,0)[cc]{$0$}} \put(226.00,29.00){\circle*{3.00}} \put(226.00,54.00){\circle*{3.00}} \put(29.00,67.00){\circle*{3.00}} \put(119.00,67.00){\circle*{3.00}} \put(138.00,27.00){\circle*{3.00}} \put(132.00,42.00){\circle*{3.00}} \put(23.00,35.00){\circle*{3.00}} \put(209.00,67.00){\circle*{3.00}} \put(46.00,29.00){\circle*{3.00}} \put(5.00,20.00){\line(1,2){23.67}} \put(5.00,20.00){\line(6,5){18.00}} \put(5.00,20.00){\line(5,1){41.00}} \put(29.00,67.00){\line(-1,-5){6.33}} \put(23.00,35.00){\line(4,-1){23.00}} \put(95.00,20.00){\line(1,2){23.67}} \put(185.00,20.00){\line(1,2){23.67}} \put(185.00,20.00){\line(5,1){41.00}} \put(209.00,67.00){\line(4,-3){17.00}} \put(226.00,54.33){\line(0,-1){25.33}} \put(185.00,20.00){\line(6,5){41.00}} \put(119.00,67.00){\line(1,-2){19.00}} \put(95.00,20.00){\line(6,1){44.00}} \put(95.00,20.00){\line(5,3){38.00}} \end{picture} \end{center} {\bf Proof:} Using $K:=K_{Y_\Sigma}= -\sum_{i=0}^{s+1}E_i$ (cf.\ \cite{Oda}, (2.1)) we have \[ (E_j\cdot K) = - (E_j\cdot E_{j-1}) -(E_j)^2 - (E_j\cdot E_{j+1})\,. \] On the other hand, as in the proof of Proposition \zitat{MR}{2}, we know that \[ 0 = (E_j\cdot E_{j-1}) \,u^{j-1} + (E_j)^2 \, u^j + (E_j\cdot E_{j+1}) \, u^{j+1}\,. \] Combining both formulas yields the final result \[ (E_j\cdot K)\, u^j = (E_j\cdot E_{j-1}) \,(u^{j-1}-u^j) + (E_j\cdot E_{j+1}) \,(u^{j+1}-u^j)\,. \vspace{-3ex} \] \hspace*{\fill}$\Box$\\ \par {\bf Remark:} The previous lemma together with Proposition \zitat{MR}{2} illustrate again the fact that all P-resolutions (and we just need the fact that the canonical divisor is relatively ample) are dominated by the maximal resolution.\\ \par \neu{P-2} In \cite{Ch-CQS} Christophersen has defined the set \[ K_{e-2}:=\{(k_2,\dots,k_{e-1})\in I\!\!N^{e-2}\,|\; [k_2,\dots,k_{e-1}] \mbox{ is well defined and yields } 0\,\} \] of chains representing zero. To every such chain there are assigned non-negative integers $q_1,\dots,q_e$ characterized by the following mutually equivalent properties: \begin{itemize} \item $q_1=0$, $\,q_2=1$, and $\;q_{i-1} + q_{i+1} = k_i\, q_i \quad (i=2,\dots,e-1)$; \item $q_{e-1}=1$, $\,q_e=0$, and $\;q_{i-1} + q_{i+1} = k_i\, q_i \quad (i=2,\dots,e-1)$; \item $q_e=0$ and $\,[k_i,\dots,k_{e-1}]= q_{i-1}/q_i$ with $\mbox{gcd}(q_{i-1},q_i)=1 \;(i=2,\dots,e-1)$. \end{itemize} (The two latter properties do not even use the fact that the continued fraction $[k_2,\dots,k_{e-1}]$ yields zero.)\\ \par {\bf Remark:} The elements of $K_{e-2}$ correspond one-to-one to triangulations of a (regular) $(e-1)$-gon with vertices $P_2,\dots,P_{e-1},P_\ast$. Then, the numbers $k_i$ tell how many triangles are attached to $P_i$. The numbers $q_i$ have an easy meaning in this language, too.\\ \par Finally, for a given $Y_\sigma$ with embedding dimension $e$, Christophersen defines \[ K(Y_\sigma):=\{(k_2,\dots,k_{e-1})\in K_{e-2}\,|\; k_i\leq a_i\}\,. \vspace{1ex} \] \par {\bf Theorem:} {\em Each P-resolution of $Y_\sigma$ (i.e.\ the corresponding subdivision $\Sigma$ of $\sigma$) is given by some $\underline{k}\in K(Y_\sigma)$ in the following way: \vspace{0.5ex}\\ (1) $\Sigma$ is built from the rays that are orthogonal to $w^i/q_i - w^{i-1}/q_{i-1}\in M_{I\!\!R}$ (for $i=3,\dots,e-1$). In some sense, if the occuring divisions by zero are interpreted well, $\Sigma$ may be seen as dual to the Newton boundary generated by $w^i/q_i\in\sigma^{\scriptscriptstyle\vee}$ ($i=1,\dots,e$). \vspace{0.5ex}\\ (2) The affine lines $[\langle \bullet,\, w^i\rangle = q_i]$ form the ``roofs'' of the $\Sigma$-cones. In particular, the (possibly degenerate) cones $\tau^i\in\Sigma$ correspond to the elements $w^1,\dots,w^e\in E$ The ``roof'' over the cone $\tau^i$ has length $\ell_i:= (a_i-k_i)\,q_i$ (the lattice structure $M\subseteq M_{I\!\!R}$ induces a metric on rational lines). In particular, $\tau^i$ is degenerated if and only if $k_i=a_i$. The Milnor number of the T-singularity $Y_{\tau^i}$ equals $(a_i-k_i-1)$. }\\ \par \neu{P-3} {\bf Proof:} According to the notation introduced in \zitat{P}{1}, the fan $\Sigma$ consists of (non-degenerate) cones $\tau^j:=\langle u^{j-1}, u^j \rangle$ with $j=1,\dots,s+1$. (Except $u^0=(1,0)$ and $u^s=(-q,n)$, their generators $u^j$ are primitive lattice points (i.e.\ $\in N$) contained in $\mbox{int}\Delta\subseteq \sigma$.) \vspace{1ex}\\ {\em Step 1: \quad For each $\tau^j$ there are $w\in E, d\inI\!\!N$ such that $\langle u^{j-1}, w \rangle = \langle u^j, w \rangle = d$.}\\ First, it is very clear that there are a primitive lattice point $w\in M$ and a non-negative number $d\inI\!\!R_{\geq 0}$ admitting the desired properties. Moreover, since $u^j\in N$, $d$ has to be an integer, and Reid's Lemma \zitat{P}{1} tells us that $w\in \sigma^{\scriptscriptstyle\vee}$. It remains to show that $w$ belongs even to the Hilbert basis $E\subseteq \sigma^{\scriptscriptstyle\vee}\cap M$.\\ Denote by $\ell$ the length of the line segment $\overline{u^{j-1}u^j}$ on the ``roof'' line $[\langle \bullet,\,w\rangle =d]$. Since $\tau^j$ represents a T-singularity, we know from (7.3) of \cite{Homog} (cf.\ \zitat{I}{1} of the present paper) that $d|\ell$. In particular, $\overline{u^{j-1}u^j}$ contains the $d$-th multiple $d\cdot u$ of some lattice point $u\in\tau^j\cap M$ (w.l.o.g.\ not belonging to the boundary of $\sigma$). Hence, $\langle u,\, w\rangle =1$ and $u\in \mbox{int}\,\sigma\cap M$, and this implies $w\in E$.\ \vspace{1ex}\\ {\em Step 2:} Knowing that each of the cones $\tau^1,\dots,\tau^{s+1}\in\Sigma$ is assigned to some element $w\in E$, a slight adaption of the notation (a renumbering) seems to be very useful: Let $\tau^i=\langle u^{i-1},\, u^i\rangle$ be the cone assigned to $w^i\in E$, and denote by $d_i,\, \ell_i$ the height and the length of its ``roof'' $\overline{u^{i-1}u^i}$, respectively. Some of these cones might be degenerated, i.e.\ $\ell_i=0$. This it at least true for the extremal $\tau^1$ and $\tau^e$ coinciding with the two rays spanning $\sigma$. Here we have even $d_1=d_e=0$; in particular $u^0=u^1=(1,0)$ and $u^{e-1}=u^e=(-q,n)$.\\ Since $d_i|\ell_i$, we may introduce integers $k_i\leq a_i$ yielding $\ell_i=(a_i-k_i)\,d_i$. For $i=2,\dots,e-1$ they are even uniquely determined. \vspace{1ex}\\ {\em Step 3:} Using the following three ingrediences \begin{itemize} \item[(i)] $\langle u^{i-1}, \, w^i\rangle = \langle u^i, \, w^i\rangle = d_i\quad (i=1,\dots,e)\,$, \item[(ii)] $w^{i-1} + w^{i+1} = a_i\, w^i\quad (i=2,\dots,e-1;\;$ cf.\ \zitat{CQS}{3}), and \item[(iii)] $\langle u^i - u^{i-1},\, w^{i-1} \rangle = \ell_i = (a_i-k_i)\,d_i$ (since $\{w^{i-1},w^i\}$ forms a $Z\!\!\!Z$-basis of $M$), \end{itemize} we obtain \[ \begin{array}{rcl} d_{i-1} + d_{i+1} &=& (a_i\,d_i + d_{i-1}) - (a_i\,d_i - d_{i+1})\\ &=& (a_i\,d_i + \langle u^{i-1}, w^{i-1} \rangle) - \langle u^i, \, a_i\, w^i- w^{i+1} \rangle\\ &=& a_i\,d_i + \langle u^{i-1}, w^{i-1} \rangle - \langle u^i, w^{i-1} \rangle\\ &=& a_i\,d_i + \langle u^{i-1}-u^i, \,w^{i-1} \rangle\\ &=& a_i\,d_i - (a_i-k_i)\,d_i \; = \; k_i\,d_i\quad (\mbox{for } i=2,\dots,e-1)\,. \end{array} \] In particular, $k_i\geq 0$ (and even $\geq 1$ for $e>3$). Moreover, since $\{w^{i-1},w^i\}$ forms a basis of $M$ and $u^{i-1}\in N$ is primitive, we have $\mbox{gcd}(d_{i-1},d_i)=1$. Hence, $d_i=q_i$ (both series of integers satisfy the second of the three properties mentioned in the beginning of \zitat{P}{2}). Finally, the third of these properties yields $[k_2,\dots,k_{e-1}]=q_1/q_2 = d_1/d_2 = 0$, i.e.\ $\underline{k}\in K_{e-2}$. \vspace{1ex}\\ The reversed direction (i.e.\ the fact that each $K(Y_\sigma)$-element indeed yields a P-resolution) follows from the above calculations in a similar manner. \hfill$\Box$\\ \par {\bf Remark:} Subdividing each $\tau^i$ further into $(a_i-k_i)$ equal cones (with ``roof'' length $q_i$ each) yields the so-called M-resolution (cf.\ \cite{M}) assigned to a P-resolution. It is defined to contain only T$_0$-singularities (i.e.\ T-singularities with Milnor number 0); in exchange, $K_{\tilde{Y}|Y}$ does not need to be relatively ample anymore. This property is replaced by ``relatively nef''.\\ \par {\bf Examples:} (1) The continued fraction $[1,2,2,\dots,2,1]=0$ yields $q_1=q_e=0$ and $q_i=1$ otherwise. In particular, the ``roof'' lines equal $[\langle \bullet,\,w^i\rangle =1]$ (for $i=2,\dots,e-1$) describing the RDP-resolution of $Y_\sigma$. The assigned M-resolution equals the minimal resolution mentioned at the end of \zitat{CQS}{3}. \vspace{1ex}\\ (2) Let us return to Example \zitat{MR}{3}: The embedding dimension $e$ of $Y_\sigma$ is $6$, the vector $(a_2,\dots,a_{e-1})$ equals $(2,3,2,3)$, and, except the trivial RDP element mentioned in (1), $K(Y_\sigma)$ contains only $(1,3,1,2)$ and $(2,2,1,3)$.\\ In both cases we already know that $q_1= q_6=0$ and $q_2=q_5=1$. The remaining values are given by the equation $q_3/q_4=[k_4,k_5]$, i.e.\ we obtain $q_3=1$, $q_4=2$ or $q_3=2$, $q_4=3$, respectively. \vspace{0.5ex}\\ Hence, in case of $(1,3,1,2)$ the fan $\Sigma$ is given by the additional rays through $(0,1)$ and $(-4,11)$. For $\underline{k}=(2,2,1,3)$ we need the only one through $(-1,4)$.\\ \par

9602

alg-geom/9602002

en

https://arxiv.org/abs/alg-geom/9602002

alg-geom/9602002

Israel Vainsencher

Israel Vainsencher

Flatness of families induced by hypersurfaces on flag varieties

13 pages, LaTeX

We answer a question posed by S. Kleiman concerning flatness of the family of complete quadrics. We also show that any flat family of hypersurfaces on Grassmann varieties induces a flat family of intersections with the corresponding flag variety.

alg-geom

\section*{Introduction} Let ~${\bf S}$~ be the variety of complete quadrics, ~${\s^{nd}}$~ the open subset of nondegenerate quadrics and ~${\ff{}}$~ the scheme of complete flags in ~$\p n$. Let ~$\raise2pt\hbox{$\varphi$}_0:{\s^{nd}}\mbox{${\,\rightarrow\,}$} \hbox{\bf Hilb\hskip1pt}({\ff{}})$~ be the morphism that assigns to each non\-degen\-erate quadric the locus of its tangent flags. We prove the following. {\bf Theorem. } \it ~$\raise2pt\hbox{$\varphi$}_0$~ extends to a morphism {}~$\raise2pt\hbox{$\varphi$}:{\bf S} \mbox{${\,\rightarrow\,}$} \hbox{\bf Hilb\hskip1pt}(\ff{})$\rm. This answers affirmatively a question S. Kleiman asked in ([K], p.362). We first show that ~${{\bf S}}$~ parametrizes a flat family that restricts, over ~${\s^{nd}}$, to the family of the graphs of the Gauss map (point $\mapsto$ tangent hyperplane) of nondegenerate quadric hypersurfaces. The family pertinent to Kleiman's question is obtained by composing the family of graphs with the appropriate flag bundle (point $\in$ line $\subset \dots \subset $ hyperplane). Our proof of flatness for the completed family of graphs relies on Laksov's description [L] of Semple--Tyrrell's ``standard'' affine open cover of {\bf S}. The space of complete conics has recently reappeared as a simple instance of Kontsevich's spaces of stable maps (cf. [P]). It is also instrumental for the counting of rational curves on a K3 surface double cover of the plane (cf. [V]). Complete quadric surfaces play a role in Narasimhan--Trautmann [NT] study of a compactification of a space of instanton bundles. We also show that \em any flat family of hypersurfaces on Grassmann varieties induces a flat family of subschemes of the corresponding flag variety \rm (cf. \rf{prop}). This statement was first obtained as an earlier attempt to answer Kleiman's question. We observe that for the case of quadric hypersurfaces the family described in the proposition does not induce the family of tangent flags. In fact, for conics it yields a double structure on the graph of the Gauss map.(cf. \S\ref{fim} for details). \section{The tangent flag to a smooth quadric} Write $x =(x_1,\dots,x_{n+1})$ (resp. $y =(y_1,\dots,y_{n+1})$) for the vector of homogeneous coordinates in $\p n$ (resp. $\pd n$ ). Let {}~$\ff{0,n-1}\subset\p n\times\pd n$~ be the incidence correspondence ``point $\in$ hyperplane''. It is the zeros of the incidence section {}~$x\cdot{}y$~ of $\mbox{${\cal O}$}_{\p n}(1)\otimes\mbox{${\cal O}$}_{\pd{\,\, n}}(1)$. Let ~$\mbox{\large$\kappa$}\subset\p n$~ denote a smooth quadric represented by a symmetric matrix $a$. The Gauss map ~$\gamma:\mbox{\large$\kappa$}\mbox{${\,\rightarrow\,}$}\pd n$~ is given by $x\mapsto y=x\cdot a$. Hence we have $$ \gamma^*(\mbox{${\cal O}$}_{\pd{\,\, n}}(1))=\mbox{${\cal O}$}_{\p n}(1)_{|\mbox{\large$\kappa$}}. $$ The tangent flag ~$\widetilde{\kp}\subset\ff n$~ of ~$\mbox{\large$\kappa$}$~ is equal to the restriction of the flag bundle $$ \ff n\mbox{${\,\rightarrow\,}$}\ff{0,n-1}\subset\p n\times\pd n \vspace{-1pt} $$ over the graph ~${\bf\Gamma}_{\mbox{$\kappa$}}$~ of ~$\gamma$. Consequently, flatness of the family ~$\{\widetilde{\kp}\}$~ of tangent flags is equivalent to flatness of the family of graphs ~$\{{\bf\Gamma}_{\mbox{$\kappa$}}\}$. The latter will be handled in \S\ref{graphs}. We proceed to compute the Hilbert polynomial of the graph ${\bf\Gamma}$ of the Gauss map of a general quadric hypersurface ~$\mbox{\large$\kappa$}\subset\p n$. \vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx} Notation as above, the Hilbert polynomial {}~$\mbox{\raise3pt\hbox{$\chi$}}\big(\mbox{${\cal O}$}_{{\bf\Gamma}}(\mbox{${\cal L}$}^{\otimes t})\big)$~ with respect to $$ \mbox{${\cal L}$}=\big(\mbox{${\cal O}$}_{\p n}(1)\otimes{}\mbox{${\cal O}$}_{\pd{\, n}}(1)\big)_{|{\bf\Gamma}} $$ is equal to $$ {2\,t+n\choose n}-{2(t-1)+n\choose n}. $$ \label{hilbgen}\el \vspace{-15pt}\vskip10pt\noindent{\bf Proof.\hskip10pt} We have $\mbox{${\cal L}$}\cong\mbox{${\cal O}$}_{\p n}(2)_{|\mbox{\large$\kappa$}}$ under the identification ${\bf\Gamma}\cong\mbox{\large$\kappa$}$. Thus we may compute $$ \ba{cl}\mbox{\raise3pt\hbox{$\chi$}}(\mbox{${\cal L}$}^{\otimes t}) &= \mbox{\raise3pt\hbox{$\chi$}}\big(\mbox{${\cal O}$}_{\p n}(2t)\big)_{|\mbox{\large$\kappa$}}\\ \noalign{\vskip5pt} &=\mbox{\raise3pt\hbox{$\chi$}}\big(\mbox{${\cal O}$}_{\p n}(2t)\big) - \mbox{\raise3pt\hbox{$\chi$}}\big(\mbox{${\cal O}$}_{\p n}(2t-2)\big)\\ \noalign{\vskip7pt} &={2\,t+n\choose n}-{2(t-1)+n\choose n}. \end{array} $$\vspace{-30pt}\vskip-7pt\phantom{|}\hfill\mbox{$\Box$} \section{Hilbert polynomial of loci of rank 1 matrices} The image of the Segre imbedding $\p n\times\p n\mbox{${\,\rightarrow\,}$}\p N$ is the variety of matrices of rank one. The image $\Delta$ of the diagonal $\p n \mbox{${\,\rightarrow\,}$} \p n \times\p n\mbox{${\,\rightarrow\,}$}\p N$ is the subvariety of $symmetric$ matrices of rank one. It's Hilbert polynomial is easily found to be $$ \hbox{dim\hskip2pt}(H^0(\Delta,\mbox{${\cal O}$}_{\p N}(t))~=~{2t+n\choose n}. $$ The bi-homogeneous ideal $I_{\Delta}$ of the diagonal is generated by the 2$\times2$ minors of the matrix \begin{equation}}\def\ee{\end{equation}\label{2x2} \left [\ba{cccc} x_{{1}}&x_{{2}}&\dots&x_{{n+1}}\\\noalign{\medskip} y_{{1}}&y_{{2}}&\dots&y_{{n+1}} \end{array}\right]. \ee Write $$ S=k[x_1,\dots, x_{{n+1}},y_1,\dots ,y_{{n+1}}] $$ for the polynomial ring in $2n+2$ variables, and let $S_{i,j}$ denote the space of bi-hom\-oge\-neous polynomials of bi-degree $(i,j)$. We have $$ \hbox{dim\hskip2pt}_k\,S_{t,t}\big/(I_{\Delta})_{t,t}~=~{2t+n\choose n}. $$ Quite generally, for a closed subscheme $X\subseteq \p m\times\p n$ defined by a bi-homogeneous ideal $I\subseteq S$ we have (cf. [KTB], p. 189) $$ H^0\big(X,\mbox{${\cal O}$}_{\p m}(t)\otimes{}\mbox{${\cal O}$}_{\p { n}}(t)_{|X}\big)~=~ S_{t,t}\big/(I)_{t,t}~~\hbox{ for all }~t >>0. $$ Indeed, the homomorphism \begin{equation}}\def\ee{\end{equation}\label{rs}\ba{ccc} R=k[z_{i,j}]\big/\id{z_{i,j}z_{r,s}-z_{r,j}z_{i,s}}&\mbox{${\,\longrightarrow\,}$} & S\\ z_{i,j}&\mapstochar\longrightarrow& x_iy_j \end{array}\ee maps $R_t$ isomorphically onto $S_{t,t}$. Let the bi-homogeneous ideal $I\subseteq S$ be generated by polynomials of some fixed bidegree ($d,d$). Its inverse image via \rf{rs} generates a homogeneous ideal $I^\#\subseteq R$. We have $$ (R/I^\#)_t\widetilde{\mbox{${\,\longrightarrow\,}$}}S_{t,t}/I_{t,t}. $$ Now for $t>>0$ we may write $$ (R/I^\#)_t~=~H^0\big(X,\mbox{${\cal O}$}_{\p N}(t)_{|X}\big)~=~ H^0\big(X,\mbox{${\cal O}$}_{\p m}(t)\otimes{}\mbox{${\cal O}$}_{\p {n}}(t)_{|X}\big). $$ Let $L(I)$ denote the monomial ideal of initial terms of $I$ with respect to some bi-graded monomial order. Then we have the equality of Hilbert functions, $$ \raise2pt\hbox{$\varphi$}_I(i,j)~=~\raise2pt\hbox{$\varphi$}_{L(I)}(i,j). $$ This is rather standard: let $f_1,\dots,f_k$ be linearly independent forms of bidegree $(i,j)$ in $I$. Replacing if needed each $f_\mu$ by $f_\mu-cf_\nu$ for suitable $c\in k$, we may assume their initial terms $L\,f_\mu \ne L\,f_\nu $. Hence the initial terms $Lf_1, \dots, Lf_k$ are linearly independent monomials in $L(I)_{i,j}$. This shows that $ \raise2pt\hbox{$\varphi$}_I(i,j) \leq \raise2pt\hbox{$\varphi$}_{L(I)}(i,j). $ Conversely, pick monomials \,$g_1> \dots> g_k$\, in $L(I)_{i,j}$. We have each $g_\mu = Lf_\mu$ for some $f_\mu \in I_{i,j}$. It follows that $f_1, \dots, f_k$ are linearly independent. \vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx}\label{k0} Let ~${\bf\Gamma}_0$~ be the subscheme of $\p n\times\pd n$ defined by the ideal $$ \id{x_iy_j\mbox{$\,|\,$} 1\leq i<j\leq n+1}+\id{\sum x_iy_i}. $$Then we have $$ \raise2pt\hbox{$\varphi$}_{{\bf\Gamma}_0}(t)={2\,t+n\choose n}-{2(t-1)+n\choose n}. $$ \el \vskip10pt\noindent{\bf Proof.\hskip10pt} The whole point is to notice that the $ x_iy_j$ span the ideal of initial terms of $I_\Delta$ with respect to a suitable order.\footnote {I'm indebted to P. Gimenez for his precious help on this matter.} ~In fact, the set of $2\times2$ minors of \rf{2x2} is known to be a (universal) Gr\"obner basis for $I_\Delta$ (see Sturmfel [BS]). By the above discussion, we may write $$ \raise2pt\hbox{$\varphi$}_{L(I_\Delta)}(t)=\raise2pt\hbox{$\varphi$}_{I_\Delta}(t)={2t+n\choose n}. $$ One checks at once that $\sum x_iy_i$ is a nonzero divisor mod $L(I_\Delta)$ (see \rf{rmk} (i)). Therefore $$ \raise2pt\hbox{$\varphi$}_{{\bf\Gamma}_0}(t)=\raise2pt\hbox{$\varphi$}_{L(I_\Delta)}(t)-\raise2pt\hbox{$\varphi$}_{L(I_\Delta)}(t-1).\vspace{-10pt} $$\phantom{|}\hfill\mbox{$\Box$} We will deduce flatness for the ``completed'' family of Gauss maps from the fact that the above Hilbert polynomial at the special point {}~${\bf\Gamma}_0$~ coincides with the generic one \rf{hilbgen}. \section{Semple-Tyrrell-Laksov cover} Let ${\bf U}_n$ denote the group of lower triangular unipotent ($n+1$)-matrices. Thus, ${\bf U}_n$ is isomorphic to the affine space $\af{n(n+1)/2}$ with coordinate functions ~$u_{i,j},\,1\leq j\leq i-1,\,i=2\dots n+1$. These are thought of as entries of the matrix, $$ u~=~\left [\begin {array}{lllll} 1&0&0&\cdots&0\\ \noalign{\medskip}u_{{2,1}}&1&0&\cdots&0\\ \noalign{\medskip}u_{{3,1}}&u_{{3,2}}&1&\cdots&0\\ \noalign{\medskip}\cdots&\cdots&\cdots&\cdots&\cdots\\ u_{{n+1,1}}&u_{{n+1,2}}&u_{{n+1,3}}&u_{{n+1,n}}&1 \end {array}\right ]. $$ Let $d_1,\dots,d_n$ be coordinate functions in $\af n$. Put \begin{equation}}\def\ee{\end{equation}\label{d1} d^{(1)}=\left [\begin {array}{lllll} 1&0&0&\cdots &0\\ \noalign{\medskip}0&d_{{1}}&0&\cdots &0\\ \noalign{\medskip}0&0&d_{{1}}d_{{2}}&\cdots &0\\ \noalign{\medskip}\cdots&\cdots&\cdots&\cdots&\cdots\\ 0&0&0&\cdots &d_{{1}}d_{{2}}\cdots d_{{n}} \end {array}\right ]. \ee For a matrix $A$ let it's $i$th adjugate be the matrix $\wed i A$ of all $i\times i$ minors. We denote by $d^{(i)} $ the matrix obtained from $\wed i d^{(1)}$ by removing the common factor $d_1^{i-1}d_2^{i-2}\cdots d_{i-1}$. $E.g.,$ for $n=3$ we have $$\ba{ll} d^{(1)} &= {\rm diag}(1,\,d_1,\,d_1d_2,\, d_1d_2 d_3)\\ \noalign{\medskip}d^{(2)} &= {\rm diag}(d_1,\,d_1d_2,\,d_1d_2d_3,\,d_1^2d_2,\,d_1^2d_2d_3,\,d_1^2d_2^2d_3) \big/(d_1) \\ &= {\rm diag}(1,\, d_2,\, d_2d_3,\, d_1d_2,\, d_1d_2d_3,\, d_1d_2^2d_3) \\\noalign{\medskip} d^{(3)} &= {\rm diag}(1 ,\, d_3 ,\,d_2d_3 ,\,d_1d_2d_3). \end{array}$$ The map ~$ {\bf U}_n\times\af n\mbox{${\,\rightarrow\,}$} {\bf S}\subset{ \prod_{i=1}^{i=n}\ps{ S_2(\bigwedge\hskip-8pt\raise6pt\hbox{$^i$}\,\,k^{n+1*})}}$~ defined by sending $(u,\,d)$ to $$ \big(u\,d^{(1)}\,u^t,\,(\wed2\,u)\,d^{(2)}\,\wed2u^t,\dots,(\wed nu)\,d^{(n)}\, \wed nu^t\big) $$ is an isomorphism onto an affine open subset ${\bf S}^0$ of {\bf S}. The variety of complete quadrics may be covered by translates of ${\bf S}^0$ (cf. Laksov [L]). Let ${\bf S}^0_d \cong{\bf U}_n\times\af n\!_{d}$ be the principal open piece defined by $d_1d_2\cdots d_n\ne0$. It maps isomorphically onto an open subvariety of $\s^{nd}$. \section{Graph of the Gauss map}\label{graphs} The variety ~$\s^{nd}$~ of nondegenerate quadrics parametrizes a flat family of graphs of Gauss maps. For a nondegenerate quadric represented by a symmetric matrix $a\in\s^{nd}$ the Gauss map is given by $x\mapsto y=x\cdot a$. We define $\mbox{\hbox{\rm I\hskip-2pt K}}^{nd}\subset\s^{nd}\mbox{$\times$}\p n\mbox{$\times$}\pd n$ by the bi-homogeneous ideal generated by the incidence relation {}~$x\cdot y$~ together with the 2$\times$2 minors of the 2$\times(n+1)$ matrix with rows ~$y,\,x\cdot z$,~ where \,$z$\, denotes the generic symmetric matrix. Clearly ~$\mbox{\hbox{\rm I\hskip-2pt K}}^{nd}\mbox{${\,\rightarrow\,}$}\s^{nd}$ is a map of ${\bf GL}_{n+1}-$homogeneous spaces. Now write $a = vc^{(1)}v^t$ with $v\in{\bf U}_n,\, c\in\af n\!_d$ \,($c^{(1)}$\, as in \rf{d1}), and put $x'=xv$, $y'=y(v^{-1})^t$. We have $y=xa$ iff $y'= x'c^{(1)}$. Let \begin{equation}}\def\ee{\end{equation} {\mbox{\hbox{\rm I\hskip-2pt K}}}^0_d ~\subset~ {\bf S}^0_d \times\p n\times\pd n. \label{g0d} \ee be defined by ~$x\cdot y$~ together with the 2$\times$2 minors of the 2$\times(n+1)$ matrix \begin{equation}}\def\ee{\end{equation}\label{x'd} \left [\begin {array}{ccccc} x'_{{1}}&d_1x'_{{2}}&d_1d_2x'_{{3}}&\dots& d_1\cdots d_nx'_{{n+1}}\\ \noalign{\medskip}y'_{{1}}&y'_{{2}}&y'_{{3}}&\dots&y'_{{n+1}} \end {array}\right ] \ee where we put $x'_j=\sum_iu_{ij}x_i$ and likewise $y'_j$ denotes the $j$th entry of $y(u^{-1})^t$. Thus ${\mbox{\hbox{\rm I\hskip-2pt K}}}^0_d$ is the total space of the family of Gauss maps parametrized by ${\bf S}^0_d$. Note that ${\mbox{\hbox{\rm I\hskip-2pt K}}}^0_d \mbox{${\,\rightarrow\,}$} {\bf S}^0_d$ is a smooth quadric bundle. Its fibre over $(I,(1,\dots,1)) \in {\bf U}_n \times \af n_{d}$ is equal to the quadric given by \,$\sum x_i^2$\, inside the ``diagonal'' \,$y_1=x_1, \dots,y_{{n+1}}= x_{{n+1}}$\, of ~$\p n\times\pd n.$ Let \begin{equation}}\def\ee{\end{equation}\label{gg0} {\mbox{\hbox{\rm I\hskip-2pt K}}}^0 ~\subset~ {\bf S}^0 \times\p n\times\pd n \ee be defined by ~$x\cdot y$~ together with the ideal \begin{equation}}\def\ee{\end{equation}\label{J} \ba{cl}J=&\langle{}x'_1y'_2-d_1y'_1x'_2,\dots,\,x'_1y'_{n+1}- d_1\cdots{}d_ny'_1x'_{n+1},\\ &\,\,\,x'_2y'_3-d_2y'_2x'_3,\dots,\,x'_ny'_{n+1}-d_ny'_nx'_{n+1} \rangle \end{array}\ee obtained by cancelling all $d_i$ factors occurring in the above 2$\times$2 minors. We obviously have {}~$ {\mbox{\hbox{\rm I\hskip-2pt K}}}^0_{\,|\,{\bf S}^0_d} = {\mbox{\hbox{\rm I\hskip-2pt K}}}^0_d$. We will show that ~$ {\mbox{\hbox{\rm I\hskip-2pt K}}}^0$~ is the scheme theoretic closure of ~$ {\mbox{\hbox{\rm I\hskip-2pt K}}}^0_d$ in ${\bf S}^0 \mbox{$\times$} \p n\mbox{$\times$}\pd n$ (cf. \rf{clos}). \section{A torus action} Notation as in \rf{d1}, imbed ~\gm~ in ${\bf GL}_{n+1}$ by sending $c=(c_1,\dots,c_n)\in\gm$ to $c^{(1)}={\rm diag}(1,\,c_1,\,c_1c_2,\dots)$. We let ~\gm~ act on ${\bf S}^0$ by $$ c\cdot (v,b) = (c^{(1)}\, v\, (c^{(1)})^{-1},~(c_1^2b_1, \dots c_n^2b_n)). $$ This action is compatible with the natural action of ${\bf GL}_{n+1}$ on the space $\ps{S_2(k^{n+1*})}$ of quadrics, $i.e.,$ for a symmetric matrix ~$a(v,b)\, :=\, v\, b^{(1)}\, v^t$~ as above, we have $$ \ba{cl} c^{(1)}\cdot{}a(v,b)&=c^{(1)}\, a(v,b)\,(c^{(1)})^t = c^{(1)} \,v\,b^{(1)}\,v^t \,(c^{(1)})^t \\\noalign{\medskip}&=c^{(1)}\,v\,(c^{(1)})^{-1}\,c^{(1)}\,b^{(1)} \,c^{(1)}\,((c^{(1)})^t)^{-1}\,v^t\, (c^{(1)})^t \\\noalign{\medskip}&=c^{(1)}\,v\,(c^{(1)})^{-1}\,(c^{(1)})^2\,b^{(1)} \,((c^{(1)})^t)^{-1}\,v^t\,(c^{(1)})^t \\\noalign{\medskip}&=a(c\cdot(v,b))\,. \end{array}$$ It can be also easily checked that ~\gm~ acts compatibly on ${\bf S}^0\times\p n\times\pd n$ and ${\mbox{\hbox{\rm I\hskip-2pt K}}}^0$ is invariant. Indeed, let $((v,b),x,y)\in{\mbox{\hbox{\rm I\hskip-2pt K}}}^0$. Pick $c\in\gm$. We have $$ c\cdot((v,b),x,y)=( (c^{(1)}\, v\, (c^{(1)})^{-1},~(c_1^2b_1, \dots c_n^2b_n)),~x\,(c^{(1)})^{-1}, \,y\,(c^{(1)})^{t}). $$ Now $x'=xv$ changes to $$ x'' ~=~ (x\, (c^{(1)})^{-1})\, (c^{(1)}\, v\, (c^{(1)})^{-1}) ~=~ x\,v\, (c^{(1)})^{-1} ~=~ x'\, (c^{(1)})^{-1} $$ so that the first row $x'\,b^{(1)}$ in \rf{x'd} (evaluated at $((v,b),x,y)$) changes to $$ x''\,(b^{(1)}\,(c^{(1)})^2) ~=~ x'\, (c^{(1)})^{-1} \,(b^{(1)}\,(c^{(1)})^2) ~=~ x'\, (b^{(1)}\, c^{(1)}). $$ Similarly, $y' ~=~ y\,(v^{-1})^t$ changes to $$ y'' ~=~ (y\,(c^{(1)})^{t})\, ((c^{(1)}\, v\, (c^{(1)})^{-1})^{-1})^t ~=~ y\,(v^{-1})^t\,(c^{(1)})^{t}) ~=~ y'\, c^{(1)}. $$ Therefore \rf{x'd} changes to the matrix with rows $x'\, (b^{(1)}\, c^{(1)})$ and $y'\, c^{(1)}$. Thus evaluation of \rf{J} at $c\cdot((v,b),x,y)$ and at $((v,b),x,y)$ differ only by nonzero multiples. \vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx} The orbit of $(I,0)\in{\bf S}^0$ is the unique closed orbit. \el \vskip10pt\noindent{\bf Proof.\hskip10pt} Conjugation of \,$v\in {\bf U}_n$\, by the diagonal matrix \,$c^{(1)}$\, replaces each entry \,$v_{ij},\,j<i$\, by $$ \ba{cl}(c^{(1)}\,v\,(c^{(1)})^{-1})_{ij} &= c^{(1)}_{ii}\,(v\,(c^{(1)})^{-1})_{ij} = c^{(1)}_{ii}\, v_{ij}\,((c^{(1)})^{-1})_{jj}\\ &= v_{ij}\,c^{(1)}_{ii}/c^{(1)}_{jj}= v_{ij}\,\,c_{i-1}\cdots{}c_{j}. \end{array}$$ Thus, letting \,$c\ar0$, we see that ($I,0$) is in the orbit closure {}~$\overline{\gm\cdot(v,b)}$ \vspace{-10pt}\phantom{|}\hfill\mbox{$\Box$} \section{Proof of the theorem} \vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx}\label{cc0} Notation as in \rf{gg0} , the family ~${\mbox{\hbox{\rm I\hskip-2pt K}}}^0\mbox{${\,\rightarrow\,}$}{\bf S}^0$~ is flat. \el \vskip10pt\noindent{\bf Proof.\hskip10pt} Since ~${\mbox{\hbox{\rm I\hskip-2pt K}}}^0 \mbox{${\,\rightarrow\,}$} {\bf S}^0$~ is equivariant for the ~$\gm-$action, it suffices to check that the Hilbert polynomial of the fiber over the representative $(I,0)$ of the unique closed orbit is right, $i.e.,$ coincides with the generic one (cf. Hartshorne [H], thm.9.9, p.261). Evaluating \rf{J} at $(I,0)$ yields the monomial ideal \rf{2x2}. We are done by virtue of \rf{hilbgen} and \rf{k0}.\vspace{-25pt} \phantom{|}\hfill\mbox{$\Box$} \vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx} Let $f:X\mbox{${\,\rightarrow\,}$} Y$ be a flat, surjective morphism of schemes. If $U\subseteq Y$ is open and schematically dense in $Y$ then $f^{-1}U$ is open and schematically dense in $X.$ \el \vskip10pt\noindent{\bf Proof.\hskip10pt} We may assume $X,\,Y$ affine. Let $A\subseteq B$ be a flat ring extension and let $a\in A$ be such that ${\rm Spec}\,A_a$ is schematically dense in ${\rm Spec}\,A$. This means that every element in ker\,($A\mbox{${\,\rightarrow\,}$} A_a$) is nilpotent. Flatness implies ker\,($B\mbox{${\,\rightarrow\,}$} B_a$)=ker\,($A\mbox{${\,\rightarrow\,}$} A_a)\otimes{}B$. Hence ${\rm Spec}\,B_a$ is schematically dense in ${\rm Spec}\,B$. \vspace{-25pt} \phantom{|}\hfill\mbox{$\Box$} \vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx}\label{clos} Notation as in \rf{gg0} and \rf{g0d}, we have that ~$\mbox{\hbox{\rm I\hskip-2pt K}}^0$~ is equal to the scheme theoretic closure of ~$\mbox{\hbox{\rm I\hskip-2pt K}}^0_d$. \el \vskip10pt\noindent{\bf Proof.\hskip10pt} In view of \rf{cc0}, we may apply the previous lemma to ~$\mbox{\hbox{\rm I\hskip-2pt K}}^0\mbox{${\,\rightarrow\,}$}{\bf S}^0\supset{\bf S}^0_d.$ \vspace{-25pt} \phantom{|}\hfill\mbox{$\Box$} \vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx} Let $G$ be an algebraic group and let $$ \ba{ccc}X^0 &\subset &X \\ \downarrow~&&\downarrow\\ Y^0&\subset &Y \end{array}$$ be a commutative diagram of maps of ~$G-$varieties. Let \,$\overline{X}$,~ $\overline{Y}$ denote the closures of $X^0,\,Y^0.$ If ~$\overline{X}\mbox{${\,\rightarrow\,}$}\overline{Y}$~ is flat over a neighborhood of a point in each closed orbit then {}~$\overline{X}\mbox{${\,\rightarrow\,}$}\overline{Y}$~ is flat. \el \vskip10pt\noindent{\bf Proof.\hskip10pt} Immediate. \phantom{|}\hfill\mbox{$\Box$} We may now finish the proof of the theorem. Let ~$\mbox{\hbox{\rm I\hskip-2pt K}}\subset{\bf S}\mbox{$\times$}\p n\mbox{$\times$}\pd n$~ be the scheme theoretic closure of ~$\mbox{\hbox{\rm I\hskip-2pt K}}^0$. We have {}~$\mbox{\hbox{\rm I\hskip-2pt K}}\cap\big({\bf S}^0\mbox{$\times$}\p n\mbox{$\times$}\pd n\big) = \mbox{\hbox{\rm I\hskip-2pt K}}^0$ flat over ~${\bf S}^0$~ by \rf{cc0}. The latter is a neighborhood of a point in the unique closed orbit of ~${\bf S}$. Now apply the previous lemma to ~$G={\bf GL}_{n+1}$, $X={\bf S}\mbox{$\times$}\p n\mbox{$\times$}\pd n$, $Y={\bf S}$, $Y^0={\bf S}^{nd}$, $X^0=\mbox{\hbox{\rm I\hskip-2pt K}}^{nd}$. Finally, since the family of tangent flags is defined by the fibre square, $$ \ba{ccc} \widetilde{\mbox{\hbox{\rm I\hskip-2pt K}}}& \mbox{${\,\longrightarrow\,}$} &\ff n \, \mbox{$\times$} \,{\bf S} \\ \downarrow& &\downarrow\\ \mbox{\hbox{\rm I\hskip-2pt K}} & \mbox{${\,\longrightarrow\,}$} &\ff{0,n-1}\,\mbox{$\times$}\,{\bf S} \end{array} $$ the composition ~$\widetilde{\mbox{\hbox{\rm I\hskip-2pt K}}}\mbox{${\,\rightarrow\,}$}\mbox{\hbox{\rm I\hskip-2pt K}}\mbox{${\,\rightarrow\,}$}{\bf S}$~ is flat. \section{Final remarks}\label{fim} \begin{exx}\em \label{rmk} (i) The primary decomposition of the monomial ideal in \rf{k0} can be checked to be given by $$ \id{x_1,\,x_2,\dots, x_n} \cap \cdots \cap \id{x_1,\dots, x_i,\,y_{i+2}, \dots, y_{n+1}} \cap \cdots \cap \id{y_2,\,y_3, \dots, y_{n+1}}. $$ Thus enlarging it to include the nonzero divisor \,$x\cdot{}y$\, we see that the special fiber \,${\bf\Gamma}_0$ \, presents no imbedded component. (ii) The example of $\p n$ acted on by the stabilizer of a point, blown up at that point might clarify why we were not able to show directly that the closure of ~$\mbox{\hbox{\rm I\hskip-2pt K}}^{nd}$~ is flat over ~${\bf S}$. (iii) For $n=1$ we may write the following global equations for $\mbox{\hbox{\rm I\hskip-2pt K}}$. Let $z,\,w$ be a pair of symmetric 3\vez3 matrices of independent indeterminates. Then ~$\mbox{\hbox{\rm I\hskip-2pt K}} \subset \p5 \mbox{$\times$} \pd5 \mbox{$\times$} \p2 \mbox{$\times$} \pd2$~ is given by the 2\vez2 minors of the 2\vez3 matrices with rows $x\cdot z,\,y$ and $x,\,y\cdot w$, in addition to the incidence relation $x\cdot y$ together with the equation {}~$3z\cdot w={\rm trace}(z\cdot{}w)I$~for ${\bf S}\subset\p5 \mbox{$\times$} \pd5$. It would be nice to give a similar description for higher dimension. (iv) Still assuming $n=1$, put $$ {\bf\Gamma}=\{(P,\,\ell,\,\mbox{\large$\kappa$},\,\mbox{\large$\kappa$}')\in\p2\mbox{$\times$}\pd2\mbox{$\times$}\p5\mbox{$\times$}\pd5\mbox{$\,|\,$} P\in\mbox{\large$\kappa$}\cap\ell,\,\ell\in\mbox{\large$\kappa$}'\}. $$ It is easy to check that ~${\bf\Gamma}_{|{\bf S}}=\mbox{\hbox{\rm I\hskip-2pt K}}$~ as sets. Furthermore, ${\bf\Gamma}$ may be endowed with a natural scheme structure such that ${\bf\Gamma}\mbox{${\,\rightarrow\,}$}\p5\mbox{$\times$}\pd5$ is flat and with Hilbert polynomial of its fibers equal to $4t$. Thus, ${\bf\Gamma}_{|{\bf S}}\mbox{${\,\rightarrow\,}$}{\bf S}$ is a family of double structures of genus one on the fibers of $\mbox{\hbox{\rm I\hskip-2pt K}}$. \end{exx} In fact, we have the following. \vskip15pt\begin{exx}{\bf{Proposition. }}}\def\ep{\rm\end{exx}\label{prop} Any flat family of hypersurfaces on Grassmann varieties induces a flat family of subschemes of the corresponding flag variety. \ep Before considering the general case, we describe the situation in the projective plane. Thus, let $$ {\ff2}\subset\p2\times\pd2 $$ be the incidence correspondence ``point ~$\in$~ line''. Let ~$f_0$~ (resp. ~$f_1$) denote a curve in ~$\p2$~ (resp. ~$\pd2$). Set $$ {\bf\Gamma}_{{\underline{f}}}:=(f_0\times f_1)\cap\ff2. $$ \noindent Then ~${\bf\Gamma}_{{\underline{f}}}$~ is easily seen to be regularly imbedded of codimension 2 in ~${\ff2}$~ (cf. \rf{claim}). Moreover, its Hilbert polynomial with respect to the ample sheaf ${\cal O}_{\p2}(1) \otimes {\cal O}_{\pd{\,\,2}}(1)$ restricted to ~${\ff2}$~ depends only on the degrees, say ~$d_0,\,d_1$~ of ~$f_0,f_1$. In fact, the Koszul complex that resolves the ideal of ~$f_0\times f_1$~ in $\p2 \times \pd2$~ restricts to a resolution of ~${\bf\Gamma}_{{\underline{f}}}$~ in ~${\ff2}$. One finds the Hilbert polynomial, \begin{equation}}\def\ee{\end{equation} \label{xi} \xi\hskip-2pt(t)=(d_0+d_1)t- d_0 d_1(d_0+d_1-4)/2 . \ee Therefore, as in the final argument for the proof of \rf{cc0}, the parameter space of pairs ~$(f_0,f_1)$, call it ~${\bf T}$ (=$\p{n_0}\times\p{n_1}$~ for suitable ~$n_0,n_1)$, carries a flat family of curves on ~${\ff2}$.\, Precisely, let $$ {\bf W}\!_0\subset \p2\times\p{n_0}\hbox{ and } {\bf W}\!_1\subset \pd2\times\p{n_1} $$ denote the total spaces of the universal plane curve parametrized by $\p{n_i}$. Then $$ {\bf\Gamma}:=({\bf W}\!_0\times\hskip-.38cm\raise-.25cm\hbox{$_{\p2}$} {\bf W}\!_1)\bigcap{\ff2}\longrightarrow {\bf T} $$ is a flat family of curves in ~${\ff2}$, with fiber ~${\bf\Gamma}_{{\underline{f}}}$. \vskip10pt For the proof of \rf{prop} we let ~$\gr{r,n}$~ denote the grassmannian of projective subspaces of dimension \,$r$\, of ~$\p n$. Recall that the dimension of the variety of complete flags $\ff n\subset\prod\gr{i,n}$~ is $$ \hbox{dim\hskip2pt}{\ff n}=1+\cdots+n. $$ The proposition is an easy consequence of the following. \vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx} \label{claim} Let ~$f_0,f_1,\dots,f_n$~ be arbitrary hypersurfaces of points, lines, \dots, hyperplanes in the appropriate grassmannians of subspaces of ~$\p{n+1}$. Then the intersection $$ {\bf\Gamma}_{{\underline{f}}}:= (f_0\times\cdots\times f_n)\cap\ff{n+1} $$ is of codimension ~$n+1$~ in ~${\ff{n+1}}$. \ep \vskip10pt\noindent{\bf Proof.\hskip10pt} We shall argue by induction on ~$n$. We may assume all ~$f_i$~ irreducible. Let ~$n=1$. Pick a line ~$h\in f_1$. Set $$ h^ {(0)}=\{P\in\p2\mbox{$\,|\,$}{}P\in{}h\}. $$ The fiber ~$({\bf\Gamma}_{{\underline{f}}})_{h} \simeq h^ {(0)}\cap f_0$~ is zero dimensional unless ~$h^{(0)}=f_0$. This occurs for at most one ~$h\in{}f_1$, hence ~${\bf\Gamma}_{{\underline{f}}}$~ is 1--dimensional (otherwise most of its fibres over ~$f_1$~ would be at least 1--dimensional). For the inductive step, we set for ~$h\in \pd{n+1}$, \begin{equation}}\def\ee{\end{equation}\label{hr} h^ {(r)}=\{g\in\gr{r,n+1}|g\subseteq h\}\simeq\gr{r,n}. \ee If the intersection $$ f'_r =h^{(r)}\cap f_r $$ were proper for all ~$r$ and ~$h\in f_n$~ then we would be done by induction. Indeed, we have $$ ({\bf\Gamma}_{{\underline{f}}})_h \simeq(f'_0 \times\cdots\times f'_{n-1})\cap\ff n. $$ By the induction hypothesis, this is of the right dimension $$ 1+\cdots +n-n =1+\cdots +(n-1). $$ Since ~$h$~ varies in the ~$n-$dimensional hypersurface ~$f_n$~ of $\gr{n,n+1}=\pd{n+1}$, we would have $$ \hbox{dim\hskip2pt} {\bf\Gamma}_{{\underline{f}}}=\big(1+\cdots +(n-1)\big)+n=\big(1+\cdots +(n+1)\big) -(n+1) $$ as desired. However, just as in the case ~$n=1$, it may well happen that the intersection ~$h^{(r)}\cap f_r$~ be \it not \rm proper for some ~$h,r$. Thus it remains to be shown that, whenever \hbox{dim\hskip2pt}$({\bf\Gamma}_{{\underline{f}}})_h$~ exceeds the right dimension, say by ~$\delta$, the hyperplane ~$h$~ is restricted to vary in a locus of codimension at least ~$\delta$~ in ~$f_n$. This is taken care of by the lemma below. \vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx}\label{lema} Notation as in \rf{hr}, for ~$r=0,\dots,n$~ we have $$ \hbox{dim\hskip2pt} \{ h\in\pd{n+1}\ |\ h^{(r)}\subseteq\ f_r\} \leq r. $$ \el \noindent {\it Proof}. Let ~$\ff{r,n}\subset\pd{n+1}\times\gr{r,n+1}$~ be the partial flag variety. Form the diagram with natural projections, $$ \ba{lcr}&\ff{r,n}&\\\noalign{\vskip5pt} \raise6pt\hbox{$\pi_n$}\! \mbox{\huge$\swarrow$} & & \mbox{\huge$\searrow$} \raise6pt\hbox{$\pi_r$}~~~~~\\ \noalign{\vskip3pt}\pd{n+1} && ~~\gr{r,n+1} \end{array}$$ For ~$g_r\in\gr{r,n+1}$, set $$ g_r^ {(n)}=\{h\in\pd{n+1}\ |\ g_r\subseteq h\}. $$ We have ~$g_r^ {(n)} \simeq\p{n-r}$~ whence it hits any subvariety of ~$\pd {n+1}$~ of dimension ~$\geq r+1$. In other words, for any subvariety ~${\bf Z}\subseteq \pd{n+1}$~ such that \hbox{dim\hskip2pt} ~${\bf Z}\geq r+1$, we have $$ \ba{ccl}\pi_r\pi^ {-1}_n{\bf Z} &=& \{g_r\ |\ \exists\, h\in {\bf Z}\hbox{ s.t. } h\supseteq g_r\}\\ &=& \{g_r\ |\ g_r^{(n)}\cap{\bf Z}\ne\emptyset\}\\ &=&\gr{r,n+1}. \end{array}$$ The lemma follows by taking {\bf Z}$= \{ h\in\pd{n+1}\ |\ h^{(r)}\subseteq\ f_r\}$. Indeed, if \hbox{dim\hskip2pt}{\bf Z}$\geq r+1$, then for all ~$g_r\in\gr{r,n+1}$~ there exists $h\in{\bf Z}\hbox{ s.t. } h\supseteq g_r$, so ~$g_r\in h^{(r)}\subseteq f_r$, contradicting that ~$f_r$~ is a hypersurface of ~$\gr{r,n+1}$. \phantom{|}\hfill\mbox{$\Box$}(for \rf{lema}) Continuing the proof of \rf{claim} we consider the stratification of $f_n$~ by the condition of improper intersection of ~$f_r$~ with ~$h^ {(r)}$, namely, $$ \matrix {f_{n,0}&=&\{h\in f_n\ |\ h^{(0)}\subseteq f_0\},&\cr f_{n,1}&=&\{h\in f_n\ |\ h^{(1)}\subseteq f_1\}&\hskip-.25cm\setminus& \hskip-.75cm f_{n,0}, \cr &\vdots&&&\cr f_{n,n}&=&\{h\in f_n\ |\ h^ {(n)}\subseteq f_n\}&\hskip-.25cm \setminus&\hskip-.15cm \bigcup\hskip-.45cm\raise-.25cm\hbox{$_{j<n}$}f_{n,j}.} $$ We will be done if we show $$ \hbox{dim\hskip2pt}({\bf\Gamma}_{{\underline{f}}})_h\ \leq 1+\cdots+n- r\quad\forall\ h \in f_{n,r}. $$ We have already seen that ~$\hbox{dim\hskip2pt}({\bf\Gamma}_{{\underline{f}}})_h=1+\cdots +n-1$~ for ~$h$~ in $f_{n,n}$. Also, for ~$r=0$, the desired estimate holds because we have $({\bf\Gamma}_{{\underline{f}}})_h\subseteq (\ff{n+1})_h\simeq\ff n$~ and ~$\hbox{dim\hskip2pt} \ff n=1+\cdots+n.$~ Let ~$r>0$~ and pick a hyperplane ~$h \in f_{n,r}$. Then the intersections, $$ f'_i = h^ {(i)}\cap f_i, $$ are proper for ~$i=0,\dots,r-1$, whereas for the subsequent index, we have $$ h^ {(r)}\cap f_r = h^ {(r)}\simeq \gr{r,n}.\vspace{-10pt} $$ Thus, we may write, $$ ({\bf\Gamma}_{{\underline{f}}})_h\hookrightarrow \big(f'_0\times\cdots\times f'_{r-1} \times \gr{r,n }\times\cdots\times\gr{n-1,n}\big)\bigcap\ff n. $$ By the induction hypothesis the intersection above is of dimension dim\,$\ff n-r$~ in view of the following easy {\bf Remark.\ }\em The validity of \rf{claim} for a given $n$~ implies properness of the ``partial'' intersection \vspace{-10pt} $$ (f_0\times\cdots\times \gr{r,n+1}\times \cdots\times f_n)\cap {\ff{n+1}}, $$ where one (or more) of the hypersurfaces $f_r\subset\gr{r,n+1}$~ is replaced by the corresponding full grassmannian\rm. \phantom{|}\hfill\mbox{$\Box$}(for \rf{claim}) (Feb.2'96) \vskip15pt \centerline{\bf REFERENCES} \vskip2pt\parskip2pt\baselineskip13pt \begin{itemize}}\def\ei{\end{itemize} \item[]{[H] }R. Hartshorne, {\it Algebraic Geometry}, GTM \# 52 Springer--Verlag (1977). \item[]{[K] }\ S.L. Kleiman with A. Thorup, {\it ``Intersection theory and enumerative geometry: A decade in review'', in} Algebraic geometry: Bowdoin 1985, S. Bloch, ed., AMS Proc. of Symp. Pure Math, {\bf 46-2}, p.321-370 (1987). \item[]{[KT]} S.~L.~Kleiman \& A.~Thorup, {\it Complete bilinear forms}, in Algebraic Geometry, Sundance, 1986, eds. A. Holme and R. Speiser, pp. 253-320, Lect. Notes in Math. 1311, Springer--Verlag, Berlin, (1988). \item[]{[KTB]} $\underline{\hskip2.6cm}$, {\it A Geometric Theory of the Buchsbaum--Rim Multiplicity}, J. Algebra \bf167\rm, 168-231 (1994). \item[]{[L] } D. Laksov {\it Completed quadrics and linear maps}, in Algebraic geometry: Bowdoin 1985, S. Bloch, ed., AMS Proc. of Symp. Pure Math., {\bf 46-2}, p.371-387 (1987). \item[]{[NT]} M. S. Narasimhan \& G. Trautmann, \it Compactification of $M_{\p{\!3}}(0,2)$~ and Poncelet pairs of conics\rm, Pacific J. Math. \bf145-2\rm, p.255-365 (1990). \item[]{[P]} R. Pandharipande, {\it `` Notes On Kontsevich's Compactification Of The Moduli Space Of Maps}, Course notes, Univ. Chicago (1995). \item[]{[BS]} B. Sturmfels, {\it ``Gr\"obner basis and convex polytopes''}, Lectures notes at the Holiday Symp. at N. Mexico State Univ., Las Cruces (1994). \item[]{[V]} I. Vainsencher, \it``Conics multitangent to a plane curve''\rm, in preparation. \ei \vskip10pt \parskip0pt\baselineskip10pt\obeylines \noindent Departamento de Matem\'atica \noindent Universidade Federal de Pernambuco \noindent Cidade Universit\'aria 50670--901 Recife--Pe--Brasil \noindent email: [email protected] \enddocument

9602

alg-geom/9602011

en

https://arxiv.org/abs/alg-geom/9602011

alg-geom/9602011

Yekutieli Amnon

Amnon Yekutieli

Residues and Differential Operators on Schemes

35 pages, AMSLaTeX, final version (minor changes), to appear in Duke Math. J

Beilinson Completion Algebras (BCAs) are generalizations of complete local rings, and have a rich algebraic-analytic structure. These algebras were introduced in my paper "Traces and Differential Operators over Beilinson Completion Algebras", Compositio Math. 99 (1995). In the present paper BCAs are used to give an explicit construction of the Grothendieck residue complex on an algebraic scheme. This construction reveals new properties of the residue complex, and in particular its interaction with differential operators. Applications include: (i) results on the algebraic structure of rings of differential operators; (ii) an analysis of the niveau spectral sequence of De Rham homology; (iii) a proof of the contravariance of De Rham homology w.r.t. etale morphisms; (iv) an algebraic description of the intersection cohomology D-module of a curve.

alg-geom

\section{Introduction} Suppose $X$ is a finite type scheme over a field $k$, with structural morphism $\pi$. Consider the twisted inverse image functor $\pi^{!} : \msf{D}^{+}_{\mrm{c}}(k) \rightarrow \msf{D}^{+}_{\mrm{c}}(X)$ of Grothendieck Duality Theory (see \cite{RD}). The {\em residue complex} $\mcal{K}^{{\textstyle \cdot}}_{X}$ is defined to be the Cousin complex of $\pi^{!} k$. It is a bounded complex of quasi-coherent $\mcal{O}_{X}$-modules, possessing remarkable functorial properties. In this paper we provide an explicit construction of $\mcal{K}^{{\textstyle \cdot}}_{X}$. This construction reveals some new properties of $\mcal{K}^{{\textstyle \cdot}}_{X}$, and also has applications in other areas of algebraic geometry. Grothendieck Duality, as developed by Hartshorne in \cite{RD}, is an abstract theory, stated in the language of derived categories. Even though this abstraction is suitable for many important applications, often one wants more explicit information. Thus a significant amount of work was directed at finding a presentation of duality in terms of differential forms and residues. Mostly the focus was on the dualizing sheaf $\omega_{X}$, in various circumstances. The structure of $\omega_{X}$ as a coherent $\mcal{O}_{X}$-module and its variance properties are thoroughly understood by now, thanks to an extended effort including \cite{Kl}, \cite{KW}, \cite{Li}, \cite{HK1}, \cite{HK2}, \cite{LS} and \cite{HS}. Regarding an explicit presentation of the full duality theory of dualizing complexes, there have been some advances in recent years, notably in the papers \cite{Ye1}, \cite{SY}, \cite{Hu}, \cite{Hg} and \cite{Sa}. In this paper we give a totally new construction of the residue complex $\mcal{K}^{{\textstyle \cdot}}_{X}$, when $k$ is a perfect field of any characteristic and $X$ is any finite type $k$-scheme. The main idea is the use of {\em Beilinson Completion Algebras} (BCAs), which were introduced in \cite{Ye2}. These algebras are generalizations of complete local rings, and they carry a mixed algebraic-analytic structure. A review of BCAs and their properties is included in Section 1, for the reader's convenience. Given a point $x \in X$, the complete local ring $\widehat{\mcal{O}}_{X, x} = \mcal{O}_{X, (x)}$ is a BCA, so according to \cite{Ye2} it has a {\em dual module} $\mcal{K}(\mcal{O}_{X, (x)})$. This module is a canonical model for the injective hull of the residue field $k(x)$. If $(x,y)$ is a saturated chain of points (i.e.\ $y$ is an immediate specialization of $x$) then there is a BCA $\mcal{O}_{X, (x,y)}$ and homomorphisms $\mrm{q} : \mcal{K}(\mcal{O}_{X, (x)}) \rightarrow \mcal{K}(\mcal{O}_{X, (x,y)})$ and $\operatorname{Tr} : \mcal{K}(\mcal{O}_{X, (x,y)}) \rightarrow \mcal{K}(\mcal{O}_{X, (y)})$. The dual modules $\mcal{K}(-)$ and the homomorphisms $\mrm{q}$ and $\operatorname{Tr}$ have explicit formulas in terms of differential forms and coefficient fields. Set $\delta_{(x,y)} := \operatorname{Tr} \mrm{q} : \mcal{K}(\mcal{O}_{X, (x)}) \rightarrow \mcal{K}(\mcal{O}_{X, (y)})$. Define a graded quasi-coherent sheaf $\mcal{K}_{X}^{{\textstyle \cdot}}$ by \[ \mcal{K}_{X}^{q} := \bigoplus_{\operatorname{dim} \overline{\{x\}} = -q} \mcal{K}(\mcal{O}_{X, (x)}) \] and a degree $1$ homomorphism \[ \delta := (-1)^{q+1} \sum_{(x,y)} \delta_{(x,y)} . \] It turns out that $(\mcal{K}_{X}^{{\textstyle \cdot}}, \delta)$ is a residual complex on $X$, and it is canonically isomorphic to $\pi^{!} k$ in the derived category $\msf{D}(X)$. Hence it is the residue complex of $X$, as defined in the first paragraph. The functorial properties of $\mcal{K}_{X}^{{\textstyle \cdot}}$ w.r.t.\ proper and \'{e}tale morphisms are obtained directly from corresponding properties of BCAs, and therefore are reduced to explicit formulas. All this is worked out in Sections 2 and 3. An $\mcal{O}_{X}$-module $\mcal{M}$ has a dual complex $\operatorname{Dual} \mcal{M} := \mcal{H}om^{{\textstyle \cdot}}_{\mcal{O}_{X}}(\mcal{M}, \mcal{K}_{X}^{{\textstyle \cdot}})$. Suppose $\mrm{d} : \mcal{M} \rightarrow \mcal{N}$ is a differential operator (DO). In Theorem \ref{thm3.1} we prove there is a dual operator $\operatorname{Dual}(\mrm{d}) : \operatorname{Dual} \mcal{N} \rightarrow \operatorname{Dual} \mcal{M}$, which commutes with $\delta$. The existence of $\operatorname{Dual}(\mrm{d})$ does not follow from formal considerations of duality theory; it is a consequence of our particular construction using BCAs (but cf.\ Remarks \ref{rem3.2} and \ref{rem3.3}). The construction also provides explicit formulas for $\operatorname{Dual}(\mrm{d})$ in terms of differential operators and residues, which are used in the applications in Sections 6 and 7. Suppose $A$ is a finite type $k$-algebra, and let $\mcal{D}(A)$ be the ring of differential operators of $A$. As an immediate application of Theorem \ref{thm3.1} we obtain a description of the opposite ring $\mcal{D}(A)^{\circ}$, as the ring of DOs on $\mcal{K}_{A}^{{\textstyle \cdot}}$ which commute with $\delta$ (Theorem \ref{thm3.2}). In the case of a Gorenstein algebra it follows that the opposite ring $\mcal{D}(A)^{\circ}$ is naturally isomorphic to $\omega_{A} \otimes_{A} \mcal{D}(A) \otimes_{A} \omega_{A}^{-1}$ (Corollary \ref{cor3.6}). Applying Theorem \ref{thm3.1} to the De Rham complex $\Omega^{{\textstyle \cdot}}_{X/k}$ we obtain the {\em De Rham-residue complex} $\mcal{F}_{X}^{{\textstyle \cdot}} = \operatorname{Dual} \Omega^{{\textstyle \cdot}}_{X/k}$. Up to signs this coincides with El-Zein's complex $\mcal{K}_{X}^{{\textstyle \cdot},*}$ of \cite{EZ} (Corollary \ref{cor4.3}). The fundamental class $\mrm{C}_{Z} \in \mcal{F}_{X}^{{\textstyle \cdot}}$, for a closed subscheme $Z \subset X$, is easily described in this context (Definition \ref{dfn4.4}). The construction above works also for a formal scheme $\mfrak{X}$ which is of formally finite type over $k$, in the sense of \cite{Ye3}. An example of such a formal scheme is the completion $\mfrak{X} = Y_{/X}$, where $X$ is a locally closed subset of the finite type $k$-scheme $Y$. Therefore we get a complex $\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}} = \operatorname{Dual} \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}/k}$. When $X \subset \mfrak{X}$ is a smooth formal embedding (see Definition \ref{dfn5.3}) we prove that the cohomology modules $\mrm{H}^{q}(X, \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}})$ are independent of $\mfrak{X}$. This is done by analyzing the $E_{1}$ term of the {\em niveau spectral sequence} converging to $\mrm{H}^{{\textstyle \cdot}}(X, \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}})$ (Theorem \ref{thm5.3}). Here we assume $\operatorname{char} k = 0$. The upshot is that $\mrm{H}^{q}(X, \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}) = \mrm{H}_{-q}^{\mrm{DR}}(X)$, the De Rham homology. There is an advantage in using smooth formal embeddings. If $U \rightarrow X$ is any \'{e}tale morphism, then there is an \'{e}tale morphism $\mfrak{U} \rightarrow \mfrak{X}$ s.t.\ $U = \mfrak{U} \times_{\mfrak{X}} X$, so $U \subset \mfrak{U}$ is a smooth formal embedding. From this we conclude that $\mrm{H}_{{\textstyle \cdot}}^{\mrm{DR}}(-)$ is a contravariant functor on $X_{\mrm{et}}$, the small \'{e}tale site. Previously it was only known that $\mrm{H}_{{\textstyle \cdot}}^{\mrm{DR}}(-)$ is contravariant for open immersions (cf.\ \cite{BlO} Example 2.2). Suppose $X$ is smooth, and let $\mcal{H}^{p}_{\mrm{DR}}$ be the sheafification of the presheaf $U \mapsto \mrm{H}^{p}_{\mrm{DR}}(U)$ on $X_{\mrm{Zar}}$. Bloch-Ogus \cite{BlO} give a flasque resolution of $\mcal{H}^{p}_{\mrm{DR}}$, the {\em arithmetic resolution}. It involves the sheaves $i_{x*} \mrm{H}^{q} \Omega^{{\textstyle \cdot}}_{k(x)/k}$ where $i_{x} : \{x\} \rightarrow X$ is the inclusion map. Our analysis of the niveau spectral sequence shows that the coboundary operator of this resolution is a sum of Parshin residues (Corollary \ref{cor5.2}). Our final application of the new construction of the residue complex is to describe the {\em intersection cohomology $\mcal{D}$-module} $\mcal{L}(X, Y)$, when $X$ is an integral curve embedded in a smooth $n$-dimensional variety $Y$ (see \cite{BrKa}). Again we assume $k$ has characteristic $0$. In fact we are able to describe all coherent $\mcal{D}_{Y}$-submodules of $\mcal{H}^{n-1}_{X} \mcal{O}_{Y}$ in terms of the singularities of $X$ (Corollary \ref{cor6.10}). This description is an algebraic version of Vilonen's work in \cite{Vi}, replacing complex analysis with BCAs and algebraic residues. It is our hope that a similar description will be found in the general case, namely $\operatorname{dim} X > 1$. Furthermore, we hope to give in the future an explicit description of the Cousin complex of $\operatorname{DR} \mcal{L}(X, Y) = \Omega^{{\textstyle \cdot}}_{Y / k} \otimes \mcal{L}(X, Y)$. Note that for $X=Y$ one has $\mcal{L}(X, Y) = \mcal{O}_{X}$, so this Cousin complex is nothing but $\mcal{F}^{{\textstyle \cdot}}_{X}$. \medskip \noindent {\bf Acknowledgments.}\ I wish to thank J.\ Lipman and S.\ Kleiman for their continued interest in this work. Thanks also to P.\ Sastry, R.\ H\"{u}bl, V.\ Hinich, V.\ Berkovich, H.\ Esnault, K.\ Smith and K.\ Vilonen for helpful conversations, and thanks to the referee for valuable advice on Sections 4 and 7. \section{Review of Beilinson Completion Algebras} Let us begin by reviewing some facts about Topological Local Fields (TLFs) and Beilinson Completion Algebras (BCAs) from the papers \cite{Ye1} and \cite{Ye2}. A semi-topological (ST) ring is a ring $A$, with a linear topology on its underlying additive group, such that for every $a \in A$ the multiplication (on either side) $a : A \rightarrow A$ is continuous. Let $K$ be a field. We say $K$ is an $n$-dimensional local field if there is a sequence of complete discrete valuation rings $\mcal{O}_{1}, \ldots, \mcal{O}_{n}$, where the fraction field of $\mcal{O}_{1}$ is $K$, and the residue field of $\mcal{O}_{i}$ is the fraction field of $\mcal{O}_{i + 1}$. Fix a perfect field $k$. A {\em topological local field} of dimension $n$ over $k$ is a $k$-algebra $K$ with structures of semi-topological ring and $n$-dimensional local field, satisfying the following parameterization condition: there exists an isomorphism of $k$-algebras $K \cong F((s_{1}, \ldots, s_{n}))$ for some field $F$, finitely generated over $k$, which respects the two structures. Here $F((s_{1}, \ldots, s_{n})) = F((s_{n})) \cdots ((s_{1}))$ is the field of iterated Laurent series, with its inherent topology and valuation rings ($F$ is discrete). One should remark that for $n = 1$ we are in the classical situation, whereas for $n \geq 2$, $F((s_{1}, \ldots, s_{n}))$ is not a topological ring. TLFs make up a category $\msf{TLF}(k)$, where a morphism $K \rightarrow L$ is a continuous $k$-algebra homomorphism which preserves the valuations, and the induced homomorphism of the last residue fields is finite. Write $\Omega^{\mrm{sep}, {\textstyle \cdot}}_{K / k}$ for the separated algebra of differentials; with the parameterization above $\Omega^{\mrm{sep}, {\textstyle \cdot}}_{K / k} \cong K \otimes_{F} \Omega^{{\textstyle \cdot}}_{F[\, \underline{s}\, ] / k}$. Then there is a functorial residue map $\operatorname{Res}_{L / K} : \Omega^{\mrm{sep}, {\textstyle \cdot}}_{L / k} \rightarrow \Omega^{\mrm{sep}, {\textstyle \cdot}}_{K / k}$ which is $\Omega^{\mrm{sep}, {\textstyle \cdot}}_{K / k}$-linear and lowers degree by $\operatorname{dim} L / K$. For instance if $L = K((t))$ then \begin{equation} \label{eqn10.1} \operatorname{Res}_{L / K} \left( \sum_{i} t^{i} \mrm{d} t \wedge \alpha_{i} \right) = \alpha_{-1} \in \Omega^{\mrm{sep}, {\textstyle \cdot}}_{K / k} . \end{equation} TLFs and residues were initially developed by Parshin and Lomadze, and the theory was enhanced in \cite{Ye1}. The notion of a {\em Beilinson completion algebra} was introduced in \cite{Ye2}. A BCA is a semi-local, semi-topological $k$-algebra, each of whose residue fields $A / \mfrak{m}$ is a topological local field. Again there is a parameterization condition: when $A$ is local, there should exist a surjection \[ F((\underline{s})) [[\, \underline{t}\, ]] = F((\underline{s})) [[ t_{1}, \ldots, t_{m} ]] \twoheadrightarrow A \] which is strict topologically (i.e.\ $A$ has the quotient topology) and respects the valuations. Here $F((\underline{s}))$ is as above and $F((\underline{s})) [[\, \underline{t}\, ]]$ is the ring of formal power series over $F((\underline{s}))$. The notion of BCA is an abstraction of the algebra gotten by Beilinson's completion, cf.\ Lemma \ref{lem10.1}. There are two distinguished kinds of homomorphisms between BCAs. The first kind is a {\em morphism of BCAs} $f : A \rightarrow B$ (see \cite{Ye2} Definition 2.5), and the category they constitute is denoted $\msf{BCA}(k)$. A morphism is continuous, respects the valuations on the residue fields, but in general is not a local homomorphism. For instance, the homomorphisms $k \rightarrow k[[ s, t ]] \rightarrow k((s))[[t]] \rightarrow k((s))((t))$ are all morphisms. $\msf{TLF}(k)$ is a full subcategory of $\msf{BCA}(k)$, consisting of those BCAs which are fields. The second kind of homomorphism is an {\em intensification homomorphism} $u : A \rightarrow \widehat{A}$ (see \cite{Ye2} Definition 3.6). An intensification is flat, topologically \'{e}tale (relative to $k$) and unramified (in the appropriate sense). It can be viewed as a sort of localization or completion. Here examples are $k(s)[[t]] \rightarrow k((s))[[t]]$ and $k(s,t) \rightarrow k(s)((t)) \rightarrow k((s))((t))$. Suppose $f : A \rightarrow B$ is a morphism of BCAs and $u : A \rightarrow \widehat{A}$ is an intensification. The Intensification Base Change Theorem (\cite{Ye2} Theorem 3.8) says there is a BCA $\widehat{B} = B \otimes_{A}^{(\wedge)} \widehat{A}$, a morphism $\widehat{f} : \widehat{A} \rightarrow \widehat{B}$ and an intensification $v : B \rightarrow \widehat{B}$, with $v f = \widehat{f} u$. These are determined up to isomorphism and satisfy certain universal properties. For instance, $k((s))[[t]] = k(s)[[t]] \otimes_{k(s)}^{(\wedge)} k((s))$. According to \cite{Ye2} Theorem 6.14, every $A \in \msf{BCA}(k)$ has a dual module $\mcal{K}(A)$. The module $\mcal{K}(A)$ is a ST $A$-module. Algebraically it is an injective hull of $A / \mfrak{r}$, where $\mfrak{r}$ is the Jacobson radical. $\mcal{K}(A)$ is also a right $\mcal{D}(A)$-module, where $\mcal{D}(A)$ denotes the ring of continuous differential operators of $A$ (relative to $k$). For a ST $A$-module $M$ let $\operatorname{Dual}_{A} M := \operatorname{Hom}_{A}^{\mrm{cont}}(M, \mcal{K}(A))$. The dual modules have variance properties w.r.t.\ morphisms and intensifications. Given a morphism of BCAs $f : A \rightarrow B$, according to \cite{Ye2} Theorem 7.4 there is an $A$-linear map $\operatorname{Tr}_{f} : \mcal{K}(B) \rightarrow \mcal{K}(A)$. This induces an isomorphism $\mcal{K}(B) \cong \operatorname{Hom}_{A}^{\mrm{cont}}(B, \mcal{K}(A))$. Given an intensification homomorphism $u : A \rightarrow \widehat{A}$, according to \cite{Ye2} Proposition 7.2 there is an $A$-linear map $\mrm{q}_{u} : \mcal{K}(A) \rightarrow \mcal{K}(\widehat{A})$. It induces an isomorphism $\mcal{K}(\widehat{A}) \cong \widehat{A} \otimes_{A} \mcal{K}(A)$. Furthermore $\operatorname{Tr}$ and $\mrm{q}$ commute across intensification base change: $\operatorname{Tr}_{\widehat{B} / \widehat{A}} \mrm{q}_{B / \widehat{B}} = \mrm{q}_{\widehat{A} / A} \operatorname{Tr}_{B / A}$. In case of a TLF $K$, one has $\mcal{K}(K) = \omega(K) = \Omega^{p, \mrm{sep}}_{K/k}$, where $p = \operatorname{rank} \Omega^{1, \mrm{sep}}_{K/k}$. For a morphism of TLFs $f :K \rightarrow L$ one has $\operatorname{Tr}_{f} = \operatorname{Res}_{f}$, whereas for an intensification $u : K \rightarrow \widehat{K}$ the homomorphism $\mrm{q}_{u} : \Omega^{p, \mrm{sep}}_{K/k} \rightarrow \Omega^{p, \mrm{sep}}_{\widehat{K}/k}$ is the canonical inclusion for a topologically \'{e}tale extension of fields. \begin{exa} \label{exa10.1} Take $L := k(s,t)$, $\widehat{L} := k(s)((t))$, $A := k(s)[[t]]$, $\widehat{A} := k((s))[[t]]$, $K := k(s)$ and $\widehat{K} := k((s))$. The inclusions $L \rightarrow \widehat{L}$, $K \rightarrow \widehat{K}$ and $A \rightarrow \widehat{A}$ are intensifications, whereas $K \rightarrow A \rightarrow \widehat{L}$ and $\widehat{K} \rightarrow \widehat{A}$ are morphisms. Using the isomorphism $\mcal{K}(A) \cong \operatorname{Hom}_{K}^{\mrm{cont}}(A, \Omega^{1, \mrm{sep}}_{K / k})$ induced by $\operatorname{Tr}_{A / K}$, we see that for $\alpha \in \Omega^{2, \mrm{sep}}_{\widehat{L} / k}$ the element $\operatorname{Tr}_{\widehat{L} / A}(\alpha) \in \mcal{K}(A)$ is represented by the functional $a \mapsto \operatorname{Res}_{\widehat{L} / K}(a \alpha)$, $a \in A$. Also for $\phi \in \mcal{K}(A)$ the element $\widehat{\phi} = \mrm{q}_{\widehat{A} / A}(\phi) \in \mcal{K}(\widehat{A})$ is represented by the unique $\widehat{K}$-linear functional $\widehat{\phi} : \widehat{A} \rightarrow \Omega^{1, \mrm{sep}}_{\widehat{K} / k}$ extending $\phi$. \end{exa} \begin{rem} The proof of existence of dual modules with their variance properties in \cite{Ye2} is straightforward, using Taylor series expansions, differential operators and the residue pairing. \end{rem} Let $A$ be a noetherian ring and $\mfrak{p}$ a prime ideal. Consider the exact functor on $A$-modules $M \mapsto M_{(\mfrak{p})} := \widehat{A}_{\mfrak{p}} \otimes_{A} M$. For $M$ finitely generated we have $M_{(\mfrak{p})} \cong \lim_{\leftarrow i} M_{\mfrak{p}} / \mfrak{p}^{i} M_{\mfrak{p}}$, and if $M = \lim_{\alpha \rightarrow} M_{\alpha}$, then $M_{(\mfrak{p})} \cong \lim_{\alpha \rightarrow} (M _{\alpha})_{(\mfrak{p})}$. This was generalized by Beilinson (cf.\ \cite{Be}) as follows. \begin{dfn} \label{dfn10.1} Let $M$ be an $A$-module and let $\xi = (\mfrak{p}_{0}, \ldots, \mfrak{p}_{n})$ be a chain of prime ideals, namely $\mfrak{p}_{i} \subset \mfrak{p}_{i + 1}$. Define the {\em Beilinson completion} $M_{\xi}$ by recursion on $n$, $n \geq -1$. \begin{enumerate} \item If $n = -1$ (i.e.\ $\xi = \emptyset$), let $M_{\xi} := M$ with the discrete topology. \item If $n \geq 0$ and $M$ is finitely generated, let \[ M_{(\mfrak{p}_{0}, \ldots, \mfrak{p}_{n})} := \lim_{\leftarrow i}\, (M_{\mfrak{p}_{0}} / \mfrak{p}_{0}^{i} M_{\mfrak{p}_{0}})_{(\mfrak{p}_{1}, \ldots, \mfrak{p}_{n})} . \] \item For arbitrary $M$, let $\{ M_{\alpha} \}$ be the set of finitely generated submodules of $M$, and let \[ M_{(\mfrak{p}_{0}, \ldots, \mfrak{p}_{n})} := \lim_{\alpha \rightarrow}\, (M_{\alpha})_{(\mfrak{p}_{0}, \ldots, \mfrak{p}_{n})} . \] \end{enumerate} \end{dfn} A chain $\xi = (\mfrak{p}_{0}, \ldots, \mfrak{p}_{n})$ is {\em saturated} if $\mfrak{p}_{i + 1}$ has height $1$ in $A / \mfrak{p}_{i}$. \begin{lem} \label{lem10.1} If $\xi= (\mfrak{p}, \ldots )$ is a saturated chain then $A_{\xi}$ is a Beilinson completion algebra. \end{lem} \begin{proof} By \cite{Ye1} Corollary 3.3.5, $A_{\xi}$ is a complete semi-local noetherian ring with Jacobson radical $\mfrak{p}_{\xi}$, and $A_{\xi} / \mfrak{p}_{\xi} = K_{\xi}$, where $K := A_{\mfrak{p}} / \mfrak{p}_{\mfrak{p}}$. Choose a coefficient field $\sigma : K \rightarrow \widehat{A}_{\mfrak{p}} = A_{(\mfrak{p})}$. By \cite{Ye1} Proposition 3.3.6, $K_{\xi}$ is a finite product of TLFs, and by the proof of \cite{Ye1} Theorem 3.3.8, $\sigma$ extends to a lifting $\sigma_{\xi} : K_{\xi} \rightarrow A_{\xi}$. Sending $t_{1}, \ldots, t_{m}$ to generators of the ideal $\mfrak{p}$, we get a topologically strict surjection $K_{\xi} [[ t_{1}, \ldots, t_{m} ]] \twoheadrightarrow A_{\xi}$. \end{proof} \section{Construction of the Residue Complex $\mcal{K}^{{\textstyle \cdot}}_{X}$} Let $k$ be a perfect field, and let $X$ be a scheme of finite type over $k$. By a {\em chain of points} in $X$ we mean a sequence $\xi = (x_{0}, \ldots, x_{n})$ of points with $x_{i + 1} \in \overline{\{x_{i}\}}$. \begin{dfn} For any quasi-coherent $\mcal{O}_{X}$-module $\mcal{M}$, define the Beilinson completion $\mcal{M}_{\xi}$ by taking an affine open neighborhood $U = \operatorname{Spec} A \subset X$ of $x_{n}$, and setting $\mcal{M}_{\xi} := \Gamma(U, \mcal{M})_{\xi}$ as in Definition \ref{dfn10.1}. \end{dfn} These completions first appeared as the local factors of Beilinson's adeles in \cite{Be}, and were studied in detail in \cite{Ye1}. According to Lemma \ref{lem10.1}, if $\xi = (x_{0}, \ldots, x_{n})$ is saturated, i.e.\ $\overline{\{x_{i + 1}\}} \subset \overline{\{x_{i}\}}$ has codimension $1$, then $\mcal{O}_{X, \xi}$ is a BCA. We shall be interested in the covertex maps \[ \begin{array}{rcl} \partial^{-} : & \mcal{O}_{X,(x_{0})} & \rightarrow \mcal{O}_{X,\xi} \\ \partial^{+} : & \mcal{O}_{X,(x_{n})} & \rightarrow \mcal{O}_{X,\xi} \end{array} \] which arise naturally from the completion process (cf.\ \cite{Ye1} \S 3.1). \begin{lem} \label{lem1.1} $\partial^{+}$ is flat, topologically \'{e}tale relative to $k$, and a morphism in $\msf{BCA}(k)$. $\partial^{-}$ is an intensification homomorphism. \end{lem} \begin{proof} By definition $\partial^{-} = \partial^{1} \circ \cdots \circ \partial^{n}$ and $\partial^{+} = \partial^{0} \circ \cdots \circ \partial^{0}$, where $\partial^{i} : \mcal{O}_{X,\partial_{i} \xi} \rightarrow \mcal{O}_{X,\xi}$ is the $i$-th coface operator. First let us prove that $\partial^{0} : \mcal{O}_{X,\partial_{0} \xi} \rightarrow \mcal{O}_{X,\xi}$ is a morphism of BCAs. This follows from \cite{Ye1} Theorem 3.3.2 (d), since we may assume that $X$ is integral with generic point $x_{0}$. By part (b) of the same theorem, $\partial^{n} : \mcal{O}_{X,\partial_{n} \xi} \rightarrow \mcal{O}_{X, \xi}$ is finitely ramified and radically unramified (in the sense of \cite{Ye2} Definition 3.1). Now according to \cite{Ye1} Corollary 3.2.8, $\partial^{i} : \mcal{O}_{X,\partial_{i} \xi} \rightarrow \mcal{O}_{X,\xi}$ is topologically \'{e}tale relative to $k$, for any $i$. We claim it is also flat. For $i=0$, $\mcal{O}_{X,\partial_{0} \xi} \rightarrow (\mcal{O}_{X,\partial_{0} \xi})_{x_{0}} = (\mcal{O}_{X, x_{0}})_{\partial_{0} \xi}$ is a localization, so it's flat. The map from $(\mcal{O}_{X, x_{0}})_{\partial_{0} \xi}$ to its $\mfrak{m}_{x_{0}}$-adic completion $\mcal{O}_{X, \xi}$ is also flat (these rings are noetherian). For $i > 0$, by induction on the length of chains, $\mcal{O}_{X,\partial_{0} \partial_{i} \xi} \rightarrow \mcal{O}_{X,\partial_{0} \xi}$ is flat, and hence so is $(\mcal{O}_{X, x_{0}})_{\partial_{0} \partial_{i} \xi} \rightarrow (\mcal{O}_{X, x_{0}})_{\partial_{0} \xi}$. Now use \cite{CA} Chapter III \S 5.4 Proposition 4 to conclude that \[ \mcal{O}_{X,\partial_{i} \xi} = \lim_{\leftarrow j} (\mcal{O}_{X, x_{0}} / \mfrak{m}_{x_{0}}^{j})_{ \partial_{0} \partial_{i} \xi} \rightarrow \lim_{\leftarrow j} (\mcal{O}_{X, x_{0}} / \mfrak{m}_{x_{0}}^{j})_{ \partial_{0} \xi} = \mcal{O}_{X, \xi} \] is flat. \end{proof} \begin{exa} \label{1.1} Take $X = \mbf{A}^{2} := \operatorname{Spec} k[s,t]$, $x := (0)$, $y := (t)$ and $z := (s,t)$. Then with $L := k(x) = \mcal{O}_{X, (x)}$, $\widehat{L} := k(x)_{(y)} = \mcal{O}_{X, (x, y)}$, $A := \mcal{O}_{X, (y)}$, $\widehat{A} := \mcal{O}_{X, (y, z)}$ $K := k(y)$ and $\widehat{K} := k(y)_{(z)}$ we are exactly in the situation of Example \ref{exa10.1}. \end{exa} \begin{dfn} Given a point $x$ in $X$, let $\mcal{K}_{X}(x)$ be the skyscraper sheaf with support $\overline{\{ x \}}$ and group of sections $\mcal{K}(\mcal{O}_{X,(x)})$. \end{dfn} The sheaf $\mcal{K}_{X}(x)$ is a quasi-coherent $\mcal{O}_{X}$-module, and is an injective hull of $k(x)$ in the category $\msf{Mod}(X)$ of $\mcal{O}_{X}$-modules. \begin{dfn} \label{dfn1.1} Given a saturated chain $\xi = (x, \ldots, y)$ in $X$, define an $\mcal{O}_{X}$-linear homomorphism $\delta_{\xi} : \mcal{K}_{X}(x) \rightarrow \mcal{K}_{X}(y)$, called the coboundary map along $\xi$, by \[ \delta_{\xi} : \mcal{K}(\mcal{O}_{X,(x)}) \xrightarrow{ \mrm{q}_{\partial^{-}} } \mcal{K}(\mcal{O}_{X,\xi}) \xrightarrow{ \operatorname{Tr}_{\partial^{+}} } \mcal{K}(\mcal{O}_{X,(y)}) . \] \end{dfn} Throughout sections 2 and 3 the following convention shall be used. Let $f: X \rightarrow Y$ be a morphism of schemes, and let $x \in X$, $y \in Y$ be points. We will write $x \mid y$ to indicate that $x$ is a closed point in the fiber $X_{y} := X \times_{Y} \operatorname{Spec} k(y) \cong f^{-1}(y)$. Similarly for chains: we write $(x_{0}, \ldots, x_{n}) \mid (y_{0}, \ldots, y_{n})$ if $x_{i} \mid y_{i}$ for all $i$. \begin{lem} Suppose $x \mid y$. Then $f^{*} : \mcal{O}_{Y,(y)} \rightarrow \mcal{O}_{X,(x)}$ is a morphism of BCAs. If $f$ is quasi-finite then $f^{*}$ is finite, and if $f$ is \'{e}tale then $f^{*}$ is an intensification. \end{lem} \begin{proof} Immediate from the definitions. \end{proof} \begin{lem} \label{lem1.2} Suppose $f: X \rightarrow Y$ is a quasi-finite morphism. Let $\eta = (y_{0}, \ldots, y_{n})$ be a saturated chain in $Y$ and let $x \in X$, $x \mid y_{n}$. Consider the \textup{(}finite\textup{)} set of chains in $X$: \[ \Xi := \{ \xi = (x_{0}, \ldots, x_{n})\ \mid \ \xi \mid \eta \textup{ and } x_{n} = x \}. \] Then there is a canonical isomorphism of BCAs \[ \prod_{\xi \in \Xi} \mcal{O}_{X,\xi} \cong \mcal{O}_{Y,\eta} \otimes_{\mcal{O}_{Y,(y_{n})}} \mcal{O}_{X,(x)} . \] \end{lem} \begin{proof} Set $\widehat{X} := \operatorname{Spec} \mcal{O}_{X,(x)}$ and $\widehat{Y} := \operatorname{Spec} \mcal{O}_{Y,(y_{n})}$, so the induced morphism $\widehat{f} : \widehat{X} \rightarrow \widehat{Y}$ is finite. Let us denote by $\xi, \widehat{\xi}, \widehat{\eta}$ variable saturated chains in $X, \widehat{X}, \widehat{Y}$ respectively. For any $\widehat{\eta} \mid \eta$ one has \begin{equation} \label{eqn1.1} \prod_{\widehat{\xi} \mid \widehat{\eta}} \mcal{O}_{\widehat{X}, \widehat{\xi}} \cong \mcal{O}_{\widehat{Y}, \widehat{\eta}} \otimes_{\mcal{O}_{Y, (y_{n})}} \mcal{O}_{X, (x)}, \end{equation} by \cite{Ye1} Proposition 3.2.3; note that the completion is defined on any noetherian scheme. Now from ibid.\ Corollary 3.3.13 one has $\mcal{O}_{X, \xi} \cong \prod_{\widehat{\xi} \mid \xi} \mcal{O}_{\widehat{X}, \widehat{\xi}}$, so taking the product over all $\xi \in \Xi$ and $\widehat{\eta} \mid \eta$ the lemma is proved. \end{proof} \begin{dfn} \label{dfn1.2} Given an \'{e}tale morphism $g : X \rightarrow Y$ and a point $x \in X$, let $y := g(x)$, so $g^{*} : \mcal{O}_{Y, (y)} \rightarrow \mcal{O}_{X, (x)}$ is an intensification. Define \[ \mrm{q}_{g} : \mcal{K}_{Y}(y) \rightarrow g_{*} \mcal{K}_{X}(x) \] to be the $\mcal{O}_{Y}$-linear homomorphism corresponding to $\mrm{q}_{g^{*}} : \mcal{K}(\mcal{O}_{Y, (y)}) \rightarrow \mcal{K}(\mcal{O}_{X, (x)})$ of \cite{Ye2} Proposition 7.2. \end{dfn} \begin{prop} \label{prop1.1} Let $g : X \rightarrow Y$ be an \'{e}tale morphism. \begin{enumerate} \rmitem{a} Given a point $y \in Y$, the homomorphism $1 \otimes \mrm{q}_{g} : g^{*} \mcal{K}_{Y}(y) \rightarrow$ \blnk{3mm} \linebreak $\bigoplus_{x \mid y} \mcal{K}_{X}(x)$ is an isomorphism of $\mcal{O}_{X}$-modules. \rmitem{b} Let $\eta = (y_{0}, \ldots, y_{n})$ be a saturated chain in $Y$. Then \[ (1 \otimes \mrm{q}_{g}) \circ g^{*}(\delta_{\eta}) = (\sum_{\xi \mid \eta} \delta_{\xi}) \circ (1 \otimes \mrm{q}_{g}) \] as homomorphisms $g^{*} \mcal{K}_{Y}(y_{0}) \rightarrow \bigoplus_{x_{n} \mid y_{n}} \mcal{K}_{X}(x_{n})$. \end{enumerate} \end{prop} \begin{proof} (a)\ Because $\mcal{K}_{Y}(y)$ is an artinian $\mcal{O}_{Y,y}$-module, and $g$ is quasi-finite, we find that \[ g^{*} \mcal{K}_{Y}(y) = \bigoplus_{x \mid y} \mcal{O}_{X, (x)} \otimes_{\mcal{O}_{Y, (y)}} \mcal{K}_{Y}(y)\ . \] Now use \cite{Ye2} Proposition 7.2 (i). \medskip \noindent (b)\ From Lemma \ref{lem1.2} and from \cite{Ye2} Theorem 3.8 we see that for every $\xi \mid \eta$, $f^{*} : \mcal{O}_{Y,\eta} \rightarrow \mcal{O}_{X, \xi}$ is both a finite morphism and an intensification. By the definition of the coboundary maps, it suffices to verify that the diagram \bigskip \noindent \[ \begin{CD} \mcal{K}(\mcal{O}_{Y,(y_{0})}) @>{\mrm{q}}>> \mcal{K}(\mcal{O}_{Y,\eta}) @>{\operatorname{Tr}}>> \mcal{K}(\mcal{O}_{Y,(y_{n})}) \\ @V{\mrm{q}}VV @V{\mrm{q}}VV @V{\mrm{q}}VV \\ \bigoplus_{x_{0} \mid y_{0}} \mcal{K}(\mcal{O}_{X,(x_{0})}) @>{\mrm{q}}>> \bigoplus_{\xi \mid \eta} \mcal{K}(\mcal{O}_{X,\xi}) @>{\operatorname{Tr}}>> \bigoplus_{x_{n} \mid y_{n}} \mcal{K}(\mcal{O}_{X,(x_{n})}) \end{CD} \] \medskip \noindent commutes. The left square commutes by \cite{Ye2} Proposition 7.2 (iv), whereas the right square commutes by Lemma \ref{lem1.2} and \cite{Ye2} Theorem 7.4 (ii). \end{proof} \begin{dfn} \label{dfn1.3} Let $f: X \rightarrow Y$ be a morphism between finite type $k$-schemes, let $x \in X$ be a point, and let $y := f(x)$. Define an $\mcal{O}_{Y}$-linear homomorphism \[ \operatorname{Tr}_{f} : f_{*} \mcal{K}_{X}(x) \rightarrow \mcal{K}_{Y}(y) \] as follows: \begin{enumerate} \rmitem{i} If $x$ is closed in its fiber $X_{y}$, then $f^{*} : \mcal{O}_{Y,(y)} \rightarrow \mcal{O}_{X,(x)}$ is a morphism in $\msf{BCA}(k)$. Let $\operatorname{Tr}_{f}$ be the homomorphism corresponding to $\operatorname{Tr}_{f^{*}} : \mcal{K}(\mcal{O}_{X,(x)}) \rightarrow \mcal{K}(\mcal{O}_{Y,(y)})$ of \cite{Ye2} Theorem 7.4. \rmitem{ii} If $x$ is not closed in its fiber, set $\operatorname{Tr}_{f} := 0$. \end{enumerate} \end{dfn} \begin{prop} \label{prop1.2} Let $f : X \rightarrow Y$ be a finite morphism. \begin{enumerate} \rmitem{a} For any $y \in Y$ the homomorphism of $\mcal{O}_{Y}$-modules \[ \bigoplus_{x \mid y} f_{*} \mcal{K}_{X}(x) \rightarrow \mcal{H}om_{\mcal{O}_{Y}}(f_{*} \mcal{O}_{X}, \mcal{K}_{Y}(y)) \] induced by $\operatorname{Tr}_{f}$ is an isomorphism. \rmitem{b} Let $\eta = (y_{0}, \ldots, y_{n})$ be a saturated chain in $Y$. Then \[ \delta_{\eta} \operatorname{Tr}_{f} = \operatorname{Tr}_{f} \sum_{\xi \mid \eta} f_{*}(\delta_{\xi}) : \bigoplus_{x_{0} \mid y_{0}} f_{*} \mcal{K}_{X}(x_{0}) \rightarrow \mcal{K}_{Y}(y_{n}) . \] The sums are over saturated chains $\xi = (x_{0}, \ldots, x_{n})$ in $X$. \end{enumerate} \end{prop} \begin{proof} (a)\ By \cite{Ye1} Proposition 3.2.3 we have $\prod_{x \mid y} \mcal{O}_{X, (x)} \cong (f_{*} \mcal{O}_{X})_{(y)}$. Now use \cite{Ye2} Theorem 7.4 (iv). \medskip \noindent (b)\ Use the same diagram which appears in the proof of Proposition \ref{prop1.1}, only reverse the vertical arrows and label them $\operatorname{Tr}_{f}$. Then the commutativity follows from \cite{Ye2} Theorem 7.4 (i), (ii). \end{proof} In \cite{Ye1} \S 4.3 the notion of a system of residue data on a reduced scheme was introduced. \begin{prop} \label{prop1.3} Suppose $X$ is a reduced scheme. Then $(\{ \mcal{K}_{X}(x) \}, \{ \delta_{\xi} \},$ \linebreak $\{ \Psi_{\sigma}^{-1} \})$, where $x$ runs over the points of $X$, $\xi$ runs over the saturated chains in $X$, and $\sigma : k(x) \rightarrow \mcal{O}_{X,(x)}$ runs over all possible coefficient fields, is a system of residue data on $X$. \end{prop} \begin{proof} We must check condition (\dag) of \cite{Ye1} Definition 4.3.10. So let $\xi = (x, \ldots, y)$ be a saturated chain, and let $\sigma : k(x) \rightarrow \mcal{O}_{X,(x)}$ and $\tau : k(y) \rightarrow \mcal{O}_{X,(y)}$ be compatible coefficient fields. Denote also by $\tau$ the composed morphism $\partial^{+} \tau : k(y) \rightarrow \mcal{O}_{X,\xi}$. Then we get a coefficient field $\sigma_{\xi} : k(\xi) = k(x)_{\xi} \rightarrow \mcal{O}_{X,\xi}$ extending $\sigma$, which is a $k(y)$-algebra map via $\tau$. Consider the diagram: \bigskip \noindent \[ \begin{CD} \mcal{K}(\mcal{O}_{X,(x)}) @>\mrm{q}>> \mcal{K}(\mcal{O}_{X,\xi}) @>=>> \mcal{K}(\mcal{O}_{X,\xi}) @>\operatorname{Tr}>> \mcal{K}(\mcal{O}_{X,(y)}) \\ @V{\Psi_{\sigma}}VV @V{\Psi_{\sigma_{\xi}}}VV @V{\Psi_{\tau}}VV @V{\Psi_{\tau}}VV \\ \operatorname{Dual}_{\sigma} \mcal{O}_{X,(x)} @>{\mrm{q}_{\sigma}}>> \operatorname{Dual}_{\sigma_{\xi}} \mcal{O}_{X,\xi} @>{h}>> \operatorname{Dual}_{\tau} \mcal{O}_{X,\xi} @>{\operatorname{Tr}_{\tau}}>> \operatorname{Dual}_{\tau} \mcal{O}_{X, (y)} \end{CD} \] \medskip \noindent where for a $k(\xi)$-linear homomorphism $\phi : \mcal{O}_{X,\xi} \rightarrow \omega(k(\xi))$, $h(\phi) = $ \blnk{3mm} \linebreak $\operatorname{Res}_{k(\xi) / k(y)} \phi$ (cf.\ \cite{Ye2} Theorem 6.14). The left square commutes by \cite{Ye2} Proposition 7.2 (iii); the middle square commutes by ibid.\ Theorem 6.14 (i); and the right square commutes by ibid.\ Theorem 7.4 (i), (iii). But going along the bottom of the diagram we get $\operatorname{Tr}_{\tau} h \mrm{q}_{\sigma} = \delta_{\xi, \sigma / \tau}$, as defined in \cite{Ye1} Lemma 4.3.3. \end{proof} \begin{lem} \label{lem1.5} Let $\xi = (x, \ldots, y)$ and $\eta = (y, \ldots, z)$ be saturated chains in the scheme $X$, and let $\xi \vee \partial_{0} \eta := (x, \ldots, y, \ldots, z)$ be the concatenated chain. Then there is a canonical isomorphism of BCAs \[ \mcal{O}_{X,\xi \vee \partial_{0} \eta} \cong \mcal{O}_{X,\xi} \otimes^{(\wedge)}_{\mcal{O}_{X,(y)}} \mcal{O}_{X,\eta} \] \textup{(}intensification base change\textup{)}. \end{lem} \begin{proof} Choose a coefficient field $\sigma: k(y) \rightarrow \mcal{O}_{X,(y)}$. This induces a coefficient ring $\sigma_{\eta}: k(\eta) \rightarrow \mcal{O}_{X,\eta}$, and using \cite{Ye2} Theorem 3.8 and \cite{Ye1} Theorem 4.1.12 one gets \[ \mcal{O}_{X,\xi \vee \partial_{0} \eta} \cong \mcal{O}_{X,\xi} \otimes^{(\wedge)}_{k(y)} k(\eta) \cong \mcal{O}_{X,\xi} \otimes^{(\wedge)}_{\mcal{O}_{X,(y)}} \mcal{O}_{X,\eta} . \] \end{proof} \begin{lem} \label{lem1.3} \begin{enumerate} \item Let $\xi = (x, \ldots, y)$ and $\eta = (y, \ldots, z)$ be saturated chains and let $\xi \vee \partial_{0} \eta = (x, \ldots, y, \ldots, z)$. Then $\delta_{\eta} \delta_{\xi} = \delta_{\xi \vee \partial_{0} \eta}$. \item Given a point $x \in X$ and an element $\alpha \in \mcal{K}_{X}(x)$, for all but finitely many saturated chains $\xi = (x, \ldots)$ in $X$ one has $\delta_{\xi}(\alpha) = 0$. \item \textup{(}Residue Theorem\textup{)} Let $x,z \in X$ be points s.t.\ $z \in \overline{\{ x \}}$ and $\operatorname{codim}(\overline{\{ z \}},$ \linebreak $\overline{\{ x \}}) = 2$. Then $\sum_{y}\ \delta_{(x,y,z)} = 0$. \end{enumerate} \end{lem} \begin{proof} Using Lemma \ref{lem1.5} we see that part 1 is a consequence of the base change property of traces, cf.\ \cite{Ye2} Theorem 7.4 (ii). Assertions 2 and 3 are local, by Proposition \ref{prop1.1}, so we may assume there is a closed immersion $f : X \rightarrow \mbf{A}_{k}^{n}$. By Proposition \ref{prop1.2}, we can replace $X$ with $\mbf{A}_{k}^{n}$, and so assume that $X$ is reduced. Now according to Proposition \ref{prop1.3} and \cite{Ye1} Lemma 4.3.19, both 2 and 3 hold. \end{proof} \begin{dfn} For any integer $q$ define a quasi-coherent sheaf \[ \mcal{K}_{X}^{q} := \bigoplus_{\operatorname{dim} \overline{\{x\}} = -q} \mcal{K}_{X}(x) . \] By Lemma \ref{lem1.3} there an $\mcal{O}_{X}$-linear homomorphism \[ \delta := (-1)^{q+1} \sum_{(x,y)} \delta_{(x,y)} : \mcal{K}_{X}^{q} \rightarrow \mcal{K}_{X}^{q+1} , \] satisfying $\delta^{2} = 0$. The complex $(\mcal{K}_{X}^{{\textstyle \cdot}}, \delta)$ is called the {\em Grothendieck residue complex} of $X$. \end{dfn} In Corollary \ref{cor2.2} we will prove that $\mcal{K}_{X}^{{\textstyle \cdot}}$ is canonically isomorphic (in the derived category $\msf{D}(X)$) to $\pi^{!} k$, where $\pi : X \rightarrow \operatorname{Spec} k$ is the structural morphism. \begin{rem} A heuristic for the negative grading of $\mcal{K}_{X}^{{\textstyle \cdot}}$ and the sign $(-1)^{q+1}$ is that the residue complex is the ``$k$-linear dual'' of a hypothetical ``complex of localizations'' $\cdots \prod \mcal{O}_{X,y} \rightarrow \prod \mcal{O}_{X,x} \rightarrow \cdots$. Actually, there is a naturally defined complex which is built up from all localizations and completions: the Beilinson adeles $\underline{\mbb{A}}^{{\textstyle \cdot}}_{\mrm{red}}(\mcal{O}_{X})$ (cf.\ \cite{Be} and \cite{HY1}). $\underline{\mbb{A}}^{{\textstyle \cdot}}_{\mrm{red}}(\mcal{O}_{X})$ is a DGA, and $\mcal{K}_{X}^{{\textstyle \cdot}}$ is naturally a right DG-module over it. See \cite{Ye4}, and cf.\ also Remark \ref{rem5.3}. \end{rem} \begin{dfn} \label{eqn1.4} \begin{enumerate} \item Let $f:X \rightarrow Y$ be a morphism of schemes. Define a homomorphism of graded $\mcal{O}_{Y}$-modules $\operatorname{Tr}_{f} : f_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$ by summing the local trace maps of Definition \ref{dfn1.3}. \item Let $g : U \rightarrow X$ be an \'{e}tale morphism. Define $\mrm{q}_{g} : \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow g_{*} \mcal{K}_{U}^{{\textstyle \cdot}}$ by summing all local homomorphisms $\mrm{q}_{g}$ of Definition \ref{dfn1.2}. \end{enumerate} \end{dfn} \begin{thm} \label{thm1.1} Let $X$ be a $k$-scheme of finite type. \begin{enumerate} \rmitem{a} $(\mcal{K}_{X}^{{\textstyle \cdot}}, \delta)$ is a residual complex on $X$ \textup{(}cf.\ \cite{RD} Chapter \textup{VI \S 1)}. \rmitem{b} If $g : U \rightarrow X$ is an \'{e}tale morphism, then $1 \otimes \mrm{q}_{g} : g^{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{U}^{{\textstyle \cdot}}$ is an isomorphism of complexes. \rmitem{c} If $f : X \rightarrow Y$ is a finite morphism, then $\operatorname{Tr}_{f} : f_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$ is a homomorphism of complexes, and the induced map \[ f_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{H}om_{\mcal{O}_{Y}}(f_{*} \mcal{O}_{X}, \mcal{K}_{Y}^{{\textstyle \cdot}}) \] is an isomorphism of complexes. \rmitem{d} If $X$ is reduced, then $(\mcal{K}_{X}^{{\textstyle \cdot}}, \delta)$ is canonically isomorphic to the complex constructed in \cite{Ye1} \S \textup{4.3}. In particular, if $X$ is smooth irreducible of dimension $n$, there is a quasi-isomorphism $\mrm{C}_{X} : \Omega^{n}_{X/k}[n] \rightarrow \mcal{K}^{{\textstyle \cdot}}_{X}$. \end{enumerate} \end{thm} \begin{proof} Parts (b), (c), (d) are immediate consequences of Propositions \ref{prop1.1}, \ref{prop1.2} and \ref{prop1.3} here, and \cite{Ye1} Theorem 4.5.2. (Note that the sign of $\delta$ in \cite{Ye1} is different.) As for part (a), clearly $\mcal{K}_{X}^{{\textstyle \cdot}}$ is a direct sum of injective hulls of all the residue fields in $X$, with multiplicities $1$. It remains to prove that $\mcal{K}_{X}^{{\textstyle \cdot}}$ has coherent cohomology. Since this is a local question, we can assume using part (b) that $X$ is a closed subscheme of $\mbf{A}_{k}^{n}$. According to parts (c) and (d) of this theorem and \cite{Ye1} Corollary 4.5.6, $\mcal{K}_{X}^{{\textstyle \cdot}}$ has coherent cohomology. \end{proof} From part (b) of the theorem we get: \begin{cor} The presheaf $U \mapsto \Gamma(U, \mcal{K}_{U}^{{\textstyle \cdot}})$ is a sheaf on $X_{\mrm{et}}$, the small \'{e}tale site over $X$. \end{cor} \begin{dfn} \label{dfn1.6} For an $\mcal{O}_{X}$-module $\mcal{M}$ define dual complex \[ \operatorname{Dual}_{X} \mcal{M} := \mcal{H}om^{{\textstyle \cdot}}_{\mcal{O}_{X}}(\mcal{M}, \mcal{K}_{X}^{{\textstyle \cdot}}) . \] \end{dfn} Observe that since $\mcal{K}_{X}^{{\textstyle \cdot}}$ is complex of injectives the derived functor $\operatorname{Dual}_{X} : \msf{D}(X)^{\circ} \rightarrow \msf{D}(X)$ is defined. Moreover, since $\mcal{K}_{X}^{{\textstyle \cdot}}$ is dualizing, the adjunction morphism $1 \rightarrow \operatorname{Dual}_{X} \operatorname{Dual}_{X}$ is an isomorphism on $\msf{D}_{\mrm{c}}^{\mrm{b}}(X)$. We shall sometimes write $\operatorname{Dual} \mcal{M}$ instead of $\operatorname{Dual}_{X} \mcal{M}$. \section{Duality for Proper Morphisms} In this section we prove that if $f : X \rightarrow Y$ is a proper morphism of $k$-schemes, then the trace map $\operatorname{Tr}_{f}$ of Definition \ref{dfn1.3} is a homomorphism of complexes, and it induces a duality in the derived categories. \begin{prop} \label{prop2.1} Let $f: X \rightarrow Y$ be a proper morphism between finite type $k$-schemes, and let $\eta = (y_{0}, \ldots, y_{n})$ be a saturated chain in $Y$. Then there exists a canonical isomorphism of BCAs \[ \prod_{\xi \mid \eta} \mcal{O}_{X,\xi} \cong \prod_{x_{0} \mid y_{0}} \mcal{O}_{X,(x_{0})} \otimes^{(\wedge)}_{ \mcal{O}_{Y,(y_{0})}} \mcal{O}_{Y,\eta}\ , \] where $\xi = (x_{0}, \ldots, x_{n})$ denotes a variable chain in $X$ lying over $\eta$. \end{prop} \begin{proof} The proof is by induction on $n$. For $n=0$ this is trivial. Assume $n=1$. Let $Z := \overline{\{ x_{0} \}}_{\mrm{red}}$, so $\mcal{O}_{Z, x_{1}}$ is a $1$-dimensional local ring inside $k(Z) = k(x_{0})$. Considering the integral closure of $\mcal{O}_{Z, x_{1}}$ we see that $k((x_{0}, x_{1})) = k(x_{0})_{(x_{1})} = k(x_{0}) \otimes \mcal{O}_{Z, (x_{1})}$ is the product of the completions of $k(x_{0})$ at all discrete valuations centered on $x_{1} \in Z$ (cf.\ \cite{Ye1} Theorem 3.3.2). So by the valuative criterion for properness we get \begin{equation} \label{eqn2.3} \prod_{(x_{0}, x_{1}) \mid (y_{0}, y_{1})} k(x_{0})_{(x_{1})} \cong \prod_{x_{0} \mid y_{0}} k(x_{0}) \otimes_{k(y_{0})} k(y_{0})_{(y_{1})}. \end{equation} For $i \geq 1$ the morphism of BCAs \[ \prod_{x_{0} \mid y_{0}} (\mcal{O}_{X, x_{0}} / \mfrak{m}_{x_{0}}^{i}) \otimes_{\mcal{O}_{Y, (y_{0})}} \mcal{O}_{Y, (y_{0}, y_{1})} \rightarrow \prod_{(x_{0}, x_{1}) \mid (y_{0}, y_{1})} \mcal{O}_{X, (x_{0}, x_{1})} / \mfrak{m}_{(x_{0}, x_{1})}^{i} \] is bijective, since both sides are flat over $\mcal{O}_{X, x_{0}} / \mfrak{m}_{x_{0}}^{i}$, and by equation (\ref{eqn2.3}) (cf.\ \cite{Ye2} Proposition 3.5). Passing to the inverse limit in $i$ we get an isomorphism of BCAs \begin{equation} \label{eqn2.4} \prod_{x_{0} \mid y_{0}} \mcal{O}_{X, (x_{0})} \otimes^{(\wedge)}_{ \mcal{O}_{Y,(y_{0})}} \mcal{O}_{Y,(y_{0}, y_{1})} \cong \prod_{(x_{0}, x_{1}) \mid (y_{0}, y_{1})} \mcal{O}_{X,(x_{0}, x_{1})}\ . \end{equation} Now suppose $n \geq 2$. Then we get \[ \begin{array}{l} \prod_{x_{0} \mid y_{0}} \mcal{O}_{X,(x_{0})} \otimes^{(\wedge)}_{ \mcal{O}_{Y,(y_{0})}} \mcal{O}_{Y, \eta} \\[2mm] \begin{array}{rclc} \blnk{25mm} & \cong & \prod_{(x_{0}, x_{1}) \mid (y_{0}, y_{1})} \mcal{O}_{X,(x_{0}, x_{1})} \otimes^{(\wedge)}_{\mcal{O}_{Y,(y_{0}, y_{1})}} \mcal{O}_{Y, \eta} & \text{(i)} \\[2mm] & \cong & \prod_{(x_{0}, x_{1}) \mid (y_{0}, y_{1})} \mcal{O}_{X,(x_{0}, x_{1})} \otimes^{(\wedge)}_{\mcal{O}_{Y,(y_{1})}} \mcal{O}_{Y, \partial_{0} \eta} & \text{(ii)} \\[2mm] & \cong & \prod_{\xi \mid \eta} \mcal{O}_{X,(x_{0}, x_{1})} \otimes^{(\wedge)}_{\mcal{O}_{X,(x_{1})}} \mcal{O}_{X, \partial_{0} \xi} & \text{(iii)} \\[2mm] & \cong & \prod_{\xi \mid \eta} \mcal{O}_{X, \xi} & \text{(iv)} \end{array} \end{array} \] where associativity of intensification base change (\cite{Ye2} Proposition 3.10) is used repeatedly; in (i) we use formula (\ref{eqn2.4}); in (ii) we use Lemma \ref{lem1.5} applied to $\mcal{O}_{Y, \eta}$; in (iii) we use the induction hypothesis; and (iv) is another application of Lemma \ref{lem1.5}. \end{proof} The next theorem is our version of \cite{RD} Ch.\ VII Theorem 2.1: \begin{thm} \label{thm2.1} \textup{(Global Residue Theorem)}\ Let $f : X \rightarrow Y$ be a proper morphism between $k$-schemes of finite type. Then $\operatorname{Tr}_{f}: f_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$ is a homomorphism of complexes. \end{thm} \begin{proof} Fix a point $x_{0} \in X$, and let $y_{0} := f(x_{0})$. First assume that $x_{0}$ is closed in its fiber $X_{y_{0}} = f^{-1}(y_{0})$. Let $y_{1}$ be an immediate specialization of $y_{0}$. By Proposition \ref{prop2.1} we have \[ \prod_{x_{1} \mid y_{1}} \mcal{O}_{X, (x_{0}, x_{1})} \cong \mcal{O}_{X, (x_{0})} \otimes^{(\wedge)}_{\mcal{O}_{Y, (y_{0})}} \mcal{O}_{Y, (y_{0}, y_{1})}, \] so just as in Proposition \ref{prop1.2} (b), we get \[ \delta_{(y_{0}, y_{1})} \operatorname{Tr}_{f} = \sum_{x_{1} \mid y_{1}} \operatorname{Tr}_{f} \delta_{(x_{0}, x_{1})} : f_{*} \mcal{K}_{X}(x_{0}) \rightarrow \mcal{K}_{Y}(y_{1}) . \] Next assume $x_{0}, y_{0}$ are as above, but $x_{0}$ is not closed in the fiber $X_{y_{0}}$. The only possibility to have an immediate specialization $x_{1}$ of $x_{0}$ which is closed in its fiber, is if $x_{1} \in X_{y_{0}}$ and $Z := \overline{\{ x_{0} \}}_{\mrm{red}} \subset X_{y_{0}}$ is a curve. We have to show that \begin{equation} \label{eqn2.5} \sum_{x_{1} \mid y_{0}} \operatorname{Tr}_{f} \delta_{(x_{0}, x_{1})} = 0 : f_{*} \mcal{K}_{X}(x_{0}) \rightarrow \mcal{K}_{Y}(y_{0}) . \end{equation} Since $\mcal{K}_{Z}(x_{0}) \subset \mcal{K}_{X}(x_{0})$ is an essential submodule over $\mcal{O}_{Y, y_{0}}$ it suffices to check (\ref{eqn2.5}) on $\mcal{K}_{Z}(x_{0})$. Thus we may assume $X = \overline{\{ x_{0} \}}_{\mrm{red}}$ and $Y = \overline{\{ y_{0} \}}_{\mrm{red}}$. After factoring $X \rightarrow Y$ through a suitable finite radiciel morphism $X \rightarrow \tilde{X}$, and using Proposition \ref{prop1.2}, we may further assume that $K = k(Y) \rightarrow k(X)$ is separable. Now \[ \operatorname{Tr}_{f} \delta_{(x_{0}, x_{1})} = \operatorname{Res}_{k((x_{0}, x_{1})) / K} : \Omega^{n+1}_{k(X) / k} \rightarrow \Omega^{n}_{K / k} . \] Since $\Omega^{n+1}_{k(X) / k} = \Omega^{1}_{k(X) / k} \wedge \Omega^{n}_{K / k}$, it suffices to check that \[ \sum_{x_{1} \in X} \operatorname{Res}_{k((x_{0}, x_{1})) / K} = 0 : \Omega^{1}_{k(X) / k} \rightarrow K . \] Let $K'$ be the maximal purely inseparable extension of $K$ in an algebraic closure, and let $X' := X \times_{K} K'$. So \[ k((x_{0}, x_{1})) \otimes_{K} K' \cong \prod_{(x'_{0}, x'_{1}) \mid (x_{0}, x_{1})} k((x'_{0}, x'_{1})) \] where $(x'_{0}, x'_{1})$ are chains in $X'$. According to \cite{Ye1} Lemma 2.4.14 we may assume $k = K = K'$. Since now $K$ is perfect, we are in the position to use the well known Residue Theorem for curves (cf.\ \cite{Ye1} Theorem 4.2.15). \end{proof} \begin{cor} \label{cor2.1} Let $f : X \rightarrow Y$ be a morphism between $k$-schemes of finite type, and let $Z \subset X$ be a closed subscheme which is proper over $Y$. Then $\operatorname{Tr}_{f}: f_{*} \Gamma_{Z} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$ is a homomorphism of complexes. \end{cor} \begin{proof} Let $\mcal{I} \subset \mcal{O}_{X}$ be the ideal sheaf of $Z$, and define $Z_{n} := \mbf{Spec}\, \mcal{O}_{X} / \mcal{I}^{n+1}$, $n \geq 0$. The trace maps $\mcal{K}_{Z_{0}}^{{\textstyle \cdot}} \rightarrow \cdots \rightarrow \mcal{K}_{Z_{n}}^{{\textstyle \cdot}} \rightarrow \cdots \rightarrow \mcal{K}_{X}^{{\textstyle \cdot}}$ of Proposition \ref{prop1.2} induce a filtration by subcomplexes $\Gamma_{Z} \mcal{K}_{X}^{{\textstyle \cdot}} = \bigcup_{n=0}^{\infty} \mcal{K}_{Z_{n}}^{{\textstyle \cdot}}$. Now since each morphism $Z_{n} \rightarrow Y$ is proper, $\operatorname{Tr}_{f}: f_{*} \mcal{K}_{Z_{n}}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$ is a homomorphism of complexes. \end{proof} \begin{thm} \textup{(Duality)}\ \label{thm2.2} Let $f: X \rightarrow Y$ be a proper morphism between finite type $k$-schemes. Then for any complex $\mcal{M}^{{\textstyle \cdot}} \in \mathsf{D}^{\mrm{b}}_{\mrm{c}}(X)$, the homomorphism \[ \operatorname{Hom}_{\mathsf{D}(X)} (\mcal{M}^{{\textstyle \cdot}}, \mcal{K}_{X}^{{\textstyle \cdot}}) \rightarrow \operatorname{Hom}_{\mathsf{D}(Y)} (\mrm{R} f_{*} \mcal{M}^{{\textstyle \cdot}}, \mcal{K}_{Y}^{{\textstyle \cdot}}) \] induced by $\operatorname{Tr}_{f} : f_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$ is an isomorphism. \end{thm} \begin{proof} The proof uses a relative version of Sastry's notion of ``residue pairs'' and ``pointwise residue pairs'', cf.\ \cite{Ye1} Appendix. Define a residue pair relative to $f$ and $\mcal{K}_{Y}^{{\textstyle \cdot}}$, to be a pair $(\mcal{R}^{{\textstyle \cdot}}, t)$, with $\mcal{R}^{{\textstyle \cdot}}$ a residual complex on $X$, and with $t: f_{*} \mcal{R}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$ a homomorphism of complexes, which represent the functor $\mcal{M}^{{\textstyle \cdot}} \mapsto \operatorname{Hom}_{\mathsf{D}(Y)} (\mrm{R} f_{*} \mcal{M}^{{\textstyle \cdot}}, \mcal{K}_{Y}^{{\textstyle \cdot}})$ on $\mathsf{D}^{\mrm{b}}_{\mrm{c}}(X)$. Such pairs exist; for instance, we may take $\mcal{R}^{{\textstyle \cdot}}$ to be the Cousin complex $f^{\triangle} \mcal{K}_{Y}^{{\textstyle \cdot}}$ associated to the dualizing complex $f^{!} \mcal{K}_{Y}^{{\textstyle \cdot}}$ (cf.\ \cite{RD} ch.\ VII \S 3, or ibid.\ Appendix no.\ 4). A pointwise residue pair relative to $f$ and $\mcal{K}_{Y}^{{\textstyle \cdot}}$, is by definition a pair $(\mcal{R}^{{\textstyle \cdot}}, t)$ as above, but satisfying the condition: for any closed point $x \in X$, and any coherent $\mcal{O}_{X}$-module $\mcal{M}$ supported on $\{ x \}$, the map $\operatorname{Hom}_{\mcal{O}_{X}} (\mcal{M}, \mcal{R}^{{\textstyle \cdot}}) \rightarrow \operatorname{Hom}_{\mcal{O}_{Y}} (f_{*} \mcal{M}, \mcal{K}_{Y}^{{\textstyle \cdot}})$ induced by $t$ is an isomorphism. By the definition of the trace map $\operatorname{Tr}_{f}$, the pair $(\mcal{K}_{X}^{{\textstyle \cdot}}, \operatorname{Tr}_{f})$ is a pointwise residue pair. In fact, $k \rightarrow \mcal{O}_{Y, (f(x))} \rightarrow \mcal{O}_{X, (x)}$ are morphisms in $\mathsf{BCA}(k)$, and by \cite{Ye2} Theorem 7.4 (i),(iv) we get \[ \operatorname{Hom}_{\mcal{O}_{X}} (\mcal{M}, \mcal{K}_{X}^{{\textstyle \cdot}}) \cong \operatorname{Hom}_{\mcal{O}_{Y}} (f_{*} \mcal{M}, \mcal{K}_{Y}^{{\textstyle \cdot}}) \cong \operatorname{Hom}_{k} (\mcal{M}_{x}, k). \] The proof of \cite{Ye1} Appendix Theorem 2 goes through also in the relative situation: the morphism $\operatorname{Tr}_{f} : f_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$ in $\mathsf{D}(Y)$ corresponds to a morphism $\zeta: \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{R}^{{\textstyle \cdot}}$ in $\mathsf{D}(X)$. But since both $\mcal{K}_{X}^{{\textstyle \cdot}}$ and $\mcal{R}^{{\textstyle \cdot}}$ are residual complexes, $\zeta$ is an actual, unique homomorphism of complexes (cf.\ \cite{RD} Ch.\ IV Lemma 3.2). By testing on $\mcal{O}_{X}$-modules $\mcal{M}$ as above we see that $\zeta$ is indeed an isomorphism of complexes. So $(\mcal{K}_{X}^{{\textstyle \cdot}}, \operatorname{Tr}_{f})$ is a residue pair. \end{proof} Let $\pi: X \rightarrow \operatorname{Spec} k$ be the structural morphism. In \cite{RD} \S VII.3 we find the twisted inverse image functor $\pi^{!} : \mathsf{D}^{+}_{\operatorname{c}}(k) \rightarrow \mathsf{D}^{+}_{\operatorname{c}}(X)$. \begin{cor} \label{cor2.2} There is a canonical isomorphism $\zeta_{X} : \mcal{K}_{X}^{{\textstyle \cdot}} \stackrel{\simeq}{\rightarrow} \pi^{!} k$ in $\mathsf{D}(X)$. It is compatible with proper and \'{e}tale morphisms. If $\pi$ is proper then \[ \operatorname{Tr}_{\pi} = \operatorname{Tr}_{\pi}^{\mrm{RD}} \mrm{R} \pi_{*}(\zeta_{X}) : \pi_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow k \] where $\operatorname{Tr}_{\pi}^{\mrm{RD}} : \mrm{R} \pi_{*} \pi^{!} k \rightarrow k$ is the trace map of \cite{RD} \S \textup{VII.3}. \end{cor} \begin{proof} The uniqueness of $\zeta_{X}$ follows from considering closed subschemes $i_{Z} : Z \hookrightarrow X$ finite over $k$. This is because any endomorphism $a$ of $\mcal{K}_{X}^{{\textstyle \cdot}}$ in $\mathsf{D}(X)$ is a global section of $\mcal{O}_{X}$, and $a=1$ iff $i_{Z}^{*}(a) = 1$ for all such $Z$. Existence is proved by covering $X$ with compactifiable (e.g.\ affine) open sets and using Theorem \ref{thm2.2}, cf.\ \cite{Ye1} Appendix Theorem 3 and subsequent Exercise. In particular $\zeta_{X}$ is seen to be compatible with open immersions. Compatibility with proper morphisms follows from the transitivity of traces. As for an \'{e}tale morphism $g : U \rightarrow X$, one has $g^{*} \mcal{K}_{X}^{{\textstyle \cdot}} \cong \mcal{K}_{U}^{{\textstyle \cdot}}$ by Theorem \ref{thm1.1} (b), and also $g^{*} \pi^{!} k = g^{!} \pi^{!} k = (\pi g)^{!} k$. Testing the isomorphisms on subschemes $Z \subset U$ finite over $k$ shows that $g^{*}(\zeta_{X}) = \zeta_{U}$. \end{proof} \section{Duals of Differential Operators} Let $X$ be a $k$-scheme of finite type, where $k$ is a perfect field of any characteristic. Suppose $\mcal{M}, \mcal{N}$ are $\mcal{O}_{X}$-modules. By a differential operator (DO) $D : \mcal{M} \rightarrow \mcal{N}$ over $\mcal{O}_{X}$, relative to $k$, we mean in the sense of \cite{EGA} IV \S 16.8. Thus $D$ has order $\leq 0$ if $D$ is $\mcal{O}_{X}$-linear, and $D$ has order $\leq d$ if for all $a \in \mcal{O}_{X}$, the commutator $[D, a]$ has order $\leq d-1$. Recall that the dual of an $\mcal{O}_{X}$-module $\mcal{M}$ is $\operatorname{Dual} \mcal{M} = \mcal{H}om_{\mcal{O}_{X}}^{{\textstyle \cdot}}(\mcal{M}, \mcal{K}^{{\textstyle \cdot}}_{X})$. In this section we prove the existence of the dual operator $\operatorname{Dual}(D)$, in terms of BCAs and residues. This explicit description of $\operatorname{Dual}(D)$ will be needed for the applications in Sections 5-7. For direct proofs of existence cf.\ Remarks \ref{rem3.2} and \ref{rem3.3}. \begin{thm} \label{thm3.1} Let $\mcal{M},\mcal{N}$ be two $\mcal{O}_{X}$-modules, and let $D : \mcal{M} \rightarrow \mcal{N}$ be a differential operator of order $\leq d$. Then there is a homomorphism of graded sheaves \[ \operatorname{Dual}(D) : \operatorname{Dual} \mcal{N} \rightarrow \operatorname{Dual} \mcal{M} \] having the properties below: \begin{enumerate} \rmitem{i} $\operatorname{Dual}(D)$ is a DO of order $\leq d$. \rmitem{ii} $\operatorname{Dual}(D)$ is a homomorphism of complexes. \rmitem{iii} Functoriality: if $E : \mcal{N} \rightarrow \mcal{L}$ is another DO, then $\operatorname{Dual}(E D) = \operatorname{Dual}(D)\, \operatorname{Dual}(E)$. \rmitem{iv} If $d = 0$, i.e.\ $D$ is $\mcal{O}_{X}$-linear, then $\operatorname{Dual}(D)(\phi) = \phi \circ D$ for any $\phi \in \mcal{H}om_{\mcal{O}_{X}}^{{\textstyle \cdot}}(\mcal{N}, \mcal{K}^{{\textstyle \cdot}}_{X})$. \rmitem{v} Adjunction: under the homomorphisms $\mcal{M} \rightarrow \operatorname{Dual} \operatorname{Dual} \mcal{M}$ and $\mcal{N} \rightarrow \operatorname{Dual} \operatorname{Dual} \mcal{N}$, one has $D \mapsto \operatorname{Dual}(\operatorname{Dual}(D))$. \end{enumerate} \end{thm} \begin{proof} By \cite{RD} Theorem II.7.8, an $\mcal{O}_{X}$-module $\mcal{M}'$ is noetherian iff there is a surjection $\bigoplus_{i = 1}^{n} \mcal{O}_{U_{i}} \twoheadrightarrow \mcal{M}'$, for some open sets $U_{1}, \ldots, U_{n}$. Here $\mcal{O}_{U_{i}}$ is extended by $0$ to a sheaf on $X$. One has $\mcal{M} = \lim_{\alpha \rightarrow} \mcal{M}_{\alpha}$, where $\{ \mcal{M}_{\alpha} \}$ is the set of noetherian submodules of $\mcal{M}$. (We did not assume $\mcal{M},\mcal{N}$ are quasi-coherent!) So \[ \mcal{H}om_{\mcal{O}_{X}}(\mcal{M}, \mcal{K}^{q}_{X}) \cong \lim_{\leftarrow \alpha} \mcal{H}om_{\mcal{O}_{X}}(\mcal{M}_{\alpha}, \mcal{K}^{q}_{X}) . \] Since the sheaf $\mcal{P}^{d}_{X/k}$ of principal parts is coherent, and $D : \mcal{M}_{\alpha} \rightarrow \mcal{N}$ induces \[ \bigoplus_{i = 1}^{n} (\mcal{P}^{d}_{X/k} \otimes \mcal{O}_{U_{i}}) \twoheadrightarrow \mcal{P}^{d}_{X/k} \otimes \mcal{M}_{\alpha} \rightarrow \mcal{N} , \] we conclude that the module $\mcal{N}_{\alpha} := \mcal{O}_{X} \cdot D(\mcal{M}_{\alpha}) \subset \mcal{N}$ is also noetherian. Therefore we may assume that both $\mcal{M}, \mcal{N}$ are noetherian. We have \[ \mcal{H}om^{{\textstyle \cdot}}_{\mcal{O}_{X}}(\mcal{M}, \mcal{K}^{{\textstyle \cdot}}_{X}) = \bigoplus_{x} \mcal{H}om_{\mcal{O}_{X}}(\mcal{M}, \mcal{K}_{X}(x)) , \] and $\mcal{H}om_{\mcal{O}_{X}}(\mcal{M}, \mcal{K}_{X}(x))$ is a constant sheaf with support $\overline{ \{ x \} }$ and module \[ \mcal{H}om_{\mcal{O}_{X, x}}(\mcal{M}_{x}, \mcal{K}_{X}(x)) = \operatorname{Hom}_{A}(M, \mcal{K}(A)) = \operatorname{Dual}_{A} M , \] where $A := \widehat{\mcal{O}}_{X, x} = \mcal{O}_{X, (x)}$ and $M := A \otimes \mcal{M}_{x}$. Note that $M$ is a finitely generated $A$-module. $D : \mcal{M}_{x} \rightarrow \mcal{N}_{x}$ induces a continuous DO $D : M \rightarrow N = A \otimes \mcal{N}_{x}$ (for the $\mfrak{m}$-adic topology). According to \cite{Ye2} Theorem 8.6 there is a continuous DO \[ \operatorname{Dual}_{A}(D) : \operatorname{Dual}_{A} N \rightarrow \operatorname{Dual}_{A} M . \] Properties (i), (iii), (iv) and (v) follows directly from \cite{Ye2} Theorem 8.6 and Corollary 8.8. As for property (ii), consider any saturated chain $\xi = (x, \ldots, y)$. Since $\partial^{-} : \mcal{O}_{X, (x)} \rightarrow \mcal{O}_{X, \xi}$ is an intensification homomorphism, and since $\partial^{+} : \mcal{O}_{X, (y)} \rightarrow \mcal{O}_{X, \xi}$ is a morphism in $\mathsf{BCA}(k)$ which is also topologically \'{e}tale, we see that property (ii) is a consequence of \cite{Ye2} Thm.\ 8.6 and Cor.\ 8.12. \end{proof} Let $\mcal{D}_{X} := \mcal{D}\textit{iff}_{\mcal{O}_{X} / k}( \mcal{O}_{X}, \mcal{O}_{X})$ be the sheaf of differential operators on $X$. By definition $\mcal{O}_{X}$ is a left $\mcal{D}_{X}$-module. \begin{cor} \label{cor3.1} If $\mcal{M}$ is a left \textup{(}resp.\ right\textup{)} $\mcal{D}_{X}$-module, then $\operatorname{Dual} \mcal{M}$ is a complex of right \textup{(}resp.\ left\textup{)} $\mcal{D}_{X}$-modules. In particular this is true for $\mcal{K}_{X}^{{\textstyle \cdot}} = \operatorname{Dual} \mcal{O}_{X}$. \end{cor} \begin{cor} \label{cor3.2} Suppose $\mcal{M}^{{\textstyle \cdot}}$ is a complex sheaves, where each $\mcal{M}^{q}$ is an $\mcal{O}_{X}$-module, and $\mrm{d} : \mcal{M}^{q} \rightarrow \mcal{M}^{q + 1}$ is a DO. Then there is a dual complex $\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}}$. \end{cor} Specifically, $(\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}}, \mrm{D})$ is the simple complex associated to the double complex $(\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}})^{p, q} := \mcal{H}om_{\mcal{O}_{X}}(\mcal{M}^{-p}, \mcal{K}_{X}^{q})$. The operator is $\mrm{D} = \mrm{D}' + \mrm{D}''$, where \[ \begin{split} \mrm{D}' & := (-1)^{p+q+1} \operatorname{Dual}(\mrm{d}) : (\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}})^{p, q} \rightarrow (\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}})^{p + 1, q} , \\ \mrm{D}'' & := \delta : (\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}})^{p, q} \rightarrow (\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}})^{p, q + 1} . \end{split} \] It is well known that if $\operatorname{char} k = 0$ and $X$ is smooth of dimension $n$, then $\omega_{X} = \Omega^{n}_{X/k}$ is a right $\mcal{D}_{X}$-module. \begin{prop} \label{prop3.1} Suppose $\operatorname{char} k = 0$ and $X$ is smooth of dimension $n$. Then $\mrm{C}_{X} : \Omega^{n}_{X/k} \rightarrow \mcal{K}^{-n}_{X}$ \textup{(}the inclusion\textup{)} is $\mcal{D}_{X}$-linear. \end{prop} \begin{proof} It suffice to prove that any $\partial \in \mcal{T}_{X}$ (the tangent sheaf), which we view as a DO $\partial : \mcal{O}_{X} \rightarrow \mcal{O}_{X}$, satisfies $\operatorname{Dual}(\partial)(\alpha) = - \mrm{L}_{\partial}(\alpha)$, where $\mrm{L}_{\partial}$ is the Lie derivative, and $\alpha \in \Omega^{n}_{X/k}$. Localizing at the generic point of $X$ we get $\partial \in \mcal{D}(k(X))$ and $\alpha \in \omega(k(X))$. Now use \cite{Ye2} Definition 8.1 and Proposition 4.2. \end{proof} \begin{rem} \label{rem3.1} Proposition \ref{prop3.1} says that in the case $\operatorname{char} k = 0$ and $X$ smooth, the $\mcal{D}_{X}$-module structure on $\mcal{K}^{{\textstyle \cdot}}_{X}$ coincides with the standard one, which is obtained as follows. The quasi-isomorphism $\mrm{C}_{X} : \Omega^{n}_{X/k} \rightarrow \mcal{K}^{{\textstyle \cdot}}_{X}[-n]$ identifies $\mcal{K}^{{\textstyle \cdot}}_{X}[-n]$ with the Cousin complex of $\Omega^{n}_{X/k}$, which is computed in the category $\msf{Ab}(X)$ (cf.\ \cite{Ha} Section I.2). Since any $D \in \mcal{D}_{X}$ acts $\mbb{Z}$-linearly on $\Omega^{n}_{X/k}$, it also acts on $\mcal{K}^{{\textstyle \cdot}}_{X}[-n]$. \end{rem} \begin{rem} \label{rem3.2} According to \cite{Sai} there is a direct way to obtain Theorem \ref{thm3.1} in characteristic $0$. Say $X \subset Y$, with $Y$ smooth. Then $\mcal{H}om_{\mcal{O}_{X}}^{{\textstyle \cdot}}(\mcal{M}, \mcal{K}^{{\textstyle \cdot}}_{X}) \cong \mcal{H}om_{\mcal{O}_{Y}}^{{\textstyle \cdot}}(\mcal{M}, \mcal{K}^{{\textstyle \cdot}}_{Y})$. Now by \cite{Sai} \S 2.2.3 any DO $D : \mcal{M} \rightarrow \mcal{N}$ of order $\leq d$ can be viewed as \[ D \in \mcal{H}om_{\mcal{O}_{Y}}(\mcal{M}, \mcal{N} \otimes_{\mcal{O}_{Y}} \mcal{D}^{d}_{Y}) \subset \mcal{H}om_{\mcal{D}_{Y}}( \mcal{M} \otimes_{\mcal{O}_{Y}} \mcal{D}_{Y}, \mcal{N} \otimes_{\mcal{O}_{Y}} \mcal{D}_{Y}) \] (right $\mcal{D}_{Y}$-modules). Since $\mcal{K}^{q}_{Y}$ is a $\mcal{D}_{Y}$-module (cf.\ Remark \ref{rem3.1}), we get \[ \mcal{H}om_{\mcal{O}_{Y}}(\mcal{M}, \mcal{K}^{q}_{Y}) \cong \mcal{H}om_{\mcal{D}_{Y}}(\mcal{M} \otimes_{\mcal{O}_{Y}} \mcal{D}_{Y}, \mcal{K}^{q}_{Y}) \] and so we obtain the dual operator $\operatorname{Dual}(D)$. I thank the referee for pointing out this fact to me. \end{rem} \begin{rem} \label{rem3.3} Suppose $\operatorname{char} k = p > 0$. Then a $k$-linear map $D : \mcal{M} \rightarrow \mcal{N}$ is a DO over $\mcal{O}_{X}$ iff it is $\mcal{O}_{X^{(p^{n}/k)}}$-linear, for $n \gg 0$. Here $X^{(p / k)} \rightarrow X$ is the Frobenius morphism relative to $k$, cf.\ \cite{Ye1} Theorem 1.4.9. Since $\operatorname{Tr}$ induces an isomorphism \[ \mcal{H}om_{\mcal{O}_{X}}^{{\textstyle \cdot}}(\mcal{M}, \mcal{K}^{{\textstyle \cdot}}_{X}) \cong \mcal{H}om_{\mcal{O}_{X^{(p^{n}/k)}}}^{{\textstyle \cdot}}( \mcal{M}, \mcal{K}^{{\textstyle \cdot}}_{X^{(p^{n}/k)}}) \] we obtain the dual operator $\operatorname{Dual}(D)$. \end{rem} Let us finish this section with an application to rings of differential operators. Given a finitely generated (commutative) $k$-algebra $A$, denote by $\mcal{D}(A) := \operatorname{Diff}_{A/k}(A,A)$ the ring of differential operators over $A$. Such rings are of interest for ring theorists (cf.\ \cite{MR} and \cite{HoSt}). It is well known that if $\operatorname{char} k = 0$ and $A$ is smooth, then the opposite ring $\mcal{D}(A)^{\circ} \cong \omega_{A} \otimes_{A} \mcal{D}(A) \otimes_{A} \omega_{A}^{-1}$, where $\omega_{A} = \Omega^{n}_{A / k}$. The next theorem is a vast generalization of this fact. Given complexes $M^{{\textstyle \cdot}}, N^{{\textstyle \cdot}}$ of $A$-modules let $\operatorname{Diff}^{{\textstyle \cdot}}_{A / k}(M^{{\textstyle \cdot}}, N^{{\textstyle \cdot}})$ be the complex of $k$-modules which in degree $n$ is $\prod_{p} \operatorname{Diff}_{A / k}(M^{p}, N^{p+n})$. Let $\mcal{K}^{{\textstyle \cdot}}_{A} := \Gamma(X, \mcal{K}^{{\textstyle \cdot}}_{X})$ with $X := \operatorname{Spec} A$. By Corollary \ref{cor3.1}, it is a complex of right $\mcal{D}(A)$-modules. \begin{thm} \label{thm3.2} There is a natural isomorphism of filtered $k$-algebras \[ \mcal{D}(A)^{\circ} \cong \operatorname{H}^{0} \operatorname{Diff}_{A / k}^{{\textstyle \cdot}}( \mcal{K}_{A}^{{\textstyle \cdot}}, \mcal{K}_{A}^{{\textstyle \cdot}}) . \] \end{thm} \begin{proof} First observe that since DOs preserve support, $\operatorname{Diff}_{A / k}(\mcal{K}_{A}^{p}, \mcal{K}_{A}^{p-1}) = 0$ for all $p$. This means that every local section $D \in \operatorname{H}^{0} \operatorname{Diff}_{A / k}^{{\textstyle \cdot}}(\mcal{K}_{A}^{{\textstyle \cdot}}, \mcal{K}_{A}^{{\textstyle \cdot}})$ is a well defined DO $D : \mcal{K}_{A}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{A}^{{\textstyle \cdot}}$ which commutes with the coboundary $\delta$. Applying $\operatorname{Dual}$ and taking $0$-th cohomology we obtain a DO \[ D^{\vee} = \operatorname{H}^{0} \operatorname{Dual}(D) : \operatorname{H}^{0} \operatorname{Dual} \mcal{K}_{A}^{{\textstyle \cdot}} \rightarrow \operatorname{H}^{0} \operatorname{Dual} \mcal{K}_{A}^{{\textstyle \cdot}} . \] But $\operatorname{H}^{0} \operatorname{Dual} \mcal{K}_{A}^{{\textstyle \cdot}} = A$, so $D^{\vee} \in \mcal{D}(A)$. Finally according to Theorem \ref{thm3.1} (v), $D = D^{\vee \vee}$ for $D \in \operatorname{H}^{0} \operatorname{Diff}_{A / k}^{{\textstyle \cdot}}(\mcal{K}_{A}^{{\textstyle \cdot}}, \mcal{K}_{A}^{{\textstyle \cdot}})$ or $D \in \mcal{D}(A)$. \end{proof} Recall that an $n$-dimensional integral domain $A$ is a Gorenstein algebra iff $\omega_{A} = \mrm{H}^{-n} \mcal{K}_{A}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{A}^{{\textstyle \cdot}}[-n]$ is a quasi-isomorphism, and $\omega_{A}$ is invertible. \begin{cor} \label{cor3.6} If $A$ is a Gorenstein $k$-algebra, there is a canonical isomorphism of filtered $k$-algebras \[ \mcal{D}(A)^{\circ} \cong \operatorname{Diff}_{A / k}(\omega_{A}, \omega_{A}) \cong \omega_{A} \otimes_{A} \mcal{D}(A) \otimes_{A} \omega_{A}^{-1} . \] \end{cor} \begin{rem} In \cite{Ho}, the right $\mcal{D}(A)$-module structure on $\omega_{A}$ was exhibited, when $X = \operatorname{Spec} A$ is a curve. Corollary \ref{cor3.6} was proved there for complete intersection curves. \end{rem} \section{The De Rham-Residue Complex} As before $k$ is a perfect field of any characteristic. Let $X$ be a $k$-scheme of finite type. In this section we define a canonical complex on $X$, the De Rham-residue complex $\mcal{F}^{{\textstyle \cdot}}_{X}$. As we shall see in Corollary \ref{cor4.3}, $\mcal{F}^{{\textstyle \cdot}}_{X}$ coincides (up to indices and signs) with the double complex $\mcal{K}^{{\textstyle \cdot}, *}_{X}$ of \cite{EZ}. According to Theorem \ref{thm3.1}, if $\mcal{M}^{{\textstyle \cdot}}$ is a complex of sheaves, with each $\mcal{M}^{q}$ an $\mcal{O}_{X}$-module and $\mrm{d} : \mcal{M}^{q} \rightarrow \mcal{M}^{q+1}$ a DO, then $\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}}$ is a complex of the same kind. \begin{dfn} \label{dfn4.1} The {\em De Rham-residue complex} on $X$ is the complex \[ \mcal{F}_{X}^{{\textstyle \cdot}} := \operatorname{Dual} \Omega^{{\textstyle \cdot}}_{X / k} . \] of Corollary \ref{cor3.2}. \end{dfn} Note that the double complex $\mcal{F}_{X}^{{\textstyle \cdot} {\textstyle \cdot}}$ is concentrated in the third quadrant of the $(p,q)$-plane. \begin{prop} \label{prop4.2} $\mcal{F}_{X}^{{\textstyle \cdot}}$ is a right DG module over $\Omega^{{\textstyle \cdot}}_{X/k}$. \end{prop} \begin{proof} The graded module structure is clear. It remains to check that \[ \mrm{D}(\phi \cdot \alpha) = (\mrm{D} \phi) \cdot \alpha + (-1)^{p+q} \phi \cdot (\mrm{d} \alpha) \] for $\phi \in \mcal{F}_{X}^{p,q}$ and $\alpha \in \Omega^{p'}_{X/k}$. But this is a straightforward computation using Theorem \ref{thm3.1}. \end{proof} \begin{prop} \label{prop4.3} Let $g: U \rightarrow X$ be \'{e}tale. Then there is a homomorphism of complexes $\mrm{q}_{g} : \mcal{F}_{X}^{{\textstyle \cdot}} \rightarrow g_{*} \mcal{F}_{U}^{{\textstyle \cdot}}$, which induces an isomorphism of graded sheaves $1 \otimes \mrm{q}_{g} : g^{*} \mcal{F}_{X}^{{\textstyle \cdot}} \stackrel{\simeq}{\rightarrow} \mcal{F}_{U}^{{\textstyle \cdot}}$. \end{prop} \begin{proof} Consider the isomorphisms $g^{*} \Omega^{{\textstyle \cdot}}_{X/k} \cong \Omega^{{\textstyle \cdot}}_{U/k}$ and $1 \otimes \mrm{q}_{g} : g^{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{U}^{{\textstyle \cdot}}$ of Theorem \ref{thm1.1}. Clearly $1 \otimes \mrm{q}_{g} : g^{*} \mcal{F}_{X}^{p,q} \rightarrow \mcal{F}_{U}^{p,q}$ is an isomorphism. In light of \cite{Ye2} Theorem 8.6 (iv), $\mrm{q}_{g} : \mcal{F}_{X}^{{\textstyle \cdot}} \rightarrow g_{*} \mcal{F}_{U}^{{\textstyle \cdot}}$ is a homomorphism of complexes. \end{proof} Let $f: X \rightarrow Y$ be a morphism of schemes. Define a homomorphism of graded sheaves $\operatorname{Tr}_{f} : f_{*} \mcal{F}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{F}_{Y}^{{\textstyle \cdot}}$ by composing $f^{*} : \Omega^{{\textstyle \cdot}}_{Y/k} \rightarrow f_{*} \Omega^{{\textstyle \cdot}}_{X/k}$ with $\operatorname{Tr}_{f} : f_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$ of Definition \ref{dfn1.3}. \begin{prop} \label{prop4.1} $\operatorname{Tr}_{f}$ commutes with $\mrm{D}'$. If $f$ is proper then $\operatorname{Tr}_{f}$ also commutes with $\mrm{D}''$. \end{prop} \begin{proof} Let $y \in Y$ and let $x$ be a closed point in $f^{-1}(y)$. Then $f^{*}: \mcal{O}_{Y,(y)} \rightarrow \mcal{O}_{X,(x)}$ is a morphism in $\msf{BCA}(k)$. Applying \cite{Ye2} Cor.\ 8.12 to the DOs \[ \mrm{d} f^{*} = f^{*} \mrm{d} : \Omega^{p}_{Y/k, (y)} \rightarrow \Omega^{p+1}_{X/k, (x)} \] we get a dual homomorphism \[ \operatorname{Dual}_{f^{*}}(\mrm{d} f^{*}) = \operatorname{Dual}_{f^{*}}(f^{*} \mrm{d}) : \operatorname{Dual}_{\mcal{O}_{X,(x)}} \Omega^{p+1}_{X/k, (x)} \rightarrow \operatorname{Dual}_{\mcal{O}_{Y,(y)}} \Omega^{p}_{Y/k, (y)} , \] which equals both $\operatorname{Tr}_{f} \operatorname{Dual}_{X}(\mrm{d})$ and $\operatorname{Dual}_{Y}(\mrm{d}) \operatorname{Tr}_{f}$. The commutation of $\mrm{D}''$ with $\operatorname{Tr}_{f}$ in the proper case is immediate from Thm.\ \ref{thm2.1}. \end{proof} Of course if $f : X \rightarrow Y$ is a closed immersion, then $\operatorname{Tr}_{f}$ is injective, and it identifies $\mcal{F}_{X}^{{\textstyle \cdot}}$ with the subsheaf $\mcal{H}om_{\Omega^{{\textstyle \cdot}}_{Y / k}}(\Omega^{{\textstyle \cdot}}_{X / k}, \mcal{F}_{Y}^{{\textstyle \cdot}})$ of $\mcal{F}_{Y}^{{\textstyle \cdot}}$. Just as in Corollary \ref{cor2.1} we get: \begin{cor} \label{cor4.1} Let $f : X \rightarrow Y$ be a morphism of schemes, and let $Z \subset X$ be a closed subscheme which is proper over $Y$. Then the trace map $\operatorname{Tr}_{f}: f_{*} \Gamma_{Z} \mcal{F}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{F}_{Y}^{{\textstyle \cdot}}$ is a homomorphism of complexes. \end{cor} Suppose $X$ is an integral scheme of dimension $n$. The canonical homomorphism \begin{equation} \operatorname{C}_{X} : \Omega^{n}_{X/k} \rightarrow \mcal{K}_{X}^{-n} = k(X) \otimes_{\mcal{O}_{X}} \Omega^{n}_{X/k} \end{equation} can be viewed as a global section of $\mcal{F}_{X}^{-n, -n}$. \begin{lem} \label{lem4.3} Suppose $X$ is an integral scheme. Then $\mrm{D}' (\operatorname{C}_{X}) = \mrm{D}'' (\operatorname{C}_{X}) = 0$. \end{lem} \begin{proof} By \cite{Ye1} Section 4.5, $\mrm{D}''(\operatorname{C}_{X}) = \pm \delta (\operatorname{C}_{X}) = 0$. Next, let $K := k(X)$. Choose $t_{1}, \ldots t_{n} \in K$ such that $\Omega^{1}_{K / k} = \bigoplus K \cdot \mrm{d} t_{i}$. Taking products of the $\mrm{d} t_{i}$ as bases of $\Omega^{n - 1}_{K / k}$ and $\Omega^{n}_{K / k}$, we see from \cite{Ye2} Theorem 8.6 and Definition 8.1 that $\operatorname{Dual}_{K}(\operatorname{C}_{X}) = 0$. \end{proof} \begin{prop} \label{prop4.4} If $X$ is smooth irreducible of dimension $n$, then the DG homomorphism $\Omega^{{\textstyle \cdot}}_{X/k} \rightarrow \mcal{F}^{{\textstyle \cdot}}_{X}[-2n]$, $\alpha \mapsto \mrm{C}_{X} \cdot \alpha$, is a quasi-isomorphism. \end{prop} \begin{proof} First note that $\mrm{D}(\mrm{C}_{X}) = 0$, so this is indeed a DG homomorphism. Filtering these complexes according to the $p$-degree we reduce to looking at $\Omega^{p}_{X/k}[n] \rightarrow \mcal{F}^{p-n, {\textstyle \cdot}}_{X}$. That is a quasi-isomorphism by Theorem \ref{thm1.1} part d. \end{proof} \begin{cor} \label{cor4.3} The complex $\mcal{F}^{{\textstyle \cdot}}_{X}$ is the same as the complex $\mcal{K}^{{\textstyle \cdot}, *}_{X}$ of \cite{EZ}, up to signs and indexing. \end{cor} \begin{proof} If $X$ is smooth of dimension $n$ this is because $\mcal{F}^{{\textstyle \cdot}}_{X} \cong \Omega^{{\textstyle \cdot}}_{X/k}[n] \otimes \mcal{K}^{{\textstyle \cdot}}_{X}$ is the Cousin complex of $\bigoplus \Omega^{p}_{X/k}[p]$, and $\mrm{D}'$ is (up to sign) the Cousin functor applied to $\mrm{d}$. If $X$ is a general scheme embedded in a smooth scheme $Y$, use Proposition \ref{prop4.1}. \end{proof} \begin{dfn} \label{dfn4.4} Given a scheme $X$, let $X_{1}, \ldots, X_{r}$ be its irreducible components, with their induced reduced subscheme structures. For each $i$ let $x_{i}$ be the generic point of $X_{i}$, and let $f_{i} : X_{i} \rightarrow X$ be the inclusion morphism. We define the fundamental class $\operatorname{C}_{X}$ by: \[ \operatorname{C}_{X}:= \sum_{i=1}^{r} \operatorname{length}(\mcal{O}_{X, x_{i}}) \operatorname{Tr}_{f_{i}}(\operatorname{C}_{X_{i}}) \in \mcal{F}_{X}^{{\textstyle \cdot}} . \] \end{dfn} The next proposition is easily verified using Propositions \ref{prop4.1} and \ref{prop4.3}. It should be compared to \cite{EZ} Theorem III.3.1. \begin{prop} \label{prop4.7} For any scheme $X$, the fundamental class $\operatorname{C}_{X} \in$ \linebreak $\Gamma(X, \mcal{F}_{X}^{{\textstyle \cdot}})$ is annihilated by $\mrm{D}'$ and $\mrm{D}''$. If $X$ has pure dimension $n$, then $\operatorname{C}_{X}$ has bidegree $(-n,-n)$. If $f: X \rightarrow Y$ is a proper, surjective, generically finite morphism between integral schemes, then $\operatorname{Tr}_{f}(\operatorname{C}_{X}) = \operatorname{deg}(f) \operatorname{C}_{Y}$. If $g: U \rightarrow X$ is \'{e}tale, then $\mrm{C}_{U} = \mrm{q}_{g}(\mrm{C}_{X})$. \end{prop} \begin{rem} In \cite{Ye4} it is shown that $\mcal{F}_{X}^{{\textstyle \cdot}}$ is a right DG module over the DGA of Beilinson adeles $\mcal{A}_{X}^{{\textstyle \cdot}} = \underline{\mbb{A}}^{{\textstyle \cdot}}_{\mrm{red}}( \Omega^{{\textstyle \cdot}}_{X / k})$. Now let $\mcal{E}$ be a locally free $\mcal{O}_{X}$-module of rank $r$, and let $Z \subset X$ be the zero locus of a regular section of $\mcal{E}$. According to the adelic Chern-Weil theory of \cite{HY2} there is an adelic connection $\nabla$ on $\mcal{E}$ such that the Chern form $\mrm{c}_{r}(\mcal{E}; \nabla) \in \mcal{A}_{X}^{2r}$ satisfies $\mrm{C}_{Z} = \pm \mrm{C}_{X} \cdot \mrm{c}_{r}(\mcal{E}; \nabla) \in \mcal{F}_{X}^{{\textstyle \cdot}}$. \end{rem} \section{De Rham Homology and the Niveau Spectral Sequence} Let $X$ be a finite type scheme over a field $k$ of characteristic $0$. In \cite{Ye3} it is shown that if $X \subset \mfrak{X}$ is a smooth formal embedding (see below) then the De Rham complex $\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X} / k}$ calculates the De Rham cohomology $\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}}(X)$. In this section we will show that the De Rham-residue complex $\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}$ of $\mfrak{X}$ calculates the De Rham homology $\mrm{H}^{\mrm{DR}}_{{\textstyle \cdot}}(X)$. This is done by computing the niveau spectral sequence converging to $\mrm{H}^{{\textstyle \cdot}}(X, \mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}})$ (Theorem \ref{thm5.3}). We will draw a few conclusions, including the contravariance of homology w.r.t.\ \'{e}tale morphisms (Theorem \ref{thm5.1}). As a reference for algebraic De Rham (co)homology we suggest \cite{Ha}. Given a noetherian adic formal scheme $\mfrak{X}$ and a defining ideal $\mcal{I} \subset \mcal{O}_{\mfrak{X}}$, let $X_{n}$ be the (usual) noetherian scheme $(\mfrak{X}, \mcal{O}_{\mfrak{X}} / \mcal{I}^{n+1})$. Suppose $f : \mfrak{X} \rightarrow \mfrak{Y}$ is a morphism between such formal schemes, and let $\mcal{I} \subset \mcal{O}_{\mfrak{X}}$ and $\mcal{J} \subset \mcal{O}_{\mfrak{Y}}$ be defining ideals such that $f^{-1} \mcal{J} \cdot \mcal{O}_{\mfrak{X}} \subset \mcal{I}$. Such ideals are always available. We get a morphism of (usual) schemes $f_{0} : X_{0} \rightarrow Y_{0}$. \begin{dfn} \label{dfn5.2} A morphism $f : \mfrak{X} \rightarrow \mfrak{Y}$ between (noetherian) adic formal schemes is called {\em formally finite type} (resp.\ {\em formally finite} or {\em formally proper}) if the morphism $f_{0} : X_{0} \rightarrow Y_{0}$ is finite type (resp.\ finite or proper). \end{dfn} Obviously these notions are independent of the particular defining ideals chosen. \begin{exa} \label{exa5.1} If $X \rightarrow Y$ is a finite type morphism of noetherian schemes, $X_{0} \subset X$ is a locally closed subset and $\mfrak{X} = X_{/ X_{0}}$ is the formal completion, then $\mfrak{X} \rightarrow Y$ is formally finite type. Such a morphism is called {\em algebraizable}. \end{exa} \begin{dfn} A morphism of formal schemes $\mfrak{X} \rightarrow \mfrak{Y}$ is said to be {\em formally smooth} (resp.\ {\em formally \'{e}tale}) if, given a (usual) affine scheme $Z$ and a closed subscheme $Z_{0} \subset Z$ defined by a nilpotent ideal, the map $\operatorname{Hom}_{\mfrak{Y}}(Z, \mfrak{X})$ \linebreak $\rightarrow \operatorname{Hom}_{\mfrak{Y}}(Z_{0}, \mfrak{X})$ is surjective (resp.\ bijective). \end{dfn} This is the definition of formal smoothness used in \cite{EGA} IV Section 17.1. We shall also require the next notion. \begin{dfn} A morphism $g: \mfrak{X} \rightarrow \mfrak{Y}$ between noetherian formal sche\-mes is called {\em \'{e}tale} if it is of finite type (see \cite{EGA} I \S 10.13) and formally \'{e}tale. \end{dfn} Note that if $\mfrak{Y}$ is a usual scheme, then so is $\mfrak{X}$, and $g$ is an \'{e}tale morphism of schemes. \begin{dfn} \label{dfn5.3} A {\em smooth formal embedding} of $X$ (over $k$) is a closed immersion of $X$ into a formal scheme $\mfrak{X}$, which induces a homeomorphism on the underlying topological spaces, and such that $\mfrak{X}$ is of formally finite type and formally smooth over $k$. \end{dfn} \begin{exa} If $X$ is smooth over $Y = \operatorname{Spec} k$ and $X_{0}, \mfrak{X}$ are as in Example \ref{exa5.1}, then $X_{0} \subset \mfrak{X}$ is a smooth formal embedding. \end{exa} Let $\xi = (x_{0}, \ldots, x_{q})$ be a saturated chain of points in the formal scheme $\mfrak{X}$. Choose a defining ideal $\mcal{I}$, and let $X_{n}$ be as above. Define the Beilinson completion $\mcal{O}_{\mfrak{X}, \xi} := \lim_{\leftarrow n} \mcal{O}_{X_{n}, \xi}$ (which of course is independent of $\mcal{I}$). \begin{lem} \label{lem5.2} Let $\mfrak{X}$ be formally finite type over $k$, and let $\xi$ be a saturated chain in $\mfrak{X}$. Then $\mcal{O}_{\mfrak{X}, \xi}$ is a BCA over $k$. If $\mfrak{X} = X_{/ X_{0}}$, then $\mcal{O}_{\mfrak{X}, \xi} \cong \mcal{O}_{X, \xi}$. \end{lem} \begin{proof} First assume $\mfrak{X} = X_{/ X_{0}}$. Taking $\mcal{I}$ to be the ideal of $X_{0}$ in $X$, we have \begin{multline*} \hspace{1cm} \mcal{O}_{\mfrak{X}, \xi} = \lim_{\leftarrow n} (\mcal{O}_{X} / \mcal{I}^{n})_{\xi} \cong \lim_{\leftarrow m,n} \mcal{O}_{X, \xi} / (\mcal{I}^{n} \mcal{O}_{X, \xi} + \mfrak{m}_{\xi}^{m}) \\ \cong \lim_{\leftarrow m} \mcal{O}_{X, \xi} / \mfrak{m}_{\xi}^{m} = \mcal{O}_{X, \xi} . \hspace{1cm} \end{multline*} Now by \cite{Ye3} Proposition 1.20 and Lemma 1.1, locally there is a closed immersion $\mfrak{X} \subset \mfrak{Y}$, with $\mfrak{Y}$ algebraizable (i.e.\ $\mfrak{Y} = Y_{/ Y_{0}}$). So there is a surjection $\mcal{O}_{\mfrak{Y}, \xi} \rightarrow \mcal{O}_{\mfrak{X}, \xi}$, and this implies that $\mcal{O}_{\mfrak{X}, \xi}$ is a BCA. \end{proof} One can construct the complexes $\mcal{K}_{\mfrak{X}}^{{\textstyle \cdot}}$ and $\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}$ for a formally finite type formal scheme $\mfrak{X}$, as follows. Define $\mcal{K}_{\mfrak{X}}(x) := \mcal{K}(\mcal{O}_{\mfrak{X}, (x)})$. Now let $(x,y)$ be a saturated chain. Then there is an intensification homomorphism $\partial^{-}: \mcal{O}_{\mfrak{X}, (x)} \rightarrow \mcal{O}_{\mfrak{X}, (x,y)}$ and a morphism of BCAs $\partial^{+}: \mcal{O}_{\mfrak{X}, (y)} \rightarrow \mcal{O}_{\mfrak{X}, (x,y)}$. Therefore we get a homomorphism of $\mcal{O}_{\mfrak{X}}$-modules $\delta_{(x,y)}: \mcal{K}_{\mfrak{X}}(x) \rightarrow \mcal{K}_{\mfrak{X}}(y)$. Define a graded sheaf $\mcal{K}_{\mfrak{X}}^{{\textstyle \cdot}} = \bigoplus_{x \in \mfrak{X}} \mcal{K}_{\mfrak{X}}(x)$ on $\mfrak{X}$, as in \S 1. Let $\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}/k}$ be the complete De Rham complex on $\mfrak{X}$, and set $\mcal{F}_{\mfrak{X}}^{p, q} := \mcal{H}om_{\mcal{O}_{\mfrak{X}}}( \widehat{\Omega}^{-p}_{\mfrak{X}/k}, \mcal{K}_{\mfrak{X}}^{q})$. \begin{prop} \label{prop5.5} Let $\mfrak{X}$ be a formally finite type formal scheme over $k$. \begin{enumerate} \item $\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}$ is a complex. \item If $g : \mfrak{U} \rightarrow \mfrak{X}$ is \'{e}tale, then there is a homomorphism of complexes $\mrm{q}_{g} : \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}} \rightarrow g_{*} \mcal{F}_{\mfrak{U}}^{{\textstyle \cdot}}$, which induces an isomorphism of graded sheaves $1 \otimes \mrm{q}_{g} : g^{*} \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}} \stackrel{\simeq}{\rightarrow} \mcal{F}_{\mfrak{U}}^{{\textstyle \cdot}}$. \item If $f : \mfrak{X} \rightarrow \mfrak{Y}$ is formally proper, then there is a homomorphism of complexes $\operatorname{Tr}_{f} : f^{*} \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}} \rightarrow \mcal{F}_{\mfrak{Y}}^{{\textstyle \cdot}}$. \end{enumerate} \end{prop} \begin{proof} 1.\ Let $X_{n} \subset \mfrak{X}$ be as before. Then one has $\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}} = \bigcup \mcal{F}_{X_{n}}^{{\textstyle \cdot}}$, so this is a complex.\\ 2.\ Take $U_{n} := \mfrak{U} \times_{\mfrak{X}} X_{n}$; then each $U_{n} \rightarrow X_{n}$ is an \'{e}tale morphism of schemes, and we can use Proposition \ref{prop4.3}.\\ 3.\ Apply Proposition \ref{prop4.1} to $X_{n} \rightarrow Y_{n}$. \end{proof} \begin{prop} \label{prop5.1} Assume $\mfrak{X} = Y_{/X}$ for some smooth irreducible scheme $Y$ of dimension $n$ and closed set $X \subset Y$. Then there is a natural isomorphism of complexes \begin{equation} \label{eqn5.3} \mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}} \cong \underline{\Gamma}_{X} \mcal{F}^{{\textstyle \cdot}}_{Y} . \end{equation} Hence $\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}} \cong \mrm{R} \underline{\Gamma}_{X} \Omega^{{\textstyle \cdot}}_{Y / k}[2n]$ in the derived category $\mathsf{D}(\mathsf{Ab}(Y))$, and consequently \[ \mrm{H}^{-q}(X, \mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}) \cong \mrm{H}^{2n-q}_{X}(Y, \Omega^{{\textstyle \cdot}}_{Y / k}) = \mrm{H}^{\mrm{DR}}_{q}(X) . \] \end{prop} \begin{proof} The isomorphism (\ref{eqn5.3}) is immediate from Lemma \ref{lem5.2}. But according to Proposition \ref{prop4.4}, $\mcal{F}^{{\textstyle \cdot}}_{Y}$ is a flasque resolution of $\Omega^{{\textstyle \cdot}}_{Y/k}[2n]$ in $\mathsf{Ab}(Y)$. \end{proof} We need some algebraic results, phrased in the terminology of \cite{Ye1} \S 1. Let $K$ be a complete, separated semi-topological (ST) commutative $k$-algebra, and let $\underline{t} = (t_{1}, \ldots, t_{n})$ be a sequence of indeterminates. Let $K[[\, \underline{t}\, ]]$ and $K((\underline{t}))$ be the rings of formal power series, and of iterated Laurent series, respectively. These are complete, separated ST $k$-algebras. Let $T$ be the free $k$-module with basis $\alpha_{1}, \ldots, \alpha_{n}$ and let $\bigwedge_{k}^{{\textstyle \cdot}} T$ be the exterior algebra over $k$. \begin{lem} \label{lem5.1} \textup{(``Poincar\'{e} Lemma'')}\ The DGA homomorphisms \[ \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K / k} \rightarrow \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K[[\, \underline{t}\, ]] / k} \] and \[ \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K / k} \otimes_{k} \bigwedge\nolimits_{k}^{{\textstyle \cdot}} T \rightarrow \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K((\underline{t})) / k},\ \alpha_{i} \mapsto \operatorname{dlog} t_{i} \] are quasi-isomorphisms. \end{lem} \begin{proof} Since \[ \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K[[\, \underline{t}\, ]] / k} \cong K[[\, \underline{t}\, ]] \otimes_{k[\, \underline{t}\, ]} \Omega^{{\textstyle \cdot}}_{k[\, \underline{t}\, ] / k} \] the homotopy operator (``integration'') of the Poincar\'{e} Lemma for the graded polynomial algebra $k[\, \underline{t}\, ]$ works here also. For $K((t))$ (i.e.\ $n = 1$) we have \[ \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K((t)) / k} \cong \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K[[t]] / k} \oplus \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K[t^{-1}] / k} \wedge \operatorname{dlog} t \] so we have a quasi-isomorphism. For $n > 1$ use induction on $n$ and the K\"{u}nneth formula. \end{proof} \begin{lem} \label{lem5.4} Suppose $A$ is a local BCA and $\sigma, \sigma' : K \rightarrow A$ are two coefficient fields. Then \[ \mrm{H}(\sigma) = \mrm{H}(\sigma') : \mrm{H} \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K/k} \rightarrow \mrm{H} \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{A/k} . \] \end{lem} \begin{proof} Choosing generators for the maximal ideal of $A$, $\sigma$ induces a surjection of BCAs $\tilde{A} = K [[\, \underline{t}\, ]] \rightarrow A$. Denote by $\tilde{\sigma} : K \rightarrow \tilde{A}$ the inclusion. The coefficient field $\sigma'$ lifts to some coefficient field $\tilde{\sigma}' : K \rightarrow \tilde{A}$. It suffices to show that $\mrm{H}(\tilde{\sigma}) = \mrm{H}(\tilde{\sigma}')$. But by Lemma \ref{lem5.1} both of these are bijective, and using the projection $\tilde{A} \rightarrow K$ we see they are in fact equal. \end{proof} Given a saturated chain $\xi = (x, \ldots, y)$ in $X$ and a coefficient field $\sigma: k(y) \rightarrow \mcal{O}_{X, (y)}$, there is the {\em Parshin residue map} \[ \operatorname{Res}_{\xi, \sigma}: \Omega^{{\textstyle \cdot}}_{k(x)/k} \rightarrow \Omega^{{\textstyle \cdot}}_{k(y)/k} \] (cf.\ \cite{Ye1} Definition 4.1.3). It is a map of DG $k$-modules of degree equal to $-(\text{length of } \xi)$. \begin{prop} \label{prop5.2} Let $\xi = (x, \ldots, y)$ be a saturated chain in $X$. Then the map of graded $k$-modules \[ \operatorname{Res}_{\xi} := \mrm{H}(\operatorname{Res}_{\xi, \sigma}) : \mrm{H} \Omega^{{\textstyle \cdot}}_{k(x)/k} \rightarrow \mrm{H} \Omega^{{\textstyle \cdot}}_{k(y)/k} \] is independent of the coefficient field $\sigma$. \end{prop} \begin{proof} Say $\xi$ has length $n$. Let $L$ be one of the local factors of $k(\xi) = k(x)_{\xi}$, so it is an $n$-dimensional topological local field (TLF). Let $K := \kappa_{n}(L)$, the last residue field of $L$, which is a finite separable $k(y)$-algebra. Then $\sigma$ extends uniquely to a morphism of TLFs $\sigma: K \rightarrow L$, and it is certainly enough to check that \begin{equation} \label{eqn5.4} \mrm{H}(\operatorname{Res}_{L/K; \sigma}) : \mrm{H} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{L/k} \rightarrow \mrm{H} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{K/k} \end{equation} is independent of $\sigma$. After choosing a regular system of parameters $\underline{t} = (t_{1}, \ldots, t_{n})$ in $L$ we get $L \cong K((\underline{t}))$. According to Lemma \ref{lem5.1}, $\mrm{H}(\sigma)$ induces an isomorphism of $k$-algebras \begin{equation} \label{eqn5.5} \mrm{H} \Omega^{{\textstyle \cdot}}_{K/k} \otimes_{k} \bigwedge\nolimits_{k}^{{\textstyle \cdot}} T \cong \mrm{H} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{L / k} . \end{equation} But by Lemma \ref{lem5.4} this isomorphism is independent of $\sigma$. The map (\ref{eqn5.4}) is $\mrm{H} \Omega^{{\textstyle \cdot}}_{K/k}$-linear, and it sends $\bigwedge^{p}_{k} T$ to $0$ if $p < n$, and $\operatorname{dlog} t_{1} \wedge \cdots \wedge \operatorname{dlog} t_{n} \mapsto 1$. Hence (\ref{eqn5.4}) is independent of $\sigma$. \end{proof} The topological space $X$ has an increasing filtration by families of supports $\emptyset = X_{-1} \subset X_{0} \subset X_{1} \subset \cdots$, where \[ X_{q} := \{ Z \subset X \mid Z \text{ is closed and } \operatorname{dim} Z \leq q \} . \] We write $x \in X_{q} / X_{q-1}$ if $\overline{\{x\}} \in X_{q} - X_{q-1}$, and the set $X_{q} / X_{q-1}$ is called the $q$-skeleton of $X$. (This notation is in accordance with \cite{BlO}; in \cite{Ye1} $X_{q}$ denotes the $q$-skeleton.) The {\em niveau filtration} on $\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}$ is $\mrm{N}_{q} \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}} := \underline{\Gamma}_{X_{q}} \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}$. Let us write $X^{q} / X^{q + 1} := X_{-q} / X_{-q - 1}$ and $\mrm{N}^{q} := \mrm{N}_{-q}$, so $\{\mrm{N}^{q} \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}\}$ is a decreasing filtration. \begin{thm} \label{thm5.3} Suppose $\operatorname{char} k = 0$ and $X \subset \mfrak{X}$ is a smooth formal embedding. Then in the niveau spectral sequence converging to $\mrm{H}^{{\textstyle \cdot}}(X, \mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}})$, the $E_{1}$ term is \textup{(}in the notation of \cite{ML} Chapter \textup{XI):} \[ E_{1}^{p,q} = \mrm{H}^{p+q}_{X^{p} / X^{p + 1}}(X, \mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}) \cong \bigoplus_{x \in X^{p} / X^{p + 1}} \mrm{H}^{q - p} \Omega^{{\textstyle \cdot}}_{k(x)/k} , \] and the coboundary operator is $(-1)^{p + 1} \sum_{(x,y)} \operatorname{Res}_{(x,y)}$. \end{thm} \begin{proof} We shall substitute indices $(p, q) \mapsto (-q, -p)$; this puts us in the first quadrant. Because $\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}$ is a complex of flasque sheaves, one has \[ E_{1}^{-q, -p} = \mrm{H}^{-p - q}_{X^{-q} / X^{-q + 1}}(X, \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}) \cong \bigoplus_{x \in X_{q} / X_{q-1}} \mrm{H}^{-p} \mcal{H}om^{{\textstyle \cdot}}_{\mcal{O}_{\mfrak{X}}} \left( \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}/k}, \mcal{K}_{\mfrak{X}}(x) \right) \] (the operator $\delta$ is trivial on the $q$-skeleton). Fix a point $x$ of dimension $q$ and let $B := \mcal{O}_{\mfrak{X}, (x)}$. Then $\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}/k, (x)} \cong \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B / k}$ and by definition \[ \mcal{H}om^{{\textstyle \cdot}}_{\mcal{O}_{\mfrak{X}}} \left( \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}/k}, \mcal{K}_{\mfrak{X}}(x) \right) \cong \operatorname{Dual}_{B} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B / k} . \] Choose a coefficient field $\sigma: K = k(x) \rightarrow B$. By \cite{Ye2} Theorem 8.6 there is an isomorphism of complexes \[ \Psi_{\sigma} : \operatorname{Dual}_{B} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B / k} \stackrel{\simeq}{\rightarrow} \operatorname{Dual}_{\sigma} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B/k} = \operatorname{Hom}^{\mrm{cont}}_{K; \sigma} (\Omega^{{\textstyle \cdot}, \operatorname{sep}}_{B/k}, \omega(K)) . \] Here $\omega(K) = \Omega^{q}_{K/k}$ and the operator on the right is $\operatorname{Dual}_{\sigma}(\mrm{d})$ of \cite{Ye2} Definition 8.1. According to \cite{Ye3} \S 3, $k \rightarrow B$ is formally smooth; so $B$ is a regular local ring, and hence $B \cong K[[\, \underline{t}\, ]]$. Put a grading on $\Omega^{{\textstyle \cdot}}_{K [\, \underline{t}\, ] / k}$ by declaring $\operatorname{deg} t_{i} = \operatorname{deg} \mrm{d} t_{i} = 1$, and let $V_{l} \subset \Omega^{{\textstyle \cdot}}_{K [\, \underline{t}\, ] / k}$ be the homogeneous component of degree $l$. In particular $V_{0} = \Omega^{{\textstyle \cdot}}_{K / k}$. Since $\mrm{d}$ preserves each $V_{l}$, from the definition of $\operatorname{Dual}_{\sigma}(\mrm{d})$ we see that \[ \operatorname{Dual}_{\sigma} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B/k} = \bigoplus_{l=0}^{\infty} \operatorname{Hom}_{K}(V_{l}, \omega(K)) \] as complexes. Because the $K$-linear homotopy operator in the proof of Lemma \ref{lem5.1} also preserves $V_{l}$ we get $\mrm{H} \operatorname{Hom}_{K}(V_{l}, \omega(K)) = 0$ for $l \neq 0$, and hence \begin{equation} \label {eqn5.2} \mrm{H}^{-p} \operatorname{Dual}_{\sigma} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B/k} \cong \mrm{H}^{-p} \operatorname{Hom}_{K} (\Omega^{{\textstyle \cdot}}_{K/k}, \omega(K)) \cong \mrm{H}^{q-p} \Omega^{{\textstyle \cdot}}_{K/k} \end{equation} (cf.\ proof of Lemma \ref{lem4.3}). It remains to check that the coboundary maps match up. Given an immediate specialization $(x,y)$, choose a pair of compatible coefficient fields $\sigma: k(x) \rightarrow \mcal{O}_{\mfrak{X}, (x)} = B$ and $\tau: k(y) \rightarrow \mcal{O}_{\mfrak{X}, (y)} = A$ (cf.\ \cite{Ye1} Definition 4.1.5). Set $\widehat{B} := \mcal{O}_{\mfrak{X}, (x,y)}$, so $f : A \rightarrow \widehat{B}$ is a morphism of BCAs. A cohomology class $[\phi] \in \mrm{H}^{-p} \operatorname{Dual}_{B} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B / k}$ is sent under the isomorphism (\ref{eqn5.2}) to the class $[\beta]$ of some form $\beta \in \Omega^{q-p}_{k(x)/k}$, such that $\mrm{d} \beta = 0$ and on $\sigma(\Omega^{{\textstyle \cdot}}_{k(x)/k}) \subset \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B/k}$, $\phi$ acts like left multiplication by $\beta$. So for $\alpha \in \Omega^{p}_{k(y)/k}$, \begin{eqnarray*} \operatorname{Tr}_{A / k(y)} \operatorname{Tr}_{\widehat{B} / A} \phi f \tau (\alpha) & = & \operatorname{Res}_{k((x,y)) / k(y); \tau}(\beta \wedge \tau(\alpha)) \\ & = & \operatorname{Res}_{k((x,y)) / k(y); \tau}(\beta) \wedge \alpha . \end{eqnarray*} This says that under the isomorphism \[ \mrm{H}^{-p} \operatorname{Dual}_{A} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{A / k} \cong \mrm{H}^{q-p-1} \Omega^{{\textstyle \cdot}}_{k(y)/k} , \] the class $\delta_{(x,y)}([\phi])$ is sent to $\operatorname{Res}_{(x,y)}([\beta])$. \end{proof} \begin{rem} Theorem \ref{thm5.3}, but with $\operatorname{R} \underline{\Gamma}_{X} \Omega^{{\textstyle \cdot}}_{Y / k}$ instead of $\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}$ (cf.\ Proposition \ref{prop5.1}), was discovered by Grothendieck (see \cite{Gr} Footnotes 8,9), and proved by Bloch-Ogus \cite{BlO}. Our proof is completely different from that in \cite{BlO}, and in particular we obtain the formula for the coboundary operator as a sum of Parshin residues. On the other hand the proof in \cite{BlO} is valid for a general homology theory (including $l$-adic homology). Bloch-Ogus went on to prove additional important results, such as the degeneration of the sheafified spectral sequence $\mcal{E}^{p,q}_{r}$ at $r=2$, for $X$ smooth. \end{rem} The next result is a generalization of \cite{Ha} Theorem II.3.2. Suppose $X \subset \mfrak{Y}$ is another smooth formal embedding. By a morphism of embeddings $f : \mfrak{X} \rightarrow \mfrak{Y}$ we mean a morphism of formal schemes inducing the identity on $X$. Since $f$ is formally finite, according to Proposition \ref{prop5.5}, $\operatorname{Tr}_{f} : \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}} \rightarrow \mcal{F}_{\mfrak{Y}}^{{\textstyle \cdot}}$ is a map of complexes in $\mathsf{Ab}(X)$. \begin{cor} \label{cor5.5} Let $f : \mfrak{X} \rightarrow \mfrak{Y}$ be a morphism of embeddings of $X$. Then $\operatorname{Tr}_{f} : \Gamma(X, \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}) \rightarrow \Gamma(X, \mcal{F}_{\mfrak{Y}}^{{\textstyle \cdot}})$ is a quasi-isomorphism. If $g : \mfrak{X} \rightarrow \mfrak{Y}$ is another such morphism, then $\mrm{H}(\operatorname{Tr}_{f}) = \mrm{H}(\operatorname{Tr}_{g})$. \end{cor} \begin{proof} $\operatorname{Tr}_{f}$ induces a map of niveau spectral sequences $E^{p,q}_{r}(\mfrak{X}) \rightarrow E^{p,q}_{r}(\mfrak{Y})$. The theorem and its proof imply that these spectral sequences coincide for $r \geq 1$, hence $\mrm{H}^{{\textstyle \cdot}}(\operatorname{Tr}_{f})$ is an isomorphism. The other statement is proved like in \cite{Ye3} Theorem 2.7 (cf.\ next corollary). \end{proof} \begin{cor} \label{cor5.1} The $k$-module $\mrm{H}^{q}(X, \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}})$ is independent of the smooth formal embedding $X \subset \mfrak{X}$. \end{cor} \begin{proof} As shown in \cite{Ye3}, given any two embeddings $X \subset \mfrak{X}$ and $X \subset \mfrak{Y}$, the completion of their product along the diagonal $(\mfrak{X} \times_{k} \mfrak{Y})_{/X}$ is also a smooth formal embedding of $X$, and it projects to both $\mfrak{X}$ and $\mfrak{Y}$. Therefore by Corollary \ref{cor5.5}, $\mrm{H}^{q}(X, \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}})$ and $\mrm{H}^{q}(X, \mcal{F}_{\mfrak{Y}}^{{\textstyle \cdot}})$ are isomorphic. Using triple products we see this isomorphism is canonical. \end{proof} \begin{rem} \label{rem5.2} We can use Corollary \ref{cor5.1} to {\em define} $\mrm{H}^{\mrm{DR}}_{{\textstyle \cdot}}(X)$ if some smooth formal embedding exists. For a definition of $\mrm{H}^{\mrm{DR}}_{{\textstyle \cdot}}(X)$ in general, using a system of local embeddings, see \cite{Ye3} (cf.\ \cite{Ha} pp.\ 28-29). \end{rem} \begin{rem} \label{rem5.3} In \cite{Ye4} it is shown that $\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}$ is naturally a DG module over the adele-De Rham complex $\mcal{A}^{{\textstyle \cdot}}_{\mfrak{X}} = \underline{\mathbb{A}}^{{\textstyle \cdot}}_{\mrm{red}}(\widehat{\Omega}^{{\textstyle \cdot}}_{X/k})$, and this multiplication induces the cap product of $\mrm{H}_{\mrm{DR}}^{{\textstyle \cdot}}(X)$ on $\mrm{H}^{\mrm{DR}}_{{\textstyle \cdot}}(X)$. \end{rem} The next result is new (cf.\ \cite{BlO} Example 2.2): \begin{thm} \label{thm5.1} De Rham homology $\mrm{H}^{\mrm{DR}}_{{\textstyle \cdot}}(-)$ is contravariant w.r.t.\ \'{e}tale morphisms. \end{thm} \begin{proof} The ``topological invariance of \'{e}tale morphisms'' (see \cite{Mi} Theorem I.3.23) implies that the smooth formal embedding $X \subset \mfrak{X}$ induces an ``embedding of \'{e}tale sites'' $X_{\mrm{et}} \subset \mfrak{X}_{\mrm{et}}$. By this we mean that for every \'{e}tale morphism $U \rightarrow X$ there is some \'{e}tale morphism $\mfrak{U} \rightarrow \mfrak{X}$, unique up to isomorphism, s.t.\ $U \cong \mfrak{U} \times_{\mfrak{X}} X$ (see \cite{Ye3}). Then $U \subset \mfrak{U}$ is a smooth formal embedding. From Proposition \ref{prop5.5} we see there is a complex of sheaves $\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}}}$ on $X_{\mrm{et}}$ with $\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}}} |_{U} \cong \mcal{F}^{{\textstyle \cdot}}_{\mfrak{U}} \cong g^{*} \mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}$ for every $g : \mfrak{U} \rightarrow \mfrak{X}$ \'{e}tale (cf.\ \cite{Mi} Corollary II.1.6). But by Corollary \ref{cor5.1}, $\mrm{H}^{\mrm{DR}}_{{\textstyle \cdot}}(U) = \mrm{H}^{{\textstyle \cdot}}(U, \mcal{F}^{{\textstyle \cdot}}_{\mfrak{U}})$. \end{proof} Say $X$ is smooth irreducible of dimension $n$. Define the sheaf $\mcal{H}^{p}_{\mrm{DR}}$ on $X_{\mrm{Zar}}$ to be the sheafification of the presheaf $U \mapsto \mrm{H}^{p}_{\mrm{DR}}(U)$. For any point $x \in X$ let $i_{x} : \{x\} \rightarrow X$ be the inclusion. Let $x_{0}$ be the generic point, so $X_{n} / X_{n - 1} = \{ x_{0} \}$. According to \cite{BlO} there is an exact sequence of sheaves \[ 0 \rightarrow \mcal{H}^{p}_{\mrm{DR}} \rightarrow i_{x_{0}\, *} \mrm{H}^{p} \Omega^{{\textstyle \cdot}}_{k(x_{0}) / k} \rightarrow \cdots \rightarrow \bigoplus_{x \in X_{q} / X_{q - 1}} i_{x*} \mrm{H}^{p + q - n} \Omega^{{\textstyle \cdot}}_{k(x)/k} \rightarrow \cdots \] called the {\em arithmetic resolution}. Observe that this is a flasque resolution. \begin{cor} \label{cor5.2} The coboundary operator in the arithmetic resolution of $\mcal{H}^{p}_{\mrm{DR}}$ is \[ (-1)^{q + 1} \sum_{(x,y)} \operatorname{Res}_{(x,y)} \] where $\operatorname{Res}_{(x,y)}$ is the Parshin residue of Proposition \textup{\ref{prop5.2}}. \end{cor} \begin{proof} Take $\mfrak{X} = X$ in Theorem \ref{thm5.3}, and use \cite{BlO} Theorem 4.2. \end{proof} \section{The Intersection Cohomology $\mcal{D}$-module of a Curve} Suppose $Y$ is an $n$-dimensional smooth algebraic variety over $\mbb{C}$ and $X$ is a subvariety of codimension $d$. Let $\mcal{H}_{X}^{d} \mcal{O}_{Y}$ be the sheaf of $d$-th cohomology of $\mcal{O}_{Y}$ with support in $X$. According to \cite{BrKa}, the holonomic $\mcal{D}_{Y}$-module $\mcal{H}_{X}^{d} \mcal{O}_{Y}$ has a unique simple coherent submodule $\mcal{L}(X,Y)$, and the De Rham complex $\operatorname{DR} \mcal{L}(X,Y) = \mcal{L}(X,Y) \otimes \Omega^{{\textstyle \cdot}}_{Y^{\mrm{an}}}[n]$ is the middle perversity intersection cohomology sheaf $\mcal{IC}^{{\textstyle \cdot}}_{X^{\mrm{an}}}$. Here $Y^{\mrm{an}}$ is the associated complex manifold. The module $\mcal{L}(X,Y)$ was described explicitly using complex-analytic methods by Vilonen \cite{Vi} and Barlet-Kashiwara \cite{BaKa}. These descriptions show that the fundamental class $\mrm{C}_{X/Y}$ lies in $\mcal{L}(X,Y) \otimes \Omega^{d}_{Y/k}$, a fact proved earlier by Kashiwara using the Riemann-Hilbert correspondence and the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber (see \cite{Br}). Now let $k$ be any field of characteristic $0$, $Y$ an $n$-dimensional smooth variety over $k$, and $X \subset Y$ an integral curve with arbitrary singularities. In this section we give a description of $\mcal{L}(X,Y) \subset \mcal{H}_{X}^{n-1} \mcal{O}_{Y}$ in terms of algebraic residues. As references on $\mcal{D}$-modules we suggest \cite{Bj} and \cite{Bo} Chapter VI. Denote by $w$ the generic point of $X$. Pick any coefficient field $\sigma : k(w) \rightarrow \widehat{\mcal{O}}_{Y, w} = \mcal{O}_{Y, (w)}$. As in \cite{Hu} Section 1 there is a residue map \[ \operatorname{Res}^{\mrm{lc}}_{w, \sigma} : \mrm{H}_{w}^{n-1} \Omega^{n}_{Y/k} \rightarrow \Omega^{1}_{k(w) / k} \] (``lc'' is for local cohomology) defined as follows. Choose a regular system of parameters $f_{1}, \ldots, f_{n-1}$ in $\mcal{O}_{Y, w}$, so that $\mcal{O}_{Y, (w)} \cong k(w)[[f_{1}, \ldots, f_{n-1}]]$. Then for a generalized fraction, with $\alpha \in \Omega^{1}_{k(X) / k}$, we have \[ \operatorname{Res}^{\mrm{lc}}_{w, \sigma} \gfrac{\sigma(\alpha) \wedge \mrm{d} f_{1} \wedge \cdots \wedge \mrm{d} f_{n - 1}} {f_{1}^{i_{1}} \cdots f_{n - 1}^{i_{n - 1}}} = \begin{cases} \alpha & \text{if } (i_{1}, \ldots, i_{n-1}) = (1, \ldots, 1) \\ 0 & \text{otherwise} . \end{cases} \] Let $\pi : \tilde{X} \rightarrow X$ be the normalization, and let $\tilde{w}$ be the generic point of $\tilde{X}$. For any closed point $\tilde{x} \in \tilde{X}$ the residue field $k(\tilde{x})$ is \'{e}tale over $k$, so it lifts into $\mcal{O}_{\tilde{X}, (\tilde{x})}$. Hence we get canonical morphisms of BCAs $k(\tilde{x}) \rightarrow \mcal{O}_{\tilde{X}, (\tilde{x})} \rightarrow k(\tilde{w})_{(\tilde{x})}$, and a residue map \[ \operatorname{Res}_{(\tilde{w}, \tilde{x})} : \Omega^{1}_{k(w) / k} \rightarrow \Omega^{1, \mrm{sep}}_{k(\tilde{w})_{(\tilde{x})} / k} \rightarrow k(\tilde{w}) . \] Define \[ \operatorname{Res}^{\mrm{lc}}_{(\tilde{w}, \tilde{x})} : \mrm{H}_{w}^{n-1} \Omega^{n}_{Y/k} \xrightarrow{\operatorname{Res}^{\mrm{lc}}_{w, \sigma}} \Omega^{1}_{k(w) / k} \xrightarrow{\operatorname{Res}_{(\tilde{w}, \tilde{x})}} k(\tilde{x}) . \] We shall see later that $\operatorname{Res}^{\mrm{lc}}_{(\tilde{w}, \tilde{x})}$ is independent of $\sigma$. Note that $\mrm{H}_{w}^{n-1} \Omega^{n}_{Y/k} = (\mcal{H}_{X}^{n-1} \Omega^{n}_{Y/k})_{w}$. \begin{thm} \label{thm6.6} Let $x \in X$ be a closed point and let $a \in (\mcal{H}_{X}^{n-1} \mcal{O}_{Y})_{x}$. Then $a \in \mcal{L}(X, Y)_{x}$ iff $\operatorname{Res}^{\mrm{lc}}_{(\tilde{w}, \tilde{x})}(a \alpha) = 0$ for all $\alpha \in \Omega^{n}_{Y / k, x}$ and $\tilde{x} \in \pi^{-1}(x)$. \end{thm} This is our algebraic counterpart of Vilonen's formula in \cite{Vi}. The proof of the theorem appears later in this section. Fix a closed point $x \in X$. Write $B := \mcal{O}_{Y, (w, x)}$ and $L := \prod_{\tilde{x} \in \pi^{-1}(x)} k(\tilde{x})$. \begin{lem} There is a canonical morphism of BCAs $L \rightarrow B$, and $B \cong L((g))[[f_{1}, \ldots, f_{n - 1}]]$ for indeterminates $g, f_{1}, \ldots, f_{n - 1}$. \end{lem} \begin{proof} Because $\mcal{O}_{\tilde{X}, (\tilde{x})}$ is a regular local ring we get $\mcal{O}_{\tilde{X}, (\tilde{x})} \cong k(\tilde{x})[[g]]$. It is well known (cf.\ \cite{Ye1} Theorem 3.3.2) that $k(w)_{(x)} = k(w) \otimes \mcal{O}_{X, (x)} \cong \prod k(\tilde{w})_{(\tilde{x})}$, hence $k(w)_{(x)} \cong L((g))$. Choose a coefficient field $\sigma : k(w) \rightarrow \mcal{O}_{Y, (w)}$. It extends to a lifting $\sigma_{(x)} : k(w)_{(x)} \rightarrow \mcal{O}_{Y, (w, x)} = B$ (cf.\ \cite{Ye1} Lemma 3.3.9), and $L \rightarrow B$ is independent of $\sigma$. Taking a system of regular parameters $f_{1}, \ldots, f_{n - 1} \in \mcal{O}_{Y, w}$ we obtain the desired isomorphism. \end{proof} The BCA $A := \mcal{O}_{Y, (x)}$ is canonically an algebra over $K := k(x)$, so there is a morphism of BCAs $L \otimes_{K} A \rightarrow B$. Define a homomorphism \[ T_{x} : \mcal{K}(B) \xrightarrow{\mrm{Tr}} \mcal{K}(L \otimes_{K} A) \cong L \otimes_{K} \mcal{K}(A) . \] Since $A \rightarrow L \otimes_{K} A \rightarrow B$ are topologically \'{e}tale (relative to $k$), it follows that $T_{x}$ is a homomorphism of $\mcal{D}(A)$-modules. Define \[ V(x) := \operatorname{Coker} \left( K \rightarrow L \right) . \] Observe that $V(x) = 0$ iff $x$ is either a smooth point or a geometrically unibranch singularity. We have $V(x)^{*} \subset L^{*}$, where $(-)^{*} := \operatorname{Hom}_{k}(-, k)$. The isomorphism $L^{*} \cong L$ induced by $\operatorname{Tr}_{L / k}$ identifies $V(x)^{*} \cong \operatorname{Ker}(L \xrightarrow{\operatorname{Tr}} K)$. Since $\Omega^{n}_{Y / k}[n] \rightarrow \mcal{K}^{{\textstyle \cdot}}_{Y}$ is a quasi-isomorphism we get a short exact sequence \begin{equation} \label{eqn6.1} 0 \rightarrow (\mcal{H}_{X}^{n-1} \Omega^{n}_{Y/k})_{x} \rightarrow \mcal{K}_{Y}(w) \xrightarrow{\delta\ } \mcal{K}_{Y}(x) \rightarrow 0 . \end{equation} Also we see that $\mcal{K}(A) = \mcal{K}_{Y}(x) \cong \mrm{H}^{n}_{x} \Omega^{n}_{Y / k}$. Now $\mcal{K}_{Y}(w) = \mcal{K}(\mcal{O}_{Y, (w)}) \subset \mcal{K}(B)$. Because the composed map \[ \mcal{K}_{Y}(w) \xrightarrow{T_{x}} L \otimes_{K} \mcal{K}_{Y}(x) \xrightarrow{\operatorname{Tr}_{L / K} \otimes 1} \mcal{K}_{Y}(x) \] coincides with $\delta$, and by the sequence (\ref{eqn6.1}), we obtain a homomorphism of $\mcal{D}_{Y, x}$-modules \begin{equation} \label{eqn6.11} T_{x} : (\mcal{H}_{X}^{n-1} \Omega^{n}_{Y/k})_{x} \rightarrow V(x)^{*} \otimes_{K} \mrm{H}^{n}_{x} \Omega^{n}_{Y / k} . \end{equation} \begin{thm} \label{thm6.3} The homomorphism $T_{x}$ induces a bijection between the lattice of nonzero $\mcal{D}_{Y, x}$-submodules of $(\mcal{H}_{X}^{n-1} \Omega^{n}_{Y/k})_{x}$ and the lattice of $k(x)$-submodules of $V(x)^{*}$. \end{thm} The proof of the theorem is given later in this section. In order to globalize we introduce the following notation. Let $Z$ be the reduced subscheme supported on the singular locus $X_{\mrm{sing}}$, so $\mcal{O}_{Z} = \prod_{x \in X_{\mrm{sing}}} k(x)$. Then $\mcal{V} := \bigoplus_{x \in X_{\mrm{sing}}} V(x)$ and $\mcal{H}_{Z}^{n} \mcal{O}_{Y} = \bigoplus_{x \in X_{\mrm{sing}}} \mcal{H}_{ \{x\} }^{n} \mcal{O}_{Y}$ are $\mcal{O}_{Z}$-modules. Using $\Omega^{n}_{Y / k} \otimes$ to switch between left and right $\mcal{D}_{Y}$-modules, and identifying $V(x)^{*} \cong V(x)$ by the trace pairing, we see that Theorem \ref{thm6.3} implies \begin{cor} \label{cor6.10} The homomorphism of $\mcal{D}_{Y}$-modules \[ T := \sum_{x} T_{x} : \mcal{H}_{X}^{n-1} \mcal{O}_{Y} \rightarrow (\mcal{H}_{Z}^{n} \mcal{O}_{Y}) \otimes_{\mcal{O}_{Z}} \mcal{V} \] induces a bijection between the lattice of nonzero coherent $\mcal{D}_{Y}$-submodules of $\mcal{H}_{X}^{n-1} \mcal{O}_{Y}$ and the lattice of $\mcal{O}_{Z}$-submodules of $\mcal{V}$. \end{cor} Since $\mcal{H}_{ \{x\} }^{n} \mcal{O}_{Y}$ is a simple $\mcal{D}_{Y}$-submodule, as immediate corollaries we get: \begin{cor} \label{cor6.1} $\mcal{H}_{X}^{n-1} \mcal{O}_{Y}$ has a unique simple coherent $\mcal{D}_{Y}$-submodule \blnk{2mm} \linebreak $\mcal{L}(X,Y)$, and the sequence \begin{equation} \label{eqn6.7} 0 \rightarrow \mcal{L}(X,Y) \rightarrow \mcal{H}_{X}^{n-1} \mcal{O}_{Y} \xrightarrow{T} (\mcal{H}_{Z}^{n} \mcal{O}_{Y}) \otimes_{\mcal{O}_{Z}} \mcal{V} \rightarrow 0 \end{equation} is exact. \end{cor} \begin{cor} \label{cor6.2} $\mcal{H}_{X}^{n-1} \mcal{O}_{Y}$ is a simple coherent $\mcal{D}_{Y}$-module iff the singularities of $X$ are all geometrically unibranch. \end{cor} According to Proposition \ref{prop4.7} the fundamental class $\mrm{C}_{X/Y}$ is a double cocycle in $\mcal{H}om(\Omega^{1}_{Y/k}, \mcal{K}^{-1}_{Y})$, so it determines a class in $(\mcal{H}_{X}^{n-1} \mcal{O}_{Y}) \otimes_{\mcal{O}_{Y}} \Omega^{n-1}_{Y/k}$. \begin{thm} \label{thm6.2} $\mrm{C}_{X/Y} \in \mcal{L}(X,Y) \otimes_{\mcal{O}_{Y}} \Omega^{n-1}_{Y/k}$. \end{thm} This of course implies that if $\alpha_{1}, \ldots, \alpha_{n}$ is a local basis of $\Omega^{n-1}_{Y/k}$ and $\mrm{C}_{X/Y} = \sum a_{i} \otimes \alpha_{i}$, then any nonzero $a_{i}$ generates $\mcal{L}(X,Y)$ as a $\mcal{D}_{Y}$-module. The proof of the theorem is given later in this section. \begin{rem} As the referee points out, when $k = \mbb{C}$, Corollary \ref{cor6.10} follows easily from the Riemann-Hilbert correspondence. In that case we may consider the sheaf $\mcal{V}$ on the analytic space $X^{\mrm{an}}$, given by $\mcal{V} := \operatorname{Coker}(\mbb{C}_{X^{\mrm{an}}} \rightarrow \pi^{\mrm{an}}_{*} \mbb{C}_{\tilde{X}^{\mrm{an}}})$. Now $\mcal{IC}_{X^{\mrm{an}}} \cong \pi^{\mrm{an}}_{*} \mbb{C}_{\tilde{X}^{\mrm{an}}}[1]$. The triangle $\mcal{V} \rightarrow \mbb{C}_{X^{\mrm{an}}}[1] \rightarrow$ \linebreak $\mcal{IC}_{X^{\mrm{an}}} \xrightarrow{+1}$ is an exact sequence in the category of perverse sheaves, and it is the image of (\ref{eqn6.7}) under the functor $\operatorname{Sol} = \mrm{R} \mcal{H}om_{\mcal{D}_{Y^{\mrm{an}}}}((-)^{\mrm{an}}, \mcal{O}_{Y^{\mrm{an}}}[n])$. Nonetheless ours seems to be the first purely algebraic proof Theorem \ref{thm6.3} and its corollaries (but cf.\ next remark). \end{rem} \begin{rem} When $Y = \mbf{A}^{2}$ (i.e.\ $X$ is an affine plane curve) and $k$ is algebraically closed, Corollary \ref{cor6.2} was partially proved by S.P.\ Smith \cite{Sm}, using the ring structure of $\mcal{D}(X)$. Specifically, he proved that if $X$ has unibranch singularities, then $\mcal{H}_{X}^{1} \mcal{O}_{Y}$ is simple. \end{rem} \begin{exa} \label{exa6.1} Let $X$ be the nodal curve in $Y = \mbf{A}^{2} = \operatorname{Spec} k [s, t]$ defined by $f = s^{2} (s+1) - t^{2}$, and let $x$ be the origin. Take $r := t / s \in \mcal{O}_{Y, w}$, so $s = (r+1)(r-1)$. We see that $\tilde{X} = \operatorname{Spec} k[r]$ and $r + 1, r - 1$ are regular parameters at $\tilde{x}_{1}, \tilde{x}_{2}$ respectively on $\tilde{X}$. For any coefficient field $\sigma$, \[ \operatorname{Res}^{\mrm{lc}}_{w, \sigma} \gfrac{\mrm{d} s \wedge \mrm{d} t}{f} = \operatorname{Res}^{\mrm{lc}}_{w, \sigma} \gfrac{- \mrm{d} (r + 1) \wedge \mrm{d} f} {(r + 1)(r - 1) f} = \frac{- \mrm{d} (r + 1)}{(r + 1)(r - 1)} \] and hence \[ \operatorname{Res}^{\mrm{lc}}_{(\tilde{w}, \tilde{x}_{1})} \gfrac{\mrm{d} s \wedge \mrm{d} t}{f} = \operatorname{Res}_{(\tilde{w}, \tilde{x}_{1})} \frac{- \mrm{d} (r + 1)}{(r + 1)(r - 1)} = 2 . \] Likewise $\operatorname{Res}^{\mrm{lc}}_{(\tilde{w}, \tilde{x}_{2})} \gfrac{\mrm{d} s \wedge \mrm{d} t}{f} = - 2$. Therefore $\gfrac{\mrm{d} s \wedge \mrm{d} t}{f} \notin \mcal{L}(X, Y) \otimes \Omega^{2}_{Y / k}$. The fundamental class is $\mrm{C}_{X / Y} = \gfrac{\mrm{d} f}{f}$, and as generator of $\mcal{L}(X, Y) \otimes \Omega^{2}_{Y / k}$ we may take $\gfrac{\mrm{d} s \wedge \mrm{d} f}{f}$. \end{exa} Before getting to the proofs we need some general results. Let $A$ be a BCA over $k$. The fine topology on an $A$-module $M$ is the quotient topology w.r.t.\ any surjection $\bigoplus A \twoheadrightarrow M$. The fine topology on $M$ is $k$-linear, making it a topological $k$-module (but only a semi-topological (ST) $A$-module). According to \cite{Ye2} Proposition 2.11.c, $A$ is a Zariski ST ring (cf.\ ibid.\ Definition 1.7). This means that any finitely generated $A$-module with the fine topology is separated, and any homomorphism $M \rightarrow N$ between such modules is topologically strict. Furthermore if $M$ is finitely generated then it is complete, so it is a complete linearly topologized $k$-vector space in the sense of \cite{Ko}. \begin{lem} \label{lem6.3} Let $A$ be a BCA. Suppose $M$ is a countably generated ST $A$-module with the fine topology. Then $M$ is separated, and any submodule $M' \subset M$ is closed. \end{lem} \begin{proof} Write $M = \bigcup_{i=1}^{\infty} M_{i}$ with $M_{i}$ finitely generated. Suppose we put the fine topology on $M_{i}$. Then each $M_{i}$ is separated and $M_{i} \rightarrow M_{i+1}$ is strict. By \cite{Ye1} Corollary 1.2.6 we have $M \cong \lim_{i \rightarrow} M_{i}$ topologically, so by ibid.\ Proposition 1.1.7, $M$ is separated. By the same token $M / M'$ is separated too, so $M'$ is closed. \end{proof} \begin{prop} \label{prop6.1} Let $A \rightarrow B$ be a morphism of BCAs, $N$ a finitely generated $B$-module with the fine topology, and $M \subset N$ a finitely generated $A$-module. Then the topology on $M$ induced by $N$ equals the fine $A$-module topology, and $M$ is closed in $N$. \end{prop} \begin{proof} Since $A$ is a Zariski ST ring we may replace $M$ by any finitely generated $A$ module $M'$, $M \subset M' \subset N$. Therefore we can assume $N = B M$ and $M = \bigoplus_{\mfrak{n} \in \operatorname{Max} B} M \cap N_{\mfrak{n}}$. So in fact we may assume $A,B$ are both local. Like in the proof of \cite{Ye2} Theorem 7.4 we may further assume that $\operatorname{res.dim}(A \rightarrow B) \leq 1$. Put on $M$ the fine $A$-module topology. Let $\bar{N}_{i} := N / \mfrak{n}^{i} N$ and $\bar{M}_{i} := M / (M \cap \mfrak{n}^{i} N)$ with the quotient topologies. We claim $\bar{M}_{i} \rightarrow \bar{N}_{i}$ is a strict monomorphism. This is so because as $A$-modules both have the fine topology, $\bar{M}_{i}$ is finitely generated and $\bar{N}_{i}$ is countably generated (cf.\ part 1 in the proof of \cite{Ye1} Theorem 3.2.14). Just as in part 2 of loc.\ cit.\ we get topological isomorphisms $M \cong \lim_{\leftarrow i} \bar{M}_{i}$ and $N \cong \lim_{\leftarrow i} \bar{N}_{i}$, so $M \rightarrow N$ is a strict monomorphism. But $M$ is complete and $N$ is separated, so $M$ must be closed. \end{proof} Given a topological $k$-module $M$ we set $M^{*} := \mrm{Hom}_{k}^{\mrm{cont}}(M,k)$ (without a topology). \begin{lem} \label{lem6.2} Suppose $M$ is a separated topological $k$-module. Then: \begin{enumerate} \item For any subset $S \subset M^{*}$ its perpendicular $S^{\perp} \subset M$ is a closed submodule. \item Given a closed submodule $M_{1} \subset M$, one has $M_{1}^{\perp \perp} = M_{1}$. \item Suppose $M_{1} \subset M_{2} \subset M$ are closed submodules. Then there is an exact sequence \textup{(}of untopologized $k$-modules\textup{)} \[ 0 \rightarrow M_{2}^{\perp} \rightarrow M_{1}^{\perp} \rightarrow (M_{2} / M_{1})^{*} \rightarrow 0 . \] \end{enumerate} \end{lem} \begin{proof} See \cite{Ko} Section 10.4, 10.8. \end{proof} Let $M,N$ be complete separated topological $k$-modules, and $\langle -,- \rangle : M \times N \rightarrow k$ a continuous pairing. We say $\langle -,- \rangle$ is a {\em topological perfect pairing} if it induces isomorphisms $N \cong M^{*}$ and $M \cong N^{*}$. \begin{prop} \label{prop6.3} Assume $k \rightarrow A$ is a morphism of BCAs. Then the residue pairing $\langle -,- \rangle_{A/k} : A \times \mcal{K}(A) \rightarrow k$, $\langle a, \phi \rangle_{A/k} = \operatorname{Tr}_{A / k}(a \phi)$, is a topological perfect pairing. \end{prop} \begin{proof} We may assume $A$ is local. Then $A = \lim_{\leftarrow i} A / \mfrak{m}^{i}$ and $\mcal{K}(A) = \lim_{i \rightarrow} \mcal{K}(A / \mfrak{m}^{i})$ topologically. Let $K \rightarrow A$ be a coefficient field, so both $A / \mfrak{m}^{i}$ and $\mcal{K}(A / \mfrak{m}^{i}) \cong \mrm{Hom}_{K}(A / \mfrak{m}^{i}, \omega(K))$ are finite $K$-modules with the fine topology. By \cite{Ye1} Theorem 2.4.22 the pairing is perfect. \end{proof} From here to the end of this section we consider an integral curve $X$ embedded as a closed subscheme in a smooth irreducible $n$-dimensional variety $Y$. Fix a closed point $x \in X$, and set $A := \mcal{O}_{Y, (x)}$ and $K := k(x)$. Choosing a regular system of parameters at $x$, say $\underline{t} = (t_{1}, \ldots, t_{n})$, allows us to write $A = K [[\, \underline{t}\, ]]$. Let $\mcal{D}(A) := \mrm{Diff}^{\mrm{cont}}_{A/k}(A,A)$. Since both $K [\, \underline{t}\, ] \rightarrow A$ and $\mcal{O}_{Y,x} \rightarrow A$ are topologically \'{e}tale relative to $k$, we have \[ \mcal{D}(A) \cong A \otimes_{K} K [ \textstyle{\frac{\partial}{\partial t_{1}}}, \ldots, \textstyle{\frac{\partial}{\partial t_{n}}}] \cong A \otimes_{\mcal{O}_{Y,x}} \mcal{D}_{Y,x} \] (cf.\ \cite{Ye2} Section 4). Define $B := \mcal{O}_{Y, (w, x)}$. Since $A \rightarrow B$ is topologically \'{e}tale relative to $k$, we get a $k$-algebra homomorphism $\mcal{D}(A) \rightarrow \mcal{D}(B)$. In particular, $B$ and $\mcal{K}(B)$ are $\mcal{D}(A)$-modules. Define $L := \prod_{\tilde{x} \in \pi^{-1}(x)} k(\tilde{x})$ as before. \begin{lem} \label{lem6.1} The multiplication map $A \otimes_{K} L \rightarrow B$ is injective. Its image is a $\mcal{D}(A)$-submodule of $B$. Any $\mcal{D}(A)$-submodule of $B$ which is finitely generated over $A$ equals $A \otimes_{K} W$ for some $K$-submodule $W \subset L$. \end{lem} \begin{proof} By \cite{Kz} Proposition 8.9, if $M$ is any $\mcal{D}(A)$-module which is finitely generated over $A$, then $M = A \otimes_{K} W$, where $W \subset M$ is the $K$-submodule consisting of all elements killed by the derivations $\frac{\partial}{\partial t_{i}}$. Note that $\Omega^{1, \mrm{sep}}_{B / k}$ is free with basis $\mrm{d} t_{1}, \ldots, \mrm{d} t_{n}$. Thus it suffices to prove that \begin{equation} \label{eqn6.5} L = \{ b \in B\ |\ \ \textstyle{\frac{\partial}{\partial t_{1}}} b = \cdots = \textstyle{\frac{\partial}{\partial t_{n}}} b = 0 \} = \mrm{H}^{0} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B / k} . \end{equation} We know that $B \cong L((g))[[f_{1}, \ldots, f_{n - 1}]]$, so $B$ is topologically \'{e}tale over the polynomial algebra $k [g, f_{1}, \ldots, f_{n - 1}]$ (relative to $k$), and hence $\mrm{d} g, \mrm{d} f_{1}, \ldots$, \linebreak $\mrm{d} f_{n - 1}$ is also a basis of $\Omega^{1, \mrm{sep}}_{B / k}$. It follows that $\mrm{H}^{0} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B / k} = L$. \end{proof} \begin{proof} (of Theorem \ref{thm6.3})\ Set $\mcal{M} := \mcal{H}_{X}^{n-1} \Omega^{n}_{Y/k}$ and define $M := A \otimes_{\mcal{O}_{Y, x}} \mcal{M}_{x}$. Tensoring the exact sequence (\ref{eqn6.1}) with $A$ we get an exact sequence of $\mcal{D}(A)$-modules \begin{equation} \label{eqn6.2} 0 \rightarrow M \rightarrow \mcal{K}(B) \xrightarrow{\delta\ } \mcal{K}(A) \rightarrow 0 . \end{equation} The proof will use repeatedly the residue pairing $\langle -,- \rangle_{B/k} : B \times \mcal{K}(B) \rightarrow k$. By definition of $\delta$ (cf.\ Definition \ref{dfn1.1} and \cite{Ye2} Section 7) we see that $M = A^{\perp}$. Consider the closed $k$-submodules $A \subset A \otimes_{K} L \subset B$ (cf.\ Proposition \ref{prop6.1}). Applying Lemma \ref{lem6.2} to them, and using $V(x) = L / K$ and $\mcal{K}(A) \cong A^{*}$, we get an exact sequence of $\mcal{D}(A)$-modules \[ 0 \rightarrow (A \otimes_{K} L)^{\perp} \rightarrow M \xrightarrow{T'} V(x)^{*} \otimes_{K} \mcal{K}(A) \rightarrow 0 . \] Keeping track of the operations we see that in fact $T' = T_{x}|_{M}$. Put the fine $A$-module topology on $M$ and $\mcal{K}(A)$, so $M \rightarrow V(x)^{*} \otimes_{K} \mcal{K}(A)$ is continuous. By \cite{Ye1} Proposition 1.1.8, $\mcal{M}_{x} \rightarrow M$ is dense. Since $\mcal{K}(A)$ is discrete we conclude that $\mcal{M}_{x} \rightarrow V(x)^{*} \otimes_{K} \mcal{K}(A)$ is a surjection of $\mcal{D}_{Y, x}$-modules. Thus any $K$-module $W \subset V(x)^{*}$ determines a distinct nonzero $\mcal{D}_{Y, x}$-module $\mcal{N}_{x} \subset \mcal{M}_{x}$. Conversely, say $\mcal{N}_{x} \subset \mcal{M}_{x}$ is a nonzero $\mcal{D}_{Y, x}$-module. On any open set $U \subset Y$ s.t.\ $U \cap X$ is smooth the module $\mcal{M}|_{U}$ is a simple coherent $\mcal{D}_{U}$-module (by Kashiwara's Theorem it corresponds to the $\mcal{D}_{X \cap U}$-module $\Omega^{1}_{(X \cap U) / k}$). Therefore the finitely generated $\mcal{D}_{Y, x}$-module $C$ defined by \begin{equation} \label{eqn6.4} 0 \rightarrow \mcal{N}_{x} \rightarrow \mcal{M}_{x} \rightarrow C \rightarrow 0 \end{equation} is supported on $\{x\}$. It follows that $C \cong \mcal{K}(A)^{r}$ for some number $r$. Tensoring (\ref{eqn6.4}) with $A$ we get an exact sequence of $\mcal{D}(A)$-modules \[ 0 \rightarrow N \rightarrow M \rightarrow C \rightarrow 0 \] with $N \subset M \subset \mcal{K}(B)$. By faithful flatness of $\mcal{O}_{Y, x} \rightarrow A$ we see that $\mcal{N}_{x} = \mcal{M}_{x} \cap N$. We put on $M,N$ the topology induced from $\mcal{K}(B)$, and on $C$ the quotient topology from $M$. Now $\mcal{K}(B)$ has the fine $A$-module topology and it is countably generated over $A$ (cf.\ proof of Proposition \ref{prop6.1}), so by Lemma \ref{lem6.3} both $M,N$ are closed in $\mcal{K}(B)$. Using Lemma \ref{lem6.2} and the fact that $M^{\perp} = A$ we obtain the exact sequence \[ 0 \rightarrow A \rightarrow N^{\perp} \rightarrow C^{*} \rightarrow 0 , \] with $N^{\perp} \subset B$. We do not know what the topology on $C$ is; but it is a ST $A$-module. Hence the identity map $\mcal{K}(A)^{r} \rightarrow C$ is continuous, and it induces an $A$-linear injection $C^{*} \rightarrow A^{r}$. Therefore $C^{*}$, and thus also $N^{\perp}$, are finitely generated over $A$. According to Lemma \ref{lem6.1}, $N^{\perp} = A \otimes_{K} W$ for some $K$-module $W$, $K \subset W \subset L$. But $N$ is closed, so $N = (N^{\perp})^{\perp}$. \end{proof} \begin{proof} (of Theorem \ref{thm6.2})\ For each $\tilde{x} \in \pi^{-1}(x)$ define a homomorphism \[ T_{(\tilde{w}, \tilde{x})} : \mcal{K}(B) \xrightarrow{\mrm{Tr}} \mcal{K}(L \otimes_{K} A) \cong L \otimes_{K} \mcal{K}(A) \rightarrow k(\tilde{x}) \otimes_{K} \mcal{K}(A) , \] so $T_{x} = \sum T_{(\tilde{w}, \tilde{x})}$. From the proof of Theorem \ref{thm6.3} we see that the theorem amounts to the claim that $T_{\tilde{x}} (\mrm{C}_{X/Y}(\alpha)) = 0$ for every $\tilde{x}$ and $\alpha \in \Omega^{1}_{Y / k, x}$. But $\mrm{C}_{X/Y}$ is the image of $\mrm{C}_{X} \in \mcal{H}om(\Omega^{1}_{X/k}, \mcal{K}^{-1}_{X}(w))$, so we can reduce our residue calculation to the curve $\tilde{X}$. In fact it suffices to show that for every $\alpha \in \Omega^{1}_{X / k, x}$ one has $\operatorname{Res}_{(\tilde{w}, \tilde{x})} \alpha = 0$. Since $\alpha \in \Omega^{1}_{\tilde{X} / k, \tilde{x}}$ this is obvious. \end{proof} \begin{proof} (of Theorem \ref{thm6.6})\ According to \cite{SY} Corollary 0.2.11 (or \cite{Hu} Theorem 2.2) one has \[ \operatorname{Res}^{\mrm{lc}}_{(\tilde{w}, \tilde{x})} = (1 \otimes \operatorname{Tr}_{A / K}) T_{(\tilde{w}, \tilde{x})} : \mrm{H}^{n - 1}_{w} \Omega^{n}_{Y / k} \rightarrow k(\tilde{x}) , \] which shows that $\operatorname{Res}^{\mrm{lc}}_{(\tilde{w}, \tilde{x})}$ is independent of $\sigma$. Now use Theorem \ref{thm6.3}. \end{proof} \begin{prob} What is the generalization to $\operatorname{dim} X > 1$? To be specific, assume $X$ has only an isolated singularity at $x$. Then we know there is an exact sequence \[ 0 \rightarrow \mcal{L}(X,Y) \rightarrow \mcal{H}_{X}^{d} \mcal{O}_{Y} \xrightarrow{T} \mcal{H}_{ \{x\} }^{n} \mcal{O}_{Y} \otimes_{k(x)} V(x) \rightarrow 0 \] for some $k(x)$-module $V(x)$. What is the geometric data determining $V(x)$ and $T$? Is it true that $T = \sum T_{\xi}$, a sum of ``residues'' along chains $\xi \in \pi^{-1}(x)$, for a suitable resolution of singularities $\pi : \tilde{X} \rightarrow X$? \end{prob}

9602

alg-geom/9602023

en

https://arxiv.org/abs/alg-geom/9602023

alg-geom/9602023

Mitchell Rothstein

Mitchell Rothstein

Sheaves with connection on abelian varieties

31 pages, AMSLaTeX amsart12. Author's address: [email protected]

The Fourier-Mukai transform is lifted to the derived category of sheaves with connection on abelian varieties. The case of flat connections (D-modules) is discussed in detail.

alg-geom

\section{Introduction}\label{introduction} \bigskip Let $X$ and $Y$ be abelian varieties over an algebraically closed field $k$, dual to one another, and let $\text{Mod}({\mathcal{O}}_X)$ and $\text{Mod}({\mathcal{O}}_Y)$ be their respective categories of quasicoherent ${\mathcal{O}}$-modules. Mukai proved in \cite{Muk} that the derived categories $D\text{Mod}({\mathcal{O}}_X)$ and $D\text{Mod}({\mathcal{O}}_Y)$ are equivalent via a transform now known as the Fourier-Mukai transform, \begin{equation} {\mathcal{S}}_1({\mathcal{F}})=\alpha_{2*}({\mathcal{P}}\otimes \alpha_1^*(-1)^*({\mathcal{F}}))\ , \end{equation} where ${\mathcal{P}}$ is the Poincar\'e sheaf and $\alpha_1$ and $\alpha_2$ are the projections from $X\times Y$ to $X$ and $Y$ respectively. A few years earlier, Krichever \cite{K} rediscovered a construction due originally to Burchnall and Chaundy \cite{BC}, by which the affine coordinate ring of a projective curve minus a point may be imbedded in the ring of formal differential operators in one variable. The construction involves the choice of a line bundle on the curve, and Krichever took the crucial step of asking, in the case of a smooth curve, how the imbedding varies when the line bundle moves linearly on the Jacobian. The answer is now well-known, that the imbeddings satisfy the system of differential equations known as the KP-hierarchy. In fact, the Krichever construction is an instance of the Fourier-Mukai transform, with the crucial addition that the transformed sheaf is not only an ${\mathcal{O}}_Y$-module but a ${\mathcal{D}}_Y$-module, where ${\mathcal{D}}_Y$ is the sheaf of linear differential operators on $Y$, \cite{N1} \cite{N2} \cite{R}. This example serves as the inspiration for the present work, which is concerned with the role of the Fourier-Mukai transform\ in the theory of sheaves on $Y$ equipped with a connection. The main point is that in the derived category, all sheaves on $Y$ with connection are constructed by the Fourier-Mukai transform\ in a manner directly generalizing the Krichever construction. The connection need not be integrable, though the paper focuses mostly on that case. The basic idea is the following. Set \begin{equation} {\mathfrak{g}} = H^1(X, {\mathcal{O}})\ . \end{equation} Then there is a tautological extension \begin{equation}\label{taut seq} 0 \longrightarrow {\mathfrak{g}}^* \otimes {\mathcal{O}} \longrightarrow {\mathcal{E}} \overset{\mu}{\longrightarrow} {\mathcal{O}} \longrightarrow 0 \end{equation} given by the extension class $1 \in \text{End}({\mathfrak{g}}^*) = \text{Ext}^1({\mathcal{O}}, {\mathfrak{g}}^* \otimes {\mathcal{O}})$. Now let ${\mathcal{F}}$ be any quasicoherent sheaf of ${\mathcal{O}}_X$-modules, and tensor the sequence \eqref{taut seq} with ${\mathcal{F}}$: \begin{equation}\label{split it!} 0 \longrightarrow {\mathfrak{g}}^* \otimes {\mathcal{F}} \longrightarrow {\mathcal{E}} \otimes {\mathcal{F}} \overset{\mu_{{\mathcal{F}}}}{\longrightarrow} {\mathcal{F}} \longrightarrow 0. \end{equation} We refer to any splitting of sequence \eqref{split it!} as a {\it splitting on ${\mathcal{F}}$.} Let $\text{Mod}(\o_X)_{sp}$ denote the category of pairs $({\mathcal{F}},\psi)$, where ${\mathcal{F}}$ is a sheaf on $X$ and $\psi:{\mathcal{F}}\longrightarrow {\mathcal{E}}\otimes {\mathcal{F}}$ is a splitting, with the obvious morphisms. Let $\text{Mod}(\o_Y)_{cxn}$ denote the category of quasicoherent sheaves on $Y$ equipped with a connection. In section \ref{mainthm} we use the Fourier-Mukai transform\ to establish an equivalence of bounded derived categories: \begin{equation}\label{equiv1} D^b\text{Mod}(\o_X)_{sp}\ \ \leftrightarrow\ \ D^b\text{Mod}(\o_Y)_{cxn}\ . \end{equation} Note that the extension class of \eqref{split it!} belongs ${\mathfrak{g}}^*\otimes H^1(X,{\mathcal{E}}nd({\mathcal{F}}))$ and is therefore a linear map \begin{equation} H^1(X,{\mathcal{O}})\longrightarrow H^1(X,{\mathcal{E}}nd({\mathcal{F}}))\ . \end{equation} It is easily seen to be the map on cohomology induced by the map ${\mathcal{O}} \longrightarrow {\mathcal{E}}nd({\mathcal{F}})$. Thus \begin{Prop}\label{affine space} Given an ${\mathcal{O}}$-module ${\mathcal{F}}$, there is a splitting on ${\mathcal{F}}$ if and only if the natural map \begin{equation} {\mathcal{O}} \longrightarrow {\mathcal{E}}nd({\mathcal{F}}) \end{equation} induces the $0$-map \begin{equation}\label{map} H^1(X, {\mathcal{O}})\overset{0}{\longrightarrow} H^1(X, {\mathcal{E}}nd({\mathcal{F}}))\ \ . \end{equation} Moreover, if (\ref{map}) holds, then the set of such splittings is an affine space over \hbox{${\mathfrak{g}}^*\otimes H^0(X,\cal End({\mathcal{F}}))$.} \end{Prop} The intuitive idea behind the equivalence \eqref{equiv1} is the following. Let $g=\text{dim}(Y)$. Let ${\mathcal{U}}_1,...,{\mathcal{U}}_k$ be an affine open cover of $X$ and for $i=1,...,g$, let $\{c(i)_{m,n}\}\in Z^1(\{{\mathcal{U}}\},{\mathcal{O}})$ be a 1-cocycle, such that the classes $[c(1)],...,[c(g)]$ form a basis for $H^1(X,{\mathcal{O}})$. Let $\xi_1,...,\xi_g$ denote this basis, and let $\omega^1,...,\omega^g$ denote the dual basis for $H^1(X,{\mathcal{O}})^*$. In light of proposition \ref{affine space}, a splitting on ${\mathcal{F}}$ amounts to a collection of endomorphisms \hbox{$\psi(i)_n\in \Gamma({\mathcal{U}}_n,{\mathcal{E}}nd( (-1_X)^*({\mathcal{F}}))$} such that \begin{equation} \psi(i)_n-\psi(i)_m=\text{multiplication by}\ -c(i)_{n,m} \ \ .\label{cobdry} \end{equation} Let ${\mathcal{G}}={\mathcal{S}}_1({\mathcal{F}})$. Then the collection of endomorphisms $\psi=\{\psi(i)_n\}$ endows ${\mathcal{G}}$ with a connection in the following way. For each $n$, there is a connection $\nabla_n$ relative to ${\mathcal{U}}_n$ on ${\mathcal{P}}\vert_{{\mathcal{U}}_n\times Y}$, such that on the overlaps, \begin{equation}\label{overlaps} \nabla_n-\nabla_m=c_{nm}\ \ . \end{equation} Therefore, one gets a connection relative to $X$ on $\alpha_1^*(-1_X)^*({\mathcal{F}})\otimes{\mathcal{P}}$ by defining \begin{equation}\label{relcxn} \nabla(\phi\otimes\sigma)=\phi\otimes\nabla_n(\sigma)+ \sum_i\omega^i\psi(i)_n(\phi)\otimes \sigma \end{equation} for $\phi\in(-1_X)^*({\mathcal{F}})$ and $\sigma\in{\mathcal{P}}$. Now one applies $\alpha_{2*}$ to produce a connection on ${\mathcal{G}}$. In the Krichever construction, $X$ is the Jacobian of a smooth curve $C$ with a base point $P$, and ${\mathcal{F}}$ is ${\mathcal{O}}_C(*P)$, regarded as a sheaf on $X$ by the abel map. The case where $X$ is an arbitrary abelian variety and ${\mathcal{F}}$ is of the form ${\mathcal{G}}\otimes{\mathcal{O}}_X(*D)$ for a coherent sheaf ${\mathcal{G}}$ and an ample hypersurface $D\subset X$ has been studied in \cite{N1} and \cite{N2}. Now consider the curvature tensor. To each object $({\mathcal{F}},\psi)\in \text{Ob Mod}(\o_X)_{sp}$ we associate a section \begin{equation} [\psi, \psi]\in \wedge^2 {\mathfrak{g}}^* \otimes End({\mathcal{F}})\ , \end{equation} simply by taking the commutator $[\psi(i)_n,\psi(j)_n]$, which, by \eqref{cobdry}, is independent of the chart. Applying the Fourier-Mukai transform to morphisms, one has \begin{equation} {\mathcal{S}}_1([{\psi, \psi}]) \in \wedge^2 {\mathfrak{g}}^* \otimes End({\mathcal{S}}_1({\mathcal{F}}))\ . \end{equation} Letting $[\nabla,\nabla]$ denote curvature, one has (Proposition \ref{curvature}), \begin{equation}\label{curvature.intro} {\mathcal{S}}_1([\psi,\psi])= [\nabla_{\psi},\nabla_{\psi}]\ . \end{equation} In particular, ${\mathcal{S}}_1$ restricts to a functorial correspondence \begin{equation} ({\mathcal{F}},\psi)\ \text{with}\ [\psi,\psi]=0\ \ \rightsquigarrow\ \ {\mathcal{D}}\text{-module structure on}\ {\mathcal{S}}_1({\mathcal{F}})\ . \end{equation} We prove that this also induces an equivalence of bounded derived categories (theorem \ref{equiv2}). The main point regarding the integrable case is the following. Let \begin{equation} X^{\natural} \overset{\pi}{\longrightarrow} X \end{equation} denote the ${\mathfrak{g}}^*$-principal bundle associated to the extension ${\mathcal{E}}$. It is known that $X^{\natural}$ is the moduli space of line bundles on $Y$ equipped with an integrable connection. For a discussion of $X^{\natural}$ in greater generality, see \cite{Me}, \cite{Ros}, and \cite{S}. Let ${\mathcal{A}}=\pi_*({\mathcal{O}}_{X^{\natural}})$. Since $\pi$ is an affine morphism, the category of ${\mathcal{O}}_{X^{\natural}}$-modules is equivalent to category of $\cal A$-modules. Then the subcategory of $\text{Mod}(\o_X)_{sp}$ whose objects satisfy $[\psi, \psi]=0$ is precisely $\text{Mod}({\mathcal{A}})=\text{Mod}({\mathcal{O}}_{X^{\natural}})$ (Proposition \ref{curvature}). Thus we have an equivalence of categories \footnote{The referee informs us that this equivalence also appears in an unpublished preprint by Laumon \cite{L}} \begin{equation}\label{equiv2.intro} D^b\text{Mod}({\mathcal{O}}_{X^{\natural}})\ \ \leftrightarrow\ \ D^b\text{Mod}({\mathcal{D}}_Y)\ . \end{equation} The outline of the paper is as follows. In section \ref{mainthm} we prove the basic equivalence theorem. By way of illustration, section \ref{Mat's thm} offers a new proof of a theorem of Matsushima on vector bundles with a connection. In section \ref{integrable case} the equality \eqref{curvature.intro} is established and the equivalence \eqref{equiv2.intro} is proved. Section \ref{some examples} gives some examples. In particular, we establish the formula \begin{equation} \hat{{\mathcal{A}}} = {\mathcal{D}}_{\{0\}\rightarrow Y}\ . \end{equation} Sections \ref{duality} and \ref{char var} contain general results about coherence, holonomicity and the characteristic variety. Section \ref{krich} illustrates the theory in the case of the Krichever construction. In sections \ref{nak} and \ref{PDOs} we refine and extend several results of Nakayashiki on characteristic varieties of BA-modules and commuting rings of matrix partial differential operators. This last topic is a natural setting for the further study of integrable systems; some brief remarks on this relationship are included at the end. Further applications to nonlinear partial differential equations will appear in a future work. \vskip 12pt \noindent Acknowledgements: The author wishes to thank Robert Varley for many valuable discussions. \section{First equivalence theorem}\label{mainthm} We adopt the following sign conventions for the Fourier-Mukai transform: \begin{align} \text{Mod}({\mathcal{O}}_X)&\overset{{\mathcal{S}}_1}\longrightarrow\text{Mod}({\mathcal{O}}_Y)\notag\\ {\mathcal{S}}_1({\mathcal{F}})&=\alpha_{2*}({\mathcal{P}}\otimes \alpha_1^*(-1)^*({\mathcal{F}}))\ , \end{align} \begin{align} \text{Mod}({\mathcal{O}}_Y)&\overset{{\mathcal{S}}_2}\longrightarrow\text{Mod}({\mathcal{O}}_X)\notag\\ {\mathcal{S}}_2({\mathcal{G}})&=\alpha_{1*}({\mathcal{P}}\otimes \alpha_2^*({\mathcal{G}}))\ . \end{align} The fundamental result in \cite{Muk} is \vskip.1in \begin{Fund} The derived functors \begin{align} D^b\text{Mod}({\mathcal{O}}_X)&\overset{R{\mathcal{S}}_1}{\longrightarrow} D^b\text{Mod}({\mathcal{O}}_Y)\notag\\ D^b\text{Mod}({\mathcal{O}}_Y) &\overset{R\cal S_2}{\longrightarrow} D^b\text{Mod}({\mathcal{O}}_X)\end{align} are defined, and \begin{align} R{\mathcal{S}}_1 R{\mathcal{S}}_2 &= T^{-g}\\ R{\mathcal{S}}_2 R{\mathcal{S}}_1 &= T^{-g}\ , \end{align} where $T$ is the shift autormophism on the derived category, \begin{equation} (TF)^n = F^{n+1}.\notag \end{equation} \end{Fund} \begin{pf} \cite[p. 156]{Muk} \end{pf} The other key result of [Muk] for our purposes is that the Fourier-Mukai transform\ exchanges tensor product and Pontrjagin product. Letting $\beta_1$ and $\beta_2$ denote the projections from $Y \times Y$ to $Y$ and denoting the group law on $Y$ by $m$, the Pontrjagin product is defined by \begin{equation} {\mathcal{F}} * {\mathcal{G}} = m_*(\beta^*_2({\mathcal{F}}) \otimes \beta^*_1({\mathcal{G}}))\ . \end{equation} \begin{Funda} \begin{align}R{\mathcal{S}}_2({\mathcal{G}}_1 \overset{R}{\underset{=}{*}} {\mathcal{G}}_2) &= R{\mathcal{S}}_2({\mathcal{G}}_1) \overset{L}{\underset{=}{\otimes}} RS_2({\mathcal{G}}_2)\label{exch2} \\ R{\mathcal{S}}_1 ({\mathcal{F}}_1 \overset{L}{\underset{=}{\otimes}} {\mathcal{F}}_2) &= T^g(R{\mathcal{S}}_1({\mathcal{F}}_1) \overset{R}{\underset{=}{*}} R{\mathcal{S}}_1({\mathcal{F}}_2))\ .\label{exch1} \end{align} \end{Funda} \begin{pf} \cite[p. 160]{Muk} \end{pf} We want to apply this to the following situation. Let ${\mathcal{I}} \subset {\mathcal{O}}_Y$ be the ideal sheaf of $0 \in Y$, and let $k(0)$ be the skyscraper sheaf at the origin with fiber $k$. Then $R^0{\mathcal{S}}_2(k(0))={\mathcal{O}}_X$ and $R^i{\mathcal{S}}_2(k(0))=0$ for $i>0$. Thus the Fourier-Mukai transform\ takes the short exact sequence \begin{equation}\label{nptext} 0 \longrightarrow {\mathfrak{g}}^* \otimes_k k(0) \longrightarrow {\mathcal{O}}/{\mathcal{I}}^2 \longrightarrow k(0) \longrightarrow 0 \end{equation} to a short exact sequence of vector bundles on $X$, and it is easy to see that it is precisely the sequence \eqref{taut seq}. Thus by theorem Mukai 2, if ${\mathcal{F}} \overset{\psi}{\longrightarrow} {\mathcal{E}} \otimes {\mathcal{F}}$ is a splitting on ${\mathcal{F}}$, $\psi$ induces a morphism \begin{equation} R{\mathcal{S}}_1({\mathcal{F}}) \overset{{\mathcal{S}}_1(\psi)}{\longrightarrow} T^{-g}({\mathcal{O}}/{\mathcal{I}}^2) * R{\mathcal{S}}_1({\mathcal{F}})\ . \end{equation} (We identify objects in an abelian category with complexes concentrated in degree 0.) If we now take $g^{th}$ cohomology, we get \begin{equation}\label{The point} {\mathcal{S}}_1({\mathcal{F}}) \overset{H^g{\mathcal{S}}_1(\psi)}{\longrightarrow} {\mathcal{O}}/{\mathcal{I}}^2 * {\mathcal{S}}_1({\mathcal{F}}). \end{equation} The point is this. If ${\mathcal{G}}$ is any ${\mathcal{O}}_Y$-module, there is a prolongation sequence \begin{equation}\label{c} 0 \longrightarrow \Omega^1 \otimes_{{\mathcal{O}}}{\mathcal{G}} \longrightarrow j({\mathcal{G}}) \overset{\nu_{{\mathcal{G}}}}{\longrightarrow} {\mathcal{G}} \longrightarrow 0, \end{equation} such that a splitting of $\nu_{{\mathcal{G}}}$ is precisely a connection on ${\mathcal{G}}$. As a sheaf of abelian groups, \begin{equation} j({\mathcal{G}}) = {\mathcal{G}} \oplus (\Omega^1 \otimes_{{\mathcal{O}}} {\mathcal{G}})\ , \end{equation} with ${\mathcal{O}}$-module structure \begin{equation} f (\phi, \omega \otimes \psi) =(f\phi, f \omega \otimes \psi + df \otimes \phi)\ \ . \end{equation} Thus a connection on ${\mathcal{G}}$ is a splitting of \eqref{c}. Since $Y$ is an abelian variety, there is a characterization of $j({\mathcal{G}})$ in terms of the Pontrjagin product. \begin{Lem} \label{pont} For any ${\mathcal{O}}_Y$-module ${\mathcal{G}}$, \begin{equation} j({\mathcal{G}}) = ({\mathcal{O}}/{\mathcal{I}}^2) * {\mathcal{G}}\ . \end{equation} \end{Lem} \begin{pf} Let $Y_1\subset Y\times Y$ denote the first order neighborhood of the diagonal, and let $\pi_i: Y_1 \longrightarrow Y$, $i = 1, 2$, denote the two projections. Then \begin{equation} j({\mathcal{F}})=\pi_{2*}\pi_1^*({\mathcal{F}})\ . \end{equation} (This holds for any variety.) Let $\tilde{Y} = \text{Spec}({\mathcal{O}}/{\mathcal{I}}^2)$, the first order neighborhood of $0$ in $Y$. Then $Y_1$ may be identified with $Y \times \tilde{Y}$ in such a way that $\pi_1$ corresponds to projection onto the first factor and $\pi_2$ corresponds to the group law, $\tilde m$. Let $\iota:\tilde Y\to Y$ denote the inclusion map. Then \begin{align} {\mathcal{O}}/{\mathcal{I}}^2&*{\mathcal{G}}= m_*(\beta_2^*\iota_*({\mathcal{O}}_{\tilde Y})\otimes\beta_1^*({\mathcal{G}}))\notag\\ &= m_*(1\times\iota)_*(\tilde\beta_2^*({\mathcal{O}}_{\tilde Y})\otimes \tilde\beta_1^*({\mathcal{G}}))\notag\\ &=\tilde m_* \tilde\beta_1^*({\mathcal{G}}))=j({\mathcal{G}})\ . \end{align} \end{pf} Combining this lemma with the map \eqref{The point}, we see that a splitting on ${\mathcal{F}}$ induces a splitting of the prolongation sequence of ${\mathcal{S}}_1(\cal F)$, i.e., a connection on ${\mathcal{S}}_1({\mathcal{F}})$. So we have a functor \begin{equation} \text{Mod}(\o_X)_{sp} \overset{S_1}{\longrightarrow} \text{Mod}(\o_Y)_{cxn}\ . \end{equation} We will check later that this description is equivalent to the one given in the introduction. Conversely, if we apply ${\mathcal{S}}_2$ to a splitting of the prolongation sequence, ${\mathcal{G}}\overset{\tau}{\longrightarrow} j({\mathcal{G}}) = {\mathcal{O}}/{\mathcal{I}}^2 * {\mathcal{G}}$, theorem Mukai 2 gives a splitting \begin{equation} {\mathcal{S}}_2({\mathcal{G}}) \overset{\psi}{\longrightarrow} {\mathcal{E}} \otimes {\mathcal{S}}_2({\mathcal{G}})\ . \end{equation} So we have \begin{equation} \text{Mod}(\o_Y)_{cxn} \overset{S_2}{\longrightarrow} \text{Mod}(\o_X)_{sp}\ . \end{equation} The categories $\text{Mod}({\mathcal{O}}_X)_{sp}$ and $\text{Mod}({\mathcal{O}}_Y)_{cxn}$ are abelian. Moreover, objects in either $\text{Mod}({\mathcal{O}}_X)_{sp}$ or $\text{Mod}({\mathcal{O}}_Y)_{cxn}$ may be resolved by a \v Cech resolution with respect to an affine open cover of $X$ or $Y$. Thus the derived functors \begin{equation} D^b\text{Mod}({\mathcal{O}}_X)_{sp} \overset{RS_1}{\longrightarrow} D^b\text{Mod}({\mathcal{O}}_Y)_{cxn} \end{equation} \begin{equation} D^b\text{Mod}({\mathcal{O}}_Y)_{cxn} \overset{RS_2}{\longrightarrow} D^b\text{Mod}({\mathcal{O}}_X)_{sp} \end{equation} exist. The main result of this section is \begin{Thm}\label{equiv} \begin{align} RS_1 RS_2 &= T^{-g} , \\ RS_2 RS_1 &= T^{-g} . \end{align} \end{Thm} \begin{pf} Let $\zeta$ denote the functor $T^g RS_1 RS_2$. Let $\underline{for}$ denote the forgetful functor from $D^b\text{Mod}({\mathcal{O}}_Y)_{cxn}$ to $D^b\text{Mod}({\mathcal{O}}_Y)$. Then \begin{equation}\underline{for}\ \zeta = \underline{for} \end{equation} by Mukai's theorem. In particular, for any object $({\mathcal{F}},\nabla) \in \text{Ob Mod}(\o_Y)_{cxn}$, $H^i\zeta({\mathcal{F}},\nabla) = 0$ for $i > 0$. Thus $\zeta({\mathcal{F}},\nabla)=({\mathcal{F}},\nabla')$ for some new connection $\nabla'$. Let ${\mathcal{F}} \overset{\tau}{\longrightarrow} j({\mathcal{F}})$ denote the splitting associated to $\nabla$, and let ${\mathcal{F}} \overset{\tau'}{\longrightarrow} j({\mathcal{F}})$ denote the splitting associated to $\nabla'$. Let $ \psi$ denote the corresponding splitting on ${\mathcal{S}}_2({\mathcal{F}})$. Then $\psi$ is the $0^{th}$ cohomology of \begin{equation} R{\mathcal{S}}_2({\mathcal{F}}) \overset{R\cal S_2(\tau)}{\longrightarrow} {\mathcal{E}} \overset{L}{\underset{=}{\otimes}} R{\mathcal{S}}_2({\mathcal{F}})\ , \end{equation} from which it follows that $\tau'$ is the $0^{th}$ cohomology of \begin{equation} \begin{CD} {\mathcal{F}} @>{T^gR{\mathcal{S}}_1 R{\mathcal{S}}_2(\tau)}>> j({\mathcal{F}})\ . \end{CD} \end{equation} Thus $\tau = \tau'$, again by Mukai's Theorem. Similarly, if $({\mathcal{G}}, \psi)$ is an object in $\text{Mod}({\mathcal{O}}_X)_{sp}$, \begin{equation} T^gRS_2 RS_1({\mathcal{G}}, \psi) = ({\mathcal{G}}, \psi)\ . \end{equation} The next lemma then completes the proof of the theorem. \end{pf} \begin{Lem} \label{bootstrap} Let $C_1$ and $C_2$ be abelian categories, and let \begin{equation} D^bC_1 \underset{F_2}{\overset{F_1}{\rightarrow}} D^bC_2 \end{equation} be $\delta$-functors. If $F_1$ and $F_2$ are isomorphic when restricted to the subcategory $C_1\subset D^bC_1$, then they are isomorphic. \end{Lem} \begin{pf} This follows by induction on the cohomological length of an object in the bounded derived category, using \cite[lemme 12.6, p.104]{Bo} and the triangle axiom TR3, \cite[p.28]{Bo}, once it is noted that the constructions used there are functorial. \end{pf} \noindent{\bf Remark}\ \ Let $\gamma_i$, $\gamma_{i,j}$ denote the projections on $X\times Y\times Y$. The key to Mukai's theorem is the elementary formula \begin{equation} \label{key} \gamma^*_{1,2} ({\mathcal{P}}) \otimes \gamma^*_{1,3}({\mathcal{P}}) = (1\times m)^*({\mathcal{P}})\ . \end{equation} This formula also plays a crucial but hidden role in theorem \ref{equiv}, which we would like to make explicit. Let $\tilde{Y} = \text{Spec}({\mathcal{O}}/{\mathcal{I}}^2)$. Then ${\mathcal{P}}|_{X \times \tilde{Y}}$ is a line bundle on $X \times \tilde{Y}$, trivial on $X \times \{0\}$. Set $\tilde{{\mathcal{P}}} = {\mathcal{P}}|_{X \times \tilde{Y}}$, and let $\tilde{\alpha}_1: X \times \tilde{Y} \rightarrow X$ be the projection. Then ${\mathcal{E}} = \tilde{\alpha}_{1*}(\tilde{{\mathcal{P}}})$ and $\mu: {\mathcal{E}} \rightarrow {\mathcal{O}}$ is the morphism which restricts a section of $\tilde{{\mathcal{P}}}$ to $X \times \{0\}$. Let $\tilde\gamma_i$, $\tilde\gamma_{i,j}$ denote the projections on $X\times Y\times \tilde Y$. Then we get an infinitesimal form of \ref{key}, \begin{equation}\label{key2} \tilde{\gamma}^*_{1,2} ({\mathcal{P}}) \otimes \tilde{\gamma}^*_{1,3}(\tilde{{\mathcal{P}}}) = (1\times \tilde m)^*({\mathcal{P}}) \ . \end{equation} If ${\mathcal{G}}$ is a sheaf on $X\times Y$, then a connection on ${\mathcal{G}}$ relative to $X$ is an isomorphism \begin{equation} \tilde{\gamma}_{1,2}^*({\mathcal{G}}) \approx (1\times \tilde m)^*({\mathcal{G}}) \end{equation} restricting to the identity on $X \times Y$. Thus \eqref{key2} says that $\tilde{\gamma}^*_{1,3}(\tilde{{\mathcal{P}}})$ is the obstruction to endowing ${\mathcal{P}}$ with a connection relative to $X$. Given a sheaf ${\mathcal{F}}$ of ${\mathcal{O}}_X$-modules, a splitting on ${\mathcal{F}}$ is precisely what is needed to cancel this obstruction. Indeed, a splitting may be regarded as an isomorphism \begin{equation} \tilde{\alpha}_1^*({\mathcal{F}})\overset{\psi}{\to} \tilde{{\mathcal{P}}}\otimes\tilde{\alpha}_1^*({\mathcal{F}}) \end{equation} restricting to the identity on $X$. Applying $(-1_X,-1_{\tilde Y})$, we get \begin{equation} \tilde\alpha_1^*(-1)^*({\mathcal{F}})\longrightarrow \tilde{{\mathcal{P}}}\otimes\tilde\alpha_1^*(-1)^*({\mathcal{F}})\ \ . \end{equation} If we then apply $\tilde{\gamma}_{1,2}^*$ to ${\mathcal{P}}\otimes\alpha_1^*(-1)^*({\mathcal{F}})$ we find \begin{align} \tilde\gamma_{1,2}^*({\mathcal{P}}\otimes&\alpha_1^*(-1)^*({\mathcal{F}}))= \tilde\gamma_{1,2}^*({\mathcal{P}})\otimes\tilde\gamma_{1,3}^* \tilde\alpha_1^*(-1)^*({\mathcal{F}})\notag\\ &\approx \tilde\gamma_{1,2}^*({\mathcal{P}})\otimes\tilde\gamma_{1,3}^*(\tilde{{\mathcal{P}}}) \otimes\tilde\gamma_{1,3}^*\tilde\alpha_1^*(-1)^*({\mathcal{F}}) \notag\\ &=(1\times\tilde m)^*({\mathcal{P}})\otimes\tilde\gamma_{1,3}^*\tilde\alpha_1^*(-1)^*({\mathcal{F}})\notag\\ &=(1\times\tilde m)^*({\mathcal{P}}\otimes\alpha_1^*(-1)^*({\mathcal{F}}))\ \ , \label{cxnalt} \end{align} which is a relative connection on ${\mathcal{P}}\otimes\alpha_1^*(-1)^*({\mathcal{F}})$. Then apply $\alpha_{2*}$ to get a sheaf with connection on $Y$, and this is our functor $S_1$. \section{Matsushima's Theorem}\label{Mat's thm} As an application of theorem \ref{equiv}, we will give a new proof of Matsushima's Theorem on the homogeneity of vector bundles admitting a connection \cite{Mat}. The key is the following lemma, which is of interest in its own right. Here we assume $char(k)=0$. \begin{Lem} \label{fin supp} Let ${\mathcal{F}}$ be a coherent ${\mathcal{O}}_X$-module with a splitting. Then ${\mathcal{F}}$ is finitely supported. \end{Lem} \begin{pf} The proof is similar to that of lemma 3.3 in \cite{Muk}. Assuming $\text{dim(supp}({\mathcal{F}}))>0$, let $C$ be a curve contained in $\text{supp}({\mathcal{F}})$, and let $\tilde C\overset{\pi}\longrightarrow C$ be its normalization. Let ${\mathcal{F}}'=\pi^*({\mathcal{F}})$. Then we get a non-zero vector bundle ${\mathcal{F}}''$ on $\tilde C$ upon taking the quotient of ${\mathcal{F}}'$ by its torsion part. Let ${\mathcal{E}}'$ denote the pullback of ${\mathcal{E}}|_C$ to $\tilde C$. Then any splitting on ${\mathcal{F}}$ induces a splitting \begin{equation} {\mathcal{F}}'\overset{\psi'}\longrightarrow {\mathcal{E}}'\otimes {\mathcal{F}}'\ . \end{equation} Since $Tor( {\mathcal{E}}'\otimes {\mathcal{F}}')= {\mathcal{E}}'\otimes Tor({\mathcal{F}}')$, we get a splitting of the sequence \begin{equation}\label{can't split} 0 \longrightarrow {\mathfrak{g}}^* \otimes {\mathcal{F}}'' \longrightarrow {\mathcal{E}}' \otimes {\mathcal{F}}'' \longrightarrow {\mathcal{F}}'' \longrightarrow 0. \end{equation} But the extension class of the sequence \eqref{can't split} is the bottom arrow in the commutative diagram \begin{equation} \begin{matrix} H^1(\tilde{C}, {\mathcal{O}}) \\ a\nearrow \qquad\qquad \searrow b \quad \\ H^1(X, {\mathcal{O}}) \overset{e}{\longrightarrow} H^1(\tilde{C}, {\mathcal{E}}nd({\mathcal{F}}'')) \end{matrix}\ , \end{equation} where $b$ is induced by the natural inclusion ${\mathcal{O}}\overset{\beta}\longrightarrow \cal End({\mathcal{F}}'')$ and $a$ is the derivative of the natural map $Pic(X)\to Pic(\tilde C)$. In particular, $a$ is not the $0$-map. Moreover, in characteristic $0$, $\beta$ splits by the trace, so $b$ is injective. This is a contradiction, for now $e\ne 0$ so that \eqref{can't split} does not split. \end{pf} We then have \begin{Thm}[Matsushima] Any vector bundle on $Y$ admitting a connection is homogeneous. \end{Thm} \begin{pf} Let ${\mathcal{G}}$ be such a vector bundle. Then the cohomology sheaves $R^i{\mathcal{S}}_2({\mathcal{G}})$ are ${\mathcal{O}}_X$-coherent and admit splittings. By lemma \ref{fin supp}, they are all finitely supported. As in \cite[example 3.2, p. 158]{Muk}, we then have $R^i{\mathcal{S}}_2({\mathcal{G}})=0$ for $i\ne g$, and ${\mathcal{G}}$ is then the Fourier-Mukai transform\ of a finitely supported sheaf. By \cite[3.1]{Muk}, ${\mathcal{G}}$ is homogeneous. \end{pf} \noindent{\bf Remark}\ \ The converse of the statement above is also part of Matsushima's theorem. That the converse can be proved by the Fourier-Mukai transform\ is already noted in \cite[prop 5.9]{N2}. \section{Curvature tensor and the integrable case}\label{integrable case} If $({\mathcal{G}},\nabla)$ is a sheaf with connection on $Y$, then its curvature tensor is a linear map \begin{equation} [\nabla,\nabla]:\wedge^2({\mathfrak{g}})\longrightarrow End({\mathcal{G}})\ \ . \end{equation} Before explaining how the curvature can be read off from the transform of ${\mathcal{G}}$, we want to show that the functor $S_1$ has the \v Cech description given in the introduction. Let $\psi_1$ and $\psi_2$ be two splittings on a given sheaf of ${\mathcal{O}}_X$-modules ${\mathcal{F}}$, and let $\eta= \psi_1-\psi_2$. By proposition \ref{affine space}, $\eta$ is a map \begin{equation} \eta:{\mathfrak{g}}\longrightarrow End({\mathcal{F}})\ . \end{equation} Applying the Fourier-Mukai transform\ to $End({\mathcal{F}})$, we get \begin{equation} {\mathcal{S}}_1(\eta):{\mathfrak{g}}\longrightarrow End({\mathcal{S}}_1({\mathcal{F}}))\ . \end{equation} Denoting the two connections on ${\mathcal{S}}_1({\mathcal{F}})$ by $\nabla_1$ and $\nabla_2$, it is easy to check that \begin{equation}\label{compat} \nabla_2= \nabla_1-{\mathcal{S}}_1(\eta)\ . \end{equation} Now let ${\mathcal{U}}\overset{\iota}\longrightarrow X$ be an affine open subset, equipped with a section $\rho\in\Gamma({\mathcal{U}},{\mathcal{E}})$ such that $\mu(\rho)=1$. Then $(\iota_*({\mathcal{O}}_{{\mathcal{U}}}),\rho)$ is an object in $\text{Mod}(\o_X)_{sp}$. The corresponding object in $\text{Mod}(\o_Y)_{cxn}$ is a connection on $\alpha_{2*}({\mathcal{P}}|_{{\mathcal{U}}\times Y})$. Since functions on ${\mathcal{U}}$ act as endomorphisms of $(\iota_*({\mathcal{O}}_{{\mathcal{U}}}),\rho)$, this connection is linear over $\alpha_{2*}\alpha_1^{-1}({\mathcal{O}}_{{\mathcal{U}}})$. So we in fact have a connection relative to ${\mathcal{U}}$ on ${\mathcal{P}}|_{{\mathcal{U}}\times Y}$. Call this connection $\nabla^{\rho}$. If ${\mathcal{F}}$ is any ${\mathcal{O}}_X$-module, $\rho$ induces a splitting on ${\mathcal{F}}|_{{\mathcal{U}}}$. Then if $\psi$ is any other splitting on ${\mathcal{F}}$, it must take the form \begin{equation} \psi(\kern .5em\cdot\kern .5em)=\kern .5em \cdot \kern .5em\otimes\rho+ \sum\limits_{i=1}^g\omega^i \psi(i)^{\rho}(\kern .5em\cdot\kern .5em)\ , \end{equation} for some endomorphisms $\psi(i)^{\rho}$. Again the corresponding sheaf with connection on $Y$ is the direct image of a sheaf with relative connection on $X\times Y$, and by \eqref{compat} it has the form of \eqref{relcxn}. It is now easy to read off the curvature tensor of $S_1({\mathcal{F}},\psi)$. There is an exact sequence \begin{align} 0 \longrightarrow \wedge^2{\mathfrak{g}}^* \otimes {\mathcal{O}} &\longrightarrow \wedge^2 {\mathcal{E}} \overset{\mu_2}{\longrightarrow} {\mathfrak{g}}^* \otimes {\mathcal{O}} \longrightarrow 0,\notag\\ \mu_2(\rho_1 \wedge \rho_2) &= \mu(\rho_1) \rho_2 - \mu(\rho_2) \rho_1. \end{align} If we iterate $\psi$ and then skew-symmetrize, we get a map \begin{equation} {\mathcal{F}} \overset{[\psi, \psi]}{\longrightarrow} \wedge^2({\mathcal{E}}) \otimes{\mathcal{F}}. \end{equation} One sees easily that $(\mu_2 \otimes 1) \circ [\psi, \psi] = 0$, so \begin{equation} [\psi, \psi] \in \wedge^2 {\mathfrak{g}}^* \otimes End({\mathcal{F}})\ .\end{equation} As noted in the introduction, in terms of the family of endomorphisms $\psi(i)_n$, one simply has \begin{equation}\label{bracket} [\psi, \psi]=\sum \omega^i\wedge\omega^j[\psi(i)_n,\psi(j)_n]\ . \end{equation} Applying the functor ${\mathcal{S}}_1$ to morphisms, we get ${\mathcal{S}}_1([\psi, \psi]) \in \wedge^2 {\mathfrak{g}}^* \otimes End({\mathcal{S}}_1({\mathcal{F}}))$. Set $S_1({\mathcal{F}},\psi)=({\mathcal{S}}_1({\mathcal{F}}),\nabla_{\psi})$. Let $[\nabla,\nabla]$ denote curvature. \begin{Prop}\label{curvature} $[\nabla_{\psi},\nabla_{\psi}]=\cal S_1([\psi,\psi])$. \end{Prop} \begin{pf} First we claim that the relative connections $\nabla_n$ on ${\mathcal{P}}\vert_{{\mathcal{U}}_n\times Y}$ are integrable. Indeed, from \eqref{overlaps}, the curvature tensor of $\nabla_n$ is independent of the index $n$. To show that it is 0, we have only to consider the case where $0 \in {\mathcal{U}}_n$ and note that all connections on the trivial bundle ${\mathcal{O}}_Y$ are integrable. Denoting the relative connection on ${\mathcal{P}}\otimes\alpha^*_1(-1_X)^*({\mathcal{F}})$ by $\nabla=\sum \omega^i \nabla(i)$, \begin{align} [\nabla(i), \nabla(j)]&=[\nabla(i)_n\otimes 1 - 1\otimes\psi(i)_n, \nabla(j)_n\otimes 1 - 1\otimes\psi(j)_n]\notag\\ &=1\otimes[\psi(i)_n, \psi(j)_n]\ \ . \end{align} \end{pf} Similarly, given $({\mathcal{G}},\nabla)$ in $\text{Ob Mod}(\o_Y)_{cxn}$, we have $\cal S_2([\nabla,\nabla])\in \wedge^2({\mathfrak{g}}^*)\otimes End({\mathcal{S}}_2({\mathcal{G}}))$. Setting $S_2({\mathcal{G}},\nabla)=({\mathcal{S}}_2({\mathcal{G}}),\psi_{\nabla})$, one proves \begin{Prop} $[\psi_{\nabla},\psi_{\nabla}]={\mathcal{S}}_2([\nabla,\nabla])$. \end{Prop} Turning now to the integrable case, consider the group extension \begin{equation} 0 \longrightarrow {\mathfrak{g}}^* \longrightarrow X^{\natural} \overset\pi{\longrightarrow} X \longrightarrow 0 \end{equation} mentioned in the introduction. The morphism $\pi$ being affine, $X^{\natural}$ is characterized by $\pi_*$ of its structure sheaf. Define a sheaf ${\mathcal{A}}$ of ${\mathcal{O}}_X$-modules as follows. For each affine open ${\mathcal{U}} \subset X$ and each $\rho \in {\mathcal{E}}({\mathcal{U}})$ such that $\mu(\rho) = 1$, introduce independent variables $x^{\rho}_1, \cdots, x^{\rho}_g$. Then \begin{equation} {\mathcal{A}}|_{{\mathcal{U}}}= {\mathcal{O}}|_{{\mathcal{U}}}[x^{\rho}_1, \cdots, x^{\rho}_g]\ . \end{equation} Glue these sheaves together by the rule that if $\tilde{\rho} =\rho + \operatornamewithlimits\sum\limits^g_{i=1} \omega^i f_i$, then \begin{equation} x^{\rho}_i = x^{\tilde{\rho}}_i + f_i \end{equation} as sections of ${\mathcal{A}}$. \begin{Def} $X^{\natural}=\text{Spec}({\mathcal{A}})$. \end{Def} \begin{Prop}\label{subcat} The full subcategory of $\text{Mod}(\o_X)_{sp}$ whose objects $({\mathcal{F}}, \psi)$ satisfy $[\psi, \psi] = 0$ is canonically isomorphic to the category of ${\mathcal{O}}$-modules on $X^{\natural}$, which is to say the category of ${\mathcal{A}}$-modules on $X$. \end{Prop} \begin{pf} Let $\rho$ be as above. Since ${\mathcal{A}}$ is locally ${\mathcal{O}}[x^{\rho}_1, \dots, x^{\rho}_g]$, to give an ${\mathcal{A}}$-module structure to an $\cal O_X$-module ${\mathcal{F}}$ is to choose, for every $\rho$, a set of commuting endomorphisms $\psi^{\rho}_1, \dots, \psi^{\rho}_g \in\Gamma({\mathcal{U}},{\mathcal{E}}nd({\mathcal{F}}))$, such that if \begin{equation} \tilde{\rho} = \rho + \sum \omega^i f_i\ , \end{equation} then $\psi^{\rho}_i - \psi^{\tilde{\rho}}_i=$ multiplication by $f_i$. This is clearly the same as lifting ${\mathcal{F}}$ to an object $({\mathcal{F}}, \psi) \in \text{Ob Mod}(\o_X)_{sp}$ such that $[\psi, \psi] = 0$. \end{pf} Now we may restrict our functors $S_1$ and $S_2$ to the subcategories $\text{Mod}({\mathcal{O}}_{X^{\natural}})\subset\text{Mod}(\o_X)_{sp}$ and $\text{Mod}({\mathcal{D}}_Y)\subset\text{Mod}(\o_Y)_{cxn}$ respectively, to get functors \begin{align} \text{Mod}({\mathcal{O}}_{X^{\natural}})&\overset{S_1}\longrightarrow \text{Mod}(\cal D_Y)\notag\\ \text{Mod}({\mathcal{D}}_Y)&\overset{S_2}\longrightarrow\text{Mod}({\mathcal{O}}_{X^{\natural}})\ \ . \end{align} We then get our second equivalence theorem, whose proof is the same as that of the first. \begin{Thm}\label{equiv2} The derived functors \begin{equation} D^b\text{Mod}({\mathcal{O}}_{X^{\natural}}) \overset{RS_1}{\longrightarrow} D^b\text{Mod}({\mathcal{D}}_Y) \end{equation} \begin{equation} D^b\text{Mod}({\mathcal{D}}_Y) \overset{RS_2}{\longrightarrow} D^b\text{Mod}({\mathcal{O}}_{X^{\natural}}) \end{equation} exist, and satisfy \begin{align} RS_1 RS_2 &= T^{-g} , \notag\\ RS_2 RS_1 &= T^{-g} .\notag \end{align} \end{Thm} \section{Some examples}\label{some examples} \subsection{} Using theorem \ref{equiv2}, one easily recovers the observation that $X^{\natural}$ is the moduli space of degree-0 line bundles on $Y$ equipped with a connection \cite{Me}. Indeed, let $\sigma:Z\to X^{\natural}$ be any morphism and let $\sigma'=\pi\circ\sigma$. Let ${\mathcal{P}}_{\sigma'}$ be the line bundle on $Z\times Y$ induced by $\sigma'$. Then the ${\mathcal{D}}_Y$-module $S_1(\sigma_*({\mathcal{O}}_Z))$ is in fact the direct image of a flat connection relative to $Z$ on ${\mathcal{P}}_{\sigma'}$. In particular, we have the 1:1 correspondence $$\text{ points of}\ X^{\natural} \leftrightarrow \text{line bundles with connection on}\ Y\ .$$ \subsection{} Let $J\subset\text{Sym}({\mathfrak{g}})$ be an ideal. Then $\text{Spec}(\text{Sym}({\mathfrak{g}})/J)$ is a subvariety of the fiber $\pi^{-1}(0)$, and we may apply $S_1$ to its structure sheaf. This gives a ${\mathcal{D}}$-module structure on ${\mathcal{O}}_Y\otimes_k\text{Sym}({\mathfrak{g}})/J$. Writing sections of ${\mathcal{D}}$ in the standard format with functions on the left and invariant differential operators on the right, make the identification \begin{equation} {\mathcal{D}}\simeq{\mathcal{O}}_Y\otimes_k\text{Sym}({\mathfrak{g}})\ . \end{equation} Then ${\mathcal{O}}_Y\otimes_kJ$ is a left ideal, and $S_1( k(0)\otimes_k\text{Sym}({\mathfrak{g}})/J)$ is the quotient ${\mathcal{D}}$-module. In particular, \vskip 12pt \noindent ${\mathcal{D}}_Y$ is the Fourier-Mukai transform\ of the structure sheaf of the subgroup ${\mathfrak{g}}^* =\pi^{-1}(0)\subsetX^{\natural}$. \subsection{}\label{B itself} Going the other way, we may ask which ${\mathcal{D}}_Y$-module transforms to ${\mathcal{A}}$. By theorem \ref{equiv2} this is the same as asking for the Fourier-Mukai transform\ of ${\mathcal{A}}$. Observe first that there is an important filtration $\{\cal A(m)\}$ on ${\mathcal{A}}$ coming from the local identifications of ${\mathcal{A}}$ with a polymonial algebra over ${\mathcal{O}}$. One has exact sequences \begin{equation}\label{filtration} 0 \rightarrow {\mathcal{A}}(m) \rightarrow {\mathcal{A}}(m+1) \rightarrow Sym^{m+1}({\mathfrak{g}}) \otimes_k {\mathcal{O}}_X \rightarrow 0\ \ . \end{equation} We then have \begin{Lem} $R^i{\mathcal{S}}_1({\mathcal{A}})=0$ for $i\ne g$, and $R^g{\mathcal{S}}_1({\mathcal{A}})$ is supported at the origin.\end{Lem}\begin{pf} By \eqref{filtration} the question reduces to the corresponding assertion about ${\mathcal{O}}$. The latter is proved in \cite[p.126]{Mum}.\end{pf} It remains to identify the ${\mathcal{D}}_Y$-module $R^gS_1({\mathcal{A}})$. If ${\mathcal{G}}$ is an ${\mathcal{O}}_Y$-module, we may form ${\mathcal{D}} \otimes_{{\mathcal{O}}} {\mathcal{G}}$, using right multiplication by functions as the ${\mathcal{O}}$-module structure on ${\mathcal{D}}$. Then ${\mathcal{D}} \otimes_{{\mathcal{O}}} {\mathcal{G}}$ has a left ${\mathcal{D}}$-module structure. In particular, define \begin{equation} {\mathcal{D}}_{\{0\}\rightarrow Y}\overset{def}={\mathcal{D}}\otimes k(0)\ . \end{equation} \begin{Prop}\label{fm of B} $R^gS_1({\mathcal{A}}) = {\mathcal{D}}_{\{0\}\rightarrow Y}$. \end{Prop} \begin{pf*}{First proof} By theorem \ref{equiv2}, it suffices to prove that $R^0S_2({\mathcal{D}}_{\{0\}\rightarrow Y}) = {\mathcal{A}}$. Explicitly, $\cal D_{\{0\}\rightarrow Y}$ is the sheaf of $k$-vector spaces $\text{Sym}({\mathfrak{g}})(0)$, with the following ${\mathcal{D}}$-module structure. Given $L \in \text{Sym}({\mathfrak{g}})$ and $\xi \in {\mathfrak{g}}$, $\xi$ acts on $L$ by $\xi \cdot L = \xi L$. To give the action of ${\mathcal{O}}_Y$, we need only consider $f \in {\mathcal{O}}_{Y,0}$. Expand the differential operator $f L$ in the form $\sum \xi^I g_I$,\ $g_I \in {\mathcal{O}}_{Y, 0}$. Define $$ :f L: = \sum \xi^I g_I(0)\in\text{Sym}({\mathfrak{g}})\ . $$ Then $f \cdot L = :f L:$\ . Let ${\mathcal{F}} = R^0S_2({\mathcal{D}}_{\{0\}\rightarrow Y})$. We want to prove that ${\mathcal{F}}$ is a free rank-1 ${\mathcal{A}}$-module. We have the global section $1 \in\text{Sym}({\mathfrak{g}}) = H^0(Y, {\mathcal{D}}_{\{0\} \rightarrow Y})$. Let ${\mathcal{U}} \subset X$ be an open subset and let $\sigma$ be a meromorphic section of ${\mathcal{P}}$ on ${\mathcal{U}} \times Y$ such that $\sigma(0) = 1$, where $\sigma(0) = \sigma|_{{\mathcal{U}} \times \{0\}}$. Since $\sigma$ is regular on the support of $\alpha^*_2({\mathcal{D}}_{\{0\}\rightarrow Y})$, there is a well-defined section $$ \sigma\otimes 1 \in \Gamma({\mathcal{U}} \times Y, {\mathcal{P}} \otimes \alpha^*_2({\mathcal{D}}_{\{0\}\rightarrow Y})) = \Gamma({\mathcal{U}}, {\mathcal{F}})\ . $$ If $f$ is a meromorphic function on ${\mathcal{U}} \times Y$, regular on ${\mathcal{U}} \times \{0\}$, then $$ f \sigma \otimes 1 = \sigma \otimes :f 1 :\ = f(0) \sigma \otimes 1\ . $$ Thus $\sigma \otimes 1$ is independent of $\sigma$, and defines a global section of ${\mathcal{F}}$. Now let $\rho \in \Gamma({\mathcal{U}}, {\mathcal{E}})$ satisfy $\mu(\rho) = 1$. Then $\rho$ determines both a relative connection $\nabla$ on ${\mathcal{P}}|_{{\mathcal{U}} \times Y}$ and a set of sections $x(1), \dots, x(g) \in \Gamma({\mathcal{U}}, {\mathcal{A}})$, such that ${\mathcal{A}}|_{{\mathcal{U}}} = {\mathcal{O}}[x(1),\dots, x(g)]$. By definition of the ${\mathcal{A}}$-module structure on ${\mathcal{F}}$, \begin{align} &x(i_1) \dots x(i_k)(\sigma \otimes L)\notag\\ = &x(i_1) \dots x(i_{k-1})(\nabla_{i_k}(\sigma) \otimes L - \sigma\otimes \xi_{i_k} L)\ . \end{align} It now follows by induction on $k$ that $\sigma\otimes 1$ generates ${\mathcal{F}}$ freely as a ${\mathcal{A}}$-module. \end{pf*} A second proof will be given in section \ref{duality}. \section{Coherence, Duality and Holonomicity}\label{duality} \subsection{} If ${\mathcal{G}}$ is an ${\mathcal{O}}_Y$-module, we may form ${\mathcal{D}}\otimes_{{\mathcal{O}}} {\mathcal{G}}$ using right multiplication by functions as the ${\mathcal{O}}_Y$-module structure on ${\mathcal{D}}$. This leaves us with left multiplication by elements of ${\mathcal{D}}$ to give a left ${\mathcal{D}}$-module structure. Similarly, if ${\mathcal{F}}$ is an ${\mathcal{O}}_X$-module, we may form the ${\mathcal{A}}$-module ${\mathcal{A}}\otimes{\mathcal{F}}$. Note that \begin{align}\label{adjoint} Hom_{{\mathcal{D}}}({\mathcal{D}}\otimes_{{\mathcal{O}}} {\mathcal{G}},\ \cdot\ )&=Hom_{{\mathcal{O}}}( {\mathcal{G}},\ \underline{for}(\cdot)\ )\ ; \label{adjoint1}\\ Hom_{{\mathcal{A}}}({\mathcal{A}}\otimes{\mathcal{F}},\ \cdot\ )&=Hom_{{\mathcal{O}}}( {\mathcal{F}},\ \underline{for}(\cdot)\ )\ , \label{adjoint2}\end{align} where $\underline{for}$ is the forgetful functor. \begin{Thm}\label{tensor} \begin{equation}RS_2({\mathcal{D}}\overset{L}{\underset{=}{\otimes}}(\cdot))= {\mathcal{A}}\overset{L}{\underset{=}{\otimes}}R{\mathcal{S}}_2(\cdot)\ . \end{equation} \begin{equation}RS_1({\mathcal{A}}\overset{L}{\underset{=}{\otimes}}(\cdot))= {\mathcal{D}}\overset{L}{\underset{=}{\otimes}}R{\mathcal{S}}_1(\cdot)\ . \end{equation} \end{Thm} \begin{pf} By theorem \ref{equiv2}, it suffices to prove the first equality. Let ${\mathcal{F}}_1$ be an object in $D^b\text{Mod}({\mathcal{A}})$ and let ${\mathcal{F}}_2$ be an object in $D^b\text{Mod}({\mathcal{O}}_X)$. Using \eqref{adjoint1}, \eqref{adjoint2}, Mukai's theorem and theorem \ref{equiv}, \begin{align} Hom({\mathcal{A}}&\overset{L}{\underset{=}{\otimes}}{\mathcal{F}}_2,{\mathcal{F}}_1)\notag\\ &=Hom({\mathcal{F}}_2,\underline{for}({\mathcal{F}}_1))\notag\\ &=Hom(R\cal S_1({\mathcal{F}}_2),\underline{for}(RS_1({\mathcal{F}}_1)))\notag\\ &=Hom({\mathcal{D}}\overset{L}{\underset{=}{\otimes}}R{\mathcal{S}}_1({\mathcal{F}}_2),RS_1({\mathcal{F}}_1))\notag\\ &=Hom(RS_2({\mathcal{D}}\overset{L}{\underset{=}{\otimes}}R\cal S_1({\mathcal{F}}_2)),T^{-g}({\mathcal{F}}_1))\notag\\ &=Hom(RS_2({\mathcal{D}}\overset{L}{\underset{=}{\otimes}}T^gR{\mathcal{S}}_1({\mathcal{F}}_2)),{\mathcal{F}}_1)\ . \end{align} Writing ${\mathcal{F}}_2$ as $R{\mathcal{S}}_2({\mathcal{G}})$ for some ${\mathcal{G}}$, we get \begin{equation} Hom({\mathcal{A}}\overset{L}{\underset{=}{\otimes}}R{\mathcal{S}}_2({\mathcal{G}}),{\mathcal{F}}_1)= Hom(RS_2({\mathcal{D}}\overset{L}{\underset{=}{\otimes}}{\mathcal{G}}),{\mathcal{F}}_1)\ . \end{equation} The theorem is proved. \end{pf} \noindent{\bf Remark}\ This result appears to conflict with the interchange of tensor and pontrjagin product stated in theorem Mukai2. Note, however, that in the definition of ${\mathcal{D}}\otimes_{{\mathcal{O}}} {\mathcal{G}}$, one is using the right ${\mathcal{O}}$-module structure on ${\mathcal{D}}$ to define the tensor product and the left ${\mathcal{O}}$-module structure on the resulting sheaf. \vskip 12pt We digress to give a \begin{pf*}{Short proof of Proposition \ref{fm of B}} By theorem \ref{tensor}, \begin{align} RS_2({\mathcal{D}}_{\{0\}\rightarrow Y}) &=RS_2({\mathcal{D}}\otimes k(0))\notag\\ &={\mathcal{A}}\otimes R{\mathcal{S}}_2(k(0))\notag\\ &={\mathcal{A}}\otimes {\mathcal{O}}_X={\mathcal{A}}\ . \end{align} \end{pf*} A more important corollary is the following. Let $D^b_{coh}\text{Mod}({\mathcal{A}})$ (resp. $D^b_{coh}\text{Mod}({\mathcal{D}}_Y)$) denote the subcategory of complexes with ${\mathcal{A}}$-coherent (resp. ${\mathcal{D}}$-coherent) cohomology. \begin{Thm} The functors $RS_1$ and $RS_2$ restrict to equivalences between the categories $D^b_{coh}\text{Mod}({\mathcal{A}})$ and $D^b_{coh}\text{Mod}({\mathcal{D}}_Y)$. \end{Thm} \begin{pf} We give the proof in one direction, the other direction being the same. The category $D^b_{coh}\text{Mod}({\mathcal{A}})$ of complexes with coherent cohomology is generated by sheaves of the form ${\mathcal{A}}\otimes{\mathcal{F}}$, where ${\mathcal{F}}$ is a coherent ${\mathcal{O}}_X$-module. It is well-known that $R\cal S_1({\mathcal{F}})$ the belongs to $D^b_{coh}\text{Mod}({\mathcal{O}}_Y)$. By theorem \ref{tensor}, \begin{equation} RS_1({\mathcal{A}}\otimes{\mathcal{F}})= {\mathcal{D}}\overset{L}{\underset{=}{\otimes}}R{\mathcal{S}}_1({\mathcal{F}})\ . \end{equation} This completes the proof, for ${\mathcal{D}}\overset{L}{\underset{=}{\otimes}}(\cdot)$\ maps $D^b_{coh}\text{Mod}({\mathcal{O}}_Y)$ to $D^b_{coh}\text{Mod}({\mathcal{D}}_Y)$. \end{pf} \subsection{} Given a complex ${\mathcal{F}} \in \text{Ob}\ D^b_{coh}\text{Mod}({\mathcal{D}}_Y)$, its dual complex is \begin{equation} \Delta^{{\mathcal{D}}_Y}({\mathcal{F}}) = R {\mathcal{H}}om_{{\mathcal{D}}_Y}({\mathcal{F}}, T^g({\mathcal{D}}_Y))\ . \end{equation} Note that $\Delta^{{\mathcal{D}}_Y}({\mathcal{F}})$ is naturally a complex of right ${\mathcal{D}}$-modules, but we regard it as a complex of left ${\mathcal{D}}$-modules, using the antiinvolution \begin{align} {\mathcal{D}} &\longrightarrow {\mathcal{D}}\notag\\ L = \sum f_I \xi^I &\mapsto \sum(-1)^{|I|} \xi^I f_I \ \ . \end{align} Given ${\mathcal{F}} \in \text{Ob}\ D^b\text{Mod}({\mathcal{A}})$, define $\Delta^{{\mathcal{A}}}{\mathcal{F}} = R {\mathcal{H}}om({\mathcal{F}}, T^g({\mathcal{A}}))$. In particular, \begin{align} \Delta^{\cal A}({\mathcal{A}}\overset{L}{\underset{=}{\otimes}}{\mathcal{F}}) &={\mathcal{A}}\overset{L}{\underset{=}{\otimes}}\Delta^{{\mathcal{O}}_X}({\mathcal{F}})\ ;\\ \Delta^{{\mathcal{D}}}({\mathcal{D}}\overset{L}{\underset{=}{\otimes}}{\mathcal{F}}) &={\mathcal{D}}\overset{L}{\underset{=}{\otimes}}\Delta^{{\mathcal{O}}_Y}({\mathcal{F}})\ . \end{align} \begin{Prop}\label{grothendieck} \begin{align} RS_2\Delta^{{\mathcal{D}}_Y} &= T^{-g}\Delta^{{\mathcal{A}}}RS_2\ ,\\ \Delta^{{\mathcal{D}}_Y} RS_1 &= RS_1\Delta^{{\mathcal{A}}}T^{-g}\ . \end{align} \end{Prop} \begin{pf} By theorem \ref{equiv2} it suffices to prove the first equality. It also suffices to consider the case where ${\mathcal{F}}$ is an object in $D^b\text{Mod}({\mathcal{D}}_Y)$ of the form ${\mathcal{F}} = {\mathcal{D}} \otimes_{{\mathcal{O}}}{\mathcal{G}}$, for some object ${\mathcal{G}} \in \text{Ob}\ D^b\text{Mod}({\mathcal{O}}_Y)$. Then $\Delta^{{\mathcal{D}}}({\mathcal{F}}) = {\mathcal{D}} \otimes \Delta^{{\mathcal{O}}_Y}({\mathcal{G}})$. By \cite[p. 161]{Muk} \begin{equation} R{\mathcal{S}}_2 \Delta^{{\mathcal{O}}_Y} = T^{-g}\Delta^{{\mathcal{O}}_X}R{\mathcal{S}}_2\ . \end{equation} Then \begin{align} &RS_2\Delta^{{\mathcal{D}}_Y}({\mathcal{F}}) = RS_2({\mathcal{D}} \otimes \Delta^{{\mathcal{O}}_Y}({\mathcal{G}}))\\ &= {\mathcal{A}} \otimes R {\mathcal{S}}_2\Delta^{{\mathcal{O}}_Y}({\mathcal{G}})\ \ \ \ \text{(by theorem \ref{tensor})}\notag\\ &= {\mathcal{A}} \otimes T^{-g} \Delta^{{\mathcal{O}}_X} R\cal S_2({\mathcal{G}})\notag\\ &= {\mathcal{A}} \otimes T^{-g}R {\mathcal{H}}om_{{\mathcal{O}}}(R{\mathcal{S}}_2({\mathcal{G}}), T^g({\mathcal{O}}_X))\notag\\ &= T^{-g}R {\mathcal{H}}om_{{\mathcal{A}}}({\mathcal{A}} \otimes R{\mathcal{S}}_2({\mathcal{G}}), T^g({\mathcal{A}}))\notag\\ &= T^{-g} R {\mathcal{H}}om_{{\mathcal{A}}}(RS_2({\mathcal{F}}), T^g({\mathcal{A}})) = T^{-g}\Delta^{{\mathcal{A}}} RS_2({\mathcal{F}})\ . \end{align} \end{pf} \subsection{} For the rest of this section we assume $char(k)=0$. Recall that a ${\mathcal{D}}$-module ${\mathcal{F}}$ is said to be holonomic if its characteristic variety has the least possible dimension, namely $g$. \begin{Prop}\label{Borel} Let ${\mathcal{F}}$ be a coherent ${\mathcal{D}}$-module. Then ${\mathcal{F}}$ is holonomic if and only if $H^i(\Delta^{{\mathcal{D}}}({\mathcal{F}})) = 0$ for $i \ne 0$. \end{Prop} \begin{pf} \cite[p. 230]{Bo}\end{pf} Since holonomicity of a ${\mathcal{D}}$-module is a local condition, one expects it to be encoded globally when one takes the Fourier transform. \begin{Thm} Let ${\mathcal{F}} \in \text{Ob}\ D^b\text{Mod}({\mathcal{A}})$ be a complex such that the cohomology of $RS_1({\mathcal{F}})$ is concentrated in a single degree $i$ and $R^iS_1({\mathcal{F}})$ is ${\mathcal{D}}$-coherent. Then $R^iS_1({\mathcal{F}})$ is holomonic if and only if the cohomology of $RS_1\Delta^{{\mathcal{A}}}({\mathcal{F}})$ is concentrated in degree $g-i$. \end{Thm} \begin{pf} Let $\hat{{\mathcal{F}}}=R^iS_1({\mathcal{F}})$. Regarding $\hat{{\mathcal{F}}}$ as a complex in degree $0$,\hfill\break $H^j(\Delta^{{\mathcal{D}}}(\hat{{\mathcal{F}}}))=H^j(\Delta^{{\mathcal{D}}}T^iRS_1({\mathcal{F}}))$. Then by proposition \ref{Borel}, $\hat{{\mathcal{F}}}$ is holonomic if and only if $H^j(\Delta^{{\mathcal{D}}}T^iRS_1({\mathcal{F}})) = 0$ for $j \ne 0$. By proposition \ref{grothendieck}, \begin{align}\label{holonomic} &H^j(\Delta^{{\mathcal{D}}}T^iRS_1({\mathcal{F}})) = H^j(\Delta^{{\mathcal{D}}}RS_1 T^i({\mathcal{F}}))\notag\\ &= H^j(RS_1\Delta^{{\mathcal{A}}}T^{i-g}({\mathcal{F}}))= H^j(RS_1T^{g-i}\Delta^{\cal A}({\mathcal{F}}))\ . \end{align} This vanishes for $j\ne 0$ if and only if $R^lS_1\Delta^{{\mathcal{A}}}({\mathcal{F}})$ vanishes for $l\ne g-i$. \end{pf} We leave the detailed study of this condition to a future work. \section{Characteristic Variety}\label{char var} In many important examples, it is possible to be quite explicit about the characteristic variety of the transform of a coherent ${\mathcal{A}}$-module. Let $\{{\mathcal{A}}(m)\}$ be the filtration in example \ref{B itself}. If ${\mathcal{F}}$ is a coherent ${\mathcal{A}}$-module, then one has the usual notion of good filtration with respect to $\{{\mathcal{A}}(m)\}$. As the Fourier-Mukai transform\ exchanges local data for global data, it is worth noting that good filtrations exist globally. \begin{Prop} Let $Z$ be a projective scheme of finite type over $k$, and let ${\mathcal{A}}$ be a sheaf of ${\mathcal{O}}_Z$-algebras. Regarding ${\mathcal{A}}$ as an ${\mathcal{O}}$-module by letting ${\mathcal{O}}$ act on the right, assume ${\mathcal{A}}$ is quasicoherent. If ${\mathcal{L}}$ is an ample line bundle on $Z$ and ${\mathcal{M}}$ is a sheaf of coherent left ${\mathcal{A}}$-modules, then there is presentation of ${\mathcal{M}}$ of the form \begin{equation}\label{presentation} ({\mathcal{A}}\otimes {\mathcal{L}}^{n_2})^{r_2}\longrightarrow ({\mathcal{A}}\otimes {\mathcal{L}}^{n_1})^{r_1}\longrightarrow{\mathcal{M}}\longrightarrow 0\ . \end{equation} \end{Prop} \begin{pf} The proof is the same as for the case ${\mathcal{A}}={\mathcal{O}}$.\ (cf. \cite[p. 122]{H} ) \end{pf} In particular, \begin{Cor} If ${\mathcal{F}}$ is a sheaf of coherent ${\mathcal{A}}$ (resp. ${\mathcal{D}}_Y$) modules, then ${\mathcal{M}}$ has a global good filtration by coherent ${\mathcal{O}}_X$ (resp. ${\mathcal{O}}_Y$) modules. \end{Cor} Let ${\mathcal{F}}$ be a coherent ${\mathcal{A}}$-module, and let $\{{\mathcal{F}}_m\}$ be a good filtration. It follows from \eqref{filtration} that \begin{equation} Gr{\mathcal{A}}=Sym({\mathfrak{g}})\otimes{\mathcal{O}}_X\ . \end{equation} Thus, for each $\xi\in{\mathfrak{g}}$ we have a homogeneous map of degree $1$ \begin{equation} Gr{\mathcal{F}}\overset{\dot\xi}\longrightarrow Gr{\mathcal{F}}\ . \end{equation} Identifying ${\mathcal{A}}$ with ${\mathcal{O}}[x(1),...,x(g)]$ on a sufficiently small open set, we get a commutative diagram \begin{equation}\label{comm} \begin{CD} {\mathcal{F}}_m @>{x(i)}>> {\mathcal{F}}_{m+1} \\ @VVV @VVV \\ Gr_m{\mathcal{F}} @>{\dot{\xi}_i}>> Gr_{m+1}{\mathcal{F}} \end{CD} \ \ . \end{equation} Set $R^jS_1({\mathcal{F}},\psi)=(R^j{\mathcal{S}}_1({\mathcal{F}}),\nabla^j)$, and set $\nabla^j=\sum \omega^i\nabla^j(i)$. It follows from the explicit formula for $S_1$ that for all $j$, the diagram \begin{equation}\label{comm transform} \begin{CD} R^j{\mathcal{S}}_1({\mathcal{F}}_m) @>{\nabla^j(i)} >> R^j{\mathcal{S}}_1({\mathcal{F}}_{m+1}) \\ @VVV @VVV \\ R^j{\mathcal{S}}_1(Gr_m{\mathcal{F}}) @> {R^j{\mathcal{S}}_1(\dot{\xi}_i)}>> R^j\cal S_1(Gr_{m+1}{\mathcal{F}}) \end{CD} \end{equation} commutes. Let ${\mathcal{K}}_m$ denote the kernel of the natural map \begin{equation} {\mathfrak{g}}\otimes Gr_m{\mathcal{F}}\longrightarrow Gr_{m+1}{\mathcal{F}}\ . \end{equation} Let us say that the filtered ${\mathcal{A}}$-module ${\mathcal{F}}$ satisfies {\it filtered W.I.T with index $i$} (cf. \cite[p.156]{Muk}) if $R^j{\mathcal{S}}_1({\mathcal{F}})=0$ for $j\ne i$ and the same is also true of ${\mathcal{F}}_m$ and ${\mathcal{K}}_m$ for $m$ sufficiently large. Following Mukai, we denote the surviving cohomology sheaf of $RS_1({\mathcal{F}})$ by $\hat{{\mathcal{F}}}$. Then we have \begin{equation} 0\longrightarrow \hat{{\mathcal{K}}}_m\longrightarrow {\mathfrak{g}}\otimes \widehat{Gr_m{\mathcal{F}}}\longrightarrow \widehat{Gr_{m+1}{\mathcal{F}}}\longrightarrow 0 \end{equation} for $m$ sufficiently large. By the preceding remarks, we therefore have \begin{Prop}\label{good filtration} Let ${\mathcal{F}}$ be a filtered ${\mathcal{A}}$-module satisfying filtered W.I.T.. Then 1. $\{\hat{{\mathcal{F}}_m}\}$ is a good ${\mathcal{D}}$-filtration on $\hat{{\mathcal{F}}}$. 2. $\widehat{Gr_m{\mathcal{F}}}=Gr_m\hat{{\mathcal{F}}}$ for $m$ sufficiently large. \end{Prop} Assume ${\mathcal{F}}$ satisfies filtered W.I.T. with index $i$, and let ${{\mathcal{I}}}(\hat{{\mathcal{F}}})\subset Sym({\mathfrak{g}})\otimes{\mathcal{O}}_Y$ denote the characteristic ideal sheaf of $\hat{{\mathcal{F}}}$. Fix an affine open subset ${\mathcal{U}}\subset Y$, let $A=\Gamma(\cal U,{\mathcal{O}}_Y)$ and let $I= \Gamma({\mathcal{U}},{\mathcal{I}}(\hat{{\mathcal{F}}}))$. This ideal may be described as follows. We have a map \begin{equation} \text{Sym}({\mathfrak{g}}) \overset{gr\psi}{\longrightarrow} H^0(X,{\mathcal{E}}nd (Gr\cal F)) \end{equation} coming from the $Gr{\mathcal{A}}$-module structure on $Gr{\mathcal{F}}$. Redefining the filtration if necessary, we may assume that ${\mathcal{K}}_m$ and ${\mathcal{F}}_m$ satisfy W.I.T. for all $m$. Since ${\mathcal{U}}$ is affine, proposition \ref{good filtration} gives \begin{align} \Gamma({\mathcal{U}},Gr\hat{{\mathcal{F}}})&=H^i(X\times\cal U,{\mathcal{P}}\otimes\alpha_1^*(Gr{\mathcal{F}}))\notag\\ &=H^i(X,\alpha_{1*}({\mathcal{P}}|_{X\times\cal U})\otimes(Gr{\mathcal{F}}))\ . \end{align} It is clear from the construction that the $\text{Sym}({\mathfrak{g}})$-module structure on $\Gamma({\mathcal{U}},Gr\hat{{\mathcal{F}}})$ is given by the composition \begin{align} \text{Sym}({\mathfrak{g}})\overset{gr\psi}{\longrightarrow} H^0(X,{\mathcal{E}}nd (Gr{\mathcal{F}}))\rightarrow H^0(X,&\cal End(Gr{\mathcal{F}} \otimes \alpha_{1*}({\mathcal{P}}|_{X\times{\mathcal{U}}})))\notag\\ &\rightarrow \text{End}(H^i(X, Gr{\mathcal{F}} \otimes \alpha_{1*}({\mathcal{P}}|_{X\times{\mathcal{U}}})))\ . \end{align} Putting this together with the $A$-module structure, we have a map \begin{equation} A\otimes \text{Sym}({\mathfrak{g}}) \overset{\lambda_{{\mathcal{U}}}}{\longrightarrow} \text{End}(\Gamma({\mathcal{U}},Gr\hat{{\mathcal{F}}}))\ . \end{equation} Thus we have \begin{Thm} \label{char} \begin{equation} \Gamma({\mathcal{U}},{\mathcal{I}}(\hat{{\mathcal{F}}}))=\sqrt{\text{ker}(\lambda_{{\mathcal{U}}})}\ . \end{equation} \end{Thm} We will make use of this result in section \ref{nak}. \section{The Krichever Construction}\label{krich} Let us briefly explain how the Krichever construction fits into the present framework. Assume now that $X=Y=Jac(C)$, where $C$ is a smooth curve of positive genus. Pick a base point $P\in C$ and let $C\overset a\longrightarrow X$ be the associated abel map. Taking ${\mathcal{F}}=a_*({\mathcal{O}}_C(*P))$, it is easy to see that ${\mathcal{F}}$ admits a ${\mathcal{A}}$-module structure. It suffices to take representative cocycles $\{c_{nm}(i)\}$ for a basis of $H^1(C,{\mathcal{O}})$ with respect to an open cover $\{{\mathcal{U}}_n\}$, and solve the equations \begin{equation} c_{nm}(i)=f_n(i)-f_m(i) \end{equation} with $f_n(i)\in\Gamma({\mathcal{U}}_n,{\mathcal{O}}(*P))$. (Note that we are identifying ${\mathcal{O}}(*P)$ with its endomorphism sheaf.) An important ingredient here is the {\it flag} on $H^1(C,{\mathcal{O}})$ coming from the natural maps \begin{equation} H^0(C,{\mathcal{O}}(nP)/{\mathcal{O}})\longrightarrow H^1(C,{\mathcal{O}})\ . \end{equation} Denoting the images of these maps by $V_n$, we have a sequence of subsheaves of ${\mathcal{D}}$, \begin{equation} {\mathcal{O}}_Y={\mathcal{D}}_0\subset{\mathcal{D}}_1\subset ...\ , \end{equation} where ${\mathcal{D}}_n$ is the ${\mathcal{O}}$-algebra generated by the vector fields belonging to $V_n$. In particular, \begin{equation} {\mathcal{D}}_1={\mathcal{O}}[\xi]\ , \end{equation} where $\xi$ is a basis of the one-dimensional space $V_1$. Now we may also filter ${\mathcal{A}}$ by subalgebras \begin{equation} {\mathcal{O}}_X={\mathcal{A}}_0\subset{\mathcal{A}}_1\subset ...\ \end{equation} in the same way, so that if we set \begin{equation} {\mathcal{A}}_i(m)={\mathcal{A}}_i\cap{\mathcal{A}}(m)\ , \end{equation} then \begin{equation} Gr_i{\mathcal{A}}=Sym(V_i)\otimes{\mathcal{O}}\ . \end{equation} Theorem \ref{equiv2} extends to this more general situation: $$\text{The Fourier-Mukai transform\ gives an equivalence of categories}$$ $$D^b\text{Mod}({\mathcal{A}}_i) \leftrightarrow D^b\text{Mod}({\mathcal{D}}_i)\ .$$ Now there is an essentially canonical ${\mathcal{A}}_1$-module structure on ${\mathcal{O}}(*P)$. Indeed, let $z$ be a local parameter at $P$. Then $a^*({\mathcal{A}}_1)$ is the subsheaf of ${\mathcal{O}}(*P)[x]$ characterized by the following growth condition: If ${\mathcal{U}}$ is a neighborhood of $P$, then \begin{equation} \Gamma({\mathcal{U}},a^*({\mathcal{A}}_1))=\{ \sum f_ix^i \in \Gamma({\mathcal{U}},{\mathcal{O}}(*P)[x])\ | \ \text{for all}\ j, \sum\limits_{i\ge j} \binom ij \frac{f_i}{z^{i-j}}\in \Gamma({\mathcal{U}},{\mathcal{O}})\ \}\ .\end{equation} Then ${\mathcal{O}}(*P)$ is in fact a sheaf of ${\mathcal{A}}_1|_C$-algebras under \begin{align} {\mathcal{A}}_1|_C&\longrightarrow {\mathcal{O}}(*P)\notag\\ f(x)&\mapsto f(x=0)\ . \end{align} The key to the Krichever construction is \begin{Prop} Let ${\mathcal{G}}=\widehat{{\mathcal{O}}(*P)}$ regarded as ${\mathcal{D}}_1$-module. Then ${\mathcal{G}}|_{X-\Theta}$ is {\it canonically} isomorphic to ${\mathcal{D}}_1$. \end{Prop} This proposition is simply a translation into the language of this paper of well-known results. The canonical generator is the Baker-Akhiezer function. The discussion given here is similar to that of \cite{R}. The important point is that \begin{equation} H^0(C,{\mathcal{O}}(*P))=End_{{\mathcal{A}}_1}({\mathcal{O}}(*P))=End_{{\mathcal{D}}_1}({\mathcal{G}})\ . \end{equation} Thus, if we let $\chi$ denote the canonical generator of ${\mathcal{G}}|_{X-\Theta}$, then for all $f\in H^0(C,{\mathcal{O}}(*P))$, there exists a unique $L_f\in H^0(X-\Theta,{\mathcal{D}}_1)$ such that \begin{equation} f\chi=L_f\chi\ . \end{equation} This is the Burchnall-Chaundy representation of $H^0(C,{\mathcal{O}}(*P))$, done for all line bundles at once. The famous result is that the $L_f$'s satisfy the KP-hierarchy. \section{Further Examples}\label{nak} Nakayashiki has studied generalizations of the Krichever construction in which the curve and point are replaced by an arbitrary variety together with an ample hypersurface. In this section we will illustrate the results of section \ref{char var} using a somewhat more general version of his examples. In particular we will obtain some refinements of his results about characteristic varieties. Let $Z$ be a smooth projective variety, and take $X$ to be its albanese variety. Assume for simplicity that the albanese map \begin{equation} Z\overset a\longrightarrow X \end{equation} is an imbedding. Let ${\mathcal{A}}_Z=a^*({\mathcal{A}})$. Then an ${\mathcal{A}}_Z$-module is the same as an ${\mathcal{A}}$-module supported on $Z$, so we have a functor $${\mathcal{A}}_Z\text{-modules} \longrightarrow {\mathcal{D}}_Y\text{-modules}\ .$$ Let $D\subset Z$ be an ample hypersurface. As in the Krichever construction, ${\mathcal{O}}(*D)$ may be given the structure of an algebra over ${\mathcal{A}}_Z$, making $\widehat{{\mathcal{O}}(*D)}$ a ${\mathcal{D}}$-module. (Note that ${\mathcal{O}}(*D)$ is W.I.T. of index 0.) Nakayashiki refers to such ${\mathcal{D}}$-modules as BA-modules. In general, when one has a splitting on a sheaf ${\mathcal{R}}$, there is an induced map \begin{equation} {\mathfrak{g}} \longrightarrow H^0(X,{\mathcal{E}}nd({\mathcal{R}})/{\mathcal{O}}) \end{equation} by virtue of \eqref{cobdry}. In the present example, this is a map \begin{equation} {\mathfrak{g}} \overset{\psi'}{\longrightarrow} H^0(Z, {\mathcal{O}}(*D)/{\mathcal{O}}) \end{equation} splitting the natural map \begin{equation} H^0(Z, {\mathcal{O}}(*D)/{\mathcal{O}}) \longrightarrow H^1(Z, {\mathcal{O}}) = {\mathfrak{g}}\ . \end{equation} Conversely, let ${\mathfrak{g}} \overset{T}{\rightarrow} H^0(Z, {\mathcal{O}}(rD)/{\mathcal{O}})$ be any left inverse of the natural map $H^0(Z, {\mathcal{O}}(rD)/{\mathcal{O}}) \rightarrow H^1(Z, {\mathcal{O}}) = {\mathfrak{g}}$. Then there is a splitting $\psi$ on ${\mathcal{O}}(*D)$ such that $\psi' = T$. We say that the splitting is {\it associated} to $T$. Equivalently, let $D_r$ denote the scheme $(D, {\mathcal{O}}/{\mathcal{O}}(-rD))$ and let ${\mathcal{M}} = \cal O(rD)/{\mathcal{O}})$, which we regard as a line bundle on $D_r$. Then we think of $\psi$ as being associated to the rational map \begin{equation} D_r\longrightarrow \Bbb P({\mathfrak{g}}^*) \end{equation} induced by the linear system $T({\mathfrak{g}})$. Now let ${\mathcal{O}}(*D)$ be endowed with a ${\mathcal{A}}_Z$-module structure associated to a fixed map $T$. From $T$ we get a map \begin{equation} {\mathfrak{g}} \otimes {\mathcal{O}}_{D_r} \overset{t}{\longrightarrow} {\mathcal{M}}\ , \end{equation} from which we may construct a Koszul complex (where ${\mathcal{O}} = {\mathcal{O}}_{D_r})$ \begin{equation} 0 \rightarrow \wedge^g {\mathfrak{g}} \otimes {\mathcal{M}}^{-g + 1} \cdots \overset{\wedge_3 t}{\longrightarrow} \wedge^2 {\mathfrak{g}} \otimes {\mathcal{M}}^{-1} \overset{\wedge_2 t}{\longrightarrow} {\mathfrak{g}} \otimes {\mathcal{O}} \overset{t}{\longrightarrow} {\mathcal{M}} \rightarrow 0. \end{equation} Now let ${\mathcal{H}}$ be a sheaf of coherent ${\mathcal{O}}_Z$-modules, and set \begin{equation} {\mathcal{F}} = {\mathcal{H}} \otimes {\mathcal{O}}(*D)\ , \end{equation} regarded as a ${\mathcal{A}}_Z$-module. We make the simplifying hypothesis that the injections \begin{equation} {\mathcal{O}}(krD) \longrightarrow {\mathcal{O}}((k+1)rD) \end{equation} induce injections \begin{equation} {\mathcal{F}}_k \longrightarrow {\mathcal{F}}_{k+1}\ , \end{equation} where ${\mathcal{F}}_k= {\mathcal{H}} \otimes {\mathcal{O}}(krD)$. This is equivalent to \vskip 14pt \noindent{\bf Assumption} \begin{equation} {\mathcal{T}}\kern -.2em or^1({\mathcal{H}},{\mathcal{O}}_D)=0\ . \end{equation} This gives us a filtration on ${\mathcal{F}}$, with associated graded sheaf \begin{equation} Gr{\mathcal{F}}_k={\mathcal{H}}|_{D_r}\otimes{\mathcal{M}}^k\ , \end{equation} upon which ${\mathfrak{g}}$ acts through the map $T$. We want this to be a good filtration. \begin{Lem}\label{base locus} The following are equivalent. \begin{enumerate} \item ${\mathfrak{g}} \otimes {\mathcal{H}} \rightarrow {\mathcal{M}} \otimes {\mathcal{H}} \rightarrow 0$ is exact. \item The baselocus of $T({\mathfrak{g}})$ does not meet the support of ${\mathcal{H}}$. \item The complex $(\wedge^i{\mathfrak{g}} \otimes {\mathcal{M}}^{1-i}) \otimes {\mathcal{H}}$ is exact \item If we set ${\mathcal{R}}_j = \text{ker}(\wedge_j t)$, then \begin{equation} 0 \rightarrow {\mathcal{R}}_j \otimes {\mathcal{H}} \rightarrow \wedge^j{\mathfrak{g}} \otimes {\mathcal{M}}^{1-j} \otimes {\mathcal{H}} \rightarrow {\mathcal{R}}_{j-1}\otimes {\mathcal{H}} \rightarrow 0\notag \end{equation} is exact for all $j$. \item The filtration $\{{\mathcal{F}}_k\}$ is a good ${\mathcal{A}}_Z$-filtration. \end{enumerate} \end{Lem} \begin{pf} The equivalence of 1 and 2 follows from Nakayama's lemma. If 2 holds, then the Koszul complex is exact at points in the support of ${\mathcal{H}}$, from which 3 follows. Since 1 is part of 3, the first three statements are equivalent. Then 4 follows because $\wedge^j {\mathfrak{g}} \otimes {\mathcal{M}}^{1-j} \rightarrow {\mathcal{R}}_{j-1} \rightarrow 0$ is exact at all points in $\text{supp}({\mathcal{H}})$, and the ${\mathcal{R}}_j$'s are all projective. But 1 is part of 4, so 1 through 4 are equivalent. Now the filtration ${\mathcal{F}}_k$ is good if and only if \begin{equation}\label{also action} {\mathfrak{g}} \otimes Gr_k{\mathcal{F}} \rightarrow Gr_{k+1}{\mathcal{F}}\longrightarrow 0 \end{equation} is exact for large $k$. But $Gr_k{\mathcal{F}} = {\mathcal{H}}|_{{\mathcal{D}}_r} \otimes {\mathcal{M}}^k$, so 1 and 5 are equivalent. \end{pf} Assume now that $\{ {\mathcal{F}}_k\}$ is a good $\cal A_Z$-filtration. \begin{Lem} ${\mathcal{F}}$ satisfies filtered W.I.T. with index $0$. \end{Lem} \begin{pf} It is clear that ${\mathcal{F}}_k$ satisfies W.I.T. with index $0$ if $k$ is sufficiently large. Let ${\mathcal{K}}_k$ denote the kernel of \eqref{also action}. Since ${\mathcal{M}}$ is invertible, \begin{equation} {\mathcal{K}}_k = {\mathcal{K}}_1 \otimes {\mathcal{M}}^k\ . \end{equation} Since ${\mathcal{M}}$ is ample on $D_r$, there exists $k$ such that \begin{equation} H^i(D_r, {\mathcal{K}}_j \otimes {\mathcal{L}}) = 0 \end{equation} for any line bundle ${\mathcal{L}}$ which is the pullback of a degree-0 line bundle on $X$, any $j \ge k$ and all $i > 0$. Thus for large $k$, ${\mathcal{K}}_k$ also satisfies W.I.T. with index $0$. \end{pf} This puts us in the position to apply theorem \ref{char}. Let ${\mathcal{U}}\subset Y$ be affine open and let $A=\Gamma(\cal U,{\mathcal{O}}_Y)$. Let $\epsilon_1$ and $\epsilon_2$ denote the projections on $D_r\times Y$. We must study the graded $A\otimes \text{Sym}({\mathfrak{g}})$-module \begin{equation} H\overset{\text{def}}= \bigoplus_{k=0}^{\infty}H^0(D_r,{\mathcal{H}}|_{D_r}\otimes \epsilon_{1*}({\mathcal{P}}|_{D_r\times{\mathcal{U}}})\otimes{\mathcal{M}}^k)\ . \end{equation} Let \begin{equation} S=H^0(D_r,\bigoplus_{k=0}^{\infty} {\mathcal{M}}^k)\ . \end{equation} Then $H$ is a graded $S$-module. Moreover, we have the map \begin{equation}\label{linear map} {\mathfrak{g}} \overset{T}{\longrightarrow} H^0(Z, {\mathcal{O}}(rD)/{\mathcal{O}})= H^0(D_r, {\mathcal{M}})\ , \end{equation} which induces \begin{equation}\label{induced} Sym({\mathfrak{g}}) \longrightarrow Sym( H^0(D_r, {\mathcal{M}}))\ . \end{equation} Then $Sym({\mathfrak{g}})$ acts on $H$ through the composition of \eqref{induced} with the natural homomorphism \begin{equation} Sym( H^0(D_r, {\mathcal{M}}))\longrightarrow S\ . \end{equation} If we apply Proj to the composite map $Sym({\mathfrak{g}})\longrightarrow S$, we recover the rational map \begin{equation} D_r\overset{\Psi}{--\rightarrow}\Bbb P({\mathfrak{g}}^*)\ . \end{equation} associated to the linear map $T$, \eqref{linear map}. Moreover, applying Proj to the graded $A\otimes S$-module $H$, we get the sheaf \begin{equation} \tilde{H}=(\alpha_1^*({\mathcal{H}})\otimes{\mathcal{P}})|_{D_r\times{\mathcal{U}}} \end{equation} on $D_r\times {\mathcal{U}}$. Thus, the consideration of $H$ as an $A\otimes Sym({\mathfrak{g}})$-module may be viewed on the sheaf level as taking the direct image of $(\alpha_1^*({\mathcal{H}})\otimes{\mathcal{P}})|_{D_r\times{\mathcal{U}}}$ under the map $\Psi\times 1$. (Recall that by lemma \ref{base locus}, $\Psi$ is defined on the support of ${\mathcal{H}}|_{D_r}$.) The discussion above gives us the main result of this section. \begin{Thm}\label{charac var} Let ${\mathcal{G}}=\hat{{\mathcal{F}}}$, where ${\mathcal{F}}=\cal H\otimes{\mathcal{O}}_Z(*D)$, ${\mathcal{T}}or^1({\mathcal{H}},{\mathcal{O}}_D)=0$, and the ${\mathcal{A}}_Z$-module structure on ${\mathcal{O}}(*D)$ is associated to $\Psi:D_r--\to \Bbb P({\mathfrak{g}}^*)$. Then the characteristic variety of ${\mathcal{G}}$, viewed as a subvariety of $\Bbb P({\mathfrak{g}}^*)\times Y$, is the support of the sheaf \begin{equation} Gr{\mathcal{G}}=(\Psi\times 1)_* ((\alpha_1^*(\cal H)\otimes{\mathcal{P}})|_{D_r\times Y})\ . \end{equation} \end{Thm} In \cite{N2} the case $Z=X$ is considered, and it is proved that the codimension of the characteristic variety is $dim(X)-dim Supp ({\mathcal{H}})$. The more general theorem \ref{charac var} yields quite detailed information in this case, and in particular has Nakayashiki's result as a corollary. We are now dealing with a smooth, ample hypersurface $D\subset X$. Then there is a natural class of ${\mathcal{A}}$-module structures on ${\mathcal{O}}(*D)$. These are described in \cite{N2} in terms of factors of automorphy, but may also be seen somewhat more geometrically. Let ${\mathfrak{h}}$ be the space of vector fields on $X$. We have the {\it Gauss map} \begin{equation} D\overset\Psi\longrightarrow \Bbb P({\mathfrak{h}}^*)\ . \end{equation} The normal bundle to $D$ is $\Psi^*({\mathcal{O}}(1))$, and thus we have a linear map \begin{equation} {\mathfrak{h}} \overset{\lambda}{\longrightarrow} H^0(D, {\mathcal{N}})\ . \end{equation} The composition of $\lambda$ with the canonical map $H^0(D, {\mathcal{N}})\to{\mathfrak{g}}$ is an isomorphism. Thus one may consider ${\mathcal{A}}$-module structures on ${\mathcal{O}}(*D)$ associated to the Gauss map. That is, we can choose the ${\mathcal{A}}$-module structure in such a way that the induced map \begin{equation} {\mathfrak{g}} \overset{\psi'}{\longrightarrow} H^0(X, {\mathcal{O}}(*D)/{\mathcal{O}}) \end{equation} is precisely the composition \begin{equation} {\mathfrak{g}}\simeq{\mathfrak{h}}\overset{\lambda}{\longrightarrow} H^0(D,{\mathcal{N}})\subset H^0(X, {\mathcal{O}}(*D)/{\mathcal{O}})\ . \end{equation} We will call such an ${\mathcal{A}}$-module structure {\it canonical}. In the language of this paper, Nakayashiki's ${\mathcal{D}}$-modules are obtained as the Fourier-Mukai transform\ of ${\mathcal{A}}$-modules of the form ${\mathcal{H}}\otimes{\mathcal{O}}(*D)$, where ${\mathcal{O}}(*D)$ has a canonical ${\mathcal{A}}$-module structure. However, one may as well consider a more general class of ${\mathcal{A}}$-module structures, namely those associated to any $g$-dimensional linear system $V\subset H^0(D,{\mathcal{N}})$ which is basepoint-free and maps isomorphically onto ${\mathfrak{g}}$. The example of the Gauss map shows that this is the generic situation. As a corollary of theorem \ref{charac var}, we have \begin{Thm} Let $D\subset X$ be a smooth ample hypersurface. Let $V\subset H^0(D,{\mathcal{N}})$ be a $g$-dimensional basepoint-free linear system mapping isomorphically onto ${\mathfrak{g}}$, and let $\Psi:D\to \Bbb P({\mathfrak{g}}^*)$ be the corresponding morphism. Let ${\mathcal{H}}$ be a coherent ${\mathcal{O}}_X$-module such that ${\mathcal{T}}\kern -.2em or^1({\mathcal{H}},{\mathcal{O}}_D)=0$\ , and set ${\mathcal{G}}=\widehat{\cal H\otimes{\mathcal{O}}(*D)}$. Give ${\mathcal{G}}$ the ${\mathcal{D}}$-module structure induced by a ${\mathcal{A}}$-module structure on ${\mathcal{O}}(*D)$ associated to $V$. Then the characterisic variety of ${\mathcal{G}}$ is \begin{equation} ss({\mathcal{G}})=\Psi(Supp({\mathcal{H}}|_D))\times Y \ .\end{equation} (In particular, $ codim(ss({\mathcal{G}})=dim(X)-dim Supp ({\mathcal{H}})$\ .) \end{Thm} \begin{pf} Because ${\mathcal{N}}$ is ample, the morphism $\Psi$ is finite. Therefore, tensoring ${\mathcal{H}}|_D$ with a line bundle has no effect on the support of its direct image. The result then follows from theorem \ref{charac var}.\end{pf} \section{Some remarks on commuting rings of matrix partial differential operators}\label{PDOs} In some sense, the origins of the present subject date to work of Burchnall and Chaundy on commuting rings of ordinary differential operators \cite{BC}. Such rings are always of dimension one. As Nakayashiki has observed in \cite{N2}, the Fourier-Mukai transform\ allows one to represent the ring $H^0(X,{\mathcal{O}}(*D))$, $X$ an abelian variety and $D$ a smooth ample hypersurface, by matrix valued partial differential operators in $g$ variables, the size of the matrix being the $g$-fold self-intersection number, $D^g$. We want to offer some further observations on this question. Consider again the data $(Z,D,{\mathcal{H}})$ in section \ref{nak}. Fix an integer $r$ and a subspace $V\subset H^0(Z,{\mathcal{O}}(rD)/{\mathcal{O}})$ mapping isomorphically onto its image under the natural map \begin{equation} H^0(Z,{\mathcal{O}}(rD)/{\mathcal{O}})\longrightarrow {\mathfrak{g}}\ . \end{equation} As in section \ref{krich}, we get a subsheaf ${\mathcal{A}}_1\subset {\mathcal{A}}$ by imitating the construction of ${\mathcal{A}}$, replacing ${\mathfrak{g}}$ by $V$ throughout. Similarly, we have a subsheaf ${\mathcal{D}}_1\subset {\mathcal{D}}_Y$ generated over ${\mathcal{O}}_Y$ by the vector fields belonging to the image of $V$. As in section \ref{nak}, we have a rational map \begin{equation} D_r\overset{\Psi}{--\rightarrow}\Bbb P(V^*)\ . \end{equation} Assuming now that $supp({\mathcal{H}})$ does not meet the baselocus of $\Psi$, we can introduce a coherent ${\mathcal{A}}_1$-module structure on ${\mathcal{F}}={\mathcal{H}} \otimes {\mathcal{O}}(*D)$, filtered by the submodules ${\mathcal{H}} \otimes {\mathcal{O}}(krD)$ exactly as before. We now have \begin{align} Gr_1{\mathcal{A}} &=Sym(V)\otimes{\mathcal{O}}_Z\\ D_r&=Spec({\mathcal{O}}/{\mathcal{O}}(-rD))\\ {\mathcal{M}}&={\mathcal{O}}(rD)/{\mathcal{O}}\ \text{(thought of as a line bundle on $D_r$)}\\ Gr{\mathcal{F}}&= \bigoplus_{l=0}^{\infty}{\mathcal{H}}|_{D_r}\otimes{\mathcal{M}}^l\ , \end{align} where the $Sym(V)$-module structure on $Gr{\mathcal{F}}$ is defined by the inclusion \begin{equation}\label{inclusion} V\longrightarrow H^0(D_r,{\mathcal{M}})\ . \end{equation} Set $ {\mathcal{G}}=\hat{{\mathcal{F}}}$, regarded as a sheaf of ${\mathcal{D}}_1$-modules. We will examine conditions under which ${\mathcal{G}}$ is free in a neighborhood of a given line bundle ${\mathcal{L}}\in Pic^0(Z)$. In order to have the equality \begin{equation} \widehat{Gr{\mathcal{F}}}=Gr\hat{{\mathcal{F}}}\ \end{equation} in a neighborhood of ${\mathcal{L}}$, we make the assumptions \begin{align} &\text{For all}\ k\ge 0\ , i>0\ ,\ H^i(Z,{\mathcal{L}}\otimes{\mathcal{F}}_k)=0\ .\\ &\text{There exists}\ j\ge 2\ \text{such that for all}\ i\ne j\ , \ H^i(Z,{\mathcal{L}}\otimes{\mathcal{F}}_{-1})=0\ . \end{align} For ${\mathcal{G}}$ to be free over ${\mathcal{D}}_1$ in a neighborhood of $\cal L$, it is sufficient that the fiber of $Gr{\mathcal{G}}$ at ${\mathcal{L}}$ be free over $Sym(V)$. We therefore have \begin{Prop}\label{generically free} For ${\mathcal{G}}$ to be free over ${\mathcal{D}}_1$ in a neighborhood of ${\mathcal{L}}$, it is sufficient that $$ \bigoplus_{l=0}^{\infty}H^0(D_r,({\mathcal{L}}\otimes{\mathcal{H}})|_{D_r}\otimes{\mathcal{M}}^l) $$ be freely generated as a module over $Sym(V)$. \end{Prop} Regarding the hypothesis of this proposition, we have the following purely algebraic lemma. Let $M$ be an arbitrary finitely generated graded $Sym(V)$-module, with $M_j=0$ for $j<0$. For all $j$ we have a complex \begin{equation}\label{complex} \wedge^2(V)\otimes M_{j-1}\overset{\beta_j}\longrightarrow V\otimes M_j\overset{\alpha_j}\longrightarrow M_{j+1}\ , \end{equation} where $\alpha_j$ is the action of $V$, and $\beta_j$ is defined by \begin{equation} v\wedge w\otimes m\mapsto v\otimes wm-w\otimes vm\ . \end{equation} \begin{Lem}\label{free} $M$ is free over $\text{Sym}(V)$ if and only if the sequence \eqref{complex} is exact for all $j$. \end{Lem} \begin{pf} If $M = \overset{\ell}{\underset{i=1}{\oplus}} \text{Sym}(V)[n_j]$, where \begin{equation} \text{Sym}(V)[n_i]_{j} = \text{Sym}^{j + n_i}(V)\ , \end{equation} then the exactness of the complexes \eqref{complex} follows from the well-known fact that the natural complex \begin{equation} \label{always exact} \wedge^2(V) \otimes \text{Sym}^k(V) \rightarrow V \otimes \text{Sym}^{k+1}(V) \rightarrow \text{Sym}^{k+2}(V) \end{equation} is always exact. Conversely, suppose that \eqref{complex} is always exact. For all $j$, choose a subspace $U_j \subset M_j$ complementary to the image of $\alpha_{j-1}$. What we must show is that for all $j$, the natural map \begin{equation} \overset{j}{\underset{i=1}{\oplus}} \text{Sym}^i(V) \otimes U_{j-i} \overset{\gamma_j}{\longrightarrow} M_j \end{equation} is injective, for then $M$ is isomorphic to \begin{equation} \oplus U_j \otimes_k \text{Sym}(V)[-j]\ . \end{equation} Given $\ell$, assume $\gamma_j$ is injective for $j < \ell$. Then $M_{\ell-1} \approx \overset{\ell - 1}{\underset{i=0}{\oplus}} \text{Sym}^i(V) \otimes U_{\ell - 1 - i}$ and we have a commutative diagram \begin{equation} \begin{matrix} V \otimes \left(\operatornamewithlimits{\oplus}\limits^{\ell - 1}_{i=0} \text{Sym}^i (V) \otimes U_{\ell-1-i}\right)\\ \qquad \qquad \qquad \downarrow \delta \qquad\qquad \searrow \alpha_{\ell - 1}\\ \operatornamewithlimits{\oplus}\limits^{\ell}_{i=1} \text{Sym}^i (V) \otimes U_{\ell - i} \overset{\gamma_{\ell}}{\longrightarrow}M_{\ell} \end{matrix} \end{equation} Since $M_{\ell - 2} \approx \overset{\ell - 2}{\underset{i=0}{\oplus}} \text{Sym}^i(V) \otimes U_{\ell - 2 - i}$, the exactness of \eqref{always exact} implies that $\text{Ker}(\delta) = \text{Im}(\beta_{\ell - 1})$. Thus $\text{ker}(\delta) = \text{Ker}(\alpha_{\ell - 1})$, which says that $\gamma_{\ell}$ is injective. \end{pf} The geometric interpretation of this lemma is the following. Let $S$ be a scheme over $k$, $V$ a finite dimensional vector space over $k$, ${\mathcal{M}}$ a line bundle on $S$, and $V \overset{T}{\longrightarrow} H^0(S, {\mathcal{M}})$ a linear map. As in section \ref{nak}, we associate to $T$ a Koszul complex \begin{equation} \dots \rightarrow \wedge^3 V \otimes {\mathcal{M}}^{-2} \overset{\wedge_3 t}{\longrightarrow} \wedge^2 V \otimes {\mathcal{M}}^{-1} \overset{\wedge_2 t}{\longrightarrow} V \otimes {\mathcal{O}} \overset{t}{\longrightarrow} {\mathcal{M}} \rightarrow 0\ . \end{equation} Let ${\mathcal{R}}_i = \text{ker}(\wedge_i t)$. Let ${\mathcal{H}}$ be a sheaf of ${\mathcal{O}}_S$-modules. Then we have complexes \begin{equation} \label{exact1} 0 \rightarrow {\mathcal{R}}_1 \otimes {\mathcal{H}} \rightarrow V \otimes {\mathcal{H}} \rightarrow {\mathcal{M}} \otimes {\mathcal{H}} \rightarrow 0 \end{equation} \begin{equation} \label{exact2} 0 \rightarrow {\mathcal{R}}_2 \otimes {\mathcal{H}} \rightarrow \wedge^2 V \otimes {\mathcal{M}}^{-1} \otimes {\mathcal{H}} \rightarrow {\mathcal{R}}_1 \otimes {\mathcal{H}} \rightarrow 0. \end{equation} As in lemma \ref{base locus}, these sequences are exact if and only if \begin{equation} \label{exact0} V \otimes {\mathcal{H}} \rightarrow {\mathcal{M}} \otimes {\mathcal{H}} \rightarrow 0 \end{equation} is exact. We therefore have \begin{Thm} Assume \eqref{exact0} is exact. Consider the graded $\text{Sym}(V)$-module \begin{equation} M = \underset{j\ge 0}{\oplus} H^0(S,{\mathcal{H}} \otimes \cal M^j)\ . \end{equation} Then $M$ is free over $\text{Sym}(V)$ if and only if \begin{align} 0 \rightarrow H^1(S, {\mathcal{R}}_2 \otimes {\mathcal{H}} \otimes \cal M^j)& \rightarrow \wedge^2 V \otimes H^1(S, {\mathcal{H}} \otimes {\mathcal{M}}^{j-1}) \ \text{is exact for all}\ j\ge 1\ ,\text{and}\\ H^0(S,{\mathcal{R}}_1\otimes\cal H)=0\ .\end{align} \end{Thm} \begin{pf} The maps $\alpha_j$ and $\beta_j$ in \eqref{complex} are given in this case by tensoring \eqref{exact1} and \eqref{exact2} with ${\mathcal{M}}^i$ and taking cohomology: \begin{equation} \begin{matrix} 0 \rightarrow H^0(S, {\mathcal{R}}_1 \otimes {\mathcal{H}}\otimes {\mathcal{M}}^j) \rightarrow V \otimes M_j \overset{\alpha_j}{\longrightarrow} M_{j+1}\\ \qquad \quad \uparrow \qquad \nearrow \beta_j\\ \wedge^2(V) \otimes M_{j-1} \end{matrix} \end{equation} Therefore, \eqref{complex} is exact for $j\ge 1$ precisely when \begin{equation} H^1(S, {\mathcal{R}}_2 \otimes {\mathcal{H}} \otimes {\mathcal{M}}^j) \rightarrow \wedge^2V \otimes H^1(S, {\mathcal{H}} \otimes {\mathcal{M}}^{j-1}) \end{equation} is injective. The second condition appears because we have defined $M$ as a sum over nonnegative $j$, but have not assumed that $H^0(S,{\mathcal{H}} \otimes {\mathcal{M}}^{-1})=0$. The theorem then follows from lemma \ref{free}. \end{pf} Let us put all these ingredients together. We have a map \begin{equation} V\overset{T}{\longrightarrow} H^0(D_r, {\mathcal{M}})\ , \ \ \cal M = {\mathcal{O}}(rD)/{\mathcal{O}}\ , \end{equation} giving us subsheaves ${\mathcal{A}}_1\subset {\mathcal{A}}$, ${\mathcal{D}}_1\subset {\mathcal{D}}$, and a ${\mathcal{A}}_1$-module structure on ${\mathcal{O}}(*D)$. We have a sheaf ${\mathcal{H}}$ such that ${\mathcal{T}}or^1({\mathcal{H}}, {\mathcal{O}}_D) = 0$ and $\text{supp}({\mathcal{H}})$ does not meet the baselocus of $\text{Im}(T)$. We have ${\mathcal{G}}=\widehat{{\mathcal{H}}\otimes{\mathcal{O}}(*D)}$ and ${\mathcal{F}}_k={\mathcal{H}}\otimes{\mathcal{O}}(krD)$. The Koszul complex associated to $T$ is therefore exact on the support of ${\mathcal{H}}|_{D_r}$, and we can apply the previous theorem in combination with theorem \ref{generically free}. \begin{Thm}\label{cohomological} For ${\mathcal{G}}$ to be free over ${\mathcal{D}}_1$ in a neighborhood of ${\mathcal{L}} \in \text{Pic}^0(Z)$, the following conditions are sufficient: \begin{enumerate} \item For all $k\ge 0$,\ $i>0$,\ $H^i(Z,{\mathcal{L}}\otimes{\mathcal{F}}_k)=0$. \item There exists $j\ge 2$ such that for all $i\ne j$\ , $H^i(Z,{\mathcal{L}}\otimes{\mathcal{F}}_{-1})=0$. \item $0 \rightarrow H^1(D_r, {\mathcal{R}}_2 \otimes{\mathcal{L}}\otimes {\mathcal{H}} \otimes {\mathcal{M}}^j) \rightarrow \wedge^2 V \otimes H^1(D_r, {\mathcal{L}}\otimes{\mathcal{H}} \otimes \cal M^{j-1})$\ is exact for all $j\ge 1$ \item $H^0(D_r,{\mathcal{R}}_1\otimes\cal L\otimes {\mathcal{H}})=0$.\end{enumerate} \end{Thm} Nakayashiki's embedding may now be recovered. As in section \ref{nak}, we take $Z$ to be $X$ itself, and we assume $D$ is smooth. We take an $\cal A$-module structure on ${\mathcal{O}}(*D)$ associated to a $g$-dimensional basepoint-free linear system $V\subset H^0(D,{\mathcal{N}})$ mapping isomorphically onto ${\mathfrak{g}}$. We take ${\mathcal{H}}={\mathcal{O}}(D)$. This affects only the filtration, not the ${\mathcal{A}}$-module structure. Thus \begin{equation} {\mathcal{F}}_k={\mathcal{O}}((k+1)D)\ . \end{equation} \begin{Thm} Set ${\mathcal{G}}=\widehat{{\mathcal{O}}(*D)}$. Then ${\mathcal{G}}$ is a free ${\mathcal{D}}$-module in a neighborhood of any ${\mathcal{L}} \ne {\mathcal{O}}$. \end{Thm} \begin{pf} By theorem \ref{cohomological}, it suffices to verify the following: \begin{enumerate} \item For all $i>0$, $k>0$, $H^i(X,{\mathcal{L}}(kD))=0$. \item There exists $j\ge 2$ such that for all $i\ne j$, $H^i(X,{\mathcal{L}})=0$. \item For all $j\ge 1$,\ $0\to H^1(D,{\mathcal{R}}_2\otimes {\mathcal{L}}\otimes {\mathcal{N}}^{j+1})\to \wedge^2(V)\otimes H^1(D,{\mathcal{L}}\otimes {\mathcal{N}}^j)$\ is exact, where $\cal R_j$ are the kernel sheaves of the Koszul complex $\wedge^j(V)\otimes\cal N^{1-j}$. \item $H^0(D, {\mathcal{R}}_1\otimes{\mathcal{L}}\otimes{\mathcal{N}})=0$. \end{enumerate} Items 1 and 2 are well-known. See, for example, \cite[sec. 13, sec. 16]{Mum}. It then follows from the exact sequence \begin{equation} 0 \rightarrow {\mathcal{L}}((k-1)D) \rightarrow {\mathcal{L}}(kD) \rightarrow {\mathcal{L}} \otimes {\mathcal{N}}^k \rightarrow 0 \end{equation} that for all $k$ and all $0<i<g-1$,\ $H^i(D, {\mathcal{L}} \otimes {\mathcal{N}}^k) = 0$. Since ${\mathcal{R}}_{g-1}={\mathcal{N}}^{1-g}$, it follows by descending induction that \begin{equation}\label{descent} H^i(D, {\mathcal{R}}_j \otimes {\mathcal{L}} \otimes \cal N^k) = 0 \text{ for $0<i<j$ and all $k$.} \end{equation} In particular, 3 holds. Taking $k=1$ we get a stronger statement, also by descending induction: \begin{equation} H^i(D, {\mathcal{R}}_j \otimes {\mathcal{L}} \otimes \cal N) = 0 \text{ for $i<j$.} \end{equation} Thus 4 holds. \end{pf} {}From the standpoint of integrable systems, the relevant feature of a $\cal D$-module is its endomorphism ring. As we saw in section \ref{krich}, if $C$ is a curve embedded in its Jacobian, $\widehat{{\mathcal{O}}(*P)}$ is a ${\mathcal{D}}_1$-module, where ${\mathcal{D}}_1 = {\mathcal{O}}[\xi]$. Extending this ${\mathcal{D}}_1$-module structure to a ${\mathcal{D}}$-module structure, we have \begin{equation} H^0(C,{\mathcal{O}}(*P)) = \text{End}_{{\mathcal{A}}}({\mathcal{O}}_C(*P)) = \text{End}_{{\mathcal{D}}}(\widehat{{\mathcal{O}}_C(*P)})\ . \end{equation} It is the analysis of this endomorphism ring which leads to the $KP$-hierarchy. Indeed, having trivialized $\widehat{{\mathcal{O}}_C(*P)}$ as a ${\mathcal{D}}_1$-module, its ${\mathcal{D}}$-endomorphisms are then differential operators in one variable with $g-1$ parameters, satisfying certain nonlinear differential equations. More generally, one may hope to associate dynamics to the endomorphism ring of a ${\mathcal{D}}$-module coherent over a proper subalgebra, ${\mathcal{O}}[\xi_1, \dots, \xi_n] \subset {\mathcal{D}}$, $n < g$. Those modules which are free over the smaller algebra provide a natural starting point for such an investigation. Note that the presence of any nontrivial endomorphisms in such a setting is already a strong condition on the ${\mathcal{D}}$-module, but one which can easily be satisfied by the methods presented here. Such examples will be the object of study in the sequel. Finally, by way of advertisement, we mention \vskip 12pt \noindent{\bf Example: The Fano Surface.} \vskip 12pt If $Z \subset \Bbb C \Bbb P^4$ is a smooth cubic hypersurface, then its family of lines\hfill\break \hbox{$S = \{\ell \in \text{Gr}(2, 5)\ |\ \ell \subset Z\}$} is a smooth surface \cite{CG}. For generic $s \in S$ corresponding to a line $\ell_s$, the set \hbox{$\{t \in S\ |\ \ell_t \cap \ell_s = \{pt\}\}$} is a smooth ample hypersurface $D_s$, the incidence divisor. We have isomorphisms \begin{equation} \text{Alb}(S) \approx \text{Pic}^0(S) \approx J(Z)\ , \end{equation} where $J(Z)$ is the intermediate Jacobian of $Z$. The albanese map is identified with the assignment $s \rightsquigarrow D_s$, which gives an embedding $S \subset \text{Pic}^0(S)$. The dimension of $\text{Pic}^0(S)$ is 5. Setting ${\mathcal{N}} =$ normal bundle of $D_s$ in $S$ and $T_s(S) =$ tangent space to $S$ at $s$, we have an isomorphism $T_s(S) \approx H^0(D_s, {\mathcal{N}})$. The image of the natural map $H^0(D_s, {\mathcal{N}}) \rightarrow H^1(S, {\mathcal{O}})$, gives a subspace \begin{equation} \Bbb C^2 \approx V \subset H^1(S, {\mathcal{O}}) \approx \Bbb C^5. \end{equation} Set ${\mathcal{D}}_1={\mathcal{O}}[\xi_1,\xi_2]\subset{\mathcal{D}}$, where $\xi_1$ and $\xi_2$ are a basis for $V$. Then the conditions of theorem \ref{cohomological} are fulfilled with ${\mathcal{H}} = {\mathcal{O}}(2D)$. Thus $\widehat{{\mathcal{O}}(*D)}$ is a ${\mathcal{D}}$-module, locally free as a $\cal D_1$-module at a generic point. The rank of $\widehat{{\mathcal{O}}(*D)}$ as a ${\mathcal{D}}_1$-module is the degree of the map $D_s \rightarrow \Bbb P^1$ corresponding to the linear system $H^0(D_s, {\mathcal{N}})$. This degree is also 5. Thus we have a representation of $H^0(S, {\mathcal{O}}(*D))$ as $5 \times 5$ matrix partial differential operators in two variables, with $3 (= \dim(H^1(S, {\mathcal{O}})) - \dim(V))$ parameters.

9602

alg-geom/9602013

en

https://arxiv.org/abs/alg-geom/9602013

alg-geom/9602013

V. Batyrev

Victor V. Batyrev and Yuri Tschinkel

Rational points on some Fano cubic bundles

8 pages., LaTeX

We consider some families of smooth Fano hypersurfaces $X_{n+2}$ in ${\bf P}^{n+2} \times {\bf P}^3$ given by a homogeneous polynomial of bidegree $(1,3)$. For these varieties we obtain lower bounds for the number of $F$-rational points of bounded anticanonical height in arbitrary nonempty Zariski open subset $U \subset X_{n+2}$. These bounds contradict previous expectations about the distribution of $F$-rational points of bounded height on Fano varieties.

alg-geom

\section{Cubic bundles} Let $X_{n+2}$ be a hypersurface in ${\bf P }^n \times {\bf P }^3$ $(n \geq 1)$ defined by the equation $$ P({\bf x},{\bf y}) = \sum_{i =0}^3 l_i({\bf x})y_i^3 = 0$$ where $$P({\bf x},{\bf y}) \in {\bf Q } [ x_0, \ldots, x_n, y_0,\ldots, y_3 ]$$ and $l_0({\bf x}), \ldots, l_3({\bf x})$ are homogeneous linear forms in $x_0, \ldots, x_n$. Put $k = \max ( n+1, 4)$. We shall always assume that any $k$ forms among $ l_0({\bf x}), \ldots, l_3({\bf x})$ are linearly independent. It is elementary to check the following statements: \begin{prop} The hypersurface $X_{n+2}$ is a smooth Fano variety containing a Zariski open subset $U_{n+2}$ which is isomorphic to ${\bf A}^{n+2}$. \end{prop} \begin{prop} Let $U_P \subset {\bf P }^n$ be the Zariski open subset defined by the condition \[ \prod_{i =0}^3 l_0({\bf x}) \neq 0. \] Then the fibers of the natural projection $\pi \,:\, X_{n+2} \rightarrow {\bf P }^n$ over closed points of $U_P$ are smooth diagonal cubic surfaces in ${\bf P }^3$. \end{prop} \noindent By Lefschetz theorem, we have: \begin{prop} The Picard group of $X_{n+2}$ over an arbitrary field containing ${\bf Q }$ is isomorphic to ${{\bf Z }}\oplus {\bf Z }$. \label{pic} \end{prop} \section{Heights on cubic surfaces} Let $F$ be a number field, ${\rm Val}(F)$ the set of all valuations of $F$, $W$ a projective algebraic variety over $F$, $W(F)$ the set of $F$-rational points of $W$, $D$ a very ample divisor on $W$, and ${ \gamma} = \{ s_0, \ldots, s_m\}$ a basis over $F$ of the space of global sections $ \Gamma (W, {\cal O}(D))$. The {\em height function associated with $D$ and ${ \gamma}$} $$H(W, D,{ \gamma},x)\; : \; W(F) \rightarrow {\bf R }_{>0} $$ is given by the formula \[ H(W, D,{ \gamma},x) = \prod_{v \in {\rm Val}(F)} \max_{i=0, \ldots, m} | s_i(x) |_v, \] where $|\cdot |_v \,:\, F_v \rightarrow {{\bf R }}_{>0}$ is the multiplier of a Haar measure on the additive group of the $v$-adic completion of $F$. \begin{dfn} {\rm Let $Z \subset W$ be a locally closed algebraic subset of $W$, $B$ a positive real number. Define \[ N(Z, D,{ \gamma},B) := {\rm Card}\{ x \in W(F) \cap Z \mid H(W, D,{ \gamma},x) \leq B \}.\]} \end{dfn} \noindent The following classical statement is due to A. Weil: \begin{theo} Let ${ \gamma'} = \{ s_0', \ldots, s_m' \}$ be another basis in $ \Gamma (W, {\cal O}(D))$. Then there exist two positive constants $c_1,c_2$ such that \[ c_1 \leq \frac{H(W, D,{ \gamma},x)}{H(W, D,{ \gamma'},x)} \leq c_2 \] for all $x \in W(F)$. \label{weil} \end{theo} For a smooth projective variety $W$ we denote by $-K_W$ the anticanonical divisor on $W$. \begin{theo} Let $Y \subset {\bf P}^3$ be a smooth cubic surface defined over $F$, ${ \gamma} = \{ s_0, s_1, s_2, s_3 \}$ the basis of global sections of ${\cal O}(-K_Y)$ corresponding to the standard homogeneous coordinates on ${{\bf P }}^3$. Assume that $Y$ can be obtained by blowing up of $6$ $F$-rational points in ${\bf P }^2$. Then for any nonempty Zariski open subset $U \subset Y$ one has \[ N(U, -K_Y, { \gamma}, B) \geq c B (\log B)^3 \] for all $B > 0$ and for some positive constant $c$. \label{cub} \end{theo} \noindent {\em Proof.} Assume that $Y$ is obtained by blowing up of $p_1, \ldots, p_6 \in {\bf P }^2(F)$. By \ref{weil}, we can assume without loss of generality that $p_1 = (1:0:0)$, $p_2 = (0:1:0)$ and $p_3 = (0:0:1)$. Denote by $Y_0$ the Del Pezzo surface obtained by blowing up $p_1, p_2, p_3$. Let $f \,: \, Y \rightarrow Y_0$ be the contraction of exceptional curves $C_4, C_5, C_6 \subset Y$ lying over $p_4,p_5,p_6$. Let $V$ be the $10$-dimensional space over $F$ of all homogeneous polynomials of degree $3$ in variables $z_0, z_1, z_2$. We identify $ \Gamma (Y, {\cal O}(-K_Y))$ with the subspace in $V$ consisting of all polynomials vanishing in $p_1, \ldots, p_6$. Analogously, we identify $ \Gamma (Y_0, {\cal O}(-K_{Y_0}))$ with the subspace in $V$ consisting of all polynomials vanishing in $p_1,p_2,p_3$. Let $ \gamma_0 = \{ s_0, \ldots, s_6 \} \subset V$ the extension of the basis ${ \gamma}$ to a basis of the subspace $ \Gamma (Y_0, {\cal O}(-K_{Y_0})) \subset V$. The surface $Y_0$ is an smooth equivariant compactification of the split $2$-dimensional algebraic torus over $F$ $$({\bf G}_{m})^2 = {\bf P }^2 \setminus \{ l_{12}, l_{13}, l_{23} \}$$ where $l_{ij}$ denotes the projective line in ${\bf P }^2$ through $p_i$ and $p_j$. Since $Y_0$ is a smooth toric variety, the main theorem in \cite{BaTschi1} shows that the following asymptotic formula holds: \begin{equation} N(({\bf G}_{m})^2, -K_{Y_0}, \gamma_0', B) = c_0 B (\log B)^3(1 + o(1)),\;\; B \rightarrow \infty, \label{f0} \end{equation} where $c_0$ is some positive constant and \[ \gamma_0' = \{ z_0z_1z_2, z_1^2z_2, z_1 z_2^2, z_2^2z_0, z_2z_0^2, z_0^2z_1, z_0 z_1^2 \}. \] Let $U$ be any nonempty Zariski open subset in $Y$. We denote by $U_0$ a nonempty open subset in $U$ such that the restriction of $f$ on $U_0$ is an isomorphism and $f(U_0)$ is contained in $({\bf G}_{m})^2 \subset Y_0$. Since \[ \prod_{v \in {\rm Val}(F)} \max_{i=0, \ldots, 3} | s_i(x) |_v \leq \prod_{v \in {\rm Val}(F)} \max_{i=0, \ldots, 6} | s_i(x) |_v \] holds for every $F$-rational point $x \in U_0$, we obtain \begin{equation} N(U_0, -K_Y, \gamma, B) \geq N(U_0, -K_{Y_0}, \gamma_0, B) \label{f1} \end{equation} for any $B >0$. By \ref{weil}, there exists a positive constant $c_3$ such that \begin{equation} N(U_0, -K_{Y_0}, \gamma_0, B) \geq c_3 N(U_0, -K_{Y_0}, \gamma_0', B). \label{f2} \end{equation} On the other hand, \begin{equation} N(U_0, -K_{Y_0}, \gamma_0', B) = N(({\bf G}_{m})^2, -K_{Y_0}, \gamma_0', B) - N(Z, -K_{Y_0}, \gamma_0', B), \label{f3} \end{equation} where $Z = ({\bf G}_{m})^2 \setminus U_0$. Let $Z_1, \ldots, Z_l$ be the irreducible components of $Z$, $\overline{Z}_i$ the closure of $Z_i$ in $Y_0$ $(i =1, \ldots, l)$. It is known that \begin{equation} N(Z_i, -K_{Y_0}, \gamma_0',B) \leq c_4 B^{2/({\rm deg}\, \overline{Z}_i)} \label{f4} \end{equation} holds for some positive constant $c_4$, where ${\rm deg}\,\overline{Z}_i$ denotes the degree of $\overline{Z}_i$ with respect to the anticanonical divisor $-K_{Y_0}$. Since every irreducible curve $C \subset Y_0$ with ${\rm deg}\, C = 1$ is a component of $Y_0 \setminus ({\bf G}_m)^2$, we have ${\rm deg}\,\overline{Z}_i \geq 2$; i.e., \begin{equation} N(Z_i, -K_{Y_0}, \gamma_0',B) \leq c_4 B \label{f5} \end{equation} holds for all $i = 1, \ldots, l$. It follows from the asymptotic formula (\ref{f0}) combined with (\ref{f1}), (\ref{f2}) and (\ref{f3}) that there exists a positive constant $c$ such that \[ N(U_0, -K_Y, { \gamma}, B) \geq c B (\log B)^3 \] holds for all $B > 0$. This yields the statement, since $U_0$ is contained in $U$. \hfill $\Box$ \begin{coro} Let $Y$ be a smooth diagonal cubic surface in ${\bf P }^3$ defined by the equation \[ a_0 y_0^3 + a_1 y_1^3 + a_2 y_2^3 + a_3 y_3^3 = 0 \] with coefficients $a_0, \ldots, a_3$ in a number field $F$ which contains ${{\bf Q }}(\sqrt{-3})$. Assume that there exist numbers $b_0, \ldots, b_3 \in F^*$ such that $a_i = b_i^3$ $(i =0, \ldots, 3)$. Then for any nonempty Zariski open subset $U \subset Y$ one has \[ N(U, -K_Y, \gamma, B) \geq c B (\log B)^3 \] for all $B > 0$ and some positive constant $c$. \label{cubic1} \end{coro} \noindent {\em Proof.} It follows from our assumptions on the coefficients $a_0, \ldots, a_3$ and on the field $F$ that all $27$ lines on $Y$ are defined over $F$. Hence, $Y$ can be obtained from ${{\bf P }}^2$ by blowing up of $6$ $F$-rational points. Now the statement follows from \ref{cub}. \hfill $\Box$ \section{Rational points on $X_{n+2}$} We have the natural isomorphism \[ \Gamma (X_{n+2}, {\cal O}(-K_{X_{n+2}}) ) \cong \Gamma ({\bf P }^3, {\cal O}(1)) \otimes \Gamma ({\bf P }^n, {\cal O}(n)).\] Let $\{ t_0, \ldots, t_m \}$ be a basis in $ \Gamma ({\bf P }^n, {\cal O}(n))$ and $\{ s_0, s_1, s_2, s_3 \}$ the standard basis in $ \Gamma ({\bf P }^3, {\cal O}(1))$. Denote by $ \gamma$ the basis of $ \Gamma (X_{n+2}, {\cal O}(-K_{X_{n+2}}) )$ consisting of $s_i \otimes t_j$ $(i=0,\ldots, 3; \; j = 0, \ldots, m)$. \begin{theo} Let $n \geq 3$. Then for any nonempty Zariski open subset $U \subset X_{n+2}$ and for any field $F$ containing ${{\bf Q }}(\sqrt{-3})$, one has \[ N(U, -K_{X_{n+2}}, \gamma, B) \geq c B (\log B)^3 \] for all $B > 0$ and some positive constant $c$. \label{t1} \end{theo} \noindent {\em Proof.} Consider the projection $\pi\,:\,U\rightarrow {\bf P }^n$. Let $U'$ be a Zariski open subset in $\pi (U) \cap U_P$ such that the image of $U'$ under the dominant (rational) mapping \[ \psi \;:\; {\bf P }^n - - \rightarrow {\bf P }^3 \] \[ \psi(x_0 : \ldots : x_n) = (l_0({\bf x}): \ldots : l_3({\bf x})) \] is Zariski open in ${\bf P }^3$. Denote by $ \varphi $ the finite morphism \[ \varphi \; : \; {\bf P }^3 \rightarrow {\bf P }^3, \] \[ \varphi (z_0 : \ldots : z_3) = (z_0^3 : \ldots : z_3^3). \] Since ${\bf P }^3(F)$ is Zariski dense in ${\bf P }^3$, there is a point $p \in {\bf P }^3(F) \cap \varphi ^{-1}(\psi(U'))$. Since $U'(F) \cap \psi^{-1}( \varphi (p))$ is Zariski dense in $\psi^{-1}( \varphi (p))$, there exists $q \in U'(F) \cap \psi^{-1}( \varphi (p))$. Therefore, the fiber of $\pi $ over $q$ is a diagonal cubic surface $Y_q$ such that and $U \cap Y_q \subset Y_q$ is a nonempty Zariski open subset. It remains to apply \ref{cubic1}. \hfill $\Box$ \begin{theo} Let $n =2$. Then there exists a number field $F_0$ depending only on $X_{n+2}$ such that for any nonempty Zariski open subset $U \subset X_{n+2}$ for any field $F$ containing $F_0$ one has \[ N(U, -K_{X_{n+2}}, \gamma, B) \geq c B (\log B)^3 \] for all $B > 0$ and some positive constant $c$. \label{t2} \end{theo} \noindent {\em Proof.} Let $U'$ be a Zariski open subset in $\pi (U) \cap U_P$. We have the linear embedding \[ \psi \;:\; {\bf P }^2 \hookrightarrow {\bf P }^3 \] defined by $l_0({\bf x}), \ldots, l_3({\bf x})$. Then $ \varphi ^{-1}(\psi({\bf P }^2))$ is a smooth diagonal cubic surface $S \subset {\bf P }^3$ defined over ${\bf Q }$. Let $F_0$ be a finite extension of ${\bf Q }(\sqrt{-3})$ such that $S(F_0)$ is Zariski dense in $S$. Then there exists a point $p \in S(F_0)$ such that $q = \varphi (p)$ is contained in $U'$. Therefore, the fiber of $\pi$ over $q$ is a diagonal cubic surface $Y_q$, and $U \cap Y_q \subset Y_q$ is a nonempty Zariski open subset. It remains to apply \ref{cubic1}. \hfill $\Box$ \begin{theo} Let $n = 1$. Then for any nonempty Zariski open subset $U \subset X_{n+2}$, there exists a number field $F_0$ $($which depends on $U$$)$ such that for any field $F$ containing $F_0$, one has \[ N(U, -K_{X_{n+2}}, \gamma, B) \geq c B (\log B)^3 \] for all $B > 0$ and some positive constant $c$. \label{t3} \end{theo} \noindent {\em Proof.} Let $U'$ be a Zariski open subset in $\pi (U) \cap U_P$. We have the linear embedding \[ \psi \;:\; {\bf P }^1 \hookrightarrow {\bf P }^3 \] defined by $l_0({\bf x}), \ldots, l_3({\bf x})$. Then $ \varphi ^{-1}(\psi({\bf P }^1))$ is an algebraic curve $C \subset {\bf P }^3$ which is a complete intesection of two diagonal cubic surfaces defined over ${\bf Q }$. Let $F_0$ be a finite extension of ${\bf Q }(\sqrt{-3})$ such that there exists an $F_0$-rational point $p \in C(F_0) \cap \varphi ^{-1}(U')$. Then the fiber of $\pi$ over $q = \varphi (p)$ is a diagonal cubic surface $Y_q$ and $U \cap Y_q \subset Y_q$ is a nonempty Zariski open subset. It remains to apply \ref{cubic1}. \hfill $\Box$ \section{Conclusions} The following statement, inspired by the Linear Growth conjecture of Manin (\cite{manin1}) and by extrapolation of known results (circle method, flag varieties, toric varieties), has been expected to be true \cite{bat.man,franke-manin-tschinkel}: \begin{conj} Let $X$ be a smooth Fano variety over a number field $E$. Then there exist a Zariski open subset $U \subset X$ and a finite extension $F_0$ of $E$ such that for all number fields $F$ containing $F_0$ the following asymptotic formula holds \[ N(U, -K_X, \gamma, B) = c B (\log B)^{t-1} (1 + o(1)), \;\; B \rightarrow \infty ,\] where $t$ is the rank of the Picard group of $X$ over $F$. \label{conjecture} \end{conj} Some lower and upper bounds for $N(U, -K_X, \gamma, B)$ for Del Pezzo surfaces and Fano threefolds have been obtained in \cite{manin1,MaTschi}. The conjecture \ref{conjecture} was refined by E. Peyre who proposed an adelic interpretation for the constant $c$ introducing Tamagawa numbers of Fano varieties \cite{peyre}. This refined version of the conjecture has been proved for toric varieties in \cite{BaTschi,BaTschi1}. The statements in Theorems \ref{t1}, \ref{t2}, \ref{t3} and the property \ref{pic} show that Conjecture \ref{conjecture} is not true for Fano cubic bundles $X_{n+2}$ $(n \geq 1)$.

9602

alg-geom/9602022

en

https://arxiv.org/abs/alg-geom/9602022

alg-geom/9602022

Chikashi Miyazaki

Chikashi Miyazaki and Wolfgang Vogel

Towards a theory of arithmetic degrees

LaTeX, 14 pages

The aim of this paper is to start a systematic investigation of the arithmetic degree of projective schemes as introduced by D. Bayer and D. Mumford. One main theme concerns itself with the behaviour of this arithmetic degree under hypersurface sections. The notion of arithmetic degree involves the new concept of length-multiplicity of embedded primary ideals. Therefore it is much harder to control the arithmetic degree under a hypersurface section than in the case for the classical degree theory. Nevertheless it has important and interesting applications. We describe such applications to the Castelnuovo-Mumford regularity and to Bezout-type theorems.

alg-geom

\section{Introduction} The aim of this paper is to start a systematic investigation of the arithmetic degree of projective schemes as introduced in \cite{BM}. One main theme concerns itself with the behaviour of this arithmetic degree under hypersurface sections, see Theorem 2.1. The classical intersection theory only considers the top-dimensional (or isolated) primary components. However, the notion of arithmetic degree involves the new concept of length-multiplicity of embedded primary ideals as considered in \cite{BM}, \cite{EH}, \cite{H}, \cite{K}, \cite{STV}. Therefore it is much harder to control the arithmetic degree under a hypersurface section than in the case for the classical degree theory. We describe in \S 3 an upper bound for the arithmetic degree in terms of the Castelnuovo-Mumford regularity, see Theorem 3.1. In addition, we generalize Bezout's theorem via iterated hypersurface sections, see Theorem 4.1. We conclude in \S 5 by studying two examples. \section{Arithmetic degree and hypersurface sections} Before stating our main result of this section we need to introduce the concept of the length-multiplicity of (embedded) primary components. Let $K$ be an arbitrary field and $S$ the polynomial ring $K[x_0,\cdots,x_n]$. Let ${\sf m}=(x_0,\cdots,x_n)$ be the homogeneous maximal ideal of $S$. Let $I$ be a homogeneous ideal of $S$. \vspace{5mm} \noindent{\bf Definition (\cite{BM}):} Let $\sf p$ be a homogeneous prime ideal belonging to $I$. For a primary decomposition $I=\cap {\sf q}$ we take the primary ideal $\sf q$ with $\sqrt{{\sf q}}={\sf p}$. Let $J$ be the intersection of all primary components of $I$ with associated prime ideals ${\sf p}_1$ such that ${\sf p}_1\rixrel{\neq}{\subset}{\sf p}$. If the prime ideal $\sf p$ is an isolated component of $I$, we set $J=S$. We define the length-multiplicity of $\sf q$ denoted by $\mbox{mult}_I({\sf p})$, to be the length $\ell$ of a maximal strictly increasing chain of ideals $${\sf q}\cap J =: J_\ell\subset J_{\ell-1}\subset\cdots\subset J_1\subset J_0 := J$$ where $J_k$, $1\leq k\leq \ell-1$, equals ${\sf q}_k\cap J$ for some $\sf p$-primary ideal ${\sf q}_k$. Despite the non-uniqueness of embedded components, the number $\ell=\mbox{mult}_I({\sf p})$ is well-defined. \vspace{5mm} For a finitely generated graded $S$-module $M$, let $H(M,\ell)$ be the Hilbert function of $M$ for all integers $\ell$, that is, $H(M,\ell)$ is the dimension of the vector space $[M]_\ell$ over $K$. It is well-known that the Hilbert function of $M$ is a polynomial in $\ell$ for $\ell$ large enough. We denote this polynomial by $P(M,\ell)$. We set $\Delta(H(M,\ell))=H(M,\ell)-H(M,\ell-1)$, $\Delta^0 H(M,\ell)=H(M,\ell)$, and $\Delta^r(H(M,\ell))= \Delta^{r-1}(\Delta H(M,\ell))$ for all integers $r\geq 2$. Moreover, we set $$\Delta_\tau(H(M,\ell))=H(M,\ell)-H(M,\ell-\tau)$$ for all integers $\tau\geq 1$. Further, the Hilbert polynomial $P(M,\ell)$ is written as $$P(M,\ell) = \frac{e}{d!}\ell^d + (\mbox{lower order terms}), \quad e\neq 0.$$ Then we define $h$-dim $M=d$ (homogeneous dimension) and degree of $M$ by $\deg M := e$. Also, we write, for any ideal $I$ of $S$, $\dim I$ and $\deg I$ for $h$-dim $S/I$ and $\deg S/I$ respectively. In case $P(M,\ell)=0$, we define $h$-dim $M=-1$ and $\deg M=\sum_{\ell\in{\bf Z}} H(M,\ell)$. In particular, $\dim {\sf m}=-1$ and $\deg {\sf m}=1$. \vspace{5mm} \noindent{\bf Definition (\cite{BM}):} For an integer $r\geq -1$, we define \begin{eqnarray*} \mbox{arith-deg}_r (I) & = & \sum_{\stackrel{{\sf p} \mbox{\ \scriptsize is a prime ideal}}{\mbox{\scriptsize such that} \dim{\sf p}=r}} \mbox{mult}_I({\sf p})\cdot\deg {\sf p}\\ & = & \sum_{\stackrel{{\sf p}\in \mbox{\scriptsize Ass } S/I}{\mbox{\scriptsize such that } \dim{\sf p}=r}} \mbox{mult}_I(\sf p)\cdot\deg{\sf p} \end{eqnarray*} \vspace{3mm} \noindent{\bf Definition (\cite{H}):} Let $I$ be a homogeneous ideal of $S$. Let $r$ be an integer with $r\geq -1$. We define the ideal $I_{\geq r}$ as the intersection of all primary components $\sf q$ of $I$ with $\dim{\sf q}\geq r$. \vspace{5mm} \noindent{\bf Remark :} Although primary decomposition is not uniquely determined, the ideal $I_{\ge r}$ does not depend on the choice of primary decomposition of $I$ and is again a homogeneous ideal. \vspace{5mm} The aim of this section is to prove the following theorem. \vspace{5mm} \noindent{\bf Theorem 2.1:}{\em\ Let $r$ be an integer with $r\geq 0$. Let $I$ be a homogeneous ideal of $S$. Let $F$ be a homogeneous polynomial of $S$ with $\mbox{degree }(F)=\tau\geq 1$. Assume that $F$ does not belong to any associated prime ideal $\sf p$ of $I$ with $\dim {\sf p}\geq r$. Then we have $$\mbox{arith-deg}_{r-1}(I,F) - \mbox{arith-deg}_{r-1}(I_{\geq r+1},F)\geq\tau\cdot \mbox{arith-deg}_r (I)$$ and the equality holds if and only if $F$ does not belong to any associated prime ideal $\sf p$ of $I$ with $\dim{\sf p}=r-1$.} \vspace{5mm} \noindent{\bf Corollary 2.2:}{\em\ Under the above condition, $$\mbox{arith-deg}_{r-1}(I,F)\geq\tau\cdot\mbox{arith-deg}_r(I)$$ and the equality holds if and only if $F$ does not belong to any associated prime ideal $\sf p$ of $I$ with $\dim{\sf p}=r-1$ and the ideal $(I_{\geq r+1},F)$ has no associated prime ideals of dimension $(r-1)$.} \vspace{5mm} \noindent{\em Proof.} Corollary 2.2 follows immediately from Theorem 2.1. \vspace{5mm} We note that Corollary 2.2 and Lemma 3 of \cite{KS} yield Theorem 2.3 of \cite{STV}. \vspace{5mm} We want to consider generic hyperplane sections. The following useful lemma is obtained from \cite{BF}, (4.2) and \cite{F}, (5.2), which was pointed out to us by H. Flenner. \vspace{5mm} \noindent{\bf Lemma 2.3:}{\em\ Let $I$ be a homogeneous ideal of $S$. We set $A=S/I$. Let $h=0$ be the defining equation of a generic hyperplane of ${\bf P}_K^n$ ($K$: infinite field). Then we have $$\mbox{Ass}(A/h)\verb+\+\{{\sf m}\}\rixrel{=}{\subset} \bigcup_{{\sf p}\in \mbox{\scriptsize Ass } A} \mbox{Min} (A/({\sf p},h)),$$ where $\mbox{Min}(A/({\sf p},h))$ is the set of minimal primes belonging to $({\sf p},h)$.} \vspace{5mm} \noindent{\bf Corollary 2.4:}{\em\ Let $r$ be an integer $\geq 1$. Let $H$ be a generic hyperplane in ${\bf P}_K^n$, given by $h=0$. Then we have: $$\mbox{arith-deg}_r (I) = \mbox{arith-deg}_{r-1} (I,h).$$ } \vspace{5mm} \noindent{\em Proof.} Corollary 2.4 follows immediately from Theorem 2.1 and Lemma 2.3. \vspace{5mm} We note that Corollary 2.4 is stated in \cite{BM}, page 33 without proof. \vspace{5mm} Before we turn to the proof of Theorem 2.1, two technical results are needed. First we state a more or less known result describing a different characterization of the arithmetic degree, which is purely algebraic and, in fact, serves as the definition in \cite{H}. \vspace{5mm} \noindent{\bf Lemma 2.5:}{\em\ Let $r$ be a non-negative integer. Let $I$ be a homogeneous ideal of $S$. Then we have $$\mbox{arith-deg}_r (I) = \Delta^r (P(S/I,\ell) - P(S/I_{\geq r+1},\ell))$$ for all integers $\ell$.} \vspace{5mm} \noindent{\bf Lemma 2.6:}{\em\ Let $r$ be an integer with $r\geq 1$. Let $I$ be a homogeneous ideal of $S$ and $F$ a homogeneous polynomial of $S$ with $\deg(F)=\tau\geq 1$. Assume that $F$ does not belong to any associated prime ideal ${\sf p}$ of $I$ with $\dim {\sf p}\geq r+1$. Then we have \begin{eqnarray*} \lefteqn{\mbox{arith-deg}_{r-1}(I,F)-\Delta^{r-1}P(S/(I_{\geq r+1},F),\ell) + \Delta^{r-1}P(S/(I,F)_{\geq r},\ell)} \\ & & \qquad\qquad= \tau\cdot\mbox{arith-deg}_r (I) + \Delta^{r-1} P([0:F]_{S/I},\ell-\tau) \end{eqnarray*} for all integers $\ell$.} \vspace{5mm} \noindent{\em Proof.} From the exact sequences $$0\rightarrow [0:F]_{S/I}(-\tau)\rightarrow S/I(-\tau)\stackrel{F}{\rightarrow}S/I \rightarrow S/(I,F)\rightarrow 0$$ and $$0\rightarrow S/I_{\geq r+1}(-\tau)\stackrel{F}{\rightarrow}S/I_{\geq r+1}\rightarrow S/(I_{\geq r+1},F)\rightarrow 0,$$ we have $$\Delta_\tau P(S/I,\ell)=P(S/(I,F),\ell)-P([0:F]_{S/I},\ell-\tau)$$ and $$\Delta_\tau P(S/I_{\geq r+1},\ell)=P(S/(I_{\geq r+1},F),\ell)$$ for all integers $\ell$. Note that $P(S/I,\ell) - P(S/I_{\ge r+1},\ell)$ is a numerical polynomial of degree $r$, see (2.5). Thus we have \begin{eqnarray*} \lefteqn{\mbox{arith-deg}_{r-1}(I,F)-\Delta^{r-1}P(S/(I_{\geq r+1},F),\ell) + \Delta^{r-1}P(S/(I,F)_{\geq r},\ell)}\\[2mm] & \qquad\qquad= & \Delta^{r-1}(P(S/(I,F),\ell)-P(S/(I,F)_{\geq r},\ell))\\ & & -\Delta^{r-1}P(S/(I_{\geq r+1},F),\ell)+\Delta^{r-1}P(S/(I,F)_{\geq r},\ell)\\[2mm] & \qquad\qquad= & \Delta^{r-1}(\Delta_\tau P(S/I,\ell)+P([0:F]_{S/I},\ell-\tau))\\ & & -\Delta^{r-1}\Delta_\tau P(S/I_{\geq r+1},\ell)\\[2mm] & \qquad\qquad= & \Delta_\tau (\Delta^{r-1}(P(S/I,\ell)-P(S/I_{\geq r+1},\ell)))\\ & & +\Delta^{r-1}P([0:F]_{S/I},\ell-\tau)\\[2mm] & \qquad\qquad= & \tau\cdot\Delta^r(P(S/I,\ell)-P(S/I_{\geq r+1},\ell)) + \Delta^{r-1}P([0:F]_{S/I},\ell-\tau)\\[2mm] & \qquad\qquad= & \tau\cdot \mbox{arith-deg}_r(I)+\Delta^{r-1}P([0:F]_{S/I},\ell-\tau), \end{eqnarray*} by Lemma 2.5. Hence the assertion is proved. \vspace{5mm} The following lemma is used in the proof of Theorem 2.1 and Lemma 4.3. \vspace{5mm} \noindent{\bf Lemma 2.7:}{\em\ Let $I$ be a homogeneous ideal of $S$. Let $r$ be an integer. Let $F$ be a homogeneous polynomial of $S$ with degree $(F)\geq 1$ such that $F$ does not belong to any associated prime ideal $\sf p$ of $I$ with $\dim{\sf p}\geq r$. Then we have $${(I_{\geq u},F)}_{\geq r} = (I,F)_{\geq r}$$ for all integers $u=-1,0,\cdots,r+1$.} \vspace{5mm} \noindent{\em Proof.} We want to prove that $${(I_{\geq u},F)}_{\sf p} = (I,F)_{\sf p}$$ for all prime ideals $\sf p$ with $\dim{\sf p}\geq r$. We may assume that $F\in{\sf p}$. If $\sf p$ does not contain any primary component $\sf q$ of $I$ with $\dim({\sf q}) <u$, the proof is done. Now assume that there is a primary component $\sf q$ of $I$ with $\dim({\sf q})<u$ such that ${\sf p}\rixrel{=}{\supset}{\sf q}$. Since $r\leq\dim({\sf p}) \leq\dim({\sf q})<u$, we see that $u=r+1$ and ${\sf p}=\sqrt{\sf q}$. Thus we have that $\sf p$ is an associated prime ideal of $I$ with $\dim{\sf p}=r$ and $F\in{\sf p}$, which contradicts the hypothesis. \vspace{5mm} \noindent{\em Proof of Theorem 2.1.} First we prove the case $r\geq 1$. Applying (2.5) and (2.6), we have \begin{eqnarray*} \lefteqn{\mbox{arith-deg}_{r-1}(I,F)-\mbox{arith-deg}_{r-1}(I_{\geq r+1},F)}\\ & \quad= & \tau\cdot\mbox{arith-deg}_r(I) + \Delta^{r-1}P(S/(I_{\geq r+1},F),\ell)\\ & & -\Delta^{r-1}P(S/(I,F)_{\geq r},\ell) + \Delta^{r-1}P([0:F]_{S/I},\ell-\tau)\\ & & -\Delta^{r-1}(P(S/(I_{\geq r+1},F),\ell)-P(S/(I_{\geq r+1},F)_{\geq r},\ell))\\ & \quad= & \tau\cdot\mbox{arith-deg}_r(I) + \Delta^{r-1}[P(S/(I_{\geq r+1},F)_{\geq r},\ell) -P(S/(I,F)_{\geq r},\ell)]\\ & & + \Delta^{r-1}P([0:F]_{S/I},\ell-\tau). \end{eqnarray*} By the assumption, $[0:F]_{S/I}$ has at most $(r-1)$-dimensional support, which means $\Delta^{r-1}P([0:F]_{S/I},\ell-\tau)\geq 0$. Further, $\Delta^{r-1}P([0:F]_{S/I},\ell -\tau)=0$ if and only if $F$ does not belong to any associated prime ideal $\sf p$ with $\dim{\sf p}=r-1$. On the other hand, $S/(I_{\geq r+1},F)_{\geq r}=S/(I,F)_{\geq r}$ by Lemma 2.7. Therefore we have $$\mbox{arith-deg}_{r-1}(I,F) - \mbox{arith-deg}_{r-1}(I_{\geq r+1},F)\geq\tau\cdot \mbox{arith-deg}_r(I)$$ and the equality holds if and only if $F$ does not belong to any associated prime ideal $\sf p$ with $\dim{\sf p}=r-1$. Next we prove the case $r=0$. Now we see that \begin{eqnarray*} \mbox{arith-deg}_{-1}(I,F) & = & \mbox{length}_S(I,F)_{\geq 0}/(I,F)\\ & = & \sum_{\ell=0}^{N}(\dim_K{[S/(I,F)]}_\ell - \dim_K{[S/(I,F)_{\geq 0}]}_\ell)\\ & = & \sum_{\ell=0}^N(\dim_K{[S/I]}_\ell-\dim_K{[S/I]}_{\ell-\tau}\\ & & + \dim_K {\left[[0:F]_{S/I}\right]}_{\ell-\tau} - \dim_K{[S/(I,F)_{\geq 0}]}_\ell) \end{eqnarray*} for large $N$. Similarly, we see that \begin{eqnarray*} \mbox{arith-deg}_{-1}(I_{\geq 1},F) & = & \sum_{\ell=0}^N\left(\dim_K{[S/I_{\geq 1}]}_\ell - \dim_K[S/I_{\geq 1}]_{\ell-\tau}\right.\\ & & \left. - \dim_K{[{S/(I_{\geq 1},F)}_{\geq 0}]}_\ell\right) \end{eqnarray*} for large $N$. Hence we have \begin{eqnarray*} \lefteqn{\mbox{arith-deg}_{-1}(I,F) - \mbox{arith-deg}_{-1}(I_{\geq 1},F)}\\ & \qquad= & \sum_{\ell=0}^N(\dim_K{[I_{\geq 1}/I]}_\ell - \dim_K {[I_{\geq 1}/I]}_{\ell-\tau})\\ & & - \sum_{\ell=0}^N\dim_K {[{(I_{\geq 1},F)}_{\geq 0}/{(I,F)}_{\geq 0}]}_\ell + \sum_{\ell=0}^N\dim_K{[{[0:F]}_{S/I}]}_{\ell-\tau}\\ & \qquad= & \sum_{\ell=N-\tau+1}^N\dim_K{[I_{\geq 1}/I]}_\ell - \sum_{\ell=0}^N\dim_K {[{(I_{\geq 1},F)}_{\geq 0}/{(I,F)}_{\geq 0}]}_\ell\\ & & + \sum_{\ell=0}^N\dim_K{[{[0:F]}_{S/I}]}_{\ell-\tau} \end{eqnarray*} for large $N$. By the assumption, $\sum_{\ell=0}^N\dim_K{[{[0:F]}_{S/I}]}_{\ell-\tau} =\mbox{length}_S({[0:F]}_{S/I}) \geq 0$ for large $N$, and ${[0:F]}_{S/I} =0$ if and only if $F$ is a non-zero-divisor of $S/I$. On the other hand, we see $\dim_K{[I_{\geq 1}/I]}_\ell =P(I_{\geq 1}/I,\ell)=\mbox{arith-deg}_0 (I)$ for large $\ell$. Further, we have ${(I_{\geq 0},F)}_{\sf p} = (I,F)_{\sf p}$ for all prime ideals $\sf p$ with $\dim{\sf p}=0$ by Lemma 2.7. Hence we have $$\mbox{arith-deg}_{-1}(I,F) - \mbox{arith-deg}_{-1}(I_{\geq 1},F)\geq\tau\cdot \mbox{arith-deg}_0 (I)$$ and the equality holds if and only if $F$ is a non-zero-divisor of $S/I$. This completes the proof of Theorem 2.1. \section{Castelnuovo-Mumford regularity} Bayer and Mumford \cite{BM} give a bound for the arithmetic degree in terms of the Castelnuovo-Mumford regularity. The aim of this section is to describe improved bound on this degree. Let $m = m(I)$ be the Castelnuovo-Mumford regularity (see, e.g., \cite{BM}, \cite{EG}, \cite{M}) for a homogeneous ideal $I$ of the polynomial ring $S = K[x_0,\cdots,x_n]$. Then our main result is the following theorem. \vspace{5mm} \noindent{\bf Theorem 3.1:}{\em\ Let $I$ be a homogeneous ideal of $S$. Let $m = m(I)$ be the Castelnuovo-Mumford regularity of $I$. Then we have, for any integer $r\geq 0$ $$\mbox{arith-deg}_r (I)\leq \Delta^r P(S/I,\ell)$$ for all integers $\ell\geq m-1$.} \vspace{5mm} We want to give two corollaries. The first one shows that (3.1) improves the bound given in \cite{BM}, Proposition 3.6. \vspace{5mm} \noindent{\bf Corollary 3.2:}{\em\ For all $r \geq 0$, we have \begin{eqnarray*} \mbox{arith-deg}_r (I) & \leq & \Delta^r P(S/I,m-1)\\ & \leq & \left(\begin{array}{c}m+n-r-1\\n-r\end{array}\right)\\ & \leq & m^{n-r} \end{eqnarray*}} \noindent{\bf Corollary 3.3:}{\em\ Let $t=\mbox{depth } S/I$. Then we have, for an integer $r\geq 0$, $$\mbox{arith-deg}_r (I)\leq\Delta^r H(S/I,\ell)$$ for all $\ell\geq m+r-t-1$ if $r-t$ is even, and for all $\ell\geq m+r-t$ if $r-t$ is odd.} \vspace{5mm} Before embarking on the proof of Theorem 3.1 and the corollaries we need two lemmas. The first one follows from \cite{S} Nr.79 (see also \cite{SV}, Proof of Lemma I.4.3). \vspace{5mm} \noindent{\bf Lemma 3.4:}{\em\ Let $I$ be a homogeneous ideal of $S$ and $t=\mbox{depth } S/I$. Then we have \renewcommand{\labelenumi}{(\theenumi)} \renewcommand{\theenumi}{\alph{enumi}} \begin{enumerate} \item $P(S/I,\ell) = H(S/I,\ell) - \sum_{i=0}^d (-1)^i\dim_K {[H_{\sf m}^i(S/I)]}_\ell$ for all integers $\ell$. \item $P(S/I,\ell) = H(S/I,\ell)$ for all $\ell\geq m-t$. \end{enumerate} } \vspace{5mm} \noindent{\bf Lemma 3.5:}{\em\ Let $I$ be a homogeneous ideal of $S$. Then we have $$\Delta^r P(I,\ell)\geq 0$$ \vspace{-10mm} \noindent for all $\ell\geq m-1$ and $r\geq 0$. } \vspace{5mm} \noindent{\em Proof.} For a generic hyperplane $H$ given by $h=0$, we can take an exact sequence $$0\rightarrow I(-1)\stackrel{h}{\rightarrow} I\rightarrow I_H\rightarrow 0 ,$$ where $I_H=(I,h)/h$. From the exact sequence, we have $$\Delta P(I,\ell) = P(I_H,\ell)$$ for all $\ell$ and $I_H$ is $m$-regular. Repeating this step, we see that $$\Delta^r P(I,\ell) = P(I_{H_1\cap\cdots\cap H_r},\ell)$$ for all $\ell$ and for generic hyperplanes $H_1,\cdots,H_r$ defined by $h_1=0,\cdots,h_r=0$, resp., where $$I_{H_1\cap\cdots\cap H_r} = (I,h_1,\cdots,h_r)/(h_1,\cdots,h_r),$$ and that $I_{H_1\cap\cdots\cap H_r}$ is $m$-regular. So ${(I_{H_1\cap\cdots\cap H_r})}_{\geq 0}$ is also $m$-regular and \linebreak $\mbox{depth}_S S/{(I_{H_1\cap\cdots\cap H_r})}_{\geq 0}\geq 1$. Therefore we have \begin{eqnarray*} \Delta^r P(I,\ell) & = & P(I_{H_1\cap\cdots\cap H_r},\ell)\\ & = & P(S,\ell) - P(S/I_{H_1\cap\cdots\cap H_r},\ell)\\ & = & P(S,\ell) - P(S/{(I_{H_1\cap\cdots\cap H_r})}_{\geq 0},\ell)\\ & = & H(S,\ell) - H(S/{(I_{H_1\cap\cdots\cap H_r})}_{\geq 0},\ell)\\ & = & H({(I_{H_1\cap\cdots\cap H_r})}_{\geq 0},\ell)\geq 0 \end{eqnarray*} for $\ell\geq m-1$, by (3.4), (b). \vspace{5mm} \noindent{\em Proof of Theorem 3.1.} Without the loss of generality, we may assume\linebreak that $I$ is a saturated ideal. First we prove the case $r=0$. By Lemma 2.5, we have $$(\ast)\qquad\qquad\mbox{arith-deg}_0(I)=P(S/I,\ell) - P(S/I_{\geq 1},\ell).$$ Now we want to show that $I_{\geq 1}$ is $m$-regular. From the short exact sequence $$0\rightarrow I_{\geq 1}/I\rightarrow S/I\rightarrow S/I_{\geq 1}\rightarrow 0$$ and the fact that $I_{\geq 1}/I$ has at most 1-dimensional support and by Grothendieck's vanishing theorem $H_{\sf m}^i(I_{\ge 1}/I) = 0$ for $i \ge 2$, we have $$0\rightarrow H_{\sf m}^1 (I_{\geq 1}/I)\rightarrow H_{\sf m}^1(S/I)\rightarrow H^1_{\sf m} (S/I_{\geq 1})\rightarrow 0$$ and $H_{\sf m}^i(S/I)\cong H_{\sf m}^i (S/I_{\geq 1})$ for $i\geq 2$. Thus we have $I_{\geq 1}$ is $m$-regular. Hence $$P(S/I_{\geq 1},\ell) = H(S/I_{\geq 1},\ell)\geq 0$$ for all $\ell\geq m-1$, by (3.4), (b). Therefore we have from $(\ast)$ $$\mbox{arith-deg}_0 (I)\leq P(S/I,\ell)$$ for all $\ell\geq m-1$. Now let us assume $r>0$. By Corollary 2.4 we see $$\mbox{arith-deg}_r (I) = \mbox{arith-deg}_0 (I,h_1,\cdots,h_r)$$ for generic hyperplanes $h_1,\cdots,h_r$. Thus we have $$\mbox{arith-deg}_r (I)\leq P(S/(I,h_1,\cdots,h_r),\ell)$$ for all $\ell\geq m-1$. On the other hand, we see \begin{eqnarray*} P(S/(I,h_1,\cdots,h_r),\ell) & = & \Delta P(S/(I,h_1,\cdots,h_{r-1}),\ell) \\ & & \qquad \vdots\\ & = & \Delta^r P(S/I,\ell) \end{eqnarray*} for all $\ell$. Hence the assertion is proved. \vspace{5mm} \noindent{\em Proof of Corollary 3.2.} By Lemma 3.5, we have $$\Delta^r P(S/I,\ell)\leq\Delta^r P(S,\ell)$$ for all $\ell\geq m-1$. On the other hand, $\Delta^r P(S,\ell) = \left(\begin{array}{c} n+\ell-r\\n-r\end{array}\right)$. Hence the assertion follows from Theorem 3.1. \vspace{5mm} \noindent{\em Proof of Corollary 3.3.} By Lemma 3.4, we see that $$\Delta^r P(S/I,\ell) = \Delta^r H(S/I,\ell)$$ for all $\ell\geq m+r-t$, and that $$\Delta^r P(S/I,m+r-t-1)=\Delta^r H(S/I,m+r-t-1)-(-1)^{r-t}\dim_K {[H_{\sf m}^t(S/I)]}_{m-t-1}$$ because $$P(S/I,m-t-1) = H(S/I,m-t-1) - (-1)^t[H_{\sf m}^t(S/I)]_{m-t-1}.$$ Hence the assertion follows from Theorem 3.1. \section{Bezout-type results} The aim of this section is to state properties of arithmetic degree under iterated hyperplane sections, and Bezout-type results. Our Theorem 4.1 describes a Bezout's theorem in terms of the arithmetic degree. \vspace{5mm} \noindent{\bf Theorem 4.1:}{\em\ Let $I$ be a homogeneous ideal of $S := K[x_0,x_1,\cdots,x_n]$. Let $r\geq 0$ and $s\geq 1$ be integers with $r+1\geq s$. Let $F_1,\cdots,F_s$ be homogeneous polynomials of $S$ such that $F_i$ does not belong to any associated prime ideal ${\sf p}$ of $(I,F_1,\cdots,F_{i-1})$ with $\dim {\sf p}\geq r-i+1$, for all $i=1,\cdots,s$. Then we have \renewcommand{\labelenumi}{(\theenumi)} \renewcommand{\theenumi}{\roman{enumi}} \begin{enumerate} \item $\mbox{arith-deg}_{r-s} (I,F_1,\cdots,F_s)\geq\left[\prod_{i=1}^s \mbox{degree} (F_i)\right]\cdot\mbox{arith-deg}_r (I)$; \item We have equality in (i) if and only if $({(I,F_1,\cdots,F_{i-1})}_{\geq r-i+2},F_i)$ has no $(r-i)$-dimensional primes and $F_i$ does not belong to any associated prime ideal ${\sf p}$ of $(I,F_1,\cdots,F_{i-1})$ with $\dim {\sf p}=r-i$, for all $i=1,\cdots,s$; \item Assume that there is an integer $t$ with $-1\leq t\leq r+1$ such that $F_i$ does not belong to any associated prime ideal ${\sf p}$ of $(I_{\geq t},F_1,\cdots,F_{i-1})$ with $\dim {\sf p}\geq r-i+1$, for all $i=1,\cdots,s$, and $(I_{\geq t},F_1,\cdots,F_s)$ has no $(r-s)$-dimensional associated prime ideals. Then we have equality in (i) if and only if $F_i$ does not belong to any associated prime ideal ${\sf p}$ of $(I,F_1,\cdots,F_{i-1})$ with $\dim {\sf p}=r-i$, for all $i=1,\cdots,s$. \end{enumerate}} \vspace{5mm} \noindent{\em Proof.} (i) and (ii) follow from (2.2). In order to prove (iii) we need Lemma 4.2 and Lemma 4.3 below. First we replace the ideal $I$ of (4.2) by the ideal $I_{\geq t}$ of (iii). Then Lemma 4.3 shows that we can apply (ii) of (4.1). This provides our result (iii). \vspace{5mm} We note that special cases of (4.1) describe generalizations of classical results in the degree theory (see, e.g., \cite{FV}, \cite{V}). \vspace{5mm} We prove the two lemmas. \vspace{5mm} \noindent{\bf Lemma 4.2:}{\em\ Let $I$ be a homogeneous ideal of $S$. Let $r$ and $s$ be integers with $1 \leq s \leq r+1$. Let $F_1, \cdots, F_s$ be homogeneous polynomials of $S$ with\linebreak degree $(F_i) \geq 1$, $i=1,\cdots,s$, such that $F_i$ does not belong to any associated prime ideal $\sf p$ of $(I,F_1,\cdots,F_{i-1})$ with $\dim{\sf p}\geq r-i+1$, for all $i=1,\cdots,s$. If the ideal $(I,F_1,\cdots,F_s)$ has no $(r-s)$-dimensional associated prime ideals, then the ideal $({(I,F_1,\cdots,F_{i-1})}_{\geq r-i+2},F_i)$ has no $(r-i)$-dimensional associated prime ideals, for all $i=1,\cdots,s$.} \vspace{5mm} \noindent{\em Proof.} It is easy to see that the ideal $(I,F_1,\cdots, F_i)$ has no $(r-i)$-dimensional associated prime ideals. By Theorem 2.1, we have $$\mbox{arith-deg}_{r-i} {(I,F_1,\cdots,F_i)}_{\geq r-i} - \mbox{arith-deg}_{r-i} ({(I,F_1,\cdots,F_{i-1})}_{\geq r-i+2},F_i)\geq 0.$$ This shows that $\mbox{arith-deg}_{r-i}({(I,F_1,\cdots,F_{i-1})}_{\geq r-i+2},F_i)=0$. Thus the assertion is proved. \vspace{5mm} \noindent{\bf Lemma 4.3:}{\em\ Let $I$ be a homogeneous ideal of $S$. Let $r$ and $s$ be integers with $1\leq s\leq r+1$. Let $t$ be an integer with $-1\leq t\leq r+1$. Let $F_1,\cdots,F_s$ be homogeneous polynomials of $S$ with degree $(F_i)\geq 1$, $i=1,\cdots,s$, such that $F_i$ does not belong to any associated prime ideal $\sf p$ of $(I,F_1,\cdots,F_{i-1})$ with $\dim{\sf p}\geq r-i+1$, for all $i=1,\cdots,s$. Then we have $${(I_{\geq t},F_1,\cdots,F_{i-1})}_{\geq r-i+2}= {(I,F_1,\cdots,F_{i-1})}_{\geq r-i+2}$$ for $i=1,\cdots,s$.} \vspace{5mm} \noindent{\em Proof.} By Lemma 2.7, we have \begin{eqnarray*} {(I,F_1,\cdots,F_{i-1})}_{\geq r-i+2} & = & {({(I,F_1,\cdots,F_{i-2})}_{\geq r-i+3}, F_{i-1})}_{\geq r-i+2}\\ & = & \cdots=\ {(\cdots{({(I,F_1)}_{\geq r},F_2)}_{\geq r-1},\cdots,F_{i-1})}_{\geq r-i+2}\\ & = & {(\cdots{({(I_{\geq t},F_1)}_{\geq r},F_2)}_{\geq r-1},\cdots,F_{i-1})}_{\geq r-i+2}. \end{eqnarray*} On the other hand, we see $${(I_{\geq t},F_1,F_2,\cdots,F_{i-1})}_{\geq r-i+2}\subset {(\cdots{({(I_{\geq t},F_1)}_{\geq r},F_2)}_{\geq r-1},\cdots,F_{i-1})}_{\geq r-i+2}$$ and $${(I_{\geq t},F_1,F_2,\cdots,F_{i-1})}_{\geq r-i+2}\supset {(I,F_1,\cdots,F_{i-2})}_{\geq r-i+2}.$$ Therefore we have ${(I_{\geq t},F_1,\cdots,F_{i-1})}_{\geq r-i+2} = {(I,F_1,\cdots,F_{i-1})}_{\geq r-i+2}$ for all\linebreak $i=1,\cdots,s$. \vspace{5mm} \section{Some examples} The first example sheds some light on Theorem 2.1 and Corollary 2.2 in case that $F$ has degree one and is a non-zero-divisor on $S/I$. It shows that we have no equality in Corollary 2.2 even under these assumptions. \vspace{5mm} \noindent{\em Example 1:} Let $S=K[x_0,x_1,x_2,x_3,y_1,y_2,\cdots,y_r]$ be a polynomial ring, where $r$ is a non-negative integer. Take ${\sf q}=(x_0x_3-x_1x_2,x_0^2,x_1^2,x_0x_1)\subset S$, which is a primary ideal belonging to $(x_0,x_1)$ (cf. \cite{SV}, Claim 1 on page 182). We set $I={\sf q}\cap (x_0^2,x_1,x_2)$ and $F(x_0,x_1,x_2,x_3)=x_3+G(x_0,x_1,x_2)$, where $G(x_0,x_1,x_2)$ is a linear form. Then we have by (2.1) $$\mbox{arith-deg}_{r-1}(I,F) - \mbox{arith-deg}_{r-1}(I_{\geq r+1},F) =\mbox{arith-deg}_r (I).$$ Now we will show that $$\mbox{arith-deg}_{r-1}(I,F)>\mbox{arith-deg}_r (I).$$ For simplicity we assume that $G=0$, that is, $F=x_3$. Clearly, $\mbox{arith-deg}_r (I)\linebreak=1$. On the other hand, \begin{eqnarray*} (I,x_3) & = & (x_0^2,x_1^2,x_0x_1,x_0x_2x_3-x_1x_2^2) + (x_3) \\ & = & (x_0^2,x_1,x_3)\cap (x_0^2,x_1^2,x_2^2,x_0x_1,x_3). \end{eqnarray*} Hence $\mbox{arith-deg}_{r-1}(I,x_3)=2$. Also, we have \begin{eqnarray*} (I_{\geq r+1},x_3) & = & ({\sf q},x_3)\\ & = & (x_0^2,x_1^2,x_0x_1,x_1x_2,x_3)\\ & = & (x_0^2,x_1,x_3)\cap (x_0^2,x_1^2,x_2,x_3,x_0x_1). \end{eqnarray*} Hence $\mbox{arith-deg}_{r-1}(I_{\geq r+1},x_3)=1$. We note that $$\mbox{arith-deg}_{r-1}(I,x_3)>\mbox{arith-deg}_r (I)$$ even in the case that $x_3$ is a non-zero-divisor on $S/I$. \vspace{5mm} The second example shows that the bound of Theorem 3.1 is sharp and improves the result of \cite{BM}, Proposition 3.6 (see Corollary 3.2). \vspace{5mm} \noindent{\em Example 2:} Take $I=(x_0^2x_1,x_0x_2^2,x_1^2,x_2,x_2^3) =(x_0^2,x_2)\cap(x_1,x_2^2)\cap(x_0^2, x_1^2, x_0x_2^2,x_2^3)\linebreak\subset S := K[x_0,x_1,x_2]$. We get $m=5$, $P(S/I,\ell)=4$ for all $\ell\geq 4$. We consider the case $r=0$ in (3.1) and (3.2). Then we have $$4 = \deg I = \mbox{arith-deg}_0 (I)\leq P(S/I,m-1)=4 <\left(\begin{array}{c} 5+2-0-1\\2-0\end{array}\right)=15.$$ \vspace{5mm}

9602

alg-geom/9602004

en

https://arxiv.org/abs/alg-geom/9602004

alg-geom/9602004

Eriko Hironaka

Alexander Stratifications of Character Varieties

30 pages with 2 figures. (Revised Sept. 25, 1996) LaTeX2e

There is a natural stratification of the character variety of a finitely presented group coming from the jumping loci of the first cohomology of one-dimensional representations. Equations defining the jumping loci can be effectively computed using Fox calculus. In this paper, we give an exposition of Fox calculus in the language of group cohomology and in the language of finite abelian coverings of CW complexes. Work of Simpson, Arapura and others show that if $\Gamma$ is the fundamental group of a compact K\"ahler manifold, then the strata are finite unions of translated affine subtori. If follows that for K\"ahler groups the jumping loci must be defined by binomial ideals. We discuss properties of the jumping loci of general finitely presented groups and apply the ``binomial criterion" to obtain new obstructions for one-relator groups to be K\"ahler.

alg-geom

\section{Introduction} Let $X$ be homotopy equivalent to a finite CW complex and let $\Gamma$ be the fundamental group of $X$. One would like to derive geometric properties of $X$ from a finite presentation $$ \langle\ x_1,\dots,x_r\ : \ R_1,\dots,R_s\ \rangle $$ of $\Gamma$. Although the isomorphism problem is unsolvable for finite presentations, Fox calculus can be used to effectively compute invariants of $\Gamma$, up to second commutator, from the presentation. In this paper, we study a natural stratification of the character variety $\chargp{\Gamma}$ of $\Gamma$, associated to Alexander invariants, which we will call the {\it Alexander stratification.} We relate properties of the stratification to properties of unbranched coverings of $X$ and to the existence of irrational pencils on $X$ when $X$ is a compact K\"ahler manifold. Furthermore, we obtain obstructions for a group $\Gamma$ to be the fundamental group of a compact K\"ahler manifold. This paper is organized as follows. In section 2, we give properties of the Alexander stratification as an invariant of arbitrary finitely presented groups. We begin with some notation and basic definitions of Fox calculus in section 2.1. In section 2.2, we relate the Alexander stratification to jumping loci for group cohomology and in section 2.3 we translate the definitions to the language of coherent sheaves. This allows one to look at Fox calculus as a natural way to get from a presentation of a group to a presentation of a canonically associated coherent sheaf, as we show in section 2.4. Another way to view the Fox calculus is geometrically, by looking at the CW complex associated to a finitely generated group. We show how the first Betti number of finite abelian coverings can be computed in terms of the Alexander strata in section 2.5. In section 3, we relate group theoretic properties to properties of the Alexander stratification. Of special interest to us in this paper are torsion translates of connected algebraic subgroups of $\chargp{\Gamma}$, we will call them {\it rational planes}, which sit inside the Alexander strata. In section 4, we show how these rational planes relate to geometric properties of $X$. For example, in 4.1 we show that the first Betti number of finite abelian coverings of $X$ depends only on a finite number of rational planes in the Alexander strata. This follows from a theorem of Laurent on the location of torsion points on an algebraic subset of an affine torus. When $X$ is a compact K\"ahler manifold, we relate the rational planes to the existence of irrational pencils on $X$ or on a finite unbranched covering of $X$. This gives a much weaker, but simpler version of a result proved by Beauville \cite{Beau:Ann} and Arapura \cite{Ar:Higgs} which asserts that when $X$ is a compact K\"ahler manifold the first Alexander stratum is a finite union of rational planes associated to the irrational pencils of $X$ and of its finite coverings (see 4.2). Simpson in \cite{Sim:Subs} shows that if $X$ is a compact K\"ahler manifold, then the Alexander strata for $\pi_1(X)$ are all finite unions of rational planes. Since the ideals defining the Alexander strata of a finitely presented group are computable and rational planes are zero sets of binomial ideals, one can test whether a group could not be the fundamental group of K\"ahler manifold in a practical way: by computing ideals defining the Alexander strata and showing that their radicals are not binomial ideals. In section 4.3 we use the above line of reasoning to obtain an obstruction for a finitely presented group of a certain form to be K\"ahler. It gives me pleasure to thank G\'erard Gonzalez-Sprinberg and the Institut Fourier for their hospitality during June 1995 when I began work on this paper. I would also like to thank the referee for helpful remarks, including a suggestion for improving the example at the end of section 4.3. \vspace{12pt} \section{Fox Calculus and Alexander Invariants} \subsection{Notation.} For any group $\Gamma$, we denote by $\mathrm{ab}(\Gamma)$ the abelianization of $\Gamma$ and $$ \mathrm{ab} : \Gamma \rightarrow \mathrm{ab}(\Gamma) $$ the abelianization map. By $F_r$, we mean the free group on $r$ generators $x_1,\dots,x_r$. For any ring $R$, we let $\Lambda_r(R)$ be the ring of Laurent polynomials $R[t_1^{\pm 1},\dots,t_s^{\pm 1}]$. When the ring $R$ is understood, we will write $\Lambda_r$ for $\Lambda_r(R)$. Note that $\Lambda_r(R)$ is canonically isomorphic to the group ring $R[\mathrm{ab}(F_r)]$ by the map $t_i \mapsto \mathrm{ab}(x_i)$. Let $\mathrm{ab}$ also denote the map $$ \mathrm{ab} : F_r \rightarrow \Lambda_r(R) $$ given by composing the abelianization map with the injection $$ \mathrm{ab}(F_r) \rightarrow R[\mathrm{ab}(F_r)] \cong \Lambda_r(R). $$ A finite presentation of a group $\Gamma$ can be written in two ways. One is by $$ \langle \ F_r\ : \ {\cal R} \ \rangle, $$ where ${\cal R} \subset F_r$ is a finite subset. Then $\Gamma$ is isomorphic to the quotient group $$ \Gamma = {F_r}/{N({\cal R})}, $$ where $N({\cal R})$ is the normal subgroup of $F_r$ generated by ${\cal R}$. The other is by a sequence of homomorphisms $$ F_s \mapright{\psi} F_r \mapright{q} \Gamma, $$ where $q$ is onto and the normalization of the image of $\psi$ is the kernel of $q$. Let $\chargp{\Gamma}$ be the group of characters of $\Gamma$. Then $\chargp{\Gamma}$ has the structure of an algebraic group with coordinate ring ${\Bbb C}[\mathrm{ab}(\Gamma)]$. (One can verify this by noting that that the closed points in $\mathrm{Spec}({\Bbb C}[\mathrm{ab}(\Gamma)])$ correspond to homomorphisms from $\mathrm{ab}(\Gamma)$ to ${\Bbb C}^*$.) A presentation $\langle F_r \ : \ {\cal R} \rangle$ of $\Gamma$ gives an embedding of $\chargp{\Gamma}$ in $\chargp{F_r}$. The latter can be canonically identified with the affine torus $({\Bbb C}^*)^r$ as follows. To a character $\rho \in \chargp{F_r}$ we identify the point $(\rho(x_1),\dots,\rho(x_r)) \in ({\Bbb C}^*)^r$. The image of $\chargp{\Gamma}$ in $({\Bbb C}^*)^r$ is the zero set of the subset of $\Lambda_r({\Bbb C})$ defined by $$ \{\ \mathrm{ab}(R) - 1\ :\ R \in {\cal R} \ \} \subset {\Bbb C}[\mathrm{ab}(F_r)] \cong \Lambda_r({\Bbb C}). $$ Given any homomorphism, $\alpha : \Gamma' \rightarrow \Gamma$ between two finitely presented groups, let $\chargp{\alpha} : \chargp{\Gamma} \rightarrow \chargp{\Gamma'}$ be the map given by composition. Let $\alpha_{\mathrm{ab}} : \mathrm{ab}(\Gamma) \rightarrow \mathrm{ab}(\Gamma')$ be the map canonically induced by $\alpha$ and let $\chargp{\alpha}^* : {\Bbb C}[\mathrm{ab}(\Gamma')] \rightarrow {\Bbb C}[\mathrm{ab}(\Gamma)]$ be the linear extension of $\alpha_{\mathrm{ab}}$. Then it is easy to verify that $\chargp{\alpha}$ is an algebraic morphism and $\chargp{\alpha}^*$ is the corresponding map on coordinate rings: $\chargp{\alpha}^*(f) (\rho) = f(\chargp{\alpha}(\rho)),$ for $\rho \in \Gamma$ and $f \in {\Bbb C}[\mathrm{ab}(\Gamma')]$. In \cite{Fox:CalcI}, Fox develops a calculus to compute invariants, originally discovered by Alexander, of finitely presented groups. The calculus can be defined as follows: fix $r$ and, for $i=1,\dots,r$, let $$ D_i : F_r \rightarrow \Lambda_r({\Bbb Z}) $$ be the map given by \begin{eqnarray*} D_i(x_j) &=& \delta_{i,j}, \mbox{ and}\cr D_i(fg) &=& D_i(f) + \mathrm{ab}(f) D_i(g). \end{eqnarray*} The map $$ D = (D_1,\dots,D_r) : F_r \rightarrow \Lambda_r({\Bbb Z})^r $$ is called the {\it Fox derivative} and the $D_i$ are called the $i$th partials. Now let $\Gamma$ be a group with finite presentation $$ \langle F_r\ : \ {\cal R} \rangle $$ and let $q : F_r \rightarrow \Gamma$ be the quotient map. The {\it Alexander matrix} of $\Gamma$ is the $r \times s$ matrix of partials $$ M(F_r,{\cal R}) = \left [\ (\chargp{q})^* D_i(R_j)\ \right ]. $$ For any $\rho \in \chargp{\Gamma}$, let $M(F_r,{\cal R})(\rho)$ be the $r \times s$ complex matrix given by evaluation on $\rho$ and define $$ V_i(\Gamma) = \{\ \rho \in \chargp{\Gamma}\ | \ \mathrm{rank}\ M(F_r,{\cal R})(\rho) < r-i\ \}. $$ These are subvarieties of $\chargp{\Gamma}$ defined by the ideals of $(r-i) \times (r-i)$ minors of $M(F_r, {\cal R})$. We will call the nested sequence of algebraic subsets $$ \chargp{\Gamma} \supset V_1(\Gamma) \supset \dots \supset V_r(\Gamma) $$ the {\it Alexander stratification} of $\Gamma$. One can check that the Tietze transformations on group presentations give different Alexander matrices, but don't effect the $V_i(\Gamma)$. Hence the Alexander stratification is independent of the presentation. Later in section 2.4 (Corollary 2.4.3) we will prove the independence by other methods. \subsection{Jumping loci for group cohomology.} For any group $\Gamma$, let $C^1(\Gamma,\rho)$ be the set of {\it crossed homomorphisms} $f : \Gamma \rightarrow {\Bbb C}$ satisfying $$ f(g_1g_2) = f(g_1) + \rho(g_1) f(g_2). $$ Then $C^1(\Gamma,\rho)$ is a vector space over ${\Bbb C}$. Note that for any $f \in C^1(\Gamma,\rho)$, $f(1)=0$. Here are two elementary lemmas, which will be useful throughout the paper. \begin{lemma} Let $\alpha : \Gamma' \rightarrow \Gamma$ be a homomorphism of groups and let $\rho \in \chargp{\Gamma}$. Then right composition by $\alpha$ defines a vector space homomorphism $$ T_\alpha : C^1(\Gamma,\rho) \rightarrow C^1(\Gamma',\chargp{\alpha}(\rho)). $$ \end{lemma} \proof Take any $f \in C^1(\Gamma,\rho)$. Then, for $g_1,g_2 \in \chargp{\Gamma'}$, \begin{eqnarray*} T_\alpha(f)(g_1g_2) &=& f(\alpha(g_1g_2))\cr &=&f(\alpha(g_1)\alpha(g_2))\cr &=&f(\alpha(g_1))+\rho(\alpha(g_1))f(\alpha(g_2)) \cr &=&T_\alpha(f)(g_1) + \chargp{\alpha}(\rho)(g_1)(T_\alpha(f))(g_2). \end{eqnarray*} Thus, $T_\alpha(f)$ is in $C^1(\Gamma',\chargp{\alpha}(\rho))$. \qed \begin{lemma} Let $g,x \in \Gamma$ and let $f \in C^1(\Gamma,\rho)$, for any $\rho \in \chargp{\Gamma}$. Then $$ f(gxg^{-1}) = f(g)(1 - \rho(x)) + \rho(g) f(x). $$ \end{lemma} \proof This statement is easy to check by expanding the left hand side and noting that $$ f(g^{-1}) = - \rho(g)^{-1} f(g), $$ for any $g \in \Gamma$.\qed Let $$ U_i(\Gamma) = \{\ \rho \in \chargp{\Gamma} \ | \ \dim C^1(\Gamma,\rho) > i\ \}. $$ This defines a nested sequence $$ \chargp{\Gamma} \supset U_0(\Gamma) \supset U_1(\Gamma) \supset \dots. $$ In section 2.4 (Corollary 2.4.3), we will show that $U_i(\Gamma) = V_i(\Gamma)$, for all $i \in {\Bbb N}$. Define, for $\rho \in \chargp{\Gamma}$, $$ B_1(\Gamma,\rho) = \{\ f : \Gamma \rightarrow {\Bbb C} \ | \ f(g) = (\rho(g) - 1)c \ \mbox{for some constant $c \in {\Bbb C}$} \ \}. $$ Then $B_1(\Gamma,\rho)$ is a subspace of $C^1(\Gamma,\rho)$. Define $$ \mathrm{H}^1(\Gamma,\rho) = {C^1(\Gamma,\rho)}/{B^1(\Gamma,\rho)}. $$ This is the {\it first cohomology group of $\Gamma$ with respect to the representation $\rho$}. Let $$ W_i(\Gamma) = \{\ \rho\in \chargp{\Gamma}\ |\ \dim \mathrm{H}^1(\Gamma,\rho) \ge i\ \}, $$ for $i\in {\Bbb Z}_+$. We will call the $W_i(\Gamma)$ the {\it jumping loci} for the first cohomology of $\Gamma$. This defines a nested sequence $$ \chargp{\Gamma} = W_0(\Gamma) \supset W_1(\Gamma) \supset \dots. $$ If $\rho = \chargp{1}$ is the identity character in $\chargp{\Gamma}$, then $\rho(g) = 1$, for all $g \in \Gamma$. Thus, $B^1(\Gamma,\rho) = \{0\}$. Also, $C^1(\Gamma,\chargp{1})$ is the set of all homomorphisms from $\Gamma$ to ${\Bbb C}$ and is isomorphic to the abelianization of $\Gamma$ tensored with ${\Bbb C}$. Thus, $$ \dim \mathrm{H}^1(\Gamma,\chargp{1}) = \dim C^1(\Gamma,\chargp{1}) = d, $$ where $d$ is the rank of the abelianization of $\Gamma$. If $\rho \neq \chargp{1}$, then $B^1(\Gamma,\rho)$ is isomorphic to the field of constants ${\Bbb C}$, so $$ \dim C^1(\Gamma,\rho) = \dim \mathrm{H}^1(\Gamma,\rho) + 1. $$ We have thus shown the following. \begin{lemma} The jumping loci $W_i(\Gamma)$ and the nested sequence $U_i(\Gamma)$ are related as follows: \begin{eqnarray*} W_i(\Gamma) = U_i(\Gamma) &\qquad& \mbox{for $i\neq d$}\\ W_i(\Gamma) = U_i(\Gamma) \cup \{\chargp{1}\} &\qquad&\mbox{for $i = d$.} \end{eqnarray*} \end{lemma} \heading{Remark.} The jumping loci could also have been defined using the cohomology of local systems. Let $X$ be a topological space homotopy equivalent to a finite CW complex with $\pi_1(X) = \Gamma$. Let $\widetilde{X} \rightarrow X$ be the universal cover of $X$. Then for each $\rho \in \chargp{\Gamma}$, each $g \in \Gamma$ acts on $\widetilde{X} \times {\Bbb C}$ by its action as covering automorphism on $\widetilde{X}$ and by multiplication by $\rho(g)$ on ${\Bbb C}$. This defines a local system ${\Bbb C}_\rho \rightarrow X$ over $X$. Then $W_i(\Gamma)$ is the jumping loci for the rank of the cohomology group $\mathrm{H}^1(X,{\Bbb C}_\rho)$ with coefficients in the local system ${\Bbb C}_\rho$. \subsection{Coherent sheaves over the character variety.} Let $\Gamma$ be a finitely presented group and let $\dual{C^1(\Gamma,\rho)}$ be the dual space of $C^1(\Gamma,\rho)$. We will construct sheaves ${\cal C}^1(\Gamma)$ and $\dual{{\cal C}^1(\Gamma)}$ over $\chargp{\Gamma}$ whose stalks are $C^1(\Gamma,\rho)$ and $\dual{C^1(\Gamma,\rho)}$, respectively. Then, the jumping loci $U_i(\Gamma)$ defined in the previous section, are just the jumping loci for the dimensions of stalks of ${\cal C}^1(\Gamma)$ and $\dual{{\cal C}^1(\Gamma)}$. This just gives a translation of the previous section into the language of sheaves, but using this language we will show that a presentation for $\Gamma$ induces a presentation of $\dual{{\cal C}^1(\Gamma)}$ as a coherent sheaf such that the presentation map on sheaves is essentially the Alexander matrix. We start by constructing ${\cal C}^1(F_r)$ for free groups. \begin{lemma} For any $r$ and $\rho \in \chargp{F_r}$, $C^1(F_r,\rho)$ is isomorphic to ${\Bbb C}^r$, and has a basis given by $\langle x_i \rangle_\rho$, where $$ \langle x_i \rangle_\rho (x_j) = \delta_{i,j}. $$ \end{lemma} \proof By the product rule, elements of $C^1(F_r,\rho)$ only depend on what happens to the generators of $F_r$. Since there are no relations on $F_r$, any choice of values on the basis elements determines an element of $C^1(F_r,\rho)$. \qed Let $$ E_r = \bigcup_{\rho \in \chargp{F_r}} C^1(F_r,\rho) $$ be the trivial ${\Bbb C}^r$-vector bundle over $\chargp{F_r}$ whose fiber over $\rho \in \chargp{F_r}$ is $C^1(F_r,\rho)$. For each generator $x_i$ of $F_r$, define $$ \langle x_i \rangle : \chargp{F_r} \rightarrow \bigcup_{\rho \in \chargp{F_r}} C^1(F_r,\rho), $$ by $\langle x_i \rangle (\rho) = \langle x_i \rangle_\rho$. The maps $\langle x_1 \rangle, \dots, \langle x_r \rangle$ are global sections of $E_r$ over $\chargp{F_r}$. Let ${\cal C}^1(F_r)$ be the corresponding sheaf of sections of the bundle $E_r \rightarrow \chargp{F_r}$. The module $M_r$ of global sections of ${\cal C}^1(F_r)$ is a free $\Lambda_r$-module of rank $r$, generated by $\langle x_1 \rangle, \dots, \langle x_r \rangle$, and ${\cal C}^1(F_r)$ is the sheaf associated to $M_r$ (in the sense of \cite{Hart:AG}, p.110). Fix a presentation $$ F_s \mapright{\psi} F_r \mapright{q} \Gamma, $$ of $\Gamma$. This induces maps on character varieties $$ \cd { &\Gamma&\mapright{\chargp{q}} &\chargp{F_r}&\mapright{\chargp{\psi}} &\chargp{F_s}\cr &&&\Vert&&\Vert\cr &&&({\Bbb C}^*)^r&&({\Bbb C}^*)^s. } $$ Let ${\cal C}^1(F_r)_\Gamma$ and ${\cal C}^1(F_s)_\Gamma$ be the pullbacks of ${\cal C}^1(F_r)$ and ${\cal C}^1(F_s)$ over $\chargp{\Gamma}$. These are the sheafs associated to the modules: $$ M_r(\Gamma) = M_r \otimes_{{\Bbb C}[\mathrm{ab}(F_r)]} {\Bbb C}[\mathrm{ab}(\Gamma)] \cong {\Bbb C}[\mathrm{ab}(\Gamma)]^r $$ and $$ M_s(\Gamma) = M_s \otimes_{{\Bbb C}[\mathrm{ab}(F_s)]} {\Bbb C}[\mathrm{ab}(\Gamma)] \cong {\Bbb C}[\mathrm{ab}(\Gamma)]^s, $$ respectively. Let $$ {\cal T}_\psi : {\cal C}^1(F_r)_\Gamma \rightarrow {\cal C}^1(F_s)_\Gamma $$ be the homomorphism of sheaves defined by composing sections by $\psi$. For any $\rho \in \chargp{\Gamma}$, the stalk of ${\cal C}^1(F_r)_\Gamma$ over $\rho$ is given by $C^1(F_r,\chargp{q}(\rho))$. Since $q \circ \psi$ is the trivial map, the stalk of ${\cal C}^1(F_s)_\Gamma$ over $\rho$ is given by $C^1(F_s,\chargp{1})$. For any $\rho \in \chargp{\Gamma}$, the map on stalks determined by ${\cal T}_\psi$ is the map $$ ({\cal T}_\psi)_\rho : C^1(F_r,\chargp{q}(\rho)) \rightarrow C^1(F_s,\chargp{1}) $$ defined by $({\cal T}_\psi)_\rho(f) = f\circ\psi$. Let $M_\Gamma(F_r,{\cal R})$ be the sub ${\Bbb C}[\mathrm{ab}(\Gamma)]$-module of $M_r(\Gamma)$ given by the kernel of the map \begin{eqnarray*} M_r(\Gamma) &\rightarrow& M_s(\Gamma)\\ f\otimes g &\mapsto& (f \circ \psi) \otimes g \end{eqnarray*} Let ${\cal C}^1(\Gamma)$ be the kernel of ${\cal T}_\psi$. That is, ${\cal C}^1(\Gamma)$ is the sheaf associated to $M_{\Gamma}(F_r,{\cal R})$. \begin{lemma} The stalk of ${\cal C}^1(\Gamma)$ over $\rho \in \chargp{\Gamma}$ is isomorphic to $C^1(\Gamma,\rho)$. \end{lemma} \proof We need to show that the kernel of $({\cal T}_\psi)_\rho$ is isomorphic to $C^1(\Gamma,\rho)$. Let $$ (T_q)_\rho : C^1(\Gamma,\rho) \rightarrow C^1(F_r,\chargp{q}(\rho)) $$ be the homomorphism given by composing with $q$ as in Lemma 2.2.1. Since $q$ is surjective, it follows that $(T_q)_\rho$ is injective. The composition $T_\psi \circ (T_q)_\rho$ is right composition by $\psi \circ q$, which is trivial, so the image of $(T_q)_\rho$ lies in the kernel of $\Psi$. Now suppose, $f \in C^1(F_r,\chargp{q}(\rho))$ is in the kernel of $\Psi$. Then $f$ is trivial on $\psi(F_s)$. Since $\chargp{q}(\rho)$ is trivial on $\psi(F_s)$, Lemma 2.2.2 implies that $f$ is trivial on the normalization of $\psi(F_s)$ in $F_r$. Thus, $f$ induces a map from $\Gamma$ to ${\Bbb C}$ which is twisted by $\rho$.\qed \begin{lemma} Let $\alpha : \Gamma' \rightarrow \Gamma$ be a homomorphism of groups and let $\chargp{\alpha} : \chargp{\Gamma} \rightarrow \chargp{\Gamma'}$ be the corresponding morphism on character varieties. Let ${\cal C}(\Gamma)$ and ${\cal C}(\Gamma')$ be the sheaves associated to $\Gamma$ and $\Gamma'$ and let ${\cal C}(\Gamma')_\Gamma$ be the pullback of ${\cal C}(\Gamma')$ over $\chargp\Gamma$. Then the map ${\cal T}_\alpha : {\cal C}(\Gamma) \rightarrow {\cal C}(\Gamma')$ defined by composing sections by $\alpha$ is a homomorphism of sheaves. \end{lemma} \proof The statement follows from Lemma 2.2.1.\qed \begin{corollary} There are exact sequences of sheaves $$ 0 \rightarrow {\cal C}^1(\Gamma)\ \mapright{}\ {\cal C}^1(F_r)_\Gamma \ \mapright{{\cal T}_\psi}\ {\cal C}^1(F_s)_\Gamma $$ and $$ \dual{{\cal C}^1(F_s)}_\Gamma\ \mapright{\dual{{\cal T}_\psi}}\ \dual{{\cal C}^1(F_r)}_\Gamma \rightarrow \dual{{\cal C}^1(\Gamma)} \rightarrow 0. $$ \end{corollary} We have seen that the modules of holomorphic sections of ${{\cal C}^1(F_r)}$ and ${\cal C}^1(F_s)$ are freely generated over ${\cal C}[\mathrm{ab}(\Gamma)]$ of ranks $r$ and $s$, respectively. Similarly, the dual sheaves $\dual{{\cal C}^1(F_r)}$ and $\dual{{\cal C}^1(F_s)}$ are freely generated. This gives $\dual{{\cal C}^1(\Gamma)}$ the structure of a coherent sheaf. In section 2.4 we will show that the Alexander Matrix gives a presentation for global sections of $\dual{{\cal C}^1(\Gamma)}$. \subsection{Jumping loci and the Alexander stratification.} In this section, we show that for a given group $\Gamma$, the jumping loci $U_i(\Gamma)$ defined in 2.2 is the same as the Alexander stratification $V_i(\Gamma)$. For any group $\Gamma$, there is an exact bilinear pairing $$ ({\Bbb C}\Gamma)_\rho \times C^1(\Gamma,\rho) \rightarrow {\Bbb C} $$ and the pairing is given by $$ ({\Bbb C}\Gamma)_\rho = {\Bbb C}\Gamma/{\{g_1g_2 - g_1 - \rho(g_1)g_2 | g_1,g_2 \in\Gamma\}}, $$ where $$ [g,f] = f(g). $$ The pairing determines a ${\Bbb C}$-linear map $$ \Phi[\Gamma]_\rho : ({\Bbb C}\Gamma)_\rho \rightarrow \dual{C^1(\Gamma,\rho)}, $$ where, for $g \in ({\Bbb C}\Gamma)_\rho$ and $f \in C^1(\Gamma,\rho)$, $\Phi[\Gamma]_\rho(f) (g) = [g,f] = f(g)$. \begin{lemma} Let $\alpha : \Gamma' \rightarrow \Gamma$ be a group homomorphism. For each $\rho \in \chargp{\Gamma}$, we have a commutative diagram $$ \cd { &({\Bbb C}\Gamma')_{\chargp{\alpha}(\rho)} &\mapright{\Phi[\Gamma']_{\chargp{\alpha}(\rho)}} &\dual{C^1(\Gamma',\chargp{\alpha}(\rho))}\cr &\mapdown{\alpha}&&\mapdown{\dual T_\alpha}\cr &({\Bbb C}\Gamma)_\rho &\mapright{\Phi[\Gamma]_\rho}&\dual{C^1(\Gamma,\rho)} } $$ where $\dual T_\alpha$ is the dual map to $T_\alpha : C^1(\Gamma,\rho) \rightarrow C^1(\Gamma', \chargp{\alpha}(\rho))$. \end{lemma} \proof For $g \in ({\Bbb C}\Gamma')_{\chargp{\alpha}}(\rho)$ and $f \in C^1(\Gamma,\rho)$, the pairing $[,]$ gives $$ [g,T_\alpha(f)] = {T_\alpha}(f)(g) = f(\alpha(g)) = [\alpha(g),f]. $$ \qed Let $\dual{M_r}$ be the global holomorphic sections of $\dual{{\cal C}^1(F_r)}$. Define $$ \Phi : {\Bbb C} F_r \rightarrow \dual{M_r} $$ by \begin{eqnarray*} \Phi(x_i) &=& \dual{\langle x_i \rangle}\cr \Phi(g_1g_2) &=& \Phi(g_1) + \mathrm{ab}(g_1)\Phi(g_2)\qquad \mbox{for}\ g_1,g_2 \in F_r, \end{eqnarray*} where $$ \dual{\langle x_i \rangle}_\rho : C^1(F_r,\rho) \rightarrow {\Bbb C} $$ is given by $$ \dual{\langle x_i \rangle}_\rho(\langle x_j \rangle_\rho) = \delta_{i,j}. $$ Define, for any $\rho \in \chargp{F_r}$ and $g \in {\Bbb C} F_r$, with image $g_\rho$ in $({\Bbb C} F_r)_\rho$, $\Phi_\rho(g_\rho) = \Phi_\rho(g)(\rho) \in \dual{C^1(F_r,\rho)}$, where $$ \Phi_\rho(g)(\rho)(f) = f(g) $$ for all $f \in C^1(F_r,\rho)$. Then $\Psi_\rho = \Psi[F_r]_{\rho}$. Since $\dual{M_r}$ is generated freely by the global sections $$ \dual{\langle x_1 \rangle},\dots,\dual{\langle x_r \rangle} $$ as a $\Lambda_r({\Bbb C})$-module, we can identify $\dual{M_r}$ with $\Lambda_r({\Bbb C})^r$. Thus, the map $\Phi$ is the extension of the Fox derivative $$ D : F_r \rightarrow \Lambda_r({\Bbb Z})^r $$ in the obvious way to ${\Bbb C} [F_r] \rightarrow \Lambda_r({\Bbb C})^r$. Let ${\cal D}_r({\cal R})$ be the sub $\Lambda_r$-module of $\Lambda_r({\Bbb C})^r$ spanned by $\Phi({\cal R})$. For $\rho \in \chargp{\Gamma}$, let ${\cal D}_r({\cal R})(\rho)$ be the subspace of ${\Bbb C}^r$ spanned by the vectors obtained by evaluating the $r$-tuples of functions in $\Phi({\cal R})$ at $\rho$. \begin{lemma} Let $\langle F_r : {\cal R} \rangle$ be a presentation for $\Gamma$. For each $\rho \in \chargp{\Gamma}$, the dimension of $C^1(\Gamma,\rho)$ is given by $$ r - \dim({\cal D}_r({\cal R})(\rho)). $$ \end{lemma} \proof Let $$ F_s \mapright{\psi} F_r \mapright{q} \Gamma $$ be the sequence of maps determined by the presentation. Then, for each $\rho \in \chargp{\Gamma}$, by Corollary 2.3.4, there is an exact sequence $$ \dual{C^1(F_s,\chargp{1})}\ \mapright{\dual{{\cal T}_\psi}} \ \dual{C^1(F_r,\chargp{q}(\rho))} \ \mapright{\dual{{\cal T}_q}} \ \dual{C^1(\Gamma,\rho)} \mapright{} 0. $$ By Lemma 2.4.1, the following diagram commutes: $$ \cd{ &({\Bbb C} F_s)_{\chargp{1}} &\mapright{\Phi[F_s]_{\chargp{1}}} &\dual{C^1(F_s,\chargp{1})}\cr &\mapdown{\psi} &&\mapdown{\dual{T_\psi}}\cr &({\Bbb C} F_r)_{\chargp{q}(\rho)} &\mapright{\Phi[F_r]_{\chargp{q}(\rho)}} &\dual{C^1(F_r,\chargp{q}(\rho))}\cr &\mapdown{q} &&\mapdown{\dual{T_q}}\cr &({\Bbb C}\Gamma) &\mapright{\Phi[\Gamma]_\rho} &\dual{C^1(\Gamma,\rho)} } $$ Thus, \begin{eqnarray*} \dim C^1(\Gamma,\rho) &=& \dim C^1(F_r,\chargp{q}(\rho)) - \dim (\mbox{image} (\dual{{\cal T}_\psi})). \end{eqnarray*} Since $\Phi[F_s]_{\chargp{1}}$ is onto \begin{eqnarray*} \mbox{image}(\dual{{\cal T}_\psi}) &=& \mbox{image}(\Phi[F_s]_{\chargp{1}} \circ \dual{{\cal T}_\psi})\cr &=& \mbox{image}(\Phi[F_r]_{\chargp{q}(\rho)} \circ \psi) \end{eqnarray*} For any $\rho$, $C^1(F_r,\chargp{q}(\rho))$ is isomorphic to ${\Bbb C}^r$. Putting this together, we have \begin{eqnarray*} \dim C^1(\Gamma,\rho) &=& r - \dim \Phi[F_r]_{\chargp{q}(\rho)}({\cal R})\cr &=& r - \dim {\cal D}_r({\cal R})(\rho). \end{eqnarray*} \qed \begin{corollary} For any finitely presented group $\Gamma$, the jumping loci $U_i (\Gamma)$ for the cohomology of $\Gamma$ is the same as the Alexander stratification $V_i(\Gamma)$. \end{corollary} \subsection{Abelian coverings of finite CW complexes.} In this section we explain the Fox calculus and Alexander stratification in terms of finite abelian coverings of a finite CW complex. The relations between homology of coverings of a $K(\Gamma,1)$ and the group cohomology of $\Gamma$ are well known (see, for example, \cite{Bro:Coh}). The results of this section come from looking at Fox calculus from this point of view. Let $X$ be a finite CW complex and let $\Gamma = \pi_1(X)$. Suppose $\Gamma$ has presentation given by $\langle x_1,\dots,x_r : R_1,\dots,R_s \rangle$. Then $X$ is homotopy equivalent to a CW complex with cell decomposition whose tail end is given by $$ \dots \supset \Sigma_2 \supset \Sigma_1 \supset \Sigma_0, $$ where $\Sigma_0$ consists of a point $P$, $\Sigma_1$ is a bouquet of $r$ oriented circles $S^1$ joined at $P$. Identify $F$ with $\pi_1(\Sigma_1)$ so that each $x_i$ is the positively oriented loop around the $i$-th circle. Each $R_i$ defines a homotopy class of map from $S^1$ to $\Sigma_1$. The 2-skeleton $\Sigma_2$ is the union of $s$ disks attached along their boundaries to $\Sigma_1$ by maps in the homotopy class defined by $R_1,\dots,R_s$. Let $\alpha : \Gamma \rightarrow G$ be any epimorphism of $\Gamma$ to a finite abelian group $G$. Let $\tau_\alpha : X_\alpha \rightarrow X$ be the regular unbranched covering determined by $\alpha$ with $G$ acting as group of covering automorphisms. Our aim is to show how Fox calculus can be used to compute the first Betti number of $X_\alpha$. Choose a basepoint $1P \in \tau_\alpha^{-1}(P)$. For each $i$-chain $\sigma \in \Sigma_i$ and $g\in G$, let $g \sigma$ be the the component of its preimage which passes through $gP$. For each generating $i$-cell in $\Sigma_i$, there are exactly $G$ copies of isomorphic cells in its preimage. Thus $X_\alpha$ has a cell decomposition $$ \dots \supset \Sigma_{2,\alpha} \supset \Sigma_{1,\alpha} \supset \Sigma_{0,\alpha}, $$ where the $i$-cells in $\Sigma_{i,\alpha}$ are given by the set $\{g \sigma \ :\ g\in G,\sigma \mbox{ an $i$-cell in $\Sigma_i$}\}$. With this notation if $\sigma$ attaches to $\Sigma_{i-1,\alpha}$ according to the homotopy class of mapping $f: \partial\sigma \rightarrow \Sigma_{i-1}$, where $\partial\sigma$ is the boundary of $\sigma$, then $g\sigma$ attaches to $\Sigma_{i-1,\alpha}$ by the map $f' : \partial g\sigma \rightarrow \Sigma_{i-1,\alpha}$ lifting $f$ at the basepoint $gP$. Let $C_i$ be the $i$-chains on $X$ and let $C_{i,\alpha}$ be the $i$-chains on $X_\alpha$. Then there is a commutative diagram for the chain complexes for $X$ and $X_\alpha$: $$ \cd { &\dots&\mapright{}&C_{2,\alpha} &\mapright{\delta_{2,\alpha}}&C_{1,\alpha} &\mapright{\delta_{1,\alpha}}&C_{0,\alpha}&\mapright{\epsilon} &{\Bbb Z}\cr &&&\mapdown{\tau_\alpha}&&\mapdown{\rho_\alpha} &&\mapdown{\tau_\alpha}&&\cr &\dots&\mapright{}&C_2&\mapright{\delta_2}&C_1 &\mapright{\delta_1}&C_0,&& } $$ where the map $\epsilon$ is the augmentation map $$ \epsilon(\sum_{g \in G} (a_g g)) = \sum_{g\in G} a_g. $$ Let $\langle x_1 \rangle_\alpha,\dots, \langle x_r \rangle_\alpha$ be the elements of $C_{1,\alpha}$ given by lifting $x_1,\dots,x_r$, considered as loops on $\Sigma_1$, to 1-chains on $\Sigma_{1,\alpha}$ with basepoint $1P$. Then $C_{1,\alpha}$ can be identified with ${\Bbb C}[G]^r$, with basis $\langle x_1\rangle, \dots, \langle x_r \rangle$ and $C_{0,\alpha}$ can be identified with ${\Bbb C}[G]$, where each $g \in G$ corresponds to $gP$. The above commutative diagram can be rewritten as \begin{eqnarray} \cd { &\dots&\mapright{}& {\Bbb Z}[G]^s &\mapright{\delta_{2,\alpha}}&{\Bbb Z}[G]^r &\mapright{\delta_{1,\alpha}}&{\Bbb Z}[G] &\mapright{\epsilon}&{{\Bbb Z}} \cr &&&\mapdown{\tau_\alpha}&&\mapdown{\rho_\alpha} &&\mapdown{\tau_\alpha}&& \cr &\dots&\mapright{}&{{\Bbb Z}}^s&\mapright{\delta_2} &{{\Bbb Z}}^r &\mapright{\delta_1}&{{\Bbb Z}.}&& } \end{eqnarray} For any finite set $S$, let $|S|$ denote its order. The map $\epsilon$ is surjective, so we have the formula \begin{equation} \begin{array}{rl} b_1(X_\alpha) &= \mathrm{nullity} (\delta_{1,\alpha}) - \mathrm{rank} (\delta_{2,\alpha})\\ &= (r-1)|G| + 1 - \mathrm{rank}(\delta_{2,\alpha}), \end{array} \end{equation} where $b_1(X_\alpha)$ is the rank of ${\ker{\delta_{1,\alpha}}}/ {\mathrm{image} (\delta_{2,\alpha})}$ and is the rank of $H_1(X_\alpha;{\Bbb Z})$. We will rewrite this formula in terms of the Alexander stratification. \begin{lemma} The map $\delta_{1,\alpha}$ is given by $$ \delta_{1,\alpha} (\sum_{i=1}^r f_i \langle x_i \rangle_\alpha) = \sum_{i=1}^r f_i\chargp{q_\alpha}^*{(t_i-1)}. $$ \end{lemma} \proof It's enough to notice that the lift of $x_i$ to $C_{1,\alpha}$ at the basepoint $1P$ has end point $\chargp{q_\alpha}^*(t_i)P$. \qed We will now relate the map $\delta_{2,\alpha}$ with the Fox derivative. Recall that $\Sigma_1$ equals a bouquet of $r$ circles $\wedge_r S^1$. Let $\tau : {\cal L}_r \rightarrow \wedge_r S^1$ be the universal abelian covering. Then ${\cal L}_r$ is a lattice on $r$ generators with $\mathrm{ab}(F_r)$ acting as covering automorphisms. The vertices of the lattice can be identified with $\mathrm{ab}(F_r)$. Let $K_\alpha = \ker(\alpha \circ q) \subset F_r$ and let $\widetilde{K_\alpha}$ be its image in $\mathrm{ab}(F_r)$. Then $\Sigma_{1,\alpha} = {{\cal L}_r}/{\widetilde{K_\alpha}}$ and we have a commutative diagram $$ \cd { &{\cal L}_r &\mapright{\eta_\alpha}&\Sigma_{1,\alpha}\cr &\mapdown{\tau}&&\mapdown{\tau_\alpha}\cr &\wedge_r S^1&=&\Sigma_1 } $$ where $\eta_\alpha : {\cal L}_r \rightarrow \Sigma_{1,\alpha}$ is the quotient map. Let $(\eta_\alpha)_* : C_1({\cal L}_r) \rightarrow C_1(\Sigma_{1,\alpha})$ be the induced map on one chains. Then identifying $C_1({\cal L}_r)$ with ${\Bbb Z}[\mathrm{ab}(F_r)]^r$ and $C_1(\Sigma_{1,\alpha})$ with ${\Bbb Z}[G]^r$, we have $(\eta_\alpha)_* = {(\chargp{q_\alpha}^*)}^r$. Choose $1\tilde P \in \tau^{-1}(P)$. Let $C_1({\cal L}_r)$ be the 1-chains on ${\cal L}_r$. Let $\langle x_1 \rangle,\dots, \langle x_r \rangle$ be the lifts of $x_1,\dots,x_r$ to $C_1({\cal L}_r)$ at the base point $1\tilde P$. This determines an identification of $C_1({\cal L}_r)$ with $\Lambda({\Bbb Z})^r$ and determines a choice of homotopy lifting map $ \ell : \pi_1(\Sigma_1) \rightarrow C_1({\cal L}_r)$. \begin{lemma} The identifications $F_r = \pi_1(\Sigma_1)$ and $\Lambda_r({\Bbb Z}) = C_1({\cal L}_r)$, make the following diagram commute $$ \cd { &\pi_1(\Sigma_1) &\mapright{\ell} &C_1({\cal L}_r)\cr &\Vert &&\Vert\cr &F_r &\mapright{D} &\Lambda_r({\Bbb Z}). } $$ \end{lemma} \proof By definition, both maps $\ell$ and $D$ send $x_i$ to $\langle x_i \rangle$, for $i=1,\dots,r$. We have left to check products. Let $f, g \in F_r$, be thought of as loops on $\wedge_r S^1$. Then the lift of $f$ has endpoint $\mathrm{ab}(f)$. Therefore, $\ell(fg) = \ell(f) + \mathrm{ab}(f)\ell(g)$. Since these rules are the same as those for the Fox derivative map, the maps must be the same. \qed \begin{corollary} Let $\Gamma$ be a finitely presented group with presentation $\langle F_r : {\cal R} \rangle$. Let $\alpha : \Gamma \rightarrow G$ be an epimorphism to a finite abelian group $G$. Let $M(F_r,{\cal R})_\alpha$ be the matrix $M(F_r,{\cal R})$ with $\chargp{q}_\alpha^*$ applied to all the entries. Then $$ \cd { &C_{2,\alpha} &\mapright{\delta_{2,\alpha}} &C_{1,\alpha} \cr &\Vert &&\Vert\cr &{\Bbb Z}[G]^s &\mapright{M(F_r,{\cal R})_\alpha} &{\Bbb Z}[G]^r. } $$ \end{corollary} \proof Let $\sigma_1,\dots,\sigma_s$ be the $s$ disks generating the $2$-cells $C_2$. For each $i=1,\dots,s$ and $g \in G$, let $g\sigma_i$ denote the lift of $\sigma_i$ at $gP$. Let $R_1,\dots,R_s$ be the elements of ${\cal R}$. By Lemma 2.5.2, the boundary $\partial\sigma_i $ maps to $D(R_i)$ in $C_1({\cal L}_r)$. Thus, the boundary of $g \sigma_i$ equals $g D(R_i)$, and for $g_1,\dots,g_s \in {\Bbb Z}[G]$, $$ \delta_{\alpha,2}(\sum_{i=1}^s g_i \sigma_i) = \sum_{i=1}^s g_i D(R_i). $$ This is the same as the application of $M(F_r,{\cal R})_\alpha$ on the $s$-tuple $(g_1,\dots,g_s)$. \qed We now give a formula for the first Betti number $b_1(X_\alpha)$ in terms of the Alexander stratification in the case where $G$ is finite. Tensor the top row in diagram (1) by ${\Bbb C}$. Then the action of $G$ on ${\Bbb C}[G]$ diagonalizes to get $$ {\Bbb C}[G] \cong \bigoplus_{\rho \in \chargp{G}} {\Bbb C}[G]_\rho, $$ where ${\Bbb C}[G]_\rho$ is a one-dimensional subspace of ${\Bbb C}[G]$ and $g \in G$ acts on ${\Bbb C}[G]_\rho$ by multiplication by $\rho(g)$. The top row of diagram (1) becomes \begin{eqnarray*} \bigoplus_{\rho\in\chargp{G}} {\Bbb C}[G]_\rho^s \ \mapright{\delta_{\alpha,2}} \ \bigoplus_{\rho \in \chargp{G}} {\Bbb C}[G]_\rho^r \ \mapright{\delta_{\alpha,1}} \ \bigoplus_{\rho \in \chargp{G}} {\Bbb C}[G]_\rho \ \mapright{\epsilon} \ {\Bbb C}. \end{eqnarray*} The map $\delta_{\alpha,2}$ considered as a matrix $M(F_r,{\cal R})_\alpha$, as in Lemma 2.5.3, decomposes into blocks $$ M(F_r,{\cal R})_\alpha = \bigoplus_{\rho \in \chargp{G}} M(F_r,{\cal R})_\alpha(\rho), $$ where, if $M(F_r,{\cal R})_\alpha = [f_{i,j}]$, then $M(F_r,{\cal R})_\alpha(\rho) = [f_{i,j}(\rho)]$. We thus have the following formula for the rank of $M(F_r,{\cal R})_\alpha$: \begin{eqnarray} \mathrm{rank} (M(F_r,{\cal R})_\alpha) &=& \sum_{\rho\in\chargp{G}} \mathrm{rank}(M(F_r,{\cal R})_\alpha(\rho)). \end{eqnarray} Recall that the Alexander stratification $V_i(\Gamma)$ was defined to be the zero set in $\chargp{\Gamma}$ of the $(r-i) \times (r-i)$ ideals of $M(F_r,{\cal R})$. For any $\rho \in \chargp{G}$, $M(F_r,{\cal R})_\alpha(\rho) = M(F_r,{\cal R})(\chargp{\alpha}(\rho)) = M(F_r,{\cal R})(\chargp{q_\alpha}(\rho))$, since $\widetilde{\alpha}(f) (\rho) = f(\chargp{\alpha}(\rho))$ and $\widetilde{q_\alpha}(f)(\rho) = f(\chargp{q_\alpha}(\rho))$. We thus have the following Lemma. \begin{lemma} For $\rho \in \chargp{G}$, $\chargp{\alpha}(\rho) \in V_i(\Gamma)$ if and only if $\mathrm{rank} (M(F_r,{\cal R})_{\alpha}(\rho)) < r-i$. \end{lemma} For each $i=0,\dots,r-1$, let $\chi_{V_i(\Gamma)}$ be the indicator function for $V_i(\Gamma)$. Then, for $\rho \in \chargp{G}$, we have \begin{eqnarray} \mathrm{rank} (M(F_r,{\cal R})_{\alpha}(\rho)) = r - \sum_{i=0}^{r-1} \chi_{V_i(\chargp{\Gamma})}(\chargp{\alpha}(\rho)). \end{eqnarray} \begin{lemma} For the special character $\widehat{1}$, $$ \mathrm{rank} (M(F_r,{\cal R})_\alpha(\chargp{1})) = r - b_1(X) $$ and $\mathrm{rank} (M(F_r,{\cal R})_\alpha(\chargp{1})) = r$ if and only if $\chargp{\Gamma} = \{\chargp{1}\}$ and $\Gamma$ has no nontrivial abelian quotients. \end{lemma} \proof The group $G$ acts trivially on $\Lambda_{\alpha, \widehat{1}}$. Thus, in the commutative diagram $$ \cd { &\Lambda_{\alpha,\widehat{1}}^s &\mapright{M(F_r,{\cal R})_\alpha(\chargp{1})} &\Lambda_{\alpha,\chargp{1}}^r &\mapright{\delta_{\alpha}(\chargp{1})} &\Lambda_{\alpha,\chargp{1}}\cr &\mapdown{}&&\mapdown{} &&\mapdown{}\cr &({\Bbb C})^s&\mapright{\delta_2}&({\Bbb C})^r &\mapright{\delta_1}&{{\Bbb C}} } $$ the vertical arrows are isomorphisms. We thus have \begin{eqnarray*} \mathrm{rank} (M(F_r,{\cal R})_{\alpha}(\chargp{1})) &=& \mathrm{rank} (\delta_2)\\ &=& r- b_1(X). \end{eqnarray*} \qed \begin{proposition} Let $\Gamma$ be a finitely presented group and let $\alpha : \Gamma \rightarrow G$ be an epimorphism where $G$ is a finite abelian group. Let $\chargp{\alpha} : \chargp{G} \hookrightarrow \chargp{\Gamma}$ be the inclusion map induced by $\alpha$. Then $$ b_1(X_\alpha) = b_1(X) + \sum_{i=1}^{r-1} | V_i(\Gamma) \cap \widehat{\alpha} (\widehat{G} \setminus \widehat{1}) |. $$ \end{proposition} \proof Starting with formula (2) and Corollary 2.5.3, we have \begin{eqnarray*} b_1(X_\alpha) &=&(r-1)|G| + 1 - \mathrm{rank} (M(F_r,{\cal R})_\alpha)\\ &=& r- \mathrm{rank} (M(F_r,{\cal R})_\alpha(\chargp{1})) + \sum_{\rho\in \chargp{G} \setminus \chargp{1}} (r-1) - \mathrm{rank} (M(F_r,{\cal R})_\alpha(\rho)). \end{eqnarray*} By Lemma 2.5.5, the left hand summand equals $b_1(X)$ and by (4) the right hand side can be written in terms of the indicator functions: $$ b_1(X_\alpha) = b_1(X) + \sum_{\rho \in \chargp{G} \setminus \chargp{1}} \sum_{i=1}^{r-1} \chi_{V_i(\chargp{\Gamma})}(\chargp{\alpha}(\rho)) $$ and the claim follows. \qed \begin{corollary} Let $\Gamma = \pi_1(X)$ be a finitely presented group and $\alpha : \Gamma \rightarrow G$ an epimorphism to a finite abelian group $G$, as above. Then $$ b_1(X_\alpha) = \sum_{i=1}^r |W_i(\Gamma) \cap \chargp{\alpha} (\chargp{G})|. $$ \end{corollary} \heading{Example.} We illustrate the above exposition using the well known case of the trefoil knot in the three sphere $S^3$: $$ \epsffile{trefoil} $$ One presentation of the fundamental group of the complement is $\Gamma = \langle x,y : xyx y^{-1}x^{-1}y^{-1} \rangle$. Then $\Sigma_1$ is a bouquet of two circles and $F = \pi_1(\Sigma_1)$ has two generators $x,y$ one for each positive loop around the circles. The maximal abelian covering of $\Sigma_1$ is the lattice ${\cal L}_2$. Now take the relation $R = xyxy^{-1}x^{-1}y^{-1} \in F$. The lift of $R$ at the origin of the lattice is drawn in the figure below. $$ \epsffile{fig1} $$ Note that the order in which the path segments are taken does not matter in computing the 1-chain. One can verify that $D(R)$ is the 1-chain defined by $$ (1-t_x+t_xt_y)\langle x \rangle +(-t_xt_y^{-1} + t_x - t_x^2)\langle y \rangle. $$ Thus, the Alexander matrix for the relation $R$ is $$ M(F_r,{\cal R}) = \left[\begin{array}{c}1-t + t^2\cr -1 + t - t^2\end{array}\right]. $$ Here $t_x$ and $t_y$ both map to the generator $t$ of ${\Bbb Z}$ under the abelianization of $\Gamma$. The Alexander stratification of $\Gamma$ is thus given by \begin{eqnarray*} V_0(\Gamma) &=& \chargp{\Gamma} = {\Bbb C}^*;\cr V_1(\Gamma) &=& V(1-t+t^2);\cr V_i(\Gamma) &=& \emptyset \qquad \mbox{for $i \ge 2$}. \end{eqnarray*} Note that the torsion points on $V_1(\Gamma)$ are the two primitive $6$th roots of unity $\exp{(\pm{2\pi}/6)}$. Now let $\alpha : \Gamma \rightarrow G$ be any epimorphism onto an abelian group. Then since $\mathrm{ab}(\Gamma) \cong {\Bbb Z}$, $G$ must be a cyclic group of order $n$ for some $n$. This means the image of $\chargp{\alpha} : \chargp{G} \rightarrow {\Bbb C}^*$ is the set of $n$-th roots of unity in ${\Bbb C}^*$. Let $X_n$ be the $n$-cyclic unbranched covering of the complement of the trefoil corresponding to the map $\alpha = \alpha_n$. By Proposition 2.5.6, $$ b_1(X_n) = \left \{ \begin{array}{ll}3 & \mbox{if $6 | n$}\cr 1 & \mbox{otherwise.}\end{array}\right. $$ \section{Group theoretic constructions and Alexander invariants.} \subsection {Group homomorphisms.} Let $\Gamma$ and $\Gamma'$ be finitely presented groups and let $\alpha : \Gamma' \rightarrow \Gamma$ be a group homomorphism. In this section, we look at what can be said about the Alexander strata of the groups $\Gamma$ and $\Gamma'$ in terms of $\alpha$. \begin{lemma} The homomorphism $$ T_\alpha : C^1(\Gamma,\rho) \rightarrow C^1(\Gamma', \chargp{\alpha}(\rho)) $$ given by composition with $\alpha$ induces a homomorphism $$ \widetilde{T_\alpha} : H^1(\Gamma,\rho) \rightarrow H^1(\Gamma',\chargp{\alpha}(\rho)). $$ \end{lemma} \proof It suffices to show that if $f$ is an element of $B^1(\Gamma,\rho)$, then $T_\alpha(f)$ is an element of $B^1(\Gamma',\chargp{\alpha}(\rho))$. For any $f \in B^1(\Gamma,\rho)$, there is a constant $c \in {\Bbb C}$ such that for all $g \in \Gamma$, $$ f(g) = (1-\rho(g))c. $$ Then, for any $g' \in \Gamma'$, \begin{eqnarray*} T_\alpha(f) (g') &=& (1 - \rho(\alpha(g'))c\\ &=& (1 - \chargp{\alpha}(\rho)(g'))c. \end{eqnarray*} Thus, $T_\alpha(f)$ is in $B^1(\Gamma',\chargp{\alpha}(\rho))$. \qed The following lemma follows easily from the definitions. \begin{lemma} If $\alpha : \Gamma' \rightarrow \Gamma$ is a group homomorphism, then (1) implies (2) and (2) implies (3), where (1), (2), and (3) are the following statements. \begin{description} \item{(1)} $\widetilde{T_\alpha} : H^1(\Gamma,\rho) \rightarrow H^1(\Gamma',\chargp{\alpha}(\rho))$ is injective; \item{(2)} $\dim H^1(\Gamma,\rho) \leq \dim H^1(\Gamma',\chargp{\alpha}(\rho)$, for all $\rho \in \chargp{\Gamma}$; and \item{(3)} $\chargp{\alpha}(W_i(\Gamma)) \subset W_i(\Gamma')$. \end{description} \end{lemma} \begin{proposition} If $\alpha : \Gamma' \rightarrow \Gamma$ is an epmiorphism, then $$ \widetilde{T_\alpha} : H^1(\Gamma,\rho) \rightarrow H^1(\Gamma',\rho) $$ is injective. Furthermore, $$ \chargp{\alpha}(V_i(\Gamma)) \subset V_i(\Gamma'). $$ \end{proposition} \proof To show the first statement we need to show that if $T_\alpha(f) \in B^1(\Gamma',\chargp{\alpha}(\rho))$ for some $\rho \in \chargp{\Gamma}$, then $f \in B^1(\Gamma,\rho)$. If $f \in C^1(\Gamma,\rho)$ and $T_\alpha(f) \in B^1(\Gamma',\chargp{\alpha}(\rho))$, then for some $c \in {\Bbb C}$ and all $g' \in \Gamma'$ we have $$ T_\alpha(f) = (1 - \chargp{\alpha}(\rho)(g'))c. $$ Take $g \in \Gamma$. Since $\alpha$ is surjective, there is a $g' \in \Gamma'$ so that $\alpha(g') = g$. Thus, \begin{eqnarray*} f(g) &=& f(\alpha(g'))\\ &=& T_\alpha(f)(g')\\ &=& (1- \chargp{\alpha}(\rho)(g'))c\\ &=& (1 - \rho(\alpha(g'))c\\ &=& (1 - \rho(g))c. \end{eqnarray*} Since this holds for all $g \in \Gamma$, $f$ is in $B^1(\Gamma,\rho)$. The second statement follows from Lemma 3.1.2, Lemma 2.2.3 and Corollary 2.4.3, since $\chargp{\alpha}$ is injective and sends the trivial character to the trivial character. \qed \begin{proposition} If $\alpha : \Gamma' \rightarrow \Gamma$ is a monomorphism whose image has finite index in $\Gamma$, then, for any $\rho \in \chargp{\Gamma}$, $$ \widetilde{T_\alpha} : H^1(\Gamma,\rho) \rightarrow H^1(\Gamma',\chargp{\alpha}(\rho)) $$ is injective. \end{proposition} \proof We can assume that $\Gamma'$ is a subgroup of $\Gamma$. Take any $\rho \in \chargp{\Gamma}$. We can think of $\chargp{\alpha}(\rho)$ as the restriction of the representation $\rho$ on $\Gamma$ to the subgroup $\Gamma'$. The map $\widetilde{T_\alpha}$ is then the restriction map $$ \mathrm{res}^\Gamma_{\Gamma'} : H^1(\Gamma,\rho) \rightarrow H^1(\Gamma',\chargp{\alpha}(\rho)) $$ in the notation of Brown (\cite{Bro:Coh}, III.9). Furthermore, one can define a {\it transfer map} $$ \mathrm{cor}^\Gamma_{\Gamma'} : H^1(\Gamma',\chargp{\alpha} (\rho)) \rightarrow H^1(\Gamma,\rho) $$ with the property that $$ \mathrm{cor}^\Gamma_{\Gamma'}\circ \mathrm{res}^\Gamma_{\Gamma'} : H^1(\Gamma,\rho) \rightarrow H^1(\Gamma,\rho) $$ is multiplication by the index $[\Gamma:\Gamma']$ of $\Gamma'$ in $\Gamma$ (see \cite{Bro:Coh}, Proposition 9.5). This implies that $\mathrm{res}^\Gamma_{\Gamma'}$ is injective. \qed Note that Proposition 3.1.4 does not hold if $\alpha(\Gamma)$ does not have finite index. For example, let $\alpha : F_{1} (= {\Bbb Z}) \hookrightarrow F_{2}$ be the inclusion of the free group on one generator into that free group on two generators, sending the generator of $F_{1}$ to the first generator of $F_{2}$. Then for any $\rho \in \chargp{F_{2}}$, $$ \dim \ H^1(F_{2},\rho) = 2 > 1 = \dim \ H^1(F_{1},\chargp{\alpha}(\rho)). $$ \subsection{Free products.} In this section, we treat free products of finitely presented groups. The easiest case is a free group. Since there are no relations, it is easy to see that $$ V_i(F_r) = \chargp{F_r} = ({\Bbb C}^*)^r $$ for $i=1,\dots,r-1$ and is empty for $i \ge r$. Thus, $$ W_i(F_r) = \left\{\begin{array}{ll} ({\Bbb C}^*)^r \qquad &\mbox{if $i=1,\dots,r-1$,}\\ \{\chargp{1}\} &\mbox{if $i = r$} \end{array}\right. $$ and is empty for $i > r$. \begin{proposition} If $\ \Gamma = \Gamma_1 * \dots * \Gamma_k$ is a free product of $k$ finitely presented groups, then $$ V_i(\Gamma) = \sum_{i_1 + \dots + i_k} V_{i_1}(\Gamma_1) \oplus \dots \oplus V_{i_k}(\Gamma_k). $$ \end{proposition} \proof We first do the case $k=2$. Suppose $\Gamma$ is isomorphic to the free product $\Gamma_1 * \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are finitely presented groups with presentations $\langle F_{r_1},{\cal R}_1 \rangle$ and $\langle F_{r_2},{\cal R}_2 \rangle$, respectively. Suppose ${\cal R}_1 = \{R_1,\dots,R_{s_1}\}$ and ${\cal R}_2 = \{S_1,\dots,S_{s_2}\}$. Then, setting $r = r_1 + r_2$ and noting the isomorphism $F_r \cong F_{r_1} * F_{r_2}$, $\Gamma$ has the finite presentation $\langle F_r,{\cal R} \rangle$ where ${\cal R} = \{ R_1,\dots,R_{s_1},S_1,\dots,S_{s_2}\}$. The character group $\chargp{F_r}$ splits into the product $\chargp{F_r} = \chargp{F_{r_1}} \times \chargp{F_{r_2}}$. Thus, each $\rho \in \chargp{\Gamma}$ can be written as $\rho = (\rho_1,\rho_2)$, where $\rho_1 \in \chargp{F_{r_1}}$ and $\rho_2 \in \chargp{F_{r_2}}$. The vector space ${\cal D}_r({{\cal R}})(\rho)$ splits into a direct sum ${\cal D}({{\cal R}})(\rho) = {\cal D}({{\cal R}_1}) (\rho_1) \oplus {\cal D}({{\cal R}_2})(\rho_2)$ so we have $$ \dim {\cal D}({{\cal R}})(\rho) = \dim {\cal D}({{\cal R}_1})(\rho_1)+ \dim {\cal D}({{\cal R}_2})(\rho_2). $$ The rest follows by induction. \qed \subsection{Direct products.} In this section we deal with groups $\Gamma$ which are finite products of finitely presented groups. \begin{lemma} Let $\Gamma$ be the direct product of free groups $F_{r_1}\times\dots\times F_{r_k}$. Let $q_i : \Gamma \rightarrow F_{r_i}$ be the projections. Let $r = r_1 + \dots + r_k$ and let $m = \max\{r_1,\dots,r_k\}$. Then $$ V_i(\Gamma) = \left\{\begin{array}{ll} \bigcup_{i<r_j}\chargp{q_j} (\chargp{F_{r_j}}) &\qquad\mbox{if $1 \leq i < m$;}\\ \{ \chargp 1 \} &\qquad\mbox{if $m \leq i < r$;}\\ \emptyset &\qquad\mbox{if $i \ge r$.} \end{array}\right. $$ \end{lemma} \proof We know from section 3.2 that $$ V_i(F_{r_j}) = \left\{ \begin{array}{ll} \chargp{F_{r_j}}&\qquad\mbox{for $i < r_j$;}\\ \emptyset&\qquad\mbox{for $i \ge r_j$.} \end{array} \right. $$ By Proposition 3.1.3, the epimorphisms $q_j : \Gamma \rightarrow F_{r_j}$ give inclusions $$ \chargp{q_j}(\chargp{F_{r_j}}) \subset V_i(\Gamma) $$ for all $j$ such that $i < r_j$. This gives the inclusion $$ \bigcup_{i<r_j} \chargp{q_j}(\chargp{F_{r_j}}) \subset V_i(\Gamma) $$ for all $i < m$. Let $x_{i,1},\dots,x_{i,r_i}$ be the generators for $F_{r_i}$, for $i=1,\dots,k$. Let $F_r = F_{r_1}*\dots*F_{r_k}$. For $i,j=1,\dots,k$, $i < j$, $\ell = 1,\dots,r_i$ and $m = 1,\dots,r_j$, let $R_{i,\ell,j,m} = [x_{i,\ell},x_{j,m}]$. Let $$ {\cal R} = \{\ R_{i,\ell,j,m} \ : \ i \neq j\ \}. $$ Then $\langle F_r,{\cal R} \rangle$ is a presentation for $\Gamma$. Let $\Lambda_r$ be the Laurent polynomials in the generators $t_{i,\ell}$, $i=1,\dots,k$, $\ell = 1,\dots,r_i$ and associate this to the ring of functions on $\chargp{F_r} = \chargp{\Gamma}$ by sending $x_{i,\ell}$ to $t_{i,\ell}$. We have $$ D(R_{i,\ell,j,m}) = (1- t_{j,m}) \langle x_{i,\ell} \rangle + (t_{i,\ell} - 1) \langle x_{j,m}\rangle. $$ It immediately follows that $M(F_r,{\cal R})(\chargp{1})$ is the zero matrix, so $\chargp{1} \in V_i(\Gamma)$ for $i<r$ and $\chargp{1} \not\in V_i(\Gamma)$ for $i \ge r$. Now consider $\rho \in \chargp{F_r} = \chargp{\Gamma}$ with $\rho \neq \chargp{1}$. We will show that if $\rho \in \chargp{q_i}(F_{r_i})$ then $\rho \in V_n(\Gamma)$ for $n < r_i$ and $\rho \not\in V_n(\Gamma)$ for $n \ge r_i$. If $\rho \not\in \chargp{q_i}(F_{r_i})$ for any $i$, then we will show that $\rho \not\in V_1(\Gamma)$. Let $\rho_{i,\ell}$, $i=1,\dots,k$ and $\ell=1,\dots,r_i$, be the component of $\rho$ corresponding to the generator $t_{i,\ell}$ in $\Lambda_r$. For each $i=1,\dots,k$, let $s_i = r_1 + \dots + \widehat{r_i} + \dots + r_k$. Take $\rho \in \chargp{q_i}(F_{r_i})$. We know from Proposition 3.1.3 that $\rho \in V_n(\Gamma)$ for $n < r_i$. Also, $\rho_{j,m} = 1$, for all $j=1,\dots,\hat i,\dots,k$. Since $\rho \neq \chargp{1}$, $\rho_{i,\ell} \neq 1$ for some $\ell$. Consider the $s_i \times s_i$ minor of $M(F_r,{\cal R})(\rho)$ with rows corresponding to the generators $\langle x_{j,m}\rangle$ and columns corresponding to generators $R_{i,\ell,j,m}$, where $j=1,\dots,\hat{i},\dots,k$ and $m=1,\dots,r_j$. This is the $s_i \times s_i$ matrix $$ (1-\rho_{i,\ell})I_{s_i} $$ where $I_{s_i}$ is the $s_i \times s_i$ identity matrix. Thus, rank $M(F_r,{\cal R})(\rho) \ge s_i$. This means that $\rho \not\in V_n(\Gamma)$ for $n \ge (r-s_i) = r_i$. Now take $\rho \not\in \chargp{q_i}(F_{r_i})$ for any $i$. Then, for some $i$ and $j$ with $i\neq j$, and some $\ell$ and $m$, we have $\rho_{i,\ell} \neq 1$ and $\rho_{j,m} \neq 1$. Consider the minor of $M(F_r,{\cal R})(\rho)$ with columns corresponding to all generators except $x_{i,\ell}$, and rows corresponding to relations $R_{i,\ell,j',m'}$, where $j'=1,\dots,\hat i,\dots,k$ and $m'=1,\dots,r_{j'}$, and $R_{i,\ell',j,m}$, where $\ell' = 1,\dots,\hat \ell,\dots,r_i$. This is the $r-1 \times r-1$ matrix $$ \left [ \begin{array}{ll} \pm(1-\rho_{i,\ell})I_{s_i} & 0\\ 0 & \pm(1-\rho_{j,m})I_{r_i-1} \end{array} \right ] $$ which has rank $r-1$. Thus, $\rho$ is not in $V_1(\Gamma)$. \qed \begin{corollary} Let $\Gamma$ be the direct product of finitely presented groups $$ \Gamma = \Gamma_1 \times \dots \times \Gamma_k $$ with $r_1,\dots,r_k$ generators, respectively. Let $$ P = F_{r_1} \times \dots \times F_{r_k}. $$ Then $$ V_i(\Gamma) \subset V_i(P) $$ for each $i$ and, in particular, $$ V_i(\Gamma) \subset\{\chargp{1}\} $$ if $\max\{r_1,\dots,r_k\} \le i$. \end{corollary} \proof This follows from Lemma 3.1.2 and Proposition 3.1.3. \qed In particular, if $\Gamma$ is abelian, we have the following result. \begin{corollary} If $\Gamma$ is an abelian group, then $$ V_i(\Gamma) = \left \{ \begin{array}{ll} \{\chargp{1}\}&\qquad\mathrm{if} \ 1 \leq i < \mathrm{rank} (\Gamma)\cr \emptyset&\qquad\mathrm{otherwise.} \end{array} \right . $$ Here $\mathrm{rank}(\Gamma)$ means the rank of the abelianization of $\Gamma$. \end{corollary} \section{Applications.} Let $X$ be any topological space homotopy equivalent to a finite CW complex with fundamental group $\Gamma$. In this section, we will study the role that rational planes in the Alexander strata $V_i(\Gamma)$ and the jumping loci $W_i(\Gamma)$ relate to the the geometry of $X$. \subsection{Betti numbers of abelian coverings.} Let $X$ be homotopy equivalent to a finite CW complex. Let $\Gamma = \pi_1(X)$. We will relate the first Betti number of finite abelian coverings of $X$ to rational planes in the jumping loci $W_i(\Gamma)$. Let $\alpha : \Gamma \rightarrow G$ be an epimorphism onto a finite abelian group $G$. Assume that $\Gamma$ is generated by $r$ elements. Then by Corollary 2.5.7, we have $$ b_1 (X_\alpha) = \sum_{i=1}^r |W_i(\Gamma) \cap \chargp{\alpha}(\chargp{G})|. $$ Since $G$ is finite, all points in $\alpha(\chargp{G})$ have finite order. Thus, to compute $b_1(X_\alpha)$ for finite abelian coverings $X_\alpha$, we need only know about the torsion points on $W_i(\Gamma)$. The position of torsion points $\mathrm{Tor}(V)$ for any algebraic subset $V \subset ({\Bbb C}^*)^r$ is described by the following result due to Laurent \cite{Laur:Equ}. \begin{theorem}(Laurent) If $V \subset ({\Bbb C}^*)^r$ is any algebraic subset, then there exist rational planes $P_1,\dots,P_k$ in $({\Bbb C}^*)^r$ such that $P_i \subset V$ for each $i = 1,\dots,k$ and $$ \mathrm{Tor}(V) = \bigcup_{i=1}^k \mathrm{Tor}( P_i). $$ \end{theorem} \noindent From this theorem it follows that, to any finitely presented group $\Gamma$, we can associate a collection of finite sets of rational planes ${\cal P}_i$, such that $$ \mathrm{Tor}(V_i(\Gamma)) = \bigcup_{P \in {\cal P}_i} \mathrm{Tor}(P). $$ We thus have the following. \begin{corollary} The rank of co-abelian, finite index subgroups of a finitely presented group $\Gamma$ depends only on the rational planes contained in the Alexander strata $V_i(\Gamma)$. \end{corollary} \subsection{Existence of irrational pencils.} Let $X$ be a compact K\"ahler manifold. An {\it irrational pencil on $X$} is a surjective morphism $$ X \rightarrow C_g, $$ where $C_g$ is a Riemann surface of genus $g \ge 2$. In this section, we will discuss the relation between properties of the Alexander stratification for $\Gamma = \pi_1(X)$ and the existence of irrational pencils on $X$. Let $\Gamma_g$ be the fundamental group of $C_g$. Then $\Gamma_g$ has presentation $\langle F_{2g}, R_g \rangle$, where $R_g$ is the single element $$ [x_1,x_{g+1}][x_2,x_{g+2}]\dots[x_g,x_{2g}]. $$ The Fox derivative of $R_g$ is given by $$ D(R_g) = \sum_{i=1}^g (t_i - 1)\langle x_i \rangle + \sum_{i=g+1}^{2g} (1-t_i)\langle x_i \rangle. $$ Thus, we have $$ V_i(\Gamma_g) = \left\{ \begin{array}{ll} \chargp{\Gamma_g}\cong({\Bbb C}^*)^{2g} &\qquad\mbox{if $1 \leq i < 2g-1$;}\cr \{\chargp 1\} &\qquad\mbox{if $i=2g-1$;}\cr \emptyset &\qquad\mbox{if $i>2g-1$.} \end{array} \right. $$ and for the jumping loci $$ W_i(\Gamma_g) = \left\{ \begin{array}{ll} \chargp{\Gamma_g}\cong ({\Bbb C}^*)^{2g} &\qquad\mbox{if $1\leq i<2g-1$;}\cr \{\chargp{1}\} &\qquad\mbox{if $2g-1 \le i \le 2g$;}\cr \emptyset &\qquad\mbox{if $i > 2g$.} \end{array} \right. $$ Given an irrational pencil $X \rightarrow C_g$, the Stein factorization gives a map $$ X \rightarrow C_h \rightarrow C_g, $$ where the map from $C_h$ to $C_g$ is a finite surjective morphism and $X$ has connected fibers. Then $h \ge g$ and there is a surjective group homomorphism $$ \pi_1(X) \rightarrow \Gamma_h. $$ By Proposition 3.1.3, this implies that there is an inclusion $$ W_i(\Gamma_h) \rightarrow W_i(\pi_1(X)), $$ for all $i$. We can thus conclude the following. \begin{proposition} If $X$ has an irrational pencil of genus $g$, then for some $h \ge g$, $W_i(\pi_1(X))$ contains an affine subtorus of dimension $2h$, for $i=1,\dots,2h-2$. \end{proposition} The question arises, do the maximal affine subtori in $W_i(\pi_1(X))$ all come from irrational pencils? This was answered in the affirmative by Beauville \cite{Beau:Ann} for $W_1(\pi_1(X))$ (see also \cite{G-L:HighOb},\cite{Ar:Higgs} and \cite{Cat:Mod}.) This shows that the irrational pencils on $X$ only depend on the topological type of $X$ (see also \cite{Siu:Strong}). Now suppose $V \subset W_i(\Gamma)$ is a translate of an affine subtorus by a character $\rho \in \chargp{\Gamma}$ of finite order. Then, since $\Gamma$ is finitely generated, the image of $\rho$ is finite in ${\Bbb C}^*$. Let $\widetilde X \rightarrow X$ be the finite abelian unbranched covering associated to this map. Then the corresponding map on fundamental groups $$ \alpha : \pi_1(\widetilde X) \rightarrow \pi_1(X) $$ has image equal to the kernel of $\rho$. Thus, $\chargp{\alpha}(\rho)$ is the trivial character in $\chargp{\pi_1(\widetilde X)}$ and $\chargp{\alpha}(V)$ is a connected subgroup, i.e., an affine subtorus of $\chargp{\pi_1(\widetilde X)}$. As we discuss in the next section, a theorem of Simpson shows that all the jumping loci $W_i(\pi_1(X))$ are finite unions of rational planes. This leads us to the following question: \heading{Question.} Can all the rational planes in the jumping loci $W_i(\pi_1(X))$ be explained by irrational pencils on $X$ or on finite abelian coverings of $X$? \vspace{12pt} \subsection{Binomial criterion for K\"ahler groups.} If $\Gamma$ is a group such that there is an isomorphism $\Gamma \cong \pi_1(X)$ for some compact K\"ahler manifold $X$, we will say that $\Gamma$ is {\it K\"ahler}. A {\it binomial ideal} in $\Lambda_r({\Bbb C})$ is an ideal generated by {\it binomial} elements of the form $$ t^\lambda - u $$ where $\lambda \in {\Bbb Z}^r$, $t^\lambda = t_1^{\lambda_1}\dots t_r^{\lambda_r}$ and $u \in {\Bbb C}$ is a unit. The following is straightforward. \begin{lemma} If $V \subset ({\Bbb C}^*)^r$ is a rational plane then $V$ is defined by a binomial ideal where the units $u$ are roots of unity. \end{lemma} In (\cite{Ar:Higgs}, Theorem 1), Arapura shows that $W_i(\Gamma)$ is a finite union of unitary translates of affine tori. Simpson (\cite{Sim:Subs}, Theorem 4.2) extends Arapura's result, showing that the $W_i(\Gamma)$ are actually translates of rational tori. \begin{theorem} (Simpson) If $\Gamma$ is K\"ahler, then $W_i(\Gamma)$ is a finite union of rational planes for all $i$. \end{theorem} \begin{corollary} If $\Gamma$ is K\"ahler, then any irreducible component of $V_i(\Gamma)$ is defined by a binomial ideal. \end{corollary} \proof By Lemma 2.2.3, $V_i(\Gamma)$ equals $W_i(\Gamma)$ except when $i$ equals the rank of the abelianization of $\Gamma$. Suppose the latter holds. Then, again by Lemma 2.2.3, $V_i(\Gamma)$ is $W_i(\Gamma)$ minus the identity character $\chargp{1}$. But $V_i(\Gamma)$ is a closed algebraic set, so $\chargp{1}$ is an isolated component of $W_i(\Gamma)$. Thus, since $W_i(\Gamma)$ is a finite union of rational planes, so is $V_i(\Gamma)$. The rest follows from Lemma 4.3.1. \qed \heading{Remark.} Stated in terms of the ideals of minors (also known as Alexander ideals or fitting ideals) of an Alexander matrix, Simpson's theorem implies a property of the radical of these ideals for K\"ahler groups. Subtler and interesting questions can be asked about the fitting ideals themselves. We leave this as a topic for further research. \vspace{12pt} Let $R_g$ be the standard relation for $\pi_1(C_g)$, where $C_g$ is a Riemann surface of genus $g$. It is possible from Corollary 4.3.3 to make many examples of nonK\"ahler finitely presented groups. For example, we have the following Proposition (cf. \cite{Ar:SurFun}, \cite{Gro:Sur}, \cite{Sim:Subs}). \begin{proposition} Let $g \ge 2$ and let $$ \Gamma = \langle x_1,\dots,x_{2g} : S_1,\dots,S_s \rangle, $$ where $$ S_i = u_{i,1}R_g u_{i,1}^{-1} \dots u_{i,k_i}R_g u_{i,k_i}^{-1}, $$ for $i=1,\dots,s$. Let $$ p_i = \mathrm{ab}(u_{i,1}) + \dots + \mathrm{ab}(u_{i,k_i}) $$ considered as a polynomial in $\Lambda_r$. Then, if $\Gamma$ is K\"ahler, the set of common zeros $V(p_1,\dots,p_s)$ must be defined by binomial ideals. \end{proposition} \proof The Fox derivative $D: F_{2g} \rightarrow {\Bbb Z}[\mathrm{ab}(F_{2g})]^{2g}$ takes each $S_i$ to $$ D(S_i) = (\mathrm{ab}(u_{i,1}) + \dots + \mathrm{ab}(u_{i,k_i})) D(R_g). $$ Thus, the $i$th row of the Alexander matrix $M(F_{2g},{\cal R})$ equals $M(F_{2g},R_g)$, considered as row vector, multiplied by $p_i$. It follows that the rank of $M(F_{2g},{\cal R})$ is at most 1 and equals 1 outside of the set of common zeros of $p_1,\dots,p_s$ and the point $(1,\dots,1)$. The rest is a consequence of Corollary 4.3.3. \qed \heading{Example.} Fix $g \ge 3$, and let $\Gamma$ be given by $$ \Gamma = \langle x_1,\dots,x_{2g} : S_1,S_2 \rangle, $$ where $$ S_1 = x_1 R_{2g} x_1^{-1} \dots x_g R_{2g} x_g^{-1} $$ and $$ S_2 = x_{g+1} R_{2g} x_{g+1}^{-1} \dots x_{2g} R_{2g} x_{2g}^{-1}. $$ Then \begin{eqnarray*} D(S_1) &=& (t_1 + \dots + t_g) D(R_g)\\ D(S_2) &=& (t_{g+1} + \dots + t_{2g}) D(R_g) \end{eqnarray*} which implies that $V_1(\Gamma)$ contains $\chargp{1}$ and the points in $$ V(t_1 + \dots + t_g) \cap V(t_{g+1} + \dots + t_{2g}). $$ This is isomorphic to the product of the hypersurface in $({\Bbb C}^*)^g$ defined by $V = V(t_1 + \dots + t_g)$ with itself. Since $g \ge 3$, this hypersurface is not defined by a binomial ideal. Thus, $\Gamma$ is not K\"ahler.